H
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THE
THEORY OF STRAINS
GIRDERS AND SIMILAR STRUCTURES,
WITH
OBSERVATIONS OK THE APPLICATION OF THEORY TO PRACTICE
AND
TABLES OF THE STRENGTH AND OTHER PROPERTIES OF MATERIALS.
BY
BINDON B. STONEY, M.A.,
MEMBER OF THE INSTITUTION OF CIVIL ENGINEERS, AND ENGINEER TO THE DUBLIN PORT AND DOCKS BOARD.
Prius qukm incipias, consulto ; et ubi consulueris, mature facto opus est.
tlj mtnunras Illustrations ingratab- an
NEW EDITION— ENLARGED AND REVISED.
D. VAN NOSTRAND,
23 MURRAY STREET,
NEW YORK.
1873.
PREFACE TO THE FIRST EDITION.
THE following pages have been written at various times during
such brief intervals of leisure as the author could spare from his
professional duties. They are for the most part the result of
experience combined with theory ; it is therefore hoped that they
may supply the student with what has long been a want in
Engineering literature, namely, a Handbook on the TJieory of
Strains and the Strength of Materials, giving practical methods
for calculating the strains which occur in girders and similar
structures. The theory of transverse strain has, indeed, been
incidentally treated by writers on Mechanical Philosophy; their
researches, however, have been confined to strains in plain girders,
or to a few brief remarks on the more elementary forms of trussing,
which, without further development, are of little practical use, and
but too frequently afford a pretext for the ill-concealed contempt
which so-called practical men sometimes entertain for theoretic
knowledge.
A thorough acquaintance with the theory of strains and the
strength and other properties of materials forms the basis of all
sound engineering practice, and when this is wanting, even natural
constructive talent of a high order is frequently at fault, and the
result is either excess and consequent waste of material, or, what
is still more disastrous, weakness in parts where strength is
essential. The time has gone by when practical sagacity formed
the sole qualification for high engineering success. Before the
IV PREFACE.
improvement of the steam engine gave rise to a new profession
there were indeed some memorable names on the roll of engineers,
generally self-taught mechanics, whom great natural ability had
raised to pre-eminence in their profession ; but practice which was
formerly excusable, or even worthy of the highest commendation,
would, now that knowledge has increased, be properly described as
culpable waste, arising either from prejudice or ignorance.
The usual resource of the merely practical man is precedent, but
the true way of benefiting by the experience of others is not by
blindly following their practice, but by avoiding their errors as
well as extending and improving what time and experience have
proved successful. If one were asked what is the difference between
an engineer and a mere craftsman, he would well reply, that the
one merely executes mechanically the designs of others, or copies
something which has been done before without introducing any new
application of scientific principles, while the other moulds matter
into new forms suited for the special object to be attained ; and
though experience and practical knowledge are essential for this, he
lets his experience be guided and aided by theoretic knowledge, so
as to arrange and proportion the various parts to the exact duty
they are intended to fulfil.
Then prove we now with best endeavour
What from our efforts yet may spring ;
He justly is despised who never
Did thought to aid his labours bring.
For this is art's true indication,
When skill is minister to thought ;
When types that are the mind's creation
The hand to perfect form has wrought.
The well-educated engineer should combine the qualifications of
the practical man and of the physicist, and the more he blends these
PREFACE. V
together, making each mould and soften what the other would
seem to dictate if allowed to act alone, the more will his works be
successful and attain the exact object for which they are designed.
The engineer should be a physicist, who, in place of confining his
operations to the laboratory or the study, exerts his energies in a
wider field in developing the industrial resources of nature, and
compelling mere matter to become subservient to the wants and
comforts and civilization of the human race.
CONTENTS.
CHAPTER I.
INTEODUCTOKY.
ART. PAGE
1. Strain — Tension — Compression— Transverse strain — Shearing-strain— Torsion, 1
2. Unit-strain — Inch-strain — Foot-strain, - 2
3. Elasticity — Cubic elasticity — Linear elasticity, 2
4. Elastic stiffness and Elastic flexibility, 3
5. Ductility — Toughness — Brittleness, - 3
6. Set — Influence of duration of strain, 4
7. Hooke's law — Law of elasticity — Limit of elasticity, 4
8. Coefficient of elasticity, E — Table of coefficients, - - 5
9. Mechanical laws — Resolution of forces, 8
10. The Lever, 8
11. Equality of moments, - 9
12. Beam — Girder — Semi -girder, - 9
13. Flanged girder — Single-webbed girder — Double-webbed or Tubular girder —
Box girder— Tubular bridge, - 10
CHAPTER II.
FLANGED GIRDERS WITH BRACED OR THIN CONTINUOUS
WEBS.
14. Transverse strain — Shearing-strain, - 11
15. Horizontal strains in braced or thin continuous webs may be neglected, 12
CASE I. — FLANGED SEMI-GIRDER LOADED AT THE EXTREMITY.
16. Flanges — At any cross section the horizontal components of strain in the
flanges are equal and of opposite kinds — Strength of flanged girders varies
directly as the depth and inversely as the length, - - 13
17. Girder of greatest strength — Areas of horizontal flanges should be to each
other in the inverse ratio of their ultimate unit-strains, - - 15
18. Shearing -strain — The web should contain no more material than is requisite
to convey the shearing-strain — The quantity of material in the web of
girders with parallel flanges is theoretically independent of the depth, 15
19. Girder of uniform strength — Economical distribution of material, - 16
20. Flange-area of semi-girder of uniform strength when the depth is constant, - 16
21. Depth of semi -girder of uniform strength when the flange-area is constant, - 16
Till CONTENTS.
CASE II.— FLANGED SEMI-GIRDER LOADED UNIFORMLY.
ART. PAGE
22. Flanges, 17
23. Web— Shearing-strain, 19
24. Flange-area of semi -girder of uniform strength when the depth is constant, - 19
25. Depth of semi-girder of uniform strength when the flange-area is constant, - 19
26. Strain in curved flange, - 20
27. Semi-girder loaded uniformly and at the extremity also, shearing-strain, - 20
28. Flange-area of semi-girder of uniform strength loaded uniformly and at the
end when the depth is constant, - - 21
29. Depth of semi-girder of uniform strength loaded uniformly and at the end
when the flange-area is constant, - - 21
CASK HI. — FLANGED GIRDER SUPPORTED AT BOTH ENDS AND LOADED AT AN
INTERMEDIATE POINT.
30. Flanges, 22
31. Maximum flange-strains occur at the weight, 23
32. Concentrated rolling load, maximum strains in flanges are proportional to
the rectangle under the segments, - 23
33. Weight at centre, 23
34. Web, shearing strain, 24
35. Single fixed load, flange-area of girder of uniform strength when the depth
is constant, - 25
36. Single fixed load, depth of girder of uniform strength when the flange-area
is constant, 26
37. Concentrated rolling load, shearing-strain, - 26
38. Concentrated rolling load, flange-area of girder of uniform strength when the
depth is constant, - 27
39. Concentrated rolling load, depth of girder of uniform strength when the
flange-area is constant, - 27
40. Concentrated rolling load, strain in curved flange— Section of curved flange, 28
CASE IV.— FLANGED GIRDER SUPPORTED AT BOTH ENDS AND LOADED AT
IRREGULAR INTERVALS.
41. Flanges, - 29
42. Web, shearing-strain, ..... - 30
CASE V.— FLANGED GIRDER SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY.
43. Flanges, .....-- 32
44. Strains at centre of girder, - - 33
CONTENTS. ix
ART. PAGE
45. A concentrated load produces the same strain in the flanges as twice the load
uniformly distributed, 35
46. Web, shearing-strain, . 35
47. Flange-area of girder of uniform strength when the depth is constant, - 36
48. Depth of girder of uniform strength when the flange-area is constant, 37
49. Suspension bridge — Curve of equilibrium, - 37
CASE VI. — FLANGED GIKDEE SUPPORTED AT BOTH ENDS AND TRAVERSED BY A
TRAIN OF UNIFORM DENSITY.
50. Passing train of uniform density — Shearing-strain — Flanges, 38
51. Maximum strains in web occur at one end of a passing train, - 39
52. Uniform load and passing train, shearing-strain, - - 40
53. Maximum strain in flanges occur with load all over, 41
54. Area of a continuous web calculated from the shearing-strain — Quantity of
material in a continuous web, - - 41
55. Depth and length for calculation, 42
CHAPTER III.
TRANSVERSE STRAIN.
56. Transverse strain, 43
57. Neutral surface, - 43
58. Neutral axis — Centres of strain — Resultant of horizontal forces in any cross
section equals cipher, 44
59. Moment of resistance, M, — Bending moment, 45
60. Coefficient of rupture, S, — Semi-girder loaded at the extremity, - - 46
61. Semi-girder loaded uniformly, - 47
62. Girder supported at both ends and loaded at an intermediate point, - 47
63. Girder supported at both ends and loaded at the centre, - 47
64. Girder supported at both ends and loaded uniformly, 47
65. Table of coefficients of rupture, 47
66. Strength of stones, even of the same kind, is very variable, - 51
67. Strength of similar girders — Limit of length, 53
68. Neutral axis passes through the centre of gravity — Practical method of finding
the centre of gravity, - - 54
CHAPTER IV.
GIRDERS OF VARIOUS SECTIONS.
69. Moment of resistance, M, 56
71. M for sections symmetrically disposed above and below the centre of gravity, 57
X CONTENTS.
ART. PAGE
72. M for a solid rectangle, - 58
73. M for a solid square with one diagonal vertical, - - 58
74. M f or a circular disc, - 58
75. M f or a circular ring of uniform thickness, - - 58
76. M for an elliptic disc with one axis horizontal, - 59
77. M for an elliptic ring with one axis horizontal, 59
78. Two classes of flanged girders, 60
79. M for the section of a flanged girder or rectangular tube, neglecting the web, 60
80. M for the section of a flanged girder or rectangular tube, including the web, 61
81. M for the section of a flanged girder or rectangular tube with equal flanges,
including the web, - 61
82. M for the section of a square tube of uniform thickness, either with the sides
or one diagonal vertical, - - 62
CASE I.— SEMI -GIRDERS LOADED AT THE EXTREMITY.
84. Solid rectangular semi -girders, 63
85. Geometrical proof, 63
86. Solid square semi-girders with one diagonal vertical — Solid square girders
with the sides vertical are T414 times stronger than with one diagonal
vertical, - 64
87. Rectangular girder of maximum strength cut out of a cylinder, - 65
88. Solid round semi-girders, - 66
89. Solid square girders are 1'7 times as strong as the inscribed circle, and 0'6
times as strong as the circumscribed circle, 66
90. Hollow round semi-girders of uniform thickness, - 66
91. Centre of solid round girders nearly useless, - 67
92. Hollow and solid cylinders of equal weight, - 67
93. Solid elliptic semi-girders, 68
94. Hollow elliptic semi -girders, - - 68
95. Flanged semi-girder or rectangular tube, taking the web into account, 69
96. Flanged semi-girder or rectangular tube with equal flanges, - 69
97. Square tubes with vertical sides, - 70
98. Square tubes with diagonal vertical — Square tubes of uniform thickness with
vertical sides are 1'414 times stronger than with one diagonal vertical, 70
99. Square tubes of uniform thickness with vertical sides are 1'7 times as strong
as the inscribed circle of equal thickness, and 0'85 times as strong as the
circumscribed circle of equal thickness — Square and round tubes of equal
thickness and weight are of nearly equal strength, 70
100. Value of web in aid of the flanges, - 71
101. Plan of solid rectangular semi-girder of uniform strength, depth constant, - 71
102. Elevation of solid rectangular semi-girder of uniform strength, breadth
constant, - - 72
CONTENTS. XI
ART. PAGE
103. Solid round semi -girder of uniform strength, 72
104. Hollow round semi -girder of uniform strength, - 72
CASE II. — SEMI-GIRDERS LOADED UNIFORMLY.
106. Solid rectangular semi-girders, 73
107. Solid round semi-girders, - 74
108. Hollow round semi-girders of uniform thickness, - 74
109. Flanged semi-girders or rectangular tubes, taking the web into account, - 74
110. Plan of solid rectangular semi -girder of uniform strength, depth constant, - 74
111. Elevation of solid rectangular semi-girder of uniform strength, breadth
constant, - 75
112. Solid round semi -girder of uniform strength, - 75
113. Hollow round semi-girder of uniform strength, - 75
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED AT AN
INTERMEDIATE POINT.
115. Solid rectangular girders, - 76
116. Solid round girders, - 77
117. Hollow round girders of uniform thickness, - - 77
118. Flanged girders or rectangular tubes, taking the web into account, 77
119. Plan of solid rectangular girder of uniform strength, depth constant, - 79
120. Elevation of solid rectangular girder of uniform strength, breadth constant, 79
121. Solid round girder of uniform strength, - - 79
122. Hollow round girder of uniform strength, - - 79
123. Concentrated rolling load, plan of solid rectangular girder of uniform
strength when the depth is constant — Elevation of same when the breadth
is constant,
CASE IV. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY.
125. Solid rectangular girders,
126. Solid round girders, -
127. Hollow round girders of uniform thickness,
128. Flanged girders or rectangular tubes, taking the web into account,
129. Plan of solid rectangular girder of uniform strength when the depth is
constant, -
130. Elevation of solid rectangular girder of uniform strength when the breadth
is constant,
131. Discrepancy between experiments and theory — Shifting of neutral axis —
Longitudinal strength of materials derived from transverse strains
erroneous, -
132. Transverse strength of thick castings much less than that of thin castings, 86
Xll CONTENTS.
CHAPTER V.
BRACED GIRDERS WITH PARALLEL FLANGES AND WEBS
FORMED OF ISOSCELES BRACING.
ART. PAGE
133. Object of bracing, - - 87
Definitions.
134. Brace, 88
135. Apex, 88
136. Bay, 88
137. Counterbraced brace, . - 88
138. Counterbraced girder, 88
139. Symbols of compression and tension, + and — , - 88
140. Axioms, - - 88
CASE I.— SEMI-GIRDERS LOADED AT THE EXTREMITY.
145. Web, - 89
146. Flanges, - 90
147. Strains in braced webs may be deduced from the shearing- strain, - -91
CASE II.— SEMI- GIRDERS LOADED UNIFORMLY.
148. Web, - 92
149. Strains in intersecting diagonals, - 92
150. Increments of strain in flanges,
151. Resultant strains in flanges, 93
152. General law of strains in horizontal flanges of braced girders, - 93
153. Lattice web has no theoretic advantage over a single system — Practical
advantage of lattice web — Long pillars, - 94
154. Multiple and single triangulation compared — Lattice semi-girders loaded
uniformly, - - 95
155. Girder balanced on a pier, - 96
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED AT AN
INTERMEDIATE POINT.
156. Web, - - 96
157. Flanges, - - 97
158. Concentrated rolling load, - - 98
169. Lattice girder traversed by a single load, - - - - 99
CASE IV. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY.
160. Web, first method, ----.... 100
CONTENTS. xiii
ART. PAGE
161. Flange-strains derived from a diagram, . - 100
162. Web, second method, • ... 101
163. Increments of strain in flanges, - - - 102
164. Strains in flanges calculated by moments, - - 103
165. Girder loaded un symmetrically, - - 103
166. Girder loaded symmetrically, - 103
167. Strains in end diagonals and bays, - - 103
168. Strains in intersecting diagonals — General law of strains in intersecting
diagonals of isosceles bracing with parallel flanges, • • - 103
CASE V.— GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED BY A TRAIN OP
UNIFORM DENSITY.
169. Web, - - 104
170. Maximum strains in web — Strains in intersecting diagonals, - 105
171. Permanent load — Absolute maximum strains, - - 106
172. Web, first method, - - 106
173. Flanges, - 107
174. Counterbracing, - - 108
175. Permanent load diminishes counterbracing, - 108
176. Web, second method, - 108
CASE VI. — LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY.
177. Approximate rule for strains in lattice web, - 111
178. Web— Flanges, - - 112
CASE VII. — LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED
BY A TRAIN OF UNIFORM DENSITY.
179. Web, first method, - - 113
180. End pillars, - - 114
181. Ambiguity respecting strains in lattice bracing, - - - 116
182. Flange- strains calculated by moments, - - - - 117
183. Web, second method, - 118
CHAPTER VI.
GIRDERS WITH PARALLEL FLANGES CONNECTED BY
VERTICAL AND DIAGONAL BRACING.
184. Introductory, - - 122
XIV CONTENTS.
CASE I. GIRDERS SUPPORTED AT BOTH ENDS AND LOADED AT AN
INTERMEDIATE POINT.
ART. PAGE
186. Single moving load, - 123
187. Trussed beam— Gantry, - - 123
CASE II. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY.
188. Web — Flanges, - - 124
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED BY A
TRAIN OF UNIFORM DENSITY.
189. Web, - 125
CASE IV.— LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED
BY A TRAIN OF UNIFORM DENSITY.
190. Web, - 127
191. End pillars — Ambiguity respecting strains in faulty designs, - 127
CHAPTER VII.
BRACED GIRDERS WITH OBLIQUE OR CURVED FLANGES.
192. Introductory— Calculation by diagram, - - 129
CASE I. — BENT SEMI-GIRDERS LOADED AT THE EXTREMITY.
193. Derrick crane, - 129
194. Wharf crane, . - 132
195. Bent crane, - - 133
196. Calculation by moments, - 135
197. Lattice webs not suited for powerful bent cranes, • -136
CASE II. — THE BRACED SEMI-ARCH.
198. Swivel bridge, - - - . - 136
199. Single triangulation, - - 137
200. Example, - - 137
201. Lattice semi-arch — Triangular semi-girder, - - 139
202. Inverted semi-arch, - ... - 139
CASE III. — CRESCENT GIRDER.
203. Suitable for roofs— Flanges, - ... 140
CONTENTS. XV
ART. PAGE
204. Example 1, - . 141
205. Example 2 — Flange-strains nearly uniform with symmetric loading, - 142
206. Ambiguity in the strains of a crescent girder when resting on more than two
points, - - - 143
CASE IV BOW-STRING GIRDER.
207. Concentrated load, - - 144
208. Passing load — Example — Little counterbracing required in bowstring
girders of large size, - - 144
209. Calculation by moments, - - 146
210. Uniformly distributed load, little bracing required — Absolute maximum
strains, - - 146
211. Single triangulation, second method of calculation, - - 147
212. Inverted bowstring, or fish-bellied girder — Bow and invert, or double bow, 149
CASE V. — THE BRACED ARCH.
213. Law of the lever applicable to the braced arch, - -149
214. Strains in the braced arch loaded symmetrically resemble those in the semi-
arch — Portions of the flanges liable to tensile strains from unequal loading, 152
215. Calculation by moments — Calculation of strains in a latticed arch impracti-
cable except when the load is symmetrical, • -152
216. Flat arch or arch with horizontal flanges, - - 152
217. Rigid suspension bridge, - - 153
218. Triangular arch, - - 153
219. Cast-iron arches, - - 153
CASE VI. — THE BRACED TRIANGLE.
220. The common A roof, - 154
221. The A truss, - - 157
CASE VII. — THE SUSPENSION TRUSS.
222. Suited for domed roofs, - 161
CHAPTER VIII.
DEFLECTION.
Class I. — Girders whose sections are proportioned so as to produce uniform strength.
223. Deflection curve circular in girders of uniform strength. Amount of
deflection not materially affected by the web, - - - 164
224: Formula for the deflection of circular curves— Deflection of similar girders
when equally strained varies as their linear dimensions, - - 166
XVI CONTENTS.
Class II. — Girders whose section is uniform throughout their length.
CASE I.— SEMI-GIBDERS OF UNIFORM SECTION LOADED AT THE EXTREMITY.
ART. PAGE
227. Solid rectangular semi-girders — Deflection of solid square girders is the
same with the sides or one diagonal vertical, - - - - 172
228. Solid round semi -girders, - - 172
229. Hollow round semi -girders of uniform thickness, - - 172
230. Semi-girders with parallel flanges, - - - 173
231. Square tubes of uniform thickness with the sides or one diagonal vertical, - 173
CASE II.— SEMI-GIRDERS OF UNIFORM SECTION LOADED UNIFORMLY.
233. Deflection of a semi-girder loaded uniformly equals three-eighths of its
deflection with the same load concentrated at its extremity, - - 174
CASE III. — GIRDERS OF UNIFORM SECTION SUPPORTED AT BOTH ENDS AND
LOADED AT THE CENTRE.
235. Solid rectangular girders, - -176
236. Solid round girders, - - 176
237. Hollow round girders of uniform thickness, - 176
238. Girders with parallel flanges, • 177
CASE IV. — GIRDERS OF UNIFORM SECTION SUPPORTED AT BOTH ENDS
AND LOADED UNIFORMLY.
241. Central deflection of a girder loaded uniformly equals five-eighths of its
deflection with the same load concentrated at the centre, - - 178
242. Solid rectangular girders, - 178
243. Solid round girders, - - 179
244. Hollow round girders of uniform thickness, - 179
245. Girders with parallel flanges, - 179
246. Discrepancy between coefficients of elasticity derived from direct and from
transverse strain, - - - 179
CHAPTER IX.
CONTINUOUS GIKDERS.
247. Continuity — Contrary flexure — Points of inflexion, - - - 181
248. Passing load, . . 182
249. Experimental method of finding the points of inflexion — The depth of a
girder does not affect the position of the points of inflexion, - - 183
250. Practical method of fixing the points of inflexion — Economical position of
points of inflexion, - - - - 185
CONTENTS. XV11
ART. PAGE
CASE I. — CONTINUOUS GIRDERS OP TWO EQUAL SPANS, EACH LOADED
UNIFORMLY THROUGHOUT ITS WHOLE LENGTH.
251. Pressures on points of support — Points of inflexion — Deflection, - - 187
252. Both spans loaded uniformly, - 190
CASE II. — CONTINUOUS GIRDERS OP THREE SYMMETRICAL SPANS LOADED
SYMMETRICALLY.
253. Pressure on points of support — Points of inflexion — Deflection, - - 191
254. Three spans loaded uniformly, - 193
255. Maximum strains in flanges, - 194
256. Maximum strains in web — Ambiguity in calculation, - 195
257. Permanent load, shearing strain, - - 195
258. Advantages of continuity — Not desirable for small spans with passing loads,
or where the foundations are insecure, - - 196
CASE III. — GIRDERS OF UNIFORM SECTION IMBEDDED AT BOTH ENDS AND
LOADED UNIFORMLY.
259. Strain at centre theoretically one-third, and strength theoretically once and
a half that of girders free at the ends, - - - - 197
CASE IV. — GIRDERS OF UNIFORM SECTION IMBEDDED AT BOTH ENDS AND
LOADED AT THE CENTRE.
260. Strain at centre theoretically one-half, and strength theoretically twice that
of girders free at the ends, - - - 198
CHAPTER X.
QUANTITY OF MATEEIAL IN BRACED GIRDERS.
CASE I.— SEMI-GIRDERS LOADED AT THE EXTREMITY, ISOSCELES BRACING.
261. Web, - 200
262. Flanges, - 201
CASE II.— SEMI-GIRDERS LOADED UNIFORMLY, ISOSCELES BRACING.
263. Web, length containing a whole number of bays, - - 201
264. Web, length containing a half -bay, - 202
265. Flanges, - - 202
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED AT AN
INTERMEDIATE POINT, ISOSCELES BRACING.
266. Quantity of material in the web is the same for each segment, • - 202
267. Flanges, ...... - 203
XVlii CONTENTS.
ART. PAGE
CASE IV. — GIBBERS SUPPORTED AT BOTH ENDS AND LOADED UNIFORMLY,
ISOSCELES BRACING.
268. Web, the length containing an even number of bays, - 203
269. Web, the length containing an odd number of bays, • - 204
270. Flanges, - - 204
CASE V.— SEMI-GIRDERS LOADED AT THE EXTREMITY, VERTICAL AND
DIAGONAL BRACING.
271. Web, - 205
CASE VI.— BOWSTRING GIRDERS UNIFORMLY LOADED.
272. Flanges, - - 205
274. Quantity of material in the bracing independent of depth — Weights of
railway girders up to 200 feet span are nearly as the squares of their
length, - - 207
CHAPTER XL
ANGLE OF ECONOMY.
275. Angle of economy for isosceles bracing is 45°, - - 209
276. Angle of economy for vertical and diagonal bracing is 55°, - 209
277. Isosceles more economical than vertical and diagonal bracing, - - 210
278. Trigonometrical functions of 0, - 210
279. Relative economy of different kinds of bracing — Continuous web theoretically
twice as economical as a braced web, - - 211
CHAPTER XII.
TORSION.
280. Twisting moment, - - - - 212
281. Solid round, square, or polygonal shafts — Coefficient of torsional rupture, T- 213
282. Hollow shafts of uniform thickness, - 213
283. Coefficients of torsional rupture for solid round shafts, - - 214
284. Moment of resistance of torsion, - - 215
285. Solid round shafts, - - - - 217
286. Hollow round shafts, . - 217
287. Solid square shafts, - - 218
CHAPTER XIII.
STRENGTH OF HOLLOW CYLINDERS AND SPHERES.
288. Hollow cylinders— Elliptic tubes - - 220
CONTENTS. XIX
ART. PAOB
289. Cylinder ends _ 222
290. Hollow spheres, . 223
CHAPTER XIV.
CRUSHING STRENGTH OF MATERIALS.
291. Nature of compressive strain, . 224
292. Flexure — Crushing — Buckling — Bulging — Splintering, - - 225
293; Crushing strength of short pillars — Angle of fracture, - 225
CAST-IRON.
294. Crushing strength of cast-iron, - 228
295. Hardness and crushing strength of thin castings greater near the surface
than in the heart — Crushing strength of thin greater than that of thick
castings. - - 232
296. Hardness and crushing strength of thick castings at the surface and in the
heart not materially different, - 232
WROUGHT-IRON.
297. Crushing strength of wrought-iron — 12 tons per square inch is the limit of
compressive elasticity of wrought-iron, - 233
298. Crushing strength of steel — 21 tons per square inch is the limit of compres-
sive elasticity of steel, - - 233
VARIOUS METALS.
299. Crushing strength of copper, brass, tin, lead, aliTminium bronze, zinc, - 235
TIMBER.
300. Crushing strength of timber — Wet timber not nearly so strong as dry, - 236
STONE, BRICK, CEMENT AND GLASS.
301. Crushing strength of stone and brick, - 237
302. Mode of fracture of stone, - - 239
303. Crushing strength of rubble masonry, - 240
304. Crushing strength of Portland cement, mortar and concrete, - 240
305. Crushing strength of glass, - - 243
CHAPTER XV.
PILLARS.
306. Very long thin pillars, - 244
307. Long solid rectangular pillars— Long solid round pillars — Long hollow
round pillars — Strength of long pillars depends on the coefficients of
elasticity, - - 246
XX CONTENTS.
ART. PAGE
308. Strength of similar long pillars are as their transverse areas — Weights of
long pillars of equal strength and similar in section, but of different lengths,
are as the squares of their lengths, - 246
309. Weight which will affect a very long pillar is very near the breaking weight, 247
310. Pillars divided into three classes according to length, - 248
Long pillars which fail by flexure; length, if both ends are flat and flrmly
bedded, exceeding 30 diameters for cast-iron and timber, and 60
diameters for wrouyht-iron.
311. Long pillars with flat ends firmly bedded are three times stronger than
pillars with round ends, - - 249
312. Strength of pillars with one end round and the other flat is a mean between
that of a pillar with both ends round and one with both ends flat, - 249
313. A long pillar with ends firmly fixed is as strong as a pillar of half the length
with round ends, - - 250
314. Hodgkinson's laws apply to cast-iron, steel, wrought-iron and wood, - 250
315. Position of fracture in long cast-iron pillars, - 251
316. Discs on the ends add but little to the strength of flat-ended pillars, - 251
317. Enlarging the diameter in the middle of solid pillars increases their strength
slightly, - 251
318. Enlarging the diameter in the middle or at one end of hollow pillars does
not increase their strength, - 251
319. Solid square cast-iron pillars yield in the direction of their diagonals, - 251
320. Long pillars irregularly fixed lose from two-thirds to four-fifths of their
strength, - - 251
321. Strength of similar long pillars is as their transverse area, - - 252
CAST-IKON PILLARS.
322. Hodgkinson's rules for solid or hollow round cast-iron pillars whose length
exceeds 30 diameters, - 252
323. Hodgkinson's rules for solid or hollow round cast-iron pillars of medium
length ; i.e., pillars whose length is less than 30 diameters, with both ends
flat and well-bedded, - 256
324. A slight inequality in the thickness of hollow cast-iron pillars does not
impair their strength materially — Rules for the thickness of hollow cast-
iron pillars, - 257
325. + and H shaped pillars, - 258
326. Relative strength of round, square and triangular solid cast-iron pillars, - 258
327. Gordon's rules for pillars, - 259
328. Solid or hollow round cast-iron pillars, - 259
329. Solid or hollow rectangular cast-iron pillars, - 261
WROUGHT-IRON PILLARS.
330. Solid wrought-iron pillars, ------- 262
CONTENTS XXI
ART. PAGE
331. Solid wrought-iron pillars stronger than cast-iron pillars when the length
exceeds 15 diameters, - - 265
332. Pillars of angle, tee, channel and cruciform iron, - - 265
333. Eesistance of long plates to flexure, - 269
334. Strength of rectangular wrought-iron tubular pillars is independent of their
length within certain limits, - 269
335. Crushing unit-strain of wrought-iron tubes depends upon the ratio between
the thickness of the plate and the diameter or breadth of the tube — Safe
working-strain of rectangular wrought-iron tubes, - 269
336. Solid steel pillars, - 279
TIMBER PILLARS.
337. Square is the strongest form of rectangular timber pillar — Hodgkinson's
rules for solid rectangular timber pillars, - - 280
338. Eondelet's and Brereton's rules for timber pillars, - - 281
STONE PILLARS.
339. Influence of the height and number of courses in stone columns, - - 283
340. Crushing strength of rollers and spheres, - - 283
BRACED PILLARS.
341. Internal Bracing— Example, - 284
342. Each bay of a braced pillar resembles a pillar with rounded ends — Com-
pression flanges of girders resemble braced pillars, - 286
343. Strength of braced pillars is independent of length within certain limits —
Working strain, - - 287
CHAPTER XVI.
TENSILE STRENGTH OF MATERIALS.
344. Nature of tensile strain, - 288
CAST-IRON.
345. Tensile strength, - - 288
346. Cold-blast rather stronger than hot-blast iron — Mixtures stronger than
simple irons, - 290
347. Re-melting, within certain limits, increases the strength and density of cast-
iron, - 290
348. Prolonged fusion, within certain limits, increases the strength and density
of cast iron, - 292
349. Tensile strength of thick castings of highly decarbonized iron greater than
that of thin ones — Annealing small bars of cast-iron diminishes their
density and tensile strength, - - - 294
350. Indirect pull greatly reduces the tensile strength of cast-iron, - - 296
XXll CONTENTS.
ART. PAGE
351. Cast-iron not suited for tension, - - 296
WROUGHT-IRON.
352. Tensile strength of wrought-iron — Fractured area — Ultimate set, - - 296
353. Tensile strength of wrought-iron, mean results, - - 303
354. Kirkaldy's conclusions, - - 304
355. Strength of iron plates lengthways is 10 per cent, greater than crossways —
Removing skin of wrought-iron does not injure its tensile strength, - 307
356. Bar and angle iron are tougher and stronger than plates — Boiler plates —
Ship plates — Hard iron unfit for ship -building, - -308
357. Large forgings not so strong as rolled iron — Annealing reduces the tensile
strength of small iron, but increases its ductility — Annealing injurious to
large forgings — Very prolonged annealing injurious to all wrought-iron —
Excessive strain renders iron brittle, - - 308
IRON WIRE.
358. Tensile strength of iron wire — Annealing iron wire reduces its tensile
strength, - - 309
STEEL.
359. Tensile strength, ultimate set and limit of elasticity of steel, - 311
360. Steel plates often deficient in uniformity and toughness — Punching as
compared with drilling greatly reduces the tensile strength of steel plates ;
strength generally restored by annealing — Annealing equalizes different
qualities of steel plates, - - 316
STEEL WIRE.
361. Tensile strength of steel wire, - 319
VARIOUS METALS AND ALLOYS.
362. Tensile strength of various metals and alloys, - 319
363. Gun metal or bronze — High temperature at casting injurious to bronze, - 320
364. Alloys of copper and tin, - - 321
TIMBER.
365. Tensile strength of timber, - - 321
366. Lateral adhesion of the fibres, - 324
STONE, BRICK, MORTAR, CEMENT, GLASS.
367. Tensile strength of stone, - - 325
368. Tensile strength of Plaster of Paris and Lime mortar, - - - 326
369. Tensile strength of Portland cement and Cement mortar — Organic matter
or loam very injurious to Cement mortar, - 326
370. Tensile strength of Roman cement — Natural cements generally inferior to
the artificial Portland, - - - - - - - 330
CONTENTS. XX111
AET. PAGE
371. Tensile strength of Keene's, Parian and Medina cements, - - 333
372. Adhesion of Plaster of Paris and mortar to brick or stone, - - 334
373. Grant's conclusions, - - 336
374. Tensile strength of glass — Thin plates of glass stronger than stout bars —
Crushing strength of glass is 12 tunes its tensile strength, - 337
375. Tensile strength of cordage, - 338
376. Strength and weight of cordage— --English rule — French rule, - 340
377. Working strain of cordage, - - 340
378. Stud-link or Cable chain — Close-link or Crane chain — Long open-link or
Buoy chain — Middle-link chain, - - 340
379. Tensile strength of stud-chain, - - 341
380. Government Proof-strain for Stud-chain, - - 342
381. Close-link chain — Proof -strain, - - 345
382. Long open-link chain — Admiralty proof-strain — Trinity proof-strain —
French-proof, - - 346
383. Working strain of chains should not exceed one-half the proof-strain, - 348
384. Comparative strength of stud and open-link chain, - - 348
385. Weight and strength of bar-iron, stud-chain, close-link chain and cordage, - 349
WIRE KOPE.
386. Tensile strength of round iron and steel wire ropes and hemp rope, - 350
387. Tensile strength of flat iron and steel wire ropes and flat hemp rope, - 353
388. Safe working load of wire rope, - - 353
MISCELLANEOUS MATERIALS.
389. Tensile strength of bone, leather, whalebone, gutta-percha and glue, - 853
CHAPTER XVII.
SHEAKING-STKAIK
390. Shearing in detail — Simultaneous shearing, - 356
391. Shearing strength of cast-iron, - - 357
392. Experiments on punching wrought-iron, - 357
393. Experiments on shearing wrought-iron, - • 358
394. Shearing strength of wrought-iron equals its tensile strength, - 360
395. Shearing strength of rivet steel is three-fourths of its tensile strength, - 361
396. Shearing strength of copper,
397. Shearing strength of fir in the direction of the grain— Shearing strength of
oak treenails, ... - 361
XXIV CONTENTS.
CHAPTER XVIII.
ELASTICITY AND SET.
ART. PAGE
398. Limit of elasticity — Set — Hooke's Law of elasticity practically true, - 364
CAST-IRON.
399. Decrement of length and set of cast-iron in compression — Coefficient of
compressive elasticity, - - 365
400. Hodgkinson's formulae for the decrement of length and set of cast-iron in
compression, - 368
401. Increment of length and set of cast-iron in tension — Coefficient of tensile
elasticity, - - 368
402. Hodgkinson's formulae for the increment of length and set of cast-iron in
tension, - - 370
403. Coefficients of tensile, compressive and transverse elasticity of cast-iron
different, - - 370
404. Increment of length and set of cast-iron extended a second time — Relaxa-
tion of set — Viscid elasticity, - 371
405. Set of cast-iron bars from transverse strain nearly proportional to square of
deflection, - - 371
WROUGHT-IRON.
406. Decrement of length of wrought-iron in compression — Coefficient of com-
pressive elasticity — Elastic limit, - - 372
407. Increment of length and set of wrought-iron in tension — Coefficient of tensile
elasticity— Elastic limit — Effects of cold-hardening and annealing on the
elasticity of iron, - - 373
408. Elastic flexibility of cast-iron twice that of wrought-iron — Law of elasticity
truer for wrought-iron than for cast-iron, - - 379
409. Stiffness of imperfectly elastic materials improved by stretching — Practical
method of stiffening wrought-iron bars — Limit of elasticity of wrought-
iron equals 12 tons per square inch — Proof -strain should not exceed the
limit of elasticity, - - 379
410. Experiments on elasticity liable to error — Sluggish or viscid elasticity, - 380
STEEL.
411. Law of elasticity true for steel— Coefficient and limit of elasticity of steel, - 381
TIMBER.
412. Limit of elasticity of timber not accurately defined — Coefficient of elasticity
depends on the dryness of the timber, - - 382
STONE.
413. Vitreous materials take no set, - - 382
CONTENTS. XXV
CHAPTER XIX.
TEMPERATURE.
ART. PAGE
414. Arches camber, suspension bridges deflect, and girders elongate from eleva-
tion of temperature — Expansion rollers, - - 384
415. Alteration of length from change of temperature — Coefficients of linear
expansion, • 385
416. Expansibility of timber diminished, or even reversed by moisture, - - 389
417. Moisture increases the expansibility of some stones — Raising the tem-
perature produces a permanent set in others,
418. A change of temperature of 15° C. in cast-iron, and 7'50 C. in wrought-iron,
are capable of producing a strain of one ton per square inch — Open-work
girders in the United Kingdom are liable to a range of 45° C., - - 390
419. Tubular plate girders are subject to vertical and lateral motions from
changes of temperature — Open-work girders are nearly quite free from
these movements, - - 391
420. Transverse strength of cast-iron not affected by changes of temperature
between 16° F. and 600° F., - 392
421. Tensile strength of plate-iron uniform from 0° F. to 400° F., - 392
CHAPTER XX.
FLANGES.
422. Cast-iron girders, - - 393
423. Cellular flanges, - - 394
424. Piled flanges — Long rivets not objectionable, - 395
425. Punching and drilling tools, - 396
426. Position of roadway — Compression flange stiffened by the compression
bracing of the web, - 397
427. Waste of material in flanges of uniform section — Arched upper flange —
Waste of material in continuous girders crossing unequal spans, - - 398
428. An excess of strength in one flange does not increase the strength of braced
girders, though it may slightly increase the strength of girders with
continuous webs, - - 399
429. Bearing surface on the abutments — Working load on expansion rollers, - 399
CHAPTER XXI.
WEB.
430. Plate web — Calculation of strains, - ... 400
431. Ambiguity respecting direction of strains in continuous webs — Bracing
generally more economical than plating — Minimum thickness of plating
in practice — Relative corrosion of metals, - 400
XXVI CONTENTS.
ART. PAGE
432. Plating more economical than bracing near the ends of very long girders —
Continuous webs more economical in shallow than in deep girders, - 402
433. Greater proportion of a continuous web available for flange -strains in
shallow than in deep girders, - -404
434. Deflection of plate girders substantially the same as that of lattice girders, 404
435. Webs of cast-iron girders add materially to their strength, - 405
436. Minute theoretic accuracy undesirable, - - 405
437. Multiple and single systems of triangulation compared — Simplicity of
design desirable — Ordinary sizes of iron, - - - 406
438. Testing small girders by a central weight equal to half the uniform load is
inaccurate, . 407
439. Connexion between web and flanges — Uniform strain in flanges— Trough
and M-shaped girders — Rivets preferable to pins — Limit of length of
single- webbed girders, - - 407
CHAPTER XXII.
CROSS-BRACING.
440. Weather bracing — Maximum force of wind — Pressure of wind may be
considered as uniformly distributed for calculation, 411
441. Rouse's table of the velocity and force of wind — Beaufort scale, - - 412
442. Cross-bracing must be counterbraced — Best form of cross-bracing — Initial
strain advantageous, - - 413
443. Strains produced in the flanges by cross-bracing — End pillars of girders
with parallel flanges and bow of bowstring girders are subject to transverse
strain, - - 414
CHAPTER XXIII.
CROSS-GIRDERS AND PLATFORM.
444. Maximum weight on cross-girders— Distance between cross-girders, - 416
445. Rail girders or keelsons— Economical distance between the cross-girders—
Weight of single and double lines— Weight of snow, - - 417
446. Regulations of Board of Trade, - - - 419
447. Roadways of public bridges — Buckled-plates, - 420
CHAPTER XXIV.
COUNTERBRACING.
448. Permanent or dead load— Passing or live load, - - 424
449. Passing loads require centre of web to be counterbraced —Large girders
require less counterbracing in proportion to their size than small ones, - 425
CONTENTS. XXV11
AKT. PAGE
450. Counter-bracing of vertical and diagonal bracing — Large bowstring girders
require little counterbracing, - -426
CHAPTER XXV.
DEFLECTION AND CAMBER
451. Deflection curve of girders with horizontal flanges of uniform strength is
circular, - • - 428
452. Deflection an incorrect measure of strength, - 428
453. Camber ornamental rather than useful — Permanent set after construction, - 429
454. Loads in rapid motion produce greater deflection than stationary or slow
loads — Less perceptible in large than small bridges — Deflection increased
by road being out of order — Railway bridges under 40 feet span require
extra strength in consequence of the velocity of trains, - - 429
455. Effect of centrifugal force, - - - 433
456. Practical methods of producing camber and measuring deflection, - - 434
CHAPTER XXYI.
DEPTH OF GIRDERS AND ARCHES.
457. Depth of girders generally varies from one-eighth to one-sixteenth of the
span — Depth determined by practical considerations, - - 435
458. Economical proportion of web to flange — Practical rules, - - 437
459. Depth of iron and stone arches, - - - 438
CHAPTER XXVII.
CONNEXIONS.
460. Appliances for connecting iron-work — Strength of joints should equal that
of the adjoining parts — Screws, - - 442
461. Bolts or pins— Proportions of eye and pin in flat links — Upsetting and
bearing surface, - - - 442
462. Rivets in single and double shear — Proportions of rivets in tension and
compression joints— Hodgkinson's rules for the strength of single and
double riveting — Injurious effect of punching holes — Relative strength of
punched and drilled holes, - - - - 446
463. Covers — Single and double covers compared — Lap-joint,
464. Tension joints of piles — Compression joints of piles require no covers if the
plates are well butted — Cast- zinc joints, .... - 451
465. Various economical arrangement of tension-joints, - - 452
466. Contraction of rivets and resulting friction of plates— Ultimate strength of
rivet-joints not increased by friction, • • 454
XXVill CONTENTS.
ART. PAGE
467. Girder-makers, Boiler-makers and Shipbuilders' rules for riveting — Chain-
riveting, - - ... 455
468. Adhesion of iron and copper bolts to wood — Strength of clenches and
forelocks, - - 458
469. Adhesion of nails and wood screws, - - - 462
CHAPTER XXVIII.
WORKING STRAIN AND WORKING LOAD.
470. Working strain — Fatigue — Proof Strain — English rule for working strain —
Coefficient of safety, .... - - 466
471. Effects of long-continued pressure on cast-iron pillars and bars, - - 467
472. Effects of long-continued impact and frequent deflections on cast-iron bars, - 470
473. Working strain of cast-iron girders — Rule of Board of Trade — Working
strain of cast-iron arches — French rule— Proving cast iron, - 473
474. Working load on cast-iron pillars, - - 477
WROUGHT-IRON.
475. Effects of repeated deflections on wrought-iron bars and plate girders, - 478
476. Net area only available for tension — Allowance for the weakening effect of
punching — Rule of Board of Trade for wrought-iron railway bridges —
Tensile working strain of wrought-iron — French rule for railway bridges, 482
477. Gross area available for compression — Compressive working strain of
wrought-iron — Flanges of wrought-iron girders are generally of equal
area, - - 484
478. Shearing working strain — Pressure on bearing surfaces— Knife edges, - 485
479. Working strain of boilers — Board of Trade rule — French rule, • • 486
480. Working strain of engine-work, - - - 487
481. Examples of working strain in wrought-iron girder and suspension bridges, 488
482. Strength and quality of materials should be stated in specifications — Proof
strain of chains and flat-bar-links— Admiralty tests for plate -iron, - 491
STEEL.
483. Working strain for steel— Steel pillars— Admiralty tests for steel plates, - 493
TIMBER.
484. English, American and French practice — Permanent working strain-
Temporary working strain, 495
485. Short life of timber bridges— Risk of fire, - - 496
486. Working load on piles depends more upon the nature of the ground than
upon the actual strength of the timber — Working load at right angles to .
the grain, - - ..... 490
CONTENTS. xxix
FOUNDATIONS, STONE, BRICK, MASONRY, CONCRETE.
ART. PACE
487. Working load on foundations of earth, clay, gravel and rock, - 498
488. Working load on rubble masonry, brickwork, concrete and ashlar-work, - 500
WORKING LOAD ON RAILWAYS.
489. A train of engines is the heaviest working load on 100-foot railway girders —
Three-fourths of a ton per running foot is the heaviest working load on 400-
foot girders — Weight of engines— Girders under 40 feet liable to con-
centrated working loads, - - 503
490. Standard working loads for railway bridges of various spans, - 510
491. Effect of concentrated loads upon the web, - - 513
492. Proof load of railway bridges — English practice — French Government rule, 514
WORKING LOAD ON PUBLIC BRIDGES AND ROOFS.
493. Men marching in step and running cattle are the severest loads on suspen-
sion bridges — A crowd of people is the greatest distributed load on a
public bridge — French and English practice — 100 Ifes. per square foot
recommended as the standard working load on public bridges — Public
bridges sometimes liable to concentrated loads as high as 12 tons on one
wheel, - - 515
494. Weight of roofing materials and working loads on roof s — Weight of snow —
Pressure of wind against roofs, - - 517
CHAPTER XXIX.
ESTIMATION OF GIRDER-WORK.
495. Theoretic and empirical quantities — Allowance for rivet holes in parts in
tension generally varies from one-third to one-fifth of the net section, - 525
496. Allowance for stiffeners in parts in compression varies according to their
sectional area — Large compression flanges seldom require any allowance
for stiffening — Compression bracing requires large percentage?, - - 525
497. Allowance for covers in flanges varies from 12 to 15 per cent, of the gross
section — Estimating girder-work a tentative process, - - 526
EXAMPLE 1.
498. Double-line lattice bridge 267 feet long, - - 527
499. Permanent strains — Strains from train-load — Economy due to continuity, - 529
EXAMPLE 2.
500. Single-line lattice bridge 400 feet long, - - 529
EXAMPLE 3.
501. Single-line lattice bridge 400 feet long, as in Ex. 2, but with higher unit-
strains, --....--- 532
XXX CONTENTS.
ART. PAGE
502. Great economy from high unit-strains in long girders — Steel plates, - 533
503. Suspension principle applicable to larger spans than girders, - 534
EXAMPLE 4.
504. Single-line lattice bridge 400 feet long, with increased depth, - 534
505. Weights of large girders do not vary inversely as their depth, - 536
EXAMPLE 5.
506. Single-line lattice bridge 480 feet long, - - 536
507. Waste of material in defective designs, - - 538
EXAMPLE 6.
508. Single-line lattice bridge 480 feet long, as in Ex. 5, but with higher unit-
strains, - - 538
509. Great economy from high unit-strains in large girders, - - 540
EXAMPLE 7.
510. Single-line lattice bridge 480 feet long, as in Ex. 5, but with increased
depth, - - 540
511. Weights of large girders do not vary inversely as their depth, - - 542
EXAMPLE 8.
512. Single-line lattice bridge 600 feet long, - - 542
EXAMPLE 9.
513. Single-line lattice bridge 600 feet long, as in Ex. 8, but with higher unit-
strains, -•-.-- . 544
514. Great economy from high unit-strains in very large girders, - 546
EXAMPLE 10.
515. Single-line lattice bridge 600 feet long, as in Ex. 8, but with increased
depth, - 546
516. Weights of large girders vary inversely in a high ratio to their depth, - 548
EXAMPLE 11.
517. Counter-bracing required for passing loads cannot be- neglected in small
bridges — Single-line lattice bridge 108 feet long, - - 548
518. Error in assuming the permanent load uniformly distributed in large girders —
Empirical percentages open to improvement, • - - 550
519. Fatigue of the material greater in long than in short bridges, - 551
GIRDERS UNDER 200 FEJET IN LENGTH.
520. Flanges nearly equal in weight to each other, and web nearly equal in
weight to one flange, ...... - 552
CONTENTS. XXXI
ART. PAGE
521. Anderson's rule — Weights of lattice and plate girders under 200 feet in
length, - - 553
522. Weights of similar girders under 200 feet span vary nearly as the squares of
their lengths — No definite ratio exists between the lengths and weights of
very large girders, ... - - 557
CHAPTER XXX.
LIMITS OF LENGTH OF GIKDERS.
523. Cast-iron girders in one piece rarely exceed 50 feet in length — Compound
girders advisable for greater spans if cast-iron is used, - - - 558
524. Practical limit of length of wrought-iron girders with horizontal flanges
does not exceed 700 feet, - - ... 553
CHAPTER XXXI.
CONCLUDING OBSERVATIONS.
525. Hypothesis to explain the nature of strains in continuous web, - - 561
526. Strains in ships, - - -563
527. Iron and timber combined form a cheap girder — Timber should be used
in large pieces, not cut up into planks — Simplicity of design most desirable
in girder work, •- •'- - - - - . 564
APPENDIX.
528. Boyne Lattice Bridge, general description and detailed weights of girder-
work, - ..... 567
529. Working strains and area of flanges, ... 571
530. Points of inflexion — Pressure on points of support, • - 572
531 . Maximum strains in the flanges of the side spans - - 572
532. Maximum strains in the flanges of the centre span, - - 573
533. Maximum strains in the flanges over the piers, .... 573
534. Points of inflexion fixed practically — Deflection — Camber, • - 574
535. Experiments on the strength of braced pillars, - - - 577
536. Experiments on the effect of slow and quick trains on deflection, - - 581
537. Newark Dyke Bridge, Warren's Girder, - - 582
538. Chepstow Bridge, Gigantic Truss, - - 583
539. Crumlin Viaduct, Warren's Girder, - - 584
540. Public Bridge over the Boyne, Lattice Girder, - - 585
541. Bowstrintj Bridge on the Caledonian Railway, - - - - 587
542. Charing Cross Lattice Bridge, - 587
543. Conway Plate Tubular Bridge, - - 588
544. Brotherton Plate Tubular Bridge, - - 591
545. Size and weights of various materials, - - 591
THE
THEORY OF STRAINS IN GIRDERS
AND
SIMILAR STRUCTURES.
CHAPTER I.
INTRODUCTORY.
1. Strain— Tension — Compression — Transverse strain —
Shearing-strain — Torsion. — On the application of force aft
bodies change either form or volume, or both together. Forces
considered with reference to the internal changes they tend to
produce in any solid are termed strains* and may be classified as
follows : —
Tensile strains,
Compressive do.,
Transverse do.,
Shearing do.,
Torsional do.,
This five-fold division is made for convenience merely, for the
strength of any material, in whatever manner it may be employed,
depends ultimately on its capability of sustaining strains which tend
either to tear its parts asunder or to crush them together. It is
therefore of essential importance to know the ultimate resistance
to tension or compression which each material possesses, and thence
deduce those strains which may be safely imposed in practice.
To this end various experimenters have devoted their attention ;
* It will be useful for the student to know that some writers apply the term
stress to what I have termed strain in the text, that is, to the combination of internal
forces or reactions which the particles of any body exert in resisting the tendency of
external forces to produce alteration of form, and they apply the term strain to what I
call deformation, that is, to alteration of form resulting from stress.
producing
fracture
tearing asunder,
crushing,
breaking across,
cutting asunder,
twisting asunder.
2 INTRODUCTORY. [CHAP. I.
in the United Kingdom, none with more perseverance or success
than the late Eaton Hodgkinson, Esq., to whose life-long labours
we are mainly indebted for the physical investigations on which
calculations of the strength of structures are based.
a. Unit-strain— Inch-strain— Foot-strain. — Wherever English
measures are used, tensile and compressive forces are measured by
the number of tons or pounds strain on the square inch or square
foot. It will be convenient, however, to have some short expression
for the strain on the unit of sectional area, irrespective of any
particular measure of length or weight, and I have adopted
the term Unit-strain to denote this quantity, and the words
Inch-strain or Foot-strain to express the strain per square inch or
square foot, as the case may be. The unit-strains of tension and
compression are represented indifferently by the symbol /, unless
it be desirable to distinguish them, in which case the unit-strain of
compression is represented by the symbol /'. Thus, if F be the
total strain in any bar whose area — a, we have
F = af. (1)
Ex. 1. If the crushing uuit-strain of cast-iron be 42 tons per square inch, what
weight will crush a short solid pillar 9 inches in diameter ?
Here, a = ?-^ ^_5 = 63 "6 inches,
/ = 42 tons.
Answer. F = af = 63'6 X 42 = 2,671 tons.
Ex. 2. If the tearing unit-strain of beech be 11,500 pounds per square inch, what
force in tons will tear asunder a tie-beam 1 5 inches square ?
Here, a = 15X15 = 225 square inches,
/= 11,500 Ibs.
Ansiver. F = 225 X l1^'00 = 1,155 tons.
3. Elasticity— Cubic elasticity — Linear elasticity. — Besides
the strains of tension and compression another matter claims
attention, namely, the alteration of length or, in other words, the
elongation and shortening of the material subject to strain. Elasticity
is the property which all bodies under the influence of external
force possess to a greater or less degree of perfection of returning
CHAP. I.] INTRODUCTORY. 3
to their original volume or form after the force has been with-
drawn. Thus we have Cubic elasticity or elasticity of volume,
and Linear elasticity or elasticity of form. Fluids possess elasticity
of volume, but not of form. Solids possess both, but linear elasticity
alone demands our attention in questions relating to the strength
of materials.
4. Elastic stiffness and Elastic flexibility are correlative
terms which express the strength or weakness of the elastic reaction
of the fibres of any elastic solid, whether that reaction be due to
tensile or compressive strains, applied separately or in combination
so as to produce flexure or torsion. Thus, glass is elastically stiff,
indian-rubber elastically flexible. In general, the terms Stiffness
and Flexibility are not restricted to elastic solids, but express merely
the relative amount of resistance to change of form, whether the
material returns to its original shape or not after the force is
withdrawn. In this sense copper is stiffer than lead, but neither
is elastic, or but very slightly so. Elasticity should not, as in
popular language, be confounded with a wide range of elastic
flexibility. Glass, for instance, is both stiff and elastic, whereas
indian-rubber, though very flexible, is less perfectly elastic than
glass, that is, it returns with less exactness to its original form
after being strained. Again, a thin spring of tempered steel is
both elastic and flexible. In popular language, however, indian-
rubber is said to be more elastic than glass or steel, because its
range of elastic flexibility exceeds that of either.
5. Ductility — Toughness — Brittleness. — Ductility is the re-
verse of elasticity and is the property of retaining a permanent
change of form after the force which produced it has been removed,
and the wider the range over which a body can be altered in shape
the more ductile it is said to be. Gold, for instance, is one of the
most ductile of metals, as it can be drawn out into extremely fine
wire or hammered into leaves of extraordinary thinness. Toughness
consists in the union of tenacity with ductility. Brittleness is
incapability of sustaining rapid changes of form without fracture,
and is opposed to toughness. Low-Moor iron, for instance, is
tough ; a bar of it can be twisted into a knot without breaking ;
4 INTRODUCTORY. [CHAP. I.
but highly tempered steel is brittle ; though more tenacious than
iron, it breaks short without any sensible change of length ; it is
not ductile ; it will not stretch under strain. Sealing-wax also is
brittle ; though more ductile than iron under prolonged pressure, it
is not tenacious and will not bear a sudden change of shape with-
out fracture. Accurately speaking we may doubt if there is such
a thing as a perfectly elastic solid, for Mr. Hodgkinson's investi-
gations seem to prove that there is no strain, however slight, which
will not after its removal leave a permanent, though perhaps to
ordinary tests an inappreciable, alteration of length in any of the
materials on which he experimented. In other words, every
material is more or less ductile.* This view, however, is not held
by some authorities.
6. Set — Influence of duration of strain. — When the unit-
strain is considerable the defect of elasticity becomes very apparent
in some materials, especially in ductile metals, for they do not
return to their original length when released from strain, but are
sensibly elongated or shortened, as the case may be, by a certain
amount which varies according to the nature of the material and
the force applied. This residual elongation or shortening is called
the Set, and is not sensibly increased by subsequent applications
of the same unit-strain which first produced it. It should be
observed, however, that the ultimate set is not instantaneously pro-
duced on the application of force. Iron, and possibly all materials,
take time, more or less prolonged, to adapt themselves to new con-
ditions of strain. Hence, a rapidly applied force may snap a
brittle bar without producing any very perceptible change in its
length.
V. Hooked law— Law of elasticity— Limit of elasticity.— It
is evident that the elastic reaction of any material is equal to the
force producing extension or compression, and it has been proved
by experiment that the following law of uniform elastic reaction,
expressed by Hooke in the phrase " ut tensio sic vis," and generally
* Report of the Commissioners appointed to inquire into the application of Iron to
Railway Structures, 1849, App. A, p. 1. Also, Experimental Researches on the Strength
and other Properties of Cast-iron, by E. Hodgkinson, pp. 381, 409, 486.
CHAP. I.] INTRODUCTORY. 5
known as the Law of elasticity, though perhaps not accurately true
of any solid, is practically true of the materials used in construction.
When any material is strained either by a tensile or a compressive
force, the elastic reaction of the fibres (equal to the applied force) is
proportional to their increment or decrement of length, provided the
alteration of length does not exceed a certain limit beyond which
the above stated law ceases to apply, and the change of length,
no longer regular, increases for each additional unit of strain
more rapidly than the reaction due to the elasticity of the fibres ;
this produces set and ultimately rupture. Experience has proved
that the safe working strain of any material does not exceed
its sensible limit of uniform elastic reaction, generally called the
limit of elasticity ; indeed, it generally lies considerably within it.
The limit of elasticity may also be defined to be the greatest strain
that does not produce an appreciable set. It will be seen hereafter
that with some materials, such as glass, there is no limit of elasticity
short of rupture, as they are elastic up to the breaking point and
apparently take no set when the strain is removed.
8. Coefficient of elasticity, E — Table of coefficients. — The
coefficient of elastic reaction, or briefly, the Coefficient of elasticity, *
is represented by the symbol E, and is the weight (in Ibs.)
requisite to elongate or shorten a bar whose transverse section
equals a superficial unit (one square inch) by an amount equal to
its length, on the imaginary hypothesis that the law of elasticity
holds good for so great a range. In assuming that the coefficient
of elasticity is the same for compression and extension I have
followed Navier,t but some writers on the strength of materials
seem to overlook the fact that, if the law of elasticity be rigidly
exact, a given force of compression will shorten any material by the
same proportion of its original length that an equal tensile force
will extend it. In practice the coefficient of elastic compression
will generally be found to differ slightly from that of elastic
tension.
If a bar whose length = I be extended or compressed within
* Called also the Modulus of elasticity.
t Resume des Lemons donne"es a VEcole des Fonts et Chaussees, p. 41.
6
INTRODUCTORY.
[CHAP. i.
the limits of elasticity by a strain of / Ibs. per square inch, the incre-
ment or decrement of length X is expressed by the following relation,
whence,
(2)
Ex. How much will an inch-strain of 5 tons stretch a bar of wrought-iron whose
length equals 10 feet ?
Here (see table following), E = 24,000,000 Ibs.,
/= 5 tons,
I = 10 feet.
Answer.
=fl= 5 X 2,240 X 10 X 12 = .Q56
E 24,000,000.
It is obvious that the coefficient of elasticity should be deduced
from experiments in which the applied unit-strain lies within the
limit of elastic reaction. It should also be noted whether the
material has been previously stretched by excessive strain ; other-
wise the results will be anomalous. The following table contains
the coefficients of elasticity of various materials, derived chiefly
from experiments on transverse strain : —
Description of Material.
Coefficient of
Elasticity in fts.
per square inch.
E
Authority.
METALS.
Brass (cast),
8,930,000
Tredgold.
Gold (drawn), -
11,564,000
Wertheim.
Do. (annealed), -
7,943,000
do.
Gun metal (copper 8, tin 1),
9,873,000
Tredgold.
Iron (cast, from transverse strain),
18,400,000
do.
Do. (do., from direct tension or compression), -
Do. (wrought), -
12,000,000
24,000,000
Hodgkiiison.
do.
Lead (cast),
720,000
Tredgold.
Platina thread, -
24,240,000
Wertheim.
Do. (annealed), -
2-2,070,000
do.
Silver (drawn), -
10,465,000
do.
Do. (annealed), - ...
10,155,000
do.
Steel, - ...
29,000,000
Young.
Do., -
31,000,000
Fairbairn.
Tin (cast),
4,608,000
Tredgold.
Zinc (cast),
13,680,000
do.
TIMBER.
Acacia (English growth),
1,152,000
Barlow.
Ash, • ...
1,644,800
do.
Beech, -
1,353,600
do.
CHAP. I.]
INTRODUCTORY.
Description of Material.
Coefficient of
Elasticity in R>8.
per square inch.
Authority.
TIMBER— continued.
Birch (American black),
1,477,000
Barlow.
Do. (common), -
1,644,800
do.
Box (Australia),
2,155,200
Trickett.
Deal (Christiana),
1,589,600
Barlow.
Do. (Memel), -
1,603,600
do.
Elm,
699,840
do.
Fir (Mar Forest),
645,360
do.
Do. (do., another specimen),
869,600
do.
Do. (New England), -
2,191,200
do.
Do. (Riga),
1,328,800
do.
Do. (do., another specimen),
990,400
do.
Do. (Memel, across the grain), -
42,500
Bevan.
Do. (Scotch, do.),
24,600
do.
Greenheart,
2,656,400
Barlow.
Iron bark (Australia), -
1,669,600
Trickett.
Larch, -
616,320
Barlow.
Do. (another specimen),
1,052,800
do.
Mahogany (Honduras),
1,596,000
Tredgold.
Norway spar,
1,457,600
Barlow.
Oak (Adriatic), -
974,400
do.
Do. (African), -
2,305,400
do.
Do. (Canadian), -
2,148,800
do.
Do. (Dantzic),' -
1,191,200
do.
Do. (English), -
1,451,200
do.
Do. (do. inferior),
873,600
do.
Pine (Pitch), -
1,225,600
do.
Do. (Red), - - -
1,840,000
do.
Do. (do.),
1,200,000
Clark.
Do. (American yellow),
1,600,000
Tredgold.
Poon, -
1,689,600
Barlow.
Spotted gum (Australia),
Stringy bark (do.),
1,942,000
1,375,600
Trickett.
do.
Teak, -
2,414,400
Barlow.
STONES.
Marble (White),
2,520,000
Tredgold.
Quartz Rock (Holyhead, across lamination), - ,
Do. ( do., parallel to lamination),
4,598,000
545,000
Mallet,
do.
Slate (Welsh), -
15,800,000
Tredgold.
Do. (Westmoreland), -
12,900,000
do.
Do. (Scotch), -
15,790,000
do.
Do. (Portland), -
1,533,000
do.
MISCELLANEOUS.
Whalebone,
820,000
Tredgold.
Bone of Beef, -
2,320,000
Bevan.
Barlow, Barlow on the Strength of Materials.
Bevan, Philosophical Magazine, 1826, Vol. Ixviii., pp. Ill, 181.
Clark, The Britannia and Conway Tubular Bridges, p. 463.
Fairbairn, Report of British Association, 1867.
Hodgkinson, Report of Commissioners appointed to inquire into the application of
Iron to Railway Structures, 1849, pp. 108, 172.
8 INTRODUCTORY. [CHAP. I
Mallet, Philosophical Transactions, 1862, p. 671.
Tredgold, Tredyold on the Strength of Cast-iron.
Young, idem.
Wertheim, Resistance des Mattriaux, par M. Morin, p. 46.
9. mechanical laws — Resolution of forces. — The investiga-
tion of transverse strains may be reduced to the three following
fundamental laws in mechanics : —
If three forces acting at the same point balance (are in equilibrium),
three lines parallel to their directions will form a triangle the sides of
which are proportional to the forces. Also, If two out of three forces
which balance meet, the third passes through their point of inter-
section.
Hence, it follows that, if we know the magnitude and direction of
two intersecting forces, we can find both the magnitude and direction
Fig. i. of their resultant ; and if the directions
of any two components into which a
single known force is resolved be given,
the amount of these components can be
found. Thus, the weight W, Fig. 1,
is supported by an oblique tie and a
horizontal strut. The weight and the
strains in the tie and strut meet at A,
and may be represented by the triangle
h t s. Let the sides of the triangle be as the numbers 3, 4 and 5 ;
then, if W := 3 tons, t will sustain a tension of 5 tons, and s a
thrust or compression of 4 tons. Calling the angle the tie makes
with the vertical line 9, the relation between these three forces
may be algebraically expressed as follows : —
1O. The I<eveT. — If a weight rest upon a beam supported by
two props at its extremities, these props react with two upward
pressures whose sum is equal to the weight, and by the principle of
the lever the portion of the weight sustained by either prop is to the
whole weight as the remote segment is to the whole beam.
Thus, in Fig. 2, if W = 10 tons and the segments are as 3 : 2, the
CHAP. I.]
INTRODUCTORY.
Fig. 2.
reaction of the left abut-
ment, R = 4 tons; that
of the right, R' = 6 tons.
Calling the segments m
and w, these relations may
be algebraically expressed
as follows : —
R + R' =
R =
W,
R' =
m
W.
m -f- n " m + n
It is obvious that this principle is not affected by any bracing of
the beam within itself, provided it merely rests on the points of
support.
11. Equality of moments. — When any number of forces acting
in the same plane on a rigid body balance (are in equilibrium), the sum
of the moments of the forces tending to turn it in one direction round
any given point is equal to the sum of the moments of those tending to
turn it in the opposite direction. Also, when any number of forces
acting in the same plane have a single resultant, the sum of the
moments of each force round a given point is equal to the moment
of their resultant*
Thus, in Fig. 2, taking moments round the right abutment,
R X m + n = W n ; the amount of R' vanishes, since R' passes
through the point round which the moments are taken.
On these three mechanical laws — the Resolution of Forces, the
law of the Lever and the Equality of Moments — are founded all the
following investigations of the strength of materials when subject
to transverse strain.
13. Beam — Girder — Semi-girder. — The term Beam is
generally applied to any piece of material of considerable scantling,
whether subject to transverse strain or not; as for example,
" Collar-beam," " Tie-beam," " Bressummer-beana;" the two former
being subject to longitudinal strains of compression and tension
respectively, and the latter to transverse strain. The term Girder
is, however, restricted to beams subject to transverse strain and
* The moment of a force round a given point is the product of the force by the
perpendicular let fall on its direction from the point.
10 INTRODUCTORY. [CHAP. I.
exerting a vertical pressure merely on their points of support.
This term was originally applied to the main beams of floors, but
has now become universally adopted by engineers. A Semi- girder
is a cantilever, that is, a beam fixed at one extremity only and
subject to transverse strain ; in addition to its vertical pressure
it exerts a tendency to overthrow the wall or other structure to
which it is attached.
13. Flanged girder — Single- webbed girder — Iftouble-
webbed or Tubular girder — Box girder — Tubular bridge.—
In the term Flanged girder are included not only iron girders
of the ordinary I form, but also all girders which consist of one or
two flanges united to a vertical web, whether the latter be con-
tinuous as in plate girders, or open-work as in lattice and bowstring
girders. Flanged girders are again subdivided into Single-webbed
and Double-webbed or Tubular. A single-webbed girder is one
whose flanges are connected by a single vertical web. Thus, we
have "Single-webbed cast-iron girders," " Single- webbed plate
girders," " Single-webbed lattice girders," " Single-webbed bow-
string girders," &c. A Double-webbed or Tubular girder is one
whose flanges are connected by a double vertical web, continuous
or open-work as the case may be. Small tubular girders formed
of continuous plates are sometimes called Box girders. A Tubular
bridge is merely a tubular girder of such large dimensions that the
roadway passes through the tube.
In the following theoretic investigations all girders are assumed
to be horizontal and without weight, unless otherwise stated.
CHAP. IT.] FLANGED GIRDERS, ETC. 11
CHAPTER II.
FLANGED GIRDERS WITH BRACED OR THIN CONTINUOUS WEBS.
14. Transverse-strain — Shearing-strain. — The formulae
investigated in this chapter are, unless otherwise expressed,
applicable to all flanged girders whose webs are formed of bracing,
or if continuous, yet so thin that the transverse strength of the web
as an independent rectangular girder may be neglected without
sensible error. Our knowledge of the strains in this vertical web
when continuous is still imperfect. Analogy indeed leads us to
conclude that they follow laws similar to those which hold good in
braced girders, but in the absence of experimental proof this is to
a certain degree conjecture — a conjecture, however, which I feel
confident my readers will share after they have had the patience to
read through this book.
The mode in which a load affects a girder may be thus analysed.
From experience we learn that the load bends the girder downwards
and develops longitudinal strains of tension and compression in the
flanges. If the semi-girder, represented in Fig. 3, be supposed
divided into vertical slices or transverse sections of small thickness,
the weight tends to shear or separate the section on which it imme-
diately rests from the adjoining one. The lateral connexion of the
sections, however, prevents this separation, and the second section
is drawn down by a vertical force equal to the weight which tends
to shear it from the third section and so on. Thus, a vertical force
equal to the weight is transmitted from section to section as far as the
point of support. This vertical strain has been aptly named the
Shearing-strain ; but few writers, until the last few years, have
noticed the practical results which follow from the fact that this
force can be communicated from section to section only through the
medium of some diagonal strain. Respecting the exact directions of
the strains which this shearing force develops in a continuous web
12 FLANGED GIRDERS WITH [CHAP. II.
we know nothing positively ; it is probable that they assume various
directions crossing each other like close lattice-work, some vertical,
some diagonal, perhaps some curved. However this may be, we
know that certain of them must be diagonal, since the weight,
which is a vertical force, produces strains in the flanges, which are
longitudinal, through the medium of the web, which in fact fulfils
the part of bracing in a lattice girder. The reader will perceive
that we have really three sets of forces to deal with, namely,
horizontal, vertical, and diagonal forces. The latter, however, may
be resolved into horizontal and vertical components, and thus we
have at present only horizontal and shearing forces to consider,
recollecting that the shearing-strain of any transverse section of a
girder means the total vertical strain transmitted through that section,
including in the term shearing strain the vertical components of
diagonal strains.
15. Horizontal strains in braced or thin continuous
webs may be neglected. — When the vertical web of a girder
with horizontal flanges is open-work like latticing, the shearing-
strain is altogether transmitted through the bracing, the flanges
being capable of conveying strains in the direction of their length
only ; but when the web is continuous, as in a plate-girder, there can
be no doubt that a certain amount of shearing-force acts upon the
flanges also, so inconsiderable, however, that we may practically
neglect it. If, however, one or both flanges are curved, the whole or
a considerable portion of the shearing-strain is conveyed through
that part of the flange which is sloped, the amount depending upon
its angle of inclination. In this case the web has less duty to
perform than if the flanges were horizontal, and its sectional area
may therefore be reduced. It will also be observed that the
diagonal strains developed by the shearing force in a continuous
web have horizontal components within the web itself, and con-
sequently, a continuous web aids the flanges to a certain extent,
for those parts of the web which adjoin the flanges share the
horizontal strains in the latter, and this flange action of the web is
greater the thicker the web is. When, however, the web is very
thin, the total amount of this flange action of the web is small
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 13
compared with the strain in the flanges themselves and may
therefore be neglected without introducing any serious error. In
this chapter all horizontal strains in the web are neglected.
CASE I. — FLANGED SEMI-GIRDER LOADED AT THE EXTREMITY.
Fig. 3.
16. Flanges — At any cross section the horizontal compo-
nents of strain in the flanges are equal and of opposite
kinds — Strength of flanged girders varies directly as the
depth and inversely as the length.
Let W = the weight,
I = the distance of any cross section A B from W,
d = the depth of the girder at this cross section,
T = the horizontal strain of tension in the top flange
at A,
C = the horizontal strain of compression in the
bottom flange at B.*
The segment A B W is held in equilibrium by the weight W,
the horizontal forces of tension and compression in the flanges at
A and B, and the shearing and horizontal strains in the web at
A B. Since these forces balance, the sum of the moments of those
which tend to turn A B W round any point in one direction is
equal to the sum of those which tend to turn it round the same
point in the opposite direction (11). If the point lie in the cross
section A B, the moment of the shearing force will be cipher,
since its direction passes through this point. Neglecting the
* When the flanges are oblique, T and C represent the horizontal components of
their longitudinal strains. The vertical components are a portion of the shearing-
strain.
14 FLANGED GIRDERS WITH [CHAP. II-
horizontal strain in the web when continuous, and taking moments
round A and B successively, we obtain the following relations : —
Wl = Td=Cd (3)
whence,
T = C (4)
that is, at any cross section the horizontal component of tension in
one flange is equal to the horizontal component of compression in
the other.
If F represent the horizontal strain in either flange indifferently,
we have from eq. 3
* = " (5)
Eq. 5 proves that the weight which a flanged girder is capable of
supporting varies directly as the depth and inversely as the length.
When both flanges are horizontal, we have from eq. 4
«/=«'/ (6.)
where a and/ represent the sectional area and unit-strain of the
upper flange, and a' and f those of the lower flange. Hence, when
both flanges are horizontal, the unit-strains in the flanges are to
each other inversely as the areas.
Ex. 1. A semi-girder, 9 inches deep, supports 7 tons at its extremity ; what is the
strain in each flange at 12 feet from the load ?
Here, W = 7 tons,
1 = 12 feet,
d = 9 inches.
^«-(Eq.6). F = Wi = I^li2LL2 = 112 tons.
If 4 tons per square inch be a safe working strain in the flanges, the sectional area
112
of each flange should = -— = 28 square inches.
Ex. 2. If the flange be 15 inches wide and 1^ inches deep, what will be the
inch-strain ?
Here, a = 22'5 square inches,
F = 112 tons.
F 112
Answer. /= — = -__ = 5 tons inch -strain nearly.
d 22'5
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 15
Ex. 3. A wrought-iron semi-girder is 7 feet long and 11 inches deep, and each
flange is 4 inches wide and ^ an inch thick ; what weight at the end will break it
across, the tearing inch-strain of wrought-iron being 20 tons ?
Here, F = af = 4 X '5 X 20 = 40 tons,
d = 11 inches,
Z= 7 feet.
Answer (Eq. 5). W = *£ = 40* !1 = 5'24 tons.
I 7 X 12
17. Girder of greatest strength — Areas of horizontal
flanges should he to each other in the inverse ratio of their
ultimate unit-strains. — The distribution of a given amount of
material in the flanges, so as to produce the girder of greatest
strength, occurs when both flanges are simultaneously on the
point of rupture, for if either flange contain more material than
is required to sustain its proper strain when the other gives way,
it can spare some of the surplus material to strengthen the other.
When both flanges are on the point of rupture, / and f are
the ultimate unit-strains of tension and compression, and since
—, — ~f, it follows that, to ensure the greatest strength with a
given amount of material in a girder with horizontal flanges, the
sectional areas of the flanges should be to each other inversely as
their ultimate unit-strains — a result amply confirmed by experience.
is. Shearing-strain — The weh should contain no more
material than is requisite to convey the shearing-strain —
The quantity of material in the weh of girders with
parallel flanges is theoretically independent of their depth.—
The shearing-strain is the same at each vertical section of the semi-
girder and equals W (14). If the flanges are parallel this strain is
transmitted from section to section of the web (15), which should
therefore have the same sectional area throughout and be suffi-
ciently strong to transmit the shearing-strain to the wall or point
of support. The web should also for economical reasons contain no
more material than is requisite to transmit the shearing-strain, for
any surplus material, if placed in the flanges, would increase the
strength of the girder more than if it were to remain in the web,
since its leverage to sustain horizontal strains would be thereby
increased. This will appear clearer when the reader has perused
16 FLANGED GIRDERS WITH [CHAP. II.
the succeeding chapters. From these considerations it follows that
the quantity of material required in the web of a girder with parallel
flanges is theoretically independent of the depth.
19. Girder of uniform strength — Economical distribution
of material. — A girder of uniform strength is one in which all
parts, both flanges and web, are duly proportioned to the strain
which they have to bear, i.e., are equally capable of sustaining
the particular strain which is transmitted through them. If such
a girder were perfect, there is no reason why any one part should
fail before another, since the train in each part is the same
sub-multiple of the ultimate or breaking-strain of that part. The
girder of uniform strength is obviously the most economical also in
its proportions, for no part has a wasteful excess of material ; the
tensile or compressive unit-strain is constant throughout the entire length
of each flange respectively, and the shearing-unit-strain in each section
of the web is the same as in every other section.
50. Flange-area of semi-girder of uniform strength when
the depth is constant. — From eq. 6 we have when both flanges
are horizontal,
/-s
where / and a express the unit-strain and sectional area of either
flange indifferently at a distance I from the extremity.
In a girder of uniform strength / is constant for all values of I,
and the quantity -, to which f is
Fie. 4.— Plan. J a
proportional (since by hypothesis the
depth d is uniform), will be constant
for every value of I ; consequently a,
that is, the area of each flange, will
vary as /, and if the depth of the
flange be uniform, its breadth will
vary as /, and the plan of the flange
will be triangular, as in Fig. 4.
51. Depth of semi-girder of uniform strength when the
flange-area is constant. — If, however, one flange be sloped, /
CHAP. II.] BRACED OB THIN CONTINUOUS WEBS. 17
Fig. 5.-Elevation. and a in e<l- 7 aPPty to the horizontal
flange only ; hence, if its sectional
area and unit-strain remain uniform,
d will vary directly as /, and the side
elevation of the girder will be trian-
gular as in Fig. 5. The strain in the
oblique flange exceeds that in the
horizontal flange in the ratio of their
lengths (9). This is due to the shearing-strain, which is entirely
transmitted through the oblique flange in addition to a horizontal
strain of the same amount as that in the horizontal flange, and the
longitudinal strain in the oblique flange is their resultant. In this
case the web has no duty to perform and may therefore be omitted,
the girder becoming the simplest form of truss, viz., a triangle.
CASE II. — FLANGED SEMI-GIRDER LOADED UNIFORMLY.
Fig. 6.
83. Flanges. — Let w = the load per unit of length,
I = the distance of any cross section A B
from the end of the girder,
d = the depth of the girder at this cross
section,
W = wl = the load on A Ct
F = the total horizontal strain exerted by
either flange at A or B, that is, the
horizontal component of the longitu-
dinal strain if the flange is oblique.
The forces which keep A B C in equilibrium are the weights
uniformly distributed along A C, the horizontal strains of tension
18 FLANGED GIRDERS WITH [CHAP. II.
and compression in the flanges at A and B, and the shearing and
horizontal strains in the web at the plane of section A B. If the
web be continuous and very thin, we may, as in the previous case,
neglect the moments of the horizontal strains in the web as insig-
nificant compared with those of the other horizontal forces. The
sum of the moments round A or B of each weight in the length I
is equal to the sum of the weights multiplied by the distance of their
centre of gravity from A or B (11), that is, their collective moments
= wl -=. Equating this to the amount of the horizontal strain in
either flange round A or B, we obtain the following relations: —
« = fd <8>
(9)
'-S-S <•»>
Ex. 1. A cast-iron semi-girder, 8 feet long and 13 inches deep, supports a uniform
load of 1 ton per running foot ; what area should the top flange have at the abutment
in order that its inch-strain may not exceed 1'5 tons ?
Here, w = 1 ton per foot,
I = 8 feet,
d = 13 inches,
/= 1-5 tons.
From eq. 10, F = ^ = 1 X 8 X 8 X 12 = ^ ^
2d 2 X 13
Answer (eq. 1). « =fe JL = ^ = 19'7 inches.
Ex. 2. The lattice-bridge at the Boyne Viaduct is in three spans. Each side span is
140 feet 11 inches long and 22 feet 3 inches deep. The permanent load supported by
one main girder of a side span equals 0'68 tons per running foot, and the gross sectional
area of its lower flange over each pier is 127 inches. On one occasion an extraordinary
load in the centre span depressed it to such an extent as to raise the ends of the side
spans off the abutments, thus forming each side span into a semi-girder. What was
the compressive inch-strain in the lower flange at the piers ?
Here, w = 0'68 tons per foot ?
I = 140-92 feet,
d= 22-25 feet,
a = 127 inches.
Antwer (eq 10). / = ^ = ^- = '68 X 140'92 X 14°'92 = 2-4 tons inch-strain.
' 7 a 2ad 2 X 127 X 22'25
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 19
53. Web — Shearing-strain. — When a semi-girder is uniformly
loaded the shearing-strain at any cross section is equal to the sum
of the weights between it and the extremity of the girder, since
this is the pressure transmitted through that section to the wall (14).
The shearing -strain therefore equals ivl, and varies directly in propor-
tion to the distance from the extremity of the girder, that is, directly
as the ordinates of a triangle. When the flanges are parallel, nearly
all the shearing-strain passes through the web, and its sectional area
should for economical reasons vary in this ratio also, for any excess
of material in the web beyond that required to transmit the
shearing-strain is valuable only for horizontal strains, and would
act with greater leverage, and therefore with greater effect, if
placed in the flanges.
54. Flange-area of semi-girder of uniform strength when
the depth is constant. — From eq. 10 we have, when both flanges
are horizontal
^_VW
7 ~ lad ~
where a and / represent the area and unit-strain of either flange
indifferently at a distance I from the extremity. If the girder be
of uniform strength, the unit-strain in each flange will be uniform
I*
throughout its length, and the quantity — , to which / is propor-
Flg 7._ Plan. tional, will be constant, that is, the
sectional area of each flange will
vary as I2. Hence, if the depth of
the flange be uniform, its breadth
will vary as I2, and the plan of
the flange will, if symmetrical, be
bounded by two parabolas whose
common vertex is at A, Fig. 7,
with the axis perpendicular to the
length of the girder.
35. Depth of semi-girder of uniform strength when the
flange-area is constant. — If one flange be horizontal and the
other curved, / and a, in eq. 11, apply to the horizontal flange only;
hence, if its sectional area be constant and if the girder be of
A
20
FLANGED GIRDERS WITH
[CHAP. II.
8.— Elevation. uniform strength, d will vary as Z2,
and the side elevation of the girder
will be bounded by a parabola whose
vertex is at A, Fig. 8, with its axis
vertical. In this case it may be shown
that the whole shear ing -strain passes
through the curved flange, and the
web has no duty to perform unless
the load rest upon the horizontal flange, in which case pillars,
represented by vertical lines (or suspension rods if Fig. 8 be
inverted with the weights beneath), are requisite for conveying
the pressure of each successive weight to the curved flange.
26. Strain in curved flange. — The longitudinal strain in
the curved flange is the resultant of the shearing -strain and a
Fig. 9- horizontal compression, the latter being equal to
the tension in the horizontal flange. If therefore,
the lines A 1, A 2, A 3, &c., Fig. 9, represent the
shearing-strains at different points, and if the
horizontal line A B represent F (or the uniform
horizontal compression), then the sloped lines B 1,
B2, B 3, &c., will represent the longitudinal strains
in the curved flange at these several points (9).
87. Semi-girder loaded uniformly and at the extremity
also* shearing-strain. — If, in addition to a uniformly distributed
load, the semi-girder support a weight W at its extremity, the
shearing-strain at any section will equal W + wl Consequently,
when the flanges are parallel, the area of the web should increase in
arithmetical ratio as it approaches the wall and may be represented
by the ordinates of a truncated triangle. If, for instance, the line
A B, Fig 10, represent the length of a
uniformly loaded semi-girder, and if A C
represent the whole distributed load, that
is, the shearing-strain at the wall, then the
ordinates of the triangle ABC will repre-
sent the shearing-strain at each point.
Now, let an additional weight W be
Pig. 10. — Shearing-strain.
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 21
suspended from the end of the girder at B, then, if B E represent this
weight, the ordinates of the rectangle A D E B will represent the
shearing- strains produced by it alone ; and when the girder supports
both it and the uniform load, the collective shearing-strains are
represented by the ordinates of the trapezium C D E B.
28. Flange-area of semi-girder of uniform strength loaded
uniformly and at the end when the depth is constant. —
When both flanges are horizontal and the semi-girder supports a
uniformly distributed load in addition to the weight W at its
extremity, we have from eqs. 7 and 11,
Where a and / represent the area and unit-strain of either flange
indifferently at a distance I from the extremity. If the semi-girder
be of uniform strength, / will be constant and a will vary as
I (2W + wl), and, if the depth of the flange be uniform, its breadth
will vary in the same ratio. Consequently, the plan of the flanges
will, if symmetrical, be bounded by a pair of parabolas, differing
however, from Fig. 7 in the position of their vertices.
S9. Depth of semi-girder of uniform strength loaded
uniformly and at the end when the flange-area is constant. —
If, however, one flange be horizontal and the other curved, / and
a, in eq. 12, apply to the horizontal flange only; hence, if its area
be uniform, d will vary as I (2W + wl\ and the elevation of the
girder will be bounded by a parabola.
Ex. A semi-girder, U'7 feet long and 22'25 feet deep, supports a uniformly
distributed load of 1'82 tons per running foot in addition to a weight of 161-6 tons at
the extremity. What is the inch-strain on the net section of the tension flange
at the point of support, its gross area being 132'6 inches, but reduced by rivet-holes
to the extent of f ths ?
Here, W = 161'6 tons,
1= 447 feet,
d = 22-25 feet,
w = 1'82 tons per foot,
a — 7 X 132'6 = 103-13 square inches.
. 12). / = ' *
22 FLANGED GIRDERS WITH [CHAP. II.
CASE III. — FLANGED GIRDER SUPPORTED AT BOTH ENDS AND
LOADED AT AN INTERMEDIATE POINT.
Fig. 11.
3O. Flanges. — Let W = the weight, dividing the girder into
segments containing respectively m
and n linear units,
/ = m + n =: the length of the girder,
d = the depth at any given cross section
AB,
x = the distance of this cross section from
the end of the segment in which it
occurs,
F = the horizontal strain exerted by either
flange at A or B, that is, the hori-
zontal component of the longitudinal
strain if the flange be oblique.
On the principle of the lever (1O), the reaction of the left abutment
= v W, and A B C is held in equilibrium by this reaction of the left
abutment, the horizontal flange-strains at A and B, the shearing-
strain in the cross section A B, and the horizontal strains in the web
when continuous. Neglecting these latter when the web is thin,
and taking the moments of the other forces round A or B, we
obtain the following relations : —
*-W*=Fd (13)
W=™ <">
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 23
31. Maximum flange-strains occur at the weight. —
If the cross section be taken at the weight, x = m, and eqs. 14
and 15 become ... _ Fdl ,.. „.
mn
x attains its greatest value when it equals m ; hence, comparing
eqs, 15 and 17, we find that the horizontal strain at any point in
either flange attains its greatest value when the weight rests there.
32. Concentrated rolling: load, maximum strains in flanges
are proportional to the rectangle under the segments. — If
W is a rolling load and the flanges are parallel, the maximum
strain at any point in either flange occurs when the load is
passing that point and is proportional to mn, that is, to the
rectangle under the segments.
33. Weight at centre. — This rectangle attains its greatest value
when the weight is at the centre, in which case eqs. 16 and 17 become
W = (18)
• Ad
Ex. 1. A cast-iron girder is 26 feet long and 274 inches deep, and the area of the
bottom flange = 16 X 3 = 48 inches. If the tearing inch-strain of cast-iron be 7 tons,
what weight laid on the middle of the girder will break it across by tearing the
bottom flange, omitting any strength which may be derived from the web ?
Here, I = 26 feet,
d = 27'5 inches,
f = 7 tons inch-strain,
a = the area of the bottom flange = 48 inches,
F = fa = 7 X 48 = 336 tons.
Answer (eq. 18). W = — = 4 X 336 X 27'5 _ 118.g tong nearly-
Ex. 2. In an experiment made by Mr. G. Berkley,* a small double-flanged cast-iron
girder was broken by 18 tons in the centre. The following were the dimensions :—
Effective length, I = 57 inches,
Total depth, d = 5' 125 inches,
Area of top flange, a, = 2'33 X 0'31 = 072 sq. inches,
Area of bottom flange, a2 = 6'67 X 0'66 = 4'4 sq. inches,
Thickness of web, = 0'266 inches.
* Proc., I. C. E., Vol. xxx., p. 254.
24 FLANGED GIRDERS WITH [CHAP. II.
What was the inch-strain in each flange at the centre of the girder at the moment
of fracture ?
An,, (a,. 19). Ingrain in top flange/- = = — ,** tons.
7 W *57 V 1 8
Inch-strain in lower flange/ = |£L = — 4.4 5.195 = U'37 tona-
It is not recorded which flange failed first, but as the tensile strength of the metal
was proved by direct experiment to be very high, namely, 13 '9 4 tons per square inch,
and as the inch-strain in the bottom flange fell considerably short of this, the girder
probably failed by the crushing of the top flange, the inch-strain in which, however,
was so unusually high, even for cast-iron, that this flange no doubt derived considerable
aid from the web.
Ex. 3. In an experiment recorded by Sir William Fairbairn,* a girder, cast from
a mixture of Gartsherrie, Dundyvan and Haematite Irons, 27 feet 4 inches long, 18
inches deep, and whose lower flange was 10 inches wide and 1$ inch thick, was
broken by a weight of 29£ tons in the centre. What was the inch-strain at the centre
of the lower flange at the moment of rupture, supposing that it derived no aid from the
web which was f inch thick ?
Here, I = 27'33 feet,
d = 1-5 feet,
a = 15 sq. inches,
W =29-5 tons.
Answer (eq. 19). f = = — • = 8*96 tons.
4ad 4 X 15 X 1'5
34. Web, shearing-strain. — The shearing-strain in each seg-
ment is uniform throughout that segment and equals the pressure
which is transmitted through it to the abutment (14). Thus, in Fig.
11, the shearing-strain at A B = ^-W = the reaction of the
I
left abutment. This shearing-strain is uniform throughout the left
segment, while that in the right segment is also uniform and equals
y W. When both flanges are horizontal, nearly all the shearing-
strain is transmitted through the web (15), and each segment
should have its web of uniform area adequate to sustain a
shearing-strain equal to the reaction of the adjacent abutment.
This may be represented graphically as follows: — let the line
* Application of Iron to building purposes, p. 171.
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS,
25
Fig- 12.
FE
A F represent the length
of the girder, divided
by W into the segments
m and n, and let the
ordinate A B represent
the reaction of the left
abutment, = W, and let
represent the reaction of the right abutment. = y W ; then the
ordinates of the rectangle A B C W will represent the shearing-
strains at each point in the left segment, and those of the rectangle
W D E F will represent the shearing-strains at each point in the
right segment. The sectional area of the web should therefore be
proportional to these ordinates when both flanges are horizontal.
When a single weight is at the centre of the girder, the rectangles
become equal, and, if both flanges are horizontal, the section of the
web should be uniform throughout its whole length, as it sustains
a uniform shearing-strain = -^-.
2
35. Single fixed load, flange-area of girder of nniform
strength when the depth is constant. — When both flanges
are horizontal, we have from eq. 15,
/= Z™ (20)
adl
where / and a represent the unit-strain and area of either flange at
a distance x measured from the abutment. When the girder is of
uniform strength, / is constant throughout each flange, and a will
Fig. 13.— Plan. vary as or. Hence, if the
depth of the flange be
uniform, its width will
vary as #, and the plan
of the flange will be two
triangles united at their
bases, as in Fig. 13.
Ex. 1. A girder (see Fig. 11), 50 feet long and 4 feet deep, supports a load of 16 tons
26 FLANGED GIRDERS WITH [CHAP. II.
at 9 feet from one end ; what should be the area of the top flange in the middle of
the girder so that the inch-strain may not exceed 4 tons ?
Here, W = 16 tons,
I = 50 feet,
d = 4 feet,
/ = 4 tons inch-strain,
n = 9 feet,
x = 25 feet.
. 20). • = =
Ex. 2. What is the strain in either flange at the load ?
Here, m = 41 feet.
. 17). F = H! = 4X* = 29-5 tons.
Ex. 3. What is the shearing-strain in each segment ?
Answer. The segments are respectively 9 and 41 feet long, and the shearing-strain
throughout the shorter segment =~X 16 = 13'12 tons, and that throughout the
50
longer segment = — X 16 = 2'88 tons.
50
36. Single fixed load, depth of girder of uniform strength
when the flange-area Is constant. — If, however, one flange be
horizontal and the other sloped, / and a, in eq. 20, apply to the
horizontal flange only, and if its area be uniform, d will vary as or,
Fig. 14.— Elevation. and the elevation of the
girder will be a triangle
whose apex is at the
weight, Fig. 14. In this
case the shearing-strain
is transmitted through
the oblique flange; the web may therefore be omitted and the
girder becomes the simplest form of truss. The longitudinal
strain in the oblique flange may be calculated according to the
principle explained in 9. When the weight rests upon the hori-
zontal flange, a strut h is required of sufficient strength to support
W and transmit its weight to the apex.
37. Concentrated rolling load, shearing-strain. — If the
weight be a rolling load, the shearing-strain in either segment varies
directly as the length of the other segment (34). Consequently, it
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS.
27
attains its greatest value at each point just as the weight passes,
when it suddenly changes both in amount and in the direction in
which it is transmitted, to the right or left abutment as the case
may be. In this case the maximum shear ing- strain at each section
is proportional to its distance from the farther abutment and, if both
flanges be horizontal, the area of the web should increase in the same
Fig. 15.— Shearing-strain. ratio also — i.e., as the ordinates of
the figure ABODE, Fig. 15, in
which the horizontal line A B re-
presents the length of the girder,
and each of the vertical lines A E
and B C represents the weight of
the passing load.
38. Concentrated rolling load, flange-area of girder of
uniform strength when the depth is constant. — In the case
of a single load traversing a girder both of whose flanges are
horizontal, we have at the place the weight is passing, from
eq. 17,
~~~,\A/
(21)
mnW
adl
where a and / represent the area and maximum unit-strain of either
flange at the weight, and m and n represent the lengths of the two
segments into which the weight divides the girder at the moment
of passing. If the girder be of uniform strength, / will be constant
throughout each flange, and a will vary as the rectangle inn.
Tig. 16.— Plan. Hence, if the depth of
the flange be uniform, its
breadth will vary as mn
also, and the plan of the
flange, if symmetrical, will
be formed by the overlap
of two parabolas whose
vertices are at A A, Fig. 16.
39. Concentrated rolling load, depth of girder of uniform
strength when the flange-area is constant. — If, however, one
flange be horizontal and the other curved, / and a apply to the
28
FLANGED GIRDERS WITH
[CHAP. II,
Fig. 17.— Elevation. horizontal flange only,
and, if its section be
uniform, d will vary as
ran. Hence, the elevation
of the curved flange will
be a parabola whose axis is vertical and its vertex at A, Fig. 17.
4O. Concentrated rolling: load, strain in curved flange —
Section of curved flange. — The maximum longitudinal strain at
any point in the curved flange of Fig. 17, i.e., the strain when the
weight rests over that point, may be thus obtained. Eq. 17 proves
that the horizontal component of this longitudinal strain is equal to
the strain in the horizontal flange at the same cross section ; it is
therefore a known quantity, and the longitudinal strain may be
found from it as follows: — Let the line A B, Fig. 18, represent F,
Fig. 18. i.e., the horizontal component ; draw A C
parallel to the tangent of the curve at the
given point, and draw B C perpendicular
to A B ; then A C will represent the maxi-
mum longitudinal strain at the given point,
and B C will represent its vertical component, or that portion of
the shearing-strain which is transmitted through the curved flange
(9); the remainder of the shearing-strain passes through the
web, which indeed prevents the girder from assuming a form
similar to Fig. 14, a result that would occur were the curved
flange flexible like a chain and the web absent.
From what has just been stated it appears that the longitudinal
strain in the curved flange from a single rolling load = F sec 9
where 0 represents the inclination of the flange to a horizontal
line, and its sectional area should increase therefore as it approaches
the abutments in proportion to seed, since, by hypothesis, F is
constant.
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 29
CASE IV. — FLANGED GIRDER SUPPORTED AT BOTH ENDS AND
LOADED AT IRREGULAR INTERVALS.
Fi<?s. 19 and 20.
41. Flanges. — When several weights rest upon a girder, the
strain at any point in either flange is equal to the sum of the strains
due to each weight acting separately. An example in which
numbers are mixed with symbols will illustrate the method of
calculation better than symbols alone. Let the girder represented
in Fig. 19 be divided into any convenient number of equal parts or
units of length, say 10 ; and let it be loaded with any number of
weights of different magnitudes, say 4, placed at irregular intervals,
as in the figure.
Let Wn W4, W8, W9. = the several weights,
/ = the length of the girder (divided into
10 units),
d = the depth at any given cross section A B,
measured in the same units as /,
F = the horizontal strain exerted by either
flange at A or B, that is, the horizontal
component of the longitudinal strain if
the flange be oblique.
30 FLANGED GIRDERS WITH [CHAP. II.
On the principle of the lever, the reaction of the right abutment
= 1<W1+4W4+8W8+9W9),
and the segment A B C is held in equilibrium by the reaction
of the right abutment acting upwards, the weights W8 and W9
pressing downwards, the horizontal flange-strains at A and B, the
shearing-strain in the cross section A B, and the horizontal strains
in the web when continuous. Neglecting the latter when the web
is thin, and taking moments round A or B, we have
Fd = A(Wj + 4 W4 + 8 W8 + 9 W9) - 2 W8 - 3 W9
arranging, we have
F = 1(4 W, + 16 W4 + 12 W8 + 6 W9).
If the weights are of equal magnitude, this becomes
C_38W_3.8W
~ld~ ~~dT
43. Webj shearing-strain. — Bearing in mind the definition
given in 14, it will be apparent that the, shearing-strain at any cross
section r= those portions of the weights in the left segment which are
conveyed to the right abutment minus those portions of the weights in
the right segment which are conveyed to the left abutment. Thus,
in the foregoing example,
the shearing-strain at A B = j(W, + 4 W4 — 2 W8 — W9).
The shearing-strain may also be derived from another considera-
tion as follows. The vertical forces acting on the right segment
ABC are: — the reaction of the right abutment acting upwards,
the weights W8 and W9 pressing downwards, and the shearing-
strain at A B. The only other forces are horizontal, namely, the
horizontal components of the flange-strains at A and B ; consequently,
the vertical forces must balance each other, for otherwise there
would be motion, and we may therefore define the shearing-strain
at any cross section to be the algebraic sum of the external
CHAP. II.] BEACED OR THIN CONTINUOUS WEBS. 31
forces on either side of the section, forces acting upwards being
positive and those acting downwards being negative. For example,
we have the shearing-strain at A B = the reaction of the right
abutment minus the intermediate weights W8 and W9
= }(Wt + 4W4 + 8 W8 + 9 W9) - (W8 + W9)
- * (V^ + 4 W4 - 2 W8 - W9)
as before. If the weights are of equal magnitude, this becomes
9 W
~ = 02 W.
6
The shearing-strain with irregular loading may be represented
graphically as follows : — Using the same example as before, let the
line A M , Fig. 20, represent the length of the girder, and let the
ordinates A B and M L represent to a scale of weights the shear-
ing-strains at the ends, that is, the reactions of the abutments;
then Bd will equal the sum of all the weights ; mark off Ba, ab,
be and cd respectively equal to Wn W4, W8 and W9, and
draw horizontal lines through these points till they intersect
vertical lines drawn through the weights. The ordinates of the
stepped figure ABCDEFGH I KLM, indicated by lines of
shading, will represent the shearing-strains in the web, and the
line E F shows where they part to the right and left.
Ex. A girder, 267 feet long and 22 feet 3 inches deep, supports three locomotives,
weighing 25 tons each, at points whose distances from the left abutment are respectively
19, 75 and 230 feet. What are the flange-strains and the shearing-strain at 180 feet
from the left abutment ?
Answer. The reaction of the right abutment = x 25 = 30'34 tons,
2b'7
and the strain in either flange at 180 feet from the left abutment = 80'34 X "^ X 5°
= 62-45 tons. The shearing-strain at the same point = 30-34—25 = 5 '34 tons.
32 FLANGED GIRDERS WITH [CHAP. II.
CASE V. — FLANGED GIRDER SUPPORTED AT BOTH ENDS AND
LOADED UNIFORMLY.
Fig. 21.
43. Flanges. — Let I =r the length of the girder,
d = the depth of the girder at any given
cross section A B,
w = the load per unit of length,
W = wl = the whole load,
F = the horizontal strain exerted by either
flange at A or B, that is, the hori-
zontal component of the longitudinal
strain if the flange be oblique,
m and n = the segments into which the section
A B divides the girder.
The forces which keep A B C in equilibrium are the reaction of
the right abutment, = -TT-, the weights uniformly distributed
along AC, = urn, the horizontal strains of compression and tension
in the flanges at A and B, the shearing-strain in the plane of section
A B, and the horizontal strains in the web when continuous.
Neglecting these latter forces when the web is thin, and taking
the moments of the remainder round either A or B, we have (II) —
^n-wn^ = Fd (22)
whence
_ ___ wmn __ mnW ^ox
and
W = ^ (24)
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 33
Ex. A wrought-iron plate girder, 50 feet long and 4 feet deep, supports a uniformly
distributed load of 32 tons ; what is the strain in either flange at 9 feet from one end ?
Here, W = 32 tons,
I = 50 feet,
d — 4 feet,
m = 9 feet,
ri = 41 feet.
If 4 tons per square inch be a safe strain, the area of the flange should -—
= 7'4
square inches.
44. Strains at centre of girder. — At the centre of the girder
i eq. 23,
VW id*
m = n —— and we have from eq. 23,
and
W = *™ (26)
Ex. 1. A segment of either side span of the Boyne Viaduct, 101 '2 feet long and 22'25
feet deep, supports a uniform load of 1"68 tons per running foot ; what is the strain at
the centre of either flange ?
Here, I = 101 "2 feet,
d = 22-25 feet,
w = 1*68 tons per running foot.
(eq. 25). F = = '2 X = 96-6 tons.
Sd 8 X 22-25
Ex. 2. The Conway tubular bridge is 412 feet long from centre to centre of bearings,
and 23 '7 feet deep from centre of top cells to centre of bottom cells at the centre of
the bridge. The weight of wrought-iron in one tube, 412 feet long, is 1,147 tons, which,
however, is not quite uniformly distributed, as the sectional area of the tube is greater
at the centre than at the ends in the ratio of — . Making an extra allowance for
this, and adding the weight of the permanent way and the light galvanized iron roof,
we may assume the total permanent load to be equivalent to 1,250 tons uniformly
distributed. What is the permanent strain in either flange at the centre of the girder
from this dead load ?
The gross area of the top flange at the centre of the bridge is 645 square inches ;
that of the bottom or tension flange is 536 square inches. If we assume that the
34 FLNAGED GIRDERS WITH [CHAP. II.
weakening effect of rivet holes in the tension flange is equivalent to the aid which
the continuous webs gives the flange, which is the same thing as if we suppose
the gross area of the flange available for tension, we have the permanent tensile
inch-strain at the centre of the lower flange = -'„„- = 5'067tons. The collective
oob
area of the two sides, i.e., of the web, at the centre of the bridge, is 257 square
inches, and it will be shown in Chap. IV. that a continuous web theoretically aids the
flanges as much as if one-sixth of its area were added to each flange. Assuming
then that — — , = 43 square inches, are added to the compression flange, we have its
6
permanent inch-strain = — — = 3*948 tons. These calculations, it will be
DID + 43
observed, are based on the hypothesis that the web gives its full theoretical aid to
the flanges, which is much too liberal an allowance to make in reality. A train-
load of | ton per running foot, = 309 tons uniformly distributed over one line of way,
will increase the permanent unit-strains by nearly one-fourth, or more accurately,
the inch-strain in the tension flange at the centre of the bridge will = 6 32 tons and
that in the compression flange will = 4'924 tons.
Ex. 3. What are the flange-strains in one of the Conway tubes from the permanent
load at the quarter-spans where the depth from centre to centre of cells = 22 '25
feet?
Here, w = 1,250 tons,
I = 412 feet,
d = 22-25 feet,
n = — ,
4
. c mn\N 3ZW 3X412X1250 „,««.
* 23)' F = Tar = m ~ 32 X 22-25 = 2'17° """•
The gross area of the top flange at each quarter-span = 566 square inches, that of
the bottom or tension flange = 461 square inches. If we assume, as before, that the
aid which the continuous sides theoretically give the tension flange compensates
for the weakening effect of rivet holes, we have the permanent tensile inch-strain in
the lower flange at each quarter-span = -1— = 4707 tons.
The area of both sides of the tube together at each quarter-span = 241 square
inches, and if we assume, as before, that one-sixth of this, or the full theoretic amount,
aids the compression flange, we have its permanent inch-strain at each quarter-
span = . * — = 3 '5 81 tons. On comparing the unit-strains in the flanges at the
quarter-spans with those at the centre of the tube we find that they are nearly equal,
and that the girder is therefore, as regards the flanges, a girder of very nearly
uniform strength.
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 35
Ex. 4. One of the large tubes of the Britannia Bridge is 470 feet long from centre
to centre of bearings, and 2 7 '5 feet deep from centre to centre of flange cells at the
middle of the span, and its weight is 1,587 tons. What was the strain in either
flange at the centre while it was an independent girder and before it was connected
with the other tubes ?
(eq. 25). F =
The gross areas of the top and bottom flanges at the centre of the span are
respectively 648 and 585 square inches, and if we concede, as before, that the theoretic
aid which the webs give the tension flange is a sufficient compensation for the
weakening effect of rivet holes, we have the inch-strain in the lower or tension
flange = J = 5795 tons.
585
The area of both sides at the middle of the span = 302 square inches, and adding,
as before, the full theoretic proportion of one-sixth in aid of the compression flange, we
o QQA
have the compressive unit-strain in the upper flange = — - - = 4'856 tons. The
648 X 50
student is cautioned that it is not safe practice to assume what has been claimed by
some advocates of continuous versus braced webs, and which has been conceded
above, namely, that so large a proportion as one-sixth of the web really aids each
flange, especially in large plate girders such as the tubular bridges. Hence, the unit-
strains in examples 2, 3, and 4 are doubtless below the reality.
45. A concentrated load produces the same strain in the
flanges as twice the load uniformly distributed. — Comparing
eqs. 17 and 23, we find that the horizontal strain at any point in
either flange from a single weight resting there is double that
which would be produced by the same load uniformly distributed.
This, however, does not apply to the web.
46. Web5 shearing-strain. — When the load is symmetrically
arranged on each side of the centre, the shearing-strain at the centre
of the girder is cipher, and at any other cross section it equals the sum
of the weights between it and the centre. This will appear evident
from the consideration that the shearing-strain at any section is the
pressure which is transmitted to the abutment through that section
(14). Hence, with a uniformly distributed load, the shearing-strain
is proportional to the distance from the centre of the girder, where
W
it is cipher, and increases towards the ends, where it equals-—,
as the ordinates of a triangle. This may be represented graphically,
FLANGED GIRDERS WITH
[CHAP. II.
Fig. 22.— Shearing strain.
as in Fig. 22, where the line A B
represents the length of the girder,
and the ordinates A C and B E
represent the reactions of each
W
abutment, =r — ; connecting C and
E with the centre at D, the ordinates of the figure A C D E B will
represent the shearing-strains at each point along the girder. When
both flanges are horizontal, the sectional area of the web ought for
economical reasons to vary in the ratio of these ordinates, for any
surplus material would be more valuable for sustaining horizontal
strains if placed in the flanges, as its leverage would be thereby
increased.
Ex. 1. What is the shearing-strain in the web at each end of the girder in the first
example in 44?
Answer. Shearing-strain = ^-= 1>68 + 101'2 = 85 tons.
Ex. 2. The iron work of one of the Conway tubes, 400 feet long in the clear span,
weighs 1,112 tons ; adding 400 tons for weight of permanent way, roof and a passing
train, we have a total load of 1,512 tons, of which one-fourth, = 378 tons, is the
shearing-strain at each end of each side where the web is about 19 feet high and | inch
thick. Consequently, its gross section = 142 '5 square inches, but as the vertical edges
of the plates are pierced by one-inch rivet holes, three inches apart centres, their net
section is one-third less, or 95 square inches, and the shearing-strain at the joints when
a heavy train is passing is about 4 tons per square inch of net section. In this
example no credit has been given to the outside plates of the cellular flanges, which
doubtless contribute their quota of strength to withstand shearing-strain.
47. Flange-area of girder of uniform strength when the
depth is constant. — From eq. 23 we have, when both flanges are
horizontal,
/=5 ^ (27)
where a and / represent the area and unit- strain of either
flange at any section which divides the girder into segments
containing m and n linear units. If the girder be of uniform
strength, / will be constant throughout each flange (19), and a
will vary as mn. Hence, if the depth of the flange be uniform,
CHAP. II.]
BRACED OR THIN CONTINUOUS WEBS.
37
Fig. 23.— Plan.
its width will vary as
\>nn, and the plan of the
iange will, if symmetrical,
je formed by the overlap
)f two parabolas whose
ertices are at A A, Fig.
4». Depth of girder of uniform strength when the flange-
area is constant. — If, however, the depth of the girder vary
while the area of the horizontal flange remains uniform, d will
vary as mn. Hence, the elevation of the curved flange will be
Fig. 24.— Elevation. a parabola whose axis is
vertical, with its vertex
at A, Fig. 24. In this
case it may be shown that
the whole shearing-strain
passes through the curved flange, and that therefore no web is
required for diagonal strains. When, however, the load rests upon
the horizontal flange, pillars, represented by vertical lines (or
suspension rods, if Fig. 24 be inverted), are required to convey
the vertical pressure of each weight to the curved flange. The
longitudinal strain in the curved flange increases towards the points
of support and may be found by the method explained in 186.
49. Suspension bridge — Curve of equilibrium. — The hori-
zontal flange, Fig. 24, prevents the ends of the curved flange from
approaching each other ; the same effect may be produced by
fastening the ends of the curved flange to the abutments, in which
case, the load being suspended below the curved flange, we have
the suspension bridge for a uniform horizontal load. The curve
which an unloaded chain of uniform section assumes from its own
weight is the catenary, which, however, differs but slightly from
a parabola when the ratio of the deflection to the span does not
exceed that commonly adopted for suspension bridges, viz., ^5
If Fig. 24 be inverted and the horizontal flange replaced by
solid abutments, to keep the arch from spreading, we have the
arch of equilibrium for a uniform horizontal load, and when the
38 FLANGED GIRDERS WITH [CHAP. II.
arch has merely its own weight to support, the inverted catenary
becomes the arch of equilibrium. Every change in the position
of a load alters the form of the curve of equilibrium, whose
horizontal component is uniform throughout the whole curve; for
it is obvious that, if the horizontal strain at one point of a
flexible chain exceed that at another point, the intermediate
portion will move towards that side on which the stronger
pull is exerted, so as to conform to the position of equilibrium.
A suspension bridge, being flexible, accommodates itself to each
change of load, assuming at each moment the position of equili-
brium for the particular load to which it is temporarily sub-
jected ; but neither the rigid flanges of a girder, nor the voussoirs
of a stone arch, can thus suit themselves to the changing position
of the load. The web of the former, and the spandril walls of the
latter, are therefore requisite to enable a rigid structure to sustain
a variable load without fracture, which they do by converting
what would otherwise be transverse strains in the arch or flanges
into longitudinal ones.
CASE VI. — FLANGED GIRDER SUPPORTED AT BOTH ENDS AND
TRAVERSED BY A TRAIN OF UNIFORM DENSITY.
5O. Passing: train of uniform density — Shearing-strain —
Flanges. — When a distributed rolling load, such as a railway
train, traverses a girder, the shearing-strain throughout the un-
loaded segment may be found as follows. Let the train be of
uniform density per running foot, and its total length not less
than that of the girder.
Fig. 25. — Shearing-strain.
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 39
Let / = the length of the girder,
d — the depth of the girder at A, in front of the train,
iv = the weight of the train per unit of length,
m and n =. the segments into which the front of the train divides
the girder, n being the loaded segment,
R z= the reaction of the left or unloaded abutment, i.e., the
shearing-strain in the segment m.
F — the horizontal strain exerted by either flange at A,
that is, the horizontal component of the longitudinal
strain if the flange be oblique.
The girder is held in equilibrium by the upward reaction of each
abutment and the downward pressure of the train. This latter =
wn, which we may conceive collected at its centre of gravity whose
distance from the right abutment = ~ (11). Taking moments round
this abutment, we have Rl = wi£. Hence,
' R =^S (28)
This is the shearing- strain throughout the unloaded segment,
since it is transmitted through every section between the front of
the train and the left abutment (14). As the train moves forward,
the shearing- strain in front increases as the square of the loaded
segment, and varies therefore as the ordinates of a parabola, the
ordinates being represented by the vertical lines of shading in
Fig. 25, with the vertex at B.
The flange-strain in front of the train may be easily found by
taking moments round either flange at A^ when we have
51. Maximum strains in web occur at one end of a passing:
train. — It can be easily proved that the shearing-strain at
any point A is greater when the load covers the longer segment
than when it covers the whole girder. In the latter case the
load is uniformly distributed all over, and the shearing-strain
at A -w(n— m) (46), but when the load covers the greater segment
40 FLANGED GIRDERS WITH [CHAP. II.
only, the shearing-strain at A = — — r. Subtracting the former
*2(in-\-n)
from the latter quantity, we obtain the following result. The
shearing-strain at the end of a passing train of uniform density
covering the greater segment exceeds that produced by a
load of equal density, but extending over the whole girder, by
a quantity equal to ^^, where ra represents the shorter and un-
loaded segment. It will be observed that this excess is equal to
the shearing-strain throughout the unloaded segment whenever
the train covers the lesser segment only.
Ex. A railway girder is 90 feet in length, and the heaviest train weighs 1£ tons
per running foot; what is the maximum shearing-strain from this train at 15 feet
from one end ? This will occur when the train covers the greater segment, and we
have
I = 90 feet,
m — 15 feet,
n — 75 feet,
w = 1-25 tons.
An^er <«,. 28). R=-£=««<«X«=»7*5 ton,
58. Uniform load and passing; train, shearing-strain. —
Let D Ej Fig. 26, represent a railway girder, and let the
Fig. 26.-Shearinff-strain. ordinates D A and E C represent
the shearing-strains at its extre-
mities from a load uniformly
distributed over its whole length,
such as the permanent bridge-
load. Draw A B and C B to the
centre of D E and the ordinates of
the figure D A B C E will repre-
sent the shearing- strains at each
point due to this uniformly distributed load (46). Again, let D E
and E H represent the shearing-strains at the extremities from the
greatest rolling load of uniform density (say engines), when covering
the whole girder. Draw the parabolas D G H and E G F, and the
ordinates of the figure D F G H E will represent the greatest
shearing-strains due to this maximum rolling load. The ordinates
CHAP. II.] BRACED OR THIN CONTINUOUS WEBS. 41
of the two figures combined, namely A B C H G F, will represent
the greatest possible shearing-strains to which the girder is liable
whatever may be the position of the rolling load.*
53. Maximum si rain in flang-es occur with load all over. —
The horizontal strains in the flanges attain their greatest value
when the load covers the whole girder, for the strain at each point
equals the sum of those produced by each weight acting separately,
and is consequently diminished by the removal of any one weight ;
the same result may be obtained by comparing equations 23 and 29,
when we find that the flange-strain in front of a train is less than
when the train covers the whole girder in the ratio ofy, where n
I
represents the segment covered by the train.
54. Area of a continuous web calculated from the
shearing-strain — Quantity of material in a continuous web. —
When the flanges are parallel, the theoretic area of a continuous web
may be calculated from the shearing-strain by the following rule : —
0 ,. ! PI Shearing-strain
{Sectional area of web = — T>— —
Unit- strain
in which the unit-strain is the safe unit-strain for shearing. This
gives the minimum thickness, which, however, is often much less
than a due regard for durability requires ; neither does this rule give
an adequate idea of the additional material required for stiffening
the web against buckling, of which more hereafter.
Ex. A single-webbed plate girder, 50 feet long and 4 feet deep, supports a uniformly
distributed load of 32 tons ; what is the theoretic thickness of the web, if 4 tons per
square inch be a safe shearing unit-strain ? The shearing-strain at each end =16 tons,
and the theoretic section of the web = ^ = 4 square inches ; but as the depth of
the girder is 4 feet, the thickness of the web would be only ?\ = TVth inch, which is
altogether too thin for safe practice. The second example in 46, however, shows that
the rule is applicable to the Conway tubular bridge.
On comparing 34, 37, 46, and 53, we find that when a girder
with parallel flanges and a continuous web is loaded in the manner
described below, where
/ — the length, and
/ = the safe unit-strain for shearing force,
* Appendix to Paper on Lattice Beams. By W. B. Blood, Esq., Proc. I. C. E.,
Vol. xi.. p. 9.
42
FLANGED GIRDERS, ETC.
[CHAP. II.
the theoretic quantity of material in the web should be as fol-
lows:—
Kind of load.
Theoretic quantity of
material in a continuous Web.
Proportional numbers
VW
Fixed central load . . . = W
12
3VW
Concentrated rolling load = W
IT
18
vw
Uniformly distributed load — W
W
6
7VW
Distributed rolling load . = W
W
7
55. Depth and length for calculation. — In calculating the
flange-strains of girders with continuous webs, the extreme depth
may be taken as the depth for calculation whenever the web is
neglected; but when a continuous web is taken into account, or
when the web is formed of bracing, the depth may be measured
from the upper to the lower intersection of the web with the
flanges, at which points the flanges are assumed to be concentrated.
Girders with cellular flanges are, however, exceptions to the fore-
going rule, as in these the depth for calculation is measured from
centre of upper cells to centre of lower cells.
The length for calculation should be measured from centre to
centre of bearings, which may be called the effective length of a
girder, and will always be greater than the clear span and less than
the total length.
Ex. The depth of the Boyne lattice girder for calculation is measured from root to
root of flange angle irons, and equals 22-25 feet — see plate IV. The extreme depth of
the Conway tube at the centre is 25'42 feet, but as the cellular flanges are each 1*75 feet
deep, the depth for calculation is 23'67 feet. The extreme length of the Conway tube
is 424 feet, the clear span between the supports is 400 feet, and the effective length
for calculation is 412 feet, the bearings at each end being 12 feet in length.
CHAP. III.] TRANSVERSE STRAIN. 43
CHAPTER III.
TRANSVERSE STRAIN.
56. Transverse strain. — Let Fig. 27 represent a semi-girder
of any form whatever of cross section, loaded at the extremity with
the weight W, and let I = the distance of W from any plane of
section A B. We know from experience that whenever a semi-
Pig. 27, girder such as that described is
subject to transverse strain, deflection
takes place, the upper edge being
extended and the lower edge com-
pressed. This longitudinal elongation
and shortening are not confined to
the outside fibres merely, but affect
those in the interior of the girder,
their change of length becoming less and less in direct proportion
as their distance from the edge increases, as is proved by the lines
A B and W D remaining straight after deflection. Experiments
also prove that the amount of deflection is proportional to the
bending weight, provided the limits of elastic reaction of the extreme
upper and lower fibres are not exceeded (?).*
5*. Mentral surface. — The surface of unaltered length, N S, at
or near the centre of the girder, where extension ceases and com-
pression begins, is called the Neutral surface — a term calculated to
produce a false impression that this part of a girder is free from
all strain, whereas, as has been already stated (14), the weight,
which is a vertical force, could not produce longitudinal strains in
the fibres except through the medium of certain diagonal strains,
which, as will be shown hereafter, act probably with their greatest
intensity in the vicinity of the neutral surface. The Neutral surface
of any girder is, therefore, that surface along which the resultant
* Morin, pp. 122, 138.
44 TRANSVERSE STRAIN. [CHAP. III.
of the horizontal components of all the diagonal forces equals cipher;
and according to this definition it may be said to exist in diagonally
braced girders, in those at least in which the systems of triangulation
are numerous. The reader will find his physical conceptions of
these diagonal strains much clearer after he has studied the action
of diagonal bracing in succeeding chapters.
58. Neutral axis— Centres of strain— Resultant of horizontal
forces in any cross section equals cipher.- The line at X, per-
pendicular to the plane of the figure, and formed by the intersection
of the neutral surface with any cross section of the girder, is called
the Neutral line, or more generally, the Neutral axis of that
particular section. The Neutral axis of any section is, therefore,
the line of demarcation between the horizontal elastic forces of
tension and compression exerted by the fibres in that particular
section of the girder. For these tensile and compressive forces we
may substitute their resultants.
Let T = the resultant of the horizontal tensile forces above the
neutral axis,
C = the resultant of the horizontal compressive forces below
the neutral axis,
& = the distance between the points of application of these
resultants,
called the Centres of strain, or for distinction's sake, the Centres of
tension and compression. The segment A B W D is held in equi-
librium by the weight W, the horizontal resultants T and C, and
the shearing-strain at the section A B. Taking moments round
the centres of compression and tension successively, we have
Wl = T$ = C& (30)
whence
T = C (31)
Thus, in every girder of whatsoever form, the resultant of all the
horizontal forces in any cross section equals cipher, or in other words,
the horizontal forces in any cross section balance each other, a result
which has been already proved in the case of flanged girders (eq. 4).
We may arrive at the same conclusion from the following
consideration. Suppose a loaded girder to rest on rollers at both
CHAP. III.] TRANSVERSE STRAIN. 45
ends so as to be perfectly free to move in a horizontal direction.
If we consider the forces acting at any cross section we find that
they may be resolved into three series, the first of which is vertical,
viz., the shearing-strain ; the second is horizontal, tending to thrust
the segments apart, and the third is likewise horizontal, tending
to draw them together. These horizontal forces must balance;
otherwise the girder would separate at the section, since by
hypothesis the segments are free to move horizontally on the points
of support.
59. Moment of resistance, M. — Bending; moment. — The
sum of the moments of the horizontal elastic forces in any transverse
section round any point whatsoever is called the Moment of forces
resisting rupture, or more briefly, the Moment of resistance of that
particular section.* Representing the moment of resistance by the
symbol M, we have for a semi-girder loaded at the extremity,
VW = M (32)
where I = the distance of W from the transverse section. It will
be observed that the moment of resistance of any particular section
is constant, no matter round what point the moments of the
horizontal forces may be taken, since the sum of the tensile forces
is equal to the sum of the compressive forces, so that they form a
couple. The product VW is called the Sending moment of the
weight, and eq. 32 may be expressed in general terms as follows : —
The moments of the external forces on either side of any given section
of a girder which tend to produce rotation round any point in that
section are equal to the moments of the horizontal elastic reactions in
the same section which resist rotation, or briefly, the bending moment
round any section = the moment of resistance.
The general case of a girder of any form of cross section is similar
to that of a flanged girder whose flanges are at the centres of
horizontal strain, and the formula in the several cases of flanged
girders in the previous chapter would be applicable to this general
case, if we only knew the resultants of the horizontal tensile and
compressive strains and also the distance between their points of
application.
* Called also the Moment of rupture.
46 TRANSVERSE STRAIN. [CHAP. III.
GO. Coefficient of rupture, S. — The following method is
frequently adopted for calculating the breaking weight of solid
rectangular or solid round girders, though applicable to other forms
also, and possesses the advantage of being founded on general
reasoning independently of any assumption relating to the laws of
elastic reaction or of direct experiments on the tensile and com-
pressive strength of materials, which generally require special
apparatus and are therefore less easily made than experiments on
transverse strength. We have just seen (eq. 30), that the relation
between the weight, length, horizontal elastic forces and distance
between the centres of strain of a semi-girder fixed at one end and
loaded at the other, is expressed by the equation
in which F represents indifferently the sum of the horizontal elastic
forces, either above or below the neutral axis, and is therefore
proportional in girders of similar section to the number of horizontal
fibres in the girder, that is, to its sectional area ; & = the distance
between the centres of strain, and is evidently proportional to the
depth, and I = the length. Hence, we obtain the following
relations for a
Semi-girder loaded at the extremity. —
W = (33)
(34)
ad
in which W = the breaking weight,
a = the sectional area,
d = the depth,
I = the length,
and S is a constant, which must be determined for each material by
finding experimentally the breaking weight of a girder of known
dimensions and similar in section to that whose strength is required.
The constant S is called the Coefficient of transverse rupture, or more
briefly, the Coefficient of rupture* of that particular material and
* Sometimes called the Modulus of rupture.
CHAP. III.] TRANSVERSE STRAIN. 47
section .from which it is derived, and equals the breaking weight of
any semi-girder of similar section in which the quantity ~ = 1 .
By reasoning similar to that adopted in the several cases of
Chapter II., we have the following formulas for girders supported
and loaded in various ways : —
61. Semi-girder loaded uniformly.
w = ^ (35)
It
1\N
S = l^L (36)
*2ad
68. Girder supported at both ends and loaded at an
intermediate point, the segments containing m and n linear units,
and I representing the length, = m + n.
W = ^ (37)
mn
8 = (38)
aal
63. Girder supported at both ends and loaded at the centre.
W = ^? (39)
s =
64. Girder supported at both ends and loaded uniformly.
w _ 8^S
S = ™
65. Table of coefficients of rupture. — These formulae,
though generally restricted in practice to solid rectangular and solid
round girders, are also applicable to girders of any form, provided
they are similar in section to the experimental girder from which
the coefficient S for that form is derived. In each class we must
obtain the coefficient of rupture for its particular section by expe-
rimentally breaking a model girder. This has been done for certain
forms of section and the results are given in the following tables
which contain the values of S, or the coefficients of rupture, which
48
TRANSVERSE STRAIN.
[CHAP. in.
in the case of square or round sections are the breaking weights of
solid semi-girders whose length, .depth, and breadth are each one
inch, fixed at one end and loaded at the other. Hence, when using
these coefficients in the preceding equations, all the dimensions
should be in inches. The reader may easily satisfy himself that the
value of S is constant for all rectangular sections of the same depth
from the consideration, that any number of rectangular girders of
equal depth placed side by side have the same collective strength as
the single girder which they would become if united laterally. Hence
W
has the same value for the multiple girder as for one of its com-
a
ponent girders, and therefore, fVom eq. 34, S is the same in both.
Value of
MATERIAL.
S
Authority.
in tons.
CAST-IRON.
Small rectangular bars (not exceeding one inch in width),
3-40
Clark
Large rectangular bars (three inches wide), -
2-25
»>
Small round bars,
2-00
Circular tubes of uniform thickness, -
2-85
,,
Square tubes of uniform thickness,
3-42
»
WROUGHT-IRON.
New rectangular bars whose deflection limits their utility,
Rectangular bars previously strained by bending them while
hot and straightening them when cold, and employed in
3-82
»
the direction in which they were straightened,
5-58
„
New round bars,
2-25
„
Circular welded tubes of uniform thickness (boiler tubes),
5-23
PI
Circular riveted tubes of plate iron with transverse joints
double riveted, - - -
3-26
„
Rolled I girders with flanges of equal area, about
4-60
—
"T* iron, with the flange above, about
4-00
—
Do., with the flange below, about
3-83
—
STEEL (Rectangular bars).
Hammered Bessemer steel for tyres, axles and rails,
9-53
Kirkaldy
Rolled Bessemer steel for tyres, axles and rails,
8-57
j>
Hammered crucible steel for tyres and axles,
11-00
)>
Rolled crucible steel for axles,
8-80
H
Average of a large number of specimens of Cast, Bes- }
semer, and Shear steel, strained only as far as the >
6-00
Fairbairn
limit of elasticity, - )
Clark, Britannia and Conway Tubular Bridges, pp. 436, 743.
Fairbairn, Report of the British Association for 1867.
Kirkaldy, Experiments on Steel by a Committee of Civil Engineers, 1868.
CHAP. III.] TRANSVERSE STRAIN.
WOOD.
SOLID RECTANGULAR GIRDERS AND SEMI-GIRDERS.
49
DESCRIPTION OF WOOD.
Initials of
Experimenters.
Specific
Gravity.
Value of S
inlbs.
Acacia, _
B.
710
1,867
Ash, English, .....
B.
760
2,026
„ American,
D.N.
626
1,795
„ „ swamp, -
D.
925
1,165
„ „ black,
D.
533
861
Beech, English,
B.
696
1,556
„ American, white, -
D.
711
1,380
„ „ red,
D.N.
775
1,739
Birch, Common, - - -
B.
711
1,928
„ American, black,
B.D.N.
670
2,061
„ „ yellow, -
D.
756
1,335
Box, Australian, -
T.
1,280
2,445
Bullet Tree, Demerara,
B.Y.
1,052
2,692
Cabacally,
B.
900
2,518
Canada Balsam, - -
D.
548
1,123
Cedar, Bermuda, - ...
N.Y.
748
1,443
N.
756
2,044
„ American, white, -
D.
354
'766
„ of Lebanon,
D.
330
1,493
Crab Wood, Demerara,
Y.
648
1,875
Deal, Christiana, -
B.
689
1,562
Elm, English,
B.D.
579
782
,, Canada Rock, ....
D.N.
725
1,970
Fir, Mar Forest, -
B.
698
1,232
„ Spruce,
M.
503
1,346
„ „ American, black,
D.
772
1,036
Greenheart, Demerara, - -
B.Y.
985
2,615
Hemlock, ... -
D.
911
1,142
Hickory, American, ...
D.N.M.Y.
831
2,129
„ Bitter Nut,
D.
871
1,465
Iron Bark, Australia,
T.
1,211
2,288
Iron Wood, Canada,
D.
879
1,800
Kakarally, Demerara,
Y.
1,223
2,379
Larch, - ...
B.D.M.
556
1,335
„ American, or Tamarack, -
D.
433
911
Lignum Vitse,
N.
1,082
2,013
Locust, Demerara, - -
B.
954
3,430
Mahogany, Nassau,
M.N.Y.
668
1,719
Mangrove, Bermuda, black,
N.
1,188
1,699
it >» white,
N.
951
1,985
Maple, soft Canada,
D.
675
1,694
Norway Spar,
B.
577
1,474
Oak, Adriatic,
B.M.
855
1,471
„ African,
B.D.M.N.
988
2,523
,, American, live,
N.
1,160
1,862
„ „ red,
D.N.
952
1,687
„ „ white, ....
B.D.M.N.
779
1,743
„ Dantzic, - -
B.M.
720
1,518
„ English, .....
B.D.M.N.
829
1,694
„ Italian,
M.
796
1,688
„ Lorraine,
M.
796
1,483
„ Memel, ...
M.
727
1,665
50
TRANSVERSE STRAIN.
[CHAP. in.
DESCRIPTION OF WOOD.
Initials of
Experimenters.
Specific
Gravity.
Value of S
in tbs.
Pine, American red,
B.D.M.N.Y.
576
1,527
pitch,
B.D.
740
1,727
„ white,
D.N.Y.
432
1,229
„ yellow,
B.D.M.
508
1,185
Archangel,
M.
551
1,370
Dantzic,
M.
649
1,426
„ MemeL,
M.
601
1,348
„ Prussian,
M.
596
1,445
„ Riga,
B.M.
654
1,383
„ Virginian,
M.
590
1,456
Poon,
B.M.
673
1,954
Sneezewood, South Africa,
N.
1,066
3,305
Spotted Gum, Australia, -
T.
1,035
2,006
Stringy Bark, Australia, -
T.
937
1,818
Teak,
B.M.N.
729
2,108
Wallaba, Demerara,
Y.
1,147
1,643
Yellow Wood, West Indies,
N.
926
2,103
The coefficients for wood are chiefly taken from the Professional Papers of the
Corps of Royal Engineers, Vol. v. The initial letters refer to the following experi-
menters :— B, Barlow ; D, Denison ; M, Moore ; N, Nelson ; T, Trickett ; Y,
Young ; two or more letters signify that the tabulated number is the mean result
of the experimenters whom they represent.
The reader should observe that the foregoing values of S for
timber are derived from selected samples of small scantling,
perfectly free from knots and other imperfections that cannot be
avoided in large timber, and the few experiments recorded on the
latter indicate that the values of S must be reduced to very little
more than one-half ('54 times,) those given in the table when
applied to girders of large size, such as occur in ordinary practice.
STONE.
SOLID RECTANGULAR GIRDERS AND SEMI-GIRDERS.
DESCRIPTION OF STONE.
Value of S
in tt>s.
Authority.
GRANITES.
Ballynocken, Co. Wicklow, coarse and loosely aggre-
gated,
Golden Hill, Blessington, Co. Wicklow, coarse,
Golden Ball, Co. Dublin, largely crystalline, -
Killiney, Co. Dublin, felspathic,
Kingstown, Co. Dublin,
Newry, Co. Down, syenitic,
Taylors' Hill, Galway, felspar red and greenish,
109
76
182
270
346
340
407
Wilkinson
n
»
»
CHAP. III.]
TRANSVERSE STRAIN.
51
DESCRIPTION OF STONE.
Value of S
in ttjs.
Authority.
SANDSTONES AND GRITS.
Green Moor, Yorkshire blue stone,
„ „ white stone, -
Caithness, Scotland, -
Irish sandstones from various localities,
335
359
857
57 to 1,095
G. Rennie
?>
Wilkinson
LIMESTONES.
Listowel Quarry, Kerry,
Ballyduff Quarry, Tullamore, King's County,
Woodbine Quarry, Athy, Co. Kildare,
Finglass Quarry, Co. Dublin, -
414
351
283
291
Wilkinson
»
»>
SLATES.
Valencia Island, Kerry, on edge of strata,
„ „ on bed of strata,
Glanmore, Ashford, Co. Wicklow, on bed of strata, -
Killaloe, Tipperary, on bed of strata, -
„ „ on edge of strata,
Welsh slate, -
1,091
951
1,097
1,233
1,037
1,961
Wilkinson
5>
5>
G. Rennie
BASALTS AND METAMOKPHIC ROCKS.
Hornblende Schist, Glenties, Donegal,
Moore Quarry, Ballymena, Antrim, crystalline, horn-
blendic and felspathic,
556
531
Wilkinson
5>
Wilkinson, Practical Geology and Ancient Architecture of Ireland.
G. Rennie, Barlow on the Strength of Materials, p. 187.
The foregoing table contains a very small selection from Mr.
Wilkinson's experiments on the transverse strength of Irish
stones, and in addition to these the reader will find in his book
a vast number of most valuable details relating to the crushing
strength and other properties of building materials throughout
Ireland.
66. Strength of stones, even of the same kind, is very
variable. — Mr. Wilkinson's experiments were made on stones
14 inches long, with sides 3 inches square ; the distance between
the bearings was exactly 12 inches, and the pressure was applied
on the top in the centre of each stone by a saddle one inch
wide. "The result of these experiments shows the average
strength of the principal rocks to be in the following order: — -
Slate rock, basalt, limestone, granite, and sandstone. The great
variation which exists in the different rocks, and even in the
52 TRANSVERSE STRAIN. [CHAP. ITI.
quality of the same kind of stone, serves to show the caution
which should be used in their selection and the value to be
attached to the records of actual experiments."
Ex. 1. In an experiment made by the author, a wrought-iron bar, 4 inches deep and
| inch wide, had a weight of 1,568 Ibs. hung from one end, the other end being rigidly
fixed. It commenced bending at 2 ft. 8 in. from the load, at a part which had been
previously softened in the fire and allowed to cool slowly. What is the value of S ?
Here, W = 1,568 Ibs.,
I = 32 inches,
d = 4 inches,
a = 3 square inches.
A~~ <e,. 34). S = » = - = 1-86 tons.
Comparing this with the tabular value of S for " new rectangular bars whose deflection
limits their utility," it would appear that the useful strength of bars rendered
ductile by annealing is only one-half that of new bars fresh from the rolls. This result
is confirmed by two of Mr. Hodgkinson's experiments on annealed wrought-iron bars
heated to redness and allowed to cool slowly. — See Appendix to Report of the Commis-
sioners on the Application of Iron to Railway Structures, pp. 45, 46.
Ex. 2. The teeth of a cast-iron wheel are 3'5 inches long, 2'3 inches thick, and 7
inches wide ; what is the breaking weight of a tooth ?
Here, I = 3' 5 inches,
d = 2-3 inches,
a = 16*1 square inches,
S = 2-25 tons.
Antwer (eq. 33). W = "-** = 16-1 X 2-8 X 2-25 = 23.8 tong
I o'o
Ex. 3. A round wrought-iron shaft, 5 feet long and supported at the extremities,
sustains a transverse strain of 30 tons at 14 inches from one end ; what should its
diameter be when on the point of yielding ?
Here, W = 30 tons,
I = 5 feet,
m = 14 inches,
n =• 46 inches,
S = 2-25 tons.
j 14 X 46 X 30 vd3
From eq. 38, ad = — — - = = 143'1 inches; but ad = " whence
to OU X ^ /O 4
Ex. 4. In an experiment made by Mr. Anderson, a piece of memel fir, 2 inches deep
and 1|| inches wide, was securely fixed at one extremity, the projecting part being 2
feet long. It sustained a load of 504'5 Ibs. at the end for twenty hours without
breaking right across. This load, however, upset the timber on the lower or
CHAP. III.] TRANSVERSE STRAIN. 53
compression side next the fulcrum. What is the value of S derived from this
experiment ?
Here, W = 504'5 Ibs.
I = 24 inches,
d = 2 inches,
6 = 1'94 inches.
This value of S exceeds that given in the table, namely, 1,348 Ibs. The piece of
memel in this experiment was, however, remarkably straight-grained and well
seasoned, and consequently above the average.
Ex. 5. A horizontal gaff of red American pine, 15 inches square, is hinged to a
mast at the inner end and suspended by a chain 9 feet from the outer end. What
weight will it safely bear at the extremity ? In this example the outer segment is a
semi-girder 9 feet long, and we have
a = 15 X 15 inches,
d = 15 inches,
I = 9 X 12 inches,
S = 1,527 Ibs.
Awer (eq. 33). W = ™® = 15X15X15X1,527 m ^ tong>
I 9 X 12 X 2,240
For temporary purposes, and if the timber be perfectly sound, one-fourth of this, or 5 '3
tons, will be the safe quiescent load. If, however, the load, though temporary, is
hoisted up and down and therefore liable to produce jerks, one-sixth, or 3'5 tons,
will be the safe load, but if the timber be exposed to the weather and in frequent
strain, one-tenth, or 2'13 tons, will be the proper working load.
67. Strength of similar girders— Limit of length. — It appears
from the foregoing investigations that the strength of similar girders
varies as the square of their linear dimensions, for — , in eqs. 33 to
CL
42, is constant in similar girders, and consequently the breaking
weight W varies as the area a. The weight of the girder itself,
however, varies as aZ, i.e., as the cube of its linear dimensions. If
this weight, which we shall call G, equal -th of the breaking
weight, we have the breaking weight of girders loaded uniformly
(eqs. 35 and 41),
W = K^? = nG
L
in which K = 2 for a semi-girder and 8 for a girder supported at
54 TRANSVERSE STRAIN. [CHAP. III.
both ends. The breaking weight W of a similar girder n times
longer is as follows : —
W = "'K^S = ««Q
where n3G is the weight of the second girder. Hence, if the
weight of any girder is -th of its breaking weight, a similar girder
n times longer will break from its own weight. This defines the
theoretic limit of length of similar girders. The same idea may
be usefully expressed in the following terms: — The unit-strains
of similar girders from their own weight will vary directly as any of
their linear dimensions. From this it also follows that, the weights
of similar girders are as the cubes of their unit-strains.
Ex. 1. The Conway tubular girder, 412 feet long, sustains from its own weight a
tensile inch- strain of nearly 5 tons in the lower flange at the centre of the bridge ;
what is the length of a similar girder whose tensile inch-strain is 7 tons ?
Answer. Length = 412 X 7 = 577 feet.
0
Ex. 2. The weight of the Conway tube is 1,147 tons ; what will be the weight of the
larger girder ?
Answer. Weight = 1,147 X ^ = 3,147 tons.
68. Neutral axis passes through the centre of gravity-
Practical method of finding: the neutral axis. — If the law of
uniform elastic reaction hold good in girders subject to transverse
strain, the horizontal elastic reaction exerted by each fibre will be
in proportion to the extension or compression of the fibre, that is,
in direct proportion to its distance from the neutral axis (56). Its
amount will also be proportional to the sectional area of the fibre,
and if the variable distance from the neutral axis be called y, and the
sectional area dff (differential of ff), then the elastic force of the fibre
may be represented by ydff multiplied by a constant, and F, or the
sum of the horizontal elastic forces on either side of the neutral axis,
r, taken within proper limits and multiplied by the same
constant. This integral for the horizontal elastic forces on the
upper side of the neutral axis is equal to the similar expression for
CHAP. III.]
TRANSVERSE STRAIN.
55
the horizontal elastic forces on the lower side (eq. 31). Now this
equality is also the condition which determines the position of the
centre of gravity of the section. Hence, it follows that, when the
fibres are not strained beyond the limit of uniform elastic reaction,
the neutral axis of any cross section of a girder passes through its
centre of gravity, and we have the following practical rule for
finding the position of the neutral axis where the section is unsym-
metrical, as in T" iron, or in girders with unequal flanges. Cut
a model of the cross section of the girder out of card-board or thin
sheet metal and find its centre of gravity by means of a plumb-bob
or by balancing it on a knife-edge. This will give the position of
the neutral axis of the girder quite accurately enough for practical
purposes.
Fig. 28.
56 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
CHAPTEK IV.
GIRDERS OF VARIOUS SECTIONS.
69. moment of resistance. — The following method of inves-
tigating the strength of girders of any form whatsoever of cross
section is based on the assumption that the law of uniform elastic
Fig' 29- reaction is true, that is, that the
horizontal fibres exert forces which
are proportional to their change of
length, and therefore directly pro-
portional to their distance from the
neutral axis, an hypothesis which
is sensibly true so long as the
strains do not exceed those which
are considered safe in practice, and which lie considerably within
the limits of uniform elastic reaction (56). Suppose a girder com-
posed of longitudinal fibres of infinitesimal thickness, and let us
consider the horizontal elastic forces developed by the weight W
in any cross section A B,
Let M = the moment of resistance of the section A B (59),
d = the depth of the girder,
y = the variable distance of any fibre in the section A B,
either above or below the neutral axis,
)3 = the breadth of the section at the distance y from the
neutral axis, and consequently a variable, except in
the case of rectangular sections,
/ = the horizontal unit-strain exerted by fibres in the same
section at a given distance c from the neutral axis,
c = a known distance, either above or below the neutral
axis, of fibres which exert the horizontal unit-
strain /.
According to our assumption, the unit-strain in any other fibres at
CHAP. IV.] GIEDEKS OF VARIOUS SECTIONS. 57
a distance y from the neutral axis will be^. Let the depth of
the latter fibres = dy (differential of y) ; then the total horizontal
force exerted by the fibres in the breadth ft will = ^ ftydy. The
moment of this force round the neutral axis = '-$y*dy, and the
integral of this quantity will be the sum of the moments of all the
horizontal elastic forces in the section A B round its neutral axis,
that is, the moment of resistance of the section in question (59).
Representing this as before by the symbol M, we have
M = tfdy (43)
in which the integral must be taken within proper limits for each
form of cross section and may be readily found for those sections
which occur in practice in the following manner.*
7O. Let hl = the distance of the top of the girder above the
neutral axis,
7i2 = the distance of the bottom of the girder below the
neutral axis.
The expression for the moment of resistance becomes
in which ft, if variable, must be expressed in terms of y.
91. M for sections symmetrically disposed above and
below the centre of gravity. — When the material is symmetri-
cally disposed above and below the centre of gravity, the neutral
axis bisects the depth (68), and if d = the depth, we have ht = 7i2
d ,
= , and
The values of M for the usual forms of cross section are as
follows, recollecting that / = the unit-strain in fibres whose distance
from the neutral axis = c.
* The reader will recognize the integral ^By*dy as that which expresses the Moment
of Inertia of the cross section round its neutral axis, represented by the symbol I.
58 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
93. M for a solid rectangle.
Let b = the breadth,
d = the depth.
In the case of a rectangle, j3 = b and is therefore constant, and we
have from eq. 45,
93. M for a solid square with one diagonal vertical.
Let a = the semi-diagonal,
b = the side of the square.
The variable breadth |3, expressed in terms of y, = 2 (a — y) ; sub-
stituting this value in eq. 45, we have
Integrating and reducing,
M-#-*2 (47)
" 3c ~ 12c
The moment of resistance of a square, it will be observed, is the
same whether the sides or one diagonal be vertical.
94. M for a circular disc.
Let r = the radius.
The variable breadth jg, expressed in terms of y, becomes 2 VV2 — y2 ;
substituting this value in eq. 45, we have
M =
Integrating and reducing,
M=^ (48)
95. M for a circular ring of uniform thickness.
Let r = the external radius,
rl = the internal radius.
The moment of resistance of a ring is equal to that of the external
circle minus that of the internal one, and we have from eq. 48,
M=(r«-V) (49)
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 59
If t = the thickness of the ring, rl = r — t ; whence, by substitu-
tion in eq. 49,
M = 4r3£ —
If the thickness be small compared with the radius the last three
terms may be neglected, and we have
M = *J£ (50)
76. M for an elliptic disc with one axis horizontal.
Let b = the horizontal semi-axis,
d = the vertical semi-axis.
#2 y2
The equation of an ellipse whose origin is at the centre is j-2 +^ = 1 '•>
hence, the variable |3 = 2x = 2 -j^d* — y* ; substituting this
value of j(3 in eq. 45, we have for the moment of resistance of an
elliptic disc round its horizontal axis,
Integrating and reducing,
MTbfd3 /F,1N
= — r — (vl)
4c
77. M for an elliptic ring with one axis horizontal.
Let b = the external horizontal semi-axis,
b , = the internal horizontal semi-axis,
d = the external vertical semi-axis,
dl = the internal vertical semi-axis.
If the ring be of uniform thickness, as generally occurs in practice,
both the external and internal curves cannot be true ellipses ; when
however the ring is thin, we may assume that the ellipse passing
through the extremities of the internal axis is equidistant from the
external ellipse, and that the moment of resistance of the ring is
equal to that of the external minus that of the internal ellipse ;
whence (eq. 51), we have for the moment of resistance of an elliptic
ring round its horizontal axis,
M = ^ (bd* - Vi3) (52>
60 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
If t = the thickness of the tube, bl =r b — t and dl = d — t; sub-
stituting these values in eq. 52, expanding, and neglecting the
terms in which the higher powers of t occur, we have when the
thickness of the tube is small compared with its axis-minor,
M = (36 + d) (53)
9*8. Tiro classes of flanged girders. — The term "flanged
girder," as has been already remarked (13), includes rectangular
tubes and braced girders as well as the ordinary single-webbed
plate girder. The sides of a tube, the braced web of a lattice
girder, and the continuous web of a plate girder — all perform the
same duty of conveying the vertical pressure of the load (shearing-
strain) to the points of support, developing at the same time
longitudinal strains in the flanges. It is obvious, therefore, that
the sides of the tube are equivalent to the web of the single-
webbed girder, which is the form best suited for calculating the
moment of resistance.
Flanged girders may be subdivided into two classes.
1st. Those in which the web is formed of bracing, or, if con-
tinuous, yet so thin that the horizontal strains developed in it
are insignificant compared with those in the flanges. This class
has been already investigated in Chapter II.
2nd. Those in which the web is continuous and so thick that the
horizontal strains in it are of considerable value, in which case the
web acts as a thin rectangular girder, enabling the flanged girder
to sustain a greater load than is due merely to the sectional area
of its flanges. In either case it will be sufficiently accurate for
practical purposes if we suppose the mass of each flange concentrated
at one point or centre of strain, which may be assumed to coincide
with the intersection of the web and flanges (55).
79. M for the section of a flanged girder or rectangular
tube* neglecting the web. —
Let at = the area of the upper flange,
a2 = the area of the lower flange,
a3 = the area of the web above the neutral axis,
a4 = the area of the web below the neutral axis,
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 61
h{ = the height of the web above the neutral axis,
A2 = the height of the web below the neutral axis,
d = h{ + h2 = the depth of the web,
A = al + a2 = the area of both flanges together.
From eq. 44 the moment of resistance of the flanges alone
If we neglect the web, the neutral axis passes through the centre of
gravity of the two flanges (68), and we have 7^ = ^- and h2 =T^5
hence, by substitution,
M = ^jp (55)
80. M for the section of a flanged girder or rectangular
tube, including the web. — When, however, the horizontal strains
in the web are too considerable to be safely neglected, the moment
of resistance of the web, derived from eq. 44, must be added to that
just obtained for the flanges (eq. 54), when we have
M ={{ (a, + |) A,« + (a, + |) V} (56)
81. M for the section of a flanged girder or rectangular
tube with equal flanges, including the web. — If the flanges
have equal areas, the neutral axis will be in the middle of the
depth, in which case hl = h2 = ^, and eq. 56 becomes
M = ^(6a + a') (57)
where a = the area of either flange,
a! = the area of the web.
The moment of resistance of a rectangular tube With flanges of
equal area may also be obtained from eq. 46 by subtracting the
moment of resistance of the inner from that of the outer rectangle
as follows : —
(58)
where b = the external breadth,
bl = the internal breadth,
62 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
d = the external depth,
dl = the internal depth.
83. M for the section of a square tube of uniform thickness*
either with the sides or one diagonal vertical. — From eqs.
46 or 47,
where b = the external breadth of the tube,
bl = the internal breadth of the tube.
If t = the thickness of the tube, bl = b — 2£; substituting this
value in eq. 59, expanding, and neglecting the terms in which the
higher powers of t occur, we have when the thickness is small
compared with the breadth of the tube,
M = y£2 (60)
When the value of M is known for any particular section of
girder we can easily find the value of the weight W in terms of /,
or vice versa, as explained in the following cases : —
CASE I. — SEMI-GIRDERS LOADED AT THE EXTREMITY.
Fig. 30.
83. Let W = the weight at the extremity,
/ = the distance of W from any cross section A B,
M = the moment of resistance of the section A B.
The forces which keep the segment A B W in equilibrium are the
weight W, the shearing-strain at A B, and the horizontal elastic
forces developed in the same section. Taking the moments of all
these forces round the neutral axis we have (eq. 32),
VW=M (61)
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 63
84. Solid rectangular semi-girders. —
Let b = the breadth,
d = the depth,
From eqs. 46 and 61,
where /= the unit-strain in fibres whose distance from the neutral
axis = c*
If, however* /= the unit-strain in the extreme fibres, c = |, and
we have
W = -^- (62)
Ex. A piece of teak, 2 inches deep and l|f inches wide, is fixed as a semi-girder at
one extremity ; what weight hung 2 feet from the point of attachment will break it
across, the crushing inch-strain of dry teak being 12,000 Ibs. ?
Here, I = 2 feet,
6 = 1-94 inches,
d = 2 inches,
/= 12,000 Ibs.
Answer. W -/^2 -, 12>°°0 X l'»* X 2 X 2 _ ^^
61 6 X 24
The crushing strength of teak being considerably less than its tearing strength, rupture
will probably ensue from the crushing of the fibres on the compressed side.
85. Geometrical proof. — Eq. 62 may be easily deduced from
geometrical consideration as follows : —
Let the rectangle A B C D, Fig. 31, represent in an exaggerated
-p- 31 degree the side view of any small
transverse slice whose breadth be-
fore deflection = A B. Suppose
the upper edge after deflection
stretched out to the length A6,
and the lower edge compressed to
Cd; then the lines of shading in the
two little triangles will represent
* When W = the breaking load, the unit-strain / has been called by some writers the
modulus of rupture of the material, but when W is the working load, it has been called
the working modulus. This must not, however, be confounded with the coefficient or
modulus of rupture, S, and it is better to restrict the expression to the latter coefficient.
64 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
the alteration of length of the intermediate fibres, N S being
the neutral surface which divides the section into equal parts (56).
The sum of the horizontal forces exerted by the fibres in either the
upper or the lower half of the section is equal to the product of the
half section by the mean unit-strain of the fibres, and if /=the unit-
strain in the extreme fibres, then*^ is the mean unit-strain of all the
fibres, for it equals the unit-strain exerted by the fibres lying
mid-way between the neutral surface and either the upper or the
lower edge. The total strain of tension in the upper half and
that of compression in the lower half are, therefore, each equal to
9 * -Q-, where b and d represent the breadth and depth of the sec-
tion. Moreover, since the horizontal elastic forces in the various
fibres are proportional to the lines of shading in the two triangles
(?), the centres of tension and compression (58) coincide with
their centres of gravity, and their distance apart therefore = \d.
Hence, taking moments round either centre of strain, we have as
before,
86. Solid square semi-girders with one diagonal vertical —
Solid square girders with the sides vertical are 1*414 times
stronger than with one diagonal vertical. — If one diagonal is
vertical, we have from eqs. 47 and 61,
where/ = the unit-strain in fibres whose distance from the neutral
axis = c.
If, however, / = the unit-strain in the extreme fibres, c = — =,
and we have,
W = g (63)
Comparing eqs. 62 and 63, we find that the transverse strength of
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 65
a solid square girder with the sides vertical =-?-^= l*414timesthe
6
strength of the same girder with the diagonal vertical.*
The strength of square semi-girders in the direction of their
Fig. 32. diagonals may be investigated in a different
manner as follows. Let Fig. 32 represent
a cross section of the girder, and let the
line A B represent the shearing force acting
downwards. We may conceive this replaced
by its components A C and A D parallel to
the sides of the girder. Since the section
AB
is square, each component will equal — ==..
Now the force AC will produce tension in the side parallel to
A D, and the force A D will produce tension in the side parallel to
A C ; the corner will therefore sustain twice the strain that either
component alone would produce, that is, it will sustain a strain
2AB
which would be produced by a force equal to -—==, = 1-41 4 A B,
acting in the direction of one side, which result agrees with that
already obtained.
87. Rectangular girder of maximum strength cut out of a
cylinder. — It is sometimes required to cut a rectangular girder of
maximum strength out of round timber.
Let D = the diameter of the log,
b = the breadth of the girder of
maximum strength,
d = its depth.
From eq. 62, the strength of a rectangular
girder is maximum when bd* is maximum,
or, since d2 = D2 — &2, when 6D2 — b3 is
maximum. Equating the differential co-
efficient of this quantity to cipher, we have
* Barlow's experiments on battens of elm, ash and beech, 2 inches square and 36 inches
long, do not corroborate the theory in the text, for the strength of the elm was the
same whether fixed erect or diagonally, whereas it was found that ash and beech were
both a little weaker in the latter position. — Strength of Materials, p. 143.
66 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
>'=TD'
from which we derive the following rule. Erect a perpendicular, p,
at one-third of the length of the diameter, and from the point where
this perpendicular intersects the circumference draw two lines, b
and d, to the extremities of the diameter ; then b2 = — ^ D2 *
o
88. Solid round semi-girders.
Let r = the radius.
From eqs. 48 and 61,
w* =-"£?.
4c
where / = the unit-strain in fibres whose distance from the neutral
axis = c.
If, however, / = the unit-strain in the extreme fibres, c = r, and
W=1 (64)
89. Solid square girders are 1*7 times as strong as the
inscribed circle, and O-6 times as strong as the circumscribed
circle. — Comparing eqs. 62 and 64, we find that the strength of a
solid square girder is 1*7 times that of the solid inscribed cylinder,
whereas its strength is only — - =0*6 times that of the solid cir-
cumscribed cylinder.f
90. Hollow round semi-girders of uniform thickness.
Let r = the external radius,
rl = the internal radius.
From eqs. 49 and 61,
where / = the unit-strain in fibres whose distance from the neutral
axis = c.
* Euclid, Book vi. ; Cor., prop. 8.
t In Barlow's experiments on very fine specimens of Christiana deal, the breaking
weight of girders 4 feet long and 2 inches square, supported at the ends and loaded in
the middle, was 1,116 Ibs. The breaking weight of round girders of the same length
and 2 inches in diameter was 772 Ibs. The ratio of these breaking weights = 1*45, not
17, which the theory in the text gives. — Barlow, p. 142.
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 67
If, however, / = the unit-strain in the extreme fibres, c = r, and
W=^—-.4) (65)
If, moreover, the thickness of the tube be small compared with
the radius, we have from eqs. 50 and 61,
W = (66)
where t represents the thickness of the tube.
Ex. A tubular crane post of plate iron is 11 feet high and 2 '4 feet diameter at the
base. The load on the crane produces a bending-strain equivalent to 20 tons acting at
right angles to the top of the post ; what should be the thickness of the plating at the
base in order that the inch-strain may not exceed 3 tons ?
Here, W = 20 tons,
I = 11 feet,
r = 1-2 feet,
f = 3 tons per square inch.
. 66). , =l=__i__= 1.35inche..
91. Centre of solid round girders nearly useless. — The
centre or core of a cylindrical girder may be removed without
sensibly diminishing its transverse strength ; for it appears, from eqs.
64 and 65, that the strengths of two cylinders of equal diameters,
r4
one solid and the other hollow, are as — ^ -- f , in which T and rl are
the external and internal radii ; let r rr nr^ then the ratio becomes
-^ — =-; if, for example, n = 2, the loss of strength in the hollow
cylinder amounts to only T^th of that of the solid cylinder while
the saving of material amounts to ^th. For this, among other
reasons, cylindrical castings, such as crane posts, should be made
hollow.
98. Hollow and solid cylinders of equal weight. — It may
also be shown that the transverse strength of a thin hollow cylinder
is to that of a solid cylinder of equal weight as the diameter of the
former is to the radius of the latter. By eqs. 66 and 64, the ratio
of the strength of a hollow to that of a solid cylinder =— 3-,in
ri
68 GIRDERS OP VARIOUS SECTIONS. [CHAP. IV.
which r and t represent the radius and thickness of the hollow
cylinder, and r{ represents the radius of the solid cylinder ;
since by hypothesis the two cylinders are of equal weight, we have
2rt = T*!2; whence, by substitution, the ratio of strength becomes
— , that is, as the diameter of the hollow cylinder is to the radius of
ri
the solid cylinder.
93. Solid elliptic semi-girders.
Let b = the horizontal semi-axis,
d = the vertical semi-axis.
From eqs. 51 and 61, we have,
VW = rf^*
4:0
where / = the unit-strain in fibres whose distance from the neutral
axis = c.
If, however, f =. the unit-strain in the extreme fibres, c = d, and
W =- (67)
94. Hollow elliptic semi-girders.
Let b = the external horizontal semi-axis,
Z>, = the internal horizontal semi-axis,
d = the external vertical semi-axis,
dl — the internal vertical semi-axis,
From eqs. 52 and 61 we have
W/=5£(foP— Mi3)
where f = the unit-strain at the distance c from the neutral
axis.
If, however, / = the unit-strain in the extreme fibres, c = d,
and
W = g (M»_ V,1) (68)
If, moreover, the thickness of the tube is small compared with
the shorter axis, we have from eqs. 53 and 61,
(69)
where t = the thickness of the tube.
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 69
95. Flanged semi-girder or rectangular tube* taking the
web into account.
Let flj = the net area of the top flange,
«2 — the area of the bottom flange,
az — the area of the web above the neutral axis,
«4 = the area of the web below the neutral axis,
A, — the distance of the top flange above the neutral axis,
h 2 = the distance of the bottom flange below the neutral axis,
/ = the unit-strain in fibres whose distance from the neutral
axis — c.
From eqs. (56) and (61), we have
w = •••»•*'• + Bi+ (70)
96. Flanged semi-girder or rectangular tube with equal
flanges. — If the flanges are equal, we have from eqs. 57 and 61,
WZ = 4? (6a + a')
where d = the depth of web,
a = the area of either flange,
a' = the area of the web,
/ = the unit-strain in fibres whose distance from the
neutral axis = c.
If/ = the unit-strain in either flange, c = r, and we have
W=^(a + !" '
l\ b>
In the case of a rectangular tube with equal flanges, the following
equation, derived from eqs. 58 and 61, may be used instead of
eq. 71,
where b — the external breadth,
bi = the internal breadth,
d = the external depth,
c/j = the internal depth,
/ = the unit-strain in the extreme fibres, in which case
d
c = -.
70 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
97. Square tubes with vertical sides. — If the tube is square
with vertical sides of uniform thickness, we have from eq. 72,
W = ^(i'-V) (73)
If, moreover, the thickness of the tube is small compared with its
breadth, we have from eqs. 60 and 61,
W = (74)
where t = the thickness of the side of the tube.
98. Square tubes with diagonal vertical — Square tubes
of uniform thickness with vertical sides are 1*414 times
stronger than with one diagonal vertical. — If one diagonal
of the square tube is vertical, the sides being of equal thickness, we
have from eqs. 59 and 61,
W*=X<6._V)
where / = the unit-strain at the distance c from the neutral
axis.
Iff = the unit-strain in the extreme fibre, c = — =, and we have
If, moreover, the thickness of the tube is small compared with its
breadth, we have from eqs. 60 and 61,
W = 'i^l' (76)
where t = the thickness of the side of the tube.
Comparing eqs. 73 and 75, we find that the strength of a square
tube of uniform thickness, with the sides vertical, equals 1'414 times
the strength of the same tube with the diagonal vertical.
99. Square tubes of uniform thickness with vertical sides
are 1*7 times as strong as the inscribed circle of equal
thickness5 and O*85 times as strong as the circumscribed
circle of equal thickness — Square and round tubes of equal
thickness and weight are of nearly equal strength. —
Comparing eqs. 74 and 66, we find that the strength of a square
tube with vertical sides is to that of a round tube of equal thickness
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 71
and whose diameter equals the side of the square (inscribed circle)
1 fi
as — - —. 1*7 ; whereas the strength of the square tube with verti-
y*4ij
cal sides is to that of a round tube of equal thickness but whose
diameter equals the diagonal of the square (circumscribed circle,) as
o
— — = 0'85. It also appears that the strength of the circumscribed
circle is twice that of the inscribed circle of equal thickness. If
square and round tubes are of equal thickness and weight, their
peripheries will be equal, that is, 45 = 2vr, or b = -r ; substituting
2
this value for b in eq. 74, and comparing the result with eq. 66, we
find that the relative strength of square tubes with vertical sides
and round tubes of equal weight and thickness = — = 1-0472, or
o
very nearly a ratio of equality, the square tube being very slightly
stronger than the other. When semi-girders are subject to trans-
verse strain in various directions like crane posts, the round tube is
generally preferable to a square tube of equal weight, as the latter
is much weaker in the direction of the diagonals (98). Nevertheless,
rectangular tubes of plate iron, with strong angle iron in the
corners, form very efficient crane posts.
100. Value of web in aid of the flanges. — The strength of a
girder with equal flanges and continuous web, in which full credit
is given to the web for the horizontal strains which it sustains, is
equal to the strength derived from the flanges alone plus that
derived from the web acting as an independent rectangular girder.
Eqs. 5 and 71 prove that a continuous web, in a girder with
flanges of equal area, does theoretically as much duty in aid of the
flanges as if one-sixth of the web were added to each flange and the
web were made of bracing. In girders with unequal flanges, the
centre of gravity, and therefore the neutral surface, is closer to the
large flange ; consequently the small flange will derive more benefit
from a continuous web than the large one.
101. Plan of solid rectangular semi-girder of uniform
strength^ depth constant. — From eq. 62, the unit-strain in the
72
GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
Fig. 34.— Plan.
extreme fibres of a solid rectangular
r/w
semi-girder/ ==——. Iftheserni-
girder be of uniform strength (19),
f will be constant, and consequently
the quantity — , to which / is pro-
DCb
portional, will also be constant.
Hence, if the depth of the girder
be uniform, b will vary as /, that is
the plan of the girder will be triangular, Fig. 34.
I O3. Elevation of solid rectangular semi-girder of uniform
Fig. 35.— Elevation. strength, breadth constant. — If,
however, the breadth be uniform,
d* will vary as Z, and if the top
of the girder be horizontal the
bottom will be bounded by a para-
bola whose vertex is at W and its
axis horizontal, Fig. 35.
103. Solid round semi-girder of uniform strength. — From
eq. 64, the unit-strain in the extreme fibres of a solid round semi-
girder/ = — 3 . If its strength be
uniform, r3 will vary as I, and the
semi-girder will be a solid formed by
the revolution of a cubic parabola
round a horizontal axis, Fig. 36. The
beak of an anvil is a rude approxi-
mation to this form of semi-girder.
104. Hollow round semi-girder of uniform strength. —
From eq. 66, the unit-strain in the extreme fibres of a thin round
/ W
tube/= 2 ' ^ ^s strength be uniform, / will be constant
and r^t will vary as I. When the thickness is constant, rz will
vary as Z, and a hollow semi-girder, formed by the revolution of
a parabola round a horizontal axis, will result. This, for instance,
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 73
is the theoretic form for a hollow crane post of plate iron ; the cir-
cumscribing cone, however, is preferable in practice, as it is more
easily constructed.
CASE II. — SEMI-GIRDERS LOADED UNIFORMLY.
Fig. 37.
105. Let I = the distance of any cross section, AB, from the
extremity of the girder,
w = the load per linear unit,
W = wl = the sum of the weights resting on AC,
M = the moment of resistance of the section AB.
The forces which keep ABC in equilibrium are the weights
uniformly distributed along AC, the shearing-strain at AB, and
the horizontal elastic forces developed in the same section. Taking
the moments of all these forces round the neutral axis of the section
A B, arid recollecting that the sum of the bending moments of the
separate weights is equivalent to the moment of a single weight
equal to their sum and placed at their centre of gravity (11), we
have (59),
W- = — = M (77)
2 2
106. Solid rectangular semi-girders. — From eqs. 46 and 77,
we have
W=/g° (78)
in which b and d represent the breadth and depth of the girder, and
/ = the unit-strain in the outer fibres at top and bottom, in which
d
case c = -.
74 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
1O7. Solid round semi-girders. — From eqs. 48 and 77,
w =
where r =: the radius, and / = the unit-strain in the extreme fibres
at top and bottom, in which case c = r.
1O8. Hollow round semi-girders of uniform thickness.—
From eqs. 49 and 77,
W = g(r'-r,') (80)
in which r represents the external, and rl the internal radius, and
/ = the unit-strain in the extreme fibres at top and bottom. If,
moreover, the thickness, t, is inconsiderable compared with the
radius, we have from eqs. 50 and 77,
W = (81)
1O9. Flanged semi-girders or rectangular tubes* taking
the web into account. — From eqs. 56 and 77,
W = ^< a/+?U«+ «. + £ V* (82)
where al = the net area of the top flange,
a2 = the area of the bottom flange,
az = the area of the web above the neutral axis,
a4 = the area of the web below the neutral axis,
7*i = the distance of the top flange above the neutral axis,
/j2 — the distance of the bottom flange below the neutral axis,
/ = the unit-strain in fibres whose distance from the neutral
axis rr c.
If the flanges are equal and iff •=. the unit-strain in either flange,
in which case c = ^, we have from eqs. 57 and 77,
Wzr -
where a = the area of either flange,
a' •=. the area of the web,
d = the depth of the web.
11O. Flan of solid rectangular semi-girder of uniform
CHAP. IV.J GIRDERS OF VARIOUS SECTIONS. 75
strength, depth constant. — From eq. 78, the unit-strain in the
outer fibres of a solid rectangular semi-girder loaded uniformly,
" bd*
W
in which w represents the load on the unit of length, = ~.
When the strength of the girder is uniform throughout its whole
length (19), the quantity — , to
uCL
which / is proportional, is constant,
and, if d be uniform, b will vary as
Z2, and the plan of the girder will,
if symmetrical, be bounded by two
parabolas whose common vertex is
at A with the axis vertical, Fig.
38.
111. Elevation of solid rectangular semi-girder of uniform
strength^ breadth constant. — If, however, the breadth be uni-
form, d will vary as /, and the elevation of the girder will be
triangular.
118. Solid round semi-girder of uniform strength. — From
eq. 79, the unit-strain in the extreme fibres of a solid round semi-
girder loaded uniformly,
2W2
/ = : *r*
If the strength be uniform, r3 will vary as Z2, and the semi-girder
will be a solid formed by the revolution of a semi-cubic parabola
round a horizontal axis.
113. Hollow round semi-girder of uniform strength. — From
eq. 81, the unit-strain in the extreme fibres of a thin round tube,
If the strength be uniform, r2t will vary as /2. Hence, if t be
constant, r will vary as /, and the tube will be conical.
The strength of semi-girders of other sections loaded uniformly
may be obtained by multiplying the corresponding values of W in
the previous case by 2.
76 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
AT AN INTERMEDIATE POINT.
Fig. 39.
114. Let W = the weight, dividing the girder into segments
containing respectively m and n linear units,
I = m + n = the length of the girder,
x — the distance of any cross section A B from that
end of the girder which is remote from W,
M = the moment of resistance of the section A B.
On the principle of the lever, the reaction of the left abutment =
-j W, and the segment A B C is held in equilibrium by this reaction,
the shearing-strain at AB, and the horizontal elastic forces developed
in the same section. Taking the moments of all these forces round
the neutral axis of the section A B, we have (59),
" W x = M (84)
When / = the unit-strain in the extreme fibres at top or bottom,
c = the distance of the top or bottom from the neutral axis, and we
have the following expressions for the strength of each class of girder.
115. Solid rectangular girders. —
Let b = the breadth,
d = the depth.
From eqs. 46 and 84,
W = (85)
bnx
If both the weight and cross section are at the centre of the girder,
x = n = , and
W = (86)
ol
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 77
116. Solid round girders. — From eqs. 48 and 84,
W = 2^ (87)
in which r — the radius.
If both the weight and cross section are at the centre,
W = *£-3 (88)
117. Hollow round girders of uniform thickness. — From
eqs. 49 and 84,
tvfl
"<-V)
where r and rl represent the external and internal radii.
If both the weight and cross section are at the centre,
(UO)
If the thickness, t, is inconsiderable compared with the radius, we
have from eqs. 50 and 84,
(91)
nx
If, moreover, the weight and cross section are at the centre,
W = (92)
Ex. A cylindrical tube of riveted boiler-plate, 0*095 inch thick, 27 feet long between
supports, 24"2 inches diameter, and weighing 0'4295 tons, was torn through a riveted
joint in the bottom by a weight of 4'857 tons at the centre (Clark, p. 92). What was
the tearing-strain per square inch in the bottom plate ?
Here, W = 4'857 +.0-21475 = 5'072 tons,
I = 27 feet,
r = 121 inch,
t - 0-095 inch.
VW 5-072 X 27 X 12
Answer (eq. 92). /= ^^ = — — 5 — — = 94 tons.
4 X 3-1 41 6 X 12T|X 0-095
11§. Flanged girders or rectangular tubes* taking the web
into account. — From eqs. 56 and 84,
(93)
78 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
where al = the area of the upper flange,
a2 — the net area of the lower flange,
a3 = the area of the web above the neutral axis,
a 4 = the area of the web below the neutral axis,
Aj = the height of the web above the neutral axis,
A2 = the height of the web below the neutral axis,
/ = the unit-strain in fibres whose distance from the neutral
axis = c.
Ex. What is the unit-strain of compression in the upper flange at the centre of the
girder described in Ex. 2 (33), supposing the web taken into account? From a
full-sized card-board section of the girder it appears that the centre of gravity, that
is, the neutral axis of the section, (68), is 3 '57 inches below the intersection of the
upper flange with the web, and we have,
a i = 0-72 square inches,
a3 = 4'4 square inches,
a3 = 3-57 X *266 = -95 square inches,
a4 = 0-585 X '266 = '156 square inches,
Aj = 3'57 inches,
A2 = 0-585 inches,
c = 3-57 inches,
I = 57 inches,
I
n = x = 2>
W = 18 tons at the centre.
From eq. 93., 18 tons = 3^7^-57 { ( 72 + ^) X(3'57)+ ( *4 + ^p)x(W) }
Solving this equation for the unit-strain in the compression flange, we have,
Answer. /'= 61'5 tons per square inch.
Comparing this with Ex. 2 (33), we see that taking the web into account has
reduced the inch-strain in the compression flange from 69'5 to 61'5 tons, or 8 tone
per square inch.
If the flanges are equal and/= the unit-strain in either flange, we
have from eqs. 57 and 84,
«=*(•+*)
in which a = the area of either flange,
a' = the area of the web,
d = the depth from centre to centre of flange.
If, moreover, the weight and cross section are at the centre,
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS.
79
119. Plan of solid rectangular girder of uniform strength,
depth constant. — From eq. 85, the unit-strain in the extreme
fibres of a solid rectangular girder,
/=
When the strength of the girder is uniform, the quantity —, to
Fig. 40._pian. whicn f fc proportional,
will be constant. Hence,
if the depth, d, is uniform,
b will vary as x, and the
plan of the girder will be
two triangles joinedattheir
bases, Fig. 40.
ISO. Elevation of solid rectangular girder of uniform
strength,, breadth constant. — If, however, the breadth be uni-
Fig. 41.— Elevation. forni) ^2 WJU yarv ^ ^
and if the top of the
girder is horizontal, the
bottom will be bounded
by two parabolas which
intersect underneath the weight, with horizontal axes and their
vertices at the extremities of the girder, Fig. 41.
181. Solid round girder of uniform strength. — From eq. 87,
the unit-strain in the extreme fibres of a solid round girder,
If the strength be uniform, r3 will vary as x, and the girder will
be formed by two spindles joined at their base, each spindle being
produced by the revolution of a cubic parabola round its axis.
122. Hollow round girder of uniform strength. — From eq.
91, the unit-strain in the extreme fibres of a thin hollow cylinder,
In a girder of uniform strength, the quantity -^-, to which / is
proportional, will be constant; hence, if t be uniform, r2 will vary
80 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
as x, and the girder will be formed by two hollow spindles joined
at their bases, each spindle being generated by the revolution of
a. parabola round its axis. This, for instance, is the form which
the hollow axis of a transit instrument should theoretically have,
though a double cone is preferred in practice from its greater facility
of construction.
Id3. Concentrated rolling: load, plan of solid rectangular
girder of uniform strength when the depth is constant —
Elevation of same when the breadth is constant. — If W be
a single moving load, the maximum strain at each point will occur
as the load passes that point, for x attains its greatest value when
it equals in\ hence, from eq. 85, the unit-strain in the extreme
fibres of the section where the weight occurs,
(96)
If the strength of the girder be uniform, — will be a constant
bd
quantity, and if d be uniform, b will vary as the rectangle under the
Fig. 42.— Plan. segments ; hence, the plan
of the girder, if symmetri-
cal, will be bounded by
two overlapping parabolas
whose vertices are at A A,
Fig. 42. If, however, the
breadth be uniform, d2 will
vary as mn and the elevation of the girder will be a semi-ellipse,
Fig. 43.
Fig. 43.— Elevation.
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 81
CASE IV. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
UNIFORMLY.
Fig. 44.
134. Let I = the length of the girder,
w = the load per linear unit,
W = wl = the whole load,
m and n = the segments into which any given cross section
A B divides the girder,
M = the moment of resistance of the section A B.
The forces which hold A B C in equilibrium are the reaction of
the right abutment, = ^ the weights uniformly distributed along
A C, = wn, the shearing-strain at A B, and the horizontal elastic
forces in the same section. Taking the moments of all these forces
round the neutral axis of A B, we have (59),
Multiplying the left side of the equation by , we have
= M (98)
When / rr the unit-strain in the extreme fibres at top or bottom
of the section, c = the distance of the top or bottom from the
neutral axis, and we have the following expressions for the strength
of each class of girder.
13d. Solid rectangular girders.
Let b = the breadth,
d = the depth.
G
82 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
From eqs. 46 and 98,
W = J™ (99)
6mn
If the cross section is at the centre, m = n = -, and
(100)
186. Solid round dirders.— From eqs. 48 and 98,
W = 5^ (101)
2mn
in which r = the radius.
If the section is at the centre, m = n = -, and
W = 2 (102)
. Hollou round girders of uniform thickness.
Let r = the external radius,
rl = the internal radius.
From eqs. 49 and 98,
w = <•"—•'> ' (103)
At the centre of the girder m = n = 3, and
,') (104)
If the thickness, i, is inconsiderable in comparison with the radius,
we have from eqs. 50 and 98,
W = (105)
mn
If, moreover, the plane of section is at the centre,
W = W (106)
188. Flanged girders or rectangular tubes* taking the
web into account. — From eqs. 56 and 98,
W = ~ M «, + 7 V + «, + ^ V h (W7)
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 83
Where a{ = the area of the upper flange,
«2 = the net area of the lower flange,
a3 = the area of the web above the neutral axis,
a4 = the area of the web below the neutral axis,
Aj = the height of the web above the neutral axis,
7t2 = the height of the web below the neutral axis,
/ = the unit-strain in fibres whose distance from the
neutral axis = c.
If the flanges are equal, and if/ = the unit-strain in either flange,
c = -)t and we have from eqs. 57 and 98,
W = . + (108)
mn \ 6/
in which a = the area of either flange,
a' — the area of web,
d = the depth of the web.
At the centre, m = n = -, and eq. 108 becomes
(109)
139. Plan of solid rectangular girder of uniform strength
when the depth is constant. — From eq. 99, the unit-strain in
the extreme fibres of a solid rectangular girder,
3mnW
' =
When the strength of the girder is uniform, and the material conse-
quently disposed in the most economical manner, the unit-strain/will
be uniform (19), and the quantity^. to which it is proportional, will
QCL
_ Fig. 45.— Plan. _ be constant. Hence, if
the depth, d, be uniform,
b will vary as ran, and the
plan of the girder, if sym-
metrical, will be formed
by the overlap of two
parabolas whose vertices
are at A A, Fig. 45.
84 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
130. Elevation of solid rectangular girder of uniform
strength when the breadth is constant. — If, however, the
46.— Elevation. breadth be uniform, d*
will vary as mn, and the
elevation of the girder
will be a semi-ellipse,
Fig. 46.
131. Discrepancy between experiments and theory-
Shifting of neutral axis — Longitudinal strength of materials
derived from transverse strains erroneous. — The student
will naturally conclude that the formulas investigated in the present
and preceding chapters should give identical, or nearly identical,
results when they are applied to the same girder ; that, for instance,
the breaking weight of a solid rectangular semi-girder, calculated
by eq. 33, should closely agree with its breaking weight calculated
by eq. 62 ; and, if our theory were complete, this would no doubt
be the case. To test its accuracy, let us compare these two equa-
tions, when we obtain this result,
that is, the value of S for solid rectangular girders of any given
material should equal one-sixth of the ultimate tearing or crushing
strength of that material, according as it yields by tearing or crush-
ing. In many instances, however, this will be found to be far
from the truth ; for example, the value of S for small rectangular
bars of cast-iron = 3'4 tons (65), and 6 times this, = 20'4 tons, far
exceeds the tensile strength of ordinary cast-iron, which is about 7 or
8 tons per square inch. It must, indeed, be confessed that the law
of elasticity ceases to be applicable when we approach the limits of
rupture ; and that the formulae for solid girders investigated in the
present chapter give their breaking weight much under what it really
is for many materials, and this discrepancy will probably be found
more marked in those whose ultimate tearing strain differs widely
from their ultimate crushing strain. Greater confidence, however,
may be placed in the formulae relating to hollow and flanged girders.
Mr. Hodgkinson endeavours to explain this discrepancy by a
change in the position of the neutral axis as soon as the limit of elastic
CHAP. IV.] GIRDERS OF VARIOUS SECTIONS. 85
reaction of the horizontal fibres has been passed, and gives some
reasons for this hypothesis derived from experiments on cast-iron,
in his Experimental researches on the strength of Cast-iron, p. 384.
This seems a plausible hypothesis, for if the neutral axis of a solid
rectangular cast-iron girder approach its compressed edge as the
weight increases, and after the limit of tensile elasticity has been
passed by the fibres along the extended edge, we shall have a larger
proportion than one-half the girder subject to tension, and conse-
quently the total horizontal tensile strain may exceed that derived
from our theory, which assumes that the neutral axis always passes
through the centre of gravity of the cross section (6§). Mr.
Hodgkinson concludes from his experiments that the neutral axis
of a rectangular girder of cast-iron divides the depth in the pro-
portion of £ or -J- at the time of fracture, that is, that the compressed
section is to the extended section nearly in the inverse proportion
of the compressive to the tensile strength of the material. This
view is corroborated by experiments made by Duhamel,* who found
that sawing through the middle of small timber girders to Jths of
their depth from the upper or compression surface, and inserting a
thin hardwood wedge in the gap, did not diminish their ultimate
strength, and also by similar experiments made by the elder
Barlow, f which seem to indicate that the neutral axis in rectangular
girders of timber is very nearly at f ths of the depth, and in rec-
tangular bars of wrought-iron at about Jth of the depth from the
compressed surface at the time of fracture.
Mr. W. H. Barlow, however, controverts Mr. Hodgkinson's
conclusions in two papers which will be found at page 225 of the
Philosophical Transactions for 1855, and at page 463 of the
Transactions for 1857. In the former of these papers Mr. Barlow
gives the results of micrometrical measurements on two cast-iron
rectangular girders, each 7 feet long, 6 inches deep and 2 inches
thick, which he subjected to transverse strain ; his inference from
these experiments is that the neutral axis does not shift its position,
and this view seems in accordance with experiments made long ago
by Sir D. Brewster who transmitted polarized light through a little
* Morin, p. 120. f Strength of Materials, pp. 126, 133.
86 GIRDERS OF VARIOUS SECTIONS. [CHAP. IV.
rectangular glass girder 6 inches long, 1'5 inch broad, and O28 inch
thick ; when this was bent by transverse pressure, the neutral surface
remained in the centre, and colours due to strain were developed
above and below it in curved lines, which may perhaps aid the
physicist in investigating the strains in continuous webs.* Unless,
however, the tensile and compressive elasticities of glass are
materially different near the point of rupture, as they are in cast-
iron when approaching its limit of tensile strength, this experi-
ment does not throw much light on the subject. The whole
question, it must be confessed, is one of great difficulty, and may
require numerous experiments before it can be satisfactorily solved.
One practical inference, however, is of great importance, namely, that
the tearing and crushing strengths of materials derived from experi-
ments on the transverse strength of solid girders are often erroneous,
and have even led astray men of such capacity as Tredgold.
133. Transverse strength of thick castings much less
than that of thin castings. — In some experiments made by
Captain (now Colonel Sir Henry) James, as a member of the
Royal Commission for inquiring into the application of iron to
railway structures, it was found that the central part of bars of
iron planed was much weaker to bear a transverse strain than bars
cast of the same size.f He states that "it was found by planing
out f-inch bars from the centre of 2-inch square and 3-inch square
bars, that the central portion was little more than half the strength
of that from an inch bar, the relation being as 7 to 12." In page
111 of the same report, Mr. Hodgkinson showed that rectangular
bars of cast-iron, cast 1, 2, and 3 inches square, laid upon supports
4J feet, 9 feet, and 13^ feet asunder, were broken by weights of
447 ft>s., 1394 ft>s., and 3043 Ibs. respectively. These weights,
divided by the squares of the lengths, should give equal results ; the
quotients, however, were as 447, 349, and 338 respectively. Mr.
Hodgkinson attributed this falling off and deviation from theory
partly to the defect of elasticity, which he had always found in
cast-iron, but principally to the superior hardness of the smaller
castings, t
* Encycl. Metrop., Art. Light, par. 1090. f Iron Report, 1849, App. B., p. 250.
IPhil. Trans., 1857, p. 867.
CHAP. V.] GIRDERS WITH PARALLEL FLANGES, ETC. 87
CHAPTEK V.
GIRDERS WITH PARALLEL FLANGES AND WEBS FORMED
OF ISOSCELES BRACING.
133. Object of bracing:. — The primary object of bracing is to
convert transverse strains into others which act in the direction of the
length of the material employed and tend either to shorten or extend
it, according as the material performs the function of a strut or tie.
This object is attained by dividing the structure into one or more
triangles ; for since the triangle, or some modification of it, is the
only geometric figure which possesses the property of preserving its
form unaltered so long as the lengths of its sides remain constant,
it is, therefore, that which is best adapted for structures in which
rigidity is essential for stability. Hence, three pieces at least are
required to constitute a braced structure. Take, for instance, the
common roof truss which is an example of one of the simplest
forms of bracing, Fig. 47. The weight W is transmitted through
47. a pair of struts S and S',
to the walls. As, how-
ever, the oblique thrust of
the struts would tend to
overthrow the walls, it is
necessary to connect their
feet by a tie-beam T.
The strains in the different parts may be derived from the principle
enunciated in 9.
The class of girders which I purpose investigating in this chapter
is that in which the flanges are parallel and connected by diagonals
which form one or more systems of isosceles triangles. This class
of bracing includes girders whose web consists of a single system
of triangles, such as " Warren's" girder, as well as girders whose
web consists of two or more systems of equal-sided triangles, such
as isosceles " Lattice" girders.
88 GIRDERS WITH PARALLEL FLANGES [dlAP. V.
Definitions,
134. Brace. — The term Brace includes both struts and ties.
135. Apex. — The intersection of a brace with either flange is
called an Apex.
136. Bay. — The portion of a flange between two adjacent apices
is called a Bay.
137. Counterbraced brace. — A brace is said to be counterbraced
when it is capable of acting either as a strut or as a tie.
138. Counterbraced girder. — A girder is said to be counter-
braced when it is rendered capable of supporting a moving load.
This may be effected either by counterbracing the existing braces,
or by adding others
139. Symbols. — The symbol +> placed before a number which
represents a strain, signifies that the strain is compressive; the
symbol — , signifies that the strain is tensile.
Axioms.
140. The strain in each brace or bay is uniform throughout its
length and acts in the direction of the length only. This will be
obvious if we consider a braced girder to be an assemblage or
framework of straight bars connected with each other by pins
passing through their extremities merely.
141. A brace cannot undergo tension and compression simul-
taneously.
142. If several weights, acting one at a time, produce in any brace
strains of the same kind, either all tensile or all compressive, the
strain produced by all these weights acting together will equal in
amount the sum of those produced by each weight acting separately.
143. If several weights, acting one at a time, produce in any brace
strains of different kinds, some tensile, some compressive, the strain
resulting from all these weights acting together will equal the algebraic
sum of all the strains; in other words, their resultant will equal the
difference between the sum of the tensile and the sum of the compressive
strains.
144. A uniformly distributed load may without sensible error be
assumed to be grouped into weights resting on the apices, each apex
CHAP. V.] AND WEBS OF ISOSCELES BRACING.
89
supporting a weight equal to the load resting on the adjoining half
bays. This view is evidently correct if each bay be connected with
the adjoining bays and diagonals by a single pin at their intersection,
as in " Warren's" girder. In this case each loaded bay is a short
girder covered by a uniform load, the vertical pressure of which is
transferred to the adjoining diagonals. In addition to the transverse
strain, each bay sustains a longitudinal strain which it transmits to
the adjacent bays, from which, however, it derives no aid to its
transverse strength on the principle of continuity. In practice, the
cross girders, on which the flooring rests, generally occur at the
apices, so that no bay is subject to transverse strain except from its
own weight.
CASE 1.— SEMI-GIRDERS LOADED AT THE EXTREMITY.
Fig. 48.
145. Web. — Let W = the load at the extremity of the girder,
5 = the strain in any diagonal,
F = the strain in any given bay of either flange,
n = the number of diagonals between the
centre of the given bay and the weight,
6 = the angle which the diagonals make with
a vertical line.
The weight W is supported by the first diagonal and the upper
flange, the former sustaining compression, the latter tension. At a
three forces meet and balance ; namely, the weight, the horizontal
tension of the upper flange and the oblique thrust of diagonal 1 ;
90 GIRDERS WITH PARALLEL FLANGES [CHAP. V
their relative amounts may therefore be represented by the sides of
the triangle abc (9). Hence, the tension in the first bay of the
upper flange is to W as ac is to cb, that is, F = Wfcmfl, and the
compression in the first diagonal is to W as ab is to cb, tha.t is,
2 = WsecO. The tension of ad is transmitted throughout the upper
flange to its connexion with the abutment, but the compression in
diagonal 1 is resolved at b into its components in the directions of
diagonal 2 and the lower flange, producing tension in the former and
compression in the latter. Thus, there are three forces in equilibrium
meeting at 6, and their relative amounts may be represented to the
same scale as before by the sides of the triangle edb ; whence, the
tension in diagonal 2 equals the compression in diagonal 1, and
the compression in the first bay of the lower flange equals twice
the tension in the first bay of the upper flange, = 2Wtan6.
In this way it may be shown that all the diagonals are strained
equally, but by forces alternately tensile and compressive, while the
flanges receive at each apex equal increments of strain each equal
to 2WtanO. The general formulae for the strain in any diagonal is
therefore
2 = WsecO (110)
Ex. If 6 = 45°, sece = 1'414, and we have 2 = 1'414 W.*
146. Flanges. — Since the flanges receive at each apex successive
increments of strain, each equal to 2Wfcm0, the resultant strains
in the successive bays, being the sum of these successive increments,
increase as they approach the abutment in an.arithmetic progression
whose difference = 2WtanO ; they are, therefore, for any given bay
proportional to the number of diagonals between it and the load,
and we have,
F = nWtanO (111)
where n represents the number of diagonals between the centre of
any given bay and the weight (SO).
Ex. In the last bay of the upper flange of Fig. 48, n~7, and if 6 = 45°, tanQ = 1,
and we have F = 7 W .
* See the table in Chap. xi. for the numerical values of the tangents and secants of
different angles.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 91
The tension in the last diagonal may be resolved at g into a
vertical force pressing downwards through the abutment, and a hori-
zontal force tending to pull the abutment towards the weight. The
relative amounts of these three forces may be represented by the sides
of the triangle fgh ; whence, the vertical pressure = W, and the
horizontal force = Wfcm0; the latter, added to the tension in the
last bay of the upper flange, gives the total horizontal force exerted
by the upper flange to pull the abutment towards W. It will be
observed that the horizontal thrust of the lower flange against the
abutment is equal and opposite to the pull of the upper flange, so
that they form a couple whose tendency is to overturn the abutment
on its lower edge next the weight.
147. Strains in braced webs may be deduced from the
shearing-strain. — When the flanges are parallel and the bracing
consists of a single system of triangulation, the strain in any brace
is equal to the shearing-strain multiplied by seed. Hence, the
strains in the bracing might be deduced from the shearing-strain
in the web calculated in the manner explained in 18. The method
of the resolution of forces just described is, however, better cal-
culated to give a correct perception of the properties of diagonal
bracing, and it has, moreover, the advantage of being applicable to
lattice girders as well as those whose bracing consists of a single
system of triangles.
CASE II. — SEMI-GIRDERS LOADED UNIFORMLY.
Fig. 49.
92 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
14 §. Web. — Let W = the weight of so much of the load as
covers one bay, i.e., the weight resting
on each apex of the loaded flange (144),
n = the number of these weights between
any given diagonal and the outer end
of the girder,
2 = the strain in the given diagonal,
F = the strain in any bay of either flange,
0 = the angle which the diagonals make with
a vertical line.
W
The weight on the apex farthest from the abutment equals — -,
z
since it is assumed to support the load spread over the outer half bay,
while the load spread over the half bay next the abutment is assumed
to rest on the apex in contact with the abutment and may therefore
be neglected. If each weight be supposed acting alone, it would,
as in Case I., produce strains of equal amount, but of opposite kinds,
in each diagonal between its point of application and the abutment,
without 'affecting that part of the girder which lies outside it ; con-
sequently, when the whole load is applied, each diagonal sustains
the sum of the strains produced by the several weights which occur
between it and the outer end of the girder (835 143) and we have
2 = nWsecO (112)
Ex. The value of n for diagonal 5 is 2| ; if 0 = 45°, sect) = 1*414, and we have
2 = 3-535 W.
149. Strains in intersecting; diagonals. — When the apex of
any pair of diagonals supports a weight, W, the strain in that
diagonal which is nearer the abutment exceeds that in the more
remote by W«#c0. But when an apex does not support a weight
(those, for instance, in the lower flange of Fig. 49), the strains in
the two diagonals are equal in amount but of opposite kinds.
150. Increments of strain in flanges. — In the case of semi-
girders loaded uniformly, the increments of strain at the apices
increase as they approach the wall in an arithmetic ratio whose
difference = 2WtanO, and the resultant strains in each bay conse-
quently increase in a much more rapid ratio, viz., as the square of
their distance from the outer end of the girder (see eq. 11).
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 93
151. Resultant strains in flanges. — The resultant strains in
the bays may be represented by equations if desirable. For the
loaded flange,
F = { m (m — 1) + i } WtanO (113)
For the unloaded flange,
F = m2Wfcm0 (114)
where m represents the number of the bay measured along its own
flange from the outer end of the girder. These equations are
obtained by summation ; their proof will afford instructive practice
to the student.
153. General law of strains in horizontal flanges of braced
girders. — The strains in the flanges may also be derived from the
following law, which is applicable to all braced girders or semi-
girders with horizontal flanges, no matter how loaded, or whether
the bracing be isosceles, or the triangulation be single or lattice.
The increment of strain developed in the flange at any apex is equal to
the algebraic sum (i.e., the resultant,) of the horizontal components of
the strains in the intersecting diagonals. Keeping this in our recol-
lection, we may readily exhibit on a rough diagram — first, the strains
in the diagonals; secondly, their horizontal components at. the
apices ; and lastly, the successive sums of these components, that
is, the total strains in the several bays of each flange.
Ex. Let Fig. 50 represent such a diagram, the load being on the upper flange.
Let W = 10 tons,
e = 30°,
SecO = 1-154,
Tan0 = 0-577.
Fig. 50.
GIRDERS WITH PARALLEL FLANGES [CHAP. V.
The horizontal numbers attached to the diagonals are the coefficients n in eq. 112 ;
these multiplied by Wsecfl give the strains in each diagonal (see the numbers written
alongside). The horizontal numbers at each apex are obtained by adding the coefficients
of the two intersecting diagonals, and when multiplied by Wtan6 give the horizontal
components of the strains in the diagonals, i.e., the increments of flange-strain at each
apex (see the vertical numbers at each apex). Finally, the successive additions of these
increments give the resultant strains in each bay (see the vertical numbers at the
centre of each bay). These may be checked by eqs. 113 and 114 ; thus, in the 3rd bay
of the upper flange, F = (3X2 + £)X10X '577 = 37'5 tons, which differs merely in
the decimals from the number obtained by the diagram.
153. Lattice web has no theoretic advantage over a single
system — Practical advantage of lattice web — JLong pillars. —
If two or more systems of triangulation be substituted for the single
system just described, we have a lattice girder; and here I may
remark that lattice bracing has no theoretic advantage over a single
system of triangulation ; its advantages are entirely of a practical
nature, consisting in the frequent support which the tension diago-
nals give to those in compression, and which both afford the flanges.
Long pillars are serious practical difficulties, owing to their tendency
Fi. 51- to flexure, and lattice tension bars subdivide the struts,
which would otherwise be long unsupported pillars, into
a series of shorter pillars and hold them in the direction
of the line of thrust. That this does not injuriously affect
the tension diagonals will be evident, when we reflect that
the longitudinal strain produced in a tension diagonal by
the deflection of a strut crossing it at right angles, in
the plane of the girder, bears the same ratio to the weight
on the strut, as twice the versine of the deflection curve
bears to the length of the half strut — an amount quite
inappreciable in practice. If, for instance, a strut adc,
Fig. 51, be ten feet long, and if its central deflection under
strain, bd, equal half-an-inch (an amount much greater
than occurs in practice), the transverse force in the
direction of bd, which will sustain the thrust due to deflection, is to
the longitudinal pressure as -=- , that is, it is only ^th of the
weight passing through the pillar ; so that in most cases a stout
wire in tension would be sufficiently strong to keep the pillar from
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 95
deflecting in the plane of the girder. Again, if the force requisite
to resist the tendency of a strut to deflect at right angles to the plane
of the girder were supplied altogether by a tension brace, the longi-
tudinal strain in that brace would equal the weight on the strut, but
it does not follow that this strain is developed in the tension brace.
In fact, the force with which the ends of the tension brace are pulled
asunder is practically independent of the strut, for the increase in
the strain on the tension brace is only due to the difference between
the lengths be and dc. These considerations show that a mode-
rate lateral force will keep a long pillar from bending, and the
apprehension of long compression bars yielding by flexure in the
plane of the girder, or producing undue strains in the tension bars,
need not deter us from applying lattice bracing to girders exceeding
in length any girder bridge hitherto constructed. They also
explain the otherwise anomalous strength and rigidity of plate
girders and lattice girders whose webs are formed merely of thin
plates or thin bars. Such modes of construction are, however,
more or less defective. The struts should be formed of angle, ~|~>
or channel iron, or the material should be thrown into some other
form than that of a thin bar, which is quite unsuitable for resisting
flexure at right angles to the plane of the web. A very effective
method of stiffening thin compression bars has been applied to
tubular lattice girders. It consists of a species of light internal
cross-bracing between the opposite compression bars of the double
web ; this stiffens them at right angles to the plane of the web,
while the tension braces keep them from deflecting in the plane of
the web (see Plate IV.)
154. multiple and single triangulation compared — Lattice
semi-girders loaded uniformly. — The effect of latticing, com-
pared with a single system of triangulation, is, as far as theory is
concerned, merely to distribute the load over a greater number of
apices, and consequently to reduce the strain in each of the original
diagonals in proportion to the increased number of systems ; for,
since the several systems are, as we have just seen, practically inde-
pendent of each other, each diagonal sustains the strain due to those
weights alone which are supported on the apices of the system to
96 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
which it belongs. Eq. 112 will, therefore, give the strain in any
brace of a lattice semi-girder loaded uniformly, observing that the
coefficient n will now express the number of those weights alone
which are supported by that system to which the brace in question
belongs, and which occur between it and the outer end of the semi-
girder. The strains in the flanges of a lattice semi-girder increase
less abruptly than when one system of triangulation is adopted, and
are most conveniently obtained by a diagram similar to Fig. 50.
155. Girder balanced on a pier. — The case of a girder balanced
midway on a pier is obviously included in the preceding cases, since
each segment is a semi-girder.
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
AT AN INTERMEDIATE POINT.
Fig. 52.
156. Web. — Let W = the weight, dividing the girder into seg-
ments containing respectively m and n
bays,
I = m + n = the number of bays in the span,
S = the strain in any diagonal,
F = the strain in any bay of either flange,
0 = the angle which the diagonals make with
a vertical line,
x •=. the number of diagonals between any
bay and either abutment, measured from
the centre of the bay.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 97
On the principle of the lever (1O), the reaction of the right
abutment = y W, and that of the left abutment = y W. Since the
flanges are capable of transmitting strains in the direction of
their length only (14O), they cannot transfer vertical pressures
to the abutments ; ~ W must therefore be transmitted through the
diagonals on the right side of W to the right abutment, while
j W pass through the diagonals on the left side of W to the left
abutment. These quantities are in fact the shearing-strains
described in 34, that is, they are the vertical components of the
strains in the diagonals of each segment. The actual strain in any
diagonal is to its vertical component as the length of the diagonal
is to the depth of the girder, or, calling the angle of inclination of
a diagonal to a vertical line 6, we have the strain in each diagonal
in the right segment,
2=^Wwc0 (115)
in the left segment,
(116)
The diagonals which intersect at the weight are both subject to the
same kind of strain, while the strains in the diagonals of each segment
are alternately tensile and compressive. If the weight be at the
centre of the girder all the diagonals will be equally strained.
157. Flanges. — The tensile strain in the second diagonal, cd,
is resolved at c into its components in the directions of the top
flange and the first diagonal. The former = -jWtanO, and is
transmitted throughout the flange" as far as W, receiving at the
intervening apices successive increments of strain each equal to
—Wtand. At W these horizontal strains are met and balanced by
a similar series of horizontal increments developed at each apex to
the right of W and acting in the opposite direction to the first series.
The strains in the lower flange may be found in a similar manner,
98 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
for the thrust of the first diagonal, ac, is resolved at a into a vertical
pressure on the abutment, = yW, and a horizontal tensile strain
in the lower flange which acts as a tie. As these three forces which
meet at a balance, their relative amounts may be represented by the
sides of the dotted triangle abc; hence, the horizontal strain in
the first bay of the lower flange =r jWtanO, which is transmitted
throughout the flange as far as the bay underneath W, receiving
at each intervening apex successive increments each equal to
-^NtanQ. Beneath W these strains are met and balanced by the
reverse series generated at the several apices in the right segment.
The resultant strain in any bay of either flange equals the sum
of the increments generated at the several apices between it
and the abutment of the segment in which it occurs. If the bay
be in the right segment and x be measured from the right abutment,
F = ™\NtanQ (lit)
If the bay be in the left segment and x be measured from the left
abutment,
F = ™\NtanQ (118)
The maximum strains in the flanges occur at W and are represented
by the equation
F = ~pWto»0 (119)
Ex.— See Fig. 52.
Let 6 = 30°,
1 = 8,
m = 5-5,
n = 2*5,
seed = 1154,
tanO = 0-577.
From eqs. 115 and 116, the strains in each diagonal of the right segment = 0*7934 W,
and those in each diagonal of the left segment = 0*3606 W. From eq. 118 the coin-
pressive strain in bay A = 1*4425 W, and the tensile strain in bay B = 1*9834 W.
158. Concentrated rolling: load. — If the weight be a rolling
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 99
load, the strains in the diagonals will vary according to its position,
changing from tension to compression and vice versa, as it passes each
apex (37). If the upper flange supports the load, the maximum
compression in any diagonal occurs when the weight is passing its
upper extremity, and the maximum tension when passing the adjoin-
ing apex at that side to which the diagonal slopes downwards. If
the lower flange supports the load, the maximum tensile strain in
any diagonal occurs when the weight is passing its lower end, and
the maximum compressive strain when passing the adjoining apex
on that side to which the diagonal slopes upwards. The maximum
strain in any bay of the unloaded flange occurs when the moving
load is in the vertical line passing through that bay, as may be seen
from eqs. 117 or 118, for mx and nx are at their maximum when
they become mn (38). The maximum strain in any bay of the
loaded flange occurs when the passing load rests on the adjoining
apex on the side next the centre, for the product mn, in eq. 119, is
greater for this apex than for the adjoining apex on the side remote
from the centre.
159. Lattice girder traversed by a single load. — In this
case the strains in the diagonals may be calculated by eqs. 115
and 116, for the reasoning by which these equations were deduced
is equally applicable to lattice girders. It will also be observed
that only one system of triangulation is strained at a time, i.e.,
supposing the load to rest on a single apex, which, however, is
seldom the case, as generally two or more adjacent apices are
loaded together.
CASE IV. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
UNIFORMLY.
Fig 53.
100 GIRDERS WITH PARALLEL FLANGES [CHAP. V-
160. Web. — Let W = the weight of so much of the load as
covers one bay, i.e., the weight resting
on each apex of the loaded flange,
/ = the number of bays in the span,
n = the number of weights between any given
diagonal and the centre of the girder,
2 = the strain in the given diagonal,
F = the strain in any bay of either flange,
0 = the angle which the diagonals make
with a vertical line.
If the load be uniformly distributed so that an equal weight rests
upon each apex, the strains in the diagonals gradually increase from
the centre toward the ends. Any two diagonals equally distant
from the centre sustain all the intermediate load. If they are tension
diagonals, the weight is suspended as it were between them; if
they are compression diagonals it is supported by them as oblique
props. Each diagonal conveys, therefore, to the abutment the pres-
sure of the weights between it and the centre, and the sum of these
weights constitutes its vertical component or shearing-strain (46).
Hence, we have for a uniform load,
S = n\NsecQ (120)
161. Flange-strains derived from a diagram. — The strain
in the flanges may be derived from the law stated in 158 by the
aid of a rough diagram, as explained in the following example : —
Ex. 1. Let Fig. 54 represent one-half of a girder 80 feet long, with the bracing
formed of 8 equilateral triangles, and supporting a uniform load of half a ton per
running foot. From these data we have
W = 5 tons,
0 = 30°,
1 = 8,
tan6 = 0-577,
sec9= 1-154,
Wton0 = 2-885tons,
Wsec0 = 5-770 tons.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 101
Fig. 54.
The horizontal numbers attached to the diagonals are the coefficients n, in eq. 120;
these, multiplied by Wsecfl, give the strains in the several diagonals (see the numbers
written alongside them). The horizontal numbers at each apex are the sums of the
coefficients attached to the intersecting diagonals ; these multiplied by Wtand give the
horizontal components of the strains in the diagonals, that is, the increments of flange-
strain at each apex (see the numbers written in a vertical direction at each apex).
Finally, the successive additions of these increments give the resultant strains in the
flanges (see the numbers written in a vertical direction at the centre of each bay).
Ex. 2. Let Fig. 53 represent a girder 80 feet long, with the bracing formed of 8
right-angled triangles, and supporting a uniform load of half a ton per running foot.
Here W = 5 tons,
6 = 45°,
l=S,
tan0 = 1,
sec0=l-414,
\NtanQ = 5 tons,
Wsecfl = 7-07 tons.
The strains in tons are as follows : —
DIAGONALS,
1
2
3
4
5
6
7
8
Strains in tons (eq. 120),
-24-7
+247
-177
+17-7
—10-6
+10-6
-3-5
+ 3-5
FLANGES, .
A
B
C
D
E
F
G
H
Strains in tons, .
+17-5
+47-5
+67-5
+77-5
-35
-60
-75
— 80
168. Web, second method. — The strains in the diagonals may
also be obtained by forming a table of the strains which each weight
102
GIRDERS WITH PARALLEL FLANGES [CHAP. V.
would produce if acting separately, and then taking as the resultant
strain from all acting together the sum or difference of the tabu-
lated strains, according as they are of the same or opposite kinds.
Thus, diagonal 4, Fig. 53, is subject to compressive strains from all
the weights except the first ; the resultant strain is therefore found
by subtracting the tensile strain produced by the first weight from
the sum of the compressive strains produced by the remaining six
weights (143). This method, as applied to the first example in 161,
is exhibited in the annexed table, the numerals in the first column
of which represent the diagonals, and the letters in the upper row
the weights, in order of position. The numbers found at the inter-
section of a diagonal with a weight represent in tons the strain
produced in that diagonal by the weight in question (see eq. 115).
The last column contains the strains which the load produces when
distributed uniformly all over. These are obtained by adding
algebraically the several horizontal rows, and the strains thus
obtained should agree with those derived from eq. 120.
1
Wi
Ws
W3
W4
W3
We
W7
Strains in
diagonals due
to a
uniform load.
1
2
3
4
5
6
7
8
Tons.
-5-1
+ 5-1
+ 0-7
-07
+ 07
-07
+ 0-7
-0-7
Tons.
-4-3
+ 4-3
— 4-3
+ 4-3
+ 1-4
- 1-4
+ 1-4
- 1-4
Tons.
-3-6
+ 3'6
-3-6
+ 3-6
-3-6
+ 3-6
+ 2-2
-2-2
Tons.
- 2'9
+ 2-9
-2-9
+ 2-9
— 2-9
+ 2-9
-2-9
+ 2-9
Tons.
- 2-2
+ 2-2
— 2-2
+ 2-2
-2-2
+ 2-2
— 2-2
+ 2-2
Tons.
~ 1-4
+ 1-4
— 1-4
+ 1-4
- 1-4
+ 1-4
- 1-4
+ 1-4
Tons.
-•72
+ •72
-72
+ •72
- 72
+ •72
- -72
+ •72
Tims.
- 20-2
+ 20-2
- 14-4
+ 14-4
- 87
+ 87
- 2-9
+ 2-9
It will be observed that, when once the strain produced by W7
in diagonal 1 is obtained, all the other strains may be derived
from it by addition.
163. Increments of strain in flange*. — The flanges receive
successive increments of strain at each apex as they approach the
centre where the maximum strains occur. These increments
diminish as they approach the centre in an arithmetic progression
whose difference = 2Wta«0. Hence, the strains in the bays
might be expressed by an equation ; they may, however, be more
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 103
conveniently found by the aid of a rough diagram, as already
described in 161.
164. Strains in flanges calculated by moments. — The strains
in any given bay may also be obtained by taking moments round
the apex immediately above or below it. To obtain the strain in
bay C, Fig. 53, for example, take moments round the apex a. The
left segment of the girder is held in equilibrium by the reaction of
the left abutment (= 17'5 tons), the two first weights, Wj and W2,
the horizontal tension in C, and the strains at a. Taking moments
round the latter point, we have
Fd = 17-5 X '2-5b — 5(1-5 + 0-5)6,
where F = the strain in the flange at C,
b =. the length of one bay,
d = the depth of the girder.
If 0 = 45°, b = 2d, and we have F = G7'5 tons, as in ex. 2, (161).
This method is, it will be perceived, merely a modification of
that described in 43. It is sometimes convenient for checking
results obtained by the resolution of forces.
165. tcirder loaded unsymmetrically. — If the load be distri-
buted in an unsymmetrical manner, the strains produced by each
weight acting separately should first be tabulated, and then the
resultant strains may be obtained as indicated in 163.
166. ftJirder loaded symmetrically. — If the central part of a
symmetrically loaded girder be free from load, the central braces
may be removed without affecting the strength of the structure.
If, for example, the girder represented in Fig. 53 support only
Wlf W2, W6, W7, the' braces in the interval, 5, 6, 7, 8, S', V, 6',
5;, may be removed. If the central weight alone be wanting, then
braces 7, 8, 8', 7', may be removed.
167. Strains in end diagonals and bays. — When the load is
symmetrical, each of the end diagonals sustains a strain equal to
one-half the load multiplied by secO, and the extreme bays of the
longer flange sustain a strain equal to one-half the load multiplied
by tanO. Consequently, when 9 = 45°, the strain in each of these
extreme bays equals half the load.
16§. Strains in intersecting diagonals — General law of
104
GIRDERS WITH PARALLEL FLANGES [CHAP. V.
strains in intersecting: diagonals of isosceles bracing: \\lth
parallel flanges. — When two diagonals intersect at a loaded
apex of a girder loaded uniformly, the strain in that diagonal
which is more remote from the centre exceeds that in the other
by WsecQ. The following law is applicable to all girders with
parallel flanges and isosceles bracing whether single or lattice;
when two diagonals intersect at an unloaded apex, no matter how the
load may be distributed, the strains in the two diagonals are equal in
amount, but of opposite kinds.
CASE V.— GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED
BY A TRAIN OF UNIFORM DENSITY.
Fig. 55.
169. Web. — Let W = the weight of so much of the uniformly
distributed load as covers one bay,
i.e., the permanent load resting on
each apex,
W' = the weight of so much of the passing
load as covers one bay, i.e., the passing
weight on each apex,
I = the number of bays in the span,
n = the number of apices loaded by the
passing load between any given dia-
gonal and either abutment,
2 = the strain in the given diagonal due to
the permanent load,
2' = the maximum strain in the given diagonal
due to the passing load,
0 = the angle the diagonals make with a
vertical line.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 105
The strains in the diagonals vary according to the position of
the passing train, not only in amount, but also in kind. If, for
instance, Wt alone rests upon the girder, diagonal 4 is subject
to tension. If now W2 be added, its tendency will be to produce
compression in diagonal 4, that is, a strain of an opposite kind
to that produced by W1? and the true strain which this diagonal
sustains, when both weights rest upon the girder, is equal to the
difference of the strains produced by each weight acting separately
(143). The third, fourth, fifth, sixth, and seventh weights tend
to increase the compression in diagonal 4, while the first weight
alone tends to produce tension. Consequently, the maximum
compression in this diagonal takes place when all the weights
except the first rest upon the girder, and the maximum tension
occurs when all the weights are removed except the first. The
same result may be arrived at in any particular case by means
of a table of strains, such as that in 168, where we find at the
intersection of diagonal 4 and Wu that this weight produces a
tension of O7 tons in the diagonal, while each of the remaining
weights produces compression. When all the weights rest upon
the girder, the first and last produce no effect on diagonal 4, since
the strains due to these weights are equal and have opposite signs.
In fact, these weights are supported exclusively by the flanges and
the last pair of diagonals at each end, and, as far as they alone are
concerned, all the intermediate diagonals might be omitted. If,
however, Wx be removed, the eighth part of W7 is transmitted to
the left abutment, and consequently increases the compression in
diagonal 4 by the strain found in the table at the intersection of
W7 and 4. If, on the other hand, W7 be removed, the eighth part
of Wj is transmitted to the right abutment, diminishing the com-
pression in diagonal 4 by the strain found at the intersection of Wj
and 4. In a similar manner we find from the table that any other
diagonal, 7 for instance, sustains the greatest amount of compression
when the first, second, and third weights alone rest upon the girder,
and the greatest tension when these are removed and the other
weights remain.
17O. maximum strains in web — Strains in intersecting:
106 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
diagonals. — The maximum strain in any diagonal occurs when the
passing train covers only one segment (51) ; and in general terms, the
maximum tensile strain in any diagonal occurs when the passing train
covers the segment from which the diagonal slopes upwards, and the
maximum compressive strain when it covers the segment towards which
the diagonal slopes upwards. When a pair of diagonals meet at the
unloaded flange, the strains in the two diagonals are equal in amount
but of opposite kinds, and the maximum tensile strain in one is equal
to the maximum compressive strain in the other, and vice versa (168).
171. Permanent load — Absolute maximum strains. — In all
the foregoing investigations the weight of the girder and roadway
has been left out of consideration, but in practice the perma-
nent load materially modifies the strains, especially in bridges of
large span where the ratio of the permanent to the passing load
is considerable. If the supported load be uniformly distributed,
its weight may be added to that of the structure, provided the
latter be also uniform, and the calculations made for their com-
bined weights as already explained for uniform loads. But when
the load moves, the strains in the bracing produced by the weight
of the permanent structure will be increased or diminished, or
even a strain of an opposite kind produced, according to the
position of the passing load. In order to obtain the absolute
maximum strains to which the bracing is liable under these cir-
cumstances, we must calculate — first, the strains produced by the
permanent structure alone, and afterwards the maximum strains,
both tensile and compressive, due to the passing load alone. These
latter, when added to, or subtracted from, the strains produced by
the permanent load, according as they are of the same or opposite
kinds, will give the absolute maximum strains to which each brace
is liable in any position of the passing load.
178. Web, first method. — Perhaps the simplest method of
obtaining the strains in the diagonals from a passing train is by
forming a table of strains produced by each weight acting sepa-
rately, as in 163. Then adding, first the tensile, and afterwards
the compressive, strains in each horizontal row, we obtain the
required maximum strains of each kind.
CHAP. V.] AND WEBS OF ISOSCELES BRACING.
107
Ex. The following example of a girder of eight bays will illustrate this method of
calculating the absolute maximum strains when the bridge is traversed by a load of
uniform density whose length is not less than the span. Let Fig. 55 represent a railway
girder, 80 feet long and 5 feet deep, the bracing of which is formed of 8 right-angled
isosceles triangles, with, the roadway attached to the upper flange. Let the permanent
bridge-load equal half a ton per running foot, and the greatest passing train of uniform
density equal one ton per foot ; we then have
W = 5 tons from the permanent load,
W = 10 tons from the passing train,
l = S,
0=45°
tanO = 1,
sec0 = l-414,
W*ec0 = 7-07 tons,
W
-y-sec0 = 177 tons,
(W + W) tanB = 15 tons.
S*
ga
Wi
w»
Ws
W4
Ws
We
w'7
C'
T
2
C
T
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
i
-12-4
-10-6
-8-9
-7-1
-5-3
-3-5
— 1-8
...
-49-6
-24-7
-74-3
2
+12-4
+ 10-6
+8-9
+7-1
+5-3
+3-5
+1-8
+49-6
+24-7
+74-3
3
+ 1-8
-10-6
-8-9
-7-1
-5-3
-3-5
-1-8
+ 1-8
-37-2
-17-7
-54-9
4
- 1-8
+ 10-6
+8-9
+7-1
+5-3
+3-5
+ 1-8
+37-2
- 1-8
+ 17-7
+54-9
5
+ 1-8
+ 3-5
-8-9
-7-1
— 5-3
-3-5
-1-8
+ 5-3
— 26-6
— 10-6
— 37-2
6
- 1-8
- 3-5
+8-9
+7-1
+5-3
+3-5
+ 1-8
+26-6
- 5-3
+ 10-6
+37-21 ...
7
+ 1-8
+ 3-5
+5-3
-7-1
-5-3
-3-5
-1-8
+ 10-6
-17-7
- 3-5
+ 7-l! -21-2
8
- 1-8
- 3-5
—5-3
+ 7-1
+ 5-3
+3-5
+1-8
+ 17-7
— 10-6
+ 3-5
+21-2
- 7-1
The numbers in the first column represent the diagonals, and the seven first
letters in the upper row the passing weights, in order of position. The numbers
found at the intersection of a diagonal with a weight represent the strains pro-
duced in the diagonals by the passing load resting, on each apex separately ; these
are derived from eqs. 115 and 116. The columns marked C' and T' contain the
maximum strains of compression and tension which the passing load can produce ;
they are obtained by adding, first the compressive, and afterwards the tensile, strains
in each row in the first part of the table. The column marked 2 contains the strains
due to the uniform permanent load; these are derived from eq. 120. Finally, the
two last columns, marked C and T, contain the absolute maximum strains which the
combination of permanent and passing loads can produce ; these are obtained by adding
algebraically column 2 to columns C' and T' respectively. If one ton per foot be
the greatest passing load to which the girder is liable, the strains in the bracing can
never exceed these absolute maximum strains.
173. Flanges. — The maximum strains in the flanges occur when
the passing load covers the whole girder (53).
108
GIRDERS WITH PARALLEL FLANGES [CHAP. V.
In owe example this occurs when the girder supports a uniformly distributed load of
1'5 tons per running foot, equivalent to 15 tons at each apex. The strains in the
several bays are given in the following table ; they are obtained by the aid of a diagram,
as described in 161.
Bays,
A
B
C
D
E
F
G
H
Strains
in tons,
+ 52-5
+ 142-5
+ 202-5
+ 232-5
- 105
— 180
— 225
— 240
174. Count erb racing1. — On examining the two last columns of
the table in 17S, it will be seen that diagonals 7 and 8 are the only
braces which are liable to both tensile and compressive strains.
Consequently, the four central diagonals alone require to be coun-
terbraced (137); whereas, if the permanent load had been left
out of consideration, all the diagonals except the extreme pair
at each end would require counterbracing ; and if, on the other
hand, the strains from the passing load had been calculated on
the supposition of its being a uniformly distributed, in place of a
passing load, none of the diagonals would require counterbracing.
175. Permanent load diminishes connterbracing;. — In bridges
of large span, the permanent load will materially diminish the
amount of counterbracing that would be required if the passing
load alone had to be provided for; and when the span is very
large, it will be more accurate to consider the permanent load as
resting, part on the upper, and part on the lower flange. In small
spans this nicety of calculation may be neglected, since the cross
road-girders and roadway, with the flange to which they are
attached, form the greater portion of the permanent load.
176. Web, second method. — The maximum strains in the
diagonals due to a passing train of uniform density may be expressed
by equations similar to those given in the preceding cases, for which
purpose it is necessary to divide girders into two classes.
Class A.
Girders in which the extreme apices of the loaded flanges are
each distant one whole bay from the abutments, as in Fig. 56.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 109
Fig. 56.
From eq. 115 the strain in any diagonal from the passing
weight at —
W
The 1st apex = —secO,
• i
W
2nd apex =' 2 S- secO,
i
W
3rd apex = 3 ~
W'
nth apex — n — secQ,
where n represents the number of loaded apices between the
diagonal and one abutment. The maximum strain is equal to the
sum of these separate strains ; hence,
S' = (1 + 2 + 3 + . . . n) ^-sec6,
or by summation,
„ _n(n + 1) W'
(121)
Class B.
Girders in which the extreme apices of the loaded flange are
each distant one half -bay from the abutment, as in Fig. 57.
Fig. 57.
110 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
The strain in any diagonal from the passing weight at —
W'
The 1st apex = _ sect),
W
2nd apex = 3 — y- secO,
£1
W'
3rd apex = 5 -^
Wy
nth apex = (2n — 1) —
Adding these together, we have the strain due to the passing load,
2' =
or by summation,
2' = (1 + 3 + 5 + • • • 2w — 1) __
-w2 W
S' = yX^-«««. (122)
Eq. 122 proves that the strains in the diagonals produced by a
passing load are proportional to the square of the loaded segment
(50).
Ex. The following example of a girder of 8 bays with equilateral triangles, belong-
ing to Class A, will illustrate this method of calculating the maximum strains produced
by a passing train of uniform density sufficiently long to extend over the whole bridge.
Let the girder be 80 feet long, the permanent load 0'5 tons per running foot, and the
passing load of greatest density (say engines) one ton per foot ; we then have, using the
same notation as before,
W = 5 tons from the permanent load,
W = 10 tons from the passing train,
1 = 8,
6 = 30°,
tan0 = 0-5773,
sec6 = 1-154,
Wscc = 5-77 tons
W
—sece = 1-442 tons
(W+W') tone = 8 -66 tons.
CHAP. V.] AND WEBS OF ISOSCELES BRACING.
Ill
Diagonals
n(n + l)
2
C'
T*
C
T
2
Tons.
Tons.
Tons.
Tons.
Tons.
1
— 28
- 20-2
...
-40-4
-60-6
2
- 0
+ 20-2
+ 40-4
...
H-60'6
...
3
— 21
- 14-4
+ 1-4
— 30-3
...
-447
4
- 1
+ 14-4
+ 30-3
+ 1-4
+ 447
...
5
- 15
— 87
+ 4-3
— 21-6
...
-30-3
6
- 3
+ 87
+ 21-6
- 4-3
+ 30-3
...
7
— 10
- 2-9
+ 87
— 14-4
+ 5-8
-17-3
8
— 6
+ 2-9
+ 14-4
- 87
+ 17-3
— 5-8
The numerals in the first column represent the diagonals (see Fig. 55). The second-
column contains the coefficients for each diagonal, n ^n — - in eq. 121, n being mea-
sured alternately from the right and the left abutment. Column 5 contains the
strains produced by the permanent bridge-load ; these are calculated by eq. 120.
Columns C' and T' contain the maximum strains produced by the passing load;
these are calculated by the aid of the second column and eq. 121 (see ISO).
Finally, the two last columns contain the absolute maximum strains of either kind in
the bracing, taking both permanent and passing loads into consideration ; these are
obtained by adding columns C' and T' algebraically to column 2. The strains in the
flanges are as follows (161) : —
Bays,
A
B
C
D 1 E
F
G
H
1
Strains in tons,
+30-3
+82-3
+117-0
+134-2
-60-6
—103-9
—129-9
—138.6
CASE VI. — LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND
LOADED UNIFORMLY.
177. Approximate rule for strains in lattice web. — It has
been already shown (154) that the effect of increasing the number of
diagonals, so as to form a lattice girder, is merely to distribute the
load over a greater number of apices and thus diminish the strain
in each diagonal in proportion to the increased number of systems.
This suggests the following approximate rule for finding the strains
112 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
in the bracing of lattice girders. Calculate the strains on the supposi-
tion that there is only one system of triangles. These divided by the
number of systems will give the strains in the corresponding lattice dia-
gonals. As, however, more exact methods of calculation are of easy
application, they are preferable to a rule which is merely approximate.
178. Web — Flanges. — In the case of a uniform load the strains
in the bracing may be calculated by eq. 120, observing that the
Fig. 58.
coefficient n will represent in a lattice girder the number of those
weights which occur between any given diagonal and the centre
of the girder, and which rest only on the apices belonging to
its own system of triangulation. This assumes that the strains
from weights belonging to different systems, but at equal distances
on opposite sides of the centre, such as W5 and Wn in Fig. 58,
do not pass through the intermediate diagonals, but merely through
the flanges and those diagonals of their respective systems which
occur between them and the abutments. This is the simplest way
of calculating the strains due to a uniform load, but they may also
be calculated for each system separately (163), in which case the
strains in the diagonals will differ somewhat from those obtained by
the first method. The strains in the flanges are most conveniently
obtained by the aid of a diagram of strains (161).
Ex. The following example of a lattice girder, 80 feet long and 10 feet deep, with
four systems of right-angled triangles, i.e., 16 bays, will illustrate the mode of calcula-
tion (see Fig. 58). If the uniform load equal half a ton per running foot, we have,
W = 2'5 tons = the weight on each apex,
6 = 45°,
Wm-0 = 3-535 tons,
W tand = 2'5 tons,
n = the number of weights belonging to its own system
between any given diagonal and the centre of the girder.
CHAP. V.] AND WEBS OF ISOSCELES BRACING.
Fig. 59.
113
The numbers attached to the diagonals in the preceding diagram of strains are the
coefficients n, in eq. 120 ; these multiplied by Wsecfl give the strains in the diagonals,
as in the following table, the upper row of which represents the diagonals in order of
position (see Fig. 58), and the lower row the corresponding strains in tons : —
Diagonals.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Strains
in tons.
+7-1
+7-1
+7-1
+5'3
+3-5
+3-5
+3-5
+1-8
»
••
"
—1-8
—3-5
—3-5
3-5
-5'3
-7-1
The horizontal numbers at the apices are obtained by adding the coefficients of the
intersecting diagonals. These numbers multiplied by WtanQ are the increments of
strain in the flanges (see the vertical figures at each apex). Finally, the successive
additions of these increments give the resultant strains in the flanges in tons (see
the vertical figure at the centre of each bay).
CASE VII. — LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND
TRAVERSED BY A TRAIN OF UNIFORM DENSITY.
179. Web, first method. — Perhaps, the simplest method of
obtaining the strains in the case of a passing train is to tabulate
the strains produced by each weight separately, and thence infer
what condition of the load will produce the maximum strains in
each diagonal (168).
114 GIRDERS WITH PARALLEL FLANGES [CHAP. V
Ex. The following example of a lattice girder, 80 feet long and 10 feet deep, with
4 systems of right angled triangles, will illustrate this method (see Fig. 60) :—
Fig. 60.
Let the permanent bridge-load equal half a ton per running foot and the passing
train equal one ton per running foot. From these data we have,
W = 2' 5 tons at each apex from the permanent load,
W = 5'0 tons at each apex from the passing train,
I = 16 = the number of bays in the span,
0 = 45°,
Wscc0= 3-535 tons,
disced = 0-442 tons,
W tan0 = 0-47 tons.
The upper row in the table on p. 115 represents the passing weights, and the first
column represents the diagonals. The next fifteen columns contain the strains pro-
duced in the diagonals by each weight acting separately ; these are derived from eqs.
115 and 116. The next two columns, marked C' and T', contain the maximum strains
of compression and tension produced by the passing load ; these are obtained by adding
the strains of compression and tension in each row separately. The column headed 5
contains the strains produced by the permanent load ; it is copied from the previous
example in 138. Finally, the last two columns, marked C and T, contain the
absolute maximum strains which the combined passing and permanent loads can
produce ; these are obtained by adding column 2 to columns C' and T' successively.
From this table it appears that diagonals 9, 10, and 11 are subject to both com-
pression and tension ; consequently, the six central diagonals require counterbracing.
The maximum strains in the flanges occur when the passing load extends uniformly
over the whole girder (53) ; they may be obtained by means of a diagram of strains
as explained in 178. In this example the flange-strains are three times greater than
in the example in 1J8, for the passing load per running foot equals twice the per-
manent load.
18O. End pillars. — The end pillars of lattice girders are some-
times subject to transverse strain from the horizontal components
of the diagonals which intersect them midway between the flanges.
This transverse strain is, however, of slight amount, as it is merely
a differential quantity, being due to the excess of the strain in the
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 115
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116 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
tension diagonals over those in compression, or vice versa. In
Fig. 60, for example, the vertical component of the diagonals
meeting at c is transmitted through the lower half of the pillar to
the abutment in addition to any pressure which it may receive
from the upper half. Their horizontal component, however, tends
to deflect the pillar outwards or inwards, according as the strain
in the compression or tension diagonal is in excess, and this trans-
verse strain converts the pillar into a vertical girder whose abutments
are the flanges. This excess does not attain its greatest value
when the girder is uniformly loaded ; for since the load is on the
upper flange, the tension in diagonal IT equals the compression in
diagonal 3, and, on examining the preceding table, we find that the
greatest excess of strain in diagonal 1 over that in diagonal 3 occurs
when all the apices of the system to which the former diagonal
belongs are loaded, while those of the latter are free from load.
This of course is a condition of load which is very unlikely to occur
in practice, but it is quite possible that passing weights may rest
on two apices of the first system, say Wj and W5, while the apices
belonging to the other system are free from load. This might
occur, for instance, if a pair of engines or heavy wagons were
to cross with a proper interval between them. If this were
to occur in our example, the horizontal component of the strain
in diagonal 1 would = O5 + H) W/tanfl = g-1 tons. The pillars
ought accordingly to be designed with adequate strength to meet
such transverse strains, as well as those of compression in the
direction of their length.
181. Ambiguity respecting strains in lattice bracing.—
When a lattice girder contains three or more systems of triangles, a
slight ambiguity may occur respecting the strains if the load be dis-
posed on both sides of the centre. Take for example W7 and W9,
Fig. 60, which belong to different systems, but rest on apices equally
distant from the centre ; the whole of W7 may be transmitted to
the left abutment through diagonals 7, 13', 3 and 17', and the
whole of W9 to the right abutment through diagonals 7', 13, 3' and
17, without producing strains in the other diagonals of either
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 117
system, which indeed might be safely removed as far as these
weights are concerned. The method of calculation described in
178 assumes this to be the case. But again, y^ths of W7 may
be transmitted to the right abutment, and -^ths to the left, through
the diagonals of its own system, and similarly with W9 (1O). This
is assumed to be the case for the passing load in the example in 179.
Hence, there is a slight ambiguity respecting the strains, asthey may
go in either way, or partly in one, partly in the other, just as it is
impossible to say how much pressure is transmitted through any
one leg of a four-legged table. If, however, the girder be strong
enough to sustain the strain in whichever way it can be conveyed the
safety of the structure is secured, and practically there is a very
slight difference in the resulting strains whichever method of calcu-
lation is adopted. It may be thought that the " principle of least
action" will necessarily determine the direction of the strains, i.e.,
that they will take that direction in which the work done is a mini-
mum ; practically, however, a slight inaccuracy in the exact length
of the bars will doubtless determine the direction they will take. It
ought also to be admitted that a structure will stand as long as it
has not exhausted the whole of its possible conditions of stability,
and it is therefore sufficient assurance that any structure will stand
if we prove that a certain state of stability can be realised.
188. Flange-strains calculated by moments. — When cal-
culating the strain in any bay of a lattice girder by the method
of moments (164), we must not neglect the moments of the strains
in the diagonals. That part of the girder represented in Fig. 60,
for instance, which is to the left of a line drawn through bays a and b,
is held in equilibrium by the reaction of the left abutment, the weights
Wj, W2, and W3, the horizontal forces at a and b, and the oblique
forces in diagonals 4, 5, 13' and 14'. The moments of the former
pair of diagonals are opposed to those of the latter pair, but they
seldom balance exactly. Hence, the strains in two bays vertically
over each other are rarely precisely the same in value, but differ by
an amount equal to the horizontal component of the strains in the
diagonals which are intersected by a line joining them ; this, indeed,
is true whether the bays lie vertically over each other or not, and
118 GIKDEKS \VITH PARALLEL FLANGES [CHAP. V.
is merely a modification of the law stated in 58. Again, it would
be erroneous to expect that the strains in the bays of braced
girders when uniformly loaded must necessarily agree precisely
with those obtained by eqs. 23 or 25. In some cases it happens
that they do so agree, but in general they are only close approxima-
tions. This arises from our assuming that the load in braced girders
is concentrated at the apices, in place of being uniformly distributed.
In Fig. 60, for instance, the load on the extreme half -bays is assumed
to rest directly over the pillars, while that on the two central half -bays
is assumed to rest exactly on the central apex ; consequently, these
portions of the load are neglected in calculating the central strains
in the flanges by the method of moments. If, however, the moments
be calculated on the supposition that these loads act at their centres
of gravity, i.e., at a distance from the pillars equal to a quarter-bay,
and at a distance from the centre also equal to a quarter-bay, the
strain at the centre will agree with that obtained by eq. 25.
183. Web3 second method. — The strains in the bracing of
lattice girders subject to passing loads of uniform density may be
expressed by an equation obtained in the following manner : —
Let W — the passing weight on each apex,
I = the number of bays in the span (= 16 in Fig. 61),
k = the number of systems of triangles, i.e., the number
of bays in the base of one of the primary triangles
(= 6 in Fig. 61),
2' = the maximum strain which any given diagonal sustains
from the passing load,
n = the number of bays between the given diagonal and
one of the abutments, measured along the loaded
flange,
p = the integral number of times that its own system
occurs between the given diagonal and the same
abutment, measured also along the loaded flange
(= the integral part of^),
iC
6 =. the angle which the diagonals make with a vertical
line.
CHAP. V.] AND WEBS OF ISOSCELES BRACING. 119
Fig. 61.
Suppose the load traversing the upper flange of Fig. 61 ; diagonal a
sustains the maximum compressive strain when W3 and W9 rest
upon the girder, and in general, each brace will sustain the maxi-
mum strain when the passing load covers only one segment — which
segment may be easily seen by inspection (17O) — but the strain it
sustains is due to those weights alone which rest on the apices of its
own system. If, for example, there are n bays between the top of
diagonal a and the left abutment, then, on the principle of the lever,
the portion of W9 which is transmitted to the right abutment
through a = jW1 ; and of W3 = ~ W. The maximum com-
pressive strain in diagonal a is equal to the sum of these quantities
_
multiplied by secO, and equals (n + n — k)-j-secO ; and in general,
the maximum strain in any given diagonal due to the passing load,
_ _ _ _ W
S' = (n + n — k + n — 2k + n — 3k + . . . . n — pk)-j-secO,
or summing these up,
(123)
The maximum tension man the maximum compression in b (17O),
and 2' will represent compressive or tensile strains according as the
load traverses the upper or lower flange.
Ex. Let Fig. 62 represent a lattice girder 80 feet long and 5 feet deep, whose
bracing consists of two systems of right angled triangles with the load traversing the
upper flange.
120 GIRDERS WITH PARALLEL FLANGES [CHAP. V.
Fig. 62.
Let the permanent bridge-load equal half a ton per running foot, and the heaviest
passing train of uniform density equal one ton per foot. Then we have,
W = 2'5 tons at each apex from the permanent load,
W = 5 tons at each apex from the passing train,
a = 45°,
Wsece = 3-54 tons,
—sec6 = 0'44 tons,
an6 = 7*5 tons.
The strains in tons are given in the following table, the numbers in the first column
of which represent the diagonals in Fig. 62. The 2nd, 3rd, and 4th columns are the
coefficients in eq. 123, from which the maximum strains produced by the passing
load, columns C' and T', are derived. The strains produced by the permanent
bridge-load (column 2) are obtained from eq. 120, observing that the coefficient
n in that equation now represents the number of weights belonging to its own system
which occur between any given diagonal and the centre of the girder (158). The
last two columns, C and T, give the absolute maximum strains due to both permanent
and passing loads ; these are obtained by adding columns C' and T' successively to
column 2.
Diagonals.
n
P
(«-f)(P + D
C'
T'
2
C
T
Tons.
Tons.
Tons.
Tons.
Tons.
1
15
1
64
+ 28-2
+ 14-2
+ 42-4
...
2
14
7
56
-H24-6
...
+ 12-4
+ 37-0
3
13
6
49
+ 21-6
— "b-4
+ 10-6
+ 32-2
...
4
12
6
42
-f 18-5
— 0-9
+ 8-9
+ 27-4
5
11
5
36
+ 15-8
— 1-8
.+ 7-1
+ 22-9
...
6
10
5
30
+ 13-2
— 2-6
+ 5-3
+ 18-5
7
9
4
25
+ 11-0
— 4-0
+ 8-5
+ 14-5
— "b-5
8
8
4
20
+ 8-8
— 5-3
+ 1-8
+ 10-6
— 3-5
9
7
3
16
+ 7-0
— 7-0
+ 7-0
— 7-0
10
6
3
12
+ 5-3
— 8-8
—"l-8
-f 3-5
— 10-6
11
5
2
9
+ 4-0
— 11-0
— 3-5
+ 0-5
— 14-5
12
4
2
6
+ 2-6
— 13-2
_ 5-3
...
— 18-5
13
3
1
4
+ 1-8
— 15-8
- 7-1
— 22-9
14
2
1
2
+ 0-9
— 18-5
— 8-9
...
— 27-4
15
1
0
1
+ 0-4
— 21-6
— 10-6
...
— 32-2
16
0
0
0
...
— 24-6
— 12-4
...
— 37-0
CHAP. V.] AND WEBS OF ISOSCELES BRACING.
121
The maximum strains in the flanges occur when the passing load covers the whole
girder. They are most conveniently obtained by the aid of a diagram, as described in
1 98, and are given in the following table, the letters in the upper rows of which refer
to the bays in Fig. 62. The figures in the lower row represent the strains in tons.
Bays,
A
B
C
D
E
F
G
H
Strains in tons,
+ 26-3
+ 78-8
+ 123-8
+ 161-3
+ 191-3
+ 213-8
+ 228-8
+ 236-3
Bays,
1
J
K
L
M
N
O
P
Strains in tons,
— 30
— 82-5
— 127-5
— 165
— 195
— 217-5
— 232-5
— 240
The compressive strain in each of the end pillars is equal to the vertical component
(shearing-strain) of the end tension diagonal plus the load resting on the last
half -bay ; it reaches its maximum when the girder is loaded all over, in which case
it equals 26*25 + 375 = 30 tons on each pillar.
122
GIRDERS WITH PARALLEL FLANGES [CHAP. VI.
CHAPTER VI.
GIRDERS WITH PARALLEL FLANGES CONNECTED BY VERTICAL
AND DIAGONAL BRACING.
184. Introductory. — In the preceding chapter our attention
was confined to that form of braced web which consists of isosceles
triangles. There is, however, another class of bracing in common
use which consists of right-angled triangles, the braces being alter-
nately vertical and oblique. Besides its employment in the webs
of girders, this species of bracing is extensively used in scaffolding
and for stiffening the platforms of suspension bridges, but more
especially for horizontal cross-bracing between the flanges of large
girder bridges, so as to strengthen them against side pressure,
whether arising from the wind or other sources. The ordinary
form of plate girder is, as will be shown hereafter, a modification
of this form of bracing. Since the verticals may act as struts, and
the diagonals as ties, or vice versa, each of the following cases might
be subdivided ; this, however, is unnecessary, as in each case it will
be evident on inspection whether any given brace is designed to
act as a strut or a tie.
CASE I. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED AT
AN INTERMEDIATE POINT.
Fig. 63.
CHAP. VI.] AND VERTICAL AND DIAGONAL BRACING. 123
185. Let W = the weight, dividing the girder into segments
containing respectively ra and n bays,
I = m + n = the number of bays in the span,
0 = the angle between the diagonal and vertical
braces,
2 = the strain in a diagonal brace,
2' = the strain in a vertical brace.
On the principle of the lever, -j W is transmitted to the right
abutment through the bracing of the right segment (1O). Hence,
the strain in each vertical of the right segment,
2' = ™W (124)
Similarly in the left segment,
2' = **W (125)
These strains in the verticals are identical with the shearing-
strain of 34. The strains in the diagonals are the same as in Case
III. of the preceding chapter, that is, they equal the foregoing strains
in the verticals multiplied by secO (see eqs. 115 and 116). The
strains in the flanges may be found by the aid of a rough diagram of
coefficients in the diagonals (153), or more simply, by adding the
m
successive increments at the apices, each of which is equal to \NtanO
77
or -j WtanO, according as the apex lies to the right or left of W.
186. Single moving- load. — If the load move, the girder
must be counterbraced (138); this may be effected either by
counterbracing the existing braces, or by adding a second series of
diagonals. In the latter case there will always be certain braces
not acting when the load is in any given position ; thus, when the
weight rests as represented in Fig. 63, and the verticals are in
compression, the dotted diagonals are free from strain.
187. Trussed beam. — The trussed beam of the gantry or
travelling crane, Fig. 64, is a familiar example of vertical and
diagonal bracing. It is, however, seldom counterbraced by the
124
GIRDERS WITH PARALLEL FLANGES [CHAP. VI.
Fig. 64.
dotted diagonals ; hence, when the weight rests on a, the tension rod
cde tends to straighten itself and thrust b upwards. This is counter-
acted by the stiffness of the horizontal beam, abe, which is generally
formed of a whole balk of timber. Fig. 64 when counterbraced is
a simple form of girder for small bridges.*
CASE II. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
UNIFORMLY.
Fig. 65.
188. By reasoning similar to that used in Case IV. of the preced-
ing chapter, it may be shown that each brace sustains a strain which
is due to all the weights between it and the centre of the girder.
Let W = the weight resting on each apex,
n = the number of weights between any given brace and
the centre of the girder,
0 = the angle between the diagonal and vertical braces,
2 = the strain in a diagonal,
2' = the strain in a vertical.
* The railway bridge over the Wye, near Chepstow, erected by the late Mr. Brunei,
is an example of this truss on a gigantic style. (See Clark on the Tubular Bridges,
p. 101). The road, however, is attached to the lower flange, but in small bridges it is
usual to place the truss upwards, like Fig. 64 inverted, for this arrangement leaves
greater headway beneath, and as the truss forms part of the hand-rail, it answers a
double purpose.
CHAP. VI.] AND VERTICAL AND DIAGONAL BRACING.
125
The strain in each vertical equals the shearing-strain of 46, that is,
S' = nW (126)
The strain in each diagonal,
2 = nWsecO (127)
The increment of strain at each apex = nWtanO where n = the
number of weights between the diagonal which intersects that apex
and the centre ; the successive additions of these increments will
give the resultant strains in the several bays.
CASE III.— GIRDERS SUPPORTED AT BOTH ENDS AND TRAVERSED
BY A TRAIN OF UNIFORM DENSITY.
Fig. 66.
Fig. 67.
189. Web. — When the load traverses the upper flange, each
vertical, if acting as a strut (Fig. 66), sustains the maximum
strain when the passing load rests on its own apex and on those
between it and the farther abutment: if acting as a tie (Fig. 67),
when its own apex is free from load and those between it and the
farther abutment are loaded.
When the load traverses the lower flange, each vertical, if acting
as a strut (Fig. 66), sustains the maximum strain when its own apex
is free from load and those between it arid the farther abutment
are loaded ; if acting as a tie (Fig. 67), when its own apex and those
between it and the farther abutment are loaded.
126
GIRDERS WITH PARALLEL FLANGES [CHAP. VI.
The maximum strain in any diagonal, if in tension (Fig. 66),
occurs when the load rests on each apex between it and the abut-
ment from which it slopes upwards; if in compression (Fig. 67),
when the load rests on each apex between it and the abutment from
which it slopes downwards (l?O).
Let W = the passing weight on each apex,
n — the number of weights resting on the girder in the
foregoing cases of maximum strain,
I = the number of bays in the span,
0 — the angle between the diagonal and vertical braces,
S := the maximum strain in a diagonal,
2' = the maximum strain in a vertical.
The maximum strain in any vertical is represented by the follow-
ing arithmetical series : —
_
n) W
~T
Similarly, the maximum strain in any diagonal,
n(l+n) W
— 2 -- • —j-secv
(128)
(129)
The absolute maximum strains in girders subject to both fixed
and passing loads are found by tabulating the strains produced by
each class of load separately, and then adding or subtracting them
according as they are of the same or of opposite kinds
CASE IV. — LATTICE GIRDERS SUPPORTED AT BOTH ENDS AND
TRAVERSED BY A TRAIN OF UNIFORM DENSITY.
Fig. 68.
CHAP. VI.] AND VERTICAL AND DIAGONAL BRACING. 127
190. Web. — In this form of latticing the verticals are generally
constructed so as to act as struts and the diagonals as ties, in
which case the dotted diagonals are theoretically unnecessary.
Let W — the passing weight on each apex,
I = the number of bays in the span (= 10 in Fig. 68),
k = the number of systems of right-angled triangles, i.e.,
the number of bays in the base of one of the primary
right-angled triangles (= 2 in Fig. 68),
T = the maximum tensile strain which any given diagonal
sustains from the passing load,
n = the number of bays between the foot of the given
diagonal and that abutment from which it slopes
upwards,
p = the. integral number of times that its own (right-angled)
system occurs between the foot of the diagonal and
the same abutment, = the integral part of 7,
0 = the angle between the diagonal and vertical braces.
It may be shown by reasoning similar to that employed in 183,
that the maximum tensile strain in any diagonal,
T = (n - Pjfj. (p + 1)^' sect) (130)
The maximum compression in any vertical equals the maximum
tension in one of the conterminous diagonals divided by secO. If
the load traverses the upper flange, take the diagonal intersecting
at bottom on the side remote from the centre. If the load traverse
the lower flange, take the diagonal intersecting it at top on the side
next the centre.
191. End pillars — Ambiguity respecting: strains in faulty
designs. — In this form of latticing the end pillars are subject to a
severer transverse strain than in the isosceles latticing described in
the preceding chapter (18O). In the present case the end pillars
must be made sufficiently strong to sustain the horizontal com-
ponents of all the diagonals which intersect them between the flanges.
This inconvenience may be remedied by introducing short diagonal
struts, such as a, a, Fig. 68, which will relieve the end pillars of a
128 GIRDERS WITH PARALLEL FLANGES [CHAP. VI.
certain, though indefinite, amount of transverse strain, and at the
same time diminish the compression in the bay c and the vertical d.
Both diagonals and verticals are occasionally constructed so as to
act indifferently either as struts or ties ; in such designs calculation
is at fault, for the strains may pass through the isosceles system
of triangles alone, or through the vertical and diagonal system
alone, or partly through one and partly through the other. In
such designs there will generally be found a certain waste of
material.
CHAP. VII.] BRACED GIRDERS, ETC. 129
CHAPTER VII.
BRACED GIRDERS WITH OBLIQUE OR CURVED FLANGES.
193. Introductory — Calculation by diagram. — The class of
braced girders to which our attention has been directed in the two
preceding chapters is characterized by the parallelism of the flanges.
We have seen that the strains in each part vary according to the
position of the load, and that they may be calculated by simple
formulae with a degree of accuracy which leaves nothing further
to be desired. I now propose investigating braced girders, one or
both of whose flanges are oblique or curved. The A truss and the
bowstring girder may be taken as the chief representatives of this
class, which also includes cranes of various kinds, crescent girders
and the braced arch. Formulae for strains are unsuited to this
species of bracing on account of the various inclinations of the
several parts of the structure. Instead, we have recourse to
carefully constructed diagrams in which strains are represented
to scale, by the aid of which, however, a degree of accuracy is
attainable which is practically nearly as perfect as that obtained
by the application of formulae to the girders described in previous
chapters.*
CASE I. — BENT SEMI-GIRDERS LOADED AT THE EXTREMITY.
193. Derrick crane. — The derrick crane, Fig. 69, consists of a
revolving post P, a jib J, a chain or tie-bar T, and two back-stays,
one of which is shown at B, the other, lying in a plane at right angles
to that of the figure, is not represented, being hidden by the post.
The derrick crane is generally made of wood. It is simple in con-
* Curved flanges are assumed to be polygonal, i.e , formed of straight lines joining
the apices (144).
K
130 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
struction and easily erected. Hence, it is well adapted for temporary
•works, as also for quarries or other situations where the back-stays'
Fig. 69 do not interfere with
the traffic. At the
peak A, three forces
meet, viz., the down-
ward pull of W, the
tension of the tie-bar
T, and the oblique
thrust of the jib J.
Since these three
forces are in equili-
brium, their relative
amounts may be represented by the sides of the triangle PTJ (9).
Hence, the tension of the tie-bar = =W, and the compression of
If the chain pass along T, and so over a pulley at b down to the
chain barrel bolted to the foot of the post, it relieves the tie-bar
of an amount of tension equal to that in the chain, namely, W
divided by the number of falls in the hanging part of the chain.*
If, however, the chain pass along the jib, the compression of the
latter is increased by an amount equal to the tension of the chain.
The tension in T being known, the strains in the post and back-
stays, which are its components, are easily found. This operation
is most conveniently performed by the aid of a skeleton diagram
(Fig. 69) drawn accurately to, scale. Let the jib and one back-stay
lie in the same plane. Lay off be by scale to represent the tension
in Tf = pWj, and draw cd parallel to B; then cd, measured by
the same scale, will represent the tension in the back-stay, and bd
the compression of the post. In this case the second back-stay is
* This is not accurately true, for the friction of the blocks, pulleys, &c., increases or
diminishes the tension of the chain, according as the weight happens to be raised or
lowered.
CHAP. VII.] OR CURVED FLANGES. 131
free from strain, but when the jib does not lie in the same plane with
either back-stay, both back-stays are subject to strain; to a less
degree, however, than in the case already considered, as will appear
from the following considerations. Let Fig. 70 represent a plan of
Fig. 70. the crane, bh and bk being the
horizontal projections of the
back-stays, arid &A that of
the tie-bar and jib; let be
represent the horizontal com-
ponent of the tension in the
tie-bar (equal ce in Fig. 69),
then bf and by will represent
the horizontal components of
the strains in the back-stays, and hence, the strains in the back-
stays can be found. It is obvious, however, that either bf or bg
will attain its maximum when the tie-bar lies in the same plane
with one of the back-stays. Hence, the former case, in which the
jib and one back-stay lie in the same plane, is sufficient for us to
consider when calculating the requisite strength of the stays.
The strain in the post attains its greatest value when the plane
of the tie-bars and jib bisects the angle between the back-stays, for
then the sum of 6/and bg is maximum, and consequently, the sum
of the vertical components of the strains in the stays is maximum
also. But the strain transmitted through the post is equal to the
sum of these vertical components -j- or — the vertical component of
the tension in the tie-bar, according as the latter slopes downwards
or upwards from the head of the post. The back-stays act some-
times as struts, sometimes as ties, and when the jib is swung round,
so as to lie alongside one of the back-stays, the latter will sustain
its maximum compression, equal to the maximum tension produced
when the jib and stay lie in the same plane. The radius of the
circle described by the jib, or the range of the derrick, is generally
capable of adjustment by lengthening or shortening the tie-bar, which
is then a chain attached to a small auxiliary crab-winch fastened to
the post near the wrorking barrel, in which case the working chain
passes along the jib. This form of derrick is convenient for setting
132 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
masonry, as its range is equal to a circle described by the jib when
nearly horizontal, in which position moreover the crane is most
severely strained.
194. W harf crane. — The wharf crane, unlike the derrick crane,
has no back-stays. Consequently, the post is subject to transverse
Fig. 71. strain from the oblique pull
of the tie-bar; it is in fact
a semi-girder fixed in the
ground and loaded at the
extremity. The strains in
the tie-bar and jib are cal-
culated in the same way as
for the derrick crane. The
bending moment (59) of
the post attains its greatest
value at its intersection with tiie ground, and equals the horizontal
component of the tension in T multiplied by the height of the post
above ground. It may, however, be more conveniently found as
follows : —
The whole crane above ac (the ground line,) is a bent semi-girder
held in equilibrium by the weight and the elastic forces at a (in this
case vertical). Taking moments round either the centre of tension
or the centre of compression at a (58), we have the bending moment
= W?-, where r = the radius of the circle described by the jib. From
this it follows that the transverse strain at a is not affected by increas-
ing the height of the post, which, however, diminishes the strains in
the jib and the tie-bar, and is so far attended with advantage ; neither
is it affected by raising or lowering the peak of the jib in the same
vertical line. It also follows that the transverse strain on the post
is increased when the weight is farther out than the circle described
by the jib, for the leverage of W is then increased and attains its
greatest value when the chain is at right angles to the jib. If the
post be fixed in the ground, the frame, to which the jib, tie-bar and
wheehvork are attached, is generally suspended by a cross head from
the top of the post which forms a pivot round which the cross-head
turns. In this form of crane the weight is transmitted from the pivot
CHAP. VII.] OR CURVED FLANGES. 133
through the whole length of the post in addition to the longitudinal
strains to which as a semi-girder it is liable, and the section of the
post should theoretically be circular (99), since it may be equally
strained in all directions.* When the post revolves on its axis, the
jib and wheelwork are bolted to it and all move together on a pivot
at the toe-plate b. In this case the post should be double-flanged.
The underground portion is subject to a vertical compression equal
to the weight (viz., the difference of the vertical components of the
strains in the jib and tie-bar,) in addition to the longitudinal strain
derived from its acting as a semi-girder. When the post moves round
its axis, friction rollers may be advantageously placed between the
post and a curb plate which is secured to the masonry at a.
To find the amount and direction of the pressure at the toe, join
b with a point c vertically beneath W. The whole structure is
balanced by three forces, viz., the weight W> tne horizontal pressure
against the curb plate at a, and the pressure on the toe at b. The
two former forces pass through c ; consequently, the latter intersects
them at the same point (9). Hence, the sides of the triangle abc
represent the relative amounts of these forces, and we have the
horizontal component of the oblique pressure at b equal QW. The
vertical components equals W, which is otherwise evident.
195. Bent crane. — This form of semi-girder has been adopted
for wharf cranes where head-room is required close to the
post. The flanges may be equi-distant, as in Fig. 72, though a
more pleasing form is produced by bringing them closer together
as they approach the peak.f
The weight W is supported by diagonal 1 and the first bay in the
lower flange E, producing tension in the former and compression
* Square tubular posts built of boiler plates with angle iron at the corners form
very simple and efficient posts for small cranes not exceeding four or five tons.
t Tubular cranes of this form were first made with plate webs by Sir Wm.
Fairbairn (Proc. Inst. M. E., Part I., 1857), and the braced web was first adopted by
William Anderson, Esq., in a six-ton crane erected for the Government at the Pigeon
House Fort, near Dublin. Mr. Anderson also designed a very fine twenty -ton bent
crane, with plate webs, for the Russian Government, 60 feet high, and 31 '"6" radius.
(Trans. In&t. C. E. of Ireland, Vols. vi. and vii.)
134
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
in the latter. The tension of diagonal 1 is resolved at d into its
components in the direction of A and diagonal 2. The resultant
of the strains in diagonal 2 and E, found by a triangle of force, is
resolved at g into its components in the directions of the third
diagonal and F. In a similar manner the resultant of the strains
in diagonal 3 and A is resolved into its components in diagonal 4
and B, and so on throughout the girder.
Fig. 72.
An example (see Fig. 72) will illustrate this fully, 'and the student
is recommended to work it out for himself by the aid of a diagram
drawn accurately to a scale of not less than five feet to one inch.
The strains may be represented to a scale of ten tons to one inch,
though in many cases a larger scale will be found preferable. * The
flanges are equi-distant, forming quadrants of two circles whose radii
are respectively 20 and 24 feet. The inner flange is divided into four
equal bays, on which stand equal isosceles triangles, and a weight of
10 tons is suspended from the peak. Draw ab vertically and equal
to 10 tons measured on the scale representing strains, and draw be
parallel to E so as to meet the diagonal 1 produced ; be and ac
represent the strains in E and diagonal 1, and measure on the scale
* Rolling parallel rules, 15 or 18 inches in length, will be found useful for laying off
parallel lines of strain.
CHAP. VII.]
OR CURVED FLANGES.
135
of strains + 10*8 tons and — 13-1 tons respectively. Next, take
de equal 13!1 tons (= ac), and draw ef parallel to diagonal 2, so as to
meet A produced ; ef and df represent the strains in diagonal 2 and
A, and measure + 18'8 tons and — 21*7 tons respectively. Next,
produce diagonal 2 so that gh may equal 18'8 tons (= ef), and
draw hi parallel to E and equal 1O8 tons (= be) ; ig is the resultant
of the strains in diagonal 2 and E, and is transmitted through F
and diagonal 3. Draw ik parallel to F ; ik and kg will represent
the strains in F and diagonal 3, and measure + 3O5 tons and — 5 '4
tons respectively. Proceeding in this manner, we obtain the strains
given in the following table : —
BRACING, .
1
2
3
4
5
6
7
8
Strains in tons, .
—13-1
+ 18-8
—5-4
+21-4
+ 3-2
+ 20-5
+11-2
+8-8
FLANGES, .
A
B
C
D
E
F
G
H
Strains in tons, .
—217
—397
—51-3
—49-0
+10-8
+30-5
+ 45-3
+52-8
196. Calculation by moments. — It is prudent to check the
calculation by diagram by computing the strains in some of the
bays by the method of moments. That portion of the crane which
extends above B/, for instance, is held in equilibrium by the tension
in B, the weight W, and the forces which meet at /. Taking
moments round the latter point, we obtain the strain in B. In this
example, B/ measures 3*55 feet, and the horizontal distance of I
from W measures 14*12 feet; hence, we have
3-55 X strain in B = 14'12 X 10 tons;
whence, the strain in B = 39*8 tons, which agrees closely with the
former result. When only one system of triangulation is adopted,
the strains in the flanges may be obtained in this manner by
moments, and those in the diagonals may afterwards be obtained
by decomposing the strains in the flanges. This method is perhaps
more simple in practice than that first described, and has a farther
advantage that errors do not accumulate.
136
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
197. Lattice webs not suited for powerful bent cranes. —
The chief merit claimed for the bent crane is the large amount of
head -room it allows underneath the jib, which enable boilers or other
bulky articles to be brought close up to the peak. This merit,
however, is balanced, and in many cases more than balanced, by the
greater simplicity of the ordinary wharf crane. The lattice web is
not well suited for bent cranes exceeding 10 tons, as the diagonal
bars become so wide, and leave so little open space, that plating
may be advantageously substituted for bracing.
CASE II. — THE BRACED SEMI-ARCH.
. 73.
198. Swing1 bridge. — This form of semi-girder is a modification
of the previous case, in which the radius of the upper flange becomes
infinite; it is suitable for swing bridges, in which case the end
next the abutment is prolonged backwards with parallel flanges
and loaded at the inner extremity with a counterpoise weight to
balance the projecting part. This backward continuation resembles
the semi-girder described in Case I., Chap. V. In order to obtain
the maximum strains when a concentrated load or a passing train
traverses the girder, we must first calculate the strains produced
by the weight on each apex separately, and tabulating these, we
can find what position of the load, if it be concentrated, or what
weights, if there are several, will produce maximum strains in each
part of the structure, and the methods of calculation described in
the preceding case are applicable to this one also.
CHAP. VII.] OR CURVED FLANGES. 137
199. Single triang;nlation. — When, however, there is but one
system of triangles in the bracing, the following plan is more simple
in practice, and as errors do not accumulate, it is less liable to
inaccuracy. Suppose a weight resting on the extremity of the
girder; on examining the forces which hold any portion CaWj
in equilibrium, we find that two of them, viz., the weight and
the horizontal tension in C pass through Wj ; consequently, the
third force, viz., the resultant of the strains in bay G and diagonal
6 also passes through Wj (9). In the same way it can be shown
that the resultants at each of the other lower apices pass through
Wr If the weight rest on any other apex, W2 for example, the
resultant strains produced by it at each lower apex pass through
W2 ; or, to express this more generally, the resultant strain at each
apex in the lower flange from a weight at any apex in either flange
will pass through the intersection of the horizontal flange with a
vertical line drawn through the weight, provided there be but one
system oftriangulation. Again, since the horizontal flange transmits
no vertical strains, the weight must be conveyed to the wall through
these resultant strains at each lower apex. Their vertical com-
ponents are in fact the shearing-strain and equal to the weight;
hence, knowing both their directions and their vertical components,
we can find their amounts. Thus, the resultant strain at a from Wj
may be found as follows : — Draw a vertical line ab, equal (by a scale
of strains) to Wj, and draw be horizontally till it meet Wja produced ;
ac is the required resultant, and may be resolved into its components
in bay G and diagonal 6. The strain in the latter may next be
resolved at W4 in the directions of bay D and diagonal 7. The
former component is the increment of horizontal strain at the apex,
and when added to the sum of the preceding increments gives the
resultant strain in D. The strains in the other parts may be
obtained in a similar manner.
SOO. Example. — The following example, Fig. 73, in which the
strains have been worked out on a diagram drawn to a scale of 5
feet to one inch, will be found useful practice for the student. The
projecting portion of the girder is 40 feet long, and 10 feet deep at the
wall, with a circular lower flange which has a horizontal tangent two
138
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
feet below the extremity of the girder. Consequently, the versine of
the arch is 8 feet, and its radius 104 feet. The load is uniform and
equal to one ton per running foot, which for calculation is supposed
collected into weights of 10 tons at each upper apex except the
outer one, which has only 5 tons, or the load which rests on half a
bay. The strains have been calculated for each weight separately.
W,
W2
w,
W4
Uniform
Load.
Max.
Compn.
Max.
Tension.
1
Tons.
+ 127
Tons.
Tons.
Tons.
Tons.
+ 127
Tons.
+ 127
Tons.
2
- 8-0
...
...
...
- 8-0
...
- 8-0
LCING.
3
4
+ 5-9
- 0-3
+ 19-0
- 9-8
...
...
+ 24-9
-101
+ 24-9
— 101
5
5
+ 0-2
+ 7-3
+ 14-0
.„
+ 21-5
+ 21-5
...
6
+ -27
— 11
— 7-5
...
— 5-9
+ 27
— 8-6
7
— 2-3
+ 0-9
+ 6-3
+ 117
+ 16-6
+ 18-9
— 2-3
8
+ 3-1
+ 17
— 2-8
- 7-2
— 5-2
+ 4-8
— 10-0
A
— 117
•M
...
...
— 117
B
— 24-1
— 16-1
...
...
— 40-2
C
-247
— 297
— 9-9
...
— 64-3
FLANGES.
D
E
— 21-6
+ 19-3
— 30-8
— 18-5
— 6-2
— 771
+ 19-3
F
+ 25-2
+ 25-2
...
...
+ 50-4
G
+ 23-9
+ 31-8
+ 15-9
...
+ 71-6
H
+ 21-4
+ 321
+ 21-4
+ 107
+ 85-6
The reader will perceive that the strain produced in bay H by W4
is half that produced by W3, and one-third of that produced by
W2, and in general, the strains produced by the different weights
in any given bay will be sub-multiples of the strain produced by the
most remote weight, for they are proportional to the leverage of the
weights round the apex above or below the given bay. This check
CHAP. VII.] OR CURVED FLANGES. 139
on the accuracy of the work is, however, applicable only in the case
of a single system of triangulation. The strains in girders of this
form are not always such as might perhaps be expected at first
sight; Wj, for instance, produces compression in both diagonals 6
and 8, and in bay D a strain of less amount than in bay C. These
apparent anomalies occur when the resultant" at the lower apex, ac
for example, passes altogether above the lower flange.
SOI. Lattice semi-arch — Triangular semi-girder. — When
two or more systems of triangulation are introduced, the strains in
one system produce strains in the others in consequence of the
curvature of the arched flange, and this renders the calculations
more tedious than would otherwise occur. This remark applies
to all arched girders with lattice webs. In this particular case the
calculations would be much simpler if the girder were triangular
with a straight lower flange, since each bay would communicate its
strain directly to the adjoining bay without affecting the diagonals
at their junction, but this form of semi-girder has the disadvantage
of being somewhat unsightly in appearance, which in some cases
might prevent its adoption, whatever merits, and they are con-
siderable, it may possess in other respects.*
SOS. Inverted semi-arch. — When head-room beneath is re-
quired, we may invert the girder represented in Fig. 73, so that
it will resemble one-half of a suspension-bridge. By so doing we
change the strains in kind, but not in amount.
* A large iron swing bridge, a drawing of which appeared in the Illustrated London
Keios for October 12, 1861, has been constructed at Brest, in France ; it is formed of
two triangular semi-girders with vertical and diagonal bracing.
140 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
CASE III. — CRESCENT GIRDER.
Tig. 74.
SOS. Suitable for roofs — Flanges. — Frequent modifications
of the crescent girder occur in the roofs of our railway stations and
crystal palaces, to which its graceful outline and lightness of
appearance impart an air of elegance which no other form possesses
to the same degree. It may also be employed for bridges where
greater headway is required beneath the centre than at the
abutments. I shall, however, merely investigate the strains pro-
duced by a load symmetrically disposed on both sides of the
centre, such as a roof principal generally sustains. When the
girder is subject to a partial or a passing load, the more general
method of calculating the strains due to each weight separately,
and which is investigated in the next case, becomes necessary.
The horizontal strains at the centre of the flanges are equal and of
opposite kinds ; their amount depends upon the central depth of the
girder and may be found by the method of moments as follows: —
Let W = the load symmetrically distributed,
I = the span,
d = the central depth from flange to flange = Z>H,
I' = the distance of the centre of gravity of each half load
measured from the centre of the girder,
T = the tension at the centre of the lower flange,
C = the compression at the centre of the upper flange.
CHAP. VII.] OR CURVED FLANGES. 141
The half girder, a&H, is held in equilibrium by the reaction of the
CW\
= -Q- ) , by the left half load (which we may conceive
collected at its centre of gravity), and by the horizontal strains of
compression and tension at b and H. Taking moments round each
of these latter points successively, we have — f - — /' J = Td = Cc/;
whence,
T = C = W<*-2*'> (131)
This, which is merely a particular form of eq. 25, proves that the
strains at the centre do not depend upon the height of the lower
flange above the chord line, but upon the depth of the girder from
flange to flange. The method of calculating the strains in other
parts of the girder consists in working by the resolution of forces
from either abutment, whose reaction is a known quantity,
towards the centre. The following examples, which have been
worked out on a diagram drawn to a scale of 5 feet to one inch,
and with strains represented by 4 tons to one inch, will explain this
clearly.
SO4. Example 1. — The span of the girder, Fig. 74, is 80 feet ; the
versines of the flanges respectively 1.0 and 16 feet; both flanges are
circular and each flange is divided into equal bays, with the excep-
tion of the extreme bays of the lower flange, which are each half as
long again as the other bays. The load is supposed equal to 8 tons
distributed, so that each apex sustains a weight of one ton ; hence,
the reaction of each abutment equals 4 tons, of which, however,
half a ton is at once balanced by the weight of the first half
bay of the roof which rests directly on the wall-plate. Conse-
quently, the resultant of the forces in A and E = 3'5 tons pressing
downwards on the wall. Draw ac = 3*5 tons, and draw cd parallel
to E until it meets A produced. The lines ad and cd represent
the strains in A and E, and measure by scale + 12*25 tons and
— 10'43 tons respectively. Next, lay off ef = ad, and draw fg
vertically equal to one ton, that is, equal to the weight at the first
apex. The line eg is the resultant of the strain in A and the weight
142
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
at e, and the strains in B and diagonal 1 are its components, and
can therefore be found by resolving eg in their directions. Similarly,
the resultant of E and diagonal 1 may be resolved in the directions
of F and diagonal 2. At h we must find the resultant of three
forces, viz., the strain in B, the strain in diagonal 2, and the
weight resting on the apex. From this resultant the strains in C
and diagonal 3 are derived, and so on to the centre. The follow-
ing table contains these strains : —
BRACING, .
1
2
3
4
5
6
Strains in tons, .
-2-4
—1-05
-1-36
-0-91
-1-04
—1-0
FLANGES, .
A
B
C
D
E
F
G
H
Strains in tons, .
+12-3
+13-5
+13-1
+12-9
-10-4
-117
-12-2
—12-2
The accuracy of the work may be checked by comparing the
strain in H with the central strain in the flanges obtained by the
method of moments. As the distance of the centre of gravity of
the half load from the centre of the girder is unknown, the most
convenient method for obtaining the leverage of the weights is by
accurately measuring on the diagram the distance of each weight
from the centre. Doing this, and taking moments round the centre
of either flange, we have
6-15 F = 40 X 3-5 tons— (31-4 + 21-6 + 1M)
whence, the strain at the centre of either flange,
F = 12-34 tons,
in place of 12*2 tons, an amount of discrepancy which is im-
material. The central depth by which F is multiplied has been
obtained by measurement, and is, it will be observed, slightly in
excess of 6 feet, arising from the central bay of the lower flange
being a straight line, and therefore slightly farther from the upper
flange than the arc of which it is the chord.
305. Example 2. — Flange-strains nearly uniform with
symmetric loading:. — The girder, represented in Fig. 75, has the
CHAP. VII.J
OR CURVED FLANGES.
143
same span, depth and versine as the preceding example, but the
mode of bracing is similar to that described in Chapter VI. Each
flange is divided into eight equal bays and every alternate brace is
nearly radial to the lower flange.
Fig. 75.
The strains due to a load of one ton at each apex of the upper
flange are as follows: —
BBACING, .
1
2
3
4
5
6
7
Strains in tons, .
—1-75
+0-6
—1-65
+0-45
—17
+0.2
—1-4
FLANGES, .
A
B
C
D
E
F
G
H
Strains in tons, . . | +18*7
+12-7
+12-6
+12-6
—11-8
—12-3
—12-6
—127
The horizontal strain at the centre of either flange equals 12*68
tons. Checking this as before by the method of moments, we
have
6 F = 40 X 3-5 tons — (31-4 + 21 -6 + 11-1)
whence, the strain at the centre of either flange,
F - 12-65 tons.
In the previous examples it will be observed that the strains are
nearly uniform throughout the flanges, and that the bracing has
comparatively little work to do. Hence, the crescent girder seems
well fitted for large roofs, the loading of which, with the exception
of wind pressure, is generally symmetrically distributed.
SOG. Ambiguity in the strains of a crescent girder when
resting on more than two points. — This class of girder is
144 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
occasionally constructed with equi-distant flanges, in which case it
is essential for accurate calculation that the girder rest on two
points only, either the extremities of the inner, or the extremities
of the outer, flange ; otherwise we cannot say how much pressure
any one point sustains, just as the pressure on any one leg of a
four-legged table is indefinite. The girder in fact becomes an
arched rib and partakes of the uncertainty of the arch as regards
the direction of the line of thrust.
CASE IV BOWSTRING GIRDKIt.
Fig. 76.
SO7. Concentrated load. — Let a single weight W3 rest upon
one of the apices which divides the girder into segments con-
taining respectively m and n segments. On the principle of the
lever, the pressure on the right abutment = - — W3, and that
m + n
on the left = - —Wo. This latter quantity is the resultant of
in + n J
the strains in bays A and F, which can therefore be obtained from
it by a diagram of strains. Again, the strains in B and diagonal
1 may be derived from that in A, and by resolving the strain in
diagonal 1 in the directions of diagonal 2 and bay G, we obtain
the strain in the former and the horizontal increment of strain
developed at the first apex of the lower flange. This increment,
added to the strain in F, gives the total strain in G. The resultant
of the strains in B and diagonal 2 is also the resultant of those in
C and diagonal 3, which can therefore be derived from it, and so on.
SOS. Passing load — Example — Little coiinterbracing re-
quired in bowstring: girders of large size. — When the load is a
concentrated passing load or a train, we must tabulate the strains
CHAP. VII.]
OR CURVED FLANGES.
145
produced by the weight on each apex separately, and thence deduce
what position of the load produces maximum strains. It will be
found that the maximum strains in the flanges occur when the train
covers the whole girder, and that they are of nearly uniform mag-
nitude throughout each flange, while the maximum strains in the
diagonals increase as they approach the centre, just the reverse of
what occurs in the webs of girders with horizontal flanges. The
following example, Fig. 76, will illustrate fully the mode of calcu-
lating the strains in this important form of girder. They have
been worked out on a diagram drawn to a scale of 5 feet to one inch.
The span is 80 feet, divided into 8 equal bays, and the bow is a
circular arc whose versine equals 10 feet, but, as there is no apex at
the crown, the central depth of the inscribed polygon, measured by
scale, equals 9*85 feet in place of 10 feet. The load is supposed to
traverse the lower flange and to be of uniform density, equal to one
ton per running foot, which is equivalent to 10 tons at each apex.
W,
W2
W3
w,
ws
W6
w,
Uniform
Load.
Max.
Compn
Max.
Tensn-
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
1
—0-39
— 0-8
— 1-2
- 1-6
— 2-0
— 2-3
— 27
—11-0
...
—11-0
2
+0-23
+ 0-5
+ 07
+ 0-9
+ 11
+ 1-4
—11-4
— 6-6
+ 4-8
—11-4
i
3
-0-5«
- 11
— 1-7
- 2-2
— 2-8
— 3-4
+ 4-8
— 7-0
+4-8
—11-8
i
4
5
+0-51
—0-90
+ i-o
— 1-8
+ 1-5
— 2-7
+ 2-0
— 3-6
+ 2-6
— 4-5
— 8-6
+ 47
— 4-3
+ 2-4
— 5-3
— 6-4
+7-6
+71
—12-9
—13-5
6
+0-88
+ 1-8
+ 2-6
+ 3-5
— 6-9
— 4-6
— 2-3
— 5-0
+8-8
—13-8
7
—1-40
— 2-S
- 4-2
— 5-6
+ 4-2
+ 2-8
+ 1-4
— 5-6
+8-4
—14-0
A
+2-82
+ 5-6
+ 8-5
+11-3
+141
+16-9
+197
+78-9
B
+3-08
+ 6-2
+ 9-2
+12-3
+15-4
+18-5
+21-6
+86-3
C
+3-47
+ 6-9
-flO'4
+13-9
+17-3
+20-8
+10-4
+83-2
8
D
+4-11
+ 8-2
+12-3
+16-4
+20-5
+137
+ 6-8
+82-0
\
E
+5-11
+10-2
+15-3
+20-4
+15-3
+10-2
+ 51
+81-6
&
F
—2-52
— 5-0
— 7-6
—101
—12-6
—151
—17-6
—70-5
G
—3-01
— 6-0
— 9-0
—12-0
—15-0
—181
—131
—76-2
H
-3-62
— 7-2
—10-9
—14-5
—181
-15-9
- 7-9
—781
1
—4-46
— 8-9
—13-4
-17-8
—171
—11-4
— 57
—78-8
146 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
On examining the foregoing table we observe that, when the
permanent (uniform) load is equal to, or less than, the passing load,
a large number of the diagonals require counterbracing ; in this
example, for instance, diagonals 4, 5, 6, 7, and their counterparts at
the other side of the centre, require counterbracing. If, however,
the permanent load be much greater than the passing load, it may
happen that the diagonals will always be in tension and thus relieve
the engineer of one difficulty in large girders, namely, that of
providing against flexure in long struts. Hence, the bowstring
girder seems well suited for large spans. On examining the table
we also find that all the intermediate strains are multiples of those
in the columns under either Wt or W7. They agree also in sign
with their sub-multiples. This arises from the reaction of each
abutment being directly proportional to the length of the remote
segment, and indicates a speedy method of filling up the table, viz.,
by calculating on a diagram the strains produced by the two
extreme weights and thence deriving those due to all the inter-
mediate weights.
8O9. Calculation by moments. — When there is only one
system of triangulation, the work may be checked by calculating
the strains in some of the bays by the method of moments. Thus,
in the central bay E, the strain
35x40—10x60
F = - — pr^ — - = ol*2 tons compression,
y°oD
a close approximation to the amount in the table, as the discrepancy
is only 0'4 tons, or ^ J^rd of the whole. Having found the strains
in the flanges by the method of moments, the strains in any pair of
intersecting diagonals may be found by decomposing the strains in
the two adjoining bays.
210. Uniformly distributed load . little bracing: required-
Absolute maximum strains. — If a uniform horizontal load be
suspended by vertical rods from a circular bow, the diagonal bracing
will scarcely come into action, and the tension throughout the string
will be very nearly uniform, for a small arc of a circle differs but
slightly from the parabola which a chain (inverted arch) assumes
when loaded uniformly per horizontal foot (49). In this case the
CHAP. VII.] OR CURVED FLANGES. 147
horizontal component of strain is nearly uniform throughout the
bow and equals the compression at the crown, or the tension in the
string. The vertical component at the springing is equal to the
half load, and at any other point it equals the half load supported
above the level of that point. The longitudinal compression at any
point in the bow is the resultant of these horizontal and vertical
components, and would be strictly tangential to the curve if it were
a parabola, i.e., the curve of equal horizontal thrust for a uniform
horizontal load. The bow forms a considerable item of the total
weight of a bridge of large span, and the annexed method of
calculating the strains will be found more accurate than one which
supposes the whole permanent load resting on the lower flange : —
1°. Calculate the maximum strains in both flanges and bracing
produced by the passing load of greatest uniform density,
as already explained.
2°. Calculate the strains produced by the permanent load which
rests on the lower flange, including in this the string, road-
way and bracing. These may be obtained by proportion
from the strains produced by the passing load when the
latter covers the whole bridge.
3°. Calculate the (nearly) uniform strain produced throughout
the bow and string by the weight of the former (eq. 25).
If greater accuracy is required the longitudinal strains in
the bow may be obtained by the method explained in 36.
Having these arranged in a tabular form, we can easily find the
absolute maximum strains which each part sustains. The 2nd and
3rd of the foregoing calculations may be replaced by the method
described in the preceding case for calculating the strains due to a
permanent load, without however simplifying the operation in
practice.
811. Single t riaiis ulal ion . second method of calculation. —
When the bracing of a bowstring girder consists of a single system
of triangulation, as in Fig. 76, the strains may be calculated by a
method similar to that described in 199. Suppose, for example,
that W3 alone rests upon the girder, dividing the lower flange into
segments containing respectively m and n bays ; the segment abc is
148
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
held in equilibrium by three forces, viz., the reaction of the right
abutment, the horizontal tension at c, and the resultant of the strains
in K and diagonal 10. The two former meet at a ; consequently,
the third, the resultant at b, passes through the same point (9).
Again, since the lower flange is horizontal, it cannot convey a vertical
m
pressure to the abutment ; hence, — — — W3 ( = the reaction of the
abutment,) must be conveyed through the bow and diagonals to the
right abutment, forming the vertical component of the resultant at
each upper apex. This suggests the following method of calculating
the strains. Draw bd vertically equal to
m -f- n
W3, and draw de
horizontally till it meets ba produced ; be represents the resultant at
b, and hence we can find its component in K and diagonal 10, or
in L and diagonal 11. The same reasoning will apply if all the
apices to the left of W3 are loaded, in which case diagonals 10 and
11 will sustain the maximum strains of tension and compression
which a passing train can produce in them. At the several apices
in the bow over the unloaded segment resultant strains will be
developed, each of which will pass through a and have the same
vertical component, viz., the reaction of the right abutment, provided
there be but one system of triangles. In the case of the train, bd
5) = W, since there
o
will represent
are 5 loaded apices in the left segment and 8 bays in the span.
This operation must be repeated at each apex of the bow.
The maximum strains in the diagonals of the example in 3O8
are calculated by this method and are given in the annexed table.
They agree closely with those previously obtained : —
DIAGONALS.
Maximum
compresition.
Maximum
tension.
Tons.
Tons.
1
— 11 0
2
+ 4-7
— 11-4
3
+ 4-8
— 11-8
4
+ 7-6
— 12-8
fi
+ 7-1
— 13-6
6
-f 8-7
— 13-6
7
+ 8-4
— H-0
CHAP. VII.]
OR CURVED FLANGES.
149
318. Inverted bowstring:, or fish-bellied g-irder — Bow and
invert, or double-bow. — The methods of calculating the strains
of the bowstring girder are also applicable to its inverse — the fish-
bellied girder, i.e., the arc in tension with a horizontal flange in
compression, as well as the lenticular girder compounded of the
two, i.e.,, a bow and invert connected by bracing, such as the Royal
Albert Bridge, Saltash. Examples of these forms are, however,
comparatively rare, except in cast-iron girders and beams of steam
engines, but the fish-bellied girder is sometimes used for cross
road-girders.
CASE V. — THE BRACED ARCH.
Fig. 77.
313. Law or the lever applicable to the braced arch. —
Properly speaking, the braced arch is not a girder, since it
exerts an oblique thrust against the abutments (13), but it
resembles a girder in so many respects that the investigation of its
strains may fitly be considered in this chapter. In the braced
arch the upper flange is usually horizontal and supports the
roadway. Both flanges are in general subject to compression
throughout their whole length, and the lower one exerts an
oblique pressure against the abutments. In this respect the
braced arch resembles its prototype, the stone arch, while it also
resembles the girder in its capability of sustaining transverse
strain. The horizontal components of the pressures against the
abutments are equal and in opposite directions ; equal — since, if
the horizontal reaction of one abutment exceed that of the other,
150 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
the arch will move towards that side which exerts the weaker
thrust, a thing manifestly impossible. We may therefore conceive
a horizontal tie substituted for the horizontal reaction of the
abutments, and the arch will then follow the laws of girders,
exerting a vertical pressure only on the points of support. The
principle of the lever (1O) is, consequently, applicable to this form
of bracing, and hence we can find the direction and amount of
the thrust against either abutment for each position of the load.
Theoretically, the lower flange of the arch represented in Fig. 77
should not be continued across the crown of the arch, for if
it were, the strains in every part would be uncertain, since
the central bay of this flange would be subject to tensile
strains of indefinite amount, varying with the load and tem-
perature, and modifying therefore to an unknown extent the
horizontal reaction of the abutments. To illustrate this, let us
suppose for a moment that the reaction of the abutments is
replaced by a tie-bar ; we then have three unknown horizontal
forces, viz., compression in the top flange, tension in the lower
flange at the crown, and tension in the tie-bar ; also three known
vertical forces, viz., the weight and the vertical reaction of each
abutment. Now, it is evident that we cannot determine the three
unknown forces by the method of moments from these data, and
we must therefore get rid of the difficulty by supposing the lower
flange discontinued at the crown, which, indeed, is not far from the
truth in practice, for the two flanges generally merge into one,
and the less in depth is the line of junction of the two semi-
arches, i.e., the depth of the arch at the crown, the nearer will the
following theory and practice agree.
Let us now consider the effect of a single weight W6. The left
semi-arch is subjected to two forces only, viz., the pressure of the
other semi-arch at the crown and the reaction of the left abutment
at a. Since equilibrium exists, these forces are equal and opposite ;
consequently, the reaction of the left abutment acts in the direction
aW4. Again, the whole arch is balanced by the weight W6 and
the reactions of the abutments. The weight and the reaction of
the left abutment intersect at b ; consequently, that of the right
CHAP. VII.]
OR CURVED FLANGES.
151
abutment passes through the same point (9). Resolving W6 in the
directions ba and be, we obtain these reactions, and once they are
known, we can work from the abutments towards the weight by
the resolution of forces and thus find the strains produced by W6
throughout the arch. Performing similar operations for each weight,
and tabulating the results, we can obtain the maximum strains of
each kind produced in every part of the structure. Those produced
in the arch represented in Fig. 77, by weights of 10 tons at each
apex, are given in the following table. The arch is 80 feet in span
with a rise or versine of 8 feet, and the depth measured from the
springing to the upper flange is 10 feet. The upper flange is
divided into 8 equal bays, and the bracing consists of a series of
isosceles triangles of which these bays form the bases.
W,
W2
W3
W4
W5
W6
W7
Uniform
Load.
Max.
Compn-
Max.
Tens"-
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
Tons.
i
+3-2
+ 6-4
+ 9-6
+127
— 9-6
— 6-4
— 3-2
+127
+ 31-9
—19-2
2
—2'0
- 4-0
— 6-0
— 8-0
+ 6-0
+ 4-0
+ 2-0
— 8-0
+ 12-0
—20-0
3
+1-5
+ 3-0
+ 4-5
+ 5-9
+14-5
— 3-0
— 1-5
+24-9
+29-4
— 4-5
BRACING.
4
5
6
— 0-07
+ 0-05
+07
- o-i
+ o-i
+ 1-4
— 0-2
+ 015
+ 21
— 0-3
+ 0-2
+ 27
— 97
+ 7-1
— 3-2
+ o-i
+ 13-9
- 8-9
+ 0-07
— 0-05
— 07
—10-2
+21-4
- 5-9
+ 0-2
+21-5
+ 6-9
—10-4
— 01
—12-8
7
—0-6
— 1-2
— 1-8
— 2-3
+ 27
+ 7-5
+12-3
+16-6
+ 22-5
— 5-9
8
+0-8
+ 1-6
+ 2-4
+ 3-1
— 0-6
- 4-4
— 8-0
- 5-1
+ 7-9
—13-0
9
—07
- 1-4
— 2-1
— 2-8
— 0-4
+ 3-6
+ 6-8
+ 3-0
+10-4
- 7'4
A
+2-0
+ 4-0
+ 6-0
+ 8-1
+24-0
+16-0
+ 8-0
+68-1
+681
...
B
—11
— 2-2
— 3-3
- 4-4
+17-1
+22-4
+11-2
+397
+507
—11-0"
C
—1-2
— 2-4
- 3-6
- 47
+ 3-8
+12-6
+11-3
+15-8
+277
— n-9
1
D
—0-4
— 0-8
— 1-2
— 17
+ 0-3
+ 2-3
+ 4-2
+ 27
+ 6-8
- 41
3
E
+4-8
+ 97
+14-5
+19-3
—14-5
- 97
— 4-8
+19-3
+48-3
—29-0
*,
F
+6-3
+12-6
+18-9
+25-2
+ 6-3
—12-6
— 6-3
+50-4
+69-3
—18-9
G
+6-0
+12-0
+18-0
+23-9
+13-9
+ 3-9
— 6-0
+717
+777
— 6-0
H
+5-3
+107
+16-0
+21-4
+16-0
+107
+ 5-3
+ 85-4
+85-4
...
152 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
914. Strains in the braced arch loaded symmetrically re-
semble those in the semi-arch — Portions of the flanges liable
to tensile strains from unequal loading-. — On examining the
preceding table it will be observed that the strains produced in the
right semi-arch by Wlt W2, and W3 are sub-multiples of those pro-
duced by W4 ; this arises from the circumstance, that the reactions
of the right abutment from the weights on the left semi-arch act
all in the same direction, viz., cW4, and are proportional to the
distance of each weight from the left abutment. Hence, having
calculated the strains produced by W4, we can deduce thence the
strains produced by the three other weights. On comparing this
table with that in 8OO, we find that the strains produced by a
symmetrical load in the diagonals and lower flange of the braced
arch and semi-arch are identical. If the weight of the structure
be small compared with that of the moving load, some of the
bays may sustain tensile strains from the latter. These are the
end bays of the upper flange and the central bays of the lower
flange.
815. Calculation by moments — Calculation of strains in
a latticed arch impracticable., except when the load is
symmetrical. — When there is only one system of triangulation,
the strains may be calculated by the method of moments in the
manner already explained in 2O9, and it is always desirable thus to
check calculations made by the aid of diagrams. When there are
two or more systems of triangulation, that is, when the web is
latticed, the strength may be calculated by working out the strains
from the weights towards the abutments, provided the load is dis-
posed symmetrically on each side of the centre, but when the
weights are distributed in an irregular manner this is not possible,
and accurate calculation seems out of the question, for then more
than two braces meet at the abutment, and we cannot say how the
reaction of the abutment, when decomposed, is divided between
them.
816. Flat arch, or arch with horizontal flanges. — If the
radius of the lower flange be infinite, both flanges will be horizontal,
and this flat arch will resemble girders of the ordinary form. Fig.
57, but with their lower flanges severed at the centre so as to exert
CHAP. VII.J OR CURVED FLANGES. 153
a lateral thrust against the abutments. When the load is uniform,
this thrust will equal the central compression in the upper flange.
This modification of the braced arch possesses some qualities which
merit our attentive consideration. In the first place the quantity
of material required for its lower flange is less than in girders of the
usual form, for the increments of strain increase as they approach
the abutments, and it is therefore more economical to convey
them from, than towards, the centre ; and again, the heavier parts
of the lower flange are near the abutments instead of near the
centre, which is a matter of some importance in very large girders
whose own weight forms the greater portion of the total load.
81?. Rigid suspension bridge. — When inverted, the braced
arch becomes a rigid suspension bridge. Other modifications might
be suggested, such as the crescent girder inverted, with a horizontal
roadway suspended beneath. The railway bridge over the Donau
Canal in Vienna, 83*44 metres long, is constructed on this latter
system. There are two suspension chains on each side formed of
flat links and equi-distant, one above the other, with bracing
between ; a trussed platform for the rails is suspended beneath by
vertical rods in the usual manner. The chains being equi-distant,
and therefore hung from four points, there must be an ambiguity
in the strains, as already explained in 3O6.
318. Triangular arch. — If the lower flange of the braced
arch be formed of two straight bars meeting at the centre like
the letter A, so that the arch becomes two braced triangles, the
calculations as well as the construction will be much simplified,
especially where multiple systems of bracing are employed. This
arrangement has some great practical merits, its chief objection
being the inelegance of its outline, which, however, will be an
immaterial objection in many situations.
319. Cast-iron arches. — The spandrils of cast-iron arches
frequently consist of vertical or radial struts without any diagonal
bracing whatever. This form of arch resembles the common
suspension bridge inverted ; and since the spandrils do not brace
the flanges together so as to change their transverse into longitu-
dinal strains, but resemble in their action the rungs of a ladder
154 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
placed on its side, it is necessary to make the flanges sufficiently
deep to act as girders and sustain the transverse strain when the
moving load causes the line of thrust to pass outside the rib or
curved flange (49). Unless very massive, iron arches with vertical
spandrils may be expected to be more subject to vibration and
deflection than those with braced spandrils.
CASE VI. — THE BRACED TRIANGLE.
S3O. The common A roof. — In the common A roof, the span of
which seldom exceeds 40 feet, each pair of rafters is kept from
- 78- exerting a lateral thrust
against the wall by a
tie-beam, which is often
placed a few feet above
the wall-plate for the
sake of the head-room
which this arrangement
allows. Consequently,
each pair of rafters with their tie-beams constitutes a simple truss
which supports so much of the roof as lies between two adjacent
pairs of rafters.
Let W = the weight uniformly distributed over each pair of
rafters,
I = the span of the roof,
V = the length of each rafter,
d = the height of the ridge above the tie-beam, i.e., the
depth of the truss;
h — the height of the ridge above the wall-plates,
T = the tension in the tie-beam.
Each rafter is held in equilibrium by the uniformly distributed weight
W
of the roof (equivalent to -^ acting downwards at the middle of
2
Cw\
= -3- J, the
CHAP. VII.] OR CURVED FLANGES. 155
horizontal thrust of the opposite rafter at the ridge and the hori-
zontal tension of the tie-beam. Taking the moments of these forces
round the ridge, we have,
_ /W
whence, V = -—7
OCl
By taking moments round the foot of the rafter it may be shown
that the horizontal thrust of the rafters against each other at the
ridge — T. This investigation of the horizontal strains in a simple
trussed girder is, it will be perceived, merely a repetition of that
given in 43 (eq. 25). Each rafter is subject to transverse strains
as a girder and to longitudinal compression as a pillar. The trans-
verse strains are produced by the components of W and of T at
/W
right angles to the rafter. The former =: -^distributed uniformly.
The latter = y,T = , applied at the intersection of the rafter
I QClL
and tie-beam. Hence, the transverse strength of the rafter may
be calculated by eqs. 100 and 85, or perhaps, more conveniently by
eqs. 41 and 37. The longitudinal component of W compresses
the rafter like a pillar, and accumulates gradually from the
ridge, where it equals cipher, to the wall-plate, where it equals
—^7. The longitudinal component of T =. sr/ — i~^77>» ^ com-
presses that part of the rafter which lies between the ridge and
tie-beam, and is balanced by the longitudinal component of the
thrust of the opposite rafter at the ridge. When the tie-beam is
placed high, for the sake of room beneath, d is shortened and T
increased in the same proportion. The transverse strain and
deflection of the rafter is, however, increased in a higher ratio,
for not only is the component of T at right angles to the rafter
increased, but its bending moment also, in consequence of its
acting nearer to the centre of the rafter and farther from the
wall-plate, which acts the part of an abutment. When rafters are
in danger of sagging from their great length, a horizontal collar-
beam is attached midway between the ridge and the tie-beam.
156 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
This collar-beam resists the tendency of the rafters to approach
each other and is subject to compression, in which case each
rafter is a continuous girder supported at both ends and at the
collar-beam, and subject to a transverse pressure from the roofing
/W
material equal to -^ distributed uniformly. If the tie-beam connect
the feet, and the collar-beam the centres, of each pair of rafters,
f ths of this pressure is sustained by the collar-beam, the remaining
f ths being supported by the thrust of the opposite rafter and the
5/W
reaction of the wall-plate (eq. 169). Hence, . ., is the pressure
oZL
against the collar-beam, measured at right angles to the rafter ;
resolving this horizontally, we have the longitudinal compression
5/W
of the collar-beam = -. ^y-. A collar-beam increases the tension of
the tie-beam, and this tension may be found when the strain in the
collar-beam is known by taking moments round the ridge.
The foregoing investigation is only an approximation to the
truth. The longitudinal strains produced in the rafter by the
forces acting at its ends will modify the longitudinal strains due to
the transverse forces, and an accurate investigation would be very
complicated, if not altogether impracticable, for we cannot say how
much of these longitudinal strains pass through the tension fibres
or lower side of the rafter, and how much pass through its compres-
sion fibres or upper side. If there be any tendency in the rafter to
sag, the probability is that they will pass altogether through the
compression fibres, and therefore the upper side of the rafter should
be strong enough to sustain the longitudinal strains produced by
the end forces in addition to the longitudinal strain due to the
transverse components of the load and tie-beam ; but in general it
is unnecessary to take these longitudinal compression strains into
consideration, for when rafters fail they commonly give way on the
under side which is in tension. Of course, if the sag be very
considerable, so that a line joining the ridge and wall -plate passes
above the rafter, the longitudinal compression will increase the strain
in the tension flange in proportion to the vcrsine of the deflection.
CHAP. VII.] OR CURVED FLANGES. 157
SSI. The A truss. — Fig. 79 represents a simple form of
braced triangle, often used for iron roofs where the span does
not exceed 40 feet. The strains in the several parts may be
conveniently obtained by finding the reaction of either abutment
and working thence towards the centre, as explained in the
following example, which has been calculated by the aid of a
diagram drawn to a scale of 5 feet = 1 inch, and with a scale of
weights of 1 ton = 1 inch.
Fig. 79.
The span is 40 feet, the depth of the truss 8 feet, and the
height of the ridge above the wall-plate 10 feet. The load is
8 tons uniformly distributed, for which we may substitute its
equivalent, namely, the load on a whole bay, or 2 tons, con-
centrated at each apex, and the load on half a bay, or 1 ton, at
each abutment. The reaction of the left abutment = 4 tons, of
which 1 ton is immediately balanced by the weight, Wn concentrated
there, leaving 3 tons to be resolved in the directions of A and C,
the strains in which are respectively + 10' 35 tons and — 9 '38
tons. The vertical pressure of W2 is supported by A and F, and
when resolved in their directions produces + 0'9 and + T78 tons
respectively; the former being a downward thrust is opposed to
the upward thrust already existing in A; consequently, the dif-
ference, = -f-9'45 tons, is the thrust transmitted upwards through
B. At a we have two known forces, namely, the tension in C
and the thrust in F ; finding their resultant, and decomposing it
again in the directions of D and E, we have the strains in these
158
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
bars rr — 4*64 tons and — 5'06 tons respectively. The following
table gives the strains in the left half truss in a collected form.
FLANGES AND BRACING.
A
B
C
D
E
F
Strains in tons.
+ 10-35
+ 9-45
— 9-38
— 4-64
— 5-06
+ 178
The accuracy of the work may be checked by the method of
moments as follows. The external forces acting on the left half-
truss are the reaction of the left abutment acting upwards and the
weights W15 W2 and W3 acting downwards. The internal forces
which resist these are the thrust of the opposite half-truss at the
ridge and the pull of the central tie-rod below ; taking moments
round the ridge, and calling the tension in the tie-rod T, we have,
4 x 20 — (1 x 20 + 2 x 10) = T x 8
whence, T = 5 tons, which shows that a trifling error of '06 tons
has been made in the calculation by diagram.
Fig. 80 represents another form of braced triangle suited for spans
between 30 and 60 feet. The method of calculation is so similar
to that just described that an example is unnecessary. In both
trusses the most important part of the bracing is in tension, and
they have therefore a light and graceful appearance.
Figs. 80 and 81.
The form of truss represented in Fig. 81 may be used for spans
CHAP. VII.]
OR CURVED FLANGES.
159
between 50 and 100 feet, and, if desirable, the secondary trussing
may be carried out to a much greater extent than in the figure, so
as to cover far wider spans. A braced triangle of the type
represented in Fig. 82 may also be used up to very large spans
indeed. Different modes of calculating the strains have been
suggested, but the method of working by the resolution of forces
from either abutment towards the centre seems the most satis-
factory, as illustrated in the following example, which has been
calculated by the aid of a diagram drawn to a scale of 5 feet to one
inch.
Fig. 82.
The span and depth are 60 feet and 15 feet respectively, and the
load distributed uniformly over the rafters, i.e., the upper flange, =
12 tons, which is equivalent to 2 tons concentrated at each of the
apices and 1 ton at each abutment. The upward reaction of the left
abutment =. 6 tons, of which 1 ton is at once balanced by Wp and
the remaining 5 tons, being decomposed in the directions of A and
D, produce a thrust of + 11-19 tons in A, and a pull of — 10 tons
in D. At the next apex, W2 (—2 tons,) is supported by A and F
in equal proportions, as they form the sides of an isosceles triangle,
and its components in their directions are each = + 2'24 tons;
that in the direction of A reduces its upward thrust to + 8 '95
tons which is transmitted onwards through B, while the thrust in
F produces a tension of — 1 ton in G and reduces the pull in D so
that a tension of only — 8 tons is transmitted through E. At
W3 we have its downward pressure (=2 tons,) added to the
downward pull of G (= 1 ton,) which gives a total vertical
160
BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
pressure of 3 tons at this apex ; this, when resolved in the directions
of H and B, produces a tension of — 2"83 tons in H and reduces
the upward thrust in B so that only + 6' 71 tons is transmitted
through C. Resolving the downward thrust in H in the directions
of E and I , we obtain a pull of — 2 tons in I , to which should be
added a corresponding pull from the right half of the truss, so
that the total tension in I = — 4 tons. We may check the
accuracy of the calculation by finding the strain in C by the
method of moments, as follows. The segment, Wt C<2, is held in
equilibrium by the external bending forces, namely, the upward
reaction of the left abutment, the downward pressures of Wn W2
and W3, and the resisting forces in the structure itself, namely,
the thrust in C and the various forces meeting at a; taking
moments round a, and measuring the distance Ca by scale, = 13*43
feet, we can find the thrust in C by the following equation,
F X 13-43 = 6 x 30 — (1 X 30 + 2 x 20 + 2 X 10)
Where F represents the strain in C ; hence,
F = l-S3 = 6-7t°nS'
or nearly exactly the same as before.
The following table gives the strains in a collected form.
FLANGES AND BRACING A
B
C
D
E
F
G
H
i
Strains in tons.
+ 1119
+8-95
+671
—10-0
—8-0
+2-24
—1-0
+2-83
-4-0
The roofing material generally rests directly on laths and purlins,
which are again supported by the upper, or oblique, flange. Con-
sequently, unless the purlins rest directly over an apex, each bay
of the upper flange is subject to a transverse strain from the
pressure of the purlins which cross it, in addition to a longitudinal
thrust which it sustains as a member of the truss, and its strength
must be made sufficient to bear this double strain. It will be
seen hereafter that its continuity across the apices adds materially
to the strength of the rafter.
The arrangement of the bracing may be varied so as to put the
CHAP. VII.] OR CURVED FLANGES. 161
verticals in compression and the diagonals in tension, and sometimes
the tie is raised at the centre so as to form a low triangle and give
more head-room beneath; this of course diminishes the effective
depth of the truss, but it has the advantage of shortening the
length of the struts.
CASE VII. — THE SUSPENSION TRUSS.
332. Suited for domed roofs. — This form of truss is gene-
rally employed for supporting low-domed roofs resting on circular
walls, in which case the trusses intersect each other at the centre
Fig. 83.
and have a common central strut beneath the crown of the dome.
Each half of the bow, or upper flange, is strengthened by a
secondary truss D E F. At first sight there seems some ambiguity
about the strains, inasmuch as three braces intersect at the abut-
ment, and we cannot say how the reaction of the latter is distributed
among them. On a little consideration, however, the matter is
simple; let us confine our attention to the external forces which
keep the secondary truss, A B C D E F, in equilibrium, and taking
their moments round the centre of the roof, we have the moment
of the tension in the string K equal to the upward moment of the
reaction of the left abutment minus the downward moments of
W2 and W3. We can thus find the tension in the string, and
knowing this and the reaction of the abutment, we can readily
find their resultants in A and D, and from these again derive the
strains in the other braces. The following example will illustrate
this clearly. It has been worked out by the aid of a diagram drawn
M
162 BRACED GIRDERS WITH OBLIQUE [CHAP. VII.
to a scale of 5 feet to one inch. Let Fig. 83 represent a suspension
truss, 80 feet in span, 5 feet in depth from the crown to a horizontal
line joining the wall-plates, and 15 feet in total depth. The bow
is divided into 6 equal bays, and the secondary truss has been
formed by making D a horizontal line, and the short struts G and I
parallel to the radial line which would pass through the centre of
B ; thus A = B = C = E, and G = I, and D = F. Let the weight
of a sector of the circular roof supported by the half-truss,
A B C L K, = 9 tons, which is divided among the apices in pro-
portion to the area of the sector supported by each bay and,
assuming that the sector is a triangle, we shall have the weights at
the several apices as follows : —
W! = 2f tons,
3 _ „
w4= i „
Since Wj rests directly on the wall-plate, we may leave it out of
consideration in calculating the longitudinal strains in the truss,
though it will be necessary to consider it subsequently when
calculating the transverse strength of A as an independent girder
supporting directly its proper share of distributed roof-load. The
secondary truss, A B C D E F, is held in equilibrium by
1°. The oblique pull in the tie K,
2°. The upward reaction of the abutment, = W2 + W3 + W4
= 6J tons,
3°. The downward pressures of W2 and W3,
4°. W4, the thrust of the central strut L, and that of the opposite
half-truss, — all three intersecting at the crown.
If we take moments round the crown we get rid of the three
latter forces, but to do this we must find by scale
the leverage of K round the crown = 14*58 feet,
do. W2 do. = 26-85 feet,
do. W3 do. = 13-45 feet.
Taking moments round the crown, we have the
tension in K = 6*8 X 40-(4 x 26-85 + 2 x 18-43) =?.93 tons
14-58
CHAP. VII.]
OR CURVED FLANGES.
163
We now know two of the forces meeting at the abutment, namely,
the upward reaction of the abutment, — 6| tons, and the tension
in K, = 7*93 tons. Finding the resultant of these, and decom-
posing it in the directions of A and D, we find the compression in
A = + 21 tons, and the tension in D = — 12-86 tons. At W2
four forces meet, namely, the thrust in A, the weight W2, the
thrust in G and the thrust in B. As we know the two former
forces we can find their resultant, and decomposing it in the
directions of G and B, we find the strains in these equal to + 2*25
tons and + 20*44 tons respectively. At «, four forces meet, namely,
the tension in D, the thrust in G, and the tensions in E and H.*
The two former are known, and finding their resultant and decom-
posing it, we get the strain in E, = — 9 '6 tons, and that in
H, = — 3-2 tons. Proceeding thus, we find all the strains which
are given in the following table.
FLANGES AND BRACING.
A
B
c
D
E
F
G
H
i
K
L
Strains in tons.
+ 21
+20-44
+17-06
-12-86
-9-6
— 9'63
+2-25
-3-2
+ 1-15
— 7-93
+3-8
The compression in the central pillar, L, is that due to both
sides of the primary truss, and should equal the vertical resultant
of the strains in the tie bars K and K'. This will be a check on
the accuracy of the work. This form of truss is that generally
used for supporting the roofs of gasholders; the truss, however,
does not come into action unless the holder is empty, for when it is
charged with gas (the pressure of which sometimes reaches 5
inches of water), the upward pressure of the gas is greater than the
weight of the roof and lifts both it and the sides of the gasholder,
and an explosion would, no doubt, sometimes occur, were it not
for the domed shape of the roof which resists internal pressure like
the ends of an egg-ended boiler.
* The diagonal H is required because W2 exceeds W3 ; in practice, however, it is
generally omitted.
164 DEFLECTION. [CHAP. VIII.
CHAPTER VIII.
DEFLECTION.
CLASS 1. — Girders whose sections are proportioned so as to produce
uniform strength.
223. Deflection carve circular in girders of uniform
strength — Amount of deflection not materially affected by
the web. — The equations generally used for calculating the deflec-
tions of loaded girders are based on the assumption that the section
of the girder is uniform throughout its entire length, that is, that
there is the same amount of material at the centre as at the ends.
In scientifically constructed girders, however, this is not the case.
Each part is duly proportioned to the maximum strain which can
pass through it, so that no material is wasted ; and when this occurs
in a girder with horizontal flanges and a uniformly distributed
load, that is, the load which produces the maximum strain in the
flanges, these latter will, as has been already shown in 4?, taper
from the centre, where their section is greatest, towards the ends
as the ordinates of a parabola. The girder is then said to be of
uniform strength, because the unit-strain in each flange is uniform
throughout the whole length of the flange and no part has an
excess of material, or is unduly strained beyond the rest (19). Now,
as the contraction and elongation are according to Hooke's law
proportional to the unit-strain, so long as it does not exceed the
limits which are considered safe in practice (7), the contraction per
running foot of the upper flange will be uniform throughout its
length, and the extension per running foot of the lower flange will
likewise be uniform throughout its length; and this uniform
contraction and elongation must produce a circular deflection, since
the circle is the only curve that is due to a uniform cause. At first
sight it may be thought that the continuous web of the plate girder,
CHAP. VIII.] DEFLECTION. 165
or the braced web of the lattice girder, will seriously affect the
amount of the deflection curve ; but it can be readily shown by
carefully constructed diagrams, in which the alterations of length
due to the load are drawn to a highly exaggerated scale, that the
construction of the web has scarcely any influence on the curvature
so long as the unit-strains in the flanges are unaltered in amount by
the method of construction, and it is only when this is the case that
a fair comparison can be instituted between the rival girders.
Fig. 1, Plate I., represents one-half of a diagonally braced girder
of the simplest form, namely, a girder with one system of triangles
before the load rests upon it. Every part is then in its normal
state, and the girder will be horizontal. Now, suppose that a
uniform load deflects it and shortens each bay of the top, or com-
pression, flange by a certain quantity, while it lengthens each bay of
the lower, or tension, flange to a similar extent ; and further, let us
suppose that the diagonals are alternately shortened and lengthened
by equal amounts, according as they are struts or ties. Fig. 2 now
represents the girder ; the deflection curve forms a segment of a
circle whose centre is at A, a little to the left of the vertical line
drawn through the middle of the girder. Next, suppose that the
flanges are compressed and extended as in Fig. 2, but that the
diagonals remain of their original length as in Fig. 1, that is, that
their length is not affected by the load. Fig. 3 is the result,
which it will be perceived, is circular and differs but slightly
from Fig. 2, having its centre, however, at B, in the vertical
line drawn through the middle of the girder. It may at first
seem strange that A, the centre of Fig. 2, is not in the vertical
line passing through the middle of the girder. This is due to
the circumstance that, with a uniform load, the two central dia-
gonals, d and d', are subject to the same strain, either both lengthened
or both shortened, while all the other diagonals are alternately
lengthened and shortened. Hence, a very slight angle is produced
at the centre, as shown in Fig. 4, where the flanges are unaltered
as in Fig. 1, while the diagonals are alternately lengthened and
shortened as in Fig. 2. Considering, however, the exaggerated
scale of the diagrams, Fig. 4 is practically horizontal when compared
1(56 DEFLECTION. [CHAP. VIII.
with Figs. 2 or 3, and the chief effect of this common change in
the length of the two central diagonals is to throw the centre of
each half of the girder in Fig. 2 a little to the right or left of the
middle line. These diagrams give very interesting results ; they
show that the curvature of flanged girders is practically independent
of change of form in the web, and almost entirely due to the
shortening of the upper, and the elongation of the lower, flange ;
and a further inference may be derived from them, viz., that
deflection is practically unaffected by the nature of the web,
whether it be formed of plates or lattice bars, provided that the
unit-strains in the flanges are not increased or diminished by a
different formation of web. Consequently, if there be two girders
of equal length and depth, one a lattice, the other a plate girder,
having the same unit-strains transmitted throughout their respective
flanges, they will both deflect to the same extent.
SS4. Formula for the deflection of circular curves —
Deflection of similar girders when equally strained varies
as their linear dimensions. — The circumstance of the curve of
a loaded girder of uniform strength being circular enables us to
find a very simple equation for calculating its deflection.
Let adbgeh, Fig. 84, represent a girder supported at both ends
and of uniform strength for the load, which generally occurs when
the load is uniformly distributed.
Fig. 84.
CHAP. VIII.] DEFLECTION. 167
Let / = adb = the length of the girder,
d = de = the depth,
R = af = the radius of curvature,
X — geh — adb = the difference in length of the flanges after
deflection,
D = cd = the central deflection.
Since the deflection is very small compared with the radius of
curvature, we may assume cf = af — R, and ab = adb = I; then
(Euclid, prop. 35, book iii.),
r
By similar triangles, R = —
X
whence, by substitution, D = — (132)
8#
in which the value of X is known, as it depends on the coefficients
of elasticity of the flanges and the strains to which they are subject.
This equation for the deflection curve confirms the previous inves-
tigation, for the depth, d, is the only quantity in the equation
which can be affected by a change in the length of the diagonals,
and it is obvious that a slight change in the value of d will not
affect that of D to any appreciable extent.
It follows from equation 132 that when similar girders sustain
the same unit-strains in their flanges, their deflections will vary
directly as any of their linear dimensions.
Ex. 1. The length and depth for calculation of the Conway tubular bridge are
respectively 412 feet and 237 feet, and it appears from ex. 2 (44) that the inch-strains
in the lower and upper flanges at the centre of the bridge from the permanent load are
5 '067 tons and 3 '9 48 tons respectively ; what is the central deflection on the supposition
that the flanges are of uniform strength, which is very nearly true ? The coefficient of
elasticity of wrought-iron is 24,000,000 Ibs. = 10,714 tons per square inch; consequently,
it contracts or extends T^fr^th of its length for each ton per square inch, and we have
the following data : —
I = 412 feet,
d = 237 feet,
A-t O
\ = -5±f_ (5-067 + 3-948) = '347 feet.
Answer (eq. 132). D = — = '!i7_* A1^ = 754 feet = 9'048 inches.
8tt 8 X 237
168 DEFLECTION. [CHAP. VIII.
The mean deflection of the two tubes immediately on removal of the platform was 8 '04
inches, and 8 '98 inches after taking a permanent set due to strain. When the permanent
way was added and after 12 month's use, the deflection of the second tube in the
month of January was 10 '3 inches. The deflection in hot weather would doubtless be
somewhat less. The deflection, from additional weight placed at the centre, was '01104
inch for each ton. (Clark, p. 662.)
Ex. 2. The length and depth for calculation of one of the large tubes of the Britannia
bridge are respectively 470 and 27 '5 feet, and from ex. 4 (44), the inch-strains at the
centre from the weight of the tube as an independent girder were 5795 and 4'856 tons
in the lower and upper flanges respectively. What was the central deflection ? Using
the same coefficient of elasticity as before, we have,
I = 470 feet,
d = 27-5 feet,
A = L (5795 + 4-856) = "467 feet.
Answer (eq. 132). D = = ' =
The mean deflection of the two tubes of the up line, immediately on removing the
platform, was 1175 inches ; the mean deflection after being raised was 12'57 inches.
(Clark, p. 673.)
Ex. 3. A wrought-iron girder of uniform strength is 84 feet long and 7 feet deep.
A certain load produces a deflection of 1*2 inches at the centre ; what are the unit-
strains in the flanges from this load ? From equation 132, we have,
.0 VX
The inch-strains in both flanges together = a * -j _ 3.55 tongj whicil when
84 X 12
divided between the two flanges inversely as their sectional areas, will give the inch-
strain in each flange due to the given load.
CLASS 2. — Girders whose section is uniform throughout their length.
885. The following investigations are based on the law of uniform
elastic reaction, and are therefore only applicable to girders whose
strains lie within the limits of elasticity (31).
Let W = the bending weight,
M = the moment of resistance of the horizontal elastic
forces at any given cross section of the girder (59),
x = the horizontal distance of the same section from the
left abutment,
y = the vertical distance of any fibre in the section, either
above or below the neutral axis,
CHAP. VIII.] DEFLECTION. 169
j3 = the breadth of the section at the distance y from the
neutral axis, and consequently a variable, except in
the case of rectangular sections,
/ = the horizontal unit-strain exerted by fibres in the given
section at a distance c from its neutral axis,
c = the distance from the neutral axis of horizontal fibres
which exert the unit-strain /,
I = the moment of inertia of any cross section round its
neutral axis, and consequently, a constant quantity
throughout the whole length of the girder when the
latter is of uniform section,
R = the radius of curvature,
E = the coefficient of elasticity.
It has already been shown (eq. 43) that M, the moment of the
horizontal elastic forces of any cross section round its neutral axis,
may be expressed by the equation,
provided the horizontal fibres are not strained beyond their limit
of elastic reaction. When the girder is of uniform section
throughout its length, the integral \ fiy*dy, being a definite integral,
will be a constant throughout the girder, and as it happens to
express the moment of inertia of the cross section round its neutral
axis (69), we may substitute for this integral the symbol I, when
we have
M = U (133)
G
In order to transform this equation into one involving the co-
ordinates .of the deflection curve, we must substitute for the three
variables, M, / and c, their values in terms of the co-ordinates x
and y. Let us first deal with / and c.
Fig. 85 represents a deflected semi-girder, whose neutral surface
isNS.
Let ab = a unit of length,
& and fc' r= the increment and decrement in length of a
linear unit of the extreme fibres after deflection.
170 DEFLECTION. • [CHAP. VIII.
Fig. 85.
When the horizontal strains do not exceed the limits of elasticity,
we have the following relation,
/ E
c=R
Substituting this in eq. 133, we have the moment of resistance,
M = 1 1 (134)
From the principles of the differential calculus we know that, where
the deflection is small compared with the length of the curve,
1 d*
whence, by substitution in eq. 134, we have,
M=-Elg (135)
in which M is a positive or negative moment according as the
upper flange is in compression or tension, y being measured down-
wards. This equation expresses the moment of resistance of the
horizontal elastic forces at any section of a girder in terms of
the ordinates of the deflection curve, the coefficient of elasticity, and
the moment of inertia of the cross section round its neutral axis.
In order to solve eq. 135, there still remains before integration to
substitute for the variable M its value in terms of the ordinates of
CHAP. VIII.] DEFLECTION. 171
the deflection curve, which may be derived from the leverage of
the weight, observing that the moments of forces are to be taken as
positive or negative according as they tend to compress or extend
the upper flange. To effect this substitution we must consider
each case separately, and after integration, the value of I, which is
a different constant for each form of section, may be obtained by
multiplying the values of M, already determined in (31) and the
succeeding articles, by >(eq. 133).
CASE I. — SEMI-GIRDERS OF UNIFORM SECTION LOADED AT THE
EXTREMITY.
SS6. Let W = the load at the extremity,
I = the length of the semi-girder,
x = the abscissa of the deflection curve measured
from the fixed end,
y — the ordinate of the deflection curve measured
downwards,
D = the deflection at the extremity,
M = the moment of resistance of the horizontal
elastic forces at any given section, whose
distance from the fixed ends = x (59),
I = the moment of inertia of any cross section,
E = the coefficient of elasticity.
Taking moments round the neutral axis of the given section, we
have,
M =__W(J — x)
Substituting this in eq. 135, we have,
Integrating,
E I ^ = W fix — ~ \ + constant.
The constant = 0, for when as — 0, also = 0, since the tangent
172 DEFLECTION. [CHAP. VIII.
of the curve is horizontal at the fixed end. Integrating again, and
determining that the new constant = 0, from the consideration that
y •=. 0 when x = 0, we have,
(136)
This is the equation of the deflection curve, y being the deflection
at any point whose distance from the fixed end equals x.
At the extremity where x = /, y •=. D, and we have,
EID = W^
o
whence, D =L (137)
837. Solid rectangular semi-girders — Deflection of solid
square girders is the same with the sides or one diagonal
vertical. — Let b = the breadth and d = the depth. From eqs.
46, 133, and 137,
Comparing eqs. 46 and 47, we find that the deflection of solid
square girders is the same whether the diagonal or one side be
vertical. Their strength, however, is not the same (86).
Ex. The piece of Memel timber, described in Ex. 4 (66), deflected 0'66 inch from
a load of 336 Ibs. hung at its extremity ; what is the value of E ?
Here, W = 336 Ibs.,
I = 24 inches,
6 = 1'94 inches,
d = 2 inches,
D = 0-66 inch.
Answer (eq. 138). E = = 1,800,000 Ibs.
888. Solid round semi-girders. — Let r = the radius. From
eqs. 48, 133, and 137,
D = i™3 (139)
889. Hollow round semi-girders of uniform thickness. —
Let t rr the thickness of the tube, supposed small in proportion to
its radius r. From eqs. 50. 133, and 137,
D = -— (140)
CHAP. VIII.] DEFLECTION. 173
S3O. Semi-girders with parallel flanges. — When the web
is formed of bracing, or if continuous, is yet so thin that we may
safely neglect the support it gives the flanges, we have from eqs.
55, 133, and 137,
E = gSS (U1)
where A = al + «2 = the sum of the areas of the two flanges, and
d = the depth of the web.
When the web is taken into account and the flanges are of equal
area,
let a = the area of either flange,
a' = the area of the web.
From eqs. 57, 133, and 137,
D = ^
S31. Square tabes of uniform thickness, with the sides
or one diagonal vertical. — From eqs. 59, 133, and 137,
4VW»
D = E(M-V)
where b and bl are the external and internal breadths.
If the thickness of the tube be small compared with the breadth,
we have from eqs. 60, 133, and 137,
D =
in which t represents the thickness of one side.
CASE II. — SEMI-GIRDERS OF UNIFORM SECTION LOADED
UNIFORMLY.
233. Let Z = the length of the semi-girder,
x — the abscissa of the deflection curve measured from
the fixed end,
y — the ordinate of the deflection curve measured
downwards,
w = the load per unit of length,
W = wl = the whole load,
174 DEFLECTION. [CHAP. VIII.
D = the deflection at the extremity,
M = the moment of resistance of the horizontal elastic
forces at any given section, whose distance from
the fixed end = x (59),
E = the coefficient of elasticity.
Taking moments round the neutral axis of the given section, we
have,
M = -£(*-*)•
Substituting this in eq. 135, we have,
c I #y _ w (l -y
* d^~ JV"
Integrating,
E I -j- — — -^ (I — ,v)3 + constant.
ax o
When x = 0, ~ = 0 also ; hence, the constant equals -TT . Substi-
ax b
tuting this value and integrating again,
E I y = --£ (I — #)4 H -- ^ -- h constant.
Determining the second constant by the consideration that y = 0
when x •=. 0, we have,
w ,, ... t wlzx wl*
_ ,
Ely =
At the extremity where x = /, y = D, and we have,
n w'* wp
D = 8EI = 8EI
333. Deflection of a semi-girder loaded uniformly equals
three-eighths of its deflection with the same load concen-
trated at its extremity. — Comparing eqs. 145 and 137, we see
that the deflection of a semi-girder loaded uniformly is to its deflec-
tion with the same load concentrated at the extremity as |. Hence,
to obtain the deflections of the various classes of semi-girders in the
case of a uniform load, we have merely to multiply the formula? in
the preceding case by f , recollecting that W will now represent the
uniformly distributed load.
CHAP. VIII.] DEFLECTION. 175
CASE III. — GIRDERS OP UNIFORM SECTION SUPPORTED AT BOTH
ENDS AND LOADED AT THE CENTRE.
S34. Let I = the length of the girder,
x — the abscissa of the deflection curve measured
from the left end of the girder,
y — the ordinate of the deflection curve measured
downwards,
W rr the load at the centre,
D rz the deflection at the centre,
M = the moment of resistance of the horizontal elastic
forces at any given section whose distance from
the left end = x (59),
E = the coefficient of elasticity.
Taking moments round the neutral axis of the given section, we
have ..
Substituting this in eq. 135, we have,
Integrating,
... . dy ,
E I ~ = -- — constant.
dx 4
To determine the constant, we must recollect that the tangent of
the curve is horizontal at the centre; hence, -~ = 0 when x = -,
dx '2
W/2
and the constant = —-— ; substituting this,
7 — . . -
dx 4 \4
Integrating again, and observing that the second constant = 0
from the consideration that y = 0 when x — 0, we have,
which is the equation of the deflection curve.
176 DEFLECTION. [CHAP. VIII.
At the centre where x — ~, y = D, and we have,
835. Solid rectangular girders. — From eqs. 46, 133, and
146,
W/3
D = ra? <147>
in which b and d represent the breadth and depth of the girder.
Ex. From the mean of five experiments made by Mr. Hodgkinson on Blaenavon •
cast-iron, No. 2,* it appears that the breaking weight and ultimate deflection of a
rectangular bar 13 feet 6 inches between points of support, 3 inches wide and 14 inch
deep, are respectively 819 Ibs. and 10'46 inches ; what is the value of the coefficient of
transverse elasticity at the limit of rupture ?
Here, W = 819 Ibs.
I = 13-5 feet,
6 = 3 inches,
d = 1'5 inches,
D = 10'46 inches.
Ans. (eq. 147). E = ^ = 819 X (13'5 X 12)3 = 8 200,000 Ibs. per square inch.
4Dbd3 4 X 10-46 X 3 X (l'5)s
The deflection of the same bar when loaded with 260 Ibs., which was within the limit of
elasticity, was 2 inches. What was its coefficient of elasticity within this limit ?
Here, W = 260 Ibs.
D = 2 inches.
The reader should be informed that this coefficient of transverse elasticity of Blaenavon
iron is less than that of average cast-iron, especially when mixed.
836. Solid round girders. — From eqs. 48, 133, and 146,
in which r represents the radius.
837. Hollow round girders of uniform thickness. — From
eqs. 50, 133, and 146,
W/3
D = (149)
in which t represents the thickness of the tube, supposed small in
proportion to its radius r.
* See Report of Com. p. 69.
CHAP. VIII.] DEFLECTION. 177
838. CJirders with parallel flanges.— When the vertical web
is formed of bracing, or if continuous, yet so thin that it affords
but slight assistance to the flanges in sustaining horizontal strains,
its stiffness as an independent girder may be neglected, and we have
from eqs. 55, 133, and 146,
° - raSS? (150)
in which A = a{ + a2 = the sum of the areas of the top and bottom
flanges, and d = the depth of the web.
When the web is taken into account, and the flanges are of equal
area, from eqs. 57, 133, and 146,
W/3
D ~ '
in which a = the area of one flange and a' = that of the web.
839. The deflections of girders of other forms of section may be
obtained in a similar manner from eqs. 133 and 146 by substi-
tuting for M the corresponding values given in Chap. IV.
CASE IV. — GIRDERS OF UNIFORM SECTION SUPPORTED AT BOTH
ENDS AND LOADED UNIFORMLY.
84O. Let I — the length of the girder,
w — the load per linear unit,
W = wl = the whole load,
x = the abscissa of the deflection curve measured from
the left end of the girder,
y = the ordinate of the deflection curve measured
downwards,
D = the deflection at the centre,
M rr the moment of resistance of the horizontal elastic
forces at any given section whose distance from
the left end = as (59),
E = the coefficient of elasticity.
N
178 DEFLECTION. [CHAP. VIII.
Taking moments round the neutral axis of the given section, we
have,
M =™(la; — X*)
Substituting this in eq. 135, we have
Elg = _!(fa_,.) (152)
Integrating,
When x — p -~ = 0, and the constant becomes ^7 5 substituting
this, c I dy _ w Ix3 la;2 Z
' -~~~~
Integrating again, and observing that the second constant = 0
from the consideration that y = 0 when x = 0,
Eh = ~(z*-2W + l3*)
which is the equation of the deflection curve.
At the centre where x = , y — D, and we have,
n 5VW3
D:=384EI=384ET
341. Central deflection of a girder loaded uniformly equals
five-eighths of its deflection with the same load concentrated
at the centre. — Comparing eqs. 153 and 146, we find that the
central deflection of a girder loaded uniformly is -jj-ths of the
deflection if the same load were concentrated at the centre. This
has been corroborated by experiments by M. Dupin on rectangular
girders of oak.*
848. No I id rectangular girders. — From eqs. 46, 133, and
153,
n 5VW3
= *-*
where b and d represent the breadth and depth of the girder.
* Morin, p. 140.
CHAP. VIII.] DEFLECTION. 179
Comparing eqs. 46 and 47, we find that the deflection of solid
square girders is the same, whether one side or the diagonal be
vertical. The former, however, is theoretically 1*414 times stronger
than the latter (86).
343. Solid round girders. — From eqs. 48, 133, and 153,
(155)
96 Err4
where r represents the radius of the cylinder.
344. Hollow round girders of uniform thickness. — From
eqs. 50, 133, and 153,
D= = - (156)
384 EvrH
where r — the radius, and t — the thickness of the tube, supposed
small in comparison with the radius.
345. Girders with parallel flanges. — When the web is formed
of bracing, or if continuous, yet so thin that its strength as an
independent girder may be neglected, we have from eqs. 55, 133,
and 153,
D_ 5AW4 5AVW3
~~»
where A = at + a2 = the sum of the areas of top and bottom
flanges, and d := the depth of the web.
If the web be taken into account and if the flanges have equal
areas, from eqs. 57, 133, and 153,
p_ 5^4 _ 5W/3
" 32 E(6a + a')d* ~ 32 E(6a + a')d*
where a = the area of one flange, and a! = that of the web.
346. Discrepancy betwreen coefficients of elasticity derived
from direct and from transverse strain. — The coefficients of
elasticity derived from experiments on transverse strain do not
always agree with those derived from direct longitudinal tension
or compression ; they vary also with different forms of cross section,
as exhibited in the following table, which contains the coefficients
of transverse elasticity of cast and wrought-iron girders of the more
usual forms of cross section.
180
DEFLECTION.
[CHAP. viii.
MATERIAL.
Value of E, the coefficient of transverse
elasticity per square inch.
CAST-IRON.
Ibs.
tons.
1.
Rectangular bars of simple cast -irons,
15,200,000
= 6,785
2.
Do. do. mixed do.,
18,892,000
= 8,434
3.
Rectangular, circular, or elliptical tubes, .
12,215,000
= 5,453
4.
Double-flanged girders, ....
13,200,000
= 5,893
WKODGHT-IRON.
! 16,360,000 )
( 7,304
5.
Double-flanged rolled beams, for floors, &c.,
to
= to
21,570,000 )
( 9,630
6.
Single-webbed double-flanged plate girders,
riveted, ......
14,316,000
= 6,391
7.
Tubular plate girders, ....
23,610,000
= 10,541
8.
Conway tubular bridge, ....
18,754,000
= 8,372
1. Experimental Researches, p. 404.
2. 3. 4. 6. 7. 8. Morin, pp. 260, 264, 269, 299, 322, 323.
5. Idem, p. 293, and Mr. W. Anderson.
CHAP. IX.] CONTINUOUS GIRDERS. 181
CHAPTER IX.
CONTINUOUS GIRDERS.
347. Continuity — Contrary flexure — Points of inflexion. —
A girder is said to be continuous when it overhangs its bearings, or
is sub-divided into more than one span by one or more intermediate
points of support. When a loaded girder is balanced on a single
pier at or near its centre, like the beam of a pair of scales, the
upper flange is subject to tension, the lower one to compression,
and the girder becomes curved with the convex flange uppermost.
If, however, the same girder be supported at its extremities, the
pier being removed, the strains in the flanges are reversed, the
upper flange being now compressed and the lower one extended,
and in this case the convex flange is underneath. If, while in this
latter position, we replace the central pier so as to form two spans,
the girder becomes continuous and partakes of the nature of both
the independent girders ; each flange is in part extended, in part
compressed, and the curve becomes a waved line. Let Fig. 86
represent a continuous girder of two spans uniformly loaded.
The central segment B B' resembles the independent girder in
the first case, namely, when balanced over a pier; the extreme
segments, AB, B'A', resemble it in the second case, since one
end of each rests upon an abutment and the other end is sup-
ported by the central segment, which thus sustains besides its
own proper load an additional weight suspended from each
182 CONTINUOUS GIRDERS. [CHAP. IX.
extremity, equal to the half load on each of the end segments.
The points B, B', where the curvature alters its direction, are
called the points of contrary flexure, or more briefly, the points of
inflexion. The curves of the end and central segments have
common tangents at these points, and here the strains in the
flanges change from tension to compression, and vice versa.
Exactly at these points the strains in the flanges are cipher;
consequently, the flanges might be severed there without altering
the conditions of equilibrium in any respect. In fact, a continuous
girder may be regarded as formed of independent girders connected
merely by chains at the points of inflexion. In braced girders the
bracing acts as the chain, in others the continuous web.
848. Passing load. — For the investigation of the strains in a
continuous girder it is necessary — first, to find the points of inflexion,
and afterwards to calculate the strains in the separate segments on
the principles already laid down for independent girders. A passing
load complicates the question, for its effect is to alter the position of
the points of inflexion, and consequently the lengths of the component
segments ; if, for instance, a passing train covers the left span, its
deflection will be increased and that of the right span diminished,
or even altogether removed, if the passing load be sufficiently
heavy to lift the right end off the abutment A'. The effect of
this partial loading on the points of inflexion will be to bring B
nearer to, and remove B' farther from, the central pier, and this is
that disposition of the load which gives the greatest length to the
segment A B ; it is necessary, therefore, in the case of a passing
load to find this new position of the points of inflexion and
calculate the strains in A B as an independent girder of this
maximum length. Of course, the same calculations will suit B'A'
when it is of maximum length, that is, when the right span
only is loaded. The central segment, B B', becomes of maximum
length when the load is uniformly distributed over the whole
girder, and the points of inflexion have to be determined under
this condition of the load also. Having thus calculated the
strength of each part when subject to the load which produces the
maximum strain in the flanges of that part, we may assume that
CHAP. IX.J CONTINUOUS GIRDERS 183
there is sufficient strength for any other disposition of the load,
since the motion of the points of inflexion is restricted within
these limits. The reaction of either abutment is equal to half
the load on the adjacent segment; thus, the reaction of the left
abutment equals half the load resting upon A B. The reaction of
the pier equals the load resting upon the central segment, B B7, plus
the sum of the reactions of the two abutments.
349. Experimental method of finding: the points of in-
flexion— The depth of a girder does not affect the position
of the points of inflexion. — The following method of finding the
points of inflection depends partly on theory, partly on experiment,
and is applicable to continuous girders containing any number of
spans. Take a long rod of clean yellow pine or other suitable
material to represent the continuous girder, and let it be supported
at intervals corresponding to the spans of the real girder. Next, load
this model uniformly all over, or each span separately, or in pairs, or
make any other disposition of the load which can occur in practice.
Now, it is clear that, if the model and its load be a tolerably
accurate representation of the girder and its load, the points of
inflection of the former will correspond with those of the latter ;
they might therefore be at once obtained by projecting the curves
of the model on a vertical plane. It is difficult, however, to do
this so as to determine the points of inflection with the requisite
accuracy, for the exact place where the curvature alters is never
very precisely defined to the eye. The pressures on the points of
support may, however, be measured with considerable accuracy,
taking the precaution of keeping them all in the same horizontal
line, as a slight error in their level would seriously affect the
curvature and lengths of the component segments. We shall
assume therefore that the reactions of the points of support have
been thus found experimentally.*
Let Fig. 87 represent a continuous girder containing any
number of spans, each loaded uniformly, and let 0,0,0, &c., represent
* It is a safe precaution to measure the pressures on the points of support with the
rod turned upside down as well as erect, and then take the mean measurement as the
true result.
184 CONTINUOUS GIRDERS. [CHAP. IX.
successive points of inflection, the intervals between which are
called segments.
Fig. 87.
Let R1? R2, R3, &c. = the reactions of the successive points of
support as found by experiment,
/, £', &c. = the lengths of the successive spans,
w, wr, &c. = the loads per linear unit on each span,
a, 6, c = the lengths of certain parts of the girder,
as represented in the figure,
Q = the centre of the third segment.
RU the reaction of the left abutment, is equal to half the load on
the first segment a, whence, R t = — , and
a = —l (159)
w
This equation gives the distance of the first point of inflexion
from the left abutment, since Rj is known from experiment.
R2, the reaction of the first pier, is equal to the load resting on
the girder as far as Q minus the reaction of the first abutment ;
that is, R2 = wl + w'b — Rp whence,
I = R- + »;~^ (160)
Again, taking moments round either flange at Q, which is now a
known point, we have,
Fd = R,(J + b) + R,b-wl (1 +!>}- —
in which F = the strain in either flange at Q, and d = the depth
of the girder; but from eq. 25 we have,
CHAP. IX.] CONTINUOUS GIRDERS. 185
c being the length of the third segment, as marked in the figure ;
substituting this value for Fd and arranging, we have,
The distance of the second point of inflexion from the first pier
= b — y and so on. It will be observed that the depth of the
girder does not enter into these equations, and therefore does not
affect the position of the points of inflexion.
85O. Practical method of fixing: the points of inflexion —
Economical position of points of inflexion. — I shall here briefly
describe a method by which the points of inflexion of braced girders
may be fixed in any particular bay at will, so that there may be no
uncertainty respecting their position, or so that they may, if
desirable, be made to assume that position which is most advan-
tageous for economy in the flanges.
Let Fig. 88 represent a continuous lattice girder capable of free
horizontal motion on the points of support. Suppose that the point
of inflexion, as determined by theory, is at a, but that it is desirable
to fix it at ft, that is, to make that part of the upper flange which
lies between a and b subject to tension in place of compression.
This may be effected by severing the flange at 6, and lowering the
end of the girder on the left abutment slightly, so as just to separate
the parts at b. The left segment, c6, will then assume the condition
of an independent girder supported at one extremity by the abut-
ment and at the other by the oblique forces in diagonals d and e.
The upper flange from c to b will undergo compression, from b to
some corresponding point in the second span, tension. Further,
186 CONTINUOUS GIRDERS. [CHAP. IX.
the operation of fixing the point of inflexion in the upper flange
determines its position in the lower one also, for, when the former
is severed at 6, the only horizontal forces acting upon the seg-
ment cbf are the strains in the lower flange at / and the horizontal
component of the strains in diagonals d and e. This component
must therefore be exactly equal and opposite to the strain at /,
otherwise, the left segment, c6/, will move either to the right or
left, since by hypothesis it is free to move horizontally on the
abutment (58). Hence, it is evident that the point of inflexion
in the lower flange is not far from /, probably not farther than
the adjoining bay. Its position is determined by the condition
that the horizontal component of the strains in 'the diagonals inter-
sected by a line joining the points of inflexion in the two flanges
is equal to cipher. Thus, by leaving any particular bay in one of
the flanges of a continuous girder of two spans permanently severed,
we have the point of inflexion in that span fixed under all conditions
of the load ; and when this is determined, we can find the strains in
the flanges over the pier, and thence deduce the position of the
point of inflexion in the second span. If the severed flange be
united when any given load rests upon the girder, though the point
of inflexion will move with every change of load, yet it will return
to its original position whenever a similar load rests on the girder
in the same position as when the flange was first severed.
If there be three spans, the central span may have both points of
inflexion fixed independently of each other, and these again will
determine the corresponding points in the side spans. The operation
is safe in practice, as was proved at the Boyne Viaduct, where the
points of inflexion in the centre span were fixed by severance in
those bays in which theory had previously indicated their probable
existence.* The most economical arrangement in theory for the
flanges of a large girder of one span uniformly loaded consists in
forming points of inflexion at the quarter-spans. In this case the
end segments of the upper flange must be held back by land chains,
as in suspension bridges, while those of the lower flange exert a
* See Description of the Boyne Viaduct in the Appendix.
CHAP. IX.] CONTINUOUS GIRDERS. 187
horizontal thrust against the abutments like the flat arch (316).
The two extreme segments of the girder thus form semi-girders,
while the central segment is an independent girder suspended
between them by the web.
The following theoretic investigations respecting continuous
girders are based on the assumption that the material is perfectly
elastic, and that the girder is of uniform section throughout its
whole length.
CASE I. — CONTINUOUS GIRDERS OF TWO EQUAL SPANS, EACH
LOADED UNIFORMLY THROUGHOUT ITS WHOLE LENGTH.*
Fig. 89.
251. Pressures on points of support — Points of inflexion —
Deflection. — Let I— AB = BC = the length of each span,
w = the load per linear unit of AB,
w' = the load per linear unit of BC,
RU R2, R3 = the reactions of the three points of support
A, B and C, respectively,
x — A/i =: the horizontal distance of any point P
from the left abutment,
y = hP = the deflection at that point,
M = the moment of resistance of the horizontal
elastic forces at P (59),
|3 = the inclination to the horizon of the tangent
to the curve at B,
* See Mr. Pole's paper on the " Investigation of general formulae applicable to the
Torksey bridge," Proc. Inst. C. E., Vol. ix., p. 261.
188 CONTINUOUS GIRDERS. [CHAP. IX.
I = the moment of inertia of any cross section
round its neutral axis, and consequently, a
constant quantity throughout the whole
length of the girder when the section of the
latter is uniform from end to end,
E = the coefficient of elasticity.
The forces which hold the segment A P in equilibrium are the
reaction of the left abutment, Rl ; the load wx uniformly distributed
over AP; the vertical shearing-strain at P, and the horizontal
elastic forces at the same place. Taking the moments of these
forces round the neutral axis at P, we have,
M = R,x — ^L (162)
Substituting for M its value in eq. 135,
Integrating this, and determining the constant by the consideration
that j? = tanfi when x = I, we have,
Integrating again, and determining the second constant by the
consideration that y = 0 when x = 0, we have,
(163)
which is the equation of the deflection curve from A to B.
At the point B, x = I and y = 0; substituting these values in
eq. 163, we have,
ton'3:=2lET(3W-8R') (164)
Applying a similar process to the second span, and remember-
ing that the angle |8 must in this case have a contrary sign, we
have,
CHAP. IX.] CONTINUOUS GIRDERS. 189
»
Again, taking moments round B, we have,
Rtl-^ = R3l-^ (166)
also,
RI + R2 + Rs = (w + w'}1 (167)
By solving these last four simultaneous equations we obtain the
reactions of the points of support, as follows : —
Rl = l2=*l (168)
R, = 1(10 + 1*0* (169)
Rz = lw'-wl (170)
At the points of contrary flexure the horizontal forces become
cipher. Hence, the distance of the point of inflexion in the left
span from A may be obtained from eq. 162, by making M = 0 and
substituting for Rl its value in eq. 168, as follows: —
x = ™± = lw-w'l (171)
w 8w
Similarly, the distance of the point of inflexion in the right span
measured from C,
The deflection y, in the left span, may be derived from eq. 163 by
substituting for tanQ its value in eq. 164, as follows: —
<173)
The value of I for each form of cross section may be obtained
from 31 and the succeeding articles by the aid of eq. 133.
The maximum strains in the flanges occur over the pier, and half
way between the abutments and the points of inflexion, and when
the latter are known, may be easily determined on the principles laid
down in the second and fourth chapters for calculating the strains
in independent girders; see eqs. 12 and 23 for girders with
braced webs; or 70, 82 and 107 for girders with continuous
webs.
190 CONTINUOUS GIRDERS. [CHAP. IX.
S5S. Both spans loaded uniformly. — If both spans have the
same load per running foot, w = u?', and we have
R! = R3 = jjfirf (174)
R2 = ^wl (175)
The distance of each point of inflexion from the near abutment,
x = |j (176)
Ex. The Torksey bridge is a continuous girder bridge in two equal spans, and was
erected by Mr. Fowler to carry the Manchester, Sheffield and Lincolnshire Railway
over the river Trent. Each span is 130 feet long in the clear, with a double line of
railway between two double-webbed plate main girders with cellular top flanges.
These main girders are 25 feet apart, with single-webbed plate cross-girders, 14 inches
in depth and 2 feet apart, attached to the lower flanges. The extreme depth of each
main girder is 10 feet. The depth from centre to centre of flanges is 9 feet 4f inches,
or ^th of each span. The gross sectional area of each top flange at the centre of
each span is 51 inches, and the net area of each lower flange is about 55 inches. The
thickness of each side of the web at the centre of each span is £ inch, increasing to -|
inch at the abutments and central pier.
The load on each span of 130 feet was estimated as follows : —
Tons. Tons.
Rails and chairs, 8
Timber platform, 15
Cross -girders, 27 177
Ballast, 4 inches thick, 35
Two main girders, . . . - . .92
Rolling load, as agreed upon by Mr. Fowler and Capt.
Simmons (Government Inspector), . . . .195
Total distributed load, 372 tons.
The strength of the Torksey bridge as a continuous girder was calculated by Mr.
Pole from the following data : —
The length of each span = 130 feet = 1,560 inches.
The total distributed load on the first span = 400 tons, or for each girder
200 tons.
The distributed load on the second span = 164 tons, or for each girder 82
tons.
The coefficient of elasticity is taken equal to 10,000 tons for a bar one inch
square.
CHAP. IX.] CONTINUOUS GIRDERS. 191
By eqs. 168, 169, and 170, the pressures of one main girder on the points of
support are as follows : —
R! = 82-375 tons.
R2 = 176-250 tons.
K3 = 23-375 tons.
By eq. 171, the distance of the point of inflection in the loaded span is 22 feet 11
inches from the centre pier. The moment of inertia = 372,500 by Mr. Pole's calcu-
lation. The distance of the top plates from the neutral axis = 64 inches ; that of the
bottom plates from the same axis = 56 inches, and the maximum strains in the
flanges of the longer segment, 107 feet long, are 4'55 tons compression per square
inch of gross area in the top flange, and 4 tons tension per square inch of net area in
the bottom flange. The deflection, with 222 tons distributed over one span, was 1-26
inches.
CASE II. — CONTINUOUS GIRDERS OF THREE SYMMETRICAL
SPANS LOADED SYMMETRICALLY.*
Fig. 90.
S53. Pressure on points of support — Points of inflexion —
Deflection. — Let Q be the centre of the centre span,
AB = CDr=/r= the length of each side span,
AQ = nl,
w = the load per linear unit on each side span,
w' = the load per linear unit on the centre span,
R! = the reaction of either abutment, A or D,
R2 = the reaction of either pier, B or C,
x •=. A/i •=. the horizontal distance of any point P from
the left abutment.
y = hP = the deflection at this point,
M = the moment of resistance of the horizontal elastic
forces at P (59),
* For the elegant investigation in 853 and 354 the author is indebted to William
B. Blood, Esq., sometime Professor of Civil Engineering in Queen's College, Galway.
192 CONTINUOUS GIRDERS. [CHAP. IX.
j3 = the inclination to the horizon of a tangent to the
curve at B or C,
I = the moment of inertia of any cross section round
its neutral axis, and consequently, a constant
quantity throughout the whole length of the
girder when the section of the latter is uniform,
E = the coefficient of elasticity.
It can be shown by the same process of reasoning as that adopted
in 251 that the equation of equilibrium for any point P in the
side span, AB, is
M=R,«-^ (177)
whence, as before,
8R') (178)
The equation of equilibrium for any point in the centre span is
M = R^+ R2(tf_Z)_W^-|)-|'(#_02 (179)
Substituting for M its value in eq. 135,
Integrating, and determining the constant by the consideration that
-j- — tanB when x = /, we have,
ax
El | = El tanfi + |fa(— 0 + ^(*-/)'-RA±^i(«'-P)
+ R,l(x — l) (180)
which is the equation of the deflection curve from B to C.
Since ~ = 0 when x — nl, we haye,
ax
_(»_1)R,} (181)
also
R1 + R2 = / {w + (n—l) w'} (182)
CHAP. IX.] CONTINUOUS GIRDERS. 193
From eqs. 178, 181, and 182, we obtain the reactions of the points
of support, as follows : —
(l-5n — 1-125) tg — (n—
i 3n-2
(l-5n — 0-875) l(> + (n* — 2n + l)w'
R* = l 3n-2
The distance of the point of inflexion in either side span from the
abutment is obtained from eq. 177 by making M = 0.
9R
x = ±^i (185)
w
The distances of the points of inflexion in the centre span from A
are obtained from eq. 179 by making M = 0, substituting for R,
its value in eq. 182, and solving the resulting quadratic, as follows : —
The equation for the deflection of the side spans is the same as
eq. 173. That for the deflection at the centre of the centre span
where x = nl, is obtained by integrating eq. 180 and determining
the constant by the consideration that y = 0 when x = /, as
follows : —
+ (n — 1)2 + £ I ten/3/ (n — 1) (187)
The value of I for each form of cross section may be obtained from
71 and the following articles by the aid of eq. 133.
354. Three spans loaded uniformly. — If the girder be loaded
uniformly throughout the three spans, w = wf, and the pressures
on the point of support become
- + 0-125)
-- <189)
194 CONTINUOUS GIRDERS. [CHAP. IX.
The distance of the point of inflexion in each side span from the
abutment is as before : —
(190)
w
The distances of the points of inflexion in the centre span from A
are as follows : —
(191)
If the radicle in eqs. 186 or 191 vanish, there will be no strain
at Q, and the centre span will be cambered throughout. If the
value of R! in eqs. 183 or 188 be negative, the ends of the girder
will be lifted off the abutments, owing to the excess of load on the
centre span.*
255. Maximum strains in flanges. — The maximum strains in
the flanges occur as follows : — in the side spans when the passing
load covers both side spans, leaving the centre span free from load ;
in the centre span, when the passing load covers it alone, leaving
both side spans free from load; and over either pier, when the
passing load covers the centre span and the adjacent side span,
leaving the remote side span free from load. When the lengths
of the component segments are determined, the strains in the
flanges may be calculated by eqs. 12 and 23 if the girders are
diagonally braced, or by eqs. 70, 82 and 107 if they are plate
girders. The hypothesis of the load being symmetrically disposed
on either side of the centre prevents us from finding the points of
inflexion when the segment over either pier is of maximum length ;
we have, however, a close approximation to its maximum length in
the case of a passing load covering all three spans, and if desirable,
a small extra allowance may be made for greater security. When
the maximum length of the segment over either pier is thus deter-
mined, the calculation for the strains in its flanges are made as
indicated in previous chapters, recollecting that each of these pier
segments supports not only its own proper load, but also the weight
of half the adjoining segments with their load, suspended from its
extremities by the vertical web.
* The reader is referred to the description of the Boyne lattice bridge in the Appendix
for a practical example of the application of the foregoing formulae.
CHAP. IX.] CONTINUOUS GIRDERS. 195
S56. Maximum strains in web — Ambiguity in calculation. —
Though we obtain by these means the maximum strains of either
kind to which the flanges are subject, it does not follow that we have
also got the maximum strains in the web. Let o, for example, in
Fig. 90, be the point of inflexion when the segment Ao is of maximum
length. Now this segment does not remain of this maximum length
while a train is passing from A to B, that is, while the maximum
strains are being produced in the web of Ao ; the point of inflexion
is much closer to A when the train first comes upon the bridge
(especially if the centre span happens to be traversed at the same time
by another train), and gradually moves forward towards B as the
train advances. It is incorrect therefore to calculate the maximum
strains in the web on the hypothesis that Ao is the length of the
segment while the load advances. The maximum strain in a diagonal,
at P for instance, takes place when the load covers A P, but the
point of inflexion is then really nearer A than the point o is, and the
maximum strain in the diagonal at P is therefore greater than if we
assume the segment constant in length during the advance of the
train. A similar or even greater uncertainty occurs in the centre
span, for there neither end of the segment is fixed.
857. Permanent load, shearing-strain. — When a continuous
girder supports a fixed load, the strains in the web are not modified
at the points of inflexion. The horizontal strains in the flanges
change from tension to compression, or vice versa, at these points,
but the vertical or diagonal strains are transmitted through the web
just as if no points of inflexion existed. The effect of contrary
flexure is merely this ; the horizontal increments of strain developed
in the flanges pull from the piers in place of thrusting towards the
centres of the component segments, and vice versa. Hence, when a
continuous girder of three, five, or any uneven number of spans,
is symmetrically loaded, the strains throughout the web of the
centre span are the same as if the centre span were an independent
girder supported at its extremities. This perhaps will be made
clearer from the consideration that the shearing-strain at any section
in the centre span, when the points of inflexion are symmetrical,
is equal to the weight between the section and the centre of the
196 CONTINUOUS GIRDERS. [CHAP. IX.
span, and this is the case whether there be any point of inflexion
or not. Thus, the shearing-strain at any point/, Fig. 90, is equal
to the load on fo' + that on o'Q; but if the central span were an
independent girder, resting on abutments at B and C and uniformly
loaded, the shearing-strain at / would equal the load on /Q, that
is, it would be the same as before.
859. Advantages of continuity — \o< desirable for small
spans with passing loads,, or where the foundations are
insecure. — The advantage of continuity arises from two causes;
first, from the smaller amount of material required in the flanges ;
secondly, from the removal of a certain portion of their weight from
the central part of each span to a position nearer the piers. The latter
is but a trifling advantage in continuous girders of moderate spans, say
under 150 feet, which' support heavy passing loads, for the part so
removed forms but a small proportion of the total weight. In the case
of a fixed load, however, the saving from this cause is considerable ;
but when the load is a passing train the advantages of continuity
are liable to be over-rated, especially in girders of small spans, for
on a little reflection it will be evident that, when the points of
inflexion move under the influence of the passing load, a greater
amount of material is required than if their position remained
stationary, and this moreover introduces the necessity of providing
for both tension and compression in those parts of the flanges which
lie within the range of the points of inflexion ; this latter objection
is perhaps of little consequence when wrought-iron is the material
employed. A subsidence of any of the points of support of a con-
tinuous girder will cause a change of strain whose amount it is quite
impossible to foresee, and which may seriously injure the structure
or perhaps render it dangerous. Hence, continuous girders should
be avoided where the foundations of the piers are insecure. In
bridges of large span, where the permanent load constitutes the
greater portion of the whole weight, the advantage of continuity is
very considerable. The position of each point of inflexion alters but
little with a passing load, and a considerable portion of the per-
manent weight, Avhich would otherwise rest at, or near the centre,
of each span, is brought close to the points of support.
CHAP. IX.] CONTINUOUS GIRDERS. 197
CASE III. — GIRDERS OF UNIFORM SECTION IMBEDDED AT BOTH
ENDS AND LOADED UNIFORMLY.
Fig. 91.
859. Strain at centre theoretically one-third* and strength
theoretically once and a half, that of girders free at the
ends. — When both ends of a girder are built into a wall so as to be
rigidly imbedded there, the tangent to the girder at its intersection
with the wall is horizontal, and the strains closely resemble those
which occur in the centre span of a continuous girder of three
spans when the load is so disposed that the tangents over the piers
are horizontal.
Let I = the span from wall to wall,
w = the load per linear unit,
M' = the moment of resistance of the horizontal elastic forces
at the intersection of the girder with the wall (59),
M = the moment of resistance of the horizontal elastic forces
at any cross section P,
x and y = the co-ordinates of P, measured from a as origin,
I = the moment of inertia of any cross section round its
neutral axis,
E = the coefficient of elasticity.
Taking moments round P (eq. 135),
M=-Elg = ^-^-M' (192)
Integrating, and determining that the constant = 0 from the con-
sideration that ~- — 0 when x = 0,
dx
_ . dy w.x3
E I -f- — —.
dx b
198 CONTINUOUS GIRDERS. [CHAP. IX.
Making x — /, we have -j- — 0, and
Substituting this value in eq. 192, we have,
72
At the points of inflexion, M = 0, and we have a* — Ix + ^ = 0,
o
whence,
x = I (g±-if ) = -211 J or -789Z (193)
The length of the middle segment = -578Z, and if the girder
be a flanged girder, the central strain in either flange (eq. 25)
(•578)2W2
— —
.
— 77-7 — = nj-j, ln which d = the depth of the
girder. This central strain is just ^rd of what it would be were
the ends merely resting on the wall, in place of being built therein.
From eq. 12, we find that the strain in either flange at the wall
= js^, which is just double the strain at the centre of the flanges,
and f rds of what would be the central strain from the same load
if the girder were merely resting on the walls. From this it
follows, that the strength of a girder of uniform section imbedded
firmly at both ends and loaded uniformly is theoretically once and a
half that of the same girder merely supported at the ends, and that
the points of greatest strain are at the intersections with the wall.
CASE IV. — GIRDERS OF UNIFORM SECTION IMBEDDED AT BOTH
ENDS AND LOADED AT THE CENTRE.
S6O. Strain at centre theoretically one-half, and strength
theoretically twice* that of girders free at the ends. —
Let W = the load at the centre of the girder, and let the other
symbols remain as before.
Taking moments round P (eq. 135),
"=-El = .-M' (194)
CHAP. IX ] CONTINUOUS GIRDERS. 199
Integrating, and determining that the constant = 0 from the
consideration that ~ = 0 when x = 0,
El$.= •!'•-&•
dx 4
Making x = ~, we have -^- — 0, and
Substituting this value in eq. 194, we have,
At the points of inflexion M = 0, and we have their distance
from the walls,
x = l- (195)
The length of the middle segment = ~, and its central strain is
just J of what it would be if the ends of the girder were not
imbedded in the wall but merely resting thereon. The strain at
the wall also is equal to the central strain; consequently, the
strength of a girder of uniform section imbedded firmly at both
ends and loaded at the centre is theoretically twice that of the
same girder merely supported at the ends. Mr. Barlow's experi-
ments on timber, however, do not corroborate this theory, as he
found the strength of an imbedded beam loaded at the centre to
be only 1-| times that of a free beam, and fracture always took
place at the centre, the ends being comparatively little strained.*
Our theory is doubtless defective in supposing that the horizontal
fibres at the wall are in the same state of strain as if the girder
were really a continuous girder in three spans, for in the latter
case the girder is bent downwards in each of the side spans,
whereas, when imbedded in the walls, the ends which correspond
to these side spans are horizontal, and consequently, the points of
inflexion are really nearer to the walls than in a truly continuous
girder.
* Strength of Materials, pp. 32, 136.
200
QUANTITY OF MATERIAL
[CHAP. x.
CHAPTER X.
QUANTITY OF MATERIAL IN BRACED GIRDERS.
CASE I. — SEMI-GIRDERS LOADED AT THE EXTREMITY, ISOSCELES
BRACING.
861. Web.
Let W = the weight at the extremity,
I = the length of the semi-girder,
d = its depth,
6 = the angle the diagonals make with a vertical line,
/ — the unit-strain,
Q = the cubical quantity of material in the diagonals,
Q' =: the cubical quantity of material in either flange.
Fig. 92. The cubical quantity of material
required for the diagonal bracing is
equal to the sum of the products of
the length and section of each brace.
When the triangles are isosceles and
the load is a single weight, the sec-
tion, if proportional to the strain, is
the same for all the diagonals, and
the quantity of material is therefore
equal to the product of their aggre-
gate length by their common section.
The line ATB, Fig. 92, is equal in
length to the sum of the several
diagonals; expressing its length in
terms of / and 0, we have
A B = l.cosecO
The section of each brace is equal to the total strain passing
CHAP. X.] IN BRACED GIRDERS. 201
WsecO
through it divided by the unit-strain, = — ^ — (eq. 110). Multi-
plying this by the foregoing value for the length, we have,
W7
Q= ~ secQ . cosecQ (196)
S63. Flanges. — The quantity of material in the flanges is most
conveniently deduced from the principles stated in Chapter II. as
VW
follows: — The sectional area of either flange at the wall = -^
Clf
(eq. 7), and when the girder is of uniform strength gradually
diminishes towards the extremity as the ordinates of a triangle (SO).
Hence, the quantity of material in one flange equals its sectional
area at the wall multiplied by „, and we have,
(197)
CASE II. — SEMI-GIRDERS LOADED UNIFORMLY, ISOSCELES
BRACING.
. Web, length containing a whole number of bays. —
Let W = the total weight resting on the girder,
n = the number of bays in the longest flange, supposed a
whole number, and the other symbols as in Case I.
When the bracing is formed of isosceles triangles the length of
one bay equals 2d.tan9, whence,
l = 2nd.tanO. (198)
The quantity of material that the weight at any given apex
would require in the bracing, if it alone were supported by the
girder, may be obtained from eq. 196 by substituting for W and
(W\
= — ) , and the distance of the weight
from the wall. The quantity required for the whole load is equal
to the sum of the quantities required for the separate weights.
Hence, recollecting that the weight on the last apex equals half
202 QUANTITY OF MATERIAL [CHAP. X.
that on each of the other apices (144), we have, when there is no
half bay in the length, that is, where n is a whole number,
Q = 2d.tan9 (1 + 2 + 3 + . . . n) — secB.cosecO
W
= -f-nd.tanQ.secO.cosecO.
Substituting for nd.tanO its value in eq. 198, we have,
W/
Q = ^secO.cosecB (199)
864. Web3 length containing: a half-bay. — When the length
contains a half-bay, the quantity of material in the bracing, derived
from eq. 196,
W7 W/72
Q = ^ secO.cosecO + -^- sec*0.tanO. (200)
*J *Jl
365. Flanges. — From eq. 11 the area of either flange at the
VW
wall = X-TJ, and diminishes towards the extremity as the ordinates
*J(*
of a parabola, but from the well-known properties of the parabola
the area of A B C, Fig. 7, equals one-third of the circumscribed
rectangle. Hence, the quantity of material in either flange equals
its area at the wall multiplied by ~, that is,
o
W/2
CASE III. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
AT AN INTERMEDIATE POINT, ISOSCELES BRACING.
366. Quantity of material in the web is the same for each
segment. — Let W = the weight resting on the girder,
/ = its length, and the other symbols as in Case I.
Let the weight divide the girder into segments containing
respectively m and n linear units, as in Fig. 52. The strains
throughout the girder will in no respect be altered if we conceive
it inverted, resting on a pier at W, and loaded with — W at the
CHAP. X.] IN BRACED GIRDERS. 203
right extremity, and with -, W at the left. Each segment will then
become a semi-girder loaded at its extremity. Hence, the quantity
of material in the bracing of each segment = — -^— seed . cosecO
(eq. 196). The quantity of the material in the bracing of both
segments together is equal to twice this, that is,
Q = ^™ secO . cosecd (202)
JL
If the weight be at the centre, equation 202 becomes
W7
4 = ^«00.MiMa (203)
S67. Flanges. — From eq. 20, the sectional area of either flange
?7i7?^^»
at the point where the weight rests = . , , , and diminishes gra-
dually towards each extremity as the ordinates of a triangle (35).
Hence, the quantity of material in one flange equals its area at the
weight multiplied by ^, and we have,
(204)
If the weight be at the centre, eq. 204 becomes,
W/2
* = (205)
CASE IV. — GIRDERS SUPPORTED AT BOTH ENDS AND LOADED
UNIFORMLY, ISOSCELES BRACING.
868. Well, length containing an even number of bays. —
Let W = the total weight on the girder,
I = the length, and the other symbols as in Case I.
In order to avoid unnecessary minuteness in this case I shall first
assume that the number of bays in the half-length is a whole number,
in other words, that the length contains an even number of bays.
Let us consider each half of the girder by itself; the vertical
forces which act upon each half are the upward reaction of its
204 QUANTITY OF MATERIAL [CHAP. X.
abutment, and the downward pressure of the weights between the
abutment and the centre. The former pressure, if acting alone,
would require a certain amount of material for the bracing, obtained
by eq. 196, while the weights, leaving the reaction of the abutment
out of consideration, would require an amount of material which may
be obtained from eq. 199. The latter forces tend to relieve the
strain produced by the reaction of the abutment; consequently,
the true quantity of material required is equal to the difference of
the amounts which would be required were each set of forces to act
independently of the other. Hence, subtracting eq. 199 from 196,
and bearing in mind that W and I have twice the value they had
in the semi-girder, we have the quantity of material in the web of
the whole girder,
W7
Q = ^ secO.cosecO (206)
that is, half the quantity that would be required if all the weight
were concentrated at the centre.
269. Web, the length containing: an odd number of bays.—
If the half-length contain a half- bay, the quantity of material in the
bracing is obtained by subtracting eq. 200 from eq. 196, that is,
W7 W^72
Q = ^sec6.cosecO—~-sec*0.tanO (207)
37O. Flanges. — From eq. 25 the sectional area of either flange
VW
at the centre of the girder zr — -, and diminishes towards either end
oja
as the ordinates of a parabola (4?). But the area of Fig. 23 equals
two-thirds of the circumscribed rectangle ; hence, the quantity of
material required for either flange equals its central section multiplied
2
by ^/, and we have,
o
W/2
V - Jg (208)
which is two-thirds of the quantity that would be required if all
the weight were concentrated at the centre.
CHAP. X.]
IN BRACED GIRDERS.
205
CASE V. — SEMI-GIRDERS LOADED AT THE EXTREMITY,
VERTICAL AND DIAGONAL BRACING.
Fig. 93. 2*1. Web. — When every alter-
nate brace is vertical, as in Fig. 93,
we must divide the material in the
web into two parts, namely, that in
the vertical, and that in the diagonal
bracing.
Let Q = the quantity of material
in the diagonals,
Q" = the quantity of material
in the verticals, and
the other symbols as
before.
The quantity of material required for the diagonal bracing is as
before (eq. 196),
W7
(209)
Q = — - secO . cosecO
The strain transmitted through each vertical = W; hence, its
W
sectional area — — r. Multiplying this by the aggregate length
of the verticals (= l.cotd), we have,
Q" = cotB.
(210)
CASE VI. — BOWSTRING GIRDERS UNIFORMLY LOADED.
. Flanges. — When a bowstring girder is uniformly loaded,
the strains are nearly uniform and equal throughout both flanges
(31O) ; hence, we can find a close approximation to the quantity
of material by multiplying the length of each flange by its sectional
area.
Let W — the total weight uniformly distributed over the girder,
I = the length of the string,
nl = the length of the bow,
206
QUANTITY OF MATERIAL
[CHAP. x.
d = the depth of girder at the centre,
Q' = the quantity of material in the string,
Q" = the quantity of material in the bow,
/ = the unit-strain.
The strain at the centre of either flange = -77-7 (eq. 25) ; hence,
VW
the sectional area of the flange = — ,; multiplying this latter quan-
tity by the respective lengths of the string and bow, we have
WZ2
«' = ^ (2H)
('212')
j, ,/. \H*.UJ
S?3. The following table contains the corresponding values of
y and n, the depth being expressed in fractional parts of the length
d
1
n
i
1-158
*
1-073
*
1-040
r'o
1-027
TV
1-019
t\
1-014
TV
1-010
n, or the ratio of the length of the bow to the length of the string,
is thus found.
Let X = the half span = ^ »
r = the radius of the bow,
0 = the angle the bow subtends at the centre of the
circle.
CHAP. X.]
IN BRACED GIRDERS.
Fig. 94.
207
_ length of bow _ rO
~~
(a)
also,
whence,
again,
whence,
Substituting in eq. (a) these values for r and 6, we have,
X2 + cZ2 -i d _ (\ , d\ _\d ,n „.
n — .tan 5T %5 Xl" X v/1**)
whence we can obtain the values of n corresponding to different
values of y.
874. Quantity of material in the bracing independent of
depth — Weights of railway girders up to 2OO feet span are
nearly as the squares of their length. — The reader will observe
that the depth of the girder does not enter into those equations
which express the quantity of material required in the bracing,
whereas it enters into the denominator of those which express
the quantity of material in the flanges. Hence, we conclude
that altering the depth of braced girders does not affect the
amount of bracing (18) ; but the quantity of material in the
flanges varies inversely as the depth, and consequently, the deeper
a girder is made the greater will be the economy, theoretically
speaking. In practice, the additional material required to stiffen
long struts generally defines the limit to which this increase of
depth can be judiciously extended; but of this in succeeding
chapters.
208 QUANTITY OF MATERIAL IN BRACED GIRDERS. [CHAP. X.
It will also be observed that, when the ratio of depth to length
is constant, the quantity of material varies as VW, or if W varies as
/, as Z2. Consequently, when such girders are of small weight
compared to the load, and when the latter is proportional to the
length, the weight of the girders will vary very nearly as the
square of their length — which rule is approximately true for rail-
way girders up to 200 feet span.
CHAP. XI.] ANGLE OF ECONOMY. 209
CHAPTER XL
ANGLE OF ECONOMY.
375. Jingle of Economy for Isosceles bracing: is 45°. — On
examining those equations in the last chapter which express the
quantity of material required for the vertical web of girders whose
bracing consists of isosceles triangles, we find that they may all be
expressed by one general equation,
Q = KsecO.cosecO
in which K for each case is a constant quantity depending upon
the length, weight, and unit-strain. Q is therefore proportional to
2
the variable quantity secO.cosecO, or to its equivalent, . „, which
is a minimum when 0 = 45°. This proves that the angle of 45°
is the most economical inclination for the diagonals of isosceles
bracing, and it is to be observed that certain of the diagonals
beingin compression, and therefore practically requiring a greater
amount of material to stiffen them than others, does not materially
affect this conclusion ; for, let the compression diagonals take m
times the quantity of material they would require on the supposition
that they were subject to tension in place of compression, then,
since every alternate diagonal is in compression when the load is
stationary, the foregoing expression becomes
Q = --tl ^secO.cosecO
but the variable part of this expression is secO.cosecO as before,
and therefore the angle of economy is 45°.*
8*6. Angle of economy for vertical and diagonal bracing
is 55°. — The angle of economy in girders with vertical and
diagonal bracing differs from that in girders whose webs are formed
of isosceles triangles. From eqs. 209 and 210. we find that the
quantity of material in the bracing may be expressed as follows : —
Q + Q" rr K (secO.cosecO + cotO).
* Mr. Bow first drew attention to the fact that 45° is the angle of economy for
isosceles bracing ; see his Treatise on Bracing. Edinburgh, 1851.
210
ANGLE OF ECONOMY.
[CHAP. xi.
It is necessary to equate the differential coefficient of the bracketed
part of this equation to cipher in order to find the value of 9 which
makes Q + Q/; a minimum. Doing so, we have,
cosec9.sec9.tan9 — sec9.cosec9.cotB — cosec29 = 0,
dividing by cosec9.sec9 and transposing,
tan9 = 2cot9
whence,
tan9 = VT, and 9 = 54° 44' 8-2" = 55° nearly,
which therefore is the angle of economy for this form of bracing,
and has moreover the merit of forming lozenge-shaped openings,
which have a more agreeable appearance than square ones.
577. Isosceles more economical than vertical and diagonal
bracing^. — The superior economy of the isosceles over the vertical
and diagonal system of bracing will be now apparent, for the quan-
tity of material required in the latter exceeds that in the former by
an amount never less than Q", and exceeds Q" when 9 differs
from 45°.
578. Trigonometrical functions of 9. — The following table
contains the value of different trigonometrical functions of 9.
Angle
of
bracing, 0.
sec6.
secQ.cosecQ.
cote.
secQ.cosecQ + cotQ.
tanQ.
20°
1-064
311
2747
5-857 .
•364
25°
1-103
2-61
2-144
4-754
•466
30°
1-154
2-31
1-732
4-041
•577
35°
1-221
2-13
1-428
3-557
•700
40°
1-305
2-03
1-192
3-222
•839
45°
1-414
2-00
1-000
3-000
1-000
50°
1-515
2-03
•839
2-869
1-192
55°
1-743
2-13
•700
2-829
1-428
60°
2-000
2-31
•577
2-886
1732
65°
2-369 '
2-61
•466
3-076
2-144
70°
2-924
311
•364
3-474
2747
CHAP. XI.]
ANGLE OP ECONOMY.
211
S79. Relative economy of different kinds of bracing —
Continuous web theoretically twice as economical as a
braced web. — By means of this table we can at once compare the
relative economy of different descriptions of bracing as follows :
Values of 9.
Value of Q.
Comparative quantities
of material
required in web.
Isosceles bracing, - - 1
? = 45°
Q = 2-00 K
100
Ditto (Warren's girder),
9 = 30°
Q = 2-31 K
115-5
Vertical and diagonal bracing,
3 = 55°
QXQ" = 2-83 K
141-5
From this it appears, that equilateral bracing ("Warren's girder")
requires 15^ per cent., and vertical and diagonal bracing of the
best form requires 41^ per cent., more material in the web than
isosceles bracing at an angle of 45°.
If we compare equations 203 and 206 with the equations in 54
which represent the theoretic quantity of material in a continuous
web, we find that the most economical form of braced web, namely,
isosceles bracing at an angle of 45°, requires just double the
quantity of material that the continuous web requires if made as
thin as theory alone would indicate. In practice, however, the
braced web is generally the most economical, as will be shown
hereafter in the chapter on the web.
212
TORSION.
[CHAP, xii,
CHAPTER XII.
TORSION.
Fig. 95.
38O. Twisting moment. — Let one end of a horizontal shaft
be rigidly fixed and let the free end have a lever, L, attached at
right angles to the axis. A weight, W, hung at the end of this
lever, will twist the shaft round its axis and fibres, such as a&,
originally longitudinal and parallel to the axis, will now assume a
spiral form, ad, like the strands of a rope. Radial lines, such as
cb, in any cross section, will also have moved through a certain
angle, bed, which experiments prove to be proportional,
1°. to ab, the distance of the section from the fixed end,
2°. to L, the length of the lever,
3°. to W, the weight,
provided the shaft be not twisted beyond its limit of elastic
reaction. If we consider any two consecutive transverse sections
of the shaft, we find that the one more remote from the fixed end
will be twisted round a little in advance of the other, and this
movement tends to wrench asunder the longitudinal fibres by one
of the sections sliding past the other. This wrenching action, it
will be observed, closely resembles shearing from transverse pressure
CHAP. XII.] TORSION. 213
(14). It is clear that, the farther the fibres are from the axis the
greater will be the arc through which they are twisted, and the
greater, therefore, will be their elastic resistance to wrenching, and
the greater also will be the leverage which they will exert, and we
may conceive, at least in shafts of circular, polygonal, or square
sections, the elastic reactions of the fibres replaced by a resultant
equal to their sum and applied in a linear ring round the axis,
whence, we have the twisting moment of the weight,
WL= F&
where F = the annular resultant of all the elastic reactions,
& =. the mean distance of this annular resultant from the
axis of the shaft.
F is proportional in shafts of different sizes, but similar in section,
to the number of fibres in the cross section, that is, in solid shafts
to the square of the diameter, and & is evidently proportional to
the diameter. Hence, we obtain the following relations.
881. Solid round, square^ or polygonal shafts — Coefficient
of torsi on a I nipt are, T. —
W = ™ (214)
d=<\^ (215)
where W — the breaking weight by torsion,
L = the length of the lever, measured from the centre of
the shaft,
d = the diameter of the shaft, if round ; or its breadth, if
square or polygonal,
and T is a constant, which must be determined for each material
by finding experimentally the breaking weight of a shaft of known
dimensions and similar in section to that whose strength is required.
The constant, T, may be called the Coefficient or modulus of
torsional rupture of that particular material and section from
which it is derived, and equals the breaking weight of a shaft of
d3
similar section in which the quantity j- = 1.
888. Hollow shafts of uniform thickness. — The number of
214
TORSION.
[CHAP. xii.
fibres in the cross section of a hollow shaft is proportional to the
product of the diameter by the thickness, and we have,
W = T<^ (216)
where t = the thickness of the tube and the other symbols are as
before.
883. Coefficients of torsional rupture for solid round
shafts. — The following table contains the values of T, or the
coefficients of torsional rupture, for solid round shafts ; these are
the breaking weights of shafts one inch in diameter and whose
length, L, is also one inch ; hence, in using these coefficients in the
preceding equations, all the dimensions should be in inches.
COEFFICIENTS OF TORSIONAL BUPTUBE FOB SOLID BOUND SHAFTS.
MATERIAL.
Initial of
Experimenter.
Value of T
in Ibs.
Cast-iron, _......
D
5,400
Wrought-iron,
9,800
Steel, Bessemer,
K
15,000
Do., Crucible, hammered,
K
17,000
Ash,
B
274
Elm, -
B
274
Larch,
B
190 to 333
Oak,
B
B
B
451
98 to 157
118
Spruce Fir,
B. Bouniceau, Eanlcines Machinery and Millwork, p. 479,
D. Dunlop, Tredgold on the Strength of Cast-iron, p. 99.
K. Kirkaldy, Experiments on Steel and Iron by a Committee of Civil Engineers.
Ex. 1. From experiments made by Mr. Kirkaldy for a "Committee of Civil
Engineers," it appears that 3,300 Ibs. at the end of a 12-inch lever will twist asunder
a round bar of Bessemer steel T382 inch in diameter ; what is the value of T ?
CHAP. XII.] TOKSION. 215
Here, W = 3,300 Ibs.,
L = 12 inches,
d = 1*382 inches.
Answer (eq. 214). T = ™= = S.300XU =
rf3 1-382]3
Ex. 2. What should be the diameter of a wrought-iron screw-propeller shaft, the
length of the crank being 13 inches and the pressure 15,000 Ibs., taking 8 as the factor
of safety ?
Here, W = 15,000 Ibs.,
L = 13 inches,
T = 9,800 Ibs.
Answer (eq. 215). d = \J -y- == ^159 = 5 '42 inches,
Ex. 3. What should be the diameter of a wrought-iron crane shaft, the radius of the
wheel being 16 inches, and the pressure at its circumference 300 Ibs., taking 10 as the
factor of safety ?
Here, W = 300 Ibs.,
L = 16 inches,
T = 9,800 Ibs.
Answer (eq. 215). d = 4=— = -V^4'9 = 17 inches.
284. Moment of resistance of torsion. — The following more
exact method of investigating torsional strain resembles that applied
to transverse strain in 69, and, like it, is based on the assumption
that the law of uniform elastic reaction is true, that is, that the
fibres exert elastic forces which resist twisting in proportion to
their change of length, and (in circular sections at least) directly
therefore as their distance from the central axis. Suppose the
shaft composed of longitudinal fibres of infinitesimal thickness, and
let us confine our attention to any given cross section represented
by Fig. 96.
Fig. 96.
216 TORSION. [CHAP. xn.
Let W = the weight producing torsion at the end of the lever L,
L = the length of the lever, measured from the axis of
the shaft,
p = the distance of any fibre in the given cross section,
measured radially from the axis,
/ = the torsional unit-strain exerted by fibres in the
same section at a distance c from the axis, that is,
the resistance of the fibres to being twisted or
shorn asunder referred to a unit of sectional surface,
c = the distance from the axis at which the unit-strain/is
supposed to be exerted,
0 = the angle between the line p and a horizontal diameter
of the section,
r = the radius vector of the curve which bounds the given
section.
According to our assumption the torsional unit-strain exerted by fibres
at the distance p from the axis will = — ; if the thickness of a little
0
element of these fibres measured radially = dp (differential of p,) and
if its width = pd0, the area of the element, shaded in the figure,
will = pdpdfl, and the resisting force exerted by it will = - p2dpd0 ;
the moment of this round the axis = - p3dpd0, and the integral of
0
this, within proper limits, is the sum of the moments round the axis
of all the elastic forces in the given section which resist torsion,
called the Moment of resistance to torsion of that particular section,
and this balances W L, or the twisting moment of W. We can
obtain the moment of resistance of the little triangle in the figure
by integrating the foregoing expression from p = 0 to p =r. Doing
this, we find the moment of resistance of the little triangle = £• r*dO,
and therefore the moment of resistance of the whole section can be
obtained by integrating this from 0 = 0 to 9 = Sir, as follows,
(217)
CHAP. XII.] TORSION. 217
885. Solid round shafts. — In the case of round shafts the
radius vector r is constant, whence, from eq. 217,
WL = 5^ ' (218)
If / — the torsional unit-strain exerted by fibres at the circum-
ference, c = r, and we have,
(219)
886. Hollow round shafts. — The moment of resistance of a
ring is equal to that of the outer circle minus that of the inner
one, whence, from eq. 218,
Where r = the external radius,
rl •=. the internal do.
If / = the torsional unit-strain exerted by fibres at the circum-
ference, c = r, and we have,
-V) (220)
If t = the thickness of the ring, TI —r — t, whence, by substitution,
W L =
If the thickness be small compared with the radius, the last three
terms may be neglected, and we have,
W L = 2vfr2t = 6'28/r2* (221)
We may perhaps get a clearer conception of the strains in a hollow
round shaft by imagining the tube to be formed of a series of
diagonal bars forming right-handed coils in one direction, and crossed
by other bars forming left-handed coils in the opposite direction, so
as to produce a spiral lattice tube, in which, however, the bars in
each series are so close together as to touch each other, side by side,
and thus form two continuous tubes. The effect of twisting this
double tube will be to extend one set of coils and compress the other
in the direction of their length, and this will tend to make the
tension coils collapse inwards towards the axis of the tube, and
force the compression coils outwards, but these tendencies, being
218
TORSION.
[CHAP. xn.
equal and in opposite directions, will balance each other. We may go
further and imagine the coils springing at an angle of 45° from any
given cross section of the tube, and therefore at right angles to each
other, and if we suppose that the same piece of material can sustain
without injury strains of tension and compression passing through
it at right angles to each other, we have the section opposed to
.,,
either tension or compression =r
where r = the radius of the tube,
t — the thickness of the tube.
If / = the unit-strain of tension or compression indifferently, we
have the twisting moment of the weight,
W L = 2irfr*t
which is the same as equation 221.
3§7. Solid square shafts.—
Fig. 97. Let a = half the side of the square.
The radius vector r = asecO as far as one
quarter extends, that is, from 9 = 0 up to 8
= — ; hence, carrying the integral over the
triangle ABC, and multiplying by 8 to com-
plete the whole section, we have from eq. 217,
W L =
sec*0 . dO =
sec*0 .
and finally,
. dtanQ =
-. 8/«4
o
if / — the torsional unit-strain exerted by the extreme fibres
in the corners, c = \/2a, and we have,
If d = the side of the square, eq. 222 becomes,
W L = |^ = 0-236/tf3 (223)
Comparing eqs. 219 and 222, we find that the moments of resistance
CHAP. XII.] TORSION. 219
to torsion of the solid square shaft and the solid inscribed circle
are in the ratio of — — • = 1*2.
ST
The foregoing theory of the strength of square shafts is based
on the hypothesis that the ratio - is a constant quantity at different
c
points of the cross section, but this is true for circular sections
only, and Professor Rankine gives the following equation for the
strength of solid square shafts on the authority of M. de St. Venant,
who has investigated the subject theoretically with great care.
W L = 0-281/d3 (224)
This, it will be observed, makes the strength of a solid square
shaft nearly 20 per cent, higher than eq. 223.
220 STRENGTH OF HOLLOW [CHAP. XIII.
CHAPTER XIII.
STRENGTH OF HOLLOW CYLINDERS AND SPHERES.
8§§. Hollow cylinders — Elliptic tubes. — The strains in
hollow cylinders from fluid pressure, either within or without, may
be investigated as follows.
Fig. 98. Let d = the diameter of the cylinder,
t = the thickness of metal,
p =. the fluid pressure on each unit of
surface (generally in Ibs. or tons
per square inch),
/ = the tangential unit-strain, either of
tension or compression, according
as p is internal pressure tending to burst the cylinder,
or external pressure tending to make it collapse.
Let Fig. 98 represent a thin slice or cross section of a cylinder,
the thickness of the slice being one unit measured at right angles
to the plane of the paper, and let A B represent an imaginary
plane through the diameter. Suppose the lower half of the fluid
below this plane converted into a solid like ice — an hypothesis
which will not affect the conditions of equilibrium in any way —
then, the pressure exerted by the upper half of the fluid on the
surface, A B, of the lower half is obviously equal to pd, and this
pressure tends to separate the upper half of the cylinder from the
lower half by tearing the metal at A and B. Hence, the tensile
strain at either A or B = pd, that is,
2ft = pd (225)
The compressive strain due to external pressure, of the same
intensity as before, is equal and opposite to the tensile strain just
found, for we may conceive the solidified half cylinder removed and
a strong plate A B substituted for it, in which case the pressure on
the under surface of the plate will balance that on the outside of the
upper semi-cylinder as before. The same result may be arrived at
CHAP. XIII.] CYLINDERS AND SPHERES. 221
in another way. Let a cylinder subject to internal pressure, as in
the first case, be immersed in a larger vessel, and let fluid be forced
into the latter until its pressure equals that within the cylinder, in
which case the previous tangential tensile strain due to internal
pressure will be cancelled, since the pressures inside and out
balance each other. Now, let the fluid inside the cylinder be
withdrawn and, the balance being destroyed, a tangential com-
pressive strain will result, equal and opposite to the tensile strain
which existed before the cylinder was immersed.
Ex. What should be the thickness of the plates of a cylindrical boiler, 6 feet in
diameter and worked to a pressure of 50 Ibs. steam per square inch, in order that the
working tensile strain may not exceed 1*67 tons per square inch of gross section ?
Here, d = 72 inches,
p = 50 Ibs. per square inch of surface,
/ = 1'67 tons = 3741 Ibs. per square inch of section.
Supposing the material equally capable of resisting tension and
compression, the strength of a cylinder subject to external pressure,
like the flue of a Cornish boiler, is theoretically the same as if it
were subject to an equal internal pressure. Practically, however,
the strength is much less, owing to the flue not being a perfect
circle in cross section. If the outside shell be not a perfect circle,
the tendency of internal pressure will be to render it more so,
whereas, with the flue, the tendency will be to increase the defect
and cause collapse, and Sir William Fairbairn has deduced from
an extensive series of experiments the following empirical rule for
calculating the strength of wrought-iron tubes, such as boiler flues,
within the limits of length which occur in ordinary practice.*
p = 806,300^ (226)
Let
where p = the collapsing pressure in Ibs. per square inch of surface,
t — the thickness of the metal in inches,
/ = the length of the tube in feet,
d = the diameter in inches.
* Useful Information for Engineers, 2nd series.
222 STRENGTH OF HOLLOW [CHAP. XIII.
Ex. What is the collapsing pressure of a flue 10 feet long, 36 inches in diameter,
and composed of ^ inch iron plates ?
Here, t = 0'5 inch,
Id = 36 X 10 = 360,
logp — log 806,300 + 219 log 0-5 — log 360
= 5-9064967 + 219 X 1-69897 — 2-5563025 = 2'6909385.
Answer, p = 491 Ibs.
491
The safe working pressure for a land boiler would be =82 Ibs.; for an ordinary
6
4.Q1
marine boiler in which salt water is used, Ili = 61 Ibs.
8
It will be observed that the strength varies inversely as the
length, and Sir William Fairbairn found that "by introducing
rigid angle or T iron ribs (in practice from 8 to 10 feet apart,)
round the exterior of the flue, we vertically decrease the length
and increase the strength in the same proportion. Two or three
such rings on the flues of boilers, constructed of plates equal in
thickness to those of the shell, will usually render the resistance to
collapse equal to the bursting pressure of any other part of the
boiler." It was also found that the ordinary longitudinal lap-
joints in boiler flues were weaker than butt joints in the ratio of
about 7 to 10, and Sir William Fairbairn recommends that tubes
required to resist external pressure should be formed with longitu-
dinal butt joints with covering strips outside riveted to both plates.
Elliptical tubes are obviously very weak for resisting external
pressure, and it appears from Sir William Fairbairn's experiments
that their strength is the same as that of the osculating circle at
the flattest part of the ellipse ; thus, if a and b are the major and
minor semi-axes of the ellipse, the diameter of the cylinder of
2a2
equal strength will equal -j- - If, for example, the ellipse be 6 X 4
feet, the diameter of the cylinder of equal strength will equal
2-±* = 9 feet.
2
889. Cylinder ends. — The flat ends of cylinders sustain a
total pressure equal to their area multiplied by the pressure per
unit of surface, that is,
total end pressure = — ~- (227)
CHAP. XIII.] CYLINDERS AND SPHERES. 223
where p = the pressure per square unit of surface,
d = the diameter.
This end pressure is sustained by the rivets or bolts which connect
the ends of the cylinder to the sides, and if t = the thickness
of metal in the latter, the longitudinal tensile unit-strain in the
cylinder,
/=Sf + r*=^ (228)
Comparing this with eq. 225, we find that the longitudinal unit-
strain in a cylinder is one-half the tangential unit-strain. If the
cylinder be a boiler with internal flues, the end area is diminished
by the sectional area of the flues, which latter moreover support a
large share of the end pressure, so that the longitudinal unit-
strain in the shell is greatly reduced. Stay rods connecting the
ends above the flues reduce this longitudinal strain still more, so
that little anxiety need be felt about the transverse joints of the
shell giving way. The longitudinal joints of the shells of high-
pressure boilers are generally double-riveted and the cross joints
either single or double-riveted.
39O. Hollow spheres. — We may conceive, as in the case of
the cylinder already investigated, an imaginary plane passing
through the centre of the sphere and dividing it into two equal
parts. The fluid pressing on the surface of this plane tends to
tear asunder the sphere along the circle formed by its intersection
with the plane. Hence, if
d = the diameter of the sphere,
t = the thickness of metal,
p = the fluid pressure per square unit of surface,
/ z= the tangential unit-strain,
wehave'
reducing,
4/£ = pd (229;
Comparing this with eq. 225, we find that a sphere is twice as strong
as a cylinder of the same diameter and thickness of metal, and that
therefore the ends of egg-ended boilers are their strongest part.
224 CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
CHAPTER XIV.
CRUSHING STRENGTH OF MATERIALS.
891. Mature of compressive strain. — In most of the foregoing
theoretic investigations it has been tacitly assumed that the tensile or
compressive strength of any material is proportional to its sectional
area, whatever that may be. This, however, is not always true of
compressive strains, and one of the first difficulties which the
student encounters, when seeking to reduce theory to practice, is
the necessity of providing in struts or pillars not only against
absolute crushing of the material, which in reality rarely occurs,
but more especially against flexure and buckling, to resist which a
greater amount of material is generally required than theory alone
might seem to indicate. To understand the matter clearly we
must recollect that the mode in which a pillar fails varies greatly,
according as it is long or short in proportion to its diameter. A
very short pillar— a cube, for instance, of wrought -iron, timber, or
stone — will bear a weight nearly sufficient to upset, to splinter, or
to crush it into powder; while a still shorter pillar — such as a
penny, or other thin plate of ductile metal — will often bear an
enormous weight, far exceeding that which the cube will sustain,
the interior of the thin plate being prevented from escaping from
beneath the pressure by the surrounding particles. Alluding to
his experiments on copper, brass, tin, and lead, Mr. Rennie
observes : — " When compressed beyond a certain thickness, the
resistance becomes enormous,"* and I have observed the same
thing in a very marked degree when experimenting on cubes of
cast zinc which slowly spreads out like a plastic material as the
strain increases. We can thus conceive how stone or other materials
in the interior of the globe withstand pressures that would crush
them into powder at the surface, merely because there is no room
* Phil. Trans., 1818, p. 126.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 225
for the particles to escape from the surrounding pressure. A long
thin pillar on the other hand, such as a walking cane, will yield by
flexure long before it is crushed, and if the bending be carried so
far as to break the pillar, the fracture will resemble that due to
transverse strain. Hence, it is convenient to subdivide the results
of compressive strain into flexure and crushing.
393. Flexure— Crushing;— Buckling;— Bulging- — Splintering.
— Flexure is the bending or deflection of a pillar whose length is
very considerable in proportion to its thickness or diameter.
Crushing may be subdivided into buckling, bulging, and
splintering.
(a.) Buckling is the undulation, wrinkling, or crumpling up, usually
of a thin plate of a malleable material. Buckling is frequently
preceded by flexure ; when, for instance, long tubes of plate-iron
are compressed longitudinally, they first deflect, and finally fail by
the buckling or puckering of a short piece on the concave side.
(b.) Bulging is the upsetting or spreading out under pressure of
ductile or fibrous materials, such as lead, wrought-iron and timber,
also of many semi-ductile crystalline metals, such as cast-brass or
zinc.
(c.) Splintering is the splitting off in fragments of highly
crystalline, fibrous, or granular materials, such as cast-iron, glass,
timber, stone and brick ; the splintering of granular and vitreous
materials is often abrupt and terminates in their being crushed to
powder, while even the most crystalline metals are to some extent
ductile and therefore bulge slightly before they splinter. Again,
some materials, such as glass, form numerous prismatic splinters ;
others, like cast-iron, form two or more wedge-shaped or pyramidal
splinters, the plane of separation being oblique to the line of
pressure.
393. Crushing strength of short pillars — Angle of frac-
ture.— It has been found by experiment that the strength of short
pillars of any given material, all having the same diameter, does
not vary much, provided the length of the pillar is not less
than one, and does not exceed four or five diameters ; and
the weight which will just crush a short prism whose base equals
226 CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
one square unit (generally a square inch), and whose height is not
less than one or one and a half, and does not exceed four or five
diameters, is called the crushing strength of the material experi-
mented upon. When the height of a solid prism lies within these
limits "fracture is (generally) caused by the body becoming
divided diagonally in one or more directions. In this case the
prism, in cast-iron at least, either does not bend before fracture,
or bends very slightly ; and therefore the fracture takes place by
the two ends of the prism forming cones or pyramids, which split
the sides and throw them out ; or, as is more generally the case in
cylindrical specimens, by a wedge sliding off, starting at one of
the ends, and having the whole end for its base ; this wedge being
at an angle which is constant in the same material, though different
in different materials (see Plate II.). In cast-iron the angle is such
that the height of the wedge is somewhat less than f of the
diameter. In timber, like as in iron and crystalline bodies generally,
crushing takes place by wedges sliding off at angles with their
base which may be considered constant in the same material;
hence, the strength to resist crushing will be as the area of
fracture, and consequently as the direct transverse area, since the
area of fracture would, in the same material, always be equal to
the direct transverse area, multiplied by a constant quantity."* In
other words, eq. 1 is applicable to short pillars, and their crushing
strength is equal to their transverse section multiplied by the
crushing unit-strain of the material. If the length exceeds four
or five times the diameter, "the body bends with the pressure,
and though it may break by sliding off as before, the strength
is much decreased. In cases where the length is much greater
than as above, the body breaks across, as if bent by a transverse
pressure." f
From the foregoing observations the reader will perceive that
the crushing unit-strain of any material should be derived from
experiments on prisms whose height is not less than the length
of the wedge, nor so great that the prism will deflect. Mr.
* Experimental Researches on the Strength of Cast-iron, by E. Hodgkinson, pp. 319, 323.
t Idem, p. 321.
PLATE II.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 227
Hodgkinson seems to have preferred prisms whose height equalled
two diameters, and in Table I. it will be seen that prisms of
cast-iron, whose height equalled one diameter, generally bore more
than those whose height equalled two diameters. If, however,
the material, like glass and some limestones, do not form wedge-
shaped but longitudinal splinters, it seems probable that, within
considerable limits, the height of the specimen will not materially
affect its crushing strength. Experimenters on stone have gene-
rally used cubes; Mr. Hodgkinson's practice, however, seems
preferable. If the length of pillars never exceeded four or five
diameters, all we need do to arrive at the strength of any given
pillar would be to multiply its transverse area in square units by
the tabulated crushing strength of that particular material. It
rarely happens, however, that pillars are so short in proportion to
their length, and hence, we must seek some other rule for cal-
culating their strength when they fail, not by actual crushing, but
by flexure. If we could insure the line of thrust always coinciding
with the axis of the pillar, then the amount of material required
to resist crushing merely would suffice, whatever might be the
ratio of length to diameter. But practically it is impossible to
command this, and a slight error in the line of thrust produces a
corresponding tendency in the pillar to bend. With tension-rods,
on the contrary, the greater the strain the more closely will the
rod assume a straight line, and, in designing their cross section, it
is only necessary to allow so much material as will resist the
tensile strain. This tendency to bend renders it necessary to
construct long pillars, not merely with sufficient material to resist
crushing, supposing them to fail from that alone, but also with
such additional material, or bracing, as may effectually preserve
them from yielding by flexure. In masonry, heavy timber framing,
or similar massive structures, the desired effect is produced by
mere bulk of material, which insures the line of thrust always
lying at a safe distance within the limits of the structure. In
hollow pillars the same result is obtained by removing the material
to a considerable distance from the line of thrust, which, though
it may deviate slightly from the axis of the pillar, yet will not
228
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
pass beyond its circumference. When the pillar is neither tubular
nor solid, one of the forms of section, represented in Fig. 99 is
generally adopted.
Fig. 99.
However, before treating about flexure, it seems desirable to
give the crushing strengths of short prisms of various materials
and afterwards show how these are modified by increasing the
length of the prism.
CAST-IRON.
394. Crushing strength of cast-iron. — Table I. contains the
results of experiments by Mr. Hodgkinson " on the crushing
strength of cylinders of cast-iron of various kinds ; the diameters
of the cylinders being turned to £ inch each, and the heights being
f and 1^ inches respectively. In both cases the height was so
small that the specimen could not bend before crushing. Before
each experiment was commenced, a very thin sheet of lead was laid
over and under the specimen, to prevent any small and unavoid-
able irregularity between its flat surface and those of the parallel
steel discs between which it was to be crushed." *
TABLE I.— CRUSHING STRENGTH OP CAST-IRON.
Description of iron.
Height
of
specimen.
Crushing weight
per square inch of
section.
Mean.
inch.
Ibs. tons.
Its. tons.
Low Moor iron, No. 1 -
l|
64534 = 28-809
56445 = 25-198
60489 = 27-004
Do. No. 2 -
li
99525 = 44-430
92332 = 41-219
95928 = 42-825
Clyde iron, No. 1 -
1*
92869 = 41-459
88741 = 39-616
90805 = 40-537
* Report of the Commissioners appointed to inquire into the application of iron to
railway purposes, 1849, App. A., pp, 12, 13.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 229
TABLE I.— CRUSHING STRENGTH OP CAST-IRON— continued.
Description of iron.
Height
of
specimen
Crushing weight
per square incli of
section.
Mean.
Clyde iron, No. 2 -
inch.
4
Ibs. tons.
109992 = 49-103
102030 = 45-549
Ibs. tons.
106011 = 47-326
Do. No. 3 -
4
107197 = 47-855
104881 = 46-821
106039 = 47-339
Blaenavon iron, No. 1
4
90860 = 40-562'
80561 = 35-964
85710 = 38-263
Do. No. 2— 1st sample -
4
117605 = 52-502
102408 = 45717
110006 = 49-109
Do. No. 2 — 2nd sample -
2.
4
68559 = 30-606
68532 = 30-594
68545 = 30-600
Calder iron, No. 1 -
4
72193 = 32-229
75983 = 33-921
74088 = 33-075
Coltness iron, No. 3
4
100180 = 44723
101831 = 45-460
101005 = 45-091
Brymbo iron, No. 1
4
74815 = 33-399
75678 = 33-784
75246 = 33-592
Brymbo iron, No. 3
4
76133 = 33-988
76958 = 34-356
76545 = 34-171
Bowling iron, No. 2
4
76132 = 33-987
73984 = 33-028
75058 = 33-508
Ystalyfera Anthracite iron, No. 2 -
4
99926 = 44-610
95559 = 42-660
97742 = 43-635
Yniscedwyn Anthracite iron, No. 1 -
4
83509 = 37-281
78659 = 35-115
81084 = 36198
Do. No. 2 -
4
77124 = 34-430
75369 = 33-646
76246 = 34-038
86284 — 38'519
Mr. Monies Stirling's iron, 2nd quality*
4
125333 = 55-952
119457 = 53-329
122395 = 54-640
Do. 3rd qualityf
4
158653 = 70-827
129876 = 57-980
144264 = 64-403
* Composed of Calder No. 1 hot-blast, mixed and melted with about 20 per cent,
of malleable iron scrap.
f Composed of No. 1 hot-blast Staffordshire iron from Ley's Works, mixed and
melted with about 15 per cent, of common malleable iron scrap.
230
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
Table II. contains the '* crushing weights of short cylinders of
different kinds of cast-iron, cut from the bars, 2| inches diameter
previously used (in experiments on pillars), and now turned to be
| inch diameter nearly, and 1 £ inch high. The results are means
from three or four experiments on each kind of iron. The specimens
were usually cut out of the iron between the centre and the
circumference of the bar, denominated the medium part. In several
cases they were cut out of the centre of the bar, and sometimes out
of the circumference."*
TABLE II. — CRUSHING STRENGTH OF CAST-IRON.
Description of iron.
Diameter
of
specimen.
Crushing weight
per square inch
of section.
Medium,
Old Park iron, No. 1.
inch.
747
K>s. tons.
88070 = 39-32
Centre, -
Old Park iron, No. 1.
747
74653 = 33-33
Medium,
Derwent iron, No. 1.
•747
97160 = 43-37
Medium,
Coltness iron, No. 1.
•747
63048 = 2814
Medium,
Blaenavon iron, No. 1.
•748
70909 = 31-66
Medium,
Level iron, No. 1.
•749
68217 = 30-45
Medium,
Carron iron, No. 1.
•750
68509 = 30-58
Medium,
London Mixture.
•749
80923 = 36-08
Medium,
Calder iron, No. 1.
•750
84648 = 37-79
Medium,
Portland iron, No. 1.
•748
94802 = 42-32
Philosophical Transactions, 1857, p. 889.
CHAP. XIV.j CRUSHING STRENGTH OF MATERIALS. 231
TABLE II.— CRUSHING STRENGTH OP CAST-IRON— continued.
Description of iron.
Diameter
of
specimen
Crushing weight
per square inch
of section.
Old Hill iron, No. 1.
Medium,
inch.
749
Ibs. tons.
54761 = 24-45
Low Moor iron, No. 2.
Medium,
•748
77489 = 34-59
Low Moor iron, No. 2.
Centre,
•742
66407 = 29-65
Blaenavon iron, No. 3.
Medium,
•737
83517 = 37-28
Blaenavon iron, No. 3.
Centre,
•747
76643 = 34-22
Second London Mixture.
Medium. From 2^ inch pillar, as all above have been,
•747
95338 = 42-56
Second London Mixture.
Centre. From 2g inch pillar, as all above have been,
•747
78451 = 35-02
Second London Mixture.
Medium. From 1| inch pillar,
•750
111080 = 49-59
Second London Mixture.
Centre. From 1J inch pillar, ------
•750
104071 = 46-46
Low Moor iron, No. 2.
From a hollow pillar 4 inches diameter and ^ inch thick.
The height of the first two specimens was 72 inch, and
of the last 1'502 inch,
(•421
87502 = 39-06
Low Moor iron, No. 2.
From the thin ring of a hollow pillar about 34 inches dia-
meter. Height of specimens '53 inch,
j-299
115993 = 5178
Low Moor iron, No. 2.
From the thin ring of a hollow pillar about 3J inches dia-
meter. Height of Specimens '53 inch,
I -296
110212 = 49-20
Mean of the foregoing 22 irons, 84200 = 37'6
232 CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
From the experiments recorded in the two foregoing tables it
appears that the average crushing strength of simple cast-irons does
not exceed 38 tons per square inch; the strength of mixtures,
however, is higher and may in general be taken at 42 tons per
square inch, though occasionally it reaches 50 tons. Repeated
meltings seem to have the effect of increasing the crushing strength
of cast-iron (See Chap. XVI.).
395. Hardness and crashing: strength of thin casting's
greater near the surface than in the heart — Crushing*
strength of thin greater than that of thick casting's. — Mr.
Hodgkinson found that " of the different irons tried in the experi-
ments on pillars, whether solid or hollow, the external part of the
casting was always harder than that near to the centre, and the iron
of the external ring of a hollow casting was very hard, the hardness
increasing with the thinness. Thus, in solid pillars 2J inches
diameter of Low Moor iron, No. 2 (Table II.), the crushing force
per square inch of the central part was 29'65 tons, and that of the
intermediate part near to the surface was 34'59 tons, whilst the
external ring, -J inch thick, of a hollow cylinder 4 inches diameter,
of which the outer crust had been removed, was crushed with 39 '06
tons per square inch; and external rings of the same iron, thinner
than half an inch, required from 49'2 to 51'78 tons per square inch
to crush them. These facts show the great superiority of hollow
pillars over solid ones of the same weight and length."* Hence,
removing the skin of a thin casting reduces its strength to resist
compression.
396. Hardness and crushing* strength of thick casting's at
the surface and in the heart not materially different. — " To
ascertain whether the internal strength of larger pillars varied in the
same manner as that of smaller ones, a cylindrical casting was made
5 inches diameter and 15 inches long. It was cast vertically, from
Blaenavon iron. Through the axis of this cylinder, a slab, 15 inches
long, 5 inches broad, and about 1 inch thick, was taken. Across the
middle of this slab five cylinders, 1 J inch long and J inch diameter,
were obtained at equal distances from each other, the middle one
Phil. Trans., 1857, p. 890.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 233
being in the centre, and the outer ones as near as possible to the
sides. Comparing the results of the experiments (on crushing these
cylinders) it appears that the external part of the casting was some-
what stronger than the internal. But the variation was only from
62 to 66 (62,444 to 65,739 Ibs. per square inch), and therefore
much less than was obtained from the smaller masses." From this
and other experiments on small cylinders cut out of a slab of
Derwent iron, No. 1, cast 9 inches square and 12 inches long, "it
appears that the difference of hardness between the external and
internal parts of a large casting is much less than in a small one, and
may frequently be neglected."* For the safe working strain on
cast-iron see Chap. XXVIII.
WROUGHT-IRON.
397. Crushing: strength of wrought-iron — 13 tons is the
limit of compressive elasticity of wrought -iron. — The crush-
ing strength of wrought-iron varies with the hardness of the iron,
but ordinary wrought-iron is completely crushed, i.e., bulged, with
a pressure of from 16 to 20 tons per square inch, and when the
pressure exceeds 12 or 13 tons, Mr. Hodgkinson found that "in most
cases it cannot be usefully employed, as it will sink to any degree,
though in hollow cylinders it will sometimes bear 15 or 16 tons per
square inch." f The point at which compressive set sensibly com-
mences, that is, the limit of compressive elasticity, is about 12 tons
per square inch. For the safe working strain in practice see Chap.
XXVIII.
STEEL.
898. Crushing- strength of steel — 31 tons is the limit of
compressive elasticity of steel. — The following table contains
the results of experiments on the crushing weights of cylinders of
cast-steel by Major Wade,t U.S. Army: —
* Phil. Trans., 1857, pp. 891, 892.
f Com. Rep., p. 121.
J Reports of Experiments on the Strength and other Properties of Metals for Cannons,
by Officers of the Ordnance Department, U.S. Army, p. 258. Philadelphia, 1856.
234 CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
TABLE III.— CRUSHING STRENGTH OF CAST-STEEL.
Kind of cast-steel.
No.
of
sam-
ple.
Length.
Diameter.
Crushing
weight per
square inch
of section.
inch.
inch.
fcs.
Not hardened,
1
1-021
•400
198,944
Hardened ; low temper ; chipping chisels,
2
•995
•402
354,544
Hardened ; mean temper ; turning tools,
3
1-016
•403
391,985
Hardened ; high temper ; tools for turning hard steel,
4
1-005
•405
372,598
NOTE — All the samples of steel tested were cut from the same bar. No. 1 remained
unchanged, as made at the steel factory. Nos. 2, 3, and 4, were all hardened, and the
temper afterwards drawn down in different proportions.
Table IV. contains the results of experiments made by Mr.
Kirkaldy for the " Steel Committee," on the crushing strength of
carefully turned cylinders of steel 1-382 inches in diameter (= 1/5
square inches area), and whose height equalled 4 diameters, the
steel being intended for tyres, axles, and rails.*
TABLE IV. — LIMIT OF COMPRESSVIE ELASTICITY OP CRUCIBLE AND BESSEMER
STEEL BARS.
KindofsteeL
Crushing weight per square inch at
which sensible set commenced, i.e.,
Limit of compressive elasticity.
Crucible steel, hammered,
tons.
22-92
Do. rolled,
1875
Bessemer steel, hammered,
21-79
Do. rolled,
18-08
Mean, - - 20 '38
Shorter cylinders of the same kinds of steel of the same sectional
area, but only one diameter in height, were subjected to a crushing
weight of 200,000 Ibs. per square inch, the result being that they
* Experiments on Steel and Iron by a Committee of Civil Enyineers, 1868-70.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 235
bulged but did not crack ; the average contraction of length (ultimate
compressive set) under this strain was for crucible steel 32 per cent.,
and for Bessemer steel 38 per cent., of the original length. From
31 experiments made subsequently by the same committee at
Woolwich Dockyard, on the compression of bars of crucible,
Bessemer, and cast-steel, 10 feet long and 1J inches diameter, the
maximum and minimum limits of compressive elasticity were 27 and
15 tons respectively, and the average was 2T35 tons per square inch,
which agrees sufficiently closely with the mean of the experiments
in Table IV. to allow us to assume 21 tons to be the practical limit
of compressive elasticity of average steel.
The reader will find in Chap. XVI. additional experiments by
Sir William Fairbairn on the crushing strength of various kinds
of steel. For the safe working load see Chap. XXVIII.
VARIOUS METALS.
399. Crashing: strength of copper^ brass, din. lead,
aluminium-bronze, zinc. — The following table contains the results
of experiments by Mr. G. Rennie on the crushing strength of £ inch
cubes of different metals.*
TABLE V. — CRUSHING STRENGTH OF VARIOUS METALS.
Description of metal.
Crushing weight
on a £ inch cube.
Cast-copper crumbled with
.
fts.
7318
Fine yellow brass reduced -fVth,
with -
3213
Do. do. i,
with -
10304
Wrought-copper reduced -reth,
with -
3427
Do. do. |th,
with -
6440
Cast-tin do. -j^n*
with -
552
Do. do. |rd,
with -
966
Cast-lead do. £,
with -
483
Alluding to these ductile metals, Mr. Rennie observes: — " The
experiments on the different metals give no satisfactory results.
The difficulty consists in assigning a value to the different degrees
* Phil. Trans., 1818, p. 125.
236
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
of diminution. When compressed beyond a certain thickness, the
resistance becomes enormous." The crushing weight of aluminium
bronze, according to Professor Rankine, is 59 tons per square inch.
In my own experiments I found that cast-zinc will spread out to
any degree under severe pressure, but it will bear 5 or 6 tons per
square inch without any very appreciable change of shape.
TIMBER.
3OO. Crushing; strength of timber — Wet timber not nearly
so strong- as dry. — The following table contains the results of
experiments by Mr. Hodgkinson on the crushing strength of various
kinds of timber, " the force being applied in the direction of the
fibre."*
TABLE VI. — CRUSHING STRENGTH OF TIMBER.
Description of wood.
Crushing weight per
square inch of section.
Wood in the
ordinary state
of dryness.
Wood
very dry.
Ibs.
Ibs.
Alder,
6,831
6,960
Ash, -
8,683
9,363
Baywood,
7,518
7,518
Beech,
7,733
9,363
Birch, American,
...
11,663
Birch, English,
3,297
6,402
Cedar,
5,674
5,863
Crab, -
6,499
7,148
Deal, red,
5,748
6,586
Deal, white,
6,781
7,293
Elder,
7,451
9,973
Elm, -
...
10,331
Fir, Spruce,
6,499
6,819
Hornbeam,
4,533
7,289
Larch (fallen two months),
3,201
5,568
Mahogany,
8,198
8,198
Oak, Quebec, -
4,231
5,982
Oak, English, -
Oak, Dantzic (very dry),
6,484
10,058
7,731
Pine, pitch,
6,790
6,790
Pine, yellow (full of turpentine),
5,375
5,445
Pine, red,
5,395
7,518
Plum,' wet,
3,654
...
Plum, dry,
8,241
10,493
Poplar,
3,107
5,124
Sycamore,
7,082
...
Teak, -
...
12,101
Walnut,
6,063
7,227
Willow,
2,898
6,128
* Phil. Trans., 1840, p. 429.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS.
237
" The results in the first column were in each case a mean from
about three experiments upon cylinders of wood turned to be one
inch diameter, and two inches long, flat at the ends. The wood
was moderately dry, being such as is employed in making models
for castings. The second column gives the mean strength, as
before, from similar specimens, after being turned and kept drying
in a warm place two months longer. The lengths of these latter
specimens were, in some instances, only one inch, which reduction
would increase the strength a little. But the great difference
frequently seen in the strength, as given by the two columns, shows
strongly the effect of drying upon wood, and the great weakness of
wet timber, it not having half the strength of dry" — a consideration
of much importance in works under water. For the safe working
load on timber see Chap. XXVIII.
STONE, BRICK, CEMENT, AND GLASS.
3O1. Crushing- strength of stone and brick. — The following
table contains the crushing strength of stone and brick. For the
safe working load see Chap. XXVIII.
TABLE VII. — CRUSHING STRENGTH OP STONE AND BRICK.
Description of stone.
Specific
gravity.
Crushing
weight
per
square
inch.
Authority.
GRANITES.
fts.
Aberdeen, blue kind, -
2-625
10914
Rennie.
Peterhead, hard close grained,
...
8283
u
Cornish,
2-662
6356
?>
Killiney, Co. Dublin, very felspathic,
,
10780
Wilkinson.
Kingstown, do., grey colour,
.
10115
39
Blessington, Co. Wicklow, coarse and loosely
aggregated, -
§
3630
n
Ballyknocken. Co. Wicklow, coarse, micaceous,
,
3173
n
Newry, slightly syenitic,
.
13440
Mount Sorrel granite,
2-675
12861
Fairbairn.
SANDSTONES AND GRITS.
Arbroath pavement, -
...
7884
Buchanan.
Caithness do.
...
6493
>5
Dundee sandstone or Brescia, -
2-530
6630
Rennie.
Craigleith white freestone,
2-452
5487
M
Bramley Fall, near Leeds (with and against strata)
Derby Grit, a red friable sandstone, -
2-506
2-316
6059
3142
»
M
Ditto, from another quarry,
Yorkshire paving (with and against strata), -
2-428
2-507
4345
5714
Red sandstone, Runcorn (17 feet per ton),
...
2185
L. Clark.
Quartz rock, Holyhead (across lamination),
...
255dO
Mallet.
Ditto (parallel to lamination),
...
14000
»
238
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
TABLE VII.— CRUSHING STRENGTH OF STONE AND BRICK— continued.
Crushing
Description of stone.
Specific
gravity
weight
per
square
Authority.
inch.
OOLITES.
ibs.
Portland stone,
2-423
3729
Rennie.
Ditto, another specimen,
2-428
4570
»
MARBLES.
Marble, statuary,
...
3216
i»
Ditto, white statuary, not veined,
2760
6058
M
Ditto, white Italian, veined,
2-726
9681
)J
Ditto, black Brabant, -
2-697
9219
JJ
Ditto, Devonshire red, variegated,
...
7428
»>
Ditto, Kilkenny black,
...
15120
Wilkinson.
Ditto, black Galway, from Menlo quarry,
...
20160
»
LIMESTONES.
Limestone, compact, -
2-584
7713
Rennie.
Ditto, black compact, Limerick,
2-598
8855
, ,
Ditto, Purbeck,
2-599
9160
,,
Ditto, magnesian, Anston, stone of which
Houses of Parliament are built,
3050
Fairbairn.
Ditto, Anglesea (13^ cubic feet per ton),
7579
L. Clark.
Ditto, Listowel quarry, Kerry,
Ditto,Ballyduff quarry near Tullamore, King's Co.
Ditto, Woodbine quarry near Athy, Kildare, -
*••
18043
11340
14350
Wilkinson.
»
>»
Ditto, Finglas quarry, Co. Dublin,
16940
Chalk,
...
501
Rennie.
SLATES.
Valencia Island, Kerry,
...
10943
Wilkinson.
Killaloe quarry, Tipperary, on bed of strata, -
...
26495
n
Do. do. on edge of strata, -
...
15225
n
Glanmore, Ashford, Wicklow, on bed of strata
...
21315
»
Do. do. on edge of strata
...
12740
»
BASALTS AND METAMORPHIC ROCKS.
Whinstone, Scotch, -
...
8270
Buchanan.
Felspathic greenstone, from Giant's Causeway,
...
17220
Wilkinson.
Hornblendic greenstone, Galway, Co. Galway,
24570
,,
Moore quarry, Ballymena, Antrim, crystal-
line and hornblendic,
20552
Grauwacke, from Penmaenmawr,
2748
16893
Fairbairn.
BRICKS.
Pale red,
2-085
562
Rennie.
Red brick,
2-168
808
tt
Yellow-face baked Hammersmith paviors,
...
1002
»
Yellow-faced burnt Hammersmith paviors,
...
1441
»»
Fire brick, Stourbridge,
...
1717
Buckley Mountain brick, N. Wales, -
2130
L. Clark.
Brickwork set in cement (bricks not of a hard
description)
...
521
?>
Buchanan, Practical Mechanic's Journal, Vol. ii., p. 285.
L. Clark, The Britannia and Conway Tubular Bridges, p. 365.
Fairbairn, Useful information for Engineers, second series, p. 136.
Mallet, Philosophical Transactions, 1862, p. 671.
Rennie, Philosophical Transactions, 1818, p. 131.
Wilkinson, Practical Geology and Ancient Architecture of Ireland.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS.
239
The following table gives the results of experiments made by
Mr. Grant with a hydraulic press on the crushing strength of
various kinds of brick.*
TABLE VIII. — CRUSHING STRENGTH OP COMMON BRICK AND BRICKS MADE OF
PORTLAND CEMENT.
Description of brick.
1
s
Breadth
1
1
Area ex-
posed to
pressure.
Weight.
Crushing
weight
per brick.
Dry.
Wet.
ins.
ins.
ins.
ins.
K>8.
fts.
tons.
Gault clay, pressed, -
875
4-125
275
36-09
5-13
6-47
40-04
Gault clay, wire cut,
9-00
4-125
275
37125
5-86
6-85
3270
Gault clay, perforated,
9-00
4-375
2-625
39-375
4-95
576
46-40
Suffolk brimstones, -
9-00
4-5
2-625
40-5
618
714
34-94
Stock, ---
9-00
4-125
2-625
37-125
5-0
5-57
3874
Fareham red, - -
875
4-125
2-625
36-09
6.55
7-52
90-40
Staffordshire blue (pressed, with
frog),_ -
875
4125
275
36-09
7-82
7-90
111-04
Staffordshire blue(rough, without
frog), ---.
875
4-125
275
36-09
775
7-81
117-92
Portland Cement bricks, neat,
compressed, and kept in air
12 months, ...
9-00
4-5
3-0
40-50
9-51
976
96-60
Do. kept in water 12 months, -
132'62
Portland Cement and sand, 1 to
4, compressed and kept in
air 12 months, -
...
...
...
...
879
9-51
43-60
Do. kept in water 12 months, -
29'92
Portland Cement and sand, 1 to
6, compressed and kept in
air 12 months, -
8'43
9'38
30-28
Do. kept in water 12 months, -
11-24
3O8. Mode of fracture of stone. — In Mr. Clark's experiments
"the sandstones gave way suddenly, and without any previous
cracking or warning. After fracture the upper portion generally
retained the form of an inverted square pyramid, very symmetrical,
the sides bulging away in pieces all round. The limestones formed
perpendicular cracks and splinters a considerable time before they
crushed." Mr. Rennie observes, "it is a curious fact in the
rupture of amorphous stones, that pyramids are formed, having
for their base the upper side of the cube next the lever, the
action of which displaces the sides of the cubes, precisely as if a
wedge had operated between them." Mr. Wilkinson remarks,
" The results of the (one inch) cubes experimented on show the
strongest stones to be the basalts, primary limestones, and slates.
Of the limestones, the primary limestones and compact hard calp
are the strongest; and the light dove-coloured and fossiliferous
* Proc. Inst. C.E., Vol. xxxii.
240
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
limestones are among the weakest. The strength of the sand-
stones, like their mineral aggregation, is very variable."
The strength of stones, though bearing the same name and pre-
senting the same lithological characters, is so variable in different
localities, that, when any building of importance is proposed, it is
prudent to test the strength of the stone by actual experiment
rather than trust to books for the information required. In my
own experiments, I find that with granite and limestones the
first crack may be expected to take place with from one-half to
two-thirds of the ultimate crushing weight.
303. Crashing; strength of rubble masonry. — Professor
Eankine states that " the resistance of good coursed rubble masonry
to crushing is about four-tenths of that of single blocks of the stone
that it is built with. The resistance of common rubble to crushing
is not much greater than that of the mortar which it contains."*
For the safe working load on masonry see Chap. XXVIII.
304. Crashing; strength of Portland cement, mortar and
concrete. — The following table contains the results of experi-
ments by Mr. Grant on the crushing strength of Portland cement
and cement mortar, f
TABLE IX. — CRUSHING STRENGTH OF PORTLAND CEMENT AND CEMENT MORTAR.
Description of cement or mortar.
Crushing
weight
per
square
inch.
Portland cement, neat,
3795
1 Portland Cement to 1 pit sand,
rf
2491
ditto 2 ditto,
rS -g
2004
ditto 3 ditto,
ll
1436
ditto 4 ditto,
" M
CO
1331
ditto 5 ditto,
959
Portland cement, neat,
5388
1 Portland Cement to 1 sand,
m
3478
ditto 2 ditto, -
•73 "§
2752
ditto 3 ditto, -
2 i
2156
ditto 4 ditto, -
*"• w
1797
ditto 5 ditto, -
1540
Portland cement, neat,
5984
1 Portland Cement to 1 pit sand,
aj
4,561
. ditto 2 ditto,
r£j -g
3647
ditto 3 ditto,
rt o
2393
ditto 4 ditto,
H
2208
ditto 5 ditto,
1678
Civil Engineering, p. 387.
t Proc. Inst. C. E., Vol. xxv.
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS.
241
In these experiments the specimens were made into bricks
9 X 4'25 X 2*75 inches, and exposed to the pressure of a hydraulic
press on their flat surface of 9 X 4-25 inches = 38'25 square
inches. The results would doubtless have been somewhat different
if they had been cubes. Each specimen showed signs of giving
way with considerably less pressure than that which finally crushed
it, the average ratio of the weight which produced the first crack
to that which finally crushed it being nearly as ^.
The following table gives the strength of lime mortar 18 months
old, on the authority of Rondelet.*
TABLE X.— CRUSHING STRENGTH OF LIME MORTAR 18 MONTHS OLD.
Crushing
weight
Description of mortar.
per
square
inch.
fcs.
Mortar of lime and river sand,
436
The same, beaten,
596
Mortar of lime and pit sand, -
578
The same, beaten,
800
Mortar of cement and pounded tiles, -
677
The same, beaten,
929
Mortar made with pounded sandstone,
417
Mortar made with puzzolana from Naples and
Rome mixed,
521
The same, beaten,
758
Fifteen years later these experiments were repeated, when
mortars of lime and sand were found to have increased in strength
about Jth, and mortars of cement or puzzolana about Jth.
The following tables give the results of some of Mr. Grant's
experiments with a hydraulic press on the crushing strength of
concrete blocks, made of Portland cement and ballast in various
proportions, set and kept in air for one year, also set and kept in
water for the same time.f
* Navier, Application de la Mticaniqiie, p. 8.
f Proc. I. C. K, Vol. xxxii.
242
CRUSHING STRENGTH OF MATERIALS. [CHAP. XIV.
TABLE XI.— PORTLAND CEMENT CONCRETE BLOCKS OF BALLAST, set and kept in Air
for One Year, also set and kept in Water for the same time.
Size of Block— 12" X 12" X 12". Compressed.
Weight in tt>s.
Weight of each
Block in tt>s.
Crushed at tons.
Propor-
Remarks.
tions*
Cement.
Sand and
Gravel.
Water.
Kept in
Air.
Kept in
Water.
Air.
Water.
Itol
59-36
66-96
16-00
137-60
147-25
107-0*
170-50
* Exceptional.
2 1
42-64
96-40
12-00
142-60
152-50
149-0
160-0
3
32-00
108-56
10-00
145-25
152-25
113-0
115-50
4
25-84
116-96
8-80
145-75
152-50
103-0
108-50
5
21-28
120-24
8-00
14210
150-95
89-0
99-50
6
1808
122-48
8-00
141-56
150-00
80-50
91-0
7
15-84
125-04
7-60
14170
150-20
75-0
80-50
8
14-08
127-04
7-60
142-30
150-80
61-50
76-0
9
12-64
128-64
7-20
142-10
151-50
54-0
68-50
10 „
11-36
128-88
6-80
142-00
150-00
48-50
48-0
Size of Block— 6" X 6" X 6". Compressed.
1 tol
7-42
8-37
2-00
17-50
18-04
38-0
33-60
2 „ 1
5-33
12-05
1-50
1778
18-97
43-0
34-50
3 „ 1
4-00
13-57
1-25
18-28
19-35
30-0
35-50
4 1
3-23
14-62
1-10
18-28
18-71
30-0
28-00
5 1
2-66
15-03
1-00
18-26
18-98
24-50
35-50
6 1
2-26
15-31
1-00
17-90
18-60
20-40
19-60
7 1
1-98
15-63
•95
17-85
18-85
16-50
16-0
8 1
1-76
15-88
•95
17-86
18-90
13-50
13-50
9 1
1-58
16-08
•90
17-78
19-0
12-0
11-00
10 1
1-42
16-11
•85
17-68
18-70
10-50
10-50
Size of Block— 6" X 6" X 6". Not Compressed.
1 to
7-12
8-04
1-92
16-44
17-60
30-0
37-50
2 „
4-90
11-09
1-38
17-57
18-03
38-50
36-00
3 „
3-56
12-11
1-11
17-75
18-98
24-0
28-00
4 „
2-85
12-92
•97
17-84
18-28
28-0
27-00
5 „
2-33
13-18
•87
17-90
18-73
24-0
23-50
6 „
2-00
13-49
•88
17-35
18-30
18-20
17-00
7 „
1-77
14-02
•85
17-32
17-90
14-0
12-50
8 „
1-60
14-51
•85
17-38
17-95
12-50
11-00
9 „ 1
1-43
14-59
•80
17-40
17-97
10-0
9-00
10 „ 1
1-26
14-35
•75
17'20
17-50
8-0
7-00
It will be observed that the concrete which was compressed
was considerably stronger than that not compressed. In my
own practice I always have concrete carefully rammed, and
when it forms the matrix for large rubble stone the concrete is
compressed between the stones with iron tamping tools having T
CHAP. XIV.] CRUSHING STRENGTH OF MATERIALS. 243
shaped ends about 5 inches long. This permits it to be mixed
stiff with but little water, and, when thus solidly rammed, the
stones will generally break sooner than the concrete in which
they are imbedded. In one of Mr. Grant's experiments a twelve-
inch cube of concrete, made with blue Lias lime and Thames
ballast 1 -f- 6, 10 months old and kept in water, bore 6 tons per
square foot, or 93 ft>s. per square inch. A similar cube of Lias
concrete, but made with Bramley Fall chippings 1 + 6, in place
of ballast, and also kept in water 10 months, bore 20'4 tons per
square foot, or 317 ft>s. per square inch.* For the safe working
load on concrete see Chap. XXVIII.
3O5. Crashing: strength of glass. — The following table con-
tains the crushing strength of glass from experiments by Sir Wm.
Fairbairn and Mr. Tate.f
TABLE XII.— CRUSHING STRENGTH OP ANNEALED GLASS BARS.
Kind of Glass.
Sp.
gravity.
Crushing weight
per square inch.
Best flint glass annealed rod, drawn out when molten
to about f inch diameter, -
Common green glass ditto ditto,
White crown glass ditto ditto,
3-0782
2-5284
2-4504
ft>s. tons.
27582 = 12-313
31876 = 14-227
31003 = 13-840
" The specimens were crushed almost to powder from the violence
of the concussion, when they gave way; it, however, appeared
that the fractures occurred in vertical planes, splitting up the
specimen in all directions ; cracks were noticed to form some time
before the specimen finally gave way ; then these rapidly increased
in number, splitting the glass into innumerable irregular prisms
of the same height as the cube ; finally, these bent or broke, and
the pressure, no longer bedded on a firm surface, destroyed the
specimen." Seven cubes were also cut from the centre of large
lumps of glass, and crushed. Their resistance was less than that
of the drawn rods in the ratio of f , possibly because they were
less perfectly annealed than the drawn rods, and also because the
external skin of the latter gave them some extra strength (895).
* Proc. I. C. E., Vol. xxv., p. 110.
t Philosophical Transactions, 1859, p. 213.
244
PILLARS.
[CHAP. xv.
CHAPTER XV.
PILLARS.
306. Very long thin pillars.— The law Fig. 100.
which determines the flexure of very long thin
pillars may be investigated theoretically as
follows: — Let Fig. 100 represent a pillar of
uniform section throughout, not fixed at the
ends, very long in proportion to its breadth, and
just on the point of failing from flexure.
Let W = the deflecting weight,
D = the lateral deflection at the centre.
M = the moment of resistance of the
longitudinal elastic forces (59),
b = the breadth of the pillar,
d = its diameter or least lateral dimension,
/ = its length,
/ = the longitudinal unit-strain in the extreme fibres in a
horizontal section across the middle of the pillar,
X — the difference in length between the convex and the
concave edges of the pillar,
C = the resultant of all the longitudinal forces of compres-
sion in the concave side at the plane of section,
T = the resultant of all the longitudinal forces of tension
in the convex side at the plane of section,
E = the coefficient of elasticity.
The upper half of the pillar is held in equilibrium by three sets
of vertical forces — viz., the weight, acting in the chord-line of the
curve; the longitudinal tensile strains in the convex side at the
middle section; and the longitudinal compressive strains in the
concave side, also at the middle section. When the pillar is very
long in proportion to its width, and the deflection therefore
CHAP. XV.] PILLARS. 245
considerable, even though the curvature be small,* we may assume
D equal to the distance from the chord-line to either the centre
of tensile or the centre of compressive strains. Taking moments
round either of these points indifferently, we have
W D = M nearly, (a)
Again, assuming that the deflection curve is a circle, from which it
can differ but slightly, we have from eq. 132,
D~M nearlv' (6)
whence, by substitution in eq. (a), we have,
W = «f (o)
Further, recollecting that X is equal to the contraction of the
concave plus the extension of the concave edge, we have from eq. 2,
Substituting this in eq. (c), we have
W = ^jp- (230)
Replacing M by its values in 71 and the succeeding sections and
d 2c
recollecting that the ratio -, in eq. 230 is equal to the ratio -j
in the 46th and succeeding equations, we obtain the following values
for the strength of long pillars f of various sections: —
* Mr. Hodgkinson's experiments show that this investigation is not applicable to
pillars whose length is less than fifty diameters if made of cast-iron, or eighty
diameters if made of wrought-iron.
t Calling the diameter unity, it may be shown that the lateral deflection of a very
long pillar per unit of its length = Jth of the shortening of the concave side, or £th of
the extension of the convex side, per linear unit, in the following manner : —
Let R = the radius of curvature,
5 = the lateral deflection of a unit of length,
\' — the longitudinal shortening or extension per linear unit,
and the other symbols as before ;
from (6), D = £_ or, since d = unity, = ^— -
od 4
also, S = ,andD =
246 PILLARS. [CHAP. xv.
307. I,o n^- solid rectangular pillars — I, cms; solid round
• pillars — Long hollow round pillars — Strength of long* pillars
depends on the coefficient of elasticity. — From equations 46
and 230 we have for long solid rectangular pillars,
W = 2-f^ (231)
where d = the shortest side.
From equations 48 and 230 we have for long solid round pillars,
W = ™ (232)
where d = the diameter of the pillar.
From equations 49 and 230 we have for long hollow round pillars
w = ^-^) (233)
where d = the external diameter,
dl = the internal diameter.
These equations prove that the strength of very long square or
round pillars varies as the fourth power of their diameter divided
by the square of their length, and the longer the pillar is in pro-
portion to its diameter, the closer will these equations represent the
truth ; in such pillars the neutral surface will not lie far from the
central axis, and the deflecting weight, W, will be small compared
to that which would crush a very short pillar of the same diameter.
It is also to be observed that the strength of very long pillars
depends, not on the strength of the material, but on E, which
represents its stiffness and capability of resisting flexure. This
theoretic result agrees with the fact that, although a short round
pillar of cast-iron will bear a much greater weight than a similar
pillar of wrought-iron, because the crushing strength of cast-iron is
from two to three times greater than that of wrought-iron, yet a
solid wrought-iron pillar over 26 diameters in length will support a
greater weight than a similar one of cast-iron, because the coefficient
of elasticity of wrought-iron is considerably higher than that of
cast-iron (338).
308. Strength of similar long pillars are as their trans-
Terse areas — Weights of long pillars of equal strength and
similar in section, but of different lengths* are as the squares
CHAP. XV.] PILLARS. 247
of their lengths. — These equations also prove that the strengths
of similar long pillars are as the squares of any linear dimension,
that is, as their transverse areas ; while their weights are as the
cubes of any linear dimension. Further, if the strengths of long
pillars of similar section remain constant while their lengths vary,
their transverse areas will vary as their lengths, and their weights
therefore will vary as the squares of their lengths.
3O9. Weight which will deflect a very long pillar is very
near the breaking weight. — It appears from eq. (b) that, if a
very long pillar be bent in different degrees, D will vary as X, that
is, as/C?); and, from eq. (a), W =: ^y, which is constant, since M
also varies as /; hence it follows, that W, the weight which keeps
the pillar bent, is nearly the same whether the flexure be greater or
less. This statement would be accurately true were it not that
the assumptions on which eqs. (a) and (b) are based and the law of
elasticity are only approximate. It will, however, agree very closely
with experiment when the pillar is long enough to allow D to be
considerable, even though the curvature be small. From this it
follows, that any weight which produces moderate flexure in a very
long pillar will also be very near the breaking weight, as a trifling
additional load will bend the pillar very much more, and strain
the fibres beyond what they can bear. This theoretic result is in
accordance with the following observation of Mr. Hodgkinson : —
" From the first experiment on long hollow pillars with rounded
ends, it was evident that so little flexure of the pillar was necessary
to overcome its greatest resistance (and beyond this a smaller weight
would have broken it), that the elasticity of the pillars was very
little injured by the pressure, if the weight was prevented from
acting upon the pillar after it began to sink rapidly, through its
greatest resistance being overcome."*
As all the longitudinal forces at the middle of the pillar balance,
we have the following equation : —
C = T + W.
This enables us to predict how a very long pillar will fail, whether
* Phil. Trans., 1840, p. 411.
248 PILLARS. [CHAP. xv.
by the convex side tearing asunder, or by the concave side crushing.
A long wrought-iron pillar, for instance, may be expected to fail
on the concave side, because its power to resist compression, i.e.,
bulging, is less than that to resist extension. A long pillar of cast-
iron, on the contrary, will probably fail by the convex side tearing
asunder, because the compressive strength of cast-iron greatly
exceeds its tenacity. This is corroborated by Mr. Hodgkinson's
experiments on long hollow cast-iron pillars which " seldom gave
way by compression."*
31O. Pillars divided into three classes according: to
length. — Our knowledge of the laws of the resistance of pillars to
flexure, though perhaps not so satisfactory in a theoretic point of
view as might be desired, is, however, owing to Mr. Hodgkinson's
able investigations, aided by the liberality of Sir William Fairbairn,
the late Mr. R. Stephenson and the Royal Society, practically far
enough advanced to enable us to predict with considerable accuracy
the strength of pillars of the usual forms. The results of these
investigations are here given; the reader who desires more
detailed information respecting the experiments, is referred to Mr.
Hodgkinson's original papers,f in which he divides pillars into three
classes according to length : —
1°. Short pillars, whose length (if cast-iron, under four or five
diameters) is so small compared with their diameter that they fail
by actual crushing of the material, not by flexure ; the strength of
these has been already investigated in the previous chapter.
2°. Long flexible pillars, whose length is so great (if cast-iron,
thirty diameters and upwards when both ends are flat, fifteen
diameters and upwards when both ends are rounded,) that they fail
by flexure like girders subject to transverse strain, the breaking
weight being far short of that required to crush the material when
in short pieces.
* Phil. Trans., 1840, p. 409.
•^Report of the British Association, Vol. vi. — Philosophical Transactions, 1840
and 1857. — Experimental Researches on the strength and other properties of Cast-iron.
By E. Hodgkinson, F.R.S. London, 1846.— Report of the Commissioners appointed to
inquire into the application of Iron to Railway Sti-uctures, 1849.
CHAP. XV.] PILLARS. 249
3°. Medium, or short flexible pillars, whose length is such that,
though they deflect, yet the breaking weight is a considerable
portion of that required to crush short pillars. This class includes
all pillars which are intermediate in length between those in the
first two classes, and they may be said to fail partly by flexure and
partly by crushing.
In the following remarks the passages in inverted commas are
verbatim extracts from Mr. Hodgkinson's writings.
LONG PILLARS WHICH FAIL BY FLEXURE; LENGTH, IF BOTH
ENDS ARE FLAT AND FIRMLY BEDDED, EXCEEDING 30 DIA-
METERS FOR CAST-IRON AND TIMBER, AND 60 DIAMETERS
FOR WROUGHT-IRON.
311. Long: pillars with flat ends firmly bedded are three
times stronger than pillars with round ends. — "In all long
pillars of the same dimensions, the resistance to fracture by flexure
is about three times greater when the ends of the pillars are flat
and firmly bedded, than when they are rounded and capable of
turning."— Exp. Res., p. 332. From this it follows, that pillars like
the jib of a crane would be much stronger if their ends were fixed ;
there is, however, a practical advantage sometimes in having them
jointed for the purpose of altering the range or height of the jib.
313. Strength of pillars with one end round and the other
flat is a mean between that of a pillar with both ends
round and one with both ends flat. — " The strength of a pillar,
with one end round and the other flat, is the arithmetical mean
between that of a pillar of the same dimensions with both ends
rounded, and with both ends flat. Thus, of three cylindrical pillars,
all of the same length and diameter, the first having its ends
rounded, the second with one end rounded and one flat, and the
third with both ends flat, the strengths are as 1, 2, 3, nearly." —
Exp. Res., p. 332. This law applies to medium as well as to long
pillars, but in the medium pillars the strength of those with flat
ends varies from 3 to 1/5 times that of those with rounded ends, or
less according as we reduce the number of times which the length
exceeds the diameter.— Phil Trans., 1840, pp. 389, 421.
250
PILLARS.
[CHAP. xv.
313. A long pillar with ends firmly fixed is as strong as a
pillar of half the length with round ends. — " A long uniform
pillar, with its ends firmly fixed, whether by discs or otherwise, has
the same power to resist breaking as a pillar of the same diameter,
and half the length, with the ends rounded or turned so that the
force would pass through the axis." — Exp. Res., p. 332.
Of this fact Mr. Hodgkinson offers the following explanation : —
" Suppose a long uniform bar of cast-iron were bent by a pressure
at its ends so as to take the form A.bcde/3, where all the curves
Fi 101 ^c, cde, efB, separated by the straight line AceB,
would be equal, since the bar was supposed to be
uniform. The curve having taken this form, suppose
it to be rendered immovable at the points b and /, by
some firm fixings at those points. This done, it is
evident we may remove the parts near to A and B,
without at all altering the curve bcdef of the part of
the pillar between b and /, and consider only that part.
The part bf, which alone we shall have to consider,
will be equally bent at all the points b,d,f. The
points c and e too are points of contrary flexure, con-
sequently the pillar is not bent in them. These points
are unconstrained except by the pressure which forces
them together, and the pillar might be reduced to
any degree in them, provided they were not crushed
or detruded by the compressing force. These points
may then be conceived as acting like the rounded ends
of the pillars, and the part cde of the pillar, with its
ends c and e rounded, will be bearing the same weight as the whole
pillar bcdef of double the length with its ends, If, firmly fixed." —
Phil Trans., 1857, p. 855.
314. Hodgkinson's laws apply to cast-iron, steel, wronght-
iron, and wood. — " The preceding properties were found to exist
in long pillars of steel, wrought-iron and wood," as well as cast-iron.
They apply only to pillars whose length is so great in proportion to
their diameter that the breaking unit-strain of the pillar is far short
(for cast-iron not exceeding one-fourth) of the crushing unit- strain
of the material.— Exp. Res., pp. 333, 341.
PLATE III
CHAP. XV.] PILLARS. 251
315. Position of fracture in long; cast-iron pillars. —
Long uniform cast-iron pillars with both ends round break in one
place only — the middle ; those with both ends flat in three — at the
middle and near each end ; those with one end round and one flat,
at about one-third of the distance from the round end. Plate III.
represents the curves indicating the form of flexure in each class
of pillar.— Phil. Trans., 1857, p. 858.
316. Discs on the ends add but little to the strength of
flat-ended pillars. — Cast-iron pillars with discs on their ends are
somewhat stronger than those with merely flat ends, but the
difference of strength is trifling.— Phil. Trans., 1840, p. 391.
317. Enlarging: the diameter in the middle of solid pillars
increases their strength slightly. — " In all the (solid cast-iron)
pillars with rounded ends, those with increased middles were
stronger than uniform pillars of the same weight, the increase
being about one-seventh of the weight borne by the former." This
increase of strength was more marked in pillars with rounded ends
than in those with discs, for " in the pillars with discs, those with
the middle but little increased had no advantage, with regard to
strength, over the uniform ones. But the pillars with the middle
diameter half as great again as the end ones bore from one-eighth
to one-ninth more than uniform pillars of the same weight with
discs upon the ends." — Phil. Trans., 1840, p. 395.
318. Enlarging the diameter in the middle or at one
end of hollow pillars does not increase their strength. —
In hollow (cast-iron) pillars of greater diameter at one end than the
other, or in the middle than at the ends, it was not found that any
additional strength was obtained over that of uniform cylindrical
pillars."— .Efcp. Res., p. 349.
319. Solid square cast-iron pillars yield in the direction
of their diagonals. — Solid "square (cast-iron) pillars do not bend
or break in a direction parallel to their sides, but to their diago-
nals, nearly." — Exp. Res., p. 331.
320. Long pillars irregularly fixed lose from two-thirds to
four-fifths of their strength. — "A (long) pillar irregularly
fixed, so that the pressure would be in the direction of the diagonal,
is reduced to one-third of its strength, the case being nearly
252 PILLARS. [CHAP. xv.
similar to that of a (long) pillar with rounded ends, the strength of
which has been shown to be only Jrd of that of a pillar with flat
ends." — Exp. Res., p. 350. And in two experiments on long solid
cast-iron pillars with the ends formed so that the pressure would not
pass through the axis, but in lines one-fourth of the diameter and
one-eighth of the diameter respectively from one side, the breaking
weights were little more than one-half that of a pillar of the same
dimensions with the ends turned so that the force would pass
through the axis, that is, their strength was reduced to about }th
of that of a similar flat-bedded pillar. — Phil. Trans., 1840, pp.
413, 449.
331. Strength of similar long* pillars is as their transverse
area. — The strength of similar long pillars is nearly as the area
of their transverse section. As derived from Mr. Hodgkinson's
experiments on cast-iron, the strength varied as the 1'865 power
of the diameter or any other linear dimensions. — Exp. Res., p. 346.
This has already been proved theoretically in 308.
CAST-IRON PILLARS.
333. Hodgkinson's rales for solid or hollow round cast-
iron pillars whose length exceeds 3O diameters. — The fol-
lowing formulae have been deduced by Mr. Hodgkinson from his
experiments to represent the breaking weights of pillars with both
ends flat and well bedded, and whose lengths exceed 30 diameters.*
If the ends are rounded or otherwise insecurely bedded, the
breaking weight given by the formulas must be divided by 3 (311).
Let W = the breaking weight in tons,
/ = the length of the pillar in feet,
d = the external diameter in inches,
d t = the internal diameter of hollow pillars in inches,
W = a coefficient varying with the quality of the cast-iron,
and derived from experiment.
Long solid round pillars of cast-iron.
W = m^ (234)
* Plat. Tram., 1857, pp. 862, 872.
CHAP XV."| PILLARS. 253
Long hollow round pillars of Low Moor cast-iron, No. 2.*
W = 42-347^^ (235)
Ex. What is the breaking weight of a solid round cast-iron pillar 10 feet long and 2
inches in diameter ? From table I., m = 42*6 tons,
23.5
Answer (eq. 234), W = m
= *2'6
= 11-3 tons.
If the pillar be not very securely fixed at the ends, the breaking weight will
= y_? = 3-77 tons, and the safe load in practice will be £th of this = '63 tons,
O
provided the pillar is not subject to vibration, in which case the safe load will be only
TVth = 0-314 tons.
The three following tables contain the values of the coefficient
m, derived from experiments on solid pillars of cast-iron 10 feet
long and 2 J inches diameter, with their ends flat ; also the powers
of diameters and lengths of pillars. — Phil. Trans., 1857, pp. 872
and 850.
TABLE I. — COEFFICIENTS m in eq. 234 (representing the strength of a pillar 1 foot
long and 1 inch in diameter.
Description of iron.
Value of
coefficient m.
Old Park iron, No. 1.
Stourbridge — cold blast,
a*.
111858 =
tons.
49-94
Derwent iron, No. 1.
Durham — hot blast,
105079 =
46-91
Portland iron, No. 1.
Tovine, Scotland— hot blast,
104C98 =
46-47
Calder iron, No. 1.
Lanarkshire— hot blast, -
104137 =
46-49
London mixture.
One-half old plate iron, and one-half Calder iron,
92862 =
41-46
Level iron, No. 1.
Staffordshire — hot blast,
94202 =
42-05
* "The pillars from this iron were cast 10 feet long, and from 2 4 to 4 inches
diameter, approaching in some degree, as to size, to the smaller ones used in practice."
—Proc. Roy. Soc., Vol. viii., p. 319.
254 PILLARS. [CHAP. xv.
TABLE I.— COEFFICIENTS m in eq. 234 — continued.
Description of iron.
Value of
coefficient m.
Coltness iron, No. 1.
Edinburgh— hot blast, -
90119 = 40-23
Carron iron, No. 1.
County of Stirling— hot blast, -
89949 = 4016
Blaenavon iron, No. 1.
South Wales— cold blast,
86114 = 38-44
Old Hill iron, No. 1.
Staffordshire — cold blast,
75270 = 33-60
Second London mixture.
One-third No. 1 best Scotch pig-iron, and two-thirds old metal, -
104623 = 46-21
Low Moor iron, No. 2.
Yorkshire- cold blast, -
90674 = 40-48
Blaenavon iron, No. 3.
South Wales— cold blast,
92329 = 41-22
Mean of 13 irons,
95486 = 42-6
TABLE II.— POWERS OF DIAMETERS, OB
i-o3-5 = i-oooo
4-253.5 = 158-26
6'83'5 = 819-94
1-253'5 = 2-1837
4-33'5 =164-87
6-93-6 = 862-92
l-5»-s = 4-1335
4.43.5 = 178.68
7-03-4 = 907-49
1-758.5 = 7-0898
4-53'6 = 193-305
7-13'5 = 953-68
2.03-5 = n.314
4-63'6 =208-76
7-23-5 = 1001-53
2-13-5 = 13-4205
4-73'5 = 225-08
7'253'5 = 1026-08
2.23.s = 15.7935
4-753'5 = 233-58
7-33-6 = 1051-07
2-253-5 = 17-086
4.83'5 = 242-295
7.43.5 =iio2-33
2-33* = 18-452
4-93'6 =260-43
7-53'5 =1155-35
2-43.5 = 21-416
5.03.5 =279-51
7-63-» = 1210-17
CHAP. XV.J PILLARS.
TABLE II.— POWERS OF DIAMETERS, OR
255
5— continued.
2-53.5 = 24705
5-13.5 = 299-57
7'73'5 =1266-83
2-63-* = 28-340
6-23'5 =320-635
7-753'5 = 1295-85
2-73'5 = 32-3425
5-253'5 = 331-56
7-83-' = 1325-35
2-753'5 = 34-488
5-33'5 = 342-74
7'93-5 =1385-78
2-83-5 = 36-733
5-43-5 =365-91
8-03-5 =1448-15
2-93.5 = 41.533
5-53'5 = 390-18
8-253.5 = 1612-83
3.03.5 = 46.765
5.63.5 = 415.58
8-53'5 =1790-47
313.5 = 52-4525
5.73.5 =442-14
8-753-5 = 1981-66
3-23-5 = 58-617
5.753.5 = 455.87
9.03.5 = 2187-00
3-2535 = 61-886
5-83-* =469-89
9-253-5 = 2407-11
3-33'5 = 65-283
5-93.5 =498-86
9-53'5 =2642-61
3.43.5 = 72-473
6.03-5 =529-09
9-753-5 = 2894-12
3-53"5 = 80-212
6-13-5 =560-60
10-03'5 =3162-28
3-63'5 = 88-5235
6-23'5 =593-43
10-253-5 = 344773
3.73.5 = 97-433
6-253'6 = 610-35
10-53'5 =3751-13
3.753.5 = 102-12
6-33-5 = 627-61
10-753.5 = 4073-14
3-83'5 =106-965
6-43'5 =66318
11-03-5 = 4414.43
3.93-5 = 117-15
6-53-5 =700-16
H-253.5 = 4775-66
4.03.5 _ 128-00
6-63-5 = 738-59
11-53.5 =5157-54
4-13-5 =139-55
6-73.5 =778-51
11-753'5 = 5560-74
4.23.5 = 151-835
6-753.5 = 799-03
12-03-5 = 5985-96
TABLE III. — POWERS OF LENGTHS, OR I1'63.
11-63 = 1-
7i1>63 = 26-6901
5 6i-63 _ 91-7731
2>-63 = 3-0951
8i-63 = 29-6508
171-63 = 101-305
2ii.63 = 4.4529
9»-63 = 35-9265
IS'-63 = 111-197
31-63 = 5-9939
lQi-63 = 42-6580
191-63 = 121-442
41.53 = 9-5798
Hi-63 = 49-8276
201'63 = 132-032
51.63 =13-7823
121-63 = 57.4203
2H-63 _ 142-961
6i.63 =18-5518
131-63 = 65-4226
22i.63 _ 154-223
6|i-63 _ 19-8282
141.53 _. 73-8225
231-63 _ 165-812
71'63 = 23-8512
15' -63 - 82-6093
24»-63 = 177-723
256 PILLARS. [CHAP. xv.
323. Hodgkinson's rales for solid or hollow round cast-
iron pillars of medium length* i.e., pillars whose length
is less than 3O diameters* with both ends flat and well
hedded. — " The formulae above (eqs. 234, 235) apply to all (cast-
iron) pillars whose length is not less than about 30 times the
external diameter ; for pillars shorter than this, it will be necessary
to modify the formulae by other considerations, since in these
shorter pillars the breaking weight is a considerable proportion of
that necessary to crush the pillar. Thus, considering the pillar
as having two functions, one to support the weight, and the other
to resist flexure, it follows that when the material is incompressible
(supposing such to exist), or when the pressure necessary to break
the pillar is very small, on account of the greatness of its length
compared with its lateral dimensions, then the strength of the
whole transverse section of the pillar will be employed in resisting
flexure; when the breaking pressure is half of what would be
required to crush the material, one half only of the strength may
be considered as available for resistance to flexure, whilst the other
half is employed to resist crushing ; and when, through the short-
ness of the pillar, the breaking weight is so great as to be nearly
equal to the crushing force, we may consider that no part of the
strength of the pillar is applied to resist flexure." — Exp. Res., p.
337. Acting on this view, Mr. Hodgkinson devised the following
formula for the ultimate strength of medium pillars of cast-iron
and timber whose length is less than 30 diameters, with both ends
flat and well bedded.
where W = the breaking weight in tons derived from the formulae
for long pillars, on the hypothesis that the pillar
yields by flexure alone,
c = the crushing weight of a short length of the pillar, i.e.,
its sectional area multiplied by the crushing unit-
strain of the material in tons,
W' — the real breaking weight of the medium pillar in tons,
from the combined effects of flexure and crushing
CHAP. XV.] PILLARS. 257
Ex. 1. What is the breaking weight of a solid pillar of Blaenavon iron, No. 3, 9 feet
long and 6 inches in diameter, with flat ends carefully bedded, and whose crushing
strength = 37 '3 tons per square inch ?
From Table I., m = 41 '2 tons,
c = 37-3 X 28-3 = 1056 tons,
from eq. 234, W = 41 '2 — = 605 tons.
36
Answer, (eq. 236). Breaking weight, W = *05 * *056 = 457 tons.
605 ~r 792
If intended for a warehouse, the greatest load in practice should not exceed £th of
this, = 76 tons, and that only when the ends are adjusted with the greatest care, so as
to have a very uniform bearing ; when this is not the case the effect will be the same
as if the ends were rounded, in which case the breaking weight will be much less
(313), probably only — . = — Z = 228'5 tons, of which £th, or the safe working
load, will = 38 tons.
Ex. 2. What is the breaking weight of a hollow flat-bedded pillar of the same iron,
of the same height and external diameter, and whose internal diameter = 4 inches ?
On examining Table II. (294), we find that the crushing strength of Blaenavon
iron, No. 3, medium, = 3 7 '3 tons per inch, while that of Low Moor, No. 2,
medium, = 34'6 tons. We may therefore assume that the coefficient in eq. 285 for
hollow cylinders of Blaenavon iron is the same as that for Low Moor.
Here, c = 37'3 X 157 = 586 tons,
from eq. 235, W = 42'35 529~128 = 472 tons nearly.
36
Answer, (eq. 236). Breaking weight, W = i^ *|||j=303 tons,
of which £th, or the working load, = 50'5 tons, i.e., when the ends are fitted with
W
extreme care ; otherwise, — = 25'25 tons, is a sufficient load in ordinary practice.
334. A slight inequality in the thickness of hollow cast-
iron pillars does not impair their strength materially —
Roles for the thickness oi'liollcnv cast-iron pillars. — Referring
to castings of unequal thickness, Mr. Hodgkinson remarks : —
" In experiments upon hollow pillars it is frequently found that
the metal on one side is much thinner than that on the other ; but
this does not produce so great a diminution in the strength as
might be expected, for the thinner part of a casting is much
harder than the thicker, and this usually becomes the compressed
side."— Phil. Trans., 1857, p. 862.
In practice, neither the excess or want of thickness should
exceed 25 per cent, of the average thickness; if, for instance, a
258 PILLAKS. [CHAP. xv.
hollow pillar is specified to be 1 inch in thickness, then in no
place should the metal be less than J inch or more than 1£ inch
thick. General Morin gives the following rule, based on the
founder's experience, for the minimum thickness of ordinary hollow
cast-iron pillars* : —
Height of pillar in feet, 7 to 10 10 to 13 13 to 20 20 to 27
Minimum thickness in inches, '5 '6 -8 I'O
Another practical rule is to make the thickness of metal in no
case less than T^th of the diameter of the pillar.
3S5. + and H shaped pillars. — A cast-iron pillar of the +
form of section, " as in the connecting rod of a steam engine, the
ends being movable, is very weak to bear a strain as a pillar, and
indeed less than half the strength of a hollow cylindrical pillar
of the same weight and length, rounded at the ends." — Phil. Trans.,
1857, p. 893.— Emp. Res., p. 350.
A cast-iron pillar of the H form of section with rounded ends
was found to be " considerably stronger than the preceding, but
much weaker than a hollow cylinder of the same weight." Their
relative strengths, according to Mr. Hodgkinson's experiments, were
in the following proportions, all the pillars being of the same weight
and length and rounded at the ends. — Phil. Trans., 1840, pp. 413,
449.
Hollow cylindrical pillar, . . . .100
H shaped pillar, ..... 75
-J- shaped pillar, ..... 44
336. Relative strength of i omul, square, and triangular
solid cast-iron pillars. — From a comparison of Mr. Hodgkinson's
experiments it appears that long solid square cast-iron pillars are
about 50 per cent, stronger than solid cylindrical pillars of the same
length and of diameters equal to the sides of the squares, whereas
their area, i.e., their weight, is only 27 per cent, greater. This is equi-
valent to saying that the breaking unit-strain of a long solid square
cast-iron pillar is I1 178 times that of the inscribed circular pillar of
* Resistance des MaMriaux, p. 110.
CHAP. XV.] PILLARS. 259
equal length.— Phil Trans., 1840, pp. 431, 437. Solid triangular
pillars of cast-iron with flat ends are somewhat stronger than those
with either circular or square sections. — Phil. Trans., 1857, p. 893.
Their relative strengths, according to Mr. Hodgkirison's experiments,
were in the following proportions, all the pillars being of the same
weight and length : —
Long solid round pillar, . . . .100
„ square ,, ... 93
,, triangular ,, . . . .110
From this it follows that for practical purposes the round pillar
is the most economical form of solid cast-iron pillar, since the
shape of the triangle will generally prohibit its use.
337. Gordon's rules for pillars. — Professor Gordon has de-
duced from Mr. Hodgkinson's experiments the following convenient
formulas for the strength of pillars ; —
Let/ = the breaking weight per square unit of section, i.e., the
breaking unit-strain,
r =r the ratio of length to diameter,
a and b = constants depending on. the material and the section of
the pillar.
1°. Pillars with both ends flat and bedded with extreme care.
/= (237)
2°. Pillars with both ends jointed or imperfectly fixed.
338. Solid or hollow round cast-iron pillars. — The values
of the coefficients in Gordon's formula? for solid or hollow cast-iron
pillars are as follows : —
a = 36 tons,
The following table has been calculated from these equations, and
260
PILLARS.
[CHAP. xv.
shows at a glance the breaking weight per square inch of solid or
hollow round cast-iron pillars of various ratios of length to diameter.
TABLE IV.— FOB CALCULATING THE STRENGTH OF SOLID OB HOLLOW ROUND
CAST-IRON PILLARS.
Ratio of length to diameter.
5
10
15
20
25
30
35
40
45
50
55
r,o
65
70
75
SO
Breaking
weight in
tons per
square inch.
Both ends flat and bedded
with extreme care.
33-9
28-8
23
18
14
11
8'9
7-2
5-9
5-0
4-2
3'fi
3-1
27
2-4
2'1
Both ends jointed or im-
perfectly fixed.
28-8
18
11
7-2
5-0
3-6
2-7
2-1
1-7
1-4
1-15
•97
•83
•72
•03
•55
Ex. 1. What is the breaking weight of a solid round cast-iron pillar, 10 feet long and
2 inches in diameter ? Here, the ratio of length to diameter =60, and, if both ends are
flat and bedded with extreme care or otherwise securely fixed, the corresponding break-
ing weight per square inch = 3'6 tons ; multiplying this by the sectional area, we have,
Answer, Breaking weight = 3'1416 X 3'6 = 11'3 tons,
which agrees with the example in 333 calculated by Hodgkinson's rule.
If the ends are jointed or imperfectly fixed, we have,
Answer, Breaking weight = 31416 X *97 = 3 '05 tons,
and the working load should in general not exceed one-sixth of this, = '51 tons.
Ex. 2. What is the breaking weight of a hollow round cast-iron pillar 9 feet long, 6
inches external, and 4 inches internal, diameter ? Here, the ratio of length to diameter
= 18, and, if both ends are flat and bedded with extreme care, the corresponding
breaking weight per square inch = 20 tons ; multiplying this by the sectional area, =
15'7 square inches, we have,
Answer, Breaking weight = 15 7 X 20 = 314 tons.
If the ends are jointed, or are not flat bedded with extreme care, the breaking weight
per square inch = 8'5 tons and we have
Answer, Breaking weight = 157 X 8'5 = 133'45 tons,
of which one-sixth, = 22*24 tons, will be the safe working load when free from vibration,
as in a grain store ; if the pillar supports a factory floor with machinery in motion,
one-eighth, = 16' 68 tons, will be a sufficient load ; but if the pillar forms a moving part
of an engine, then one-tenth, = 13'34 tons, or even less, will be the proper working load.
The reader will observe that Gordon's rule in this example gives results which agree
tolerably closely with the 2nd example in 333 calculated by Hodgkinson's rule.
Ex. 3. What is the breaking weight of a solid round cast-iron pillar, 9 feet long and
6 inches in diameter, with both ends solidly imbedded ? Here, the ratio of length to
diameter = 18, and the corresponding breaking weight per square inch is 20 tons, and
we have,
Answer, Breaking weight = 8-1416X6X6X20 = 565>5 tong>
CHAP XV.]
PILLAKS.
261
This, it will be observed, is nearly 24 per cent, higher than the 457 tons in example 1,
(323) ; no doubt, because Professor Gordon's rule applies to average mixed irons, which
are in general stronger than simple irons, such as Blaenavon.
339. Solid or hollow rectangular cast-iron pillars. — It
appears from Mr. Hodgkinson's experiments that the breaking
unit-strain of a long solid square cast-iron pillar is T178 times
that of the inscribed circular pillar of equal length (3S6), and,
guided by this, we may modify Gordon's formulae to suit rectangular
pillars by making r = the ratio of length to least breadth, and
The following table has been calculated on this basis, and gives
the breaking weight per square inch of solid or hollow rectangular
cast-iron pillars of various ratios of length to breadth.
TABLE V.— FOB CALCULATING THE STRENGTH OF SOLID OB HOLLOW RECTANGULAR
CAST-IRON PILLARS.
Eatio of length to least breadth.
5
10
15
'JO
25
30
35
40
45
50
55
GO
Go
70
75
80
Breaking
weight in
tons pei-
square iii ch.
Both ends flat and bed-
ded with extreme care.
34-3
30
24-8
20
16
12-9
10-4
8-G
7-1
6-0
5-1
4'4
3-8
3-3
2-9
2-6
Both ends jointed or im-
perfectly fixed.
30
20
12-9
8-6
6-0
4-4
3-3
2'6
2-1
1-7
1-4
1-2
1-0
•90
•78
•69
Ex. 1. What is the breaking weight of a solid cast-iron pillar, 10 feet long and 2
inches square? Here, the ratio of length to breadth = 60, and, if both ends are
securely fixed, the corresponding breaking weight per square inch = 4'4 tons ; multi-
plying this by the area, we have,
Answer, Breaking weight = 4 X 4'4 = 17'6 tons.
If the ends are imperfectly fixed, we have,
Answer, Breaking weight = 4 X 1'2 — 4'8 tons.
Of which, in general, one-sixth, = '8 tons, will be the proper working load.
Ex. 2. What is the breaking weight of a hollow cast-iron pillar, 9 feet long, 6 inches
square, with metal one inch thick ? Here, the ratio of length to breadth = 18, and, if
both ends are flat and bedded with extreme care, the corresponding breaking weight per
square inch = 21 '84 tons. Multiplying this by the area, = 20 square inches, we have,
Answer, Breaking weight = 20 X 21 '84 = 436'8 tons.
If the ends are not very carefully bedded, the breaking weight per square inch = 10'02
tons, and we have,
Answer, Breaking weight = 20 X 10'02 = 200'4 tons,
262
PILLARS.
[CHAP. xv.
of which one-sixth, = 33 '4 tons, will be the safe working load for ordinary warehouses,
when free from vibration.
For the safe working load on cast-iron pillars see Chap. XXVIII.
WROUGHT-IRON PILLARS.
33O. Solid wrought-iron pillars. — Professor Gordon's for-
mula in 327 may be applied to solid rectangular wrought-iron
pillars by giving the coefficients the following values,
a = 16 tons I = ^
The following table has been calculated from these formula, and
gives the breaking weight per square inch of solid rectangular
wrought-iron pillars of various ratios of length to least breadth.
TABLE VI. — FOB CALCULATING THE STRENGTH OF SOLID EECTANGULAE
WROUGHT-IRON PILLARS.
Ratio of length to least breadth,
5
10
15
20
25
30
35
40
45
50
55
60
G5
70
75
80
Breaking
weight in
tons per
square inch.
Both ends flat and
bedded with ex-
treme care, -
15-8
15-5
15-
14-1
13-2
12-3
11-3
10-4
9-5
8-7
77
7-3
6-6
G-OS
5'6
5-1
Both ends jointed or
imperfectly fixed,
15-5
14-1
12-3
10-4
8-7
7'3
6-1
5-1
4-3
37
3-2
276
2-4
2-1
1-9
1-7
Ex. 1. What is the breaking weight of a solid square pillar of wrought-iron, 10
feet long and 2 inches square ? Here, the ratio of length to breadth = 60, and the
corresponding breaking weight per square inch, if both ends are very securely fixed, =
7 '3 tons ; multiplying this by the sectional area, we have,
Answer, Breaking weight = 4 X 7'3 = 29'2 tons.
If the ends are jointed or imperfectly fixed, we have,
Answer, Breaking weight = 4 X 276 = 11'04 tons,
of which one-fourth, = 2 76 tons, will be the safe working load if the pillar be free
from vibration, but if liable to shocks like the jib of a crane, one-sixth, = 1'84 tons,
will be enough. If, however, the bar forms a moving part of machinery, such as the
connecting rod of a steam engine, one-twelfth, = '92 tons, will generally be a sufficient
load.
Ex. 2. What is the breaking weight of a rectangular pillar of wrought-iron, 10 feet
long, and whose sectional area = 4X3 inches, with the ends securely riveted to a fixed
structure ? Here, the ratio of length to least breadth = 40, and the corresponding
breaking weight per square inch = 10*4 tons ; multiplying this by the area, we have,
Answer, Breaking weight = 4 X 3 X 10'4 = 124'8 tons.
CHAP. XV.]
PILLARS.
263
Of this, one-fourth, = 31 '2 tons, will be sufficient in practice for a stationary load, and
that only when the ends are rigidly secured.
The following table, arranged in a convenient form by Mr. G.
Berkley, M.I.C.E., contains the results of experiments on the
compressive strength of solid rectangular wrought-iron bars, with
their ends perfectly flat and well-bedded, which were made under
Mr. Hodgkinson's supervision during the experimental inquiry
respecting the Britannia and Conway tubular bridges.*
TABLE VII.— HODGKINSON'S EXPERIMENTS ON SOLID RECTANGULAR
WROUGHT-IRON PILLARS.
Form of section.
Length.
Least
breadth.
Sectional
area.
Ratio of
length to
least
breadth.
Breaking
weight.
Breaking
weight per
square inch
of area.
Hi
inches.
90-0
inches.
1-024
sq. ins.
1-049
88-0
fts.
10,236
tons 4*57
tt)S.
9,753
tons 4-354
'l!
60-0
1-024
1-0486
58-6
18,106
17,268
» 7'7°9
ll
30-0
1-023
1-0475
29-3
26,530
„ 11-843
25,327
1*023
•C
15-0
1-023
1-0465
14-6
36,162
„ 16-144
34,554
11
7'5
1-023
1-0465
7-3
50,946
» "'744
48,682
» "733
l!
3-75
1-023
1-0465
3-65
Bore 23-549
tons, = 22-5
tons per sq.
in., without
fracture.
E*8
120-0
•503
1-498
238-56
1,222
'545
8,157
» -363
mm
120-0
•766
2-306
156*6
7,793
3,379
* Proc. Inst. C. E., Vol. xxx.
264
PILLARS.
[CHAP. xv.
TABLE VII.— HODGKINSON'S EXPERIMENTS ON SOLID KECTANGULAR
WROUGHT-IRON PILLARS — continued.
Form of section.
Length.
Least
breadth.
Sectional
area.
Ratio
of length
to least
breadth.
Breaking
weight.
Breaking
weight per
square inch
of area.
Mi
inches.
120-0
inches.
•995
sq. ins.
2-975
120-0
His.
12,735
tons 5-685
4,280
tons 1*91
fc
120-0
1-51
4-53
80-0
46,050
„ 20-558
10,165
•• 4-537
™t
90-0
•5023
1*498
179-0
3,614
„ 1-613
2,410
» 1-076
3-OOS >,>
90-0
•9955
2-9915
90-0
29,619
9,912
>• 4-425
s-ec
90-0
90-0
60-0
1-53
•995
•507
4-59
5*8307
1-511
59-0
90-0
118-0
91,746
n 40-958
54,114
8,469
» 378
19,087
9,280
» 4-H3
5,604
„ 2-502
^^$^§^^^S
V
S'M
60-0
•507
1-498
119-28
8,496
» 379*
5,653
V
60-0
•767
2-309
78-0
29,955
12,969
» 5-79
A
JMM
60-0
60-0
30-0
•995
•996
•5026
2-995
5-8166
1-5011
60-0
60-0
60-0
54,114
„ 24-158
102,946
25,299
„ 11-294
18,067
„ 8-066
17,698
,» 7'901
16,853
» 7'5 *4
K^^$§^^:12
mj
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M 39^58
29,655
CHAP. XV.] PILLAES. 265
I have made the following abstract from the foregoing experi-
ments in order to show how closely they corroborate Gordon's
formula? when applied to solid rectangular wrought-iron pillars.
TABLE VIII. — TABLE DERIVED FROM HODGKINSON'S EXPERIMENTS ON SOLID
RECTANGULAR WROUGHT-IRON PILLARS CAREFULLY BEDDED.
Proportion of length to least breadth, .
7
15
30
40
60
80
90
120
160
180
Breaking weight per square inch in tons,
22
15
12
10
7'5
5
4'3
2-2
1-5
1
The breaking unit-strain of solid round wrought-iron pillars is
probably from 15 to 20 per cent, less than those given in Table VI.
for rectangular pillars.
331. Solid wrought-iron pillars stronger than cast-iron
pillars when the length exceeds 15 diameters. — Comparing
Tables V. and VI. which represent the relative strengths of solid
rectangular cast and wrought-iron pillars, we find that a cast-iron
pillar with round ends is stronger than one of wrought-iron when
the length is under 15 diameters, but above that ratio, wrought-iron
is the stronger of the two, thus corroborating the theoretic result
previously arrived at in 3O7.
333. Pillars of angle,, tee* channel and cruciform iron.—
Mr. Unwin has deduced from experiments made by Mr. Davies of
the Crumlin Works the following values for the coefficients of
Gordon's formulae in 337, when applied to pillars of angle, tee,
channel and cruciform wrought-iron.*
a = 19 tons, 6 = -L
In each of these sections the least diameter for calculation is to be
measured in that direction in which the pillar is most flexible. This
may be found by taking the shortest diameter of a rectangle or
triangle circumscribed about the section. The following tables
exhibit the results of Mr. Davies' experiments reduced to a con-
venient form by Mr. Berkley .f
* Iron Bridges and Roofs, p. 50.
t Proc. Inst. C. K, Vol. xxx.
) 42
CO
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li
CHAP. XV.] PILLARS. 269
333. Resistance of long plates to flexure. — An isolated
plate under compression may be regarded as a wide rectangular
pillar, or as a number of square pillars placed side by side, and it
will therefore follow the laws of pillars so far as deflection at right
angles to its plane is concerned. Hence, the ultimate resistance
of long unsupported plates to flexure is theoretically as the cube
of the thickness multiplied by the breadth and inversely as the
square of the length. Mr. Hodgkinson found that this closely
agreed with his experiments on plates whose length exceeded 60
times their thickness, and which were so long that they failed by
flexure with strains not exceeding 9 tons per square inch (see
Table VII.).* If, however, the plates form the sides of a tube,
this rule does not apply, since in that case they yield by buckling
or wrinkling of a short length and not by flexure, being held in
the line of thrust by the adjacent sides which enable them to bear
a greater unit-strain than if not so supported along their edges.
334. Strength of rectangular wronght-iron tubular pil-
lars is independent of their length within certain limits. —
When the length of a rectangular wrought-iron tubular pillar does
not exceed 30 times its least breadth, it fails by the bulging or
buckling of a short portion of the plates, not by flexure of the
pillar as a whole, and within this limit the strength of the tube
seems nearly independent of its length. It is quite possible that
the ratio of length to breadth of rectangular wrought-iron tubes
might be considerably greater than 30 without very materially
affecting their strength, but the recorded experiments do not
extend sufficiently far to determine this point.
335. Crushing unit-strain of wrought-iron tubes depends
upon the ratio between the thickness of the plate and the
diameter or breadth of the tube — Safe working-strain of
rectangular wrought-iron tubes. — The crushing unit-strain of
a wrought-iron tubular pillar is generally greater the thicker the
plates are in proportion to the diameter or breadth of the tube,
and in most of the experimental rectangular tubes which sustained
a compression of 10 tons per square inch or upwards the thickness
* Com. Rep., p. 119.
270 PILLARS. [CHAP. xv.
of the plate was not less than one- thirtieth of the breadth of the
tube. In the last experiment recorded in Table XII., a square
tube, 8 feet long, 18 inches in breadth, and made of J-inch plates
united by angle-irons in the corners, sustained a compressive strain
of 13' 6 tons per square inch. Unfortunately there were no further
experiments made on tubes thus strengthened at the angles. From
this and other experiments, but especially from one made during
the construction of the Boyne Viaduct to test the strength of a
braced pillar, and which is described in the appendix at the end
of this volume, I infer that the strongest form of rectangular cell
to resist buckling is one in whose angles the chief part of the
material is concentrated, making the sides of plating or lattice
work to withstand flexure of the angles, in which case the sides
act the part of the web, and the angles act as the flanges of a
girder.
From what has been said we may conclude that a rectangular
plate-iron tubular pillar, whose length does not exceed 30 times its
least breadth and whose greatest breadth does not exceed 30 times
the thickness of the plates, will sustain a breaking weight of not
less than 12 tons per square inch, especially if the corners are
strengthened by stout angle-iron. When the ends of such pillars
are properly fixed, as in the compression flange of a girder,
experience sanctions a working-strain of 4 tons per square inch in
ordinary girder-work, and 3 tons in crane-work where shocks may
be expected.
I have deduced the foregoing conclusions respecting tubular
pillars chiefly from experiments conducted under Mr. Hodgkinson's
supervision during the experimental inquiry respecting the Con-
way and Britannia tubular bridges. The following tables exhibit
the results of these experiments reduced to a convenient form by
Mr. G. Berkley,* and the reader can judge for himself how far
the experiments warrant the foregoing conclusions.
* Proc. lust. C. K, Vol. xxx.
CHAP. XV.]
PILLARS.
271
Breaking weight
per square inch
of area.
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272
PILLARS.
[CHAP. xv.
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per square inch of
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CHAP. XV.]
PILLARS.
273
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274
PILLARS.
[CHAP. xv.
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CHAP. XV.]
PILLARS.
275
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[CHAP. xv.
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CHAP. XV.] PILLAKS. 279
STEEL PILLARS.
336. Solid Steel Pillars. — Mr. B. Baker gives the following
values for the co-efficients in Gordon's formulas in 337, when
applied to solid steel pillars.*
Solid f Mil<l Steel . . a = 30 tons . . b =
round •<
pillars. ( Strong Steel . . a = 51 tons . . b =
Solid f Mild Steel . . a = 30 tons . . b =
rectangular •<
pillars. ( Strong Steel. . a = 51 tons . . b =
Ex. 1. What is the breaking weight of a mild cast-steel pillar, 10 feet long and 2
inches in diameter, securely fixed at both ends ? Here, the ratio of length to diameter
= 60, and we have, from eq. 237, the inch-strain,
30
f= , 60_X_60 = 8<4tons;
' 1400
multiplying this by the sectional area, we have,
Answer, Breaking weight = 3'1416 X 8-4 = 26'39 tons.
If the pillar is jointed at the ends, we have from eq. 238,
30
/= --4X60^60 = 2'658t°nS;
1400
multiplying this by the area as before, we have,
Answer, Breaking weight = 3 '141 6 X 2'658 = 8 '35 tons,
of which one-fourth, = 2 '09 tons, will be a sufficient load when the pillar is free from
vibration or shocks.
Ex. 2. What is the breaking weight of a mild cast-steel pillar, 1 0 feet long and 2
inches square, securely fixed at both ends ? Here, the ratio of length to breadth = 60,
and we have, from eq. 237, the inch-strain,
30
/=— 60X60 = 12-245 tons;
' 2480
multiplying this by the sectional area, we have,
Answer, Breaking weight = 4 X 12-245 = 49 tons nearly.
If the pillar is jointed at the ends, we have from eq. 238,
30
f= 60^60 = 4'4°5t0nS;
620
multiplying this by the area as before, we have,
Answer, Breaking weight = 4 X 4'405 = 17'62 tons,
of which one-fourth, = 4'405 tons, will be a sufficient load for pillars free from shocks.
* Strength of Beams, pp. 207, 209.
280 PILLARS. [CHAP. xv.
TIMBER PILLARS.
337. Square is the strongest form of rectangular timber
pillar — Hodgkinson's rules for §olid rectangular timber
pillars. — It appears from Hodgkirison's experiments that the
strength of long round or square timber pillars is nearly as the fourth
power of the diameter or side divided by the square of the length.
Also, " of rectangular pillars of timber it was proved experimentally
that the pillar of greatest strength, where the length and quantity of
material are the same, is a square."*
Hodgkinson gives the following rules for the strength of timber
pillars with both ends flat and well bedded and whose lengths
exceed 30 diameters.f
Let W = the breaking weight in tons,
/ = the length of the pillar in feet,
d = the breadth in inches,
Long square pillars of Dantzic oak (dry). —
W = 10-95 ~ (239)
Long square pillars of Red deal (dry). —
W = 7-8 j, (240)
Long square pillars of French oak (dry) 4 —
W=6-9^ (241)
When timber pillars are less than 30 diameters in length, they
come under the class of medium pillars, and their strength may be
calculated by eq. 236, the value of W being computed by one of
the equations just given. To find the strength of a rectangular
pillar, find as above the breaking weight of a square pillar whose
side is equal to the short side of the rectangle ; this multiplied by
the ratio of the long to the short side will give the breaking weight
of the rectangular pillar.
Ex. 1. What is the breaking weight of a pillar of white deal, 9 feet long, 11 inches
wide and 3 inches thick ? Looking at the table in 3OO, we find that the crushing
* Exp. Res., p. 351.
t Phil. Trans., 1840, pp. 425, 426.
£ The crushing strength of French oak, according to Rondelet, = 6,336 fts. per
square inch.— Phil. Trans., 1840, p. 427.
CHAP. XV.] PILLARS. 281
strength of white deal is about 1*2 times that of red deal, from which we may conclude
that the strength of a long square pillar of white deal, derived from eq. 240, is as
follows : — ,.
W = 1-2 X 7-8 1
From this, the breaking weight of a pillar 9 feet long and 3 inches square =
1-2 X 7 '8 ~2 = 9-36 tons, and we have for a pillar 11 inches wide,
Answer, Breaking weight = 11 X 9'36 = 34'32 tons.
3
o t .00
If the pillar be not very securely fixed at the ends, its breaking weight will = — — =
3
11-44 tons (311), of which £th, = 2'86 tons, will be a sufficient working load for
temporary purposes; and £th, = 1'43 tons, for permanent use where protected from
the weather.
Ex. 2. What is the breaking weight of a strut of red deal, 26 feet long and 13 inches
square ? If the strut were long enough to give way chiefly by flexure (over 30 diameters
in length), its breaking weight, from eq. 240, would be
W = 7-8 ljj| = 329-5 tons,
and if the strut were short enough (under 10 to 15 diameters in length), to give way by
crushing alone, its breaking weight would equal its sectional area multiplied by the
tabulated crushing strength of red deal in the table in 3OO, that is,
2240
As the strut is a medium-sized pillar, we have the true breaking weight, from eq. 236,
that is, provided the ends are very carefully bedded ; but if they are liable to rough
adjustment, as in the cross struts of a cofferdam, from which this example has been
taken, the breaking weight will probably be about £ the above, = 109 tons (313), and
the safe working load for this kind of temporary work will be one-fourth of this again,
= 27'25 tons.
33§. Rondelet's and Brereton's rules for timber pillars. —
Rondelet deduced the following rule from his experiments on the
compression of oak and fir.* Taking the force which would crush
a cube as unity, the force requisite to break a timber pillar with
fixed ends whose height is —
12 times the thickness, will be |
24 „ „ „ J
36 „ „ „ - - J
48 „ „ „ - - i
60 „ „ „ TV
• ^ ?i »> » ^i
* Navier ; Application de la Mecanique, p. 200.
282
PILLARS.
[CHAP. xv.
Rondelet also found that timber pillars do not begin to yield by
flexure until their length is about ten times their least lateral
dimensions. This rule is easily applied, as illustrated by the
following examples : —
Ex. 1. What is the breaking weight by Rondelet's rule of a white deal pillar, 9 feet
long, 11 inches wide, and 3 inches thick, with the ends very carefully secured ? From
the table in 3OO the crushing strength of white deal = 6781 R>s. per square inch, and
the crushing strength of a very short length of the pillar is therefore 11 X 3 X 6781,=
223,773 R>s. As the length of the plank is 36 times its least width, we have according
to Rondelet's rule,
Answer, Breaking weight = 223>778 = 74,591 Ibs. = 33'3 tons,
3
which differs but slightly from its strength calculated by Hodgkinson's rule in ex. 1,
337.
Ex. 2. What is the breaking weight of a red deal strut 26 feet long and 13 inches
square, with both ends securely fixed ? In ex. 2, 339, we found that the breaking
weight of a short length of the strut was 434 tons, and as the real length = 24 diameters,
Rondelet's coefficient is £ ; consequently we have,
A ft A
Answer, Breaking weight = -— = 217 tons,
which is almost identical with the strength calculated by Hodgkinson's rule in the
example referred to.
Mr. R. P. Brereton states that " in experiments made with large
timbers, with lengths of from ten to forty times the thickness, he
had found that timber 12 inches square and 10 feet long bore a
weight of 120 tons; when 20 feet long it bore 115 tons; when 30
feet long 90 tons ; and when 40 feet long it carried 80 tons."*
Plotting the curve of Mr. Brereton's experiments we get the
following : —
TABLE XIV. — FOR CALCULATING THE STRENGTH OF RECTANGULAR PILLARS OP
FIR OR PINE TIMBER.
Ratio of length to least breadth
10
15
20
25
30
35
40
45
50
Breaking weight in tons per
square foot of section,
120
118
115
100
90
84
80
77
75
This is probably the most useful rule yet published for the
strength of large pillars of soft foreign timber with their ends
* Proc. Inst. C. E., Vol. xxix., p. 66.
CHAP. XV.] PILLAKS. 283
adjusted in the ordinary manner, that is, without any special
precautions.
Ex. 1. What is the breaking weight of a red deal strut, 26 feet long and 13 inches
square ? Here, the ratio of length to side is 24, and the breaking weight in the table
for this ratio is 103 tons per square foot ; consequently, for 13 inches square,
Ansiver, Breaking weight = 18 X 1B X 103 = 121 tons, nearly.
1 ^i /\ 1 2t
This answer, it will be observed, approximates very closely to the 109 tons obtained by •
Hodgkinson's rule in ex. 2, 332.
Ex. 2. A pillar of ordinary memel timber, 20 feet long and 13 inches square,
was broken in a proving machine with 136 tons. What is its breaking weight
computed by the foregoing rule? Here, the ratio of length to side is 18 '5, and the
corresponding breaking weight from the table = 116 tons per square foot.
Answer, Computed breaking weight = 13 X 13 Xo116 = 136 tons.
12 X 12
STONE PILLARS.
339. Influence of the height and number of courses in
stone columns. — From Rondelet's experiments it would appear
that when three cubes of stone are placed on top of each other,
their crushing strength is little more than half the strength of a
single cube.* Vicat, however, attributes this result to imperfect
levelling and the absence of mortar or cement in the joints, and
he found from experiments on plaster prisms carefully bedded,
that the strength of a monolithic prism, whose height is A, being
represented by unity, we have the strength of prisms : —
Of 2 courses and of the height h = O930
Of 4 „ „ 2/i = 0-861
Of 8 „ „ 4A = 0-834
even without the interposition of mortar. He concludes that the
division of a column into courses, each of which is a monolith, with
carefully dressed joints and properly bedded in mortar, does not
sensibly diminish its resistance to crushing ; but he intimates that
this does not hold good when the courses are divided by vertical
joints.f
340. Crushing: strength of Rollers and Spheres. — From
M. Vicat's experiments it appears that the strength of cylinders
employed as rollers between two horizontal planes is proportional
* Morin, p. 72. t Idem, p. 76.
284
PILLARS.
[CHAP. xv.
to the product of their axis by the diameter, and that the strength
of spheres to resist crushing is proportional to the square of their
diameter. If the strength of a cube be represented by unity, that
of the inscribed cylinder standing on its base will be 0*80 ; that of
the same cylinder on its side will be O32 ; and that of the inscribed
sphere will be 0-26.*
BRACED PILLARS.
341. Internal Bracing — Example. — One of the chief practical
difficulties which occur in bridges of large span is the combination
of lightness with stiffness in long struts, such as the compression
bars of the web. The internal bracing represented in Fig. 102 is
a modification of the bracing so familiar in scaffolding. It is now
in common use for the compression bars of lattice girders, and the
bracing of iron piers, and as it unites the requisite qualities of
strength and lightness in an eminent degree, it is worth devoting
some space to investigating the nature of the strains in this form
of pillar.
The diagram represents the cross section and side elevation of a
Fig. 102.
lattice tubular girder of simple construction. The tension diagonals
(marked T,) intersect the compression diagonals (marked C,) at
moderate intervals, and keep them from deflecting, especially in
the plane of the girder. It is obvious, however, that long com-
pression bars, even though formed of angle or tee iron, have but
little stiffness in themselves, and we cannot trust to the tension bars
* Morin, pp. 75, 82.
CHAP. XV. | PILLARS. 285
keeping them in the line of thrust at right angles to the plane of
the girder, for the tension bars may not always be in a sufficient
state of strain (153). Hence, it is desirable, at least in long pillars,
to connect each pair of compression bars by internal cross-bracing,
as shown in the section. The strains to which a braced pillar is
subject may be investigated in the following manner, which, though
rude, is yet sufficiently approximate for practical purposes : —
Let Fig. 103 represent a pillar which has become deflected, either
from the weight resting more on one side than on the other, or
from defective construction, or from accident.
Fi£- 103- Let W = the weight resting on one
side,
D = ab = the lateral deflection
in the interval of two bays,
I = Wa = ao = the length of
one bay,
R = the radius of curvature of
the deflected pillar,
P = the resultant of the strains
in Wa and ac, i.e., the
nearly horizontal pres-
sure produced on the two
braces intersecting at a, in
consequence of the weight
being transmitted through
a curved pillar.
At the apex, a, three forces balance, viz., the nearly vertical pres-
sures (each = W,) in the two adjacent bays, and their resultant P.
Hence, we have P = — -, — ; but D = JR> therefore,
P = ^ (242)
The pillar may therefore be regarded as a girder, each of whose
flanges is subject to a longitudinal pressure equal to W, in addition
to having a weight P resting on each apex. Hence, the strains in
the bracing may be found by the methods already explained in
286 PILLARS. [CHAP. xv.
Chapters V. and VI. If the pillar have a tendency to assume an
S form, the strains developed in the internal bracing in one loop of
the curve may, to some extent, neutralize those produced in the
other. If, however, the pressure on one side exceed that on the
other by any known or assumed quantity, then their difference of
length, and the corresponding deflection, may be obtained as
explained in the chapter on deflection, but in practice, errors of
workmanship will almost always exceed the amount of deflection
produced by a difference of pressure and experience must dictate
the requisite allowance. Let, for example, a pillar with internal
bracing, composed of two systems of right-angled triangles, similar
to that represented in Fig. 102. be 30 feet long and two feet wide,
and let each bay be 2 feet in length, in which case there willbe 15
bays in each side, and let the total load on the pillar = 40 tons, or
20 tons on a side. Now, suppose that the maximum error of
workmanship amounts to half an inch of lateral deflection in the
centre of the pillar, in which case R will equal 2,700 feet, then
the pressure P, produced at each apex by a vertical pressure of 20
tons on each side of the pillar, is as follows : —
As there are 14 apices in each system of bracing, i.e., 1 on each
side, the strain in each of the end braces = - — —
= 328-6 Ibs. (eq. 120). We thus see that the strain in the
internal bracing is comparatively trifling, and that the difficulty of
providing against flexure in long compression braces is not so
formidable as might have been supposed. It will be observed that
the internal bracing develops longitudinal strains in the side bars
at each apex. These increments ar^ however, insignificant com-
pared with the pressure due to the weight.
343. Each bay of a braced pillar resembles a pillar with
rounded ends — Compression flanges of girders resemble
braced pillars. — In braced pillars the side bars must be made stiff
enough to resist flexure for the length of one bay between the apices
of the internal bracing. Each bay cannot, however, be regarded as
CHAP. XV.] PILLARS. 287
a pillar of this length firmly fixed at the ends, but rather as one
with rounded ends, since it might assume a waved form like the
letter S, consecutive bays deflecting in opposite directions. This
remark also applies to the compression flanges of girders. The
vertical webs preserve them from deflecting in a vertical plane ; the
cross-bracing between the flanges performs the same service in a
horizontal plane, and the compression diagonals, especially if they
are braced pillars, also convey a large share of rigidity from the
tension flanges and roadway to the compression flanges. The
failure of the latter, therefore, as far as flexure is concerned, is thus
generally confined to the short length of one bay.
343. Strength of braced pillars Is independent of length
within certain limits — Working strain. — From Hodgkinson's
experiments on plate-iron tubular pillars, it seems highly probable
that the strength of braced pillars is also within considerable limits
independent of their length, for internal bracing will generally be
made somewhat stronger than theory alone might require (334).
In my own practice I adopt 4 tons per square inch of gross
section (excluding, of course, the cross bracing,) for the working-
strain of wrought-iron braced pillars in ordinary girder-work. In
crane-work, where shocks may occur, 3 tons per square inch is
enough. In both cases the ends of the pillar are supposed to be
firmly fixed by construction.
CHAPTER XVI.
TENSILE STRENGTH OF MATERIALS.
344. Nature of tensile strain. — The tendency of tensile strain
is to draw the material into a straight line between the points of
attachment, and, unless its shape alters very suddenly or the mode
of attachment is defective, so as to produce indirect strain, each
transverse section will sustain a uniform unit-strain throughout its
whole area ; eq. 1 is, therefore, applicable to ties without any other
practical correction than this, that if the material be pierced with
holes, such as rivet or bolt holes in iron, or knots in timber, the
effective area for tension in any transverse section is not the gross,
but the net area which remains after deducting the aggregate area
of all the holes or imperfections which occur in that particular
transverse section.
CAST-IRON.
345. Tensile strength. — The following table contains the
results of Mr. Hodgkinson's experiments on the tensile strength
of various kinds of British cast-iron.* Those samples whose
specific gravity are given are the same irons as those whose
crushing strengths have been already stated in Table I., 894.
TABLE I.— TENSILE STRENGTH OF CAST-IRON.
Description of iron.
Specific
gravity.
Tearing weight
per square inch
of section.
Carron iron (Scotland),
No. 2, hot-blast, -
1
Ibs. tone.
13,505= 6-03
Ditto,
do., cold-blast, ....
1
16,683= 7-45
Ditto,
No. 3, hot-blast, -
1
17,755= 7-93
Ditto,
do., cold-blaHt, ....
I
14,200= 6-35
Devon iron (Scotland),
No. 3, hot-blast, - - -
21,907= 9-78
* Experimental Researches on the Strength and other Properties of Cast-Iron, by
Eaton Hodgkinson, p. 310. Also, Report of the Commissioners appointed to inquire
into the application of Iron to Railway Structures, 1849, p. 9.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 289
TABLE I. — TENSILE STRENGTH OF CAST-IRON — continued.
Description of iron.
Specific
gravity.
Tearing weight
per square inch
of section.
Buffery iron (near Birmingham), No. 1, hot-blast,
5
Ibs. tons.
13,434= 6'OU
Ditto, do., cold-blast,
1
17,466= 7-80
Coed-Talon iron (North Wales), No. 2, hot-blast,
i
16,676= 7-45
Ditto, do., cold-blast,
1
18,855= 8-40
Low Moor iron (Yorkshire), No. 3,
14,535= 6-50
16,542= 7-39
Low Moor iron, No. 1, -
7-074
12,694= 5-667
Ditto, No. 2,
7-043
15,458= 6-901
Clyde iron (Scotland), No. 1, -
7-051
16,125= 7-198
Ditto, No. 2, -
7-093
17,807= 7-949
Ditto, No. 3,
7-101
23,468=10-477
Blaenavon iron (South Wales), No. 1,
7-042
13,938= 6-222
Ditto, No. 2, first sample, -
7-113
16,724= 7-466
Ditto, No. 2, second sample,
7-051
14,291= 6-380
Calder iron (Lanarkshire), No. 1,
7-025
13,735= 6131
Coltness iron (Edinburgh), No. 3,
7-024
15,278= 6-820
Brymbo iron (North Wales), No. 1,
7-071
14,426= 6-440
Ditto, No. 3, -
7-037
6-989
15,508= 6-923
13,511= 6-032
Ystalyfera Anthracite iron (South Wales), No. 2,
7-119
14,511= 6-478
Yniscedwyn Anthracite (South Wales), No. 1, -
7-034
13,952= 6-228
Yniscedwyn Anthracite, No. 2, ...•».
7-013
13,348= 5-959
Mr. Morries Stirling's iron, denominated 2nd quality,*
7-165
25,764=11-502
Mr. Morries Stirling's iron, denominated 3rd quality, f
7-108
23,461=10-474
* Composed of Calder, No. 1, hot blast, mixed and melted with about 20 per cent,
of malleable iron scrap.
f Composed of No. 1, hot-blast, Staffordshire iron, from Ley's works, mixed and
melted with about 15 per cent, of common malleable iron scrap.
U
290 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
From these experiments it appears that the average tensile
strength of simple British irons is 7 tons per square inch. The
strength of mixed irons, however, often reaches 9 or 10 tons, while
that of some American cast-iron is nearly double of this.
346. Cold-blast rather stronger than hot-blast iron —
mixtures stronger than simple irons. — On comparing the
tenacity of hot and cold-blast iron in the first part of the foregoing
table, it will be observed that, with one exception, the cold-blast
irons are rather stronger than the hot-blast irons of the same make.
This is confirmed by experiments made in the United States, where,
since 1840, hot-blast iron has been condemned for ordnance pur-
poses.* The following are the conclusions which the late Mr.
Robert Stephenson deduced from a series of experiments on the
transverse strength of cast-iron bars, made preparatory to the com-
mencement of the high level bridge at Newcastle.
1. Hot-blast is less certain in its results than cold-blast.
2. Mixtures of cold-blast are more uniform than those of hot-
blast.
3. Mixtures of hot and cold-blast give the best results.
4. Simple samples do not run so solid as mixtures.
5. Simple samples sometimes run too hard, and sometimes too
soft for practical purposes .f
Having regard to the fact that hot-blast is now in general
use, and that it seems to improve some kinds of iron, probably
those of a hard nature, the best plan for the engineer to adopt
is to specify the test which he requires the iron to stand and
let the founder bear the responsibility of producing the required
result.
347. Re-melting within certain limits, increases the
strength and density of cast-iron. — Re-melting cast-iron seems
to have an important effect in increasing its density as well as in
* Report on the Strength and other Properties of Metals for Cannon. By Officers of
the Ordnance Department, U.S. Army. Philadelphia, 1856, p. 338.
f Rep. of Iron Com., App., p. 389.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
291
improving its tensile and transverse strength, as appears from the
following experiments by Major Wade on proof bars of No. 1
Greenwood pig-iron thrice re-melted :* —
TABLE II. — EXPERIMENTS ON THE TENSILE AND TRANSVERSE STRENGTH OF
RE-MELTED CAST-IRON.
Density.
Tearing weight
per square inch.
Coefficient of
transverse rupture,
S
R>s.
fts.
Crude pig-iron, -
7-032
15,129
5,290
Do. re-melted once,
7-086
21,344
6,084
Do. do. twice,
7-198
30,107
7,322
Do. do. three times,
7-301
35,786
9,448
In summing up the results of his experiments on re-melting
cast-iron, Major Wade observes, " the softest kinds of iron will
endure a greater number of meltings with advantage than the
higher (more decarbonized) grades, and it appears that when iron is
in its best condition for casting into proof bars of small bulk, it is
then in a state which requires an additional fusion to bring it up to
its best condition for casting into the massive bulk of cannon. In
selecting, and preparing iron for cannon, we may, therefore, proceed
by repeated fusions, or by varying the proportions of the different
grades, until the maximum tenacity in proof bars is attained ; the
iron will then be in its best condition for being again melted and
cast into cannon."
Experiments made by Sir William Fairbairn, for the British
Association, though on a much more limited scale than those by
Major Wade, also prove the advantage to be derived from repeated
fusions.f One ton of No. 3 Eglinton hot-blast iron was melted
18 times successively, each time under similar conditions of fusion,
Rep. on Metals for Cannon, pp. 242, 249.
Application of Iron to Building Purposes, p. 60.
292
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
and proof bars, 5 feet long and 1 inch square, were cast each time,
and broken by transverse strain, the distance between the supports
being 4 feet 6 inches. The results are given in the following
table:—
TABLE III.— EXPERIMENTS ON THE TRANSVERSE AND CRUSHING STRENGTH OP
KE-MELTED CAST-IRON.
No. of
meltings.
Specific
gravity.
Mean breaking weight
of bars exactly 1 in.
square, and 4 feet 6
inches between
supports.
Mean ultimate
deflection.
Power to resist
impact.
Crushing
weight per
square inch.
fts.
inches.
tons.
1
6-969
490-0
1-440
705-6
44-0
2
6-970
441-9
1-446
630-9
43-6
3
6-886
401-6
1-486
596-7
4M
4
6-938
413-4
1-260
520-8
40-7
5
6-842
431-6
1-503
648-6
411
6
6771
438-7
1-320
579-0
41-1
7
6-879
449-1
1-440
646-7
40-9
8
7-025
491-3
1-753
861-2
41-1
9
7-102
546-5
1-620
885-3
55-1
10
7-108
566-9
1-626
921-7
577
11
7-113
651-9
1-636
1066-5
69-8
12
7-160
692-1
1-666
1153-0
73-1
13
7-134
634-8
1-646
1044-9
66-0*
14
7'530
603-4
1-513
912-9
95-9
15
7-248
371-1
0-643
238-6
767
16
7-330
351-3
0-566
198-5
70-5
17
Lost.
...
...
...
...
18
7-385
312-7
0:476
148-8
88-0
In these experiments it will be observed that the transverse
strength increased up to the 12th melting, after which it fell off in
a marked degree.
348. Prolonged fusion, within certain limit*, increases
the strength and density of cast-iron. — The improvement due
to prolonged fusion is shown by the following experiments by
Major Wade on Stockbridge iron of the 2nd fusion.f
* The cube did not bed properly upon the steel plates, otherwise it would have
resisted a much greater force — probably 80 or 85 tons per square inch,
t Rep. on Metals for Cannon, pp. 40, 44.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
293
TABLE IV.— EXPERIMENTS ON PROLONGED FUSION.
Density.
Tearing weight per
square inch.
Coefficient of
transverse rupture,
Iron in fusion | hour,
7-187
K>s.
17,843
ibs.
7,126
Do. do. 1 „
7-217
20,127
8,778
Do. do. 14 „
7-250
24,387
10,083
Do. do. 2 „
7-279
34,496
11,614
In some experiments made at Woolwich Arsenal by Mr. F. J .
Bramwell, it was found that fusion for 3| hours increased the tensile
strength of No. 1 Acadian cold-blast iron, from Nova Scotia, from
7'5 to 1OS tons per square inch, or nearly 50 per cent. This
when cooled was re-melted with an equal proportion of the original
No. 1 iron and the tensile strength of bars cast immediately upon
re-melting was 11 tons, and after 4 hours fusion, 1S'5 tons per square
inch.*
On this subject Mr. Truran makes the following observa-
tions f: — " The composition of the original grey pig-iron doubtless
influences, in a very great measure, the amount of improvement
obtained with different periods of fusion. A refining of the iron
takes place; and the quantity of alloyed matters oxidized and
removed will vary with the character of the pig-iron. Carbon is a
principal ingredient in cast-iron ; and a long exposure, equally with
repeated meltings, offers a ready method of burning it away. The
reverberating column of gases in the re-melting furnace contains a
proportion of free oxygen, which combines with the carbon to form
carbonic acid ; but since the oxygen is in contact only with the
surface of the metal, its removal requires numerous fusions, or the
maintenance in fusion for a long period. Repeated fusions of the
* Proc. I. C. K, Vol. xxii., p. 559.
t The Useful Metals and their Alloys, pp. 215, 217. London : 1857.
294 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
iron are attended with a heavy waste of material, which goes far to
compensate for the increase of strength. The tensile strength,
as influenced by the size of the masses and rapidity of cooling,
varies with the condition of the iron previous to casting. If the
refining process, by lengthened fusion or numerous re-meltings, be
carried too far, the resulting product will be of a hard, brittle
quality ; and when cast into small articles, be chilled to that extent
as to be incapable of working with steel cut ting- tools. Cast into
larger articles, however, and cooled more slowly, a maximum
tenacity may be developed, and the texture of the iron be found of
a character to bear cutting-tools on its surface. Continuing the
operation too long also produces a thickening of the molten iron,
until it is of too great a consistence for the proper filling of the
moulds, and the prevention of air cavities in the body of the
casting. The burning away of the carbon is attended with a loss
of fluidity ; and this defect occurring, there is no remedy short of
introducing further portions of the original crude iron, to restore,
by mixing, a certain degree of fluidity."
349. Tensile strength of thick castings of highly decar-
bonized iron greater than that of thin ones — Annealing
small bars of cast-iron diminishes their density and tensile
strength. — It has been already shown (138) that the transverse
strength of thin castings exceeds that of thick ones, and it might
naturally be thought that this was always due to greater tensile
strength in the smaller castings. This, however, seems to be
disproved by the following experiments by Major Wade, of the
United States army, who found that small castings in vertical dry
sand moulds had a less tensile strength than large gun castings
similarly moulded and cast at the same time.* The diminution of
tensile strength in the small bars amounted to nearly 5 per cent.,
while their transverse strength was 14 per cent, greater than that of
bars cut from the guns, as is shown in the following table : —
* Report on Metals for Cannon, p. 45.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
295
TABLE V. — COMPARISON OF PROOF BARS CUT FROM THE BODY OF THE GUN, WITH
THOSE CAST AT THE SAME TIME IN SEPARATE VERTICAL DRY SAND MOULDS,
SHOWING THE DIFFERENCE IN THE SAME IRON, CAUSED BY SLOW COOLING IN LARGE
MASSES, AND MORE RAPID COOLING IN SMALL CASTINGS.
Guns.
Coefficient of
transverse rupture,
Tearing weight per
square inch.
Specific gravity.
Bar cut
from gun.
Bar cast
separate.
Bar cut
from gun.
Bar cast
separate.
Bar cut
from gun.
Bar cast
separate.
6-pounder gun,
Ibs.
8,415
Ibs.
9,880
ffis.
30,234
Ibs.
29,143
7-196
7-263
6-pounder gun,
9,233
9,977
31,087
30,039
7-278
7-248
8-inch gun, -
8,575
10,176
26,367
24,583
7-276
7-331
Mean,
8,741
10,011
29,229
27,922
7-|50
7-281
Proportional,
1-000
1-145
1-000
•955
1-000
1-004
" These results," observes Major Wade, " show that the transverse
strength is augmented by rapid cooling in small castings, and that
the tensile strength is increased by slow cooling in large masses.
The differences in specific gravity are less marked ; but it is some-
what higher in the small castings cooled rapidly." This conclusion,
however, must be qualified by further statements of the same author
at pp. 234 and 268 ; where, in allusion to similar experiments, he
says : — " Such results happen only in cases where the iron is very
hard. As a general rule, the tenacity of the common sorts of
foundry iron is increased by rapid cooling. In this case the
condition of the iron when cool was too high — that is to say, the
process of decarbonization had been carried too far — for a maximum
strength, when cooled rapidly, in small mass.es; although it was
in its best condition for casting into a large mass, where it must
cool slowly. As iron of high density, when cast into bars of small
bulk, is liable to become unsound and to contain small cavities, this
cause may account, in some measure, for the diminished tensile
strength in bars of high density." Major Wade found that
annealing small bars of cast-iron invariably diminished both their
density and tenacity.* American cannon iron, the reader will
* Report on Metals for Cannon, p. 234.
296 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
observe, is much stronger and denser than ordinary English cast-
iron, the mean tensile strength of a large number of American
guns cast in 1851 being 37,774 Ibs., or nearly 17 tons per square
inch.*
350. Indirect pull greatly reduces the tensile strength
of cast-iron. — Mr. Hodgkinson found " that the strength of a
rectangular piece of cast-iron, drawn along the side, is about one-
third, or a little more, of its strength to resist a central strain."!
In proving specimens of cast-iron in a testing machine it is essential
that the strain pass exactly through the axis of the specimen,
otherwise the apparent will be much less than the real tensile
strength.
351. Cast-iron not suited for tension. — Cast-iron is liable to
air-holes, internal strains from unequal contraction in cooling and
other concealed defects which often seriously reduce its effective
area for tension and, as its tenacity is only about one-third of that
of wrought-iron, the latter material or steel should be preferred
for tensile strains whenever practicable. For these reasons cast-
iron is seldom used in the form of a tie-bar. It frequently occurs,
however, in tension in the lower flanges of girders with continuous
webs, for the safe working strain in which see Chap. XXVIII.
WROUGHT-IRON.
353. Tensile strength of wronght-iron — Fractured area —
Ultimate set. — We are indebted to Mr. David Kirkaldy for an
exceedingly valuable series of experiments on the tensile strength
of wrought-iron and steel, made by means of a lever testing machine
at the works of Messrs. Robert Napier and Sons, Glasgow.} The
following tables contain abstracts of the more important results of
these experiments. The column headed " Tearing weight per
square inch of fractured area" gives the breaking weight per square
inch of the area when reduced by the specimen drawing out under
proof. The ratio of this to the " tearing weight per square inch of
* Report on Metah for Cannon, p. 276.
t Ex. Res., p. 312.
J Experiments on Wrouyht-iron and Steel, by David Kirkaldy, Glasgow, 1863.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
297
original area" indicates the quality of the iron, whether ductile or
the reverse. The soft and ductile irons draw out to a small
" fractured area," and consequently have a very high unit-strain
referred to it, whereas the hard irons stretch but little under proof,
and therefore have a comparatively low unit-strain referred to the
same standard. The last column, headed " Ultimate elongation
or tensile set after fracture," gives the ratio of the increment
of length after fracture to the original length before fracture,
in the form of a percentage of the latter. The figures in this
column are greater or less according as the material is more or less
ductile, and consequently, this " set after fracture" is a test of the
toughness and ductility of the iron under proof. In my own
practice I find that the " set after fracture" is more easily measured
than the " fractured area," and that it is a very convenient test of
the ductility and toughness of the iron.
TABLE VI.— TENSILE STRENGTH OF WROUGHT-IRON BARS.
•
NOTE.— All the pieces were taken promiscuously from engineers' or merchants' stores,
except those marked samples, which were received from the makers.
District.
Names of the Makers
or Works.
Description.
Tearing
weight
per
square
inch of
original
area.
Tearing
weight
per
square
inch of
fractured
area.
Ultimate elongation,
or tensile set after
fracture.
Low MOOR,
Rolled bars, 1 inch square,
fts.
60,364
fts.
117,147
per
cent.
24-9
Do.
Rolled bars, 1 inch round,
61,798
131,676
26-5
Do.
Rolled bars, }% inch,
for rivets,
60,075
125,775
20-5
1
Do.
Planed from 1 inch
square bars,
60,245
114,410
23-8
1
Do.
Forged from 1J inch
round bars,
66,392
115,040
20-2
BOWLING,
Rolled bars, 1 inch round,
62,404
114,220
24-4
Do.
Turned from 1J inch
round bars,
61,477
120,229
26-0
FARNLEY,
Rolled bars, 1 inch round,
62,886
127,425
25'6
298 TENSILE STRENGTH OF MATERIALS. [CHAP- XVI-
TABLE VI. — TENSILE STRENGTH OP WROUGHT-IRON BARS — continued.
District
Names of the Makers
or Works.
Description.
Tearing
weight
per
square
inch of
original
area.
Tearing
weight
per
square
inch of
fractured
area.
IS
Ibs.
Ibs.
per
>
J. BRADLEY and Co., ©
T Rolled bars, 1 inch
57,216
146,521
cent.
30-2
(Charcoal)
J2 1 round,
Do. B. B., Scrap, -
S I Rolled bars, 1 inch
59,370
123,805
26-6
CQ L round,
Do. S C ^
Rolled bars, f inch, for
56,715
112,336
22-5
<0
rivets,
•|
Do. do.
Rolled bars, finch round,
62,231
97,575
22-2
1'
G. B. THORNEYCROFT & Co.,
Rolled bars, \% inch, for
59,278
99,595
22-4
•s
TN S
rivets,
-S
02
LORD WARD, L^ W.R-O
Rolled bars, -^ inch, for
59,753
95,724
18-6
rivets,
MALINSLEE, $& BEST,
Rolled bars, f inch X
56,289
88,300
21-4
1 inch,
BAGNALL, ^ J. B.
Rolled bars, 1£ inch
55,000
75,351
17'3
round,
Do. do.
Do. do., turned down
55,381
80,638
19-1
to 1 inch,
1
ULVERSTON RIVET, =£^
Rolled bars, f inch round,
53,775
104,680
21-6
"1
BEST,
~
MERSEY Co., BEST,
Forged from | inch
60.110
86,295
16-9
N
square bars,
GOVAN, Ex. B. BEST,
Rolled bars, f inch square,
56,655
99,000
19-1
Do. do.
Rolled bars, f inch round,
57,591
95,248
17-3
Do. do.
Rolled bars, 1^ inch
58,358
97,821
23-8
round,
Do. do.
Rolled bars, 1 inch round,
59,109
98,527
22-3
Do. do.
Rolled bars, £ inch round,
58,169
101,863
19-2
1
Do. do.
Rolled bars, f inch round,
57,400
92,880
17-6
-a,
GOVAN, B. BEST,
Rolled bars, 1£ inch
60,879
84,770
17-0
rt
round,
3
Do. do.
Rolled bars, 1 inch round,
62,849
88,550
19-1
Do. do.
Rolled bars, | inch round,
61,341
•96,442
20-0
Do. do.
Rolled bars, J inch round,
64,795
97,245
17-3
Do do.
Rolled bars, f inch round,
59,548
95,706
16-9
GOVAN, -X-
Rolled bars, 1J inch
58,326
78,139
167
round,
Do. do.
Rolled bars, 1 inch round,
59,424
79,373
16-4
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 299
TABLE VI. — TENSILE STRENGTH OF WROUGHT-IRON BARS — continued.
District.
Names of the Makers
or Works.
Description.
Tearing
weight
per
square
inch of
original
area.
Tearing
weight
per
square
inch of
fractured
area.
Ultimate elongation,
or tensile set after
fracture.
fts.
tt)S.
per
cent.
GOVAN, -X-
Rolled bars, % inch round,
63,956
88,512
15-8
Do. do.
Rolled bars, % inch round,
61,887
95,319
18-8
GLASGOW, B. BEST,
Rolled bars, 1 inch round,
58,885
97,548
23-2
Do. do.
Rolled bars, }$ inch
58,910
97,559
21-3
round,
Do. do.
Forged from 1 inch
59,045
80,053
20-9
rolled bars,
Do. do.
Rolled bars, 1J inch
54,579
85,012
20-3
round,
Do. do.
Do., do., turned down
55,533
86,590
21-3
to 1 inch,
Do. do.
Do., do., forged down
56,112
81,508
18-6
to 1 inch,
Do. do.
Rolled bars, finch round,
59,300
99,612
20-0
GLASGOW BEST KIVET, -
Rolled bars, % inch round,
57,092
96,205
237
-§
COATBRIDGE, BEST ElVET,
Rolled bars, % inch round,
61,723
96,267
21-6
1
ST. ROLLOX, BEST KIVET,
Rolled bars, -^ inch
56,981
77,383
16-6
"s
round,
8
R. SOLLOCH E. BEST,
Rolled bars, ^ inch, for
57,425
96,959
177
J,
rivets,
•J3 (
<^> GOVAN, <^^>
Rolled bars, 1£ inch
57,598
114,866
24-8
M
round,
1
Do. do.
Do., do., turned down
57,288
116,869
25-6
to 1 inch,
H
Do. do.
Do., do., forged down
57,095
112,705
231
to 1 inch,
Do. do.
Rolled bars, 1 inch round,
58,746
113,700
25-2
Do. do.
Rolled bars, % inch round,
58,199
116,549
21-4
DEMDYVAN (Common), -
Rolled bars, 1% inch
51,327
54,100
6-3
round,
Do. do.
Do., do., turned down
55,995
63,280
11-1
to 1 inch,
Do. do.
RoUed bars, 1J inch,
54,247
60,856
7'3
forged down,
Do. do.
Rolled bars, 1 inch round,
53,352
58,304
6-8
BLOCHAIRN, B. BEST,
Rolled bars, 1 inch round,
56,141
90,313
21-3
BLOOHAIRN, BEST RIVET, -
Rolled bars, f inch round,
59,219
89,279
19-4
PORT DUNDAS, Ex. B. BEST,
Rolled bars, 1% inch
54,594
85,563
20-6
round,
GOVAN, Puddled Iron, -
Rolled bars, J X 2iinch,
46,771
48,057
3'4
forged down,
300
TENSILE STRENGTH OF MATERIALS.
CHAP. XVI.
TABLE VI. — TENSILE STRENGTH OP WROUGHT-IRON BARS — continued.
Description.
at.
Tearing
Tearing
11
weight
weight
|-c5
41
Names of the Makers
or Works.
per
square
inch of
per
square
inch of
|8§
"^0
* " 2
C
"C
original
fractured
E 2
"ao
area.
area.
S
Is
fi>S.
ibs.
per
cent.
1|
YSTALTFERA, Puddled Iron,
Rolled bars, f X 2f inch,
forged down,
29,626
29,818
0-6
Do. do.
Do., do., strips cut off,
38,526
39,470
2-0
/
HAMMERED SCRAP IRON, -
— —
53,420
94,105
24-8
BUSHELED IRON FROM
TURNINGS,
— —
55,878
72,531
16-6
Cut out of a CRANK SHAFT
>
of Hammered Scrap
Iron, 14" wide, and re-
Length of shaft,
47,582
59,003
21-8
1
o>
duced to the required
shape in the lathe, not
Across shaft, -
44,75S
50,971
16-8
g
on the anvil,
J
W
^j Lengthways, -
43,759
56,910
20-5
Do. do.
J Crossways,
38,487
42,059
8-4
HAMMERED ARMOUR PLATE,
16' 6"X3' 9"X4£", cut
j Crossways,
38,868
44,611
11-7
off the end and turned
down,
1
Do.
36,824
39,085
6-4
1
1
Per ECKMAN AND Co., R F.
Gothenburg,
^
I
' Strips cut off,
47,855
121,065
27-8
02
Do. do.
Forged round, -
48,232
150,760
26-4
1
PRINCE DEMIDOFF, CC ND,
"1 '
Strips cut off,
49,564
73,118
13-3
I
Do. Do.
>**
.Forged round, -
56,805
77,632
15-3
SWEDISH, OC
-
•
48,933
141,702
17-0
Do. & <& W
43,509
77,349
15-3
Strips cut off,
Do. £& C
f
42,421
63,632
15-2
i
RUSSIAN, tO P3
ji
-|
•
59,096
68,047
6-0
£
SWEDISH, OC
43
"
50,262
188,731
187
Do. $3 C
i
41,251
98,510
14-8
- Forged down,
Do. $3 & W
44,230
83,851
15-8
RUSSIAN, POPS
•
,
.
51,466
67,907
7-5
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
301
TABLE VII. — TENSILE STRENGTH OP ANGLE IRON.
NOTE. — All the pieces were taken promiscuously from engineers' or merchants'
stores, except those marked samples, which were received from the makers.
District.
Names of the Makers or Works.
jj
H
Tearing
weight
per
square
inch of
original
area.
Tearing
weight
per
square
inch of
fractured
area.
Ultimate
elongation,
or tensile
set after
fracture.
Yorkshire, -
FARNLEY,
T96
Ibs.
61,260
fta
104,468
per cent.
20-9
GLASGOW Best Scrap, -
1
56,094
71,764
15-0
GLASGOW Best Best,
T96
55,937
70,706
15-4
Lanarkshire,
Do. do.
1
55,520
62,373
8'5
Do. do.
A
53,300
65,770
12-8
Do. do.
f
51,800
64,962
127
ALBION & Best,
1
56,157
69,367
14-0
ALBION Best,
1
52,159
67,695
141
Staffordshire, i
Do. do.
tt
51,467
60,675
11-2
EAGLE Best Best,
1
54,727
71,441
137
EAGLE, -
n
50,056
58,545
8-8
Durham,
CON SETT Best Best,
CONSETT Ship Angle Iron,
i
&
53,548
50,807
65,554
58,201
12-6
5-8
TABLE VIII.— TENSILE STRENGTH OP WROUGHT-IRON STRAPS AND BEAM IRON.
NOTE.— All the pieces were taken promiscuously from engineers' or merchants'
stores, except those marked samples, which were received from the makers.
Tearing
Tearing
weight
weight
Ultimate
.1
per
per
elongation,
District.
Names of the Makers or Works.
iM
square
square
or tensile
2
inch of
inch of
set after
H
original
fractured
fracture.
area.
area.
fts.
fts.
per cent.
GLASGOW, Ship Beam, -
I 4
55,937
67,606
1079
Lanarkshire,
DUNDYVAN, Ship Strap,
itt
55,285
63,635
8-03
MOSSEND, Ship Strap, -
T9*l
45,439
50,459
518
Staffordshire,
THORNEYCROFT, Ship Strap,
k
52,789
59,918
8-03
S. Wales, -
DOWLAIS, Ship Beam, -
*
41,386
45,844
4-82
302
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE IX.— TENSILE STRENGTH OP WROUGHT-IRON PLATES.
NOTE.— All the pieces were taken promiscuously from engineers' or merchants'
stores, except those marked samples, which were received from the makers. L denotes
that the strain was applied lengthways of the plate ; C, crossways.
District.
Names of the Makers
or Works.
Thick.
eS
1
1
Tearing
weight
per square
inch of
original
area.
Tearing
weight
per square
inch of
fractured
area.
Ultimate
elongation,
or tensile
set after
fracture.
LOWMOOB,
A
L
C
fta.
52,000
50,515
fts.
64,746
57,383
per cent.
13-2
9-3
BOWLING,
1
L
C
52,235
46,441
61,716
50,009
11-6
5-9
Yorkshire, !
FARNLEY,
1
L
C
56,005
46,221
68,763
53,293
14-1
7'6
Do.
i
L
C
58,487
54,098
70,538
59,698
10-9
5-9
Do.
f
|L
|c
58,437
55,033
83,112
68,961
17-0
11-3
CONSETT,
1
L
C
51,245
46,712
59,183
52,050
8-93
6-43
Durham, >
Do. Best Best, -
A&tf
L
C
49,120
46,755
55,472
50,000
8-0
5-2
Do. do.
T?6 & A
L
C
53,559
45,677
62,346
48,358
11-5
4-0
J. BRADLEY & Co.,
S. C. ft
i
L
C
55,831
50,550
67,406
55,206
12-5
5-5
Do. L F do. -
I to*
L
C
56,996
51,251
66,858
56,070
13-0
5-9
Do. „ do. -
|»i
L
C
55,708
49,425
65,652
54,002
107
5-1
T. WELLS, Best Best
ft
A to A
L
C
47,410
46,630
51,521
48,348
4-0
3-4
Staffordshire,
KBM
TV
L
C
46,404
44,764
51,896
47,891
6-1
4-3
MALINSLEE, Best
1
L
C
52,572
50,627
62,131
55,746
8-6
5-8
G. B. THORNEYCROFT,
Best D W Best, -
it
L
C
54,847
45,585
62,747
47,712
11-2
4-6
J. WELLS & B. Best, -
«*«
IL
|c
45,997
49,311
51,140
54,842
67
7-0
LLOYDS, FOSTER, & Co.,
Best,
A to A
L
C
44,967
44,732
49,162
48,344
5-3
4-6
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 303
TABLE IX. — TENSILE STRENGTH OF WROUGHT-IRON PLATES — continued.
District.
Names of the Makers
or Works.
Thick.
a
1
8
tn
Tearing
weight
per square
inch of
original
area.
Tearing
weight
•per square
inch of
fractured
area.
Ultimate
elongation,
or tensile
set after
fracture.
Shropshire,
SNEDSHILL §& Best, -
A to A
L
C
Rs.
52,362
43,036
fts.
61,581
45,300
per cent.
9-6
2-8
MOSSEND, Best Best, -
1
L
C
43,433
41,456
46,038
43,622
3-3
2-9
GLASGOW, Best Boiler,
fto-B
L
C
53,849
48,848
60,522
52,252
9-3
4-6
Do. Ship,
T36 tO «
L
C
47,773
44,355
49,816
45,343
3-65
2-11
Do. Best Best, -
TVto^
L
C
45,626
41,340
48,208
42,430
4-34
2-37
Lanarkshire,
Do. do.
itof
L
C
53,399
41,791
59,557
43,614
8-95
2-63
Do. Best Scrap,
1
L
50,844
58,412
10-5
Makers' stamp uncertain,
-fctott
L
C
47,598
40,682
53,182
43,426
5-9
2-5
GOVAN, Best, -
ito|
L
C
43,942
39,544
45,886
40,624
3-4
1-4
<^> GOVAN <^>
H
L
C
54,644
49,399
66,728
54,020
11-6
6-5
353. Tensile strength of wrought-iron, mean results. —
The following short table contains the mean results of Mr.
Kirkaldy's experiments on the tensile strength of wrought-iron : —
TABLE X. — TENSILE STRENGTH OP WROUGHT-IRON, MEAN EESULTS.
188 bars, rolled, -
fta.
- 57,555
tons.
= 25f
72 angle -iron and straps,
54,729
= m
167 plates, lengthways,
160 plates, crossways,
50,737
- 46,171
= 22-65
= 20-6
21f
In my own experience I find that the common brands of plate-
iron which are manufactured for girder-work and ship-building are
304 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
about 10 per cent, weaker than the mean results in the foregoing
table, and that their set after fracture, lengthways, rarely exceeds
5 per cent, of the total length ; also that Staffordshire and North
of England iron are generally tougher than Scotch iron.
354. Hirkaldy's conclusions. — Mr. Kirkaldy sums up the
results of his experimental inquiry in the following concluding
observations, which the student should study carefully : —
1. The breaking strain does not indicate the quality, as hitherto assumed.
2. A high breaking strain may be due to the iron being of superior quality, dense,
fine, and moderately soft, or simply to its being very hard and unyielding.
3. A low breaking strain may be due to looseness and coarseness in the texture, or
to extreme softness, although very close and fine in quality.
4. The contraction of area at fracture, previously overlooked, forms an essential
element in estimating the quality of specimens.
5. The respective merits of various specimens can be correctly ascertained by com-
paring the breaking strain jointly with the contraction of area.
6. Inferior qualities show a much greater variation in the breaking strain than
superior.
7. Greater differences exist between small and large bars in coarse than in fine
varieties.
8. The prevailing opinion of a rough bar being stronger than a turned one is
erroneous.
9. Rolled bars are slightly hardened by being forged down.
10. The breaking strain and contraction of area of iron plates are greater in the
direction in which they are rolled than in a transverse direction.
11. A very slight difference exists between specimens from the centre and specimens
from the outside of crank shafts.
12. The breaking strain and contraction of area are greater in those specimens cut
lengthways out of crank shafts than in those cut crossways.
13. The breaking strain of steel, when taken alone, gives no clue to the real qualities
of various kinds of that metal.
14. The contraction of area at fracture of specimens of steel must be ascertained as
well as in those of iron.
15. The breaking strain, jointly with the contraction of area, affords the means of
comparing the peculiarities in various lots of specimens.
16. Some descriptions of steel are found to be very hard, and, consequently, suitable
for some purposes ; whilst others are extremely soft, and equally suitable for other uses.
17. The breaking strain and contraction of area of puddled-steel plates, as in iron
plates, are greater in the direction in which they are rolled; whereas in cast-steel
they are less.
18. Iron, when fractured suddenly, presents invariably a crystalline appearance ;
when fractured slowly, its appearance is invariably fibrous.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 305
19. The appearance may be changed from fibrous to crystalline by merely altering
the shape of specimen, so as to render it more liable to snap.
20. The appearance may be changed by varying the treatment, so as to render the
iron harder and more liable to snap.
21. The appearance may be changed by applying the strain so suddenly as to render
the specimen more liable to snap, from having less time to stretch.
22. Iron is less liable to snap the more it is worked and rolled.
23. The " skin" or outer part of the iron is somewhat harder than the inner part, as
shown by appearance of fracture in rough and turned bars.
24. The mixed character of the scrap-iron used in large forgings is proved by the
singularly varied appearance of the fractures of specimens cut out of crank shafts.
25. The texture of various kinds of wrought-iron is beautifully developed by im-
mersion in dilute hydrochloric acid, which, acting on the surrounding impurities,
exposes the metallic portion alone for examination.
26. In the fibrous fractures the threads are drawn out, and are viewed externally,
whilst in the crystalline fractures the threads are snapped across in clusters, and are
viewed internally or sectionally. In the latter cases the fracture of the specimen is
always at right angles to the length ; in the former it is more or less irregular.
27. Steel invariably presents, when fractured slowly, a silky fibrous appearance ;
when fractured suddenly, the appearance is invariably granular, in which case also
the fracture is always at right angles to the length ; when the fracture is fibrous, the
angle diverges always more or less from 90°.
28. The granular appearance presented by steel suddenly fractured is nearly free of
lustre, and unlike the brilliant crystalline appearance of iron suddenly fractured ; the two
combined in the same specimen are shown in iron bolts partly converted into steel.
29. Steel which previously broke with a silky fibrous appearance is changed into
granular by being hardened.
30. The little additional time required in testing those specimens, whose rate of
elongation was noted, had no injurious effect in lessening the amount of breaking
strain, as imagined by some.
31. The rate of elongation varies not only extremely in different qualities, but also
to a considerable extent in specimens of the same brand.
32. The specimens were generally found to stretch equally throughout their length
until close upon rupture, when they more or less suddenly drew out, usually at one
part only, sometimes at two, and, in a few exceptional cases, at three different places.
33. The ratio of ultimate elongation may be greater in short than in long bars in
some descriptions of iron, whilst in others the ratio is not affected by difference in the
length.
34. The lateral dimensions of specimens forms an important element in comparing
either the rate of, or the ultimate, elongations — a circumstance which has been hitherto
overlooked.
35. Steel is reduced in strength by being hardened in water, while the strength is
vastly increased by being hardened in oil.
36. The higher steel is heated (without, of course, running the risk of being burned)
the greater is the increase of strength, by being plunged into oil.
X
306 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
37. In a highly converted or hard steel the increase in strength and in hardness is
greater than in a less converted or soft steel.
38. Heated steel, by being plunged into oil instead of water, is not only considerably
hardened, but toughened by the treatment.
39. Steel plates hardened in oil, and joined together with rivets, are fully equal in
strength to an unjointed soft plate, or the loss of strength by riveting is more than
counterbalanced by the increase in strength by hardening in oil.
40. Steel rivets, fully larger in diameter than those used in riveting iron plates of
the same thickness, being found to be greatly too small for riveting steel plates, the
probability is suggested that the proper proportion for iron rivets is not, as generally
assumed, a diameter equal to the thickness of the two plates to be joined.
41. The shearing strain of steel rivets is found to be about a fourth less than the
tensile strain.
42. Iron bolts, case-hardened, bore a less breaking strain than when wholly iron,
owing to the superior tenacity of the small proportion of steel being more than coun-
terbalanced by the greater ductility of the remaining portion of iron.
43. Iron highly heated and suddenly cooled in water is hardened, and the breaking
strain, when gradually applied, increased, but at the same time it is rendered more
liable to snap.
44. Iron, like steel, is softened, and the breaking strain reduced, by being heated
and allowed to cool slowly.
45. Iron subject to the cold-rolling process has its breaking strain greatly increased by
being made extremely hard, and not by being " consolidated," as previously supposed.
46. Specimens cut out of crank-shaft are improved by additional hammering.
47. The galvanizing or tinning of iron plates produces no sensible effects on plates
of the thickness experimented on. The result, however, may be different, should the
plates be extremely thin.
48. The breaking strain is materially affected by the shape of the specimen. Thus
the amount borne was much less when the diameter was uniform for some inches of
the length than when confined to a small portion— a peculiarity previously unascer-
tained, and not even suspected.
49. It is necessary to know correctly the exact conditions under which any tests are
made before we can equitably compare results obtained from different quarters.
50. The startling discrepancy between experiments made at the Koyal Arsenal, and
by the writer, is due to the difference in the shape of the respective specimens, and not
to the difference in the two testing machines.
51. In screwed bolts the breaking strain is found to be greater when old dies are
used in their formation than when the dies are new, owing to the iron becoming harder
by the greater pressure required in forming the screw thread when the dies are old
and blunt than when new and sharp.
52. The strength of screw-bolts is found to be in proportion to their relative areas,
there being only a slight difference in favour of the smaller compared with the larger
sizes, instead of the very material difference previously imagined.
53. Screwed bolts are not necessarily injured, although strained nearly to their
breaking point.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 307
54. A great variation exists in the strength of iron bars which have been cut and
welded ; whilst some bear almost as much as the uncut bar, the strength of others is
reduced fully a third.
55. The welding of steel bars, owing to their being so easily burned by slightly over-
heating, is a difficult and uncertain operation.
56. Iron is injured by being brought to a white or welding heat, if not at the same
time hammered or rolled.
57. The breaking strain is considerably less when the strain is applied suddenly in-
stead of gradually, though some have imagined that the reverse is the case.
58. The contraction of area is also less when the strain is suddenly applied.
59. The breaking strain is reduced when the iron is frozen ; with the strain gra-
dually applied, the difference between a frozen and unfrozen bolt is lessened, as the
iron is warmed by the drawing out of the specimen.
60. The amount of heat developed is considerable when the specimen is suddenly
stretched, as shown in the formation of vapour from the melting of the layer of ice on
one of the specimens, and also by the surface of others assuming tints of various shades
of blue and orange, not only in steel, but also, although in a less marked degree, in
iron.
61. The specific gravity is found generally to indicate pretty correctly the quality of
specimens.
62. The density of iron is decreased by the process of wire -drawing, and by the
similar process of cold rolling, instead of increased, as previously imagined.
63. The density in some descriptions of iron is also decreased by additional hot-
rolling in the ordinary way ; in others the density is very slightly increased.
64. The density of iron is decreased by being drawn out under a tensile strain,
instead of increased, as believed by some.
65. The most highly converted steel does not, as some may suppose, possess the
greatest density.
66. In cast-steel the density is much greater than in puddled-steel, which is even
less than in some of the superior descriptions of wrought-iron.
The foregoing extracts afford the reader but a meagre idea of
Mr. Kirkaldy's laborious researches, and the student who seeks
more detailed information regarding his experiments, or the instru-
ments and method he adopted in testing specimens, is referred to
his book on the subject.
355. Strength of iron plates lengthways 1O per cent,
greater than crossways — Removing skin of wrought-iron
does not injnre its tensile strength. — From Table X. it appears
that the average strength of wrought-iron plates drawn in the
direction of their length is about ten per cent, greater than when
drawn across the grain. The "set after fracture" is also much
greater in the direction of the fibres. This agrees with Mr. Clark's
308 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
experiments* as well as with my own experience. With reference
to the effect of removing the outer skin or glaze on rolled iron,
Mr. Kirkaldy observes, " The generally received opinion, that by
removing the * skin' the relative strength was greatly reduced, or
that a rough bar was much stronger than one turned to the same
diameter, is proved to be erroneous." f
356. Bar and angle iron are tougher and stronger than
plates — Boiler plates — Ship plates — Hard iron unfit for ship-
building.— Both bar and angle iron are tougher and stronger than
plate iron, and from Table X. it appears that bars of ordinary
sizes are nearly 14 per cent, stronger than plates; perhaps this
does not apply to bars of large section, say three inches in diameter
and upwards. The great demand for iron ships has given rise to
the manufacture of a cheap quality of plate iron called "ship" or
"boat" plates; this iron is generally inferior in strength and
toughness to "boiler" plates, and is often so hard and brittle that
its set after fracture does not exceed two or three per cent, of the
length, even with the grain, while its tensile strength is frequently
less than eighteen tons per square inch. There can be no greater
mistake than to suppose that hard iron is fit for ships. Iron plates
which are tough and ductile like copper will, when struck, often
escape with a mere dint or bulge, whereas hard iron under the same
circumstances will crack or tear, especially along a line of rivet holes.
357. Large forgings not so strong as rolled iron —
Annealing reduces the tensile strength of small iron, but
increases its ductility — Annealing injurious to large forgings
— Very prolonged annealing injurious to all wrought-iron —
Excessive strain renders iron brittle. — It is generally believed
that large forgings are less tenacious than small ones. About
this, however, there is some difference of opinion, and the sub-
ject requires further experiments before it can be definitively
settled, t Large forgings certainly require greater manufacturing
skill than small ones, and it is probable that large forgings, such as
* Clark on the Tubular Bridges, p. 377.
t Expts., p. 27.
t See discussion on Mr. Mallet's paper on the Coefficients of Elasticity and Rupture
in Massive Forgings.— Proc. Imt. C. E., Vol. xviii., p. 296.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 309
shafts for marine engines, are somewhat weaker in tensile strength
than bar or plate iron to which the rolling process imparts a fibrous
structure; this view seems to be confirmed by Mr. Kirkaldy's
experiments on hammered iron in Table VI. Annealing small
iron reduces its tensile strength (354 44), though it increases
its ductility and toughness, which are sometimes more important
qualities. For instance, it is a good practice to anneal old crane
chains which have become brittle by overstraining, and thus
render them less liable to snap from sudden jerks. Annealing large
forgings is injurious, as it produces a crystalline structure, the
reverse of fibrous, and very prolonged annealing of small sized iron
seems to have a similar bad effect.* If an iron bar be torn asunder
several times in succession, its tensile strength each time will
apparently increase, because it first gives way at the weakest point,
next time at the second weakest, and so on ; but though several
applications of the tearing strain do not diminish its ultimate
strength to resist a steady pull, they take the ductility or stretch
out of the iron and render it hard and brittle and therefore liable
to snap from sudden shocks. For the safe working load of wrought-
iron see Chap. XXVIII.
IRON WIRE.
358. Tensile strength of iron wire — Annealing; iron wire
reduces its tensile strength. — From Mr. Telford's experiments
it appears that the strength of iron wire ^th inch diameter = 36
tons per square inch.f The strength of the iron wire used by
Mr. Roebling at the Niagara Falls suspension bridge was nearly
100,000 Ibs. (= 44'6 tons) per square inch. This wire measures
18*31 feet per lb., and is " small No. 9 Gauge, 60 wires forming
one square inch of solid section. "J
The following table contains the results of experiments made by
M. Seguin on iron- wire of different sizes and qualities. §
* Morin, p. 47.
f Barlow on the Strength of Materials, p. 283.
J Papers and Practical Illustrations of Public Works of Recent Construction, both
British and American. Weale : 1856. pp. 16, 18.
§ RtsumZ des lemons sur Vapplication de la Mecanique. Par M. Navier. Bruxelles,
1839, p. 30.
310
TENSILE STKENGTH OF MATERIALS. [CHAP. XVI.
TABLE XI. — TENSILE STRENGTH OP IRON WIRE.
Description of Wire.
Diameter.
Tearing weight per
square millimetre.
Iron wire from Bourgogne, No. 8, unequally
annealed,
Idem, No. 7, carefully annealed, -
millimetres.
1-172
1-062
kilogrammes.
38-2
361
Idem, No. 18, not annealed,
3-366
58-8
Idem, No. 7, not annealed, -
1-062
737
Fil de 1'Aigle, employed for carding,
0-2294
89-8
Passe-perle, rather soft,
0-5917
857
Wire from a factory in Besancon —
.
No. 1, soft,
0-6188
86-1
2, soft,
0-7078
87-0
3, brittle, -
07327
80-8
4, brittle, -
0-838
76-6
5, v&ry brittle,
0-9115
72-3
6 -
1-022
761
7 -
1-08
71-2
8, very brittle,
1123
67-3
9, rather brittle,
1-293
69-8
10, very soft,
1-435
64-8
11, very soft,
1-476
58-6
12 -
1-691
55-5
13 -
1-8
57-2
1 4, very soft, without elasticity,
2-072
49-3
15
2-226
51-9
16, very soft,
2-489
63-9
17, flawed, -
2-695
681
18' -
3-087
84-0
19 -
3-492
78-2
20 -
4-14
65-7
21 -
4-812
62-5
22, very brittle,
5-449
677
23, soft,
5-942
62-6
NOTE. — A millimetre equals very nearly "04 = ^th inch ; and kilogrammes per square
millimetre may be converted into tons per square inch by multiplying by 0'635.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
311
That annealing iron wire seriously impairs its tensile strength
may be inferred from the foregoing experiments.
STEEL.
359. Tensile strength* ultimate set and limit of elasticity
of steel. — The following table contains the results of experiments
on the tensile strength and other properties of steel bars 50 inches
long and 1-382 inch diameter (= 1-5 sq. inch), made by Mr.
Kirkaldy for the " Steel Committee," the samples being carefully
turned down from two-inch square bars.*
TABLE XII. — TENSILE STRENGTH AND LIMIT OF ELASTICITY OF STEEL BARS.
Kind of Steel.
Tearing weight
per square inch.
Ultimate
elongation, or
tensile set
after fracture.
Limit of
tensile
elasticity.
CRUCIBLE STEEL.
tons.
per cent.
tons.
f Tyres,
35-51^
9-17
20-62
Hammered, « Axles,
1
40-9*1 38-19
8-72
25-56
lllails,
38-14J
2-96
19-64
Boiled, Axles,
30-62
10-56
18-75
BESSEMER STEEL.
f Tyres,
35-09^
11-1
23-30
Hammered, -I Axles,
33-47 133-93
12-1
21-87
lllails,
33-24 J
12-8
21-43
f Tyres,
32-09^
18-8
19-19
Rolled, J Axles,
32-22 1 31-99
19-0
17-85
iRails,
81-67 J
16-0
20-09
Mean,
33-68
12-12
20-83
Table XIII. contains the results of additional experiments made
* Experiments on Steel and Iron ly a Committee of Civil Engineers, 1868-70.
312
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
by the same Committee at Woolwich Dockyard on various
descriptions of steel bars 10 feet long and 1J inch diameter.
TABLE XIII. — TENSILE STRENGTH AND LIMIT OF ELASTICITY OP STEEL BARS.
Kind of Steel
What the steel was
intended for.
Tearing
weight
per square
inch.
Ultimate
elongation,
or tensile
set after
fracture.
Limit of
tensile
elasticity.
Crucible cast steel from Swedish
bar iron, chisel temper,
tons.
5276
per cent.
5'29
tons.
26-00
Crucible cast steel,
...
51-01
7'29
25-50
Cast steel,
Tyres
43-48
474
26-00
Ditto,
Piston rods, &c. -
41-85
1-12
27-00
Crucible steel, -
...
40-54
4-13
20-50
Ditto,
Gun barrels
38-51
7-95
16-83
Hammered crucible cast steel, -
...
37-05
13-54
25-00
Crucible steel, -
...
35-47
9-63
20-00
Bessemer steel, -
(Faggoted, ham-)
( mered & rolled)
35-40
11-13
19-60
Cast steel,
Piston rods, &c. -
33-65
0-89
26-75
Rolled crucible cast steel,
...
34-43
2-02
20-50
Bessemer steel, -
...
3419
11-90
20-00
Ditto,
...
33-63
11-48
17-50
Ditto,
Tyres and axles -
33-66
13-61
16-50
Mean, -
...
38-97
7-48
21-97
Table XIV. gives the results of experiments by Sir William
Fairbairn on the mechanical properties of steel.*
* Brit. Asa. Rep., 1867.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
313
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314
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
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CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
315
Tables XV. and XVI. contain the principal results of Mr.
Kirkaldy's experiments on the tensile strength of steel bars and
plates.* His " conclusions" respecting steel will be found in 354.
TABLE XV.— TENSILE STRENGTH OF STEEL BARS.
NOTE. — All the pieces were taken promiscuously from engineers' or merchants'
stores, except those marked samples, which were received from the makers.
District.
Names of the Makers
or Works.
Description.
Tearing
•weight
per square
inch of
original
area.
Tearing
weight
per square
inch of
fractured
area.
Ultimate
elonga-
tion, or
set after
fracture.
R)S.
Ibs.
percent.
T. TURTON AND SONS,
o , ^
132,909
139,124
5'4
Cast Steel for Tools
^ s §
(from Acadian Iron),
^J3 °
THOMAS JOWITT, Cast
^ *"" °»
132,402
151,857
5-2
Steel for Tools,
£ £ **
Do. do., Cast Steel for
J %
124,852
150,243
7-1
Chisels,
""3 ^ ^
Do. do., Cast Steel for
^ $ a
115,882
147,670
13-3
Drifts,
8 r£ C«
T. JOWITT, Double Shear
S h
118,468
147,396
13-5
Steel,
8 ^ q
BESSEMER (tool), samples,
'o'l * ^
111,460
143,327
5-5
3J
WILKINSON, © Blister
Ifll
104,298
132,472
97
jg
Steel,
* 111
•1
T. JOWITT, Cast Steel
i- g a &
101,151
142,070
10-8
CQ
for Taps,
•^
T. JOWITT, Spring Steel,
Forged from f inch
rolled bars,
72,529
95,490
18-0
Moss AND GAMBLES,
Cast Steel for Rivets,
Rolled bars, f inch
round,
107,286
158,013
12-4
NAYLORS, VICKERS, AND
Rolled bars, f inch
106,615
158,785
87
Co., Cast Steel for
round,
Rivets,
SHORTKIDGE, HOWELL,
Rolled bars, •& inch,
90,647
142,920
137
AND Co., Homo-
for rivets,
geneous Metal,
Do., do.,
Forged, -
89,724
121,212
11-9
t(
MERSEY Co., Puddled
Forged, -
71,486
110,451
19-1
- 1
Steel,
BLOCHAIRN, Puddled
Rolled bars,
70,166
84,871
11-3
i
Steel,
Do., do.,
Forged from slabs,
65,255
80,370
12-0
3
Do., do.,
Forged from rolled
62,769
71,231
9-1
bars,
4
KRUPP, Dusseldorf, Cast
Rolled bars, round,
92,015
139,434
15-3
1
Steel for Bolts,
Expts. on Wrought Iron and Steel.
316
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XVI. — TENSILE STRENGTH OF STEEL PLATES.
NOTE. — All the pieces were taken promiscuously from engineers' or merchants'
stores, except those marked samples, which were received from the makers. L denotes
that the strain was applied lengthways of the plate ; C, crossways.
District
Names of the Makers or Works.
Thick.
C
3
1i
Tearing
weight
per square
inch of
original
area.
Tearing
weight
per square
inch of
fractured
area.
Ultimate
elonga-
tion, or
tensile
set after
fracture.
inch.
fte.
Ibs.
oercent.
T.TURTON AND SONS, Cast Steel,
i
P
94,289
96,308
100,063
111,811
5-71
9-64
i.
NAYLOR, VICKERS, AND Co.,
Cast Steel,
Moss AND GAMBLES, Cast Steel,
A &T%
1 L
110
L
1C
81,719
87,150
75,594
69,082
104,232
112,018
105,554
112,546
17-50 .
17-32
19-82
19-64
1
SHORTRIDGE, HOWELL, AND Co.,
Homogeneous Metal,
*
L
C
96,280
97,150
114,106
114,300
8-61
8-93
Do., do.,
I
C
96,989
113,305
14-4
Do., Second Quality, -
I
He
72,408
73,580
81,823
78,245
5-93
3-21
1
MERSEY Co., Puddled Steel (Ship
Plates),
***
L
C
101,450
84,968
109,552
91,746
279
1-25
MERSEY Co., Puddled Steel
" Hard,"
k
1C
102,593
85,365
107,827
89,116
4-86
3-30
s
Do. "Mild," do.,
i
1 1 T
rSl | JLj
1C
77,046
67,686
88,240
73,634
6-16
572
Do. do. (Ship Plates), -
A
L
71,532
77,520
3-57
il
BLOCHAIRN, Puddled Steel, -
A
!{c
102,234
84,398
108,079
87,877
3-60
2-68
i(
Do., do. (Boiler Plates),
A
L
C
96,320
73,699
107,614
76,646
8-22
4-14
36O. Steel plates often deficient in uniformity and tough-
ness— Punching as compared with drilling; greatly reduces
the tensile strength of steel plates; strength generally
restored by annealing — Annealing equalizes different quali-
ties of steel plates. — From the foregoing table it appears that
the difference of strength lengthways and crossways is often much
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 317
greater in steel than in iron plates, amounting to nearly 20 per
cent, in some specimens. The reader will also observe that the
ultimate tensile set of steel plates is in general small compared
with that of the tougher kinds of iron in Table IX. This
indicates the direction to which manufacturers of steel should
direct their attention, as for many purposes, especially shipbuild-
ing, toughness and ductility are quite as essential as great tensile
strength (356). Sometimes steel plates are so brittle as to fly
in pieces under the hammer, or split in punching, and thick plates
are said to possess this undesirable quality to a greater degree than
thin ones, and occasionally they fly without any apparent cause
whatever shortly after they have been riveted in place. Com-
plaints also are made of want of uniformity of texture, some
plates of a lot being all that could be desired, while others of the
same lot may be hard and brittle. Owing to this uncertainty
the manufacture of steel plates seems still in a transition state,
and consequently, engineers and shipbuilders have not made use of
the material to the extent to which its superior tensile strength
seems to destine it.
It appears from papers on the treatment of steel, read at the
annual meeting of the Institution of Naval Architects in April,
1868, that steel plates, such as are now sometimes used in ship-
building, may be obtained of a tensile strength of from 30 to 35
tons per square inch. Punching, as compared with drilling,
reduced the strength of Bessemer steel plates 33 per cent. It
was found, however, that annealing these punched Bessemer
plates restored them to their original strength. In other experi-
ments on mild puddled steel plates the loss of strength from
punching was 21 per cent., and there was no benefit from subse-
quent annealing. With mild crucible steel plates the loss of
strength from punching was 7 per cent., and the gain of annealed
over unannealed was 14 per cent. Annealing was also recom-
mended to equalize the strength of steel, as in a batch of plates
sent in by the same manufacturer the plates sometimes greatly
differ, and a bath of molten lead was recommended as a cheap and
certain mode of annealing. It was also stated that enlarging the
318 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
die when punching steel, so as to give the die a large clearance, as
much as ^th inch, round the punch and make a taper hole, gave
a great advantage with Bessemer steel, amounting to 25 per cent.,
but in experiments on iron plates it was found that a greater
clearance than the usual one of 7^th inch rather injured the iron.
Mr. Krupp says with regard to the treatment of cold cast-steel
boiler plates: — "In working the plates cold, all sharp turns,
corners, and edges must be avoided or removed. The surfaces of
cuts and rivet-holes must, before bending and riveting, be worked
and rounded off as neatly as possible, so that no rough and serrated
places remain after cutting and punching." He also recommends
as a general rule that the plates should be thoroughly and equally
annealed at a dark-red heat after every large operation, and that
they should certainly have such annealing at the conclusion of all
operations. The directions given by him as to bending hot are
as follows : — " The plates should be heated, preparatory to bend-
ing, to a heat not exceeding a bright cherry-red. Also the greatest
possible portion of the surface should be heated, and not merely
the edge, and even, where practicable, the whole plate should be
equally heated. By this means the strains which arise from local
heating and cooling, and which are much greater in cast-steel
plates, on account of their higher absolute and reflex density, than
in iron, are, by the general heating of the plate, more equably
distributed. The thickest and toughest plates can be broken by
local heating, bending and cooling. Bends which cannot be com-
pleted in one, or at most in two consecutive heatings, must be
made gradually and equably over the whole extent to be operated
on." In bending, for example, to an angle of 90°, the whole
plate should first be bent through about one-third of the angle,
then through another third, and finally to the complete angle : —
" After the whole of these operations, the plate is to be equably
annealed at a dark-red heat, which will thus equalize the strains
caused by the previous working."* For the safe working-strain of
steel see Chap. XXVIII.
* Reed on Shipbuilding.
CHAP. XVI.] TENSILE STEENGTH OF MATERIALS.
319
STEEL WIRE.
361. Tensile strength of steel wire. — In experiments made
for the Atlantic Telegraph the strength of steel wire '095 inch
diameter was 1950lbs., while that of special charcoal wire of the
same size was 750 fibs.*
VARIOUS METALS AND ALLOYS.
363. Tensile strength of various metals and alloys. — The
following table contains the tensile strength of various metals and
alloys by several experimenters.
TABLE XVII. — TENSILE STRENGTH OP VARIOUS META.LS AND ALLOTS.
Description of Metal.
Specific
gravity.
Initials of
Experi-
menters.
Tearing weight
per
square inch.
Aluminium Bronze,
RK.
Ibs. tons.
73,000 = 32-59
Brass, Fine Yellow Cast,
—
R.
17,968 = 8-02
Do., Wire,
—
D.
91,325 = 40-77
Copper, Wrought, reduced per hammer, -
—
R.
33,792 = 15-08
Do., do., in bolts,
—
K.
47,936 = 21-40
Do., Cast, -
—
R.
19,072= 8-51
Do., do., Lake Superior,
8,672
W.
24,252 = 10-82
Do., Sheet, -
—
N.
30,016 = 13-4
Do., Wire, not annealed,
8,741
M. D.
77,504 = 34-6
Do., do., annealed, -
8,741
M. D.
32,144 = 14-35
Gun Metal or Bronze, hard,
—
R.
36,368 = 16-23
Do., mean of 83 gun-heads,
8,523
W.
29,655 = 13-24
Do., mean of 5 breech-squares,
8,765
W.
46,509 = 20-76
Do., mean of 32 small bars cast in same
moulds with guns,
Do., small bars cast ( ir°n moulds' '
separately in j clay do>)
8,584
8,953
8,313
W.
W.
W.
42,019 = 1876
37,688 = 16-82
25,783 = 11-51
Do., in finished guns,
—
w.j
23,108 10-3 to
to 52,192 " 23-3
* Fairbairn's Useful Information for Engineers, third series, p. 282.
320 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XVII. — TENSILE STRENGTH OP VARIOUS METALS AND ALLOTS— continued.
•J
Description of Metal. •
Specific
gravity.
Initials of
Experi-
menters.
Tearing weight
per
square inch.
fts. tons
Yellow Metal, Patent, -
—
K.
49,185 = 21-9
Lead, Cast,
—
R.
1,824 = 0-81
Do., Sheet, -
—
N.
1,926= 0-86
Soft Solder, 2 parts tin to 1 lead by weight,
—
RK.
7,500= 3-35
Tin, Cast,
—
R.
4,736= 2-11
Do., Banco,
7,297
W.
2,122 = 0-95
Do.,
—
M. D.
2,845= 1-27
Zinc, Cast,
—
S.
2,993 = 1-336
D. Dufour, Application de la Mecanique, Navier. Brussels, 1839, p. 35.
M. D. Minard et Desormes, idem, pp. 34, 36.
N. Navier, idem, p. 36.
K. Kingston, Barlow on the Strength of Materials, p. 211.
R. Rennie, Philosophical Transactions for 1818, p. 126.
RK. Rankine's Machinery, p. 464.
S. Stoney.
W. Wade, Reports on Metals for Cannon, pp. 281, 288, 289, 290, 295.
Gun-metal or bronze — High temperature at casting:
injurious to bronze. — The proportion of tin to copper in the
bronze gun-metal on which Major Wade experimented was 1 to 8,
and the great diversity in its tenacity seems attributable to defective
homogeneity in the alloy, some parts containing more tin than others,
and consequently having a smaller tenacity. A high temperature
at casting is injurious to the quality of bronze, as it seems to
facilitate the separation of the metals, and small bars are stronger
than large castings, probably because the former solidify more
suddenly and are thereby not allowed a sufficient time for a division
of the alloy into separate compounds. Bronze guns are cast on
end in flask moulds, with the breech downwards, and a large extra
head of metal above the muzzle to ensure sufficient liquid pressure.
Breech-squares, being at the bottom of the moulds, are subject to
CHAP. XVI.] TENSILE STRENGTH OP MATERIALS.
321
a much higher pressure than the gun-heads which are at the top,
and they are consequently both stronger and denser than the latter.
The small bars cast in the gun mould are stronger than those cast
separately, probably in consequence of their being under greater
pressure, and because they were fed, as they solidified, from the
mass of the gun with which they communicated. Major Wade
also attributes their superiority to the annealing process they
underwent after solidification, from the proximity of the large
mass of the gun.*
364. Alloys of copper and tin. — The following table contains
the results of experiments made by Robert Mallet, Esq., F.R.S.,
on the physical properties of certain alloys of copper and tin.f
TABLE XVIII.— PHYSICAL PROPERTIES OF ALLOYS OP COPPER AND TIN.
COPPER AND TIN.
Chemical
Constitution.
Composition by
weight per cent.
Specific
gravity.
Tearing
weight
per square
inch.
Commercial Title.
10 Cu + Sn
84-29 + 15-71
8-561
tons.
16-1
Gun Metal.
9 Cu + Sn
82-81 + 17-19
8-462
15-2
Gun Metal. «*
8 Cu + Sn
81-10 + 18-90
8-459
177
Gun Metal, tempers best.
7 Cu + Sn
78-97 + 21-03
8-728
13-6
Hard Mill Brasses, &c.
Cu + Sn
34-92 + 65-08
8-056
1-4
Small bells, brittle.
Cu + 3 Sn
15-17 + 84-83
7-447
31
Speculum Metal of Authors.
Sn
0 + 100
7-291
2-5
Tin.
NOTE. — " The ultimate cohesion was determined on prisms of 0'25 of an inch square,
without having been hammered or compressed after being cast. The weights given
are those which each prism just sustained for a few seconds before rupture."
TIMBER.
365. Tensile strength of timber. — The following table con-
tains the results of experiments by various authorities on the
* Report on Metals for Cannon, pp. 296, 299.
f On the Construction of Artillery, p. 82.
322
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
tensile strength of timber drawn in the direction of the fibres.
For the safe working-strain see Chap. XXVIII.
TABLE XIX. — TENSILE STRENGTH OP TIMBER LENGTHWAYS.
Description of Wood.
Tearing weight
per
square inch.
Authority.
Alder,
ft>8.
13,900
Muschenbroeck.
Apple,
19,500
Be van.
Ash,
16,700
Do.
Do.
17,000
Barlow.
Beech,
11,500
Do.
Do.
17,300
Muschenbroeck.
Do.
22,000
Bevan.
Birch,
15,000
Do.
Box,
20,000
Barlow.
Cane,
6,300
Bevan.
Cedar,
11,400
Do.
Chesnut, Spanish, -
13,300
Rondelet.
Do.
10,500
Bevan.
Do., Horse,
12,100
Do.
Cypress, -
6,000
Muschenbroeck.
Deal, Christiana, -
12,900
Bevan.
Elder,
10,000
Muschenbroeck.
Elm,
14,400
Bevan.
Fir,
12,000
Barlow.
Hawthorn, -
10,000
Bevan.
Holly,
16,000
Do.
Jugeb,
18,500
Muschenbroeck.
Laburnum, -
10,500
Bevan.
Lance Wood,
23,400
Do.
Larch,
10,220
Rondelet.
CHAP. XVI.] TENSILE STRENGTH
TABLE XIX.— TENSILE STRENGTH OF
OF MATERIALS. 323
TIMBER LENGTHWAYS— continued.
Description of Wood.
Tearing weight
per
square inch.
Authority.
Larch,
fts.
8,900
Bevan.
Lemon,
9,250
Muschenbroeck.
Lignum Vitae,
11,800
Bevan.
Locust-tree,
20,100
Muschenbroeck.
Mahogany, -
8,000
Barlow.
Do.
16,500 to 21,800
Bevan.
Maple,
17,400
Do.
Mulberry, -
10,600
Do.
Do.
12,500
Muschenbroeck.
Oak, English,
10,000
Barlow.
Do., do.
14,000 to 19,800
Bevan.
Do., French,
13,950
Rondelet.
Do., Black Bog, -
7,700
Bevan.
Orange,
15,500
Muschenbroeck.
Pear,
9,800
Barlow.
Pine, Pitch,
7,650
Muschenbroeck.
Do., Norway,
14,300
Bevan.
Do., do.
7,287
Rondelet.
Do., Petersburg,
13,300
Bevan.
Plane,
11,700
Do.
Plum,
11,800
Muschenbroeck.
Pomegranite,
9,750
Do.
Poplar,
5,500
Do.
Do.
7,200
Bevan.
Quince,
6,750
Muschenbroeck.
Sycamore, -
13,000
Bevan.
Tamarind, -
8,750
Muschenbroeck.
324 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XIX. — TENSILE STRENGTH OF TIMBER LENGTHWAYS — continued.
Description of Wood.
Tearing weight
per
square inch.
Authority.
Teak,
fts.
15,000
Barlow.
Do., old, -
8,200
Bevan.
Walnut,
8,130
Muschenbroeck.
Do.
7,800
Bevan.
Willow,
14,000
Do.
Yew,
8,000
Do.
Barlow, Barlow on the Strength of Materials, p. 23.
Muschenbroeck, idem, p. 4.
Bevan, Philosophical Magazine, 1826, Vol. Ixviii., pp. 270, 343.
Rondelet, Tredgold's Carpentry, 4th edition, p. 41.
Comparing the foregoing table with Table VI. (3OO), we see
that the tensile strength of most kinds of wood is much greater
than their compressive strength.
366. Lateral adhesion of the fibres. — The following table
gives the lateral adhesion of the fibres, that is, the tensile strength
of timber across the grain, in which direction it is much weaker
than lengthways.
TABLE XX.— TENSILE STRENGTH OF TIMBER CROSSWATS.
Description of Wood.
Tearing weight
per
square inch.
Authority.
Fir, Memel,
ibs.
540 to 840
Bevan.
Do., Scotch,
562
Do.
Larch,
970 to 1,700
Tredgold.
Oak,
2,316
Do.
Poplar,
1,782
Do.
Bevan, Philosophical Magazine, 1826, Vol. Ixviii., p. 112.
Tredgold, Tredgold's Carpentry, p. 42.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
325
STONE, BRICK, MORTAR, CEMENT, GLASS.
367. Tensile strength of stone. — As stone is rarely employed
in direct tension, there are but few experiments on its tensile
strength, and it would be desirable to have these corroborated.
TABLE XXL— TENSILE STRENGTH OF STONE.
Name of Material.
Tearing weight
per
square inch.
Authority.
Arbroath Pavement,
ibs.
1,261
Buchanan.
Caithness do. ...
1,054
Do.
Craigleith Stone, -
453
Do.
Hailes,
336
Do.
Humbie, -
283
Do.
Binnie,
279
Do.
Kedhall, -
326
Do.
Whinstone,
1,469
Do.
Marble, White,
722
Do.
Do., do.
551
Hodgkinson.
Buchanan, Practical Mechanics' Journal, Vol. i., pp. 237, 285.
Hodgkinson, Tredgold on the Strength of Cast-iron, p. 287.
326
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
36§. Tensile strength of Plaster of Paris and Lime
mortar. —
TABLE XXII.— TENSILB STRENGTH OP PLASTER OP PARIS AND LIME MORTAR.
Name of Material.
Tearing weight
per
square inch.
Authority.
Plaster of Paris,
tt>s.
71
Eondelet.
Mortar of Quartzose Sand and eminently Hydraulic
Lime, well made, -
136
Vicat.
Mortar of Quartzose Sand and ordinary Hydraulic
Lime, well made, -
85
Do.
Mortar of Quartzose Sand and ordinary Lime, well
made,
51
Do.
Mortar badly made,
21
Do.
Eondelet, Navier's Application de la Mgcanique, p. 13.
Vicat, idem.
369. Tensile strength of Portland cement and cement
mortar — Organic matter or loam very injurious to cement
mortar. — The following tables showing the tensile strength of
cements and cement mortar are taken from Mr. Grant's valuable
papers on the Strength of Cement in the Proceedings of the Insti-
tution of Civil Engineers, Vols. xxv. and xxxii. Proof samples of
cement are generally made into ^^-shaped bricks with rounded
shoulders and 1J inches square, = 2*25 square inches area, at the
waist; these are immersed in water as soon as the cement sets,
and they remain immersed till the time of testing.
Artificial Portland cement is made of chalk and clay in certain
definite proportions, carefully mixed together in water. The
mixture is then run off into reservoirs where it settles, and, after
attaining sufficient consistency to handle, it is artificially dried
and calcined in kilns at a high temperature, the calcination being
carried to the verge of vitrification. The calcined cement is
ground in the ordinary way between millstones, and for the sake of
economy its fineness should be such that not more than 10 per cent.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
327
is stopped by a sieve the meshes of which are ^th of an inch in
diameter, for the coarser particles act to a great degree like inert
grains of sand and consequently reduce the value of the cement.
TABLE XXIII.— METROPOLITAN MAIN DRAINAGE— PORTLAND CEMENT,
SEVEN DAY TESTS, from 1866 to 1871.
Names of Manufacturers and Agents.
Quantity
in
bushels.
Average
weight per
bushel.
Number
of
tests.
Average breaking
weight
on area = 2-25 square
inches.
ibs.
fts.
Formby, -
31,581
118-27
550
862-01
Booth,
12,464
11975
80
846-50
Lee and Co.,
512
120-00
10
839-00
Burham Brick and Cement Com-
pany, -
320,716
113-54
3,705
825-73
Casson and Co., Agents, -
5,200
114-50
50
816-80
Knight, Bevan, and Sturge,
19,429
114-52
820
803-38
Eobins and Co. (Limited),
68,880
118-00
620
795-31
White and Co., -
60
119-00
10
791-70
Burge and Co., Agents, -
4,500
113-00
30
789-30
Hilton, -
103,453
117-17
1,300
786-99
Beaumont, Agent,
40
116-00
10
765-00
Lavers, Agent,
12,002
116-17
160
706-97
Weston, -
600
120-00
10
666-40
Young and Son, Agents,
200
117-00
10
655-80
Coles and Shadbolt,
240
107-00
10
580-00
Tingey,
6,300
115-50
100
564-27
Harwood and Hatcher, Agents, -
Generally,
3,040
11778
30
408-03
806-63 = 358-5
per square inch.
589,217
115-23
7,505
NOTE — 1 cubic foot = '779 bushels.
1 bushel = 1-283 cubic feet.
328
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XXIV.— Results of Experiments with Portland Cement, weighing 112fts.
per bushel, mixed with different proportions of Sand, showing the Breaking Weight
on a sectional area of 2'25 square inches.
1 Month.
6 Weeks.
2 Months.
6 Months.
12 Months.
Proportion of
sand to cement.
fts.
Ibs.
K>s.
fts.
fts.
306-0
383-0
407-5
505-5
541'0
3tol
403-5
397-5
411-0
479-0
554-5
4tol
Broke wind- )
ing up. j
246-0
269-5
439-0
482-0
5tol
133-5
189-5
221-0
273-0
319-0
6tol
159-0
186-0
215-0
280-5
368-0
7tol
103-0
143-0
140-5
282-5
352-5
8tol
Organic matter or loam in the sand are very detrimental to
the strength of cement mortar, and clean sharp sand, quite free
from argillaceous matter, will give the best result. Portland cement
bears a much greater proportion of coarse than of fine sand, and
cement mortar should be mixed rapidly and not be triturated under
edge stones, as is a common practice with lime mortar. It is also
very essential that bricks or porous stone, which are to be set in
cement, should be previously well soaked in water, as dry materials
absorb moisture from the mortar and prevent it from setting
properly.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
329
TABLE XXV.— Results of Experiments with Portland Cement weighing 123 Ibs. to
the imperial bushel, gauged neat, and with an equal proportion of clean Thames
Sand. The whole of the specimens were kept in water from the time of their
being made till the time of testing.
Age.
On area = 2'25 square inches.
Neat Cement.
1 of Cement to
1 of Sand.
Average breaking
test of 10
experiments.
Average breaking
test of 10
experiments.
7 Days
fts.
817-1
fts.
353-2
1 Month -
935-8
452-5
3 Months -
1055-9
547-5
6 Ditto -
1176-6
640-3
9 Ditto -
1219-5
692-4
12 Ditto -
12297
716-6
2 Years
1324-9
790-3
3 Ditto
1314-4
784-7
4 Ditto
1312-6
818-1
5 Ditto
1306-0
821-0
6 Ditto
1308-0
819-5
7 Ditto
1327-3
863-6
330
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XXVI.— Southern Outfall Works, Crossness. Summary of Portland
Cement Tests, from 1862 to 1866, showing generally increase of strength with
increased specific gravity.
Number of
bushels.
Average
weight
per
bushel
Tearing weight
on
area = 2-25
square inches ;
7 days old.
Number of
bushels.
Average
weight
per
bushel.
Tearing weight
on
area = 2-25
square inches ;
7 days old.
fts.
B)S.
fts.
Ibs.
1,800
106
472-6
12,500
119
777-9
5,800
107
592-3
18,530
120
732-3
26,166
108
6501
15,144
121
705-6
37,036
109
646-6
5,000
122
716-6
20,820
110
708-3
5,428
123
673-6
6,900
111
693-8
13,400
124
819-9
13,812
112
687-5
5,400
125
816-2
10,610
113
. 701-5
1,800
126
657-2
24,224
114
699-7
1,800
127
864-6
16,240
115
705-5
3,600
128
916-6
27,400
116
768-3
1,820
129
920-2
26,800
117
718-4
1,800
130
913-9
23,306
118
6441
3$O. Tensile strength of Roman cement — Natural cements
generally inferior to the artificial Portland. — The following
tables contain the results of Mr. Grant's experiments on the tensile
strength of Roman cement. This cement is much weaker than
Portland, and inferior qualities are apt to vegetate and crumble
away, especially if mixed with loamy sand. Roman cement is a
natural cement, derived from argillo-calcareous, kidney-shaped
stones, called " Septaria," belonging to the Kimmeridge and London
clay, generally gathered on the sea-shore near the mouth of the
Thames, though sometimes dug out of the ground. Natural
cements are found in various places at home and abroad and, though
generally inferior in strength to artificial Portland, are very useful
in their way.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
331
TABLE XXVII.— Results of Experiments with neat Eoman Cement, manufactured
by Messrs. J. B. WHITE and BROTHERS.
Time kept immersed
in water.
On Area — 2-25 square inches.
Minimum
breaking
test.
Maximum
breaking
test.
Average
breaking
test.
fts.
fts.
fts.
7 Days
170
240
202-0
14 Ditto
160
190
173-0
21 Ditto
170
205
186-5
1 Month -
246
291
260-3
3 Months -
307
344
322-5
6 Ditto
442
502
472-7
9 Ditto
313
520
4711
12 Ditto
596
680
643-1
2 Years -
577
610
546-3
3 Ditto
522
647
603-8
4 Ditto
600
658
632-2
5 Ditto
582
662
627-4
6 Ditto
603
711
666-4
7 Ditto
646
780
708-7
332
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
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CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
333
371. Tensile strength of Keene's3 Parian, and Medina
cements. — The following tables contain the results of Mr. Grant's
experiments on the tensile strength of Keene's, Parian and Medina
cements. The two former are chiefly used for internal decoration.
Keene's cement is made by soaking plaster of Paris in alum water,
then re-burning and grinding it ; Parian cement is made by mixing
gypsum with borax in powder, then calcining the mixture and
grinding it. Medina is a natural cement with rather more lime
than Roman cement, and is inferior in strength to Portland cement,
which, as already stated, is an artificial mixture of chalk and clay.
Quick-setting Medina is useful for pointing the joints of marine
masonry which have been set in Portland cement. It hardens
rapidly and prevents the rising tide from washing the slower
setting Portland out of the joints before it has had time to harden
sufficiently to resist the action of water in motion.
TABLE XXIX.— Kesults of 120 Experiments with Keene's Cement, manufactured by
Messrs J. B. WHITE and BROTHERS ; and Parian Cement, manufactured by Messrs.
FRANCIS and SONS.
Average breaking test on area = 2-25 square inches.
Age and time
immersed in water.
Keene's Cement.
Parian Cement.
In water.
Out of water.
In water.
Out of water.
Ibs.
fts.
Ebo.
fcs.
7 Days -
543-9
546-0
5951
642-3
14 Ditto -
486-9
585-8
600-8
671-2
21 Ditto -
503-0
579-4
543-4
696-6
1 Month
490-2
584-2
544-3
746-7
2 Months
454-7
648-4
500-7
725-6
3 Ditto -
508-8
720-5
521-1
853-7
334
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XXX.— Results of 100 Experiments with Medina Cement, manufactured by
Messrs. FRANCIS, BROTHERS, 1864.
Age and time Immersed
in water.
On area = 2-25 square inches.
Minimum
breaking
test.
Maximum
breaking
test.
Average
breaking
test.
fts.
fefc
R>s.
7 Days
83
100
921
ditto (2nd Series)
195
235
211-0
14 Days
238
335
303-4
21 ditto
274
332
298-0
1 Month
210
346
306-0
3 Months
420
468
448-8
6 ditto
376
438
412-4
9 ditto
438
507
457-2
12 ditto
456
527
476-9
2 Years
235
328
276-0
3 ditto
200
342
275-5
4 ditto
236
430
287-8
5 ditto
245
395
307-0
6 ditto
309
475
365-0
7 ditto
335
440
377-5
. Adhesion of Plaster of Paris and Mortar to brick or
stone. — Kondelet states that the adhesive strength of plaster of
Paris to brick or stone is about two-thirds of its tensile strength,
and that its adhesion is greater for millstone and brick than for
limestone, and diminishes greatly with time ; he also states that
the adhesion of lime mortar to stone or brick exceeds its tensile
strength and increases with time.*
The following table gives the results of experiments by Mr.
Grant on the tensile strain required to separate bricks cemented
together in blocks of 4, one on top of the other, with Portland
cement and lime mortars, at the end of 12 months, f
* Navier, Application de la Mtcanique, p. 13. f Proc. Inst. C. E., Vol. xxxii.
m
$1 d 4! d d ^1 d il
riJJ
25
rlO.
me and San
1 to 2.
Dorking.
10 -g CO CM CO CO 00
r-IHri— IrHi— li— ICOi— 1
1 to 2.
Blue Lias.
OO. 00. CO. CM . O. O5. CM . OS .
TH * "o * b-* CO* CM * O5* rH* (M*
1
.9
3
"S
•^ 3 b- "o OOCO CO rH b- OO OCO b- b- "o "o
1
i
d
3
XOO 1—100 OOCO OO5 1OO5 «OXO COCO OOCO
CO XO 05 ^ OOCO COCO <N XO CO-* 00 7-1 00 ^H
b- b- b- 0 005 0500 00 <N CM i-l r-l O O5 OO
H
1
3
00-* b- ffl CO AH <=>~^ COO 0500 10 CO 050
ooco cOjj coco coco i— i oo os<n coxo osos
O5<M OQ COXO OO5 1OXO •* i— 1 XOO <MO
e
%
•d
3
Q <p . .
b^ O ^^ COCO -* CO t^ 0 CO »-l -* O ~& CO
COb- OO <NOO (N<M XOO OO5 OOCO b-i— 1
3
TH CO CO CO CM
cb(M CO CO (Mi— 1 CO-* -*XO Ob- OOO5 iHOO
b--* 1OO OO<M -*OO COXO O5CO OCM O-*
10 CO XO TH 05 b- CO TH <M TH 05 (M OCO b- O
j-H r-l i-H i-l <M <M <N <M CM <M <N CN CM r-l i-H rH
i
TH TH OO <N OO TH TH
AH CO O O5 GO 7-1 10 i-l CO IO 00 O5 CO O5 TH r-l
COCO THCO OOO <NCN OOTH XOTH XOb- b-TH
r-l i-l CN r-l COCO COCO (N CO TH TH (N <M r-{ r-t
J
^
tj . £. g . ^ . b-^ g e Ot rH,
j
*
cb* cb- xb* b-* xb* b-* b-* b-*
,
b
CO CO XO OO IO C^ XO
i— i ; °o; 05; T— i ; o; xa- oo* b-«
£
s
xb* xb" TH" cb" xb" cb* b-* b-*
*c
0
!J|
n
XO XO XO
O5 CN b- CM O5 05 OS
o; T— i ; eo; xo* i— i* o* o* O*
I
<
Is}*
1
«O* b-" O5* O* b-' CO* CO* CO*
COCOCOTHCOCOCOCO
Description of Brick.
1 i t j . * ll !i
fill J It II
1 1 1. | I | jS jl
336 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
" The pressed gault bricks show the lowest amount of ad-
hesiveness ; partly because of their smooth surface, and partly
because in making them some oily matter is used for lubricating
the dies of the press through which they are passed before being
burnt. In the case of the perforated gault bricks the cement-
mortar seems to act as dowels, and the results are consequently
high. The Suffolk and the Fareham red bricks, which each
absorb about a pound of water per brick, adhere much better
than the Staffordshire, which are not absorbent. This shows the
importance of thoroughly soaking bricks which are to be put
together with cement, as dry bricks deprive the cement-mortar of
the moisture which is necessary for its setting." Mr. Robertson
found that the adhesion of first-class hydraulic mortar, made of
blue Lias lime and ground in mortar pans for forty minutes,
to blue vitrified Staffordshire bricks, not too highly glazed, was
40 tbs. per square inch, after six months ; while to the hardest
grey-stocks, although watered, as in practice, the adhesion was only
36 ft>s., or 10 per cent. less. To soft "place" bricks, the adhesion
was only 18 Ibs., or 55 per cent, less than to blue bricks.*
3?3. Grant's conclusions. — The following conclusions are the
result of Mr. Grant's numerous experiments on cement during the
execution of the Southern Metropolitan Main Drainage Works : —
1. Portland cement, if it be preserved from moisture, does not, like Roman cement,
lose its strength by being kept in casks, or sacks, but rather improves by age ; a great
advantage in the case of cement which has to be exported.
2. The longer it is in setting, the more its strength increases.
3. Cement mixed with an equal quantity of sand is at the end of a year approximately
three-fourths of the strength of neat cement.
4. Mixed with two parts of sand, it is half the strength of neat cement.
5. With three parts of sand, the strength is a third of neat cement.
6. With four parts of sand, the strength is a fourth of neat cement.
7. With five parts of sand, the strength is about a sixth of neat cement.
8. The cleaner and sharper the sand, the greater the strength.
9. Very strong Portland cement is heavy, of a blue-grey colour, and sets slowly.
Quick setting cement has, generally, too large a proportion of clay in its composition,
is brownish in colour, and turns out weak, if not useless.
10. The stiff er the cement is gauged, that is, the less the amount of water used in
working it up, the better.
* Proc. Inst. C. E., VoL xvii, p. 420.
CHAP. XVI.] TENSILE STRENGTH OP MATERIALS.
337
11. It is of the greatest importance, that the bricks, or stone, with which Portland
cement is used, should be thoroughly soaked with water. If under water, in a quiescent
state, the cement will be stronger than out of water.
12. Blocks of brick-work, or concrete, made with Portland cement, if kept under
water till required for use, would be much stronger than if kept dry.
13. Salt water is as good for mixing with Portland cement as fresh water.
14. Bricks made with neat Portland cement are as strong at from six to nine months
as the best quality of Staffordshire blue brick, or similar blocks of Bramley Fall stone,
or Yorkshire landings.
15. Bricks made of four parts or five parts of sand to one part of Portland cement
will bear a pressure equal to the best picked stocks.
16. Wherever concrete is used under water, care must be taken that the water is
still. Otherwise, a current, whether natural or caused by pumping, will carry away the
cement, and leave only the clean ballast.
17. Roman cement, though about two-thirds the cost of Portland, is only about
one-third its strength, and is therefore double the cost, measured by strength.
18. Roman cement is very ill adapted for being mixed with sand.
374. Tensile strength of glass — Thin plates of glass
stronger than stout bars — Crushing strength of glass is
13 times its tensile strength. —
TABLE XXXII.— TENSILE STRENGTH OP GLASS.
Description of Glass.
Tearing weight
per square inch.
Authority.
•
Bbs. tons.
Glass Tubes and Rods, -
3,527 = 1-57
Navier.
Annealed Flint Glass Rod,
2,413 = 1-07
Fairbairn and Tate.
Common Green Glass Rod,
2,896 = 1-29
Do.
White Crown Glass Rod,
2,546 = 114
Do.
Fairbairn and Tate, Philosophical Transactions, 1859, p. 216.
Navier, Resume des lefons sur I'application de la Mecanique, p. 37.
In their experiments on the resistance of thin glass globes to
internal pressure, Sir William Fairbairn and Mr. Tate found that
the tenacity of glass in the form of thin plates is 5,000 ft>s. per
square inch, or about twice that of glass in the form of bars, on
which they observe : — " The tensile strength is much smaller in
the case of glass fractured by a direct strain in the form of bars,
338
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
than when burst by internal pressure in the form of thin globes.
This difference is, no doubt, mainly due to the fact that thin
plates of this material generally possess a higher tenacity than
stout bars, which, under the most favourable circumstances, may
be but imperfectly annealed." " The ultimate resistance of
class to a crushing force is about 12 times its resistance to
to o
extension"* (3O5).
CORDAGE.
335. Tensile strength of cordage. — The following table
gives the sizes, weights, and strength of different kinds of best
Bower cables employed in the British Navy.f The strength was
determined by the chain-testing machine in Woolwich Dockyard,
in which the strain is measured by levers.
TABLE XXXIIL— TENSILE STBENGTH OP BOWER CABLES.
Best Bower hempen cables, 100 fathoms.
Number of
threads in each.
Tearing weight
by experiment.
Circumference.
Weight.
Inches.
Cwt. qrs. ibs.
Cwt. qrs. R»s.
23
96 2 27
2,736
114 0 0
22
21
89 0 12
80 0 22
2,520 )
2,268 J
89 0 0
18
58 2 6
1,656
63 0 0
1*1
38 0 21
1,080
40 0 0
The next table " shows the mean results of 300 trials made by
Captain Huddart. It shows the relative strength or cohesive
power of each kind of rope, taking as a standard of comparison -f^th
of a circular inch, equal to an area of -078 or nearly J^th of a
square inch. It shows that ropes formed by the warm register are
stronger than those made up with the yarns cold; because the
heated tar is more fluid, and penetrates completely between every
fibre of hemp, and because the heat drives off both air and moisture,
* Phil. Trans., 1859, pp. 216, 246.
t Barlow on the Strength of Materials, p. 260.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
339
so that every fibre is brought into close contact by the twisting
and compression of the strand ; the tar thus fills up every interstice,
and the rope becomes a firmly agglutinated elastic substance almost
impermeable to water. But, although rope so made is both
stronger and more durable, it is less pliable, and therefore the cold
registered rope is more generally used for crane work, where the
rope must be wound round barrels, or passed through pulleys."*
TABLE XXXIV.— TENSILE STRENGTH OF TARRED HEMP ROPE.
Size of
Ropes.
Tearing weight,
made by the old method.
Tearing weight,
made by the register.
5
s
Of common
staple Hemp.
OS'S
H2|!
£il
Of the best
Petersburg
Hemp.
per TO of a
circular inch
in area.
Cold
Register.
Per T^ of a
circular inch
in area.
•I
a 'S>
£5
«!
~£l|
£1.2
in.
in.
R>S.
Bfc
fts.
R>s.
ft&
1T)S.
fta.
fcs.
3
0-95
5,050
561
6,030
670
7,380
935
8,640
960
H
1-11
6,784
554
8,669
707
11,165
911
11,760
906
4
1-27
8,768
548
10,454
653
13,108
819
15,360
960
*i
1-43
10,308
504
12,440
614
16,325
806
19,440
960
5
1-59
13,250
530
15,775
631
20,500
820
24,000
960
54
1-75
15,488
512
18,604
614
24,805
820
29,040
960
6
1-91
18,144
504
21,616
600
24,520
820
33,120
920
64
2-07
20,533
486
23,623
559
34,645
820
40,554
959
7
2-24
22,932
468
27,342
558
40,188
819
47,040
960
H
2-39
24,975
444
30,757
546
46,125
820
54,000
960
8
2-54
26,880
421
32,000
500
52,480
820
61,430
960
NOTE. — j^o-th of a circular inch = '078, or nearly ^th of a square inch.
The proof-strain of rope which is given in Table XXXVII. is
about one-half its tearing weight.
* Grlynn's Rudimentary Treatise on the Construction of Cranes and Machinery,
pp. 93, 94.
340 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
376. Strength and weight of Cordage — English rule-
French rule. — By the old ropemakers' rule the -square of the
girth in inches multiplied by four gave the ultimate or breaking
strength of the rope in cwts., and it was a good rule for small
cordage, up to 7 inches in girth. The square of the girth divided
by four was considered to represent the weight of a fathom in
pounds.* The old ropemakers' rule for strength is equivalent to
2-51 tons per square inch of section. The French rule, as given
by Morin,t allows 2- 79 tons per square inch for the tearing weight
of tarred hemp cordage.
377. "Working strain of Cordage. — Cordage rapidly deterio-
rates by use and exposure to the weather, and when passed
round barrels or pulleys the outer strands are subject to greater
strains than those next the barrel. For this reason, as well as in
order to diminish useless work, the diameters of pulleys and barrels
should be made as large as practicable. Experience alone can
estimate the proper allowance to be made for wear and friction,
which latter is sometimes excessive in badly made blocks, and after
deducting this allowance from the original tearing strength, one-
fourth of the remainder is a sufficient load for continued strain,
and one-third for merely temporary purposes, though workmen
often apply one-half. A common practical allowance for friction
in ordinary tackles is one-third of the theoretic amount; if, for
example, the tackle consists of an upper and lower block with
three pulleys in each block, there will be 6 parts to the rope and
W
the theoretic pull on each part will = -~- ; the foregoing rule,
1/33W
however, makes the pull on each part — — ~ — , and the rope
should therefore be one-third stronger than if friction had not
existed.
CHAINS.
378. Stud-link or Cable chain. — Close-link or Crane
chain — Long open-link or Buoy chain — middle-link chain. —
* Glynn's Rudimentary Treatise, p. 92.
t Resistance des Materiaux, p. 41.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 341
Stud-link chain is chiefly used for ships' cables, and derives its
name from the cast-iron stud or stay which is inserted across the
shorter diameter of each oval link to keep the sides from closing
together under heavy strains. It also prevents the chain from
kinking, to which long links without stays are liable. Short or
close-link chain, called also rigging or crane chain, is that in common
land use. It is well adapted for crane work where flexibility is
essential to enable the chain to pass freely round barrels and
pulleys. Long open-link chain without studs is used for permanent
mooring cables, where flexibility is a secondary object, and where
lightness is desirable, as in the case of light-ships or beacon buoys.
Middle-link chain is occasionally used ; its link is intermediate in
length between those of the close and open-link chains.
The standard proportions of the links of the different kinds of
chain are as follows, in terms of the diameter of the bar of iron : —
Extreme length. Extreme width.
Stud-link, - 6 diameters. - 3' 6 diameters.
plose-link, 5 do. - 3'5 do.
Open-link, - 6 do. - 3*5 do.
t Middle-link, - 5'5 do. - 3'5 do.
' End-links, - 6'5 do. - 4'1 do.
End-links are the links which terminate each 15-fathom length
of chain ; they are longer and wider than the common links in
order to allow the joining shackles to pass through, and they
require therefore to be made of stouter iron, generally 1*2 diame-
ters of the common links.
379. Tensile strength of stud-chain. — The following table
contains the results of experiments on the tensile strength of stud-
chain made by Mr. William Smale, leading man of the test house
in Her Majesty's Dockyard, Woolwich.* Mr. Smale found that
the average tearing weight of good round bars of one inch diame-
ter was 19 tons, = 24*19 tons per square inch of section, their
greatest strength being about 20 tons, = 25'46 tons per square inch
of section.
* Report from the Select Committee on Anchors, &c. (Merchant Service), 1860.
Appendix, pp. 151, 152.
342
TENSILE STRENGTH OF MATERIALS. LCHAP- XVI<
TABLE XXXV. — TENSILE STRENGTH OP STUD-CHAIN.
Size
of
Chain.
Length
of each
piece.
Number
of
pieces
tested.
Mean
tearing
weight.
Govern-
ment
proof
strain.
Ratio of
tearing
to proof
strain.
Area
of
Bar.
Tearing
weight
per square
inch of
each side
of link.
in.
ft. in.
tons.
tons.
sq. in.
tons.
i
24 0
6
9-58
7-00
1-37
•307
15-6
I
».
6
13-51
10-125
1-33
•442
15-3
1
„
6
24-25
18-00
1-35
•785
15-4
H
U
»
6
6
6
29-54
59-58
74-125
2275
40-50
55-125
1-30
1-47
1-34
•994
1-767
2-405
14-9
16-9
15-4
Manufactured
by various
contractors
for the
Government.
1!
»
6
92-88
63-25
1-47
2.761
16-8
2
n
3.
99-54
72-00
1-38
3-141
15-8
i
2 0
20
20-38
13-75
1-48
•601
16-9
If
Single links
30
78-70
55-125
1-42
2-405
16-3
( Made in
] Woolwich
( Dockyard.
Mean
—
—
1-39
15-9
Messrs. Brown, Lenox, & Co., inform me that they have found
by experience that the average breaking strain of stud-link chain,
up to 2£ inches, is from 900 to 1,000 ibs. per circular £th of an
inch of the diameter of the bar — equivalent to from 16*37 to 18*19
tons per square inch of each side of the link. This is for cables of
good quality, much chain being made of a description of iron that
will stand the proof and but little more. Hence, stud-chain is
about f rds as strong as bar iron of the same sectional area as both
sides of the links together ; in other words, the bar loses about 33
per cent, of its strength by being converted into a link.
Ex. A one-inch stud-chain contains 64 circular £ths, and, if of good quality, its
tearing weight should equal 64 X 900 = 57,600 Ibs. = 25'7 tons. The tearing weight
of two round bars of good iron, each one inch diameter, should equal 2 X 19 = 38 tons.
3§O. Admiralty Proof-strain for Stud-chain. — By the Chain
Cable and Anchor Act of 1871 it is enacted' that a maker of or
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 343
dealer in chain cables or anchors shall not sell, consign, or contract
to sell or consign, nor shall any person purchase or contract to
purchase any chain cable whatever, or any anchor exceeding
168 Ibs., which has not been previously tested and duly stamped,
and where any chain cable is brought to a tester for the purpose
of being proved, he shall test every fifteen fathoms of it in the
manner following ; that is to say,
1°. He shall select and cut out a piece of three links from
every such fifteen fathoms and shall test that piece by subjecting
it to the appropriate breaking strain mentioned in the second
schedule to this Act (see the last column in Table XXXVI.) : —
2°. If the piece so selected fail to withstand such breaking strain,
he shall select and cut out another piece of three links from the
same fifteen fathoms, and shall test such piece in like manner : —
3°. If the first or second of such pieces of any fifteen fathoms
of cable withstand the breaking strain, he shall then, but not
otherwise, test the remaining portion of that fifteen fathoms of
cable by subjecting the same to the tensile strain mentioned in
the Act of 1864 (see the Admiralty proof -strain in the 7th column
of Table XXXVI.) :—
4°. He shall not stamp a chain cable as proved which has not
been subjected to the breaking and tensile strains in accordance
with the provisions of this section, or has not withstood the same.
For stud-chain the Admiralty proof -strain equals 6 30 Ibs. per
circular ith of an inch of the diameter of the bar, equivalent to
11-46 tons per square inch of each side of the link. Hence, this
proof-strain for stud-chains is about two-thirds of the ultimate
strength of cables of good quality, and one -half the strength
of good round bar iron — i.e., the Government proof of a stud-
chain is equal to the ultimate strength of the single bar of which
it is made, supposing this equals 23 tons per square inch, = 18'064
tons per circular inch.
Ex. A one-inch stud-chain has 1'57 square inches of area in both sides of the link
together, and 1'57 X 11'46 = 18 tons = the proof-strain. The ultimate strength of
O
good chain should reach -X 18 = 27 tons, and the breaking weight of the single bar
should not be less than 18' 064 tons, = 23 tons per square inch, and the iron should be
tough and fibrous with a "set after fracture" of not less than 15 per cent.
344
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
The following table gives the proof-strains and weight per 100
fathoms of stud-chain cables for Her Majesty's Naval Service,
also the appropriate breaking strain referred to in the Act of
Parliament.
TABLE XXXVI.— ADMIRALTY PROOF-STRAIN AND APPROPRIATE BREAKING-STRAIN
FOR CHAIN CABLES.
Common Links.
Weight of
100 fathoms of Cable
Diameter
of the bar
of which
the chain
is made.
Mean
length
6 diameters
of the bar;
not to be
over more
than
one-tenth
Mean
width
30
diameters
of the bar;
not to be
under more
than
Stay Pins,
one
diameter
of the bar
at the ends ;
0-6 do. at
the centre.
Weijrht of
each not to
in 8 lengths,
including 4 swivels
and
8 joining shackles,
not to be
exceeded by more
than one fifteenth
part for sizes 2J inch
and upwards,
and not more than
Weight of 100
fathoms,
with the allowance
added.
Admiralty
Proof-
strain,
equal to
&JO Ibs.
cir^lar
Jthinch.
Appro-
priate
breaking
strain.
of a
one- tenth
exceed,
one-twentieth part
diameter.
of a
for sizes under
diameter.
21 inch.
inch.
inch.
inch.
OZS.
cwts. qrs. fts.
cwts. qrs. fts.
tons.
tons.
2f
16*
9-9
72
363 0 0
387 0 22
136$
190-5
24
15
9-0
54-69
300 0 0
320 0 0
112$
157-5
2|
Hi
8-55
47'5
270 3 0
288 3 6
101*
141-9
2J
134
8-1
40
243 0 0
259 0 22
911
127-5
2|
12|
7-65
33-584
216 3 0
227 2 9
8U
113-7
2
12
7-2
28
192 0 0
201 2 11
72
100-8
1|
H|
6-75
23
168 3 0
177 0 21
63J
88-5
If
10i
6-3
18-76
147 0 0
154 1 11
551
77-0
If
9|
5-85
15
126 3 0
133 0 9
47*
66-5
1|
9
5-4
11-81
108 0 0
113 1 17
401
60-75
11
H
4-95
9
90 3 0
95 1 4
34
51-0
U
H
4-5
6-836
75 0 0
78 3 0
28|
42-0
11
6*
4-05
4-983
60 3 0
63 3 4
221
35-5
i
6
3-6
3-5
48 0 0
50 1 16
18
27-0
i
H
3-15
2-344
36 3 0
38 2 10
13f
20-5
I
4
2-7
1-473
27 0 0
28 1 11
10|
15-0
H
H
2-475
1-137
22 2 21
23 3 8
81
12-75
1
H
2-25
•854
18 3 0
19 2 21
7
10-5
tV
8|
2-025
•622
15 0 21
15 3 22
54
8-25
k
3
1-8
•437
12 0 0
12 2 11
44
6-75
*
2|
1-575
•293
9 0 21
9 2 16
34
5-25
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
345
The "appropriate breaking strains" of the heavier chains are
almost exactly 16 tons per square inch of each side of the link ;
for the smaller sizes they are about one ton higher.
Cables generally weigh the full weight allowed, the iron being
rolled a little full to allow for waste in the manufacture. Those
for the merchant service are usually made in lengths of 15 fathoms
each, with joining shackles connecting the lengths together.
381. Close-link chain — Proof-strain. — The Admiralty proof-
strain for close-link chain is 420 Ibs. per circular Jth of an inch of
the diameter of the bar, or two-thirds of the proof for stud-chains ;
this is equivalent to 7*64 tons per square inch of each side of the
link, or nearly one-half the breaking weight of the chain. The
following table gives the proof-strain and weight per 100 fathoms
of close -link chain, the extreme length of links not to exceed 5
diameters of the bar ; it also gives the size and weight of rope of
equal strength.
TABLE XXXVII.— ADMIRALTY PROOF-STRAINS FOR CLOSE-LINK CHAIN.
Diameter of
Chain.
Average weight
per 100
fathoms.
Proof-strain, equal
to 420 Ibs. per
circular |th inch.
Girth of Rope of
equal strength.
Weight of Rope
per fathom.
inches.
cwt.
tons.
inches.
tt>S.
If
155
31|
—
—
14
125
27
—
—
If
104
22|
—
—
11
86
18f
_
—
li
70
l«i
—
—
i
56
12
10
22
tt
50
10|
94
19i
I
42
9|
9
174
it
35
n
8*
15
1
32
6|
H
12
li
25
51
7
101
1
21
4|
6i
8£
346 TENSILE STRENGTH OP MATERIALS. [CHAP. XVI.
TABLE XXXVII. — ADMIRALTY PROOF-STRAINS FOR CLOSE-LINK CHAIN— continued.
Diameter of
Chain.
Average weight
per 100
fathoms.
Proof-strain, equal
to 420 Ibs. per
circular Jth inch.
Girth of Rope of
equal strength.
Weight of Rope
per fathom
ft
16
N
«i
7
*
13
3
4|
5
ft
10
2*
4
M
1
7
n
8|
81
ft
5
ii
2i
1|
}
3
i
2
—
ft
2
8|cwt.
H
—
The rope of the foregoing table " is such as is now generally
made by machinery at most of the large rope works, but was
formerly known as ' Patent Rope,' in which every yarn is made to
bear its part of the strain ; but if common hand-laid rope be used,
the proof -strain must be reduced one-fourth, and in actual work
the load should not, at any time, exceed one-half the proof."*
It will be observed that the diameter of a close-link chain is
approximately one-tenth of the girth of hemp rope of equal
strength.
383. Long; open-link chain — Admiralty proof-strain —
Trinity proof-strain — French proof. — The links of open-
link chain are not oval like those of a stud-chain, but parallel-
sided, and the open-link chain of the same length of link as the
stud-chain is lighter by the weight of the studs. As already
observed, it is suited for moorings of a permanent character, such
as those of mooring buoys, beacon buoys, or light-ships, which are
seldom shifted, and where, consequently, flexibility in passing round
chain barrels is a secondary object. Besides its comparative light-
ness, open-link chain has another advantage over either close-link
or stud-chain, for each 15-fathom length of the two latter requires
long end links for the purpose of connecting it by joining shackles
to the adjoining lengths, and if either of these chains break, a whole
* Glynn on the Construction of Cranes, p. 92.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
347
length must be taken out, since there is not room for a shackle to
pass through the ordinary close-link or stud-link. When, how-
ever, a long-link chain breaks, the links adjoining the fracture can
be connected together without taking out a whole 15-fathom
length, as a shackle will generally pass through any of the common
links. The old Admiralty proof for large open long-link chain
without studs was 315 ibs. per circular Jth of an inch, or one-half
the proof of stud-chain, as shown in the following table ; the links
were generally of great length.
TABLE XXXVIII.— ADMIRALTY PROOF-STRAINS FOR PENDANT AND BRIDLE CHAINS.
Diameter
of iron.
Proof strain
equal to 315 Ibs.
per circular
|th inch.
Inches.
Tons.
84
110
95
3
81
74.
Permanent deflection or
2|
2f
24
68
62
56
collapsion
to exceed
of an inch
of link not
one quarter
2
36
The following are the proofs which the Elder Brethren of the
Trinity House require in testing open-link chains such as are used
for mooring light-ships and beacon buoys, as well as close-link
rigging or crane chains : — The chains are subjected, in lengths of
15 fathoms, to a strain of 466 fos. per circular Jth inch of the
diameter of the bar, (equivalent to 8*47 tons per square inch of
each side of the link, or about one-half the breaking weight of
the chain.) This test — which was determined after numerous
experiments — is the highest strain to which open-linked chain
can be subjected without altering the shape of the link, and
is comparatively much more severe than the usual test for chain
without studs. In addition to the foregoing limited proof-strain,
test pieces, 4 feet long, are cut out of each size of chain and the
quality of the iron is ascertained by testing the iron in one link of
each length. The remainder of each fc«ir-feet length is then torn
asunder to test the welding, and its breaking weight must not be
348 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
less than 16 tons per square inch of each side of the link, or
880 Ibs. per circular £th inch of the diameter of the bar. The
lengths of chain from which the test pieces are taken are then
made good and re-proved as before.
In the French Marine the proof for stud-chains fth inch in
diameter and upwards equals 1 0*8 tons per square inch of the bar.
For chains less than |th inch, without studs, the proof is 8' 9 tons
per square inch.*
383. Working-strain of chains should not exceed one-
half the proof-strain. — Mr. Glynnf states that chains " may
safely be worked to half the strain to which they have been
11 A d
proved, but not to more." This for stud-chain — - - — 5 '73
2
tons per square inch of each side of the link, or about one-third of
the ultimate strength of good chain and one-fourth of that of round
bar-iron. For close-link chain this rule allows — — , = 3'82 tons
per square inch of each side of the link, or about one-fourth of the
ultimate strength of common chain and one-sixth of that of bar-
iron. When, however, chains are liable to shocks, as in cranes,
one-third of the proof-strain, = 2*55 tons per square inch of each
side of the link, will be a sufficient working load.
384. Comparative strength of stud and open-link chain. —
I am indebted for the following practical observations to the
courtesy of Messrs. Brown, Lenox, & Co., the eminent manu-
facturers of anchors and chains : — " We are not of opinion that
studs increase the strength of chain, or enable it to bear a heavier
ultimate breaking strain than if made without them, both descrip-
tions being made of the same length of link. The object of their
being used is to prevent collapse of the link, which in open-link
chain takes place at a strain considerably below the breaking
weight, and, of course, renders the chain unserviceable. They
thereby enable chains, made with them, to be used for heavier
strains than open-link chain, but do not add to their ultimate
* Morin, Resistance des Mattriaux, p. 42.
f Rudimentary Treatise on the Construction of Chains, p. 91.
CHAP. XVI.] TENSILE STRENGTH OP MATERIALS. 349
strength — indeed, from the experiments we have tried, and the
experience we have had, we are inclined to believe that the link
without stay-pins almost invariably breaks at a higher strain than
stud-chains. The proof for studded chain is the higher, only
because a sufficient proof cannot be given to open-link chain before
the link spoils its form and becomes rigid. The stay prevents
collapse, by which the link is prevented elongating so much, and
taking its natural position before its utmost power is exhausted
and a break ensues. The link, if sound in the workmanship, will
nearly always break near the stay-pin, which is caused by the nip
across the stay-pin. If made without stays, it will collapse until
it is rigid, and the iron will reach as near as possible the direct
line of the strain, or right through the centre of the chain; the
sides of the links will incline inwards, and the break will ensue at
the nip across the crown of the next link."
385. Weight and strength of liar-iron, stud-chain, close-
link chain, and cordage. — The weight of a stud-chain in ibs.
per foot is very nearly equal to 9 times the square of the diameter
of the bar ; for instance, a two-inch stud-chain weighs 36 Ibs.
per foot nearly. Stud-chain is about 3J times as heavy as the
bar of which it is made: — thus, one fathom of 1J inch stud-
chain weighs about 125 Ibs. — a bar 21 feet long would weigh about
124 Ibs. Close-link chain is about 4 times as heavy as the bar: —
thus, one fathom of 1 J chain weighs about 140 ibs. — a bar 24 feet
long would weigh about 141 ibs. Close-link chain is about 12 per
cent, heavier than stud-chain made with stay-pins of Government
dimensions ; large and heavy stays are introduced by some manu-
facturers into ordinary cables, thereby greatly increasing the useless
weight of cast-iron, and enabling the chain to be sold cheaper by
weight. The following table shows at a glance the relative weights
and strength of bar-iron, stud-chain, close-link chain, and hemp
cordage.
350 TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XXXIX. — WEIGHTS AND STRENGTH OF BAB-IRON, CHAIN AND CORDAGE.
jSi-g
|i"Sa
f*l $
ftp
— £ e.3
— h7, k.
Weight of
100
fathoms :
(d =. dia-
meter in
Tearing weight
per square inch.
ight of equal
ne ultimate si
ength on the
•oni the same
Safe Working
strain
per square inch.
ight of eqnal
me useful si
.ength straini
mfe workinj
iame load.
inches.)
Relative we
of the sar
i.e., each 1
rupture f i
lip*
m
tons.
tons.
tons.
Bar-iron, best quality,
0-70c*2
24
100
6-0
100
Stud-chain,
2'45rf2
16 ) on each
262
5'73 ) on each
184
[ side of
> side of
Close-link chain,
2'80cZ2
16 jlink
300
3-82 ) link
314
Hemp Cordage,
O'llcJ2
2-51
150
0-63
150
WIRE ROPE.
386. Tensile strength of round iron and steel wire ropes
and hemp rope. — The following table shows the strength of iron
wire rope and hemp rope, by the eminent American Engineer,
J. A. Roebling, Esq.* The breaking weight is given in the
American ton of 2,000 Ibs.
TABLE XL.— STRENGTH OP ROUND IRON WIRE ROPE AND HEMP ROPE, BY
J. A. ROEBLING, C.E.
Circumference
of Wire rope
in inches.
Trade
number.
Circumference
of Hemp rope
of equal strength
in inches.
Tearing
weight in tons
of 2,000 Ibs.
6-62
1
15|
74
6-20
2
"i
65
5-44
3
13
54
4-90
4
12
43-6
Fine Wire,
4-50
3-91
5
6
10|
H
35
27-2
3-36
7
8
20-2
2-98
8
7
16
2-56
9
6
11-4
2-45
10
5
8-64
Memoranda on the Strength of Materials, by J. K. Whildin, New York, p. 9.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
351
TABLE XL.— STRENGTH OF ROUND IRON WIRE KOPE AND HEMP ROPE, BY
J. A. HOBBLING, C.E. — Continued.
Circumference
of Wire rope
in inches.
Trade
number.
Circumference
of Hemp rope
of equal strength
in inches.
Tearing
weight in tons
of 2,000 fts.
I
4-45
11
lOf
36
4-00
12
10
30
3-63
13
»i
25
3-26
14
H
20
2-98
15
n
!6
2-68
16
84
12-3
2-40
17
&i
8-8
212
18
5
7'6
Coarse Wire,
1-9
19
4-75
5-8
1-63
20
4
4-09
1-53
21
3-3
2-83
1-31
22
2-80
213
1-23
23
2-46
1-65
1-11
24
2-2
1-38
0-94
25
2-04
1-03
0-88
26
1-75
0-81
\
078
27
1-50
0-56
352
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
TABLE XLL — WEIGHT, STRENGTH, AND WORKING LOAD OF HEMP AND ROUND IRON
AND STEEL WIRE ROPES, AS STATED BY THE MAKERS, MESSRS. NEWALL AND Co.
OF GATESHEAD-ON-TYNE.
HEMP.
IRON.
STEEL.
Equivalent Strength.
Circum-
ference.
Inches.
Lbs.
Weight per
fathom.
Circum-
ference.
Inches.
Lbs.
Weight per
fathom.
Circum-
ference.
Inches.
Lbs.
Weight per
fathom.
Working
Load.
Cwt.
Tearing
weight.
Tons.
2|
2
1
1
—
—
6
2
—
—
14
H
1
1
9
3
3|
4
it
2
—
—
12
4
—
—
if
24
H
U
15
5
44
5
if
3
—
—
18
6
—
—
2
34
11
2
21
7
54
7
2|
4
If
24
24
8
—
—
2*
44
—
—
27
9
6
9
2|
5
If
3
30
10
—
—
24
54
—
—
33
11
64
10
2|
6
2
34
36
12
. —
—
2f
64
21
4
39 t
13
7
12
21
7
2*
44
42
14
—
—
3
74
—
—
45
15
74
14
N
8
21
5
48
16
—
—
S|
84
—
—
51
17
8
16
3|
9
24
54
54
18
—
—
34
10
2|
6
60
20
84
18
3|
11
2|
64
66
22
—
—
3|
12
—
—
72
24
94
22
3f
13
»i
8
78
26
10
26
4
14
—
—
84
28
—
—
4i
15
3|
9
90
30
11
30
4|
16
—
—
96
32
—
—
44
18
34
10
108
36
12
34
41
20
8f
12
120
40
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS.
353
387. Tensile strength of flat iron and steel wire ropes
and flat hemp rope.
TABLE XLII.— WEIGHT, STRENGTH AND WORKING LOAD OF FLAT HEMP ROPE
AND FLAT IRON AND STEEL WlRE KOPES, AS STATED BY THE SAME MAKERS.
HEMP.
IRON.
STEEL.
Equivalent Strength.
Size in
inches.
Lbs.
Weight per
fathom.
Size in
inches.
Lbs.
Weight per
fathom.
Size in
inches.
Lbs.
Weight per
fathom.
Working
Load.
Cwts.
Tearing
weight.
Tons.
4 +14
20
2J+ £
11
—
—
44
20
5 +1J
24
24+ »
13
—
—
52
23
54+11
26
21+1
15
—
—
60
27
5|+14
28
3 + „
16
2 + *
« 10
64
28
6 +14
30
3i+ „
18
2H-4
11
72
32
7 +lf
36
34+ „
20
» „
12
80
36
8i+2|
40
31+ «
22
24+ 4
13
88
40
84+2i
45
4 + „
25
21+ 1
15
100
45
9 +2£
50
4i+ 1
28
3 + „
16
112
50
94+2|
55
4*+,,
32
3^+ 5}
18
128
56
10 +2J
60
4!+,,
34
3i+,,
20
136
60
388. Safe working load of wire rope. — From Table
XLL, the safe working load of round hemp or wire rope is a
little more than one-seventh of their tearing weight; and from
Table XLIL, the working load of flat hemp and wire rope is about
one-ninth of their tearing weight; and Messrs. Newall and Co.
state that " round rope in pit-shafts must be worked to the same
load as flat ropes." It also appears from Table XLI. that the
length at which a round iron wire rope .will break from its own
weight is 26,880 feet; the working limit of length therefore,
supposing the rope has only its own weight to support, is under
4,000 feet.
MISCELLANEOUS MATERIALS.
389. Tensile strength of hone, leal her. whalehone5 g:utta-
percha* glue. — From Bevan's experiments it appears that the
2 A
354
TENSILE STRENGTH OF MATERIALS. [CHAP. XVI.
tensile strength of bones of horses, oxen and sheep varies from
33,000 to 42,500 Ibs. per square inch.*
The following are the results of Mr. H. Towne's experiments
on the tensile strength of single leather belts.f
Tearing weight
per inch wide.
Through the lace holes, 210
Through the rivet holes, 382
Through the solid part, 675
The thickness being *219 inch, the tensile strength of the solid
leather was 3,082 Ibs., = 1*376 tons per square inch. The strengths
of new and partially used belts were found to be nearly identical.
The maximum working strain may vary from one-fourth to one-
third of the tearing weight, i.e., from 52 to 70 ft>s. per inch wide of
ordinary single belting, but the former is the safer rule. Helvetia
leather belting, manufactured by a peculiar process by Messrs.
Norris and Co., of Shadwell, London, from fresh Swiss ox hides,
is stated to be stronger and more flexible than ordinary tanned
English belting, as shown by the following table, which contains
the results of Mr. Kirkaldy's experiments. J
TABLE XLIIL— TENSILE STRENGTH or LEATHER BELTING.
English Belting.
Helvetia Belting.
Double, 12 inches,
Ibs.
14,861
Ibs.
17,622
7 „
6,193
11,089
6 „
5,603
10,456
4 „
4,365
6,207
2 „
2,942
4,237
Single, 10 „
8,846
11,888
5. „
4,060
5,426
4 „
3,248
3,948
' „ 3* „
3,007
3,377
Phil. Mag., 1826, p. 181. f Engineer, Aug., 1868, p. 145.
I The Engineer, Aug., 1872, p. 125.
CHAP. XVI.] TENSILE STRENGTH OF MATERIALS. 355
Professor Rankine states that the tenacity of raw hide is about
once and a half that of tanned leather, and that the tenacity of
whalebone is 7,700 Ibs. per square inch.* Mr. Box states that
the tensile strength of gutta-percha is 1,680 Ibs., = -75 ton, per
square inch, and that in belting it will bear about 400 Ibs. per square
inch.f
Bevan found that the adhesion of common glue to dry ash
timber amounted to 715 tbs. per square inch when the glue was
freshly made and the season was dry; when the glue had been
frequently melted and in the winter season, the adhesion varied
from 350 to 560 Ibs. per square inch. The tensile strength of
solid glue was 4,000 Ibs. per square inch.t
* Machinery, p. 475.
f Box on Millgearing, p. 69.
J Phil. Mag., 1826, Vol. Ixviii., p. 112.
356 SHEARING-STRAIN. [CHAP. XVII.
CHAPTER XVII.
SHEARING-STRAIN.
39O. Shearing: In detail — Simultaneous shearing*. — The
nature of shearing-strain* in the vertical web of girders has been
already investigated in the second chapter, and we have frequent
examples of the same kind of strain, though on a smaller scale, in
rivets or similar connexions which sustain forces tending to cut
them across at right angles to their length. For example, the
rivet joining the blades of a pair of scissors is subject to a shearing-
strain equal to the pressure applied to the handles, plus the
resistance of the fabric which is being cut. The latter also is
subject to a shearing-strain, differing, however, in character from
that which the rivet sustains in consequence of the inclination of
the blades which sever only a short length of the fabric at a time.
Machines for shearing metals act on this principle, their cutting
edges being generally set at an acute angle to each other, so that
they shear plates in detail, and thus diminish the effort exerted at
each instant of time ; in punching machines, however, the whole
circumference of the hole is cut at the first effort, and subsequent
pressure is merely necessary to overcome friction and push out the
burr. The shearing-strains which occur in engineering structures
generally resemble that which rivets sustain, where the whole
transverse area simultaneously resists shearing. In this case it is
clear that the strength of the rivets is proportional to their sectional
area ; in other words, if F and / represent the total and the unit
shearing-strains, eq. 1 will apply to shearing as well as to tensile
and compressive forces, provided always that the cutting edges
bear simultaneously over the whole surface of the rivet or material
under strain.
* Called Detrusion by some authors.
CHAP. XVII.]
SHEARING-STRAIN.
357
391. Shearing strength of cast-iron. — The shearing strength
of cast-iron, according to Professor Rankine, is 27,700lbs. = 12-37
tons per square inch. In my own experiments I have found its
shearing strength equal to 8 or 9 tons per square inch, which
is substantially the same as its tensile strength.
392. Experiments on punching wronght-iron. — Table I.
exhibits the results of experiments made at Bristol by Mr. Jones,
" on the force required for punching different sized holes in different
thicknesses of plates, up to 1 inch diameter and 1 inch thickness ;
the force was applied by means of dead weights with a pair of
levers giving a total leverage of 60 to 1, so that 1 cwt. in the scale
gave a pressure of 3 tons on the punch ; the weights were added
gradually by a few Ibs. at a time until the hole was punched."*
TABLE I.— EXPERIMENTS ON PUNCHING PLATE IRON.
Diameter
of hole.
Thickness
of plate.
Sectional area
cut through.
Total pressure
on Punch.
Pressure
per square inch
of area cut.
inch.
inch.
square inch.
tons.
tons.
0-250
0-437
0-344
8-384
24-4
0-500
0-625
0-982
26-678
27-2
0-750
0-625
1-472
34-768
23-6
0-875
0-875
2-405
55-500
23-1
1-000
1-000
3-142
77-170
24-6
Table II. contains experiments by Mr. C. Little on punching
holes in hammered scrap iron with Eastwood's hydraulic shearing
press, the force applied being measured by weights hung on the
end of the force-pump handle. This method of measurement is
not so accurate as that by direct leverage, since the friction of the
press is rather an uncertain element in the calculation.!
* Proc. Inst. Mech. Eng., 1858, p. 76.
f Idem, p. 73.
358 SHEARING-STRAIN. [CHAP. XVII.
TABLE II. — EXPERIMENTS ON PUNCHING HAMMERED SCRAP IRON.
No. of
experi-
ment.
Diameter
of
Punch.
Sectional area cut.
Pressure on Punch.
Remarks.
Thickness
•and
circumference.
Area.
Total.
Tons
per square
inch of area
cut.
ins.
inches.
sq. ins.
tons.
tons.
1
2
1
1
0-51X314
0-98X3-14
1-60
3-08
35-8
69-3
22-4
22-6
[ 22-5 mean.
3
2
0-52X6-28
3-27
597
18-3
4
5
2
2
0-57X6-28
1-06X6-28
3-58
6-66
70-5
132-8
19-7
19-9
19'4 mean.
6
2
1-52X6-28
9-55
186-7
19-5
393. Experiments on shearing: wrought-iron. — Table III.
contains experiments, also by Mr. Little, with Eastwood's hydraulic
shearing press, on the force required to shear bars of hammered
scrap and rolled iron presented edgeways and flatways to the cutter.
TABLE III.— EXPERIMENTS ON SHEARING HAMMERED SCRAP BARS AND ROLLED IRON.
Sectional area cut.
Pressure on Cutters.
Direc-
No of
tion
experi-
ment.
of
shear-
ing.
Thickness
and
breadth.
Area.
Total.
Tons
per square
inch of
Remarks.
area cut.
inches.
sq. ins.
tons.
tons.
|
7
Flat
0-50X3-00
1-50
33-4
22-3
\
! 22-7 mean.
8
Edge
0-50X3-00
1-50
34-6
23-1
1
9
Flat
1-00X3-00
3-00
69-2
23-1
® •
1 <IP
10
Edge
1-00X3-00
3-00
68-1
22-7
If 1
[-21 '5 mean. l*g
11
Flat
1-00X3-02
3-02
59-7
19-8
1 3
12
Edge
1-00X3-02
3-02
62-1
20-6
13
Edge
1-80X5-00
10-20
210-6
20-6
Flanged tyre. S
CHAP. XVII.] SHEARING-STRAIN. 359
TABLE III. — EXPERIMENTS ON SHEARING WROUGHT-!RON — continued.
No. of
experi-
ment.
Direc-
tion
of
shear-
ing.
Sectional area cut.
Pressure on Cutters.
Remarks.
Thickness
and
breadth.
Area.
Total.
Tons
per square
inch of
area cut.
14
Flat
inches.
0-56X3-00
sq. ins.
1-68
tons.
21-2
tons.
12-6
j
15.
Edge
0-56X3-00
1-68
33-2
19-7
' 1
16
Flat
0-90X3-37
3-03
27-4
9-0
I -S
17
Edge
0-87X3-32
2-89
57-4
19-8
! i
18
Flat
1-06X3-02
3-20
50-2
157
8
19
Edge
1-06X3-02
3-20
67-5
21-1
f i
20
Flat
1-52X3-03
4-61
83-7
18-2
I f
21
Edge
1-53X3-03
4-64
93-3
20-1
} 3
22
Flat
1-39X4-50
6-25
89-7
14-3
)
23
Edge
1-38X4-50
6-21
111-2
17-9
i 1
24
Flat
1-73X5-30
917
153-1
16-7
}
25
Edge
1-73X5-30
9-17
207-0
22-6
1 1
26
Flat
1-56X6-00
9-36
140-0
15-0
I ^
27
Edge
1-56X6-00
9-36
172-3
18-4
1
28
Square
3-10X310
9-61
165-1
17-2
Hammered iron. __.
29
Square
3-10X3-10
9-61
155-5
16-2
Kolled iron. ^
30
Flat
1-80X5-00
10-20
99-3
9-7
Flanged tyre.
31
Edge
1-80X5-00
10-20
185-5
18-2
Flanged tyre.
32
Edge
1-70X5-25'
10-57
179-5
17-0
Flanged tyre.
"In the above experiments of shearing (Nos. 7 to 13 inclusive),
cutters with parallel edges were used ; but when the ordinary cutter
with edges inclined to one another at an angle of 1 in 8 were em-
ployed (Nos. 14 to 32 inclusive), the force required in shearing
was diminished, and considerably so in the case of the thinner
sections when sheared flatways; and as bars are usually sheared
flatways, a decided advantage is shown in favour of inclined over
360 SHEARING-STRAIN. [CHAP. XVII.
parallel cutters. The force in tons per square inch of section cut
with the bars
Flatways. Edgeways.
tons. tons.
3 XU inch was 18'2 and 20'1 or 10 per cent, less flatways.
4£Xli „ 14-3 „ 17-9 „ 20
3 XI „ 157 „ 21-1 „ 26
5*X1| „ 167 „ 22-6 „ 26
6 Xl£ „ 15-0 „ 18-4 „ 18
" A trial was also made of the force required to shear some hard
railway tyres If inch thick, and the result was 185 tons total
edgeways, and 99 tons flatways (Nos. 30 and 31). A 3 inch square
bar of rolled iron was also tried, and the force required was 155
tons total, against a total of 165 tons required for a hammered bar
of the same section (Nos. 28 and 29)."*
During the construction of the Britannia and Conway tubular
bridges several experiments were made by means of a lever on the
shearing strength of bars of rivet iron Jth inch diameter. " The
mean result from these experiments gives 23*3 tons per square
inch as the weight requisite to shear a single rod of rivet iron of
good quality. The ultimate tensile strength of these same bars
was also found to be 24 tons; hence their resistance to single
shearing was nearly the same as their ultimate resistance to a
tensile strain." Two plates fth inch thick were also "riveted
together by a single rivet |th inch diameter, and the rivet was
sheared by suspending actual weights from the plate; the rivet
thus sustained 12-267 tons, or 20'4 tons per square inch. Three
plates were then united by a similar rivet, and the rivet was
sheared in two places by the centre plate. The ultimate weight
suspended from the rivet was 26'8 tons, or 22*3 tons per square
inch of section. "f
394. Shearing strength of wrought-iron equals its tensile
strength. — From these various experiments on punching and
shearing, we may infer that the shearing strength of wrought-iron
is practically equal to its tensile strength, and that the safe shearing
* Proc. Inst. Mech. Eng., 1858, p. 74.
t Clark on the Tubular Bridges, p. 392.
CHAP. XVII.] SHEAKING-STRAIN. 361
unit-strain for wrought-iron rivets or bolts is practically the same
as the safe tensile unit-strain in the plates they connect, i.e., about
5 tons per square inch of section in ordinary girder-work.
395. Shearing strength of rivet steel is three-fourths
of its tensile strength. — From Mr. Kirkaldy's experiments it
appears that the shearing strength of rivet steel is 63,796 Ibs.,
= 28-48 tons per square inch, the tensile strength of the bar
employed being 86,450 Ibs., = 38'59 tons per square inch of area.*
Hence, the shearing strength of rivet steel is about three-fourths
of its tensile strength. The tensile strength of some rivet steel
used in one of H.M. ships was 35*93 tons per square inch.f The
heads of steel rivets are very apt to fly off, and Lloyd's committee
have prohibited their use in shipbuilding.
396. Shearing strength of copper. — From experiments by
Mr. Joseph Colthurst on punching plates of wrought-iron and
copper with a lever apparatus, it appears that the force required
to punch copper is two-thirds of that required to punch iron. " It
was observed, that duration of pressure lessened considerably the
ultimate force necessary to punch through metal, and that the use
of oil on the punch reduced the pressure about 8 per cent."t
397. Shearing strength of fir in the direction of the
grain — Shearing strength of oak treenails. — From Mr.
Barlow's experiments on the resistance of fir to drawing out, i.e.,
shearing, in the direction of the grain, it appears that this
amounts to 592 Ibs. per square inch, or nearly one-twentieth of
the tensile strength of the timber lengthways. §
The following table contains experiments by Mr. Parsons of
H.M. dockyard service, on the "transverse strength of Treenails
of English oak, used as fastening for planks of 3 and of 6 inches in
thickness, and subjected to a cross strain." ||
* Experimental Inquiry, p. 71.
f Eeed on Shipbuilding, p. 382.
I Proc. Inst. of C. £., Vol. i., p. 60.
§ Barlow on the Strength of Materials, p. 23.
Murray on Shipbuilding in Iron and Wood, p. 94.
362 SHEARING-STRAIN. [CHAP. XVII.
TABLE IV.— STRENGTH OF TREENAILS OP ENGLISH OAK.
DIAMETER OF THE TREENAILS.
Number
of the
1 inch.
IJinch. IJinch. 1| inch.
ex-
THICKNESS OF THE PLANK.
3 inches.
6 inches.
3 inches.
6 inches.
3 inches.
6 inches.
3 inches.
6 inches
T. C.
T. C.
T. C.
T. C.
T. C.
T. C.
T. C.
T. 0.
1
1 8
1 7
1 14
2 8
2 0
3 12
3 0
5 10
2
1 7
1 15
2 2
2 2
2 6
2 10
2 10
3 13
3
1 2
1 8
1 17
2 19
2 15
2 10
4 0
4 0
4
1 5J
1 8
2 2
2 2
2 4
3 12
2 8
3 8
5
2 12
1 3
2 2
1 15
2 18
2 5
3 10
4 0
6
2 2
1 7
2 9
2 10
2 6
2 5
3 10
5 8
7
2 4
1 10
2 8
2 10
3 7
2 5
3 5
3 12
8
1 6
2 3
2 7
2 0
2 5
3 0
3 5
3 13
9
1 8
1 8
2 12
2 10
3 0
4 0
4 6
4 13
10
1 2
2 3
2 10
2 15
3 0
4 10
3 8
4 0
11
2 0
2 0
2 7
2 0
3 9
2 18
4 0
3 8
12
1 8
1 7
2 10
2 0
4 2
3 0
4 10
5 0
13
1 16
2 8
2 17
2 0
3 2
8 18
4 2
5 5
Average
1 11
1 13
2 6
2 6
2 16
3 2
3 10
4 6
Total
Shearing
force in
1-6
2-3
2-95
3-9
tons.
Tons per
square
inch of
2-04
1-88
1-67
1-62
section.
" In all these experiments where the treenails were evidently
good, they gave way gradually. In some of the rejected experiments,
however, the treenails certainly did break off suddenly, but then
CHAP. XVII.] SHEARING-STRAIN. 363
they were evidently, on examination, either of bad or over-seasoned
material. In the experiments on treenails, the plank generally
moved about half an inch previous to the fracture of the treenail."
From these experiments Professor Rankine deduces,
1. That the shearing strength of English oak treenails across the
grain is about 4,000 Ibs. per square inch of section.
2. That in order to realize that strength, the planks connected
by the treenails should have a thickness equal to about three times
the diameter of the treenails.*
* Civil Engineering, p. 459.
364 ELASTICITY AND SET. [CHAP. XVIII.
CHAPTER XVIII.
ELASTICITY AND SET.
398. Limit of Elasticity — Net — Hooke's law of elasticity
practically true. — It has been already stated in 5 that Mr.
Hodgkinson's experiments led him to infer the non-existence of a
definite elastic limit within which, if the particles of a substance be
displaced, they will return exactly to their original relative positions
after the disturbing force is removed. The opposite view was held
by Professor Robison, whose opinions are also entitled to great
respect. In the article on the " Strength of Materials" in the
Encyclopaedia Britannica, he writes as follows : — " It is a matter of
fact that all bodies are in a certain degree perfectly elastic ; that
is, when their form or bulk is changed by certain moderate com-
pressions or distractions, it requires the continuance of the changing
force to continue the body in this new state ; and when the force
is removed, the body recovers its original form. We limit the
assertion to certain moderate changes. For instance, take a lead
wire of one-fifteenth of an inch in diameter and ten feet long ; fix
one end firmly to the ceiling, and let the wire hang perpendicular ;
affix to the lower end an index like the hand of a watch ; on some
stand immediately below, let there be a circle divided into degrees,
with its centre corresponding to the lower point of the wire ; now
turn this index twice round, and thus twist the wire. When the
index is let go, it will turn backwards again, by the wire untwisting
itself, and make almost four revolutions before it stops ; after which
it twists and untwists many times, the index going backwards and
forwards round the circle, diminishing, however, its arch of twist
each time, till at last it settles precisely in its original position.
This may be repeated for ever. Now, in this motion, every part
of the wire partakes equally of the twist. The particles are
stretched, require force to keep them in their state of extension
and recover completely their relative positions. These are all the
CHAP. XVIII.] ELASTICITY AND SET. 365
characters of what the mechanician calls perfect elasticity. This
is a quality quite familiar in many cases, as in glass, tempered steel,
&c., but was thought incompetent to lead, which is generally
considered as having little or no elasticity. But we make the
assertion in the most general terms, with the limitation to moderate
derangement of form. We have made the same experiment on a
thread of pipe-clay, made by forcing soft clay through the small
hole of a syringe by means of a screw, and we found it more elastic
than the lead wire; for a thread of one-twentieth of an inch
diameter and seven feet long allowed the index to make two turns,
and yet completely recovered its first position. But if we turn the
index of the lead wire four times round and let it go again, it
untwists again in the same manner, but it makes little more than
four turns back again ; and after many oscillations, it finally stops
in a position almost two revolutions removed from its original
position. It has now acquired a new arrangement of parts, and
this new arrangement is permanent like the former ; and what is
of particular moment, it is perfectly elastic. This change is
familiarly known by the denomination of a set."*
Whatever opinion the reader may hold regarding the existence
or non-existence of a definite elastic limit, experiments prove that
Hooke's Law of Elasticity, namely, that the elastic reaction of the
fibres is proportional to their increment or decrement of length,
according as they are subject to tension or compression, is for all
practical purposes substantially true of most of the materials used
in construction over a very considerable range of strain, extending
in some cases even to the breaking weight of the material (7).
CAST-IRON.
399. Decrement of length and set of cast-iron in com-
pression— Coefficient of compressive elasticity. — We are in-
debted to Mr. Hodgkinson for some valuable experiments on the
decrements of length and compressive sets of eight bars of cast-iron,
each 10 feet long and 1 inch square nearly. The first pair of bars
were Low Moor iron No. 2 ; the second pair, Blaenavon iron No. 2 ;
* Enc. Brit., 8th Ed., Vol. xx., p. 749, Art. " Strength of Materials."
366
ELASTICITY AND SET.
[CHAP. xvin.
the third pair, Gartsherrie iron No. 3; and the fourth pair, a
mixture of Leeswood iron No. 3 and Glengarnock iron No. 3, in
equal proportions. Table I. contains the mean of these experiments
reduced to a convenient unit-strain by Mr. Clark, and I have
added in the last column the coefficients of compressive elasticity
per square inch, obtained by dividing the original length, viz.,
120 inches, by the decrements of length per ton in the second
column (8).*
TABLE I.— DECREMENTS OP LENGTH AND COMPRESSIVE SETS OF A CAST-IRON BAR
10 FEET LONG AND 1 INCH SQUARE.
Tons
per square
inch.
Decrements of length
per ton.
Total Decrements
of length.
Sets.
The coefficient
of Compressive
Elasticity
per square inch.
inch.
inch.
inch.
tons.
1
•020338
•020338
•000510
5900
2
•021038
•042077
•002452
6704 \
3
•021618
•064855
•004340
6551
4
•021369
•085479
•006998
5615
4
5
•021594
•107872
•009188
5557
£
0
OO
6
•021752
•130513
•011798
5517
CO
7
•021950
•153654
•015243
5467
3
8
9
•022154
•022374
•177235
•201373
•018572
•024254
5416
5363
II
a
3
10
•022477
•224774
•028126
5339
rH
•***
11
12
•022567
•022802
•248237
•273632
•032023
•037653
5317
5262
I
s
13
•023014-
•299187
•043318
5214
14
•023523
•329330
•052640
5101 ,
15
•023539
•353092
•060905
5098
16
•024409
•390558
•080256
4916
17
•024805
•421695
•086298
4838
* Rep. of Iron Com., App., p. 63 ; and Clark on the Tubular Bridges, p. 312.
CHAP. XVIII.] ELASTICITY AND SET. 367
Mr. Hodgkinson makes the following remarks on these experi-
ments : — " The great difficulty of obtaining accurately the decre-
ments and sets from the small weights in the commencement of
the experiments, rendered those decrements and sets, particularly
the latter, very anomalous ; it was found, too, that some of the
bars which had been strained by 16 or 18 tons had become very
perceptibly undulated. It has not been thought prudent, there-
fore, to draw any conclusion from bars which have been loaded
with more than 14 to 16 tons ; and it may be mentioned that the
results from 2 to 14 tons are those only which ought to be used in
seeking for general conclusions."* (See the mean value of E' in
the last column.)
The results of Table I. are exhibited graphically in Fig. 104,
where the longer curve refers to the total decrements of length,
and the shorter one to the sets. The ordinates represent the
weights in column 1, and the abscissas the total decrements of
length and sets in columns 3 and 4 respectively of Table I.
Fig. 104.
DECREMENT OP LENGTH AND SET OF CAST-IRON IN COMPRESSION.
Rep. of Iron Com., App., p. 64.
368 ELASTICITY AND SET. [CHAP. XVIII.
The uniformity of the curve of decrements shows that there is
no abrupt alteration in the compressive elasticity of cast-iron as -
far as 17 tons per square inch and possibly up to a higher amount.
400. Hodgkinson's formulae for the decrement of length
and set of cast-iron in compression. — The following formula
was deduced by Mr. Hodgkinson from his experiments on the four
different irons just described to express the relation between the
load and the corresponding decrements of length in cast-iron bars
1 inch square and of any length.*
\' = I {-012363359 — VH)00152853 — -000000001 9 12 12 W} (243)
Where V = the decrement of length in inches,
I = the length in inches,
W = the weight in Ibs. compressing the bar.
Mr. Hodgkinson expressed the compressive set of bars of Low
Moor cast-iron 10 feet long by the following equationf : —
Compressive set in inches = -543X'2 + -0013. (244)
401. Increment of length and set of cast-iron in tension —
Coefficient of tensile elasticity. — The following table shows the
increments of length and tensile sets of cast-iron bars 10 feet long
and 1 inch square, reduced by Mr. Clark from Mr. Hodgkinson's
experiments " upon round bars of iron, united together at the ends,
so that the whole length, exclusive of the couplings, was 50 feet,
except in two instances, where the length was 48 feet 3 inches.
There were nine experiments upon these connected lengths, and
the experiments were upon four kinds of cast-iron — Low Moor
No. 2, Blaenavon No. 2, Gartsherrie No. 3, and a mixture of
iron, composed of Lees wood No. 3 and Glengarnock No. 3, in
equal proportions. There were two experiments upon each of the
simple irons, and three upon the mixture, and the mean results
were afterwards reduced to those of 10 feet and 1 square inch
exactly." " The bars were suspended vertically, and acted upon
directly by weights attached at their lower ends."! I have added
in the last column the coefficients of tensile elasticity, obtained by
* Rep. of Iron Com., App., p. 109.
f Idem, p. 123.
J Idem, pp. 59, 51 ; and Clark on the Tubular Bridges, p. 379.
CHAP. XVIII.]
ELASTICITY AND SET.
369
dividing the original length, viz., 120 inches, by the increments of
length per ton in the second column.
TABLE II. — INCREMENTS OF LENGTH AND TENSILE SETS OF A CAST-IKON BAR
10 FEET LONG AND 1 INCH SQUARE.
Tons,
per square
inch.
Increments of
length per Ton.
Total increments
of length.
Sets.
The coefficient of
Tensile Elasticity
per square inch.
inch.
inch. inch.
tons. Ibs.
1
•01976
•01976
•000579
6073 = 13,603,520
*
•02027
•04155
•001860
5920 = 13,260,800
II
3
•02171
•06515
•003954
5528 = 12,382,720
II
4
5
•02318
•02479
•09274
•12397
•007543
•012619
5177 = 11,596, 480 Mil
§*[
4841 = 10,843,840 ~
6
•02727
•16363
•020571
4400 = 9,856,000
6k
•02815
•18297
•023720
4263 = 9,549,120
The mean increment of length per ton for the first 3 tons per
square inch equals '0001715 of the length. The results of Table
II. are exhibited graphically in Fig. 105, where the longer curve
refers to the total increments of length and the shorter one to the
sets. The ordinates represent the weights in column 1, and the
abscissas the total increments of length and sets in columns 3
and 4.
Fig. 105.
INCREMENT OF LENGTH AND SET OF CAST-IRON IN TENSION.
The uniformity of the curve of increments shows that there is
2 B
370 ELASTICITY AND SET. [CHAP. XVIII.
no abrupt change in the tensile elasticity of cast-iron up to 6'5 tons
per square inch, and possibly up to the limit of rupture, the mean
of which for the 4 irons experimented on was 7'014 tons per square
inch.
By the aid of Tables I. and II. we can easily find approximately
the decrement, increment, or set of cast-iron bars of any section.
Ex. The compression flange of a new cast-iron girder, 40 feet long, which has not
been previously strained, will be shortened by an inch-strain of 6 tons by an amount
equal to 40 X 0-0130513 = 0'522052 inch, and its set, or residual decrement of length
after the load has been removed, will equal 40 X 0'0011798 = 0'047192 inch. If the
whole of this set were permanent, which however is problematical, the flange would be
permanently shortened by this amount, and on any subsequent application of the same
load its new decrement of length would = 0'522052 — 0'047192 = 0'474860 inch.
4O3. Hodgkinson's formula* for the increment of length
and set of cast-iron in tension. — The following formula was
deduced by Mr. Hodgkinson from his experiments on the ex-
tension of the four different irons just described, to express the
relation between the load and the corresponding increments of
length in cast-iron bars 1 inch square and of any length.*
X = J{-00239628-V-000005 74215 - '000000000343946 W} (245)
Where A. = the increment of length in inches,
I = the total length in inches,
W = the weight in ft>s. extending the bar.
The tensile set of bars 10 feet long is as follows : —
Tensile set in inches = -0193?, + -64X2 (246)
4O3. Coefficients of tensile, compressive and transverse
elasticity of cast-iron different. — On comparing Tables I. and
II. it will be observed that, though the mean of the coefficients of
compressive elasticity up to 14 tons, and of tensile elasticity up
to 5 tons, per square inch are substantially the same, namely,
1 2, 000, 000 lb s. per square inch, the several coefficients themselves
differ materially, especially as they approach the limit of tensile
strength ; for instance, at 6 tons per square inch the coefficient of
compressive elasticity is 1'25 times that of tensile elasticity. The
coefficients of transverse elasticity derived from experiments on a
* Rep. of Iron Com., App., pp. 60, 108.
CHAP. XVIII.] ELASTICITY AND SET. 371
moderate and on the ultimate deflection of a rectangular bar of
Blaenavon iron, broken by transverse pressure, are also different,
though they closely approach the limiting coefficients of tensile
elasticity in Table II. See ex. in 835, also 246.
404. Increment of length and set of cast-iron extended
a second time — Relaxation of set — Viscid elasticity. — Mr.
Hodgkinson made a second series of experiments on the extension
of some parts of the coupled bars which were strained nearly to
their breaking point, but had escaped actual rupture at the first
trial.* Their total increments of length on the second trial,
though very nearly the same as before, were slightly less for the
higher loads. It might perhaps be supposed that bars once
stretched would not again take a set, provided the second load did
not exceed that previously applied. This, however, was not the
case, for the barstook sets again, though in general less than
before, their mean ultimate set being nearly half that on the first
trial. It is very probable that cast-iron, and also other materials,
recover a portion of the set when the strain producing it is
relaxed for some time — in fact, that there exists a sort of sluggish
elasticity, due perhaps to a certain viscidity of the material.
Possibly, constant repetitions or long continuation of strain would
render the set permanent. Experiments alone can settle these
points, which, however, have more interest for the physicist than
practical importance for the engineer.
405. Set of cast-iron bars from transverse strain nearly
proportional to square of deflection. — The set of cast-iron
bars subject to transverse strain is nearly proportional to the
square of their deflection, though somewhat less, and may be
expressed approximately by the following formula deduced by Mr.
Hodgkinson from his experiments on rectangular bars of Blaenavon
cast-iron bent transversely by a load in the middle. |
D2
Transverse set in inches = -^—= (-47)
ol'O
in which D represents the deflection of the bar in inches.
* Rep. of Iron Com,., App., p. 61.
t Ibid., p. 69.
372
ELASTICITY AND SET.
[CHAP, xviii.
WROUGHT-IRON.
4O6. Decrement of length of n ronght-iron in compres-
sion— Coefficient of compressive elasticity — Elastic limit. —
The following table contains the results of experiments by Mr.
Hodgkinson on the compression of two wrought-iron bars 10 feet
long and 1 inch square nearly, the weights increasing at first by 2
tons and afterwards by 1 ton at a time.*
TABLE III. — DECREMENTS OF LENGTH OF WROUGHT-IRON BARS 10 FEET LONG
AND 1 INCH SQUARE NEARLY.
Bar 1.
Area of section = 1 -025 X V025 = 1-0506
square inches.
Bar 2.
Area of section = 1-016 X 1-02 = 1-0368
square inches.
Weight
compressing
Bar.
Total
Decrements
of length.
Decrements
per ton.
Weight
compressing
Bar.
Total
Decrements
of length.
Decrements
per ton.
Ibs.
inches.
inches.
fta.
inches.
inches.
5098
•028
—
5098
•027
—
9578
•052
•012
9578
•047
•010
14058
•073
•0105
14058
•067
•010
16298
•085
•012
—
—
—
18538
•096
•on
18538
•089
•on
20778
•107
•Oil
20778
•100
•on
23018
•119
•012
23018
•113
•013
25258
•130
•on
25258
•128
•015
27498
•142
•012
27498
•143
•015
29738
•154
•012
29738
•163
•020
31978
•174
•020
31978
•190
•027
34218
•214
•040
in £ hour.
•261
•071
—
—
—
31978
•269
—
—
—
—
in J hour.
•282
—
—
—
—
repeated.
•328
—
In the foregoing experiments the total decrements of length
* Rep. of Iron Com., App., p. 122.
CHAP. XVIII.] ELASTICITY AND SET. 373
increase with considerable uniformity in proportion to the weight,
until the pressure reaches the elastic limit of about 12 tons per inch,
after which irregular bulging begins, the amount of which, no doubt,
will depend on the quality of the iron, the hard and brittle irons
bulging less than the tough and ductile kinds. The mean decrement
of length per ton per square inch within this elastic limit = '0000964
= th of the original length. Hence, the coefficient of
10,o7o
compressive elasticity of bar iron from Hodgkinson's experiments
= 10,376 tons = 23,243,179 ft>s. per square inch.* In several
experiments made by the " Steel Committee" on the compression
of iron bars 10 feet long and 1J inch diameter, the mean limit of
compressive elasticity was 12*32 tons per square inch, and the mean
decrement of length within this limit was '00007725, = rarrrvtk
1.2, y4o
of the original length for each ton, which makes the coefficient of
compressive elasticity of these particular bars = 12,945 tons =
29,000,000 Ibs. per square inch, or very nearly equal to that of
steel, f
4O7. Increment of length and set of wrought-iron in
tension — Coefficient of tensile elasticity — Elastic limit —
Effects of cold-hardening and annealing on the elasticity
of iron. — Table IV. contains the results of experiments by Mr.
Hodgkinson on the extension and set of two bars of annealed
wrought-iron of the quality denominated " best," reduced to the
standard of bars 10 feet long and 1 inch square ; their real dimen-
sions were as follows :J —
Bar 1. Bar 2.
Length, - - 49 feet 2 inches, - 50 feet.
Mean diameter, - '517 inch, - - -7517 inch.
Mean area of section, -2099 square inch, -44379 square inch.
* Rep. of Iron Com., App., p. 172.
f Expts. on Sted and Iron.
£ Rep. of Iron Com., App., pp. 47, 49.
374
ELASTICITY AND SET.
[CHAP. xvni.
TABLE IV.— INCREMENTS OP LENGTH AND TENSILE SETS OP Two ANNEALED
"BEST" WRODGHT-TBON BARS, 10 FEET LONG AND 1 INCH SQUARE.
Barl.
Bar 2.
Weight per
square inch of
section.
Total
Increments of
length.
Sets.
Weight per
square inch of
section.
Total
Increments of
length.
Sets.
fl)3.
inches.
inches.
fl>s.
inches.
inches.
—
—
—
1262
•00520
—
2668
•00986
—
2524
•01150
—
5335
•02227
—
3786
•01690
•00050
8003
•03407
•000305
5047
•02240
•00060
10670
•04556
•000407
6309
•02772
•00050
13338
•05705
•000509
7571
•03298
•00045
16005
•06854
•000610
8833
•03790
•00050 ?
18673
•07993
•000813
10095
•04300
•00050 ?
21340
•09193
•001525
11357
•04854
—
24008
•10485
•003966
12619
•05370
•00070
26676
•12163
•009966
13880
•05950
—
29343
•15458
•031424
15142
•06480
—
32011
•26744
—
16404
•06980
—
—
•28271
in 5 minutes.
•13566
17666
•07530
•00130
34678
•5148
•36864
18928
•08170
—
37346
1-0995
1-01695
20190
•08740
•00270
Repeated.
1-1949
1-02966
21452
•09310
—
40013
•220
in 5 minutes.
1-093
22713
•09920
•00410
Repeated and
left on.
1-411
after 1 hour.
—
23975
•10570
—
»
1-424
after 2 hours.
—
25237
•11250
•00680
M
1-433
after 3 hours.
—
26499
•12040
—
»
1-434
after 4 hours.
—
27761
•12880
•0120
»
1-436
after 5 hours.
—
29023
•14500
—
»»
1-437
after 6 hours.
30285
•1991
CHAP. XVIII.]
ELASTICITY AND SET.
375
TABLE IV. — INCREMENTS OF LENGTH AND TENSILE SETS OP Two ANNEALED
WROUGHT-IRON BARS, 10 FEET LONG AND 1 INCH SQUARE— continued.
Barl.
Bar 2.
Weight per
square inch of
section.
Total
• Increments
of length.
Sets.
Weight per
square inch of
section.
Total
Increments
of length.
Sets.
Ibs.
inches.
inches.
fts.
inches.
inches.
Repeated and
1-443
30285
•2007
left on.
after 7 hours.
after 5 minutes.
j}
1-443
—
~m
•2018
•0736
after 8 hours.
after 10 minutes.
99
1-443
•2054
•0774
after 9 hours.
after 15 minutes.
1-443
Repeated.
•2080
•0796
after 10 hours.
•
nearly, after 20
minutes.
42681
2-148
1-983
)t
•2096
•0814
in 5 minutes.
after 1 hour.
Repeated.
2-339
H
•2366
•1082
in 6 minutes.
after bearing the
weight 17 hours
2-383
2-212
31546
•242
•1083
in 10 minutes.
after 1 hour.
after 5 minutes.
Repeated.
2-428
after 46 hours.
2-237
Repeated.
•2449
after 5 minutes.
•1111
45348
2-580
2-377
32808
•5506
•4141
after 5 minutes,
Repeated.
2-605
after 1 hour.
—
Repeated.
•7024
after 5 minutes.
•5635
2-606
Jf
•7966
•6558
after 2 hours.
after 10 minutes.
2-606
2-403
JJ
1-014
•866
after 19 hours.
after about Jhour.
48016
2-975
2733
34070
1-346
—
after 5 minutes.
after 10 minutes.
after 1 minute.
Repeated.
3-019
after 1 hour.
—
n
1-400
after 2 minutes.
—
j)
3-029
n
1-600
1-44
after 11 hours.
50684
4-195
'3-941
Repeated.
1-65
—
in 10 minutes.
in 10 minutes.
after 1 minute.
Repeated.
4-226
—
»
1-786
after 1 hour or less.
1-628
99
4-227
35332
2-04
1-874
in 7 hours.
after 5 minutes.
4-227
Repeated.
2-18
2-01
in 12 hours.
after 5 minutes.
r
Broke at one of the "weld-
2-254
2-08
53351 = \
ings" where there was a slight
"
23-817 tons, t
defect; perhaps a rather smaller
weight would have broken it.
36594
2-54
after 6 minutes.
—
37856 = )
o-qqi
16-9 tons, j
£t Otft
The loop at the lower end
of the rod having broken,
the experiment was discon-
tinued.
376
ELASTICITY AND SET.
[CHAP. xvin.
From these experiments Mr. Hodgkinson inferred that the
coefficient of tensile elasticity = 27,691,200lbs. = 12,362 tons per
square inch.* The limit of tensile elasticity, it will be observed,
lies between 11 and 12 tons per square inch.
The relation between the weights and corresponding increments
of length of the first bar in the foregoing table are exhibited
graphically in Fig. 106, in which the ordinates represent the
weights per square inch of section, and the abscissas the corre-
sponding increments of length.
Fig. 106.
INCREMENT OP LENGTH OP WROUGHT-IRON IN TENSION.
The following table is given by Mr. Clark at p. 373 of his work
on the Britannia and Conway tubular bridges. Though not
expressly so stated, it is probably reduced from Mr. Hodgkinson's
experiment on Bar 1 in Table IV.
* Rep. of Iron Com., App.j p. 172.
CHAP. XVIII.]
ELASTICITY AND SET.
377
TABLE V.— INCREMENT OP LENGTH AND TENSILE SET OF A NEW WROUGHT-IRON
BAR, 10 FEET LONG AND 1 INCH SQUARE.
Computed
Corresponding
extension
extension in
Tons
per
square
inch.
Observed extension
in terms of the
length.
assumed
uniform at
16o8ooo
of the length
fractional parts
of the length
computed at
1OOOOO
Observed set
in terms of
the length.
Observed set
in fractional
parts of
the length.
per ton
per ton per
per square inch.
square inch.
1
•0000689
•00008
1250TJ
2
•000156
•00016
6-T5S
3
•000238
•00024
~3T$~5
•00000213
*i^m
4
•000319
•00032
3"TT5
•00000283
JTOTTT
5
•000399
•00040
TSTTo"
•00000356
TffToTc)
6
•00048
•00048
if OFT
•00000427
vnVra
7
•00056
•00056
TT8-6
•00000497
ToToo^
8
•00064
•00064
TsW
•00000650
f3^j¥TB
9
•00072
•00072
T3V?
•00001201
'B"5"2T5
10
•00080
•00080
T-T50
•00001334
T^TST
11
•000896
•00088
Ti3^
•00003392
27?-8^
12
•00102
•00096
W??y
•00008368
TT9"5"0~
13
•00128
•00104
»ih
•0002598
3"ff4"S'
j -00218 j
14
< in ten minutes >
•00112
If1!
•0011075
^T
( '00231 )
15
•00416
•00120
a
cS
•002976
rsr
16
•00443
•00128
JQ
•0003175
l.
j -00934 }
1
17
< in ten minutes >
•00136
1 fl)
•008750
TT*
( -01015 )
?5
( _ -01024 )
**
18
< in ten minutes ;
•00144
w §
•009170
T^ir
( '01212 )
II
j -01785 )
19
< in ten minutes >
•00152
1 *>?
•018590
i
( '02017 )
Tl
! -02124 )
||
20
in ten minutes >
•00160
^ &
•019790
^
•02146 )
sl
j ^ '02429 )
1 «
21
< in ten minutes >
•00168
S3 "*"
•022310
1
( -02472 )
:§
( -03400 J
•s
22
< in ten minutes >
•00176
c
•031933
JL
( -03425 )
A
378 ELASTICITY AND SET. [CHAP. XVIII.
The foregoing tables and the diagram show that the increment
of length of annealed wrought-iron in tension increases with great
uniformity in proportion to the weight, and nearly equals '00008,
of the length for each ton per square inch up to 11 or
12 tons, after which the law suddenly changes, and rapid and
rather irregular stretching begins, the amount depending, no doubt,
on the quality of the iron, i.e., its hardness or ductility.
Mr. Barlow also made several experiments on bars of wrought-iron,
from which he inferred that its limit of tensile elasticity is about
10 tons per square inch, and that it extends "000096 = -- -r~th
J- \J j^r L I
of its length for each ton within this limit.* In experiments made
by the " Steel Committee" on 10 feet lengths of iron bars, 1J
inches diameter, the mean limit of tensile elasticity was 12 '7 tons
per square inch, and the mean increment of length within this
limit was '0000784 = , »Rgth of the original length for each ton
1Z, 1 00
per square inch.
General Morin also made some experiments on fine charcoal
iron wire, and found that the process of hardening wire by cold
drawing increased its limit of elasticity to about 19 tons per square
inch, while the coefficient of elasticity remained the same as that
of ordinary bar iron, viz., 12,473 tons per square inch. Annealing
iron wire had the effect of reducing its coefficient of tensile elasticity
to 10,009 tons per square inch.f We may conclude from these
various experiments that the elastic limit and the coefficient of
elasticity of wrought-iron vary considerably with the quality and
condition of the iron, but for practical purposes we may generally
adopt 12 tons as the limit of elasticity, and 24,000,000 fibs., = 10,714
tons per square inch, as the coefficient of elasticity of ordinary plate
and bar-iron, either in tension or in compression, though sometimes
it may reach 29,000,000 Ibs. ; the former is equivalent to an
* Strength of Material, p. 315.
f Proc. Inst. C. E., Vol. xxx., p. 261.
CHAP. XVIII.] ELASTICITY AND SET. 379
alteration of jTrynth — -000093 of the original length for each
ton per square inch.
408. Elastic flexibility of cast-iron twice that of wrought-
iroii — Law of elasticity truer for wrought than for cast-
iron. — Comparing the coefficients of elasticity of cast and wrought-
iron, we find that the elastic flexibility of cast-iron is nearly twice
as great as that of wrought-iron, that is, the alteration of length
from the same unit-strain is nearly twice as great in cast as in
wrought-iron ; in other words, wrought-iron is nearly twice as stiff
as cast-iron. On this account a girder of cast-iron will deflect
nearly twice as much as a similar one of wrought-iron, provided
the flanges of both girders are subject to the same unit-strains. It
will also be observed that Hookes' law of the proportionality of the
loads to the changes of length they produce is less exact for cast
than for wrought-iron within the limits of elasticity.
409. Stifftiess of imperfectly elastic materials improved
by stretching — Practical method of stiffening wrought-iron
bars — Limit of elasticity of wrought-iron equals 12 tons per
square inch — Proof-strain should not exceed the limit of
elastictiy. — When an imperfectly elastic material has received a
permanent set from the application of any weight which is sub-
sequently removed, the material becomes more perfectly elastic
than before within the range of strain which first produced the set,
and its alteration of length per unit of strain is less than at first.
When, for instance, a girder is tested for the first time, its deflection
exceeds that produced by a subsequent application of the same load.
Hence, the common practice of " stretching" girders by heavy
loads before their final inspection. In compound structures, such
as lattice girders, some of the initial deflection may, perhaps, be
attributed to the separating or closing together of the numerous
joints on the first application of a heavy load, though probably the
greater portion is due to the straightening of parts in tension
originally constructed a little out of line. The ultimate deflection
of a bar of soft wrought-iron subject to transverse strain is very
considerable, and when the useful load which such a bar will carry
is determined by the amount of deflection rather than by its
380 ELASTICITY AND SET. [CHAP. XVIII.
breaking weight, its useful strength, i.e., its stiffness, may be much
increased by giving it a considerable camber when at a dull red
heat, and afterwards straightening it when cold. Such a bar, as
far as deflection in the direction in which it was straightened is
concerned, is stronger than before.* For practical purposes the
limit of elasticity of wrought-iron, as already stated, does not exceed
12 tons per square inch, and though higher strains than this may
not in the least diminish its ultimate strength, yet they will take the
" stretch" out of the iron and may thus render what was originally
tough and ductile metal so hard and brittle as to be seriously
injured for many purposes. A tough quality of iron will evidently
sustain sudden shocks with greater impunity than brittle iron, and
previous over-straining may perhaps thus explain the unexpected
rupture of chains with suddenly applied loads considerably below
their statical breaking weight. For instance, sudden jerks from
surging may double the usual safe working strain of a chain and
thus strain it temporarily beyond its limit of elastic reaction. This
frequently repeated will produce permanent elongation and render
the chain brittle until it has been annealed (357). These con-
siderations show that the proof-strain of wrought-iron should not
exceed its limit of elasticity.
41O. Experiments on elasticity liable to error — Sluggish
or viscid elasticity. — Scientific conclusions derived from experi-
ments on the elasticity of materials in which the effect of previous
strain is overlooked are evidently worthless, and it should be recol-
lected that time ought to be allowed after each experiment in order
to let the material adjust itself to the new condition of strain,
especially when the load approaches the limits of rupture, in which
case the deformation, or change of form, may continue for a con-
siderable time after the load is laid on, especially if aided by
vibration. Referring to the Britannia and Con way Tubular Bridges
Mr. Clark observes, " In all the tubes a considerable time elapsed
before they attained a deflection which remained constant. Time
is an important element in producing the ultimate permanent set
* Clark on the Tubular Bridget, p. 449.
CHAP. XVIII.] ELASTICITY AND SET. 381
in any elastic material; but when the permanent set due to the
strain is once attained, the continuance of the same strain induces
no further deflection, which is confirmed by the fact, that no sub-
sequent change has occurred in the deflection of the Con way
Bridge from two years of use, nor has any increase in the versed
sine of the Menai Suspension-bridge taken place in twenty-five
years, where the strain is greater than in the plates of the Conway
Bridge, and liable to be considerably varied from the oscillation
which occurs in gales of wind. The permanent strain in the
Britannia Bridge is under three-fifths of that in the Suspension
Bridge. The effect of time in producing permanent elongation has
been also observed at the High Level Bridge (Newcastle-upon-
Tyne), where the wrought-iron tie-chains, which resist the thrust
of the arches, although under much less strain than the above,
continued to extend for a considerable period before they attained
a set at which they remained constant. These motions are so
extremely minute that they are only ascertainable in large rigid
structures, where they are measured by the corresponding increase
of deflection."*
The residual set, after the strain has been removed, also takes
time to adjust itself to a permanent condition, and some crude
experiments of my own tend to prove that the set of wrought-iron
relaxes to a considerable extent, even after the lapse of several days
after the strain has been removed.
STEEL.
411. I^aw of elasticity true for steel — Coefficient and limit
of elasticity of steel. — Numerous experiments made by the
" Steel Committee" prove that the law of elasticity applies to
steel with great exactitude within the limit of elastic reaction
which for practical purposes is about 2 1 tons per square inch both
for tension and compression (898 and 359). Within this limit the
mean decrement of length per ton per square inch from compression
= -0000743 = th of the original length, and the mean
lo,4:oy
* Clark on the Tubular Bridges, p. 671.
382 ELASTICITY AND SET. [CHAP XVIII.
increment from extension = '0000764 =: -, ^"nuTT^1 of the original
length. Taking the mean of these, the coefficient of either tensile
or compressive elasticity = 13,274 tons = 29,733,760 Ibs. per
square inch. From Sir William Fairbairn's experiments on deflec-
tion under transverse strain, the coefficient of transverse elasticity
= 31, 000,000 fts. (359). For practical purposes we may assume
30,000,000 ft>s., = 13,393 tons per square inch, as the coefficient
of elasticity of steel, which is 25 per cent, greater than the
usual coefficient for wrought-iron, though the latter sometimes
approaches 29,000,000 Ibs., or very closely that of steel.
TIMBER.
418. Limit of elasticity of timber not accurately de-
fined — Coefficient of elasticity depends on the dryness of
the timber. — Experiments on timber by MM. Chevandier and
Wertheim lead them to form the following conclusions.*
1°. The density of timber appears to vary but slightly with age.
2°. The coefficient of elasticity, on the contrary, diminishes
beyond a certain age and depends on the dryness and aspect as
well as the nature of the soil in which the trees grow, northerly
aspects and dry soils raising the coefficient.
3°. The coefficient of elasticity is not sensibly affected by cutting
trees before or after the sap is down.
4°. Properly speaking, there is no true limit of elasticity, as
there is always a permanent set along with an elastic elongation.
5°. The limit of elasticity rises with the dryness of the timber, and
wet timber takes a permanent set more readily than dry timber.
6°. In timber artificially dried in a stove, the limit of elasticity
coincides nearly with the limit of rupture, i.e., such timber takes
scarcely any permanent set.
7°. Artificial drying greatly increases the stiffness of timber.
STONE.
413. Vitreous materials take no set. — It is stated by Dr.
Robinson that " hard bodies of an uniform glassy structure, or
* Morin, Resistance des Afattriaux, p. 37.
CHAP. XVIII.] ELASTICITY AND SET. 383
granulated like stones, are elastic through the whole extent of their
cohesion, and take no set, but break at once Avhen overloaded."*
It may be doubted whether this is true of all granulated bodies
like stones, for Mr. Mallet, referring to his experiments on crushing
small cubes of quartz and slate rock from Holyhead, 0*707 inch
upon each edge, observes, " the per-saltum way in which all the
specimens of both rocks yield, in whatever direction pressed, is
another noteworthy circumstance. The compressions do not con-
stantly advance with the pressure, but, on the contrary, the rock
occasionally suffers almost no sensible compression for several
successive increments of pressure, and then gives way all at once
(though without having lost cohesion, or having its elasticity per-
manently impaired), and compresses thence more or less for three
or four or more successive increments of pressure, and then holds
fast again, and so on. This phenomenon is probably due to the
mass of the rock being made up of intermixed particles of several
different simple minerals, having each specific differences of hard-
ness, cohesion, and mutual adhesion, and which are, in the order
of their resistances to pressure, in succession broken down, before
the final disruption of the whole mass (weakened by these minute
internal dislocations) takes place. Thus it would appear that the
micaceous plates and aluminous clay-particles interspersed through
the mass give way first. The chlorite in the slate, and probably
felspar-crystals in the quartz-rock, next, and so on in order, until
finally the elastic skeleton of silex gives way, and the rock is
crushed. It is observable, also, that this successive disintegration
does not occur at equal pressures, in the same quality and kind of
rock, when compressed transverse and parallel to the lamination. "f
Hookes' law probably applies up to the limit when the first crush-
ing of the weakest ingredient occurs. What takes place afterwards
corresponds with the intermittent way in which wrought-iron in
tension stretches once the limit of elasticity has been passed.
* Encyc. Metr., 8th ed., art. " Strength of Materials," Vol. xx., p. 756.
f Phil. Trans., 1862, p. 669.
384 TEMPERATURE. [CHAP XIX.
CHAPTER XIX.
TEMPERATURE.
414. Arches camber* suspension bridges defied, and
girders elongate, from elevation of temperature— Expansion
rollers. — Changes of temperature affect bridges very differently
according to their mode of construction. An increase of tem-
perature causes the crowns of iron arches which are confined
between fixed abutments to rise, and the spandrils to extend
lengthways, chiefly along their upper flange or horizontal member ;
hence, room for longitudinal expansion should be provided by
leaving a vertical space between the ends of the spandrils and the
masonry of the abutments above springing level. When iron
arches extend over several spans, the spandrils of the different
spans should not be rigidly connected together like continuous
girders, for then their expansion may cause a dangerous crushing
strain along the vertical line of junction and throughout the
horizontal member, a portion of which strain will, no doubt, be
transmitted to the ribs themselves. When, therefore, it is con-
sidered desirable to connect together the spandrils of consecutive
iron arches, this should be effected by sliding covers, or some
similar contrivance, which, though they restrain lateral motion, yet
will allow perfect freedom for changes of length. The rise in the
crown of one of the cast-iron arches of South wark Bridge was
observed by Mr. Rennie to be about 1*25 inches for a change of
temperature of 50°F; the length of the chord of the extrados is
246 feet and its versed sine is 23 feet 1 inch; accordingly, the
length of the arch, which is segmental, is 302O8 inches.* The
cast-iron bridge of Charenton, whose span and versed sine are 35 and
* Trans. Inst. C. K, VoL iii., p. 201.
CHAP. XIX.] TEMPERATURE. 385
4 metres respectively, has been observed to rise 14 millimetres
('55 inch) on the side exposed to the west from an elevation of
14°C. in the temperature of the air.*
Stone arches are affected in a similar way to iron arches. With
increased temperature the crown rises and joints in the parapets
open over the crown, while others over the springing close up.
The reverse takes place in cold weather ; the crown descends, joints
over the springing open and those over the crown close. When
stone or iron arches are of large span these movements from changes
of temperature will generally dislocate to a certain degree the
flagging and pavement of the roadway above. This is very con-
spicuous in Southwark Bridge.
An increase of temperature causes suspension bridges to deflect,
just the reverse of what happens with arches. Girders, which exert
only a vertical pressure on the points of support, extend longi-
tudinally under the same influence, and on this account it is usual
in long bridges to provide expansion rollers, or, if the span be
moderate, sliding metallic surfaces, under one end of each main
girder. It may be questioned, however, whether sliding surfaces
remain long in working order, and some engineers prefer timber
wall-plates beneath the ends of the girder, even when the span
reaches 150 feet. In place of being supported by rollers, which are
apt to set fast, girders are sometimes hung from suspension links, the
pendulous motion of the links affording the requisite longitudinal
movement due to change of temperature.! The chains of suspen-
sion bridges are generally attached to saddles which rest on rollers
on top of the towers ; the object of these, however, is rather to
compensate for unequal loading than for changes of temperature.
415. Alteration of length from change of temperature —
Coefficients of linear expansion. — The coefficient of linear
* Morin, Resistance des Matgriaux, p. 116.
•\" Expansion rollers were placed under one end of each principal of the roof over
the New-street Station, Birmingham, 212 feet span ; the other end was attached to
cast-iron columns. The rollers did not move, but the columns rocked 0'01917 inches
for each degree Fahrenheit. — (Proc. Inst. C.E., Vol. xiv., p. 261.) Expansion rollers
were also placed under one end of each of the crescent-shaped principals of the old
Lime-street Station, Liverpool, 1534 fee* sPan> but ^d not act.— (Idem, Vol. ix., p. 207.)
2 c
386
TEMPERATURE.
[CHAP. xix.
expansion of any material is the fractional part of its length at zero
centigrade which it elongates or shortens from a change of one unit
of temperature, generally 1°C. The alteration of length for other
changes of temperature is expressed by the following equation :—
X = nkl (248)
Where I = the length of the bar at 0°C.,
k = the coefficient of linear expansion of the material for
one degree centigrade,
n = the number of degrees through which the temperature
of the bar is raised or lowered,
\ — the increment or decrement of length due to a change
of temperature equal to n degrees.
Ex. The total length of the Britannia wrought-iron tubular bridge is 1,510 feet, and
an increase of temperature of 26°F. caused an increase of length of 3| inches, what is
the coefficient of linear expansion of the tube for 1°C. ? — (Clark, p. 715.)
Here, I = 1510 feet = 18120 inches,
n = 26°F. = 14-44°C.,
A = 3-25 inches.
3-25
Answer, Jc = ± =
= 0-00001 2421 inch,
14-44X18120
which, it will be observed, agrees closely with the coefficient of expansion of wrought-
iron in the table below.
The following table contains the coefficients of linear expansion
of various materials for one degree centigrade.
TABLE I.— COEFFICIENTS OF LINEAR EXPANSION FOR 1°C.
Description of Material.
Authority.
Coefficients
of linear
expansion
for 1°C.
METALS.
Antimony,
Smeaton,
•000010833
Bismuth,
Do.
•000013917
Brass (supposed to be Hamburg plate brass), -
Ramsden,
•000018554
Do. (English plate, in form of a rod),
Do.
•000018928
NOTE. — One degree Fahrenheit = $ths of one degree centigrade. To convert a
given temperature on Fahrenheit's scale to the corresponding temperature centigrade,
subtract 32°., and multiply the remainder by $. Thus, the temperature of 86°F. =
30°C., but a range of 86°F. = 48°C., nearly.
CHAP. XIX.] TEMPERATURE. 387
TABLE L— COEFFICIENTS OF LINEAR EXPANSION FOB 1°C.— continued.
Description of Material.
Authority.
Coefficients
of linear
expansion
for 1°C.
METALS.
Brass (English plate, in form of a trough),
Ramsden,
•000018949
Do. (cast),
Smeaton,
•000018750
Do. (wire),
Do.
•000019333
Copper, -
Laplace & Lavoisier,
•000017122
Do.
Do.
•000017224
Gold (de depart)
Do.
•000014661
Do. (standard of Paris, not annealed),
Do.
•000015516
Do. ( do. annealed),
Do.
•000015136
Iron (cast),
Ramsden,
•000011094
Do. (from a bar cast 2 inches square),
Adie,
•000011467
Do. ( do. | an inch square),
Do.
•000011022
Do. (soft forged),
Laplace & Lavoisier,
•000012204
Do. (round wire),
Do.
•000012350
Do. (wire),
Troughton,
•000014401
Lead, -
Laplace & Lavoisier,
•000028484
Do.,
Smeaton,
•000028667
Palladium, - ...
Wollaston,
•000010000
Platina, -
Dulong & Petit, -
•U00008842
Do., -
Troughton,
•000009918
Silver (of Cupel),
Laplace & Lavoisier,
•000019097
Do. (Paris standard), -
Do.
•000019087
Do., -
Troughton,
•000020826
Solder (white ; lead 2, tin 1), -
Smeaton,
•000025053
Do. (spelter ; copper 2, zinc 1),
Do.
•000020583
Speculum metal,
Do.
•000019333
Steel (untempered),
Laplace & Lavoisier,
•000010788
388 TEMPERATURE. [CHAP. XIX.
TABLE I.— COEFFICIENTS OP LINEAR EXPANSION FOR 1°C. — continued.
Description of Material
Authority.
Coefficients
of linear
expansion
for 1°C.
METALS.
Steel, (tempered yellow, annealed at 65°C.), -
Laplace & Lavoisier
•000012396
"Do. (blistered),
Smeaton,
•000011500
Do. (rod),
Ramsden,
•000011447
Tin (from Malacca),
Laplace & Lavoisier
•000019376
Do. (from Falmouth),
Do.
•000021730
Zinc,
Smeaton,
•000029417
TIMBER.
Baywood, in the direction of the grain, dry,
Deal, do. do. do.
Joule,
Do. - •
•00000461 to
•00000566
•00000428 to
•00000438
STONE, BRICK, GLASS, CEMENT.
Arbroath pavement,
Adie,
•000008985
Brick (best stock),
Do. -
•000005502
Do. (fire),
Do. -
•000004928
Caithness pavement,
Do. -
•000008947
Cement (Roman),
Do. -
•000014349
Glass (English flint),
Laplace & Lavoisier,
•000008117
Do. (French, with lead),
Do. -
•000008720
Granite (Aberdeen grey),
Adie,
•000007894
Do. (Peterhead red, dry), -
Do. -
•000008968
Do. ( do. moist),
Do.
•000009583
Greenstone (from Ratho),
Do. -
•000008089
Marble (Carrara, moist),
Do. -
•000011928
Do. ( do. dry), -
Do. -
•000006539
Do. (black Galway),
Do. -
•000004452
Do. ( do. softer specimen, containing more
fossils),
Do. -
•000004793
CHAP. XIX.] TEMPERATURE. 389
TABLE I. — COEFFICIENTS OF LINEAR EXPANSION FOB 1°C. — continued.
Coefficients
Description of Material.
Authority.
of linear
expansion
for 1°C.
STONE, BRICK, GLASS, CEMENT.
Marble (Sicilian white, moist),
Adie,
•000014147
Do. ( do. dry), -
Do.
•000011041
Sandstone (from Craigleith quarry),
Do. -
•000011743
Slate (from Penrhyn quarry, Wales), -
Do.
•000010376
Adie ; Dixoris Treatise on Heat, p. 35.
Dulong and Petit ; Pouillet, Elements de Physique, p. 221.
Joule ; Proc. Roy. Soc., Vol. ix., No. 28, p. 3.
Laplace and Lavoisier ; Dixon's Treatise on Heat, p. 29.
Ramsden ; idem, p. 27.
Smeaton ; Pouillet, Elements de Physique, p. 221.
Troughton ; idem.
Wollaston ; idem.
416. Expansibility of timber diminished, or even reversed,,
by moisture. — Mr. Joule found that moisture occasioned a
marked diminution in the expansibility of timber by heat. After
a rod of bay-wood on which he experimented " had been immersed
in water until it had taken up 150 grains, making its total weight
882 grains, its coefficient of expansion was found to be only
•000000436. Experiments with the rod of deal, weighing when
dry 425 grains, gave similar results ; when made to absorb water
its coefficient of expansion gradually decreased, until, when it
weighed 874 grains, indicating an absorption of 449 grains of
water, expansion by heat ceased altogether, and on the contrary, a
contraction by heat equal to '000000636 was experienced.*
417. Moisture increases the expansibility of some stones —
Raising1 the temperature produces a permanent set in
others. — "In the case of greenstone, and some descriptions of
marble, the effect of moisture was to increase the amount of
* Proc. Roy. Soc., Vol. ix., No. 28, p. 3.
390 TEMPERATURE. [CHAP. XIX.
expansion; in other instances no effect of this kind was perceptible.
Mr. Adie also found that in white Sicilian marble a permanent
increase in length was produced every time that its temperature
was raised, the amount of increase diminishing each time."*
418. A. change of temperature of 15°C. in cast-iron, and
7*5°C. in \vrou ght-iron3 are capable of producing a strain of
one ton per square inch — Open-work girders in the United
Kingdom are liable to a range of 45°C. — The alteration of
length of a cast-iron bar within the range of three tons tension and
seven tons compression per square inch, which include the ordinary
limits of working strain, is about '000175 of the original length
for each ton per square inch, and its coefficient of linear expansion
for 1°C. = -000011467 according to Adie; consequently a change
of temperature of about 15°C. (= 27°F.) is capable of developing
a force equal to one ton per square inch. Again, if we assume
that the alteration of length of a bar of wrought-iron for both
tensile and compressive strains = '000093 of its length for each
ton per square inch, its coefficient of expansion for 1°C. being
•000012204, a change of temperature of about 7'5°C. (= 13'5°F.)
is capable of developing a force equal to one ton per square inch.
Hence, a given change of temperature will develop twice as much
force in wrought as in cast-iron. The range of temperature to
which open-work bridges through which the air has free access are
subject in this country seldom exceeds 45°C. (= 81°F.), for which
range wrought-iron alters '000549, or nearly T^o*n °f '^s original
length. This change of length is nearly equivalent to that which
would be produced by a strain of 6 tons per square inch. The
range of temperature of cellular flanges may, however, exceed that
mentioned above, as Mr. Clark mentions that the temperature of
the Britannia Tubular Bridge, before it was roofed over, differed
" widely from that of the atmosphere in the interior, for the top
during hot sunshine has been oberved to reach 120°F., and even
considerably more ; and, on the other hand, a thermometer on the
surface of the snow on the tube has registered as low as 160F."f
* Dixon's Treatise on Heat, p. 34.
t Britannia, and Conway Tubular Bridges p. 71
CHAP. XIX.] TEMPERATURE. 391
A familiar instance of the contractile force of wrought-iron in
cooling is exhibited in the tires of wheels. " An ingenious appli-
cation of this force was also made in the case of a gallery in the
Conservatoire des Arts et Metiers in Paris, whose walls were
forced outwards by some horizontal pressure. To draw them
together M. Molard, formerly director of the Museum in that
establishment, had iron bars passed across the building, and
through large plates of metal bearing on a considerable surface of
the external walls. The ends of these bars were formed into
screws, and provided with nuts, which were first screwed close
home against the plates. Each alternate bar was then elongated
by means of the heat of oil lamps suspended from it, and when
expanded the nuts were again screwed home. The lamps being
removed, the bars contracted, and in doing so drew the walls
together. The other set of bars was then expanded in the same
manner, their nuts screwed home, and the wall drawn in through
an additional space by their contraction. And this series of
operations was repeated until the walls were completely restored
to the vertical, in which position the bars then served permanently
to secure them."*
419. Tabular plate girders are subject to vertical and
lateral motions from changes of temperature — Open-work
girders are nearly quite free from these movements. — In
addition to the longitudinal movements to which all girders are
subject from changes of temperature, tubular plate girders move
vertically or laterally whenever the top or one side becomes hotter
than the rest of the tube. Referring to the Britannia Tubular
Bridge, Mr. Clark states that " even in the dullest and most rainy
weather, when the sun is totally invisible, the tube rises slightly,
showing that heat as well as light is radiated through the clouds.
On very hot sunny days the lateral motion has been as much as 3
inches, and the rise and fall 2 inches and T%ths."t These vertical
and lateral motions have not been much observed in lattice or
open-work girders; no doubt because the air and sunshine have
*
* Dixon's Treatise on Heat, p. 121.
f Tubular Bridges, p. 717.
392 TEMPERATURE. fCHAP. XIX.
free access to all parts and thus produce an equable temperature
throughout the whole structure.
430. Transverse strength of cast-iron not affected by
changes of temperature between 16°F. and 6OO°F. —
It appears from Sir William Fairbairn's experiments on the trans-
verse strength of cast-iron at various temperatures from 16°F.
upwards, that its strength " is not reduced when its temperature
is raised to 600°F., which is nearly that of melting lead; and
it does not differ very widely, whatever the temperature may be,
provided the bar be not heated so as to be red hot."*
431. Tensile strength of plate-iron uniform from OF. to
4OO°F. — It also appears from Sir William Fairbairn's experiments
on wrought-iron at various temperatures that the tensile strength
of plates is substantially uniform between 0°F. and 400°F. This
result is corroborated by the experiments of the committee of the
Franklin Institute appointed to report on the strength of materials
employed in the construction of steam boilers. Sir Wm. Fairbairn
also found that the strength of the best bar-iron was increased
about one-third when the temperature reached 320°F., after which
it again diminished.! This, however, seems anomalous, and further
confirmation would be desirable.
* Hodgkinson's Exp. Res., p. 378.
•f* Useful Information for Engineers, second series, pp. 114, 124.
CHAP. XX.] FLANGES. 393
CHAPTER XX.
FLANGES.
422. Cast-iron girders. — The compression flange of cast-iron
girders is frequently made stronger than is theoretically necessary
for the purpose of rendering it sufficiently stiff to resist side
pressure, vibration, or other disturbing causes ; in a word, to resist
flexure. As the average crushing strength of cast-iron is about
5 times its tensile strength, theory indicates the most economical
proportion of the compression to the tension flange, when both are
horizontal, to be also 1 to 5 (l?), whereas it is generally made
much stronger than this, its area being sometimes one-third of
that of the tension flange. Hence, cast-iron girders rarely fail in
the compression flange and it is a common practice to calculate
their strength, as well as that of wrought-iron girders, from the
leverage of the tension flange by the following well-known modifi-
cation of eq. 18: —
W = ^ (249)
in which W = the breaking weight at the centre in tons,
a = the net area of the tension flange in square inches,
d = the depth of the web at the centre in inches,
/ = the length between bearings in inches,
c = a coefficient depending on the material.
For cast-iron double-flanged girders the coefficient c = 4 x 7
= 28, the average tensile strength of simple cast-irons being about
7 tons per square inch. For wrought-iron box girders with equal
flanges, c = 4 X 20 = 80, the tensile strength of ordinary plate
iron being about 20 tons per square inch. This equation omits any
strength derived from the vertical web acting as an independent
rectangular girder (1OO) ; it gives, therefore, too low a result when
394 FLANGES. [CHAP. xx.
the area of the web forms a large portion of the total cross section,
or when the tensile strength of cast-iron exceeds 7 tons; on the
other hand, the formula will give too high a result with narrow
plate girders which, if unsupported, generally fail by bending
sideways.
423. Cellular flanges. — The closed cell was for some years
a favourite form for the compression flange of tubular plate girders,
whereas the tension flange was generally made of one or several
plates riveted together so as to form practically one thick plate.
Fig. 107.
The adoption of the cell in this instance arose from the impression
that it was better adapted than other forms of pillar for resisting
flexure, and so no doubt it was when used as a pillar without
extraneous support. Its connexion with the continuous web, how-
ever, prevents the flange from deflecting in a vertical direction,
for at each point along its length it is held rigidly in the direction
of the thrust, nor can it escape from this without separating from
the side plates, and it is obvious that a very moderate force will
hold a pillar in the line of thrust when the flexure is of trifling
amount (153). It should also be kept in view that the stiffness of
a long unsupported plate to resist flexure is proportional to the cube
of its thickness (333), and consequently, if the top and bottom
plates of the cell be riveted together, we have a plate 8 times as
stiff as either separately. If to these we add the central plate and
CHAP. XX.] FLANGES. 395
the upper half of each side of the cell (so as to leave the depth
of girder measured from the centre of the cell to the lower flange
unaltered) and the spare angle irons, we have a top flange at least 3
times as thick and therefore 27 times as stiff to resist vertical flexure
as the unsupported top of the original cell. Though we do not
thoroughly know the laws which govern the buckling of the sides
of a tube (335), it is evident that the pile of plates possesses a
superiority over the cell in this respect. It is, moreover, clear that
the lateral stiffness of the flange is scarcely, if at all, affected by
using one thick plate of the same width and sectional area as the
cell, for, regarding the pile as a girder on its side, we have the
adjacent parts of the double web performing the duty of flanges in
place of the sides of the cell. One great objection to the cell is
this ; a large extent of surface is exposed to corrosion and is at the
same time difficult of access and therefore liable to be neglected ;
at the best its preservation is costly, and depends on the amount of
care which the painter may feel inclined to bestow on an irksome task,
for the proper completion of which he feels but little responsibility
since his work is rarely inspected, while during its tedious and
unhealthy performance he is obliged to assume an unnatural and
fatiguing posture.*
434. 1'iled flanges — Long; rivets not objectionable. —
When several plates are built into one pile it may be objected that
great length of rivet is required, and that the workmanship is in
consequence less sound ; but this objection has no real value so far
as the riveting is concerned. In parts of the Britannia Tubular
Bridge rivets passed through six layers of iron of an aggregate
thickness of nearly 3J inches,f and in the Boyne Viaduct many
rivets passed through six and seven plates, and in some parts even
nine. As I had forgotten the exact method of manipulating these
long rivets at the Boyne Viaduct, I obtained from Mr. Colville,
* A painful soreness of the eyes and tendency to faint are experienced in close
cells whenever the stifling vapour of new lead paint is not removed by constant
currents of fresh air passing through them. Hence, when the ventilation is defective,
the painter must come out at short intervals to breathe the fresh air.
•\- Britannia and Comvay Tubular Bridges, p. 575.
396 FLANGES. [CHAP. xx.
the intelligent superintendent of the iron-work, the following
details : —
" The longest rivet we had was about 8 inches long and the
holes must be well rimed out. The rivets were kept cool, head
and point, by dipping in water, and the body of the rivet made
very hot, which enabled the workmen to use the cup tool and the
heavy hammer at once. Some of the long rivets I had cut out
after being riveted, to see what they looked like, and I must say
they filled better than I expected, being at top of the piers, which
was very difficult to get to. I see no difficulty in riveting such
thickness as was at the Boyne Bridge, but it must be with care in
the heating of the rivets and using about a 14 Ib. hammer and cup
tools. Common light riveting hammers would only upset the
rivet at the point and would not fill in the body in such thickness
as 4J to 5 inches." Mr. Clark made some experiments on rivets 12
inches long, most of which "broke at the head in cooling, and it was
found necessary to cool the centre part of the rivet artificially
previous to inserting them, the head and tail alone remaining red-
hot. In this manner the contraction was avoided and the rivets
remained sound." This seems to be the reverse of the practice at
the Boyne Bridge, but it is probable that in Mr. Clark's experi-
ments the heads of the rivets were damaged by prolonged hammer-
ing with light hammers, as he inserted some red-hot rivets 8 feet
long in some castings of great strength, which, therefore, could not
yield to the tension, and these rivets on cooling remained in all
cases perfectly sound and had merely undergone a permanent
extension proportionate to the temperature.*
435. Punching and drilling tools. — Careful attention is doubt-
less required in punching plates so that the holes in the successive
layers may coincide, and without proper precaution much trouble
and expense would be incurred in subsequent riming out the holes,
but this labour may, to a great extent, be avoided by using accurate
templates, or when the magnitude of the work warrants such an
outlay, by punching machines similar to the Jacquard machine used
* The Tubular Bridyes, p. 395.
CHAP. XX.] FLANGES. 397
at the Conway Bridge, and subsequently at the Boyne Viaduct and
Canada Works, and constructed expressly for the purpose of pro-
ducing accurate repetitions of any required pattern.* Drilling tools
for boring several holes at once have been introduced with much
success, as at Charing-cross Bridge. Such tools will often repay
their first cost by the saving of manual labour in punching and
plating, besides insuring more accurate work, but for ordinary
girder- work the common punching machine is the cheapest tool.
436. Position of roadway — Compression flange stiffened
by the compression bracing: of the web. — The roadway is
generally attached to one or other of the flanges, but is sometimes
placed midway. The latter position is objectionable, since we then
lose the advantage of horizontal rigidity which the roadway imparts
to the flange to which it is attached. Moreover, less material is
generally required for forming the connexion between the cross-
girders and the main girders at the flanges than elsewhere. When
local circumstances do not determine the level of the road it may at
first sight appear desirable to connect it with the upper or compres-
sion flanges, so as to stiffen them against horizontal flexure, and
this is generally the best position with shallow girders, as it allows
the load to be placed more immediately over the longitudinal axis
of each girder and thus dispenses with heavy cross-girders, which
is often a very important saving, besides removing any tendency to
unequal strain which a one-sided load on the lower flanges might
produce. But with large and deep girders, independently of the
theoretic consideration that the lower the centre of gravity the
more stable the structure, some slight counterbalancing advantage
results from connecting the road with the lower flange, as the
expense of a parapet is saved and there is a greater appearance of
security when a train travels through, instead of over, a tubular
bridge. When the roadway is attached to the lower flanges and the
depth of girder is not sufficient to admit of cross-bracing between the
upper flanges, the horizontal stiffness of the road is communicated
to the upper flanges by the internal bracing of the compression
* For a description of this machine see Part 121 of the Civil Engineers' and Architects'
Journal.
398 FLANGES. [CHAP. xx.
braces when the web is a double-latticed web like Fig. 102, or
by vertical angle-iron frames when the web is plated, and in the
latter case triangular gussets are sometimes introduced to connect
these stiffening frames with the cross-girders. The cross-girders
are also occasionally prolonged like cantilevers and their extremities
connected by raking struts with the upper flanges, as is usual in
the parapets of wooden bridges.
437. Waste of material in flanges of uniform section —
Arched upper flange — Waste of material in continuous
girders crossing unequal spans. — It frequently happens that
the flanges have a greater sectional area near their ends than theory
requires, in order to preserve the symmetry of the flange through-
out its entire length and avoid injudicious thinning of the material.
This source of loss does not exist in the bowstring girder, as in it
the strain is nearly uniform throughout each flange. A compromise
may be effected between the bowstring girder and that with parallel
flanges by arching the upper flange, as in Fig. 108. In this form
of girder the strains near the ends of each flange are increased and
Fig. 108.
thus the extra material is utilized at the same time that the strains
in the end braces are diminished in consequence of the oblique
flange taking a share of their shearing strain. The mode of calcu-
lation is the same as for the bowstring girder. For a similar cause
to that just mentioned there is sometimes a waste of material in the
flanges of continuous girders of uniform depth crossing spans of
very unequal length. In this case the segments over the smaller
spans are much deeper in proportion to their length than those over
the larger spans, and hence a considerable waste of material may
arise from carrying the general design of the flanges symmetrically
throughout.
CHAP. XX.] FLANGES. 399
438. I n excess of strength in one flange does not increase
the strength of braced girders, though it may slightly in-
crease the strength of girders with continuous webs. — If
the flanges of a braced girder be well proportioned, both flanges
will fail simultaneously with the breaking load, and any increase of
strength in one flange only does not increase the strength of the
girder, but rather diminishes its useful strength by the excess of
dead weight. When, however, the web is continuous, an increase
of strength is produced by enlarging one of the flanges beyond its
due proportion for the following reason: — The unit-strain in the
re-enforced flange is less than before; consequently, there is less
alteration in its length from strain and the neutral surface ap-
proaches closer to it than if the flanges were duly proportioned ;
hence, a larger proportion of the web aids the weaker flange. The
useful strength of the girder, however, is not necessarily increased,
since the extra strength thus obtained may merely suffice to sup-
port the extra weight of the re-enforced flange (1OO).
439. Bearing surface on the abutments — Working load
on expansion rollers. — The area of bearing surface of a girder
on the abutments should be sufficient to prevent un.due crushing
of the wall-plates on top of the abutments. A common rule for
cast-iron girders is to make the length of bearing on the abutment
equal to the depth of the girder at the middle, say Jjth of the
span. It does not seem desirable to put a greater pressure on
cast-iron expansion rollers than 2 or 3 tons per linear inch, and
where the length of a girder does not exceed 150 feet, creosoted
timber wall-plates will generally be found preferable to rollers or
metallic sliding beds, both of which are apt to become rigid (414).
400 WEB. [CHAP. xxi.
CHAPTER XXI.
WEB.
430. Plate web — Calculation of strains. — In lattice girders
the flanges and the compression braces are intersected at short
intervals and thus divided into short pillars as far as their
tendency to flexure in the plane of the girder is concerned ; this
support is carried to its extreme limit in plate girders, the charac-
teristic feature of which is the continuity of the vertical connexion
(single or double, as the case may be) between the flanges. As the
thin webs of plate girders are ill adapted to resist buckling or
flexure under compression, it is usual to stiffen them by vertical T
or angle irons reaching from flange to flange, like the frames of a
ship. On a little consideration it will be obvious that these
stiffening frames make the web more rigid at short intervals in
vertical lines; thus this method of constructing plate girders
resembles the vertical and diagonal bracing investigated in the
sixth chapter, and the strains in the web may be approximately
calculated in the manner there described, though they are more
frequently obtained from the shearing-strain, as explained in 54.
If these frames are placed diagonally in place of vertically, the web
will resemble the class of bracing investigated in the fifth chapter
and should be treated accordingly.
431. IntlHK'iiH.v respecting1 direction of strains in con-
tinuous webs — Bracing1 generally more economical than
plating — Minimum thickness of plating in practice — Relative
corrosion of metals. — Besides these compressive strains acting
in directions more or less defined, there exist in the web of every
plate girder diagonal tensile strains which cross the stiffening frames
and whose directions are not so clearly defined and doubtless vary
to some extent with every position of the load. It thus appears
CHAP. XXI.] WEB. 401
that some portions of the web of plate girders are simultaneously
sustaining tension and compression and it might therefore seem at
first sight that a continuous web is more economical than one
formed of diagonal bracing, since in the former arrangement
the same piece of material performs a double duty, which in the
diagonal system requires two distinct braces (S79). Theoretically
this view is correct if it be conceded that one and the same portion
of material is capable of sustaining without injury both tensile and
compressive strains transmitted through it simultaneously at an
angle with each other and, in the absence of direct experiment,
there seems some reason for believing this to be the case within the
limits of strain which are considered safe in practice. For instance,
the shell and ends of a cylindrical boiler with internal flue are
subject to tensile strains, the former in two directions at right angles
to each other, the latter in various directions, while the flue is
subject to tension longitudinally and compression transversely.
Again, experiments on the strength of riveted joints have not
indicated any source of weakness in the plates other than that due
to the reduction of area by the rivet holes or the mode of punch-
ing, and if moderate compression does reduce tensile strength,
closely riveted joints, such as those of boilers, would be perceptibly
weakened by the compression caused by the contraction of the
rivets in cooling. Further, in experiments on the tensile strength
of iron bars, their ends are frequently grasped by powerful nippers
which compress them sufficiently to prevent the bar slipping
through, and it seldom breaks where thus compressed, rupture
generally taking place near the centre. It seems, therefore, reason-
able to infer that a moderate strain of either kind does not affect
the ultimate strength of iron to sustain a strain of the other kind
at right angles to the former. However this may be, practical
reasons prevent plate-iron webs from being so economical as those
formed of bracing, except in small or shallow girders, or girders
which sustain unusually heavy loads and in which therefore the
shearing strain is exceptional, or near the ends of girders of very
large span ; for unless the plating be reduced in thickness to
the extent which theory indicates as sufficient, but which is quite
2 D
402 WEB. [CHAP. xxi.
unsuitable for practical reasons, the bars of the braced web will
require so much less material than the continuous web of a plate
girder as to make the former really the more economical.
One quarter inch may be assumed to be the minimum thickness
that experience sanctions for the plating of permanent structures.
A thinner plate than this may with care last for years, but few
engineers would wish to risk the stability of any important
structure on the chance of such frequent attention to prevent
corrosion as so great a degree of tenuity would require. Indeed,
T5^ is quite thin enough for ordinary practice, and | or -| inch if a
girder is within the influence of air charged with salt, as when
railway bridges cross tidal estuaries. Mr. Mallet gives the relative
oxidation of certain metals in moist air as follows :* —
Cast-iron, - '42
Wrought-iron, - - *54
Steel, -56
He also states at p. 27 of his third report to the British -Associa-
tion in 1843 on the action of air and water upon iron, that in one
century the depth of corrosion of Low Moor Plates, as deduced
from his experiments, would be —
Inch.
In clear sea water, - 0*215
In foul sea water, - 0'404
In clear fresh water only, - 0'035
433. Plating; more economical than bracing: near the ends
of very long" girders — Continuous webs more economical in
shallow than in deep girders. — When the span is of great
extent the opens between the braces towards the ends become
smaller from the increased width of the bars and therefore nearly
equal to their overlap ; hence, there is a certain length of girder
beyond which it may be found more economical to form the ends
of the web of continuous plating and the intermediate portion of
diagonal bracing. The length of girder at whose extremities the
same amount of material is required for the web, whether formed
of bracing or of plates, depends, among other things, on the ratio
* On the Construction of Artillery, p. 138.
CHAP. XXI.] WEB. 403
of depth to span. In large railway girders, in which this ratio is
frequently about 1 to 15, the span beyond which it becomes more
economical to substitute plating near the ends in place of bracing
lies between 300 and 400 feet. Take, for instance, the single-line
railway bridge of 400 feet span, whose weight is calculated in
Example 4, in the chapter on the estimation of girder-work.
The length is 400 feet and the depth is 26'67 feet, or l-15th of
the length, and the maximum weight, including the permanent
load, which the bridge has to support is 1,490 tons distributed
uniformly. One-fourth of this, or 372*5 tons, is the shearing-strain
supported by the web at each end of each main girder. Now, if the
bracing be at an angle of 45°, which is the angle of economy, the
strain in the end diagonals will equal the shearing-strain multiplied
by T414, = 526*7 tons, requiring, at 4 tons per square inch, a
gross section of 131*7 square inches.* If the iron be half-inch
thick, the width of the end diagonal will equal 263 inches, as in
Fig. 109, in which for simplicity only one system of triangulation
is represented, since the overlap will be the same whether one or
several systems be adopted.
Fig. 109.
It is evident that the overlap of the bars considerably exceeds the
open spaces. This example, therefore, has attained the span beyond
which it would be more economical to employ plating for the end
portions of the web. If §-inch plating be considered sufficiently
thick the limit would of course happen sooner. If, however, the
depth were greater than 1-1 5th of the length, the limit would be
* In consequence of the rivet holes, 4 tons per square inch of gross section is for
tensile strain assumed equivalent to 5 tons per square inch of net section.
404 WEB. [CHAP. xxi.
greater than in our example. It is obvious also, from what has
just been stated, that the relative economy of plate webs is greater
in shallow than in deep girders ; for, if bracing were used, the
opens between the braces would be much smaller in the former
than in the latter case, and consequently, if these opens be filled
up by continuous plating, there will be less waste of material in
the shallow than in the deep girder.
433. Greater proportion of a continuous web available
for flange-strains in shallow than in deep girders. — That
plate girders derive from the continuity of the web some increase
of strength over that due to the sectional area of the flanges is
certain (1OO), but the amount of horizontal strain which a thin web
is capable of transmitting is, in large girders, generally too indefinite
to .admit of any considerable reduction in the area of the flanges on
this account and is, therefore, practically of slight importance, for
it seems unlikely that horizontal strains of compression can be
transmitted with much energy through the thin continuous web
of a deep girder, except in that portion which is close to the flange
and therefore stiffened against buckling by its connexion therewith.
In shallow plate girders, however, such as those used for the cross-
girders of bridges, deck-beams of ships, fire-proof floors, &c., the
web generally forms a large portion of the whole section, possesses
considerable strength by itself, and is therefore available for hori-
zontal as well as vertical strains. These considerations show that
the flanges of a shallow plate girder derives a greater percentage
of aid from the web than those of a deep girder.
434. Deflection of plate girders substantially the same
as that of lattice girders. — From these considerations it would
also appear that the deflection of plate girders is little, if at all, less
than that of lattice girders, the length, depth and flange-area being
the same in both ; for if their flanges be subject to the same unit-
strains, their deflections will be alike (333). Even assuming that
the web does relieve the flanges of horizontal strain to the full
extent which theory indicates, the deflection will not be very
materially diminished thereby, for it appears from eq. 151 that
a continuous web is for horizontal strain equivalent to only Jth
CHAP. XXI.] WEB. 405
of its area placed in each flange. Plate girders, it is true, are
generally thought to be stiffer than those with braced webs, and
closely latticed girders than those with only one or two systems
of triangulation, but I am not aware of accurate comparative
experiments on this subject. It is quite possible that when the
compression flange has but few points supported by intersecting
braces it may assume under strain a slightly undulating line, and
therefore be a little shorter than a similar flange held straight at
short intervals by close latticing or a plate web; this would of
course increase the deflection.
435. Webs of cast-iron girders often add materially to
their strength. — The webs of cast-iron girders are usually made
much stronger than is required for the mere transmission of the
shearing-strain. Hence, they rarely require stiffening ribs, and the
web should add to the strength of such girders, calculated merely
from the leverage of either flange round the other as a fulcrum, by
an amount nearly equal to the breaking weight of the web taken
separately. Stiffening ribs are generally to be avoided in cast-iron
girders, as they have been found to cause rupture in some instances
from unequal Contraction of the metal.
436. Minute theoretic accuracy undesirable. — In construct-
ing wrought- iron girders of small span, say under 30 or 40 feet,
it is generally more economical to make the lattice bars of one, or
at most of two sizes throughout, even though they might be safely
reduced in section as they approach the centre. This arises from
the expense and trouble of having different templates and a stock
of bars of various sizes. It is, therefore, cheaper to have a slight
excess of material than go to the nicety of sizes which would be
theoretically strong enough. For a similar reason 2J inches may
be assumed to be the minimum useful width for a lattice bar of
ordinary railway girders. When of less width it is generally
necessary to swell out the rivet holes in the forge, so as to avoid
reducing the effective section of the bar and, independently of the
bad effect sometimes produced by heating the iron, this process is of
course more expensive than cold punching. One result of all this is
that the central bracing is generally stronger than theory requires.
406 WEB. [CHAP. xxi.
437. JInltiple and single systems of triangnlatioii com-
pared— Simplicity of design desirable — Ordinary sizes of
iron.— This leads to another consideration, viz., the number of
systems employed in bracing. It has been already stated in 153
that the practical advantage of a multiple over a single system of
triangulation consists in the more frequent support given to the
compression bars by those in tension, and by both to the flanges,
thus subdividing the parts which are subject to compression into a
number of short pillars and restraining them from deflection, chiefly
in the plane of the girder. It may also be urged in favour of close
latticing, that if an accident, such as an engine running off the line,
occurs on a bridge with the braces few and far apart, that in such a
case the safety of the whole structure is menaced by the fracture of
a single bar, whereas a closely latticed or plate girder is not only
freer from this danger, but affords greater security in case of one
bar being originally defective, while to the public eye it has the
semblance of greater safety, a consideration not altogether to be
despised. The number of systems adopted will also depend on the
distance between the cross-girders which generally occur at an
apex, and on the practical consideration of what sized material is
the most economical ; and this again will depend on two things, the
first cost of iron of small and large scantlings and the subsequent
cost of workmanship, which latter item varies much according to
the simplicity or complexity of the design. No definite rule can
be laid down for all cases, but one consideration of importance
should not be overlooked in seeking after apparent economy at the
outset. The larger the scantlings and the more simple the method
of construction, the smaller is the surface exposed to atmospheric
influences and the more easily detected is any corrosion or decay.
The chief advantage of masonry is its permanent character. No
rust or decay in it requires constant attention or painting and, if
well executed at the outset, masonry truly deserves the title of
permanent.
It will be useful to recollect that bars or strips are not rolled
wider than 9 inches; when a greater width than this is required
narrow plates with shorn edges must be used. Plates exceeding 4
CHAP. XXI.] WEB. 407
feet in width, or 15 feet in length, or containing more than 32
square feet, or weighing more than 4 cwt., are generally charged
extra ; also T or angle iron, the sum of whose sides exceeds 9 or 1 0
inches. Plates can be rolled up to 7 feet wide, or 30 feet long, or
60 square feet in area, but such sizes are very costly ; they increase
in thickness by sixteenths of an inch, and are generally called sheet
iron when less than -f^ inch thick. Ordinary angle iron can be got
in lengths of from 30 to 36 feet, and up to 6 X 6 X J inches.
438. Testing small girders by a central weight equal to
half the uniform load is inaccurate. — Small girders are fre-
quently tested by a central weight equal to half the uniform or
passing load which they are expected to carry with safety. Though
convenient, this is not altogether a fair trial of the web. Let W =
the proof load in the centre, and 2W =: the uniform load. The
web of a girder designed to support a central load, W, should be of
uniform strength, for it sustains throughout a shearing- strain equal
W
to -^-(34). The web of a girder designed for a uniform load,
2W, should increase from the centre where the shearing-strain is nil,
towards the ends where the strain = W, in proportion to the dis-
tance from the centre (46) ; and the web of a girder designed to
support a passing load of the same density as the uniform load
should increase from the centre towards the ends, where the shear-
ing-strain — W, in the ratio of the square of the distance from the
further end (5O). Consequently, the strain in the centre of the
W
web from a passing load = -j-. It is obvious, therefore, that the
web near the centre is subject to a much greater strain from a
central load than from a uniform or passing load of twice its weight,
whereas at the ends the reverse of this takes place. The impor-
tance of these remarks may be practically lessened by the con-
siderations referred to in 436.
439. Connexion between web and flanges — Uniform strain
in flanges — Trough and HH-shaped flanges — Rivets pre-
ferable to pins — Limit of length of single-webbed girders. —
In wrought-iron girders the shearing area of the rivets con-
necting each brace with the flanges should equal the net section
408 WEB. [CHAP. xxi.
of the brace ; otherwise there is a risk of its separating from the
flanges at a much lower strain than would destroy the brace. If
the web be a continuous plate, the shearing area of the connecting
rivets should equal its theoretic horizontal section, i.e., the horizontal
net section of a plate whose thickness is that which theory demands;
in practice, however, the plate area is generally considerably in
excess of what theory requires and hence the rivet area seldom
equals its horizontal net section. The trough-shaped section, such
as that represented in Plate IV., is a favourite form for the flanges
of tubular braced girders as it affords great facilities for attaching
the bracing to the flanges. Objections have been raised to the
trough with deep vertical plates on the ground that the unit- strain
is not constant throughout its whole area, the unconnected edges
of the vertical plates being subject to a severer unit-strain than the
horizontal plates in consequence of each brace giving off its hori-
zontal component of strain at a point which generally lies nearer
the free edge of the vertical plate than the centre of gravity of
the whole section. Let us confine our attention to the upper or
compression flange, as similar reasoning applies to that in tension.
This tendency to excessive local strain is sometimes supposed to
show itself by a slight undulation or buckling of the free edge of
the vertical plate endeavouring to escape from the line of thrust.
This buckling, however, is not necessarily a sign of excessive local
compression, but rather of defective stiffness in the lower part of
the plate, for if it were stiffened laterally so that it could not
escape from the line of thrust, and if the unit-strain along this
edge were greater than that in the horizontal plates, the result
would be that the whole flange would camber from the shortening
of its lower edge. This, however, does not take place, and hence
it is reasonable to suppose that the strain is not very unequally
distributed throughout the whole section. Undulation certainly is
a defect and proves that the plate is not standing up to its work,
and therefore not subject to excessive compressive strain ; it rather
indicates that a small portion of the vertical plate at each apex on
the side remote from the centre may be in tension, pulling, instead
of thrusting, the flange towards the centre. Vertical plates ought
CHAP. XXI.] WEB. 409
therefore to be thick enough to resist buckling, say yjtii of their
depth (335), or else be stiffened by an angle iron along their free
edges. The weight of the trough itself, acting as a series of short
girders between the apices, tends to produce local tension in the
lower edges of the vertical plates, and so far counteracts excessive
coinpressive strain, and the whole flange being held at short
intervals by the bracing resembles a long thin pillar inside a tube ;
the pillar may undulate slightly and press here and there against
the sides' of the tube, but the compressive strain may for all
practical purposes be considered as being distributed uniformly
throughout the whole section of the pillar. The H section of
flange also has its advocates, who maintain that it is free from the
objections alleged to lie against the trough section. The practical
convenience of the latter, however, will probably enable it to hold
its ground against its rival. The student who wishes to learn the
views of eminent engineers on this subject is referred to the
discussions on " The Charing Cross Bridge" and " Uniform Stress
in Girder Work," in the 22nd and 24th Vols. of the Proceedings
of the Institution of Civil Engineers. The main bracing is some-
times connected to the vertical plates by pins, like those of sus-
pension bridges. Judging, however, from the experience gained at
the Crumlin viaduct — where riveting was substituted for pins, after
some years' wear and vibration had loosened the latter* — it seems
generally desirable jto make rigid connexions, and for this purpose
riveting is at once the most convenient and effective method.
Moreover, pins evidently do not form so firm a termination for a
strut as riveting, a matter of great importance in long pillars (311).
The braces should intersect somewhere in the vertical plate. In
very faulty designs they are sometimes arranged so that they do
not intersect each other in the flange, but would, if produced,
meet considerably outside it, in which case the flange is subject to
an injurious cross-strain and is liable to become broken-backed from
the compression braces thrusting it upwards while the tension
braces pull it down, or vice versa. In some instances this has
* The Engineer, November, 1866, p. 384.
410 WEB. [CHAP. xxi.
produced disastrous results. When the vertical plate is deep
enough to give a choice of position, the apex may either be in the
middle or rather closer to the upper edge, the latter position being
perhaps the better of the two.
The length of single- webbed girders rarely exceeds 150 feet.
Indeed, a double web seems desirable when the span exceeds 40
feet, as there can be no doubt that it contributes greatly to ,the
stiffness of the flange plates to be bound by angle iron along both
edges when their width exceeds 18 or 20 inches, and, regarding
the whole flange as a long unsupported pillar, it is obvious that its
resistance to lateral flexure is far greater when the angle irons are
along the edges than when they are central.
CHAP. XXII.] CROSS-BRACING 411
CHAPTER XXII.
CROSS-BRACING.
44O. Weather-bracing? — Maximum force of wind — Pres-
sure of wind may be considered as uniformly distributed
for calculation. — Cross-bracing generally fulfils two functions;
it acts as a horizontal web, holding the compression flanges at
short intervals in the line of thrust and thus preserving them
from lateral flexure to which all long pillars are liable; it also
braces the whole structure in a horizontal plane, stiffening it
against vibration and strengthening it to resist the side pressure of
the wind just as the vertical web enables the main girders to sus-
tain the downward pressure of the load. When the roadway is
attached to the lower flange and the depth of the main-girders is
not sufficient to admit of cross-bracing between the upper flanges,
the latter must be made sufficiently wide to resist any tendency
they may have to deflect sideways under longitudinal compression
and their lateral stiffness may be calculated by the laws of pillars,
though they are much aided by the internal bracing of latticed
webs or the angle iron stiffening frames of plate webs, which
convey a large share of rigidity from the roadway to the upper
flanges. Under these circumstances the roadway and cross-bracing
between the lower flanges have to resist the greater portion of
the lateral pressure of the wind whose maximum force in this
country may, for the purpose of calculation, be assumed equivalent
to a uniform pressure of 25 Ibs. per square foot of side surface
exposed to its influence. The pressure of the wind is not always,
as might be supposed, uniformly exerted along the whole length of
a girder. With reference to the effect of violent gales on the
Britannia Bridge, Mr. Clark remarks: — "The blow struck by the
gale was not simultaneous throughout the length of the tube, but
impinged locally and at unequal intervals on all parts of the length
which presented a broadside to the gale."* A little further on he
remarks : — " The tube, however, on no occasion attained any serious
* The Tubular Bridges, p. 455.
412
CROSS-BRACING.
[CHAP. xxii.
oscillation, but appeared, to some extent, permanently sustained in
a state of lateral deflection, without time to oscillate in the opposite
direction." Hence, the effect of wind may be assumed to be not very
different from that of a uniformly distributed load; as a precau-
tionary measure, however, it is desirable to make the central weather-
bracing somewhat stronger than would be requisite if the pressure
wrere really uniform.
441. Rouse's table of the velocity and force of wind —
Beaufort scale. — The following table of the velocity and corres-
ponding pressure of the wind by Mr. Rouse is given by Smeaton
in the Philosophical Transactions for the year 1759 :—
TABLE I.— KOUSE'S TABLE OF THE VELOCITY AND FORCE OP WIND.
Velocity of the Wind.
Perpendicular
force on a
square foot,
in tbs.
avoirdupois.
Common appellations of the Wind.
Miles
per
hour.
Feet per
second.
1
1-47
•005
Hardly perceptible.
2
3
2-93
4-40
•020
•044
> Just perceptible.
4
5
5-87
7-33
•079
•123
> Gentle pleasant gale.
10
15
14-67
22-00
•492
1-107
> Pleasant brisk gale.
20
25
29-34
36-67
1-968
3-075
> Very brisk.
30
35
44-01
51-34
4-429
6-027
> High winds.
40
45
58-68
66-01
7-873
9-963
| Very high.
50
73-35
12-300
A storm or tempest.
60
88-02
17715
A great storm.
80
117-36
31-490
A hurricane.
100
14670
49-200
A hurricane that tears up trees, and carries build-
ings before it, &c.
CHAP. XXII.] CROSS-BRACING. 413
The following table contains the Beaufort scale which is used
in the Navy to represent the force of the wind, but it conveys no
information respecting its actual pressure or velocity and is there-
fore of little use for scientific purposes.
TABLE II.— BEAUFORT SCALE.
0. Calm.
1. Light air, steerage way.
2. Light breeze, ship in full sail will go 1 to 2 knots.
3. Gentle breeze, dc. 3 to 4 do.
4. Moderate breeze, do. 5 to 6 do.
5. Fresh breeze, ship will carry royals.
6. Strong breeze, single reefed topsails and topgallant sails.
7. Moderate gale, double reefed topsails, jib, &c.
8. Fresh gale, triple reefed topsails, &c.
9. Strong gale, close reefed topsails and courses.
10. Whole gale, will scarcely bear close reefed main topsail and
reefed foresail.
11. Storm, storm staysails only.
12. Hurricane, which no canvas could withstand.
443. Cross-bracing; must be cownterbraced — Best form of
cross-bracing: — Initial strain advantageous. — As the wind
may blow on either side of a bridge it is necessary to counterbrace
the cross-bracing throughout; hence, the description of bracing
described in Chap. VI., with transverse struts and diagonal ties, is
well suited for cross-bracing and, in order to make it stiff and come
into action before much lateral movement takes place, it is de-
sirable to put a small initial strain on the diagonals. This will
tend also to stiffen the whole structure against lateral vibration
from loads in motion. The initial strain may be produced by
coupling screws, cotters, or similar appliances. When the design
does not admit of these the transverse struts may be first riveted
in place, and then the diagonals may be riveted while they are
temporarily expanded by heat ; when cold the whole will be in a
state of slight strain. The same effect may be produced in small
tubes by laying them on their side so that the cross-bracing may
be in a vertical plane ; a few weights will then stretch one system
414 CROSS-BRACING. [CHAP. XXII.
of diagonals, and when thus strained the second series may be
riveted in place; after the removal of the weights the required
degree of initial strain will be produced if the operation has been
carefully performed. The sagging of the horizontal tension bars of
cross-bracing from their own weight will also aid in producing the
required amount of stiffness, provided the bars be supported in a
horizontal position while riveting up.
The absence of the initial strain alluded to was strongly marked
in the Britannia Bridge, for Mr. Clark remarks : — " The effect of
pressure against the side of the tube is very striking; a single
person, by pushing against the tube, can bend them to an extent
which is quite visible to the eye ; and ten men, by acting in unison,
and keeping time with the vibrations, can easily produce an
oscillation of 1^ inch, the tube making 67 double vibrations per
minute."* A severe storm on the 14th of January, 1850, pro-
duced oscillations not exceeding one inch. This, however, was
before the two tubes were connected together, side by side.
443. Strains produced in the flanges by cross-bracing: —
End pillars of girders with parallel flanges and bow of
bowstring girders are subject to transverse strain. — When
there are both upper and lower cross-bracings, each has to sustain
one-half the pressure of the wind ; consequently, in every gale the
compression flange on the weather, and the tension flange on the
lee side have their normal strains somewhat increased, while those
in the other flanges are diminished to the same extent. This
increase and diminution of strain are, however, generally insigni-
ficant compared to the strains produced by the load and are, of
course, less in open-work girders than in those with solid sides
which present a larger unbroken surface to the action of the wind.
When cross-bracing occurs between the upper flanges, the
pressure of the wind against the upper half of the girder is
transmitted to the abutments or piers through the end pillars
which form the terminations of the web immediately over the
points of support, at least so much of it as is not conveyed by the
web stiffeners to the lower flanges and thence to the abutments.
* The Tubular Brides, p. 717.
CHAP. XXII.] CROSS-BRACING. 415
These pillars are, therefore, semi-girders as well as pillars, for they
are subject not only to vertical compression from the shearing-
strains in the main bracing, but to lateral pressures at top tending
to overthrow them, which are nearly equal in amount to one-half
the total pressure of the wind. Thus, if there be two main girders
and four end pillars, each of the latter sustains a transverse pressure
at top nearly equal to one-eighth of the pressure of the wind. It
is, therefore, desirable to fix the lower ends of these pillars very
securely by means of strong iron gussets attached to the masonry,
or, if these be inadmissible from the longitudinal expansion of the
bridge, to a cross road-girder,, which may be made stronger and
stiffer than usual for this purpose, so as to resist the racking action
of the wind.
The bowstring girder, with roadway attached to the string, does
not admit of cross-bracing between the bows throughout their
entire length, but only near the centre where there is sufficient
headway for carriages beneath. The ends of the bows are, con-
sequently, subject to transverse strains similar to those just described
in the case of the end pillars of girders with horizontal flanges.
416 CROSS-GIRDERS AND PLATFORM. [CHAP. XXIII.
CHAPTER XXIII.
CROSS-GIRDERS AND PLATFORM.
444. Maximum weight on cross-girders — Distance be-
tween cross-girders. — The cross-girders of railway bridges sup-
port the platform, ballast, sleepers and rails ; and when the interval
between them does not exceed that between two adjacent axles of
a locomotive, say 6 or 7 feet, the greatest load which each cross-
girder has to support is determined by the weight resting on one
pair of driving-wheels, which rarely, if ever, exceeds 16 tons, or
8 tons per wheel. Consequently, if the effect of the rails, sleepers
and platform in spreading the load over several girders be neglected,
each cross-girder, however close they may be together, ought to be
capable of sustaining 16 tons if the bridge be made for a single
line, and twice this if made for a double line, in addition to the
dead weight of platform, ballast and permanent way, and as a train
of ordinary locomotives and tenders, that is, the load of maximum
density, does not exceed 1J tons per running foot, it would
obviously be the most economical arrangement to place the cross-
girders, at all events, not closer together than the above stated
distance of 6 or 7 feet.* It may, perhaps, be supposed that cross-
girders placed at shorter distances need not be so strong in con-
sequence of the rails, sleepers and platform distributing the load
over several cross-girders, and this, no doubt, is to a certain extent
correct, and numerous bridges have been constructed on this
principle. Government Inspection is now, however, more critical
* The cross-girders of the Boyne Viaduct are 7 feet 5 inches apart, equal to the
diagonal of the square formed by the lattice bars of the main -gird era. The interval
between those of the Britannia and Conway Tubular Bridges is 6 feet.
CHAP. XXIII.] CROSS-GIRDERS AND PLATFORM. 417
than formerly, and each cross-girder should be strong enough to
sustain the load on the driving wheels of the heaviest engine
which can come on the line, inasmuch as the sleepers may decay,
joints may occur in the rails close to a cross-girder, or the platform
may require renewal and perhaps be altogether removed for this
purpose.
445. Rail-girders or keelsons — Economical distance be-
tween the cross-girders — Weight of single and double lines —
Weight of snow. — When the cross-girders are farther than 3
feet apart (the distance between centres of sleepers) the rails may
be supported by shallow longitudinal girders resting on the cross-
girders or framed in between them, and in certain cases, especially
when the levels permit the cross-girders to be of great depth,
these rail-girders may be economically made of considerable length,
with the cross-girders placed at long intervals apart, in some cases
20 feet asunder ; but care must be taken not to strain the lattice
bars of the main girders beyond their safe limit by bringing too
great a local pressure on those which intersect at the ends of the
cross-girders. The rail-girders may be conveniently made of
plating or lattice work, similar in general design to the main girders
of small bridges and framed in between the cross-girders. In
some cases these rail-girders run above the cross-girders in un-
broken lines from end to end of the bridge like the keelsons of
a ship. This arrangement requires greater depth from soffit of
bridge to rail than the former, and cannot therefore be so fre-
quently adopted. Mr. Win. Anderson has shown the great
economy of placing the cross-girders 12 feet apart or upwards,
especially with double line bridges, by means of the following data
and estimate based thereon.*
Maximum weight of engine, - - 34 tons,
Maximum load on driving wheels, - 16 tons,
Wheel base, - - 12 feet,
Depth of cross-girders, - - 75 th of span.
* Trans. Inst. of C. E. of Ireland, Vol. viii., 1866.
2 E
418
CROSS-GIRDERS AND PLATFORM. [CHAP. XXTII.
SINGLE LINE.
Span.
Total
load on
girders.
Net area
of bottom
flange.
Weight
of
girders.
Weight per
ft. run of
bridge.
feet.
tons.
sq. in.
tt>s.
ftS.
Cross-girders, 3 feet apart,
14
17-26
6-3
1,206
402
Cross -girders, 12 feet apart, -
14
29-35
10-93
1,700
)
> 268-2
Longitudinal rail-girders,
12
19-54
10-8
1,518
1
DOUBLE LINE.
Cross-girders, 3 feet apart,
25£
35-00
11-4
3,654
1,218
Cross girders, 12 feet apart,
25£
58-64
19-2
4,704
)
645
Longitudinal rail-girders,
12
38-64
21-6
3,026
I
The permanent load of the roadway per running foot, including
cross-girders 3 feet apart, sheeting, ballast, sleepers and rails for a
single-line bridge, 14 feet wide between main girders (Irish gauge
5' 3"), he estimates as follows: —
SINGLE LINE BRIDGE.
Weight in tons
per running foot of bridge.
Cross-girders, 3 feet apart, *18
Sheeting of 4-inch planks and bolts for same, "10
Rails, chairs, spikes, and sleepers (permanent way), '06
Ballast (from 3 to 4 inches deep), - '20
0-54 tons,
which is equivalent to a load of 86'4lbs. per square foot of
platform. This O54 ton is the permanent load of roadway for a
single line per running foot, and is exclusive of main girders and
cross-bracing, which vary with the span. The similar permanent
load of roadway for a double line, 25J feet between main girders,
is about 1*2 ton per running foot, or a little more than double that
for a single line, which, however, may be reduced to about 1 ton
by placing the cross-girders from 10 to 12 feet apart with rail-
girders between.
In cold countries the weight of snow should not be left out of
consideration. This has been estimated in America as high as
30 Ibs. per square foot over the whole surface of the bridge.
CHAP. XXIII.] CROSS-GIRDERS AND PLATFORM.
419
In bridges of moderate span it is generally more economical to
place the main-girders immediately beneath the rails; they then
act as rail-girders and thus dispense with cross-girders. When,
however, there is but little head-room beneath the rails, a modi-
fication of the trough girder may be adopted, such as that designed
by Mr. Anderson for one of the bridges on the Dublin, Wicklow
and Wexford Railway, and represented below.
Fig. 110.
Half Longitudinal Section and half Elevation of Bridge.
Fig. 111.
Cross Section of Bridge.
Each rail is carried between a pair of plate girders connected
by short cast-iron saddles on which the sleeper and rail are laid
and to which they can be securely bolted. The girders are thus
accessible in every part for cleansing and painting without dis-
turbing the permanent way, and at the same time no water can
lodge in any part of the structure.*
446. Regulations of Board of Trade.— The following are
the regulations of the Board of Trade respecting the cross-girders
and platforms of railway bridges.
1. The heaviest engines in use on railways afford a measure
of the greatest moving loads to which a bridge can be subjected.
* Tram. Inst of C. E. of Ireland, Vol. viii., p. 45.
420 CROSS-GIRDERS AND PLATFORM. [CHAP. XXIII.
This rule applies equally to the main and the transverse girders.
The latter should be so proportioned as to carry the heaviest
weights on the driving wheels of locomotive engines.
2. The upper surfaces of the wooden platforms of bridges and
viaducts should be protected from fire.
3. No standing work should be nearer to the side of the widest
carriage in use on the line than 2 feet 4 inches at any point
between the level of 2 feet 6 inches above the rails and the level
of the upper parts of the highest carriage doors. This applies to
all arches, abutments, piers, supports, girders, tunnels, bridges,
roofs, walls, posts, tanks, signals, fences and other works, and to
all projections at the side of a railway constructed to any gauge.
4. The intervals between adjacent lines of rails, or between lines
of rails and sidings, should not be less than 6 feet.
447. Roadways of public bridges — Buckled-plates. —
The roadways of iron public bridges are generally formed in one
of the four following ways.
1°. Brick arches spring between the lower flanges of the longi-
tudinal or cross-girders as the case may be, and their haunches
are levelled up with concrete, over which the pavement is laid.
Sometimes a thin layer of tar asphalt is spread over the concrete
to prevent surface water from percolating through the brickwork.
The span of the arches, that is, the distance between the girders,
may vary from 4 to 8 feet, and iron cross-ties are required at
moderate intervals to bind the girders together and prevent them
from spreading sideways under the thrust of the arches. The
weight of a square foot of this roadway, exclusive of girders and
cross-ties, may be estimated as follows :—
Ibs. fts.
Brickwork, 4£ inches deep, - - 36 if 9 inches deep, 72
Concrete, averaging 4 inches deep, - 47 if 6 do. 70' 5
Asphalt, \ inch deep, 7 7
Pavement and sand, 9 inches deep,
or 12 inches of broken stone, - 110 - 110
200 259-5
CHAP. XXIII.] CROSS-GIRDERS AND PLATFORM. 421
2°. Arched wrought-iron flooring plates, f to \ inch thick, are
riveted to the upper flanges of the longitudinal girders and their
haunches are levelled up with asphalt or concrete, over which the
pavement or broken stone is laid as before. These arched plates
also require cross- ties to prevent the outside girders from spread-
ing, but the plates themselves may often be made to take an
important share in the structure by strengthening the upper, or
compression, flanges of the girders, and thus economizing material.
The weight per square foot of this roadway, excluding cross-ties,
may be estimated as follows : —
tbs. Ibs.
Arched plates, 20 to 26
Asphalt, averaging 3 inches deep, - - 42 if 4 inches, 56
Pavement or broken stone as before, - 110 - - - 110
172 192
3°. Flat cast-iron plates, f to 1 inch thick with stiffening ribs
on the upper surface, are bolted to the upper flanges of the
longitudinal girders and then levelled up with asphalt to the top
of the ribs, 3 or 4 inches deep, over which the pavement or broken
stone is laid as before. The weight per square foot of this road-
way is from 20 to 30 fts. more than in the last case, but no cross-
ties are required.
4°. Wrought-iron buckled-plates, £ to T5¥th inch thick, are bolted
or riveted to the upper flanges of the longitudinal girders and
levelled up with concrete or asphalt, over which the broken stone
or pavement is laid as before. Angle or tee iron covers are riveted
to the cross joints of the plates and support them at frequent
intervals like short cross-girders. The weight per square foot of
this roadway, including the angle or tee iron covers, is closely the
same as in case 2.
The following data respecting Mallet's buckled-plates are
derived from the trade circular.
The resistance of square buckled-plates is directly as the thick-
ness and inversely as the clear bearing. A buckled-plate, bolted
or riveted down all round, gives double the resistance of the same
422 CROSS-GIRDERS AND PLATFORM. [CHAP. XXIII.
plate merely supported all round, and if two opposite sides be
wholly unsupported, its resistance is reduced in the ratio of 8 to 5.
Within the limit of " safe load" the resistance is nearly the same,
whether it be upon the crown or uniformly diffused. The stiffness
at any point of the plate, as against unequal loading, is as the
square of the thickness nearly, and inversely as the curvature.
The curvature (unless for special object) should never exceed that
which will just prevent the "crippling load" bringing the plate
down flat, by compression of the material ; less than 2 inches
versed-sine of curvature has been found sufficient for £ inch
buckled-plates 4 feet square. A 3 foot square buckled-plate, of
ordinary Staffordshire iron £ in. thick, 2 in. width of fillet, 1J in.
curvature, supported only all round, requires upwards of nine
tons diffused over about half the superficies at the crown to cripple
it down, and double this, or eighteen tons to cripple it, if firmly
bolted or riveted down to rigid framing all round. A similar
plate of soft puddled steel has been found to bear nearly double
the preceding, or thirty-five tons to the square yard. Mr. Thomas
Page, C.E., has proved the buckled-plates of the floor of West-
minster new bridge — each averaging 7 feet by 3 feet, £ inch thick,
and 3J inch curvature — by lowering upon the crown of each a
block of granite of seventeen tons weight, which they sustained
without injury. In structures exposed to impulsive loads, such as
railway or other bridge flooring, one-sixth of these "crippling
loads" should not be exceeded for the safe load, nor one-fourth
for quiescent loading. The size of buckled-plates formed of one
single rolled plate is only limited by the breadth, to which sheet
or plate iron can be rolled, at market prices ; and the sizes that
have been found most advantageous for the majority of purposes
are plates of 3 feet and of 4 feet square, or of those widths by the
full length of the sheet. Square plates of either of the two
ordinary market sizes are always to be preferred, on the ground
of economy in prime cost, and in application, and facility in being
obtained promptly from the makers. Square plates produce a
stronger floor, with a given weight of iron, than any rectangular
plate ; the resistance of the latter being that nearly of a square
CHAP. XXIII.] CROSS-GIRDERS AND PLATFORM.
423
plate, whose side is equal to the longer dimension. If rectangular
plates be used the longer edge should not be much more in length
than twice the shorter. Economy is always consulted by sup-
porting each plate all round — one pair of opposite fillets resting on
the girders or joists of the structure, and the joints of the cross
fillets supported by an angle iron above, thus forming a lap plate.
TABLE OF STRENGTH, WEIGHT, AND COST OP BUCKLED-PLATES.
No.
Thickness of Plate.
Weight
per square
yard of
Buckled-
Plate,
excluding
the angle
iron at the
cross-
joints.
Weight of
an equal
surface
(1 square
yard) of
Corrugated
Plate of
correspond-
ing
thickness.
Safe passive
load,
uniformly
diffused per
square yard,
for three
feet square
Buckled-
Plates.
Safe im-
pulsive
load,
uniformly
diffused
per square
yard, for
three feet
square
Buckled-
Plates.
Cost per
superficial
yard of
Buckled-
Plate,
at £13
per ton.
Nearest
number
of square
yards in
one ton of
Buckled-
Plates.
B.W.G. inch.
No. 18 = -048
No. 16 = -066
No. 12 = -107
1-8
3-16
1-4
5-16
3-8
17-3
23-6
387
45-0
67-5
90-0
112-5
135-0
Ibs.
207
28-3
46-4
54-0
81-0
108-0
135-0
162-0
tons.
0-27
0-43
0-64
1-0
2-5
4-5
6-2
9-0
tons.
0-20
0-32
0-48
075
1-7
3-0
4-7
6-8
8. d.
2 2
2 10
4 7
5 3
7 11
10 6
13 2
15 8
sq. yards.
129
95
57
49
33
24
20
16
NOTE. — The safe loads in columns 5 and 6 may be taken at double for buckled-
plates of puddled steel.
Nos. 1, 2, and 3 — Applicable to roofing, iron house building, and fireproofing,
flooring, &c.
Nos. 4 and 5 — For the lighter class of bridge and other floors.
Nos. 6 and 7 — For the heavier floors of railway and other bridges, and viaducts :
No. 6 is that adopted for the new bridge at Westminster, London : No. 7 for
bridges in India.
No. 8 — Has not hitherto been found necessary in any structures, however
heavy.
The working loads on public bridges are given in Chapter
XXVIII.
424 COUNTERBRACING. [CHAP. XXIV.
CHAPTER XXIV.
COUNTERBRACING.
448. Permanent or dead load — Passing: or live load. —
The strains in the web of a braced girder are constant both in
amount and kind so long as the load remains stationary. If,
however, the load changes its position the strain will alter in
amount, and perhaps in kind also, and it is to meet this latter
change in the character of the strain that counterbracing is re-
quired. Now, a certain portion of the load which every girder
sustains is fixed and consists of what I have elsewhere called the
" permanent load," or " dead load," including in this term the
weight of the whole superstructure, viz., the main girders, cross-
girders, cross-bracing, platform, rails, sleepers and ballast. This
permanent load produces definite strains in the bracing which
remain constant, both in amount and kind, until a further load
comes upon the bridge. Let us consider the effect of a moving or
"live" load of uniform density, say a train of carriages, traversing
a girder with horizontal flanges, and we may chiefly confine our
attention to the strains developed in the bracing at either end of
the train, as it has been shown in 51 and 1?O, that the maximum
strains in the bracing from train-loads occur at these points. As
the advancing train approaches the centre of the girder the normal
strains in the bracing between the centre and the front of the train
are diminished, or even reversed, by the passing load. In the
latter case each brace attains its maximum reverse strain as the
front of the train passes it and counterbracing must be provided
accordingly. During the same period, i.e., while the train advances
towards the centre, the permanent strains in the second half-girder
are receiving gradual increments of their own kind, but each brace
in this half does not attain its state of maximum strain until the
CHAP. XXIV.] COUNTERBRACING. 425
train has crossed the centre and is so far advanced that its front is
passing that particular brace, after which the strain again diminishes
till the other end of the train is passing, when the strain is either
at its minimum, or, if altered, attains its maximum of the reverse
kind to that produced by the permanent load, in which case there-
fore the brace requires counterbracing.
449. Passing loads require the centre of the web to be
counterbraced — fcarge girders require less counterbracing
in proportion to their size than small ones. — The permanent
load is usually disposed symmetrically on either side of the centre ;
consequently, the normal strains in the bracing near the centre
are less in amount than in other parts, and it is in the central
braces alone that strains of a reverse character are produced
by a moving load, requiring counterbracing for some distance
on either side of the centre. It is evident that the heavier
the permanent load is, the less will be the amount of counter-
bracing required for a given passing load. It has been already
shown in 5O that the shearing-strain (to which the strain in the
w'n'
bracing is proportional) at the end of a passing train = — ^~- where
£v
w' = the passing load per linear unit,
I = the length of the girder,
n = the length covered by the advancing load.
But the shearing-strain at the same point from the permanent load
where w — the permanent load per linear unit, and n and I are as
before, n being supposed less than ^. Now, if n be proportional to
I in girders of different lengths, the shearing- strain from the
passing load will vary as w'l, and that from the permanent load
as ivl', and, since w increases in large girders as some high power
of the length, while wf may be considered constant for girders of
all sizes, the shearing-strain due to the permanent load will bear
a considerably greater ratio to that from the passing load in long
than in short girders. Consequently, the proportion which the
426 COUNTERBRACING. [CHAP. XXIV.
counterbracing bears to the whole amount of material diminishes
rapidly with the span of the girder. The counterbracing termi-
nates where the two shearing-strains are equal, and the point where
this occurs may be determined by equating them to each other and
solving the resulting equation for n as follows : —
Arranging according to powers of n,
w'n2 + 2wln — id2 = 0
solving for n,
— in "i v in" -4- ww
-') («0)
W
If, for example, w = w/,
w = J (— 1 + \/2) = -414J
45O. Counterbracing of vertical and diagonal bracing —
Large bowstring girders require little connterbracing. —
Girders with vertical and diagonal bracing, such as that inves-
tigated in Chapter VI., may be counterbraced either by making
the bracing near the centre capable of acting indifferently as struts
and ties, or by adding a second system of diagonals crossing the
first. If this counterbracing be carried throughout the whole
length of the girder (as in cross-bracing), it is possible by tighten-
ing it up to produce an initial strain in the bracing proper, in
which case the effect of a load will be to diminish the strain in the
counterbracing, which, however, will relapse into its former state
of strain as soon as the load is removed (44S).
I cannot close these observations on counterbracing without
drawing attention to one important merit which bowstring
girders possess. When the load is uniformly distributed the
strains in the bracing are tensile, for the lower flange and load
are merely suspended from the bow, which differs but slightly from
the curve of equal horizontal thrust and therefore requires but little
bracing to keep it in form. Hence, compressive strains are produced
in the bracing only under the influence of passing loads ; and in
large girders, where the permanent load of string and roadway
CHAP. XXIV.] COUNTERBRACING. 427
is great compared with the passing load, it may happen that the
compressive strains produced by the latter do not exceed the
tensile strains which the bracing sustains in its normal state. If,
for instance, the permanent load of the lower flange and roadway
in the example worked out in 808 were twice as heavy as the
passing load, the strains in all the diagonals would be tensile under
all circumstances ; even if the permanent load were only once and a
half as heavy as the passing load, diagonal 6 alone would sustain
slight compression. In this case the difficulty of providing against
flexure in long compression bars does not arise, and the only part
of the structure subject to compression is the bow, which from its
large sectional area can be economically constructed of a form
suited to resist buckling or flexure.
428 DEFLECTION AND CAMBER. [CHAP. XXV.
CHAPTER XXV.
DEFLECTION AND CAMBER.
451. Deflection curve of girders with horizontal flanges
of uniform strength is circular. — It has been already shown in
Chap. VIII. that the deflection curve of girders with horizontal
flanges of uniform strength, that is, girders whose flanges vary in
sectional area so that they are subject to the same unit-strain
throughout the whole length of each flange respectively, is circular
and easily calculated by a simple formula (eq. 132). When, how-
ever, the flanges are of uniform section throughout their whole
length, and their strength therefore excessive near the ends, the
deflection will be somewhat less, and may be calculated by the
method explained in SS6 and the following articles. When the
strength of a girder is not uniform, there is of course a certain
waste of material, which, however, cannot always be avoided,
although some methods of construction — the cellular flanges of
tubular bridges for instance — are more liable to this objection than
others, as they cannot in practice be tapered off towards the ends in
accordance with theory.
453. Deflection an incorrect measure of strength. — Since
the deflection depends not only on the unit-strains in the flanges,
but also on the proportion of length to depth, on the coefficient of
elasticity of the material, and to some extent on the mode of con-
struction, the popular rule by which the strength is estimated from
the deflection alone, though possessing the merit of simplicity, is
extremely vague and liable to lead to false conclusions unless when
comparing girders of the same length, depth, and material. The
deflection of any particular girder, however, is sensibly proportional
to the load, provided the strains are within the elastic limit, which
they always are in safe practice.
CHAP. XXV.] DEFLECTION AND CAMBER. 429
453. Camber ornamental rather than useful — Permanent
set after construction. — As the amount of deflection is in
practice very small compared with the length of a girder, no appre-
ciable diminution of strength is produced merely by the change
from a horizontal line to the deflection curve, for deflection, unless
so excessive as to change the vertical reaction of the abutments
into an oblique one, is the result, not the cause, of increased
strain. A downward curve, or even a truly horizontal line is,
however, less pleasing to the eye than a slight camber ; hence, it
is desirable to give an initial camber somewhat in excess of the
calculated deflection, so that when the girder is loaded no per-
ceptible sag may suggest the idea of weakness, even though
imaginary. It should also be borne in mind that the various parts
of a built girder are put together free from strain and are fre-
quently a little out of line ; consequently, when a large girder first
supports its own weight, and again, but in a less degree, when it
is tested with a heavy load for the first time, there is a certain
slight motion from the closing up or stretching out of the various
parts accommodating themselves to their new state. A permanent
set is the result, which, however, is not necessarily indicative of
weakness, provided it is not increased by subsequent loads, which
should only produce a temporary deflection. This congenital set
sometimes nearly doubles the calculated deflection.
454. Loads in rapid motion produce greater deflection
than stationary or slow loads — Less perceptible in large
than small bridges — Reflection increased by road being
out of order — Railway bridges under 4O feet span re-
quire extra strength in consequence of the velocity of
trains. — The Commissioners appointed to inquire into the ap-
plication of iron to railway structures "carried on a series of
experiments to compare the mechanical effect produced by weights
passing with more or less velocity over bridges, with their effect
when placed at rest upon them. For this purpose, amongst other
methods, an apparatus was constructed, by means of which a car
loaded at pleasure with various weights was allowed to run down
an inclined plane; the iron bars which were the subject of the
experiment were fixed horizontally at the bottom of the plane, in
430 DEFLECTION AND CAMBER. [CHAP. XXV.
such a manner that the loaded car would pass over them with the
velocity acquired in its descent. Thus the effects of giving different
velocities to the loaded car, in depressing or fracturing the bars,
could be observed and compared with the effects of the same loads
placed at rest upon the bar. This apparatus was on a sufficiently
large scale to give a practical value to the results ; the upper end
of the inclined plane was nearly 40 feet above the horizontal
portion, and a pair of rails, 3 feet asunder, were laid along its whole
length for the guidance of the car, which was capable of being
loaded to about 2 tons ; the trial bars, 9 feet in length, were laid
in continuation of this railway at the horizontal part, and the
inclined and horizontal portions of the railway were connected by
a gentle curve. Contrivances were adapted to the trial bars, by
means of which the deflections produced by the passage of the
loaded car were registered ; the velocity given to the car was also
measured, but that velocity was, of course, limited by the height of
the plane, and the greatest that could be obtained was 43 feet per
second, or about 30 miles an hour. A great number of experiments
were tried with this apparatus, for the purpose of comparing the
effects of different loads and velocities upon bars of various
dimensions, and the general result obtained was that the deflection
produced by a load passing along the bar was greater than that
which was produced by placing the same load at rest upon the
middle of the bar, and that this deflection was increased when the
velocity was increased. Thus, for example, when the carriage
loaded to 1,120 ft>s. was placed at rest upon a pair of cast-iron bars,
9 feet long, 4 inches broad, and 1J inch deep, it produced a
deflection of -f$ ths of an inch ; but when the carriage was caused
to pass over the bars at the rate of 10 miles an hour, the deflection
was increased to y^ths, and went on increasing as the velocity was
increased, so that at 30 miles per hour the deflection became 1J
inch ; that is more than double the statical deflection. Since the
velocity so greatly increases the effect of a given load in deflecting
the bars, it follows that a much less load will break the bar when
it passes over it than when it is placed at rest upon it, and
accordingly, in the example above selected, a weight of 4,150ibs. is
CHAP. XXV.] DEFLECTION AND CAMBER. 431
required to break the bars if applied at rest upon their centres ;
but a weight of 1,778 ibs. is sufficient to produce fracture if passed
over them at the rate of 30 miles an hour. It also appeared that
when motion was given to the load, the points of greatest deflection,
and, still more, of the greatest strains, did not remain in the centre
of the bars, but were removed nearer to the remote extremity of
the bar. The bars, when broken by a travelling load, were always
fractured at points beyond their centres, and often broken into four
or five pieces, thus indicating the great and unusual strains they
had been subjected to." * These experiments show that a load in
rapid motion causes greater deflection than the same load at rest
or moving slowly, especially when the moving load is very large
compared with the dead weight of the girder. The increase,
however, is generally slight in railway practice, and the greater the
weight of the structure is to that of the passing train the less will
be the increment of deflection due to rapid motion. The difference
of deflection caused by a locomotive crossing the central span of
the Boyne Viaduct, 264 feet in the clear between supports, at a
very slow speed and at 50 miles an hour was scarcely perceptible,
and did not exceed the width of a very fine pencil stroke, but the
increase of deflection is more marked in bridges of small span, as
appears from the following experiments made on the Godstone
Bridge, South Eastern Railway, by the Commissioners appointed
to inquire into the application of iron to railway structures.! The
Godstone is a cast-iron girder bridge, 30 feet in span, with two
lines of railway.
Tons.
Weight of two girders, - - 15
Weight of platform between these girders, - 10
Weight of half the bridge, i.e., dead load, - 25
Weight of engine, -
Weight of tender, - -32
Moving load,
* Iron Com. Report, p. XL f Idtm, App., p. 250.
432 DEFLECTION AND CAMBER. [CHAP. XXV.
Velocity in feet per second.
Deflection in decimals of an inch.
o,
•19
22 == 15 miles per hour, -
•23
40 = 27-3 do.
do.
•22
73 = 49-8 do.
do.
•25
Similar results were obtained from the Ewell Bridge, upon the
Croydon and Epsom Line. The span of the Ewell Bridge is 48
feet, the dead weight of one-half is 30 tons, and the statical
deflection due to an engine and tender, weighing 39 tons, was
rather more than one-fifth of an inch. " This was slightly but
decidedly increased when the engine was made to pass over the
bridge, and at a velocity of about 50 miles per hour an increase of
one-seventh was observed. As it is known that the strain upon a
girder is nearly proportional to the deflection, it must be inferred
that in this case the velocity of the load enabled it to exercise the
same pressure as if it had been increased by one-seventh, and placed
at rest upon the centre of the bridge. The weight of the engine
and tender was 39 tons, and the velocity enabled it to exercise a
pressure upon the girder equal to a weight of about 45 tons."*
The fact of slightly increased deflection from rapidly moving
loads is also confirmed by Mr. Hawkshaw's experiments with an
engine and tender run at a speed of about 25 miles an hour over
five compound iron girder bridges on the Wakefield and Goole
Railway. These girders varied in span from 55 feet 7 inches to
88 feet 6 inches, and were therefore less affected by rapid loads
than the smaller bridges just described. Mr. Hawkshaw inferred
that " where the road is in good order the deflection is not much
increased by speed, but that where the road is out of order, then
there is an increase of deflection." For instance, the road im-
mediately leading on to one of the bridges in question "was
considerably depressed in level, so that in running the train over
the bridge at speed the whole weight of the train had to be
* Iron Com. Report, p. xiv.
CHAP. XXV.] DEFLECTION AND CAMBER. 433
suddenly lifted, and this of course had to be sustained by the girders
as well as the ordinary weight of the train."*
The conclusions of the Commissioners, as given at p. xviii. of
their report, is as follows : — " That as it has appeared that the effect
of velocity communicated to a load is to increase the deflection
that it would produce if set at rest upon the bridge; also that
the dynamical increase in bridges of less than 40 feet in length is
of sufficient importance to demand attention, and may even for
lengths of 20 feet become more than one-half of the statical
deflection at high velocities, but can be diminished by increasing
the stiffness of the bridge ; it is advisable that, for short bridges
especially, the increased deflection should be calculated from the
greatest load and highest velocity to which the bridge may be liable ;
and that a weight which would statically produce the same
deflection should in estimating the strength of the structure, be
considered as the greatest load to which the bridge is subject."
455. Effect of centrifugal force. — Centrifugal force produces
a very slight but appreciable increase of pressure when the load
passes rapidly across girders which, though ordinarily level, become
deflected by the load, and still more so if they happen to have
been built originally hollow in place of being level or cambered.
The increased pressure due to this cause is expressed by the fol-
lowing well known equation : —
P - "2W (251)
* — n~ \ )
9
Where P = the pressure due to centrifugal force,
R = the radius of curvature in feet,
W = the load,
v = the velocity in feet per second,
g = the acceleration due to gravity = 32 feet per second.
Ex. 1. A girder bridge 200 feet in span is deflected 0'25 foot below the horizontal line
by a certain load, W, at rest ; what is the increased pressure due to centrifugal force
if W traverses the bridge at the rate of 60 miles an hour ?
Here, v= 60X528° = 88 feet per second.
60X60
R= 100X100 = 20,000 feet.
* Iron Com. Report, App., p. 412.
2 P
434 DEFLECTION AND CAMBER. [CHAP. XXV.
Ex. 2. If the span were only 100 feet, and the deflection and velocity as before, we
would have R = 5,000 feet, or ^th of its former value, whence,
\A/
Answer, P = -0484W = |L nearly.
456. Practical methods of producing: camber and measur-
ing- deflection. — The deflection of a girder supported at both
ends is the result of the lower flange being extended while the
upper one is shortened, and camber may be produced by the reverse
of this, that is, by making the bays of the upper flange slightly
longer than those of the lower one when the girder is in process
of construction (833).
When small girders are under proof, their deflection may be
conveniently measured, unless there happens to be a strong wind,
by means of a fine wire fastened to one end of the girder and
passing over a pulley attached to the other end, where a small
weight will keep it in a state of constant tension. The deflections
should be read on a scale attached to the girder itself; when
measured from an object fixed outside the girder they cannot be
depended on, owing to the supports on which the ends of the
girder rest being compressed by the weight of the testing load.
When great accuracy is not required the deflection of a girder
bridge from passing loads may be measured by means of two
wooden rods, the bottom of one of which rests on the surface of
the ground beneath the bridge, while the top of the second rod is
pressed upwards against the soffit of the girder, so that they over-
lap each other midway ; a pencil line is then ruled across both
rods, and when the upper one is depressed by a passing load its
line will descend slightly, the distance between the two lines giving
the deflection of the girder.
CHAP. XXVI.] DEPTH OF GIRDERS AND ARCHES. 435
CHAPTER XXVI.
DEPTH OF GIRDERS AND ARCHES.
457. Depth of girders generally varies from one-eighth
to one-sixteenth of the span — Depth determined by practical
considerations. — The depth of large girders, with the exception
of triangular trusses, seldom exceeds l-8th, or is less than l-16th of
the span. For many years the common rule for cast-iron girders
was to make the depth 1-1 5th of the span and this established a
precedent for wrought-iron girders, but modern practice has with
great advantage increased the ratio, so that from l-8th to 1-1 2th
are now common proportions for braced girders. As the leverage
of the flange is directly as the depth, while the quantity of material
in the web is theoretically independent of it, it might be inferred
that the deeper the girder the greater the economy (S?4). The
practical limit, however, is defined by the extra material required
to stiffen long compression bars or thin deep plate webs, nor should
we overlook the necessity of having sufficient thickness in the web
for durability and sufficient material in the compression flange to
keep it from flexure or buckling. The following table contains
the principal dimensions of some important Bowstring bridges,
which are generally made deeper than girders with horizontal
flanges.
436
DEPTH OF GIRDERS AND ARCHES. [CHAP. XXVI.
1
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vertical axis 12'2£
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imber on Iron Bridge
Supplement to Bridges.
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CHAP. XXVI.] DEPTH OF GIRDERS AND ARCHES. 437
458. Economical proportion of web to flange — Practical
roles. — When a given quantity of material is to be distributed in
the most advantageous manner, the thinner the web and the more
the material is concentrated in the flanges, the stronger will the
girder be, provided the web retains sufficient material for trans-
mitting the shearing-strain ; but when, as is frequently the case in
small girders, the girder derives a considerable portion of its
strength from the web acting as an independent rectangular girder,
its thickness being determined from practical considerations, there
is a certain depth, depending on the thickness of the web and
the relation between the flanges, which will produce a girder of
maximum strength. If the flanges are of equal area this depth
may be found as follows : —
Let I = the length of the girder,
b = the thickness of the web, as determined by practical con-
siderations,
d rr the depth of the girder,
a = the area of either flange,
a' = bd = the area of the web,
A = 2a-j-a' = the total sectional area, which is a given
quantity.
From equation 71, we have for the weight which an equal-
flanged semi-girder loaded at the end will support,
in which / is the unit-strain in either flange. W is maximum
when d (a+ . ) is maximum, and in order to find what value of
V b /
d will produce this result we must equate the differential coefficient
of d («+7r) to cipher, first substituting for a and a' their values
in terms of d and the constant A, as follows : —
-
~ l2 3
Equating the differential coefficient of the term within the bracket
to cipher, we have,
438 DEPTH OF GIRDERS AND ARCHES. [CHAP. XXVI.
whence,
^ = ^A (252)
The depth therefore should be such that the web may contain Jths
of the whole amount of material.
The thickness of the webs of wrought-iron plate girders for
railway or public bridges should not be less than T5^ inch (431),
while those of cast-iron girders generally vary from 1 to 2 inches.
The following rule for the minimum thickness of cast-iron webs is
given by M. Guettier, a skilful French founder.*
Length of girder. Minimum thickness of cast-iron Webs.
4 metres, - 20 millimetres = 0*8 inches.
5 „ 25 „ = 1-0 „
6 „ 30 „ = 1-2 „
8 „ 35 „ = 1-4 „
StiiFening ribs are sometimes formed at right angles to the webs
of cast-iron girders, so as to act as brackets to the flanges, but
they are apt to shrink unequally in cooling and produce dangerous
cracks in the casting.
459. Depth of iron and stone arches. — The two following
tables contain the principal dimensions of some important iron and
stone arched bridges. See also the tables relating to cast-iron
arches and wrought-iron roofs in Chap. XXVIII.
* Morin, Resistance des Materiaux, p. 277.
CHAP. XXVI.] DEPTH OF GIRDERS AND ARCHES.
439
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440
DEPTH OF GIRDERS AND ARCHES. [CHAP. XXVI.
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CHAP. XXVI.] DEPTH OF GIRDERS AND ARCHES.
441
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442 CONNEXIONS. [CHAP. xxvn.
CHAPTER XXVII.
CONNEXIONS.
46O. Appliances Tor connecting: iron-work — Strength of
joints should equal that of the adjoining: parts — Screws. —
One general rule applies to all jointed structures, namely, that the
strength of the whole is limited by that of its weakest part, and
accordingly the strength of joints should not be less than that of
the parts which they connect. The usual appliances for connecting
iron- work may be divided into four classes : —
1. Screws. 3. Gibs and cotters.
2. Bolts or pins. 4. Rivets.
The strain to which the above-mentioned connectors are subject
is generally a shearing-strain, and as the strength of iron to resist
shearing is practically equal to its tensile strength (394), the
strength of an iron rivet, bolt, cotter, or screw, is measured by the
product of the area subject to shearing multiplied by the tearing
unit-strain of the iron. The thread of a screw which is subject
to longitudinal tension may be " stripped" or shorn off by the nut ;
in the case of V threaded screws both nut and screw may be
stripped simultaneously midway between the base and vertex of
the thread, and the shearing area is approximately measured by
the circumference of the screw at base of thread multiplied by
half the length grasped by the nut ; in the case of square threads
the shearing area is the same. From this it follows that the
length of the nut should be at least one-half the effective or net
diameter of the screw. In practice it is generally made equal to
1 or 1£ times the gross diameter and the diameter of a nut, or
bolt-head, or rivet-head is seldom less than twice that of the bolt.
461. Bolts or pins — Proportions of eye and pin in flat
links — Upsetting- and bearing: surface. — A bolt or pin is the
CHAP. XXVII.] CONNEXIONS. 443
simplest appliance for connecting together two pieces of iron, and
as the principal considerations connected with a bolt joint also
apply to other and more complex forms, I shall devote a short
space to its investigation. Take, for example, the joint of a
suspension bridge, the chains of which are formed of long flat
links connected by pins passing through eyes formed at their
ends. Such a joint may fail in six ways.
Fig. 112.
1. By the link tearing through the eye at cd, for want of
sufficient material to withstand the longitudinal tensile strain.
Hence, the sectional area at cd should theoretically equal that of
the shank at ab, but in practice it may be somewhat greater, as the
strain is less direct round the eye than in the body of the link.
2. By the end of the link being split along one or two lines,
such as gh and ik, for want of sufficient area to resist the shearing
action of the pin. Hence, the combined areas at gli and ik
should theoretically equal that of the shank at ab, but in practice
be considerably greater, as this part of the eye acts as a short girder
whose abutments are ce and fd ; this causes the outer circumference
near h and k to be in severe tension and, therefore, very liable to
tear, especially when the " reed" of the iron is open, as is frequently
the case with bar and angle iron.
3. By the pin being shorn across. This arises from its diameter
being too small. Hence, if the pin be iron and in double-shear,
its area should in no case be less than one-half that of the shank
at ab (394).
4. By the pin bending. This also arises from its diameter being
too small to afford the requisite stiffness, but ultimate failure may
444 CONNEXIONS. [CHAP. xxvu.
generally be prevented by the links being kept from spreading
asunder by a head and nut on the pin, at the loss, however, of
freedom of motion.
5. By the link tearing through the shoulder at Im, in consequence
of the curvature or change of form being too abrupt to permit the
lines of strain in the shank bending gradually round the eye.
6. By the crown of the eye being upset between g and i. This
arises from the bearing surface of the pin being too small in
proportion to the longitudinal strain, in which case there is an
excessive pressure on each superficial unit at the crown of the eye,
whereby the material there is upset, and the sides of the eye at
e and / become first unduly attenuated and then torn, the rent
extending from the inside towards the circumference. Sir C. Fox
has drawn attention to this latter source of failure in a valuable
communication to the Royal Society, in which the following
remarks occur :* — " If the pin be too small, the first result on the
application of a heavy pull on the chain will be to alter the
position of the hole through which it passes, and also to change it
from a circular to a pear-shaped form, in which operation the
portion of the metal in the bearing upon the pin becomes thickened
in the effort to increase its bearing surface to the extent required.
But while this is going on, the metal round the other portions of
the hole will be thinned by being stretched, until at last, unable
to bear the undue strains thus brought to bear upon it, its thin
edge begins to tear, and will, by the continuance of the same strain,
undoubtedly go on to do so until the head of the link be broken
through, no matter how large the head may be ; for it has been
proved by experiment that by increasing the size of the head,
without adding to its thickness (which, from the additional room it
would occupy in the width of the bridge, is quite inadmissible), no
additional strength is obtained. The practical result arrived at
by the many experiments made on this very interesting subject is
simply that, with a view to obtaining the full efficiency of a link,
the area of its semi-cylindrical surface bearing on the pin must be a
* "On the Size of Pins for connecting Flat Links in the Chains of Suspension
Bridges."— Proc. Roy. Soc., VoL xiv., No. 73, p. 139.
CHAP. XXVII.] CONNEXIONS. 445
little more than equal to the smallest transverse sectional area of its
body; and as this cannot, for the reasons stated, be obtained by
increased thickness of the head, it can only be secured by giving a
sufficient diameter to the pins. That as the rule for arriving at
the proper size of pin proportionate to the body of a link may be
as simple and easy to remember as possible, and bearing in mind
that from circumstances connected with its manufacture the iron
in the head of a link is perhaps never quite so well able to bear
strain as that in the body, I think it desirable to have the size of
the hole a little in excess, and accordingly for a 10 inch link I
would make the pin 6f inch in diameter, instead of 6^ inch, that
dimension being exactly two-thirds of the width of the body, which
proportion may be taken to apply to every case (where the body
and heads are of uniform thickness). As the strain upon the iron
in the heads of a link is less direct than in its body, I think it right
to have the sum of the widths of the iron on the two sides of the
hole 10 per cent, greater than that of the body itself. As the pins,
if solid, would be of a much larger section than is necessary to
resist the effect of shearing, there would accrue some convenience,
and a considerable saving in weight would be effected, by having
them made hollow and of steel." Mr. G. Berkley also has made
several valuable experiments on the strength of links, from which
he concludes that the diameter of the pin should equal f ths of the
width of the shank, while Mr. Brunei in his latest practice adopted
the same proportion of pin as Sir C. Fox, but made the curve of
the shoulder exceedingly gradual — the radius being 7 '6 times the
width of the shank — with the object of deflecting the lines of
strain along the shank as gradually as possible before passing
round the eye, the experiments which were made for the Chepstow
and Saltash bridges having led to the belief that strength depended
more upon the shape of the shoulder than upon excess of metal
about the eye.*
The following table gives these and other proportions adopted
by the foregoing authorities in a concise form : —
* Proc. Inst. G. E.% Vol. xxx., pp. 220 and 271.
446 CONNEXIONS. [CHAP. xxvu.
TABLE I.— PROPORTIONS OP THE EYES OP FLAT BAR LINKS.
Fox.
Berkley.
Brunei.
Shank, A, -
1-00
i-oo
i-oo
Diameter of pin, B, -
•66
75
•66
End of eye, C, -
—
i-oo
•60
Sides of eye, D + D
MO
1-25
1-21
Width of shoulder, E, -
—
1-00
—
Radius of shoulder, R, -
—
1-50
7'60
The sides of the eye of an ordinary forged tie rod have usually a collective
area equal to 1'5 or 2 times that of the rod.
Fig. 113.
463. Rivets in single and double shear — Proportions of
rivets in tension and compression joints — Hodgkinson's
rules for the strength of single and double riveting —
Injurious effect of punching holes — Relative strength of
punched and drilled holes. — The strength of a riveted joint,
so far as the rivets are concerned, is proportional to the number of
shears to which they are subject, a rivet in double-shear, Fig. 114,
being twice as strong as a rivet in single-shear, Fig. 115; so that
to make the joints of equal strength, the single-shear joint must
have twice as many rivets as the other.
CHAP. XXVII.]
CONNEXIONS.
Fig. 114.
Double-Shear.
447
Fig. 115.
Single-Shear.
When a joint connects plates in tension, the aggregate shearing
area of the rivets on each side of the joint line multiplied by the safe
shearing unit-strain of the rivets should equal the total working strain
transmitted through the plates. It thus happens in girder- work
that the collective shearing area of the rivets of a well proportioned
tension joint is nearly equal to the effective plate area, i.e., the net
area of the plates after deducting rivet holes (394). In practice the
rivet area is generally made about l-10th greater, in order to com-
pensate for any inequality in the distribution of the strain among
the rivets. In steel plating the rivet area, if the rivets are steel,
should be one-third greater than the net area of the plates, but the
heads of steel rivets are very apt to fly off (395). When a joint
connects compression plates whose ends do not butt closely against
each other, the thrust is transmitted through the covers and tends
to shear the rivets across exactly in the same manner as when a
tensile strain is transmitted, and the foregoing rule applies here also.
If, however, the compression plates have their ends planed square
and then brought very carefully into close contact so as to form
a "jump" joint, a short cover and one, or at most two, transverse
448 CONNEXIONS. [CHAP. xxvn.
rows of rivets on each side of the joint line will suffice, as the use
of the cover in this case is merely to keep the plates in line but not
to transmit the thrust. A jump compression joint is erroneously
supposed to be stronger than one in which the plates are slightly
apart with the covers and rivets duly proportioned as for a tension
joint, and engineers are sometimes over-exacting in this respect,
expecting water-tight joints when the contractor gets only 18s. or
20s. per cwt. for the girder. A real jump joint with the plates
butting along their whole width is rare, as the process of riveting
generally draws the plates slightly apart and an interval of a
hundredth of an inch is theoretically as bad as a quarter inch. A
little caulking of the edges, however, makes all smooth to the eye,
and the so-called "jump" joint passes muster. A practical remedy
for this is described in 464.
With respect to the ordinary method of riveting in transverse
rows, each row containing the same number of rivets, Mr.
Hodgkinson deduced from his experiments that " the strength of
plates however riveted together with one row of rivets, was reduced
to about one-half the tensile strength of the plates themselves ; and
if the rivets were somewhat increased in number, and disposed
alternately in two rows, the strength was increased from one-half
to two-thirds or three-fourths at the utmost."*
Reducing these conclusions to a convenient standard, we have the
following rule for the relative strength of lap-joints : —
Strength of the unpunched plate, - 100
Strength of a double-riveted joint, - 66
Strength of a single-riveted joint, - 50
Nearly all experimenters on the subject agree, and my own
experience corroborates the fact, that punching reduces the tensile
strength of iron to a greater degree than the aggregate area of
the metal punched out, and a close examination of the border of
each hole shows that it has been subject to a certain degree of
violence, which in most cases has injuriously affected the fibre of
the iron. Drilling does not damage the metal surrounding the
* Iron Com. Rep., App , p. 116.
CHAP. XXVII.] CONNEXIONS. 449
hole, and it is therefore preferred where the nature of the work
will permit the extra cost of drilling over punching. Mr. Maynard
inferred from his experiments that drilled plates are 19 per cent,
stronger than punched plates. There can be little doubt, however,
that the exact percentage will depend — 1°., on the condition of the
punching tool, i.e.t the maintenance of the proper proportion of
size between the punch and die; and 2°., on the quality of the
iron — a tough coppery iron, like Low Moor, suffering less injury
from punching than a hard brittle iron, and thick plates suffering
more than thin ones. Mr. Maynard was also led to the conclusion
that rivets in drilled holes were 4 per cent, weaker than rivets in
punched holes, because the sharp edges of the drilled plates have a
tendency to shear off the rivets cleaner than those in the punched
plates, and he finally concluded that the difference is 15 per cent,
in favour of drilled work when compared with punched work.
463. Covers — Single and double covers compared — Lap-
joint. — The strength of the covers of tension joints, and compression
joints where the plates do not butt closely, should equal that of the
plates ; hence, a single cover should resemble a short length of the
plate and each side of a double cover be at least half as thick as the
plate.
As the quantity of material required for covers forms a very
considerable percentage of the plates (12 per cent, and upwards,
depending on the length of the plates), it is of great importance
that the joints be as few as possible and arranged in the very
best manner. This is more especially the case in large girders,
where every ton of useless weight requires perhaps several tons in
the main girders for its support, as will be shown in a succeeding
chapter. For this reason large plates, with few joints, though
they may cost extra per ton, will often make a cheaper girder
than plates of ordinary sizes with more numerous joints (43 ?). In
the usual method of cover riveting, two or three transverse rows
of rivets are placed on each side of the joint line, each row
containing the same number of rivets, and the effective area of the
plate, if in tension, is reduced by the aggregate section of the
rivet holes in any one row. Hence, it would appear that the fewer
2 G
450 CONNEXIONS. [CHAP. xxvu.
rivet holes there are in each transverse row the less is the plate
weakened and the more is its material economized. But this again
requires several successive rows of rivets in order to provide
sufficient rivet area, thus introducing the necessity of long covers,
which may more than counterbalance the saving in the plates.
The size of the plates therefore will determine to some extent the
economical length of the covers as well as the transverse pitch of
the rivets.*
The few experiments described in 393 seem to indicate that
rivets in single-shear will not withstand so great a unit-strain as
rivets in double-shear; this, however, requires confirmation, and
good experiments on the strength of various forms of rivet joints
are much wanted. From those recorded by Sir William Fairbairn
in the appendix to the first series of " Useful Information for
Engineers," it appears that, so far as the plates are concerned, a
single-cover or lap-joint with only one transverse row of rivets in the
lap is considerably weaker (in the experiments about 25 per cent,
less) than a double-cover joint of the same theoretic strength, i.e.,
with the same net area of plates taken across the rivet holes. This
arises from the distortion of the single cover or lap-joint which,
yielding in its effort to assume a straight line between the points of
traction, bends the plates slightly and makes them liable to tear
across the line of rivet holes. When, however, a single-cover or
lap-joint had two or more transverse rows of rivets in the lap its
strength was not less than that of a double-cover joint of equal plate
area. If the plates are kept in a straight line by being riveted to
an angle iron or web, like the flange plates of a girder, it is still
more likely that the strength of a single-cover joint will be fully
equal to that of a double-cover joint of the same theoretic strength,
but whenever convenient, the double-cover should be adopted from
economical motives, as it gives double-shear to the rivets, and need
therefore be only half as long as a single cover with the same rivet
area. The common lap-joint represented first in Fig. 115, is,
however, an exception to this, as the lap need not be longer than
half the single cover represented beneath it.
* The "pitch" is the distance measured from centre to centre of rivets.
CHAP. XXVII.] CONNEXIONS. 451
464. Tension Joints of wiles — Compression joints of piles
require no covers if the plates are well butted — Cast-zinc
joints. — I have already advocated the piling of plates over each
other when a large flange area is required, and I have shown that
long rivets form no practical objection to this arrangement
(433, 484). When several plates are riveted together their joints
are generally arranged in steps, and the length of each cover equals
the lap of one plate multiplied by the number of plates + 1.
Thus, in Fig. 116, the pile consists of three plates and the length
. 116.
of each cover equals four laps. The length of lap is generally twice
the longitudinal pitch of the riveting. The thickness of the covers
of tension piles should be somewhat greater than half that of one
plate, for it is clear that when a joint occurs in an upper or lower
plate, more than half the tension in that plate will be thrown into
the nearest cover. Hence, it is a good rule to make the covers of
tension piles not less than f ths of the thickness of a single plate.
If a pile of several plates be in compression and closely fitted
so as to butt against each other, no covers will be required, and
great economy will result from this in very large girders, so much
so as amply to repay the extra expense of planing the ends of the
plates and bringing them carefully into close contact. To ensure
this, however, requires considerable attention, for the riveting
process has, as already observed, a tendency to open the joints
slightly, but cast-zinc, which is a very hard substance, may be
usefully employed for running into the compression joints of
wrought as well as cast-iron, provided they are sufficiently open
to let the molten metal flow freely. The joints of the cast-iron
voussoirs of the Bridge of Austerlitz in Paris, finished in 1806,
were thus formed,* and in my own practice I have used cast-zinc
for filling up the irregular intervals between the ends of the arched
ribs of a cast-iron bridge of 96 feet span and the wall-plates from
* Enc. Brit., 8th Ed., art. " Iron Bridges," Vol. xii., p. 581.
452 CONNEXIONS. [CHAP. xxvu.
which they sprang ; in the latter case accurate fitting would have
been extremely difficult, if not impossible, and a very satisfactory
and close joint was made by slightly warming the parts with a fire
of chips " to expel the cold air," as the workmen say, before
pouring in the molten zinc. The heat probably expels moisture and
assists the flowing of the metal into the narrower crevices. I have
also used cast-zinc very successfully for securing crane posts (both
cast and wrought-iron) in their foundation plates, where it ensures
close contact without the cost of fitting. The following description
of this method of forming the joints of a cast-iron arch of 133 feet
span on the Pennsylvania Central Railroad occurs at p. 244 of
Haupt on Bridge Construction : — " The joints were separated to
the distance of one-fourth of an inch, and filled with spelter (cast-
zinc) poured into them in a melted state ; this was very conve-
niently done by binding a piece of sheet-iron around each joint,
and covering it with clay. The material introduced being nearly
as hard as the iron itself, and filling all the inequalities of the
surface, rendered the connexion perfect." If the space between
two plates be very narrow, the joint should be placed in a vertical
position so that gravity may aid the flow of the metal, and a little
tin added to the zinc is said to render the latter more fluid.
465. Various economical arrangements of tension joints. —
The following method of riveting reduces the tensile strength of
the parts connected less than that in common use, and possesses the
merit of being applicable to plates as well as bars. Its peculiarity
consists in diminishing the number of rivets in each row as they
recede from the joint-line, and at the same time slightly increasing
the thickness of the cover or covers beyond that of the parts
connected. Fig. 117 represents this arrangement applied to a bar
or narrow plate with double covers. There are eight different ways
in which the joint may fail. 1°. By the bar tearing at a, where its
area is reduced by only one rivet hole. 2°. By both covers tearing
at b, where each is weakened by two rivet holes; this, however,
is compensated for by their united area being somewhat greater
than that of the bar. 3°. By the bar tearing at b at the same time
that the rivet at a is double-shorn. 4°. By the rivets on one side
CHAP. XXVII.]
CONNEXIONS.
453
of the joint line double-shearing. 5°. By the rivets on the alternate
half-faces single-shearing. 6°. By the rivets on one half-face single-
shearing while the opposite cover tears at b. 7°. By both covers
tearing at a simultaneously with the rivets double-shearing at b.
8°. By both covers tearing at a simultaneously with the bar
tearing at b. If, for example, the plates are 7 inch x \ inch,
connected by two T5(-th inch covers with yf th inch rivet holes, the
net area of the plate at a is 3*1 square inches nearly; the double-
shearing area of the rivets at one side of the joint line equals 3'1
inches, and the net area of both covers together at b is 3*36 inches.
Fig. 117.
Finally, the net area of the plate at b together with the double-
shearing area of the rivet at a equals 3-7 inches. This joint is
therefore tolerably well proportioned, while the effective strength
of the plates is really reduced by only one rivet hole, viz., that at a.
A similar plan of joint is applicable to broad plates, Fig. 118.
Fig. 118.
When this mode of riveting is applied to a pile of plates, the
454 CONNEXIONS. [CHAP. xxvu.
extra thickness of the covers should be sufficient to compensate for
the reduction in the strength of the whole pile caused by the
close transverse riveting at the joints.
When bars or plates are lap-jointed the arrangement proposed
by Mr. Barton, and represented in Fig. 119, is an excellent one.
Fig. 119.
The diagonal joint running obliquely across the plate is another
useful arrangement, and it appears from experiments instituted by
Mr. J. G. Wright that the strength of a single-riveted diagonal
lap-joint at 45° was 64*7 per cent, of that of the solid plate, whereas
the strength of a similar straight joint was only 48'2 per cent., the
increase in strength of the diagonal joint being 34 per cent, over
the other, that is, the diagonal single-riveted joint was nearly as
strong as an ordinary straight double-riveted joint.*
466. Contraction of rivets and resulting: friction of plates —
Ultimate strength of rivet-joints not increased by friction. —
Rivets contract in cooling and draw the plates together with such
force that the friction produced between their surfaces is generally
sufficient to prevent them from slipping over each other so long as
the strain lies within limits which are not exceeded in practice,
and when this occurs the rivets are not subject to shearing strain.
From experiments made during the construction of the Britannia
Tubular Bridge it appears that the value of this friction is rather
variable.f In one experiment with a Jth inch rivet passing through
three plates, and therefore in double-shear, it amounted to 5*59
tons, in another with a |th inch rivet and two plates lap-jointed
with T5^th inch washers next the rivet heads it reached 4*73 tons,
while in a third experiment with three plates and |th inch rivet
with ^ inch washers next the rivet heads, making the shank of the
* Proc. Inst. M. K, 1872, p. 77. t Clark on the Tabular Bridges, p. 393.
CHAP. XXVII.] CONNEXIONS. 455
rivet 2J inch long, the middle plate supported 7*94 tons before it
slipped. In these experiments the hole in one or both plates was
oval and the sliding took place abruptly. Though the friction of
riveted plates may be sufficient to convey the usual working- strain
without subjecting the rivets to shearing, it does not follow, nor
do experiments indicate, that the ultimate strength of a rivet joint
is increased by this friction. It is an interesting fact, however,
that rivets in ordinary girder-work and plating are subject to a
tensile and not a shearing strain.
467. Girder-makers', Boiler-makers' and Shipbuilders'
rales for riveting' — Chain-riveting. — Joints may fail by each
rivet splitting or shearing out the piece of plate in front of itself.
Consequently, the minimum theoretic distance of the rivets from
the edge of an iron plate or from each other lengthways should be
determined by the consideration that the shearing area of the
plate (along two lines) between each rivet and the one behind it, or
between each rivet in the first row and the edge of the plate, be
not less than that of the rivet. If, for example, the rivets in
Fig. 117 be f inch and the plates J inch thick, the shearing area
of each rivet (in double-shear) equals 1 square inch nearly,* and
the distance of the edge of the rivet holes from the joint line should
theoretically not be less than 1 an inch. Practically, however, this
is insufficient, for punching tends to burst the edges of the holes if
placed so close to each other or to the edge of the plate, especially
if the plate be thick or of brittle quality, and in boilers the dis-
tance between the holes and the edge of the plate is usually about
once the diameter of the rivet. If the distance exceed this it is
difficult to make the seam steam-tight by caulking. In girder-
work, which does not require caulking like a boiler, this distance is
seldom less than 1J times the diameter of the rivet, and the pitch
may vary from 2^ to 5 or even 7 inches, but should not exceed 15
times the thickness of a single plate, from 6 to 12 times being
common practice. The rivets in ordinary girder-work range from
* Rivet holes are generally punched from J^nd to -j^th inch larger than the nominal
size of the rivet, in order that the latter when red hot may pass freely through the
hole. Hence, the area of a f inch rivet, after riveting, is nearly half a square inch.
456
CONNEXIONS.
[CHAP. xxvu.
j to 1 inch and occasionally l£ inch in diameter. The rivet holes
in first-class work are now frequently bored out with drilling
machines, so as to avoid the weakening effect of punching on the
plates. The great majority of girder-work, however, will probably
always be done by the punch, as it does not pay to have the holes
drilled unless in large girders where there are frequent repetitions
of the same pattern (435). The following table shows the usual
practice in boiler-work.
TABLE II.— RULES FOB BOILER RIVETING.
Thickness
of
plate.
Diameter
of
rivet.
Length of
rivet
from head.
Central
distance of
rivets.
(Pitch).
Lap in
single
joints.
Lap in
double
joints.
Equivalent
length of
head.
inch.
inch.
inch.
inch.
inch.
inch.
inch.
A = '19
1 = '38
I
Htoli
1*
2A
»
i = -25
\ --= '50
i*
H to If
H
2*
i
A -'31
1 = '63
i|
1| to If
11
31
f
1 = '38
f = "75
1|
lg- to 2J-
2i
3|
1
4 = '50
«= 'SI
2|
2* to 21
2|
8|
H
A = '56
1 = '88
*i
24 to 2J
24
4s1
if
1 = '63
18= -94
21
2J to 2|
2|
4|
1*
^=•69
1 = 1-00
3
2|to3
3
5
1|
f = '75
1J = 1-13
3*
3 to 34
8*
5|
1|
NOTE.— The equivalent length of head given in the last column is intended for bat
heads, such as are usual in boilers, but if the rivets have cup heads like those in
Fig. 117, as is usual in girder- work, the equivalent length of head must be about
one-half more than the amount given in the last column. The pitch in girder-work
is generally from once and a-half to twice that in column 4.
The boiler-maker's rule is nearly as follows: — For plates less
than half an inch thick, the diameter of the rivet = twice the
thickness of the plate. For plates more than half an inch thick, the
diameter of the rivet = once and a half the thickness of the plate.
The pitch of single joints = 2| to 3 diameters, and that for double
joints — 3J to 4 diameters of the rivet. The lap for single joints =
3 diameters, and that for double joints = 5 diameters of the rivet.
CHAP. XXVII.]
CONNEXIONS.
457
Lloyd's rules for the dimensions of rivets in ship-building are as
follows : —
TABLE III— LLOYD'S RULES FOR SHIP RIVETING.
Thickness of Plates
Rivets to be
in inches,
^6
fe
TV
A
i"
tt
if
if
il
i£
if
J of an inch
larger in dia-
meter in the
Diameter of Rivets
stem, stern-post
in inches,
\ 1
3
4
I
1
and keel.
" The rivets not to be nearer to the butts or edges of the plating,
lining pieces to butts, or of any angle iron, than a space not less
than their own diameter, and not to be farther apart from each
other than four times their diameter, or nearer than three times
their diameter, and to be spaced through the frames and outside
plating, and in reversed angle iron, a distance equal to eight times
their diameter apart. The overlaps of plating, where double
riveting is required, not to be less than five and a half times the
diameter of the rivets; and where single riveting is admitted, to
be not less in breadth than three and a quarter times the diameter
of the rivets." The Liverpool rules differ somewhat from Lloyd's
and are as follows : —
TABLE IV.— LIVERPOOL RULES FOR SHIP RIVETING.
1
Thickness of Plates in
inches,
I5e
A
A
T86
T3a
ft
H
if
il
if
if
if
Diameter of Rivets in
inches,
TF
H
is
it
if
it
if
it
T*
if
i*
«
Breadth of lap in seams
in inches,
Single-riveting, - If
2i
2i
2|
—
—
—
—
—
—
—
—
Double-riveting, -
3
81
8}
44
H
4|
5*
5k
5|
6
6|
6J
Breadth of butt strip,
Double-riveting, -
73
8
8
10
10
10|
"i
Hi
12i
13
13f
H4
Treble-riveting, -
9
114
1U
134
13|
15
16
16
17
18
19
H
458 CONNEXIONS. [CHAP. xxvu.
" Rivets to be four diameters apart, from centre to centre, longi-
tudinally in seams and vertically in butts, except in the butts where
treble riveting is required, where the rivets in the row farthest
from the butt may be spaced eight diameters apart, centre to centre.
Rivets in framing to be eight times their diameter apart, from
centre to centre, and to be of the size required in the above table.
All double or treble riveting in butts of plates to be in parallel
rows, or what is termed chain riveting. It is recommended that
the necks of all rivets be bevelled under the head so as to fill the
countersink made in punching, and their heads should be no thicker
than two-thirds the diameter of the rivet." It will be observed
that the pitch may be one-third as great again in water as in
steam joints.
The term " chain-riveting" is applied to riveting in several
transverse rows, the rivets being placed longitudinally one behind
the other like the links of a chain. It merely means that both the
longitudinal and transverse rows of rivets form straight lines, in
place of the rivets being zigzag.
46§. Adhesion of iron and copper bolts to wood — Strength
of clenches and forelocks. — The shearing strength of oak
treenails has been already given in 397. The two following tables
are also the results of Mr. Parson's experiments.* " The first of
these tables exhibits the adhesion of iron and copper bolts, driven
into sound oak, with the usual drift, not clenched, and subject to
a direct tensile strain. By drift is meant the allowance made to
insure sufficient tightness in a fastening; it is therefore the quantity
by which the diameter of a fastening exceeds the diameter of the
hole bored for its reception."
* Murray on Shipbuilding, p. 94.
CHAP. XXV1I.J
CONNEXIONS.
459
TABLE V. — TABLE OF THE ADHESION OF IRON AND COPPER BOLTS DRIVEN INTO
SOUND OAK WITH THE USUAL DRIFT, NOT CLENCHED, AND SUBJECTED TO A DIRECT
TENSILE STRAIN.
Diameter
of the
Bolt.
Number
of the
experiment.
Iron.
Copper.
• Length of the Bolt driven into the Wood.
Four
inches.
Six
inches.
Four
inches.
Six
inches.
inches.
1
tons. cwts.
1 13
tons. cwts.
tons. cwts.
0 184
tons. cwts.
4
2
3
2 0
2 2
—
0 18
0 19
—
4
1 13
—
0 18
—
1
2 6
2 12
1 7
2 2
2
2 4
2 11
1 8
2 2
3
2 4
2 16
1 10
2 2
4
2 0
2 10
1 13
2 0
1
3 2
3 12
2 10
2 15
2
3 4
4 0
1 17
3 10
3
3 0
4 0
2 2
3 1
4
2 10
4 0
2 5
2 15
1
3 2
5 5
3 0
4 5
1
2
3
3 0
3 1
4 8
4 8
3 6
3 6
3 18
3 15
4
3 1
5 0
2 9
3 5
1
3 3
6 0
3 10
5 5
3
4
2
3
3 2
3 10
6 0
5 0
3 10
3 10
5 5
5 8
4
3 10
6 0
3 18
4 18
1
4 10
6 2
4 0
4 13
i
2
3
5 12
3 10
5 10
6 11
4 0
4 5
4 13
4 19
4
4 10
6 4
4 2
4 19
460
CONNEXIONS.
[CHAP, xxvii.
TABLE V. — TABLE OP THE ADHESION OF IRON AND COPPER BOLTS DRIVEN INTO
SOUND OAK WITH THE USUAL DRIFT, NOT CLENCHED, AND SUBJECTED TO A DIRECT
TENSILE STRAIN— continued.
Diameter
of the
Bolt.
Number
of the
experiment.
Iron.
Copper.
Length of the Bolt driven into the Wood.
Four
inches.
Six
inches.
Four
inches.
Six
inches.
inches.
1
tons. cwts.
5 0
tons. cwts.
7 2
tons. cwts.
4 2
tons. cwts.
5 19
1
2
3
4 7
4 11
8 1
6 5
4 8
3 15
5 0
6 5
4
4 0
7 0
4 10
5 0
" In Riga fir the adhesion was, on an average, about one-third
of that in oak, and in good sound Canada elm it was about three-
fourths of that in oak.
" The following table exhibits the strength of clenches and of
forelocks as securities to iron and copper bolts, driven six inches,
without drift, into sound oak, either clenched or forelocked on
rings, and subjected to a direct tensile strain. It gives the
diameter of the bolt on which the experiment was made, as well as
the number of the experiment : —
TABLE VI. — TABLE OF THE STRENGTH OF CLENCHES AND OF FORELOCKS, AS
SECURITIES TO IRON AND COPPER BOLTS, DRIVEN SIX INCHES, WITHOUT DRIFT,
INTO SOUND OAK, EITHER CLENCHED OR FORELOCKED ON RlNGS, AND SUBJECTED
TO A DIRECT TENSILE STRAIN.
Diameter
of the Bolt.
Number of the
experiment.
Iron.
Copper.
Clench.
Forelock.
Clench.
Forelock.
inch.
tons. cwts.
tons. cwts.
tons. cwts.
tons. cwts.
1
1 16
0 16
1 0
0 8
I
2
1 13
1 9
0 14
0 20
0 19
1 0
0 8
0 7
4
1 9
0 18
1 0
0 6 .
CHAP. XXVII.]
CONNEXIONS.
461
TABLE VI.— TABLE OP THE STRENGTH OF CLENCHES AND OP FORELOCKS, AS
SECURITIES TO IRON AND COPPER BOLTS, DRIVEN SIX INCHES, WITHOUT DRIFT,
INTO SOUND OAK, EITHER CLENCHED OR FORELOCKED ON RlNGS, AND SUBJECTED
TO A DIRECT TENSILE STRAIN — Continued.
Diameter
of the Bolt.
Number of the
experiment.
Iron.
Copper.
Clench.
Forelock.
Clench.
Forelock.
inch.
tons. cwts.
tons. cwts.
tons. cwts.
tons. cwts.
1
3 0
1 15
2 10
1 4
1
2
3
3 0
2 16
1 8
1 9
2 10
2 5
1 0
1 2
4
2 15
1 14
2 9
1 4
1
4 15
2 11
3 10
1 18
JL
2
3
4 10
4 5
2 15
2 10
3 15
4 0
1 18
2 4
4
4 12
2 12
4 10
1 16
1
5 18
3 15
6 0
2 13
1
2
3
6 8
6 8
3 6
3 0
5 15
6 5
2 10
2 16
4
6 0
3 7
5 10
2 10
1
7 10
3 10
7 0
• —
1
2
3
7 10
8 0
3 15
3 10
7 0
7 5
—
4
8 15
3 15
7 8
—
1
11 11
5 1
7 16
—
7
2
11 15
5 10
7 16
—
8
3
8 11
4 6
7 12
—
4
8 6
4 15
7 5
—
1
12 0
5 18
7 1
—
1
2
3
12 3
11 3
6 18
5 12
7 1
7 14
—
4
11 1
5 2
8 14
—
462 CONNEXIONS. [CHAP. xxvu.
" In the experiments on the clenches, the clenches always gave
way; but with the forelocks it as frequently occurred that the
forelock was cut off as that the bolt broke ; and in the cases of the
bolt breaking, it was invariably across the forelock hole. Accord-
ing to the tables, the security of a forelock is about half that of
a clench. It appears an anomaly that the strength of a clench on
copper should be equal to that of one on iron. But, in con-
sequence of the greater ductility of copper, a better clench is
formed on it than on iron. Generally the thickness of the fractured
clench in the copper was double that in the iron. With rings of
the usual width for the clenches, the wood will break away under
the ring, and the ring be imbedded for two or more inches before
the clench will give way. With the inch copper-bolts, all the
rings under the clenches turned up into the shape of the frustrum
of a cone, and allowed the clench to slip through at the weights
specified.
" Experiments with ring-bolts were made to ascertain the
strength of the rings in comparison with the clenches. The rings
were of the usual size, viz. : the iron of the ring one-eighth inch
less in diameter than that of the bolt. It was found that the
rings always carried away the clenches, but that they were drawn
into the form of a link with perfectly straight sides. The rings
bore, before any change of form took place, not quite one-half the
weight which tore off the clenches. It appears that the rings are
well proportioned to the strength of the clenches."
469. Adhesion of nails and wood-screws. — " The following
abstract of Mr. Bevan's experiments exhibits the relative adhesion
of nails of various kinds, when forced into dry Christiana Deal, at
right angles to the grain of the wood."*
* Tredyold's Carpentry, p. 189.
CHAP. XXVII.] CONNEXIONS. 463
TABLE VII. — ADHESION OP NAILS OF VARIOUS KINDS IN DRY CHRISTIANA DEAL.
Kind of Nails.
Number to the
pound
avoirdupois.
Inches long.
Inches
forced into
the wood.
Pounds
required to
extract.
Fine sprigs,
4,560
0-44
o-o
22
Ditto,
3,200
0-53
0-44
37
Threepenny brads,
618
1-25
0-50
58
Cast-iron nails, -
380
1-00
0-50
72
Sixpenny nails, -
73
2-50
1-00
187
Ditto,
—
—
1-50
327
Ditto,
—
—
2-00
530
Fivepenny,
139
2-00
1-50
320
u The force required to draw the same sized nail from different
woods averaged as under: —
TABLE VIII.— RELATIVE ADHESION OF SAME NAIL IN DIFFERENT WOODS.
Kind of Wood.
Weight in ttis. required
to draw a sixpenny
nail, driven in
one inch.
Dry Christiana deal,
187 fts.
Dry oak, -
507 „
Dry elm, -
327 „
Dry beech,
667 „
Green sycamore, -
312 „
Dry Christiana deal, driven in endways,
87 „
Dry elm, driven in endways,
257 „
*' It was further desirable to ascertain the degree of dependence
that might be placed on nailing two pieces together, and Mr. Bevan
kindly undertook to make some trials. Two pieces of Christiana
deal, seven-eighths of an inch thick, were nailed together with two
sixpenny nails ; and a longitudinal force in the plane of the joint,
and consequently at right angles to the direction of the nails, was
applied to cause the joint to slide; it required a force of 712 Ibs.,
464 CONNEXIONS. [CHAP. xxvu.
and the time was 15 minutes ; the nails curved a little and were
then drawn. Another experiment was made in the same manner
with dry oak, an inch thick, in which the force required was
1,009 Ibs. ; the sixpenny nails curved, and were drawn by that
force. Dry sound ash, an inch thick, joined in the same manner
by two sixpenny nails, bore 1,220 Ibs. 30 minutes without sensibly
yielding; but when the stress was increased to 1,420 Ibs. the pieces
separated with an easy and gradual slide ; curving and drawing the
nails as before, one of which broke.
" The following experiments on the force necessary to draw screws
of iron, commonly called wood screws, out of given depths of wood,
were made by Mr. Bevan. The screws he used were about two
inches in length, ^j diameter at the exterior of the threads, -f^fo
diameter at the bottom, the depth of the worm or thread being
TUUo' an(^ the number of threads in one inch zz 12. They were
passed through pieces of wood, exactly half an inch in thickness,
and drawn out by the weights stated in the following tables : —
TABLE IX.— RELATIVE ADHESION OF SCREWS IN DIFFERENT WOODS.
Kind of Wood.
Weight required to
draw out screws
passed through half-
inch boards.
Dry beech,
460 fts.
Ditto ditto,
790 „
Dry sound ash,
790 „
Dry oak, -
760 „
Dry mahogany, -
770 „
Dry elm, -
665 „
Dry sycamore,
830 „
" The weights were supported about two minutes before the
screws were extracted. He found the force required to draw
similar screws out of deal and the softer woods about half the
above.
" The force necessary to cause pieces screwed together to slide
CHAP. XXVII.] CONNEXIONS. 405
at the joining, was also determined; the pieces being joined by
two screws ; the resultant of the force coinciding with the plane of
the joint, and in line with the places of the screws. With Chris-
tiana deal, seven-eighths of an inch thick, joined by two screws
one and five-eighths of an inch in length, and five-fortieths of an
inch in diameter within the worm, a load of 1,009 Ibs. gradually
applied broke both the screws at the line of joint, after elongating
the interior of the hole and sliding about six-tenths. With very
dry seasoned oak, 1 inch thick, two screws one and five-eighths
long, and six-fortieths diameter within the thread, bore 1,009 Ibs.
for ten minutes without any signs of yielding: with 1,137 Ibs. both
screws broke in two places; each screw about two-tenths of an
inch within each piece of wood ; the holes were a little elongated.
With dry and sound ash, 1 inch thick, with screws 2^ inches
long, passing one quarter of an inch through one of the pieces, the
diameter at bottom of the worm seven-fortieths ; the load began
with was 1,224 fibs. ; gradually increased for two hours to 2,661 Ibs. ;
they produced a slow and moderate sliding, not separation, the
screws being neither drawn nor broken; but probably would, if
not removed on account of night coming on, and putting an end to
the experiment."
2 H
466 WORKING STRAIN AND [CHAP. XXVIII.
CHAPTER XXVIII.
WORKING STRAIN AND WORKING LOAD.
4*O. Working strain — Fatigue — Proof strain — English
role for working strain — Coefficient of safety. — The work-
ing strain is the strain to which any material is subject in actual
practice, but the term, unless accurately defined, is somewhat
ambiguous, as it is applied to strains which the material sustains
on rare occasions from extraordinary loads, as well as to those to
which it is liable in ordinary every-day use. For instance, a
railway girder may sustain a constant strain of 3^ tons per square
inch from the permanent bridge-load, which rises to 4| tons when
an ordinary train passes, but reaches a maximum of 5 tons with a
train of the greatest possible density, such as locomotives ; or again,
the chains of a suspension bridge may sustain only 2J tons per
square inch from the permanent or dead weight of the structure,
while a dense crowd of people may occasionally raise this to 6 tons
per square inch. In such cases we have three classes of strains.
1°. The permanent strain, due to the permanent or dead weight
of the structure itself, and from which the material suffers what
has been termed fatigue. 2°. The ordinary working strain, due to
ordinary live loads added to the dead weight of the structure.
3°. The maximum working strain, due to the greatest load possible
in practice added to the dead weight of the structure, and it is
this latter maximum strain which defines the strength of any
structure, and which therefore we have to consider in this chapter.
The proof load of a bridge is generally equal to the greatest load
possible in practice, but the proof strain of separate parts of a
structure, such as the individual links of a suspension bridge, is
frequently 50 per cent, over their intended maximum working
CHAP. XXVIII.] WORKING LOAD. 467
strain when in the structure. As might have been anticipated,
different opinions are held respecting the safe unit-strain for each
kind of material. English practice generally makes the working
strain some sub-multiple of the tearing or crushing strength of the
material, while General Morin and others recommend the working
strain to be such that the resulting alteration of length shall in no
case exceed one -half that which corresponds to the limit of elasticity.
Neither rule should be adopted to the exclusion of the other, but
as we know the limit of elasticity of but few materials (in fact only
wr ought-iron and steel), and as those which are not ductile seem
to have no very definite limit at all (see Chap. XVIII.), the
common English rule seems more generally applicable, and it has
the sanction of extensive experience in its favour. The term
factor, or coefficient of safety is applied to the ratio of the
breaking to the working strain. If, for instance, the tearing
inch-strain of plate-iron is 20 tons, and the working inch-strain
5 tons, the coefficient of safety will be 4.
CAST-IRON.
471. Effect of long: continued pressure on cast-iron pillars
and bars. — To determine the effect of long continued pressure
upon cast-iron, Sir Wm. Fairbairn had four pillars cast of Low-Moor
iron ; the length of each was 6 feet, and the diameter 1 inch, and
they were rounded at the ends. The first was loaded with 4 cwt.,
the second with 7 cwt., the third with 10 cwt., and the fourth with
13 cwt. These weights are respectively 30, 52, 75, and 97 per
cent, of the weight which had previously broken another pillar of
the same dimensions when the weight was carefully laid on without
loss of time. The pillar loaded with 13 cwt. bore the weight
between five and six months and then broke ; that loaded with 10
cwt. was increasing slightly in flexure at the end of three years ;
when first taken its deflection was '230 inch, and after each
succeeding year it was '380, '380, and "409. The other pillars,
though a little bent, did not alter. In these experiments we see
that a cast-iron pillar bore a steady load of one-half its breaking
weight for three years without alteration, while the deflection of
468 WORKING STRAIN AND [CHAP. XXVIII.
another pillar with three-fourths of its breaking weight was in-
creasing slightly at the end of the same period.*
To ascertain how far cast-iron bars might be trusted with per-
manent loads, Sir Win. Fairbairn made the following experiments
also : — " He took bars, both of cold and hot blast iron (Coed Talon,
No. 2), each 5 feet long, and cast from a model 1 inch square ; and
having loaded them in the middle with different weights, with
their ends supported on props 4 feet 6 inches asunder, they were
left in this position to determine how long they would sustain the
loads without breaking. They bore the weights, with one excep-
tion, upwards of five years, with small increase of deflection, though
some of them were loaded nearly to the breaking point." After
that time, however, less care was taken to protect them from
accident, and three others were found broken. They were examined,
and had their deflections taken occasionally, which are set down in
the following Table, which contains the exact dimensions of the
bars, with the load upon each.f
* Experimental Researches by E. Hodgkinson, p. 351.
f Idem, p. 374.
CHAP. XXVIII.]
WORKING LOAD.
469
TABLE I.— EXPERIMENTS ON THE STRENGTH OF C AST-IRON BARS TO RESIST LONG-
CONTINUED TRANSVERSE STRAIN.
•§
1
g
§
1
|
1
§
|
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tg82
'to °°
III
111
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Joo
P— o
§
III
111
-wOO
§ O O
££
2 s
13
o S
•a
~ ,
1
° !3
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f||
2|l
^ J£
w
''^•s
f||
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lo
"H^-s
S **5
«**£
^"^•s
*' *«£
^"®5
*'"s*
05 "" .a
II
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gg-S
III
IP
.111
||1
ill
C ^3 TJ
l&g
Ifl
Date of
H
I
|««
M
|o«
M
|PW
|QM
|ai
I fi
1
observation.
w
w
H
w
w
M
w
H
w
Fah.
Deflections with
a permanent
load of 280 fts.
aid upon each.
Deflectionswith
a permanent
load of 336 fts.
laid upon each.
Deflections
with a perma-
nent load of
392 fos. laid
upon each.
Deflections
with a perma-
nent load of
448 fts. laid
upon each.
1837.
Mar. 6
1-267
1-684
1-715
1-964
1-410
This broke with
9
„ H
49°
•916
•930
1-043
1-064
1-270
1-270
1-454
1-461
1-694
1-694
1-758
1-760
2-005
2-005
1-413
•413
392tt)S. ; other
hot blast bars
were tried, but
„ 17
2-010
•413
they were suc-
April 15
May 31
47°
62°
•930
•932
1-078
1-082
1-271
1-274
1-475
1-481
1-716
1-725
1-767
1-775
2-014
Broke
•422
•424
cessively bro-
ken with 448
fts.
Aug. 22
70° !
•937
1-086 ! 1-288
1-504
1-737
1-783
after
•438
Nov. 18
45°
•942
1-083
1-286
1-499
1-724
1-773
ing the
1-431
1838.
weight
Jan. 8
38°
•941
1-086
1-288
1-502
1-722
1-773
37 days
1-430
Mar. 12
51°
•945
1-091
1-298
1-505
1-801
1-784
1-439
June 23
78°
•963
1-107
1-316
1-538
1-824
1-803
1-457
1839.
Feb. 7
54°
•950
1-093
1-293
1-524
1-815
1-784
1-433
July 5
72°
•959
1-104
1-305
1-533
1-824
1-798
1-446
Nov. 7
50°
•955
1-102
1-303
1-531
1-824
1-796
1-445
Dec. 9
39°
•956
1-102
1-303
1-531
1-823
1-796
1-445
1840.
Feb. 14
50°
•955
1-104
1-305
1-531
1-824
1-797
1-446
April 27
63°
•954
1-116
1-309
1-519 i
1-818
1-802
1-445
June 6
61°
•951
1-112 j
1-303
1-520
1-825
1-798
1-445
Aug. 3
74°
•953
1*115
1-305
1-523
1-826
1-801
1-447
Sept. 14
55°
•1-047
1115
1-305
*1-613 i
1-826
1-802
1-447
1841.
Nov. 22
50° i
1-045
1-115
1-306
1-620
1-829
1-804
1-449
1842.
April 19
58° ;| -
—
1-308
1-620 |
1-828
1-812
1-449
On these experiments Mr. Hodgkinson made the following
observations : — " Looking at the results of these experiments, and
the note upon the first and fourth, it appears that the deflection
in each of the beams increased considerably for the first twelve or
* After August 3, 1840, a body seems to have fallen upon the bars of the 1st and
4th Experiment, and this may have increased their deflections.
470 WORKING STRAIN AND [CHAP. XXVIII.
fifteen months ; after which time there has been, usually, a smaller
increase in their deflections, though from four to five years have
elapsed. The beam in experiment 8, which was loaded nearest to
its breaking weight, and which would have been broken by a few
additional pounds laid on at first, had not, perhaps, up to the time
of its fracture, a greater deflection than it had three or four years
before; and the change in deflection in Experiment 1, where the
load is less than frds of the breaking weight, seems to have been
almost as great as in any other ; rendering it not improbable that
the deflection will, in each beam, go on increasing till it becomes a
certain quantity, beyond which, as in that of Experiment 8, it will
increase no longer, but remain stationary (41O). The unfortunate
fracture of this last beam, probably through accident, has left this
conclusion in doubt." Mr. Hodgkinson inferred from these
experiments that cast-iron girders might be safely trusted with
one-third of their breaking weight. This conclusion, however, he
seems to have subsequently modified, when a member of the Iron
Commission in 1849, which reported in favour of not less than
one-sixth.
473. £ITects of long-continued impact and frequent de-
flections on cast-iron bars. — The Commissioners appointed to
inquire into the application of iron to railway structures, reported
as follows on the effects of long-continued impacts and frequent
deflections of cast-iron bars : — " A bar of cast-iron, 3 inches square,
was placed on supports about 14 feet asunder. A heavy ball was
suspended by a wire 1 8 feet long, from the roof, so as to touch the
centre of the side of the bar. By drawing this ball out of the
vertical position at right angles to the length of the bar, in the
manner of a pendulum, to any required distance, and suddenly
releasing it, it could be made to strike a horizontal blow upon the
bar, the magnitude of which could be adjusted at pleasure either
by varying the size of the ball or the distance from which it was
released. Various bars (some of smaller size than the above) were
subjected by means of this apparatus to successions of blows,
numbering in most cases as many as 4,000; the magnitude of
the blow in each set of experiments being made greater, or smaller,
CHAP. XXVIII.] WORKING LOAD. 471
as occasion required. The general result obtained was, that when
the blow was powerful enough to bend the bars through one-half
of their ultimate deflection (that is to say, the deflection which
corresponds to their fracture by dead pressure), no bar was able to
stand 4,000 of such blows in succession; but all the bars (when
sound) resisted the effects of 4,000 blows, each bending them
through one-third of their ultimate deflection.
" Other cast-iron bars, of similar dimensions, were subjected to
the action of a revolving cam, driven by a steam-engine. By this
they were quietly depressed in the centre, and allowed to restore
themselves, the process being continued to the extent, even in some
cases, of an hundred thousand successive periodic depressions for
each bar, and at a rate of about four per minute. Another con-
trivance was tried by which the whole bar was also, during the
depression, thrown into a violent tremor. The results of these
experiments were, that when the depression was equal to one-third
of the ultimate deflection, the bars were not weakened. This
was ascertained by breaking them in the usual manner with
stationary loads in the centre. When, however, the depressions
produced by the machine were made equal to one-half of the
ultimate deflection, the bars were actually broken by less than
nine hundred depressions. This result corresponds with and con-
firms the former.
" By other machinery, a weight equal to half of the breaking
weight was slowly and continually dragged backwards and forwards
from one end to the other of a bar of similar dimensions to the
above. A sound bar was not apparently weakened by ninety-six
thousand transits of the weight.
"It may, on the whole, therefore, be said, that as far as the
effects of reiterated flexure are concerned, cast-iron beams should
be so proportioned as scarcely to suffer a deflection of one-third of
their ultimate deflection. And as it will presently appear, that the
deflection produced by a given load, if laid on the beam at rest, is
liable to be considerably increased by the effect of percussion, as
well as by motion imparted to the load, it follows that to allow the
greatest load to be one-sixth of the breaking weight, is hardly a
472 WORKING STRAIN AND [CHAP. XXVIII.
sufficient limit for safety even upon the supposition that the beam
is perfectly sound.
" In wrought-iron bars no very perceptible effect was produced
by 10,000 successive deflections by means of a revolving cam, each
deflection being due to half the weight which, when applied stati-
cally, produced a large permanent flexure.
*' Under the second head, namely, the inquiry into the mechanical
effects of percussions and moving weights, a great number of ex-
periments have been made to illustrate the impact of heavy bodies
on beams. From these, it appears, that bars of cast-iron of the
same length and weight struck horizontally by the same ball (by
means of the apparatus above described for long-continued impact),
offer the same resistance to impact, whatever be the form of their
transverse section, provided the sectional area be the same. Thus
a bar, 6 X 1£ inches in section, placed on supports about 14 feet
asunder, required the same magnitude of blow to break it in the
middle, whether it was struck on the broad side or the narrow one,
and similar blows were required to break a bar of the same length,
the section of which was a square of three inches, and, therefore,
of the same sectional area and weight as the first.
" Another course of experiments tried with the same apparatus
showed, amongst other results, that the deflections of wrought-iron
bars produced by the striking ball were nearly as the velocity of
impact. The deflections in cast-iron are greater than in proportion
to the velocity.
" A set of experiments was undertaken to obtain the effects of
additional loads spread uniformly over a beam, in increasing its
power of bearing impacts from the same ball falling perpendicularly
upon it. It was found that beams of cast-iron, loaded to a certain
degree with weights spread over their whole length, and so attached
to them as not to prevent the flexure of the bar, resisted greater
impacts from the same body falling on them than when the beams
were unloaded, in the ratio of two to one. The bars in this case
were struck in the middle by the same ball, falling vertically through
different heights, and the deflections were nearly as the velocity
of impact."*
* Rep. of Iron Com., p. x.
CHAP. XXVIII.] WORKING LOAD. 473
473. Working; strain of cast-iron girders — Rale of
Board of Trade — Working- strain of cast-iron arches —
French rule — Proving cast-iron. — The reader will observe that
the Commissioners considered one-sixth of the breaking strain
hardly a sufficient limit of safety for cast-iron girders when liable
to percussion arid deflection from moving loads. This inference
was, no doubt, influenced by their experiments on bars which were
much lighter in proportion to their trial loads than ordinary bridge
girders are compared with the loads which traverse them. As a
general rule, one-sixth of the breaking strain may be taken as the
safe working strain for cast-iron girders which are liable to vibra-
tion, as in railway or public bridges, but when the load is stationary
and free from all vibration, such as water tanks, one-fourth of the
breaking strain is safe. When, however, cast-iron girders are
liable to sudden severe shocks, as in crane posts or machinery,
their working strain should not exceed one-eighth of their breaking
strain. The railway department of the Board of Trade has laid
down the following rule for the guidance of engineers in the con-
struction of railways: — " In a cast-iron bridge the breaking weight
of the girders should be not less than three times the permanent
load due to the weight of the superstructure, added to six times
the greatest moving load that can be brought upon it." Notwith-
standing this rule, engineers will do well not to design cast-iron
girders for railway bridges of less strength than six times the total
maximum load, that is, six times the permanent load added to six
times the greatest moving load. The reader who desires detailed
information respecting the practice of our most eminent engineers
during the reign of cast-iron is referred to the evidence attached
to the " Report of the Commissioners appointed to inquire into
the application of iron to railway structures" in 1849. It seems
certain that the transverse strength of thick rectangular cast-iron
bars is less than that of thin ones (13S), ,but it does not neces-
sarily follow that the strength of large flanged girders is diminished
by the massiveness of the casting, or that they are relatively
weaker than smaller girders of similar section, for the quality of the
iron will, no doubt, materially influence their strength (3483 349).
474 WORKING STRAIN AND [CHAP. XXVIII.
Experiments on a large scale can only decide these questions,
which, however, have less importance now than when the Iron
Commission sat in 1849, as it is very unlikely that large cast-iron
girders will be employed in important works when wrought -iron
is available.
Cast-iron can be readily got, on specification, to stand from 7J
to 9 tons per square inch in tension; consequently, the rule of
one-sixth allows an inch-strain of from 1 J to 1J tons for the usual
safe tensile working-strain in the lower flanges of cast-iron girders,
but this material is quite unfitted for tie-bars for the reasons
referred to in 35O and 351. Cast-iron will safely bear 6 or 7 tons
per square inch in compression, provided it be in a form suited to
resist flexure ; but the effects of flexure will seriously diminish the
safe unit-strain for pillars or unbraced cast-iron arches, in which the
line of pressure may vary so as to alter the calculated unit-strain
very materially, perhaps as much as 50, or even 100 per cent. In
practice, the safe working-strain of cast-iron arches rarely exceeds 3
tons per square inch. For instance, the calculated working strain
in the Severn Valley Bridge carrying the Coalbrookdale Railway,
200 feet span and 20 feet rise, is between 2J and 3 tons per square
inch,* while that of the centre arch of South wark Bridge, 240
feet span, is about 2 tons per square inch.
The French ministerial limit of working strain for cast-iron in
tension is one kilogramme per square millimetre (= 0'635 tons
per square inch), and in compression five kilogrammes per square
millimetre (= 3-175 tons per square inch), and the following
table, prepared by M. Poir^e, engineer of Fonts et Chaussees,
illustrates some of the best French practice in cast-iron arches. f
* Proc. Inst. C. E,, Vol. xxvii., p. 109.
t Morin's Resistance des Mattriaux, p. 114.
CHAP. XXVIII.]
WORKING LOAD.
475
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476 WORKING STRAIN AND [CHAP- XXVIII.
The direct tensile strength of cast-iron may be tested in the
manner described in 483, but it is also usual to prove its trans-
verse strength by breaking small rectangular bars made of the
same metal and at the same time as the principal castings. The
following tests were applied in the case of the cast-iron sleepers
provided for the Great Indian Peninsula Railway. " The mixture
of metal is to be such as will produce the strongest and toughest
castings, and is to be approved as such by the consulting engineer.
The contractor must cast twice each day, from the same metal as
that used in the sleepers, two duplicate bars 3' 6" X 2" X I",
and two duplicate castings of the form shown on the contract
drawing, and exactly \" square for a length of \\" in the middle.
One of the two bars must be tested on edge, on bearings 3 feet
apart, by placing weights on the centre thereof, to ascertain its
elasticity and breaking weight; and one of the two castings must
be tested in a suitable machine of approved construction to ascer-
tain the tensile strength of the iron. The company's inspector
will reject all sleepers cast on any day when each of the bars will
not bear 30 cwt. placed on the centre without breaking, or when
each bar does not deflect at least O29 of an inch before fracture,
and when each casting will not bear a tensional strain of 11^ tons
per square inch of section. Three sleepers will also be tested each
day by a weight of 3{ cwt. falling through 5' 6", the same having
previously been subjected to blows from the same weight falling
through 2' 0", 2' 6", 3' 0", 3' 6", 4' 0", 4' 6", and 5' 0" suc-
cessively after the sand foundation (which shall not be more than
24 inches thick under the centre of the sleeper and laid on a cast-
iron bed plate 8 inches thick, and weighing 2 tons,) has been well
consolidated to the satisfaction of the consulting engineer or his
inspector; and whenever every sleeper so tested does not bear
these blows without cracking, or showing other signs of failure,
the day's make will be rejected. Immediately after every sleeper
is cast, it must be protected in a manner which will satisfy the
company's engineer, that the process of cooling will proceed so
slowly, that its strength will not in any degree be diminished by
too rapid or unequal cooling."* Some engineers consider this proof
* Proc. Inst. C. K, Vol. xxx., p. 225.
CHAP. XXVIII.] WORKING LOAD. 477
rather high, and specify that test bars, 2x1 inch, placed edgeways
on bearings 3 feet apart, shall support a weight on the centre
of 25 cwt., as it appears that sleepers can be obtained which would
stand better, as far as blows went, without using so high a bar test
as that above described. It is a singular fact that there is an
excess of about 16 per cent, in the weight that a 2-inch X 1-inch
test-bar will support when cast on edge and proved as cast, over
that which it will support when proved with the underside as cast
placed at the top as proved, and 8 per cent, over the weight which
the same test-bar will support if cast on its side or end, and proved
on edge.* Hence, cast-iron girders should be cast with the tension
flange downwards in the sand.
474. Working load on cast-iron pillars. — Owing to the
want of recorded information it is difficult to assign what propor-
tion of the breaking weight eminent engineers have considered to be
the safe working load for cast-iron pillars. The opinions elicited by
the Commissioners appointed to inquire into the application of iron
to railway structures throw little or no light on the matter, as the
evidence was chiefly confined to the strength of girders under
transverse strain. Navierf gives l-5th of the breaking weight as
the safe load in practice. Francis, J an American engineer, also
gives l-5th; while Morin§ adopts l-6th. My own experience
leads me to recommend that cast-iron pillars supporting loads free
from vibration, such as grain, should in general not be loaded with
more than l-6th of their calculated breaking weight. In factories
or stores, where strong vibrations from machinery occur, the
working load should not exceed l-8th ; and if the pillar be liable
to transverse strains, or severe shocks, like those on the ground
floors of warehouses where loaded waggons or heavy bales are apt
to strike against them, the load should not exceed l-10th of the
breaking weight, or even less when the strength of the pillar
depends rather on the transverse strain to which it is liable than
* Proc. Inst. C.E., Vol. xxx., pp. 228, 267.
•f- Application de la Mecanique, p. 204.
J On the Strength of Cast-iron Pillars, p. 17. New York, 1865.
§ Resistance des Mate">'iaux, p. 106.
478 WORKING STRAIN AND [CHAP. XXVIII.
the weight it has to support. For instance, the pressure of wind
against a light open shed, supported by pillars, may produce a
transverse strain which will be very severe compared with that due
to the mere weight of the roof. The same thing may occur if heavy
rolling goods, such as casks or loaves of sugar, are piled up against
the pillar in such a manner as to cause horizontal pressure like
that of a liquid. It is also necessary to take into consideration the
foundations on which the pillars rest, for if these yield unequally,
one pillar may sustain much more than its proper share of load.
Wrought-iron is gradually superseding cast-iron for struts in
machinery; when, however, cast-iron is adopted, it is well that
the working load should, at all events, not exceed l-10th of the
calculated breaking load. In all these cases it is essential to con-
sider carefully whether the pillar is flat bedded or very securely
fixed at the ends, as a slight imperfection in this respect, either
immediate or prospective, will reduce the strength to one-third in
long pillars, and somewhat less in medium pillars, and if there is
any doubt whatever on this point it will be only common prudence
to assume in the calculations that the pillar is imperfectly bedded
(3113 318). The reader will find practical rules for the thickness
of hollow cast-iron pillars in 334, and examples of calculation from
388 to 389.
WROUGHT-IRON.
475. Effects of repeated deflections on wronght-iron bars
and plate girders. — Sir Henry James and Captain Galton made
some experiments in Portsmouth Dockyard for determining the
effects produced by repeated deflections on wrought-iron bars.*
These experiments were made with cams caused to revolve by
steam machinery, which alternately depressed the bars and allowed
them to resume their natural position for a great number of times.
Two cams were used ; one was toothed on the edge so as to com-
municate a highly vibratory motion to the bar during the deflection ;
the other, a step cam, first gently depressed the bar and then
released it suddenly when the full deflection had been obtained.
The depressions were at the rate of from four to seven per minute,
and the following table gives the principal results : —
* Rep. of Iron Com., App. B., p. 259.
CHAP. XXVIII.]
WORKING LOAD.
479
TABLE III.— EXPERIMENTS ON REPEATED DEFLECTIONS OP WROUGHT-IRON BARS,
2 INCHES SQUARE AND 9 FEET LONG BETWEEN POINTS OP SUPPORT.
No. of
experiment.
Amount of
deflection in
inches.
Number of
depressions.
Permanent set
in inches.
Remarks.
1
•833
100,000
0-015
Toothed cam.
2
•83
10,000
o-
Step cam.
3
1-00
10,000
0-06
Do.
4
2-00
10
0-30
Do.
50
0-54
Do.
100
0-69
Do.
150
0-84
Do.
200
0-98
Do.
300
1-84
Do.
The following experiments were made for the purpose of com-
parison to determine the deflections due to statical loads at the
centre of a similar bar.
TABLE IV. — EXPERIMENTS ON A WROUGHT-IRON BAR, 2 INCHES SQUARE AND
9 FEET LONG BETWEEN POINTS OF SUPPORT, SHOWING THE STATICAL WEIGHTS
DUE TO GIVEN DEFLECTIONS, THE WEIGHTS BEING APPLIED AND THE DEFLECTIONS
MEASURED AT THE CENTRE.
Deflections
in inches.
Weights
in tbs.
Permanent
set.
Remarks.
•333
507
0
After the bar had 1,950 Ibs. on,
•666
•833
926
1,121
0
0
it suddenly gave way, and
although it did not break, no
further weight could be applied
1-00
1,364
0-054
with certainty.
1-80
1,950
0-86
In these experiments two things are worthy, of note; first, the
largest deflection which did not produce a permanent set appears to
be that due to rather more than one-half the statical weight which
crippled the bar : secondly, 10,000 depressions with the step cam,
causing a deflection of 1 inch, produced almost exactly the same
480 WORKING STRAIN AND [CHAP. XXVIII.
permanent set as the statical weight due to the same deflection of
1 inch.
With the view of arriving " at the extent to which a bridge or
girder of wrought-iron may be strained without injury to its
ultimate powers of resistance, and to imitate as nearly as possible
the strain to which bridges are subjected by the passage of heavy
railway trains," Sir William Fairbairn caused a weighted lever to
be lifted off and replaced alternately, by means of a water-wheel,
upon the centre of a wrought-iron single-webbed plate girder of
the usual construction, with double angle-irons and flange-plates
riveted on top and bottom respectively. The dimensions of the
girder were as follows:* —
Extreme length, - 22 feet.
Length between supports, - 20 feet.
Extreme depth, - 16 inches.
Weight of girder, - - 7 cwt. 3 qrs.
Square inches.
Area of top flange, 1 plate, 4 inches X i inch, - 2*00
„ „ 2 angle-irons, 2 x 2 X T5ff. - 2*30
4-30
Area of bottom flange, 1 plate, 4 inches X J inch, TOO
„ „ 2 angle-irons, 2 X 2 X T3ff, 1 '40
2-40
Web, 1 plate, 15 J X J inch, - - 1-90
Total sectional area in square inches, - - 8*60
The area of the | inch rivet holes in the bottom flange, two in
each angle-iron and two in the plate, is equal to *625 square
inches, which reduces the effective flange area for tension from 2*4
to 1*775 square inches. The web being continuous gave some aid
to the flanges, but as it was composed of 9 short plates with
vertical joints and single-riveted covering strips, the amount of
aid given to the tension flange probably did not exceed one-half
the theoretic aid of a perfectly continuous web (1OO), that is, it
probably equalled one-twelfth of the gross area of the web, or
* Useful Information for Engineers, third series, p. 301.
CHAP. XXVIII.]
WORKING LOAD.
481
0*158 square inches; adding this to the net area of the bottom
flange, we have a total of 1'775 + 0-158 = 1'933 square inches
available for tension, and assuming the tearing strength of the iron
to have been 20 tons per square inch, and the depth for calculation
to be taken from inside to inside of the angle-iron flanges, which
measures 14J inches, we have the breaking weight in the centre,
from eq. 18, as follows : —
... 4Fd 4 x (20 x 1-933) x 14-75 0 _
W=-r = - -^^ - = 9-5 tons.
The compression flange, it will be observed, was much stronger than
that in tension, and hence it may be supposed that a larger fraction
than one-twelfth of the web should be added to the lower flange
(488). The extra strength on this account must, however, have
been very small and could scarcely raise the breaking weight beyond
10 tons. Sir William Fairbairn, however, calculated the breaking
weight at 12 -8 tons by an empirical formula derived from the
model tube at Millwall. The following table contains a summary
of the experiments with the corresponding tensile strains, cal-
culated on the supposition that 10 tons was the true statical
breaking weight at the centre, and that 20 tons per square inch
was the tearing strength of the iron.
TABLE V. — EXPERIMENTS ON REPEATED DEFLECTIONS OP A SINGLE-WEBBED PLATE-
IRON GIRDER, 16 INCHES DEEP AND 20 FEET LONG BETWEEN POINTS OF SUPPORT.
Tensile
+2
strain
*]
Weight
on middle
No. of changes.
Deflection.
per square
inch of
Remarks.
l|
of girder.
net area .
K
of bottom
«
flange.
tons.
inches.
tons.
1
2-96
596,790
0-17
5-92
Above half a million changes,
working continuously for two
months, night and day, at the
rate of about eight changes per
minute, produced no visible
alteration.
2
3-50
403,210
0-23
7-00
One million changes and no
apparent injury.
3
4-68
5,175
0-35
9-36
Permanent set of '05 inches ;
broke by the tension flange
tearing across a short distance
from the middle. None of the
rivets loosened or broken.
2 i
482
WORKING STRAIN AND [CHAP- XXVIII.
Girder repaired by replacing the broken angle-irons on each side, and putting a
patch over the broken plate equal in area to the broken plate itself.
Tensile
.
strain
•si
Weight
on middle
No. of changes.
Deflection.
per square
inch of
Remarks.
o "C
of girder.
net area
S5 p.
of bottom
H
flange.
tons.
inches.
tons.
4
4-68
158
9-36
Apparatus accidentally set in
motion ; took a large but un-
measured set.
5
3-58
25,742
0-22
716
6'
2-96
3,124,100
0-18
5-92
No increase of deflection or per-
manent set.
7
4-00
313,000
0-20
8-00
Broke by failure of the tension
flange as before, close to the
plate riveted over the previous
fracture. Total number of
changesafterrepair=3,463,000.
These experiments seem to indicate that a constantly repeated
tensile strain of 6 or 7 tons per square inch will not injure
wrought-iron, but, as the actual breaking weight of the girder was
not determined after each experiment, we cannot be quite certain
whether the strength was really impaired or not by the lesser
strains. To carry out the experiment scientifically would have
required several girders to be broken by dead weight — one when
new, as a standard for comparison ; and each of the others after a
few million changes of the same amount in any one girder, but of
different amounts in successive girders.
436. Net area only effective for tension — Allowance for
the weakening: effect of punching — Rale of Board of Trade
for w nought-iron railway bridges — Tensile working strain
of wrought-iron — French rule for railway bridges. — The
reader will recollect that the whole area of a riveted plate is not
available for tension, but only the unpierced portion which lies
between the rivet holes in any line of transverse section ; this is
called the net area of the plate, and on this net area alone the
working tensile strain should be calculated. The effective tensile
area of a punched plate is, indeed, somewhat less than its net area,
CHAP. XXVIII.] WORKING LOAD. 483
for the tearing strength of iron is generally injured by punching,
especially if there be too great a clearance between the punch and
die, or if the iron be brittle and, though it is not the practice, it
would be more correct to diminish the gross section by the sum
of the rivet holes multiplied by a factor greater than unity,
perhaps 1/1, or 1*2. It may, perhaps, be supposed more accurate
to add a constant quantity, say Jth inch, to the diameter of each
hole in place of adding a percentage, but it is probable that the
weakening effect of punching is greater the thicker the plate,
and as thick plates have generally larger rivet holes than thin
ones, the percentage allowance will be more accurate in practice-
Good experiments on this subject are much wanted. Meantime,
the weakening effect of punching affords an argument in favour of
drilling holes, especially in hard and brittle materials. Punching
will probably do little injury to soft and ductile iron, or to mild
steel, especially when the latter is subsequently annealed (463).
The following rule has been laid down by the Board of Trade for
the strength of railway bridges. " In a wrought-iron bridge the
greatest load which can be brought upon it, added to the weight
of the superstructure, should not produce a greater strain on any
part of the material than 5 tons per square inch." This rule is
now confined to parts in tension, in which case the 5 tons is com-
puted on the net area only, while the usual limit of strain in the
compression flanges is 4 tons per square inch of gross area, and, as
the tearing and crushing strengths of ordinary plate iron are re-
spectively 20 and 16 tons per square inch, the foregoing rules are
equivalent to stating that one-fourth of the breaking strain is the
maximum safe working strain for wrought-iron girders which are
subject to vibration like railway bridges, and this is now the
recognized English practice. When wrought-iron girders support
a dead load, like water tanks or grain lofts, they will safely bear
one-third of their breaking strain, but when liable to sudden
severe shocks, as in gantries or cranes, the working strain should
not exceed one-sixth of the computed breaking strain.
The safe tensile working strain for ordinary bar, angle, or tee
iron in girder- work is generally the same as for plates, namely, 5
484 WORKING STRAIN AND [CHAP. XXVIII.
tons per square inch of net section, but bar iron of extra quality,
such as the links of suspension bridges, will safely bear 6 tons per
square inch. Special care is taken with the manufacture of this
class of iron, and it is customary to prove each link individually to
a strain of from 8 to 10 tons per square inch before it is admitted
into the suspension chain, the tearing strength of the iron being not
less than 24 tons per square inch. For merely temporary purposes
wrought-iron will bear safely a tensile strain of 9 tons per square
inch, unless when subject to violent shocks, in which case 6 tons
will be sufficient.
The French rule for wrought-iron railway bridges is that in no
part shall the strain, either of tension or compression, exceed 6
kilogrammes per square millimetre, i.e., 3*81 tons per square inch
of gross section.
427. Gross area available for compression — Com press! ve
working strain of wrought-iron — Flanges of wrought-iron
girders are generally of equal area. — The total sectional area
of a riveted plate is available for compression (flexure being duly
provided against), since the thrust is transmitted through the rivet
just as if it were a portion of the solid plate, for, if the rivet head
be properly hammered up, its shank will upset and fill the hole
completely. Even supposing that the rivet do not perfectly fill
the hole, an exceedingly small motion of the parts, which must
take place before crushing commences, will cause the strain to pass
through the shank. In practice, however, the longitudinal con-
traction of each rivet in cooling will produce an amount of friction
between the surfaces riveted together which is generally sufficient
to resist any movement so long as the strain lies within the usual
working limits (466). The crushing strength of wrought-iron is
generally taken at 16 tons per square inch (897), and the safe limit
of compressive working strain in girder- work is, according to
ordinary English practice, 4 tons per square inch over the gross
area, provided the section is so large that it can without extra
material be put into a form suitable for resisting flexure or
buckling. This is generally the case with the compression flanges
of girders. When, however, a thin sheet, like the web of a plate
CHAP. XXVIII.] WORKING LOAD. 485
girder, sustains compression, or when the theoretic section of a
strut is small, as in the compression bars of a braced web, it is
necessary to add additional material to prevent flexure or buckling.
Angle, tee, or channel iron are suitable for plate stiffeners or for
short struts; for long struts the plan of internal cross-bracing,
represented in Plate IV., may be advantageously adopted, the
cross-bracing, of course, not being measured as effective area to
resist crushing, since it merely keeps the sides in line, but sustains
none of the longitudinal thrust, and in small scantlings it will be
prudent to limit the maximum compressive working strain to 3
tons per square inch. The working strain of wrought-iron pillars,
when subject to shocks, like the jib of a crane, should not exceed
l-6th of the computed breaking weight ; with quiescent loads l-4th
is a safe rule. The reader is referred to 33O and the following
articles for the mode of calculating the strength of wrought-iron
pillars of various sections.
When wrought-iron arches have braced spandrils, the ribs are
free from transverse strain and will safely bear as high longitu-
dinal strains as the flanges of girders, but if the spandrils are
not braced, the line of pressure in the ribs may vary under the
influence of passing loads and thus double, or even treble the normal
working strain (819). The extreme compressive strains, produced
by the most unfavourable combination of circumstances in the
wrought-iron arched ribs of the Victoria Railway Bridge, in
four spans of 175 feet each, which was designed by Mr. John
Fowler, are said in no case to exceed 4J tons per square inch.*
The flanges of wrought-iron girders are generally made of equal
or nearly equal area, for the deduction for rivet holes in the tension
flange is compensated by the higher unit-strain in the net area
between the holes which is effective for tensile strain.
428. Shearing: working- strain — Pressure on bearing- sur-
faces— Knife edg-es. — The shearing strength of wrought-iron is
substantially the same as its tensile strength (394), from which it
follows that the shearing working strain of iron rivets or bolts in
ordinary girder- work may equal 5 tons per square inch of section,
* Proc. Inst, C.E., Vol. xxvii., p. 67.
486 WORKING STRAIN AND [CHAP. XXVIII.
but, as already stated in 468, the rivet area of a tension joint is
usually about 10 per cent, in excess of what this rule allows, in
order to compensate for accidental inequalities in the distribution
of strain among the rivets. When calculating the area of a plate
web from the total shearing-strain in the manner described in 54,
it is a safe rule to adopt 4 tons per sectional inch of web as the
maximum shearing unit-strain, but this rule gives no idea of the
amount of material requisite for stiffening the web, and which can
only be determined by experience in each separate case (43O).
The bearing surface of a round bar, such as the pin or bolt of a
flat link, is measured by the product of its diameter by the length
of bearing, and it appears from the experiments referred to in
461, that the statical working pressure on a bearing surface of
wrought-iron may equal 1*5 times the safe tensile strain, that is,
it may equal 7*5 tons per square inch of bearing surface. The
pressure of rivets in double-shear against the middle plate, sup-
posing friction does not affect the bearing pressure (466), is often
double of this, and the pressure of the links of a chain against
each other must also be far greater. The rule of the Board of
Trade for the steel knife edges of public chain-testing machines
requires that the pressure shall not exceed 5 tons per linear inch
of knife edge. In my own practice I have frequently put a
pressure of 10 tons on each linear inch, and occasionally 17 tons,
and found no bad effects.
479. Working-strain of boilers — Hoard of Trade role-
French rale. — The working load of fresh water boilers should
not exceed one-sixth of their bursting pressure, though locomo-
tives are occasionally worked (very unsafely) to one-fourth.
One-seventh of the bursting pressure seems a proper working
load for salt water boilers, as they are liable to greater hardship
than fresh water boilers. The following table will illustrate these
rules in a convenient form, applied to parts in tension ; the strains
are given in tons per square inch of gross area. The method of
calculating the strength of boiler flues is explained in Chap. XIII.
CHAP. XXVIII.]
WORKING LOAD.
487
TABLE VI.— TENSILE WORKING-STRAIN OF BOILERS.
"Best best" boiler plate.
Common boiler plate.
Tearing
strain
per
square
inch of
gross
area.
Working strain
per square inch
of gross area.
Tearing
strain
per
square
inch of
gross
area.
Working strain
per square inch
of gross area.
Fresh
water
boilers.
Salt
water
boilers.
Fresh
water
boilers.
Salt
water
boilers.
tons.
tons.
tons.
tons.
tons.
tons.
Wrought-iron plates, unpunched,
22
—
—
20
—
-
Do. do., single -riveted,
(strength = 50 per cent, of that of
the unpunched plate),
11
1-833
1-57
10
1-667
1-43
Do. do., double-riveted,
(strength = 70 per cent, of that of
the unpunched plate),
15-4
2-567
2-20
14
2-333
2-00
Some engineers allow for single-riveted joints one-fifth greater
working strain than is given in the table, in consequence of the
additional strength supposed to be derived from the plates break-
ing joint with each other, but I am not aware of any experiments
which support this view. The oral rule of the Board of Trade
Surveyors for marine boilers is that their tensile working strain
shall not exceed 6,000 Ibs., = 2*678 tons, per square inch of gross
section ; for example, the working pressure of a cylindrical boiler
of J inch plates, 12 feet in diameter, and double-riveted along the
longitudinal joints, should not exceed 62*5 Ibs. per square inch.
General Morin states that according to a French royal decree
the working strain of plate-iron in boilers shall not exceed 1*9 tons
per square inch.*
48O. Working strain of engine- work. — In engine and wheel-
work it is generally safe practice to proportion the moving parts so
that their working strain shall not exceed one-tenth or one-
twelfth of that which would break or cripple them ; for instance,
the working strain of screw bolts in engine-work is generally
limited to about 4,000 Ibs. per square inch of net section, and the
same rule is applied to piston and connecting rods when in tension
* Resistance des Mattriaux, p. 20.
488 WORKING STRAIN AND [CHAP. XXVIII.
merely; when in compression, one ton, or 2,240 Ibs. per square
inch, is an ordinary rule, though, properly speaking, the safe
working strain will depend on the strength of the rod to resist
flexure, and will therefore vary, like that of other pillars, with the
ratio of length to diameter.
481. Examples of working: strain in wroug-ht-iron girder
and suspension bridg-es. — The following tables contain examples
of the working strains in some important wrought-iron girder and
suspension bridges. Several of the suspension bridges in Table
VIII. have toll-gates which prevent the occasional load from
reaching so high as 80 Ibs. per square foot of platform. There
are also regulations to prevent horses or vehicles from going faster
than a walking pace. See " Working Load on Public Bridges "
near the end of this chapter.
CHAP. XXVIII.]
WORKING LOAD.
489
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WORKING STRAIN AND [CHAP. XXVIII.
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CHAP. XXVIII.] WORKING LOAD. 491
483. sirens;* h and quality of materials should be stated
in specifications — Proof strain of chains and flat liar links —
Admiralty tests for plate-iron. — In drawing- up specifications
for girders, ships, or boiler-work, it is well to specify the tearing
strength and quality of the materials. Plates may be tested by
tearing asunder samples of the following shape [7^ ^ in a
proving machine, several of which are now to be found throughout
the kingdom. The amount of elongation of wrought-iron or steel
under tensile strain is a test of toughness, a most desirable
quality for many purposes, though of little importance in the
compression flanges of girders. In my own practice I require the
tensile set after fracture (ultimate elongation,) of ship plates and
tension plates of girders to be not less than 5 per cent, of their
original length, when torn with the grain ; at right angles to the
grain the set is generally much less, perhaps only 1 or 2 per cent.
I also require their tensile strength to be not less than 20 tons
per square inch with the grain, and 18 tons across the grain (3533
3533 356). In proving cast-iron, care should be taken to round
off the arrises of the pin-holes by which the sample is suspended,
so that the strain may pass accurately through its axis (35O).
Chains are now tested in proving machines sanctioned by the Board
of Trade (38O to 383), and it is customary also to prove all the
flat bar links of suspension bridges to 9 or 10 tons per square
inch, but the proof strain should in no case exceed the limit of
elasticity, say 12 tons per square inch, lest the ductility of the
iron be impaired and brittleness result (4O9).
The following are the Admiralty tests for wrought-iron ship
plates : —
PLATE-IKON (FIKST CLASS).
B.B.
Tensile strain per ( Lengthways, - -22 tons,
square inch. ' Crossways, - 18 „
FORGE TEST (HOT).
All plates of the first class, of one inch in thickness and under, should be of such
ductility as to admit of bending hot, without fracture to the following angles : —
Lengthways of the grain, - 125 degrees.
Across, - - - - - - - - 90 „
492 WORKING STRAIN AND [CHAP. XXVIII.
FORGE TEST (COLD).
All plates of the first class should admit of bending cold without fracture, as
follows : —
With the grain.
1 in. and \% of an inch in thickness to an angle of 15 degrees.
I » IS „ „ 20 „
I „ -H „ „ 25 „
I, A „ 4 „ „ 35 „
TV i, I „ „ 50 „
A » i >, „ 70 „
T3F „ under, „ „ 90 „
Across the grain.
I in., }%, |, and |§ of an inch in thickness to an angle of 5 degrees.
!> T%> » 4 » » 15 „
TV „ I „ „ 20 „
AM* „ „ 30 „
& „ under, „ „ 40 „
PLATE-IRON (SECOND CLASS).
B.
Tensile strain per ( Lengthways, - - 20 tons.
square inch. ' Crossways, - - 17 „
FORGE TEST (HOT).
All plates of the second class of one inch in thickness and under, should be of such
ductility as to admit of bending hot, without fracture, to the following angles : —
Lengthways of the grain, - - 90 degrees.
Across, - - 60 „
FORGE TEST (COLD).
All plates of the second class should admit of bending cold without fracture, as
follows : —
With the grain.
1 in. and \% of an inch in thickness to an angle of 10 degrees,
i „ tt „ „ 15 „
I » . tt „ „ 20 „
I. A » 4 » '„ 30 „
•h » i » „ 45 „
A » i » ,, 55 „
fe „ under, „ „ 75 „
Across the grain.
f in. and ^ of an inch in thickness to an angle of 5 degrees.
8» A » 4 „ ,, 10 „
TS » I » „ 15 „
tk » i » „ 20
A „ under, „ 30
CHAP. XXVIII.] WORKING LOAD. 493
Plates, both hot and cold, should be tested on a cast-iron slab, having a fair surface,
with an edge at right angles, the corner being rounded off with a radius of £ an inch.
The plate should be bent at a distance of from 3 to 6 inches from the edge.
It is intended that all the iron shall stand the forge tests herein named, when taken
in four feet lengths, across the grain ; and the whole width of the plate, along the grain,
whenever it may be necessary to try so large a piece ; but a smaller sample will
generally answer every purpose.
All plates to be free from lamination and injurious surface defects.
One plate to be taken indiscriminately for testing from every thickness of plate,
sent in per invoice, provided they do not exceed fifty in number. If above that
number, one for every additional fifty, or portion of fifty.
Where plates of several thicknesses are invoiced together, and there are but few
plates of any one thickness, a separate test for plates of each thickness need not be
made ; but no lot of plates of any one thickness must be rejected before one of that
lot has been tested.
" The sample pieces cut from the plate, after having their edges
planed, are secured one by one to the cast-iron slab, about 3 or 4
inches from its edge, and are then bent down by moderate blows
from a large hammer. The result may be greatly affected by
humouring and coaxing on the part of the hammer-man. By
striking the iron in the direction of the fibre the workman can
make an inferior iron bend with less symptoms of distress than a
better iron may exhibit when used more roughly. The same
leniency may be shown to the iron by bending it under a steady
pressure instead of by blows. The blows should, therefore, be
delivered not too lightly, and about square to the surface, and the
first signs of fracture should be observed and recorded. The
samples for the hot test are heated until they assume an orange
colour, and are then bent down to the prescribed angles in the
same way as in the cold test."*
STEEL.
483. Working strain for steel — Steel pillars — Admiralty
tests for steel plates. — We cannot yet infer from extensive
practice what is the safe working strain for steel. Probably one-
fourth of the tearing strain, or 8 tons per square inch, is a safe
tensile working strain for mild steel plates such as those described
in 36O. The most important steel girder bridge which has come
* Keed on Shipbuilding, pp. 385, 395.
494 WORKING STRAIN AND [CHAP. XXVIII.
under my notice is that constructed of puddled steel by Major
Adelskold, of the Royal Swedish Engineers, for the Herljunga
and Wenersborg Railway in Sweden. The girder is an inverted
bowstring, carrying the railway in one span of 137J feet over
a rapid torrent. " The dimensions are calculated for a strain of
8 tons per square inch, every portion having been tested to 16
tons per square inch before being put in place."* The crushing
strength of steel is so high that 12, or even 15 tons, per square
inch is perhaps a safe compressive working strain when the material
is not permitted to deflect, but when in the form of a solid pillar,
the strength of mild steel seems to be only about If times that of
wrought-iron (336). Experiments are, however, still wanting to
determine this, and, until such are made, it will scarcely be safe to
adopt for steel pillars a higher load than 50 per cent, above that
which a similar section of wrought-iron would safely carry. The
Admiralty tests for steel plates for shipbuilding are as follows : —
Tensile strain per ( Lengthways, - 33 tons.
square inch. ' Crossways, - 30 „
The tensile strength is in no ease to exceed 40 tons per square inch.
FOBGE TEST (Hor).
All plates of one inch in thickness and under, should be of such ductility as to admit
of bending hot, without fracture, to the following angles : —
Lengthways of the grain, - 1 40 degrees.
Across the grain, - 110 „
FORGE TEST (COLD).
All plates should admit of bending cold, without fracture as follows : —
With the grain.
1 inch in thickness to an angle of 30
I » „ » 40
I „ „ » 50
I » » » 60
k » » » 70
& ,, „ „ 75
I » „ » 80
T^T » » „ 85
£ „ and under, „ 90
Across the grain.
Degrees.
1 inch in thickness to an angle of 20
i » » „ 25
I „ „ „ 30
I » »» » 35
k » „ » 40
A » » » 50
ff „ » ,, 60
A » n » 65
£ „ and under, „ 70
The edges should be drilled or sawn, and not punched, in cutting the sample from the
plate. In other respects they should be treated as already described for wrought-iron.f
* The Engineer, VoL xxii., p. 240, 1866.
+ Reed on Shipbuilding, p. 399.
CHAP. XXVIII.] WORKING LOAD. 495
Steel rivets are very brittle and their heads frequently fly off,
and accordingly it is usual to unite steel plates with iron rivets, of
much larger size, however, than would be required for iron plates
of the same thickness.
TIMBER.
484. F.iisrlisli , American and French practice — Permanent
working: strain — Temporary working; strain. — The use of
timber in important structures is now so rare in the United
Kingdom that it is difficult to assign the working strain which
English engineers consider safe. At the Landore viaduct, con-
structed by the late Mr. Brunei of creasoted American pine
in compression, with wrought-iron in tension, the timber was
generally calculated to bear 373 Ibs. per square inch, though
in some parts of the structure the strain was allowed to reach
560 Ibs., or 50 per cent, more.* At the Innoshannon lattice
timber bridge, erected by Mr. Nixon on the Cork and Bandon
railway, the ordinary working strains in the flanges were 484 Ibs.
compression, and 847 ibs. tension per square inch. After 16
years' life this bridge was so decayed that it became unsafe and
was replaced by a wrought-iron structure in 1862. f In America
large timber bridges are still common, and General Haupt, a
distinguished American engineer, "has not considered it safe to
assign more than 800 Ibs. per square inch as a permanent load, and
1,000 ft>s. as an accidental load,"t and in a paper on American
timber bridges, read by Mr. Mosse at the Institution of Civil
Engineers in 1863, it is stated that about 900 Ibs. per square inch
is usually considered by American engineers to be the limit of safe
compression for timber framing. § Navier and Morin, distinguished
French authorities, recommend that the working strain of timber
should not exceed one-tenth of the breaking strain || and, owing to
its liability to decay, this rule seems safe practice for structures
* Proc. Inst. C.E., Vol. xiv., p. 500.
t Trans. Inst, C.E. of Ireland, Vol. viii., p. 1.
J Haupt on Bridge Construction, p. 62.
§ Proc. Inst. C.E., Vol. xxii., p. 310.
II Navier, p. 103, and Morin, pp. 51, 64, 68.
496 WORKING STRAIN AND [CHAP. XXVIII.
which are exposed to the weather, but when timber is under cover
one-eighth of the breaking strain is a safe working load. For
merely temporary purposes a strain of one-fourth of the breaking
weight is probably safe, provided there are no shocks, as Mr. Barlow,
referring to tensile strain, states that he " left more than three-
fourths of the whole weight hanging for 24 or 48 hours, without
perceiving the least change in the^state of the fibres, or any
diminution of their ultimate strength."* With reference to
transverse strain, however, Tredgold states that " one-fifth of the
breaking weight causes the deflection to increase with time, and
finally produces a permanent set,"f and the reader should recol-
lect that the coefficients of rupture of timber, tabulated in 65,
were derived from selected samples of small size and require
therefore to be reduced to about one-half when applied to ordinary
timber of large size. The method of calculating the strength of
timber pillars has been already described in 337 and 338.
485. Short life of timber bridges— Risk of fire.— In the
paper on American timber bridges already referred to, Mr. Mosse
states that they do not last in good condition more than 12 or 15
years, the timber being generally unseasoned and shrinking much
after being framed. When covered in to protect them from the
weather " and cared for, any shrinkage of the braces being im-
mediately remedied, it is believed these bridges will remain in
good condition double the usual time, or about twenty -five years."
Some of the old Continental bridges, however, lasted much longer
than this, but fire seems to be as common an agent of destruction
as time in America, where doubtless, the long dry summers give it
every advantage.
486. Working load on piles depends more npon the
nature of the ground than upon the actual strength of the
timber — Working load at right angles to the grain. — As
piles in foundations beneath masonry are buried in the ground,
which itself supports an uncertain share of the weight of the
superstructure, it is impossible to say exactly what weight rests on
the pile and how much on the surrounding soil. The piles in the
* Barlow on the Strength of Materials, p. 24. f Tredgold's Carpentry, p. 57.
CHAP. XXVIII.] WORKING LOAD. 497
foundations of the High Level Bridge at Newcastle, erected by
Mr. R. Stephenson, were 40 feet long and driven through sand and
gravel till they reached the solid rock. One of these foundation
piles was tested with a load of 150 tons, which was allowed to
remain several days, and upon its removal no settlement whatever
had taken place. The piles are four feet from centre to centre,
filled in between with concrete made of broken stone and Roman
cement, and the utmost pressure that can come upon a single pile
is 70 tons, supposing none of the weight to be carried by the inter-
vening planking and concrete.* The piles in the Royal Border
Bridge, erected by Mr. Stephenson over the river Tweed, in 1850,
are American elm driven from 30 to 40 feet into gravel and sand ;
the pressure on each of these is also 70 tons, neglecting any
support derived from the intervening soil,f and this is the severest
load on piles I find recorded.
Assuming the piles in these two instances to be 15 inches square,
and that no part of the weight was supported by the ground
between the piles, the pressure does not exceed =^« = 45 tons per
square foot, or 700 Ibs. per square inch ; if, however, the piles
were only 12 inches square, the pressure is nearly 1100 Ibs. per
square inch. Some of the uprights in the lofty scaffolding on
which the land spans of the Britannia Bridge were built carried 28
tons per square foot, or 435J Ibs. per square inch. The horizontal
timbers, however, were somewhat compressed under this load.J The
working load on timber piles, surrounded on all sides by the ground,
may vary, according to Rondelet, from 427 to 498 fibs, per square
inch,§ and Professor Rankine || says: — " It appears from practical
examples that the limits of the safe load on piles are as follows : —
" For piles driven till they reach the firm ground, 1000 Ibs. per
square inch of area of head (= 64'3 tons per square foot).
" For piles standing in soft ground by friction, 200 Ibs. per square
inch of area of head" (= 12*85 tons per square foot).
* Encycl. Brit., Art. " Iron Bridges," Vol. xii., Part iii., p. 604.
t Proc. Inst. C.K, Vol. x., p. 224. £ Clark on the Tubular Bridges, p. 549.
§ Morin's Resistance des Materiaux, p. 71. II Manual of Civil Engineering, p. 602.
2 K
498 WORKING STRAIN AND [CHAP XXVIII.
Professor Rankine's rule is based on sound principles, for the
nature of the ground, and the resistance which it offers to the pene-
tration of the piles, have generally more to do with their safe work-
ing load than the strength of the timber has. As far as the latter
alone is concerned, we might safely load piles surrounded by the
ground with l-5th of the crushing weight of wet timber, which,
according to Hodgkinson's experiments, is equivalent to a load of
about l-10th of the crushing weight of dry timber (3OO). When,
however, loaded piles project above the surface of the ground
they act in the capacity of pillars, and their strength accordingly
should exceed that of piles surrounded by earth. The safe work-
ing load of timber at right angles to the grain is about one-third
of that lengthways. For instance, 300 Ibs. per square inch is a
sufficient load for pine or fir cross-sleepers, and, if we estimate that
the pressure from the driving wheel is equal to 8 tons when the
engine is running, the bearing surface of the rail in a cross-sleeper
road should not be less than from 50 to 60 square inches. Three-
fourths of this will probably be sufficient if the sleepers are made
of hard wood. A similar rule applies to timber wall-plates, such
as those which support the ends of girders.
FOUNDATIONS, STONE, BRICK, MASONRY, CONCRETE.
487. Working load on foundations of earth, clay, gravel
and rock. — Professor Rankine states that " the greatest intensity
of pressure on foundations in firm earth is usually from 2,500 Ibs.
to 3,500fts. per square foot, or from 17 Ibs. to 23 Ibs. per square
inch," and that " the intensity of the pressure on a rock foundation
should at no point exceed one-eighth of the pressure which would
crush the rock."* Foundations should be placed sufficiently deep
to protect them from the influence of frost or running water, nor
should it be forgotten that excavations and pumping operations in
the neighbourhood of buildings frequently cause subsidence of the
foundations and superstructure. The following table contains a
few examples of heavy pressures on foundations.
* Civil Engineering, pp. 380, 377.
CHAP. XXVIII.]
WORKING LOAD.
499
Observations.
Lattice girders resting on cast-iron disc piles,
the discs being 2 feet 6 inches in diameter and
sunk by forcing water down the centre of the
pile. Rubble stone tipped in round the piles
prevents the sand from being scoured away
by the current.
Bowstring girders resting on cast-iron cylinders
8 feet in diameter, and filled with concrete
and masonry. A large mound of rubble stone
tipped in round the cylinders.
Lattice girders resting on cast-iron cylinders
14 feet diameter below the ground and 10 feet
diameter above, filled with Portland cement
concrete and brickwork, and sunk from 50
to 70 feet below Trinity high water into the
solid London clay. Friction of cylinder not
taken into account. Bridge supposed loaded
all over with locomotives. The friction in
sinking each cylinder amounted to 150 tons.
If this be taken into account, it would
reduce the pressure on the clay by about 1
ton per square foot.
Cast-iron cylinders filled with concrete and
brickwork. Friction not taken into account.
Bridge .supposed loaded all over with loco-
motives.
A
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500 WORKING STRAIN AND [CHAP. XXVIII.
4§§. Working? load on rabble masonry* brickwork* con-
crete and ashlar-work. — The crushing strength of building
materials has been already given in Chap. XIV. The working
load on rubble masonry, brickwork, or concrete, rarely exceeds one-
sixth of the crushing weight of the aggregate mass, and this seems
a safe practical limit. General Morin, however, states that mortar
should not be subject to a greater pressure than one-tenth of its
crushing weight.* The ashlar voussoirs of an arch, where the line
of thrust may vary considerably from the calculated direction,
should not be subjected to a greater (calculated) pressure than one-
twentieth of that which would crush the stone. It is safe to apply
the same rule to all ashlar-work, as it is very difficult, if not
impossible, to command a perfectly uniform pressure throughout
the whole bed of each stone, and a slight inequality in the line of
pressure may cause splintering or flushing at the joints. Vicat's
experiments on plaster prisms (339) and the examples of pressure
given in the following table, seem to show that the weight on
stone columns may sometimes reach as high as one-tenth of the
crushing strength of the stone. This, however, is a much severer
load than is usual in modern practice and cannot be recommended
as very safe.
Ex. What is the safe load per square foot for brickwork in cement, similar to that
whose crushing weight is given at p. 238. Here, the crushing weight = 521 Ibs. per
square inch = 33*5 tons per square foot, and we have,
OO. K
Answer, Safe working load = — - =5*6 tons per square foot.
6
* Resistance des Materiaux, p. 51.
CHAP. XXVIII.]
WORKING LOAD.
501
l| |p f:slisl »
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502
WORKING STRAIN AND [CHAP. XXVIII.
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CHAP. XXVIII.] WORKING LOAD.
TABLE XII. — EXAMPLES OF WOKKING LOADS ON
503
Load
No.
Name
of the Structure.
Date.
Engineer.
Material.
per
square
Observations.
foot.
tons.
1
Charing Cross
1863
Hawkshaw.
Concrete made
8
See Ex. 3, Table
Bridge.
of Portland
IX.
cement and
Thames
gravel, 1+7.
2
Chimney at
1867
Concrete base
2
See Ex. 7, Table
West Cum-
3 feet thick,
XI. Pressure
berland
made with
on ground = 1'6
Haematite
hydraulic
tons per square
Iron Works.
lime.
foot.
3
Base of St.
Strong con-
3
450 feet below the
Rollox
crete or
summit.
chimney,
beton, 6
Glasgow.
feet thick.
1 Proc. Inst. C.E., Vol. xxii., p. 515. 2 Trans. Inst. Eng. in Scotland, Vol. xi., p. 157.
3 Eankine's Civil Engineering, p. 378.
WORKING LOAD ON RAILWAYS.
489. A train of engines is the heaviest working1 load on
lOO-foot railway girders — Three-fourths of a ton per running
foot is the heaviest working load on 4OO-foot girders —
Weight of Engines — Girders under 4O feet liable to concen-
trated working loads. — A train of locomotives, the weight of
which generally varies from 1 to 1J tons per running foot, is the
heaviest rolling load to which a single-line railway bridge is liable,
but it rarely happens in practice that girders are subject to a uniform
load of this density, except in short bridges whose length does not
exceed that of two engines with their tenders, which may collectively
cover from 80 to 100 feet of line. We may therefore safely
assume that the maximum strain to which the flanges of railway
girders 100 feet in length are subject, does not exceed that due to
the permanent bridge-load plus a train-load of from 1 to !£• tons
(according to size of engines), per running foot on each line of way.
In longer bridges than 100 feet, the train-load per running foot will
be less, and in bridges of 400 feet span or upwards, the greatest
occasional load can scarcely exceed f ton per running foot on
504 WORKING STRAIN AND [CHAP. XXVIII.
each line, as this is a denser load than that of an ordinary goods
train.*
Until lately it has been usual to take one ton per running foot
on each line as the ruling load for engines. This, however, is
scarcely safe practice, since many engines now exceed this, as shown
by the following tables, for the first of which I am indebted to
A. M'Donnell, Esq., Locomotive Superintendent of the Great
Southern and Western Railway, Ireland, and for the second to
J. Ramsbottom, Esq., late Locomotive Superintendent of the
London and North Western Railway.
* The following memorandum shows the weight of a train of wagons loaded with
sulphur ore on the Dublin, Wicklow and Wexford Railway : —
" Weight of mineral engine loaded, 27 tons.
tender do. 17 do.
Length of engine and tender, buffer to buffer, 44 feet.
Wagon, empty 4 tons, loaded 12 tons ; length 18 feet, out to out of buffers. Two
other descriptions of wagons, one 12 feet, and the other 14 feet 6 inches long, taking
one ton less and weighing about 5 cwt. less. A mineral train of engine, 20 wagons
and van, will weigh about 280 tons and its length will be about 400 feet when
buffers are close up ; when running, somewhat longer."
CHAP. XXVIII.] WORKING LOAD.
505
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506
WORKING STRAIN AND [CHAP. XXVIII.
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r1,£il.s.|.s _.|.g_.|
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loi.iMi,
f«fi g °ft § °fl § °0
_CO 00 <N
|3 '«
L L
00
I I
I I I
~c
JfflJ
tsl
I, ".
CM <M
I I
1° ^^
CO t>- CO OS
_g <M <M (M
»O 00 000 CO 10 (NO 00t»
co
o oo
»0 Oi 00 O CO OS
OO CO IO CO XO CO
O CO COO U5 00 O rH ^ C^ COO V£5
CO t- COOS 00 OS COCO (N t- COCO CO
IS
o o
00 i-"
CO O rH O
I— I
OS <M rH (^
O CO
rH OS
(NO xH CO
00 O OS <M
CO t>. (N rH l^ Wi OS
rH f— I
OS O CO t^ CO O CO
00 OS OS
-^ OS OS O
OS O OS OS
O CO
I— I
CO t~
O ^X 00 rH
t- at co o
CO O O »O 00 -* U5
i— I rH rH
1O CO COt^ CO CO CO
CO ^
CO CO
CO
co r>-
oo oo
t^ o
I I
CHAP. XXVIII.]
WORKING LOAD.
507
CO
CO
CO CO •*
C5 O rH rH
.9 °
4^ O
.S °
-*i OO
I I
311
is*
' '
1 ' tl
S £ a *
4 4
« «
2 ^
508
WORKING STRAIN AND [CHAP. XXVIII.
CNJ (M rH
CO CO <N CO CO
00-*
i-H r- 1
OS O OS
3 rH
CO OO OO O O O
<N <M rH (M O O
& O
5 rH
no O
rH rH
rH O OS OS
O CO O O
is-
Sll
OO CO 00
•f ' t • -•£ ~w ••£ 't 0 n*
a>oa>oa>koo<i>
^"U8Ug °U
o -g g -g
o S o g
w rg a, ,q
® r4 P r 13 ,_5 ^
s s ^ _s s ^
> •§ >
3, OS JQ O
o ib o °°
r<i) ^-; To U)
B- BhT
.go go
tiD C^
v^ (-1 H
I
CHAP. XXVIII.]
WORKING LOAD.
509
Occasional monster engines occur on some railways, generally
where the gradients are unusually steep, as illustrated in the
following table : —
TABLE XVI. — EXAMPLES OF MONSTER ENGINES ON VARIOUS RAILWAYS.
1
Eailway.
No.
of
Wheels.
Wheel
base.
Weight.
Observations.
ft. in.
tons.
1
North London,
4
—
42
Four wheels coupled.
2
Oldham, -
6
—
49
Goods engine with 6 wheels cou-
pled ; gradient 1 in 27.
3
Brecon & Merthyr,
6
12 0
38
Tank engine ; gradient 1 in 38.
4
Vale of Neath, -
8
—
56
Tank engine with 8 wheels coupled;
afterwards altered into engine
with tender in consequence of
the destruction to the permanent
way ; gradient 1 in 47.
5
Mauritius Eailway,
8
15 6
47
Tank engine with eight wheels
coupled, 4 feet diameter; gra-
dients 1 in 27.
6
Northern Railway
12
19 8
674
Tank engine with 4 outside cylin-
of France,
ders; wheels coupled together
as in two separate six-wheeled
coupled engines.;
7
Semmering,
8
—
55*
—
8
Giovi,
8
—
55£
Four cylinders.
9
Cologne Minden, -
—
11 2
32
—
10
Rhenish, -
—
13 . 0
39|
—
11
Do.
—
11 0
29
—
1 2 Proc. Inst. C.E., Vol., xxvi., p. 343, 383. 5 Ibid., p. 384.
3 Ibid., p. 335. 6 7 8 Ibid., pp. 373, 343.
4 Ibid., pp. 372, 374. 9 10 » Ibid., Vol. xxv., p. 436.
It has been already shown in 454 that railway bridges under
40 feet span require extra strength in consequence of high-speed
trains increasing their deflection, but besides this they are liable to
heavier statical strains than those due to uniform loads of 1, l£,
or 1J tons per running foot on each line, and their strength should
accordingly be greater in proportion than that of girders which
exceed this span. If, for instance, a six-wheeled engine, 24 feet
510 WORKING STRAIN AND [CHAP. XXVIII.
long and weighing 32 tons on a twelve-feet wheel base, rest on the
centre of a bridge 32 feet in length, the strain in the flanges is
obviously greater than would occur if 42'7tons(z= 32 X 1^) were
distributed uniformly over the whole length of the bridge. A
40-foot bridge would, it is true, have the weight of only one such
engine on the centre at a time, and if the load on the middle pair
of wheels equal 1 6 tons, and that on the leading and trailing pairs
(6 feet on either side of the centre), equal 8 tons respectively, the
equivalent load concentrated at the centre of the bridge is 27*2
tons, or 54*4 tons distributed. If there were three such engines in
a row, the pressure might be slightly increased by the weight on
the leading and trailing wheels of the extreme engines, each of
which would have one pair of wheels, or 8 tons, resting on the
bridge within 2 feet of the abutments. This is equivalent to Tti
tons concentrated at the centre, or 3*2 tons distributed over the
bridge. Adding this to the 54'4 tons due to the central engine, we
have a total weight equivalent to a distributed load of 57*6 tons, or
1'44 tons per running foot. This arrangement of engines produces
the greatest strain at the centre of the flanges. Again, two such
engines might stand with their buffers in contact at the centre of the
40-foot bridge, and, though their outer ends would project beyond
each abutment, their collective wheel base would cover only 36 feet
of the bridge. This arrangement of engines produces greater
strains than the former near the ends of the flanges. Indeed, these
end strains will in some cases slightly exceed those given by the
following rules, but this is compensated for by the flanges being
generally made heavier near the ends than theory requires (437).
49O. Standard working: loads for railway bridges of
various spans. — The following tables are intended to give the
results of the preceding observations in a concise form. They are
based on six assumptions : —
1. The working load for railway bridges 400 feet in length and
upwards does not exceed | ton per running foot on each line.
2. No more locomotives than will cover 100 feet in length follow
each other without interruption; hence, the working load per
foot diminishes as the span increases from 100 feet up to 400 feet.
CHAP. XXVIII.] WORKING LOAD. 511
3. Engines may be arranged on bridges less than 100 feet long
so as to produce greater strains than would be due to the engine
load if it were of uniform density ; hence, the equivalent working
load per foot increases as the span diminishes from 100 feet
downwards.
4. Bridges less than 40 feet in span are subject to concentrated
loads from single engines, as well as to extra deflection from high-
speed trains.
5. The standard locomotive is assumed to be 24 feet long and to
have 6 wheels with a 1 2 feet base ; to have half its weight resting
on the middle wheels, and one-fourth on the leading and trailing
pairs respectively, which are supposed to be at equal distances on
either side of the middle wheels.
6. Standard Engines are assumed to weigh 24 tons, 30 tons, and
32 tons, according to their construction. This makes the standard
load 1 ton, 1 J ton, or 1^ ton per foot run of single line, according
to the weight of the engines which work it, but it is safest to take
the higher standards for the railways in Great Britain, as they are
so interlaced that engines may pass from one line to another, and it
is quite possible that we have not yet arrived at the limit of weight.
BRIDGES FROM 40 TO 400 FEET IN LENGTH.
If the standard working load (the heaviest engine) on a 100-foot
bridge weigh 1 ton per foot, while that on a 400 -foot bridge weighs
•75 tons per foot, the difference (= '25 ton per foot) must be
gradually distributed among the intervening 300 feet; in other
•25
words, the difference for each 10 feet in length = -^ = '0083 tons.
ou
The differences for the other standards may be found in a similar
way, and the following table contains the values of the working
loads corresponding to the three standards for bridges of various
lengths between 40 and 400 feet.
512
WORKING STRAIN AND [CHAP. XXVIII.
TABLE XVII.— WORKING LIVE LOADS FOB EAILWAY BRIDGES
FROM 40 TO 400 FEET IN LENGTH.
Length
of bridge
in feet.
Working load in tons per running foot of single line,
when the
standard load on a
100-foot bridge =
1 ton per foot.
when the
standard load on a
100-foot bridge =
1£ ton per foot.
when the
standard load on a
100-foot bridge =
1£ ton per foot.
40
1-05
1-35
1-45
50
1-04
1-33
1-43
60
1-03
1-32
1-41
70
1-03
1-30
1-39
80
1-02
1-28
1-37
90
1-01
1-27
1-35
100
1-00
1-25
T33
120
•98
1-22
1-30
140
•97
118
1-26
160
•95
1-15
1-22
180
•93
1-12
1-18
200
•92
1-08
1-14
250
•88
1-00
1-04
300
•83
•92
•94
350
•79
•83
•85
400
•75
•75
•75
BRIDGES UNDER 40 FEET IN LENGTH.
Bridges under 40 feet in length should be strong enough to
support a standard engine resting at the centre of the bridge.
The following is an approximate method of calculating the value
of the working load corresponding to each standard. First, find
what load concentrated at the centre of the bridge will produce a
strain in the centre of the flanges equivalent to that due to the
standard engine; twice this may be taken as the equivalent
uniformly distributed load, which again, divided by the span, gives
the working load per running foot required, as contained in the
following table : —
CHAP. XXVIII.]
WORKING LOAD.
513
TABLE XVIII.— WORKING LIVE LOADS FOE RAILWAY BRIDGES
UNDER 40 FEET IN LENGTH.
Length
of bridge
in feet.
Working load in tons per running foot of single line,
when the
standard load on
a 100-foot bridge
= 1 ton per foot.
when the
standard load on
a 100-foot bridge
= 1| ton per foot.
when the
standard load on
a 100-foot bridge
= Ik ton per foot.
12
2-0
2-5
2-67
16
1-88
2-34
2-5
20
1-68
2-1
2-24
24
1-5
1-87
2-0
28
1-35
1-68
179
32
(l-22
1-53
1-62
36
I'll
1-39
1-48
It will be prudent to adopt the highest standard for railway
bridges under 40 feet, since loads in rapid motion have a much
greater effect on these short bridges than on longer and heavier
ones, and if velocities of 50 miles an hour are anticipated, it will
be well to add from 10 to 20 per cent, to the above tabulated
working loads of bridges under 40 feet (454). Short railway
girders are so light in proportion to the passing load, that it is a
good plan to bed them on thick timber wall plates, which act
as elastic cushions and prevent the masonry of the abutments
from being shaken to pieces by the vibration of heavy trains.
491. Effect of concentrated loads upon the web. — The
weight of a heavy engine may, as already explained, be concen-
trated within a short wheel base and thus produce a great local
pressure on one or two cross-girders, which they again will transmit
to one or two points in each main girder. It might even happen
in a lattice girder that the intervals of the bracing and cross-
girders were such as to throw the load from several successive
pairs of wheels on one system of diagonals, which would thus be
liable to excessive strain. We have, it is true, some compensation
for this; first, in the rigidity of the flanges, platform, sleepers,
and rails, all of which help to distribute the weight ; and secondly,
2 L
514 WORKING STRAIN AND [CHAP. XXVIII.
in the fact that the bracing of the central parts of small girders
is for practical reasons generally made stronger than theory
requires (436), and it will generally be found sufficient to calculate
the web strains on the supposition that the passing load is of
uniform density, and equal in weight per running foot to the
working loads given above.
493. Proof load of railway bridges — English practice —
French Government rule. — No definite rule has been yet made
by the Board of Trade for the proof load of railway girder bridges,
but it is a common practice on the inspection of any important
bridge to load each line with as many engines and tenders as the
bridge will hold, and measure the corresponding deflection. This
proof is generally assumed to vary from 1 ton per running foot on
the longer bridges to 1 J ton on the shorter ones ; but when a bridge
exceeds a certain span, say 150 feet, it is obviously unreasonable to
cover it with heavy engines, and ballast wagons may be used along
with two or three engines so as to bring the proof load more in
accordance with Table XVII.
The following are the French Ministerial regulations for the
proof loads of wrought-iron railway bridges : —
a. For bridges under 20 metres each span, a dead load of 5,000
kilogrammes per running metre of each line (= T5 tons per running
foot).
b. For bridges exceeding 20 metres each span, a dead load of
4,000 kilogrammes per running metre of each line (= l"2 tons per
running foot), but in no case less than 100,000 kilogrammes.
c. In addition to the foregoing proof by dead weight, a train
composed of two engines (each weighing with its tender at least
60 tons), and wagons (each loaded with 12 tons), in sufficient
number to cover at least one span, is driven across at a speed of
from 20 to 35 kilometres (12 to 22 miles) per hour.
d. A second trial is made by driving at a speed of from 40 to
70 kilometres (25 to 43 miles) per hour a train composed of two
engines (each with its tender weighing 35 tons), and wagons loaded
as in ordinary passenger trains, in sufficient number to cover at
least one span.
CHAP. XXVIII.] WORKING LOAD. 515
e. For bridges with two lines the trains are made to traverse each
line, at first in parallel, and then in opposite directions so that the
trains may meet at the centre.
WORKING LOAD ON PUBLIC BRIDGES AND ROOFS.
493. Hen marching: in step and running- cattle are the
severest loads on suspension bridges — A crowd of people is
the greatest distributed load on a public bridge — French
and English practice — 1OO Ibs. per square foot recom-
mended for the standard working load on public bridges —
Public bridges sometimes liable to concentrated loads as
high as 18 tons on one wheel. — It is generally considered
that infantry marching in step will strain suspension bridges
far more severely than any other form of passing load. The
actual dead weight of troops on the march is said to be about
35 Ibs. per square foot, but this statical load does not represent
the true strain due to troops marching in step ; on this subject
Drewry came to the following conclusions : — " 1st, That any body
of men marching in step, say at three to three and a half miles
per hour, will strain a bridge at least as much as double their
weight at rest ; and, 2nd, that the strain they produce increases
much faster than their speed, but in what precise ratio is not
determined. In prudence, not more than one-sixth of the num-
ber of infantry that would fill a bridge, should be permitted to
march over it in step; and if they do march in step, it should
be at a slow pace. The march of cavalry, or of cattle, is not so
dangerous; first, because they take more room in proportion to
their weight; and secondly, because their .step is not simul-
taneous."* Referring to the Niagara Falls Suspension Bridge
Mr. Roebling observes — " In my opinion a heavy train, running at
a speed of twenty miles an hour, does less injury to the structure
than is caused by twenty heavy cattle under a full trot. Public
processions marching to the sound of music, or bodies of soldiers
keeping regular step, will produce a still more injurious effect."f
A crowd of people constitutes the greatest distributed load on a
* Drewry on Suspension Bridges, p. 190.
t Papers and Practical Illustrations of Public Works, p. 29. Weale, London.
516 WORKING STRAIN AND [CHAP. XXVIII.
public bridge, and 15 adults are generally estimated to weigh
1 ton, which gives an average of 149*3 fts. to each adult. Different
statements, however, have been made respecting the number of
people that can stand in a given space, and in order to test this
I packed twenty-nine Irish artisans and one boy, taken from a
forge and fitting shop, and weighing collectively 4,382 fts. or
146 fts. per individual, on a weigh-bridge 6' I" X V 10" = 29'4
square feet. In this experiment the men overhung the edges of
the weigh-bridge to a slight extent and gave too high a result,
and accordingly, on another occasion I packed 58 Irish labourers,
weighing 8,404 Ibs. or 145 fts. a man, in the empty deck-house
of a ship, 9' 6" X 6' 0" = 57 square feet ; this gives a load of
147'4 fts., or very nearly one man per square foot, and is, I believe,
a perfectly reliable experiment. Such cramming, however, could
scarcely occur in practice except in portions of a strongly excited
crowd, but I have no doubt that it does occasionally so occur.
The standard proof load for suspension bridges in France was
formerly 200 kilogrammes per square metre, = 41 fts. per square
foot.* This may be a sufficient standard for bridges with gate-
keepers at the ends to prevent overcrowding, but it is obviously
insufficient for bridges which are free to the public, especially in
the vicinity of towns, and modern French practice seems to have
raised the standard to 82 fts. per square foot.f Drewry adopted
70 fts. per square foot of platform as the greatest load that a public
bridge would sustain if covered with people.! Tredgold and Pro-
fessor Rankine estimate the weight of a dense crowd at 120 fts. per
square foot,§ and the late Mr. Brunei is said to have used 100 fts.
in his calculations for Hungerford Suspension Bridge. Mr.
Hawkshaw adopted 80 fts. per square foot for the footpaths of
Charing Cross Bridge, | and (in conjunction with Mr. W. H.
Barlow) 70 fts. for the Clifton Suspension Bridge,f where there are
* Drewry on Suspension Bridges, p. 113.
t Trans. Soc. ofEny.for 1866, p. 197.
J Drewry on Suspension Bridges, p. 189.
§ Tredgold's Carpentry, p. 169, and Rankine's Civil Engineering, p. 466.
II Proc. Inst. C. £., Vol. xxii., p. 534.
U Idem, Vol. xxvi., p. 248.
CHAP. XXVIII.] WORKING LOAD. 517
toll-gates and regulations that carriages and horses shall cross at a
walking pace. In my own practice, I adopt 100 ft>s. per square
foot as the standard working load distributed uniformly over the
whole surface of a public bridge, and 140 ft) s. per square foot for
certain portions of the structure, such, for example, as the foot-
paths of a bridge crossing a navigable river in a city, which are
liable to be severely tried by an excited crowd during a boat race,
or some similar occasion. Public bridges are also subject to con-
centrated loads at single points of quite as severe a character as
those to which railway bridges are liable ; if, for instance, a marine
boiler, a large cannon, an iron girder, a heavy forging or casting
be conveyed across a public bridge, the weight resting on a single
pair of wheels may reach or even exceed 16 tons. For example,
the crank shaft of H.M. armour-plated ship Hercules — weighing,
shaft and lorry, about 45 tons on four wheels — wras refused a
passage across Westminster iron bridge in 1866 for fear of injury
to the bridge, and had to be conveyed across Waterloo stone
bridge,* and I am informed that even much lighter weights are
habitually sent round by the stone bridge. It is necessary there-
fore to make not only the main ribs and cross-girders, but every
part of the sheeting or platform on which the road material rests,
strong enough to bear heavy local loads, which, as we have seen
in the foregoing instance, may sometimes reach nearly 12 tons on
a single wheel.
494. Weight of roofing materials and working load on
roofs — Weight of snow — Pressure of wind against roofs. —
The following table contains the weights of various roofing
materials, exclusive of framing, which is given separately.
* Engineer, Vol. xxii., p. 298, Oct., 1866.
518
WORKING STRAIN AND [CHAP. XXVIII.
TABLE XIX.— WEIGHTS OF VARIOUS ROOFING MATERIALS.
Kind of covering.
Lbs. per square
foot of roof
surface.
Copper,
Lead, -
Zinc, 13 to 16 zinc gauge,
Corrugated iron, 20 to 16 B. W. G., -
Slating, first quality, -
Do., second quality,
Rendering of Mortar 4 inch thick,
Stone slate,
Plain tiles,
Pantiles,
Thatch of straw,
Ordinary timber framing for slated roofs,
Boarding £ inch thick,
Do. IJ do.,
I inch glass, exclusive of sash, bars, or frames,
1-0
6 to 8
1-5 to 2
2-5 to 4
6 to 7
8 to 9
5 to 6
24
18
6-5
6-5
5 to 6
2-5
4-2
3-5
CHAP. XXVIII.
WORKING LOAD.
519
The following table gives the size and weight of Welsh slating,
and the number of squares (100 square feet) of roof each mil. of
1,200 slates will cover, 4 inches being allowed for lap.
TABLE XX.— WEIGHT OF WELSH SLATING.
Kind of slate.
Weight per mil. of 1,200.
1,200 will cover
squares of roof.
1st quality.
2nd quality.
in. in.
cwt.
cwt.
Princesses, 24 X 14
70
90
12
Duchesses, 24 X 12
60
81
10
Marchioness, 22 X 12
55
70
9
Countess, 20 X 10
40
53
7
Viscountess, 18 X 10
36
47
6
Ladies, 16 X 10
31
42
54
Do. 16 X 8
25
33
4
Do. 14 X 12
33
44
61
Do. 14 X 8
22
27
3£
Doubles, 13 X 10
25
31
4
Do. 13X7
17k
21
24
Do. 12 X 8
18k
22
24
Queens-ton slates are from 27 to 36 inches long and of various
breadths; 20 cwt. will cover: — 1st quality, 3 to 3J squares; 2nd
quality, 2J to 3 squares.
The following table contains particulars of some large station
roofs.*
* Proc. Inst. C.E., Vols. ix., xiv., xxvii., xxx.
520
WORKING STRAIN AND [CHAP. XXVIII.
'P9J9AOO TOJB JO ?OOJ
arenbs .led SUI.IOAO.I
jo )\i3\9M.
1 1
•uoissaadtuoD
ut qu jo B3Jy
•an jo «aiy
•UMOJO OJ
aojj pajns
'pdpuud jo q;d»a
•UMOJO 01 9!}Bld
n^M mojj paansuara
iBdpuiad jo ouisaaA
Is
•siBdpupd jo S8.i}uao
s
CO
I 2
"IBdpuud
JO UBdS JB91
1
-t» O r" 0) g
o u o^^S 9d
Iflllf
sniji
^^iD^ S "o
£«««'. 1
S i
'a
II
I|
B>§
I
CHAP. XXVIII.] WORKING LOAD. 521
When the weight of the covering per square foot, and the
distance of the principals apart, are constant for roofs of different
spans, the weights of the principals will vary nearly as the squares
of the spans (S74), and if estimated per square of ground, directly
as the spans ; acting on this rule, Mr. W. H. Barlow states that
with an ordinary truss, the distance between the principals being
30 feet, and the covering being boarding, slating and glass, the
weight of metal required in the principals can be expressed
approximately in tons per square of ground covered (100 square
feet), by dividing the span in feet by 320, which gives the fol-
lowing weights for different spans : —
Span of roof in feet. Weight of Principals in tons,
per square of ground covered.
80 - -250
120 - -375
160 - -500
200 - -625
240 - -750
The previous remarks apply more especially to large roofs whose
principals are far apart. In smaller roofs, say under 120 feet span,
it is unusual to place the principals farther apart than from 8 to
12 feet, and Mr. Henderson states the results of his experience
regarding these in the following terms.*
" If a roof was to be covered with slates, either laid upon iron
laths, or upon boarding, for ordinary spans, the principals would be
fixed 8 feet apart, from centre to centre ; whilst if the roof was to
be covered with corrugated iron, either painted or galvanized, the
principals would be 12 feet apart, from centre to centre, and purlins
of T iron would be used to carry the corrugated iron. The
distance of 8 feet apart for the principals, in the former case, was
fixed by the fact of that being the greatest limit to which it was
safe to go with the ordinary L iron laths, in one case, and 1^ inch
boarding in the other. The distance of 12 feet apart, for the
principals of roofs covered with corrugated iron was arrived at, by
that being about the limit to which purlins of T iron 4 inches deep
* Proc. Inst. C.K, Vol. xiv., p. 268.
522 WORKING STRAIN AND [CHAP. XXVIII.
could be applied, and from the fact of the same strength which
would suffice for principals, placed at distances of 8 feet apart for a
slated roof, being also sufficient when placed at 12 feet apart, if the
roof was covered with corrugated iron, on account of that covering,
with its supports, being so much lighter than a covering of slates
with their supports, that expression, * supports,' being intended to
apply only to the laths and the boarding, or purlins, as the case
might be.* The four descriptions of coverings, including every-
thing except the principals themselves, might be stated to be of the
following values per square (in the year 1855): —
"1st. A covering consisting of L iron laths and slating, including
the laths, slates, gutters, skylights, louvre standards and blades,
rain-water pipes, glass, and painting complete, at £5 10s. per square.
" 2nd. A covering consisting of 1^ inch beaded boarding, grooved
and tongued with iron tongues, including the boarding, slates,
gutters, skylights, louvre standards and blades, rain-water pipes,
glass and painting complete, at £5 17s. 6d. per square.
" 3rd. A covering consisting of T iron purlins and corrugated
sheet iron No. 18 B.W.G., painted with four coats on each side,
including the purlins, the sheet iron covering, the skylights, the
louvre standards and blades, rain-water pipes, glass, and painting
complete, at £6 12s. 6d. per square.
" 4th. A covering consisting of T iron purlins, and corrugated
galvanized sheet iron No. 18 B.W.G., including the purlins, the
sheet iron covering, the skylights, the louvre standards and blades,
rain-water pipes, glass, and painting complete, at £7 per square.
" The whole of the above calculations were based upon the case
of a roof of 60 feet span in the clear, from centre to centre of the
shoes, with one-third of the entire surface of covering glazed, and
with a raised louvre over the centre, for ventilation. For roofs of
60 feet square, such as the above covering was intended for, the
* " This is perhaps not quite correct, because, although the principals and covering are
much lighter, yet in order to make a fair comparison, the same strength ought to be
provided for wind and weather ; but the truth is, that corrugated iron covering has
generally been introduced with a view to economy, and the principals have been made,
even comparatively, somewhat lighter and not so strong as for slated roofs."
CHAP. XXVIII.] WORKING LOAD. 523
principals placed in the one case 8 feet apart, from centre to centre,
and in the other case 12 feet apart, from centre to centre, would
weigh about 18 cwt. and cost about £25. In the one case each
principal would serve for about five squares of roofing, measured on
plan, and in the other case for about seven squares and a half.
It therefore followed, that the weight per square, in the one case,
would be about 3 cwt. 2 qrs. 24 ft>., and in the other case, about
2 cwt. 1 qr. per square, whilst the cost, in the one case, would be
£5 per square, and in the other case, a little more than £3 10s.
The average weight of covering, if slating was used, would be
about 9 cwts. per square, and if galvanized iron was used it would
not exceed 5J cwts. per square. The foregoing facts, in reference
to covering, might be considered to hold good for all cases, where
a similar description of roofing was used, with principals 8 feet
apart in the one case, and 12 feet apart in the other, and, of course,
it would be understood that these dimensions were given as the
extreme limits. If the principals were fixed further apart, the
strength of the supports of the covering must be increased, and
that would augment the expense. For instance, taking a roof
where the principals were fixed 24 feet apart, from centre to
centre, the purlins would have to be increased in strength to such
an extent as would double the price per square for the purlins
themselves, but the expense of the other part of the covering would
not be altered. As already stated, for a roof of 60 feet span, the
principals themselves would weigh 18 cwts. each, and these prin-
cipals might be used either 8 feet apart or 12 feet apart, according
to the covering adopted. For roofs of greater spans the weight of
the principals would increase as the squares of the span (the load per
superficial foot and the pitch of the rafter being the same), so that
the weight of a principal, for a roof of 120 feet span, would be
72 cwts., but of course some trifling alterations in the weight might
arise from variations in the details and connexions."
Morin states that snow weighs ten times less than water, and
that it may accumulate on roofs to half a metre, or nearly 20
inches in depth, when it will weigh 10 ft>s. per square foot.* Mr.
* Resistance des MaUriaux, p. 382.
524 WORKING STRAIN, ETC. [CHAP. XXVIII.
Zerah Colburn estimates that the weight of saturated snow on
bridges in America is equal to 6 inches of water, or 30 ibs. per
square foot over the whole floor of a bridge.* The maximum
pressure of wind against bridge girders has been already given in
44O as equivalent to a horizontal pressure of 25 ibs. per square
foot of vertical surface. The slope of a roof must greatly diminish
this, and it will be sufficient to assume the maximum eifort of the
wind against a sloped or curved roof to be equivalent to a down-
ward pressure of 20 ibs. per square foot, acting separately on each
side. For ordinary roofs in the English climate it will be
sufficiently accurate if we calculate their strength on the suppo-
sition that they are liable to the following loads: —
1°. A uniform load of 40 Ibs. per square foot of ground surface,
distributed over the whole roof.
2°. A uniform load of 40 Ibs. per square foot of ground surface
distributed over the weather side of the roof, and 20 ibs. on the
other side which is away from the wind. This 40 ibs. will generally
cover the weight of slates, boarding or laths, purlins, framing or
principals, snow and wind for roofs under 100 feet in span. For
roofs exceeding 100 feet in span, we may assume that the total
load is increased by 1 ib. per additional 10 feet — thus, the load for
calculation on a 200 feet roof will be —
1°. A uniform load of 50 ibs. per square foot of ground, dis-
tributed over the whole roof.
2°. A uniform load of 50 ibs. per square foot of ground plan
distributed over one half the roof, and 30 Ibs. on the other. When
the strength of roof is calculated by the foregoing rules, the
working strain in iron tie rods may be as high as 7 tons per
square inch of net area, unless they are welded, or unless their
section is very small, in either of which cases 5 tons will be
enough.
* Proo. last. C. K, Vol. xxii., p. 546.
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 525
CHAPTER XXIX.
ESTIMATION OF GIRDER-WORK.
495. Theoretic and empirical quantities — Allowance for
rivet holes in parts in tension generally varies from one-
third to one-fifth of the net section. — Chapter X. contains
formulas for calculating the theoretic amount of material required
for braced girders with horizontal flanges, when their length, depth,
load and unit-strain are known. In order to render these formulae
of practical use in estimating girder- work, certain large additions,
derived from experience, must be added to the theoretic quantities.
If, for instance, the girder be made of wrought-iron, the formulas
are based on the supposition that the material is in one continuous
piece whose whole section is equally effective for resisting strain.
This is not the case in reality, for rivet holes in parts subject to
tension, stiffeners in those subject to compression, covers, packing,
rivet heads and waste — all require certain additions to the theoretic
quantities which experience alone can supply. When the general
design is arranged, it is easy to estimate the increased percentage
of material arising from the weakening effect of rivet holes in parts
subject to tension (476). In girder-work the allowance for rivet
holes generally varies from one-third to one-fifth of the net sectional
area according to the design ; the larger allowance of one-third may
be required for the tension diagonals of small girders ; a medium
allowance of one-fourth for the tension diagonals of large girders
and the tension flanges of small ones ; and an allowance of one-
fifth for the tension flanges of large girders.
496. Allowance for stiffeners in parts in compression
varies according' to their sectional area— Large compression
flanges seldom require any allowance for stiffening — Com-
pression bracing requires large percentages. — The additional
percentage of material required to withstand flexure or buckling in
526 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
parts subject to compression is not so easily estimated. It will
generally be found to diminish in proportion as the area of the
part increases, for when the area is considerable, a stiff form of
cross section may be given with little or no extra material. This
is frequently the case with the compression flange, especially in
large girders. Long compression braces, however, require much
extra stiffening and the amount of this varies within considerable
limits. In the Boyne Lattice Bridge the extra material required
to stiffen the various compression braces varied from 60 to 128
per cent, of the theoretic amount (calculated at 4 tons per square
inch) which would have been required to resist crushing merely, if
flexure had been left out of consideration, the higher percentages
being required in the central diagonals whose scantlings were
small, since they had to sustain but slight strains. In bridges
above 250 feet span, with two main girders and a double line of
railway, a sufficiently close approximation will generally be made
if we assume the extra quantity of material to resist flexure in the
compression bracing equal to as much again as the theoretic
quantity calculated by the formulae, but when the bridge is
designed for a single line of railway this percentage is insufficient ;
perhaps, in this case twice the theoretic quantity would generally
be a safe allowance, as the extra quantity required for stiffening
the compression bracing of a single-line bridge is not widely
different from that required for the double line.
497. Allowance for covers in flanges varies from 13 to
15 per cent, of the gross section — Estimating? girder- work
a tentative process. — The allowance for covers will also vary
much with the design, long flange-plates requiring fewer covers
than short ones (463 to 465). In the piled flanges of the Boyne
lattice girders, the covers formed about 12 per cent., or nearly
l-8th, of the plates and angle iron. In the cellular flanges of the
Conway tubular bridge, the covers of the compression flange formed
5 per cent, of the plates and angle iron, and those of the tension
flange 28 per cent.; adding both flanges together, the covers
formed about 15 per cent, of the plates and angle iron.*
* Clark on the Tubular Bridges, p. 586.
CHAP. XXIX.] ESTIMATION OP GIRDER- WORK. 527
The process of estimating the quantities in any proposed bridge
is tentative and depends upon experience, for it is necessary to
assume a weight for the permanent bridge-load, and then make the
calculations with the various practical allowances above mentioned.
Now, the resulting weight from this calculation may not agree
with that which has been assumed. In this case the first estimate
gives an approximation for a second calculation, and even a third
may be necessary where great nicety is required. The following
examples will illustrate this method of forming estimates: —
EXAMPLE 1.
498. Double-line lattice bridge 867 feet IOIIR.— I shall
select for the first example a wrought-iron lattice bridge for a
double line of railroad of the same length, depth and width as the
central span of the Boyne Lattice Bridge, the weight of which is
given in detail in the appendix. As the Boyne Bridge is a con-
tinuous girder in three spans, its central span, of course, requires
less material than a bridge of equal dimensions which has not the
same advantage of continuity.
Let I =: 267 feet = the length measured from centre to
centre of end pillars (55),
d = ^~ = 22-25 feet = the depth,
i'Z
9 = 45° = the angle of the bracing, whence
sec0. cosecfl = 2 (878),
/ = 5 tons tensile inch-strain of net section,
/' = 4 tons compressive inch-strain of gross section,
and let the width of platform between the main girders equal 24
feet as in the Boyne Bridge. Let the maximum passing load equal
1 ton per running foot on each line, = 534 tons when covering
both lines together, and let us assume that the permanent bridge-
load equals 490 tons, which gives the total load supported by the
girders as follows : —
W = 534 + 490 - 1024 tons.
With this load uniformly distributed, the theoretic quantities of
material (eqs. 206 and 208) are as follows, 4'6 cubic feet of wrought-
iron being assumed equal to 1 ton.
528 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
Tons.
Tension bracing = * r = 94*93 cubic feet,* - 20*64
Compression bracing (= fths of the tension bracing), 25*80
Tension flange I = ^ X tension bracing, eq. 208 J , 82*56
Compression flange (= |ths of the tension flange), - 103*20
Total theoretic weight, - 232 -2O
The true quantities are obtained from the foregoing by adding
the percentages derived from experience, as follows: —
Tons. Tons.
Theoretic tension bracing, - - 20 '64
Rivet holes, say Jth of net section, - 5*16
Theoretic compression bracing, - - 25*80 .
o l*bU
'80 )
'80 )
Add as much again for stiffening, - 25
Theoretic tension flange, - - 82'56 j
Rivet holes, say }th of net section, - 16*51 )
Covers of tension flange, say Jth of flange, - 12*38
Theoretic compression flange, - 103*20
Covers of compression flange, say Jth of flange, - 12*90
304*95
Rivet heads, packings, waste (437, 436), say 10 per cent., 30*49
"Weight of iron in the main girders, - 335-44
35 cross-girders, 7 feet 5 inches apart, each
1-32 tons (see Appendix, "Boyne Viaduct"), 46*20 )
Cross-bracing, do. do. 17-66 j
Weight of iron between end pillars, - 399-3O
6-inch planking of platform 24 feet wide,
= 3,204 cubic feet, © 50 cubic feet per ton, 64*08
Longitudinal timbers under rails, 12 inches
X 6 inches = 534 cubic feet, - 10'68
Barlow rails, 356 yards, @ 100 ibs. per yard, - 15'89
90*65
Permanent bridge-load between end pillars, 489-95
* NOTE. — The theoretic quantity of material in the tension bracing is only one-half
that given by eq. 206, which represents the quantity for the whole web.
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 529
being O05 tons less than that assumed. In order to obtain the
total weight of wrought-iron in the bridge, we must add the weight
of the 4 end pillars with their 2 lower cross-girders and 2 top
cross-girders and gussets (443), say 30 tons in all, to the weight
of iron between the end pillars ; this makes the total weight of
wrought-iron in the structure = 399' 30 + 30 = 429 -3O tons.
In this example we find that 335*44 tons of iron are required in
the main girders to support themselves and an additional load of
688*56 tons uniformly distributed. Consequently, each ton of
additional load uniformly distributed requires /»OO.K/. = 0*487 tons
of iron in the main girders, and if an additional load of 100 tons of
ballast were spread over the platform, we should add 48' 7 tons of
iron to the main girders to support the weight of this ballast
without the unit-strains being increased.
499. Permanent strains — Strains front train-load —
Economy due to continuity. — The permanent inch-strains, that
is, the inch-strains due to the permanent bridge-load of 489*95 tons,
are 2*39 tons tension and 1*91 tons compression; those due to the
main girders alone, weighing 335'44 tons, are 1/64 tons tension and
1*31 tons compression, and those due to a train-load of one ton per
running foot on each line uniformly distributed are 2*61 tons tension
and 2 '09 tons compression. The actual weight of iron in the main
girders of the long central span of the Boyne Bridge = 297*41
tons; the difference between this and our example = 335*44 —
297*41 = 38*03 tons, which represents the saving effected in the
central span of the Boyne Bridge by its connexion over the piers
with the side spans. As, however, this connexion causes a certain
loss of material in the shorter side spans, the total amount of
economy produced by continuity is probably less than that above
stated (858, 481).
EXAMPLE 2.
500. Single-line lattice bridge 4OO feet long.— A wrought-
iron lattice bridge for a single line of railway, 400 feet long from
2 M
530 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
centre to centre of end pillars, 25 feet deep and 14 feet wide
between main girders, with the bracing at an angle of 45°. Using
the same symbols as before, we have,
/ = 400 feet,
d = 4 = 25 feet,
lo
0 = 45°,
/ = 5 tons tensile inch-strain of net section,
f •=. 4 tons compressive inch-strain of gross section.
Let the maximum train load equal f ton per running foot (49O),
and assuming that the permanent bridge-load equals 1300 tons, we
have the total distributed load,
W = 300 + 1300 = 1600 tons.
The theoretic quantities with their empirical percentages are as
follows (eqs. 206, 208).
Tons. Tons.
1600 X 400
Theoretic tension bracing =
4 X 0 X 144:
IP. fppf npr tnn . 4.8-2 1
60-4
4 X 5 X 144
222-2 cubic feet, @ 4-6 cubic feet per ton, - 48'3
Rivet holes, say one-fourth of net section, - 121
Theoretic compression bracing (= fths of the
theoretic tension bracing), - 60'4
Add twice as much for stiffening - - 120*8
1600x400x16
„,, . n
Theoretic tension flange =
12x5x144
hATl 9.^7'fi ^
309-1
1,185-18 cubic feet, @ 4-6 cubic feet per ton, 257-6 )
Rivet holes, say £th of net section, - 51"5 )
Covers, say Jth of flange, - 38'6
Theoretic compression flange (= |ths of the
theoretic tension flange), - 32 2 '0
Covers, say ^th of flange, - 40'5
951-8
Rivet heads, packings, waste, say 10 per cent., - 95'2
Iron In main girders, 1O47 O
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 531
Cross-girders = 400 X 0*18 tons (445),
Cross-bracing, say,
Weight of iron between end pillars, - 115 4-0
Platform, rails, sleepers and ballast = 400 X 0'36
tons (445), 144-0
Permanent bridge-load between end pillars, - 1298 -O
being 2 tons less than that assumed. If the 4 end pillars and
cross-girders over the abutments weigh 40 tons, the total weight
of wrought-iron in the bridge = 1,154 + 40 = 1194 tons.
From this estimate it appears that 1047 tons of iron are required
in the main girders to support themselves and an additional load of
553 tons uniformly distributed; consequently, each ton of additional
load uniformly distributed requires for its support -^=^- = 1*89 tons
OOo
in the main girders. If, for instance, the maximum train-load be
1 ton in place of f ton per running foot, this uniformly distributed
load will amount to 400 tons in place of 300 tons, that is, 100 tons
more than has been assumed, and this will require 100 X 1*89 =
189 tons extra iron in the main girders for its support, and the
increased total load on the bridge will be 289 tons, or nearly three
times the useful addition. The iron in the flanges, including the
10 per cent, for rivet heads, packings and waste, weighs 781-2 tons ;
the iron in the web, also including the percentage for rivet heads,
&c., weighs 265'8 tons ; consequently, each ton of useful load uni-
781*2
formly distributed requires ,,» =1*41 tons of iron in the flanges,
and -.. , = 0'48 tons in the webs. The inch-strains due to the
553
permanent bridge-load of 1,300 tons between the end pillars are 4'06
tons tension and 3*25 tons compression, while those due to a uni-
formly distributed train-load of f ton per running foot are 0'94
tons tension and 0'75 tons compression.-
532 ESTIMATION OF GIRDER- WORK. [CHAP. XXIX.
EXAMPLE 3.
5O1. Single-line lattice bridge 4OO feet long, as in Ex. 2,
bat with higher unit-strains. — A wrought-iron lattice bridge
of the same dimensions as the last, but in place of the inch-strains
being 5 and 4 tons let
/ = 6 tons tensile inch-strain of net section,
f = 5 tons compressive inch-strain of gross section.
Assuming that the permanent bridge-load = 960 tons, we have the
total distributed load,
W = 300 + 960 = 1,260 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
1260 X 400
Theoretic tension bracing = ^ ^- — ^TJ =
145-83 cubic feet, @ 4-6 feet per ton, - 31'7 j
Rivet holes, say £ of net section, - - 7' 9 )
Theoretic compression bracing (= fths of the
theoretic tension bracing), - 38*0 )
Add three times as much for stiffening,* - 1 14*0 )
1260x400x16
Theoretic tension flange = — ^ — ~ — . . =
777-8 cubic feet, @ 4'6 cubic feet per ton, 1691 j
Rivet holes, say Jth of net section, - 33' 8 )
Covers, say Jth of flange, - 2 5 '4
Theoretic compression flange (= fths of the
theoretic tension flange), - 202*9
Covers, say Jtli of flange, - 25*4
648-2
Rivet heads, packings, waste, say 10 per cent., - 64*8
Iron in main girders, - 713 "O
* In this example I allow three times, in place of twice the theoretic amount,
because the extra quantity of material required for stiffening the compression bracing
is but slightly affected by the adoption of higher unit-strains.
CHAP. XXIX.] ESTIMATION OF GIRDER- WORK. 533
Cross-girders, as in last example, -
Cross-bracing, say,
Weight of iron between end pillars, -
Platform, rails, sleepers and ballast, as in last,
Permanent bridge-load between end pillars, - 959 -O
being 1 ton less than that assumed. If the four end pillars and
cross-girders over abutments weigh 35 tons, the total weight of
wrought-iron in the bridge = 815 + 35 = 85O tons.
The main girders in this example, weighing 713 tons, support
themselves and an additional load of 547 tons uniformly distributed.
Consequently, each ton of useful load uniformly distributed re-
713
quires for its support ^-y = 1*304 tons in the main girders. The
inch-strains due to the permanent bridge-load of 960 tons between
6 x 960 , 5 X 960
end pillars = 9 „ = 4'57 tons tension, and — = 3'81
tons compression, while those produced by a uniformly distributed
train-load of f ton per running foot are T43 tons tension and 1-19
tons compression.
5O2. Great economy from high unit-strains in long:
girders — Steel plates. — Comparing this with the preceding
example, we find a saving in the main girders equal to 1,047 —
713 = 334 tons, or nearly 47 per cent, of the lighter bridge.
The saving may even be greater than this, since I have neglected
any reduction in the weight of the cross-girders due to higher
unit-strains. These two examples illustrate the great economy
produced in large girders by adopting high unit-strains. In
place of the weights of the main girders being in the inverse
ratio of the unit-strains, as might be supposed at first sight, we
find that they vary in a much higher ratio, at least in large
bridges where the main girders form a large proportion of the total
load (62). Economy from the adoption of high unit-strains will be
chiefly marked in the flanges and tension bracing, owing to the
necessity of having a certain amount of material to stiffen the
compression bracing, no matter how high the ultimate crushing
534 ESTIMATION OF GIRDER- WORK. [CHAP. XXIX.
strength of the material may be. Even a better method of riveting
or jointing may produce a very important saving in a large girder,
by not requiring so may holes in the tension plates, or such large
covers at the joints. Mild steel plates, which are now manufactured
at a cost not much exceeding that of the better kinds of iron, but
about once and a half as strong as the latter, will, doubtless, enable
the engineer to construct girders over spans which have been
hitherto impracticable. The tensile strength of steel is known ; it
is to be hoped that satisfactory experiments will be made to deter-
mine its stiffness, that is, its strength to resist flexure when in the
form of long pillars — an essential element in its application to
girder-work (483).
5O3. Suspension principle applicable to larger spans than
girders. — We are now in a position to understand how suspension
bridges can be built over spans far exceeding those to which rigid
girders are applicable, for not only are there no compressive strains
in the webs of suspension bridges, but the compression flange of
the girder is superseded by land chains, and the structure between
the piers is thus relieved of the weight of one flange. Moreover,
the material used is generally of such an excellent quality that it is
capable of sustaining with safety a higher unit-strain than ordinary
plate-iron (476), and there is also a less percentage of material
required for the joints of suspension chains, as pins passing through
eyes in the ends of long bar links supersede the ever-recurring
rivets of plated work and the whole intermediate shank of the link
is thus available for tension without waste.
EXAMPLE 4.
50-1. Single-line lattice bridge 4OO feet long, with in-
creased depth. — The preceding example illustrates the great
economy effected in large girders by the adoption of high unit-
strains. Let us now examine the result of a slight increase of
depth, all the other dimensions and the unit-strains remaining the
same as in Example 2, but in place of the depth being 25 feet, i.e.,
one-sixteenth of the length, let
d = ~ = 26-67 feet,
lo
CHAP. XXIX.] ESTIMATION OF GIRDER- WORK. 535
Assuming, the permanent bridge-load to be 1,190 tons, we have the
total distributed load,
W = 300 + 1190 =1490 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
™ .. • v • 1490X400
Theoretic tension bracing = -A — ^ — =-T-J =
4: X 0 X -1-4:4:
207 cubic feet, @ 4- 6 feet per ton, - - 45 '0 j «g>2
Rivet holes, say £th of net section, - 11*2 )
Theoretic compression bracing (= fths of the
theoretic tension bracing), - - 56*2 j _ .
Add for stiffening the same as in Ex. 2,* - 120'8 )
„,, ,. 1490x400x15
1 neoretic tension nange = — ^ — * — =rn — =
12x5x144
1034-7 cubic feet, @ 4-6 feet per ton, - 225-0 j
Rivet holes, say |-th of net section,- - 45'0 )
Covers, say Jth of flange, - 33'7
Theoretic compression flange ( = fths of the
theoretic tension flange), - - . - 281 '2
Covers, say Jth of flange, - 35*1
852-2
Rivet heads, packings, waste, say 10 per cent., 85'2
Iron in main girders, - - 937-4
Cross-girders, as in Ex. 2, - 72-0
Cross-bracing, do., 35'0
Weight of iron between end pillars, - - 1O44-4
Platform rails, sleepers, and ballast, as in Ex. 2, - 144*0
Permanent bridge-load between end pillars, • 1188-4
* In place of adding, as usual, twice the theoretic amount for stiffening, viz.,
2X56-2 = 112-4 tons, I have assumed that this example requires the same quantity
as Ex. 2, for though the load in this example is less, yet the length of the compression
bracing is greater than in Ex. 2, and the assumption in the text, therefore, will
be probably not far from the truth.
536 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
being 1-6 tons less than that assumed. If the four end pillars and
cross-girders over the abutments weigh 40 tons, the total weight
of wrought-iron in the bridge = 1044-4+40 = 1O84-4 tons.
The main girders in this example, weighing 937*4 tons, support
themselves and 552'6 tons uniformly distributed. Consequently,
each ton of useful load uniformly distributed requires for its support
=r 1'7 tons nearly in the main girders. The inch-strains due
OO^'D
to the permanent bridge-load of 1190 tons between end pillars
5x1190 ,4x1190
= . = 4 tons tension, and .. , = 3'2 tons compression.
The inch-strains due to the main girders, weighing 937'4 tons,
5x937-4 ,4x937-4
= — 1490— = 3'14 tons tension, and — = 2'52 tons com-
pression. The inch-strains due to a train-load of f tons per
5 x 300
running foot over the whole bridge = ^ , =1-0 ton tension,
and 1 - = 0*8 tons compression.
505. Weights of large girders do not vary inversely as
their depth. — Comparing this with Ex. 2, the saving of material
in the main girders = 1047 — 937*4 = 109'6 tons. We find
therefore that the weights of the girders in these two examples
are inversely as the 1'7 power of the depths, but this particular
proportion is accidental (8*4).
EXAMPLE 5.
506. Single-line lattice bridge 48O feet long. — A wrought-
iron lattice bridge for a single line of railway, 480 feet long from
centre to centre of end pillars, 30 feet deep, and 14 feet wide
between main girders. Using the same symbols as in Ex. 1, we
have,
I = the length = 480 feet,
d = the depth = ^ = 30 feet,
6 = 45°= the angle the diagonals make with a vertical line,
/ = 5 tons tensile inch-strain of net section,
/' = 4 tons compressive inch-strain of gross section.
CHAP. XXIX.] ESTIMATION OP GIRDER-WORK. 537
Let the maximum passing load = J ton per running foot (489),
and assuming that the permanent bridge-load weighs 2760 tons,
we have the total distributed load,
W = 360 + 2760 = 3120 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
Theoretic tension bracing =
cubic feet, @ 4-6 feet per ton, - - 113-0 ]
Rivet holes, say £th of net section, - 28'3 j
Theoretic compression bracing ( = |ths of the
theoretic tension bracing), - - 141*3 j
Add twice as much for stiffening,* - 282*6 j
3120x480x16
Theoretic tension flange —
12x5x144
. fiH9-Q •>
723-5
2773-3 cubic feet, @ 4'6 feet per ton, - - 602*9 )
Rivet holes, say Jth of net section, - 120'6 j
Covers, say Jth of flange, - 90*4
Theoretic compression flange ( = f ths of the
theoretic tension flange), - - 753'6
Covers, say Jth of flange, - 94'2
2226-9
Rivet heads, packings, waste, say 10 per cent., 222- 7
Iron in main girders, - - 2449-6
Cross-girders =480x0-18 tons (445), - 86'4
Cross-bracing,t - 50*4
Weight of iron between end pillars, - - 2586-4
* This allowance for stiffening is probably excessive.
f The quantity of cross-bracing is proportional to VW (eq. 206), where W represents
the pressure of the wind against the side of the bridge ; if this pressure be assumed
proportional to the product of length and depth, which is the case in plate girders,
the quantity of cross-bracing in similar girders will vary as I3. As, however, the side
surface of similar lattice girders does not in general increase so rapidly as Z2, and as
also the empirical percentages are somewhat less in large than in small bridges, it will
probably be nearer the truth to assume that the quantity of cross-bracing is proportional
to the square of the length. If, therefore, a bridge 400 feet long (Ex. 2,) requires 35
tons, one 480 feet long will require 35 X || = 50'4 tons.
538 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
Weight of iron between end pillars, - 2586-4
Platform, rails, sleepers, and ballast, = 480 X
0-36 tons (445), 172-8
Permanent bridge-load between end pillars, 2759*2
being 0'8 less than that assumed. If the weight of the four pillars
and cross-girders at the ends be assumed equal to 70 tons, the
total weight of wrought-iron in the bridge will equal 70 +
2586-4 = 2656-4 tons.
The inch-strains due to the permanent bridge-load of 2760
tons between end pillars are — .... , — = 4*42 tons tension, and
Q = 3'54 tons compression. The inch-strains due to the
OL'ZO
main girders, weighing 2449*9 tons, are - — *>-i9r\ ~ = 3*92 tons
4x2449*6
tension, and - = 3' 14 tons compression. The inch-
oL'20
strains due to a train-load of f ton per running foot over the whole
5x360 ,4x360
bridge = ~ „ = 0'576 tons tension, and ^ Q = 0'46 tons
compression.
5O7. Waste of material in defective designs. — In this
example, 2449'7 tons of iron in the main girders support themselves
and an additional load of 670'4 tons uniformly distributed over the
bridge. Consequently, each ton of useful load requires for its
2449-6
support = 3' 6 5 tons of iron in the main girders. This
illustrates the great waste of material produced by defective designs
for large bridges, since every ton of iron uselessly added involves
the necessity of adding 3*65 other tons for its support, making
collectively upwards of 4J tons which might be saved were the
design skilfully planned.
EXAMPLE 6.
5O§. Single-line lattice bridge 48O feet long-, as in Ex. 5,
but with higher unit-strains. — A wrought-iron lattice bridge of
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK 539
the same dimensions as the last, but in place of the inch-strains
being 5 and 4 tons respectively,
Let / — 6 tons tensile inch-strain of net section,
/' = 5 tons compressive inch-strain of gross section.
Assuming that the permanent bridge-load equals 1710 tons, we
have the total distributed load,
W = 360 + 1710 = 2070 tons.
The quantities are as follows (eq. 206, 208).
Tons. Tons.
„,, , • 2070x480
Theoretic tension bracing = - — - — rTT =
4 x b X 144
287-5 cubic feet, @ 4-6 feet per ton, - - 62-5 j „«,
Rivet holes, say ^th of net section, - 15*6 )
Theoretic compression bracing ( = Jths of the
theoretic tension bracing), - - 75'0 | aor\.r\
Add three times as much for stiffening,* - 225"0 j
. fl 2070x480x16
Theoretic tension flange = — y^ — g — , . * —
1533-3 cubic feet, @ 4-6 feet per ton, - 333-3 j
Rivet holes, say J-th of net section, - 66-7 j
Covers, say Jth of flange, - 50'0
Theoretic compression flange ( = f- ths of the
theoretic tension flange), - 400*0
Covers, say Jth of flange, - 50'0
1278-1
Rivet heads, packings, waste, say 10 per cent., 127-8
Iron in main girders,
Cross-girders, as in last example, -
Cross-bracing, say,
Weight of iron between end pillars, 1537-3
Platform, rails, sleepers and ballast, as in last, 172*8
Permanent bridge-load between end pillars, - 171O-1
being O'l ton greater than that assumed. If the four pillars and
* See note to Ex. 3, p. 532.
540 ESTIMATION OF GIRDER- WORK. [CHAP. XXIX.
cross-girders at the ends weigh 50 tons, the total weight of
wrought-iron in the bridge will equal 50 + 1537'3 = 1587-3
tons.
In this example, the main girders, weighing 1405-9 tons, support
themselves and an additional load of 664*1 tons uniformly dis-
tributed. Consequently, each ton of useful load requires for its
support - = 2-117 tons in the main girders. The inch-
strains due to the permanent bridge-load of 1710 tons between
6 x 1710 . , 5 X 1710
end pillars = 2070 = tension, and
tons compression. The inch-strains due to the main girders,
weighing 1405'9 tons, are mc\ ------- = ^'^ ^ons tensi°n>
5 X 1405-9 . r™ . , ,
= 3'4 tons compression. The inch-strains due to a
uniformly distributed train-load of f ton per running foot over the
6x360 , 5x360 n Q_
whole bridge are 9^70 — *"0^ tons tension, and -o'OTO =
tons compression.
509. Great economy from high unil-sf rains in large
girders. — The economy effected in large girders by the adoption
of high unit-strains is very marked in this example. Compared
with the preceding example, the saving amounts to 2656'4 —
1587-3 = 1069-1 tons, or nearly 68 per cent, of the lighter bridge
(508, 67).
EXAMPLE 7.
510. Single-line lattice bridge 48O feet long, as in Ex. 5,
but with increased depth. — The previous example illustrates the
great economy in large bridges due to the use of a material capable
of sustaining high unit-strains with safety. We shall now examine
the effect of a slight increase of depth, all the other dimensions and
the unit-strains remaining the same as in Ex. 5. In place of the
depth being 30 feet, or Jffth of the length, let
d - l- '= 32 feet.
15
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 541
Assuming the permanent bridge-load to be 2435 tons, we have the
total distributed load,
W = 360 + 2435 = 2795 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
Theoretic tension bracing = j^-5g X 4,9,° -
4 X 5 X 144
465-8 cubic feet, @ 4-6 feet per ton, - - 101*3 j
Eivet holes, say Jth of net section, - - 25'3 j 126'6
Theoretic compression bracing ( = |ths of the
theoretic tension bracing), - 126 '6 )
Add for stiffening the same as in Ex. 5,* - 282'6 j 4°9'2
Theoretic tension flange = 480x15
12 x 5 X 154
2329 cubic feet, © 4-6 feet per ton, - - 506'3 |
Eivet holes, say }th of net section, - - 101'3 j 6°7'6
Covers, say Jth of flange, - - 76-0
Theoretic compression flange ( = |ths of the
theoretic tension flange), - 6 32 '9
Covers, say Jth of flange, - 79' 1
1931-4
Rivet heads, packings, waste, say 10 per cent., 193-1
Iron in main girders, - - 2124*5
Cross-girders, as in Ex. 5, - 86*4
Cross-bracing, do., 50*4
"Weight of iron between end pillars, - 2261-3
Platform, rails, sleepers and ballast, as in Ex. 5, - 172' 8
Permanent bridge-load between end pillars, - 2434-1
being 0*9 ton less than that assumed. If the four pillars and
cross-girders at the ends weigh 70 tons, the total weight of
wrought-iron in the bridge will equal 70 + 2261-3 = 2331*3 tons.
The main girders, weighing 2124*5 tons, support themselves and
* See note to Ex. 4, p. 535.
542 ESTIMATION OF GIRDER-WORK. [ CHAP. XXIX.
670'5 tons uniformly distributed. Consequently, each ton of
2124'5
useful load uniformly distributed requires for its support „_,. - =r
3*17 tons in the main girders. The inch-strains due to the per-
5 X 2434
manent bridge-load of 2434 tons between end pillars =
4 x 2434
4-35 tons tension, and — caz — = 3'48 tons compression. The
inch-strains due to the main girders, weighing 2124'5 tons n
5 X 2124-5 , 4 x 2124-5
- = 3-8 tons tension, and 0r,nK = 3'04 tons com-
pression. The inch-strains due to a train-load of f ton per running
foot over the whole bridge = — - — = 0*64 tons tension, and
£ i Jo
4 x 360
TOK~~ = V9**- tons compression.
511. Weights of large girders do not vary inversely as
their depth. — Comparing this with Ex. 5, the saving effected in
the main girders by a slight increase of depth = 2449*6 — 2124'5
— 325*1 tons. We find also that the weights of the girders in
these two examples are inversely as the 2*2 power of their depths
(505).
EXAMPLE 8.
519. Single-line lattice bridge 6OO feet long. — A wrought-
iron bridge for a single line of railway, 600 feet long between
centres of end pillars, 37-5 feet deep, and 14 feet wide between
main girders. Using the same symbols as in Ex. 1, we have,
/ = 600 feet,
d = ^ = 37-5 feet,
lo
e = 45°,
/ = 5 tons tensile inch-strain of net section,
f' = 4: tons compressive inch-strain of gross section.
Let the maximum passing load = | ton per running foot, and
assuming that the permanent bridge-load weighs 9100 tons, we
have the total distributed load,
W = 450 + 9100 = 9550 tons.
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 543
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
Theoretic tension bracing =
o
1989 6 cubic feet, @ 4-6 feet per ton, - 432-5 |
Rivet holes, say £th of net section, - - 108*1 }
Theoretic compression bracing ( = |ths of the
theoretic tension bracing), - 540*6 )
Add as much again for stiffening,* - 540'6 )
Theoretic tension flange = ^0x600x16
12x5x144
10,611 cubic feet, @ 4-6 feet per ton, 2306-7 )
Rivet holes, say Jth of net section. - 461-3 )
Covers, say Jth of flange, - ... 346-0
Theoretic compression flange ( = |ths of the
theoretic tension flange), - ... 2883*4
Covers, say Jth of flange, ..... 360'8
7980-0
Rivet heads, packings, waste, say 10 per cent., - 798'0
Iron in main girders, - - 877 O-O
Cross-girders = 600x018 tons (445), - - 108'0
Weight of iron between end pillars, - 8886-O
Platform, rails, sleepers, and ballast = 600 X
0-36 tons (445), 216'0
Permanent bridge-load between end pillars, - 91O2-O
being 2 tons in excess of that assumed. No allowance has been
made for cross-bracing, for the sectional area of the flanges is so
great that they would probably extend over the whole space
between the main girders so as to form a tubular bridge, and
thus supersede the usual cross-bracing formed of cross-girders
* The quantity of material in the web is so large that it can be thrown into a form
suitable for resisting flexure without much extra stiffening ; I have therefore added
only half the percentage for stiffening that was adopted in most of the preceding cases.
544 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
and diagonal tension bars. If the four' pillars and cross-girders
at the ends be assumed equal to 200 tons, the total weight of
wrought-iron in the bridge will equal 200 + 8886 = 9O86 tons.
In this example, 8778 tons of iron in the main girders support
themselves and an additional load of 772 tons uniformly dis-
tributed over the bridge. Consequently, each ton of useful load
ft T7R
requires for its support -=-,=^- = 1T37 tons of iron in the main
/ (4
girders. The inch-strains due to the permanent bridge-load of
9100 tons between end pillars are — = 4*76 tons tension,
4x9100
and - . A = 3'81 tons compression. The inch-strains due to
5 X 8778
the main girders, weighing 8778 tons, are — — = 4*6 tons
tension, and —7^7: — = 3*67 tons compression. The inch-strains
due to a train-load of J ton per running foot over the whole bridge
5x450 AOQK . 4x450
= 0-235 tons tension, and -KK— = 0188 tons com-
pression.
EXAMPLE 9.
513. Single-line lattice bridge 6OO feet long, as in Ex. 8,
bat with higher unit-strains. — A wrought-iron bridge of the
same dimensions as the last, but in place of the inch-strains being
5 and 4 tons,
Let / = 6 tons tensile inch-strain of net section,
/ = 5 tons compressive inch-strain of gross section.
Assuming that the permanent bridge-load = 3800 tons, we have
the total distributed load,
W - 450 + 3800 = 4250 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
Theoretic tension bracing = =
4x6x144
737-8 cubic feet, @ 4-6 feet per ton, - - 160-4
Rivet holes, say £th of net section, - -401
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 545
Tons. Tons.
Theoretic compression bracing ( = f- ths of the
theoretic tension bracing), - 192-5 ]
Add twice as much for stiffening, - 385*0 j •*•'"*
Theoretic tension flange = 4215?XR60°1?/6 =
12x6x144
3935-2 cubic feet, @ 4'6 feet per ton, - 855'5 |
Rivet holes, say Jth of net section, - - 171*1 )
Covers, say Jth of flange, - 128*3
Theoretic compression flange ( = f ths of the
theoretic tension flange), - - 1026'6
Covers, say Jth of flange, - - - 128*3
3087-8
Rivet heads, packings, waste, say 10 per cent., - 308*8
Iron in main girders, 3396-6
Cross-girders, as in last example, - 108'0
Cross-bracing,* - 78*8
Weight of iron between end pillars, 3583-4
Platform, rails, sleepers and ballast, as in last
example, - 216'0
Permanent bridge-load between end pillars, - 3799 -4
being 0*6 tons less than that assumed. If the four pillars and
cross-girders at the ends weigh 100 tons, the total weight of
wrought-iron in the bridge will equal 100 X 3583'4 = 3683-4 tons.
In this example the main girders, weighing 3396'6 tons, support
themselves and an additional load of 853*4 tons uniformly
distributed. Consequently, each ton of useful load requires for its
f\ O A f* f*
support _ = 3'98 tons in the main girders. The inch-strains
*
due to the permanent bridge-load of 3800 tons between end pillars
= 5-36 tons tension, and ^|^ =4-47 tons com-
* See note to Example 5, p. 537.
2 N
546 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
pression. The inch-strains due to the main girders, weighing
3396-6 tons, are ~ = 4-8 tons tension, and
4-0 tons compression. The inch-strains due to a uniformly dis-
tributed train-load of £ ton per running foot over the whole bridge
°*64 tons tension» and = °'53 tons
sion.
514. Great economy from high unit-strains in very large
girders.— The economy due to the adoption of high unit-strains
in girders of great size, whose permanent weight forms by far the
larger portion of the total load, is very conspicuous in this example.
Compared with the preceding example, the saving amounts to
9086 — 3683-4 = 5402-6 tons, or nearly 147 per cent, of the
lighter bridge (5OS, 5O9).
EXAMPLE 10.
515. Single-line lattice bridge. 6OO feet long, as in K\.
8, but with increased depth. — Let us now examine the effect
of a slightly increased proportion of depth to span. In Ex. 8, the
depth is -j^th of the length ; let the proportion now be ^th, and
retaining all the other dimensions and unit-strains as before, we have,
I = 600 feet,
d = 1= = 40 feet,
lo
e = 45°,
/ = 5 tons tensile inch-strain of net section,
/ = 4 tons compressive inch-strain of gross section.'
Let the passing load equal J ton per running foot, and assuming
the permanent bridge-load to equal 6800 tons, we have the total
distributed load,
W = 450 + 6800 = 7250 tons.
The quantities are as follows (eqs. 206, 208).
Tons. Tons.
r™ , . 7250x600
Theoretic tension bracing = T — = — =-=-7 =
4x5x144
1510-4 cubic feet, @ 4-6 feet per ton, - 328' l
41U'D
Rivet holes, say Jth of net section, - - 82
;
CHAP. XXIX.] ESTIMATION OP GIRDER-WORK. * 547
Tons. Tons.
Theoretic compression bracing ( = f ths of the
theoretic tension bracing), . 410-5 )
Add for stiffening the same as in Ex. 8,* - 540' 6 f
Theoretic tendon flange = ™g^ =
7552-1 cubic feet, @ 4'6 feet per ton, 1641-8 )
Rivet holes, say }th of net section, - 328'4 )
Covers, say Jth of the flange, - - - 246*3
Theoretic compression flange ( = fths of the
theoretic tension flange), - - 2052-2
Covers, say Jth of the flange, - 256 5
5886-8
Rivet heads, packings, waste, say 10 per cent., - 588'7
Iron in main girders, - 6475-5
Cross-girders, as in Ex. 8, 108*0
Weight of iron between end pillars, - 6583-5
Platform, rails, sleepers and ballast, as in Ex. 8, •- 216*0
Permanent bridge-load between end pillars, - 6799-5
being 0*5 tons less than that assumed. If the four pillars and
cross-girders at the ends weigh 160 tons, the total weight of
wrought-iron in the bridge will equal 160 + 6583*5 = 6743-5
tons.
The main girders, weighing 6475 '5 tons, support themselves
and 774-5 tons uniformly distributed. Consequently, each ton
of useful load uniformly distributed requires for its support
_., = 8-36 tons in the main girders. The inch-strains due
to the permanent bridge-load of 6800 tons between end pillars
5*68QO
= 4-69 tons tension, and = 3-75 tons com-
iZOU
* See note to Ex. 4, p. 535.
548 ' ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
pression. The inch-strains due to the main girders, weighing
5x6475-5 , 4x6475-5
6475*5 tons, are — — — ^'47 tons tension, and
= 3-57 tons compression. The inch-strains due to a uniformly
distributed train-load of j ton per running foot over the whole
5x450 A 01 , 4x450 A 0
bridge are „..-,. = 0*31 tons tension, and —^^^ = 0'248 tons
compression.
516. Weights €>f very large girders vary inversely in a
high ratio to their depth. — From this example we see that very
considerable economy is effected in girders of great size, whose
permanent weight forms the larger portion of the total load, by
increasing the ratio of depth to length, even in a slight degree.
Compared with Example 8, the saving in the main girders = 8778
— 6475"5 = 2302'5 tons, and the weights of these girders are in-
versely as the 4-7 power of their depths (511).
EXAMPLE 11.
517. Connterbracing required for passing loads cannot be
neglected in small bridges — Single-line lattice bridge 1O§
feet long. — The examples given in the preceding pages are those
of large bridges, exceeding 250 feet in span, in which the per-
manent bridge-load forms such a large portion of the total load
that I have neglected the extra material required for counter-
bracing the web so as to enable it to meet the maximum strains
produced by the passing load when in motion. This is allowable,
since the empirical additions for stiffening the compression bracing
are probably in excess of those actually required in large girders.
In short girders, however, it is necessary to make some allowance
in the bracing for the load being in motion, in place of being
uniformly distributed, and there is, moreover, a greater propor-
tional waste both in the flanges near the ends, and in the web near
the centre, than in large girders (487, 436). Hence, the allowance
for waste, &c., will be more than 10 per cent. The following
example of a wrought-iron lattice bridge for a single line of rail-
way, 108 feet long, 9 feet deep, and 14 feet wide between main
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK. 549
girders, will illustrate this. Using the same symbols as in Ex. 1,
we have,
I - 108 feet,
d = ~ = 9 feet,
0 = 45°,
f — 5 tons tensile inch-strain of net section,
/' = 4 tons compressive inch-strain of gross section in the
flanges, and 3 tons in the bracing (4W).
Let the maximum passing load = 1*32 tons per running foot (49O),
and assuming that the permanent bridge-load = 105 tons, we have
the total distributed load,
W= 143 + 105 = 248 tons.
The quantities are as follows (eqs. 206, 208).
248 X 108 Tons- Tons-
Theoretic tension bracing =
•± A. *J * -L^t
ton. - - 2-02 }
2-69
4x5x144 ~
9-3 cubic feet, @ 4'6 feet per ton, - - 2'02
Rivet holes, say Jrd of net section, - -67
Theoretic compression bracing, ( = f rds of the
theoretic tension bracing), - 3' 3 7 \
Add twice as much for stiffening and counter- V 10*11
bracing, - - 6*74 )
„,, • n 248x108x12
Iheoretic tension flange = -^-^ — = — ., . . =
12 x 5 x 144
37-2 cubic feet, ® 4-6 feet per ton, - - 8'09 j
Rivet holes, say Jth of net section, - 2'02 )
Covers, say ^th of the flange,* - T68
Theoretic compression flange ( = f ths of the
theoretic tension flange), - 10-11
Covers, say |th of the flange, T68
36-38
Rivet heads, packings and waste, say 25 per cent., - 9'09 •
Iron in main girders, - 45 '47
* In large girders it is important to diminish the dead load as much as possible, and
it is therefore worth paying extra for large plates so as to diminish the percentage for
covers. This, however, is not the case with small girders : hence, the percentage of
covers is larger in this than in the preceding examples.
550 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
Iron in main girders, .......
Cross-girders = 108x0*18 tons (445), -
Cross-bracing, say,
Iron between end pillars, 65-91
Platform, rails, sleepers and ballast = 108x0-36
tons (445), - 38-88
Permanent bridge-load between end pillars, - 1O4-79
being 0'2 1 ton .less than that assumed. If the four end pillars
weigh 1-5 ton, the total weight of wrought-iron in the bridge
will equal 65'91 +1*5 = 67-41 tons.
In this example, the main girders, weighing 45*47 tons, support
themselves and an additional load of 202*53 tons uniformly dis-
tributed over the bridge. Consequently, each ton of useful load
45*47
uniformly distributed requires for its support U>AL>.KQ == 0'2245 ttons
of iron in the main girders. The inch -strains in the flanges, due
to the permanent bridge-load of 105 tons, are ~.~ = 2*12 tons
tension and . = 1*7 tons compression. The inch-strains due
5 x 45*47
to the main girders alone, weighing 45*47 tons, are — ^j~ — = 0*92
4x45*47
tons tension, and — TQ — = 0*73 tons compression. The inch-
strains in the flanges, due to a uniformly distributed train-load
of 1*32 tons per running foot over the whole bridge, are
a ,.. = 2*88 tons tension, and = 2'3 tons compression.
518. Error in assuming the permanent load uniformly dis-
tributed in large girders — Empirical percentages open to
improvement. — In the foregoing examples it has been tacitly
assumed that the weight of the main girders is uniformly dis-
tributed. This is erroneous, because there is a preponderance of
material in the flanges at the centre. It is true that the amount of
bracing, both in the web and in the horizontal bracing, increases
CHAP. XXIX.] ESTIMATION OF GIRDER- WORK. 551
towards the ends and £hus to a great degree compensates for the
variation of section in the flanges. Still, the difficulty remains in
the case of very large girders whose own weight forms the greater
portion of the total load, and this preponderance of flange weight
near the centre is the chief reason why single girders are less
economical than continuous ones when the span is very great.
The empirical percentages adopted in the foregoing examples
may perhaps be objected to, and it must be confessed that they are
liable both to criticism and to correction from future experience.
I have, however, made the most of the few recorded facts on which
dependence can be placed, and would here suggest to my brother
engineers that they should, as opportunity occurs, place on record
in a tabular form the detailed weights of wrought-iron and steel
girders, in order that this branch of our practice may attain that
amount of precision that such statistical information alone can
supply. In furtherance of this object I have added in the
Appendix the detailed weights of the Boyne lattice bridge, which
I collected when Resident there, also the details of the Conway
tubular plate bridge and a few others. The examples in the
present chapter indicate the direction in which improvements in
constructive detail may be sought with most prospect of success.
In very large girders this is a matter of great importance, for even
a very slight diminution of any of the empirical percentages may
effect a large amount of economy.
519. Fatigue of the material greater in long than in short
bridges. — Though the maximum unit-strains may be the same in
two bridges, one long and the other short, the permanent unit-
strains, that is, the fatigue of the material from the permanent load
(47O), will be much higher in the bridge of great span. Thus, com-
paring Examples 2 and 11, we find that the fatigue, or permanent
inch-strains, of a railway bridge 400 feet long, are 4*06 tons tension
and 3'25 tons compression, while the corresponding inch-strains
of a bridge 108 feet long, are 2'12 tons tension and 1-7 tons
compression. If iron possessed unlimited viscidity, that is, the
property of slowly and continuously changing shape, like pitch,
under prolonged strains of moderate extent, it seems reasonable to
552 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
suppose that the longer bridge would fail sooner than the short one,
in consequence of its progressive deflection increasing more rapidly.
Experience does not favour this hypothesis, for though experiments
render it probable that all ductile metals will change shape to
an unlimited extent under enormous pressure, in this respect
resembling plastic clay, it seems equally certain that no continuous
deformation takes place in structures whose unit-strains are kept
well within the limits of elasticity (41O). Again, it is conceivable,
nay probable, that severe fatigue (especially if aided by vibration),
may so alter the constitution of iron as to weaken parts in tension,
either by rendering them brittle or by actually diminishing their
tensile strength (4O9). If this were the case within the limits of
strain which occur in practice, the longer bridge should still fail
first. If, on the other hand, large fluctuations in the amount of
strain affect the molecular condition of iron injuriously, and
produce a tendency to rupture, then the short bridge should fail
sooner. The experiments recorded in Chap. XXVIII. will prevent
anxiety in either case when the working strains do not exceed
those in usual practice (471, 478, 475).
GIRDERS UNDER 200 FEET IN LENGTH.
52O. Flanges nearly equal in weight to each other* and
web nearly equal in weight to one flange. — When an iron
lattice girder of the ordinary proportions of length to depth does
not exceed 200 feet in span, the flanges are very nearly equal in
weight to each other (477), and the web is very nearly equal in
weight to one flange. Moreover, the quantity of material in the
compression flange is nearly equal to its theoretic central area
multiplied by its length ; for though, in correct practice, the section
of the flange is reduced towards the ends, it so happens that the
empirical allowance for covers, rivet heads, packings and waste,
that is, the difference between the actual and the theoretic flange,
is closely compensated for by assuming that the flange carries its
theoretic central area uniformly throughout the whole length.
Hence, we have the following empirical formula for the weight of
material in the main girders, which will be found convenient in
practice.
CHAP. XXIX.] ESTIMATION OP GIRDER-WORK. 553
O 7 f)
G = ^ = g al, nearly. (253)
where G = the weight of the main girders and end pillars in tons,
a = the theoretic area of both compression flanges
together at the centre, in square feet,
/ = the length in feet,
4-6 = the number of cubic feet of wrought-iron in one ton.
For girders loaded uniformly we have (eq. 25),
_VW
~8/d
whence, by substitution in eq. 253,
WZ2
(254)
where W = the total distributed load in tons, including the weight
of the girder,
I = the length in feet,
d = the depth in feet,
/= the working strain in tons per square foot of gross
section.*
Ex. In Ex. 11, for instance, G = 248X(108X108) = 46.5 tonS) wycn is but very
slightly less than the former result.
591. Anderson's rule — Weights of lattice and plate
girders under SOO feet in length. — I am indebted to William
Anderson, Esq., for the following simple rule, derivable from eq.
254, for approximate estimates of railway bridges under 200 feet
in length, whose depth is TLth of their length, and whose working
inch-strains are 5 tons tension and 4 tons compression. Multiply the
total distributed load in tons by 4, and the product is the weight of the
main girders, end pillars and cross-bracing in Ibs. per running foot.
Ex. 1. The total distributed load in Ex. 11 equals 248 tons; hence, 4X248=
992 Rs. = the weight of main girders, end pillars and cross-bracing per running foot,
and their total weight = 992Xl08 = 47'8 tons, which agrees very closely with the
992X108
11 weigut =
former result.
* The reader will recollect that the usual tensile working strain of iron, namely, 5
tons per square inch of net section, practically requires the same sectional area as the
usual compressive working strain of 4 tons per square inch of gross section (-4I5"SI).
554 ESTIMATION OF GIRDER-WORK. [CHAP. XXIX.
The following table contains the weights of wrought- iron lattice
girders for railway bridges up to 200 feet in length, calculated by
the foregoing rule for the three different standard working loads
described in 49O. In making use of this table, the reader will
bear in mind the following conditions : —
a. The working strains in the flanges are 5 tons per square
inch of net section for tension, and 4 tons per square inch
of gross section for compression.
b. The proportion of depth to length = T^.
c. The dead weight of cross-girders, platform, ballast, sleepers,
and rails = 0'54 tons per running foot of single line (445).
d. The weight of main girders for a double-line bridge is twice
that given in the table for a single-line bridge.
e. It is probable that the weights in the table for the longer
bridges, say above 140 feet, are rather in excess of truth,
and that those for the shorter bridges, say under 60 feet,
are slightly under the truth.
CHAP. XXIX.] ESTIMATION OF GIRDER-WORK.
555
TABLE I. — WEIGHTS OP SINGLE-LINE WROUGHT-IRON LATTICE RAILWAY GIRDERS,
THE DEPTH BEING j^TH OF THE LENGTH.
Length of bridge
from centre to
centre of bearings.
Weight of Main girders, End pillars, and Cross-bracing,
when the
standard load on
a 100- foot
bridge = 1 ton
per foot.
when the
standard load on
a 100-foot
bridge = 1J ton
per foot.
when the
standard load on
a 100-foot
bridge = 1J ton
per foot.
feet.
tons.
tons.
tons.
12
0-7
0-8
0-84
16
114
1-36
1-44
24
2-19
2-59
2-73
32
3-4
4-0
4-2
40
4-9
5-8
6-2
60
11-3
13'4
14-0
80
20-8
24-3
25-5
100
33-5
39-0
40-7
120
497
57-6
60-2
140
70-5
80-3
84-0
160
95-4
108-2
112-6
180
125-4
141-6
146-7
200
162-2
180-0
1867
Ex. 2. What is the weight of iron required for a single-line lattice girder bridge,
140 feet long between bearings, whose depth = 11 feet 8 inches, and whose working
inch-strains are the ordinary ones of 5 and 4 tons tension and compression respectively,
the standard load being 1| tons per foot on a 100-foot bridge ? From the table we find
that the weight of the main girders, the end pillars and cross-bracing equals 80 '3 tons,
adding to this the weight of the cross-girders, supposed 3 feet apart, namely, 140 X '18
= 25-2 tons (445), we have the total weight of iron = 105'5 tons.
The following table has been constructed by Mr. Baker, by
taking as far as possible the weights of girders actually erected,
calculating missing links in the series, rectifying the curves, and
interpolating.*
Baker on the Strength of Beams, p. 319.
556
ESTIMATION OF GIKDER-WORK. [CHAP. XXIX.
TABLE II.— WEIGHTS OP WRODGHT-IRON PLATE GIRDERS, the depth being l-10th
of the length, and the working strain 4'5 tons per (gross ?) square inch in tension.
r
feet.
Load in cwt. per foot run (exclusive of the weight of the girders).
10
15
20
25
30
35
40
50
60
70
80
Weight of Girders in cwt.
10
5-6
6-5
7'4
8-3
9-2
10-2
11-0
12-8
14-6
16-2
17-5
15
10-0
11-9
13-5
15-2
167
18-4
20-2
23-4
26-4
29-2
32-0
20
17-8
20-5
23-4
26-3
29-0
317
34-5
40-3
45-6
50-5
55-5
25
26-6
31-0
35-4
39-3
44-2
48-0
52-0
60-8
68-6
76-0
83-5
30
38
44
50
56
62
68
74
86
97
108
118
35
51
58
•66
74
82
89
97
113
129
145
160
40
65
75
85
95
105
115
125
145
166
187
209
45
82
94
106
118
130
142
154
180
207
236
264
50
101
115
130
144
159
173
188
220
254
290
325
55
118
135
152
169
187
204
222
259
298
340
382
60
138
157
177
196
217
237
258
302
348
395
441
65
159
181
204
227
251
275
298
348
400
453
507
70
198
225
253
282
312
342
372
435
500
565
630
75
228
260
292
326
360
394
429
500
575
650
726
80
258
294
331
369
407
446
484
566
650
735
822
85
291
333
375
416
460
502
546
637
732
829
928
90
326
373
420
467
515
563
612
712
818
927
1040
95
365
417
470
523
576
630
686
800
920
1043
1172
100
406
464
522
581
641
701
764
892
1028
1167
1310
110
495
565
636
708
780
855
930
1090
1260
1430
1610
120
595
677
762
848
934
1020
1112
1305
1510
1720
1940
130
705
800
900
1000
1100
1200
1310
1540
1780
2000
2300
140
823
940
1059
1178
1298
1417
1546
1810
2085
2376
2686
150
950
1090
1230
1370
1510
1650
1800
2100
2410
2740
3100
160
1095
1255
1414
1574
1732
1896
2066
2415
2782
3172
3585
170
1250
1430
1610
1790
1970
2160
2350
2750
3180
3630
4100
180
1426
1626
1826
2036
2240
2450
2670
3140
3630
4130
4670
190
1614
1832
2060
2280
2510
2750
3010
3550
4100
4670
5270
200
1810
2050
2300
2550
2800
3070
3370
3980
4600
5230
5900
CHAP. XXIX.] ESTIMATION OF GIRDER- WORK. 557
538. Weights of similar girders under SOO feet span
vary nearly am the square of their length — \o definite ratio
exists Between the lengths and weights of very large
girders. — An analysis of the foregoing tables shows that the ratio
of the weights of similar railway girders from 40 to 200 feet in
length vary between the square and the 2*3 power of their lengths
(874). In Example 2, the main girders, 400 feet long, weigh
1047 tons, and in Example 5, a similar pair of main girders, 480
feet long, weigh 2449'6 tons. These weights are nearly as the 5th
power of the lengths. Again, comparing Examples 3 and 6, which
differ from the two former merely in having higher unit-strains,
we find the weights of the main girders, which are 713 tons and
1405*9 tons respectively, are nearly as the 4th power of the lengths.
These comparisons show that no definite ratio exists between the
lengths and weights of very large girders, and any argument based
on such an assumption must be altogether fallacious.
558 LIMITS OF LENGTH OF GIRDERS. [CHAP. XXX.
CHAPTER XXX.
LIMITS OF LENGTH OF GIRDERS.
523. Cast-iron girders in one piece rarely exceed 5O feet
in length — Com pound girders advisable for greater spans if
cast-iron is used.— Cast-iron girders in one piece rarely exceed
50 feet in length, though this is by no means the possible limit of
length of single castings, for Mr. Hawkshaw has employed cast-iron
in single girders of 86 feet span,* and Sir Wm. Fairbairn mentions
a bridge with girders, each 76 feet long in one casting, that were
made in England and erected on the Haarlem Railway in Holland.f
When cast-iron girders are required of greater length than 40 or
50 feet, it is advisable to truss them with wrought-iron, as cast-
iron is ill-suited for resisting tension (351). Disastrous results
have sometimes attended the use of compound girders, and they
acquired a very bad reputation at one time, but the fault lay not
so much in the combination of the two materials as in the mode of
combination, which sometimes betrayed sad ignorance of the
elementary principles on which girders should be constructed, the
depth of the trussed girder having been in some instances con-
siderably less at the centre than at the ends.
534. Practical limit of length of wronght-iron girders
with horizontal flanges does not exceed ?OO feet. — Vested
interests and local peculiarities generally determine the spans of
large bridges and it may therefore seem useless to attempt
solving the question, " What is the practical limit of length of a
girder?" Curiosity on this subject is, however, natural, and I
may therefore claim indulgence for devoting a short space to
investigating a question which, indeed, is not altogether devoid of
* Proc. Inst. C. E., Vol. xiii., p. 474.
t On the Application of Iron to Building Purposes, p. 27.
CHAP. XXX.] LIMITS OF LENGTH OF GIRDERS. 559
practical utility. When the dimensions, weight and unit-strains of
any given girder are known, we can find the length of a similar
girder which will barely support itself; for it has been already
shown in 67, that if the weight of a given girder equals -th of its
breaking weight, a similar girder n times longer will just break
with its own weight. Thus, in the first example in the previous
chapter, a pair of girders whose depth equals 1-1 2th of their length,
267 feet long and weighing 335*44 tons, sustain from their own
weight 1*64 tons tension and 1*31 tons compression per square
inch ; supposing the tensile and compressive strength of plate iron
to be 20 tons and 16 tons per square inch respectively, these work-
ing strains are equal to the breaking strains divided by 12-2.
Hence, a similar girder 12'2 times longer, or 3257 feet in length,
will just break down from its own weight. Now, the length of a
similar girder whose working strains are only one-fourth of its
3257
ultimate strength will be — j— =814 feet nearly, which therefore
is the extreme possible limit of an iron lattice girder whose depth
equals 1-1 2th of its length, whose inch-strains are 5 tons tension and
4 tons compression, and whose empirical percentages are similar to
those in the first example of the preceding chapter. The practical
limit is of course far short of this and probably does not exceed
650 feet.
Again, in Ex. 4, the main girders, 400 feet long, whose depth
equals 1-1 5th of their length and which weigh 937*4 tons, sustain
3' 14 tons tension and 2'52 tons compression per square inch from
their own weight. As these strains are equal to the ultimate
strength of ordinary plate iron divided by 6*35, a similar girder 6 '35
times longer, or 2540 feet in length, will just break down from its
own weight. Hence, the length of a similar girder whose working
strains from its own weight are l-4th of its ultimate strength
will be -^j— = 635 feet, which therefore is the limiting length of
an iron lattice girder whose length equals 15 times its depth, whose
inch-strains are 5 tons tension and 4 tons compression, and whose
560 LIMITS OF LENGTH OF GIRDERS. [CHAP. XXX.
empirical percentages are similar to those adopted in the fourth
example of the preceding chapter. The practical limit probably
does not exceed 500 feet.
Again, in Ex. 9, the main girders, 600 feet long, whose depth
equals l-16th of their length and which weigh 3396'6 tons, sustain
4*8 tons tension per square inch from their own weight. This
equals the ultimate tensile strength of ordinary plate iron divided
by 4'16; hence, a similar girder 4'16 times longer, or 2496 feet in
length, will just break down from its own weight, and the length
of a similar girder whose working tensile inch-strain from its own
weight is 6 tons, or Q.QaQ of its ultimate strength, will be •=•
O'OOO O'OOO
749 feet. This therefore is the limiting length of an iron lattice
girder whose tensile inch-strain is 6 tons, whose depth equals
1-1 6th of the length and whose empirical percentages are the same
as those adopted in Ex. 9 of the preceding chapter. The practical
limit is, doubtless, below 600 feet.
From these few examples we may reasonably infer that, even
with the most careful attention to proportion and economy, the
practical limit of length of wrought-iron girders with horizontal
flanges does not exceed 700 feet. For girders of greater span steel
must be employed.
CHAP. XXXI.] CONCLUDING OBSERVATIONS. 561
CHAPTER XXXI.
CONCLUDING OBSERVATIONS.
535. Hypothesis to explain the nature of strains in con-
tinuous webs. — The reader who has perused the foregoing pages
with even slight attention has probably arrived at the conclusion
that diagonal strains are not confined to braced girders, but are also
developed in every structure which is subject to transverse strain.
This follows at once from the mechanical law, that a force cannot
change its direction unless combined with another force whose
direction is inclined to that of the former. Thus, a vertical pres-
sure cannot produce horizontal strains in the flanges without
developing diagonal ones at the same time in the web. The
following hypothesis will perhaps give a clearer conception of the
nature of the strains in continuous webs. It is offered, however,
merely as a conceivable condition of these strains.
Fig. 120.
Let Fig. 120 represent part of a closely latticed girder whose
neutral surface, or surface of unaltered length is N S. The strain in
each diagonal of an ordinary lattice girder is uniform throughout its
entire length (14O). Now, suppose that horizontal stringers are
attached to the lattice bars at their first intersections next the flanges,
and let us confine our attention to the upper one marked c. As
soon as the girder deflects under a load, this stringer will become
compressed; consequently, it will relieve the upper flange of a
certain portion of the horizontal strain which the flange would
2 o
562
CONCLUDING OBSERVATIONS. [CHAP. XXXI.
sustain were the stringer absent, and the unit-strain in the stringer
will be to that in the flange as jrr>. The part of each diagonal
above the stringer will also be relieved of a certain portion of
its strain, depending on the horizontal component it yields up to
the stringer. Now, conceive similar stringers attached at each
horizontal row of lattice intersections above and below the
neutral surface, in which case each stringer will sustain horizontal
unit-strains directly proportional to its distance from the neutral
surface where they are cipher, while, on the other hand, the strains
in the diagonals will diminish as they approach the flanges, their
decrements of strain being cipher at the neutral surface and
increasing towards the flanges in the direct ratio of their distance
from the neutral surface, provided the stringers are all of
equal area. We thus see that the diagonal strains, and therefore
the shearing strain in solid girders, or in girders with con-
tinuous webs, act with greatest intensity in the neighbourhood of
the neutral surface where the horizontal strains are nil, while they
act with least intensity at the upper and lower edges where the
horizontal strains are most intense. This theory agrees with an
instructive experiment made by Mr. Brunei on a single-webbed
plate girder, 66 feet long between bearings and 10 feet deep at the
centre, in which the web, formed of £ inch plates with vertical lap
joints, gave way by several of these joints near one end tearing
open in the neighbourhood of the neutral surface.*
Fig. 121.
* Clark on the Tubular Bridges, p. 437.
CHAP. XXXI.] CONCLUDING OBSERVATIONS.
563
When a single weight rests upon a girder with a continuous web,
it sends off strains radiating out from the weight in all directions,
as represented in Fig. 121, and we may conceive that this first
series of diagonal strains are resolved at every point along their
length into diagonal and horizontal strains, as in the lattice girder ;
this second series of diagonal strains being again resolved in a
similar manner, and so on, and thus we have horizontal and diagonal
strains interlacing at various angles in all girders except those in
which they are forced to take definite directions by means of the
bracing, but there will probably exist certain lines of maximum
strain, either straight or curved, whose directions will vary according
to the position and amount of the weight, as well as the flexibility
of the material. The student may make some instructive experi-
ments on this subject by the aid of a model girder formed by
stretching a web of drawing paper over a light rectangular frame
of timber, which will represent the flanges and end pillars. By the
aid of little movable wooden struts, to represent verticals, he can
vary the directions of the lines of strain to a very considerable
extent.
It is not at first sight easy to see how strains are transmitted
through the neutral surface, for the particles there are apparently
undisturbed in form. It is conceivable, however, that particles
which are spherical when free from strain may become elongated
by tension in one direction and shortened by compression at right
angles to it, so as to assume an oval shape, while horizontal lines
parallel to the neutral surface, N S, retain their original length, as
represented in Fig. 122.
Fig. 122.
5S6. Strains in Ships.— An iron ship is a large tubular
564 CONCLUDING OBSERVATIONS. [CHAP. XXXI.
I
structure, more or less rectangular in section, underneath which the
points of support are continually moving, so that, when the waves
are high and far apart, the deck and bottom of the vessel are
alternately extended and compressed in the same way that the
flanges of a continuous girder are near the points of inflexion when
traversed by a passing train. The sides of a ship are formed of
continuous plating with vertical frames at short intervals, and form
very efficient webs; the bottom also is, from its large area, fully
adequate to its duty as a flange. The sides and bottom flange
of the girder are therefore fully developed, but the upper iron
flange is sometimes altogether wanting, or else sadly out of pro-
portion to the remainder of the structure. This deficiency is
properly remedied, either by attaching what are technically called
stringers to the topsides, or better still, by making the upper deck
entirely of iron with a thin sheeting of planks resting on the iron.*
Deck stringers are horizontal plates which run continuously fore
and aft beneath the planking of the deck. They are seldom more
than 3 or 4 feet in width, but in some few cases extend as far as the
hatchways. Similar stringers are occasionally riveted to the sides
underneath each of the lower decks, and when stringers in the
same plane on opposite sides of the ship are connected by diagonal
tension braces, the latter, in conjunction with the deck beams, form
very efficient cross-bracing, and greatly increase the strength and
stiffness of the ship when labouring in a heavy sea. Bulkheads act
as gussets or diaphragms, and stiffen the ship transversely by
preventing any racking motion from taking place in the direction
of their diagonals.
537. Iron and timber combined form a cheap girder —
Timber should be used in large pieces* not cut up into
planks — Simplicity of design most desirable in girder-
work. — Within certain limits of length, one of the cheapest forms
of girder is one made of timber in compression with wrought-iron
in tension (1873 881). The earlier types of wooden lattice bridges
had little or no iron in their composition and were characterized by
* The author has built several iron vessels in which tar asphalt is substituted for
the timber sheeting.
CHAP. XXXI.] CONCLUDING OBSERVATIONS. 565
the small scantlings of the parts, the closeness of the latticing, and
in many cases, a want of stiffness both vertically and laterally.
This defect was, no doubt, often due to insufficient flange area, but
may also be attributed to the small size of the scantlings, and
consequent multiplicity of joints. The remedy is obvious. Timber
in compression should be used in bulk, and not cut up into thin
planks. Laminated arches, it is true, are an apparent exception to
this rule, but in reality a laminated beam possesses the aggregate
section of its component parts which are bound together so that
they act as one solid piece. Even when used in tension, it may be
doubtful economy to use several thin planks where one of larger
section would suffice. The liability to decay from moisture
lodging in the numerous joints is another serious objection to
close timber latticing, though this is sometimes diminished by the
protection of a roof extending over the whole bridge (485).
In conclusion, it may not be amiss to say a few words on
designing girders. Simplicity and consequent facility of construc-
tion should never be lost sight of. Complicated arrangements are
to be deprecated, whether designed to affect some saving more
apparent than real, or, as one is sometimes tempted to conjecture,
from a craving after novelty. The various parts of girder work
should, as much as possible, be repetitions of the same pattern,
easily put together and accessible for preservation or repair. Hence,
as a rule, closed cells, difficult forgings, curved forms where straight
ones would effect the object equally well, and a great variety of
sizes to meet excessive theoretic refinement, are to be carefully
avoided.
APPENDIX.
BOYNE LATTICE BRIDGE.
538. General description and detailed weights of girder-
work. — The Boyne Viaduct carries the Dublin and Belfast
Junction Railway across the valley of the River Boyne near
Drogheda, and consists of several lofty semi-circular stone arches
on the land, and a wrought-iron lattice bridge in three spans over
the water, the surface of which is about 90 feet below the girders,
so that vessels of considerable tonnage can sail beneath. The
girder- work is formed of two lattice double- webbed main girders,
having their top flanges connected by cross-bracing, and the lower
flanges connected by cross-girders and diagonal ties, so as collectively
to form an openwork tubular bridge for a double line of railway,
as shown in cross-section in Plate IV. Each main girder is a
continuous girder, 3 feet wide and 550 feet 4 inches long, in three
spans. The centre span is 267 feet from centre to centre of
bearings, and 264 feet long between bearings. Each side span is
140 feet 11 inches long from centre to centre of bearings, and 138
feet 8 inches long between bearings. The flanges are horizontal
throughout, and the depth of girder, measured from root to root of
angle irons, is 22 feet 3 inches, or l-12th of the centre span and
TTTrrth of each side span. Each of the terminal pillars is 18 inches
b'o4
broad in elevation and has a bearing surface of 3 X 1*5 = 4-5
square feet ; each of the pillars at the ends of the centre span is
3 feet broad in elevation and has a bearing surface of 3x3 = 9
square feet. The cross-girders are 7 feet 5 inches apart from
centre to centre and correspond with the intersections of the lattice
bars, which are placed at an angle of 45° and form squares of 5
feet 3 inches on the side. The quantities of material in the
girder- work are as follows : —
* For further description, see Proc. Inst. C.E., Vol. xiv. ; also, Proc. Inst. C.E. of
Ireland, Vol. ix.
568
BOYNE LATTICE BRIDGE.
[APP.
TABLE I.— WEIGHT OP WROUGHT-IRON IN BACH SIDE SPAN, 140 FEET 11 INCHES
BETWEEN CENTRE OP BEARINGS AND 30 FEET WIDE FROM OUT TO OUT.
Two BOTTOM FLANGES.
Two TOP FLANGES.
Plates and angle iron,
Covers,
Packings,
Rivet heads,
Plates and angle iron,
Covers,
Packings, -
Rivet heads,
TWO DOUBLE-LATTICED WEBS.
Tension diagonals, -
Compression do.,
Rivet heads at intersections,
CROSS-BRACING.
6 lattice cross-girders connecting top flanges,
Horizontal diagonal tension bars (top and bottom) and a longitu-
dinal angle iron stiffener along the centre at top,
Rivet heads,
CROSS-GIRDERS.
18 lattice road-girders, including end gussets,
Iron between end pillars,
Platform planking, -
Longitudinal sleepers (double line),
Rails and joint plates (Barlow's), -
Permanent load on one side span, -
equal to 1'36 tons per running foot for the double line.
Tons.
27-45
3-57
6-38
2-44
27-10
3-84
6-40
2-25
10-96
27-70
0-13
3-70
5-36
o-io
Tons.
39-84
39-59
38-79
916
29-40
2-45
8-56
40-41
191-51
APP.]
BOYNE LATTICE BRIDGE.
56!)
TABLE II.— WEIGHT OF WROUGHT-IRON IN THE CENTRE SPAN, 267 FEET BETWEEN
CENTRES OF BEARINGS AND 30 FEET WIDE FROM OUT TO OUT.
Two TOP FLANGES.
Two BOTTOM FLANGES.
Plates and angle iron,
Covers, -
Packings,
Rivet heads,
Plates and angle iron,
Covers, - ...
Packings, »- .
Rivet heads,
TWO DOUBLE-LATTICED WEBS.
Tension diagonals,
Compression do., • ...
Rivet heads at intersections,
CROSS-BRACING.
11 lattice cross-girders connecting top flanges, -
Horizontal diagonal tension bars (top and bottom) and a longitu-
dinal angle-iron stiff ener along the centre at top, - - 10 '6 9
Tons.
79-09
9-38
11-83
518
8219
9-85
11-90
5-18
30-80
51-76
•25
677
Tons.
105-48
10912
82-81
Rivet heads,
CROSS-GIRDERS.
35 lattice road-girders, including end gussets,
Iron between end pillars, -
Platform planking,
Longitudinal sleepers (double line),
Rails and joint plates (Barlow's),
Permanent load on centre span, -
equal to 1'64 tons per running foot for the double line.
•20
17-66
4613
55-57
4-62
16-20
361-20
76-39
437-59
570 BOYNE LATTICE BRIDGE. [APP.
TABLE III.— WEIGHT OP WROOOHT-IRON IN THE PILLARS AND CROSS-GIRDERS
OVER SUPPORTS.
PILLARS, &c., OVER EACH LAND ABUTMENT.
Tons. Tons.
2 terminal pillars at end of one side span, - 6'38
1 lattice cross-girder connecting heads of pillars, - 3 '40
1 lattice cross-girder and gussets connecting feet of pillars, - 3'45
PILLARS, &c., OVER SOUTH RIVEB PIER.*
2 pillars at south end of centre span, - - 15 '30
1 lattice cross-girder connecting heads of pillars, - 5*24
1 lattice cross-girder connecting feet of pillars, - 1'09
2 gussets between pillars and pier,
PILLARS, &c., OVER NORTH RIVER PIER.
2 pillars at north end of centre span, - - 15' 30
1 lattice cross-girder connecting heads of pillars, - 5 '2 4
1 lattice cross-girder connecting feet of pillars, - - 5'02
13-23
24-06
25-56
TABLE IV.— SUMMARY OP WROUGHT-IRON.
Tons.
One side span, - - 151*10
Second do., - 151 '10
Centre span, - 361 "20
Pillars, &c., over one land abutment, - - 13'23
Do. „ second do. - 13'23
Do. „ south river pier, 24'06
Do. „ north river pier, - 25 "56
Total weight of wrought-Iron in the 3 spans, - 739-48
550 feet 4 inches in total length, equal to T344 tons per running foot for
the double line of railway.
* The pillars are firmly secured to this pier ; rollers are used on the north pier and
on both abutments.
APP.] BOYNE LATTICE BRIDGE. 571
TABLE V.— WEIGHT OP SOLE-PLATES, HOLLERS AND WALL-PLATES.
OVER TWO ABUTMENTS. Tons. cwts. qrs. tt>s.
4 planed cast-iron sole-plates riveted to feet of pillars, - — 17 2 16
4 planed cast-iron wall -plates resting on the masonry, - 2 11 0 0
2 sets of 4-inch wrought-iron rollers and frames over the
north abutment, - - - — 10 2 0
2 sets of 4^-inch wrought-iron rollers and frames over
the south abutment, - - — 12 2 26
OVER SOUTH RIVER PIER.
2 cast-iron sole-plates riveted to feet of pillars, - - — 19 0 12
2 cast-iron wall-plates resting on the masonry, - 5 4 0 0
OVER NORTH RIVER PIER.
2 planed cast-iron sole-plates riveted to feet of pillars, - — 19 0 12
2 planed cast iron wall-plates resting on the masonry, - 4 13 0 16
2 sets of 5-inch chilled cast-iron rollers and wrought-iron
frames, •- - - - - 1 15 0 16
Total weight of sole-plates, rollers and
wall-plates, - - - 18 2 1 14
5S9. Working? strains and area of flanges. — The strains
produced by the permanent bridge-load, plus one ton of train-load
per running foot on each line of way, do not exceed 5 tons tension
per square inch of net area, i.e., after deducting the rivet holes, and
4 tons compression per square inch of gross area. The gross
sectional area of the top flange of each main girder in the centre
of the centre span = 11 3*5 square inches; the gross area of the
bottom flange at the same place =127 square inches, and its net
area = 99 square inches ; over the piers, between the centre and
side spans, the gross area of the top flange of each main girder
= 132*6 square inches, and its net area = 103'4 square inches; the
gross area of the bottom flange at the same place = 127 square
inches. At the points of inflexion in the centre span, about 40
feet from the piers measured towards the centre of the bridge, the
gross area of each flange = 68*5 square inches.
572 BOYNE LATTICE BRIDGE. [APP.
53O. Points of inflexion — Pressures on points of support. —
The points of inflexion may be obtained by the method explained
in 853,. as follows : —
Fig. 123.
Let Q be the centre of the centre span, and o and o' the points
of inflexion.
Let / = A B = C D = 141 feet nearly,
A Q = n/, whence n = ..... = 1*95 nearly,
w = the load per running foot on either side span,
wf = the load per running foot on the centre span,
R! = the reaction of either abutment, A or D,
R2 = the reaction of either pier, B or C.
When the bridge supports its own weight only,
w = 1*36 tons and w' = 1*64 tons.
CASE 1.
531. maximum strains in the flanges of the side spans. —
These occur when the passing load covers both side spans and the
centre span is unloaded (855) ; in which case, assuming that the
maximum train-load is equivalent to one ton per running foot on
each line of way, we have,
w = 3- 36 tons and wr = 1-64 tons.
From equations 183 and 184 the pressures on the points of
support are as follows :—
R, = 170 tons. R2 = 523 tons.
BOYNE LATTICE BRIDGE. 573
The positions of the points of inflexion, obtained from equations
185 and 186, are as follows: —
Ao = 101-2 feet. Bo' = 53-2 feet.
The strain in each of the four flanges midway between A and o,
i.e., in the centre of the first segment, is 96*6 tons (eq. 25).
CASE 2.
533. Maximum strains in the flanges of the centre span. —
These occur when the passing load covers the centre span alone,
in which case,
w = 1'36 tons and w' = 3*64 tons.
The pressures on the points of support are as follows : —
RI = — 24-6 tons. R2 = 704 tons.
R! being negative, signifies that a load of 24-6 tons is required at
each end to prevent the girder from rising off the abutments (854),
and this was actually the case when the bridge was proved with
one ton per running foot on each line of the centre span, the side
spans being unloaded. The girder was temporarily tied down to
the abutments by bolts secured to the masonry, but the bolts drew
out and the ends of the girder rose more than an inch above their
normal position on the rollers. The weight of a locomotive at
each end, however, soon brought them down again. With the
lighter working loads which occur in practice this rising off the
abutments does not occur. The position of the points of inflexion
in the central span is as follows : —
Bo' = 40-3 feet,
and the strain in each of the four flanges in the centre at Q = 355
tons (eq. 25). At this place the net area of each lower flange = 99
355
square inches and the tensile inch-strain therefore = -^- = 3'6
tons.
CASE 3.
533. Maximum strains in the flanges over the piers. — The
maximum strains over a pier occur when the centre span and the
adjacent side span are loaded, and the remote side span is unloaded.
We have, however, no formula for this condition of load, but we
574 BOYNE LATTICE BRIDGE. [APP.
have a close approximation to it when the passing load covers all
three spans (855), in which case,
w = 3'36 tons and wr = 3*64 tons.
The pressures on the points of support are as follows : —
R! = 107 tons. R2 = 853 tons.
The positions of the points of inflexion are as follows (eqs. 185
and 186):-
Ao = 63-4 feet. ' Bo' = 44-7 feet.
The strain in each of the four flanges over the piers = 406*4 tons
(eq. 12). The net area of each upper flange at this place = 103'4
square inches and the tensile inch-strain therefore = .TTTTT = 3'93
10o'4
tons.
534. Points of inflexion fixed practically — Deflection —
Camber. — The points of contrary flexure in the centre span were
practically fixed in the manner described in 85O. Two joints in
the upper flange, 170 feet apart and equi-distant from the piers,
were selected for section. The rivets were cut out and drifts tem-
porarily inserted in their place. These drifts were then cautiously
struck out with a light hammer, and a slight closing of the joints
proved that a certain amount of compression had previously existed
in place of perfect freedom from strain. The extreme ends of the
side spans were then lowered, one an inch, the other half an inch,
which caused the joints to open about ^th of an inch. In this
condition it was obvious that no strain was transmitted through
the joints, and they were then finally riveted up, the altered
levels of the extreme ends of the side spans being maintained
by rollers of the proper diameter placed beneath the terminal
pillars. Tables VI. and VII. contain the deflections produced by
various conditions of load during the first, or Engineer's testing,
and the second, or official testing of the bridge by the G overnment
Inspector (4O9).
APP.]
BOYNE LATTICE BRIDGE.
l
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576
BOYNE LATTICE BRIDGE.
[APP.
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APP.] BOYNE LATTICE BRIDGE.
577
Each span was built on the platform with a camber in order that
the sky-line might be nearly horizontal when the bridge was finished
(453). The camber at the centre of the centre span at different
periods was as follows : —
TABLE VIII. — CAMBER AT CENTRE OF THE CENTRE SPAN.
Inches.
During construction on the platform, - . . 3.43
After wedges were struck and bridge was self-supporting, - - - 1*56
After fixing points of inflexion and lowering the ends of the side spans, - 1-80
After the second, or official testing of the bridge, - - - 0'84
After four months' traffic, - - - - - . - 0'90
535. Experiments on the strength of braced pillars. —
The following experiments were made at the Boyne Viaduct in
1854, to determine the strength of one of the compression diagonals
of the web which were made of flat bar iron similar to the tension
diagonals, but with the addition of internal angle irons and cross-
bracing riveted between them as already described in 341. The
theory of braced pillars was then imperfectly understood, and
it was determined to test by direct experiment whether this
arrangement of internal cross-bracing would enable a bar, thin in
proportion to its length, to sustain an endlong pressure like a pillar,
such as the compression diagonals should sustain in the bridge.
Accordingly, the following experiments were made on one of the
smaller compression diagonals which occur near the centre of the
centre span, the author being present and recording the results.
EXPERIMENT 1.
The first experimental pillar resembled Fig. 1, Plate V., in every
respect, except the lower portion, which was formed as shown in
Fig. 4. This pillar, which was 31' 6" in length with 4" X J" side
bars, was erected in the midst of some timber scaffolding which had
been used for a stone hoist. The testing weight was suspended
below the wooden framing on which the pillar stood by long sus-
pender rods which were attached to cross pieces of timber resting
2 P
578
BOYNE LATTICE BRIDGE.
[APP.
on the top of the pillar (see Figs. 2 and 3). By this arrangement
the pressure was made to pass more accurately through the axis of
the pillar than if the testing weight had been heaped up on top ; it
was also more convenient to load at the lower level. Cross bars
/,/,/, were attached to the sides at the same intervals as the latticing
in the main girders, and were connected at their ends to the scaffold-
ing, so as to represent the tension diagonals in the bridge ; and here
I may again remind the reader that the chief advantage of a multiple
over a single system of triangulation consists in the more frequent
support given by the tension bars to those in compression, as well
as by both to the flanges; the parts in compression are in fact
subdivided into short pillars, and thus prevented from deflecting in
the plane of the girder (153). A cord was stretched vertically, in
order to get the lateral deflections during the experiment. These
were taken at three points, A, B, C, Fig. 1, and the symbols +
or — placed before a deflection in the table signifies that it was in
the direction of the same sign engraved at the sides of the figure.
TABLE IX. — LATERAL DEFLECTIONS OP A BRACED PILLAR.
Date.
Tons.
A.
B.
C.
REMABKS.
1854.
inches.
inches.
inches.
Nov. 16
5
+ 0-03
+ 0-05
+ 0-03
9)
10
o-o
o-o
+ 0-05
15
— 0-03
+ 0-03
+ 0-05
M
20
— 0-05
o-o
o-o
99
25
— 0-05
o-o
o-o
»
30
— 0-05
— 0-03
— 0-04
374
— 0-06
— 0-07
— 0-06
M
40
— 0-05
— O'Ol
— 0-05
With 40 tons, the side bar at a, Fig.
4, bent slightly at right angles to
the plane of the figure. The deflec-
tion at B seems anomalous; probably
a mistake for O'lO.
99
42£
— o-io
— o-io
— 0-13
With 42£ tons, the lower part of the
pillar at b, b, became slightly curved,
with the convex side towards the
— side.
Nov. 17
424
— o-io
— o-io
— 0-13
Left on all night ; no change in the
morning.
n
45
— o-io
— 0-15
— 0-16
„
47i
...
...
...
The side bars gave way, as shown in
Fig. 5.
Looking at Fig. 4, it will be seen that about 8 inches in length
APP.]
BOYNE LATTICE BRIDGE.
570
of each side bar near the ends of the pillar were left without internal
angle iron, and when the weight amounted to 47J tons, this part
yielded sideways, as shown in Fig. 5. The area of the two side-
bars at the part which failed amounted to 5 square inches; con-
sequently, the compressive strain which passed through them at
the moment of yielding equalled 9J tons per square inch.
EXPERIMENT 2.
The pillar in the first experiment failed, as indeed had been
anticipated, by the upper part moving sideways past the lower, as
if connected to it by hinges. The pillar was taken down, the
injured part removed, and the length thus reduced to 28' 6". The
repaired pillar, Fig. 1, was then replaced within the scaffolding and
the following table contains the observations recorded, which include
the contraction in length of each side under compressive strain.
These latter observations were made by the aid of wooden rods
suspended at each side from near the top of the pillar. Each rod
was 24' 8J" in length from the point of suspension to the index at
the lower end, and it will be observed that the contraction of one
side exceeds that of the other in a very anomalous manner, which
can only be explained by supposing that the timber framing yielded
more beneath one side than the other and thus caused a greater
strain of compression to pass through that side of the pillar which
contracted most.
TABLE X.— LATERAL DEFLECTIONS AND YEBTICAL CONTRACTION OF A BRACED PILLAR.
Date.
Tons.
A.
B.
C.
Rod on
+side.
Rod on
— side.
OBSERVATIONS.
inches.
inches.
inches.
inches.
inches.
1854.
Nov. 25
30
+0-03
+0-04
+0-01
0-05
0-25
At 30 tons, the side bar at c bulged
outwards slightly, with a tendency
to increase; also a slight hollow
was produced at d; to remedy
this bulging (which seemed to be
caused by the unequal compression
of the timber packing, that on the
-f- side yielding more than that
opposite), a strut was placed
against c, and the weight was
blocked up until the 27th ; wedges
also were driven between the
wooden packings underneath, in
order to tighten them up.
580 BOYNE LATTICE BKIDGE.
TABLE X. — LATERAL DEFLECTIONS, &o.— continued.
[APP.
Date.
Tons
A.
B.
C.
Rod on
-(-side.
Rod on
—side.
OBSERVATIONS.
inches.
inches.
inches.
inches.
inches.
Nov. 27
0
o-o
0-20
Load removed, and bulging at c re-
moved as nearly as possible by
means of a screw-jack which was
left in position ; opposite side simi-
larly blocked out from staging,
and blocks were placed at similar
positions at top of pillar, as there
appeared a tendency of top to
move over to — side.
,.
30
...
...
...
0-05
0-25
„
35
...
...
...
0-06
0-27
»
40
o-o
—o-oi
—0-03
0-08
0-31
Left hanging on all night, wind so
strong as to make deflections un-
certain.
Nov. 28
40
o-o
—0-37
—0-07
0-09
0-31
The hollow at d still well marked
and a tendency to deflect towards
+ side, at the centre of the pillar.
M
45
o-o
—o-oi
—0-07
0-09
0-34
Wind in gusts; 45 tons left hanging
on one hour.
»
50
+ 0-07
+0-06
—0-03
010
0-40
Wind much abated ; no visible
change.
»
55
...
...
...
010
0-44
Wind so strong as to prevent deflec-
tions being taken. No visible
change.
M
60
...
...
010
0-49
No visible change.
Nov. 29
624
...
...
...
0105
Oil
012
0-50
0-50
0-53
Left hanging on all night.
No visible change this morning.
The buckle at centre strongly
marked.
65
...
...
M
70
+015
+014
+0-03
012
0-56
Wind much abated.
„
72*
...
...
...
013
0-60
»
75
....
...
...
014
0-65
No visible change or upsetting of
any part
H
jj
Nov. 30
H
H
774
80
...
...
...
013
0-69
Left hanging on all night.
In morning.
Broke down as the additional ton
was being laid on, parts b and e,
Fig. 1, giving way. At e, both
sides of the pillar bent and the
internal lattice was completely
distorted, the |_ iron being broken
away from the side bar (see Fig. 2).
014
015
0-78
0-795
824
83*
...
...
...
APP.] BOYNE LATTICE BRIDGE. 581
The sectional area of that part of the pillar which was subject
to compression, namely, the side bars and the angle irons, was
7-5 inches. The compression therefore equalled 11 tons per square
inch at the period of failure. For a very short portion at c, where
the bracing ended, the angle irons of the lower cell and that to
which the internal lattice bars were connected were not in one con-
tinued piece, and the whole weight passed through the unsupported
side bars, which were, however, a little thicker here than elsewhere
from a weld having been made at that point, so that the area of
both side bars together equalled 6 square inches ; this short length
was therefore subject to a compression of nearly 14 tons per square
inch. If we wish to compare the economy of this form of pillar
with a tubular one, we must add the cross area of the lattice bars to
that of the side bars and angle irons, in order to obtain the strain per
sectional inch of material in the whole pillar. The cross area of the
lattice bars = 2 inches nearly ; adding this to the area of the side
bars and angle irons, we have the total sectional area of the braced
pillar rr 9J inches, and the compression per square inch of material
employed — 8*7 tons. This is a favourable result when compared
with those arrived at by Mr. Hodgkinson in his experiments on
tubes subject to compression, for if the same amount of iron were
thrown into the form of a plated tube, it would have such thin sides
that the ultimate crushing inch-strain would probably fall very far
short of 87 tons (335). We may regard the lattice pillar as one
side of a tube, in the corners of which the chief part of the material
is collected and the sides of which are formed of bracing, connecting
and holding the corner pillars in the line of thrust.
536. Experiments on the effect of sfow and quick trains
on deflection. — The following experiments were made at the
Boyne Viaduct to try the effect of slow and quick trains on
vibration and deflection : —
April bth, 1855.— The lateral oscillation at the centre of the
centre span from an engine and tender going at the rate of from
30 to 50 miles an hour equalled 0'05 inch on each side, i.e., the
total oscillation equalled 01 inch. That from a slow engine
was scarcely perceptible. The deflection at the centre of the
centre span, measured on the same side as the line on which
582 NEWARK DYKE BRIDGE. [APP.
the engine and tender travelled, both for quick and slow speeds
equalled -25". The same deflection was produced when the engine
was brought to a stand at the centre of the centre span. If any
difference of deflection with different speeds was perceptible, those
deflections which were produced by rapid travelling exceeded the
others by a very small amount, perhaps the width of a fine pencil
stroke, but for all practical purposes they were identical. On
starting the engine from rest at the centre of the bridge the
deflection was momentarily increased to a very slight extent. There
were about five quick trains, of which one travelled at 48 and the
others 50 miles an hour, and about as many slow ones (454).
NEWARK DYKE BRIDGE, WARREN'S GIRDER.*
537. — This bridge carries the Great Northern Railway across
the Newark Dyke, a navigable branch of the river Trent. It is a
skew girder bridge, formed of a single system of equilateral triangles
on Warren's principle. Each girder consists of a hollow cast-iron
top flange, and a bottom flange, or tie, of wrought-iron flat bar
links, connected together by diagonal struts and ties, alternately of
cast and wrought-iron, which divide the whole length into a series
of equilateral triangles, 18 feet 6 inches long on each side. There
are two main girders to each line, between which the train travels
on a platform attached to the lower flanges. The length from
centre to centre of points of supports is 259 feet, and the clear
span between the abutments is 240 feet 6 inches. The depth
from centre to centre of flanges is 16 feet, or nearly l-16th of
the length. The permanent weight of bridge for a single line of
railway, consisting of two main girders, top and bottom cross-
bracing, platform, &c., is as follows:—
Tons. Cwts.
Wrought-iron, - 106 5
Cast-iron, - - 138 5
244 10
Platform, rails, handrail and cornice, - 56 0
Total permanent weight for one line of way, 3OO 1O
* "Description of the Newark Dyke Bridge."— Proc. Inst. C.E., Vol. xii.
APP.] CHEPSTOW BRIDGE. 583
With a load of one ton per running foot the central deflection
amounted to 2f inches. The strain with this load, whether tensile
or coinpressive, is said not to exceed 5 tons per square inch on any
part.
CHEPSTOW BRIDGE, GIGANTIC TRUSS.*
538. — This bridge was erected by Mr. Brunei to carry the
South Wales Railway across the river Wye near Chepstow. It
consists of two gigantic trusses, one for each line of way, 305
feet long and about 50 feet deep, and resembling Fig. 64, p. 124,
with this exception, that the roadway is attached to the lower
flange. The compression flange of each truss is a round plate-
iron tube, 9 feet in diameter and f th inch thick, with stiffening
diaphragms at intervals, and supported by cast-iron arched standards,
or end pillars, which rest on the piers. The side girders are plate
girders which are divided by the truss into three spans. The
weight of iron in one bridge for a single line of railway is as
follows : —
Tons.
298 feet run of tube and butt plates, 127J-
Hoops of ditto over piers, 7£
Side and bottom plates for attachment of main
chains, - 15
Side plates for attachment of counterbracing chains, 2£
Stiffening diaphragms, 26 feet apart, 4£
Rivet heads, &c., 4f
Total weight of one tube (top flange), - 16 1J
Main chains, eyes, pins, &c.,
Counterbracing chains, eyes, pins, &c., -
Vertical trusses, - - 182
Total weight of side-bracing, -
Encyc. Brit., Art. "Iron Bridges," and Clark on the Tubular Bridges, p. 101.
584 CRUMLTN VIADUCT. [APP.
Tons.
Side girders, cross-girders, &c., - - 130
Saddles, collars, &c., at points of suspension, - 22
152
Total weight of iron for one line of railway, - 46O
CRUMLIN VIADUCT, WARREN'S GIRDER.*
539. — The Crumlin Viaduct is situated on the Newport section
of the West-Midland Railway about five miles from Pontypool.
The structure is divided by a short embankment into two distinct
viaducts of exactly similar construction. The larger viaduct has
seven, the smaller three openings of 150 feet from centre to centre
of piers. The girders are " Warren's Patent" of 148 feet clear
span, but not connected together as in continuous girders. The
compression flange is a rectangular plate-iron box or tube, and the
tension flange is formed of flat wrought-iron bars; both flanges
increase in sectional area from the ends towards the centre. The
diagonals form a series of equilateral triangles of angle and bar iron,
the section of those in compression being in the form of a cross.
The length of each side of the triangle is 16 feet 4 inches. The
maximum tensile strain in the diagonals from the permanent load
plus a train-load of one ton per running foot was 6*65 tons per
square inch of net section when the bridge was first made, the
maximum tensile strain in the lower flange from the same load
was 5 '75 tons per square inch of net section, and in no part did
the maximum compression strain from the same load exceed 4-31
tons per square inch of gross section. The viaduct has four girders,
two to each line of railway with the road above the girders. The
weights for a single line, 150 feet long, were as follows when the
bridge was first made, but a very large amount of additional
material appears to have been added subsequently for the purpose
of strengthening it.f
* Trans* Inst. C. E. of Ireland, Vol. vii., p. 97 ; and Humber on Bridges, 1st ed.
f Engineer, 1866, Vol. xxii., p. 384.
APP.] PUBLIC BRIDGE OVER THE BOYNE. 585
Tons. Cwts.
A pair of main girders, - 37 18
Cross-bracing of do., - - 3 3
Platform, - 18 1
Permanent way, - 15 3
Hand-railing, - - 9 0
Total permanent weight for one line of way, 83 5
The tension flange of one girder weighs 5'97 tons, of which
1-5 ton, or one-fourth, was required to make the connexions of the
flange.
PUBLIC BRIDGE OVER THE BOYNE, LATTICE GIRDER.*
54O. — This bridge crosses the river Boyne at the Obelisk near
Drogheda. The main girders are double-webbed lattice girders,
128 feet long, and 10 feet 8 inches deep, or l-12th of the length.
The clear span between the abutments is 120 feet, and the clear
width of the roadway, between the inside planes of the lattice bars,
is 16 feet 8 inches. Sufficient strength is provided in the main
girders to sustain a total load of 3 cwt. per super foot of roadway
when the iron in tension is strained up to 5 tons per square inch of
net section, and that in compression up to 4 tons per square inch of
gross section. The cross-girders are shallow plate girders about 3J
feet apart and capable of supporting a load of 5 cwt. per super foot,
the additional strength being given to meet the contingency of a
very heavy load resting on each girder in succession with the same
working strains as above. The roadway is supported on buckled
plates resting on the cross-girders; these plates weigh 67Jft>s. per
square yard 'and have a versine of 2-J- inches, four plates being laid
in the width of the bridge. A layer of wooden chips, sand and
coal tar was first laid so as to cover a little over the level of the
crown of the buckled plates and upon this was laid asphalt 8 inches
deep, consisting of broken stones, sand and coal tar.
The following table gives the actual weight, the theoretic weight,
* Trans. Inst. C.E. of Ireland, Vol. ix., p. 67.
586
rUBLIC BRIDGE OVER THE BOYNE.
[APP.
and the percentage of material practically required over the
theoretic weight, i.e., the loss of iron due to rivet holes, cover
plates, stiffeners and waste.
TABLE XL— SUMMARY OF MATERIALS IN THE BOYNE OBELISK BRIDGE,
120 FEET SPAN.
Top flange, in compression,
Actual
weight.
Theoretic
weight.
Percentage lost.
382 cwt.
302 cwt.
26 per cent.
Bottom do., in tension,
382
242
58
End pillars, in compression,
41 „
12 „
242
Latticing, in compression,
152
75 „
114
Do. in tension,
Total for Main girders,
Hand-rail bars, ....
1U „
71 „
60
1,071 cwt.
26 „
702 cwt.
52J per cent.
35 cross-girders,
315
70 cast-iron chairs, under ends of last,
44 „
Buckled plates and side plates for
retaining asphalt in place,
210
Asphalt, -
Total weight of bridge,
754 „
2,420 cwt. - 121 tons.
The weight of the main girders is 8'5 cwt. per foot run and that
of the roadway 10*8 cwt., forming a total of 19*3 cwt. per foot run.
The weight per square foot of roadway surface is *52 cwt. for the
main girders, and '65 cwt. for the roadway, the total being 1*17 cwt.
per square foot. This leaves a balance of 3 — 1*17, = I1 83 cwt.
per square foot, for the greatest load, say dense crowds, which in
a country bridge can scarcely exceed 100 Ibs. per square foot (493).
There is, therefore, a considerable margin for deterioration of the
iron, which is a wise precaution in a country bridge that is not
likely to be painted frequently.
APP.] CHARING-CROSS LATTICE BRIDGE. 587
BOWSTRING BRIDGE ON THE CALEDONIAN RAILWAY.*
541. — This bridge was erected by Mr. E. Clark to carry the
Caledonian Railway over the Monkland Canal. The arch is partly
wrought-iron and partly cast-iron, and the tie or lower flange
consists of wrought-iron plates. The total length of the girders is
148 feet, and the depth is 15 feet or about l-10th of the length.
The whole weight of the girders for a double line is 128 tons.
CHARING-CROSS LATTICE BRIDGE, f
543. — This bridge was erected by Mr. Hawkshaw to carry the
Charing-Cross Railway across the Thames on the site of the
Hungerford Suspension Bridge, the chains of which were removed
to Clifton. It comprises nine independent spans, six of 154 feet
and three of 100 feet. The leading particulars of one of the 154
feet spans are as follows. The main girders are wrought-iron
lattice tubular girders, the web consisting of two systems of nearly
right-angled triangles. The tension diagonals are Howard's patent-
rolled suspension links, and the compression diagonals are forged
bars, varying in thickness from 2J to 3 inches, and united in
pairs by zigzag internal cross-bracing. The flanges are formed of
horizontal plates in piles, with four vertical ribs attached by angle
irons to the horizontal plates, the two outer ribs being 2 feet deep
and the two inner ones 21 inches deep. The flanges therefore
resemble the usual trough section, but with 4, in place of 2 ver-
tical ribs (439). The diagonals have enlarged ends with eyes,
and are attached to the vertical ribs by turned pins of puddled
steel. In addition to the diagonals already mentioned, there are
vertical bars 1 inch thick connecting each pin in the upper flange
with that in the flange directly beneath ; these vertical bars form
diagonals to the squares made by the diagonal bracing and are
superfluous (191). The extreme length of the main girders is 164
feet, their extreme depth is 14 feet, and the depth from centre to
centre of pins is 10 feet 9 inches, but the distance between the
* Encyclopaedia Britannica, Art. " Iron Bridges," p. 605.
f Proc. Inst. C. E., Vol. xxii. ; and Trans. Soc. Eng. for 1864.
588 CONWAY PLATE TUBULAR BRIDGE. [APP.
centres of gravity of the flanges is 12 feet 9 inches, or nearly
1-1 2th of the clear span, and this seems to have been assumed to
be the correct depth for calculating the working strains, which
with 1J ton per foot on each line, are stated to be 5 tons tension
per square inch of net section, and 4 tons compression per square
inch of gross section. The cross-girders are attached to the under
sides of the lower flanges, and project beyond them with cantilever
ends which support footpaths 7 feet wide. These cross-girders are
1 1 feet apart and correspond with the apices of the diagonals in the
lower flanges. There are four lines of railway and the width in the
clear between the main girders is 46 feet 4 inches. The weight of
iron in one main girder, including the end pillars, is as follows : —
Tons. Cwts. Qrs.
Top flange, 70 4 2
Bottom do., 67 15 2
Web, 46 0 0
End pillars, 600
Weight of iron in one main girder, 19O O O
Taking the rolling load at 1J tons per foot of single line, the
maximum distributed load on each main girder is nearly as
follows: —
Tons.
Rolling load on two lines = 156 X 2J tons, - 390
One main girder, deducting end pillars, - 184
One half the cross-girders and cantilevers, - 67
Rails for two lines, 7
Timber in the half platform and longitudinals under rails, 41
Load of people on one footpath at 100 Ibs. per square foot, 48 J
Total distributed load on one girder, - 737 J
The foregoing load is exclusive of cornice, hand-rail, fish-plates,
bolts, spikes, chairs for rails, hoop-iron tongue and bolts for planking
and ballast.
CONWAY PLATE TUBULAR BRIDGE.*
543. — The Conway bridge was erected by Mr. Robert Stephenson
* Clark on the Tubular Bridges.
APP.]
CONWAY PLATE TUBULAR BRIDGE.
589
to carry the Chester and Holyhead Eailway over the river Con way,
in North Wales. It consists of two wrought-iron plate tubular
bridges placed side by side, with one line of railway in each tube.
The entire length of each tube is 424 feet, the clear span is
400 feet, and the effective length for calculation 412 feet. The
external depth at the centre is 25 feet 6 inches, or nearly 1-1 6th of
the length, thence it diminishes gradually towards the ends where
it is 22 feet 6 inches. The external width is 14 feet 9 inches; the
clear width inside is about 12 feet 6 inches. The tubes are placed
9 feet apart and are not connected in any way.
TABLE XII.— TABULAR STATEMENT OF WROUGHT-IRON WORK IN THE CONWAY
BRIDGE — ONE TUBE, SINGLE LINE, LENGTH 424 FEET.
Upper Flange.
Sides.
Lower Flange.
Summary.
tons.
tons.
tons.
tons.
Plates, -
239
201
242
682
Angle and T-iron,
115
146
59
320
Covers, -
15
22
77
114
Rivet heads,
Total,
23
24
17
64
392
393
395
1180
Plates, 58 per cent. ; angle and T-iron, 27 per cent. ; covers, 10 per cent. ; rivet-
heads, 5 per cent. ; total, 100.
The following is an analysis of the wrought-iron in one tube
412 feet long, i.e., 6 feet longer at each end than the clear span.
This was the length of the tube when floated into its place between
the abutments ; 6 feet were afterwards added to each end.
TOP FLANGE.
Plates and angle-iron in compression,
Plates and angle-iron acting as covers,
Transverse keelsons,
Eivet-heads,
Tons. Cwts.
336 0
17 8
7 0
22 7
382 15
Per cent.
87-5
4-5
2-0
6-0
100-0
590
CONWAY PLATE TUBULAR BRIDGE.
SIDES.
Tons. Cwts.
Plates acting as sides, - - 163 0
Covers and proportion of T-iron acting as
covers, - 90 10
Gussets, stiffeners, arid projecting rib of
T-iron engaged in stiffening the sides, 101 16
llivet-heads, 23 15
379 1
LOWER FLANGE.
Plates and angle-iron in tension,
Plates and angle-iron acting as covers,
Transverse keelsons,
Rivet-heads,
[APP.
Per cent.
43-0
24-0
27-0
6-0
100-0
Tons. Cwts.
Per cent.
279 9
72-5
76 6
20-0
14 0
3-5
15 17
4-0
385 12
100-0
This makes the total weight of wrought-iron in 412 feet of
one tube = 1147*4 tons, or 2 '78 tons per running foot for each
line. The weight of wrought-iron in each tube, 400 feet long in
the clear, is 1112 tons.
Summary of cast-iron work in the Conway Bridge for both
lines : —
Tons.
Castings fixed in the ends of tubes, - - 201
Bed-plates, rollers, &c., - 108
Castings fixed in the masonry, - 325
Total weight of castings for both tubes, 634
The working inch-strains, as already given in Table VII. (481),
are 6' 32 tons tension and 4'924 tons compression with a train-load
of J ton per foot uniformly distributed.
The mean deflection of the two tubes, immediately after the
removal of the platform on which they were built, was 8 '04 inches
APP.] BKOTHERTON PLATE TUBULAR BRIDGE. 591
which became 8'98 inches after they took a permanent set due to
the strain (41 0). The deflection, from additional weight placed at
the centre, is "01104 inch per ton. The difference of deflection due
to change of temperature, between noon and midnight on the 5th
of July, 1848, was 1'56 inches (419).
BROTHERTON PLATE TUBULAR BRIDGE.*
544. — The Brotherton bridge, on the York and North Midland
Railway is a tubular plate bridge with one line of railway in each
tube. The span is 225 feet, the depth 20 feet or 1-1 1th of the
span nearly, and the width of each tube between the side plates
is 11 feet.
The weight of one tube is as follows : —
Wrought-iron between the bearings, - 198 tons.
Wrought-iron on the bearings, - - 13 ,,
Cast-iron on the bearings, - 14J ,,
Cast-iron in rollers and plates, - 9^ „
Total weight of iron for one line of railway, 235 tons.
The top flange is composed of a single plate in thickness, and
no cells whatever have been used either in top or bottom.
545. Size and weights of various materials. — The following
tables refer chiefly to the size and weights of various materials, and
will be found useful for reference.
* Encycl. Brit., Art. " Iron Bridges," p. 609.
592
WIRE AND SHEET METAL GAGES.
[APP.
TABLE XIII.— VALUES OP GAGES FOR WIRE AND SHEET METALS IN GENERAL
USE, EXPRESSED IN DECIMAL PARTS OP THE INCH.*
Birmingham Wire
Gage for Wire,
and for Sheet Iron
and Sheet SteeL
Birmingham
Metal Gage for
Sheet Metals, Brass,
Gold, Silver, Zinc,
&c.
Lancashire Gage for round Steel Wire, and also for
Pinion Wire.
The smaller sizes are distinguished by numbers.
The larger by letters, and called the Letter Gage.
Mark. Size.
Mark. Size.
Mark. Size.
Mark. Size.
Mark. Size.
0000 — '454
1 — '004
80 — '013
40 — -096
A — -234
000 — '425
2 — -005
79 — -014
39 _ -098
B — -238
00 — -380
3 — -008
78 — -015
38 — -100
C — -242
0 — '340
4 — -010
77__-Ol6
37 — -102
D — -246
1 — -300
5 — -012
76 — -018
36 — -105
E — -250
2 — -284
6 — -013
75 — -019
35 — -107
F — -257
3 — -259
7 — '015
74 — -022
34 — -109
G — -261
4 — -238
8 — -016
73 — -023
33 — -111
H — -266
5 — -220
9 — -019
72— -024
32 — -115
I — -272
6 — -203
10 — -024
71 — '026
31 — -118
J — -277
7 — "180
11 — -029
70 _ -027
30 — -125
K — -281
8 — -165
12 — -034
69 — -029
29—134
L — -290
9 — -148
13 — -036
68 — -030
28 — -138
M — -295
10 — -134
14— -041
67 — -031
27 — -141
N — -302
11 — -120
15 _ -047
66 — -032
26 — -143
0 — -316
12 — -109
16 — -051
65 — -033
25 — "146
P — -323
13 — '095
17_ -057
64 — -034
24— -148
Q — -332
14 _ -083
18 — -061
63 — -035
23 — -150
B — -339
15 — -072
19_ -064
62 — -036
22 — '152
S — '348
16 _ -065
20 — -067
61 — -038
21 — -157
T — '358
17 — '058
21 — -072
60 — -039
20— -160
U — '368
18 — -049
22 — -074
59- -040
19— '164
V — -377
19 _ -042
23 — -077
58 — -041
18 — -167
W — '386
* From Holtzapffel's Mechanical Manipulation.
APP.]
WIRE AND SHEET METAL GAUGES.
593
TABLE XIII. — VALUES OP GAUGES FOE WIRE AND SHEET METALS IN GENERAL
USE, EXPRESSED IN DECIMAL PARTS OP THE INCH — continued.
Birmingham Wire
Gauge for Iron Wire,
and for Sheet Iron
and Sheet Steel.
Birmingham Metal
Gauge for Sheet
Metals, Brass, Gold,
Silver, Zinc, <fec.
Lancashire Gauge for round Steel Wire, and also for
Pinion Wire.
The smaller sizes are distinguished by numbers.
The larger by letters, and called the Letter Gauge.
Mark. Size.
Mark. Size.
Mark. Size.
Mark. Size.
Mark. Size.
20 — -035
24 — -082
57 _ -042
17 — '169
X — -397
21 — '032
25 — '095
56_ -044
16 — -174
Y — -404
22 — -028
26 — '103
55— -050
15 — -175
Z — -413
23 — -025
27 — 113
54 _ -055
14 — -177
A 1 — -420
24 — -022
28 — -120
53 — -058
13 — -180
Bl — -431
25 _ -020
29— '124
52 — -060
12 - -185
C 1 — -443
26 — -018
30 — -126
51 — "064
11 — -189
D 1 — -452
27 — '016
31 — -133
50 — -067
10 — -190
El — -462
28 — -014
32 — -143
49 — -070
9 — -191
F 1 — -475
29- -013
33 — -145
48 — -073
8 —192
G 1 — -484
30 — -012
34 — -148
47 _ -076
7 — 195
H 1 — -494
31 — -010
35 — -158
46 _ -07 8
6 — 198
32 — -009
36 _ -167
45— '080
5 — '201
33 _ -008
44 _ -084
4— -204
34 — -007
43 — '086
3 — '209
35 — '005
42 — -091
2 — -219
36_ -004
41 _ -095
1 — -227
Column 1 refers to the gauge commonly called the Birmingham Wire Gauge, which is
employed for iron, brass and other wires, for black steel wire, for sheet iron, sheet
steel and various other materials.
The gauge referred to in the second column is called the Birmingham Metal Gauge
or the Plate Gauge, and is employed for most of the sheet metals, excepting iron and
steel.
8 Q
594 WEIGHT OF METALS. [APP.
TABLE XIV. — WEIGHT OF A SUPERFICIAL FOOT OF VARIOUS METALS IN LBS.
THICKNESS BY THE BIRMINGHAM WIRE GAUGE.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Wrought-)
Iron, }
Copper, -
Brass, -
12-50
14-50
1375
12-00
13-90
13-10
11-00
12-75
12-10
10-00
11-60
11-00
8-74
10-10
9-61
812
9-40
8-93
7-50
8-70
8-25
6-86
7-90
7'54
6-24
7-20
6-86
5-62
6-50
6-18
5-00
5-80
5-50
4-38
5-08
4-sa
3-75
4-34
4-12
3-12
3-60
3-43
2-82
3-27
3-10
Wrought-)
Iron, j
Copper, -
Brass, •
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
2-50
2-90
2-75
2-18
2-52
2-40
1-86
2-15
2-04
1-70
1-97
1-87
1-54
1-78
1-69
1-40
1-62
1-54
1-25
1-45
1-37
112
1-30
1-23
1-00
1-16
110
•90
1-04
•99
•80
•92
•88
•72
•83
•79
•64
•74
•70
•56
•64
•61
•50
•58
•55
THICKNESS IN PARTS OF AN INCH.
T6
i
tV
i
A
1
A
4
1
I
1
1
Wrought-)
Iron, j
2-5
5-0
7-5
10-0
12-5
15-
17'5
20-
25'
30-
35-
40-
Copper, -
2-9
5-8
8-7
11-6
14-5
17-2
20-0
23-2
28-9
34-3
40-4
46-2
Brass, -
2-7
5-5
8-2
11-0
13-7
16-4
19-0
21-8
27-4
32-5
37-9
43-3
Lead,
3-7
7-4
11-1
14-8
18-5
22-2
25-9
29'6
37-0
44-4
51-8
59-2
Zinc,
2-3
4-7
7-0
9-4
11-7
14-0
16-4
18-7
23-4
28-1
32-8
37-5
It is useful to recollect that a square foot of plate-iron, | inch thick, weighs 10 fts.
APP.]
WEIGHT OF METALS.
599
TABLE XV. — WEIGHT OP A LINEAL FOOT OP ROUND AND SQUARE BAR IRON
LBS.— (Molesworth).
Breadth
or diam.
in inches.
Square
Bars.
Round
Bars.
Breadth
or diam.
n inches.
Square
Bars.
Round
Bars.
Breadth
or diam.
in inches.
Square
Bars.
Round
Bars.
*
•209
•164
u
5'25
4-09
3
30-07
23-60
A
•326
•256
If
6-35
4-96
81
35-28
27-70
t
•470
•369
ii
7-51
u-90
8i
40-91
32-13
ft
•640
•502
it
8-82
6-92
3f
46-97
36-89
i
•835
•656
if
10-29
8-03
4
53-44
41-97
T96
1-057
•831
if
11-74
9-22
*J
60-32
47-38
1
1-305
1-025
2
13-36
10-49
44
67-63
53-12
«
1-579
1-241
2i
15-08
11-84
4f
75-35
59-18
1
1-879
1-476
2J
16-91
13-27
5
83-51
65-58
it
2-205
1-732
2f
18-84
14-79
5|
92-46
72-30
i
2-556
2-011
24
20-87
16-39
5i
101-03
79-35
if
2-936
2-306
2|
23-11
18-07
5|
110-43
86-73
1
3-34
2-62
2f
25-26
19-84
6
120-24
94-43
H
4-22
3-32
21
27-61
21-68
—
—
—
To convert into weight of
steel X 1-02, for copper X 1
other metals, multiply tabular No. for cast-iron by '93, for
•15, for brass X 1'09, for lead X T47, for zinc X '92.
596
WEIGHT OF TIMBER.
[APP.
TABLE XVI. — SPECIFIC GRAVITY AND WEIGHT OF A CUBIC FOOT OF DIFFERENT
WOODS/
Kind of Wood, and state.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Kind of Wood, and state.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Abele, dry, -
•oil T.
32-00
Chestnut (horse), dry, -
•596 T.
37-28
Acacia (false), green, -
•820 E.
51-25
Do. do., another
•483 T.
30-18
Do., dry, -
•791 H.
49-43
specimen, dry,
Do., dry,
•748 T.
4675
Cocoa wood,
1-040 M.
65-00
Do., (three-thorned), -
•676 H.
42-25
Cork, -
•240 M.
15-00
Alder, -
•800 M.
50-00
Cowrie,
•579
36-20
Do., dry,
•555 E.
34-68
Crab tree, meanly dry,
•765 P.
47-81
Almond tree, -
1-102 H.
68-87
Cypress,
•655 H.
40-93
Apple tree,
•793 M.
49-56
Do. (Spanish),
•644 M.
40-25
Apricot tree, -
•789 H.
49-31
Deal, white. See fir.
Arbor vitse (Chinese), -
•560 H.
35-00
Do., yellow. See pine.
Ash (heart -wood), dry,
•845 P.
52-81
Ebony (American),
1-331 M.
83-18
Do., dry,
•832 W.
52-00
Do. (Indian),
1-209 M.
75-56
Do., young wood, dry,
•811 T.
50-68
Do. -
1-108 R.
69-25
Do. -
•800 J.
50-00
Elder tree,
•695 M.
43-43
Do. -
•760 B.
47-50
Elm, green,
•940 C.
58-75
Do. (old tree), dry, -
•753 T.
47-06
Do. -
•693 S.
44-41
Do., dry,
•690 E.
43-12
Do., seasoned,
•588 C.
3675
Bay tree,
•822 M.
51-37
Do. -
•553 B.
34-56
Beech (meanly dry), -
•854 P.
53-37
Do. (common), dry, -
•544 E.
34-00
Do. -
•852 M.
53-25
Do., wych, young tree,
•763 E.
47-68
Do. -
•720 H.
45-00
green,
Do. -
•696 B.
43-50
Do. do., dry,
•684 T.
42-75
Do., dry,
•690 E.
43-12
Filbert tree, •
•600 M.
37-50
Birch, dry,
•720 E.
45-00
Fir (Norway spruce), -
•512 T.
32-00
Box (Dutch), -
1-328M.
83-00
Do. (white American
•465 T.
29-06
Do., dry,
1-030 J.
64-37
spruce),
Do. -
1-031 P.
64-43
Do. (silver green),
•531 Wi.
33-20
Do ffr°m
1-024 B.
64-00
Do., dry,
•403 Wi.
25-22
Do' ' \to
•960 B.
60-00
Do. (Scotch). See
Do., dry,
•950 W.
59-37
pine.
Do., Turkey, -
•949 K.
59-31
Fustic,
•817 R.
51-06
Brazil wood (red),
1-031 M.
64-43
Hazel,
•606 M.
37-87
Canary wood, -
•723 R.
45-18
Hickery,
•929 S.
58-06
<~^edar (Indian),
1-315 M.
82-18
Hornbeam,
•760 H.
47-50
Do. (Canadian),
•753 C.
47-06
Jasamine (Spanish),
•770 M.
48-12
Do. (Virginian red\dry,
•650 T.
46-62
Juniper wood, -
•556
34-75
Do. (Palestine),
•596 M.
37-25
Laburnum,
•843 T.
52-70
Do. (American),
•560 M.I 35-00
Lance wood, -
1-038 L.
64-87
Do. do., seasoned,
•453 C.
28-31
Do. do., dry,
•943 R.
58-93
Cedar of Libanus,
•603 H.
37-68
Larch, green, -
•858 Wi.
5363
Do. do., dry, -
•486 T.
30-37
Do. (redwood),seasoned
•640 T.
40-00
Cherry tree,
•741 H.
46-31
Do., dry,
•612WL
38-31
Do. do., dry,
•672 T.
42-00
Do., dry,
•496 T.
31-00
Chestnut (sweet), green,
•875 E.
54-68
Do. (white wood),
•364 T.
22-75
Do. -
•685 H.
42-81
seasoned,
Do. do., dry, -
•606 T.
37-95
Lemon tree,
•703
43-93
Do., another specimen,
•535 T.
33-45
Letter wood, -
1-286 C.
80-37
dry,
Lignum vitae, -
1-333 M.
83-31
Do. (horse), -
•657 H.
41-06
Do.
1-327 P.
82-93
* Tredgold's Carpentry, p. 298.
APP.]
WEIGHT OF TIMBER.
597
TABLE XVI.— SPECIFIC GRAVITY AND WEIGHT OP A CUBIC FOOT OP DIFFERENT
WOODS — continued.
Weight
Weight
Kind of Wood, and state.
Specific
gravity.
of a
cubic
foot in
Kind of Wood, and state.
Specific
gravity.
of a
cubic
foot In
pounds.
pounds.
Lime tree,
•604 M.
37-75
Pine (planted Scotch),
•529 T.
33-06
Do.
•564 H.
3525
dry,
Do.
•480 T.
30-00
Do. (Scotch), dry,
•429Wi.
26-81
Logwood,
Mahogany (Spanish), dry
•913 P.
•852 T.
57-06
53-30
Do.(Memel),dry £°m
•553
•544 T.
34-56
34-00
Do. dry,
Do. (Honduras), dry, -
•816 W.
•560 T.
51-00
35-00
Do. (Riga), dry, f™m
•480
•466 T.
30-00
29-12
Maple (Norway),
•795 L.
49-68
Do. (Weymouth), dry,
•460 T.
2875
Do. dry,
•755 P.
47-18
Do. (American), dry, -
•368 T.
23-00
Do. (common), dry, -
•624 T.
3275
Plane (occidental), dry,
•648 E.
40-50
Medlar tree, -
•944 M.
59-00
Do. (oriental),
•538 H.
33-62
Mulberry tree (Spanish),
•897 M.
56-06
Plane tree (common).
Oak (live), half seasoned,
l'216Ch.
76-03
See sycamore.
Do. (English green), -
1-113 C.
69-56
Plum tree,
•785 M.
49-06
Do. (French green), -
l'063Bu.
66-43
Do.
•663 P.
41-43
Do. (Irish bog),
1-046 C.
65-37
Poona (seasoned),
•635 C.
39-95
Do. (evergreen),
•994 H.
62-25
Poplar (Spanish, white),
•529 M.
33-06
Do. (Adriatic),
•993 B.
62-06
Do. (black), dry,
•421 T.
26-31
Do. (black bog), dry, -
•965 R.
60-31
Do. (Lombardy), dry,
•374 E.
24-37
Do. (white American),
•908 Ch.
56-75
Quince tree, -
•705 M.
44-00
half seasoned,
Sassafras,
•482 P.
30-12
Do. (Quercus sessili-
•879 T.
54-97
Satin wood,
•952 R.
59-50
flora),
Saul (Bengal), seasoned,
•994 L.
62-12
Do. (American white),
•840 H.
52-50
Service tree,
•742 H.
46-37
Do. ( Provence), sea-
•828 D.
5175
Sissoo (Bengal), seasoned,
•889 L.
55-52
soned.
Stinkwood ^seasoned), -
•681 C.
42-56
Do. (Quercus robur),
•807 T.
50-47
Sycamore,
•645 H.
40-31
dry,
Do., dry, -
•590 E.
36-87
Do. (English), seasoned,
•777 C.
48-56
Teak, dry, ' -
•832 Ch.
52-00
Do. (Dan tzic), seasoned,
•755 T.
47-24
Do.
•745 B.
46-56
Do. (American), red, -
•752 L.
47-00
Do., seasoned,
•657 C.
41-06
Do. (Riga), dry,
•688 T.
43-00
Tulip tree,
•477 H.
29-81
Do. (English), from an
•625 T.
39-06
Vine,
1-237 M.
77-31
old tree, dry,
Walnut tree, green,
•920 E.
57-50
Olive tree,
•927 M.
57-93
Do. (American),
•735 H.
45-93
Orange tree, -
•705 M.
44-06
Do. (French),
•671 M.
41-93
Pear tree, dry,
•708 T.
44-25
Do., dry,
•616 T.
38-50
Do.
•646 B.
40-37
Willow, green,
•619 E.
38-68
Pine (American pitch),
•936 T.
58-5
Do drv \ from
•568
3550
dry,
Do-' dr^' j to
•404 T.
25-25
Do. (do.), seasoned,
•741 C.
46-31
Yellow wood (seasoned),
•657 C.
41-06
Do. (pinaster), green,
•837 Wi.
52-35
Yew (Spanish),
•807 M.
50-43
Do. (Scotch), green, -
•816 Wi.
51-08
Do. (Dutch),
•788 M.
49-25
Do. (Mar Forest),
•696 B.
43-50
Do. -
•788 H.
48-62
The letters following the specific gravities refer to the authorities — B., Barlow ;
Bu., Buffon ; C., Couche ; Ch., from Chapman on Preservation of Timber; E., Ebbels ;
H., from Rondelet's table ; J., Jurin; L., Layman; M., Muschenbroek; P., Philosophical
Transactions, Vol. i., Lowthorp's Abridgement; R., Ralph Tredgold ; S., Scoresby; T.,
Tredgold ; W., Watson (Bishop) ; Wi., Wiebeking.
598 WEIGHTS OF VARIOUS MATERIALS. [APP.
TABLE XVII. — SPECIFIC GRAVITY AND WEIGHT OF A CUBIC FOOT OF VARIOUS MATERIALS. '
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Air (atmospheric),
•0012
•075
Coal (Newcastle
1-269 Th.
79-31
Alabaster. See gypsum.
caking),
Ballast, drained,
97-4
Concrete, Ballast and
4-464
140-00
•ROC,«H i from
3-00
187-50
Portland Cement,
Basalt, j to
2-478
154-87
Copper (British sheet),
8-785 Ha.
549-06
Do. (Fairhead),
2-95 K.
184-37
Do. (British cast),
8-607 Ha.
537-93
Do. (Derbyshire),
Do. (Giant's Cause-
2-921 W.
2-90 K.
182-56
181-25
Earth (common), j ^om
1-520
1-984
95-00
124-00
way),
Do. (loamy or strong),
2-016
126-00
Do. do.
2-864 Br.
179-00
Do. (rammed),
1-584 Pa.
99-00
Do. (Rowley rag), -
2-478 K.
154-87
Do. (loose or sandy), -
1-520
95-00
Bees' wax (yellow),
•965
60-31
Firestone. See stone.
1-800
112-50
Bismuth (cast),
9-822
613-87
Flint \ from
2-580
161-25
Bitumen, of Judea,
1-104
69-00
mt' j to
2-630 Th.
164-37
Bone, Beef, -
2-08f
Do. (black Cambridge)
2-592 W.
162-00
Brass (wire drawn),
8-544
534-00
Freestone. See stone.
Do. (plate), -
8-441 W.
527-56
Glass, white flint,
3-000
187-50
Do. (cast), -
8-100 P.
506-25
Do., plate,
2-760
172-50
Brick (common), j *™m
1-557
2-000
97-31
125-00
Do., crown, -
Gold, pure cast,
2-520
19-361 Br.
157-50
1210-06
Do. (red), -
2-168 Re.
135-50
Do., standard,
17724 Th.
110775
Do. (pale red),
2-085 Re.
130-31
Cranitp i fr°m
2-999
187-47
Do., -
1-857 Be.
116-06
' j to
2-538 K.
158-62
Do. (common London
1-841 T.
115-06
Do. (Guernsey),
2-999 W.
187-47
stock),
Do. (Aberdeen gray),
2-664 R.
166-5
Do. paving (English
1-653 R.
103-31
Do. (Cornish),
2-662 Re.
166-37
clinker),
Do. (do.),
2-653 R.
165-81
Do. (Dutch clinker), -
1-482 R.
92-62
Do. (Aberdeen red), -
2-643 R.
165-18
Do. (Welsh fire),
2-408 T.
150-50
Do. (Cornish), -
2-624 T.
164-00
Brickwork, about
95-00
Gravel,
1-749 P.
109-32
Broken stone. See stone.
Gunpowder (solid),
1-745
109-06
Cement (Roman) and
1-817 T.
113-56
Do. (shaken), -
•922
57-62
sand in equal parts,
Gypsum (plaster stone),
2-286 W.
142-87
Do., alone (cast),
Pliallr i from
1-600 R.
2-315
100-00
144-68
Iron (bar), j ^om
7-600
7-800 K.
475-00
487-50
alk' j to
2-657 Th.
166-06
Do., hammered,
7-763 M.
48518
Do. (C ambridge clunch)
2-657 W.
166-06
Do., not hammered, -
7-600 M.
475-00
Do. (Dorking),
1-169 R.
116-81
f)n /._afv j from
7-600
475-00
Charcoal from birch, -
•542 K.
33-87
Do. (cast), j to
7-200 Th.
450-00
Do. from fir, -
•441 K.
27-56
Do. (horizontal ditto),
7-113 Re.
444-56
Do. from oak,
•332 K.
20-75
Do. (vertical castings),
7-074 Re.
442-12
Do. from pine,
•280 K.
17-50
Ivory, -
1-826 P.
114-12
Clay(potter's), j £om
1-800
2-085 K.
112-50
130-31
Lead (milled), -
Do. (cast), -
11-407 Th.
11-352 Br.
712-93
709-50
Do. (common),
1-919 Be.
119-93
Do., black. See Plum-
Do., with gravel,
2-560
160-00
bago.
Do., puddling,
113-35
Lime, quick, -
•843 Be.
52-68
Do., slate. See slate.
Limestone. See stone
Coke, -
•744 K.
46-50 •
and marble.
Coal (Kilkenny),
1-526 K.
95-37
Loam. See earth,
Do. (Glasgow splint),
Do. (Cannel),
1-290 Th.
1-272 Th.
80-62
79-50
Marble, j £°m
2-840
2-580
177-50
161-25
* Tredgold's Carpentry, p. 300.
+ Bevan, Phil. Mag. 1826, p. 181.
APP.]
WEIGHTS OF VARIOUS MATERIALS.
TABLE XVII.— SPECIFIC GRAVITY AND WEIGHT OF A CUBIC FOOT OF VARIOUS MATERIALS—
continued.
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
font in
pounds.
Marble, Parian white,
2-837 K.
177-31
Plumbago,or black lead,
2-267
141-68
Do., veined white,
2-726 Re.
170-37
Porphyry (green),
2-875
179-68
Do., Carrara white, -
2-717 K.
169-81
Do. (red),
2-793
174-56
Do., do. blue, -
Do., Italian black, -
2-713 K.
2712 K.
169-56
169-50
Potstone,
3-000
2-768 K.
187-50
173-00
Do., Derbyshire entro-
chal,
2-709 R.
169-31
Puzzolana,
2-570
2-850 K.
160-62
178-12
Do., Saxon gray,
2-700 K.
168-75
Quartz (crystallized), -
2-655
165-93
Do., Brabant black, -
2-697 Re.
168-56
Roe-stone. See stone.
Do., Derbyshire black,
2-690 W.
168-12
Road-grit. See sand.
Do., Namur black, -
2-682 R.
167-62
Road metal. See stone.
Do., Sienna yellow, -
Do., Pallion brown
2-677 K.
2-586 R.
167-31
161-62
Rubble masonry,
j from
jto
145-00
160-00
figured,
Sand (pure quartz), -
2-750
171-87
TV/T i \ from
1-600
100-00
Do., river,
1-886 Be.
117-87
Marl, j to
2-870 Th.
179-37
Do., River Thames
1-638 T.
102-37
Mercury (fluid),
13-568 Br.
848-00
(best),
Mortar,
1-715 Be.
10718
Do., pit (clean but
1-610 T.
100-62
Do. of river sand three
1-615 Ro.
100-93
coarse),
parts, of lime in
Do., pit (fine-grained
1-523 T.
95-18
paste two parts,
Do., do., do., well beat
1-893 Ro.
118-31
and clean),
Do., scraped from
1-494 T.
93-37
together,
London roads
Do. of pit sand three
1-588 Ro.
99-25
(road-grit),
parts, of lime in
Do., pit (very fine
1-480 T.
92-50
paste two parts.
grained),
Do., do., do., well beat
1-903 Ro.
118-93
Do., River Thames
1-454 T.
90-87
together,
(inferior),
Do. of pounded tile
1-457 Ro.
91-06
Sandstone. See stone.
three parts, of
Serpentine, Anglesey
2-683 R.
167-68
quick-lime two
green,
parts,
Do., blackish green, -
2-574 K.
160-87
Do., do., do., well beat
1-663 Ro.
103-93
Do., dark reddish
2-561 K.
160-06
together,
brown,
Do., common, of chalk
1-550 R.
96-87
Silver, pure cast,
10-474 Br.
654-62
lime, and sand,
Do., standard,
10-312 Th.
644-50
dry,
Slate, Welsh, -
2-888 K.
180-50
Do., the lining of an
antique reservoir
1-549 Ro.
96-81
Do., Anglesey,
Do., Westmoreland,
2-876 K.
2-791 W.
179-75
174-43
near Rome,
pale blue,
Do., from the interior
1-414 Ro.
88-37
Do., do., dark blue, -
2-781 W.
173-81
of an old wall,
Do., do., pale greenish
2-768 W.
173-00
Rome
blue,
Do., lime, sand, and
1-384 R.
86-50
Do., do., blackish blue,
2-758 W.
172-37
used for floors,
plastering, dry,
Oolite. See stone, roe.
Peat, hard,
1-329
83-06
Do., Welsh rag,
Do., Westmoreland,
fine grained pale
2-752 K.
2-732 W.
172-00
170-75
Pebble (English),
Pewter,
2-609
7-248
163-06
453-00
blue,
Do., Cornwall, greyish
2-512 K.
157-00
Pitch, -
Plaster (cast), -
1-150 P.
1-286 Be.
71-87
80-37
blue,
Stone, Bath (roe-stone),
2-494 K
155-87
Platina pure, -
21-531 Th.
1345-68
Do., do.
1-975 R.
123-43
600
WEIGHTS OF VARIOUS MATERIALS.
[APP.
TABLE XVII. — SPECIFIC GRAVITY AND WEIGHT OP A CUBIC FOOT OF VARIOUS MATERIALS —
continued.
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
foot in
pounds.
Name of the Substance.
Specific
gravity.
Weight
of a
cubic
foot in
pounds
Stone, blue lias (lime-
2-467 R.
154-18
Stone, Portland (roe-
2-423 Re.
151-43
stone),
stone),
Do., Bromley-fall
2-506 Re.
156-62
Do., do., do.,
2113 R.
132-06
(sandstone),
Do., pumice,
•629 R.
39-31
Do., do.,
2-261 R.
141-31
Do., Purbeck,
2-680 W.
167-50
Do., Bristol stone, -
2-510
156-87
Do., do.,
2-599 Re.
It52'48
Do., Burford (dry
2 049 P.
128-06
Do., Roach Abbey
1-893 R.
118-31
piece),
(rnagnesian lime-
Do., Caen (calcareous
2108 R.
13175
stone),
sandstone),
Do. (Tottenhoe cal-
1-800 T.
112-50
Do., Clitheroe lime-
2-686 W.
167-87
careous sandstone),
stone,
Do., Woodstock flag-
2-614 K.
163-37
Do., Collalo, white
2-423 Re.
151-43
stone,
(sandstone),
Do., Yorkshire paving,
2-507 Re.
156-68
Do., do.,
2-040 R.
127-50
Do., do., do.,
2-356 R.
147-25
Do.. Craigleith, sand-
2-452 Re.
153-25
Stone, limestone broken
1-44
90-00-
stone,
to go through a
Do., do.,
2-360 R.
147-50
two-inch ring,
Do., Derbyshire (red
2-346 Re.
146-62
Stonework, mean
107-00?
friable sandstone),
weight according
Do., Dundee,
2-530 Re.
158-12
to Belidor, about
Do., Ho.,
2-517 T.
157-31
Shingle,
1-424 Pa.
89-00
Do. (grindstone),
2-143
133-93
Steel - !fr°m
7-780
486-25
Do., Heading-stone,
2-029 P.
126-81
\ to
7-840 Th.
490-00
lax kind,
Syenite (Mount Sorrel),
2-621
163-81
Do., Hilton (sand-
2-177 R.
136-06
Tile (common plain), -
1-853 R.
116-15
stone),
Do., -
1-81 5 Be.
113-43
Do., Kentish rag,
2-675 R.
167-18
Tin, hammered,
7-299 Br.
456-18
Do., Ketton (roe-stone)
2-494 K.
155-87
Do., pure cast,
7-291 Br.
455-68
Do., do.,
2-058 R.
128-62
Toadstone (Derbyshire),
2-921 W.
18-2-56
Do., Kincardine (sand-
2-448 T.
153-00
Tufa (Roman),
1-217 Ro.
76-06
stone),
Water, sea, -
1-027 Th.
64-18
Do., Limerick (black
2-598 Re.
162-37
Do., rain,
1-000
62-50
compact limestone),
Wheat,
•64
48-00
Do., Pennarth (lime-
2-653 W.
165-81
Whinstone (Scotch), -
2-760 W. .
172-50
stone),
Wood ashes, -
•933 P.
58-32
Do., Portland (roe-
2-461 W.
153-81
Wood petrified,
2-341 P.
146-31
stone),
Zinc, -
7-028 W.
439-25
Part of the letters of reference are explained in a note to the preceding table. The rest
are as follows: — Be., Belidor; Br. Brisson ; Ha., Hatchet; K., from Kirwan's Mineralogy;
Re., Rennie, Phil. Magazine, Vol. liii. ; Ro., Rondelet ; Th., from Dr. Thomson's System of
Chemistry, 5th edition ; Pa., Pasley, Course of Military Instruction.
APP.] TONS CONVERTED INTO LBS. AVOIRDUPOIS. 001
TABLE XVIII.— FOR CONVERTING TONS INTO LBS. AVOIRDUPOIS.
Tons.
Lbs.
Tons.
Lbs.
Tons.
Lbs.
Tons.
Lbs.
0-05
112
12
26,880
42
94,080
72
161,280
010
224
13
29,120
43
96,320
73
163,520
0-15
336
14
31,360
44
98,560
74
165,760
0-20
448
15
33,600
45
100,800
75
168,000
0-25
560
16
35,840
46
103,040
76
170,240
0-30
672
17
38,080
47
105,280
77
172,480
0-35
784
18
40,320
48
107,520
78
174,720
0-40
896
19
42,560
49
109,760
79
176,960
0-45
1,008
20
44,800
50
112,000
80
179,200
0-50
1,120
21
47,040
51
114,240
81
181,440
0-55
1,232
22
49,280
52
116,480
82
183,680
0-60
1,344
23
51,520
53
118,720
83
185,920
0-65
1,456
24
53,760
54
120,960
84
188,160
070
1,568
25
56,000
55
123,200
85
190,400
075
1,680
26
58,240
56
125,440
86
192,640
0-80
1,792
27
60,480
57
127,680
87
194,880
0-85
1,904
28
62,720
58
129,920
88
197,120
0-90
2,016
29
64,960
59
132,160
89
199,360
0-95
2,128
30
67,200
60
134,400
90
201,600
1
2,240
31
69,440
61
136,640
91
203,840
2
4,480
32
71,680
62
138,880
92
206,080
3
6,720
33
73,920
63
141,120
93
208,320
4
8,960
34
76,160
64
143,360
94
210,560
5
11,200
35
78,400
65
145,600
95
212,800
6
13,440
36
80,640
66
147,840
96
215,040
7
15,680
37
82,880
67
150,080
97
217,280
8
17,920
38
85,120
68
152,320
98
219,520
9
20,160
39
87,360
69
154,560
99
221,760
10
22,400
40
89,600
70
156,800
100
224,000
11
24,640
41
91,840
71
159,040
101
226,240
602 TONS CONVERTED INTO LBS. AVOIRDUPOIS. [APP.
TABLE XIX. — FOE CONVERTING LBS. AVOIRDUPOIS INTO TONS.
Lbs.
Tons.
Lbs.
Tons.
Lbs.
Tons.
Lbs.
Tons.
Lbs.
Tons.
0
o-ooo
775
0346
23,000
10-268
54,000
24-107
85,000
37-946
25
0-011
800
0-357
24,000
10-714
55,000
24-554
86,000
38-393 .
50
0-022
825
0-368
25,000
11-161
56,000
25-000
87,000
38-839
75
0-033
850
0-379
26,000
11-607
57,000
25-446
88,000
39-286
100
0-045
875
0-390
27,000
12-054
58,000
25-893
89,000
39-732
125
0-056
900
0-402
28,000
12-500
59,000
26-339
90,000
40-178
150
0-067
925
0-413
29,000
12-946
60,000
26-786
91,000
40-625
175
0-078
950
0-424
30,000
13-393
61,000
27-232
92,000
41-071
200
0-089
975
0-435
31,000
13-839
62,000
27-678
93,000
41-518
225
o-ioo
1,000
0-446
32,000
14-286
63,000
28-125
94,000
41-964
250
0-112
2,000
0-893
33,000
14-732
64,000
28-571
95,000
42-411
275
0123
3,000
1-339
34,000
15178
65,000
29-018
96,000
42-857
300
0-134
4,000
1-786
35,000
15-625
66,000
29-464
97,000
43-303
325
0-145
5,000
2-232
36,000
16-071
67,000
29-911
98,000
43-750
350
0156
6,000
2-678
37,000
16-518
68,000
30-357
99,000
44-196
375
0167
7,000
3-125
38,000
16-964
69,000
30-804
100,000
44-643
400
0-179
8,000
3-571
39,000
17-411
70,000
31-250
101,000
45089
425
0190
9,000
4-018
40,000
17-857
71,000
31-696
102,000
45-535
450
0-201
10,000
4-464
41,000
18-303
72,000
32-143
1\) 3,000
45-982
475
0-212
11,000
4911
42,000
18-750
73,000
32-589
104,000
46-428
500
0-223
12,000
5-357
43,000
19-196
74,000
33-036
105,000
46-875
525
0-234
13,000
5-804
44,000
19-643
75,000
33-482
106,000
47-321
550
0-246
14,000
6-250
45,000
20-089
76,000
33-929
107,000
47768
575
0-257
15,000
6-696
46,000
20-535
77,000
34-375
108,000
48-214
600
0-268
16,000
7-143
47,000
20-982
78,000
34-821
109,000
48-660
625
0-279
17,000
7-589
48,000
21-428
79,000
35-268
110,000
49-107
650
0-290
18,000
8-036
49,000
21-875
80,000
35-714
111,000
49554
675
0-301
19,000
8-482
50,000
22-321
81,000
36-161
112,000
50-000
700
0-313
20,000
8-929
51,000
22-768
82,000
36-607
113,000
50-446
725
0-324
21,000
9-375
52,000
23-214
83,000
37-054
114,000
50-893
750
0-335
22,000
9-821
53,000
23-660
84,000
37-500
115,000
51-339
APP.]
CHANNEL IRON SECTIONS.
60S
TABLE XX. — CHANNEL IRON SECTIONS OP VARIOUS THICKNESSES,
IN PROPORTION TO THEIR SIZE.
Base.
Sides.
Base.
Sides. Base.
Sides. Base.
Sides.
inch.
inch.
nch.
inch.
inch.
inch.
inch.
inch.
1
ft by ft
2
2 by 2
4
1| by If
«=tb
2| by 2|
1
1 „ 1
24
u „ n
4
If H If
Si
3 ,,3
!
1 „ 1
2f
1T36 » lT36
4
3 ,,3
7
2 ,,2
**
H » it
21
If ,, If
*f
2 „ 2
7
2f „ 2f
1
I H I
2i
H » U
4f
2i „ 2t
7
2f „ 2|
1
1 „ i
2|
U » li
*i
U „ U
7
3 ,,3
i
i „ 1
2|
if „ if
44
If „ If
7
34 „ 3^
1
1 „ 1
3
I „ 1
4^
If „ If
74
2 ,,2
i
l „ l
3
i „ i
44
2 „ 2
74
2i „ 2J
i
U » H
3
H » H
44
24 „ 2i
7*
3 ,,3
li
1 „ 1
3
if „ if
*i
3 ,,3
7|
2 ,,2
li
1 „ 1
3
U » H
4|
2^ „ 2A
71
2f ,, 2|
li
i ,, 4
3
if » if
5
H „ li
8
3| „ 3|
U
i „ 1
3
2 ,,2
5
2 ,,2
8
3^ „ 31
u
f „ 1
3
3 „ 3
5
2i „ 2J
8
3| „ Si
u
i ,,i
3J
U » H
5
21 „ 2|
8
4 „ 4
it
14 » n
3i
li „ U
5
2| „ 2|
8
4* » *i
H
i „ i
3i
2 ,,2
5
3O
j? 5
8i
2i „ 2J
I*
2 ,,2
34
H » H
Bi
H „ u
84
li » U
it
1 „ 1
34
If „ If
6
2ft » 2ft
9
2ft „ 2ft
If
il „ H
3f
2i „ 2i
6
24 „ 24
9
3f „ 31
2
1 „ !
4
1 „ 1
6
3 „ 3
9*
31 „ 3f
2
i „ i
4
U » U
6
34 „ 3J
H
3ft „ 3ft
2
H » f
4
if ,, if
6
4 „ 4
10
•°'i - :;^
2
U „ i*
4
U » H
6i
U „ H
604
ROLLED IRON GIRDERS.
TABLE XXI.— ROLLED IRON GIRDERS.
[A PP.
Depth.
Width
of
Flanges.
Approximate
weight
per foot.
Depth.
Width
of
Flanges.
Approximate
weight
per foot.
inch.
inch.
Dto.
inch.
inch.
R)S.
193
61 by 6;}
97
10
54 by 54
16
6 „ 6
70
10
5 „ 5
34 to 36
16
54 „ 5£
60
10
44 „ 41
31 „ 42
151
51 „ 54
60 to 71
10
4 „ 4
29 „ 39
15
«i „ 51
59
9|
4f „ 4|
38 „ 42
15
5 „ 5
70
»ft
3| ,, 3|
20 „ 28
14
6 „ 6
60
9J
44 „ 4J
28 „ 36
18j
6 „ 6
54 to 56
91
31 „ 3|
24
13f
51 ,, 5£
54 „ 62
9|
3| „ 3|
24 to 30
12ft
8i „ 81
150
91
34 „ 3*
21 „ 29
12
10 „ 10
118 to 120
9s1
3| „ 3|
23 „ 28
12
6 „ 6
57 „ 65
9
54 ,, 5J
38 „ 40
12
51 „ 61
60
9
5 „ 5
32 „ 364
12
5 „ 5
41 to 60
9
5 „ 3
28 „ 32
11
3 „ 3
30
9
4i „ 44
32 „ 35
10|
2| „ 2|
27 to 33
9
4 „ 4
30 „ 35
10ft
2« „ 2i|
29
9
3| „ 3|
10|
51 „ 54
35 to 37
9
3J „ 8J
25
10f
8ft „ 3^
42
9
2* „ 21
22
10*
5| „ 5J
66
8|
3| „ 3|
23 to 27
10*
5J „ 5J
35 to 45
8|
3 „ 3
20 „ 28
10T35
6ft » «ft
85
H
5 „ 5
30 „ 32
10
8 „ 8
62 to 63
84
41 „ 4|
28 „ 29
10
6 „ 6
84
41 „ 4J
APP.] ROLLED IRON GIRDERS. 605
TABLE XXI.— ROLLED IRON GIRDEBS— continued.
Depth.
Width
of
Flanges
Approximate
weight
per foot.
Depth.
Width
of
Flanges.
Approximate
weight
per foot
inch.
inch.
R)8.
inch.
inch.
ih.
84
4 by 4
32
7
3 by 3
19 to 22
84
3n
» 3
25 to 34
7
2fi „ 2|
15
84
24 „ 24
17 „ 27
7
24 „ 24 )
14 to 18
84
4 „ 4
45
7
2| „ 2| )
8
5 „ 5
29 to 34
7
2| „ 14
9
8
44 „ 44
28 „ 35
7
24 „ 2i
14 to 18
8
41 „ 41
33 „ 34
6i
34 „ 3|
15 „ 18
8
4 „ 4
21 „ 30
8i
3 „ 3
18
8
3| ,, 3|
24 „ 27
6i
24 » 24
11 to 13
8
34 „ 3£
26 „ 30
8|
2 „ 2
8
3 „ 3
27
6i
If „ If
124 „ 15
8
2* „ 24
15 to 20
6
6 „ 6
29 „ 32
8
2J „ 2i
20
6
5 „ 5
25 „ 31
7|
2| „ 2J
24
6
44 „ 44
24
74
4f „ 4|
27 to 30
6
4* » 4TV
24
Mr
2| „ 14
9 „ 11
6
4 „ 4
16 to 19
n
5| „ 64
42 „ 45
6
3| „ 3|
21
7
7 „ 7
46
6
34 „ 34
17
7
5 „ 5
6
3i „ Si
17
7
44 „ 4
27
6
3 „ 3
13 to 22
7
4 „ 4
25
6
24 „ 24
13 „ 15
7
31 „ 24
20 to 26
6
2 „ li
12 „ 14
7
3| „ 3|
19 „ 25
54
34 „ 34
18 „ 20
7
34 „ 34
23 „ 25
54
3 „ 3
11 „ 15
7
3* „ 31
21 „ 22
54
2f „ 2|
7
si „ 34
19 „ 22
54
2i „ 2i
9 „ 13
606 ROLLED IRON GIRDERS.
TABLE XXI. — ROLLED IRON GIRDERS — continued.
[APP.
Depth.
Width
of
Flanges.
Approximate
weight
per foot.
Depth.
Width
of
Flanges.
Approximate
weight
per foot.
inch.
inch.
fti.
inch.
inch.
R>s.
5k
6i
2 by 2 )
IS ,, If)
9 to 13
44
*i
2i by 21
30
?? 0
11
25
5^
2£ „ . 2£
15 „ 18
4i
2i „ 2J
12 to 15
5
5 „ 5
24
4
4 „ 4
19 „ 29
5
4* „ 4£
22 to 24
4
3 „ 3
10
5
2| „ 2f
13
4
2§ „ 2|
8
5
5
5
IS „ If
Si „ Si )
3 „ 3 I
8 to 11
11 „ 16
4
4
4
2 „ 2 1
IS ,, IS i
If „ If
7 to 8
6 „ 8
4|
4 „ 3
15 „ 18
4
U » H
7
4f
4 „ 2f
14 „ 18
35
2i „ 2i
11
4|
3$ „ Si
14 „ 18
H
ii » 14
7 to 9
4S
S| „ 2jf
14 „ 18
8|
if » if
4£ „ 6
4|
2| „ 2f
10
3
3 „ 3
9 „ U
4|
IS ,, If
8 to 10
n
iiV » 1A
6
*f
3| „ 2S
15 „ 18
n
1 „ 1
4
4f
Si „ 2|
14 „ 18
2i
f „ S
2S to 4
4f
IS „ IS
9 „ 12
li
U « 4
2
*4
4 „ 4
15 „ 16
li
W » -U
IS to 2
44
3S „ IS
14 „ 16
if
4 » i
S
4*
Si „ Si
15 „ 18
APP.]
DECK BEAM IRON.
TABLE XXII.— DECK BEAM IRON.
607
Depth
of
Beam.
Width
of
Flange.
Width
of
Bulb.
Average
weight per
lineal foot.
Depth
of
Beam.
Width
of
Flange.
Width
of
Bulb.
Average
weight per
lineal foot
inch.
inch.
inch.
tbs.
inch.
inch.
inch.
tt>8.
16
«i
»i
60 to 63
S
6J
If
31
15
6|
8*
56 „ 59
8
51
IS
26 to 28
14
6J
3i
55 „ 58 8
4i
2*
32 „ 33
13
12
6*
6i
8i
8*
54 „ 57
54 „ 56 ;
8
7
4
5
2^
2
24
22
11
64
2i
42 „ 44
7
5
If
22 „ 25
10
6
2*
35 „ 37
7
H
If
25 „ 27
10
4i
3
30 „ 32
7
4
2*
21
9
6*
2
35 „ 37
6
5
li
18 to 20
9
6i
H
43 „ 45
6
4
n
18 „ 20
9
54
2
31 „ 33
6
4
2i
20
9
54
If
31 „ 33
5
4J
3
22 to 23
9
4i
3
28 „ 29
5
4
U
15 „ 16
BJ
5
If
31 „ 33
5
4
1|
14
8J
5i
li
29 „ 30
4
34
li
12 to 13
TABLE XXIIL— PLAIN BULB BEAM IRON OF VARIOUS THICKNESSES,
IN PROPORTION TO THE DEPTH.
&, 7, 74,
I 9, 9J, 9}, 10, 11, 12, 12J inches deep.
608
ANGLE IRON.
[APP.
TABLE XXIV. — ANGLE IRON SECTIONS OF VARIOUS THICKNESSES,
IN PROPORTION TO THE SIZE.
EQUAL SIDED ANGLE IRON.
iuch.
inch.
inch.
inch.
1 by |
If by 1|
2| by 2i
44 by 44
i » 4
U » H
2| „ 2f
44 » 4|
1 „ 1
If „ li
3 „ 3
4f „ 4|
43 „ f
If „ 1|
84 » 3|
5 „ 5
3 » I
2 „ 2
3i „ 3i
54 » «4
i „ i
2* „ 2i
84 „ 84
54 „ 54
U » i*
21 „ 2|
3| ,, 3|
6 „ 6
IA » iT36
2^ „ 24
4 „ 4
8 „ 8
U « U
UNEQUAL SIDED ANGLE IRON.
inch.
inch.
inch.
inch.
1 by 4
H by 1J
If by 14
2f by 14
A » I
If » A
1* » 1!
2| „ 2|
A » TV
1| „ 1
2 „ |
24 „ 11
1 » 44
11 ' „ H
2 „ 1
2£ „ 14
1 » T7S
IA » IA
2 „ li
24 » H
« » A
ii „ i
2 „ 14
24 „ 2
1 „ 4
u » u
2 „ If
2^ „ 2}
1 „ 1
H » 1
24 „ I
2| „ 1
1
if » A
24 » H
24 „ 11
IA » 3j
i! „ i
24 » H
2| „ Is1
H » i
is » u
24 » H
21 » If
H „ i
H » if
2| » 2
21 „ 2
APP.] ANGLE IRON.
TABLE XXIV.— ANGLE IRON SECTIONS— continued.
609
UNEQUAL SIDED ANGLE IRON.
inch.
inch.
inch.
inch.
2| by 24
3g by If
5 by 11
64 by 24
3 „ 14
3f „ If
5 „ 2
61 „ 31
3 „ 2
03 O I
5 „ 24
64 » 4
3 „ 24i
Si „ 2}
5 ,,3
64 „ 54
3 „ 21
3J „ 2|
5 „ 31
7 „ 3
3 „ 21
31 „ 2|
5 „ 34
7 „ 31
34 „ U
4 „ 11
5 „ 4
7 „ 4
34 „ 2
4 „ 2
5 „ 41
7 „ 5
34 „ 2|
4 „ 2i
5 „ 41
7 „ 54
31 „ If
4 „ 24
5| „ 4|
74 ,, 31
3^ „ 1|
4 „ 2|
54 „ 3
8 „ 24
31 „ 2
4 „ 3
51 ., ' 34
8 „ 3
31 „ 21
4 „ 84
54 n 4
8 „ 31
31 „ 21
4 „ 31
51 „ 44
8 „ 4
31 „ 2f
4 ,, 34
51 „ 44
8 „ 41
31 „ 3
44 » 3
5f „ 3^
8 „ 6
34 „ 11
4i „ 34
6 „ 24
84 „ 44
31 „ 14
41 „ 31
6 „ 3
8| „ If
34 „ if
4ft >. 2ft
6 „ 31
9 „ 3
34 „ 2
4| „ 2T%
6 „ 4
9 „ 34
34 „ 21
41 „ 24
6 „ 44
9 „ 44
34 „ 24
44 » 3
6 „ 5
10 „ 34
Og jj -^4^
44 „ 31
6 „ 54
10 „ 4
34 „ 3
44 „ 4
6,\ „ 3|
12 „ 3J
8ft „ 2f
4| „ 3|
61 „ If
12 „ 8
3| „ 11
2 R
610
ANGLE IKON.
TABLE XXV.— ANGLE IRON.
[APP.
ROUND BACKED.
L
inch.
2| by 2|
inch.
4 by 2
^- — -a
inch.
1 by i
2<^ ,, 2i
2i „ 24
4 „ 21
4 „ 3
1 » 1
2^ „ 2T%
4 „ 4
14 „ n
2| „ 2|
4i ,, 3J
U „ H
2| „ 2|
4i „ 4i
i§ „ if
2J „ 21
4i „ 21
H » U
3 „ 3
4i „ 3
if „ 14
3i „ 3i
44 „ 41
if „ if
3J „ 8i
4| „ 4|
2 „ 2
3| „ 8|
5 „ 5
2| „ 2J
Si „ 2i
6 „ 2i
2T3<r » 2T'«
3A „ 3
6 „ 34
2i „ 24
34 ,, 8J
7-A » 2^
SQUARE ROOT.
inch.
§ by i
I „ 1
1 „ 1
inch.
U by 1J
If » A
If „ If
|
inch.
§ by i
i » I
1 „ i
li » i
T76 „ 1
i „ i
U » i
i »
i „ i
14 » 14
4 „ i
i „ i
if » i
A » TV
H „ H
if „ if
i „ 1
U » i
if „ i
i » T'«
U „ if
H „ if
« » W
U » i
2 „ i
APP.]
ANGLE IRON.
TABLE XXV.— ANGLE IRON— continued.
611
SQUARE ROOT.
inch.
inch.
inch.
2 by 1
21 by 21
3 by 3
2 „ 11
24 ,, 24
81 „ 24
2 „ 2
2| „ 2|
3S „ 22
21 „ 11
BULB ANGLE.
inch.
n
f\
4 by 21
^=o
^=3
4 „ 3
inch.
inch.
4* „ 2^
2 by 11
21 by 11
44 ,, 81
3 „ 2
2* „ 2
5 „ 3
5 „ 21
3 „ 2
5 „ 3J
6 „ 34
3 „ 21
54 „ 4
6i „ 81
34 „ 24
6 „ 3J
4 „ 2
6 „ 6
TABLE XXVI.— Z IKON SECTIONS.
TOP.
BOTTOM.
Top.
Depth.
Bottom.
Thick.
Top.
Depth.
Bottom.
Thick.
inch.
81
inch.
64
inch.
81
inch.
4
inch.
H
inch.
3
inch.
21
inch.
iV
03
^4:
6
2f
4
2
2g
2
ft
24
**
21
4
2
24
2
T^T to TV
24
4
21
i
f
If
TV
i
2
4
2
176
g
U
4
A
H
81
21
TV
i
if
i
i
3
3
21
T7*
612
TEE IRON.
[APP.
TABLE XXVII. — "T IRON SECTIONS OP VARIOUS THICKNESSES,
IN PBOPORTION TO THE SIZE.
Table. Leg.
Table. Leg.
Table. Leg.
TiU.le. Leg.
inch. inch.
inch. inch.
inch. inch.
inch. inch.
10 by 10
6 by 5}
54 by 6
44 by 4£
8 „ 4§
6 „ 5
54 » 31
44 „ 4
8 „ 4
6 „ 4i
64 » 34
44 „ 3f
7* „ 4
6 „ 44
54 » 2f
44 ,, 3|
7 „ 7
6 „ 4
5J „ 8J
4^ „ 3i
7 „ 6
6 „ 34
5J „ 3
41 „ 3§
7 „ 5i
6 „ 8J
5 „ 8
4 „ 3
7 „ 6*
6 „ 3J
5 „ 6
44 „ 2|
7 „ 5
6 „ 3
5 „ 5
41 „ 24
7 „ 44
6 ,, 21
5 „ 4
4^ „ 24
7 „ 84
5| „ 84
5 „ 3|
4^ „ 2
6| „ 5
5f „ 5
5 „ 34
4i „ IS
6f „ 4
5f „ 4
5 „ 3
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6| „ 3|
5| „ 8«
5 „ 2f
4§ „ 15
6i „ 44
6| „ 3^
5 „ 24
4] „ 4J
6* „ 8|
5S „ 84
& „ 24
4* „ 4
64 „ 3
54 „ 4|
4| „ 4J
44 » 32
6| „ 2|
51 „ 4|
4| „ 3f
44 „ 34
6i „ 8
54 „ 4f
4| „ 3|
4| „ 3
64 „ 6
5i » 3}
4f „ 3
44 » 24
6i „ 3|
54 „ 3|
4S „ 11
4| „ 2
64 „ 34
5J „ 3
4| „ If
4i « U
6 „ 6f
6J „ 2f
4g „ 34
4J „ 4
6 „ 6
5i „ 2TV
44 „ 5
4 -,,6
APP.] TEE IRON.
TABLE XXVII.— J IRON SECTIONS— continued.
613
Table. Leg.
Table. Leg.
Table. Leg.
Table. Leg.
inch. inch.
inch. inch.
inch. inch.
inch. inch.
4 by 5
3J by 2
2i by 2g
2 by 1
4 „ 44
3 „ 6i
24 „ 4
If „ 3
4 „ 4
3 „ 6
24 „ 34
If » 2g
4 „ 34
3 „ 5ls
24 „ Si
If „ If
4 „ 3
3 „ 5
24 „ 3
H » U
4 „ 2T%
3 „ 44
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ift » 24
4 )> *18
3 „ 34
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if „ i|
4 „ 2i
3 „ Si
24 „ H
if „ u
4 „ 2
3 „ 3
24 „ If
if » if
4 „ li
3 „ 2f
24 „ U
H » 2i
37 Q
g^ ,, O
3 „ 2f
2| „ 1
H „ 2
3J „ 2|
3 „ 24
2i „ 3
14 » 14
si „ si
3 „ 2
2i „ 2|
14 „ If
3| „ 2
3 „ If
2i „ 2i
li » li
3| „ ItV
3 „ If
2i » 2
14 » 1
34 „ 44
3 „ li
2i „ li
if „ H
34 „ 4
21 „ 34
2 „ 4
if „ if
34 „ 34
2| „ 2|
2 „ 3^
l| „ li
34 „ 3i
2f „ 4
2 „ 3
li » H
34 „ 3
2f „ 34
2 „ 2^
li » 1
34 „ 2£
2f „ Si
2 „ 2i
li ,, 3
Si „ 2
2f „ 3
2O_3
j> -^To
U „ 2
34 „ 14
2| „ 2|
2 „ 2
li » U
Si „ 4
2— 2§
2 „ If
1J ,. 1J
Si „ Si
2| „ If
2 „ 14
li „ li
3i „ If
2f „ 14
2 „ H
i.i „ li
614
TEE IRON.
TABLE XXVII.— T IKON SECTIONS— continued.
[APP.
Table. Leg.
Table. Leg.
Table. Leg.
Table. Leg.
inch. inch.
inch. inch.
inch. inch.
inch. inch.
li by 1
1 by l^
i by |
f by 1J
U » I
1 „ U
i „ n
f M 1
H „ U
1 » 1
1 » 1
1 „ 1
U „ 1
1 „ i
1 „ 2^
1 „ i
1A » «
1 ,, A
f » 2
i „ i
TE IV.
PLATE V.
VQADU^T
EXPERIMENT
1
I
I
INDEX
ART.
A truss, - 220, ±J1
Alloys, coefficients of linear expansion, - -415
— crushing strength, - - :J'.<'.»
— tensile strength, 362 to 364
Aluminium bronze, 299, 362
Angle-iron pillars, -
— ordinary sizes of, 437, 545
— tensile strength, - 352, 353
Angle of bracing, trigonometrical functions, - 278
— economy for bracing, - -275, 276
— fracture from crushing, - 293, 302
Annealing cast-iron, - 349
— chains, - . 357, 409
— copper wire, - - 362
— glass,
— gun metal,
— steel, - 360
— wrought-iron, - - 354, 357, 358, 409
Antimony, coefficient of linear expansion, - - 415
Apex, • 135
Arch, braced, - 213
cast-iron, - 219, 459, 473
-flat, - - 216
— laminated
— stone,
— triangular, -
— wrought-iron,
Arches, how affected by changes of temperature, -
Ashlar work, working load,
Axioms, -
Ballast, weight of - 445» 545
Bay, - - '136
Beam,
Bearing surface, -
.
Bell metal, tensile strength,
616 INDEX.
ART.
Belting leather, tensile strength and working strain, - - 389
Bending moment, - - - 59
Bent crane, - - 195
— girders, - 192
Birmingham wire gage, - - 545
Bismuth, coefficient of linear expansion, - - 415
Board of Trade regulations respecting railway bridges, 446, 473, 492
Boilers, strength of, - 288
— working load on, - - 479
Boiler -maker's rules for riveting, - - 467
Bolts and pins, strength and adhesion in timber, - 460, 461, 468
Bone, 8, 389
Bow and invert, or double-bow girder, - - 212
Bowstring girder, - - 207, 272, 443, 450, 457
— at Saltash, - 212
— on the Caledonian railway, - - 541
— quantity of material in, - - 272
Box girder, - - 13
Boyne viaduct, description and details, - - 528 to 536
Brace, «• - 137
Braced arch, - 213
-pillars, 341, 535
— semi-arch, - - 198
— triangle, - 220
Bracing. (See "Angle of Economy" " Counterbracing," "Cross-bracing," "Lat-
tice," " Web")
Brass, coefficient of elasticity, - - 8
— coefficient of linear expansion, - 415
— crushing strength, - - 299
— tensile strength, - - 362
— wire, tensile strength, - 362
Brewster, experiment on glass girder, - - 131
Brick, coefficient of linear expansion, - 415
— crushing strength, - - 301
— working load, - - 488
Bridges. (See "Appendix," "Cast-iron," "Lattice," "Public," "Railway" "Steel,"
"Suspension" "Swing" "Timber," "Tubular," " Wrought-iron")
Brittleness, - 5
Bronze. (See " Gun metal.")
— aluminium, - 299, 362
Brotherton plate tubular bridge, description, - 544
Buckled-plates, - 447
Buckling, - 292
Bulging, - • - 292
INDEX. 617
ART.
Cables. (See " Chains," " Cordage.")
Camber, practical method of producing, . 455
— ornamental rather than useful,
Cast-iron, annealing, effect on strength,
— arches, 219, 459, 473
— coefficient of elasticity, - - 8, 246, 399 to 405
do. linear expansion, - - 415
do. transverse rupture, 65
do. torsional rupture, - - 283
— cold and hot blast, relative strength, - 346
— compound girders of cast and wrought-iron, - 523
— corrosion,
— crushing strength, - 294
— deflection, 246, 472
— effect of changes of temperature, 418, 420
— elastic flexibility twice that of wrought-iron, - - 408
— girders, - 132, 422, 435, 458, 523
— indirect pull reduces the tensile strength, - 350
— mixtures stronger than simple irons,
- pillars, - 322 to 329, 471, 474
— prolonged fusion, effect on tensile strength, - - 348
- proof strain, 473, 482
— re-melting, effect on tensile strength,
set, - - 399 to 405
— shearing strength,
— Stirling's toughened,
— tensile strength,
— working strain and working load, - 473, 474
relative strength of thin and thick castings, - 132, 295, 296, 349
Cellular flanges,
Cement. (See " Keene, Medina, Parian, Portland, Roman")
Centres of strain, -
Centrifugal force, effect on deflection,
Chains, - 357> 378 *° 385
flat-link, -
-proof-strain,
- weight, -
— working-strain,
Chain-riveting,
Channel-iron pillars,
- sizes of,
Charing-cross Lattice Bridge, description, -
Chepstow Truss Bridge, description,
Clay, working-load,
618 INDEX.
ART.
Clenches and forelocks, strength of, - - 468
Coefficient of elasticity, E, - 8
— linear expansion, - - 415
— transverse rupture, S, - 60 to 66
— torsional rupture, T,
— safety, - 470
Cold and hot-blast iron, relative strength, - - 346
Collar-beam, - 220
Columns, stone, - 339, 448
Compound girders of cast and wrought -iron, - - 523
— of timber and wrought-iron, - 187, 527
Compressive strain, subdivisions of, - 292
— symbol of, +, - 139
Concrete, crushing strength,
- working load, - 488
Connexions. (See " Joints.") - 460 to 469
Continuous girders, 247 to 260, 427, 499
— ambiguity respecting strains in webs, - 256
( not desirable for small spans with passing loads, or where
(. foundations are insecure, - - 258
— of two equal spans, each loaded uniformly, - - 251
— of three symmetrical spans, loaded symmetrically, - - 253
Contrary flexure. (See "Inflexion.")
Conway Plate Tubular Bridge, description, - 543
Copper, coefficient of linear expansion, - - 415
— crushing strength, - - 299
— shearing strength, - - 396
— tensile strength, - - 362
— weight and specific gravity, - 362
— wire, - 362
Copper-bolts, adhesion of in timber, - 468
Cordage, tensile strength, - - 375 to 377, 381, 386 to 388
weight, - - 375, 376, 381, 385 to 387
— working strain, - 377, 386 to 388
Corrosion of metals, - 431
Cotters, - - 460
Counterbraced brace, - 137
— girder, - - 138
Counterbracing, - 174, 175, 186, 187, 208, 448 to 450, 517
Covers, allowance for in estimating girder-work, - - 497
— strength and proportions of, • - 463 to 465
Crane, bent, - 195
- derrick, - - 193
lattice 197
INDEX. 619
ABT.
Crane, travelling, or gantry, - - - - 187
— tubular, - - - 195
— wharf, - . . ij)4
— working- strains, - - - 473 to 484
Crescent girder, - . 203
Cross-bracing, - - 440 to 443
Cross-girders, ... . - 444 to 447
Cruciform-iron pillars, - . 332
Crumlin Viaduct, description, - • - 539
Crushing strength of materials, - - 291 to 305
Crushing, subdivisions of, . 292
Cubic elasticity, ... 3
Curve of equilibrium, - -49
Cylinders and spheres, strength of, - - 288
Deflection, . 223 to 246, 434, 451 to 456
— effect of centrifugal force on, - - - 4">fi
— experiments on deflection, - 454, 471, 472, 475, 534, 536
— method of measuring deflection of girders, - - 456
— not affected by nature of web, - 223, 434
— of small bridges increased by loads in rapid motion, - - 454, 489
— of continuous girders, • 251, 253, 534
— of girders of uniform section, - - 225
do. of uniform strength, 223, 224, 451
— of lattice and plate girders nearly alike, • • J23, 434
— of similar girders, - - 224
Depth of girders and arches, - 18, 274, 457 to 459
— for calculation,
— weights of girders do not vary inversely as their depths, - 505, 511, 516
Derrick Crane, - 193
Detrusion,
Diagonals. (See "Bracing," " Web")
— law of strains in intersecting diagonals,
Diagram, calculation by, -
Drilling tools, - - 425
Drilling preferable to punching, -
Ductility, - -5, 356, 357
E, coefficient of elasticity,
Earth, working pressure on,
Economy, angle of,
— relative economy of different kinds of bracing,
Elastic flexibility and elastic stiffness,
Elasticity and set, - - - - 3 to 8, 398 to 413
620 INDEX.
ART.
Elasticity, cubic, - - 3
— coefficients of, E, 8, 399 to 413
— law of elasticity (Hooke's law), 7, 393
limit of, ... 7, 398 to 413
— linear,
— modulus of,
— sluggish or viscid, - - - 404, 410
— tensile, compressive and transverse elasticity often different, - 8, 246, 403
Ellipse, moment of resistance of, - 76, 77
Elliptic semi-girders, - 93, 94
Engine-work, working strain,
Engines, weight of, • 489, 490
Equality of moments, - 11
Equilibrium, curve of,
Estimation of girder-work, - 495 to 522
Expansion from heat, coefficients of linear, - 415
— — effect on girders, arches and suspension bridges, - - 414
- rollers, 340, 414, 429
F, symbol which represents the total strain, - 2
/, symbol which represents the unit-strain of tension or compression, - - 2
/', symbol which represents the unit-strain of compression,
Factor or coefficient of safety, - 470
Fatigue of materials, - - 470, 519
Fish-bellied girders, or inverted bowstring, - 212
Flanges, - - 17, 100, 152, 422 to 429, 439, 443, 477, 496, 497, 520
Flexibility, 4
Flexure, - - 292
Foot-strain, - 2
Forelocks, strength of, - - 468
Forgings, tensile strength of, - 352, 354, 357
Foundations, working load on, - - 487
Fractured area, - - 352
French rules for working strain, - 473, 476, 479
— proof load and working load of bridges, - 492, 493
— proof strain for chains and ropes, - 376, 382
Friction due to riveting, - - 466
Gages for wire and sheet metals, - - 545
Gantry or travelling crane, - 187
Gasholder roof, • - 222
Girder, - - - 12
— arched, - 213
— bowstring. (See " Bowstriny.")
INDEX. 621
ABT.
Girder, box, . - - - ] :j
— cast-iron. (See " Cast-iron.")
— compound cast and wrought-iron, - . 623
— compound timber and iron, . 187, 527
— continuous, . 247, 427
— crescent, - . 203
— cross, - . 444 to 447
— curved, - - 192
— deflection, . 223, 451
-depth, - . .j;,7
— double-bow, - . - 212
— double -webbed, or tubular, - 13
- elliptic, - 93, 94
— estimation of, - - 495 to 522
— fish-bellied, - 212
— imbedded at both ends and loaded uniformly, - - 259
do. and loaded at the centre, - - - 260
— lattice. (See " Lattice girder.")
-limit of length, 67,524
— of uniform strength, - 19
— plate. (See "Plate girders")
— proving. (See " Proof load. ' ')
— quantity of material. (See " Quantity.")
— rail girder, or keelson,
rectangular girder of maximum strength cut out of a cylinder, • - 87
— road girder,
— similar girders. (See "Similar girders")
— single-webbed,
— temperature, effect on girders, - 414, 418, 419
— timber. (See " Timber")
- triangular, 201, 218, 220
— trough,
-trussed, - • 187
-tubular. (See " Plate" girder.") -
— Warren's,
— weight of girders under 200 feet in length,
— with parallel flanges and isosceles bracing,
do. do and vertical and diagonal bracing, -
— working loads on. (See "Public bridges" "Railway bridges" "Jtooft.")
— wrought-iron. (See " Wrought-iron")
- loaded at an intermediate point, -
loaded at the centre,
- loaded uniformly, 43, 124, 160, 177, 188, 2<
do. and traversed by a train of uniform density, -
622 INDEX.
ART.
Girder, loaded unsymmetrically, - . 41, 155
— traversed by a concentrated rolling load, 32, 37 to 40, 54, 123, 158, 186, 491
— traversed by a train of uniform density, - - 50, 169, 189, 190, 489, 490
Girder-work, estimation of - 494 to 522
Glass, coefficient of linear expansion, - - . - 415
— crushing strength, - - - - 305
— elasticity of, - . - 413
— girder, Brewster's experiments on, - . 13J
— tensile strength, - - 374
— weight and specific gravity, - 305, 545
Glue, tensile strength and adhesion to timber, - - . 339
Gold, coefficient of linear expansion, - - - 415
— weight and specific gravity, - _ 545
Government inspection of railway bridges. (See " Board of Trade.")
Gravel, working load on, - . 437
Gun metal or bronze, annealing, effect on strength, - - 363
— coefficient of elasticity, - g
— high temperature at casting injurious to strength, - - 363
— tensile strength, - . . 362, 363
Gutta-percha, tensile strength and working-strain, - - 389
Heat. (See " Temperature")
Homogeneous metal, tensile strength, - . 359
Hooke's law of elasticity, . . 7^ 393
Horizontal bracing, - . . . 440
Hot and cold -blast cast-iron, relative strength, - - - 346
Impact, effect of long continued impact on cast-iron bars, - - 472
Inch-strain, . - - 2
Inertia, moment of, .... 69 225
Inflexion, points of, • ..... 247
— economical position of, - ... 250
— experimental method of finding, - - - 249
— not affected by depth of girders, - - - 249
— practical method of fixing, - - - 250, 534
Initial strains in bracing, method of producing, ..... 442
Internal bracing, - ..... 34^
Inverted bowstring, . . . - 212
Iron. (See "Angle-iron," " Cast-iron," "Plates," " Tee-iron,' " Wrouyht-iron")
Isosceles bracing, ... . ---133
Joints, - 439? 460 to 469
— bolts and pins, 460, 461, 468
cast-zinc, - ---.,.. 454
INDEX.
AIT.
Joints, clenches and forelocks, ... . . 4^3
— jump. • 462, 464
- lap, . 462 to 465
— in piles of plates, - 423, 424, 464
nails and bolts, - - 468, 469
— riveted. (See "Rivets.")
— screws, - - - 354, 460, 469
— treenails, - - 897
Keelsons, or rail girders, ... - 445
Keene's cement, tensile strength, - - 371
Knife edges, working load on, - - 478
Laminated beams, - • - 527
Lancashire gage for steel wire,
Lattice bridges, description of. (See Appendix.)
Lattice crane, -... . -197
Lattice girders, ambiguity respecting strains, - - 181, 191, 215
— curved or oblique,
— deflection, - 223, 224
effect of temperature, - 418, 419
— effect of concentrated load, - 445, 491
— end pillars subject to transverse strain, - 180, 191,443
— estimation of quantities, - 495 to 522
— loaded uniformly, - 177
— timber, - - ^-1
traversed by a passing train, - - 179, 190
— traversed by a single load,
— weight of, - 521
Lattice pillars, - - 341
— semi-arch,
- semi-girder, 154, 197, 201
Lead, coefficient of elasticity,
— coefficient of linear expansion,
— crushing strength, -
— elasticity, -
— tensile strength,
— weight and specific gravity,
Leather, tensile strength, -
Length for calculation, -
-limit of, - 67,524
Lever, law of the,
Lime. (See " Concrete," "Mortar")
Limestone. (See " Stone")
624 INDEX
ART.
Limit of elasticity, 7, 393 to 413
- length of girders, . 67, 524
Linear expansion from heat, coefficients of, - - - . - 415
Liverpool and Lloyd's rules for ship riveting, - ... 457
Locomotive. (See "Engine")
M, moment of resistance to rupture, - - - - 59, 69 to 82
Machinery, working strain, - .... 480
Masonry, crushing strength, - - - 393
— working strain, . . . - 488
Mechanical laws, - .... 9
Medina cement, tensile strength, - ..... 371
Modulus of elasticity, - .... 8
— of rupture, - ... 60
Moment of inertia, • .... 69, 225
— of resistance to rupture, M> - - - 59, 69
— of rupture, . . - 59
— of resistance to torsion, - .... 284
Moments, equality of, - - - 11
— strains calculated by, - - 164, 196
Mortar, adhesion of, - ...... 372
crushing strength, - - - - - - 304
— tensile strength, - 368, 369, 370
— weight and specific gravity, - - - - 545
— working load, - .... 437
Nails, bolts and screws, adhesion of, - - 468, 469
Neutral axis, - ... 58
— passes through centre of gravity of section, - 68, 131
— practical method of finding, - 68
— shifting of, - - - 131
Neutral line, - ...... 58
Neutral surface, - - - - . . . . - 57
Newark Dyke Bridge, description, . - - 537
Obelisk Bridge over the Boyne, description, .... 549
Oblique or curved girders, . - 192
Palladium, coefficient of linear expansion, - . - 415
Parian cement, tensile strength, --•-... 371
People, crowd of, the greatest distributed load on a public bridge, - - 493
Piles of plates, . - 423, 424, 464
Piles, working load on timber, - - - - 486
Pillars, 306
INDEX.
ART.
Pillars, angle -iron, - ...
braced, - . - 341 to 343, 535
— cast-iron, - - 322 to 329, 47 1
— channel iron, • • • • 332
— cross shaped, +, - - 325
— discs add little to the strength of flat-ended pillars, - - - 316
— effect of long-continued pressure on the strength of pillars, - 471
— end pillars of girders, - 180, 191, 443
— effect of enlarged diameter in the middle or at one end, - - 317
— H -shaped, . 325
— Hodgkinson's laws, - 311 to 326
— Gordon's rules, - - 327
— lattice. (See "Braced," above.}
— long flexible pillars which fail by flexure, - 306, 310, 311
— medium, or short flexible pillars, which fail partly by flexure, partly by
crushing, - - - 310, 323
— short pillars which fail by crushing, - 293, 310
— similar long pillars, strength of, - - 308, 321
— steel, - - 336, 483
— stone, - - 339, 488
— strain passing outside axis of pillar reduces its strength greatly, - - 320
— strength of very long pillars depends on their coefficient of elasticity, - 307
— tee-iron, - - 332
— theory of very long pillars,
— three classes of pillars,
timber, - - 337, 338, 484 to 486
— triangular, - 326
-tubular, - 334, 335, 423
— weight which will deflect a very long pillar is very near the breaking
weight, -
— wrought-iron, - 330 to 335, 477, 535
Pins, ... - 439, 460 to 469
Plaster of Paris, adhesion to brick and stone,
— tensile strength,
— weight of cast plaster, -
Plates, boiler,
— friction of riveted plates,
— ordinary sizes of,
-piled,
— resistance to flexure,
ship,
— strength. (See " Wrought-iron.")
— temperature, effect on strength, -
— ten per cent, stronger lengthways than crossways,
'2 S
626 INDEX.
ART.
Plate girders, calculation of strains. (See " Web.") 54, 100, 430 to 435
— deflection same as that of lattice girders of equal length, - 223, 434
— effects of temperature, - - 419
— examples. (See "Appendix.")
— weights of, - 521
Platform of bridges, - 426, 444 to 447
Platina, coefficient of linear expansion, - - 415
— weight and specific gravity, - - - 545
Points of inflexion or contrary flexure. (See " Inflexion.")
Portland cement, crushing strength, - - 301, 304
- tensile strength, - 369
— (See " Concrete," " Mortar")
Proof strain and proof load, - - 409, 438, 470, 482, 483, 492, 493
— (See " Chains," " Cast-iron,'' " Wrought-iron.")
Public bridges, working load, ... - 493
weight of roadway, - 447, 540
Punching experiments, - - 392, 396
— injurious effect on plates, - 462,476
tools, ..... - 425
Quantity of material in bowstring girders, • • 272,450
— in girders with horizontal flanges - - 18, 54, 261
— in different kinds of bracing compared, - - 279
— theoretic and empirical, - - 495 to 522
Rail girders, or keelsons, - - 445
Railway bridges, estimation of, - • - 495 to 522
— proof load, - 492
— roadway, - 444, 445
— rules of Board of Trade, 446, 473, 476, 492
— rules of French Government, - 473, 476, 492
— under 40 feet in length require extra strength, - 454, 490, 491
— weight of bridges under 200 feet span, - - 620, 521
(444 to 446, 473 to 478,
-working strain and working load, - | 481, 489 to 492
Resistance to rupture, moment of, M, - 69, 69 to 82
— to torsion, moment of, . - 284
Resolution of forces, - 9, 639
Riveting, chain, - ... 467
Rivets, - - 394, 395, 424, 439, 460 to 467
— boiler-makers' rules, ..... - 467
— friction due to contraction, - 466
— girder-makers' rules, - - 462 to 467, 476
INDEX. 627
ART.
Hi vets, long rivets not objectionable, - . . . -424
— preferable to pins for girder- work, . . - 439
— snip-builders' rules, ...... 457
- steel, 395, 462, 483
— working strain, • • • 462
Rivet-holes, allowance for weakening effect of, - 462, 476, 495
— drilled in first-class work, .... 425, 462, 467
Roadway, - - - 426, 444 to 447
Rollers and spheres, crushing strength, - - 340
— expansion, under ends of girders, 340, 414, 429
Roman cement, tensile strength, - - 370
— coefficient of linear expansion, - • - -415
Roof A, - 220
— cost of, . - '494
— arched, . 203
working load on, « - • 494
Roofing materials, weight of, - - 494
Rope. (See " Cordage " and " Wire.")
Rubble masonry, crushing strength,
working load, - - 488
Rupture, coefficient or modulus of S, • 60 to 66
moment of, M, - • - - 59
S, coefficient or modulus of transverse rupture, - • « - 60 to 66
Safety, coefficient or factor of, - - 470
Screws, strength of • 354, 460, 469
— adhesion in wood, - 469
Semi-arch, braced, • 198
— inverted, - 202
— lattice, - 201
Semi-girder,
— loaded at the extremity, - 16, 83, 145, 226
— loaded uniformly, - - 22, 105, 148, 232
loaded uniformly and at the extremity also,
— triangular, -
Set, - - - 6, 298 to 413
Set, relaxation of, • 404, 410
— . ultimate set after fracture, -
vitreous materials take no set,
Shearing experiments,
in detail,
— simultaneous, -
( 14, 15, 18, 23, 34, 37, 42, 46, 50 to
- strain in girders, ^ ^ ^ 43J> 4?8
628 INDEX.
Shearing strength. (See " Cast-iron" " Copper," " Rivets," "Steel," " Timber,"
" Treenails," " Wrought-iron")
Ship-builders' rules for riveting, - - 467
Ship plates. (See " Plates.")
Ships, strains in, • - 526
Silver, coefficient of linear expansion, - -415
Similar girders, deflection of - - 224
— .limit of length, - 67,524
— strength of, - 67
— weight of, - 67, 274, 522, 524
Snow, weight of, - 445, 494
Solder, tensile strength, - - 362
— coefficient of linear expansion, - - 415
Specific gravity, alloys, - - 362, 364, 545
bricks, - - 301,545
— cast-iron, 345, 347, 348, 349, 545
— glass, • 305, 545
— stone, - • - 301, 545
— tables and weights of various materials, - 545
— timber, - 65, 545
— wrought-iron, - - 354, 545
(See " Weight.")
Speculum metal, tensile strength and specific gravity, - - 364
— coefficient of linear expansion, - - 415
Spheres, strength of hollow, - - 290
Spheres and rollers, crushing strength, - - 340
Splintering, • 292
Steel, annealing improves and equalizes strength of steel plates, - - 360
— coefficient and limit of elasticity, - - 8, 298, 359, 411
— coefficient of linear expansion, - - 415
— coefficient of transverse rupture, • • 65
— coefficient of torsional rupture, - - - 283
corrosion of, ••-•--• - 431
— crushing strength, - - 298, 483
girders, - - 483, 502
— pillars, 336, 483, 502
— proving, - - 482, 483
— punching reduces strength, - • - 360
- rivets, . 354, 395, 483
shearing strength, ....... 364, 395
— ship plates, - - 360, 483
— tensile strength, - 354, 359, 483
ultimate set after fracture, - - 359
wire, ......... 361
INDEX.
ABT.
Steel wire rope, - . 386, S87
— working strain, •--..... 433
Stiffness, elastic, •• ......4
Stone arches, - - ... . . . . 459^ 433
coefficients of elasticity, - - . . . . 8, 413
— coefficients of linear expansion, - • « 415, 417
— coefficients of rupture, -•-.... (55
— columns, ... . 339? 443
— crushing strength, ..... 301
— elasticity of stone not always apparently in accordance with Hooke's
law, - . 413
tensile strength, - ..... 367
working load, - - 488
Strain, centres of, - 58
— classification of, - - 1
— crushing, - 291
— — inch-strain, - 2
— foot-strain, • 2
— shearing. (See " Shearing") - 14, 390
— tensile, - - 344
— torsional, - - 280
— unit, - 2
Strut. (See " Pillar.")
Suspension chains, proof strain, - 476, 481, 482
— proportions of eye and pin,
— working strain, - 476, 481
Suspension bridges, 49, 217, 414, 481, 503
— rigid, - 217
— temperature, effect of,
— working load, - 481, 41
-truss, - 222
Swivel or swing bridge, -
Symbols + and — , - 139
T, coefficient or modulus of torsional rupture, - - 281, 283
Tee-iron pillars, -
Temperature, coefficients of linear expansion of various materials,
— effect on cast-iron, - -418, 420
— effect on girders and bridges, • 414, 418, 419
— effect on wrought-iron,
— effect on stone bridges, - - 414, 417
— effect on suspension bridges,
— effect on timber, -
Tenders, weight of, - 489
630 INDEX.
ART.
Tensile strength of materials, - - 344
Timber, adhesion of bolts and screws in timber, ' - 468, 469
adhesion of glue to, - 364
— coefficients and limit of elasticity, 8, 412
— coefficients of linear expansion, - - 415, 416
do. of transverse rupture, - 65
do. of torsional rupture, - 283
— crushing strength at right angles to the fibre, - - 486
crushing strength lengthways, - - 300
— girders, - 484, 485, 527
— lateral adhesion of the fibres, - - 366
- piles, - 486
pillars, - 337, 338, 484 to 486
— shearing strength, - 397
— should be used in large scantlings, - 5 27
— tensile strength, • - 365
— wet timber not nearly so strong as dry to withstand crushing, - - 300
— working strain, - ... 484 to 486
Tin, coefficient of elasticity, • 8
— coefficient of linear expansion, - 415
— crushing strength, - - 299
— tensile strength, - - 362, 364
Torsion, - 1, 280
Torsional rupture, coefficient or modulus of T, - - 281, 283
Toughness, - - 5
Trade, Board of Trade regulations respecting railways, - - 446, 473, 476, 492
Travelling crane, or gantry, • - 187
Treenails, strength of, • 397
Triangular arch, -
— girder, - 220
— semi-girder, - - 201
Trigonometrical functions of 0, the angle of economy, - - 278
Trough girders, - - 4 45
— section of flange, - - 439
Trussed girders, - - 187, 523
Tubular bridges and tubular girders, - 13
— examples of. (See " Appendix.")
— effect of changes of temperature on, - - 418, 419
— effect of wind, - - 442, 443
Tubular pillars, - - 334, 335
Twisting moment, - ... 280
Uniform strength,
Unit-strain, ... - 2
INDEX. i;:;i
ART.
Unit-strain, economy from high unit-strains in large girders, - 502, 509, 61 4
Upsetting of iron under pressure, . 473, 486
Vertical and diagonal bracing, - - - 184
Warren's girder, - - 133
— economy, relative, . . . 279
— example. (See " Appendix.")
Web, - 430 to 439, 625
— ambiguity respecting strains in, - 181, 191, 206, 215, 256, 431
angle of economy in braced webs, - - 275
— braced- generally more economical than plated webs, - - 279,431
— quantity of material in, - - 18, 54, 261 to 274, 495 to 622
— continuous or plate, nature and calculation of strains, 15, 64, 430, 431, 525
do. minimum thickness in practice, - 431
do. more economical in shallow than in deep girders, 482, 433
do. more economical than bracing near the ends of very
long girders, - - - - 432
do. value of in aid of flanges, - 15, 78, 100, 433, 435
Weight of ballast, 445, 545
— chains, 380, 381, 385
— cordage, - 375, 376, 381, 386, 387
— cross-girders, - - 445
— engines and tenders, - - 489, 490
— girders under 200 feet in length, - - 274, 521
— people, - -I1.';}
— permanent way, - 445
— roadway, - 445, 447
— roofing materials, - - 494
— snow, ... - 445, 494
- timber, 65, 546
— various existing bridges. (See "Appendix.")
— various materials,
— wire rope, - - 386, 387
(See " Specific gravity .")
Whalebone, tensile strength, - - 389
Wharf crane, - - 194
Wind, force of, - 440, 441, 494
Wire, copper, tensile strength,
gages, ... - ,M:>
— iron, tensile strength,
— rope, tensile strength and weight,
— do., working load,
Wood. (See " Timber.")
<;;••',:> INDEX.
ART.
Wood screws, adhesion of, - 469
. , ( 335, 343, 377, 378, 383, 386, 387, 388, 429, 446,
Working strain and working load, j ^Q ^ ^
Wrought-iron, annealing, effect on strength, - 357, 358, 407
boiler plates, 356, 479
_ coefficient and limit of elasticity, - - 8, 297, 406 to 410
— coefficient of linear expansion, - - 415
— coefficient of transverse rupture, - 65
do. of torsional rupture, - - 283
— r- corrosion of, - 423, 431
— crushing strength, - 297
— deflection, - 475
— elastic flexibility half that of cast-iron, - - 408
— elastic limit, - 297, 406 to 410
— forgings, tensile strength, - 357
— Kirkaldy's conclusions, - - 354
— ordinary sizes of, - 437, 545
pillars, - - 331 to 335
plates. (See " Plates")
— practical method of stiffening bars, - 409
— proving, ----- - 409, 482
— punching experiments, - 392, 396
— removing skin not injurious to strength, - - 355
— set after fracture, - 352
— shearing strength, - 392 to 395, 478
— ship plates. (See "Plates")
— temperature, effect of , 418, 419, 421
— tensile strength, - 352, 353
— toughness very valuable, - 356, 360, 482
— wire, tensile strength, - - 358
- working strain, - 475 to 482, 494
Yellow metal, tensile strength, - 362
Zinc, coefficient of elasticity,
— coefficient of linear expansion, - 415
— crushing strength, - - 299
— joints, - 464
— tensile strength, - . - 362
— weight and specific gravity, - - 545
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