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REFERENCE.
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No.
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tf 187^
University of California.
I 76
A TREATISE
ON THE
RESISTANCE OF MATERIALS,
5lntr an
ON THE
PRESERVATION OF TIMBER.
BY
DE YOLSON WOOD,
14
'PROFESSOR OF CIVIL ENGINEERING IN THE UNIVERSITY OF MICHIGAN.
^ Library*
California-
NEW YORK:
JOHN WILEY & SON,
15 ASTOE, PLACE.
Entered according to Act of Congress, in the year 1871,
By JOHN WILEY & SON, ' In the Office of the Librarian of Congress, at Washington.
TIIK NKW YOHK PRINTING COMPANY,
206, 207, 209, 211, and 213 East Twelfth Street.
PREFACE.
THIS book contains the substance of my lectures to the Senior Class in Civil Engineering, in the University, during the past few years, on the Resistance of Materials. The chief aim has been to present the theories as they exist at the present time. The subject is necessarily an experimental one, and any theory which has not the results of ex periments for its foundation is valueless. I have therefore presented the results of a few experiments under each head, as they have been obtained in various parts of the world, that the student may judge for himself whether the theory is well founded or not. It is hoped that this part of the work will be valuable to the practical man.
The descriptive parts are given more fully here than they were in the lectures, because they can be consulted more profitably on the printed page than they could in the manuscript, and will be examined more by the general reader than the mathematical part. But, on the other hand, the mathematical part is much more condensed here than it was in the class-room. This was done so as to keep the work in as small a space as possible ; and also because a student is supposed to have time for deliberate study, and can take time to overcome his dif ficulties and secure his results. It is intended, however, in the next edition, to publish an appendix, in order to explain the more difficult mathematical operations of the text.
I have taken special pains to make frequent references to other books and reports from which I have secured information. This will enable any one to verify more fully the positions which have been taken, and will be convenient for those who desire to secure a more thorough knowledge of any particular topic.
I do not deem it necessary to indicate those topics which are wholly original. To the reader who has never before given the subject any attention, all will be new ; and the well-informed reader will readily detect what is original.
A large amount of labor and study has been given to this subject in nearly all civilized countries, and yet the theories in regard to re sistance from transverse stress are not very satisfactory. In regard to the strength of rectangular beams, the " Common Theory," as I
iv PREFACE.
have called it, is sufficiently correct for ordinary practical purpose?, especially if the modulus of rupture, as determined by direct experi ment upon rectangular beams of the same material, be used. Bar low's " Theory of Flexure " appears to be more nearly correct in theory when applied to rectangular beams and beams of the I section, or other forms which are symmetrical in reference to the neutral axis. But when the sections are irregular none of the theories can be relied upon for securing correct results. Whatever theory may yet do for us, it is quite evident that no theory will ever be devised, of practical value, which will be applicable to the infinite variety of forms of beams which are or may be used in the mechanic arts. That I may not be misunderstood upon this point, I will be more specific. We know that our present theories do not always give correct results, and that the more irregular the form the greater the discrepancy between the actual and computed strengths of a beam. Now, if a theory is ever devised which will take into account all the conditions of strains in a beam, I think it will be too complicated to be of practical value to the mechanic. I do not desire by this remark to disparage theory. Theories are valuable. Without them we would make little or no progress. Fortunately for the engineer, it is not the mathematically exact result that he desires, but the reliable result. He does not so much desire to know that one pound more of load will break his structure, as he does that he may depend upon it to carry from four to six times the load which he intends to put upon it. The theories, as now developed, are safe guides to the mechanic and engineer ; still we learn to depend more and more upon direct experiment. The theory also in regard to the deflection of beams under a transverse strain, has recently received a modification, due to a consideration of the effect of transverse shearing ; but the modification is sustained both from mathematical and experimental considerations. May not more careful experiments yet teach us that it must be still further modified on account of the longitudinal shearing strain?
The author will be pleased to receive the results of experiments which have been made in this country, so that if this work is revised in the future, it may be made more profitable to the engineering profession.
Axx ARBOR, MICH., Sept., 1871.
TABLE OF CONTENTS.
INTRODUCTION.
NO. OF THE
ARTICLE. PAOK
1. General Problems 1
2. Definition of Certain Terms 2
3. Stresses produce Elastic and Ultimate Resistances 3
4. General Principles of Elastic Resistance 3
5. Coefficient of Elasticity , 4
C. Proofs of the Laws of Elastic Resistance 5
CHAPTER I.
TENSION.
7. Experiments on the Elongation of Wrought Iron 7
8. Graphical Representation of Results 8
9. Elongation of Cast Iron , 10
10. Graphical Representation 13
11. Tables of Experiments on Cast Iron 12
12. Coefficient of Elasticity of Malleable Iron 14
13. Elasticity of Wood Longitudinally 15
14. Elasticity of Wood Radially 16
15. Remark on the Coefficient of Elasticity 17
1C. Elongation of a Prismatic Bar by a Weight 17
17. Elongation of a Prismatic Bar when the weight of the bar is con
sidered 10
18. Work of Elongation 20
19. Vertical Oscillations 21
20. Viscocity of Solids 22
21. Modulus of Strength 23
22. Strength of a Prismatic Bar 24
23. Strength of a Prismatic Bar when its weight is considered 25
24. Bar of Uniform Strength 25
25. Strength of a Closed Cylinder 26
26. Strength of Glass Globes '. 30
27. Experiments on Riveted Plates 31
28. Strength of Rolled Sheets of Iron in different directions 33
29. Strength of Wrought Iron at various temperatures : . 36
30. Effect of Severe Strains on the Tenacity of Iron 40
31. Effect of Repeated Rupture 41
VI CONTENTS.
NO. OF THE
AKTICLE. PAOE
82. Strength of Annealed Iron 41
83. Strength of Metals Modified by Treatment 42
84. Effect of Prolonged Fusion on Cast Iron 43
35. Remelting Cast Iron— On Strength of 43
36. Cooling Cast Iron — Affects Strength of 44
37. Strength Modified by Various Circumstances 44
38. Safe Limit of Loading 44
CHAPTER II.
COMPRESSION.
39. Elastic and Ultimate Resistance 46
40. Compression of Cast Iron 47
41. Compression of Wrought Iron 48
42. Graphical Representation of Results 49
43. Comparative Compression of Cast and Wrought Iron 49
44. Compression of other Metals 50
45. Modulus for Crushing 50
46. Modulus of Strain 51
47. Resistance of Cast Iron to Crushing 52
48. Resistance of Wrought Iron to Crushing 53
49. Resistance of Wood to Crushing 53
50. Resistance of Cast Steel to Crushing 54
61. Resistance of Glass to Crushing 54
52. Strength of Pillars 55
53. Formulas for the Weight of Pillars 58
54. Irregularities in the Thickness of Cast Pillars 60
55. Experiments of the N. Y. C. R. R. Co. on Angle Irons 60
56. Buckling of Tubes 66
57. Collapse of Tubes 66
58. Results of Experiments on Collapsing 69
59. Law of Thickness to resist Collapsing 69
60. Formula for Thickness to resist Collapsing 72
61. Resistance of Elliptical Tubes to Collapsing 72
62. Strength of very long Tubes 73
63. Strength from External and Internal Pressure Compared 73
64. Resistance of Glass Globes to Collapsing 73
CHAPTER ILL
THEORIES OP FLEXURE AND RUPTURE FROM TRANSVERSE STRESS.
65. Remark upon the Subject 75
66. Galileo's Theory 75
67. Robert Hooke's Theory 76
68. Marriotte's and Leibnitz's Theory 76
69. James Bernouilli'a Theory 76
70. Parent's Theory 77
CONTENTS. Yll
NO. OF THE
ABTICLK. PAQE
71. Coulomb's Theory 77
72. Young's " Modulus of Elasticity " 77
73. Navier's Developments of Theory 78
74. The Common Theory 78
75. Barlow's Theory 79
76. Remarks upon the Theories 82
77. Position of the Neutral Axis found Experimentally 83
78. Position of the Neutral Axis found Analytically 84
CHAPTER IV.
SHEARING STRESS.
79. Examples of Shearing1 Stress 89
80. Modulus of Shearing Stress 89
81. A Problem of a Tie Beam 91
82. A Problem of Riveted Plates 92
83. Longitudinal Shearing in a Bent Beam 92
84. Transverse Shearing in a Bent Beam 93
85. Shearing Resistance to Torsion 95
CHAPTER V.
FLEXURE.
86. General Equation of the " Elastic Curve " 96
87. Moment of Inertia of a Rectangle and of a Circle 98
88. GENERAL STATEMENT OF THE PROBLEMS 99
89. Beams Fixed at one end Load at the Free End 99
90. Beams Fixed at one end and Loaded Uniformly 101
91. Previous Cases Combined 102
92. Beams Supported at their Ends and Loaded at any Point 102
93. Beams Supported at their Ends and Loaded Uniformly 105
94. The two preceding Cases Combined 105
95. Examples 106
96. Deflection according to Barlow's Theory 108
97. Beams Fixed at one end, Supported at the other, and Loaded at any
Point I 108
98. Beams Fixed at one end, Supported at the other, and Loaded Uni
formly 113
99. Beams Fixed at both ends and Loaded at the Middle 115
100. Beams Fixed at both ends and Loaded Uniformly 116
101. Table of Results 118
102. Remarks upon the Results 118
103. Modification of the Formulas for Deflection to include shearing re
sistance 119
104. Unsolved Problems 121
105. Deflection of Beams having Variable Sections 122
106. Beams subjected to Oblique Strains 124
VI 11 CONTKXTS.
NO. OF THE
AKTICLE. PAOE
107. Floxure of Columns 120
108. Definition of %i Graphical Methods " 129
109. General Expression for the Deflection of Beams found by the Graphi
cal Method 129
110. Solution of Case I. by the Graphical Method 131
111. Solution of Case II. by the Graphical Method 133
112. Solution of Case III. by the Graphical Method 134
113. Solution of Case IV. by the Graphical Method 134
114. Remark in regard to other Cases 135
115. Moment of Inertia of a Rectangle by the Graphical Method 136
110. Moment of Inertia of a Triangle by the Graphical Method 130
117. Moment of Inertia of a Circle by the Graphical Method 137
118. Remark in regard to the Moment of Inertia of other Surfaces 139
CHAPTER VI.
TRANSVERSE STRENGTH.
119. Strength of Solid Rectangular Beams 140
120. Definition of the Modulus of Rupture 142
121. Practical Formulas 144
122. Relative Strength of a Beam 145
123. Examples 145
124. Relation between Strain and Deflection 146
125. Strength of Hollow Rectangular Beams 147
126. Strength of Double T Beams 148
127. True Value of d 150
128. Experiments of Baron Von Beber for determining the Thickness of
the Vertical Web 151
129. ANOTHER GRAPHICAL METHOD 154
130. Strength of a Square Beam with its Diagonal Vertical 154
131. Treatment of Irregular Sections 150
132. Formula of Strength according to Barlow's Theory 158
133. Strength of a Beam Loaded at any number of Points 159
134. Strength of a Beam when Loaded Uniformly over a portion of its
length 1«0
135. Example of Oblique Strain 162
136. GENERAL FORMULA OK STRENGTH 163
137. Strength of a Rectangular Beam determined from the General For
mula 164
138. Rectangular Beam, with its Sides Inclined 164
139. Strongest Rectangular Beam which can be cut from a given Cylin
drical one 166
140. Strength of Triangular Beams 166
141. Strongest Trapezoidal Beam which can be cut from a Triangular one. 168
142. Moment of Resistance of Cylindrical Beams 109
143. Moment of Resistance of Elliptical Beams. 171
144. Moment of Resistance of Parabolic Beams 171
145. Strength according to Barlow's Theory 172
CONTENTS. IX CHAPTER VII.
BEAMS OF UNIFORM RESISTANCE.
NO. OF THE
ARTICLE. PAGE
146. General Expression 173
147. Beams Fixed at one end and Loaded at the Free End 173
148. Beams Fixed at one end and Uniformly Loaded 174
149. The two preceding Cases Combined 175
150. Beams Fixed at one end, and the weight of the Beam the only Load. 176
151. Beams Supported at their Ends 178
152. Beams Fixed at their Ends 179
153. Effect of Transverse Shearing Stress on Modifying the form of the
Beams of Uniform Resistance, and Tabulated Values of Shearing
Stresses. 180
154. Unsolved Problems of Beams of Uniform Resistance 183
155. Best form of Cast-Iron Beam, as found Experimentally 184
15"6. Hodgkinson's Formula for the Strength of Beams of the "Type
Form " 187
157. Experiments on T Rails '. 187
158. Remark on Rolled Wrought Iron Beams 188
CHAPTER VIII.
TORSION.
159. How Torsive Strains are Produced 189
160. Angle of Torsion 189
161. Value of the Coefficient of Elastic Resistance to Torsion 191
162. Torsion Pendulum 192
163. Rupture by Torsion 193
164. Practical Formulas 194
165. Results of Wertheim's Experiments 196
CHAPTER IX.
EFFECTS OF LONG-CONTINUED STRAINS AND OF SHOCKS — CRYSTALLIZATION.
166. General Remark 198
167. Hodgkinson's Experiments 199
168. Vicat's Experiments 199
169. Fairbairn's Experiments 199
170. Roebling's Observations 202
OFT REPEATED STRAINS 203
171. Effect of Shocks 206
172. Crystallization 210
CONTENTS. CHAPTER X.
LIMITS OF SAFE LOADING FOR MECHANICAL STRUCTURES.
NO. OF THE
ABTICLE.
173. Risk and Safety
174. Absolute Modulus of Safety
175. Factor of Safety
176. Safe Load as determined by the Elastic Limit
177. Examples of Existing Structures
APPENDIX I.
ON THE PRESERVATION OF WOOD.
APPENDIX II.
TABLE OF THE PROPERTIES OF MATERIALS.
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A TEEATISE
THE RESISTANCE OF MATERIALS.
Library.
INTKODUCTIOK
1. IN PROPORTIONING ANY MECHANICAL STRUCTURE,
there are at least two general problems to be considered :---
1st. The nature and magnitude of the forces which are to be applied to the structure, such as moving loads, dead weights, force of the wind, etc. ; and,
2d. The proper distribution and magnitude of the parts which are to compose the structure, so as to successfully resist the applied forces.
These problems are independent of each other. The former may be solved without any reference to the latter, as the struc ture may be considered as composed of rigid right lines. The latter depends principally upon the mechanical properties of the materials which compose the structure, such as their strength, stiffness, and elasticity, under various circumstances.
The mechanical properties of the principal materials — wood, stone, and iron * — have been determined with great care and expense by different experimenters, both in this and foreign countries, to which reference will hereafter be made.
* The properties of mortars have been thoroughly discussed by Gen. Q. A. Gilmore in his work on Limes, Mortars, and Cements. 1862. 1
2 TIIE RESISTANCE OF MATERIALS.
£. DEFIMTIOXS OF TERMS.
STRESSES are the forces which are applied to bodies to bring into action their elastic and cohesive properties. These forces cause alterations of the forms of the bodies upon which they act.
STRAIN is a name given to the kind of alterations produced by the stresses. The distinction between stress and strain is not always observed ; one being used for the other. One of the definitions given by lexicographers for stress, is strain • and in asmuch as the kind of distortion at once calls to mind the manner in which the force acts, it is not essential for our pur pose that the distinction should always be made.
A TENSILE STRESS, or Pull, is a force which tends to elongate
ft O
a piece, and produces a strain of extension, or tensile strain.
A COMPRESSIVE STRESS, or Push, tends to shorten the piece, and produces a compressive strain.
TRANSVERSE STRESS acts transversely to the piece, tending to bend it, and produces a bending strain. But as a compressive stress sometimes causes bending, we call the former a transverse strain, for it thus indicates the character of the stress which produces it. Beams are generally subjected to transverse strains.
TORSIVE STRESS causes a twisting of the body by acting tan- gentially, and produces a torsive strain.
LONGITUDINAL SHEARING STRESS, sometimes called a detrv- xive strain, acts longitudinally in a fibrous body, tending to draw one part of a solid substance over another part of it ; as, for instance, in attempting to draw the piece A B, Fig. 1, which has a shoulder, through the mortise C, the part forming the shoulder will be forced longitudinally off from the body of the piece, so that the remaining part may be drawn through.
TRANSVERSE SHEARING STRBSE ie a force which acts trans versely, tending to force one part of a solid body over the adja-
INTRODUCTION. 3
cent part. It acts like a pair of shears. It is the stress which would break a tenon from the body of a beam, by acting per pendicular to the side of the beam and close to the tenon. It is the stress which shears large bars of iron transversely, so often seen in machine-shops. The applied and resisting forces act in parallel planes, which are very near each other.
SPLITTING STRESS, as when the forces act normally like a wedge, tending to split the piece.
•$. THE EFFECT OF THESE STRESSES IS TWOFOLD:—
1st. Within certain limits they only produce change of form ; and, 2d, if they be sufficiently great they will produce rupture, or separation of the parts ; and these two conditions give rise to two general problems under the resistance of materials, the former of which we shall call the problem of ELASTIC RESIST ANCE ; the latter, ULTIMATE RESISTANCE, or RESISTANCE TO RUPTURE.
4. GENERAL PRINCIPLES OF ELASTIC RESISTANCES.—
To determine the laws of elasticity we must resort to experi ment. jBars or rods of different materials have been subjected to different strains, and their effects carefully noted.
From such experiments, made on a great variety of materials, and with apparatus which enabled the experimenter to observe very minute changes, it has been found that, whatever be the physical structure of the materials, whether fibrous or granu lar, they possess certain general properties, among which are the following : —
1st. That all bodies are elastic, and within very small limits they may be considered perfectly elastic ; i. e., if the particles of a body be displaced any amount within these limits they will, when the displacing force is removed, return to the same posi tion in the mass that they occupied before the displacement. This limit is called the limit of perfect elasticity*
* Mr. Hodgkinson made some experiments to prove that all bodies are non- elastic. (See Civil Eng. and Arch. Jour, vol.vi., p. 354.) He found that the limits of perfect elasticity were exceedingly small, and inferred that if our
4 THE RESISTANCE OF MATERIALS.
2d. The amount of displacement within the elastic limit ia directly proportional to the force which produces it. It follows from this, that in any prismatic bar the force which produces compression or extension, divided by the amount of extension or compression, will be a constant quantity.
3d. If the displacement be carried a little beyond this limit the particles will not return to their former position when the displacing force is removed, but a part or all of the displace ment will be permanent. This Mr. Ilodgkinson called a set, a tenn which is now used by all writers upon this subject.
4th. The amount of displacement is not exactly, but nearly, proportional to the applied force considerably beyond the elastic limit.
5th. Great strains, producing great sets, impair the elasticity.
5. COEFFICIENT (OR MODULUS*) OF ELASTICITY.
If a prismatic bar, whose section and length are unit}-, be compressed or elongated any amount with in the elastic limit, the quotient obtained by dividing the force which produces the displacement by the amount of compression or extension is called the COEFFICIENT OF ELASTICITY. This we call E. Let K= section of a prismatic bar (See Fig. 2), fcits length,
Pio. 2.
powers of observation were perfect in kind and infinite in degree, we should find that no body was perfectly elastic even for the smallest amount of dis placement. And although more recent experiments have indicate I the same result in cast-iron, yet the most delicate experiments have failed to thoroughly establish it. I have, therefore, accepted the principle of perfect elasticity, which, for the purposes of this work, is practically, if not theoretically, correct. It does not appear from Mr. Hodgkinson's report how soon the effect was observed after the strain was removed. If he had allowed considerable time the set might have disappeared, as it is evident that it takes time for the dis placed particles to return to their original position.
• The terms cofjfiricnt and modulm are used indiscriminately for the con stants which enter equations in the discussion of physical problems, and are sometimes called physical comtant*. The! modulus of elasticity, as used by moat writers on Analytical Mechanics, is the ratio of the force of restitution to
INTRODUCTION. 5
and A=the elongation or compression caused by a force, P,
which is applied longitudinally. Then -= force on a unit of section, and -=the elongation or compression for a unit of length.
Hence, from the definition given above, we have -p P x PI m
•k = K ~ T = KX * * * - I1)
From this equation E may be easily found. It will here after be shown that the coefficient is not exactly, but is nearly the same for compression as for tension.
For values of E, see Appendix, Table 1.
6. PROOFS OF THE LA.WS GIVEN IN ARTICLE FOUR.—
Article 5 has preceded these proofs, so as to show how the results of experiments may be reduced by equation (1). The 1st and 2d laws seem first to have been proved by S. Gravesend, since which they have been confirmed by numerous experimenters. One of the most extensive and reliable series of experiments upon various substances for engineering purposes is given in " The Report of Her Majesty's Commissioners, made under the direction of Mr. Eaton Hodgkinson." The results of his experiments are pub lished in the Reports of the British Association, and in the 5th volume of the Proceedings of the Manchester Literary and Philosophical Society, from which extracts have been made and to which we shall have occasion to refer. The experiments were made not only to prove these laws but several others, principally relating to transverse strength.
Barlow made many experiments, the results of which are given in his valuable work on the " Strength of Materials." The series of experiments on iron which had been commenced and so ably
that of compression. It relates to the impact of bodies, and, as thus denned, depends upon the set. But the coefficient of elasticity depends neither upon impact nor set. Another term should therefore be used, or else a distinction should be made between the terms coefficient and modulus, so that the former shall apply to small displacements, and the latter to the relative force of resti tution. For this reason I have used the former in this work, and avoided the latter when applied to elasticity.
6 THE RESISTANCE OF MATERIALS.
conducted by Mr. Ilodgkinson were continued by Mr. Fairbairn. The latter confined his experiments mostly to transverse strength, the results of which are given in his valuable work on " Cast and Wrought Iron." A valuable set of experiments has been made in France at "le Conservatoire des Arts et Metiers."'
In this country several very valuable sets of experiments have been made, among the most important of which are the experi ments of the Sub-Committee of the Franklin Institute, the re sults of which are published in the HHh and 20th volumes of the Journal of that Society, commencing on the 73d page of the former volume. The experiments were made upon boiler iron, but they develojnjd many properties common to all wrought iron. They were conducted with great care and scientific skill. The report gives a description of the testing machine; the manner of determining its friction and elasticity ; the modifica tions for use in high temperature ; the manner of determining the latent and specific heats of iron ; and the strength of differ ent metals under a variety of circumstances.
Another very valuable set of experiments was made by Cap- taiii T. J. Rodman and Major W. Wade, upon "Metals for Cannon, under the direction of the United States Ordnance Department," and published by order of the Secretary of War.
Numerous other experiments of a limited character have been made, too many of which have been lost to science be cause they were not reported to scientific journals, and many others were of too rude a character to be very valuable.
The results of these experiments will form the basis of our theories and analysis.
* See " Morin's Resistance des Matcriaux," p. 126.
EXPERIMENTS ON WROUGHT IRON.
CHAPTEE I.
TENSION.
7. TAKING THE PHENOMENA IN THEIR NATURAL, OR-
DEK, the first thing which claims attention is the elastic re sistance due to tension, or, as it is sometimes called, a pull, or elongating force.
EXPERIMENTS ON WEOUGIIT IRON.
Experiments for determining the total elongation and permanent elongation "pro duced by different weights acting by extension on a tie of wrought iron of the best quality, by Eaton Hodgkinson.
Weight in |
Elongation per |
metre of length. |
Coefficient of |
centimetre. P. |
Total. A. |
Permanent. |
per square metre. E. |
Kil. 187 429 |
M. 0 000082117 |
Mill. |
Kil. 22 824 500 000 |
874 930 |
0 000185261 |
20 216 200 000 |
|
562.400 749.456 937.430 1124.813 1312.283 1499.720 1687.219 1874.645 2063.580 2249.627 2403.653 2824.564 |
0.000283704 0.000379467 0.000475113 0.000570792 0.000665647 0.000760311 0.000873265 0.001012911 0.001283361 0.002227205 0.004287185 0.009156490 0 009950970 |
0.00254 0.0033894 0.0042398 0.00508 0.0067705 0.0100879 0.0330283 0.0829955 0.2616950 3.0709900 8.4690700 8 5748700 |
19 824 100 000 19 704 000 000 19 729 909 000 19 706 000 000 19 714 600 000 19 320 300 000 19 320 700 000 18398100000 16 079 200 000 *5 606 590 666 2 866 380 000 |
2812.033 Repeated after 1 hour |
0.010492805 0.011750313 |
9.1023600 |
2 681 520 000 |
' 2 |
0 011858889 |
||
' 3 |
0 011933837 |
||
' 4 |
0.011942168 |
||
1 5 |
0 011958835 |
||
' 6 |
0 011967149 |
||
' 7 |
0 012027114 |
||
' 8 |
0 012027014 |
||
8 THE RESISTANCE OF MATERIALS.
EXPERIMENTS ON WROUGHT IRON. Continued.
Weight in |
Elongation per i |
nctro of length. |
Coefficient of elasticity |
centimet-'e. P. |
Total. A. |
Permanent. |
per square metre. E. |
Kil. Repeated after 9 hours |
M. 0 012027114 |
Mill. |
Kil. , |
it U JO U |
0 012027114 |
||
2999.500 2999 500 |
0.017888263 0 019478898 |
16.5145 |
1 676 820 000 |
0.01984831 |
18.4212 |
||
0 02022006 |
18 8886 |
||
3186.97S |
0.02148590 0 02169401 |
19.7954 |
1 483 290 000 |
0 02170242 |
|||
0 02170242 |
22 0119 |
||
8374.440 |
0.02477441 0 02514184 |
22.7087 |
1 362 020 000 |
0 02522512 |
|||
3561.900 |
0.03493542 0 03519357 |
32.8201 |
1 019 580 000 |
0 03520190 |
|||
3745 3d |
|||
This table is given in French units because it was more con venient.*
8. THE RESULTS OF THESE EXPERIMENTS may 1)6 1*C-
presented graphically by taking, as has been done in Fig. 3. the total elongations or the permanent elongations for abscissas and the weights for ordi nates.
* To reduce the French measures to English we have the f ollowing relations : — LINEAR MEASURE.
3.2808992 feet = 1 metre. 0.0328089 feet = 1 centimetre. 0.0032808 feet = 1 millimetre. 0.0393696 in. = 1 millimetre,
WEIGHT.
2.20462 Ibs. avoird. = 1 kilogramme.
1422.28 Ibs. pr. sq. in. = 1 kilog. to the sq. millimetre.
0.00142228 Ibs. sq. in. = 1 kilog. pr. sq. metre.
Hence to reduce the above quantities to English units, multiply the numbers in the first column by 14.2228 to reduce them to pounds avoirdupois per square inch ; those in the second column by 3. 28089 + to reduce them to feet ; the third by 0.03930+ to reduce them to inches ; and the fourth by 0.00142228 to reduce them to pounds per square inch.
TENSION.
9
When the construction is made on a large scale it makes the results of the experiments very evident.
axil UAtuiinia- duuu tion of Fig. 3 shows : — 1st, That to a load of 1499.T2 kil. pr. square 24( centimetre, the 2000 total elongations are practically proportional to ^200 the loads ; 3d. That with- 80° in the same lim- AOQ its the perma nent elongations t are nearly pro- |
- |
3flt |
^ |
^ |
^ |
- |
||||
^ |
>L^ |
|||||||||
t |
/? |
^^ |
||||||||
~7/_ |
||||||||||
/ |
||||||||||
j |
||||||||||
— |
||||||||||
•pP* 004 '016 -028 -03G FIG. 3. |
portional to the loads, and that they are exceedingly small ;
3d. That beyond the load of 14.997 kil. to 22.00 kil. per square millimetre, the total and permanent elongations increase very rapidly and more than proportional to the loads ;
4th. That near and beyond 22.49 kil. per square millimetre, the total elongations become sensibly proportional to the loads, but in a much greater ratio than that which corresponds to small loads. For the loads near rupture the elongations are a little inferior to that indicated by the new proportion.
5th. Beyond 14.99 kil. per square millimetre, the permanent elongations increase much more rapidly than the total elonga tions. We also observe that the permanent elongations increase with the duration of the load, although very slowly. The latter
property will be more particularly noticed hereafter.
p 6th. Finally, the values — of the loads per square metre to
A
the elongation per metre, and which is called the coefficient of elasticity r, is sensibly constant when the elongations are nearly proportional to the loads ; and that the mean value is E = 19,816,440,000 kil. per square metre ;
— 28,283,000 Ibs. per square inch. The first value of E, in the table, is much larger, and may
10 THE RESISTANCE OF MATERIALS.
have resulted from an erroneous measurement of the exceed ingly small total elongation. From the experiments made on another bar, Ilodgkinson found
E = 19,359,458,500 kil. per sq. metre ;
= 27,700,000 pounds pr. square inch ; which is but little less than the preceding.
Mr. Ilodgkinson infers from these experiments that the small est strains cause a permanent elongation. But Morin for cibly remarks * that none of these experimenters appear to have verified whether time, after the strains are removed, will not cause the permanent elongations to disappear. Also that the deflections of the machine cannot be wholly eliminated, and hence appear to increase the true result. In practice such small permanent elongations may be omitted.
The preceding example has, for a long time, been given to show the law of relation between the applied force and the total and permanent elongations ; but we should not expect to find exactly the same results for all kinds of iron. Even wrought iron has such a variety of qualities, depending upon the ore of which it is made, and the process of manufacture, that it cannot be expected that the above results will always be applicable to it. Only a wide range of experiments will determine how far they may generally be relied upon.
It is found, however, that the GENERAL RESULTS of extension, of set, of increased elongation with the duration of the stress within certain limits, and of the increase of set with the in crease of load, are true of all kinds of iron.
EXPERIMENTS TPON CAST IRON. 9. THE FOLLOWING EXPERIMENTS UPON CAST IRON sllOW
that the numerical relation between the applied force and the extension is somewhat different from the preceding. The expe riments were made under the supervision of Captain T. J. Hod man : — f
u The specimens had collars left on them at a distance of thirty-five inches
* Morin's Resistance des Matcriaux, p. 10.
f Experiments on Metals for Cannon, by Capt. T. J. Rodman, p. 157.
For a full description of the testing apparatus, with diagrams, see Major Wade's Report on the Strength of Materials for Cannon, pp. 305-315. The machine consists principally of a very substantial frame and levers resting on knife edges.
TENSION.
11
apart, the space between the collars being accurately turned throughout to a uniform diameter.
" The space between the collars was surrounded by a cast-iron sheath, eight- tenths of an inch less in length than the distance between the collars ; it was put on in halves and held in position by bands, and was of sufficient interior diameter to move freely on the specimen.
" When in position, the lower end of the sheath rested on the lower collar of the specimen, the space between its upper end and the upper collar being sup plied with and accurately measured by a graduated scale tapered 0.01 of an inch to one inch.
u The upper end of the sheath was mounted with a vernier, and the scale was graduated to the tenth of an inch.
" This afforded means of measuring the changes of distance between the collars to the ten-thousandth part of an inch, and these readings divided by the distance between the collars gave the extension per inch in length as recorded in the following table : — •
TABLE
Shoiving tlie extension and permanent set per inch in length caused by the under mentioned weights, per square inch of section, acting upon a solid cylinder 35 inches long and 1.866 inches diameter. (Cast at the West Point Foundry in 1857.)
Weight per square inch of section. |
Extension per inch of length. |
Permanent set per inch in length. |
Coefficient of elasticity. |
P. |
A. |
E. |
|
Ibs. |
in. |
in. |
|
1,000 |
0.0000611 |
0. |
16,366,612 |
2,000 |
0.0000794 |
0. |
25,189,168 |
3,000 |
0.0001089 |
0. |
27,548,209 |
4,000 |
0.0001771 |
0. |
22,586,674 |
5,000 |
0.0002129 |
0. |
23,489,901 |
6,000 |
0.0002700 |
0.0000014 |
22 222 222 |
7,000 |
0.0003328 |
0.0000029 |
21 J033] 653 |
8,000 |
0.0003986 |
0.0000043 |
20,070,245 |
9,000 |
0.0004557 |
0.0000071 |
19,749,835 |
10,000 |
O.OOO-llOO |
0.0000109 |
19,607,843 |
11,000 |
0.0005500 |
0.0000157 |
20.000,000 |
12,000 |
0.0006414 |
0.0000257 |
18,693,486 |
13,000 |
0.0007100 |
0.0000300 |
18,309,859 |
14,000 |
0.0007700 |
0.0000357 |
18,181,181 |
15,000 |
0.0008557 |
0.0000477 |
17,529,507 |
16,000 |
0.0009243 |
0.0000529 |
17,310,397 |
17,000 |
0.0010014 |
0.0000643 |
16,977,231 |
18,000 |
0.0010900 |
0.0001014 |
16,537,614 |
19,000 |
0.0012271 |
0.0001471 |
15,483,660 |
20,000 |
0.0013586 |
0.0002014 |
14,721,109 |
21,000 |
0.0015386 |
0.0002900 |
13,648,771 |
22,000 |
0.0017043 |
0.0003986 |
12,908,523 |
23,000 |
0.0019529 |
0.0005529 |
11,265,246 |
24,000 |
0.0022786 |
0.0007529 |
10,532,344 |
25,000 |
0.0026037 |
0.0010843 |
9,601,720 |
26,000 |
0.0032186 |
8,078,046 |
12
THE RESISTANCE OF MATERIALS.
1O. I K.I 1C i: 4 IS A GRAPHICAL, REPRESENTATION OF
THE ABOVK TABLE, constructed in the same way as Figure 3. Experiments were made upon many other pieces, from which I have selected four, and called them A, B, C, and D, a gra phical representation of which is shown in Figure 5. The right hand lines represent extensions, the left hand sets.
A is from an inner specimen of a Fort Pitt gun, No. 335, and the others from different cylinders which were cast for the purpose of testing the iron.
From these we observe :—
1st. That for small elongations the ratio of the stresses to the elongations is nearly constant.
2d. There does not appear to be a sudden change of the rate of increase, as in Mr. Ilodgkinson's example, but the ratio gra dually increases as the strains increase.
3d. The sets at first are invisible, but they increase rapidly as the strains approach the breaking limit.
It appears paradoxical that the first and second experiments in the preceding table should give a less coefficient than the third, but the same result was observed in several cases.
11. THE FOLLOWING TABLES ARE THE RESULTS SOIflE i:\IT1C I 'irvrs IVIADE.BY flit. HODGKINSON t—
OF
TENSION. 13
'Direct longitudinal extension of round rods of cast iron, fifty feet long.
1 |
^ |
Weights per sqnare inch laid on, |
ll§ |
1. |
|||
'c8 g |
with their corresponding ex |
"a "^tj |
.§ § |
||||
NAME OF IRON. |
°. G |
s J |
tensions and sets. |
§ cu 3: |
"S'a |
||
!^'i |
C « |
ft &,„ |
al |
||||
» |
« M |
ifrfl |
IS |
||||
0) |
Weights. |
Extension. |
Seta. |
sfl |
|||
Ibs. |
in. |
in. |
Iba. |
in. |
|||
Low Moor No 2 |
2 |
1.058 |
2,117 |
0. 0950 |
0. 00345 |
16,408 |
1.085 |
6,352 |
0.3115 |
0.0250 |
|||||
10,586 |
0.5740 |
0.4425 |
|||||
14,821 |
0.9147 |
0.12775 |
|||||
Blaneavon Iron, No. 2 . . |
2 |
1.0685 |
2,096 |
0.0942 |
0.00268 |
14,675 |
0.9325 |
6,289 |
0.3065 |
0.01675 |
|||||
10,482 |
0.5770 |
0.0575 |
|||||
13,627 |
0.8370 |
0.11475 |
|||||
Gartsherrie Iron, No. 2 . |
2 |
1.062 |
2,109 |
0.0922 |
0.001 + |
16,951 |
1.167 |
6,328 |
0.3117 |
0.01450 |
|||||
10,547 |
0.5862 |
0.0475 |
|||||
14,766 |
0.9452 |
0.11352 |
In these experiments the ratio of the extensions is somewhat greater than that of the weights. The value of E, as computed for the first weights which are given, and the corresponding extensions, is a little more than 13,000,000 pounds per square inch.
Extension of cast-iron rods, ten feet long and one inch square.
Weights P. |
Extensions. Ae. |
Sets. |
p Ae. |
BMlSI Jbf&ti |
Ibs. |
in. |
in. |
||
1053.77 |
.0090 |
.00022 |
117086 |
- -h |
1580.65 |
.0137 |
.000545 |
115131 |
- ^ |
2167.54 |
.0186 |
.00107 |
113309 |
-T*¥ |
3161.31 |
.0287 |
.00175 |
110150 |
+ uio |
4215.08 |
.0391 |
.00265 |
107803 |
+ Y5T |
5262.85 |
.0500 |
.00372 |
105377 |
+ T^U |
6322.62 |
.0613 |
.00517 |
103142 |
-1- TT-T |
7376.39 |
.0734 |
.00664 |
100496 |
+ Tk |
8430.16 |
.0859 |
.00844 |
98139 |
+ 75<T |
9483.94 |
.0995 |
.01062 |
95316 |
+ -rk |
10537.71 |
.1136 |
.01306 |
92762 |
+ T^T |
11591.48 |
.1283 |
.01609 |
90347 |
-«rhr |
12645.25 |
.1448 |
.02097 |
87329 |
-Tk |
13699.83 |
.1668 |
82133 |
+ j&T |
|
14793.10 |
.1859 |
! 02410 |
79576 |
- «Lo |
14 THE RESISTANCE OF MATERIALS.
Let P = the elongating force and
\ = the total elongation in inches due to P. Then Ilodgkinson found, from an examination of the table, that the empirical formula
P = 116117Ae - 201905*;
represented the results more nearly than equation (1). This for mula reduced to an equivalent o^p for I in inches (observing that the bar was 10 feet long), becomes
P =-- 13,934,000 _Ae — 2,907,432,000^
Although this equation gives the elongations for a greater range of strains than equation (1) for this particular case, yet the law represented by it is more complicated, and hence would make the discussions under it more difficult, without Yielding
V
any corresponding advantage. It is the equation of a parabola in which P is the abscissa and Ae the ordinate.
We also see that when the elongations are Yen- small, the
A2
quantity -*- will be very small, and the second term may be C
omitted in comparison with the first, in which case it will be re duced to equation (1). The coefficient in the first term is the coefficient of elasticity, hence it is nearly 14,000,000 Ibs. for extension.
MALLEABLE IRON.
1<J. ACCORDING TO BARLOW'S EXPERIMENTS malleable iron may be elongated T7Vir °^ ^ length without endangering its elasticity.* To ascertain this, the strains were removed from time to time, and it was found that the index returned to zero for all strains less than 9 or 10 tons. The mean extension per ton (of 2,240 Ibs.) per square inch, for four experiments, was 0.00009565 of its original length. Hence the mean value of the coefficient of elasticity is
E = -= - _22_42_ = 23,418.000 Ibs. A O.OU009565
* Journal Frank. Lost., voL xvi., 2d Series, p. 126.
TENSION. 15
ELASTICITY OF WOOD. 13. EXPERIMENTS BY MESSRS. €HEVANDIER ANI* WER-
THEiM.-The following are some of the results of the recent experiments of Messrs. Chevandier and Wertheim on the resis tance of wood. These experimenters have drawn the follow ing principal conclusions : —
1. The density of wood appears to vary very little with age.
2. The coefficient of elastiok}r diminishes, on the contrary, be
yond a certain age ; it depends, likewise, upon the dry- ness and the exposure of the soil to the sun in which the trees have grown ; thus the trees grown in the northern exposures, north-eastern, north-western, and in dry soils, have always so much the higher coefficient as these two conditions are united, whereas the trees grown in muddy soils present lower coefficients.
3. Age and exposure influence cohesion.
4. The coefficient of elasticity is affected by the soil in which
the tree grows.
5. Trees cut in full sap, and those cut before the sap, have not
presented any sensible differences in relation to elasticity.
6. The thickness of the woody layers of the wood appeared to
have some influence on the value of the coefficient of elasticity only for fir, which was greater as the layers wTere thinner.
7. In wood there is not, properly speaking, any limit of elasti-
ticity, as every elongation produces a set. It follows from this circumstance that there is no limit of elasticity for the woods experimented upon by Messrs. Chevan dier and Wertheim, but in order to make the results of their ex periments agree with those of their predecessors, the authors have given for the value of the limit of elasticity the load under which it produces only a very small permanent elongation ; the limit which they indicate in the following table for loads under which the elasticity of wood commences to change, corresponds to a permanent elongation of 0.00005, its original length.
1C
THE RESISTANCE OF MATERIALS.
TABLE CONTAINING TIIE MEAN RESULTS OF THE EXPERIMENTS OF
MESSRS. CllEVANDIER AND WERTIIEEM.
S^S |
sis* |
?S*B |
-5 - |
||
111 |
iNi |
f'^t |
§| |
||
Species. |
•c |
^ |
•Ssj-s |
w |
a| |
« |
fe»l |
llll |
03 |
sf |
|
sill |
I sill |
iill |
«'S. j> |
||
Kilogr. |
Kilopr. |
Kilogr. |
|||
0.717 |
1261.9 |
3.188 |
7.93 |
s^» |
|
Fir |
0.493 |
1113.2 |
2.153 |
4.18 |
•gg . |
Yoke Elm |
0756 |
1085.3 |
1.282 |
299 |
|
Birch |
0 812 |
997.2 |
1.617 |
430 |
-M* -9 |
Beech |
0.823 |
980.4 |
2.317 |
3.57 |
•2 "^ S |
Oak from pedunculate acorn " " sessile acorn |
0.808 0.872 |
977.8 921.8 |
u 2.349 |
6.49 5.66 |
III |
White Pine |
0.559 |
564.1 |
1.633 |
2.48 |
|
Elm ... |
0.723 |
1165.3 |
1.842 |
699 |
.2 o ^< |
0.692 |
1163.8 |
1.139 |
6.16 |
gSjj |
|
Ash |
0.697 |
1121.4 |
1.246 |
6.78 |
'5 o £ |
Alder |
0.601 |
1108.1 |
1.121 |
4.54 |
|
Aspen |
0.602 |
1075.9 |
1.035 |
7 20 |
_ |
Maple |
0.674 |
1021.4 |
1.068 |
3.58 |
rj O - ~ — |
Poplar. . |
0.477 |
517.2 |
1.007 |
1.97 |
H « |
14. ELASTICITY OF WOOD, TANGENTIALLY AND RATI-
AL.LY.-The same observers have also determined the coefficient of elasticity and the cohesion of wood in the direction of the radius and in the direction of the tangent to the woody layers. An examination of the following Table shows that the resis tance in the direction of the radius is always greater than the resistance in the direction of the tangent to the woody layers ; the relation between the coefficients of elasticity in the two cases varying nearly from 3 to 1.15.
TENSION.
Library*
California
MEAN RESULTS OF THE EXPERIMENTS OF MESSRS. CIIEVANDIER AND WERTHEIM.
^PECIES. |
IN THE DIRECTION OF RADIUS. |
IN THE DIRECTION OF THE TAN GENT TO THE LAYERS. |
||
Coefficient of Elasticity, E, per square millime tre. |
Cohesion, or load, per square millimetre, capa ble of producing rupture. |
Coefficient of Elasticity, E, per square millime tre. |
Cohesion, or load, per square millimetre, capa ble of producing1 rupture. |
|
Yoke Elm |
Kilogr. 208.4 134.9 157.1 188.3 81.1 269.7 111.3 121.6 94.5 97.7 170.3 |
Kilogr. 1.007 0.522 0.716 0.582 0.823 0.885 0.218 0.345 0.220 0.256 ii |
Kilogr. 103.4 80.5 72.7 129.8 155.2 159.3 102.0 63.4 34.1 28.6 152.2 |
Kilogr. , 0.608 0.610 0.371 0.406 1.063 0.752 0.408 0.366 0.297 0.196 1.231 |
Sycamore |
||||
Maple |
||||
Oak |
||||
Birch |
||||
Beech |
||||
Ash |
||||
Elm |
||||
Fir |
||||
Pine . . . |
||||
Locust |
The highest coefficient of elasticity in this table is for beech, and this is less- than 400,000 pounds per square inch.
lo. REMARK. — The value #/E, which is used in practice, i& not the coefficient of perfect elasticity, but it is that value which is nearly constant for small strains. In determining it, no ac count is made of the set. If the total elongations were propor tional to the stresses which produce them, we would use the value of E found by them, even if the permanent equalled the total elongations. But in practice the permanent elongations will be small compared with the total for small stresses.
APPLICATIONS.
16. TO FIND THE ELONGATION OF A PRISMATIC BAR SUBJECTED TO A LONGITUDINAL STRAIN WHICH IS WITH IN THE ELASTIC LIMITS.
From (1) we have
A =
_ EK
which is the required formula.
2
18
THE RESISTANCE OF MATERIALS.
Also from (1) we have
P = ?EK
(3)
Fio. 6.
Equations (1), (2), and (3) are equally applicable to compressive strains, as will hereafter he shown. If in (3) we make K = l and *=l we shall have P = E ; hence, the coefficient of elasticity may he de fined to he a force which will elongate a lar whose section is unity, to double its original length, pro vided the elasticity of the material does not change. But there is no material, not even a perfectly elastic body, — as air and other gases, — whose coefficient of elasticity will not change for a perceptible change of volume. The material may not lose its elasticity, but equation (1) only measures it for small displacements. To illustrate further, let it be observed that, according to Mariotte's law, the volumes of a gas are inversely proportional to the compressive (or exten sive) forces ; double the force producing a compression of half the volume ; four times the force, one-fourth the volume, and so on, — the compressions being a fractional part of the original volume ; but in equation (2), A is a linear quantity, so that if one pound produces an extension (or compression) of one inch, two pounds would produce an extension of two inches, and
so on.
Examples.— \. If the coefficient of elasticity of iron be 25,000,000 Ibs., what must be the section of an iron bar 60 feet long, so that a weight of 5,000 Ibe. shall elongate it i inch ?
PI From (1) we obtain K = • -- which by substitution becomes
- = 0. 288 square inches. /
_ 5,000.12x60 K = 25,000,000 x
2. How great a weight will a brass wire sustain, whose diameter is 1 inch ; coefficient of elasticity is 14,000,000 Ibs., without elongating it more than ^ of its length? Ans. 13,744.5 Ibs.
TENSION.
19
* « • REQUIRED THE ELONGATION (OR COMPRESSION) OF A PRISMATIC BAR WHEN ITS WEIGHT IS CONSIDERED.
Let I = the whole length of the bar before elon gation or compression,
x = variable distance = A5,
dx = 1)0 = an element of length,
w = weight of a unit of length of the bar,
W = weight of the bar, and
Pj = the weight sustained by the bar. Then (I — x) w + P, = P — the strain on any section, as be.
Hence, from equation (2), we have FlG 7
EK
•. the total length will become,
(5)
wT' If Px = 0, X = T =
Wl 2FK
one-
half of what it would be if a weight equal to the whole weight of the bar were concentrated at the lower end.
REQUIRED THE ELONGATION (OR COMPRESSION) OF A CONE IN A VERTICAL POSITION, CAUSED BY ITS OWN WEIGHT WHEN IT is SUS PENDED AT ITS BASE (OR RESTS ON ITS BASE).
Take the origin at the apex before extension, Fig. 8, and let K = any section,
K0 = the upper sectien,
I = the length or altitude of
the cone,
x = the length or altitude of any portion of the cone, and
B = the weight of a unit of volume. Then, because the bases of similar cones are as the squares
a?2 of their altitudes, K = K0 -™
FIG. 8.
20 THE RESISTANCE OF MATERIALS.
The volume of the cone whose altitude is x
X s*X 3?
„ K^=/o K.-
and the weight of the same part
/
(from equation (2)) X=
from which it appears that the total elongation is independent of the transverse section, and varies as the square of the length.
18. THE WORK OF ELONGATION. — If P be the force which does the work, and x the space over which it works, then the general expression for the work is
............ (6)
o
To apply this to the prism, substitute P from Eq. (3) in (6), and make dx = d\ and we have
/EK 1
i E^ 7,
U = / —r d\ =
which is the same result that we would have found by suppos ing that P was put up by increments, increasing the load gradu ally from zero to P.
Example. — If the coefficient of elasticity of wrought iron be 28,000,000 Ibs., and is expanded O.OOOOOC98 of its length for one degree F., how much work is done upon a prismatic bar whose section is one inch, and length 30 feet, by a change of 20 degrees of temperature ?
Walls of buildings which were sprung outward have been drawn into an erect position by heating and cooling bars of iron. Several rods were passed through the buildings, and extending from wall to wall, were drawn tight by means of the nuts. Then a part of them were heated, thus elongating them, and the nuts tightened; after which they were allowed to cool, and the contraction which resulted drew the walls together. Then the other rods were treated in a similar manner, and so on alternately.
TENSION.
21
19. VERTICAL, ©SCIL,L,ATIONS._If a bar Aa, Fig. 9, with a weight, P, suspended from its lower end, be pressed down by the hand, or by an additional weight from a to £, and the addi tional force be suddenly removed, the end of the bar on returning will not stop at «, but will move to some point above, as c, a dis tance ac = ab. From a principle in Mechanics, viz. , that the living force equals twice the work, we are enabled to determine all the circumstances of the escillation when the weight of the bar is neglected. The weight P elongates the bar so that its lower ex tremity is at «, at which point we will take the origin of co-or dinates.
Fig. 9. Let X = ab = the elongation caused by the additional force,
x = ad — any variable distance from the origin,
v = the velocity at any point, as d, and
M = the mass of the weight P. If the weight of the rod be very small compared with P, the vis viva is
= — «2 very nearly.
The work for an elongation equal to X, is by Eq. (7),
Wr t L tvwv
EK
for half an oscillation PI
and the time for a whwe oscmation is
m
PI / X
— TTA / - - -
(8)
hence the oscillations will be isochronous. ... ^ I ~
It is evident that by applying and removing the force at regular intervals, the amplitude of the oscillations may be increased and possibly produce rupture. In this way the Broughton suspension bridge was broken.*
As a second example take the case in which P is applied suddenly to the end of the rod. It is evident that the total elongation will be greater than X, — the permanent elongation. For the fundamental equation we may use another
* Mr. E. Hodgkinson, in the 4th volume of the Manchester PhilosopJiicol Transactions, gives the circumstances of the failure, from this cause, of the suspension bridge at Broughton, near Manchester, England. And M. Navier, in his theory of suspension bridges (Fonts Suspendus, Paris, 1823), states that the duration of the oscillation of chain bridges may be nearly six seconds.
•¥r
ctt
22 THE RESISTANCE OF MATERIALS.
J
principle in Mechanics, which might have been used in the preceding problem, viz., that the mass multiplied by the acceleration equals the moving force. The
EK*
resisting force for an elongation x is -y (See Eq. (3) ), and the moving force is
P
P, whose mass = — ; hence
^ M ^=P -55*.
d*t't~ ' I
= «•
e=4/-
x — x Hence, the amplitude is twice the permanent elongation. If x = 2 A we have
t = T -i/ — „-= = * / — . Investigations of this kind give rise to a divi sion of the subject called Resilience of JPmwa.
The investigations are interesting, but the results are of little use beyond those which have already been indicated. From the last problem we see that a weight suddenly applied produces twice the strain that it would if applied gradually.
As additional exercises for the student, I suggest the following : Suppose the weight be applied with an initial velocity. Suppose ti weight P is attached to one end, and the weight P' is placed suddenly upon it ; or it falls upon it. To find the velocity at any point in terms of t, — also A in terms of t.
If a weight W is suspended at the end, and another weight W, falls from a height A, giving rise to a velocity fl, we have for the common velocity of the
TIT
bodies after impact, if both are non-elastic, V = J _, and the ?i8 viva of
both will be
W 'TJ* »FK
MV" = ' which equals^— X-, or twice the work.
•*' X -V(W. + W) V EK* This is only an approximate value, for the inertia of the wire is neglected.
2O. VISCOSITY OF SOLIDS. — Experiments show that the pri n- ciple of equal amplitudes, referred to in the preceding article, is not realized in practice. This is more easily observed in trans verse vibrations. The amplitudes grow rapidly less from the first vibration, and the diminution cannot be fully accounted for by the external resistance of air. Professor Thompson of Eng-
TENSION. 23
land has shown that there is an internal resistance which opposes motion among the particles of a body, and is similar to that resistance in fluids which opposes the movement of particles among themselves. He therefore called it viscosity* He proved :
1st. That there was a certain internal resistance which he called Viscosity, and which is independent of the elastic pro perties of metata;
2d. That this force does not affect the co-efficient of elasticity.
The law between molecular friction and viscosity was not discovered.
The viscosity was always much increased at first by the in crease of weight, but it gradually decreased, and after a few days became as small as if a lighter weight had been applied. Only one experiment was made to determine the effect of con tinual vibration ; and in that the viscosity was very much in creased by daily vibrations for a month.
This latter fact, if firmly established, will prove to be highly important ; for it shows that materials which are subjected to constant vibrations, such as the materials of suspension bridges, have within themselves the property of resisting more and more strongly the tendency to elongate from vibration. Experi ments will be given hereafter which tend to confirm this fact, when the vibrations are not too frequent or too severe.
But the true viscosity of solids has been fully proved by Mr. Tresca, a French physicist, who showed that when solids are subjected to a very great force, the amount of the force depending upon the nature of the material, that the particles in the immediate vicinity of pressure willow over each other, so as to resemble the flowing of molasses, or tar, or other viscous fluids. Thus, the true viscosity differs entirely in its character from the property recognized by Professor Thompson.
RESISTANCE TO RUPTURE BY TENSION. 21. MODULUS OF STRENGTH. — Many more experiments have been made to determine the ultimate resistance to rupture by tension, than there have to determine the elastic resistance. In the earlier experiments the former was chiefly sought, and more recently all who experimented upon the latter also deter mined the former.
* Civ. Eng. Jour., vol. 28, p. 322.
24 TITE RESISTANCE OF MATERIALS.
The force which is necessary to pnll asunder a prismatic bar whose section is one square inch, when acting in the direction of the axis of the bar, is called the modulus of strength. This we call T. It expresses the tenacity of the material, and is sometimes called the absolute strength and sometimes modulus of tenacity.
S2. I OK Ml I. A FOR Till: HODI M S OF STRENGTH ; OT the
force necessary to break a prismatic bar, when acted upon by a tensile strain.
Let K=the section of the bar in inches, T=the modulus of tenacity, and P:=the required force.
It is proved by experiment that the resistance is proportional to the section ; hence
P=TK ... (9)
.-. T=? (10)
Iv
""From (10) T may be found. In (10) if P is not the ultimate resistance of the bar, then will T be the strain on a unit of section. From (9) we have
K=? (11)
which will give the section.
The following are some of the values of T which have been found from experiment by the aid of equation (10).
Cohesive force or Tenacity in pounds per square inch.
Ash (English} 17,000
Oak (English] 9,000 to 15,000
Pine (pitch) 10,500
Cast Iron * 14,800 to 16,900
Cast Iron ( Weisbach <& Overman) . 20,000
Wrought Iron 50,000 to 65,000
Steel wire 100,000 to 120,000
Bessemer steel f 120,000 to 129,000
" " i . . . . ; 72,000 to 101,000
Bars of Crucible Steel § . . . . 70,000 to 134,000
The most remarkable specimen of cast steel for tenacity which
* Hodgkinson, Bridges. Weale, sup., p. 25. f Jour. Frank. Inst. Vol. 84, p. 366.
| Also experiments by Wm. Fair bairn, Van Nostrand's EC. En. Mag., Vol. I., p. 273. § Do. p. 1009.
TENSION.
25
is on record, was manufactured in Pittsburgh, Pa. It was tested at the Navy Yard at Washington, D. C., and was found to sustain 242,000 pounds to the square inch ! *
For other values see the Appendix.
23. A vertical prismatic bar is fixed at its upper end, and a weight Pl is suspended at the other • what must he the upper section^ at A, Fig. 79so as to resist n times all the weight below it, the weight of the bar being considered ?
Let <J = the weight of a unit of volume of the bar, and the other notation as before.
ThenKT = n
If n — 1 K =
p
_L — of,
T
; and if 2 1 = T, K= oo, or no .section is pos-
sible, and I — _ is the corresponding length of the bar.
a
BAR OF UNIFORM STRENGTH. Suppose a bar is fixed at its upper extremity, Fig. 1O, and a weight P, is sus pended at its lower extremity ; it is required to find the form of the bar so that the horizontal sections shall be proportional to the strains to which they are subjected — the weight of the bar being considered.
Let <J = weight of ak unit of volume, W = weight of the whole bar, K0 = ?l= the section at B (Eq. (11) ),
Kj = the upper section, K — variable section, and x = variable distance from B upwards. Also let the sections be similar :
Then P = Pt + J /*K dx— strain on any section, ^^--\ as D C. But TK is the ability to resist this strain ; FIG- 10- .'. Pj -f- <J flLdx =1 TK. Differentiate this and we have J K 6fo= TdK or ^_ fa _ _ — which by integrating gives
I
Tpj = :N"ap.logK + C (12a)
* Am. K. E. Times (Boston), Vol. 20, p. 206.
THE RESISTANCE OF MATERIALS.
But for x = 0, we have K = K0 .-. C = — Nap. log K0 = - Nap. log tJ. Hence, Eq. (I2a) becomes.! x = Nap. log K Or,
.
( K
°T
,
passing to exponetials, gives «T =
' 0
. K .-K.i-.>
. . JV — iv0e _—
(13)
A
For the upper section K = K, and x—l:. K, = £je* (14)
W equals,
Example. What must be the upper section of a wrought-iron shaft of uni form resistance 1,000 ft. long, so that it will safely sustain its own weight and 75,000 Ibs.
Let T = 10,000 Ibs., and
2 — 0.27 Ibs. per cubic inch.
Then Eq. (11) gives K0 = 7.5 sq. inches, and
equation (14) gives K, = 10.37 inches.
In these formulas the form of section does not appear. For tensile strains, the strength is practically independent of the form, but not so for compression. When it yields by crushing, the influence of form is quite perceptible, but not so much so as when it yields by bending under a compressive strain. The latter case will be considered under the head of flexure.
STRAINS IN A
Fio. 11.
CLOSED CYLINDER.
If a closed cylinder is subjected to an internal pressure, it will tend to burst it by tearing it open along a rectilineal element, or by forcing the head off from the cylinder, by rup turing it around the cylinder. First, consider the latter case. The force which tends to force the head off is the total pressure upon the head, and the resisting section is the cylindrical annulus.
TENSION. 27
Let D = the external diameter, d = the internal diameter, p — the pressure per square inch, and t = the thickness of the cylinder. Then fard*p=tke pressure upon the head ;
— 6^)= the area of the cylindrical amiulus ; 2— eT)— the resistance of the annulus ; and, 2t=D-d.
Hence, for equilibrium,
J^=i<rT(D2-6r)
or, <Fp=2Tt(T> + d)=4:T(t* + dt) - - - (16)
which solved gives t=( — l + \/l-f£p " ' " C^)
Next consider the resistance to longitudinal rupturing. As it is equally liable to rupture along any rectilinear element, suppose that the cylinder is divided by any plane which passes through the axis. The normal pressure upon this plane is the force wrhich tends to rupture it, and for a unit of length is
&+.
and the resisting force is
hence, for equilibrium,
2T£,
pd - - - (18)
The value of t from (18) divided by that of t from (16) gives
the ratio — -j — ? and since D always exceeds d, this ratio is greater
than 2 ; hence there is more than t\vice the danger of bursting a boiler longitudinally that there is of bursting it around an annulus when the material is equally strong in both directions. The last equation was established by supposing that all the cylindrical elements resisted equally, but in practice they do not ; for, on account of the elasticity of the material, they will be compressed in the direction of the radius, thus enlarging the internal diameter more than the external, and causing a corre sponding increase of the tangential stress on the inner over the outer elements. In a thick cylindrical annulus it is necessary to consider this modification.
28
Till; KK-I-T \NVI-; <>F MATKKI ALP.
To find the VARYING LAW OF TANGENTIAL STRAINS, let D and d be the external and internal diameters before pressure, and D + 2 and d+y the corresponding diameters after pressure. Then, as a first approximation — which is near enough for prac tice — suppose that the volume of the annulus is not changed, and we have
or, ~Ds=dy nearly ........ (19)
But the strain upon a cylindrical filament varies as its elon gation divided by its length; see equation (3). Hence the strain on the external ammlus, compared with the internal, is as
« z y
I —JT ~ or as D to d
which combined with (19) gives
d I jya to ^ or as d* to D', or as r* to IT
where r and R are radii of the annulus.
Hence, tfie strain varies inversely as the square of the dis tance from the axis of the cylinder.
To FIND THE TOTAL RESISTANCE, let
x = the variable distance from the axis of the cylinder,
T = the modulus of rupture, or of strain, and
t = the thickness of the annulus.
Then Tdx is the strain on an element at a distance r from the axis of the cylinder, or otherwise upon the inner surface of the cylinder ; and according to the principle above stated,
r'
T— tdx is the strain on any element, and the total strain on both x
sides is
/K f=21
r
If t = r, this becomes
(20)
Tt
which compared with equation (18) shows that when the thick ness equals the radius, the resistance is only half what it would
TENSION. 29
be if the material were non-elastic. In (20) if t is small com pared with T*, it becomes 2T£ nearly, which is the same as equation (18).
If the ends of the cylinder are capped with hemispheres, the stress upon an elementary aimulus at the inner surface is %*Trdx.* Proceeding as before, and we find that the total stress necessary to force the hemispherical heads off is
which is also the stress necessary to force asunder a sphere by internal pressure, when the elasticity is considered.
If cylinders are formed by riveting together plates of iron, their strength will be much impaired along the riveted section. The condition of the riveted joint will doubtless have much more to do with the strength than the compressibility of the material, and will hereafter be considered.
* T. J. Kodman says the resistance on any elementary annulus is ^^xdx (Exp. on metal for cannon, p. 44) ; but it appears to me that, to make his expression correct, T must be the modulus at any element considered, and hence variable, whereas it should be constant. The strain on any elementary
r-2
annulus whose distance is x from the centre of the sphere, is T2*rdx, —2 —
dx 2n-r3T — ; and the total resistance is the integral of this expression between
05
the limits of r and r+t.
30
TIIE RESISTANCE OF MATERIALS.
RESISTANCE OF GLASS GLOBES TO INTERNAL PRES SURE.
EXPERIMENTS OF WM. FAIRBAIRN.
Description of the glass. |
Diameter in inches. |
Thickness in inches. |
Bursting pres sure in Ibs. per square inch. |
Bursting pres sure in Ibs. per square inch of section. |
Flint-glass, . . . |
4.0 x 3.98 4 0 x 3.98 |
0.024 0 025 |
84 93 |
3504 3720 |
4 4.5 x 4.55 6 |
0.038 0.056 0.059 |
150 280 152 |
3947 5025 3804 |
Mean |
4132 |
||||
Green-glass |
4.95 495 |
x 5.0 x 5 0 |
0.022 0020 |
90 85 |
5113 5312 |
4.0 4.0 |
x 4.05 x 4.03 |
0.018 0.016 |
84 82 |
4666 5126 |
|
Mean |
5054 |
||||
Crown-glass . . . |
4.2 4 05 |
x 4.35 x 4 2 |
0.025 0 021 |
120 126 |
5040 6000 |
5.9 6.0 |
x 5.8 x 6.3 |
0.016 0.020 |
69 86 |
6350 6450 |
|
Mean |
..59601 |
The following table exhibits the tensile strength of cylindrical glass bars according to the experiments of Fairbairn : —
Description of the glass. |
Area of specimen in inches. |
Breaking weight in Ibs. |
Tenacity per square "inch. |
Annealed flint-glass. . . Green -glass |
(0.255 ( 0.196 0.220 |
583 254 639 |
1?^<5 2540 'j<'.n; |
0.229 |
588 |
2546 |
|
TENSION.
31
As might have been anticipated, the tenacity of bars is much less than globes ; for it is difficult to make a longitudinal strain without causing a transverse strain, and the latter would have a very serious effect ; it is also probable that the outer portion of the annealed glass is stronger than the inner, and there is a larger amount of surface compared with the section, in globes than in cylinders.
RIVETED PLATES.
RIVETED PL.ATES are used in the construction of boil ers, roofs, bridges, ships, and other frames. It is desirable to know the best conditions for riveting, and the strength of riveted plates compared with the solid section of the same plates. The common way of riveting is to punch holes through both plates, into which red-hot bolts or rivets are placed, and headed down while hot. The process of punching strains, and hence weakens, the material. A better way is to bore the holes in the plates, and then rivet as before. The holes in the separate plates should be exactly opposite each other, so that there will be no side strain on the plates caused by driving the rivets home, and to secure the best ef fects of the rivets them selves. They are some times placed in single and sometimes in double rows, and experiment shows that the latter possesses great advantage over the former. Experiments have been made upon plates of the form shown in Fig. 12, both with lap and butt-joints, and with single and double rows of rivets.*
* Lond. Phil. Transactions, part 2d, 1850, p. 677.
-
FIG. 12.
Library. J
•
32
THE RESISTANCE OF MATERIALS.
Table dwwing Hie strength of single and double riveted plates.
Cohesive strength of the plates iu Ibs. per square inch. T. |
Strength of single-riveted joints of equal section to the plates, taken through the line of riv ets. Breaking weight in Ibs. per square inch. |
Strength of double-riveted joint* of equal section to the plates, taken through the line of riv ets. Breaking weight in Ibs. per square inch. |
57,724 61,579 68,322 50,983 51,130 49,281 |
45,743 36,606 43,141 43,515 40,249 44,715 |
52,352 48,821 58,286 54,594 53,879 53,879 |
Mean. .54,836 |
42,328 |
53,635 |
It will be observed that in double-riveting there is but little loss of strength, while there is considerable loss in single-rivet ing. In the preceding experiments the solid section of the plates, taken through the centre of the rivet-holes, was used ; but, as Fairbairn justly remarks, we must deduct 30 per cent, for metal actually punched out to receive the rivets. But as only a few rivets came within the limits of the experiments, and as an extensive combination of rivets must resist more effectually, and as something will be gained by the friction between the plates, it seems evident that we may use more than 60 per cent, of the strength of riveted plates as indicated above. Fairbairn says we may use the following proportions : —
Strength of plates 100
Strength of double -riveted plates 70
Strength of 'single-riveted plates 56
Size and distribution of rivets. — The best size of the rivets, the distance between them, and the proper amount of lap of the plates, can be determined only by long experience, aided by experiments. Fairbairn gives the following table as the results of his information upon this important subject, to make the joint steam or water tight : —
TENSION.
33
Table showing the strongest forms and best proportions of riveted joints, as deduced from expei*iments and actual practice. ( Useful Information for Engineers, 1st Series, p. 285.)
Thickness of plates in inches, t. |
Diameter of the rivets in inches, d. |
Length of rivets from the head in inches. 1. |
Distance of rivets from cen tre to centre in inches, a. |
Quantity of lap in single joints in inches, b. |
Quantity of lap in double- riveted joints in inches, c. |
tV to A |
2 t |
4*t |
6t |
6t |
lot |
A |
u |
u |
5t |
||
A |
« |
u |
u |
5H |
8H |
Atott |
lit |
u |
4t |
4it |
8|t |
STRENGTH OF IRON IN DIFFERENT DIRECTIONS OF THE ROLLED SHEET.*
In obtaining specimens for these experiments, care was gen erally taken to have them cut in different directions of the roll ing, longitudinally and transversely, and in some cases diag onally, to that direction. The table will be found to indicate the direction of slitting in each case, and the comparison con tained in the table is given to show what information the in quiry has elicited.
The comparison is made principally on the minimum strength of each bar, being that which can alone be relied on in practice ; for if the strength of the weakest point in a boiler be overcome, it is obviously unimportant to know that other parts had greater strength. In one case, however, two bars, one cut across the direction of rolling, and the other longitudinally, were, after be ing reduced to uniform size, broken up cold, with a view to this question. The result showed that the length-strip was 7TV per cent, stronger than the one cut crosswise, considering the tenacity of the latter equal to 100. Of the other sets, embracing about 40 strips cut in each direction, it appears that some kinds of boiler iron manifest much greater inequality in the two direc tions than others. It is in certain cases not much over one per cent., and in others exceeds twenty, and as a mean of the whole series it may be stated to amount to six per cent, of the strength of the cross-cut bars. The number of trials on those cut diag onally is not perhaps sufficiently great to warrant a general de duction ; but, so far as they go, they certainly indicate that the strength in this direction is less than either of the others.
* Experiments of Franklin Institute.
34 THE RESISTANCE OF MATERIALS.
Had we compared the mean instead of the least strength of bars as given in the table, the result would not have differed materi ally in regard to the relative strength in the respective directions.
The boiler-iron manufactured by Messrs. E. II. & P. Ellicott, which was tried in all these modes of preparation of specimens, gave the following results :
1. When tried at original sections, seven experiments on length-sheet specimens gave a mean strength of 55285 Ibs. per square inch, the lowest being 44399 Ibs., and the highest 59307 Ibs. Fourteen experiments on cross-sheet specimens gave a mean of 53890 Ibs., the lowest result being 50212 Ibs., the highest 58839 Ibs. ; and six experiments on strips cut diagonally from the sheet exhibited a strength of 53850 Ibs., of which the lowest was 51134 Ibs., and the highest 58773 Ibs.
2. When tried by filing notches on the edges of the strips, to remove the weakening effect of the shears, the length-sheet bars gave, at fourteen fractures, a mean strength of 63946 Ibs., vary ing between 56346 Ibs. and 78000 Ibs. per square inch. The cross-sheet specimens tried after this mode of preparation exhibited, at three trials, a mean strength of 60236 Ibs., vary ing from 55222 Ibs. to 65143 Ibs. ; and the diagonal strips, at four trials, gave a mean result of 53925 Ibs., the greatest differ ence being between 51428 Ibs. and 56632 Ibs. per square inch.
3. Of strips reduced to uniform size by filing, four compara ble experiments on those cut lengthwise of the sheet gave a mean strength of 63947 Ibs., of which the highest was 67378 Ibs., and the lowest 60594 Ibs.
Cross-sheet specimens, tried after the same preparation, ex hibited, at thirty-three fractures, a mean of 50176 Ibs., of which the highest was 65785 Ibs., and the lowest 52778 Ibs. No bar cut diagonally was reduced to uniform size.
From the foregoing statements it appears that by filing in notches and filing to uniformity, we obtain results 63946 Ibs. and 63947 Ibs. for the strength of strips cut lengthwise, differing from each other by only a single pound to the square inch, and that by these two modes of preparation the cross-sheet speci mens gave respectively 60236 Ibs. and 60176 Ibs., differing by only 60 Ibs. to the square inch. This seems to prove that by both methods of preparing the specimens the accidental weak ening effect of slitting had been removed by separating all that
TENSION.
35
portion of the metal on which it had been exerted. Hence we may infer that the differences between length-sheet and cross- sheet specimens are really and truly ascribable to a difference of texture in the two directions, which will be seen to amount, in the case of filing in notches, to 6.15 per cent., and in that of filing to uniformity, to 6.26 per cent, of strength of cross-sheet specimens.
Table of the comparative mew of the strength of specimens of ten different sorts of boiler and one of bar iron, in the longitudinal, transverse, and diago nal direction of the rotting, as deduced from the least strength of each specimen, and the average minimum of each sort of iron, in each direction in which it was tried.
_J |
, |
£ |
- |
.A |
. |
? |
1 |
1 |
5 |
fc |
"a |
i |
I |
«H E |
•Ej> . |
id |
C) |
fd |
1 . |
|
Is |
C'rS |
|| |
1 |
.fli |
11 |
c 8 |
•3 o |
|l |
0 |
I1 1 |
f g |
f |
|
* |
OQ |
|
fc |
OQ |
m |
|
2 |
58977 |
125 |
57182 |
Tilted. |
||
3 |
53828 |
130 |
Tilted. |
57789 |
||
4 |
47167 |
133 |
do. |
53176 |
||
6 |
52280 |
135 |
do. |
47738 |
||
8 |
50103 |
137 |
do. |
50358 |
||
Mean |
53324 |
51191 |
Mean |
57182 |
55882 |
49048 |
42 |
51653 |
Puddled. |
142 |
44399 |
||
48 |
44102 |
do. |
143 |
53135 |
||
44 |
53836 |
do. |
146 |
60594 |
||
46 |
59262 |
H'd pla.* |
148 |
52468 |
||
48 |
59418 |
do. |
149 |
52228 |
||
49 |
575(55 |
do. |
150 |
56869 |
||
51 |
H'd pi. |
59656 |
151 |
53811 |
||
68 |
H'd pla. |
56062 |
152 |
56073 |
||
56 |
Puddled. |
57926 |
154 |
51134 |
||
58 |
do. |
50570 |
157 |
52102 |
||
59 |
48.308 |
Puddled. |
160 |
53862 |
||
60 |
58684 |
do. |
162 |
50212 |
||
61 |
52869 |
do. |
164 |
56346 |
||
62 |
57612 |
do. |
167 |
5(5682 |
||
64 |
Puddled. |
45392 |
169 |
54361 |
||
65 |
do. |
51255 |
171 |
55612 |
||
68 |
57929 |
H'd pla. |
174 |
51425 |
||
70 |
47638 |
do. |
||||
71 |
H'd pla. |
54634 |
Mean |
54253 |
53646 |
52568 |
73 |
do. |
52657 |
||||
74 |
do'. |
49351 |
||||
Mean |
54074 |
53049 |
226 |
49053 |
||
227 |
53699 |
|||||
228 |
40643 |
|||||
i 229 |
46473 |
|||||
230 |
49368 |
|||||
Mean |
49368 |
47467 |
|
|||
Hammered and rolled into plates.
36
THE RESISTANCE OF MATERIALS.
The specimens from 42 to 74 were partly puddled iron, and partly Juniata blooms, hammered and rolled into plate. The length and cross-sheet specimens of these two kinds must be compared separately.
All the experiments on No. 228 (cross) and 230 (length) were made at ordinary temperatures with a view to this comparison.
29. TENSILE STRENGTH OF WROUGHT IRON AT VARIOUS TEMPERATURES.
Mr. Fairbaim has made experiments upon rolled plates of iron, and rods ojprivet iron, at various temperatures. The for mer were broken in the direction of the fibre and across it. The specimen when subjected to experiment was surround ed with a vessel into which freezing mixtures were placed to produce the lower temperatures, and oil heated by a fire underneath to produce the high temperatures. The experi ments were made upon Staffordshire plates, which are inferior to several other kinds in common use. The following table gives a summary of the results : — Table showing the Resistance of Staffordshire Plates at Different Temperatures.
1 |
a |
2-r |
Sf |
|||
1 |
2 m |
1 |
~-S |
5* |
||
« |
jj |
•sl |
fj |
* £ |
C8 |
Remarks. |
«-l |
£ |
§1 |
II |
*f| |
||
o |
H |
t>.S |
i |
ht « SJ |
||| |
|
1 |
0° |
0.6868 |
33,660 |
49,009 |
49,009 |
With. |
2 |
60 |
0.7825 |
31,980 |
40,357 |
Across. |
|
3 |
60 |
0.6400 |
27,780 |
43,406 |
44,498 |
Across. |
4 |
60 |
0.6368 |
31,980 |
50,219 |
With. |
|
5 |
110 |
0.6633 |
29,460 |
44,160 |
Across.* |
|
6 |
112 |
0.6800 |
28,620 |
42,088 |
42,291 |
With. |
7 |
120 |
0.8128 |
37,020 |
40,625 |
With. |
|
8 |
212 |
0.8008 |
31,980 |
39,935 |
With. |
|
9 |
212 |
0.6633 |
30,300 |
45,680 |
45,005 |
Across. |
10 |
212 |
0.6800 |
33,660 |
49,500 |
With. |
|
11 |
270 |
0.6432 |
28,690 |
44,020 |
44,020 |
With. |
12 13 |
340 340 |
0.6400 0.6800 |
31,980 28,620 |
49,968 42,088 |
j- 46,018 |
With.f Across. |
14 |
395 |
0.6666 |
30,720 |
46,086 |
46,086 |
With. |
15 16 |
Scarcely red Dull red |
0.6200 0.6076 |
23,520 18,540 |
38,032 80,513 |
j. 34,272 |
Across. Across, t |
* Too high ; fracture very uneven.
f Too low ; tore through the eye.
\ Too high; the specimen broke with the first strain.
TENSION.
37
The mean values given in the sixth column of this Table exhibit a remarkable degree of uniformity in strength for all temperatures, from 60 degrees to 395 degrees. The single ex ample at 0 degrees gives a higher value than the mean of the others, but not higher than for some of the specimens at higher temperatures. At red heat the iron is very much weakened. This fact should be noticed in determining the strength of boiler-flues, as they are often subjected to in tense heat when not covered with water.
The experiments upon rivet iron were made with the same machine, and in the same manner, the results of which are shown in the following table : —
Table showing the Results of Experiments on Rivet Iron at Different Tem peratures.
1 |
1 |
| |
•«.S i |
|l |
||
« |
|J 1 |
1 |
I J |
Remarks. ( |
||
H |
£ |
> |
bc^ |
6D 0 |
ft aS |
|
"o |
P. S |
| |
S |
f* O LJ • |
SjjLf |
|
£ |
1 |
1 |
1 |
ill |
a*! |
|
17 |
-30° |
0.2485 |
15,715 |
63,239 |
63,239 |
Too low. |
18 19 |
+60 60 |
0.2485 |
15,400 15,820 |
61,971 63,661 |
I 62,816 |
Too low. Too low. |
20 |
114 |
17,605 |
70,845 |
70,845 |
||
21 |
212 |
20,545 |
82,676 |
|||
22 |
212 |
0.1963 |
14,560 |
74,153 |
[ 79,271 |
|
23 |
212 |
0.2485 |
20,125 |
80,985 |
) |
|
24 25 |
250 270 |
0.1963 0.2485 |
16,135 20,650 |
82,174 83,098 |
i 82,636 |
|
26 27 |
310 325 |
0.1963 0.1963 |
15,820 17,185 |
80,570 87,522 |
| 84,046 |
|
28 |
415 |
0.2485 |
20,335 |
81,830 |
^ 83 943 |
|
29 |
435 |
21,385 |
86,056 |
r »j,y4d |
||
30 |
Red heat. |
8,965 |
36,076 |
35,000 |
Too high. |
|
From this Table we see that there is a gradual increase of strength from 60 degrees to 325, where it appears to attain its maximum. The increase is a very important amount, being about 30 per cent.
It is a little remarkable that the specimen at minus 30 de grees is stronger than the mean of the two at 60 degrees ; but we observe, as before, that it is not as strong as some of the single specimens at higher degrees.
38
THE RESISTANCE OF MATERIALS.
Mr. Johnson, when in the employ of the Navy Department, in 1844, made some experiments to determine the effects of thermo-tension upon different kinds of iron.* lie took two bars of the same kind of iron, and of the same size, and broke one while cold. He then subjected the other to the same ten sion when heated 400 degrees, after which the strain was re lieved, and the bar was allowed to cool, and the permanent elongation noted, after which it was broken by an additional load. It will thus be seen that the experiments were not con ducted in the same way as those by Fairbairu. The following table gives the results of his experiments :
TJw Results of Experiments on Thermo-Terwion, at 400° Temperature.
MJ |
Mlf |
o |
||||
KIND OF IRON. |
1 |
c- 5 |
dn r 6 "5 oo |
I |
!§ |
|
i |
111 |
1 |
i:!1! |
o c 3 |
5> 13 |
|
• |
&f £ |
• |
c-c ?S * |
o |
H3 |
|
Tons. |
Tons. |
Inches. |
Per cent. |
Per cent. |
Per cent. |
|
Tredegar, round. . . |
60 |
71.4 |
1.91 |
6.51 |
19.00 |
25.51 |
Tredegar, round . . . |
60 |
72.0 |
1.91 |
(6.51) |
20.00 |
26.51 |
Tredegar, square bar |
60 |
67.2 |
1.69 |
6.77 |
12.00 |
18.77 |
Tredegar, r'nd, No 3 |
58 |
68.4 |
1.15 |
5.263 |
17.93 |
23.19 |
Salisbury, round . . . |
105.87 |
121.0 |
3.59 |
3.73 |
14.64 |
18.37 |
M |
can |
5.75 |
16.64 |
22.40 |
||
Remarks. — From the two former sets of experiments, pp. 36 and 37, it appeal's that the strength of the iron was in creased by an increase of temperature at the time the bar was broken, and by the latter that it was not only increased, but, by being subjected to severe tension while at a high temper ature, the increased strength was not lost by cooling. It hardly seems probable that this increased strength would be retained indefinitely, and hence it is important to know how long it was after the piece was cooled before it was broken.
These results are confirmed by the experiments of the com mittee of the Franklin Institute, as shown by the following table. See Journal of the Franklin Institute, vol. xx., 3d series, p. 22.
Senate Doc., No. 1, 28th Cong., 3d Sett., 1844-5, p. 639.
TENSION.
39
ABSTRACT OF A TABLE
Of the comparative view of the influence of higli temperatures on the strength of iron, as exhibited by 73 experiments on 47 different specimens of that metal, at 46 different temperatures, from 212° to 1317° Fahr., compared with tlie strength of each bar ichen tried at ordinary temperatures, the number of expe riments at the latter being 163.
No. of the experi ment. |
Temperature observed at moment of fracture. |
Strength at ordinary temperature. |
Strength at the tem perature observed. |
1 |
212° |
56736 |
67939 |
2 |
214 |
53176 |
61101 |
3 |
394 |
68356 |
71896 |
9 |
440 |
49782 |
59085 |
10 |
520 |
54934 |
58451 |
15 |
554 |
54372 |
61680 |
20 |
568 |
67211 |
76763 |
23 |
574 |
76071 |
65387 |
40 |
722 |
57133 |
54441 |
45 |
824 |
59219 |
• 55892 |
50 |
1037 |
58992 |
37764 |
58 |
1245 |
54758 |
20703 |
59 |
1317 |
54758 |
18913 |
Remark. — According to these experiments, as shown in the 4th column, the strength increases with the temperature to 394 degrees, when it attains its maximum ; although in some cases the strength was increased by increasing the temperature to 568 degrees. By comparing the 3d and 4th columns we see that the strength is greater for all degrees from 212° to 574° than it is at ordinary temperatures, but above 574° it is weaker. The experiments on Salisbury iron showed that the maximum tenacity was 15.17 per cent, greater than their mean strength when tried cold. The committee above referred to determined the maximum strength of about half the specimens used in the preceding table by actual experiment, and calculated it for the others ; and from the results derived the following empirical formula for the diminution in strength below the maximum for high degrees of heat : —
D5 = c(& - SO)13
in which D is the diminution after it has passed the maximum, 0 the temperature Fahrenheit, and c a constant.
40
THE RESISTANCE OF MATERIALS.
This formula appears to be sufficiently exact for all tem peratures between 520° and 1317°.
EFFECT OF SEVERE STRAINS UPON THE CTI/TITTIATE
TENACITY OF IRON RODS. — Thomas Loyd, Esq., of England, took 20 pieces of If S. C. ^> bar iron, each 10 feet long, which were cut from the middle ot as many rods. Each piece was cut into two parts of 5 feet each, and marked with the same letter. A, B, C, etc., were first broken, so as to get the average breaking strain. A2, 132, tfcc., were subjected to the constant action of three-fourths the breaking weight, previously found, for five minutes. The load was then removed, and the rods afterwards broken.
Results of the Experiments.*
FIRST. |
SECOND. |
||
Mark on the bare. |
Breaking weight in tons (gross). |
Mark. |
Breaking weight in tons. |
A |
33.75 |
A 2 |
33.75 |
B |
30.00 |
B 2 |
33.00 |
C |
33.25 |
C 2 |
33.25 |
D |
32.75 |
D 2 |
32.25 |
E |
32.50 |
E 2 |
32.50 |
F |
33.25 |
F 2 |
33.00 |
G |
32.75 |
G 2 |
33.00 |
H |
33.25 |
H2 |
33.50 |
I |
33.50 |
I 2 |
32.75 |
J |
33.50 |
J 2 |
33.25 |
K |
32.25 |
K2 |
32.50 |
L |
32.25 |
L 2 |
81.50 |
M |
30.25 |
M2 |
32.75 |
N |
34.25 |
N 2 |
34.00 |
0 |
31.75 |
0 2 |
32.50 |
P |
29.75 |
P 2 |
31.00 |
Q |
;>,::.r,o |
Q2 |
33.75 |
B |
33.75 |
R 2 |
33.75 |
S |
33.00 |
S 2 |
33.25 |
T |
:tt.w |
T 2 |
31.00 |
Mean |
32.57 |
32.81 |
|
We here see that a strain of 25 tons, or three-fourths the breaking weight, did not weaken the bar.
Fairbairn, Useful Information for Engineers, First Series, p. 313.
TENSION.
These experiments indicate that a frame or bridge may be subjected to a severe strain of three-fourths of its strength for a short time without endangering its ultimate strength.
3 1 . EFFECT OF REPEATED RUPTURE. The following eX-
periments were made at Woolwich Dockyard, England. The same bar was subj ected to three or four successive ruptures by tensile strains. They show the remarkable fact, that while great strains impair the elasticity, as shown by Hodgkinson, yet they do not appear to diminish the ultimate tenacity. This fact is important, for it shows that iron, which has been broken by tension in a structure, may safely be used again for any strain less than that for which it was broken.
Table shamng the effect of repeated Fracture cm Iron Bars.
First breakage. |
Second breakage |
Third breakage. |
Fourth breakage. |
Reduced |
|||||
from |
|||||||||
Mark |
sectional |
||||||||
Tons. |
Stretch in 54 inches. |
Tons. |
Stretch in 36 inches. |
Tons. |
Stretch in 24 inches. |
Tons. |
Stretch in 15 inches. |
area of 1.37 sqr. inches to |
|
In. |
In. |
In. |
|||||||
A |
33.75 |
0.9125 |
35.50 |
0.200 |
|||||
B |
33.75 |
0.9250 |
35.25 4 |
0.225 |
37.00 |
1.00 |
38.75 |
1.25 |
|
E |
32.50 |
0.9250 |
34.75 |
0.125 |
|||||
P |
33.25 |
1.0500 |
35.50 |
0.112 |
37.25 |
0.62 |
40.40 |
1.18 |
|
G- |
32.75 |
0.8500 |
35.00 |
0.125 |
37.50 |
40.41 |
1.25 |
||
H |
33.75 |
1.0025 |
36.25 |
0.187 |
|||||
I |
33.50 |
0.8375 |
34.50 |
0.62 |
36.50 |
1.50 |
|||
J |
33.50 |
0.9250 |
36.00 |
0.025 |
36.75 |
1.12 |
41.75 |
1.25 |
|
L |
32.25 |
Defect' e |
36.50 |
0.150 |
37.75 |
41.00 |
0.31 |
1.25 |
|
M |
30.25 |
Defect' e |
36.50 |
.62 |
37.75 |
0.60 |
38.50 |
0.06 |
1.25 |
Mean |
32.95 |
35.57 |
37.21 |
40.16 |
1.24 |
||||
Mean pr. sq. in. |
24.04 |
25.93 |
27.06 |
29.20 |
0.90 ' |
We thus see that while the section is reduced 10 per cent., the strength is apparently increased over 20 per cent. It is not, however, safe to infer that the strength is actually increased, for it is probable that it broke the first time at the weakest point, and the next time at the next weakest point, and so on.
We also observe that the total elongations are not proportional to the tensile strains, which is in accordance with the results of other experiments.
ANNEALING.
. ANNEALING is aprocess of treating metals so as to make
42 THE RESISTANCE OF MATERIALS.
them more ductile. To secure this, the metals are subjected to a high heat and then allowed to cool slowly. Steel is softened in this way, so that it may be more easily worked. Campin * says that steel should not be overheated for this purpose. Some bury the heated steel in lime ; some in cast-iron borings ; and some in saw-dust. He (Campin) says the best plan is to put the steel into an iron box made for the purpose, and fill it with dust-charcoal, and plug the ends up to keep the air from the steel ; then put the box and its contents into a fire until it is heated thoroughly through, and the steel to a low red heat. It is then removed from the fire, and the steel left in the box until it is cold. Tools made of annealed steel will, in some cases, last much longer than those made of unannealed steel.
But it appears from the following table that it weakens iron to anneal it.
Table of the strength of Wrought Iron Annealed at Different Temperatures.
Strength at or- |
Strength at |
Strength after |
Ratio of di |
||
No. of com parisons. |
dinary temp, before auneal- |
Temperature at which an- the annealing annealing and nealiug took place. temperature, cooling. |
minution of strength. |
||
ing. |
|||||
1 |
57,133 |
1037' |
37,704 |
55.678 |
0.025 |
5 |
53,774 |
1155 |
21,907 |
45,597 |
0.152 |
10 |
52,040 |
1245 |
20,703 |
38,843 |
.253 |
15 |
48,407 |
Bright welding heat. |
38,676 |
.201 |
|
17 |
73,880 |
Low welding heat. |
53,578 |
.275 |
|
18 |
76,986 |
Bright welding heat. |
50,074 |
.349 |
|
19 |
89,162 |
Low welding heat. |
48,144 |
.460 |
|
33. Ilia: STRENGTH OF I1COIV AND STEEL ALSO DEPENDS
largely upon the processes of their manufacture, and their treatment afterwards. The strength of wrought iron de pends upon the ore of which it is made ; the manner in which it is smelted and puddled, the temperature at which it is hammered, and the amount of hammering which it receives in bringing it into shape. The same remark applies to cast steel. If the former is hammered when it is comparatively cold, it will weaken it, especially if the blows are heavy ; but the latter, steel, may be greatly damaged, or even rendered worthless by excessive heat, and it is greatly improved by hammering when comparatively cold. For the effect of tempering on the crushing strength, see Article 50.
* Campin's " Practical Mechanics," p. 364.
TENSION.
34. PROLONGED FUSION OF CAST IRON. — Cast iron is also subjected to great modifications of strength on account of the manipulations to which it is or may be subjected in its manufacture and preparations for use. The strength in some cases is greatly increased by keeping the metal in a fused state some time before it is cast. Major "Wade made experiments upon several kinds of iron, all of which were increased in strength with prolonged fusion (see Rep. p. 44), one example of which is given in the following
Table shoioing the Effects of Prolonged Fusion.
Tensile Strength in Ibs. per sq. in. |
|||
Iron |
in fusion |
-fa hour |
17 843 |
tt It |
1 " |
20 127 |
|
u |
u u |
1 ^ hour |
24,387 |
u |
u a |
2 hours |
34,496 |
35. EFFECT OF REUIELTING CAST IRON. But the great est effect was produced by remelting. The density, tenacity, and transverse strength were all increased by it, within certain limits. For instance, a specimen of No. 1 Greenwood pig- iron gave the following results : — (Rep., p. 279.)
Table showing the Effects of Remdting.
No. 1 Greenwood iron. |
Density. |
Tensile Strength. |
Crude pig-iron |
7.032 |
14,000 |
" remelted. once |
7 086 |
20 900 |
" twice |
7 198 |
30 229 |
three times |
7301 |
35,786 |
But there is a point beyond which, remeltings will weaken the iron. Mr. Fairbairn made an experiment in which the strength of the iron was increased for twelve remeltings, and then the strength decreased to the eighteenth, where the experiment ter minated. In some cases no improvement is made by remelting, but the iron is really weakened by the process ; so that it be comes necessary to determine the character of each iron under the various conditions by actual experiment.
The laws which govern Greenwood iron were so thoroughly
44 THE RESISTANCE OF MATERIALS.
determined that the results which will follow from any given course of treatment may be predicted with much certainty. (Rep., p. 245.)
By mixing grades Nos. 1, 2, and 3, and subjecting them to a third fusion, one specimen was obtained whose density was 7.304, and whose tenacity was 45,970 pounds, which is the strongest specimen of cast iron ever tested. (Rep., p. 279.)
As a general result of these experiments, Major AVade re marks (p. 243), " that the softest kinds of iron will endure a greater number of meltings with advantage than the higher grades. It appears that when iron is in its best condition for casting into proof bars (that is, small bars for testing the metal) 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."
36. THE MANNER OF COOLING also affects the strength. It was found that the tensile strength of large masses was in creased by slow cooling ; while that of small pieces was increased by rapid cooling. (Rep., p. 45.)
37. THE MODULUS OF STRENGTH IS MODIFIED, WC thllS
see, by a great variety of circumstances ; and hence it is im possible to assign any arbitrary value to it for any material, that will be both safe and economical ; but its value must be determined, in any particular case, by direct experiment, or something in regard to the quality of the material must be known before its approximate value can be assumed.
O 38. SAFE LIMIT OF LOADING — Structures should not be strained so severely as to damage their elasticity. According Article 9, it appears that a weight suddenly applied will produce twice the elongation that it will if applied gradually or by increments. Hence, structures which are subjected to shocks by sudden applications of the load, should be so propor tioned as to resist more than double the load as a constant dead-weight without straining it beyond the elastic limit.
This method of indicating the limits, suggested by M. Pon- celet, is perfectly rational ; but, unfortunately, the elastic limits have not been as closely observed and as thoroughly determined by experimenters as the limit of rupture. The latter was for-
TENSION. 45
merly considered more important, and hence furnished the basis for determining the safe limit of the load. Observations on good constructions have led engineers to adopt the following values as mean results for permanent strains on bars :—
For wood, TV ) The ]oad which would
For wrought iron, | to U d
For cast iron, i to -J- )
Further observations will be made upon this subject in the latter part of this volume.
LlhJL^^
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THE RESISTANCE OF MATERIALS.
CHAPTER II.
COMPRESSION.
38. RESISTANCE TO COMPRESSION also divides itself into two general problems — elastic and ultimate. The law of elastic resistance for compression may be as readily found as that for ten sive resistance ; but the law of resistance to crushing is very com plex. If the pieces which are subjected to this stress are long, they will bend under a heavy stress, unless they are confined, and when they bend they break partly by bending and partly by crush ing. If the pieces are very short, compared with their diameter, they may be crushed without being bent ; but even in this case, with granular substances, the yielding is more or less peculiar, dividing off in pieces at certain angles with the line of pressure. The results of some experiments will now be given, which will enable us to test the prevailing theories upon this subject.
ELASTIC RESISTANCE. TABLE
Showing the compression, permanent set, and coefficient of elasticity * of a solid *'•>"*• vw cylinder 10 inches long and 1.382 inch diameter.
Weight per square inch of section in IDS. |
Compression per inch of length. |
Permanent set per inch of length. |
Coefficient of elasticity. |
1,000 |
0.000000 |
0. |
11,111,000 |
2,000 |
0.000170 |
0. |
ll..v,M.(»<M) |
3,000 |
0.000255 |
0.000005 |
11,843,100 |
4,000 |
0.000320 |
0.000015 |
12,500,000 |
5,000 |
0.000385 |
0.000025 |
12,987.000 |
6,000 |
0.000455 |
0.000030 |
13,189,000 |
7,000 |
0.000505 |
0.000035 |
13,861.300 |
N.noo |
0.000575 |
0.000045 |
13,813,000 |
9,000 |
0.000645 |
0.000055 |
13,952.000 |
10,000 |
0.000705 |
0.000070 |
14,196000 |
15,000 |
0.001035 |
0.000170 |
14,492,000 |
20,000 |
0.001395 |
0.000300 |
14,3:*7,000 |
25,000 |
0.001826 |
0.000495 |
13,687,900 |
30,000 |
0.002380 |
0.000820 |
12,602,300 |
* The author computed the coefficients of elasticity from the other data of the table.
COMPRESSION.
. COMPRESSION OF CAST-IRON. — Captain T. J. Rodman, in his report upon metals for cannon, page 163, has given the results of experiments upon a piece of cast-iron, which was taken from the body of the same gun as was the specimen re ferred to on page 11 of this work, the results of which are given on the preceding page.
We observe that the coefficient of elasticity is much less for the first strains than for those that follow. It thus appears that this metal resists more strenuously after it has been somewhat compressed than at first. The coefficient of elasticity is con siderably less than for the corresponding piece, as given on page 11. The difference is very much greater than that found by Mr. Hodgkinson in the specimens which he used in his experi ments, lie took bars 10 feet long, and about an inch square, and fitted them nicely in a groove so that they could not bend, and occasionally, during the experiment, they were slightly tapped to avoid adherence. The metal was the same kind as that used in the experiment recorded on page 13.
TABLE
Giving the results of experiments by Mr. Ilodgkinson on bars of cast-iron 10 feet
lonp.
Pressure per square inch of section. P. |
Compression per inch of length. |
Coefficient of elasticity per square inch. |
Error in parts of P of the formula P=170,763AC -30,318 A* |
|
Total. Ac |
Permanent. |
|||
Ibs. |
in. |
in. |
Ibs. |
|
2064.74 |
0.0001561 |
0.00000391 |
13,231,300 |
- £ |
4129.49 |
0.0003240 |
0.00001882 |
12,764,910 |
8 1 ~5 |
(5194.24 |
0.0004981 |
0.00003331 |
12,442,300 |
+ -h |
8258.98 |
0.0006565 |
0.00005371 |
12,585,100 |
+ -nVo |
10823.73 |
0.00082866 |
0.00007053 |
12,467,100 |
+ lfl |
12388.48 |
0.00100250 |
0.00009053 |
12,357,200 |
+ *i« |
14453.22 |
0.00128025 |
0.00011700 |
12,253,700 |
+ -iis |
1(5517.97 |
0.00136150 |
0.00014258 |
12,141,200 |
+ lisr |
18582.71 |
0.00154218 |
0.00017085 |
12,058,100 |
+ T*T |
20647.40 |
0.00171866 |
0.00020685 |
12,021,800 |
+ £* |
24776.95 |
0.00208016 |
0.00036810 |
11,920,000 |
-Tfrr |
28906.45 |
0.00247491 |
0.00045815 |
11,687,400 |
-rhr |
33030.80 |
0.0029450 |
0.00050768 |
11,222,750 |
+ 1/4 |
37159.65 |
0.003429 |
48
THE RESISTANCE OF MATERIALS.
Iii this case the highest coefficient of elasticity results from the smallest strain which is recorded. The difference in this respect between this example and the preceding one results doubtless from the internal structure of the iron. The coeffi cient in both these cases is much less than that found for other kinds of cast-iron, as is shown in the table of resistances in the Appendix.
Mr. llodgkinson proposed the empirical formula, — P= 170,763XC — 36,318X2,— to represent the results of the experi ments ; and although it may represent more nearly the results of a greater range of strains than equation (3), yet there is no advantage in its use in practice.
I O. COMPRESSION OF WROUGHT, IRON.
Mr. llodgkinson also made experiments upon bars of wrought iron in precisely the same manner as upon those of cast iron, the results of which are given in the following
TABLE
Giving the results of experiments by Mr. E. HodgJdnson on bars of wrought iron, each of which was ten feet long.*
Weight producing the compression. |
1st Bar. Section=l.G25 x 1.025 sq. in. |
SdBar. Section= 1.016x1. 02 sq. in. |
||
Amount of Compression. |
Value of E. |
Amount of Compression . |
Value of E. |
|
IDS. |
inch. |
Ibs. |
inch. |
Ibs. |
5098 |
0.028 |
20,796,500 |
0.027 |
21,864,000 |
9578 |
0.052 |
21,049,000 |
0.047 |
23,595,000 |
14058 |
0.073 |
21,979,000 |
0.067 |
24,273,000 |
16298 |
0.085 |
21,343,000 |
||
18538 |
0.096 |
22,156,000 |
0.089 |
24,108.000 |
20778 |
0.107 |
22,160,000 |
0.100 |
24,038,000 |
23018 |
0.119 |
23,587,000 |
0.113 |
23,587,000 |
25258 |
0.130 |
22,095,000 |
0.128 |
23,679,000 |
27498 |
0.142 |
22,111,000 |
0.143 |
22,259,000 |
29738 |
0.152 |
21,!»38,000 |
0.163 |
21,139,000 |
31978 |
0.174 |
20,979,000 |
0.190 |
19,478,000 |
In \ hour. |
0.261 |
|||
Again after £ |
||||
hour. |
0.269 |
|||
Then repeated. |
0.328 |
* The coefficients of elasticity were computed by the author.
COMPEES8ION.
GRAPHICAL, REPRESENTATION. TllGSG tWO
are graphically represented in Fig. 13. It is seen from the tables that the compressions are quite uniform for a large range of strains, and hence equation (2), page 17, is applicable to com- pressive strains when within the elastic limits. In the case of the wrought-iron bars, the first one attains its maximum coeffi cient of elasticity for a strain somewhat less than one-half its ultimate resistance to crushing, and the second bar at about one- third its ultimate resistance.
30000
20000
10000
2000
I
0-02 in.
0-10
FIG. 13.
-20
43. COMPARATIVE RESISTANCE OF CAST AND WROUGHT
IRON. — The coefficient of elasticity is a measure of the com pressibility of metals. Hence, an examination of the two preceding tables shows that of the specimens used in these ex periments, the cast iron was compressed nearly twice as much as the wrought iron for the same strains. An examination of the table of resistances, in the appendix, shows that for a mean
50 THE RESISTANCE OF MATERIALS.
value wrought iron is compressed about two-thirds as much as cast iron for the same strain. The same ratio evidently holds for tension. This is contrary to the popular notion, that cast iron is stiffer than wrought iron ; for it follows from the above that a cast-iron bar may be stretched more, compressed more, and bent more, than an equal wrought iron one with the same force under the same circumstance, and in some cases, the changes will be twice as great. One reason why cast is con sidered stiffer than wrought iron probably is, that wrought iron does not fail suddenly as a general thing, but it can be seen to bend for a long time after it begins to break ; while cast iron, on account of its granular structure, fails suddenly after it begins, and the bending which has previously taken place is not noticed. It is not safe to trust to such general observations for scientific or even practical purposes, but careful observations must be made, so that all the circumstances of the case may be definitely known. It will hereafter be shown that the ultimate resistance to crushing of cast iron is double that of wrought iron, and yet Fairbairnand other English engineers have justly insisted upon the use of wrought iron for tubular and other bridges. For, without considering the comparatively treacherous character of cast iron when heavily loaded, it appears that within the elastic limits (and the structure should not be loaded to exceed that), a wrought iron structure is stiffer than a cast iron one of the same dimensions, and will sustain more for a given compression, extension, or deflection.
44. COMPRESSION OF OTHER MATERIALS. — All materials are compressible as well as extensible, and it is generally assumed that their resistance to compression, within the elastic limits, is the same as for extension ; but, as has been seen in the previous articles, this is not rigorously correct. Indeed the same piece resists differently under different circumstances, depending upon its temperature, the duration of the strain, and the suddenness with which the force is applied. But these changes are not great, and the mean value of the coefficient of elasticity is sufficiently exact for practical cases.
ULTIMATE STRENGTH.
45. MODULUS FOR CRUSHING. — The modulus of resistance
COMPKESSION. 51
to crashing is the pressure which is necessary to crush pieces of a material whose length does not exceed from one to five times its diameter, and whose section is unity. The value thus found we call C. It is found by experiment that the re sistance of all substances used in the mechanic arts varies very nearly as the section under pressure. Hence, if
P = the crushing force, and
K = the section under pressure, we have
P = CK (22)
46. MODULUS OF STRAIN.— If the force P is not sufficient to crush the piece, we have for the strain on a unit of section
c,=|. ....
It is necessary to use short pieces in determining the value of C, because long pieces will bend before breaking, and will not be simply crushed, but will break more like a beam.
THE RESISTANCE OF MATERIALS.
47. RESISTANCE TO CRUSHING OF CAST-IRON.
TABLE
Of the results of experiments on tJie tensile and crushing resistance* of cast iron of various kinds, made by Eaton Hodykin«oit,*
Description of the iron. |
Tensile Strength per square inch. |
Height of Specimen. |
Crushing strength per square inch. C. |
Ratio of tenacity to crushing. T:C. |
|
Lbs. |
Inch. |
Lbs. |
Mean. |
||
Low Moor Iron, No. 1 |
12,094 |
( i* |
64,534 56,445 |
1:5-084) 1 : 4-440 \ |
1 : 4-7C5 |
" " No. 2 |
15,458 |
lit |
99,525 92,332 |
1 : 0-438 ) :5-973f |
1:6-205 |
Clyde Iron, No. 1 |
10,125 |
U |
92,869 88,741 |
: 5-750) :5-503f |
1:5-631 |
u " No. 2.... |
17,807 |
U |
109,992 102,030 |
: 0-177 ) : 5-729 f |
1:51(53 |
" u No. 3 |
23,408 |
14 |
107,197 104,881 |
: 4-568 ) : 4-469 f |
1:4-518 |
Blaenavon Iron, No. 1 |
13,938 |
M |
90,860 80,561 |
: 0-519 ) :5'780f |
1:6-149 |
" u No. 2 |
10,724 |
U |
117,605 102,408 |
: 7-032 ) |
1 : 6-577 |
" . " No. 3 |
14,291 |
M |
68,559 68,532 |
: 4-797 ) : 4-795 J |
1:4-796 |
Calderlron No. 1 |
13,735 |
liJ |
72,193 75,983 |
:5'25G) :5-532f |
1:5-394 |
Coltness Iron, No. 3. . |
15,278 |
I n |
100.180 101,831 |
:6-557) 1: 0-005 J |
1:6-611 |
Brymbo Iron, No. 1 . . |
14,426 |
i i* |
74,815 75,678 |
1:5-180) 1: 5-204 f |
1:5-216 |
u u No. 3.. |
15,508 |
H |
76,133 76,958 |
1 : 4-909 ) 1:4-903 j" |
1: 4 -936 |
Bowling, No. 2 |
13,511 14,511 |
' H j * ( H |
76,132 73,984 99,926 95,559 |
1:5-470 f |
1:5-555 1:6-735 |
Ystalyf ea No 2 . |
|||||
(Anthracite) |
|||||
Yniscedwyn, No. 1 ... (Anthracite) |
13,952 |
U |
83,509 78,659 |
1:5-985) 1 : 5-038 f |
1:5*1 |
No. 2... |
13,348 |
M |
77,124 75,369 |
1 : 5-778 ) 1: 5-040 f |
1:5-712 |
Stirling, 2d quality.. |
25,764 |
M |
125,333 119,457 |
1:4-865) 1:4-637)" |
1:4-751 |
" 3d quality.. |
23,461 |
-U |
158,653 129,876 |
1:6-762) 1:5-538) |
1:6-149 |
Mean . |
10,303 |
{:: |
88,800 94,730 |
Mean ratio |
1:5-64 |
* Supplement to Bridges, by Geo. R. Brunei] London.
and Wm. T. Clark. John Weale,
COMrEESSION. 53
In this table the ratio of resistances range from less than 4£ (Clyde, No. 3) to more than 7 (Blaenavon, No.l). The same ex perimenter once obtained the ratio of 8.493 from a specimen of Carron iron, No. 2, hot blast ;* and the mean of several ex periments, made at the same time, gave 6.594. Hence we have, as the mean result of a large number of experiments, that the crushing resistance of cast iron is about 6 times as great as its
o o
tenacity ; but the extremes are from 4 J to 8-J- times its tenacity.
48. RESISTANCE OF WROUGHT IRON TO CRUSHING. —
Comparatively few experiments have been made to determine how much wrought iron will sustain at the point of crushing, and those that have been made give as great a range of results as those for cast iron. Wrought iron being fibrous, does not in dicate the point of yielding as distinctly as cast iron and other erranulons substances.
o
Ilodgkinson gives C= 65000 f Kondulet " C = 70800 J Weisbach " 0=72000 §
Eankine " C = 30000 to 40000 |
-------- ...... ii __ ^ , Q
Hence it appears that the crushing resistance1 of wrought is from \ to f as much as its tenacity.
RESISTANCE OF woo» TO CRUSHING. — The resistance of wood to crushing depends as much upon its state of dryuess, and conditions of growth and seasoning, as its tenacity does. The following are a few examples : —
Kind of Wood. |
Moderately Dry. |
Very Dry. |
Ash . : |
8 680 |
9360 |
Oak (English} |
6,480 |
10,058 |
Pine (Pitch) |
6 790 |
6 790 |
These results, compared with the corresponding numbers in
* Resistance des Materiaux, Morin, p. 95.
f Vose, Handbook of Railroad Construction, p. 127.
^ Mahan's Civil Engineering, p. 97.
§ Weisbach Mech. and Eng1., vol. i., p. 215.
| Rankin's Applied Mech., p. 633.
5-4
THE RESISTANCE OF MATERIALS.
article 22, show that these kinds of wood will resist from nearly 2 times as much to tension as to compression.
to
RESISTANCE OF CAST STEEL TO CRUSHING. - Major
"Wade found the following results from experiments upon the several samples, all of which were cut from the same bar and treated as indicated in the table.*
Specimen. |
Length. |
Diameter. |
Crushing in Ibs. per sq. inch. |
Not Hardened |
1-021 |
0-400 |
198,944 |
Hardened, low temper |
0-995 |
0-402 |
354,544 |
mean k * |
1-010 |
0-403 |
391,985 |
u high " for tools for turning hard steel. . . |
1-005 |
0-405 |
372,598 |
Ln )3|JT>~u-o _
4rr "4 VY7j"C" 51 . RESISTANCE OF GLASS TO CRUSHING. We OW6 hlOSt
of our knowledge of the strength of glass to Wm. Fairbairn and T. Tate, Esq. According to their experiments we have the following results for the crushing resistance of specimens of glass whose height varied from one to three times their diameter.
MEAN CRUSHING RESISTANCE OF CUT-GLASS CUBES AND ANNEALED GLASS CYLINDERS.
Weight per Square Inch.
Cubes. |
Cylinders. |
|
Flint Glass |
Ibs. 13,130 |
Ibn. 27582 |
Green Glass |
20,200 |
31,870 |
Crown Glass |
21,867 |
31,003 |
18,401 |
30,153 |
|
The ratio of the mean of the resistances is as 1 to 1 '6 nearly.
The cylinders were cut from round rods of glass, and hence retained the outer skin, which is harder than the interior, while the cubes were cut from the interior of large specimens. This
* Report on Metals for Cannon, p. 258.
COMPRESSION. 55
may partially account for the great difference in the two sets of experiments. The cubes gave way more gradually than the cylinders, but both fractured some time before they entirely failed. The cylinders failed very suddenly at last, and were divided into very small fragments. The specimens had rubber bearings at their ends, so as to produce an uniform pressure over the whole section.
L STRENGTH OF PILLARS. — The strength of pillars for incipient flexure has been made the subject of analysis by Euler and others, but practical men do not like to rely upon their results. Mr. llodgkinson deduced empirical formulas from ex periments which were made upon pillars of wood, wrought iron, and cast iron. The experiments wrere made at the expense of "Wm. Fairbairn, and the first report of them was made to the Royal Society, by Mr. llodgkinson, in 1840. The following are some of his conclusions : —
1st. In all long pillars of the same dimensions, when the force is applied in the direction of the axis, the strength of one which has flat ends is about three times as great as one with rounded ends.
2d. The strength of a pillar with one end rounded and the other flat, is an arithmetical mean bet\veen the two given in the preceding case of the same dimensions.
3d. The strength of a pillar having both ends firmly fixed, is the same as one of half the length with both ends rounded.
4th. The strength of a pillar is not increased more than ^th by enlarging it at the middle.
To determine general formulas, bars of the same length and different sections were first used ; then others, having constant sections and different lengths ; and formulas wrere deduced from the results. The formulas thus made were compared with the results of experiments on bars whose dimensions differed from the preceding. The following are the results of some of his
*/"" » 3^y^Tr%/K-» / //
,
j
.V,
THE RESISTANCE OF MATERIALS.
EXI'EKIMKNTS ON SQUARE PILLARS.
Lenpth of the l«u-rt. |
Side of the square. |
Crushing weight. |
Exponent of the side. |
Feet |
Inches. |
Lbn. |
|
10 |
0-766 1-51 |
1,948) 23,025 f |
3-57 |
10 |
1-00 1-50 |
4,225 f 23,025 f |
417 |
7i |
1-02 1-53 |
10.236 ) 45,873 f |
3-69 |
71 |
0-50 1-00 |
583 ) 9,873 f |
4-08 |
5 |
0-50 1-00 |
1,411 I 18,038 f |
367 |
0-502 |
4,216 | |
o-no |
|
"1 |
1-00 |
27,212 f |
£ Oy |
21 |
0-502 0-76 |
4,216 ) 15,946 f |
3-28 |
Mean |
3-591 |
The fourth column is computed as follows : — Suppose that the strengths are as the x power of the diame ters, then for the first bar we have ^ fy % /% b /,' /
= 23025 .0.766
log. 1.987
The others are computed in the same way. An examination of the table shows that whe^the square^ section is the same the strength varies inversel^asl the length. Thus, of two bars whose cross section is one square inch, the one five feet long is nearly four times as strong as the one ten feet long.
Let I = length of one, I — " of other, d = diameter of first one, d' = " of the second one, and *to = the power of the length.
u ,73.59
Then the strength of the first one is, P= constant x
" " " second is, P' = constant x
COMPRESSION.
57
in which substitute the values from any two experiments. Thus if we take from the table
I' = 10 feet, d! = 1 inch, P' = 4225 Ibs., and
I = 5 feet, d = 1 inch, and P = 18038 Ibs., we have
18038
4225
log. 4.2694
/. y = -V - = 2.094. log. 2
Proceed in a similar way with each of the others and take the mean of the results for the power to be used. In this way was formed the following
TABLE
For the absolute strength of columns.
In which P = crushing- weights in gross tons,
d — the external diameter, or side of the column in inches, dl = the internal diameter of the hollow in inches, and I — the length in feet.
Kind of Column.
Both ends rounded, the
length of the column ex
ceeding fifteen times its
diameter.
Both ends flat, the length of
the column exceed ing thirty
times its diameter.
Solid Cylindrical Columns of ) cast iron ]
Hollow Cylindrical Columns ) of cast iron )
Solid Cylindrical Columns of ) wrought iron f
Solid Square Pillar of Dant- zic oak
Solid square Pillar of red ) dry deal [
P = 44.16-
P = 13
«z8;!_V76
P = 44.34
I1
3.65 3.55
d— di
P =
£3.76
P = 133. 75-
£3.
P = 10.95 jt P = 7.81 7j
The above formulas apply only in cases where the length is so great that the column breaks by bending and not by simple crushing. If the column be shorter than that given, in the
58 THE RESISTANCE OF MATERIALS.
table, and more than four or five times its diameter, the strength is found by the following formula :
in which P — the value given in the preceding table,
K = the transverse section of the column in square
inches, C = the modulus for crushing in tons (gross) per
square inch, and
AV= the strength of the column in tons (gross).* Experiments have been made upon steel pillars which gave good results.f
53. WEIGHT OF PILLARS. — From the first formula of the preceding table we find
The area of the cross section is % * d2, and the volume ki
inches =- if. * d* I - i
Cast iron weighs 450 pounds to the cubic foot, hence the
450 450
weight = T^-OO x 3 x *-#2 x I =-&rfi x 3.141C x
which by reduction gives
weight = 0.0152803 (?. ZS<58)T W . _ \ (24.) If P is given in pounds, this coefficient must be divided by
If the pillar is hollow the section of the iron is J *• (d? — d*), .and if n is the ratio of the diameters, so that dl=nd this be comes
12
J * d*(l — 7i9) ; and its volume in inches = -r * d2 (I — if) I;
* James B. Francis, C. E., has published a set of tables which gives the strength of cast-iron columns, of given dimensions, by means of equation (23), and also by those in the above table, i... f London Builder, No. 1211.
COMPRESSION. 450
and its weight in pounds = - x 3 x * d? (I — n2) L
59
If the value of d from the second equation of the first column in the preceding table, be substituted in the preceding equation, we find the
weight in pounds = 25 __ ?r_ 32 2240 x
(P.*-'
Proceeding in this way with each of the cases given above and we form the following :
TABLE
Of the weights in pounds of pillars in terms of their lengtJis in feet, and crushing forces in pounds. -1- ^
Kind of Pillar.
Solid Cylindrical Column of cast iron.
Hollow Cylindrical Col umns of cast iron,
if 71=0-98 if 71=0-95 if 71=0-925 if 7i=0-90 if 7i=0 -875 if 71=0-85 if 7i=0-80 71=0-75
Solid Cylindrical Columns of Wrought Iron,
Square Column of Dant- zic Oak.
Weight in pounds.
Both ends rounded. 1 > 15 d.
0-0101645
(l_ft*-Ta)
0--001349014 (P.Z3-58) 0-002549688 (P. Is 04)03000033
0--005649247
.jQ -00599670 (P. Zs-M)
(Cubic foot weighs 47.24 pounds. )
Both ends flat. Z > 30 d.
0-OG37
A •n(VJ-~ ~,Y^> \r \J\Mj t CTwU<O
1— n*
-9-000060628 (P.^
0-00120664
0-00152392 (P.I3- 0-00165855 (P.Z1- 0-00189914 0-00211346
0-00201664 (P.I3 -i^600547291 :
60 THE RESISTANCE OF MATERIALS.
If the thickness of the metal (t) and the external diameter are given, n may be found as follows: d—2t= internal diame ter, hence n='~*='\. — *. For instance, if the external diameter is 6 inches, and the thickness f of an inch, the internal diame ter is 5J inches and 71=^=0.875.
The iron used in the preceding experiments was Low Moor No. 2, whose strength in columns is about the mean of a great variety of English cast iron, the range being about 15 percent, above and below the values given above.
«54. CONDITION OF THE CASTING. — Slight inequalities in the thickness of the castings for pillars does not materially af fect the strength, for, as was found by Mr. Ilodgkinson, thin castings are much harder than thicker ones, and resist a greater crushing force. In one experiment the shell of a hollow column resisted about 60 per cent, more per square inch than a solid one.* But the excess or deficiency of thickness should not in any case exceed 25 per cent, of the average thickness. f Thus, if the average thickness is one inch, the thickest part should not exceed 1J inch, and the thinnest part should not be less than £ of an inch.
It is also found that in large castings the crushing strength of the part near the surface does not much exceed that of the internal parts.
55. EXPERIMENTS MADE BY THE NEW YORK CENTRAL,
RAILROAD COMPANY. — The immediate object of these experi ments was to determine the relative values of different sorts and forms of wrought iron of lengths greatly exceeding their dia meters, when subjected to longitudinal compression. The pieces wrere not in all cases broken, nor even materially altered in form by the compressions to which they were subjected, the experiments being generally discontinued as soon as the pro gressive rate of flexure due to a regularly increased load was ascertained.
The testing machine used in the experiments was designed by C. Hilton, and was made at the Company's carpenter shop, at Al bany, by order of the Chief Engineer ; its arrangement and all its principal details were afterwards found to be exactly similar to
* Phil Trans., 1857, p. 890. f Stoney on Strains, vol. ii., p. 206.
COMPRESSION.
61
those of the machine used for the same purpose by Mr. Eaton Hodgkinson, at Manchester, England (for a description and draw ing of which see " Tredgold on the Strength of Wrought and Cast Tron," Weale, London, 1847), with this difference, that the ma chine made at Albany was of wood, while that used by Mr.
Hodgkinson was of iron.
EXPERIMENT No. 1.
Made upon a bar of English Crown Iron from the works of Hawkes, Crawsliay & Co., Gateshead, planed at both ends and perfectly straight, exactly $ feet in length, and of cross section as sketched in Fig. 14.
lio. 11.
Weight applied in Ibs. |
Deflection in parts of an inch. |
Remarks. |
2,568 |
.0 |
|
5,468 |
.0 |
|
8,468 |
.033 |
In direction C D. |
10,448 |
.0625 |
t i |
11,738 |
.0833 |
41 |
12,778 |
.15625 |
« |
13,778 |
.1875 |
1C |
15,208 |
.1875 |
« |
16,758 |
.25 |
<( |
20,928 |
.3125 |
On the removal of the above recorded load, the bar immediately resumed its original form, having taken no appreciable set. Being once more placed in the machine the results were :—
Lbs. |
Deflections in A B. |
Deflections in C D. |
20,248 25,968 30,348 J |
.100 .300 Bar bent doi |
.400 .600 ible in C D. |
THE RESISTANCE OF MATERIALS.
EXPERIMENT No. 2.
Made upon a bar of English Crown Iron from ihe same works, planed at both ends and perfectly straight, exactly 8 feet in lengthy and of the cross section sluncn in Fig. 15.
FIG. 15.
Weight applied in Ibs. |
Deflection in parts of an inch. |
Remarks. |
8,688 |
.0 |
|
5,468 |
.0 |
|
8,178 |
.05 |
In direction C D. |
10,918 |
.075 |
|
13,668 |
.1 |
|
16,468 |
.125 |
|
19,218 |
.156 |
|
21,968 |
.1875 |
|
24,678 |
.218 |
|
27,478 |
.25 |
|
30,248 |
.3125 |
No more than 30,248 Ibs. was placed upon this bar, the bar being required for use, and the quality of the iron being very soft and easily bent.
No appreciable set was found upon the removal of the above load. On being placed a second time in the machine, and sub jected to a load of 19,218 Ibs., the flexure was observed to be .125 inch, this weight being left on for 42 hours ; on removal a permanent set was observed of .01 inch.
63
EXPERIMENT No. 3.
Made upon a bar of English Crown Iron from the same works, 8 planed at the ends, and cross section sketched below in Fig 16.
. 16.
Weights applied in Ibs. |
Deflection in parts of an inch. |
Remarks. |
||
2,568 |
.0 |
Flexure not appreciable. |
||
5,468 |
.144 |
In direction C D. |
||
8,178 |
.168 |
|||
10,888 |
.192 |
|||
13,638 |
.242 |
|||
'-16,438 |
.288 |
|||
19,188 |
.314 |
|||
21,938 |
.360 |
|||
24,648 |
.408 |
|||
27,448 |
.480 |
|||
30,218 |
.612 |
The bar was found, on removing the weights, to be perfectly straight. When replaced in the machine the results were as follows : — •
Weights. |
Inches in C D. |
Inches in A B. |
16,510 |
.5625 |
.1875 |
24,338 |
.625 |
.21875 |
27,638 |
.6875 |
.25 |
f 30,418 |
.75 |
.3125 |
33,188 |
.875 |
.3750 |
35,948 |
1.000 |
.3750 |
"With 35,948 Ibs. the bar bent double after four minutes.
G-i
THE RESISTANCE OF MATERIALS.
EXPERIMENT Xo. 4.
Made upon a piece of English Crown Iron from the same work*, of the section thown in Fig. 17, planed at both ends and exactly 5 feet in length.
Fig. 17.
Weight in Ibs. |
Deflection in A B, in parts of an inch. |
Deflection in C D, in parts of ar. inch. |
2,508 |
.00 |
.0 |
3,038 |
.015 |
.0 |
4,038 |
.025 |
.020 |
5,468 |
.060 |
.050 |
8,228 |
.083 |
.083 |
10,908 |
.1 |
.1 |
13,678 |
.1 |
.125 |
16,448 |
.1 |
.142 |
19,248 |
.1 |
.150 |
21,998 |
.083 |
.166 |
24,708 |
.083 |
.166 |
On the application of 27,508 Ibs. the bar assumed a new form, as shown in the figure ; the deflection in the new direction is designated as taking place in x.
Weight in Ibs. |
Deflection in A B, in puts of an inch. |
Deflection in C D, in parts of an inch. |
Deflection in ar, in parts of an inch. |
27,508 30,288 33,058 35,818 39,228 |
0.083 .083 .083 .083 Bar bro |
.166 .166 .166 .225 ke at y. |
.ON::, .1 .125 .1875 |
COMPRESSION.
65
EXPERIMENT No. 5.
A Bar of Angle Iron, 5 feet in length, planed at both ends and quite straigJit, of the cross section shown in Fig. 18, furnished by the Albany Iron Works, Troy, of ordinary quality.
FIG. 18.
Weight in Ibs. |
Deflection A B. |
Deflection C D. |
2,568 |
.04 |
.05 |
3.038 |
.06 |
.08 |
4,038 |
.08 |
.15 |
5,468 |
.375 |
.25 |
8,228 |
.500 |
.375 |
10,968 |
.5625 |
.375 |
13,678 |
.5625 |
.375 |
16,448 |
.625. |
.375 |
19,248 |
Broke or bent double. |
The general deflection in the direction of the arrow could not be observed until 8,228 Ibs. were applied, when it was succes sively —
.156, .25, .33, .5.
No. 6.
Made upon the bar used in Experiment No. 2. It being pre sumed that the previous experiment had somewhat weakened this bar, it was determined to break it. The following weights were used: —
Weight in Ibs.
Deflection in C D.
22,048 30,398 35,928
.126
.5625
Bar bent nearly double.
THE RESISTANCE OF MATERIALS.
#6. COMPRESSION OF TtiBEs.-BtirKi.iNG. — Wrought iron tubes when subjected to longitudinal compressive stresses may yield by crushing like a block, or by bending like a beam, or by buckling. The first takes place when the tube is very short ; the second, when it is long compared with the diameter of the tube ; and the last, for some length which it is difficult to assign, inter mediate between the others.
The appearance of a tube after it has yielded to buckling i> shown in Figs. 19 and 20.
The experiments heretofore made do not indicate a specific law of resistance to buckling ; but the following general facts appear to be established : —
1. The resistance to buckling is al ways less than that to crushing ; and is nearly independent of the length.
2. Cylindrical tubes are strongest ; and next in order are square tubes, and then rectangular ones.
3. Rectangular tubes, Q ^], are not as strong as tubes of this form Q ^]. The
FIG. 19.
Fiu. 20.
tubes in bridges and ships are generally rectangular or square.
COLLAPSE OF TUBES.
•57. THE RUPTURE OF TUBES wliich are subjected to great external normal pressure is called " a collapse." The Hues of a steam-boiler are subjected to such an external pres sure, and in view of the extensive use of steam powrer, the subject is very important. The true laws of resistance to collapsing were unknown until the subject was investigated by Wm. Fairbairn. Experiments were carefully made, and the results discussed by him with that scientific ability for wliich he is so noted. They were published in the Transactions of the Royal Society, 1858, and republished in his " Useful Information for Engineers," second series, page 1.
* Civ. Eng. and Arch. Jour., vol. xxviii., p. 28.
COMPRESSION.
67
The tubes were closed at each end and placed in a strong cylindrical vessel made for the pur pose, into which water was v forced by a hydraulic press, thus enabling him to cause any desirable pressure upon the outside of the tube. In order to place the tube as nearly as possible in the condition of a flue in a steam-engine, a pipe which communicated with the external air was inserted into one end of the tube. This pipe permitted the air to escape from the tube during collapse.
The vessel, pipe, tube, and their connections were made practically water-tight, and the pressure indicated by gauges.
Fig. 21 shows the appearance and cross-section at the middle of the short tubes after the collapse ; and Fig. 22 of a long one. Although no two tubes appeared exactly alike after the collapse, yet the examples which I have selected are good types of the appearances of thirty tubes used in the experiments.
The tubes in all cases collapsed suddenly, causing a loud report. In the first and second tubes the ends were supported by a rigid rod, so as to pre vent their approaching each other when the sides were com pressed. The following tables give the results of the experiments : —
FIG. 22.
FIG. 21.
V\
G8
THE RESISTANCE OF MATERIALS.
TABLE I.
Mark. |
No. |
Thickness of Plate, inches. t. |
Dinmeter in inches. a. |
Length in inches. L. |
Pressure of Collapse, 11*. pr. sq. in. of Surface. P. |
Product of I*ressure and Length. P. L, |
Product of the Pressure, Length, and Diameter. P.L. d.=p. |
A B C D E F |
1 2 3 4 5 6 |
0043 u • u n (( |
4 |
19 19 40 38 60 60 |
170 137 65 65 43 140* |
3230 2603 2(500 2470 2580 2800 |
10412 10400 9880 10320 |
Mean |
2714 |
10253 |
|||||
G H J K L M |
7 8 9 10 11 12 |
1 I 1 « |
6 |
30 29 59 30 30 30 |
48+ 47f 32 52 65 85J |
1440 1263 1888 1560 1950 V |
11328 9360 11700 |
Mean |
1620 |
10796 |
|||||
N O P |
13 14 15 |
« « u |
8 a a |
30 39 40 |
39 32 31 |
1170 1248 1240 |
9360 9984 9920 |
Mean |
1219 |
9754 |
|||||
Q R |
16 17 |
({ |
10 u |
50 30 |
19 33 |
950 990 |
9500 9900 |
Mean |
970 |
9700 |
|||||
S T V |
18 19 20 |
(( tt ii |
12-2 12 u |
68* 60 30 |
11-0 12-5 22 |
643-7 750 662 |
7850 9000 7920 |
Mean |
685^3 |
8256 |
* This tube had two solid rings soldered to it, 20 Inches apart, thus practically reducing it to three tubes, as shown in Fig. 23.
FIG. 23.
t The ends of both were fractured, causing collapse, perhaps before the outer shell had at tained its maximum. J A tin ring had been left in by mistake, thus causing increased resistance to collapsing.
COMPRESSION. 69
58. DISCUSSION OF RESULTS — By comparing the tubes of the same diameter and thickness, but of different lengths, we see that the long tubes resist less than the short ones ; hence, the strength is an inverse function of the length, and an ex amination of the seventh column shows that it is nearly a sim ple inverse function of the length. The first of the 4-inch tubes is so much stronger than the others, it may be neglected in determining the law of resistance, although it differs from a mean of all the others by less than £ of the mean. An exami nation of the several cases indicates that wre may safely assume that the resistance to collapsing varies inversely as the lengths of the tubes*
The mean of the results for the several diameters in the last column shows that the resistance diminishes somewhat more rapidly than the diameter increases ; but this includes the error, if any, of the preceding hypothesis. As the power of the diameter is but little more than unity, it seems safer to con clude, for all tubes less than 12 inches in diameter, as Fair- bairn does, that the resistance of tubes to collapsing varies in versely as their diameters.
59. L.AW OF THICKNESS — Experiments were also made to determine the law of resistance in respect to the thickness. Comparatively few experiments were made of this character, but these few gave remarkably uniform results. One of the
* A more exact law may be found as follows : — Let P — the compressing force per square inch ; C = a constant for any particular diameter and thickness, I = the length, and n the unknown power. Then
C P = — for one case.
o
PI == — for another.
M/v
• n - '• 1^1
ft
By means of this equation, and any two experiments in which the thickness and diameter are the same, n may be found, and by using several experiments a series of values may be found from which the most probable result can be obtained. But in this case the mean result is so near unity, there is no prac tical advantage secured by finding it.
-/< ///
*r*ftty
V </
70
THE RESISTANCE OF MATERIALS.
tubes (No. 24), was made with a butt joint, as shown in Fig. 24, and the others with lapW joints, as in Fig. 25.
FIG. 24. FIG. 25.
The following are the results of the experiments
TABLE II.
Mark. |
No. |
Thickness. t. |
Diameter. d. |
Length in inches. L. |
Pressure per square inch. P. |
P. L. |
Product. L. P. <L — P. |
w X Y Z JJ |
21 22 23 24 33 |
0-25 0-25 014 014 0125 |
9 18f 9 9 144 |
37 61 37 37 60 |
(450) 420 263 378 125 |
Uncoil 25620 9694 13986 7500 |
apsed. 480375 89046 125874 108750 |
Tubes NOB. 23 and 24 were exactly alike in every respect except their joints ; and it appears that the butt joint, No. 24, is 1*41 times as strong as the lap joint, a gain of 41 per cent. But this is a larger gain than is indicated in other cases ; for instance, No. 33, which is also a lap joint, offers a greater re sistance as indicated in the last column, than No. 23, although the former is not as thick as the latter. Still it seems evident that butt joints are stronger than lap joints, for with the former the tubes can be made circular, and there is no cross strain on the rivets, conditions which are not realized in the latter.
The resistance of the 23d is so small compared with others, it is rejected in the analysis.
We observe that the resistance varies as some power of the thickness ; if then C and n be two constants to be determined by experiment, and we use the notation given above, we shall have for the pressure of collapse of one tube.
6 / '"M^ C^^^L^^L <2^t^ fa, A<^ ecW
* "i^*"~" y I '
n-, it i t
COMPRESSION. 71
and for another tube
<7*5W
Pl = 5^T •••*Pi^=l>i=^" (25)
Hence we have
log. < - log. *,
(26)
(27)
TO FIND THE CONSTANTS U AND O.
The mean of the mean of the values of p from Table I. is jp = |[10253+10T96+9754:+9TOO+8256]=9T52and #=0-04:3.
Using these values and others taken from the preceding tables, and the following values may be found for n : —
In equation (26) make^? = 480375, t = 0-25, jpl= 9752, t,= 0-043; and we get
Jog. 480375 -log. 9752
Similarly, taking^? = 480375,= 0-25,^= 10253, ^= 0-043 ; and we get
log. 480375 -log. 10253
log. 0-25 - log. 0-043 The mean value of j? for all but the 12-inch tubes in Table I. is
p = i (10253 + 10796 4- 9754 + 9700) = 10125 ; hence, using^ = 125874, t — 0-^,4, p,— 10125, £1= 0-043 ; and we get
_ Iog.l25874-log. 10125 _ §
log. 0-14 - log. 0-043
and taking p =108750, t = 0-125, ^=10125 and t,= 0-043; we get
log. 108750 -log. 10125 Tog. 0-125 - log. 0-043 : and the mean of these results is, n = 2-18.
Fairbairn made it 2*19 by including some data which I have rejected as paradoxical ; I have also given more weight to those
72 TIIE RESISTANCE OF MATERIALS.
cases which gave nearly uniform results. The difference, how ever, of 0-01 is too small to seriously affect practical results.
To determine the constant, Cy substitute the proper values taken from the preceding tables in equation (27), and we have for four cases the following :—
= ^98,900. = 9,864,300.
The mean of which is (7= 9,604,150. Calling C ^ 9,600,000 and equation (&fa becomes:
P= 9,600,000 --^ - >V' ^^ (28)
If L be given in feet, so that L = 121^, we have
P = 800,000-^4- ......... C29)
(I. Jy,
The coefficient, 9,600,000, applies only to the kind of iron used ; but the exponent, 2*18, is supposed to be constant for all kinds of iron.
OO. roic 'i i i. \ FOK THICKNESS TO RESIST COLLAPSING.
—Equation (28) readily gives the following expression forjind- \L ing the thickness in inches of a tube to resist cottapriny :—
l^^+i:!^- 1-203 - (30)
"
Ol. ELLIPTICAL TI'BES. — Experiments made upon etiipti- oal tube* showed that the preceding formula would give the strength, if the diameter of the circle of curvature at the extremity of the minor axis is substituted for d. The diameter of curva ture is -^-, in which a is the major and b the minor axis. b
Experiments made upon tubes in which the ends were not connected by internal rods, showed that the resistance was in-
MTM'ly as their length.
Zo 6&<i-(&
COMPRESSION. 73
VERY LONG TUBES. — Some experiments were made upon a tube 35 feet long and one 25 feet long. Sufficient pres sure was applied to distort them, but not to collapse them, and it was found that equation (28) erred by at least 20 per cent,, giving too small an amount. It was, however, very evident that the length was still a very important element in the strength.
63. COMPARISON OF STRENGTH FROM EXTERNAL, AND
INTERNAL PRESSURE. — Let p be the internal pressure per square inch at which the tube is ruptured, then for tubes of the same thickness and diameter we have from equations (18) and (29), by calling T = 30,000 Ibs.,
p 1 L
F^T^^.^n* /. A _/
If p = P, then L = I $,3;j t^*.
Ift = 0-25, then we find L = 3-56 feet, that is, a tube whose thickness is J of an inch, and whose length is 3'56 feet, is equal- ly strong whether subjected to internal or external pressure.
If the tube is so thick that the unequal stretching of the fibres must be considered, then equation (20) must be compared with equal ion (29), in which case we have :> — T? 7*
P- _T _***
_
P~ 800,000 (r+*)*i-i8 o
Ifp = P, T = 40,000 Ibs., and 2;> = d = 4 inches ; /
then<|£-< *+*»•" =* I*
If t = t inch, L = 5-504 feet. If t = I " L = 15 feet.
»«•***•"
64. RESISTANCE OF GLASS GLOBES TO COLLAPSING. -
Fairbairn also determined that glass globes and cylinders fol lowed the same general law of resistance. For globes of flint glass lie found :
^=28,300,000^ ,,:,' (32)
and for cylinders of flint glass :
Pi==740,000 j^ ..... (33)
provided that their length is not less than twice, nor more than
*,
-t t <a^£ ^v^
<- '
74 THE RESISTANCE OF MATERIALS.
six times their diameter. Dividing equation (33) by (28) gives P, 0-0770,
P "JO-IS
P
If t = 0-043 in., p-1 = 0-896 ; or the glass cylinder is nearly -^
as strong as the iron one. If they are equally strong, P = P, .•.* = 0-0373 of an inch.
FLEXURE AND RtTPTURE FROM TRANSVERSE STRESS.
75
CHAPTER III.
Library.
California
THEORIES OF FLEXURE AND RUPTURE FROM TRANSVERSE
STRESS.
65. REMARK. — The ancients seem to have been entirely ignorant of the laws which govern the strength of beams. They made some rude experiments to determine the absolute strength of some solids, especially of stone. They may have recognized some general facts in regard to the strength of beams, such as that a beam is stronger with its broad side ver tical than with its narrow side vertical, but we find no trace of any law which was recognized by them. This department of science belongs wholly to modern times. A very brief sketch of the history of its development is given below.*
GO. GALILEO'S THEORY. — Galileo was the first writer, of whom we have any knowledge, who endeavored to establish the mathematical laws which govern the strength of beams. f He assumed —
1st. That none of the fibres wrere elongated or compressed.
2d. When a beam is fixed at one end, and loaded at the other, it breaks by turning about its lower edge, B, Fig. 26 ; or if it be sup ported at its ends and loaded at the middle of the length, it would turn about the upper edge ; hence every fibre resists tension.
3. Every fibre acts with equal energy. From these he readily deduced,— that, when one end is firmly fixed in a wall or other
* For a more complete history, see introduction to " Resistance des Corps Solides," par Navier. 3d edition. Paris, 1864. f Opere di Galileo. Bologne, 1856.
76 THE RESISTANCE OF MATERIALS.
immovable mass, the total resistance of the section is equal to the sum of all the fibres, or the transverse section, multiplied by the resistance of a unit of section, multiplied by the dis tance of the centre of gravity from the lower edge. Hence, in a rectangular beam, if
T = the tenacity of the material, b = the breadth, and d = the depth of the beam ;
the moment of resistance is
(34)
67. ROBERT HOOK i:-x THEORY. — Kobert Ilooke was one of the first, and probably the first, to recognize the compressi bility of solids when under pressure. In 1678 he announced his famous principle, Ut tensio sic vis ; which he gave in an anagram in 1676, and stated as the basis of the theory of elasti city that the extensions or contractions were proportional to the forces which produce them, and also that when a bar was bent the material was compressed on the concave side and extended on the convex side.
68. MARRIOTTE'S AND LEIBNITZ'S THEORY. - Mamotte,
in 1680, investigated the subject, and finally stated the follow ing principles: —
1st. The material is extended on the convex side and com pressed on the concave side.
2d. In solid rectangular sections the line of invariable fibres (or neutral axis) is at half the depth of the section.
3d. The elongations or compressions increase as their distance from the neutral axis.
4th. The resistance is the same whether the neutral axis is at the middle of the depth or at any other point.
5th. The lever arm of the resistance is $ of the depth.
We here find some of the essential principles of the resist ance to flexure, as recognized at the present day ; but the two last are erroneous. As hereafter shown, the neutral axis is at half the depth, and the lever arm is f of £ the depth.
Leibnitz's theory, given in 1684, was the same as Marriotte's.
69. JAMES BERNOUILLI'S THEORY was essentially the same
FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 77
as Marriotte's, except that he stated that extensions and compres sions were not proportional to the stresses. " For," said he, " if it is true, a bar might be compressed to nothing with a finite force." On this point see Article 16. He was the first to give a correct expression for the equation of the elastic curve.
70. PARENT'S THEORY — Parent, a French academician of great merit, but of comparatively little renown, published, in 1713, as the result of his labors, the following principles, in addition to those of his predecessors: —
1st. The total resistance of the compressed fibres equals the total resistance of the extended fibres.
2d. The origin of the moments of resistance should be on the neutral axis.
By the former of these principles the position of the neutral axis may be found, when the straining force is normal to the axis of the beam ; and by the latter he corrected the error of Marriotte and Leibnitz ; showing that the ratio of the absolute to the relative strength is as six times the length to the depth, in stead of three, as will be shown hereafter.
71. COULOMB, IN 1773, PUBLISHED the most scientific work on the subject of the stability of structures wilich had appeared up to his time. He deduced his principles from the fundamental equations of statics, and generalized the first of the principles of Parent, which is given above, by saying that the algebraic sum of all the forces must be zero on the three rectangular axes. This establishes the position of the neutral axis when the applied forces are oblique to it, as well as when they are normal. He also remarked, that if the proportionality of the compressions and extensions do not remain to the last, or to the point of rupture, the final neutral axis will not be at the centre of the section.
MODULUS OF ELASTICITY.— In 1807 Thomas Young introduced the term modulus of elasticity, which wre have defined as the coefficient of elasticity in Article 5. After this several writers, among them Duhamel, Navier in his early writings, and Barlow in his first work, stated the erroneous principle, that the sum of the MOMENTS of the resistances to compression equalled those for tension.
78 THE RESISTANCE OF MATERIALS.
73. IN 1821 NAVIER PUBLISHED the lectures wllicll he
had given to V&cole des Fonts et Chaussees, in which he estab lished more clearly those principles of elastic resistance, and resistance to rupture, which have since his day been accepted by nearly all writers. He was the first to show that when the stress is perpendicular to the axis of the beam, the neutral axis parses through the centre of gravity of the transverse sections. His most important modifications in the analysis was in making ds = dx, or otherwise, considering that for small deflections the tangent of the angle which the neutral axis makes with the original axis of the beam is so small compared with unity that it may le neglected • and also, that the lever arm of the force remains constant during flexure. These principles we have used in Chapter V. He resolved many problems not before attempted, and became an eminent author in this department of science.
74. THE COHUION THEORY. — The theories of flexure and of rupture which result from these numerous investigations, I will call, for convenience, the common theory. It consists of the following hypotheses : —
1st. The fibres on the convex side are extended, and on the concave side are compressed, and there are no strains but .cj^in- pression and extension.
2d. Between the extended and compressed fibres (or elements) there is a surface which is neither extended nor compressed, but retains its original length, and which is called the neutral surface, or in reference to a plane of fibres it is called the neutral axis.
3d. The strains are proportional to their distance from the neutral axis.
4th. The transverse sections which were normal to the neu tral axis of the beam before flexure, remain normal to the iieu- tral axis during flexure.
5th. A beam will rupture either by compression or extension when the modulus of rupture is reached.
6th. The modulus of rupture is the strain at the instant of rupture upon a unit of the section which is most remote from the neutral axis on the side which first ruptures. This is called R
The remainder of this article properly belongs to Chapter
FLEXUKE AND KUPTUEE FROM TRANSVERSE STRESS. 79
YL, but it is given here so that the reasons for Barlow's theory may be understood.
If a beam ruptures on the convex side, it appears that it ought to break when its tenacity (T) is reached ; but it is found by experiment that in this case R always exceeds T. Similarly, it would seem that if it failed by crushing on the concave side, as in the case of rectangular cast-iron beams, R ought to equal C, but experiment shows that in thia_case JJ-exeeeds C ; and gen erally the value of R is always &&twee» those of T and C for the same material; b^ing groatc^^Jt^4hes
The values of R in the tables were deduced from experiments upon rectangular beams, as will hereafter be shown ; and hence, if the common theory is correct, R should equal the value of the lesser resistance, whether it be for com pression or extension ; but it does not. This discrepancy be tween theory and the results of experiment * has led Barlow to investigate the subject further, and it has resulted in a new theory which he calls " Resistance to Flexure " — an ex pression which I consider unfortunate, as it does not express his idea. " Longitudinal Shearing " would express his idea better, as will appear from the following article :—
75. BARLOWS THEORY. — According to the common theory
* Mosley's Mech. and Arch. , p. 557. ' ' The elasticity of the material has been supposed to be perfect up to the instant of rupture, but the extreme fibres are strained much beyond their elastic limits before rupture takes place, while the fibres near the neutral axis are but slightly strained, and hence the law of pro portionality is not maintained, and the position of the neutral axis is changed,
TJT
and the sum of the moments is not accurately — - (see equation 171). To de
termine the influence of these modifications we must fall back upon experiment, and it has been found in the case of rectangular beams that the error will be
corrected if we take — T (= B) instead of T, where m is a constant depend
ing upon the material."
Weisbach, vol. ii., 4th ed., p. 68, foot-note says, "Excepting as exhibiting approximately the laws of the phenomena, the theory of the strength of mate rials has many practical defects. "
In the Report of the Ordnance Department, byMaj. Wade, p. 1, it says: — " A trial was made with cylindrical bars in place of square ones. These gen erally broke at a point distant from that pressed, and the results were so ano malous that the use of them was soon abandoned. The formula by which the strength of round bars is computed appears to be not quite correct, for the unit of strength in the round bars is uniformly much higher than in the square bars cast from the same iron."
80
THE RESISTANCE OF MATERIALS.
the resistance at a section is the same as if the fibres acted in dependently of each other, and the transverse section remained normal to the neutral axis. But Barlow correctly considers that in order to keep the transverse sections normal to the neutral axis, the consecutive planes of fibres must slide over each other, and to this movement they offer a resistance.
He presented his view to the Royal Society (Eng.), in 1855, and it has since been published in the Civil Engineer and Architects Journal, vol. xix., p. 9, and vol. xxi., p. 111.* The subject is there discussed in a very able and thorough manner, and although he may have failed to establish his theory, by not taking into account all the incidents which exist at the instant
O
of rupture, yet the results of his analysis seem to agree more nearly with the results of experiment than those obtained by any other theory heretofore proposed.
It is admitted in this theory that a beam will rupture when the stress upon any fibre equals its tenacity, or its resistance to compression, as the case may be. But, on the other hand, when the adjacent fibres are unequally strained, as they are in the case of flexure, it requires a greater stress to produce this strain than it would if the fibres acted independently, ac cording to the previously assumed law. This, Barlow makes evident from the following example : —
If a weight, P, Fig. 27, is suspended on a prismatic bar, BCEF, all the fibres will be equally strained, and hence equally elon gated.
But if the bar ABCD be substituted for the former, and the weight P acts upon a part of the section, as shown in the figure, it is evi dent that all the fibres will not be equally strained, and hence will not be equally elon gated ; and if the force P was just sufficient to rupture the bar FBCE, it will not be sufficient to rupture the bar ABCD, although P acts directly upon the same section, for the cohe-
* Civ. Eng. and Arch. Jour., Vol. xix., p. 9, Barlow says that the strength of a cast-iron rectangular bar, as found from existing theory, cannot be recon-
.d: - 1
^/'
FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 81
sion of the particles along FE will not permit the fibres next to that line to be elongated as much as if the part AFED were removed ; and these fibres will act upon those adjacent, and so on, till they produce an effect upon BC. From this we see that it takes a greater weight than P acting upon the section EC to produce a strain T per unit of section, when the part ADEF is added. It is also evident that if the section of ABCD is twice as great as FBCE, it will not take twice P to rupture the fibres on the side BC.
A phenomenon similar to this takes place in transverse strain. One side is compressed and the other elongated ; arid the fibres less strained aid those which are more strained by virtue of the cohesion which exists between them, and it takes a greater force to cause a strain, T, longitudinally upon the fibres than it would if there were no cohesion.
There is, then, at the time of the rupture of a beam, a tensile strain on the extended fibres, and a compressive strain on the other fibres, and a longitudinal shearing strain between the fibres, due to cohesion. These remarks will, I trust, enable the reader to understand the difference between the " Common Theory " and " Barlow's."
Barlow's Theory consists of the following hypotheses :—
1st. The fibres or elements on the convex side are extended, and on the concave side compressed.
2d. There is a neutral surface, as in the common theory.
3d. The tensive and compressive strains on a fibre are pro portional to the distance of the fibre from the neutral axis.
4th. That in addition to these there is a " Resistance to flexure " or longitudinal shearing strain, which consists of the following principles : —
a. It is a strain in addition to the direct extensive and com pressive forces, and is due to the lateral cohesion of the adja-
ciled with the results of experiment if the neutral axis be at the centre of the sections. He then proceeded to show by experiment that the neutral axis is at the centre, and then remarked that the formula commonly used for a beam
2 Tbd2 supported at the ends and loaded in the middle, or W = ^ — , — did not give
half the actual strength if T is the tenacity of the iron. He then pro ceeds to point out a new element of strength, which he cal|s "Resistance to Flexure."
!~ar-*»— a — — -»— :
82 THE RESISTANCE OF MATERIALS.
cent surfaces of fibres or particles, and to the elastic reaction which ensues when they are unequally strained.
b. It is evenly distributed over the surface, and consequently within the limits of its operation its centre of action will be at the centre of gravity of the compressed or of the ex tended section. This force for solid beams Barlow calls $, and for T °r I sections, or open-built beams, it is easily deduced from the following principle :—
c. It is proportional to and varies with the inequality of strain between the fibres nearest the neutral axis and those most remote.
From this it appears that if d' is the depth of the horizontal flanges of the I section, and dl the distance of the most remote fibre from the neutral axis, then the resistance to flexure of the
d' flanges will be <P -y and similarly for other forms.
5. Sections remain normal to the neutral axis during flex ure.
6. Rupture of solid beams takes place when the strain on a unit of section is T-f- p, or C + £, whichever is smaller, or rather, whichever value is first reached.
V
76. REMARKS UPON THE THEORIES. — For scientific pur poses it is desirable to determine the correct theory of the strength of beams, but the phenomena are so complex that it is not probable that a single general theory can be found which will be applicable to all the irregular forms of beams used in practice. Although Barlow's theory appears plausible, yet according to principle c the resistance to flexure, <?>, can not be uniform over the surface, as stated in principle &, because the proportionality of the elongations and compressions do not continue up to the point of rupture. The common theory is faulty beyond what has already been said in the I section ; for in the upper and lower portions the strains on all the fibres are not proportional to their distances from the neutral axis, to realize which the material should be continuous ; and Barlow's theory is defective in the same case, on account of the peculiar strains upon the fibres at the angles where the parts join. For rupture, then, we can use these theories to ascertain general facts,
FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 83
and make the results safe in practice by using a proper coeffi cient of safety ; but for flexure the common theory is suffi ciently exact if the elastic limit is not passed, and this is for tunate as the conditions of stability should be founded on the elastic properties rather than on the ultimate strength of the material. For the rupture of rectangular beams the common theory will be sufficiently exact if the value of R is used instead of T or C in the formulas.
POSITION OF THE NEUTRAL AXIS. 77. POSITION FOUND EXPERIMENTALLY. According to
Galileo's, Marriotte's, and Leibnitz's theories, the neutral axis is on the surface opposite the side of rupture.
Professor Barlow made the following experiments : — He took a cast-iron beam and drilled holes in its side, into which were fit ted iron pins. He carefully measured the distance between the pins, before and after flexure, by means of a micrometer, and thus found that in solid cast-iron beams bent by a normal pressure the neutral axis passes through the centre of the sections (Civ. Eng. Jour., vol. xix., p. 10). He also made the same kind of an experiment on a solid rectangular wrought-iron beam, and with the same result (Civ. Eng. Jour., vol. xxi., p. 115).
Some years previous to the preceding experiments, he took a bar of malleable iron and cut a trans verse groove in one side, into which he nicely fitted a rectangular key. When it was bent, the fibres on the concave side were compressed, and the groove made narrower, so that the key would no longer pass through, and thus he showed that the neutral axis was between -J and -J- the depth of the beam from the compressed side (Barlow's Strength of Materials, p. 330 ; Jour. Frank Inst., vol. xvi., 2d series, p. 194).
Experiments made at the Conservatoire des Arts et Metiers, in 1856, on double T sections, show that it passes through the centre of the sections (Morin, Resistance des Materiaux, p. 137). And experiments made at the same time on rectangular wooden beams showed that it passed at or very near the centre of gravity of the sections.
In these experiments the elasticity of the material was not seriously damaged by the strains. To render them complete,
' eM^l *-"~^ <w-£-
84 THE RESISTANCE OF MATERIALS.
the strains should have been carried as near to the point of nip ture as possible.
'
78. POSITION DETERMINED ANALYTICALLY* - We knOW
from statics that the algebraic sum of all the forces on each of the rectangular axes must be zero for equilibrium ; hence, if the deflecting forces are normal to the axis of the beam, the sum of the resistances to compression must equal those for tension.
1st. Suppose that the coefficient of elasticity for compression equals that for tension. Then will the compressions and exten sions be equal at equal distances from the neutral axis. In Fig. 28, let Rc be the strain on a unit of fibres most remote from the neutral axis on the compressed side, and dc = the distance of the most remote fibre on the same side ; then,
— L= s = strain at a unit's distance from the neutral axis. dc
-r i A ' */' r V < " -L 'rk « kfr*-* ' A^ ' / / -T1 * Vfc^x i
Let &„ A*,, »„ &c., be the sections of fibres on one side of the
neutral axis, at distances of
^u 2A? 2/3? &c-j fr°m the axis, and
k ', k", k"1 ', &c., and y', ?/", y'" , &c., corresponding quan tities on the other side.
=s (k'y +k"y" +k"'y"f + &c.), or, %1+^a + %,H-&c.-(>?:y+^y/-f^/y// + ifcc.) = 0, or, *ky = 0 ........ (35)
or the neutral axis passes through the centre of gravity of the sections.*
If the resistance to compression is greater than for tension, the neutral axis will be nearer the compressed side than when they are equal.
2. Suppose that the coefficient of elasticity is not the same for tension as for compression.
* The analytical expression for the ordinate to the centre of gravity is
&c. k'y' + k"y" + &c., or — - '
IP
/
/t_ —
FLEXURE AND RUPTURE FROM TRANSVERSE STRESS.
85
Let Fig. 28 represent the beam. Suppose that the sections
CM and EF were parallel be fore deflection. If through K,
O 7
the point where EF intersects the neutral axis, KH is drawn parallel to CM, the ordinates between EF and KH will re present the elongations on one side, and the compressions on the other, for those fibres whose original length was LN-
Let 1= LIST,
A == Jce = the elongation of a fibre at Jc ; p = a pulling or pushing force which would produce A • y = 7$k = distance of any fibre from the neutral axis ; k — section of any fibre ; E^ — coefficient of elasticity for tension ; and Ec = " " compression.
From equation (3) we have,
FIG. 28.
P =
I
But A is directly proportional to its distance from the neutral axis ; hence, if c be a constant quantity, whose value may or may not be known, we shall have A = cy
(36)
Or, if we adopt the same notation as in the preceding case, we shall have for the total force tending to produce extension,
-j' j— viyi
Similarly for compression 0E,
&c.)
(37)
&c.) - (38)
Placing these equal to each other and we have, £,(%, + %2 + %, + &c.) = Ec (k'yr + k"y" H- or, in the language of the integral calculus,
E, v^dydx — Ec s_ ydydx,
(39)
86
THE RESISTANCE OF MATERIALS.
in which y is an ordinate and x an ab.wissa. Equation (30) enables us to find the position when the form of section is known. In most cases, however, the reduction is not easily made. Example. — Suppose the sections are rectangular.
Let b = AC, d = AB,
E<
— = a, and
EC]
y = AE for the superior limit. Then equation (39) becomes
r*> ry rb rd-y
I I ydydx = I j ydydx, »/fl ^ 0 J 0 J 0
which reduced becomes
(40)
a = X), y = 0 a — 0, y = d.
If y is known in equation (40), the ratio of the coefficients may easily be found; for, we have from (40)
(41)
3d. Suppose that the deflecting force is not perpendicular to the axis, and Ec = E, = E.
Let 0 — the angle which P makes with the axis of the beam Fig. 30;
P, = P cos f = the com ponent of P in the direction . — IA of the axis of the beam; . \ -!B-
P, = P sin t = the com ponent of P perpendicular^ to the axis of the beam ;
h = the distance of the neuti al axis from the centre of gravity of the section AB, and K = the transverse section.
Fio. 30.
FLEXURE AND RUPTURE FROM TRANSVERSE STRESS. 87
The whole force of compression equals the whole force of extension, equations (37) and (38).
cE r/Jt cE rr°
.*. P cos 8 + -y- JJ y ay ax = -j-JJ y dy dx
But the ordinate to the centre of gravity is (see foot-note on page 84),
--
.-. P cos 6 = ~Kh L
PZ or A — --pjr- cos d ....... (42)
If 6 = 90°, h = 0 as before found.
If & = 0 there is no neutral axis, for the force coincides with the axis of the beam. The equation would show the same re
sult, if the value of c — - — -, equation (45), were substituted
y i
in the formula, for then p would be infinite, for 0=0, and h becomes infinite.
4th. Let the law of resistance be according to Barlow's theory of flexure, and the deflecting forces normal to the axis of the beam.
Using the same notation as before, also
dl = the distance of the most remote fibre frcan the neutral axis, and
Q = the coefficient of longitudinal shearing stress.
/y y dx — the resistance to shearing for tension,
r°
and <p I y dx = the resistance to shearing for compression,
-y and, proceeding as we did to obtain equation (39), we have
T *2' ' T r°
Examples. — Let the sections be rectangular, b — the breadth, d = the depth. Then (43) becomes
*di = ~ (d - dtf + t (d-d,)
or,
Td .:dl=id- or, dl = - — '
S3
THE RESISTANCE OF MATERIALS.
the former only of which is admissible.
If the section is a double T> as in Fig. 31, with the notation as in the figure, </> will be used in find ing the resistance of the vertical rib, and according
of the lower flange, and
d'
to Article 75, <j>- — -
(t — Ct
— of
the upper flange.
id,
d
FIG. 31.
It appears from these several cases that the neutral axis passes near the centre of gravity in most practical cases, and it will be assumed that it passes through the centre unless other wise stated.
SHEARING STRESS. 89
(
CHAPTER 1Y.
[SHEARING STRESS.
79. GENERAL STATEMENT. — Two kinds of shearing stress are recognized — longitudinal and transverse — both of which have been defined in Article 2. Materials under a variety of cir cumstances are subjected to this stress — such as, rivets in shears ; the rivets in riveted plates; pins and bolts in spliced joints; beams subjected to transverse strains; bars which are twisted; and, in short, all pieces which are subjected to any kind of distor- sive stress in which all parts are not equally strained. In the first examples above enumerated, all parts of the section are supposed to be equally strained. Shearing may take place in detail, as when plates or bars of iron are cut with a pair of shears, when only a small section is operated upon at a time ; or it may be so done as to bring into action the whole section at a time, as in the process of punching holes into metal, where the whole surface of the hole which is made is supposed to resist uniformly.
80. MODULUS OF SHEARING. — The modulus of resistance to shearing is the resistance which the material offers per unit of section to being forced apart when subjected to a shearing stress.
This we call Ss. The resistance for both kinds of shearing has been found to vary directly as the section ; so that if K = the area of the section subjected to this stress the total re sistance will be
K.&.
The value of Ss has been found for several substances, the principal of which are as follows : —
90
THE RESISTANCE OF MATERIALS.
METALS.
Fine'cast steel *
Rivet steel f -
Wrought iron *
Wrought-iron plates punched \
Wrought iron hammered scrap punched §
Cast iron
Copper [
WOOD. With the filres.
White pine Spruce
Hemlock **
Oak
Locust
Ss in Ibs. per square inch.
92,400
64,000 50,000
51,000 to 61,000
44,000 to 52,000
30,000 to 40,000
33,000
480 470 592 540 780 1,200
Across the fibres.
Eed pine 500 to 800
Spruce 600
Larch ft 970 to 1,700
Treenails, English oak J J 3,000 to 5,000
It will be seen from these results that the shearing strength of wrought iron is about the same as its tenacity ; of cast steel it is a little less than its tenacity ; of cast iron it is double its tenacity, and about £ its crushing resistance ; and of copper it is about § its tenacity.
The following table, which gives the results of some experi ments upon punching plate iron, illustrates the law of resistance, and gives the value of 8s for that material.
* Weisbach Mech. and Eng., vol. i., p. 407.
f Kirkaldy's Exp. Inq., p. 71.
i Proc. Inst. Mech. Eng. England, 1858, p. 76.
§ Proc. Inst. Mech. Eng. England, 1858, p. 73.
| Stoney on Strains, vol. ii., p. 284.
^[ Barlow on the Strength of Materials, p. 24.
** Engineering Statics, Gillespie, p. 33.
ft Tredgold's Carpentry, p. 42.
\\ Murray on Shipbuilding Wood and Iron, p. 94.
SHEARING STRESS.
91
TABLE
Of Experiments on Punching Plate Iron. :
Diameter of the |
Thickness of the |
Sectional area cut |
Total pressure on |
Pressure per square |
hole. |
plate. |
through. |
the punch. |
inch of area. |
Inch. |
Inch. |
Square inch. |
Tons. |
Tons. |
0-259 |
0-437 |
0-344 |
8-384 |
24-4 |
0-500 |
0-625 |
0-982 |
26-678 |
27-2 |
0-750 |
0-C25 |
1-472 |
34-768 |
23-6 |
0-875 |
0-875 |
2-405 |
55-500 |
23-1 |
1-000 |
i-ooo |
3142 |
77-170 |
24-6 |
These results give for the value of Ss from 51,000 Ibs. to 61,000 Ibs. The total resistance varies nearly as the cylindrical surface of the hole.
APPLICATIONS.
81. PROBLEM OF A TIE-REAM. — To find the relation be tween the distance AB, Fig. 32, and the depth of a rectangu lar beam below the notch, so that the total shearing strength shall equal the total tenacity.
If.-. |
|
FIG. 32. |
Let h = AB = the distance of the bottom of the notch from the end,
d = the remaining depth of the beam,
k = the section of AB,
K = the section belowf A,A.
T — the modulus of tenacity, and
Ss = the modulus of shearing strength : Then the condition requires that
__
K d 8s
8s
* Proceedings Inst Mech. Eng., 1858, p. 76.
X
'- * j^
92 THE KESISTA^CE OF MATERIALS. V"
« ~ ^ rt_.
T 1°000
F.r,,mpk.— For Oak ~ = - - = 15^ nearly ; hence AB should be about 15fr o* toO
times the remaining depth.
8 £. RIVETED PLATES — Given the diameter of the rivets ; it is required to find tfie distance between them from centre to centre, so that tJie strength of the rivets for a single row sJiall equal the strength of the remaining iron in the plates. Let d = the diameter of the rivets,
c = the distance between them from centre to centre,
k = the section of the rivet,
K = the remaining section of the plate, and
t = the thickness of the plate.
For iron T = &s ; hence, proceeding as above, and we have k ±*d*- 0.785W
x^^z^r1 •••«= —-+«
'.. — If t = i inch, and d = i inch thenc = 1.2854, inch,
and- - =
c
Ift = ± inch, and d = f inch ; then c = 0.8238 and - = 0.544, which ia
c
nearly the value given by Fairbairn for the strength of single riveted plates. See Article 27. To insure this strength the rivet should fit tightly in the hole.
83. LONGITUDINAL, SHEARING IN A RENT BEAM. - "When
a beam is subjected to a transverse stress, we have already seen, Articles 74 and 75, that the fibres are unequally strained, and hence are unequally elongated and compressed. This can not be done without producing a shearing stress between the adjacent elements or fibres, as shown in Figs. 27 and 28. This shearing strain rarely overcomes the cohesion of the particles, but if they were held only by friction it might overcome that. To illustrate this latter idea, suppose several boards from ordinary lumber are placed upon each other, and the whole supported at the ends in any convenient way. When in this condition draw several straight lines across the pile, perpen dicular to the central board. Then deflect the whole by a weight at the middle, or in any other convenient manner, and it will be observed that the lines are no longer straight, but bro ken, and the general direction does not remain normal to the axis or central board. In the experiment the top layer, instead
t si ••
a^i^t^u^f *Ar /tceX OuJis fa { Kt-vv^
^ < <</
^t^e*~ c •:si\*A'}
SHEARING STEESS. 93
of being shortened as in a solid beam, retains its length by overcoming the friction between the top board and the one im mediately under it. The friction, whether it be much or little, represents the shearing stress in a beam.
The elongations and compressions of the fibres in a bent i beam being proportional to their distances from the neutral axis. Article 74, it follows that the shearing stress is evenly dis- ^ tributed over the cross section ; and that, beginning at the ^L axis, the total shearing stress increases uniformly with the dis
tance from the axis. In a beam which is bent by forces pendicular to the axis the shearing resistances to compression - - >4fa and tension form a couple whose arm is the distance between r. the centre of the compressed section and the centre of the , extended section. This resistance in bent beams is generally elastic. The coefficient of elasticity for this case for fibrous bodies has not been determined.
84. TRANSVERSE SHEARING IN BENT BEAMS. Quite
analogous to the preceding case is that of transverse shearing in a beam which is bent by external forces. Referring to Fig. 28, in order that the weight P should be sustained by the horizon tal beam, there is necessarily a vertical force, or a vertical com ponent of forces in the beam, and it is the same at all sections between A and B. This is easily shown by the principles of mechanics.
In order to simplify the problem, suppose that all the bend ing forces are in a plane, and let P, Pn P2, &c., be the bending forces,
F, F1? Fu, &c., be the forces in a beam, each of which is the re sultant of all the forces concurring at that point, "> *u *2> &c-j the angles which P, P1? &c., make with the axis
of x, a, #0 #2, &c., the angles which F, F,, F2, &c., make with the
axis of #, and y an axis perpendicular to x. Then the principles of statics give the following equations : sP cos * + sF cos a = 0, zP sin « + sF sin a = 0,
s(Py cos a, — Yx sin *) -f- s (Fy cos a — F^ sin a) = 0. Let x coincide with the axis of the beam, and let all the forces be vertical ; or * — 90° or^0° ; then ^^
r Ftju UH^y'far'«i ±v'i&A\*jtc£ M &**fai<
t ^ri*^(<LeuJ!Ls
:*
94: THE RESISTANCE OF MATERIALS.
(1) ZF COS a = 0
(2) - z + P + zF sin a = 0
(3) z ± Pa? + zFycosa - zF^sina = 0
The first of these equations shows that the sura of the resist ing forces parallel to the axis is zero ; or that the total compres sion equals the total tension. This is equation (35) in another form. The second shows that the sum of the bending forces equals the sum of the vertical components of the resisting forces. If we let Ss represent a strain as well as a modulus, this equa tion becomes zP = zF sin a = Ss, which is the result sought.
This is as far as it is necessary to carry the investigation in this connection ; but it may be well to show the use of the third equation. If we use a resultant moment for each of the above sets of moments the equation becomes
P 'x' — x" zF sin a = Y'y' , OxM a 14 ffir^l^t^- or, PV — x" Ss = Fy ; but Ss = zP = P', ? ( .-. P' (xr - x") = Fy ;
/3L- f e CM ' /3 * i £ ~~^~~ (l
hence the shearing stress forms a couple wiui the applied force, —or resultant of applied forces. This equation under the form «. 7\^'.
zP^ = zFy
is an essential one in Articles 86 and 136.
Examples of transverse shearing stress. The second of equations (44a), as reduced is,
1. Let a beam be uniformly loaded over its whole length, and supported at its ends as in Fig. 42,
and let w = the load on a foot of length, I = the length of the beam,
V = $wl — the amount sustained at each support, x = any distance from either end ; then
wx = the load on the length .r; and the expression for the shearing stress becomes
which is the equation of a straight line (see Fig. 100). Its value is greatest for x = 0, for which it is \id — $ W ; and is zero for x = 41
2. Suppose the beam is supported at its ends, and has a weight at the middle of its length.
Let P = the weight, and the other notation as before ; then V = |P, and >' 9 — |P - 0 to the middle, and beyond the middle Ss = iP — P = — iP ; and hence it is constant over its whole length.
rT,^
SHEARING STRESS.
95
3. If the beam sustains a uniform load, and also a uniformly increasing load from one end, as in Fig. 98, in which Wi is the total load which
we have &
= V - wx - W^ .' . ^
85. SHEARING RESISTANCE TO TORSION. — When a piece is twisted there is a tendency in one section to slip over the adjacent one, and the corresponding resistance constitutes a shearing strain. It is least at the axis, and increases gradu ally as we proceed from it.
96
FLEXURE.
CHAPTER v.
FLEXURE.
ELASTIC CURVE.
WIIKN a beam is bent by a transverse strain, equilibrium is established between the external and internal forces ; or, to be more specific, all the external forces to the right or left of any transverse section are held in equilibrium by the elastic resistances of the material in the section. When in this state the curve assumed by the neutral axis is called the elastic curve.
To rind the general equa tion of the elastic curve, let " FlG Fig. 33 represent a beam,
fixed at one end, or supported in any manner, and deflected by a weight, P, or by any number of forces. AB is the neutral axis. Take the origin of coordinates at B (or at anj other point on the neutral axis), and let x be horizontal and coincide with the axis of the beam before flexure, y vertical and u perpendicular to the plane of xy. The transverse sections CM and EF being consecutive and parallel before flexure, will meet after flexure, if sufficiently prolonged, in some point, as o. Through N draw KII parallel to CM^hen will ke be the elongation of a fibre whose original length was ck. We have the following notation :—
dx = LN — the distance between consecutive sections,
y = N<? = any ordinate of the surface,
u — "Na or N#',
1) = NN' — the limiting value of u, f (y,u) = equation of the transverse section,
* Several of the more important probbms of this chapter are solved in Arti cles 93 to 103, without the use of the calculus.
FLEXURE. 97
dy du = the transverse section of a fibre, •/, •_ N N ' — limiting value-of ^, p = ON — the radius of curvature at ~N9 p = the force necessary to elongate any fibre an amount equal to A when applied in the direction of its length,
1 = the moment of inertia of the section,
E = the coefficient of elasticity of the material, which is sup posed to be the same for extension and compression,
2 P# = a general expression for the moment of applied forces.
We suppose that the strain is within the elastic limit, and establish the algebraic equation on the condition that the sum of the moments of the applied or deflecting forces equals the sum of ? the moments of the resisting forces. We also assume that the neutral axis coincides with the centre of the transverse sections ' of the beam.
By the similarity of the triangles LON and &Ne, we have
Ne : : LN : ke, or f : y : : dx : A
- - - - (45)
The force necessary to produce this elongation is (see equa tion (3) ),
which becomes, by substituting A from (45), J?.
TJI
p — — ydydu ....... (46)
and the moment of this force is found by multiplying it by y •
E ••- py = -> y' dy du - - (47)
The total moment of all the resisting forces to extension and compression is found by integrating (47) so as to include the whole transverse section, and this will equal the sum of the moments of the applied forces :
p rt r+y r$ /*o
/ / ydydujr I I tfdydu L^/o c/0 i/O •/—¥ -J
THE RESISTANCE OF MATERIALS.
9
,- . - ;T ... , (48)
The quantity ^Ltfdy du, which depends upon the form of
the transverse section and nature of the material, is called the moment of flexure.
The quantity ft y*dy duy when taken between limits so as to
include the whole transverse section, is called the moment of inertia of the surface.* Calling this I and equation (48) becomes
El
j = XPX, --- .* , * : P - :- - (49)
which is the equation of the elastic curve.
An exact solution of equation (49) is not easily obtained in practice, except in a few very simple cases ; but when the deflec tion is small an approximate solution, which is generally com paratively simple and always sufficiently exact, is easily found.
cFydx (Ty
= -p- nearly, since for small deflections
-r- (which is tlie tangent of the angle which the tangent line, to
the curve makes with the axis of x) is small compared with unity, and hence may be omitted. Hence equation (49) becomes
Elg = zP*, ......... (50)
which is the general approximate equation of the neutral axis.
87. THE MOMENT OF INERTIA t of all transverse sec tions of a prismatic beam, is constant, and hence I is constant for prismatic beams.
* See Appendix. f See Appendix.
t
FLEXURE.
For a rectangle, as Fig. 34, we have 1= / / ifdydu = Jil& - (51) •
99
/b (* + \d I y*dydu J-\d
Fio. 34.
For a circle, the origin of coordinates being at the centre ;
/V /^2?r .'.!=/ / r*drd*sm*t
JoT Jo
Fio. 35.
- - (52)
SPECIAL CASES OF PRISMATIC BEAMS.
88. REQUIRED THE EQUATION OF THE NEUTRAL AXIS, AMOUNT OF DEFLECTION, AND SLOPE OF THE CURVE OF A PRISMATIC BEAM, WHEN SLIGHTLY DEFLECTED, AND SUBJECTED TO CERTAIN CONDITIONS AS FOLLOWS I
89. CASE i. — SUPPOSE A HORIZONTAL BEAM is FIXED AT ONE
EXTREMITY AND A WEIGHT P RESTS UPON THE FREE EXTREMITY; REQUIRED THE EQUATION OF THE NEUTRAL AXIS AND THE TOTAL DEFLECTION.
Fio. 37.
FIG. 36.
The beam may be fixed by being imbedded firmly in a wall, as in Fig. 36, or by resting on a fulcrum and having a weight ap plied on the extended part, which is just sufficient to make the curve horizontal over the support, as in Fig. 37. The latter
•
100 THE RESISTANCE OF MATERIALS.
case more nearly realizes the mathematical condition of fixed ness. In either case let
I = AB = the length of the part considered, i — the inclination of the curve at any point, and A = BC — the total deflection.
Take the origin of coordinates at the free end, A; x horizontal, y vertical and positive downwards. • The moment of P on any section distant x from A is Pa;, which is the second member of equation (50) in this case. Hence the equation becomes
• * Elg = P* -.. - -*'V *'\V (53) Multiply both members by the dx and integrate, and we have El^iPrf + C, - - - - - - - -(64)
CttL ,
AVhcn the deflections are small, the length of the beam re mains sensibly constant, hence for the point B, x = l\ and at
the fixed end ~- = 0. Substitute these values in equation (54), ttx
and we find C, = - 4 PF, and (54) d
The integral of equation (55) is
yVflgirf-sroO + c.
But the problem gives y = 0 for x = 0 /. Ca = 0;
•••y = &-
•• $
which is the equation of the neutral axis, and may be discussed like any other algebraic curve.
The greatest slope is at A, to find which make x = 0 in equa tion (55)
P£» •}_
/. tang i (at the free end) = — ^_
' 2LI Sj B
The greatest distance between the curve and the axis of x is
at B, to find which make x — I in equation (56), and we have FT8
A=~ ........ (5T)
^ #*^ /— -
ZZ:.-
£
'
•;••
FLEXURE. 101
•
If y were positive upward, everything else remaining the same, the second member of equation (53) would have been negative, for it is a principle in the differential calculus that when the curve is concave to the axis of x> the second differen tial coefficient and the brdinate must have contrary signs. This would make tang i and A positive. It will be a good exercise for the student to solve this and other problems by taking the origin of coordinates at different points, only keeping so horizontal and y vertical. For instance, take the origin at B ; at C ; at the point where the free end of the beam was before deflection ; at the middle of the beam ; or at any other point.
Example.— If I = 5 ft. , I - 3 in. , d = 8 in.1, E = 1,600,000 Ibs. , and P = 5,000 Ibs. ; required the slope at the free end and at the middle, and the maximum deflection.-X j&]&± c( < ' - ^ £, Z'% /fV^
90. CASE II. SUPPOSE THAT THE BEAM IS FIXED AT ONE END,
IS FREE AT THE OTHER, AND HAS A LOAD UNIFORMLY DISTRIBUTED A ST *J. *
OVER ITS WHOLE LENGTH. — The beam may be fixed as before, as shown in Figs. 38 and 39.
FIG. 38.
FHJ. 39.
Let w = the load on a unit of length. This load may be the weight of the beam, or it may be an additional load. W = wl =.the total load. Take the origin at A.
Then wx = the load on a distance a?, and
\wx* — the moment of this load on a section distant
x from A. Hence equation (50) becomes
EIf=iW - - - (58)
w
i(a(l"4?a'> - <
/ ^rrj^jju
'• \
102 THE RESISTANCE OF MATERIALS.
8EI - / *
In which ~ = 0 fora? = Z /. 6? = — ^y-,
y = 0 fora; = 0 /. C\ = 0, and y = A for x = l.
If the origin of coordinates were at the fixed end, 2 Pa? in the J first case would be P (I — x\ and in the second -= (I — »)*. The
student may reduce these cases and find the constants of inte gration. This case may he further modified for practice by taking the origin of coordinates at- different points.
9 1 . CASE HI. — LET THE BEAM BE FIXED AT ONE END AND A
LOAD UNIFORMLY DISTRIBUTED OVER ITS WHOLE LENGTH, AND A WEIGHT ALSO APPLIED AT THE FREE END. — This is a combination of the two preceding cases, and is represented by Figs. 36 and 37, in which the weight of the beam is the uniform load.
and A- -
hence the deflection of a beam fixed at one end and free at the other, and uniformly loaded, is f as much as for the same weight applied at the free end.
. CASE iv. — LET THE BEAM BE SUPPORTED AT ITS ENDS AND A WEIGHT APPLIED AT ANY POINT. — Figs. 4:0 and 41 represent the case.
Pia. 40.
Fio. 41.
.
Let the reaction of the supports be Y and V,. Take the
FLEXURE. 103
origin at A over the support, and let AD = c = the abscissa of the point of application of P.
Then, Y =^f? P, and Y, = j P.
1 • V , , J
The case is the same as if a beam rested on a support at D, and weights equal to Y and V, were suspended at the ends. For the part AD, equation (50) becomes : —
• ^Elg=-V«=-^P»; - -;-:-. (63)
dy -P(l-c) •'•= ~x- C"
+ (G, = 0); • (65)
in the last of which, y — 0 f or x = 0 /. C, = 0 as indicated.
For the part DB, the origin of coordinates remaining at A, we have :-
El = - V«+P(«-o) - Fc~= -V>(l-x) ; (66)
C". --.. (68)
To find the constants, make x = e in equations (64) and (67) and place them equal to each other ; do the same with (65) and (68) ; and also observe that in (68) y — 0 for x = I. These con ditions establish the three following equations : —
PC' C'-27EI(<!
1 ( ' I T1IE RESISTANCE OF MATERIALS.
From these we find
C"
iv
GEI Hence, for the part AD we have
or, -F- =
6EU
i'
6EIZ
(09)
- '- » (70)
To find the maximum deflection, if c is greater than J7, make ,/,, = 0 in (69) and find x~ then substitute the value thus found in
equation (7()). If c <4/make ^ = 0 in equation (67) and substi tute the value thus found in equation (68).
If D is at the middle of the length, make c = \l in equations (63), (69), and (70): and we have for the curve AD
El' -I IV an
,/ 2 • j V< )
-<
and A = —
- -
(if x = i? in (72) ) - - (73)
The greatest stiv.-s is at the centre, and the nuixinium ino- incnt is found by makini; 25 = II in the >ec..nd member of e<]ua- ti«»nuli. Hence, M<. //- moment
FLEXUKE. 105
111 this case the curve DB is of the same form as AD, but its equation will not be of the same form unless the origin of co ordinates be taken at the other extremity of the beam.
93. CASE v. — SUPPOSE THAT A BEAM is SUPPORTED AT OR
NEAR ITS EXTREMITIES, AND THAT A LOAD IS UNIFORMLY DISTRIB UTED OVER ITS WHOLE LENGTH.
No account is made of the small por tion of the beam (if any) which projects beyond the supports. The distance be tween the supports is the length of the beam which is considered.
Let the notation be the same as in the FIG. 43.
preceding cases.
Then Y= \wl = £W= the weight sustained by each support ; ~Vx = \wlx = the moment of V on any section, as c ; wx is the load on #, and the lever arm of this load is the distance from its centre to the section c, or \x ; hence its moment is ^wz?, and the total moment is the difference of the two moments. Hence equation (50) becomes
-M 3"x fo + <**); (74)
>~ - - (T3)
In these equations ^=0 for x = $, .'. C, =
cLx ,
and y = 0forx = 0, /. <7a = 0. 94. CASE vi. — LET THE BEAM BE SUPPORTED AT ITS ENDS,
UNIFORMLY LOADED, AND ALSO A LOAD MIDWAY BETWEEN THE SUPPORTS.
This case is a combination of the two preceding ones, and
106 THE RESISTANCE OF MATERIALS.
may be represented by Fig. 40 ; for the weight of the beam may be the. uniform load. Hence,
(77)
Experiments on the deflection of beams are generally made in accordance with this case. If the beam be rectangular, we have from equation (51),
I = To^> which in (78) gives
In making an experiment to determine E, the beam is weighed, and that portion of it which is between the supports and unbal anced will be W, and all the quantities except E may be directly measured. If E be known, we may measure or assume all but one of the remaining quantities, and solve the equation to find the remaining quantity, as the following examples will illus trate : —
Examples. — 1. If a rectangular beam, 5 feet long, 3 inches wide, and 3 inches deep, is deflected -fo of an inch by a weight of 3,000 Ibs. applied at the middle ; required the coefficient of elasticity. E = 20,000,000.
2. If b = 2 inches, d = 4 inches, and I = 6 feet, the weight of the beam 144 Ibs., and a weight P=10,000 IDS, placed at the middle of the beam deflects it i an inch ; required E. / Y- /'//, L 1 0. E = 14,580,000 Ibs.
3. A joist, whose length is 16 feet, breadth 2 inches, depth 12 inches, and co efficient of elasticity 1,600,000 Ibs., is deflected £ inch by a weight in the middle ; required the weight ; the weight of the beam being neglected.
Ans. P = 1,562 Ibn.
4. An iron rectangular beam, whose length is 12 feet, breadth Ifc inches, co efficient of elasticity 24,000,000 Iba, has a weight of 10,000 Ibs. suspended at the middle ; required ite depth that the deflection may be ^ of its length.
Ans. 8.8 in.
5. A rectangular wooden beam, 6 inches wide and 30 feet long, is supported at its ends. The coefficient of elasticity is 1,800,000 Ibs. ; the weight of a
FLEXURE. 1 07
cubic foot of the beam is 50 Ibs. ; required the depth that it may deflect 1 inch from its own weight. & j V '£**/€<&£'• •4—
How deep must it be to deflect 4^ of its length ? £ JT faA*£sbjL*t, t 6. A cylindrical beam, whose diameter is 2 inches, length 5 feet, weight of a cubic inch of the material 0.25 lb., is deflected | of an inch by a weight P = 3,000 Ibs. suspended at the middle of the beam. Required the coefficient of elasticity. VK"
To solve this substitute I = far* (equation (52)) in equation (S^. This gives
E = .
7. Required the depth of a rectangular beam which is supported at its ends, and so loaded at the middle that the elongation of the lowest fibre shall equal of its original length. (Good iron may safely be elongated this amount.)
Equations (49) and (73a) become — = £PZ . •. f=^. In this substitute the value of I, equation (51), and it becomes
= __ . By .ito -problem find p = 700d 3P2
_ 2100PZ
8. Required the radius of curvature at the middle point of a wooden beam, when P = 3,000 Ibs. ; I = 10 ft. ; b = 4 in. ; d = 8 in. ; and B rr 1,000,000 Ibs.
9. Let the beam be iron, supported at its ends. Let b = 1 in., d = 2 in., I = 8 ft., E = 20,000,000 Ibs. Required the radius of curvature at the middle when the deflection is £ of an inch. Use eqs. (49) and (73) for P at the middle.
from which it appears that it is independent of the breadth and depth.
10. The centrifugal force caused by a load moving over a deflected beam
may be found from the expression ™—, in which m is the mass of the moving
P
load, v its velocity in feet per second, and p th« radius of curvature of the beam. (See Mechanics.)
11. All these problems may be applied to beams fixed at one end, and P ap plied at the .free end, or for a load uniformly distributed over the whole length, by using the equations under Cases I. , II. , and III.
According to equation (79) the deflection varies as the cube of the length ; and inversely as the breadth and cube of the depth, and directly as the weight applied. A
A^}
108 TIIE RESISTANCE OF MATERIALS.
96. BABI,OW»S THEORY has not, to my knowledge, been
applied to flexure, but it may In- well to inquire what effect it would have. In the common tln-i.ry, it i- a— iimed that the total • is expended in elongating and compressing the fibres j but, according to Barlow's theory, u portion of the, force is absorbed in drawing (so to speak) one fibre over the adjacent one ; hence the deflection should be less by this theory than by the common one.
An experiment made by Mr. Hatcher, England, showed that it was less. (See Mosley's Mechanics and Engineering, p. 514.)
To find E by this theory, <£ will represent & fractional part of the strain (not of the ultimate resistance).
Then <£ fTy dy dx is the moment of resistance to longitudinal
shearing.
Hence we have
Or for a rectangular beam su i 'ported at the ends, we have, by combining the general moments of equations (71) and (74) and using y positive upwards : —
ftP+iW- \w*)* - Ifcfy = El (81)
<j> is /v/v/ xirmll fin* small deflections but, whatever its value, we see that E found by this method will be less than that found by the common theory ; and hence less than that gi\en by the method in Article 7.
97. CASE vii. LET TUT: T.KAM in: rrxrn AT ONI: i-:.\Tin:>nTY,
SUPPORTED AT THE OTIIi:i:, AM) JIAM-J A WKioHT,!', AIM'LIKD AT A. NY PODsT.
A j) e i A
A
i
Fio. 44.
FLEXUKE. 109
The beam may be fixed by being encased in a wall, Fig. 43, or by extending it over a support and suspending a weight on the extended part sufficient to make the beam horizontal over the support, Fig. 44 ; or by resting a beam whose length is %l on three equidistant supports, and having two weights, each equal to P, resting upon it at equal distances from the central
v,
PIG. 45.
support, Fig. 45. In the latter case each half of the beam fulfils the condition of the case.
Let I = AB, Fig. 43, be the part considered, Y = the reaction of the support, nl = AD — the abscissa of P, and f =2 the deflection of the beam at D.
Take the origin at A, the fixed end. We may consider that the curve DB is caused by the reaction of Y, while all the forces at the left of P hold the beam for Y to produce its effect. Similarly the curve AD is produced by the reaction Y and the weight P, while all the forces at the left of them hold the beam. In all cases we may consider that the applied forces on one side of the transverse section are in equilibrium^with the resisting forces of tension and compression in the section. It is well also to observe that the origin of moments is at the centre of the trans verse section, wiiile the origin of coordinates may be at any point.
clfu For the curve AD we have, observing that -~ = 0 for x = 0,
and y = 0 for x = 0 : —
El = -P(nl - x) -V(l - x\ - (82)
110 THE RESISTANCE OF MATERIALS.
For the point D, we have, by making x = nly J = tan« = OT - (n - K)V] ^ ; (85)
y =/= [KP-(K-K)T1 If • (86)
For the curve DB, observe that 3- = tang^ for a; = n£, and
W*C
y =y for a; = wZ, using for their values (85) and (86) in deter mining the constants in the following equations, and we have : —
Elg=-V (?-*), ,--:- - -- >..
//^•S {T$\
Ely = (*» - ^)P»V - V( -- -). - , - (89)
To find the reaction Y, observe that y = 0, for aj = I in (89), and we obtain : —
0 = (3 - n)Pn'? - .•.V = -Jnf(3-n)P. (90)
By substituting this value of Y in the preceding equations, they become completely determined. For the curve AD we shall have : —
El = P[nZ - x- K(3 - n)(l - x}~\ ; (91)
-flOL - (92)
and for the curve DB : —
EI 2? =-*ptt'<8
FLEXURE. Ill
r ~ ~
= [2P~ (3 ~ w) (2fe - ^« • : - (95)
The points of greatest strain in these curves are where the sum of the moments of applied forces is greatest, and this is greatest when the second members of (91) and (94) are greatest. Neither of these expressions have an algebraic maximum, and hence we must find by inspection that value of x which will give the greatest value of the function within the limits of the problem. Equation (91) has two such values, one for x = 0, the other for x = nl, and equation (94) has one such for x = nl, which value will reduce (91) and (94:) to the same value.
Making x = 0 in (91) gives for the moment of maximum strain,
z?x = $Pl[2n-3ri'+ns] ------ (97)
For the moment of strain at P, make x = nl, in (91) or (94), and we have
sPaj = i Pfo1 [-3 + 4n- - <] - - - ' - - (98)
To find where P must be applied so that the strain at the point of application shall be greater than if applied at any other point, we must find the maximum of (98) : —
.-. D» = 0 = - 6n + 12n-f- 4rf - .>*,,, - . - - (99) .•.71 = 0.634 + .......... (100)
or the force must be applied at more than -ffo of the length of the beam from the fixed end. This value of n in (98) gives,
.174
Equation (99) has two values of n, but the other is not within the limits of the problem.
The position of the weight, which will give a maximum strain at the fixed end, is found by making (97) a maximum. Pro ceeding in the usual way, we find : —
"=1±* |/3 = 0.422+ ....... (101)
which in (97) gives, sP# = PI x 0.181 .... (102)
and in (98) tPx =Ylx 0.131 + ^fr~
To find where P must be applied so that the strain at the
1 1 - THE RESISTANCE OF MATERIALS.
point of application will equal the strain at the fixed end, make equations (97) and (98) equal to each other, and find n. This gives,
n = \ 3. 4141+ ; - - - - '-" - -' -I:'^: (103) ( 0. 5858 + .
But n = 0.5858 -f- is the only practical value.
To find where P must be applied so that the curve at that
point shall be horizontal, make -j- = 0, and x = nl in. (95).
(1.
This gives n = 1 3.4141 ( 0.5858
which are the same as the preceding values of n. To find the corresponding deflection, make x = nl, and n = 0.5858 -f, in (93), and we find
P73
A = o.oo98
~ - v-r v - k -I.ijt1 •-. - •
For n < 0.5858, tang i is + )
n > 0.5858, tang i is — V > ^- ;-; - (105) n= 0.5858, tang i is 0 )
Tofiiid the maximum deflection when n = O.G34, make -- = 0 in
dx
(92) or (95), according as the greater deflection is to the right or left of P. But, according to (105), it belongs to the curve AD ; hence use (92). Making n = 0.634 in (92), placing it equal zero, and solving gives,
x = 0. 60457 ; which in (93) gives,
P/3
y = A = 0.00957 ~. ..... (106)
To find where P must be applied so as to give an absolute' maximum deflection ; first find the nl-.-ci-Mi of the point of maximum deflection, when P ifi applied at any point by making
= 0 in (92), and thus find
which, substituted in (93) gives the corresponding maximum
FLEXURE. 113
deflection. Then find that value of n which will make the expression a maximum.
The point of contra-flexure in the curve AD is found by
ctfii making -r4= 0 in (91) (see Dif. Cal.) which gives,
Cvdj
_ 3ln* - n3l - 2nl
: 3n*-n^2~ If n = J-, x = T3TZ.
The second member of (91) is the moment of applied forces, and as it is naught at the point of contra-flexure, it follows -that at that point there is no bending stress, and hence no elongation or compression of the fibres, tut only a transverse shearing stress, the value of which is determined in Article 153.
If a beam rests upon three horizontal equidistant supports, and two weights, each equal P, are placed upon it, one on each side of the central support and equidistant from it, it fulfills the condition of a beam fixed at one end and supported at the other, as before stated, and the amount which each support will sustain for incipient flexure may easily be found from the pre ceding equations.
The three supports will sustain 2P, and the end supports each sustain V = %n\3 - ri)P. (See Eq. (90).)
Hence, the central support sustains
y = 2P-?i'(3-7i)P.
If n = i, Y = A P, and V - ff P.
CASE viii. — LET THE BEAM BE FIXED AT ONE ENDJ SUP
PORTED AT THE OTHER, AND UNIFORMLY LOADED OVER ITS WHOLE LENGTH.
FIG. 46.
Take the origin at A, Figs. 45 and 46, and the notation the same as in the preceding cases, then equation (50) becomes
(108)
,.>
t-
114 THE RESISTANCE OF MATERIALS.
Integrating gives d-ji = gjJ^^+^El ^ ~ ^'
in which _^ = 0 f or x = I, and y = 0 for x = 0. dx
If Y = 0, these equations become the same as those under CASE II.
In equation (110) y is also zero, for x = I ; for which values
This value substituted in equations (108), (100), and (110) gives :—
= -J- wx (4 x — 3Z) ; (112)
:*). (H4)
The point of maximum deflection is found by placing equa tion (113) equal zero and solving for x. This gives
x = **~^l = 0.4215Z ;
• and this in (114) gives
y = A =0.0054^- (115)
There are two maxima strains ; one f or x = I ; the other for x = %L The former in (112) gives
and the latter gives
The point of contra-flexure is found from equation (112) to be at x — %l, at which point the longitudinal strains are zero, and there is only transverse shearing. (See Article 84.)
If the beam is supported by three props, which are in the same horizontal, Fig. 46, then each part is subjected to the same conditions as the single beam in Fig. 45. llence, if W is the load
FLEXURE.
115
on half the beam, each of the end props wrill sustain Y = f "W, (Eq. (Ill) ), and the middle prop will sustain 2W— f W = fW.
Such are the teachings of the " common theory." But the mathematical conditions here imposed are never realized. It is impossible to maintain the props exactly in the same horizontal. As they are elastic they will be compressed, and as the central one will be most .compressed, the tendency will be to relieve the strain on it and throw a greater strain upon the end supports. If the supports be maintained in the same horizontal, the results above deduced will be practically true for very small deflec tions, but will be somewhat modified as the strains approach the breaking limit.
9f>. CASE ix. — LET THE BEAM BE FIXED AT BOTH ENDS AND
A WEIGHT REST UPON IT AT ANY POINT.
To simplify the case, sup pose that the weight rests at the middle of the length.
Let the beam be extend ed over one support and a weight, Pj rest at C, sufficient to make the curve horizontal over the support A. "We have Y = P, + iP.
Let AC = ql.
Then for the curve AD we have,
FIG. 47.
To find P, observe that = 0 for x = \l\
This reduces the preceding equations to the following
:-4a?) - - - - (117)
(118)
and by integrating again, we find : —
116
THE RESISTANCE OF MATERIALS.
For x = $ in (110), y = A = j • (120)
There is no algebraic maximum of the moment of strain as given in the second member of equation (117), but inspection shows that within the limits of the problem the moment is greatest for x = 0 or x = %L These in (117) give the same value, with contrary signs ; hence the moment of greatest strain is
*Px=±fPl (121)
The moment is zero for x = \l.
1OO. CASE x. — LET THE BEAM BE FIXED AT BOTH ENDS AND
A LOAD UNIFORMLY DISTRIBUTED OVER ITS WHOLE LENGTH.
FIG. 48.
The notation being the same as before used, we have
Y = P, + Let ql = AC. The equation of moments is
El =
x)
Integrating, and observing that -j- = 0 for x = 0 ; also y = 0 for x = 0, and we have
But -/- = 0 for x =• I ; also y = 0 for a? = I ;
/.pt=±
FLEXTJKE. 117
which substituted in the previous equations give : —
1 W73
(125)
Making -^-^ — 0 we find for the points of contra-flexure
_ ( 0.788 ~ ( 0.211
.7887Z 2113Z
at which points there is no longitudinal strain, but a transverse shearing strain. (See Article 84.)
The maximum moments are for oj = 0 and x = \l. For x — 0, the second member of Eq. (122) gives TyW7. (126) Force = JZ, « « jj-W^.
Hence the greatest strain is over the support, at which point it is twice as great as at the middle. If W— P, we see that the strain over the support is f as great in this case as in the former.
-X
118 THE RESISTANCE OF MATERIALS.
IOI . RESULTS < OI.M:< i i:i>.
;NO. OF TUB CASK. |
CONDITION OF THE UKAM. |
HOW LOADED. |
GENERAL MOMENT OF |
MAXIMt'M MOMENT OF STRESS. |
ElELATIVB M»MI ST. |
RELATIVE MAX. DKKI.KC- TION OR CO EFFICIENT or I* |
El |
||||||
I. |
LOAD AT FREE END. |
Px. Eq. (53). |
K |
24 |
JP. Eq. (57). |
|
FIXED AT ONE END. |
||||||
II. |
UNIFORM LOAD. |
Eq.™58). |
» |
12 |
iw. Eq. (61). |
|
IV. |
SUPPORTED AT THE ENDS. |
AT THE MIDDLE. |
Eq. (71). |
-,,. |
6 |
^8(73). |
V. |
UNIFORM. |
Eq. (74).' |
im |
3 |
JB?. (76). |
|
VII. |
FIXED AT ONE END AND SUP PORTED AT THE OTHER. |
AT 0.634? FROM FIXED END Eq. (100). |
For AD Eq. (91). ForDB Eq. (94). |
|(2f3-3)Pf. |
4 + |
p — _2 nearly. Bq. (104). |
VIII. |
UNIFORM. |
8 Eq.(\l2). ' |
fWl Eq. (116). |
3 |
Eq. (115). |
|
IX. |
FIXED AT BOTH ENDS. |
AT THE MIDDLE. |
Eq. (117).' |
JB?. (181). |
3 |
P Eq.l(l20). |
W |
W |
|||||
X. |
UNIFORM. |
Eq. (122). |
TVWJ. Eq. (126). |
2 |
Eq. "(125). |
REMARKS. — It will be seen that the greatest strains in the 1st and 2d cases are as 2 to 1 ; and the same ratio holds in the 4th and 5th cases ; but in the 9th and 10th the ratio is as 3 to 2. This is peculiar, and further remarks are made upon
FLEXURE. 119
it in Article 119. The maximum strains in Cases VII. and VIII. do not occur at the points of maximum deflection.
Although the moment in the 1st case is to that in the 2d as 2 to 1, yet the deflections are as 8 to 3 ; and in the 4th and 5th cases the deflections are as 8 to 5.
A comparison of Cases IV. and IX. shows the advantage of fixing the ends of the beam. The same remark applies .to Cases V. and X. In the former cases the strain is only one-half as great when the beam is fixed at the ends as when it is supported, and in the latter two-thirds as great.
o
Other interesting results may be seen by examining the table.
1O3. MODIFICATION OF THE FORMULAS FOR DEFLEC TION. — It will be observed that the general form of the expres sion for the maximum deflection of rectangular beams is
PZ3 A = constant x
Ebd3
Prof. "W. A. Norton, of New Haven, Ct., has made experi ments to test the correctness of this expression. (See Van fflos- trancPs Eclectic Engineering Magazine, vol. 3, page 70.) Ac cording to his experiments,^^ beams supported at their ends and loaded at the middle, the expression should be
p/3
For the pine sticks which he used he found the mean value of C to be
C = 0.0000094. £
A consideration of transverse shearing stress, in combina tion with the stretching and compressing of the fibres, leads to an expression of this form. For, as we have before seen, the strain is evenly distributed over the whole transverse section, and hence the deflection will vary inversely as the area, or as bd ; it is also uniform over its whole length, and equal £P (see Example 2, Article 84) ; and hence the amount of deflection will vary as JP; and the total deflection at the middle will
120 TIIE RESISTANCE OF MATERIALS.
evidently vary as the length ; or, in this case, as %l. Hence,
ip \i the total deflection due to transverse shearing is, C — j^- =
C PZ
2 -y-r, which is the same form as that given by Professor Norton.
The same form of expression is also reached, in a more circuitous way, by Weisbach, in his Mechanics of Engineering, 4th edi tion, vol. 1, page 522 of the recent American edition.
In Professor Norton's formula \ C is the reciprocal of the co efficient of elasticity to transverse shearing of white pine / hence the coefficient is 425,531 pounds. The mean value of E in the above experiments was found to be 1,427,965 Ibs. ; and hence, in this case, the reciprocal of ^C is a little more than J^ of E. Weisbach, in the reference above given, says : " The coefficient of elasticity for transverse shearing is generally assumed to be equal to £E."
If the load is uniformly distributed over the whole length, the shearing stress on any section, distant x from the end, is £ wl — wx. (See Example 1, Article 84.) Hence the deflection for an element of length of a rectangular beam due to this cause, is
and for a distance x this becomes by simple integration, C -*
and for half the length, make x = $1, and the expression be comes
t* wr
from which we see that the same load, distributed uniformly over the whole length, produces half as much deflection due to transverse shearing as the same load concentrated at the middle. Equation (76) when corrected for this effect becomes
°r'A = from which we see that if the depth be constant the deflection
FLEXURE. 121
due to transverse shearing will be more apparent compared with that due to the other cause, as the piece is shorter. If the piece is very long, the effect due to C is comparatively small. If
p- = ^E, as assumed by Weisbach, the deflection becomes
Wl
A =
SEbd
If I — d the quantity in [ ] becomes f + 12 ; or the effect due to C is ^8. of that due to E.
If I = 20d, the effect due to C is -fa nearly of that due to E.
1O4. ADDITIONAL PROBLEMS WHICH ARE PURPOSELY LEFT UNSOLVED.
1. Suppose that a beam is supported at its extremities, and has two forces at any point between. In this case the curve between the support and the nearest force will have one equa tion ; the curve between the forces another ; and the remaining part a third.
2. In the preceding case, if the forces are equal and equidis tant from the supports, the curve between the forces wrill be the arc of a circle.
3. Suppose that the beam is uniformly loaded and rests on four supports.
4. Suppose that the beam is supported at its extremities and has a load uniformly increasing from one support to the other.
5. Suppose that the beam is uniformly loaded over any por tion of its length.
6. Suppose that ft has forces applied at various points. These problems will suggest many others.
Y. Suppose that a beam is supported at several points, and loaded uniformly over its whole length.
Let "W = the weight between each pair of supports,
Yn Y3, Y8, &c., be the reactions of the supports, counting
from one end, and let the distances between the supports be equal.
122 THE RESISTANCE OF MATERIALS.
Then we have : —
No. of Sup- ! '•'-• 2 |
v, |
TT |
V9 |
V,, |
||||||
iW |
*w |
Frac |
tional |
parts |
of W. |
|||||
3 |
1 |
ia 8 |
1 |
|||||||
4 |
A |
TO |
it |
A |
||||||
5 |
H |
11 |
H |
II |
H |
|||||
6 |
it |
II |
H |
H |
H |
H |
||||
7 |
to |
H4& |
iSf |
ttf |
W |
HI |
to |
|||
8 |
A6/ |
H* |
Hi |
HI |
HI |
HI |
in |
fA |
||
9 |
Hl |
m |
HI |
181 |
181 |
111 |
HI |
IH |
IH |
|
10 |
m |
m |
H* |
III |
HI |
W |
HI |
IB |
IH |
209 |
If the beams and props were perfectly rigid, all but the end ones would sustain W, and the end ones each -J W.
It may be shown that, for any number of equidistant props, the inclination at the end may be found from the equation
which for 10 props becomes
._153 - 265 25EI ;
and the maximum deflection for any number of props is
A = '" ~™
1O5. BEAMS OF VARIABLE SECTIONS.
For these I is variable, and its value must be substituted in
FLEXURE.
123
equation (50) before the integration can be performed. As an
example, let the beam be
fixed at one extremity, and
a weight, P, be suspended at
the free extremity, Fig. 50.
Let the breadth be constant,
and the longitudinal vertical
sections be a parabola. Then
all the transverse sections
will be rectangles.
Let I = the length, F'G- 50-
~b =. the breadth, and d = the depth at the fixed extremity.
If y is the whole variable depth at any point, we have, from the equation of the parabola,
cF —pi, :. p = 27, in which p is the parameter
(-J-?/)2 = of the parabola.
From equation (51) we have
(127)
I = -fjby*, m which substitute ?/, from equation (127), and we
— ** - ri2N
a Wl (M&)
The equation of moments is, see equation (50),
cC'y El -j-i — P^, in which substitute I, from equation (128), and
we have
12PZ1
-x~~b
Multiply by the dx and integrate, observing that j- = Of or a? — I and we have dy_
124 THE RESISTANCE OF MATERIALS.
Integrating again gives
y is zero for x — 0.
y = A for x = I
Eb(F~
If, in equation (57), we substitute I = comes
- - (129) (Eq. (51)), it be-
which is one-half that of (129) ; hence the deflection of a prismatic beam is one-half that of a parabolic beam of the same length, breadth, and greatest depth, when fixed at one end and free at the other, and has the same weight suspended at the free end.
In a similar manner the equation of the curve may be found for any other form of beam, if the law of increase or decrease of section is known. Several examples may be made of beams of uniform strength, which will be given in Chapter VII.
1O6. BEAMS SUBJECTED TO OBLIQUE .STRAINS.— Let the
beam be prismatic, fixed at one end, a fid support a weight, P, at the free end; the beam being so inclined that the direction of the force shall make an obtuse angle with the axis of the beam, as in Fig. 51.
Let PI = P sin 6 = component of P per pendicular to the axis of the beam, and P3 = P cos 0 = component parallel to
the axis of the beam. Take the origin at the free end, the axis of x being parallel to the axis of the beam? and y perpendicular to it.
Then equation (50) becomes Fia. 51.
or, £-t = —p-x + q-y P,
(130)
in which p* = ~ ; and q* = — . The complete integral of (130) is (see Appen-
El
Er
dix III.)
FLEXURE.
Q* „ — «
125
* e
+ —
The conditions of the problem give dy
= 0 for x = I-
u*o
y — 0 for x = 0 ; and these combined with the preceding equation
give : —
gi
ql
0 = ft + <7a ;
From which Ci and C2 may be found, and the equation becomes completely known.
We also have y = A for x — £;
ql —ql p3
*"¥ '
Next, suppose that the force makes an acute angle with the axis of the beam, as in Fig. 52.
For the sake of variety, take the origin at A, the fixed end, x, still coin ciding with the axis of the beam before flexure. Using the same notation as in the preceding and other cases, we have
-y)
(131)
The complete integral is vs
A -y =
-4-(Z-z) (132)
FIG. 52.
in which A and B are arbitrary constants. From the problem we have y — 0 f or x = 0 ;
-/-=0foraj = 0; and
dx
y — A f or x = I ;
by means of which the equation becomes completely known.
From these examples we see how easily the problem is com plicated. One difficulty in applying these cases in practice is in determining the value of I. Before it can be determined, the position of the neutral axis must be known. According to Article 78, 3d case, it appears that the neutral axis does not coincide with the axis of the beam. Indeed, according to the same article, it is not parallel to the axis, and hence I is varia ble, and the equations above are only a secondary approxima tion ; the first approximation being made in establishing equa-
12G
THE RESISTANCE OF MATERIALS.
tion (50), and the next one in assuming I constant. All writ ers, within the author's acquaintance, who have investigated this and similar cases, beginning with Navier, have assumed that I is constant, and that the neutral axis coincides with the axis of the beam. These assumptions may be admissible in any prac tical case were extreme accuracy is not desired. Many other practical examples might be given, the solutions of which are more difficult than the preceding; but enough have been given to illustrate the methods.
FIG.
1O7. FLEXURE OF COLUMNS.
—If a weight rests upon the axis of a perfectly symmetrical and homogeneous column, we see no reason why it should bend it ; but in practice we know that it will bend, however symmetrical and homogeneous it may be, and however carefully the weight may be placed upon it. If the weight be small, the deflection may not be visible to the unaided eye. If the weight is not so heavy as to crush the column, an equilibrium will be established between the weight and the elastic resistances within the beam. Let the col umn rest upon a horizontal plane, and the weight P on the upper end be vertically over the lower end. Take the origin of coor dinates at the lower end of the column, Fig. 53. x being ver-
O / t">
tical, and y horizontal. They must be so taken here, because x was assumed to coincide with the axis of the beam when equa tion (50) was established. Then y being the ordinate to any point of the axis of the column after flexure, the moment of P is Py, which is negative in reference to the moment of re sisting forces, because the curve is concave to the axis of x, in which case the ordinate and second differential coefficient must have contrary signs (Dif. Cal.). Hence we have,
(133)
FLEXURE. 127
Multiply by dy and integrate (observing that dx is constant), and find
-v- = 0 for y = A = the maximum deflection. These
PA"
values in the preceding equation give C( = -^y, which being substituted in the same equation and reduced gives
dy /**=?
But y = 0 for a? = 0 /. <72 = 0. Hence the preceding gives y = A sin ^^ a> (134)
But y — 0 for a? = £ Therefore, if n is an integer, these val- ues reduce (134) to
P
This value of P reduces (134) to
y=A6m«,r7 v Library.
which is the equation of the curve. It is dependent" the length of the column and the maximum deflectionTTSF n = 1, the curve is represented by a, Fig. 54 ; if n = 2, by 5 ; if n = 3, by c.
If n = 1, equation (135) becomes
*"
P = ?EI - - (136)
which is the formula to be used in practice. We see that the
128 THE RESISTANCE OF MATERIALS.
resistance is independent of the deflection. If the column is cylindrical, I = i * r* (see equation (52) ) ;
/. P = ^ (137)
hence the resistance varies as the fourth power of the radius (or diameter), and inversely as the square of the length. If the column is square, I = ^b* (equation (51)),
*-'E V •••P== 12-Xjr (133)
These formulas, according to Xavier * and "Weisbach,t should be used only when the length is 20 times the diameter for cylindrical columns, or 20 times the least thickness for rectangular columns ; and Kavier says that for safety only -^ of the calcu lated weight should be used in case of wood, and J to -J- in case of iron ; but Weisbach says they should have a twenty-fold security.
Examples 1. — What must be the diameter of a cast-iron column, whose length is 12 feet, to sustain a weight of 30 tons (of 2,000 Ibs. each) ; E = 1C. 000, 000 Ibs. ; and factor of safety ^-0-. Ans. d = 7.52 in.
2. If the column be square and the data the same as in the preceding exam ple, equation (138) gives
=</"
12 X 00,000 X (12 X 12)-' X 20 _ „ . ,
= G.G inches.
(81416)* X 16,000,000 In the analysis of this problem I have followed the method of Navier ; but as it is well known that the results are not relied upon by practical men, I have given only one case. For other cases see Appendix III. There are some reasons for the failure of the theory which are quite evident, but it is not easy to remedy them ; and for this reason the empyrical formulas of Article 52 are much more satisfactory. It will be observed that the law of strength, as given in the formulas in that article, are the same as those given in equations (137) and (138) for .wooden columns, and nearly the same as for iron ones. The chief difference is in the coefficients, or constant factors. In the analysis it was as sumed that the neutral axis coincides with the axis of the beam, but it is possible for the whole column to be compressed, although much more on the concave than on the convex side, in which
* Navier, Resume des Le9ons, 1839, p. 204.
f Weisbach's Mechanics and Engineering. Vol. 1, p. 219. 1st Am. ed.
FLEXURE. 129
case the neutral axis would be ideal, having its position entirely outside the beam on the convex side. In this case, if the ideal axis is parallel to the axis of the beam, it does not affect the form of equation (136), but it does affect the value of I ; and hence the values of equations (137) and (138). The problem of the flexure of columns is then more interesting as an analytical one than profitable as a practical one.
GRAPHICAL METHOD.
1O8. THE GRAPHICAL METHOD consists in representing quantities by geometrical magnitudes, and reasoning upon them, with or without the aid of algebraic symbols. This method has some advantages over purely analytical processes; for by it many problems which involve the spirit of the Differential and Integral Calculus may be solved without a knowledge of the processes used in those branches of mathematics ; and in some of the more elementary problems, in which the spirit of the Calculus is not involved, the quantities may be directly presented to the eye, and hence the solutions may be more easily retained. It is distinguished, in this connection, from pure geometry by being applied to problems which involve mechanical principles, and to use it profitably in such cases requires a knowledge of the elementary principles of mechanics as well as of geometry.
But graphical methods are generally special, and often re quire peculiar treatment and much skill in their management. It is not so powerful a mode of analysis as the analytical one, and those who have sufficient knowledge of mathematics to use the latter will rarely resort to the former, unless it be to illus trate a principle or demonstrate a problem for those who cannot use the higher mathematics. A few examples will now be given to illustrate the method.
1O9. GENERAL PROBLEM OF THE DEFLECTION OF
BEAMS. — To find the total deflection of a prismatic beam which is bent by a force acting normal to the axis of the beam without the aid of the Calculus.
Let a beam, AB, Fig. 55, be bent by a force, P, in which case the fibres on the convex side will be elongated, and those on the concave side will be compressed. Let AB be the 9
130
TIIE RESISTANCE OF MATERIALS.
axis.
ON:
Take two sections normal to the neutral axis at L and N, which are in definitely near each other. These, if prolonged, will meet at some point as O. Draw KN parallel to LO. Then will ke, = A, be the distance between KN and EN at &, and is the elonga tion of the fibre at k. Let eN = y, then from the similar triangles fcNe and LON we have
LN LN
FIG. 55.
LN: fo = A =
ON
If, now, we conceive that a force p, acting in the direction of the fibres, or, which is the same thing, acting parallel to the axis of the beam, is applied at k to elongate a single fibre, we have, from equation (3) and the preceding one,
. ke E — '~ ' = A
in which A# is the transverse section of the fibre. As the sec tion turns about N on the neutral axis, the moment of this force is
E . T-?
which is found by multiplying the force by the perpendicu lar y.
This is the moment of a force which is sufficient to elongate or compress any fibre whose original length was LN, an amount equal to the distance between the planes KN and EN measured on the fibre or fibre prolonged. Hence, the sum of all the moments of the resisting forces is
E
in which z denotes summation ; and in the first member means that the sum of the moments of all the forces which elongate and compress the fibres is to be taken ; and in the second mem-
FLEXURE. 131
her it means that the sum of all the quantities if &a included in the transverse section is to be taken. The quantity, Sy1 A# is called the moment of inertia, which call I.
But the sum of the moments of the resisting forces equals the sum of the moments of the applied forces. Calling the latter sPX, in which X is the arm of the force P, and we have
In the figure draw U tangent to the neutral axis at L, and N0 tangent at N. The distance ab, intercepted by those tan gents on the vertical through A, is the deflection at A due to the curvature between L and N. As ~LN is indefinitely short, it may be considered a straight line, and equal x ; and U — LC very nearly for small deflections ; and LC = X. (L stands for two points.)
By the triangles OLX and aU, considered similar, we have
OJNT :x :: U:at
in which substitute ON from equation (139) and we have , ~Xx sPX
which is sufficiently exact for small deflections. If, now, tan gents be drawn at every point of the curve AB, they will divide the line AC into an infinite number of small parts, the sum of which will equal the line AC, the total deflection. But the expression for the value of each of these small spaces will be of the same form as that given above for ab, in which P, E, and I are constant.
This is as far as we can proceed with the general solution. We will now consider
PARTICULAK CASES.
114). CASE 1.— L.ET THE BEAM BE FIXED AT ONE END, AND A LOAD, P, BE APPL.IED AT THE FIIEE END. - TlllS is a
part of Case I., page 99, and Fig. 37 is applicable. The moment
132 THE RESISTANCE OF MATERIALS.
of P, in reference to any point on the axis, is PX. Hence is simply PX, which, substituted in equation (140), gives
= £-IsX'-* <U1>
This equation has been deduced directly from the figure. It now remains to find the sum of all the values of X2#, which result from giving to X all possible values from X = 0 to X = I*. To do this, construct a figure some property of which represents the expression, but which has not necessarily any relation to the problem which is being solved. If X be used as a linear quantity, Xa may be an area and X3«E will be a small volume. These conditions are represented by a pyramid, Fig. 56, in which
AB = I = the altitude, and the base BCDE is a square, whose sides, BC and CD, each = I. Let Icde be a section parallel to the base, and make another section infinitely near it, and call the distance between the two sections x.
Then Al = X = Ic = cd,
Xa = area bcde, and Xa« = the volume of the la mina bcde,
which is the expression sought. The sum ^ f~~ ~~s of all the laminae of the pyramid which Fl°- 5f)-
are parallel to the base is limited by the volume of the pyramid, and this equals the value of the expression sXa# between the limits 0 and I. The volume of the pyramid is the area of the base (= F) multiplied by one-third the altitude (-£/), or ^Z3, which is the value sought.
Hence, AC = ^g-j
wliich is the same as equation (57).
The value of Xa# may also be found by statical moments as follows : — Let ABC, Fig. 57, be a triangle, whose thick-
* This by the calculus becomes / x" dx — {^
FLEXUKE. 133
ness is unity, and which is acted upon by gravity (or any other system of parallel forces which is the same c
on each unit of the body). Take an infinitely thin strip, be, perpendicular to the base, and let AB = I = BC,
Ab = X = be, and
p — the weight of a unit of volume. Then X# — the area of the infinitely thin strip be, and
= the weight of the strip be, and
*x — the moment of the strip, when A is taken as the origin of moments. If the weight of a unit of volume be taken as a unit, the moment becomes X2a?, which is the quantity sought, and the value of sX2^ from 0 to Zis the moment of the whole triangle ABC. Its area is -JZ2, and its centre of gravity \l to the right of A. Hence the moment is \l? as before found.*
111. CASE II. — LET THE BEAM BE FIXED AT ONE END, AND UNIFORIttLY LOADED OVER ITS WHOLE LENGTH. -
This is the same as a part of Case II., page 101, and Fig. 39 is applicable.
Let X be measured from the free end, and w — the load on a unit of length ; then ^X = the load on a length X, and •J-X = the distance of the centre of gravity of the load from
the section which is considered.
Hence the moment is ^X2, which equals sPX, and equation (140) becomes
, w ao = ^pi-j X a?, and
w x=l AC = FT 2X3^ = the total deflection.
To find the value of sX3#, observe, in Fig 56, that Xa# is the volume of the lamina bcde, and this multiplied by the alti tude of A— bede, which is X, gives X3#, the expression sought. Hence the sum sought is the volume of the pyramid A— BCDE,
X = I
* This may be written 2 XQ# = J3Z3.
134: TTIE RESISTANCE OF MATERIALS.
multiplied by the distance of the centre of gravity of the pyra mid from the apex ; or,
where TV is the total load on the beam.
112. CASE III. — LET THE BEAItt BE SUPPORTED AT ITS ENDS AND LOADED AT THE MIDDLE BY A WEIGHT P, as in
Fig. 40. The reaction of each support is £P, and the moment is J-PX, and equation (140) becomes
oJ^^jX's.
But in this case the greatest deflection is at the middle, and the limits of zX2# are 0 and \l. Hence, in Fig. 56, let the altitude of the pyramid be ££, and each side of the base also £/, and the volume will be
±lx$lx I of
\C-
~
48E.I' which is the same as equation (73).
113. CASE IV.— LET THE BEAItt BE SUPPORTED AT ITS ENDS AND UNIFORMLY LOADED, AS IN FIG. 42.
w being the load on a unit of length, the reaction of each support is %wl, and its moment at any point of the beam is $wHi. On the length X there is a load 10X, the centre of which is at J-X from the point considered ; hence its moment is 3, and the total moment is the difference of these moments ;
and equation (140) becomes
o$= ~
*jji<. _
and the total deflection at the middle is,
FLEXURE. 135
The values of the terms within the parentheses have already been found, and by subtracting them we have
__ ~384 E.I
114. REMARK ABOUT OTHER CASES. - This method,
which appears so simple in these cases, unfortunately becomes very complex in many other cases, and in some it is quite pow erless. To solve the 9th and 10th cases, pages 115 and 116, ne cessitates an expression for the inclination of the curve, so that the condition of its being horizontal over the support may be imposed upon the analysis. But the 9th case may be easily solved if we find by any process that the weight which must be suspended at the outer end of the beam to make it horizontal over the support is |P£ divided by AC,* Fig. 4:7. For, the reaction of the support is JP -f- P ;
- (fp+P,)X
KI
p
and the deflection at the centre - i (teXas — 4sX2a) taken
between the limits 0 and \l.
The part sX# is the area of a triangle whose base and alti tude are each JZ, /. sX# = %F, and sX2# between the limits 0
P I3 and is
All these expressions contain I, the value of which remains to be found by the graphical method.
* This " AC " refers to Fig. 47. f This " AC " refers to Fig. 55. ,
136
THE RESISTANCE OF MATERIALS.
rectangle
115. MOMENT OF INERTIA OF A RECTANGLE. — Required
the moment of inertia of a rectangle about one end as an oasis.
Let ABCD, Fig. 58, be a rectangle. Make BG perpendicu lar to and equal AB, and complete the wedge G - ABCD.
Let A« = the area of a very small sur face at E, and
y = AE = EF, then y A« = the volume of a very small prism EF, and this multi plied by y gives
y*A# — the moment of inertia of the elementary area at E, which is also the statical moment of the prism EF, and ^y* Act = I = the moment of inertia of the
ABCD.
Hence the moment of inertia of the rectangle is represented by the statical moment of the wedge G — ABCD. If AB = d = BG, and AD = 6, then the volume of the wedge is
Id x $d = W and the moment = $<F x fi = $hF - - - - (143)
If the axis of moments passes through the centre of the rec tangle, and parallel to one end, we have BE = GB = \d in Fig. 59. Hence the moment of inertia of the rectangle =
2x5xi^xi^x|of^= TV Id* which is the same as equation (50).
116. THE MOMENT OF INERTIA OF A TRIANGLE about
an axis parallel to the base and passing through the vertex is, in a similar way, the statical moment of the pyramid ABCDE. Fig. 60.
Let b = CB = base of the triangle, and
d = AB = BD = CE = altitude of the triangle and pyramid and sides of the base of the pyramid. The volume of the pyramid = J bcF.
The centre of gravity is J/Z from the apex, consequently the statical moment is J bcF x $d = tyd3.
FLEXURE.
137
But in a triangular beam the neutral axis passes through the centre of gravity of the triangle, and it is desirable to find the moment of inertia about an axis which passes through the centre and parallel to the base.
This may be done as in the preceding Article ; but it may be
FIG. 59.
FIG. 60.
more easily done by using the formula of reduction, which is a? follows: — The moment of inertia of a figure about an axis passing through its centre equals the moment of inertia about an axis parallel to it, minus the area of the figure multiplied by the square of the distance between the axes. (See Appendix III.) This ffives for the moment of inertia of a triangle about an
O O
axis passing through its centre and parallel to the base
\bd* - \bd x (W = ^V ^ (143)
117. THE MOMENT OF INERTIA OF A CIRCLE maybe
represented in the same way, but it is not easy to find the vol
ume of the wedge, or the position of its centre of gravity,
except by analysis which is more tedious than that required to
find the moment directly, as was done in equation (51). But it
may be found practically, by those who can only perform
multiplication, as follows : — Make a w^edge-shaped piece out of
wood, or plaster of Paris, or other convenient material, the base
of which is the semicircle required, and the altitude is the
radius of the circle ; then find its volume by im
mersing it in a liquid and measuring the amount of
water displaced. Then determine the distance of
the centre of gravity of the wedge from the centre
of the circle by balancing it on a knife edge, holding
the edge of the knife under the base of the wedge,
and parallel to the edge, ab, of the wedge, keeping the side
138
THE RESISTANCE OF MATERIALS.
vertical, and measuring the distance between the edge ab and the line of support. Then the statical moment is the product of the volume multiplied by the horizontal distance of the centre from the edge. Its value for the whole circle, or for both wedges, is \^r\
There are, however, many methods of calculating the moment of inertia of a circle without using the Calculus. The following method, which the author has devised, appears as simple as any of the known methods : —
The moment of inertia of a circle is the same about all its diameters. Hence the moment about X in the figure, plus the moment about Y, equals twice the moment about X. The distance to any point A is P, and equals ^ tf + if ; or f — x* + if ; and if ~Ati be an elementary area, as be fore, we have
Fio606
' + V Atf#a = V A« f,
the latter of which is called the polar moment of inertia, in reference to an axis perpendicular to the plane of the circle, and passing through its centre C. To find the value of J A# pa, take a triangle whose base and altitude are each equal to r, the radius of the circle, and revolve it about the axis through C, and construct an infinitely small prism on the element A# as a base. We have f = CA = AB, Fig. 60 c.
A« p = volume of the small prism AB.
A = /\af=t\\e moment of AB, the form of the quantity sought.
f is the product of the volume of the solid generated by the triangle, multiplied by the abscissa of its'centre of gravity from C. The solid is what re mains of a cylinder after a cone has been taken out of it, the base of the cone being the upper base of the cylinder, and the apex of which is at the centre of the base of the cylinder. Hence the volume of the solid is the volume of the cylinder, less the volume of the cone ; or vr* x r — w/1" x lr = far*.
Hence
FIG. 60 c.
FLEXURE. 139
If now the solid be divided into an infinite number of pieces, by planes which pass through its axis, each small solid will be a pyramid, having its vertex at C, and the abscissa to the centre of gravity of each is %r from C. Hence we finally have
equals 2
3 — *T4 which
118. M01WENT OF INERTIA OF OTHER SURFACES. - The
general method indicated in the preceding articles is applicable to surfaces of any character, and with careful manipulation ap proximations may be made which will be very nearly correct, and, as we have seen above, in some cases exact formulas may be found.
140
THE RESISTANCE OF MATERIALS.
CHAPTER VI.
TRANSVERSE STRENGTH.
119. STRENGTH OF RECTANGULAR HE A. US. The tll6-
ories which have been advanced from time to time to explain the mechanical action of the fibres, have been already given in Chapter IV. Both the common theory, and Barlow's theory of " the resistance to flexure " will be considered in this chap ter.
First, consider the common theory, according to which the neutral axis passes through the centre of gravity of the trans verse sections, and the strain upon the fibres is directly propor tional to their distance from the neutral axis.
Continuing the use of the geometrical method, let Fig. 61 represent a rectangular beam which is strained by a force P applied at any point. Let de be on the neutral axis, and al) repre sent the strain upon the lowest fibre. Pass a plane, de—cb, and the wedge so cut off represents the strains on the lower side, and the similar wedge on the other side represents the strains on the upper side.
Let R — the strain upon a FIO. «i.
unit of fibres most remote from the neutral axis on the side which first ruptures, on the hypothesis that all the fibres of the unit are equally strained, and b = the breadth and d = the depth of the beam.
Let ab = R ; then, the total resistance to compression = £R£ x \d = JR&/, = the volume of the lower wedge ; and the mo-
TRANSVERSE STRENGTH. 141
ment of resistance is this value multiplied by the distance of the centre of gravity of the wedge from de, which is f of \d = \d\ consequently the 'moment is
and as the moment of resistance to tension is the same, the total moment of resistance is
iKM, •=-£.( (145)
which equals the moment of the applied or bending forces.
If the beam be fixed at one end and loaded by a weight, P, at the free end, w^e have for the dangerous section, or that most liable to break,
PI = ±Rbd\
In rectangular beams the dangerous section will be where the sum of the moments of stresses is greatest, the maximum values of which for a few cases are given in a table on page 118. Using those values, and placing them equal to I'RbcF, and we have for solid rectangular beams at the dangerous section, the following formulas : —
FOR A BEAM FIXED AT ONE END AND A LOAD, P, AT THE FREE
END; PZ = |K&F; (146)
AND FOR AN UNIFORM LOAD ; $Wl — ^Rbd* - - (147)
FOR A BEAM SUPPORTED AT ITS ENDS AND A LOAD, P, AT THE
MIDDLE ; %Pl = £R&? ; - (148)
AND FOR AN UNIFORM LOAD ; $Wl. = ^Rbd* ; - (149)
AND FOR A LOAD AT THE MIDDLE, AND ALSO AN UNIFORM LOAD ;
t(2P + W)l = iRfof (150)
FOR A BEAM FIXED AT BOTH ENDS AND A LOAD, P, AT THE
MIDDLE; £PZ = £Rfof; - - (151)
AND FOR AN UNIFORM LOAD, END SECTION ; -^Wl = ^~RM' ; (152)
MIDDLE SECTION ; &WI — £R&F - - (153)
These expressions show that in solid rectangular beams the strength varies as the breadth and square of the depth, and hence breadth should be sacrificed for depth. In all the cases, except for a beam fixed at the ends, it appears that a beam will support twice as much if the load be uniformly distributed over the whole length as if it be concentrated at the middle of the length. The case in which a beam is fixed at both ends and loaded at the middle has given rise to considerable discussion,
142 THE RESISTANCE OF MATERIALS.
for it is found by experiment that a beam whose ends are fixed in walls or masonry will not sustain as much as is indicated by the formula, and also that it requires considerably more load to break it at the ends than at the middle, but the analysis shows that it is equally liable to break at the ends or at the middle. But it should be observed that there is considerable difference between the condition of mathematical fixedness, in which case the beam is horizontal over the supports, and that of imbedding a beam in a wall. For in the latter case the deflec tion will extend some distance into the wall.
Mr. Barlow concludes from his experiments that equation (151) should be
$>l = ^b<? - - - - (154)
and this relation is doubtless more nearly realized in practice than the ideal one given above. In either case, it appears that writers and experimenters have entirely overlooked the effect due to the change of position of the neutral axis, which must take place. It has been assumed that the neutral axis coincides with the axis of the beam, and that its length remains unchanged during flexure ; but if the ends of the beam are fixed, the axis must be elongated by flexure, or else approach much nearer the concave than the convex side, or both take place at the same time, in which case the moment of resistance will not be Jll/W. The phenomena are of too complex a character to admit of a thorough and exact analysis, and it is probably safer to accept the results of Mr. Barlow in practice than depend upon theoreti cal results.
12O. MODULUS OF RUPTURE. — When a beam is support ed at its ends, and loaded uniformly over its whole length, and also loaded at the middle, we find from equation (150)
_ W
in which W may be the weight of the beam. Beams of known dimensions, thus supported, have been broken by weights placed at the middle of the length, and the corresponding value of R has been found for various materials, the results of which have been entered in the table in Appendix IV. This is called the MODULUS OF HUPTURE, and is defined to be the strain upon a,
TRANSVERSE STRENGTH. 143
square inch of fibres most remote from the neutral axis on the side which first ruptures. It would seem from this definition that R should equal either the tenacity or crushing resistance of the material, depending upon whether it broke by crushing or tearing, but an examination of the table shows the paradoxical result that it never equals either, but is always greater than the smaller and less than the greater. For instance, in the case of cast iron : —
The mean value of T = 16,000 Ibs. " " C = 96,000 "
" Rz= 36,000 " nearly,
hence R is about 2J times T, and a little over -J of C. For English oak —
T = 17,000 Ibs. ;
C = 9,500 Ibs. ; and
R = 10,000 Ibs. ;
hence R exceeds C, and is more than half of T. For ash —
T = 17,000 Ibs. ; C = 9,000 Ibs. ; and R = 12,000 Ibs. ; hence R = 1-J- C and about f T.
These discrepancies have long been recognized, and the cause has generally been attributed to a departure from the law of perfect elasticity and a movement of the neutral axis away from the centre of the beam in the state bordering on rupture ; but as the laws of these variations were not assigned, their influence could not be analyzed. (See Articles 74 and 75.)
The tabulated values of R being found from experiments up on solid rectangular beams, they are especially applicable to all beams of that form, and they answer for all others that do not depart largely from that form ; but if they depart largely from that form, as in the case of the i (double T) section, or hollow beams, or other irregular forms, the formulas Avill give results somewhat in excess of the true strength ; and in such cases Bar low's theory gives results more nearly correct.
But if, instead of R, we use T or C, whichever is smaller, in the formulas which we have deduced, and suppose that the neutral
144 THE RESISTANCE OF MATERIALS.
axis remains at the centre of the beam, we sJiall always le on tJie safe side / but there would often be an excess of strength, as, for instance, in the case of cast iron the actual strength of the beam would be about twice as strong as that found by such a computation.
The difficulty is avoided, practically, by using such a small fractional part of R as that it will be considered perfectly safe. This fraction is called the coefficient of safety. The values com monly used for beams are the same as for bars, and are given in Article 38.
Experiments should be made upon the material to be used in a structure, in order to determine its strength; but in the absence of such experiments the following mean values of II are used :—
850 to 1,200 Ibs. for wood, 10,000 to 15,000 Ibs. for wrought iron, and 6,000 to 8,000 Ibs. for cast iron.
. PRACTICAL, FORMULAS.
If R =: 1,000 for wood, and
12,000 for wrought iron,
we have for a rectangular beam, supported at its ends and loaded at the middle of its length, '/ M>
666 W - P = — j -- for wooden beams ; and
8000 &?.
1 = -- j --- tor wrougnt-iron beams. t>
The length of the beam, and the load it is to sustain, are gen erally known quantities, and the breadth and depth are required ; but it is also necessary to assume one of the latter, or assign a relation between them. P\>r instance, if the depth be n times the breadth, the preceding formulas give
7< 5=V;»=1 for wood; (156)
and b = V/r 5 andrf = V " f°r wrouSht iron >
TKANSVEKSE STRENGTH. 145
THE RELATIVE STRENGTH OF A BEAM Under
the various conditions that it is held is as the moment of the applied forces ; hence, all the cases which have been con sidered may, relatively, be reduced to one, by finding how much a beam will carry which is fixed at one end and loaded at the free end, equation (146), and multiplying the results by the fol lowing factors : —
FACTORS.
Beam fixed at one end and loaded at the other - 1
" " " " uniformly loaded - - 2
Beam supported at its ends and loaded at the middle 4
« " " " uniformly loaded 8
Beam fixed at one end and supported at the other,
and uniformly loaded - - - - __8
Beam fixed at both ends and loaded at the middle - - 8 " " " " uniformly loaded - 12
If it is required to know the breadth of a beam which will sustain a given load, find 5, from equation (146) ; and for a beam in any other condition, divide by the factors given above for the corresponding case.
If the depth is required, find d from equation (146), and di vide the result for the particular case desired by the square root of the above factors.
EXAMPLES.—
1. A beam, whose depth is 8 inches, and length 8 feet, is supported at its ends, and required to sustain 500 pounds per foot of its length ; required its breadth so that it will have a factor of safety of -fa R being 14,000 pounds.
From equation (146) we have,
QPl 6x500x8x8x12
and by examining the above table of factors we see that this must be divided bv 8 . . •. Ans. = 3ia4- inches.
2. If I - 10 feet, P at the middle = 2,000 Ibs., b = 4 inches, E = 1,000 Ibs., required d.
3. If a beam, whose length is 8 feet, breadth is 3 inches, and depth 6 inches,
10
14C THE RESISTANCE OF MATERIALS.
is supported at its ends, and is broken by a weight of 10,000 pounds placed at the middle, and the weight of a cubic foot of the beam is 50 pounds; required the value of R. Use equation (150).
4. If R = 80,000 Ibs., I = 12 feet, b = 2 inches, d — 5 inches, how much will the beam sustain if supported at its ends and loaded uniformly over its whole length, coefficient of safety i ? Ana W = 9,259 Ibs.
5. A wooden beam, whose length is 12 feet, is supported at its ends ; re quired its breadth and depth so that it shall sustain one ton, uniformly distri buted over its whole length. Let R = 15,000 Ibs., coefficient of safety j1,,, and depth = 4 times the breadth. Ans. b = 2.08 inches.
inches.
6. A beam is 2 inches wide and 8 inches deep, how much more will it sustain with its broad side vertical, than with it horizontal ?
7. A wrought-iron beam 12 feet long, 2 inches wide, 4 inches deep, is supported at its ends. The material weighs £ Ib. per cubic inch ; how much load will it sustain uniformly distributed over its whole length, R = 54,000 Iba. ? Ans. Without the weight of the beam, 15,712 Ibs.
8. A beam is fixed at one end ; I — 20 feet, b—\\ inch, R = 40,000 Ibs. ; weight of a cubic inch of the beam ^ Ib. Required the depth that it may .sustain its own weight and 500 Ibs. at the free end. Ans. 4. 05 inches.
9. The breadth of a beam is 3 inches, depth 8 inches, weight of a cubic foot of the beam 50 pounds, R — 12,000; required the length so that the beam shall break from its own weight when supported at its ends.
Ans. I- 175. 27 feet,
RELATION BETWEEN STRAIN AND DEFLECTION.
—When the strain is within the elastic limit we may easily find the greatest strain on the fibres corresponding to a given deflec tion. For instance, take a rectangular beam, supported at its ends and loaded at the middle of its length, and we have from equation (148)
-
and from equations (73) and (51) A
A =
= J „. » , which becomes, by substituting P frouj the preeed
{'M f % d&£~~?5
g3 A~v ^^^ r.
A = iE^
TRANSVERSE STRENGTH.
147
p 6Ed R=-7rA
(158)
Examples. — 1. If I — 6 feet, b = li inch, d = 4 inches, coefficient of elas ticity = 23,000,000 Ibs. is supported at its ends and loaded at the middle so as to produce a deflection at the middle of A = f inch ; required the greatest strain on the fibres. Also required the load.
2. On the same beam, if the greatest strain is R — 12,000 Ibs., required the greatest deflection.
3. If the beam is uniformly loaded, required the relation between the greatest strain and the greatest deflection.
. 4. Generally, prove that R = constant x --- A .
125. HOLLOW RECTANGULAR REAMS — If a rectangular beam has a rectangular hollow, both symmetrically placed in reference to the neutral axis, as in Fig. 62, we may find its strength by deducting frojao.. the strength of a solid rectangular beam the slpbhgtk-o£ a solid beam of the same size as the hollow. But in. this case, when the beam ruptures at &, the strain at V will be less than R. As the strains increase di rectly as the distance of the fibres from the neutral axis, we have, if d and d! are the depths of the outside and hollow parts re spectively,
d' \d : \d' : : R : strain at V — R -j.
Fl°- 62-
If 5 = the breadth of the hollow, the stress on that part, if it
TJ U ' J' A' /-I ^K\
TJ U ' J' A' /-I ^K\
were solid, would DC, according to equation (145),
which, taken from equation (145), gives for the resistance of a hollow rectangular beam,
If the hollow be on the outside, as in Fig. 63, a forming an II section, the result is the same.
_
FIQ. 63,
L>r4A4+ , i
• fr&lw
;«6-
^^ ^
148
THB RESISTANCE OF MATERIALS.
IF THE UPPER AND LOWER FLANGES ARE UN EQUAL it forms a double T, as in Fig. 64. Let the notation be as in the figure, and also d\ equal the distance from the neutral axis to the upper element, and x the distance from the neutral axis to the lower element.
Fio. 64.
To find the position of the neutral axis, make the statical moments of the surface above it equal to those below it. This gives
d' I' (d, - i d') + i V" (d, - dj = d" I" (x - i d") + i V"
'l 41 <
We also have dt = d - 0 = d' 4 d" + d"' - x
These equations will give x and dr
Constructing the wedges as before, and the resistance to com pression is represented by the wedge whose base is 1)' dl and altitude R, minus the wedge whose base is (&' — V") (dl — d')
d-d and altitude - — j — R. Hence the resistance to compression is
- R (V
£**$- */,.& -<?r~
*
TRANSVERSE STRENGTH. 149
The centre of gravity is at f the altitude, or ^dl for the for mer wedge, and f (dl — d) for the latter, and if the volumes be multiplied by these quantities respectively, it will give for the moment of resistance to compression
Next consider the resistance to tension. Since the strains on the elements are proportional to their distances from the neu tral axis, therefore
dl : x : : ~R : strain at the lower side of the section — — j #,
ctl
and similarly,
dl : (x — d"} : : R : strain at the opposite side of the lower
K
flange = -y- (x —d").
Hence the tensive strains will be represented by a wedge whose base is l}"x and altitude j-x, minus a wedge whose base is( ~b" —
P-' K
by and altitude -T (x —\d"). Hence the moment of resistance is
The total moment of resistance is the sum of the two moments, or * J I}' d* ~ ^' ~ ?-'") ~ d' + V> " ~ J///
. - (102)
For a single T make ~b" and d" = 0 in the above expression.
The method which has here been applied to rectangular beams may be applied to beams of any form ; but it often re quires a knowledge of higher mathematics to find the volume of the wedge, and the position of its centre of gravity ; or resort must be had to ingenious methods in connection with actual wedges of similar dimensions.
•v y^^z^xi^ixu-^t^-c--
*M 5&. ^
150 THE RESISTANCE OF MATERIALS.
197. TRUE VALUE OF dl AND AN EXAMPLE. — In this
and similar expressions
dl = the distance from the neutral axis to the fibre most re mote from it ON THE SIDE •WHICH FIRST RUPTURES.
di is usually taken as the distance to the most remote fibre, without considering whether rupture will take place on that side or not ; but this oversight may lead to large errors.
For example, let the dimensions of a cast- iron double T beam be as in Fig. 65, and 228 inches between the supports. Required the load at the middle necessary to break it.
The position of the neutral axis is found from equations (160) and (161) to be 7.96 inches from the lower side, and 11.54 inches from the upper. As cast iron will resist from four to six times as much ^compression as/ff
©tension — this beam will rupture on the lower side first ; hence dl in the equation = 7.96 inches. As the value of R is not known, take a mean value = 36,000 Ibs. The moment of the rupturing force — neglecting the weight of the beam — is ^ IV, which placed equal to expression (162) and re duced gives
A f\p AAA
P = o x ~£a ~ X i?672 = 132,0:0 Ibs. =58.9 tons gross. — — ^ T.vt)
Had we used dl = 11.54, it would have given P = 40.4 tons. Such beams actually broke with from 50 to 54 tons ; or, in cluding the weight of the beam, with a mean value of 52£ tons.
By reversing the problem, and using 52£ tons for P, we find that R is a little more than 32,000 pounds. Had this value of R been used in the first solution, and dl made equal 11.54, it would have given for P a little more than 36 tons, which would be the strength if the beam were inverted. If the upper flange were smaller or the lower larger, the discrepancy would have been greater.
The strain upon a fibre in the upper surface is to the strain upon
y^ one in the lower surface aaf^ Jto^cJ hence, if the material resists
more to compression than to tension (as cast iron), it should
be so placed that the small flange shall resist the former, and
^"^ Terr
TRANSVERSE STRENGTH. 151
the large one the latter. If a cast-iron beam be supported at its ends, the smaller flange should be uppermost, and as it re sists from four to six times as much compression as tension, the neutral axis should be from four to six times as far from the upper surface as from the lower, for economy. Using the same notation as in Fig. 64 and we have,
d^ greatest compressive strain ^t£*^ ^£-
x ~ greatest tensive strain and for economy we should have,
d^ ultimate compressive strength x ~ ultimate tensile strength
The ultimate resistance of wrought iron is greater for ten sion than for compression ; hence, if a wrought-iron beam is supported at its ends, the heavier flange should be uppermost.
The proper thickness of the vertical web can be determined only by experiment, and this has been done, in a measure, by Baron von Weber, in his experiments on permanent way.
1S8. EXPERIMENTS OF BARON VON WEBER for deter mining the thickness required for the central web of rails.
Baron von Weber desired to ascertain what was the mini mum thickness which could be given to the web of a rail, in order that the latter might still possess a greater power of re sistance to lateral forces than the fastenings by which it was
FIG. 65a.
secured to the sleepers. For this purpose a piece of rail 6 feet in length, rolled, of the best iron at the Laurahutte, in Silesia, was supported at distances of 35.43 in., and loaded nearly to the limit of elasticity (which had been determined previously by ex-
//a.
THE RESISTANCE OF MATERIALS.
periments on other pieces of the same rail), and the deflections were then measured with great care by an instrument capable of registering 1-1000 in. with accuracy. This having been done, the web of the piece of rail was planed down, and each time that the thickness had been reduced 3 millimetres the vertical deflection of the rail under the above load was again tested, and the rail was subjected to the following rough but practical ex periments. The piece of rail was fastened to twice as many fir sleepers by double the number of spikes which would be em ployed in practice, and a lateral pressure was then applied to the head of the rail by means of a lifting-jack, until the rail began to cant and the spikes were drawn. The same thing was then done by a sudden pull, the apparatus used being a long lever fas tened to the top of the rail, as shown in Fig. 65a. The lifting- jack and the lever were applied to the ends of the rail, and the
FIG. 65b.
web of the latter had, in each case, to resist the whole strain re quired for drawing out the spikes. The results of the experi ments made to ascertain the resistance of the rail to vertical flexure with different thicknesses of web, and under a load of 5,000 Ibs., were as follows :—
TRANSVERSE STRENGTH.
153
Thickness of web. In.
15 millimetres = 0.59 12 « 0.47 -
9 " 0.35 -
6 " 0.24
3 " 0.12 -
Vertical deflection. In.
- - - 0.016
0.016 0.019
0.0194
- - - 0.022
These results showed ample stiffness, even when the web was reduced in thickness to 0.12 in. To determine the power of resistance of the rail to lateral flexure, an impression of the sec tion was taken in lead each time that the spikes were drawn.
The forces applied in these experiments were very far greater than those occurring in practice, yet it was found that with the web 12, 9, and even 6 millimetres thick, no distortion took place, and only when the thickness of the web was reduced to 3 milli metres (0.12 in.) was a slight permanent lateral deflection of the head caused just as the spikes gave way. The section shown in Fig. 65b had then been reduced to that shown in Fig. 65c.
"9851it
FIG. 65c.
Next, a rail, with the web reduced to 3 mill. (0.12 in.) in thickness, was placed in the line leading to a turn-table on the
154: THE RESISTANCE OF MATERIALS.
Western Railway of Saxony, where it has remained until the present time, 1870, receiving the shocks due to engines passing to and from the turn-table more than one hundred times daily. It follows from these experiments that the least thickness ever given to the webs of rails in practice is more than sufficient, and that if it were possible to roll webs J in. thick, such webs would be amply strong, if it were not that there would be a chance of their being torn at the points where they are traversed by the fish-plate bolts. Baron von Weber concludes that webs f in. or J in. thick are amply strong enough for rails of any ordinary height, and that, in fact, the webs should be made as thin as the process of rolling and as the provision of sufficient bearing for the fish-plate bolts will permit.
129. ANOTHER GRAPHICAL ittETHOD. — If manipulating processes are to be used for determining the strength, the fol lowing method possesses many advantages over the former.
Since the strains vary directly as their dis tance from the neutral axis, the triangle ABC (Fig. 66), in the rectangle BCDE, represents the compressive strains if each element of the shaded part has a strain equal to R; and its moment is R times the area multiplied by the distance of the centre of gravity of the triangle from the neutral axis ; or,
R x (b x \ of \d) x f of $d = ^EbcT,
and the moment of tensile resistance is the same, hence the total moment is double this, or ^HbcF, as found by the preced ing process.
130. IF A SQUARE BEAltt HAVE ONE OF ITS DIAGONALS
VERTICAL, (Fig. 67), the neutral axis will coincide with the other diagonal. Take any element, as ab, and project it on a line cdj which passes through A and is parallel to BC, and draw the lines Ocand ()//, and note the points f and <j where they intersect the line ab. If the element were at cd, the strain upon it would be R, multiplied by the area of cd, or simply R.cd ; but because
TRANSVERSE STRENGTH.
155
the strains are directly proportional to the distances of the ele ments from the neutral axis, the strain on ab is ^&.fg. Proceed in this way with all the elements and construct the shaded figure. The strains on the upper part of the figure ABC, which
FIG. 67.
begin with zero at BC, and increase gradually to R, at A, will be equivalent to the strains on the shaded figure AC, if the strain is equal to R on each unit of its surface. Hence the total strain on each half is the area of the shaded part AO, multiplied by R, and the moment of the strain of each part is this product multiplied by the distance of the centre of the shaded part from the axis BC.
By similar triangles we have
Aa : ab : : AB : BC, and cd= ab \fg : : AO : x : : AB : Ba or AB - Aa ;
x being the distance of fg from O. From these eliminate ab, and find
hence the curve which bounds the shaded figure is a parabola which is tangent to AB, and whose axis is parallel to BC. Let d = one side of the square, then
= AO, and
= the widest part of the shaded figure. The area of a parabola is two-thirds the area of a circumscribed rectangle.
156 THE RESISTANCE OF MATERIALS.
Hence the area of AO is
and the moment is
2 =
K
and the moment of both sides, multiplied by R, is
i;
64/2
(163)
If I = d in equation (145) and the result compared with the above, we find :—
The strength of a square beam with its side vertical : strength of the same beam with one of its diagonals vertical : : \/2 : 1 or as 7 : 5 nearly.
So that increased depth merely is not a sufficient guarantee of increased strength. The reason why the strength is dimin ished when the diagonal is vertical, is because there is a very small area at the vertex where the strain is greatest, but when a side is horizontal the whole width resists the maximum strain.
131. IRREGULAR SECTIONS. — This method is applicable to irregular sections, as shown by the following example.
Fio. 68.
Let Fig. 68 be a cross section of a beam. In a practical case it may be well to make an exact pattern of the cross
TRANSVERSE STRENGTH. 157
section, of stiff paper or of a tliin board of uniform thick ness. To find the position of the neutral axis, draw a line on the pattern which shall be perpendicular to the direc tion of the forces which act upon the beam, that is, if the forces are vertical the line will be horizontal. In a form like Fig. 68 this line will naturally be parallel to the base of the figure. Then balance the pattern on a knife-edge, keeping the base of the figure (or the line previously drawn) parallel to the knife-edge, and when it is balanced the line of support will be the neutral axis. Proceed to construct the shaded part as shown in the figure, by projecting any element, as ab on the line cd, and drawing cO and d 0, and noting the intersections f and g, the same as in Fig. 67. The elements on the lower side must be projected on a line mn, which is at the same distance from the neutral axis as the most remote element on the upper side. The area of the shaded part above the neutral axis should equal that below, because the resistance to extension equals that for compression. The area of the shaded part may be found ap proximately by dividing it into small rectangles of known size, and adding together the full rectangles and estimating the sum of the fractional parts. Or, the shaded part may be cut out and carefully weighed or balanced by a rectangle of the same mate rial, after which the sides of the rectangle may be carefully measured and contents computed. The area of the rectangle would evidently equal the area of the irregular figure.
The ordinate to the centre of gravity of each part may be determined by cutting out the shaded parts and balancing each of them separately on a knife-edge, as before explained, keeping the knife-edge parallel to the neutral axis. The distance be tween the line of support and the neutral axis will be the ordi nate to the centre of gravity. The moment of resistance is then found by multiplying the area of each shaded part by the dis tance of its centre of gravity from the neutral axis, and multi plying the sum of the products by R.
These mechanical methods may be managed by persons who have only a very limited knowledge of mathematics, and if skilfully and carefully done will give satisfactory results. It does not, however, furnish such an uniform, direct and exact mode of solution as the analytical method which is hereafter explained.
158 THE RESISTANCE OF MATERIALS.
FORMULA OF STRENGTH ACCORDING TO BARLOW'S
TiiEORY.-Either of the above methods may be used. One part of the expression for the strength is of the same form as that found by the common theory; but instead of R we must use T, or C — the former if it ruptures by tension, the latter if by crushing. The other resistance, <£, for solid beams is evenly dis tributed over the surface. For example, take a rectangular beam, Fig. 61, and the resistance to longitudinal shearing on the upper side is <£ b x \d = £ <£ bd, and its moment is -J <£ bd x % of \d = -J- <f> bd?, and for both sides, J <f> bd1. Hence, according to Barlow's theory, the expression for the strength of a rectan gular beam is
[ J <£ + fT] bcT for cast iron, and
[t <t> + £C] bd* for wrought iron and wood - - (164)
If the beam is supported at its ends and loaded at the middle, we have
iPZ = [J <f> + -J-T] bd1 for cast iron ..... (165)
The volume which represents the resistance due to </> is always a prism, having for its base the surface of the figure and </>, or some fraction of </>, for its altitude. If the second method of illustration be used, it will take two figures to fully illustrate the strains. For instance, if the section be as in Fig. 68, the moment of the shaded part will be multiplied by T or C, as the case may be. To find the remaining part of the moment, find the area of each part of the transverse section, also the distance of the centre of gravity of each part from the neutral axis. Then, to find the moment of re sistance due to longitudinal shear ing, multiply the area of each part by the distance of its centre of grav ity from the neutral axis, add the products and multiply the sum by <f>. This is true for solid sections ; but for hollow beams, T and II sections, where there is an abrupt angular change from the flange to the verti cal part of the beam, the factor <f> requires a modification. For instance, take the simple case of a single T, Fig. 69, in which
TRANSVERSE STRENGTH. 159
the breadth of the T is b' and its depth d', and the other nota tion as in the figure.
The resistance of the upper part is represented by the prism whose base is fcc, and whose altitude is <£, plus the prism whose
base is d' (J'-J), and whose altitude is — </>. The resistance of
#?
the lower part is <£ bdv. The total moment of this resistance is — -f
To this add the moment of resistance for direct extension and compression, the expression for which is of the same form as for common theory, and we have for the total moment : —
+ W- b) (x- K) + &
(x-d')3] (166)
From numerous experiments made upon cast-iron beams having a variety of cross sections, Barlow found that $ varied nearly as T, that practically it was a fraction of T, the mean value of which was 0.9T. For wrought iron he found <j> = 0.53T
z= 0.8C nearly.
Peter Barlow, F.K.S., father of W. II. Barlow, F.RS., the latter of whom proposed the " theory of flexure," in an article in the Civ. Eng. Jour., Yol. xxi., p. 113, assumes that </> = T.
From the above it is inferred that the practical mean values of </> are : —
16,000 Ibs. for cast iron. 30,000 Ibs. for wrought iron. 8,000 Ibs. for wood.'
Examples. — 1. How much will a beam whose length is 12 feet, breadth 2 inches, depth o inches, sustain, if supported at its ends, and uniformly loaded over its whole length, and C = 50,000 Ibs., <£ = 30,000 Ibs., and coefficient of safety £ ? Am.— 11,000 Ibs. nearly.
~2. If 0 = T = 16,000 Ibs., b = 2 inches, d = 5 inches, I = 8 feet ; required the uniform load which it will sustain with a coefficient of safety of £.
~~3. Ifb — d = 2 inches, 1=6 feet, R= 50,000 Ibs., is broken by an uni form load of 10,000 pounds, required 0.
BEAUIS LOADED gAT ANY NTJlttBER OF POINTS. -
If the beam is loaded otherwise than has heretofore been sup posed, it is only necessary to find the moment of all the forces
160
THE RESISTANCE OF MATERIALS.
n |
n4 i |
||
"T |
p,"' |
'• fc |
in reference to the centre of a section and place the algebraic sum equal to the moments of resistance. Those which act in opposite directions will have contrary signs.
For instance, if a beam, AB, Fig. 70, rests upon two supports, and has weights, P,, Pa,
P,, &c., resting upon it at v, v,
distances respectively of n,, 7ia, 7is, etc., from one support, and m,, raa, mt, &c., from the other, the sum of the moments of the forces on any section C whose distance is x from the support A, is FIO TO
Ytaj — P,0 —/I,) — P9(jj - n,) &c., to include all] the terms of P in which n is less than x. This equals $WjcF for rectan gular beams.
Yt, the reaction of one support, is readily found by taking the moments of all the external forces about B, and solving for
= ?,»*, + Pama + P3m, + &c. = SPm
Vt, thus :-
Similarly Y3 = also, Y. 4-
. = 2 P.
134. A PARTIAL UNIFORM LOAD.— Let -the beam be loaded uniformly over any portion of its length, as in Fig. 71. Let I = AB — length of beam ; 2a = DE = length of the
uniform load ; x = AF = the distance to
any section ;
w = the load on a unit of length ; Y = the reaction of the support A ; C the centre of the load ; I, = AC ; I =CB.
s\
- 2,<^ +•% ****-"*- — /r
- __ - „ £)
=-0-W. £ - ' V<u£_
X_ TRANSVERSE STRENGTH. 161
-&OL + £\
Then AD = ^-0, and DF = a? - Z> + a. Load on DF = w (x — lr + #),
By the principle of moments
The moment of stress at F is
Vx - %w (x - It + af ty -Cc(v
- - (167)
That value of a? which will make equation (167) a maximum, gives the position of the dangerous section. Differentiate, place , v equal zero, and make Z, +£, = £, and solve for a?, and find
If I, =«* = !, 3 ~ l^^~2f^€/- + %^-2/f. ll < \l, x >Zj ; Zj > -i^ aj < ll ;
so that the maximum strain is at the centre of the loading only when the centre of the loading is over the centre of the beam ; and in all other cases it is nearer the centre of the beam than the centre of the loading is.
The maximum strain is found by substituting the value of x equation (168) in equation (167).
The following interesting facts are also proved.
Let A D = y /. a = 1^ — y which in equation (168) reduces it to
vj »
+
which is a maximum for y = 0 ; hence so far as A D is con cerned, equation (168&) is a maximum when one end of the load is over the support, and for this case the equation becomes
$'
which is a maximum for IY = J I or 2^ = %l, or the load must 11
162
THE RESISTANCE OF MATERIALS.
extend to the middle of the beam. equation (108) becomes
Making a = ll = J Z, and
and these values of Z, and x in equation (167) give for the max imum moment of stress,
(169)
in which TV is the load on half the beam.
Equation (167) gives the stress at the middle of the load, by making a = lv = % I and x = J I. This gives i W J for the stress at the middle of the loading ; hence, the maximum stress is 1£ times the stress at the middle of the loading when the load extends from the one support to the middle of the beam.
135. OBLIQUE STRAINS. — If the force be inclined to the axis, as in Figs. 72 and 73, let t = the angle which P makes with a normal section.
Fio. 72.
Fio. 73.
Then, P cos 6 = normal component,
P sin 6 = longitudinal component.
If K = the transverse section, then
P sin 9
—Tr— = the tension or compression upon a unit or sec tion which arises directly from the longitudinal component. This tends directly to diminish R in the formula whether * be obtuse or acute. If the beam be fixed at one end and free at the other, as in Fig. 72, the equation of moments becomes : —
Psin ~TT Jd,
— - fa\jwl ^'l
dL, CU^j
~) 7^
r W_y. ./ .
rfc
TEANSVERSE STRENGTH. 163
which for rectangular beams becomes
P sin
136. GENERAL, FORMULA. — The preceding methods are easily understood, and are perhaps sufficient for the more simple cases ; but for the purposes of analysis a general formula is bet ter, by means of which a direct analytical solution may be made for special cases.
Let R = the modulus of rupture, as explained in article 120 ; x and u horizontal coordinate axes, the former coinciding
with the axis of the beam, and y a vertical axis ; Then ~Rdudy = the resistance of a fibre which is most remote
from the neutral axis. Let d: = distance between the neutral axis and the most remote
fibre ; then, according to the common theory, since
the strains vary as the distance from the neutral axis
di : y : : Hdudy : resistance of any fibre — -j y dy du
ul
E
•*• ~j y1 dydu — the moment of resistance of any fibre,
0| and the sum of all the moments of resistance of any section is
^r1
y
I* C
J J
which is called the moment of rupture, and must equal the sum of the moments of straining forces ;
(171)
The second member of this equation involves the character of the material (R) and the form of the transverse sections (~) ; the
latter of which may be determined by analysis, and the former by experiment. The second member shows that for economy the material should be removed as much as possible from the neutral axis. A few special cases will now be given.
~
"J /'
v
<J
1C4
THE RESISTANCE OF MATERIALS.
. LET THE m \ M BE RECTANGULAR, I the breadth, and d the depth, as in Fig. 61,
Then I =
y'dy du =
.'. -j I = $ II for which is the same as expression (145).
138. IF THE SIDES OF THE BEA3I ARE INCLINED to the
FIG. 74.
Fio. 75.
direction of the force, as in Fig. 74, let i be the inclination of the side to the horizontal ; then
I = jLybd^irfi + Fees1*)* di — Jdisim
inV 4- J'
308*^ "I JOS£ J
d&iui +
Tliis expression has an algebraic minimum,! but not an alge braic maximum. By inspection, however, we find that the practical maximum is found by making 2' = 90°, if d exceeds />. Hence, a rectangular beam is strongest when its broad side is parallel to the direction of the applied forces.
Hence, the braces between joists in flooring, as in Fig. 75, not
* See Appendix III.
\ See an article by the author in the Journal of Franklin Institute, VoL LXXV., p. 200.
TRANSVERSE STRENGTH. 165
only serve to transmit the stresses from one to another, but also to strengthen them by keeping the sides vertical. If i = 90°, equation (172) becomes |K^a - - (173) If b = d and i = 45°, equation (172) reduces to
62
(which is the same as equation (163)), and if J = d, and i = 0° or 90°, it becomes
Hence, the strength of a square beam having a side vertical is to the strength of the same beam having its diagonal vertical, as
1: V*,
or -y/2 to 1 or as 7 to 5 nearly,
In establishing equation (172) it was assumed that the neutral surface was perpendicular to the direction of the applied forces, which is not strictly true unless the forces coincide with the diagonal ; for in other cases there is a stronger tendency to deflect sidewise than in the direction of the depth. In this case, as soon as the beam is bent there is a tendency to torsion. Both these conditions make the beam weaker than when the sides are vertical. If the tendency to torsion be neglected, the case may be easily solved ; but as the result shows the advantage of keep ing the sides vertical, the solution is omitted.
1 39. THE STRONGEST RECTANGUL, AR BEAM which Can
be cut from a cylindrical one has the breadth to the depth as 1 to •/ 2, or nearly as 5 to 7. Let x = AB — the breadth, y = AC = the depth, and D = AD — the diameter.
and equation (173) becomes
(Da -
166
THE RESISTANCE OF MATEKIALS.
which by the Differential Calculus is found to be a maximum for
/. x : y : : 1 : \/2 or nearly as 5 to 7.
Example*. — How much stronger is a cylindrical beam than the strongest rectangular one which can be cut from it ? (For the strength of a cylindrical beam, see equation (180)).
An*.— About 53 per cent.
How much stronger is the strongest rectangular beam that can be cut from a cylindrical one, than the greatest square beam which can be cut from it ?
14O. TRIANGULAR BEAMS — If the base is perpendicular to the neutral axis, as in Fig. 77 ;
Let d = AD = the altitude, and I = BC = the base.
Take the origin of coordinates at the cen tre of gravity of the triangle, y vertical and u horizontal.
Then, by similar triangles,
i b : y : : d
FIG. 77.
'.y = ^-f^-.'.du=~dy
We also have
/» ly'du =
' tJ J ™™ ^t-w.- — -z — /£
- / f
Us ''"'
/!*/
(175)
in which A is the area of the triangle.
If the base is parallel to the neutral axis, as in Fig. 77 #, then, by similar triangles,
d : J b : : %d — y : u b
TRANSVERSE STRENGTH.
167
We also have
(176)
Equations (173) and (175) show that a triangular beam which has the same area and depth as a rectangular one, is only half as strong as the rectangular one.
Some authors have said that a triangular beam is twice as strong with its apex up as with it down, but this is not always the case. If the ultimate resistance of the material is the same for tension as for compression, the beam will be equally strong with the apex up or down.
If the beam is made of cast iron, and supported at its ends, it will be about 6 times as strong with the apex up as down ; but if the beam be fixed at one end, and loaded at the free end, it will be about 6 times as strong with the apex down as with it up.
141, TRAPEZOIDAL BEAM. — Required the strongest trap ezoidal beam which can be cut from a given triangular one.\ c Let ABC be the given triangle,
ABED the required trapezoid, d = CG- = the longest altitude,
w, and v = DE.
IJ is the neutral axis of the trapezoid, which passes through its centre of gravity II. We may then find : —
* This is more easily solved by taking the moment about an axis through the J^ vertex and parallel to the base, and using the formula of reduction. See Ap pendix.
f See an article by the author in the Journal of Franklin Institute, vol. xli. , thi^d Beries, p. 198. ^ ^ fit ^ vL _ ^ *> £ ^ ^ J
3 d" ^V ^
+»J
3*y
r*
1G8 THE RESISTANCE OF MATERIALS.
^ = tjx
* + Vv - 8ft V + 85V
S*
- 8b V
which is to be a maximum. By the Calculus we find, after re duction, that
for a maximum, which solved gives
v = 0.13093& or 0.136 nearly, and hence w = 0.130936? or 0.13d .... (ITS)
which substituted in (177) gives
Rj = 0.545625 ^ ..... (179)
Dividing equation (179) by equation (176) gives 1.09125 ; hence from (178) and (179) we infer that if the angle of the prism le taken off 0.13 of its depth, the remaining trapezoidal beam will be 1.091 times as strong as the triangular one, which is a gain of over 9 per cent.
In order to explain this paradox it must be granted that the condition does not require that the beam shall be broken in two, but that a fibre shall not be broken — in other words, the beam shall not be fractured. The greatest strain is at the edge, where there is but a single fibre to resist it ; but, after a small portion of the edge is removed, there are many fibres along the line DE, each of which will sustain an equal part of the greatest strain.
If the triangular beam were loaded so as to just commence fracturing at the edge, the load might be increased 9 per cent. and increase the fracture to only thirteen-hundredths of the depth ; but if the load be increased 10 per cent, it will break the beam in two.
These results are independent of the material of which the
TRANSVERSE STRENGTH. 169
beam is made. If the beam be cut off £ the depth, its strength is found from equation (177) to be
0.465608
12 '
which is 0.93101 of equation (176).
Mr. Couch found, for the mean of seven experiments on tri angular oak beams of equal length, that they broke with 306 pounds. The mean of two experiments on trapezoidal oak beams, made from triangular beams of the same size as in the preceding experiments, by cutting off the edge one-third the depth when the narrow base was upward, was 284.5 pounds. This differs by less than half a pound of 0.931 times 306 pounds.
\. CYLINDRICAL BEAMS. — The moment of inertia of a circular section in which r is the radius, is
+ r r+v*
I
J-r
IL <L 3
--=
KI /. -j — iR*r3 (180)
If polar co-ordinates are used, we have
dudy — gdgdff),
where f is a variable radius and <f> a variable angle. Also y = { sin <j>
/T /*2* / ^sin2 <bd$d<b t/0
= %f\ JQ i(l — cos 20) d<b = ^irr4. as before. '
V-T- >
For a circular annulus we have
^- ^( -n)-
7
170
THE RESISTANCE OF MATERIALS.
r* X
^
By comparing equations (180) and (!•&>) we see that the strength of a cylindrical beam is to that of a circumscribed
/ *
rectanular one as -, : or
-:1.
» /
•"I a 1 *•— i I
Also the strength of a cylindrical beam is to that of a square one of the same area as ^Adf to ^RAd (d' being the diameter of the circle),
or as 1 : (£-3, = IV*") or as 1 : 1.18 nearly.
143. ELLIPTICAL III: \ TBS.
Let b = the conjugate axis, and
d = the transverse axis ; then if d is vertical (Fig. 80), we have I = -fa *ld? and dl = \d.
If b is vertical (Fig. 81), we have
Fio. W.
141. PARABOLIC. BEA31S. '
Fio. 82.
Fio. 83.
If b = the base, and
d = the height of the parabola, and if d is vertical (Fig. 82), we have
TKANSVEKSE STRENGTH. 171
If I is vertical (Fig. 83), then
I = -gV^3, and dv = \l.
145. ACCORDING TO BARLOWS THEORY W6 have
+ tfjydydu = SP* (isi)
which must be integrated between the proper limits to include the whole section.
If the neutral axis is at the centre of the sections, and the beam is rectangular, we have
T
which reduced gives
hence, if $ has any ratio to T, the law of resistance in solid rec tangular beams is the same as for the common theory only,
If $ = T, this becomes
172 THE RESISTANCE OF MATERIALS.
CHAPTER VII.
BEAMS OF UNIFORM RESISTANCE.
146. GENERAL EXPRESSION. — If beams are so formed that they are equally liable to break at every transverse section, they are beams of uniform resistance, and are generally called beams of uniform strength. The former term is preferable, be cause it applies with equal force to all strains less than that which will produce rupture. In such a beam the strain on the fibre most remote from the neutral axis is uniform throughout the whole length of the beam. The analytical condition of such a beam is: The sum of the moments of the resisting forces must vary directly as the sum of the moments of the applied forces ; hence equation (171) is applicable; or
(182)
which must be true for all values of x. But to obtain practical results it is necessary to consider
PARTICULAR CASES. 147. BEAMS FIXED AT ONE END AND LOADED AT THE FREE
END. — Required the form of a beam of uniform resista/nce when it is fixed at one end and loaded at the free end. 1st. Let the sections be rectangular, and
y = the variable depth, and
u = the variable width.
Then I = -fauy* (see equation (51)), d* = iy, and
= P« = the variable load.*
* For 2P.r use the general moments as given in the table in Article 101, so far as they are applicable.
BEAMS OF UNIFORM RESISTANCE.
Hence equation (182) becomes
173
(183)
a. Let the breadth be constant ; or u = ~b ; then (183) be comes
Px = |I%2, (184)
which is the equation of a parabola, whose axis is horizontal and parameter is |ry. See Fig. 84.
FIG. 84.
FIG. 85.
b. Suppose the depth is constant, or y = d. Then (183) be comes
Px = £R^X (185)
which is the equation of a straight line ; hence the beam is a wedge, as in Fig. 85.
c. If the sections are rectangular and similar, then
u : y : : 1> : d I
and equation (183) becomes
K5
which is the equation of a cubical parabola.
2d. Let the sections be circular. Then I — -fairy* (equation (52), in which y is the diameter of the circle), and d1 = %y\ hence (182) becomes
FIG. 86.
which is also the equation of a cubical para bola, as shown in Fig. 86. 3d. Let the transverse sections be rectangular, and I con-
174
THE RESISTANCE OF MATERIALS.
stant, the breadth and depth both being variable, then equation (182) becomes
P£ = R-^^ = K^- (186)
2^ (>y
in which c is a constant, = bd*, ft and d being the breadth and depth at the fixed end. Equation (186) is the equation of the vertical longitudinal sections, and is the equation of an hyperbola
Fio. 87.
Fio. 88.
referred to its asymptotes. See Fig. 87. If the value of y from this equation be substituted in the equation uif = <?, it gives
216PV W
v
IS-
\.*\
z.n
which is the equation of the horizontal longitudinal sections ; hence they are cubical parabolas, as in Fig. 88. For x and u = 0,
y == oo, and for x = I, u = b =,-
4th. If the breadth is the wth power of the depth, and the sec tions are rectangular, then u = y*, and equation (183) becomes
which is the general equation of parabolas.
148. BEATVIS FIXED AT ONE END AND UNIFORMLY
>.— Required the form of a learn, of uniform resistance
BEAMS OF UNIFORM RESISTANCE.
175
when it is fixed at one end and uniformly loaded over its whole length" the weight of the beam being neglected.
The origin of co-ordinates being still at the free end, we have
wx = the load on a length x, and Jmzr* = the moment of the load (equation (53)). Hence, for rectangular sections, equation (182) becomes
%wx? - %Ruy* (188)
a. If the breadth is constant, or u = b in (188), it becomes
which is the equation of a straight line ; and hence the beam will be a wedge, as in Fig. 89.
' FIG. 89.
. Let the depth be constant ; or y = d in (188)
a parabola whose axis is perpendicular to the axis of the beam, as in Fig. 90.
c. Let the sections be similar ; —
then d : b : : y : u = ^y,
.*. equation (188) becomes \wy? = -J-jR-^8 ; —
CL
a semi-cubical parabola, as in Fig. 91.
FIG. 90.
Fio. 91.
176 THE RESISTANCE OF MATERIALS.
d. Let I be constant, or -fauif = -fabd*. Then equation (182) becomes
bd*
±wx* = |K --- ; — an hyperbola of the second order. J
149. PREVIOUS CASES COMBINED. — Required the form of the beam of uniform resistance when it is iixed at one end and loaded uniformly, and also loaded at the free end.
The moment of applied forces is Px+^wx*; hence equation (182) becomes, for rectangular beams,
Pa? + ±wz* = $Ruy*.
Hence, if the depth is constant, PJJ -f %wx* = %Rud* ; — a parabola ;
Hence, if the breadth is constant, P# + \wj? = £R%* ; — an ellipse ;
Hence, if the sections are similar, ~Px + %wx* — -J-ll -ry1 ; — a
a i
semi-cubical parabola.
150. AVEIGHT OF THE BEAItt CONSIDERED. - Required
the form of the beam of uniform resistance wJien the weight of the beam is the only load / the beam being fixed at one end and free at the other.
a. Let the sections be rectangular and the breadth constant.
Let x = AB ; Fig. 92,
b = the breadth, and
8 = the weight of a unit of
volume.
Then fydx = the area of ADC, and
= tlie weight of ADC ; the limits of the integration
F,0. 92.
being 0 and x. If F is the centre of gravity of ADC ; we have, from the
/* ^J
principles of mechanics, the distance AF='-;.--7— *
BEAMS OF UNIFOEM RESISTANCE. 177
The moment of the applied forces is the weight of ADC
multiplied by the distance BF = x — AF. Hence, equation (182) becomes
which reduced gives 2K
which is the equation of the common parabola, the axis being vertical.
If there is a single curve, -~- is its parameter ; but if two
T>
curves, as in the figure, -* is the parameter of each. 1). Let the depth be constant. In a similar way we find
This solved gives
in which C and C' are constants of integration, and involve the position of the origin of co-ordinates and direction of the curve at a known point.
c. Let the beam ~be a conoid of revolution, as in Fig. 93.
We have, as before
which reduced gives
FIG. 93.
a 10. ya. v
which is the equation of the common parabola.
d. Suppose, in the preceding cases, that an additional load, P, is applied at the free end.
Some of the equations which result from this condition can- 12
178
THE RESISTANCE OF MATERIALS.
not be integrated in finite terms, and hence the curves cannot be classified.
151. HE: Mis SUPPORTED AT THEIR ENDS.
A. Let the beam be supported at its ends and loaded at the middle point.
For this case, equation (182) becomes, for rectangular sec tions,
JP<B = iEWy' (192)
a. If the breadth is constant, we have
which is the equation of the common parabola.
—•
©
Fio. 94.
FJG. 95.
The beam consists of two parabolas, having their vertices, one at each support, as in Fig. 94.
b. If the depth is constant, wre have
±Px = lTUl*y; - (193)
a wedge, as in Fig. 95.
E. If the beam is uniformly loaded^ we have from equations (74) and (182),
fro (fa — x*) = $R?ty* — if rectangular, and if the breadth is constant, \w (fa — a?) = -J-1%2 ; - - - (194)
an ellipse, Fig. 96.
If the depth is constant, fro (fa — a?2) = fR^X a parabola, Fig. 97.
FIG. 96.
FIG. 97.
C. Let the beam have an uniform load and also an uni formly increasing load from one end to the other, as in Fi^.
BEAMS OF UNIFOEM KESISTANCE. 179
<
Let W = the weight of the uni- v
form load,
"VV, = the weight of the uni formly increasing load, and V = the reaction of the sup
port at the end which has the least load. Then V=$W+$Wl.
Let x be reckoned from A, then the load on x is
W W, 2
T* + y^
and the moment of this reaction and load on a section which is at a distance x from A is
(195)
which equals ^R by* for rectangular beams of uniform breadth. To find the point of greatest strain, make the first differential coefficient of (195), equal to zero. We thus find
If W = 0, this gives
x =
When "W" = 0, this becomes the case of water pressing against a vertical surface.
BEAMS FIXED AT THEIR ENDS. If the beam IB
fixed at its ends and loaded at the middle with a weight, P, we have, from equations (117) and (182), wiien the breadth is uni form,
which is the equation of a parabola. The beam really consists
of four double parabolas with their
vertices tangent to each other, as in
Fig. 99. The vertices are iZ from .—4-^ A
the end.
If the load were uniform we would obtain, in a similar way, a beam com- FIG. 99.
posed of four wedges. These are di rect deductions from theorv, but it is evident that there is some-
ISO THE RESISTANCE OF MATERIALS.
thing wanting, for a beam like Fig. 90 has no transverse strength. The same result, though not quite so glaringly appa rent at first sight, exists in all the cases which we have discussed. For instance, in figures 85, 86, and 87 the sections at the free end must have a finite value to resist the shearing stress, and the beams must be enlarged, as determined in the next article. If the section is reduced to naught, it can sustain no weight. In the present case, there is neither tension nor compression at A and B, as was shown in articles 99 and 100 ; but there is a transverse shearing stress at those points, and there must be suf ficient transverse section to resist it. The same remark applies to the preceding cases, and the forms must all be modified to meet this condition, as is shown in the next article.
] 53. EFFECT OF. TRANSVERSE SHEARING STRESS Oil
modifying the forms of the beams of uniform resistance.
The value of the transverse shearing stress is given in Article 84. For instance, in the case of a beam uniformly loaded, it is V — wx = fywl —wx = %w(l — 2#) at any point in the length. This quantity, divided by the product of the breadth and modu les of strength for transverse shearing, gives the depth neces sary to sustain this force. Take, for example, case A, Article 134:. The load being uniform, we have S« = fyo(l — 2x) as given above, which is the equation of a straight line, Fig. 100, in which
Fio. 100. Fio. 101.
AB = $wl ~- (b x modulus of shearing).
Hence we would at first thought naturally infer that the form of the beam of uniform strength in this case would be found by adding the ordinates of the straight line, AC, to the correspond ing ordinates of the ellipse, thus giving Fig. 101. But as soon as this is done the equation of moments is changed ; for the lever arm of the force is increased, and the moment of resisting
BEAMS OF UNIFORM RESISTANCE. 181
forces is greater. To avoid this difficulty we would add the section which is necessary for sustaining the shearing stress, to the side of the beam. But in all those cases where the depth, as found by moments, is zero, this method is imprac ticable, for the thickness to be added would be infinite. It seems, then, that to solve this case theoretically, we must add some arbitrary quantity to the depth as found by moments, which quantity shall increase the section so as to fully resist the moment of the applied forces, and, in addition thereto, PARTLY resist the shearing stress, and then a section must le added to the side of the beam which shall sustain the remain der of the shearing stress.
Tabulated values of shearing stresses for several of the cases which have been considered. The values in the fourth column of the following table may be found according to the principles given in Article 84, or they may be found by taking the first differential coefficient of the moments of applied forces.*
* The third of equations (42a) is £P# — £F sin. a x x = EF,y, and since the lever arms, x and y, of the forces are always linear quantities, we may enter under the sign £ and differentiate them. This gives LPdx — £F sin adx —
dx (SPi)e^, or — [£P — £F sin a] = £F, which, combined with the second of
(42«), gives £P=EF sin a=&s. Hence we have this simple rule : Ss is the first differential coefficient of the moments oftlie applied forces.
When the bending moment has an algebraic maximum, the abscissa of the point of greatest bending stress may be found by making the first differential coefficient of the moment of the stress equal to zero, and solving for x ; hence, in this case, the bending moment is greatest where the shearing stress is zero,
''w
T~.
182 THE RESISTANCE OF MATERIALS.
A TABLE OF MOMENTS AND SHEARING STRESS.
Number of the CascV Seepage^ |
Condition of the beam. |
General momenta of the Applied Forces. See page 130. |
Shearing Stress. Br |
I. |
Fixed at one end and P at the free end. |
PZ. Eq. (35). |
P. |
II. |
Fixed at one end and uniformly loaded. |
iirz2. Eq. (40). |
wx. |
IV. |
Supported at the ends and P at the mid dle. |
iPx. Eq. (44). |
ip. |
V. |
Supported at the ends and uniformly load ed. |
:v« "7*f \wlx-\wx\ Eq. (48). |
$wl— wx. |
VII. |
Fixed at one end, sup ported at the other, and P anywhere. |
For the part AD, Fig. 43, Eq. (04). For the part DB, Fig. 43, Eq. (67). |
(-l + K-(3-7i))P. i»9(3— n)P. |
VIII. |
Fixed at one end, sup ported at the other, and uniformly load ed. |
^(^r2— 3£r). Eq. (87). |
>M, |
.DC |
Fixed at the ends and P at the middle. |
W-4x)P. Eq. (94). |
*. |
X. |
Fixed at the ends and uniformly loaded. |
£<P-«*+<W). Eq. (102). |
\id— MCX. |
Art. 150. |
Fixed at one end, and the weight of the beam the load. |
Sbxfyfa-lbfxydx. |
Mfydx. |
C. Art. 151. |
Supported at the ends, uniform load, also load uniformly in creasing. |
(|W+iW,)aJ-~- - WV |
-«*-?-¥ |
BEAMS OF UNIFORM RESISTANCE. 183
If a beam is supported at its ends, and loaded with several
weights Pt, P2, P3, etc., as in Fig. 102, we may readily find the shearing stress at any point by article 84. It is there shown that the shearing stress FfG. 102 = .TP, where 2,'P equals the
algebraic sum of all the ver tical forces, including the reaction at the abutment. Hence, we have for the shearing stress
between the end and P, == Y ;
between P, and P2 = Y - P, ;
between P2 and P3 = Y - Px — P2 ;
between P3 and P4 = Y - - P, - P2 - P3 ; etc. If the weights are equal to each other = P, we have P = Pt = P2 =P3, etc. ; and if there are n of them, and they are sym metrically placed in reference to the centre of the beam, we have
Y = JwP.
If n is even, we have, at the centre of the beam, the transverse shearing stress = &iP — %nP = 0 - - (197) ;
and if n is odd, there will be a weight at the centre, and each side of the central weight we have
transverse shearing stress = %n~P — J(?i±l)P — ± JP - (198).
154. UNSOLVED PROBLEMS — Many practical problems in re gard to the resistance of materials cannot be solved according to any known laws of resistance. Some of these have been solved experimentally, and empirical formulas have been de duced from the results of the experiments, which are sufficiently exact for practical purposes, within the range of the experi ments. The resistance of tubes to collapsing, the strength of columns, and the proper thickness of the A7ertical web of rails, are such problems which have been solved experimentally. The following problems are of this class, and have not been solved. The first four are taken from the Mathematical Monthly, Yol. 1., page 148.
1
1. Required a formula for the strength of a circular flat iron
184
THE RESISTANCE OF MATERIALS.
plate of uniform thickness, supported throughout its circumfer ence and loaded uniformly.
2. Required the strength of the same plate if the edges are bolted down.
3. Required the equation of the curve for each of the pre ceding cases, that they may have the greatest strength with a given amount of material.
4. In the preceding problems, suppose that the plate is square.
5. Required the form of a beam of uniform strength which is supported at its ends, the weight of the beam being the only load. Suppose, also, that it is loaded at the middle.
The latter part of this problem has received an approximate solution under certain conditions, as will be seen from the fol lowing experiments.
15*>. BEST FORM OP CAST-IRON BEAM AS FOUND EXPERI-
3IENTAL.L.Y. — Cast-iron beams were first successfully used for building purposes by Messrs. Boulton and Watt. The form of the cross-section of the beams which they used is shown in Fig. 103. More recent experiments show that this is a good form, but not the best.
About 1822 Mr. Tredgold made an experiment upon a cast-iron beam of the form shown in Fig. 104, to determine its deflection. lie recommended this form for beams.
Mr. Fairbairn has justly the credit of making the first series of experiments for determining the best form of the If cam. These experiments were prose cuted by himself for a few years, beginning about 1822, and continued still later by Mr. Ilodgkinson.
The experiments quickly indicated that the lower flange should be considerably the largest.
Fig. 103.
Fig. 104.
BEAMS OF UNIFORM RESISTANCE. 185
The following experiments were made by Mr. Ilodgkinson (Fairbairn on Cast and "Wrought Iron, p. 11).
Fig. 105.
Fio\ 105 shows the elevation and cross-section of a beam whose dimensions are as follows : —
Area of top rib =1. 75 x 0.42 =0.735 inches. Area of bottom rib =1.77 x 0.39 = 0.690 « Thickness of vertical rib, 0.29 "
Depth of the beam,- - 5.125 "
Distance between the supports, 54.00 " Area of the whole section, - - 2.82 square inches. Weight of the beam, - 36J pounds.
Breaking weight, 6,678 pounds.
The form of the fracture is shown at b n r. It broke by tension.
EXPERIMENT IV.
Dimensions. Inches.
Thickness at A = 0.32 « « B = 0.44 « " C = 0.47 « « FE = 2.27 " " DE = 0.52
Depth of the beam = 5.125 R* ioe.
Area of the section = 3.2 square inches. Distance between the supports = 54 inches. Weight of casting = 40| Ibs. Deflection with 5,758 Ibs. = 0.25 inches.
« " 7,138 " =0.37 "
Breaking weight 8,270, Ibs.
186 THE RESISTANCE OF MATERIALS.
FIG. 107.
Dimensions in inches :—
Area of top rib = 2.33 x 0.31 = 0.72. " " bottom rib = 6.67 x 0.66 = 4.4.
Ratio of the area of the ribs = 6 to 1.
Thickness of vertical part = 0.266.
Area of section, 6.4.
Depth of beam, 5-J.
Distance between the supports, 54 inches.
"Weight of beam, 71 Ibs.
This beam broke by compression at the middle of the length with 26,084 Ibs.
It is probable that the neutral was very near the vertex n, or about £ the depth.
EXPERIMENT 21.
pr
n<
Fio. 108.
This was an elliptical beam, Fig. 108.
Dimensions in inches : —
Area of top rib = 1.54 x 0.32 = 0.493.
" " bottom rib = 6.50 x 0.51 = 3.315. Ratio of ribs, 6£ to 1. Thickness of vertical part = 0.34. Depth of beam, 5J-. Area of the section, 5.41. Distance between supports, 54 inches. Weight of beam, 70f Ibs.
BEAMS OF UNIFORM RESISTANCE. 187
Broke at the middle by tension with 21,009 Ibs. Form of fracture 1) n r ; l> n = 1.8 inches.
As these beams have all the same depth and rested on the same supports, 4 feet 6 inches- apart, their relative strengths will be approximately as the breaking weight f divided by the area of the cross section.
In Experiment 1, 6,678 ~ 2.82 = 2,368 Ibs. per square inch. " « 14,8,270 ^3.2 =2,584 "
" " 19,26,084-^6.4 =4,075 « «
" " 21, 21,009 ~ 5.41 = 3,883 " «
It is evident from these experiments, that when the vertical rib is thin, the area of the lower rib should be about 6 times that of the upper. In the 19th experiment it has already been observed that the beam broke at the top, and in the 21st it broke at the bottom, although the lower flange was larger in proportion to the upper than in the preceding case, and the comparison shows that they were about equally well propor tioned. They should be so proportioned that they are equally liable to break at the top and bottom.
A beam proportioned so as to be similar to either of the two last forms above mentioned may be called a " TYPE FORM."
156. HODGKiNSON's FORMULAS for the strength of cast- iron beams of the TYPE FORM.
Let TF — the breaking weight in tons (gross).
a — the area of bottom rib at the middle of the beam. d = the depth of the beam at the middle. and I = the distance between the supports. Then according to Mr. Hodgkinson's experiments we have
W= 26 y- when the beam was cast with the bottom
i
rib up, and
TF"=24-- when the beam is cast on its side.
157. EXPERIMENTS ON T RAILS.— Experiments on T bars, supported at their ends and loaded at the middle, gave the fol lowing results : — *
* Mahan's Civ. Eng., pp. 88 and 89; Barlow on the Strength of Materials, p. 183.
L
188 THE RESISTANCE OF MATERIALS.
Hot blast bar, rib upward, J_ broke with - 1,120 pounds.
" " downward, T broke with 364
Cold blast " " upward, J_ broke with. 2,352 "
" " downward, J broke with - 980 "
The ratio of the strengths is nearly as 3 to 1, but according to the table in Article 47, we might reasonably expect a higher ratio. If a greater number of experiments would not liave given a higher ratio, we would account for the discrepancy by supposing that the neutral axis moved before rupture took place, or that the ratio of the crushing strength and tenacity is less for comparatively thin castings than for thick ones. It is known that the crushing strength of thin castings is proportion ately stronger than thick ones. Ilodgkinson found that for castings 2, 2-£, and 3 inches thick, the crushing strengths were as 1 to 0.780 to 0.756 ; and Colonel James found a greater in crease — being as 1 to 0.794 to 0.624. See also Article 37.
158. WROUGHT-IRON ii i-:.%. TIN. — The treacherous character of cast-iron beams, on account of the internal structure of the metal, and the unseen cracks and flaws which may exist, has led to the introduction of solid wrought-iron beams. When cast- iron beams were first used, it was practically impossible to manufacture solid wrought-iron ones, but the great improve ments which have been made since then in the processes of manufacturing, have not only made their construction possible, but they have enabled the manufacturer to produce them so cheaply as to bring them within the means of those who desire such articles. At Trenton and Pittsburg they make rolled beams from a single pile,* but it is stated that by this method they can make beams only about nine inches in depth. At Buffalo and Pho3nixville they use Mr. John Griffin's patent, which consists in rolling the flanges separately, piling the plates for the web between them, and then rolling and welding the whole together. By this method they can make beams at least twenty inches deep, and of any desired length. There is no attempt to make them of uniform strength. They are of the double T (I) pattern, and of uniform section throughout.
* Jour. Frank. Inst., Vol. 80, p. 231.
TORSION. 1S9
CIIAPTEE VIII.
TORSION.
159* TORSIVE STRAINS are very common in machinery. In all cases where a force is applied at one point of a shaft to turn (or twist) it, and there is a resisting force at some other point, the shaft is subjected to a torsi ve strain. The wheel and axle is a familiar case in which the axle is subjected to this strain. To produce torsion without bending, two equal and par allel forces, acting in opposite directions, and lying in a plane which is perpendicular to the axis of the piece, must be so applied to the section that the arms of the forces shall be equal. In other words, mechanically speaking, a couple whose axis coincides with the axis of the piece, must be applied to the piece. If only a single force, P, is applied, as in Fig. 109, the piece is pushed sidewise at the same time that it is twisted ; but the amount of twisting is the same as if the force, P, were divided into two, each equal ^P, and each of these acted on opposite sides of the axis and in opposite directions, and at a distance from the axis equal AB, Fig. 109. For, the moment of the couple thus formed, is -|Px2x AB=P.AB, which is the moment of P.
16O. THE ANGI^E OP TORSION is the angle through which a, fibre whose length is unity, and which is situated at a unit's distance from the axis, is turned by the twisting force. It depends for its value, in any case, upon the elastic resistance to torsion, as well as upon the dimensions of the piece and the twisting force. The analysis by which its value is determined is founded upon the following hypotheses, which are approxi mately correct.
190
TITE RESISTANCE OF MATERIALS.
1st. The resistance of any fibre to torsion varies directly as its distance from the axis of the piece.
2d. The angular amount of torsion of any fibre between any two sections, or the total angle of torsion, varies directly as the distance between them.
It is found by experiment, that these hypotheses are suffi ciently exact for cylinders and regular polygonal prisms of many sides. They assume that transverse sections which were plane before twisting, remain so while the piece is twisted, but in reality the fibres which were parallel to the axis before being twisted are changed to helices, and this operation pro duces a longitudinal strain upon the fibres ; and this, in turn, changes the transverse sections into warped surfaces.*
To find the angle of torsion : — Let I = AD = the length of the piece,
Fig. 109.
a = AB = the lever arm of P. P = the twisting force. * = aAL = the total angle of tor
sion, or
angle
through
Fig. 109.
which Aa has been twist ed.
6 - 7 = " The Angle of Torsion," f/
— supposed to be small.
y(£, <p) == the equation of a transverse section, and G = the coefficient of the elastic resistance to torsion, which is the force necessary to turn one end of a unit of area and unit of length of fibres through an angle unity, the vertex of the angle being on the axis of torsion, one end of the fibres being fixed and the twisting force being applied directly to the other end, and acting in the direction of a tangent to the arc of the path described by the free end.
As a unit of fibres cannot be placed so that all of them will be at a unit's distance from the axis, we must suppose that the resistance of a very thin annulus, which is at a unit's distance, is proportional to that of a unit of section ; or the resistance
* Resume des Lemons, Navier, Paris, 1864, p. 270 and several other pages following.
TORSION. 191
of an element at a units' distance from the axis is G multiplied by its area ; which expressed analytically is
Grpdpd<f>,
and according to the first law
Gp*dpd(f> = the resistance of any
fibre whose length is unity, to being twisted through an angle unity ; and the moment of resistance = Gp3dpd<f> for an angle unity ; and for any angle 6 the moment is, according to the second law,
Gdp'dpd<f>
and the total moment equals the moment of the applied force, or moments of the applied forces ; hence
Pa = G&p'dpdf = GI;J, where T.p is the polar moment of inertia of the section.
r r:
I jP dpd<j>
Jo Jo
/ / p3dpd<j> = £-7iT4 (199)
ttffr 7r*r*
2Pa
or, 6 = ~ — 4 GTJT
161. THE VALUE OF THE COEFFICIENT G may be found from equation (200). M. Cauchy found analytically on the con dition that the elasticity of the material was the same in all directions, that G = f E.* M. Duleau found experimentally that G is less than J- E, and nearly equal -J- E, f and M. Wert- heim found G = f E nearly. $ M. Duleau's experiments gave the following mean values for G : f
Pounds.
Soft iron -•'-•- 8,533,680
Iron bars 9,480,917
English steel - - 8,533,680
Forged steel (very fine) ----- 14,222,800 Cast iron - 2,845,600
* Resume des Lemons, Navier, Paris, 1856, p. 197. f Resistance des Materiaux, Morin, p. 461. $ L'Engineer, 1858, p. 52.
192
THE RESISTANCE OF MATERIALS.
Pounds.
Copper ......... 6,209,670
Bronze - 1,516,150
Oak ..... 568,912
Pine - 615,472
Example. — If an iron shaft whose length is 5 feet, and diameter 2 inches, is twisted through an angle of 7 degrees by a force P = 5,000 Ibs. , acting on a lever, a — 6 inches, required G. The 7 degrees is first reduced to axe by multi
plying it by ~Q, which gives a = ^, and Eq. (200) gives,
2~ Tf
1G3. TORSION^ ABlflnriiUM.—If a prism is suspended from its l\ upper end, and supports an arm at its lower end, and two weights each equal \ W are fixed on the arm at equal distances from the prism, and the prism be twisted and then left free to move, the torsional force will cause an angular movement of the arm until the fibres are brought to their normal position, after which they will be carried forward into a new position by the inertia of the moving mass in the weights | W until the torsional resistance of the prism brings them to rest, after which they will reverse their movement, and an oscillation will result. The conditions of the oscillation may easily be inves tigated if the prism is so small that its mass may be neglected.
For, equation (200) readily gives :
V
from which it appears that the torsional force P varies as the space (ao) over which it moves.
It is a principle of mechanics that the moving force varies directly as the product of the moving mass multiplied by the acceleration. Hence, if x = (flo), the variable space, and t = the variable time, and M = the mass moved, and observing that t and x are inverse functions of each other, and the above prin ciple of mechanics gives the following equation : —
«•*-*—•?•
Multiplying both members by the dx, gives
W dxffix _ irGr4 g df '' 2la* a
where W is the weight of the mass moved, and g is the acceleration due to gravity. The oscillations commence at the extremity of an arc whose length
TORSION. 193
is s, at which point the velocity is zero. The integral of the last equation between the limits s and x is
" r -Vi * _ I TrWto2 * LS1D T--2G~
which is the time of half an oscillation. For a whole oscillation :
This is essentially the theory of Coulomb's torsion pendulum. A torsion pendulum was used by Cavendish in 1778 to determine the density of the earth. (See Royal Philosophical Transactions: London, Vol. 18, p. 388.) He found the mean density of the earth by this method to be 5.48 times that of water. This is considered the most reliable of all the known methods, but the results of other methods exceed the value given above by a small amount only, thereby confirming this result and showing that the mean density of the earth is about 5£ times that of water.
1 63. RUPTURE BY TORSION — The resistance which .a bar offers to a twisting force is a torsional shearing resistance, and in regard to rupture, the equation of equilibrium is founded upon the following principles : —
1st. The strain upon any fibre varies directly as its distance from the axis of torsion ; and,
2d. The sum of the moments of resistances of the fibres equals the sum of the moments of the twisting forces.
Let S = the MODULUS OF TORSION, that is, the ultimate resist ance to torsion of a unit of the transverse section which is most remote from the axis of torsion. It is the ultimate shear ing resistance to torsion, but may be used for any shearing strain which is less than the ultimate,
dt •=. the distance of the most remote fibre from the axis of torsion,
f (0? 0) — the equation of the section,
P — the twisting force, and
a = the lever arm of P.
Ip = the polar moment of inertia of a section.
Then pdpdQ = dA. = the area of an element of the section ; • 13
194 THE RESISTANCE OF MATERIALS.
= the shearing strain of the most re mote element ; and, by the first prin- ciple given above,
= the shearing strain of any element, which is at a unit's distance from the axis of torsion, according to the first principle above ; and from the same principle we have
— p'dpcfy = the shearing strain of any element, 1/1 ,, ; ( and this, multiplied by the distance of
the element, p, from the axis, gives
Q
— p\lpd<f> = the moment of resistance to torsion. - di Hence, according to the second principle we have
A.= lP - (20!) .
For circular sections, we have already found, Eq. (199),
lp = i- TT/.
For square sections, whose sides are &, we may find .* Ip = -J- £', and A =1>V±
iUu< '*-
164. PRACTICAL FORMULAS.— Equations (199) and (201) give for cylindrical pieces, observing that d, = r, J>
9 T*/7
. P* = ^s,-,.s = -_^. >(202)^
(/ &_ ^L- ""7^
If cylindrical pieces are twisted off by forces which form a coujtKe, and P, «, and r measured, the value of S may be found from equation (202). Cauchy found S = % H,f which is con sidered sufficiently ex^efc when a proper coefficient of safety is used. Calling S = 25,000 pounds for iron, and using about a
We have AH/A = ff^+y*] <?A = /zVA +//KA, that is, the polar
moment equals the sum of the rectangular moments, the origin being the same in both cases. In this case the origin being at the centre of the square, we
r /* r
have / ortfA = / y'dfA .: Ip = 2/y?efA =°2 x &P (see Eq.l^)/ *)
.., f Resume des Lemons, Navier. Paria, 1856, pp. 193-203, and p. 507.
v
f'j fo,£.c»-^~4-4--y. I,>CL r-c^-^ — w
\
....*
v
_ TORSION. 319*^™
five-fold security ; and S = 8,000 pounds for wood, and using
about a ten-fold security, and we may use for
. ROUND IRON SHAFTS (wrought f\ ^
or cast), diameter 9^=^^J: SQUARE IRON SHAFTS (wrought
or cast), side of the square-j£= -J^f/Pa SQUARE WOODEN SHAFTS, •
side of the squares T' = -J- ft Pa)
The dimensions given by these formulas are unnecessarily large for a steady strain, but shafts are frequently subjected to sudden strains, amounting sometimes to a shock, and in these cases the results are none too large.
Practical formulas may also be established on the condition that the total angle of torsion shall not exceed a certain amount. Making G = | E, and solving (200) in reference to /-, and we have for cylindrical shafts,
£Y_ 2/"L ^ : .'J5U
r _ 4/16 Pal
~~V 3?rE* ' and similarly for square shafts,
= .
v^^
In these expressions P should not be so great as to impair the elasticity, — say for a steady strain P should not exceed the values given by equation (203).
If «° is given in degrees, it is reduced to arc by multiplying
^ ^ 180 S° tllat " ~ 180*° ' nence tlie Precediiig equations be come : for cylindrical iron shafts,
= 3-14v-w-; -
C A and for square iron shafts,
Examples— \. A round iron shaft fifteen feet long, is acted upon by a-, weight P = 2,000 Ibs. applied at the circumference of a wheel which is on the
196 THE RESISTANCE OF MATERIALS.
shaft, the diameter of the wheel being two feet ; what must be the diameter of the shaft so that the total angle of torsion shall be 2 degrees ? If the shaft is cast-iron E = 10,000,000, and
2000x12x15x12 -
2. A round wooden shaft, whose length is 8 feet, is attached to a wheel whose diameter is 8 feet. A force of 200 pounds is applied at the circumfer ence of the wheel, what must be the diameter of the shaft so that the total angle of torsion shall not exceed 2 degrees ?
> =,.»„»,..
For further information upon this subject see u Resistance des Materiaux," Navier ; Paris, 1856, pp. 237-509, and the exhaustive articles of Chevandier and Wertheim in " Annales des Chemie et Physique," Vol. XL. and Vol. L.
165. RESULTS OF WERTHEIItt'S EXPERIMENTS. A few
years since M. G. Wertheim presented to the French Academic des Sciences an exhaustive paper upon the subject of torsion, the substance of which was published in the Annales de Chimie et de Physique, Yol. XXIII., 1st Series, and Yol. L., 3d Series. These articles would make a volume by themselves, and hence we will content ourselves at this time with presenting his
CONCLUSIONS.
When a body of three dimensions is subject to torsion the following facts are observed :—
1st. The torsion angle will consist of two parts, one tempo rary, the other permanent ; the latter augments continually, though not regularly.
2d. The temporary displacements augment more and more rapidly than the moments of the applied couples, and the increase of the mean angle, which in hard bodies continues until rup ture, in soft bodies continues only to the point where the body commences to suffer rapid and continuous deformation.
3d. The temporary angles are not rigorously proportional to the length, and, all else being equal, the disproportionality in creases in measure as the bar becomes shorter.
4th. In all homogeneous bodies, torsion caused a diminution of the volume, which is proportional to the length and square •of the angle of torsion, and each point of the body, instead of
TORSION. 1 97
describing an arc of a circle, follows the arc of a spiral. The condensation of the body increases from the centre to the cir cumference.
5th. In bodies with three angles of elasticity, the change of volume and resistance to torsion are functions of the three axes, and the relation between them may be such that the volume will augment.
6th. Circular or turning vibrations of great amplitude are difficult to produce, and as small angles of torsion only are used, the preceding conclusions apply to this case.
7th. Rupture produced by torsion usually takes place at the middle of the length of the prism ; it commences at the dan gerous points, and operates by slipping in hard bodies and by elongation in soft ones.
8th. With regard to the influence of the figure and absolute dimensions of the transverse sections of the bodies, we derive the following conclusions : —
9th. In homogeneous circular cylinders the diminution of the volume is equal to the original volume multiplied by the prod uct of the square of the radius, and the angle of torsion for a unit of length (the angle being always very small). Further, under torsion the radius of the cylinder equals the primitive radius multiplied by the sine of the angle of inclination of the helicoidal fibres. This last gives a means of calculating the diminution of volume. But in reality the twisted cylinder takes the form of two frustra of cones joined at the smaller bases ; and although this does not sensibly affect the theoretical results for long cylinders, yet it deprives our formulas of all their value in ordinary practical cases.
198 THE RESISTANCE OF MATERIALS.
CHAPTER IX.
EFFECT OF LONG-CONTINUED STRAINS— OF OFT-REPEATED STRAINS, AND OF SHOCKS— REMARKS UPON THE CRYSTAL LIZATION OF IRON.
EFFECT OF LONG-CONTINUED STRAINS. .
166. GENERAL EFFECT. — The values of the coefficients of elasticity and the moduli! of tenacity, crushing, and of rupture were determined from strains which were continued for a short time — generally only a few minutes — or until equilibrium was apparently established ; and yet it is well known that if the strain is severe, the distorsion, whether for extension, compres sion, or bending, will increase for a long time ; and as for rup ture, it always takes time to break a piece, however suddenlv rupture may be produced. By sudden rupture we only mean that it is produced in a very short time.
The increased elongation due to a prolonged duration of the strain beyond a few minutes, will affect the coefficient of elas ticity but very slightly, for the strains which are used in deter mining it are always comparatively small, and the greater part of the effect is produced immediately after the stress is applied. Still, if the distortion should go on indefinitely, no matter how slowly, the elasticity, and hence the coefficient, would be greatlv modified by a very great duration of the stress, however small the stress may be ; and at last rupture would take place. If the basis of this reasoning be well founded, we might reasonablv fear the ultimate stability of all structures, and especially those in which there are members subjected to tension. But the con tinued stability of structures which have stood for centuries, teaches us, practically at least, that in all cases in which the strain is not too severe, equilibrium is established in a short time between the stresses and strains, and in such cases the piece will sustain the stress for an indefinitely long time.
TORSION.
100
. IIODGKINSON'S EXPERIMENTS TllC results of the
experiments which are recorded in Article XL., page 48, show that in one case the compression increased with the duration of the strain for three-fourths of an hour. In the case of exten sion 011 another bar, as shown in Article VII., page 7, it ap pears that the same weight produced an increased elongation for nine hours ; but during the last, or tenth hour, there was no increase over that at the end of the ninth hour.
In both these cases the strain was more than one-half that of the ultimate strength.
1<58. VICAT»S EXPERIMENTS. — M. Yicat took wrought- iron wire and subjected it to an uniform stress for thirty-three months. The elongations produced by the several weights were measured soon after the weights were applied, and total lengths determined from time to time during the thirty-three months. It was found for all but the first wire, as given in the following table, that the increased elongations after the first one were very nearly proportional to the duration of the stress. (Annales de Ohemie et Physique, Vol. 54, 2d series.)
TABLE Of the Result* of M. VicaVs Experiments on Wrought-iron Wire.
Amount of Strain. |
IIJ |
Increased Elongation after 33 months. |
S o ^ **** 18*3 |
||
i of its ultimate tensile strain. . . |
No additional increase. |
|
£ of its ultimate tensile strain . . . |
s 8 g | |
0.027 of an inch per foot. |
$ of its ultimate tensile strain . . . |
&t &^ |
0.040 of an inch per foot. |
f of its ultimate tensile strain . . . |
£ "3 £ |
O.OG1 of an inch per foot. |
* FAIRBAIRN'S EXPERIMENTS. — Fairbairn made ex periments upon several bars of iron, which were subjected to a transverse strain, the results of some of which are recorded in the following tables. (See Cast and Wrought-iron, by Wm. jFairbairn.) The bars were four feet six inches between the supports, and weights were applied at the middle, and permitted
200
THE RESISTANCE OF MATERIALS.
to remain there several years, as indicated by the tables. The deflections were noted from time to time, and the results were recorded.
TABLE I.
In icJiicJi the Weight Applied was 336 pounds.
TEMPERATURE. |
•H |
3-s |
ast, — ileflec- in incheK. |
ii |
«l |
si |
|| |
If! |
|
" |
6 |
™ |
78° |
March 11, 1837. . June 3, 1838 |
1.270 1 316 |
1.461 1 538 |
Cold-blast, 0 661 • 1 |
72° |
July 5, 1839 |
1 305 |
i :>:•,:{ |
Hot-blast |
61° |
June 6, 1840 |
1.303 |
1 520 |
0 694 • 1 |
50° 58° |
November 22, 1841 . April 19, 1842 |
1.306 1 308 |
1.620 1 620 |
|
Mean |
1 301 |
1 548 |
||
Previous to taking the observations in November and April the hot-blast bar had been disturbed.
In regard to this experiment Mr. Fairbaini remarks : — " The above experiments show a progressive increase in the deflec tions of the cold-blast bar during a period of five years of 0.031 of an inch, and of 0.087 of the hot-blast bar." The numerical results are found by comparing the first deflection with the mean of all the observed deflections. But an examination of the table shows that the greatest deflection, which was ob served in both cases, was at the second observation, which was about a year and a quarter after the weight was applied, and during the next two years the deflections decreased 0.015 of an inch for the cold-blast, and 0.018 of an inch for the hot-blast bar. After this the deflections appear to increase for the cold- blast bar 0.005 of an inch the next two years. Considering all the particulars of these experiments it does not seem safe to conclude that the deflections would go on increasing indefinitely
TORSIOX.
201
with a continuance of the load. Admitting that the small in crease of deflections during the last two years are correct and not due to errors of observation, and we see no reason why the deflections would not be as likely to decrease after a time as they were after the first year.
TABLE II.
In which the Bar was Loaded with 392 pounds.
* |
4 |
2% |
||
i |
•8J |
<c « |
•a 9 |
|
TEMPERATURE. |
0 |
ll |
Jl tS =3 |
;% |
sS |
|S |
3 a 4i.2 |
.2 c'3 |
|
c« -^ |
||||
ft |
° |
w |
^ |
|
March 6 1837 |
1 684 |
1 715 |
||
78' |
June 23, 1838. . |
1 824 |
1 803 |
For cold-blast |
72° |
July 5, 1839 |
1 824 |
1.798 |
0 771 • 1 |
61° |
June 6, 1840 |
1 825 |
1.798 |
For hot-blast, |
50° |
November 22, 1841 . |
1.829 |
1.804 |
0.805 : 1 |
58° |
April 19, 1842 |
1.828 |
1.812 |
|
1.802 |
1.788 |
|||
Here we see a general increase in the deflections from year to year, being very regular in the cold blast, and quite irregular in the hot blast. But we observe that the increase is exceed ingly small after the first year, being only 0.004 of an inch in the cold blast bar, and 0.009 of an inch in the other.
202
THE RESISTANCE OF MATERIALS. TABLE III.
I |
TJ |
!l |
ill |
|
TEMPERATURE. |
£ ^f |
|||
•g |
5.9 |
"£ C |
C ^ -5 |
|
il |
11 |
|1| |
||
G |
o |
r* |
" |
|
March 6, 1837 |
.410 |
' 00 |
||
78° |
June 23, 1838 |
.457 |
Is |
Cold-blast |
72° |
July 5, 1839 |
.446 |
||
61° |
June 0 1840 . |
445 |
'> |
0 881 • 1 |
50° |
November 22. 1841 . |
.449 |
'©£"3 |
|
58° |
April 19, 1842 |
449 |
£ -^ 9 |
|
1-1 .2 o |
||||
Mean. |
1 442 |
|||
"We find from this table, as from Table I., that the maximum deflection was observed about a year and a quarter after the weight was applied, and that it decreased during the next two years, after which it slightly increased. The deflections were the same at the two last observations. These changes took place under the severe strain of more than four-fifths of the breaking weight. These experiments indicate that for a steady strain which is less than three-fourths of the ultimate strength of the bar, the deflection will not increase progressively until rupture takes place, but will be confined within small limits.
17O. ROEBLINGS OBSERVATIONS. — The old Moiiongahcla bridge in Pennsylvania, after thirty years of severe service, was removed to make place for a new structure. The iron which was taken from the old structure was carefully examined and tested by Mr. Iloebling, and found to be in such good condition that it was introduced by him into the new bridge.*
lie also found that the iron in another bridge over the Alle- ghany river was in good condition after forty-one years of service.
* Roebling's Report on the Niagara Railroad Bridge, 1800, p. 17; Jour. Frank. Inst., 18CO, Vol. LXX., p. 301.
TORSION. 203
OFT-REPEATED STRAINS.
Xearly all kinds of structures are subjected to greater strains at certain times than at others, and some structures, as bridges and certain machines, are subject to almost constant changes in the strains. Loads are put 011 and removed, and the operation constantly repeated. The only experiments to which we can refer for determining the effect of a load which is placed upon a bar and then removed, and the operation of which was fre quently repeated, are those of Win. Fairbairn, made in I860.* The beam was supported at its ends, and the weight which pro duced the strain was raised and lowered by means of a crank and pitman, as shown in Fig. 110.
The {rearing; was connected
C5 c5
with a water-wheel, which was
kept in motion day and night,
and the number of changes of
the load were registered by an automatic counter. The beam
was 20 feet clear span and 16 inches deep. The dimensions of
the cross section were as follows : —
Top— Plate, 4 x i= 2.00 sq. inches
Angle irons, 2 x 2 x ^= . . . 2.30 " "
Bottom— Plate, 4 x i= f .00 " "
AngleIrons,2x2xfV=1.40 " " Web— Plate, 15J x £= 1.90 «
Total 8.60 " "
Weight of beam, 1 cwt. 3 qr. 3 Ibs. Probable breaking weight, 9.6 tons. '~3r~V'Z
First Experiment. — Beam loaded to J the breaking weighf: —
Total applied load 5,809 Ibs. - £ J? fa
Half the weight of the beam 434 « „
Strain on the bottom flange 4.3 tons
Margin of strength by Board of
Trade v....,...:i:^ 3.4
* t!iT. Eng.'and Arch. Jour., Vol. 'xXIlt, p. 257/and Vol. XXIV., p. 237.
204
THE BE3ISTANCE OF MATERIALS.
TABLE
Of the Results of Experiments made upon a Beam which was Supported at its Ends, and a Weight repeatedly but gradually Ajyplied at the Middle.
DATE. |
No. of Changes. |
1 *l f o |£ |
DATE. |
No. of Changes. |
Donoction at Cen tre of Beam. |
1800. March 21 |
0.17 |
1800. April 13 |
268 328 |
0 17 |
|
22 |
10,540 |
0.18 |
14 |
281 210 |
0 17 |
23 |
15,610 |
0 10 |
17 |
321 015 |
0 17 |
24 |
27,840 |
20 |
343 880 |
0 17 |
|
26 |
46 100 |
0 10 |
25 |
390 430 |
0 10 |
27 |
57,790 |
0 17 |
27 |
408 204 |
0 10 |
28 |
72,440 |
0.17 ! |
28 |
417,940 |
0 10 |
29 |
85,960 |
0.17 |
May 1 |
449 280 |
0 16 |
30 |
97,420 |
0.17 |
3 |
408 600 |
0 10 |
31 |
112 810 |
0 17 |
5 |
489 709 |
0 10 |
April 2 |
144350 |
0 10 |
7 |
512 181 |
0 16 |
4 |
165,710 |
0 18 |
9 |
536 355 |
0 16 |
7 |
202,890 |
0.17 |
if |
500 529 |
0.16 |
10 |
235,811 |
0.17 |
14 |
596.790 |
0.1(5 |
At this point, after half a million of changes, the beam did not appear to be damaged. At first it took a permanent set of 0.01 of an inch, which did not appear to increase afterwards, and the mean deflection for the last changes was less than for the first. For the last seventeen days the deflection was uni form, but for the first seventeen days it was variable.
The moving load was now increased to one-third the break ing weight, = 7,406 Ibs., with the following results : —
1 |
1 1-1 |
•8 1 |
1 |
||
c |
a |
||||
DATE. |
-C |
g |
DATE. |
J |
G |
i'S |
S3 |
f ^ |
3 |
||
d^ |
1 |
C " |
1 |
||
fc |
« |
55 |
G |
||
1800. |
1860. |
||||
May 14 |
022 |
June 7 |
217 300 |
||
15 |
12,623 |
0 22 |
9.' |
236 460 |
0 21 |
17 |
36,417 |
0.22 |
12. |
264 220 |
0 21 |
19 |
53 770 |
0 21 |
16 |
292 600 |
0 21 |
22 |
85,820 |
022 |
21 |
327 000 |
0 °2 |
26 |
128,300 |
022 |
23 |
350 000 |
0 23 |
29 |
161,500 |
022 |
25 |
375 650 |
0 25 |
31 |
177,000 |
0.22 |
26.. |
403 210 |
0 23 |
June 4. . |
194,500 |
0.21 |
0.23 |
TORSION.
205
The beam had now received 1,000,000 changes of the load, but it remained uninjured. The moving load was now in creased to 10,050 Ibs. — or one-half the breaking weight — and it broke with 5,175 changes. The beam was then repaired by riveting a piece on the lower flange, so that the sectional area was the same as before, and the experiment was continued. One hundred and fifty-eight changes were made with a load equal to one-half the breaking weight ; and the load was then reduced to two-fifths the breaking weight, and 25,900 changes made. Lastly, the load was reduced to one-third the breaking weight, with the following results :-T—
DATE. |
No. of Changes of Load. |
Deflection in Inchs. |
DATE. |
No. of changes of Load. i |
Deflection in Inchs. |
1860. Augnst 13 .... |
25,900 |
0 18 |
1860. Dec. 22 |
929,470 |
0.18 |
16 |
46 326 |
29 |
1,024,500 |
||
20 |
71 000 |
1861. |
|||
24 . |
101,760 |
Jan. 9 |
1,121,100 |
||
25 |
107 000 |
19 |
1,278,000 |
||
31 |
135,260 |
26 |
1,342,800 |
||
Sept. 1 |
140,500 |
Feb. 2 |
1,426,000 |
||
8 |
189,500 |
11 |
1,485,000 |
||
15 |
242 860 |
16 |
1,543,000 |
||
22 |
277 000 |
23 |
1,602,000 |
||
30 |
320,000 |
March 2 |
1,661,000 |
||
October 6 |
375 000 |
9 .... |
1,720,000 |
0.18 |
|
13. |
429 000 |
13 |
1,779,000 |
017 |
|
20 |
484 000 |
23 |
1 829,000 |
||
27 |
538 000 |
80 ... |
1,885,000 |
||
November 3 |
577 800 |
April 6 |
1,945,000 |
||
10 |
617,800 |
13 |
2,000,000 |
||
17 |
657 500 |
20 |
2 059,000 |
||
23 |
712 300 |
27 .. |
2,110,000 |
||
December 1 ... |
768,100 |
May 4 |
2,165,000 |
||
8 |
821,970 |
11 |
2,250,000 |
||
15 |
875,000 |
018 |
June |
2,727,754 |
o.ii |
The piece had now received nearly 4,000,000 changes in all, but the 2,727,000 changes after it was once broken and re paired did not injure it. The changes were not very rapid. During the first experiment they averaged about 11,000 per day, or less than eight per minute, and during the last experi ment the highest rate of change appears to have been less than
206 THE RESISTANCE OF MATERIALS.
eleven per minute, which is very slow compared with the stroke of some machines. Tilting-hammers often run from ten to twenty times this speed.
SHOCK — CRYSTALLIZATION.
171. SHOCKS — When a weight is applied to another body suddenly it produces a " shock " upon the materials which com pose the bodies. We cannot, practically, tell how frequently or with what force bodies must come in contact with each other in order to produce, a " shock ; " but theoretically any small body which is suddenly arrested in its movement, or suddenly devi ated in its course by another body, produces a shock. Mass is necessary to the production of a shock, and the masses must im pinge upon each other. If a force could manifest itself inde pendent of matter, arresting the movement of it, however sud denly, by another force, would not produce a shock. Also, changing the movement of a mass by such a force, however sudden, will not produce a shock. These ideas are approxi mately realized in the movements of steam, air, and other gases. Steam impinges against air without producing shock, practically speaking. A moving piston (in some machines) is brought to rest by the reaction of steam, or by a steam cushion, without producing shock. The alternate expansions and contractions of a piston-rod, or pitman, or other similar piece in steam machin ery, which are caused by the alternate pull and push of the moving force, do not produce a shock. The pieces may be "shocked" on account of working with loose connections, but that cause is not here considered. In the first example above cited there is very little mass in the moving or resisting bodies ; in the next one the motion of the moving mass is changed, and may be brought quite suddenly to rest by the action of a highly elastic medium which has but little mass ; and in the last exam ple the particles are contiguous and are only slightly moved in reference to each other, as the forces of extension and compres sion are transmitted through the bar. The particles are not permanently displaced in reference to each other, as they are liable to be by a blow or "shock."
Shocks are practically prevented in many cases by the
TORSION. 207
introduction of elastic substances which possess considerable mass. Thus steel, rubber, and wooden springs in vehicles and in certain machines are familiar examples. But elasticity alone is not a sufficient protection, for, as has been previously observed, all bodies are elastic. When masses are used for springs they must be so arranged as to operate through a perceptible space in bringing the moving body to rest, or in changing its velocity a perceptible amount.
Springs, however elastic, will not always prevent a shock, al though they may greatly relieve it. Thus, the springs under a car will not prevent the shock which always follows when the car- wheel strikes the end of a rail, although the shock is not as severe as it would be if the body of the car were rigidly con nected with the axle. So, too, the springs between the buffers on a car and the body of the car will not prevent one buffer striking another so as to produce a " shock." In these cases the springs may prevent the shock from being transmitted in a large degree to the body of the car. The springs in certain forge-hammers operate in a similar way to prevent a " shock " upon the working parts of the machinery.
" Shocks " are very injurious to machinery, and hence should, so far as possible, be avoided. All machines in which " shocks " are necessary, or incidental, or accidental, such as steam forge- hammers, morticing machines, stone-drilling machines, and the like, are much more liable to break than those that operate by a steady pull and push. Metals are so liable to break under such circumstances that many have supposed that the internal structure is changed, and the metal becomes more or less crys tallized.
The strength of the metal which is subjected to shocks is also greatly modified by the temperature — the lower the temperature the more damaging is the shock. It has been shown in Article 29, that wrought iron is somewhat stronger at a low tempera ture under a steady strain than at a higher temperature. Not withstanding this is contrary to the " popular notion," it has been further confirmed by the very careful experiments of the " Committee appointed by His Majesty the King of Sweden," and reported by Knut Styffe, Director of the Royal Technolog-
208 THE RESISTANCE OF MATERIALS.
ical Institute of Stockholm.* Their first conclusion was : — f " The absolute strength (tenacity) of iron and steel is not dimin ished by cold, but that even at the lowest temperature which ever occurs in Sweden, it is at least as great as at the ordinary temperature — or 00° Fahr."
These results are confirmed by the more recent experiments of Joule of England, and by several other experimenters.
But it is generally supposed that machinery, railroad iron, tyres on locomotives, and tools, break much more frequently with the same usage when very cold than they do when warm. Is this a mere, notion ?
Is it because breakages are more annoying in cold than in warm weather, and hence make a more lasting impression upon the minds of those who have to deal with them ; so that they think they occur more frequently? Impressions are not safe guides in scientific investigations. Our observations on the use of out-door machinery in cold and warm weather lead us to believe that they do break much more frequently with the same usage in winter than in summer. The same fact, in regard to the breaking of rails on railroads, was admitted by Styffe ; but after arriving at the conclusion which he did in regard "to the effect of cold upon the absolute strength of iron, he concluded that the cause of the more frequent breakages was due to the more rigid and non-elastic foundation caused by the frozen ground. But Sandberg, the translator of StyfFe's work, thought that iron when subjected to shocks might not give the same relative strength at different temperatures that it would when subjected to a steady strain. He therefore instituted another series of experiments to satisfy himself upon this important point, and aid in solving the problem. The following is an abstract of his report : —
The supports for the rails in the experiments were two large granite blocks which rested upon granite rocks in their native bed. The rails were supported near their ends on these blocks. They were broken by a ball which weighed 9 cwt., which was permitted to fall five feet the first blow, and the height increased
* The Elasticity, Extensibility, and Tensile Strength of Iron and Steel. By Knut Styffe. Translated by Christcr P. Sandberg. London, f Ibid., p. 111.
EFFECT OF SHOCKS. 209
one foot at each succeeding fall, and the deflection measured after each impact. A small piece of wrought iron was placed on the top of each rail to receive the blow, so as to concentrate its effect.
The rail was thus broken into two halves, and each part was afterwards broken at different temperatures. As the experi ments were not made till the latter part of the winter, the lowest temperature secured was only 10° Fahr. Fourteen rails were tested: — Seven of which were from Wales; five from France ; and two from Belgium. From these the experimenter drew the following conclusions : — *
1. " That for such iron as is usually employed for rails in the three principal rail-making countries (Wales, France, and Bel gium), the breaking strain, as tested by sudden blows or shocks, is considerably influenced by cold ; such iron exhibiting at 10° F., only one-third to one-fourth of the strength which it pos sesses at 84° F.
2. u That the ductility and flexibility of such iron is also much affected by cold, rails broken at 10° F. showing on an av- erage^a permanent deflection of less than one inch, whilst the other halves of the same rails, broken at 84° F., showed less than four inches before fracture."
This seems to be the fairest and most conclusive experiment upon this point that we have met with, but it is not satisfactory to all, or else they are ignorant of the experiment, for there has been of late considerable discussion upon the subject in the scientific journals. Some take the experiments of Fairbairn and Joule as conclusive upon the point, and attribute the cause of the failures, in many cases, to an inferiority of the iron, and in the case of tyres to an over-stretching of the metal when it is put on the wheel. By many, the presence of phosporous is considered especially detrimental to iron which is subject ed to shocks in cold weather. But until the fact is estab lished that cold iron is weaker than warm iron, when subjected to shocks, it is worse than useless to speculate upon the cause. More experiments are needed on this point, in which the quality of the metal, and all the conditions of the experiments should be definitely known.
* Styffe's work on iron and steel, translated by Sandberg, p. 157.
14:
210 THE RESISTANCE OF MATERIALS.
The following experiments, by John A. Roebling,* bear upon this subject, although they are not conclusive, for it is not re ported that the same metal was tested when warm : —
<v The samples tested were about one foot long, and were re duced at the centre to exactly three-fourths of an inch square, and their ends left larger, were welded to heavy eyes, making in all a bar three feet long. These were covered with snow and ice, and left exposed several days and nights. Early in the morning, before the air grew wanner, a sample inclosed in ice, was put into the testing-machine and at once subjected to a strain of 20,000 pounds, the bar being in a vertical position, and left free all around. The iron was capable of resisting 70,000 Ibs. to 80,000 Ibs. per square inch. A stout mill-hand struck the reduced section of the piece, horizontally, as hard as he could, with a billet one and a half inches in diameter and" two feet long. The samples resisted from three to one hundred and twenty blows. With a tension of 20,000 Ibs. some good sam ples resisted 300 blows before breaking."
The finest and best qualities of iron, or those that have the highest coefficient elasticity will resist vibration best. It is generally supposed that good iron will resist concussions much better than steel. Sir AVilliam Armstrong, of England, says : — " The conclusion at which I have long since arrived, and which I still maintain, is, that although steel has much greater tensile strength than wrought iron, it is not as well adapted to resist conclusive strains. It is impossible, then, that the vibratory action attending concussion is more dangerous to iron than to steel. The want of uniformity is another serious objection to the general use of steel in such cases." This has been used as an argument against the use of steel rails, but practically this has proved not to be a serious difficulty. So many ele ments must be considered in the use of steel rails, aside from fracture, that the problem must be solved by itself, and princi pally upon other grounds.
1 7£. CRYSTALLIZATION. — A crystal is a homogeneous in organic solid, bounded by plane surfaces, systematically arranged. The quartz crystal is a familiar example. Different substances
* Jour. Frank. Inst., vol. xl., 3d series, p. 361.
CRYSTALLIZATION.
crystallize in forms which are peculiar to themselves. Metals, under certain circumstances, crystallize ; and if they are broken when in this condition the fracture shows small plane surfaces, which are the faces of the crystals. It is found in all cases that crystallized iron is weaker than the same metal in its or dinary state. By its ordinary state we mean that wrought iron is fibrous, and cast iron and steel are granular in their appearance.
Iron crystallizes in the cubical system.* Wholer, in break ing cast-iron plates readily obtained cubes, when the iron had long been exposed to a white heat in the brickwork of an iron smelting furnace.
Augustine found cubes in the fractured surface of gun barrels which had long been in use.
Percy found on the surface and interior of a bar of iron, which had been exposed for a considerable time in a pot of glass-making furnace, large skeleton octahedra. (He seems to differ from the preceding in regard to the form of the crystals.)
Prof. Miller, of Cambridge, found Bessemer iron to consist of an aggregation of cubes.
Mallet says : — " The plans, of crystallization group themselves perpendicular to the external surfaces."
Bar iron will become crystalline if it is exposed for a long time to a heat considerably below fusion. Hence we see why large masses which are to be forged may become crystalline, on account of the long time it takes to heat the mass. Forging does not destroy the crystals, and forging iron at too low a temperature makes it tender, while steel at too high a tempera ture is brittle. The presence of phosphorous facilitates crystal lization. Time, in the process of breaking iron, will often determine the character of the fracture. If the fracture is slow, the iron will generally appear fibrous ; but if it be quick, it will appear more or less crystalline. Many mechanics have noticed this result. At Shoeburyness armor-plates were shat tered like glass under the impact of shot at a velocity of 1,200 feet to 1,600 feet per second. The iron was good fibrous iron.
Many engineers are of the opinion that oft-repeated and long-
* See Osborn's Metallurgy, pp. 83-86.
212 THE RESISTANCE OF MATERIALS.
continued shocks will change fibrous to crystalline iron. Opin ions, however, are divided upon this subject. In view of the immense amount of machinery and other constructions, parts of which are constantly subjected to shocks, the importance of the subject can hardly be overestimated.
William Fairbaini says: — * "We know that in some cases wrought iron subjected to continuous vibration assumes a crys talline structure, and that then the cohesive powers are much deteriorated; but we are ignorant of the causes of this change."
The late Robert Stephenson f referred to a beam of a Cor nish engine which received a shock eight or ten times a minute, equal to about fifty tons, for a period of twenty years without apparent change. These shocks were not very frequent, and would not be considered as detrimental as if they occurred a few times each second. He also says : — " The connecting-rod of a certain locomotive engine that had run 50,000 miles, and re ceived a violent jar eight times per second, or 25,000,000 vibra tions, exhibited no alteration." In all the cases investigated by him of supposed change of texture, he knew of no single in stance where the reasoning was not defective in some important link. These are not fair examples of shocks, as the vibrations referred to seem to be only changes from tension to compression, arid the reverse.
Mr. Brunei accepted the theory of molecular change, for a time, as due to shocks, but afterwards expressed great doubts as to its correctness, and thought that the appearance depended more upon the manner of breaking the metal than upon any molecular change.
Fairbaini has speculated a little upon the probable cause of the internal change when it takes place. In his evidence before the Commissioners appointed to inquire into the application of iron to railway structures, he says : — " As regards iron it is evident that the application and abstraction of heat operates more powerfully in effecting these changes than probably any other agency ; and I am inclined to think that we attribute too much influence to percussion and vibration, and neglect more obvious causes which are frequently in operation to produce
* Civ. Eng. and Arch. Jour., vol. lit p. 257. \ Am. R. Times, March 0, 1809, Boston.
CRYSTALLIZATION. 213
the change. For example, if we take a bar of iron and heat it red hot, and then plunge it into water, it is at once converted into a crystallized instead of a fibrous body ; and by repeating this process a few times, any description of malleable iron may be changed from a fibrous to a crystalline structure. Vibration, when produced by the blows of a hammer or similar causes, such as the percussive action upon railway axles, I am willing to admit is considerable; but I am not prepared to accede to the almost universal opinion that granulation is produced by those causes only. I am inclined to think that the injury done to the body is produced by the weight of the blow, and not by the vibration caused by it. If we beat a bar with a small ham mer, little or no effect is produced ; but the blows of a heavy one, which will shake the piece to the centre, wrill probably give the key to the cause which renders it brittle, but probably not that which causes crystallisation. The fact is, in my opinion, we cannot change a body composed of a fibrous texture to that of a crystalline character by a mechanical process, except only in those cases where percussion is carried to the extent of pro ducing considerable increase of temperature. We may, how ever, shorten the fibres by continual bending, and thus render the parts brittle, but certainly not change the parts which were originally fibrous into crystals.
For example, take the axle of a car or locomotive engine, which, when heavily loaded and moving with a high velocity, is severely shocked at every slight inequality of the rails. If, under these circumstances, the axle bends — however slightly — it is evident that if this bending be continued through many thousand changes, time only will determine when it will break. Could we, however, suppose the axle so infinitely rigid as to re sist the effects of percussion, it would then follow that the in ternal structure of the iron will not be injured, nor could the assumed process of crystallization take place."
The late John A. Roebling, the designer and constructor of the Niagara Railway Suspension Bridge, in his report on that structure in I860,* says he has given attention to this subject for years, and as the result of his observation, study, and experi ment, gives as his view that " a molecular change, or so-called
* Jour. Frank. lust., vol. xl., 3d series, p. 361.
214 THE RESISTANCE OF MATERIALS.
granulation or crystallization, in consequence of vibration or tension, or both combined, has in no instance been satisfactorily proved or demonstrated by experiment." " I further insist that crystallization in iron or any other metal can never take place in a cold state. To form crystals at all, the metal must be in a highly heated or nearly molten state." Notwithstanding these positive statements, he still hesitates to express a decided opinion which will cover the whole field of investigation. Still further
O
on he states that he is witnessing the fact daily that vibration and tension combined will greatly affect the strength of iron without changing its fibrous texture. Wire ropes and iron bars will become weakened as the vibration and tension to which they are subjected increase.
Certain machines in which the working parts are subjected to frequent shocks, more or less severe, are constantly failing, and the general impression is that the failure is due to crystal lization. In speaking of the rock-drilling engines used in Iloosac Tunnel, Mass., which were driven by compressed air, the committee say : * — " Gradually they begin to fail in strength ; the incessant and rapid blows — counted by millions — to which they are subjected, appearing to granulate or disintegrate por tions of the metals composing them." Having had some experi ence with this class of machines, I know something of the diffi culties which surround them.f During the winter of 1866-7 my assistant in the University, Professor S. W. Hobinson, and myself made several experimental machines, in the use of which we learned many essential conditions which must be observed in order to avoid frequent breakages in the use of iron which is subjected to frequent and long-continued shocks. As first designed, the breakages in the several working parts were ex ceedingly numerous, the remedy for which was not in making those parts larger and stronger, for that only aggravated the evil in most cases, but in arranging the moving parts so that they would be moved and brought to rest with as little shock as possible, and then making them as light as possible consistent
* Annual Report of the Commissioners on the Troy and Greenfield Railroad and Hoosac Tunnel. House Doc. , No. 30, p. 5, Boston, Mass.
f Notice of, by B. H. Latrobe, C. E. Sen. Doc. No. 20, 18G8, p. 31, Boston, Mass.
EFFECT OF SHOCKS. 215
with strength. But at the rapid rate at which we ran them, which was from 300 to over 500 blows per minute, it was not easy to comply with these conditions. No especial difficulty was experienced from the general shock caused by the striking of the tool upon the rock ; but the chief difficulty arose from the blow or shock to which the working pieces were directly subjected in operating them. At last, however, according to the report of the superintendent of the Marmora Iron Mine, Out., at which place they have been in use for some time, we so far overcame all the difficulties as to make it a decidedly prac tical machine. In the experiments we found that it was a bad condition to subject a piece to a blow crosswise of its length ; that is, perpendicular to its length.* It was also found that a piece struck obliquely would sustain a much greater number of blows than if struck perpendicularly. Many pieces evidently broke slowly, and was analogous to breaking a piece of tough iron on an anvil by comparatively light blows. If the blows were so severe as to start a crack in the piece, it would ulti mately break if the blows were continued sufficiently long. Several of the broken pieces were critically examined to see if they were crystallized, but there were no indications of any change in the internal structure of the metal.
All sharp angles in pieces which are shocked should, if pos sible, be avoided • for in the process of manufacture they are liable to be rendered weaker at such points, and if they are equally strong so far as manufacture is concerned, a greater strain, at the instant they are shocked, is liable to fall at such points, thus rendering them relatively weaker there. At least it is found that such pieces are more liable to break at the angle. Hence, in the construction of direct-action rock-drills, direct-action steam-hammers, and similar percussion machines, the steam piston is not only generally made solid with the rod, but it is connected with it by a curve. In other words, the rod is more or less gradually enlarged into a piston. At first, much difficulty was experienced in this regard with steam-hammers,
* I have seen it published that a small hammer was made to strike a blow upon the side of a bar which was suspended vertically, the blows being repeated night and day for nearly a year, when the bar broke ; but as the force of the blow and size of the bar were not given, I have thought that the statement was too indefinite to be of any scientific value.
216
THE RESISTANCE OF MATERIALS.
for in some cases the piston broke away from the rod, and slipped down over it.
In certain steam forge-hammers the piston-rod is liable to break where it joins the hammer. In this case the worst pos sible arrangement is to make a rigid connection between the hammer and rod. The author once saw a rod three and one- half inches in diameter nicely fitted into a conical hole in a 400- pound hammer, the taper of the rod and hole being slight, so that it would hold by friction when once driven into^place. The connection was practically rigid. The rod broke twice with ordinary use inside of twenty days. Probably it would have lasted a long time if the blow could always have been exactly central ; but in ordinary use it would very naturally be sub jected to cross strains by making a blow wrhen the material was under one edge of the hammer. By a repetition of these cross strains, rupture might have been produced without any crystal lization.
Fio. 112.
This difficulty is practically overcome in several ways, one of the most common of which is to place blocks of wood or other slightly elastic body between the end of the piston-rod and the hammer. Morrison overcame the difficulty by making the rod so large that it (the rod) became the hammer, and a small block of iron or steel was fitted into the forward end of the rod to serve avs a face for the hammer.
A Mr. Webb, of England, proposed to overcome the difficulty by a device which is shown in Fig. 112.
CRYSTALLIZATION. 217
Keferring to the figure, it will be seen that the piston-rod, which is for the main part of its length 4 in. in diameter, is en larged at the lower end to 6f in. in diameter, and is shaped spherically. This spherical portion of the rod is embraced by the annealed steel castings, B B, which are secured in their place in the hammer-head by the cotters, A, and the whole thus forms a kind of ball-and-socket joint, which permits the hammer head to swivel slightly on the rod without straining the latter. Mr. Webb first applied this form of hammer-rod fastening to a five-ton Nasmyth hammer with a 4 in. rod. With the old mode of attachment, with a cheese end, this hammer broke a rod every three or four weeks when working steel, while a rod with the ball-and-socket joint, which was put in in No vember, 1867, has been working ever since, that is, to some time in 1869, without giving any trouble. The inventor has also ap plied a rod thus fitted to a five-ton Thwaites and Carbutt's ham mer with equal success.
These examples seem to indicate that if iron is crystallized on account of shocks, the progress of the change may be slackened by a judicious arrangement of the pieces and by proper connec tions. But it does not follow that because a metal breaks so frequently when subjected to shock, that it has become crystal lized. It has been observed by those who have made the experi ment, that a piece of bar iron which is broken by heavy blows, when the piece is so supported as to bound with each stroke, will present a crystalline fracture ; but if the same bar be broken by easy blows near the place of the former fracture, it will pre sent a fibrous texture ; showing that in the former case the in ternal structure was not changed, unless it were in the immedi ate vicinity of the fracture. In such cases the appearance of the s^irface of the fracture does not indicate the true state of the internal structure. One reason why metals fail which are subjected to concussions, without crystallizing, is ; — an excessive strain is brought upon some point, thus impairing the elasticity and weakening the resisting powers, in which case, if the strains be repeated sufficiently long, rupture must ultimately take place. Or, if the concussion be sufficiently severe and local, it may dis place the particles, and thus begin a fracture. This frequently takes place in the case of anvils, hammers, hammer-blocks, and the like.
218 TIIE RESISTANCE OF MATERIALS.
If a bar is bent by a blow sufficiently great to produce a set, and the bar be bent back by another blow, and so on, the bar being bent alternately to and fro, rupture would probably take place at some time, however remote. It is often difficult to determine the strain which falls at a particular point of a piece when it is subjected to a shock, but if we could determine its exact amount, we might find it to be sufficiently large to account for rupture by shocks, without considering any mysterious change in the internal structure of the metal.
Note. Since the above was written, several articles have appeared in the scientific journals, giving the results of obser vations and experiments upon the strength of iron at low tem peratures, and they all confirm the position above stated, — that iron will not resist shocks as effectually at very low temperatures as it will at ordinary temperatures.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 219
CHAPTER X.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES.
173.— RISK AND SAFETY.— We have now considered the breaking-strength of materials under a variety of conditions, and also the changes produced upon them when the strains are within the elastic limits. In a mechanical structure, in which a single piece, or a combination of pieces, are required to sus tain a load, it is desirable to know how small the piece, or the several pieces, may be made to sustain a given load safely for an indefinite time ; or, how much a given combination will sus tain safely. The nature of the problem is such that an exact limit cannot be fixed. Materials which closely resemble each other do not possess exactly the same strength or stiffness ; and the conditions of the loading as to the amount or manner in which it is to be applied, may not be exactly complied with. Exactness, then, is not to be sought ; but it is necessary to find a limit below which, in reference to the structure, or above which, in reference to the load, it is not safe to pass.
It is evident that to secure an economical use of the material on the one hand, and ample security against failure on the other, the limit should be as definitely determined as the nature of the problem will admit ; but in any case we should incline to the side of safety. No doubt should be left as to the stability of the structure. There is no economy in risk in permanent structures. Risk should be taken only in temporary, or experi mental, structures ; or where risk cannot, from the nature of the case, be avoided.
U> 174. ABSOLUTE MODULES OF SAFETY. — In former times,
one of the principal elements which was used for securing safety in a structure, was to assume some arbitrary value for the resistance of the material, such value being so small that
220 THE RESISTANCE OF MATERIALS.
the material could, in the opinion of the engineer, safely sus tain it. This is a convenient mode, but very nnphiloBOphical, although still extensively used. The plan was to determine, as nearly as possible, what good materials would sustain for a long period, and use that value for all similar materials. But it is evi dent, from what has been said in the preceding pages, that some materials will sustain a much larger load than the average, while others will not sustain nearly so much as the average. In all such cases the proper value of the modulas can only be determined by direct experiment. In all important structures the strength of the material, especially iron and steel, should be determined by direct experiment.
The following values are generally assumed for the modulas of safety.
Pounds per square inch.
Wrought, iron for tension or compression, from 10,000 to 12.000
Cast iron, for tension, from 3,000 to 4,000
Cast iron, for compression, from 15,000 to 20,000
Wood, tension or compression, from 850 to 1 ,200
f granite, from 400 to 1,200
) quartz, from 1,200 to 2,000
Stone, compression j 84andstonei from 300 to 600
[limestone, from 800 to 1,200
The practice of French engineers,* in the construction of bridges, is to allow 3.8 tons (gross) per inch upon the gross sec tion, both for tension and compression of wrought iron.
The Commissioners on Railroad Structures, England, estab lished the rule that the maximum tensile strain upon any part of a wrought iron bridge should not exceed five tons (gross) per square inch, f
In most cases the effective section is the section which is sub jected to the strain considered.
1 75. FACTOR OF SAFETY. — The next mode, and one which is also largely in use, is to take a fractional part of the ultimate strength of the material, for the limit of safety. The recipro cal of this fraction is called the factor of safety. It is the ratio of the ultimate strength to the computed strain, and hence is the
* Am. R. R. Times, 1871, p. 6.
f Civ. Eng. and Arch. Jour. Vol. xxiv., p. 327.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 221
factor by which the computed strain must be multiplied to equal the actual strength of the material, or of the structure.
Experiments and theory combine to teach that the factor of safety should not be taken as small as 2. See articles 19, 166, 167, and 168.
Beyond this the factor is somewhat arbitrarily assumed, de pending upon the ideas of the engineer. For instance, the fol lowing values were given to the -Commissioners on Railway Structures, in England.*
Factors. Messrs. May and Grissel ...... ......................... 3
Mr. Brunell ....................................... 3 to 5
Messrs. Rasbrick, Barlow and others ...................... 6
Mr. Ilawkshaw ........................................ 7
Mr. Glyn ........................................... 10
The following values are also given by others : —
Factors. Bow, for wrought-iron beams .......................... 3.5
Weisbach, for wrought iron f ........................ 3 to 4
Vicat, for wire suspension bridges .............. more than 4
Eankine, for wire bridges steady fT .............. f to 4
movin load ............... 6 to 8
Fink, iron - truss bridges j *or Poste and ^f8 ........ 5 to «
( for cast-iron chords ............. 7
Fairbairn, for cast iron beams ^ ...................... 5 to 6
C. Shaler Smith, compression of cast iron ................. 5
Rankine and others, for cast-iron beams ............... 4 to 6
Mr. Clark in Quincy Bridge, lower chord .............. 6 to 7
Washington A. Roebling, for suspension cables ............. 6
Morin, Yicat, Weisbach, Rondelet, Navier, Barlow, and many others say that for a wooden frame it should not be less than ......................................... 10
For stone, for compression ................ •. ....... 10 to 15
From the experiments which are recorded in Article 170, Fairbairn deduced the following conclusions in regard to beams
* Civ. Eng. and Arch. Jour. , Vol. xxiv, p. 327. f Weisbach, Mech. and Eng. Vol. 1, p. 201. J Fairbairn, Cast and Wrought Iron, p. 58.
222 THE RESISTANCE OF MATERIALS.
and girders, whether plain or tubular. * " The weight of the girder and its platform should not in any case exceed one-fourth the breaking weight, and that only one-sixth of the remaining three-fourths of the strength should be used by the moving load." According to this statement the maximum load, including the live and dead load, may equal, but should not exceed,
i + i of f == I
of the breaking 1< >ad. Hence the factor of safety must not be less than 2.C6 when the above conditions are fulfilled. This value is, however, evidently smaller than is thought advisable by most engineers.
The rule adopted by the Board of Trade, England, for rail road bridges is f u to estimate the strain produced by the greatest weight which can possibly come upon abridge throughout every part of the structure which should not exceed one-fifth the ulti mate strength of tlie metal? They also observed that ordinary road bridges should be proportionately stronger than ordinary railroad bridges.
176. RATIONAL, L.IMIT OF SAFETY. — It is evident that materials may be strained any amount within the elastic limit. Their recuperative power — if such a term may properly be used in connection with materials — lies in their elasticity. If that is damaged the life of the material is damaged, and its powers of resistance "are weakened. As we have seen in the preceding pages, there is no known relation between the coefficient of elas ticity, and the ultimate strength of materials. The coefficient of elasticity may be high and the modulus of strength comparatively low. In other words, the limit of elasticity of some metals may be passed by a strain of less than one-third their ultimate strength, while in others it may exceed one-half their ultimate strength. We see, then, the unphilosophical mode of fixing an arbitrary modulus of safety, or even a factor of safety, when they are made in reference to the ultimate strength. But an examination of the results of experiments shows that the limit of elasticity is rarely passed for strains which are less than one- third of the ultimate strength of the metal, and hence, according
* Civ. Eng. and Arch. Jour., Vol. xxiv., p. 329. f Civ. Eng. and Arch. Jour., Vol. xxiv., p. 226.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 223
to the views of the engineers given in the preceding article, the factor of safety is generally safe. But if the limit of elas ticity were definitely known it is quite possible that a smaller factor of safety might sometimes be used.*
This method of determining the limit has been recognized by some writers, and the propriety of it has been admitted by many practical men, but the difficulty of determining the elastic limit has generally precluded its use. The experiments which are necessary for determining it are necessarily more delicate than those for determining the ultimate strength.
There is also a slight theoretical objection to its use. The limit of elasticity is not a definite quantity, for it is not pos sible to determine the exact point where the material is over strained. But this is not a fatal objection, for the limit can be determined within small limits.
In regard to the margin that should be left for safety, much depends upon the character of the loading. If the load is simply a dead weight, the margin may be comparatively small ; but if the structure is to be subjected to percussive forces or shocks, it is evident, as indicated in articles 19 and 171, that the margin should be comparatively large, not only on account of the indeterminate effect of the force, but also on account of the effect of such a force upon the resisting powers of the material. In the case of railroad bridges, for instance, the vertical posts or ties, as the case may be, are generally subjected to more sudden strains due to a passing load, than the upper and lower chords, and hence should be relatively stronger. The same remark applies to the inclined ties and braces which form the trussing; and to any parts which are subjected to severe local strains.
The frames of certain machines, and parts of the same machines, are subjected to a constant jar while in use, in which cases it is very difficult to determine the proper margin which is consistent with economy and safety. Indeed, in such cases, economy as well as safety generally consists in making them
* James B. Eads, in his Report upon the Illinois and St. Louis Bridge, for 1871, states that he tested samples of steel which were to be used in that structure, which showed limits of elastic reaction of 70,000 to 93,000 pounds per square inch.
224 THE RESISTANCE OF MATERIALS.
excessively strong, as a single breakage might cost much more than the extra material necessary to fully insure safety.
The mechanical execution of a structure should be taken into consideration in determining the proper value of the mar gin of safety. If the joints are imperfectly made, excessive strains may fall upon certain points, and to insure safety, the margin should be larger. No workmanship is perfect, but the elasticity of materials is favorable to such imperfections as necessarily exist ; for, when only a portion of the surface which is intended to resist a strain, is brought into action, that por tion is extended or compressed, as the case may be, and thus brings into action a still larger surface. But workmanship which is so badly executed as to be considered imperfect would fail before all its parts could be brought into bearing.
17O. EXAMPLES OF STRAINS THAT HAVE BEEN USED
IN PRACTICAL CASES. The margin of safety that has been used in various structures may or may not serve as guides in designing new structures. If the margin for safety is so small that the structure appeal's to be insecure and gives indi cations of failure, it evidently should not be followed. It serves as a warning rather than as a guide. If the margin is evidently excessively large, demanding several times the amount of material that is necessary for stability, it is not a guide. Any engineer or mechanic, without regard to scientific skill or economy in the use of materials, may err in this direc tion to any extent. But if the margin appears reasonably safe, and the structure has remained stable for a long time, it serves as a valuable guide, and one which may safely be followed under similar circumstances. Structures of this kind are practical cases of the approximate values of the inferior limits of the factors of safety. The following are some practical examples : —
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 225
IRON TRUSSED BRIDGES.
NAME OF THE BKIDGE. |
TENSION. Tons per square inch. |
COMPRESSION. Tons per square inch. |
Passaic (Lattice) |
5* to 6 4 5 5 5 5 Pounds per square inch. 12,000 7,000 to 12,000 9,251 10,000 11,375 Factor of safety, 5 |
4i to 5i »f 4 5 4 Pounds per square inch. 12,000 V to f the strength 8,902 Factor of safety, 5 711 Factor of safety, 5 |
Place de 1'Europe (Lattice) |
||
Canastota (N. Y. C. R. R ) (Lattice) |
||
Newark Dyke ( Warren Girder) |
||
Boyne Viaduct (Lattice) . . . |
||
Charring Cross (Lattice) |
||
St. Charles, Mo. ( Whipple Truss) * |
||
Louisville, Ky. (Fink Truss) |
||
Keokuk and Hannibal j- |
||
Quincy Bridge \ .... |
||
Kansas City Bridge § |
||
Hannibal Bridge (Quadrangular Truss) | . |
WOODEN BRIDGES.
NAME OF THE BRIDGE.
MAXIMUM STRAIN.
Cumberland Valley R. R. Bridge |
635 pounds per square inch. |
Portage Bridge (N Y & E M R) . ., |
Factor of safety 20 |
* R. R. Gazette, July 8, 1871, p. 169.
f R. R. Gazette, July 15, 1871, p. 178. Pivot span 376 feet 5 inches ; longest pivot span yet constructed.
\ Report of Chief Engineer Clark.
§ Calculated from the Report of Chief Engineer 0. Chanute, pp. 106 and 136.
I The tensile strength of the material ranged from 55,000 Ibs. to 65,000 Ibs. per square inch.— R. R. Gazette, July 15, 1871, p. 169.
15
226
THE RESISTANCE OF MATERIALS.
CAST-IRON ARCHES/
NAME OF THE ARCH. |
SPAN. Feet. Inches. |
VERMEI> Feet. |
SINE. Inches. |
STRAIN PER HyUARE INCH IN TONS. |
|
Austerlitz |
10G 152 102 137 197 120 |
0 2 5 9 10 0 |
10 10 11 15 16 20 |
7 1 4 0 5 0 |
2.78 1.46 1.37 1.90 2.37 3.00 |
Carrousal |
|||||
St Denis |
|||||
Ilhone |
|||||
Westminster |
|||||
STONE ARCJIES.f
NAMK OF THE ARCH. J |
1 .S Q g. |
Versed sine in feet. |
Pressure per square inch in jKmnds at the key. |
Factor of safety at the point of greatest strain. |
Wellington . . . |
100 |
15 |
175- |
11 3 |
Waterloo (9 Arches) . . . |
120 128 |
35 32 |
151 172 |
20.0 11.6 |
Taaf(&utlt Wales).... Turin |
140 147 |
35 18 |
244 293 |
8.0 10.2 |
London |
152 |
38 |
215 |
14.0 |
200 |
42 |
349 |
8.G |
|
CAST-STEEL ARCH.
NAME OF ARCH.
SPAN.
Feet.
FACTOR OF SAFETY.
Illinois and St. Louis Bridge.
515
0 + t
* Irwin on Iron Bridges and Roofs.
f Cresy's Encyclopaedia.
\ Report of the Engineer, p. 33.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 227
SUSPENSION BRIDGES.
NAME OF THE BBIDGE. |
02 |
Strain in tons per square inch. From Bridge. |
Strain in tons per square inch. Bridge and Load. |
Factor of safety. |
Menai |
580 |
4 21 |
800 |
q O* |
Hammersmith . |
422± |
5 38 |
0 Qfi |
q q* |
Pesth |
066 |
5 01 |
S 1 1 |
30* |
Chelsea |
384 |
4 36 |
ft 07 |
30* |
Clifton |
702;}: |
2 90 |
n AQ |
64-* |
Niagara |
821 |
6 70 |
840 |
K qj. |
Suspension Aqueduct, Pitts- } burgh, Pa. 7 spans, each, [ Cincinnati Bridge | . . |
1GO 1 057 |
9 1 |
11 7 |
4.0 69 |
East River |
1 600:}: |
6 A |
||
Highland (proposed) |
1 665§ |
.... |
6n |
|
TUBULAR BRIDGES.
FOB WEIGH |
r OF BETDGE |
||
AND |
LOAD. |
||
SPAN. |
|||
NAME OF BRIDGE. |
Feet. |
||
Tension. |
Compression. |
||
Tons. |
Tons. |
||
Conway |
400 |
6.85 |
5.03 |
Britannia (Central span). . ... |
460 |
3.00 |
|
Penrith (Tubular Girder) |
4.75 |
4.25 |
|
* Tensile strength, 70,000 Ibs. per square inch.
f Tensile strength, 100,000 Ibs. per square inch.
t Engineer's Report. Suspending ties, factor of safety, 8.
§ Jour. Frank. Last., vol LXXXVII., p. 165.
| Report of the Chief Engineer, J. A. Roebling.
THE RESISTANCE OF MATERIALS.
STO1TE FOUNDATIONS.
FACTOR OF SAFETY. |
|
Pillars of the Dome of St Peter's (Rome) . .... |
16 |
" " St. Paul's (London) |
14 |
" " " St Geneva- re (l*aris) |
7 6 |
Pillars of the Church Touissaint (Angei'x) |
10 |
Merchants' Shot Tower (Baltimore)* |
4 8 |
31 |
|
Lower courses of the Piers of Neuillv Bridge (Paris) |
15 8 |
Foundation of St. Charles' Bridge (Missouri) |
12 to 14 |
10 to 20 |
|
177. PROOF LOAD. The proof load is a trial load. It is intended as a practical test of a theoretical structure.
It generally exceeds the greatest load that it is ever intended to put upon the structure. Some structures, especially steam boilers, are subjected to excessively severe trial strains, often being two or even three times that to which they are to be subjected in actual use. " This is done," they say, " to insure against failure." But such severe strains do no good, and often do great damage, for they may overstrain the material and thus weaken it. For instance, if a structure is proportioned to carry 10,000 pounds per square inch, but on account of the imperfection of the material or imperfect workmanship, some point is required to carry 20,000 pounds per square inch ; and if the structure is tested to 20,000 pounds per square inch, that point would be obliged to carry nearly 40,000 pounds per square inch, an amount which it might not be able to sustain for a long time. We say nearly 40,000 pounds ; for it should be remembered that where the workmanship is imperfect, thereby throwing greater strains on certain points than was intended, the elasticity of the materials permits those members to be compressed or extended more than they otherwise would be, and thus tends
* Strength of materials, J. K. Whttdin, p. 23.
f " In the stone work the pressures vary from 8 to 26 tons per square foot. Stone used is granite, selected samples of which have borne a crushing strain of 600 tons per square foot. Some will not bear over 100 tons per square foot. The general average is necessarily much less than that of the best specimens. " — Statement of tlie Chief Engineer, Washington A. Roebling.
LIMITS OF SAFE LOADING OF MECHANICAL STRUCTURES. 229
to bring into action the other members which at first were more lightly strained. In this way there is a tendency to bring about an equilibrium of strains on all those parts which were calculated to carry equal amounts.
According to the * principles which have been discussed in the preceding pages, it is evidently better for the struc ture, and should be more satisfactory, to apply a moderate proof load for a long time than an excessive one for a short time.
APPENDIX I.
PRESERVATION OF TIMBER.
(The Graduating Thesis of Mr. H. W. Lewis, of the class of 18G6, and more recently engineer on the Missouri Valley Railroad, forms the basis of this article. I have added to it such new matter as I find in the Graduating Thesis of Mr. A. B. Raymond, of the class of 1871. — AuTiioil.)
1. THE IMPORTANCE OF THIS SUIJJECT maybe shown by many familiar examples in practical life.
Although iron is coming more and more extensively into use, yet the amount of wood which is used at the present time in mechanical structures, and which will, in the nature of things, be used for a long' time to come, is enormous. For instance: in 1885 there was sold in Chicago alone 900,000,000 pieces of lath, 2,000,000,000 of shingles, and 5,000,000,000 feet of lumber.*
In the matter of railroad ties alone, any process which could be easily and cheaply applied, which would double their life, would literally save millions to the country. This may be shown by an approximate calculation, thus : — Al lowing only 2,000 sleepers to the mile, at a cost of fifty cents each, and admit ting that the average life of American sleepers is only seven years, f and that it costs ten cents to treat each tie in some way so as to make it last fourteen years, then the saving at the end of seven years is $(500 per mile. For ten cents at compound interest at ten per cent, for seven years amounts to twenty cents, which from fifty cents leaves thirty cents as the net saving on one ; and on 2,000 it amounts to $600.
There are in the United States about 45,000 miles of railroad ; and hence, if the above conditions could be realized of all of them, the annual saving would be about $3,400,000 ! Other uses of timber would show a correspond ing saving.
2. CLASSIFICATION OF CONDITIONS.— Timber may be subjected to the following conditions : —
It may be kept constantly dry ; at least, practically.
It may be constantly wet in fresh water.
It may be constantly damp.
It may be alternately wet and dry.
It may be constantly wet in sea water.
3. TOIBER KEPT CONSTANTLY DRY will last for centuries. The roof of Westminster Hall is more than 450 years old. In Stirling Castle are carv ings in oak, well preserved, over 300 years old ; and the trusses of the roof of
* Hunt's Merchants'1 Magazine.
t New American Cyclopedia, vol. xiii., p. 734.
232 APPENDIX.
the Basilica of St. Paul, Rome, were sound and good after 1,000 years of service.* The timber dome of St. Mark, at Venice, was in good condition 850 years after it was built, f
Artificial preservatives seem to be unnecessary under this condition.
4. TIMBER KEPT CONSTANTLY WET IN FUESH WATER, under such conditions as to exclude the air, is also very durable. The pillars upon which dwellings of the Canaries rest were put in their present place in 1402, and they remain sound to the present time.:}: The utensils of the lake dwellings of Switzerland are supposed to be at least 2,000 years old.g
The piles of the old London Bridge were sound 800 years after they were driven. The piles of bridge built by Trajan, after having been driven more than 1,000 years, were found to be petrified four inches, the rest of the wood being in its ordinary condition. §
Beneath the foundation of Savoy Place, London, oak, elm, beech, and chestnut piles and planks were found in a perfect state of preservation after having been there 050 years. §
While removing the old walls of Tunbridge Castle, Kent, there was found, in the middle of a thick stone wall, a timber curb which had been enclosed for 700 years. §
It is doubtful if artificial preparations would have prolonged the life of the timber in these cases.
5. TI3IBEU IN DAMP SITUATIONS.— Timber, in its native state, under these circumstances, is liable to decay rapidly from the disease called "dry rot." In dry rot the germs of the fungi are easily carried in all directions in a structure where it has made its appearance, without actual contact between the sound and decayed wood being necessary ; whereas the communication of the disease resulting from wet rot takes place only by actual contact. The fungus is not the cause of the decay, but only converts corrupt matter into new forms of life. |
There are three conditions which are at our command for prolonging the life of timber in damp situations : —
1st. Thoroughly season it;
2d. Keep a constant circulation of air about it ; and
3d. Cover it with paint, varnish, or pitch.
The first condition is essential, and may be combined with either or both of the others.
By seasoning we do not mean simply drying so as to expel the water of the sap, but also a removal or change of the albuminous substances. These are fermentable substances, and when both are present they are ever ready, under suitable circumstances, to promote decay. The cellulose matter of the woody fibre is very durable when not acted upon by fermentation, and it is this that we desire especially to protect.
* The London Bulkier, vol. iL, p. filfi.
t Modern Carpentry, Silloway, p. 40.
$ Journal oft/ie Frank. Iwt., 1870.
§ Moilern Carpentry, Silloway, p. 39.
| " There is no reason to believe that f unpri can make use of organic compounds in any other than a etate of decomjKwition." — Cari>enter'8 Comp. Physiology, p. 1G5. (See also EncyclopoKdia Lri- t* i H irii on this subject.)
APPENDIX. 233
Unseasoned timber which is surrounded by a dead air decays very rapidly. The timber of many modern constructions is translated from the forests and enclosed in a finished building in a few weeks, and unless it is subject to a free circulation of air it inevitably decays rapidly. *
Thorough ventilation is indispensable to the preservation of even well-sea soned naked wood in damp localities. The rapid decomposition of sills, sleep ers, and lower floors is not surprising where neither wall-gratings nor venti lating flues carry off the moisture rising from the earth, or foul gases evolved in the decay of the surface mould. In the close air of cellars, and beneath buildings, the experiments of Pasteur detected the largest percentage of fungi spores. Remove the earth to the foot of the foundation, and fill in the cavity with dry sand, plaster-rubbish, etc., or lay down a thick stratum of cement to exclude the water, and provide for a complete circulation of air, and lower floors will last nearly as long as upper ones.f
A covering of paint, pitch, varnish, or other impervious substance upon un- dried timber is very detrimental, for by it all the elements of decay are re tained and compelled to do their destroying work. The folly of oiling, paint ing, or charring the surface of unseasoned wood is therefore evident. Owing to this blunder alone, it is no unusual thing to find the painted wood-work of older buildings completely rotted away, while the contiguous naked parts are perfectly sound.
While an external application of coal tar promotes the preservation of dry timber, nothing can more rapidly hasten decay than such a coating upon the surface of green wood. But this mistake is often made, and dry rot does the work of destruction. \ Carbonizing the surface also increases the durability of dry, but promotes the decay of wet timber. Farmers very often resort to one of the latter methods for the preservation of their fence-posts. Unless they discriminate between green and seasoned timber, these operations will prove injurious instead of beneficial.
There are numerous methods for promoting the process of seasoning. Some have in view simply drying, a process which is important in itself, but which will not in itself prevent decay in damp situations unless the moisture be permanently excluded. Some dry with hot air, and some with steam. In the latter case, if the steam be superheated the process is very rapid, but it seems to damage the life of the timber.
Others have in view the expulsion of the albuminous substance. Water-soak ing the logs and afterwards drying the lumber, seems to be a cheap and quite effectual mode. But there are many patented processes for securing this end, or for changing the albuminous substances ; and in many cases the latter end is not only secured, but the salts which are used act directly upon the cellulose and lignite of the wood, thereby greatly promoting its durability.
* For an account of the rapid destruction of the floors and joists of the Church of the Holy Trinity, Cork, Ireland, by dry rot, see Civil Engineer's Journal, vol. xii., p. 303. For an ac count of the decay of floors, studs, &c., in a dwelling, see the London Builder, vol. vi., p. 34.
"In some of the mines in France the props seldom last more than fifteen months. " — Annalea des Mines.
t The Builder, vol. xi., page 46.
J According to Col. Berrien, the Michigan Central Eailroad bridge, at Niles, was painted, before seasoning, with "Ohio fire-proof paint," forming a glazed surface. About five years after, it was so badly dry-rotted as to require rebuilding.
234 APPENDIX.
The f ollowing are the principal processes which have been used : — Mr. P. W. Barlow's patent* provided for exhausting the air from one end of the log, while one or more atmospheres press upon the other end. This artificial aerial circulation through the wood is prolonged at pleasure. However excel lent in theory, this process is not practicable.
By another method, the smoke and hot gases of a coal fire are conveyed among the lumber, placed in a strong draft. Some writers recommend the re moval of the bark one season before felling the tree. All good authorities agree that the cutting should take place in the winter season. \
Kyan's process, which consists in the use of corrosive sublimate, was pat ented in 1833. His specific solution:}: was one pound of chloride of mercury to four gallons of water. Long immersion in the liquid in open vats, or great pressure upon both solution and wood, in large wrought-iron tanks, is neces sary for the complete injection of the liquid. The durability of well kyanized timber has been proved, but the expensiveness of the operation will long for bid its extensive adoption.
For " Burnettizing," ^ a solution of chloride of zinc — one pound of salt to ten gallons of water — is forced into the wood under a pressure of 150 Ibs. per square inch.
Boucherie employs a solution || of sulphate of copper one pound to water twelve and a hilf gallons, or pyrolignite of iron one gallon to six gallons of water. He enclosed one end of the green stick in a close-fitting collar, to which is at tached an impervious bag communicating through a flexible tube with an elevated reservoir containing the salt liquid. Hydrostatic pressure soon expels the sap at the opposite end of the log. When the solution makes its appear ance also the process is completed.
He finds the fluid will pass along the grain, a distance of 12 'feet, under a lower pressure than is required to force it across the grain, three-fourths of an inch. The operation is performed upon green timber with the greatest fa cility.^
In 1846, eighty thousand sleepers of the most perishable woods, impregnated, by Boucherie's process, with sulphate of copper, were laid down on French railways. After nine years' exposure, they were found as perfect as when laid.** This experiment was so satisfactory that most of the railways of that empire at once adopted the system. We would suggest washing out the sap with water, which would not coagulate its albumen. The solution would appropri ately follow.
Both of the last-named processes are comparatively cheap. The manufac turing companies of Lowell. Massachusetts, have an establishment for "Bur nettizing" timber, ff in which they prepare sticks fifty feet in length. Under
* Civ. Eng. Jour., vol. xix., p. 422.
t Experiments detailed in the Common show conclusively that winter-cut pine is stronger and more durable than that cut at any other season of the year. — .-!««. f!c. Discovery for liSfil, p. 340.
"Oak trees felled in the winter make the best timber."— The JluiUler, 1859, page 138.
J Civ. Eng. Jour., vol. v., page 202.
§ Civ. Eng. Jour., vol. xiv., p. 471. Invented by Burnett in 1838.
I Civ. Eng. Jour., vol. xx., p. 405.
1 As a modification of this method he also cut a channel in the wood throughout the circumfer ence of the tree, fitted a reservoir thereunto, and poured in the liquid. The vital forces speedily disseminated tke solution throughout the tree.
** Jour, of the Frank. fn»t., vol. xxxii., pp. 2, 3.
tt yew American Cyclopedia.
APPENDIX. 235
a pressure of 125 pounds per square inch they inject from two to eight ounces of the salt into each cubic foot of wood. The cost, in 1861, was from $5 to $0 per 1,000 feet, board measure.* Boucherie's method must be still cheaper. It costs less than creosoting by one shilling per sleeper, f
An American engineer, Mr. Hewson, for injecting railroad sleepers, proposes a vat deep enough for the timbers to stand in upright. The pressure of the surrounding solution upon the Idwer ends of the sticks will, he thinks, force the air out at their upper extremities, kept just above the surface of the solution, after which the latter will rise and impregnate the wood. In 1859 he estimated chloride of zinc at 9 cents per pound, sulphate of copper at 14 cents per pound, and pyrolignite of iron at 23 cents per gallon. He found the cost of impregnating a railway tie with sufficient of those salts to prevent decay, to be : for the chloride of zinc 2 '8 cents, for blue vitriol o'24 cents, for pyrolignite of iron 7 '5 cents.:}:
By Earle's process the timber is boiled in a solution of one part of sulphate of copper, three parts of the sulphate of iron, and one gallon of water to every pound of the salts. A hole was bored the whole length of the piece before it was boiled. It was boiled from two to four hours, and allowed to cool in the mixture.
Ringold and Earle invented the following process : — A hole was made the whole length of the piece, from one-half to two inches in diameter, and boiled from two to four hours in lime-water. After the piece was dried the hole was filled with lime and coal tar. Neither of these methods was very successful.
A Mr. Darwin suggests that the piece be soaked in lime-water, and after wards in sulphuric acid, so as to form gypsum in the pores.
BethelFs process consists in forcing dead oil into the timber. This is called creosoting. § He inclosed the timber and dead oil in huge iron tanks, and sub jected them to a pressure varying between 100 and 200 pounds per square inch, at a temperature of 120° F. about twelve hours. From eight to twelve pounds of oil are thus injected into each cubic foot of wood. Lumber thus- prepared is not affected by exposure to air and water, and requires no painting. |[ A large number of English railway companies have already adopted the system.^" Eight pounds of oil per cubic foot is sufficient for railway sleepers.**
One writer has said that if creosote has ever failed to prevent decay, it has been because of an improper treatment, or because the oil was deficient in car bolic acid.
Sir Robert Smirke was one of the first architects to use this process, and when examined before a Committee on Timber, stated that this process does not
* The Philadelphia, Wilmington and Baltimore Railroad Company have used the process since I860 with complete success. The Union Pacific Railroad Company have recently erected a large building for this purpose. Their cylinder is 75 feet long, (51 inches in diameter, and capable of holding 250 tics. They " Burnettize" two batches per day. — Report on Pacific Railroad, by Col. Simpson, 1865.
t Jour. Frank'Inst., vol. xxxii., pp. 2, 3.
J Ibid., vol. xxvii., p. 8.
§ " Creosote from coal undoubtedly contains two homologous bodies, CisHoOa and CuIIsOa, the first being carbolic and the second crysilic acid." — Ure's Diet, of Arts, Manu., and Mines, vol. ii., p. 623.
B Ure's Diet, of Manu. and Mines.
^ The Great Western, North-Eastern, Bristol and Exeter, Stockton and Darlington, Manchester and Birmingham, aad London and Birmingham. — Ure's Diet, of Manu. and Mines.
** Jour. Frank. Inst., vol. xliv., p. 275.
236 APPENDIX.
diminish the strength of the material which is operated upon. He afterwards said, " I cannot rot creosoted timber, and I have put it to the severest test I could apply."
The odor of creosote makes it objectionable for residences and public buildings.
Mr. S. Beer, of New York City, invented a mode of preserving timber by boiling1 it in borax with water. But this process has been objected to on the ground that it is not a good protection against moisture.
Common salt is known to be a good preservative in many cases. According to Mr. Bates's opinion,* it answers a good purpose in many cases if the pieces to which it is exposed are not too large.
6. TIMBER AI/TEKNATELY WET AND DRY.-The surface of all timber exposed to alternations of wetness and dryness, gradually wastes away, be coming dark-colored or black. This is really a slow combustion, but is com monly called wet rot, or simply rot. Other conditions being the same, the most dense and resinous woods longest resist decomposition. Hence the su perior durability of the heart-wood, in which the pores have been partly filled with lignine, over the open sap-wood, and of dense oak and lignurn-vitas over light poplar and willow. Hence, too, the longer preservation of the pitch-pine and resinous u jarrah " of the East, as compared with non-resinous beech and ash.
Density and resinousness exclude water. Therefore our preservatives should increase those qualities in the timber. Fixed oils fill up the pores and increase the density. Staves from oil-barrels and timbers from whaling ships are very durable. The essential oils resinify, and furnish an impermeable coat ing. But pitch or dead oil possesses advantages over all known substances for the protection of wood against changes of humidity. According to Professor Letheby,f dead oil, 1st, coagulates albuminous substances ; 2d, absorbs and appropriates the oxygen in the pores, and so protects from eremacausis ; 3d, resinifies in the pores of the wood, and thus shuts out both air and moisture ; and 4th, acts as a poison to lower forms of animal and vegetable life, and so pro tects the wood from all parasites. All these properties specially fit it for im pregnating timber exposed to alternations of wet and dry states, as, indeed, some of them do, for situations damp and situations constantly wet. Dead oil is distilled from coal-tar, of which it contains about .30, and boils between 390° and 470' Fahr. Its antiseptic quality resides in the creosote it contains. One of the components of the latter, carbolic acid (phenic acid, phenol), CiaHnO..., the most powerful antiseptic known, is able at once to arrest the de cay of every kind of organic matter.^ Prof. Letheby estimates this acid at £
* Report of the Commissioner of Agriculture. t Civ. Eng. Jour., vol. xxiii., p. 216.
£ " I have ascertained that adding one part of the carbolic acid to five thousand parts of a strong
solution of glue will keep it perfectly sweet for at least two years Hides and
skins, immersed in a solution of one part of carbolic acid to fifty parts of water, for twenty -four hours, dry in air and remain quite sweet''— Prof. Grace Calvert, Ann. Sc. /Hscor., 1865, p. 55.
"Carbolic acid is sufficiently soluble in water for the solution to possess the power of arresting or preventing sjwntaneous fermentation. Saturated solutions act on animals and plants as a viru lent poison, though containing only five per cent, of the acid.1'— Civ. Eng. Jour., vol. xxii., p. 216.
" Parasites and other worms are instantly killed by a solution containing only one-half per cent.
of acid, or by exposure to the air containing a email portion of the acid By
examining the action on a leaf, we find the albumen is coagulated. All animals with a naked skin, and those that live in water, die sooner than those that live in air and have a solid envelope." — Dr. I. Lemaire, Ann. Sc. Discov., 1855, p. 238.
APPENDIX.
237
to 6 per cent, of the oil. Chrysilic acid CMH,,O,., the homologue of carbolic acid, and the other component of creosote, is not known to possess preservative properties.
Creosoting, or Bethell's process, is the most valuable of all the well-tried pro cesses in this case. For railway sleepers eight pounds of oil per cubic foot of timber is sufficient. * If the timber is dry, a coating of coal-tar, paint, or resin ous substance, is valuable.
A Mr. Heinmann, of New York City, proposes the following process, which appears to be very promising : —
The sap is first expelled and then the timber is injected with common rosin. The latter is introduced while in a liquid state, under high pressure, while in vessels especially constructed for the purpose.
In an experiment made by Prof. Ogden, one cubic foot of green wood ab sorbed 8.90 pounds of rosin, while a cubic foot of well-seasoned wood absorbed only 2.06 pounds. The strength of the timber was increased by this process, as is shown by the following experiment : —
Woo] |
3 TBEATED WITH |
ROSIN. |
Wooi |
IN ITS NATUKAI |
J STATE. |
Breaking Weight. |
Quality. |
Grain. |
Break! ng?~T Weight. |
Quality. |
Grain. |
Pounds. |
Pounds. |
||||
163.5 |
Checked. |
Straight. |
98.5 |
Sound. |
Slant. |
193 |
Sound. |
u |
103.0 |
||
171.5 |
n |
(( |
110.0 |
Straight. |
|
72.5 |
Checked. |
Cross. |
57.5 |
1 1 |
|
57.5 |
u |
Slant. |
40.00 |
n |
|
57.5 |
u |
u |
40.0 |
a |
|
121.0 |
1 1 |
u |
71.0 |
n |
|
155.5 |
u |
Cross. |
84.0 |
u |
It is found by experiment that wood thus treated is not as flammable as air- dried wood. This is accounted for from the fact that a kind of inflammable slag is deposited over the surface immediately after the rosin begins to burn.
The chief advantages which are claimed for this method are more theoretical than practical, as it has not yet had sufficient time to test its practical merits, and it may, like many other processes, disappoint the hopes of its strongest ad vocates and well-wishers.
7. TIMBER CONSTANTLY WET IN SALT WATER.— We have not to guard against decay when timber is in this situation. Teredo navalis, a mollusk of the family Tubicolaria, Lam , soon reduces to ruins any unprotected sub marine construction of common woods. We quote from a paper read before the "Institute of Civil Engineers," England, illustrating the ravages of this animal : —
" The sheeting at Southend pier extended from the mud to eight feet above low-water mark. The worm destroyed the timber from two feet below the
* Jout\ Frank. Inxt., vol. xliv., p. 275.
238 APPENDIX.
surface of the mud to eight feet above low-watermark, spring- tide; and out of 38 fir-timber piles and various oak-timber piles, not one remained perfect after being up only three years." ' Specimens of wood, taken from a vessel that had made a voj-age to Africa, are in the museum, and show how this rapid de struction is effected.
" None of our native timbers are exempt from these inroads. Robert Stephen- son, at Bell Rock, between 1814 and 1843, f found that green heart oak, beef- wood, and bullet-tree were not perforated, and teak but slightly so. Later experiments show that the u jarrah " of the East, also, is not attacked. \ The cost of these woods obliges us to resort to artificial protection.
4' The teredo never perforates below the surface of the sea-bottom, and proba bly does little injury above low- water mark. Its minute orifice, bored across the grain of the timber, enlarges inwards to the size of the finger, and soon be comes parallel to the fibre. The smooth circular perforation is lined through out with a thin shell, which is sometimes the only material separating the ad jacent cells. The borings undoubtedly constitute the animal's food, portions of woody fibre having been found in its body. § While upon the surface only the projecting siphuncles indicate the presence of the teredo, the wood within may be absolutely honey-combed with tubes from one to four inches in length.
'• It was naturally supposed that poisoning the timber would poison or drive away the teredo, but Kyan's, and all other processes employing solutions of the salts of metals of alkaline earths, signally failed. This, however, is not surprising. The constant motion of sea- water soon dilutes and washes away the small quantity of soluble poison with which the wood has been injected. If any albuminate of a metallic base still remains in the wood, the poisonous properties of the injection have been destroyed by the combination. More over, the lower vertebrates are unaffected by poisons which kill the mammals. Indeed, it is now known that certain of the lower forms of animal life live and even fatten on such deadly agents as arsenic. |
" Coatings of paint or pitch are too rapidly worn away by marine action to be of much use, but timber, thoroughly creosoted with ten pounds of dead oil per cubic foot, is perfectly protected against teredo navalis. All recent au thorities agree upon this point. In one instance, well authenticated, the mol- lusk reached the impregnated heart-wood by a hole carelessly made through the injected exterior. The animal pierced the heart-wood in several directions, but turned aside from the creosoted zone. *[[ The process and cost of ' ' creo- soting" have already been discussed."
A second destroyer of submarine wooden constructions is limnoria terebrans, (or L. perforata, Leach) a mollusk of the family Assellotes, Leach, resembling the sow-bug. It pierces the .hardest woods with cylindrical, perfectly smooth, winding holes, -^th to -j^-.th of an inch in diameter, and about two inches deep.** From ligneous matter having been found in its viscera, some have concluded that the limnora feeds on the wood, but since other mollusks of the same ge-
* Civ. Eng, Jour., vol. xii., p. 382.
t The Builder for 18(52, p. 511.
£ Civ. Eng. Jour., vol. xx., p. 17.
$ C/r/ Eng. Jour., vol. xii., p. 382. Also Diet, Univ. cfHist. Xatur. tome xii.
j British and Foreign Medical Kevitio.
^ Cic. Eng, Jour., vol. xii., p, 191.
** Diet. Univ. tfllixt. Xatur.
APPENDIX. 239
mis, Pholas, bore and destroy stone-work, the perforation may serve only for the animal's dwelling. The lumnoria seems to prefer tender woods, but the hardest do not escape. Green-heart oak is the only known wood which is not speedily destroyed.* At the harbor of Lowestoft, England, square fourteen- inch piles were, in three years, eaten down to four inches square, f
While all agree that no preparation, if we except dead oil, has repelled the limnoria, an eminent engineer has cited three cases in which that agent afford ed no protection.^:
We do not find that timber impregnated with water-glass has been tested against this subtle foe. The experiment is certainly worthy of a trial.
A mechanical protection is found in thickly studding the surface of the tim ber with broad-headed iron nails. This method has proved successful. Oxy- dation rapidly fills the interstices between the heads, and the outside of the timber becomes coated with an impenetrable crust, so that the presence of the nails is hardly uecessaiy.
In conclusion, we cannot but express surprise that so little is known in this country concerning preservative processes. Their employment seems to excite very little interest, and the very few works where they are being tested at tract hardly any attention. Those railroads which have suspended their use assign no reasons, and those upon which the timber is injected publish no re ports concerning the advantages of their particular methods. Even the Na tional Works, upon which Kyan's process was formerly employed, have laid it aside, and now subject lumber to dampness and alternations of wetness and dry- ness, without any preparation beyond seasoning. When sleepers cost fifty cents and creosoting thirty cents each, it is cheaper to hire money at seven per cent. , compound interest, than to lay new sleepers at the end of seven years. Allowing any ordinary price for the removal of the old and laying down the new ties, the advantage of using Bethell's process seems evident. If some cheaper method will produce the same effects, the folly of neglecting all means which aim at increasing the durability of the material is still more palpable.
* Civ. En<j< Jour. , vol. xxv., p. 206, t Ibid., vol. xvi., p. 7(i. t Ibid., vol. xxv., p. 206. _
240
AI'I'KMMX.
8. THE FOl/l.OWINC; is A S1TM.1IAKY of the different processes that have been invented from time to time, with the names of their inventors : —
Numcs of Inventors.
Chemicals Used.
Manner of using them.
Bethell . .
Kyan
Margery . Burnett . . Ransome . LeGras... Margary. .
Payne . . .
Bouchere. .
Gemini. . . . Heinmann. ,
Earle
Ringold . . .
Tregold
S. Beer. .
Creosote, or pitch-oil
iChloride of mercury
'Sulphate of copper
Chloride of zinc
; Liquid silicate of pota-ssa
Manganese, lime, and creosote . . [Solution of acetate of copper. . . Sulphate of iron, carbonate of )
soda f
Pyrolignate of iron, sulphate {
of copper (
.JTar
By injection.
j Rosin, or colophony. ! Sulphate of copper. .
'Lime
i Sulphate of iron .! Borax. .
Dorset and Blythe
Huting and j
Boutigny {
Vernet. .,
Same as Bouchere.
Oil of schist, tar, pitch, and )
shellac \
Arsenic
Salt . .
boiling.
By means of a vacuum and
injection. By immersion, and by fire
or burning. By saturation. External application.
APPENDIX II.
TABLE
Of the Mechanical Properties of the Materials of Construction.
NOTE. — The capitals affixed to the numbers in this table refer to the following authorities :
B. Barlow. Report of the Commissioners of I the Navy, etc.
Be. Bevan.
Bn. Buchanan.
Br. Belidor, Arch. Hyclr.
Bru. Brunei.
C. Couch. . 01. Clark.
D. Darcel, Annales for 1858.
D. W. Daniell and Wheatstone. Report on the stone for the Houses of Parliament.
E. Eads.
F. Fairbairn.
G. Grant.
H. Hodgkinson. Report to the British Asso ciation of Science, etc.
Ha. Haswell. Eng. and Mech. Pocket-Book, 1809.
J. Journal of Franklin Institute, vol. XIX., p. 451.
K. Kirwan.
Ki. Kirkeldy. La. Lame.
M. Mischembroeck. Introd. ad. Phil. Nat. I. Ma. Mallet. Mi. Mitis. Mt. Mushet. Pa. Colonel Pasley. R. Roudelet. L'Art de Batir, IV. Ro. Roebling.
Re. Reunie. Phila. Trans., etc. S. Styffe. On Iron and Steel. T. Thompson. Te. Telford.
Tr. Tredgold. Essay on the Strength of Cast Iron.
W. Watson. Wa. Major Wade. Wn. Wilkinson.
* Calculated from the experiments of Fair- bairn and Hodgkinson.
i* |
03 A |
||||
§1 |
El |
*s |
|||
"-* 5 |
ft» . |
"8 |
|||
NAMES OF MATERIALS. |
%£ |
"C.S |
&e| 0 |
§ £« |
If |
11 |
1.1 |
IK |
11 |
11 |
|
METALS. |
|||||
Antimony — |
|||||
Cast |
281-25 |
1,000 M. |
|||
Bismuth |
013 '87 |
3,250 M. |
|||
Brass — |
|||||
Cast .... |
525-00 |
17 908 Re |
10 304 Re |
0 170,000 |
|
Wire-drawn |
534 • 00 |
14,230,000 |
|||
Copper — Cast |
537-93 |
19,072 |
29,272 Re. |
||
Sheet |
549-00 |
32,184 |
|||
Wire-drawn |
500-00 |
(51,228 |
|||
In Bolts |
48,000 |
||||
Iron. |
|||||
Cast Iron. |
|||||
Old Park |
48,240 T. |
18,014,400 T. |
|||
Carron, No. 2 — |
|||||
Cold Blast |
441-02 |
10,083 H. |
100,375 H. |
38.550 H. |
17,270,500 H.* |
Hot Blast |
440-37 |
13,505 H. |
108,540 H. |
37,503 H. |
10,085,000 H.* |
Carron, No. 3 — |
|||||
Cold Blast |
44-3-37 |
14,200 H. |
115,442 H. |
35,980 F.* |
10,240,906 F. |
Hot Blast. |
441-00 |
• 17,775 H. |
1:33,440 H. |
42,687 F.* |
17,873,100 F. |
16 |
APPENDIX TABLE— Continued,
NAMES OF MATERIALS. |
W t'ipht of one cubic ft. in Ibs. £ |
j K if ii |
Crushing Force per square inch in 11)8. 0. |
Modulus of Rupture. R. |
Coefficient of Elasticity. E. |
Iron. Cast Iron. Devon, No. 3 — Cold Blast |
455 -a3 451 81 442-4-3 437-37 434-06 435-50 439-37 436-00 |
86,288 H.* 43,497 H.* 37,503 H.* 35,316 H.* 33,453 F.* 33,696 H.* 34,587 F.* 29,889 F.* 36.693 F.* 33,850 F.* \- |
22. 907.700 H. 22,478,660 H. 15,381.200 H. 13,730,500 II. 14,313.500 F. 14,322,500 F. 13,981,000 F. 11,974,500 F. 14,003.550 F. 13,294,490 F. 27,548,000 Wa. 31,359,000 S. 31,819,000 S. 29,215,000 F. 30,278,000 F. 31,901,000 F. 29,000,000 1 5 ll |
||
Hot Blast |
29,107 H. 17,466 H. 13,434 H. 18,855 H. 16,676 H. |
145,435 H. 93,366 H. 86,397 H. 81,770 H. 82,739 H. |
|||
Buffery, No. 1 — Cold Blast Hot Blast |
|||||
Coed Talon, No. 2— Cold Blast |
|||||
Hot Blast Elsicar, No. 1— Cold Blast |
|||||
Milton, No. 1— Hot Blast |
|||||
Muirkirk, No. 1— Cold Blast |
444-56 434 -5U |
||||
Hot Blast |
|||||
Morris Stirling's 2d quality . . |
25,764 -•! |
119,000 14,000 to 34.000 Wa. 45,970 Wa. |
|||
(See also p. 52.) Gun Metal— |
|||||
595-00 |
1 |
f |
|||
Steel. Hammered Cast Steel, from |
91.000 | c 142,000 f b' 171,000 S. ;i40,945 S. 88,415 F. 108,099 F. 93,616 F. 120,000 |
||||
F. Krupp |
|||||
Bessemer Steel, from Hogbo, marked 10 |
|||||
Bessemer Steel, Eng. Mean of |
485-37 488-70 492-50 486-25 |
225,668 F. 225,568 F. |
|||
Navlor, Vickers & Co. Cruci ble Steel |
|||||
Mushefs Steel- Soft |
|||||
Cast Steel- Soft |
198,944 Wa. 381,985 Wa. |
||||
Not Hardened |
|||||
Razor Tempered Steel Wire Rope- Fine Wire |
490-00 |
150,000 40,000 Bo. 242,100 See page 25. 57,300 La. 57,300 La. 67,200 Bra 60,480 La. 71.680 R. 80,000 Te. 98,000'iv; 134,000 La. 203,000 L». |
|||
> See page 53. |
|||||
Cast Steel . . |
|||||
WroucjM Iron. English |
481-20 475-50 487-00 |
||||
In Bars \ |
|||||
English, in wire 1-10 inch duim |
\ |
||||
Uu.-«i:m. in wire; diam. l-2li to 1-30 inch |
j |
APPENDIX. TABLE— Continued.
243
NAMES OF MATERIALS. |
II tM S ®«*° §i £! |
? k- !i si |
Crushing Force per square inch in Ibs. C. |
Modulus of Rupture. R. |
•£ !f 8 * ow |
Wrought Iron. Rolled in sheets and cut cross- |
40 320 Mi |
||||
31 360 Mi |
1 |
||||
In chains, oval links, iron % |
48 160 Br |
g |
|||
73 600 Ha |
|||||
Lake Superior and Iron Mountain Charcoal Bloom. |
90 000 Ha |
1" |
|||
Missouri Iron, bar |
47,909 J. 52 099 J |
S |
|||
Salisbury Ct., 40 exp |
58 009 J |
||||
Centre Co. Pa., 15 exp Phillipsburgh Wire, Pa. ( 0-333 in., 13 exp... |
58,400 J. 84,186 J. |
0 I o* |
|||
Diam •< 0 190 in., 5 exp |
73 888 J |
« |
|||
(0.156 in.. 5 exp.... Mean of 188 rolled bars Mean of 167 plates length- |
|
89.162 J. 57,557 Ki. 50 737 Ki |
s £ £ |
||
Mean of 160 plates crosswise . |
46,171 Ki. |
||||
Low Moor, bars |
60,304 Ki. |
||||
Swedish forged |
41,000 Ki. |
||||
Hammered Bessemer Iron, from Hbgbo |
50,000 Ki. |
32 320 000 S |
|||
Low Moor Rolled Puddled |
31 976 000 S |
||||
Rolled Iron, Swedish, char coal heath |
65 000 S |
27 000 000 S |
|||
Lead, cast, English Lead Wire |
717-45 705'12 |
1,824 Re. 2 581 M |
|||
Silver standard. |
644'50 |
40 902 M |
|||
Tin cast |
455 '68 |
5,322 M. |
4 608 000 Tr |
||
439 ' 25 |
13 680 000 Tr |
||||
STONE — NATURAL AND Aim- FICIAL. Granites. |
164 |
10,914 Re. |
|||
166 |
6,356 Re. |
||||
10,780 Wn. |
|||||
166 |
12 286 F |
||||
Sandstones. |
6,493 Bn. |
||||
158 |
6,630 Re. |
||||
Derby Grit, a red, friable |
148 |
3,142 Re. |
|||
156 |
4,345 Re. |
||||
Limestones. Limestone, Magnesian (G-raf- |
| |
17,000 E. |
\ |
Same as .Wt. |
|
ton, 111.) |
} |
( |
APPENDIX. TABLE — Contimifd.
NAMES OF MATKIUAUB. |
Weight of one cubic ft. in Ibs. 6 |
Tenacity per aq. inch in Ibs. T. |
Crushing Fore* p«>r square inch in Ibs. C. |
Modulus of Rupture. R. |
Coefficient of Elasticity. |
Limestone*. Limestone, compact |
162 |
7,713 Re. |
|||
Limestone, Kerry, Listowel |
18,043 Wn. |
||||
Chalk ' |
501 Re. |
||||
Other Stones. Alabaster (Oriental), white.. |
170 |
8,216 Re. |
|||
Do white Italian, veined. |
165 |
9, (581 Re. |
1.062 |
25,200,000 T. |
|
168 |
9,219 Re. |
2,664 |
|||
Portland Stone (Oolite) Valentia, Kerry (slate stone).. |
151 |
3,792 Re. 10,943 Wn. |
|||
Green Stone, from Giant's Causeway |
17.220 Wn. |
||||
Quartx Rock,Holyhead(across |
25,500 Ma. |
||||
Quartz Rock (parallel to lami nation) |
14,000 Ma. |
||||
120 |
|||||
158 |
2.010 Re. |
||||
Artificial Stone. Brick red .... |
135'5 |
280 |
9 - i;. . |
^ |
~~~ ^ |
Brick pale red . . |
130-31 |
300 |
5<i2 Re. |
// *^ |
|
j 800 to |
1 n J |
r- »J |
|||
Bire Brick Stourbridge |
| 4.000 Ha. 1,717 Re. |
Vv -* |
jlOt*Cl |
||
2, 177 Ha. |
v^ |
||||
Bricks set in cement (bricks |
521 Cl. |
Jf pollf^tT |
|||
j 500 to |
\L- |
* v/amori- |
|||
Cement, Portland, with sand |
{ 92 to |
( 800 Ha. \ |
|||
Cement, Portland, with no |
(427 to |
) I |
|||
1 ill |
(1,000 to |
||||
Chalk |
116-81 |
334 Re. |
|||
Glass, Plate |
153-31 |
9420 |
|||
Mortar |
107 |
6Q |
j 120 to |
\ |
|
TIMBER. Acacia, English \ld«T |
47-37 50-00 |
16,000 Be. 14,186 M. |
'"6,859'H." |
11,202 B. |
1,152,000 B. |
Apple Tree |
49-56 |
19,500 Be. |
|||
43-12 |
j 8,683 H. |
1 fU4 <^)(1 Tl |
|||
Ash -j yery ,jrv |
55-81 |
1 9.363 H. |
|||
51-37 |
12,396 |
7,158 H. |
|||
67 '51 |
20,886 B. |
2,601,600 B. |
|||
TI^., j Ordinary Beech1 Very dry... |
53-37 45 12 |
15.784 B. 17,850 B. |
7.733 H. 9,363 H. f |
9,336 B. |
1,353,600 B. |
APPENDIX. TABLE— Continued.
245
NAMES ov MATERIALS. |
Weight of one cubic ft. in Ibs. 8 |
£ !*- I'- ii |
Crushing Force per square inch in Ibs. C. |
Modulus of Rupture. R. |
Coefficient of Elasticity. E. |
TIMBER. Birch, common |
49-50 40-50 |
15,000 |
j 4,5.33 H. ( | 6,402 H. j 11 663 H |
10,920 B. 9 624 B |
1,562,400 B. 1 257 600 B |
Box: dry. |
60-00 |
19 891 B |
10,299 H. |
||
Bullet Tree (Berbice) .... |
64 '31 |
15 636 B |
2 610 600 B |
||
Cane |
25-00 |
6 300 Be |
|||
Cedar Canadian |
56 '81 |
11 400 Be |
5 674 H |
||
Crab Tree |
47 '80 |
6 499 H |
|||
Deal- Christiana Middle |
43 -62 |
12 400 |
9 864 B |
1 672 000 B |
|
Norway Spruce English. |
21-25 29'37 |
17,600 7 000 |
|||
Red |
5,748 H. |
||||
White. |
6,741 H. |
||||
Elder |
43-43 |
10 230 |
8 467 H |
||
Elm, seasoned |
36-75 |
13 489 M. |
10,331 H. |
6,078 B. |
699 840 B |
Fir- New England |
34-56 |
6,612 B. |
2 191 200 B |
||
Riga |
47-06 |
j ii,549' to |
5,748 to |
6,648 B. |
1,328,800 |
Hazel .... |
53-75 |
1 12,857 B. IS 000 Be |
6,586 H. |
7,572 B. |
869,600 B. |
63 -87 |
24 696 |
||||
Larch — Green |
32-62 |
10 220 B |
3,201 H. |
4,992 B. |
897 600 B |
Dry Lignum-vitse |
35-00 76-25 |
8,900 B. 11 800 M |
5,568 H. |
6,894 B. |
1,052,800 B. |
Mahogany, Spanish |
50-00 |
16,500 |
8,198 H. |
||
Maple, Norway Oak- English |
49-56 58-37 |
10,584 17 300 M. |
j 4,684 to |
10,032 B. |
1,451,200 B. |
Canadian |
54-50 47-34 |
10,253 12780 |
1 9,509 H. j 4,231 to | 9,509 H. |
10,596 B. 8,742 B. |
2,148,800 B. 1,191 200 B |
Adriatic |
62-06 |
8298 B |
974 400 B |
||
African Middle |
60-75 |
13,566 B. |
2,2a3,200 B. |
||
Pear Tree |
41-31 |
7,518 H. |
|||
Pine- Pitch |
41-25 |
7 818 M. |
9,792 |
1,225.600 B. |
|
Red |
41-06 |
5,375 H. |
8,946 B. |
1,840,000 B. |
|
28-81 |
5,445-H. |
1,600,000 Tr. |
|||
Plum. Tree |
49-06 |
11,351 |
j 3,657 to |
||
Poplar Teak dry |
23-93 41-06 |
7,200 15,000 B. |
| 9,3(>7.H. j 3,107 to 1 5,124 H. 12,101 H. |
14,722 B. |
2,414.400 B. |
Walnut |
41 • 93 |
8 130 M. |
6,635 H. |
306,000 |
|
Willow, dry |
24-37 |
14,000 Be. |
ERRATA.
Page 14, line 9, for P = 13, 934,000y = 2,907,432,000^, read,
P = 13,934,000y - 2,907,432,000^'
" 22, " 5, for cFt, read, dt-.
" " "at the middle of the page, for Resiliance of Prisms, read, Kesi- fance of Prisms.
" " 10 from the bottom, for - >— A2, read, -y-A2.
6 6
26, " 3, for exponetials, read, exponentials.
36, " 10, for rods or rivet iron, read, rods of rivet iron.
53, "2, for No. 1, read, No. 2.
56, " 3 from the bottom, for «, read, y.
68, " at the bottom of the table, for Mean 6852, read, 685 '2.
72, " 15, for equation (26), read, equation (24).
96, " 1, for 82, read, §6.
KMVY or CAI.iKnRNIA MHKARY
THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW
JAN 31 1916
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