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The Concrete 
Engineer's Handbook : 


A CONVENIENT REFERENCE BOOK 
For All Persons Interested In 


Cement, Plain and Reinforced Concrete, 
Building Construction, Architecture, 
Concrete Blocks, Mill Building, 
Office Building, Fireproof 


Houses, Etc. = 


BY 


International Correspondence Schools 
SCRANTON, PA. 


ia) 
, aN ~ \ — Ist Edition, 12th Thousand, 2d Impression 


\ : 


Weoaairvoi, PA. 
#) 
Z-) INTERNATIONAL TEXTBOOK COMPANY 


Z 1912 


— 


CopyRiGHt, 1911, By 
INTERNATIONAL TEXTBOOK COMPANY 
ENTERED AT STATIONERS’ HALL, LONDON 
Att RicHTs RESERVED 


PRINTED IN THE UNITED STATES 


7 

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a Ne eS ee Ss Ko n —— _ See 


PREFACE 


The publishers have not attempted in this 
work to produce a condensed cyclopedia cover- 
ing the broad field of concrete engineering, but 
they have aimed to present to the public a 
handy reference book convenient to carry in 
the pocket—a pocketbook in reality—and con- 
taining rules, formulas, tables, and diagrams 
that are often used and needed by architects, 
concrete engineers, inspectors, superintendents, 
foremen, carpenters, contractors, draftsmen, 
designers, house owners and prospective house 
owners; in fact, every one engaged in any pro- 
fession or trade connected with building of con- 
crete, or in any way interested therein. 

The aim of the publishers has been to select 
from the vast amount of material on hand only 
that portion which is most likely to be used in 
connection with daily work, or which will be 
most frequently consulted. Although the treat- 
ment of some subjects is of necessity brief, it 
has been the aim to so distribute the space 
available that it would cover the more important 
subjects as fully as possible. It has been found 
impracticable to give tables for the strength of 
concrete beams and slabs, because the stresses 
employed are subject to city regulations. Until 
the large cities adopt uniform requirements, such 
tables cannot come into general use. The tables 
selected throughout this work are those most in 
demand, and the applications of the rules and 


iii 


iv PREFACE 


formulas are shown in many cases by practical 
examples and solutions, together with expla- 
nations. 

The material is conveniently arranged for 
ready reference. Particular attention has been 
given to the weights of various materials. Most 
tables on weights are incomplete and it is often 
necessary to search some time before the weight 
of any particular material can be ascertained. 
It is hoped that this volume will obviate tedious 
work of this kind. 

The section on tests of cement will be found to 
contain useful information that is usually found 
only in special treatises on the subject. 

INTERNATIONAL CORRESPONDENCE SCHOOLS, 

July, 1911. Scranton, Pa. 


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INDEX 


A 


Accelerated test, 213. 
tests for soundness, 360. 
Accidental load, 56. 
Aggregate, Definition of, 
2 


other than sand, 231. 
Selection of, 232. 
Size of, 231. 
Specifications for, 368. 
Aggregates, Comparative 
value of different, 234. 
Allowable live load on floors 
in different cities, 63. 


Analysis of cement, 229. 


Angles having equal legs, 
oo of standard, 


54. 
having unequal legs, Prop- 
erties of standard, 158. 
or arcs, Measure of, 2 
Apparatus for fineness tests, 


for specific-gravity tests, 
225 


Arc, Length of, 51. 
Arch, Reinforced-concrete, 


306. 
ee eas of angles and, 


Area of circle, 50. 
of circular segment, 52. 
of ellipse, 51. 
of parallelogram, 51. 
of rectangle, 51. 
of ring, 52. 
of sector of circle, 51. 
of peuece of circular ring, 


of surface of cone, 53. 

of surface of cylinder, 52. 

of surface of frustum of 
cone, 53. 

of surface of frustum of 
cylinder, 53. 


Area of surface of frustum 
of rectangular pyramid, 


of surface of parallelo- 
piped, 55. 

of surface of prism, 55. 

of surface of rectangular 
pyramid, 54. 

of surface of sphere, 53. 

of. trapezoid, 51. 

of trapezium, 51. 

of triangle, 51. 

Areas and weights of square 

and round bars, 253 
Irregular, 55. 
of circles, Circumferences 
and, 10. 

Argillaceous sand, 197. 

Australian woods, Table of 
weight of, 79. 

Avoirdupois weight, 3. 

Axis by means of the prin- 
ciple of moments, Lo- 
cating neutral, 97. 

Neutral, 97, 143. 


B 


Bar, Columbian, 262. 
Corrugated, 257. 
De Mann, 255. 
Diamond, 260 
Hyatt, 255. 
iron, Plain, 254. 
iene 258. 
ahn cup, 257. 
Kahn trussed, 260. 
Longitudinal column, 250. 
Main reinforcement, 250. 
Monolith steel, 261. 
of special construction, 


Quad, 255. 
Ransome, 256. 
Rerolled, 254. 
Shear, 250. 


vi INDEX . 


Bar, Shrinkage, 250. 

Siamese, 255. 

Spiral, 256. 

Splice, 250. 

Square-twisted, 256. 

Square-twisted lug, 257. 

Staff, 255. 

Table of properties of cor- 
rugated, or Johnson, 258. 

Table of properties of 
Diamond, 260. 

Table of properties of 
Kahn cup, 257. 

Table of properties of 
Ransome, , 

Table of properties of 
square-twisted lug, 258. 

Table of properties of 
Thacher, 259. 

Table of properties of 
universal, 259. 

Tension, 250. 

Thacher, 255, 259. 

ie-, 250. 
Trus-Con, 261. 
Trussed, 250. 
263 


Unit, 255. 
Universal, 258. 
Weights and areas of a 
square and round, 253. 
Barrett specifications for 
waterproofing, 345. 
Batch mixers, 347. 
Beam and girder, Bearings 
for a concrete, 332. 
Concrete T, 298. 
Continuous, 99. 
Continuous concrete, 294. 
Deflection of, 174 
Formula for design of a, 


167. 
Roca for deflection of 


172. 
Lends suddenly applied 
on, 176. 
Plain concrete, 282. 
Rectangular concrete, 291. 
Reinforced-concrete, 290. 
Restrained, 100. 

Shear on, 104. 

Simple, 99. 


Beam sockets, 328. 
Weight of, 170. 
Beams, Bending moments 
of, 110. 
Continuous, 120. 
Diagram for bending mo- 
ment on, 114. 
fixed at both ends, 100. 
Forces acting on, 99. 
Formulas for maximum 
shear and bending mo- 
ment of, 116 
Homogeneous, 129. 
Loads on, 100 
over equal spans, Bend- 
ing moments for con- 
tinuous, 120. 
over equal spans, Reac- 
tions for continuous, 


Positive and negative 
shear on, 105. 
oe age of standard I, 


Reactions on, 100. 
reinforced at top and 
bottom, 295. 
Shear diagrams of, 107. 
Styles of, 99. 
Bearing value of foundation 
soils, 311, 313. 
Bearings for concrete beams 
and girders, 
sire: -moment diagram, 
4. 


moment for continuous 
beams over equal spans, 


-moment formulas, 167. 
moment of beams, For- 
mulas for maximum 
shear and, 116. 
moments, 110. 
stress in Sceee bt Table 
of, 284. 
stress of various materials, 
Table of, 283. 
Bigelow socket, 328. 
Bituminous concrete, 230. 
membranes, 345 
Boiler, Instructions for start- 
ing and managing a, 255. 


Boiling test, 218. 
Bolts, all forms 
clamp, 338. 
Bond and shear, 300. 

Brace for wall forms, 341. 
Brayton reinforcement, 280. 
Brick facing, 327. ; 
masonry, Crushing 
strength of, 288. 
Briquet, Form for, 217. 
Methods of making a, 218. 
r Storage of, 218. 
4 Table of tensile strength 
3 of a cement, 221 


with 


Building details, 322. 
laws, 357. 
material, Strength of, 124. 
material, Table of weight 


of, 59. 
Built-up section, 98. 
P C 


Calcareous sand, 197. 
Calcination, 192. 
Cantilever foundations, 318. 
Capacity, Measure of, 5. 
Carlin cube mixers, 347. 
re column, Wood and, 
-iron columns, 183. 
-iron columns, Dimensions 
of standard connections 
to, 190. 
; -iron columns, Safe load 
E on, F 
Cement, 194. : 
Accelerated test for, 213. 
Accelerated tests for 
j soundness of, 360. 
; and sand, 192. 
Apparatus for fineness test 
for, 223. 
; Apparatus 


eat Ld 4 


Say ory 


} specific- 
_ gravity tests on, 225. 
| Boiling test for, 213. 
R 
3 
* 


for 


briquet, Form for, 217. 
briquet, Methods 
making, 218. 
briquets, Storage of, 218. 
briquets, Table of tensile 
strength of, 221. 


of 


INDEX 


vii 


Cement, Chemical analysis 
of, 229. 
Constancy of volume of, 


229. 

Fineness of, 229. 

for waterproofing, 344. 

Hard set of, 222. 

Improved, 195. 

Initial set of, 222. 

Inspection of, 229. 

mortar, Materials required 
per cubic yard of, 203. 

Measurement of expan- 
sion of, 211. 

Method of making fine- 
ness test on, 224. 

Method of making specific- 
gravity test on, 226. 

Mixed, 195. 

Molds for, 218. 

mortar, 201. 

mortar, Table of tensile 
strength of, 204. 

-mortar tests, Sand for,217. 

mortars, Lime and, 199 

Natural, 194. 

Normal consistency of, 
216. 

Normal test of, 211. 

Packages for, 228. 

Portland, 194. 

Properties of, 195, 

Protection of, 229. 

Puzzolan, 194. 

Results of tests for sound- 
ness of, 215. 

Results of time of setting 
tests on, 223. 

Sand, 195. 

Secondary tests for, 221. 

Set of, 229. 

Slag, 195. 

specifications, 227, 367. 

Steam test for, 214. 

Table of requirements of 
high-grade, 228 

-testing machine, 219. 

Tests for fineness of, 223. 

Tests for specific gravity 
of, 225. 

Tests for tensile strength 
of, 215 


viii INDEX 


Cement, Tests of natural and 
slag, 
Tests on, 208. 
setae -of- setting test for, 


Weight of hydraulic, 196. 
Cementing material, 192. 
Center of gravity, 95. 
of grauy of plane figures, 
9 


Centering for concrete, 333. 
Pigaer rp loaded columns, 
2 


Channels, Properties of 


standard, 151. 
Chatelier flask, Le, 225. 
Chemical analysis of cement, 


29. 

Cinder concrete, 230. 

Circle, Area of, 10, 50. 
Circumference of, 10, 50. 
Chord of, 52. 

Diameter of, 50. 
Radius of, 50. 
Sector of, 51. 
Segment of, 52. 

Circular ring, ‘Surface of, 53. 
ring, Volume of, 

Circumference of circle, 50. 
and areas of circles, 10. 

Cities, Unit working values 

of concrete allowed by 
various, 240. 
a bolts, Wall forms for, 


Clamping device for forms, 


Clay, Waterproofing by add- 
ing lime or, 343 
Clinton wire cloth, 268. 
Coarse sand, 198 
ockburn mixer, 352. 
Cold yoga Concreting in, 


weather, Laying mortar in, 


Collapsible forms, 336. 
Coloring of mortar, 208. 
Columbian bar, 262. 
Column, Cast-iron, 187. 
Centrally loaded, 285. 
Concrete, 285. 


Column, Dimensions of 
standard connections to 
cast-iron, 190. 

- Eccentrically loaded, 288. 

Eccentrically loaded con- 
crete, 306. 

footings, Placing rein- 
forcement in, 314. 

formulas, 187. 

reinforcement, Empirical 
rules for straight, 304. 

reinforcement, Hooped, 


reinforcement, Straight, 


rods, Longitudinal, 250. 
Safe load on a cast-iron, 


8. 
Spread footings for out- 

side, 
Table of constants for a 
rectangular wooden, 


ties, 250. 

Wood and cast-iron, 179. 
Common limes, 193 
stp erin , Granulometric, 


of forces, 88. 
Compressive strength of 
brick masonry, 288. 

eins of concrete, 233, 


strength of stone and 
stone masonry, 287. 
stress, 122. 


‘Concentrated load, 100. 


Concrete allowed by various 
cities, Unit working 
values of, 240. 

arches, Reinforced-, 306. 

beam, ‘Continuous, 294. 

beam, Rectangular, 291. 

beams and girders, Bear- 
ing for, 332. 

beams, Plain, 282. 

beams reinforced at top 
and bottom, 295. 

Bituminous, 230. 

by weight, Proportioning, 


235. 
Cinder, 230. 


INDEX ix 


Concrete columns, 285, 302. 
columns, Eccentrically 
loaded, 306. 
Comparative value of dif- 
» ferent aggregates used 
in, 234. 
Compressive strength of, 


233. 
Cost of, 357, 364. 
Crushing strength of, 286. 
po 285 
Dry, 236. 
Effect of fire on, 237. 
Effect of thermal changes 
in, 238. 
Effect of vibration on,238. 
Fastenings in, 327. 
for waterproofing, Mixing 


of, 342. 

Fuller’s rule for quantities 
for, 242 

Lime, 230. 

Materials used for, 230. 

measurer and feeder, Gil- 
braith, 353. 

Medium, 285. 

Methods of measuring in- 
gredients for, 242. 

mixer, Cockburn, 352. 

mixer, Continuous, 351. » 

mixer, Cube, 347 

mixer, Drake, 351. 

mixer, Quantitative, 353. 

mixer, Ransome, 348. 

mixer, Smith, 349. 

mixer, Starting and opera- 
ting, 356 

mixers, 347. 

mixers, Operation of, 354. 

Mixing and working of, 242. 

Mixing of, 246. 

Moduli of rupture of, 284. 

Plain, 230. 

Properties of, 237. 

Quantities for, 244. 

Specifications for, 357, 367. 

Stone, 230. 

T beams, 298. 

oe or gderarags strength 
of, ° 

Transverse strength of, 
284. 


Concrete, Usual proportions 
of material for, 235 
Water for, 236. 
Waterproofing of, 342. 
Wet, 2 
with new, Joining of old, 
247 


Working stresses of, 238. 
Concreting at high temper- 
atures, 246. 
in freezing weather, 247. 
Cone, Frustum of, 53. 
Surface of, 53. 
Volume of, 53. 
Connection to cast-iron col- 
umns, Dimensions of 
standard, 190. 
Consistency, Normal, 216. 
Constancy. of volume, 210. 
of volume of cement, 229. 
Construction and finish of 
form work, 333. 
Cornice, 343. 
Continuous beam, 99, 120. 
beams over equal spans, 
ae moments for, 


beams over equal spans, 
Reactions for, 119. 
concrete beam, 294. 
inserts, 330. 
mixer, 351. 
Conversion of inches to fest, 
Table of, 14 
tables, 6. 
Chord, Length of, 52. 
Cornice construction, 343. 
Cornices and eaves, 325. 
Corrugated bar, 257. 
ohnson, bar, Table of 
properties of, 
Cost data, 364. 
of concrete, 357 
Crushing strength. of brick 


masonry, 288. 
strength of concrete, 286. 
strength of stone and 


stone masonry, 287. 
Cube mixers, 347. 
mixers, Carlin, 347. 
root, 27. 
Cubes and squares, 28. 


x INDEX 


Cubic measure, 2. 
Cu » 4 ings reinforcement, 


Cup bar, Kahn, 257. 
bar, Table of properties of 
Kahn, 257. y 
Curing, Definition of, 236. 
Cylinder, Frustum of, 53. 
Surface of, 52. A 
Volume of, 52. 


Damp concrete, 285. 
Dead load, 56. 
Decimal equivalents, 14. 
Decimals of a foot for each 
fraction of an inch, 14. 
Deflection of beams, 174. 
of Sen Formulas for, 


De Mann bar, 255. 
tit = of beams, Formulas 
or, : 
of concrete structural 
members, 282. 
of footings, Formulas for, 


Details, Building, 322. 
ape borg Bending moment, 


Shear, 107. 
Diameter of circle, 50. 
Diamond bar, 260. 

bar, Table of properties of, 


Dimension of standard con- 
nections to cast-iron 
columns, 190. 

of standard T rails, Prop- 
erties and principal, 166. 

Disposition of loads, 8 

Distributed load, 100. 

Double reinforced - concrete 
beam, 295. 

Drake mixer, 351. 

Dry concrete, 236. 

measure, 3. 

Dryer, Sand, 199. 


E 
Eaves and cornices, 325. 
seasrat waged loaded column, 


Eccentrically loaded concrete 
columns, 306 

Elastic properties, 123. 

Electrically welded fabric, 


268. j 
Elements of usual sections, 


Ellipse, Area of, 51. 
Perimeter of, 51. 

Elongation, 122. 
Ultimate, 124. 


*Eminently hydraulic lime, 
194. 


Empirical rules for straight 
reinforcement, 304. 
Equivalents, Decimal, i} 

of Fee to feet, Table of, 
Rate sen: and involution, 
Expanded metal, 263. 
metal, Herring-bone, 266. 
metal, Kahn, 265. 
metal, Table of properties 
of, 264. 
metal, ce of properties 
of Kah 
sea pe ‘Meaiipeivialt of, 


F 
Binion Electrically welded, 


Lock-woven wire, 267. 
Tie-locked wire, 267. 
Fabricated system of rein- 
forcement, 277. 
Facing, Brick, 327. 
Factor of safety, 124. 
Failure, rere tie to re- 
sist lines of, 2 
Farm products, Table of 
weight of, 71. 
Fastenings in concrete, 327. 
Fat lime, 193. 
Feebly hydraulic lime, 194. 
Ferroinclave, 267 
Field inspection, 208. 
operations, 347 
Fine sand, 198. 
Fineness of cement, 229. 
test, Apparatus for, 223 


“Ss 


ll Mi tli ee 


Ne a 


INDEX xi 


Fineness, Test for, 223. 
ee, ee of making, 


Finish of form work, Con- 
struction and, 333. : 
Fire on concrete, Effect of, 


Fixed at both ends, Beam, 


100. 
Flask, Le Chatelier, 225. 
Floor and test load, 368 
systems, Forms for, 334. 
Floors in different cities, 
Table of allowable live 
loads on, 63. 
Foot for each fraction of an 
inch, Table of decimals 


of a, 14. ! 
a Formula for design 
of a, : 
for outside columns, 
Spread, 315. 
Placing teinforcement in 
a column, 314. 
Spread, 314. 
Force of wind, Table of 
velocity and, 82. 
Representation of a, 87. 
Forces, 87. 
acting on beams, 99. 
Composition of, 88. 
Moment of, 93. 
Parallelogram of, 88. 
Resolution of, 91. 
Resultant of several, 89. 
Triangle of, 89. 
Form, Braces for wall, 314. 
Clamping devices ‘for a, 


340. 
Collapsible, 336. 
constructed of planks, 


for floor systems, 334. 
Partly collapsible, 337. 
Plank holders 8 a, 340. 
Spandrel wall, 341 

bi clamp bolts, Wail, 


38. 
with wire ties, Wall, 338. 


work, 
work construction and 
. finish, 333. 


Formula, Bending moment, 
167. 


for cast-iron columns, 187. 
for deflection of beams, 


for design of beams, 167. 
for long posts, 181 
Formulas, 16. 
for maximum shears and 
bending moments of 
beams, 116. 
Foundation, Cantilever, 318. 
soils, Bearing value of, 
311,313. 
Foundations, 311. 
Frames, Pin-connected gir- 


der, 278. 
Freezing weather, Concret- 
ing in, 247. 
weather, Laying mortar 
in, 206 


Frustum of cone, 53. 
of cylinder, 53. 
of prism, Volume of, 55. 
of rectangular pyramid, 
Surface of, 54 
of rectangular pyramid, 
Volume of, 54. 
bias rule for quantities, 


G 
Gabriel system of reinforce- 
ment, 
Gilbraith measurer and 
feeder, 353. 
Girder frames, Pin-con- 
nected, 278. 


Girders, Bearings for con- 
crete beams and, 332. 
sae? aia composition, 


Gravity, Center of, 95. 
Grouting, 207. 
Gyration, Radius of, 142. 


H 


Hancock insert, 328. 

Hard set, 222. 

Heat changes in concrete, 
Effect of, 238. 


xii INDEX 


Helix, oan of, 52. 
Herring- bone expanded 


metal, 266. 
High-carbon steel, 252. 
temperatures, Concreting 
at, 246. 


Homogeneous beams, 129. 

Hooke’s law, 123. 

Hooped column reinforce- 
ment, 302, 305. 

Hot Ne sa Concreting in, 


Hyatt bar, 255. 
Hydrated lime, 193. 
Hydraulic cement, Weight 


of; 196 
lime, 193. 
vE 
I beams, Properties of, 148. 
Imperviousness, Effect of 


strength and, 234. 
Improved cement, 195. 
Inch, Table of decimals of a 

aot for fraction of an, 


Inertia, Moment of, 129. 
Ingredients for concrete, 
Methods of measuring, 
242. 
Proportioning of, 234. 
Initial set, 222. 
Insert, Continuous, 330. 
Hancock, 328. 
Inspection, Field, 208. 
of cement, 229. 
Integral method of water- 
proofing, 342. 
Involution and evolution: 
lron column, Wood and 
cast-, 179. 
columns, Cast-, 187. 


column, Dimensions of 
standard connections to 
cast-, 190. 

column, Safe load on 
cast-, 


Plain bar, 254. 
Irregular areas, 55. 


Jennings-Steinmetz socket, 
328. 


Johnson bar, 258. 

_ bar, Table of properties 
of, 258. 

Joining old and new work, 
247. 


K 


Kahn cup bar, Table of 
properties of, 257. 

expanded metal, Table of 
properties of, 265 

Stes of reinforcement, 


Ths 
trussed bar, 260. 


L 
Lath, Herring-bone metal, 


Law, Hooke’s, 123. 
for steel, 361. 
for walls, 361. 
Laws, Building, 357. 
Laying mortar in freezing 
weather, 206. 
Le Chatelier flask, 225. 
Legs, Properties of standard 
ener having equal, 


Properties of standard 
angles having unequal, 


158. 
Length of arc, 51. 
of chord, 52. 
of helix, 52. 
of spiral, 54. 
Lengths, Measure of, 5. 
Lime and cement mortars, 
199. 
Common, 193. 
concrete, 230. 
Fat, 193. 
Hydrated, 193. 
Hydraulic, 193. 
Meager, 193. 
mortar, 200. 
or clay, Waterproofing by 
adding, 343. 
Poor, 193. 


Pe ae ee hlmvehlCh eh Stt:™~S~; SL 


INDEX xiii 


Lime, Rich, 193. 
Slaked, 193. 
Linear measure, 1. 
Lintel and spandrel construc- 
tion, 322. 
Liquid measure, 3. 
Live load, 56, 61. 
loads on floors in different 
cities, Table of, 63. 
Load, Accidental, 56. 
Concentrated, 100. 
Dead, 56. 
Distributed, 100. 
Floor and test, 368. 
from floor to floor, Re- 
duction of live, 86. 
Live, 56, 61. 
on cast-iron columns, 
Safe, 188. 
on floors in different cities, 
Table of allowable live, 


63. 
Snow, 56. 
Suddenly applied, 176. 
Table of wind, 
Uniform, 100. 
Wind, 56. 
Wind and snow, 81. 
Loading testing machine, 
Rate of, 220. 
Loads, Disposition of, 85. 
in structures, 56. 
on beam, 100. 
Locating neutral axis by 
means of the principle 
of moments, 97. 
sugges wire fabric, 


Long posts, 181. 
-ton weight, 3. 
ce are column rods, 


Loop truss, Cummings, 
278 


Loose-rod system of rein- 
forcement, 269. 
Low temperature, 
mortar at, 206. 
Lug De r, Square-twisted, 


Laying 


bar, Table of properties of 
twisted, 258. 


M 


Machine, Rate of loading 
testing, 220. 
Testing, 219. 
Main reinforcing rods, 250. 
Masonry, Crushing strengch 
of brick, 288 
Crushing strength of 
stone, 287. 
Material, Cementing, 192. 
required per cubic yard of 
mortar, 203. 
Strength of building, 124. 
Table of weight of build- 


ing, 59. 
Table of weights of mis- 
cellaneous, 68. 
used for concrete, 230. 
i 2 fas proportions of, 


Mathematical tables, 10. 

Mathematics, 16. 

Matrix, Definition of, 230. 

Maximum shears and bend- 
ing moments of beams, 
Formulas for, 116. 

McCarty separator, 340. 

Meager lime, 193. 

Measure, Cubic, 2. 


Liquid, 3. 

of angles or arcs, 2. 

of capacity, 5. 

of length, 5. 

of surface, 5. 

of volume, 5. 

of weight, 6. 

Square, 2. 

Surveyor’s, 1. 

Surveyor’s square, 2. 
Measurement of expansion, 


of moments, 93. 
Measurer and feeder, Gil- 
braith, 353. 
Measures, Weights and, 1. 
Measuring ingredients, 
Method of, 242. 
Mechanics, 87. 
Medium concrete, 285. 


xiv «INDEX 


Medium steel, 252. 

Membrane method of water- 
proofing, 345. 

Mensuration, 49. 

Merchandise in bulk, Table 
of weight of, 64. 

Merrick system of reinforce- 
ment, 274. 

Metal, Expanded, 263. 

Herring-bone expanded, 


Kahn expanded: 265. 
reinforcement, Sheet-, 267. 
Table of properties of 
expanded, 264. 
Table of properties of 
Kahn expanded, 265. 
Metallic reinforcement, 
Characteristics of, 252. 
Metals, Table of ultimate 
strength of, 126. 
Tables of weights of vari- 
ous, 67 

Metric capacity, 5. 
conversion tables, 6. 
lengths, 5. 
surface, 5. 
system, 4. 
volume, 5. 
weight, 6. 

Mild, or soft, steel, 252. 

Miscellaneous materials, 

Table of weights of, 68. 
reinforcement, 280. 

Mixed cement, 195. 

Mixer, Carlin cube, 347. 
Cube, 347. 
Cockburn, 352. 
Concrete, 347. 
Continuous, 351. 
Drake, 
so aay of concrete, 

Quantitative, 353. 

Ransome, 34 

Smith, 349. : 

ae and operating a, 


Mixing and working of con- 
crete, 242. 
concrete for waterproof- 
ing, 342 


Mixing of concrete, 246. 
Mixtures, Sand and its, 197. 
Moduli of rupture of con- 
crete, Table of, 284. 
aoe of rupture, 124, 


Section, 145. 
Molds for mortar, 218. 
Moment, Diagram for bend- 
ing, 114 
for continuous beams over 
equal spans, Bending, 
120. 

formulas, Bending, 167. 

Measurements of, 93. 

of beams, Formulas for 

maximum shear and 
bending, 116. 

of forces, 93. 

of inertia, 129. 

Resultant, 95. 

Moments, Bending, 110. 

Locating neutral axis by 

means of the principle 
of, 97. 

Negative and positive, 94. 
Monolith steel bar, 262. 
Mortar briquets, Method of 

making, 

briquets, Storage of, 218. 

Cement, 201. 

Coloring of, 208. 

Consistency of, 216. 

in freezing weather, Lay- 

ing, 206. 

Lime, 200. 

Lime and cement, 199. 

Materials required per 

cubic yard of, 2! 

Molds for, 218. 

Retempering of, 206. 

Shrinking in, 207. 

Table of tensile strength 

of cement, 204. 

test, Sand for, 217. 

Mushroom system of rein- 
forcement, 275. 


N 


Natural and slag cement, 
Tests of, 227. 


.---. 


”_ Oe eee 


——— os ee S.”,:CmCrC( .mhlhl eo 


INDEX xv 


Natural cement, 194. 
Needle, Vicat, 222. 
Negative and positive mo- 
ments, 94. 
shear, Positive and, 105. 
Neutral axis, 97, 143 
axis by means of the prin- 
ciple of moments, Lo- 
cating, 97. 
New, Joining of old concrete 
with, : 
Normal consistency, 216. 
tests, 211. 
strc pressure, Table of, 


oO 
Old and new work, Joining, 
247. 


Operation of mixers, 354. 
Ordinarily hydraulic lime, 


Outside columns, Spread 
footings for, 315. 
Pr 


Package for cement, 228. 
Paraffin for waterproofing, 


Parallelogram, Area of, 51. 
of forces, 88. 
Duke vie ta Surface of, 


Volume of, 55. 
Percentage of voids, 197. 


_ Perimeter of ellipse, 51. 


Philippine wood, Table of 
weights of, 79. 
Joie girder frames, 


Pit sand, 197. 
Plain bar iron, 254. 
concrete, 230. 


Plane figures, Center of 
vity of, 96. 
pe om constructed of, 


holders for forms, 340. 
Polygons, Regular, BD. 
Poor lime, 193. 
Portland cement, 194. 
perment, Specifications for, 


Positive and negative mo- 
ments, 

and negative shear, 105. 
Post, Long, 1 

hort, 

Table of constants for a 
rectangular wooden, 
184. 

Wooden, 179. 

Powers, roots, and recipro- 
cals, 31. 

Preparation of sand, 199. 

Pressure, Table of normal 
wind, 83. 

Primary tests, 210. 

Principal dimensions of 
standard T rails, 166. 

Prism, Surface of, 55. 

Volume of, 55. 

Volume of frustum of, 55. 
Prismoid, Volume of, 53. 
Properties and principal 

dimensions of standard 
T rails, 166. 

of rolled ‘steel shapes, 146. 

of sections, 129. 

of standard angles having 
equal legs, 154 

of standard angles having 
unequal legs, 158. 

of standard channels, 151. 

of standard I beams, 148. 

of T bars, 164. 

of Z bars, 162. 

Property, "Plastic, 123. 

of cement, 195. 

of concrete, 237. 

of sand, 197. 

Proportion of materials, 
Usual, 235. 
at ca by weight, 


of ingredients, 234. 
Proportions of ingredients 
for concrete, 244. 
Protection of cement, 229. 
Purpose and classification of 
cement tests, 210. 
Puzzolan cement, 194. 
Pyramid, Surface of frus- 
tum of rectangular, 54. 
Surface of rectangular, 54. 


Xvi INDEX 


Pyramid, Volume of frus- 
tum of rectangular, 54. 
Volume of rectangular, 54. 


Q 


uad bar, 255. 

uantitative mixer, 353. 

uantities for concrete, 
Table of, 244. 

Fuller’s rule for, 242. 


R 


Radius of circle, 50. 

of gyration, 142. 

il, Properties and prin- 
cipal dimensions of 
standard, 166. 

Ransome bar, 256. 
bas. — of properties of, 


mixer, 348. 

Rate of loading testing 
machine, 220. 

Reactions for. continuous 


beams over equal spans, 
119. 


on beams, 100. 
Reciprocals, 30. 

Powers, roots, and, 31. 
Rectangle, Area of, 51. 
weceaesar concrete beam, 


pyramid, Surface of, 54. 
pyramid, Surface of frus- 
tum of, 54. 
pyramid, Volume of, 54. 
pyramid, Volume of frus- 
tum of, 54 
wooden posts, Table of 
eonstants for, 184. 
Reduction of live loads from 
floor to floor, 86. 
Regular polygons, 55. 
Reinforced at top and bot- 
o> Concrete beam, 


-concrete arches, 306. 

-concrete beam, 290. 

-concrete column, 302. 
sags iE Brayton, 


Reinforcement, Characteris- 

tics of metallic, 252. 

Cummings, 278. 

defined, Parts of steel, 250. 

Elements of steel, 248. 

Empirical rules for 
straight column, 304. 

si system of, 


Gabriel system of, 274. 
Hooped column, 302, 305. 
in column footings, Pla- 
cing, 314. 

Kahn system of, 271. 
Loose-rod system of, 269. 
Merrick system of, 274. 
Mushroom system of, 275. 
of PO ere construction, 


gpa oat girder, 278. 

Plain bar, 254. 

Shear frame, 279. 

Sheet-metal, 267. 

Square- and _ triangular- 
mesh wire, 268. 

Straight column, 302. 

Structural shapes used for 
steel, 263. 

to resist lines of failure, 
251 


Trussit, 267. 
Types of steel, 254. 
Unit system of, 277. 
Visintini, 281. 

Reinforcing rods, Main, 250. 
sted. Specifications for, 


Representation of a force, 87. 
Rerolled bars, 254 
Resolution of forces, 91. 
Restrained beams, 100. 
Resultant moment, 95. 

of several forces, 89. 
Retempering of mortar, 206. 
Rich lime, 193. 
Ring, Area of, 52. 
River sand, 197. 
Rod, 1» vongitudinal column, 


Main reinforcement, 250. 
of eigen construction, 


INDEX 


Rod, Plain iron, 254. 
Shear, 250. 
Shrinkage, 250. 
Slab, 250. 
Splice, 250. 
Tension, 250. 
Tie-, 250. 
Trussed, 250. 
Weights and areas of a 
square and round, 253. 
Rolled sections, Values for 
standard, 136. 
shapes, Pro; mes Paar 146. 
Roof y dara f, 56: 
Root, Cube, 2 
Square, i 
Roots, and reciprocals, Pow- 


ers, 31. 
Round bars, Weights and 
areas of square and, 


253. 
Rule, ior quantities, Fuller’s 
Rules yar operating mixers, 
Rupture, Modulus of, 124, 
179. 


of concrete, Table of 
moduli o 


Ss 


Safe load on cast-iron col- 


h of ‘concrete, 328. 
Safety factor, 124. 
Sampling, 209. 
Sand and cement, 192. 
and its mixtures, 197. 
Argillaceous, 197. 
Calcareous, 197. 
cement, 195. 
Coarse, 198. 
dryer, 199. 
Fine, 198. 
for mortar test, 217. 
Granulometric composi- 
tion of, 198. 
sas op Consistency of, 


Percentage of voids in, 
197. 


XVii 


Sand, Pit, 197. 
, Preparation of, 199. 
Properties of, 197. 
River, 197. 
Silicious, 197. 
a, 197. 
Secondary tests, 210, 221. 
Section, Built-up, 98. 
modulus, 145. 
Rootes. Elements of usual 


Properties of, 129. 

Values for rolled, 136. 
Sector of circle, 51. 
Segment of circle, 52. 
Selection of aggregates, 232. 
Separator, McCarty, 340. 
Set, Hard, 222. 

Initial, 222. 

of cement, 229. 
hier test, Results of time- 

oO - 

Shapes, Properties of rolled 


ste 

Shear and bending moment 
of beams, Formulas for 
maximum, 116. 


and bond, 300. 
bar, 250. 
diagram, 107. 


frame reinforcement, 279. 
Negative and _ positive, 


105. 
Vertical, 104. 
Shearing stress, 122. 
a? pace reinforcement, 


-steel socket, 330. 
Short posts, 180. 
Shot machine, 219. 
Shrinkage bar, 250. 
in mortars, 207. 
Siamese bar, ig 
Side of triangle, 
Silicious me, HE “gr 
Simple beam, 9 
Size of nota yodl 231, 
Slab rods, 250. 
Slag cement, 195. 
cement, Tests of natural 


and, 227. 
Slaked lime, 193. 


XVil 


Smith mixer, 349. 
Snow and wind load, 81. 
load, 56. 
Socket, Beam, 328. 
Bigelow, 328. 
Jennings-Steinmetz, 328. 
Sheet-steel, 330. 
Unit, 328. 
Soft, or mild, steel, 252. 
Soil, Bearing value of a 
foundation, 311, 313. 
Soundness, Accelerated tests 


or, 360. 
Results of tests for, 215. 
Tests for, 210. 

Span, Definition of, 99. 

Spandrel and lintel construc- 
tion, 322. 

wall forms, 341. 

Spans, Bending moment for 
continuous beams over 
equal, 120. 

Reactions for continuous 
beams over equal, 119. 
cian apa herksten Bars of, 


Specific-gravity tests, Appa- 
tatus for, 225. 
gravity, Tests for, 225. 
-gravity tests, Method of 
making, 226 
Sees for aggregates, 


for cement, 227, 367. 
for concrete, 367. 
for floor and test load, 
368. 
for reinforcing steel, 368. 
for Sige 7 omen ait Bar- 
rett, 345. 
Sphere, Surface of, 53. 
Volume of, 
Spiral bar, 256. 
Length of, 54.° 
Splice rod, 250. 
Spread footing for outside 
columns, 315 
footings, 314. 
Square and -round bars, 
My ay and areas of, 


measure, 2. 


INDEX 


Square measure, Surveyor’s, 
-mesh wire reinforcement, 
268 : 


root, 26. 
-twisted bars, 256. 
-twisted lug bar, 257. 
wooden posts, Table of 
constants for, 184. 
Squares and cubes, 28. 
Staff bar, 255 


Standard angles having 


equal legs, Properties 

of, 154. 

angles 
legs, Pro 


c hannels, 


having unequal 
Slee of, 158. 
roperties of, 


pesaieal to cast-iron 
columns, Dimensions of, 
190. : 
I he egg Properties of, 
rolled sections, Values for, 


rolled steel shapes, Prop- 
erties of, 146. 

T rails, Properties and 
peoctpat dimensions of, 
16 


Starting and managing boil- 
ers, Instructions for, 


and operating a mixer, 
356 


Steam test, 214. 
Stearate, Waterproofing 
with meta ic; 3 
Steel angles, having equal 
legs, Properties of, 154. 
angles having unequal 
legs, Properties of, 158. 
bar, Monolith, 262. 
ware es Properties of, 


High carbon, 252. 

I beams, Properties of, 148. 

Laws for, 

Medium, "252. 

Plain bar, 254. 

reinforcement, Character- 
istics of, 252. 


oe 
i , 


INDEX xix 


Steel reinforcement defined, 


of, ; 
reinforcement, Elements 
of, 248. F 
teinforcement of special 
construction, 255. 
reinforcement, Structural 
shapes used for, 263. 
oe uaa Types of, 


' shapes, Properties of 


rolled, 1 


socket, Sheet-, 330. 

Soft, or mild, "252. : 

Specifications for fein- 
forcing, 368. 

T bars, Properties of, 164. 

T rails, Properties and 
pepcipat dimensions of, 


z bars, Properties of, 162. 


Stirrup, 250 
Stone and stone masonry, 


Crushing strength of, 
287 


concrete, 230. 


Storage of briquets, 218. 
Straight column reinforce- 


ment, 302 


Strain, Unit, 122. 
Strain and stresses, 121. 
Strength and impervious- 


ness, Effect of, 234. 
of brick masonry, 288. 
of building material, 124. 
of cement _  briquets, 
Table of tensile, 221. 
of concrete, Compressive, 


of concrete, Crushing, 


of concrete, Safe, 238. 

of concrete, Table of 
transverse, 284. 

of concrete, Table of 
ultimate, 241. 

of metals, Table of ulti- 
mate, ‘ 

of stone and stone ma- 
sonry, Crushing, 278. 


Sltrength of various ma- 
terials, Table of trans- 
verse, 283. 

of wood, Ultimate, 128. 
Tensile, 122. 

Tests for tensile, 215. 
i, Result of tensile-, 


Ultimate, 124. 
Stress, Compressive, 122. 
Shearing, 122. 
Unit, 122. 
Stresses and strains, 121. 
of concrete, Working, 238. 
SAE CoureRs, Terra-cotta, 


Structural members, Design 
of concrete, 282 

shapes used for steel rein- 
forcement, 263. 

—- aneyee, Properties of, 


Structures, Loads in, 56. 
Suddenly applied load, 176. 
ullivan pressed steel plank 
holder, 340. 
Superficial ‘method of water- 
proofing, 344 
Surface, Measure of, 5. 
of circular ring, 53. 
of cone, 53. 
of cylinder, 52. 
of frustum of cone, 53. 
of frustum of cylinder, 53. 
of frustum of rectangular 
pyramid, 54. 
of amir sae Area of, 


of prism, 55. 
of rectangular pyramid, 


of sphere, 53. 
Surveyor’s measure, 1. 
square measure, 2. 
Sylvester process of water- 
proofing, 343. 
System, Metric, 4. 


4 


T bars, Properties of, 164. 
beams, Concrete, 298. 


xx 


fasteners, 327. 

-headed bolt for continu- 
ous insert, 330 

rail, Properties and prin- 
cipal dimensions of 
standard, 166. 


Table of allowable live loads 


on floors in different 
‘cities, 63. 

of areas and weights of 
square and round bars, 
253. 

of avoirdupois weight, 3. 

of bearing value of differ- 
ent foundation soils, 


of bending moments for 
continuous beams over 
equal spans, 120. 

of circles, 10. 

of comparative value of 
different aggregates 
used for concrete, 234. 

of compressive strength of 
concrete made of dif- 
ferent-sized stone, 233. 

of constants for rectangu- 
lar wooden posts, 184. 

of crushing strength of 
concrete, 286. 

of crushing strength of 
brick masonry, 288. 

of crushing strength of 
stone and stone ma- 
sonry, 287. 

of cubic measure, 2. 

of decimal equivalents, 14. 

of decimals of a foot for 
“0 fraction of an inch, 

4, 

of dimensions of standard 
connections to cast-iron 
columns, 190. 

of dry measure, 3. 

of elements of usual sec- 
tions, 130. 

of formulas for deflection 
of beams, 172. 

of formulas for maximum 
shears and bending mo- 
ments of beams, 116. 


INDEX 


_T-headed bolt for concrete | Table of linear measure, 1. 


of liquid measure, 3. 

of live loads, 62. 

of long-ton weight, 3. 

of materials required per 
cubic yard of mortar, 


of measure of angles or 
arcs, 

of measure of capacity, 5. 

of measure of lengths, 5 

of measure of surface, 5. 

of measure of volume, 5 

of. measure of weight, 6. 

of moduli of rupture of 
concrete, 284. 

of moduli of rupture of 
various materials, 283. 

of powers, roots, and 
reciprocals, 31. 

of properties of standard 
channels, 

of properties of corru- 
eae or Johnson, bar, 


of properties of diamond 
bar, i 

of properties of expanded 
metal, 264 

of properties of Kahn cup 
bar, 

of properties of Kahn 
expanded metal, 265. 

of properties of Ransome 
bars, 

of properties of square- 
twisted lug bar, 258. 

of properties of standard 
angles having equal 
legs, 154. 

of properties of standard 
angles having unequal 
legs, 158. 

of properties of standard 
T rails, 

of properties of T bars, 164. 

of properties of Thacher 
bar, 259. 

of properties of universal 
bar, 

of properties of Z bars, 
162. 


INDEX xxi | 


Table of quantities for con- 
crete, 244 

of reactions for continuous 

beams over equal spans, 


19. 

of reduction of live loads 
from floor to floor, 86. 

of requirements of high- 
grade cement, 228. 

of safe loads on cast-iron 
columns, 188. 

of square measure, 2. 

of standard I beams, 148. 

of surveyor’s measure, 1. 

of surveyor’s square meas- 
ure, 2. 

of tensile strength of ce- 
ment briquets, 221, 

of tensile strength of ce- 
ment mortar, 204. 

of troy weight, 3. 

of ultimate strength of 
concrete, 241. 

of aoe strength of 


metals, 
of unit a a values of 
concrete allowed by 


various cities, 240. 

of velocity and force of 
wind, 82. 

of ultimate strength of 
wood, F 

of values for standard 
rolled sections, 136. 

of weight of Australian 
wood, 79. 

of weight of building 
material, 59. 

of weight of hydraulic 
cement, 196. 

of ‘weight of merchandise 
in bulk, 64. 

of weight of Philippine 
wo 

of weight of roof trusses, 


of weights of farm prod- 
ucts, 71. 

of weights of various 
metals, 67. 

of weights of oo 
ous materi 


Table of weights of wood, 74. 
Tables, Conversion, 6. 
Mathematical, 10. 
Temperature changes in con- 
crete, Effect of, 238. 
Laying: mortar at tlow, 206. 
Tensile strength, 
strength of Coane bri- 
quets, Table of, 221. 
strength of cement mortar, 
Table of, 204. 
sah test, Results of, 


strength, ae for, 215. 
Tension bar, 250 
Terra-cotta string - courses, 


327. 

Test, Accelerated, 213, 360. 
Apparatus for fineness, 
Apparatus for specific- 

gravity, 225. 
Boiling, 213. 


for fineness, 223. 

for soundness, 210. 

se ena, Results of, 

for specific gravity, 225. 

for tensile strength, 215. 

loads, Floor and, 368. 

Method of making specific- 
gravity, 226. 

Method of making the 
fineness, 224. 

Normal, 211. 

of natural and slag ce- 
ment, 227 

on cement, 208. 

Primary, 210. 

Per ae. and classification 


Result of tensile- strength, 
Results of time-of-setting, 
223. 


Sand for mortar, 217. 

Secondary, 210, 221. 

Steam, 214. 

Time-of-setting, 221. 
Testing machine, 219. 

Seay one Rate of loading, 


“Xxii INDEX 


Thacher bar, 255, 259. 
ig aia of properties of, 


Thermal changes in concrete, 
Effect of, 238 
Tie-bar, 250. 
Column, 250. 
2 a wire fabric, 267. 


1 forms with wire, . 


38. 
Time-of-setting test, 221. 
MsetOing test, Results of, 


3: 

Ton weight, Table of long-, 3. 
Transverse strength of con- 
crete, Table of, 284. 

strength of various ma- 
terials, Table of, 283. 
Trapezium, Area of, 51. 
Trapezoid, Area of, 51. 
Triangle, Area of, 51. 
Dimensions of, 50. 
of forces, 89. 
Triangular-mesh wire rein- 
forcement, 268 
Troy weight, 3. 
Trussed bar, Kahn, 260. 
bars, 250. 
Trus-Con bar, 261. 
Trusses, Weight of roof, 56. 
Trussit reinforcement, 267. 
Twisted bars, Square-, 256. 
lug bar, Square-, 257. 
lug bar, Table of proper- 
ties of square-, 258. 


U bar, 263. 
Ultimate elongation, 124. 
strength, 124. 
strength of concrete, 241. 
se i ha metals, Table 
fe) 
strength of wood, Table 


128. 
Uniform load, 100. 
Unit bar, 255. 
socket, 328. 
strain, 122. 
stress, 122. 
system of reinforcement, 
277. 


Unit working values of con™ 
crete allowed by vari- 
ous cities, 240. 
Universal bar, 258. 
jg ener of properties of, 


Vv 


Velocity and force of wind, 
Table of, 
Vertical shear, 104. 
Vibration on concrete, Effect 
of, 238. 
Vicat needle, 222. 
Visintini reinforcement, 281. 
Voids, Definition of, 230 
Percentage of, 197. 
Volume, Constancy of, 210. 
Measure of, 5. 
of cement, Constancy of, 
229. 
of circular ring, 53. 
of cone, 53 
of cylinder, 52. 
of frustum of cone, 53. 
of frustum of cylinder, 


53. 
of frustum of prism, 55. 
of frustum of rectangular 
pyramid, 54. 
of parallelopiped, 55. 
of prism, 55. 
of prismoid, 53. é 
of z TOCHRRUIAT pyramid, 


of sphere, 53. 
of wedge, 53. 


Ww 


Wall forms, Braces for, 341. 
forms, Spandrel, 341. 
forms with clamp bolts, 


forms with wire ties, 338. 
Walls, Laws for, 361 
Water for concrete, 236. 
Waterproofing, Barrett spe- 
cifications for, 345. 
by adding lime or clay, 


Cements for, 344. 


INDEX 


Waterproofing, 
method o 

Membrane ‘‘iathed of, 345. 
Mixing concrete for, 342. 
of concrete, 342. 
Paraffin for, 344. 
Superficial method of, 344. 
Sylvester process of, 343. 
Waxes for, 344. 
with metallic stearates, 


Integral 


44, 
Waxes for waterproofing, 
344 


Weather, Laying mortar in 
freezing, 206. 
Wedge, Volume of, 53. 
Weight, Avoirdupois, 3: 
Measure of, 6. 
of Australian wood, Table 
of, 79. 
of beams, 170. 
of building material, Table 
of, 59. 
of farm products, Table of, 


of fires cement, 196. 
of merchandise in bulk, 
« Table of, 64. 

of miscellaneous materials, 
Table of, 68. 

of Philippine woods, Table 
of, 79. 

of roof trusses, 56. 

of various metals, Table 
of, 67. 

of wood, Table of, 74. 

Proportioning by, 235. 

Table of long-ton, 3. 


XXiii 


Weight, Troy, 3. 
Weights and areas of square 
and round bars, 
and measures, l. 
Mise! fabric, Electrically, 


Wet concrete, 285. 

Wind and snow load, 81. 
load, 56. 
pressure, Table of normal, 


Table of velocity and 
force of, 82. 

Wire cloth, Clinton, 268. 
fabric, Lock-woven, 267. 
fabric, Tie-locked, 267. 
reinforcement, Square- and 

triangular-mesh, 268. 
nas forms with, 


Nivee and cast-iron column, 


Table ‘of weight of Austra- 
lian, 79. 
Table ‘of weight of Philip- 
pine, 79. 
Table of weights of, 74. 
Ultimate strength of, 128. 
Wooden posts, 179. 
posts, Table of constants 
for rectangular, 184. 
Working of concrete, Mixing 
and, 242. 
stresses of concrete, 238. 


Z 
T bars, Properties of, 162. 


The Concrete Engineer’s 


Handbook 


USEFUL TABLES 


WEIGHTS AND MEASURES 


LINEAR MEASURE 


BS inches (Gin?)s os ots ho PD IO0b i os cok oc ae he caewteee 
SB OCCr oie cnc Oo ee 1 VAIO as had oe werk nee 
Wsth PARAS Se ose ato crecce aiscee ord POO ac avesicid os, slerataelareretor tele 
MO" F008 oS ke si sons vote mL POTIONG::. 5 2 eure dte oe ees 
S~ furlongs: 3.7. cs ss cee ee miles Fk ee hats ee 
in. it. yd. rd. fur. mi. 
36= 3= 1 
198= 165= 55= 1 
7,920= 660= 220= 40=1 
63,360 = 5,280 = 1,760 =320=8= 1 


SURVEYOR’S MEASURE 


(le Gs SO 1k SC: Seen eee gee 
ES OS eg Se PEN Bf . :: SUNS See ERE. A 


4 rods 
100 links Ie Aaa eiehs is Moree, Se CAGED dacs, heri ehe a aie nie ch. 
66 feet 


OO CHAM koe gael we oe MIG, Bor ve oe eclea curee ik 
1 mi. =80 ch. =320 rd. =8,000 li. = 63,360 in. 


2 ; USEFUL TABLES 
SQUARE MEASURE 


144 square inches (sq. in.). =1 square foot..........sq. ft. 
9 square feet...........=1 square yard......... sq. yd. 
304 square yards......... .=1 square rod.......... sq. rd. 
.160 square rods.......... eet BOTS is “sh cscaiwre 4, Re cued ERE 
640 acres... 25..0.....=1 square mile; 2.6... sq. mi. 
sq.mi.A sq. rd. sq. yd. sq. ft. sq. in. 


1=640 = 102,400 = 3,097,600 = 27,878,400 = 4,014,489,600 


SURVEYOR’S SQUARE MEASURE 


625 square links (sq. li.)....=1 square rod.......... sq. rd. 
LG equiare TOGSR. osm os =1 square chain........sq. ch. 
Meee he ata: ee <= lacre. . i eidia bis soe 

640 acres. Nae =] sue. aie Vict sq. mi. 
36 square ‘niles 6 | mi. , 

SUAS) 6 ice 1 tOWNERID: £25 cess. a eed Tp. 


1 sq. mi. =640 A.=6,400 sq. ch. =102,400 sq. rd. 
= 64,000,000 sq. li. 


The acre contains 4,840 sq. yd., or 43,560 sq. ft., and is 
equal to the area of a square measuring 208.71 ft. on a side. 


CUBIC MEASURE . 
1,728 cubic inches (cu. in.) =1 cubic foot...........cu. ft. 


27° cubic feet.......5.:. =1 cubic yard..... 2%. cu. yd. 
Pao voubic feet... 6... 5's Sons we NCOP sos ae eae seit sik so OCS 
24? cubic feet........... asl PORCt en pace es eared se es 


1 cu. yd.=27 cu. ft. = 46,656 cu. in. 


MEASURE OF ANGLES OR ARCS 


GO Seconds (") 2. etal oid PE ie iss ke ee le ee 
60 minutes...............=1 degree. . > AR A 4 
90 degrees. 2). on... ed et ele or ‘ igdraanie” oO 
360 degrees.... Hivve ee = icirele,« Es ae ee 


1 cir. = 360° = 21,600’=1, aes 5,000” 


USEFUL TABLES 3 
AVOIRDUPOIS WEIGHT 


437.5 grains EAE ....-=1 ounce. inet ne ary IOs 
16 ounces. . Venger’ Phin tonne: oS 24,61: 
100 pounds.. =] hundredweight Gives ett cwt. 
20 cwt., or 2.000 Ib.. at ton. seks 


1 T.=20 cwt. = 2,000 tice —32,000 oz. = ia 000, 000 ¢ gr. 
The avoirdupois pound contains 7,000 gr. 


LONG-TON TABLE 


16 ounces. . Rete -=1 pound.. seks Se 
112 ae : =] himdesiweisht:. . -cwt. 
20 cwt., or 2,240 Ib.. Dts Sh wes eon Sok Rukh ae 
TROY WEIGHT 
Se oradins (O°) 26.0 voce se sts send hae scattered tstete gee pwt. 
20 pennyweights........... =} ounce. . cates phate detous enn Oe 
12 ounces..... .=1 pound.. A Pees | ° 


1 Ib. eaty OZ. 7. =240 pwt. =5, 760 Pe 
DRY MEASURE 


Ptnts CDG). scnate occ eee ee QUAL b sete ce a piesa aes 
PA CIUAT RG eee ahs Sines tise cies of peck:... <.(4:.. “s pk 
4 DOCKS. 5335.50. «'s see) bushel ie yeas weaeces bu. 


i ha. pe as =32 qt. =64 pt. 

The U.S. struck bushel contains 2,150.42 cu. in. =1.2444 
cu. ft. By law, its dimensions are those of a cylinder 18} in. 
in diameter and 8 in. deep. The heaped bushel is equal to 
1} struck bushels, the cone being 6 in. high. The dry gallon 
contains 268.8 cu. in., being 4 struck bushel. 

For approximations, the bushel may be taken at 1} cu. ft.; 
or 1 cu. ft. may be considered # bushel. 

The British bushel contains 2,218.19 cu. in. = 1.2837 cu. ft. 
= 1.032 U. S. bushels. 


LIQUID MEASURE 


Be GINS (SP) ois osc. 0'0,.5 5.5100. A DUR ora otis A ineio sees, bese 
PVPS sas esaad bs ae on to CGURTEs Ss wo ores casea we oatetlts 
Mr erCHUAT EG a. eons, Nec aie a4 oa LION a org tented ena oe oe 
314 gallons. . ave .=1 barrel.. ditawe Sa ceeee 
2 barrels, or © 63 mations: =] ideation © . -hhd. 


1 hhd.=2 bbl. =63 gal. = 252 qt. =504 pt. =5, 016 gi. 


4 USEFUL TABLES 


The U. S. gallon contains 231 cu. in. =.134 cu. ft., nearly; 
or 1 cu. ft. contains 7.481 gal. The following cylinders con- 
tain the given measures very closely: 


Diam. Height Diam. Height 
Gill... ... .12 in. 3 in. Gallon.... 7 in. 6 in. 
Pint. .....34 in. 3 in. 8 gallons 14 in. 12 in. 
ual. cc.s)s 34 in. 6 in. 10 gallons 14 in. 15 in. 


When water is at its maximum density, 1 cu. ft. weighs 
62.425 lb. and 1 gal. weighs 8.345 lb. 

For approximations, 1 cu. ft. of water is considered equal 
to 74 gal., and 1 gal. as weighing 84 lb. 

The British imperial gallon, both liquid and dry, con- 
tains 277.274 cu. in. =.16046 cu. ft., and is equivalent to the 
volume of 10 lb. of pure water at 62° F. To reduce British to 
U.S. liquid gallons, multiply by 1.2. Conversely, to convert 
U. S. into British liquid gallons, divide by 1.2; or, increase 
the number of gallons one-fifth. 


THE METRIC SYSTEM 


The metric system is based on the meter, which, according 
to the United States Coast and Geodetic Survey Report of 
1884, is equal to 39.370482 in. The value commonly used 
is 39.37 in., and is authorized by the United States govern- 
ment. The meter was originally intended to be one ten- 
millionth of the distance from the pole to the equator, 
measured on a meridian passing near Paris. This distance 
was carefully calculated, and a standard meter bar made 
and deposited among the archives of France, at Paris. It 
has since been discovered that the original calculations were 
at fault and the standard meter is somewhat short of one 
ten-millionth of the earth’s quadrant. Nevertheless, the 
error is so small that it was not considered necessary to 
change the standard to make it correct, and the original 
meter length is still preserved. 

There are three principal units—the meter, the liter (pro- 
nounced ‘‘lee-ter’’), and the gram, the units of length, . 
capacity, and weight, respectively. Multiples of these units 


USEFUL TABLES 5 


are obtained by prefixing to the names of the principal units 
the Greek words deca (10), hecto (100), and kilo (1,000); the 
submultiples, or divisions, are obtained by prefixing the 
Latin words deci (zs), centi (xs), and milli (xgo0). These 
prefixes form the key to the entire system. The abbrevi- 


ations of the principal units of these submultiples begin with . 


a small letter, and those of the multiples begin with a capital 
letter. 


MEASURES OF LENGTH 


10 millimeters (mm.).......=1 centimeter............ cm. 
10 centimeters........4.... = ] decimeter.............dm. 
10 Gecimetets:.. i005 is 6c.cc ed meter: 5. sce sare HEE Ae 


10‘ meters ii. 6... Ssed ca os OE detameter. 65022... DRG 
10 decameters.............=1 hectometer...........Hm. 
10 hectometers............ =1 kilometer... ......... Km. 


MEASURES OF SURFACE (NOT LAND) 
100 square millimeters 


(sq. mm.)............=1 square centimeter. ..sq. cm. 


100 square centimeters. ....=1 square decimeter...sq. dm. 
100 square decimeters......=1 square meter........sq. m. 


MEASURES OF VOLUME 
1,000 cubic millimeters 


(cu. mm.)...........=1 cubic centimeter... .cu. cm. 
1,000 cubic centimeters..... =1 cubic decimeter ....cu. dm. 
1,000 cubic decimeters...... =] cubic meter......... cu. m. 


MEASURES OF CAPACITY 


10 miliititers: (ml.).. 2... cenfiliter:. oo 5605. oes sels 
EO Centahters.:5 . 5 20c.s occ osic SE 2 OCHMEOR. cae ec sce 26 pGke 
MA ECTILOTS Fo vcarine: cic vie 31 wv 0 EL AIROP. Lainie wercle sue soa Oky 
POPHCOES otc tec ast satis arose e DL ORRNGED, cao cacn aenk secs 
10 decaliters: ...4.2255.6.... <= 1 Heotoliter. osc cxreess 2 Hl. 
10 peatediters. Se ee eae =1 kiloliter.. rete Fans <6 


The liter is equal to the volume ae oa 1 cu. ben 


- 


6 USEFUL TABLES 
MEASURES OF WEIGHT 


10 milligrams (mg.)........=1 centigram............. -Cg. 
IO\centigrams. 65) .ss6066 om. =) decigram:: 2.3. cantdhienewaes 
MO Gecivrarmms: =.\fr, bes deed gram...) 2s sh ee 
40 GF808 Ail athicen wii = Lidetagram:.. is 4.2 as 
ID decagrams. ose via. 5.08 =] hectogram............Hg. 
10: hectograms............. =1 kilogram........ Kg. or kilo 
1,000 kilograms............ Cay We ic) + Ee Be pay aes Pes bal 


The gram is the weight of 1 cu. cm. of pure 3 distilled ‘ents 
at a temperature of 39.2° F.; the kilogram is the weight of 
1 liter of water; the ton is the weight of 1 cu. m. of water. 

CONVERSION TABLES 

By means of the tables on pages 8 and 9, metric measures 
can be converted into English, and vice versa, by simple addi- 
tion. All the figures of the values given are not required, 
four or five digits being all that are commonly used; it is 
only in very exact calculations that all the digits 
are necessary. Using table, proceed as follows: 1,828.8 
Change 6,471.8 ft. into meters. Any number, as 121.92 
6,471.8, may be regarded as .6,000+400+70+1 21.336 
+.8; also, 6,000=1,0006; 400=100X4; etc. .3048 
Hence, looking in the left-hand column of the .2438 
upper part of the table, page 8, for figure 6 (the 1,972.6046 
first figure of the given number), we find opposite 
it in the third column, which is headed Feet to Meters, the 
number 1.8287838. Now, using but five digits and increas- 
ing the fifth digit by 1 (since the next is greater than 5), we 
get 1.8288. In other words, 6 ft. = 1.8288 m.; hence, 6,000 ft. 
= 1,000 X 1.8288 = 1,828.8, simply moving the decimal point 
three places to the right. Likewise, 400 ft.=121.92 m.; 
70 ft. =21.336 m.; 1 ft.=.3048 m.; and .8 ft.=.2438 m. 
Adding as shown above, we get 1,972.6046 m. 

Again, convert 19.635 kilos into pounds. The 22.046 
work should be perfectly clear from the explana- 19.8416 


tion given above. ‘The result is 43.2875 lb. 1.3228 
The only difficulty in applying these tables lies .0661 
in locating the decimal point. It may always be .0110 


found thus: If the figure considered lies to the 43.2875 
left of the decimal point, count each figure in 


———— 


USEFUL TABLES P 7 


order, beginning with units (but calling unit’s place zero) 
until the desired figure is reached; then move the decimal 
point to the right as many places as the figure being considered 
is to the left of the unit figure. Thus, in the first case above, 
6 lies three places to the left of 1, which is in unit’s place; 
hence, the decimal point is moved three places to the right. 
By exchanging the words ‘‘right’”’ and ‘“‘left,’’ the statement 
will also apply to decimals. Thus, in the second case above, 
the 5 lies three places to the right of unit’s place; hence, the 
decimal point in the number taken from the table is moved 
three places to the left. 


USEFUL TABLES 


CONVERSION TABLE 


ENGLIisH MEAsuRES INTO METRIC 


Metric Metric Metric Metric 
Eng- 
ia Inches to Feet to Pounds to | Gallons to 
Meters Meters los Liters 
1 .0253998 .38047973 .4535925 3.7853122 
2 .0507996 .6095946 -9071850 7.5706244 
3 .0761993 .9143919 1.3607775 | 11.3559366 
4 .1015991 1.2191892 1.8143700 | 15.1412488 
5 .1269989 1.5239865 | 2.2679625 | 18.9265610 
6 .1523987 1.8287838 | 2.7215550 | 22.7118732 
7 .1777984 2.1335811 3.1751475 | 26.4971854 
8 .2031982 2.4383784 | 3.6287400 | 30.2824976 
9 .2285980 2.7431757 | 4.0823325 | 34.0678098 
10 .2539978 3.0479730 | 4.5359250 | 37.8531220 
Metric Metric Metric Metric 
Eng- Square Square Cubic Pounds per 
lish Inches eet Feet Square Inch 
to to to to Kilo per 
Square Square Cubic Square 
Meters Meters Meters eter 
1 .000645150 | .092901394 | .028316094 108 aes 
2 .001290300 | .185802788 | .056632188 | 1,406.16482 
3 .001935450 | .278704182 | .084948282 | 2,109.24723 
4 .002580600 | .371605576 | .113264376 | 2,812.32964 
5 .003225750 | .464506970 | .141580470 | 3,515.41205 
6 003870900 | .557408364 | .169896564 | 4,218.49446 
7 004516050 |. .650309758 | .198212658 | 4,921.57687 
8 005161200 | .748211152 | .226528752 | 5,624.65928 
9 .005806350 | .836112546 | .254844846 | 6,327.74169 
10 .006451500 | .929013940 | .283160940 | 7,030.82410 


USEFUL TABLES 


CONVERSION TABLE 
Metric MEasurRES INTO ENGLISH 


English English English English 
Metric 
Meters to Meters to Kilos to Liters to 
Inches Feet Pounds Gallons 
1 39.370432 | 3.2808693 | 2.2046223 .2641790 
2 78.740864 | 6.5617386 | 4.4092447 .5283580 
3 118.111296 | 9.8426079 | 6.6138670 -7925371 
4 157.481728 | 13.1234772 8.8184894 | 1.0567161 
5 196.852160 | 16.4043465 | 11.0231117 | 1.3208951 
6 236.222592 | 19.6852158 | 13.2277340 | 1.5850741 
7 275.593024 | 22.9660851 | 15.4323564 | 1.8492531 
8 314.963456 | 26.2469544 | 17.6369787 | 2.11343822 
9 354.333888 | 29.5278237 | 19.8416011 | 2.3776112 
10 393.704320 | 32.8086930 | 22.0462234 | 2.6417902 
English English English English 
4 Metric Square Square Cubic — eed 
Meters Meters Meters M 2 pas 
eter to 
to to to Pounds per 
Square uare Cubic Squ ri 
Inches eet Feet tack 
1 1,550.03092; 10.7641034, 35.3156163 .001422310 
2 3,100.06184 21 ‘5282068 70.6312326 .002844620 
3 4,650.09276 32.2923102 105.9468489| .004266930 
4 6,200.12368 43.0564136) 141.2624652 .005689240 
5 7,750.15460| 53.8205170| 176.5780815) .007111550 
6 9,300.18552) 64.5846204' 211.8936978 .008533860. 
7 |10,850.21644| 75.3487238 247.2093141)| .009956170 
* 8 |12,400.24736|) 86.1128272 282.5249304| .011378480 
9 |13, '950.27828| 96. 8769306 317.8405467 .012800790 
10 15, 500.30920 107.6410340 353.1561630 .014223100 


10 USEFUL TABLES 


MATHEMATICAL TABLES 
CIRCUMFERENCES AND AREAS OF CIRCLES FROM 


1-64 TO 100 
Diam. | Circum. area ‘Diam. | Circum. Area 
oy 0491 .0002 4 12.5664 | 12.5664 
a 0982 .0008 4 12.9591 | 13.3641 
is 1963 .0031 4 13.3518 | 14.1863 
i 3927 .0123 43 13.7445 | 15.0330 
.0276 4 14.1372 | 15.9043 
+ 7854 0491 4 14.5299 | 16.8002 
Ye 0767 4 14.9226 | 17.7206 
¥ 1.1781 1104 4 15.3153 | 18.6555 
1.3744 1503 5 15.7080 | 19.6350 
q 1.5708 1963 5 16.1007 
1.7671 2485 5 16.4934 | 21.6476 
a 1.9635 3068 5# 16.8861 | 22.6907 
4h 2.1598 3712 5 17.2788 | 23.7583 


12.1737 | 11.7933 9 28.2744 | 63.6174 


USEFUL TABLES 


TABLE—(Continued) 


il 


Diam Circum Area Diam. | Circum. | Area 
9 28.6671 | 65.3968 19 61.2612 | 298. 
9 29.0598 | 67.2008 19 62.0466 | 306.355 
9% 29.4525 | 69.0293 20 62.8320 | 314.160 
9 29.8452 | 70.8823 204 63.6174 | 322.063 
9 30.2379 | 72.7599 204 64.4028 | 330.064 
9 .6306 | 74.6621 207 65.1882 | 338.164 
9 31.0233 76.589 21 65.9736 | 346.361 
10 31.4160 78.540 214 66.7590 | 354.657 
104 32.2014 82.516 214 67.5444 | 363.051 
HH 32.9868 86.590 212 .3298 | 371.543 
10 33.7722 90.763 22 69.1152 | 380.134 
11 34.5576 95.033 224 69.9006 | 388.822 
114 35.3430 99.402 224 0.6860 | 397.609 
11 36.1284 | 103.869 222 71.4714 | 406.494 
11 36.9138 | 108.43 23 72.2568 | 415.477 
12 37.6992 | 113.098 23+ 73.0422 | 424.558 
124 38.4846 | 117.859 234 73.8276 | 433.737 
12 39.2700 | 122.719 232 74.6130 | 443.015 
12 40.0554 | 127.677 24 75.3984 | 452.390 
13 8 132.733 244 6.1838 | 461.864 
13} 41.6262 | 137.887 244 76.9692 | 471.436 
13 42.4116 | 143.139 243 77.7546 | 481.107 
13 43.1970 | 148.490 25 8.5400 | 490.875 
14 .9824 | 153.938 25 79.3254 .742 
14} 44.7678 | 159.485 25 80.1108 | 510.706 
14 45.5532 | 165.130 25 80.8962 | 520.769 
14 46.3386 | 170.874 26 81.6816 0.930 
15 47.1240 | 176.715 26 82.4670 | 541.190 
15} 47.9094 | 182.655 26 83.2524 | 551.547 
15 6 188.692 26 84.0378 62.003 
15 49.4802 | 194.828 27 84.8232 | 572.557 
16 50.2656 | 201.062 2734 85.6086 | 583.209 
16} 51.0510 | 207.395 274 86.3940 | 593.959 
16 51.8364 | 213.825 272 87.1794 4.80 
16 52.6218 | 220.354 28 87.9648 | 615.754 
17 53.4072 | 226.981 28} 88.7502 | 626.798 
17} 54.1926 | 233.706 284 89.5356 | 637.941 
17 54.9780 | 240.529 287 90.3210 | 649.182 
17 55.7634 | 247.450 29 91.1064 | 660.521 
18 56.5 254.470 293 91.8918 | 671.959 
18} 57.3342 | 261.587 2 92.6772 4 
184 .1196 | 268.803 29 93.4626 | 695.128 
182 58.9050 | 276.117 30 94.2480 | 706.860 
19 59.6904 | 283.529 304 95.0334 | 718.690 
194 60.4758 | 291.040 304 95.8188 | 730.618 


12 USEFUL TABLES 
TABLE—(Continued) 

Diam. | Circum Area Diam. | Circum. Area 
303 96.6042 742.645 | 42 131.947 | 1,385.450 
31 97.3896 754.769 | 424 132.733 | 1,401.990 
314 98.1750 766.992 | 424 133.518 | 1,418.630 
314 98.9604 779.313 | 422 134.303 | 1,435.370 
312 99.7458 791.732 | 43 135.089 | 1,452.200 
32 100.5312 804.250 | 434 135.874 | 1,469.140 
32} 101.3166 16.865 | 434 6.660 | 1,486.170 
32 102.1020 829.579 | 432 137.445 | 1,503.300 
32 102.8874 842.391 44 .230 | 1,520.530 
33 103.673 855.301 44} 139.016 | 1,537.860 
334 104.458 868.3 444 139.801 | 1,555.29 
334 105.244 881.415 | 44? 140.587 | 1,572.81 
33% 106.029 894.620 | 45 141.372 | 1,590.43 
34 106.814 907.922 | 454 142.157 | 1,608.16 

-84} 107.600 921.323 | 454 142.943 | 1,625.97 
34 108.385 934.822 | 452 143.728 | 1,643.89 
34 109.171 420 | 46 144.514 | 1,661.91 
35 109.956 962.115 | 464 145.299 | 1,680.02 
354 110.741 975.909 | 464 146.084 | 1,698.23 
354 111.527 89.800 | 462 146.870 | 1,716.54 
352 112.312 | 1,003.790 | 47 147.655 | 1,734.95 
36 113.098 | 1,017.878 | 474 441 | 1,753.45 
36+ 113.883 | 1,032.065 | 474 149.226 | 1,772.06 
364 114.668 | 1,046.349 | 472 150.011 | 1,790.76 
362 115.454 | 1,060.732 | 48 150.797 | 1,809.56 
37 116.239 | 1,075.213 | 484 151.582 | 1,828.46 
374 117.025 | 1,089.792 | 484 152.368 | 1,847.46 
374 117.810 | 1,104.469 | 48? 153.153 | 1,866.55 
372 118.595 1,119.244 49 53. 1,885.75 
38 119.381 1,134.118 491 154.724 | 1,905.04 
38} 120.166 | 1,149.089 | 494 155.509 | 1,924.43 
384 120.952 | 1,164.159 | 493 156.295 | 1,943.91 
38} 121.737 | 1,179.327 | 50 157.080 | 1,963.50 
39 122.522 | 1,194.593 | 504 158.651 | 2,002.97 
394 123.308 | 1,209.958 | 51 160.222 | 2,042.83 
| 124.093 | 1,225.420 | 514 161.792 | 2,083.08 
39 124.879 | 1,240.981 52 3 2,123.72 
40 125.664 { 1,256.640 | 524 164.934 | 2,164.76 
40+ 126.449 12725 53 166.505 | 2,206.19 
404 127.235 | 1,288.250 | 534 168.076 | 2,248.01 
403 128.020 | 1,304.210 |] 54 169.646 | 2,290.23 
41 128.806 | 1,320.260 | 544 171.217 | 2,332.83 
41} 129.591 1,336.410 55 172.788 | 2,375.83 
41 130.376 | 1,352.6 554 174.359 | 2,419.23 
41 131.162 | 1,369.000 | 56 175.930 | 2,463.01 


USEFUL TABLES 13 
TABLE—(Continued) 

Diam. | Circum -Area Diam. | Circum. Area 
564 177. 2,507.19 784 246.616 | 4,839.83 
57 179.071 | 2,551.76 79 248.186 | 4,901.68 
574 80. 2,596.73 794 249.757 | 4,963.92 
58 182.213 | 2,642.09 80 251.328 | 5,026.56 
584 183.784 | 2,687.84 804 252.899 | 5,089.59 
59 185.354 | 2,733.98 81 254.470 | 5,153.01 
594 186.925 | 2,780.51 814 256.040 | 5,216.82 
60 188.496 | 2,827.44 82 257.611 5,281.03 
604 190.067 | 2,874.76 824 259.182 | 5,345.63 
61 191.638 | 2,922.47 83 260.753 | 5,410.62 
614 193.208 | 2,970.58 834 262.324 | 5,476.01 
62 194.779 | 3,019.08 84 263.894 | 5,541.78 
624 196.350 | 3,067.97 | .844 265.465 | 5,607.95 
63 197.921 | 3,117.25 85 267.036 | 5,674.51 
634 199.492 | 3,166.93 854 268.607 | 5,741.47 
64 201.062 | 3,217.00 86 270.178 | 5,808.82 
644 202.633 | 3,267.46 864 271.748 | 5,876.56 
65 204.204 | 3,318.31 87 273.319 | 5,944.69 
654 205.775 | 3,369.56 874 274.890 | 6,013.22 
66 207.346 | 3,421.20 88 6.46 6,082.14 
664 208.916 | 3,473.24 884 278.032 | 6,151.45 
67 210.487 | 3,525.66 89 279.602 | 6,221.15 
674 212.058 | 3,578.48 894 281.173 | 6,291.25 
68 213.629 | 3,631.69 90 282.744 | 6,361.74 
684 215.200 | 3,685.29 904 284.315 | 6,432.62 
69 216.770 | 3,739.29 91 285.886 | 6,503.90 
694 218.341 | 3,793.68 914 287.456 | 6,575.56 
70 219.912 | 3,848.46 92 289.027 | 6,647.63 
704 221.483 | 3,903.63 924 290.598 | 6,720.08 
71 223.054 | 3,959.20 93 292.169 | 6,792.92 
714 224.624 | 4,015.16 934 293.7 6,866.1 
72 226.195 | 4,071.51 94 295.310 | 6,939.79 
724 227.766 | 4,128.26 944 296.881 7,013.82 
73 229.337 | 4,185.40 95 298.452 | 7,088.24 
73% | 230.908 | 4,242.93 954 300.023 7,163.04 
74 232.478 | 4,300.85 96 301.594 | 7,238.25 
744 234.049 | 4,359.17 964 303.164 7,313.84 
75 235.620 | 4,417.87 97 304.735 | 7,389.83 
754 237.191 | 4,476.98 974 >| 306.306 | 7,466.21 
76 238.762 | 4,536.47 98 307.877 | 7,542.98 
764 240.332 | 4,596.36 984 9. 7,620.15 
as 241.903 | 4,656.64 99 311.018 | 7.697.71 
774 243.474 | 4,717.31 994 312.589 7,775.66 
78 245.045 | 4,778.37 | 100 314.160 | 7,854.00 


14 


USEFUL TABLES 
DECIMAL EQUIVALENTS 


1-64 | .015625 | 17-64 | .265625 | 33-64) .515625 | 49-64! .765625 
1-32 | .031250|] 9-32/| .281250]17-32| .531250] 25-32) .781250 
3-64 | .046875]| 19-64 | .296875 | 35-64} .546875 | 51-64) .796875 
1-16 | .0625 - 312500} 9-16) .562500} 13-16 | .812500 
5-64 | .078125} 21-64 | .328125 | 37-64] .578125 | 53-64} .828125 
3-32 | .093750] 11-32 | .843750 | 19-32 | .593750 127-32 | 843750 
7-64 | .109375| 23-64 | .359375 |] 39-64! .609375 155-64 | .859375 
1-8 |.125000] 8-8 |.375000}] 5-8 | .625000] 7-8 | .875000 
9-64 | .140625] 25-64 | .390625 | 41-64 | .640625 157-64 | .890625 
5-32 | .156250| 13-32] .460250 | 21-32! .656250 |29-32 | .906250 
11-64 |.171875| 27-64| .421875 | 43-64] .671875 |59-64 | .921875 
3-16 | .187500] 7-16} .487500} 11-16) .687500 }15-16 | .937 
13-64 | .203125] 29-64) .453125 | 45-64! .703125 161-64| .953125 
7-32 |.218750]| 15-32 8750 | 23-32) .718750 {31-32 750 
15-64 | .234375]| 31-64 | .484375| 47-64) .7343875 |63-64| .984375 
1 250000} 1-2 3-4 |.7 1 BT 
DECIMALS OF A FOOT FOR EACH 1-32 INCH 
Inch 0” Sh 2” 3” 4” 5* 
0 .0833 1667 | .2500 | .3333 4167 
ay .0026 | .0859 1693 | .2526 | .8359 | .4193 
vs .0052 | .0885 1719 | .2552 | .3385 | .4219 
¥ .0078 | .0911 1745 | .2578 | .8411 | .4245 
.0104 | .0937 1771 .2604 | .3437 | .4271 
fs 0130 | .0964 1797 | .2630 | .3464 | .4297 
is 0156 | .0990 1823 | .2656 |! .3490 4323 
az .0182 | .1016 1849 | .2682 | .3516 | .4349 
4 .0208 | .1042 1875 | .2708 | .3542 | .4375 
= .0234 | .1068 901 2734 | .3568 | .4401 
¥s .0260 | .1094 1927 760 | .3594 | .4427 
4 .0286 | .1120 1953 2786 | .3620 | .4453 
2 .0812 | .1146 | .1979 | .2812 | .3646 | .4479 
AR .0339 1172 | .2005 | .2839 | .3672 | .4505 
6 .0365 1198 | .2031 .2865 | .8698 | .4531 
AB .0391 1224 | .2057 | .2891 .3724 | .4557 
Py .0417 1250 | .2083 | .2917 | .38750 | .4583 
Ay .0443 1276 2109 | .2943 | .3776 | .4609 
Ys .0469 1302 2135 | .2969 | .3802 | .4635 
43 .0495 1328 2161 -.2995 | .8828 | .4661 
3 0521 .1354 2188 | .3021 0804 | .4688 
§4 .0547 | .1380 2214 | .3047 | .8880 | .4714 
so .0573 | .1406 2240 | .8073 | .38906 | .4740 
33 .0599 | .1432 2266 | .8099 | .8932 4766 


USEFUL TABLES 
DECIMALS OF A FOOT FOR EACH 1-32 INCH—(Continued) 


15 


Inch 0” 1” 2" 3” 4” 5” 
0625 | .1458 | .2292 | .3125 | 3958 | .4792 
ii 0651 | .1484 | .2318 | .3151 | .3984 | .4818 
i 0677 | .1510 | .2344 | .3177.| .4010 | .4844 
Fy] 0703 | .1536 | .2370 | .3203 | .4036 | .4870 
% (0729 | .1562 | .2396 | .3229 | .4062 | .4896 
rt) 0755 | .1589 | .2422 | .3255 | .4089 | .4922 
33 0781 | .1615 | .2448 | 3281 | .4115 | .4948 
iu 0807 | .1641 | .2474 | 3307 | .4141 | .4974 
DECIMALS OF A FOOT FOR EACH 1-32 INCH 
Inch 6” | 7” 8” | 9” 10” | 11” 
0 .5000 | .5833 | .6667 | .7500 | .8333 | .9167 
ce 5026 | .5859 | .6693 | .7526 | .8359 | .9193 
=a 5052 | .5885 | .6719 | .7552 | .8385 | .9219 
* 5078 | .5911 | .6745 | .7578 | .8411 | .9245 
4 5104 | .5937 | .6771 | .7604 | .8437 | .9271 
dr 5130 | .5964 | .6797 | .7630 | .8464 | .9297 
Ys 5156 | .5990 | .6823 | .7656 | .8490 | .9323 
a 5182 | .6016 | .6849 | .7682 | .8516 | .9349 
4 5208 | .6042 | .6875 | .7708 | .8542 | 9375 
& 5234 | .6068 | .6901 | .7734 | .8568 | .9401 
Ys 5260 | .6094 | .6927 | .7760 | .8594 | .9427 
Ht 5286 | .6120 | .6953 | .7786 | .8620 | .9453 
3 5312 | .6146 | .6979 | .7812 | .8646 | .9479 
# 5839 | .6172 | .7005 | .7839 | .8672 | .9505 
a 5365 | .6198 | .7031 | .7865 | .8698 | .9531 
ii 5391 | .6224 | .7057 | .7891 | .8724 | .9557 
$ .5417 | .6250 | .7083 | .7917 | .8750 | .9583 
a 5443 | .6276 | .7109 | .7943 | .8776 | .9609 
¥s 5469 | .6302 | .7135 | .7669 | .8802 | .9635 
bey 5495 | .6328 | .7161 | .7995 | .8828 | .9661 
5 5521 | .6354 | .7188 | .8021 | .8854 | .9688 
# 5547 | .6380 | .7214 | :8047 | .8880 | 9714 
rm 5573 | .6406 | .7240 | .8073 | .8906 | .9740 
34 5599 | .6432 | .7266 | .8099 | .8932 | .9766 
3 5625 | .6458 | .7292 | 8125 | .8958 | .9792 
rr} 5651 | .6484 | .7318 | .8151 | .8984 | .9818 
i 5677 | .6510 | .7344 | .8177 | .9010 | 9844 
.5703 | .6536 | .7370 | .8203 | .9036 | .9870 
% 5729 | .6562 | .7396 | .8229 | .9062 | .9896 
ra) 5755 | .6589 | .7422 | 18255 | .9089 | .9922 
8 ‘5781 | .6615 | .7448 | .8281 | .9115 | :9948 
ae 5807 | .6641 | .7474 | -8307 | .9141 } .9974 


16 MATHEMATICS 


MATHEMATICS 


FORMULAS 
={+[- :(Vx /+):-]}= 

The term formula, as used in mathematics and in techni- 
cal books, may be defined as a rule in which symbols are used 
instead of words; in fact, a formula may be regarded as a 
shorthand method of expressing a rule. 

Most persons having no knowledge of algebra regard for- 
mulas with distrust; they think that a person must be a good 
algebraic scholar in order to be able to use formulas. This 
idea, however, is erroneous. As a rule, no knowledge of 
any branch of mathematics except arithmetic is required to 
enable one to use a formula. Any formula can be expressed 
in words, and when so expressed it becomes a rule. 

Formulas are much more convenient than rules. They 
show at a glance all the operations that are to be performed; 
they do not have to be read three or four times, as is the case 
with most rules, to enable one to understand their meaning; 
they take up less space, both in the printed book and in 
one’s notebook, than rules; in short, whenever a rule can 
be expressed as a formula, the formula is to be preferred. 
It is the intention in the following pages to show ‘how to 
use such formulas as are likely to be encountered in ‘‘hand- 
books,”’ or other works of like nature. 

The signs used in formulas are the ordinary signs indica- 
tive of operations and the signs of aggregation. All these 
signs are used in arithmetic, but, to refresh the reader’s 
memory, their nature and uses will be explained before . 
proceeding further. 

The signs indicative of operations are six in number, viz.: 
+,—, X, +, |, and Vv. 

The sign (+) indicates addition, and is called plus; when 
placed between two quantities, it indicates that the two 
quantities are to be added. Thus, in the expression 25+17, 
the sign (+) shows that 17 is to be added to 25. 


MATHEMATICS 17 


The sign (—) indicates subtraction, and is called minus; 
when placed between two quantities, it indicates that the 
quantity on the right is to be subtracted from that on the 
left. Thus, in the expression 25—17, the sign (—) shows 
that 17 is to be subtracted from 25. 

The sign (X) indicates multiplication, and is read times, or 
multiplied by; when placed between two quantities, it indi- 
cates that the quantity on the left is to be multiplied by that 
on the right. Thus, in the expression 25X17, the sign (xX) 
shows that 25 is to be multiplied by 17. 

The sign (+) indicates division, and is read divided by; 
when placed between two quantities, it indicates that the 
quantity on the left is to be divided by that on the right. 
Thus, in the expression 25+ 17, the sign (+) shows that 25 is 
to be divided by 17. 

Division is also indicated by placing a straight line 
between the two quantities. Thus, 25 | 17, 25/17, and 23 all 
indicate that 25 is to be divided by 17. If both quantities 
are placed on the same horizontal line, the straight line 
indicates that the quantity on the left is to be divided by 
thaton the right. When one quantity is below the other, the 
straight line between indicates that the quantity above the 
line is to be divided by the one below it. 

The sign (WV) indicates that some root of the quantity to 
the right is to be taken; it is called the radical sign. To 
indicate what root is to be taken, a small figure, called the 
index, is placed within the sign, this being always omitted 
when the square root is to be indicated. Thus 25 indicates 


that the square root of 25 is to be taken; ~25 indicates that 
the cube root of 25 is to be taken; etc. 


NOTE.—As the term ‘‘quantity’’ is a very convenient one to use, it 
will be defined. In mathematics the word guantity is applied to any- 


_ thing that is to be subjected to the ordinary operations of addition, 


subtraction, multiplication, etc., when it is desired not to be more ex- 
plicit and not to state exactly what the thingis. Thus, the terms ‘‘two 
or more numbers,’’ or “‘two or more quantities’? may be used. How- 
ate Head word quantity is more general in its meaning than the word 
number. 


The signs of aggregation are four in number, viz.: : 
(), (], and { 3 respectively called the vinculum, the paren- 
thesis, the brackets, and the brace. They are used when it is 


18 MATHEMATICS 


desired to indicate that all the quantities included by them 
are to be subjected to the same operation. ‘Thus, suppose 
that the sum of 5 and 8 is to be multiplied by 7, and that 
the addition is to precede the multiplication. Any one 
of thefoursignsof aggregation may be employed toindicate 


the operation. Thus, 5+8X7, (5+8) X7,[5+8]X7, {5+8} 


X7 The vinculum is placed above the quantities that are 
to be treated as one quantity and subjected to the same 
operations. 

While any one of the four signs may be used as shown 
above, custom has restricted their use somewhat. The vincu- 
lum is rarely used except in connection with the radical sign. 


Thus, instead of writing {/(5+8), {[5+8], or 1 {5+8} 
for the cube root of 5 plus 8, all of which would be correct, 


the vinculum is nearly always used, 45 +8. 

In cases where only one sign of aggregation is needed 
(except, of course, when a root is to be indicated), the 
parenthesis is always used. Hence, (5+8) x7 would be the 
usual way of expressing the product of 5 plus 8 by 7. 

If two signs of aggregation are needed, the brackets and 
parenthesis are used, so as to avoid having a parenthesis 
within a parenthesis, the brackets being placed outside. For 
example, [(20—5)+3]X9 means that the difference between — 
20 and 5 is to be divided by 3, and this result multiplied by 9. 

If three signs of aggregation are required, the brace, 
brackets, and parenthesis are used, the brace being placed 
outside, the brackets next, and the parenthesis inside. For 
example, { [(20—5)+3]x9—21} +8 means that the quo- 
tient obtained by dividing the difference between 20 and 5 
by 3 is to be multiplied by 9, and that 21 is to be subtracted 
from the product thus obtained, and the result divided by 8. 

Should it be necessary to use all four signs of aggrega- 
tion, the brace would be put outside, the brackets next, 
the parenthesis next, and the vinculuminside. For example, 
{\(20—5+3)x9-21]+8}%12. The reason for using the 
brace in this last instance will be explained, as it is not 


. generally understood. 


MATHEMATICS 19 


When several quantities are connected by the various 
signs indicating addition, subtraction, multiplication, and 
division, the operation indicated by the sign of multiplication 
must always be performed first. Thus, 2+3X4 equals 
14, 3. being multiplied by 4 before adding to 2. Similarly, 
10+2X5 equals 1, since 2X 5 equals 10, and 10+ 10 equals 1. 
Hence, in the preceding case, if the brace were omitted, the 
result would be 4; whereas, by inserting the brace, the 
result is 36. 

Following the sign of multiplication comes the sign of 
division in its order of importance. For example, 5—9+3 
equals 2, 9 being divided by 3 before subtracting from 5. The 
signs of addition and subtraction are of equal value; that is, 
if several quantities are connected by plus and minus signs, 
the indicated operations may be performed in the order in 
which the quantities are placed. 

There is one other sign used, which is neither a sign of 
aggregation nor a sign indicative of an operation to be per- 
formed; it is (=), and is called the sign of equality; it means 
that all on one side of it is exactly equal to all on the other 
side. For example, 2=2, 5—3=2, 5X (14—9) =25. 

Having described the signs used in formulas, the formulas 
themselves will now be explained. First, consider the well- 
known rule for finding the safe load that a rectangular 
white-oak post will carry, which may be stated as follows: 

From unity subtract 1 one-hundredth of the dividend 
obtained by dividing the length of the post in inches by the 
least dimension of its cross-section in inches. Multiply the 
remainder by 1,000 times the area of the post section in square 
inches. The result is the safe load the post will carry. 

This rule is rather complicated, and it can be greatly 
simplified by putting it in the form of a formula. 

An examination of the rule will show that three quantities 
(viz., the area of the section, the length, and the least 
dimension of the cross-section) are involved. Hence, the 
rule might be expressed as follows: 

Safe load = 1,000 X area of section in square inches 
x ( << length in inches ) 
100 X least dimension of cross-section in inches 


20 _ MATHEMATICS 


This expression could be shortened by representing each 
quantity by a single letter. Thus, representing the safe load 
by W, the area of the cross-section of the post in square 
inches by A, the length of the post in inches by L, the least 
dimension of its cross-section in inches by D, and substi- 
tuting these letters for the quantities that they represent, 
the preceding expression would reduce to 

Bik, 
W=1,000X AX (i~Taep): 
a much simpler and shorter expression. This last expression 
is called a formula. 

As a further example, consider the rule as explained on 
page 300 for finding the safe resisting moment of a rein- 
forced-concrete beam, which is as follows: 

Multiply the area of the steel reinforcement in square inches 
by the distance from the center of the steel to the top of the beam. 
Multiply the product by 13,760. The result is the safe resisting 
moment of the beam in inch-pounds. 

In this case, let M represent the safe resisting moment 
in inch-pounds; a, the area of the steel, in square inches; 
and d, the distance, in inches, from the center of the rein- 
forcement to the top of the beam. Then, the preceding 
rule may be expressed by the following formula: 

M = 13,760 XdXa 

The formula just given shows, as was stated in the begin- 
ning, that a formula is really a shorthand method of express- 
ing a rule. It is customary, however, to omit the sign of 
multiplication between two or more quantities when they 
are to be multiplied together, or between a number and a 
letter representing a quantity, it being always understood 
that when two letters are adjacent with no sign between 
them, the quantities represented by these letters are to be 
multiplied. Bearing this fact in mind, the formula just 
given can be further simplified to 

M = 13,760da 

The sign of multiplication, evidently, cannot be omitted 
between two or more numbers, as it would then be impossible 
to distinguish the numbers. A near approach to this, how- 
ever, may be attained by placing a dot between the numbers 


MATHEMATICS 21 


that are to be multiplied together, and this is frequently done 
in works on mathematics when it is desired to economize 
space. In such cases it is customary to put the dot higher 
than the position occupied by the decimal point. Thus, 2°3 
means the same as 2X3; 542*749-1,006 indicates that the 
numbers 542, 749, and 1,006 are to be multiplied together. 

It is also customary to omit the sign of multiplication in 
expressions similar to the following: aX Vb+c, 3X (bc), 
(b+c) Xa, etc., writing them avb+c, 3(b+c), (b+c)a, etc. 
The sign is not omitted when several quantities are included 
by a vinculum, and it is desired to indicate that the quantities 
so included are to be multiplied by another quantity. 
For example, 3Xb+c, b+cXa, Vb+cXa, etc., are always 
written as here printed. 

Before proceeding further, it may be well to explain one 
other device used by formula makers, and which is apt to 
puzzle one who encounters it for the first time. It is the 
use of what mathematicians call primes and subs., and what 
printers call superior and inferior characters. As a rule, 
formula makers designate quantities by the initial letters of 
the names of the quantities. For example, they represent 
moment by M, stress by s, length by /, etc. This practice 
is to be commended, as the letter itself serves in many cases 
to identify the quantity that it represents. Some authors 
carry the practice a little further and represent all quantities 
of the same nature by the same letter throughout the book, 
always having the same letter represent the same thing. 
Now, this practice necessitates the use of the primes and 
subs. above mentioned when two quantities have the same 
name, but represent different things. Thus, consider the 
word moment as applied to beams. The safe moment equals 
the ultimate moment divided by the factor of safety. If it 
is decided to designate all moments by M, it will be necessary 
to make a distinction between the M that refers to safe 
moment and the M that refers to ultimate moment. This 
may be effected by designating the safe moment as Ms, and 
the ultimate moment as Mx. The formula may then be 


written M.= , in which F equals the factor of safety. 


22 MATHEMATICS 


The main thing to be remembered is that when a formula is 
given in which the same letters occur several times, all like letters 
having the same primes or subs. represent the same quantities, 
while those which differ in any respect represent different 
quantities. Thus, in the formula 


R= i +Io+I3+I4 

Gb AE. aes 
R represents the radius of gyration of a section composed of 
four parts; J;, Jo, J3, and I4 represent the moment of inertia 
of the respective parts about the same axis around which R is 
taken; and A represents the total area of the section. 

It is very easy to apply the above formula when the 
values of the quantities represented by the different letters 
areknown. All that is required is to substitute the numerical 
values of the letters, and then perform the indicated oper- 
ations. Thus, suppose that the values of Jj, Io, Iz, and I4 
are 4.2, 3.6, 7.5, and 9.8, respectively, and that A is equal 
to 8 sq. in. Then, the value of R may be found by substi- 
tuting in the above formula; thus, 


nf Pate er SSS UE Apes , : 
R=\ 3 = 5 = ¥8.1375 =1.77 in. 


Attention is called to one or two other facts relating to 
formulas. 


160 sometimes occur, the heavy line 


Expressions arenes to = 660 


25 ; 
indicating that 160 is to be divided by the quotient obtained 
by dividing 660 by 25. If both lines were light, it would be 


impossible to tell whether 160 was to be divided by = or 


whether Ba irensbo be divided: Sy BS aks Inkter rahe 
160 


were desired, the expression would be written a In every 
case the heavy line indicates that all above it is to be divided 
by all below it. 


In an expression like ct 


660° 
+35 


the heavy line is not neces- 


MATHEMATICS ° 23 


sary, because it is impossible to mistake the operation that 
Re to bs petioanek. But, ance 74 Seat Os 


25 25 
175-900 is substituted for 7+0°, the heavy line becomes 
necessary in order to make the resulting expression clear. 


Thus, 


160. _—«:160_~—_—i160 
7880 175+660 835 
25 25 25 


Fractional exponents are sometimes used instead of the 
radical sign; that is, instead of indicating the square, cube, - 
fourth root, etc. of some quantity, as 37, by 37, 437, 437, 
etc., those roots are indicated 372, 373, 374, etc. Should 
the numerator of the fractional exponent be some quantity 
other than 1, this quantity, whatever it may be, indicates 
that the quantity affected by the exponent is to be raised to 
the power indicated by the numerator; the denominator is 
always the index of the root. Hence, instead of expressing 
the cube root of the square of 37 as 4372, it may be expressed 
37%, the denominator being the index of the root; in other 
words, 4372 =373. Likewise, Va +a%b)3 may also be writ- 

_ten (1 +a%)3, a much simpler expression. 

Several examples showing how to apply some of the more 
difficult formulas will now be given. 

The area of any segment of a circle that is less than (or 
equal to) a semicircle is expressed by the formula 

wa CG =h) 
. 860 2 . 
in which A =area of segment; r =3.1416; r=radius; E=angle 
obtained by drawing lines from the center to the extremities 
of arc of segment; c=chord of segment; and h=height of 
segment. 

EXAMPLE.—What is the area of a segment whose chord is 
10 in. long, angle subtended by chord is 83.46°, radius is 
7.5 in., and height of segment is 1.91 in.? 

SoLtuTion.—Applying the formula just given, 

2 

A ="rE_¢ a _ 3.1416 TK 88 6 10 7 5 — 1.91) 

= 40.968 — 27.95 = 13.018 sq. in., nearly 


24 MATHEMATICS 


The area of any triangle may be found by means of the 
following formula, in which A=the area, and a, b, and ¢ 
represent the lengths of the sides: 


a2+b2—¢2 
1-bNo-(=5=) 
ExAaMPLE.—What is the area of a triangle whose sides are 
21 ft., 46 ft., and 50 ft. long? 
SoLtuTion.—In order to apply the formula, assume that 
a represents the side that is 21 ft. long; b, the side that is 50 ft. 
long; and c, the side that is 46 ft. long. Then substituting 
in the formula, 


-) fa PBR 2 _ 504 —( P+502—462\2 
Ams Ve ( )- ay 2x50 4 


_50 441 +2,500—2,116 \2 825 
2 Vsti - ( 100 ) =25 441 (3) 


= 25 V441 — 8.25? = 25 V441 — 68.0625 = 25 V372.9375 
= 25 X 19.312 = 482.8 sq. ft., nearly 
The above operations have been extended much further 
than was necessary in order to show every step of the process. 
The Rankine-Gordon formula for determining the least 
load in pounds that will cause a long column to break is 


in which P=load (pressure)in pounds; S= ultimate strength, 
in pounds per square inch, of material composing column; 
A=area of cross-section of column, in square inches; gq=a 
factor (multiplier) whose value depends on the shape of the 
ends of the column and on the material composing the column; 
1=length of the column, in inches; G=least radius of gyration 
of cross-section of column. 

The meaning of the term G is explained on page 142. 

ExAMPLeE.—What is the least load that will break a hollow 
steel column whose outside diameter is 14 in., inside diam- 
eter 11 in., length 20 ft., and whose ends are flat? 


SoLuTIonN.—For steel, S=150,000, and g= for flat- 


1 
25,000 


MATHEMATICS 25 


ended steel columns. The area of the cross-section A 
= .7854 (d,2—d>?), d, and dz being the outside and inside diam- 
2 
eters, respectively; 1=20X12=240 in.; and Gute 
Substituting these values in the formula, 
Pus SA  _ 150,000 X .7854(142— 112) 
as 2 - 1 2402 
1+4¢3 1 +95 000 *T2@ +10 
16 

150,000 X 58.905 _ 8,835.750 _ 
-- tue es 


INVOLUTION AND EVOLUTION 


By means of the table on pages 31-48 the square, cube, 
square root, cube root, and reciprocal of any number may be 
obtained correct always to five significant figures, and in the 
majority of cases correct to six significant figures. 

In any number, the figures beginning with the first digit 
(a cipher is not a digit) at the left and ending with the last 
digit at the right, are called the significant figures of the 
number. Thus, the number 405,800 has the four significant 
figures, 4, 0, 5, and 8; and the number .000090067 has the 
five significant figures 9, 0, 0, 6, and 7. 

The part of the number consisting of its significant figures 
is called the significant part of the number. Thus, in the 
number 28,070, the significant part is 2807; in the number 
.00812, the significant part is 812; and in the number 170.3, 
the significant part is 1703. 

In speaking of the significant figures or of the significant 
part of a number, the figures are considered in their proper 
order, from the first digit at the left to the last digit at the 
right, but no attention is paid to the position of the decimal 
point. Hence, all numbers that differ only in the position of 
the decimal point have the same significant part. For example, 
.002103, 21.03, 21,030, and 210,300 have the same significant 
figures 2, 1, 0, and 3, and the same significant part 2103. 

The integral part of a number is the part to the left of the 
decimal point. 


26 MATHEMATICS 


It will be more convenient to explain first how to use the 
table for finding square and cube roots. } 

Square Root.—First point off the given number into periods 
of two figures each, beginning with the decimal point and 
proceeding to the left and right. The following numbers 
are thus pointed off: 12703, 1/’27'03; 12.703, 12.70/30; 
220000, 22/00/00; .000442, .00’04’42. 

Having pointed off the number, move the decimal point 
so that it will fall between the first and second periods of the 
significant part of the number. In the preceding numbers, 
the decimal point will be placed thus: 1.2703, 12.703, 
22., 4.42. 

If the number has only three (or less) significant figures, 
find the significant part of the number in the column headed u; 
the square root will be found in the column headed Vx or ~ 
Vi0n, according to whether the part to the left of the 
decimal point contains one figure or two figures. Thus, 
4.42 =2.1024, and V22= 10 X 2.20=4.6904. The decimal 
point is located in all cases by reference to the original 
number after pointing off into periods. 

There will be as many figures in the root preceding the decimal 
point as there are periods preceding the decimal point in the 
given number; tf the number is entirely decimal, the root is 
entirely decimal, and there will be as many ciphers following the 
decimal point in the root as there are cipher periods following 
the decimal point in the given number, 

Applying this rule, ‘220000=469.04 and .000442 
= .021024. 

The operation when the given number has more than 
three significant figures is best explained by an example. 

EXxampPLe.—(a) ¥3.1416=? (6) V2342.9=? 

SoLuTION.—(a) Since the first period contains only one 
figure, there is no need of moving the decimal point. Look 
in the column headed n?2 and find two consecutive numbers, 
one a little greater and the other a little less than the given 
number; in the present case, 3.1684= 1.782 and 3.1329 = 1.772. 
The first three figures of the root are therefore 177. Find the 
difference between the two numbers between which the 
given number falls, and the difference between the smaller 


MATHEMATICS 27 


number and the given number; divide the second difference 
by the first difference, carrying the quotient to three decimal 
places and increasing the second figure by 1 if the third 
is 5 or a greater digit. The two figures of the quotient 
thus determined will be the fourth and fifth figures of the 
root. In the present example, dropping decimal points, 
in the remainders, 3.1684—3.1329 =355, the first difference; 
3.1416 — 3.1329 =87, the second difference; 87 +355 = .245+, 
or .25. Hence, V3.1416=1.7725. 

(b) 2342.9=? Pointed off into periods, the number 
appears as 23’42.90, and by moving the decimal point, the 
number appears as 23.4290; the first three figures of the root 
are 484; the first difference is 23.5225—23.4256=969; the 
second difference is 23.4290 — 23.4256 = 34; 34+969 = .035+, 
or .04. Hence, V¥2342.9 = 48.404. 

Cube Root.—The cube root of a number is found in the 
same manner as thesquare root, except the given number is 
pointed off into periods of three figures each. The following 
numbers would be pointed off thus: 3141.6, 3/141.6; 
67296428, 67'296/428; 601426.314, 601’426.314; .0000000217, 
.000’000’021’700. 

Having pointed off, move the decimal point so that it will 
fall between the first and second periods of the significant 
part of the number, as in square root. In the above 
numbers the decimal point will be placed thus: 3.1416, 
67.296,428, 601.426314, and 21.7. 

If the given nimber has but three (or less) significant 
figures, find the significant part of the number in the 
column headed u; the cube root will be found in the column 
headed Vn, 410, or W100, according to whether one, 
two, or three figures precede the decimal point after it has 
been moved. Thus, the cube root of 21.7 will be found oppo- 
site 2.17, in column headed Vi0n, while the cube root of 2.17 
will be found in the column headed Vn, and the cube root 
of 217 in the column headed +100n, all on the same line. 
If the given number contains more than three significant 
figures, proceed exactly as described for square root, but use 
the column headed n3. 

ExampLe.—(a) ¥,0000062417=? (b) 4%50932676=? 


28 MATHEMATICS 


SoLtuTion.—(a) Pointed off into periods, the number 
appears as 000’006’241’700, and by moving the decimal 
point the number appears as 6.2417. The number falls 
between 6.22950 = 1.848 and 6.33163 =1.853; the first differ- 
ence is 10213; the second difference is 6.24170—6.22950 
= 1220; 1220+10213=.119+, or .12, the fourth and fifth 
figures of the root. The decimal point is located by the rule 
previously given; hence, 40000062417 =.018412. 

(b) ~50932676=? As the number contains more than 
six significant figures, reduce it to six significant figures by 
replacing all after the sixth figure with ciphers, increasing 
the sixth figure by 1 when the seventh is 5 or a greater 
digit. In other words, the first five figures of 450932700 and 
of 450932676 are the same. Pointed off into periods, the | 
number appears as 50’932’700, and by moving the decimal 
point, the number appears as 50.9327, which falls between 
50.6530 =3.703 and 51.0648=3.71%; the first difference 
is 4118; the second difference is 2797; 2797+ 4118=.679+, 


or .68. The integral part of the root evidently contains 
three figures; hence ~V50932676=370.68, correct to five 
figures. 


Squares and Cubes.—If the given number contains but 
three (or less) significant figures, the square or cube is found 
in the column headed u? or 3, opposite the given number in 
the column headed u. If the given number contains more 
than three significant figures, proceed in a manner similar to 
that described for extracting roots. To square a number, 
place the decimal point between the first and second signifi- 
cant figures and find in the column headed Wn or Vi0n two 
consecutive numbers, one of which shall be a little greater 
and the other a little less than the given number. The 
remainder of the work is exactly as heretofore described. To 
locate the decimal point, employ the principle that the square 
of any number contains either twice as many figures as the 
number squared or twice as many less one. If the column 
headed V¥10” is used, the square will contain twice as many 
figures, while if the column headed Nn is used, the square will 
contain twice as many figures as the number squared, less 
one. If the number contains an integral part, the principle 


MATHEMATICS 29 


is applied to the integral part only; if the number is wholly 
decimal, there will be twice as many ciphers following the 
decimal in the square or twice as many plus one as in the 
number squared, depending on whether V10n or Vn column 
is used. For example, 273.42? will contain five figures 
in the integral part; 4516.2? will contain eight figures in the 
integral part, all after the fifth being denoted by ciphers; 
.002945322 will have five ciphers following the decimal point; 
.0524362 will have two ciphers following the decimal point. 

EXxAMPLE.—(a) 273.422=? (6) .0524386?2=? 

SoLuTION.—(a) Placing the decimal point between the first 
and second significant figures, the result is 2.7342; this 
number occurs between 2.73313 = V7.47 and 2.73496 = V7.48 
in the column headed Vu. The first difference is 2.73496 
— 2.73313=183; the second difference is 2.73420— 2.73313 
=107; and 107+183=.584+, or .58. Hence, 273.42? 
=74,758, correct to five significant figures. 

(b) Shifting the decimal point to between the first and 
second significant figures, the number 5.2436 is obtained. 
This falls between 5.23450 =: V27.4 and 5.24404= 27.5. The 
first difference is 954; the second difference is 910; 910+954 
=.953+, or .95. Hence, .0524362=.0027495, to five sig- 
nificant figures. 

A number is cubed in exactly the same manner, using the 
column headed ©n, Vi0n, or Vi00n, according to whether 
the first period of the significant part of the number contains 
one, two, or three figures, respe tively. If the number con- 
tains an integral part, the number of figures in the integral 
part of the cube will be three times as many as in the given 
number if column headed ¥100n is used; it will be three 
times as many less 1 if the column headed 410n is used; 
and it will be three times as many less 2 if the column 
headed Wn is used. If the given number is wholly decimal, 
the cube will have either three times, three times plus 
one, or three times plus two, as many ciphers following the 
decimal as there are ciphers following the decimal point in 
the given number. 

ExAMPLE.—(a) 129.6843=? (6) .764423=? (c) .0324253=? 

SoLuTION.—(a) Placing the decimal point between the 


30 MATHEMATICS 


first and second significant figures, the number 1.29684 is 
found between 1.29664= 2.18 and 1.29862= 12.19. The 
first difference is 198; the second difference is 20; and 20+198 
=.101+, or .10. Hence, the first five significant figures are 
21810; the number of figures in the integral part of the cube 
is 3X3—2=7; and 129.6848=2,181,000, correct to five sig- 
nificant figures. 

(b) 7.64420 occurs between 7.64032 = 1446 and 7.64603 
= 447. The first difference is 571; the second difference 
is 888; and 388+571=.679+, or .68. Hence, the first five 
significant figures are 44668; the number of ciphers following 
the decimal point is 3X 0=0; and .764423 = .44866, correct to 
five significant figures. 

(c) 3.2425 falls between 3.24278= 134.1 and 3.23961 
= 34.0. The first difference is 317; the second difference 
is 289; 289+317=.911+, or .91. Hence, the first five sig- 
nificant figures are 34091; the number of ciphers following 
the decimal point is 3X 1+1=4; and .0324253 = .000034091, 
correct to five significant figures. 

Reciprocals.—The reciprocal of a number is 1 divided by 
the number. By using reciprocals, division is changed into 
multiplication, since a+b=t—axe, The table gives the 
reciprocals of all numbers éxpressed by three significant 
figures correct to six significant figures. By proceeding in a 
manner similar to that just described for powers and roots, | 
the reciprocal of any number correct to five significant 
figures may be obtained. The decimal point in the result 
may be located as follows: If the given number has an 
integral part, the number of ciphers following the decimal 
point in the reciprocal will be one less than the number of 
figures in the integral part of the given number; and if the 
given number is entirely decimal, the number of figures in 
the integral part of the reciprocal will be one greater than 
the number of ciphers following the decimal point in the 
given number. For example, the reciprocal of 3370= .00029- 
6736 and of .00348 = 287.356. 

When the number whose reciprocal is desired contains 
more than three significant figures, express the number to 


POWERS, ROOTS, AND RECIPROCALS 31 


3 


r—) 
| 


alkane anlanll alll anll aall aalh aula 


Serre Creo Seeee seas 


j ‘01489 | 3.20936 | 1.00990 | 2.17577 | 4.68755 | 970874 
‘0816 | 1.12486 | 1.01980 | 3.22490 | 1.01316 | 2.18278 | 4.70267 | .961539 
1025 | 1.15763 | 1.02470 | 3.24037 | 1.01640 | 2.18976 | 4.71769 | .952381 


-1236 | 1.19102 | 1.02956 | 3.25576 | 1.01961 | 2.19669 | 4.73262 


392 x a é - 
-4161 | 1.68516 | 1.09087 | 3.44964 | 1.05970 | 2.28305 | 4.91868 | .840336 
-4400 | 1.72800 | 1.09545 | 3.46410 | 1.06266 | 2.28943 | 4.93242 | .833333 


4641 | 1.77156 | 1.10000 | 3.47851 | 1.06560 | 2.29577 | 4.94609 | .826446 
4884 | 1.81585 | 1.10454 | 3.49285 | 1.06853 | 2.30208 | 4.95968 | .819672 
1.5129 | 1.86087 | 1.10905 | 3.50714 | 1.07144 | 2.30835 : 97319 | .813008 


1.5625 | 1.95313 | 1.11803 | 3.53553 ~1.07722 | 2.32080 5.00000 -800000 


1.5876 | 2.00038 | 1.12250 | 3.54965 | 1.08008 | 2.32697 | 5.01330 | .793651 


1.6129 | 2.04838 | 1.12694 | 3.56371 | 1.08293 | 2.33310 | 5.02653 | .787402 

1.6384 | 2.09715 | 1.13137 | 3.57771 | 1.08577 | 2.33921 | 5.03968 | .781250 

1.6641 | 2.14669 | 1.13578 | 3.59166 | 1.08859 | 2.34529 | 5.05277 | .775194 
-6900 6 5.06580 


1.7161 2.24809 1.14455 | 3.61939 | 1.09418 | 2.35735 | 5.07875 | .763359 


1.7956 | 2.40610 | 1.15758 | 3.66060 | 1.10247 | 2.37521 | 5.11723 | .746269 
1.8225 | 2.46038 | 1.16190 | 3.67423 | 1.10521 | 2.38110 | 5.12993 | 740741 


1.8496 | 2.51546 | 1.16619 | 3.68782 | 1.10793 | 2.38696 | 5.14256 | .735294 
1.8769 | 2.57135 | 1.17047 | 3.70135 | 1.11064 | 2.39280 | 5.15514 | .729927 
1.9044 | 2.62807 | 1.17473 | 3.71484 | 1.11334 | 2.39861 | 5.16765 | .724638 
1.9321 | 2.68562 | 1.17898 | 3.72827 | 1.11602 | 2.40439 | 5.18010 | .719425 

-9600 | 2.74400 | 1.18322 | 3.74166 | 1.11869 | 2.41014 | 5.19249 | .714286 


1.9881 | 2.80322 | 1.18743 | 3.75500 | 1.12135 | 2.41587 | 5.20483 | .709220 
2.0164 | 2.86329 | 1.19164 | 3.76829 | 1.12399 | 2.42156 | 5.21710 | .704225 
2.0449 | 2.92421 | 1.19583 | 3.78153 | 1.12662 | 2.42724 | 5.22932 | .699301 
2.0736 | 2.98598 | 1.20000 | 3.79473 | 1.12924 | 2.43288 | 5.24148 | .694444 
2.1025 | 3.04863 | 1.20416 | 3.80789 | 1.13185 | 2.43850 | 5.25359 | .689655 


2.1316 | 3.11214 | 1.20830 | 3.82099 | 1.13445 | 2.44409 | 5.26564 | .684932 


2.1609 | 3.17652 | 1.21244 | 3.83406 | 1.13703 | 2.44966 | 5.277 680272 


2.1904 | 3.24179 | 1.21655 | 3.84708 | 1.13960 | 2.45520 | 5.28957 | .675676 
2.2201 | 3.30795 | 1.22066 | 3.86005 | 1.14216 | 2.46072 | 5.30146 | .671141 


SEE55 GESES SSREAS KESSS SSENS GRGSA 


2.2500 | 3.37500 | 1.22474 | 3.87298 | 1.14471 | 2.46621 | 5.31329 | .666667 


32 POWERS, ROOTS, AND RECIPROCALS 


dn | Vion | *i00n 


3 
3 
to 
3 
on) 
+ 
| 
3 


2.2801 | 3.44295 | 1.22882 | 3.88587 | 1.14725 | 2.47168 | 5 
2.3104 | 3.51181 | 1.23288 | 3.89872 | 1.14978 | 2.47713 | 5.33680 
2.3409 | 3.58158 | 1.23693 | 3.91152 | 1.15230 | 2.48255 | 5.34848 
F . a 1.15480 | 2. 5.36011 
2.4025 | 3.72388 | 1.24499 | 3.93700 | 1.15729 | 2.49332 | 5.37169 


2.4336 | 3.79642 | 1.24900 | 3.94968 | 1.15978 | 2.49866 | 5.38321 


oe agBge 
ee 
3 

s Ned 
2 
nw 
8 
_ 
w 
z 
w 
s 
s 
@ 
w 
& 
Fs 


& 
2 
oo 
: 
7) 
s 
meee 
3 
a 
2 
S 
on 


2.7225 | 4.49213 | 1.28452 | 4.06202 | 1 118167 | 2.54582 5.48481 


2.7556 | 4.57430 | 1.28841 | 4.07431 | 1.18405 | 2.55095 | 5.49586 
4,65746 | 1.29228 | 4.08656 | 1.18642 | 2.55607 | 5. 

2.8224 | 4.74163 | 1.29615 | 4.09878 | 1.18878 | 2.56116 | 5.51785 
2.8561 | 4.82681 | 1.30000 | 4.11096 | 1.19114 | 2.56623 | 5.52877 
2.8900 | 4.91300 | 1.30384 | 4.12311 | 1.19348 | 2.57128 | 5.53966 


5.00021 | 1.30767 | 4.13521 | 1.19582 | 2.57631 | 5.55050 


- « 


& aRSs2 see 
i) 
g 
hs 
mH 
BF 
oO 
=] 

el ad 5 : 
be 
3 
a 
—J 
= 
S 
i) 
2 
g 
or 
& 
z 


<ytes Se toys 
= 
i] 
Je) 


ananm SSB 
ns 
NS 
ay 


3.2400 | 5.83200 | 1.34164 | 4.24264 | 1.21644 | 2.62074 | 5.64622 
3.2761 | 5.92974 | 1.34536 | 4.25441 | 1.21869 | 2.62558 | 5.65665 


3.4969 | 6.53920 | 1.36748 | 4.32435 | 1.23201 | 2.65428 | 5.71848 
“65: 


3.6864 | 7.07789 | 1.38564 | 4.38178 | 1.24289 | 2.67773 | 5.76900 
3.7636 | 7.30138 | 1.39284 | 4.40454 |.1.24719 | 2.68700 8.78806 | « 


er © 


SSSSR SESSE SSSLE BERR Suara 
co 
& 
Ks 
a 
Ke 
5 
ie 
& 
rs 
& 
or 
é 
= 
& 
bo 
s 
or 
3 


4.0000 | 8.00000 | 1.41421 | 4.47214 | 1.25992 | 2.71442 | 5.84804 


a a 


ROOTS, AND RECIPROCALS 


3 


4100 n 


1 
n 


. 


twrbhry 
B32 &FB82 


. 


& RBS SSAA Smawe SB 


. 


nw 
~7 


ar) 


S6G55 SRESE SESeR RERES SBE 


* 


RRR NHN HD HHH HHD HNHWNHNHHD HNHWHMHNH HHH HH HMw ds Hobbs bh fo to bo hb 


12.9779 


13.1443 
13,3121 
13.4813 


3489 
14.5268 
14.7061 


1.52971 
1.53297 


1,53623 


5.85777 


6.24025 
6.2 

6.25732 
6.26583 
6.27431 
6.28276 


6.29119 
6.29961 


497512 


‘476191 
473984 
"471698 
"467290 
465116 


POWERS, ROOTS, AND RECIPROCALS 


3 


n2 n3 Nn | i0On| Wn | Vion! *i00n ze 


tno 
noe 


. 


SSSSR SRRSA SSSAS BES 


SBS See oiO ODD ED DedmDi Ww waa: 
DAD Geese Sats KeSoe Sana anann 


2 | 15.8133 | 1.58430 | 5,00999 | 1.35902 | 2.92791 | 6.30799 | .398406 
3504 | 16.0030 | 1.58745 | 5.01996 | 1.36082 | 2.93179 | 6.31636 | .396825 
4009 | 16.1943 | 1.59060 | 5.02991 | 1.36262 | 2.93567 | 6.32470 | .89525T 
-4516 | 16.3871 | 1.59374 | 5.03984 | 1.36441 | 2.93953 | 6.33303 | .893701 
6.5025 | 16.5814 | 1.59687 | 5.04975 | 1.36620 | 2.94338 | 6.34133 | .392157 


6.5536 | 16.7772 | 1.60000 | 5.05964 | 1.36798 | 2.94723 | 6.34960 | .390625 


6.9696 | 18.3997 | 1.62481 | 5.13809 | 1.38208 | 2.97761 | 6.41507 | .378788 
7.0225 | 18.6096 | 1.62788 | 5.14782 | 1.38383 | 2.98137 | 6.42316 | .377359 


7.0756 | 18.8211 | 1.63095 | 5.15752 | 1.38557 | 2.98511 | 6.43123 | .375940 
7.1289 | 19. 0342 1.63401 | 5.16720 | 1.38730 | 2.98885 | 6.43928 | .374532 


7.5625 | 20.7969 | 1.65831 | 5.24404 | 1.40102 | 3.01841 | 6.50296 | 363636 


7.6176 | 21.0246 | 1.66132 | 5.25357 | 1.40272 | 3.02206 | 6.51083 | .362319 
7.6729 | 21.2539 | 1.66433 | 5.26308 | 1.40441 | 3.02571 | 6.51868 | .361011 
7.7284 | 21.4850 | 1.66733 | 5.27257 | 1.40610 | 3.02934 | 6.52652 | .859712 
7.7841 | 21.7176 | 1.67033 | 5.28205 | 1.40778 | 3.03297 | 6.53434 | 858423 
7.8400 | 21,9520 | 1.67332 | 5.29150 | 1.40946 | 3.03659 | 6.54213 | .357142 


7.8961 | 22.1880 | 1.67631 | 5.30094 | 1.41114 | 3.04020 | 6.54991 | .355872 
7.9524 | 22.4258 | 1.67929 | 5.31037 | 1.41281 | 3.04380 | 6.55767 | .354610 
0089 | 22.6652 | 1.68226 | 5.31977 | 1.41448 | 3.04740 | 6.56541 | .353357 
8.0656 | 22.9063 | 1.68523 | 5.32917 | 1.41614 | 3.05098 | 6.57314 | .352113 
8.1225 | 23.1491 | 1.68819 | 5.33854 | 1.41780 | 3.05456 | 6.58084 | .350877 


8.1796 | 23.3937 | 1.69115 | 5.34790 | 1.41946 | 3.05813 | 6.58853 | .349650 
8.2369 | 23.6399 | 1.69411 | 5.35724 | 1.42111 | 3.06169 | 6.59620 | .348432 
8.2944 | 23.8879 | 1.69706 | 5.36656 | 1.42276 | 3.06524 | 6.60385 | .847222 
8.3521 | 24.1376 | 1.70000 | 5.37587 | 1.42440 | 3.06878 | 6.61149 | .346021 
8.4100 | 24.3890 | 1.70294 | 5.38516 | 1.42604 | 3.07232 | 6.61911 | .344828 


8.4681 | 24.6422 | 1.70587 | 5.39444 | 1.42768 | 3.07585 | 6.62671 | .343643 
8.5264 | 24.8971 | 1.70880 | 5.40370 | 1.42931 | 3.07936 | 6.63429 | .342466 
8.5849 | 25.1538 | 1.71172 | 5.41295 | 1.43094 | 3.08287 | 6.64185 | .341297 

6436 | 25.4122 | 1.71464 | 5.42218 | 1.43257 | 3.08638 | 6.64940 | .840136 
8.7025 | 25.6724 | 1.71756 | 5.43139 | 1.43419 | 3.08987 | 6.65693 | 338983 


8.7616 | 25.9343 | 1.72047 | 5.44059 | 1.43581 | 3.09336 | 6.66444 | .337838 
8.8209 | 26.1981 | 1.72337 | 5.44977 | 1.43743 | 3.09684 | 6.67194 | .336700 
8.8804 | 26.4636, 1.72627 | 5.45894 | 1.43904 | 3.10081 | 6.67942 | 835571 

4448 


8.9401 | 26.7309 | 1.72916 | 5.46809 | 1.44065 | 3.10378 | 6.68688 | .33 
9.0000 | 27.0000 | 1.73205 | 5.47723 | 1.44225 | 3.10723 | 6.69433 | .333333 


ss 


POWERS, ROOTS, AND RECIPROCALS 35 


3 


n2 ms | Nn |Nion| *n | Vion|Vioon| — 


= 58882 Reese 


OPO COCO COCKS UNO OO Mm ms 


gogecoce gocoeoc@ce coco eecn te se co 0 co co 
SSSSS RFSSR SSGNS ARSE SSRIS GSE 


Py 


commence cococm eee co erences 
SSE55 SEER 


9.0601 | 27.2709 | 1.73494 | 5.48635 | 1.44385 | 3.11068 | 6.70176 | .832226 
9.1204 | 27.5436 | 1.73781 | 5.49545 | 1.44545 | 3.11412 | 6.70917 | .331126 
9.1809 | 27.8181 | 1.74069 | 5.50454 | 1.44704 | 3.11755 | 6.71657 | .3830033 

a 28. ° z 1.44863 | 3.12098 | 6.72395 | .328947- 
9.3025 | 28.3726 | 1.74642 | 5.52268 | 1 45022 | 3.12440 | 6.73132 | .327869 


9.3636 | 28.6526 | 1.74929 | 5.53173 | 1.45180 | 3.12781 | 6.73866 | .326797 
9.4249 | 28.9344 | 1.75214 | 5.54076 | 1.45338 | 3.13121 | 6.74600 | .325733 
9.4864 | 29.2181 | 1.75499 | 5.54977 | 1.45496 | 3.13461 | 6.75331 | 324675 
9.5481 | 29.5036 | 1.75784 | 5.55878 | 1.45653 | 3.13800 | 6.76061 | .323625 
9.6100 | 29.7910 | 1.76068 | 5.56776 | 1.45810 | 3.14138 | 6.76790 | .322581 


9.6721 | 30.0802 | 1.76352 | 5.57674 | 1.45967 | 3.14475 | 6.77517 } 321543 
9.7344 | 30.3713 | 1.76635 | 5.58570 | 1.46123 | 3.14812 | 6.78242 | .320513 


9.8598 | 30.9591 | 1.77200 | 5.60357 | 1.46434 | 3.15484 | 6.79688 | .318471 
9.9225 | 31.2559 | 1.77482 | 5.61249 | 1.46590 | 3.15818 | 6.80409 | .317460 


9.9856 | 31.5545 | 1.77764 | 5.62139 | 1.46745 | 3.16152 | 6.81128 | .316456 


10.1761 | 32.4618 | 1.78606 | 5.64801 | 1.47208 | 3.17149 | 6.83277 | .313480 
10.2400 | 32.7680 | 1.78885 | 5.65685 | 1.47361 | 3.17480 | 6.83990 | .312500 


10.3041 | 33.0762 | 1.79165 | 5.66569 | 1.47515 | 3.17811 | 6.84702 | .311527 
10.3684 | 33.3862 | 1.79444 | 5.67450 | 1.47668 | 3.18140 | 6.85412 | .310559 
10.4329 | 33.6983 | 1.79722 | 5.68331 | 1.47820 | 3.18469 | 6.86121 | .309598 
10.4976 | 34.0122 | 1.80000 | 5.69210 | 1.47973 | 3.18798 | 6.86829 | .308642 
10.5625 | 34.3281 | 1.80278 | 5.70088 | 1.48125 | 3.19125 | 6.87534 | 307692 


10.6276 | 34.6460 | 1.80555 | 5.70964 | 1.48277 | 3.19452 | 6.88239 | .306749 
10.6929 | 34.9658 | 1.80831 | 5.71839 | 1.48428 | 3.19779 | 6.88942 | .305810 
10.7584 | 35.2876 | 1.81108 | 5.72713 | 1.48579 | 3.20104 | 6.89643 | .304878 
10.8241 | 35.6129 | 1.81384 | 5.73585 | 1.48730 | 3.20429 | 6.90344 | .303951 
10.8900 | 35.9370 | 1.81659 | 5.74456 | 1.48881 | 3.20753 | 6.91042 | .303030 


10.9561 | 36.2647 | 1.81934 | 5.75326 | 1.49031 | 3.21077 | 6.91740 | .302115 
11.0224 | 36.5944 | 1.82209 | 5.76194 | 1.49181 | 3.21400 | 6.92436 | .301205 
11.0889 | 36.9260 | 1.82483 | 5.77062 | 1.49330 | 3.21723 | 6.93130 | 300300 
11,1556 | 37.2597 | 1.82757 | 5.77927 | 1.49480 | 3.22044 | 6.93823 | .299401 
11.2225 | 37.5954 | 1.83030 | 5.78792 | 1.49629 | 3.22365 | 6.94515 | .298508 


11.2896 | 37.9331 | 1.83303 | 5.79655 | 1.49777 | 3.22686 | 6.95205 | .297619 

11.3569 | 38.2728 | 1.83576 | 5.80517 | 1.49926 | 3.23005 | 6.95894 | .296736 

11.4244 | 38.6145 | 1.83848 | 5.81878 | 1.50074 | 3.23325 | 6.96582 | .295858 

11.4921 | 38.9582 | 1.84120 el 1.50222 | 3.23643 | 6.97268 | .294985 
5.8309 


6281 | 39.6518 | 1.84662 | 5.83952 | 1.50517 | 3.24278 | 6.98637 | .293255 
1.6964 | 40.0017 | 1.84932 | 5.84808 | 1.50664 | 3.24595 | 6.99319 | .292398 
11.7649 | 40.3536 | 1.85203 | 5.85662 | 1.50810 | 3.24911 | 7.00000 | .291545 
11. 40.7076 | 1.85472 | 5.86515 | 1.50957 | 3.25227 | 7.00680 | .290698 
11.9025 | 41.0636 | 1.85742 | 5.87367 | 1.51108 | 3.25542 | 7.01358 | 289855 


11.9716 | 41.4217 | 1.86011 : 88218 | 1.51249 | 3.25856 | 7.02035 | .289017 
12.0409 | 41.7819 | 1.86279 | 5.8906 . 

12,1104 | 42.1442 | 1.86548 | 5.89915 | 1.51540 | 3.26482 | 7.03385 | .287356 
12.1801 | 42.5085 | 1.86815 | 5.90762 | 1.51685 | 3.26795 | 7.04058 | .286533 
12.2500 | 42.8750 | 1.87083 | 5.91608 | 1.51829 | 3.27107 | 7.04730 | .285714 


=a 
_ 
o 
i 
S 
~ 
ao 
iJ 
a 
_ 
3 
a 
& 
-~ 
_ 
a 
E 
= 


36 


POWERS, 


ROOTS, AND RECIPROCALS 


3 


n2 


<n 


<i0 n 


wwe switcte tite cote Cote to to te 
SBRBSSB R2BS2 SBeas arase 


SS RESRA SSara aks 


Resse See 


Ed 


> co co 60 6 
Biss 


3 


12.3201 


14,2129 


14.7456 
14.8225 


14.8996 
14,9769 


54.8720 


55.3063 
55.7430 
56.1819 
56.6231 
57.0666 


57.5125 


1.91050 
1.91311 


1.51974 


1 re 


1.56322 


3.27418 


8.31714 


3.32017 
3.32319 
3.32621 
3.32922 


3.34716 


3.35014 
3.35310 
5607 


7.27479 


7.28108 
7.28736 
363 


7.33723 


7.34342 
7.34960 
7.85576 
7.36192 
7.36806 


251889 


POWERS, ROOTS, AND RECIPROCALS 37 
: 1 
n? n3 Nn 41002 = 

16.0801 | 64.4812 | 2.00250 7.37420 | .249377 
16.1604 | 64.9648 | 2.00499 7.38032 | 248756 
6.2409 | 65.4508 | 2.00749 7.38644 | 248139 
16.3216 | 65.9393 | 2. 739254 | 247525 
16.4025 | 66.4301 | 2.01246 739864 | 1246914 

16.4836 | 66.9234 | 2.01494 7.40472 | .2 
16.5649 | 67.4191 | 2.01742 7.41080 | .245700 

16.6464 | 67.9173 | 2.01990 7.41686 | .2 
16.7281 | 68.4179 | 2.02237 7.42291 | 1244499 
16.8100 | 68.9210 | 2.02485 7.42896 | 1243902 
16.8921 | 69.4265 | 2.02731 7.43499 | 243309 
16.9744 | 69.9345 | 2.02978 7.44102 | 242718 
17.0569 | 70.4450 | 2.03224 7.44703 | 242131 
17.1396 | 70.9579 | 2.03470 7.45304 | 241546 
17.2225 | 71.4734 | 2.03715 7.45904 | 240964 
17.3056 | 71.9913 | 2.03961 7.46502 } .240385 
17.3889 | 72.5117 | 2.04206 7.47100 | .239808 
17.4724 | 73.0346 | 2. 7.47697 | 239234 
17.5561 | 73.5601 | 2.04695 7.48292 | _238664 
17.6400 | 74.0880 | 2.049 "48887 | .238095 
17.7241 | 74.6185 | 2.05183 7.49481 | .237530 
1 75.1514 | 2.05426 7.50074 | 1236967 
17.8929 | 75.6870 | 2.05670 7.50666 | 236407 
17.9776 | 76.2250 | 2.05913 7.51257 | .235849 
18.0625 | 76.7656 | 2.06155 7.51847 | 1235294 
18.1476 | 77.3088 | 2.06398 7.52437 | 284742 
18.2329 | 77.8545 | 2.06640 7.53025 | .234192 
18.3184 | 78.4028 | 2.06882 7.53612 | 1233645 
18.4041 | 78.9536 | 2.07123 7.54199 | |233100 
4.30 | 18.4900 | 79.5070 | 2.07364 7.54784 | 1232558 
4.31 | 18.5761 | 80.0630 | 2.07605 7.55369 | 232019 
4.32 | 18.6624 | 80.6216 | 2.07846 7.55953 | 231482 
4.33 | 18.7489 | 81.1827 | 2.08087 7.56535 | 230947 
4.34 | 18.8356 | 81.7465 | 2.08327 757117 | .230415 
4.35 | 18.9225 | 82.3129 | 2.08567 7.57698 | (229885 
4.36 | 19.0096 | 82.8819 | 2. 7.58279 | 229358 
4.37 | 19.0969 | 83.4535 | 2.09045 7.58858 | |228833 
4.38 | 19.1844 | 84.0277 | 2.09284 7.59436 | 228311 
4.39 | 19.2721 | 84.6045 | 2.09523 7.60014 | .227790 
4.40 | 19.3600 | 85.1840 | 2.09762 "60590 | |227273 
4.41 | 19.4481 | 85.7661 | 2.10000 7.61166 | .226757 
4.42 | 19.5364 | 86.3509 | 2.10238 7.61741 | 1226244 
4.43 | 19.6249 | 86.9383 | 2.10476 7.62315 | 225734 
4.44 | 19.7136 | 87.5284 | 2.10713 7.62888 | 1225225 
4.45 | 19.8025 | 88.1211 | 2.10950 7.63461 | .224719 
4.46 | 19.9916 | 88.7165 | 2.11187 7.64082 | .224215 
4.47 | 19.9809 | 89.3146 | 2.11424 7.64603 | 223714 
4.48 | 20.0704 | 89.9154 | 2.11660 7.65172 | 223214 
4.49 | 20.1601 | 90.5188 | 2.11896 7.65741 | 222717 
4.50 | 20.2500 | 91.1250 | 2.12132 "222292 


38 POWERS, ROOTS, AND RECIPROCALS 


SS a BS, 15 
n? n3 Nn | ~vVion| Wn | Mion|*i00n| — 


3 


20.3401 | 91.7339 | 2.12368 | 6.71565 | 1.65219 | 3.55953 | 7.66877 | .221730 

20.4304 | 92.3454 | 2.12603 | 6.72309 | 1.65341 | 3.56215 | 7.67443 | .221239 

20.5209 | 92.9597 | 2.12838 | 6.73053 | 1.65462 | 3.56478 | 7.68009 | .220751 

20.6116 | 93.5767 | 2.13073 | 6.73795 | 1.65584 | 3.56740 | 7.68573 | .220264 

20.7025 | 94.1964 | 2.13307 | 6.74537 706 | 3.57002 | 7.69137 | .219780 
8 


SEBh Basta SkseHd SeRke aeeee seeee ares 
Nw 
rent 
a3 | 


21.0681 | 96.7026 | 2.14243 | 6.77495 F ; 865 
21.1600 | 97.3360 | 2.14476 | 6.78233 | 1.66310 | 3.58305 | 7.71944 | .217391 


21,2521 | 97.9722 | 2.14709 | 6.78970 | 1.66431 | 3.58564 | 7.72503 | .216920 
21.3444 | 98.6111 | 2.14942 | 6.79706 | 1.66551 | 3.58823 | 7.73061 | .216450 
21.4369 | 99.2528 | 2.15174 | 6.80441 | 1.66671 | 3.59082 | 7.73619 | .215983 
21.5296 | 99.8973 | 2.15407 | 6.81175 | 1.66791 | 3.59340 | 7.74175 | .215517 
21.6225 | 100.545 | 2.15639 | 6.81909 | 1.66911 | 3.59598 | 7.74731 | .215054 


21.7156 | 101.195 | 2.15870 | 6.82642 | 1.67030 | 3.59856 | 7.75286 | .214592 

bp en ys 6.83374 | 1.67150 | 3.60113 | 7.75840 | .214133 

‘ 2.1 d F ‘ 

21.9961 | 103.162 | 2.16564 | 6.84836 | 1.67388 | 3.60626 | 7.76946 | .213220 

22.0900 | 103.823 | 2.16795 | 6.85565 | 1.67507 | 3.60883 | 7.77498 | .212766 

22.1841 | 104.487 | 2.17025 | 6.86294 geben 3.61138 | 7.78049 | .212314 
21 


i 


22.5625 | 107.172 | 2.17945 | 6.89202 | 1.68099 | 3.62158 | 7.80245 | 210526 


22.6576 | 107.850 | 2.18174 | 6.89928 | 1.68217 | 3.62412 | 7.80793 | .210084 © 
22.7529 | 108.531 | 2.18403 | 6.90652 | 1.68334 | 3.62665 | 7.81339 | .209644 
22.8484 | 109.215 | 2.18632 | 6.91375 | 1.68452 | 3.62919 | 7.81885 | .209205 
22.9441 | 109.902 | 2.18861 | 6.92098 | 1.68569 | 3.63171 | 7.82429 | .208768 
23.0400 | 110.592 | 2.19089 | 6.92820 | 1.68687 | 3.63424 | 7.82974 | .208333 


23.1361 | 111.285 | 2.19317 | 6.93542 | 1.68804 | 3.63676 | 7.83517 | .207900 


23.4256 | 113.380 | 2.20000 | 6.95701 | 1.69154 | 3.64431 | 7.85142 | .206612 
23.5225 | 114.084 | 2.20227 | 6.96419 | 1.69270 | 3.64682 | 7.85683 | .206186 


23,6196 | 114.791 | 2.20454 | 6.97137 | 1.69386 | 3.64932 | 7.86222 | .205761 
23.7169 | 115,501 | 2.20681 | 6.97854 | 1.69503 | 3.65182 | 7.86761 | .205339 
23.8144 | 116.214 | 2.20907 | 6.98570 | 1.69619 | 3.65432 | 7.87299 | .204918 
23.9121 | 116.930 | 2.21133 | 6.99285 | 1.69734 | 3.65682 | 7.87837 | .204499 
24.0100 | 117.649 | 2.21359 | 7.00000 | 1.69850 | 3.65931 | 7.88374 | .204082 


24,1081 } 118.871 | 2.21585 | 7.00714 | 1.69965 | 3.66179 | 7.88909 | .203666 
24.2064 | 119.095 | 2.21811 | 7.01427 | 1.70081 | 3 66428 | 7.89445 | .203252 
24.3049 | 119.823 | 2.22036 | 7.02140-| 1.70196 | 3.66676 | 7.89979 | .202840 
24,4036 | 120.554 | 2 22261 | 7.02851 | 1.70311 | 3.66924 | 7.90513 | .202429 
24,5025 | 121.287 | 2.22486 | 7.03562 | 1.70426 | 3.67171 | 7.91046 | .202020 


24.6016 | 122.024 | 2.22711 | 7.04273 | 1.70540 | 3.67418 | 7.91578 | .201613 
24,7009 | 122.763 | 2.22935 | 7.04982 | 1.70655 | 3.67665 | 7.92110 | .201207 
24.8004 | 123.506 | 2.23159 | 7.05691 | 1.70769 | 3.67911 | 7.92641 | .200803 
24.9001 | 124.251 | 2.23383 | 7.06399 | 1.70884 | 3.68157 | 7.93171 | .200401 
25.0000 | 125.000 | 2.23607 | 7.07107 | 1.70998 | 3.68403 | 7.93701 | .200000 


: 


. 


TREAD RPP R A PPR PD 
SSSR FLSSS SSELS GS 


POWERS, 


ROOTS, AND RECIPROCALS 


39 


3 


N10 n 


<n 


<0 n- 


S 
= 


NNN ONO THON VON OVO 
SSERa Keawem SSIS GESeH 


5.27 


AN NNN NNN ONTO rE 
SSG55 GESEA SSSR KEBES SSS 


29. 5936 


056 
143.878 


2.28692 
2.28910 


2.31517 
2.31733 
2.31948 
2.32164 
2.32379 


2.32594 


7.07814 
7.08520 


rf ‘21110 


7.21803 
7.22496 
7.23187 
7.23878 
7.24569 


7.25259 
7.25948 
7.26636 
7.27324 
7.28011 


7.37564 


7. 41620 


1.71112 
1.71225 
1.71339 
1.71452 
1.71566 


1.71679 
1.71792 
1.71905 
1.72017 
1.72130 


1.76517 


3.78210 
3.78442 
3.78675 
3.78907 
3.79139 


80065 
3.80295 


8.12310 
8.12814 
8.13319 
8.13822 
8.14325 


8.18824 
8.19321 


"194175 
.193798 
1193424 
-193050 
"192678 
"192308 
.191939 


181818 


| ROOTS, AND RECIPROCALS 


3 


n2 


Cron cr cr or Cr or or cn or Or Orn Oren Cron cn on 


SSNS AESee SEZLS ARSVRZ SSAAS AKSTA SBS 


PHMNMAM NANT HNN THN HON TO HE HE 


5 RZSES 


Peary 
oO 
a 


S33 


> 
a 


. 


8 a2aes 


g 


30.3601 
30.4704 
30.5809 
30.6916 
30.8025 
30.9136 
31.0249 
31.1364 
31.2481 
31.3600 


31.4721 


34, noe 


174.677 


eos _ 


193, 101 
194.105 
195,112 


196.123 
197.137 
198.155 
199.177 
200.202 


205.379 


206.425 
207.475 


: 35797 
2.36008 
2.36220 


2.36432 
2.36643 


2.38747 
2.38956 
2.39165 
2.39374 
2.39583 
2.39792 


2.44745 


2.44949 


7.61577 
7.62234 


1.77051 


1.77157 
1.77263 


1.78108 


1.78213 
1.78318 
1.78422 
1.78527 
1.78632 


1.78736 


1.79876 


1.81712 


3.83495 
3.83721 


3.83948 
3.84174 


3.87532 


8.21303 
8.21797 


8.22290 
8.22783 
8.23275 
8.23766 
8.24257 


8.24747 


8.41554 
8.42025 
8.42494 
8.42964 
8.43433 


173913 
173611 


172117 
171821 
-171527 
171233 
-170940 


POWERS, ROOTS, AND RECIPROCALS 41 
5 | 
n na n3 Nn 4n | 40n|*i00n = 
6.01 | 36.1201 | 217.082 | 2.45153 1.81813 | 3.91704 | 8.43901 | .1 
6.02 | 36.2404 | 218.167 | 2.45357 1.81914 | 3.91921 | 8.44369 | .166118 
6.03 | 36.3609 | 219.256 | 2.45561 1.82014 | 3.92138 | 8.44836 | .165888 
6.04 | 36.4816 | 220.349 | 2.45764 1.82115 | 3.92355 | 8.45308 | .165568 
6.05 | 36.6025 | 221.445 | 2.45967 1.82215 | 3.92571 | 8.45769 | .165289 
6.06 | 36.7236 | 222.545 | 2.46171 1.82316 | 3.92787 | 8.46235 | .165017 
6.07 | 36.8449 | 223.649 | 2.46374 1.82416 | 3.93003 | 8.46700 | 164745 
6.08 | 36.9664 | 224.756 | 2.46577 1.82516 | 3.93219 | 8.47165 | 1164474 
6.09 | 37.0881 | 225.867 | 2.46779 1.82616 | 3.93434 | 8.47629 | .164204 
6.10 | 37.2100 | 226.981 | 2.46982 1.82716 | 3.93650 | 8.48093 | .163934 
6.11 | 37.3321 | 228.099 | 2.47184 1.82816 | 3.93865 | 8.48556 | .163666 
6.12 | 37.4544 | 229.221 | 2.47386 1.82915 | 3.94079 | 8.49018 | .168399 
6.13 | 37.5769 | 230.346 | 2.47588 183015 | 3.94294 | 8.49481 | .163132 
6.14 | 37.6996 | 231.476 | 2.47790 1.83115 | 3.9 8.49942 | .162866 
3.15 | 37.8225 | 232.608 | 2.47992 1.83214 | 3.94722 | 8.50404 | .162602 
6.16 | 37.9456 | 283.745 | 2.48193 1.83818 | 3.94936 | 8. 162388 
6.17 | 38.0689 | 234.885 | 2.48395 '83412 | 3.95150 | 8.51324 | .162075 
6.18 | 38.1924 | 236.029 | 2.48596 1.83511 | 3.95 ‘51784 | 161812 
6.19 | 38.3161 | 237.177 | 2.48797 1.83610 | 3.95576 | 8.52248 | .161551 
6.20 | 38. 238,328 | 2.48998 1.83709 | 3.95789 | 8.52702 | .161290 
6.21 | 38.5641 | 239.483 | 2.49199 1.88808 | 8.9 8.53160 | .161031 
6.22 | 38.6884 | 240.642 | 2.49399 1.83906 | 8.96214 | 8.53618 | .160772 
6.23 | 38.8129 | 241.804 | 2.49600 1.84005 | 8.96426 | 8.54075 | .160514 
6.24 | 38.9376 | 242.971 | 2.49800 1.84103 | 3.96639 | 8.54532 | .160256 
6.25 | 39.0625 | 244.141 | 2.50000 1.84202 | 8.96850 | 8.54988 | .160000 
.26 | 39.1876 | 245.314 | 2.50200 1.84300 | 3.97062 | 8.55444 | .159744 
"27 | 39.3129 | 246.492 | 2.50400 1.84398 | 8.97273 | 8.55899 | .159490 
"28 | 39.4384 | 247.678 | 2.50599 1.84496 | 3.97484 | 8.56354 | .169236 
6.29 | 39.5641 | 248.858 | 2.50799 1.84594 | 3.97695 | 8.56808 | .158988 
6.30 | 89.6900 | 250.047 | 2.50998 1.84691 | 3.97906 | 8.57262 | .158730 
6.31 | 39.8161 | 251.240 | 2.51197 1.84789 | 3.98116 | 8.57715 | .158479 
6.32 | 39.9424 | 252.436 | 2.51396 1.84887 | 8.98326 | 8.58168 | .158228 
6.33 | 40.0689 | 253.636 | 2.51595 1.84984 | 8.98536 | 8.58620 | 157978 
6.34 | 40.1956 | 254.840 | 2.51794 1.85082 | 8.98746 | 8.59072 | .157729 
6.35 | 40.8225 | 256.048 | 2.51992 1.85179 | 3. 8.59524 | .157 
6.36 | 40.4496 | 257.259 | 2.52190 1.85276 | 8.99165 | 8.59975 | .157238 
6.37 | 40.5769 | 258.475 | 2.52389 1.85373 | 3.99374 | 8.60425 | .156986 
6.38 | 40.7044 | 259.694 | 2.52587 1.85470 | 3. 8.60875 | .156740 
6.39 | 40.8321 | 260.917 | 2.52784 1.85567 | 3.99792 | 8.61825 | .156495 
6.40 | 40.9600 | 262.144 | 2.52982 1.85664 | 4. 8.61774 | .156250 
6.41 | 41.0881 | 263.375 | 2.53180 1.85760 | 4.00208 | 8.62222 | .156006 
6.42 | 41.2164 | 264.609 | 2.53377 1.85857 | 4.00416 | 8.62671 | .155763 
6.43 | 41.3449 | 265.848 | 2.58574 1.85953 | 4.00624 | 8.63118 | .155521 
6.44 | 41.4736 | 267.090 | 2.53772 1.86050 | 4.00882 | 8.63566 | .155280 
6.45 | 41.6025 | 268.336 | 2.53969 1.86146 | 4.01039 | 8.64012 
6.46 41.7316 | 269.586 | 2.54165 1.86242 | 4.01246 | 8.64459 | .154799 
647 | 41.8609 | 270.840 | 2.54362 1.86388 | 4.01453 | 8.64904 | .154560 
6.48 | 41.9904 | 272,098 | 2.54558 1.86434 | 4.01660 | 8.65350 | .154821 
6.49 | 42.1201 | 273.359 | 2.54755 1.86530 | 4.01866 | 8.65795 | .154088 
6.50 | 42.2500 | 274.625 | 2.54951 1.86626 | 4.02073 | 8.66239 | |153846 


42 POWERS, ROOTS, AND RECIPROCALS 


n2 n3 Vn | ion| Wn | Vi0n|*i00n 


3 


42.3801 | 275.894 | 2.55147 | 8.06846 | 1.86721 | 4.02279 | 8.66683 
42.5104 | 277.168 | 2.55343 | 8.07465 | 1.86817 | 4.02485 | 8.67127 
42.6409 | 278.445 | 2.55539 | 8.08084 | 1.86912 | 4.02690 | 8.67570 
42.7716 | 279.726 | 2.55734 | 8.08703 | 1.87008 | 4.02896 | 8.68012 
42,9025 | 281.011 | 2.55930 | 8.09321 | 1.87103 | 4.03101 | 8.68455 


43.0336 | 282.300 | 2.56125 | 8.09938 | 1.87198 | 4.03306 | 8.68896 
43.1649 | 283.593 | 2.56320 | 8.10555 | 1.87293 | 4.03511 | 8.69338 
43,2964 | 284.890 | 2.56515 | 8.11172 | 1.87388 | 4.03715 | 8.69778 
43.4281 | 286.191 | 2.56710 | 8.11788 | 1.87483 | 4.03920 | 8.70219 
43,5600 | 287.496 | 2.56905 | 8.12404 | 1.87578 | 4.04124 | 8.70659 


43.6921 | 288.805 | 2.57099 | 8.13019 | 1.87672 | 4.04328 | 8.71098 
43.8244 | 290.118 | 2.57294 | 8.13634 | 1.87767 | 4.04532 | 8.71537 
43.9569 | 291.434 | 2.57488 | 8.14248 | 1.87862 | 4.04735 | 8.71976 
44,0896 | 292.755 | 2.57682 | 8.14862 | 1.87956 | 4.04939 | 8.72414 
44,2225 | 294,080 | 2.57876 | 8.15475 | 1.88050 | 4.05142 | 8.72852 


44,3556 | 295.408 | 2.58070 | 8.16088 | 1.88144 | 4.05345 | 8.73289 
44,4389 | 296.741 | 2.58263 | 8.16701 | 1.88239 | 4.05548 | 8.73726 
44.6224 | 298.078 | 2.58457 | 8.17313 | 1.88333 | 4.05750 | 8.74162 
44,7561 | 299.418 | 2.58650 | 8.17924 | 1.88427 | 4.05953 | 8.74598 
44,8900 | 300.763 | 2.58844 | 8.18535 | 1.88520 | 4.06155 | 8.75034 


45.0241 | 302.112 | 2.59037 | 8.19146 | 1.88614 | 4.06357 | 8.75469 
45.1584 | 303.464 | 2.59230 | 8.19756 | 1.88708 | 4.06558 | 8.75904 
45.2929 | 304.821 | 2.59422 | 8.20366 | 1.88801 | 4.06760 | 8.76338 
45.4276 | 306.182 | 2.59615 | 8.20975 | 1.88895 | 4.06961 | 8.76772 
8.77205 


45.6976 | 308.916 | 2.60000 | 8.22192 | 1.89081 | 4.07364 | 8.77638 
45.8329 | 310.289 | 2.60192 | 8.22800 | 1.89175 | 4.07564 | 8.78071 
45.9684 | 311.666 | 2.60384 | 8.23408 | 1.89268 | 4.07765 | 8.78503 
46.1041 | 313.047 | 2.60576 | 8.24015 | 1.89361 | 4.07965 | 8.78935 
46.2400 | 314.432 | 2.60768 | 8.24621 | 1.89454 | 4.08166 | 8.79366 


46.3761 | 315.821 | 2.60960 | 8.25227 | 1.89546 | 4.08365 | 8.79797 
46.5124 | 317.215 | 2.61151 | 8.25833 | 1.89639 | 4.08565 | 8.80227 
46.6489 | 318.612 | 2.61343 | 8.26438 | 1.89732 | 4.08765 | 8.80657: 
46.7856 | 320.014 | 2.61534 | 8.27043 | 1.89824 | 4.08964 | 8.81087 
46.9225 | 321.419 | 2.61725 | 8.27647 | 1.89917 | 4.09164 | 8.81516 


47.0596 | 322.829 | 2.61916 | 8.28251 | 1.90009 | 4.09362 | 8.81945 
47.1969 | 324.243 | 2.62107 | 8.28855 | 1.90102 | 4.09561 | 8.82373 
47.3344 | 325.661 | 2.62298 | 8.29458 | 1.90194 | 4.09760 | 8.82801 
47.4721 | 327.083 | 2.62488 | 8.30060 | 1.90286 | 4.09958 | 8.83229 
47.6100 | 328.509 | 2.62679 | 8.30662 | 1.90378 | 4.10157 | 8.83656 


47.7481 | 329.939 | 2.62869 | 8.31264 ; 1.90470 | 4.10355 | 8.84082 
47.8864 | 331.374 | 2.63059 | 8.31865 | 1.90562 | 4.10552 | 8.84509 
48.0249 | 332.813 | 2.63249 | 8.32466 | 1.90653 | 4.10750 | 8.84934 
48.1636 | 334.255 | 2.63439 | 8.33067 | 1.90745 | 4.10948 | 8.85360 
48.3025 | 335.702 | 2.63629 | 8.33667 | 1.90837 | 4.11145 | 8.85785 


48.4416 | 837.154 | 2.63818 | 8.34266 | 1.90928 | 4.11342 | 8.86210 
48.5809 | 338.609 | 2.64008 | 8.34865 | 1.91019 | 4.11539 | 8.86634 
48.7204 | 340.068 | 2.64197 | 8.35464 | 1.91111 | 4.11736 | 8.87058 
48.8601 | 341.532 | 2.64386 | 8.36062 | 1.91202 | 4.11932 | 8.87481 
49.0000 | 343.000 | 2.64575 | 8.36660 | 1.91293 | 4.12129 | 8.87904 


NAARAM APAMA MAAA® AAAAM ABAAH ARHHR PAADHR PHARA BAAAHA HAI2aH 
SSSSE FLESL SSSRR SLSSS SBS AKRSNA SSRAFZ HSPBSS SSRSAIF ARFSSS 
& 
or 
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a 
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POWERS, ROOTS, AND RECIPROCALS 43 


3 


n? nt | va |vion| % | *on|*ioon| 2 


49.1401 | 344.472 | 2.64764 | 8.37257 | 1.91384 | 4.12325 | 8.88327 | .142653 
49.2804 | 345.948 | 2.64953 | 8.37854 | 1.91475 | 4.12521 | 8.88749 | .142450 
49.4209 | 347.429 | 2.65141 | 8.38451 | 1.91566 | 4.12716 | 8.89171 | .142248 


aaaasa 


SN ANa8a As9g88 ANAS AAA 


49.5616 Lay tn 2.65330 | 8.39047 | 1.91657 | 4.12912 | 8.89592 | .142046 


49.7) B 2.65518 | 8.39643 | 1.91747 | 4.13107 | 8.90013 | .141844 
49.8426 | 351.896 | 2.65707 | 8.40238 | 1.91838 | 4.13303 | 8.90434 | .141643 
P 2.65895 | 8 1.91929 | 4.13498 | 8 141443 
50.1264 | 354.895 | 2.66083 | 8.41427 | 1.92019 | 4.13695 | 8.91274 | .141243 
50.2681 | 356.401 | 2.66271 | 8.4: 1.92109 | 4.13887 -141044 
50.4100 | 357.911 | 2.66458 | 8.42615 | 1.92200 | 4.14082 | 8.92112 | . 
8 3 


52.1284 | 376.367 | 2.68701 | 8.49706 patois 4.16402 | 8.97110 | .138504 


52.4176 | 379.503 | 2.69072 | 8.50882 | 1.93455 | 4.16786 | 8.97988 | .138122 
52.5625 | 381.078 | 2.69258 | 8.51469 | 1.93544 | 4.16978 | 8.98351 | .137931 


52.7076 | 382.657 | 2.69444 | 8.52056 | 1.93633 | 4.17169 | 8.98764 | .137741 


SB Beuish BERGE REERE BESS8 ReRSS 


48 REBRE BRE 


SSE55 SEESE S83 


52.8529 | 384.241 | 2.69629 | 8.52643 | 1.93722 | 4.17361 | 8.99176 | .137552 


53.5824 | 392.223 | 2.70555 | 8.55570 | 1.94165 | 4.18315 | 9.01233 | .136612 


1 s . : 4. 08156 | . 
2500 | 421.875 | 2.73861 | 8.66025 1.95748 | S216 | 9.08560 -133333 


POWERS, 


ROOTS, AND RECIPROCALS 


3 


ne 


Nn 


1007 


SS SSS93 sae88 sees sees 
Be SSRIS SFBVS SSSAS SRESA 


1.73 


Bae 


& 
© 


&esse & 


SCNAAT ATTA Aaa 


S8eee 


58.5225 


58.6756. 
58.8289 
58.9824 
59.1361 
59.2900. 


59.4441 
59.5984 
59.7529 
59.9076 
60.0625 


60.2176 
60.3729 
60.5284 
60.6841 


61.6225 


61.7796 
61.9369 


63.5209 


423.565 


447.697 


449.455 
451,218 
452.985 
454,757 
456,533 


458.314 


472.729 
474,552 


476.380 
478.212 
480.049 


491.169 
493.039 


494.914 


502.460 


504.358 
506.262 
508.170 
510.082 
512,000 


2.74044 
2.74226 
2.74408 
2.74591 
2.74773 


2.74955 


2.77669 
2.77849 
2.78029 
2.78209 
2.78388 


2.78568 
2.78747 
2.78927 
2.79106 
2.79285 


2.79464 
2.79643 
2.79821 
2.80000 
2.80179 


2.80357 
2.80535 
2.80713 
2.80891 
2.81069 


2.81247 
2.81425 


2.81957 
2.82135 


2.82843 


8.71780 


8.72353 
8.72926 
8.73499 
8.74071 
8.74643 


8.75214 


8.80341 
8.80909 
8.81476 
8.82043 
8.82610 
8.83176 


8.83742 


8.91628 |. 


8.92188 
8.92749 
8.93308 
8.93868 
8.94427 


1,96523 


1.97040 


1.97126 
1.97211 
1.97297 


1.97895 


1.97980 
1.98065 
1.98150 
1.98234 
1,98319 


1.98911 


4.22651 


4.22838 
4.23024 
4.23210 
4.23396 
4,23582 


4,23768 
4.23954 


4.25248 


4.28177 
4.28359 


4.29265 
4.29446 
4.29627 
4.29807 
4.29987 


4.30168 
4.30348 
4.30528 
4.30707 
4.30887 


9.08964 
9.09367 
9.09770 
9.10173 
9.10575 


9.14179 
9.14577 


9.14976 
9.15374 
9.15771 
9.16169 
9.16566 


9.16962 
9.17359 
9.17754 
9.18150 
9.18545 


9.18940 
9.19335 
9.19729 
9.20123 
9.20516 


9.22087 
9.22479 


9.22871 
9.23262 
9.23653 
9.24043 
9.24433 


9.24823 


9.28318 


POWERS, ROOTS, AND RECIPROCALS 45 
rat 1 

n n3 Nn |Vi0n| Wn | MOn| oon 2 
8.01 513.922 | 2.83019 | 8.94986 | 2.00083 | 4.31066 | 9.28704 | .124844 
8.02 615.850 | 2.83196 | 8.95545 | 2.00167 | 4.31246 | 9.29091 | 124688 
8.03 517.782 | 2.83373 | 8.96103 | 2.00250 | 4.31425 | 9.29477 | .124533 
8.04 519.718 | 2.83549 | 8. 2.00333 | 4.31604 | 9.29862 | .124378 
8.05 521.660 | 2.83725 | 8.97218 | 2.00416 | 4.31783 | 9.30248 | 1124224 
8.06 523.607 | 2.83901 | 8.97775 | 2.00499 | 4.81961 | 9.80633 | .124070 
8.07 25.558 | 2.84077 | 8.98332 | 2.00582 | 4.32140 | 9.31018 | .123916 
8.08 527.514 | 2.84253 | 8.98888 | 2. 4.32818 | 9.31402 | .123762 
8.09 529.475 | 2.84429 | 8.99444 | 2.00747 | 4.82497 | 9.31786 | .123609 
8.10 531.441 | 2.84605 | 9.00000 | 2. 4.32675 | 9.32170 | .123457 
8.11 533.412 | 2.84781 | 9.00555 | 2.00912 | 4.32858 | 9.32553 | .123305 
8.12 535.387 | 2.84956 | 9.01110 | 2.00995 | 4.33031 | 9.32936 | 123153 
8.13 537.368 | 2.85132 | 9.01665 | 2.01078 | 4.33208 | 9.33319 | .123001 
8.14 539.353 | 2.85307 | 9.02219 | 2.01160 | 4.33386 | 9.33702 | .122850 
8.15 641.343 | 2.85482 | 9.02774 | 2.01242 | 4.33563 | 9.34084 | 1122699 
8.16 543.338 | 2.85657 | 9.03827 | 2.01325 | 4.83741 | 9.84466 | .122549 
8.17 545.339 | 2.85832 | 9.03881 | 2.01407 | 4.33918 | 9.34847 | .122399 
8.18 547.343 | 2.86007 | 9.04434 | 2.01489 | 4.34095 | 9.35229 | 1122249 
8.19 549.353 | 2.86182 | 9.04986 | 2.01571 | 4.34272 | 9.35610 | .122100 
8.20 551.368 | 2.86356 | 9.05539 | 2.01653 | 4.3 9.35: 121951 
8.21 553.388 | 2.86531 | 9.06091 | 2.01735 | 4.34625 | 9.36370 | .121808 
8.22 565.412 | 2.86705 | 9.06642 | 2.01817 | 4.34801 | 9.36751 | 121655 
8.23 557.442 | 2.86880 | 9.07193 | 2.01899 | 4.34977 | 9.37130 | .121507 
8.24 559.476 | 2.87054 | 9.07744 | 2.01980 | 4.35153 | 9.37510 | 1121359 
8.25 561.516 | 2.87228 | 9.08295 | 2.02062 | 4.35829 | 9.37889 | {121212 
8.26 563.560 | 2.87402 | 9.08845 | 2.02144 | 4.35505 | 9.38268 | .121065 
8.27 565.609 | 2.87576 | 9.09395 | 2.02225 | 4.35681 | 9.38646 | 120919 
8.28 567.664 | 2.87750 | 9.09945 | 2.02307 | 4.35856 | 9.39024 | .120773 
8.29 569.723 | 2.87924 | 9.10494 | 2.02388 | 4.36032 | 9.39402 | |120627 
8.30 571.787 | 2.88097 | 9.11043 | 2.02469 | 4.36207 | 9.39780 | {120482 
8.31 573.856 | 2.88271 | 9.11592 | 2.02551 | 4.86382 | 9.40157 | .120337 
8.32 575.930 | 2.88444 | 9.12140 | 2.02632 | 4.36557 | 9.40534 | .120192 
8.33 578.010 | 2.88617 | 9.12688 | 2.02713 | 4.36732 | 9.40911 | .120048 
8.34 580.094 | 2.88791 | 9.13236 | 2.02794 | 4.36907 | 9.41287 | .119904 
8.35 582.183 | 2.88964 | 9.13783 | 2.02875 | 4.37081 | 9.41663 | .119761 
8.36 584.277 | 2.89137 | 9.14330 | 2.02956 | 4.87255 | 9.42039 | .119617 
8.37 586.376 | 2.89310 | 9.14877 | 2.03037 | 4.37480 | 9.42414 | .119474 
8.38 588.480 | 2.89482 | 9.15423 | 2.03118 | 4.37604 | 9.42789 | 119332 
8.39 590.590 | 2.89655 | 9.15969 | 2.03199 | 4.37778 | 9.43164 | .119190 
8.40 592.704 | 2.89828 | 9.16515 | 2.03279 | 4.37952 | 9.43539 | .119048 
8.41 594,823 | 2.90000 | 9.17061 | 2.03360 | 4.38126 | 9.43913 | .118906 
8.42 596.948 | 2.90172 | 9.17606 | 2.03440 | 4.38299 | 9.44287 | |118765 
8.43 599.077 | 2.90345 | 9.18150 | 2.03521 | 4.38473 | 9.44661 | .118624 
8.44 601.212 | 2.90517 | 9.18695 | 2.03601 | 4. 9.45034 | .118483 
8.45 603.351 | 2. 9.19239 | 2.03682 | 4.38819 | 9.45407 | 118343 
8.46 605.496 | 2.90861 | 9.19783 | 2.03762 | 4.38992 | 9.45780 | .118203 
8.47 7.645 | 2.91033 | 9.20826 | 2.03842 | 4.39165 | 9.46152 | 118064 
8.48 609.800 | 2.91204 | 9.20869 | 2.03923 | 4.39338 | 9.46525 | 117925 
8.49 611.960 | 2.91376 | 9.21412 | 2.04003 | 4.39511 | 9.46897 | 117786 
8.50 614.125 | 2.91548 | 9.21954 | 2.04083 | 4.39683 | 9.47268 | .117647 


ROOTS, AND RECIPROCALS 


3 


Vn 


N10 n 


Vn 


P 90 90 G0 go 


00 GO 00 GO OO 


ae monn Konno or 
ZSS2 SSSIS APSA 


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SSR2aa & 


Nad URas 


PON COOMA OP ON 


Resse Sezne 9 Sp & Go be Oo se saaa 


Om Cobo 


{oc 00 GD GOOD GD ODED BODG~O MaMmGMD WMMMO MoDG~MM 


SSeS8 


72.4201 


13.2736 
73.4449 
73.6164 
73.7881 
73.9600 


74,1321 


74.9956 
75.1689 
75.3424 
75.5161 
75.6900 


76.3876 
76.5625 


16.7376 


77.6161 
17.7924 
77.9689 
78.1456 
718.3225 


78.4996 
18.6769 
78.8544 
79.0321 
79.2100 
79.3881 


644.973 
647.215 


649.462 
651.714 


681.472 
683.798 


695.506 
697.864 
700.227 
702.595 
704.969 


107.348 
709.732 
712,122 
714.517 
716.917 


719,323 
721.734 
724,151 
726.578 
729,000 


2.91719 


2.94279 
2.94449 
2.94618 
2.94788 
2.94958 


2.95127 


2.95804 
2.95973 


2.97489 
2.97658 


2.98161 
2.98329 


2.98496 
2.98664 
2.98831 
2.98998 
2.99166 


2.99333 
2.99500 


9.22497 
9.23038 
9,23580 
9.24121 
9.24662 


9.25203 


9.32738 
9.33274 


9.38616 
9.39149 
9.39681 
9.40213 
9.40744 


9.41276 


2.04163 
2.04243 
2.04323 
2.04402 
2.04482 


2.04562 
2.04641 
2.04721 
2.04801 
2.04880 


2.04959 
2.05039 
2.05118 
2.05197 
2.05276 


2.05750 
2.05828 


2.06924 
2.07002 
2.07080 
2.07157 
2.07235 


2.07313 
2.07390 
2.07468 
2.07545 
2.07622 


2.07700 


4.41571 
4,41742 
4.41913 
4.42084 
4.42254 


4.42425 
4.42595 
4.42765 
4.42935 
4.43105 


4.44627 
4.44796 


4.44964 
4.45133 
4.45301 


4.45469 
4,45637 


4.47476 
4.47642 
4.47808 
4,47974 
4,48140 


1 
9.48752 
9.49122 


9.53175 
9.53542 
9.53908 
9.54274 
9.54640 


9.55006 
9.55371 
9.55736 
9.56101 
9.56466 


9.56830 
9.57194 
9.57557 


9.57921 
9.58284 


9.58647 


116279 
116144 


115473 
115340 


11111 


/ 


POWERS, ROOTS, AND RECIPROCALS 47 


3 


n2 n3 Wn | Vion| *n | MOn|Moon| = 


i=) 
1 


www 
eae 


-48306 | 9.65847 | .110988 
-48472 | 9.66204 | .110865 
9. 1 


48803 | 9.66918 | .110620 


"9025 | 741.218 | 3.00832 | 9.51315 | 2.08393 | 4.49968 | 9.67274 | .110497 


82.0836 | 743.677 | 3.00998 | 9.51840 | 2.08470 | 4.49134 | 9.67630 | .110375 


WwWWwWO Www 
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dad el ire leer 
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82.2649 | 746.143 | 3.01164 | 9.52365 | 2.08546 | 4.49299 | 9.67986 | .110254 


J 9. - . 108342 
é .03974 | 9.61249 | 2.09841 | 4.52089 | 9.73996 | .108225 
791.453 | 3.04138 | 9.61769 | 2.09917 | 4.52252 | 9.74348 | .108108 


85.7476 | 794.023 | 3.04302 | 9.62289 | 2.09992 | 4.52415 | 9.74699 | .107991 
85.9329 | 796.598 | 3.04467 | 9.62808 | 2.10068 | 4.52578 | 9.75049 | .107875 
86.1184 | 799.179 | 3.04631 | 9.63328 | 2.10144 | 4.52740 | 9.75400 | .107759 
86.3041 | 801.765 | 3.04795 | 9.63846 | 2.10219 | 4.52903 | 9.75750 | .107643 
86.4900 | 804.357 | 3.04959 | 9.64365 | 2.10294 | 4.53065 | 9.76100 | .10752T 


. . 


Beeb 


#8ee 


& GESSE SERRE & 


86.6761 | 806.954 | 3.05123 | 9.64883 | 2.10370 | 4.53228 | 9.76450 107411 
Ns 4.53390 


J J A z . 4.54038 -106838 
87.7969 | 822.657 | 3.06105 | 9.67988 | 2.10821 | 4.54199 | 9.78543 | .106724 


843.909 | 3.07409 | 9.72111 | 2.11419 | 4.55488 | 9.81320 | :105820 


89.4916 | 846.591 | 3.07571 | 9.72625 | 2.11494 | 4.55649 | 9.81666 | .105708 


~ 
=a 


ses 


849.278 | 3.07734 | 9.73139 | 2.11568 | 4.55809 | 9.82012 | .105597 
’ 9.73653 | 2. A 16 
90.0601 | 854.670 | 3.08058 °. 74166 | 2.11717 | 4.56130 | 9.82703 | .105374 


90.2500 | 857.375 | 3.08221 | 9.74679 | 2.11791 | 4.56290 | 9.83048 | .105263 


48 POWERS, ROOTS, AND RECIPROCALS 


vion| %m | tion|*ioon| + 


9.75192 | 2.11865 | 4.56450 | 9.83392 | 105158 


3 
4 
3 
a 


22 
8 
S 
3 
S 
& 
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Ra 
3 


e 4.56930 
91.2025 | 870.984 | 3.09031 | 9.77241 | 2.12162 | 4.57089 | 9.84769 | .104712 


91.3936 | 873.723 | 3.09192 | 9.77753 | 2.12236 | 4.57249 | 9.85113 | .104603 
91.5849 | 876.467 | 3.09354 | 9.78264 | 2.12310 | 4.57408 | 9.85456 | .104493 
91.7764 | 879.218 | 3.09516 | 9.78775 | 2.12384 | 4.57568 | 9.85799 | .104384 
91.9681 | 881.974 | 3.09677 | 9.79285 | 2.12458 | 4.57727 | 9.86142 | .104275 
92.1600 | 884.736 | 3.09839 | 9.79796 | 2.12532 | 4.57886 | 9.86485 | .104167 

F 4 -10000 | 9. 2.12605 | 4.58045 | 9.86827 | .104058 
92.5444 | 890.277 | 3.10161 | 9.80816 | 2.12679 | 4.58203 | 9.87169 | .103950 


. ts 03842 
92.9296 | 895.841 | 3.10483 | 9.81835 | 2.12826 | 4.58521 | 9.87853 | .103734 
93.1225 | 898.632 | 3.10644 | 9.82344 | 2.12900 | 4.58679 | 9.88195 | .103627 


WOOHOO OOWOOO Owoowoo 


Ba2z Reese sesse are 
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g 
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93.8961 | 909.853 | 3.11288 | 9.84378 | 2.13194 | 4.59312 | 9.89558 | .103199 
‘ 2.13267 | 4.59470 | 9.89898 | .103093 


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: 9. . < 
94,8676 | 924.010 | 3.12090 | 9.86914 | 2.13560 | 4.60101 | 9.91257 | .102669 
95.0625 | 926.859 | 3.12250 | 9.87421 | 2.13633 | 4.60258 | 9.91596 | .102564 


95.2576 | 929.714 | 3.12410 | 9.87927 | 2.13706 | 4.60416 Hp ben 102459 
A 9.92274 


95.8441 | 938.314 | 3.12890 | 9.89444 | 2.13925 | 4.60887 | 9.92950 | .102145 
96.0400 | 941.192 | 3.13050 | 9.89949 | 2.13997 | 4.61044 | 9.93288 | [102041 


96.2361 | 944.076 | 3.13209 | 9.90454 | 2.14070 | 4.61200 | 9.93626 | .101937 
96.4324 | 946.966 | 3.13369 | 9.90959 | 2.14143 | 4.61357 | 9.93964 | .101833 
96.6289 | 949.862 | 3.13528 | 9.91464 | 2.14216 | 4.61513 | 9.94301 | .101729 
96.8256 | 952.764 | 3.13688 | 9.91968 | 2.14288 | 4.61670 | 9.94638 | .101626 
97.0225 | 955.672 | 3.13847 | 9.92472 | 2.14361 | 4.61826 | 9.94975 | .101523 


97.2196 | 958.585 | 3.14006 | 9.92975 | 2.14483 | 4.61983 | 9.95311 | .101420 
97.4169 | 961.505 | 3.14166 | 9.93479 | 2.14506 | 4.62139 | 9.95648 | .101317 
‘ 964.430 | 3.14325 | 9.93982 | 2.14578 | 4.62295 | 9.95984 | .101215 
97.8121 | 967.362 | 3.14484 | 9.94485 | 2.14651 | 4.62451 | 9.96320 | .101112 
98.0100 | 970.299 | 3.14643 | 9.94987 | 2.14723 | 4.62607 | 9.96655 | .101010 


98.2081 | 973.242 | 3.14802 | 9.95490 | 2.14795 | 4.62762 | 9.96991 | .100908 
98.4064 | 976.191 | 3.14960 | 9.95992 | 2.14867 | 4.62918 | 9.97826 | .100807 
98.6049 | 979.147 | 3.15119 | 9.96494 | 2.14940 | 4.63073 | 9.97661 | .100705 
98.8036 | 982.108 | 3.15278 | 9.96995 | 2.15012 | 4.63229 | 9.97996 | .1 

9.97497 | 2.15084 | 4.63384 | 9.98331 | .100503 


99.2016 | 988.048 | 3.15595 | 9.97998 | 2.15156 | 4.63539 | 9.98665 | .100402 
4009 9.98499 | 2.15228 | 4.63694 | 9.98999 | .100301 


© CLwowLo 
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99.8001 | 997.003 | 3.16070 | 9.99500 | 2.15372 | 4.64004 | 9.99667 | .100100 
3.16228 | 10.0000 | 2.15443 | 4.64159 | 10.0000 | .100000 


Seoews woos 


ss 
S 
s 
s 
3 


MATHEMATICS 49 


six significant figures (adding ciphers, if necessary, to make 
six figures) and find between what two numbers in the 


column headed B the significant figures of the given number 


fall; then proceed exactly as previously described to deter- 
mine the fourth and fifth figures. 


EXxAMPLE.—(a) The reciprocal of 379.426 = ? (b) — 7004853 4692 > =? 
SoLuTion.—(a) .379426 falls between 878788 = 5, nial 
380228-= 57. The first difference is 380228—378788 


= 1440; the second difference is 380228 — 379426 =802; 802 
+ 1440 = .557, or .56. Hence, the first five significant figures 
are 26356, and the tect of 379.426 is .0026356, to five 
significant figures. 


(b) .469200 falls between .469484 = si and .467290 


= sir The first difference is 2194; the second difference 
i . <3 C 1 a 
is 284; 284+ 2194=.129+,or.13. Hence, 9004602 ~ 2131.3, 


correct to five significant figures. 


MENSURATION 
In the following formulas, unless otherwise stated, the 
letters have the meanings here given: 
D=larger diameter; 
d=smaller diameter; 
R=radius corresponding to D; 
r=radius corresponding to d; 
p=perimeter of circumference; 
C=area of convex surface=area of flat surface that can 
be rolled into the shape shown; 
S=area of entire surface=C-+area of the end or ends; 
A=area of plane figure; 
m7=3.1416, nearly=ratio of any circumference to its 
diameter; 
V =volume of solid. 
The other letters used will be found on.the cuts. 


50 MATHEMATICS 


CIRCLE 


p=nd=3.1416d 
p=2nr =6.2832r 


p=2VrA =3.5449 VA 


A 4A 
ac aaae 

ay QUES aeee) 

Te Siete oe 
d=2 A =1.1284VA 


Satie De 
"=n 6.9832 1°92? 


r= (4 =.5642Va - 


A =72 = .7854d2 
A=nr2=3.1416r2 


_?r_pd 
A= 2a. 4 
TRIANGLES 
D=B+C E+B+C=180° 
B=D-C E'+B+C=180° 


"L, EISE B'=B 


The above letters refer to angles. 
sa a right triangle, c being the’ hypotenuse, 
= ae 
a= Nc2 
b= V2 
C= 
an acute angle of an 
nie triangle. 
= Va?+b?—Qbe + 
“pa Na2— ¢2 
c=length of side opposite an obtuse 
angle of an oblique-angled 
_triangle. 
c= Va2+b2+ 2Qhe 
h= Va2— 
For a triangle inscribed in a semicircle; 
i. e., any right triangle, 


MATHEMATICS 51 


@26 4 dik 
,—-eP_@ 
Cte 
a:b+e=e:a=h:c 
6 th thie 
For any triangle, 
bh 
A=y = tbh 
b a2+b2—¢2\2 
saree ee 


RECTANGLE AND PARALLELOGRAM 


Ao 
x TRAPEZOID 
/ A=th(a+b) 
. TRAPEZIUM 


Divide into two triangles and a trapezoid. 
A=bh'+4a(h' +h) +4ch; 
or, A=34[bh’+ch+a(h’+h)] 
Or, divide into two triangles by draw- 
ing a diagonal. Consider the diagonal 
as the base of both triangles, call its 
length /; call the altitudes of the triangles 4, and 2; then 
A=3(hi +h) 


ELLIPSE 


pean Pte (D—d)? 
AWA 2 8.8 
MMM A = {Dd=.7854Dd 


SECTOR a, 
A=3lr 
rrE 
A= 360 7 .008727r2E fi 


1=length of arc 


*The perimeter of an ellipse cannot be exactly determined without 
a very elaborate calculation, and this formula is merely an approxima- 
tion giving fairly close results. 


- MATHEMATICS 


SEGMENT 
A =4lr—c(r—h)] 


= sag at) 


RING 
wT 
A =7(D?—d?) 


CHORD 
c=length of chord 
c2 + 4h2 ¢2 


c=22hr—h2 
p= S28 
3 


, approximately 


HELIX 
Tc construct a helix: , 
1=length of helix; fC 
n=number of turns; Pe e 
t= pitch. a pR ES 2 


to 

t= ee 32d. 

l=nWNnr2d2+2 

Oe seeks 

rick +2 
CYLINDER 
C=ndh 
S=2rrh+2rr2 


=ndh +50 


V=arh= adh 


p*h 
V= "7, = .0796p% 


MATHEMATICS 53 


FRUSTRUM OF CYLINDER 
h=4 sum of greatest and least heights 
C=ph=tdh 


S=ndh+ 7 +area of elliptical top 


V=Ah=30h 
CONE 
Wi C=3rdl=nrl 
h S=nrl + ar2= ar V2 + h2+ wr? 
i md? h _.7854d*h _ p%h 


— 9 amie bar Seine Goes 6 
FRUSTUM OF CONE 
C=4(P+p)=51(D +d) 


S=5(D +d) +43(D?+d?)] 


V=5(D2+Dd +a) Xth 
= .2618h(D?-+ Dd +?) 


SPHERE 
S = 1d? = 4nr? = 12.5664r? 
V =}a3 = grr? = .5236d3 = 4.18887 


CIRCULAR RING 


D=mean diameter; 
R=mean radius. 

S = 4n2Rr =9.8696Dd 
V =2n2Rr2 = 2.4674Dd?2 


WEDGE 

V =twh(a+b+c) 
- Ao PRISMOID 

A prismoid is a solid having two parallel plane ends, the 


edges of which are connected by plane triangular or quadri- 
lateral surfaces. 


54 MATHEMATICS 


A=area of one end; 
a=area of other end; 
m=area of section midway between ends; 
l=perpendicular distance between ends. 
V=4l(A+at+4m) 
The area m is not in general a mean between the areas of 
the two ends, but its sides are means between the correspond- 
ing lengths of the ends. 


Approximately, V= Ate 


REGULAR PYRAMID | 


P=perimeter of base; 
A=area of base. 


To obtain area of base, divide it into triangles, and find 
their sum. 

The formula for V applies to any pyramid whose base is 
A and altitude h. 


FRUSTUM OF REGULAR PYRAMID 


a =area of upper base; 
A=area of lower base; 
p=perimeter of upper base; 
P=perimeter of lower base. 
C=4l(P+>) 
S=4l(P+p)+A+t+a 
V=3n(A+a+ VAa) 
The formula for V applies to the frustum of any pyramid. 


LENGTH OF SPIRAL 


I= (2<7) n=number of coils; 
="n\—> 1 =length of spiral; 
1 =7(R2—1?) Bar 


' MATHEMATICS 55 
PRISM OR PARALLELOPIPED 


C=ph 

S=ph+2A ey a> 0 Ge 

eet amy, qf 
For prisms with regular polygon as 


bases, p=length of one side X number of sides. 
To obtain area of base, if it is a polygon, divide it into tri- 
angles, and find sum of partial areas. 
FRUSTUM OF PRISM 
If a section perpendicular to the edges is a tri- 
angle, square, parallelogram, or regular polygon, 


yasum of lengths of edges 
~ number of edges 


REGULAR POLYGONS 
Divide the polygon into equal triangles and find the sum of 


the partial areas. Otherwise, square the length of one side 
and multiply by proper number from the following table: 


Kit 
HM 


X area of right section. 


Name No. Sides Multiplier 
Triangle... 2. cc03°8 .433 
— DOUATG. na er ve 4 1.000 
(/ \ Pentagon. ......5-..°8 1.720 
Hexagon......... 6 2.598 
\ y), Heptagon. «i: ¢,.::6 7 3.634 
——— Octagon... 14065. 4.828 
NOnagOns ssc. 9 6.182 
Decagon......... 10 7.694 


IRREGULAR AREAS 

Divide the area into trapezoids, triangles, parts 
of circles, etc., and find the sum of the partial areas. 

If the figure is very irregular, the approximate 
area may be found as follows: Divide the figure 
into trapezoids by equidistant parallel lines b, c, d, 
etc. The lengths of these lines being measured, 
then, calling a the first and m the last length, and . 
y the width of strips, 


area=(F"+b-+c-+etc. +m) 


56 LOADS IN STRUCTURES 


LOADS IN STRUCTURES 


The loads that a structure has to carry may be divided 
into five classes namely, the dead load, the live load, the 
accidental load, the snow load, and the wind load. 

The dead load is the weight of the materials of which the 
structure is composed, as the weight of the beams, columns, 
walls, floors, and the like. 

The live load is the weight of the various articles in the 
building that are not part of the structure itself, such as 
furniture, material stored, if the building is a storehouse, 
the occupants of the building, light machinery, etc. 

The accidental load is one whose application is doubtful 
or whose magnitude is great and its effect local. As exam- 
ples of such loads may be classed the load that would be 
generated if the ropes of an elevator broke and the safety 
device was suddenly brought into action, and the load that 
' would be created on a stairway in a mill if there was a panic 
among the employes, as during a fire. As an accidental 
load may also be classed such loads as would be caused by 
an especially heavy office safe or piece of machinery. Many 
engineers no longer consider an accidental load. The load 
that would come from an extra-large crowd, as on a stair- 
way in a fire, they class as part of the liveload. Such loads 
as would occur from a large safe, they usually class as part 
of the dead load, because the safe would be too heavy to be 
moved. Therefore, only that part of the structure in the 
vicinity of the safe need be strengthened. 

The snow and wind loads are, as the names imply, loads that 
are caused respectively by snow and wind. 


DEAD LOADS 
WEIGHT OF ROOF TRUSSES 


The dead load must be calculated for each member of a 
structure by means of tables that give the weights of build- 
ing materials, such as those on pages 59-61. However, it 


LOADS IN STRUCTURES 57 


is often desirable to know the approximate weight of roof 
trusses before the actual size of the members has been cal- 
culated. In such cases, the following empirical formula 
may be used: ; 

W=aDL ( 1+ a) 
in which W is the approximate weight of the truss, in pounds; 
a, a constant—for wood .5, for steel .75; D, the distance, in 
feet, from center to center of trusses; and L, the span of the 
truss, in feet. 

It is sometimes desirable to know the approximate weight 
of roof truss per square foot of roof surface. It will be noted 
that this is not the same thing as per square foot of hori- 
zontal surface, since the roof slants. If x is the angle 
between the rafter and the chord, L the span of the truss 
in feet, and w the approximate weight of the truss per square 
foot of roof surface, then 

y= eo +L) cos x 
10 
in which a is .5 for wooden trusses and .75 for steel trusses. 

ExaMPpLeE.—Determine the weight, per square foot, that 
must be added to the weight of a roof covering to provide for 
the weight of the principals, the steel trusses in this case 
having a span of 72 ft. and a rise of 18 ft. 

SoLuTION.—The angle between the rafter and the chord 


may be found from the formula tan x= ——=H8=.5, corre- 


_ sponding to an angle x of 26° 34’. Substituting the values 
of a, L, and cos x in the preceding formula, 


.75 (10+72) cos 26° 34’ .75xX82xX.8944 
w= 10 = 0 = 5.5 lb. 


The following table has been calculated from the preceding 
formula. This table gives the weight that must be added 
to a square foot of roof covering to provide for the weight of 
the principals, or trusses. The slope of the roof is in each 
case given in terms of its “‘pitch,’’ which is here taken as 
the ratio of rise to span. Thus, a $ pitch is such a slope that 
the rise of the truss is one-third the span of the truss. 


LOADS IN STRUCTURES 


58 


OF'9 b0'9 co'G | SZZ hb | OSO'F | FFL | 98TO'S| O08 

$0°9 02'S 0gs | 809% | Ges'e | 9Ec's | 60GS'z| GZ 

69'S LE°¢ 66% | Shor | OON'S | SzE'E | E897! O02 

veg €0°9 89> | LL6E | SLES | OTE | SSIgs) gg 

86'P OL'P Lev | GIZ’s | OST'S | 216s | S4tES| O09 

oOF 98°F 90°F | LHP'S | G26S | HOLS | TOST'S| g¢ ee any oe oe 2 COST 
LF c0'P PLE | G8I'ES | 00LS | 96'S | FZI0'S| OG 

16'S 69'S eve | L166 | SLPS | 88S | LFPST| GPF 

9c°¢ cee GUS | 699% | OSS'S | O80 | OLZ9T| OF 

06'S 60'S 18% | 988% | S20 | SLZ8'T | S609 | gE 

S8'% 89'S OS'S | Ié1@ | OOST | F99T | 9TFPET| OF | 

LQ°F Z0'F bLZS | S8lE | O0OLS | 96F'S | FZIO'SZ! OS |) 

80°F 08's ros | GOO'S | OSG'S | LEES | 9006T| Gz 

62'S | 8s 68§ | 868% | OOPS | SIZS | S88ZT| OL 

9¢o°E ges oS | gos | OSSS | O8OS | OLLO'T ¢9 

oes el'é 16% | GZS | COOLS | IFET | Zg9C'T 09 

80'S 16% OLS | 862° | OS6T | GOST | FESFT| g¢ ee a oer ae poom 
C8 89'S OS'S | IZtS | OOST | FOOT | 9TFET! OS 

19% 9F'% 62'S | SFET | OSOT | SSS | 86201) Ft 

LEG bo 80° | 89ZT | OOST | Z8ET | OSITT| OF 

ELS 10% LST | 169°C | OSET | SST | Z900'T cE 

06'T 6LT 99'T | FIPT | OOST | GOTT | HF68 0€ |. 

Wud $ UU | [MU F YU F MW F Yd F UW T SSHIL JO JozoVIVyy) 
7 79297 
uedg er 


aoRjng jJooy jo yooy orenbg sed spunog ERTS yam 


SassnuL fOOU AO LHOIGM ALVWIXOUddV 


LOADS IN STRUCTURES 
WEIGHT OF BUILDING MATERIALS PER CUBIC FOOT 


59 


Name of Material 


Average Weight 


Pounds Pounds 
per per 
Cubic Inch} Cubic Foot 
Asphalt-pavement eeope an. Acree 130 
Bluestone. . ae 160 
Brick, best pressed. . 150 
Brick, common and hard. r25 
Brick, paving. Ae 150 
Brick, soft, inferior. . 100 
Brickwork, in lime mortar (average) 120 
Brickwork, i in cement mortar (aver- 13h 
Brickwork, pressed brick, thin joints 140 
Cement, Portland, packed. . poe tay, A 100 to 120 
Cement, natural, packed. . 75 to 95 
Concrete, LE, ee as 105 
Concrete, eden Bee Sas 140 
Concrete, slag. . 135 
Concrete, stone. . ee 140 
Concrete, reinforced (average). a 150 
‘Earth, dry and loose.. 72 to 80 
Earth, dry and moderately rammed 90 to 100 
Be ipetriok: one Seed 2c Re 150 
Granite: 2655. 165 to 170 
Gravel.. eS LZ tor1 25 
Iron, cast. . Save Sis: > .260 450 
Iron, wrought. . 207 480 
Limestone. . 146 to 168 
Marble. . -168 
Masonry, squared granite. or lime- 
stone. 165 
Masonry, granite or limestone rubble 150 
Masonry, granite or limestone dry 
LUDDIey, esis oy . :- Nfs 138 
Masonry, sandstone. . 145 
Mineral wool. . j 12 
Mortar, hardened. . 90 to 100 
Prick, ground, loose, ‘or small 
ee . 53 
Ouie eee ground, thoroughly 
shaken. sekere 75 
Sand, pure quartz, dry.. 90 to 106 


60 LOADS IN STRUCTURES 


WEIGHT OF BUILDING MATERIALS PER CUBIC FOOT 


(Continued) 


Average Weight 


Name of Material 


Pounds Pounds 
er Per 
Cubic Inch} Cubic Foot 
Sandstone, chew hed 139 to 151 
Slate.. Mins 160 to 180 
Snow, fresh fallen. 5 to 12 
Steel, structural. . .283 489.6 
Terra cotta. 110 
2 os Saba masonry work. . 112 
Tile. . Suita Nite 110 to 120 


WEIGHT OF BUILDING MATERIALS PER SQUARE FOOT 


Average 
Weight 
Name of Material Pounds 
per 
Square 
Foot 
SINGE6 aca 2.91 
NoraSisiwi 2.36 
a ce ee ae 
: : : Oy Secas ae 54 
Corrugated (24-in.) galvanized iron Nae Bat >. 127 
No: 26 <<. .99 
NO 275.6 .93 
NG 28.6356 .86 
Corrugated galvanized iron, No. 20, severaae 
Rona gs of side lap, unboarded. 24 
per roofing, 16-oz., standing seam. 14 
Reh and pitch, without bape: ee 3 
tell 4 in. thick. . 12 
Hemlock sheathing, ‘1 in. thick. . 2 
Lead, about 4 in. thick.. 6 to8 
Lath-and- plaster ceiling (ordinary) .. Seats 6 to 8 
Mackite, 1 in. thick, with plaster. . Ais PSs 10 


LOADS IN STRUCTURES 


61 


WEIGHT OF BUILDING MATERIALS PER SQUARE FOOT 


(Continued) 
Average 
Weight 
c Pounds 
Name of Material Per 
uare 
oot 
Neponset roofing felt, two layers. . ete faye 4 
Spruce sheathing, 1 in. thick.. Sphate.e te eG 2 
¢ in. eid ey 1.81 
¥e,in. thick. . bree 2.71 
?} SRG oss oe 3.62 
Slate, single thickness{ $ in. thick...... 5.43 
in. thick 7.25 
$ in. thick. . 9.06 
# in. thick.. : ol «10:87 
hingles, common, 6 in. X 18 in., 5 in. ‘to weather. 2 
Sruene of glass, ts to }in., including f frame....| 4to10 
Slag roof, four-ply.. Ate 4 
Steel roofing, standing seam. 1 
Tiles, Spanish, 144 in. X 103 i in., 7 hj in. to ‘weather 84 
Tiles, plain, 104 in. x6} in. xs in., ee in. to 
weather. ; 18 
White-pine “sheathing, 1 in. thick. . 2 
Yellow-pine sheathing, 1 in. thick. . 3 
Gravel roof and four-ply felt.. 54 
Gravel roof and five-ply felt. . : 6 
peers, three-ply ready (asphalt, rubberoid, reat 
3 to 
Bodie ‘wooden, ‘with 12- to 16- ft. span. Ay 2 
Chestnut or maple sheathing, 1 in. thick........ 4 
Ash, hickory, or oak anest eet 1 in. thick.. 5 
Sheet iron, ae in. apaaen Seas 3 
Thatch.. ; dcusvera eats + kik 6.5 


LIVE LOADS 


If the live load consists of heavy material, such as mer- 
chandise in a warehouse, and if the amount of this material 
and its location are known, the live load is usually calculated 
from the table of weights of materials given on pages 64-81. 
If, however, the structure is to be used as a dwelling, a hotel, 


62 LOADS IN STRUCTURES 


or a warehouse for general merchandise the character of 
which is yet unknown, it is customary to assume the live 
load to be a certain number of pounds per square foot of 
floor space and to design the structure for this load. The 
following table gives the loads per square foot of aoe space 
often employed: 


LIVE LOADS PER SQUARE FOOT IN BUILDINGS 


Character of Building © Pounds 
Dwellings. . ¢ 8b bea ASG ER otis. che 70 
Offices. . Loe 7 
Hotels and apartment houses... ye 7 
Theaters. . Sashes 120 
Churches. . eek 120 
Ballrooms and drill ‘halls. rae 120: 
Factories...... PRR from 150 up 
Warehouses...........0sscsecceeee from 150 to 250 up 


The load of 70 lb. will probably never be realized in 
dwellings; but inasmuch as acity house may at times be used 
for some purpose other than that of a dwelling, it is: not 
generally advisable to use a lighter load. In a country 
‘house, a hotel, or a building of like character, a live load 
of 40 lb. per sq. ft. of floor surface is ample for all rooms 
not used for public assembly. For assembly rooms, a live 
load of 100 1b. will be sufficient. If the desks and chairs 
are fixed, as in a schoolroom or a church, a live load of 
more than from 40 to 50 lb. will never be attained. Retail 
stores should have floors proportioned for a live load of 
100 lb. and upwards. Wholesale stores, machine shops, etc. 
should have the floors proportioned for a live,load of not 
less than 150 lb. per sq.ft. The floors of printing houses and 
binderies should be proportioned for a live load of at least 
250 lb. per sq. ft. Special provision should be made in such 
floor systems tor heavy presses, trimmers, and cutters, and 
the beams should be proportioned for twice the static load 
likely to occur from such machines. The static load in 
factories seldom exceeds from 40 to 50 Ib. per sq. ft. of floor 


LOADS IN STRUCTURES 63 


surface; therefore, in the majority of cases, a live load of 
100 Ib. is ample. The conservative rule is, in general, to 
assume loads not less than those just given, and to propor- 
tion the beams so as to avoid excessive deflection, Stiffness 
is as important a factor as strength. 


ALLOWABLE LIVE LOADS ON FLOORS IN DIFFERENT 
CITIES 


Pounds per Square Foot 


Character of Building 


New : Phila- 
Vani Chicago delphia Boston 


Gerretse for gs as- 

semb 

Sulidings “for ‘ordinary 
stores, light manufac- 
turing, and light storage| 120 100 120 

Dwellings, apartment 
houses, tenement houses, 
and lodging houses. . 60 40 70 50 

Office buildings, first floor} 150 100 100 100 

Office eae above 


90 100 120 150 


first floo 
Public buildings. ‘except 


schools. . 150 
Roofs, pitch less than 20°. . 50 25 30 25* 
Roofs, pitch more than 20° 30 25 30 25* 
Schools or places of in- 

struction. 75 80 


Stables or carriage ‘houses 
less than 500 sq. ft. in 
Grea .. 75 40 

Stables or carriage “houses 
more than 500 sq. ft. 
in area . 75 100 

Stores for heavy ‘materials, 
Bercnoes, and _fac- 

sake tdetec a) DO 150 250 

Sidewalks. Shea otal COU. 


NOTE.~In the table the values given for roofs are for snow and 
wind loads. Inthe last column, the roof loads marked with the aste- 
risk (*) do not include the wind load; the building laws of Boston 
require that a proper allowance for the wind load exerting a presfure 
of 30 lb. persq. ft. of vertical surface shall be made in designing roofs- 


64 LOADS IN STRUCTURES 


In proportioning the live loads on floors, the engineer 
cannot always exercise his own judgment, for if the building 
is to be erected in a large city, the live load must comply 
with the building laws. As such laws are not uniform in the 
several cities the table on page 63 is given to show the stip- 
ulated live loads in the four largest cities in the United 
States. 


. WEIGHTS OF MERCHANDISE, IN BULK, FOR CALCU- 
LATING LIVE LOADS 


Measure- Bi dere aog 
ments eights 
; g . 9 ® 
Name of Material Ee. 23 a3 ahsisze 
oj Px} 2 pS | 
8 Oe, 5 ee 5 oe Bom 
ay Oo ay A 
Cotton, etc.: 
Bale of commercial cotton | 8.10/44.20| 515 64 | 12 
Bale of.compressed cotton | 4.10/21.60) 5 134 | 25 
Bale of American Cotton 
Co.. -| 4.00/11.00|} 263 66 | 24 
Bale of Planters Compress 
Co. . .| 2.30] 7.20) 254 | 110 | 35 
Bale of jute... Seite ai gre Gia alata 2.40} 9.90} 300 | 125 | 30 
Bale of jute lashings...... 2.60/10.50| 450 | 172 | 43 
Bale of manila...........| 3.20/10.90} 280 88 | 26 
Bale of hemp............| 8.70|34.70} 700 81 | 20 
Bale OF Sisal Sie, Sais ss 5.30/17.00| 400 75 | 24 


Cotton Goods: 


Bale of unbleached jeans. .| 4.00/12.50|} 300 75 | 24 
Piece of duck. . -| 1.10) 2.30) 75 33 
Bale of brown sheetings. .| 3.60)10.10) 235 65 | 23 
Case of bleached sora wae 4.80)11.40) 330 69 | 29 
Case of quilts... 2 . | 7.20)19.00} 295 41 16 
Bale of print cloths....... 4, 9.30) 175 44] 19 
Case of prints............ 4.50/13 420 93 | 31 
Bale of ebitignte 3.30) 8.80) 325 99 | 37 
Burlaps.... 130 30 
cle bagging. . 1.40} 5.30) 100 71} 19 
Wheat in bags. . 4.20) 4.20) 165 39 ; 39 
Flour in barrels on side. 4.10) 5.40) 218 53 | 40 


: | 


LOADS IN STRUCTURES 


WEIGHTS OF MERCHANDISE, IN BULK, FOR CAL- 
CULATING LIVE LOADS—(Continued) 


65 


Measure- Approximate 
ments eights 
N f Material g, |¥ go (8 
ateri “ad 
tu “3 ° 3 
Bam | BO") SE |B) Om 
aS a |s 
Grain—Continued 
Flour in barrels on end. 3.10) 7:10) 218; 70] 31 
COPED AGS sa. oc reste hoc .60} 3.60; 112} 31] 31 
Corn meal in barrels.....| 3.70} 5.90} 218; 59 | 37 
Oats inibags: «. .biccuaans 3.30} 3.60} 96; 29 | 27 
Bale of clover hay........ 5.00/20.00} 284; 57 | 14 
Bale of clover Beye snlenal 
compressed . 12751 -6.25|) 225). Flo} 24 
Bale Gt Straw aces oceoen es 1276) 3225) 100) “52°19 
Bale of tow . Bi lousicte Selo ES TOL Oncol LON “SOulnace 
Bale of excelsior ero cel Lt Olcoseol LOO -S7-41) ao 
Rags in Bales: 
gs on Be linen. 8.50/39.50} 910] 107 | 23 
White cotton.. Suse Oe 140700! “7151 “78-) “18 
ISTOWit COLLON sic soo oes: 7.60/30.00} 442; 58 15 
Paper ee: ...-| 7.50;384.00| 507; 68 15 
ee A ee 16.00\65.00} 450; 28 Z 
oolen Cee 7.50/30.00} 600} 80 | 20 
He ore 2.80/11.00} 400; 143 | 36 
Wool: 
Bale, East India. . 3.00)/12.00} 340] 113 | 28 
Bale, Australian. . ..-| 5.80/22.00] 385; 66] 15 
Bale, South American. ..-| 7.00/34.00/1,000;} 143 | 29 
Bale, Oregon. ...| 6.90133.00] 482; 70 | 15 
Bale, Cali ifornia. 7.50133.00} 550/ 73 17 
Bag of woo | ie 5.00/30.00} 200! 40 7 
Sack of scoured wool. . 5 
Woolen Goods: 
Case of flannels. . se aac] 1.00]/12..70| - 220) 40°) 17 
Case of flannels, heavy .--| 7.10115.20} 330; 46/1 22 
Case of dress goods....... 5.50/22.00) 460; 84] 21 
Case of cassimeres........ 10.50/28.00} 550] 52 | 20 
of underwear........ .30/21.00} 350} 48 | 17 
Case of blankets. . .. .110.30/385.001 450) 44); 13 
Case of horse blankets... . 4.00)14.00} 250 18 


66 


LOADS IN STRUCTURES © 


Were OF MERCHANDISE, IN BULK, FOR CAL- 
CULATING LIVE LOADS—(C ontinued) 


Measure- ND Enea 
ments eights 
g, i2 g 
Name of Material Bo jf. .| 9 |Be|2. 
<iao 8:5 9 Ss |wss mAs 
Boe ee| Ce leaelede 
Ba" |gO™| Me | eam 20% 
— AY AY 
Miscellaneous: 
Box of tins ie ae 16 139) 51 | 278 
Crate of crockery......... 9.9 |39.6 | 1,600) 162 40 
Casket of crockery. . 13.4 |42.5 600; 45 14 
Bale of leather........... 728) /9I222 190} 26 16 
Bale of goat skins........j11.2 |16.7 300| 27 18 
Bale of raw hides........| 6.0 |30.0 400! 67 13 
en ot raw hides, com- 
ahtitathioekes Wheres 6.0 |30.0 700! 117 23 
Bale of sole leather....... 12.6 | 8.9 200} 16 22 
Barrel of granulated sugar | 3.0 | 7.5 317| 106 42 
Barrel of brown sugar. 3.0 | 7.5 339| 113 45 
Hogshead of bleaching 
powder. . .111.8 |89.2 | 1,200} 102 31 
Hogshead of soda ash . 10.8 |29.2 | 1,800) 167 62 
Box of indigo. . te 3.0 | 9.0 385| 128 43 
Box of sumac....... hea Gots4. 4 160} 100 39 
Caustic. soda in iron drum.' 4.3 | 6.8 600} 140 88 
Barrel of starch..... eis. oc 11055 250! 83 24 
Barrel of pearl alum...... 3.0 |10.5 350] 117 33 
Box of extract logwood ..} 1.1 8 57| 52 71 
Barrel of lime............ 3.6 | 4.5 225) 63 a 
Barrel of Portland cement. 3.8 376 aa 
Vis, 
Barrel of natural cement . 3.8 282). = 
80 
Barrel of slag cement..... 3.8 330 3 
Barrel of English visgieer 
cement. 3.8 | 5.5 400} 105 73 
Barrel of plaster. . 3.7 | 6.1 325] 88 53 
Barrel of rosin. Jidetea dpe Oe OO 430} 148 48 
Barrel of lard oil......... 4.3 {12.3 422} 98 34 
Books in library... 30 
134 
Crowd of mean.) 2520.08. to 
157 


67 


LOADS IN STRUCTURES 


LEP ** (OUIZ O8F “qyBnorM & Mogae | 
8S TI ueayssun J, OcF * *4S€0 ‘uoly 
Sor “* ‘ULL OOF‘T ee Ree ee Oh de “wan pH] 
06h * *[294S Rete ioe ee -psoururey aid * Bony 
e¢c9 ew ‘IOATIS $02'T ta hat “yeres $Z% Kee 9 v3 
LS9 : ‘ paxourey “JOATIS ggg oe Bare S OP as! 
eel @ 9 9-03 290 ‘OITM ‘umnuly}eld ZSG Cy ek ee an te ae ee re "4880 170000) 
STP'T ‘*peyjor ‘umutyeTg 09¢ - *47eqQo°) 
O22'T --posouruiey “wnueld OCP "wUNTWOIY 
8hg “PAIN 9FS ‘ "ezuolg 
88S * -ummuepq AoW bZS “-potfos ‘ssvig 
9F8 a 209 ye ‘AIMoIOW £0¢E "480 ‘ssvig 
66F ‘osouvsueyy E19 “YyNusig 
60 p'4: lace * ‘UINISOUse SIP wieasee Bey eee | 
Zz oe “qsB9 pets sentcien ‘pray LOT we eee ae ee ‘uInuIUnyy 
spunog spunod 
yooy sIqng yOOy oIqn 
; od faire A [eq jo owen Sod Sito [ee Jo owey 
eseIDAY aBeIIAY 


STVLAW SNOTAVA AO SLHOIGM 


LOADS IN STRUCTURES 


62 ee ee ee “pros ‘jauueo ‘TeOD 06 04 (ay *9S00 ‘pueytog ‘2Uata) 
£8 veteeeessss*oros ‘snourUingziq ‘Te0D AGL-OVOOT AS ive sri “‘purlywod ‘yusuled 
ec ae  pexes ‘snourumyiq ‘Teo) LOT pres _xBlog 
og eee ‘uaxo1q ‘snourunyiq * ‘Te0d cg eer ee * JOBE] ‘100g 
7 rn pros ‘oqroesryque ‘Teo. ISI Pe Oe ee ae er ee “ayeseg 
8g uexeys Ajeyese pour ‘oqroesyjzue & Te0d 08 rec gia ‘qyeydsy 
$C “waxorq, ‘oqfoeryue Teo") OF irae Goals Vals ieamips oath oie soysy 
Gg sees tees ‘JoyUYD Z61 yim "*"£are4s ‘sojsoqsy 
61T eee eco ee oee hee ee 8103300 ‘ARID SIP * ee ee wenn ‘AuowUy 

OST 04 A eee eo cececoescece “<TeurIpso ‘heID 9¢ - “quae ) 10d RZ ‘eTuOW Wy 

RRS eaten "“uaxeYs [JPM “snp eso ouroryD 6 6F "poom ‘JoyooTy 

SI re cereee ‘ourd wol} yeooreyd 9°6F +m S52 *-ureis ‘ [OYoOol[y 

IZ ce ae | > tte) wo} yeoorey) ZG eee ee we eee Sah epaderell ‘joyooly 

82 ewe eo eo oe Seep tistey uwtory yeooreyg 69T “* “ee we ** MOTTOA ‘leyseqely 

$e trees esceeee se eno WOsy TeOOlEYy) IZI Trrctsterss ress *saqnys ‘ayseqery 

Z9 re eusedureyy GIL ee ‘oLmnydms ‘PPy 

9CI oo Oe ewe ce ee te Bete eeecrene aed) 16 ‘‘oroydsoyd * (PPV 

SIOSGG [09884 4 Be ee 8 + Sg ‘ZeIs ‘quaures) 9L ‘ “OLtqia ‘PPV 

00T 9108 |""°*'’*'** °° ot pexoed ‘Beys ‘yuouted PO ee do as “(u0TyoospAy) oryeLmur ‘PPV 

6901 Gp |" °° °° ******‘asooy ‘TemnzeU ‘yuaTIE, 6 “opony ‘PHY 

C6 01 CL ***pexyoed ‘Temyzeu ‘yueuled 99 ‘ooo ‘phy 
spunog spunodg 

yoo s1qng yoo s1qnag 

aor 5 Pe nl [elsozeyy Jo owen jeer Mage TeHoyeW Jo oureNy 

aseloAy aseloAy 


STVIMALVA SQOANVTIAOSIN AO SLHOIDM AOVAAAV 


69 


LOADS IN STRUCTURES 


eeoeo ee ees 


Sr Nain teehee ae sy LL CPO LO! LS OOr 
sone ess ease 2) 2 PSRs SUAS 
Trrrrtescceseceses Burgi ‘sadeg 
‘Buiddvim ‘rodeg 
** 1 *3J00q ‘pesopuayeosedns Jodeg 
eee . ee eeeee *preoqMelys ‘rodeg 
eoeoe ee eeee ov eeeee "smou ‘sodeq 
“eplueul ‘todeg 
a ee ed PrIvOg-JOY} eI] ‘rode 
eee re eeee **300q, peleapusyeo ‘rodeg 
*(971ZeUWIaY) UOIT 4jos ‘210 

" * @}HeuseUT) uost prey ‘aI0 
sislerereleWieie ste) sie si4igiest e040 oyeu ‘HO 
eoeere reer eer ev eeeese ourjuadin3 ‘TO 
tneis sis esncieserw aa $610 bi 6:0 “QATTO ‘10 
"***peesuyy ‘TO 
cece “ToqN 
eevee eon sce ee er re ee eeeese -eyyyden 
Freee rere ee eter er cece es -oUOASTTTAL 
(bie: SIN A aU N ioe etelacuienele gedeteie Gis 4/6 014 BOT 
eeeeveea oe ee * O1[Se yl 
**pouroyeo ‘ayisouse yy 
cooere ee ee eevee ‘ayeuogieo ‘vIsousey[ 
Trees ceessssapid ur ‘afos ‘s0y}8eT 
40D 6:8. 6 6-8 8 OO. 8:e 816 *AIOAT 
eee ve eveeorsrerveeeeeeeeene *sSB[SUIST 


eeeevovecs Te ce eccesssssaqqns Btpuy 


eee vere eoeee se ee ceeeers ‘90T 


ee @ owe Sere Eee sisie)2 2° i*/° sO PUO TA UsO LH 


eeere eo eo ne ee ee oe 


eeeeore ee oe oe oe oe 


oe ee oe ee esr eeee 


ee ee ew ores 


OOT °F 06 


66 97 28 
08 93 ZL 


rts wom a te ty eee Ae Se et a ho ik * ‘ato eq BUlOH 


‘ummsd Ax) 

eee ee eee **eyoied %44N+) 

0 Ge O00 ea. 6.07 Cie) Ble 01 @.6 prjos ‘rapmoduns) 
656 5), 6:8) 50 10.6 0: 4 08'S: 6 “uaxeys ‘Iopmodunr) 


* *@SOO] ‘sepmoduns) 


PRR ee oe oe “* “yeyeur uns) 


: tees eeeeeeeseorqere UNS 
a6 rele. Greveisco'e) 9.6) 0i8iie9'='4ie 9u0}spulss) 
O36" 60 S708 668, mt 9SOO0] ut ‘sstour) 
eee vere ee ee eeeee uOUTUIOS ‘sstaur) 
eee . eee ee quit ‘sseyy 
* ee uouwlul09 ‘SSB]L) 
ce cece cece ce veces SCE Nt i 
eee CW ek ak . ‘+ redsppy 
SiS) 6 646 6G 16, O'S Rise’ do 86 oe Ss ‘1943 
Soe: CIEE goa 
. "*phnur asuap oF!] ‘qyed 
- pr Burmoy | ‘qos ayy ‘Yweq 
"AjozeI9 pout 
powues “UIBO] uOUIWIOD ‘qyeg 
*uayeVys ‘UIBO] UOUIUWIOD ‘yyseg 
eri: asoo] ‘UIBOT UOUTUIOD ‘YWIeW 
wie isieiecsless\alvie(ainis's!siefs uaul JO PMOID 
eae "SUfOHS Uy ‘ured u03407¢ 
cece ccoee *“umpunsoy 
. ei oan rad. sites Undo 


0:6, 6 OS 60710106: 0, Bed 0. 5's. asOo] ‘ayog 


70 


LOADS.IN STRUCTURES 


Z9 es ee @e “SOUL M, 90L O} 06 D6 BaD ~ pees 
18 "* guoqeyey M. idk: Sac aie “ peLtp- T1984 ‘eppuy 250M ‘4T8S 
89 o> * +p UL A CP see eee *OSIBOO “4TeS 
0g 0} 0Z *qeod Io ‘ ty I&t see ee . * ** “ee eewe *Ioyodyfes 
OLT se eeees ely, PZL * *9U04SU944OR 
e9 nek ess eT oP -- erueyy ‘odor 
gg "xo JO > outed ‘MOTB IL, 69 . “UTSO 
ISt yoorq ‘oye y CoOL “ond | ‘wourUt0S ‘2y7en. 
CSI ‘Inydms Lg **eu04s soruIng 
OOT Be lst bie Se ats S-s' se De le ae Ole, resnsg OLT SA Rae *AtAYdIOg 
6S **to0euLIEdS OFT RES eS ee *osequini[d 
LEP sourz Io ‘raqjeds 08 see eee meee 4880 ‘sq3eq jo J04SeId 
OLT *-guo0ysde0g aL ee ee "youd 
ZI Cov es ce eresssscvesene uwOUIWIOO *I0G OIL * *snroydsoyd 
PLI SESE RA ren rp ae **-o9RIG a eo umndo130g 
ZOL ff ee. . eenee *saTeys og O} 0z eer ee ew ee eee posseidur0o ‘AIp ‘qeog 
spunog spunog 
yoo o1qnag 4yOO,F oIqn 
eeiea [evozey jo owen : oS aioe . [eHoyeWy JO owen 
eseIIAY aBvIOAY 


(panu1iu0)—STVIMALVW SNOANVTTAOSIN AO SLHOIAM AOVUAAAV 


LOADS IN STRUCTURES 


ge “surazs UFTA sodvir) OF ; *SpOIIBY) 
re ; ‘ saLIteqasoor) OF é “yopout un ‘odnopeiue?) 
CF \(paasuly) peasxe|y SF ; "pees AreuB) 
gg “UO4yNUT JO IBY OF “oseqqey 
6g “S80y Jo ey 6S Pe Regd tie 
8g goed 20 78a 68 Sine * “qeoq Mong 
89 “S337 og oe ae = paos UIOD-WOO1 Eg 
Ze ‘*sque1n) Il sees "sseiZ aulolg 
62 ““sq1sFoquvs) 91 jana _Uelg 
CZ Soren or Gaicre * * paast0}}0/) Il so “paas sseis-on|g 
ge ay eey areas “ peajoqun ‘Teaut WIOD rE “‘sottsoqenig 
Lg Ct ey ee Ce De ge Oa *pestoq ‘jeour AOD Se * “SoLlieq soe 
CP ea *porpeys ‘u10d 0s - pos poomsvssog 
gg aan toa ‘paysnyun ‘qoo a4} UO UO) bP . "S}00G 
9¢ "rte tts" paysny ‘qoo ey} wo W105 19 : : * *XBMSoog 
8P ** paas JOAO[) 1g -“poqlous ‘s0nme5 ‘subg 
$9 a ‘Iaplg SP “our ‘suvog 
es Srey aioe tcrets ey. ge wh “Age 
ep eeeeeeeeee seganuasayy Ze * "pees addy 
OF see ee cece eeeeees ssatraUy 0Z “poup ‘sojddy 
og e aceiecaie arent * sao) ge ‘sojddy 
spunog spunodg 
yoo, a1qng [elioyep Jo oweN yoo s1qng [eHoyeR JO owen 
sod 7310 \\ Jad 74310 M 


SLOnNdOdd WUVA JO SLHSIGM AOVAAAV 


LOADS IN STRUCTURES 


72 


ee qoo 9y4 uo ‘uroodog 6S ed ‘°° "DIeyT 
9c ee u1oodog 08 Ce toyse[d puey 
4 ee eee oe SEUSS So ag 
QP ereeene seog ZS eee cw eo ee ee eens ****sspis wosuyO 
68 TIAN LS eae 91 ste tesesesss “Daas SsBis-aAT WeLeIT 
SI dae : ‘synuveg CF se eres*-OZTeur IO ‘UIOD UeIpUy 
9% FE eS ‘pojeod pue pep ‘sayoved 68 Bnet ee "pees sseis ueiiesunyy 
OF ata se ayitt ‘souveg OF coe eee cece se * “ysIpesasioH] 
8g eee **sdrusieg 16 ee *AQUOPT 
92 = Br tac ge 6h ee Autwoyy 
he " "pees Ssvis-psreyoIO OF ee sqynu ALIOYOIG 
CP “SU ce eee ere ee ee ee eer ee ee ee eee poos dwey 
9Z *s72@0 9 Bae 2 4les s 9.906 09-08: - BELIOSSRI sre Ef 
tZ ee ee --pavisny oF ee MOU ul ‘JaAO]O ‘ABH 
8Z Be sy ‘pavcuseq asounde ft TTA ‘74 setters -Dagsarduro0d ‘JOA0]O ‘AeYy 
OF eee eevee ee THN FI eee ee we weeeree “sereq ur *IQAO ) ‘ACT 
¢9 “ATEN 98°9€ - ae “SeTeq passe. u09 
Ze ge oak emanp Cin -21qn0p eoupuryas ul ‘eyyeyre ‘ACE 
se “@STB09 “ssul]PPlAL €9°SS ‘ *sayeq pesseidur0o 
Lg “Teo “ejqnop semMsuejzoes Ul ‘eyyeyte ‘ACE 
x4 “MPN FS PT OF SST)" ig aps ul “exreste ‘ABH 
$9 * -QUIrT oF sete eeeee "SBABNE) 

spunodg spunog 
yOOy 9Iqny TeboyeW JO owen yoo oIqng TeBoqyeyT JO oureNy 
Jod 1y310 Jod Jy 310M 


(panuuuod)—SLondodd WAVA dO SLHOIGM AOVAAAV 


LOADS IN STRUCTURES 


“eo 
- -sqnuyeM 


er +++ pags SSBIB-IOAT2A 


‘sdiuin 

* *sooPeUIOL, 

“ paas Ay,OULL 
“sole 


Bios. 91d 8 pee (050.14) 616 85.050 ‘poos auvo-se8NS 


“Sottaq MeI}S 


Sar gs sesianeeses (agg ue Bens 


(doy ye 


ae ae Siete seis S5eisi eis OTT wotetisel, 
e «6, @6 sre ee cece ss saggms ‘S90}810g 
oe eens stresses soqruMm ‘se0zepOd 


—— 


2 Sere oy = 


LOADS IN STRUCTURES 


74 


*soTqe} os 


94} UI WOATS 


SON[LA OY] 0} Poppe oq ISNuUI pooM 94} UT peurejuOS o1Njslou Jo oSejuao10d oyY “*TS-64 S9Sed uo spoom ueI[eIIsnYy 
pue ourddryiyg 10} sojqe} oyeredeas 90g ‘*poom uae13 10j ou pue poom Aip AIdA IO} ae S2TqQe} BuUIMO]T[O] 943 


pue Sty} Ul Sz4SsTAM oY L 


*SUIEIUOD FI 9IN}S[OU! yo JUNOUIe Oy} UO AJasIe] Spusdap poomM jo }YSIOM OY L—ALON 


CZ eek MSN VE Bow Sera SSeS SOUL IOREE ‘repedg LP eowoeve ee ee ee ee wee eens q90MS ‘qolig 

&Z% Riese ASS SACS tS Fee OURO. \TEDery ce Te a es Be ele Te per ‘yorg 

GS eee ee one eee “OPITTM erusOHTe) ‘repag 1g eeeoee ee eevee "OUI M 10 ‘roded ‘yollg 

GZ Sil ed I SASS Tae ate cea ad 88 yf edjeyeg 91 Pr ee ee ee ee) ereqeyyogd 

LZ ie. & 0. 6 6 4 8 eee ‘ueeq uevIpUy Io ‘edjeyeg aP ee eo e@ceeeneeeseseee yooog 

9¢ SENSI SIE UH ae MES BP RE oy Ew 1g eeeeeeeoe ere eeeesreee ee eee 901} Avg 

cE eoenrevoeeee “91OWUIBOAS Io ‘poomuoyyng 8% eevee see nese ee era ere eee eee poomsseg 

CZ beeen rere eeee ee ees saniraqaing 2z a hee We re Se ala are ooqueg 

iste) Rie h SSR = se A Se SS. oes SOOT OMe LZ Ua eh 2s See [9 8-8: 8 ay SRE ee 

1% cr eseecereseese ss saggms ‘oAoHOng 6g eo ee ee eros oo ee sees reese ou M ‘Sy 

8% ARAN, Sas SNR OL. 8 SORE), OOM TEL Se eeneeee re eoee ree eee ‘por ‘ysy 

LS ore ersneeo ee ee eee * *youely ‘poomxog cE eaeesreere ee ee oe eeeeeeee uoso1l ‘UsV 

es se ee Stay Es 1) 5 9-8 OTT ‘poomxog 6 eoerneeroereeaoeee eer eeeeree ee usool ‘UsV 

£9 . oe ee eree *por ‘uelizelg ‘poomxog PP . ° Pa ee ee ee en]{q ‘UsyV 

9% eee ‘o[deu poAavoa]-yse Io ‘rape xog 6 eseeeeeweeoeeeen ee eeeeeees yoryq ‘UsV 

69 o4 EP oes ee we eeseee (9014 I9A9J) wns anid 61 eee oe ee ere ew ee ewww ee ee “SYA soqiy 

CF “ee ee eo werner (poomuolt) yooeq, eng Z sesevceenvnenee ee eeeen ee eeeeeeer eddy 

OP BLY ER are tae ast eee 6 ok SONA: Osteh oP exe sveseveeveeeeeeverveoeeeeeee Jeply 
spunog spunodg 
OO dIqng yoo. oIqn 

yod Fut Ah eel jo wey en Pichtok eel JO oweN 

esvIBAY aBVIIAY 


Aud ‘SdqOOM AO LHOIGM 


LOADS IN STRUCTURES 


a eres CR eee 
; ese 

eet hai fice of tf. eeeeoe 
eee ecceoe ‘OJOIIg Ooreon 
“-ysruedg ouluse 

wan "9013 enyso 


eed ‘AIOYOUT 
“gnu 1 39900ut ‘ALOMOY 


* *yooyweyy 


"uO MEPL 
pote? (your uvonowy) soeseunnoey] 
cheat *700MS ‘une) 
‘mos ‘ums 

* "0033.00 ‘uins) 

* -qreayuees£) 

Sa) may oie) “ouTyM ‘It 

Nalaou B<e) ‘per ‘It 

“ptuso} fey Jo ‘pad ‘It 

‘JOATIS yeois ‘It 

“mresyeq ‘Il 


pitas: “**901} Joq TI y 
oe Srreesss BUT ‘WTA 
“OTT “WA 

‘ Areddys ‘Wy 

“ytoo ‘WTA 

**90r4 Joply 


: “oureseq 
genoa 


SLD uy ASAE e! Mise ele eis \616. 5) Ripe preq ‘ssoid Ag 


‘aa4} Jaquinond 
“ORIG ‘poomuozj}0) 
* *poomuo309 

me) (ere Leele@) 

‘poom e000) 
“uoINID 
‘‘uidexury) 
“qnuysey) 

“yoeiq. PIM ‘Artay) 
“MOTOA ‘repag 
“pyran teped 


ee evece “(@rya socure) (o7ty ‘repedg 


“ysod 10 ‘az “Iepad 


ee Oe ee pes ‘repad 


"*psojfxQ Hog ‘seped 
“oursoyed ‘repeg 
‘rodrunt ‘repad 

cee eeee ‘ueipuy ‘repag 


LOADS IN STRUCTURES 


76 


Ge tier. payee eh (seqeyg peyUuy ur9yseq Ze "*9JOS JO ‘JOATIS ‘aide 
pue, Tesj099-qy10N) bell ta Sor mide) 8E “por ‘ojdey 
FP **ystuedg * mise) 0€ **UO0SII0O ‘aide 
CP ‘pel ‘VO ee BE Ri ‘Auesoyeyyy 
0s ysod ‘yeVO gS vceneaea ‘Auesoyey 
FP uid ‘yeOQ Ze * UBOrXe A, ‘Auesoyey 
6¢ (89383 cv *AuesoyeW 
wsayanog oun t ur span oATT mike) Iv “20ye 
Tg » PRIORI ‘OAT mite) &h " “SyOIpEN 
OF "MOO ‘120 SP Bae RICE, 2 ne ‘+ Kauoy * ‘ySno0'T 
OF “-4nuysoyp Ride) SNS eae ee Sauk MOTpOA . Jo ‘yoelq “sno07T 
oF “mq 120 88 SPURT ‘uapury 
cv ““yoeIq “AeVO €8 ‘“eyA wnUsry 
9€ “-yoeIq 3 #9 ) pax ‘AlTOqIN cv Oe] 
Lg “BIOW eh - euospeyy ‘mney 
6¢ 901} [OSSTP] OF + erusoyttes) ‘Ppaney 
LY ae ** gmbsoyy 9F BS maieasin'y ote “qore’T 
6S “JeTpoW 8& B. Sebel | 
€¢ 90d} O17Se ec ** poomaouer’y 
Ph pA 10 + eens ‘aide 1g “*** UMUINge’T 
spunodg spunog 
yoo orqn OO y dIqn 
Sod aiiae ne ool, Jo oureN : te fe agin eer], JO oweN 
eselIAy esvIoAYy 


(panunuo))—Kuad ‘SdOOM JO LHOIGM 


77 


LOADS IN STRUCTURES 


"*poom diny 
"9074 dnp 


tere sees eyopreure yy 
‘*eBIurOfITeg ‘alouredAG 
"* poomuoz3ng Io ‘aIOWUIBoAG 
: *(eIquIn[og ysHig pues 
surequnoyy _Axo0y) eu sonidg 
(S981 
poy _UTEqWON) | fh ie ‘agonidg 
eG ‘sonidg 
aie eds ine: “Gy ‘wresteq) [suis ‘aonidg 
sigue st e@MJON ‘oonidg 
"sepanod ‘gonidsg 
“oe. ‘gonidg 


Fite t ee ee sere esse ess DOOM JOTIOY ~ 


Se ene hr bree te bg ane eee oe 
She oe agna S's 0. ofa eo: 41.9 _ 8023 OUM, 
see se cens *oyoeiqen 


‘9013 pare Io i abnak ‘rejdog .- 


--uadse q}00}3- OBIE] Jo ‘rejdog 
*u00g 


"90d} _oyeuvisaw0d 

5 “poomyo0g 

90s} WINId 

“MOBUIT 

--(surwyunoyy Geen) ipa ‘ould 
“(eIqUIN[OD YsyWg 

pue saieis oyloeg) om ‘auld 
* (S84B4S UreyseayyION 

pue yesjueg- “U3JON) - aaa Can ‘aut 

“leans ‘ould 

* -eurjored) JO ‘Feoy- qioys ‘ould 

oyid ‘ ‘auld 

“uoseIO ‘ould 

ARASH ‘ould 

*“useyqjou ‘ould 


ce sauta nlareras “eysu09H 4 so “feay- sug] ‘ ould 


* ATIOTIOT ‘oulg 
‘Liney ‘oulg 
Ce ee} ‘ueqnyg ‘auld 
- “7m ‘outg 
* ‘wOUTUTS Jad 
: *“Ieag 
vamedae> ‘oyjouryed 
* woysuryse M ‘ued 
‘pooma[ppeg 
* 901} VBURIO 
peas ‘asuelOC 
(oesalieg « oyut eemares ysHig 
Woly 4sBvog sy19eq) i ‘eo 


ee eee 


78 


LOADS IN STRUCTURES 


&% * "9033 enysof to ‘eo0n K GZ sr oq ‘Qnuye A 
os “ystuedg ‘MO 9¢ Cie Ne RC ee ee a "‘UBISIOg ‘qnuye 
6P “yong ‘MOK SP ee “* “UeITeIT ‘qnuyle Mm 
Zo . *eInie xX 9g ee “Ysijsuq ‘qnuye M 
1B “yoeiq ‘MOTIELM, ce "Tr r ttt tts SBISSBOII) ‘qnuye M 
9% " * POOMa7TY M Se. ae "eid ‘qnuye MA 
92 “eqyese M €8 *901} OUIA 
Spunod spunog 
yooy s1qng 4oo,F dIqn 
Foc vuse Ry sol], JO oWIeN i oe eis sol], JooweN 
BBPIIAY WBBIOAY 
(panuyuo))—xXad ‘SdOOM JO LHOIGM 
; 


LOADS IN STRUCTURES 


79 
WEIGHT OF PHILIPPINE WOODS, DRY 
Sati dgoalt wa 
Name of Tree Cubic eet Name of Tree Cubie poet 
Pounds Pounds 
Acle 37 EANGIR SS ee ca 44 
Amuguis 43 Lumbayao..... 35 
Apitong., Ack 22.s.+ 41 Macaasin....... 44 
Aranga.... 154 Malasontol..... 40 
Balacat.... Malugay....... 40 
Balacbacan ..... 34 Mayapis........ 25 
Bansalaguin..... 53 Molave......... 49 
BSAHUVO S355 o0eusc) « 33 Buse t.ie si 36 
Batitinan : 49 Palo Maria..... 39 
1 3 9 PR err Be 49 Sacat 37 
WSIANEAS 2: ic ids0 58 27 WaSalt 36 iis css 55 
Dungon.. 49 Supa;.... 45 
Guijo.... 43 Tanguile....... 30 
BTL oc setetsiek ccvsjes 6 47 TIMNGalos wics-4 05-0 48 
Lauan by 2s age ee §2 
WEIGHT OF AUSTRALIAN WOODS, DRY 
Ae bi beac 
Name of Tree hha a Wer 
Pounds 
Acacia dealbata (silver wattle)................ 57 
Acacia decurrens (common wattle) . 47 
Acacia implexa.. 44 
Acacia melanoxylon. (blackwood; lightwood). . 47 
Acacia mollissima (silver wattle). . 50 
Acacia pycnantha (golden wattle)... 52 
Acacia salicina. . Te weittnre Bees 48 
Araucaria cunninghamii (pine). ilasatniwWa sitet ere 45 
Aster argophyllus (musk tree)................. 40 
Banksia integrifolia (coast honeysuckle tree) . 50 
Banksia marginata (common honeysuckle tree) . 38 
Banksia serrata (heath honeysuckle tree). . : 50 
Callitris verrucosa anata PRE pine, or 
cypress). . shes ; : 43 


NOTE.—On account of the unsettled nomenclature of Australian 
woods, this table gives botanical names, with common names, so far as 


possible, in parenthesis after the botanical name. 


80 LOADS IN STRUCTURES 


WEIGHT OF AUSTRALIAN WOODS, DRY—(Continued) 


Name of Tree 


Castanaspermum australe (black bean) ........ 
Casuarina torulusa (forest oak) ............000. 
Casuarina quadravalvis (drooping she seat 
Cedrela australis (cedar). ............8.cceeees 
Ceratopetalum apetalum (coachwood) . ss wee ey 
Dacrydium cupressinum (rimu)............... 
' Dissiliaria baloghioides (teak) ....... ra re 
Dysoxylon muelleri (red bean) ................ 
gelpsonp dm amygdalina regnans (mountain ash 
ppermint tree).. ; 
ed yptus botryoides (biue’ gum, Gippsland 
mahogany, or bastard mahogany).. ee ep'dsns 
Eucalyptus corymbosa (bloodwood). .......... 
Eucalyptus corynocalyx es ced ene STS 
Eucalyptus diversicolor (karri Shee SSLa ae 2 
Eucalyptus globulus (blue gum).. Ree re 
Eucalyptus gomphocephala (tuart).. 5 
Racaoee goniocalyx (bastard box, ‘spotted 


m).. 
inapalspties haewastowa (spotted gum).. ; 
Eucalyptus a eicaaonge (canary hia ig white box, 
or gray ; 
Eucalyptus laxetioieas (slaty gum). sake arelenad. seats 
Eucalyptus leucoxylon (iron bark, red seebieeo 
or black iron bark).. ee 
Eucalyptus longifolia Gwollybute tree). Ae eT ae 
Eucalyptus maculata pote PUM) ck cows oe ha'as 
Eucalyptus marginata (jarrah)........--+-+++-- 
Eucalyptus melliodora root es DOK). cna taas a5 va 
Eucalyptus microcorys (tallow wood).. ; 
Eucalyptus obliqua (messmate, stringy ‘bark).. 
Eucalyptus pilularis (blackbutt, or flintwood). . 
Eucalyptus piperita sem coring ate stringy 
bark tree)..... ’ 
Eucalyptus resinifera (mahogan SM Slain Abe eee 
Eucalyptus robusta (swamp m oan). SEAS hICAr 
Eucalyptus rostrata (red gum tree). ate BATS 
Eucalyptus saligna (gray gum)...............: 
Eucalyptus siderophloia (iron bark). . 
.Eucalyptus sieberiana (iron bark, "gumtop 
stringy bark, mountain ashes)............... 


woke 
eight per 
Cubic Foot 


Pounds 


SI2SSS FS SSRERSSS 


LOADS IN STRUCTURES 


81. 


WEIGHT OF AUSTRALIAN WOODS, DRY—(Continued) | 


Sha 
Name of Tree Cube Rad 
Pounds 

Eucalyptus tereticarnis (flooded gum).......... 68 
Eucalyptus viminalis (manna sees tree, a Croaeroe 

gum, or white gum tree)..... 43 
Eugenia smithii (myrtle) . 57 
Exocarpus cupressiformis (native cherry ‘tree).. 50 
Fagus cunninghamii (evergreen Deech or native 

myrtle). . aly idroian a7 45 
Hakea leucoptera (water tree). 51 
Heterodendron preroliieas;: ate 53 
Lomatia fraseri . : 42 
Melalenca decussata. . 59 
Melalenca econ ape 62 
Myporum insulare.. 51 
Myrsine variabilis. . 45 
Panax murrayi (palm panax).. 22 
Pimelea microcephala. 55 
Pittosporum bicolor (white wood)... 48 
Pomaderris apetala (hazel).. 48 
Prostanthera lasianthas {mmiet tree).. 51 
ean acuminatum (native peach ‘or quan- xi 
Bantatasn persicarium (native sandalwood). cracciece 47 
Senecio bedfordii (native dogwood)........... 56 
Syncarpia laurifolia (turpentine) .............. 63 
Tristania conferta (brush or white ‘bpr).- Bratend cleats 67 
Tristania neriifolia yan a) hs oe 63 
Viminaria denudata . : 39 


SNOW AND WIND LOADS 


In calculating the weights on roofs, the snow load must be 
included. When the rise of the roof is under 12 in. per ft. 
of horizontal distance, the snow load is estimated at 12 lb. 
per sq. ft.; for roofs of a rise of more than 12 in. per ft. of 
horizontal distance, assume the snow load to be 8 lb. persa. ft. 
In northern climates, such as that of Canada, snow loads 
50% greater than the preceding should be assumed. 

The wind pressure depends on the velocity with which 
the air is moving. United States government tests have 


82 LOADS IN STRUCTURES 


determined that the pressure per square foot on a vertical 
surface is approximately represented by the formula 
p= .00492V2, 
in which p is the pressure, in pounds per square foot, of ver- 
tical surface, and V is the velocity of wind, in miles per hour. 
Careful records, extending over a period of years, show 
that the velocity of the wind seldom attains 100 mi. per hr.— 
probably not more than once in the lifetime of a structure. 
The following table was calculated by means of the pre- 
ceding formula. Though the table indicates that for 100 mi. 
an hr. the pressure per square foot is nearly 50 lb., modern 
practice often allows only 40 lb. per sq. ft. for large surfaces. 


VELOCITY AND FORCE OF WIND, IN POUNDS PER 
SQUARE FOOT, ON A VERTICAL SURFACE 


Force 
Miles | Feet | Feet oe 
Strength of Wind pr per per ae 
our |Minute | Second Square 
Foot 
Hardly perceptible........ 1 88 |. 1.47 .005 
J bl 5 364 440 ‘bas 
ust perceptible......... he : } 
4 352 5.78 .079 
Gentle breeze............. 5 440 7.33 .123 
. 10 880 14.67 .492 
Pleasant breeze........:6. 15 1,320 22.00 | 1.107 
Briskwale, . bo. « 52.w:do asnwde 20 ,760 29.33 | 1.968 
25 2,200 36.67 | 3.075 
FRIGH RVIIG Fo patel RF a ck hele 30 2,640 44.00 | 4.428 
35 3,080 51.33 | 6.027 
40 3,520 58.67 | 7.872 
Very high wind........... 45 3,960 66.00 | 9.963 
PRETO N ee an geal oars ae Se 50 4,400 73.33 | 12.300 
60 5,280 88.00 | 17.712 
Great stormyn.. ss sete ee bee's 70 6,160 | 102.67.) 24.108 
80 7,040 | 117.33 | 31.488 
Hurricane or cyclone......} 100 8,800 | 146.67 | 49.200 


Curved and flat surfaces not in a vertical plane are sub- 
jected to less pressure than flat vertical surfaces. The pres- 


LOADS IN STRUCTURES 83 


sure on a cylindrical surface is about one-half the pressure 
on a flat surface having the same 
width as the diameter of the cyl- 
inder and the same height. 

If p’, Fig. 1, represents the 
direction and strength of the 
wind pressure against the roof abc, 
it isthe normal component p that 
must be ascertained in order to 
calculate the total pressure nor- 
mal to the roof, or to determine 
the stresses in the members of a roof frame or truss. The 
other component py is acting upwards and in a direction 
parallel with the slope. The latter force is not taken into 
consideration. The wind, which is usually supposed to 
exert a horizontal pressure of 40 lb., strikes the roof at an 
angle; consequently, the pressure p, normal to the slope, is 
less than 40 lb. 

The full discussion of the relation between p’ and »p is 
somewhat more complex than the one given here, however. 
What has been said shows in a general way why »p is more 
nearly equal to »’ when a roof is steep than when a roof is 
flat. In the design of roof trusses, a horizontal wind pres- 
sure of 40 Ib. is usually assumed. 


NORMAL WIND PRESSURE FROM HORIZONTAL PRES- 
SURE OF 40 LB. PER SQ. FT. 


Fic. 1 


Horizontal Per Pitch, Wind Pressure 
Rise per Sere Aa Proportion | Normal to Slope 
poe ‘Horizontal of Rise Pounds per 
Inches to Span Square Foot 
4 18° 26’ 4 23.00 
4.8 21° 48’ 26.11 
6 26° 34’ 29.82 
8 33° 41’ 33.93 
12 45° QO’ 37.71 
16 53° 8! 39.02 
18 56° 19’ 39.33 
24 63° 26/ 1 39.75 


84 LOADS IN STRUCTURES 


All necessary data for calculating the wind pressure on a 
roof with any one of the customary pitches and a horizon- 
tal wind pressure of 40 lb. per sq. ft. are given in the 
table on page 83. 

The diagram shown in Fig. 2 facilitates the finding of the 
normal pressure » for the usual slopes and for horizontal 
wind pressures of 20, 30, and 40 lb. per sq. ft. 


70 


i Pitch Lid dee ba ae eo 
60° 


Pitch 4 a 4 
& eeure 


<0 


N 


¥ genes 


-t 7. i.e SEUU EEE ADELE ~ £02 


30 
Wind Pressure Normil to Slope of Roof in Pounds Per Square Foot 
Fic. 2 


The values of the normal pressure for a given slope and a 
horizontal wind pressure of 20, 30, or 40 lb. may be found 
as follows: Assume that the normal pressures on a roof hav- 
ing an angle with the horizontal of 40° is to be determined. 
Proceed along the horizontal line marked 40° until it inter- 
sects the curve marked 20 lb., which represents a horizontal 
wind pressure of 20 lb. The point of intersection indicates 
the normal pressure p, the value of which is found by draw- 
ing an imaginary vertical line to the base line, which is 
marked off in pounds of pressure per square foot. It is 


- 


LOADS IN STRUCTURES 85 


found that the normal pressure p amounts to 18.2 lb. per 
sq. ft. Proceeding in the same manner, it is found that for 
horizontal pressures of 30 and 40 lb., the normal pressures 
are 27.3 and 36.4 lb. per sq. ft., respectively. 

In Fig. 3, the normal force » has been resolved into its 
two components, p, and py, the former acting in a horizontal 
direction and the latter in a 
vertical one. The force pz 
tends to push the roof in a 
direction parallel with the 
wind, while the force p, tends 
to depress the roof or, in some 
cases, to press it sidewise. In 
open sheds, where the wind is 
liable to strike the inner, far Fic. 3 
side of the shed roof, the effect of the force », must be con- 
sidered, as its tendency would be to lift the roof. 


DISPOSITION OF LOADS 

In warehouses where all floors are likely at any one time 
to be fully loaded, the beams, girders, columns, and founda- 
tions are always proportioned for the entire live and dead 
loads. However, where the building exceeds four or five 
stories in height and is used for any other purpose except 
storage, as, for instance, a modern office building, it is cus- 
tomary to assume that certain members, while proportioned 
for the entire dead load, carry only a certain percentage of 
the live load. 

In an office building, or similar structure, it is highly 
improbable that all the floors or all parts of the same floor 
will be fully loaded at the same time, and in view of this fact 
it is considered good practice, while proportioning the floor- 
beams for the full live load, to calculate only say 90% of the 
live load on the girders and columns. It is customary to 
proportion the columns supporting the roof and the top 
floor for the full live load. The live loads on the columns, 
in each successive tier, from the floor above is reduced 10% 
until 50% of the live load is reached, when such reduced 
loads are used for all the remaining floors to the basement. 


86 LOADS IN STRUCTURES 


The economy obtained by this disposition of the live load is 
observed from the following table, which gives the dis- 
tribution of the assumed live loads on the columns in the 
several tiers of an eighteen-story office building. 


REDUCTION OF LIVE LOADS FROM FLOOR TO FLOOR 


Floors a a, = .90a za za on AGS 
2a 
Roof 20 20.00 20 20.00 

18 60 60.00 80 80.00 

17 60 54. 140 134.00 4.3 

16 60 48 .60 200 182.60 8.7 

15 60 43.74 260 226 .34 12.9 

14 60 39.37 320 265.71 17.0 

13 60 35.43 380 301.14 20.8 

12 60 31.89 440 333 .03 24.3 

11 60 30.00 500 363 .03 27.4 

10 60 30.00 393 .03 29.8 
9 60 30.00 620 423 .03 31.8 
8 60 30.00 680 453 .03 33.4 
7 60 30.00 740 483 .03 34.7 
6 60 30.00 513.03 35.9 
5 60 30.00 860 543.03 36.9 
4 60 30.00 920 573.03 37.7 
3 60 30.00 980 603 .03 38.5 
2 60 30.00 1,040 633 .03 39.1 
1 60 30.00 1,100 663 .03 39.7 


The following may serve to explain the data given in the 
table: a represents the live load on each floor, in pounds 
per square foot; a; the live load on each floor, in pounds 
per square foot, reduced by 10%, as a;=.90 a; 2a, the sum 
of all live loads, in pounds per square foot, on a column 
from all floors above, if no reduction is made; and 2a,, the 
sum of all live loads, in pounds per square foot, on a column 
from all floors above, if 10% reduction is made. 

The theoretical percentage of saving resulting from the 
reduction of 10% on the upper floors is found by the for- 


za— za 
—. These percentages of saving are given in 


ul 
mula Sa 


the last column of the table. 


MECHANICS 87 


It should be understood that each column of a building 
supports a given floor area, and that the load coming on each 
column will depend on the extent of this area multiplied by 
- the live load, in pounds per square foot of floor. Each 
column carries not alone this load, but also the loads trans- 
mitted directly from column to column. Thus, the column 
supporting the fifteenth floor supports also four other 
columns above with all their loads. 

While this system of graduating the live loads on the 
columns from floor to floor is generally practiced, the amount 
of reduction at each floor is a matter that depends on the 
judgment of the designer. The percentage of reduction is 
often fixed by city building laws, with bbe us the designer 
must comply. 


MECHANICS 


FORCES 


Two forces may be compared when the three following 
facts about each force are known: (1) The point of applica- 
tion, or point at which the force acts; (2) the direction of 
the force or line along which it acts; and (3) the magnitude 
of the force when compared with a given standard. 

In engineering work in America, the unit of force is always 
taken as the pound. 

Representation of a Force.—A force may be represented 
by aline. Thus, in Fig. 1, let A be the point of application 
of the force, let the length of the line AB 
Tepresent its magnitude and line of action 
to any convenient selected scale, as, for Fie. 1 
instance, 1 in. equals 10 lb., and let the arrowhead 
indicate the direction in which the force acts. Then the 
line AB fulfils the three required conditions in regard to 
point of application, direction, and intensity, and the force is 
fully represented. 


z 


& 


88 MECHANICS 


COMPOSITION OF FORCES 

If several forces act on a body, and if they are replaced by 
a single force that has the same effect in moving the body 
through space as the several forces combined, the single force 
is called the resultant of the several forces; and, conversely, 
the several forces are called the components of the single 
force. The process of finding the resultant when the various 

components are known is called the composition of forces. 
_  Parallelogram of Forces.—When 
*/ “two forces act on a body at the 
same time and at the same point, 
but at different angles, their final 
effect may be obtained as follows: 
In Fig. 2, let A’ be the com- 


SO1n. 


4 ~— c mon point of application of the 
bcs hae two forces, and let AB and 
Fic: 2 AC represent the magnitude and 


direction of the forces. Let, for instance, the line AB repre- 
sent the distance that the force AB would cause the body to 
move in a certain length of time; similarly, let AC represent 
the distance that the force AC would cause the body to move 
in the same length of time, when both forces are acting 
separately. A fundamental law of mechanics states that the 
motion is proportional to the force applied, and, therefore, 
while AB and AC represent the magnitude of the forces to 
some scale, they are also proportional to the distances these 
forces would move the same body in the same length of time. 
The force AB, acting alone, would carry the body to B. If 
the force AC were now to act on the body, it would carry it 
along the line BD, parallel to AC, to a point D, at a distance 
from B equal to AC. Join C and D, then CD is parallel to 
AB and ABDC isa parallelogram. Draw the diagonal AD. 
The body will stop at D, whether the forces act separately 
or together, but if they act together, the path of the body 
will be along AD, the diagonal of the parallelogram. More- 
over, the length of the line AD represents the magnitude of 
a force, which, acting at A in the direction AD, would cause 
the body to move from A to D; in other words, AD, measured 
to the same scale as AB and AC, represents the magnitude 


MECHANICS 89 


and direction of the combined effect of the two forces AB 
and AC. 

The force represented by the line AD is the resultant of 
the forces AB and AC. Suppose that the scale used was 
50 lb. to the inch; then, if AB=50 lb. and AC=623 Ib., the 
length of AB would be 50+50=1 in., and the length of AC 
would be 62.5+50=1}4in. If the line AD measures 1? in., 
the magnitude of the resultant, which it represents, would be 
13 X 50 =87}3 Ib. 

Therefore, a force of 874 Ib., acting on a body at A, in the 
direction AD, will produce the same result as the combined 
effects of a force of 50 lb. acting in the direction AB and a 
force of 624 lb. acting in the direction AC. 

Triangle of Forces.—Let Fig. 3 represent a parallelogram 
of forces. AB and AC are the two component forces and 
AD is theresultant. Now, to find this resultant AD without 
drawing the entire parallelogram, first lay off AB, and then 
from the point B lay off BD ee 
equal to AC and parallel to its 
line of action; then, draw AD, 
which completes the triangle 
ABD and makes it unnecessary 
to draw AC and CD. Or, 
the force AC could be drawn 
first, and then from C, the line Fic. 3 
CD, thus completing the triangle ACD and not drawing AB 
and BD. Ineither case, it is seen that, if desired, the result- 
ant of two forces can be found by drawing a triangle instead 
of a parallelogram. 

Resultant of Several Forces.—When three or more forces 
act on a body at a given point, their resultant may be found 
as follows: Find the resultant of any two forces; treat this 
resultant as a single force, and combine it with a third force 
to find a second resultant. Combine this second resultant 
with a fourth force, to find a third resultant, etc. After all 
the forces have been thus combined, the last resultant will 
be the resultant of all the forces, both in magnitude and 
direction. The order in which the forces are taken is 
immaterial. 


90 MECHANICS 


ExamPLe.—Find the resultant of all the forces acting on. 
the point O, Fig. 4, the length of the lines being proportional 


to the magnitude of the forces, 

SoLuTion.—Draw OE parallel and equal to AO, and EF 
parallel and equal to BO; then OF is the resultant of these 
two forces, and its direction is from O to F. Consider OF as 
replacing OF and EF, and draw FG parallel and equal to 
CO; OG will be the resultant of OF and FG; but OF is the 
resultant of OE and EF; hence, OG is the resultant of OE, EF, 
and FG, and likewise of AO, BO, and CO. The line FG, 
parallel to CO, could not be drawn from the point O to the 
right of OE, for in that case it would be opposed in direction 
to OF; but FG must have the same direction as OF. 


For the same reason, draw GL parallel and equal to DO. 
Join O and L, and OL will be the resultant of all the forces 
AO, BO, CO, and DO (both in magnitude and direction) 
acting at the point O. If L’O is drawn parallel and equal to 
OL, and having the same direction, it will represent the effect 
produced on the body by the combined action of the forces 
AO, BO, CO, and DO. For brevity, the terms forces AO, 
BO, etc. and resultants OF, OG, and OL have been used in 
this solution. It should be remembered, however, that these 
are merely lines that represent the forces in magnitude and 
direction, 


MECHANICS 91 


In Fig. 4 the forces OE, EF, etc., all point in the same 
direction; that is, a body at O acted on by these forces in 
succession would move from Otol. The resultant therefore 
acts from O to L, and not from L to O, This line of reason- 
ing will give the direction of the resultant in a force diagram. 
In Fig. 4, the forces were taken in the order AO, BO, CO, and 
DO. However, the magnitude and direction of the resultant 
OL would be the same, no matter in what order the forces 
were taken. 

RESOLUTION OF FORCES 

Since two forces can be combined to form a single redeteat 
force, a single force may also be treated as if it were the 
resultant of two forces whose joint action on a body will be 
the same as that of a single force. Thus, in Fig. 5, the force 
OA may be resolved into two forces, OB’ and B’A. 

It will be observed that one resultant force may have an 
innumerable number of combinations of components. 
Instead of OB’ and B’A, 

Fig.5,OB” or B’” A orOB'”’ 

and B’”’A may be taken as 

components. It is cus- 

tomary, however, to make 32” 

OB’ and B’A perpendicu- 
lar to each other, as pe SN 
shown in the illustra- ? < 
tion. Fic. 5 

Frequently, the positiony magnitude, and direction of @ 
certain force are known, and it is desired to know the effect 
of the force in some direction other than that in which it 
acts. Thus, in Fig. 6, suppose 
that OA represents, to some 
scale, the magnitude, direction, 
and line of action of a force 
acting on a body at A, and 
weet pee ece eee ee eo 4 that it is desired to know what 

Fic. 6 effect OA produces in the 
direction BA. From A draw a 
line AB in the required direction; from O draw a line per- 
pendicular to AB. Then BA is the component required. 


on 


oo $ 
o°* 
a 


92 MECHANICS 


It is necessary, of course, that OB be at right angles to BA, 
so that all the effect of OA in the required direction may be 
represented by BA and none of it by OB. Thus, OB’ and 
B’A, although components of OA, and although one of them 
is in the required direction, would not be a correct solution 
of the problem because, besides B’A, OA exerts some more 
effect in the line BA, namely, a part of OB’ is in that direc- 
tion with an amount BB’. 

ExaMPLE.—If a body weighing 200 Ib. rests on an inclined 
plane whose angle of inclination to the horizontal is 18°, 
what force does it exert perpendicular to the plane, and what 
force does it exert parallel to the plane, tending to slide it 
downwards? 

SoLutTion.—Let ABC, Fig. 7, be the plane, the anette A 
being 18°, and let W be the weight. Draw a vertical line 


Fic. 7 Fic. 8 


FD=200 1b., to represent the magnitude of the weight. 
Through F draw FE parallel to AB, and through D draw DE 
perpendicular to EF, the two lines intersecting at E. FDis 
now resolved into two components, one FE tending to pull 
the weight down the incline, and the other ED acting as a 
perpendicular pressure on the plane. On measuring FE 
with the same scale by which the weight FD was laid off, 
it is found to be about 61.8 lb., and the perpendicular pres- 
sure ED on the plane is found to measure 190.2 Ib. 

_ As it is often necessary to resolve a force into two com- 
ponents at right angles to each other, a simple method of 
solution is employed. In Fig. 8, let OA be the force it is 
desired to resolve into two components at right angles to. 


MECHANICS 93 


each other. Through O as a center draw an indefinite 
line XX’ in the required direction of one of the components, 
and through the same center O draw the indefinite line Y Y’ 
in the direction of the other component. Then XX’ and 
YY’ are at right angles to each other, since the components 
to be found are to be at right angles to each other. The 
line XX’ is commonly called the XX’ axis, or simply the 
X axis, and the line YY’, the YY’ axis or the Y axis. From 
A draw a line perpendicular to the X axis, as Aa’, and also 
a line perpendicular to the Y axis, as Aa”. Then Qa’ is the 
component force in the direction of XX’ and Oa” is the other 
component in the direction of YY’. 

When the components Oa’ and Oa” are horizontal and 
vertical, as in the present case, they are called, respectively, 
the horizontal component and the vertical component of the 
force OA. ; 


‘ 


MOMENTS OF FORCES 


DEFINITIONS AND MEASUREMENTS 


In Fig. 1, W is a weight that tends to fall—that is, to act 
downwards—with a force of 10,000 lb. It is well known 
that if some fixed point, as a, not in the line along which 
the weight W acts, is con- 
nected with the line of action 4 
of W by a rigid arm, so that jj Hi 
W pulls on one end of this | 
arm while the other end is |i//-% 
firmly held at a, the pull of as 
W will tend to turn, or rotate, [im 
the arm about the point a. 

It is also known that the 
tendency to rotate is directly 
proportional to the magni- 
tude of the force, provided the 
arm remains of the same length, and directly proportional 
to the length of the arm, if the force remains constant. In 
general, therefore, the rotative- effect is proportional to the 


Fic. 1 


94 MECHANICS 


product of the magnitude of the force and the length of the 
lever arm. This product is called the moment of the force 
with respect to the point in question. Thus, in Fig. 1, the 
moment of the force W with respect to the point a is the 
product obtained by multiplying the magnitude, 10,000 Ib., 
by the perpendicular distance, 10 ft., from the point a to the 
line of action of W. 

The point a, Fig. 1, that is assumed as the center around 
which there is a tendency to rotate, is called the center, or 
origin, of moments. : 

The perpendicular distance from the center of moments .- 
to the line along which the force acts, is the lever arm of the 
force, or the leverage of the force. 

Since the unit of force is the pound, and the ordinary unit 
of length is the foot, the unit of moment will be a derived 
unit, the foot-pound (abbreviated to jt.-lb.), and moments 
will usually be expressed in foot-pounds. In Fig. 1, for 
example, the moment of the force W with respect to the 
point ‘a is 10,000 X 10=100,000 ft.-Ib. 

The moment of a force may be expressed in inch-pounds 
(in.-lb.), foot-pounds, or foot-tons (ft.-T.), depending on the 
unit of measurement used to designate the magnitude of the 
force and the length of its lever arm. For instance, if the 
magnitude of a force is measured in pounds, and the lever 
arm through which it acts in inches, the moment will be in 
inch-pounds; again, if a force of 10 T. acts through a lever 
arm of 20 ft., the moment of the force is 10 X 20 = 200 ft.-T. 

Positive and Negative Moments.—In order to distinguish 
between the directions in which there is a tendency to 
produce rotation, the signs + and — may be used. Thus 
if a force tends to produce right-hand rotation, that is, rota- 
tion in the same direction as the hands of a clock, it is called 
positive, and its moment takes the sign +; a force that tends 
to produce rotation in the opposite direction is called neg- 
ative, and its moment takes the sign —. The selection of 
one direction for positive and another for negative is merely 
an arbitrary distinction to show that the directions are 
opposite. It has, however, been adopted by engineers, and 
should always be used as given. 


MECHANICS 95 


Resultant Moments.—In Fig. 2 is shown a lever com- 
posed of two arms at right angles to each other, and free to 
turn about the center C. A force A acts on the horizontal 
arm in such a manner that it tends to produce left-hand 
rotation, its moment being 10X5=50 ft.-lb., which, since 
it tends to produce left-hand rotation, will be ealled negative. 
Another force B, whose moment with respect to the center C 
is 12X3=36 ft.-lb., tends to produce right-hand rotation. 
B-12 Bt: 


i. 


\ 


—5 ft. 


pe Bowes 


Fic. 2 


Therefore, —50+36=—14 ft.-lb., which is the resultant 
moment, and has the same turning effect as the two moments 
already given. 

If, instead of the two forces just considered, there is a body 
that is acted on by any number of forces whose moments 
about a given center are known, the resultant moment of 
these forces will be the algebraic sum of the moments of the 
given forces. 

CENTER OF GRAVITY 


The center of gravity of a body, or of a system of bodies, 
or forces, is that point at which the body or system may 
be balanced, or it is the point at which the whole weight 


96 MECHANICS 


of the body or bodies may be considered as concentrated. 
If the body or system were suspended from any other point 
than the center of gravity, and in such a manner as to be 
free to turn about the point of suspension, it would rotate 
until the center of gravity reached a position directly under 
the point of suspension. 

Center of Gravity of Plane pistes: —If the plane figure 
has one axis of symmetry, this axis passes through its center 
of gravity. If the figure has two axes of symmetry, its center 
of gravity is at their point of intersection, 

The center of gravity of a triangle lies on a line drawn from 
a vertex to the middle point of the opposite side, and at a 
distance from that side equal to one-third the length of the 
line; or it is at the intersection of lines drawn from the ver- 
texes to the middle of the opposite sides. The perpendicular 
distance of the center of gravity of a triangle from the base 
is equal to one-third the altitude. 

The center of gravity of a parallelogram is at the inter- 
section of its two diagonals; Soerenee ae it is midway 
between its sides. 

The center of gravity of an auseulae jour-sided figure may 
be found as follows: First divide it, by a diagonal, into two 
triangles and join the centers of gravity of the triangles by a 
straight line; then, by means of 

the other diagonal, divide the fig- 

ee) ure into two other triangles, and 

a ! join their centers of gravity by 

e another straight line; the center 

of gravity of the figure is at the 

intersection of the lines joining 

the centers of gravity of the two 
sets of. triangles. 

Another method by which to 

A locate the center of gravity of an 

irregular four-sided figure is illus- 

Fic. 3 trated in Fig. 3. Draw the diag- 

onals ac and bd, and from their intersection e¢, measure the 

distance to any vertex, as ae. From the opposite vertex, 

lay off this distance, as at cf. Then from f, draw a line to 


MECHANICS . 97 


one of the other vertexes, as fd, and bisect this line as at g. 
Connect g and b and lay off one-third of its length from g 
at the point hk, This point is the center of gravity of the 


The distance of the center of gravity of the surface of a half 
circle from the center is equal to the radius multiplied by 
.424, 

Neutral Axis.—The neutral axis of a flat or plane figure 
may be considered as any line passing through the center of 
gravity. This definition of neutral axis is correct for use in 
the mechanics of ordinary materials. However, in speaking 
of the neutral axis of reinforced-concrete beams, something 
else is meant, This will be explained later. The location 
of the horizontal neutral axis, that is, a horizontal line 
through the center of gravity of a plane figure, is of great 
importance in engineering. Of course, if the center of 
gravity is located, the horizontal neutral axis can be located 
easily; but, as a rule, the horizontal neutral axis is located 
without first locating the center of gravity. 

Locating the Neutral Axis by Means of the Principle of 
Moments.—A convenient method of locating the neutral 
axis is based on the principle that the moment of any plane 
figure, with respect to any given line as an axis or origin of 
moments, is equal to the product of its area by the perpen- 
dicular distance from the center of gravity of the figure to the 
given axis, 

Let M represent the moment; A, the area of the figure or 
section; andc the perpendicular distance from center of gravity 
of figure to given axis. Then, M 

M=AXce andc= A 


If necessary, the figure may be subdivided; then the 
moment of the figure is equal to the sum of the moments of 
its separate parts with respect to the same axis. Designa- 
ting the areas of the subdivisions by the letters a, a’, a”, etc. 
and their moments by m, m’, m”, etc., then the preceding 
formula becomes 


m+m’+m”, etc. 
c= 
at+a’+a’”, etc. 


98 MECHANICS 


Built-Up Section.—Fig. 4 shows a section of the rafter 
member of a large roof truss formed of a 3” X16” web-plate 
and a 3” X12” flange plate, the two joined by two 4” X 4” X }” 
angles. It is desired to know the distance from the neutral 
axis of the section to the top edge of the flange plate. By 
means of the principles given, the centers of gravity of the two 
rectangular plates are easily located as shown. The centers 
of gravity of the angles might also be located by applying the 
rule given in the preceding paragraph; this, however, is 
unnecessary because the position of the center of gravity 


can be obtained directly by referring to the tables of the 
properties of rolled sections furnished by the various steel 
manufacturers. By referring to any of these tables, the 
center of gravity of a 4”X4”X4” angle is found to be 1.18 
in, from the back of a flange, thus giving the distance 1.18 + 
.875=1.555, or about 144, in. from the top edge of the flange 
plate to the axis through the centers of gravity of the angles, 
From the same tables it is also found that the area of the 
section of a 4” 4” 4” angle is 3.75 sq. in. 

The area of the section of the flange plate is 3X12=4.5 
sq. in., and of the web-plate, 3 16=6 sq. in.; the area of the 
whole section is therefore 2X 3.75+4.5+6=18 sq. in. 


MECHANICS 99 


The moments of the areas of the separate sections, with 
respect to the line ab, are as follows: 


Flange plate, 45X %&= 84 
Two angles, 2X3.75X14#=11:70 
Web-plate, 6X82 =50.25 

Total, 62.79 


The distance c from the top edge of the flange plate to the 
neutral axis de of the section is therefore 62.79 +18=3.48 in. 


FORCES ACTING ON BEAMS 
STYLES OF BEAMS 


A beam that has two points of support, one at each end, 
as shown in Fig. 1, is known asa simple beam, The distance 
between the points of sup- 
port is called the span. 

A beam that has more & 
than two supports, as Fic. 1 
shown in Fig 2, is known 
asa continuous beam. The 
point of support of the 
middle support is con- 
sidered to be at its cen- 
ter, as at a. Continuous 
beams, except in rein- 
forced-concrete work, have 
not found much favor 
among engineers in the Fic. 3 
last few years. 

A cantilever beam is 
one with only one sup- 
port, which is at the 
middle, or any part of 
a beam that projects 
out beyond its support. A common form of cantilever is 
shown in Fig.3. This beam projects from a wall or some 
other solid structure and has no support at its outer end. 


Fic. 2 


Fic. 4 


100 3 MECHANICS 


_ When a beam is rigidly held, or fixed, at both ends, as 
shown in Fig, 4, it is called a restrained beam, or, more com- 
monly, a beam fixed at both ends. 


LOADS ON BEAMS 


The forces due to the weights that a beam supports are 
known as loads. If the whole load is applied at one point, 
or practically one point, it is called a concentrated load; if it 
extends over a portion of the beam, it is called a distributed 
load; and if it is equally distributed over the beam, so that 
each unit of length has the same load, it is called a uniform 
load. 

There are certain methods by which such loads may be 
represented graphically. These methods may best be illus- 
trated by referring to Fig. 5, which shows a simple beam, 
Starting at the left, the 
point of the arrow 
shown at R; is the 
point of support. The 

Fic. 5 arrow a represents a 
concentrated load. At bb’ is shown a uniformly distributed 
load, of a certain number of pounds per foot of beam, that 
extends from b tob’. Atc,d, and f are shown other concen- 
trated loads similar to the one at a. From ¢ to é’ is a distrib- 
uted load, represented by the shaded triangle under ee’, that 
starts from nothing at e and increases to a given number of 
pounds per foot of beam ate’. At gg” is a distributed load, 
_ starting from nothing at g and increasing to a maximum 
at g’, and then decreasing to nothing at g”. At Ro is shown 
the right-hand support. 


REACTIONS 


A beam with any loads it may carry is held up by the 
supports; that is, the beam presses on its supports at the ends, 
The supports resist this pressure and prevent the beam from 
falling. This upward force exerted by each support is 
known as the reaction. 

There are two facts susceptible to proof in regard to a 
beam: (1) The resultant of all the forces acting on a body 


MECHANICS 101 


or beam must be zero, and (2) the resultant moment of all 
the forces about any point must be zero. 

The forces acting on a beam are the forces due to loads on 
the beam and the weight of the beam itself, if that is con- 
sidered, and then there are the reactions of the supports on 
the beam. Now, all the loads act vertically downwards and 
the reactions act vertically upwards. They are therefore 
parallel; and as they are opposite, the sum of the loads must 
equal the sum of the reactions. 

A beam with two supports will be considered. The sum 
of the reactions is known from the loads, but the value of 
each reaction is not known. To find this, it is necessary to 
resort to the second condition, namely, that the resultant 
moment of all the forces about any point must be zero. 
Any point can be assumed, but it is found convenient to 
choose the reaction of one support as the point about which 
to take moments, The moment of the reaction at the point 
is zero, because the arm of the moment is zero. It is there- 
fore necessary that the sum of the moments of the loads about 
one point of support and the moment of the other reaction 
about the same point shall equal zero, 


ass. =: 
~) : J 
i 3 3 $ 
] — + 27 ft-—-— 
| — —— — —-19 st- — + — — 
aan p——8 ft— 
{ i rt 
ere RESET oa 
z b, 
Fic. 6 


A practical example will now be considered. In Fig. 6 
let it be required to find the reactions R, and R2 at the points 
of support aand b. (In all the subjoined problems, R; and 
Rz represent the reactions.) The center of moments may 
be taken at either R; or Ro. Let the point b be taken in 
this case. The three loads are forces acting in a downward 


102 MECHANICS 


direction; the sum of their moments, in foot-pounds, with 
respect to the assumed center, may be computed as follows: 


8,000X 5= 40000 
6,000 x 19=114000 
2,000X27= 54000 


Total, 208000 


This is the total negative moment about the pointb. The 
positive moment about this point is of course R, X30. 
Since the resultant moment is zero, the positive moment 
must equal the negative moment, or 208,000=30 R; or 
R; = 208,000 +30 =6,9334 lb. The sum of all the loads is 
2,000+ 6,000+ 8,000 = 16,000 lb. This is also the sum of 
the reactions. Therefore, R2=16,000—6,9334 =9,066% Ib. 
In this problem, the weight of the beam itself has been 
neglected. 

EXAMPLE 1.—What is the reaction at Rz in Fig. 7? 

SoLuTION.—In computing the moment due to a uniform, 
or evenly distributed, load, as at a, the lever arm is always 
considered as the distance from the center of moments to 


<_- 20 ft. $ 
$-127¢ sono te parirunnt no bok ety 
LIA zy Yi st tintzorm tout 


_ 10 ft: ! 
30 ft. 

Ry Rez 

Fic. 7 


the center of gravity of the load. The amount of the uni- | 
form load a is 3,000 X 10 =30,000 1b., and the distance of its 
center of gravity from R; is 13 ft. Therefore, the moments 
of the loads on this beam about Rj, in fopspeunes, are as 
follows: 
30,000 X 13=3 90000 
4,000 4= 16000 
9,000 xX 20=180000 


Total, 586000 


MECHANICS 103 


This is the sum of the moments of all the loads about R,, 
asacenter. The leverage of the reaction Re is 30 ft. Hence, 
the reaction at Re is 586,000 +30 =19,5334 Ib. 


000 1b. per running ft. 
6 ft. 


ExAamMPLE 2.—A beam is loaded as shown in Fig. 8. Com- 
pute the reactions R; and Ro. 

SoLuTion.—Consider, say Ro, as the center of moments. 
Then, the negative moments of the loads about Ro, in foot- 
pounds, are 

20,000 X 27=5 40000 
2,000X12= 24000 
3,000X 8= 24000 

Total, 588000 

Now, the positive moments of the loads about Re, in foot- 
pounds, are 

1,000 X6X3=18000 

5,000X6 =380000 

. Total, 48000 

The resultant moment of the loads about Rz is negative 
and is —588,000+ 48,000 = —540,000 ft.-lb.; 540,000+30 

= 18,000 lb., which is the value of Rj. 

The sum of the loads is 


20000 

2000 

3000 

6X1,000= 6000 

5000 

36000 
R2=36,000 — 18,000 = 18,000 1b. 


104 MECHANICS 


VERTICAL SHEAR 


In any beam, as for instance the one shown in Fig. 9, there 
are forces—either loads or reactions—acting both upwards 
and downwards. In this figure, the left-hand reaction Ry 


mes 


R, RF, 
Fic. 9 


acts upwards and tends to push the end of the beam up. 
On account of the strength of the beam, however, the end 
remains stationary. If the beam were suddenly cut on the 
line ab, the left-hand portion of the beam would move up 
in relation to the right-hand portion. This action of the 
forces on a beam in tending to make the surfaces at any 
imaginary section slide past each other, from its similarity 
to a shearing action, is called shear. 

Consider now the beam shown in Fig. 10. Since the loads 
are symmetrically applied, each reaction is equal to 40 1b. or 
one-half the total load on the beam. Considering therefore, — 


—-- 3} — —— 9 


S < < < s s 

— 3 3 A) a} ~) 

‘EA yada le ie 
R, Ry 


Fic. 10 


any transverse section of the beam between R; and the point 
of application of the load u, it is evident that the part of the 
beam at the left of this transverse section is subjected to an 
upward thrust of 40 lb., while the part at the right is sub- 
jected to an equal downward thrust. The result is a shear- 


MECHANICS 105 


ing action at this place, the magnitude of which is equal to 

the reaction R,. This shearing action is resisted by the 
strength of the fibers of the beam at the section under con- 
sideration. 

When the point of application of is reached, the effect 
of the upward force R; is partly balanced by the downward 
force of 10 lb. due to the load m. Any section between the 
points of application of m and m is therefore subject to a 
shearing stress equal to the difference between the reaction R; 
and the load n, or 40—10=30 1b. In the same way, it fol- 
lows that the shearing stress for any section between m and o 
is 40—(10+15)=15 lb. For any section, as cd, between 
the points of application of o and p, the shearing stress is 
40—(10+15+15)=0. In other words, it is a section in 
which there is no shear. — 

Positive and Negative Shear.—For convenience, it is cus- 
tomary to call the reactions, or forces, acting in an upward 
direction, positive, and the loads, or downward forces, nega- 
tive. Since the difference between the sums of the positive 
and negative numbers representing a given set of values is 
called their algebraic sum, it follows that the shear for any 
section of a. beam is equal to the algebraic sum of either 
reaction and the loads between this reaction and the trans- 
verse section under consideration. In speaking of the shear 
at a certain section of a beam as being positive or nega- 
tive, it is simply meant that the resultant of the forces 
acting on the portion to the left of the section under con- 
sideration is either positive or negative. 

If a transverse section of a simple beam is taken near 
the left reaction and the forces acting on the part of the 
beam at the left are considered, it will be seen that their 
resultant acts upwards. The shear at this section is there- 
fore called positive shear. If, however, a section near the 
right reaction is taken, the resultant of the forces at the left 
of this section is found to act downwards, and in conse- 
quence the shear is called negative. It is also evident that 
there is a section between the two where the resultant of 
the forces changes from positive to negative. At such a 
section the shear is said to change sign. 


106 MECHANICS 


It can readily be seen that in a cantilever beam the con- 
ditions are somewhat reversed; that is, in a beam having 
one support at the middle, the shear in the part to the left 
of the section taken to the left of the support is negative, 
while if the section is taken to the right of the support, it is 
positive. Like in a simple beam, however, there is an 
intermediate point where the shear changes sign. 


3 
~ 

. 
$ ds 
~~ 3s 
$ =e 


Fic. 11 


ExamMPLe.—At what point in the beam loaded as shown 
in Fig. 11 does the shear change sign? 

SoLuTIon.—Compute the reaction R; as follows: With 
the center of moments at Re, the moments of the loads, in 
foot-pounds, are 


9,000 X 10 90000 
4,000 X 26 =104000 
3,000 X 10K 17=510000 


Total, 704000 


The reaction at R, is therefore 704,000 +30=23,4662 Ib. 
Proceeding from R;, the first load that occurs is c of 4,000 Ib. 
Then, 23,4662—4,000=19,4662 1b. The next load that 
occurs on the beam is the uniform load of 3,000 lb. per 
running ft. There being altogether 30,000 lb. in this load, 
it is evident that the load will more than counteract the 
remaining amount of the reaction R,; the point where the 
change of sign occurs must consequently be somewhere in 
that part of the beam covered by the uniform load. The 
load being 3,000 lb. per running ft., if the remaining part 


MECHANICS 107 


of the reaction, 19,4663 1b., is divided by the 3,000 Ib., the 
result will be the number of feet of the uniform load required 
to counteract the remaining part of the reaction, and this 
will give the distance of the section, beyond which the result- 
ant of the forces at the left becomes negative, from the edge 
of the uniform load at a; thus, 19,466%+3,000=6.49 ft. 
The distance from R, to the edge of the uniform load is 8 ft. 
The entire distance to the section of change of sign of the 
shear is, therefore, 8+ 6.48 = 14.48 ft. from Rj. 

Shear Diagram.— It is sometimes necessary to plot the 
shear of the forces acting on a beam in a diagram known as 
the shear diagram. In order to illustrate how this may be 


—_1f.-— -—— 


Renee Bft—|- 3 ft 


d 
oe be 


(a) | 


Fic. 12 


done the beam shown in Fig. 12 (a), the span of which is 
12 feet, will be considered. Since the beam is symmetrically 
loaded, each reaction is equal to one-half the sum of the 
400 + 1,200+ 400 
2 
=1,000 lb. The shear at R; is equal to the reaction, or 
1,000 Ib. To plot the diagram, proceed as follows: Draw 
a line ab to any convenient scale to represent the length of 


loads; that is, each reaction is equal to 


108 MECHANICS 


the beam, This line will also represent the base, or datum, 


line of the shearing forces, positive values being laid off 
above and negative values below the base line. The shear 
at R, being positive and equal to 1,000 lb., draw from a 
upwards a vertical line ac equal to 1,000 Ib., according to 
any convenient scale. From c draw to the same scale as ab 
a horizontal line cd equal to 3 ft. Ordinates drawn from 


— —25 ft — 
3 ft. Tih —<«érn— B Ua aia ft 
ii tere | | * 

g s 


TD 


6852 1b. 


Z 


Y, 


Fic. 13 


any point on the line ab to cd will be of equal lengths, show- 
ing that the shear retains the value of 1,000 lb. from the 
reaction R; to the first load of 400 lb. At this point the 
shear is reduced by the load of 400 1b.; therefore, from d, 
draw downwards a vertical line de of a length corresponding 
to 400 lb., as shown in the diagram. 

The shear is then uniform until the central load is reached, 
and may be represented by the distance of the line ef from 


MECHANICS 109 


the line ab, namely, 600 lb. At this point the shear is 
reduced by 1,200 lb., shown on the diagram by the line fg, 
and becomes negative. At the third load, counting from 
the left, the shear is still further reduced by 400 lb., and is 
represented by the line 4z in the diagram. From this load 
on to the right-hand reaction the shear is negative in sign 
and equal to 1,000 lb. This right-hand reaction drawn 
from j to b closes the diagram. 

It may now be seen that to find the shear at any point ina 
beam it is necessary to draw an ordinate from the corre- 
sponding point on the line ab and to measure the length of 
the part included in the shaded diagram with the correct. 
scale. For instance, if it is desired to ascertain the shear 
and its sign at the point k, draw a perpendicular line km 
crossing the shear diagram. The length /m measured by 
the scale selected for the shear will show that the shear at 
this point is 600 lb., and, as lm is located below the base 
line ab, the shear is negative. 

As a general example, the shear diagram of the beam 
shown in Fig. 138 may be plotted. As the load here is not 
symmetrical, it is first mecessary to calculate the reactions 
R, and Ry. The positive moments about the left-hand end 
of the beam, in foot-pounds, are: 


500X 3= 1500 
(1,000 X6)X 8= 48000 
200xX 9= 1800 
1,000X15= 15000 
5,000 X 21=105000 


Total, 171300 


The total negative moment is R2X25. Therefore, Re 
=171,300+25=6,852 1b. The sum of the loads is 500 
+ (1,000 x 6) + 200+ 1,000+5,000 = 12,700 Ib. R, is there- 
fore equal to 12,700—6,852 =5,848 Ib. 

The plotting of the shear diagram shown in Fig. 13 may 
now be started. Draw the line ab equal to the length of the 
beam; ac equal to R;, or 5,848 lb.; cd, horizontally, equal 
to 3 ft.; de, vertically downwards, equal to 500 Ib.; and éf, 
horizontally, equal to 2 ft. From a on ab lay off 9 ft., as 


110 MECHANICS 


at p. The shear at this point, just to the left of the con- 
centrated load, is +5,848—500—(1,0004)= +1,348 Ib. 
This value laid off above ab gives the point g. Draw fg. 
From g lay off vertically downwards 200 1b. to the point h, 
From a, on the line ab, lay off 11 ft., as at g. The shear at q 
is +5,848—500— (1,000 x 6)—200= —852 1b. From gq lay 
off, vertically, 852 lb. to 7, downwards in this case because 
the shear is negative. Join h and 7. If correctly drawn, 
hi should be parallel to fg. From 7 draw 7 horizontal and 
equal to 4 ft.; from 7 draw jk vertically downwards, equal 
to 1,000 lb.; from k& draw kl horizontally, equal to 6 ft.; 
from / draw lm vertically downwards, equal to 5,000 Ib.; 
and from m draw mn horizontally, equal to 4 ft. Then, if 
the diagram is correctly constructed, bn should be vertical and 
equal to Ro, or 6,852 1b. Then acde...mnb is the shear 
diagram, and ordinates drawn from any point on ab across 
the diagram will, when measured by the proper scale, indi- 
cate the shear at that section, which is positive if above the 
line ab and negative if below it. 


BENDING MOMENTS 
In order to illustrate the method of calculating the bend- 
ing moment, or the moment of a force that tends to bend 
a beam, the beam shown in Fig. 14 will be considered. At 
the point a is a joint, or hinge. It is evident that when loads 


w 
Pie | 
1 
ee ase 
—— saree mat Re 
Fic. 14 


are applied, as shown by the arrows, the beam will bend 
at the joint and take the position indicated by the dotted 
lines; or, to be more exact, will bend still farther until it 
falls off the supports entirely. 

The loads and reactions (and weight of the beam itself, if 
this is considered) are what cause the beam to bend. Con- 


MECHANICS 111 


sidering the portion of the beam to the left of the joint, this 
portion moves clockwise, and the moment that moves it is 
therefore positive. The magnitude of the moment tending 
to move the left-hand part of the beam is R,l—Wh, since 
the load W acts in the opposite direction to R,. 

It will thus be seen that no matter in what part of the 
beam the joint a is placed, the beam will collapse. There- 
fore, it is evident that in any beam carrying loads, these 
loads and the reactions exert a moment at any transverse 
section that tends to bend the beam. It is only on account 
of its own strength that a beam does not break. A beam is 
therefore designed to withstand the bending moment of the 
forces acting on it, and for this reason the engineer must at 
all times be able to find this moment. With a simple beam 
the bending moment (considering, as usual, the part to the 
left of the section) is always positive, while with cantilever 
beams it is negative. 


ee oft: 3ft---+-—6ft.—_— 
00 lb. per ft. $ 
3 


6 
24 ft. 
R, R2 


Pre 15 


As an illustration, the bending moment around various 
sections of the beam shown in Fig. 15 will be considered. 
It is of course first necessary to find the reactions, so that 
the moments may be calculated. To find Re, take moments 
about R;. The positive moments, in foot-pounds, are 

1,000X3= 3600 
(800 X9)xX103= 75600 
2,000X18= 36000 


Total; 115200 
The span is 24 ft. Therefore, Ro=115,200 +24 =4,800 Ib. 
The sum of the loads is 1,200+ (800 X 9) +2 000 = 10,400 Ib. 
Therefore, R; = 10,400 — 4,800 = 5,600 lb 
The bending moment at any section of the beam may now 
‘be found. For example, find the bending moment around 


112 MECHANICS 


point a directly under the first load. The moment here, 
on the left-hand part of the beam, is R,xX3 = 5,600X3 
= 16,800 ft.-lb. f 

Then find the moment on the left-hand portion of the beam 
around point b. Here the positive moment is R, x 104, or 
5,600 X 104 = 58,800 ft.-lb. There are, however, two neg- 
ative moments acting, and these must be subtracted from 
the positive moment in order to get the resultant moment. 
One of these moments is that due to the concentrated load 
of 1,200 1b., and the other is that due to the part of the dis- 
tributed load to the left of the pointb. The negative moment 
of the concentrated load is therefore 1,200 X 74 =9,000 ft.-Ib. 
The portion of the distributed load considered is 800 44} 
=3,600 lb. Its moment arm may be considered to extend 
from the point b to the center of the portion under consid- 
eration, or 44+-2=24ft. Its moment is therefore 3,600 x 24 
= —8,100 ft.-lb. The resultant moment of the left-hand 
section about b is therefore 58,800 —9,000—8,100= + 41,700 
ft.-Ib. 

The bending moment about ¢ is 5,600 18—1,200X15 
— (800 X 9) X 74 = 100,800 — 18,000 — 54,000 = + 28,800 ft.-lb. 

The bending moment of the forces acting on a beam, 
tending to break it at any section, may be calculated from 
either end of the beam. It is customary to call the bending 
moment positive if it tends to turn the left-hand part in the 
direction of the hands of a clock, and negative if it tends to 
turn it in the opposite direction. However, the moment 
may be calculated from either end, remembering that if it 
is calculated from the right-hand end the bending moment 
acts in an opposite direction and will therefore receive a sign 
opposite to that given the left-hand end of the beam. 

As has been stated, the reason a beam does not break is 
because its strength at any transverse section is sufficient tc 
resist the moment of the forces about that section. It has 
also been shown that the bending moment is different at 
different points along a beam. It is therefore important to 
find out around what point the forces acting on a beam 
exert their maximum moment, and then, by the method 
already given, to find this maximum moment. 


MECHANICS 113 


The point where the shear on a beam changes sign, 
that is, changes from positive to negative, or from nega- 
tive to positive, is the point around which the maximum 
bending moment occurs. The point where the shear on a 
cantilever beam changes sign, that is, the point around which 
the moment is maximum, is always at the point of support. 

Sometimes, the shear will change from positive to negative 
or vice versa two or more times on a beam, and each sec- 
tion, when the shear is zero, must therefore be investigated . 
to determine around which point the maximum moment 
occurs. Fig. 16 shows an example of this kind. To solve 
this example, it is first necessary to find the reactions. The 
moments of the loads about R;, in foot-pounds, are as follows: 

500 2=1000 
800X 5=4000 
300 x 14=4200 

Total, 9200 

R2X10 = 9,200 ft.-lb.; R.=9,200+10= 920 1lb.; and Ry 
= 500+ 800+ 300 — 920 =680 Ib. 


veri 
I 


At R;, the shear is + 680 lb.,and at 2 ft. from the left-hand 
end the shear changes from +680 to 680—500= +180 Ib. 
At 5 ft. from the left-hand end, the shear changes from 
+180 to 180—800= —6201b. This, therefore, is one place 
where the shear changes sign. Under the reaction Re, the 
shear changes from —620 to 920—620= +300 Ib., and this 
is therefore another point where the shear changes sign. 

There are therefore two places to be investigated for 
maximum bending moment, one 5 ft. from the left-hand end 
and the other 10 ft. from the left-hand end. The bending 
moment about the point 5 ft. from the left-hand end is 


114 MECHANICS 


680 X 5—500 X3=+1,900 ft.-lb., and that about the point . 
10 ft. from the left-hand end is 680 10—500 X8—800X5 
= —1,200 ft.-lb. It is thus seen that the greater of the 
two maximum bending moments occurs about the point 
5 ft. from the left-hand end. | 

Bending-Moment Diagrams.— The bending moments that 
act at various points of a loaded beam may be repre- 
sented graphically in the same manner in which the shear- 
ing forces were shown. For example, a beam 50 ft. long 


s 

» i) 

$ S 

~- 

SW, W. 

= 1 s 2 

~ ~) 
A . 
S ) 
~ ~ 

) 

3 : 
~ bo) 
~ " 
t) o 
te < 


zi ———-~------ 


Pus. 17 


supports two concentrated loads of the magnitudes and in 
the positions indicated in Fig. 17. Now, assuming that 
Fig. 17 (a) is drawn to scale, draw the horizontal line ab and 
produce the lines indicating the reactions and concentrated 
loads until they intersect line ab at a, c,d, and b. By cal- 
culating the moments about the left-hand reaction, the 
moment for the load W, is found to be 10,000 X 10= 100,000 
ft.-lb. and that for We, 6,000 25=150,000 ft.-lb. The 


100,000 X 150,000 
reaction Rp» is therefore 50 =5,000 lb., and 


reaction R; is 10,000 +6,000 — 5,000 = 11,000 Ib. 


MECHANICS 115 


The bending moment at W; is 11,000 X 10 =110,000 ft.-lb., 
and that at We is (11,000 « 25) — (10,000 x 15) = 125,000 ft.-Ib. 

Lay off the line cc; to any convenient scale to represent 
the bending moment at W,, and at dd; to represent that at 
We. Connect points a, c,, dj, and b, as shown, and the dia- 
gram is complete. 

As another illustration, assume that it is desired to plot 
the bending-moment diagram of the beam shown in Fig. 18. 
This beam is 50 ft. long and has a uniform load of 1,000 lb. 
per running ft. 

First, divide the beam into any number of smaller parts, 
each, for instance, 10 ft. long, and then find the bending 


= 
LY 


— 


R,=25000 Lb. 


moment for each of these parts. In this instance, the 
reactions R; and Rp» are each equal to one-half the total load, 
or 50,000+2=25,000 lb. The bending moment at c is 
(25,000 X 10) — (1,000 X 10 * 5) = 200,000 ft.-lb.; at d it is 
(25,000 X 20) — (1,000 X 20 X 10) = 300,000 ft.-lb. Inasimilar 
manner, it will be found that the bending moments at points e 
and f are 300,000 and 200,000 ft.-lb., respectively. 

On laying off these bending moments at ¢)C2, d,do, etc. and 


" connecting points a,c)... b, the diagram is complete. The 


FORMULAS FOR MAXIMUM SHEAR AND BENDING 
MOMENTS ON BEAMS 


: Maximum | Maximum 
Case Method of Loading Shear Moment 
—— J ——__— 
I i w| Ww Wl 
wi 
II Ww <2 
\ Wasi net 
Il = at 3 
S wy 2W1 
3 
wi 
Vv Ww. ep 
VI \ “dln = WwW we 
VII © i 
ALR eg Ae Fe eae id ue 
eon, ae ; ; 
roe bs Watches Sees ape PE a 
ee Pe ee 
x | et ony = “ 
a, — : ; 


116 


TABLE—(Continued) 


| Maximum 


Case | Method af Loading a 
‘a. @ 2 8 
XII “ a 
XI b a ud - 
as Ww 
: 2 52W1 
XIV = “405 
w 
XV i HW Wi 
a ae | 
=e EE Ys sw aw 
N - ‘ 4 
5 NS Ww Wl 
cibad Ni % = oy ie 
XVIII 
XIX 


xX 


117 


118 | MECH ANICS 


outline of this diagram, however, is only approximately cor- 
rect. To find its true form, the beam should be divided into 
an infinite number of parts, in which case the lines acy, c,d}, 
dé, 171, and 7;b would be parts of a curve, shown dotted, 
instead of a series of short, broken lines. 


MOMENT TABLES 

The methods of finding the shear and maximum moment 
occurring in simple and in cantilever beams have been 
explained. In these calculations the weight of the beam 
itself, if of wood or steel, is often omitted. The various 
unusual loadings were also considered, and the method of 
finding the maximum bending moment was explained. 
These methods will be found usefiil if such cases ever arise. 
However, in practice the loads on the beams are often more 
uniform, and if the weight of the beam itself is neglected, 
the formulas given in the accompanying table may be used. 

Beams with fixed ends or even with one end fixed and one 
simply supported are also given in the table. 

In each case in the table, the total load on the beam, in 
pounds, is denoted by W. If there are two separate and 


WwW 
equal loads on the beam, each one is called 4) so as to make 


the total load W. If the load is uniformly distributed over 
the entire length of the beam, the load per foot will be 
W +length of beam, in feet. The length of the beam is 
denoted by/. If J is taken in inches, the moment will be in 
inch-pounds; but if taken in feet, the moment will be in 
foot-pounds. 

The usual method of indicating loading is adopted. A 
simple support under a beam is shown by the conventional 
arrow, while a cantilever or a beam fixed at the ends is 
shown as in Case I or XVII, respectively. Cases XV and 
XVI indicate a beam fixed at one end and supported at the 
other. 

The maximum shear on a cantilever or on a beam fixed 
at the end depends somewhat on the method of holding it in 
the wall and the length that extends into the wall. How- 
ever, the maximum shears given in the table are correct 


MECHANICS «119 


for usual cases. In Case XX, two values are given fér 
both shear and bending moment, and the maximum of the 
two values obtained must be the one used. The sign of 
the shears or bending moments has not been put in the table, 
but it can be told by inspection whether they are positive 
or negative, as every-day experience should enable the engi- 
neer to determine whether the loads will bend the beam up 
or down. 

The point of maximum shear is marked on the beam at 
the point a, while maximum bending moment is at the 


REACTIONS FOR CONTINUOUS BEAMS OVER EQUAL 


SPANS 
(Coefficients of W) 
Num- Number of Each Support 
Spans ah 2d | 3d! 4th} 5th eth | 7th| 8th| 9th| 10th 
2 % | 4e) # 
3 vo | 33 | 46 | ws 
4 33 | 38 | 38 | | 38 
5 48 | 43 | 34 | 34 | 43 | 3 
6 | tor 43d /482| 18F| 402 | 108 | woe 
(Moc rabcraberabesabcsabeeabe sibs y3 
8 | 385/443 |558| 388 | 338 | F848 | HS | 398 | 388 
9 580 $30) $86 355/348 | 345 | $38| 343| $93) 33 


= 


point b. In some cases, either of these values may reach 
its maximum at two or more places, in which case it is so 
marked. 

The following example will serve to show the use of the 
table: 

ExaAMPLE.—A simple beam on a span of 13 ft. 24 in. carries 
a uniformly distributed load of 85 lb. per ft. What is the 
maximum shear and the maximum bending moment devel- 
oped? 


120 MECHANICS 


*SoLuTION.—The length of the beam, reduced to feet, is 
13.2083. The total load W is therefore 85 13.2083 
=1,122.7055, say 1,123 lb. Referring to Case XI, the max- 


W 1123 
imum shear = A a =561.51lb. Likewise, the maximum 


bending moment is 
wl 43 1,123 X 13.2083 


8 8 


= 1,854,115, say 1,854, ft.-Ib. 


CONTINUOUS BEAMS 


When a single beam extends over three or more supports 
‘it is said to be continuous. The bending moments produced 
are very different from those in an ordinary beam. For this 
reason, the treatment of this class of beams must be con- 
sidered separately. 


BENDING MOMENTS FOR CONTINUOUS BEAMS OVER 


EQUAL SPANS 
(Coefficients of W1) 
saa: Number of Each Support 
ber of 
Spans! 1<¢ | 2d | 3d | 4th! 5th| 6th| 7th| sth| 9th| 10th 
2 0 |4/)0 
3 0 \zvo\rs| 0 
4 | 0 | ds \as| Hs | 0 
5 0 | as \se| ds | as | 0 
6 0 |Yor|\rer|roq|1eq|fox| 0 
7 | 0 |site rede] ree || ave |r| 0 
g | 0 | atelaite| aie |aie| aie | atte) ates 0 
9 O | sehr state | seo | sAe'o | so | sto | s*z#o| sao) 0 


The first of the tables, on page 119, gives the coefficients 
for the reactions of the beam at its supports. The value 
given in the table multiplied by W, which represents the load 
per span, will give the reaction. By knowing the reactions 


STRESSES AND STRAINS 121 


and the loads per foot, the shear at any section may be found. 

The coefficient from the second table multiplied by WI, 
which represents the load per span and the length of span, 
will give the negative bending moment over each support. 
As the maximum moment occurs over a support, it is seen, 
by referring to the table, where this moment will be. If 
the coefficient given in this table is multiplied by the load in 
pounds and the span in feet, the result will be in foot-pounds; 
but if the span is taken in inches, the result will be in inch- 
pounds. 

To illustrate the method of using the tables, the following 
example will be assumed: 

EXxaMPLE.—A beam of three spans is supported by four 
reactions. Each span is 11 ft. The beam carries a load of 
2,000 lb. per ft. (a) What will be the amount of the end 
reactions? (b) What is the greatest bending moment pro- 
duced? 

SoLuT:on.—(a) The load W supported by each span is 
2,000 X 11=22,000 lb. From the first table, the reaction at 
each end support is equal to y45 W. Substituting these 
values, the reaction at each end is equal to 39% 22,000 
==8,800 lb. 

(b) The greatest bending moment occurs at the inter- 
mediate supports; from the second table, this moment is 
equal to yy WI, or yp X 22,00 X 11 = 24,200 ft.-Ib. 


STRESSES AND STRAINS 
STRESS 

It is evident that the weight of the materials composing 
a building and its contents produces forces that must be 
resisted by the different members of the structure. The 
action of these forces has a tendency to change the relative 
position of the particles composing the members, and this 
tendency is, in turn, resisted by the cohesive force in the 
materials, which acts to hold the particles together. 

The internal resistance with which the force of cohesion 
opposes the tendency of an external force to change the 
relative position of the particles of any body subjected to a 
load is called a stress; or, stress may be defined as the load 


122 STRESSES AND STRAINS 


that produces an alteration in the form of a body, and this 
alteration of form is called the strain. — 

In accordance with the direction in which the forces act 
with reference to a body, the stress produced may be either 
tensile, compressive, or shearing. 

Tenstle stress is the effect produced when the external 
forces act in sucha direction that they tend to stretch a body. 

Compressive stress is the effect produced when the tendency 
of the forces is to compress the body. 

Shearing stress is when the forces act so as to produce a 
tendency for the particles in one section of a body to slide 
over the particles of the adjacent section. 

When a beam is loaded in such a manner that there is in it 
a tendency to bend, it is subjected to a bending stress. In 
this case, there is a combination of the three stresses already 
mentioned (tension, compression, and shear) in different 
parts of the beam. 

The unit stress (called, also, the imtensity of stress) is the 
name given to the stress per unit of area; or, it is the total 
stress in a tie-rod, column, or the like divided by the area 
of the cross-section. 

Let P represent the total stress, in pounds; A, the area of 
cross-section, in square inches; and s, the unit stress, in 
pounds per square inch. Then, 


P 
om FOF P=As 


STRAIN 


When a body is stretched, shortened, or in any way — 


» deformed through the action of a force, the deformation is 
called a strain. Thus, if a rod were elongated + in. by a 
load of 1,000 1b., the strain would be yy in. Within certain 
limits, to be given hereafter, strains are proportional to the ~ 
stresses producing them. 

Unit strain is the strain per unit of length or of area, but 
is usually taken per unit of length and called, for tension, the 
elongation per unit of length. If the unit of length is taken 
as 1 in., the unit strain is equal to the total strain divided by 
the length of the body in inches. 


STRESSES AND STRAINS 123 


Let / represent the length of the body, in inches; e, the 
' deformation, in inches; and gq, the unit strain. Then, 


e 
q=7, OF e=lq 


ELASTIC PROPERTIES 

It can be proved by experiment that when a certain unit 
stress is created in a substance, a certain definite unit strain 
is developed. If the unit stress is doubled, it will be found 
that the unit strain also has doubled; that is, the alteration 
of shape or the strain in a body is proportional to the force 
applied to that body. This experimental fact is known as 
Hooke’s law. 

When a certain stress is created in a body a certain strain 
is produced. When the stress is removed, the body returns 
to its original shape, provided the unit stress has not been 
too great. For each substance, however, there is a certain 
maximum unit stress that the substance will stand and still 
return to its originai shape after the external forces are 
removed. This unit stress is called the elastic limit of the 
material. If a body is strained beyond the elastic limit, 
it will maintain a permanent distortion, or set, even after 
the strain forces are removed. Hooke’s law, which is almost 
exact for most materials below the elastic limit, does not hold 
good for these materials above the elastic limit, as the 
strain increases much more rapidly than the stress. Thus, 
if the unit stress is doubled beyond the elastic limit, the unit 
strain will be more than doubled. The unit stress that is 
so great that the strain increases greatly with very little 
increment of stress is called the yield point. For all prac- 
tical purposes, with many materials the yield point com- 
mences at the elastic limit. 

To restate Hooke’s law, the ratio of the unit stress to the 
unit strain for any substance is constant below the elastic 
limit. This ratio of unit stress to unit strain is called the 
modulus of elasticity, or coefficient of elasticity, which will be 
represented by the symbol E. It is 


Ss 
E=- 
q 


124 STRESSES AND STRAINS 


From this equation, when s and E are known, g may be 
found; that is, if the modulus of elasticity of a certain sub- 
stance and the unit stress are known, the unit strain can 
readily be found. 


STRENGTH OF BUILDING MATERIALS 

The ultimate strength of any material is that unit stress 
which is just sufficient to break it. 

The ultimate elongation is the total elongation produced - 
in a unit of length of the material by a unit stress equal to 
the ultimate strength of the material. 

The modulus of rupture differs from the ultimate tensile 
or compressive stress, but is a quantity, something like it, 
that is used in calculating the strength of beams. Its use 
is discussed under the heading Homogeneous Beanis. 


; FACTOR OF SAFETY 

A value that is taken for the ultimate strength of any 
material is an average value of a number of experiments 
made on the material. As it is impossible to get two sam- 
ples of the same material exactly alike, so is it also impos- 
sible to get two samples with the same ultimate strength. 
It is therefore customary in design to avoid stressing a 
material up to its ultimate strength or even up to its 
elastic limit. 

The factor of safety, or, as it is sometimes called, the 
safety factor, is the ratio of the ultimate strength of the 
material to the load that, under usual conditions, the material 
is called on to carry. Suppose that the load required to 
rupture a piece of steel is 5,000 lb., and that the load it is 
called on to carry is 1,000 lb.; then, the factor of safety 
may be obtained by dividing the 5,000 lb. by the 1,000 Ib. 
Thus, 5,000+1,000=5, which is the factor of safety of this 
material. 

Another factor to be considered in selecting materials is 
deterioration, which is due to various causes. In metals, 
there is corrosion on account of moisture and gases in the 
atmosphere. 

Wood is subject to decay from either dry or wet rot, 
caused by local conditions; it may, like iron and steel, be 


STRESSES AND STRAINS 125 


subjected to fatigue, produced by constant stress due to the 
load it may have to sustain. 

The preceding reasons are sufficient to require the factors 
now adopted by conservative constructors in all engineering 
work. These factors are 3 to 5 for structural steel, and 
6 to 10 for cast iron. 

Stone and brick are very unreliable and a high factor of 
safety should therefore be used. Usually, the factors 
employed are 10 for compressive stresses, 15 for tensile 
stresses, and from 10 to 20 for bending stresses. Some 
engineers, however, do not use such high factors. Stone- 
work or brickwork is even more unreliable than stone or 
brick themselves, but the same factors are often employed. 
Usually, however, the strengths of stonework and brickwork, 
instead of being given as ultimate strength, are arranged in 
tables to give the safe, or allowable, stress per square inch or 
per square foot. This value embodies its own factor of safety. 

Concrete is another material that sometimes requires a 
high factor of safety. In reinforced-concrete work a factor 
varying from 4 to 6 is usually employed, although some 
engineers use a higher one. 

Ordinances in various cities throughout the country 
govern the allowable load that building materials should 
carry. 

A structure withstanding shocks or containing rapidly 
vibrating machinery is more liable to fail than one in which 
the loads are quiet, even if the latter has the greater loads. 
Therefore, in designing a structure to withstand vibrating 
loads or shocks, the designer, to make the structure safe, 
uses a larger factor of safety. Usually, this factor averages 
from two to three times the ordinary factors, except those 
for timber, which need not as a rule be increased. 

The average ultimate strength of various metals and 
woods are given in the accompanying tables. 


STRESSES AND STRAINS 


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STRESSES AND STRAINS 


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HOMOGENEOUS BEAMS 129 


HOMOGENEOUS BEAMS 


PROPERTIES OF SECTIONS 


MOMENT OF INERTIA 


The strength and the stiffness of a beam depend on vari- 
ous factors. For instance, the load that a beam can bear 
depends on the material of the beam, on the manner in which 
the load is applied, and on the length and cross-section of 
the beam. As to the area of the cross-section, the strength 
does not depend on that area itself, because, as every-day 
experience shows, a plank used as a beam will sustain a 
greater load when placed edgewise than when placed on its 
broad side, although it has the same area in both cases. 
The strength of a beam depends on the manner in which 
the area is distributed or disposed with respect to the neutral 
axis of the cross-section. The effect of the cross-section 
is measured by a quantity that depends on such disposition 
or distribution of area, and is called the moment of inertia 
of the cross-section with respect to the neutral axis. The 
moment of inertia of a plane figure has really nothing to do 
with the property of matter known as inertia, and as here 
used it is simply the name for a certain constant depending 
on the shape of the cross-section of a beam. It is not a 
moment as the word is generally understood, and it might 
have been called anything so far as the structural engineer 
is concerned, but it has received its name from some relations 
in higher mathematics where it is derived. This quantity 
must be considered simply as a constant that has been found 
suitable for use in certain formulas for the design of beams, 
which will be explained later. 

The following table, entitled Elements of Usual Sections, 
contains, in the column marked J, the moments of inertia 
of the shapes of beam sections commonly used in practice. 
The axis in each case passes through the center of gravity, 
and is designated neutral axis in the table for reasons that 
will be explained later. The square may be regarded as a 


. £ 
oe ZY oe’ I; —-= 97 
pees ve Soren a 2 £ 
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HOMOGENEOUS BEAMS 135 


particular case of a rectangle whose base b and altitude d 
are equal. The value of the area A is given in the second 
column. The distance c, given 
in the third column, is the dis- 
tance of the most remote part 
of the figure from the neutral 
axis. The section modulus and 
radius of gyration, the character 
of which will be explained later, 
are given in the last two columns. 
EXxAMPLE.—What is the area, 
the distance from the neutral axis 
to the extremities of the section, and the moment of inertia 
about the neutral axis of the section illustrated in Fig. 1? 
SoLuTion.—Referring to the table, it will be seen that }; 
=4 in., b>=104 in., andd=7in. Therefore, the area is 
b+b, en un Sent 
2 
Likewise, the distance from the neutral axis to the longer 
parallel side is 


b+2b,'d 10.5+2X4 7 
= X -—=2.977 in. 
Oth Ss WOSME- 8 , 


The distance c, which is the distance of the neutral axis 
from the shorter parallel side, is 
b+2b d 44+2X10.5 7 
— X—=4.023 in. 
eet 8 1 eee S 
As a check, c; +c should equal d, or 7in.; thus 2.977+ 4.023 
=7.000 in., which indicates that the preceding solution is 
correct. 
‘The moment of inertia is found by the formula 
__6?+,4bb; +b;? 
36 (b+5;) 
Substituting values for the letters in this formula, 
10.5°+4X10.5X4+4 
= X 73 =193.348 
36 X (10.5 + 4) 
In general, it one of the values of ¢ and ¢; is found by means 
of the formula, the other may be found by subtracting the 


Bicod 


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140 HOMOGENEOUS BEAMS 


known one from d. Thus, after having calculated c; in this 
example, c would be equal to d—cy =7—2.977 =4.023 in., 
which value corresponds to the one found by means of the 
formula. 

Many of the beams used in building construction are made 
of steel. These steel beams are rolled with various cross- 
sectional shapes, from which they derive their names. 
These shapes are standard; that is, they are rolled to con- 
form to certain sizes that have been adopted by many of the 
large steel companies. In the first column of the accom- 
panying table, entitled Values for Standard Rolled Sections, 
are shown the cross-sections of the various structural shapes. 
The first section shown is known as an angle with equal legs; 
the second, as an angle with unequal legs; the third, as a 
channel; the fourth, as a bulb beam; while the fifth, sixth, 
and seventh sections are known, respectively, as anI beam, 
a T bar, and a Z bar. 

In this table is given the moment of inertia for each sec- 
tion about at least two axes, both of which pass through 
the center of gravity of the section; that is, the moments of 
inertia are given always with respect to neutral axes. The 
moments of inertia with respect to different neutral axes are, 
in general, different, and as a rule there is one neutral axis 
about which the moment of inertia is less than it is about 
any other. With the channel, bulb beam, I beam, and T bar, 
the smallest moment of inertia is about the axis y’—y’, or the 
vertical axis. However, with both styles of angles and with 
the Z bar, there is another moment of inertia about a neu- 
tral axis, not horizontal nor vertical, that is smaller than the 
moment of inertia about any other neutral axis; that is, in 
certain work, more particularly in the design of columns, it 
is necessary to know about what neutral axis the moment of 
inertia will be smallest. It is also necessary to know what 
this moment of inertia will be. Therefore, in the section 
of the angle with equal legs, the section of the angle with 
unequal legs, and the Z-bar section, the moment of inertia is 
also given for the oblique axis giving the smallest moment 
of inertia. The position of this neutral axis is found by 
higher mathematics. With the angle with equal legs, this 


3) elle 


HOMOGENEOUS BEAMS 141 


oblique neutral axis y’-y’ makes an angle of 45° with the 
horizontal. With the angle section with unequal legs and 
with the Z-bar section, this oblique neutral axis y’—y” makes 
an angle a with a vertical line. Instead of giving this angle 
direct, it is more convenient to give a value for the trigono- 
metric tangent of twice the angle, and this is done in the 
last column of the table. 

The formulas given in the table are long and difficult to 
use. The values of the properties of various structural- 
steel sections have therefore been calculated for all the 
standard sizes of angles, channels, etc. that are ordinarily 
manufactured. The tables containing these values will be 
found on page 147, where their use will be explained. 

In the two tables just given, the moment of inertia is 
taken about the neutral axis. Sometimes, however, it is 
desirable to find the moment of inertia of the section about 
some axis other than the neutral axis. This may be accom- 
plished as follows: 

Let A represent the area of any section; J, the moment 
of inertia with respect to an axis through the center of grav- 
ity of the section; Ix, the moment of inertia with respect to 
any other axis parallel to the former; and », the distance 
' between axes. Then, 

Ix=I+Ap? 


EXxAMPLE.—Determine the moment of inertia of a triangle 
with respect to its base (see fourth section of the table entitled 
Elements of Usual Sections). 

SoLuTIon.—The distance between the base and the neutral 
axis c, is 4 d, the area is 4 bd, and J=y, bd. Then, by the 
preceding formula, 

bd? bd? ba 
= 2=—_ — ae 
Ix= 95 bd? +4 bd X (4 d) a6 Tame 

Many sections may be regarded as built up of simpler 
parts; they are called compound sections. For example, a 
hollow square consists of a large square and a smaller square; 
the T shown in the table entitled Elements of Usual Sections 
consists of two slender rectangles, one horizontal and one 
- vertical. 


142 HOMOGENEOUS BEAMS 


The moment of inertia of a compound section with respect 
to any axis may be found by adding, algebraically, the 
moments of inertia, with respect to the same axis, of the 
component parts of the figure. - 

ExaMPLE.—Find the moment of inertia of the Z bars 
shown in Fig. 2, about the neutral axis X’X, the dimensions 
being as shown. 

SoLuTION.—The figure may be divided into the three rect- 
angles efgh, e’f'g'h’, and jgig’. The moment of inertia of 

efgh about an axis 
i = 1, through its center of 
e gravity and parallel to 
I X'X is yy a'f; that of 
- t+ e’f'g'h’ about an axis 
through its center of — 
gravity and parallel to 
X'X is also yy a’. The 
distance between this 
Lt gt! = axis and the X’X axis 
be e is 4 (b—2. The mo- 
ment of inertia of the 
rectangle efgh and also 
Fic. 2 of the rectangle e’}’g’h! 
; about the axis X’X 
is then, 7 a’f#+a’t [4 (0—f)]?. The moment of inertia 
of the rectangle jgig’ about the axis X’X is 7 tb?. The 
moment of inertia of the entire figure is, therefore, 


2{ rr a’+a't[t (b—OP} +r 8 
Expanding and reducing this expression, 
ab3— a’ (b — 2t)8 


oh 
=> 


2 


. 


+ 
- 


| 

| 
HE 

| 


| 
| 
> 
| 
| 
| 
| 
x Sear 


&, 
sf-- 
y 


T= 


RADIUS OF GYRATION 


The radius of gyration of a section with respect to an axis 
is a quantity whose square multiplied by the area of the 
section is equal to the moment of inertia of the section with 
respect to the same axis. If 7 and J denote, respectively, 


_HOMOGENEOUS BEAMS 143 


the radius of gyration and the moment of inertia of a sec- 


tion, and A the area in square inches, then, 
Arz=T, 


fi 
h =< [— 
whence rT y A 


The last column of the table entitled Elements of Usual 
Sections gives radii of gyration corresponding to the mo- 
ments of inertia given in the fourth column. 

The radius of gyration of a section, or figure, may be found 
directly from its moment of inertia by means of the formula 
just given. For example, the radii of gyration for the 
rectangle and the hollow square in the table just referred to 
are found as follows: 

For the rectangle, 

r= Vib bd? + bd --2—= 280d 
Vi2 


For the hollow square, 


a? + a;? 
r= Vis (at— a4) + (@2— 0,2) = ery 
NEUTRAL AXIS 
If, in a cantilever loaded as shown in Fig. 3, any point x 
on the center line ab is taken as a center of moments. and a 


WH 


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S 
i 
1 
Y i 
Yj ‘pes : 
1." Pe SR Oe Vee. ET’ 
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Fic. 3 
section made by a vertical plane cd through this center is 
considered, it will be evident that the moment of the force 
due to the downward thrust of the load tends to turn the 
end of the beam to the right of cd around the centerx. The 


144 HOMOGENEOUS BEAMS 


measure of this tendency is the product of the weight W 
multiplied by its distance from cd, and, since this tendency 
is the moment of a force that tends to bend the beam, it is 
called the bending moment. 

A further inspection of Fig. 3 will show that through the 
bending action of the load the upper part of the beam is 
subjected to a tensile stress, while the lower part is sub- 
jected to a compressive stress. 

Fig. 3 also shows that the greater the distance of the par- 
ticles in the assumed section above or below the center x, 
the greater will be their displacement. In other words 


ae WY 
at Sit REA | IS 41 8H 
I f' Ze 


D’ 


Fic. 4 


since the stress in a loaded body is directly proportional to 
the strain, or relative displacement of the particles, it fol- 
lows that the stress in a particle of any section is propor- 
tional to its distance from the center line ab, and that the 
greatest stress is in the particles composing the upper and 
lower surfaces of the beam. 

In Fig. 4, let ABCD represent a cantilever. A force F 
acts on it at its extremity A. This force will tend to bend 
the beam into something like the shape shown by A’BCD’. 
It is evident from what has preceded that the effect of the 
force F in bending the beam is to lengthen the upper fibers 
and to shorten the lower ones. Hence, the upper part A’B 
is now longer than it was before the force was applied; that 
is, A’B is longer than AB. It is also evident that D/C is 
shorter than DC. Further consideration will show that there 
must be a fiber, as SS”, that is neither lengthened nor short- 


HOMOGENEOUS BEAMS 145 


ened when the beam is bent; that is, SS”=S’S”. A line 
drawn through this fiber SS” when the beam is straight is 
called the neutral line, and the horizontal plane in which this 
line lies is known as the neutral surface of the beam. The 
neutral line corresponds to the center line ab, Fig. 3, on 
which the center of moments x was taken. 

The line s,s2, which passes through any beam section, as 
abcd, Fig. 4, at the neutral line, perpendicular to the direc- 
tion in which the beam bends, is called the neutral axis. 
It is shown in works on mechanics that the neutral axis 
always passes through the center of gravity of the cross-section 
of a beam made of uniform material. 

Thus, in a beam, it is evident that the neutral axis of any 
section is a horizontal line at which the fibers composing 
the beam are neither stretched nor compressed. The axis 
is horizontal because it is at right angles to the direction of 
the load, which, in structural problems, usually acts down- 
wards. It so happens that in a beam made of one material, 
this line will pass through the center of gravity of the sec- 
tion. It is for this reason that the neutral axis has been 
spoken of as a line through the center of gravity of a section. 
It happens, however, that in many beams made of two 
materials, as, for example, concrete and steel, the neutral 
axis—that is, the horizontal line along which the particles 
of the beam are neither stretched nor compressed—does not 
pass through the center of gravity of the section. 


SECTION MODULUS 

The modulus of a section of a beam is equal to the moment 
of inertia of the section about its neutral axis divided by 
the distance from the outermost fiber in that section to the 
neutral axis. This fiber may be either above or below the 
neutral axis, and it is immaterial whether the beam is a 
simple beam, a cantilever, or any other kind of beam. If 
the moment of inertia of a section about its neutral axis is 
represented by J, the distance from the neutral axis to the 
outermost fiber by c, and the section modulus by S, then 

iT 


c 


146 HOMOGENEOUS BEAMS 


EXAMPLE.—Prove the formula for the modulus of a section 
as shown in Fig. 5 to be correct as given in the table on 
page 133. 

SoLutTion.—According to the table, the moment of inertia 

2 

of F the section is aneEar® 3. Itisa question whether 
¢ or c, is the longer. However, on examining the values 
L—» given for these two quantities 
‘eawieoe in the table, it is seen that they 
rN et are the same, with the exception 
i pa ‘’ of the first part of the numerator. 
Uy = For cj, this part of the numer- 
% ator is b+2b,, while for ¢ it is 
Wi b:+2b: “In Fig 5, bas: greater 
than b;; therefore, b:+2b is 
a na, 5 greater than b+2b,, and it fol- 
lows that c is greater thanc,. Accordingly, the formula is 

I bh 4bbi + Oi) 5 b+2b a 


~¢€ B86(b+b) bth 3 
_ Cabby +b?) B b+b) 3_ B+ Abi +b? oy 
36 (6+b:) (b+ 2b) d 12 (b; +20) 


which is the same as the one given in the table. 


PROPERTIES OF ROLLED-STEEL SHAPES 


The table of values for standard rolled sections already 
given shows various cross-sections of steel beams. Each 
one of these beams is made in various sizes and weights. 
On account of the confusion that ensued from using beams 
of different sizes from different manufacturers, the latter 
agreed to adopt standard sizes, which are known as American 
standard. Therefore, knowing the dimensions of the section 
of any beam that is rolled according to the American stand- 
ard, the properties of the section may be found by using the 
table already given. This table, however, while quite accu- 
rate, is inconvenient to use, because the formulas it con- 
tains are long and complicated. The following tables of 
properties of standard sections are therefore given to diminish 
the labor required in beam design. 


i 


HOMOGENEOUS BEAMS 147 


The first of the following tables gives the properties of 
standard I beams. The bevel on the inside of the flange 
is always made 1 to 6. The same bevel is also used for 
channels. 

The tables dealing with I beams and channels give two 
columns dealing with the deflection, or sag, of beams. The 
use of these columns will be explained on page 174. The 
other values given in the tables will be readily understood. 


Fic. 6 


Frequently, steel beams are built up of several steel sec- 
tions riveted together. To find the strength of a beam of 
this kind it is first necessary to determine the moment of 
inertia of the entire section about its neutral axis. This is 
done by the methods previously described on page 97, but 
it may probably be best illustrated by an actual problem. 


PROPERTIES OF STANDARD I BEAMS 


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154 


PROPERTIES OF STANDARD ANGLES HAVING EQUAL LEGS 


HOMOGENEOUS BEAMS 


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155 


HOMOGENEOUS BEAMS 


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HOMOGENEOUS BEAMS 157 
REROCCO AWVOWOrPrOOCCOW 
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158 


PROPERTIES OF STANDARD ANGLES HAVING UNEQUAL LEGS 


HOMOGENEOUS BEAMS 


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159 


HOMOGENEOUS BEAMS 


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160 


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161 


HOMOGENEOUS BEAMS 


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164 


PROPERTIES OF T BARS 
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HOMOGENEOUS BEAMS 


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165 


HOMOGENEOUS BEAMS 


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HOMOGENEOUS BEAMS 
PROPERTIES AND PRINCIPAL DIMENSIONS OF 


166 


STANDARD T RAILS 


Axis 1-1 


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HOMOGENEOUS BEAMS 167 


Sometimes, the sections are not symmetrical and the neutral 
axis itself must be found. As an example, consider the 
beam shown in Fig. 6 (a). It is composed of anI beam and 
a channel riveted together. Sucha beam is often used to 
carry the track of a traveling crane. 

The first problem is to locate the neutral axis cd, Fig. 6 (6). 
Assume any axis, as ab, about which to take ordinary 
moments of the areas. The moment of each area about ab 
is equal in each case to the product of the area and the dis- 
tance from its center of gravity to ab. The areas and loca- 
tion of the centers of gravity of the beam and column 
sections can be obtained from the tables. Adding these 
moments, the work is as follows: 


Moment of I beam=20.59X9 .......=185.31 
Moment of channel= 11.76 X (18+.78) = 220.85 
Total.. .406.16 16 


The total area “ee the eon is 20. 59 ve 11. 76= 32.35 sq. in. 
Therefore, the distance from the line ab to the neutral 


Amar 
3935 °° 


It now remains, by the method given, to find the moments 
of inertia of the channel and the I beam about the axis cd 
and to add them together. This can be done with the help 
of the tables, as follows: The moment of inertia of the 
channel about cd is 9.39+11.76X (5.44+.78)?= 464.36. 
Likewise, the moment of inertia of the I beam about the 
axis cd is 921.2+20.59 x (12.56 —9)2=1,182.15; therefore, 
the total moment of inertia is 464.36 + 1,182.15 = 1,646.51 in*. 


axis cd is 


FORMULAS FOR DESIGN 
BENDING-MOMENT FORMULAS 
In any homogeneous beam the maximum stress at any 
section of the beam developed by the loads may be found 
by the following formula: 
Bast 
c 


in which M is the bending moment at that point, in inch- 


168 HOMOGENEOUS BEAMS 


pounds; J, the moment of inertia of the section referred to 
the inch; c, the distance, in inches, from the neutral axis to 
- the most remote fiber; and s, the maximum stress, in pounds 
per square inch. This stress, of course, occurs in the most 
remote fiber. 

This is the general equation for beams. It expresses the 
bending moment about any section, caused by the loads 
and reactions, in terms of the maximum unit stress at the 
section and the dimensions of the beam at that section 


IT J 
(nat is, ) . Thestresss may be either tension or compres- 
Cc 


sion, but it is always the maximum stress in the section. 

It is evident that if a beam is going to break, the break 
will occur at that section about which the external bending 
moment is maximum. If this section holds, every other 
part of the beam will be strong enough. Therefore, it is 
the section of a beam about which the external bending 
moment is maximum that is always investigated. 


I 
The quantity a is equal to the section modulus; therefore, 


the formula is often written 
M=sS, 
in which S is the section modulus. 

In the preceding tables the moments of inertia, etc. are 
given with the inch as a base, and s is usually measured in 
pounds per square inch. Therefore, in the preceding for- | 
mulas M must be given in inch-pounds and not in foot- 
pounds. 

EXAMPLE,— What is the maximum stress in a simple beam 
12 ft. long and uniformly loaded with 1,000 Ib. per ft. of 
length? The beam is rectangular in section, 10 in. broad 
and 14 in. deep. 

SoLuTiIon.—The total load on the beam is 1,000 12 
=12,000 lb. The maximum bending moment, in foot- 


Wl 
pounds, is 7%. In this case, W=12,000 Ib. and /=12 ft. 


Therefore, the maximum moment, which shall be called M, 


HOMOGENEOUS BEAMS 169 


12, 12 
equals =X = 18,000 ft.-lb. To change this to inch- 


pounds, multiply by 12. Thus, 12X 18,000 = 216,000 in.-Ib. 


bd2 10X14x14 
The section modulus of the beam is — 6 —— = 326.67. 


Substituting these values in the formula, 216,000 =s x 326.67, 
and s= 216,000 + 326.67 = 661.22 lb. per sq. in. Ans. 


MODULUS OF RUPTURE 

Assume that the load on a beam is increased until it 
breaks. From the loading that causes failure, M can be 
found, while S can be found from the shape of the beam sec- _ 
tion. Substituting these values in the formula M=Ss, a 
value for s will be obtained. This value of s, which corre- 
sponds to the moment that causes the beam to break, is 
called the modulus of rupture, or ultimate unit bending stress, 
of the material. Since the formula M=Ss is strictly cor- 
rect only for values of s below the elastic limit, the modulus 
of rupture will not agree exactly with either the ultimate 
unit tensile or the ultimate unit compressive strength of 
the material. The modulus of rupture is a valuable constant, 
however, because by substituting it in the formula M=Ss 
the breaking moment of a beam can be ascertained. 

In designing a beam, of course the beam is not intended 
to develop the ultimate unit bending stress, but only a 
fraction of it, depending on the factor of safety used. For 
timber, a factor of 6 is usually employed; for wrought iron, 
one of 4 is sufficient; while for cast iron, from 6 to 10 is 
employed. For structural steel, the factor of safety gen- 
erally used is about 4. With structural steel, however, 
instead of dividing the modulus of rupture by the factor of 
safety it is customary to use certain approved unit working 
bending stresses. For light roof construction, this value is 
often taken as high as 18,000 to 20,000 lb. per sq. in In 
ordinary building construction, 16,000 lb. is usually employed 
and in bridge work, 12,500 lb. per sq. in. is often used. 

ExAMPLE.—Design a rectangular white-oak beam to carry 
a safe load of 2,000 1b. located at the center, the span being 
11 ft. 5 in. 


i aie ey 


170 HOMOGENEOUS BEAMS 


SoLuTion.—The maximum external bending moment is 
WI 2,000 X 187 
4 4 
for white oak is 7,000, and a factor of safety of 6 will be used. 
‘Therefore, s=7,000+6=1,167 lb. per sq. in. Substituting 
these values in the formula, 68,500=SX1,167, and 


=68,500 in.-lb, The modulus of rupture 


bd? 
S=68,500+1,167=58.71. S= ae therefore, either the 


breadth or the depth of the beam may be assumed and the 
other dimensions found. It will also be noted that in the 
value of S the breadth is involved only as a first power, 
while the depth is squared. Therefore, to design an econom- 
ical beam, the better plan is to make the beam narrow and 
as deep as possible. Of course, there are practical considera- 
tions that govern this matter, such as obtaining commercial 
sizes of material and the like. 

In the problem at hand, let is be assumed that the beam 


bd? bX 10? 
will be 10 in.deep. Then, d=10 in., S=58.71 er ae ~ 


6X 58.71 
and pa = 8.523 in. The next larger size of com- 


mercial timber is 4 in. x 10 in., and is therefore the size to be 
used. 

The modulus of rupture of metals and timbers will be 
found in the two tables on pages 126 and 128. 


WEIGHT OF BEAMS 


Every beam, whether it is made of wood, steel, or stone, 
has a certain weight, and the question is whether it should 
be considered or not. Neglecting the weight of the beam 
itself in beam design does not make so much difference on 
’ a short span with heavy loads as it does on a long span with 
comparatively light loads. Just when the weight of the 
beam itself should be considered and just when it should 
not, is a matter of experience, and no set rule can be laid 
down. Usually, however, if the weight of the beam is less 
than 5% of the load it is intended to carry, its weight may 
be neglected. 


HOMOGENEOUS BEAMS 171 


Since, in some cases, it has been decided that the weight 
of the beam itself must be taken into account, the methods 
of attaining these results will be considered. As the weight 
of the beam cannot be obtained until its size is known, and 
as the size of the beam cannot be found until the total bend- 
ing moment is known, this problem can be solved only by 
trial. The following example will serve to illustrate the 
method to be pursued: 

ExaMPLe.—Calculate the size of I beam required to carry, 
besides its own load, a uniformly distributed load of 960 lb. 
per ft. over a span of 20 ft. 

‘SoLuTION.—The total load on the beam, exclusive of its 
own weight, is 960 X 20=19,200 1b. The maximum bend- 
WI 19,200 x 20 X 12 


8 
wl ae 
M =Ss=-— = 576,000. Giving s a value of 16,000 and 


ing moment is = 576,000 in.-lb. 


neglecting the weight of the beam, 16,000 x S=576,000, or 
S =576.000+ 16,000=36. On consulting the table on 
page 148, it will be seen that the value of S here found corre- 
sponds to that of a 12-in. 31.5-lb. 1 beam. This beam would 
satisfy the requirements if the weight of the beam itself were 
left out of consideration, but as it is necessary to provide for 
this additional load, the next larger size may be chosen and a 
trial calculation made to see whether it will support the com- 
bined load. This beam is a 12-in. one, weighing 35 Ib. per ft.; 
hence, the weight of the beam is 35 X 20=700 1b. From the 


Wl 
preceding formula 3 the maximum bending moment due 


700 X 20x 12 
to the weight of the beam alone is —_— = 21,000 in.-ib. 


The sum of this moment and that of the external load is 
576,000 + 21,000 =597,000 in.-lb.=M. M=Ss, or 597,000 
= 16,000 x S; therefore, S=597,000+ 16,000 =37.31. As the 
value given for S in the table is greater than this, the beam 
selected is of ample strength. 

As was stated, it is usually considered safe to neglect 
the weight of the beam itself in calculations of beam design. 


172 


HOMOGENEOUS BEAMS 


FORMULAS FOR DEFLECTION OF BEAMS 
Case Method of Loading eae 
: are Sey we 
pS 3 EI 
‘ we 
8 El 
III Malas 
15 EI 
Gente at See aie, soos 
iy akan arabe 1 WR 
A 4 @D 8 El 
Eel wey2t—x) V3xQl—2) 
Vv yi b y 27 IEI 
-— — ——1— — 
= a We (3p — 422) 
VI V] a8 4 7 48 El 
= 2 “ 
5WB 
vil 384 El 
3WB 


Vill 


320 EI 


. 


: HOMOGENEOUS BEAMS 173 
TaBLE—(Continued) 
Case Method of Loading sense 
| KK Y Wwe 
ms ae =p 60 EI 
Y 
x rw 
3,600 EI 
XI _3 WP 
322 EI 
5 WB 
II 5WR 
4 926 EI 
Yy aap gh ee 
ee oe We 
ere - ® 192 El 
XIV Wwe 
384 EI 
For overhang: 
—— sa (3 xl—4 x)? 
vied opr = — ae 
= = isne 
ri For part between sup- | 
| | 3 ports: 
eer 


174 HOMOGENEOUS BEAMS 


As the amount of load a beam must carry, particularly the 
live load, is very uncertain at best, the addition of a slight 
weight due to the weight of the beam itself is seldom con- 
sidered to be a factor of great importance. When beams 
carry floors, it is customary to find the weight of the floor 
per square foot and then multiply this value by the distance 
between beams and by the span to get the total load on the 
beam, Many engineers assume that the weight of the 
beams themselves add a certain weight per square foot to 
the weight of the floor. This added weight is assumed to 
be 8 lb. for wooden beams and 6 1b. for steel beams. Thus, 
if all the materials composing a floor, exclusive of beams, 
were calculated to weigh 12 lb. per sq. ft., a weight of 20 lb. 
per sq. ft. would be taken to constitute the total load of a 
floor supported by wooden beams, while 18 lb. per sq. ft. 
would be taken for the total load of one supported by steel 
beams. This method, while not absolutely accurate, is 
one way of estimating the weight of the beams in a floor. 
In concrete beams and stone beams, however, the weight of the 
beam ttself must almost always be considered. 


‘DEFLECTION 


Deflection is the name applied to the distortion or bend- 
ing produced in a beam when subjected to bending stresses. 
The measure of the deflection at any point on a beam is the 
vertical displacement of the point from its original position. 

Stiffness is a measure of the ability of a body to resist 
bending; this property is very different from the strength 
of the material or its power to resist rupture. 

The stiffness of a beam does not depend so much on the 
elasticity of the material of which it is composed as on its 
length of span. This property of stiffness is as important 
in building construction as mere strength, and the two 
should be considered together; thus, the floor joists of a 
building may be strong enough to resist breaking, but they 
may also be so long as to lack stiffness, in which case the floor 
will be springy and will vibrate from persons walking on it. 
If there is a plastered ceiling on the under side of the joists 
of such a floor, the deflection of the joists may cause the 


HOMOGENEOUS BEAMS 175. 


plaster to crack and fall into the room below. The allow- 
able deflection of a plastered ceiling is usually placed at 
xéo of the span, or wy in. for each foot of span. Where 
stiffness is lacking in the rafters of a roof, they will be liable 
to sag, thereby causing unsightly hollows in the surface, in 
which moisture and snow may lodge. 

The amount of deflection that exists in beams loaded and 
supported in different ways may be calculated by the for- 
mulas given in the accompanying table. In using these for- 
mulas, all the loads should be expressed in pounds and the - 
lengths in inches. The modulus of elasticity is denoted 
by E, and the moment of inertia of the section by J. 

ExampLe.—A 10-in. 35-lb. steel I beam supported at the 
ends must sustain a uniformly distributed load of 10,000 lb. 
The span of the beam is 20 ft., and its moment of inertia is 
146.4. There is to be a plastered ceiling on its under side, 
the allowable deflection of which is #% in. for each foot of 
span. Will the deflection of the beam be excessive? 

SoLuTION.—The formula of the deflection of a beam of this 


5 
character, from the table, is adic The modulus of elas- 
384 EI 


ticity of structural steel is 29,000,000. Substituting the 
values of the example in the formula, the deflection equals 
5 X 10,000 < 2403 

384 X 29,000,000 x 146.4 © 

Since the allowable deflection is ss of the span, the total 

allowable deflection is s45X 240=% in. This is greater than 

the calculated deflection, and the beam therefore satisfies 
the required conditions. 

The values for N and N’, the coefficients of deflection for 

uniform and center loads, respectively, given in the tables 

containing the properties of sections of I beams, channels, 


42, or about zs in. 


5 WB 
and Z bars, were obtained from the formulas N = ———— 
384 EI 
We ., . : 
and N’=———,, in which W equals 1,000 Ib.; J, 12 in.; E, 


48 EI 
29,000,000; and J, the moment of inertia about the axis 1-1. 
Therefore, these coefficients represent the deflection, in 


176 HOMOGENEOUS BEAMS 


inches, of a beam 1 ft. long having a load of 1,000 lb. Mul- 
tiplying the proper coefficient by the cube of the span, in 
feet, and by the number of 1,000-lb. units in the given load, 
will give the deflection of a beam for any load and span. 

EXAMPLE.—What is the deflection of a 20-in, 65-Ib. 
I beam that carries a center load of 28,000 lb. and has a span 
of 20 ft.? 

SoLuTION.—The amount of deflection is obtained by mul- 
tiplying the coefficient of deflection for beams with center 
- loads (column 138 in the table of properties of I beams) 
by the cube of the span, in feet, and the number of 
1,000-lb. by units in the load. Hence, the deflection equals 
00000106 x 203 x 28992 — .237 in. 


SUDDENLY APPLIED LOADS 


In the formulas and investigations so far discussed, it 
has been assumed that the loads on the beams were laid 
gently in place. This, however, is not always the case, for 
the load may be suddenly or almost instantaneously applied, 
or it may even be dropped on the beam. Of course, in design- 
ing such beams, a large factor of safety may be employed, 
but if the load is dropped or very suddenly applied, this 
method is at best a matter of guesswork and experience. 

The investigation of sudden loads divides itself naturally 
into two classes. The first includes loads that are not 
raised above a beam and whose weight is suddenly applied 
to the beam. The second class of loads includes those that 
fall vertically on a beam, as when heavy boxes or crates are 
dropped on the beams of a floor. 

As the problems of the first class are the simplest to solve, 
they will be taken up first. When a load is placed suddenly 
on a beam, the stress produced is twice as great as if the 
same load had been at rest; that is, a beam to sustain a sud- 
denly applied load should have twice the transverse strength 
required to sustain the same load at rest. 

Often, a problem occurs concerning suddenly applied loads 
in which the beam has a quiet load, which is the dead load, 
and a suddenly applied load, which is the live load. Sucha 
problem should be solved as follows: 


HOMOGENEOUS BEAMS 177 


ExAMPLE.—Design an I beam to carry a uniformly dis- 
tributed load of 140 lb. per ft. on a span of 12 ft., and also 
a centrally concentrated, suddenly applied load of 3,700 1b. 

SoLUTION.—The bending moment due to the uniformly 

WI (14012) x12 


distributed load is “tie 3 =2,520 ft.-lb., ‘or 
2,520 X 12 =30,240 in.-lb. The concentrated load, if gently 
3,700 X 12 


applied, would cause a bending moment of ri 


=11,100 ft.-lb., or 11,100 X 12=133,200 in.-lb. Since, how- 
ever, the load is suddenly applied, it will produce stresses 
equivalent to twice this bending moment, or 133,200 2 
= 266,400 in.-lb. The total moment that the beam must 
be designed to withstand is therefore 266,400+30,240 
= 296,640=Ss. Since s=16,000, then, S=296,640~+ 16,000 
=18.54. Referring to the table on page 148, it will be found 
that a 9-in., 21.5-lb. I beam is required. 

The other class of loads referred to are those which drop 
on a beam from a distance above it. It is customary in 
considering the effect of a falling concentrated load to deter- 
mine the statical or quiet load concentrated at the center 
that would produce the same stress, and then to design the 
peam for this statical load. The formula used to accom- 
plish this is aE 

wiew(1+ Pts): 
in which W, is the static load, in pounds, concentrated at the 
center, that would produce the same stress in the beam as 
the falling load; W, the falling load, in pounds, that strikes 
the beam in the center of the span; h, the distance, in inches, 
that the load falls; d, the deflection of beam, in inches, pro- 
duced by load W statically applied; and a, the constant. 

The value of d is found as previously explained, while a is 
found by the formula 1 

ta 14.499 V2 
+. Ww 


in which W2 is the combined weight, in pounds, of beam and 
dead load that it supports; and W is the falling load. 


178 HOMOGENEOUS BEAMS 


From the construction of the formulas, it will be noted 
that the size of a beam required to sustain a certain falling 
load cannot be found direct. The size of beam must be 
assumed; then the formulas are used to ascertain whether 
the beam will meet the requirements. 

ExamPLe.—A 12-in., 40-lb. I beam carries, besides its 
own weight, a uniform load of 260 lb. per ft. The span is 
10 ft. Ifaload of 400 lb. drops on the beam from a distance 
of 18 in., will it develop a unit stress beyond the safe unit 
stress of 12,500 1b.? 

SoLuTION.—The total static load per foot on the beam 
is 260+ weight of beam per foot =260+40=300 lb. per ft. 
The total static load on the beam, therefore, is 300 X span 
=300 X10=3,000 lb. The deflection due to the falling load 
of 400 lb., according to column 18, of the table on page 000 
is .00000505 x 103 xk .4=.00202 in. The constant a thus 


equals 1 ae 1 9140 
1+.489 Like slates 
WwW 400° 
Therefore, 
2 ah 2X .2142« 18 
wiaw(1+ i) a. 
= 25,117 Ib. 


The maximum bending moment due to this load is 
25,117 X 10 ; 
————— = 62,792.5 ft.-lb., or 62,792.5 X 12 = 753,510 in.-lb. 


The maximum bending moment due to the static or dead 
_ 8,000 < 10 ; 
load is ue gore =3,750 ft.-lb., or 3,750 XK 12 =45,000 in.-lb. 
The total bending moment of both the static and sudden 
load is therefore 753,510+45,000=798,510 in.-lb.=Ss. 
From the table on page 148, S=41. Therefore, 798,510 

=41s, and s=798,510+41=19,476 lb. per sq. in. 

This is greater than 12,500, which was assumed as the 
allowable unit stress. Even if 16,000 lb. were taken as the 
allowable unit stress, the actual stress would still be too 
large and a beam of larger size would have to be assumed. 


WOOD AND CAST-IRON COLUMNS 1757. 


WOOD AND CAST-IRON COLUMNS 


WOODEN POSTS 
MATERIALS 


The kinds of timber usually employed for columns are the 
long-leaf and short-leaf yellow pines, red pine, white and 
red oak, spruce, hemlock, cypress, fir, and redwood. The 
timber generally preferred is the yellow pine, the long-leaf 
variety being stronger and more durable than the short- 
leaf. The disadvantage of this wood is that the resinous sap 
makes it very inflammable. 

The compressive strength of timber varies greatly, accord- 
ing to the amount of moisture it contains, a decrease in 
moisture resulting in increased strength. The percentage 
of moisture in wood is usually reckoned from the dry weight. 
Thus, if a certain piece of timber that weighed 165 lb. when 
green weighs 100 lb. when kiln dried, it would be said that 
in this instance the wood when green contained 65% of 
moisture. 

The drying of green wood does not effect an increase in 
strength until the moisture is decreased to a value amounting 
to about 20 to 30% of the weight of the dry material; that 
is, the strength of a piece of green wood when being dried 
remains constant until the moisture remaining in the piece 
is reduced to from about 20 to 30% of the weight of the dry 
wood, and from this stage the strength starts to increase. 

Experiments made by Tieman, the results of which are 
given in the Proceedings of the American Society for Testing 
Materials, show, for instance, the following variations in 
strength of long-leaf pine: If the strength of the green 
wood is taken as 1, then the strength of air-dried wood con- 
taining 12% moisture is 2.4, and that of kiln-dried wood 
containing 34% moisture is 2.9. 

Thus, it is important when consulting tables of strength 
of timber to know the percentage of moisture contained in 


180 WOOD AND CAST-IRON COLUMNS 


the samples tested. Generally, this percentage amounts 
to about 18% in ordinary commercial stock. 

The table on page 128 gives the ultimate compressive 
strengths of the more common kinds of timber, together 
with their moduli of elasticity. 


SHORT POSTS 


A post, or column, may in its elementary form be con- 
sidered as a cubical or rectangular block, as shown in Fig. 1. 
If the post does not exceed in length from six to ten times 
the smallest dimension of 
its cross-sectional area, it~ 
is designated asa short post, 
or column. 

The load that a short 
post may safely carry may 
be estimated by multiply- 
ing its sectional area, in 
square inches, by the safe 
unit compression of the 
material parallel to the 
grain. The ultimate unit 
values for compression are 
given in the table on page 
128. A factor of safety 
of. 5 is generally used, but 
in some instances it may 

Fic. 1 be good practice to use 6. 
The proper factor of safety to choose is usually governed 
by the conditions to be met. 

ExamPpLe.—A short post of Georgia yellow pine is 12 in. 
square and 6 ft. long. What safe load will it support while 
standing on end? The factor of safety is 5. 

SoLuTIon.—The ultimate strength of Georgia yellow pine 
is 5,000 lb.; hence, the safe unit compressive stress is 5,000+5 
=1,0001b. The area of the post is 12 in. x 12 in.=144 sq. in. 
Therefore, the safe load is 144 x 1,000 = 144,000 Ib. 

Posts under compression develop more strength if the end 
surfaces are true and level. The tendency then is to resist 


WOOD AND CAST-IRON COLUMNS 181 


compression equally, and not to crush at one place before 
the remainder of the section can be brought under compres- 
sion, as would be the case if the bearing surfaces were uneven 
and rough. 
LONG POSTS 

If the length of a post is over ten times its dinrabter or 
the width of its narrowest side, it is termed a long post. A 
post of this length, if not secured against 
yielding sidewise, is liable to bend before 
breaking, as shown in Fig. 2 (a). In 
this case, the compressive stress is not. | 
uniformly distributed over the cross- 
sectional area of the post, but will de- 
crease from a maximum value at the 
concave side of the post to a minimum 
value at the convex side. Or, if the 
bending proceed far enough, the com- 
pressive stress at the convex side may 
‘change into one of tension. In some 
cases, the post will split, as shown in 
Fig. 2 (b), the two halves bending inde- 
pendently. 

The formula generally used for long 
square or rectangular wooden columns 
with flat ends, deducted from elaborate 


tests made on full-sized specimens at the (b) 
Watertown Arsenal, is: ’ Fic. 2 
Si 
*s~T00d 


in which u is the ultimate strength of post per square inch of 
sectional area; s, the ultimate compressive strength of mate- 
rial, in pounds per square inch; /, the length of post, in 
inches; and d, the dimensions of least side of post, in inches. 

EXaMPLE.—A white-pine post with flat ends is 10 in. 
square and 20 ft. long. Using a factor of safety of 6, what 
safe load will the post support? 

SoLuTIon.—The ultimate compressive strength of white 
pine parallel to the grain is 3,500 lb. per sq. in. Inserting 
the several values in the formula, 


182 WOOD AND CAST-IRON COLUMNS 


3,500 X 240 
u= 3,500 — poaian la = 2,660 lb. 
100 x 10 


Since the factor of safety is 6, the safe bearing value per 
square inch of sectional area is 2,660+6=4434 lb. The 
area of the post being 100 sq. in., the safe load is 100 x 4434 
= 44,333 Ib. 

The column formulas in general use do not give a direct 
method of calculating the dimensions of a post that will 
safely support a given load. The usual method of obtain- 
ing this information is to assume values for the dimen- 
sions of the post; substitute these values in the formula, 
and then solve for u, the ultimate average compression per 
square inch of sectional area of post. If the assumed dimen- 
sions give a value of u that is satisfactory for the given con- 
ditions, they are accepted as ‘correct. If, however, the 
resulting value of u is smaller than desirable, it shows that 


the sectional area is too small. Larger dimensions must - 


then be chosen and the solution repeated until a satisfactory 
result is obtained. 

If, on the contrary, the value of u is much greater than the 
required ultimate strength per square inch of the post sec- 
tion, a smaller cross-sectional area is chosen and the corre- 
sponding value of wis found. After a few trials, a size that 
gives a satisfactory stress for the given conditions is found. 

ExaMpLeE.—Design to the nearest inch a white-oak post 
that is to be 15 ft. long and that is to carry a load of 40,000 Ib. 
with a factor of safety of 5. The post is to be square in 
cross-section. 

SoLUTION.—Since a factor of safety of 5 is to be used, 
the required post must crush under a load of 5X 40,000 
= 200,000 lb. The ultimate compressive strength of white 
oak, from the table on page 128, is 5,000 lb. As a trial, 
first try a 7” X7” post. 

Substituting the correct values in the preceding formula, 


z 5,000 x 15 x 12 . 
u= 5,000 — = 3,714 lb. per sq. in. 
100 X 7 
The ultimate strength of the post is therefore 3,714 


X7X7=181,986 lb. But the required’ ultimate strength is 


i 
2 


WOOD AND CAST-IRON COLUMNS 183 


200,000 lb. Therefore, a 7”X7” post is not strong enough. 
Next, try a 10”X10” post. Thus, 
5,000 X 15 x 12 


100 x 10 
The ultimate strength of this post is therefore 4,100 x 10 
xX 10=410,000 lb. Since this value is greater than 200,000, 
the post is safe; but perhaps a smaller post would also fill the 
requirements. Therefore, try an 8” X8” post. Thus, 


5,000 X 15 X 12 ; 
u=5,000 — = 3,875 lb. per sq. in. 
100 x8 


The ultimate strength of this post is therefore 3,875 X8X8 
= 248,000 lb. Since this result is greater than the required 
ultimate strength, the post is strong enough. Moreover, 
it has been found that a 7”X7” post is not strong enough. 
Therefore, unless the timber be cut to fractions of an inch, 
the 8” X 8” post is the smallest post that will fulfil the require- 
ments. 

If the formula just given is transposed so as to read 


u=5, = 4,100 lb. per sq. in. 


l oe 
u=s | 1————], the factor | 1———— } may be calculated 
100 d 100 d 


for various values of / and d and arranged in the form of a 
table, as shown on pages 184-186, that will prove of great 
assistance when using the formula. 

In this table, the first column contains various lengths 
of posts, in inches, and in the top horizontal row are values 
of the least thickness of a post, in inches. In order to find 
the factor for a given post, first find the nearest length in 
the first column; then proceed from this value to the right 
until a point is reached that is below the least dimension 
in width. The value found at this point is the factor desired. 

ExampLe.—A 12” X14” Georgia-pine post is 18 ft. long. 
What safe load will it carry? 

SoLutTion.—According to the table, the ultimate strength 
per square inch for Georgia pine is 5,000 lb. The length of 
the post, in inches, is 18 X 12=216, and the area of the post 
is 12X14=168 sq. in. According to the table, the factor 


l 
is 821. Inserting this value in the f la; st ——— 
is nserting this value in the formula, u s( 100 :) 


9T 


oT 


lat 


&T 


él 


006° | 068" | 8Z8" | E98" | EFS 
G68" | €88" | 698" | 08 

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¥68" | I88" | 798 

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£68" 

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€ a T 


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769° | 0S9" 
002’ | Lg9° 
902° | £99" 
SIZ | TL9° 
6IL' | 629° 
ZL | 989° 
T@Z' | 869° 
sez | O02" 
PPL | LOL’ 
OSL: | FIL 
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292 | 6ZL" 
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618° | $62" 
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ges’ | PIS" 
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WOOD AND CAST-IRON COLUMNS 187 


=5,000 X .821=4,105. The ultimate strength of the post 
is therefore 168 X 4,105 =689,640 lb. - With 5 as a factor of 
safety, the safe strength of the post is 689,640+5 
= 137,928 lb. : 


CAST-IRON COLUMNS 
COLUMN FORMULAS 


Cast-tron columns are most frequently used in buildings 
of moderate height, but in some cases they have been used 
in buildings of sixteen, and even more, stories. The uncer- 
tain strength of cast iron has compelled the adoption of a 
low unit stress per square inch, or, in other words, a high 
factor of safety. 

The safe loads that cast-iron columns will carry can be 
obtained by the use of the table on pages 188-189. The 
first column of this table gives the external diameters of 
various columns, and the second column, the several thick- 
nesses that the column of a given diameter is likely to 
possess. 

ExampLe.—A hollow, cylindrical column is 18 ft. long, 
and has an external diameter of 14 in. and an internal one 
of 114 in. (a) What safe load will it support? (6) What 
load will it support if the internal diameter is 11 in.? 


, 14—11.5 
SoLuTION.—(a) The thickness of the metal is — 3 


=1.25 in. Proceeding in the table from 14 in the first 
column to the value 1} in the second column, and then con- 
tinuing to the right until the column headed 18 is reached, 
the value 386 is found. This shows that the column will 
support a safe load of 386,000 lb. 

(6) In a column having an internal diameter of 11 in., 
the thickness of the metal is =u in. Proceeding 
to the right from the value 14 in the second column of the 
table, the value 454 is found in the column headed 18; hence, 
the safe load for this column is 454,000 Ib. 

It is thus evident that a difference in the thickness of the 


WOOD AND CAST-IRON COLUMNS 


188 


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139 


WOOD AND CAST-IRON COLUMNS 


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DIMENSIONS, IN INCHES, OF STANDARD CONNECTIONS 


b 


WOOD AND CAST-IRON COLUMNS 191 


metal of 1.5—1.25=.25 in. gives an increase in strength of 
454,000 — 386,000 = 68,000 Ib. 
The table may also be used for calculating the strength 
of columns when a factor of safety other than 8 is to be used. 
Let W be safe load given in the table: W; the safe load 
corresponding to another factor of safety; and 7, the new 
factor of safety. Then, 


W:Wi=7:8, 
8SW 
and W,=— 
f 
_ Wit 
8 


ExamPLe.—A hollow, cast-iron column of 13-in. external 
diameter and 1}-in. metal is 18 ft. long. Assuming that the 
factor of safety is 6, find the safe load. 

SoLUTION.—From the table W=343,000; hence, accord- 
ing to the formula, 


8 X 343,000 
W.= ara or = 457,000 1b., nearly. 


Many concerns have their own standard designs for column 
connections and brackets. These are usually embodied in 
tables that give the required dimensions. The accompany- 
ing table gives the standard dimensions of brackets on 
cast-iron columns for I-beam connections, both for double 
and for single beams. The top surface of the shelf should 
have a pitch away from the column of 4 in. to the foot to 
allow for the deflection of the beam. The top surface of 
the shelf should have a pitch away from the column of } in. 
to the foot to allow for the deflection of the beam. As the 
holes in the column are cored, it will usually be necessary 
to have the beams drilled in the field in order to insure 
alinement. 

In this table the values given in the columns marked 
A, B, C, etc. are the various dimensions for brackets, these 
dimensions being represented by corresponding letters in the 
figures accompanying the table. Thus, for a 12-in. I-beam 
connection, the distance from the bottom of the beam flange 


192 SAND AND CEMENT 


to the center of the outside bolt ot the vertical lug should 
be 3 in.; the pitch of the bolts, 3 in.; the projection of the 
bracket beyond the column, 44 in.; the depth of the ver- 
tical leg of the bracket, 7? in.; etc. 


SAND AND CEMENT 


aS 


CEMENTING MATERIALS 


Any substance that becomes plastic under certain treat- 
ment and subsequently reverts to a tenacious and inelastic 
condition may, in a broad sense, be termed a cement, How- 
ever, nearly all the cementing materials employed in build- 
ing construction are obtained by the heating, or calcination, 
as it is called, of minerals composed wholly or in part of lime. 
The different composition of these minerals, as well as the 
properties of the calcined products, enables the various 
resulting substances to be classified as. limes, hydraulic 
cements, plasters, and miscellaneous cements. Although all 
these materials have cementing properties, the term cement 
is commonly used to apply only to the group made up of 
hydraulic cements, hydraulic meaning that these substances 
possess the ability to set, or become hard, under water. 

Limes and hydraulic cements (commonly called simply 
cements) are composed essentially of oxide of calcium, or 
lime, generally called quicklime, with which may be com- 
bined certain argillaceous, or clayey, elements, notably silica 
and alumina, it being to these elements that the hydraulic 
properties of certain of these materials are due. The 
quantity of silica and alumina present in these substances 
enables them to be classified as common limes, hydraulic limes, 
and cements. 

The ratio of the quantity of sflica and alumina present in 
these materials to the quantity of lime is called the hydraulic 
index. In common limes, this index is less than y%; in 
hydraulic limes, it lies between 7% and 4%; and in cements, 
it exceeds y%. These limes merge into each other so grad- 
ually, however, that it is often difficult to distinguish the 
dividing line between them. 


SAND AND CEMENT 193 


LIMES 

The commercial varieties of lime may be classified as 
common, hydrated, and hydraulic. The common limes, 
also called quicklimes, may be subdivided into rich, or fat, 
lime, and meager, or poor, lime. 

Common Limes.—The grade of common lime known as 
fat, or rich, lime is almost pure oxide of calcium, CaO, and 
contains only about 5% .of impurities. It has a specific 
gravity of about 2.3 and a great affinity for water, of which 
it absorbs about one-quarter of its weight. This absorp- 
tion is accompanied by a great rise in temperature, by the 
lime bursting, and by the giving off of vapor. The lime 
finally crumbles into a powder. This powder occupies 
from two and one-half to three and one-half times as much 
volume as the original lime, the exact amount depending on 
its initial purity. When the lime is in this plastic state, it is 
said to be slaked. It is then unctuous and soft to the touch, 
and from this peculiarity it derives the name of fat or rich. 

Meager, or poor, lime consists of from 60 to 90% of pure 
lime, the remainder being impurities, such as sand or other 
foreign matter. These impurities have no chemical action 
on the lime, but simply act as adulterants. Compared with 
fat lime, poor lime slakes more slowly and evolves less heat. 
The resulting paste is also thinner and not so smooth, 
greatly resembling fat slaked lime mixed with sand. Poor 
lime is not so good for building purposes as fat lime, nor 
has it such extensive use. 

Hydrated Lime.—The class of lime called hydrated lime 
(calcium hydrate) is merely thoroughly slaked fat lime dried 
in the form of a fine powder, Ca(OH)»s. It is used exten- 
sively in conjunction with cement for making mortar, and 
also in the sand-lime brick industry. 

Hydraulic Limes.—Limes that contain enough quick- 
lime to slake when water is added, and enough clay or 
sand to form a chemical combination when wet, thus giving 
them the property of setting under water, are called hydraulic 
limes. 

Limes of this class are made by burning limestones con- 
taining from 5 to 30% of clay or sand. They are often 


194 SAND AND CEMENT 


considered as divided into three classes, namely, feebly 
hydraulic, ordinarily hydraulic, and eminently hydraulic, in 
proportion to the quantity of argillaceous materials present. 
The slaking qualities vary from slaking in a few minutes 
with considerable heat after water is added, in the feebly 
hydraulic, to slaking only after many hours, with practically 
no evolution of heat and without cracking or powdering, in 
the eminently hydraulic. The time of setting under water 
also varies from setting as hard as soap in 2 years, with the 
feebly hydraulic, to becoming as hard as stone in 3 or 4 days, 
with the eminently hydraulic. If carbonate of magnesia is 
present in the lime, it reduces the energy of the slaking, 
but increases the rapidity of the setting and the ultimate 
strength when set. 


CEMENTS 


Cement may be divided into four general classes: Port- 
land, natural, puzzolan (also called pozzuolana), and mixed. 
The relative importance of each cement is indicated by the 
order in which it is named. 

Portland cement may be defined as the product resulting 
from the process of grinding an intimate mixture of cal- 
careous (containing lime) and argillaceous (containing clay) 
materials, calcining (heating) the mixture until it starts to 
fuse, or melt, and grinding the resulting clinker to a fine 
powder. It must contain not less than 1.7 times as much 
lime by weight as it does of those materials which give the 
lime its hydraulic properties, and must contain no materials 
added after calcination, except small quantities of certain 
substances used to regulate the activity or the time of 
setting. 

Natural cement is the product resulting from the burning 
and subsequent pulverization of an argillaceous limestone 
or other suitable rock in its natural condition, the heat of 
burning being insufficient to cause the material to start to 
melt. 

Puzzolan cement is a material resulting from grinding 
together, without subsequent calcination, an intimate 
mixture of slaked lime and a puzzolanic substance, such as 


SAND AND CEMENT 195 


blast-furnace slag or volcanic scoria. The only variety of 
puzzolan cement employed at all extensively in American 
practice is slag cement. This cement is made by grinding 
together a mixture of blast-furnace slag and slaked lime. 
The slag used for this purpose is granulated, or quenched, 
in water as soon as it leaves the furnace, which operation 
drives off most of the dangerous sulphides and renders the 
slag puzzolanic. 

Mixed cements cover a wide range of products obtained 
by mixing, or blending, the foregoing cements with each 
other or with other inert substances. Sand cements, im- 
proved cements, and many second-grade cements belong 
to this class. Mixed cements. however, are of comparatively 
little importance. 

Properties of Cements——The hydraulic cements differ 
from the limes in that they do not slake after calcination, 
and that they set, or harden, under water. They can be 
formed into a paste with water without any sensible increase 
in volume and with little, if any, disengagement of heat. 
They do not shrink appreciably in hardening, so that the sand 
and broken stone with which they are mixed are employed 
merely through motives of economy and not, as with limes, 
of necessity. 

The color uf the different grades of cement is variable, 
but in certain cases it is distinctive. Portland cement is a 

-dark-bluish or greenish gray; if it is a light yellow, it may 
indicate underburning. Natural cement ranges in color 
from a light straw, through the grays, to a chocolate brown. 
Slag cement is gray with usually a tinge of lilac. In general, 
however, the color of cement is no criterion of its quality 

Cement is packed either in wooden barrels or in cloth or 
paper bags, the latter being the form of package most com- 
monly employed. A barrel of Portland or of slag cement 
contains the equivalent of 4 bags, while but 3 bags of natural 
cement equals a barrel. The average weights of the various 
cements are given in the table on page 196. 

In proportioning mortar or concrete by volume, the com- 
mon assumption is that a bag of Portland cement occupies 
OG, CU, iit: 


596 SAND AND CEMENT 
AVERAGE WEIGHTS OF HYDRAULIC CEMENTS 


Net Net Weight per Cubic Foot 
Kind of | Weight | Weight Pounds 

Cement of Bag of Barrel 
Pounds Pounds Packed Ines 


Portland ... 94 376 100-120 70-90 
Natural. ... 94 . 282 75-95 45-65 
SOG fos tk 823 330 80-100 55-75 


Portland cement may be distinguished by its dark color, 
heavy weight, slow rate of setting, and greater strength. 
Natural cement is characterized by lighter color, lighter © 
weight, quicker set, and lower strength. Slag cement is 
somewhat similar to Portland, but may be distinguished 
from it by its lilac-bluish color, by its lighter weight, and by 
the greater fineness to which it is ground. 

Portland cement is adaptable to any class of mortar or 
concrete construction, and is unquestionably the best mate- 
rial for all such purposes. Natural and slag cements, how- 
ever are cheaper, and, under certain conditions, may be sub- 
stituted for the more expensive Portland cement. All 
heavy construction, especially if exposed, all reinforced- 
concrete work, sidewalks, concrete blocks, foundations of 
buildings, piers, walls abutments, etc. should be made 
with Portland cement. In second-class work, as in rubble 
masonry, brick sewers, unimportant work in damp or wet 
situations, or in heavy work in which the working loads 
will not be applied until long after completion, natural 
cement may be employed to advantage. Slag cement is 
best adapted to heavy foundation work that is immersed 
in water or at least continually damp. This kind of cement 
should never be exposed directly to dry air, nor should it be 
subjected either to attrition or impact. 


SAND AND CEMENT 197 


SAND AND ITS MIXTURES 


SAND 


Sand is an aggregation of loose grains of crystalline struc- 
ture, derived from the disintegration of rocks. It is called 
stlicious, argillaceous, or calcareous, according to the char- 
acter of the rock from which it is derived. Sand is obtained 
from the seashore, from the banks and beds of rivers, and 
from land deposits. The first class, called sea sand, contains 
alkaline salts that attract and retain moisture and cause 
efflorescence in brick masonry. This efflorescence is not at 
first apparent but becomes more marked as time goes on. 
It can be removed temporarily at least by washing the stone- 
work in very dilute hydrochloric acid. The second, termed 
river sand, is generally composed of rounded particles, and 
may or may not contain clay or other impurities. The third, 
called pit sand, is usually composed of grains that are more 
angular; it often contains clay and organic matter. When 
washed and screened, it is a good sand for general purposes. 

Sand is used in making mortar because it prevents excess- 
ive shrinkage and reduces the quantity of lime or cement 
required. Lime adheres better to the particles of sand than 
it does to its own particles; hence, it is considered that sand 
adds strength to lime mortar. On cement mortar, on the 
contrary, sand has a weakening effect. 

Properties of Sand.—The weight of sand is determined by 
merely filling a cubic-foot measure with dried sand and 
obtaining its weight. Dry sand weighs from 80 to 120 lb. 
per cu. ft.; moist sand, however, occupies more space and 
weighs less per cubic foot. The weight of sand is more or 
less dependent on its specific gravity and on the size and shape 
of the sand grains, but, other things being equal, the heaviest 
sand makes the best mortar. 

The specific gravity of sand ranges from 2.55 to 2.80. 
For all practical purposes the specific gravity may be 
assumed to be 2.65 with little danger of error. 

By percentage of voids is meant the amount of air space 
in the sand. Structurally, it is one of the most important 


198 SAND AND CEMENT 


properties of sand. The greater these voids, the more 
cement paste will be required to fill them in order to give a 
dense mortar. The percentage of voids may be determined 
by observing the quantity of water that can be introduced 
into a vessel filled with sand, but it is best computed as 
follows: 


- 100 X weight per cubic foot 
percentage of voids=100— sin wna daan HX =— 
62.5 X specific gravity 


EXAMPLE.—What is the percentage of voids in a sand 
having a specific gravity of 2.65 and weighing 105 lb. per 
cu. ft.? : 

SoLuTION.—Substituting in the formula, the percentage 


of voids is 100 X 105 


62.5 X 2.65 

The percentage of voids depends principally on the size 
and shape of the sand grains and the gradation of its fine- 
ness, and hence will vary from 25 to 50%. Sand contain- | 
ing over 45% of voids should not be used to make mortars. 

The shape of the grains of sand is of chief importance in 
the influence that the sand exerts on the percentage of voids. 
Obviously, a sand with rounded grains will compact into a 
more dense mass than one whose grains are angular or flat 
like particles of mica. Therefore the more nearly the grains 
approach the spherical in shape, the more dense and strong 
will be the mortar. This fact is contrary to the common 
opinion on the subject. 

The fineness of sand is determined by passing a dried 
sample through a series of sieves having 10, 20, 30, 40, 50, 
74, 100, and 200 meshes, respectively, to the linear inch: 
The result of this test, expressed in the amount of sand 
passing each sieve, is known as the granulometric composition 
of the sand. Material that does not pass a 4-in. screen is 
not considered to be sand, and should be separated by 
screening. Sand that is practically all retained on a No. 30 
sieve is called coarse, while 80 or 90% of sand known as 
fine will pass through this sieve. Fine sand produces a 
weaker mortar than coarse sand, but a mixture of fine and 
coarse sand will surpass either one. 


=100—63.4=36.6 


SAND AND CEMENT 199 


The purity, or cleanness, of sand may be roughly ascer- 
tained by rubbing it between the fingers and observing how 
much dirt remains. To determine the percentage of the 
impurities more accurately, a small dried and weighed sam- 
ple is placed in a vessel and stirred up with water. The 
sand is allowed to settle, the dirty water poured off, and the 
process repeated until the water pours off clear. The sand 
is then dried and weighed. The loss in weight gives the 
quantity of impurities contained in the sand.. The presence 
of dirt, organic loam, mica, etc. is decidedly injurious and 
tends to weaxen the resulting mortar. Clay or fine min- 
eral matter in small proportions may actually result in 
increased strength, but excessive quantities of these mate- 
rials may be a possible source of weakness. The best mod- 
ern practice limits the quantity of ampurities found by this 
washing cest to 5%. 

It is advisable, prior to the selection of a sand, to deter- 
mine what its strength will be when made into mortar. 

Preparation of Sand.—Sand is prepared for use by (1) 
screening. to remove the pebbles and coarser grains the 
fineness of the meshes of the screen depending on the kind 
of work in which the sand is to be used; (2) washing, to 
remove salt, clay, and other foreign matter; and (3) drying 
if necessary. When dry sand is required, it is obtained by 
evaporating the moisture either in a machine, called a sand 
dryer, or in large, shallow, iron pans supported on stones, 
with a wood fire placed underneath. 


LIME AND CEMENT MORTARS 


Mortars are composed of lime or cement and sand mixed 
to the proper consistency with water. The proportions of 
the ingredients depend on the character of the work in 
which the mortar is to be used. 

In proportioning mortar, the quantities of the separate 
ingredients are usually designated by a ratio, such as 1-1, 
1-2, 1-8, etc. Thus, 1-2 signifies that 1 part of lime or 
cement is used to 2 parts of sand; etc. For great accuracy 
these measurements should be made by weight, but they are 
usually made by volume, which is almost the same thing. 


200 . SAND AND CEMENT 


Lime Mortars.—In lime mortar, besides effecting an econ- 
omy, the presence of sand is necessary to prevent the shrink- 
age that would otherwise occur during the hardening of the 
paste. 

When a mortar is made of lime and sand, enough lime 
should be present to just cover completely each grain of 
sand. An excess of lime over this quantity will cause the 
mortar to shrink excessively on drying, while a deficiency 
of lime will produce a weak and crumbly mortar. The cor- 
rect quantity of lime depends on the character of the ingre- 
dients, the method of treatment, and, to some extent on 
the judgment of the builder. The mixtures employed vary 
from 1-24 to 1-5. Building laws in many municipalities 
require the use of a 1-3 mixture, and for most materials this 
proportion will be found satisfactory, although for rich, fat 
limes a 1-34 or a 1-4 mixture is sometimes preferable. 

In mixing lime mortar,a bed of sand is made in a mortar 
box, and the lime distributed as evenly as possible over it, 
first measuring both the lime and the sand in order that 
the proportions specified may be obtained. The lime is 
then slaked by pouring on water, after which it should be 
covered with a layer of sand, or, preferably, a tarpaulin, 
to retain the vapor given off while the lime is undergoing 
the chemical reaction of slaking. Additional sand is then 
used, if necessary, until the mortar attains the proper pro- 
portions. 

Care should be taken to add just the proper quantity of 
water to slake the lime completely to a paste. If too much 
water is used, the mortar will never attain its proper strength, 
while if too little is used at first, and more is added during 
the process of slaking, the lime will have a tendency to chill, 
thereby injuring its setting and hardening properties. 
Rather than make up small batches, it is considered better 
practice to make lime mortar in large quantities and to keep 
it standing in bulk so that it can be used as needed. 

Lime mortar is employed chiefly for brickwork of the 
second class and its use is continually decreasing as that of 
cement increases. It is absolutely unsuitable for any 
important construction, because it possesses neither strength 


SAND AND CEMENT 201 


nor the property of resisting water. It cannot be used 
in damp or wet situations, nor should it ever be laid in cold 
weather, as it is very susceptible to the action of frost, being 
much injured thereby. Moreover, since it hardens by the 
action of dry air, only the exterior of lime mortar ever 
becomes fully hardened, so that anything like a concrete 
with lime as a matrix is impossible. However, for second- 
class brickwork, such as is commonly used in the walls of 
smaller buildings, lime mortars are economical and suf- 
ficiently good. 

The strength of lime mortars is extremely variable, depend- 
ing on the ingredients themselves and on their treatment, 
environment, etc. Moreover, it is unsafe to figure a lime- 
mortar joint as possessing much strength, since only a part 
of the joint is hardened and capable of developing any 
strength at all. The tensile strength of thoroughly hard- 
ened 1-3 lime mortars averages from 40 to 70 lb. per sa. in., 
and the compressive strength from 150 to 300 lb. 

Cement Mortars.—The sand for all mortars should be 
clean, of suitable size and granulometric composition. For 
structures designed to withstand heavy unit stresses, or for 
those intended to resist either the penetration of moisture 
or the actual pressure of water, the selection of the sand 
should be most carefully made. Generally, it is not advi- 
sable to use asand containing over 5% of loam by the wash- 
ing test, nor one that soils the fingers when it is rubbed 
between them. Very fine sand, such as is found on the sea- 
shore, should not be employed in mortar unless it is intended 
simply for pointing or for grouting. 

A simple method of determining the best sand for cement 
mortar is to prepare mixtures of the cement, sand, and 
water, using the same quantities in each case, and fhen to 
place each mixture in a measure; that mixture giving the 
least volume of mortar may be considered to contain the 
most desirable sand for use. 

Limestone screenings, brick dust, crushed cinders, etc., 
are.sometimes substituted for sand in making mortars, and, 
if care is taken in their selection, they may prove economical 
and entirely suitable for certain purposes. 


202 | SAND AND CEMENT 


The theory of the composition of a correctly proportioned 
mortar is that the cement paste will just a little more than 
fill all the voids between the particles of sand, thus giving 
an absolutely dense mortar at the least expense. The cor- 
rect proportion of cement to sand, therefore, is more or less 
variable, depending on the granulometric composition of the 
sand. Since, however, Portland-cement paste that has set 
weighs nearly as much as sand, and since the average sand 
contains about 30 to 40% of voids, it is evident that 1-3 
mixtures most nearly approach the best and most economical 
proportion. 

Mortars, however, are made in proportions varying from 
1-1 to 1-8. The richer mixtures are used for facing, point- 
ing, waterproofing, granolithic mixtures, etc., the 1-2 mix- 
ture being usually made for such purposes. The leaner 
mixtures are used for rough work, filling, backing, etc., 
but should never be employed where either much strength 
or much density is desired. Natural-cement mortars are 
commonly made 1 part of sand less than Portland-cement 
mortars intended for the same purpose; that is, where a 
1-3 Portland-cement mortar would be used, a 1-2 natural 
mortar would be required, although natural-cement mortars 
should be decreased by about 2 parts of sand to equal the 
strength of Portland. In other words, a 1-4 Portland 
mortar closely equals the strength of a 1-2 natural mortar. 
Puzzolan cements are usually proportioned the same as 
Portlands. 

Cements are commonly proportioned by volume, the unit 
volume of the cement barrel being assumed. Ifa1-3 mortar 
is desired, a box having a capacity of 10.8 cu. ft. is filled 
with sand and mixed with 4 bags or 1 bbl. of cement. A 
box 3 ft. 3x4 in. square and 1 ft. deep will have a capacity 
of very nearly 10.8 cu. ft. and, besides, makes a convenient 
size of box for actual work. 

For general purposes, the mortar should be of a plastic 
consistency—firm enough to stand at a considerable angle 
yet soft enough to work easily. Wet mortars are easiest to 
work and are the strongest. However, they are subject to 
greater shrinkage, are slower setting, and are more easily 


SAND AND CEMENT 203 


attacked by frost. Dry mortars, on the other hand, are 
often friable and porous. 

In the following table are given the quantities of materials 
required to produce 1 cu. yd. of compacted mortar. The 
proportions are by volume, a cement barrel being assumed 
to contain 3.6 cu. ft. 


MATERIALS REQUIRED PER CUBIC YARD OF MORTAR 


i a i ee 


; : Portland Cement} Loose Sand 

Kind of Mixture Barrels Cubic Yards 
} 1-1... 4.95 .65 
- 4-2.. 3.28 88 
er i—3. 2.42 1.01 
1-4... 1.99 1.06 
: 1-5. . 1.62 1.11 
1-6.. 1.34 1.15 
1-7... 1.18 1 i) Wf 
1-8.. 1.05 1.18 


XAMPLE.—How much cement and sand will be required 

to obtain 8.5 cu. yd. of 1-3 Portland-cement mortar? 
SoL_uTion.—According to the table, 1 cu. yd. of a 1-3 Port- 
land-cement mortar requires 2.42 bbl. of cement; therefore, 
8.5 cu. yd. will require 8.5 2.42=20.57 bbl. of cement. 
Also, since 1 cu. yd. of a mixture of this kind requires 1.01 cu. 
yd. of sand, the quantity of sand required will be 8.51.01 

=8.59 cu. yd. 

Mortar that is to be mixed by hand is prepared on a plat- 
form or in a mortar box. The sand is first measured by 
[ means of a bottomless box with handles on the sides. After 
filling the box, the sand is struck off level, the box lifted up, 
and the sand spread in a low, flat pile. The required num- 
ber of bags of cement are then emptied on the sand and 
spread evenly over it. The pile is then mixed with shovels, 
working through it not less than four times. After this 
operation, the dry mixture is formed into a ring, or crater, 
and the water intended to be used is poured into the center. 
The material from the sides of the basin is then shoveled 


204 SAND AND CEMENT 


into the center until the water is entirely absorbed, after 
which the pile is worked again with shovels and hoes until 
the mixture is uniform and in a plastic condition. 

Another method of mixing, where a mortar box is used, 
is to gather the mixed dry materials at one end of the box 
and pour in the water at the other end drawing the mixture 
into the water with a hoe, a little at a time, and hoeing until 
a plastic consistency is obtained. 

Properties and Uses of Cement Mortars.—The strength of 
a mortar is measured by its resistance to tensile, compressive, 


TENSILE STRENGTH OF CEMENT MORTARS 


Tensile Strength, in Pounds 
per Square Inch 

Proportions 

Portland Cement | Natural Cement 

Cement Sand 
ae Parts 7 da. 28 da.|3 mo.| 7 da. |28 da.|3 mo. 

1 1 450 | 600 | 610 | 160 | 245 | 280 
it 2 280 | 380 | 395 | 115 | 175 | 215 
1 3 170 | 245 | 280 85 | 130 | 165 
1 4 125 | 180 | 220 60 | 100 | 135 
1 5 80 | 140 | 175 40 75 | 110 
1 6 50 | 115 | 145 25 
1 Z 30 95 | 120 15 50 75 
1 8 20 70 | 100 10 45 65 


cross-breaking, and shearing stresses, and also by determi- 
nations of its adhesion to inert surfaces, its resistance to 
impact, abrasion, etc. There is no definitely fixed ratio 
between the strength of mortar subjected to these different 
stresses, but there is nevertheless a close relation between 
them, so that, practically, it may be assumed that if a mor- 
tar shows either abnormally high or low values in any one 
test, the same relation will develop when tested under other 
stresses. In practice, therefore, the strength of mortar is 


SAND AND CEMENT 205 


commonly determined through its resistance to tensile 
stresses, and its resistance to other forms of stress is com- 
puted from these results. 

The tensile strength of mortar has been shown to vary 
with the character of its ingredients, with its consistency, 
its age, and with many other factors. In the accompany- 
ing table is given a fair average of the tensile strength that 
may be expected from mortars of Portland and natural 
cements that are made in the field and with a sand of fair 
quality but not especially prepared. 

The strength of Portland-cement mortar increases up to 
about 3 mo.; after that period, it remains practically con- 
stant for an indefinite time. Natural-cement mortar, on 
the other hand, continues to increase in strength for 2 or 
3 yr., its ultimate strength being about 25% in excess of 
that attained in 3 mo. The strength of slag-cement mortar 
averages about three-quarters of that of Portland-cement 
mortar. 

The compressive strength of cement mortars is usually 
given in textbooks as being from eight to ten times the tensile 
strength. This value is rather high for the average mortar, 
a ratio of from 6 to 8 being one more nearly realized in prac- 
tice. The ratio increases with the age and richness of the 
mortar, and varies considerably with the quality of the sand. 
Portland-cement mortars of 1-3 mixture that are 3 mo. old 
develop, on an average, a compressive strength of about 
1,800 lb. per sq. in., while 1-2 natural-cement mortars 
average about 1,600 lb. 

The strength of mortars in cross-breaking and shear may 
be taken at about one and one-half to two times the tensile 
strength, with a fair amount of accuracy. 

The adhesion of mortars to inert materials varies both 
with the character of the mortar and with the roughness 
and porosity of the surfaces with which they are in contact. 
The adhesion of 1-2 Portland-cement mortar, 28 da. old 
to sandstone averages about 100 lb. per sq. in.; to limestone, 
75 lb.; to brick, 60 lb.; to glass, 50 lb.; and to iron or steel, 
75 to 125 Ib. Natural-cement mortars have nearly the 
same adhesive strength as those made of Portland cement. 


206 . SAND AND CEMENT 


In bricklaying and in other places in which mortar is 
employed it is frequently desired to use a material that is 
more plastic or smoother than pure cement mortar. This 
quality is usually obtained by adding from 10 to 25% of 
lime to the mortar. This addition of lime not only renders 
the mortar more plastic, and hence easier to work, but also 
increases both its adhesive strength and its density, which 
assists in making the mortar waterproof. Hydrated lime 
is to be preferred for use in cement mortar, because its 
complete slaking is assured. Hydrated lime may also be 
readily handled and measured on the work. 

Occasionally, small quantities of cement are added to lime 
mortars so as to make them set quicker and to increase their 
strength. Such mixtures, however, are not especially 
economical nor are they convenient in practice. 

Retempering of Mortar.—Mortar composed of cement, 
sand, and water soon begins to set and finally becomes hard. 
When it is desired to use this material, more water has to 
be added and the mixture worked until it again becomes 
plastic. This process is called retempering. Laboratory 
tests generally show that retempering slightly increases the 
strength of mortar, but the reworking is more thorough as 
a rule in the laboratory than would be the case in actual 
work. Any part of the hardened mortar that is not retem- 
pered is a source of weakness when incorporated in the 
building. The adhesive strength of cement, moreover, is 
greatly diminished. by this process For these reasons, 
it is generally inadvisable to permit the use of retempered 
mortars; but if they are allowed, great care should be taken 
to see that the second working is thorough and complete. 

Laying Mortar in Freezing Weather.—Frost or even cold 
has a tendency to retard greatly the set of cement mortars. 
When the temperature, moreover, is so low that the water 
with which the mortar is mixed freezes before it combines 
with the cement, it may, if care is not exercised, result in 
complete destruction of the work. A single freezing is not 
particularly harmful. because when thawing occurs, the 
arrested chemical action continues. A succession of alter- 
nate freezings.and thawings, however, is extremely injurious, 


. 


SAND AND CEMENT 207 


Nevertheless, Portland-cement mortars may be laid even 
under the worst conditions if certain precautions are ob- 
served, but mortars of natural cement should never be used 
in extremely cold weather, as they are generally completely 
ruined by freezing. 

The bad results that arise during mild frosts may be suc- 
cessfully guarded against by heating the sand and water 
and by using a quick-setting cement mixed rich and as dry 
as possible. In extremely cold weather, salt must be added 
to the water, so as to convert it into a brine that requires a 
temperature lower than 32° F. to freeze it. The common 
rule for adding salt is to use a quantity equal to 1% of the 
weight of the water for each degree of temperature that is 
expected below 33° F. Thus, at 32° F., a 1% solution would 
be used, while at 25°, an 8% solution would be required. 
Solutions greater than 12% should not be employed, and 
if a temperature below 20° F. is expected, heat must be used 
in addition to the salt. The finished work should also be 
protected with canvas or straw. Manure should not be 
used for this purpose, because the acids it contains tend to 
rot the cement. Unless the conditions are such as to make 
it imperative, it is not advisable to lay mortars during 
freezing weather. 

Shrinkage of Mortars.—Cement mixtures exposed to the 
air shrink during the process of hardening, while those 
immersed in water tend to expand. The shrinkage of 
ordinary cement mortars is slight, and when they are used 
as a bonding material it need not be considered. When 
used as a monolith, as in sidewalks, shrinkage is guarded 
against by keeping the mortar wet during setting. This 
can be done by covering with moist straw or by sprinkling 
the mixture with water. 

Grouting.— By grouting is meant the process of filling spaces 
in masonry with a thin, semifluid mixture known as grout. 
This mixture consists of cement, 1 or 2 parts of sand, and an 
excess of water. Grout can be used for filling the voids in 
walls of rubble masonry for backing arches and tunnels, and 
for filling the joints between paving brick. In fact, it can be 
used in all places where mortar cannot be laid in the ordinary 


208 TESTS ON CEMENT 


manner. When hardened, grout is weak, friable, and porous. 

Coloring of Mortars.—Colors are often used in mortars to 
effect contrasts, or to subdue the glaring tone of cement in 
sidewalks or in similar situations. Red lead weakens 
mortar and should not be used. The color of hardened 
mortar is quite different in appearance from one that is 
still wet, so that where it is important to secure the cor- 
rect tints, preliminary trials should be made until the pro- 
portions desired have been determined. 

The various materials employed to produce different colors 
in mortar, together with the quantity required per barrel of 
cement, are as follows: For gray, 2 lb. of lampblack; for 
black, 45 lb. of manganese dioxide; for blue, 19 lb. of ultra- 
marine; for red 22 lb. of iron oxide; for bright red, 22 lb. of 
Pompeian or English red; and for violet, 22 lb. of violet oxide 
of iron. 


TESTS ON CEMENT 


FIELD INSPECTION 


In order to determine correctly the structural value of a 
shipment of cement, an examination in the field is very 
necessary. A number of packages of cement should be 
weighed at intervals, and the average weight should never 
be permitted to fall below 94 lb. per bag, since mortar and 
concrete are usually proportioned on the assumption of this 
weight. Each package should also be plainly marked with 
the brand and name of the manufacturer; those not branded 
should be discarded, and, if possible, a mixture of different 
brands should be avoided. 

A possible indication of inferiority is the presence of lumps 
throughout the bulk of the material. On standing, cement 
gradually absorbs moisture from the air. At first this mois- 
ture is present in merely a minute and harmless state, but 
eventually it combines chemically with the cement; that is, 
in the same manner as when cement and water are actually 
mixed together in practice. In the first condition, lumps 
usually appear, but they are so soft that they may be readily 


TESTS ON CEMENT 209 


crushed with the fingers, and of course would be entirely 
broken up when mixed into mortar. When, however, the 
cement contains lumps that are hard and pebble-like and that 
can be crushed only with considerable effort, it indicates 
that chemical action has begun. Cement containing any 
appreciable amount of these hardened lumps is generally of 
decidedly inferior quality, and it should never be permitted 
to enter any important part of a structure. 

Storing cement too long will tend to weaken it. Cement 
from 2 to 6 mo. old is usually the safest and will produce 
the best results. 

The color of Portland cement, ranging from bluish to 
yellowish gray, affords no indication of quality except in 
cases where different shipments or different parts of the 
same shipment show a variation in color, thus pointing to a 
lack of uniformity. 


SAMPLING 

In securing a sample for testing, the essential point is to 
get one that will fairly represent the entire shipment whose 
qualities are to be determined. The common practice is to 
take a small portion of material from every tenth barrel, or, 
what is the same thing, from every fortieth bag. When tests 
are to be made, however, on a shipment of only a few barrels, 
more packages than one in ten should be opened; and when 
the shipment is large, say over 150 bbl., it should be sub- 
divided and each portion tested separately. The bags 
selected should be taken at random and from different layers 
and not all from one part of the pile. 

The cement, moreover, should be taken not only from the 
top of the packages, but from the center and sides as well. 
When the cement is contained in barrels, a sampling auger 
is used to extract the sample, a hole being bored in the staves 
midway between the heads. 

After the samples of cement have been taken from the 
packages they are thoroughly mixed in a can or basin, and 
this mixed sample is used for the various tests. To make 
a complete series of tests, the sample should contain from 6 
to 8 lb. The cement, after sampling and before testing, 


ae TESTS ON CEMENT 


must be well protected, as exposure to heat, cold, dampness, 
or any other abnormal condition may seriously affect the 
results. 


PURPOSE AND CLASSIFICATION OF TESTS 


In order that a mortar or a concrete made with cement 
shall give good results in actual construction it must possess 
two important properties, namely, strength and durability. 
The primary purpose of cement testing, therefore, is to deter- 
mine whether any particular shipment of cement possesses 
sufficient strength and durability to admit of its use in 
construction. 

A determination of the quality of cement necessitates the 
employment of several tests, which may be classified as 
primary tests and secondary tests. The former tests, which 
include tests for soundness and tensile strength, are made to 
give directly a measure of the essential qualities of strength 
and durability. Unfortunately, neither of these tests is 
capable of being made with precision. Therefore, the second- 
ary tests, which include tests to determine the time of setting, 
the fineness, the specific gravity, and the chemical analysis, 
are made to obtain additional information in regard to the 
character of the material. However, with the possible 
exception of the test of time of setting, the secondary tests 
have but little importance and only indicate by their results 
indirectly the properties of the material. 


PRIMARY TESTS 
TESTS FOR SOUNDNESS 


Soundness may be defined as the property of cement that 
tends to withstand any forces that may operate to destroy 
or disintegrate it. This property of soundness, or, as it is 
sometimes called, constancy of volume, is the most important 
requisite of a good cement. 

The most common cause of unsoundness in Portland 
cement is an excess of free or uncombined lime, which 
crystallizes with great increase of volume, and thus breaks 
up and destroys the bond of the cement. This excess of lime 


ra eee yy re 


Pree Ora ae 


a -- 


Ce Ne a eal 


TESTS ON CEMENT 211 


may be due to incorrect proportioning or to insufficient grind- 
ing of the raw materials, to underburning, or to lack of 
sufficient storing before use, called seasoning. A certain 
amount of seasoning is usually necessary, because almost 
every cement, no matter how well proportioned or burned 
it may be, will contain a small amount of this excess of lime, 
which, on standing, will absorb moisture from the air, slake, 
and become inert. 

Excess of magnesia or the alkalies may also cause unsound- 
ness, but the ordinary cement rarely contains a sufficient 
amount of these ingredients to be harmful. Sulphate of 
lime is occasionally responsible for unsoundness, but this 
ingredient usually acts in the opposite direction, tending to 
make sound a cement that otherwise might disintegrate. 

The property of soundness is determined in one or more 
of three ways: by measurements of expansion, by normal 
tests, and by accelerated tests. 

Measurements of expansion are made by forming specimens 
of cement, usually in the shape of prisms, and measuring the 
change in volume by means of a micrometer screw. At the 
present time, however, it is believed that expansion is not a 
sure index of unsoundness, so that this test is seldom 
employed. 

Normal tests consist in making specimens of cement mixed 
with water, preserving them in air or in water under normal 
conditions, and observing their behavior, The common 
practice is to make from a paste of neat, or pure, cement on 
glass plates about 4 in. square, two circular pats, about 3 in. 
in diameter, 4 in. thick at the center, and tapering to a 
thin edge. These pats are kept in moist air for 24 hr.; 
then one of them is placed in fresh water of ordinary tempera- 
ture and the other is preserved in air. The condition of the 
pats is observed 7 da. and 28 da. from the date of making, 
and thereafter at such times as may be desired. 

The most characteristic forms of failure are illustrated in 
Figs. 1 and 2. 

Fig. 1 (a) shows a pat in good condition. 

Fig. 1 (d) illustrates shrinkage cracks that are due, not to 
inferior cement, but to the fact that the pat has been ailowed 


212 TESTS ON CEMENT 


to dry out too quickly after being made. Pats must be kept 
in a moist atmosphere while hardening, or these cracks, 
indicative merely of careless manipulation will develop. 

Fig. 1 (c) shows cracks that are due to the expansion of the 
cement. This condition is common in the air pats, and is 
not indicative of injurious properties. Pats kept in water. 
however, should not show these cracks. 


(d) | (e) (f) 
IG 


Fig. 1 (d) shows cracking of the glass plate to which the pat 
is attached. This cracking is caused by expansion or con- 
traction of the cement, combined with strong adhesion to the 
glass. It rarely indicates injurious properties. 

Fig. 1 (e) illustrates blotching of the pats, the cause of 
which should always be investigated by chemical analysis 
or otherwise, which may or may not warrant the rejection of 
the material. Slag cements or cements adulterated with slag 
invariably show this blotching. 


—_— -_ Ye): 


TESTS ON CEMENT 213 


Fig. 1 (f) shows the radial cracks that mark the first stages 
of disintegration. Such cracks should never occur with 
good material. They are signs of real failure, and cement 
showing them should never be used. 

Fig. 2 shows three pats that, 
for different reasons, have left 
the glass plate on which they 
were made. The disk shown in 

(a) left the plate because of 
J lack of adhesion; the one in (b), 
through contraction; and the one 
in (c), through expansion. The 
_ condition illustrated in (a) is 
“ never dangerous in either air or 
water; that in (c) is only danger- 
Fic. 2 ous when existing in a marked 
degree; and that in (b) hardly ever occurs in water, but in 
air it often indicates dangercus properties. Air pats that 
develop the curvature shown in (b) generally disintegrate 
later. A curvature of about % in. in a 3-in. pat can be 
considered to be about the limit of safety. 

The normal pat tests are the only absolutely fair and 
accurate methods of testing cements for soundness, but the 
serious objection to them lies in the fact that frequently 
several months or even years elapse before failure in the 
cement so tested becomes apparent. To overcome this 
difficulty the accelerated tests have been devised. These 
tests are intended to produce in a few hours results that 
require months in the normal tests. 

Many forms of accelerated tests have been devised. At 
present, however, the only tests employed commercially 
are the boiling test and the steam test. 

The boiling test is made by forming specimens of neat- 
cement paste into pats, such as are employed for the normal 
tests, or preferably into balls about 14 in. in diameter. 
The specimens are allowed to remain in moist air for 24 hr. 
and are then tested. 

The form of apparatus used for the boiling test is shown 
in Fig. 3. It consists of a copper tank that is heated by a 


214 TESTS ON CEMENT 


Bunsen burner and is filled with water. The water in the 
tank is kept ata uniform height by means of a constant-level 
bottle. A wire screen placed an inch from the bottom of 
the tank prevents the specimens from coming into contact 
with the heated bottom. The test pieces, which are 24 hr. 
old, are placed in the apparatus, which is filled with water of 
a normal temperature, and heat is applied at a rate such that 
the water will come to boiling in about 4 hr. Quiet boiling 
is continued for 3 hr., after which the specimens are removed 
and examined, Care must be taken that the water employed 


Fic. 3 x 


is clean and fresh, because impure water may seriously 
affect the results.. The same water, also, should never be 
used for more than one test. A good cement will not be 
affected by this treatment, and the ball will remain firm and 
hard. Inferior cement will fail by checking, cracking, or 
entirely disintegrating. 

The steam test is made in the same way as the boiling test, 
except that instead of immersing the specimens in water, 
they are kept in the steam above the water. The apparatus 
employed is the same as that used for the boiling test. The 


ee a ea 


TESTS ON CEMENT 215 


wire screen, however, is raised so that it is an inch above 
the surface of the water; also, there must be provided a cover 
that is close enough to retain the steam without creating 
pressure. The steam test is less severe than the boiling test 
and is somewhat less accurate. 

Results of Tests for Soundness.—The peeult of the normal 
tests, if properly made and interpreted, may be considered 
reliable guides to the soundness of the material, and cement 
failing in these tests should always berejected. The acceler- 
ated tests, on the other hand, furnish merely indications, and 
are by no means infallible. A cement passing the boiling 
test can generally be assumed sound and safe for use, but, 
if failure occurs, it simply means that other tests should be 
performed with greater care and watchfulness. It often is 
advisable to hold for a few weeks cement that fails in boiling, 
so that the expansive elements may have an opportunity to 
hydrate and become inert; but if the material fulfils all the 
conditions except the boiling test, and is sound in the normal 
tests up to 28 da., it is generally safe for use. All things being 
equal, however, a cement that will pass the boiling test is 
to be preferred. 


TESTS FOR TENSILE STRENGTH 


The tensile-strength test is for the purpose of ascertaining a 
measure of the ability cf the material to withstand the loads 
that the structure must carry. This test is made by form- 
ing specimens, called briquets, of cement and cement mortar, 
and determining the force necessary to rupture them in ten- 
sion at the expiration of fixed intervals of time. Cement 
constructions are rarely called on to withstand tensile stresses, 
but if the tensile strength is known, the resistance to other 
forms of stress may be computed with a fair degree of accur- 
acy. The tensile-strength test is the most convenient for 
laboratory determinations, on account of the small size of the 
specimens and the comparatively low stress required to cause 
rupture. 

Cement is tested both neat or pure and in a mortar com- 
monly composed of 1 part of cement and 3 parts of sand. 
The period at which the briquets are broken have been fixed 


216 TESTS ON CEMENT 


by usage at 7 da. and 28 da. after making, although tests 
covering much longer periods of time are necessary in research 
or in investigative work. 

Normal Consistency.—The strength of cement and cement 
mortars varies considerably with the amount of water 
employed in making the briquets. Dry mixtures ordinarily 
give the higher results for short-time tests, and wet mixtures 
show stronger with a greater lapse of time. For testing 
purposes, therefore, it is essential that all cements be mixed, 
not with the same amount of water, but with the amount that 
will bring all the cements to the same physical condition, or 
to what is called normal consistency. Different cements 
require different percentages of water because of their vary- 
ing chemical composition, degree of burning, age, fineness, etc. 

The normal consistency of neat-cement pastes may be 
determined by the method that follows. 

This method is to form of the paste a ball about 2 in. in 
diameter and to drop this ball on a table from a height of 
about 2 ft. If the cement is of the correct consistency, the 
ball will not crack nor will it flatten to less than half its 
original thickness. The percentage of water required 
will vary from 16 to 25, depending on the characteristics of 
the material, the average cement taking about 20%. 

Consistency of Sand Mortars.—The consistency of sand 
mortars, however, cannot be obtained by the foregoing 
method, because the mixture is tooincoherent. For mortars, 
therefore, it is necessary to employ a formula by means of 
which the sand consistency can be computed when that of the 
neat paste is known. Several such formulas have been 
devised, of which the following is adaptable to the greatest 
variety of conditions. 

Let x be the per cent. of water required for the sand mix- 
ture; NV, the per cent. of water required to bring the neat 
cement to normal consistency; u, the parts of sand to one of 
cement; and S, a constant depending on the character of the 
sand. Then, _38N+Sn+1 

4(n+1) 

For crushed-quartz sand, the constant S is 30, for Ottawa 

sand, it becomes 25; and for the bar and bank sands used in 


TESTS ON CEMENT 217 


construction, it varies from 25 to 35, and must be determined 
for each particular sand. 

EXAMPLE.—How much water is required in a mixture of 
1 part of cement and 3 parts of crushed-quartz sand? The 
neat cement requires 19% of water to give normal consis- 
tency. F 

SoLutTion.—Here, N=19, S=30, andu=3. Substituting 
these values in the formula, 


_3X19+30X3+41 
4X (3+1) 


Sand for Mortar Tests.—The size, gradation, and shape 
of the particles of sand with which cement mortars are made 
have great influence on the resulting strength. There are two 
varieties of standard sand for cement testing, one an artificial 
sand of crushed quartz, the particles of which are angular in 
shape, and the other a natural sand from Ottawa, Illinois, the 
particles of which are almost 
spherical. Both sands are 
sifted to a size that will pass 
a sieve of 20 meshes to the 
inch and be retained on a 
sieve of 30 meshes, the dia- 
meters of the sieve wires be- 
ing .0112 and .0165 in., re- me is an ae 
spectively. The Ottawa sand 
will develop strengths in 1-3 
mortars about 20 to 30% 
greater than those obtained 
with crushed quartz, and itis 
theoretically the better sand 
for testing, but, at present, 
crushed quartz is more exten- 
sively employed. On most Fic. 4 
important works, tests for purposes of comparison are also 
made of the actual sand-entering the construction. 

Form of Briquet.—The form of tensile briquet, adopted as 
standard in the United States, is shown in Fig. 4. Its 
cross-section is exactly 1 sq. in. 


=9.3% 


wy — 


wt fpr 
Lit } 
St 


218 TESTS ON CEMENT 


Molds.—_Cement briquets are made in molds that come 
either single or in gangs of three, four, or five. The gang 
molds are preferable, as they tend to produce greater uni- 
formity in the results. Molds should be made of brass or of 
some other non-corrodible material; those made of cast iron 
soon rust and become unfit for use. 

Method of Making Briquets.—First, 1,000 g. of cement is 
carefully weighed and placed on the mixing table in the form 
of a crater, and into the center of this is poured the amount 
of water that has previously been determined to give the 
correct normal consistency. Cement from the sides of the 
crater is then turned into the center, by means of a trowel, 
until all the water is absorbed, after which the mass is 
vigorously worked with the hands, as dough is kneaded, 
for 14 min. When sand mixtures are being tested, 250 g. 
of cement and 750 g. of sand are first weighed and thor- 
oughly mixed dry until the color of the pile is uniform; then 
the water is added and the operation is completed by vigor- 
ous kneading. 

After kneading, the material is immediately placed into the 
molds, which should first have been wiped with oil to pre- 
vent the cement from sticking to them. The entire mold is 
filled with material at once—not compacted in layers—and 
pressed in firmly with the fingers without any ramming or 
pounding. An excess of material is then placed on the mold 
and a trowel drawn over it under moderate pressure, at each 
stroke cutting off more and more of the excess material, 
until the surface of the briquets is smooth and even. The 
mold is then turned over, and more material placed in it and 
smoothed, as before. The mixing and molding should be 
performed on a surface of slate, glass, or some other smooth, 
non-absorbent material. During the mixing the operator 
should wear rubber gloves, so as to protect his hands from 
the action of the lime in the cement. - 

Storage of Briquets.—For 24 hr. after making, the briquets 
are stored in a damp closet so that the cement can harden 
in a moist atmosphere. The damp closet is simply a tight 
box of soapstone with doors of wood lined with zinc, or some 
similar arrangement, with a receptacle for water at the 


TESTS ON CEMENT 219 


bottom and racks for holding the briquets. The briquets 
remain in the molds while in the damp closet, but at the 
expiration of 24 hr. they are removed, marked, and placed in 
clean water near 70° F. until broken. 

Testing Machines.—There are many styles of testing 
machines on the market. In Fig. 5 is shown what is called 
a shot machine, made by the Fairbanks Company. It is 


HIG oO 


constructed on the cast-iron frame a, and is operated as fol- 
lows: The cup/ is hung on the end of the beam d, the poise r 
placed at the zero mark, and the beam balanced by turning 
the weight 7. The hopper 0 is then filled with fine shot, and 
the briquet to be tested is placed in the clips kh. The hand- 
wheel p is now tightened sufficiently to cause the graduated 
beam d to rise to the stop k, and the automatic valve 7 


220 TESTS ON CEMENT 


opened so as to allow the shot to run into the cup f. The 
flow of the shot can be regulated by means of a small valve 
located where the spout joins the reservoir. When the 
briquet breaks, the beam d drops and by means of the lever ¢ 
automatically closes the valve 7. After the specimen has 
broken, the cup with its contents is removed, and the counter- 
poise g is hung in its place. The cup f is then hung on the 
hook under the large ball ¢, and the shot weighed. The 
weighing is done by using the poise 7 on the graduated 
beam d and the weights on the counterpoise g. The 
result will show the number of pounds required to break the 
specimen. A mold for a single briquet is shown at c. 

Rate of Loading Testing Machine.—The load should be 
applied in all tests at the uniform rate of 600 lb. per min. 
The briquets should be broken as soon as they are removed 
from the storage tanks and while they are still wet, because 
drying out tends to lower their strength. The average of 
from three to five briquets should be taken as the result of a 
test. 

Results of Tensile-Strength Tests.—The tensile strength of 
cement tested in the preceding manner should increase with 
age up to about 3 mo, and should then remain practically 
stationary for longer periods. The average results of tests 
of Portland cement made in the Philadelphia laboratories, 
covering a period of several years and based on over 200,000 
briquets, are given in the accompanying table. 

Specifications for strength commonly stipulate minimum 
values for the 7- and 28-da. tests, the customary require- 
ments for Portland cement being 500 lb. at 7 da. and 600 lb. 
at 28 da., when tested neat, and 170 lb. at 7 da. and 240 lb. 
at 28 da., when tested in a mortar consisting of 1 part of 
cement and 3 parts of crushed-quartz sand. When Ottawa 
sand is used, the requirements for mortar should be raised 
to 200 and 280 lb., respectively. Retrogression in strength 
of the neat briquets between 7 and 28 da. is not necessarily 
indicative of undesirable properties, but if the mortar briquets 
show retrogression, the cement should be condemned. 
Abnormally high strength in the 7-da. test of neat cement, 
say over 900 lb., may generally be taken as an indication of 


TESTS ON CEMENT 221 


TENSILE STRENGTH OF CEMENT BRIQUETS 
(Pounds per Square Inch) 


— 

3) a & ee 
He | ud | we | ae 
“B / <s | <B | <5 
; fa) 6.) Ss | Ss 
Mixture balan | Ss ee 2 

o3 8:2 pS Pr ip os 
To A a As \|As 
gh as a 
ge} ©] &| & 
PRO ore oes cries ie oe oe ae ewes et ee 12108 Pie. liven 
1 cement, 1 crushed-quartz sand ..| 360 | 590 | 695 | 700 
1 cement, 2 crushed-quartz sand ..| 210 | 370 | 455 | 465 
1 cement, 3 crushed-quartz sand ..| 105 | 210 ; 300 | 310 
1 cement, 4 crushed-quartz sand ..| 60 | 130 | 210 | 230 
1 cement, 5 crushed-quartz sand ..| 35 80 | 155 | 195 


weakness rather than of superiority, because such a condition 
is usually created by an excess of lime or of sulphates, either 
of which may be injurious. Neat cement testing from 600 
to 800 Ib. at 7 da. is generally the most desirable. 


SECONDARY TESTS 


TESTS FOR TIME OF SETTING 


The time-of-setting test is made to determine whether or 
not the cement will become hard at the time most desirable 
in actual construction. If it begins to set too soon, the 
crystallization of the particles will have begun before the 
mortar or concrete is thoroughly tamped into place. If, 
on the other hand, the cement sets too slowly the material 
is more likely to suffer from exposure to heat, cold, dampness, 
and frost; also, the progress of the work will be much delayed 
on account of the greater interval required between different 
sections, and the longer time the forms must be left up. 

In the setting of cements, two stages are recognized: 
(1) When the paste begins to harden or to offer resistance to 


t 
222 TESTS ON CEMENT 
change of form, called initial set, and (2) when the setting is 
complete, or when the mass cannot be appreciably distorted 
without rupture, called hard set. The time-of-setting test 
consists, therefore, in determining the time required for the 
cement to reach these two critical points. 

The test is made by mixing cement with the amount of 
water required to produce normal consistency, in the same 
manner as for neat tensile briquets, forming specimens, 
placing them under one of the forms of apparatus, and observ- 
ing the time that elapses between the moment the mixing 
water is added and the mo- 
ments when the paste acquires 
initial set and hard set. 

The Vicat needle, shown in 
Fig. 6, consists of a frame k, 
holding a movable rod 1, which 
carries a cap d at the upper 
end and a needle hk at the 
lower. A screw 7 holds the 
rod in any desired place. 
The position of the needle is 
shown by a pointer moving 
over a graduated scale. The 
rod with needle and cap weighs 
exactly 300 gr. and the needle 
is 1 mm. in diameter with 
the end cut off square. When 

Fic. 6 making tests of normal con- 

4 sistency, the plunger b is sub- 

stituted for the needle /, and the cap a for the cap d, the dif- 

ference in weight between the needle and plunger being com- 

pensated by the difference in the weight of the caps. The 

mold z for holding the cement paste is in the form of a trun- 

cated cone. It has an upper diameter of 6 cm., a lower dia- 

meter of 7 cm., and a height of 4 cm., and rests on a 4” 4” 
X }” glass plate j. 

After the cement paste is mixed, the mold is filled by 
forcing the cement through the large end; then, after turning 
it over and smoothing the top, it is placed on the glass plate 


TESTS ON CEMENT 223 


under the needle. The needle is lowered until it is exactly 
in contact with the surface of the paste, then quickly released 
and the depth to which it penetrates read from the graduated 
scale. JIpitial set is said to have taken place when the 
needle ceases to penetrate to within 5 mm. of the bottom 
of the specimen; and hard set takes place when the same 
needle ceases to make an impression on the surface. Trials 
of penetration are made every 5 or 10 min. until these points 
are reached. 

Time of setting varies considerably with the amount 
of mixing water employed, so that it is essential that every 
sample tested be brought exactly to normal consistency; 
otherwise, the results may be in decided error. Variations 
in temperature in both environment and in the mixing water, 
also influence the results. Standard practice requires that 
both the materials and the room in which the tests are made 
be at a temperature of as nearly 70° F. as practicable. 

Results of Time-of-Setting Tests.—In specifying results 
to be obtained in testing the time of setting, it is obvious that 
a minimum value should be stipulated for initial set and a 
maximum, as well as a minimum, for hard set. It must also 
be remembered that a cement mixed with an aggregate and 
with an excess of water in the field, will require from two to 
four times as long to set as the neat-cement paste mixed with 
little water in the laboratory. Cement, therefore, showing 
an initial set at the expiration of 20 min. with the Vicat 
needle, will rarely begin to set on the actual work in less than 
# hr. which gives ample time for mixing and placing the 
materials, and cement setting in less than 10 hr., will usually 
have hardened completely in the work in 24 or at least in 
36 hr. Specifications usually stipulate that Portland cement 
shall show initial set in not less than 20 minutes and shall 
develop hard set in not less than 1 hr. nor more than 10 hr. 
Cement reaching initial set in less than 12 or 15 min. should 
never be used for any work. 


TESTS FOR FINENESS 


Apparatus for Fineness Test.—The fineness of cement is 
important, because it affects both the strength and the sound- 


224 TESTS ON CEMENT 


ness of the product. The fineness of cement is determined 

by passing it through a series of sieves of different mesh and 

then measuring the amount retained on each. ‘Three sieves 

are commonly employed, namely, those having 50, 100, and 

200 wires to the linear inch. Sieves for cement testing should | 
never be used until they have been carefully examined and | 
found to conform to the following standard specification: 

1. Cloth for cement sieves shall be of woven brass wire 
of the following diameters: No. 50, .0090 in.; No. 100, 
.0045 in.; and No. 200, .00235 in. 

2. Mesh to count on any part of the sieve as follows: 
No. 50, not less than 48 nor more than 50 per lin. in.; No. 100, 
not less than 96 nor more than 100 per lin. in.; and No. 200, 
not less than 188 nor more than 200 per lin. in. 

3. Cloth to be mounted squarely and to show no irregu- 
larities of spacing. 

Method of Making the Fineness Test.—The method of using 
the sieves in the fineness test is to weigh out 50 g. of cement 
on a scale sensible at least to yy g. and to place it on the 
No. 200 sieve, on which it is shaken until not more than 7 g. 
passes the sieve at the end of 1 min. of shaking. The arrival 
of this stage of completion can be watched either by using a 
pan under the sieve or by shaking over a piece of paper. 
The residue remaining on the sieve is weighed, placed on the 
No. 100 sieve and the operation repeated, again weighing the 
residue. The amount remaining on the No. 50 sieve is then 
determined similarly. The process of sifting can be acceler- 
ated by placing a small quantity of coarse shot or pebbles 
on the sieves with the cement during the shaking. These 
may be separated from the cement by passing the residue 
with the shot through a coarse sieve, such as the No, 20. 

Results of Fineness Tests.—Portland cement should be 
ground to such a fineness that it will leave a residue of not 
more than 25%, by weight, on the No. 200 sieve, and not 
more than 8% on the No. 100 sieve. Of these two require- 
ments, the first is the more important, because it is only that 
part of the cement passing the finest sieve that is active in 
the setting and hardening of the material. The amount 
remaining on the No. 100 sieve is also important, because this 


TESTS ON CEMENT 225 


part is most liable to cause unsoundness in the cement, and 
although specifications do not call for tests with the No. 50 
sieve, it is usually employed for the same reason as the No. 
100 sieve. Any appreciable residue on this sieve indicates 
that the material is much more liable to unsoundness. Any 
cement failing to pass the fineness test should be watched 
more carefully in the soundness and strength tests, but if 
these tests show good results up to 28 da., the cement can, as ~ 
arule, be used safely. It must be remembered, however, that 
only that part passing the No. 200 sieve is really cement, so 
that a coarsely ground shipment is practically equivalent to 
one adulterated with weak and unsound material. 


TESTS FOR SPECIFIC GRAVITY 


The object of the spectfic gravity test is to furnish indica- 
tions of the degree of burning, the presence or absence of 
adulteration, and the amount of seasoning that the cement 
has received. When Portland cement is burned, the separate 
ingredients are in close contact and gradually combine by a 
process of diffusion. The greater the amount of this burn- 
ing the more thoroughly are the elements combined. Thus, 
by their contraction they give, in volume, a higher density 
or specific gravity. Since, to secure good cement the burn- 
ing must have been made within definite limits, it fouows 
that the specific gravity must also lie within fixed limits if 
the cement has been properly manufactured. 

The common adulterants of Portland cement, namely, 
limestone, natural cement, sand, slag, cinder, etc., all have 
specific gravities ranging from 2.6 to 2.75, while the specific 
gravity of Portland cement averages about 3.15. An appre- 
ciable amount of adulteration, therefore, is at once indi- 
cated in the results of the test. 

Seasoning is indicated because the cement on standing 
gradually absorbs water and carbonic acid from the air. 
These ultimately combine with it and thus lower the 
specific gravity. 

Apparatus for Specific-Gravity Test.—Of the many forms 
of apparatus employed for the specific-gravity test, the 
Le Chatelier flask, shown in Fig. 7, is the one most com- 


226 TESTS ON CEMEN T 


monly used. It is also the one adopted by the technical 


societies as standard. It consists of a glass flask about 
30 cm. high. The lower part up to mark a contains 120 
cu. cm., and the bulb between the marks a and b contains 
exactly 20 cu.cm. The neck of the flask above the mark b 
is hee into yy cu. cm. The funnel ¢ inserted in the 

neck is to facilitate the introduction of 
- the cement. 

Method of Making the Specific-Gravity 
Test.— The method of conducting the 
specific-gravity test is as follows: 64g. 
of cement is carefully weighed on scales 
that should have a sensibility of at least 
.005 g. The flask, Fig. 7, is filled to the 
lower mark a with benzine or kerosene, 
which has no action on the cement, and 


ce 


¢ 


° 
--20 el 


by adding the liquid a drop at a time. 
The funnel is then placed in the neck of 
the flask and the weighed cement intro- 
duced slowly through it, the last traces 
of the cement being brushed through 
with a camel’s-hair brush. The funnel 
is then removed and the height of the 
benzine read from the graduations, esti- 
mating to .01 cu. cm. The displaced 
volume is then 20 plus the reading in 
cubic centimeters, and the specific gravity 
of the cement is 64 divided by that quan- 
tity. For example, suppose that the 
reading on the flask is .54, then the dis- 

Pic. 7 placed volume will be 20 +.54=20.54 and 
the specific gravity will be 64+20.54=3.116. 

The apparatus must be protected from changes in tem- 
perature while in use; even touching the flask with the 
fingers will change the volume of the liquid noticeably. The 
flask is sometimes immersed in water during the tests to 
prevent these changes of temperature, but this precaution is 
unnecessary if proper care is exercised. 


30 cm. 


“ 


carefully adjusted precisely to the mark — 


eee Se ee 


TESTS ON CEMENT 227 


Results of Specific-Gravity Tests.—The specific gravity 
of well-burned Portland cement averages about 3.15 and 
should not fall below 3.1. If it falls below 3.1, tests should 
also be made on dried and on ignited samples to ascertain 
whether or not this condition has been produced by reason 
of excessive seasoning. As a rule, low specific gravity 
merely indicates well-seasoned cement, and if sound and 
sufficiently strong, such cement is the best sort of material 
for use, as its durability is scarcely open to question. 


TESTS OF NATURAL AND SLAG CEMENTS 


The methods of conducting tests of natural and slag 
cements are, in all important particulars, identical with those 
employed for Portland cement, although the results obtained 
and the interpretation to be put on them are often radically 
different. In the testing, the only essential difference is in 
the amount of water required by these cements to produce 
normal consistency, natural cement requiring from 23 to 
35% and slag cement taking about 18%, or an average 
of 2 or 3% less than Portland. Tests of natural cement for 
tensile strength are also frequently made on 1-1 and 1-2 
mortars, but recent practice is to test mortars of all kinds 
of cement in 1-3 mixtures. For these cements, moreover, 
the specific-gravity test has practically no significance, 
except in determining the uniformity with which the dif- 
ferent brands are made. 


CEMENT SPECIFICATIONS 


The common requirements for high-grade Portland, 

natural, and slag cements are given in the table on page 

: 228, and following the table is given a good example of a 
complete modern specification for Portland cement. 


SPECIFICATIONS FOR PORTLAND CEMENT 
Kind.—All cement shall be Portland of the best quality, 
dry and free from lumps. By Portland cement is meant the 
finely pulverized product resulting from the calcination 
to incipient fusion of an intimate mixture of properly pro- 


228 


TESTS ON CEMENT 


portioned argillaceous and calcareous materials to which no 
addition greater than 3% has been made subsequent to 


calcination. 


REQUIREMENTS FOR HIGH-GRADE CEMENTS 


P Portland Natural Slag 
Requirements Cement Cement Cement 
Specific gravity: 
Not less than........ 3.1 2.8 27 
Fineness: 
Residue on No. 100 
sieve, not over..... 8% 10% 3% 
Residue on No. 200 : 
sieve, not over..... 25% 30% 10% 
Time of setting: 
Initial, not less than .| 20 min. 10 min. 20 min 
Hard, not less than . 1 hr. 30 min. 1 hr. 
Hard, not more than. 10 hr. 3 hr. 10 hr. 
Tensile strength per sq. in. 
da., neat, not less 
THA sc. ancicae ct DOU Ep 125 lb. 350 lb 
28 da., neat, not less 
than.. 600 Ib 225 Ib. 450 lb 
7 da., 133 quartz, not 
less than. 170 lb 50 Ib. 125 lb 
28 da., 1-3 quartz, x not 
less than. . 240 Ib 110 Ib. 200 Ib 
Soundness: 
pik gpk 5 od ior i sound and | sound and | sound and 
ogee ig E ard ar hard 
Boiling test to be .. { ee ws me and 
Analysts: 
Magnesia, MgO, not 
over. 4% 4% 
Anhydrous sulphuric 
acid, SO3, not over 1.75% 
Sulphur, S, not over. . 1.38% 


Packages.—Cement shall be packed in strong cloth or 


canvas bags, or in sound barrels lined with paper, which 
shall be plainly marked with the brand and the name of the 
manufacturer. Bags shall contain 94 lb. net and barrels 
shall contain 376 lb. net. 


TESTS ON CEMENT 229 


Inspection.—All cement must be inspected, and may be 
reinspected at any time. The contractor must submit the 
cement, and afford every facility for inspection and testing, 
at least 12 da. before desiring to use it. The chief engineer 
shall be notified at once on receipt of each shipment at the 
work. No cement will be inspected or allowed to be used 
- unless delivered in suitable packages properly branded. 
Rejected cement must be immediately removed from the 
work. 

Protection.—The cement must be protected in a suitable 
building having a wooden floor raised from the ground, or 
on a wooden platform properly protected with canvas. It 
shall be stored so that each shipment will be separate, and — 
each lot shall be tagged with identifying number and date 
of receipt. A 

Quality.—The acceptance or rejection of a cement to be 
used will be based on the following requirements: 

Specific gravity: Not less than 3.1. 


Ultimate tensile strength per square inch: Pounpbs 
qioa. (ida..miarr-6.Ga7 tm water) is soccct res 500 
28 da. (1 da..in air, 27 da. in water).......... 600 


7 da. (1 da. in air, 6 da. in water), 1 part 
cement to 3 parts of standard quartz sand 170 
28 da. (1 da. in air, 27 da. in water), 1 part of 
cement to 3 parts of standard quartz sand 240 
Fineness: Residue on No. 100 sieve not over 8%, by 
weight; residue on No. 200 sieve not over 25%, by weight. 
Set: It shall require at least 20 min. to develop initial 
set, and shall develop hard set in not less than 1 hr. nor more 
than 10 hr. These requirements may be modified where the 
conditions of use make it desirable. 
Constancy of Volume: Pats of cement 3 in. in diameter, 
4 in. thick at center, tapering to thin edge, immersed in water 
after 24 hr. in moist air, shall show no signs of cracking, dis- 
tortion, or disintegration. Similar pats in air shall also 
remain sound and hard. The cement shall pass such accel- 
erated tests as the chief engineer may determine. _ 
Analysts: Sulphuric anhydride, SO,, not more than 
1.75%; magnesia, MgO, not more than 4%. 


230 PLAIN CONCRETE 


PLAIN CONCRETE 
- MATERIALS USED IN CONCRETE 


DEFINITIONS AND TERMS 


Concrete is usually made of cement, sand, and broken 
stone. The cement in a plastic state, either by itself or with 
the sand that is generally mixed with it, is called the matrix, 
and the broken stone, gravel, or other material used as a 
filler is called the aggregate. The sand is correctly classed as 
a part of the aggregate, although some engineers include it 
with the matrix. The aggregate is used to cheapen con- 
crete. Pure, or neat, cement, when wet with water, would 
in a way fulfil all the physical requirements of concrete, but 
it would be too expensive. 

In the concrete of today, hydraulic cement is used almost 
exclusively. For this reason, the term concrete, as com- 
monly used, refers only to that variety. In specifying any 
other kind of concrete, the usual custom is to mention it 
by its full name, as bituminous concrete, lime concrete, etc. 
Such varieties, however, are of comparatively little impor- 
tance. ‘ 

The term concrete, besides being restricted to hydraulic- 
cement concrete, has another restriction: the aggregate must 
not be sand alone, although it may be partly sand, A mix- 
ture of hydraulic cement, sand, and water is called by the 
special name of mortar. 

Concrete is usually named from the kind of aggregate used. 
For example, stone concrete embodies the use of broken 
stone or coarse pebbles, while in cinder concrete, the aggre- 
gate consists of cinders or broken slag. 

The proportion of cement and sand to the broken stone 
depends on the spaces between the stones, which are known 
as voids. In all instances, there must be sufficient mortar 
to fill the voids entirely and to cover all surfaces of the 
separate stones. 


— es a ee ee ee 


PLAIN CONCRETE . 231 


AGGREGATES OTHER THAN SAND 


The aggregates or broken stone used in concrete work 
should possess three qualities: (1) They should be hard 
and strong, so as to resist crushing and shearing or transverse 
stresses; (2) they should have surface texture that will per- 
mit the cement mortar to adhere to their surfaces; and 
(3) where the concrete is to be used for building construc- 
tion, such as in reinforced-concrete work, and for fireproofing, 
they should possess refractory, or fire-resisting, qualities. 
Usually, aggregates that break in such a way as to allow the 
smallest spaces, or interstices, between the particles, will 
make the strongest concrete for construction purposes 
because the voids can be most economically filled with 
cement mortar. , 

Size of Aggregates——In measuring broken stone, the size 
of the stone is determined by the size of the ring through 
which it will pass. For instance, a 2-in. stone is one that 
will pass through.a ring, or hole, that is 2 in. in diameter. 

The broken stone used in concrete work varies in size with 
the nature of the work. For foundation and mass con- 
struction, it is the custom to use broken stone of a size that 
will pass through the 2- or 24-in. ring. For filling the 
spandrels of bridges or the spaces between walls, where 
mere mass is desired, broken stone of a much larger size is 
used. 

In reinforced-concrete work, the broken stone must be 
small, owing to the narrow spaces inthe forms. For columns 
and wall work, stone that will pass through a 1- or ?-in. ring 
is suitable, while for filling the beam and girder forms, where 
numerous reinforcing rods occur, the broken stone is some- 
times so small as to pass through a 4-in. ring. 

The latest practice in making concrete is to use stone as 
it comes from the crusher, without screening it. While 
such stone, termed the rum of crusher, contains broken stone 
of a size specified, it also has smaller particles of stone and 
such stone dust as is carried along with the broken stone 
from the crusher. Where the run of crusher is used, the 
proportion of the cement and sand must be changed, because 
the stone dust takes the place of some of the sand. In 


232 PLAIN CONCRETE | 


using run of crusher the very finest dust should be washed 
or screened out as it tends to coat the large pieces and © 
to prevent the cement from adhering to them. 

Selection of Aggregates.—Usually the character of the 
aggregates used in mixing concrete depends on the avail- 
ability of the supply. Where there is much choice in the 
selection of the aggregates those which are hardest and 
which break with a cubical fracture will make the best con- 
crete although rounded pebbles are considered by some 
engineers to possess great advantages. 

The size of the aggregates has much to do with the quality 
and strength of the concrete. It can, however, be stated 
as a general proposition that the larger the stones the 
stronger will be the concrete. This fact is shown by the 
accompanying table, which gives the results of tests made 
at the Watertown Arsenal in 1898. It is interesting to note 
that the concrete becomes heavier per cubic foot, or, in other 
words, more dense, the larger the stone used. All these 
tests were made with concrete manufactured in the pro- 
portion of 1 part of cement, 1 part of sand, and 3 parts of 
broken stones, or a 1-1-3 (1 to 1 to 3) mixture, as it is usually 
expressed, The figures on cinder concrete in the table are 
added simply to give a comparison of weights, for it will 
be noted that the cinder concrete is older than the other 
concretes and therefore stronger in proportion. 

Aggregates that consist of stone of varying sizes are best 
for making concrete, owing to the fact that they pack closer. 
It is well. however, to screen all the fine particles, such as 
4-in, sizes, and use them with the sand, as otherwise they 
will not mix properly with the cement. 

Broken trap rock is the best aggregate for concrete work, 
the next in value is broken granite, while the third in order 
of merit is good clean gravel. In fact, these three aggregates ; 
can be classified together, and the item of cost only should 
influence the selection of any one of them. Marble, lime- 
stone, and slag make good aggregates, in the order named, 
but marble and limestone are objectionable if the concrete 
is to be used as a fireproofing. The poorest aggregates are 
sandstone, slate, and shale. The sandstone is inefficient 


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234 PLAIN CONCRETE 


on account of its lack of hardness and its liability to crumble, 
and from the fact that its surface is likely to be unstable. 
Slate and shale are hard, but they are of a laminated struc- 
ture and break in such flaky shapes that they will not pack 
closely. Cinders are frequently used as the aggregate for 
concrete. Cinder concrete, however, does not possess suf- 
ficient strength for structural purposes, and is generally used 
for filling or for fireproofing. 

Based on percentages of efficiency, with trap rock taken 
at 100, the following table gives a fair representation of the 
comparative values of the different aggregates: 


COMPARATIVE VALUE OF DIFFERENT AGGREGATES 
USED IN CONCRETE 


Material Value 
Trap rock.. Biers, Mate tbubats odo chpided cnied sie been Aha 100 
Granite. . ERNcis tes cus Sunt tonpasrk noes 90 
Gravel (quartz).. SRN RS KN ced oh 90 
Limestone (hard, like marble). att at 
Berane tne APOE « UP eg ier oe 75 
TSS a als i sa > eo fel rea 75 
Slate. 60 
SARA Girls, a elas Brean sore ONG DiS TA Ne IBS. Gm a be 55 
Cased oe PE pede wee Nickels Oe eink See ee 50 


PROPORTIONING OF INGREDIENTS 

Effect on Strength and Imperviousness.—The strength of 
concrete depends on the strength of the cement and the 
thoroughness with which the cement binds togethe. the 
various pieces of aggregate. The more completely the voids 
are filled, the more completely will the aggregate be held 
together. Therefore, the more solid and condensed the con- 
crete is, the less voids it will have, and the stronger it will 
be. The same is true with regard to making concrete water- 
proof: the more dense the concrete is, the more nearly water- 
proof it is. 

A mixture of 1 part of cement, 14 parts of sand, and 3 parts 
of stone, which would be considered extravagantly rich for 


ve —~ 


PLAIN CONCRETE 235 


a dry place, is probably as dense a concrete, and as good for 
waterproofing qualities, as can be made. 

When a concrete is made of cement, sand, and stone, and 
the stone is of such a size that it will pass through a 3-in. 
ring, but will not pass through a 24-in. ring, the concrete is 
weaker and requires more cement than one made with 
graded stone from 3-in. down. When the stone is graded 
in size, the stones of smaller size fill the voids between the 
larger stones and thus reduce the quantity of cement and 
sand required. 

Proportioning by Weight.—A method of proportioning 
the materials, that is simple and fairly accurate, is as fol- 
lows: A batch of concrete is mixed in known proportions. 
The same quantity of water is used that it is proposed to use 
on the work, and the mixture is rammed and tamped in the 
receptacle in a uniform manner. The receptacle should pre- 
ferably be of metal; a tin washtub, or a short section of 
12-in. pipe, capped at one end, will answer. When the 
receptacle is full, it is weighed, and if the weight of the recep- 
tacle itself has previously been found, the weight of the con- 
crete may be obtained. Various other mixtures of concrete 
are tried in the same manner, and since the denser the mixture 
the stronger it will be, the heaviest concrete is the strongest 
for the particular work. Each batch of concrete must be 
weighed and taken out of the receptacle before it has time 
to set; otherwise, some difficulty might be experienced in 
getting.it loose. ; 

Usual Proportions of Materials.—The strongest concrete 
does not always have to be used, as it may be required to 
withstand only slight stresses and be simply used for its 
weight. The strongest concrete would then be unnecessarily 
expensive. Therefore, the foregoing method for proportion- 
ing concrete is seldom employed. The engineer usually 
specifies a mixture from his own experience without testing 
the aggregates in any way, except to see that the stone is 
under the specified maximum size and that the sand is in 
large grains and free from dirt and loam. A common pro- 
portion for unimportant work is 1-3-6. This proportion 
may be used for foundations below ground, in engine bases, 


236 PLAIN CONCRETE 


in the foundations for asphalt pavements, and for similar 
purposes. A richer mixture, 1-2-4 is used in piers, in 
dams, in important reinforced-concrete work, and in other 
places where great strength is desired. 

_ Water for Concrete.—The wetter the concrete is, the easier 
it will be put in place, but mixtures that are too wet are 
not so strong as medium mixtures. The amount of water 
that will make the best mixture is such that after the con- 
crete has been put in place and rammed it will quake like 
jelly when struck with a spade, and water will come to the 
surface. If the concrete is wetter than this, the water will 
have a slight chemical effect on the cement, and, moreover, 
the sand and cement will tend to separate from the broken 
stone. 

In cinder concrete, owing to the porosity of the cinders, 
it is necessary to use a little more water, so that the cement 
will be liquid enough to fill the little cavities in each cinder. 
This precaution is indispensable when the concrete is to 
be used with steel, as otherwise the steel will be rapidly 
corroded by the action of air reaching it through the pores 
in the cinder. 

Dry Concrete.—With the advent of the concrete block, 
a great deal is heard about dry concrete. This name is given 
to concrete in which as little water as possible is mixed. 
In the concrete-block manufacturing business, the mold 
in which each block is made is required as soon as possible, 
so that it can be used over again and thus increase the 
capacity of the machine to which it belongs. For this 
reason, the concrete-block manufacturers use, often, dry 
concrete, and attempt to supply the remainder of the water 
required for the complete crystallization, or setting, of the 
cement by curing the blocks; that is, by sprinkling them 
with water for a week or so. The results of recent tests 
seem to indicate that dry concrete will show higher com- 
pression values for a limited time after it is made, but that 
the rate of increase of strength is not so great as with wet 


concrete. After 1 yr. or 6 mo., the strength of the wet con- | 


crete will be found to have attained, and perhaps surpassed, 
that of the dry mixture. 


PLAIN CONCRETE 237 


A serious objection to dry concrete is that it cannot be 
rammed to so dense a mass as wet concrete. Therefore, 
if a waterproof concrete is desired, it should be mixed wet. 


PROPERTIES OF CONCRETE 
GENERAL CHARACTERISTICS 


Corrosion of Steel in Concrete.—It has been conclusively 
proved by experience and test that steel or iron completely 
embedded in the usual mixture of concrete will not corrode 
seriously. Portland cement contains free alkali, and steel 
or iron will not rust in the presence of an alkali. Corrosion 
will occur only where the concrete has been carelessly placed 
and where voids in the concrete have exposed the metal. 

Wet concrete offers more protection to iron or steel than 
a dry mixture, because the metal is better coated with the 
cement mixture. Cinder concrete, when of a rich mixture— 
at least one sufficiently rich in cement to coat every particle 
of the cinder—will protect ironwork or steelwork as well 
as stone concrete; but, if it is not properly mixed and particles 
of the uncoated coal or cinder come in contact with the steel, 
rapid corrosion is likely to take place. 

Effect of Fire on Concrete.—Concrete is essentially a fire- 
proof material. All the ingredients of which it is composed 
are of a highly refractdry nature, the aggregates being the 
elements of the mixture that are most quickly affected by 
intense heat. This is especially true of granite and lime- 
stone aggregates, the former being likely to crack or burst 
when heated, and the latter liable to calcine. After cement 
has set, the chemical union of its particles is liable to destruc- 
tion by fire, because intense heat robs the cement of the 
water of crystallization, or dehydrates the cement, thus 
softening the material and making it crumbly. If concrete 
in a mass is subjected to intense heat, this action of dehydra- 
tion extends into the concrete for a depth of only } to 4 1ncu, 
and is not likely to penetrate farther. 

Considére, a French concrete expert, has found by experi- 
ment that a 1-3 mortar will shrink from about .05 to .15% 


238 PLAIN CONCRETE 


in setting in air, and that the shrinkage will be two to three 
times as great with neat cement. The shrinkage in concrete 
will be much less than with neat cement or cement mortar. 

The shrinkage of concrete is lessened by embedding in it 
steel rods or bars, as these, by their tensile resistance, prevent 
the shrinkage of the material in setting. By the experi- 
ments of Considére, it is found that with 1-3 mortar rein- 
forced with steel the shrinkage in setting is about one-fifth 
_ that of the same mortar without the steel reinforcement. 

Effect of Thermal ‘Changes in Concrete.—Nearly ail 
materials expand slightly as they become heated. Con- 
crete and steel also follow this law. The contraction or the 
expansion of concrete due to changes in temperature is about 
the same as that of steel. The average coefficient of expan- 
sion of a 1-2-4 concrete for each Fahrenheit degree in change 
of temperature is .0000055. Experiments made on 1-3-6 
concrete give a coefficient of expansion of 0000065, which 
is practically the same as the coefficient of steel. 

Effect of Vibration on Concrete.—The effect of constant 
vibration on concrete structures has not been definitely 
determined. Many buildings and bridges constructed of 
concrete reinforced with steel rods and bars have withstood 
heavy and constant vibration, either continuous or inter- 
mittent, for an extended period of years with no apparent 
deterioration in strength. Fresh concrete is always, how- 
ever, subject to deterioration by vibration, and the strength 
of concrete subjected to jar or shock when setting is materially 
reduced, because the process of crystallization between the 
particles, and the consequent cohesion of the mass, seems 
to be partly destroyed. 


WORKING STRESSES AND STRENGTH VALUES OF 
CONCRETE 


The ultimate strength of concrete varies so with the pro- 
portion of the mixture, manner of working, character of 
ingredients, and age of material, that it is necessary to 
assume low unit working stresses for it. 

The usual working stress for plain concrete under com- 
pression is from 250 to 300 lb. per sq. in., although, in-masses, 


eo 


PLAIN CONCRETE 239 


as in footings, a 1-24—5 concrete would safely sustain as 
much as 500 lb. per sq. in. When reinforced concrete is 
subjected to compression from loads causing bending, it is 
customary to figure the safe allowable unit compressive 
stress in the compression portion of a reinforced-concrete 
beam. at from 500 to 600 or even 750 lb. per sq. in. 

In tension, concrete has little value; in fact, it cannot be 
relied on to resist this stress. Generally, the tensile strength 
of plain concrete is about one-tenth of its compressive 
strength. 

The modulus of rupture, or the unit value for figuring the 
transverse strength of plain concrete, is much lower than 
the modulus of rupture of any of the good building stones. 
The safe unit bending stress for plain concrete, based on a 
factor of safety of 4, from values of the modulus of rupture 
obtained from recent tests made on concrete 30 da. old, is 
about 110 lb. This value is for concrete composed of 1 part 
of cement, 2 parts of sand, and 4 parts of broken stone. 
With a poorer mixture, as a 1—2—5 concrete, a safe bending 
stress of about 95 lb. should be used, while with a 1-3-5 
mixture, the safe bending stress is barely 70 lb., and this 
value shows a corresponding decrease as the mixture becomes 
leaner. 

The safe unit shearing stress of plain concrete is, in practice, 
taken at a very low figure when compared with recent tests 
giving the ultimate shearing resistance of this material. 
This low figure is probably due to the fact that few tests 
have been made to determine the value of plain concrete 
in shear; or perhaps it is due to the unreliability of concrete, 
as found in practice, to resist this stress. The conservative 
safe unit shearing stress of plain concrete is taken at 50 Ib., 
although the value may be increased for rich mixtures and 
careful workmanship to 75 lb. 

The safe grip, or bond, as it is called, of concrete on steel 
rods or bars with plain surfaces embedded in it, is taken, for 
purposes of calculation, at 50 lb. per sq. in. of the surface 
of the metal in contact with the concrete. 

All the values just mentioned are based on concrete at 
least 1 mo. old. There is great diversity of opinion regard- 


240 PLAIN CONCRETE 


ing the safe unit values of plain concrete, and there is no 
uniformity in the building laws of the several cities regard- 
ing the strength of this material. This is shown by the 
following table, which gives the working values of concrete 
allowed by the building laws of several cities: 


UNIT WORKING VALUES OF CONCRETE ALLOWED BY 
VARIOUS CITIES 


Direct - 
Compres- | Shear | Unit Cotapress- 
sion Pounds i 
Name of City Pounds Se Unde wes ding 
rT uare 
Pounds per 
ae Inch Square Inch 
New York....... ; 350 50 500 
Philadelphia...... 250 50 600 
Cleveland. . \ 400 50 500 
San Francisco . 450 75 500 
PAUSED «a: 35- 040.030 350 50 500 
TOPOUtO\. cxcee acess 340 50 500 


An average ultimate unit compressive strength of 2,000 Ib. . 
would be conservative for a 1-2-4 concrete, from 1 to 3 mo, 
old, and 1,600 lb. for a 1-3-6 concrete of the same age. 
Mixtures varying in richness between these limits would 
have proportional values. 

The tensile strength of concrete is more affected by the 
quality of the mixture than is its compressive stress. There- 
fore, a conservative ultimate-tensile-strength value of a 
1-2-4 mixture would be about 200 lb., while for a 1-3-6 
mixture, it would be about 125 Ib. 

The shearing strength of concrete is usually much less 
than the compressive strength. For a 1—2—4 mixture an 
average ultimate shearing strength of 1,480 lb. per sq. in. 
has been determined by tests, while a 1-3-5 mixture has 
given 1,180 lb., and a 1-3-6 mixture has given a value in 
shear of 1,150 Ib. per sq. in. 

The average modulus of rupture for plain concrete that was 
from 33 to 35 da. old has been found to be 439 lb. per sq. in. 


241 


PLAIN CONCRETE 


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242 PLAIN CONCRETE 


for a 1-2—4 mixture, 380 lb. per sq. in. for a 1-2-5 mixture, 
and 285 lb. per sq. in. for a 1-3-5 mixture, while a 1-3-6 
mixture gave a result of 226 lb. per sq. in. 

The values given in the accompanying table are recom- 
mended as ultimate values by W. Purves Taylor, engineer 
in charge of the municipal testing laboratory of Philadelphia. 
The figures represent values obtained from six hundred 
experiments made on concrete properly mixed with good 
Portland cement. It will be noticed that the values given 
for shear are considerably lower than those just given. The 
results obtained depend to a large extent on the method of 
testing. Some engineers prefer the lower values. 


MIXING AND WORKING OF CONCRETE 
CONCRETE MIXTURES 


Methods of Measuring Ingredients.—After deciding what 
proportions of ingredients will be used for the concrete, the 
engineer must be able to calculate the exact quantity of 
each material that he must order. An ordinary box car 
holds from 400 to 600 bags of cement. The purchaser is 
charged for the bags by the manufacturer, unless they are 
of paper, but he gets a rebate for those which are returned. 

Cement is usually measured by the barrel just as it comes 
from the manufacturer, or as 4 bags to the barrel, while 
broken stone and sand are measured loose ina barrel. Port- 
land cement, after it is taken out of its original package and 
stirred up, fills a larger volume than when packed. It is 
therefore necessary to state just how the cement is to be 
measured; and, as said before, the custom is to measure 
it by the barrel, compact. A cement barrel contains about 
3.8 cu. ft. 

Fuller’s Rule for Quantities——A practical rule has been 
devised by W. B. Fuller whereby, after the proportions of 
ingredients have been fixed, the quantity of material for a 
certain work may be obtained. It is called Fuller’s rule for 
quantities, and may be expressed in mathematical symbols 
as follows: 


PLAIN CONCRETE 243 


Let c be the number of parts of cement; s, the number of 
parts of sand; g, the number of parts of gravel or broken 
stone; C, the number of barrels of Portland cement required — 
for 1 cu. yd. of concrete; S, the number of cubic yards of 
sand required for 1 cu. yd. of concrete; and G, the number 
of cubic yards of stone or gravel’required for 1 cu. yd. of 
concrete. Then 


If the broken stone is of uniformly large size, with no 
smaller stone in it, the voids will be greater than if the stone 
were graded. Therefore, 5% must be added to each value 
found by the preceding formulas. 

ExAmMPLe.—lIf a 1-2-4 mixture is considered, what will 
be: (a) the number of barrels of cement, (b) the number of 
cubic yards of sand, and (c) the number of cubic yards of 
stone required for 1 cu. yd. of concrete. 

So_tuTion.—(a) Here c=1, s=2, and g=4. Substituting 
these values in the first formula, 


a ee. 
1+2+4 


(b) Substituting the values of C and s in the second 
formula, 


3.8 
S=— X1.57X2=.44 
27 


(c) Substituting the values of C and g in the third 
formula, 


3.8 
G=—X 1.57 X4=.88 
27 


Table of Quantities.—The table on pages 244 and 245, giving 
the quantities of ingredients for concrete of various pro- 
portions, was prepared by Edwin Thacher. It will be noted 
in this table that the difference in the character and size of 
the stone or gravel used has been taken into account. These 
values will be found to agree fairly well with values found 
by Fuller’s rule. 


PLAIN CONCRETE 


244 


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246 PLAIN CONCRETE 


_ WORKING OF CONCRETE 

Mixing of Concrete-——Concrete may be mixed either by 
hand or by machine. For small work, the concrete is mixed 
by hand in small batches, such as would be made up from 
1 of 2 bags of cement. In mixing, hand work should be 
performed on a flat, water-tight platform. The sand, after 
it has been measured, is spread over the platform in an even 
layer. Upon the sand is placed the cement, and these two 
materials are turned over with shovels at least three times, 
or until the uniform color of the mixture indicates that they 
are thoroughly incorporated. The stones, or aggregates, 
having previously been well wetted, are then placed on the 
top of the mixture of sand and cement and these materials 
are also turned at least three times, water being added after 
the first turning. The water should always be added in 
small quantities. Ifa hose is used for this purpose, it should 
be fitted with a sprinkling nozzle, as otherwise much of the 
cement is liable to be washed out of the mixture. The 
concrete, when ready for placing, should be of uniform con- 
sistency, either mealy for a dry mix or mushy for a wet mix. 

In large work, the mixing should be done by machine. 

Retempering of Concrete.—If the cement of the concrete 
has attained its initial set before being placed—that is, if 
the concrete has commenced to harden—remixing with 
water, or retempering of concrete, as it is called, should not 
be allowed; and if concrete treated in this manner has been 
deposited in the forms, it should be taken out and removed 
from the site of the operation, because concrete cannot be 
retempered properly, except in small quantities for labor- 
atory tests. 

Concreting at High Temperatures.—If the weather is 
extremely warm, the stone and sand are liable to become 
heated toa high temperature. Then, in mixing the materials, 
the water necessary for the crystallization of the cement is 
rapidly absorbed by the stone and the sand, or else rapidly 
evaporated by contact with them. Again, the extreme 
heat will hasten the setting of the cement, and this tends 
to cause the concrete to cake in the mixing machine, pro- 
ducing lumpy and inferior concrete. In order to overcome 


Ey eo 


iy 


ee rege © 


PLAIN CONCRETE 247 


such difficulties, the stone should be thoroughly wetted 
with a hose, and the sand and stone should be kept under 
cover, away from the direct rays of the sun. Likewise, the 
mixing platform or machine should be roofed over. It is 
well, also, to wet down the finished concrete work with a 
hose several times a day in extremely hot weather, and less 
frequently in moderate temperatures. 

. Concreting in Freezing Weather.—Although it is practi- 
cable to mix and place concrete at a temperature as low as 
27° F., it is not advisable to lay concrete work when the 
temperature is below 32°; neither should it be mixed and 
placed even at this temperature, if there is a possibility that 
the temperature will fall. If concrete is frozen, its setting 
is retarded and it is liable to become worthless, never properly 
setting and obtaining the requisite hardness and strength. 
There is, however, no certainty of the action of frost on 
concrete, as frozen concrete will frequently thaw out and 
set, with apparently little loss of strength. 

To prevent the freezing of concrete when the temperature 
has fallen below 32° F., salt is sometimes used in the mixture. 
The addition of 1} lb. of salt to the water used with 1 bag 
of cement will not decrease the strength of the concrete; 
or, a 10% solution of salt can be used in the water employed 
in mixing the concrete. The addition of salt, however, is 
never advisable if a surface finish is required, as it is liable 
to cause efflorescence, or a white deposit, on the surface 
causing the work to become very unsightly. 

Aggregates that are coated with ice or that have been 
exposed to severe weather for a long time should be heated 
or thawed out before being used. Concrete that is exposed 
to freezing after it was set should always be protected by 
placing over it a layer of boards and straw, or salt hay, or 


cement bags; or, where the work is in the nature of a rein- 


forced-concrete floor system, by heating the interior of the 
structure by means of salamanders or fires. 

Joining of Old Concrete With New.— New and old concrete 
can be joined only with difficulty, and the strength of such 
a connection is always uncertain. The joining of old and 
new concrete work is best done by thoroughly chipping, or 


248 STEEL REINFORCEMENT 


cutting away, the old surface, saturating it with water, and 
working into it thin coats of a 1-1 Portland-cement mortar, 
and, then, while the coating is still fresh, placing egataat it 
the new concrete. 

There are some high-grade, imported cements that, in the 
form of cement mortar, more readily adhere to old concrete 
work than the usual Portland cements. These cements are 
frequently used for patching and piecing out work already 
in place. 


ELEMENTS OF STEEL REINFORCEMENT 


THEORY OF STEEL REINFORCEMENT 

Principles of Construction.—When a beam is subjected 
to transverse stress, the portion of the beam section above 
the neutral axis is in compression, while in that portion 
below the neutral axis, tensile stresses are created. Ordi- 
narily, concrete is about ten times as strong in compression 
as itisin tension. Thus, it can readily be seen that a beam 
of plain concrete without steel reinforcement would fail 
primarily from lack of tensile resistance, without realizing 
its full compressive 
strength. In order, 
therefore, to make 
concrete an eco- 
nomical material to 
use in construction, 
its deficiency in 

Fic. 1 tensile resistance 
must be made up by embedding steel rods, bars, or some 
other form of metallic reinforcement in that portion of the 
beam section subjected to tensile stress. 

In order to explain more fully this primary principle of 
reinforced concrete, reference is made to the reinforced, 
rectangular concrete beam shown in Fig.1. The neutral line 
of the section is shown at 4 7; in the side view (a), while the 
neutral axis is represented by y y in the end view (b). When 
the concrete beam is under transverse stress, there is neither 


STEEL REINFORCEMENT 249 


tensile nor compressive stress at the neutral axis. Therefore. 
the point in the beam marked a, which is on the neutral axis, 
is subjected to zero stress. 

Imagine that the concrete is cut away below the neutral 
plane, leaving the steel reinforcing rods, or bars, exposed as 
atb. Itis evident that the strength of the beam is not much 
affected by the cutting away of the concrete in this manner, 
as the necessary tension below the neutral axis is supplied by 
the reinforcing rods of steel, while the necessary compression 


Fic. 2 


above it is furnished by the concrete, asatc,c. The amount 
of compression in each square inch of concrete above the 
neutral axis varies from zero at the axis to maximum at 
the extreme upper surface of the beam. The concrete 
below the neutral axis yy is usually so filled with very fine 
cracks that all the tension must be carried by the steel alone. 

In ordinary reinforced-concrete column _ construction, 
merely vertical rods are employed. They are tied, however, 
at intervals with wire or other ties. 


250 STEEL REINFORCEMENT 


The principle of hooped columns is best explained as follows: 
It is well known that a column of sand will not resist com- 
pression, because it will spread, and it is likewise certain 
that a cylinder of; say, very thin metal will sustain only 
a small load. However, if the cylinder is filled with sand, 
the combination, in which the tensile strength of the cylinder 
is realized with the compressive resistance of the sand, will 
result in a strong column, or post, capable of resisting con- 
siderable compression. This principle is applied to the rein- 
forcement of concrete columns by binding, or tying, together 
the concrete with cylindrical hoops, or helical, or spiral, 
windings of steel. 

Parts of Steel Reinforcement Defined. —In Fig. 2 (a) is 
shown a perspective view of a complete bay of a reinforced- 
concrete floor system, and in (b), a diagrammatic representa- 
tion of a typical system of reinforcement for a concrete 
girder and column. In (a), the heavy members A running 
between columns are commonly known as girders, and the 
lighter members B running between girders, as beams. In 
both (a) and (bd), the rods, or bars, a are the main reinforcing 
bars, or rods, of the girders. The beams, of course, have 
similar main reinforcing bars. Of these main reinforcing 
bars, several are bent up, as at b, to form trussed bars. The 
web reinforcement of the girders is shown at c, and consists 
of U-shaped pieces of iron or steel, called stirrups. The rods 
that reinforce the slab of the reinforced-concrete floor system, 
called slab rods, are shown at d, This slab reinforcement 
may consist of straight rods, expanded metal, woven-wire 
lath, or any other metallic reinforcement. The rods of the 
columns, shown at e, are called longitudinal column rods, and 
the hooped separators, or ties, shown at f, column ties. 

Any rod, or bar, used to resist shearing stresses is desig- 
nated asa shear bar. A rod, or bar, used to resist the shrink- 
age of the concrete in setting, or to provide against cracks 
due to thermal changes, is called a shrinkage rod. Shrinkage 
rods are shown at g in view (a). <A rod used to connect 
abutting beams or girders is called a tension bar or a tie- 
bar. The short rods used at the splice when longitudinal 
column rods are butted are called splice rods, or bars. 


STEEL REINFORCEMENT 251 


Members to Resist Lines of Failure.—In Fig. 3 are illus- 
trated a typical beam having the usual type of steel rein- 
forcement and the several methods of failure that might 
occur. Ata are shown cracks, or lines of failure, that would 
be caused by lack of tensile resistance in the main reinforcing 
rods b. These cracks, although usually invisible, generally 
extend from the bottom surface to the neutral axis. They 
are nearly always present in concrete, but, of course, so > long 
as the steel holds, the beam will not fail. 

If the main reinforcing rods do not extend to the bearings, 
failure by vertical shear may occur near the abutments, along 
the line x x. Ordinarily, however, failures of this kind sel- 
dom happen, because the main rods usually extend across all 


_— 


| alte 
x 
—— 
Fic. 3 


such lines of vertical shear, and add greatly to the shearing 
resistance of the beam. 

If the slab concrete is not placed at the same time that. 
the concrete of the beam section is poured, failure by shearing 
usually occurs at the junction of the beam with the slab, as 
shown atcc. The shearing resistance at this junction may 
be increased, however, by extending stirrups d into the slab. 
If the crack cc opens, it usually joins with a crack like ee 
at each end of the beam, as suggested in the preceding 
paragraph. 

The lines of failure indicated at e e are those which usually 
occur from diagonal tension stresses that cross these lines of 
failure at right angles. A beam is held against failure in this 
manner by placing stirrups in the concrete either vertically 
or obliquely. The bending up of the main reinforcing rods. 


252 STEEL REINFORCEMENT 


to form the trussed bar, as shown at ff, will also assist in 
resisting such stresses, and, besides, will provide against 
negative bending moment where tension instead of compres- 
sion is created at gg. The line of fracture shown at ee is 
typical of nearly all reinforced-concrete failures. 


METALLIC REINFORCEMENT 


CHARACTERISTICS OF METALS USED FOR 
REINFORCEMENT 


Mild, or Soft, Steel——The commercial mild, or soft, steel, 
which is an excellent material for reinforcing rods, especially 
in concrete structures subjected to sudden strain or shock, 
should have a unit ultimate tensile strength of from 52,000 
to 62,000 Ib. and an elastic limit of not less than one-half 
this amount. It should have an elongation of at least 25% 
in 8 in., and should be capable of being bent cold through 180° 
and hammered flat on itself without evidence of fracture on 
the outside circumference of the bend. 

Medium Steel.—The grade of steel known as medium more 
fully meets all the requirements of reinforced-concrete con- 
struction than any other grade. Commercial medium steel 
should have an ultimate unit tensile stress of 60,000 to 
70,000 lb., and an elastic limit of not less than one-half this 
amount. Upon testing, the elongation should be found to 
be at least 22% in a length of 8 in., and it should withstand 
bending through 180° around a diameter equal to the 
diameter or the thickness of the pieces tested. Such a 
test piece should not show fracture on the outside circum- 
ference of the bent portion. 

High-Carbon Steel.—The name high-carbon steel is applied 
to steels rolled particularly for the reinforcement of concrete. 
Such steels contain a higher percentage of carbon than either 
medium or soft steel. The product as a rule is brittle, and 
possesses a unit ultimate tensile strength of from 80,000 to 
100,000 lb., with an elastic limit of about one-half of its 
ultimate strength. The amount of elongation in 8 in. is 
sometimes as low as 5%, and seldom exceeds 15 to 18%. 
Ordinarily, it will not bend much beyond a right angle with- 


STEEL REINFORCEMENT 253 


out showing fracture. On account of the brittleness and 
general unreliability of high-carbon steel, the material should 
be carefully tested and’/inspected before being used in rein- 


forced-concrete construction. 


AREAS AND WEIGHTS OF SQUARE AND ROUND BARS 


Square Round 
‘1 e 
Size 

Inches Weight Weight 
Area per Foot Area per Foot 

Inches Poswax Inches Puasa 

ts 39 .013 0031 010 

4 0156 053 0123 042 

0352 120 0276 094 

0625 E213 0491 167 

0977 S532 0767 261 

6 .478 1104 .376 

1914 .651 503 511 

a 2500 .850 1963 . 668 

.3164 1.076 2485 .845 

.3906 1.328 . 3068 1.043 

4727 1.607 nous 1.262 

5625 1.913 .4418 1.502 

6602 2.245 -5185 1.763 

.7656 2.603 .6013 2.044 

.8789 2.989 . 6903 2.347 

1 tA 3.400 . 7854 2.670 

14 1.1289 3.838 .88 3.014 

1g 1.2656 4.303 .9940 3.379 

1 1.4102 4.795 1.1075 3.766 

1 1.5625 5.312 1.2272 4.173 

Bt ey 7 5.857 1.3530 4.600 

1 1.8906 6.428 1.4849 5.049 

1l& 2.0664 |~> 7.026 1.6230 5.518 

14 2.2500 7.650 1.7671 6.008 

175 2.4414 8.301 1.9175 6.520 

it 2.6406 8.978 2.0739 T2051 

1 2.8477 9.682 2.2365 7.604 

it 3.0625 10.413 2.4053 8.178 

1 3.2852 11.170 2.5802 8.773 

it, 3.5156 11.953 2.7612 9.388 

y 3.7539 12.763 2.9483 10.024 

2 4.0000 13 3.1416 10.681 


254 STEEL REINFORCEMENT 


There are to be had on the market twisted bars of square 
and hexagonal section. Such bars are from 8 to 25% 
stronger than the bars in their original form. The per- 
centage of increase of strength is greatest with the bars of 
small section and least with those of large section. 

Rerolled Bars.—Bars made from old steel rails can be 
obtained for reinforcing concrete. In making such bars, 
the flange, web, and bulb of the rails are cut apart, and 
rerolled into square and round sections. The square sections 
may be had twisted, the twisting being performed while the 
material is hot. , 

Much of this material has a high elastic limit and tensile 
strength, but it is inclined to be brittle and variable in 
strength and ductility. Besides, any lapping or folding of 
the original material when passed through the rolls is liable to 
develop laminations throughout the bar, tending to diminish 
its strength. 

TYPES OF STEEL REINFORCEMENT 

Plain Bar Iron.—The cheapest form of metallic reinforce- 
ment for concrete is the plain, round, rolled bar. Bars of 
this kind can be obtained in 
any part of the United States. 
The pound price of round, 
rolled bars is lower than that 
of any other form of rolled 
steel, so they offer the cheapest 
and most available material. 
For slabs, 3- to 4-in. round 
bars are used, while for beam, 


({ girder, and column reinforce- 


ment, from §- to 1}-in. bars 
are ordinarily employed. 
(e) @ The principal objection to 
Fic. 1 the use of plain, round bars in 
reinforced-concrete work is that they are not gripped, or held, 
well by the concrete. 
Plain square and flat bars are sometimes used for the 
reinforcement of concrete, though, generally, both of these 
sections, when so used, are deformed by twisting. 


STEEL REINFORCEMENT 255 


EN ENEN NEN 


a 
(é) 
Fic. 2 


In the nomenclature 
of reinforced concrete, 
round, rolled sections 
are designated as rods; 
square sections as bars; 
and rectangular sec- 
tions, as flats, or flat 
bars. 

In the table on page 
253 are given the areas 
and weights of square 
and round bars from 
ys- to 2-in. sizes. 

Bars of Special Con- 
struction.—Some early 
forms of bars used in 
reinforcement of con- 
crete are shown in Fig. 
1. That shown in (a) 
is known as the Hyatt 
bar. An early form 
of the Thacher bar is 
shown in (b). In (c) is 
shown the Staff bar. 
It consists of a flat bar, 
through which a coun- 
tersunk punch has 
been partly driven, 
thus forcing the metal 
out on the opposite 
side so as to form 
projections. The De 
Mann bar is shown in 
(d). Two forms of the 
untt bar are shown in 
(e) and (f); that in (e) 
is known as the Siam- 
ese bar and that at 
(f) as the quad bar. 


256 STEEL REINFORCEMENT 


Square-Twisted Bars.—The square-twisted bar consists of 
a square bar thatis twisted by giving it a certain number of 
turns around its axis, either while it is hot or while it is 
cold. This bar is often known as the Ransome bar. 

By twisting the bar to the screw shape, as shown in 
Fig. 2 (a), a form is obtained that has great resistance to 
pulling from a mass of concrete. 

If the square bars are twisted cold, their elastic limit and 
ultimate strength are increased from 8 to 25%. 


PHYSICAL PROPERTIES OF THE RANSOME BAR 


< sa - 

oe ‘ Hfe ais 2g 

y fee) oo le.) ee le. ble..| Be 
agea) ae io) £3 (S33) eee) 85a 

alse] wa (82/98 |Sbs| sae] SS 
Sg08).. 8» 183| ee Teese 408) gee 
95/95 e 182) 28 |e ¢|/ e286 Sq 0 
N18 8 eg | 28) eH a) SOU) ese 

Pl “s |8 | ae le sls ag 

5] > §|A = 
8 erie z 
Ay e¥ 

} 5 0625 213| 62,350 | 86,700 | 3,897 5,419 
3 1406 478) 61,8 86, 12,176 
2 2500 8 60,120 | 86,850 | 15,030 21 vis 
2 3906 |1.328) 57,890 | 85,820 | 22,612 33,520 
1 5625 |1.913) 56,720 | 85,240 | 31,905 47,948 
$|1 7656 |2.603) 56,150 | 84,730 | 42, ,86 
1 a 1.0000 |3.400)} 55,760 | 84,275 | 55,760 84,275 
14/4 1.5625 |5.312) 55,450 | 83,150 | 86,641 129° 921 


The physical properties of the Ransome bar are given in 
the accompanying table. 

Spiral Bar.—In Fig. 2 (b) is shown a type of twisted bar 
called the spiral bar. It is made by rolling steel and then 
twisting it. The twisted bars have elastic limits of from 
65,000 to 80,000 lb. The section of the bar is practically 
round, with four attached half-round beads. These beads 
assume the spiral form on twisting the bar, and the same 
advantage with regard to the bond is secured as with square- 


STEEL REINFORCEMENT 257 


twisted bars. An added advantage is assumed to exist in 
the elimination of all the sharp corners that occur in the 
square bar. 

Kahn Cup Bar.—One of the latest types of reinforcing bars 
to be commercially used is the Kahn cup bar illustrated in 
Fig. 2 (c). The ribs are connected by cross-ribs, forming 
cups, or depressions, of such a shape as to allow the concrete 
to flow into them readily. In this way, a positive mechanical 
bond in the concrete is provided. The properties of the 
Kahn cup bar are given in the accompanying table. 


PROPERTIES OF THE KAHN CUP BAR 


Ultimate 
Nominal Sectional Weicht Elastic Tensile 
Size Area per Foot Limit of | Strength 
of Bar Square Pounds Each Bar | of Each 
Inches Inches Pounds Bar 
Pounds 
2 1406 .502 5,000 9,800 
.893 9, 17,500 
3906 1.394 14,000 27,300 
5625 2.008 21,000 : 
7656 2.733 28,000 
1 1 3.570 37,000 70,000 
1k 1.2656 4.518 47.000 ; 
1} 1.5625 5.578 ; 109,400 


Square-Twisted Lug Bars.—The twisted lug bar shown in 
Fig. 2 (d) is a development of the square-twisted bar. The 
square bar in this case is rolled with rounded corners, and 
with projections, or lugs, a at intervals, thus providing 
additional mechanical bond in the concrete. The rounded 
corners of this bar eliminate the sharp angles of the ordinary 
square-twisted bar. These bars have a high elastic limit. 
Their safe working strength is based on a safe unit stress 
of 20,000 lb. The properties of twisted lug bars are given 
in the accompanying table. 

Corrugated Bar.—In Fig. 2 (e) and (f) are shown two styles 
of deformed bar of the corrugated type known as the Johnson 


258 STEEL REINFORCEMENT 
PROPERTIES OF TWISTED LUG BARS 


Size of Net Sectional Weight per Safe Working 
Bar a _ Foot Each Bar 
Inches | Square Inches Pounds Pounds 
} 0625 . 222 1,250 
1406 .492 2,810 
. 2500 .870 i 
. 3906 1.350 7,810 
. 5625 1.940 11,250 
. 7656 2.640 15,310 
if 1.0000 © 3.450 
14 1.2656 4.350 25,310 
1}; 1.5625 5.370 3 


bar, named after its inventor, A. L. Johnson. The old style 
of bar is shown in (e), and the new style in (f). High-carbon 
steel having an elastic limit of from 65,000 to 70,000 Ib. per 
sq. in. is used in making the Johnson bar. This type of bar 
was one of the first deformed bars to be manufactured. The 
following table gives the size, net section, and —_— of 
Johnson bars : 


SIZE, NET SECTION, AND WEIGHT OF CORRUGATED, 
OR JOHNSON, BARS 


Size of Bars Weight per Foot Net Section 
Inches Pounds Square Inches 
4 .78 .19 
1.56 .38 
2.25 .55 
1 2.90 .70 
14 4.56 1.10 


Universal Bar.—In Fig. 2 (g) is shown a type of flat 
deformed bar, called the Universal bar, that has been used 
to some extent for reinforcement. The net section and 
weight of the various sizes of the Universal bar are given 
in the following table. 


STEEL REINFORCEMENT 259 
NET SECTIONS AND WEIGHTS OF UNIVERSAL BARS 
N Size of Bar Weight per Foot Net Section 
Ct Inches Pounds Square Inch 
1 x1 13 .19 
2 xX 12% 1.35 -41 
3 X12 1.97 54 
4 X2 2.26 .65 
5 X 24 2.85 .80 


Thacher Bar.—The deformed bar illustrated in Fig. 2 (h), 
called the Thacher bar, was devised in order to obtain a bar 
so deformed that the net section throughout the bar would 
be uniform. By having the section uniform, the bar is of 


the same strength at every point. 


In forming this bar, an 


SIZE, WEIGHT, AND ULTIMATE TENSILE STRENGTH 
OF THACHER BARS 
(Medium Steel) 


Average 
Diameter Weight Area of Net Ultimate 
of Bar per Foot ' Section Tensile Strength 
Inches Pounds Inches for Each Bar 
Pounds 
.16 .047 3,000 
.34 210 6,400 
-61 .18 11,500 
.95 .28 17,900 
1.39 .41 ,200 
k 1.87 .55 35,200 
1 2.41 A A 45,400 
1} 3.06 .90 57,600 
14 3.74 1.10 70,400 
12 4.49 1.32 84,500 
14 5.30 1.56 99,800 
1% 6.15 1.81 115,800 
13 7.07 2.08 133,100 
1¢ 7.99 2.35 150,400 
2 9.01 2.65 169,600 


260 STEEL REINFORCEMENT 


effort is made to eliminate all sharp corners. The Thacher 
bar is rolled from medium steel and has an elastic limit of 
about 35,000 lb. The accompanying table gives the size, 
weight, and ultimate tensile strength of the Thacher bar. 


SIZE, AREA, AND WEIGHT OF DIAMOND BARS 


Pi eta coe Area of Section Weight per Foot 
Tackas Square Inches Pounds 

L .062 22 

f .14 -48 
.19 -65 

+ .25 85 
.39 1.33 

.56 1.91 

.76 2.60 

1 1.00 8.40 
14 1.56 5.31 


Diamond Bar.—One of the most recent forms of deformed 
bars is the Diamond bar, which is shown in Fig. 2 (7). This 
bar is rolled with a cross-section of constant area. The 
nominal size, area, and weight of Diamond bars are given 
in the accompanying table. 


NK Ze 


Kahn Trussed Bar.—Fig. 3 shows two styles of a deformed 
bar known as the Kahn trussed bar. In the old style of bar, 
shown in (a), the prongs are opposite each other; in the new 
style, shown in (db), they are staggered. The shapes of the bar 


STEEL REINFORCEMENT 261 


in section are shown in Fig. 4. The fins are partly sheared 
across and also in a direction parallel with the axis of the bar, 
and are bent up, as shown in Fig. 3 so as to form a grip with 
the concrete and to provide the stirrups, or web members, 
necessary to resist diagonal stresses. 

The Kahn bars are made in the several sizes indicated 
in Fig. 4. The respective sectional areas of the bars are also 


“> 


M$ 
$14 Bar-Weight=1.4 
tb. per foot-Area=0.41 SGM. 


2436 Bar-Meight =/02 
1b. per foot-Aréd=3.00 SG.li- 


3228 Bar-Meight=27 lb per 
foor —hreda0.79 SF M7. 


1x3 Bar-Weight «48 /b. 
per foot -Aréa=141 5g.la ae 
rads Bar-Weight =68 
1b. per foot-Ared=200 59.1. 
Fic. 4 


given in this figure. These bars are rolled of mild steel, as 
this metal will best stand the shearing stresses to which the 
bar is subjected in manufacture. The elastic limit of the 
metal in the bars ranges from 33,000 to 35,000 Ib. per sq. in. 

Trus-Con Bar.—Another deformed bar, known as the 
trus-con bar, is illustrated in Fig. 5. This bar is round in 


262 STEEL REINFORCEMENT 


section and has notched shoulders for holding washers. 


eee 


Fic. 5 


The washers are punched so that they will slip over the 
projections, and, then by turning, they are locked with the 
bar. The trus-con bar is made in 
' §. 8-1, 14-, and 1}-in. sizes. 

Monolith Steel Bars.—In Fig. 6 
is shown the monolith steel bar. 
This type of bar has no sharp cor- 
ners, and is not reduced in area or 
strength by deformations. It 
also has large surface area. The 
bar is grooved on the sides so 

Fic. 6 that round iron stirrups, as shown 

ata, may be inserted. These stirrups are held in place in 
the bar by swedging the flanges of the bar together. The 
monolith steel bars are made in sections equivalent to 3-, 
8-, 1-, and 14-in. square bars, and are 
rolled for stirrups of 3%s-, #s-, 3-, and 4-in. 
diameter. 

Columbian Bar.—A rolled shape known 
as the Columbian bar is extensively used 
in the construction of the Columbian fire- 
proof floor systems and reinforced-con- 
crete structures. The typical forms of 
this type of bar are shown in Fig. 7. (a) 
Mild steel is used for rolling the Colum- 
bian bar. This kind of steel has an ul- 
timate strength of from 60,000 to 70,000 Ib. 
per sq. in. and an elastic limit of one- 
half the ultimate strength. The smallest 
bar, shown in (a), is known as the 1-in. bar, while the type 
of bar shown in (db), is made in 2-, 24-, 34-, and 44-in. sizes. 
The 5-in. bar carries a double rib at the bottom. 


(bo) 
Fic. 7 


STEEL REINFORCEMENT 263 


U Bars.—In Fig. 8 is shown a section of U bar that can 
be used to advantage in reinforced-concrete construction 
either as a tension or as a compression member, 
being particularly efficient for the latter pur- 
pose. These bars are rolled from high elastic- 
limit steel, and are in some instances made from 
rerolled steel rails. They are }, ¥%, %, and 4 in. 
in thickness, weigh. from 3 to 9 lb. per ft., and can be ob- 
tained in lengths up to 60 ft. 

Structural Shapes Used as Steel Reinforcement.—The usual 
rolled-steel structural shapes have been used extensively for 
reinforced-concrete construction. They have an advantage 
in that they can be readily obtained. Structural shapes 
as metallic reinforcement for concrete work are being super- 
seded by either plain or deformed rolled bars. One advan- 
tage of the use of structural shapes is that a rigid framework 
can be built up, after the manner of skeleton construction, 
though of much lighter section, and the concrete then filled 

in around it. 
Expanded Metal. 
Among the earlier 
forms of metallic 
(a) : reinforcement for 
concrete is the dis- 
torted steel plate 
known as expanded 
metal, a familiar il- 
lustration of which 
is shown in Fig. 9 
(a). This form 
of reinforcement is 
manufactured by 
partly shearing a 
sheet of steel in 
parallel rows, as shown in Fig. 9 (b), and then pulling the ma- 
terial sidewise, thus forming a diamond mesh. In this way, 
the area of a sheet is increased about eight times, with acor- 
tesponding decrease in weight per unit area and without any 
waste of material. The steel from which expanded metal is 


Fic. 9 


STEEL REINFORCEMENT 


264 


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TIVLAW GHAGNVdXa AO SAZIS 


STEEL REINFORCEMENT 265 


manufactured has a unit tensile strength of about 65,000 Ib. 
and an average elastic limit of about 30,000 lb. It is 


7 


“) 


Fic. 10 
principally used in reinforced-concrete construction as slab 
reinforcement. The sizes of expanded metal are given in the 


accompanying table. 
Kahn Expanded Metal.—Another concrete-slab reinforce- 
ment, called Kahn expanded metal, is shown in Fig. 10. This 


PROPERTIES OF KAHN EXPANDED METAL 


eae f 
> od ~ 
o. '§ e) ® 2 
4 § Once a (e) pA S| o & 
tu oes on = 
£[ So 8 loess) BS 8 | Gegg | oy aS a 
g |A2 Gigs | Og ¢ 260 § “wo |fo 8 
© 3 o Oe ss eon BD rs) Ss 
3 om & q ee fe) lal ° af |.Ms 5 
2/\s “lo8e8| S8eR Gyr | eS | O50, 
# jeefs gem | oe 2 |Fg 
soz | | “Shr alee 
2/ 2 |,.48 | 8,640 | 33,600 17, 3), 2.43 
Sree (|- 82 5,760 22,400 25 | 1.48 
4) 4 | 124 | 4'320 | 16800 | 33 | 1.08 
5 5 |-..19 3,420 13,300 | 41 | .87 
6} 6 | .16 | 2880 | 11,200 | 49 72 
7 7 .14 2,520 9,800 | 57 62 
S| 8 .12 | 2,160 65 55 


form of reinforcement is manufactured by shearing a rolled 
- section along lines parallel with its length, and then pulling 


266 STEEL REINFORCEMENT 


the metal so sheared in a direction at right angles to the lines 
of shear. In this type of expanded metal, the main bars 


of the section represent the material available for the rein- 
forcement of the slab, while the light cross-bars act as spacing 
bars and also as shrinkage rods when embedded in concrete. 
The advantage claimed for the Kahn expanded metal is 
that it transmits the load directly to the supports without 
any tendency to elongate or distort. The properties of this 
reinforcing metal are given in the accompanying table. 


Herring-Bone Metal Lath.—In Fig. 11 is shown a type of 
expanded metal known as herring-bone metal lath. This. 


STEEL REINFORCEMENT 267 


metal is particularly interesting, because, when bent into the 
form shown in Fig. 12, it provides a self-centering material 
that is useful for light roof construction. When bent in 
this manner, the herring- 
bone expanded metal is 
known to the manufac- 
turers as trussit. The lon- 
gitudinal ribs, as at a, give 
it at least sufficient trans- 
verse resistance to allow 
the placing of a thin roof 
slab of concrete on it 
without other centering or > 
support. 

Sheet-Metal Reinforce- 
ment.—The type of 
metal reinforcement 
known as Ferroinclave is used for making floor slabs 
and for stair and roof construction. This reinforce- 
ment consists of sheet metal that is bent into grooves, as 
indicated in Fig. 13 (a), annealed sheet steel generally of 
No, 24 U. S. gauge, being used in its manufacture. The 
corrugations are made dovetailed in section so that the end 
of one sheet can slip into another, as shown. The sheets 
are 20§ in. in width and 10 ft. in length. Fig. 13 (b) shows 
’ how this type of metallic reinforcement is used in con- 

structing a slab for a light roof. 
Lock-Woven and _ Tie-Locked 
Wire Fabric.—In Fig. 14 (a) is 
illustrated the form of wire net- 
ting known as lock-woven witre 
fabric. The cross-wires of this 
(») fabric are joined by means of a 
staple of light wire, which is 
bent so as to embrace the cross- 
ing wires at their intersection. 

Another type of junction for cross-wires is shown in 
Fig. 14 (b), which illustrates the principal features of the 
tie-locked fabric. Here, the cross-wires are secured, or 


Fic. 13 


(a) 


Fic. 14 


268 


STEEL REINFORCEMENT 


locked, in position by means of a small disk, or washer, and 


4 


La 


Fic. 15 


i 


by kinking the wires. 
Electrically Welded 
Fabric.—The Clinton 
wire cloth is a fabric 
that is secured at the 
intersections by a per- 
fect electric weld, and 
it has at intervals a 
double wire that twines 
in and out, as shown 
at a, Fig. 15. 
Triangular- and 
Square-Mesh Wire 
Reinforcement. — The 
two types of wire re- 


inforcement shown in Fig. 16 are extensively manufactured 


for reinforced-concrete construction. 


The reinforcement 


shown in (a) is known as the triangular-mesh reinforcement, 
and that shown in (bd), as the square-mesh reinforcement. The 
latter consists of heavy longitudinal wires and cross-wires, 


or spacing wires. 


\\ 


— 
— 
=I) 


Fic. 16 


The cross-wires are carried through and 


(8) 


twisted around the longitudinal wires so as to form the 


rectangular spaces. 


STEEL REINFORCEMENT 269 


SYSTEMS OF STEEL REINFORCEMENT 


LOOSE-ROD SYSTEMS 


A complete floor system constructed of loose rods is shown 
in Fig. 1. The beam reinforcement consists of three rein- 
forcing rods. Two of these rods run straight through the 
entire length of the beam, as at a, while the third one is bent 
upwards at the ends, as at b. This bent member provides 
tensile resistance at the top of the beams and thus takes 
care of the negative bending moment, which occurs in all 
beams fixed at the end. The bend in such rods is usually 
made at an angle of about 30° with the horizontal. The 
rods should be straight at the center of the span for at least 
one-third the distance between the supports. 

A tie-rod c that is 4 or 5 ft. in length, and sometimes bent 
down at the ends, should be placed over the top of the beam 
juncture. 

The girder reinforcement consists of five rods, two of them 
being bent up, as shown at e, to provide against negative 
bending moment. 

In the best work, two short rods? are located transversely 
through the column. These rods tie the adjoining girders 
together and provide additional rigidity at the junction of 
the girders with the column. 

The slab rods, shown at h, are generally spaced at about 
6 inches from center to center. They should bond with the 
stirrups, or web reinforcement, of the beams, and may be 
threaded through, interlocked, or wired to them. It is 
customary to provide shrinkage rods that extend at right 
angles to the regular slab reinforcement, in order to prevent 
shrinkage cracks in the concrete. For this purpose, }-in. 
round or square rods 7 are generally used, and these are 
spaced about 2 ft. from center to center. In order to bond 
the concrete over the main girders securely, it is also good 
practice to provide over these important members rods d 
of about the same size asthe slabrods. These rods should 

fun through holes punched in the top of the stirrup, as 


270 


STEEL REINFORCEMENT 


pe 


; iN" | mill 


Fic. 1 


STEEL REINFORCEMENT 271 


illustrated at g, and should 
extend at right angles to the 
axis of the girder. Sometimes, 
similar rods are used in the 
slab over beams, as shown 
Sea at k, 
—=— _ The longitudinal reinforce- 
ment of the concrete columns 
consists of four round rods /, 
It is customary to project 
them above the concrete of 
each story about a foot and to 
splice them by lapping and 
wiring or by using pipe sockets 
m, as illustrated. Frequently, 
it is not possible to lay out 
beforehand the electric-light 
or power wiring, but if this 
installation is to be adopted 
l}-in. pipes to serve as a 
REE] passageway should be embed- 
“ ded near the center of the span 
of all beams and girders, close 
to the under side of the slab 
aa construction, as at m. 
Kahn System.— The i alite 
-—— system of reinforced-concrete 
construction is based on the 
said use of the Kahn trussed bar. 
-—— A typical monolithic construc- 
tion based on the use of the 
Kahn system is shown in 
aR Fig. 2. The 
ani 


©: of the beams 
" and girders are 
ee es ae ** shown at a,and 
Sed Peps ee dads Se re 2 
Fic. 2 _ they consist of 


272 STEEL REINFORCEMENT 


Kahn bars with the 
prongs bent upwards 
so as to form stirrups. 
The bars are usually 
placed in the forms 
on a 2-in, bed of con- 
crete, and after being 
centered, or registered 
in their proper posi- 
tions, they are secured 
by wiring or by block- 
ing. In order that 
the girders and beam 
connections over col- 


sie Ket 
i. 


perpen ah aetna oooh 


Ceniiheaes 


Idee eS ee EE 
EPCRA 3, vib a ae ke 


: = | umns may have con- 
ies tinuity, an inverted 
4 Kahn bar b is used 
b 


over such junctions. 

Kahn bars are 
spaced in each corner 
of the column, as 
shown in Fig. 3, with 
the prongs bent at 
right angles. The bars 
are usually tied to- 
gether with heavy wire 
or light rods, wound 
around them as shown 
part way up the col- 
umns in Fig, 2. 

The Kahn bar is fre- 
quently used with con- 
crete and hollow terra- 
cotta tile to form floor 
systems designed for 
light loads. This type 
of construction is illus- 
trated in Fig. 4. In 
view (a) is shown a 


ie 


ii 
(@) 
Fic. 4 


5 
rs) 
asa 


pee ved wane 


Hh 


: 
: 


paves ves pep eee 


Reo he ERS S 


Hs 


BA fetes. + Sakeer S- 


SSeS ear 
casas ow 7 


STEEL REINFORCEMENT 273 


cross-section of the construction through the beams 
or secondary members, of the floor system. Usually, 
the tiles are from 12 to 16 in. in width and from 6 to 12 in. 
in depth, the hollow part having the cross-section indicated 
at a. The tiles are spaced about 53 in. apart, and in the 
space between them is laid concrete that is reinforced with 
Kahn bars. The concrete is carried over the top of the tile 
to a depth of about 2 in. The Kahn bar shown at b in (6b) 
is provided with the usual prongs, or stirrups, which are 
bent upwards toward the abutments. The tiles c are 
serrated, or corrugated, along the sides, so as to provide 
some shearing strength in addition to the adhesion of the 
concrete. During the construction of this floor system, the 


Fic. 5 


tiles should be held temporarily in position either by placing 
blocking between them or by nailing them to the forms. 
It is not customary in this type of construction to use 
shrinkage rods in the slab that is laid over the top of the tile. 
Since the tiles are quickly laid on the centering and, con- 
sequently, minimize the quantity of concrete required for 
the construction of the floor, and, also, since the concrete 
placed in conjunction with the tiles sets more rapidly than 
when placed alone, this particular type of construction can 
be carried on with considerable rapidity. Such a floor con- 
struction as this, embodying tile having a hollow air space, 
acts as a good deadener of sound and also tends to preven: 
hcat from passing from one floor to another. 


274 STEEL REINFORCEMENT 


Merrick System.—In Fig. 5 is shown the Merrick floor 
system of reinforced-concrete construction. The space 
between girders is occupied by narrow concrete beams a, 
as shown in the transverse section (a). Between these 
beams are placed boxes made of one of the many styles 
of metal fabric on the market. These boxes run the entire 
length of the clear span of the floorbeams, as shown in the 
longitudinal section (b), and serve to make the floor lighter. 
They are indicated in both views by the heavy dotted lines b. 
Above and below the metal-fabric boxes is a layer of con- 
crete. This layer is usually made about 2 in. thick, so as 
to give a flat ceiling and a flat floor surface. 

Gabriel System.—The Gabriel system of reinforced-concrete 
construction consists of steel reinforcing bars to which are 
attached round-iron stirrups. These stirrups are formed by 
wrapping a wire a several times around the reinforcing bar 


Fic. 6 


and extending it up into the slab, as illustrated in Fig. 6, 
which shows this particular type of construction complete 
and in its several details. The wiring of the columns is 
continuous, and extends from the bottom to the top, as 
indicated in the figure, particularly in the sectional view (0). 
Several of the reinforcing rods of both the girders and the 
beams bend upwards, as shown at c and d, respectively. 
They also lap over the center of the column to form an 
additional bond in the concrete, as shown at e. In this sys- 
tem of construction a lap bar is sometimes provided at the 
top of the junction, as shown. The slab is shown reinforced 
with woven wire and shrinkage rods of round iron. 


STEEL REINFORCEMENT 275 


With the Gabriel system of floor construction, hollow tile 
is also used, as illustrated in Fig. 7. The system shown in 
view (a) is suitable only for very light construction. The 
hollow tiles a are sandwiched between reinforced-concrete 
joists, as at b, and the reinforcement of the floor consists of 
a main reinforcing bar c, around which a round-iron stirrup 
is wrapped so as to extend continuously. A similar con- 
struction is shown in view (b). Here, however, a slab of 


Fie. 7 


concrete is extended over the top of the tile, thus giving 
additional compressive strength at the top of the concrete 
beams and permitting this construction to be used for 
heavier loads. 

Mushroom System of Reinforcement.—A system of rein- 
forced-concrete construction that differs from all others in 
that no beams nor girders are used throughout is illustrated 
in Fig. 8. This system of construction, termed the mushroom 


276 STEEL REINFORCEMENT 


Fic. 8 


RANI 
iT Wit 
\\ 

\\\ 

a\\\\\W 

ji | NIN 
\\W 
\ \ i { 


Wy: \\ 
call 


) if i NO 
WEI 


iM 
VA S\ 
Vit 4; A 


y 


STEEL REINFORCEMENT . 277 


system, consists of reinforced columns that have a spréad cap 
of heavy reinforcing bars radiating from the column, as 
illustrated at a, Fig. 8. These rods are held radially by 
circular reinforcing bars b, to which they are securely wired. 
By means of this heavy reinforcement, the column is greatly 
strengthened at the top, and by spreading in this way, it 
supports a large portion of the floor slab, after the manner 
of radiating cantilevers. The flat slab that covers the 
column supports is reinforced in several directions with light 

reinforcing rods. , 


FABRICATED SYSTEMS 


Unit System.—The girder frame in the unt system is now 
made of plain bars, and the stirrups in the latest develop- 
ment are in the form of an inverted U, the construction of 
the frame being as shown in Fig. 9. 


The frame shown in view (a) is fabricated, or fastened 
together, by a shrinking process; that is, each stirrup and 
tie-rod is hot-shrunk to the main members. This girder 
frame is designed to withstand the hard usage of shipment 
and unloading. The type of frame shown in view (b) is 
designed for use in districts located so far away from the 
shops of the manufacturer as to make the freight rates for 
shipping complete frames prohibitory. 

In the frames shown in Fig. 9 there are four main rein- 
forcing rods a, although the frame may be made up with any 


278 STEEL REINFORCEMENT 


number of rods necessary to secure the desired sectional area. 
The two rods at the top are bent upwards, as shown at 3, 
and the ends ¢ are turned over so as to add to their grip in 
the concrete. The stirrups in both frames are shown at d, 
2-in. round bar iron being used for the stirrups of the frame 
shown in view (a), and light strap, about 4 in. <1 in., for the 
stirrups of the construction shown in view (b). In the 
latter frame, the section at the right of the figure shows 
the reinforcing bars held in the clamping device, which con- 
‘+sists of the casting e, the clip f, and the stirrup g. These 
parts are secured by means of a bolt h. 

Pin-Connected Girder Frame.—Another type of rod rein- 
forcement built up in the form of a girder frame is illustrated 


in Fig. 10. This reinforcement is known as the pin-con- 
nected girder frame, and it is arranged so that the frames 
forming the reinforcement of the beams and girders may be 
connected by pins and links at intersections and over column 
supports. " 
Cummings System of Reinforced Concrete.—The system of 
steel reinforcement shown in Fig. 11, known as the Cum- 
mings loop-truss girder, consists of a series of main reinforcing 
bars a, to which are attached a set of smaller bars b that turn 
up at the ends and form a loop welded at the ends. These 


STEEL REINFORCEMENT 279 


loops are secured to the main reinforcing rods by means of 
metal clips c. 


Fic. 11 


Shear-Frame System of Steel Reinforcement.—The shear- 
frame system is employed to resist the negative, or reversed, 
bending moments that occur at the points of support of 
monolithic beams and girders. This type of reinforcement 
consists of a built-up frame constructed as shown in Fig. 12. 
The shear frame consists of straight top rods a and bent 
bottom rods b. These two sets of rods are securely clamped 
together by means of flat bar-iron plates, or clips, c, which 
also act as stirrups. Additional stirrups are provided at g, 


Fic. 12 


and where the top and bottom rods join they are firmly 
fastened together by means of special clamps f. 


280 STEEL REINFORCEMENT 


This type of reinforcement provides a positive tie, or 
junction, between beams and at the intersection of beams, 
girders, and columns, insuring against failure at these points 
by providing resistance to shear and to failure by horizontal 


Fic. 13 


or oblique tension cracks. The frame may be used with any 
type of reinforcement, and is placed at the junction of. 
beams with girders, as shown in Fig. 13, and at the inter- 
section of the girder and column supports. 


MISCELLANEOUS SYSTEMS 


Brayton System.—The system of reinforced concrete 
known as the Brayton is illustrated in Fig. 14. The main 
reinforcing member is a standard, rolled-steel, I-beam 
section, and the entire reinforcement, including the steel- 
work, is put together in a manner similar to the usual-steel- 
frame construction. The structural-steel beams that form 
the reinforcement carry the dead load of the centering and 
the weight of the concrete. Thus, shores need not be used 
in the erection. 

The I-beam a forming the main reinforcing member of the 
concrete beam is shown in view (a). As shown at b, the 
stirrups extend beyond the top flange of the I beam in the 
form of loops, and are either riveted through the web or 
fastened by clipping them to the lower flange, as indicated 
in (b). The I beam is supported on a column built up of 
angles and provided with brackets d. The slab rods are 


STEEL REINFORCEMENT. 281 


fastened to the stirrups by means of clips and wiring, as 
shown in detail in view (c). 


ear 


Fic. 14 


Visintini System.—Another system of concrete construction 
is the Visintint system. A general view of a girder con- 
structed according to this system is shown in Fig. 15. The 
girders can be made of any width and breadth required to 
withstand the stresses. In France, the columns of buildings 
are also built by this system, but in the United States solid 
columns have been preferred. The beams are made along 


Fioe.. 15 


the lines of latticed steel girders. In light work, the vertical 
webs a, which are in compression, are not always reinforced, 


282 CONCRETE DESIGN 


but the slanting webs b, which are in tension, always have 
steel in them. The upper and lower chords contain each 
a set of three reinforcing rods, as shown at d and e, respect- 
ively. The beams may be made in any convenient location — 
and put in place on the job afterwards. When solid rein- 
forced-concrete columns are used with this system, these 
columns are cast in place and are provided with brackets 
to hold the principal girders. These girders are then hoisted 
on to these brackets. Part of the end of the girder is made 
solid, as shown at c, in order to resist the stresses in shear 
and direct compression due to the reaction of the support. 


DESIGN OF CONCRETE STRUCTURAL 
MEMBERS 


“PLAIN CONCRETE 
CONCRETE BEAMS 


Method of Design.—Stone and plain-concrete beams are 
designed by exactly the same method as any other kind of 
beam, except that the weight of the beam itself can hardly 
ever be neglected. The formula employed is as follows: 


in which M is the bending moment; s, the wnit stress pro- 
duced; J, the moment of inertia; c, the distance from the 
neutral axis to the most remote fiber; and S, the section 
modulus. : 


b d2 
In a beam of rectangular section, aie Se in which b is 


the breadth of the beam and d is its depth of the beam. 

The modulus of rupture for various kinds of stones and 
other materials is given in the accompanying table. 

With the values in this table, a factor of safety of from 
10 to 20 is usually employed when problems dealing with the 


CONCRETE DESIGN 283 
MODULI OF RUPTURE OF VARIOUS MINERALS 


Modulus 


° 
Fh 


Material 
Rupture 


Slate. . AS a POR fe a He Re 
Glass. . ; — 
Bluestone flagging. . 

Marble, white, Italian. . 

Marble, white, Vermont.. 

Marble, gray, ‘Vermont. . 

Granite, Quincy, Massachusetts. 
Granite, New York.. 

Granite, Connecticut. . 

Sandstone, Massachusetts. . 

Sandstone, Middletown, Connecticut. . 
Sandstone, Ohio.. 

Freestone, Little Falls, New York... 
Freestone, Belleville, New Jersey. . 
Freestone, Dorchester, Massachusetts. 
Freestone, Hubeginy.. : 
Freestone, Caen, Normandy. . 

Limestone, average value. . 

Brick, common, or Philadelphia pressed. 
Brick, best hard. . So ecegen sas 
Rubble masonry in cement. Sena ict se abe 200 


ae) 


- 


- 


- 


- 


bent ben Fr rk fen rt ret DN CD Oo 


- 


— 


sungubasszessuesuss 


safe load instead of the ultimate breaking load are being 
solved. The weight of the stone itself must almost always 
be taken into account. 

ExAMPLE.—Design a Quincy granite lintel 6 in. wide on 
a 5-ft. span to carry a uniform load, which includes its 
own weight, of 300 lb. per ft., with a factor of safety of 10. 

SoLuTIon.—In this example, W =300X5=1,500 lb. and 
1=5X12=60 in. Therefore, 


Ae OO raga in.-lb. 
8 8 
According to the table, the modulus of rupture for Quincy 
granite is 1,800, and if a factor of safety of 10 is used, the 
value of s is 1,800+10=180 lb. per sq. in. Therefore, 


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OFS OFS CLI pbs thot = G-Z-T |° °° *° a}e10U0d Jepul 
048 | 798 | css | 989 | 068 Cate % ne aoe) I Soe eesoreqsORy 
O80'T | 800'T | 9Z0'T | 888 88¢ { AIp JO yy StoM } ¢-I + ws ite n> ot ODE 
s %F' TT ‘7072 
‘ ‘ : ‘ [elioyzeur Ap jo mig 
go's | e96't | F249'T | ZIS'T | 192 |{qusiom srg save S quouleo JvoNy 
IAT | 4M9OZ| AMET) AMP | AMT! “VPT . 
AdU94SISuUO? a [eHozey, 


sesy snore, ye oinjdny jo snfnpoy; 


GLAAXONOD dO AANIdnAaA AO TINGOW 


284 


CONCRETE DESIGN 285 


Substituting these quantities in the formula M=S s, then 
11,250 =d2X180. Thus, d?=62.5 and d=7.906, say 8, in. 

The modulus of rupture of concrete is usually less than 
that of stone. 

The table on page 284 gives values found by the United 
States Geological Survey. Three degrees of wetness are 
recognized in mixing the concrete, namely, wet, medium, 
and damp. Wet concrete is such that sufficient water is 
added to make it semiliquid; damp concrete is decidedly 
granular, with little tendency to lump; while medium concrete 
is of a consistency between the other two mixtures. The 
values given in the table are mostly for 1-2-4 mixtures. 
A 1-3-6 mixture gives values at least 15% lower than the 
1-2-4 mixture. 

The factor of safety employed is sometimes 4, but usually 
6 or higher, as the strength of concrete is uncertain. 

EXAMPLE.—Design a concrete beam 12 in. wide on a 12-ft. 
‘span to carry 800 lb. at its center. The safe working stress 
is to be 100 lb. per sq. in. 


Wl 800Xx12x12 
SoLuTION.—The moment is equal to Sa = apa 


= 28,800 in.-lb. Assume the beam itself weighs 260 Ib. 
per ft. Then the moment due to the dead load is 
260 X12 X12 X12 


= 56,160 in.-lb. The total moment is 


8 
56,160 + 28,800 =84,960 in.-lb. Substituting the correct 
12x a? 
values in the formula M=Ss, then 84,960= a X 100. 


Therefore, d?= 424.8 and d=20.6, say, 21 in. 


CONCRETE COLUMNS 
Plain-concrete and stone columns may be divided into two 
classes, namely, those which are centrally loaded, and those 
which are eccentrically loaded. The height of the column 
should never be more than twelve times the least dimen- 
sion of the cross-section and even less for stone and brick. 
Column Centrally Loaded.—For a centrally loaded column 
the allowable compressive stress per square inch is multiplied 
by the area of the cross-section of the column to find the 


{ 


286 CONCRETE DESIGN 


allowable load. Thus, if it is decided to allow an intensity 
of stress of 300 lb. per sq. in., and the column is of square 
section 10 in. on a side, the allowable load will be 10 x 10 x 300 
=30,000 lb. The breaking load’'on columns between two 
and twelve times as high as the least dimension of their 
cross-section seems to be independent of their height. A 
column between these two limits, however, cannot with- 
stand as high an intensity of stress as a cube, for it is more 


ULTIMATE UNIT CRUSHING STRENGTH OF STONE CON- 
CRETE WITH PORTLAND-CEMENT MORTAR 


Proportion of Compression ~ 
Ingredients Pounds per Square Inch 
Cement Sand Stone | 7 da. | 1 mo. | 3 mo. | 6 mo. 
1 2.0 4 1,600 | 2,150 | 2,400 | 2,500 
1 2.5 5 1,480 | 1,950 | 2,250 | 2,350 
1 3.0 6 1,250 | 1,800 | 2,100 | 2,200 
1 3.5 7 1,100 | 1,660 | 1,960 | 2,080 
1 4.0 8 980 | 1,520 ,820 | 1,950 
1 4.5 9 850 | 1,400 | 1,690 | 1,840 
1 5:0 10 750 | 1,260 | 1,550 | 1,720 
1 5.5 11 650 | 1,120 | 1,420 | 1,600 
1 6.0 12 600 | 1,000 ,300 | 1,500 


NOTE.—For gravel concrete, use 75% of the figures given in the 
table. 


likely to break by shearing. For this reason, when employ- 
ing values taken from the tables on pages 286 and 287, for 
column calculations, a larger factor of safety should be used 
than with other work. This factor is usually taken as at 
least 6, and 10 or higher for masonry. 

EXAMPLE.—What is the allowable working load ona 
concrete column that is 10 ft. high and 12 in. in diameter and 
made of 1-2-4 stone concrete 6 mo. old with a factor of 
safety of 6. 

SoLutTion.—The cross-sectional area of the column is 
.7854 X 12?=113.1 sq. in. From the table the ultimate 


CONCRETE DESIGN 


ULTIMATE UNIT CRUSHING STRENGTH OF VARIOUS 
STONES AND STONE MASONRY PIERS 


287 


© +25 0 wea 
> >: 
Peta Bon 
wn 
Material RECe Material Esve 
pgs Eg 8 3 
8 ug 5 ah 
Granite, Colo.......| 15,000 | Limestone, Mar- 
Granite, Cotin:-3..% 14,000 quette, Mich....... 8,000 
Granite, NOS <2 ps'k' 16,000 | Limestone, Consho- 
Granite, Me......... 15,000 hocken, Pa... 15,000 
Granite, Minn. .....| 25,000 | Marble, Montgomery 
Granite: IN eyo 22s: 16,000 CORIPAS< (See ees 11,000 
Granite, N. H. .....| 12,000 | Marble, Lee (dolo- 
Bluestone.......... 15,000 mite), Mass........ 22,800 
Sandstone, Middle- Marble, Pleasantville 
town, Conn....... 7,000 (dolomite), Y..| 22,000 
Sandstone,  Long- Marble, ltalian. . .| 12,000 
meadow, Mass... .| 10,000 | Marble, Vt.. 0,000 
Sandstone, Hudson Slate. . oi ec dee ,000 
River, N. Y.. 12,000 | Piers, ‘ashlar, Blue- 
Sandstone, Little stone. ; 2,100 
Falls (brown), Granite, ashlar piers. 2,100 
i Micense sce ce 2} 40.000: | Piers; ashlar, lime- 
Sandstone, Ohio... . ,000 stone. 1,500 
Sandstone, Hum- Piers, ashlar, common 
melstown (brown), sandstone........ 1,050 
Be bees Se eas 12,000 | Piers rubble, cement 
Limestone, Kings- mortar. é 900 
nm, N. 12,000 | Piers, rubble, lime 
Limestone, Garrison mo sos intake 480 
Station, 'N. Y.. 18,000 
Limestone, Bedford 
(oolitic), Ind. ....| 8,000 


crushing strength of 1-2-4 concrete 6 mo. old is 2,500 Ib. 


per sq. in. 


_ 2,500 
of stress is a7 


2 
carry is 113.1X 


2000 = 47,125 Ib. 


Using a factor of safety of 6, the safe intensity 


Then the safe total load the column can 


288 


CONCRETE DESIGN 


ULTIMATE CRUSHING STRENGTH OF BRICK MASONRY 


PIERS 
(Average Age of Brickwork, 6 Months) 

Oo ud 

9 4 2 E 

Material Composition of Mortar | & he e 

HES 

Bao 2 

Oo AeA 
Wire-cut brick......... A 1 cement} 5 sand 3,000 
Dry-pressed brick....... 1 cement, 5 sand 3,400 
Dry-pressed brick....... 1 cement, 1 lime, 3 sand | 2,300 
Repressed brick. 1 cement, 5 sand 1,700 

sass ot gpa sand-struck 
rick. . 1 cement, 5 sand 1,900 
Li ie, " sand-struck 

rick. . 1 cement, 7 sand 853 
Hard, sand-struck brick. 1 cement, 1 sand 2,100 
Hard, sand-struck brick.| 1 cement, 1 lime, 3 sand | 1,500 
Hard, sand-struck brick. 1 cement, 5 sand 1,200 
Sand-lime brick......... 1 cement, 3 sand 1,100 
Sand-lime brick......... 1 lime, 3 sand 450 
Sand-lime brick......... Neat cement 1,400 
Terra-cotta work.... 1 cement, 3 sand 2,000 


Eccentrically Loaded Column.—The stress on an eccen- 
trically loaded column is computed by the following formulas: 
For circular columns, 


Oe ds 
ser he oe 
For rectangular columns, 
P GP, 
TA? id 


In these formulas, s is the stress, in pounds per square inch, 
developed in the column; P, the total load on the column, in 
pounds; A, the area of column section, in square inches; 
é, the eccentricity of eccentric part of load, in inches; P,, 
the eccentric part of load in pounds; d, the diameter of 
column, or dimensions measured in the plane of the eccen- 
tricity, in inches. 


CONCRETE DESIGN | 289 


If the total load is eccentric, the formulas just given 
reduce to the following: 
For circular columns 


P. (, ,8¢ 
s=— a 
A d 


For rectangular columns, 


P.(, , 6 
A gem ae 
A d 


According to these formulas, it is necessary first, in design- 
ing a column that will stand a given load, to select by 
inspection the section of column that seems to be about 
correct, and then to solve the equation fors. This value of s 
must be less than the allowable working stress it is pro- 
posed to use. If it is larger, a larger area of column must 
be selected, and the problem worked out again; if it is very 
much smaller, possibly too large a section for economy has 
been selected, and in this case a smaller section should be 
assumed and the equation again solved for s. It should be 
borne in mind that the height of the column must not be 
greater than twelve times the least dimension of the cross- 
section. The eccentricity should not be so great as to cause 
tension in the column. 

EXAMPLE.—Design a cylindrical column of 1-2-4 stone 
concrete, 18 ft. high, to carry with a factor of 6, a central 
load of 100,000 lb. and an eccentric load of 100,000 Ib., the 
eccentricity being 4 in. 

SoLUTION.—Since the column is 18 ft. high, it should be 
at least 18 in. in diameter. The ultimate crushing strength 
may be taken as 2,500 lb. persq. in. in6mo. The safe working 
stress would therefore be 2,500 +6=417 lb. per sq. in. 

A column 28 in. in diameter will be tried first. The area 
of the cross-section is .7854 X 282, or 615.75 sq. in. To apply 
the first formula, A = 615.75, P=200,000, P,=100 000, e=4, 
and d=28. Substituting in the formula, 


ne 8 X 4X 100,000 
615.75 615.75 X 28 


=325+186=511 Ib. per sq. in. 


290 CONCRETE DESIGN 


This stress is larger than the allowable stress, which shows 
that the column section selected is too small. If a section 
31 in. in diameter is assumed, then, 

A=.7854 X 31°=754.77 sq. in. 

Substituting in the formula, 

_ 200,000 _8x4x 100,000 


754.77 = 754.77 X 31 

Since this is less than 417 lb., a column of this diameter 
is safe. 

In a fire, a column is apt to be injured by the heat to a 
distance below the surface of 14 in. Therefore, in design- 
ing columns by the preceding methods, 14 in. should be 
added all around the column proper. 


Ss = 402 lb. per sq. in. 


REINFORCED CONCRETE 


BEAMS 


The design of reinforced concrete is not an exact science. 
The majority of the recommendations and formulas herein 
given are taken from the excellent report of the ‘Joint 
Committee.”’ The Joint Committee is a committee of mem- 
bers of the American Society of Civil Engineers, the 
American Society for Testing Materials, the American Engi- 
neering and Maintenance of Way Association, and the 
Association of American Portland Cement Manufacturers 
formed ‘“‘for the purpose of investigating current practice 
and providing definite information concerning the properties 
of concrete and reinforced concrete.” 

Only Portland cement is suitable for reinforced concrete. 

The aggregate is divided into two classes, namely, fine 
and coarse. Fine aggregate consists of sand, crushed stone, 
or gravel screenings, and which, when dry, passes through a 
screen having holes } in. in diameter. Coarse aggregate 
consists of crushed stone or gravel which is retained on a 
screen having holes 4 in. in diameter, but which passes 
through a screen having holes 1 in. in diameter or smaller. 
In both fine and coarse aggregates, a gradation of size of the 
particles is generally desirable. 


CONCRETE DESIGN 291 


Cinder concrete is not suitable for reinforced-concrete 
structures. 

For reinforced-concrete work, a mixture based on the pro- 
portion of 1-6 is generally used; that is, 1 part of cement 
to a total of 6 parts of fine and coarse aggregates, measured 
separately. The fine and coarse aggregates are often in the 
proportion of 1-2, which makes a concrete that is commonly 
called a 1-2-4 mixture. Forcolumns, richer mixtures are often 
required, and leaner mixtures can often be used in mass work. 

Moving live loads and suddenly applied loads require 
special consideration. These loads can often be taken care 
of by increasing the amount of live load used in the cal- 
culation. The weight of the beam or the floor slab itself 
should always be considered when estimating the dead load. 

The span length of beams should be taken as the length 
from center to center of supports aad not of the clear span, 
but it need not be considered longer than the clear span 
plus the depth of the beam. Brackets are not considered 
as reducing the clear span. 

In the formulas about to be given, rather than employ 
ultimate stresses and divide the result by the factor of safety, 
working, or safe, stresses should be used. 

Rectangular Beams.—The following notation will be used 
in the design of beams and columns: 

F,=tensile stress in steel, in pounds per square inch; 

F,.=compressive stress in concrete, in pounds per 

square inch; 

E;=modulus of elasticity of steel; 

* E-=modulus of elasticity of concrete; 

n=E,+E,; 

M=moment of resistance of beam, in inch-pounds; 

A=area of steel, in square inches; 

b=breadth of beam, in inches; 
d=depth of beam, in inches, from top to center of steel 
reinforcement; 
k=coefficient; 
j =coefficient; 
A 
p=ratio of area of steel to bd= bd’ 


292 CONCRETE DESIGN 


First assume values for F;, F,, and . These values are 
usually controlled by building ordinances. The following 
values are recommended as safe working values by the Joint 
Committee and are used throughout the text to serve as 
examples in working out problems used as illustrations: 
F,=16,000; F-=650; »=15 for concrete capable of devei- 
oping an average compressive stress of 2,000 pounds in 
28 days when tested in cylinders of specified shape. 

After values of F;, F,, and m are decided on, substitute 
them in the following formula and solve for . This formula 
gives the value of » that makes F s and F, reach their full 
values under the same load. 


1 
p=4tX Baa) 
F, \nF, 
Substituting the values mentioned above 
1 
= = .00769 

tities: 16,000 / 16,000 

650 \15x650 


The value of & is now found by the formula 


k= V2pn+ (pn)?— pn 


Substituting the values for p and m gives 


k= V2 x .00769 X 15 + (.00769 X 15)2— .00769 & 15 = .379 

From this value, j is found by the following formula: 

j=1-3k 

Substituting the value of & just found, 

j=1—-4X.379 =.874, or enorodiradtely. %. 

For any value of F;, F;, and n employed, the preceding 
formulas must be solved to obtain p, k, and 7 before the 
problem proper can be attacked. For example, the values 
of k and j may be taken as # and i, respectively. because 
these values are close to the values found from the values 
of F;, F;., and n, assumed. 

The resisting moment M may then be found by transposing 
in either of the following formulas: 

M M 2M 


leech Sint eae ad ans F Side cocaine 
Aja pid! ° jee 


= 


PPP ORS > 


ee 


wer » ae, 4 
ar Tr) (ere tats % 


a 


Me 


CONCRETE DESIGN 293 


The first of these will be found to be the more convenient 
to use in most cases, and it may be transposed to the fol- 
lowing: _ M=F,Ajd. 


As an example, design a girder on a 20-ft. span to carry 
a load of 700 lb. per ft. This load includes the weight of the 


girder. The total load is 20x 700=14,000 lb. The bend- 


‘ . WI 14,000 x 20 
ing moment is 3 = diay ae = 
= 420,000 in.-lb. Assume the effective depth of the girder— 
that is, the distance from the top to the center of the steel 
reinforcement, or d—to be 18 in. Substituting the correct 
values in the equation M=F;Ajd, 420,000 =16,000 x AX} 
X18, and A =1.667 sq. in. 

Since A=pbd, and since p is limited to .00769, 1.667 
= .00769 Xb X18 and b=12.04, say 123, in. 

All that now remains is to decide how much concrete 
to put below the steel. This does not affect directly the 


=35,000 ft.-lb., or 35,000 X 12 


strength of the beam. Its principal uses are to hold the steel 


in place and protect it from fire and rust. The Joint Com- 
mittee recommends a thickness of 2 in. for girders, 14 in. 
for beams, and 1 in. for slabs. Therefore, the girder just 
designed would be 123 in. broad and 20 in. in total depth, 
and it would have 1.667 sq. in. of steel 18 in. from the top. 
ExaMPLE.—Design a floor slab ona 10-ft. span tocarry a 
load of 250 lb. persq. ft. This load includes its own weight. 
SoLuTion.—Consider a section of the slab 12 in. wide. 
The total load is 250 10=2,500 lb. The maximum mo- 


Wi 2,500 10 
ment is car Ce as ft.-lb., or 37,500 in.-lb. 


A= .00769 bd, but b=12 in., as a strip that wide is consid- 
ered. Therefore, A =.00769X12d=.09d. Substituting the 
values for M, A, Fs and j in the equation for the moment, 
37,500 = 16,000 X .09 dx 4 X d; d?=29.76, and d=5.5. There- 
fore, A=.09X5.5=.495, say 4, sq. in. The slab must 
therefore be 64 in. thick and have 4 sq. in. of steel every foot, 
which must be placed 54 in. from the top. Of course, all the 
steel must not be put in one rod, but in several rods spaced 
at equal distances so that it will average 4 sq. in. per ft. 


294 CONCRETE DESIGN 


These solutions give the most economical design; that is, 
when the allowable unit stress in the steel and the allowable 
unit stress in the concrete are realized under the proposed 
load. This condition is determined by the value used for p. 

However, other values of p are sometimes used, and in such 
cases the procedure is as follows: 

Assume a value of ». Find the value of k by the for- 


mula k= V2 pn+(pn)2—pn. Then find the value of j by 
the formula j=1—%3k. Next assume values for F, and F,. 
Find M from the conditions of the problem. Assume 
values for either } or d, and find the other by means of the 
formula F;=M-~+pjbd?, Also find the same dimension by © 
the formula F,=2 M+jkbd?, and use the largest value found 

by either of these equations. 

Sometimes the problem is thus: b, d, and M are given. 
. Assume F,, then find A by the formula M=F,Ajd, using ¢ 
for 7. Solve p=A-~+bhd for p, which should, by changing }b, 
if required, be kept less than the value of p that gives the 
most economical design mentioned above. Then to check, 
find k andj as on page 292 from the value of » obtained and 
find F, and F, accurately by the formulas at the foot of 
page 292. These values must not be excessive. 

To investigate a beam already built, proceed as follows: 
Measure the value of b, d, and A. Calculate the value of p 
as well as the moment of the loads on the beam, or that are 
to be put on the beam, including the weight of the beam 
itself. Assume a value for , and find the value of k by the 


formula k= V2 pn+(pn)?—pn. Find the value of j by the 
formula 7=1—4k. Find the stress in the steel by the for- 
mula F,=M-+ Ajd, and then the stress in the concrete by 
the formula F.=2M-+jkbd?, Neither of these values should 
exceed the safe allowable limit. 

Continuous Beams.—If W is the total uniform load on a 
beam and / is its length, then, for a simple beam, the moment 
is W/+8. In building construction, many beams and floor 
slabs are continuous, and in this case the external moment 
at the center of the span is decreased and there is produced 
a negative moment over each support. Sufficient steel 


CONCRETE DESIGN 295 


should be placed over each support at the top of the beam 
to withstand this moment. This steel should extend far 
enough on each side of the support to reach the point where 
the bending moment changes sign. In many cases one- 
quarter of the span each way will be sufficient. This steel 
is often made up partly of rods bent up from the bottom of 
the beam and partly from extra rods inserted at the top. 

For both beams and slabs for interior spans of continuous 
beams the Joint Committee recommends that the bending 


Wl 
moment be taken as eT, both at the center of the span and 
over the support. In beams for end spans and the adjoining 
Wl 
support, the bending moment should be taken at to After 


the bending moment is found, the beam is designed in the 
manner already stated. 
Some engineers claim this reduction in bending moment 


Wl 
is unwarranted and use a? for all cases. No matter which 


method is followed, the beam should be reinforced over the 
support. 

Caution must be exercised in designing continuous 
T beams. At the support, the compression in the concrete 
is apt to be excessive, as it comes on the stem of the T. 
The Joint Committee allows a higher stress in the concrete 
here and recommends a value approaching 750 lb. It is 
often necessary to leave some steel at the bottom of 
T beams near the supports to assist in withstanding this 
compression. 

Beams Reinforced at Top and Bottom.—For beams rein- 
forced at the top and bottom, the preceding notation, with 
the following additional characters, is employed: 

A’=area of compressive steel at top of beam, in square 


inches; 
, 


p’ =ratio of area of compressive steel to bd ret 


F’ =compressive stress in steel, in pounds per square inch; 
d’ =distance from top of beam to top steel, in inches. 


296 CONCRETE DESIGN 


To design such a beam, assume values for p, p’, da’, d, 
and ”, Then find k by the formula 


k= \? n (o+0°7) +n2(p+p'P—n(p+p’) 
After k is found, assume a value for ), and find F, by the 


formula 
6 p'n d' a’\ | 
l3k~e+——— (e—-—) (2 
pa] same 2% (5) (14) | 


Also, find F by the formula 


1-—k 
eee, 
and F,’ by the formula @ 
tale 
d 
F,’=nF- ; 


The values of F,, Fs, and F,’ should all be within safe 
limits. FF’ will usually be low, which shows that the steel 
is not used economically. On account of this defect, double 
reinforced beams are seldom used except in plkebe where 


a ies 


Fic. 1 


the size of the beam is limited by the conditions of the 
problem. 

As an example, design a girder, as shown in Fig. 1, to 
carry besides its own weight 450 Ib. per ft. The section of 


CONCRETE DESIGN 297 


the girder is limited to that shown. The steel in the bottom 
should be kept up 2 in. to protect it from fire, so that the 
effective depth, or d, is only 18 in. In the first place, the 
12. 20 
beam itself weighs 150 LP Kya t= 250 lb. per ft. The 
total load to be carried is therefore 450 + 250 =700 Ib. per ft., 
or 700 X 30=21,000 Ib. in all. The span is here taken, for 
convenience, as the clear distance betwéen supports and 
not the distance from center to center of support, which 


wi 
is more correct. The maximum bending moment is — 
_ 21,000 X 30 


8 
First find A by the formula given for rectangular beams, 


=78,750 ft.-lb., or 78,750 X 12 =945,000 in.-Ib. 


M 
F,=—. Assume that j=} and that F;=16,000. Then, 


945,000 
AX$+18’ 

If the beam were reinforced at the bottom only, the 
allowable steel would be about .0075 x bd=.0075X12x18 
=1.62 sq. in. The excess of steel is then 3.75—1.62=2.13 
sq.in. It is for this reason that reinforcement must be put 
at the top of the beam; if it were not put there, the stress 
in the concrete at that place would be too great. The 
amount of steel required at the top is usually about two 
and one-quarter times the excess at the bottom, or, in the 
problem at hand, 2} 2.13=4.79, say 43, sq. in. Assume 
that this steel is placed 2 in. from the top. 

This completes the approximate design of the beam; that 
is, its size and the amount and location of the steel at the 
top and bottom have been assumed. It now remains to see 
whether or not the values assumed will be safe. First k 
must be found by the formula given, which is 


a’ 
k= NE n (0+) +n?(p+ p’)?—n(p+p’) 
3.75 
The values of p, p’, d, and d’ are: p=——-——-=.01736, 
12x18 


16,000 = or A=3.75 sq. in. 


298 CONCRETE DESIGN 


4.75 
| Aga aappemtek 02199, d=18, and d’=2. Assuming that 


n=15 and substituting these values in the foriaula, 


2 et i 
k= bees 15 (o1726-+.02199 =) + 15?(.01736 + .02199)? 


— 15(.01736 + .02199) = .380. 
6 M 


6 pn d’ d’ 
bd2| 3 k—k2+——— (k-—) (1-— 
[sear Z*(2-$) (1-3) | 


Substituting the values-in this equation, 
6 X 945,000 ; 

6 X .02199 « 15 

—— —(,380 — 1- 

erent? Ts) ( | 
=651 lb. per sq. in., 

which, according to the stresses used in these examples, 
is about safe. 

The stress in the steel in tension is found by the formula, 

1—k - 
| ae ral 

Substituting the correct values, 


Now, F,.= 


F,= 


12 x 18? E X .3880 — .3802+ 


1—.380 
F,=15X 651 X 330 _ = 15,930 lb. per sa. in., 


which is also safe. 

The stress in the steel in. compression is found by the 
formula d’ 
k-— 
d 

F,'=nF-; 


k 
Substituting the correct values, 


F,/=15X 651 X 380— v's =6,913 Ib. 
380 per sq. 1n. 


This value is low, but it cannot be helped. It is this fact 
that makes the double reinforced beam uneconomical. 
T Beams.—The Joint Committee makes the following 
* suggestions in regard to T beams: 
‘In beam and slab construction, an effective bond should 
be provided at the junction of the beam and slab. When 


CONCRETE DESIGN 299 


the principal slab reinforcement is parallel to the beam, 
transverse reinforcement should be used extending over 
the beam and well into the slab. 

““Where adequate bond between slab and web of beam is 
provided, the slab may be considered as an integral part of 
the beam, but its effective width shall be determined by the 
following rules: 

“1. It shall not exceed one-fourth of the span length of 
the beam. 

“2. Its overhang width on either side of the web shall 
not exceed four times the thickness of the slab. 

“3. In the design of T beams acting as continuous beams, 
due consideration should be given to the compressive stresses 
at the support.” 

The notation used for T-beam formulas is the same as 
before, with the following exceptions: 

‘b=width of flange, in inches; 

b’=width of stem or beam proper, in inches; 

t=thickness of slab, in inches. ; 

First assume values for F,;, F, and. Also, assume values 
for d, b’, andt. The value to be taken for 6 is determined 
partly by the rules just given. Assume an approximate value 
of A from the formula M=A(d—4t)F,. Solve the following 

2 ndA + bt2 


— 2nA+2 bt 
If kd is less than t, design the same as a rectangular beam; 
that is, again find k by the formula 


formula for kd: 


k= V2 pn+ (pn)? — pn where ae not El 
: bd b’d 
Then find 7 by the formula 
j=i— sk. 
and M by the formula 
M=F,Ajd, 
F jkbd? 


which must not exceed 


If kd is greater than t, proceed as follows: Find jd by 


3kd-—2i ¢t 
the formula j@d=d—— 


300 CONCRETE DESIGN 


Find F, by the formula a M 
* Ajd 
and find F, by the formula 
F Mka 
°” bt(kd —4t)jd 

These values of F, and F, must be less than the allowable 
values. 

These formulas neglect the compression in the stem. 
For approximate results, the formulas for rectangular beams 
may be used. ‘ 

As an example, design a beam to carry 5,184 Ib. per ft. 
on a span of 10 ft. The load given includes the weight of 
the beam, and the floor slab is 5 in, thick. The bending 


5,184 X 10 X 10 
: =64,800 ft.-lb., or 64,800X12 


=777,600 in.-lb. Assume that F;=16,000 lb., F,.=650 Ib., 
and =15, and that d=16 in, and b!=16 in; b is governed 
by the preceding rules. It must not exceed one-fourth the 
span, or 30 in., and its overhang must be less than four times 
the thickness of the slab, which would limit its width to 
16+2X4xX5=56in. Thereforebis takenat 30in. Solving 
for A in the formula, M=A (d—it)F,, 777,600=A (16-24) 
16,000. 

Therefore, A=3.6 sq. in.; ee x OS ee 

2X 15X3.6+2X30X5 


3X6,073-2X5 5 . 

= 6073, and jd<16— Y wiht, oe 

re A id 2x6.073-5 3 ie 
__ 777,600 


POS a5 141 =15,319 lb. per sq. in., 


which is safe, and 
_ 777,600 X 6.073 
“30 X 5(6.073 — 24)14.1 
which is also safe. 

Shear and Bond.—One method of failure of beams that is 
common is shown in Fig. 2. Such cracks are caused by 
shear or diagonal tension. They are preveuted by bend- 
ing up some of the main reinforcing rods to make truss 
rods and by the use of Y-shaped stirrups. The size and the 


moment is 


=625 lb. per sq. in., 


ie 
q 


CONCRETE DESIGN 301 


method of placing these stirrups are still matters of discus- 
sion that depend much on the experience of the designer. 
One important thing to be re- 
membered is that they arecom- 
paratively short and that /7 
there is danger of their pulling Fic. 2 
out of the concrete. For these reasons they are frequently 
securely fastened to the slab reinforcement or bent over at 
the ends. Sometimes deformed bars are used, as they give 
a more secure grip on the concrete. 
When inclined stirrups are used, they should be fastened to 
the main reinforcing bars in sucha way that they will not slip. 
An approximate formula for shear in a rectangular beam 


reinforced at the bottom is = where v=unit shearing 
stress in pounds per square inch at any point, V =total 
vertical shear at that point in pounds, and the other terms 
have the same meaning as before. The Joint Committee 
recommends for a grade of concrete mentioned above a unit 
safe shear of 40 pounds. If V is greater than this, stirrups 
must be used. The same formula may be used for T beams 
with b’ insérted instead of band jd referring to T beams. Even 
with web reinforcement, v should never exceed 120 pounds. 
No absolute rules can be given for the size and placement 


of stirrups. In many instances ? in. square or ¥’ X 1” flats 


are used, but not in all cases. The spacing of stirrups is 
greater at the center of the span than at the supports. The 
Joint Committee recommends a maximum limit of spacing 
of three-fourths the depth of the beam. The spacing at the 
supports is often one-fifth or one-sixth of the depth of the 
beam. Great care is required in the design of stirrups of 
T beams to insure that the slab and the beam proper are 
securely tied together. 

An approximate formula for ordinary rectangular beams 


Vs 
with verticalstirrupsis P= 42° where P =stress in one stirrup, 
5 7 


s=spacing of stirrups, V=proportion ot shear supposed to 
be carried by reinforcement, usually %, and j and d have the 
values already given. 


302 CONCRETE DESIGN 


The Joint Committee recommends that the bonding stress 
between plain reinforcing bars and concrete be assumed to 
be 80 lb. and in the case of drawn wire, to be 40 lb. for the 
same grade of concrete as specified in discussing beams. 
The ditference in stress in the tensile steel at two sections 
must be taken up by the bond to the concrete between these 
two sections and should be investigated. 

} COLUMNS 

Concentrically Loaded Columns.—There are two general 

methods of reinforcing concrete columns with steel. One 


Fic. 3 


method is known as straight reinforcement and the other as 
hooped reinforcement. These two styles of reinforcement are 
illustrated in Fig.3. In (a) is shown straight reinforcement. 
This consists of steel rods that stand vertically in the con- 
crete. Sometimes, the rods are placed directly in the middle 
of the column, but as a rule they are arranged around the 
outside of the column about 2 in. from the surface. These 
steel rods are tied together by wire ties, asshown ata. The 


“we 4 


PI 


CONCRETE DESIGN 303 


distance between two ties should not exceed the width, or 
diameter of the column. If the ties are spaced too far apart, 
the column is apt to fail by the reinforcement bulging. 

In Fig. 3 (b) is shown a column reinforced with hooped 
reinforcement. This type of reinforcement consists of either 
a steel spiral or a separate steel hoop that is about 2 in. from 
the surface of the column, as shown at b. 

Some columns have both styles of reinforcement just 
mentioned. 

In the design, as given here, A; is the effective area of the 
concrete. The effect of fire is to injure the concrete for a 
depth of about 14 in. from the surface. Therefore, in 
investigating a column already built, first deduct 14 in. all 
the way around the column from the total area of the con- 
crete so as to get the effective area of the concrete. For the 
same reason, after a column is designed, add a coat 14 in. on 
all sides for fire protection. The reinforcement to protect 
it from fire should be embedded at least 2 in. in the concrete. 

In a hooped column, the effective area should not only be 
limited to 14 in. from the surface, but should further be 
limited to the concrete within the hooping. Outside of the 
hooping, of course, at least 2 in. of concrete must be placed. 

In height, columns should be less than fifteen times their 
least dimension. 

Straight Reinforcement.—Let A, represent the effective 
area of cross-section of concrete, in square imches; As, the 
area of cross-section of steel in square inches; E,, the modulus 
of elasticity of concrete; E,, the modulus of elasticity of steel; 
F,, the safe compressive stress per unit area of concrete; 
F,, the safe compressive stress per unit area of steel; m, the 


E 
ratio = and W, the total load on column. 
ec 
The design formulas are as follows: 


W =F(A,+nAs) 
F,=nF, 
Any values of F, and m as may be required may be used. 
The Joint Committee uses 450 lb. for F,; and 15 for m for 


- grade of concrete as specified in discussing beams. These 


values are used here for the sake of example. 


304 CONCRETE DESIGN 


If a column is 12 in. square and has 3 sq. in. of steel, 
determine the safe load that it will carry. Deducting 13 in. 
from each surface, the column will be 81 sq.in. But3 sq. in. 
of this area is steel. Therefore, the effective area of the 
column is 81—3=78 sq. in. By the formula, W=450 
(78+15 X 3) =55,350 Ib. The stress in the steel is there- 
fore 450 X 15 =6,750 Ib. per sq. in., which is safe. 

The use of steel in a column has two advantages. In the 
first place, the formulas do not take into account the length 
of the column; the longer a column is, the more it is apt to 
bend or bulge, and the steel helps materially to resist this 
tendency. Then again, the introduction of steel permits 
the column to be made much smaller in size, and this is often 
of great advantage. 

Having shown how to investigate a column, the method 
of designing one to carry a certain load will now be con- 
sidered. Thus, assume that it is desired to design a column 
to carry a load of 40 T. The percentage of reinforcement 
does not have to be a definite amount as is the case with 
beams, but may be any amount. Suppose that it is desired 
not to have the column too large; therefore, assume that the 
reinforcement will occupy 4% as much area as the concrete. 
Let A,=area of concrete. Then, A,=4% of A-=aA;. 
Substituting the correct values in the formula it will be found 
that 80,000 = 450(A,+NA,) =450(A,+15 X 3A.) =720 Ag. 
A,=111 sq. insand A,=y5A,=4.4+ sq. in. Therefore, the 
total area=A,+A,=116 sq. in. 

If the column is square, it will be 10.77 in. on a side; 
say 11 in., to allow for chamfered corners. When 14 in. 
of fireproofing is put on, the column will be 14 in. square 
and will contain a little over 4.4 sq. in. of steel. 

Empirical Rules for Straight Reinforcement.—One or two 
more or less empirical formulas are used in designing concrete 
columns with straight reinforcement. These formulas 
mostly originate from the building laws of various cities. 
The building laws of one large city, for instance, furnish 
good examples of such formulas. They stipulate that the 
safe allowable load shall be 500 Ib. per sq. in. of column 
section, counting both concrete and steel the same. If 


CONCRETE DESIGN 305 


sufficient steel is inserted, this rule will of course be as safe 
as the formula already given. 

In a large building, the loads on the columns in the lower 
floors become very great, and as it is usually not desirable 
to increase the outside size of the column, the percentage of 
reinforcement is increased. A favorite style of reinforce- 
ment employed under such conditions is shown in Fig. 4: 
It consists of four angles riveted back to back in the center 
of the column to form a steel core. The area of this steel 
core can be increased if desired by using packing plates 
between the angles. The laws of this city then assume 
that the steel core takes all the vertical 
load and that the concrete prevents it 
from bending sidewise. The rule for 
designing such a column is, therefore, to 
allow a safe stress of 16,000 lb. per sa. in. 
on the steel and nothing on the concrete. 
Thus, suppose 160 T. is to be safely car- 
tied. Thisequals 160 x 2,000 =320,000 lb. 
The area required for the steel core, then, 
will be 320,000+16,000=20 sq. in. If four angles are to 
be used, each one will have to have an area of 20+4=5 sq. in. 
Angles 4 in. X4 in. x4 in. would be large enough. 

It is recommended that vertical rods be inserted in the 
concrete around the steel core to help stiffen it. The rule 
just given is of an empirical nature and is here presented 
to show the practice sometimes followed. 

Hooped Reinforcement.—Hooping of concrete columns 
increases the ultimate strength of the concrete contained 
inside the hooping, but has little effect below the elastic 
limit. Therefore, with hooping, a greater unit stress may be 
used, although the steel in the hooping itself does not carry 
the load applied. The amount of hooping should not be 
less than 1% of the volume of the concrete inclosed. The 
clear spacing between the bands should not exceed one- 
fourth the diameter of the inclosed column. Adequate 
means must be provided to hold the hooping in place while 
the concrete is being placed. The Joint Committee allows 
a unit working stress of 540 lb. on hooped concrete columns, 


306 CONCRETE DESIGN 


and on hooped columns that contain in addition’ more than 

1% and less than 4% of longitudinal reinforcement they 

allow a unit working stress of 650 lb. This is for the grade 

of concrete above specified. Some engineers consider these 

stresses rather high; nevertheless, in all cases the stress 

selected should be governed by the laws of the locality and 
’ the experience of the designer. 

Eccentrically Loaded Columns.—It is difficult to find a 
column that is entirely concentrically loaded. If, in a build- 
ing, the live load is transferred from one girder to another, 
the girders being carried by the same column, their deflection 
will put a twist in the column itself. Also, the outside 
columns of buildings are often eccentrically loaded.. In the 
lower floors of buildings the eccentric load due to unequal 
distribution of load, in panels, may usually be neglected 
because of the large concentric load on the column, but in 
the top floors of a building the matter may at least deserve 
attention. If the load is eccentric, the following method 
will usually give safe results: 

First, assume a section and find the stress in the concrete 
and steel due to bending as in a beam with double reinforce- 
ment. Next,-find the stress in the concrete and steel due 
to direct compression and add the two algebraically. The 
sums obtained should not exceed the allowable stress for 
columns. Sometimes it is found advisable to put more 
steel in the tension side than is used in the other side. As 
this method is approximate, low unit stresses should be used. 


ARCHES 

In reinforced-concrete arches, the arch ring is nearly always 
thinner at the crown than near the haunches. The arch in 
reinforced -concrete work is often either true parabolic or 
the curve is made up of a series of circular arcs approach- 
ing the shape of a parabola. The reinforcement for arches is 
‘usually placed in two layers—one near the intrados and 
the other near the extrados—and these two layers of 

reinforcement are usually laced together with lighter rods 
In constructing bridges of reinforced concrete, it is consid- 
ered advisable to lay all the concrete at once, but if this is 


CONCRETE DESIGN 307 


impossible, the bridge may be constructed in parallel sections 
running lengthwise of the arch. By following this method 
of construction, the arch is not materially weakened, for 
each ring may be considered as a complete unit of the arch. 

The manufacturers of Kahn bars have suggested an approxi- 
mate method of reinforced-concrete arch design. This method 
is based on an article by F. F:! Weld, C. E., published in the 
Engineering Record. It consists in using an empirical 
formula obtained from a study of many existing arches 
and original designs analyzed by more elaborate methods. 

The first step in the design is to determine the rise of the 
arch. This should be at least one-tenth of the span. 

_The curve that the arch takes, especially where uniform 
loads are expected, is often a parabola. This parabolic 
curve is usually followed by the center line of the arch ring, 
and not by the curve of the intrados. This curve may be 
drawn by plotting a sufficient number of points determined 
by the following formula: 


PB, es 
e ak ( = | 
in which Y is the rise of parabola, in feet, at any point under 
consideration; H, the rise, in feet, at the center of the arch; 
X, the horizontal distance, in feet, from vertical center line 
of arch to point under consideration; and S, the span of the 
arch, in feet. 

As an example, lay out the parabolic curve for the center 
line of an arch where the span is 88 ft. 

First, the rise must be determined; this is at least one- 
tenth of the span, which is 8.8, or practically 9, ft. Having 
the rise and the span, proceed to lay out the curve shown in 
Fig. 10 as follows: First lay off to a convenient scale the 
line ab equal to the span. At the center of ab, orc, erecta 
perpendicular, and on it measure cd equal to the rise H. 
Then, the points a, d, and b lie on the required curve. 

To obtain other points, proceed as follows: Divide the 
span into any number of convenient parts, say in this case, 
eight, because eight is an even number, which makes one 
division point at c. Then, each division is 11 ft. long. 
Now, the first division immediately to the right of c is e. 


308 CONCRETE DESIGN 


From: ¢ erect a perpendicular. Then, returning to the 
formula, it is desired to find the vertical distance to the 


Fic. 10 
parabolic curve. This distance is denoted by Y. The fol- — 
lowing values are known: H=9, X=11, and S=88. Sub- 
stituting these. values in the formula, 

2X47\-* 

Y=9x]1- 7") | = 8.4375 ft. 

Lay off ef equal to 8.4375 ft. Then f is a point on the 
curve. 

Suppose it is desired to locate the curve at the first division 
point from a, namely, g. Through g draw a perpendicular. 
Here, H=9, X =33, and S=88. Therefore, 

2 
Y=9xX E 7) | =3.9375 ft. 

88 

From g, there is plotted upwards to scale 3.9375 ft. to h, 
which is a point on the curve. Other points may be obtained 
in the same manner and the parabolic curve drawn through 
them. * ; 

Instead of using the formula to determine the location of 
points along the parabola, as just shown, it is customary 
Svan to be divided into 10 eguatl Parks 

. H ‘ t 
' 


| 

{ 

' ' 
‘ 

' 

' 


o++----7 


S\ 
z 

: 
C4) 
i 4 


Fic. 11 
to divide an arch into ten equal horizontal parts and then 
use the values given in Fig. 11. These values are derived 
from the formula on page 307, and enable a parabola to 
be plotted with ease when the span is divided into ten equal 
parts. The rise at each one of these parts is given in terms 


CONCRETE DESIGN _ 309 


of the total rise. As will be observed, these values are 
independent of the span. 

As an example, consider an arch having a span of 70 ft. 
and arise of 8 ft. In this case, H=8 and the height of the 
curve at 70+10=7 ft. from each end is .36X8=2.88 ft. 
At 14 ft. from each end, the height of the curve above the 
spring line is .64< 8=5.12 ft., and so on, until the necessary 
number of points on the curve are located. The curve may 
then be drawn, passing through these points. 

Having determined the rise to be given to the arch, the 
next step is to find the thickness of the arch ring at the 
crown. This may be found by the formula proposed by 
F. F. Weld, C. E., which is 2 follows: 


Dense 4 
10 200 400’ 


in which D is the crown thickness, in inches; S, the span of 
the arch, in feet; L, the live load per square foot; and F, the 
dead load at crown per square foot, exclusive of the weight 
of the arch itself. 

Thus, for example, take an arch whose span is 64 ft., whose 
live load is 100 Ib. per sq. ft., and whose dead load is 200 Ib. 
per sq. ft., and determine its thickness at the crown. 

In this case, S=64, L=100, and F=200. Then, 

64 100 200 
D bere 15.4 in. 

If there is no live load, the formula may still be used. 
Thus, find the thickness of an arch at the crown when the 
span is 40 ft. and the dead load at the crown is 150 lb. per 
sq. ft. Here, S=40, L=0, and F=150. Therefore, 


40 150 
. D=~ = —=10. in. 
+55 et 0.6996, say 11, in 


The arch ring is made thinner at the crown than else- 
where. The custom is to increase gradually the thickness of 
the arch ring from the crown to the abutments. The thick- 
ness of the arch ring directly above points on the spring line, 
one-quarter of the span from the abutments, is made from 
one and one-fourth to one and one-half times the thickness 
at the crown. From these points to the abutments the arch- 


310 CONCRETE DESIGN 


ring thickness usually increases more rapidly, and while no 
definite proportions can be laid down, it is usually at least 
twice as thick at the skewbacks as at the crown. 

One of the points yet to be considered is the amount 
of steel reinforcement to be used. As was said, this rein- 
forcement usually is placed in two layers, one layer near 
the intrados and the other near the extrados. If the arch 
carried only a dead load and the conditions were absolutely 
uniform, only one layer of reinforcement would be neces- 
sary; but, in actual practice, as even the changes of temper- 
ature cause large variations in stress, two layers of rein- 
forcement are used unless a very careful analysis of stresses 
has been made to prove that they are not needed. The. 
amount of steel in each layer is kept uniform throughout its 
length. The cross-sectional area ot steel in each layer is equal 
to 745 of 1% of the area of the arch-ring section at the crown. 

Thus, suppose that the thickness of an arch at the crown 
is 24 in. For a width of 1 ft., the area of the arch-ring 
section is 24*12=288 sq. in. The area of the steel in the 
top or bottom layer per foot of arch width is therefore 


288 X 1.152 sq. in. The area of a $-in. round bar is 


.6013 sq. in. Therefore, one $-in. round bar placed near the 
extrados and one }-in. round bar placed near the intrados, 
every 6 in. in width along the bridge, will be found sufficient. 

The distance from the center of the bars to the surface 
of the intrados and extrados, that is, the depth to which 
the bars are to be embedded in the concrete, is another 
question that must be decided. The nearer the reinforce- 
ment is to the surface, the more efficient it will be. On the 
other hand, the reinforcement must be embedded deep 
enough, so that it will not tear loose and so that it will be 
protected from fire and rust. As a general rule, the distance 
from the center of the steel reinforcement to the surface of 
the arch ring should be from 2 to 8 in. 

The two layers of reinforcement are laced together by light 
steel rods that are run from one layer to the other. No uniform 
method is followed in designing these shear members, as they 
are called, and the amount of material used also varies greatly. 


FOUNDATIONS | 311 


As an example of a complete problem, the following case 
is suggested: Design a reinforced-concrete arch for a 65-ft. 
span. This arch is to carry a live load of 200 lb. per sq. ft. 
and a dead load of 300 lb. per sq. ft. at the crown. 

First, the rise of the arch must be determined. This may 
be taken as one-tenth the span, or 6.5 ft. The curve of the 
arch may now be laid out. If the curve to be followed 
is a parabola, the method of constructing it is given on 
page 307. This curve may be used either as the center line 
or as the intrados of the arch, preferably the former. 

The thickness of the arch ring at the crown may now be 
determined by the formula on page 309. In this case, S=65, 
L=200, and F=300. Therefore, 

— 65 200 300 
D rer rr 16.3123 in. 

Calling this thickness 18 in., the thickness of the arch ring 
at the quarter points may be taken as 14 X 18=27 in., and 
its thickness at the haunches may be taken as 2K 18 =36 in. 

The amount of steel reinforcement required for each layer 
must now be determined. For each foot of width of the arch, 
this will be 18X12 yo0 X 74 =.864 sq. in. A 3-in. round 
rod has an area of .4418 sq. in.; therefore, one }-in. rod every 
6 in. near the extrados, and one 1}-in. rod every 6 in. near the 
intrados will be sufficient. These rods should be embedded 
2 or 3 in. in the concrete. The two layers of reinforcement 
should be either securely tied together or bonded into the 
concrete with shear or diagonal members. 


FOUNDATIONS 


BEARING VALUE OF FOUNDATION SOILS 
There is some difference of opinion regarding the safe 
_ bearing value of foundation soils, due probably to the diffi- 
culty of arriving at any experimental results that will have 
a general application. Conservative engineering practice, 
however, dictates that the greatest unit pressure on the 


PM 

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FOUNDATIONS 


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FOUNDATIONS 313 


different foundation soils shall not exceed the values given 
in the accompanying table 

The observance of the revised building laws of the several 
cities is considered good engineering practice, for they are 
usually the results of careful investigations and records of 
long experience. The following, taken from the New York 
Building Laws, is interesting and gives bearing values that 
are well within the safe limits: 


SAFE BEARING VALUES OF DIFFERENT FOUNDATION 


SOILS 
Tons per 
Material Square 
Foot 
Granite rock ies go Pea a 30 
Limestone, compact 25 
Sandstone, compact pede. 20 
Shale fecciandon: 6: or soft friable rock. . 8 to 10 
Gravel and sand, compact. . ‘| 6 to 10 
Gravel, dry and coarse, packed ‘and confined . 6 
Gravel and sand, mixed with dry clay. 4to6 
Clay, absolutely ‘dry and in thick 4 
Clay, moderately dry and in thick bedan. : 3 
Clay, soft (similar to Chicago clay).. .| 1 tol} 
Sand, compact, well-cemented, and confined . | 4 
Sand. clean and dry, in natural beds and confined 2 
Earth, solid, dry, and in natural beds 202s 4 


Where no test of the sustaining power of the soil is made, 
different soils, excluding mud, at the bottom of the footings 
shall be deemed to sustain safely the following loads to the 
superficial foot: 

Soft clay, 1 T. per sq. ft. 

Ordinary clay and sand together, in layers, wet and 
springy, 2 T. per sq. ft. 

Loam. clay, or fine sand, firm and dry, 3 T. per sq. ft. 

Very firm, coarse sand, stiff gravel, or hard clay, 4 T. per 
sq. ft., or as otherwise determined by the Commissioner of 
Buildings having jurisdiction. 


314 FOUNDATIONS 


Where a test is made of the sustaining power of the soil, — 
the Commissioner of Buildings shall be notified so that he ° 
may be present in person or by representative. The record 
of the test shall be filed in the Department of Buildings. 


SPREAD FOOTINGS 
DESIGN AND CONSTRUCTION 


- The term spread footings is applied to either wall or column 
footings that have a considerable projection beyond the 
upper tier of the footing, wall, or column base, as the case 
may be. 

The usual type of spread footing for the support of a 
column is illustrated in Fig. 1. This footing was designed 
to be used with a structural-steel column core, and is one that 
would ordinarily be used for a ten- or twelve-story building 
that is to be erected on unstable soil. : 

Placing Reinforcement in Column Footings.—The rein- 
forcing rods, or bars, are placed from 2 to 4 in. from the 
bottom of the footing, and are arranged so as to cross each 
other at right angles. No attempt is made to interlace the 
bars or rods. It is good practice, however, to wire them, 
together, for by so doing, any danger of misplacing the bars 
is avoided. Several tiers of bars, or rods, are used in heavy 
footings, although spread footings that support light loads 
‘are sometimes reinforced with expanded metal or woven- 

wire fabric. 

In designing column footings in reinforced concrete, some 
steel reinforcement is often placed in the upper part of the 
footing directly under the base of the column. This rein- 
forcement acts as a mattress to distribute the concentrated 
- load from the column and also as a bond to tie the concrete 
together. 

Two other methods of arranging the reinforcing rods in 
concrete column footings are shown in Fig. 2. In (a), the 
rods and bars are crossed at right angles; every other bar is 
made short so as to save metal. 


FOUNDATIONS 


315 


In the method of placing the steel reinforcement shown 
in Figs. 1 and 2 (a), it will be observed that the corners of 


the footings are subjected to great moment. 


order to strengthen the 
corners of the footing, 
the reinforcing rods 
are frequently ar- 
ranged as shown in 
Fig. 2 (0). 

The steel reinforce- 
ment of column _ foot- 
ings is never painted, 
even if the footings 
are to be placed in 
damp situations. 
Coating the bars will 
partly destroy the 
bond that it is neces- 
sary to maintain be- 
tween the concrete 
and steel; besides, the 
concrete is sufficient 
protection for the steel 
‘against any serious 
corrosion. In all in- 
stances, however, the 
steel bars should be 
cut off enough to allow 
the ends to be entirely 
protected. 

Spread Footings for 
Optside Columns.—In- 
stead of a continuous 
footing, ztsolated spread 
footings are frequently 
used in outside, or wall, 
columns. 


Therefore, in 


m = BL Ue aeay 
poser 
44 + 
- 
| 
11 
Ht 
Reinforcing 
Rods and Bars 
I 
t va 
1% +4 
ba oe 
ae Et 
es 
+ rt—-t- 


1 Reinforcing 
4 Mods and Bars 
* 
AA ARIE Si 
17% A, AS Ee ee 
TALS xt 
HATS pay WG-4 RK ~ 
(b) 
Fic. 2 


Such footings are illustrated in Fig.3. The position 


that the footings occupy on the building plan is shown, the 
wall footing being at a and the corner footing at b. 


316 FOUNDATIONS 


FORMULAS FOR THE DESIGN OF FOOTINGS 


The theoretical design of a spread reinforced-concrete 
footing consists first in determining the total load on the 
column to be supported by the footing, and then finding the 
required area of the bottom reinforced-concrete footing or 
layer of concrete by dividing the load by the assumed 
allowable bearing value of the soil. The next thing to find 
is the area of the base of the column where it bears on the 
upper tier, or layer, of concrete, because the size of this por- 
tion of the footing is fixed by the size of the column base. 
Aftcr the areas of the bottom and top tiers, or layers, of con- 


{ 
! 
rt 
= 
al 
b-50 


RSRP ITA, FS ag RSF ee 


Fic. 3 


crete have been ascertained, the projection of the bottom tier 
beyond the upper is known, and the bending-moment stress 
on the lower tier of concrete can be calculated. Sufficient 
steel rods may then be introduced, and the footing made of a 
depth that will resist this bending moment. 

The usual formulas for bending moments and the resist- 
ance of reinforced-concrete rectangular sections may be 
applied in determining the strength of reinforced-concrete 
spread footings. However, in office practice, it is desirable 
to use direct formulas for finding the area of steel 
reinforcement required and for determining the unit 
compression created in the upper portion of the concrete 
footing. 


FOUNDATIONS 317 


The formula for finding the area of steel reinforcement 


_ required is expressed as follows: 


Wx 
27,0002’ 
in which A is the sectional area of steel reinforcement, in 


_ square inches, required for each linear foot of spread footings 


along one side; W, the total load, in pounds, on a portion of 
projection of footing 1 ft. in width; x, the projection of 
footing, in inches; and ¢, the distance, in inches, from center 
of action of steel reinforcement to top surface of concrete 


footing. 


oe =] 


i 
+=20° 
==. —fosition f Canter Y oY Re Reinforcerrent ue 


| | | | Pressure on So $000 pe goers 
a ge saa Ad Se 


Fic. 4 


—_——_— 


The following formula is used to determine the compres- 

sion created on the concrete due to bending stress: 
Ce. 
4.5922 

The value C is the compression, in pounds per square inch 
of section, and the other values are the same as in the first 
formula. 

In order to explain the application of the preceding 
formulas, assume that the reinforced-concrete footing of a 
column is required to sustain 900,000 lb., and that, as the 
soil will safely support 4,000 lb. per sq. ft. of bearing area, 
the footing will be 900,000+ 4,000 =225 sq. ft. To obtain 
this area, the footing will have to be 15 ft. square, and the 


318 FOUNDATIONS 


design shown in Fig. 4 may be assumed in applying the 
formula. 

In this figure, the projection of the footing, or the dis- 
tance x, is 60 in., and the dimension #, or the distance from 
the center of action of the steel reinforcement to the top 
of the footing, is 20'in. The pressure on a projection of 
the footing 1 ft. in width is‘equal to the unit pressure on the 
soil, or, 4,000 X 5 =20,000 lb., which is the value of W in the 
formula. These values may be substituted in the formula 
for determining the amount of the steel reinforcement, so that 

__ 20,000 x 60 
27,000 X 20 

This area of steel is to be included in each linear foot of 
the footing course; therefore, the reinforcement may consist 
of 1-in. square twisted bars spaced practically every 6 in. 
both ways, or other bars at a spacing to give this sectional 
area may be used. 

The amount of the steel reinforcement having thus been 
determined, it remains to find out whether or not the con- 
crete in the footing course is overstressed, and the second 
formula may be applied as follows: ; 


20,000 x 60 
C= 653 Ib. 
4.59 X 20 X 20 


This result, which is the maximum compression, in pounds 
per square inch, on the concrete section, is somewhat high. 
If good concrete is used, the footing as reinforced and 
designed may be considered safe, though, in fact, the thick- 
ness of the concrete footing might be increased several 
inches. 


= 2.2 sq. in. 


CANTILEVER FOUNDATIONS 


In reinforced-concrete construction, as well as in other 
types of construction, it is frequently necessary to place a 
new building close against the walls of an adjoining building. 
In many instances the wall of the adjoining property rests 
entirely on its own lot and is not a party wall built half on 
each side of the party line; also, the adjoining building may 
be of inferior construction or may be occupied by tenants 


OO EE 
2 Ee ee ee eee ames 


a 


FOUNDATIONS 319 


engaged in manufacture. Under such conditions it is unde- 
sirable to tear out the wall and build a party wall. 

With buildings of ordinary height and load, provided the 
basement floor of the new building does not extend below 
that of the old, few difficulties are encountered in the design 
of the foundations for the new structure; but if the new 
building is to be many stories in height and requires exten- 
sive foundations along the wall lines, the problem of the 
design of the reinforced-concrete foundations becomes more 
complicated, because it is desirable to proportion the footings 
so that the center of action of the loads will coincide with the 
center of action from the pressure of the soil beneath. In 
order to accomplish this desired result in a steel structure, 
a cantilever-girder system of foundation construction would 
be employed, and a similar system can be constructed in 
reinforced concrete. 

In Fig. 1 is shown the detail drawings of a reinforced- 
concrete cantilever-foundation construction for a six-story 
building, the floors of which are designed for light manu- 
facturing purposes. In view (a) is shown a diagrammatic 
cross-section of the building, which illustrates the conditions 
of loading and the spans of the cantilever and other girders. 
The live load to be supported by the floor construction and 
the cantilever girder is 220 lb. per sq. ft., and the soil beneath 
the footings is capable of supporting safely 6,000 lb. per sq. ft. 

The details of the construction and the reinforcement 
are shown in view (b). The footing is reinforced against 
failure from transverse stress by bars a, placed near the 
bottom of the footing, and further by a mattress of rods or 
bars placed directly beneath the bearing of the foundation 
wall column, as at b. The column, or wall pier, ¢ is rein- 
forced with vertical reinforcing bars, which are tied at close 
intervals with wire or loop ties d. Particular attention is 
called to the manner in which the outside rods of the founda- 
tion pier or column run straight through and up into the 
column above, being continued as the inside rods of the 
upper tiers. The outside rods of the first-floor wall column 
are bent so as to pass obliquely through the cantilever 
bracket of the foundation pier or column and the flare at the 


FOUNDATIONS 


320 


70 


TO 


deg mats: ee 
Br aie va 


lafats twanshauir eas Wee 


ewe ms FasL 


“SK | 1 ye 

aU thir 

Pies Seee ew 
ND sel jo : 


321 


FOUNDA TIONS 


4 —_-— + -— — — 40 — 
4 
: | 
Pass Fa=s ae 7 9-7 EY 
code | enter eke 1049,9 @ seg ele x 
| 
| u 
ft she yt09 8281082,018 8109 M1 bs F SS ed . 
= ay JO MAY 207 
ye ye eee ee saemepgase: me Fe beinae ecane bap 
Nake i" 1 Ay, te e109 4 bs hp ole 
" . ; 4 ee al we “Pe 
tS evayuar, 21) seu: seer, f +. ‘a | k "De. 
rab Lh 4 a = 
1 Hes s 1 
ait —— 200 acts t-r anh Lt if | 
att itt a! 2D Oe oe = So Te 
ihe . pate no Soa Soe ee tS Baer 
sug my bs f/-06 


3 surg 1 tS, 1-6 <F 


‘— 
wayI99 Y09-" agli te Mic ® < 
a LO SL LY q Yy 
WY (SEPA AN' Pe Dare @ sm ry f a TW 
* vr] ’ bee 4 . 


UOM WY ID 42149, F/ — 


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322 BUILDING DETAILS 


bottom, as shown at e. The reinforcing rods of the canti- 
lever girder are arrangéd as shown at f, g,h,and7. The rods 
in the upper part of the girder are arranged in three rows, 
or layers, and are stopped off in length in practically the 
same manner as the flange plates of a built-up girder, because 
the bending moment is reduced toward the interior column. 
Only one set of the reinforcing rods, as at f, extends entirely 
through into the interior column, because the bending moment 
is so reduced that this set is all thatis required. The bottom 
rods, as at 7, are inserted to improve the resistance to com- 
pression in the cantilever girder beam at the bottom. © 

Both the girder and the cantilever bracket are well supplied 
with stirrups, as shown at j, The vertical reinforcement of 
the interior column is shown at k, and the ties and bracket 
reinforcement, at / and m, respectively. The lintels, or con- 
necting beams, at the end of the cantilever are shown at n, 

In Fig. 2 is shown a reinforced-concrete cantilever-founda- 
tion construction designed to carry the same load on the 
end of the cantilever as was required of the construction 
illustrated in Fig. 1. As the cantilever girders have a 
considerable projection beyond the foundation footings, they 
must be very strongly reinforced in both the top and bot- 
tom, as shown. The several details of this construction are 
worked out in the illustration, and an analysis of these will 
show that the several principal and secondary stresses 
created in the structure are amply provided for 


BUILDING DETAILS 


LINTEL AND SPANDREL CONSTRUCTION 


There are two ways of arranging the beams and girders 
in a building. These different ways bring different loads on 
the spandrels 

In the construction shown in Fig. 1 (a), one-half of the floor 
load from the beams a and b is concentrated at two points 
upon the lintel, and the lintel has this load, as well as the 
weight of the spandrel wall, tosustain. Also, as the beams a 


BUILDING DETAILS 323 


and b extend into the lintel, it must at least be equal in depth 
to the depth of these beams. Thus, the height of the window 
opening is materially reduced. 

The method of framing shown in Fig. 2 (b) is frequently 
employed. The girders, instead of extending from column 
to column, as shown at ¢ in view (a), extend from column 
to wall pier; consequently, the beams extend in a direction 
parallel with the lintel. The slab spans from the beam a 
to the lintel b. The lintel carries one-half of the slab load 
between these structuralmem- . 
bers and is required tosustain 77 
the weight of the spandrel as 


well. The advantage gained a 
by this method of construc- ls 
tion is that there are no beams 


abutting the lintel. Thus, this 
member may be reduced in 
depth to the very minimum for — 
strength required to support 
the small floor load and the 
weight of the spandrel wall. —s 

The style of lintel shown in 
Fig. 2 (a) is the one in most 
common use. The girders ® a 
are arranged, as shown in 
Fig. 1 (b), so that the lintel 
carries only part of the slab 
and the spandrel section. (b) #3 
The lintel is made as shal- 
low as possible, so as to admit Fia. 1 
maximum light by having the window head near the 
ceiling. 

When no beams are used and the slab spans from girder 
to girder, the lintel carries no load except the weight of the 
spandrel. In this case, reinforced concrete is sometimes 
dispensed with and a brick arch employed, as shown in 
Fig. 2 (b).. The arch, however, cuts off the top corners of 
the window, and thus somewhat reduces the amount of light 
that can be admitted. 


(@) 


BUILDING DETAILS 


324 


wee 


wy 


mT] 


Mf 


Uy 
ZY 


Fic. 2 


BUILDING DETAILS 325 


When the lintel supports the ends of floor beams, as shown 
in Fig. 1 (a), the construction shown in Fig. 2 (c) is used. 

Sometimes it is desired to cover the concrete lintel with a 
stone facing, as shown in (d). The stone is held to the 
concrete by means of an iron clamp at the top, as shown. 

When the span of the lintel is very long or the load to be 
carried is heavy, it is necessary to make the lintel deep. So 
as not to lower the head of the window, the lintel is run up 
into the spandrel above, as shown at (e¢). In this case, the 
floor slab joins the lintel near its lower surface and therefore 
does not assist in resisting compression. If the lintel must 
be strengthened still further, sometimes the entire spandrel is 
made of reinforced concrete, as shown in (f). 

In cases where a building is to have terra-cotta trimmings, 
the concrete lintels are often also covered with terra cotta. 
Such an arrangement is shown at (g). The brick spandrel 
is placed on top of the terra cotta and assists in holding it 


in place. 
EAVES AND CORNICES 

For ordinary factory construction, in suburban districts at 
least, where little artistic effect is desired, the simplest way 
to construct the eaves of the roof with a hanging gutter is 
shown in Fig. 3 (a). The gutter itself is often omitted and 
the rain allowed to fall to the ground, especially if there is 
no cellar and if the floor level is somewhat above grade. 
The slab reinforcing rods are shown running parallel to the 
wall. Where the overhang of the slab is considerable, the 
reinforcing rods must run at right angles to the wall and be 
bent up toward the top of the slab in the overhang so as to 
take up the reverse bending moment, as shown in (0d). 

Sometimes the gutter is constructed in the concrete by 
forming a gusset, as shown in (c). The roofing is extended 
directly over the gusset and secured by a wooden strip. 

When a parapet wall is required, the construction shown 
in (d) may be used. The flashing is secured under the coping. 
Where the parapet wall is low enough, the best plan is to 
raise the gusset so that it will come to the under side of the 
coping. In this way the flashing may be omitted altogether 
and the roofing run under the coping. Such an arrangement 


BUILDING DETAILS 


326 


Fic. 3 


BUILDING DETAILS 327 


is shown in (e). In both (d) and (e), the gusset is made of 
cinder concrete, as it is cheaper and lighter than stone 
concrete and is not required to carry a heavy load. 

In (f) is shown the method of reinforcing a cornice that 
has a heavy overhang. This construction requires special 
care, as the cornice may be very top heavy. 


BRICK FACING AND TERRA-COTTA STRING- 
COURSES 


The method of tying a brick facing to a concrete pier by 
means of copper ties is illustrated a : 

in Fig. 4. The ties, which consist 
of strips of copper about 7 in. 
thick, #? in. wide, and 7 in. long, 
are placed every seventh joint. 
These ties are spaced about 2 ft. 
apart horizontally, and are embed- 
ded in the concrete work to a 
depth of about 4 in., which gives 
them, when a §- or }-in. space is left 
between the facing and the concrete, 
a surface about 2} in. long in the 
bricks. It is customary in arran- 
ging the brick facing to allow about 
5 in. for the width of the brick, 
and the space back of the brick is 
usually flushed in with mortar as the bricks are laid 


FASTENINGS IN CONCRETE 


There are numerous devices intended to be embedded in 
the concrete work for the purpose of holding T-headed or tap 
bolts ina secure manner. These necessarily must be arranged 
in the forms before the concrete is poured. Ordinarily, these 
sockets should be tapped out for }-in. bolts, but where heavy 
machinery or hoisting apparatus is to be employed, it will be 
better to use {-in. bolts. Sockets should be provided near the 
end bearings of each beam and girder, and intermediately, 
not farther apart than 4 to 5 ft. 


328 BUILDING DETAILS 


A #-in, bolt will ordinarily test in a tensile pull to from 
15,000 to 17,000 lb. Thus, it can be considered to sustain 
skate a foal of about 4 000 Ib. 

Where it is necessary to bolt heavy suachinaey to the 
ceiling, the location should be ascertained, and heavy 
through bolts should then be embedded in the concrete work 


BEAM SOCKETS 


Styles of Sockets ——The socket shown in Fig. 5 (a) is known 
as the Unit socket. The cast-iron part a rests on the bottom 
of the form load, and is bolted to it by the bolt b, which 
passes through.a hole in the bottom of the forms. This 
socket, as shown, clamps to four reinforcing rods of the beam 
or girder in which it is placed. In (b) is shown a malleable- 
iron socket. It is tapped at a to receive a stud bolt. This 
socket is secured to the form boards by means of lugs |, 
through which nails are driven into the wood. 

The socket shown in (c) is known as the Jennings-Stein- 
metz socket. It consists of a piece of pipe swaged out or 
broken at the upper end, as indicated at a, with a solid 
wrought-iron bar or block welded to the other end. The 
solid portion of the block is tapped out for the bolt b, which 
passes through the bottom board c of the form. 

What is known as the Hancock insert is shown in (d). 
This device consists of a cast- or malleable-iron casing that 
is made in loose halves a, and is wired together through the 
lugs 6 cast on the side. Before the two halves of the insert 
are wired together, however, there is placed in the recess 
a nut c, into which a bolt can be screwed from beneath. 
The castings are arranged with lugs, or flanges, d on the face 
end and by means of these the insert may be screwed or 
nailed to the form board and thus secured in an upright and 
secure position when the concrete is placed. The flange e 
at the top of the insert furnishes additional bond, or key, 
with the concrete. The Hancock insert is made in 3-, 4-, 
and 6-in. lengths for 4-, %-, ?,- and 1-in. bolts. 

In (e) is shown the Bigelow socket. ‘This casting, which 
is made of malleable iron, is arranged with a slot a that is 


se rhuagig 


La 


‘aA so 
“ ) PA 
4° 


x 
site 


4 


aa, 


BUILDING DETAILS 


Tats Teter ee: 
SS “Sere 


329 


SSSSAAVAQRUQAH 


qr 


Fic. 


330 BUILDING DETAILS 


enlarged at b so as to receive the head of the bolt. In order 
that the casting may be securely held in the concrete work, 
it is arranged with a plate d and connecting pieces e. On 
the sides of the socket at the face are cast notches ¢ into 
which nails may be driven so that the device may be secured 
to the forms. The Bigelow socket is made for }-, 2-, -, 
is-, $-, and #-in. bolts. The smaller sizes are suitable for 
embedment in slab work, and the 75-, 4-, and ?-in. sizes may 
be used in beams and girders. 


YZ 


EE 


In (f) is shown a sheet-steel socket. It is cut, as shown 
at a, so as to fit around the beam reinforcing bars jf. In (g) 
is shown separately the cap that fits over the nut held iby 
the socket into which the stud bolt fits. 

Continuous Inserts for T-Headed Bolts.—In the construc- 
tion of factory buildings it is sometimes desirable to arrange 
a slot in the bottom of the concrete beams and girders so 


BUILDING DETAILS 331 


that a T-headed bolt can be introduced at any point along 
the entire length. To accomplish this, several methods are 
employed, the principal ones xine illustrated in Fig. 6. 


iZ Yi 


Y 
YU ty, 
YI AN 
7 se » VZZZZA LZ 
@ aS a Oa\\ ey 
: Wt TD Ys = 
| A WA . x Up 
q ees ° Za 


RS 


r 


edition a! 


en re 


oe 


ot 


FIG. 7 


Two views of one device are shown in (a) and (b). 
View (a) shows the T-headed bolt that carries the shafting 


332 . BUILDING DETAILS 


and view (b) shows the bolt that secures the continuous 
insert to the under side of the beam. 

The device shown in (c) consists of a split pipe in which 
the T-headed bolts slide. The form boards are shown in 
place in this view so that it may be seen how the split pipe 
is held in place in the forms by the strips 7. 

In (d), the slide consists of a piece of bent steel embedded 
in the concrete. As shown, bolts are also used to hold the 
steel slide in place. 

Two views of another device are shown in (e) and (f). 
View (e) shows how the slide for the T-headed bolts is held 
in place, and view (f) shows the bolt in position to support 
machinery. 


BEARINGS FOR CONCRETE BEAMS AND 
GIRDERS 


It is often necessary to increase the bearing area of con- 
crete beams, especially when they rest.on brick walls. The 
ways in which this can be done are shown in Fig. 7. 

In view (d), the end of the beam is formed with projecting 
spurs at right angles to the axis of the beam, those spurs 
being of such length as to distribute the bearing over a con- 
siderable area. The reinforcing rods of the cross-beam are 
shown at c. 

In order to save concrete, the bearing of the beams may 
be increased by the method shown in (e). This construction, 
however, is not so practical as the one shown in (d), because © 
the form is more troublesome to make and fill, and, besides, 
the masonry does not build so readily with flat beds on the 
slope as it will with the construction shown in (d). 


} 


——$————— 


——— 


LL LT 


FORM WORK 333 


FORM WORK 


- CONSTRUCTION AND FINISH OF FORM 
WORK 


In the erection of reinforced-concrete work, nothing 
requires more careful consideration than the construction 
of the form work, or molds, necessary to shape and support 
the concrete until it has thoroughly set and hardened. 
Throughout the practice of reinforced-concrete construction 
various methods of form constructions are in use. 

The greatest economy is gained by constructing the forms 
so that they can be used over and over again in the structure. 
Economy in construction can also be gained by fastening the 
form work together with a minimum amount of nailing. 
Every nail that is driven gives trouble when the forms are 
taken down to be replaced for the upper floors. In many 
constructions, wedges and clamps are used instead of nails 
or screws if the forms are to be reused. 

In some instances, both wooden and metal forms are 
coated on the side next to the concrete in order that the 
forms may be detached more readily. Coating the forms 
also serves to prevent the marking of the grain of the wooden 
forms on the finished concrete work. 

Dead oil, or crude petroleum, has been used with success 
for this purpose. It is not unusual to soap wooden forms, 
and in some cases tallow and bacon fat have been employed. 
The latter is especially recommended for coating metal 
forms, as it seems to give the best results with forms of this 
material. In some instances, wooden forms have been 
covered on the inside with paper, and even canvas has been 
used, although it is usually found that the paper adheres 
to the concrete work and is detached only with difficulty. 
It is not customary, however, to oil or coat the forms unless 
they are to be used for fine exterior work. 


334 . FORM WORK 


FORMS FOR FLOOR SYSTEMS 


Common Types of Form Work.—In Fig. 1 is shown a type © 


of form work extensively used for the construction of rein- 


Fre. 1 


forced-concrete floor systems, and in Fig. 2 is shown a per- 
spective of the forms at the intersection of a beam and 
girder, This form work is designed so that light $-in. dressed 
tongued-and-grooved material may be used extensively in its 
construction. It is arranged so that the sides of the beams 
and girders, together with the slab form boards, may be 
removed without the necessity of removing the supports 
directly underneath the beams and girders. 
The column form 
for this type. of 
construction is 
shown in Fig. 3. 
Forms Construct- 
ed of Plank.—A 
superior type of 
form for a rein- 
Fic. 2 forced-concrete Fic. 3 
floor system is 
shown in Fig. 4. The wooden forms are supported by 3” xX 4” 
studs. As it is important to bring the forms toa true level, a 
double adjustment wedge is provided at the bottom of the 
studs. The forms for the columns are made of 14- or 2-in. 


FORM WORK 335 


material. In the construction of the beam and girder forms 
|‘ 2-in, planks are generally used for the sides. In order toform 


i Fic. 4 


a chamfer on the lower edges of the beams and girders, tri- 
angular fillet pieces are nailed in the forms. It is customary 


a SSN EZ ES Es OS SA SETS 


LS TG 2 AO Te 


Fic. 5 


336 FORM WORK 


in this type of construction to make the forms for the slabs 
of }-in. plain boards, frequently using tongued-and-grooved 
material, 


Collapsible Forms.—Several attempts have been made to 
arrange the forms for a reinforced-concrete floor system so 
that they will partly collapse and thus be easy to take down 
or away from the concrete as soon as it has properly set. 

One type of collapsible form is shown in Fig. 5. In this 
construction, the side forms for the beams are embodied with 
the slab centering by the construction of a collapsible box, 
which furnishes a 
form that is the 
exact shape of the 
space between two 
adjacent beams. 
These boxes are 
arranged with a 
hinge c, and when 
in place they rest 
upon cap pieces d. 

In order to take 
down the forms, 
the struts are re- 

Fic. 7 moved, and the 

form is allowed to 

double up on the hinges c and thus release itself from the 
concrete. After the collapsible centering has been removed, 


lel ia <a 


the form work for the sides 


—— ee rey 


FORM WORK ; 337 


the planks beneath the soffit of the beams and girders may 
be retained in position by means of struts h. 

Another type of collapsible 
form is shown in Fig. 6. In 
this form, the box that supplies 


of the beams and the center- 
ing of the slabs, instead of 
being hinged, is cut through 
along the line a a and is held in 
position by the slotted iron 
plate 6 and screw e. By re- 
leasing and loosening the 


4 
Ness 
ee 1 
A 


J 
ae 
: 


are) 

pee 
J 

1 


screw, it will slip in the slot c. |Ppsiaens 
Then one-half of the form can '7%% be Mh 


be drawn downwards, so that 

it will recede from the face of Fic. 8 

the beam and thus allow the 

form to free itself from the concrete and be taken down. 
It will be observed that the sides of the collapsible forms 
make the sides for the beam forms, the battens d furnishing 
the support for the bottom form board f of the beams. 


Heavy Construction—Partly Collapsible Forms.—In Fig. 7 
are shown the details of a form construction for a reinforced- 


338 : FORM WORK 


concrete floor system that is particularly well constructed. 
It may be readily set up and taken apart. The sheathing 

se 2 of the beam and girder forms 
and the plank used for the slab 
centering are 2 in. in thickness. 


4” X 4” studs. 

Wall Forms With Wire Ties. 
A type of wall-form con- 
struction that is frequently 
used is illustrated in Fig.8. In 
order to prevent the sidés of the 
forms from spreading when 
the concrete is tamped in place, 
a wire tie c is used. This 

Fic. 11 tie is made taut by twisting 
with a bar, or stick, d. To keep the form boards the proper 
distance apart for the thickness of the wall, a block or stick 
e of wood is sometimes inserted. 

Wall-Form Construction With Clamp Bolts.—A wall-form 
construction similar to that shown in Fig. 8 is illustrated in 
Fig. 9. In this form, however, a clamp bolt a, instead of 
a wire tie, is used to prevent spreading. If a bolt of this 
character is used, it must be knocked out before the concrete 
has finally set and when the form boards are tovbe raised to 
form the next course of concrete. The bolt is preferable to 
the wire tie, because it is removed from the concrete. Wire 
ties are usually cut off close to the concrete work after the 
form boards have been re- 
moved, and as the ends fre- 
quently project, they rust and 
thus stain the wall. * 

In Fig. 10 is shown. the 
construction of a wall form 
in which a pipe separator a 
is used with the clamp bolt. 
The pipes may be driven out 
of the concrete after it has obtained its initial set, or they 
may be left in place. 


ty) 
| 
HN 
lh 


Fic. 12 


The forms are supported by 


Lore 


339 


FORM WORK 


dated ee ee 
a 


tome has 


a 


ee 


3 


Fic. 1 


340 FORM WORK : 


One of the best methods of constructing wall forms is to 
use concrete separators. The separator known as the 


ia 


McCarty separator, used in constructing the form work, is 
illustrated in Fig. 11. The separator ¢ is cast previous to 
the construction of 
the forms. 
Clamping De- 
vices and Plank 
Holders for Wall 
Forms.—Many de- 
vices that aid in 
- the construction of 
= concrete walls have 
beeninvented. One 
of the most useful 
of these devices is 
the Sullivan 
pressed-steel plank 
holder, various 
forms of which are 
shown in Fig. 12. 
These holders are 
formed from an 
iron plate by shear- 
ing and bending it 
so as to form clips. 
The application 


Fic. 15 
of this type of plank holder is illustrated in Fig. 13. 


FORM WORK 341 


Braces for Wall Forms.—If a wall is to be constructed in 
a place where there is no embankment, a double set of forms 
braced as shown in Fig. 14 (a) must be used. If the soil of 
an embankment against which a concrete wall is to be built 
is unstable, it is necessary to sheath and brace it. This may 


Fic. 16 


be accomplished in either the manner shown in Fig. 14 (b) 
or that shown in (c). 

_ Spandrel-Wall Forms.—In the molding of spandrel walls 
of buildings a special type of form construction must be used. 
A typical spandrel-wall form is shown in Fig. 15. Attention 


342 WATERPROOFING 


is called to the fact that the spandrel wall cannot be carried 
up at the time that the lintel and slab forms are filled, 
especially if wet concrete is used, Therefore, the concrete 
must first be deposited to the level of the top of the slab, 
and when this has its initial set, the balance of the wall 
may be filled. If this precaution is not taken, the hydro- 
static pressure of the wet concrete in the spandrel wall will 
force the concrete from underneath the form at the slab and 
prevent the filling of the wall form. 

Cornice Construction.—In Fig. 16 is shown a typical form. 
construction for a reinforced-concrete cornice. ‘The cornice 
work is carefully constructed of 13-in. -dressed material. 
In filling the form, the concrete is carried to the level A A, 
and the parapet wall is afterwards constructed above this 
point. 


WATERPROOFING OF CONCRETE 


CLASSIFICATION OF SYSTEMS 


There are three principal methods, or systems, employed 
in the waterproofing of concrete. They may be termed the 
integral, the superficial, and the membrane method, 

The integral method consists in adding something to the . 
concrete when it is placed, or in mixing the concrete in cer- 
tain proportions, so as to make it waterproof throughout. 
The superficial method consists in coating the concrete with 
paint or some other material. This material adheres to the 
concrete and hardens, or dries, on it. The membrane 
method consists in putting on the concrete a coating that is 
distinct from it. While the coating may adhere to the con- 
crete, it does not crack when the concrete cracks, but is in 
a distinct membrane usually strengthened by felt or some 
other fiber cloth. 


INTEGRAL METHOD 
Mixing of Concrete.—According to some authorities, if, 
concrete is properly mixed, it will be impervious to water. 
The exact mixture to use will depend on the quality of sand 


WATERPROOFING 343 


and broken stone. This most waterproof. mixture is also 
the densest mixture. Therefore, in searching for the most 
waterproof mixture of concrete, the engineer really finds 


the densest mixture. For many grades of sand and broken 


stone, a 1-14-3 mixture is used. To be impervious to water, 
concrete must be placed in a wet condition and be well 
rammed into place 

Concrete, particularly that which is very dense, becomes 
more impervious to water as it grows older. The first water 
that penetrates it carries particles of clay and other material 
that stops up its pores and gradually makes it more water- 
proof. ; 

Adding of Lime or Clay.— By hydrated lime is meant lime 
that has been slaked in water. It can be bought com- 
mercially in the form of a dry powder. The purpose of 
adding hydrated lime to concrete is to fill mechanically the 
voids in the latter. A small quantity added to the concrete 
when it is being mixed does not greatly reduce the strength 
of the latter. After the cement and lime are mixed until 
the color of the mixture is uniform, the sand, broken stone, 
and water are added as usual. The concrete should be made 
wet, and great care must be exercised in bonding old and 
new work. The amount of lime to be employed is usually 
given as a percentage, by weight, of the cement. Under 
ordinary conditions, for 1-2-4 concrete, 8% of lime will be 
found sufficient, and for 1-3-6 concrete, 17% of lime will 
be required. 

In place of hydrated lime, concrete is often waterproofed 
by mixing with it finely ground colloidal clay. It is recom- 
mended that clay equal to about 10% of the weight of the 
cement be used in a mixture. The clay must be thoroughly 
dry and well mixed with the cement. 

Sylvester Process.—The Sylvester process of waterproofing 
concrete consists in adding powdered alum and sojt soap to the 
concrete. The sand and cement are mixed together dry, 
as usual. To this mixture is added alum equal to 1% of the 
weight of the mixture. To the water to be used is added 
1%, by weight, of soft soap, and this soap is then thoroughly 
dissolved. The mortar is made wet, and the broken stone 


344 WATERPROOFING 


is added in the usual manner. Ordinary precautions should 
be taken to make the concrete dense, and as a rule it is 
mixed rather wet. 

Metallic Stearates and Other Compounds.—Besides the ,; 
materials mentioned, various other chemicals are used to’ 
make concrete waterproof. One of the most successful of 
these is calcium stearate, which is a salt of a fatty acid. 
Usually, the quantity of finely ground metallic stearate 
added to the cement to make the conctete waterproof is 
about 2% of the weight of the cement. 

Besides the metallic stearates, other substances are used 
more or less to waterproof concrete by the integral method. 
A mixture of oil and water has been used with some success, 
as has also chloride of lime. The purpose of the chemicals 
is in every case to fill the voids, and, in addition, some of 
them are water repellants. 


SUPERFICIAL METHOD 

Waxes.—To make concrete waterproof, it is sometimes 
coated with wax. ‘The wax sinks into the pores of the con- 
crete, filling the voids therein and thus preventing the per- 
colation of water. 

Paraffin is used for waterproofing with considerable suc- 
cess. A paraffin especially hardened to resist the sun’s rays 
is used. The concrete surface on which the paraffin is to be 
applied should be thoroughly dry and not very cold. In fact, 
it is better to warm the surface with a torch if possible. The 
paraffin is heated and then applied witha brush.. The hot 
paraffin is absorbed by the concrete, into which it penetrates 
for a short distance. Paraffin will resist the action of acids 
and alkalies. 

Instead of applying paraffin while hot, it may be dissolved 
in some volatile carrier, such as benzine. The dry wall is 
painted with this solution, which is readily absorbed. The 
carrier then evaporates and leaves the paraffin to fill the pores 
in the surface of the wall. If the carrier is inflammable, care 
must be taken in using it. 

Cements.—A common method of waterproofing concrete is 
to coat the surface of the concrete with an impervious cement 


Py hhte nem ieee ai a Lathe — 


Se a eo — ay 


ear 


OE — SS ee a 


WATERPROOFING 345 


coating. Sometimes this coating is simply a rich, dense 


ip. cement mortar; sometimes it contains a waterproofing com- 


pound similar to those mentioned in connection with the 
integral method; and sometimes a special waterproof cement 
is used. 

The adhesion between the coating and the concrete 
work requires the most careful consideration. The concrete 
surface on which the waterproof coating is to be placed 
must first be chipped and scraped to remove all glaze and 
to obtain a rough texture. The surface must then be thor- 
oughly washed with clean water to remove all dust, and all 
cracks must be filled with mortar. While the surface is still 
wet, it is painted with a mixture of cement and water of about 
the consistency of thick cream. This mixture is applied 
with a stiff brush and must be well rubbed into the surface. 
Before this coat dries, the first coat of waterproof mortar is 
put on, usually about fin. thick. This coat must be troweled 
into the surface with great care. Then, before this coat sets, 
another coat of waterproof mortar of about the same thick- 
ness or a little thicker is applied. 


MEMBRANE METHOD 


Bituminous Membranes.—The bituminous-membrane method 
consists in the use of a felt saturated with bituminous 
material and a bituminous binder. By a binder is meant a 
material that binds two surfaces together; that is, an adhesive 
material. By a bituminous material is meant any material 
containing a large proportion of solid or semi-solid bitumen, 
bitumen being that portion of pitch or asphalt that is soluble 
in carbon bisulphide, benzol, petroleum, ether, or other simi- 
lar solvent. 

The bituminous membrane is built up in place in suc- 
cessive layers, and should form, when finished, a practically 
homogeneous and continuous waterproof envelope. The 
felt serves to hold the bitumen in place, while the latter is 
the waterproofing material. 

Specifications for Coal-Tar Pitch and Felt Roof Over Con- 
crete.—The following specifications, known as Barrett spectfi- 


346 WATERPROOFING 


cations, will be found excellent for placing a coal-tar pitch 
and felt roof over concrete. These specifications also . 
describe how the work should be carried out. 

“There shall be used five thicknesses of approved felt 
weighing not less than 14 Ib. per 100 sq. ft., single thickness, 
not less than 200 lb. of approved pitch, and not less than 
400 lb. of gravel or 300 Ib. of slag from } to § in. in size, free 
from dirt, per 100 sq. ft. of completed roof, 


a 
UY 


Une 


Fic. 17 


“The material shall be applied as follows: (1) Coat the 
concrete a (see accompanying figure), with hot pitch } 
mopped on uniformly. (2) Lay two full thicknesses of 
tarred felt c, lapping each sheet 17 in. over the preceding one, 


FIELD OPERATIONS 347 


and mop with hot pitch d the full width of the 17-in. lap, so 
that in no case shall felt touch felt. (3) Coat the entire 
surface with hot pitch e mopped on uniformly. (4) Lay 
three full thicknesses of felt f, lapping each sheet 22 in. over 
the preceding one and mopping with hot pitch g the full 
width of the 22-in. lap between the plies, so that in no case 
shall felt touch felt. (5) Spread over the entire surface of the 
roof a uniform coat of pitch, into which, while hot, embed 
the gravel or slag h. The gravel or slag in all cases must 
be dry.”’ 

The preceding specifications are designed for roofs having 
an incline not exceeding 1 in. to the ft., and by adding the 
words “‘such nailing as is necessary shall be done so that all 
nails will be covered by at least two plies of felt,’’ the specifi- 
cations are suitable for inclines not exceeding 3 in. to the ft. 
For surfaces steeper than 3 in. to the foot, nailing strips of 
wood must be provided. These should be embedded in the 
concrete from 3 to 6 ft. apart, running at right angles to the 
pitch of the roof, and the felt nailed to these strips. 


FIELD OPERATIONS 


CONCRETE MIXERS 
BATCH MIXERS 

Cube Mixers.—One of the oldest and best-known forms 
of batch mixer is the cube mixer. In Fig. 1 is shown what is 
known as a Carlin cube mixer. The cube is driven by a belt a, 
and gears b serve to reduce the speed of rotation. The cube 
is shown in the position it occupies when being filled. The 
trap door for filling and discharging is shown open and swung 
back atc. The hopper above the cube is made in two parts. 
The lower part is hung from one of the posts of the platform, 
as shown. When the cube is revolving, this lower part can 
be swung out of the way. At d is shown the water tank, 
which feeds water into the cube through the pipe e. Below 
the cube is shown a car f into which the finished concrete is 
dumped. 


348 FIELD OPERATIONS 


ded At 


The following conditions have been found to produce excel- — 


lent results with a 4-ft. cube mixer: Each batch is mixed 


13 min. at a speed of 10 rev. per min., making a total of f 


15 turns per batch. The time allowed for dumping, cleaning 
and charging the mixer and for temporary stoppage of the 
work is about 34 min., making an average time interval 
between successive batches of 5 min., which is at the rate 
of 12 batches per hour. Allowing only ¢ cu. yd. of rammed 
concrete per batch, this gives for a 10-hr. day an output of 
105 cu. yd. 

Ransome Mixer. 
Another form of batch 
mixer, called the Ran- 
some mixer, is illustra- 
ted in two views in 
Fig. 2. This machine 
consists essentially of 
a hollow cylindrical 
drum that is mounted 
ona horizontalaxisand 
has a circular opening 
in each end. As shown 
in (a), the mixer is 
charged through one 
of these openings a. 
In the other opening, 
shown in (b), is achute 
b that receives the 
mixed concrete from 
the drum and delivers 
it to a wheelbarrow or other receptacle. Inside the 
drum are several steel blades arranged in such a manner 
as to deflect the material from side to side as the drum 
revolves. The drum is mounted on four rollers and is sup- 
ported on a truck, which is either made stationary or mounted 
on wheels, as required. The mixer is turned by power 
applied to the rim of the drum either from an engine mounted 
on the same frame or from a belt or a chain. The illustra- 
tion shows one driven by a belt. 


FIELD OPERATIONS 349 


The chute b, Fig. 2 (6), through which the mixer discharges, 
is kept in the position shown in Fig. 3 (a) while the concrete 
is being mixed. When it is time to empty the mixer, the 
handle c, Fig. 2,isturned. This turning of the handle, which 
is connected to the chute by means of a chain, as shown, tips 
the chute b, as illustrated by dotted lines in the figure. Mean- 
while, the drum is revolving and lifting masses of concrete 
up one side in the 
direction of its motion. 
These masses finally 
fall by the action of 
gravity to the bottom 
of the drum. 

The path followed 
by the concrete in the 
mixer is shown dia- 
grammatically in Fig. 
3. The blades are 
represented by short 
radial lines. When 
the chute is tipped 
down, as shown by 
dotted lines in Fig. 2 
(b), the upper end in- 
tercepts the flow of 
concrete, as shown in 
Fig. 3 (6), and the 
mass slides down the 
chute and out of the 
drum. 

The Ransome mixer Fic. 2. 
is made in several sizes, with varying capacities. 

Smith Mixer.—In Fig. 4 is illustrated the Smith mixer. 
The machine consists of a drum of double conical form that 
is supported and guided by a frame that can be tilted at will 
while the drum is revolving. The ingredients are fed in at 
one end of the drum, and after the required number of 
revolutions the mixed concrete is discharged at the other 
end by tilting the drum while it is running at full speed. 


350 FIELD OPERATIONS. 2 


The interior of the drum is provided with blades so as to 
insure thorough mixing. ‘The capacity of the mixer is from 


Fie: 3 


10 to 35 cu. yd. of concrete per hour, according to the size 
of the machine. | 


FIELD OPERATIONS 351 


CONTINUOUS MIXERS 

Drake Mixer.—One of the best known forms of continuous- 
mixing machines is the Drake mixer. This mixer consists 
of an open trough fitted with a longitudinal shaft, to which 
are fastened blades, or paddles, some of which are set at an 
inclination, so that they will not only mix the ingredients, 
but also feed the mixture toward the discharge end. The 
mixer is so set that the shaft revolves on a horizontal axis. 
The ingredients are deposited by wheelbarrows or from a 


measuring box at the upper, or feeding, end of the trough. 
The straight blade, or knife, at that end cuts or stirs the 
mass, which is then turned over by the adjacent curved 
blade, or scoop, and advanced to the next knife, where it is 
again cut and then turned over by the next scoop, and so 
passed on. This process is repeated until the end of the 
trough is reached, when the material is pushed out and falls 
into a receptacle as mixed concrete. The ingredients are 
mixed dry for about one-half the length of the trough and 


352 FIELD OPERATIONS 


then a spray from water pipes wets the material as it is cut 


Fic. 5 


and turned over. 

Fig. 5 shows a Drake 
mixer of usual design. 
The ingredients are 
dumped in at the far 
end, and when they are 
entirely mixed are de- 
livered down the chute. 
The Drake mixer is made 
in various styles. 

Cockburn Mixer.—An- 
other type of continuous 


. mixer, called the Cock- 


burn mixer, is illustrated 
in Fig. 6. This machine 
consists essentially of a 
long box of square cross- 
section, mounted on a 
substantial iron frame or 


truck and revolving on a longitudinal axis on friction rollers, 
as shown. The axis of the box is inclined slightly from the 


horizontal. The ingredients are fed in through a hopper or 
chute, at the upper end, and, as the mixer revolves, the mate- 


= 5 ere pe 


FIELD OPERATIONS . 353 


rial moves by gravity toward the lower end. The volume 
of its output depends largely on the rapidity with which 
the ingredients are fed into the mixer. 


QUANTITATIVE MIXERS 


Quantitative mixers are mixers that measure as well as mix 
the ingredients used to make concrete. They usually con- 
sist of continuous mixers similar to those just described, to 
which is added a device for measuring automatically the 


cement, and broken stone as it enters the mixer. Some- 
times the measuring apparatus and mixer proper are built as 
one machine, and sometimes the measuring apparatus is 
built separate, so that if may be used with any suitable mixer. 

Gilbert Measurer and Feeder.—A machine that can be 
applied to any continuous mixer is the Gilbert accurate 
measurer and feeder. Fig. 7 shows a view of this measurer. 
At a are shown the bottoms of the three storage bins, which 


® 


354 FIELD OPERATIONS 


are built hopper shaped. In one of these bins is stored 
_ cement, in another sand, and in the third, broken stone. 
These bins, although hopper shaped, have nobottoms. They 
are closed at the bottom by the cylinder c, which is of 
sufficient diameter to act as a practically flat bottom to the 
bin. In the front of each hopper there is a door } that may 
be adjusted by a weight, as shown in the figure. The cylin- 
der c is turned by a man at the handle d. As the cylinder 
revolves, it carries with it a layer of cement, sand, and broken 
stone. The amount of each material carried forwards is con- 
trolled by adjusting,the gate at b. As the wheel revolves, 
the materials finally slide off into a conveyer or directly into 
the mixer. The faster the operator revolves the cylinder, ~ 
the more materials he will discharge, but they will always 
be discharged in the same proportion. 


OPERATION OF MIXERS 


A concrete mixer may be operated by a steam engine, a 
gas engine, or an electric motor; also, where the water-power 
is available, it can be operated by an impulse wheel or by a 
turbine; or, if the power is available from shafting in an 
adjacent building, the mixer may be operated by a chain or 
belt drive. 

The superintendent or foreman of a reinforced-concrete 
job should take good care of the machinery in his charge. 
It*is well for him to obtain from the manufacturers written 
directions as to the use and operation of their machines. 
Usually, these directions are explicit and are based on 
experience with the particular machine in question. As an 
example, the directions published by the Ransome Machinery 
Company are given in part as follows: 

“Rules for Operating a Mixer.—If the machine is mounted 
on wheels, see that the weight is first taken off the wheels 
and carried on suitable sills, as shown in Fig. 8. The points 
of support should be beneath each of the roller shafts, 
beneath the bed of the engine, and beneath the boiler. The 
mixer frame should be carefully leveled in both directions. 
Remove the hook bolts that hold the drum to the frame. 


rar arr ae 


FIELD OPERATIONS 355 


Fill all grease and oil cups and grease carefully the traction 
rings and roller faces. See that in all cases the lubricant is. 
fed to the bearings. Use good graphite, hard oil, or grease 
in all compression cups, and screw the caps down so as to 
force the grease through the journal box. When the machine 
is in operation, a turn should be given on all compression 
cups at least once every 2 hr: 

“Instructions for Starting and Managing Boilers.—See that 
all connections with the boiler are properly made and are 
tight. Fill the boiler up to or above the second gauge with 
water, and take particular notice while it is being filled that 


wore EEE EERE) 


e ” ae 
Wes SINGS ae ke 
2 629 : stosteses 
® 2 
@ 
Fic. 8 


all handhole plates and connections around it are tight; 
particularly note that the check-valve does not leak. Build 
a slow fire in the boiler until the water becomes hot; under 
no circumstances force the fire until after steam begins to 
generate; this can be determined by leaving the top gauge- 
cocks open until steam appears. After about 10 or 15 lb. 
of steam has been raised, note whether there are any slight 
leaks appearing in the boiler or its connections. After steam 
has been raised to the pressure to be carried, try the safety 
valve and be sure that it is in good working order. It is 
advisable to lift the safety valve from its seat at least twice 


356 FIELD OPERATIONS 


a day. Always carry the water in the boiler at a height 
that will best allow the engine to operate without carrying 
over water with the steam. It is always best to carry the 
water-line in the boiler as high as possible. Never allow 
the fire-door of the boiler to be opened except when firing 
the boiler. In checking steam, always close the ash-pit doors 
and damper in the stack; if this is not sufficient to check the 
steam, the fire should be banked. When shutting the boiler 
down at night, under no circumstances allow the fire-door 
to remain open. 

“Starting and Operating the Mixer.—In starting the mixer, 
turn the machine over light a few times, meanwhile setting 
up such runways as may be required. See that the discharge 
chute is in position. Feed into the machine the amount of 
water required for the batch. Follow with stone, sand, 
and cement in the order named. Let the material remain 
in the machine about 4 min., which is long enough under 
average conditions, and then reverse the chute. Then dis- 
charge direct into wheelbarrows, buckets, or other vehicles, 
the whole batch or part, as desired. Reverse the chute and 
feed into the machine the next batch. 

“In securing results as to output, watch the delivery side 
of the machine; get all the material out at once, so that the 
next batch can be mixing. If it is necessary to discharge 
only part at a time, use the largest cart or barrow that can 
be obtained. 

‘‘An occasional inspection of the journal boxes will guard 
against undue wear, which may result in bottoming the main 
gears, with disastrous results to both pinion and spur. Also 
watch that the rollers do not wear down so as to cause 
bottoming.” 


1" 


| 


ea 


ba sah ili 8 


BUILDING LAWS : 357 


BUILDING LAWS, SPECIFICATIONS, 
AND COST 


BUILDING LAWS 


The construction of reinforced-concrete buildings is regu- 
lated by the laws of the municipality in which the work is 
carried on. These building laws differ for each city. The 
following laws are used in the City of Philadelphia. They 
are among the latest and may be considered as a good 
example. 

“The term reinforced concrete shall be understood to mean 
an approved concrete mixture, reinforced by steel or iron of 
any shape, so that the steel or iron will take up all the 
tensional stresses and assist in the resistance to compression 
and shear. 

‘Before a permit to erect any reinforced-concrete structure 
is issued, complete specifications and drawings shall be filed 
with the Bureau of Building Inspection, showing all details 
of the construction, size and position of all reinforcing rods, 
stirrups, etc., and giving the composition and proportions of 
the concrete. 

“The execution of the work shall be performed by work- 
men under the direct supervision of a competent foreman 
or superintendent. 

‘‘Reinforced-concrete construction will be accepted for 
fireproof buildings of the first class, if designed as herein- 
after prescribed; provided, that the aggregate for such 
concrete shall be clean, broken, hard stone, or clean, graded 
gravel, together with clean siliceous sand or fine-grained 
gravel; should the concrete be used for flooring between 
rolled-steel beams, clean furnace clinkers entirely free of 
combustible matter, or suitable seasoned furnace slag may 
be used; when stone is used with sand or gravel it must be 
of a size to pass through a 1-in. ring, and 25% of the whole 
must not be more than one-half the maximum size; and 
provided further, that the minimum thickness of concrete 


358 BUILDING LAWS 


surrounding the reinforcing members of reinforced-concréte 
beams and girders shall be 2 in. on the bottom and 14 in. on 
the sides of the said beams and girders. The minimum 
thickness of concrete under slab rods shall be 1 in. All 
reinforcement in columns to have a minimum protection of 
2 in. of concrete. 

“‘All the requirements herein specified for the protection 
of steel and for fire-resisting purposes shall apply to rein- 
forced-concrete flooring between rolled-steel beams, as well 
as to reinforced-concrete beams and to entire structures in 
reinforced concrete. Any concrete structure or the floor 
filling in same, reinforced or otherwise, which may be erected 
on a permanent centering of sheet metal, of metal lathing 
and curved bars or a metal centering of any other form, 
must be strong enough to carry its load without assistance 
from the centering, unless the concrete is so applied as to 
protect the centering as herein specified: for metal rein- 
forcement. 

“Exposed metal centering or exposed metal of any kind 
will not be considered a factor in the strength of any part 
of any concrete structure, and a plaster finish applied over 
the metal shall not be deemed sufficient protection unless 
applied of sufficient thickness and properly secured, as 
approved by the Chief of the Bureau of Building Inspection. 

“All concrete shall be mixed in a mechanical batch mixer 
to be approved by the Bureau of Building Inspection, except 
when limited quantities are required or when the condition 
of the work makes hand mixing preferable; hand mixing to 
be done only when approved by the Bureau of Building 
Inspection. In all mixing the material shall be measured 
for each batch. 

*“‘When hand mixing is done under the aforesaid limita- 
tions, the cement and fine gravel or coarse sand shall be 
first thoroughly mixed dry and then made into a mortar by 
gradually adding the proper amount of water. The crushed 
stone or gravel shall be spread out to a depth not to exceed 
6 in., in a tight box or upon a proper floor, and be sprinkled 
with water as directed; the mortar is then to be evenly spread 
over the crushed stone, and the whole mass turned over a 


ay ey re 


7 


BUILDING LAWS 359 


sufficient number of times to effect the thorough mixing of 
the ingredients. 

“All forms and centering for concrete shall be built plumb 
and in a substantial manner, made tight so that no part of 
the concrete mixture will leak out through cracks or holes, 
or joints, and after completion shall be thoroughly cleaned, 
removing shavings, chips, pieces of wood and other material, 
and no debris of any kind shall be permitted to remain in the 
forms. All forms to be properly supported and braced in 
a manner to safely sustain the dead load that may be imposed 
upon them during construction. 

“The reinforcing steel shall be accurately located in the 
forms and secured against displacement. 

“Concrete shall be placed immediately after mixing. 

“Whenever fresh concrete joins concrete that is set, the 
surface of the old concrete shall be roughened, cleaned, and 
spread with cement mortar, which mortar shall be mixed in 
proportions of 1 of cement to 2 of sand. 

“Concrete shall not be mixed or deposited in freezing 
Weather, unless precautions are taken to avoid the use of 
materials covered with ice or that are in any other way unfit 
for use, and that further precautions are taken to prevent the 
concrete from freezing after being put in place. All forms 
under concrete so placed to remain unt'l all evidences of frost 
are absent from the concrete and the natural hardening of 
the concrete has proceeded to the point of safety. 

‘‘Concrete laid during hot weather shall be drenched with 
water twice daily, Sunday included, during the first week. 
The broken stone, if hot and dry, must be wet before going 
to the mixer. 

‘The time at which props, or shores, may safely be removed 
from under floors and roofs will vary with the condition of 
the weather, but in no case should they be removed in less 
than 2 wk.; provided, that column forms shall not be removed 
in less than 4 da.; provided further, that the centering from 
the bottom of slabs and sides of beams and girders may be 
removed after the concrete has set 1 wk.; provided, that the 
floor has obtained sufficient hardness to sustain the dead 
weight of the said floor and that no load or weight shall be 


360 BUILDING LAWS | 


placed on any portion of the construction where the said 
centers have been removed. 

“The concrete for all girders, beams, slabs, and columns, 
shall be mixed in the proportions of 1 of cement, 2 of sand or 
fine gravel, and 4 of other aggregates, as before provided. 
The concrete used in reinforced-concrete-steel construction 
must be what is usually known as awet mixture. When the con- 
crete is placed in water it must be placed in a semidry state. 

“Only Portland cement shall be permitted in reinforced- 

concrete constructed buildings. All cement shall be tested © 
in carload lots when so delivered or in quantities equal to 
same, and report filed with the Bureau of Building Inspection © 
before using it in the work. Cement failing to meet the 
requirements of the accelerated test will be rejected. 
_ “Soundness, Accelerated Test.—Pats of neat cement will 
be allowed to harden 24 hr. ip moist air, and then be sub- 
mitted to the accelerated test as follows: A pat is exposed 
in any convenient way in an atmosphere of steam, above 
boiling water, in a loosely closed vessel, for 3 hr., after 
which, before the pat cools, it is placed in the boiling water 
for 5 additional hr. 

“To pass the accelerated test satisfactorily, the pats shall 
remain firm and hard, and show no signs of cracking, dis- 
tortion, or disintegration. 

“Such cements, when tested, shall have a minimum tensile 
strength as follows: Neat cement shall, after 1 da. in moist 
air, develop a tensile strength of at least 150 Ib. per sq. in.; 
and after 1 da. in air and 6 da. in water shall develop a tensile — 
strength of at least 500 Ib. per sq. in.; and after 1 da. in air 
and 27 da. in water shall develop a tensile strength of at 
least 600 lb. per sq. in. Cement and sand tests composed of 
1 part of cement and 3 parts of crushed quartz shall, after 
1 da. in air and 6 da. in water, develop a tensile strength of 
at least 175 Ib. per sq. in., and after 1 da. in air and 27 da. — 
in water shall develop a tensile strength of at least 240 Ib. 
per sq. in. These and other tests as to fineness, set, etc., 
made in accordance with the standard method prescribed by 
the American Society of Civil Engineers, may, from time to 
time, be required by the Bureau of Building Inspection. 


——— 


BUILDING LAWS 361 


“‘Walls.—Reinforced concrete may be used in place of brick 
and stone walls, in which cases the thickness may be two- 
thirds of that required for brick walls, as shown in the 
Schedule, Section 18 of the Act of Assembly, No. 123, of the 
Commonwealth of Pennsylvania, approved June 5, 1901, 
provided the unit stresses as set forth in these regulations 
are not exceeded. 

“Concrete walls in such cases must be reinforced in both 
directions in a manner to meet the approval of the Chief of 
the Bureau of Building Inspection. 

“Steel.—All reinforcements used in reinforced concrete 
shall be of standard grade of structural steel or iron of either 
grade to meet the Manufacturers’ Standard Specifications, 
revised February 3, 1903. 

“‘Reinforced-concrete slabs, beams, and girders shall be 
designed in accordance with the following assumptions and 
requirements: 

-“*(q) The common theory of flexure to be applied to all 
beams and members resisting bending. 

‘*(b) The adhesion between the concrete and the steel 
is sufficient to make the two materials act together. 

‘*(c) The design shall be based on the assumption of a 
load four times as great as the total load (ordinary dead load 
pius ordinary live load). 

‘‘(d) The steel to take all the tensile stresses. 

‘“‘(e) The stress-strain curve of concrete in compression 
is a straight line. 

“(f) The ratio of the moduli of elasticity of concrete to 
steel: 


Stone or gravel concrete............... 1 to12 
PERE CONCTORO oid 2 eV ati sins waa mce as el AO RO 
Caner comerete:. reer kalau 6, 2y0, 2 EOL OU 


“The allowable unit pe ER, eas upon concrete in 
compression: 
Ls. PER Sg. IN. 


Be eee re ee ane a 600 
SNEED 5 ial cas: 2 dh cca: do eel eaek Uwe eae 
Cloder concrete:. ee Juaie speieboee- 6: LOO 


“The allowable unit etieus in a Peer 


362 BUILDING LAWS 


LB. PER So. IN. 
BLOG Oto wewcers cale slate ie DG ak Me se 
Steel. . me od wey, be 
“The owalsle anit ahecseiner nivanoth upon concrete: 
Ls. PER Soa. IN. 


Stone or gravel concrete......)...6..5225.5 70 
SIAG Concrete cc .ise if o.5-1i)4 bie ve ae ee De 
Cinder chinteatins V5. wee ee 
“The allowable unit adhesive aioe of ennenete 
Ls. PER So. IN. 
Sisnb or-gravel concrete: ¢ sissies bbe d soled Oo 
Sigg conereten.. Gy sawinaee Sod sees phen 
Cinder sewichehe 2 £0 oie sacked Sracceyes 


“The allowable unit Kiceiman fates contirete in direct com- 
pression in columns: 
Ls. PER Sg. In. 


Stone or gravel concrete.................. 500 
Slag concrete. . Lotte, See ae ial, Amal . 800° 
Cinder puaibtete.; fee's Kis . 150 


‘The allowable unit pana pera habe: colienie composed of 
stone or gravel concrete shall not be over 1,000 Ib. per sq. in., 
figuring the net area of the circle within the hooping. The 
percentage of longitudinal rods and the spacing of the hoops 
to be such as to permit the concrete to safely develop the 
above unit stress with a factor of safety of 4. 

‘“When steel or iron is in the compression sides of beams, 
the proportion of unit stress taken by the steel or iron shall 
be in the ratio of the modulus of elasticity of the steel or iron 
to the modulus of elasticity of the concrete; provided that 
the rods are well tied with stirrups connecting with the lower 


rods of the beams; provided, further, that when rods are used 


in compression, the approval of the Chief of the Bureau of 
Building Inspection must be obtained. 

“In the design of structures involving reinforced-concrete 
beams and girders, as well as slabs, the beams and girders 
shall be treated as T beams, with a portion of the slab acting 
as flange in each case. The portion of the slab that may be 
used to take compression shall be dependent upon the hori- 
zontal shearing stress that may exist in the beam, and in no 


a 


a ST Pn, ear rt 


BUILDING LAWS 363 


case shall the slab portion exceed twenty times the thickness 
of the slab. 

“‘All reinforced-concrete T beams must be reinforced 
against the shearing stress along the place of junction of the 
rib and the flange, using stirrups throughout the length of 
the beam. When reinforced-concrete girders carry rein- 
forced-concrete beams, the portion of the floor slab acting 
as flange to the girder must be reinforced with bars near the 
top, at right angles to the girder, to enable it to transmit local 
loads directly to the girder and not through the beams, thus 
avoiding an integration of compressive stresses due to 
simultaneous action as floor slab and girder flange. 

‘In the execution of the work in the field, work must be so 
carried on that the ribs of all girders and beams shall be 
monolithic with the floor slabs. 

‘In all reinforced-concrete. structures special care must be 
taken with the design of joints to provide against local stresses 
and secondary stresses due to the continuity of the structures. 

‘‘Shrinkage and thermal stresses shall be provided for by 
the introduction of steel. 

‘In the determination of bending moments due to the 
external forces, beams and girders shall be considered as 
simply supported at the ends, no allowance being made for 
continuous construction over supports. Floor slabs, when 
constructed continuously, and when provided with rein- 
forcement at top of slab over the supports, may be treated 
as continuous beams, the bending moment for uniformly 


WL 
distributed loads being taken at not less than a0. In the 


case of Square floor slabs, that are reinforced in both direc- 
tions and supported as continuous beams on all four sides, 


WL 
the bending moment may be taken at "ag" In floor slabs in 


juxtaposition to the walls of the building, which are acting 
as simple beams, the bending moment shall be considered 


‘- 
as ae when reinforced in one direction. If the floor slab 


is square and reinforced in both directions, but acting as a 


364 COST DATA 


WL 
simple beam, the bending moment shall be taken as Te" 


‘“When the shearing stresses developed in any part of a 
reinforced-concrete building exceed the shearing strength as 
fixed in this section, a sufficient amount of steel shall be 
introduced in such a position that the deficiency in the 
resistance to shear is overcome. 

‘*When the safe limit of adhesion between the concrete and 
steel is exceeded, provision must be made for transmitting 
the strength of the steel to the concrete. 

“Reinforced concrete may be used with the unit stresses, 
previously given, for columns in which the ratio of the length 
to least side or diameter does not exceed fifteen. If. more 
than fifteen diameters, the allowable stress shall be decreased 
proportionally. Reinforcing rods that are introduced for 
lateral stresses must be tied together at intervals of not more 
than the least side or diameter of the columns. 

“Longitudinal reinforcing rods will not be considered as 
taking any direct compression. 

‘*The contractor must be prepared to make load tests in any 
portion of a reinforced-concrete building within a reasonable 
time after erection, and as often as may be required by the 
Chief of the Bureau of Building Inspection. The tests must 
show that the construction will sustain a load equal to twice — 
the calculated live load without signs of cracks.”’ 


COST DATA 


The cost of reinforced-concrete construction depends on 
the local cost of labor and materials. In making an approxi- 
mate estimate of the cost of a building, unit prices per cubic 
yard or per square foot of floor surface may be used. These 
prices are obtained from the cost of similar buildings already 
erected in the same locality. In making an exact estimate of 
the cost of a building, the materials can usually be accurately 
determined and their cost found from the current market 
prices. The cost of labor, including the placing and mixing 
of concrete, and the cost of form construction are usually 
more difficult to determine. The latter cost is greatly 


ey are 


COST DATA 365 


influenced by the amount of experience local seresires have 
had in this class of work. 

In assuming approximate unit prices for pe A et con- 
crete, it is customary to include in the estimate the cost of 
the forms; thus, the cost of reinforced-concrete floor and 
wall construction per cubic yard will usually be between 
$16 and $20. This price includes the steel reinforcement, 
the cost of the labor on the forms, and the cost of the con- 
crete. This cost per cubic yard increases as the thickness 
of the wall decreases. 

The cost of reinforced-concrete floor systems can be 
estimated at a square foot. The total cost per square foot 
of floor, including the steel reinforcement, form work, etc., 
will usually be found to vary between 60 and 65 cents. 

It must be understood that the preceding figures are 
subject to considerable changes in different sections of the 
country. They indicate only that such costs can be obtained 
fairly accurately for certain localities, for a period of 1 or 
2 yr. 

Sometimes the cost of a building is estimated per square 
foot of floor surface, this cost including also the cost of the 
walls, etc. Thus, if it costs in a certain locality $1.50 per 
square foot of floor to build a concrete mill building, the 
approximate cost on a proposed building two stories high, 
50 ft. wide, and 200 ft. long, can be found as follows: 
The floor space on each floor is 200 X 50=10,000 sq.ft. For 
the two floors, the surface is 20,000 sq. ft., and at $1.50 per 
square foot, the price is 20,000 x $1.50 =$30,000. This cost 
is for the building complete. It includes ordinary plumbing, 
painting, etc., such as would be necessary-.for a factory 
building. If the building were more ornamental, the unit 
cost would be taken higher—at perhaps $2 per square foot, 
or whatever the current figures in the same locality hap- 
pened to be. 

In making an exact estimate of a reinforced-concrete 
building, the following suggestions will be of interest: 

The principal steel reinforcement is carefully estimated. 
For most styles of patented bars or round and square bars, 
the manufacturers give weights that correspond to various 


366 COST DATA 


sections, as steel reinforcement is bought by the pound. 


Tables of these weights will be found on pages 253 to 265. — 


If such weights cannot be found in manufacturers’ catalogs 
or elsewhere, the following rule may be used: 

Multiply the sectional area of the steel in square inches by 
its length in feet and by 3.4. The product will be the weight 
-of the steelin pounds. Usually it is not necessary to estimate 
the quantity of steel used for stirrups, as 10% of the main 
reinforcement is a close approximation. This 10% includes 
such additional steel as would be used for column ties, 


wiring, etc. It should be noted that the different sizes of — 


steel should be kept separate, as in most localities the price 
of steel per pound increases as the size of the bar decreases. 

It is not necessary in estimating the volume of concrete 
to deduct the volume taken up by the steel reinforcements. 
After the volume of concrete has been obtained, the amount 
of sand, cement, and broken stone can be estimated from 
the table on page 244. The cost of mixing concrete is in the 
neighborhood of 30 cents per cu. yd., using labor at $1.50 
per da. In localities where the price is higher, this figure 
must be increased. Machine-mixed concrete costs from 
50 to 60% as much as hand-mixed concrete. 

The entire cost of mixing, wiring, and placing, including 
foreman’s wages, in a large eastern city amounted to $1.30 


per cu. yd. However, this value must be determined for — 


every different locality having different scales of wages. 


Sand is usually bought by the ton, but measured by the © 


cubic yard. One long ton is equal to about 1 cu. yd. of dry 
sand. The cost of steel reinforcement varies in different 
localities, and no exact figures can be given. Usually, if 
it is of the girder-frame type, a man can place in the forms 
about 1,600 Ib. of reinforcement in a day. If it consists of 


loose rods, one man can place from 800 to 1,200 Ib. per da. © 


The cost of form work cannot be given in general terms. 
The estimator must know the quality of forms to be made, 
and measure up the approximate amount of wood required. 


If possible, the forms should be used two or three times on 


the same job, and they should have a certain selling price 
after the work is completed. 


ee ee ee ae 


RP Ngee > om 


SPECIFICATIONS 367 


\ 


The amount of carpenter work on the forms is variable, 
depending on the experience of the carpenters and the 
intricacy of the work. USually, a carpenter can erect 
20 lin. ft. of column form in a day, or 40 sq. ft. of floor and 
girder form. These figures are for smooth, finished work, 
and not for rough work. 


SPECIFICATIONS 


CONDENSED SPECIFICATIONS FOR REINFORCED- 
CONCRETE WORK 


The following specification is the briefest form that can 
be used in specifying reinforced-concrete construction. It 
briefly states the requirements of the architect regarding 
the submission of the construction for approval, the char- 
acter of the concrete and the steel reinforcement, and the 
floor load upon which the calculation for the design of the 
floor system is based. 


REINFORCED-CONCRETE CONSTRUCTION 


General.—The reinforced-concrete construction shall be 
of an approved and successful type, and the contractor shall 
submit for the architect’s approval framing plans, and 
details of construction, together with schedules showing the 
amount of steel reinforcement in all the beams, girders, and 
columns, 

The contractor is to assume all responsibility for the safety 


‘and protection of the work during construction, and shall 


deliver the same in a complete and finished condition in 
compliance with the specifications and the accompanying 
plans. 

Concrete.—The concrete for beams, girders, and slabs shall 
be proportioned of 1 part Portland cement, 24 parts sand, 
5 parts broken stone or gravel. The concrete for columns 
shall be a 1-2-4 mixture. 

Cement.—The cement shall conform to the requirements 
of the standard specifications for cement adopted by the 
American Society for Testing Materials, on November 14, 
1904. 


368 SPECIFICATIONS 


Aggregates.—The sand shall be clean, and not contain over 


3% of loam. Broken stone and gravel shall be hard and 
close-grained, and free from dust and dirt; they shall be of 
such size as to pass through a ring 1 in. in diameter. 
Reinforcing Steel.—The steel for the reinforcement shall 
be manufactured by the open-hearth process. This steel 
should have an ultimate tensile strength of from 60,000 to 
70,000 Ib. per sq. in. and an elastic limit of at least half that 
amount, with an elongation of at least 20%. A bar shall 
bend cold through an angle of 180° and close down on itself 


without showing signs of fracture. High-carbon steel or ¥ 


steel with an elastic limit greater than 45,000 Ib. per sq. in. 
shall not be used. 

The beam and girder reinforcement shall consist of square- 
twisted or approved type of deformed bar, while the slab 
reinforcement may be of square-twisted bars, deformed bars, 
or of an approved form of expanded metal or woven or 
electrically welded wire fabric. The column rods may be 
of plain round or square-twisted bars. The girders and 
beams must have the rods so bent and arranged as to provide 
amply for the bending moments throughout their length and 
also to take care of the negative bending moments near the 
bearings and supports. Sufficient stirrups must be provided 


tals een 


near the bearings or throughout the length of the beams and ~ 


girders to resist the horizontal, oblique shearing, and tensile 
stresses. 
Floor and Test Loads.—The floors and columns throughout 


shall be designed to sustain a live floor load of 150 Ib. per ~ 
sq. ft. The finished floor systems shall be tested in two © 
places, to be designated. The test load shall be equal to © 


one and one-half times the amount of the live floor load. 


The test loads shall be so placed as to fully load the girders © 


and beams of one bay. 


MEMORANDA 


MEMORANDA ~~ 


MEMORANDA 


MEMORANDA 


MEMORANDA 


MEMORANDA 


MEMORANDA 


MEMORANDA 


Promotion 
Advancement in Salary 


-@ Business Suceess = 


Secured 
Through the 
Concrete Engineering, 
Complete Architectural, Architec- 
tural Drawing and Designing, 
Building Contractors’, 
and Structural Engineering 


COURSES OF INSTRUCTION 


OF THE 


International 
Correspondence Schools 


International Textbook 
* Company, Proprietors 


SCRANTON, PA., U.S.A. 


& ~) 
oo . SEE FOLLOWING PAGES 


Superintendent of Construction 
of Million-Dollar Concrete 
Building 


There is now being built in San Bernardino, 
Cal., a great precooling plant for citrus fruits 
that will cost one million dollars. The 
superintendent of construction in charge of 
this plant, which is built of reinforced con- 
crete, is Wm. L. Snook, 1643 W. Thitty-Fifth 
Street, Los Angeles, Cal., an I. C. S. trained 
man, Mr. Snook states: “For the last 
3 years I have been superintendent of con- 
struction of reinforced-concrete buildings. 
At the present time I am superintending the 
construction of the largest ice and precooling 
plant of its kind in the country.”’ 


Many Opportunities for 
Concrete Engineers 
WANTED — COMPETENT, EXPERIENCED 


reinforced concrete engineer and designer; 
only firct-class man need apply; good salary and 
steady position with large responsible firm, Ad- 
dress immediately Builders’ Material Supply 
Co /> : 


Kansas City, Mo. 


WANTED—A ca 
Who can 

gonerete. torms. fons i 
: [) . Apply 

ITT CO.; ‘Macedon, 


Address, with references, » 
Apected, No. 3871, care Engi- 


emonsl 


Se 


Field Engineer in United 
States Service 


When I enrolled I was doing odd jobs 
wherever I could get work to do. Through 
study of my Course I have filled important 
positions in the United States Reclamation 
Service, Inspector in the General Land Office, 
and other United States Government posi- 
tions, ranging in salary from $75 to $200 a 
month. I am now Field Engineer with the 
United States Reclamation Service in charge 
of the location and construction for the 
irrigation of about 25,000 acres of land on 
the Payette-Boise Project. 

I am enrolled for the Concrete Engineering 
Course. The Schools have done wonders for 
me and I highly recommend them to any one 
who wishes to learn and advance and be 
worth more to their employers. 

W. W. PEFLEy, 
215 N. 14th St., Boise; Idaho 


PASSED CIVIL SERVICE EXAMINATION 


H. W Tuorne, 1379 E. Fifty-Third St., Los Angeles, 
Cal., enrolled for the Architectural Course while employed 
as a carpenter, earning $3.50 a day. After studying, he 
gradually advanced to a high class of work and was able to 
do his own calculating. Later he took the civil service 
examination for deputy building inspector and received the 
highest average ever given in Los Angeles. He was able to 
answer the technical questions in the examination by means 
of the knowledge acquired from his Course. 


HIGH-SCHOOL STUDENT BECOMES DRAFTSMAN 


Oscar GEORGE KNEcHT, 1245 E. 5th St., Kansas City, Mo., 
enrolled for the Architectural Drawing Course in December, 
1905, when 21 years of age, while attending high school and 
working as a baker’s helper in the afternoons and evenings. 
He was then earning $6 a week. After completing a small 

art of the Course he secured a position with an architect 
y showing some of his ei np and was steadily advanced 
until he received $15 a week. e was then offered a position 
as architectural draftsman with Clark-Williams Company, 
contractors and builders, at a salary of $20 a week, which 
ition he now holds. Ina letter dated December 6, 1907, 

r. Knecht says: ‘‘My opinion of your method of instruc- 
tion is that it is a very good one. I prefer it to a college 
course because, if I were under a personal instructor I would 
have him sg solve every difficult problem and would not 
be studying them out for myself.” 


BECAME DESIGNER AND BUILDER 


L. W. Eppy, 1801 Sunnyside Ave., Fruitvale, Cal., had 
been out of the United States Navy _but a short time, and 
was working as ac nter’s helper at $2 a day, when he 
enrolled in the I. C. S$. Building Contractors’ Course. Mr. 
Eddy is now a designer and builder, earning $6 a day. He 
says this was through the help of the Course, especially in 
concrete and stair-building work. 


INCOME INCREASED $500 A YEAR 


Cuar_Les E. WERKING, Hagerstown, Ind., a foreman car- 
nter, enrolled for the I. C. S. Complete Architectural 
ourse. He says, “I never entered into anything I enjoyed 
more and gained so much.’’ Mr. Werking is now an archi- 
tect, with income increased from $750 a year to $1,250 


a year. 
FROM $6 A WEEK TO $25 A WEEE 
Harry Crossy, Hyde Park, Cincinnati, Ohio, was earn- 
ing but $6 a week when he enrolled for the Complete Archi- 
tectural Course. It was hard for him to persevere because 
of his circumstances. But he has studied until now he is 
an architectural draftsman earning $25 a week. 


5 


Now Earns $300 a Month 


M. P. Kellogg, of San Diego, Cal., enrolled 
for and studied in the I. C. S. Complete Archi- 
tectural Course. For 16 years he had been 
a carpenter and contractor. After studying 
his Course he began the practice of architec- 
ture. Mr. Kellogg has won several prizes, 
over many competitors, for the excellence of 
his designs. 

In a letter to us regarding his progress 
Mr. Kellogg says: ‘‘My success is due to 
the instruction received from the Inter- 
national Correspondence Schools. I am well 
satisfied with your system of teaching; in fact, 
I think it as good as personal instruction. I 
consider my Scholarship the best investment 
I ever made—it has been worth several times 
the cost to me. My Bound Volumes have 
been especially helpful. I shall be glad to 
answer any letters regarding the Schools and 
my Course with them. - My business now nets 
me about $300 a month.” 


MACHINIST TO STRUCTURAL DESIGNER 


Epmunp B. La Sate, Batavia., Ill, was employed as a 
machinist when he enrolled in the Structural Engineering 
Course of the I. C.S. He was ambitious, of course, to become 
a structural draftsman and designer, but the numerous books 
he bought were found to be a waste of money. After studying 
the Course, Mr. La Salle obtained work as a draftsman. 
Recently, he has accepted a position in Chicago as structural 
designer, at a salary of $130 a month. He designs standpipes 
and steel tanks; also towers for water-supply stations. 


HIS COURSE BROUGHT RAPID INCREASE 


ErNEstT N. WoopBECK, 133 Crawford Ave., Detroit, Mich., 
was learning the business of carpenter for reinforced concrete 
work when he enrolled with the I. C. S. for a Concrete Engi- 
neering Course. His studies made it possible for him to 
become layerout of work from building plans. He had only 
a seventh grade education when he enrolled, but in eighteen 
months he increased his salary nearly 40 per cent. 


NOW OWNS A PROSPEROUS BUSINESS 


Won. F. GLeason, 709 Crawford Ave., Augusta, Ga., was 
earning $2.50 a day as a carpenter when he enrolled for the 
Complete Architectural Course with the I.C.S. This enabled 
him to become superintendent of construction for a concrete 
building block and tile company. He has since established 
@ prosperous business of his own as a member of the Skinner 
& Gleason Contracting Company, builders, making a specialty 
’ of concrete construction and brick work. His income is at 
least $2,000 a year. 


TEMPORARY EMPLOYMENT MADE PERMANENT 


Harry L. ACKERLEY, 3 Richard St., West Lynn, Mass., 
says: ‘‘I first enrolled for a Course in Architecture. After 
taking the preliminary studies I applied for a position as 
draftsman and was given ‘temporary’ employment. I 
remained with the concern four years. I then enrolled for 
an Electrical Engineering Course, which has been invaluable 
to me. I am now employed by the General Electric Com- 
pany and my salary has been increased 350 per cent.” 


SALARY INCREASED 400 PER CENT. 


Mitton S. GRAWL, 1803 Erie Ave., Philadelphia, Pa., at 
the time of enrolling in the Architectural Drawing Course, 
was working in a meat store. The Course started him on 
the road to success. Today he is chief draftsman for a steel 
company in Philadelphia, with 400 per cent. increase in income 
over his salary as clerk. 


7 


Won Prizes in Five 
Contests 


The fact that my house plans have won 
prizes in five different contests argues well for 
the Course I took with the International 
Correspondence Schools. 

My early education was derived from the 
public schools. At the age of 15 I built a 
rowboat so well that I was advised to take 
up carpentering. I did so, serving my term 
with Henry Lorenz, of Matamoras, Pa., and 
working with the builders around Port Jervis. 
Then I decided to go into the building con- 
tracting business for myself, and knowing 
that a knowledge of architecture and drafting 
was necessary, I enrolled with the I. C. S., 
studying evenings, rainy days, and odd times 
until I earned my Diploma. I now do con- 
siderable planning and drafting, and I am 
satisfied that I will soon be able to give up 
all other work. The Course was a great help 
to mein many ways. I say that no mechanic 
has any chance to advance these days unless 
he takes up a good course of study along the 
line of his trade, such as the International 
Correspondence Schools give. 

C. A. WAGNER, 
Port Jervis, N. Y. 


} 


NOW OWNS A CONCRETE FACTORY 

Not being able to afford a college education, Cuas. F. CuLP, 
New Rockford, N. Dak., enrolled with the I. C. S. for the 
Complete Architectural Course, paying for it with the first 
money he earned after becoming of age. At the time, his 
wages as a carpenter were only $1.25 a day. He is now a 
presperces contractor and builder, being also the owner of a 
actory for the making of cement products. He attributes 
his success very largely to his work with the I. C. S. 


CARPENTER TO $10-A-DAY DRAFTSMAN 

ELMER B. Cutts, Hinckley, Ill., was working for his father 
as enter, earning 174 cents an hour at hard labor, when 
he took up a course of study in the I. C. S. The knowledge 
gained from the Course as placed him in the position of chief 
draftsman for the Robert Cutts Construction Company. He 
prepares plans for various dwellings, hotels, churches, and 
public buildings, earning $10 a day. 


NOW SUPERINTENDENT 

FRANK FREE, Greenfield, Ohio, was a common éarpenter 
employed some eight months a year at $2.70 a day, when he 
enrolled for the Complete Architectural Course, and after- 
ward for the Concrete Engineering Course. He is now super- 
intendent of construction for Samuel Hannaford & Sons, 
Cincinnati, Ohio, engaged on reinforced concrete and other 
construction. His aslary is $1,500 a year. 


NOW AN ARCHITECT 
Epwarp THAL, 702 Ohio Building, Toledo, Ohio, was earn- 
ing $18 a week as a paper hanger and painter when he enrolled 
with the Schools for the Complete Architectural Course. He 
is now an architect and engineer, making a specialty of rein- 
forced concrete, with an income several hundred per cent. 
larger than when he enrolled. 


NOW IS STRUCTURAL DRAFTSMAN 
C. B. GitBertT, care Philadelphia Engineering Compeanss 
Philadelphia, Pa., prior to becoming a student in the I. C. S. 
Structural Engineering Course was general utility man in 
a shoe factory, with little or no chance to rise. Before he 
had completed his I. C. S. Course he secured a position as 
structural draftsman with the Niles-Bement-Pond Company. 

His prospects are bright and his income is greater. 


CARPENTER TO CONTRACTOR 
FRANK J. BERDEL, 333 Columbus Ave., Canton, Ohio, 
writes: ‘‘I enrolled for the Building Contractors’ Course 
three years ago while employed as a carpenter, receiving 
$2.50 a day for 10 hours’ work. After studying evenings for 
one year I started contracting. Though but 25 years old my 
income is $12 a day.” 


9 


Now Reinforcement 
Contractor 


I am a pretty busy man these days, but 
I must send you a little message of appreciation 
concerning your Concrete Engineering Course. 
Two years ago I was a machinist’s helper in 
a railroad shop, earning 23 cents an hour. 
But I was ambitious and when your repre- 
sentative approached me I soon saw that I 
would make the mistake of my life if I did not 
sign up for a Course. I found your Concrete 
Engineering Course a veritable gold mine. 
I immediately secured work on a reinforced 
concrete building. Before this was completed 
I was tendered the position of steel foreman 
on a large reinforced concrete building in 
Portland, at just double my former salary. 
I am now a reinforcement contractor, bending, 
fabricating and placing steel reinforcement 
in all classes of reinforced concrete construc- 
tion. I have now four large contracts under 
way. Iam thankful for the fact that a repre- 
sentative of the I. C. S. yanked me out of my 
hole and pointed out a way for me to stay out. 

C. G. WiILForD, Jr., 
147 Front St., 
Portland, Ore. 


10 


BETTERED HIS POSITION AND SALARY 


Cuas. A. CARPENTER, 16 West 3d St., Fulton, N. Y., was 
_a steam fitter, earning $65 a month, when he enrolled with the 
Schools, at the age of 5. for a Mechanical Drawing Course and 
afterward for the Mechanical Engineering Course. He is 
master mechanic of reinforced concrete work for the Walter 
Bradley Concrete Company. His Courses have enabled hi 

to design several new machines, one of which saves the com- 
pany 25 per cent. of the cost of production. His salary has 
anhargi about 50 per cent. since he began to study his first 

ourse. 


FROM BRICKLAYER TO GENERAL CONTRACTOR 


W. C. Davis, Michigan Building, Wichita, Kans., was 
working as a bricklayer, with no prospect of advancement, 
when he enrolled with the Schools for the Complete Archi- 
tectural Course. After successfully superintending the con- 
struction of a large warehouse he became a member of the 
H. I. Ellis Construction Company, employing from 50 to 
100 men. He says that his Course enables him to superintend 
the field work of the company, especially in the line of masonry 
and concrete. His income has increased at least 300 per cent. 


PASSED A SUCCESSFUL EXAMINATION 


Otis M. TOWNSEND Eighth St., Ocean City, N. J., was an 
72 ada boy when he enrolled with the Schools’ for the 

mplete Architectural Course. He is now an architect and 
ogee employing 40 carpenters and keeping a draftsman in 
his office. 27 Aig ean enabled him to pass the state exam- 
ination On and receive a certificate as an architect. 


SALARY INCREASED 400 PER CENT. 


When C. M. Maynarp, 810 Belleville Ave., New Bedford, 
Mass., enrolled for the Building Contractors’ Course he was 
a carpenter, earning $1.75 a day. He gives the I. C. S. the 
credit for establishing him as a contractor and builder and 
increasing his income about 400 per cent. 


FROM POVERTY TO PROSPERITY 


H. L. Tuomas, Englewood, Colo., enrolled in the I. C. S. 
while out of work, without money, and with a family of six 
to support. Just one year after starting the I. C. S. Course 
he secured a position as draftsman, at $55 a month. Soon 
he got a better position at $75 a month, laying out work, 
and making plans and designs, all of which he could do without 
instruction other than that received from the Course. At 
present Mr. Thomas is with the Denver Union Water Company, 
as structural engineer. 


i 


Fireman Becomes 
Architect 


Michael O’Connor, Kansas City, Mo., a fire- 
man, became ambitious to learn architecture. 
He enrolled for the I. C. S. Complete Archi- 
tectural Course and studied during his spare 
time. Soon he drew plans for a $5,000 
dwelling and they were accepted. Then he 
tried again with equal success. Now the 
self-taught architect is on the way to fortune. 
His city is proud of him. In an article about 
his career the Kansas City Star says: 

“The Kansas City fireman that has 
equipped himself for the profession of archi- 
tect by studious application between alarms 
furnishes an example of applied ambition 
that has become rather rare in these days of 
larger privileges. In olden times, when oppor- 
tunities for learning were few, there were more 
instances of such fruitful self-help. In fact, 
many of the great men of American history, 
several Presidents among them, gained their 
learning by great sacrifice. The Kansas City 
fireman has used only time that many men 
similarly placed have wasted. He has fitted 
himself for one of the noblest of professions. 
He has furnished a ‘modern instance’ worthy 
of emulation.” 


12 


BRICKLAYER BECOMES FOREMAN, AT $7 A DAY 


A. Rousseau, Hardin, Mont., was an ordinary bricklayer, 
earning $2.50 a day, when he enrolled for the Building Con- 
tractors’ Course. The studies enabled him to draw complete 
plans for his work and for outside work. He is now foreman, 
earning $7 a day. 


HILTON & SADLER, ARCHITECTS 


L. L. H1LTon and F. E. SADLER, Architects, Janesville, Wis., 
were carpenters when they enrolled in the I. C. S. Complete 
Architectural Course. Now they, as a firm, occupy a position 
of honor in Janesville. They have designed and superintended 
the building of many schools, churches, factories, and other 
large buildings in their own and in surrounding cities. 


CARPENTER BECOMES FOREMAN 


J. F. HARRINGTON, Missoula, Mont., before enrolling in the 
I. C. S., was a journeyman carpenter, earning $4.50 a day. 
The Course of study taught him to understand building plans 
= Se he has secured the position of foreman and earns 

a day. 


TEN TIMES HIS FORMER SALARY 


When the National Convention of Carpenters met in Scran- 
ton some years ago, JOHN G. GARBART, 16 Verner Avenue, 
Ingram, Pa., visited the I. C. S. Instruction Department. 
What he saw convinced him that he needed a Course and he 
enrolled in our School of Architecture. He was then earning 
$1.50 a day. His Course has enabled him to purchase the 
business of his former employer and he now earns ten times 
what he did when he enrolled. 


ANOTHER CARPENTER BECOMES A BUILDING 
CONTRACTOR 


CHARLES F. STENGEL, Jefferson, Wis., was a carpenter when 
he started to study in the I.C.S. Asa result of partly finish- 
ing the Complete Architectural Course he is in business with 
a friend, doing two-thirds of the building work in his vicinity. 
His business is mre larger every year and he says he owes 
his success to the I. C. S. 


NOW EARNS $5,000 A YEAR ~ 


Ep. J. WEAVER, Youngstown, Ohio, considers that the work 
he did with the I.C. S. in his two Courses, Architectural Draw- 
ing and Complete Architectural, are the foundation of his 
success in his chosen profession. When he enrolled he was 
a machine hand in a planing mill, earning $2 a day. He is 
now a practicing architect, and has earned $5,000 a year for 
several years past. 


13 


Salary Increased 300 
Per Cent. 


When I began my Course in the Inter- 
national Correspondence Schools I was em- 
ployed as wheelwright at the Rodenhausen 
Wagon Works, Philadelphia, at a salary of $8 
a week. JI am nowin business for myself as 
carpenter and builder. Since enrolling with 
your institution I have been so benefited that 
my salary has increased over 300 per cent. 
Your method of instruction is clear, com- 
prehensive, and direct in its meaning. It is 
my opinion that there is no better way for a 
workman to receive an education, without 
losing time from his work, than to enroll with 
the International Correspondence Schools. 

I had attended no school since 1897, until 
I enrolled with the I. C. S., December 19, 
1900. My first Course was Mechanical Draw- 
ing. In May, 1904, I began working on the 
Complete Architectural Course, from which 
I have derived great benefits for the position 
I now hold. 

Please note that I fully realize and appre- 
ciate the interest you have taken in my 
Course. Trusting that other workmen will 
enroll in an I. C. S. Course and be benefited 
as I have been, I am, 

Henry A. Hosert, 
2620 E. Cumberland St., 
Philadelphia, Pa. 


14 


Erected the Largest Rein- 
forced Concrete Build- 
ing in the World 


Erik Holman, Room 657, Pacific Building, 
San Francisco, Cal., had worked as a carpenter 
and contractor on a small scale for some years 
before he enrolled with the I. C. S. for the 
Complete Architectural Course. Having suc- 
cessfully superintended the erection of the 
first large reinforced concrete building in Los 
Angeles, he was called upon to erect the Pacific 
Building, corner of Market and Fourth Sts., 
San Francisco, Cal., costing $1,500,000, said 
to be the largest reinforced concrete office 
building in the world. He is now a consulting 
engineer, with an average income of $300 a 
month. 


15 


From the New Jersey 
State Board of 
Architects 


The New Jersey State Board of Architects 
in its second annual report, 1904, in recom- 
mendations under the heading of ‘‘ Books for 
Study,’’ makes the following statement indors- 
ing the International Correspondence Schools’ 
Courses of study: 

“The Board having received numerous 
inquiries concerning the extent of the prep- 
aration necessary to fif one to engage in an 
examination for a certificate to practice archi- 
tecture in the State, it has been thought best 
that the following information be issued. 

“The Boa d recommends, where it is pos- 
sible, that the student complete a full course 
in architecture at some well-known university 
or other institution of learning. A diploma 
of graduation supplemented by a proper 
amount of experience is sufficient evidence to 
the Board that the applicant for a certificate 
is worthy of one and that the formal exami- 
nation may be waived. 

“The Board realizes that to many this is 
impossible and to such it can recommend the 
Courses of study conducted through the mails 
by the International Correspondence Schools, 
Scranton, Pa. Any one who has taken one 
of these Courses will be well prepared to pass 
the Board’s examination.” 


16 


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