UNIVERSITY OF CALIFORNIA
ARCHITECTURAL DEPARTMENT LIBRARY
GIFT OF
Mrs* George Beach
AN
ELEMENTARY COURSE
CIVIL ENGINEERING
FOB THE DSJt OJ"
CADETS OF THE UNITED STATES MILITARY ACADEMY.
BY
J. B. WHEELEK,
of Civil and Military Engineering in tAe United State* Military Acadcm*
Wett Point, N. r., and Brevet-Colonel U. S. Army.
FIFTH EDITION.
THIRD THOUSAND.
NEW YORK:
JOHN WILEY AND SONS,
53 EAST TENTH STKEET,
A
CkjPYRIOHTKD, 1878, B
JOHN WILEY & SONS
Press of J. J. Little & Co,
Astor Place, New York.
PREFACE.
THIS text-book is prepared especially for the cadets of the
United States Military Academy, to. be, used while pursuing^
their studies in the course of Civil Engineering laid down for
them.
The object of the book is to state concisely ffie principles of
the science of Civil Engineering, and to illustrate these prin-
ciples by examples taken from the practi.de and writings of
civil engineers of standing in their profession.
These principles and facts are widely known and are famil-
iar to all well-informed engineers ; they will, however, be new
to the beginner.
The present edition differs slightly from the one that has
been used for the past seven years. The modifications in the
text are simply those that have been suggested by the use of
the book in the class-room. The differences between the two
editions are, however, not sufficiently great to prevent a simul-
taneous use of both old and new in the same class.
J. B. W.
WEST POINT, N. Y., July, 1884.
812397
CIVIL ENGINEERING.
CONTENTS.
INTRODUCTION . xvii
PART L
Building Materials.
CHAPTER L WOOD.
4BTICLE PAGH
2. Timber, kinds of 1
3. Timber trees, structure of 2
4. Timber trees classed 2
5-9. Soft-wood trees, examples of 3
10-12. Hard-wood trees, examples of 4
13. Age and season for felling timber 5
14. Measurement of timber. 6
15. Appearances of good timber 6
16. Defects in timber 7
17. Seasoning of timber natural and artificial 7
18. Durability and decay of timber wet and dry rot 8
19-24. Durability under certain conditions and means of increasing it. 9
25. Preservation of timber in damp places 10
CHAPTER II. STONE.
26. Qualities requisite in stone for building 13
27. Stones classed natural and artificial 14
L NATURAL STONESI
28-31. Remarks on the properties of natural stone strength, hardness,
and durability 14
3&-34. Effect of heat and cold on stone 15
35. Preservation of stone 17
36. Ease of working stone 17
37. Quarrying 18
Varieties of Building Stones.
38. Silicious stones 18
89. Argillaceous stones 20
40-42. Calcareous stones, marbles, common limestones 21
IL ARTIFICIAL STONES.
1. Brick.
43. Brick 23
44. Sun-dried brick. 23
45. Burnt brick 23
CONTENTS. VU
46-49. Common brick size and manufacture 24
50. Qualities and uses of brick 26
51. Characteristics of good brick 26
52. Varieties of common brick 26
53-54. Pressed and fire bricks 27
55. Brick-making as one of the arts 27
56. Tiles. 27
2. Concrete*.
57-59. Concrete, its composition, manufacture and uses 28
60-62. Patent stones Beton Agglome're, and Ransome's patent stone 29
3. Asphaltic Concrete.
63. Asphaltic concrete, composition, manufacture, and uses 31
4. Glass.
64-65. Glass, composition and uses glazing 33
CHAPTER IIL METALS.
66. Metals used in engineering constructions 32
67. Ironandsteel 32
68-71. Cast iron varieties, appearances of good cast iron, test of
quality, indications of strength 33
72-73. Wrought iron Appearance of good wrought iron Forms-
Iron wire 34
74-78. Steel General modes of manufacture Varieties Hardening
and tempering 36
79. Durability of Iron and steel 38
80. Protection of iron- work. 39
81-85. Copper, zinc, tin, lead, alloys, 40
CHAPTER IV. UNITING MATERIALS.
86. Uniting materials 41
87. Glue 41
88. Lime, varieties of 42
89-91. Limestones, hydraulic and ordinary 42
92. Characteristics and tests of hydraulic limestone 43
93-97. Calcination of limestones Kilns, intermittent and perpetual
Object of kilns 44
98-104. Products of calcination, common lime, hydraulic lime, hy-
draulic cement, and pozzuolanas 48
105. Trass 51
106-108. Manufacture of limes and cements 51
109-113. Manufacture of slow-setting and quick-setting cements from
argillaceous limestones 53
114-115. Hydraulic cements from other stones 54
116. Scott's hydraulic cement 55
117-118. Tests for, and the storage of limes and cements. 56
119. Mortar common and hydraulic 57
120-121. Slaking lime, and the preservation of slaked lime 58
122. Sand, varieties of, and uses in mortar 6C
123-126. Manufacture of mortar, proportions of ingredients, and mani-
pulation 61
127-128. Setting of mortar, theory of 63
CONTENTS.
129-132. Adherence, hardness, strength, durability, and uses of mortar. 64
133-137. Mastics, bituminous and artificial uses 67
CHAPTER V.- PRESERVATIVES.
138-139. Paints 69
140-145. Japanning, oiling, varnishes, coal tar, asphaltum, metal cov-
erings. 70
146. Preservatives based upon chemical combinations 71
PART H.
Strength of Materials.
CHAPTER VI. STRAINS.
147. General problems 72
148-149. Strength of materials strains stress 72
150-152. Classification of strains 73
153-157. Constants weight, limit of elasticity, coefficient of elasticity,
modulus of rupture 77
Tension.
158. Elongation of a bar by a force acting in the direction of its
axis. 81
159. Tensile strength per square inch of certain building materials. 82
160. Work expended in the elongation of a bar 83
161. Elongation of a bar, its weight considered 85
162-163. Bar of uniform strength to resist elongation 86
164. Modulus of resistance to crushing
165. Values of C for certain building materials.
Shearing.
166. Kinds of shearing strains coefficient of lateral elasticity
modulus of shearing 90
167. Values of S for certain materials 92
Transverse Strain.
168-169. General equation expressing the relation between the moments
of the external forces bending the bar and the moments of
the resistances 92
170. Shearing strain produced by a bending force 98
171. Changes in form of the bar 98
172. Stress on the unit of area 99
173. ValuesofI 99
Flexure.
174. General equation of the elastic curve 100
175. Bar fixed at one end and acted on by a force at the free end
to bend it 102
176-181. Beam resting on two points of support 103
182-183. Beam having its ends firmly held down 109
CONTENTS. IX
184. Beam fixed at one end and the other end resting on a support. 113
185-186. Beam resting on three points of support 114
187-194. Theorem of three moments and applications 117
Torsion.
195. Coefficient of torsional elasticity . 125
196. Valuesof G 127
197. Rupture by twisting. 127
198. Influence of temperature 128
CHAPTER VII STRENGTH OF BEAMS.
199. General problems 128
200. Strength of beams of uniform cross-section strained by a ten-
sile force 129
201. Strength of beams of uniform cross-section under compressive
strains 129
202. Hodgkinson's formulas 130
u Gordon's formulas 131
" C. Shaler Smith's formula 132
203. Deductions made by Mr. Hodgkinson 132
204. Strength of beam to resist shearing 134
205. Strength of beam to resist rupture by bending 134
206. Formulas for maximum stress on the unit of area in the dan-
gerous section. 135
207. Safe values f or R' 136
208. Influence of form of cross-section on the strength of a beam . 137
209. Strongest beam of rectangular cross-section that can be cut
from a cylindrical piece 138
210. Beams of uniform strength 139
211-215. Beams of uniform strength to resist transverse strain 140
216. Relation between the stress on unit of area and deflection
in a beam produced by bending forces 144
217. Action of oblique forces 145
218. Strength of beams against twisting 146
219-220. Strength of a beam strained by rolling loads 147
221-222. Limits of practice and factors of safety 151
223-224. General equation between the moments of the external forces
and the moments of the resistances in curved beams 152
225-227. Method of determining the equation of mean fibre, the un-
known reactions, and the stress on unit of area 155
228. Approximate method of determining stresses in a curved
beam resting on two supports 160
229. Curved beam with ends firmly fixed 162
PART m.
CHAPTER Vin. FRAMING.
230-231. Art of construction frames carpentry 163
232. Joints 164
233-239. Joints in timber- work 165
240. Fastenings of joints 171
241. General rules for construction of joints. 172
242-248. Joints for iron-work. 173
249. Simple beams 178
CONTENTS.
ABTICLK
250. Solid-buHt beams ........................................ 178
251. Framing single beams with intermediate supports ........... 180
252. Open-built beams king and queen post trusses ............. 181
253. Necessity for braces where rigidity is required ............ 183
254-25">. Stresses in an inclined beam ............................ 183
256-257. Stresses in a triangular frame ........................... 187
253. Stresses in a jib-crane ........ .......................... 189
259. Combined triangular frames. .............................. 191
260-262. Triangular bracing ....................................... 191
263. Vertical and diagonal bracing ............................. 194
264. Angle of economy ........................................ 196
PART IV.
CHAPTER IX. MASONRY.
265. Definition of masonry 198
266. Kinds of masonry structures 198
267. General definitions 199
268. Retaining and reservoir walls and dams 199
269. Areas, lintels, and plate-bands 200
270-271. Arches and their classification 201
272-279. Cylindrical, groined, cloistered, annular arches, domes, etc. 201
Mechanics of Masonry.
279. Distribution of pressure on a surface 205
280-286. Normal pressure 205
287. Oblique pressure. 210
288. Strains on structures of first and second classes 211
289-296. Strains on retaining walls 211
297-298. Counterforts. 221
299-300. Reservoir walls and dams 222
301. Strains on structures of fourth class 224
302-303. Arches and modes of yielding 224
304-305. Conditions of stability for arches 225
306. Joints of rupture 227
307-309. Conditions of equilibrium for a full-centered cylindrical arch. 228
310. Rankine's rule for obtaining approximate value of horizontal
thrust 231
311. Curves of pressure and of resistance 233
312-316. Equation of the curve of resistance 233
317. Depth of keystone 236
318-319. Thickness of piers and abutments, and table of dimensions
for arches of small spans 237
320. Forms of cylindrical arches , 237
321. Rampant and inverted arches 238
822. Wooden arches 238
CHAPTER X. MASONRY CONSTRUCTION.
824-325. Rubble masonry 239
826-327. Ashlar masonry 241
828. Cut-stone masonry ." . . . . . 242
829. Stone-cutting. 343
830-333. Strength of masonry . . . * 243
884-336. Machinery used in constructing masonry work 248
CONTENTS. XI
ABTTCLK PAGE
337_344. Brick masonry and construction 251
345. Construction of concrete masonry 253
346_349. Construction of retaining and reservoir walls 255
350. Construction of areas, lintels, etc 257
351. Form of soffit of the arch 258
352-357. Ovals 258
358. Construction of voussoirs 263
359. Bond in arches 264
360. Oblique or askew arches , 265
361-363. Construction of arches 266
364-366. Cappings, abutments and piers, and connection 268
367. Machinery used in constructing arches 270
368-369. General remarks on the arch 272
370. General rules to be observed in constructing masonry 273
871-375. Preservation and repairs of masonry 274
876. Mensuration of masonry 276
PART V.
CHAPTER XI. FOUNDATIONS.
377. Definition of foundation 277
379. Yielding of foundations 277
380. Natural and artificial beds of foundations 278
381. Classification of soils 278
Foundations on Land.
382-383. In rock, compact earth, etc 278
384. In ordinary soils 280
385-386. In soft earths and compressible soils 280
387-393. Piles, and kinds of 282
394-397. Piles, how forced in the soil 286
398. Load allowed on piles 288
399^01. Bed of foundation made of piles 288
CHAPTER XII. FOUNDATIONS IN WATER.
402. Difficulties met with 290
403-404. Concrete beds 290
405. Beds of piles 292
406. Common caisson 292
407. Permanent caissons 294
408. Submarine armor and diving-bell 294
409. Pierre perdue 295
410. Screw piles 295
411. Well foundations 295
412. Iron tubular foundations. 296
413-414. Exclusion of water by earthen dam 297
415-418. Cofferdam 297
419-420. Caisson and crib-work dams 300
422-424. Pneumatic pile 302
425. Brunei's method at Saltash, England 306
426 Pneumatic caisson -308
427 Pneumatic caissons at L'Orient, France 308
428. Pneumatic caissons at St. Louis, Mo 310
429. Pneumatic caissons at St. Joseph, Mo. 312
CONTENTS.
AHTIOLK
430. Pneumatic caissons at New York City ...................... 314
431. Movable pneumatic caisson ............................... 315
433. Securing the bed of the foundation from injury ........... 317
PART VI.
CHAPTER XIII. BRIDGES.
434 Definitions and classification 818
435-436. Component parts of a bridge 318
437-441. Piers and abutments, fenders, ice-breakers 319
442-443. Approaches . 325
444. The frame of a bridge and classification 327
CHAPTER XIV. TRUSSED BRIDGES.
445. Definitions 328
446. Systems 329
447. External forces acting to strain the bridge 329
448. Bang-post truss 331
449. Fink's truss 332
450 Bollman's truss 332
451-453. Method of determining the strains on a triangular truss 333
454. The panel system , 338
455. Queen-post truss 339
456-457. The bowstring system 340
458-459. Compound systems 344
460. Strains produced by moving loads 345
461. Counter-braces 346
462. Length and depth of a truss 347
463^72. Description of the " graphical method " 347
473. Working, proof, and breaking loads 356
474. Wooden bridge trusses 356
475. Town's truss. 357
476. Long's truss 358
477. Burr's truss 359
479. Canal bridge truss 360
480. Howe's truss 360
481. Pratt's truss 361
488. Bridge trusses of iron 362
484. Continuity of the truss 364
CHAPTER XV. TUBULAR AND IRON PLATE BRIDGES.
485. Tubular bridges 365
486. Iron plate bridges 367
CHAPTER XVI. ARCHED BRIDGES.
487. Form of arch used in bridges 368
488. Masonry arches centres 369
489-490. Arched bridges of iron construction 370
491. Expansion and contraction 371
492. Arched bridges of steel , 371
CONTENTS.
xiii
IXTICXJE PAGH
493. Ead's patent bridge 371
494. Circumstances under which the arch may be pref e'/red to the
truss in a bridge 372
CHAPTER XVII. SUSPENSION BRIDGES.
495-496. Component parts of a suspension bridge 373
497. Towers for suspension bridges 373
498-501. Anchorages, main chains, suspension chains, and roadway. . 374
502. Oscillations and means to stiffen a bridge * ... 377
503. Suspension railroad bridge over Niagara River 378
504. Suspension bridge over tie East River, New York 381
CHAPTER XVHL MOVABLE AND AQUEDUCT BRIDGES.
505-511. Movable bridges and classification 381
512. Aqueduct bridges 383
CHAPTER XIX BRIDGE CONSTRUCTION.
513. Necessary things to be considered in advance 884
514-517. Site, water-way, and velocity of current 384
518. Design of bridge 387
519-523. Erection, machinery used, modes of erection, and cost of
construction 388
PART VIL
CHAPTER XX. ROOFS.
524. Definition of roof 390
525-526. Various forms of roofs and kinds of coverings 390
527. Frames used to support a roof 391
528. Remarks upon the weights resting on a roof 391
529-530. Rise and span, and materials used in construction of roofs. 392
531. King-post roof truss 393
532. Queen-post roof truss 394
533. Iron roof trusses 394
534. Determination of the kind and amount of stresses in the
pieces of a king -post truss 394
535. The same for a king-post framed with struts 395
536. The same for a queen-post truss 398
537-538. Strains on the parts of an iron roof truss with trussed
rafters 398
539. Strains on the parts of a roof truss, the rafters of which are
divided into three parts, and are supported at the points
of division 403
541-542. Determination by the graphical method of the stresses in
the pieces of a roof truss 407
543. Purlins 409
544. Construction of roofs 409^
CONTENTS.
PART vm.
CHAPTER XXL ROADS.
ABTZCLK
PA01
545. Definition of a road ..................................... 41C
546! Considerations to be observed in laying out a road ......... 411
547_548. Considerations governing the choice of direction of the road. 411
549-551. Grades to be adopted .................................... 412
552-557. Form and details of cross-section ......................... 413
558. Road-coverings .......................................... 416
559 Classification of ordinary roads from the kind of coverings
used ................................................ 416
560. Earth or dirt roads ...................................... 417
561. Corduroy roads ....................................... . 417
562. Plank roads ............................................ 417
563. Gravel roads ............................................ 418
664. Broken-stone roads ---- . ................................. 418
565. Macadamized roads. . . , ................................ 419
566, Telford roads ......................................... 419
667. Kinds of stone used in broken-stone roads ................. 420
568. Repairs of broken-stone roads ............................ 420
569. Essential qualities of a paved road ........................ 421
570. Roman paved roads ..................................... 421
671. English paved roads ...... . .............................. 421
672. Belgian pavement ...................................... 422
573. Cobble-stone pavement ....... ........................... 423
574. Kinds of stone suitable for paved roads ......... ......... 423
575. Wooden pavements ..................................... 423
576. Asphaltic pavements .................................... 424
577. Tram-roads ............................................ 424
CHAPTER XXII. LOCATION AND CONSTRUCTION OF ROADS.
578. Selection of route ....................................... 425
579. Reconnoissance ......................................... 425
680. Surveys ................................................ 427
581-582. Map, memoir, and estimate of cost ....................... 427
583-584. Surveys of location and construction ...................... 429
685-587. Earthwork embankments, etc ........................... 430
588. Construction in swamps and marshes ..................... 433
589. Construction of side-hill roads ........................... 433
590-594. Drainage of roads ....................................... 435
595-596. Footpaths and sidewalks ................................ 437
597-599 . Construction of tram-roads .............. ---- 438
CHAPTER XXIII. RAILROADS.
600. Definition of railroad 439
601. Direction 430
602. Grades 440
603. Curves 441
604-607. Resistances offered to traction on railroads 442
608. Formulas for total resistance 443
609-613. Tractive force used on railroads 444
614. Gauge of railroads 444
615-616. Location and construction of railroads 446
617-622. Tunnels 44?
623. Ballast . . 45C
CONTENTS. XV
ARTICU PAGE
624. Cross-ties 450
625. Bails 450
626. Coning of wheels 451
627. Elevation of outer rail on curves. 451
628-630. Crossings, switches, turn-tables, etc 452
CHAPTER XXIV. CANALS.
681. Definition of canal 453
632-637. Navigable canals, form, construction, and size 453
638-640. Locks 457
641. Lock-gates 461
642. Inclined planes 462
643. Guardlock 462
644. Lift of locks 462
645-646. Levels and water-supply 463
648-650. Feeders, reservoirs, dams, and waste-weirs 466
651. Water-courses intersecting the line of the canal 468
652. Dimensions of canals and locks in the United States 469
653-655. Irrigating canals 469
656. Drainage canals. 471
657-658. Canals for supplying cities and towns with water 479
INTRODUCTORY CHAFIER.
I. Engineering is defined to be " the science and art of
utilizing the forces and materials of nature."
It is divided into two principal branches, Civil and Military
Engineering.
The latter embraces the planning and construction of all de-
fensive and offensive works used in military operations.
The former comprises the designing and building of all works
intended for the comfort of man, or to improve the country either
by beautifying it or increasing its prosperity.
In this branch the constructions are divided into two classes,
according as the parts of which they are made are to be relatively
at rest or in motion. In the former case they are known as
structures, and in the latter as machines.
II. It is usual to limit the term civil engineering to the
planning and construction of works of the first class, and to use
the term mechanical or dynamical engineering when the
works considered are machines.
It is also usual to subdivide civil engineering into classes,
according to the prominence given to some one or more of its parts
when applied in practice, as topographical engineering, hydraulic
engineering, railway engineering, etc. By these divisions, greater
progress toward perfection is assured. Notwithstanding this sepa-
ration into branches and subdivisions, there are certain general
principles common to them all.
HI. The object of the following pages is to give in regular order
these elementary principles, common to all branches of engineering,
which the student should learn, so that he may understand the
nature of the engineer's profession, and know how to apply these
principles in practice.
XV111 INTRODUCTORY CHAPTER.
IV. A structure is a combination of portions of solid ma eriais
BO arranged as to withstand the action of any external for:es to
which it may be exposed, and still to preserve its form. These
portions are called pieces, and the surfaces where they touch and
are connected are called joints. The term solid here used is
applied to a body that offers an appreciable resistance to the action
of the different forces to which it may be subjected.
V. That part of the solid material of the earth upon which the
structure rests is called the foundation, or bed of the founda-
tion, of the structure.
VI. In planning and building a structure, the engineer should
be governed by the following conditions :
The structure should possess the necessary strength ; should
last the required time ; and its cost must be reasonable.
In other words, the engineer in projecting and executing a work
should duly consider the elements of strength, durability, and
economy.
VII. The permanence of a structure requires that it should
possess stability, strength, and stiffness. It will possess these
when the following conditions are fulfilled :
When all the external forces, acting on the whole structure, are
in equilibrium ;
When those, acting on each piece, are in equilibrium ;
When the forces, acting on each of the parts into which a piece
may be conceived to be divided, are in equilibrium ; and
When the alteration in form of any piece, caused by the exter-
nal forces, does not pass certain prescribed limits.
A knowledge, therefore, of the forces acting on the structure,
and of the properties of the materials to be used in its construc-
tion, is essential.
VIII. The designing and building of a structure f. rm three dis-
tinct operations, as follows :
1. The conception of the project or plan ;
2. Putting this on paper, so it can be understood , and
3. Its execution.
INTEODUCTOKT CHAPTER.
The first requires a perfect acquaintance with the locality whera
the structure is to be placed, the ends or objects to be attained by
it, and the kind and quantity of materials that can be supplied at
that point for its construction.
The second requires that the projector should know something
of drawing, as it is only by drawings and models accompanied by
descriptive memoirs, with estimates of cost, that the arrangement
and disposition of the various parts, and the expense of a proposed
work, can be understood by others.
The drawings are respectively called the plan, elevation, and
cross-section, according to the parts they represent. A sym-
metrical structure requires but few drawings ; one not symmetri-
cal, or having different fronts, will require a greater number.
These, to be understood, must be accompanied by written speci-
fications explaining fully all the parts.
The estimate of cost is based upon the cost of the materials, the
price of labor, and the time required to finish the work.
The third may be divided into three parts :
1. The field-work, or laying out the work ;
2. The putting together the materials into parts ; and
3. The combining of these parts in the structure.
This requires a knowledge of surveying, levelling, and other
operations incident to laying out the work ;
A knowledge of the physical properties of the materials used ;
The art of forming them into the shapes required ; and
How they should be joined together to best satisfy the condi-
tions that are to be imposed upon the structure.
ELEMENTARY COURSE
OF
CIVIL ENGINEERING.
PART I.
BUILDING MATERIALS.
1. The materials in general use by civil engineers for their
constructions may be arranged in three classes :
1st. Those which constitute the more solid components of
structures ; as Wood, Stone, and the Metals.
2d. Those which unite the solid parts together; as Glue,
Cements, Mortars, Mastics, etc.
3d. Those mixtures and chemical preparations which are
employed to protect the structure from the action of the
weather and other causes of destructibility ; as Paints,
Solutions of Salts, Bituminous Substances, etc.
CHAPTER L
WOOD.
2. The abundance and cheapness of this material in the
United States, the ease with which it could be procured and
worked, and its strength, lightness, and durability, under
favorable circumstances, have caused its very general use in
every class of constructions.
Timber, from the Saxon word timbrian, to build, is the
term applied to wood of a suitable size, and fit for building
purposes. While in the tree it is called standing timber;
after the tree is felled, the portions fit for building are cut
into proper lengths and called logs or rough timber ; when
the latter have been squared or cut into shape, either to be
r-*2***:: CIVIL ENGINEERING.
used in this form or cut into smaller pieces, the general term
timber is applied to them ; if from the trunk of the tree,
they are known as square or round, hewn or sawed, accord-
ing to the form of cross-section and mode of cutting it ; if
from the branches or roots, and of crooked shape, they are
called compass timber. The latter is used in ship-building.
The logs, being sawed into smaller pieces, form lumber,
and the latter is divided into classes known as joists, scant-
lings, strips, boards, planks, etc., and, when sawed to suit a
given bill ; as dimension stuff.
3. The trees used for timber are exogenous that is, they
grow or increase in size by formation of new wood in layers
on its outer surface.
If the trunk of a tree is cut across the fibres, the cut will
show a series of consecutive rings or layers.
These layers are of annual growth in the temperate zones,
and, by counting them, the approximate age of the tree may
be determined.
The trunk of a full-grown tree presents three distinct
parts : the bark, which forms the exterior coating ; the sap-
wood, which is next to the bark ; the heart, or inner part,
which is easily distinguishable from the sap-wood by its
greater density, hardness and strength, 'and oftentimes by its
darker color.
The heart embraces essentially all that part of the trunk
which is of use as a building material. The sap-wood
possesses but little strength, and is subject to rapid decay,
owing to the great quantity of fermentable matter contained
in it. The bark is not only without strength, but, if suffered
to remain on the tree after it is felled, it hastens the decay of
the sap-wood and heart.
VARIETIES OF TIMBER-TREES IN THE UNITED STATES.
4. The forests of our own country produce a great variety
of the best timber for every purpose. For use in construc-
tion, trees are divided into two general classes, soft wood,
and hard wood trees.
The first includes all coniferous trees, like the pines, and
also some few varieties of the leaf- wood trees; and the
other includes most of the timber trees that are non-conifer-
ous, like the oaks, etc.
The soft wood trees generally contain turpentine, and are
distinguished by straightness of fibre and by the regularity
of form of the tree. The timber made from them is more
TIMBER. 3
easily sawed or split along the grain, and much more easily
broken across the grain, than that of the second class.
The hard-wood, or non-coniferous timber, contains no tur-
pentine, and, as a class, is tough and strong.
Examples of Soft-wood Trees.
5. Yellow Pine (Pinus mitis). This tree is, perhaps,
in this country the most widely distributed of all the pines,
being found in all the States from New England to the Gulf
of Mexico. In the Southern States it is called the Spruce
Pine, and the Short-leaved Pine.
The heart-wood is fine grained and moderately resinous.
Its sap-wood decays rapidly when exposed to the weather.
The tree grows mostly in light clay soils and furnishes a
strong and durable timber extensively used in house and
ship building.
Long-leaf Pine (Pinus australis). This tree is found
from southeastern Virginia to the Gulf, and is the principal
tree where the soil is sandy and dry. Inferior growths of
it are frequently called Yellow Pine. It has but little
sap-wood. The heart- wood is fine grained, compact, and has
the resinous matter very uniformly distributed.
The timber made from it is strong and durable, being
considered superior to that of the other pines. Its quality
depends, however, on the kind of soil in which the tree
grows, being less resinous in rich soils.
Red Pine (Pinus resinosa). This tree is found in Cana-
da and the northwestern parts of the United States, and is
often wrongly called " Norway Pine. " It furnishes good,
strong and durable timber.
White Pine (Pinus strobus). This tree is found in
Canada and New England, and along the Alleghanies as far
south as Georgia, and frequently called Northern Pine.
Its timber is light, soft, free from knots, slightly resinous,
easily worked, and durable when not exposed to the weath-
er. It is used in a great variety of ways for building pur-
poses and for joiners' work.
6. Fir. The genus Fir (Abies\ commonly known as
Spruce, furnishes large quantities of timber and lumber
which are extensively used throughout the Northern States.
The lumber made from it has the defects of twisting and
splitting on exposure to the weather and of decaying rapidly
in damp situations. The common fir (Abies alba and Abies
nigra), the spruce fir found in Northern California, and the
4 CIVIL ENGINEERING.
Oregon fir \_Pinus (Abies) Douglasii] which grows to an
enormous size, all furnish timber much used in building.
7. Hemlock (Abies Canadensis) is a well-known species,
used throughout the Northern States as a substitute for pine
when the latter is difficult or expensive to procure. It is
very perishable in damp situations or when subjected to alter-
nate wetness and dryness. It has been used in considerable
quantities in positions where it is entirely submerged in
fresh water. Hemlock timber has the defects of being
shaky, full of knots, and more difficult to work than
pine.
8. Cedar. The White Cedar, called Juniper, and the
Cypress are celebrated for furnishing a very light timber
of great durability when exposed to the weather; on this
account it is much used for shingles and other exterior
coverings. The shingles made of it will last, so it is said,
for 40 years. These two trees are found in great abundance
in the swamps of the Southern States.
9. The foregoing kinds of timber, especially the pines, are
regarded as valuable building materials, on account of their
strength, their durability, the straightness of the fibre, the
ease with which they are worked, and their applicability to
almost all the purposes of constructions in wood.
Examples of Hard- wood Trees.
*10. White Oak (Quercvs alba). The bark of this tree is
light, nearly white ; the leaf is long, narrow, and deeply in-
dented ; the wood is compact, tough, and pliable, and of a
straw color with a pinkish tinge.
It is largely used in ship-building, the trunk furnishing the
necessary timber for the heavy frame- work, and the roots and
large branches affording an excellent quality of compass-tim-
ber. Boards made from it are liable to warp and crack.
This tree grows throughout the United States and Canada, but
most abundantly in the Middle States. Proximity to salt air
during the growth of the tree appears to improve the quality
of the timber. The character of the soil has a decided effect
on it. In a moist soil, the tree grows to a larger size, but
the timber loses in firmness and durability.
Live Oak (Quercus virens). The wood of this tree is of
a yellowish tinge ; it is heavy, compact, and of a fine grain ;
it is stronger and more durable than that of any other species,
and on this account is considered invaluable for the purposes
of ship-building, for which it has been exclusively reserved,
TIMBER. 5
The live oak is not found farther north than the neighbor,
hood of Norfolk, Virginia, nor farther inland than from fif-
teen to twenty miles from the sea-coast.
Post Oak (Quercus obtusiloba). This tree seldom attains
a greater diameter than about fifteen inches, and on this
account is mostly used for posts, from which use it takes its
name. The wood has a yellowish hue and close grain ; is said
to exceed white oak in strength and durability, and is there-
fore an excellent building material for the lighter kinds of
frame-work. This tree is found most abundantly in the
forests of Maryland and Virginia, and is there frequently
called Box White Oak and Iron Oak. It also grows in the
forests of the Southern and Western States, but is rarely seen
farther north than the southern part of New York.
Chestnut White Oak (Quercus prinuspalustris). This
tree is abundant from North Carolina to Florida. The tim-
ber made from it is strong and durable, but inferior to that
of the preceding species.
Water Oak (Quercus aquaticd). This tree gives a tough
but not a durable timber. It grows in the Southern country
from Virginia to as far south as Georgia and Florida.
Red Oak (Quercus rubra). This tree is found in all parts
of the United States. The wood is reddish, of a coarse tex-
ture, and quite porous. The timber made from it is gener-
ally strong, but not durable.
11. Black Walnut (Juglans nigra). The timber made
from this tree is hard and fine-grained. It has become too
valuable to be used in building purposes, except for orna-
mentation.
Hickory (Gary a tomentosa). The wood of this tree is
tough and flexible. Its great heaviness and liability to be
worm-eaten have prevented its general use in buildings.
12. There are a number of other trees, belonging to both
hard and soft woods, that produce timber inferior to those
named. They may possibly in the future be used to some
extent to furnish timber for building purposes. The Red
Cedar, Chestnut, Ash, Elm, Poplar, American Lime or Bass-
wood, Beech, Sycamore, Tamarack, etc., have all been used
to a limited extent in constructions when the other kinds
were not to be obtained.
PREPAEATION OF TIMBER.
13. Felling. Trees should not be felled for timber until
they have attained their mature growth, nor after they ex-
6 CIVIL ENGINEERING.
hibit symptoms of decline ; otherwise the timber will not pos-
sess its maximum strength and durability. Most forest treea
arrive at maturity in between fifty and one hundred years,
and commence to decline after one hundred and fifty or two
hundred years. When a tree commences to decline, the
extremities of its older branches, and particularly its top,
exhibit signs of decay. The age of a tree can, in most cases,
be approximately ascertained either by its external appear-
ances or by cutting into the centre of its trunk and counting
the rings or layers of the sap and heart.
Trees should not be felled while the sap is in circulation ;
for this substance is of such peculiarly fermentable nature,
that if allowed to remain in the fallen timber, it is very pro-
ductive of destruction of the wood. The best authorities on
the subject agree that the tree should be felled in the -win-
ter season.
The practice in the United States accords with the above,
not so much on account of the sap not being in circulation,
as for the reason that the winter season is the best' time for
procuring the necessary labor, and the most favorable for re-
moving the logs, from where they are cut, to the points where
they are to be made into rafts.
As soon as the tree is felled, it should be stripped of its
bark and raised from the ground. A short time only should
elapse before the sap-wood is taken off and the timber reduced
nearly to its required dimensions.
14:. Measuring- Timber. Timber is measured by the
cubic foot, or by board measure / the unit of the latter is a
board one foot square and one inch thick.
Appearances of Good Timber.
15. Among trees of the same species, that one which has
grown the slowest, as shown by the narrowness of its annual
rings, will in general be the strongest and most durable.
The grain should be hard and compact, and if a cut be
made across it, the fresh surface of the cut should be iirm and
shining.
And, in general, other conditions being the same, the
strength and durability of timber will increase with its weight,
and darkness of color.
Timber of good quality should be straight-grained, and
free from knots. It should be free f rom all blemishes and
defects.
TIMBER.
Defects in Timber.
16. Defects arise from some peculiarity in the growth of
the tree, or from the effects of the weather.
Strong winds oftentimes injure the growing tree by twist-
ing or bending it so as to partially separate one annual layer
from another, forming what is known as rolled timber or
shakes.
Severe frosts sometimes cause cracks radiating from the
centre to the surface.
These defects, as well as those arising from worms or age,
may be detected by examining a cross-section of the log.
SEASONING OF TIMBER.
17. Timber is said to be seasoned when by some process,
either natural or artificial, the moisture in it has been ex-
pelled so far as to prevent decay from internal causes.
The term seasoning means not only the drying of the
timber, but also the removal or change of the albuminous sub-
stances in it. These substances are fermentable, and when
present unchanged in the timber are ever ready to promote
decay.
The seasoning of timber is of the greatest importance, not
only to its own durability, but to the solidity of the structure
for which it may be used ; for, if the latter, when erected,
contained some pieces of unseasoned or green timber, their
after-shrinking might, in many cases, cause material injury,
if not complete destruction, to the structure.
Natural Seasoning consists in exposing the timber freely
to the air, but in a dry place, sheltered from the sun and high
winds.
This method is preferable to any other, as timber seasoned
in this way is both stronger and more durable than when pre-
pared by any artificial process. It will require, on an aver-
age, about two years to season timber thoroughly by this
method. For this reason, artificial methods are used to save
time.
Water Seasoning. The simplest artificial method con-
sists in immersing the timber in water as soon as cut,
taking care to keep it entirely submerged for a fortnight,
and then to remove it to a suitable place and dry it.
The water will remove the greater portion of the sap,
even if the timber is full when immersed. This method
doubtless weakens the timber to some extent, and therefore
g CIVIL ENGINEERING.
is not recommended where strength in the timber is the
most important quality.
Boiling and Steaming have both been used for seasoning
but are open to the same objection as the last method ; viz...
the impairing of the elasticity and strength of the timber.
Hot-air Process. This consists in exposing the timber in
a chamber, or oven, to a current of hot air, whose temperature
varies according to the kind and size of the timber to be sea-
soned. This is considered the best of the artificial methods.
The time required for sufficient seasoning depends upon the
thickness of the timber, ordinary lumber requiring from one
to ten weeks.
DURABILITY AND DECAY OF TIMBER.
18. Timber lasts best when kept, or used, in a dry and
well-ventilated place. Its durability depends upon its pro-
tection from decay and from the attacks of worms and insects.
The wet and dry rot are the most serious causes of the
decay of timber.
Wet Rot is a slow combustion, a decomposition of moist
organic matter exposed to the air, without sensible elevation
of temperature. The decay from wet rot is communicated
by contact, and requires the presence of moisture.
To guard against this kind of rot, the timber must not be
subjected to a condition of alternate wetness and dryness, or
even to a slight degree of moisture if accompanied by heat
and confined air.
Dry Rot is a decay arising from the decomposition of the
fermentable substances in the timber; it is accompanied by
the growth of a fungus, whose germs spread in all directions,
finally converting the wood into a fine powder. The fungus
is not the cause of decay ; it is only a morbid growth due to
the decaying fibres of the wood.
Dry rot derives its name from the effect produced and not
from the cause, and although it is usually generated in moist-
ure, it is frequently found to be independent of extraneous
humidity. Externally, it makes its first appearance as a mil-
dew, or a white or yellowish vegetation of like appearance.
An examination under a microscope of a section of apiece of
wood attacked by dry rot shows minute white threads spread-
ing and ramifying throughout the substance.
Dry rot only attacks wood which is dead, whereas wet
rot may seize the tree while it is still alive and standing.
Timber, not properly seasoned, used where there is a want
TIMBER. 9
of free circulation of air, decays by dry rot even if there be
only a small amount of moisture present. It will also decay
by dry rot, if covered while unseasoned by a coat of paint, or
similar substance.
Durability under certain Conditions, and Means of In
creasing it.
19. Timber may be subjected to the following conditions :
It may be kept constantly dry, or at least practically so
It may be kept constantly -wet in fresh water.
It may be constantly damp.
It may be alternately wet and dry.
It may be constantly wet in sea-water.
20. Timber kept constantly dry in well-ventilated posi-
tions, will last for centuries. The roof of Westminster Hall
is more than 450 years old. In Stirling Castle are carvings
in oak, well preserved, over 300 years old ; and the trusses of
the roof of the Basilica of St. Paul, Rome, were sound and
good after 1000 years of service. The timber dome of St.
Mark, at Venice, was in good condition 850 years after it was
built.
It would seem hardly worth while to attempt to increase
the durability of timber when under these conditions, except
where it may be necessary to guard against the attacks of in-
sects, which are very destructive in some localities. Damp
lime hastens the decay of timber ; the latter should therefore,
in buildings, be protected against contact with the mortar.
21 . Timber kept constantly wet in fresh water, under
such conditions as will exclude the air, is also very durable.
Oak, elm, beach, and chestnut piles and planks were found
beneath the foundation of Savoy Place, London, in a perfect
state of preservation, after having been there 650 years.
The piles of the old London Bridge were sound 800 years
after they were driven. In the bridge built by Trajan, the
piles, after being driven more than 1600 years, were found to
have a hard exterior, similar to a petrifaction, for about four
inches, the rest of the wood being in its ordinary condition.
We may conclude that timber submerged in fresh water
will need no artificial aid to increase its durability, although in
time it may be somewhat softened and weakened.
22. Timber in damp situations. Timber in damp sit-
uations is in a place very unfavorable for durability, and is
liable, as previously stated, to decay rapidly. In such situa-
10 CIVIL ENGINEERING.
tions only the most lasting material is to be employed, and
every precaution should be taken to increase its durability.
23. Timber alternately wet and dry. The surface of
all timber exposed to alternations of wetness and dry ness
gradually wastes away, becoming dark-colored or black. ' This
is wet rot, or simply " rot"
Density and resinousness exclude moisture to a great ex-
tent; hence timber possessing these qualities should be used
in such situations. Heart-wood, from its superior density, is
more durable than sap-wood ; oak, than poplar or willow.
Resinous wood, as pine, is more durable than the non-resin-
ous, as ash or beech, in such situations.
24. Timber constantly wet in sea-water. The re-
marks made about timber placed in fresh water apply equally
to this case, as far as relate to decay from rot. Timber
immersed in salt water is, however, liable to the attacks of tw,o
of the destructive inhabitants of our waters, the Limnoria
terebrans and Teredo navalis ; the former rapidly de-
stroys the heaviest logs by gradually eating in between the
annual rings ; and the latter, the well-known ship-worm, con-
verts timber into a perfectly honeycombed state by its nu-
merous perforations. They both attack timber from the level
of the mud, or bottom of the water, and work to a height
slightly above mean low water. The timber, for this dis-
tance, must be protected by sheathing it with copper, or by
thickly studding the surface with broad-headed iron nails, or
other similar device. Resinous woods resist their attacks
longer, most probably on account of the resin in the wood.
The resin after a time is washed or dissolved out, and the
timber is then speedily attacked.
An examination of' piles in the wharf at Fort Point, San
Francisco harbor, where these agents are very destructive,
showed that piles which were driven without removing the
bark, resisted to a certain extent, their destructive attacks.
Timber saturated with dead oil by the process known as
creosoting is said to offer an effective resistance.
PRESERVATION OF TIMBER.
25. The necessity of putting timber into damp places has
caused numerous experiments to be made as to the best
method of increasing its durability under such circumstances.
There are three means which may be used to increase the
durability of timber placed in damp situations, viz :
TIMBER. 11
1st To season it thoroughly.
2d. To keep a constant circulation of air about it.
3d. To cover it with a preservative.
The cellulose matter of the woody fibre is very durable
when not acted upon by fermentation, and the object of sea-
soning is to remove or change the fermentable substances, as
well as to expel the moisture in the timber, thus protecting
the cellulose portion from decay. Even if the timber be
well seasoned, thorough ventilation is indispensable in damp
situations. The rapid decay of sills and lower floors is not
surprising where there are neither wall-gratings nor venti-
lating flues to carry off the moisture and the foul gases rising
from the earth under them. The lower floors would last
nearly as long as the upper ones if the earth were removed
to the bottom of the foundation and the space filled in with
dry material, as sand, plaster, rubbish, etc., or the bottom
covered with a concrete floor to exclude the moisture, and
arrangements made to allow a free circulation of air under
the sills.
An external coating of paint, pitch, or hot oil increases
the durability of well-seasoned timber, but such a coating
upon the surface of green timber produces just the opposite
effect. The coating of paint closes the pores of the outer
surface, and prevents the escape of the moisture from with-
in, thus retaining in the wood the elements of decay.
It is not always practicable to employ the foregoing means
in damp places to preserve the timber, and other methods
have to be used. These methods are based upon the
principle of expelling the albuminous substances and replac-
ing them by others of a durable nature, or on that of chang-
ing the albuminous substances into insoluble compounds by
saturating the timber with salts of an earthy or metallic base
which will combine with the albuminous matter and make
it inert.
Some of the methods which have been proposed, or used,
are as follows :
Kyanizing. Kyan's method is to saturate the timber
with a solution of mercuric chloride, one pound of chloride
to four gallons of water.
The complete injection of the liquid is obtained either by
long immersion in the liquid in open vats, or by great pres-
sure upon both solution and wood in large wrought-iron
tanks.
The expensiveness of the process, and its unhealthiness
to those employed in it, forbid its extensive use.
12 CIVIL ENGINEEKING.
Burnettizing. Burnett's process is to use a solution of
chloride of zinc, one pound of the chloride to ten gallons of
water; the solution being forced into the wood under a pres-
sure of 150 pounds to the square inch.
Earle's Process consisted in boiling the timber in a solu-
tion of one part of sulphate of copper to three parts of the
sulphate of iron ; one gallon of water being used with every
pound of the salts. A hole was bored ^through the whole
length of the piece ; the timber was then immersed from two
to TOUT hours, and allowed to cool in the mixture.
Ringold and Earle invented the following process : A hole
from -J to 2 inches in diameter was made the whole length of
the piece, and the timber boiled from two to four hours in
lime-water. After the piece was dried, the hole was filled with
lime and coal-tar. Neither of the last two methods was very
successful.
Common Salt is known in many cases to be a good
preservative. According to Mr. Bates's opinion this method
often answers a good purpose if the pieces so treated are
not too large.
Boucherie's Process employs a solution of sulphate of cop-
per or pyrolignite of iron. One end of the green stick is en-
closed in a close-fitting collar, to which is attached a water-
tight bag communicating through a flexible tube with an
elevated reservoir containing the solution. Hydrostatic pres-
sure soon expels the sap. When the solution issues in a pure
state from the opposite end of the log, the process is complete.
It was found that the fluid will pass a distance of twelve
feet along the grain under less pressure than is necessary to
force it across the grain three-fourths of an inch. The opera-
tion is performed upon green timber with great facility.
In 1846, 80,000 railroad ties of the most perishable woods,
impregnated, by Boucherie's process, with sulphate of copper,
were laid down on French railways. After nine years' expo-
sure they were found as perfect as when laid. This experi
ment was so satisfactory that most of the railways of that
country at once adopted the process. It has been suggested
to wash out the sap with water, which would not coagulate
the albumen, and then to use the solution.
Bethel's Process. The timber is placed in an air-tight
cylinder of boiler-iron, and the air partially exhausted. Dead
oil is then admitted at a temperature of 120 Fahr., and a
pressure of about 150 pounds to the square inch is then ap
plied, and maintained from five to eight hours, according to
the size of the timbers under treatment. The oil is then
drawn off, and the timber is removed.
STONE. 13
The Seeley Process consists in subjecting the wood, while
immersed in dead oil, to a temperature between 212 and
300 Fahr. for a sufficient length of time to expel any mois-
ture present ; the water being expelled, the hot oil is quickly
replaced by cold, thus condensing the steam in the pores of
the timber, forming a vacuum into which oil is forced by at-
mospheric pressure and capillary attraction. In this process
from six to twelve pounds of oil is expended for each cubic
foot of wood.
The theory of this process is that the first part of the opera-
tion seasons the wood, destroys or coagulates the albumen,
and expels the moisture ; and that the second part fills the
wood-cells with a material that is an antiseptic and resists de-
structive agents of every kind.
Robbins's Process consists in treating timber with coal-tar
in the form of vapor.
The wood is placed in an air-tight iron chamber, with
which is connected a still or retort, over a furnace. The fur-
nace is then fired and the wood kept exposed to the heated
vapors of the coal tar from six to twelve hours ; the operation
is then considered complete.
The most improved of all these methods is Seeley's ; thig
is a modification and an improvement of Bethel's process, and
is generally known as " creosoting."
It is thought that the ancient Egyptians knew of some pro-
cess of preserving wood. Old cases, supposed to have been
2,000 years old, apparently of sycamore impregnated with
bitumen, have been found to be still perfectly sound and
strong.
CHAPTER IL
STONE.
26. The qualities required in stone for building purposes
are so various that no very precise directions can be given to
exactly meet any particular case. What would be required
for a sea-wall would not be suited to a dwelling-house. In
most cases the choice is limited by the cost. The most
essential properties of stone as a building material are
strength, hardness, durability, and ease of working.
These properties are determined by experience or actual
experiment.
14 CIVIL ENGINEERING.
27. The term Stone, or Rock, is applied to any aggregation
of several mineral substances ; as a building material, stones
may be either natural or artificial.
Natural Stones may be subdivided into three classes ; the
silicious, the argillaceous, and the calcareous, according as
silica, clay, or lime is the principal constituent.
Artificial Stones are imitations of natural stone, made by
consolidating fragmentary solid material by various means;
they may be subdivided into classes as follows:
1st. Those in which two or more kinds of solid materials
are mixed together and consolidated by baking or burning ;
as brick, tiles, etc.
2d. Those in which the solid materials are mixed with
some fluid or semi-fluid substance, which latter, hardening
afterwards by chemical combinations, binds the former firmly
together; as ordinary concrete, patent stone, etc.
3d. Those in which the solid materials are mixed with
some hot fluid substance which hardens upon cooling ; as
asphaltic concrete, etc.
I. NATURAL STONES.
GENERAL OBSERVATIONS ON THE PROPERTIES OF STONE AS A
BUILDING MATERIAL.
28. Strength, hardness, durability, and ease of working
have already been mentioned as essential properties to be
considered in selecting stone for building purposes.
It is not easy to judge of the qualities from external
appearances. In most cases stone, which has one of the
three properties first named, will have also the other two. In
general, when the texture is uniform and compact, the grain
fine, the color dark, and the specific gravity great, the stone
is of good quality. If there are cracks, cavities, presence
of iron, etc., even though it belong to a good class of stone, it
will be deficient in some of these essential qualities, and
should be rejected. A coarse stone is ordinarily brittle, and
is difficult to work ; it is also more liable to disintegrate than
that of a finer grain.
29. Strength. Among stones of the same kind, the strong-
est is almost always that which has the greatest heaviness.
As stone is ordinarily to be subjected only to a crushing force,
it will only be in particular cases that the resistance to this
strain need be considered, the strength of stone in this respect
being greater than is generally required of it. If its dura-
STONE. 15
bility is satisfactorily proved, its strength, as a rule, may be
assumed to be sufficient.
30. Hardness. This property is easily ascertained by
actual experiment and by a comparison made with other
stones which have been tested. It is an essential quality in
stone exposed to wear by attrition. Stone selected for paving,
flagging and for stairs, should be hard and of a grain too
coarse to admit of becoming very smooth under the action to
which it is submitted.
By the absorption of water, stones become softer and more
friable.
31. Durability. By this term is. meant the power to resist
the wear and tear of atmospheric agencies, the capacity to
sustain high temperature, and the ability to resist the destruc-
tive action of fresh and salt water.
The appearances which indicate probable durability are
often deceptive.
As a general rule, among stones of the same kind, those
which are fine-grained, absorb least water, and are of greatest
specific gravity, are also most durable under ordinary expo-
sures. The weight of a stone, however, may arise from a
large proportion of metallic oxide a circumstance often un-
favorable to durability.
The various chemical combinations of iron, potash, and
alumina, when found in considerable quantities in the sili-
cious rocks, greatly affect their durability. The decompo-
sition of the feldspar by which a considerable portion of the
silica is removed when the potash dissolves, leaves an excess
of aluminous matter behind. The clay often absorbs water,
becomes soft, and causes the stone to crumble to pieces.
32. Frost, or rather the alternate action of freezing and
thawing, is the most destructive agent of nature with which
the engineer has to contend. Its effects vary with the tex-
ture of stones ; those of a fissile nature usually split, while
the more porous kinds disintegrate, or exfoliate at the surface.
When stone from a new quarry is to be tried, the best indi-
cation of its resistance to frost may be obtained from an ex-
amination of any rocks of the same kind, within its vicinity,
which are known to have been exposed for a long period.
Submitting the stone fresh from the. quarry to the direct
action of freezing would seem to be the best test of it, if it
were not that there are some kinds of stone that are much
affected by frost when they are first quarried due to the
moisture present in the stone, which moisture is lost by
exposure to the air, and is never reabsorbed to the same
amount.
16 CIVIL ENGINEERING.
A test for ascertaining the probable effects of frost on
stone was invented by M. Brard, a French chemist, and may
be used for determining the probable comparative durabili-
ties of specimens. It imitates the disintegrating action of
frost by means of the crystallization of sodium sulphate. The
process may be stated briefly as follows : Let a cubical block,
about two inches on the edge, be carefully sawed from the
stone to be tested. A cold saturated solution of the sodium
sulphate is prepared, placed over a fire, and brought to the
boiling-point. The stone, having been weighed, is suspended
from a string, and immersed in the boiling liquid for thirty
minutes. It is then carefully withdrawn, the liquid is de-
canted free from sediment into a flat vessel, and the stone is
suspended over it in a cool cellar. An efflorescence of the
salt soon makes its appearance on the stone, when it must be
again dipped in the liquid. This should be frequently done
during the day, and the process be continued for about a
week. The earthy sediment found at the end of this period
in the vessel is carefully weighed, and its quantity will give
an indication of the like effect of frost. This process is
given in detail in Yol. XXXVIII. Annales de Chemie et de
Physique.
This test, having corresponded closely with their experi-
ence, has received the approval of many French architects
and engineers. Experiments, however, made by English engi-
neers on some of the more porous stones, by exposing them
to the alternate action of freezing and thawing, gave results
very different from those obtained by Brard's method.
33. The Wear of Stone from ordinary exposure is very
variable, depending not only upon the texture and constituent
elements of the stone, but also upon the locality, and the posi-
tion, it may occupy in a structure, with respect to the pre-
vailing driving rains. This influence of locality on the
durability of stone is very marked. Stone is observed to wear
more rapidly in cities than in the country, and exhibits signs
of decay soonest in those parts of a building exposed to the
prevailing winds and rains.
The disintegration of the stratified stones placed in a wall
is materially affected by the position of the strata or laminae
with respect to the exposed surface, proceeding faster when
the faces of the strata are exposed, as is the case when the
stones are not placed with their laminae lying horizontally.
Stones are often exposed to the action of high temperatures,
as in the case of great conflagrations. They are also used to
protect portions or a building from great heat, and sometimes
to line furnaces. Those that resist a high degree of heat are
STONE. 17
termed fire- stones. A good fire-stone should be infusible,
and not liable to crack or exfoliate from heat. Stones that
contain lime or magnesia are usually unsuitable. Also, sili-
cates containing an oxide of iron.
Their durability under such circumstances should be con-
sidered when selecting them for building.
The only sure test, however, of the durability of any kind
of stone is its wear, as shown by experience.
34. Expansion of Stone from Heat. Experiments have
been made in this country and Great Britain to ascertain the
expansion of stone for every degree of Fahrenheit, and the
results have been tabulated. Within the ordinary ranges of
temperature the stone is too slightly affected by expansion or
contraction to cause any perceptible change. Professor
Eartlett's experiments, however, showed that in a long line of
coping the expansion was sufficiently great to crush mortar
between the blocks.
35. Preservation of Stone. To add to the durability
of stone, especially of that naturally perishable or showing
signs of decay, various processes have been tried or proposed.
All have the same end in view ; viz., to fill the exposed
pores of the stone with some substance which shall exclude
the air and moisture. Paints and oils are used for this pur-
pose. Great results have been expected from the use of
soluble glass (silicate of potash), and also from silicate of
lime. The former, being applied in a state of solution in
water, gradually hardens, partly through the evaporation of
its water, and partly through the removal of the potash by
the carbonic acid in the air. The latter is used by filling the
pores with a solution of silicate of potash, and then introdu-
cing a solution of calcium chloride or lime nitrate ; the chemi-
cal action produces silicate of lime, filling the pores of the
natural stone. Time and experience will show if the hopes
expected from the use of these silicates will be realized.
36. Ease of Working the Stone. This property is to a
certain extent the inverse of the others. The ease with which
stone can be cut or hammered into shape implies either soft-
ness or else a low degree of cohesiveness between its particles.
It often happens that its hardness may prevent a stone, in
every other way suitable, from being wrought to a true sur-
face and from receiving a smooth edge at the angles. More-
over, the difficulty of working will increase very materially
the cost of the finished stone.
It requires experience and good judgment to strike a me-
dium between these conflicting qualities.
18 CIVIL ENGINEERING.
37. Quarrying. If the engineer should be obliged to get
out his own stone by opening a new quarry, he should pay par-
ticular attention to the best and cheapest method of getting it
out and hauling it to the point where it is to be used. In all
cases he will, if possible, open the quarry on the side of a
hill, and arrange the roads in and leading to it with gentle
slopes, so as to assist the draught of the animals employed.
The stone near the surface, not being as good as that beneath,
is generally discarded. The mass or bed of stone being ex-
posed, a close inspection will discover the natural joints or
fissures along which the blocks will easily part from each
other. When natural fissures do not exist, or smaller blocks
are required, a line of holes is drilled at short regular inter-
vals, or grooves are cut in the upper surface of a bed. Then
blunt steel wedges or pins, slightly larger than the holes, are
inserted, and are struck sharply and simultaneously with ham-
mers until the block splits off from the layer.
If large masses of stone be required, resort is had to blast-
ing 1 . This operation consists in boring the requisite number
of holes, loading them with an explosive compound, arid fir-
ing them. The success of blasting will depend upon a judi-
cious selection of the position and depth of the holes and upon
the use of the proper charges.
Instead of trusting, as is too often done, to an empirical
rule, or to no rule at all, it is well, by actual experiments on
the particular rock to be quarried, to ascertain the effect of
different charges, so as to determine the amount required in
any case, to produce the best result.
VARIETIES OF BUILDING STONES IN GENERAL USB.
SILICIOIJS STONES.
38. Silicious Stones are those in which silica is the prin-
cipal constituent. With a few exceptions, their structure ia
crystalline-granular, the grains being hard and durable. They
emit sparks when struck with a steel, and do not generally
effervesce with acids.
Some of the principal silicious stones used in building are
Syenite, Granite, Gneiss, Mica Slate, Hornblende Slate,
Steatite, and the Sandstones. For their composition, partic-
ular description, etc. see any of the manuals of mineralogy.
Syenite, Granite, and Gneiss. These stones differ but lit-
tle in the qualities essential to a good building material, and
SILICIOC8 STONES. 19
from the great resemblance of their external characters and
physical properties are generally known to builders by the
common term granite.
Granite (Syenite, Granite, and Gneiss). This stone ranks
high as building material, in consequence of its superior
strength, hardness, and durability, and furnishes a material par-
ticularly suitable for structures which require great strength.
It does not resist well very high temperatures, and its great
hardness requires practised stone-cutters to be employed in
working it into proper shapes. It is principally used in works
of magnitude and importance, as light-houses, sea-walls,
revetment-walls of fortifications, large public buildings, etc.
Only in districts where it abounds is it used for ordinary
dwelling-houses. It was much used by the ancients, especially
by the Egyptians, some of whose structures, as far as the stone
is concerned, are still remaining in good condition, after 3,000
years' exposure. Granite occurs in extensive beds, and may
be obtained from the quarries in blocks of almost any size re-
quired. Gneiss, in particular, having the mica more in layers,
presents more of a stratified appearance, and admits of being
broken out into thin slabs or blocks. A granite selected for
building purposes should have a fine grain, even texture, and
its constituents uniformly disseminated through the mass. It
should be free from pyrites or any iron ore, which will rust
and deface, if not destroy the stone on exposure to the weath-
er. The feldspathic varieties are the best, and the syenitic
are the most durable. An examination of the rock in and
around the quarry may give some idea of its durability.
Mica Slate bas in its composition the same materials as
gneiss, and breaks with a glistening or shining surface. The
compact varieties are much used for flagging, for door and
hearth stones, and for lining furnaces, as they can be broken
out in thin, even slabs. It is often used in ordinary masonry
work, in districts where it abounds.
Hornblende Slate resembles mica slate, but is tougher, and
is an excellent material for flagging.
Steatite, or Soapstone, is a soft stone easily cut by a knife,
and greasy to the touch. From the ease with which- it is
worked, and from its refractory nature, it is used for fire-stones
in furnaces and stoves, and for jambs in fire-places. Being
soft, it is not suitable for ordinary building purposes.
Sandstone is a stratified rock, consisting of grains of silicious
sand, arising from the disintegration of silicious stones, ce-
mented together by some material, generally a compound of
silica, alumina, and lime. It has a harsh feel, and every dull
shade of color from white, through yellow, red, and brown, to
20 CIVIL ENGINEERING.
nearly a black. Its strength, hardness, and durability vary
between very wide limits ; some varieties being little inferior
to good granite as a building-stone, others being very soft,
friable, arid disintegrating rapidly when exposed to the weath-
er. The least durable sand-stones are those which contain the
most argillaceous matter ; those of a f eldspathic character also
are found to withstand poorly the action of the weather. The
best sandstone lies in thick strata, from which it can be cut in
blocks that show very faint traces of stratification; that which
is easily split into thin layers, is weaker. It should be firm in
texture, not liable to peel off when exposed, and should be free
from pyrites or iron-sand, which rust and disfigure the blocks.
It is generally porous and capable of absorbing much water,
but it is comparatively little injured by moisture, unless when
built with its layers set on edge. In this case the expansion of
water between the layers in freezing makes them split or
" scale " off. It should be placed with the strata in a horizon-
tal position, so that any water which may penetrate between
the layers may have room to expand or escape. Most of the
varieties of sandstone yield readily under the chisel and saw,
and split evenly ; from these properties it has received from
workmen the name of free-stone. It is used very exten-
eively as a building-stone, for flagging, for road material ; and
Borne of its varieties furnish an excellent fire-stone.
Other varieties of silicious stones besides those named, as
porphyry, trap or greenstone, basalt, quartz-rock
(cobble-stone), buhr-stone, etc., are used for building and
engineering purposes, and are eminently fit, either as cut-
stone or rubble, as far as strength and durability are concerned.
AEGILLAOEOU8 STONES.
39. Argillaceous or Clayey Stones are those in which
clay exists in sufficient quantity to give the stone its charac-
teristic properties. As a rule, the natural argillaceous stones,
excepting roofing slate, are deficient in the properties of hard-
ness and durability, and are unfit for use in engineering con-
structions.
Roofing Slate is a stratified rock of great hardness and
density, commonly of a dark dull blue or purplish color. To
be a good material for roofing, it should split easily into even
slates, and admit of being pierced for nails without being
fractured. It should be free from everything that can on ex-
posure undergo decomposition. The signs or good quality in
slate are compactness, smoothness, uniformity of texture, clear
CALCAREOUS STONES. 21
dark color; it should give a ringing sound when struck, and
should absorb but little water. Being nearly impervious to
water, it is principally used for covering of roofs, linings oi
water-tanks, and for other similar purposes.
CALCAREOUS STONES.
40. Calcareous Stones are those in which lime (calcium
monoxide) is the principal constituent. It enters either as a
sulphate or carbonate.
Calcium Sulphate, known as gypsum in its natural
state, when burnt and reduced to a powder, is known as
plaster-of-Paris. A paste made of this powder and a little
water, soon becomes hard and compact. Gypsum is not
used as a building-stone, being too soft. The plaster, owing
to its snowy whiteness and fine texture, is used for taking casts,
making models, and for giving a hard finish to walls. Care
must be taken to use it only in dry and protected situations,
as it absorbs moisture freely, then swells, cracks, and exfoliates
rapidly.
Calcium Carbonates, or Limestones, furnish a large
amount of ordinary building-stone, ornamental stone, and form
the source of the principal ingredient of cements and mortars.
They are distinguished by being easily scratched with a
knife, and by effervescing with an acid. In texture they are
either compact or granular; in the former case the fracture is
smooth, often conchoidal ; in the latter it has a crystalline-
granular surface, the fine varieties resembling loaf-sugar.
The limestones are generally impure carbonates, and we
are indebted to their impurities for some of the most beauti-
ful as well as the most invaluable materials used for construc-
tions. Those stones which are colored by metallic oxides, or
by the presence of other minerals, furnish the numerous color-
ed and variegated marbles ; while those which contain a cer-
tain proportion of impurities as silica, alumina, etc., yield, on
calcination, those cements which, from possessing the prop-
erty of hardening under water, have received the names of
hydraulic lime, hydraulic cement, etc.
Limestones that can be made to have a smooth surface and
take a polish are known as marbles ; the coarser kinds
are called common limestones, and form a large class of
much value for building purposes.
41. Marbles. Owing to the high polish of which they
are susceptible, and their consequent value, the marbles are
mostly reserved for ornamental purposes.
22 CIVIL ENGINEERING.
They present great variety, both in color and appearance,
and the different kinds have generally received some appro-
priate name descriptive of their use or appearance.
Statuary Marble is of the purest white, finest grain, and
is free from all foreign minerals. It receives a delicate
polish, without glare, and is, therefore, admirably adapted to
the purposes of the sculptor, for whose uses it is mostly
reserved.
Conglomerate Marble. This consists of two varieties;
the one termed pudding stone, composed of rounded pebbles
embedded in compact limestone ; the other termed breccia,
consisting of angular fragments united in a similar manner.
The colors of these marbles are generally variegated, making
the material very handsome and ornamental.
Bird's-eye Marble. The name of this stone is descriptive
of its appearance after sawing or splitting, the eyes arising
from the cross-sections of a peculiar fossil (jucoides demissus)
contained in the mass.
Lumachella Marble. This is a limestone having shells
embedded in it, and takes its name from this circumstance.
Verd Antique. This is a rare and costly variety, of
a beautiful green color, the latter being caused by veins
and blotches of serpentine diffused through the lime-
stone.
There are many other varieties that receive their name
either from their appearance or the localities from which
they are obtained.
Many of these are imitated by dealers, who, by processes
known to themselves, stain the common marbles so success-
fully that it requires a close examination to distinguish the
false from the real.
Common Limestone.
42. This class furnishes a great variety of building-stones,
which present great diversity in their physical properties.
Some of them seem as durable as the best silicious stones, and
are but little inferior to them in strength and hardness ; others
decompose rapidly on exposure to the weather ; and some
kinds are so soft that, when first quarried, they can be
scratched with the nail and broken between the fingers. The
durability of limestones is materially affected by the foreign
minerals they may contain ; the presence of clay injures the
stone for building purposes, particularly when, as sometimes
happens, it runs through the bed in very minute veins
blocks of stone having this imperfection soon separate
BRICK. 23
these veins on exposure to moisture. Ferrous oxide, sulphate
and carbonate of iron, when present, are also very destructive
in their effects, frequently causing by their chemical changes
rapid disintegration.
Among the varieties of impure carbonates of lime are the
magnesian limestones, called dolomites. They are re-
garded in Europe as a superior building material ; those being
considered the best which are most crystalline, and are com-
posed of nearly equal proportions of the carbonates of lime
and magnesia. The magnesian limestone obtained from
quarries in New York and Massachusetts is not of such good
quality ; the stone obtained being, in some cases, extremely
friable.
EL ARTIFICIAL STONES.
1st BRICK.
43. A brick is an artificial stone, made by moulding tem-
pered clay into a form of the requisite shape and size, and
hardening it, either by baking in the sun or by burning in a
kiln or other contrivance, when hardened by the h'rst pro-
cess, they are known as sun-dried, and by the latter as burnt-
brick, or simply brick.
44. Sun dried Brick. Sun-dried bricks have been in use
from the remotest antiquity, having been found in the ruins
of ancient Babylon. They were used by the Greeks and
Romans, and especially by the Egyptians. At present they
are seldom employed.
They were ordinarily made in the spring or autumn, as
they dried more uniformly during those seasons ; those made
in the summer, drying too rapidly on the exterior, were apt
to crack from subsequent contraction in the interior.
It was not customary to use them until two years after they
had been made.
"Walls, known as adobes, made of earth hardened in a simi-
lar way, are found in parts of our country and in Mexico.
They furnish a simple and economical mode of construction
where the weights to be supported are moderate, and where
fuel is very scarce and expensive. This mode, however suit-
able for a southern, is not fit for our climate.
45. Burnt Brick. Bricks may be either common T>r
pressed, hand or machine made.
The qualities of a brick are dependent upon the kind of
24 CIVIL ENGINEERING.
earth used, the tempering of this earth, the moulding of the
raw brick, and the drying and burning processes.
46. Common Brick. The size and form of common bricks
vary "but little. They are generally rectangular parallelopi-
pedons, about 8 inches long, 4 inches broad, and 2f inches
thick, the exact size varying with the contraction of the clay.
Kinds of Earth. The argillaceous earths suitable for
brick-making may be divided into three principal classes, viz. :
Pure Clays, those composed chiefly of aluminum silicate,
or one part of alumina and two of silica, combined with a
small proportion of other substances, as lime, soda, magnesia,
ferrous oxide, etc.;
Loams, which are mechanical mixtures of clay and sand ;
and
Marls, which are mechanical mixtures of clay and car-
bonate of lime.
Pure clay, being made plastic with water, may be moulded
into any shape, but will shrink and crack in drying, however
carefully and slowly the operation be conducted. By mixing
a given quantity of sand with it, these defects may be greatly
remedied, while the plastic quality of the clay will not be
materially affected.
The loams oftentimes have too much sand, and are then so
loose as to require an addition of clay or other plastic mate-
rial to increase their tenacity.
Earth is frequently found containing the proper proportions
of clay and sand suitable for making bricks ; but, if it be not
naturally fit for the purpose, it should be made so by adding
that element which is lacking. The proportion of sand or
clay to be added should be determined by direct experiments.
Silicate of lime, if in any considerable quantity in. the
earth, makes it too fusible. Carbonate of lime, if present
in any considerable quantity in the earth, would render it
unfit, ^since the carbonate is converted, during the burning,
into lime, which absorbs moisture upon being exposed, would
cause disintegration in the brick.
Preparation of the Earth. The earth, being of the proper
kind, is first dug out before the cold weather, and carried
to a place prepared to receive it. It is there piled into heaps
and exposed to the weather during the winter, so as to be
mellowed by the frosts, which break up and crumble the
lumps.
In the spring the earth is turned over with shovels, and the
stones, pebbles, and gravel are removed ; if either clay or
sand be wanting, the proper amount is added.
Tempering The object of tempering is to bring the earth
BRICK. 25
into a homogeneous paste for the use of the moulder. This
is effected by mixing it with about half its volume of water,
and stirring it and kneading it either *by turning it over re-
peatedly with shovels and treading it over by horses or men
until the required plasticity is obtained, or by using the pug-
mill or a similar machine.
The plastic mass is then moulded into the proper forms by
hand or machinery.
By Hand. In the process by hand the mould used is a
kind of box, without top or bottom, and the tempered clay is
dashed into it with sufficient force to complete!} 7 fill it, the
superfluous clay being removed by striking it with a straight-
edge. The newly-made brick is then turned out on a drying-
floor, or on a board and carried to the place where it is to
dry.
47. By Machines. Bricks are now generally moulded by
machines. These machines combine the pug-mill with an
apparatus for moulding. This apparatus receives the clay as
discharged from the pug-mill, presses it in moulds, and pushes
the brick out in front ready to be removed from the frames
and carried to the drying-floor.
48. Drying. Great attention is necessary in this part of
the process of manufacture. The raw bricks are dried in the
open air or in a drying-house, where they are spread out on
the ground or floor, and are frequently turned over until they
are sufliciently hard to be handled without injury. They are
then piled into stacks under cover for further drying.
In drying bricks, the main points to be observed are to pro-
tect them from the direct action of the sun, from draughts of
air, from rain and frost, and to have each brick dry uni-
formly from the exterior inwards. The time allowed for dry-
ing depends upon the climate, the season of the year, and the
weather.
49. Burning-. The next stage of manufacture is the burn-
ing. The bricks are arranged in the kiln so as to allow the
passage of the heat around them ; this is effected by piling
the bricks so that a space is left around each. This arrange-
ment of the bricks, called setting the kiln, is to allow the heat
to be diffused equally throughout, to afford a good draught,
and to keep up a steady heat with the least amount of fuel.
A very moderate fire is next applied under the arches of
the kiln to expel any remaining moisture from the raw brick ;
this is continued until the smoke from the kiln is no longer
black. The fire is then increased until the bricks of the
arches attain a white heat ; it is then allowed to abate in some
degree, in order to prevent complete vitrif action ; and it is
26 CIVIL ENGINEERING.
thus alternately raised and lowered until the burning is com-
plete, as ascertained by examining the bricks at the top of the
kiln.' The bricks should be slowly cooled; otherwise they
will not withstand the effects of the weather. The cooling is
done by closing the mouths of the arches and the top and
sides of the kiln, in the most effectual manner, with moist clay
and burnt brick, and by allowing the kiln to remain in this
state until the heat has subsided. The length of time of burn-
ing varies, but is often fifteen days or thereabouts.
50. General Qualities and Uses. Bricks, when properly
burnt, acquire a degree of hardness and durability that ren-
ders them suitable for nearly all the purposes to which stone
is applicable ; for, when carefully made, they are in strength,
hardness, and durability but little inferior to the ordinary
kinds of building-stone. They remain unchanged under the
extremes of temperature, resist the action of water, set firmly
and promptly with mortar, and, being both cheaper and
lighter than stone, are preferable to it for many kinds of
structures, as for the walls of houses, small arches, etc.
The Romans employed bricks in the greater part of their
constructions. The scarcity of stone in Holland and the
Netherlands led to their extensive use, not only in private
but in their public buildings, and these countries abound
in fine specimens of brick-work.
51. Characteristics of good Bricks. Good bricks should
be regular in shape, with plane surfaces and sharp edges;
the opposite faces should be parallel, and adjacent faces per-
pendicular to each other.
They should be free from cracks and flaws ; be hard ;
possess a regular form, and uniform size ; and, where exposed
to great heat, infusibility.
They should give a clear, ringing sound when struck ; and
when broken across, they should show a fine, compact, uni-
form texture, free from air-bubbles and cracks.
They should not absorb more than J-g- of their weight of water.
52. From the nature of the process of burning, it will be
evident that in the same kiln must be found bricks of very
different qualities. There will be at least three varieties: 1,
bricks which are burned too much ; 2, those, just enough ; and,
3, those, not enough. The bricks forming the arches and ad-
jacent to the latter, being nearer the fire, will be burnt to
great hardness, or perhaps vitrified ; those in the interior will
be well burnt ; and those on top and near the exterior will
be under-burned. The first are called arch brick ; the sec-
ond, body, hard, or, if the clay had contained ferrous-oxide,
cherry red j and the third, soft, pale, or sammel brick.
TILES. 27
The arch bricks are very hard but brittle, and have but
slight adhesion with mortar ; the' soft or sammel, if exposed
to the weather, have not requisite strength or durability,
and can, therefore, be used only for inside work.
53. Pressed Brick. Pressed brick are made by putting
the raw bricks, when nearly dry, into moulds of proper
shape, and submitting them to a heavy pressure by machinery.
They are heavier than the common brick. All machine-
made bricks partake somewhat of the nature of pressed
brick.
54 Fire-bricks. Fire-bricks are made of refractory clay
which contains no lime or alkaline matter, and remains un-
changed by a degree of heat that would vitrify and destroy
common brick. They are ~baked rather than burnt, and their
quality depends upon the fineness to which the clay has been
ground and the degree of heat used in making them.
They are used for facing fireplaces, lining furnaces, and
wherever a high degree of temperature is to be sustained.
Bricks light enough to float in water were known to the
ancients. During the latter part of the last century M. Fab-
broni, of Italy, succeeded in making floating bricks of a ma-
terial known as agaric mineral, a kind of calcareous tufa,
called fossil meal. Their weight was only one-sixth that of
common brick ; they were not affected by the highest tem-
perature, and were bad conductors of heat.
55. Brick-making was introduced into England by the
Romans, and arrived at great perfection during the reign of
Henry YIII.
The art of brick-making is now a distinct branch of the
useful arts, and the number of bricks annually made in this
country is very great, amounting to thousands of millions.
The art of brick-making does not belong to that of the en-
gineer. But as the engineer may, under peculiar circum-
stances, be obliged to manufacture brick, the foregoing out-
line has been given.
Tiles.
56. Tiles are a variety of brick, and from their various
uses are divided into three classes, viz. : roofing 1 , paving, and
draining tiles.
Their manufacture is very similar to that of brick, the
principal differences arising from their thinness. This re-
quires the clay to be stronger and purer, and greater care tc
be taken in their manufacture.
Their names explain their use.
CIVIL ENGINEERING.
2d. CONCRETES.
57. Concrete is the term applied to any mixture of incrta*
with coarse solid materials, as gravel, pebbles, shells, or frag-
ments of brick, tile, or stone.
The term concrete was formerly applied to the mixture
made with common lime mortar ; beton, to the mixture when
the mortar used was hydraulic, i. e., will harden under water.
The proportions of mortar and coarse materials are de-
termined by the following principle: that the volume of
cementing substance should always ~be slightly in excess of the
volume of voids of the coarse materials to be united. This
excess is added as a precaution against imperfect manipula-
tion.
Concrete is mixed by hand or by machinery.
One method, by hand, used at Fort Warren, Boston Harbor,
was as follows : The concrete was prepared by lirst spread-
ing out the gravel on a platform of rough boards, in a layer
from eight to twelve inches thick, the smaller pebbles at the
bottom and the larger on the top, and then spreading the
mortar over it as uniformly as possible. The materials were
then mixed by four men, two with shovels and two with hoes,
the former facing each other, always working from the out-
side of the heap to the centre, then stepping back, and recom-
mencing in the same way, and continuing the operation until
the whole mass was turned. The men with hoes worked each
in conjunction with a shoveller, and were required to rub well
into trie mortar each shovelful as it was turned and spread.
The heap was turned over a second time, this having been
usually sufficient to make the mixture complete, to cover the
entire surface of each pebble with mortar, and to leave the
mass of concrete ready for use.
Yarious machines have been devised to effect the thorough
mixing of the materials. A pug-mill, a cylinder in an in-
clined position revolving around its axis, a cubical box revolv-
ing eccentrically, and various other machines, have been used.
58. Uses of Concrete. Concrete has been generally used
in confined situations, as foundations, or as a backing for mas-
sive walls. For many years it has been extensively employed
in the construction of the public works throughout the U nited
States, and is now extended in its application, not only to
foundations, but even to the building of exterior and partition
walls in private buildings. It has of recent years had quite
an extensive application in harbor improvements in Europe.
There are evidences of its extensive use in ancient times
PATENT STONES. 2P
in Rome ; many public buildings, palaces, theatres, aqueducts,
etc., being built of this material. It has been asserted that
the pyramids of Egypt are built of artificial stone composed
of small stone and mortar.
It is especially suitable as a building material when dry ness,
water-tightness, and security against vermin are of conse-
quence, as in cellars of dwelling-houses, magazines on the
ground, or underneath, for storage of provisions, etc.
59. Remarks. In order to obtain uniformly a good con-
crete by the use of hydraulic lime or cement, or both, it is
essential
1. That the amount of water be just sufficient to form the
cementing material into a viscous paste, and that it be sys-
tematically applied ;
2. That each grain of sand or gravel be entirely covered
with a thin coating of this paste ; and
3. That the grains be brought into close and intimate con-
tact with each other.
These conditions require more than the ordinary methods
and machinery used in making mortars, especially if a supe-
rior article be desired.
Patent Stones.
60. Various attempts from time to time have been made to
make an imitation which, possessing all the merits, and being
free from the defects, of the most useful building-stones,
would supplement, if not supersede, them. These imitations
are generally artificial sandstones.
Beton Agglomere.
61. Beton agglomere, or Coignet-Beton, is an arti-
ficial sandstone, made by M. Francois Coignet, of Paris,
France, in which the grains of sand are cemented together
by a lime paste possessing hydraulic properties.
It is made by placing the hydraulic cement with about
one-third its volume of water into a mill, and mixing until
a plastic and sticky paste is formed. This paste and per-
fectly dry sand, in suitable proportions, are then put into a
powerful mill and mixed together until a pasty powder is
formed. The pasty powder is placed in layers of from
one and a half to two inches thick, in strong moulds, and
rammed by repeated blows of an iron-shod rammer until each
30 CIVIL ENGINEERING.
layer of material is reduced to about one-third of its origi-
nal thickness. The upper surface is struck with a straight-
edge, and smoothed off with a trowel. The mould is turned
over on a bed of sand, and detached from the block. If the
block be small, it may be handled after one day; larger
pieces should have a longer time to harden.
In common practice, the cement and the sand in a dry
state are mixed with shovels, spread out on the floor, and
then sprinkled with the proper amount of water. The damp-
ened mixture is shovelled into the mill and thoroughly
mixed.
The proportions of sand and lime will vary according to
the probable uses of the stone ; 6 volumes of sand to 1 of
hydraulic lime in powder ; or, 5 of sand, 1 of hydraulic lime,
and 1 of Portland cement, are sometimes used.
The distinctive features of this beton are the very small
proportion of water used, the thorough mixing of the materi-
als, and the consolidation effected by ramming the layers.
If too much water be used, the mixture cannot be suitably
rammed ; if too little, it will be deficient in strength.
Beton agglomere is noted for its strength, hardness, and
durability, and has had quite an extensive application in
France ; 'aqueducts, bridges, sewers, cellars of barracks, etc.,
have been built with it.
Ransome's Patent Stone.
62. Among other artificial stones that are offered to the
builder are several bearing the name of Kansome, an English
engineer. The patent silicious stone, Ransome's apoenite, and
Ransome's patent stone, are all artificial sandstones, in which
the cement is a silicate of lime. They differ mostly in the
process of making.
A patent stone has been made in San Francisco and in
Chicago, and employed to some extent in those cities.
Principles of Manufacture. Dry sand and a solution of
silicate of soda, about a gallon of the silicate to a bushel of
sand, are thoroughly mixed in a suitable mill, and then
moulded into any of the forms required. These blocks or
forms are then saturated by a concentrated solution of calcium
chloride, which is forced through the moulded mass by exhaus-
tion of the air, by gravity, or by other suitable means. The
chemical reactions result in the formation of an insoluble
ASPHALTIO CONCRETE. 31
Bilicate of lirne, which firmly unites all the grains of the mass
into one solid, and a solution of sodium chloride (common
salt). The latter is removed by washing with water.
Remark. The artificial stone thus formed is uniform and
homogeneous in its texture, and said to be free from liability
to distortion or shrinkage. It is also claimed that it is not
affected by variations of climate or temperature.
3D. ASPHALTIO CONCRETE.
63. Asphaltic Concrete is a concrete in which the solid
materials are united by mastic, a mixture of powdered lime-
stone, or similar material, with artificial or natural combina-
tions of bituminous or resinous substances.
The manufacture of mastics will be described under the
head of UNITING MATERIALS ; the manufactured product may
be bought in blocks ready for use.
Asphaltic concrete is made as follows :
The mastic is broken into small pieces, not more than half
a pound each, and placed in a caldron, or iron pot, over a fire.
It is constantly stirred to prevent its burning, and as soon as
melted there is gradually added two parts of sand to each
one of the mastic, and the whole mass is constantly stirred
until the mixture will drop freely from the implement used
in stirring.
The ground having been made perfectly firm and smooth,
covered with ordinary concrete, or otherwise prepared, the
mixture is applied by pouring it on the surface to be coated,
taking care to spread it uniformly and evenly throughout.
A square or rectangular strip is first laid, and then a second,
and so on, until the entire surface is completely covered, the
surface of each square being smoothed with the float. Before
the concrete hardens a small quantity of fine sand is sifted
over it and is well rubbed in with a trowel or hand-float.
The thickness of the coating will depend upon its situa-
tion, being less for the capping of an arch than for the floor-
ing of a room, and less for the latter than for a hall or pave-
ment that is to be in constant use.
Care is taken to form a perfect union between edges of
adjoining squares, and, where two or more thicknesses are
used, to make them break joints.
A mixture of coal tar is frequently used as a substitute for
mastic.
Uses. The principal uses of asphaltic concrete are for pav
ing streets, side- walks, floors of cellars, etc.
32 CIVIL ENGINEERING.
4TH. GLASS.
64. Glass is a mixture of various insoluble silicates. Its
manufacture depends upon the property belonging to the al-
kaline silicates, when in a state of fusion, of dissolving a
large quantity of silica. The mixture hardens on cooling,
and is destitute of crystalline structure.
Uses. Glass is extensively used in building, as a roof-
covering for conservatories, ornamental buildings, railroad
depots, and other structures for which the greatest possible
light or the best-looking material is required. Other uses,
as for windows, sky-lights, doors, etc., are familiar to every
one.
65. Glazing is the art of fixing glass in the frames of win-
dows. The panes are secured with putty, a composition of
whiting and linseed-oil with sometimes an addition of white
lead. Large panes should be additionally secured by means
of small nails or brads.
CHAPTER III.
METALS.
66. The metals used in engineering constructions are Iron,
Steel, Copper, Zinc, Tin, Lead, and some of their alloys.
IRON AND STEEL.
67. Iron has the most extensive application of all the
metals used for building purposes. It is obtained from tho
ore by smelting the latter in a blast-furnace. When the fuel
used is coal, the blast is generally of hot-air; in this process,
known as the hot-blast* the air, before being forced into the
furnace, is heated high enough to melt lead.
When the metal has fused, it is separated from the other
substances in the ore, and is allowed to combine with a small
amount of carbon, from 2 to 5 per cent., forming a com-
pound known as cast-iron.
A sufficiency of cast-iron having accumulated in the fur-
CAST-IBON. 33
aace, the latter is tapped, and the molten metal running out
is received in sand in long straight gutters, which have
numerous side branches. This arrangement is called the sow
and pigs ; hence the name of pig-iron.
The iron in the pig is in a shape to be sent to market, and
in suitable condition to be remelted and cast into any re-
quired form, or to be converted into wrought or malleable
iron.
Impurities. The strength and other good equalities of the
iron depend mainly on the absence of impurities, and espe-
cially of those substances known to cause brittleness and weak-
ness, as sulphur, phosphorus, silicon, calcium, and magnesium.
CAST-IRON-.
68. Cast-iron is a valuable building material, on account
of its great strength, hardness, and durability, and the ease
with which it can be cast or moulded into the best forms for
the purposes to which it is to be applied
Varieties of Cast-iron. Cast-iron is divided into six varie-
ties, according to their relative hardness. This hardness
seems to depend upon the proportion and state of carbon in
the metal, and apparently not so much on the total amount
of carbon present in the specimen, as on the proportionate
amounts in the respective states of mechanical mixture and
of chemical combination. Manufacturers distinguish tho
different varieties by the consecutive whole numbers from 1
to 6.
No. 1 is known as gray cast-iron, and No. 6 as white
cast-iron. They are the two principal varieties.
Gray Cast-iron, of good quality, is slightly malleable when
cold, and will yield readily to the action of the file if the
hard outside coating is removed. It has a brilliant fracture of
a gray, sometimes bluish gray, color. It is softer and tough-
er, and melts at a lower temperature, than white iron.
White Cast-Iron is very brittle, resists the file and chisel,
and is susceptible of high polish. Its fracture presents a sil-
very appearance, generally fine-grained and compact.
The intermediate varieties, as they approach in appear-
ance to that of No. 1 or No. 6, partake more or less of the
properties characteristic of the extreme varieties.
iN umbers 2 and 3, as they are designated, are usually con-
sidered the best for building purposes, as combining strength
and pliability.
3
34 CIVIL ENGINEERING.
Appearances of Good Cast-iron.
69. A medium-sized grain with a close compact texture in
dicates a good quality of iron. The color and lustre present-
ed by the surface of a recent fracture are good indications of
its quality. A uniform dark-gray'color with a high metallic
lustre is an indication of the best and strongest iron. With
the same color, but less lustre, the iron will be found to be
softer and weaker. No lustre with a dark and mottled color
indicates the softest and weakest of the gray varieties.
Cast-iron, of a light-gray color and high metallic lustre, is
usually very hard and tenacious. As the color approaches to
white, and as the metallic changes to a vitreous lustre, hard-
ness and brittleness of the iron become more marked ; when
the extreme, a dull or grayish white color with a very high
vitreous lustre, is attained, the iron is of the hardest and most
brittle of the white variety.
70. Test of its Quality. The quality of cast-iron may be
tested by striking a smart stroke with a hammer on the edge
of a casting. If the blow produces a slight indentation,
without any appearance of fracture, the iron is shown to be
slightly malleable, and therefore of a good quality ; if, on
the contrary, the edge is broken, there is an indication of brit-
tleness in the material, and consequent want of strength.
71. Strength. The strength of cast-iron varies with its
density, and the density depends upon the temperature of the
metal when drawn from the furnace, the rate of cooling, the
head of metal under which the casting is made, and the bulk
of the casting.
From the many causes by which the strength of iron may
be influenced, it is very difficult to judge of the quality of a
casting by its external characters ; however, a uniform ap-
pearance of the exterior devoid of marked inequalities of sur-
face, generally indicates uniform strength ; and large castings
are generally proportionally weaker than small ones.
WROUGHT OR MALLEABLE IRON.
72. Wrought, or Malleable Iron, in its perfect condition,
is simply pure iron.
It generally falls short of such condition to a greater or less
extent, on account of the presence of the impurities referred
to in a previous paragraph. It contains ordinarily more than
one-quarter of one per cent, of carbon.
WROTJGHT-IBON. 35
It may be made by direct reduction of the ore, but it is
usually made from cast-iron by the process called pud-
dling.
Wrought-iron is tough, malleable, ductile and infusible in
ordinary furnaces. At a white heat it becomes soft enough
to take any shape under the hammer, and admits of being
welded. In order to weld two pieces together, each surface
should be free from oxide. If there be any oxide present, it
is easily removed by sprinkling a little sand or dust or borax
over the surfaces to be joined ; either of these forms with the
rust a fusible compound, which is readily squeezed out by the
hammering or rolling.
Appearances of good Wrought-iron.
73. The fracture of good wrought-iron should have a clear
gray color, metallic lustre, and a fibrous appearance. A
crystalline structure indicates, as a rule, defective wrought-
iron. Blisters^ flaws^ and cinder-holes are defects due to bad
manufacture.
Strength. The strength of wrought-iron is very variable,
as it depends not only on the natural qualities of the metal,
but also upon the care bestowed in forging, and upon the
greater or less compression of its fibres when it is rolled or
hammered into bars.
Forms. The principal forms in which wrought-iron is
sent to market are Bar-iron, Round-iron, Hoop and Sheet-
iron, and Wire.
Bar-iron comes in long pieces with a rectangular cross-
section, generally square, and is designated as 1 inch, 1J inch,
2 inch, according to its dimensions. It is then cut and worked
into any shape required.
Bars receive various other forms of cross-section, depend-
ing upon the uses that are to be made of them. The most
common forms are the T, H, I, and L, cross-sections, called
T-iron, H -iron, etc., from their general resemblance to these
letters, and one whose section is of this shape, i ', called
channel iron. The section like an inverted U is frequently
seen.
Round iron comes in a similar form, except the cross-sec-
tion is circular, and it is known, in the same way, as 1 inch, 2
inch, etc.
Hoop and Sheet-iron are modifications of bar-iron, the
thickness being very small in comparison with the width.
Corrugated iron is sheet-iron of a modified form, by which
86 CTVTL ENGINEERING.
its strength and stiffness are
greatly increased. The dis-
tance between the corruga-
tions, A B, (Fig. I.) varies,
being 3, 4, or 5 inches ; the
depth, B 0, being about one-
fourth A B.
Iron Wire. The various sizes of wire might be consid-
ered as small sizes of round-iron, distinguished by numbers
depending on the dimensions of cross-section, except that wire
is drawn through circular holes in a metal plate, while round-
iron is rolled^ to obtain the requisite cross-sections.
The numbers run from to 36 ; No. wire has a diameter
equal to one-third of an inch, and No. 36 one equal to .004
or an inch; the other numbers being contained between
these, and the whole series being known as the Birmingham
Wire Gauge.
A series in which the numbers run from to 40, the ex-
tremes being nearly the same as that just given, is sometimes
used. It is known as the American Gauge.
STEEL.
74. Steel, the hardest and strongest of the metals, is a
chemical combination of iron and carbon, standing between
wrought and cast-iron.
No sharp dividing line can be drawn between wrought-iron
and steel, based on the proportions of carbon present in the
product. The differences in their physical properties are
largely due to the process of manufacture. Many of the
properties peculiar to wrought-iron have been found to dis-
appear upon melting the iron, showing that they were the re-
sult of the manipulation to which the iron was subjected.
The term steely-iron, or semi-steel, has been applied wher
the compound contains less than 0.5 per cent, of carbon ;
steel, when containing more than this, and less than 2 per
cent. ; but when 2 per cent, or more is present, the compound
is termed cast-iron, as before stated.
75. Sieel is made from iron by various processes, which
are of two general classes ; the one in which carbon is added
to malleable iron ; the other in which a part of the carbon is
abstracted from cast-iron. Like iron, steel is seldom pure,
but contains other substances which, as a rule, affect it inju-
riously. There are, however, some foreign substances which,
introduced into the mass during manufacture, have a bene-
STEEL. 37
ficial effect upon the steel by increasing its hardness and
tenacity and making it easier to forge and weld.
76. Steel, used for building purposes, is made generally by
one of three processes :
1. By fusion of blister steel in crucibles ; as cast-steel ;
2. By blowing air through melted cast-iron ; as Bessemer
eteel; or
3. By fusion of cast-iron on the open hearth of a rever-
beratory furnace, and adding the proper quantities of malle-
able iron or scrap steel ; as Siemens-Martin steel.
77. The different kinds of steel are known by names given
them either from their mode of manufacture, their appear-
ance, from some characteristic constituent, or from some in-
ventor's process; such are German-steel, blister-steel, shear-
steel, cast-steel, tilted-steel, puddled-steel, granulated-steel,
Bessemer-steel, etc.
German-steel is produced direct from certain ores of iron,
by burning out a portion of the carbon in the cast-iron ob-
tained by smelting the ore. It is largely manufactured in
Germany, and is used for files and other tools. It is also
known as natural steel.
Blister-steel is made by a process known as " cementation"
which produces a direct combination of malleable iron and
carbon. The bars, after being converted into steel, are found
covered with blisters, from which the steel takes its name. It
is brittle, and its fracture presents a crystalline appearance.
It sometimes receives the name of bar-steel.
Shear-steel is made by putting bars of blister-steel to-
gether, heating and welding them under the forge-hammer,
or between rolls ; the product is called " Shear-steel,"
"Double," "Single," or "Half," from the number of times
the bars have been welded together. It is used for tools.
Cast-steel, known also as crucible-steel, is made by break-
ing blistered steel into small pieces, and melting it in close
crucibles, from which it is poured into iron moulds. The
resulting ingot is then rolled or hammered into bars.
Its fracture is of a silvery color, and shows a fine, homoge-
neous, even, and close grain. It is very brittle, acquires ex-
treme hardness, and is difficult to weld without a flux.
This is the finest kind of steel, and the best adapted for
most purposes in the arts ; but, from its expensiveness, it is
not much used in building.
Tilted-steel is made from blistered steel by moderately
heating the latter and subjecting it to the action of a tilt or
trip-hammer ; by this means the tenacity and density of the
steel are increased.
38 CIVIL ENGINEERING.
Puddled-steel is made by puddling pig-iron, and stopping
the process at the instant when the proper proportion of car-
bon remains.
Granulated-steel is made by allowing the melted pig-iron
to fall into water, so that it forms into grains or small lumps ;
the latter are afterwards treated so as to acquire the proper
proportion of carbon, and are then melted together.
Bessemer-steel, which takes its name from the inventor of
the process, is made by direct conversion of cast-iron into
steel. This conversion is effected either by decarbonizing
the melted cast-iron until only enough of carbon is left to
make the required kind of steel, or, by removing all the car-
bon, and then adding to the malleable iron remaining in the
furnace the necessary proportion of carbon ; the resulting
product is then immediately run into large ingots.
Siemens-Martin steel is another variety of steel obtained
directly from the cast-iron, and takes its name from the in-
ventors of the process. In this process, the carbon is not
removed by a blast of atmospheric air, as in the Bessemer
process, but by the oxygen of the iron ore or iron scales, etc.,
the oxygen being freed as a gas during combustion.
In each of the last two processes, the temperature is so
great as to melt wrought-iron with ease.
There are other kinds of steel, possessing certain character-
istics peculiar to themselves or claimed for them, but whose
process of manufacture is not publicly known.
78. Hardening and Tempering. Steel is more granular
than iron, and is much more easily melted, but the great dif-
ference between them is the capability of the steel to become
extremely hard and elastic when tempered. The quality of
the steel depends in a great measure on the operation of hard-
ening and tempering.
It is hardened by being heated to a cherry-red color, and
then being suddenly cooled by being plunged into some cold
liquid. In this way it is rendered very brittle, and so hard
as to resist the hardest file. To give elasticity, it is tem-
pered ; this is done by heating the hardened steel to a cer-
tain degree, and cooling it quickly ; the different degrees of
heat will depend upon the use to which the steel is to be put.
These qualities of hardness and elasticity .adapt it for vari-
ous uses, for which neither cast nor wrought-iron. would bo
suitable.
DURABILITIY OV IRON AND STEEL.
79. Constructions in these metals are, like those in woorl,
subject to the same general conditions. They may be ex-
PROTECTION OF IRON WORK. #$
posed to the air in a dry place, or in a damp place, be kept
alternately wet and dry, or be entirely immersed in fresh or
salt water.
Their exposure to the air or moisture, especially if an acid
be present, is followed by rusting which proceeds with
rapidity after it once begins. The corrosion is more rapid
under exposure to alternate wetness and dryness than in
either of the other cases.
Cast-iron is usually coated with a film of graphite and
ferrous silicate, produced by the action of the sand of the
mould on the melted iron ; this film is very durable, and,
if not injured, the casting will last a long time without
rusting.
Iron kept in a constant state of vibration rusts less rapidly
than in a state of rest.
Iron completely imbedded in brick-work or masonry ia
preserved from rust, and in cathedrals and other ancient
buildings it has been found in good condition after six hun-
dred years. In these cases the iron was probably protected
by the lime in the mortar, the latter being a good pre-
servative.
The rapid deterioration of iron-work when exposed to the
air and to moisture makes its protection, so as to increase its
durability, a matter of great importance.
PROTECTION OF IRON-WORK.
80. The ordinary method, used to protect iron from rust,
is to cover its surface with some material that withstands the
action of the air and moisture, even if it be for a limited time.
The following are some of the methods :
By painting." The surface of the iron is covered with a
coat of paint. Eed and white lead paints, ochreous or iron
oxide paints, silicate paints, and bituminous paints, all are
used. For this purpose, the value of the paint depends
greatly upon the quality of the oil with which it is mixed.
The painting must be renewed from time to time.
By japanning-. The iron being placed in a heated cham-
ber, or furnace, the paint is there applied, and is to some
extent absorbed by the iron, forming over it a hard, smooth,
varnish-like coating.
By the use of coal-tar. The iron is painted with coal-tar
alone or mixed with turpentine or other substances ; another
method consists in first heating the iron to about 600 Fahr.,
and then boiling it in the coal-tar.
40 CIVIL ENGINEERING.
By the use of linseed oil. The iron is heated, and the
surface while hot is smeared over with cold linseed-oil.
By galvanizing. This term, "galvanized iron/' is ap-
plied to articles of iron coated with zinc. The iron, being
thoroughly cleaned and free from scale, is dipped into a bath
of melted zinc, and becomes perfectly coated with it. This
coating protects the iron from direct action of the air and
moisture, and as long as it lasts intact the iron is perfectly
free from rust.
COPPER.
81. This metal possesses great durability under ordinary
exposure to the weather, and from its malleability and tena-
city is easily manufactured into thin sheets and fine wire.
When used for building purposes, its principal application
is in roof-coverings, gutters, and leaders, etc. Its great
expense, compared with the other metals, forms the chief
objection to its use.
ZINC.
82. This metal is used much more than copper in building,
as it is much cheaper and is exceedingly durable. Though
zinc is subject to oxidation, the oxide does not scale off like
that of iron, but forms an impervious coating, protecting the
metal under it from the action of the atmosphere, thus ren-
dering the use of paint unnecessary.
In the form of sheets, it can be easily bent into any required
shape.
The expansion and contraction caused by variations of tem-
perature are greater for zinc than iron, and when zinc is used
for roof -coverings, particular attention must be paid to seeing
that plenty oiptay is allowed in the laps.
Zinc, before it is made into sheets or other forms, is called
spelter.
TIN.
83. This metal is only used, in building, as a coating for
sheet-iron or sheet-copper, protecting their surfaces from
oxidation.
LEAD.
84. This metal was at one time much used for roof -cover-
ing, lining of tanks, etc. It ip now almost entirely super-
aeded by the other metals.
TTNTTINQ MATERIALS. 41
It possesses durability, but is wanting in tenacity ; this
requires the use of thick sheets, which increase both the
expense and the weight of the construction.
ALLOTS.
85. An alloy is a compound of two or more metals,
mixed while in a melted state. Bronze, gun-metal, bell-
metal, brass, pewter, and the various solders are some
of the alloys that have a limited application to building pur-
poses.
CHAPTER IV.
UNITING MATERIALS.
86. Structures composed of wood and iron have their dif-
ferent portions united principally by means of straps and
pins made of solid materials; in some cases, especially in
the smaller structures, a cementing material is used, as glue,
etc.
The use of straps, pins, and like methods of fastenings
will be described under the head of FRAMING.
Structures composed of stone have their different portions
united principally by cementing materials, as limes, cements,
mortars, etc.
GLUE.
67. Glue is a hard, brittle, brownish product obtained by
boiling to a jelly the skins, hoofs, and other gelatinous parts
of animals, and then straining and drying it.
When gently heated with water, it becomes viscid and
tenacious, and is used as a uniting material. Although pos-
sessing considerable tenacity, it is so readily impaired by
moisture that it is seldom used in engineering constructions,
except for joiner's work.
42 CIVIL ENGINEERING.
LIMES AND CEMENTS.
LIMES.
88. If a limestone be calcined, the carbonic acid will be
driven off in the process, and the substance obtained is gen-
erally known as lime.
This product will vary in its qualities, depending on the
amount and quality of the impurities of the limestone. As
a building material, the products are divided into three prin-
cipal classes :
1. Common or fat lime.
2. Hydraulic lime.
3. Hydraulic cement.
Common lime is sometimes called air-lime, because a paste
made from it with water will harden only in the air.
Hydraulic lime and cement are also called water limes
and cements, because a paste made from either of them
with water has the valuable property of hardening under
water.
The principal use of the limes and cements in the engineer's
art is as an ingredient in the mortars and concretes.
Varieties of Limestone.
89. The majority of limestones used for calcination are
not pure carbonates, but contain various other substances, the
principal of which are silica, alumina, magnesia, etc.
If these impurities be present in sufficiently large quan-
tities, the limestone will yield on calcination a product pos-
sessing hydraulic properties.
Limestones may therefore be divided into two classes, or-
dinary and hydraulic, according as the product obtained by
calcination does or does not possess hydraulic properties.
90. Ordinary Limestone. A limestone which does not
contain more than ten per cent, of these impurities, produces
common lime when calcined. White chalk, and statuary
marble, are specimens of pure limestone.
91. Hydraulic Limestones. Limestones containing more
than ten per cent, of these impurities are called hydraulic
limestones, because they produce, when properly calcined, a
lime having hydraulic properties.
HYDRAULIC LIMESTONES. 43
The hydraulic limestones are subdivided into silicious,
argillaceous, magnesian and argillo-magnesian, according
to the nature of the predominating impurity present in the
stone.
Physical Characters and Tests of Hydraulic Limestones.
92. The simple external characters of a limestone, as color,
texture, fracture, and taste, are insufficient to enable a person
to decide whether it belongs to the hydraulic class.
Limestones are generally of some shade of drab or of gray,
or of a dark grayish blue ; have a compact texture, even or
conchoidal fracture, a clayey or earthy smell and taste. Al-
though the hydraulic limestones are usually colored, still the
stone may happen to be white, from the combination of lime
with a pure clay.
The difficulty of pronouncing upon the class to which a
limestone belongs renders necessary a resort to chemical
analysis and experiment.
To make a complete chemical analysis of a limestone re-
quires more skill in chemical manipulations than engineers
usually possess ; but a person who has the ordinary element-
ary knowledge of chemistry can ascertain the quantity of
clay or of magnesia contained in a limestone, and (know-
ing this) can pronounce, with tolerable certainty, as to the
probabilities of its possessing hydraulic properties after cal-
cination.
Having from the proportions ascertained that the stone will
probably furnish a lime with hydraulic properties, a sample
of it should be submitted to experiment. The only apparatus
required for this purpose is a crucible that will hold about a
pint, and a mortar and pestle. The bottom as well as the top
or cover of the crucible should be perforated to give an up-
ward current of air and allow the carbonic acid to escape.
The stone to be tested is broken into pieces as nearly the
same size as possible, not exceeding three-fourths of an inch
cube, and placed in the crucible. When more than one speci-
men is to be tried, and a comparison between them made,
there should be several crucibles. Access being had to an
anthracite coal-fire in an open grate, or to any other steady
fire, the crucibles are embedded in and covered with glowing
coals, so that the top and bottom portions of their contents
will attain simultaneously a bright- red heat, each crucible
containing as nearly as possible the same quantity of stone.
If there be only one crucible, two or three of the fragments
are removed in forty-five minutes after the stone has
44 CIVIL ENGINEERING.
reached a red heat ; in forty-five minutes afterwards two or
three more are taken out, and this repeated for f our ^ and a
half and perhaps six hours, which time will be sufficient to
expel all the carbonic acid. If there be several crucibles,
they themselves may be removed in the same order. By this
means we will have some samples of the stone that are burnt
too much, some not enough, and some of a class between
them.
The specimen, if a cement, will not slake when sprinkled
with water. By reducing it to a powder in the mortar, mix-
ing it to a stiif paste with water, immersing it in fresh or salt
water, and noting the time of setting and the degree of hard-
ness it attains, an approximate value of the cement may be
obtained.
Calcination of Limestones.
93. As the object in burning limestone is to drive ofF the
water and carbonic acid from the limestone, many devices
have been used to effect it. A pile of logs burning in the
open air, on which the limestone or oyster-shells are thrown,
has been frequently used to obtain common lime. It is, how-
ever, generally manufactured by burning the limestone in a
kiln suitably constructed for the purpose.
94. Kilns are divided into two classes : 1st, the intermit-
tent kilns, or those in which the fuel is all at the bottom,
and the limestone built up over it; and, 2d, the perpetual
or draw kiln, in which the fuel and the limestone are
placed in the kiln in alternate layers. The fuel used is
either wood or coal. In the first class one charge of lime is
burned at a time, and, when one burning is complete, the kiln
is completely cleared out previous to a second ; while in the
latter class fresh layers of fuel and limestone are added at
the top as the lime is drawn out at the bottom.
The shapes given to the interiors of kilns are very different.
The object sought is to obtain the greatest possible uniform
heat with the smallest expenditure of fuel, and for this pur-
pose thick walls are necessary to prevent loss of heat by radi-
ation.
95. Intermittent Kilns. The simplest form of kiln is
that represented in Fig 2, in which wood is used for fuel. It
has a circular horizontal cross -section, and is made of ham-
mered limestone without mortar.
The cut represents a vertical section through the axis and
arched entrance communicating with the interior of a kiln
for burning lime with wood ; 0, , c, large pieces of limestone
LIME-KILNS.
45
forming the arch upon which the mass of limestone rests ; A ,
arched entrance communicating with the interior.
FIG. 2.
It is usually placed on the side of a hill, so that the top
may be accessible for charging the kiln.
The largest pieces of the limestone to be burned are
formed into an arch, , , 0, and above this the kiln is filled
by throwing the stone in loosely from the top, the largest
stones first and smaller ones afterwards, heaping them up, as
shown in the figure. The fuel is supplied through the
arched entrance, A.
The circular seems the most suitable form for the horizon-
tal sections of a kiln, both for strength and for economy of
heat. Were the section the same throughout, or the form of
the interior of the kiln cylindrical, the strata of stone, above
a certain point, would be very imperfectly burned when the
lower strata were calcined just enough, owing to the rapidity
with which the inflamed gases arising from the combustion
are cooled by coming into contact with the stone. To pro-
cure, therefore, a temperature which shall be nearly uniform
throughout the heated mass, the horizontal sections of the
kiln should gradually decrease from the point where the
flame rises, which is near the top of the dome of broken
stone, to the top of the kiln. This contraction of the hori-
zontal section from the bottom upward should not be made
46 CIVIL ENGINEERING.
too rapidly,
whence it is seen that the work expended upon the elongation
of the bar varies directly with the square of the force pro-
ducing it, with the length of the bar, and inversely with the
area of cross section and coefficient of elasticity.
TENSION.
85
Elongation of a bar, its -weight considered.
161. To determine the elongation of
a bar, under the same circumstances
as the preceding case, when its weight
is taken into consideration.
In eq. (2), the weight of the bar
being very small compared with W, it
was neglected.
To determine the elongation, con-
sidering the weight of the bar, repre-
sent (Fig. 13) by L, W, I, and A, the
same quantities as before, by a?, the
distance from A of any section as C,
by dx, the length of an elementary
portion as C D, and by w, the weight
of a unit of volume of the bar. The
volume of the portion B C, will be ex-
pressed by (L a?) A ; and its weight
by (L x) Aw. FIG. 13.
The total force acting to elongate the elementary portion
C D, will be expressed by
W -f (L x) Aw.
Substituting this for W, and dx for L in eq. (2), we have
elongation of dx =
x) Aw
,
dx.
The total length of dx after elongation will, therefore, be
, W+(L x) Aw,
EA
Integrating this between the limits x = and SB = L, there
obtains,
WL
L + 1 =
. . . . (7)
for the total length of the bar after elongation
This may be written,
flfi CIVIL ENGINEERING.
If, in this expression, we make W = 0, we have
In this, wAL is the weight of the bar; representing this
weight by W and substituting in last expressson, we have
EA
or the elongation due to the weight of the bar, is one half of
what it would be if a weight equal to that of the bar were
concentrated at the lower end.
An examination of the expression, W+ (L x) Aw, shows
that the strain on the different cross-sections varies with x,
decreases as x increases, and is greatest for x = 0, or on the
section at the top. Since the bar has a uniform cross-section,
the strain on the unit of area is different in each section.
BAB OP UNIFORM STRENGTH TO RESIST ELONGATION.
162. To determine the form a vertical bar should have, in
order to be equally strong^ throughout, when strained only ~by
a force acting in the direction of the axis of the bar, the
weight of the bar being considered.
Suppose the bar, fixed at one end aiid the applied force
producing elongation to be a weight suspended from the
other end. [Fig. 14]
From the preceding article, it is seen that if the bar has a
uniform cross-section, that the strain on each section is dif-
ferent. In order that the bar should be equally strong
throughout, the strain on each unit of area of cross-section
must be the same throughout the bar. This can only be
effected by making the area of the cross-section proportional
to the stress acting on it, or having the cross-sections variable
in size.
Represent by
A, the area of the variable cross-section ;
A', the area of cross-section at B, or the lower one ;
A", the area of cross-section at A, or the top section^;
T,, the strain allowed on the unit of area ;
W, the force applied to the bar producing elongation ;
a?, the distance, B C, estimated upwards from B.
TENSION.
87
The total force acting on any
section as C, to elongate it, is
W+w
w being the weight of the nnit
of volume of the bar.
Since T, is the strain allowed
on the unit of area, T, x A will
represent the total strain on
the section at C, and will be
equal to the force acting on this
section to elongate it. Hence,
we have
(8)
Differentiating, we have
wAdx = T^A,
which may be written
wdx_dA.
^r r= x'
Integrating, we get
wx
-TT =Nap. log A+C.
(9)
Making x = 0, we have A = A', whence
Substituting for C in eq. (9) its value obtained from the
last equation, we get
= tfap.log-^
and passing to the equivalent numbers,
W
But
A'=
which substituted above gives,
33 CIVIL ENGINEERING.
Making x = L and A becomes equal to A", hence
A"- W ^
A 7p-0
*-!
the value for the area of the section at the upper end.
Form of bar when it has a circular cross-section.
163. No particular form has been assigned to the cross sec
tion of the bar in this discussion. Let it be a circle and rep
resent the variable radius by r.
Then the area of any cross-section will be TT/**, which being
substituted for A in eq. (8), gives
W + w Cn^dx = TjTiT 9 .
Differentiating, there obtains
W7rr*dx =
hence
dr_ w
~ "2T,
which integrated gives
Nap. log. r = j^x + 0, . . (10)
which shows the relation between x and r.
Eq. (10) is the equation of a line, which line being con-
structed will represent by its ordinates the law of variation of
the different cross-sections of the bar. It also shows the kind
of line cut from the bar by a meridian plane.
The most useful application of this problem is to determine
the dimensions of pump-rods, to be used in deep shafts, like
those of mines.
COMPRESSION.
164. The strains caused by pressure acting in the direction
of the axis of the piece tend to compress the fibres and shorten
the piece.
DEPRESSION. 89
From the principle that v all bodies are elastic, it follows
that all building materials are compressible.
Within the limit of elasticity it is assumed that the resist-
ances to compression are the same as tension. They are not
really the same ; but within the elastic limit the differences
are so small, that for all practical purposes it is sufficiently
exact to consider them equal.
The coefficient of elasticity of the material is assumed the
same in both cases, and to distinguish it from the coefficients
of elasticity when the fibres are displaced in other ways, it is
sometimes called the coefficient of longitudinal elasticity,
or resistance to direct lengthening or shortening.
To ascertain the force under which a given piece would be
crushed, we first ascertain the weight necessary to crush a
piece of the same material ; and since experiment has shown
that the resistances of different pieces of the same material to
crushing are nearly proportional to their cross-sections, the
required force can be easily determined.
Assuming that these resistances are directly proportional
to the cross-sections, let W be the required force, A the area
of cross-section of given piece, and C the force necessary to
crush a piece of the same material whose cross-section is
unity.
We have, W' : C :: A:l,or
W 7 = AC, (11)
hence JP = C . (12)
A.
Many experiments have been made on different materials
to find the value of C, and the results tabulated. If the ex-
periments for finding C were not made on pieces whose
cross-sections were unity, they were reduced to unity by
means of eq. (12). The pieces used in the experiments
were short, their lengths not being more than five times their
diameter or least thickness.
This value of C, the modulus of crushing, is equal
therefore to the pressure, upon the unit of surface, necessary
to crush a piece whose length is less than five times its least
thickness, the pressure being uniformly distributed over the
cross section and acting in the direction of the length of the
piece. Experiment shows that it requires a much less press-
ure to crush apiece when the force is applied across the fibres,
than when it is applied in the direction of their length.
- 165. The following are the values of C for some of the ma-
90 CIV ILi ENGINEERING.
terials in common use, and were obtained by crushing pieces
of small size, and as a rule not longer than twice their diame
ter:
Material Crushing Forces per sq. inch, in Ibs.
Ash 4,475 to 8,783
Chestnut 5,000
Cedar 5,970
Hickory 5,492 " 11,213
Oak, white 5,800 10,058
Oak,live 6,530
Pine 5,017 8,947
Fir 6,644 9,217
Hemlock 6,817
Cast iron 56,000 105,000
Wrought iron 30,000 40,000
Cast steel 140,000 " 390,000
Brick 3,500 " 13,000
Granite 5,500 " 15,300
Rankine gives from 550 to 800 for common red brick, and
1,100 for strong red brick.
The remarks relative to the specimens of wood used to
obtain the values of T in the table on page 83 apply equally
to this case.
SHEARING STRAINS.
166. There are two kinds of simple shear ; one in which
the stress acts normally to all the fibres, like that developed
in a rivet when the plates which it fastens are strained by
tension or compression in the direction of their lengths ; and
one in which the stress acts in a plane parallel to the fibres,
either in the direction of, or across, the fibre. The former is
called a transverse shear, and the latter, detrusion.
The relations between the strains and the stresses devel-
oped by a shearing force may be expressed by equations
analogous to those used for tension.
In describing the shearing strain, the section C D (Fig. 15)
was supposed not to have rotated around any line in its
plane, but to have had a motion of translation parallel to
the plane A B, so that after the movement, any fibre, as
db t will have a new position, as ab'.
SHEARING STRAIN.
91
Suppose A B to remain fixed, and represent by
L, the original length of any fibre ab
between the two consecutive planes A B
and C D ;
y, the distance W which every point
of the plane C D has moved in the
direction of C D, relatively to the
plane A B, owing to the force causing FIG. 15.
this displacement ;
5, the amount of shearing stress in any fibre ;
a, the area of the cross-section of the fibre ;
E', a constant.
Then,
= the intensity of the shearing stress on a unit of area,
and --= the measure of displacement of the fibre per unit
of length. Hence,
- = E'-f (13)
a L
from which we get
E'=
a value analogous to that obtained for E in equation (1).
This value of E' is constant within the limit of elasticity
for each elementary fibre. If the material is homogeneous
it has the same value for all the fibres, or is constant for the
same material.
Represent by S t the total stress developed in the section
CD; by A, the area of the section ; and let the piece be of
homogeneous material. Then,
(14)
which expresses the relation between the total stress de-
veloped in the section and the shearing strain.
The constant E' is the coefficient of elasticity correspond-
ing to a transverse shearing strain, and is frequently called
the coefficient of lateral elasticity, to distinguish it from
the coefficient of longitudinal elasticity.
02 CIVIL ENGINEERING.
The shear is assumed to be distributed uniformly over the
cross-section of the material. Suppose the shear to be in-
creased until rupture takes place and let S represent the in-
tensity of the total shearing stress on the cross-section.
Then,
in which S is the modulus of shearing for the material
167. The following are some of the values of S, obtained
by experiment, for some of the building materials in use, viz. :
TRANSVERSE SHEARING.
Materials. Value of S.
Ash G,2801b8.
Cedar -. 3,400
Hickory 6,500
Oak, White 4,000
Oak, Live 8,000
Pine, Yellow. 4,500
Pine, White 2,500
Cast steel 92,400
Wrought iron 50,000 "
Cast iron 30,000
Copper 33,000 "
DETRUSION.
White pine 480 Ibs.
Spruce ^ 470
Fir 592
Hemlock 540 "
Oak 780 "
TRANSVERSE STRAIN.
168. Extraneous forces acting either perpendicularly or
obliquely to the axis of a piece that is fixed, cause cross-
strains and develop transverse stresses in the material.
In describing the nature of a cross-strain (Art. 150), it is
assumed that a consecutive section of the piece, as C D (Fig.
16), could not take a position as C' D' unless the fibres on
one side of the axis of rotation were lengthened and those
on the other side shortened. Also, that the fibres farthest
from this axis were elongated or shortened more than those
TRANSVERSE STRAIN. 93
nearest to it, and as a consequence the stresses in the fibres
were variable in their intensities throughout the cross-sec-
tion.
To determine the relations between the strains of the
fibres caused by the bending forces and the corresponding
stresses developed, a theory must be adopted relating to the
strains produced, and a law assumed for the distribution of
the stresses over the cross-section.
Suppose a piece of homogeneous material, in form of a
bar or beam, to be placed in a horizontal position and fixed
at one end, and suppose this piece to be acted upon by a sys-
tem of extraneous forces, the resultant, W, of which is per-
pendicular to the axis and intersects it at the free end.
The action of this system of extraneous forces is to bend
the piece, causing cross-strains and developing both trans-
verse and shearing stresses throughout the piece.
Neglecting the shearing stress for the present, let it be
required to determine the relations between the cross-strains
and the transverse stresses produced by the lending force, W.
The cross-sections of the piece are assumed to be uniform,
or to vary from each other by some law of continuity that is
known ; the forms of the cross-sections are similar, and for
any two consecutive sections may be considered to be equal.
The common theory for the strains, deduced from obser-
vation and experiment, is as follows, viz. :
1. That the fibres on the convex side of the piece are ex-
tended, and those on the opposite side are compressed.
2. That the strains of the fibres caused by the bending
force are either compressive or tensile.
3. That there is a surface between the compressed and ex-
tended fibres in which the fibres are neither compressed nor
extended.
4. That the strains of the fibres are proportional to their
distance from this surface, known as the neutral surface.
5. That the cross-sections of the piece normal to the
fibres before bending will remain normal to them after
bending.
6. That rupture will take place either by compression, or
by extension, of the fibres on the surface of the piece when
the stress is equal to the modulus of rupture.
The intersection of the neutral surface by the plane of
cross-section is called the neutral axis of the section.
From this theory, it follows, that the intensities of the
stresses of tension "and compression in the fibres^ are also
proportional to their distances from the neutral axis as long
as the strain is within the elastic limit. The stress devel-
94 CIVIL ENGINEERING.
oped on a cross-section to resist the action of a bending
force is, therefore, a uniformly varying one ; being least, or
zero, at the neutral axis, and greatest at the points farthest
from this axis.
To find the stress in any fibre in terms of the strain, let
A B and C D (Fig. 16) be the intersections of two consecutive
cross-sections of the piece by the plane of the axis, E F, of
the piece and the resultant, W, of the bending forces.
A (
; c'
q b
t
r
f
FIG. 16.
FIG. 17.
Let Y and Z (Fig. 17) be two rectangular co-ordinate
axes to which all points of the cross-section are referred.
Kepresent by
y and z* the co-ordinates of all points in the plane Y Z ;
a?, the distances measured on the line E F ;
dx = O'O = the distance between the sections A B and C D;
dydz = a = the cross-section of a fibre ;
A = be = the elongation of any fibre as abj
p = R, the radius of curvature.
Let the section A B remain fixed and the section C D take
some position as C'D' under the action of the bending force;
the strain being within the elastic limit.
Then, by hypothesis, the fibres above E F will be elon-
gated, and the elongation bo of any one fibre, as ab, will be
proportional to its distance, y, from the neutral axis.
Irom the similar triangles bO'c and R 0' we have
or,
be : 0' : : 50' : R,
A : dx i : : y : p,
whence
(15)
an expression for the amount of elongation of a fibre at the
distance y from the neutral axis.
The expression for the intensity of the stress developed
TRANSVERSE STRAIN. 95
in a bar to resist an elongation eqnal to I is (eq. 3) equal to
EA-j-. In this expression substituting dydz for A, the
JL/
value of A just obtained for Z, and dx for L, we obtain *
E
ydydz ..... (16)
for the intensity of the stress developed in the fibre db.
Since this expression is true for any fibre that is elongated,
the total stress on the elongated fibres of this section will be
expressed by
In like manner the total stress on the compressed fibres will
be expressed by
\ffydyte,
the negative sign being used to denote the contrary direction
of the elastic resistance of the compressed fibres.
Since the strain is within the elastic limit the beam is
strong enough to resist the action of the extraneous forces,
and the moment of resistance at the cross-section is exactly
equal and opposite to the moment (Wx) of the bending forces
at the same cross- section.
The moment of resistance to elongation of a fibre, at the
distance y from the neutral axis, is equal to the intensity of
the stress in the fibre (eq. 16) multiplied by y, and, to com-
pression, the same expression multiplied by y.
The total moment of resistance at the cross-section will be
which placed equal to W#, gives an equation expressing the
relation between the moments of the transverse stresses and
those of the extraneous forces producing bending at any
cross-section of the beam.
Let b be the greatest value of s, and d that of y (Fig.
17) and integrating expression (17) so as to include the whole
cross-section, we may write this equation as follows :
E S**
P S =
. (18)
96 CIVIL ENGINEERING.
It will be seen that the quantity under the sign of inte-
gration when integrated twice will give the moment of
inertia of the cross-section of the piece with respect to the
neutral axis. Representing this by. I and that of the extra-
neous force by M, we may write (eq. 18) as follows:
= M. (19.)
P
The first member is oftentimes called the moment of
elasticity, sometimes the moment of resistance, and
at others the moment of flexure, and the second member
is called the bending moment.
169. This equation may be verified as follows :
We know that if all the elementary masses were concen-
trated at the principal centre of gyration, the moment of
inertia would be unaltered ; also, that the forces tending to
produce rotation of the body might be concentrated at this
point without thereby changing the conditions of equilib-
rium.
Suppose the resistances offered by the fibres to rotation
concentrated at the principal centre of gyration, and equal
to P' acting with a lever arm, L We have for equilibrium,
Yk = Wx = M.
From Mechanics, we have
r
Jc = principal radius of gyration =A/-
in which m is the elementary mass, r its distance from the
axis, and A the area of cross-section.
Substituting for 2 the sign of integration, and for m its
value in terms of y and z (Fig. 17), we get,
;& =
fffdydt
A
Squaring and dividing both members by , we get
fftfdydz
~ ~
TRANSVERSE STRAIN. 97
Hence,
and
whence
J J
J J y*dydz
which is the value the force would have on the unit of area
at the principal centre of gyration, or the distance Jc from the
neutral axis, under this hypothesis.
It has been assumed that the resistances are directly pro-
portional to the distance from the neutral axis ; hence, at the
unit's distance, the force on the unit of area would be
E M
** " ffy'dyfc
and at the distance, y, the force would be
My
The strain on the unit of area at the distance, y, from the
p
axis is shown by expression (16), to be equal to y- Hence,
E My
7? =
or
which is the same result as that shown by eq. (18).
7
98 CIVIL ENGINEERING.
SHEARING STRAIN PRODUCED BY A FORCE ACTING TO BEND THE
BAR.
170. No reference was made in the preceding article to
the shearing strain produced in the bar by a^ bending force
acting at one end, for the reason, that in prismatic bars of
this kind it is rarely necessary in practice to consider this
strain.
If in this bar (Fig. 16), the section A B had been taken
consecutive to the section, at F, where the force was applied,
the action of the force would not have been to turn this
section F around a line in its plane, but to have sheared it
off from its consecutive section. This action would have been
resisted by the adhesion of the sections to each other. The
force W is supposed to act uniformly over the entire sec-
tion F, hence the resistance to shearing in the adjacent
section will be uniformly distributed over its surface and
equal to W. The resistance on the unit of surface would
therefore be -j-.
A
The adhesion of these two sections prevents their separa-
tion by this force, hence the second section is drawn down by
the force W, which tends to shear it from the third section,
and so on.
In this particular case, the action of the force W to shear
the sections off, is transmitted from section to section until
the fixed end is reached, and the shearing strain of each sec-
tion is the same and equal to W. And in general, the shear-
ing stress of any cross-section of a bar or beam placed in
a horizontal* position is equal to the sum of all the vertical
forces transmitted through arid acting at that section.
CHANGES IN FORM OF THE BAR.
171. In a bar strained by a force acting in the direction
of its axis, the lengthening and shortening of the bar have
been the only changes of form considered. There is anothei
change that invariably accompanies them. This is the con-
traction or enlargement of the area of cross-section, when the
bar is extended or compressed. When the elongation or con-
traction is small, the change in cross-section is microscopically
small ; but when these strains are very great, this change is
sensible in many materials.
TRANSVERSE STRAIN. 99
In structures, the piecea are not subjected to strains of
sufficient magnitude to allow this change of cross-section to
be observed, and hence it is neglected.
It is well to keep this change in section in mind, as by it
we are able to explain certain phenomena that are met with
in experiments, when the strains to which the specimens are
submitted pass the limits of elasticity.
STRAIN ON THE UNIT OF AREA PRODUCED BY A BENDING FORCE.
-172. Expression (16) represents the stress of extension ou
the fibre whose cross-section is dydz. Dividing this expres-
sion by the area of cross-section of the fibre, we have
in which P represents the stress on the unit of area at the
distance y from the neutral axis. Dividing through by y
and multiplying both members by I, we have
p y
whence
(22)
which formula gives for a force of deflection, the stress on a
unit of area at any point of the section.
When the bar lias a uniform cross-section, I will be con-
stant, and P will vary directly with y and M, and by giving
to y its greatest value, we find the greatest strain in any as-
sumed cross-section.
VALUES OF L
173. In bars or pieces having a uniform cross-section, the
moment of inertia for each section with reference to the neu-
tral axis is the same, and hence I is constant for each piece,
and is easily determined when the section is a known geomet
rical figure.
100
JL..
-K
CIVIL ENGINEERING.
1. When the cross-section is a rectangle (Fig. 18)
in which b is the breadth, and d the depth, the
integral taken within the limits 3 = 0, and z ~ &,
y = 4^ and y = d, gives
I = ,
FIG. 18.
2. For a cross-section of a hollow girder, like that
of (Fig. 19) in which b is the entire breadth, d the total
depth V the breadth of the hollow interior, d' its depth, the
integral gives
Fie.
the limits
+ ,and
FIG. 20.
The expression will be of the same form in the
case of the cross-section of the I-girder, (Fig.
20), in which b is the breadth of the flanges ; b f
the sum of breadths of the two shoulders ; d the
depth of the girder, and d' the depth between the
flanges.
3. When the cross-section is a circle, and the
axes of co-ordinates are taken through the centre,
of s will be + r, r ; and those of y will be
I = JTT/**.
4. For a hollow cylinder, in which r is the
exterior and r' the interior radius,
5. When the cross-section is an ellipse, and
the neutral axis coincides with the conjugate
axis, if the transverse axis be represented by <#,
and the conjugate by 5, and the limits of z and y
be taken in the same manner, as in the circle, then,
1 =
6. When the cross section is a rhombus or lozenge, in
which 5 is the horizontal and d the vertical diagonal,
FLEXURE.
174. In the preceding article on transverse strain, to sim-
plify the investigation, without affecting the accuracy of the
FLEXURE. 101
results, the bar was placejj horizontally, and no notice was
taken of the change of position of the mean fibre after the
application of the bending force.
The strain was within the limit of elasticity, and for this
force the body was regarded as perfectly elastic.
The action of the force was to bend the bar, and hence to
bend the mean fibre without lengthening or shortening it,
making it assume a curved form.
"When the bar is bent in this manner, the curve assumed
by the mean fibre is called the elastic curve or equilibrium
curve. Its equation is deduced by equating the moment of
resistance and the bending moment, and proceeding through
the usual steps.
All the external forces to the right, or to the left, of any
assumed cross-section are held in equilibrium by the elastic
resistances of the material in the section.
FT
The general equation (19), - = M, expresses the condi-
tion of equality between the moments of resistance and bend-
ing, and is the equation from which that of the curve as-
sumed by the mean fibre after flexure may be deduced.
From the calculus, we have
which, substituted in eq. (19), gives
ETv&
^' M . (23)
When the deflection is very small, -^ is very small com-
pared with unity and may be omitted ; and eq. (23) becomes
for this supposition
(24)
which is the general equation expressing the relation between
the moment of flexure and the bending moment of the ex-
102
CIVIL ENGINEERING.
traneous forces for the mean fibre of any prismatic bar, when
the deflection is small.
175. To find the equation of mean fibre of a bar
placed horizontally, fixed at one end, and strained
by a vertical force W at the other end.
Denote by (Fig. 21)
Z, the length of the bar
from the fixed end^ to
the point of application
of W , it will be equal
to the length of the
mean fibre, A B.
Let AX and AY be
the co - ordinate axes
and Y positive downwards. The bending moment of W for
any point, a?, will be W (I a?), and substituting this for M
in eq. (24), we have
FIG. 21.
= W(Z x). . . . (25)
-or*) + C. (26)
Integrating, we have
If x = 0, by hypothesis ~- = 0, and hence = 0.
Integrating eq. (26) we have
Ely = ^ (Six? or*) + C' . . (27)
Noting that for x = 0, y = 0, we have C' = 0,
hence,
y =
w
-a?) . . . (28)
which is the equation of the curve of mean fibre under these
circumstances.
Inspection of eqs. (26 and 28) will show that the greatest
slope of the curve and the greatest distance between any point
of it and the axis of X will be at B. Eqs. (25) and (28) show
that the curve is convex towards the axis of X.
Represent by f the maximum ordinate of the curve. It8
value will be obtained by making x = I, hence
(29)
STRAINS IN BEAMS. 103
If the bar had been loaded uniformly instead of by a
weight acting at its extremity; representing by w the load
on a unit of length, eq. (24) would have become for this case,
hence the equation of the curve of its mean fibre,
w
The value of the maximum ordinate in this case would
wl*
instead of W concentrated at the end as shown by eq.
(28), suppose it to have been uniformly distributed over the
W
bar, then would be the load on each unit of length in that
L
case, and substituting this in eq. (32) for w, and calling the
corresponding ordinate, f ', we have,
JVF 74
I WZ 3
f ~8ET := 8EI * ' ' (33)
Hence f \f ; ; : -J-, from which we see that concentrating
the load at the end of the bar increases the deflection nearly
three times that obtained when the load was uniformly dis-
tributed.
BEAMS OF UNIFOEM CROSS-SECTION.
BEAMS RESTING ON TWO OR MORE SUPPORTS.
176. The term bar is used to designate a piece when the
dimensions of its cross-section are not only small compared
with the length of the piece, but are actually small in them-
selves. The term beam is used when the cross-section is
of considerable size, consisting of several square inches.
A beam resting on three or more supports, or having its
ends fixed so that they will not move is called a continuous
beam. If it rests on two points of support only, and the
ends are free to move, it is a non-continuous beam. If placed
in a horizontal position, with one end fixed and the other
free, it is known as a semi-girder or cantilever.
104
CIVIL ENGINEERING.
Beam Resting on two Points of Support.
177. Let it be required to determine the bending mo
ments, shearing stress, and equation of mean fibre of a
straight beam resting in a horizontal position on two
points of support.
There are two cases : 1, when the beam is uniformly loaded ;
and, 2, when acted upon by a single force between the two
points of support.
1st CASE. The external forces acting on the beam are the
load uniformly distributed over it and the vertical reactions
at the points of support.
FIG. 22.
Let A B (Fig. 22) be the beam, A and B the points of sup
port, and A the origin of co-ordinates. A X and A Y, the axes.
Denote by 21 the distance between two points of support A B.
w = weight on unit of length.
x = abscissa of D, any section of the beam A B.
The total load on the beam is %wl and the reactions at each
point of support are respectively equal to wl.
Bending moment. Let D be any section of the beam made
by a plane passed perpendicularly to the axis, through the
point, whose abscissa is x, and let us consider all the forces act-
ing on either side of D ; in this case let it be on the side A D.
The forces acting on the beam from A to D are the
weight on this portion of the beam, and the reaction at A.
The algebraic sum of their moments will be the bending
moment of the external forces acting on this segment. Let
M be this moment and we have
wx x wl x x = wlx . . . (34)
STRAINS IN BEAMS. 105
The second member of this equation is a function of a sin-
gle variable, and may therefore be taken as the ordinate of a
line of which x is the abscissa. Constructing the different
values of the ordinate, the line may be traced. This line is a
parabola, and shows the rate of increase or decrease in the
bending moments.
The curve thus constructed may be called the curve of the
bending moments.
Shearing strain. The shearing stress in the beam at D
is equal to the algebraic sum of all the vertical forces acting
at this section, hence
S'= wx wl ...... (35)
The second member of this equation represents the ordi-
nate of a right line. Constructing the line, the ordinates will
show the rate of increase or decrease of the shearing strain
for the different sections.
By comparing equations (34) and (35) it will be seen that
which shows that the shearing stress at any section is
eqiial to the first differential coefficient of the bending 'moment
of that section taken with respect to x.
For convenience we used the segment A D, but the results
would have been the same if we had taken B D. For, sup-
pose we find the bending moment for this segment, we have
for the moment of the weight, acting to turn it around D,
And for the moment of reaction,
- wl(2l - x).
The algebraic sum of these moments will be
the same as (34), as it should be.
Equation of mean fibre. Substituting the second mem
ber of eq. (34) for M in eq. (24), we have
EI^j [ = l s wx t --wto. . . (37)
Integrating, we get
106 CIVIL ENGINEERING.
For = , -^ = 0, and we have C = %wl\
dx
Substituting this value of C, and integrating, we get
Ely = x*lx* + wl 3 x + C'.
24 6
For x = 0, y is equal 0, and hence C' 0, and we have
which is the equation of the curve of mean fibre, and may be
discussed as any other algebraic curve.
Deflection. If we represent the maximum ordinate of the
curve by/j we find
the maximum deflection, which is at the middle point of the
beam.
Equation (38) may be placed under the form,
w
O[5Z'-(a ! -Z)'] . (39)
For values of as, differing but slightly from Z, the quantity
(xl)* may be omitted without materially affecting the value
of the second member for these values. Omitting this quan-
tity, and eq. (39) reduces to
fo -*') (40)
which is the equation of a parabola. Hence, a parabola may
be constructed passing through the middle point of the curve
of mean fibre and the points of support, which nearly coin-
cides with the curve of mean fibre in the vicinity of its
middle point.
The parabola whose equation is eq. (40) differs but slightly
throughout from the curve given by eq. (38) ; for the greatest
difference between the ordinates of the two lines for the same
I __
value of , will be when x = (2 V 2), which gives
/, representing the ordinate of the curve for this value of a?,
and y", the ordinate of the parabola for the same value of
STRATA'S IN BEAMS.
107
"Whence, we get
178. 2o CASE. The external forces acting oii the beam are
the applied force, whatever it may be, and the vertical re-
actions at the points of support.
Let A B (Fig. 23) represent the beam resting on the supports,
A and B, sustaining a weight, 2W, at any point, as P, between
the points of support. JDenote the reactions at A and B by
R, and R a , A B by 2Z, A P by I'.
2YV
FIG. 23.
The reactions R, and R, will be proportional to the segments
in which the beam is divided, and this sum, disregarding the
weight of the beam, is equal to 2W. Hence,
R, : R, : 2W : : PB : AP : AB,
from which proportion we, knowing 2W and ', can determine
the values of R, and R,. Knowing these, we can obtain the
bending moment and shearing strain of any section, and the
deflection of the beam due to the force 2W.
179. The most important case of the single load is that in
which the load is placed at the centre. Suppose 2W to act at
the centre, then R 1 =R 2 = "W. Assume the origin of co-ordin-
ates and the axis of X and Y to be the same as in the first case.
Bending moment. For any section between A and C the
bending moment will be M = Wx.
Shearing strain. The shearing stress on any section will be
S' = W.
Equation of mean fibre, Substituting in second mem-
ber of eq. (24) the above value of M, we have
EI = -
(41)
Integrating, and substituting for C, its \ ;ilue, ire get
J*9 W ,
t08 CIVIL ENGINEERING.
Integrating again and substituting for C, its value, we get
y = ^ (3P% a? 3 ), . . (43)
which is the equation of so much of the mean fibre as lies be-
tween the origin, A, and the middle point, C.
The right half of the mean fibre is a curve exactly similar
in form. Assuming B as the origin and the abscissas as posi-
tive from B towards C, eq. (43) is also the equation of the
right half of the curve.
Deflection. The maximum deflection is at the centre, and is
-
~ ' El
Comparing this with the deflection at the centre in the
previous case, it is seen that the deflection produced l>y a load
uniformly distributed over the beam is Jive-eighths of that
produced by the same load concentrated and placed at the
middle point.
180. Comparison of strains produced. The bending
moment for any section, when the beam is uniformly loaded,
is, eq. (34),
nr war .
M. = -- wlx,
and when the beam is acted upon by a load at the middle
point, is, eq. (41),
M = Wx,
Both will have their maximum values for x = I.
Equating these values, we have
whence "W" = ,
2i
which shows that the greatest strain on the unit of area of
the fibres, when the load is uniformly distributed, is the same
as that which would be caused by half the load concentrated
and placed at the middle point of the beam.
Beam strained by a uniform load over its entire
length and a load resting midway between the two
points of support.
181. If a beam be uniformly loaded, and support also
load midway between the points'of support, the correspondii
a
STRAINS IN TraAMR. 109
values for the strains can be obtained by adding algebraically
the results determined for each case taken separately.
If the beam had other loads besides the one at C, we could
in the same manner find the bending moments, shearing
strains, and deflections due to their action. The algebraic
sum of the moments, ordinates of deflection, etc., would give
the results obtained by their simultaneous action.
Beam having its ends firmly held down on its sv/p-
ports.
182. In the preceding cases the beams are supposed to be
resting on supports, and not in any way fastened to them.
If the ends of the beams had been fastened firmly so that
they could not move as, for example, a beam having its ends
firmly imbedded in any manner in two parallel walls the
results already deduced would have been materially modified.
Let it be required to determine the strains and equation of
curve of mean fibre in the case where the beam has its ex-
tremities horizontal, and firmly embedded so that they shaft
not move, the beam being uniformly loaded.
If we suppose a bar fitted into a socket (Fig. 24) and acted
upon by a force to bend it, it is evident, calling Q! the force
of the couple developed at the points B and H, that the mo-
ment of the force "W", whose lever arm is I, is opposed by the
moment of resistance of the couple, B Q t and H Q t acting
through the points H and B.
FIG. 24.
Hence, we have
Q/ = AV,
f being the lever arm of the couple.
110
CIVIL ENGINEERING.
We see that Q, increases proportionally to any decrease in
I', and that these quantities themselves are unknown, although
their product must be constant and equal to the bending mo-
ment of the beam at B.
To determine the bending moment at any section or a beam
having its ends firmly held down ; let A B (Fig. 25) be the
beam before being loaded, and denote by
21 = A B = the length ;
w = the weight on unit of length ;
x the abscissa at any point, the origin of co-ordinates
being at A, and A B coinciding with axis of X, as in preced-
ing cases.
A S ,D
C'
Y FIG. 25.
The total load on the beam will be 2wZ, and the reactions
at the points of support are each equal to wl.
The bending moment of any section D, is equal to the
algebraic sum of the moments of vertical reaction at A, of
the weight on A D, and of the unknown couple acting on the
left of A.
Calling jjb the moment of the unknown couple and substi-
tuting this algebraic sum in eq. (24), we have
Integrating and noting that for x= 0,-r-= 0, we have 0=0,
and
dy wl , w
Ln this equation make x 2Z, for which-y-=0, and we find
STRAINS IN BEAMS. Ill
which is the value of the moment of the unknown couple
acting at the left point of support. It is also the value of
the one at the right point of support, B.
Writing this value for /z- in equations (44) and (45), we have
HJQ
+ ... (46)
and then by integration,
wl w
We find C'=0, and substituting, etc., we get
which is the equation of the curve of mean fibre.
Deflection. Denoting by f, the maximum value for y, and
we have
The corresponding value obtained, from eq. (38), is
. /-&
A comparison of these values of f shows that by firmly
fastening the ends of the beam to the points of support in a
horizontal position, the deflection at the centre is one-fifth of
what it was when they merely rested on the supports.
Bending moments. The curve of the bending moments is
given by the equation.
w w
which is that of a parabola.
The bending moments for x = 0, and 2Z, are both equal to
-5- Z 2 , and for x = Z, -77- . The bending moment of the
section at the middle point is therefore half that of the section
w
at A or B. Assuming a scale, lay off -g-Z 2 , below the line A 3,
on perpendiculars passing through A and B. Lay off half this
value on the opposite side of the line A B on a perpendicular
112 CIVIL ENGINEERING.
through the middle point. This gives us three points of the
curve of which one is the vertex. The perpendicular through
the middle point is the axis of the parabola, and with thr
three points already found the curve may be constructed.
This curve of bending moments cuts the axis of X in two
points, the abscissas of which are I (1 >/J), and at the
sections corresponding to them the bending moments will be
equal to 0.
d?ii
These values substituted in eq. (46) for a?, reduces-^
to zero, and an examination of this equation shows that
fT&i]
there is a change of sign in -r4 at these points. It therefore
follows that the curve of mean fibre has a point of inflex-
ion for each of these values of a?, that is, the curve changes
at these points from being concave to convex, or the reverse,
towards the axis of X.
The greatest strains on the unit of area produced by the
deflecting force, will be in the cross-sections at the ends and
middle ; the lower half of the cross-section at the middle
being extended, and the lower halves of these at the points of
the support being compressed.
Shearing strain. The expression for the shearing force is
S'= j = wx wL
dx
which is the same as eq. (35), and its values may be repre-
Bented by the ordinates of a right line which passes through
the middle point.
The uniform load concentrated and placed at the middle.
183. If instead of being uniformly loaded, the beam was
only strained by a single load, 2W, at the middle point, the
bending moment, disregarding the weight of the beam, would
be for values of x < I.
M= Wo? + IJL
and by a process similar to that just followed, we would find
to be the equation of the mean fibre from A to C.
The maximum deflection will be
STRAINS IN BEAMS.
113
which is equal to one-fourth of that obtained, with a load at
the centre, when the ends of the beam are free. It is also
seen that the deflection caused by a concentrated load placed
at the middle of the beam, is the same as that caused by
double the load uniformly distributed over the whole length.
If the beam was loaded both uniformly and with a weight,
2W, the results would be a combination of these two cases.
Seam loaded uniformly, fixed at one end, and resting on a
support at the other.
184. Let A B (Fig. 26) represent the beam in a horizontal
position, fixed at the end, A, and resting on a support at the
end B.
FIG. 26.
Adopting the notation used in previous case, we have
for the total load on the beam.
The reactions at A and B are unequal. Represent by ^
the reaction at A, and by p the moment of the unknown
couple at A. We have
I I . Elg=-B^+r +/ . . . (49)
Hence by integration,
a' t + /',0=0 (50)
Ely = - *RX+ j^tf+ /* ~, C'= (51)
The bending moment at B is equal to zero, hence for x = 21,
.. will be and eqs. (49) and (51) reduce for this value of x to
dor
= - R$l + ^(llf + /* . . . (52)
0=-
f . (53)
114 CIVIL ENGINEERING.
Combinin these we find
wl*
Hence the reaction at B is %w (2Z).
Substituting these values for R, and p in eq. (49) the bend-
ing moment at any point, shearing strain, and curve of mean
fibre can be fully determined. Placing the second member
of eq. (49) equal to zero, and deducing the values of #, these
will be the abscissas of the points of inflexion, and by placing
the second member of eq. (50) equal to 0, the abscissa cor-
responding to the maximum ordinate of deflection may be
obtained. The curve of bending moments, etc., may be de-
termined as before.
Beam resting on three points of support in the same hori-
zontal straight line.
185. Let it be required to determine the bending moments,
shearing strain, and equation of mean fibre of a single,
beam resting in a horizontal position on three points of sup-
port, each segment being uniformly loaded.
Let ABC (Fig. 27) be the beam resting on the three points,
A, B, and C.
Fig. 27.
Let us consider the general case in which the segments are
unequal in length and the load on the unit of length dif-
ferent for them.
Let I = A B, and w, the weight on each unit of its length,
l f = BC, and w' the weight on each unit of its length
II, , R 2 , R 3 , the forces of reaction at the points of support,
A, B, and C, respectively.
^Take A B C as the axis of X and A the origin of coordinates
with y positive downwards as in the other cases.
First, consider the segment A B, and let D be any section
whose abscissa is x.
Since the reactions at the points of support are unknown,
they must be determined.
STRAINS IN BEAMS. 115
We have
Integrating, we get
Mj=-*B^ + + a . . (55)
Let a represent the angle made by the curve of mean fibre
with the axis of X at B, then for x = I we havef^A = tan o>,
, VW i,
whence
EHan= iiy + i0p+0. . . (56)
Subtracting from preceding equation, member by member,
we have
-l^. (57)
Integrating eq. (57) we get
El (y-x tan )= - -J- I^a* + ^^+ i^A - 1 *Zte. (58)
the constant of integration in this case being equal to 0.
If in eq. (54) we make x = I, and denote the bending mo-
ment of the section at B by /*, we have
^-iy+2* .... (59)
In eq. (58) make x = Z, hence y = 0, and we have
El tanw-|K 1 Z 2 + ^^ + 1^-1^=0 . (60)
by omitting common factor L Combining this equation with
the preceding one and eliminating R! and reducing, we get
El tan w = -J lp -fowl? . . (61)
which expresses the relation between the tan and p.
Going to the other segment, taking C as the origin of co-
ordinates and calling x positive towards B, we may deduce
116 CIVIL ENGINEERING.
similar relations between the bending moment at B and the
tangent of the angle made by the mean fibre at B with the
axis of X. Since the beam is continuous, these curves are
tangent to each other at the point B, and the angles made by
both of them with the axis of X at that point are measured by
a common tangent line through B. Therefore, the angles are
supplements 01 each other and we may at once write the cor-
responding relation as follows,
.... (62)
Since, for equilibrium, the algebraic sum of the extraneous
forces must be equal to zero, we have
wl+wT Bi BS B3=0 . . . (63)
and since the algebraic sum of their moments with respect to
any assumed section must be equal to zero, we have for the
moments taken with respect to the section at B,
O O \ /
These last four equations contain four unknown quantities,
B!, B-j, BS, and tan co.
By combining and eliminating, their values may be found.
Combining equations (61) and (62), and eliminating tan co, we
have
+ w'l*
i/ i i/
The bending moment of any section, as D, is from equa-
tion (54)
hence for x I, we have M equal to the bending moment at
B, which has been represented by /*, or eq. (59)
from which we get
-p __ wl n wl
" ~- : "
In a similar way, the- value of BS may be found. These
values of B! and K-J substituted in eq. (63), will give the value
of Bo.
STRAINS IN BEAMS. 117
The external forces, all being known, the bending moments,
shearing strain, and equation of mean fibre may be deter-
mined as in previous examples.
186. Example.
The most common case of a beam resting on three points
of support, is the one in which the beam is uniformly
loaded throughout and the intermediate support is placed at
the middle point.
In this case, I = I' and w = ID'. Substituting these values,
in the expressions for p and Rj, we have
and R! = wZ.
The reaction at the middle point will therefore be
or
Substituting the value of E! in eq. (54) we obtain the bend-
ing moment for any section.
In the case of a beam resting on two supports, Fig. (22), and
having a weight uniformly distributed along its length, it has
been shown that each support bears one half of the distributed
load ; and that the deflection of the mean fibre at the middle
point, represented by^J is the same as the beam would take
were fths of the load acting alone at the middle point. In
the latter case the pressure upon a support, just in contact
with the beam at its middle point, would be zero ; and if the
support were to be raised so as to bring the middle of the
beam into the same right line with the extreme supports,
tho. intermediate support would evidently sustain the total
pressure at C to which the deflection was due, and which was
f ths of the entire load ; hence the reaction of the middle sup-
port will be equal to fths. This conclusion agrees with the
result determined by the previous analysis.
Each segment of the beam in this case might have been
regarded as a beam having one end fixed and the other rest-
ing on a support; a case which has already been consid-
ered.
Theorem of Three Moments.
187. From the preceding, it is seen, that the reactions at
the points of support can be determined whenever we know
the bending moments at these points. These moments are
readily found by the " theorem of three moments."
This theorem has for its object to deduce a formula express-
118 CIVIL ENGINEERING.
ing the relation between the bending moments of a beam
at any three consecutire points of support, by means of which
the bending moments at these points may be obtained, with-
out going through the tedious operations of combination arid
elimination practised in the last example.
Take any three consecutive points of support, as A, B, and
FIG. 28.
C, Fig. (28), of a beam resting on n supports. Denote by I
and I', the lengths of the segments, A B and B C, w and w f ,
the weights on each unit of length in each segment and
M L M 2 M 3 , the bending moments at these points, A, B, C.
The formula expressing the relation between these bending
moments is
V) 4- M 8 Z' = JwZ 3 + %w'l'\ (67)
In every continuous beam, whose ends are not fixed, the
bending moments at the end supports are each equal to zero.
Hence, by the application of this formula, in any given case,
as many independent equations can be formed as there are
unknown moments, and from these equations the moments
can be determined.
188. The demonstration of this theorem depends upon the
principle, that the bending moment at any point of support
whatever, and the tangent of the angle made by the neutral
fibre with the horizontal at that point, may be expressed in
functions of the first degree of the bending moment at the
preceding point of support, and the tangent of the angle
made by the neutral fibre with the horizontal at that point.
Let A B (Fig. 29) be any segment of a beam resting on n
supports, A the origin, A X and A Y the axes of co-ordinates,
and Mj and M 2 the bending moments at A and B.
FIG. 29.
The applied forces acting on the beam and the reactions
are taken vertical and in the plane of the mean fibre.
STRAINS IN BEAMS. 119
The external forces which act on the beam to the left of
the support, A, may be considered as replaced by a resultant
moment and a resultant shearing force, without disturbing
the equilibrium. This resultant moment, represented by M t ,
is equal and opposite to the moment of the internal forces
at the section through the support A ; the vertical force,
which we represent by Si, is equal and opposed to the shear-
in the angle which the neutral fibre after de-
flection makes with the axis of X, at A, and integrating, we
have
El l^L - tan 6\ = M^ + Aafej + iS^. (69)
\ax 1 JQ
\
r x
Representing the quantity I vdx by M' and integrating,
'A
we have
El (y - x tan <) = JM^ + fWdx+^S^. (70)
In these three equations, make x = I and denote by N", Q,
a?
and K what /*, M', and / Wdx become for this value of a?,
JO
and by to the angle made by the curve of mean fibre with
the axis of X at B ; noting that for x = I, El ^ = M 2 , we
Inure
M ; = M! + N + SA ]
El (tan - tan ^) = M^ + Q + iS^, J. (71)
EB tan > = JM/ + K 4- -JS/. j
120 CIVIL ENGINEERING.
Combining the first and third, and then the second and third
of these equations and eliminating S 1? we have
+ EK tan = - JM^ -f %~NP - K,
JEKtanw + fEB tan = - -JM^ + QZ - K '
In these equations, N", Q, and K depend directly upou the
applied forces, and are known when the latter are given.
But Mi, M 2 , tan > and tan o> are unknown.
An examination of equations (72) shows that Mg and tan
are functions of the first degree of Mj and tan <, whatever
be the manner in which the external forces are applied.
Let us impose the condition that the system of forces acting
on the beam shall be a load uniformly distributed over each
segment, and denote by w the load on a unit of length of the
segment A B.
For this case we have
J M dx =
and in these, by making x = Z, we have
Q =
Substituting in equations (72) these values for N, Q, and
we have
M 2 = - 2Mi - tan
\
1(73)
^ wP
which agree with the principle already enunciated.
189. To deduce formula (67), let A, B, C (Fig. 28) be any
three consecutive points of support of a beam resting on n
supports.
STRAINS IN BEAMS. 121
From the first of equations (73) we may at once write
M 8 = - 2M 2 - tan f + W\
and by considering x positive from B to A, and giving the
proper sign to tan , we write
6EI
M! = 2M 3 + -y- tan ' + fyoP.
Multiplying these respectively by I' and by I, and adding
them together, we have
M^ + 2M 2 (I + Z') + M/ = %wP+ Jw'J*
which expresses the relation between the bending momenta
for any three consecutive points of support, and is the same
as formula (67).
By a similar process we can find an equation expressing
the relation between the tangents of the angles taken at the
three points of support.
Applications of Formula (67).
190. IST CASE. Seam in a horizontal position, loaded
uniformly, resting on three points of support, the segments
being of equal length.
In this case, we have I' = I, w' = w, and M x and M 8 each
equal to zero. Substituting these values in eq. (67), we get
whence
M 2 =
The bending moment of the section at B is, eq.
whence we get for the reaction at A,
B, = |*rf,
*hich is the same value before found. The reaction at C ia
122 CIVIL ENGINEERING.
the same, and that at B can now be easily determined, from
the equation,
Knowing all the external forces acting on the beam, the
bending moment at any section, the shearing strain, etc., can
be determined.
191. 2D CASE. Beam in a horizontal position resting on
four points of support.
Ordinarily a beam resting on four supports is divided
into three unequal segments, the extreme or outside ones
being equal to each other in length, and the middle one
unequal to either.
If we suppose this to be the case, represent by A, B, C, and
D the points of support in the order given. The bending
moments at A and D are each equal to zero. To find those
at B and C, take the general formula (67) and apply it first
to the pair B C and B A, and then to the pair C B and C D, and
determine the bending moments from the resulting equa-
tions. Having found them, the reactions are easily found ;
and knowing all the forces acting on the beam, the bending
moments, shearing strains, and curve of mean fibre may be
obtained.
192. SD CASE. Beam in a horizontal position resting on
five points of support, the segments being equal in length.
When the number of supports is odd, the segments are
generally equal in length, or if unequal, they are symmetri-
cally disposed with respect to the middle point.
If the beam be uniformly loaded, it will only be necessary
to find the bending moments at the points of support of either
half of the beam, as those for corresponding points in the
other half will be equal to them.
Suppose the case of five points of support.
Let A, B, C, D, and E be the points of support, C being the
centre one. Eepresent by I the length of a segment, w the
weight on a unit of length, M 2 , M 8 , M 4 , the bending moments
at B, C, and D, and the forces of reaction at A, B, and C, by
RU -R*}, Eg respectively. From the conditions of the problem,
M 2 is equal to M^ and .the reactions at A and B are equal to
the reactions respectively at E and D.
STRAINS IN BEAMS. 128
Applying formula (67) to the first pair of segments, we have
and applying it to the second pair, BC and CD, we get
In these equations, making M^ equal to M a and combining
the equations, we find
M 2 = -fewfi, and M 3 =
The external forces acting on the first segment, AB, to turn
it around the section at B, are K! and wl. Hence we have
whence
The external forces acting to turn the segment A C or half
the beam around C are the reactions at A and B and the loads
on the two segments A B and B C.
The algebraic sum of the moments for the section at C is,
Substituting in this the value just found for R! and solving
with respect to B^ we get
The sum of the reactions is equal to the algebraic sum of
the applied forces, hence,
R! + R.J + ES + K 4 + Kg = 2Ri + 2R2 + ES =
in which substituting for R! and E.J, their values, we find
The external forces acting on the beam are now all known,
and hence the bending moments, shearing strain, etc., may be
determined.
193. 4:TH CASE. Seam in a horizontal position, resting on
R points of support , the segments being equal in length.
If the beam be uniformly loaded, it will, as in the last case,
only be necessary to find the bending moments at the points
of support of either half of the beam.
124 CIVIL ENGINEERING.
If n be even, the reaction of the %n th and (Jfi + l) 01 support
will be equal; if n be odd, the i(rc+l) will be the middle
support, and the reactions of the supports equidistant from the
middle point will be equal.
The formula for the segments would become, n being even,
-M 8 =
##**##
M ift + 4M in+ !+ M^ + 2
In the last equation, M in + 1 and M Jn + 2 would be equal
espectively to M in and M in _!.
From these equations, K^ Eg, Eg, . . . R n could be obtained.
General Exam/pie.
194. STH CASE. Beam in a horizontal position resting on
n + ~L points of support, segments unequal in length, and
uniform load on unit of length being different for each seg-
ment.
Kepresent the points of support by A x A 9 A, . . . A,, A n + 19
and the respective bending moments at these points of
support by M 1? M 2 , M 3 , . . . . M n , M + i. Kepresent the
length of the segments by ^, ,, Z 8 , . . . . l n and the respective
units of weight on the segments by w^ w^ w^ . . . . w n .
The bending moments M 1? 1^ + being those at the ex-
tremities, are each equal to zero, and therefore there are only
Ti1 unknown moments to determine. Applying eq. (67) suc-
cessively to each pair of segments, we obtain' n 1 equations
of the first degree with respect to these quantities, which
by successive eliminations give us the values of the moments,
M,, M,, ..... M n .
These equations will be of the following form :
2 ft + y M, + Z,M, = i fay + w jf)
a ft + Z.) M, + Z,M. = i (w,l,> + w,l*)
*#####
! + 2 4.! +
From these equations, the reactions at the points of sup
port can be determined, and knowing all the external forces
the strains on the beam may be calculated.
TORSION. 125
TORSION.
195. A beam strained by a system of ext/aneous forces,
among which is a couple acting in a plane perpendicular to
the axis of the piece, will be subjected to a stress of torsion
in addition to the other stresses already described.
Suppose a beam fixed at one end (Fig. 30) and a couple
applied to the free end, F, the axis of the couple intersecting
the axis of the piece, and the plane of the couple perpen-
dicular to the axis. The action of the couple will be to
twist the beam around its axis, causing a twisting strain of
the fibres and developing torsional stresses in the material.
FIG. 30. FIG. 31.
To determine the stress of torsion at any cross-section as
C D, let a be equal to the angular amount of torsion between
any two cross-sections of the beam, and ft the amount of
angular change for a unit of length.
It is assumed that the total amount of angular change of
any fibre between any two sections, or a, is directly propor-
tional to the distance between the sections, and that the
stress of torsion developed in the fibre is directly proportional
to its distance from the axis of the piece.
Let T/ = the stress of torsion in any fibre, a = the area
of cross-section of the fibre, and G = the coefficient of tor-
sional elasticity; then
-S = G& or T', = aGft.
Let be taken as the pole. (Fig. 31) Z, the fixed line,
and r and v the polar co-ordinates of points in the plane of
cross-section C D. Then
a = rdr dv.
Since the stress is assumed to be directly proportional to
the distance of the fibre from the axis, we get by substitut-
126 CIVIL ENGINEERING.
ing for a its value, and multiplying byr, the intensity of
the stress in the fibre at the distance r from the axis to he
G ft i*dr dv.
Suppose the section C D to be fixed. The twisting action
of the couple at F is transmitted from section to section of
the piece until it reaches C D, where it is opposed by the
resistance developed in the section. The moment of resist-
ance offered by the fibre at the distance r from the axis will
be the intensity of the twisting stress in the fibre multiplied
by its lever arm, r, or
G ft r z dr dv.
The total moment of the resistance developed in the cross-
section C D may be expressed as follows :
drdv, . . . . (74)
Eepresent the moment of the couple acting at the section
F by F' x A, and equating the moments, we^have
v = W\, . . (75)
This expression / / r*drdv is called the polar moment
of inertia ; that is, the moment of inertia of a cross-section
of the beam about an axis through its centre and perpendicu-
lar to the plane of cross-section.
Representing it by l p , we have
G/SI^F'A, .... (76)
Suppose the cross-section considered to be a circle, whose
radius == R, and the section in which the resistance is con-
sidered is at the distance I from the plane of the twisting
couple. Equation (76) would become for this case, by
substituting 1L for ft, and J ;rR 4 for 1^
TORSION. 127
196. General Morin, in his work on Strength of Materials,
gives the value for G for different materials.
The following are some of the values :
Wrought iron ............. G = 8,533,700 Ibs.
Cast-iron .................. G = 2,845,000 Ibs.
Cast-steel .................. G = 14,223,000 Ibs.
Copper .................... G = 6,210,000 Ibs.
Oak ....................... G = 569,0001bs.
Pine ...................... G = 616,000 Ibs.
Rupture by Twisting.
197. It is assumed that the torsional stress developed in
the fibres of a piece varies directly with the distance of the
fibre from the axis of torsion, and is greatest in the fibres
farthest from this axis. If the strain be increased until
rupture takes place, those fibres farthest from the axis will
be the ones to give way first.
The intensity of the torsional stress for any cross-section
developed in a fibre at the distance r from the axis is
G ft r*dr dv.
This expression divided by the area of cross-section of the
fibre, r dr dv, gives G ft r as the intensity of the torsional
stress on the unit of surface at the distance r from the axis.
Represent this intensity by T', and we have
T = Grftr.
Multiplying both members of this equation by I p , and di-
viding by r, we get
in which the second member is the same as the first member
of equation (76). Hence,
Ip =F'A
r
from which we get
T = ?*r, . .,". . (78)
i P,
or, an expression for the torsional stress on any unit of cross-
section of a piece strained by a twisting force.
128 CIVIL ENGINEERING.
Let d = the greatest value that.?* can have for any cross-
section. If d be substituted for r in equation (78) the result-
ing value of T" will be the stress on the unh farthest from
the axis for the cross-section considered.
Suppose F'A to be increased until rupture is produced,
then T' for this value of r d, in the section where rupture
begins, will be T^, the modulus of torsion, or
TV, = F'A x ~ (79)
*p
from which the values of the modulus of torsion may be de-
duced.
INFLUENCE OF TEMPERATURE.
198. The influence of changes in temperature, especially
in the metals, forms an important element to be considered
in determining the amount of strain on a beam.
If the beam is free to move at both ends, there will be no
strain in the beam arising from the changes of temperature ;
if the ends are fixed, there will be, and these strains must be
determined.
The elongation or contraction produced by the changes of
temperature is known for the different metals. The amount
of strain upon the unit of area will be the same as that pro-
duced by a force elongating or contracting the beam an
amount equal to that resulting from the change of tempera-
ture under consideration.
CHAPTEK YII.
STRENGTH OF BEAMS.
PROBLEMS.
199. The object of the previous discussions has been to find
the strains to which a beam is subjected by certain known
forces applied to it.
The problems which now follow are:
Knowing all the external forces acting on a beam, to de-
tennwe the form and dimensions of its cross-section, so that
STRENGTH OF BEAMS. 129
the strain on the unit of surf ace shall at no point be greater
than the limit allowed ; and knowing the form and dimen-
sions of the cross-section of a beam, to determine tJie load
which it will safely bear.
There are two cases ; one is where the cross- section is con-
stant throughout the beam ; and the other is where it varies
from one point to another.
1st CASE. BEAMS OF UNIFORM CROSS-SECTION.
200. Strength of beam strained by a tensile force.
Let W be the resultant force whose line of direction is in
the axis of the beam and whose action is to elongate it.
From the equation preceding eq. (5), we have
W
= the stress on a unit 01 cross-section.
A
Knowing the value of T for different materials, a value less
than T for the given material is assumed for the stress to be
allowed on the unit of cross-section. Assuming this value of
the stress and calling it T 1? we have
W
A = -. ..,.*. (80)
From which, knowing the form of cross-section and its area,
the problem can be solved.
Suppose the form to be rectangular, and let b be the
breadth and d the depth. Then
W
A = b x d, or bd = ;
J-i
in which, if b be assumed, d can be determined, and the con-
verse.
The solution of the reverse problem is evident. Knowing
A and T 1? the value of W, or the load which will not produce
a stress greater than T L on the unit of area, is easily deter-
mined.
201. Strength -when strained by a compressive force.
For all practical purposes, it is assumed sufficiently exact
for short pieces to apply the methods just given for tension,
substituting C t for T t ; the former being the assumed limit of
compressive stress on the unit of area. "When the pieces are
longer than five times their diameter, they bend under the
crushing load and break by bending, or by bending and
by crushing.
9
130
CIVIL ENGINEERING.
Rankine gives the following limits of proportion between
length and diameter, within which failure by crushing alone
will take place, and beyond which there is a sensible ten-
dency to give way by bending sideways.
Pillars, rods, and struts of cast iron, in which the length
is not more than five times the diameter.
The same of wrought iron, not more than ten times the
diameter.
The same of dry timber, not more than twenty times the
diameter.
202. Formulas for obtaining the strength of columns
or pillars 'whose lengths are greater than five times the
diameter of cross-section, when subjected to a compres-
sive strain.
The formulas deduced by Mr.' Hodgkinson, from a long series
of experiments made upon pillars of wood, wrought iron, and
cast iron are much used in calculating the strength of pillara
or columns strained by a force of compression.
Hodgldnsorfs Formulas.
Table for finding the strength of pillars, in which
W = the breaking weight, in tons of 2,000 pounds ;
L = the length of the column in feet ;
D = the diameter of exterior in inches ;
d = the diameter of interior in inches.
Nature of column.
Both ends being round-
ed, length of column
exceeding 15 times
its diameter.
Both ends being flat,
the length of column
exceeding 30 times
its diameter.
Solid square pillar of
red cedar (dry). . ,
Same of oak (Dantzic)
dry ,
Solid cylindrical col. of
wrought iron ....
Solid cylindrical col. of
cast iron .
Hollow cylindrical col.
of cast iron .
W =
W = 16.6^
W = U.l
W = 12.2^
!'*
W =49.6^
STRENGTH OF PILLARS.
131
If the column be shorter than that given in the table, and
more than five times its diameter, the strength may be deter-
mined by the following formula :
WAG
. . . . (81)
in which W= the breaking weight, computed from the
formulas in the above table ;
C = the modulus of crushing in tons ;
A = the cross-section in square inches ; and
W = the strength of the column in tons.
Gordons Formulas.
These are deduced from the same experiments, and are aa
follows :
SOLID PILLARS.
Cross-section a square.
Of cast iron W =
80,000 A
Of wrought iron . . W =
1 +
266 5 2
36,000 A
. (82)
HOLLOW PILLARS.
Circular in cross-section.
Of cast iron , . . W =
Of wrought iron . . W =
80,000 A 1
. . (83)
132
CIVIL ENGINEERING.
Cross-section a square.
Of cast iron . . .
Of wrought iron
_ 80,000 A
1 +
533 6 2
_ 36,000 A
(**)
6,000 I
in which,
"W = the breaking load in pounds ;
A = the area of cross-section in square inches ;
I the length of the pillar in inches ;
b = the length of one side of the cross-section ; and
d = the diameter of the outer circumference of the base.
These formulas apply to pillars with flat ends, the ends
being secured so that they cannot move laterally and the load
uniformly distributed over the end surface. In the hollow
columns, the thickness of the metal must not exceed \ of the
outer diameter.
Mr. G. Shaler Smith? s Formula.
This formula is deduced from experiments made by Mr.
Smith on pillars of both white and yellow pine, and is
/==
. . . (85)
in which 5 and I are in inches, and represent the same quanti-
ties as in the preceding formulas. W is the breaking load
on the square inch of cross-section in pounds.
203. Mr. Hodgkinson, in summing up his conclusions de-
rived from the experiments made by him on the strength of
pillars, stated that :
" 1st. In all long pillars of the same dimensions, the resist-
ance to crushing by flexure is about three times greater when
the ends of the pillars are flat than when they are rounded.
" 2d. The strength of a pillar, with one end rounded and
the other flat, is the arithmetical mean between that of a
pillar of the same dimensions with both ends round, and one
with both ends flat. Thus of three -cylindrical pillars, all of
the same length and diameter, the first having both its euda
STRENGTH OF PILLARS. 133
rounded, the second with one end rounded and one flat, and
the third with both ends flat, the strengths are as 1, 2, 3,
nearly.
" 3d. A long, uniform, cast-iron pillar, with its ends firmly
fixed, whether by means of disks or otherwise, has the same
power to resist breaking as a pillar of the same diameter, and
half the length, with the ends rounded or turned so that the
force would pass through the axis.
" 4th. The experiments show that some additional strength
is given to a pillar by enlarging its diameter in the middle
part ; this increase does not, however, appear to be more than
one-seventh or one-eighth of the breaking weight."
Similar pillars." In similar pillars, or those whose length
is to the diameter in a constant proportion, the strength is
nearly as the square of the diameter, or of any other linear
dimension ; or, in other words, the strength is nearly as the
area of the transverse section.
" In hollow pillars, of greater diameter at one end than the
other, or in the middle than at the ends, it was not found that
any additional strength was obtained over that of cylindrical
pillars.
" The strength of a pillar, in the form of the connecting
rod of a steam-engine " (that is, the transverse section pre-
senting the figure of a cross +), "was found to be very
small, perhaps not half the strength that the same metal
would have given if cast in the form of a uniform hollow
cylinder.
" A pillar irregularly fixed, so that the pressure would be
in the direction of the diagonal, is reduced to one-third of its
strength. Pillars fixed at one end and movable at the other,
us in those flat at one end and rounded at the other, break at
one- third the length from the movable end ; therefore, to
economize the metal, they should be rendered stronger there
than in other parts.
" Of rectangular pillars of timber, it was proved experimen-
tally that the pillar of greatest strength of the same material
is a square."
Long-continued pressure on pillars. "To determine
the effect of a load lying constantly on a pillar, Mr. Fairbairn
had, at the writer's suggestion, four pillars cast, all of the
same length and diameter. The first was loaded with 4 cwt.,
the second with 7 cwt, the third with 10 cwt., and the fourth
with 13 cwt. ; this last load was ^ of what had previously
broken a pillar of the same dimensions, when the weight waa
carefully laid on without loss of time. The pillar loaded
134 CIVIL ENGINEERING.
with 13 cwt. bore the weight between five and six months,
and then broke."
STRENGTH OF BEAM TO RESIST A SHEARING FORCE.
204. It has been shown that the transverse shearing stress
varies directly with the area of cross-section, and that we have
S' = AS,
in which S is the modulus of shearing. Assuming a value
which we represent by Si less than S for the given material,
and we have
W = AS t ,
in which "W" is the force producing shearing strain and B! the
limit of the shearing stress allowed on the unit of surface.
Knowing the form, the dimensions to give the cross-section
for any assumed stress are easily obtained.
TRANSVERSE STRENGTH OF BEAMS.
205. The stress on the unit of area of the fibres of a beam
at the distance y from the neutral axis, in the case of trans-
verse strain, is obtained from eq. (21),
y '
As previously stated, the hypothesis is that the stress on
the unit of area increases as y increases, and will be greatest
in any section when y has its greatest value. That unit of
area in the section farthest from the neutral axis will there*
fore be the one that has the greatest stress upon it. Now
suppose M to be increased gradually and continually. It
will at length become so great as to overcome the resistance
of the fibres and produce rupture. Since the material is
homogeneous, and supposed to resist equally well both ten-
sion and compression, the stresses on the unit of area at the
same distance on opposite sides of the neutral surface are
considered equal.
Representing by R the stress on the unit of area farthest
from the neutral surface in the section where rupture takes
place, and the corresponding value of y by y\ we have
5J=M', . . . . (86)
in which M' is the bending moment necessary to produce
rupture at this section.
TRANSVERSE STRENGTH OF BEAMS. 135
When the cross-section is a rectangle, in which I is the
breadth and d the depth, I is equal to T*ybd*, and the greatest
value of y'is r- ; substituting these values in eq. (86) we have
for a beam with rectangular cross-section,
K x 4&P=M'. ..... (87)
The first member is called the moment of rupture and
its value for different materials has been determined by ex-
periment.
These experiments have been made by taking beams of
known dimensions, resting on two points of support, and
breaking them by placing weights at the middle point.
From equation (87) we have
M/ '
in which, substituting the known quantities from the exper-
iment, the value of fi, called the modulus of rupture, is
obtained.
These values, thus obtained, are especially applicable to all
beams with a rectangular cross-section, and with sections that
do not differ materially from a rectangle. "Wliere other
cross sections are used, special experiments must be made.
206. In a beam of uniform cross-section the stresses on the
different sections vary, and that particular section at which
the moment of the external forces is the greatest is the one
where rupture begins, if the beam break. This section most
liable to break may be called the dangerous section.
In rectangular beams the dangerous section will be where
the moments of the straining forces are the greatest.
Let "W denote the total load on a beam, and I its length, we
have for the greatest moments in the following cases :
M = WZ, when the load is placed at one end of a beam, and
the other end fixed.
M= x I = 4"W7, for the same beam uniformly loaded.
2
^y ^
M = x ~ JWZ, when the load is placed at the middle
2 2
point of a beam resting its extremities on supports.
M = x J ^ = WZ, for the same beam uniformly loaded.
If a less value than that necessary to break the beam be
136 CIVIL ENGINEERING.
substituted in eq. (88) for M', the corresponding value for R,
will not be that for the modulus of rupture, but will merely
be the stress on the unit of area farthest from the neutral
axis in the dangerous section. Suppose a beam strained by a
force less than that which will produce ru-pture and find for
M the corresponding maximum value for each case. Sub-
stituting these in eq. (87), we have
(89)
in which E' is the greatest stress on the unit of area in the
dangerous section for the corresponding case x>f rectangular
beams, whose moments are given above.
The value of R for a material may be determined by find-
ing the force that will break a piece of the same material,
of a similar form, and substituting the moment of this
force for M' in eq. (86), and deducing the value of R.
Some of the values of R for pieces of rectangular cross-
section are as follows :
Material. Value of R.
Ash .............................. 12,156 Ibs.
Chestnut .......................... 10,660 "
Oak ............................... 10,590 "
Pine ............ . . . ............... 8,946
Fir ............................... 6,600 "
Cast-iron .......................... 33,000 "
The value of R is also taken as equal to eighteen times the
force required to break a piece of one inch cross-section, rest-
ing on two supports one foot apart, and loaded at the middle.
207. From the definition for R, it would seem, as before
stated, that it should be equal either to C or to T, depending
upon whether the beam broke by crushing or tearing of the
fibres. In fact, it is equal to neither, being generally greater
than the smaller and less than the greater ; as shown in the
case for cast iron, for which
The mean value of C = 96,000 pounds ;
The mean value of T = 16,000 pounds ; and
The mean value of R = 36,000 pounds.
If, ther, instead of taking R from the tables, the value of T
e calcu-
That is,
, r, na o ang rom e aes, e v
or C 6e used, taking the smaller value of the two, the calcu-
lated ^reiigth of the beam will be on the safe side.
INFLUENCE OF CROSS-SECTION. 137
the strength of the beam will be greater than that found by
calculation.
Experiments should be made upon the materials to be used
in any important structure, to find the proper value for R.
In determining the safe load to be placed on a beam, the
following values for R' may be taken as a fair average :
For seasoned timber, R' = 850 to 1,200 pounds ;
For cast iron, R' = 6,000 to 8,000 pounds ;
For wrought iron, R' = 10,000 to 15,000 pounds.
INFLUENCE OF THE FORM OF CROSS SECTION ON THE STRENGTH
OF BEAMS.
208. The resistance to shearing and tensile strains in any
section of a beam is the same for each unit of surface through-
out the section. The same has been assumed for the resist-
ance to compressive strains within certain limits. Hence so
long as the area of cross-section contains the same number of
superficial units, the form has no influence on the resistance
offered to these strains.
This is different in the case of a transverse strain.
We may write equation (21) under this form,
In this, if we suppose M to have a constant value, P will then
y
vary directly with the factor ^; that is, as this factor increases
or decreases, there will be a corresponding increase or decrease
in P.
Represent by d the depth of the beam, \d will be the
greatest value that y can have. It is readily seen, that for
any increase of %d y I will increase in such a proportion as to
decrease the value of , and hence decrease the amount of
stress on the unit of area farthest from the neutral axis.
Therefore we conclude that for two sections having the same
area, the stress on the unit of surface farthest from the neutral
d
axis is less for the one in which ^ is the greater.
This principle affords a means of comparing the relative
resistances offered to a transverse strain by beams whose cross-
sections are different in form but equivalent in area.
138 CIVIL ENGINEERING.
For example, compare the resistances offered tc a trans-
verse strain by rectangular, elliptical, and I-girders, with
equivalent cross-sections.
The values of I for the rectangle, ellipse, and I-section are
respectively,
I = ^&p, I = frirlffi, and I = T \(bd? - Vd'*\
Represent the equivalent cross-section by A, and we will
have A = M for the rectangle, A = \Trbd for the ellipse, and
A = b(dd') for the I-section. The latter is obtained by
neglecting the breadth of the rib joining the two flanges, its
area being small compared with the total area, and by regard-
ing d z = dd = d' 2 , d d being small compared with d.
Substituting these values of A in the factor !=-, we get for
the rectangle, -r . ; for the ellipse, -r^ ; for the I-section ,.
A.d J\.d Act/
Id
Hence we see that - is least for the third, and greatest for
the second, and therefore conclude that the stress on the
unit of surface farthest from the neutral axis is the least for
the I-girder, and its resistance to a transverse strain is greater
than either of the other two forms.
Since the quantity A contains b and d, by decreasing 5 and
increasing <#, within limits, the resistance of any particular
form will be increased. And hence, in general, the mass of
fibres should be thrown as far from the neutral axis as the
limits of practice will allow.
The strongest Beam that can be cut out of a Cylin-
drical Piece.
209. It is oftentimes required to cut a rectangular beam
out of a piece of round timber. The problem is to obtain the
one of greatest strength.
Denote by D the diameter of the log, by 1} the breadth,
and d the depth of the required beam.
From the value
R'- M
"
it is evident tHat the strongest beam is the one in which M*
has its maximum value.
BEAMS OF UNIFORM STRENGTH. 13!)
Representing the crosstsection of the beam and of the log
by a rectangle inscribed in a circle, we have
D being the diameter of the circle. Multiplying by 5, gives
Iff = 5D 2 - P.
In order to have bd? a maximum, D 2 35 s must be equal te
zero, which gives
and this, substituted in the expression for d 2 , gives
To construct this value of 5, draw a diameter of the circle,
and from either extremity lay off a distance equal to one-
*hird of its length. At this point erect a perpendicular to
she diameter, and from the point where it intersects the cir-
cumference draw the chords joining it with the ends of the
diameter. These chords will be the sides of the rectangle.
2d CASE. BEAMS OF VARIABLE CROSS-SECTION.
210. Beams of uniform strength. Beams which vary
in size so that the greatest stress on the unit of area in each
section shall be constant throughout the beam, form the prin-
cipal class of this second case.
In the previous discussions and problems the bar or beam
has, with but one exception, been considered as having a
uniform cross-section throughout, and in th6se discussions the
moment of inertia, I, has been treated as a constant quantity.
Since the beams had a uniform cross-section it is evident
that the greatest stress was where the moment of the exter-
nal forces was the greatest.
Finding this greatest moment of the external forces, we
determined the greatest stress and the section at which it
acted. If this section was strong enough to resist this action,
it follows that all other sections were strained less and were
larger than necessary to resist the stresses to which they
were exposed ; in other words, there was a waste of material.
The greatest stress on a unit of surface of cross-section
being known or assumed, let us impose the condition that it
shall be the same for every section of the beam. This will
140 CIVIL ENGINEERING.
necessitate variations in the cross-sections, hence I will vary
and must be determined for each particular case.
A beam is called a " solid of equal resistance " when so
proportioned that, acted on bj a given system of external
forces, the greatest stresses on the unit of area are equal for
every section.
This subject was partly discussed under the head of tension
in determining the form of a bar of uniform strength to resist
elongation. The method there used could be applied to the
case'of a beam to resist compression.
Beams of Uniform Strength to resist a Transverse
Strain.
211. Suppose the beam to be acted upon by a force produc-
ing a transverse strain, and let the cross-section be rectangular.
Let b and d represent the breadth and depth of the beam,
and we have
I = jytxP.
Substituting in eq. (21) this value of I, and giving to y its
greatest value, which is -J<#, we have
or
for the stress on a unit of surface at the distance \d from the
neutral axis in the cross-section under consideration.
The greatest stress will be found in that section for which
M is the greatest. Kepresenting this moment by M" and the
corresponding value of P' for this section by P", we have
"M "
P"
~
This value of P" is then the greatest value of the stress,
upon the unit of surface, produced by the deflecting forces
acting to bend the beam.
From the conditions of the problem, the greatest stress on
the unit of surface must be the same for every cross-section.
Eq. (90) gives the greatest stress on the unit of surface in any
cross-section. It therefore follows that for a rectangular beam
of uniform strength to resist a cross-strain, we must have
BEAMS OF UNIFOEM STRENGTH.
141
Since P" is constant, b or d, or both of them, must vary as
M varies, to make the equation a true one ; that is, the area of
cross-section must vary as M varies.
We may assume b constant for a given case, and giving
different values to M, deduce the corresponding ones for d ;
or, assuming d constant, do the same for b ; or we may assume
that their ratio shall be constant.
For the first case, b, the breadth constant, we have
(93)
For the second case, d, the depth constant, we have
M
and for the third, their ratio constant, b = rd, we have
The assumed values of M with the deduced values of d t
from eq. (93), will show the kind of line cut out of the beam
by a vertical section through the axis, when the breadth is
constant ; and the deduced values of , from eq. (94), will
show the kind of line cut out of the beam by a horizontal
section through the axis when the depth is constant. These
lines will show the law by which the sections vary from one
point to another throughout the beam,
As examples take the following cases :
212. CASE IST. A horizontal beam ftrmly fastened at one
FIG. 32.
end (Fig. 32), and the other end free to move, strained by a
load uniformly distributed along the line, A B.
142
CIVIL ENGINEERING.
Take B as the origin of co-ordinates, B A the axis of X, y
positive downwards, the axis of Z horizontal, and w the weight
on a unit of length.
The moment of the weight acting at any section as D is
-~-f substituting which for M in the expression (93) for d,
we have
which is the equation of a right line as B D, passing through
the origin of co-ordinates.
If the depth be constant, the breadth will vary from point
to point, and the different values of the ordinate may be ob-
tained by substituting this moment for M in expression (94),
and we have
3w
= Z =
which is the equation of a parabola having its vertex at B,
as in Fig. 33.
FIG. 33.
213. CASE 2D. A beam as in preceding case strained by a
load, W, concentrated and acting at -#, the weight of the
learn disregarded.
The breadth being constant, we have
or
6W
BEAMS OF UNIFORM STRENGTH.
143
which is the equation of a parabola, the vertex of which is
at B. (Fig. 34)
FIG. 34.
Suppose the depth constant ; in this case we have
6W
which is the equation of a right line, and shows that the plan
of the beam is triangular.
214. CASE 3o. /Suppose the beam resting on two supports
at its ends and- uniformly loaded.
Kepresent by 2Z the distance between the supports, by w
the load on a unit of length, and take A (Fig. 22) as the origin
of co-ordinates.
The moment of the external forces at any section, as D,
will be tyotf' wlx, which substituted in eq. (93), gives
TO o
= *-
which is the equation of an ellipse.
This moment substituted in eq. (94), gives
, _
~
~
which is the equation of a parabola.
215. In a similar way we may determine the forms of beams
of rectangular cross-section, when other conditions are im-
posed.
If we had supposed the sections circular, then I = JTTT**,
and this being substituted for I in the general expression foi
144 CIVIL ENGINEERING.
the stress on a unit of surface farthest from the neutral axis
a similar process would enable us to determine the form of
the beam.
Hence, knowing the strains to which any piece of a structure
is to be subjected, we may determine its form and dimensions
such that with the least amount of material it will successfully
resist these strains.
BELATTON BETWEEN STRESS AND DEFLECTION PRODUCED BY A
BENDING FORCE.
216. Within the elastic limit, the relation between the
greatest stress in the fibres and the maximum deflection of
the beam produced by a bending force, may be easily deter-
mined.
Take a rectangular beam, supported at the ends and loaded
at its middle point.
The third of equations (89) gives for this case
x
and solving with respect to W, we have
in which W is the load on the middle point of the beam.
The maximum deflection produced by a load, 2W, in this
case has been found, the length of beam being 2, to be
W
J- S *W
Substituting for I, W, and I, the proper values, we have
W 73
*
Solving with respect to W, and placing it equal to the value
of W obtained from eq. (89), we have
from which we get
R/= >' ;
OBLIQUE FORCES.
145
Hence, knowing the deflection and the coefficient of elas-
ticity, the greatest stress on the unit can be obtained and the
converse.
FORCES ACTING OBLIQUELY.
217. The forces acting on the beam have been supposed to
be in the plane of, and perpendicular to, the mean fibre.
The formulas deduced for this supposition are equally
applicable if the forces act obliquely to the mean fibre.
Suppose a force acting obliquely in the plane of the mean
fibre, it can be resolved into two components, one, P, perpen-
dicular, and the other, Q, parallel to the fibre. The com-
ponent P will produce deflection, and the component Q,
extension or compression depending on the angle, whether
obtuse or acute, made by the force with the piece.
The strains caused by each of the components can be deter-
mined as in previous cases.
For suppose the force applied in the plane of the axis of
a beam, at F (Fig. 35), and let x be the distance to any
tion, as K, measured on the axis of the beam E F.
FIG. 35.
FIG. 36.
Let
Z = E F, the length of the beam, and a = the angle made
by the axis E F with the vertical.
10
146 CIVIL ENGINEERING.
The bending moment at any section, as K, is equal to
sin
and its value for the dangerous section will be Wl sin or,
I being the greatest lever arm of W.
The greatest stress caused by P on the unit, at the danger-
ous section of a rectangular beam, b and d being the dimen-
sions of cross-section, will be
_ Wl sin a
6 w
The stress caused by Q on the unit will be either com-
pressive, as Fig. (35), or tensile, as Fig. (36), and its intens-
ity will be
W cos a
bd '
The total stress on the unit subjected to the greatest strain
will therefore be
"W7 sin a W cos a
~W~ ~bd~'
If a value, as R', be assumed as the limit of the stress on
the unit of material, it will be necessary to deduct from B'
the intensity of the stress caused by Q, so as to avoid de-
veloping a greater stress on the unit than that assumed, or,
we must have at the dangerous section for a rectangular
beam,
. . (97)
and in general,
STRENGTH OF BEAMS TO RESIST TWISTING.
218. Strains of torsion are not common in structures and
are prevented by distributing the loads symmetrically over
the pieces, making the resultants of the straining forces
intersect the axes of the pieces.
Whenever such a strain does exist, the intensity of the
stress may be determined by the use of formula (79). In
determining the value of T t by this formula, the experiment
must, as in the case of transverse strain, be made upon apiece
similar in form to that for which the stress is to be found.
ROLLING LOADS. 147
If the piece be circular .in cross section, formula (79) may
be placed under the form,
which gives the force necessary to produce rupture by twist-
ing.
It will be seen that the modulus of torsion is independent
of the length of the piece, being dependent upon the mo-
ment of the twisting couple and upon the form and dimen-
sions of the cross-section.
The length of the piece affects the value of the angle of
torsion, a ; the total angle being greater as I is greater. In
using formula (77) a limit should be assumed for a such that
the limit of torsional elasticity shall not be passed.
ROLLING LOADS.
219. Systems of forces, the points of application of which,
like those of stationary loads, do not move, have been the
only kinds considered in the previous discussions.
Many structures, such as bridges for example, are built to
sustain loads in motion, the load coming upon the structure
in one direction and moving off in another. A load of this
kind is called a moving, a rolling, or live load, to dis-
tinguish it from the stationary kind usually called a dead
load.
220. In determining the strength of a beam to resist the
stresses developed by a live load/it is necessary to determine
the positions the load should have that will cause the greatest
bending moment and the greatest transverse shearing strain
at any section of the learn.
Let the beam (Fig. 37) be horizontal, uniformly loaded,
and strained by a uniformly distributed live load that grad-
ually covers the entire beam. Let
2Z = A B, the length of the beam ;
w = the weight of the uniform stationary load on the
unit of length ;
w' = the weight of the rolling load per unit;
m the length of the rolling load in any one position ;
n = the length of beam not covered by the rolling load ;
Bj, KS, the reactions at the points of support.
Take the origin of co-ordinates at A, the axes X and Y as
in the previous cases, and suppose the live load to have come
on at the end A, and to occupy, in one its positions, the
space from A to D.
148
CIVIL ENGINEERING.
The reactions at the points of support, due to the uniform
load on the beam and the live load from A to D, are
EI = wl + w'm ^j , and E^ = wl + w'm jy .
A s
x*
I
ii *~ */
FIG. 37.
The bending moment at any section whose abscissa is #,
and which lies between A and D, for this position of the load,
is
M = - "Rjx + (w-+ w')-, . . (99)
and for any section between D and B, the abscissa being a?,
m). . (100)
*j 41
and, as seen, increases as m increases. The bending mo-
ment will, therefore, be greatest when m is greatest, or when
m= 21. Hence,
M r= (w -|- W'} ( lx J . . (101)
is the expression for the greatest bending moment at any
section 'of the beam, and it exists when the rolling load covers
the whole beam.
The shearing stress at any section between A and D is
S' = ( w + w') x - K! . . . (102)
and for any section between D and B is
S' = (wx + w'm) - Ej . . . (103)
ROLLING LOADS. 149
in which substituting for 1^ its value, we have
S' = w (x - 1) - w' (m - ^--#) - (104)
and
S' = w ( x - 1) + w' tg- . . . . (105)
from which the shearing stress at any section is obtained.
Let x be the abscissa of the section, D, at the end of the live
load in any one of its positions as its moves from A toward
B. Substituting x for x in eq. (105) we have
S"=w(x'-l) + w'^. . . (106)
4t
for the shearing stress at this section when the live load
extends to D.
If the rolling load extends entirely over the beam, the
shearing stress at any section is
S' = (w + w') (x - Z) . . . (107)
and for the section D,
which may be written
&" = w(x' -l)+w'(x'-l). . (108)
The values of S" at the section 0, for the positions of these
two loads, one extending to D and the other entirely over the
beam, only differ from each other in the terms, w' (x' T)
i ,m 2
and w -JT .
4:1
Since 2Z = m + n, we may write
w' j ^ w ' m *
w > (a /_ Q == ^(m - n), and w ^ == ? ^.
Bv comparing with m n, it is seen that the
J ft -ra 4- n
term w' (x' T) is less than w'^- whenever m>l,or at any
section of a beam the greatest shearing stress occurs when
the moving load covers the longer of the two segments into
which the section divides the beam.
150 CIVIL ENGINEERING.
"When the rolling load covers the longer segment, the
shearing stress is said to be a main shear ; when it covers
only the shorter segment, it is called a counter shear.
the difference in the intensity of the shearing stress, at a
given section, caused by a partial rolling ^ load and by
one that covers the beam can be shown graphically.
The term, w (x I), in equation (105), expresses the inten-
sity of the shearing stress at any section caused by the dead
load ; the term, w r j expresses the shearing stress at the
4^
sections between D and B caused by the live load. If we place
y' = w (x I) and y' = m 2 , two equations will be formed,
^tv
one that of a right line, the other a parabola, in which the
ordinates represent the shearing stresses caused by these
loads. Construct the parabola, and let A S' be the arc
determined. The ordinate D F of this arc will represent the
shearing stress at the end section D, and at all sections
between D and B produced by the live load, A D, in this
position.
When the live load covers the beam, the total shearing
stress at any section is given by equation (108). That part
of the stress produced by the live load is expressed by the
term w' (x' 1), which is the ordinate of aright line passing
through C and S'. No ordinate of this line between C and B
is equal in length to the corresponding ordinate of F S';
hence, the shearing stress in any section between C and D
is greater when the live load extends from A to the section
considered, than when it extends entirely over the beam.
Let m and x have simultaneous and equal values in
equation (105) and the equation will be that of a parabola, the
ordinates of which will express the intensity of the shearing
stress in that section coinciding with the end of the moving
load in all of its positions.
It will be seen that this parabola intersects the axis of X
between A and C, which shows that there is a section of the
beam at which there is no shearing stress when the end of the
rolling load reaches it. The expression for the distance from
the origin to this section may be obtained by placing the
second member of eq. (105) equal to zero, and solving it with
respect to x ; there results
. (109)
An equal moving load coming on the beam from B
produces a similar effect to that of the one coming from A,
LIMITS OF PRACTICE. 151
It therefore follows that there is a point of "no shear-
ing " beween B and C, and that this point, in this case, coin-
cides with the section at the rear end of the rolling load
coming from A as it rolls off the beam. These points of
" no shear " are of interest in " built " beams or beams com-
posed of several pieces.
LIMITS OF PRACTICE.
221. Until quite recently, comparatively speaking, it was
the custom of most builders, in planning and erecting a
structure, to fix the dimensions of its various parts from pre-
cedent, that is, by copying from structures already built.
So long as the structure resembled those already existing
that had stood the test of time, this method served its pur-
pose. But when circumstances forced the builder to erect
structures different from any in existence or previously known,
and to use materials in a way in which they had never before
been applied, the experience of the past could no longer be
his guide. Practical sagacity, a most excellent and useful
qualification, was not sufficient for the emergency. Hence
arose the necessity that the builder should acquire a thorough
knowledge of the theory of strains, the strength of materials,
and their general properties.
The principal object of "strength of materials " is to de
termine the stresses developed in the different parts of a struc-
ture, and to ascertain if the stresses are within the adopted
limits. And as a consequent, knowing the strains, to deter-
mine the forms and dimensions of the different parts, so that
with the least amount of material they shall successfully re-
sist these strains.
The limits adopted vary with the materials and the charac-
ter of the strain. The essential point is that the limit of
elasticity of the material should not be passed, even when by
some unforeseen accident the structure is subjected to an un-
nsual stress. The adopted limit to be assigned is easily
selected if the limit of elasticity be known ; but as the latter
is obtained with some difficulty, certain limits of practice
have been adopted.
In many cases this practice is to arbitrarily assume some
given weight as the greatest load per square inch on a given
material, and to use this weight for all pieces of the same
material. From the varying qualities of the same material it
is easily seen that this method of practice differs but little
from a " mere rule of thumb."
The most usual practice, especially for structures of im-
portance, as bridges, is to determine the breaking weights or
152 CIVIL ENGINEERING.
ultimate strength of the different parts, and take a frac-
tional part of this strength as the limit to be used. The re-
ciprocal of this fraction is called the factor of safety.
A more accurate method would be to calculate the dimen-
sions of the pieces necessary to resist the strains produced by
the maximum load, and then enlarge the parts sufficiently to
give the strength determined by the factor of safety.
When the structure is one" of great importance, actua?
experiments should be made on each kind of material used fa
its construction, so that the values deduced for the ultimate
strength shall be as nearly correct as possible.
222. These factors of safety are arbitrarily assumed, being
generally about as follows :
Material. Factor of safety.
Steel and wrought iron 3
Cast iron 6
Timber 6
Stone and brick , 8 to 10.
These are for loads carefully put on the structure.
If the materials and workmanship were perfect, these factors
could be materially reduced.
It has been shown (Art. 160) that the work expended by
the sudden application of a given force, W, is equal to that
expended by 2W if applied gradually at a uniform rate
from zero to 2W. Hence a force, W, applied suddenly to
a beam will produce the same strain on the beam as 2W
applied gradually.
A rolling load moving swiftly on a structure approximates
nearly to the case of a force suddenly applied.
Hence, for rolling loads, the factors of safety should be
doubled.
CURVED BEAMS.
223. A beam which before it is strained has a curvilinear
shape in the direction of its length is called a curved beam.
The curve given to the mean fibre is usually that of a cir-
cular or a parabolic arc.
For the purposes of discussing the strains on beams of this
class, it is supposed that:
1. The beam has a uniform cross-section ;
2. That its cross-section is a plane iigure, which if moved
along the mean fibre of the beam and normal to it, keeping
CURVED BEAMS. 153
the centre of gravity of the plane figure on the mean fibre,
would generate the solid ; and
3. That the dimensions of the cross-section in the direction
of the radius of curvature of the mean fibre are very small
compared with the length of this radius.
If the beam be intersected by consecutive planes of cross-
section, the hypotheses adopted for a straight beam subjected
to a cross strain are assumed as applicable to this case.
224. General equations. Suppose the applied forces to
act in the plane of mean fibre, let it be required to deter-
mine the relations between the moment of resistance
at any section and the moment of the external forces
acting on the beam.
Let E F (Fig. 38) be a curved beam ; the ends E and F so
arranged that the horizontal distance between them shall
remain constant.
Km
FIG. 3&
.Let A B be any cross-section. The external forces acting
on either side of this section are held in equilibrium by the
resistances developed in this section. Suppose A B to be fixed,
and let C'D' be the position assumed by the consecutive
section under the action of the external forces, on the right
of A B. The resultant of these external forces may be resolved
into two components, one normal and the other parallel to the
tangent, to the curve of the mean fibre at 0. Represent the
former by F, the latter by F, and by M, the sum of the
moments of the external forces around the neutral axis in the
section A B.
The fibre ab is elongated by an amount be, proportional to
its distance from the neutral axis.
154- CIVIL ENGINEERING.
The force producing this elongation is
Eg x l>c
or since ah may be considered equal to 0',
Ea x 1)0
OCX '
in which E is the co-efficient of elasticity and a the area oi
cross-section of the fibre, ah.
Hence, there obtains to express the conditions of equilibrium,
(110)
.Represent by p and />', the radii of curvature, R 0' and R'O'.
The triangle, #R, has its three sides cut by the right line,
R'C'. Hence the product of the segments, R 0', fo, and aR' is
equal to the product of the three segments, R R', 50', and ac.
Substituting p for R 0', p p' for R'R, and p' for #R', since
Q'b is very small in comparison with p f , and we have
p x Ic x p' = (p p') x bO' x ac.
From which we get
OG pp p p
Since ac differs from 00' by an infinitely small quantity,
"bo
the expression obtained for may be taken as the value of
> Substituting this value for ,, in the second of equa-
tions (110), we get.
E X -
This sum, S(a x JO 72 ), is the moment of inertia of the
cross-section taken with respect to the neutral axis passing
through the centre of gravity of the section. Kepresenting
this by I, equation (111) may be written
M r
which is the general equation, showing the relation existing
CURVED BEAMS.
155
between the moments of resistances of any section and the
moments of the external forces acting on that section.
225 Displacement of any point of the curve of mean
fibre. Let A B (Fig. 39) be the curve of mean fibre before
the external forces are applied to the beam.
FIG. 39.
Take the origin of co-ordinates at the highest point, C, and
the axes X and Y as shown in the figure.
Let D be any point whose co-ordinates are x and y, and
represent by the angle made by the plane of cross-section
at D with the axis of Y.
Suppose the external forces applied, and denote by x f and
y' the co-ordinates of D in its new position, and by ' the
new angle made by the plane of cross-section with the axis of
Y.
It is supposed that the displacement of the point, D, is so
slight that M remains unchanged.
From the calculus we have
-- A '
in which dz and dz' are the lengths of the elementary prism
before and after the strain measured along the mean fibre.
Since they differ by an infinitely small quantity from each
other, by making dz = dz' and substituting in equation (112)
we get
El
Integrating we obtain
.. . . (H3)
156 CIVIL ENGINEEEING.
The component force, parallel to the tangent at D, acts in
the direction of the length of the fibre. Since the points E
and F are fixed, this force produces a strain of compression
on the fibre. The length of this fibre, after compression
between the two consecutive planes, is represented by dfc',
and is
The values of cos <, sin <, cos <', and sin <' may be written
as follows :
dx dy
cos0 = ^ sin * = ^
da/ dy 1
'
Substituting, in the last two of these, the value just found
for dz ', we get
da/ _ dy'
cos = . >b and sin <' = *
If $' is very small, we may write
cos <' = cos (f> (<' <) sin <^>, and
sin <' = sin tj> + (>' ^>) cos <.
Substituting these values of cos >' and sin $', in the expres-
eions above, and solving with respect to dx' and dy ', we get
dx' = dz(l - - (cos - (<' - ^) sin <^>),
(sin < + (f - 0) cos 0).
Substituting in these for sin and cos <, their values in
terms of dz, dy, and ^a?, we get
CURVED BEAMS. 157
whence, by omitting the products of the second terms, we get
dx' dx -pTT dx (<' $) dy,
y + ( f ) dx.
r* / _ A_
Integrating, there obtains
r P r
-J T^A dx ~ I W tne lever arms of J^ and Q 2 , with
respect to the same point.
We have three equations and four unknown quantities. By
introducing the condition that the point B, shall occupy the
158
CIVIL ENGINEERING.
same position after the application of the forces as it had be-
fore, that is, befisced, a fourth equation may be obtained, and
the problem made determinate.
To express this last condition, let a^ and y t be the co-ordi-
nates of the extremity B (Fig. 40), x and y the co-ordinates
FIG. 40.
of any point as D, and $ the angle made by the tangent line at
D with the axis of X. Represent by TV the sum of the com-
ponents of the applied forces parallel to the tangent DT, and
by fjL the sum of the moments of the applied forces with re-
spect to to the section at D.
The bending moment at D will be
M = A i + Qi(^-y)-Bi( are fished by
side pieces c and d bolted to them.
is said to be fished, and is sometimes called a fish joint.
A plain joint is a good one when the onfy strain is that of
compression. It is recommended, in this case, to place h'sh-
pieces on all four of the sides of the beam, to prevent any
lateral displacement of the ends that might be caused by
shocks.
If the strain be one of tension, it is evident that the strength
166
CIVIL ENGINEERING.
is joint (Fig. 42) depends entirely upon the strength of
olts, assisted by the friction of the fish-pieces against
Such a joint would seldom be used
of this
the bolts,
the sides of the timber.
for tension.
A better fastening for the joint would be that in which
the fish-pieces were let into the upper side of the beam, as
3>
FIG. 43 Represents a joint to resist extension, iron rods or bars being used
to connect the beams instead of wooden fish-pieces.
shown in Fig. 44.
Sometimes the beam and the fish-pieces have shallow
notches made in them, into which keys or folding wedges of
FIG. 44 Represents a fished joint in which the side pieces c and d are either
let into the beams or secured by keys e, e.
hard wood as e, e (Fig. 44) are inserted.
Scarf Joints.
When the ends of the pieces overlap, the joint is called a
scarf joint. The ends of the pieces are fastened together
by bolts, to keep them in place. An example of a simple
scarf joint is shown in Fig. 45, that is sometimes used when
the beam is to be subjected only to a slight strain of exten-
FJG. 45.
e perpendicular to each other when practicable ; and the
FlG. 51 Represents a mortise and tenon joint when the
axes of the beams are oblique to each other.
thickness of the tenon d should be about one-fifth of that of
the beam A. The joint should be left a little open at c to
allow for settling of the frame. The distance from J to the
end D of the beam should be sufficiently great to resist safely
the longitudinal shearing strain caused by the thrust of the
team A against the surface ab.
Denote by
H the component of the thrust, parallel to the axifl of
the beam B D ;
I the breadth in inches of the beam B D ;
170 CIVIL ENGINEERING.
I the distance in inches from I to the end of the
beam at D ; and
S the resistance per square inch in the beam B to lon-
gitudinal shearing.
The total resistance to shearing will be S x U, hence
S x U = H, from which we have
The value of S for the given material, Art. 166, being sub-
stituted in this expression, will give the value for I, when the
Btrain just overcomes the resistance of the fibres. In this
case the factor of safety is ordinarily assumed to be at least
four. Therefore the value of I, when the adhesion of the
fibres is depended upon to resist this strain, will be
S being taken from the tables.
A bolt, ef, or strap, is generally used to fasten the ends
more securely.
In both of these cases the beam A is subjected to a strain
of compression, and is supported by B. If we suppose the
beams reversed, A to support B, the general principles for
forming the joints would remain the same.
Suppose the axes of the beams to be horizontal, and the
beam A to be subjected to a cross-strain, the circumstances
being such that the end of the beam A is to be connected with
the face of the other beam B.
In this case a mortise and tenon joint is used, but modified
in form from those just shown.
To weaken the main or supporting beam as little as possi-
ble, the mortise should be cut near the middle of its depth ;
that is, the centre of the mortise should be at or near the neu-
tral axis. In order that the tenon should have the greatest
strength, it should be at or near the under side of the joint.
Since both of these conditions cannot be combined in
the same joint, a modification of both is used, as shown in
Fig. 52.
The tenon has a depth of one-sixth that of the cross-beam
A, and a length of twice this, or of one-third the depth of the
beam. The lower side of the cross-beam is made into a shoul-
der, which is let into the main beam, one half the length of
the tenon.
Double tenons have been considerably used in carpentry
FASTENINGS. 371
As a rule they should never be used, as both are seldora in
bearing at the same time.
FIG. 52. A, the cross-beam.
B, cross-section of main beam,
i, the tenon.
III. Joints used to connect beams, the faces resting on
or notched into each other.
238. The simplest and strongest joint in this case is made
by cutting a notch in one or both beams and fastening the
fitted beams together.
If the beams do not cross, but have the end of one to rest
upon the other, a dove-tail joint is sometimes used. In this
joint, a notch trapezoidal in form, is cut in the supporting
beam, and the end of the other beam is fitted into this notch.
On account of the shrinkage of timber, the dove-tail joint
should never be used except in cases where the shrinkage in
the different parts counteract each other.
It is ajoint much used in joiner's work.
239. The joints used in timber-work are generally composed
of plane surfaces. Curved ones have been recommended
for struts, but the experiments. of Hodgkinson would hardly
justify their use. The simplest forms are as a rule the best,
as they afford the easiest means of fitting the parts together.
FASTENINGS.
The fastenings used to hold the pieces of a frame together
at the joints may be classed as follows :
1. Pins, including nails, spikes, screws, bolts, and wedges;
2. Straps and tiebars, including stirrups, suspending-rods,
etc. ; and
3. Sockets.
These are so well known that a description of them is un-
necessary.
172 CIVIL ENGINEERING.
General Rules to be observed in the Construction of Joints.
241. The following general rules should be observed in the
construction pf joints and fastenings for frames of timber :
I. To arrange the joints and fastenings so as to weaken as
little as possible the pieces which are to be connected.
II. In a joint subjected to compression, to place the abut-
ting surfaces as nearly as possible perpendicular to the direc-
tion of the strain.
III. To give to such joints as great a surface as practicable.
IV. To proportion the fastenings so that they will be equal
in strength to the pieces they connect.
Y. To place the fastenings so that there shall be no danger
of the joint giving way by the fastenings shearing or crushing
the timber.
JOINTS FOR IRON-WORK.
242. The pieces of an iron frame are ordinarily joined by
means of rivets, pins, or nuts and screws.
Riveted Joints.
243. A rivet is a short, headed bolt or pin, of iron or other
malleable material, made so that it can be inserted into holes
in the pieces to be fastened together, and that the point of
the bolt can be spread out or beaten down closely upon the
piece by pressure or hammering. This operation is termed
riveting, and is performed by hand or by machinery. By
hand, it is done with a hammer by a succession of blows.
By machinery, as ordinarily used, the heated bolt is both
pressed into the hole and riveted by a single stroke. If a ma-
chine uses a succession of blows, the operation is then known
as snap-riveting. By many it is claimed that machine
riveting possesses great superiority over that by hand, for
the reason that the rivets more completely fill the holes, and
in this way become an integral part of the structure. It is
doubtful if it possesses the advantage of superior strength to
any marked degree. It does certainly possess, however, the
advantage of being more quickly executed without damage
to the heads of the rivets.
The holes are generally made by punching, are about one-
twentieth of an inch larger than the diameter of the rivet, and
NUMBER OF RIVETS. 173
are slightly conical. The diameter of the rivet is generally
greater than the thickness of the plate through which the hole
is to be punched, because of the difficulty of punching holes
of a smaller size. Punching injures the piece when the latter
is of a hard variety of iron, and for this reason engineers often
require that the holes be drilled. Drilling seems to be the
better method, especially when several thicknesses of plates
are to be connected, as it insures the precise matching of the
rivet holes. The appearance of the iron around a hole made by
punching gives a very fair test of the quality of the iron.
When two or more plates are to be riveted, they are placed
together in the proper position, with the rivet-holes exactly
over one another, and screwed together by temporary screw-
bolts inserted through some of the holes. The rivets, heated
red-hot, are then inserted into the holes up to the head, and
by pressure or hammering, the small end is beaten down fast
to the plate. In a good joint, especially when newly riveted,
the friction of the pieces is very great, being sufficient to sus-
tain the working-load without calling into play the shearing
resistance of the rivets. In calculating the strength of the
frame, this amount of strength due to friction is not consid-
ered, as it cannot be relied on after a short time in those cases
where the frame is subjected to shocks, vibrations, or great
changes of temperature.
Number and Arrangement of Rivets.
244. The general rule determining the number is that the
sum of the areas of the cross-sections of the rivets shall be
equal to the effective sectional area of the plate after the holes
have leen punched. This rule is based on the theory that the
resistance to shearing strain in the rivet is equal to the tena-
city of the plate.
To determine the proper distance bet-ween the rivets
in the direction of any row, so that the strength of the rivets
in any single row shall be equal to the strength of the section
of the plate along this row after the holes have been punched,
let
d y be the diameter of the rivet ;
c, the distance between the centres of consecutive rivets ;
a, the area of cross-section of the rivet ;
A', the effective area, between two consecutive rivets, of the
cross-section of the plate along the row of rivets ; and
t, the thickness of the iron plate.
174 CIVIL ENGINEERING.
It has been assumed that
T = S,
and the rule requires that
TA' = S x a, or -g- = -, - 1.
We have
whence
c = ^+d, (123)
for the distance from centre to centre of tlie consecutive
rivets in any one row.
English engineers, in practice, use rivets whose diameters
are f, f, -J, 1, 1J, and 1J- inches, for iron plates J, ^, , , |,
and f inches thick, respectively, and take the distance from-
centre to centre at 2 diameters for a strain of compression,
and 2J diameters for extension. The distance of the centre
of the extreme rivet from the edge of the plate is taken be-
tween 1 and 2 diameters.
Instead of assuming the resistance to shearing in the rivet
equal to the tenacity of the iron plate, a better rule would be
to make the product arising from multiplying the sum of the
areas of the cross -section's of the rivets, by the amount of
shearing strain allowed on each unit, equal to the maximum
strain transmitted through the joint.
If the strain was one of compression in the plates and the
ends exactly fitted, the only riveting required w r ould be that
necessary to keep the plates in position. As the workman-
ship rarely, if ever, admits of so exact fitting, the rivets
should be proportioned by the rules just given.
245. The head of a rivet is usually circular in form, with
a diameter not less than twice the diameter of the rivet.
The thickness of the head at its centre should be not less
than half the thickness of the rivet.
)
ooioo
ooioo
O OiOO
<
FIG. 53.
^ 246. Various methods are used in the arrangement of the
rivets. The arrangement often used for lengthening a plate
is shown in Fig. 53. This method is known as "chain rivet-
ing."
ARRANGEMENT OF RIVETS.
175
Fig. 54 shows another method used for the same purpose,
in which the number of rivets is the same as in the previous
example, but there is a better disposition of them.
FIG. 54.
Figs. 55 and 56 show the arrangement of the rivets often
nsed to fasten ties to a plate.
r
o o
000
000
o o
(
/-v
*"S,
FIG. 55.
FIG. 56.
Figs. 57, 58, and 59 show in plan the forms of several
kinds of riveted joints.
FIG. 57.
Fig. 57 shows the single shear-joint or single lap-joint.
FIG. 58.
Fig. 58 is a plain joint fished. In this example the fish
or cover plates are placed on each side, and have a thick-
176 CIVIL ENGINEERING.
ness of half that of the plates to be connected ; sometimes
only one cover plate is used.
_s L
v - i
FIG. 59.
When several plates are to be fastened together, the method
shown in Fig. 59 is the one ordinarily used.
Eye-bar and Pin Joints.
247. A simple and economical method of joining flat bars
end to end when subjected to a strain of extension, is to con-
nect them by pins passing through holes or eyes made in the
ends of the bars.
When several are connected end to end, they form a flexi-
ble arrangement, and the bars are often termed links.
This method of connecting is called the eye-bar and pin,
or link and pin joint, and is shown in plan in Fig. 60.
PIG. 60.
The bar should be so tormed at the end that it would be
no more liable to break there than at any other point. The
following are the dimensions in the case where the head has
the same thickness as the bar.
If the width of the bar be taken as equal to. 1 .
The diameter of the eye should equal 75.
Depth of head beyond the eye should equal 1 .
Sam of the sides of the head through eye should equal 1 .25.
Radius of curve of neck should equal 1.5.
Hence, for a bar eight inches wide, the dimensions would
be as shown in Fig. 61.
SCREW-BOLTS.
177
By this rule the pin has a diameter which gives a sufficient
bearing surfaee, the important point to be considered.
FIQ. 61.
There should be a good fit between the pin and eye, espe-
cially in structures subjected to shocks, hence the conditions
of manufacture and the quality of material and workmanship
should be of the best kind.
Screw-bolt Joints.
248. The connection by nut and screw is simple and
economical.
The strength of a bolt or rod on which a screw is made,
when subjected to a shearing strain, is determined as in the
case of rivets or pins. In case of a tensile strain the strength
is measured by the area of cross-section of the spindle inside
the thread.
The resistance offered to stripping by the nut depends upon
the form of the thread and the depth of the nut. In order
that this resistance should be equal to that offered by the bolt
to being pulled apart, the length of the nut should be at least
equal to one-half the diameter of the screw.
The following proportions have been recommended by the
Franklin Institute :
Diameter of
No. of threads
Six-sided nut. Length of
Depth of
Depth of
bolt in inches.
per inch.
head.
nut.
Long diameter,
Short diameter,
i
13
1
i
ft
4
I
10
i-.V
H
*
J
8
ii
u
H
1
H
2*
8f
1A
14
3
4|
a*
84
ift
2
i
4
4*
3i
Itt
9*
84
5|
4|
2A
8
178 CIVIL ENGINEERING.
SIMPLE BEAMS.
249. One of the most common and simple use of frame*
is that in which the frame is supported at its extremities and
subjected only to a transverse strain.
When the distance between the points of support, or the
bearing, is not very great, frames are not necessary, as beams
of ordinary dimensions are strong and stiff enough to resist
the cross-strains arising from the load they support, without
bending beyond the allowed limits. The load placed upon
them may be uniformly distributed, or may act at a point ;
in either case the strains produced, and the dimensions of the
beam to resist them, can be easily determined. (Arts. 177
and 179.)
The usual method is to place the beams in parallel rows,
the distances apart depending on the load they have to sup-
port. The joists of a floor, the rafters of a roof, are exam-
ples of such cases.
The depth of a beam used for this purpose is always made
much greater than its breadth, and arrangements should be
made to prevent the beam twisting or bending laterally. It
is usual to place short struts or battens in a diagonal direction
between the joists of a floor, fastening the top of one joist
with the bottom of the next by the battens to prevent them
from twisting or yielding laterally.
SOLID BUILT BEAMS.
250. A solid beam is oftentimes required to be of a
greater size than that possessed by any single piece of tim-
ber. To provide such a beam it is necessary to use a com-
bination of pieces, consisting of several layers of timber laid
in juxtaposition and firmly fastened together by bolts, straps,
or other means, so that the whole shall act as a single piece.
This is termed a solid built beam.
FIG. 62.
"When two pieces of timber are built into one beam having
twice the depth of either, keys of hard wood are used to resist
the shearing strain along the "joint, as shown in Fig. 62.
SOLID BUILT BEAMS.
179
Tredgold gives the rule that the breadth of the key should
be twice its depth, and the sum of the depths should be equal
to once and a third the total depth of the beam.
It has been recommended to have the bolts and the keys on
the right of the centre make an angle of 45 with the axis of
the beam, and those on the left to make the supplement of
this angle.
The keys are sometimes made of two wedge-shaped pieces
(Fig. 63)/for the purpose of making them fit the notches
FIG. 63 Represents the folding wedges, a, J, let into a notch in
the beam c.
more snugly, and, in case of shrinkage in the timber, to allow
of easy readj ustment.
When the depth of the beam is required to be less than the
sum of the depths of the two pieces, they are often built into
one by indenting them, the projections of the one fitting
accurately into the notches made in the other, the two being
firmly fastened together by bolts or straps. The built beam
shown in Fig. 64 illustrates this method. In this particular
example the beam tapers slightly from the middle to the
ends, so that the iron bands may be slipped on over the ends
and driven tight with mallets.
FIG. 64 Represents a solid bnilt beam, the top part being of two pieces, b, b,
which abut against a broad flat iron bolt, a, termed a king-bott.
When a beam is built of several pieces yi lengths as well
as in depth, they should break joints with each other. The
layers below the neutral axis should be lengthened by the
scarf or fish joints used for resisting tension, and the upper
180
CIVIL ENGINEERING.
ones should have the ends abut against each other, using plain
butt joints.
Many builders prefer using a built beam of selected tim-
ber to a single solid one, on account of the great difficulty of
getting the latter, when very large, free from defects ; more-
over, the strength of the former can be relied upon, although
it cannot be stronger than the corresponding solid beam if
perfectly sound.
FRAMING WITH INTERMEDIATE POINTS OF SUPPORT.
251. If the bearing be great, the beam will bend under
the load it has to support, and to prevent this it will need in-
termediate points of support. These points of support may
be below the beam, or they may be above it.
The simplest method, when practicable, is to place at suit-
able intervals under the beam upright pieces to act as props
or shores.
When this cannot be done, but points of support can be
obtained below those on which the beam rests, inclined struts
may be usjed.
These may meet at the middle point of the beam, divid-
ing it into two equal parts. The beam is then said to be
braced, and is no longer supported at two points, but rests
on three.
The struts may be placed so as to divide the beam (Fig. 65)
into three parts, being connected with it by suitable joints.
FIG. 65.
The bearing of the beam may be reduced by placing under
it and on the points of support (Fig. 66) short pieces, termed
corbels. These, when long, should be strengthened by struts,
as shown in the figure.
In some cases the beam is strengthened by placing under
OPEN-BUILT BEAMS. 181
the middle portion a short piece, termed a straining beam
(Fig. 67), which is supported by struts.
FIG. 66 A horizontal beam, e, resting on vertical posts, a a, with
corbels, d d, and struts, e e.
These methods may be combined when circumstances re-
quire it, and the strains on the different parts can be deter-
mined. It is well to remember that placing equal beams over
FlG. 67 A horizontal beam, e, strengthened by a straining beam, /.
each other only doubles the strength, unless they are firmly
connected so as to act as one beam, in which case the combi-
nation follows the law already deduced, that is, the strength
will be four times as great.
OPEN-BUILT BEAMS.
252. An open-built beam, or truss, is a frame in which
two beams, either single or solid built, with openings between
them, are connected by cross and diagonal pieces, so that the
whole arrangement acts like a single beam in receiving and
transmitting strains.
These frames are largely used in bridge building, and their
details will be considered under that head.
The king-post truss is one of the simplest forms of frames
belonging to this class.
This truss is employed when there are no points ot support
beneath the beam which can be used, but when the middle of
the beam can be sustained by suspension from a point, above.
The arrangement consists of two inclined pieces framed
182
CIVIL ENGINEERING.
into the extremities of the beam, and meeting At an angle
above, from which the middle of the beam is ^ supported by a
third piece. This combination is shown in Fig. 68.
FIG. 68.
The construction is simple and the frame is rigid. It is
frequently employed in roofs and in bridges of short span.
In the earlier constructions the third piece, , was made of
wood, and resembled a post, hence the name of king-post.
The strain it sustains is one of tension, and in modern con-
structions an iron-rod is generally used. It would be better
if a more appropriate name were given, since the term post
conveys to the mind an impression that the strain is one of
compression.
When the suspension piece is made of timber, it may be a
single piece framed into the struts, and the foot connected
with the beam by a bolt, an iron stirrup, or by a mortise and
tenon joint ; or it may be composed of two pieces bolted
together, embracing the heads of the struts and the supported
beam. In the latter case, these pieces are called bridle-
pieces, two of which are shown in Fig. 69.
FIG. 69.
When two points of support are necessary, the arrangement
known as the queen-post truss may be used. It consists of
two struts framed into the extremities of the beam, and abut-
ting against a short straining beam (Fig. 69). The suspen-
8TEAJNS ON FRAMES. 183
sion pieces are either of iron or wood, single or double, as in
the king-post truss.
The remarks just made about the name "post" apply also
to this combination.
Both of these trusses may be inverted, thus placing the
points of support beneath the beam. This change of position
changes the character of strains on the different parts, but
does not affect their amount, which is determined in the same
way in both cases.
Points of support above and beneath may be obtained by
the use of curved beams.
METHODS OF CALCULATING STRAINS ON FRAMES.
253. It has been previously stated that to prevent a change
of form in a quadrilateral frame, secondary pieces are intro-
duced for the purpose of dividing the frame into two or more
triangular figures.
In all frames "where rigidity is essential to stability, this in-
troduction of braces is necessary, as the triangle is the only
geometrical figure which, subjected to a straining force,
possesses the property of preserving its form unaltered as
long as the lengths of its sides remain constant.
The triangular is the simplest form of frame, and will be
first used in this discussion.
254. As a preliminary step, let the strains in an inclined
beam, arising from a force acting in the plane of its axis, be
determined.
For example, take
An inclined beam with the lower end resting against an
abutment and the upper end against a vertical watt, and sup-
porting a weight, W, applied at any point.
Fig. 70 represents the case.
Denote by
Z, the length of the axis, A B, of the beam;
n x I, the distance from A to the point C, where W is ap-
plied ;
a, the angle between A B, and vertical line through C.
Disregarding the weight of the beam, the external forces
acting on it are the weight, W, and the reactions at A and B.
Suppose the reaction at B to be horizontal and represent
it by H. Kepresent the horizontal and vertical components
of the reaction at A, respectively by H' and W.
184:
CIVIL ENGINEERING.
These forces are all in the same plane, and the analytical
conditions for equilibrium are
H - H' = 0, and W - W = 0.
If
w
FIG. 70.
Taking the bending moment about A, we have
WxAD-HxBE = 0,
or, H x I cos a = W x nl sin a,
hence, H = n W tan a
The forces H, IF, W, and W act in the plane of and
obliquely to the axis, A B, and their effect is to produce de-
flection and compression of the fibres of the beam. The
strain arising from deflection will be due to the algebraic sum
of the perpendicular components, and that from compression
will be due to the sum of the parallel ones. (Art. 217.)
Resolve W and H'into components acting perpendicularly
and parallel to the axis of the beam. Represent by P and
P', and Q and Q', these components; see Fig. 71.
kd
= W = W.
=.P, Ac = Q.
= H' = nW tan a.
= P', Aw = Q'.
STRAINS ON FRAMES.
185
The perpendicular components kd and km act in opposite
directions, hence the strain arising from deflection will be due
to their difference, P P'.
FIG. 71.
The parallel components kc and kn act in the same direc-
tion, hence the strain of compression will be due to their sum,
Q + Q'.
Representing the force W, by the line A5, we find the values
of these components to be as follows :
P = W sin a ; P' = n W tan a cos a = n W sin a;
Q = W cos a ; Q' = n W tan a sin a.
Suppose the cross-section of the beam to be a rectangle of
uniform dimension, the sides of which are respectively b
and d, the plane of the latter being taken parallel to the
direction of the force, W, we have
Q 4. Q' = W cos a + n W tan a sin a,
equal to the total compression on the segment from A to C ;
this sum divided by bd will be the amount of compression
on the unit of area in any cross-section in this segment.
We also have
P - P' = (1 - n) W sin a,
for the force perpendicular to the axis of the beaml Ita
moment for any section, at the distance, a?, measured on the
line A B, and lying between A and C, will be
(1 n) W sin a x .
186 CIVIL ENGINEERING.
Substituting in the expression for R.' (Art. 206), we have
(1 _ ri\ Wo? sin a
-/ \
for the stress on the unit of area farthest from the neutral
axis in any section produced by deflection, x being the lever
arm
For the segment of the beam, B C, it is seen that the strain
of direct compression is due to the force
Q' = n W tan a sin a.
Giving values to ^, from to 1, we can place the force, W,
at any point on the axis. And knowing 5, d, and W, and
substituting them in the foregoing expressions, we obtain the
stresses in the beam.
Let us place it at the middle point, arid suppose W and a to
be given.
The value of n for the middle point is ; substituting which
in the expressions for P, Q, etc., there obtains :
W cos a 4- iW tan a sin a
for the stress of compression on the unit of cross-section ; and
(P P'fe _ JW x sin a
for the stress due to deflection on the unit of cross-section
farthest from the neutral axis. Represent these by C' and R',
respectively. To determine the greatest stress on the unit of
area in any cross-section ; first, determine R/ for the particu-
lar section and add to the value thus found that for C', and
the result will be the total stress on the unit, and hence the
maximum stress in that section.
To determine the greatest stress produced by the force,
W, upon the unit of surface of the beam : first, find the value
of R' for the dangerous section and then add to it the value
of C' for this section ; the result will be the greatest stress.
Assuming limiting values for R' and C'and knowing & and
d, the corresponding value for "W" can be deduced. Or, as-
suming R' and C' and having "W given, we can deduce values
for ~b and d.
Suppose the beam to be vertical, then a = 0, and we get
Q = W, and Q' = 0,
STRAINS ON FRAMES 187
or the compression in B C will be zero, and on A C equal to
W. "We also have H' = 0, or there is no horizontal thrust.
Suppose the beam horizontal, then a = 90, and we get IF
and Q', each equal to infinity.
From this it is seen that the compression on the beam and
the horizontal thrust at the foot both decrease as a decreases,
and the reverse.
255. Uniformly loaded. Suppose the beam to be uni-
formly loaded, and let w be the load on a unit of length of
the beam.
We have H = \wl tan a.
The corresponding values for P, P', Q, and Q' are easily
obtained.
256. Let it be required to determine the strains on a
triangular frame, and take for example,
A frame made of three beams connected at the ends by
proper joints and strained by a force acting in the plane of
their axes and at one of the angular points.
Suppose the plane of the axes ofc the three beams to be ver-
tical, and one of the sides, B C, to be horizontal, resting on
fixed points of support at B and C.
Disregarding the weight of the frame itself, suppose the
straining force to be a weight suspended from or resting on
the point A. (Fig. 72.)
Represent by
"W, the weight acting at A,
a, the angle BAD,
" CAD.
FIG. 72.
The weight, W, acts vertically downwards and is prevented
from falling by the support at A. The pressure exerted by
it at A is received by the inclined beams, A B, and A C, and is
transmitted by them to the fixed points of support at B and C.
188 CIVIL ENGINEERING.
The weight, W, is therefore the resultant force acting on the
frame, and the pressure on the inclined beams are its compo-
nents in the directions of the axes of the beams.
Kepresent by kd the weight W, and construct the parallelo-
gram kbcd. We have from the principle of the parallelo-
gram of forces :
Wsin Wsina
M> = - f . Q\ and Ac = ' t -L/QV
sin (a + p) sin (a + p)
The strains produced by these components are compressive.
Knowing the breadth and depth of the beams, the amount of
stress on the unit of cross-section can be determined ; or
assuming a limit for this stress on the unit, the values for the
breadth and depth of the beams may be deduced.
These components being transmitted along the axes of the
beams to the points of support, B and C, may be resolved at
these points into their horizontal and vertical components
respectively.
Doing so, it is seen that the horizontal components are
equal to bm and en, and are equal to each other, but act in
opposite directions. The value for these components is
. (126)
Hence, they balance each other, producing a strain of ex-
tension on the beam, B C, the amount of which on the unit of
cross-section, or dimensions of beam to resist which, may be
determined. The vertical components are respectively equal
to Am and kn, and act in the same direction. We have
_ sin /3 cos a _ sin a cos 8 .
Am = W . , , ^ and kn = W -r-, r-~. (12T)
sm(a + )' sin (a + /3) v '
They are resisted by the reactions at the points of support,
which must be strong enough to sustain these vertical pres-
sures. Adding Am to kn we find their sum is equal to W.
It is well to observe that producing kd to D, we have the pro-
portion, Am : ATI : : C D : B D. That is, the vertical through A
divides the side B C into two segments proportional to the
vertical components acting at B and C.
257. The common roof-truss, in which A B is equal in
length to A C, and the angle a equal to ft, is the most usual
form of the triangular frame.
STRAINS ON FRAMES. 189
For this case we would have
W
A5 = kc -J - - , lm = W tan a, and km = ATI = \ W.
COS CL
Kepresent by 21 the length of B C, d, the length of A D
and A, the length of A B = A C, and substituting in the fore-
going expression, we have
= en = ,
which are fully given for any assumed value for W when
either two of the quantities in the second members are
known.
If, instead of a single weight, the frame had been strained
by a uniform load distributed over the inclined pieces A B
and A C, we may suppose the whole load to be divided into
two equal parts, one acting at the middle point of A B and
the other at the middle point of A C, the discussion of which
would have been similar to that of the previous article.
If the frame be inverted (Fig. 73) the method of calculat-
ing the strains will be the same. Under this supposition the
W
FIG. 78.
strains in the inclined pieces will be tensile instead of com-
pressive, and in the horizontal piece B C will be compressive
instead of tensile, the expression for the intensities remaining
the same.
258. The jib-crane. The machine known as the jib-
crane, which is used for raising and lowering weights, is an
example of a triangular frame. Its principal parts are
a vertical post, B C ; a strut, A C ; and an arm or tie-bar, A P.
(Fig. 74.)
Ordinarily, the whole frame allows a motion of rotation
around the vertical axis, B C.
190
CIVIL ENGINEERING.
The weight, W, suspended from the frame at A is kept
from falling by resistances acting in the directions A B and
A C. There being an equilibrium of forces at A, the resultant,
W, and the direction of the resistances being known, the in-
tensities of these resistances are easily determined.
w
FIG. 74.
JRepresent W by kd, and construct the parallelogram kbdc.
kl and kc will represent the intensities of the forces acting to
keep W from falling.
From the parallelogram we have
ko W
sin /3
sin (a +
. . (128)
which, as it is seen, produces compression on the strut A C,
and a transverse shearing strain at C on the post C B. The
horizontal component of A C divided by the area of cross-
section of the post B C, gives the shearing stress on the unit
of cross-section at C.
We also have A5 = W
sm a
sin (a + /3)'
for the stress acting in the direction of A B, tending to elon-
gate it, and to produce a cross-strain on B C. The greatest
bending moment is at C. Knowing the stresses, it is a simple
problem to proportion the pieces so that the crane may be
able to lift a given weight, or to determine the greatest
weight which a given crane may lift with safety.
TRIANGULAR BRACING.
191
COMBINED TRIANGULAR FRAMES.
259. Open-built beams constructed by connecting the uppe"
and lower pieces by diagonal braces are examples of com
binations of triangular frames.
Triangular Bracing.
260. Triangular bracing with load at free end. Take
a beam of this kind and suppose it placed in a horizontal
position, one end firmly fixed, the other free to move, and
strained by a force acting at the free end. Suppose the tri-
angles formed by the braces to be equilateral (Fig. 75) and
disregard the weight of the beam.
*
FIG, 75.
Represent by "W the force acting at A, in the plane of the
axes of the pieces of the frame and perpendicular to A G.
The force W acting at A is supported by the pieces A B
and A A', and produces a stress of compression in A A' and
tension in A B. Laying off on A W the distance kd to repre-
sent W, and constructing the parallelogram kbcd, we have Ac
and ko representing the intensities of these stresses.
From the parallelogram there results
W
kc~ - , and kb = W tan a.
cos a'
The compressive force Ac is transmitted to A' and there
supported by the pieces A'B and A'B'. Resolving this force
at A' into its components acting in the directions of A'B and
A'B', we have k'd' = 2Wtan a, which produces compression
W
in A'B'. and k'b' = - . which produces tension in A'B.
cos a'
This tension A' I' is transmitted by the brace to B. Re-
192 CIVIL ENGINEERING.
solving it into its components in the directions B B' and B C,
we have
Compression on B B' = ^-^,
Tension on B C = 2W tan a.
The tension at A is transmitted through the beam to B,
hence the tension at B is equal to the sum of them, or
Tension at B = 2W tan a + W tan a = 3W tan a.
Continuing this process, we find that the force W, strains
all the diagonals equally, but by forces which are alternately
compressive and tensile, and the expression for which is
. In this case the braces numbered odd in the figure are
cos a
compressed, and those even are extended.
The stresses in the upper and lower beams are cumulative,
receiving equal increments, each equal to 2W tan a, at each
point of junction of the brace with the beam. Hence, in this
case, for the upper beam we have
W tan a for A B, 3 W tan a for B C, 5W tan a for C D, etc.,
and for the lower,
2W tan a for A'B', 4W tan a for B'C', 6 W tan a for C'D', etc.
Having determined the stresses in the different parts of
the frame produced by a force "W", it is easy to find the
greatest weight that such a frame will support, or to propor-
tion its different parts to resist the strains produced by a given
load.
The triangles taken were equilateral. If we denote by d
the altitude E'o? of one of these triangles, or depth of the
beam ; by Z, the length of one of the sides F E, or distance
between the vertices of two adjacent triangles, which we will
call a bay ; and express the values of cos a and tan a in terms
of these ; then we have cos a = , , and tan a = ^. Substituting
which in the foregoing expressions, there obtains -rW for the
stress in the diagonal, and, -jW for the increment to be
Cu
added at each point of junction.
To find the stress in any segment ; as, for example, L F.
The tension on A B is W tan a = /rjW, to which add four
TEIAKGULAE BRACING. 193
equal increments, there being four bays between A and the
segment E F, and we have, for the tensile stress in E F,
9
9W tan a, or its equal ^W.
261. Triangular Bracing Strained by a Uniform Load.
Suppose the strains on the same beam to be caused by a
weight uniformly distributed over either the upper or lower
beam of the frame.
Let A E F A 7 (Fig 76) be an open-built beam supporting a
load uniformly distributed over the upper beam A E.
Denote by w the weight distributed over any one segment.
We may, without material error, suppose the whole load
divided into a number of equal parts, each equal to that rest-
ing on the adjacent half segments, acting at the points A, B, C,
etc., where the braces are connected with the beam, A E.
Since there are four of these bays, the total load is 4w, the
action of which may be considered to be the same as that
produced by the weight w acting at each of the points B, C,
and D, and \w at A and E.
The strains on A B, A A', A'B, and A'B' are due to the weight
5 acting at A, and are determined as in the preceding case.
The strains on B C, B B', B' C, and B'C' are due to the ac-
tion of the weight w acting at B, increased by the strains due
w
to -Q- acting at A.
The strains on the remaining parts are due to the weight
acting at each vertex, increased by those transmitted from the
points to the right of them.
Hence it is seen that the stresses in each of the pieces in
any pair of diagonals are equal in amount, but different in
kind, and increase as they go from the point of application to
the points of support for each set; and* that the stresses in
the segments of the upper ond lower beams increase in the
same (Erection. The rate of increase can be easily determined.
13
194 CIVIL ENGINEERING.
METHOD OP SECTIONS.
262. The stresses in the different pieces of a frame may be
obtained by using the principle of moments, or, as it is usu-
ally called, the " method of sections." This method consists
in supposing the frame to be divided by a section cutting not
more than three pieces of the frame, and taking the inter-
section of two of these pieces as a centre of moments.
It is evident that the stresses in the two pieces passing
through the centre of moments will have no moments to op-
pose those of the extraneous forces acting to turn the frame
around the assumed centre, and that these external moments
must be held in equilibrium by the moment of the stress in
the third piece. If the moment of the stress in the third
piece, with respect to the assumed centre, be placed equal to
the bending moment of the extraneous forces with respect
to the same point, an equation will be found that must be
true for equilibrium, and which, when solved, will give the
intensity of the stress in the third piece whenever the posi-
tion of this piece and the bending moments are known.
Let it be required to find by this method the stress in the
segment E F (Fig. 75).
Intersect the frame by a vertical plane perpendicular to the
axis between x and E, and let T' be the stress in the piece
E F. This plane will cut the pieces E F, E E', and E' D', and
Do others. Assume E' to be the centre of moments.
The resultant of the stress T' is supposed to act along the
axis of the piece F E. Its moment with respect to E' will be
T' X E'x.
Since there is an equilibrium,
T' x E'x = W x Aa>, or, T' x d = W x 4|Z; hence
T' = 4J-T W, the same value before deduced.
In a similar manner, assuming E as a centre, the intensity
of the stress in E' D' may be obtained.
This method, in many cases, is a convenient one and its
use is simply a matter of choice.
Vertical and Diagonal Bracing.
263. Suppose the triangles, instead of being equilateral,
to be right-angled, as in Fig. 77, and the beam^strained by a
load, W, as in the preceding case.
The stresses in the upper and lower beams would be re-
VERTICAL AND DIAGONAL BRACING.
195
spectively tensile and compressive, and cumulative as in the
preceding case.
The expression for the equal increment would be
Wtana.
The force acting on the diagonals would be compressive
and equal to
W
- , same as in preceding case.
cos a'
The stress in the verticals would be tensile and equal to
W for each.
Representing by
A, the length of a diagonal, A A',
I, the length of a segment, A B,
d y the length of a vertical, A'B, we can write
= W-, and
cos
d
(129)
^ '
expressions more frequently used when calculating the stresses
than the expressions involving the circular functions.
If, in the preceding cases, W had acted in the opposite di-
rection, that is, pushed the point A upward instead of pulling
it down, or the same thing, the frame had been turned over
so that the upper beam became the lower, the stresses would'
have been determined in the same manner with similai
results, excepting that the inclined pieces would have been
extended instead of compressed, and the verticals compressed
instead of extended.
196
CIVIL ENGINEERING.
ANGLE OF ECONOMY.
264. It has been shown that the stress on the unit of cross-
section of a brace, strained by a force as W (Fig. 77) varies
with the angle made by the brace with the straining force.
It is plain that of two braces of the same material, for the
same stress on the unit and the same span, the more eco-
nomical brace will be the one that contains the less amount of
material ; or, for the same stress and the same amount of
material, the one that gives the wider span.
Suppose the stress on the unit of cross-section and the
span to be fixed, it is required to find the angle that a brace
shall make with the straining force so that the amount of ma-
terial in the brace shall be a minimum.
Let B C be the fixed span (Fig. 78) and 2W the intensity
of the straining force acting vertically to be transmitted by
braces to the points B and C considered as fixed. Let h =
the length of A B = A C, 21 = the length of B C, and d = the
distance A D.
FIG. 78.
The straining force produces a compressive stress in each
brace equal to Wp
Suppose the resistance offered by the brace to vary directly
with the area of its cross-section (Art. 164) and let J 2 be the area
of cross-section, and C', the assumed compressive stress al-
lowed on the unit. We can then form the following equa-
tion :
.
d
= x
. . (130)
from which we obtain
ANGLE OF ECONOMY. 197
and
W A 8
tfh = x -T, for the volume of the brace.
Substituting d* -f P in this expression for 7i 2 , we have
Volume of brace = -777- x -7 (131)
L #
The value of d = I makes this function a minimum. Hence,
it is seen that the volume of the brace is a minimum when
the angle which it makes with the straining force is equal
to 45. This angle is called " the angle of economy " of
the brace.
In this discussion, the length of the bay or span has been
fixed. A similar result would have been obtained if d, the
depth of the truss, had been fixed and the length of the bay
B C determined.
The resistance in a tie to tension varies directly with the
area of cross-section, however long the piece may be, and
therefore the angle above obtained is the true angle of econ-
omy for ties in all cases. This is not true for struts, as
experiment has shown (Art. 202) that when the diameter is
small in comparison to its length, the resistance to compres-
sion becomes also a function of its length, which latter di-
mension must be duly considered.
The angle of economy for a strut when its length exceeds
its diameter more than fifteen or thirty times can be deter-
mined by taking the formulas deduced from Hodgkinson's
experiments for finding the strength of pillars, and following
the steps just described.
Merrill, in his " Iron Truss Bridges," gives the angle of
economy for a cast-iron strut in a triangular frame at 27 51',
or the depth of the frame to be a little greater than one-fourth
of the span. In diagonal bracing with vertical ties (Art. 236)
he gives the angle of economy for the struts to be 39 49'
with the vertical.
PART IV
MASONRY.
CHAPTER IX.
Masonry is the art of erecting, structures in stone,
brick, and mortar.
It is classified, from the nature of the material used, into
stone, brick, and mixed masonry ; from the manner in which
the material is prepared, into cutstone, ashlar, rubble, and
hammered masonry ; and from the mode of laying the
blocks, into irregular and regular masonry.
MASONRY STRUCTURES.
266. Masonry structures are divided into classes accord-
ing to the kind of strains they are to sustain. Their forms
and dimensions are determined by the amount and kind of
strains they are required to resist. They may be classed as
follows :
1st. Those which sustain only their own weight ; as walls
of enclosures.
2d. Those which, besides their own weight, are required to
support a vertical pressure arising from a weight placed upon
them ; as the walls of a building, piers of arches, etc.
3d. Those which, besides their own weight, are required to
resist a lateral thrust ; as a wall supporting an embankment,
reservoir walls, etc.
4th. Those which, sustaining a vertical pressure, are sub-
jected to a transverse strain ; as lintels, areas, etc.
5th, Those which are required to transmit the pressure they
directly receive to lateral points of support; as arches.
EETAINING WALLS. 190
WALLS.
267. Definitions. In a wall of masonry the front is called
the face ; the inside or side opposite, the back ; the layer of
stones which forms the front is called the facing, and that of
the back, the "backing ; the portion between these, forming
the interior of the wall, the filling.
If a uniform slope is given to the face or back, this slope is
termed the batter.
The section made by a vertical plane passed perpendicular
to the face of the wall is called the profile.
Each horizontal layer of stone in the wall is called a course ;
the upper surface of the stone in each course, the bed or
build; and the surfaces of contact of two adjacent stones,
the joints.
When the stones of each layer are of equal thickness
throughout, the term regular coursing is applied ; if un-
equal, irregular or random coursing. The particular ar-
rangement of the different stones of each course, or of con-
tiguous courses, is called the bond.
Walls. The simplest forms of walls are those generally
used to form an inclosing fence around a given area, or to
form the upright inclosing parts of a building or room.
RETAINING WALLS.
268. A retaining -wall is the term used to designate a
wall built to support a mass of earth in a vertical position, or
one nearly so. The term sustaining is sometimes applied to
the same case. In military engineering, the term revetment
wall is frequently used to designate the same structure.
The earth sustained by a retaining wall is usually deposited
behind and against the back after the wall is built. If the
wall is built against the earth in its undisturbed position, as
the side of an excavation or cutting, it is called a face-wall,
and sometimes breast-wall.
Reservoir walls and dams are special cases of retaining
walls, where the material to be supported is water instead of
earth.
Counterforts are projections from the back of a retaining
wall, and are added to increase its strength. The projections
from the face or the side opposite to the thrust are called
buttresses.
20U
CIVIL ENGINEERING.
AREAS, LINTELS, AND PLATE-BANDS.
269. The term area is applied to a mass of masonry, usually
of uniform thickness, laid over the ground enclosed by the
foundations of walls.
The term lintel is applied to a single stone, spanning an
interval in a wall ; as over the opening for a window, door, etc.
The term plate-band is applied to the lintel when it is
composed of several pieces. The pieces have the form of
truncated wedges, and the whole combination possesses the
outward appearance of an arch whose under surface is plane
instead of being curved.
ARCHES.
270. An arch is a combination of wedge-shaped blocks,
called voussoirs or arch-stones, supporting each other by
their mutual pressures, the combination being supported at
the two ends. (Fig. 79.)
These blocks are truncated towards the angle of the wedges
by a curved surface, generally normal to the joints between
the blocks.
The supports against which the extreme voussoirs rest are
generally built of masonry.
c
*^
/ \
L
V
A H "~B
FIG. 79.
If this mass of masonry, or other material, supports two
successive arches it is called a pier; if the pier be strong
enough to withstand the thrust arising from either of the
arches alone, it is called an abutment pier; the extreme
ARCHES. 201
piers which support an embankment, generally of earth, on
one side, and an arch on the other, are called abutments.
The inner surface of the arch is called the soffit ; its outer
surface, the back. The sides of the arch are called reins ;
the end surface, the face, and sometimes the head of the
arch. The connection of the arch with the pier is called the
impost ; if the top surface of a pier is sloped to receive the
end of the arch, this surface is called a skewback.
The highest stones of a pier, or the stones on which an
arch rests, are called cushion stones ; the highest stone of
the arch is called the keystone.
The line in which the soffit of the arch intersects the pier
is called the springing line. The line of intersection of
the face of the arch with the soffit is the intrados ; with
the back of the arch, the extrados. The chord, A B (Fig.
79) is termed the span, and the height, H C, of the key-
stone above this line, is termed the rise. The length of
the arch is that of the springing line. The highest line of
the soffit, that projected at C, is called the crown. The
line in the plane of the springing lines projected at H, sym-
metrically disposed with respect to the plan of the soffit on
that plane, is the axis of the arch. The courses of stones
parallel to the head of the arch are called ring-courses.
The courses which run lengthwise of the arch are termed
string-courses. The joints between the different ring-
courses are called heading joints. Those between the
different string-courses are termed coursing orbed-joints.
A wall standing on an arch and parallel to the head is
called a spandrel- wall.
271. Classification. Arches may be classified according
to the direction of the axis with respect to a vertical or hori-
zontal plane, or according to the form of the soffit.
A right arch is one whose axis is perpendicular to the
heads. The arch is called oblique or askew, when the axis
is oblique to the heads; and rampant, when the axis is
oblique to the horizontal plane.
Arches are termed cylindrical, conical, warped, etc., ac-
cording as the soffit is cylindrical, conical, etc.
272. The cylindrical arch. The cylindrical is the most
usual and the simplest form of the arch. A section taken at
right angles to the axis is called a right section.
These arches are classified according to the shape of the
curve cut out of the soffit by the plane of right section.
If the curve be a semicircle, the arch is called a full
centre arch ; if a portion of a semicircle, a segmental arch.
202
CIVIL ENGINEERING.
When the section gives a semi -ellipse, the arch is called an
elliptical arch ; if the curve resembles a semi-ellipse, but is
composed of arcs of circles tangent to each other, the term
oval of three, five, etc., centres, according to the number of
arcs used, is applied to designate it.
273. Groined and Cloistered Arches. The intersection
of cylindrical arches having their axes in the same plane, and
having the same rise, form the arches known as groined and
cloistered.
The groined arch (Fig. 80) is made by removing from
each cylindrical arch those portions of itself which lie with-
in the corresponding parts of the other arch ; in this way,
the two soffits are so connected that the two arches open
freely into each other.
r .
n
V
M| N,
V 1
B\
\m
/B
FIG." 80 Represents the plan of the soffit and the right sections M and
N of the cylinders forming a groined arch.
aa, pillars supporting the arch,
fo, groins of the soffit.
om, mn, edges of coursing joint.
A, key- stone of the two arches formed of one block.
B, B, groin stones, each of one piece, situated below the key-stone, and
forming a part of each arch.
The curves of intersection of the soffits form the edges of
salient angles and are termed groins, hence the name of the
arch.
The cloistered arch (Fig. 81) is made by retaining in each
cylindrical arch only those portions of itself which lie within
the corresponding portions of the other arch ; thus, a portion
ARCHES.
203
of the soffit of each arch is enclosed within the other, these
portions forming a fonr-sided vaulted ceiling.
FIG. 81 Represents a horizontal section
through the walls supporting the arch and
plan of the soffit of a cloistered arch.
B, B, the walls of the enclosure or abut-
ments of the arches.
ab, curves of intersection of the soffits.
, e, groin stones.
This arch was much used in forming the ceilings of the
cells of monasteries ; from their object and use is derived the
term cloistered.
274. Annular arches. An annular arch is one that may
be generated by revolving the right section of an arch about
a line lying in the plane of the section, but not intersect-
FiQ. 82. N, right section of an annular arch.
C, plan of soffit.
ing it This line is usually vertical and also perpendicular
to the span of the arch. (Fig. 2.) The axis is curved
204
CIVIL ENGINEERING.
being described by the centre of the curve of right section.
The coursing joints are conical, and the heading joints are
plane surfaces.
275. Domes. An arch whose soffit is the surface of a
hemisphere, the half of a spheroid, or other similar surface,
is called a dome. The soffit may be generated by revolving
the curve of right section about the rise for 360, or about
the span for 180. In the first case the horizontal section at
the springing lines is a circle, in the other it is the generating
curve.
The plan may be any regular figure. Fig. 83 represents a
plan and vertical section of a circular dome.
FIG. 83. A, vertical section and elevation of a circular dome.
B, B, horizontal section and plan of its soffit.
276. Conical arches. Their name explains their con-
struction. They are but rarely used, in consequence of the
varying sizes of the voussoirs.
277. Arches with warped soffits. Arches, whose
soffits are warped surfaces, are frequently used. The partic-
ular kind of warped surface will depend* upon circumstances.
A common example of this class is an arch which has the
same rise at the heads but unequal spans. The soffit in this
case may be generated by moving a straight line so as to con-
tinually touch the curves of section of the soffit at the heads,
and at the same time to remain parallel to the plane of the
springing lines. A surface generated in this manner belongs
to the class of warped surfaces having a plane director. In
particular cases it is a conoid, hence the name of conoidal
arches is frequently applied to this kind.
DISTRIBUTION OF PBE88UBBL
205
Arches whose soffits may be thus generated possess the
advantage of having straight lines for the edges or the joints
running lengthwise in the soffit.
278. Oblique or askew arches An arch whose axis
makes an angle with the head is called oblique or askew.
In arches of this kind the chord of the arc of the head is the
span. The angle of obliquity is the angle which the axis
makes with a normal to the head.
MECHANICS OF MASONRY.
DISTRIBUTION OF PRESSURE.
279. The surface on which a structure rests is required to
support the weight of the structure, and also the load it
carries, or the thrust it may have to resist. It is necessary
for stability that the resultant line of these pressures should
pierce this surface within the limits of the base of the struct-
ure, and that all the forces acting within this area be com-
pressive. The point in which this resultant pierces the
surface is known as the centre of pressure.
Structures generally rest upon plane surfaces .and the
portion pressed is usually a simple plane figure. Since the
pressure on this surface may vary from point to point, it is
necessary to determine what the pressure is at any point of
the surface, and to find the limits within which the centre
of pressure must be to have all the forces acting upon the
surface compressive.
280. Normal pressure. Suppose a series of blocks, of
the form of rectangular parallelopipedons with equal bases,
but (Fig. 84) whose altitudes in-
crease in arithmetical progres-
sion, be placed side by side on a
given plane area, A B C D. It is
evident that the pressure on the
area A B C D, is less on that part
under block 1 than it is on the
part under block 5, and that the
pressure on any part, as B C 5,
will be directly proportional to
the altitude of the block resting
upon it
If these blocks be very thin,
that is, the width of the bases
measured in the direction of
FIG. 84.
A B be infinitely small,
and have altitudes that reach to the line E F drawn through
206
CIVIL ENGINEERING.
the middle points of the upper sides of blocks 1, 2, 3, 4 and
5, the total pressure on the area A B C D will be the same as
that produced by the five blocks. The pressure on the
units of this area will not, however, be the same, being dif-
ferent for the two cases for most of them.
The pressure on each line of the surface parallel to B C,
caused by the thin blocks, is directly proportional to the
corresponding ordinate of the trapezoid B F E A, and the
centre of pressure of each block will be found on the sur-
face A B C D directly under the centre of gravity of the
block. The centre of pressure of the entire mass will be
found on the surface directly under the centre of gravity of the
trapezoid forming the middle section of the thin blocks.
281. Uniform pressure. If the blocks were all of the
same size and of the same material, the pressure on a unit of
PIG. 85.
FIG. 86.
area would be the same for every point pressed by it, and the
centre of pressure would be directly under the centre of the
base. Assuming the form of the base of a structure to be
rectangular, the system of forces acting to produce a press-
ure that is uniformly distributed over the surface pressed
may be represented by a rectangular parallelopipedon of
homogeneous density, of which the rectangle is the base.
Suppose a rectangular surface, as A B C D (Fig. 85), to be
pressed by such a system of forces, and P to be the resultant.
NORMAL PRESSURE.
207
The centre of pressure would be at the centre, 0, of the
p
rectangle, and the pressure on each unit of area would be T .
A
282. Uniformly varying pressure. Suppose the
pressure to be zero along the line A D (Fig. 84), and to in-
crease uniformly toward B C, along which the pressure is
equal to B F. The system of forces producing this pressure
may be represented by a wedge-shaped mass of homogeneous
density, as shown in Fig. 86. The centre of pressure of any
section parallel to A B, is below its centre of gravity and to
the right of the middle point of its base at a distance equal to
one-sixth of A B. The centre of pressure of the whole mass
will therefore be on the line X X', and at a distance from
equal to one-sixth of A B.
The pressures on the different lines parallel to A D vary as
the ordinates of the triangle, N L M. The pressure on the
p
unit at 0, the centre of the rectangle, is equal to -r , the mean
-A.
pressure on the surface of the rectangle, P being -the result-
ant force.
To find the pressure F' on the unit, at the distance x
from measured on XX', we have, representing the sides of
the rectangle by 2
and for that produced by the force at V 2 to be
2A
and hence the total pressure on the unit of area due to P
acting at V, at the point whose co-ordinates are x and y, will
be
P/ Sx'x
The pressure at the different points of the base may be
determined in a similar way when the base is a circle, ellipse,
lozenge, etc.
285. General solution. It is evident that there is a ten-
dency to produce rotation about some right line in the base
whenever the resultant pressure pierces the plane of the
base in any point excepting its centre of figure. Kegarding
the base as a cross-section, this right line will be its neutral
axis.
14
210 CIVIL ENGINEERING.
And since the condition is imposed that all the forces
acting within the base shall be compressive, it is evident that
tliis neutral axis must remain outside of, or at least tangent to,
the base. If the neutral axis should intersect the base, it is
plain that the portion of the base on the same side with the
centre of pressure would be compressed, while the portion of
the base on the other side would be subjected to a strain of
extension, a condition which is not allowable.
The centre of pressure of any section is the centre of per-
cussion of the plane area representing it. Hence, the general
solution obtained from mechanics for obtaining the centres of
percussion and axes of rotation for any plane figure may be
applied to these cases.
The normal pressure upon the base is generally produced
by a uniformly distributed load, by a uniformly varying one,
or by a combination of the two, placed upon the structure.
These are the cases which have been considered.
286. Symmetrical base. In general the blocks used in
building have a plane of symmetry, and these loads above
named are symmetrically distributed with respect to this
plane and to the base of the block. It follows, therefore,
that the resultant pressure pierces the base in its axis or
middle line.
For such cases the expression for the pressure on any point
will be of the general form,
in which K is a positive coefficient depending upon the figure
of the base. We have found it equal to 3 for the rectangle ;
we would find it equal to 4 for the ellipse or circle, and 6 for
the lozenge. 20 being the longest diameter. Hence we con-
clude that the pressure is more equally distributed over a rect-
angular base than over a circular, elliptical, or lozenge-shaped
one.
In the general expression for P x it is seen that in the
rectangle if x' is greater numerically than J#, that the
corresponding values of x = = # give negative values for P,..
That is, there will be no pressure on the opposite edge ; on
the contrary, there will be tension, and the joint will open or
tend to open, along this line. If x' = %a the values of P,,
for x = a are ; that is. there is no pressure on the edge.
Hence, if the pressure is to be distributed over the entire
base, the resultant must pierce it within the limits of %a.
287. Oblique pressure. In a large number of cases,
STRAINS ON MASONRY. 211
especially in structures of the third and fifth classes, tho
resultant pressure has its direction oblique to the plane of
the base.
This resultant may be resolved at the centre of pressure
into two components, one normal to the plane of the base
and the other parallel to it. The former is the amount of
force producing pressure on the base, and is to be considered
as in the preceding cases. The latter does not produce pres-
sure, but acts to slide the base along in a direction parallel to
its plane. The effect of sliding will be alluded to in future
articles.
MASONRY STRUCTURES OP THE FIRST AND SECOND
CLASSES.
288. The strains which these structures sustain are pro-
duced by vertical forces.
For stability, the resultant pressure should pierce the plane
of the base at a distance from its middle line not greater than
one-sixth the thickness of the wall at its base.
The wall having to support a load, either its own weight
alone, or its weight with a load placed upon it, the largest
stones should be placed in the lower courses, and all the
courses so arranged that they shall be perpendicular, or as
nearly so as practicable, to the vertical forces acting on the
wall. Great care should be taken to avoid the use of con-
tinuous vertical joints.
The thickness of the wall will depend upon the load it has
to support and the manner of its construction.
STRUCTURES OF THE THIRD CLASS.
289. Retaining walls, besides supporting their own
weight, are required to resist a lateral thrust which tends to
turn them over.
Observation has shown that if we were to remove a wall
or other obstacle supporting a mass of earth against any one
of its faces, a portion of the embankment would tumble aown,
separating from the rest along a surface as B R (Fig. 89),
which may be considered a plane ; and that later more and
more of the earth would fall, until finally a permanent slope
as B S is reached.
The line B R, is called the line of rupture, the line B S
CIVIL ENGINEERING.
the natural slope, and the angle made by the natural slope
with the horizontal is termed the angle of repose. The
angle C B R is called the angle of rupture. If dry sand be
poured out of a vessel with a spout upon a flat surface, the
sand will form a conical heap, the sides of which will make
FIG. 89.
a particular angle with the horizontal, and it will be found
that the steepness of this slope cannot be increased, however
judiciously the sand may be poured, or however carefully it
is heaped up. This slope or angle of repose varies for differ-
ent earths, being as much as 55 for heavy, clayey earth, and
as little as 20 for fine dry sand.
This prism of earth C B R, which would tumble down if
not sustained, presses against the wall, producing a horizontal
thrust, and the wall should be made strong enough to resist
it.
290. Two distinct problems are presented : the first being
to ascertain the intensity of the thrust exerted against the wall
by the earth ; and the second, to determine the dimensions
of a wall of given form so as to successfully resist this thrust.
The intensity of the thrust depends upon the height of the
prism, and upon the angle of rupture.
The angle of rupture, or the tendency in the earth to slip,
is not only different for the various kinds of earth, but is
different in the same earth, according as it is dry or saturated
with water, being greater in the latter case.
The manner in which the earth \& fitted i/n, behind the wall,
affects the intensity of the thrust, the latter being less when
the earth is well rammed in layers inclining from the wall
than when the layers slope towards it.
Therefore, in calculating the amount of resistance the wall
should have, the effect produced by the maximum prism of
pressure under the most unfavorable circumstances should be
RETAINING WALLS. 213
considered. The greatest pressure that earth can pi-educe
against the back or the wall is when the friction between its
grains are destroyed, or when the earth assumes the form of
mud. The pressure under these circumstances would be the
same as that produced by a fluid whose specific gravity was
the same as earth.
291. Retaining walls may yield by sliding along the base
or one of the horizontal joints ; by bulging ; or by rotation
around the exterior edge of one of the horizontal joints.
If the wall be well built and strong enough to prevent its
being overturned, it will be strong enough to resist yielding
by the other modes.
Hence, the formulas used in determining the thickness of a
retaining wall are deduced under the supposition that the only
danger to be feai^d is that of being overturned.
Having determined the horizontal thrust of the prism of
pressure, its moment in reference to any assumed axis can be
obtained.
A wall to be stable must have the moment of its weight
about the axis of rotation greater than the moment of the
overturning force about the same line.
The term stability in this subject differs slightly in its
meaning from that previously given it. A mass is here said
to be stable when it resists without sensible change of form
the action of the external forces to which it is exposed the
variations produced by these forces being in the reactions of
the points of support and the molecular forces of the body,
and not changing in any way the form of the mass.
The excess of moment in the wall, or factor of safety, as
we have heretofore designated it, will vary in almost every
special case, being much greater for a wall exposed to shocks
than when it has to sustain a quiescent mass ; greater for a
wall poorly built, or of indifferent materials, than one of bet-
ter material and well constructed. The formulas which are
used give results which make this factor of safety at least
equal to 2, or twice as strong as strict equilibrium requires.
RETAINING WALLS, with back parallel to the face.
292. Let it be required, to find the thickness of a retaining
wall, the upper surface of the embankment being horizontal
and on a level with the top of the wall. The wall being of
uniform thickness, with vertical face and back.
CIVIL ENGINEERING.
w
w',
a,
Denote by (Fig. 90),
H, the height B C of the wall,
5. " thickness A B of the wall,
weight of a unit of volume of the earth,
" " same unit of volume of masonry,
angle C B S of the natural slope with the vertl
cal B C,
fi, " angle S B F of the natural slope with the hori
zontal.
Let it be assumed that the density and cohesion of the
earth are uniform throughout the mass. The pressure ex-
erted against the wall may then be represented by a single
FIG. 90.
resultant force acting through the centre of pressure on the
surface of the wall.
If we suppose the prism C B S to act as a solid piece, the
friction along B S would be just sufficient to prevent sliding,
and there would be no horizontal thrust. This is true for
any prism making an angle less than /3.
The horizontal thrust upon the back of the wall must there-
fore be due to a mass of earth, the lower surface of which
makes a greater angle with the horizontal than ft.
Let B R be a plane which makes an angle greater than /3,
and represent by < the angle which it makes with the natural
slope.
We may suppose two cases : one in which there is no f ric
tion existing between the prism and the plane which supports
it ; and the other, in which there is friction.
In the first case, the horizontal thrust would be equal to
that of a fluid whose specific gravity is the same aa that of
the earth, or
Hor. thrust =
the centre of pressure being f H below C.
RETAINING WALLS. 215
In the second case, the friction between the plane and
prism is considered, and if we denote by P the horizontal
component of the pressure acting to overthrow the wall, and
neglect the adhesion and friction of the earth on the back of
the wall, we have, supposing = \a,
P = -^H 2 tan 8 < . . . (137)
The moment of this force about the edge A will be
/
M = froW tan 2 < x^H.
The moment of the weight of the wall about the jame line
is M = i^I
a
Equating these moments, we have
whence,
x
' o
, .... (138)
for the value of the thickness of base to give the wall to resist
the pressure due to P.
It can be shown that the maximum prism of pressure will
be obtained when the angle of rupture, C B R, is equal to
\ (90 #), or equal to \a. This has also been proved by ex-
periment. Substituting for < this value in the expression
for J, and we get,
*aH.tanjy.|.jp
The value for P may be put under the form,
o x !" S j"^ (HO)
which is the form in which it frequently appears in other
works when treating this subject.
Suppose B R to coincide with B S, then ^ = 0, and hence
P = 0,
a conclusion already reached.
216
CIVIL ENGINEERING.
293. General case. The wall was assumed vertical in
the preceding case. The general case would be where the
back of the wall and the up-
J ..- per surface of the embank-
ment were both inclined to
the horizontal. Let B C (Fig.
91) be the back of the wall ;
C S, the upper surface of the
embankment ; B S, the line
of natural slope ; and < and
ft represent the same angles
FIG. 91. as in preceding example. The
pressure on the back of the
wall is produced by some prism as C B R. The horizontal
thrust produced by this prism is equal to its weight multiplied
by the tan <, or
P = w x area C B R X tan .
Let it be required to find the maximum prism of pressure.
This will be a maximum when the product of the area C B R
and the tan is a maximum.
Draw through C and R perpendiculars to the line of natural
slope B S. Represent the distance R L by a?, the distance C K
by #, and the distance B S by b.
The area C B R is equal to
Substituting in the expression for P, we get
P = w x i b (a x) tan >.
Eepresent the angle B S C by /3', and we can write
P = w x %b(a x) 7 - 5j.
1 b x cot ft
This expression is in terms of a single variable x. Taking
ax - - a/ 2
the factor ^ _ x c ~ t "7p? an( i differentiating, and placing the
differential coefficient equal to zero, we get
(b-x cot ft') (a-2x)-(ax-a?)(- cot ft') = 0,
whence
a?cotft'-2bx= -ab.. . .
This may be put under the form
ab bx = bx x 2 cot ft' = x(b x cot /3'),
or
ab bx x x B L.
RETAINING WALLS. 217
Whence,
area CBS area RBS = i(a!xBL) = area R B L,
and
area R B L = area C B R,
or the thrust is a maximum when the area C B R is equal to
the area B R L.
If C S is horizontal and B C is vertical, the triangle
R B L is equal to R B C only when the line B R bisects the
angle CBS. This result is the same as that of the previous
case.
Substituting in the expression for P, the area R B L for the
area C B R, we get
P = w x area R B L x tan 0.
Substituting for this area and for the tan <, their values in
terms of a?, we get
? = %wy?, (142)
for the maximum thrust.
From equation (141) we find the value of a; to be
x = b tan ft 1 - Vb tan fi' (b tan ft' -a).
We may write this value x under another form by draw-
ing the line B E from B perpendicular to B S and repre-
senting it by c. We have c = b tan /?', and substituting, we
get
x = c V c (c a).
Substituting this value of x in equation (142), we get
for the horizontal thrust, produced by the maximum prism of
pressure.
Knowing the horizontal thrust, its moment around the
edge, A, can be obtained. The moment of the wall around
the same line is easily found.
Equating these moments, the value of b can be deduced,
giving the requisite thickness for an equilibrium.
294. These examples show the general method used to de-
termine the thickness of retaining walls.
The specific gravity of the materials forming an embank-
ment ranges between 1.4 and 1.9, and that of masonry be-
rt/
tween 1.7 and 2.5. The ratio of the weights , is therefore
ordinarily between and 1. For common earth and ordinary
CIVIL ENGINEERING.
W
masonry it is usual for discussion to assume 7 = |, and a =
45. In practice it is recommended to measure the natural
slope of the earth to be used, and to weigh carefully a given
portion of the masonry and of earth, the latter being
thoroughly moistened.
In military works, the upper surface of the embankment
is generally above the top of the wall. The portion of the
embankment above the level of the top is called the surcharge,
and in fortifications rests partly on the top of the wall. "When
its height does not exceed that of the wall, the approximate
thickness of the wall may be obtained by substituting, the
sum of the heights of the wall and the surcharge, for H in
the expression for the thickness already obtained.
The manner in which earth acts against a wall to overturn
it cannot be exactly determined, hence, the thrust not being
exactly known, the results obtained are only approximations.
Nevertheless, a calculation right within certain limits is better
than a guess, and its use will prevent serious mistakes being
made.
FIG. 92.
In our discussion the cohesion of the particles of earth to
each other and their friction on the back of the wall have
been disregarded. The results therefore give a greater thick-
ness than is necessary for strict equilibrium, and hence errs
on the side of stability.
295. Among the many solutions of this problem, those given
RETAINING WALLS.
219
oy M. Poncelet, and published in No. 13 " Du Memorial de
POfficier du Genie," are the most complete and satisfactory.
In this memoir he gives a table from which the proper
thickness of a retaining wall supporting a surcharge of earth
may be obtained.
The principal parts of this table giving the thickness in
terms of the height, for surcharges whose heights vary be-
tween and twice the height of the wall, are as follows :
Kepresent by (Fig. 92).
H, the height B C of the wall ;
A, the mean height of C F of surcharge ;
a, the angle CBS made by the vertical with line of natu-
ral slope B S.
/?, the angle of natural slope with the horizontal ;
J 9 the coefficient of friction = cotan a ;
if, the distance from foot of surcharge E to D outer edge
of wall ;
Wj weight of unit of volume of earth ;
w\ weight of unit of volume of masonry.
TABLE.
1
RATIO OP HEIGHT TO THICKNESS, OB 5
B
When w=-w' and
V> = to'
I*
/=0.6
= 81
/=1.4
= 51* 25'
/= 0.6 = 31
/=1 ' 4 2y
W=0
=*
=o
-*
=
-*H
=
-*"
=0
U=i H
0.452
0.452
0.258
0.258
0.270
0.270
0.350
0.350
0.198
0.198
0.1
0.498
0.507
0.282
0.290
0.303
0.306
0.393
0.393
0.222
0.229
0.2
0.548
0.563
0.309
0.326
0.336
0.342
0.439
0.445
0.249
0.262
0.4
0.665
0.670
0.369
0.394
0.399
0.405
0.532
0.522
0.303
0.299
0.6
0.778
0.754
0.436
0.450
0.477
0.457
0.617
0.572
0.360
0.328
03
0.867
0.820
0.510
0.501
0.544
0.504
0.668
0.610
0.413
0.357
1
0.930
0.873
0.571
0.546
0.605
0.540
0.707
0.636
0.457
0.384
2
1.107
1.004
0.812
0.714
0.795
0.655
0.811
0.705
0.622
0.475
220
CIVIL ENGINEERING.
The thickness obtained by using " this table are nearly
double that of strict equilibrium. This factor of safety 01
excess of stability is that used by Vauban in his retaining
walls which have stood the test ol more than a century with
safety.
The formula,
= 0.845 (H + h) A_ x tan 5 -
will give very nearly the same values as those given in the
table.
RETAINING WALLS, face and back not parallel.
296. To transform a wall of rectangular cross-section into
one of equal stability having a batter on its face and its back
vertical, the usual form of cross-section of a retaining wall,
we may use the following formula of M. Poncelet,
V = I + n H.
(145)
in which (Fig. 93) b = the thickness, B d, of wall of rectangu-
lar cross-section,
FIG. 93.
V = the base, A B, of the equivalent wall with trapezoidal
cross-section,
H = the height B C of the wall, and n = the quotient ^-f .
D r
The base of the rectangular wall for the height, H, is ob-
tained from the previous formulas, then, knowing n, the value
of I' is obtained from formula (145).
COUNTERFORTS. 221
Tiiat is, the thickness of the equivalent trapezoidal wall at
the base is equal to the thickness of the rectangular wall in-
creased by one-tenth of the product obtained by multiplying
the height of the wall by the quotient resulting from dividing
the base of the slope by its perpendicular. This rule gives
the thickness to within T Q- of the true distance for values
of n less than $-, and within ^5- for values less than . Batters
with a slope less than J are seldom used.
297. Counterforts. Counterforts are considered to give
additional strength to a wall by dividing it into shorter
lengths, these short lengths being less liable than longer ones
to yield by bulging out or sliding along the horizontal courses ;
by the pressure being received on the back of the counterfort
instead of on the corresponding portion of the wall, thus
increasing the stability of the wall against overturning at
those points ; and by the filling being confined between the
sides of the counterforts, the particles of the filling, especially
in case of sandy material when confined laterally, becoming
packed and thus relieving the back of the wall.
Counterforts are, however, of doubtful efficiency, as they
increase the stability of the wall but slightly against rotation,
and not at all against sliding. They certainly should not be
used in treacherous foundations on account of the danger of
unequal settling.
The moment of stability of a wall with counterforts may
be found with sufficient accuracy for all practical purposes
by adding together the moments of stability of one or the
parts between two counterforts, and one of the parts aug-
mented by a counterfort, and dividing this sum by the total
length of the two parts.
Their horizontal section may be either rectangular or
trapezoidal. The rectangular form gives greater stability
against rotation, and costs less in construction; the trape-
zoidal form gives a connection between the wall and coun-
terfort broader and therefore firmer than the rectangu-
lar, a point of some consideration where, from the char-
acter of the materials, the strength of this connection must
mainly depend upon the strength of the mortar used for the
masonry.
298. Counterforts have been used by military engineers
chiefly for the retaining walls of fortifications. In regu-
lating their form and dimensions, the practice of Vauban
has been generally followed ; this is to make the horizontal
section of the counterfort trapezoidal, to make the length, ef,
of the counterfort (Fig. 94) equal to two-tenths of the height
CIVIL ENGINEERING.
of the wall added to two feet, the front, ab, one-tenth ofih*
height added to two feet, and the back, cd, equal to two-
thirds of the front, ab.
FIG. 94 Represents a section A and plan D of a
wall, and an elevation B and plan E of a trape
zoidal counterfort.
RESERVOIR WALLS AND DAMS.
299. These are retaining walls which are used to resist the
pressure of a volume of water instead of earth, and they do not
differ mathematically from the walls already discussed. Their
dimensions are therefore obtained in the same way.
Their cross-section is generally trapezoidal.
Let A B C D (Fig 95) represent the cross-section of a reser-
voir wall, with a vertical water face B C, and let the upper
surface of the water be at E F.
Represent by
A, the depth E B of the water ;
h', the height B C of the wall ;
5, b', the upper and lower bases A B and D C ;
10, the weight of unit of volume of water ;
w', the weight of unit of volume of masonry.
Lay off B H equal to one-third of B E, and draw the nori*
zontal H. This gives the direction and point of application
of the thrust on the wall produced by the pressure of the
water. Its intensity is equal to %w/i*. The weight of the wall
acts through the centre of gravity G, and is equal to \w'h'
(b + b f ). The moments around the edge at A can be deter-
mined and the values for b and b' found.
RESERVOIR WALLS.
223
The resultant E of these pressures intersects the base A B
between A and B. Stability requires that this should be so.
FIG. 95.
If the resistance to a crushing force were very great in the
surface, A B, supporting the wall, it would make no difference
how near the resultant came to the edge A. But r.s such is'
not the case, it should not come so near the edge as to pro-
duce a pressure along the latter sufficiently great to injure
the resistance of the material.
The nearer the intersection is to the middle point of the
base, the more nearly will the pressure on the foundation of
the wall be uniformly distributed over it.
It is evident, from the figure, that the batter given to the
face A D contributes greatly to the uniform distribution of the
pressure. And it is easily seen that if the outer face had
been made vertical, the resultant would have intersected the
base much nearer to the edge A, producing a far greater pres-
sure in that vicinity than in the former case.
C
FIG. 90.
300. Reservoir walls are usually constructed with both their
faces sloped Having found the thickness of the wall, as
224
CIVIL ENGINEERING.
above, the profile is easily transformed. For example, let
A B C D (Fig. 96) be a cross-section of a wall in which b and
b f have been determined by previous rule. Let M N be the
thickness at the middle point of the inner vertical face. It is
evident that if the thickness at top be, diminished by C, and
that at the base be increased by the equal quantity B P, that
the weight of the wall will remain the same, with an increase
of stability.
STRUCTURES OF THE FOURTH CLASS.
301. Structures belonging to this class sustain a transverse
strain. Since stone resists poorly a cross-strain, great caution
must be used in proportioning the different parts of these
structures. The rules for determining the strength of beams
subjected to transverse strains can be applied.
STRUCTURES OF THE FIFTH CLASS.
302. Arches are the principal structures belonging to this
class. They are used to transmit the pressure they directly
receive to lateral points of support.
Arches are generally made symmetrical, hence the condl-
tions of stability deduced for either half are equally applica-
ble to the other.
303. Modes of yielding. Arches may yield either by
sliding along one of their joints, or by turning around an edge
of a joint.
FIG. 97.
Suppose the arch to be divided into equal halves by its
plane, of symmetry, and let the right portion be removed
AECHE8.
225
(tig. 97). "We may suppose the equilibrium preserved by
substituting a horizontal force H for the half arch removed.
If the semi-arch were one single piece, the intensity of this
force, H, could be easily determined, for the conditions of
equilibrium would require the moment of the weight of the
semi-arch around the springing line at A to be just equal to
the moment of H about the same line.
The semi-arch not being a single piece, but composed of
several, mav separate at any of the joints, and therefore the
difficulty of determining the values of H is increased.
CONDITIONS OP STABILITY to prevent sliding at the joints.
304. The resistance to sliding arises from the friction of
the joints and from their adherence to the mortar.
Arches laid in hydraulic mortar, or thin arches in common
mortar, may derive an increase of stability from the adhesion
of the mortar to the joints, but in our calculations we should
disregard this increase, and depend for stability upon the
resistance due to friction alone.
It is found that friction, when the pressure is constant, is
FIG. 98.
independent of the area of the surfaces in contact, and de-
pends solely upon the nature and condition of the surfaces.
Let F be the resistance to sliding, produced by friction at
any joint I K (Fig. 98). The external forces acting on this
15
226 CIVIL ENGINEERING.
joint are the horizontal force H, and the weight of the mass
K B C I. Denote by R the resultant of these forces, and con-
struct it. This resultant pierces the plane of the joint I K at
Borne point as M, and M N will be the normal component.
Represent by P this normal component, and by S the com-
ponent parallel to the joint. We have
in which/* is the coefficient of friction determined by experi-
ment.
In order that sliding along this joint shall not take place,
we must have
S< F, orS
0.
Suppose the second case, or that the arch opens at K, and
denote by u and v the lever arms of W and H with respect
to I. We must have for stability
If we find the joints at which "W is a maximum and
y
JOINTS OF RUPTURE. 227
M
W- is a minimum, then for stability the value of H must
lie between these two values.
That is, the condition for stability against rupture by
rotation around the edge of a voussoir requires the thrust, H of
the arch to be greater than the maximum value of W-, and
less than the minimum value of W-.
J v
Joints of Rupture.
306. From observations made on the manner in which large
arches have settled, and from experiments made in rupturing
small ones, it appears that the ordinary mode of fracture is
for the arch to separate into four pieces, presenting five joints
of rupture.
Cylindrical arches in which the rise is less than half the
span, and the full centre arch, yield by the crown settling and
the sides spreading out The vertical joint at the crown
FIG- 99.
opens on the soffit, the reins open on the back, and if there
be no pier, the joints at the springing line open on the soffit
(Fig. 99).
The two lower segments revolve outwardly on the exterior
edge of the joints, leaving room for the upper segments to
revolve towards each other on the interior edges of the joints
at the reins.
This is almost the only mode of yielding for the common
cylindrical arch. If the thickness be very great compared
with the span, the rupture will take place by sliding. As a
rule, this mode of rupture never does take place for the reason
that the arch will rupture by rotation around a joint before
it will yield by sliding.
228 CIVIL ENGINEERING.
Yery light segmental arches, full-centre aruhes which are
slightly loaded at the crown and overloaded at the reins, and
pointed arches, are liable to rupture, as shown in Fig. 100.
In this case the crown rises and the sides fall in ; the open-
FIG. 100.
ing of the joints and the rupture occur in a manner exactly
the reverse of that just described. This mode of rupture is
still more uncommon than that by sliding; for all these
teasons, the condition
H x y Wx >
is in general the one applied to test the stability of the arch.
Cylindrical Arch.
307. Let it be required to find the conditions o'f equili-
brium for a full centre arch.
The strains in the arch are produced by the weight of the
arch stones, the load placed upon the arch and the reactions
at the springing lines.
The object of this discussion is to show how these external
forces may be determined and how to arrange the joints and
fix the dimensions of the voussoirs so as to resist successfully
the action of these forces.
The joints are the weak places, since the separation of the
parts at these points is not resisted by the material of which
the arch is made.
AB before stated, the arch may yield by sliding along one
of the joints or by turning around an edge. The first mode
of yielding may be prevented by giving the plane of the ioint
such a position, that its normal shall make with the resultant
pressure an angle less than the angle of friction of the ma-
terial of which the voussoirs are made.
CYLINDRICAL ARCH.
This is usually effected by making the coursing joints nor-
mal to the ring courses and to the soffit of the arch.
Since there is little danger of the arch rupturing by the
crown rising and the sides falling in, we make use of the
formula
H x y-Wx > 0.
The additional condition is imposed that the whole area of
the joint must be subjected to compression. It therefore
follows that the resultant of the external forces must pierce
the joint within its middle third.
Since the form of the arch is known, the direction of the
coursing joints chosen, and the limits of the resultant deter-
mined, it will only be necessary to find where the resultant
pierces each joint and see if the angle it makes with the nor-
mal is less than the angle of friction, and that the resultant
pierces the plane of the joint within the required limits.
Cylindrical Arch, Unloaded.
308. For simplicity, let us consider the arch to be a full
centre, the extrados and intrados being parallel and the
arch not loaded.
',-.-, -.- / ,: 's.
FIG. 101.
Let I K (Fig. 101) oe a joint of the arch whose thicknosi
in the direction of the length of the arch is unity.
Represent by
R, the radius of the extrados ;
r, the radius of the intrados ;
<. the angle made by the joint I K with the vertical ;
W and H, same as in previous case ;
g, the centre of gravity of the ring K B C I ;
to, the weight of a unit of volume of masonry
230 CIVIL ENGINEERING.
The point of application of the thrust H, at the joint B C
is somewhere above the middle of the joint, and when the
arch begins to rupture it is at C (Fig. 101). The condition
of stability for this case at the joint I K is
W-=H.
y
If the values of x and y be found in known terms, and sub-
stituted in this expression, the horizontal thrust can be
determined.
To find these values of x and y, denote by u the distance
of the centre of gravity g from 0, and by U L and u% the dis-
tances of the centres of gravity of the sectors I C and K B
from the same point. We have
M! x sector I C = w 2 x sector K B + u x ring K B C I.
The areas of the sectors are ^R 2 > and -Jrfy, hence the area
of K B C I is equal to < (E 2 -/^).
We find (Anal. Mech., par. 121, p. 96) the values of 11^ and
3 arc$ 3 arc$
Substituting for the areas, and for % and u% their values
as above, and solving with respect to u, we have
u _ 4 E 3 r 3 sin^ft
3K 2 r*' arc< *
Now x is equal to K M N\g f = r sin < Og sin <, whence
and y R r cos .
T>
Hence, by writing k for _, we have
T
H = ^. = r t w i sm 4> ffi*l) ^Q ^- cos
y cos <>
. . (146)
an expression for the horizontal thrust, in terms of R, r, w,
and , which force applied to the arch at C will prevent the
rotation of the volume K C B I around the edge K.
CTLESDEICAL ARCH.
231
Th:s expression might be differentiated with respect to ,
and that value for < obtained, which would make H a maxi-
mum. This maximum value thus found, if applied to the
arch at C, would prevent its rotation around any edge on the
soffit.
309. Instead of differentiating as suggested, it is usual in
practice to take the above expression for H, calculate the
values for every ten degrees, and select for use the greatest
of these values. This greatest value thus obtained will differ
but slightly from the true maximum.
If we assume k = 1.2, r = 10 feet, B, = 12 feet, and w =
150 pounds, and find the values of H for the different values
of $ for every ten degress from 10 to 90 ; we may tabu-
late them as follows :
Values of +.
Values of H in pounds.
10
208
20
670
30
1,127
40
1,450
60
1,625
60
1,675
70
1,662
80
1,490
90
1,285
A calculation for > = 57 gives H = 1,672, 63 gives 1,670,
and 65 gives 1,661 pounds.
The angle requiring the maximum thrust is very nearly 60.
310. The foregoing applies only to an unloaded full centre
arch, its extrados and intrados being parallel. All arches
carry loads which frequently rise above the arch to a surface
either horizontal or nearly so. It is evident that if verticals
be erected at the joints, and be produced until they meet the
upper surface of the load, that they will define and limit
the load resting on each voussoir. An analogous process to
that just given will enable the student to determine the hori-
zontal thrust in tire arch thus loaded.
232
CIVIL ENGINEERING.
Prof. Rankine gives the following rule to find the approxi-
mate horizontal thrust in a full centre arch loaded as shown
in the figure. (Fig. 102.)
FIG. 102.
The horizontal thrust is nearly equal to the weight sup*
ported between the crown and that part of the soffit whose
inclination is 45.
The approximate thrust obtained by this rule seldom differs
from the true horizontal thrust by so much as one-twentieth
part.
Represent by (Fig. 102).
R, the radius D of the extrados ;
7", the radius C of the intrados ;
GJ the distance D E, F E being horizontal ;
w, the weight of a cubic foot of masonry ;
w', the weight of a cubic foot of the load resting on the
arch ;
H, the horizontal thrust required.
Draw K making an angle of 45 with the vertical ; then,
the horizontal thrust of the arch on the pier at A is stated to
be nearly equal to the weight of the mass C K I F E, which
lies between the joint I K and the vertical plane through C ;
hence,
H = w' R (.0644 R + .7071 c) + .3927 w (R 2 -/- 2 ). (147)
for the value of the horizontal thrust.
The edge I is at the level to which it is advisable to build
the backing solid, or at least to give the blocks a bond which
will render the mass effective in transmitting the horizontal
thrust.
CURVE OF PRESSURE.
In the case of a segmental arch, Eankine takes the weight
of half the arch with its load, and multiplies it by the co-
tangent of the inclination of the intrados, at the springing
line, to the horizon ; the result is the approximate value of H.
311. Having determined the value for H for the given
arch, combine it with the external forces acting on the first
voussoir at the crown and construct their resultant. The
point in which this resultant pierces the joint will be the
centre of pressure for that joint. Do the same for the other
joints and the intensity of the resultant and the centre of
pressure for each joint are known.
If these resultants be produced, a polygon will be formed,
each angle of which will be on the resultant of the external
forces, acting on the voussoir between the two joints to which
the sides of the polygon correspond. A curve inscribed in
this polygon tangent to its sides is called the curve of
pressure of the arch, since a right line drawn through a
centre of pressure, tangent to this curve, will give the direc-
tion of the resultant pressure for this point.
If normals be drawn through the centres of pressure a
polygon will be formed whose sides give the direction of the
components producing pressure on the joints. A durve
tangent to its sides is called the curve of resistance, and
is the locus of the centres of pressure of the joints.
For stability, the curve of resistance should pierce each
joint in its middle third, and the curve of pressure should be
so situated that right lines tangent to it drawn through the
centres of pressure should make angles with the normals less
than the angle of friction.
312. Equation of the curve of resistance. Suppose
the loads on an arch to be symmetrically disposed so that
the resultant forces will lie in a vertical plane.
Equations (688) of Anal. Mechanics for this case will be
*. o
(148)
FIQ. 103.
in which H is the horizontal thrust at (Fig. 103) ; C the
compressive stress on any section, ae D ; , the length of any
portion of the curve, as 6 D ; and W the sum of the vertical
forces acting on the portion considered.
234 CIVIL ENGINEERING.
The first of equations (148) shows that the horizontal com-
ponent of the force of compression at any joint is equal to
the horizontal thrust at the crown, or is the same at every
section of the arch.
The second of these equations shows that the vertical com-
ponent of the force acting at any joint is equal to the load
between the vertical plane through the crown and the section
considered.
313. Suppose an arch loaded as
shown in figure (104) ; the material
being homogeneous and the Veight
of a unit of volume being represented
by w. Represent F by a.
FIG. 104. The weight of the volume resting
on the arch between the vertical section at D and the consecu-
tive section is
(adx + ydx)w.
Taking this between the limits, and a?, we get
( ax + J ydx W,
for the load resting on D. Substituting this in the second
of equations (148) for W, we get
(ax + jf* yda\ -0^ = 0. . (149)
w (ax
Combining this equation with the first of equations (148), we
have
whence, by differentiating, we get
.
Integrating this differential equation twice, we gel: the
equation of the curve, and find it to be a transcendental line.
314. If the load had been placed on the arch so as to be a
function of the first power of the abscissa, that is, if the load
between the origin and any section whose abscissa is a, was
then equations (148) would have taken the form
ds
CUEVE OF RESISTANCE. 235
Whence, by combination,
and by integration,
y = m af - < 152 )
which is the equation of a parabola.
315. Polar equation of the curve of resistance.
This equation is deduced by General Woodbury as follows :
Kepresent by (Fig. 105)
H,the horizontal thrust at ra; mnp,
the curve of resistance ; /', the dis-
tance, Om, from pole to the point of
application, m of the horizontal
thrust ; b, the horizontal distance
between the centre of gravity of the
segment E F I K C and the vertical
through C ; A, the area of the seg-
ment ; v, the variable angle nQm, FlQ
and 7-, the variable distance On.
For equilibrium, considering w equal to unity, we have
H (r' r cos v) = A (r sin v J),
whence r = . r + . . . . (153)
A sin v -f H cos v
Assuming any joint, the corre'sponding values of A and b
for this joint are easily calculated. These being substituted,
and H and v being known, the corresponding value of r is
deduced. The curve may then be constructed by points.
A simple inspection of the curve of resistance will
show where the weak points of the arch are, where the
heaviest strains are exerted, and where the joints tend to
open, whether on the soffit or on the back.
316. The deviation of the curve of pressure from the curve
of resistance is not great, and no material error is ordinarily
made when the points of the curve of pressure cut by the
joints are taken as the centres of pressure for the joints.
In arches with the ordinary form of voussoirs, the curve of
pressure lies below the curve of resistance, and the condition
that it shall lie within the middle third of the joints is
favorable to the stability of the arch.
236 CIVIL ENGINEERING.
When the weight of the voussoirs and the load on the
arch are determined, as in Art. 313, by considering them com-
posed of vertical laminae, the curves of pressure and of re-
sistance will coincide with each other.
Economy of material would indicate that the intrados and
extrados should be similar curves.
317. Depth of keystone. The form of the arch being
assumed, the next step is to fix its thickness or depth. The
power of the arch to resist the horizontal thrust at the crown
will depend upon the strength of the material of which it is
made and upon the vertical thickness (depth) of the key.
The pressure at the extrados of the key, which in general
is the most exposed part of the joint, should not exceed -fa
the ultimate strength of the material. Admitting that the
centre of pressure on this joint may be at one-third of the
length of the joint from the extrados, we see that in order
to keep within this limit of -j^, the mean pressure should not
exceed fa.
The celebrated Ferronnet gave a rule for determining the
thickness or depth of the key, which is very nearly expressed
by the following formula :
' d = T + ' 33 ( 154 )
dj the depth in metres ; and
7*, the radius of the semicircle, or intrados, in same unit.
Gen. Wood bury expressed this rule as follows :
d = 13 inches + -fa the span.
For arches with radius exceeding 15 metres, this rule gives
too great a thickness.
Prof. Rankine gives
d= V.12r,
in which r is the radius of curvature at the crown in feet.
His rule is, " For the depth of the keystone, take a mean pro-
portional between the radius of curvature of the intrados at
the crown and a constant whose value for a single arch is .12
feet."
He recommends, however, in actual practice, to take a
depth founded on dimensions of good examples already built.
318. Thickness of piers and abutments. The stability
of these may be considered by regarding them either as con-
tinuations of the arch itself clear to the foundation, or as walla
whose moment about the axis of rotation is greater than the
moment of the thrust of the arch.
THICKNESS OF ABUTMENTS.
237
In either case, the student will be ablej by applying the
principles already discussed, to determine the dimens.ons
necessary to give the pier, in order that its moment around
any edge shall exceed the moment of the thrust around the
same axis.
The factor of safety is taken at about 2. In piers of great
height this factor should be increased, while for small heights
it may be reduced.
319. Thickness of abutment and depth of keystone
for small arches.
The following empirical table is deduced from actual ex-
amples, and may be used for small arches if made of first-
class masonry:
TABLE.
Bpanin
feet
Thickness of Abutment for heights of
Depth of key-
stone in inches.
10 feet.
15 feet.
20 feet.
25 feet.
10
5
6
7
8
14
20
6
7
8
9
19
25
*}
7i
8|
H
20
30
7
8
9
10
21
35
7*
8|
w
10*
22
40
8
9
10
11
23
45
Si
9i
10)
111
24
50
9
10
11
12
25
If the masonry be second-class, or be roughly dressed, the
depth of the keystone should be increased about one-fourth.
Form of Cylindrical Arches.
320. As stated before, these arches may be full centre,
segmental, elliptical, or oval.
Full centre arches offer the advantages of simplicity of
form, great strength, and small lateral thrust. But where the
238
CIVIL ENGINEERING.
span is considerable, they require a correspondingly great
rise, which is often objectionable.
The segmental arch enables us to reduce the rise, but
causes a greater lateral thrust on the abutments.
The oval affords a means of avoiding both the great rise
and the great lateral thrust, and gives a curve of pleasing
appearance.
Rampant and Inverted Arches.
321. The arch in the preceding cases has been supposed to
have been upright, and either right or oblique. Rampant
arches are frequently used ; sometimes the axis is even verti-
cal. A retaining wall with a semi-circular horizontal section
would be an example. Arches are often constructed with
their soffits forming the upper side. These are frequently
used under openings, their object being to distribute the
weight equally over the substructure or along the founda-
tions. They are known as inverted arches, or inverts. The
principles already laid down for the upright arch apply
equally to them.
Wooden Arches.
322. This term, -wooden arch, is quite often Applied to a
beam bent to a curved shape, its ends being conimed so that
the beam cannot resume its original form. In this shape the
beam possesses under a load greater stiffness than when it is
straight.
A single beam may be used for narrow spans, but built
beams, either solid or open, must be used for wide ones.
FIG. 106.
The load they support rests upon the ty p of the beams, as
diown in Fig. 106, or is suspended from hem. as shown in
Fig. 107.
RUBBLE WALLS.
239
Although called arches, they are so only in form, as they
are not composed of separate pieces held in place by mutual
pressure. They are now more generally called by their
proper name, curved beams.
If we assume that the beam resists by compression alone,
the dimensions of the beam can be easily determined, in terms
of the load, of the rise, and the span.
FIG. 107.
GRAPHICAL METHOD OF INVESTIGATION.
323. The graphical method by means of the curve of equi-
librium is a method much used at the present time for obtain-
ing the strains on the different parts or the arch.
This method of investigation will be alluded to in a future
article.
CHAPTER X.
CONSTRUCTION OF MASONRY.
WALLS OF STRUCTURES.
Stone-masons class the methods of building walls of stone
into rubble work and ashlar work.
L Rubble Work.
324. The stones used are of different sizes and shapes, pre-
pared by knocking off all sharp, weak angles of the blocks with
a hammer. They are laid in the wall either dry or in mortar.
If laid without reference to their heights, tne masonry is
known as uncoursed rubble, or common rubble masonry.
240 CIVIL ENGINEERING.
In building a -wall of rubble (Fig. 108) the mason must
be careful to place the stones so that they may fit one upon
the other, filling the interstices between the larger stones by
smaller ones. Care should be taken to make the vertical
courses break joints.
If mortar is used, the bed is prepared by spreading mortar
over the top of the lower course, and in this bed the stone
is firmly imbedded. The interstices are filled with smaller
FIG. 108.
stones, or stone chippings, and mortar, and finally the whole
course grouted.
The mean thickness of a rubble wall should not be less
than one-sixth of the height ; in the case of a dry stone wall,
the thickness should never be less than two feet. It strengthens
the wall very much to use frequently in every course, stones
which pass entirely through the wall from the front to the
back. These are called throughs. If they extend only part
of this distance, they are called binders.
325. Coursed rubble, or hammered masonry. When the
stones are laid in horizontal courses, and each course levelled
throughout before another is built upon it, the work is termed
coursed rubble. As this requires the stones to be roughly
dressed, or hammered into regular forms before they are laid,
the work is frequently called hammered, or dressed rubble.
The same care should be taken in building masonry of this
kind as that required for common rubble. The mason must
be particular in making the upper and lower surfaces of each
stone parallel, and when laying the stones to keep a uniform
height throughout each course. If a stone in the course is
not high enough, other stones are laid on it till the required
height is obtained.
The different courses are not all of the same height, but
vary according to the size of the stone used. The only condi-
tion required is that each course shall be kept of the same
height throughout.
ASHLAR MASONRY.
241
At the corners, stones of large size, and more acurately
dressed, are used. These are known as quoins, and are laid
with care, serving as gauges by which the height of the course
is regulated.
n. Ashlar Work.
326. The stones in this kind of masonry are prepared by
having their beds and joints accurately squared and dressed.
They are made of various sizes depending on the kind of
wall to be built and the size of the blocks produced by the
quarry. Ordinarily they are about one foot thick, two or
three feet long, and have a width from once to twice the
thickness. They are used generally for the facing of a
wall, to give the front a regular and uniform appearance, and
where, by the regularity of the masses, a certain architectural
effect is to be produced.
Ashlar work receives different names, from the appearance
of the face of the "ashlar," and from the kind of tool used in
dressing it. If the block be smooth on its face, it is called
plane ashlar (Fig. 109) ; if fluted vertically, tooled ashlar ;
PIG. 109 Represents a wall with facing of plane ashlar.
if roughly trimmed, leaving portions to project beyond the
edges, rustic ashlar, etc., etc. Rustic ashlar is known as
rustic, rustic chamfered, rustic work frosted, rustic work
vermiculated, etc.
Ashlars are laid in fine mortar or cement. Each one should
be first fitted in its place dry, so that any inaccuracy in the
dressing may be discovered and corrected before the stone is
finally set in mortar.
To provide for a uniform bearing the stone should be ac-
curately squared. Frequently the bed is made to slant down
16
242
CIVIL ENGINEERING.
wards, from front to back, for the purpose of making close
horizontal joints in front. This weakens the stone, as the
weight is thrown forward on the edges of the stones, which
chip and split off as the work settles.
327. Walls built with ashlar facing are backed with brick
or rubble. Economy will decide which is to be used. In the
construction, throughs of ashlars should be used to bind the
backing to the facing. Their number will be proportioned to
the length of the course. The vertical courses break joints,
each vertical joint being as nearly as possible over the middle
of the stone below.
Fig. 110 Represents a section of wall with facing of ashlar and a back-
ing of rubble.
When the backing is rubble, the method of slanting the
may be allowed for the purpose of forming a better bond
between the rubble and ashlar ; but, even in this case, the
block should be dressed true on each joint, to at least one foot
back from the face. If there exists any cause which would
give a tendency to an outward thrust from the back, then,
instead of slanting off all the blocks towards the tail, it will be
preferable to leave the tails of some thicker than the parts
which are dressed.
Cut-stone Masonry.
328. Where great strength is required in the wall, each
stone is prepared by cutting it to a particular shape, so that
it can be exactly fitted in the wall ; masonry of this kind is
called cut-stone. In other words, every stone is an ashlar ;
STRENGTH OF MASONET. 243
hence the terms cut-stone and ashlar masonry are often
used one for the other.
Cut-stone masonry, when carefully constructed, is more
solid and stronger than any other class. The labor required
in preparing the blocks makes it the most expensive. It is,
therefore, restricted in its use to those structures where
great strength is indispensable.
Stone-cutting.
329. The usual method of dressing a surface is to cut
draughts around and across the stone with a chisel, and then
work down the intermediate portions between the draughts by
the use of proper tools. The latter are usually the chisel, axe,
and hammer.
No particular difficulty occurs in working a block of stone,
the faces, beds, and joints of which are to be plane or even
cylindrical surfaces ; the only difference in method for the
two being that a curved rule is used in one direction and a
straight one in another for the cylindrical surface, while for
the plane surface only one rule is used.
If the surfaces are to be conical, spherical, or warped, the
operation is more difficult. It becomes necessary to bring the
block to a series of plane or cylindrical surfaces, and then
reduce them to the required form. To show how this can be
done with the least waste of material is one of the objects of
- stereotomy .*
Strength of Masonry.
Strength. The strength of masonry will depend on the
size of the blocks, on the accuracy of the dressing, and on
the bond.
330. Size of stone. The size of the blocks varies with the
kind of stone and the nature of the quarry.
Some stones are of a strength so great as to admit of their
being used in blocks of any size, while others can only be used
with safety when the length, breadth, and thickness of the
block bear certain relations to each other.
The rule usually followed by builders, with ordinary stone,
is to make the breadth at least equal to the thickness, and
seldom greater than twice this dimension, and to limit the
length to within three times the thickness. When the breadth
or the length is considerable in comparison with the thick
244 CIVIL ENGINEERING.
ness, there is danger that the block may break, if any unequal
settling or unequal pressure should take place. As to the ab-
solute dimensions, the thickness is generally not less than one
foot, nor greater than two ; stones of this thickness, with the
relative dimensions just laid down, will weigh from 1,000 to
8,000 pounds, allowing, on an average, 160 pounds to the
cubic foot. With these dimensions, therefore, the weight of
each block will require a very considerable power, both of
machinery and men, to set it on its bed.
From some quarries the formation of the stone will allow
only blocks of medium or small size to be furnished, while
from others stone of almost any dimensions can be obtained.
331. Accuracy of dressing. The closeness with which
the blocks fit is solely dependent on the accuracy with which
the surfaces in contact are wrought or dressed ; if this part of
the work is done in a slovenly manner, the mass will not only
open at the joints with an inequality in the settling, but, from
the courses not fitting acurately on their beds, the blocks will
be liable to crack from the unequal pressure on the different
points of the block.
To comply with the first of the general principles to be
observed in the construction of masonry, we should have, in a
wall supporting a vertical pressure, the surfaces of one set of
joints, the beds, horizontal. This arrangement will prevent
any tendency of the stones to slip or slide under the action of
the weight they support.
The surfaces of the other set should be perpendicular to
the beds, and at the same time perpendicular to the face, or to
the back of the wall, according to the position of the stones in
the mass ; two essential points will thus be attained ; the angles
of the blocks at the top and bottom of the course, and at the
face or back, will be right angles, and the block will therefore
be as strong as the nature of the stone will admit.
The greater the accuracy of the dressing, the more readily
can these surfaces be made to fulfil these conditions.
When a block of cut stone is to be laid, the first point to be
attended to is to examine the dressing, by placing the block
on its bed, and seeing that the face is in its proper plane, and
that the joints are satisfactory. If it be found that the fit is
not accurate, the inaccuracies are marked, and the requisite
changes made.
332. Bond. Among the various methods used, the one
known as headers and stretchers is the most simple, and
offers, in most cases, all requisite solidity; in this method the
vertical joints of the blocks of each course alternate with the
BOND.
245
rertical joints of the courses above and below it, or break
joints with them, and the blocks of each course are laid alter-
nately with their greatest and least dimensions to the face of
the wall ; those which present the longest dimension along the
face are termed stretchers, the others headers. (Fig. 111.)
[_ 151
1 ,
1 1
1 1
FIG. Ill Represents an elevation
A, vertical section B, and horizontal
section C, of a wall arranged as
headers and stretchers.
a, stretchers.
5, headers.
1
-
1 1
! 7
I * i 6
By arranging the blocks in this manner the facing and
backing of each course are well connected ; and, if any une-
qual settling takes place, the vertical joints cannot open, as
would be the case were they continuous from the top to the
bottom of the mass, for each block of one course confines the
ends of the two blocks on which it rests in the course beneath.
FIG. 112 Represents an elevation A, and perspective views C and D, of
two of the blocks of a wall in which the blocks are fitted with indente,
and connected with bolts and cramps of metal.
246
CIVIL ENGINEERING.
333. In masonry exposed to violent shocks, the blocks of
each course require to be not only very firmly united with
each other, but also with the courses above and below them.
To effect this various means have been used. Sometimes the
stones of different courses are connected by tabling, which
consists in having the beds of one course arranged with pro-
jections (Fig. 112) which fit in corresponding indentations
of the next course. Iron cramps in the form of the letter S,
set with melted lead, are often used to confine two blocks to-
gether. Holes are, in some cases, drilled through several
courses, and the blocks of these courses are connected by
strong iron bolts fitted to the holes.
Light-houses, in exposed positions, are peculiarly liable to
violent shocks from the waves. They are ordinarily, when
thus exposed, built of masonry, are round in cross-section, and
solid up to the level of the highest tide. The stones are often-
times dove-tailed and dowelled into each other, as well as
fastened together by metal bolts and cramps.
The manner of dove-tailing the stones is shown in plan in
Fig. 113, which represents part of a course where this method
is used.
FIG. 113.
The chief use of the dove-tailing is to resist the tendency of
the stones to jump out immediately after receiving the blow
of the wave. This method was first used by Smeaton in build-
ing the Eddystone light-house. The light-house on Minot'a
Ledge, Massachusetts Bay, built under the superintendence
of General B. S. Alexander, U. S. Corps of Engineers, by
the, Light-House Board, is a good example of the bond and
metal fastenings used in such structures. (Figs. 114 and
115.)
BOND.
247
FIG 114. Vertical section showing foundation courses, metal fastenm**,
and the first story above the foundation courses.
Kio 115._Plan of twenty-second course, showing the method of
tailing the stones-
248 CIVIL ENGINEERING.
Machinery used in Constructing Walls of Stone.
334. Scaffolding 1 . In building a wall, after having raised
it as high as it can be conveniently done from the ground,
arrangements must be made to raise the workmen higher, so
that they can continue the work. This is effected by means
of a temporary structure called scaffolding.
If the wall is not used to afford a support for the scaffold-
ing, two rows of poles are planted firmly in the ground, par-
allel to the wall, and about four and a half feet apart. These
uprights in each row are from twelve to fourteen feet apart,
and from thirty to forty and even fifty feet in height, depend-
ing upon how high the wall is to be built.
Horizontal pieces are then firmly fastened to the uprights,
having their upper surfaces nearly on the same level as the
highest course of masonry laid. Cross pieces or joists are
laid on these, and upon them a flooring of boards. Upon
this platform the masons place their tools and materials and
continue the work.
As the wall rises other horizontal pieces are used, and the
joists and boards carried to the new level. Diagonal pieces
are used between the rows to brace them together, and in
each row to stiffen the supports.
The workmen ascend the scaffolding by means of ladders.
The materials are hoisted by means of machinery placed on
the scaffolding or detached from it.
335. Crane. The movable or travelling crane, which is
so arranged as to admit of being moved in the direction of
the scaffolding and across it, is often used on the scaffolding
for hoisting the stone.
Shears, which consist of two or more spars or stout pieces
of timber, fastened together near the top, and f urnished with
blocks and tackles, are sometimes used.
The kind of machinery to be used in hoisting the stone
will be determined by the size of the blocks to be lifted, the
magnitude and character of the work, and the suitability of
the site.
In the United States, the machine known as the " boom
derrick," or simply " derrick," a modified form of crane, is
much used in works of magnitude.
In the example shown in Fig. 116, the mast is held in a
vertical position by four guys, generally wire ropes, fastened
to a ring on the iron cap which is fitted to the top of the
HOISTING MACHINERY.
249
mast. Below this ring, and revolving freely on the cap, is
a wrought-iron frame containing two sheaves or pulleys.
The " boom," or derrick, has its outer end supported by a
topping-lift fastened to this wrought-iron frame. The other
end fits into an iron socket with collar, or is fastened to a
wooden frame which embraces the mast, and has a motion of
rotation around it. The wooden frame bears two windlasses
and a platform on which the men stand while working them.
Two tackles are used, one suspended from the outer end of
the boom, the other from the mast-head, the falls of both
leading over the sheaves and thence to the windlasses.
FIG. 116.
The lower blocks of the tackles are fastened to a triangular
plate from which a hook is suspended. It is seen that by
hauling upon or slacking the falls alternately, the stone sus-
pended from the triangular plate can be placed at any point
within the circle described by the outer end of the boom.
336. The blocks of stone are attached to the tackle in
various ways. Some of the most usual methods are as fol-
lows :
I. By nippers or tongs, the claws of which enter a pair of
holes in the sides of the stone.
IL By two iron pins let into holes, which they closely fit,
250
CIVIL ENGINEERING.
sloping towards each other (Fig. 117). The force applied to
the chain to lift the block, jams the pins in their holes.
FIG. 117 Represents a perspective view of the tackling for hoisting a
block of stone, A, with draughts around the edges of its faces, and th
intermediate space axed or knotted.
a, draughts around edge of block.
6, knotted part between draughts.
c, iron bolts with eyes let into oblique holes cut in the block.
d and e, chain and rope tackling.
III. By a simple contrivance made of three pieces of iron,
called a lewis (Fig. 118), which has a dove- tail shape, with
the larger end downwards, fitting in a hole of similar shape.
The depth of the hole depends upon the weight and the kind
of stone to be raised. The tapering side-pieces, n, n, of the
lewis are inserted and placed against the sides of the hole ;
the middle piece, 0, is then inserted and secured in its place
by a pin. The stone is then safely hoisted, as it is impossible
for the lewis to draw out of the hole.
FIG. 118 Represents the com-
mon iron lewis B.
n, n, side pieces of the lewis.
o, centre piece of lewis, with
eye fastened to n, n by a bolt.
P, iron ring for attaching tackling.
FIG. 119 A line attached to
the straight piece, &, admits
of the latter being drawn
out, allowing the piece, ,
to be removed.
Where it may not be convenient to reach the pin after the
Btone has been placed in position, a lewis of the form showd
in (Fig. 119) may be used.
BOND. 25 J
WALLS OF BEIOK.
337. Bricks have been referred to in a previous chapter
as artiiicial stones. It therefore follows that the general
principles enunciated for the construction of stone masonry
are the same for brick as far as they are applicable.
From the uniformity of size of brick, builders describe the
thickne&s of a wall by the number of bricks extending across
it. ^ Thus, a wall formed of one thickness of brick lying on
their broad side, with their length in the direction of the
length of the wall, is said to be " half brick thick." If the
thickness of the wall is equal to the length of one brick, the
wall is called " one brick thick," etc.
The bond used depends upon the character of the struc-
ture. The most usual kinds are known as the English and
Flemish.
338. English bond. This consists in forming each course
entirely of headers or of stretchers, as shown in Fig. 120.
Sometimes the courses of headers and stretchers occur
alternately ; sometimes only one course of headers for three
or four courses of stretchers. The effect of the stretchers is
to tie the wall together lengthwise, and the headers, cross-
I I I
I I I I I
I. .1. I. .1
I I I I
Fig. 120.
wise. The proportionate number of courses of headers to
those of stretchers depend upon the relative importance of
the transverse and longitudinal strength in the wall.
Since the breadth of a brick is nearly equal to half its
length, it would be impossible, beginning at a vertical end or
angle, to make this bond with whole bricks alone. This
difficulty is removed by the use of a half-brick, made by
cutting a brick in two longitudinally. A whole brick, useu
as a header, is placed at the corner; next to this is put a
SJ52 CIVIL ENGINEERING.
half -brick. This allows the next header to make the neces-
sary overlap, which can be preserved throughout the course.
These half-bricks are called closers.
339. Flemish "bond. This consists in laying headers and
stretchers alternately in each course.
A wall built with this bond presents a neater appearance
than one built in English bond, and is, therefore, generally
preferred for the fronts of buildings. It is not considered
as strong as the English, owing to there being, ordinarily, a
less number of headers in it.
840. Strengthening of bond. Pieces of hoop-iron or iron
lath, so thin that they may be inserted in the joints without
materially increasing their thickness, add to the strength of
the bond, especially when hydraulic mortar is used. They
are laid flat in the bed-joints, and should break joints. It is
well to nick them at intervals and bend the ends at right
angles for the length of two inches, inserting the bent ex-
tremities into the vertical joints.
This method was used by Brunei in forming the entrance
to the Thames tunnel, and is sometimes designated as
hoop iron bond.
341. Hollow masonry. Hollow brick walls are now ex-
tensively used in buildings.
The advantages of hollow walls are economy, lightness,
and, particularly, freedom from dampness.
The bricks may be hollow, being laid in the usual way, but
the usual method of forming the walls is to use ordinary brick,
and so arrange them in the walls as to leave hollow spaces
where required.
342. Strength of brick masonry. The strength of brick
masonry depends upon the same three conditions already
given for stone. Hence, all misshapen and unsound bricks
should be rejected.
With good bricks and good mortar a masonry of strength
and durability nearly equal to rtone is easily formed, and at
less cost. Its strength is largely due to the strong adhesion
of mortar to brick. The volume of mortar used is about one-
fifth that of the brick.
343. Laying the bricks. The strength of brick masonry
is materially affected by the manner in which the bricks are
laid. They should not only be placed in position, but pressed
down firmly into their beds.
As bricks have great avidity for water, it would always be
well not only to moisten them before laying, but to allow
them to soak in water several hours before they are used
CONCRETE WALLS.
By taking this precaution, the mortar between the joints will
set more firmly.
To wet the bricks before they were carried on the scaffold
would, by making them heavier, add materially to the labor
of carrying, It is suggested to have arrangements on the
scaffold where they can be dipped into water, and then
handed to the mason as he requires them. The wetting is of
great importance when hydraulic mortar or cement is used,
for if the bricks are not wet when laid, the cement will not
attach itself to them as it should.
Machinery of Construction.
34A. Scaffolding. In ordinary practice the scaffolds are car-
ried up with the walls, and are made to rest upon them. The
essential features are the same as those used for stone walls.
It would, be an improvement if an inner row of uprights were
used instead of the wall to support the framework, for the
cross-pieces, resting as they often do on a single brick in a
green wall, must exert an injurious influence on the wall.
Machinery for hoisting the bricks, mortar, etc., are used in
extensive works. For ordinary buildings the materials are
carried up by workmen by means of ladders.
WALLS OF CONCRETE.
345. Concrete masonry. "Within recent years much at-
tention has been paid to the construction of walls entirely of
concrete.
Method of construction. The concrete is moulded into
blocks, as previously described, and then laid as in stone ma-
sonry ; or it is moulded into the wall, the latter becoming a
monolithic structure.
The walls in the latter case are constructed in sections
about three feet high and ten or fifteen long. For this pur-
pose a mould is used made of boards forming two sides of a
box, the interior width of which is equal to the thickness of
the wall. Its sides are kept in place by vertical posts, which
are connected together and prevented from spreading apart
by small iron rods, as shown in Fig. 121.
The concrete is shovelled into the mould in lavers and
rammed with a pestle. As soon as the mould is filled, the
iron rods are withdrawn and the mould lifted up. A. second
254
CIVIL ENGINEERING.
section is formed in like manner on the top of the tirst, and
the process goes on until the wall reaches the required height.
if scaffolding be required in their construction, one of the
ordinary form may be used, or one like that shown in Fig. 121.
Fig. 121.
Tail's bracket scaffolding, in which the platforms are
sustained by clamping them to the wall as it is built up,
using the holes left when the iron rods are withdrawn, is an
example of one of the devices used in the construction of
concrete walls ; so also Clarke's adjustable frame, in which
the platform is supported by a frame from above, fastened to
clamps embracing the wall. Hoisting apparatus suitable for
the work is also employed.
Hollow -walls. In case the wall is required to be hollow,
a piece of board of the thickness of the required space to be
left open, and slightly wedge-shaped to admit of its being
easily removed, is laid horizontally in the mould, and the
concrete rammed in well around it. When the concrete is
filled to the top of the board, it is drawn out, leaving the re
CROSS-SECTION^ OF RETAINING WALL. 255
quired air space. At regular intervals, ordinary bricks are
laid as ties to connect together the outer and inner walls.
Fines, pipes, and other openings for heating, ventilating,
conveying water, gas, smoke, etc., are constructed in a similar
manner by using movable cores of the proper size and form.
Strength and advantages of concrete -walls. It ia
claimed that concrete walls are easier of construction,
cheaper, and stronger than brick walls of the same thickness,
and that they possess the great advantage in allowing air pas-
sages and flues to be easily constructed of uniform size and
smooth interiors.
RETAINING AND RESERVOIR WALLS.
346. Especial care should be taken, in the construction of
these walls, to secure a firm foundation, and to observe all the
precautions mentioned in previous articles for laying masonry.
Thorough drainage must be provided for, and care be
taken to keep water from getting in between the wall and the
earth. If the water cannot be kept out, suitable openings
through the masonry should be made to allow the water to
escape.
When the material at the back of the wall is clay, or is
retentive of water, a dry rubble wall, or a vertical layer of
coarse gravel or broken stone, at least one foot thick horizon-
tally, must be placed at the back of the retaining wall, be-
tween the earth and the masonry, to act as a drain.
In filling in the earth behind the wall, the earth should be
well rammed in layers inclined downward from the wall.
Especial care should be taken to allow the mortar to harden
before letting the wall receive the thrust of the earth.
Whenever it becomes necessary to form the embankment
before the mortar has had time to set, some expedient should
be employed to relieve the wall as far as possible from pres-
sure. Instead of bringing the embankment directly against
the back of the wall, dry stone or fascines may be interposed,
or a stiff mortar of clay or sand with about ^th in bulk of
lime may be used in place of the dry stone.
347. Form of cross-section of retaining -walls. The
rectangular and the trapezoidal forms are the most common.
It is usual, in the latter case, to give the face a batter, varying
between -f- and \*, and to build the back, or side in contact
with the earth, vertical, or in steps. From experiments mado
with models of retaining walls, it was shown that as the wall
256
CIVIL ENGINEERING.
gave way, the prism of earth pressing against it did not revolve
around any line, but settled suddenly and then rested until
another shock. These experiments seem to confirm the prac-
tice of building the back in steps.
In some cases the wall is of uniform thickness with eloping
or curved faces. (Figs. 122 and 123.)
FIG. 122.
FIG. 123.
It will be seen that, the weight remaining the same, the
wall with sloped or curved faces has an increase of stability
over the corresponding equivalent wall of rectangular cross-
section.
The advantage of such forms, therefore, lies in the saving
of material.
FIG. 124.
Walls with curved batter should have their bed-joints per-
pendicular to the face of the wall, so as to diminish the obli-
quity of pressure on the base. (Fig. 124.)
AREAS, LINTELS AND PLATE-BANDS.
257
348. Counterforts. Counterforts are generally placed
along the back of the wall, 15 to 18 feet apart, from centre to
centre; their construction is in every way similar to that
lecomrnended for retaining walls.
They should be built simultaneously with the wall, and be
well bonded into it.
349. Relieving arches. The name of relieving arches
is given to a range of arches resting against the back of a re-
taining wall to relieve it from the pressure, or a part of the
pressure, produced by the earth behind. (Fig. 125.)
FIG. 125.
These arches have their axes placed at right angles to the
back of the wall, and may have their fronts enclosed by the
earth, as shown in the vertical section represented in Fig. 125.
There may be one or several tiers of them.
Knowing the natural slope of earth to be retained, and
assuming the length of the arch, its height can be deduced,
or assuming the height, its length may be obtained, so that the
pressure of the earth on the wall shall not exceed a given
amount.
The distance between the centres of relieving arches is
ordinarily about 18 or 20 feet. The thickness of the
arch and piers will depend upon the weight they have to
support.
AREAS, LINTELS, ETC.
350. These structures sustaining a vertical pressure either
upwards or downwards, are subjected to a cross-strain.
Area. It happens sometimes that an upward pressure is
produced on an area by the presence of water ; this pressure
must be guarded against. The area of the new capitol at
17
258 cmr, ENGINEERING.
Albany, N. Y., is several feet thick, and was made by first
placing large flat stones over the surface, and then adding
successive layers of broken stone and concrete.
Lintels. The resistance to a transverse strain is very slight
in stone ; therefore the distance to be spanned by the lintel
should be quite small, seldom exceeding six feet.
Plate-bands. For a similar reason to that just given for
lintels, the span of a plate-band should not exceed ten feet,
and all pressure from above should be borne by some inter
posing device.
ARCHES.
351. The form of the arch is generally assumed, and the
number and thickness of the voussoirs are determined after-
wards. The curves of right section of full centre, segmental,
and elliptical arches require no further description, as the
student has already learned the method of constructing these
curves. The various ovals will be the only ones described.
Methods of Constructing Ovals.
352. The span and rise of an arch being given, together
with the directions of the tangents to the curve at the spring-
ing lines and crown ; an infinite number of curves, composed
of arcs of circles, can be determined, which shall satisfy the
conditions of forming a continuous curve, or one in which the
arcs shall be consecutively tangent to each other, and the con-
ditions that these arcs shall be tangent at the springing lines
and the crown to the assumed directions of the tangents to
the curve at those points. To give a determinate character to
fhe problem, there must be imposed, in each particular case,
certain other conditions upon which the solution will depend.
When the tangents to the curve at the springing lines and
srown are respectively perpendicular to the span and rise, the
^urve satisfying the above general conditions will belong to
Jie class of oval or basket-handle curves; when the tangents
it the springing lines are perpendicular to the span, and those
it the crown are oblique to the rise, the curves will belong to
the class of pointed or obtuse curves.
The pointed curve gives rise to the pointed or Gothic
itch.
If the intrados is to be an oval or basket-handle, and its
OVALS O THREE CENTRES.
259
rise is to be not less than one-third of the span, the oval of
three centres will generally give a curve of a form more pleas-
ing to the eye than will one of a greater number of centres ;
but if the rise is to be less than a third of the span, a curve
of five, seven, or a greater odd number of centres will give
the more satisfactory solution. In the pointed and obtuse
curves, the number of centres is even, and is usually restricted
to four.
353. Three centre curves. To obtain a determinate
solution in this case it will be necessary to impose one more
condition which shall be compatible with the two general
ones of having the directions of the tangents at the springing
lines and crown fixed. One of the most simple conditions,
and one admitting of a great variety of curves, is to assume
the radius of the curve at the springing lines. In order that
this condition shall be compatible with the other two, the
length assumed for this radius must lie between zero and the
rise of the arch ; for were it equal to zero or to the rise there
would be but one centre ; and if taken less than zero or greater
than the rise, then the curve would not be an oval.
FIG. 123.
General Construction. Let A D (Fig. 126) be the half
span, and A C the rise. Take any distance less than A C, and set
it off from D to R, along A D ; and from C to P, along A C.
Join R and P, bisect by a perpendicular. Prolong this per-
pendicular until it intersects C A produced. Then S, R, and
a point on A B, distant from A equal to A R, will be the three
centres of the required oval.
It is evident that there will be an infinite number of ovala
for the same span and rise.
260 CIVIL ENGINEERING.
For, denote by R the radius S C of the arc at the crown,
by r the radius R D at the springing line, by a the half span
A D, and by I the rise A C.
There results from the right angled triangle S A R,
SR 2 = ATS 2 + A~R*,
or
(R - rf = (R - J) 2 + (a - r)*,
from which is obtained
^ _ $ + &8 _ 2a r
2(& - r) '
which may be satisfied by an infinite number of sets of values
of B. and r.
354. To construct an oval of three centres, with the
condition that each of the three arcs shall be of 60.
Let B D be the span and A C the rise (Fig. 126). With the
radius A B describe Bba of 90 ; set off on it Bb = 60 ; draw
the lines ab, JB, and kb ; from C draw a parallel to db, and
mark its intersection c with >B; from c draw a parallel to
AZ>, and mark its intersections N and with A B, and C A pro-
longed. From N, with the radius N B, describe the arc Be;
from 0, with the radius 00, describe the arc Cc. The curve
BcC will be the half of the one satisfying the given condi-
tions, and N and two of the centres.
355. To construct an oval of three centres imposing the
condition that the ratio between the radii of the arcs at the
crown and springing line shall be a minimum.
Let A D be the half span, A C the rise (Fig. 126). Draw
D C, and from C set off on it Cd = Ca, equal to the differ-
ence between the half span and rise. Bisect the distance Dd
by a perpendicular, produced until it intersects C A prolonged.
From the points of intersection, R and S, as centres, with the
radii R D and S Q, describe the arcs D Q and Q C ; and the
curve D Q C will be the half of the one required.
For, from the triangle S A R, we get
R a 2 -f b* - 2
in which W is the greatest load, E and p the weight re-
spectively of the ram and pile, all in pounds ; h the fall of
the ram, and d the penetration of the pile at the last blow,
both in feet. He used a factor of safety of 4.
T> 7
Capt. Sanders' formula is W = -~ X ; the quantities be-
ing the same excepting that W is the safe load.
The rule used by builders is to limit the load to 1,000
pounds on the square inch of the head when the pile trans-
mits the weight to firm soil ; and to 200 pounds when it
resists by friction only.
399. Preparation of bed in compressible soil, using
common wooden piles. The piles having been driven to
the firm soil beneath, their heads are sawed off at a given
level and the whole system is firmly connected together by
longitudinal and cross pieces notched into each other and
bolted to the piles. On these piles a platform is laid ; or the
soft earth around the top of the piles is scooped out for five
or six feet in depth, and this space filled with concrete.
If a platform is to be used, it is constructed as follows :
A large beam, called a capping 1 , is first placed on the heads
of the outside rows of piles and is fastened to them by iron
bolts, or wooden pins termed treenails. Sometimes an occa-
sional tenon is made on the piles, fitting into a corresponding
mortise in the capping. Other beams are then laid resting
on the heads of the intermediate piles, with their extremities
on the cappings, and are then bolted firmly to the piles
and cappings. Another set of beams are laid at right angles
to these, and are bolted to the piles. Where the beams cross
each other, they are both notched so as to have their upper
surfaces in the same plane. The beams which have their
lengths in the direction of the longer sides of the structure
are known as string pieces, and the other set are termed
cross pieces.
A platform of thick planks is laid upon the upper surface
of the beams and is spiked to them.
The cappings are sometimes of larger size than the other
beams, in which case a rabbet is made in the inner edge so as to
have the platform flush with the upper surface of the capping.
GRILLAGE AND PLATFORM.
289
The whole construction is called a grillage and. platform.
(Fig. 144.)
a p IQ 144 Represents a grillage
and platform fitted on piles.
A, masonry,
a, a, piles.
6, string pieces.
c, cross pieces.
d, capping piece.
e, platform of plank.
/, concrete.
<7, soft soil.
?i. firm soil.
400. When the firm stratum into which the piles have been
driven underlies a soil so soft that there is doubt of the lateral
stability of the piles, the soft soil should be scooped away and
stones should be thrown between and around the piles to in-
crease their stiffness and stability. (Fig. 145.)
FIG. 145 Represents the manner of
using loose stone to sustain piles and
prevent them from yielding laterally.
A, section of the masonry.
B, loose stone thrown around the piles.
401. If the situation be such that decay in the timber is to
be expected, the more costly method of excavation must be
adopted.
The practical difficulty met when trenching in such cases, is
19
290 CIVIL ENGINEERING.
the presence of water in such quantities as to seriously impede
the work, even to the extent often of failure.
Pumps are used to keep the water out, and it may even bo
necessary to enclose the entire area by a sheet-piling. In
this case, two rows of sheet-piles are driven on each side of
the space to be enclosed, through the soft material and into
the firm stratum beneath. The soft material between the rows
is then scooped out, and its place filled with a clay puddling,
forming a water-tight dam around the space enclosed. If the
water comes from springs beneath the dam or from within
the area enclosed, this method will fail, and it may be neces-
sary to resort to some of the methods used for laying founda-
tions under water.
CHAPTER XIL
FOUNDATIONS IN WATER.
402. Two practical difficulties meet the engineer in pre-
paring beds of foundations under water. One is to make the
necessary arrangements to enable the workmen to prepare
the bed ; and the second, having prepared the bed, to secure
it against the deteriorating effects of the water and to preserve
its stability.
Preparation of the bed. The situation in which the bed
is to be prepared may be either of two kinds : one is where it
may be prepared without excluding the water from the place ;
and the other is where the water must be excluded from tho
area to be occupied before the bed can be made.
PREPARATION OP BED WITHOUT EXCLUDING THE
WATER.
403. Concrete beds. A bed of concrete is frequently used
in water. To prepare the bed, the upper layer of loose, soft
soil is removed by a dredging-machine or by other means, and
the site is made practically level. The concrete is laid within
this excavation. A conduit made of wood or iron, or a box
or contrivance which opens at the bottom when jowered in
position, may be used in laying the concrete.
FOUNDATIONS IN WATER.
291
A cylindrical conduit of boiler iron, made in sections of suit-
able lengths which can be successfully fastened on or detached
as the case requires, has been used with success. The lower
end of the conduit has the form of a frustum of a cone. The
whole arrangement is lowered or raised and moved about at
pleasure by means of a crane. The concrete is placed in the
conduit at the upper end, and by a proper motion of the crane
is spread in layers as it escapes from the lower end. By lift-
ing and dropping the apparatus the layers can be compressed.
iBags filled with concrete have been used, with a moderate
degree of success, for the same purpose.
a a
Section on A.B.
FIG. 146.
The object to be attained is to get the concrete placed in
fK>sition in as nearly as possible the same condition as when it
292 CIVIL ENGINEERING.
is made. If it be allowed to fall some distance through water,
or be placed in a strong current, the ingredients of the con-
crete are liable to be separated.
Where the site is in flowing water, it is often necessary to
provide some arrangement which, by enclosing the area of the
site, will calm the water within the enclosure, and will thus pre-
vent its inj urious effect upon the fresh concrete before it has set.
404. The arrangement shown in Figure 146 was used for this
purpose. It consisted of a framework composed of uprights
connected together by longitudinal pieces in pairs ; each pair
being notched on and bolted to the uprights, leaving an interval
through which sheet-piles were inserted. The sheet-piles
were driven into close contact with the bottom, which was
rock. The frame was put together on the shore and then
floated to its place. It was secured in position by inserting
the uprights in holes drilled in the rock. The sheet-piles
c, c'j were then inserted between the horizontal pieces b, >',
and rested on the bottom. The whole area was thus en-
closed by a wooden dam, within which the water was quiet.
The concrete was then laid on the bottom of the enclosed
space. To prevent the sides of the dam from spreading out
iron rods d, d, d , d', were used to connect them.
405. Beds made of piles. Common wooden piles are fre-
quently used to form a bed for the foundation courses of a
structure. They are driven through the soft soil into the
firm stratum beneath, and are then sawed off on a level at or
near the bottom. On these are laid a grillage and platform
or other suitable arrangement to receive the lower courses.
Where the bottom is suitable for driving piles, and there is
no danger of scour to injure their stability, this method is
economical and efficient. The foundation courses must be
placed in position by some submarine process, as by the use
of a diving-bell, or by means of a caisson.
406. Common caisson. This caisson (Fig. 147) is a water-
tight box, whose sides are ordinarily vertical, and which are ca-
pable of being detached after the caisson has been sunk in posi-
tion. The bottom of the caisson, as it is to form a part of the
foundation of the structure, is made of heavy timbers, and
conforms in its construction to that of a grillage and platform.
The size of the timbers for the bottom is determined by the
weight of the structure which is to rest on them, and for the
sides, upon the amount of pressure from the water when the
caisson rests on its bed.
The sides are generally made of scantling, covered with
thick plank. The lower ends of the scantling or uprights fit
CAISSONS.
293
into shallow mortises made in the cap pieces of the grillage.
Beams are laid across the top of the caisson, notched upon the
sides and projecting beyond them. These cross pieces are
connected with the lower beams of the grillage by long iron
bolts, which have a hook and eye joint at the lower end and a
nut and screw at the upper. After the bolts are unscrewed
at the top, they can be unhooked at the bottom, the cross
beams raised, and the sides of the caisson detached.
PIG. 147 Represents a cross-
section and interior end
view of a caisson. The
boards are let into grooves
in the vertical pieces in-
stead of being nailed to
them on the exterior.
a, bottom beams let into
grooves in the capping.
6, square uprights to sustain
the boards.
c, cross pieces resting on b.
d, iron rods fitted to hooks at
bottom and nuts at top.
e, longitudinal beams to stay
the cross pieces c.
A, section of the masonry.
B, bed made 01 piles.
/, guide piles.
In a caisson which was used in building a bridge pier, the
exterior dimensions of the principal parts were nearly as fol-
lows :
The caisson was 63 feet long, 21 feet wide, and 15 feet
deep. The cross beams on top were made 10 inches square
in cross-section, and were placed about three feet apart ; the
uprights were of the same size as the cross pieces, and were
placed about six feet apart.
Much larger caissons have been used, especially in some of
the engineering constructions in England.
The caisson is built at some convenient place where it can
be launched and towed to the position it has to occupy. The
bed having been prepared by levelling off the bottom or by
driving piles, the caisson is floated to and moored over the
spot. The masonry courses are then laid on the bottom of
the caisson, and are built up until the caisson rests on its bed.
Just before it reaches the bed, it is sometimes settled in place,
294: CIVIL ENGINEEEING.
by admitting water into the interior, and an examination
made as to its proper position. If it does not occupy its
proper place, and there is a desire to change the position of
the caisson, the gates by which the water was admitted are
Bhiit and the water is pumped out. The removal of the water
will allow it to float and a rectification of its position may
then be effected.
The caisson having been satisfactorily settled in position,
the masonry is built above the surface of the water, and the
sides are then detached and removed.
Caissons are frequently used whose sides are not detached.
This is especially the case where the sides are of a permanent
character. These might be termed permanent caissons.
407. Permanent caissons. Caissons built with brick
sides and timber bottoms were used to construct the sea-wall
at Sheerness, in England, in 1811-12. After being sunk, they
were filled with concrete.
Rankine mentions a kind that are built wholly of bricks
and cement, arid which are filled with concrete after being
sunk in place.
408. Diving-apparatus. The bed may be prepared as on
dry land, provided some apparatus be used which will admit
of the workmen executing their labors notwithstanding the
presence of the water. Submarine or diving armor and diving-
bells are devices which are frequently used for this purpose.
I. Submarine armor. This is an apparatus to be used by
a single person, and consists essentially of a metallic helmet
from which the water is excluded by atmospheric pressure.
The helmet encloses the man's head ; rests upon his shoulders
and is connected with an air and water-tight dress which he
wears. He is supplied with fresh air forced through a flexible
tube entering at the back of the helmet ; a valve opening out-
wards allows the foul air to escape. To enable him to see,
the helmet is provided with eye-holes protected by strong glass.
II. Diving-bell. The form of diving-bell, commonly used,
is that of a rectangular box with rounded corners. Holes
protected by strong glass about two inches thick are made in
the top to admit light into the interior. Fresh, air is forced
through a flexible tube into the bell by means of air-pumps.
The bell is raised and lowered by means of a crane and
windlass.
A bell, whose dimensions are four feet wide, six feet long,
and five feet high on the inside, is of convenient size for lay-
ing masonry under water.
ihe diving-bell has been much used in laying submarine
WELL FOUNDATIONS. 295
foundations where there was no scour and where the bed was
easily prepared.
409. Pierre perdue. The methods just given are appli-
cable to structures of moderate dimensions, but when the area
occupied by the bed is very considerable, these methods are
either inapplicable or require modifications. One known by
the French as pierre perdue has been frequently used. It
consists in forming an artificial island of masses of loose stone
thrown into the water, and allowing the stone to arrange them-
selves. This island is carried up several feet above the sur-
face of the water and the foundations are built upon it.
The structure should not be commenced until the bed has
fully settled. If there is any doubt about this, the bed should
be loaded with a trial weight, at least twice as great as that of
the proposed structure.
This method can not be used in navigable rivers or other
situations where it is of greater importance not to contract
the water-way.
410. Screw piles. Iron screw piles have been used with
success f oT foundations in localities where the methods already
mentioned were not practicable. They do not differ, in prin-
ciple, from the common wooden pile. Iron piles last well
both in fresh and salt water ; whereas wooden piles can not
be relied upon at all in salt water, and they will not last in
fresh water unless entirely submerged.
Iron screw piles have been much used, in the United States,
in the construction of light-houses on or near sandspits at the
entrance of our harbors and on shoal spots off the coast, where
it would be almost impossible to prepare the beds by any of
the other more usual methods.
411. Well foundations. In India, a method known as
well or block foundations has been quite extensively used,
especially in deep sandy soils. The method consists in sink-
ing a number of wells close together, filling them with
masonry, and connecting them together at top.
The method of sinking one of these wells is to construct a
wooden curb about a foot in thickness ; its cross-section
being the same as that of the well, and to place it in position
on the proposed site. On this curb a cylinder of brickwork
is built to a height of about four feet. As soon as the mortar
has set, the sand is scooped out from under the curb, and it
descends, carrying with it the masonry. When the curb has
settled about four feet, another block or height of masonry is
added, and again the sand is scooped out from under the
curb, and the whole mass descends as before. This process
296 CIVIL ENGINEERING.
is then repeated and carried on until the curb has reached the
required depth. Care must be taken to regulate the excava-
tion so that the cylinder shall sink vertically.
From the very nature of the soil, water is soon met. As
long as the water can be kept out either by bailing or by
pumping, the work proceeds with rapidity. If the water
comes in so fast that it cannot be exhausted by these means,
the sand must be scooped out by means of divers or by some
other method. Under these circumstances the excavation
proceeds slowly and with difficulty.
When the curb reaches a firm stratum, or a depth where
there is no danger of the foundations being affected by the
water, the bottom is levelled, a concrete bed made, and the
interior of the cylinder filled in solid with masonry. If the
concrete bed is made without exhausting the water, the latter
is pumped out as soon as the concrete sets, and the masonry
is then built in the usual manner.
Cylinders of boiler iron have been used in the same way as
the masonry curbs, and are an improvement upon them.
412. Iron tubular foundations. This is a general name
applied to large iron cylinders which are sunk through water
and a soft bottom to a firm soil, and used to support a given
structure in the same manner as common piles. The number
and size of the tubes depend upon the weight to be supported
and the means adopted to sink them.
The method just described for the well is frequently used
for the iron tubes. Brunei, the English engineer, in building
the Windsor Bridge, on the Windsor branch of the Great
Western Railway, employed this method in constructing the
abutments of the bridge. There were in each abutment six
cast-iron cylinders, each six feet in diameter, and they were
sunk to the proper depth by excavating the earth and gravel
for the interior with dredges and by forcing the cylinders
down by weights placed on the top of each one.
The concrete bed in the bottom was made by lowering the
concrete in bags, which were arranged so that by pulling a
rope the bags were emptied under the water in the proper
place. When a sufficient quantity had been put in and had
hardened, the water was pumped out and the cylinders filled
in the usual manner.
This method does not differ in principle from a foundation
on piles, and the same general rules apply as to the amount of
load to be supported and the depth to which the pile is to be
driven.
In some cases a clump of common piles was driven within
COFFER-DAMS. 297
the cylinder at the bottom, and the spaces filled with concrete.
In some of the recent constructions the piles extend to the top
of the cylinder.
PREPARATION OF BED, THE WATER BEING EXCLUDED.
413. There are two cases : where the water is excluded by
means of a dam, and where it is excluded by atmospheric
pressure.
I. EXCLUSION OF WATER BY DAMS.
The dams used are the common earthen or clay dam, the
common coffer-dam, and modified forms of the coffer-dam.
414. Earthen dam. In still water not more than four
feet deep, a dam made of earth or ordinary clay is usually
adopted to enclose the given area and to keep out the sur-
rounding water. This dam is made by digging a trench
around the area to be enclosed and removing the soft material
taken out ; the earth or clay is then dumped along the line of
this trench until it rises one or two feet above the surface of
the water ; as the earth is dumped in place it should be firmly
pressed down, and when practicable, rammed in layers. Any
good binding earth or loam will be a suitable material for the
dam.
The dam being finished, the water within the enclosed area
is pumped out, and the bed and foundations constructed as
already prescribed for those " on land."
415. Coffer-dam. Where the water is more than four feet
deep, and especially if in running water, the common earthen
dam would be generally too expensive a structure, even if it
could be built. In a case of this kind, and where the water
does not exceed twenty-five feet in depth, the common coffer-
dam is usually employed.
The common coffer-dam (Fig. 148) is essentially a clay
dam, whose sides are vertical and retained in position by two
rows of piling.
The common method of constructing the coffer-dam is to
drive two parallel rows of common piles around the area to
be enclosed ; the distance between the rows being equal to
the required thickness of the dam, and the piles in each row
being placed from four to six feet apart.
The piles of each row are then connected by horizontal
298
CTVTL ENGINEERING.
beams, called string or wale pieces, which are notched on
and bolted to the piles on the outside of each row, about one
foot above the highest water mark. On the inside of the
rows, and nearly opposite to the wale pieces, are placed string
pieces of about half the size, to serve as guides
to the sheet-piles.
K"^ ^48-Representa
A section of a
coffer-dam.
a, common piles.
5, wale or string
pieces.
c, cross pieces.
d, sheet piles.
A, puddling.
B, mud and loose
soil.
C, firm soil.
The two rows of piles are tied together by cross pieces
notched on and bolted to the outer wale pieces. Upon these
CIVBS pieces are laid planks to form a scaffolding for the
workmen and their tools, etc.
The sheet-piles are driven in juxtaposition through the soft
soil and in contact with the firm soil beneath. They are
about four inches thick and nine inches wide, and are spiked
to the inner string pieces. Sometimes an additional piece
known as a ribbon piece, is spiked over the sheet-piles.
These rows of sheet-piles form a coffer for the puddling
whence the name of the construction. The sheet-piles having
been driven and secured to the string pieces, the mud ancj
soft material between the rows are scooped or dredged out.
The puddling which forms the dam is then thrown in and
pressed compactly in place, care being taken to disturb the
water as little as possible during the operation. When the
top of the puddling rises to its required height, pumps are
used to exhaust the water from the enclosed area. The in-
terior space being free from water, the bed of the foundation
is prepared as on dry land .
The puddling is composed of clay mixed with sand or
COFFEE-DAMS.
gravel, or of fine gravel alone, freed from all large stones,
roots, or foreign material which may be mixed with it. The
clay is worked into a plastic condition with a moderate
amount of water, and then mixed thoroughly with a given
quantity of sand or fine gravel. Care is taken that there are
no lumps in the puddling after the mixing.
The dam is given the required strength ordinarily by
making the thickness equal to the height of the dam above
the ground or bottom 011 which it is to rest, when this
height does not exceed ten feet. For greater heights the
thickness is increased one foot for every additional height of
three feet.
This rule gives a greater thickness than is necessary to make
the dam water-tight, but adds to its stability. The stability
of the dam is sometimes still further increased by supporting
the sides of the dam by inclined struts, the upper ends of
which abut against the inner row of common piles, and the
lower ends against piles driven for that purpose into the
ground.
416. The principal difficulties met with in constructing a
coffer-dam are as follows :
First, To obtain a firm hold for the common piles ; a dif-
ficult thing to do in deep muddy or rocky bottoms ;
Second, To prevent leakage between the surface of the
ground and the bottom of the puddling ;
Third. To prevent leakage through the puddling ;
Fourth, To exhaust the water from the enclosed area after
the dam is finished.
These difficulties and the expense of construction of the
dam, increase very greatly with the depth of the water. In
deep water, the size and length of the piles and the amount
of bracing required to resist the pressure of the water render
the expense very great.
Common piles can not be efficiently used where the bottom
is rocky. In a case of this kind, the following construction
was successfully used :
Instead of the common piles, two rows of iron rods were
used. These rods were "jumped" into the rock, a depth of
fifteen inches. The sheet-piles were replaced by heavy planks
which were laid in a horizontal position and fastened to the
rods by iron rings. This method of fastening allowed the
planks to be pushed down until each one rested on the one
below it ; the plank resting on the bottom being cut to fit the
surface of the rock.
The frame was strengthened by bolting string pieces of
300 CIVIL ENGINEERING.
timber in pairs on both of its sides and by using inclined
struts upon the interior.
The puddling was of the usual kind and was put in the
darn in the way already described.
417. It will be very difficult to avoid leakage between the
bottom of the puddling and the soil on which it rests unless
the stratum of overlying soft soil be removed. It is therefore
recommended for important works that a part of the dredging
for this purpose be done before the common piles are driven.
Leakage through the puddling is mostly due to poor work-
manship. If the sheet-piles are fitted and carefully driven,
and the puddling is free from lumps and thoroughly mixed,
leakage through the dam should not occur. It is not advisa-
ble to have bolts or rods passing through the dam, as leakage
almost invariably takes place through the holes thus made.
Fine gravel alone has been proved in some cases to be a better
material for the filling than ordinary puddling.
Leakage due to springs in the bottom of an enclosed area
is the great source of trouble, and in some soils is stopped
with much difficulty. It may be necessary to fill in the whole
area with a bed of concrete, and after it has set to pump out
the water.
418. The water having been pumped out, the enclosed
space is drained into some convenient spot in the enclosure,
and arrangements are made to keep the interior dry. The
bed having been prepared, the masonry is then built to the
proper height. When it is above the surface of the water,
the dam may be removed, and as there is danger of disturbing
the bed if the piles were drawn out, it is customary to cut
them off at some point below the water line, letting the lower
ends remain as driven.
419. Caisson dams. This name was given to a coffer-dam
in which the outer row of common piles was replaced by
structures resembling caissons, which were sunk and ballasted
to keep them in position along the line which would have
been occupied by the common piles.
The character of the bottom and the nature of the stream
were such that common piles could not be used for the dam.
The caisson (Fig. 149) was a flat-bottomed boat, which hav-
ing been floated to its place was sunk gradually, by the ad-
mission of water, until it rested on the bottom. A row of
common piles was then placed in a vertical position against
each side of the caisson and lowered until they rested on the
bottom. They were then bolted in that position to the sides of
the caisson. The caisson was then heavily loaded with stones
CAISSON DAMS.
301
and other weighty materials, until a considerable weight rested
on the piles. It is observed, that instead of the piles being
held fast by being driven into the ground, they are held in
place by the sunken boat, and the whole arrangement takea
the place of the outer row of piles in the common coffer-dam.
FlG. 149 Represents a cross-section of a caisson dam.
A, cross-section of caisson. C, puddling.
D, foundation courses of the pier.
To complete the dam, a row of posts, parallel to the inner
row of piles, resting on the bottom and connected by a frame-
work with the caissons, took the place of the inner row of
piles in the common coffer-dam.
The sheet-piles were required only on the one side, the
sides of the caissons being sufficient on the other. They were
laid in a horizontal position, as shown in the figure. The
puddling was in all respects the same as that described in the
previous cases.
The masonry being finished, the loads were removed from
the caissons. They were then pumped dry and the dam re-
moved.
420. Crib-work dam. A dam in which a crib ballasted
with stone takes the place of the common piles, has been used
with success.
In- the example (Fig. 150). the cribs were built by laying
the logs alternately lengthwise and crosswise, and fastening
them together at their intersections by notching one into the
other and pinning them.
302
CIVIL ENGINEERING.
On each crib a platform was laid about midway between
the top and bottom, on which the stone was placed to sink the
crib. The cribs were floated to the place they were to occupy
and sunk gradually by loading stone on the platform. After
they had been fully settled in their place, more stones were
piled on until the required stability was secured.
FlG. 150 Represents a cross-section of a crib-work dam.
A, inner row of cribs. B, outer row of cribs. C, puddling.
Both of the preceding methods were used in constructing
the piers and abutments of the Victoria Bridge, over the
Saint Lawrence, at Montreal. A rocky bottom, covered with
boulders, prevented the driving and the use of the common
pile as in the ordinary method. There was also in the river a
swift current, which in the spring of the year brought down
large quantities of ice, the effect of which would have been
to have destroyed any ordinary caisson or common coffer-dam.
It is seen that these dams do not differ in principle from
the common coffer-dam, and that the modifications in each
case consisted in finding for the common pile a substitute
which would be stronger and equally effective.
H. EXCLUSION OF WATER FROM THE SITE BY ATMOSPHERIC
PRESSURE.
4:21. In recent years, the use of compressed air has been ex-
tensively adopted as a means for excluding the water from the
site of a proposed work, while the bed was being prepared.
There are two general methods of its application : in the
pneumatic pile and in the pneumatic caisson.
422. Pneumatic piles. Pneumatic piles are hollow verti-
cal cylinders of cast iron, from six to ten feet in diameter,
intended to be forced through soft and compressible materials
to a firm soil beneath, and to be then entirely filled with
PNEUMATIC PILES.
303
masonry or concrete or other solid material. Rankine classes
them under the head of iron tubular foundations.
Their general construction and the mode of sinking them
in the soil are shown in Fig. 151.
Fio. 151 Represents vertical sec-
tion of a pneumatic pile.
A, body of cylinder.
B, the bell.
C, elevation of air-lock.
D, vertical section of air-lock.
E, water discharge pipe.
M, windlass on inside.
N, windlass on the top.
O, O, buckets ascending and de-
scending.
W,W, iron weights.
In this example, shown in the figure, the cylinders were
cast in lengths of nine or ten feet, with flanges on the interior
at each end. These pieces were united by screw bolts passing
through holes in the flanges, the joints being made water,
tight either by an india-rubber packing or by a cement made
or iron turnings.
To sink a pile of this kind, a strong scaffolding is erected
over the site, and from which the lengths of the cylinders can
be lowered and placed in position. On this scaffold a steam-
engine is ordinarily placed, and furnishes the power required
during the operation.
The lower edge of the lowest section of the cylinder is
sharpened so that it may sink more easily through tfre soiL
304 CIVIL ENGINEERING.
The upper section, termed the "bell," is usually made of
boiler iron, with a dome-shaped or flat top. An " air-lock " is
used to pass the men and materials in and out of the cylinder.
In this example there were two air-locks, which were placed
in the top of the bell, as shown in the figure. Each lock had
at the top a trap door which opened downwards, and at the
side a door which opened into the interior of the pile. Stop-
cocks were provided in each, communicating with the ex-
ternal air and the interior of the pile, respectively ; they
could be opened or closed by persons inside the tube, within
the lock, or on the outside.
The bell was provided with a supply pipe for admission of
compressed air, a pressure gauge, a safety valve, a large escape
valve for discharging the compressed air suddenly when
necessary, and a water-discharge pipe about two or three
inches in diameter.
Windlasses placed within the cylinder and on the outside,
as seen in the figure, were used to hoist the buckets employed
in the excavation
The first operation in sinking the pile was to lower the
lowest section, with as many additional lengths united to it as
were necessary to keep the top of the cylinder two or three
feet above the surface of the water, until it rested on the
bottom. The bell and one additional length were then bolted
to the top of the pile.
The weight of the mass forced it into the soil at the bot-
tom of the river a certain distance, dependent upon the na-
ture of the soil. As soon as the pile stopped sinking, the air
was forced in by means of air-pumps worked by the steam-
engine, until all the water in the tube was expelled. Work-
men, with the proper tools, then entered the cylinder by
means of the air-locks.
To get into the pile, the men entered the lock, closed all
communications with the external air, and then opened the
stop-cock communicating with the interior of the pile ; in a
few minutes the compressed air filled the lock, the men opened
the side door and thus effected an entrance into the interior.
To pass out it was only necessary to reverse this operation.
The gearing of the hoisting apparatus was so arranged that
the buckets, when filled, were delivered alternately into the
locks, and were then hoisted out by the windlass on the out-
side.
Care was taken to guard against the uplifting force of the
compressed air within the pile. In the above example, a
heavy weight, composed of cast-iron bars resting on brackets
PNEUMATIC PILES. 305
attached to the outside of the bell, was used to resist this
action.
The workmen having descended to the bottom of the pile,
excavated the material to the lower edge ; they then took off
the lowest joint of the water discharge pipe and carried it
and their tools to the bell, and passed out of the lock. The
valve for admitting compressed air was then closed and the
large escape valve opened, allowing the compressed air to
escape. The cylinder being deprived of the support arising
from the compressed air, sank several feet into the soil, the
distance depending on the resistance offered by the soil.
When the pile had stopped sinking, the escape valve was
closed, the air forced in, and the operations just described
continued. Great care was taken to keep the pile in a verti-
cal position while sinking.
The pile, having reached the required depth, was then filled
with concrete.
The usual method of filling the pile is to perform about
one-half of the work in the compressed air and then remove
the bell and complete the rest in the open air. In filling with
concrete, it should be well rammed under the flanges and
around the joints.
423. This description of a pneumatic pile, just given, is
that of one of the piles used in the construction of a bridge
over the river Theiss, at Szegedin, in Hungary.
The river, at this point, has a sluggish current with a gra-
dual rise and fall of the water, the difference between the
highest and lowest stages of water being about twenty-six feet.
The soil of the bottom is alluvial, composed to a great depth
of alternate strata of compact clay and sand.
The piles were sunk to about thirty feet below the bottom of
the river, which latter was about ten feet deep at low water.
The excavation was carried down to within six feet of the
bottom of the pile. Twelve common piles of pine were then
driven within the cylinder, extending to a depth of twenty
feet below it. The concrete was then thrown in and rammed
in layers until its upper surface was on a level with that of
ordinary low water.
The air-locks were about six feet and a half high and two
and three-quarters in diameter.
424. In the first uses of the pneumatic piles, the cylinders
were of small size, as many being sunk as were required to
support the load, as in the use of common piles.
They wero sunk into the soil by exhausting the air from
the interior. The result following this removal of air was tint
20
306 CIVIL ENGINEERING.
the earth immediately under the pile was forced togethei with
water, into the inside of the cylinder, and the pile sank into
the opening thus made, both under its own weight and the
pressure of the atmosphere.
This process is known as Dr. Pott's, and is well adapted to
soft or sandy soils, when free from stones, roots, pieces of tim-
ber, etc. The presence in the soil of any obstacle which the
edge of the tube cannot cut through or force aside, renders
this method impracticable.
The next step was to increase the size of the pile, and in-
stead of exhausting the air, to fill it with compressed air.
The top being closed and the bottom open, all fluid matter
was driven from the interior of the pile by the compressed
air. By means of air-locks on the top of the cylinder, work-
men were enabled to descend and remove the soil and such
obstructions as prevented the pile from sinking. This pro-
cess is generally known as " Triger's."
The air being compressed in the interior of the pile, the
weight or the pressure downward was much lessened. To
increase the pressure a weight was placed on the pile.
Although many improvements have been made in the de-
tails, the arrangements just described illustrate the general
outline of all the pneumatic methods in use.
425. Pneumatic method used by Mr. Brunei. The first
improvement in the pneumatic method was that used by Mr.
Brunei in preparing the bed for the centre pier of the Koyal
Albert Bridge, at Saltash, England.
This improvement consisted in confining the compressed
air to a chamber at the bottom of a cylinder, the rest of the
space inside of the cylinder being open to the air. The air
chamber communicated with the outside air by means of a
tube, six feet in diameter, with air-locks at the upper end.
Outside of this tube, was another tube, ten feet in diameter,
connecting the dome with the outside air. (Fig. 152.)
A dome, about 25 feet high, was built in the lower portion,
BO arranged that the top of the dome should be above the
mud when the cylinder rested on the rock.
The chamber for the compressed air was annular, four
feet wide, twenty feet high, was built around the inner cir-
cumference of the lower edge and was divided into eleven
compartments by vertical and radial partitions; apertures
in the partitions afforded communications from one to the
other. An air passage at the top of the compartments con-
nected them with each other, and with the vertical tube of
six feet diameter before alluded to.
PNEUMATIC PILES.
307
The cylinder was lowered into the water exactly over the
place it was to occupy. As soon as it stopped sinking, the
annular chamber was shut off from the rest of the dome, the
air forced in, the water driven out, the workmen descended
and dug out the mud and loose soil under the edge.
FIG. 152 Represents a longitudinal section
through the axis of the cylinder. The
cylinder was 37 feet in diameter, about
100 feet high, made of boiler iron, and
weighed nearly 300 tons. The rock on
which it was to rest was about 90 feet
below the surface of the water, overlaid
with about 20 feet of loose sand and mud.
The rock surface had a slight slope, to
which the bottom of the cylinder was
made to fit.
When the rock was reached, a level bed was cut in its sur-
face and a ring of masonry built. The water was then pumped
out of the main tube and the masonry begun on the inside.
As the masonry rose, the partitions, shaft, and the dome were
removed. When the pier was above the surface of the water,
the upper part of the cylinder, about fifty feet in length, was
unbolted and taken away, it having been made in two sections
for this purpose.
As the volume of the annular chamber in which the com-
pressed air was used was small in comparison with the vol-
ume of the main cylinder, no extra weight was needed to
balance the upward pressure.
The above is a good example of the pneumatic process
combined with the principle of the coffer-dam.
308 CIVIL ENGINEERING.
426. Pneumatic caisson. The next important modifica-
tion in the pneumatic method was to combine the principle
of the diving-bell with that of the common caisson. This com-
bination is known as the pneumatic caisson and furnishes
the means now most commonly used in situations like that at
the Saltash bridge, and especially where the foundations have
to support a great pressure.
It consists essentially of three parts: 1st, The caisson;
2d, The working chamber ; and 3d, The pneumatic ap-
paratus and its communications with the working chamber.
Caisson. This does not differ in its principles of construc-
tion from the common caisson already described. The bottom
is of wood or iron, made strong enough to support the struc-
ture with its load, and forms the roof of the working chamber.
The sides are generally of wrought iron, and are not usually
detached from the bottom when the structure is finished.
Working chamber. This is below the caisson, and as just
stated, the bottom of the caisson is the roof of the chamber.
Its sides are firmly braced to enable it to resist the pressure
from both the earth and water as it sinks into the ground.
The chamber is made air and water tight.
Pneumatic apparatus and communications. Vertical
shafts, either of iron or masonry, passing through the roof of
the chamber furnish the means of communication between
the working chamber and the top of the caisson. The air-
locks may be placed in the upper end of the shaft, as in the
pneumatic pile, or at the lower end of the shaft where it con-
nects with the working chamber.
The usual supply pipes, air-pumps, discharge pipes, etc., are
required as in the other pneumatic methods.
Sinking the caisson. It is moored over the place it is tc
occupy and is sunk gradually to the bottom as an ordinary cais-
son. Air is then forced into the working chamber, driving
out the fluid matter; the earth and loose material are then dug
out, while the caisson settles slowly under its own weight and
that of the masonry until it rests on the firm soil or solid rock.
An outline description of some of the caissons recently
used will more fully illustrate their construction and the
method of sinking them.
427. Pneumatic caissons used at L'Orient, France.
These were used in laying the foundations of two of the piers
of a railroad bridge over the river Scorff, at L'Orient, in
France. The river bed consisted of mud from 25 to 45
feet deep, lying upon a hard rock. The surface of the water
was about 60 feet above the rock at mean tide, and 70 feet al
PNEUMATIC CAISSONS.
309
high tide. It was essential for the stability of the piers that
they should rest on the rock.
The caissons used were 40 feet long, 12 feet wide, and
made of boiler iron.
The thickness of the iron forming the sides of the caisson
varied according to the depth in the water, being greater for
the lower than for the middle and upper parts. The ratio of
the thickness was for the upper, middle, and lower, as 3, 4,
and 5.
The working chamber was ten feet high and communicated
with the upper chamber or bells, where the air-locks were
E laced, by two tubes for each bell ; these tubes were each two
3et and three-quarters in diameter. Each bell was ten feet
high and eight feet in diameter, and contained two air-locks
and the necessary hoisting gear ; the full buckets ascended
through one tube and descended through the other.
Fig. 153 shows the caisson used for the pier on the right
bank.
FIG. 153 Represents a vertical section
of caisson and masonry of pier during
the process of sinking.
A, the working chamber.
B, interior elevation of caisson.
C,C, elevation of the bells.
D,D, the communicating tubes.
E,E, masonry of pier, built as the caison
was sinking.
When the rock was reached, its surface was cleaned off
and a level bed made under the edges of the caisson. The
working chamber was then filled up to the roof with ma-
sonry. *
The pier was of concrete with a facing of stone masonry,
and built up as the caisson was sinking to its place.
The working chamber being filled, the tubes were with-
drawn and the spaces occupied by them filled with con
crete.
310
CIVIL ENGINEERING.
Pneumatic Caissons at St. Louis, Mo.
428 At the time the foundations of the piers of the bridge
over the Mississippi River, at St. Louis, were laid, the caissons
there used were the largest that had ever been employed for
such a purpose.
This bridge consists of three spans, supported on two piers
PlG. 154 Represents a section of the caisson used in construc-
tion of east pier of the bridge over the Mississippi
River, at St. Louis, Mo.
A, main shaft. B, air-locks. C, working chamber.
D, sides of caisson. E, side shafts. F, sand pumps.
G, discharge of sand.
and the abutments. The river at this point is 2,200 feet wide
at high water, with a bed of sand over rock. The rock slopes
from the west to the east, the upper surface of the sand being
practically level. The depth of the sand on the western shore
was about 15 feet, and on the eastern nearly 100 feet.
PNEUMATIC CAISSONS. 311
As the scour on the bottom is very great in the Mississippi
River, it was regarded as essential that the piers should rest on
the rock. To penetrate this sand and lay the foundations on
the rock, the pneumatic caisson was used.
Fig. 154 represents a section of the one used for the east
pier. There the rock was 128 feet below the high- water
mark. When the caisson was moored in position there was
above the rock 35 feet of water and 68 feet of sand.
The plan of the caisson was hexagonal, the long sides beincj
50 feet each, and the short ones 35 feet each. The sides of
the caisson were made of plate iron, three-eighths of an inch
in thickness, and built up as the caisson sank.
The bottom, which was to support the masonry, was com-
posed of iron girders, placed 5 feet apart. Iron plates,
\ inch thick, were riveted to the under side of these girders
to form the roof of the working chamber. The sides of the
caisson, prolonged below the girders, formed the sides of the
chamber, and were strongly braced with iron plates and stif-
fened by angle irons. The chamber, thus formed, was 80
feet long, 60 feet wide, and had an interior height of 9 feet.
The interior space was divided into three, nearly equal, parts
by two heavy girders of timber placed at right angles to
those of iron, and intended to rest on the sand and assist
in supporting the roof of the chamber. Openings made
through the girders allowed free communication between the
divisions.
Access to the top of the caisson was obtained by vertical
shafts lined with brick masonry, and passing through the roof
of the chamber. The air-locks were at the lower end of the
shafts and within the chamber.
As the caisson descended, the masonry pier was built
up in the usual manner, its foundation resting on the iron
girders.
In the chamber were workmen who excavated the sand,
and shovelled it under the sand-pumps. (Fig. 154.) A
pump of 3 inches diameter, working under a pressure of 150
pounds on the square inch, was capable of raising 20 cubic
yards of sand 125 feet per hour.
When the caisson reached the rock, the latter was cleared
of sand and the entire chamber then filled with concrete.
The experience acquired in sinking this caisson enabled the
engineer to make material modifications in the details of the
caissons subsequently used.
The health of the workmen was greatly affected by the
high degree of compression of the air in. which they had to
312 CIVIL ENGINEERING.
work. In some cases the pressure was as high as fifty pounds
on the square inch, and several lost their lives in consequence.
In the second pier, instead of filling the chamber entirely
with concrete when the rock was reached, the space around
the edges was only closed with concrete and the cnamber was
then filled with clean sand.
Pneumatic Caisson at St. Joseph, Mo.
429. This was used in 1.871-2 in laying the foundations of
the piers for a railroad bridge over the Missouri River, at St.
Joseph, Mo.
For a reason similar to that given in the last case, it was
decided to rest the piers on the rock below the bottom of the
river. The rock was about sixty-seven feet below the level
of high water, and was overlaid with mud and sand to depths
varying from forty to the whole distance of sixty -seven feet.
Six piers were used and were placed in depths of water vary-
ing at the low stage from zero to twenty-five feet ; the differ-
ence between high and low water being twenty-two feet.
Pockets of clay, with occasionally snags and boulders, were
met with in the sand and mud.
The caisson used for pier No. 4 was made of twelve-inch
square timber, and was at the bottom fifty-six feet long, and
twenty-four feet wide. The sides of the working chamber
were three feet thick, sloping inwards with a batter of
ig-. It was built by placing a row of timbers in a vertical
position, side by side, for the outside ; then, inside of this,
a second row was laid horizontally ; and then, for the inside,
a third row in a vertical position. The outer row extended
one foot below the middle row, and the latter one foot
below the third. A horizontal beam extending entirely
around the interior was bolted to the sides of the chamber,
one foot above the bottom of the inside row. A set of in-
clined struts rested on this beam, and abutted against strain-
ing beams framed into the roof of the chamber. The roof
was solid timber, four feet thick, on which rested the grillage
for the masonry of the pier. The grillage was made of tim-
ber, seven courses thick, each course being laid at right
angles to the one below it. The timbers of each course were
separated by a space of six inches, excepting the top course,
which was solid.
All the timber work was accurately fitted, and the whole
PNEUMATIC CAISSONS. 313
bolted together so as to form one unyielding mass. The
interior of the working chamber was calked, and was prac-
tically air-tight. The dimensions of the chamber were, on
the inside, twenty-two feet wide and fifty-four long at the
bottom : five feet wide and seven feet long at the top ; and
nine feet high at the centre. The grillage was drawn in so
that its top was of the same dimensions as the base of the
gier, being nine feet wide and twenty long, with curved star-
ngs at each end.
The air-lock was four feet in diameter and seven high,
made of plate iron, and placed in the middle of the top of
the chamber. A door in the top of the air-lock opening
downwards communicated with a vertical iron shaft three feet
in diameter; the shaft extended above the top of the ma-
sonry and allowed access to the top of the caisson. An iron
ladder in the shaft was used for ascent and descent. The
usual supply and discharge pipes passed through the grillage
to the working chamber.
The caisson was sunk by the process previously described.
The arrangement of the lower bearing surfaces of the cais-
son are regarded as worthy of notice. The lower edge of
the outside row of timbers was sharpened ; as soon as it had .
sunk one foot, the under surface of the second or horizontal
row came into play, adding a foot of bearing surface. When
the caisson had descended two feet, the bottom of the inside
or third row pressed on the soil, thus giving three feet of
bearing surface. By this arrangement the amount of bearing
surface was under the control of the engineer. If the soil
through which the caisson was sinking was variable in its na-
ture, that is, if on one side of the caisson it was soft, and on
the other it was hard, the bearing surface could be increased
on the soft side and diminished on the other. In this way
the caisson could be kept vertical while sinking.
The greater part of the material excavated was mud or
sand, and was discharged easily arid rapidly by means of
sand pumps. The clay, boulders, and snags were discharged
through the air-lock.
The caisson was sunk at the rate of from five to seven feet
in twenty-four hours.
When the caisson reached the bed rock, a wall of concrete,
six feet thick, was built on the rock under the edges, and
was solidly rammed under the three rows of timbers and up
to and including the horizontal beam supporting the struts.
Strong vertical posts were placed under the roof to assist in
supporting it. The sand pumps were then reversed, and the
314 CIVIL ENGINEERING.
chamber was filled with clean sand and gravel. A tube was
so placed as to allow the escape of the water in the sand, so
that the whole interior was compactly filled with solid mate-
rial. The sand pumps were then withdrawn, and the shafts
themselves were filled.
Caissons of the East River Bridge at New York.
430. The caissons used for the foundations of the piers in
this bridge were rectangular in form, and made of timber.
The exterior of the bottom of the chamber in the Brooklyn
caisson was 168 feet long and 102 wide. In the one on the
New York side the width was the same, but the length was
four feet greater.
Both were nine and a half feet high on the inside. The
roof of the Brooklyn caisson was a solid mass of timber, fif-
teen feet thick (Fig. 155), and of the New York caisson,
twenty-two feet thick.
FlG. 155 Represents section through water shaft of the Brooklyn caisson,
showing- method of removing boulders or other heavy materials.
The sides of the caisson had a slope of -L- - for the outer
face, and of -J- for the inner, as shown in the figure. The outer
slope was for the purpose of facilitating the descent of the
caisson into the ground. The lower edge was of cast iron,
protected by boiler iron, extending up the sides for three feet.
The sides, where they joined the roof, were nine feet thick.
The chambers were calked both on the outside and inside, to
make them air-tight. As a farther security, an unbroken
sheet of tin extended over the whole roof between the fourth
and fifth courses, and down the sides to the iron edge. The
New York chamber was, in addition, lined throughout on the
inside with a light iron plate, to protect it from fire.
MOVABLE PNEUMATIC CAISSON. 315
Each chamber was divided by five solid timber partitions
into six compartments, each from twenty-five to thirty feet
wide. Communication from one to the other was effected by
doors cut through the partitions.
The air-locks were placed in the roof, projecting into the
chamber four feet, and communicating at the top with
vertical shafts of iron, built up as the caisson descended.
The locks were eight feet high and six and a half feet in dia-
meter.
The mud and sand were discharged through pipes by the
compressed air. A pipe, three and a half inches in diameter,
discharged sand from a depth of sixty feet at the rate of one
cubic yard in two minutes, by the aid of the compressed air
alone.
The heavy materials were removed through water shafts.
These were seven and three-quarter feet in diameter, open at
the top and at the lower end, the latter extending eighteen
inches below the general level of the excavation. A column
of water, in the shaft, prevented the compressed air from
escaping.
Tne material to be removed through the water shaft was
thrown into an excavation under the lower end of the shaft ;
it was there grasped by a " grapnel bucket," which was low-
ered through the shaft, and hoisted through the water to the
top of the shaft, where it was removed.
After the caisson had reached the rock, the chamber was
filled with concrete, in the usual manner.
The great thickness of the roof, and the moderate depth of
water, enabled the engineer to dispense with the use of sides
to the caisson, as the masonry could be kept always above the
surface of the water.
Movable Pneumatic Caisson.
431. A pneumatic caisson has been successfully used in
laying the foundations of piers of bridges, which differs from
those already described, in its construction admitting of its
being moved after completion of one pier, to another place for
the same purpose. It was an iron cylinder, ten feet in dia-
meter (Fig. 156), connected at its lower end with a working
chamber, eight feet high and eighteen feet in diameter. On
the roof of the latter was another chamber, annular in form,
eighteen feet in diameter and about six feet high, so arranged
as to allow of being filled with water when any additional
weight was necessary, and being emptied of water and its
316
CIVIL ENGINEERING.
place supplied with compressed air when less weight was de-
sired. On top of this annular chamber was a similar one ar-
ranged to be loaded with iron ballast. Strong chains attached
to the roof of the working chamber and connected with a
hoisting apparatus, placed on a strong scaffolding over the
site of the pier, were used to lower and lift the cylinder, as
necessity required.
FIG. 156 Represents section of moT-
able pneumatic caisson.
B, working chamber.
A, chamber for water, or for compressed
air.
W, chamber for iron ballast,
c, c, elevation of lengths of the iron
cylinder.
Air-locks, air-pumps, and all the necessary adjuncts of a
pneumatic pile, were provided and used. Having reached the
rock or firm soil, the bed and the foundations were con-
structed as already described. As the masonry of the pier
rose, the whole apparatus was lifted by the chains and hoist-
ing apparatus, the cylinder being lightened by expelling the
water from the chamber, A, and filling the latter with com-
pressed air. The masonry of the pier having risen above the
surface of the water, the whole apparatus was removed and
used in another place.
432. Remark. It is seen that the pneumatic caisson, as
before stated, is simply a combination of the diving-bell with
the common caisson, the diving-bell being on a large scale,
and its roof being intended to form a part of the bed of the
foundation.
Experience has shown that the large caissons are more
easily managed than the small ones. The circumstances of
the case can only decide as to which is preferable, the caisson
or the pneumatic pile. Either method is an expensive one,
PROTECTING THE FOUNDATION BED. 317
and is only employed in localities where the others are not
applicable.
SECURING THE BED FROM THE INJURIOUS ACTION OF
WATER.
433. The bed of a river composed of sand or gravel is liable
to change from time to time, as these materials are moved
by currents in the river. This change, when accompanied by
an increase in depth of the river, is known as the " scour."
Sometimes a scour will occur on one side of a structure and
not on the other, producing an undermining threatening the
stability of the masonry. Where common piles have been used,
they have occasionally been washed out by this action. Even
in rocky bottoms, when of loose texture, the rock will gradu-
ally wear away under the action of currents, unless protected
It therefore becomes an important point to provide security
for the beds in all soils liable to any change. It is for this
reason that in very important structures, the foundations are
placed on the bed rock far below the possible action of cur-
rents, and so arranged that even if they should be exposed to
a scour they would be safe. This requirement has caused the
free use of the pneumatic methods.
Various expedients have been used to secure the beds where
they do not rest on the rock or on a soil below the action of
the water. A common method is to rip-rap the bed, that is,
to cover the surface of the bottom, around the bed, with frag-
ments of stone too large to be moved by the currents, and if
the soil is a sand or loose gravel, to use clay in connection
with the stone to bond the latter together.
Where the bed is made of piles, it is well to enclose the
piles by a grating of heavy timber, before throwing in the
stone. In some cases the foundations are boxed, that is, the
piles are enclosed by a sheeting of planks, or by other device,
BO as to protect them from the scour.
PART VI.
BRIDGES.
CHAPTER XIII.
434. A "bridge is a structure so 'erected over a water-course,
or above the general surface of the ground, as to afford a con-
tinuous roadway between the opposite sides of the stream, or
above the surface of the country, without obstructing those
lines of communication lying beneath.
Such a structure, thrown over a depression in which there
is ordinarily no water, is generally called a viaduct.
If the structure supports an artificial channel for conveying
water, it is known as an aqueduct; and where it crosses a
stream, it is frequently called an aqueduct-bridge.
Bridges may, for convenience of description, be classed
either from the materials of which they are made: as
masonry or stone, iron, wooden bridges, etc. ; or from the
character of the structure : as permanent, movable, float-
ing bridges, etc. ; or from the general mechanical principles
employed in arranging its parts : as arched, trussed, tubular
bridges, etc.
435. Component parts. A bridge consists of three es-
sential parts :
1st, The piers and abutments on which the superstruc-
ture rests ; 2d, the frames or other arrangements which sup-
port the roadway ; and 3d, the roadway, with the parts used
in connection with it for its preservation or to increase its
security, as the roof, parapets, etc.
Bridges are of various kinds, both in their general plan and
dimensions. The latter are dependent upon the objects of
and the circumstances requiring the erection of the bridge.
The simplest bridge is one in which the points of support
PIERS AND ABUTMENTS. 319
are so near together that two or more simple beams laid
across the stream, or across an opening to be passed over, are
sufficient for the frame ; a few planks laid upon the beams
may then form the roadway.
The supports being strong enough, the proper dimensions
for the beams and for the planking are easily determined.
This calculation for the beams is made under the hypo-
thesis that each is a simple beam, resting on two points of
support at the extremities, strained by a load uniformly distrib-
uted over it, and also by a weight acting at the middle point.
The uniform load is the weight of the structure, ordinarily
assumed to be uniformly distributed in the direction of its
length. The weight at the middle represents the heavy body
as it passes over ; as, for example, a heavily loaded wagon for
a common, and a locomotive for a railroad bridge. Having
determined what this weight shall be, its equivalent uniform
load may be obtained, and added to that already assumed ; or
if preferred, the uniform load may be replaced by its equiva-
lent weight at the middle.
If the number of these beams be represented by yi, and we
suppose that they are at equal distances apart, then the total
load on the bridge divided by n will give the load on each
beam. Then by formulas already deduced we can, knowing
the value for R, determine the proper breadth and thickness
for each beam.
436. Platform of roadway. In a common wooden bridge
the roadway is generally of planks. These are of hard wood,
from three to four inches thick, resting on longitudinal
pieces placed from two to three feet apart from centre to
centre. This thickness of plank is greater than is required
for strength, but has been found necessary to enable the road-
way to withstand the shocks, friction, and wear due to the
travel over it.
If the longitudinal pieces which rest directly on the sup-
ports are too far apart to allow the plank to rest safely upon
them, cross pieces, called roadway bearers, are placed upon
the longitudinal pieces. On these cross pieces other longitu-
dinal pieces, called joists, are placed close enough together,
and the planking is laid upon the joists.
The particular kind and width of roadway will depend
upon the character of the travel over the bridge. Knowing
these, the weight per unit of length is quickly determined.
437. Piers and abutments. Walls should be built to
support the ends of the beams. These walls may be of stone,
wood, or iron. Those placed at the ends of the bridge are
320 CIVIL ENGINEERING.
called abutments ; the intermediate ones are termed piers ;
the distance or space between any two consecutive piers is
called a span, and sometimes a bay.
If the frame of the bridge is of a form that exerts a lateral
thrust, as, for instance, in an arch, the abutments and piers
must be proportioned to resist this thrust.
As the foundations are exposed to the action of currents
of water, precaution must be taken to secure them from any
damage from this source. The piers and abutments must also
be guarded against shocks from heavy bodies and against the
damaging effects of floating ice.
438. Wooden piers and abutments. Wooden abutments
may be constructed of crib-work. The crib is ordinarily
formed of square timber or logs hewn flat on two of their
opposite sides. The logs are halved into each other at the
angles, are fastened together by bolts or pins, and are some-
times further strengthened by diagonal ties. The rectangular
space thus enclosed is filled with earth or loose stone. V ery
frequently the crib is built with three sides only. Another
way of constructing the abutment is to make a retaining wall
of timber by which the earth of the bank is held up.
The piers also are sometimes made of cribs. The cribs are
floated to the spot, sunk in place, filled with stone, and built
up to the proper height. There are serious objections to
their use for piers, and they are recommended only where no
injurious results will follow their adoption, and where it is
not expedient to employ some one of the other methods.
The pier made of piles is the most common form of the
wooden pier. It is constructed by driving piles from three
to six feet apart, in a row, parallel to the direction of the
current. The piles are then cut off at the proper distance
above the surface of the water, and capped with a heavy
piece of square timber. If the piles extend some distance
above the water, they must be stiffened by diagonal braces.
In some cases the piles are cut off, at or just below the level
of the water, so that the capping piece will always be kept
wet. Mortises are made in this cap into which uprights are
fitted ; the uprights taking the place of the upper parts of the
piles in the preceding case. Or, what is more common, a
trestle made in the form of an inverted W is fitted on this
cap, and the upper side of this trestle is capped with a square
piece of timber.
Where the bottom is hard and not liable to " scour," the
piles are dispensed with and the trestle alone is used. In
this case the piece on which the trestle rests is laid flat on the
FENDERS AND ICE-BEE AKER8.
321
bottom and is called the mud-sill. The upper part of the
trestle is capped as before, and if necessary to get additional
height another trestle is framed on top of this.
439. Fenders and ice-breakers. Wooden piers are not
constructed to resist heavy shocks from floating bodies. In
positions exposed to such shocks, fenders should be built. A
clump of piles driven on the exposed side of the pier, oppo-
site to and some distance from it, will be a sufficient protec-
tion against ordinary floating bodies when the current is
gentle. The piles should be bound together so as to increase
their resistance; this may be done by wrapping a chain
around their heads. If there is danger from floating ice, an
inclined beam (Fig. 157), protected by iron, should be used
to break up the ice as it moves towards the pier.
Elevation.
FIG. 157. Plan.
In rapid currents, where the ice is thick, a crib-work
square in plan, with one of the angles up-stream, has been
used. The crib was filled with heavy stone and the up-stream
angle was gjven a slope and was protected by a covering of
iron.
The construction shown in Fig. 158 is a good one. Its re-
sisting power is increased by filling the interior with stone.
440. Masonry piers and abutments. The methods,
described in the chapters on masonry and foundations, are
applicable to the construction of piers and abutments.
Since they are, from their position, especially liable to
damage from the action of currents, both on the soil around
them and on the materials of which they are made, particular
attention should be paid to their construction.
21
322
CIVIL ENGINEERING.
In preparing the bed, a wide footing should be given
to the foundation courses, if the soil is at all yielding, and
whenever this footing does not rest on rock, means should be
taken to secure the bed from any injurious action of the
water.
Elevation.
FIG. 158. Plan.
The piers, although they are generally built with a slight
batter, may be built vertical. The thickness given them is
greater than is necessary to support the load which is to be
placed upon them, in order that they may better resist the
shocks from heavy floating bodies and the action of the cur-
rents to which they are continually exposed.
FlG. 159. A, horizontal sections of starling.
B, same of pier.
They should be placed, if possible, so that their longest
dimensions should be parallel to the direction of the current
They should have their up and down-stream faces either
FENDERS AND ICE-BREAKERS.
323
curved or pointed, to act as cut- waters turning the current
aside, and preventing the formation of whirls, and to act as
fenders.
These curved or pointed projections are called starlings.
Of the different forms of horizontal section which have been
given them (Fig. 159), the semi-ellipse appears to be the most
satisfactory.
Their vertical outline may be either straight or slightly
curved. They are built at least as high as the highest water
line, and finished at the top with a coping stone called a
hood.
In streams subject to freshets and to floating ice, the up-
stream starlings are provided with an inclined ridge to
facilitate the breaking of the ice as it floats against and by
them. Where very large masses are swept against the piers,
FIG. 160 Represents longitudinal section, elevation, and plan of a piei
of the Potomac aqueduct bridge.
A, A, up-stream starling, with the inclined ice-breaker D, which rises from
the low-water level above that of the highest freshets.
B, down-stream starling.
E, top of pier.
F, horizontal projection of ice-breaker.
it is not unusual to detach the ice-breakers and place them in
front of the piers, as is generally done in the case of wooden
piers.
Fig 160 represents the ice-breaker planned and constructed
324: CIVIL ENGINEERING.
by Colonel Turnbull, of the Topographical Engineers, United
States Army, for the piers of the Potomac aqueduct bridge
of the Alexandria Canal, at Georgetown, D. C.
The pier was at the bottom 66.6 feet long and 17.3 thick,
and terminated by starlings whose horizontal cross-section
was circular. The pier shown in the drawing was 61 feet
high, and built with a batter of f.
The starlings were built up with the same batter, except
that the up-stream one, when at the height of 5 feet below
the level of high water, received an inclination of 45, which
it retained until 10 feet above it. From there to the top it
had the same batter as the rest of the pier. The two lower
courses of the ice-breaker were 22 inches thick, the rest being
18 inches. The stones were laid in cement, and no stone was
allowed in the ice-breaker of a less volume than 20 cubic
feet.
The ice brought down by the river 'at this point is often 16
inches thick, and the current is often six miles an hour. On
such occasions the ice is forced up the ice-breakers to a
height of 10 or 12 feet. The ice breaks by its own
weight, and passes off between the piers without doing any
harm.
Probably the ice-breakers of the International Bridge, over
the Niagara River, at Buffalo, are more severely tested than
any in our country. They are triangular in plan, have a
slope of i, and are protected by iron plating.
441. Iron piers and abutments. Until a very few years
ago all piers were made either of masonry or timber. Where
a solid bed could not be reached by excavation, piles were
driven, their tops were sawed off, and on them a grillage and
platform was placed to form the bed.
The substitution of iron for wood in many engineering
structures, soon led to the use of iron in the above class of
constructions.
Iron is used in the construction of piers and abutments in
various forms as follows :
1. As piles or columns, wholly of iron ; as screw piles.
2. As a hollow column, open at the bottom, and partly or
entirely filled with concrete ; the weight of the bridge resting
on the iron casing.
3. As a cylinder, entirely filled with masonry or concrete ;
the weight of the bridge resting on the masonry, the iron
casing serving to protect and to stiffen the column.
4. As a caisson ; the sides being left standing.
APPROACHES.
325
The precautions recommended for stone and wooden piers
are equally necessary for those made of iron.
442. Approaches. The portions of the roadway, at each
extremity of the bridge and leading to it, are termed the
approaches.
These are to be arranged so that vehicles, using the bridge,
may have an easy and safe access thereto.
The arrangement will depend upon the locality, upon the
number and direction of the avenues leading to the bridge,
upon the width of these avenues and upon their position,
whether above or below the natural surface of the ground.
When the avenue to the bridge is in the same line as its
axis, and the roadway of the avenue and of the bridge is of
the same width, the abutment is generally made as shown in
Fig. 161. The returns or short walls carried back parallel to
FIG. 161.
the axis of the road to flank the approach are called wing-
walls, and are intended to sustain the embankment as well as
to serve as a counterfort to the abutment.
FIG. 162 Represents a horizontal section of an abutment, A, with curved wing-
walls, B, B, connected with a central buttress, C, by a cross tie-wall, D.
When several avenues meet at the bridge, or it is necessary
that the width of the approach shall be greater than the
326 CIVIL ENGINEERING.
way of the bridge, the wing-walls may be given a curved
shape, as shown in Fig. 162, in this way widening the
approach.
When the soil of the river banks is bad, the foundation of
the wing-walls should be laid at the same depth as that of the
abutment. But if the soil ^s firm, they may be built in steps,
and thus save considerable expense.
The rules for the dimensions of wing- walls are the same as
for other retaining walls. A common rule is to make their
length one and a half times the height of the roadway above
the bed of the river, their thickness at bottom one-fourth their
height, and to build them up in off-sets on the inside, reduc-
ing their thickness at the top to between 2 and 3 feet.
In some cases plane-faced wing-walls are arranged so that
the faces make a given angle with the head of the bridge.
The top of the wall is given a slope to suit the locality, and
is covered by a coping of flat stones, to shelter the joints
and to add a pleasing appearance to the wall (Fig. 163).
The lower end of the coping is generally terminated by a
newel stone.
Instead of wing-walls, a single wall in the middle is used
in many cases. The plan of the abutment in such a case is
that of a T.
In case there are no wing-walls to retain the earth, the
abutment wall must be sufficiently distant from the crest
of the slope of the water-course to allow room for the slope
of the embankment. This slope of the embankment may be
the natural slope, or, if steeper, the embankment should be
revetted with dry stone or sods, as shown in Fig. 164.
It may be necessary, to avoid obstructing the communica-
WATER WINGS.
327
tions along the bank, to construct arched passage-ways under
the roadway of the approaches.
FIG. 164. Plan and elevation showing a method of arranging the em-
bankments where there are no wing- walls,
a, a', side slopes of embankment of the approach.
6, J', dry stone revetment of the slope towards the water-course.
d, d', dry stone facing of the slope of the bank,
, ', paving used on the bottom of stream.
/, /', stairs for foot passengers.
443. Water wings. When the face of the abutment pro-
jects beyond the bank, an embankment faced with stone should
connect it with points of the bank, both above and below the
bridge. These are called -water-wings, and serve to contract
gradually the water-way of the stream at this point.
Where there is danger of the banks above and below the
abutment being washed or worn away by the action of the
current, it is advised to face the slope of the bank with dry
stone or masonry, as shown in Fig. 164.
444. The frame. It is evident that the arrangement used
to support the roadway admits of the greatest differences in
form. From these differences in the forms used, many classi-
fications have been made.
328 CIVIL ENGINEERING.
According to the kind of frame, bridges may for analysis
be classed as follows :
I. Trussed Bridges j
II. Tubular Bridges j
[II. Arched Bridges and
IY. Suspension Bridges.
Considering the simple bridge to belong to the first class,
every bridge may be placed under the head of one or more
of these divisions.
CHAPTER XIY.
L TRUSSED BRIDGES.
445. A trussed bridge is one in which the frame support-
ing the roadway is an open-built beam or truss.
A truss has been defined (Art. 252) to be a frame in which
two beams either single or solid built, with openings between
them, are connected by cross and diagonal pieces so that the
whole arrangement acts as a single beam.
It generally has to sustain a transverse strain caused by a
weight which it supports. To do this in the best manner,
the axes of the pieces of which the truss is composed are kept
in the same vertical plane with the axis of the truss, or are
symmetrically disposed with reference to it.
Supposing the truss to rest on two or more points of sup-
port, in the same horizontal line, its upper and lower sides
are called chords. In some cases the upper side has been
called a straining beam, and the lower a tie. Sometimes
both beams are designated as stringers. English writers call
them booms.
Generally, both chords are straight and parallel to each
other. Both may be and are sometimes curved ; in some cases
one is curved and the other is straight.
The secondary pieces, or those connecting the chords, are
called braces, and are so arranged as to divide the frame
into a series of triangular figures. The braces are known as
struts or ties, depending upon the kind of strain they have
to sustain. The triangles may be scalene, isosceles, equila-
teral, or right angled. They may be placed so as to form a
system of single triangles, or by overlapping, form a lattice
or trellis pattern.
CALCULATING THE STRAIN ON A TRUSS. 329
446. Systems. Trussed bridges are divided into three
general systems :
1, The triangular system ; 2, The panel system ; 3, The
bowstring system.
Other subdivisions are frequently made, based upon the
^articular arrangement adopted for the braces and upon the
form given to the chords.
Special cases belonging to the systems are generally known
by the name of the inventor: as Long's truss, Howe's,
Fink's, etc.
The essential qualities in a truss are those already given for
a frame (Art. 231), viz., strength, stiffness, lightness, and
economy of material.
These qualities are dependent upon the kind of material
used in its construction, the size of the pieces, and the method
of arranging them in the frame. The latter gives rise to the
variety of trusses met with in practice.
METHODS OF CALCULATING STRAINS ON THE DIFFERENT PARTS OF
A TRUSS.
447. External forces acting on a truss. It is necessary
to know all the external forces which act on a truss, in order
to determine the strains on its different parts.
The external forces which are considered, are :
\ , The weight of the bridge ;
2, The moving or live load ;
3, The reactions at the points of support ;
4, The horizontal and twisting forces which tend to
push the frame in a lateral direction or around some line in
the direction of its length.
1. The weight of the bridge. Previous to the calcula-
tion of the strains, the weight is not known, since it is de-
pendent upon the thing which we seek, viz., the dimensions
of the parts of the bridge. An approximate weight is there-
fore assumed, being taken by comparison with that of some
similar structure already built. The strains are then
determined under the supposition that this is the weight
of the bridge and the dimensions of its parts are computed.
The weight is then calculated from these dimensions, and if
the assumed weight does not exceed very greatly that of the
one computed, the latter, and also the strains deduced there-
from, are assumed to be correct.
330 CIVIL ENGINEERING.
2. The moving load. This is any load which may pass
over the bridge, and when calculating the strains, should be
assumed at its maximum ; that is, as equal to or exceeding
slightly the greatest load which will ever be placed on the
structure. This load should be considered as occupying vari-
ous positions on the bridge, and the greatest strains in these
positions determined.
For a common road bridge, the load is assumed to be a
maximum when the bridge is covered completely with men.
This load is estimated at 120 pounds to the square foot, and
must be added to the weight of the bridge.
For a railroad bridge, the load is assumed a maximum when
a train of locomotives extends from one end of the bridge ta
the other. This load is assumed at one ton (2,240 Ibs.) to the
running foot.
Sometimes, common road bridges are liable to be crossed
by elephants, in which case it is assumed that the maximum
load is equivalent to that of 7,000 pounds supported on two
points, six feet apart.
A load applied suddenly produces on the parts of a bridge
double the strain which the same load would produce if it
were applied gradually, beginning at zero and increasing
gradually until the whole load rested on the bridge. A load
moving swiftly on the bridge approximates in its effect to
that of one applied suddenly.
Therefore, the action of a live load may be considered to
be the same as that of a load of double its weight placed care-
fully on the bridge* The latter may then be treated as any
stationary load added to the weight of the bridge ; the strains
can be determined in the usual manner.
To distinguish between these loads, it is usual to call the
weight of the bridge the permanent or dead load, and that
caused by bodies crossing the bridge the moving, the rolling,
or the^live load.
3. Reactions of the points of support. The applied
forces cause reactions at the points of support, which must be
considered in the calculations, as external forces acting on
the bridge ; their value, therefore, must be determined. No
sensible error is committed by regarding the reactions as verti-
cal for trusses whose chords are straight and parallel to each
other.
4 Forces producing lateral displacement or twist-
ing. The action of the wind on the sides of the truss tends
to push the bridge in a horizontal direction. This pressure
may be regarded as uniform over the entire extent of the
UNO-POST TRUSS. 331
surface exposed. The best authorities assume this pressure
ordinarily at forty pounds per square foot. The locality will
decide as to the exact amount, since the force of the wind
is greater in one place than in another. The wind gauge
has recorded as high as sixty pounds in this locality.
Care is taken to guard against any forces which might
produce a twisting strain, and to reduce their effect to a mini-
mum. If there be any such forces acting, their effect on the
bridge must be provided for.
The King-post Truss.
448. Excepting the triangular frame (Art. 256), the king-
post truss is the simplest of the trusses belonging to the tri-
angular system.
It is frequently employed in bridges of short span, and
where the span is so small that the beam requires support
only at its middle point.
For a single roadway, two of these frames are placed side
by side, and far enough apart to allow room for the roadway
between them. Roadway bearers are placed on the beams,
or are suspended from them, to support the joists and flooring.
Each truss will therefore be loaded with its own weight,
one-half that of the roadway, and one-half of the live load.
Knowing these weights, the strains on the different parts
are easily determined, and the dimensions of the parts cal-
culated.
To determine the amount and kind of strains on the parts,
consider the load resting on the beams as uniformly distri-
buted over them, and represent (Fig. 68), by w, the load on a
unit of length of the beam, C ; 2Z, the distance between the
points of support.
The load on the beam, C, will be %wl. The post, , is so
framed upon the inclined braces, e r e^ and into the beam, C,
that the middle point of the beam is kept in the same straight
line with its ends. C is therefore in the condition of a beam
resting on three points of support in a light line. Five-
eighths of the load 2wl, is therefore held up by the king-post
(Art. 186), and by it transmitted to the apex of the frame,
the king-post sustaining a tensile stress. The amount of this
stress being known, the dimensions required for the king-post
are easily calculated.
The stresses developed and the dimensions of the braces
are determined as in Art. 256.
332
CIVIL ENGINEERING.
FIG. 165.
If the middle point of a beam, as A B (Fig. 165), is sup-
ported at C by inclined braces resting against the abutments,
the amount and kind of
stresses in the braces are the
same as those in the king-post
truss; in this case the hori-
zontal thrust at the lower
ends of the braces, instead
of being taken up by a tie
beam, will act directly against
the abutments.
Inverted king-post. If the king-post truss be inverted,
and supported at the extrem-
ities (Fig. 166), the amount of
stress in each piece will be
the same as before. The
strains, however, will be re-
versed in kind; that on the
beam, and that on the king-post, being compression, and those
on the braces being tension.
FIG. 166.
FINK TRUSS.
449. Fink truss. This is the name by which a truss de-
vised by Mr. Albert Fink, civil engineer, is generally known
(Fig. 167). It consists of a combination of inverted king-post
trusses, as shown in the figure. There is a primary truss,
A B ; two secondary ones, A K C and C L B ; four tertiary
ones, A P D, D M C, etc.
M N
FIG 167.
The load may be upon the upper or the lower chord, as the
circumstances may require. The strains on the different
parts are easily determined, when the weights to be placed
upon the bridge are known.
If the load should be on the upper chord, there would be
no necessity for a lower chord, so far as strength is con-
cerned.
450. Bollman truss. If the braces all pass from the foot
TRIANGULAB TETT88.
333
of the posts to the ends of the chord, as in Fig. 168, the truss
thus formed is known as Bollman's truss.
FIG. 168.
The calculations for the strains do not differ in principle
from those in the preceding case. It is observed that the ties
are of unequal length in each of the triangular frames of this
truss ; excepting the one at the middle post.
There is no necessity, as in Fink's, for a lower chord if the
load is placed above the truss.
From the fact that they need but one chord, both of these
constructions are frequently called "trussed girders," to
distinguish them from the ordinary bridge-truss, which, by
the definition given for a truss, requires two chords.
I. The Triangular System.
451. The term, triangular truss, is ordinarily used to de-
signate a truss whose chords are connected by inclined braces,
so arranged as to divide the space between them into isos-
celes or equilateral triangles, as shown in Fig. 169.
& **i
In the isosceles bracing the braces are generally arranged
so as to make an angle of forty-five degrees with the vertical,
although sometimes other angles are used.
When the triangles are equilateral, the truss is known in
England and in the United States as the " Warren girder,"
and in other countries as the " Neville."
The strains on this truss may be determined by the methods
given in Arts. 260-1-2, or they may be determined by using
the reactions of the points of support when these reactions are
known. The following is an example of the latter method.
334 CIVIL ENGINEERING.
452. Let it be required to determine the strains produced
upon the different parts of a triangular truss by a weight
supported at the middle point of the truss.
The truss is supposed to be resting on firm points of sup
port at its ends, these supports being in the same horizontal
line.
Eepresent by (Fig. 169),
2W, the weight resting on the truss at the middle ;
K! ES, the reactions at the points of support ;
a, the angle RI A t B t between the brace and a vertical line.
Since the load is at the middle, the reactions due to it are
Ei = W, and E, = W 1 .
The strains in one half will be equal to the corresponding
strains in the other half. Take the right half, as shown in the
figure, and on R t = W, as a resultant, construct a parallelo-
gram of forces, the components of which are in the directions
of the pieces, A t Bj and A t A 3 . These components will be
Ty
respectively equal to and W tan a. Going to B,, and re-
cos a
solving into two components, one in the direction of
cos a
B l B,, and the other in the direction of B, A a , their values
will be 2W tan a and Performing the same operation
cos a
at A,, for the components in the directions of A Q A 3 and A a
B a , there are found the same values just determined, 2W tan a
Ty
and At B 2 , A 8 , B 8 , etc., until the point of application
cos a
of the force is reached, similar expressions for the stresses will
be found.
Hence the stress in Bj B 2 is equal to 2W tan a ; in B 2 B 8 ,
this same amount is increased by that in Bj B 2 , or 4W tan a
for the stress in B 2 B 3 ; on B s B 4 , 6W tan <*., etc. The stress
in A! A 2 is W tan a ; in A 2 A 3 , it is 2W tan a increased by
that in Aj A 2 , or in all, 3W tan a; in A 8 A 4 , 5W tan , etc.
There is no increase of the stress as we pass from one brace
W
to another, the intensity being the same for each, viz., .
An examination of the forces acting will show the
nature of the strain in each piece. The direction of the
component of the reaction along the axis of the chord is
towards its centre or middle point ; the strain is therefore
one of compression, and increases from each end toward the
middle.
DETERMINING THE STRAIN.
On the lower chord, the strain is in the opposite direction,
and is therefore tensile, increasing in amount towards the
centre, as already shown.
In the right half of the truss, the strain on the brace Aj Bj
and on those parallel to it, is compress! ve ; on those not par
allel to it, the strain is tensile.
The pieces of the other half are strained in a similar man-
ner; on the corresponding pieces, the strains are equal in
amount, and they are also of the same kind on the braces,
being tensile in those parallel to the brace ^ B 1? and com-
pressive in the others.
It will be noticed that these results are identical with those
already obtained in Art. 260.
In the above example, the load was placed at the middle
point of the truss, but if the load had been placed at any other
point, the process used to obtain the strains would be the same ;
it would only be necessary to find the corresponding values for
R! and Rj, and substitute them in the foregoing expressions.
453. Let it be required, to determine the strains produced
upon the parts of this truss by a uniform load distributed
over the lower chord.
The effect of the uniform load upon the truss may, without
material error, be considered to be the same as that produced
by a series of weights acting at the points A 1} Ag, A 3 , A 4 ,
etc., each weight being equal to that part of the uniform
load resting on the adjacent half segments.
Denote by n the number of these points thus loaded, and
by 2w, the load at each point.
Their total weight on the chord will be 2nw, and the
reactions at the points of support due to them will be, at each
support, equal to nw.
To determine the strains, proceed as before. Construct
the parallelogram on R! =nw, and determine the stresses in
A! Ao and Aj Bj, which are found to be nw tan or, and nW .
cos OL
Going to Bj, the stress in B 1 B 2 is %nw tan ar, and that in
B! AS is - . At Ag the components of 2w, acting at this
point in the direction of Ag A s and A 2 B 2 must be subtracted
from those of the transmitted forces along these lines. The
stress in Ag A 3 will therefore be %nw tan a %w tan a =
2 (n 1) w tan a. To this must be added the stress already
determined in A t A 2 , which gives the total stress in AS Ag to
be w [n + 2 (n 1) ] tan a.
336 CIVIL ENGINEERING.
The stress in A 2 B 2 is , which may be written
cos a. cos a
' W . Going to B 2 the stress in B 2 B 3 , produced by the
COS OL
strain on the brace A 2 B 2 , is 2(n 2) w tan #, to which the
stress in Bj B 2 is to be added, making the total stress 2 (n2)w
tan a + 2nw tan <*, which may be written 4(^1) w
tan a. The stress in B 8 Aj is the same as that in A 2 B 2 ,
(n %\w
or is equal to .
cos a
It is plain that the stress in any segment of the upper
chord is obtained by adding to the stress transmitted to it by
the brace with which it is connected, the respective stresses in
each of the segments preceding it ; and, that the same law
obtains for the stresses in the lower chord.
It is to be noticed that the stresses in the first pair of braces
are the same in intensity but different in kind, being compres-
sive for the first and tensile for the second, as in the last case ;
that in the next pair the intensity differs from that in the
first by ; that the stresses in the third pair differ from
J cos <*'
the second by the same quantity ; and hence, that the stress
in any pair may be obtained when that in the preceding one
Qw
is known by subtracting from it. It is noticed that those
to cos a
braces whose tops incline towards the middle point of the
truss are compressed, while those that incline from it are
extended.
It is seen that while the strains on the braces decrease from
the ends towards the middle, that it is the reverse for the
chords ; in both the upper and lower, the strains increase
from the ends to the middle.
The stresses thus determined may now be written out, as
follows :
1. The compressions on the braces, Aj B,, A 2 B 2 , A 8 B 8 , etc.,
are
nw (n %)w (n 4) w (n 6)w
cos a' cos a ' cos a ' cos a '
2. The tensions on the braces, B! A 2 , B 2 A 8 , B } A 4 , etc., are
the same in amount, viz.,
nw (n 2) w (n 4:) w
j L _ 7 etc.
coso-' cos a cos a '
DETERMINING THE STRAIN. 337
3. The compressions on the segments of the upper chord
are, for B x B 2 , B 2 B 3 , B 3 B 4 , etc.,
%nw tan a, (n l)w tan a, 6(71 2) w tan a, 8(n 3)w tan a,
etc.
4. The tensions on the segments of the lower chord are, for
AI A 2 , Aj A 3 , A s A 4 , etc.,
nw tan a, [n + 2 (n 1)] w tan a, [7i+4 (TI 2) w tan a,
[n-f 6 (n 3) ] w tan a, etc.
General term. By examining the expressions just ob-
tained for compression on the segments of the upper chord,
it is seen that a general term may be formed, from which any
one of these may be deduced upon making the proper substi-
tution. Let the segments be numbered from the ends to the
middle, by the consecutive whole numbers, 1, 2, 3, 4, etc.,
and represent the number of any segment by m. Then,
2m (n m + 1) w tan a,
will be the general term expressing the intensity of the
stress in the n^ segment.
It is seen that the term,
[n + 2 (m 1) (n m + 1)] w tan a,
will represent the amount of tension on the ra th segment oi
the lower chord.
The value of m = - , corresponds to a maximum in the
first expression, and upon substitution gives ^(/i-fl) 8 w tan a for
the maximum compression. The value of m = ~ , corre-
sponds to a maximum in the second, and upon being substi-
tituted in it gives [ (n + I) 2 1] w tan a for the maximum
tension. The quantity, n + 1, denotes the number of bays in
the lower chord, which if we represent by X, the expression,
N~ 2 w tan a,
will very nearly correspond to the maximum tension or com-
pression upon the chords.
Strains on the chords. The strains on the chords vary from
segment to segment, but are uniform throughout any one seg-
ment. If the segments were infinitely short, the strains in that
case would be a continuous function of the abscissa, and the rate
of increase could be represented by the ordinates of a
parabola. Suppose a vertical section made, cutting the truss
between A 4 and B 4 , and A. taken as the centre of moments.
22
338
CIVIL ENGINEERING.
From the principle of moments, there must be for equilib-
rium,
d x d = \wv? R! a;,
or
wx 2 2E, x
Cl = ~2d >
in which x is the distance of the centre of moments from A! ;
Cj is the stress in the upper piece B 3 B 4 ; d the distance be-
tween the axes of the chords ; w the uniform load on the
unit of length ; and Bj the reaction at the point of support
A!. This is the equation of a parabola whose axis is vertical
and whose vertex is over the middle of the truss.
Remark. The usual method of computing the strains
upon the pieces of a truss is that of adding and subtracting for
each consecutive piece, as shown in the previous methods for
calculating strains. General formulas are used in connection
with these methods to check the accuracy of the computa-
tions.
II. The Panel System.
454. If the ties of the triangular truss be pushed around
until they are vertical, we shall have the method of vertical
and diagonal bracing referred to in Article 263, and the re-
sulting truss will be a type of the system. In England this
truss is frequently called the trellis girder, and in France
the American "beam. (Fig. 170.)
BS Bs 04 Bj D9 B I
\
As
A*
170.
A3
A2
The methods already given for the determination of the
strains on the parts of a Warren truss, and on a frame where
vertical and diagonal bracing is used, can be applied to this
truss.
The space included between any two consecutive verticals
is known as a panel ; hence the name of the system.
Diagonal pieces, as shown by the dotted lines in the figure,
THE QUEEN-POST.
339
called counter-braces, are generally inserted in each panel.
Their particular use will be alluded to in another article.
The Queen-post^ or Trapezoidal Truss.
455. This is the simplest truss belonging to the panel sys-
tem, and is much used in bridges where the span is not greater
than forty or fifty feet. Its parts are most strained when
the load extends entirely from one end to the other. Suppose
this load to be uniformly distributed over the lower chord,
A! A 4 , and represent by (Fig. 171),
Z, the length of the segment A t A 2 ;
w, the weight on the unit of length ; and by
o, the angle of F^ A! Bj.
B>
FIG. 171.
Since the segments Aj A 2 , A 2 A 8 , A 3 A 4 , are ordinarily equal
to each other, 31 will be the length of the lower chord, and
3wl will be the total load on the truss. The queen-posts are
framed into the lower chord ; the latter, therefore, has four
points of support. Supposing the lower chord to be a single
beam, or so connected as to act like one piece, each post
would sustain J-J- of wl. Each weight is transmitted to the
upper end of its post, where it is held in equilibrium by two
forces, one acting in the direction of the inclined brace, and
the other in the direction of the chord B x B 2 . The components
along B t A t and B 2 A 4 are each equal to -fj - and those
COS ft,
along B x B 2 are equal to fj- wl tan a. The two latter balance
each other, producing a strain of compression on the upper
chord. The other two produce compression on the braces,
which, transmitted to the points of support, causes a strain of
tension on the lower chord and a vertical pressure on the
points of support. Knowing the amount and kind of strains,
the dimensions of the pieces can be calculated.
Instead of considering the lower chord as a beam resting on
four points of support, it is more usual to consider that one-
340 CIVIL ENGINEERING.
third of the entire load is held up by each post, and one-sixth
at each point of support at the ends ; or, if the segments are
unequal in length, to consider the weight held up by each
post to have the same proportion to the whole load that the
segments have to the entire length of the chord A x A 4 . The
remarks made upon the inverted king-post truss will apply
to this frame, if inverted.
The queen-post truss, in its present shape, will not change
its form under the action of a load uniformly distributed over
it; when loaded in this manner, the truss is said to be
balanced. If, however, the load be only partially distributed
over it, so that the resultant acts through some other point
than the middle of the truss, the truss may become distorted
by a change of figure in the parallelogram A 2 B 1 B 2 A 3 . The
truss is then said to be unbalanced.
Sometimes, a certain amount of stiffness in the joints and
of resistance to bending in the pieces, give sufficient rigidity
to the truss, and may be relied upon to prevent distortion
under light loads.
As the load moves from one point to another, a change of
form will generally take place, due to the elasticity of the
materials of which the frame is made and to the imperfection
of the joints. To prevent this change of form, diagonal
pieces are inserted, as shown in the dotted lines of the figure.
The truss is then said to be thoroughly braced.
A truss is said to be thoroughly braced when the parts are
so arranged that no distortion takes place under the action of
its usual load, whatever may be the position of the load.
A truss may be distorted and even broken, by an excessive
load, notwithstanding the use of braces, but this distortion is
excluded by the definition of a frame, given in Art. 230.
In the calculations to determine the strains, the joints of
the truss are considered to be perfect.
III. The Bowstring System.
456, The common bowstring girder is one in which the
upper chord is curved into either a circular or parabolic form and
lias its ends secured to the lower chord, which is straight (Fig.
107). The horizontal thrust of the upper beam is received by the
lower chord ; the latter therefore acts as a tie, and as a conse-
quence, the reactions at the points of support are vertical. The
intermediate space between the bow and the string is filled with
BOWSTRING GIRDERS.
341
a diagonal bracing, like that used in the triangular or panel
systems, for the purpose of stiffening the truss.
The load straining the girder may rest directly upon
the lower chord or be suspended by vertical ties from the
upper one, and the greatest stresses developed in the pieces
of the girder are found by the usual methods.
"Where the span is of considerable length, the usual practice
is to form the upper chord of a number of straight pieces,
the intersections of whose axes ace in the curve of the bow.
(Fig. 172.)
/?/
&
As A* As A2
FIG. 172.
To find the strains produced upon the parts of a truss be-
longing to this system, by a uniform load resting on the lower
chord, which is connected with the upper one by vertical ties
dividing the truss into an even number of panels of equal
horizontal length, represent by
2#, the length of the lower chord ;
jf, the rise of the curve, or depth of the truss at the centre ;
w, the weight on the unit of length of the lower chord ;
Pj, the stress in any piece of the upper chord ; and
T!, the stress in the lower chord.
Take the origin of the co-ordinates at A x , the axis of X coin-
ciding with the axis of the lower chord, and Y perpendicu-
lar to it.
Disregarding the braces, and supposing a vertical section
made on the left of A 3 , and very near to it, and B 8 taken as
the centre of moments.
Taking the moments around this point, there results
= K 1 a J - = (2a- aJ; , (155)
aj, representing the distance
Taking the curve containing the intersections BU B 2 , B 3 , etc.,
to be a parabola, its general equation when referred to the
vertex and tangent at that point is
34-2 CIVIL ENGINEERING.
The vertex being the origin, the value of y f gives
x = #, or
whence,
which being substituted for 2p in the equation of the para
bola, gives
* = ^y,ory=4#, . . . (156)
/ a
Placing the origin at A 1? the equation of the curve will
be
y = (9o-afy . . . (157)
Since A 8 B 3 is equal to y, for the value of x equal to A 1 A 3 ,
there follows from the substitution of this value of A 3 B 3 , in
equation (155),
wx (20 - x) via?
Tl= T-^T- = a/' ' (158)
Hence, the strain on the lower chord, produced by a uni-
form load, is constant throughout.
It is observed that this is the same value obtained for the
horizontal component of the thrust in Art. 228.
In the same section, taking the moments around A 3 , the
lever arm of the strain on B 2 B 8 , is A s m drawn perpendicular
to the piece and. through the centre of moments.
There results
P t x A 3 m = ~(2a - x). . . (159)
2
Through B 2 , draw a straight line parallel to the lower
chord. From the triangles B 2 B 8 j? and A 3 B 8 m, we have the
proportion,
The first term of this proportion is the length of the piece
of the upper chord in this panel, and varies in length for each
panel from A t to the centre. The second term is the hori-
zontal length of the panel and constant. Representing the
former by v, and the latter by Z, and substituting in the above
proportion, we obtain
I
v : I : : y : h*in. .*. A s m = v ,
9 v
BOWSTRING GIRDERS. 343
a v
Substituting which in equation (159), we get
< 160 >
This shows that the strain is independent of x and depend-
ent upon v the only variable present, and that it increases as
v increases, or is greatest at the points of support.
Suppose a brace to be inserted in this panel, joining Aj
and B 3 , or B 2 and Ag. A section taken midway between A 2
and Ag would cut the upper chord, the lower, and the brace.
For an equilibrium, the algebraic sum of the horizontal com-
ponents and of the vertical components of all the forces must
oe separately equal to zero.
Represent the strain on the brace by F, and the angles
made by the brace and the piece B 2 B 8 of the upper chord
with a vertical, by a and y3, respectively.
The first of these conditions of equilibrium can be ex-
pressed analytically, as follows :
P! sin - F sin a T = 0.
But P! sin = T = ^ ^, hence
2 J
F sin a = 0, or F = 0.
That is, there is no strain on the brace produced by a load
uniformly distributed over the truss.
If the load had been placed directly upon the upper chord,
there would have been no strain on the verticals.
If the triangular instead of the panel system had been
used for the bracing, its use would have been simply to trans-
mit the loads on the lower to the upper chord. Knowing
the angle of the bracing, the strain on any brace could be
easily determined.
The vertical component of Pi may be obtained as follows :
Let y' and y" be the ordiuates of the lower and upper ex-
tremities of any piece, as B 2 B 8 , of the upper chord.
Let v, the length of the piece, denote the intensity of the
strain on the piece, then y" y' would represent its vertical
component.
From the equation of the curve, we have
y" = " (2a - a"), and y' = f (20 - a/>
844 CIVIL ENGINEERING.
But " = x f + I, substituting which in the first of these
equations for x", and then from this result subtracting the
second of the equations, we get
y" -y' = f -Va-W-l).
Eepresenting the vertical component by Y, we may form
the following proportion :
v:y" - y' :: P t : Y.
Substituting for P t and y" y' , the values just found, and
solving, we find
w
V = 2 (2a - , perpendicular to
P. This is equal to the " pole distance,'' which represent by H.
Through b draw cd parallel to P, and limited by Cj and C 2
produced. From similar triangles, the following proportion
is obtained :
P : H : : cd : ab, or P x ab = H x cd,
which was to be proved.
In Fig. 178 the bending moment at 0' is R! x A 0', which,
as has just been shown, is equal to H xpp l ; at 0" the bending
moment is E-, x AO" "W^xA'O". The components of W a
are ab and be. Hence the moment of W x at 0" is H xp\p,, y
and the total moment is II xp'p\.
And as this is true for any section, it is seen that the bend-
ing moments are proportional to the ordinates drawn from
the closing line to the sides of the equilibrium polygon. And
at any section, it is equal to the product of H and the ordin-
GRAPHICAL METHOD.
353
ate of the equilibrium polygon corresponding to the section
under consideration.
1 t*'
FIG. 178.
The ordinate is measured by the scale used for the equilib-
rium polygon, and the pole distance, H, by the scale for the
force polygon. These may be drawn on the same or different
scales, whichever is the most convenient.
Representation of the shearing strain. The shearing
force between ~R^ and W ly is 1^. At TV*!, the shearing force is
Hi- W l ; at W 2 , it is R!- Wj W 2 , etc. Hence, the line, K!, 1,
2, 3, etc., represents graphically the shearing forces for all
parts of the beam.
An examination of the figure shows that the shearing force
is greatest where the bending moment is the least, and the
reverse.
470. Couples. It has been assumed, in the previous dis-
cussions and examples, that the forces were in equilibrium,
or by the addition of a single force an equilibrium could be
established.
* *
a/
FIG. 179.
If two forces form a couple, they cannot be replaced by a
single force. Let P x and P 2 be a couple (Fig. 179), and 012
the force polygon.
23
354
CIVIL ENGINEERING.
It is seen that this force polygon closes, that is, the result
ant is zero. From any point on P t draw ac and ab parallel to
C and 1 C. At J, where ab intersects P 2 or P 2 produced,
draw lines parallel to C 1 and 2 C. The lines ac and M are par-
allel. Therefore the equilibrium polygon will not close, or the
lines will intersect at an infinite distance. A result which
was to be expected. (Art. 98, Analytical Mechanics.)
The figure shows that the components of the forces I\ and
P 2 , which act in the direction of the line ab, are equal and
directly opposed to each other, and that the other two are
parallel, forming a couple. Hence, it is concluded that a
couple can be replaced by another without changing the ac-
tion of the forces.
From what has been shown, it is evident that if both the
force and equilibrium polygon close, that an equilibrium exists
among the forces. But if the force polygon closes and the
equilibrium does not, that the forces cannot be replaced by a
single force, but only by a couple.
471. Influence of a couple. Let A B (Fig. 180) be abeam
fastened at its ends and acted upon by the couple P 1 P 2 .
FIG. 180.
The beam being fastened, the reactions at A and B will
keep the couple from moving and the four forces will be
in equilibrium. Construct the force polygon, 012, and
from a pole, C, draw the lines C 0, C 1. Form the equilib-
rium polygon, a 1) c d, of the forces P t P 2 ; produce ~b a and
c d until they intersect the lines of direction of the reac-
tions ; join a and d and this will be the closing line of the
polygon. Parallel to this line draw Cg in the force polygon.
An examination of the force polygon shows that g is the
vertical reaction acting downwards at B, and g 0, the reaction
at A, acting upwards, which with the couple Pj P 2 form an
equilibrium.
The ordinates drawn from the closing line, ad, upon the
sides, ab, bo, and ad, multiplied by the pole distance give
the bending moments for the corresponding sections of the
beam.
GRAPHICAL METHOD.
355
In the preceding examples the force polygon has been given,
and from it the equilibrium polygon has been constructed.
Inversely, the equilibrium polygon being given, the force
polygon is easily constructed.
472. From the preceding demonstrations, the following
theorem may be enunciated :
Theorem. If straight lines be drawn through any as-
sumed point parallel to the sides of a polygonal frame, then
the sides of any polygon whose angles lie on these radiating
lines may be taken to represent a system of forces which, if
applied to the angular points of theframe^ will be in equi-
librium among themselves. And the converse, that if a sys-
tem of external forces acting at the angles of a frame are in
equilibrium^ that from an assumed point drawing straight
lines parallel to the sides of the frame, and then parallel to
the directions of these forces drawing straight lines whose
successive intersections are on the successive radial lines, the
distances cut off by the second set will represent the strains
on the corresponding sides of the frame.
Let ABC (Fig. 181) be a triangular frame acted upon at
the points A B C by a system of external forces which are in
equilibrium. Let Pj P 2 P 8 be the resultants of the forces act-
ing at these points, and suppose that these resultants are in
the plane ABC.
From an assumed point, P, draw the straight lines, P 1, P 2,
FIG. 181.
and P 3, respectively, parallel to the sides A B, B C, and C A.
Through an assumed point, as 0, on the line P 3, draw the line
M parallel to the direction of the force P 1? and from its point
of intersection with P 1, draw the line M N parallel to the
force P 2 .
Join N and by a straight line, and this will be parallel to
the force P 8 . The triangle, M N, will be the force polygon.
The distance, P 0, will measure the force acting along the
piece AC; P M, that along A B; and P N, that along B C.
356 CIVIL ENGINEERING.
If the external forces are parallel the polygon becomes a
straight line, which will be divided into segments by the lines
drawn parallel to the sides of the frame. Each segment will
represent the external force acting at one of the angles of
the frame, and the distances cut oft will represent the forces
acting along the adjacent pieces.
An application of these principles will enable the student
to determine graphically the strains on the different parts of
a frame, and test the accuracy of calculations already made
by other methods.
Working, Proof, and Breaking Loads.
473. Ultimate strength of a structure. The object of
the calculations made to determine the strength of a given
structure is to find the load which, placed on the structure,
will cause it to give way or break in some particular way.
This load is called the ultimate strength or breaking load
of the structure.
Working load. As the bridge must not be liable to yield
or give way under any load which it is expected to carry,
it is made several times stronger than is actually necessary to
sustain the greatest load which it will ever have to support.
The greatest load thus assumed is called the working load.
The ratio of the breaking load to the working load, or
" factor of safety," is assumed arbitrarily, limited by experi-
ence. It is usually taken from four to six for iron, and even
as high as ten for wooden bridges. It should be large enough
to ensure safety against all contingencies, as swift rolling
loads, imperfect materials, and poor workmanship.
Proof load. When the bridge is completed, it is usual to
test the structure by placing on it a load greater than it will
ever have to support in practice. A train of locomotives for
a railroad bridge, and a crowd of men, closely packed, upon
an ordinary road bridge, are examples. These loads are
known as proof loads.
A proof load should remain on the bridge but for a short
time, and should be removed carefully, avoiding all shocks.
Excessive proof loads do harm by injuring the resisting pro-
perties of the materials of which the bridge is built.
Wooden Bridge-trusses.
474-, Both the king and queen-post trusses, as stated in a
LATTICE TRUSS.
357
previous article, are frequently made entirely of wood, and
are used in bridges of short spans.
A compound truss, entirely of wood, the outline of which
is shown in Fig. 182, has been used in bridges for spans oi
considerable width.
FIG. 182.
The celebrated bridge at Schaffhausen, which consisted of
two spans, the widest being 193 feet, was built upon this
principle.
475. Town's lattice truss. This truss was made entirely
of wood, and at one time was much used in bridge construc-
tion. It belongs to the triangular system. The chords (Fig.
183) were built of beams of timber, and frequently of plank of
the same dimensions as that used for the lattice. They were
in pairs, embracing the diagonals connecting the upper and
lower chords. The diagonals were of plank, of a uniform
thickness and width, equally inclined towards the vertical and
placed at equal distances apart. They were fastened to the
chords, and to each other at their intersections, by treenails,
as shown in the figure.
I I
FIG. 183.
This truss was frequently made double. In case the lat-
tices were separated by a middle beam, as shown in the cross-
358
CIVIL ENGENEEEING.
section in Fig. 183, the chords, instead of being in pail's,
were made of three beams, placed side by side.
When the truss was of considerable depth, intermediate
longitudinal beams were used to stiffen the combination, as
shown in the figure.
This truss possessed the advantages of a simple arrange-
ment of its parts and ease of construction. It also possessed
the disadvantages of a waste of material and a faulty con-
struction by which the strength of the truss depended upon
the strength and the perfect fitting of the treenails.
n
U
U
FIG. 184 Represents a panel of Long's truss.
A and B, upper and lower chords.
C, C, uprights, in pairs.
D, main braces, in pairs.
E, counter-brace, single.
a, a, mortises where gibs and keys are inserted.
&, &, blocks behind uprights, fastened to the chord.
F, gib and key of hard wood.
476. Long's truss. This truss belongs to the panel sys
tern, and was built entirely of wood. It was one of the earlier
trusses used in the United States, and takes its name from
LONG'S AND BURR'S TRUSSES.
359
Colonel Long, of the Corps of Engineers, United States
Army, who invented it. It was one of the first trusses in
which a scientific arrangement of the parts was observed.
"
.11 the timber used in its construction had the same dimen-
sions in cross-section.
Each chord was composed of three solid-built beams, placed
side by side, with sufficient intervals between them to allow of
the insertion of the uprights. The uprights which connected
the chords were in pairs, and fastened to the chords by gibs
and keys. These gibs were inserted in rectangular holes made
in the chords, and fitted in shallow notches cut in the up-
rights. Pieces of wood wide enough to fill the space between
the beams, about three or four inches thick and two feet long,
were inserted between the beams of the chords, behind the
uprights, and fastened to the beams by treenails. These were
for the purpose of strengthening the uprights and preventing
their yielding at the notches.
The main braces were in pairs, and were joined to the up-
rights, as shown in the figure. The counter-braces were single,
and were placed between the main braces, abutting against
or fastened upon the upper surface of the middle beam of
the chords. Generally they were fastened to the main braces
by treenails at their intersections.
Fia. 185.
477. Burr's truss. This is another of the earlier wooden
trusses, much used at one time in the United States. This
truss (Fig. 185) belongs to a compound system, being com-
posed of a truss of the panel system, stiffened by solid-built
360
CIVIL ENGINEERING.
curved beams, called arch timb'ers. These arch timbers
were in pairs, embracing the truss and fastened to it at the
different intersections of the pieces of the truss with the
curved beams, as shown in the figure.
478. Other forms of wooden trusses. The trusses al-
ready named may be considered as typical trusses. There
are many others, all of which may be referred to one of the
systems already given, or a combination of those systems.
Haupt's lattice, Hall's lattice, McCallum's truss, etc., are
examples of some of the different forms of wooden bridge-
trusses.
Bridge-trusses of Wood and Iron.
479. Canal bridge. A truss composed of wood and iron,
which has been much used for common road bridges over the
New York State canals, is shown in Fig. 186.
FIG. 186.
In this truss, the chords and diagonals are of wood, and
the verticals of iron. In some cases, the lower chord is also
of iron.
480. Howe's truss. A popular truss for bridges, both
common and railroad, and one which has probably been
used more than any other, is known as the Howe truss.
(Fig. 187.)
This truss belongs to the panel system. The chords and
braces are made of wood, and the verticals of iron.
The chords are solid-built beams of uniform cross-section
throughout.
The braces are also of uniform size, the main braces being
in pairs, and the counter-braces single, &nd placed between
the main braces, as in Long's truss. Between the ends of
HOWE'S AND JPBATT'S TRUSSES.
361
the braces and the chords, blocks of hard wood or of cast iron,
inserted in shallow notches in the chords, are used as shown
in the figure. The faces of the blocks should be at right
angles to the axes of the braces.
FIG. 187.
The verticals are in pairs, and pass through the blocks and
chords, and are secured by nuts and screws at both ends, or
by heads at the ends witn a nut and screw arrangement at
the middle. By tightening the screws, the chords are drawn
towards each other, and the reverse. To prevent the edges
of the nuts from pressing in and injuring the timber, washers,
or iron plates, are placed between the nut and the wood.
Where the pressure on the block is great, an iron block or
other arrangement is placed between the block on one side
and the washer on the other, to prevent the block from
crushing into the chord.
It is seen that there is an excess of material in some of the
chords and the braces. The corresponding gain obtained in
reducing the amount of material, by proportioning the pieces
to the strains they would have to support, would not pay the
cost of extra time and labor required ; these pieces are there
fore made, as a rule, with a uniform cross-section.
There would be a gain if the verticals were proportioned to
the strains which they have to support, instead of being made
of uniform size.
It is observed that the framing is such that the diagonals
will only take a compressive strain, and the verticals a tensile
one.
481. Pratt ; s truss. If the framing of the Howe truaa
is changed so that the diagonals will only take a tensile strain,
and the verticals a compressive one, there results the trusi
CIVIL ENGINEERING.
known as Pratt's. The chords and verticals in this case are
of wood, and the diagonals are of iron.
482. There are quite a number of trusses besides those
just named, which are composed of wood and iron. Those
mentioned are typical ones, and illustrate fully the method
}f combining the two materials in the same structure.
Iron Bridge-trusses.
483. Bridge-trusses made entirely of iron, or of iron and
steel are much used at the present time.
Trusses of iron belong to the three systems already
described, viz : the triangular, the panel, and the bow-
string systems, and are generally known by the names of
their inventors.
At one time, the use of cast-iron for the compressive
members of a truss was much favored by builders in the
United States. At the present time, wrought iron or steel
is preferred to cast-iron for all the parts of a truss.
The trusses known as Fink's, Bollman's, Warren's. Jones,
WhippleX Murphy- Whipple, Linville, Post's, etc., are some
of the trusses made entirely of iron which are most frequently
seen in use in the United States.
Fink's truss. The principles of Fink's truss are given in
Art. 449. The arrangement of its parts enables the truss to
resist in the best manner the effect produced by a moving
load, or by changes of temperature. The lower extremities
of the verticals being free to move, the verticals remain
normal to the curve assumed by the chord under the strain-
ing force, and the distances of their lower ends from the con-
nection of the ties with the chord remain relatively the
same. None of its parts are therefore unequally strained by
the force producing the deflection.
Bollman's truss. The principle on which this truss is
constructed is mentioned in Art. 450. In order to avoid the
ill effects of unequal expansion or contraction of the ties pro-
duced by changes of temperature, a compensating link is
used, by means of which the pin holding the ties is enabled
to change its position as the ties contract or expand, without
straining the verticals.
Warren's truss. The principle of this truss is explained
in Article 451. It is ordinarily made entirely of wrought
iron. In some cases the braces are of cast iron, in the form
of hollow pillars, with wrought-iron ties enclosed. The brace
WHIPPLE'S AND POST'S TBTTSSES.
363
is thus composed of two distinct parts, and is better suited to
resist the strains which it has to sustain.
Jones's truss. This truss is the Howe truss in principle,
all the parts being of iron.
Whipple's truss. This truss is one of the first used
in this country made entirely of iron. It is composed of
cast and wrought iron ; the former being used for the
compression members, and the latter for the tension mem-
bers.
This truss (Fig. 188) belongs to the panel system. The
upper chord is usually made of hollow tubes of cast iron, in
sections, whose lengths are each equal to a panel distance.
FIG. 18&
The lower chord is made of links, or eye-bars, of wrought
iron, which fit upon cast-iron blocks. These blocks hold the
lower ends of the vertical pieces.
The vertical pieces are of cast iron, and are so made that
the inclined pieces can pass through the middle of them.
The parts are frequently trussed by iron rods, to prevent
bending.
The inclined pieces are wrought-iron rods, and it is seen
that each of them, excepting those at the ends, crosses two
panels.
An examination of this truss shows that the inventor has
considered economy of material in making the verticals,
struts, and the diagonals, ties. In principle it corresponds
with the Pratt truss.
Murphy- Whipple truss. This is the Pratt truss, entirely
of iron, with some of the details of Whipple's.
JLinville truss. This is Whipple's truss made entirely
of wrought iron, the verticals being wrought-iron tubular
columns.
Post's truss. This truss is composed of cast and wrought
iron. Its peculiarity lies principally in its form (Fig. 189) ;
364: CIVIL ENGINEERING.
the struts, instead of being vertical, are inclined towards the
centre of the bridge, making an angle of about 23 30' with
the vertical, as shown in the figure. The ties cross two panels,
and make an angle of 45 with the vertical. The counter-
ties make the same angle, but cross only one panel.
FIG. 189.
The inclination given to the struts was for the purpose of
obtaining the same strength with a less amount of material
than that obtained when the struts were vertical.
Lattice trusses. Lattice trusses, made entirely of iron,
are frequently used in railroad bridges. They do not differ
in principle from the lattice truss made of wood.
Continuity of the Truss.
Various opinions have been held as to the advantages
obtained in connecting the trusses over adjacent spans, so
that the whole arrangement should act as a single beam.
If the load is permanent, or the weight of the structure
is very great, compared with the moving load, it is advisable
to connect the trusses, so that they shall act as a single con-
tinuous beam.
But when this is not the case, the effect of a heavy load is
to reverse the strains on certain members of the trusses over
the adjacent spans ; a result which is to be avoided, and hence
the trusses are ordinarily not rigidly connected.
WROUGHT-IEON BRIDGES. 365
CHAPTER XV
IL- -TUBULAR AND IRON PLATE BRIDGES.
485. Bridges of this class are made entirely of wrought
iron.
A tubular girder is one which is made of iron plates,
so riveted together as to form a hollow beam. These girders
may be placed side by side and a roadway built upon them,
forming a simple bridge, which in principle, would not differ
from the simple bridge described in Art. 435.
When the tube is made large enough to allow the roadway
to pass through it, it is called a tubular bridge.
The difference in construction between the tubular bridges
and the tubular girders consists in the arrangements made to
stiffen the four sides of the tube.
The three great examples of tubillar bridges are the Bri-
tannia Bridge, across the Menai Straits, in Wales ; the Conway
Bridge, over the Conway River, in Wales ; and the Victoria
Bridge, over the St. Lawrence River, at Montreal, Canada.
The Britannia Bridge consists of two continuous girders,
each 1,487 feet long, resting on three piers and two abut-
ments. Each tube is fixed to the central pier and is free to
move on rollers placed on the other piers and abutments.
The middle spans are 459 feet each, and the shore spans are
230 feet each. The bridge is 100 feet above the surface of
the water.
The Conway Bridge consists of two tubes, separated by a
few feet, over a span of 400 feet.
The Victoria Bridge is a single tube, 6,538 feet long, rest-
ing on piers, forming twenty-four spans of 242 feet each, and
a centre span of 330 feet, or twenty-five spans in all. The
tube is made continuous over each set of two openings, the
middle of the tube being fixed at the centre pier of the open-
ing and the extremities being free to move on rollers placed
on the adjacent piers.
The centre span is level, and is about sixty feet above the
surface of the water. From the centre span the bridge slopes
downward at an inclination of
366
CIVIL ENGINEERING.
In the Con way and Britannia bridges, the tops and bottoms
are made cellular ; that is, the plates are so arranged as to
form rows of rectangular cells (Fig. 190). The joints of the
cells are connected and stiffened by covering plates on the
outside and by angle-irons within.
till
FIG. 190.
FIG. 191.
The top, A, is composed of eight cells, each of which is one
foot and nine inches wide, and one foot and nine inches high,
interior dimensions The bottom, C, is divided into six cells,
each of which is two feet and four inches in width, and one
foot and nine inches high. These dimensions are sufficiently
large to admit a man for painting the interior of the cells and
for repairs.
The sides, B, are composed of plates set up on end (Fig.
192), their edges adjoining, and connected by means of verti-
cal T-iron ribs, /*, f (Fig. 190). The horizontal joints of the
side plates are fastened by covering strips. The connection
between the sides and top and bottom is strengthened by
gussets, A, A, riveted to the interior T-irons.
WROTTGHT-IROX BRIDGES.
367
FIG. 192.
In the Victoria Bridge the top and bottom, instead of being
cellular, consist of layers of plates riveted together and stif-
fened by means of ribs (Fig. 191). The
top. A, is slightly arched, and is stiffened
by longitudinal T-irons, d, d, d, placed
about two feet three inches apart, and by
transverse ribs, 0, about seven feet apart.
The bottom, c, is stiffened by T-shaped
beams, g, which form the cross-pieces of
the roadway.
Erection, There are three methods
which have been used to place tubular
bridges in position : 1, building the tube
on the ground, and then lifting it into
place ; 2, constructing the tube, and
moving it endwise upon rollers, on the
piers ; and 3, building it in position on a scaffold.
The first of these methods was adopted for the Britannia
Bridge and the third for the Victoria Bridge.
Cambering. If the top and bottom of the tube were made
horizontal, the tube would when placed in position suffer
deflection at the middle point from its own weight. In order
that it may be horizontal after it has fully settled in position,
the tube is made convex upwards. This convexity is called
the camber of the tube or truss. The expression for maxi-
mum deflection of a beam in a horizontal position resting
upon two points of support will give the amount of camber to
give the tube. The camber given the Britannia Bridge was
eighteen inches.
^Remark. Tubular bridges of these types are not now in
much favor with the engineering profession, and few, if any,
will ever be built in the future. The same amount of mate-
rial in the form of a truss bridge will give a
better bridge.
486. Plate bridges. If we were to sup-
pose the top removed from the tubular bridge,
or to suppose the diagonals of the lattice truss
to be multiplied until the side was a con-
tinuous piece, we would obtain the plate
girder. In cross-section, the girder is x-form
(Fig. 193). Its general construction conforms
to that given for the tubular bridge.
The joints of the flanges, A and B, are con- 1 ~ < !I Tjjl ' B
nected by covering plates; the web, C, is gen-
erally of thin plate. The web and flanges are fastened by
368 CIVIL ENGINEERING.
angle-irons, D, riveted to both of them. The sides are stif-
fened by T-irons, as in the tubular bridges.
The advantages gained by using this class of bridges are
confined to shallow bridges of moderate span. When the
span exceeds sixty feet, it is more economical to use one
of the iron trusses already named.
CHAPTEE XYL
in. ARCHED BRIDGES.
487. Arched bridges are made either of masonry, of iron,
or of steel.
The form of arch most generally used is the cylindrical.
The form of soffit will be governed by the width of the
span, the highest water level during the freshets, the ap-
proaches to the bridge, and the architectural effect which
may be produced by the structure, as it is more or less ex-
posed to view at the intermediate stages between high and
low water.
Oval and segment arches are mostly preferred to the full
centre arch, particularly for medium and wide bays, for the
reasons that for the same level of roadway they afford a
more ample water-way under them, and their heads and
spandrels offer a smaller surface to the pressure of the water
during freshets than the full centre arch under like circum-
stances.
The full centre arch, from the simplicity of its construc-
tion and its strength, is to be preferred to any other arch for
bridges over water-courses of a uniformly moderate current,
and which are not subjected to considerable changes in their
water-levels, particularly when its adoption does not demand
expensive embankments for the approaches.
If the spans are to be of the same width, the curves of the
arches should be the same throughout. If the spans are to
be of unequal width, the widest should occupy the centre of
the structure, and those on each side of the centre should
either be of equal width, or else decrease uniformly from the
centre to each extremity of the bridge. In this case the
curves of the arches should be similar, and the springing
lines should be on the same level throughout the bridge.
ABCHED BRIDGES. 369
The level of the springing lines will depend upon the rise
of the arches, and the height of their crowns above the water-
level of the highest freshets. The crown of the arches should
not, as a general rule, be less than three feet above the high-
est known water-level, in order that a passage-way may be
left for floating bodies descending during freshets. Between
this, the lowest position of the crown, and any other, the rise
should be so chosen that the approaches, on the one hand,
may not be unnecessarily raised, nor, on the othe other, the
springing lines be placed so low as to mar the architectural
effect of the structure during the ordinary stages of the water.
488. Masonry arches. These may be of stone, of brick,
or of mixed masonry. The methods of construction, already
described under the heads of Foundations and Masonry, are
applicable to the construction of masonry arches used for
bridges. As the foundations and beds of the piers and abut-
ments are exposed to the action of the water, precaution
should be taken to secure them. (Art. 440.)
Centres. The centres used should be strong, so as to settle
as little as possible during the construction of the arch, and
for wide spans, should be so constructed that they can be
removed without causing extra strains on the arch. This is
effected by removing the centering from the entire arch at
the same time. Removing the centering is termed striking
the centre.
In wide spans, the centres are struck by means of an
arrangement of wedge blocks, termed striking plates. This
arrangement consists in forming steps upon the upper surface
of the beam which forms the framed support for the centre.
On this a wedge-shaped block is placed, on which rests an-
other beam, having its under surface also arranged with steps.
The struts of the rib of the centering either abut against the
upper surface of the top beam, or else are inserted into cast-
iron sockets, termed shoe-plates, fastened to this surface.
The centre is struck by driving back the wedge block. When
the struts rest upon intermediate supports between the abut-
ments, folding wedges may be placed under the struts, or else
upon the back pieces of the ribs under each bolster. The
latter arrangement presents the advantage of allowing any
part of the centre to be eased from the soffit, instead of de-
taching the whole at once, as in the other methods of striking
wedges.
Another method of striking centres is by the use of sand.
In this method, the centres rest upon cylinders filled with sand.
These cylinders are arranged so that the sand can run out
24
370
CIVIL ENGINEERING.
slowly near the bottom. When ready to strike the centre,
the sand is allowed to run out of the cylinders, and all the ribs
gradually and evenly settle down away from the soffit. The
sand having run out, the centre can then be removed in the
ordinary manner.
489. Iron arched bridges. Next to masonry, cast iron is
the material best suited for an arched bridge. It combines
great resistance to compression or strength, with durability
and economy ; qualifications already given as requisite for an
engineering structure.
Wrought iron is sometimes used for arched bridges. Where
the bridge is liable to considerable transverse strains or shocks,
wrought iron would be a better material than cast iron.
490. Construction. Instead of the soffit being a continu-
ous surface, as in the masonry arch, it is formed, in the iron
arch, of curved iron beams placed side by side at suitable
distances apart, and bound together by lateral bracing. This
lateral bracing binding the ribs together, the proper abut-
ting of the ends of the ribs, and the fastening of them upon
the bed-plates or skew-backs of the abutments, form the most
important part of the construction.
The ribs are generally made in segments, the joints being
in the direction of the radii of curvature of the under surface
of the rib. To guard against any possibility of accident, the
segments are bolted together at the joints, forming in this
way a continuous curved beam.
The form of the under surface of the rib is either parabolic
or circular, more generally the latter. The depth of the rib
is taken ordinarily at about -fath of the span.
J I
[
1 1
1
1
1 , 1
I
1
1
1
1
1
1
1 1
FIG. 194.
The rib may be solid, having a cross-section of the usual
x-shape, the upper and lower flanges being equal ; or it may
be tubular ; or it may be open-work, similar to a truss in which
the chords are curved.
The first is the usual form. The other forms have been
are frequently used, but require no particular description.
ARCHED BRIDGES. 371
Whatever be the form of cross-section of the rib, it is usual
to place above the crown a horizontal beam, generally of
wrought iron, suitably stiffened by covering plates and angle
irons. (Fig. 194.)
The connection of this beam with the curved rib is made
by a truss-work, called the spandrel filling, as shown in the
figure.
On the horizontal beams the roadway is placed.
491. Expansion and contraction. The rib is frequently
hinged at the crown and ends, and sometimes at the ends
only, to provide for the expansion and contraction of the
metals produced by changes of temperature.
It is a matter of doubt whether anything is gained by this
provision, as the friction arising from the great pressure on
the joint probably prevents the motion of rotation necessary
to relieve the arch from the increased strain.
492. Arched bridges of steel. Bridges of this class,
made of steel, do not differ in principle from those in iron.
The most noted example of the steel arch is that used in the
St. Louis and Illinois Bridge, across the Mississippi River,
at St. Louis, Missouri.
In this bridge, the portion which corresponds, in the previ-
ous descriptions, to the rib, is composed of two tubular steel
ribs placed directly one over the other and connected by a
truss-work.
The segments of each of the tubular ribs are straight
throughout their length, instead of being curved. The ends
of each segment are planed off in the direction of the radius
of curvature, and abut against the ends of the adjacent seg-
ment, to which they are joined and fastened. In this way
the tube is made continuous ; but instead of being curved, it
is polygonal, as in the case of the bowstring girder. The
tubes are connected by a truss- work, and the whole forms a
rib of the third class.
493. Bads' patent arch bridge. Captain Eads, the en-
gineer of the St. Louis Bridge, has patented an arch bridge,
the principle of which is shown in Fig. 195.
This arch is hinged at the crown, C, and springing lines,
A and B, to provide for the expansion and contraction of the
metal used in its construction. This arrangement of hinging
the arch at the crown reduces the construction to that of two
inclined beams resting against each other at C. Each beam
is a truss belonging to the triangular system and having curved
chords.
The line, A C B, is the arc of a parabola, whose vertex if
372
CIVIL ENGUNEEBING.
at C. The lines, ADC and C E B, are also arcs of parabolas.
The maximum depth of either truss must not exceed one-half
the rise of A C B.
FIG. 195.
494. Cases in which the arch may be preferred to the
truss.
The arch will usually be found to be a less expensive struc-
ture than the truss, when the banks are of rock forming good
natural abutments.
It will oftentimes be more economically employed where a
deep valley is to be spanned and where high arches can be
used.
It is to be preferred when the roadway is a very heavy one,
as in the case of a macadamized, or similar covering.
It is frequently selected in preference to a truss, from
architectural considerations.
CHAPTER XVII.
IV. SUSPENSION BRIDGES.
495. A suspension bridge is one in which the roadway
over the stream or space to be crossed is suspended from
chains or wire ropes. The chains or wire ropes pass over
towers, the ends of the chains being securely fastened or
"anchored" ia masonry at some distance behind and below
the towers. The roadway, usually of wooden planking, is
SUSPENSION BRIDGES.
373
supported by suspending rods placed at regular distances
along the chains. (Fig. 196.)
FIG. 196.
Suspension bridges are used principally for spans so great
that they can not be crossed by arches or truss-work at a
reasonable cost. Sometimes they are used, where the span is
not very great, as a roadway only for foot passengers, especi-
ally over high-banked rivers, ravines, and similar places where
the cost of a bridge of the other kinds would be out of pro-
portion to the service required.
496. A suspension bridge consists of the towers or piers,
over which the main chains or cables pass ; the anchorages,
to which the ends of the cables are attached ; the main chains
or cables, from which the roadway is suspended ; the sus-
pending rods or chains, which connect the roadway with
the main chains ; and the roadway.
497. Towers. The towers, frequently termed piers, are
made generally of masonry, although iron has sometimes been
used. The particular form of the towers will depend in a
measure upon the locality and the character of the surround-
ings. Their dimensions will depend
upon their height and the amount
of strains which they will have to
resist.
Their construction will be governed
by the rules already given for the
careful construction of masonry.
A cast-iron saddle on rollers, to allow
of free motion in the direction of the
length of the main chains, is placed FIG. 197.
on each tower. (Fig. 197.)
The main chains may be fastened to these saddles, but thej
are generally passed over them.
374 CIVIL ENGINEERING.
The strains on the towers are produced by the vertical and
horizontal components of the tensions in the cables.
The tower must be built expressly to resist the crushing
forces due to this vertical component of the tension and the
weight of the masonry.
If the saddle was not free to move, the horizontal force
tending to push the tower over would be equal to the differ-
ence of the horizontal components of the tension in the two
brandies of the main chain. But since the saddle, by means
of the rollers, is free to move, the horizontal force acting at
the top of the tower must be less than the friction of the
rollers.
498. Anchorage. If the shore or bank be of rock, a ver-
tical passage should be excavated and a strong iron plate
placed in the bottom and firmly imbedded in the sides of the
passage. Through this plate the ends of the main chains are
passed and firmly secured on the under side. After the
chains are put in place the passage should be filled with con-
crete and masonry.
If the rock is not suitable, a heavy mass of masonry should
be built of large blocks of cut stone, well bonded together
for this purpose. In this case it is advisable to construct a
passage way, so that the chains and the fastenings may be
examined at any time. This mass of masonry, or the natural
rock to which the ends of the chain are fastened, is frequently
called the abutment. Its stability must be greater than the
tension of the chains. The principles of its stability are
precisely the same as those for the abutment of an arch ; its
weight and thickness must be sufficient to prevent its being
overturned ; and its centre of resistance must be within safe
limits.
499. Main chains or cables. These may be made of
iron bars, connected by eye-bar and pin joints; of iron links,
as in common chains ; of hoop or strap iron ; of ropes or
cables of wire, and in some cases of vegetable fibre, as hemp,
flax, or bark. When of ropes or strap iron they are of
uniform cross-section ; when of links they may have variable
cross-sections.
The smallest number of cables in a suspension bridge is
two, one to support each side of the roadway. Generally
more than two cables are used, since, for the same amount of
material, they offer at least the same resistance, are more
accurately manufactured, are liable to less danger of accident,
and can be more easily put in place and replaced than a single
chain of an equal amount of material.
SUSPENSION BEIDGE8. 375
Discussions have arisen as to the respective advantages
possessed by the chain and wire cables, some engineers pre-
ferring the former to the latter, and the reverse. The wire
cable is generally adopted in the United States.
The wire cable is composed of wires, generally from th
to th of an inch in diameter, which are brought into a cylin-
drical shape by a spiral wrapping of wire. Great care is
taken to give to each wire in the cable the same degree of
tension.
The iron wires are coated with varnish before they are
bound up into the cable, and when the cable is completed
the usual precautions are taken, as in other iron-work, to
protect it from rust and the action of the weather.
If the load placed on a cable be a direct function of its
length, the curve assumed by the mean fibre of the cable will
be a catenary. If it be a direct function of the span, it will
be a parabola. But the weight resting on the main chains is
neither a direct function of the length of the cable, nor of
the span, but a function of both. The curve is therefore
neither a catenary nor a parabola. But since the roadway,
which forms the principal part of the load, is distributed very
nearly uniformly over the span, the curve approaches more
nearly the parabola and in practice is regarded as such a
curve.
Knowing the horizontal distance between the tops of the
towel's and the deflection, the corresponding length of the
cable between the two points of support may be obtained by
the operation of rectifying the curve of a parabola. (Church's
Integral Calculus, Art. 235.) The length obtained by this
method will be expressed in terms containing logarithmic
functions. For this reason approximate formulas are made
which will give the length, in most cases, near enough for
practical purposes. Rankine gives the following approxi-
mate value for the length of a parabolic arc :
v 2
s = x + f ^- (nearly). . . (162).
Where the cable is to have a constant cross-section through-
out, the area of this section must be proportioned to the
greatest tension upon the cable. This tension is greatest at
the points of support when they are of the same height, or
at the highest point when the heights are unequal.
If the main chain is made of bars or links, it may be pro-
portioned to form a chain of uniform strength, in which case
the cross-sections will be made to vary from the lowest point
376 CIVIL ENGINEERING.
to the highest, increasing in area of cross-section as the strain
of tension increases. The horizontal component, or tension
of the lowest point, is dependent upon the parameter of the
curve. It therefore follows that for the same curve and the
same load on the unit of length throughout, the horizontal
component is the same for a bridge of a span of ten as for
one of a thousand feet. And it is also plain that the wider
the span, the deflection remaining constant, the greater will
be the tension on the cable, and the reverse.
500. Suspending chains. The roadway is suspended
from the cables by wire ropes or iron rods, which are placed at
equal distances along the cable, for the purpose of distributing
the load as uniformly as possible over the cables.
If the cables are composed of links or bars, the suspending
rods may be attached directly to them. If of rope, either of
wire or of vegetable material, the suspension rod is attached
to a collar of "iron of suitable shape bent around the cable, or
to a saddle-piece resting on it.
Where there are two cables, care must be taken to dis-
tribute the load upon the cables according to their degree of
strength.
In the Hungerford Suspension Bridge the method adopted
was as follows : The suspension rod, A (Fig. 198), was at-
tached to a triangular plate, B,
which hung by the rods, C and
D, from the main chain, E and
F. By this arrangement half
of the load on the rod, A, was
supported by each of the main
chains, E and F.
The suspending rods may
be vertical or inclined. In
recent constructions they are
FIG. 198. frequently inclined inwards,
for the purpose of giving ad-
ditional stiffness to the framing. The cross-section of the
rod is constant, and is determined by the amount of strain on
the upper section.
501. Roadway. The roadway in its construction does
not differ in principle from that used for other forms of
bridges. The roadway bearers are supported by the suspen-
sion rods. On the bearers are laid longitudinal joists, and on
them the planking, or the planking is laid directly on the road-
way bearers. The latter are stiffened by diagonal ties of iron
placed horizontally between each pair of roadway bearers.
SUSPENSION BRIDGES.
377
502. Oscillations. Suspension bridges, from the nature
of their construction, are wanting in- stiffness, and hence are
peculiarly liable to both vertical and horizontal oscillations,
caused by moving loads, action of winds, etc.
These oscillations cannot be entirely prevented, but their
effect may be reduced so as to be almost harmless.
When the banks will admit of it, guy-ropes of wire may
be attached to the roadway and fastened to points of the
bank beneath the bridge. The guy-ropes directly under the
bridge will be the most effective in resisting the vertical
oscillations; those oblique to the bridge, for resisting the
horizontal.
The elder Brunei fastened the roadway to a set of chains,
whose curve was the reverse of that of the main chains. The
reversed chains had a cross-section of about one-third of the
main chains, and preserved the shape of the roadway under a
movable load even better than the guys.
Engineers have made many efforts to provide for this want
of stiffness in suspension bridges and to fit them for railroad
uses.
A heavy moving load coming on a suspension bridge,
when at a point, as M (Fig. 199), causes the roadway and
cables to assume positions similar to those indicated by the
FIG. 199.
dotted lines in the figure. To prevent this deformation, the
cables are fastened at the points of greatest change by chains,
A E and B F, attached to the piers. These are known as
Ordish's chains.
Roebling effected the same result by fastening these points
of change in the roadway to the top of the towers, by the
lines, Da, Db, etc., as shown in Fig. 200.
It is agreed at the present time that the best method of
increasing the stiffness of a suspension bridge is to use, in
addition to the chains just named, trussed parapets on each
side of the roadway. These parapets form two open-built
beams, strongly connected and braced by the roadway, and
378
CIVIL ENGINEERING.
supported at intermediate points by the attachments to the
main chains. Each end of the roadway is firmly secured to
the base of the tower.
FIG. 200.
The objection to this method is the increase of the weight
placed upon the main chains.
503. Niagara Suspension Bridge, This bridge was
planned and constructed by Roebling, and illustrates the
method of stiffening just described.
The bridge affords a passage-way over the Niagara River, a
short distance below the Falls, both for a railroad and a com-
mon road. It consists of two platforms (Fig. 201), one above
Die"
FIG. 201.
the other, and about fifteen feet apart ; the upper is for the
railroad track, and the lower, B, is for the common road. The
platforms are connected by a lattice truss-work, C, C, on each
Bide, which serves to increase its stiffness. The whole bridge
is suspended by four main wire cables, F, F, F', F', the upper
NIAGARA SUSPENSION BRIDGE.
379
two being connected with the upper platform, and the lower
two with the lower platform.
Each platform consists of a series of roadway bearers in
pairs ; the lower covered by two thicknesses of flooring-
plank, the upper by one thickness ; the portion of the latter
immediately under the railroad track having a thickness of
four inches, and the remainder on each side but two inches.
The roadway bearers and flooring of the upper platform
are clamped between four solid-built beams ; two above the
flooring, which rest on cross supports ; and two, correspond-
ing to those above, below the roadway bearers ; the upper
and lower corresponding beams, with longitudinal braces in
pairs between the roadway bearers and resting on the lower
beams, being firmly connected by screw-bolts. The rails are
laid upon the top beams, forming the railroad track, A. A
parapet, D, D, of the form of the Howe truss is placed on
each side.
The lattice-work, C, C, which connects the upper and
lower platforms, consists of vertical posts in pairs (Figs. 202),
and ot diagonal wrought-iron rods, T, T. The rods pas*
380
CIVIL ENGINEERING.
through cast-iron plates fastened above the roadway bearers
of the upper platform, and below those of the lower, and are
brought to a proper bearing by nuts and screws on each end.
A horizontal rail of timber is placed between the posts of the
lattices at their middle, to prevent flexure.
The towers (Fig. 203) are
four obelisk-shaped pillars, each
sixty feet high, with a square
base of fifteen feet on a side,
and one of eight feet at the
top.
The height of the pedestals on
the Canada side is eighteen feet,
and on the United States twenty-
eight. An arch, C, connectg
the two pedestals, under which
is a carriage-way, D, for com-
municating with the lower plat-
form.
B \^ ft ^1 |s The ma i n cables pass over
1 I I* saddles on rollers placed on tops
....XQL...L.../A'.C....l 1 of the towers, and are fastened
FIG. 203. at their ends (Fig. 204) to chains
made of iron bars attached to an
anchoring plate, D, of iron, firmly secured in an anchorage of
rock, B, and a mass of masonry, A.
20'
FIG. 204.
The upper cables are drawn in towards the axis of the
bridge to reduce the amount of horizontal oscillations.
MOVABLE BBHKJE8. 381
The following are some of its principal dimensions :
Span of the cables, 821^ feet.
Deflection of upper cables (mean temperature), 54 feet.
Deflection of lower cables " " 64 feet.
Length of upper cables u " 1,193 feet.
Length of lower cables " " 1,261 feet.
Ultimate strength of the four cables, 12,000 tons.
Permanent weight supported by the cables, 1,000 tons.
Tensile stress in the four cables, 1,810 tons.
Height of railroad track above mean stage of water, 245
feet.
After a constant use of over 25 years, this bridge had its
anchorages reinforced (1877-8), and its superstructure re-
newed (1880) by substituting iron and steel for the wood.
(Transactions of the Am. Society of Civil Engineers, July,
1881.)
504. East River Suspension Bridge. This bridge
connects the cities of New York and Brooklyn, and was
thrown open to the public in 1883. Is was* planned by
Koebling and completed under the direction of his son, Col.
W. A. Roebling. Some of its dimensions are as follows :
Length of span over the river, 1,595 feet.
Total length of bridge, 5,989 feet.
Height of bridge at the centre over the river, 135 feet.
The ultimate strength of the cables is 49,200 tons, with a
tensile stress of 11,700 tons produced by an estimated load
of 8,120 tons.
CHAPTEK XVIIL
V. MOVABLE AND AQUEDUCT BRIDGES.
505. Movable bridges. In bridges over navigable rivers
it is often necessary that one or more spans be made to move
aside to allow of the passage of vessels. The term, movable
bridge, is therefore applied to any arrangement, whatever
be its nature, by means of which the roadway can at pleasure
be made continuous or broken, between two points of a per-
manent bridge, or over a water-way. The methods used to
effect this result are various.
They may be classed under five heads :
382 CIVIL ENGINEERING.
The passage may be opened or closed ; 1, by turning a
portion of the bridge around a vertical axis ; 2, by turning
it around a horizontal axis ; 3, by making it roll forwards
and backwards in a line with the bridge ; 4, by lifting it
vertically above the passage ; and 5, by floating it from and
into place upon the water.
506. I. By turning around a vertical axis. The term,
swing-bridge, is generally applied to a bridge which turns
about a vertical axis. This form of bridge is the one most
generally used when the opening is of any size. If two open-
ings are required, the bridge rests upon a masonry pier, which
is placed midway between the openings, and which supports
a circular plate, whose diameter is equal, or nearly equal, to
the breadth of the bridge. This plate has in the centre a
pivot surrounded by a circular track with rollers. On this
pivot and rollers the bridge is revolved horizontally, being
turned by suitable machinery.
If only one opening is required, the abutment is generally
used to support the mechanism for turning the bridge, care
being taken to place the pivot far enough back from the face
of the abutment so that the bridge, when open, shall not pro-
ject beyond it.
In calculating the strains on the parts of such a bridge, the
latter is usually considered when open, as composed of two
cantilevers, each loaded with its own weight ; when closed, as
a bridge of two spans.
507. II. By turning around a horizontal axis. Where
the width of the opening is small, the moving portion of the
bridge, which may be in one or two pieces, is lifted by chains
attached to the extremities, the operation of lifting being as-
sisted by counterpoises connected with the mechanism used.
One of the simplest counterpoises is a lever revolving on a
horizontal axis above the bridge, one end of the lever being
connected with the movable end of the bridge by a chain, the
other being weighted and connected with the mechanism by
which the bridge is lifted.
508. III. By moving a portion of the bridge forward
and backward in a line with its axis. Bridges of this
kind are placed upon fixed rollers, so that they can be moved
forward or backward, to interrupt or open the communication
across the water-way. The part of the bridge that rests upon
the rollers, when the passage is closed, forms a counterpoise
to the other. The mechanism usually employed for moving
these bridges consists of tooth- work, and may be so arranged
that it can be worked by one or more persons standing on the
AQUEDUCT BRIDGES. 383
bridge. Instead of fixed rolleis turning on axles, iron balls
resting in a grooved roller- way may be used, a similar roller-
way being affixed to the frame-work beneath.
Bridges of this class are known as rolling bridges.
509. TV. By lifting. In small bridges, like those over
canals, the bridge is sometimes hung by the four corners to
chains which pass over pulleys and have counterpoises at the
other ends. A slight force applied to it raises the bridge
to the required height, allowing the boats to pass under the
bridge.
510. Y. By floating. A movable bridge of this kind may
be made by placing a platform to form a roadway upon a
boat or a water-tight box of a suitable shape. This bridge is
placed in or withdrawn from the water-way, as circumstances
may require.
A bridge of this character cannot be conveniently used in
tidal waters, except at certain stages of the water. It may
be employed with advantage on canals in positions where a
fixed bridge could not be placed, in which case a recess in
the side or the canal is made to receive the bridge when the
passage-way is opened.
511. The general term, draw-bridge, is applied to all these
movable bridges, although technically the term is confined
to bridges of the second class, or those revolving around a
horizontal axis.
Movable bridges are either simple bridges or made of
truss-work belonging to one of the three systems already
named.
The objections to using either a tubular, an arched, or a
suspension bridge for a movable bridge are apparent. Where
either of these classes is used, the passage-way can only be
kept open by constructing the bridge so that a vessel can pass
beneath it.
512. Aqueduct bridges. In aqueducts for supplying
a city with water, the volume of water conveyed is com-
paratively small, and the aqueduct bridge will present no
peculiar difficulties except those of a water-tight channel.
The latter may be made either of masonry, or of cast-
iron pipes, according to the quantity of water to be de-
livered. If formed of masonry, the sides and bottom of
the channel should be laid in the most careful manner
with hydraulic cement, and the surface in contact with the
water should receive a coating of the same material, par-
ticularly if the stone or brick used be of a porous nature.
This part of the structure should not be commenced until the
384: CIVIL ENGINEEKING.
arches have been uncentered and the heavier parts of the
structure have been carried up and have had time to settle.
The interior spandrel-filling, to the level of the masonry
which forms the bottom of the water-way, may either be
formed of solid material, of good rubble laid in , hydraulic
cement, or of concrete ; or a system of interior walls, like
those used in common bridges for the support of the roadway,
may be used to sustain the masonry of the water-way.
In aqueduct bridges of masonry, supporting a navigable
canal, the volume of water is much greater than in the
preceding case, and every precaution should be taken to
procure great solidity, and to secure the structure from acci-
dents.
Segmental arches of medium span will generally be found
most suitable for works of this character. The section of
the water-way is generally of a trapezoidal form, the bot-
tom line being horizontal. For economy, the water-way is
usually made wide enough for one boat only ; on one side is
a tow-path for the horses, and on the other a narrow foot-
path.
The principle of the suspension bridge is well adapted to
aqueduct bridges, because, as each boat displaces its own
weight of water, the only moving load is the passage of men
and horses along the tow-path.
CHAPTER XIX.
BEIDGE CONSTRUCTION.
513. Before a bridge can be constructed there are three
things to be considered, viz. ; 1st, the site ; 2d, the water-way ;
3d, the design or plan.
Before a bridge can be designed a thorough knowledge of
the site, the amount of water-way, and the particular service
required of the bridge, must all be known.
514. Site. The site may already be determined, and it
may not be in the power of the engineer to change it. If it
is in his power to locate the site within certain limits, he
will select the locality which offers the most security to the
BRIDGE CONSTRUCTION. 385
foundations and which requires the least expense to be
incurred in their construction and in that of the bridge.
in many cases it is a matter of indifference where the
stream is crossed, but a careful survey of the proposed site
should always be made, accompanied by borings. The
object of this survey is to ascertain thoroughly the natural
features of the surface, the nature of the subsoil of the bed
and banks of the water-course, and the character of the
water-course at its different phases of high and low water,
and of freshets. This information should be embodied in a
topographical map; in cross and longitudinal sections of the
water-course and the substrata of its bed and banks ; and in
a descriptive memoir which, besides the usual state of the
water-course, should exhibit an account of its changes, occa-
sioned either by permanent or by accidental causes, as from
the effects of extraordinary freshets, or from the construction
of bridges, dams, and other artificial changes either in the
bed or banks.
Having obtained a thorough knowledge of the site, the
two most essential points next to be considered are to adapt
the proposed structure to the locality, so that a sufficient
water-way shall be left both for navigable purposes and for
the free discharge of the water accumulated during high
freshets ; and to adopt such a system of foundations as will
ensure the safety of the structure.
515. Water-way. When the natural water-way of a
river is obstructed by any artificial means, the contraction, if
considerable, will cause the water, above the point where the
obstruction is placed, to rise higher than the level of that
below it. This difference of level is accompanied by an in-
crease of velocity in the current of the river at this place.
This damming of the water above the obstruction, and in-
crease of velocity in the current between the level above and
the one below the obstruction, may, during heavy freshets,
cause overflowing of the banks ; may endanger, if not
entirely suspend, navigation during the seasons of freshets ;
and expose any structure which, like a bridge, forms the
obstruction, to ruin, from the increased action of the current
upon the soil around its foundations.
If on the contrary, the natural water-way is enlarged at
the point where the structure is placed, with the view of pre-
venting these consequences, the velocity of the current
during the ordinary stages of the water will be decreased,
and this will occasion deposits to be formed which, by gradu-
ally filling up the bed of the stream, might prove, on a sudden
386 CIVIL ENGINEERING.
rise of the water, a more serious obstruction than the struc-
ture itself; particularly if the main body of the water
should happen to be diverted by the deposit from its
ordinary course, and form new channels of greater depth
near the foundations of the structure.
For these reasons, the water-way to be left after the bridge
is built should be so regulated that no considerable change
shall be occasioned in the velocity of the current through it
during the most unfavorable stages of the water.
The beds of rivers are constantly undergoing change, the
amount and nature of which depend, upon the kind of soil oi
which they are composed, and the velocity of the current.
516. The following table shows, on the authority of Du
Buat, the greatest velocities of the current close to the bed
without injury to or displacement of the material of which it
is composed :
Soft clay 0.25 feet per second.
Fine sand 0.50 " "
Coarse sand and fine gravel. . . 0.70 " "
Gravel, ordinary 1.00
Coarse gravel, 1 in. in diameter 2.25
Pebbles, 1 J- in. in diameter. . . 3.33 " "
Heavy shingle 4.00 " "
Soft rock, brick, etc 4.50 " "
T? i \ 6.00
Rock jand greater.
Knowing the material of which the bed of the river at
the site is composed, and regulating the water-way so that
the velocity of the current close to the bottom after the
bridge has been erected, during the heaviest freshets shall
not exceed the limit of safety or disturbance of the material
forming the bed, the stability of the foundations is assured.
If the velocity should exceed the limits here given, precau-
tions must be taken to protect the foundations, as heretofore
described.
517. Velocity. The velocity of a current depends upon
the slope of the bed. Since the particles of water in contact
with the earth of the sides and bottom of the stream are
retarded by friction, it follows that in any cross-section the
velocity of the particles in the centre differs from those at
the bottom and on the sides. In ordinary cases it is suffi-
ciently exact to take the least, mean, and greatest velocities
as being nearly in the proportions of 3, 4, and 5 ; and for
very slow currents they are taken to be nearly as 2, 3, and 4.
WATEB-WAY. 387
The greatest velocity may be obtained by actual measure-
ment, by means of floats, current metres, or other suitable
apparatus, or it may be calculated from the slope of the bed
or the river at and near this locality.
Having determined the greatest velocity, the mean velocity
is taken as four-fifths of it. Col. Medley, in his Treatise on
Civil Engineering, takes the mean velocity as nine-tenths
(nearly) of the surface velocity when the latter exceeds three
feet per second, and four-fifths when less than this.
Having determined the mean velocity of the natural water-
way, that of the contracted water-way may be obtained from
the following expression,
o
v = m V, (163)
8
in which 8 and v represent, respectively, the area and mean
velocity of the contracted water-way; S and V, the same
data of the natural water-way ; and m a constant, which, as
determined from various experiments, may be represented by
the number 1,045.
Giving to s a particular value, that for v may be deduced,
and may then be compared with the velocity allowable at
this locality ; or, assuming a value for #, the value of s may
be deduced, and will be the area of the contracted water-
way. The safest width, or area of water-way, in many cases
may be inconveniently great ; therefore, some risk must be
run by confining the floods to more contracted limits. To
reduce this risk as much as possible is the object of the
engineer in seeking this information. With this information,
the engineer can decide upon the number of piers, hence the
number of spans of the bridge. Knowing the nature of the
bottom, the character and kind of piers and abutments may
be selected.
518. Design or plan of bridge. Before the engineer can
complete the design of the bridge, it is necessary that he
should know what service it has to perform : whether it is to
be a common or a railroad bridge ; whether a single or double-
track one. This information being given, and the knowledge
acquired of the site and water-way being furnished him, he is
able to decide whether the structure shall be a truss, arched,
or suspension bridge ; and, knowing the facilities at the place
for the construction of the work, can prepare an estimate of
its probable cost.
In deciding on the form of bridge which shall best com-
bine efficiency with economy, there are many things to be
388 CIVIL ENGINEERING.
considered. The cost of the superstructure, or all above the
piers and abutments, increases rapidly with the length of
span. Hence, economy would, as far as the superstructure is
concerned, demand short spans. But short spans require an
increase in the number of piers. When the height is small,
the stream not navigable,- and the piers easy to build, short
spans may be used ; but, if the foundations are in bad soils,
if the river is deep, with a rapid current, or liable to great
freshets, if it is navigable and requires an unobstructed
water-way, the construction of piers will be very expensive,
and therefore it is often desirable in these cases that there
should be few or no piers in the stream ; hence, long spans
are necessary, even at great cost. Good judgment and accu-
rate knowledge on the part of the engineer will be necessary,
in order that these and similar questions should be decided
correctly.
EEECTION OF BRIDGE.
519. The bridge having been planned, its parts all prepared
and taken to the site, the abutments and piers built, the next
step is to put it in position.
There are three methods, which have already been named,
viz., building the bridge on a scaffolding in the position it is
to occupy ; building it and rolling it in position, known as
launching / and building away from the site and then float-
ing it to the spot, and lifting it in place.
520. Scaffolding. The scaffolding is, so far as principle
is concerned, the same as that already described under the
head of masonry. That used for bridge construction is
simply a rough but rigid trestling, resting on the ground, or
on piles when the scaffolding is over water. The whole
arrangement is sometimes called staging, and frequently
false-works.
By means of this scaffolding the different pieces of the
structure are lifted in place and fastened together. When
the bridge is finished the staging is removed. This method
is the one most generally used.
521. Launching. This method has been used where the
scaffolding would have been too great an obstruction to the
stream or too costly. Deep and rapid rivers or ravines,
where the bridge is erected at a very high level, or rivers
with rapid currents subject to great freshets, are cases
where scaffolding would be costly, and in some cases imprac-
ticable.
COST OF BRIDGE.
522. Floating to site and lifting in place. This
method has been used in connection with the last method.
In this method the truss or tube is placed on boats 01
pontons, and floated to the spot it is to occupy. Then, bj
cranes or other suitable lifting machinery, the truss is lifted
to its place. This was the method adopted for the Britannia
Tubular Bridge.
In tidal waters this method has been used with great
success. The truss was put together on platforms on the
decks of barges, at a sufficient height above the surface of
the water, so that at high tide the truss would be above the
level of its final position. The barges were then floated into
position at high tide, and as the tide fell the truss was de-
posited in its proper place.
523. Cost. The cost of erecting a bridge is divided gene-
rally into four parts : 1, Scaffolding ; 2, Plant ; 3, Labor ;
4, Superintendence.
Scaffolding The cost of this forms an essential part of
the estimate, and depends greatly upon the facilities for
obtaining the proper materials in the vicinity of the site.
Plant. This is a technical word used to include the tools
and machinery employed in the work. The employment of
steam in so many ways at the present time renders this item
an important one in estimating the cost.
Labor. The number of men, their wages, subsistence, and
oftentimes their transportation, have all to be considered
under this head.
Superintendence. Good foremen and able assistants are
essential to a successful completion of the work. Their wages
may be included in the last item. It is usual to allow a given
percentage on the estimate to include the cost of superintend-
ence.
Summing these four items together, the cost of erecting
the superstructure of the bridge may be estimated.
PART VII
ROOFS.
CHAPTER XX.
524. The term roof is used to designate the covering
placed over a structure to protect the lower parts of the
Duilding and its contents from the injurious effects of the
weather. It consists of two distinct parts the covering and
the frames which support the covering. By some the term
roof is applied only to the "covering," exclusive of the
frames.
525. jRoofs are of various forms angular, curved, and
flat, or nearly so.
The most common form of roof is the angular. These
vary greatly in appearance and in construction. Some of the
most common examples of the angular roof are the ordinary
gabled, the hipped, the curb or Mansard, the French roof,
etc.
Curved roofs and domes are frequently used. They cost
more than the angular roofs, if the cost of the abutments be
included. But if the abutments already exist or if for other
reasons they have to be built, the curved roof, under these
circumstances, in many cases, may be found cheaper and more
suitable.
Flat roofs are very common, especially in hot climates.
The covering of these roofs rests upon beams placed in a
horizontal position, or one that is nearly so. The slope given
them is generally about 4 with the horizontal.
These roofs are easy to construct, and are simple in plan,
but they are heavy, do not allow the water to escape freely,
and there is a waste of material in their use.
526. Coverings. The coverings of roofs are made of
boards, shingles, slates, mastics, the metals, or any suitable
ROOFS. 391
material which will stand exposure to the weather and afford
a water-tight covering. The style of the building, and the
especial object to be attained, will govern their selection.
The extent of surface covered by them is usually expressed
in square feet. Sometimes the term square is only used, in
which case it means an area of 100 square feet.
The weight of the materials used for the covering is about
as follows :
Material. Weight per square foot.
Copper 1 Ib.
Lead Tibs.
Zincv 1.5 Ibs.
Tin fib.
Iron (common) 3 Ibs.
Iron (corrugated) 3.5 Ibs.
Slates 5 to 12 Ibs.
Tiles 7 to 18 Ibs.
Boards, 1 inch thick 2 Ibs.
Shingles.. lib.
-
These are fastened directly upon the frames, or upon
pieces of scantling and boarding which rest on the frames.
527. Frames. The frames which support the covering
have their exterior shape to correspond to the form of the
roof. These frames, known generally as roof-trusses, are
tied together and stiffened by braces which may occupy
either a horizontal or inclined position, and may be either
notched upon or simply bolted to the trusses.
The trusses are placed from five to ten feet apart, depend-
ing upon the weight of the covering and the amount of load
which each truss has to support. They rest usually upon
pieces of timber called wall-plates, laid on the wall to
distribute the pressure transmitted by the truss over a larger
surface of the wall.
528. Although nearly the last part of a building which is
constructed, the roof is one of the first to be considered in
planning the building, since the thickness and the kind of
wall depend greatly upon the weight of the roof. The
weight of the roof and the size of the "pieces to be used in its
construction, when the roof is flat, are easily determined.
The pieces are simple beams, subjected only to cross-strains,
and the joints are of the simplest kind.
When the roof is curved or inclined, these determinations
are more difficult. In these roofs the strains on the parts
produced by the covering are of different kinds, and must be
392 CIVIL ENGINEERING.
determined completely, both in amount and kind, before the
dimensions of the different pieces can be fixed, and the best
form of joints and fastenings selected.
In calculating the strains on a roof-truss, we > must take
.into consideration, besides the weight of the covering and of
the truss itself, the weight of the snow, ice, or water which
may at times rest upon the covering, the effect due the action
of the wind, and such extra loads as the weight of a ceiling,
of machinery, of floors, etc., which may be supported by the
frames.
The weight of the covering varies, as has been shown,
from one pound to twenty pounds upon the square foot.
The weight of the truss increases with the span, but it is
only in very wide spans that the weight of the parts and of
the whole truss have to be considered.
The weight of snow is assumed to be about one-tenth that
of the same bulk of water. Knowing the maximum depth
of the falls of snow, an approximate weight may be deter-
mined. Six pounds per square foot is the estimated weight
of snow adopted by European engineers. A greater weight,
even as high as twenty pounds, is recommended for the
northern part of the United States.
The action of the wind is very great in some localities.
Tredgold recommends an allowance of forty pounds to the
square foot as an allowance for its effect.
529. Rise and span. These are quantities dependent
upon circumstances. The rise is dependent upon the kind of
roof, the order of architecture used for the building, and the
climate. The span is dependent upon the size of the
building.
In gabled roofs and ordinarily angled roofs, the inclina-
tion which the sides of the roof make with the horizontal is
called the pitch. In countries where heavy falls of snow
are common the pitch is ordinarily made quite steep al-
though builders are now more generally inclined to a mode-
rate pitch, even for these cases. The objections to a steep
pitch are the exposing of a greater surface of the roof to the
direct force of the wind, the waste of room, etc. The mate-
rial of which the covering is composed affects the pitch. An
ordinary roof covered with shingles should have a pitch of at
least 22-| degrees ; one covered with slate or tiles a pitch
something greater, between 23 and 30 degrees.
The style of roof and architecture affect the pitch. Gothic
styles and parts of French roofs require a pitch of 4:5 degrees,
and even of 60 degrees.
ROOFS.
393
530. Materials used in construction. Wood and iron
are the materials used for the construction of the frames.
The truss may, as in other frames, be made entirely of wood,
or entirely of iron, or of a combination of the two materials.
Wooden Roof-trusses.
531. The simplest wooden truss is the triangular frame.
The inclined pieces are called rafters and the horizontal one
is termed the tie-beam.
It is used for spans of 12 to 18 feet, and when the roof
is light. For spans of 18 to 30 feet the king-post truss
(Fig. 205) is used. Its component parts are :
FIG. 205.
1. The principal rafters. These are the inclined pieces,
B B, which abut against each other or against the king-post
at the top.
2. The tie-beam. This is the horizontal beam, A, con-
nected with the lower ends of the rafters to prevent their
spreading out under the action of the load placed on them.
3. The king-post. The upright, C, framed at the upper
end upon the rafters and connected at the lower end with the
tie-beam.
4. Purlins. These are horizontal pieces, E, E, notched upon
or bolted to the rafters to hold the frames together and to
form supports for the common rafters, F, F.
5. Common rafters. These are inclined pieces, F, F, of
smaller dimensions than the principal rafters, placed from 1
to 2 feet apart and intended to support the covering.
6. Struts. The inclined pieces, D, D, framed into the
principal rafters and king-post to prevent the rafters from
sagging at the middle.
394
CIVIL ENGINEERING.
If the king-post and struts be removed, the simple triangu-
lar truss is left.
532. Queen-post truss, This truss is employed for spans
from 30 to 45 feet long. Its parts (Fig. 206) are all shown in
the figure ; C, C, being the queen-posts.
FIG. 206.
533. Iron roof- trusses, Wooden roof -trusses have been
used for wider spans than those named, but the use of iron in
building has enabled the engineer to construct roof-trusses of
wider spans which are much lighter and present a better
appearance.
These trusses are sometimes made of wood and iron in
combination, as we have seen in bridge-trusses, but now they
are more generally made entirely of iron.
The coverings are frequently made of iron, mostly corru-
gated, and are fastened to the purlins by the usual methods
for iron -work.
DETERMINATION OF THE KIND AND AMOUNT OF STRAINS ON THE
PARTS OF A ROOF-TRUSS.
534. Amount and kind of strains upon the different
parts of the simple king-post truss. The method of
determining the amount and kind of strains on the simple
triangular frame has already been explained. (Art. 256.)
It is usual, except in very short spans and where the tie-beam
supports nothing but its weight, to support the middle point
of this piece by a king-post. To find the strains on a tri-
angular frame with a king-post, let A B and A C (Fig. 207)
be the rafters, B C, the tie-beam, and A H, the king-post. The
king-post is so framed on the rafters at A, as to hold up any
load which it has to support. It is connected with the tie-
beam in such a manner as to keep the middle point, H, in the
same straight line with B and C.
ROOFS.
395
The strains on this truss are produced most usually by a
uniform load on the rafters and a load on the tie-beam.
Denote by I, the length of either rafter ; by w, the load on
Linit of length, including the weight of the rafters ; by W',
a unit
FIG. 207.
the weight of the tie-beam, including the load it has to sup
port, as a ceiling, floor, etc., and by a, the angle ABC.
The load on one of the rafters, as A B, will be wl, and acts
through the middle point, or at a distance from B equal to \l.
The strains produced by this load are compressive on the
rafter and tensile on the tie-beam, and the amount for each
may be determined, as shown in Art. 254.
The king-post is used to prevent the sagging of the tie-
beam at its middle point, it therefore supports, besides its
own weight, -fW (Art. 186), which produces a strain of ten-
sion on the king-post and which is transmitted by it to A,
where it acts as a load suspended from the vertex of the
frame. The strains produced by it on the rafters and tie-
beam may be determined as in Ait. 256.
The strains being known in amount and kind for each piece,
can now be summed and the total amount on the different
parts determined.
535. Strains on a king-post truss framed with struts.
Let Fig. 208 represent an outline of this truss. Let D F and
F G be the struts framed in the king-post and supporting the
rafters at their middle points.
The truss is supposed to be strained by a load uniformly
distributed over the rafters.
396 CIVIL ENGINEERING.
Adopt the notation used in the previous case and repre-
sent by ft, the angle A D F. We may neglect without
material error the weight of the struts and king-post, their
weights being small compared with the load on the rafters.
The load acts vertically downwards and is equal to wl for
each rafter. Acting obliquely, it tends to compress and bend
them. Each rafter is a case of a beam resting on three points
of support, hence the pressure on either strut is due to the
action of ^wl.
Pressure on the struts. The pressure on the strut D F
arises from the action of the component of %wl perpendicu-
lar to the rafter at the point, D. Denote by Pj the pressure
on the strut in the direction of its axis. To keep the point, D,
in the same straight line with A and B, the resistance offered
by the strut must be equal to the force acting to deflect the
rafter at that point. Hence there results,
P! sin ft = %wl cos a. ... (164)
From which we find
cos a
-r-,
sin /3'
for the pressure on the strut, D F. In the same way the pres-
sure on the strut F G is obtained, which in this case is exactly-
equal in amount.
Tension on king-post. This pressure, P 1? is transmitted
through the strut to the king-post at F. Resolving this force
into its components respectively perpendicular and parallel
to the axis of the king-post, we find the component in the
direction of the axis to be P t sin (ft a).
The king-post supports the tie-beam at its middle point.
Represent as before by W, the weight of the tie-beam and
its load, and we have -|W for the pull on the king-post from
this source. Represent the total stress of tension by T l5 and
there results,
T! = 2P X sin (ft - a) + f W. . . (165)
Substituting in this for Pj, its value just found, and the
value of T t will be known.
Tension on the tie-beam. Denote by T the tension on
the tie-beam produced by the thrust along the rafters, and
by Q, the vertical reaction at B caused by the load on the
rafters.
The relation between the normal components to the rafter,
BOOFS. 397
at B, of the three forces, Q, T, and -fowl acting at that point,
may be expressed by this equation,
T sin a = Q cos a -fowl cos a. . (166)
From which the value of T can be obtained when Q is
known.
Since the truss is symmetrical with respect to a vertical
through A, the sum of the reactions at B and C, due to the
strains on the rafters, is 2Q, and is equal to the total load
placed on the rafters, which is 2wl 4- -jj-W'. Hence
2Q = 2wZ + f W,
and
Q = wl +
which, substituting in equation (166), gives,
T sin a = \%wl cos a + A^' cos a >
and
Strains on the rafters. The forces acting in the direc-
tion of the rafters produce compressive strains, and those
perpendicular, transverse strains. These are determined as
previously shown.
Size of the pieces. Having found all the strains, the
limit on the unit of cross-section may be assumed and the
dimensions of the pieces obtained.
Remark. It is well to notice, that if we substitute for
P!, its value in the expression for T 1} the tension on the king-
post, that we will get
which may be put under the form
It is seen from this value of T 15 that whenever yS is equal
to 90 or differs but slightly from it, the expression will
reduce to the form
T! = f (W' + tool cos 2 a).
536. Strains on the queen-post truss. It is easily seen
398
CIVIL ENGINEERING.
from the foregoing how the strains on this truss may be de-
termined. It is usual to suppose the truss (Fig. 209) separ-
ated into two parts ; one the primary truss, BAG, and the
other, the secondary trapezoidal truss, B D G C.
FIG. 209.
In some cases, short rafters from C to G, and B to D, are
placed in contact with the principal rafters, A C and A B,
which further strengthens the truss by the additional thickness
given to the rafters in this part of the truss, and more fully
satisfies the condition of a secondary trapezoidal truss placed
within a triangular frame to increase its strength. There
are various other modifications of this truss, but the method
of determining the strains is not affected by them.
Iron Roof-trusses.
537. The trussing already explained under the head of
Bridges enters largely into iron roof-trusses. One of the
most common forms is the one in which the rafters are
trussed.
FIG. 210.
Roof-truss with trussed rafters. A common method
of supporting the middle point of a rafter is shown in Fig.
210. In this case the lower end of the strut, instead of
abutting against a king-post, is held up by tie-rods joining it
with the ends of the rafters.
BOOF8.
It is seen from the figure that each rafter, with the strut
and tie-rod, forms a simple king-post truss inverted. The
tie-rod connecting the points, E and F, completes the truss.
This tie-rod sustains the horizontal thrust produced by the
strains on the rafters, preventing its action on the walls at the
points of support, B and C.
In this truss the rafters are equal in length, and make
equal angles with the horizon ; the struts are placed at the
middle points and perpendicular to the rafter ; and the
strains are produced by a uniform load resting on the
rafters.
Use the notation of the previous cases, and denote by a
the angle ABC; by , the angle D B E ; by 25, B C ; by d,
the height A H ; and by d', the distance A K. The truss is
symmetrical with respect to a vertical A H, through the vertex,
A. Suppose the truss cut in two along this line, A H, we may
preserve the equilibrium, upon removing the left half, by
substituting two horizontal forces, one at A and the other at K.
Suppose this done, and represent these by H and T respec-
tively. As the weight of the tie-rods and struts is small
compared with the load on the rafters, we may neglect it with-
out material error.
The reaction at B is equal to wl.
The external forces acting on the right half of the frame are
the reaction at B, the horizontal forces H and T at A and K,
and the load on the rafter including its own weight. These
forces act in the same vertical plane.
The analytical conditions for equilibrium are
H T = 0, and wl-wl = 0,
and the bending moment at B is
We find the value of H = } .
d
The external forces are now all known and the strains pro-
duced by them may be determined.
Pressure on the struts. Considering the rafter as a
single beam, there results
P! = f wl cos a,
for the pressure on either strut.
Tension on the tie-rods of the rafters. Let T t be the
tension on the tie-rod B E, and T a the tension on A E.
4:00 CIVIL ENGINEERING.
At the point, B, the normal pressure must be equal to the
normal component of the resultant of the forces, wl and T a
acting at that point, which may be expressed as follows :
cos a = wl cos a T t sin /?,
and at A, for the same reason, we have
ffWl cos a = H sin a T 2 sin /3.
These equations give, since H is known,
(169)
for the tensions on the tie-rods B E and A E.
Tension on the main tie-rod, EF, of the truss.
From the analytical condition,
H T = 0,
there results,
T = H = i-^A . . . (170)
This may be verified. The stresses, P l5 T b and T 2 , in the
pieces connected at E (Fig. 210) have been determined.
These forces with T must be in equilibrium at E. Let us
find the components of these forces in the direction of the
strut, D E, and a perpendicular to the strut at E. (Fig. 211.)
For equilibrium, we have the fol-
lowing :
\\ D (T t + T 2 ) sin ft - T sin a - P t = 0,
\ / and
r ^ ^/
XVv (T 2 TI) cos & + T cos a = 0.
/
\ T} Substituting in the first of these
pf x equations, the values of P t , T 1? and T 2 ,
FIG 211. already obtained, there results,
-J-J- wl cos a -f- H sin a f wl cos a T sin a 0, or H = T.
In a similar manner, by substitution in the 2d, it can be
shown that the condition is satisfied, or H = T.
Compression on the rafters. The compression on the
rafter at B is due to the components of the forces acting at
that point parallel to the rafter. Hence
Compression at B = wl sin a + TI cos /3,
BOOFS.
401
and
Compression at A = H cos a + T 2 cos 0. (171)
Frequently in the construction of this truss, the struts are
extended until they meet the tie-rod -joining B and C. (Fie.
212.)
PIG. 212.
In this case the stresses are the same as those just deter-
mined in the struts and rafters, but are less in the* secondary
tie-rods, because of the increase in the angle ft.
538. When the span is considerable, this method of truss-
ing is oftentimes used to increase the number of supports for
the rafter. By adding to the trussed rafter, the two struts,
bf&nd cd (Fig. 213), and the two secondary tie-rods, fD and
# D, two additional points of support are furnished to the
rafter.
FIG. 213.
The points, b and , are midway between B D and A D, divid
ing the rafter into four equal parts, and making the triangles
Bj D and D d k equal to each other and similar to B E A.
Using the previous notation, the reaction at B is wl, and the
horizontal force at A is -J - , as in previous case. The
Cu
external forces are all known.
Pressure on the struts. The struts are respectively per-
pendicular to the rafter; the normal components of the
forces acting at 5, D, and c will give the amount of pressure
on each strut, due to the load acting at these points. Repre-
sent this component at D by P 1? at b and c by P a , and at A
26
4-02 CIVIL ENGINEERING.
and B by P 8 . Since the rafter is kept by the struts in such ft
position that b, D, and c are in the same straight line with A
and B, it is an example of a beam resting on five supports, and
we have,
P 3 = -^Yzwl cos a, P 2 = f wl cos a, and P t = $%wl cos a.
This value of P 2 is the amount of pressure acting on either
of the struts, bfor cd, and the strain on them is determined.
That on D E is still to be determined.
Tension on the secondary tie-rods. Let T! be the ten-
sion on the rod, Bf, and we have,
wl cos a = wl cos a T! sin
from which we get
And in the same way we find the tension T\ on kd to be
Hsina cos a
1 i =
. -rrs* f
sm ft sin ft
Denote by T 2 , T 8 , T' 2 , and T' 8 the tensions on/D,/E, d D,
and d E respectively. Since an equilibrium exists between
the forces acting at the point f, and the same at d, the com-
ponents of these forces, taken respectively parallel and per-
pendicular to the rafter, must fulfil the following conditions :
T 2 + T 8 - T t = 0, and (T 8 - T 2 - T t ) sin/3 + P 2 = 0,at/,
and
T'j + T' 8 - T'i = 0, and (T' 8 - T', - T'J sin ft + P 2 = 0, at d.
The values of T l9 P 2 , and T^, have already been found.
The values for the others are easily deduced. They will be
as follows :
and T' s = -r (H sin a - f&wl cos a).
The strains of tension and compression on all the secondary
pieces have been obtained excepting for the strut, D E, at the
middle. This can now be determined.
KOOFS. 403
Strain on strut, D E, at the middle. This strain is due
to the pressure, P ly and the components of T 2 and T' 2 in the
direction of the strut, or
Compression on D E = P = P t + (T 2 + T' 2 ) sin .
Substituting in this for T 2 , T 2 , and P 1? their values already
found, we finally obtain,
P = $%wl cos a,
for the stress in the strut, D E.
The amount and kind of strain on each piece are now
known, and the strength of the truss may therefore be deter-
mined.
539. Roof -truss in which the rafters are divided into
three segments and supported at the points of division J>y
struts abutting against king or queen-posts.
This form of truss shown in Fig. 214 is in common use for
roofs. In this case, the rafters are trisected respectively at
the points, H, D, G, and M, by the struts H K, D F, G F, and
L F K
FIG. 214.
M L, which have their lower ends connected with and abutting
against the vertical rods at the points K, F, and L, where these
rods are fastened to the tie-rod B C.
The usual method of determining the amount of strains on
the different parts of a frame of this kind is to consider it as
formed of several triangular ones. In this particular case,
we consider the truss A B C as made up of the secondary
trusses, B H K, B D F, and B F A, on the right of A F, and
a similar set on the left of it.
The strains are supposed to arise from a uniform load over
the rafters, the weight of the vertical ties and the struts being
neglected, as in the previous cases.
In the previous examples, the rafters have been regarded
as single beams resting on two, three, five, etc., points of sup-
port, and the reactions of these points of support have been
taken as the value of the load resting upon them. This pro-
cess may be followed in this case and is to be preferred,
whenever the rafters, A B and A G, are continuous.
404: CIVIL ENGINEERING.
In most treatises on roofs, the action of the load on the
points of support is considered in a different manner. There
are two general methods. Taking either half of a truss
of the kind just described, one method supposes that each
segment of the rafter supports one-third of the entire load
on the rafter ; or, each segment is considered a beam sup-
ported at its ends and uniformly loaded. According to this
hypothesis, since %wl is the load on the segment, ^wl will
act at the points, H and D, and wl, at B and A, of the half
ABF.
The other method assumes the pressures exerted at the
four points of support to be equal to each other, that is, %wl
to be the load acting at each of the points, B, H, D, and A.
This is sometimes called " the method of equal distribution
of the load."
Adopting the first method, assuming one-third of the load
on the rafter as resting on each segment, let us first determine
the strains in the secondary truss, B H K.
Strains on B HK. By hypothesis, the pressure at H is \wl,
acting vertically downwards. The problem then is the case
of a simple triangular frame sustaining a load at the vertex.
Denote by a, the angle H B K ; since the triangle is isos-
celes, the components of %wl along the rafter and strut are
each equal to 4-^ , and develop compressive stresses in
sin a
H B and H K.
The stress transmitted to B produces a vertical pressure on
the point of support equal to \wl and a stress of tension in
B K equal to \wl cot a.
In like mariner, the stress transmitted to K produces a ver-
tical pull at that point equal to \wl which is sustained by the
tie-rod, D K, and a horizontal stress equal to and directly op-
posed to the stress of tension at B.
Strains on B D F, The problem in this case is that of the
simple triangular frame, sustaining a weight at the vertex.
The load v acting at D is %wl, increased by the pull on the
tie-rod, D K, or \wl, and is supported by the rafter B D and
the strut D F. Since these pieces do not make equal angles
with the vertical through D, the components of %wl in the
directions of these pieces are not equal. Kesolving, we
wl ,
find the one in the direction of the rafter will be -5-7- , and
bill CL
the other along the strut, i^ 5- ; being the angle D F K.
KOOF8. 405
The first of these is transmitted to B, where it produces a
vertical pressure equal to {wl, and a tensile stress in the
tie-beam equal to {wl cot a.
The other, transmitted to F, produces a pull on the king-
post equal to \wl, and a tensile stress in the tie-beam equal
to and directly opposed to that at B produced by the com-
ponent acting along the rafter.
Strains on BFA. The stress in AD is due to the assumed
load, {wl at A and the transmitted stress in the king-post,
{wl, or is equal to \wl.
Resolving this into its components along the rafter A B
and a horizontal at A, we have for the first. { -^ , and
sin a'
for the latter, { wl cot a.
The former transmitted to B produces a vertical pressure
equal to {wl, and a tensile stress in the tie-beam equal to {wl
cot a.
The horizontal component at A is balanced by an equal and
directly opposite component due to the half A C F.
Strains on the whole truss. Knowing the strains on
one-half, and the truss being symmetrical about the vertical
through A, the stresses in all the pieces can now be determined.
Summing and recapitulating, the stresses are as follows :
T D u /~ M 1 w % 1 wl . wl .wl
In B H C M=4 : -- 1-4 : -- \-{- =f-s - , compressive.
sin a sin a sin a sin a
-.
sin a 2 sm a d sm a
DA=GA=*4^-,
sm a
H K=M L=J-, andDF=G F=
6 sin sin
" D K=G L = {wl, and A F = \wl + {wl = \wl, tensile,
" B K=C L = $wl cot or, and K F = F L = \wl cot a, "
By the use of moments. These same values may be ob-
tained by using the " method of sections." To apply this
method to determine the stresses in the rafter, suppose
a vertical section of the rafter made on the right of and con-
secutive to A, and take the point F as the centre of moments.
Represent the compressive stress in the segment, A D, of
406 CIVIL ENGINEERING.
the rafter cut by this section, by C^ Its direction is paral.
lei to A B, and its lever arm, which denote byj?, will be
equal to a perpendicular let fall from F upon the rafter.
The reaction at B and the load on the rafter are known.
For equilibrium we would have,
B F
d x p = wl x B F wl x -g-,
whence
d = \wl x .
We find p to be equal to -j- , which being substituted in
this expression gives
Substituting in (172) the value of d = I sin a, we obtain
wl
vrhich is the same value already determined.
If the same method be applied to the segment H B, we
will find the value of C 3 to be equal to 4- -^ .
6 sin a
540. In the preceding roof -truss, the inclined pieces were
struts and the verticals were ties. Another form of truss is
one in which the verticals are struts and the diagonals are
ties. (Fig. 215.) The rafters are subdivided into a number
of equal segments. At each point of division, a strut is
placed, and kept in a vertical position by the main tie-beam
and the inclined tie-rods, as shown in the figure.
FIG. 215.
The methods previously explained will enable the student
to determine the kind and amount of strains on each piece of
the truss.
BOOF8.
407
541. It has been recommended to check the accuracy of
the calculations by some other method than the one used ;
the graphical method is a very convenient one for this purpose.
Let us apply this method to finding the strains in the roof-
truss referred to in Art. 539.
The load over the rafters is supposed to act as there taken,
viz., -J- at A and B, and % at H and D, each.
Assume any point, as 0. From 0, on a vertical line, lay
off, according to a scale, Ob = %wl, bh = $wl, hd = fyol,
and da = \wl. These distances represent the loads acting
at B, H, D, and A, respectively. Their sum 00 = wl, hence,
aO=wl represents the reaction at B, due to the load acting
on the half A B F of the truss. The forces at B are Ob, aO,
and the stresses in the pieces B H and B K. Through 5,
draw bf parallel to B H, and through a, draw af parallel to
B K. The polygon aObfa will represent the system of forces
acting at B, and the lines fa and bfwill represent the inten-
sities of the stresses in K B and B H, respectively, at B, and
may be taken off with the same scale used to lay off the ver-
tical forces, Ob, bh, etc.
It is seen that the forces acting at H are the weight \wl =
bh, the stress >/, and the unknown stresses in H K and H D.
Through /*, draw fa parallel to H K, and through A, draw hg
parallel to H D. The polygon, fbhqf will represent the in-
tensities of the forces acting at H.
The forces acting to strain B K and H K have been deter-
mined ; the forces acting at K in the directions of K D and
K F are unknown. Through g, draw gk parallel to K D, and
through a, draw ak parallel to K F, forming the polygon
qfgka ; the lines gk and ka, will represent the intensities of
the stresses in D K and K F.
The stresses in the pieces K D and HD being known, the
stresses in DA and DF can be determined.
408
CIVIL ENGINEERING.
In a^ similar way, the stresses in the other pieces can be
determined.
542. Application of graphical method to the roof
with trussed rafters. Let us apply the same method tc
the trussed roof of Art. 537. Instead of the frame being uni-
formly loaded over the rafters, consider it as supporting a
load W at the vertex A. (Fig. 217.)
The applied forces acting on the frame are the load "W
and the reactions at B and C. Assume a point, as 0, and lay
off on a vertical line the distance Ob to represent W. The
distances be and cO will represent the reactions at B and at
C. Through 5, draw ~bd parallel to B D, and through , the
line cd parallel to B E. The triangle bed will represent the
FIG. 217.
system of forces acting at B. Through 0, draw the line
Og parallel to AC, and through 0, draw the line eg parallel to
C F. The triangle Oeg will represent the forces acting at C.
Going to E, since the load on the truss has been supposed
to act at A, there will be no strain on D E,- and the forces at
E will be those acting in the direction B E already found, and
the unknown forces along E A and E F. Through d, draw da
parallel to E A, and through 0, draw ca parallel to E F. The
triangle cda will represent these three forces acting at E.
And in the same way, the triangle ega would represent the
strains on the pieces at F.
If there had been a force acting at E in the direction of
D E, then there would have been three unknown forces acting
at E, and we could not have solved the problem until one of
these were known.
KOOFS. 409
Purlins.
543. The pnrlins are simply beams, and are considered as
resting on two or more supports, according to the number
of frames connected by them. The strains are easily deter
mined.
CONSTRUCTION OF EOOFS.
544. The most important element of the roof is the frame.
The same rules given for frames, and the general methods
described for their construction apply to the construction of
the roof-truss.
410 CTVTL ENGINEERING.
PART VIII.
KOADS, RAILROADS, AND CANALS.
CHAPTER XXI.
ROADS.
545. A road is an open way or passage for travel, forming
a communication between two places some distance apart.
A path or track over which a person can travel on foot is
the simplest form of a road. A line, having been marked
out or " blazed" between two places, is soon beaten into a well-
defined path by constant use. A person travelling over a road
like this will tind nothing but a beaten path on the surface of
the ground, with few or no modifications of its surface, and
generally with no conveniences for crossing the streams or
rivers which intersect it.
As the travel over a road of this kind increases and beasts
of burden begin to be used for packing the merchandise,
baggage, etc., which are to be carried over the route, modifi-
cations and improvements of the path become necessary.
For convenient passage of the animals, the path must be
widened, the brush and undergrowth removed, temporary
bridges constructed or means of ferriage provided for cross-
ing streams of any considerable depth, and steep ascents and
descents must be modified and rendered practicable for the
pack-animals. The term "trail" is used to designate the
original path and also the path when improved so that it can
be used by pack-animals.
Since transportation by wheels is cheaper and more rapid
than by pack-animals, the next step will be to still further
improve the road so that vehicles on wheels can be used over
the route. This necessitates a still further widening of the
trail, a further reduction of the slopes so as to render them
practicable for carts and wagons, the providing of means to
ROADS. 411
cross the streams where they cannot be forded, and the raising
of the ground in those localities where it is liable to be over
flowed. In this condition, the trail is called a road.
As the travel over this kind of road increases, the wants and
conveniences of the community demand a further improve-
ment of the road so that the time taken in going over it and
the cost of transportation shall be reduced. This is effected
by shortening the road where possible, by reducing still further
the ascents and descents or by avoiding them, and by improv-
ing the surface of the road.
It has been proved that a horse can draw up a slope of ^
only one-half the load he can draw on a level. Hence, a level
road would enable one horse to do the work required of two
on a road with these slopes.
It has been shown that a horse can draw over a smooth,
hard road, as one of broken stone, from three to four times
as much as he can draw on a soft earthen road. It therefore
follows that an improvement of the surface will be accom
panied by a reduction both in time and cost of the transpor-
tation.
546. The engineer may be required to lay out and make
a road practicable for wagons connecting two settlements
or points, in a wild, uninhabited, and therefore unmapped
country, as is the case frequently on our frontier, or he may
be required to plan and construct a road having for its ob-
jects the reduction of time and expense of transportation,
in a country of which he has maps and other authentic
information. In either case, the general principles guiding
the engineer are the same. These may be considered under
the following heads : 1st, Direction 2d, Gradients j 3d,
Cross-Section ; 4th, Road-Coverings 5th, Location ; 6th,
Construction.
DIRECTION.
547. Other things being equal, the shortest line between
the two points is to be adopted, since it costs less to con-
struct ; costs less for repairs ; and requires less time and labor
to travel over it.
But straightness will be found of less consequence than
easy ascents and descents, and as a rule must be sacrificed to
obtain a level or to make a road less steep.
Good roads wind around hills instead of running over
them, and this they may often be made to do without increasing
their lengths. But even if the curved road, which is prac-
4:12 CIVIL ENGINEERING.
tically level, should be longer, it is the better ; for on it a
horse will draw a full load at his usual rate of speed, while
on the road over the hill, the load must be diminished or the
horse must reduce his rate of speed.
Roads often deviate from the straight line for reasons
of economy in construction, such as to avoid swampy, marshy,
or bad ground, or to avoid large excavations, or to reach
points on streams better suited for the approaches of bridges,
etc.
Great care must be exercised in deciding on the line which
the road is to follow. If the line is badly chosen, the ex-
pense of construction and repair may be so great that it may
finally be necessary to change the line and adopt a new one.
548. The considerations which should govern the selec-
tion of the line are : to connect the termini by the most
direct and shortest line ; to avoid unnecessary ascents and
descents ; to select the position of the road so that its longi-
tudinal slopes shall be kept within given limits ; and to so
locate the line that the cost of the embankments, excavations,
bridges, etc., shall be a minimum.
The wants of the community in the neighborhood of the
line oftentimes affect the direction of the line, since it may
be advisable and even more economical in the end to change
the direction so as to pass through important points which do
not lie on the general direction of the road than to leave
them off the road.
GRADIENTS.
549. Theoretically, every road should be level. If they
are not, a large amount of the horse's strength is expended in
raising the load he draws up the ascent. Experiment has
shown that a horse can draw up an ascent of y-J-g-, only 90
per cent, of the maximum load he can draw on a level ; up
an ascent of ^, he can draw about 80 per cent. ; of -g^, he
can draw only 64 per cent. ; of ^, only 50 per cent. ; and of
jijf, only 25 per cent.
These numbers are affected by the nature and condition of
the road, being different for a rough and for a smooth road,
the resistance of gravity being more severely felt on the
latter.
A level road is therefore the most desirable, but can seldom
be obtained. The question is to select the maximum slope
or steepest ascent allowable.
An ascent affects chiefly the draught of heavy loads, as has
been already shown.
GBADIENTS. 413
A descent chiefly affects the safety of rapid travelling.
550. The slope or grade of a road depends upon the kind
of vehicle used, the character of the road-covering, and the
condition in which the road is kept. From the experiments
above mentioned it would seem that the maximum grade for
ascent should not be greater than 1 in 30, although 1 in 20
may be used for short distances.
For descent, the grade should be less than the angle of
repose, or that inclination at which a vehicle at rest would
not be set in motion by the force of gravity. This angle
varies with the hardness and smoothness of the road-covering,
and is affected by the amount of friction of the axles and
wheels of the vehicles. On the best broken stone roads iii
good order, for ordinary vehicles, the maximum grade is
taken at 1 in 35.
Steeper grades than these named produce a waste of ani-
mal power in ascending and create a certain amount of dan-
ger in descending.
551. Although theoretically the road should be level, in
practice it is not desirable that it should be so, on account of
the difficulty arising of keeping the surface free from water.
A moderate inclination is therefore to be selected as a mini-
mum slope for the surface of the road. This slope is taken
at 1 in 125, and in a level country it is recommended to form
the road by artificial means into gentle undulations approxi-
mating to this minimum.
It is generally thought that a gently undulating road is
less fatiguing to a horse than one which is level. Writers
who hold this opinion attempted to explain it physiologically,
stating that as one set of muscles of the horse is brought into
play during the ascent and another during the descent, that
some of the muscles are allowed to rest, while others, those in
motion, are at work. This explanation has no foundation in
fact, and is therefore to be rejected. The principal advan-
tage of an undulating road is not the rest it gives the horse,
but the facilities which are afforded to the flowing of the
water from the surface of the road.
CROSS-SECTION.
552. The proper width and form of roadway depend upon
the amount and importance of the travel over the road.
Width. The least width enabling two vehicles to pass
with ease is assumed at 16 feet. The width in most of the
States is fixed by law.
4:14 CIVIL ENGINEERING.
In England, the width of turnpike roads approaching
large towns, on which there is a great amount of travel, is
60 feet. Ordinary turnpike roads are made 35 feet wide. Or-
dinary carriage roads across the country are given a width
of 25 feet ; for horse-roads, the width is 8 feet ; and for foot-
paths, 6-J feet.
Telford's Holyhead road is made 32 feet wide on level
ground ; 28 feet wide in moderate excavations ; and 22 feet
in deep excavations and along precipices.
In France there are four classes of main roads. The first
or most important are made 66 feet wide, the middle third of
which is paved or made of broken stone. The second class
are 52 feet wide ; the third are 33 feet wide ; and the fourth
are 26 feet wide. All these have the middle portion ballasted
with broken stone.
The Roman military roads had their width established by
law, at twelve feet when straight and sixteen when crooked.
Where a road ascends a hill by zigzags it should be made
wider on the curves connecting the straight portions ; this in-
crease of width being one-fourth when the angle included
between the straight portions is between 120 and 90, and
one-half when the angle is between 90 and 60.
553. Form of roadway. The surface of the road must
not be flat, but must be higher at the middle than at the
sides, to allow the surface water to run off freely.
If the surface is made flat, it soon becomes concave from
the wear of the travel over it, and forms a receptacle for
water, making a puddle if on level ground, and a gulley if
the ground is inclined.
The usual shape given the cross-section of the roadway
is that of a convex curve, approaching in form a segment of
a circle or an ellipse. This form is considered objectionable
for the reasons that water stands on the middle of the road ;
washes away its sides ; that the road wears unequally, and
is very apt to wear in holes and ruts in the middle ; and that
when vehicles are obliged to cross the road, they have to
ascend a considerable slope.
554. The best form of the upper surface of the roadway is
that of two inclined planes rounded off at their intersection
by a curved surface. The section of this curved surface is a
flat segment of a circle about flve feet in length.
The inclination of the planes will be greatest where the
surface of the road is rough and least where it is smoothest
and hardest. A slope of -fa is given a road with a broken
stone covering, and may be as slight as -^ for a road paved
with square blocks. The transverse slope should always
DITCHES. 415
exceed the longitudinal slope of the road, so as to prevent the
surface water from running too far in the direction of the
length of the road.
On a steep hillside, the surface of the roadway should be
a plane inclined inwards to the face of the hill. A ditch on
the side of the road next to the hill receives the surface
water.
555. Foot-paths. On each side of the roadway, foot-paths
should be made for the convenience of passengers on foot.
They should be from five to six feet wide and be raised about
six inches above the roadway. The upper surface should
have an inclination towards the " side channels," to allow the
water to flow into them and thence into the ditches. When
the natural soil is firm and sandy, or gravelly, its surface will
serve for the foot-paths ; but if of loam or clay, it should be
removed to a depth of six inches and the excavation filled
with gravel.
Sods, eight inches wide and six inches thick, should be
laid against the side slope of the foot-path next to the road, to
prevent the wash from the water running in the side chan-
nels.
Fences, hedges, etc., where the road is to be enclosed,
should be placed on the outside of the foot-paths, and outside
of these should be the ditches. (Fig. 218.)
FIG. 218. a, cross-section of roadway; ft, ft, foot-paths ; /,/, fences;
c?, d, ditches ; s, s, side drains.
556. Ditches. Ditches form an important element in the
construction of a good road.
The surface of the road has been given a form by means
of which the water falling on it is carried off into the gut-
ters or side channels of the road, whence it is conveyed by
side drains, s, s (Fig. 218), into ditches, which immediately
carry off all the water which enter them.
The ditches are sunk to a depth of about three feet below
the roadway, so that they shall thoroughly drain off the
water which may pass through the surface of the roadway.
These ditches should lead to the natural water- courses of the
country, and have a slope corresponding to the minimum lon-
gitudinal slope of the road. Their size will depend upon
circumstances, being greater where they are required to carry
416 CIVIL ENGINEERING.
away the water from side-hills or where they are made in
wet grounds. A width of one foot at the bottom will gen-
erally be sufficient.
There should be a ditch on each side of the road, on level
ground or in cuttings. One is sufficient where the road is on
the side of a hill.
557. Side-slopes. The side-slopes of the cuttings and
embankments on each side of the road vary with the nature
of the soil.
Rock cuttings may be left vertical or nearly so, Common
earth should have a slope of at least f, and sand, -J. Clay is
treacherous and requires different slopes according to its liabil-
ity to slip and the presence of water. The slope required in
each case is best determined by observing the slope assumed
by these earths in the locality of the work where exposed to
the weather.
When the road is in a deep cutting, the side slopes should
not be steeper than J, so as to allow the road, by its exposure
to the sun and wind, to be kept dry.
Whenever the side-slopes are of made earth, earth removed
and placed in position like that of an enbankment, the slopes
should be more gentle.
ROAD-COVERINGS.
558. The road-covering of a common country road, and
most generally of all the new roads in our country, is the
natural soil thrown on the road from the ditches on each side,
in many cases there are even no ditches, and the road-cover-
ing or upper surface of the roadway is the natural soil as it
exists on the hard subsoil beneath, when the soft material has
been removed by scraping or by some other method.
Roads of this kind are deficient in the qualities of hardness
and smoothness. To improve these roads, it is necessary to
cover the surface with some material, as wood, stone, etc.,
which will substitute a hard and smooth surface for the soft
and uneven earth, and which, acting as a covering, will pro-
tect the ground beneath from the action of the water that
may fall upon it.
559. Roads may be classified from their coverings as
follows :
I. EARTH ROADS.
II. ROADS OF WOOD.
III. GRAVEL ROADS.
IV. ROADS OF BROKEN STONK.
COBDTJBOY EOADS. 417
Y. ROADS PAVED WITH STONE.
VI. KOADS COVERED OE PAVED WITH OTHEB MATF.RTAT.fl.
VII. TRAM-EOADS.
L EAETH EOADS.
560. These are the most common and almost the only kind
of roads in this country. From what has been said, we know
that they are deficient in hardness and generally in smooth-
ness. In wet weather, when there is much travel of a heavy
kind over them, they become almost impassable.
The principal means of improvement for these roads are to
reduce the grades, thoroughly drain the roadway, and freely
expose the roadway to the influence of the sun and wind. In
repairing them, the earth used to fill the holes and hollows
should be as gravelly as possible and free from muck or
mould. Stones of considerable size should not be used, as
they are liable to produce lumps and ridges, making an un-
even surface disagreeable to travel upon.
H. EOADS OF WOOD.
561. Corduroy roads. When a road passes over a marsh
or soft swampy piece of ground which cannot be drained, or
the expense of which would be too great, a corduroy road
is frequently used. This kind of road is made by laying
straight logs of timber, either round or split, cut to suitable
lengths, side by side across the road at right angles to its
length.
It is hardly worthy of the name of a road, and is extremely
unpleasant to persons riding over it, but it is nevertheless
extremely valuable, as otherwise, the swamp across which it
is laid would at times be impassable.
562. Plank roads. In districts where lumber is cheap and
gravel and stone cannot be easily obtained, road-coverings of
plank have been used.
The method most generally adopted in constructing a road
of this class consists in laying a flooring or track, eight feet
wide, of boards from nine to twelve inches in width and three
inches in thickness. The boards rest upon two parallel rows
of sleepers, or sills, laid lengthwise of the road, and having
their centre lines about four feet apart, or two feet from the
axis of the road.
The boards are laid perpendicular to the axis of the road,
27
4:18 CIVIL ENGINEERING.
experience having shown that this position is as favorable to
their durability as any other and is also the most economical.
When the road is new and well made, it offers all the ad-
vantages of a good road and is a very pleasant one to use.
But when the planks become worn and displaced it makes a
very disagreeable and indifferent road.
Some years ago they were much used, but as a general
thing they are no longer built except under very peculiar
and urgent circumstances.
HI. GRAVEL ROADS.
563. These are roads upon which a covering of good gravel
has been laid.
The roadway is first prepared by removing the upper layer
of soft and loose earth, and thoroughly draining the road.
The bed is sometimes of the shape of the upper surface of the
road, but more generally it is merely made level : on this a
layer of gravel about four inches in thickness is laid, and
when compacted by the travel over it another layer is laid,
and so on until a thickness of sixteen inches at the centre
has been reached.
It is advisable to compress the bed by rolling it well with
a heavy iron roller before beginning to lay the gravel. In
some cases a bed of broken stone has been u?ed.
Gravel from the river shores is generally too clean for this
kind of road, there not being enough clayey material mixed
with it to bind the grains together. On the other hand,
gravel from pits is apt to be too dirty and requires a partial
cleansing to fit it for this purpose.
The gravel used should be sifted through screens, and all
pebbles exceeding two inches in diameter be broken into
small pieces or rejected.
The iron roller can be advantageously used to assist in
compacting the layers of gravel as they are put on the road.
A gravel road carefully made, with good side ditches to
thoroughly drain the road-bed, forms an excellent road.
Some gravel roads are very poor, even inferior to an earth
road, caused in a great measure by using dirty gravel which
is carelessly thrown on the road in spots, which cause the road
1o soon wear into deep ruts and hard ridges.
IV. ROADS OF BROKEN STONE.
564. The covering of roads of this class, both in this
country and Europe, is composed of stone broken into small
TELFOED BOADS. 4:19
angular fragments. These fragments are placed on the natu-
ral bed in layers, as in the gravel road, or they may be placed
in layers on a rough pavement of irregular blocks of stone.
565. Macadamized roads. When the stone is placed on
the natural road-bed, the roads are said to be " macadamized,"
a name derived from Mr. Me Adam, who iirst brought this
kind of road into general use in England.
The construction of this road is very similar to that just
given for a gravel road. The roadway having received its
proper shape and having been thoroughly drained, is covered
with a layer of broken stones from three to four inches thick.
This layer is then thoroughly compacted by allowing the
travel to go over it and by rolling it also with heavy iron
rollers ; care being taken to fill all the ruts, hollows, or other
inequalities of the surface as fast as they are formed. Suc-
cessive layers of broken stone are then spread over the road
and treated in the same manner, until a thickness of between
eight and twelve inches of stone is obtained. Care-is taken
that the layers, when they are spread over the surface, are
not too thick, as it will be difficult, even if it be possible, to
get the stone into that compact condition so necessary for a
good road of this kind.
566. Telford roads. This is the name given to the broken
stone roads in which the stone rests on a rough pavement
prepared for the bed. (Fig. 219.)
PIG. 219.
This pavement is formed of blocks of stone of an irregular
pyramidal shape ; the base of each block being not more
than five inches, and the top not less than four inches.
The blocks are set by the hand as closely in contact at their
bases as practicable ; and blocks of a suitable size are selected
to give the surface of the pavement a slightly convex shape
from the centre outwards. The spaces between the blocks
are filled with chippings of stone compactly set with a small
hammer.
A layer of broken stone, four inches thick, is then laid
over this pavement, for a width of nine feet on each side of
the centre ; no fragment of this layer should measure over
4-20 CIVIL ENGINEERING.
two and a half inches in any direction. A layer of broken
Btone of smaller dimensions, or of clean coarse gravel, is
spread over the wings to the same depth as the centre layer.
The road- covering, thus prepared, is thrown open to travel
until the upper layer has become perfectly compact ; care
having been taken to fill in the ruts as fast as formed with
fresh stone, in order to obtain a uniform surface. A second
layer, about two inches in depth, is then laid over the centre
of the roadway ; and the wings receive also a layer of new
material laid on to a sufficient thickness to make the outside
of the roadway nine inches lower than the centre. A coat-
ing of clean coarse gravel, one inch and a half thick, is then
spread over the surface, and the road-covering is considered
as finished.
The stone used for the pavement may be of an inferior
quality in hardness and strength to the broken stone on top,
as it is but little exposed to the wear and tear occasioned by
travelling. The surface-stone should be of the hardest kind
that can be procured.
567. Kind of stone used for broken stone roads. The
stone used for these roads should be selected from those
which absorb the least water, and are also hard and not brit-
tle. All the hornblende rocks, porphyry, compact feldspar,
and some of the conglomerates furnish good, durable road-
coverings. Granite, gneiss, limestone, and common sand-
stones are inferior in this respect, and are used only when the
others cannot be obtained.
568. Repairs. Broken stone roads to be good must be
kept in thorough repair. If the road is kept in order it will
need no repairs. The difference between " kept in order "
and " repairs " is that the latter is an occasional thing, while
the former is a daily operation. To keep the road in order
requires that the mud and dust be daily removed from the
surface of the road and that all ruts, depressions, etc., be at
once filled with broken stone.
It is recommended by some that when fresh material is
added, the surface on which it is spread should be broken
with a pick to the depth of half an inch to an inch, and the
fresh material be well settled by ramming, a small quantity
of clean sand being added to make the stone pack better.
When not daily repaired by persons whose sole business it is to
keep the road in good order, general repairs should be made
in the spring and autumn by removing all accumulations of
mud, cleaning out the side channels and other drains, and
adding fresh material where requisite.
If practi cable, the road -surf ace at all times should be kept
ROMAN ROADS. 421
free from an accumulation of mud and dust, and the surface
preserved in a uniform state of evenness by the daily addition
of fresh material wherever the wear is sufficient to call for it.
"Without this constant supervision, the best constructed road
will, in a short time, be unfit for travel, and with it the weak-
est may at all times be kept in a tolerably fair condition.
V. ROADS PAVED WITH STONE.
569. A good pavement should offer but little resistance to
the wheels, and at the same time give a firm foothold to
horses ; it should be durable, free from noise and dirt, and
so constructed as to allow of its easy removal and replace-
ment whenever it may be necessary to gain access to gas or
water pipes which may be beneath it.
570. Roman roads. The ancient paved Roman roads,
traces of which may still be seen as perfect as when first
made, were essentially dressed stone pavements with concrete
foundations resting on sub-pavements. The entire thickness
of the road-covering was about three feet, and was made as
follows :
The direction of the road was marked out by two parallel
furrows in the ground, and the loose earth from the space
between them removed. A bed of mortar was then spread
over the earth, and on this the foundation (statumeii), com-
posed of one or two courses of large flat stones in mortar,
was laid. On this foundation was placed a course of con-
crete (rudm), composed of broken stones. If the stones
were freshly broken, three parts of stone to one of lime were
used ; if the stone came from old buildings, two parts of lime
were used. On this course a third (nucleus), composed of
broken bricks, tiles, pottery, mixed with mortar, was placed.
In this layer was imbedded the large blocks of stone (sum-
ma crustd) forming the pavement. These stones were ir-
regular in form, rough on their under side, smooth on their
upper, and laid so that the upper surface should be level.
They were laid with great care and so fitted to each other as
to render the joints almost imperceptible.
When the road passed over marshy ground, the foundation
was supported by timber- work, generally of oak ; the timber
was covered with rushes, reeds, and sometimes straw, to pro-
tect it from contact with the mortar.
On each side of the roadway were paved foot-paths.
571. English paved roads. Some of the paved roads in
England are partial imitations of the Koman road. This
422 CIVIL ENGINEERING.
pavement (Fig. 220) was constructed by removing the sur-
face of the soil to the depth of a foot or more to obtain a iirm
bed. If the soil was soft it was dug deeper and a bed of
sand or gravel made in the excavation. On this a broken
stone road-covering similar to those already described was
laid. On this broken stone was spread a layer of fine clean
FIG. 220.
gravel, two and a half inches thick, on which rested the pav-
ing stones. The paving stones were of a square shape, and
were of different sizes, according to the nature of the travel
over the road. The largest size were ten inches thick, nine
inches broad, and twelve inches long ; the smallest were six
inches thick, five inches broad, and ten inches long. Each
block was carefully settled in its place by means of a heavy
rammer ; it was then removed in order to cover the side of
the one against which it rested with hydraulic mortar; this
being done, the block was replaced, and properly adjusted.
The blocks of the different courses across the roadway break
joints.
This pavement fulfils all the conditions required of a good
road -covering, presenting as it does a hard even surface to
the action of the wheels, and reposing on a firm bed formed
by the broken-stone bottoming. The mortar-joints, so long
as they remain tight, will effectually prevent the penetration
of water beneath the pavement.
572. Belgian pavement. This pavement, so named from
its common use in Belgium, is made with blocks of rough
stone of a cubical form measuring between eight and nine
inches along the edge of the cube. These blocks are laid on
a bed of sand ; the thickness of this bed is only a few inches
when the soil beneath is firm, but in bad soils it is increased
to from six to twelve inches. The transversal joints are usu-
ally continuous, and those in the direction of the axis of the
road break joints. In some cases the Hocks are so laid that
the joints make an angle of 45 with the -*ds of the roadway,
one set being continuous, the other set breaking joints. By
this arrangement of the joints, the wear upon the edges of
the blocks, by which the upper surface soon assumes a con-
vex shape, is diminished. It has been ascertained by experi-
ence, that when the blocks are laid in the usual manner, the
WOODEN PAVEMENTS. 423
wear upon the edges of the block is greatest at the joints
which run transversely to the axis.
When a bed of concrete is used, instead of or in addition
to a bed of sand, and the upper surface of the blocks is rec-
tangular instead of square, there results a pavement much
used in New York City.
573. Cobble-stone pavement. Rounded pebbles (cobble
stones) are used frequently for pavements. This pavement
is composed of round or egg-shape pebbles, from five to ten
inches long, three to six inches wide, set on end in a bed of
sand or fine gravel, and firmly settled in place by pounding
with a heavy rammer. A.f ter the stones are driven, the road-
surface is covered with a layer of clean sand or gravel, two
or three inches thick.
The objections to this pavement are its roughness; the
resistance offered to the wheels ; the noise ; the ease with
which holes are formed in the road by the stones being pressed
down in the ground by heavy loads passing over them ; the
difficulty of cleaning its surface ; and its need of frequent
repairs.
574. Kind of stones used for pavements. The fine-grained
franites which contain but a small proportion of mica, and the
ne-grained silicious sand-stones which are free from clay,
form good material for blocks for paving. Mica slate, talcose
slate, hornblende slate, some varieties of gneiss, and some
varieties of sand-stone of a slaty structure, yield excellent
materials for pavements for sidewalks and paths.
VI. ROADS OF OTHER MATERIALS.
575. "Wooden blocks have been much used recently in
paving the streets of our towns and cities. Brick, concrete,
asphalte, and even cast iron, are or have been used for road-
coverings. Roads near blast-furnaces are frequently seen
covered with the slag from the furnaces, and those near kilns
where cement is burned, with cinders and clinkers from
the kilns. Road-coverings of charcoal have been tried in
Michigan and Wisconsin.
The wooden, brick, and asphaltic pavements are the most
common of these.
Wooden pavements. Wooden pavements are the same
in principle as stone. The road-bed is formed and the
blocks of wood are placed in contact with each other upon
the surface of the road-bed as described for the blocks of
fltone pavements. The wooden blocks are parallelopipedons
424 CIVIL ENGINEERING.
in form and are laid with the grain of the wood in the direc-
tion of the depth of the road. From slight differences in the
details of construction of wooden pavements there has arisen
quite a variety of names, as the Nicolson, the bastard Nicolson,
the Stowe, the Greeley, the unpatented, etc., all using the
wooden blocks, but differing slightly in other ways.
Wooden pavements offer a smooth surface ; are easily
kept clean ; not noisy ; easy for the horses and vehicles ;
pleasant to ride upon ; and are cheaper at first cost than
stone pavements. For these reasons they have been much
used in the United States.
They are, however, slippery in wet weather ; soon wear out ;
and unfit for roads or streets over which there is a heavy travel.
True economy forbids their use except as temporary roads.
576. Asphaltic coverings. Asphaltic roads may be com-
posed of broken stone and this covered with asphaltic con-
crete, or the broken stone covered with ordinary concrete and
this overlaid with a covering of asphalte mixed with sand.
Asphaltic roads present a smooth surface which does not
become slippery by wear ; a surface free from dust and mud ;
not noisy ; and from its imperviousness to moisture forms
an excellent covering over the road-bed beneath and prevents
the escape of noxious vapors from below.
Asphaltic roads properly made are growing steadily in
favor and when they are better known will be more generally
adopted for all streets in towns and cities, over which the
travel is light.
VH. TRAM-ROADS.
577. In order that the tractive force should be a minimum,
the resistance offered to the wheels of the carriage should be
a minimum. In other words, the harder and smoother the
road, the less will be the tractive force required. But car-
riages drawn by horses require that the surface of the road
should be rough, to give a good foothold to the horses' feet.
These two opposite requirements are united only in roads
with track-ways, on which there are at least two parallel
tracks made of some hard and smooth material for the wheels
to run upon, while the space between the tracks is covered
with a different material suitable for the horses' feet. Con-
structions of this class are termed i: tram-roads " or " tram-
ways." The surface of the tracks or " trams " are made flush
with that of the road and are suitable for the wheels of ordi-
nary carriages. Their construction will be alluded to in the
next chapter.
BECONNOISSANCE. 425
CHAPTER XXIL
LOCATION AND CONSTRUCTION OP ROADS.
578. In establishing a road to afford means of communi-
cation between two given places, there are several points
which must be considered by the engineer and those inter-
ested in its construction. These are the kind of road to be
selected, the general line of direction to be chosen or located,
and the construction of the road.
The selection of the kind of road depends upon the kind
of travel which is to pass over it ; the amount of travel, both
present and prospective ; and the wants of the community
in the neighborhood of the line. The location and construc-
tion of the road depend upon the natural features of the
country through which the road must pass, and as these come
exclusively within the limits of the engineer's profession,
they alone will be considered in this chapter.
LOCATION.
579. Reconnoissance. The examination and study of the
country by the eye is termed a reconnoissance, and is usually
made in advance of any instrumental surveys, to save time
and expense. The general form of the country and the ap-
proximate position of the road may frequently be determined
by it.
A careful examination of the general maps of the country,
if any exist, will lessen the work of the reconnoissance very
much, as by this the engineer will be able to discover many
of the features which will be favorable or otherwise to the
location of the road in their vicinity.
Roads alon the bank of a large stream will have to cross
a number of tributaries. Roads joining two important
streams running nearly parallel to each other must cross high
ground or dividing ridges between the streams.
An examination of the map will show the position of the
streams, and from these the engineer may trace the general
directions of the ridges, determine the lowest and highest
points, and obtain the lines of greatest and least slopea
4:26
CIVIL ENGINEERING.
With this information the directions of the roads leading from
one valley to another may be approximately located.
It is seen (Fig. 221) that if A and B are to be joined by a
road, that the road may run direct from A to B, as shown by
the dotted line joining them, or it may go, by following the
FIG. 221.
general directions of the streams, through C, as shown by the
dotted line A C B. By the first route, the road would be
apparently shorter, but the ascents and descents would be
greater ; by the second, the road would be longer, but the
ascents and descents more gentle, and the total difference of
level to be passed over would be less.
We can draw this conclusion from the fact that the streams
have made for themselves channels which follow the lines of
gentlest slope. And that if two streams flow in the
same
direction, the high ground or ridge separating them has
the same general direction and inclination as the streams.
And if two streams approach each other near their sources,
as those at C in the figure, that this indicates a depression in
the main ridge in this vicinity.
Hence long lines of road usually follow the valleys of
streams, obtaining in this way moderate grades and crossing
the ridges by the lowest passes.
The engineer having studied thoroughly the map and made
himself acquainted with the natural features of the country
as there indicated, proceeds to make a personal examination
of the ground, to identify these natural features, and to verify
the conclusions deduced from the study of the map.
In making the examination, he goes both forwards and
backwards over the ground so as to see it from both direc-
ESTIMATE OF THE COST. 4:27
tions, and in this way verify or correct the impressions he has
received as to its nature.
By means of the reconnoissance he establishes " approxi-
mate " or " trial lines " for examination. These lines are
marked out by " blazing " if in a wooded country, or by stout
stakes driven at the important points if the country be a
cleared or open one.
580. Surveys. The surveys are divided into three classes :
preliminary surveys, surveys of location, and surveys of con-
struction.
The preliminary survey is made with ordinary instru-
ments, generally a transit and a level, and has for its object
the measurement of the length of the road, the changes of
direction of the different courses, the relative heights of the
different points or differences of level along the line, and of
obtaining the topography of the country passed over in the
immediate neighborhood of the line.
The line is run without curves, and therefore, when plot-
ted, consists of a series of straight lines of different lengths,
forming at their connection angles of varying size.
The levelling party, besides taking the measurements re-
quisite to construct a profile of the line, make cross-section
levellings at suitable points, so as to show the form of surface
of the road.
The topography on each side of the line is ordinarily
sketched in by eye ; instrumental measurements being occa-
sionally made to check the work.
581. Map and memoir. The results of these surveys
are mapped, and all the information gathered during the
survey which cannot be shown on the map is embodied in a
memoir.
From these trial lines thus surveyed, the engineer makes a
selection, being governed by the considerations mentioned in
Art. 567, viz., shortness of route, avoidance of unnecessary
ascents and descents, selection of favorable grades, and econ
omy of construction.
582. Estimate of the cost. This can be made ap-
proximately after the engineer has established the grades.
The kind of road and the character of the travel over it
generally fix the limits of its longitudinal slopes. To fix
them exactly, the engineer constructs the profiles of the dif-
ferent sections of the road and draws the "grade lines" on
these profiles, keeping their slopes within the general limit
already assumed. Thus in a profile (Fig. 222) the grade
line A "B is drawn, following the mean or general slope of the
ground, equalizing as far as possible the undulations of tho
428
CIVIL ENGINEERING.
profile above and below the grade line. The inclination of
the grade line with the horizontal is then measured, and if its
slope falls within the limit assumed, the grade is a satisfactory
one and the amounts of excavation and embankment are
nearly equal. If the inclination be found too steep, either
FIG. 222.
the top of the hill must be cut down or the length of the line
between the two points at top and bottom be increased. The
latter is the method usually adopted. Thus if the road laid
out on a straight line joining C and D (Fig. 223) requires a
\
FIG. 223.
steeper grade than the maximum grade adopted, the length
of the road between these points, C and D, may be increased
by curving it, as shown by the line C E F D. The length to
give this winding road is easily determined so that the grade
of every portion of the road shall be kept within the assumed
limit. The proper grade line having been determined and
drawn on the profiles, the height of the embankments and
the depth of the cuttings are determined.
Knowing the width of the road, the form of its surface,
and the inclination of the side slopes, the cubical contents of
the excavations and embankments may be calculated, and an
estimate of the cost made.
The comparative costs of the routes being determined and
the considerations mentioned in last article given their full
weight, the engineer selects the particular line for the road.
It is well to say that it happens often that no trial lines
SURVEYS. 429
are necessary ; the route to be followed by the road being
apparent.
583. Survey of location. The route being selected, it is
gone over again and more accurately surveyed. It is care-
fully levelled at regular intervals in the direction of its length,
and cross-levels at all important points are made. The angles
made by the changes of direction of the line are rounded off
by curves, the curves being generally arcs of circles. Ad-
vantage is taken of this survey to place the line in its best
position so as to reduce to a minimum the embankments
and excavation, and to give the best approaches to the points
where streams are to be crossed.
The line is divided into a number of divisions, and maps of
these divisions are made showing the road in plan and the
longitudinal and cross-sections of the natural ground, with
the horizontal and vertical measurements written upon them.
By these maps, the engineer can lay out the line on the
ground and can determine the amount of excavation and
embankment required for each division.
Besides these maps, detailed drawings of the road-covering,
of the bridges, culverts, drains, etc., with the written specifi-
cations explaining how the work on each must be done, should
be prepared.
The work is now in the condition that estimates of its cost
can be accurately made and its construction begun.
584. Survey of construction, The road is constructed
by contract or " day labor." Whichever method is adopted,
it is first necessary to " lay out the work." This laying out
the work forms the third class of surveys, or survey of con-
struction.
From the maps showing the location, the engineer proceeds
to mark out the axis of the road upon the ground by means
of stout pegs or stakes driven at equal intervals apart, using a
transit or theodolite to keep them in the proper line. These
stakes are numbered to correspond with the same points indi-
cated on the map.
The width of the roadway and the lines on the ground
corresponding to the side slopes of the excavations and em-
bankments, are laid out in the same manner, by stakes placed
along the lines of the cross profiles.
Besides the numbers marked on the stakes, to indicate their
position on the map, other numbers, showing the depth of the
excavations, or the height of the embankments from the sur-
face of the ground, accompanied bv the letters Cut. Fill.' to
indicate a cutting, or a filling, as the case may be, are also
added to guide the workmen. The positions of the stakes on
430 CIVIL ENGINEERING.
the ground, which show the principal points of the axis of the
road, should be laid down on the map by bearings and dis-
tances from bench-marks in their vicinity, in order that the
points may be readily found should the stakes be subsequently
misplaced.
Curves. Curves are not necessary for common roads, but
it always looks better even in a common road to join two
straight portions by a regular curve than by a bent line.
Curves are laid out by means of offsets from a chord or tan-
gent, or by angles of deflection from the tangent. The latter
method, using a transit or theodolite, is the one most com-
monly employed.
CONSTRUCTION.
585, Earth-work. This term is applied to all that relates
to the excavations and embankments, whatever be the mate-
rial excavated or handled.
Excavations. In forming the excavations, the inclination
of the side slopes demands particular attention. This incli-
nation will depend on the nature of the soil, and the action of
the atmosphere and internal moisture upon it. In common
soils, as ordinary earth formed of a mixture of clay and sand,
hard clay, and compact stony soils, although the side slopes
would withstand very well the effects of the weather with
a greater inclination, it is best to give them a slope of -J ;
as the surface of the roadway will, by this arrangement, be
better exposed to the action of the sun and air, which will cause
a rapid evaporation of the moisture on the surface. Pure
sand and gravel require a slope of -J. In all cases where the
depth of the excavation is great, the base of the slope should
be increased. It is not usual to use artificial means to protect
the surface of the side slopes from the action of the weather ;
but it is a precaution which, in the end, will save much labor
and expense in keeping the roadway in good order. The
simplest means which can be used for this purpose, consist in
covering the slopes with good sods, or else with a layer of
mould about four inches thick, and sown with grass-seed.
These means will be amply sufficient to protect the side
slopes from injury when they are not exposed to any other
causes of deterioration than the wash of the rain and the
action of frost on the ordinary moisture retained by the soil.
The side slopes form usually an unbroken surface from the
foot to the top. But in deep excavations, and particularly in
soils liable to slips, they are sometimes formed with horizontal
offsets, termed benches, which are made a few feet wide and
EMBANKMENTS. 431
have a ditch on the inner side to receive the surface-water
from the portion of the side slope above them. These benches
catch and retain the earth that may fall from the portion of
the side slope above.
In excavations through solid rock, which does not disinte-
grate on exposure to the atmosphere, the side slopes might be
made perpendicular ; but as this would .exclude, in a great
degree, the action of the sun and air, which is essential to
keeping the road-surface dry and in good order, it will be
necessary to make the side slopes with an inclination, varying
according to the locality ; the inclination of the slope on the
south side in northern latitudes being greatest, to expose bet-
ter the road-surface to the sun's rays.
Embankments. In forming the embankments, the side
slopes should be made less than the natural slope ; for the pur-
pose of giving them greater durability, and to prevent the width
of the top surface along which the roadway is made from
diminishing by every change in the side-slopes, as it would
were they made with the natural slope. To protect more
effectually the side-slopes, they should be sodded or sown in
grass seed ; and the surface-water of the top should not be
allowed to run down them, as it would soon wash them into
gullies and injure the embankment. In localities where
stone is plenty, a retaining wall of dry stone may be advan-
tageously substituted for the side-slopes.
To reduce the settling which takes place in embankments,
the earth should be laid in successive layers, and each layer
well settled with rammers. As this method is expensive, it
is seldom resorted to except in works which require great
care, and are of small extent. For extensive works, the
method usually adopted is to embank out from one end, carry-
ing forward the rork on a level with the top surface. In
FIG. 224.
this case, as there must be a want of compactness in the
mass, it is best to form the outsides of the embankment
tii-st, and to gradually fill in towards the middle, in order
that the earth may arrange itself in layers with a dip
towards the centre (Fig. 224). This arrangement will in a
432 CIVIL ENGINEERING.
great measure counteract the tendency of the earth sliding
off in layers along the sides.
586. Removal of the earth. In both excavation and
embankment, the problem is " to remove the earth from the
excavation to the embankment or place of deposit by the
shortest distance, in the shortest time, and at the least
expense." This is an important problem in practice, and
its proper solution affects very materially the cost of the
work.
The average distance to which the earth is carried to form
the embankment is called the lead, and is assumed to be
equal to the right line joining the centre of gravity of the
volume of excavation with that of the embankment. When
this lead is made the least possible, all other things being
equal, the cost of removal of the earth is a minimum.
In the execution of earthwork, it is not always advisable
to make the whole of an embankment from the adjoining
cuttings, as the lead would be too long. In such a case, a
part of the cutting is wasted, being deposited in some conve-
nient place, forming what is known as a spoil-bank. The
necessary earth required to complete the embankment is
obtained from some spot nearer to the work, and the cutting
or excavation made in supplying it is called a borrow-pit.
Means used to move the earth. The earth is loosened
by means of ploughs, picks, and shovels, and then thrown
into wheelbarrows, carts, or wagons to be removed. A
scraper drawn by a horse is frequently used to great advan-
tage.
Resort is sometimes had to blasting to loosen the soil, when
it is rock, hard clay, and even frozen earth.
The advantages of wheelbarrows over carts, and carts over
wagons, etc., depend upon circumstances. When the earth
is to be transported to a considerable distance, the wheelbar-
row becomes too expensive. By combining the cost of filling
the cart or wheelbarrow, the amount removed, and the time
occupied in transporting the earth in each case, the cost of the
two methods can be obtained and compared with each other.
Shrinkage. When embankments are made in layers com-
pacted by ramming or by being carted over, the subsequent
settling is quite small. But made in the usual way, there ia
always a certain amount of settling which follows and which
is provided for by making at first the embankment a few
inches higher than it is to be. Earth occupies a less space in
an embankment than in its natural state ; that is, a greater
number of cubic yards of excavation is required to form an
embankment than there are cubic yards in its voluma
This shrinkage of the earths is about as follows :
SIDE-HILL BOAD8. 433
G ravel shrinks about eight per cent.
Gravel and sand nine per cent.
Clay and clayey earth ten per cent
Loain and light earths twelve per cent.
On the contrary, rock occupies more space when broken
up than it does in its natural state, the percentage of its
increase in volume varying with the way the fragments are
piled together. Carelessly piled its increase of volume was
found to be about seventy-five per cent, and when care-
fully piled, fifty per cent.
587. Methods of obtaining the quantities to be exca-
vated, etc. In comparing the costs of the routes or for
rough estimates, it is sufficiently exact to take a number
of equidistant profiles, and calculate the solid contents
between each pair, either by multiplying the half sum of
their areas by the distance between them, or else by taking
the profile at the middle point between each pair, and
multiplying its area by the same length as before ; the first
of these* methods gives too large a result, and the second too
small.
Where an exact estimate is to be made, the Prismoidal
formula (Mensuration, p. 129) should be used. This formula
gives the exact contents.
588. In swamps and marshes. When the embankment
is made through a swamp or marsh, many precautions are
necessary.
If the bog is only three or four feet deep and has a hard
bottom, it is recommended to remove the soft material and
build the embankment on the hard stratum.
If it be too deep to remove the soft material, its surface,
provided it be not too soft, may be covered with some sub-
stance to form an artificial bed for the embankment. Kows
of turf with the grassy side downward have been used.
Brushwood has also been tried. %
If the swamp be deep and the material quite fluid, the first
thing to do is to drain it, and then prepare an artificial bed
for the embankment.
589. Side-tiill roads. When a road runs along the side
of a hill, it is usually made half in excavation and half in
embankment. But as the embankment is liable to slip if
simply deposited on the natural surface of the ground, the
latter .should be cut into steps or offsets (Fig. 225). A low
stone wall constructed at the foot of the embankment will add
to its stability.
If the surface of the hill be very much inclined, the side
slopes of both the excavation and the embankment should be
28
434
CIVIL ENGINEERING.
replaced by retaining walls of dry stone (Fig. 226), or of stunes
laid in mortar.
The upper wall may be dispensed with when the side hill
is of rock.
FIG. 225.
"When the road passes along the face of a nearly perpen-
dicular precipice at a considerable height, as around a pro-
jecting point of a rocky bank of a river in a mountainous
FIG. 226.
district, it may rest on a frame- work of horizontal beams let
into holes drilled in the face of the precipice and supported
at their outer ends by inclined struts beneath, the lower ends
of which rest in notches formed in the rock.
CROSS DRAINS. 435
DRAINAGE.
590. A system of thorough drainage, by which the water
rrtat filters through the ground will be cut off from the soil
beneath the roadway, to a depth of at least three feet below
the bottom of the road-covering, and by whic'i the water
falling upon the surface will be speedily conveyed off, before
it can filter through the road-covering, is essential to the good
condition of a road.
The form of the road, the side drains, and the ditches (Fig
218), are arranged and constructed with this object in view
(Art 556.) **.
591. Covered drains or ditches. As open ditches would
be soon filled by the washings of the side-slopes in certain
parts of the roads, covered drains (Fig. 227) are substituted
ror them in these places.
Fig. 227.
They may be constructed with a bottom of concrete, flag-
ging, or brick, with sides of the same material, or as shown
in the figure, and covered with flat stones, leaving open joints
of about half an inch to give free admission to the water. The
top is covered with brushwood or with fragments of broken
stone, or with pebbles and clean gravel, through which the
water will filter freely without carrying any earth or sedi-
ment into the drain.
592. Cross drains. Besides the covered drains parallel to
the axis of the road in cuttings, other drains known as cross
drains are made under the roadway. They should have a
slope along the bottom to facilitate the escape of the water.
A slope of 1 in 100 will be sufficient.
They may be constructed in the same manner as the cov-
ered drains* or trenches may be dug to the required depth
436 CIVIL ENGINEERING.
with the proper slope and filled with broken stone. On the
stone a layer of brushwood is placed and over this the road-
covering. Drains of this kind are known as blind ditches.
Any construction will be effective which will leave a small
open waterway at the bottom of the trench which will not be-
come choked with sediment.
If the road is level, the cross drains may run straight across,
but if inclined they form a broken line, in plan the shape of
the letter V, with the angular point in the centre of the road
directed towards the ascent. From their form, they are
termed cross-mitre drains.
They are placed at intervals depending upon the nature
of the soil and kind of road-covering used, in some cases as
much as sixty yards apart, in others not more than twenty
feet.
593. Catchwaters. These are broad shallow ditches con-
structed across the surface of the road so arranged that
vehicles can pass over them easily and without shock. They
are used to catch the water which runs down the length of
the road and to turn it off into the side ditches. They are
sometimes called water-tables.
They are necessary on long slopes, and in depressions where
a descent and an ascent meet, to prevent the water from cut-
ting the surface of the road in furrows. In a depression, they
are usually placed at right angles to the road ; on a slope, they
cross the road diagonally where the water is to be carried to
one side ; if to both sides, their plan is that of a V with the
angular point up the road.
The inclination of the bottom of the catch water should be
sufficient to carry off the water as fast as it accumulates in
the trench, and where the velocity of the current flowing
through them is considerable, they should be paved.
A mound of earth crossing the road obliquely is frequently
used as a substitute for the catchwater. When used it should
be arranged to allow carriages to pass over them without
difficulty and inconvenience.
594. Culverts. These structures are used to carry under
the road the water of small streams which intersect it, and
also the water of the ditches on the upper side of a road to
the lower side, or side on which the natural water-courses lie
by which the water is finally carried away.
They may be built of stone, brick, concrete, or even of
wood.
Where stone is scarce, a culvert may be built of planks or
slabs, forming a long box open at the ends. This is a tern*
porary structure unless it can be kept always wet.
SIDEWALKS. 437
A small full-centre arch of brick resting on a flooring of
concrete forms a good culvert.
The length of a culvert under an embankment will be
equal to the width of the road increased by the horizontal
distance on each side forming the base of the side-slope. At
each end, wing-walls should be built, their faces having
the same slope as that of the embankment. The ends of the
culvert must be protected against the undermining action of
the water.
The form of cross-section varies according to the circum-
stances of the case, depending greatly on the strength required
in the structure and the volume and velocity of the water
flowing through it. The dimensions of the waterway of a
culvert should be proportioned to the greatest volume of
water which it may ever be required to carry off, and should
always be large enough to allow of a person entering it to
clean it out.
595. Footpaths and sidewalks. Ordinarily, footpaths
are not provided for in our country roads. They should be,
however, and the remarks made in Art. 575 apply to their
construction.
In cities and towns, sidewalks and crossings are arranged
in all the streets. They are made of flagging-stone, brick,
wood, ordinary concrete, asphaltic concrete, etc. They
differ in construction only in degree from roads of the same
kind.
596. Sidewalk of flagging-stone. The flagstones are at
least two inches in thickness, laid on a bed of gravel. The
width of the sidewalk depends upon the numbers liable to
use them, being wider where great crowds are frequent and
less wide on streets not much used. A width of twelve feet
is sufficient for most cases.
The upper surface is not level, but has a slight slope to-
wards the street to convey the surface water to the side
channels.
The pavement of the street is separated from that of the
sidewalk by a row of long slabs set on their edges, termed
curb-stones, which confine both the flagging and paving
stones. The curb-stones form the sides of the side channels,
and should for this purpose project six inches above the out-
side paving stones, and be sunk at least four inches below
their top surface ; they should be flush with the upper sur-
face of the sidewalks, to allow the water to run over into the
side channels, and to prevent accidents from persons tripping
by striking their feet against them.
The crossings should be from four to six feet wide, and be
438 CIVIL ENGINEEEING.
slightly raised above the general surface of the pavement, to
keep them free from mud.
TEAM-KOADS.
Tram-roads are built of stone, of wood, or of iron.
597. Stone tram-roads. The best tram-roads of stone
consist of two parallel rows of granite blocks, about 4|- feet
apart from centre to centre, the upper surface of the blocks
being flush with the surface of the road. The blocks should
be from 4 to 6 feet long, 10 to 12 inches broad and 8 to 12
inches deep. Sometimes the upper surface is made slightly
concave for the purpose of retaining the wheels on the tracks.
Stone tram-roads were used by the Egyptians, traces of
them being found in the quarries which supplied stone for the
pyramids.
Tram-roads of stone have been used in England, and are
used at the present time in Italy.
The granite blocks used in the Italian tram-roads are from
4 to 6 feet long, about 2 feet broad, and 8 inches deep, laid on
a bed of gravel 6 inches thick. The space between the
" trams " is paved with cobble stones with an inclination from
the outside to the middle line. The centre is therefore lower
than the sides, forming a channel for the water, which flows
into cross drains provided to carry it off.
In a tram-road on the Holy head road, the granite blocks
were required to be not less than 4 feet long, 14 inches broad,
and 12 inches deep. The blocks were laid on a bed composed
of a rough sub-pavement, similar to that used for the Telford
road, on which was a layer three inches thick of small broken
stone, and on top of this a layer of gravel two inches thick,
compacted by a heavy roller.
The effect of this tram-road was to reduce the required
amount of tractive force to less than one-half of what was
required on the broken stone road.
598. Tram-roads of wood. Where timber is plenty, tram-
roads of wood are frequently used. They do not differ in
principle of construction from the stone tramway. Since the
wood is extremely perishable when buried in the damp
ground, tramways of wood are used only in temporary con-
structions.
. 599. Iron tram-roads. The iron tram-roads formerly used
were made by covering a wooden track with flat iron bars, so as
to increase the durability of the track and to lessen the resist-
ance offered to the wheels. To keep the wheels on the track, a
RAILROADS. 4:39
flange was placed on the side of the bar (Fig. 228). The
objections to these tramways were that the broad surface cf
the iron plate collected obstructions upon it, and that the fric-
tion of the wheels against the flange was very great.
FIG. 238. FIG. 229.
An iron plate (Fig. 229) is used quite extensively in the
United States, particularly in Philadelphia, for tracks for
street cars. The upper and narrower portion is used by the
wheels of the car, while the wider and flat portion can be
used by ordinary carriages.
CHAPTER XXEEL
RAILROADS.
600. As long as the flange attached to the bar was used to
keep the wheels on the track, the road was called a tram-road.
When the flange was removed from the bar and transferred
to the wheel, the road became changed in character and was
named a railway or railroad. The marked difference be-
tween a tram-road and a railroad is, that the former is used by
all classes of carriages, while the latter can be used only by
cars specially built tor the purpose.
A railroad may be defined to be a track formed of iron or
steel bars, called rails, placed in parallel lines, and upon
which the wheels of vehicles run.
The general principles already alluded to as governing the
location and construction of roads, apply equally to railroads,
but in a higher degree. Greater importance is attached, for
railroads, to straightness, to easy grades, and to using curves
of larger radius where a change of direction takes place, than
for any other kind of road.
601. Direction. Straightness of direction is more import-
440 CIVIL ENGINEERING.
ant for railroads than for common roads, for the reasons that
the shorter the line the cheaper is its cost, and that there i8
a greater resistance offered by curves, causing a greater ex-
penditure of tractive force.
The same considerations which govern in determining the
direction of a common road apply to the railroad, viz., cost of
construction, wants of the community, etc.
602. Grades. The question of grade is more one of
economy than of practicability. Locomotives can be made
to ascend steep grades by increasing their power and adhesion,
but as the grades increase in steepness, the effective tractive
force of the engine decreases. Thus with an ascent of 20
feet to the mile, an engine can draw about one-half the load
which it can draw on a level ; with 40 feet to the mile, about
one-third, etc.
The cost of drawing a load on a railroad varies very nearly
with the power employed. Hence it will cost nearly twice as
much to haul a load on a grade of 20 feet to the mile as it
would on a level road. This consideration will therefore
justify large expenditures in the construction of the road if
made with the view of reducing the grades.
The ruling or maximum grade adopted for the line depends
upon the motive power used to ascend the grades and upon
the avoidance of a waste of power in descending.
The steepest grade upon a given line is not necessaiily the
maximum inclination adopted for the road. It may be much
greater than the ruling grade, and will then require special
arrangements to be made to overcome it.
When the loads to be carried in one direction over the road
are much heavier than those carried in the other, the ascent
up which the heavy loads are to be carried should be made by
easy grades, while the descent may be made by steeper ones.
If the travel is equal in both directions, the ruling grades
should be equal for both slopes.
The length of grades must be considered, as it is found
more advantageous to have steep grades upon short portions
of the line than to overcome the same difference of level by
grades not so steep on longer developments.
From various experiments, it appears that the angle of
repose (Art. 550) for a railroad is about -^fa. But in de-
scending grades much steeper than this, the velocity due to
the accelerating force of gravity soon attains its greatest
limit and remains constant, from the resistance caused by the
air.
The limit of the velocity thus attained, whether the train
CURVES. 441
descends by the action of gravity alone, or by the combined
action of the motive power of the engine and gravity, can be
determined for any given load. It appears from calculation
and experiment that heavy trains, allowed to run freely
without applying the brakes, may descend grades of T ^ with-
out attaining a greater velocity than about 40 miles an hour.
Hence, the question to be considered in comparing the
advantages of different grades is one between the loss of power
and speed for ascending trains on steep grades, and the extra
cost of heavy excavations, tunnels, and embankments required
by lighter grades.
Since locomotives are not taxed to their full extent, grades
of 60 feet to the mile may be used without any practical loss
of power either in the ascent or descent.
603. Curves. Curves are necessary to enable the road to
pass around obstacles, such as hills, deep ravines, valuable
houses which cannot be removed, etc.
The objections to curves in the road are the resistances
which they offer to the motion of the cars and the dangers to
which the cars are exposed.
The resistances offered by the curves are chiefly due to the
following causes :
1. The obliquity of the moving power while passing around
the curve.
2. The friction of the flanges of the wheels against the
outer rail due to the centrifugal force.
3. The friction of the flanges against the rails due to the
parallelism of the axles.
4. The fastening of each pair of wheels to the same axle.
The danger of a car running off the track is much increased
by curves. The car is kept on the rails while going around
a curve by the flanges of the wheels and by the firmness of the
outer rails. If the resistance offered by the rails and flanges
should be overcome by the " quantity of motion " of the car,
the latter would leave the track. Hence, where sharp curves
are necessary, they should be located, if possible, near stop-
ping places, and never at those points where the speed is to be
very high or where the car will pass with great velocity, as
at the foot of a steep grade.
The minimum radius of a curve depends greatly upon the
Bpeed to be employed. In France, the minimum radius
allowed is 2,700 feet. In England, no curve less than 2,640
feet can be used without special permission of Parliament or
the Board of Trade. The minimum radius used on the Hud-
son River Railroad is 2,062 feet. On the Baltimore and Ohio
Railroad, the minimum radius is 600 feet, although when first
442 CIVIL ENGINEERING.
constructed there were several curves of 400 feet radius, and
one of 318 feet over which trains passed at a speed of 15
miles an hour.
604. Resistances of vehicles on railroads. The resist-
ance offered to the force of traction by a train of cars is due
to friction, concussion, and the atmosphere. The amount
of this resistance depends upon a variety of conditions, such as
the condition of the road, whether well or badly constructed,
in bad order, etc. ; the state of the rolling machinery ; the
climate ; the season of the year ; state of the weather, etc.
In discussing the resistance, it is assumed that the cars are
well made, the track in good order, and the weather moder-
ately calm. The amount of resistance may be determined by
means of a dynamometer between the engine and the train,
and may be expressed either as a fraction or as a certain num-
ber of pounds per ton, the latter being generally used.
That part of the resistance offered by the train due to
friction is constant at all speeds ; that due to concussion and
the atmosphere varies with the velocity, increasing with the
speed. The law of increase is not fully known.
605. On a level and straight road. The resistance
offered by a train running on a level and straight road, nearly
as possible under the conditions in ordinary practice, has been
determined by experiment to be nearly that given by the
following formula :
in which r is the resistance in pounds per ton of the enginej
tender, and train ; and v the velocity in miles per hour.
Hence it is seen, that for a train moving at the rate of 20
miles an hour, the resistance would be 10.33 pounds per ton
of the entire train.
If the road is in bad repair, the values obtained by this
formula should be increased 40 per cent. ; for strong side
winds, 20 per cent.
606. Resistance due to grades. The resistance due to
a grade is found by multiplying the whole weight of the train
by the difference of level and dividing this product by the
length of the slope. By this rule it is found that the resist-
ance per ton due to a grade of 24 feet in a mile is
2,240 x jr^ = 10.2 pounds,
or about the same as that on a level with the speed of 20
miles an hour. Therefore, if the train runs over this grade at
443
20 inik-s an hour, the resistance would be just double, or it
would require the same power to run one mile on the grade
that would draw the same load at the same speed two miles
on a level road.
607. Resistance due to curves. The resistance due to
curvature is much affected by the gauge of the road, the ele-
vation of the outer rail, the form of surface of the tires and
the size of the wheels, the speed and length of the train, etc.
Hence, experiments made to obtain this resistance will be
found to vary greatly for the same curve on different roads.
The point to be gained, however, is to find the amount of
curvature which will consume an amount of power sufficient
to draw a train one mile on a straight and level road.
It is assumed that the resistance from curvatur"e is inversely
as the radius ; that is, the resistance offered by a curve of 2
is double that of a curve of 1.
From experiments made under his direction, Mr. Latrobe
deduced the resistance upon a curve of 400 feet radius to be
double that upon a straight line.
Upon averaging a large number of experiments made for
this purpose, it is found that a radius of 574 feet, or curve of
10, offers a resistance to a train travelling at the rate of 20
miles an hour, double that on a straight and level line, at the
same speed. Hence a curve of ten degrees causes a resist-
ance of ten pounds to the ton. Knowing this resistance, that
for any other curve is easily obtained.
If we desire to make the resistance uniform upon any sys-
tem of grades and curves, it will be necessary, whenever a
curve occurs upon a grade, to reduce the latter to an amount
sufficient to compensate for the resistance caused by the
curve.
608. Mr. Scott Russell's formula. Formula (173) gives
the value of the total resistance without separating it into its
parts.
The formula of Mr. Russell and Mr. Harding gives separate
expressions for each resistance. This formula is as follows :
,. . .(174)
in which r and v are the same as in (173), "W, the weight of
the train in tons, and A, the area of rrontage of the train in
square feet.
This formula may be expressed in words, as follows:
1. Multiply the weight in tons by 6. The product will be
the amount in pounds due to friction.
2, Multiply the weight in tons by the velocity in miles pei
444 CIVIL ENGINEERING.
hour and divide the product by 3. The result will be the
amount in pounds due to concussion.
3. Multiply the square of the velocity in miles per hour by
the frontage of the train in square feet and divide the pro-
duct by 400. The result will give the resistance in pounds
due to the atmosphere.
4. Add these three results, and the sum is the total resist-
ance. Divide the total resistance by the weight, and the quo-
tient is the resistance per ton.
The foregoing results corresponded closely with the experi-
ments for speed from 30 to 60 miles per hour. At lower rates
of speed, the rule gave too great results.
Another formula has been used in which the resistance of
the atmosphere is assumed to be proportional to the volume
of the train. It is as follows :
in which 13 is the volume of the train, the other quantities
being the same as in (174).
609. Tractive force. The forces employed to draw the
cars on railroads are gravity, horses, stationary engines, and
locomotive engines.
610. Gravity. Gravity either assists or opposes the other
kinds of motive power on all inclined parts of a railroad. It
may be used as the sole motive power on grades which are
sufficiently steep. In this case the loaded cars descending the
grade draw up a train of empty ones. The connection is
made between the trains by means of a wire rope which runs
over pulleys placed along the middle of the track.
611. Horses. Horses are frequently used to draw cars on
a railroad.
The power of a horse to move a heavy load is ordinarily
assumed at 150 pounds, moving at the rate of 2$ miles an
hour for 8 hours a day. At greater speeds his power of
draught diminishes ; for example to half that load at 4 miles
an hour, etc.
The power of the horse is rapidly diminished upon ascents.
On a slope of 1 in 7 (8J) he can carry up only his own
weight (Gillespie).
612. Stationary engines, These are employed sometimes
where the speed is to be moderate, the grade steep, and the
distance short.
The power is usually applied by means of an endless wire
rope running on pulleys, like that employed where gravity
is the only motive power. And as in that case, the descent
LOCOMOTIVE ENGINES. 445
of one train is generally made to assist in the drawing up of
another to the top of the inclined plane.
613. Locomotive engines. The principal motive power
on railroads is the locomotive engine.
The locomotive is a non-condensing, righ-pressure engine,
working at a greater or less degree or expansion according to
circumstances, and placed on wheels which are connected
with the piston in such a manner that any motion of the latter
is communicated to them.
The power exerted in the cylinder and transferred to the
circumference of the driving wheel is termed " traction ; "
its amount depends upon the diameter of the cylinder, the
pressure of the steam, the diameter of the driving wheel, and
the distance, called the stroke, traversed by the piston from
one end of the cylinder to the other.
The means by which the traction is rendered available for
moving the engine and its load is the friction of the driving
wheels on the rail ; this is called the " adhesion," and its
amount varies directly with the load resting on the wheels,
and with the condition of the surface of the rails, varying
from almost nothing when ice is on the rails, up to as much
as one-fifth of the weight on the driving wheels when the
surface of the rail is clean and dry.
The speed of the engine depends also upon the rapidity
with which its boiler can generate steam. One cylinder full
of steam is required for each stroke of the piston. Each
double stroke corresponds to one revolution of the driving
wheels and to the propulsion of the engine through a space
equal to their circumference.
Steam-production, adhesion, and traction, are the three
elements which determine the ability of a locomotive engine
to do its work. The work required of the engine depends
upon the nature and amount of the traffic over the road and
the condition of the road. Hence, engines of different pro-
portions are employed on the same road, one set to haul heavy
loads at low velocities and another set to move light loads at
high rates of speed.
Stronger and more powerful engines are needed on a road
with steep grades and sharp curves than on roads with easy
grades and large curves.
Locomotive engines may be so proportioned as to run at any
speed from to 60 miles an hour ; to ascend grades even as
Bteep as 200 feet in the mile ; and to draw from 1 to 1,000
tone.
The weight and speed of the trains, and the ruling grades
of the road determine the amount of power required of the
446 CIVIL ENGINEERING.
engine. This power depends, as has just been stated, upon
the steam-producing capacity of the boiler, upon the leverage
with which the steam is applied, and upon the adhesion.
614. Gauge. The width of a railroad between the innei
sides of the rails is called the gauge.
The question as to what this width should be has been a
subject for discussion and of controversy among engineers.
The original railroads were made of the same width as the
tram-roads on which the ordinary road wagon was used. It
happened that the width of the tram-road was 4 feet 8-J- inches ;
this was adopted for the railroad, and soon became universal.
In a few cases, other widths were adopted, but the advantages
of uniformity so far exceed all other considerations, that the
width of 4 feet 8 inches is now generally adopted for main
lines or roads of the first class.
For branch lines, a still narrower gauge is recommended ;
a width of 3 feet, and even of 2 feet 6 inches, has been em-
ployed. A road of this narrow gauge costs less to construct
and admits of steeper grades and sharper curves being used.
Railroads may have either a single or a double track.
When first constructed and where the traffic is light, a single
track is used, but even then it is recommended to secure
ground sufficient for a second track when the latter becomes
necessary.
The New York Central Railroad has four tracks, two of
which are used for passenger traffic and two for movement
of freight.
LOCATION AND CONSTRUCTION OF RAILROADS.
615. Location. Location of railroads is guided by the same
principles as that of ordinary roads and is made in the same
manner. The greater importance to railroads of easy grades
and straightness justifies a greater expenditure for surveys,
which are more elaborate than those required for common
roads.
616. Construction. This may be divided into two parts :
forming the " road-bed," and the " superstructure."
The remarks already made concerning the " construction of
roads " apply to " forming the road-bed of a railroad."
The excavations and embankments are generally much
greater on railroads than for any other of the roads usually
constructed. Where, for instance, an ordinary road would
wind around a hill, a railroad would cut through it, in this
way obtaining straightness and avoiding curves.
The sides of an excavation are often supported by retain-
SHAFTS AND TUNNELS. 4:47
ing walls in order to reduce the width of the cutting at the
top.
617. Tunnels. When the depth of excavation is very
great it will frequently be found cheaper to make a passage
under ground called a tunnel.
The choice between deep cutting and tunnelling will de-
pend upon the relative cost of the two and the nature of the
ground. When the cost of the two methods would be about
equal, and the slopes of the deep cut are not liable to slips, it
is usually more advantageous to resort to deep cutting than to
tunnelling. So much, however, will depend upon local cir-
cumstances, that the comparative advantages of the two
methods can only be decided by a careful consideration of
these circumstances for each particular case. Where a choice
may be made, the nature of the ground, the length of the
tunnel, that of the deep cuts by which it must be approached,
and also the depths of the working shafts, must all be well
studied before any decision can be made. In some cases it
may be found that a long tunnel with short deep cuts will be
most advantageous in one position, and a short tunnel with
long deep cuts in another. In others, the greater depth of
working shafts may be more than compensated for by the ob-
taining of a safer soil, or a shorter tunnel.
As a general rule tunnelling is to be avoided if possible.
The dimensions and form of the cross-section will depend
upon the nature of the soil and the object of the tunnel as a
communication. In solid rock, the sides of the tunnel are
usually vertical, the top curved, and the bottom horizontal.
In soils which require to be sustained by an arch, the exca-
vation should conform as nearly as practicable to the form
of cross-section of the arch.
In tunnels through unstratified rocks, the sides and roof
may be left unsupported; but in stratified rocks there is
danger of blocks becoming detached and falling : wherever
this is to be apprehended, the top of the tunnel should be
supported by an arch.
In choosing the site of a tunnel, attention should be had,
not only to the nature of the soil, and to the shortness and
straightness of the tunnel, but also to the facilities offered for
getting access to its course at intermediate points by means of
shafts and drifts.
618. Shafts. Vertical pits which are sunk to a level with
the crown or top of the tunnel are known as shafts.
There are three kinds : trial, -working, and permanent
shafts.
Trial shafts are, in general, sunk at or near the centre line
448 CIVIL ENGINEERING.
of the proposed tunnel to ascertain the nature of the strata
through which the tunnel is to be excavated. Their dimen-
sions and shape are regulated by the uses to which they are to
be put.
Working shafts are used to give access to the tunnel, for
the purpose of carrying on the work and removing the mate-
rial excavated, for admitting fresh and discharging foul air,
and for pumping out water.
Their dimensions will be fixed by the service required of
them. Their distance apart varies between 50 and 300 yards,
although in some cases they are only from 20 to 30 yards
apart, and in others none are used.
They may be located along the centre line of the tunnel or
they may be on a line parallel to it.
Permanent shafts are generally working shafts that have
been made permanent parts of the" tunnel for the purposes of
ventilation and of admitting Jight.
619. Drifts. Small horizontal or slightly inclined under-
ground passages made for the purpose of examining the strata,
for the purpose of drainage, of affording access to the tunnel
for the workmen and for transport of materials, etc., are
termed drifts or headings.
Their least dimensions are those in which miners can con-
veniently work, or from 4f to 5 feet high and 3 feet wide.
Headings are almost always used to connect the working
shafts, running along the centre line or parallel to the line
of the tunnel. In soft ground, the heading is at or near the
bottom of the tunnel ; in rock or hard and dry material at or
near the top.
620. Laying out tunnels. The establishment of a correct
centre line for a tunnel and the fixing of the line at the bot-
tom of the shafts are most important operations and require
the utmost care.
The work is commenced by setting out, in the first place,
with great accuracy upon the surface of the ground, the pro-
file line contained in the vertical plane of the axis of the
tunnel, and at suitable intervals along this line, sinking work-
ing shafts. At the bottom of these shafts the centre line is
marked out by two points placed as far apart as possible. By
these the line is prolonged from the bottom of the shaft in
both directions.
In constructing the Hoosac Tunnel, so accurate were the
alignments, that the heading running eastward from the
central shaft for a distance of 1,563 feet met the heading
from the eastern end with an error of but five-sixteenths of
an inch; and the heading running westward for 2,056 feet
DBAINAGE AND VENTILATION. 449
met the heading from the western end with an error of but
nine-sixteenths of an inch.
An elaborate trignometrical survey was used to lay out the
Mont Cenis Tunnel, which was 7.5 miles long, with no work-
ing shafts.
621. Operation of tunnelling, The shafts and the ex-
cavations which form the entrances to the tunnel are con-
nected by a drift, usually five or six feet in width and seven
or eight feet in height, made along the crown of the tunnel
when the soil is good. After the drift is completed, the
excavation for the tnnnel is gradually enlarged ; the ex-
cavated earth is raised through the working shafts, and
at the same time carried out at the ends. The speed with
which the drift is driven determines the rate of progress of
the whole.
If the soil is loose, the operation is one of the most hazard-
ous in engineering construction, and requires the greatest pre-
cautions against accident. The sides of the excavations must
be sustained by strong rough frame- work, covered by a sheath-
ing of boards to secure the workmen from danger. When in
such cases the drift cannot be extended throughout the line
of the tunnel, the excavation is advanced only a few feet in
each direction from the bottom of the working shafts, and
is gradually widened and deepened to the proper form and
dimensions to receive the masonry of the tunnel, which is
immediately commenced below each working shaft, and is
carried forward in both directions towards the two ends of
the tunnel.
In some cases, two headings were run forward and the side
walls of the tunnel were built before the remainder of the
section was excavated.
The ordinary difficulties of tunnelling are greatly increased
by the presence of water in the soil through which the work
is driven. Pumps, or other suitable machinery for raising
water, placed in the working shafts, will, in some cases, be
requisite to keep them and the drifts free from water until an
outlet can be obtained for it at the ends, by a drain along the
bottom of the drift.
622. Drainage and ventilation of tunnels. The drain-
age of a tunnel is effected either by a covered drain under the
road-bed at the centre or by open drains at the sides.
Artificial ventilation is found not to be necessary in ordinary
tunnels, and the permanent shafts constructed for the purpose
have been considered detrimental rather than beneficial in
getting rid of the smoke. The passage of the train appears
to be the best ventilator ; the air being thoroughly disturbed
450 CIVIL ENGINEERING.
and displaced by the quick motion of the train through the
tunnel.
623. Ballast. The tops of the embankments and the bot-
tom of the excavations are brought to a height called the
k< formation level," about two feet below the intended level of
the rails. The remaining two feet, more or less, is filled up
with gravel, or gravel and sand, or broken stone, or similar
material, througn which the water will pass freely. This
layer is called the " ballast," and the material of which it is
composed should be clean and hard, so as not to pack into a
solid mass preventing the water from passing through it.
The object of the ballast, besides allowing the water to run
off freely, is to hold the Sleepers firmly in their places and to
give elasticity to the road-bed.
624. Cross ties. The cross ties or " sleepers " are of wood,
hewn flat on the top and bottom ; they are from 7 to 9 feet
long for the ordinary gauge, 6 inches deep, and from 6 to 10
inches wide. The distance between the ties depends upon
the weight of the engines used on the road and the strength
of the rail; 2 feet from centre to centre is about the
usual distance. The nearer the sleepers are to uniformity in
size and to being equidistant from each other, the more uni-
form will the pressure from the passage of the train be
distributed over the ground.
The sleepers may be of oak, pine, locust, hemlock, chest-
nut, etc. They last from 5 to 10 years, depending upon their
positions and the amount of travel over them. Their duration
may be increased by using some of the preservative means
referred to in Art. 25.
625. Rails. The rails are made of wrought iron, or of
wrought iron with a thin bar of steel forming the top surface,
or entirely of steel.
Since the rail acts as a support for the
train between the ties, and as a lateral
guide for the wheels, it must possess
strength and stiffness to a marked degree.
The top surface should be of sufficient
size and hardness to withstand the action
of the rolling loads, and the bottom surface
should be wide enough to afford a good
_ bearing upon the tie. The rail should
FIG. 230. have that form which gives the required
strength with the least amount of mate-
rial. The form of cross-section in most general use at the
present time in the United States is shown in Fig. 230. This
particular rail is 4^ inches high and 4 inches wide at the
ELEVATION OF .THE OUTEK BAIL.
bottom. The width of the head varies from 2J to 2J indies
the top surface having a convex form, circular in cross-section,
described with a radius double the height of the rail. The
thickness of the rib or stem is generally from -J to f of an
inch, although recent experiments would indicate that a less
thickness might be used with safety.
The rails are rolled in lengths varying from 15 to 21 feet,
and when laid are connected by fish-joints and fastened to
the cross-ties by spikes. The method of fastening formerly
used was to confine the ends of the rails in a cast-iron chair
which rested on the cross-ties. This method may be seen on
some of the older railroads, but is fast going out of use on
all first-class roads.
626. Coning of the 'wheels. The wheel running on the
outer rail of a curve has to pass over a greater distance than
the one running on the inner rail. Since the wheels and axles
are firmly connected, some arrangement must be made to keep
the wheels from dragging or slipping on the rails and to re-
duce the twisting strain brought on the axles. This is usually
effected by making the tread of the wheel conical instead of
cylindrical, so that the tendency of the car to press against
the outer rail brings a larger diameter upon the outer and a
smaller diameter on the inner rail. The difference between
these diameters must be proportioned to the distance to be
traversed by the wheels, and must depend, therefore, upon
the radius of the curve and the gauge. The sharper the curve,
the greater should be the difference between the diameters.
Upon many roads it is customary to widen the gauge from 4
feet 8J inches to 4 feet 9 inches on sharp curves, thus allowing
more play for the wheels and giving a greater difference in the
diameters of those parts of the wheel in contact with the rails.
As the tread of the wheel is conical, the tops of the rails
are inclined, or given a " cant n to fit this cone. The amount
of inclination depends upon the amount of conical form given
to the tread of the wheel. For the common gauge, this inclina-
tion is taken at about -fa.
627. Elevation of the outer rail. When the track is
straight, a line drawn in the cross-section made by a plane
perpendicular to the axis of the road, tangent to the upper
surfaces of the rails, is horizontal. On the curved portions
of the track the centrifugal force tends to throw the car
against the outer rail. This tendency is resisted by raising
the outer rail to a certain height above the inner one. The rule
for obtaining this height is expressed as follows :
(176)
459
CIVIL ENGINEERING.
in which h is the elevation above inner rail in inches ; v, the
velocity in feet per second; g, the gauge of the road in
inches ; and R, the radius of the curve in feet.
628. Crossings, switches, etc. To enable trains to pasa
from one track to the other, crossings are arranged as shown
in Fig. 231. The connection between the crossing and the
track is made by a switch.
FIG. 231.
The switch consists of one length of rails, movable around
one of the ends, so that the other can be displaced from the
line of the main track and joined with that of the crossing, or
the reverse, depending upon which line of rails the train is to
use. A vertical lever is attached to the movable end by
means of which the ends of the rails are pushed forward or
shoved back, making the connection with the tracks. The
handles of the lever should be so fashioned and painted that
their position may be seen from a considerable distance.
Where one line of rails crosses another, an arrangement
called a crossing-plate, or frog (Fig. 232), is used to allow
free passage of the wheels.
PIG. 232.
In order that the wheels should run smoothly on the rail
A B, the rail C D must be cut at its intersection with the
former ; for a similar reason, the rail A B must be cut at ita
intersection with C D.
A guard-rail, G G, is used to confine the opposite wheel for
short distance and prevent the wheel running on A B from
leaving the rail at the cut. This guard-rail is parallel to th
NAVIGABLE CANALS. 453
outer rail and placed about two inches from it. It extends
a short distance beyond the opening in both directions and
has its ends curved slightly, as shown in Fig. 231.
The angle between the lines of the main track and the
crossing should be very small, not greater than 3.
629. Turn-tables. When the angle is too great to use the
crossing, the arrangement called a turn-table is employed.
This cousists of a strong circular platform of wood or iron,
movable around its centfe by means of conical rollers beneath
it running upon iron roller- ways. Two rails are laid upon
the platfdrm to receive the car, which is transferred from one
track to the other by turning the platform sufficiently to place
the rails upon it in the same line with those of the track upon
which the car is to run. The greater the proportion of the
weight borne by the pivot at the centre and the less that
borne by the rollers, the less will be the friction.
630. Telegraph, mile-posts, etc. On all well managed
railroads, telegraph lines are essential to the safe working of
the road. These should be connected with every station. By
their use, the positions of the different trains at all hours are
made known.
Mile-posts, numbered in both directions, should be placed
along the sides of the road. Posts showing the grades, the
distance to crossings of roads, to bridges, etc., should be used
wherever necessary.
CHAPTER XXIV.
CANALS.
631. A canal is an artificial water-course. Canals are
used principally for purposes of inland navigation ; for irriga-
tion; for drainage; for supplying cities and towns with
water, etc.
NAVIGABLE CANALS.
632. Navigable canals may be divided into three classes ;
level canals, or those which are on the same level through-
out ; lateral canals, or those which connect two points of
different levels, but have no summit level ; and canals -with
a summit level, or those connecting two points which lie
on opposite sides of a dividing ridge.
4:54 CIVIL ENGINEERING.
I. Level canals. In canals of this class, the level of the
water is the same throughout. As in roads, straightness of
direction gives way to economy of construction, and the econ-
omical course will be that which follows a contour line,
unless a great saving may be made by using excavation or
embankment. Where changes of direction are made, the
straight portions are connected by curved ones, generally arcs
of circles, of sufficient curvature to allow the boats using the
canal to pass each other without sensible diminution in their
rate of speed.
II. Lateral canals. In these canals, the fall of water is in
one direction only. Where the difference of level between the
extreme points is considerable, the canal is divided injto a
series of levels or ponds, connected by sudden changes of
level. These sudden changes in level are overcome by means
of locks or other contrivances by which the boat is transferred
from one level to the other.
III. Canals with summit levels. These are canals in
which the points connected are lower than the intermediate
ground over which the canal has to pass, and in consequence
the fall is in both directions. As the water for the supply of
the summit level must be collected from the ground which
lies above it, it follows that the summit level should be at the
lowest point of the ridge dividing the two extremes of the
canal.
633. Form and dimensions of water-way. The general
width of a canal should be sufficient to allow two boats to
pass each other easily. Where great expense would be in-
curred in giving this width, like that of a bridge supporting a
canal, short portions may be made just wide enough for one
boat.
The depth should be such as not to materially increase the
resistance to the motion of the boat beyond what is felt in
open water. ,
The bottom of the canal is generally made horizontal. The
sides are inclined, and when of earth should not be steeper
than one upon one and a half ; if of masonry, the sides may
be vertical or nearly so. In the latter case a greater width
must be given to the bottom of the canal.
The water-way is usually of a trapezoidal form, in cross-
section (Fig. 233) with an embankment on each side, raised
above the general surface of the country and formed of the
material from the excavation for the canal.
The relative dimensions of the parts of the cross-section
may be generally stated as follows :
TOWPATH. 455
The width of the water-way, at bottom, should ue at least
twice the width of the boats used in navigating the canal.
The depth of the water-way should be at least eighteen
inches greater than the greatest draft of the boat.
FIG. 233. A, water- way. B, towpath. C, berm. D, Hide-drain. E,
puddling of clay.
The least area of water-way should be at least six times the
greatest midship section of the boat.
634. A towpath for horses is made on one of the em-
bankments and a footpath on the other. This footpath should
be wide enough to serve as an occasional towpath.
The towpath should be from ten to twelve feet wide, to
allow the horses to pass each other with ease ; and the foot-
path at least six feet wide. The height of the surfaces of
these paths, above the water surface, should not be less than
two feet, to avoid the wash of the ripple ; nor greater than
four feet and a half, for the facility of the draft of the
horses in towing. The surface of the towpath should incline
slightly outward, both to convey off the surface water in wet
weather and to give a firmer footing to the horses, which
naturally draw from the canal.
The width given to these paths will give a sufficient thick-
ness to the embankments to resist the pressure of the water
against them, and to prevent filtration through them, provided
the earth is at all binding in its composition.
635. Construction. All canal embankments should be
carefully constructed. The earth of which they are formed
should be of a good binding character, and perfectly free
from mould and all vegetable matter, as the roots of
plants, etc. In forming the embankments, the mould should
first be removed from the surface on which they are to
rest, and the earth then spread in uniform layers, from
nine to twelve inches thick, and well rammed. If the char-
acter of the earth, of which the embankments are formed, is
such as not to present entire security against filtration, a pud-
dling of clay, two or three feet thick, should be laid in the
interior of the mass, extending from about a foot below the
natural surface up to the same level with the surface of the
water. Sand is useful in stopping leakage through the holes
456
CIVIL ENGINEERING.
made in the embankments near the water surface by insects,
moles, rats, etc.
The side slopes of the embankment vary with the character
of the soil : towards the water-way they should seldom be less
than two base to one perpendicular ; from it, they may be
less. The interior slope is usually not carried up unbroken
from the bottom to the top ; but a horizontal space, termed &
bench or berm, about one or two feet wide, is left, about one
foot above the water surface, between the side slope of the
water-way and the foot of the embankment above the berm.
This space serves to protect the upper part of the interior
side slope, and is, in some cases, planted with such shrubbery
as grows most luxuriantly in moist localities, to protect more
efficaciously the banks by the support which its roots give to
the soil. The side slopes are better protected by a revetment
of dry stone, from six to nine inches thick. Aquatic plants
of the bulrush kind have been used, with success, for the
same purpose ; being planted on the bottom, at the foot of
the side slope, they serve to break the ripple, and preserve
the slopes from its effects.
Side drains must be made, on each side, a foot or two from
the embankments, to prevent the surface water of the natural
surface from injuring the embankments.
636. Slight leakage may sometimes be stopped by sprinkling
fine sand in small quantities at a time over the surface of the
water in the vicinity of the leaks. The sand settling to the
bottom gradually fills the crevices in the sides and bottom of
the canal through which the water escapes*
The leakage may be so great that it may be necessary, in
certain cases, to line the canal with masonry, concrete, or to
face the sides with sheet- piling to retain the water.
When the bottom of the canal is composed of fragments
of rock forming large crevices, or composed of marl, it ha8
been frequently found necessary to line the water-way in such
localities with masonry (Fig. 234) or with concrete.
LOCKS.
457
In a lining of this kind, the stone used was about four
inches thick, laid in cement or hydraulic mortar, and covered
with a coating of mortar two inches thick, making the entire
thickness of the lining six inches. This lining was then covered,
both at bottom and on the sides, by a layer of earth, at least
three feet thick, to protect it from the shock of the boats strik-
ing against it.
637. Size of canals. The size of a canal depends upon the
size of the boats to be used upon it. The dimensions of com-
mon canal boats have been fixed with a view of horses being
used to draw them. The most economical use of horse-power
is to draw a heavy load at a low rate of speed. Assuming a
speed of from two to two ^nd a half miles an hour, a horse
can draw a boat with its load, in all about 170 tons. This
requires a boat of the ordinary cross-section to be about twelve
feet wide, and to have a draught of four and a half feet when
fully loaded.
Boats of greater cross-section are frequently used, and are
drawn by various applications of steam as well as by horse-
power. The methods used are various, as the screw propeller,
stationary engines with endless wire ropes, etc. Canals are
sometimes made only twelve feet wide at bottom, with a
draught of four feet ; common canals are from twenty-five to
thirty feet wide at bottom, with a depth of from five to eight
feet ; ship or large canals are fifty feet wide at bottom, and
have a depth of twenty feet. These are the minimum dimen-
sions.
638. Locks. An arrangement termed a lock is ordinarily
used to pass a boat from one level to another.
A lock is a small basin just large enough to receive a boat,
and in which the water is usually confined on the sides by
FIG. 235.
two upright walls of masonry, and at the ends by two
gates ; the gates open and shut, both in order to allow the
passage of the boat and to cut off the water of the upper level
from the lower, or from the water in the lock.
A lock (Figs. 235 and 236) may be divided into three dis-
tinct parts : 1st. The part included between the two gates,
1:58 CIVIL ENGINEERING.
which is termed the chamber. 2d. The part above the
upper gates, termed the fore or head-bay. 3d. The part
below the lower gates, termed the aft or tail-bay.
Fig. 235 shows a vertical longitudinal section through the
axis of a single lock built on a foundation of concrete, and
Fig. 236 represents the plan.
P .:-. fc
.,^:::>. H
C A A
A D t\ fl
f L J ;
"~~ \
tr
13
FIG. 236.
In these figures, A is the lock-chamber ; E, E, the side
walls ; B, the head-bay ; C, the tail-bay ; and D, the lift-wall.
The lock-chamber must be wide enough to allow an easy
ingress and egress to the boats commonly used on the canal ;
a breadth of one foot greater than the greatest breadth of
the boat is deemed sufficient for this purpose. The length
of the chamber is regulated by that of the boats ; it should
be such that when the boat enters the lock from the lower
level, the tail-gates may be shut without requiring the boat
to unship its rudder.
The plan of the chamber is usually rectangular, the sides
receiving a slight batter ; as when so arranged they are found
to give greater facility to the passage of the boat than when
vertical. The bottom of the chamber is either flat or curved ;
more water will be required to fill the flat-bottomed chamber
than the curved, but less masonry will be required in its con-
struction.
The chamber is terminated just within the head-gates by
a vertical wall, the plan of which is usually curved. As this
wall separates the upper from the lower level, it is termed
the lift-wall ; it is usually of the same height as the lift of
the levels. The top of the lift- wall is formed of cut stone,
the vertical joints of which are normal to the curved face of
the wall; this top course projects from six to nine inches
above the bottom of the upper level, presenting an angular
point for the bottom of the head-gates, when shut, to rest
against. This projection is termed the mitre -sill. Various
degrees of opening have been given to the angle between the
two branches of the mitre-sill; it is, however, generally so
LOOKS. 459
determined, that the perpendicular of the isosceles triangle,
formed by the two branches, shall vary between one- fifth and
one-sixth of the base.
The side-walls sustain the pressure of the embankment
against them, and when the lock is full the pressure from the
water in the chamber. The former pressure is the greater
and the more permanent of the two and the dimensions of the
wall are determined to resist this pressure. The usual man-
ner of doing this is to make the wall four feet thick at the
water line of the upper level, to secure it against filtration ;
and then to determine the base of the batter, so that the mass
of masonry shall present sufiicient stability to resist the thrust
of the embankment. The spread and other dimensions of
the foundations will be regulated according to the nature of
the soil, as in other masonry structures.
The bottom of the chamber, as has been stated, may be
either flat or curved. The flat bottom is suitable to firm
soils, which will neither yield to the vertical pressure of the
chamber walls nor admit the water to filter from the upper
level under the bottom of the lock. In either of these cases,
where yielding or undermining may be expected, the bottom
should be an inverted arch. The thickness of the masonry
of the bottom will depend on the width of the chamber and
the nature of the soil. Were the soil a solid rock, no bottom-
ing would be requisite ; if it is of soft material, a very solid
bottoming, from three to six feet in thickness, may be neces-
sary. Great care must be taken to prevent the water from
the upper level filtering through and getting under the bot-
tom of the lock.
The lift-wall may have only the same thickness as the side'
walls, but unless the soil is very firm, it would be more pru-
dent to form a general mass of masonry under the entire
head-bay, to a level with the base of the chamber founda-
tions, of which mass the lift-wall should form a part.
The head-bay is enclosed between two parallel walls, which
form a part of the side walls of the lock. They are termi-
nated by two wing walls, m, m, at right angles with the side
walls. A recess, termed the gate-chamber, is made in the
wall of the head-bay ; the depth of this recess should be suf-
ficient to allow the gate, when open, to fall two or three
inches within the facing of the wall, so that it may be out of
the way when a boat is passing; the length of the recess
should "be greater than the width of the gate. That part of
the recess where the gate turns on its pivot is termed the
hollow quoin ; it receives what is termed the heel or quoin-
post of the gate, which is made to fit the hollow quoin. The
460 CIVIL ENGINEERING.
distance between the hollow quoins and the face of the lift-
wall will depend on the pressure against the mitre-sill, and
the strength of the stone ; eighteen inches will generally be
found sufficient.
The side walls need not to extend more than twelve inches
beyond the other end of the gate-chamber. The wing walls
may be extended back to the total width of the canal, but it
will be more economical to narrow the canal near the lock,
and to extend the wing walls only about two feet into the
banks or sides. The dimensions of the side and wing walla
of the head-bay are regulated in the same way as the chamber
walls. The top of the side walls of the lock may be from
one to two feet above the general level of the water in the
upper level.
The bottom of the head-bay is flat, and on the same level
with the bottom of the canal ; the exterior course of stones at
the entrance to the lock should be so jointed as not to work
loose.
The side walls of the tail-bay are also a part of the general
side walls, and their thickness is regulated as in the preceding
cases. Their length will depend chiefly on the pressure which
the lower gates throw against them when the lock is full, and
partly on the space required by the lockmen in opening and
shutting the gates. These walls are also terminated by wing
walls, n, n, similarly arranged to those of the head-bay. The
points of junction between the wing and side walls should, in
both cases, either be curved or the stones at the angles be
rounded off . One or two perpendicular grooves are sometimes
made in the side walls of the tail-bay, to receive stop-planks,
when a temporary dam is needed, to shut off the water of the
lower level from the chamber, in case of repairs, etc.
The gate-chambers for the lower gates are made in the
chamber walls ; the bottom of the chamber, where the gates
swing back, should be flat, or be otherwise arranged so as
not to impede the play of the gates.
The bottom of the tail-bay is arranged, in all respects, like
that of the head-bay.
639. Those parts of the lock where there is great wear and
tear, as at the angles generally, should be of cut-stone ; or
where an accurate finish, is indispensable, as at the hollow
quoins. The other parts may be of brick, rubble, concrete,
etc., but every part should be laid in cement or the best
hydraulic mortar.
The mitre-sills are generally faced with timber, to enable
them to withstand better the blows which they receive from
the gates, and to make a tighter joint.
LOOK GATES. 461
640. The locks are filled and emptied through sluices in
the head and tail-gates, opened and closed by slide valves, rr
by culverts made of masonry or iron pipe placed as shown
in the figures at c, c, c, etc. The latter is the method gene-
rally recommended. From the difficulty of repairing the
sluices when out of order, many prefer the use of valves in
the gates.
The bottom of the canal below the lock should be protected
by what is termed an apron, which is a covering of plank
laid on a grillage, or of dry stone. The length will depend
upon the strength of the current ; generally a distance of
from fifteen to thirty feet will be sufficient.
641. Lock gates. The gates may be made of wood or of
iron. Each gate is ordinarily composed of two leaves, each
leaf consisting of a framework, covered with planking or iron
plates. The frame, when of timber, consists usually of two
uprights, connected by.horizontal pieces let into the uprights
with the usual diagonal bracing.
In gates of this kind, each leaf turns about an upright,
which is called the quoin or heel-post. This post is cylin-
drical on the side next to the hollow quoins, which it exactly
fits when the gate is shut. It is made slightly eccentric, so
that when the gate is opened it may turn easily without rub-
bing against the quoin. At its lower end it rests on a pivot,
and its upper end turns in a circular collar which is strongly
anchored in the masonry of the side walls. One of the
anchor-irons is usually placed in a line with the leaf when
shut, the other in a line with it when open ; these being the
best positions to resist most effectually the strain produced
by the gate. The opposite upright, termed the mitre-post,
has one edge bevelled off, to tit against the mitre-post of the
other leaf of the gate, forming a tight joint when the gate is
shut.
A long, heavy beam, termed a balance beam from its
partially balancing the weight of the leaf, is framed upon the
quoin-post, and is mortised into the mitre-post. The balance
beam should be abou-t four feet above the top of the lock ; its
principal use being to bring the centre of gravity of the leaf
near the heel-post and to act as a lever to open and shut the
leaf.
Sometimes this bar is dispensed with, and the leaves are
supported on rollers placed under the lower side to assist the
pivot in supporting their weight. These rollers run on iron
rails placed on the floor of the gate-chamber. In these cases
the gates are ordinarily opened and shut by means of wind-
lasses and chains. This is the method generally used for
462 CIVIL ENGINEERING.
very large gates. Gates formed of a single leaf moving on a
horizontal axis are frequently used.
642. Inclined planes. Instead of locks, inclined planes
are sometimes used, by means of which the boats are passed
from one level to another. In these cases, water-tight cais-
sons or cradles, on wheels are used.
At the places where the levels are to be connected, the
canal is deepened to admit of the caisson or the cradle to run
in under the boat to be transferred. Two parallel lines of
rails start from the bottom of the lower level, ascend an in-
clined plane up to a summit a little above the upper level,
and then descend by a short inclined plane into the upper
level. Two caissons or cradles, one on each set of rails, are
connected by a wire rope, so that one ascends while the other
descends. Power being applied, the boats are transferred to
the appropriate levels.
The caissons are preferred because they balance each other
at all times on the inclined plane, whether the boats are light
or heavy, as they displace exactly their own weight of water
in the caisson. In some cases, the caissons have been lifted
vertically instead of being drawn up inclined planes.
643. Guard lock. A large basin is usually formed at
the outlet, for the convenience of commerce ; and the en-
trance from this basin to the canal, or from the river to the
basin, is effected by means of a lock with double gates, so
arranged that a boat can be passed either way, according as
the level in the one is higher or lower than that in the other.
A lock so arranged is termed a tide or guard lock, from its
uses. The position of the tail of this lock is not indifferent
in all cases where it forms the outlet to the river ; for were
the tail placed up stream, it would generally be more difficult
to pass in or out than if it were down stream.
644. Lift of locks. The vertical distance through which
a boat is raised or lowered by means of the lock is called the
" lift." This vertical distance between two levels may be
overcome by the use of a single lock or by a " flight of locks."
The lift of a single lock ranges from two to twelve feet, but
generally in ordinary canals is taken at about eight feet.
Where a greater distance than twelve feet has to be over-
come, two or more, or a flight of locks, are necessary.
In fixing the lengths of the levels and the positions of the
locks, the engineer, if considering the expenditure of water,
will prefer single locks with levels between them, to a flight
of locks.
In most cases, a flight is cheaper than the same number of
single locks, as there are certain parts of the masonry which
WATER SUPPLY. 463
can be omitted. There is also an economy in the omission of
the small gates, which are not needed in flights. It is, how-
ever, more difficult with combined than with single locks
to secure the foundations from the effects of the water, which
forces its way from the upper to the lower level under the
locks. Where an active trade is carried on, a double flight is
sometimes arranged, one for the ascending, the other for the
descending boats. In this case the water which fills one flight
may, after the passage of the boat, be partly used for the
other, by an arrangement of valves made in the side wall
separating the locks.
The engineer is not always left free to select between the
two ; for the form of the natural surface may require him to
adopt a flight at certain points. In a flight the lifts are
made the same throughout, but in single locks the lifts vary
, according to circumstances. Locks with great lifts consume
more water, require more care in their construction, and re-
quire greater care against accidents than the smaller ones,
but cost less for the same difference of level.
645. Levels. The position and the dimensions of the
levels must be mainly determined by the form of the natural
surface. By a suitable modification of its cross-section, a
level can be made as short as may be deemed desirable ; there
being but one point to be attended to in this, which is, that a
boat passing between the two locks, at the ends of the level,
will have time to enter either lock before it can ground, on
the supposition that the water drawn off to fill the lower lock,
while the boat is traversing the level, will just reduce the
depth to the draught of the boat.
646. Water supply. Two questions are to be considered :
the quantity of water required, and the sources of supply.
The quantity of water required may be divided into two
portions: 1st. The quantity required for the summit level,
and those levels which draw from it their supply. 2d. The
quantity which is wanted for the levels below those, and
which is furnished from other sources.
The supply of the first portion, which must be collected at
the summit level, may be divided into several elements: 1st.
The quantity required to fill the summit level, and the levels
which draw their supply from it. 2d. The quantity required
to supply losses, arising from accidents ; as breaches in the
banks and the emptying of the levels for repairs. 3d. The
supplies for losses from surface evaporation, from leakage
through the soil, and through the lock gates. 4. The quan-
tity required for the service of the navigation, arising from
the passage of the boats from one level to another.
4:64: OTVTL ENGINEERING.
The quantity required to fill the summit level and its de-
pendent levels will depend on their size, an element which
can be readily calculated; and upon the quantity which
would soak into the soil, which is an element of a very inde-
terminate character, depending on the nature of the soil in
the different levels.
The supplies for accidental losses are of a still less deter-
minate character.
The supply for losses from surface evaporation may be de-
termined by observations on the rain- fall of the district, and
the yearly amount of evaporation. Losses caused by leakage
through the soil will depend on the greater or less capacity
which the soil has for holding water. This element varies
not only with the nature of the soil, but also with the shorter
or longer time that the canal may have been in use ; it having
been found to decrease with time, and to be, comparatively,
but trifling in old canals. In ordinary soils it may be esti-
mated at about two inches in depth every twenty-four hours, for
some time after the canal is first opened. The leakage through
the gates will depend on the workmanship of these parts.
In estimating the quantity of water expended for the ser-
vice of the navigation, in passing the boats from one level to
another, two distinct cases require examination : 1st. Where
there is but one lock ; and 2d. Where there are several con-
tiguous locks, or, as it is termed, a flight of locks between
two levels.
To pass a boat from one level to the other from the lower
to the upper end, for example the lower gates are opened,
and the boat having entered the lock they are shut, and water
is drawn from the upper level to fill the lock and raise the
boat ; when this operation is finished, the upper gates are
opened and the boat is passed out. To descend from the
upper level, the lock is first filled ; the upper gates are then
opened and the boat passed in ; these gates are next shut, and
the water is drawn from the lock until the boat is lowered to
the lower level, when the lower gates are opened and the boat
is passed out.
Hence, to pass a boat, up or down, a quantity of water
must be drawn from the upper level to fill the lock to a height
which is equal to the difference of level between the surface
of the water in the two ; this volume of water required to
pass a boat up or down is termed the prism of lift. The
calculation, therefore, for the quantity of water requisite for
the service of the navigation, will be simply that of the
number of prisms of lift which each boat will draw from the
summit level in passing up and down.
WATEB SUPPLY. 465
An examination of the quantity of water used in passing
from one level to another, will show that the quantity required
for a flight of locks is greater than that required for isolated
locks.
The source of supply of water is the rain-fall. The rain-
water which escapes evaporation on the surface and absorp-
tion by vegetable growth, either runs directly from the surface
of the ground into streams, or sinks into the ground, flows
through crevices of porous strata and escapes by springs, or
collects in the strata, from which it is drawn by means of weUs.
647. In whatever way the water may be collected, the
measurement of the rain-fall of the district from which it
comes is of the first importance. To make this measurement,
the area of the district called the drainage area or catchment
basin, and the depth of the rain-fall for a given time must be
determined.
Drainage area. This area is generally a district of country
enclosed by a ridge or water-shed line which is continuous
except at the place where the waters of the basin find an
outlet. It may be divided by branch ridges or spurs into a
number of smaller basins, each drained by a stream which
runs into the main stream.
Depth of rain-fall. The depth is determined by estab-
lishing rain-gauges in the district and having careful obser-
vations made for as long a period as possible.
The important points to be determined are : 1. The least
annual rain-fall ; 2. The- mean annual rain-fall ; 3. The great-
est annual rain-fall ; 4. The distribution of the rain-fall
throughout the year ; 5. The greatest continuous rain-fall in
a short period.
For canal purposes, the least annual rain-fall and the
.ongest drought are the most important points to be known.
Knowing the depth of the rain-fall and the area of the
catchment basin, an estimate of the amount of water which
may be available for the canal may be made. Theoretically
considered, all the water that drains from the ground adjacent
to the summit level, and above it, might be collected for its
supply ; but it is found in practice that channels for the con-
veyance of water must have certain slopes, and that these
slopes, moreover, will regulate the supply furnished in a cer-
tain time, all other things being equal. The actual discharge
of the streams should be measured so as to find the actual
proportion of available to total rain-fall, and the streams
should be measured at the same time the rain-gauge observa-
tions are made.
The measurement of the quantity of water discharged bv a
30
4:66 CIVIL ENGINEERING.
stream is called " gauging," and to be of value should Y.e made
with accuracy and extend through some considerable time.
648. Feeders an^ reservoirs. The usual method of col-
lecting the water, and conveying it to the summit level, is
by feeders and reservoirs. The feeder is a canal of a small
cross-section, which is traced on the surface of the ground
with a suitable slope, to convey the water either into the
reservoir, or direct to the summit level. The dimensions of
the cross-section, and the longitudinal slope of the feeder,
should bear certain relations to each other, in order that it
shall deliver a certain supply in a given time. The smaller
the slope given to the feeder, the lower will be the points at,
which it will intersect the sources of supply, and therefore
the greater will be the quantity of water which it will re-
ceive. The minimum slope, however, has a practical limit,
which is laid down at four inches in 1,000 yards, or nine
thousand base to one altitude ; and the maximum slope should
not be so great as to give the current a velocity which would
injure the bed of the feeder. Feeders are furnished, like
ordinary canals, with contrivances to let off a part, or the
whole, of the water in them, in cases of heavy rains, or for
making repairs.
A reservoir is a place for storing water to be held in re-
serve for the necessary supply of the summit level. A reser-
voir is usually formed by choosing a suitable site in a deep
and narrow valley, which lies above the summit level, and
erecting a dam of earth, or of masonry, across the outlet of
the valley, or at some more suitable point, to confine the water
to be collected. The object to be obtained is to collect the
greatest volume of water, and at the same time present the
smallest evaporating surface, at the smallest cost for the con-
struction of the dam.
649. Darns. The dams of reservoirs have been variously
constructed : in some cases they have been made entirely of
earth ; in others, entirely of masonry ; and in others, of earth
packed in between parallel stone walls. It is now thought
best to use either earth or masonry alone, according to the
circumstances of the case ; the comparative expense of the
two methods being carefully considered.
Earthen darns should be made with extreme care, of the
best binding earth, well freed from everything that might
cause filtrations.
The foundation is prepared by stripping off the soil and
excavating and removing all porous materials, such as sand,
gravel, and fissured rock, until a compact and water-tight bed
is reached.
DAMS. 467
A culvert for the outlet-pipes is next built This should
rest on a foundation of concrete and should have the masonry
laid in cement or the best of hydraulic mortar. It should be
well coated with a clay puddling. Frequently the inner end
of the culvert terminates in a vertical tower, which contains
outlet-pipes for drawing water from different levels, and the
necessary mechanism by means of which the pipes can be
closed or opened. Sometimes a cast-iron pipe alone is laid
without any culvert.
The earth is then carefully spread in layers not over a foot
thick and rammed. A " puddle-wall " with a thickness at the
base of about one-third its height and diminishing to about
half this thickness at the top, should form the central part of
the dam. Care should be taken that it forms a water-tight
joint with the foundation and also with the puddle coating
of the culvert.
The dam may be from fifteen to twenty feet thick at top.
The slope of the dam towards the pond should be from three
to six base to one perpendicular ; the reverse slope need only
be somewhat less than the natural slope of the earth.
The outer slope is usually protected from the weather by
being covered with sods of grass. The inner slope is usually
faced with dry stone, to protect the dam from the action of
the surface ripple.
FIG. 237. A, body of the dam.
a, top of the waste -weir.
6, pool, formed by a stop-plank dam at c, to break the fall of the
water.
d, covering of loose stone to break the fall of the water from the
pool above.
Masonry dams are water-tigjht walls, of suitable forms
and dimensions to prevent filtration, and to resist the pressure
of water in the reservoir. The cross-section is usually that of
a trapezoid, the face towards the water being vertical, and the
exterior face inclined with a suitable batter to give the wall
sufficient stability. The wall should be at least four feet thick
468 CIVIL ENGINEERING.
at the water line, to prevent filtration, and this thickness may
be increased as circumstances may require.
650. Waste-weirs. Suitable dispositions should be made
to relieve the dam from all surplus water during wet seasons.
For this purpose arrangements should be made for cutting off
the sources of supply from the reservoir ; and a cut, termed a
waste-weir (Fig. 237), of suitable width and depth, should
be made at some point along the top of the dam, and be faced
with stone, or wood, to give an outlet to the water over the
dam. In high dams the total fall of the water should be
divided into several partial falls, by dividing the exterior
surface over which the water runs into offsets. To break the
shock of the water upon the horizontal surface of the offset,
it should be covered with a sheet of water retained by a dam
placed across its outlet.
In extensive reservoirs, in which a large surface is exposed
to the action of the winds, waves might be forced over the
top of the dam, and subject it to danger; in such cases the
precaution should be taken of placing a parapet wall towards
the outer edge of the top of the dam, and facing the top
throughout with flat stones laid in mortar.
651. Water-courses intersecting the line of the canal.
The disposition of the natural water-courses % which intersect
the line of the canal will depend on their size, the character
of their current, and the relative positions of the canal and
stream.
Small streams which lie lower than the canal may be con-
veyed under it through an ordinary culvert. If the level of
the canal and stream is nearly the same, it may be conveyed
under the canal by an inverted syphon of masonry or iron,
usually termed a broken-back culvert, or if the water of the
stream is limpid, and its current gentle, it may be received
into the canal. Its communication with the canal should be
so arranged that the water may be shut off or let in at plea-
sure, in any quantity desired.
In cases where the line of the canal is crossed by a torrent,
which brings down a large quantity of sand, pebbles, etc., it
may be necessary to make a permanent structure over the
canal, forming a channel for the torrent ; but if the discharge
of the torrent is only periodical, a movable channel may be
arranged, for the same purpose, by constructing a boat with
a deck and sides to form the water-way of the torrent. The
boat is kept in a recess in the canal near the point where it
is used, and is floated to its position, and sunk when wanted.
When the A ine of the canal is intersected by a wide water-
course, the communication between the two shores must be
IRRIGATING CANALS. 469
effected either by a canal aqueduct bridge, or by the boats
descending from the canal into the stream.
652. Dimensions of canals and their locks in the United
States. The original dimensions of the New York Erie Canal
and its locks have been generally adopted for similar works
subsequently constructed in most of the other States. The
dimensions of this canal and its locks were as follows : .
Width of canal at top 40 feet.
Width at bottom 28 "
Depth of water 4 "
Width of tow-path 9 to 12 "
Length of locks between mitre-sills 90 "
Width of locks 15
For the enlargement of the Erie Canal, the following are
the dimensions :
Width of canal at top 70 feet.
Width at bottom 42 "
Depth of water 7 "
Width of tow-path ; 14 "
Length of locks between mitre-sills 110 "
Width of lock at top 18.8 "
Width of lock at bottom 14.6 "
Lift of locks 8
Between the double locks a culvert is placed, which allows
the water to now from the level above the lock to the one
below, when there is a surplus of water in the former.
IBBIGATING CANALS,
653. Canals belonging to this class are used to bring from
its source a supply of water, which, when reaching certain
localities, is made to flow over the land for agricultural pur-
poses. This kind of canal is practically unknown in the
United States, as the farmer depends almost entirely on the
rain-fall alone for the requisite amount of moisture for his
crops.
Irrigation canals of large size have been used in India for
hundreds of years ; they are also found in Italy. Kude imi-
tations, of small size, are to be seen in Mexico, the territory
of New Mexico, lower part of California, and other parts of
the United States.
In certain parts of our country they could be used to great
470 CIVIL ENGINEERING.
advantage, and since in the future they may be used, it is
thought advisable to allude briefly to them in this treatise.
The special difference between a navigable and an irri-
gation canal is that the former requires that there should be
little or no current in the canal, so that navigation may be
easy in both directions, while the latter requires that the
canal should be a running stream, fed by continuous supplies
of water at its source, to make up the losses caused by the
amounts of water drawn off from the canal for the purposes
of irrigation.
Hence, for two canals of the same size, the navigable
canal will require a less volume of water than the irriga-
tion canal, and is more economically constructed on a low
level.
The irrigation canal should be carried at as high a level as
possible, so as to have sufficient fall for the water which is to
be used to irrigate the land on both sides of it and at con-
siderable distances from it. This irrigation is effected by
means of branch canals leading from the main one, whence
the water is carried by small channels on the fields.
654. The problem of an irrigation canal is to so connect it
with the stream furnishing the supply of water, and to so
arrange the slope of the bed of the canal, that the canal
shall not become choked with silt.
A canal opening direct into the stream which supplies it
with water, if proper arrangements are not made, will be lia-
ble to have the volume of water greatly increased in time of
freshets, and at other times have the supply entirely cut off.
In the first case, large quantities of silt would be washed into
the canal, choking it up as the water receded to its proper
level. In the second case, the supply would probably fail at
the critical period of the growing crops when water was
greatly needed.
A good selection of the point where the canal joins the
stream, and the use of sluices to govern the supply of
water, will greatly prevent the occurrence of either of these
conditions.
To prevent the silting up of the canal, the slope of the bed
is so fixed that the water shall have a uniform velocity
throughout. It is therefore seen that, as the water is drawn
off at different points for the irrigation of the land, on the
right and left of the canal, the volume of water is reduced.
The portions of the canal below these points must then be so
fixed as to preserve the same rate of motion in the water.
This is done by decreasing the width and depth of the canal,
and increasing the slope of the bed. Thus starting with a
DRAINAGE CANALS. 471
water-way 100 feet wide, 6 feet deep, having a slope of 6
inches to the mile, the width of water-way, as the water is
drawn off, may be contracted to 80, 60, 40, and 20 feet with
the corresponding depths, 5-J-, 5, 4J, and 4 feet ; to keep the
velocity uniform the bed should have slopes of 6.4, 7, 7.9, and
10.3 inches per mile.
655. An irrigation canal may be used for the purposes of
navigation. In this case the principles already laid down
for navigable canals equally apply, with the condition, how-
ever, that the velocity of the current in the canal should not
be so slight as to injure its uses as an irrigation canal, nor so
swift as to offer too great a resistance to the boats using it
as a navigable canal.
DBAINAGE CANALS.
656. Canals of this class are the reverse of irrigation canals.
They are used to carry off the superfluous water which falls
on or flows over the land.
The water-levels of canals for drainage, to be effective,
should at all times be at least three feet below the level of
the ground.
Each channel for the water should have an area and decliv-
ity, when subjected to the most unfavorable conditions, suffi-
cient to discharge all the water that it receives as fast as this
water flows in, without its water-level rising so high as to
obstruct the flow from its branches or to flood the country.
Hence, to plan such a system the greatest annual rain-fall
of the district, and the greatest fall in a short period or flood
must be known.
Where the land to be drained is below the level of high
water, the area to be drained must be protected by embank-
ments. The canals are then laid off on the plan just given,
and the water from the main canals is removed by pumping,
Drainage canals may be divided into two classes: open
and covered. Where pure water is to be removed, the former
are -used ; when filthy water, or foul materials, are to be re-
moved, the latter are used, and are known then as sewers.
Sewerage is the special name used to designate the drainage
of a city or town, in which the foul waters and refuse are
collected and discharged by sewers.
As far as the principles of construction are concerned sewers
do not differ from the works already described. Especial at-
tention must be paid to prevent the escape of the* foul gaa
*nd disagreeable odors from the drains.
4:72 CIVIL ENGINEERING.
CANALS FOE SUPPLYING CITIES AND TOWNS WITH WATER.
657. As sewers are only particular cases of drainage canals,
so canals for supplying cities with water are only particular
cases of irrigation canals, and are therefore governed by the
same general principles in their construction.
The canals of this class are usually covered, and receive the
general name of aqueducts.
658. The health and comfort of the residents of cities and
towns are so dependent upon a proper supply of water and a
good system of sewerage that the greatest care must be taken
by the engineer that no mistakes are made by him in planning
and constructing either of these systems. The principles
which regulate in deciding upon the quantity of water re-
quired, the means and purity of the supply, the location of the
reservoirs, the method of distribution, etc., form a subject
which can be considered in a special treatise only. The same
remark applies also to sewerage.