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, 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, = - 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