Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031263902 Qornell University Library arV18356 A handy book tor the cs!S"''fi°,!;,,9L^^^^^^ 3 1924 031 263 902 olin.anx PLATE I. HANDY BOOK FOK THE CALCULATION OF STEAINS IN GIKDEES AND SIIILAR STRUCTURES, AND THEIR STRENGTH; CONSISTIKO OF FORMULAE AND CORRESPONDING DIAGRAMS, WITH NUMEROUS DETAILS FOR PRACTICAL APPLICATION, ETC., ETC. By WILLIAM HUMBER, ^ssoc. Inst. C.E. AUTHOR OP "a FXtACTlCAL TREATISE ON CAST AKD WBOnOHT IBOK BRIDGE COHSTEUCTION," "a RECORD OP THE PROGRESS OP MODERN EKOIKBEBIKO," SIC., ETC NEW YORK: D. VAN N08TRAND, PUBLISHER, No. 23 MuBKAT AND 27 Wabrbn ST8. 1869. Pholo-lithographic Beprodnction BY THE New York Lithograrhing, Engraving and rriuting Co., 16 and 18 Fatk Flaee. PEEFACE. Notwithstanding that the subject of Strains has' been ably treated of again and again, it is dif&cult in submitting this little work to the public to avoid the almost stereotyped ex- pression that " the design has been to supply a want long felt in the profession ; " for the numerous volumes which have appeared on the subject have not, principally on account of their elaborate investigations, been calculated to afford that ready assistance which in the ordinary run of office and other work is being continually needed, while on the other hand, most of the general Engineering Pocket-books, not having been able to afford svifficient space to do justice to the subject, have been compelled to leave its treatment incomplete. It is hoped, therefore, that by devoting a small work, in a handy form, entirely to Bridge and Girder Calculations, without giving more than is absolutely necessary for the complete solution of prac- tical problems, both the above obstacles to quick and satisfac- tory manipulation may be overcome. One of the chief features of the present work is the exten- sive application of simply constructed diagrams to the. calcu- lation of the strains on bridges and girders, the advantages of which as a system, most ■ undeniably far outweigh its disad- vantages. The parabola (anything but a troublesome figure to draw) and a few right lines are all that are required. Tliere is, again, a more general application of the Moments of Buptore, and Shearing Forces, to open-webbed girders of all kinds, than has hitherto been attempted. It was originally intended to divide the whole work into three sections or chapters, similar to those actually adopted only for the middle portion (pp. 24 to 60), which chapters should correspond with the yanous processes in the design of a bridge, thus making the yery arrangement of the work a general gmde. But it was afterwards deemed advisable,- as win be seen, to place the Moments of Bupture, and Shearing Forces, by themselves at the commencement, as a basis upon which the remainder is principally founded. In the following pages will be found, almost necessarily, many omissions, but care has been taken as far as possible to. avoid inaccuracies. It wiU be observed that attention has been paid to the arrangement of the matter in different types, so as to facilitate as far as possible the manipulation of the contents. The work is not advanced with the pretensions of a treatise, as no investigations whatever are given, but merely their results; and these, it is hoped, in an intelligible and practical form, suited to the wants of the Engineer, Architect, Draughtsman, or Builder. CONTENTS. STRAINS IN BEAMS. PARAGRAPH PAOK 1, Inibosuotort.— Stabilit; of a loaded beam . > . ' . 1 MOMENTS OF RUPTURE, 2, 3. AbbreTiatioDS 1 i. Positive and negative moments 2 5, 6. Diagrams of tbe moments 2 7. Semi-beam loaded at extremity . . ... . .2 8. Semi-beam loaded at several points . . . . -2 9. Semi-beam uniformly loaded .I 10. Semi-beam uniformly loaded for part of its length onljr . . 3 11. Semi-beam uniformly loaded, and also with a weight at its extremity 3 12. Beam loaded at the centre 4 13. Beam loaded at any other point i 14. Beam loaded with several weights ..... 4 15. Beam loaded with two equal weights equidistant about the centre 5 16. Beam loaded with four equal weights symmetrical from the centre . . . 5 17. Beam loaded with a concentrated rolling-weight . , . 8 18. Beam loaded with two concentrated rolUng-weigbts > . 6 19. Beam uniformly loaded 6 20, 21. Beams with a load uniformly distributed oyer part of their lengths . . ' . . . . ■ . .7 22. Beam with a distributed rolling load 8 23. Points of contrary flexure 8 24. Beam of uniform section, or uniform strength, fixed at both ends and loaded at centre . . . . . .9 26. Beam of uniform section, fixed at both ends, and uniformly loaded 9 26. Beam of uniform strength, fixed at both ends, and uniformly loaded • 1" 27. Beam of uniform section, uniformly loaded, the ends being subject to known moments of rupture . - . . .10 28. Beam of uniforiu section, fixed at one end only, and uni- formly loaded . , 11 VI OOSTKHTS. PAKAOBIPH PAQB 29. Beam of uniform strength, fixed at one end only, and uni- formly loaded .11 30. Continuous beam, with uniformly distributed btationary load 12 31. Continnons beams with varying loads 12 32. Beam continuous over one pier, subject to a stationary, and « also a moving load . . . . . . .12 33. Beam continuous over two or more pierB, subject to a sta- tionary, and also a moving load 14 SHEARING F0RGE3. • 34,35. Abbreviations and. diagrams 16 36. General rule for the shearing force 16 37. Semi-beam loaded in any manner 16 38. Semi-beam uniformly loaded 17 39. Semi-beam loaded uniformly, and with a concentrated weight 17 40. Beam loaded at centre 17 41. Beam loaded at any point , 17 42. Beam loaded with a concentrated rolling weight . . .17 43. Diagram of the shearing forces produced by a concentrated weight on a beam . . ; 18 44. Beam uniformly loaded . -■. 18 45. Beam with distributed moving load 18 46. General formulae for the shearing forces in continuous beams . 19 47. Beam fixed at both ends, and uniformly loaded . . .19 48. Beam of uniform .strength, fixed at one end only, uniformly loaded 19 49. Beam of uniform section, fixed at one end only, uniformly loaded . .20 50. Continuous beams with uniformly distributed stationary loads 20 51. Beam continuous over one pier, subject to a stationary, and also a moving load. ....... 20 52. Beam continuous over two or more piers, subject to a sta- tionary, and also a moving load 21 FLAITSED GIRDERS, ARCHES, AND SUSPENSION BRIDGES. 53. Processes in the design of abridge 24 SGCTIOK I. — ^DKIERHIKAIIOir OF THK NAItTSB, FslHOIPAL DlHBK- 8I0HB, EIO., OF THB SiKVOIDBB. 54, 55. Kind of bridge and general cross-seotion . . . .24 66 — 59. General proportions 25 Section U.—OaloviiAiioii of ihb Stbaiih on thb taxioits Pabts. 60, 61. Approximate estimation of the weight of the structure , 25 .FLANGED GIRDERS WITH THIN CONTINUOUS WEBS. PABAQRAPa PAOE 62, Distinct fanctions of the flanges and the web . . .26 6S— 6{i. Strains in the flanges generaUy 26 66. Strains in the web generally 27 Oirden ivith Parallel Straight Flangta. 67—71. Flanges 27 72—74. Web 28 OirderB imth Curved or OUique Flanget. 75—78. Flanges 28 79—83. Web 28 GIRDERS WITH WEBS 01" OPEN BRACING. aXNEBAIi BCLES. 84—87. Booms 30 88—95. Web 80 CaLODLAIIOK by means of the MoUXNTS of BnPTDRB AND Shxabiho FOBOES. 96. Loads concentrated at apices 81 Girderi viiih, PwrallA Straight Boona, 97, 98. Notes 31 99. Warren girder, loaded on one boom . . . .31 100, 101. Uounter strains in the web 32 102. Warren girder, both booms loaded . . . .32 103. Girder with vertical struts and inclined ties . . .32 104. Warren girder with scalene bracing, loaded on one boom . S3 105,106. Method of applying the diagrams 33 107. Lattice girder loaded on one boom . . . .34 108. Lattice semi-girder loaded on one boom . . . .35 109, 109a. Lattice, whole or semi, girder loaded equally on both booms 36 110. Concentrated loads on lattice girders . . . .36 Oirdera mth Curved or Oblique Boona. 111. Curred or oblique whole or semi-girder, single-triangular web, loaded on one boom 37 112. Curved or oblique whole or semi-girder, single-triangular web, loaded equally on both booms . . . .37 113, 113a. Continuous girders 38 114. Fixing the points of inflection in continuous gurders . 38 115. Continuons girders with varying depths. , . .38 CaLOCLATION BT the CoMPOBIItOir AMD UlSOIiUTIOH QF FoBOES. 116. Method # . 38 117. Keaction of supports , , 39 118, 119. Composition and resolution of forces — parallelogram — triangle 39 120. More than one concentrated load on a girder . . .39 Till C0KTEKT8, PARAOBAFB _ _ PAS" Example — Btrains in a bent girder (roof principal) 121. Calculation b; the iDomentB of mptare and shearing forces 39 122. Calculation by the composition and resolntion of forces . 41- KeTBODS or CALOTTLAIIOir FOUSDID OH THE FAEALIiELOGnAH OF. FoKOXS. 123. General law of the strains in the booms . . .41 124.~ Note on General Rules for booms and bracing . . 42 125. Concentration of loads'at the apices . . .42 126. Trigonogietrical functions 42 127. Straight semi-girder, loaded in any manner . .42 128. Example 42 129. 'Straight Warren semi-girder, loaded at extremity . . 43 130. 'Straight Warren semi-girder, loaded uniformly on one boom 43 131. Straight Warren girder, loaded at any point . . .43 132. Same, with weight at centre 44 133. Straight Warren girder, with a concentrated roiling load . 44 134. 'Any Uraight Warren or lattice girder, with any load symmetrically disposed about the centre . . ^ . 44 135. Any straight Warren or lattice girder, with a uniformly distributed moving load .45 136. Example 45 187. Dead and live loads on girders , , ... .45 138. Any straight Warren, lattice, or other girder, with an nnsymmetrical load . ...... 46 139. Lattice girders with the bars fixed at their intersection . 46 140. Simple truss ; central load 46 141. Same; load not at centre 47 142. Note on the above two cases 47 143. Simple truss; load distributed . . ■ . . .47 144. Compound' truss 47 ARCH BRIDQES. Akches with Sfandril Bbaoino. 145. Uniform horizontal load 48 146. Moving load 48 Unbbaosd Arches. 147. Neutral surface or ourv^ 49 148. ][jine gf pressures 49 U8^154. Stability of arches . , 49 1S4a. Arched bp^ge pf several spans 50 SUSPBNSJON BBIDetlSB, 155. Suspension bridge of one span, nniform horizontal load , 51 156. Suspension bridge of more thftn oq? spa^, uniform hori- zontal load ,...,... 51 157._S9Spen8ion In-idge with sloping rods, uniform horizontal ^ load . ..... . . .52 158—1610. Suspension bridges with moving loi^ds . . .52 162. Abutments and Fiers ....... 53 CONTENTa. ix Section hi. — ^Distribution of Material to £ksi3i the Caloulates Straihb. PARAGRAPH PACE 163, 164. Strength of structures 6:{ 165. Uniform strength 54 166. Units 54 FRiiraiPAL Strains to be met, eto. 167. Tension . . .* ' 64 168. Compression 54 169. Breaking weights of columns 55 170. Long struts 66 171. Shearing .66 172. Coefficients of safety 66 173. Modulus of elasticity 66 JOINTS. 174, 175. General rules , . . 67 Ikon Joints and Fastevinqs. 176. Rivets 67 177, 178. Bolts 57 179. Pin joints in tension bars 57 Riveted Joinit in Tenaion. 180, 181. Effective section, &o., of a plate [,S 182.' Lap-joints 68 183. Fish-joints 69 Riveted Jtmitt in Compreiiion. 184. Lap-joints 183. Butt-joints 186. Gibs and cotters Joints in Timber Strtiotures, 187. General note .... 188. Joints in tension — fished and scarfed 189. Joints in compression 69 59 59 . 59 . 59 . 60 190. Shouldered tenon for attaching cross to main beams . 60 BEAMS OP VARIOUS SECTIONS. 191. General remarks ^ . 60 192 — 195. Designing a beam 60 196. Stability of a loaded beam 61 197. Abbreviations 61 198, 199. Neutral axis 61 200, 201. Notes «1 202. Curved beams ffl 203. Modulus of rupture 62 MouENis OF Ihekila. ahd ResistasoS' of Beams or VARIOUS SECTIONS. I-ABAORAFH PACK 204. Beam of solid rectaBgnlar section 62 20.'). Beam of hollov rectangular section . . ,62 206. Beam of solid circnlar section 62 207. Beam of hollow circular section 63 208. Beam of solid elliptical section 63 209. Beam of hdllow elliplical section 63 210. Beam with one flange 63 211. Beam with two equal flahjies . . .64 212. Beam with two unequal flanges 64 213. Beam of any section 64 214. Similar beams 64 215 — 218. Beams of rectangnlar section and of uniform strength . 65 219. How to cut the strongest and stiffest heam from a cylin- drical log 66 220. To find the centre of gravity of any cross section . . 66 DEFLECTION. 221. Definition 66 222. Camber 66 223. Girders of uniform section ..,',, 66 224. Flanged girders of uniform strength , . , .66 225. Flanged semi-girders of nniform strength . . .67 226. Continuous girders and whole girders fixed at one or both ends . 67 BBEAEING AND SAFE LOADS FOR BRIDGES AND OIRDERS. 227, 228. Breaking or safe load found from amount of material, span, &c. 229, Example I.— Flanged girder 280. Btample II. — Semi-beam of rectangular section 231. Table of the strength and elasticity of materials 232, Methods of constructing parabolas . . 67 68 68 68 70 Directions to Binder. Plate I. to form .... Frontispiece. ,, II. to face Page 58 „ III. to face 70 FORMULAE AND DIAGRAMS FOK THE CALCULATION OF GIRDERS, &o. STRAINS IN BEAMS. I. The stability of a loaded beau or girder is.fonnded on the equality that must always exist between the resultants of all the various external forces tending to cause its rupture, and the sum of the molecular reactions which resist the same. The former may be resolved — (1) hori- zemtally into strains, depending for their value upon what are known as Moments of Supture, or Bending Moments, tending to cause the failure of the beam, by tearing asunder its fibres in one part' and crushing them together in another (4) : and (2) vertically into what are known as Shear- ing Forces, due to the transmission of the vertical pressure of the load to the points of support, and tending to cause contignous vertical sections in the beam to slide over each other (171). The values of the molecular reactions are theMoments of Sesistance, for which see (804 — 91*}. MOMENTS OF KUPTURE. 9. Abbrevlatloug adopted .tn tlie Formnla;. Mx = moment of rupture at any point {x). Mai Ms = „ at points of support (A, B). Mc = „ at centre of span. I = length of clear span = distance between supports in a whole beam =. distance between W and support in a semi-beam loaded with W. Where need in any other way explanation will be made. X = horizontal distance between the left abutment (except where otherwise stated) and the point at which HI is to he found. W = concentrated load at any point (61). w = distributed stationary or dead load per unit of length, «!' = ,, moving or live „ „ (61). . Bef. = maximum deflection (2%1) for a beam of uniform section (ggs). I = momenli of inertia of the section of the beam (for value see 304— «14). E = modulus of elasticity (ITS) ; for value see 831. Other abbreviations will be explained as they occur, 3. Note, Z, z, and other horizontal distances when occurring in the same case, must be all of the same denomination ; and so also must w and w'. stuains in beaus. 4. Note.— The value of a fornrala being (^neg|^^e(!!!J) show* that the aotion of the load makes, oi tends to make the upper surface of the heam ( ^^'^''^' \ and therefore compresses together the fibres in the ( j™.. ) part, and stretches them in the ( lower \ pj^yj._ Vupper/ 5. In the Diaskaus, the . ordinates (the vertical distances from the horizontal, or other lines) to the curves, &c., as shown, thereon, correspoml to the values of th# formulae accompanying them. If the diagram be Jiawn to scale in the manner directed, the. Moments of Supture may be obtained by direct measurement. Fig. A. Fig. B. The vertical lines, as in fig. A, correspond to positive, and those as in fig. B to negative values in the formulae (4). 6. ))7hen a diagram is used in the calculation of the moments, it should be drawn immediately under, or over, and to the same horizontal scale as the outline sketch (5S, HI., and &9) of the girder itself, so that the ordi- nates in the diagram Will correspond with the several points in the length of the girder to which they apply. 7. Semi-lSeam Fixed at One Knd, and l.oaded ivith » Concen- trated Weigbt at tlie Othen Kg. 1. M.= -WZ . . . (*.) M« = (4.) Def. "3EI DiASKAU. — Let A B be the beam (5, 6). Draw A C = W J. Join C to B. Then the vertical dis- tances between A B and B will give the moments of rapture. S* Beml-Beam Fixed at One Knd, and loaded witli several Con- centrated WelBlits, W, W,, W„ (87). Fig. 2. M^=-(Wt-l-W,?, + W,Z,) . . . (4.) M, -- {W(Z-a!)-hWi (ii-a:) + W,(Z,-a:)} (4.) When (I— x) or (ij— a:), or so on, is negative, it is to be omitted. DiASKAM.— Let A B be the beam (5, 6). Draw AD = W,i„ DC = Wi«„ and AE = Wi. Join D to B, E to W, and C to F, in the manner shown. Then the vertical dis- tances between C F B and E W B wiU give the moments of rupture. MOMENTS OF Rt;t>!rtTlt£> Z 9. Beml-Beaiu Fixed at One Knd, and loaded ITnlformljr Ita Entire tensth (Bg). M i= — — g- ; ome-W/ t/to< iIik to the same load (w 2) concentrated at the end. M.= --sil-!^ 2* Def. = (4.) 8£X' DiASKAH. — Let A B be the beam (5, 6). Draw A C = wH' Draitr the parabola C B, whose vertex is at B (asa). Then the vertioal dUtanceB between A. B.and C B will give the moments of rapture, 10. 8emlBeam Fixed at One End, and Loaded wltli a IJni. formly Dlstrlbnted load, and also a Cokieentrated IiOad at Its Extremity (39). M.= -^W+ gjt .... (4.) M. = -(w(2-a!)+|(2-!F)») (4.) Fig. 5. wP BlAOluir. — Let A B be the beam (5, 6). Draw A = -7 , and A D= W I. Draw the parabola OB (as in 9) with its vertex at B (832 J. Then the vertical distances between D B and O. B wiU''=give the moments of riiptore. This is but a combination of ( T) and (9j, B 3 1%. STBAIKS IK BEAMS. Supported at Botb Enda, and Loaded at the Centre (40). Fig. 8. M^ = M„ = 0. Mc = Wl ' 2 the nearer pier) Ma: = ~s~ ^"^ l*™? me'osnred from Def. -W l^ ~48BI DiAGBAu. — ^Let A B be the heam W I B, and A K B as for three separate cases, by (IS). Produce W GniJl WL=WF + WG + WC. Produce W, DtillW, N=W, H + W, I + W; D; and so on for tlie \yeighf .W„ making W, P = W^ K + Wj J + WjE. • Join Jt to t, h to N, N to P, and F to B. Then the ordinates from A B to the poly^ gonal figure A L N P B will Bive the moments of rupture. Note. If the weights be all equal, the vertieals at the weights representing the mnments produced there by those 'weights (as WOj WjD, W^B) will all be ordinates to a parabola (S32) drawn as for (II). The following cases (15, 16) are adaptations of the above scheme to con- ditions frequently met with in practice, 15. Benin Supported at Both ISiids, nnd landed with Two EqunI Woiplits placed Kquldlstant n-oiii • the Centre.* ^'^-^ The moment fur any point between the weights is a constant quautity W (« - S) = -^2 — -= Wa = WJ. Between the weights and the sup- ports Uj: = Wa: . M^ = M, = 0. DiAQRAM. — Let AB be the beam (S, 6). At the weights erect WC and W D each equal to (W a). Join A to C, C to D, and I) to B. Then the ver- tical distances between A B and A G X> B will give the moments of rupture. 16, Beam Supported nt Both Ends, nnd l.ondcd with Four Eaunl Weiffhts ss'minctrlcally disposed nbont the Centre, f M at Wi = M at Wj =i W (2 a -^ a') ; constant from W, to Wj. M at W = M at W, = 2 W*-. DiAflSAU. — Let A B be the beam (6, «). At W and W, erect W C and WjF, each equal to (2Wa); and at Wi and Wj erect W, D and W, E, each equal to W (2 o + a'). Join A to C, C to D, D to B, K to P, and F to B. Then the vertical distances between A II and A C D E F B will g;ive the moments of rupture. * As in the caf^e of a cross girder cnrrylng a single line of railway, t Case of a crobs girder carrying a double line of railway. aiBiiKS iH beahs. II. Beam Supported at Both Ends, loaded with, a Coneentrated RoIIlUK WelEbt (49). — — — '- The mBzimom moment at any point— M^ = M. =0. DiAOKAH. — ^Let A B te the lieam (5, 6). DraT the iiarabola A D B (238), -whose ordinate at centre (C D) = -7— Then the vertical distanceB betvreen A3 and the parabola ADB will give the mazimam moments of ruptnie. 18. Beam Supported at Both Knds, and loaded with Two Welehts moTlne slmnltaneoiuly VifS-'ii- In either direction over the Let V and v, be the two weights. The valne of the maximum moinent produced at any point is M, = 7{(w+wJ (Z-a!)-Wi«} X jbeing measured from the nettrest pier. The positioii of w causing ihe ' greatest moment is when I 'w S IS X -! 5"— g/w4-'w \ ' ^^ '^ *''* *^° weights be equal ; when * = o — 4* Mi = M, = 0. DiASBAK. — Let A B be the beam (5, 6) . Draw the parabola A C B (S3S), ^ (w + w,) I. At A and B erect AD and BB = whose ordinate at centre = - Kg. IS. W, S. Join A to B and D to B. .Then the vertical distances between AFB and the paiabola A C U will give the masimnm moments of rupture. 19. Beam Bnpported at Both Knds, and loaded nnlfonnly Its entire lencth (41). % a: V) fl M. = -J- {l-x). Mc = -g-. Mx = M. = 0, one half the moment at centre, and | the deflection produced by the tame load cono^rated at.the centre. DiAOBAK. — Let A B be the beam (S, 6). pn A B draw the parabola * As tn the conpled driving wheels of 4 locomotive. MOMSHTS 07 KUPTURB. Wl* A B (23%) wbose ordinate at centre = 8 Then the vertical dis- tances between A B and the parabola A C B will give the moments of rupture. Flg.U, SO. Beam supported at Both Ends, subject to a I«ad milformly Iklstrlbated over a certain leugtb trotu one Support (15). . iet 3 tqual the length of the load, and B E, each = Join D to E, ^nd C to A and B. Then the vertical distances between DE and A.CB will give the mo- ments of rapture. The j>aint> of contrary jUxare (33) are at the intersection of A Q #n4 OB, withDE. 35. Beam of Eanal and Vniform Section, Fixed liorlzontaliy at Botii ^iM|SuA>ul Iioaded nnlformi]' Its entire I«nEtli (4T). (^^e 3a.) The length f i> identicai wilk (19), and the parts A/ and B/' mth (11), the Gontsentrated load at the ex- tremity of each semi-beam A/, Bf being Pig. 18. equal to half the distributed load on// . M. wx wl^ ■wl' Mi =M, = — -jg- . . (4.) Where M, = 0, there are the points of contramj flexure (l^S), distant from A and B respectively by '2112. 5v>l* ., , , Wk ~ 1 S36 B I ~ ""' V^i^^ "^' "/ '^ same ieaifii. if not fixed at die endsT DiASBAU. — liet AB be the beam (5, 6). On AB draw the parabola ACB (832), whose ordinate at centre CD = -5-. At A and B erect 10 STRAIH3 IN BEAMS. A K and B F respectively, each equal to 12 ■ Join E to F. Then the vertical distaneeB lietween E F and A.G B will give the momenta of rupture. The points of contrary flesaire (S3), are at the intersection of A C B .•Brith E.P. Fig. 19. !!6. Beam of Tniform Strensth (I65), Tlxed hoTlzantally at Both Ends, and loaded nntfonnly Its entire lenstta (4T). (See 30.) The length ff {= ^ij w identical vnth {19), amd the parts A/ amd Bf wtectiTely, makin(r A E = 11 ;,, and B F = M. . Join E to F. Then Uie vertical digtanoes between B F and A O B will give the moments of rapture. When S F interaeets A C B, there will be the pointt of contrary fiaatre (S3). >8. Beam ut VnironB and Knnal Beetton, anpported at One Bnd (A), and Fixed horizontally at the Other (B)*, nnirormlr leaded over Ita entire Eencth (49). Thi» east it identical v/ith the Ungthf^df(M). w X mix M, = -g- (J — a) - — g— (a being measared from the nnfixed end). M» ?= 0, M. = - -g- . (4.) Tbepoia( of eontrary flexure (83) ia where Mi = 0. 82 M at midway between A and / = . ' loa ' ' DuasAK.— Let A B be the beam (S, 6). On A B draw the panboU A C B (831), whose ordinate at centre CD = ^ . At B, the fixed end of a the beam, erect B E = ^ . = C 0. Join A to E. Then the vertical diatances 8 between A. E and A C B will e^ve the moments of rnptnre. Where A E inter- sects C B, there will be the point of contrary fltxm-e (83). 29. Beam of Vnlform Btrencth (16S), inpported at One End (A) and Fixed horlzontallr at the other .(B)*> nnlformly I«aded over tta entire Kensth (48). 2%i( cate w identical wif^ tJie length fB of (26). wa mix . Mj, = -J" (' — as) — -g~i [x being measured from the nnfixed end). M^ = 0. M, = - -g- . (4.) M at midway between A and / = "iF" The paint of amtrary flexure (23) ia where Mz => 0. Tig. 22, A/=|'- • See 39. f The valaea in this case are slightly erroneous. The oonect valae for A/ ia •783*. 12 STKAINS IN BEAMS. DjASKAM., — LetAB be the team (5, «). .On AB draw the parabola ACS (»3!S), whose ordinate at centre (C D) = ^. At B, the fixed enll of the girder, erect B E = M, = — p-. Join A to E. Then the vertical 6 distances between A E andA OB will give the moments ofnipture. Where AE intersects A C B, there is the point of contrary flexure (83). 30. Benin continuous Tor two of more Rltrlitly Pritportioned * Spans, sultject to a Stationary Load, Vulforinly Dlslributcil over its entire JLenetli (50). All such cases may he regarded as comhinations of some of the cases previously given (24 — 29). For if, in any of the latter, -the beam, instead of being fixed atone or both ends, be .continued over a support where originally fixed, and subjected to the action of a load, which shall produce at the point of support a moment equal to that produced there by the first load when the beam was fixed at that support, then the moments in the original length of the beam will remain as they were^ and will not be affected by the substitution of the continuation for the fixing. Fijj. 23. Let AF.be a beam continuous over, a number of equidistant piers, B, C, &c. If the beam be of uniform strength (l«S), the outer spans (A B and E F) should be J the length of the otUeis. If of uniform section, the outer spans .should be 789 the length of the others.. (See USA.) The end spans may be regarded as identical with (28) or (20), and the remaining spans with (24, 29, or 2G), so that the moments of rupture may be obtained from the formulae or diagrams there given. .tl. Continuous Bcains, not of Vniforin Section, subject to Taryins Loads. Tt would, perhaps, be impossible to give mathematically accurate formulse for the moments of nipture in continnous beams, witih moving loads, that would be worth anything for practicaV application. A well-known author t has even pro- nounced the case ** too complicated for investigation." The following approximations, however, may. be relied on for tafeiy vnihnvt enetravagance. 32. Beam of Vnlform Strenetb .(I6S) for the Blaxiuium Strainn, • By riphtly proportioned is meant, proportioned so that if the beam were fixed on any one of the piers instead of continuous over it, the moments iwodnoed there by the two adjacent loaded spans would be eqnal. Then Tf thin condition bfobnerveU the cabre will include beams not uniformly loaded throughout. t J. H. Latham, Esq., M. A.— "iron Bridges." MOMKNTS OF EUriDRE. 13 rontlnnoiis over one Pier, formlQe: two Evoal Spans, subject to a Fixed Load Viiiformly Oistrlbuted, and also to a Itlovlns load (Si). (See 113 A.) The greatest moment over the pier will be pvoduoed when both spans ave fallyloaded. Each spaa will then nearljr correspond to ■'.99). The greatest posiiive (4) moment will obtain in the span fully loaded, when the other spau bears only the fixed loud. Let w = fixed load per unite of length. a/= varying ,, ,, and let X be always measured from an abutment, and not from the pier. Then, X {rb + w') (I — x) (2 TO + M)*)*- 1 Maxm. positive (4) moment, Mj, = 5 — j^ . (to + w') t* Masm. negative (4) moment over pier (C) = „ Max. neg. mom. [M. = g (M + «/) (2 2 - 3 x), or 1 ^^^ ^^^^^^ ^,_ at any other/ ^^ ^^ ) be taken. P^"*. j^M« = Y (^ - =") - la '^ *" + "'>' J Any positive value of the last two, and any negative^ value of the first of these four equation^ must of course be disregarded. . m:» = Mb = 0. By making Mx = in the first and last equations, and then finding the Value of a,, the limits of deviation of the points of contrary flexure (aS) may be obtained.* * Fig 24. DiAQKAM. — Let A C B be the beam (5. «). On A C draw the parab4|('ADC SS9), whose ordinate at centre D E = [w ^ w') -;. and on C B draw the o V)P parabola C F B (asa), whose ordinate at centre F G = — . At the centre . Xt/tl.\ be not greater than (' "I"" — ), the beam will require holding down to the abutments. 14 STRAIHS IK BEIUS. pier erect C H = e'^'~t *"''> nie»s»ring from C, make C K = (2 10 + W) i» ^ — jg . Join H to A, and E to A and B. Then the vertical distances between the parabolic arc A D L and A L will give the maximum poBitive (4) moments. The vertical distances between N H and the arc K L G, or those be- tween K M and F M, whichever be the greaUr^ will give the maximum negative (4) moments. The points of corUrarg jtexttre \1SS) will approach as near the centre pier as L, and recede from it as far as M. (See foot note, p. 13.) NoTS. — The various values, &c., given above, apply equally to both the spans. The diagram ab8ve is drawn to scale, on the supposition that the rolling load is i the intensity of the flxed.load. 33.' Beam of Vnironn Strenetli (165) for the Maximiuii Strains,* eontlnuons over two op more Piers, subject to a Fixed load Vnl- rorinly Sistrlltated, and also to a.Hoviiiic load (SIS). (See 113 A.) The maxinmm momeni over any pier will obtain, wlien only the two a^jaceht spans, and every alternate span from them, are simultaneansly loaded with the greatest load, the remaining spans sustaining only the -fixed load. The maximum moment at the centre of any span will ohtain when it and the alternate spans from it are iuUy loaded, the remaining spans sustaining only the fixed load. Let w = the fixed or dead load per unit of length. . «/= themoving or live ,, „ I' = either outer span. 2 = any other span.* Then, the meximnm negative (4) moment over any pier, B or C, Maximum negative (4) moment between any two piers (t.e., in any inner span, Q, (positive valaes of which must he disregarded). Kaximom potitive (4) moments between any two piers (i.e., in any inner span, 1), (w + vf ) , . ZwP M. 2~" <^ - *) "* - 'W (negative values of '#fiich must be disregarded). For^her outer span, the maximum negative moments, a(ro + i»7 . x.P /2w i^v (positive values of which must he disregarded). * If there be but thfee spans, a modification of the values hereafter given will be necessary, wiiich see. MOMKHTS OP KDPTURE. 15 The maximum positive moments, x{w + w') SwxP M* = 2 (l - X) —327-. X being measured from the abutment. If any of the foregoing expressions for M, be made equal to 0, the value of X obtained from them will givS the positions bi the points of isontrary flexure («S).* JSIo'CE.^Jf the beam be eantinuout for three epani only, Z, as a coefficient . P /2 w vf\ xP /2w v/\ in the expression -3-1~+y1, or in sTT ( "»-+-2"l. must have a value l + l' given to it = —5 — Fig. 26. DiAGKAH. — Let ABO (6, 6) be part of the beam. On BC draw the parabola B D (2S8), whose ordinate at centre (DE) = (" +.^ . 8 On A B draw the parabola A F B ( S3S )j whose ordinate at centre (w +■ v/) I'i ' (P G) = — — g-!— • At B and C erect B H and C Jt respectively^ P /2 w v/\ each.= - [_-+ — ). Join A to H and H to J. Measuring from B and ^ S w P 0, make fi'E and C L, each = -^^— . Join A to K and K to L. The vertical distaiices between AFO P and AF give the maximum positive moments for either outer span ; and those between M N D N' Mi , and M Mi give those for any inaw span. The vertical distances between O H and O P B ^ive the maximum negative moments for the outer spans^ and those between H N and BM N, or between N' J and N'Mi C, give the maximum negative* mdments for any inner span. The_ points and P, and M and N or M, and N', show the limits of deviation of the points of comirary flexure (83). The diagram above is drawn to scale, on the supposition that the intensity of the ToUing load is one-half that of the Sxed load. * If ^__j be not greater than f-^ — ), the beam will require holding down to the- abutments. ^ t N.B. — The note given above (for the value of 2 in th^mbrmula) must be nbserved here. 16 STUAIKS IN BEAMS. S3 A. Note. — The moment of rupture at any point, produced by several loads acting simultaneously on a beam, is equal to the sum of the moments produced by the several loads acting separately. SHBAEING FORCES (I). 34. Abbr^yiations : — Let F and P — the reactions on the supports due to the total load on the beam between those supports. S Hi = the shearing force at any point x. w = distributed load per unit of length (61). W = total load concentrated a,t any point (61). X = distance from left-band support to the section at which S H is required. { = length of span. Other abbreviations Will be explaified when they 35. In the Diasbaus, the ordinates (the vertical distataces from the horizontal or other lines) to the curves, &c., as shown thereon, correspond to the values of the formulse accompanying them. If the diagram be drawn to scale in the manner as directed, the shearing -forces may be obtained by dwed measwrement. See also (6). 36. Geneial Itnle ibr deterininlne the Sliearins Force at any part or a Beam and nnder any Disti'Ilintlon of load. Fig. 26. Let it be required to find the shearing force at any point (C) of a beam. Let W = the load between A and C. W"= „ BandC. Then, SHatC = S Hx = P - W, or = F - W" ;— the greater of the two values to be talcen. At the supports, W or W" = ; so that the shearing forces there are always equal to P or P'. The above values hold good for semi-beams. 37. Semi-beam Fixed at One End, ft-co at tbe Otber, and loaded In any Manner (g). Fig. 27. ^ ' The shearing force at any point P is equal to all the load between that point and the unsup- ported extremity. SHEARIKO FOHCES. 17 »8. Semi-beam Fixed at One End only, and uniformly loaded ItK entire Leneth (9). ' S Hx = w (« - x). 5 Hi = Ml!. Diagram.— Let AB be the beam (35). At A erect A C = w I. Join C to B. Then the vertical distances between AB and B will give the shearing forces. Fig. 29. 39. Beam Fixed nt One End only, loaded uniformly its entire lenetli, and also with a Concentrated W||Bl>t nt Its tree Extremity (11). S H, = W + w (l-x). S Hi = W + M Z. DiAORAH. - Let A B be the beam (35). Make A C = w i. Join C to B (as in 38). Make A D and B E = W. Join D to E. Then the vertical distances between B and D E will give the shearing forc^. 40. Beam Supported at Both Ends, and loaded at the Centre (1%). At any point in the beam, W ' 2" DlAQBAU. — In Pig. 30, the shearing forces (35) for this case are given by the W vertical distances between C D and A B, A C and B D being each = -—•■. (43.) SH = 41. Beam Supported at Both Ends, and loaded with a Con- centrated WelEht at any Point (13). Let a and b be the distances of W from the supports A and B respec- tively. 6 SHi = P=W2==SH,, oonsUint between A and W. a S H, = F = W 7 = S H, , constant between B and W. DlASKAH. — In Fig. 30, the shearing forces (35) for this case are given by the vertical distances between A B and B P Q iS, W being supposed at Q. AB^Wy. BS=W-2... (43.) 4%. Beam Supported at Both Ends, with movlnc: in either Blrectlou (17). Concentrated load* Or, S H. = W y I S Hi = S H, = W. The greater of these two -ralnea to be taken. DlAORAX. — In Fig. 30, the verticals A E and E F at either extremity of IS STBAIMS IS BXAMS. the beam are made equal to W j B joined to B, and A toP. The vertical dis- tances between B K F and A. B give the shearing forces. 43> IHaKmiii of the Stacarlne Forces In u whole Beam with s Concentrated Iioad. Fig. 30. A B is the beam. A E and B F are equal to W. When W is in the centre of the beam (K), the shearing force for the vhole W length of the beam equals — = the ^d vertical distance between C D AB. When W is at any other point (Q), the: shearing forces in the two segments, R F, Q S, are inversely as the lengths of the segments. The vertical distances between A B and H P Q S, give those shearing farces. When W rolls from end to end of the beam, the shearing forces are as the vertical distances between A B and £ K -P. The points F and Q will always be points in either K B or A F. * 44. Beam 'Hui»pfl»rte4l, at Uoth Ends, and l^Mided nnlTormly Itit entire Leneth (19). Pis.31. SHi = SH. = P = F' = "' •2 SH« '(.--)• Pig. 32. The sign of the result to he disregarded. At mid-span, S H = 0. DiAQBAii. — Let A B be the beam (35). At A and B erect A C and B D, ' 10 1 -3 n each equal to „-. Join G and D to the mid-span, £. Then the vertical diatnnees between A B and C E D will give the shearing forces, 4.1. Beam Sniiported at Both Kiids, subject to » DIstrlbntcd I.ond atlvancine trom clher Support * (ao). The greatest shearing force will be developed at the point of junction of the loaded and unloaded segments ; in which case let x also represent the length of the load. Then, ■ .»)" ) The greater of the two values to be taken. ml At mid-span, S H = -g- « As iu the case of a railway train crossing a bridge. SIIKARING FORCES. 19 DiAORAM.— iet A B be the beam (35). At A and B erect A C and B D . w I respectively, eacU equal to ^. Draw the semi-parabola, C E B («.■*«), whose V) I vertex is at B, and ordinate B D =< y. Draw the semi-parabola A E D corresponding exactly to C E.B. Then, as the load advances from either pier (say A), tlie sheijj-ing force developed at the junction of the loadft with the nnloaded apex will be represented by the vertical distances be- tween A B and A E D. When the load covers the whole span, the shearing forces will be ^s in (44). As the loacj is liable to advance from the pier B as well as from A, the maximum shearinj; forces for all positions of the load will bo given by the vertical distances from A B to C {! 0. 46. Cencntl Foniiulie for Itetcrmlnlng the Keartlon or the Supports, and the ShenrlUB Forces In tli« r^tse at Continuous ■Icams uuil Beamii whose Extrcinilics arc Fixed, »r subject ti> the Action or known Moments of Kuptiire (ST), Let A B be a beam, subject to the action of the moments M^ , Mj , and let tlie beam be uniformly loaded. The notations as before. ^ = T+ I • wl M» — Mi ^ = T+ ~ r ■ The shearing force at any point distant x from either pier, is found by subtracting (w x) from the re-action of that pier produoed by tb^ load between A and B. The values of F, F' are the pressures on the piers produced by the load between A and B only. Should there be a load on the beam continued beyuud these points, the pressures similarly found must of course be added to those above for the total pressures on the piers (IWt). 47. Beam Fixed at noth Extremities, ami Eoaded uuiroruil.v (*.=>, SO). Exactly the same as for (44). DiAQBAU. — The same as in (44), which see. 48. Beam of Uniform Strensth {IHH), supported nt One End, and Fixed horizontally at the other, uniformly (.omled IM eutire lenslh (99). «> I At the unfixed end, S H^ = -^' 2 wl At the fixed end, S Hj, = -^s — a 2(| STRAINS W BKAHS. S Hi = w f r — x), z being measured from the unfixed end. The sign of the result to be disregarded. DlAGBAH. — Let A B (fig. 34) be the beam (35). At the unfixed end (A) w I erfft A C = „-, aud at the fixed end (B) erect B D = 2 (A G). Join o and D to E, distant I I from A, or half-way between A and the point of contrary flexure, / (23). The vertical distances between A B and G G D will give the shearing &rceg. The length A/is identical leilh A/ (fig. 22), and tlie length f B with /B (fig. 22). 49. Beam of irnifonn and Eonal Section, supported at One Knd and Fixed Uorizontally at. the Other, "'^' unlformlr Iioaded Its entire I«nsth («8). 3 w 2 At the unfixed end, S H^ = — g— • 5 w I At the fixed end, S Hj = -v— 9 fSl \ 8 Hi = JO ( -g- — a: 1, a; being measured from the unfixed end ; and the sign of the result being disregarded. DiAGBAH. — Let A B be the beam (3S). At the unfixed end (A) 3 w Z 5 w I erect A C = —5—, and at the fixed end (B) erect B D' = —3 — . Join C o o and jy to E', distant 1 1 from A, or mid-way between A and the point of contrary flexure, /' (83). Then the vertical distances between A B and C K'D' will give the shearing forces. The length A/' w identical viith A/ (fig. 21), and /' B with f B (fig. 21). '■ 80. Contlnnons Beams with Fixed nntlormly Blgtrltonted loads (SO). If the spans be rightly proportioned (see foot-note, (•) p. 12), case (48) or (49) will apply to the outer spans, and case (4T) to the remaining spans. 51. Beam of Uniform Strenicth (165) for the maximum strains, continuous for Two eunal Spans, ^'g- •''^- suhjcet to a Fixed lAad nnlt'ormir Distributed, and also to a Moving load (3«). (See 1I3A.) The maximum shearing fijrce ai either abutment will obtain when its span only sustains the moving load. The maximum shearing force at the centre pier will obtain when both spans are fully loaded. SBEAniirO TOKOKS. 21, The total maximum presanre on the centre pier, when hoth apans are folly loaded, will be twice the above maximum shearing forceat the pier. Letw' = the moving load per unit of length, the other notations as before. The following ue the maximum values* : — At oither abntment, S H^ = — - (4 u; + 6 «/), or 2 I At the centre pier, S Ho = — (is + «/), or 3 At an; other point (a;), S H, = Tj (4 w + 5wf) — x(a + jo'), The greatest value to be • I taken ;t x to be mea- Or, =(g-a)(w + w'), 1 _ \ /._ . . j> / Bured from the abut- ment, and the sign of I I the result to be disre- Or, = j2(4w — w') — wat-t I garded. As the load may be supposed to advance from either abutment right across the beam, an addition muit be made to the above values for S Hi { bemg equal to -g- when x is about equal to } I, and gradually diminishing as X gets more or less than -. (See the dotted curve in the diagram.) 3 DiAORAK. — Let A C (6g. 35) be one span of the beam (3S). At the abutment A erect AD = ^(4jo+ 5 »')• Make A E = r- (tB + w'). 12 o At C erect C F = twice A E. Join E and F to M, distant 1 1 from A. Through D draw D N parallel to E M. Sketch in a curve similar to that (dotted) in the figure, making an additional depth to the ordinates, at the point of minimum shearing force, of W -. t Then the vertical distances be- tween A and D S T F may he considertd to Bive the maximum shearing forces for either span. S9, Beam of Tnirorm atrenarth (165) for the ^axlmiiin Strains, Continuous over two or more Piers, snbJe«|'to'a Fli^ea Koad uniformly Distributed, nad also to n MovIuk' Koad (33). (See USA.) * .^)pivoximate values corresponding with the moments of rupture 32. (See 31.) t ^e last of the three valiieS fs the shearini; force in one span, when the other only is flilly loaded, the addition'however made for the load being a moving one will entirely cover it. ' t An exact expression for the value of the shearing forces dev|Blopod by a load gnidually advancing across a continuous beam of (about) uniforiA ^tf ength, would, even if it could be obtained, be most domplicatfld. The piroos^S here auggested, however, though necessarily only approximate, may be regarded as practically safe. 22 STRAINS IN BRAHS. The maximum shearing force at any pier (B or C) will obtati simulta neously vibh the maximum moment of rupture over that pier (33). For any inner span, SH. = SHc-i(g + -J-)- /'to 2 w)'\ SHx = Z(- + -— I -a;(w + ie'), X being measuredl^pm the nearer pif}'; For either outer span' — Ajb either abutment, Or, SH, = («. + «/) 2 - 32 2' At the pier, S Hb = 1 («) + W). At any other point, SH, = (2 -a:)(«'+OT') - Swl? S2l' = («' + «;') (3 -4 The greater of these va- lues tg be taken, and X tn ']be measured from the abutment. At t}ie middle of the inn.er spans, and at J I' from the abutments in the outer 8p;uis, the values obtained by the above formulse must be increased Wl ' w' I' by — g- and —7- respectively ; and this addition, gradually diminished, must be ma4.e for some distance on either aide of those points (see foot note (j:), p. Zi) as shown in the diagram below. If the bear)t be conlimious for three spang only, the above formuise must be modified as directed on the next page. Fig. 36. DiAQBAU. — Let A B C be part of the beam (3S). /' w \T For any inner span (as l).—ht B and C erect B G and C H, each= ( "j" + ""3" /• *'*''® ^ ' and C E each = — (w + «/). Join D and E to the mid-span, -and draw 6 K and K H parallel to D F and F E respectively. For either outer span (as V) : — 2 /,' ' . 3 / At B erect a perpendicular = — — (la + w'), -wljich, if I' = —, will SUKARINS FOKCUS. 23 coincide with B S. At A erect A L = 4 S B. Join D and L to M, I' Z V) I' distant J V from A. Make A = - (to + w') 55-t; . Draw N paral- lei to L M, At T and K sketch in curves,* similar to those in the figure, giving additional depths to the ordinatea there of — r- and -3- respec- . I 8 8 tively. Then the vertical distances between O T D and A B may be considered to Kive the maximum shearing forces for either outer span, und those between G y H and B C for the remaining spans. If the beam be conlitiitous for three spans only, the value given in the formal^ for S H , and S He , and in the diagram for B Q and C H, must be ,, ^ {IS w + 16 w')l V(iii)v)'\ ■ ^ ^, ^^ , . altered to ^ ^^r + S"i V "7" "*■ 'o' ) > *'"1 further, the value given to S H» for the inner span must be altered to = S Hb — a! (w + to'), in which latter expression S Hb must have the value just assigned to .1 + 1' .t. I- =-3-- See foot note (t), p. 21. 24 DESIRN OF BBinCES AKU CIRUKKS. FLANGED GIRDERS, AECHES, AND SUSPENSION BRIDGES. 53. In the nesien or » tiridee, girder, or otber similar structnre, certain parts of which are supposed to resist certain strains, * the varions processes are followed out in a progressive order. When the data do not extend beyond the amount and nature of the load, and the width, &c., of the obstacle to be crossed, the processes will be as follows : — I. Determination of the Mnd of bridge or girder (54). II. Determination of the general cross section, and major proportions of the structure (65 — 68). These enable — III. An outline sketch to he drawn (59). IV. Appraeijitate estimation of the weight of the strudure (60, 61). V. Calculation of the strains on the various parts (62 — 168) (which strains must be figured on to the outline sketch), at as many points as will be found necessary for the accurate — VI. Determination of the amount of material to retist the strains on the various parts (163 — 173). VII. Diatrilmtion of Uie matei^l in the vairious parts into a form of cross section best suited to resist the kinds of strain brought upon them (174 — 190), having regard to — a. facility of construction. b. adaptation of one })art to another in contact with it. SECTION I. DKTERMINATtOS 07 THE NATURE, PbINCIFAL DiHXNSIORS, STC, 07 THE BBISOE OB GlBDEB. (Embodying processes, I, II, III, above.) 54. The first operation (I, 53), that of determining the kind of bridge or girder, must be left entirely to the discretion of the designer ; at least no rules can be laid down for bis guidance : the several conditions which would influence the decision, such as nature of site, obstacles to be crossed, facilities of construotion, &o., being infinitely diversified. 55. Tbe areneTnl cross section of a bridge must also be left to the experience and discretion of the designer, as no definite rules can be giveu * The system of reftnrdin!^ particular members or parts of a girder as resistiug the particular strains for which they are adapted, and these alone, is perhaps the most generally received. (See 6J3 and 191.) OALCHLATION OF STRAINS. upon the questions of tlie relative advantage and economy of the many systems which have been adopted and suggested. GENKRAL PROPORTIONS. 56. Ttie ceracti(!al cases, where the action of the load is vertical, the straia on the itange increases the more it wanders from th6 horizontal position. The extra strain it soffers, however, iUocUaes proportionally the strain on the web (80, 81). FOLL-WBBBED StfcAIOHT GIRDEBSi 27 65. Nature of the strains. At any vertical section of a girder, the strains in the two flanges are of different kinds (4) : — 1. When the action of the load tends* to malce the beam convex on its lower surface, then — the upper flange is in compression (168) ; and the lower flange is in tension (161). 2. When the action of the load tends* to make the beam convex on its upper surface, then — the upper flange is in tension, and the lower flange is in compression. Strains in the Web Generally. 66. Thestrains borne' by the web are the shearing forces (I, .34), due to the transmission of the Tertieal pressure of the load to the points of sup- port. Their amount (and sometimes manner of action) are greatly modified by the longitudinal form of the adjoining flanges (64, SO). GIRDERS WITd EARALLKI. STRAIOHI FLANGES. Flanget, 6T. To And the Amount of Strain on either Flanare at any vertical section (6S), — nivld« the Moment of Uuptnre, as found from the formulie or diagrams, pp^ 2 to 15, by the depth of the srirdeF, i. e. by the distance between th6 centres of gravity of the sections of the two flanges., 68. At any vertical section, the strains on the two flanges are equal in amount, but opposite in their nature (65). 69. The strains in the flanges will vary throughout as the moments of rupture (1, jt), and therefore as the of dinatcs in the diagrains (S). TO. Bt Diagram. — If the ordinate, in the diagram for the case (s), *at any point of the girder, be made to represent on a scale of parts the strain on the flanges at that point, the strain at any other point may be MttuUVed off from the diagram. 71. The strains are either direct tension (16T), or direct compres- sion (168). * The action of the load ma girder supported at both ends, and having its lower surface concave (see Plate I.) will lessen the concavity, and so ttnd to cause convexity. 28 OAlODLlTIOIt OF SIBA.III8. Web. 19. The Strain upon any Vertical Section of the Web 1* eanal to the " Shearlnir I'orce " (1, 34) developed at that section, for tbe Tfttes eee (84 — 62). T3. If the girder have more irebs than one,* the strains as fbnnd l>y (72) mnst be divided by the number of webs for the strain in each, 74. The nature of the strain in tiie web is a shearing (proper) (17l). OlkCXttS WITH OUBTED OB OBLIQVE FLAD&^S. Flanges. 76. To find the Amount of Strain on either Fiance at any point, — Divide the moment of rupture (formulse and diagrams, pp. 2 to l6) by the depth of the girder, the vertical distance between the centres of gravity of the sections of the two flanges, and multiply the quotient by the secant (186) of the angle inhich ihejlomge, or a tangent to U at the point, makes mith a horizontal, f 76. Tf the depth of the girder vary throughout as the moments of rupture (I, 57), i. e., as the ordinates shown in the diagrams for those moments (S), then (a), the strains in the flanges mil vary as the secants (see 77, 186) of the angles ef inclvaatiim to the horiion. So that (h), if the strain at a horizontal parti be hmum, the strain at any other part may be found by multiplying the fanner by the secant (see 77, 1S6) o/ the angle of inclination of the latter. 77. IToTB. I'or the operation of muldplying by the secant of an angle there may be Bubstituted a geometrical process. Let Yig, 37, Aug be part of a curved flange. Let the value of /Moment of Rupture at Bx. i*j * t , .u- ( DepthofGl^er.tB )^ represented to scale by the horiioiUal line DE, measaring it firom some point (B!) on Oie tatigent (f £) to ABC at B. If D7 be drawn vertical, >. e , perpendicnlar to AE, then F B wiU = "°'°""°'^'"^'""*° X seoant DEF, r a wiu— uepth of Girder at B * ««»"' ui^n. (DEF being the angle of Inclination of the flange atB.) 78. The strains in the flanges are either direct compression (168), or direct tension (167). Wei. 7A. In girders with obli^ne or , or the flange in tension be inclined down j '°™ ! , the nearest point of support, j '^" ??° f the vertical component of the strain in that flange j ' j the shearing force. The sign of the result need not here be regarded. If S = strain in flange, 6 = angle which the flange or a tangent to it makes with a horizontal line, as D E P (fig. 37) ; then, S X Sin 6 = the " vertical component referred to above." 81. By «ion»ti'iicti«u.— Drawavertical A B = sheaT- '^^ ^^' ing force. If the flange in tension («.'», IS) be inclined down from, or the flange in compress-ion be inclined down to the nearest point of support (we will suppose the latter case), — Draw CD representing both in direc- tion and amount the strain in that flange, so that a horizontal drawn through D shall cut A. Then CA will be the vertical component of the strain in D. Again, if the flange in compression be inclined dovn from, or the flange in tension be inclined down to the nearest point of support (we will suppose the latter case),— -Let it be re- presented byEF, Obtain its vertical component EB, which should be added to A B, in the same way that A C was subtracted from it. Then C E will be the total resulting strain on the web. Should the amount to be subtracted exceed the sum of the original and that added to it, the difference must still be taken. SI A. The ."train taken must be that nbtainiug when the shearing force being considered is developed. For instance, with a load gradually advancfnt: across a beam supported at both ends, the maximnm shearing? foroe will be developed at the centre, when the load covers only half the span (45), at wliich time the moment of rupture, from which the 3afif?e strain is fpund, vriii be that given by (20), and not by (19), Which latter would be used when isaking the calculation for the flanges. 88. If the girder have more webs than one, the strain as found by (80 or 81) must be divided by the number of webs for the strain in each, 83. NoTK. It will obviously follow, from the abov^ rules, that in girders with curved or oblique flanges, the maximum strain in the web does not necessarily occur when the maximum shearing strain is produced. * The rules, &c., given here are not advanced as mathematically accurate, as there is much connected with the effect of curvature in the flanges on the strains in the web that hits not been satisfactorily determined. so CALCOLATIOK OF STKAIHS. GiRDBKS WITH Webs or Opek EiiAoiNa. Fig. 3Sa. OEBERAIi KULES FOK OFEN-ITEBBIID GIRDEBS. JBoomg. 84. At an; vertical section of a girder the strains in the two booms are opposite in kind. (See 65, which also applies here.) 85. When the girder is loaded at the joints (87), the strain in an; hay of either boom is constant throughout its length, and acts only in the direction of its length. 86. A bay cannot be in direct compression and tendon simnltaneously (SI). 87. Wheneaer the load or part of the load upon a girder is situated betweenihe two extremities of any bay, that bay must be considered as a loaded beam, and the strains in it calculated and provided for accord- ingly. Web. 88.. The strain on any brace is constant throughout its length, and acts only in the direction of its length. 89. A brace cannot suffer compression and tension simnltaneously. 90. If two or more strains, not all of the same kind, be acting upon a brace at the same time, the total actual strain in the brace will be equal to the algebraical sum of those strains.* 91. The Nature of the Strains in the diagonal braces of girders symmetrically loaded will be — (a) in all bars inclined down to the nearest support — compression (168). (6) in all bars inclined down from the nearest support— («Ksio» (IC7). 98. If the girder be not symmetrically loaded, some of the bars will be Eu\)ject to strains of a nature-contrary to that stated in (91). 9.t. Counter stralnH.-^With a, moving load, some of the bars will be subject to strains both of compression and tension, depending on the position of the load and the proportion it bears to the weight of the girder. The strain not according with (91) is known as the "counter strain." Note. (OS) and (93) do not apply to semi-beams or cantilevers. • Equal to tbe difference between the sum of the tensile and the sum of the cutnpressive strains. SIBAISHI WAKBEN SIKDERS. 31 94. If the load be brought only upon one boom, * an; two bars forming an apex on the unloaded boom may be termed a "pair ;"f if these make equal angles with the boom they are equally strained. 95. If both booms be loaded, then — (a) in a warren (zig-zag or single triangle) girder there will be no " pairs." + (i) in a straight latticet girder, if the load be equally divided between the two booms, bars iiitersec]bing at the mid-depth of the girder, and making equal angles with the booms, may be termed a '-'pair," and are equally strained. CALCpLATIO,^ BY MEADS OF THS MOMENTS OP BUPTURB (pp. 2 to 16) AND TUB SHEARISG FORCES (pp. 16 to 23). 9C. lonilii regularly concentrated at the apices of the diagonall, may be considered as itniformiy distributed loads (w, in % and 34) when the moments of rapture (1) and sbciiring Fig^^S^ forces are being determined, so long as the concentrated load at any apex is equal tp half the euiq of f,h^ siipposed uniformly distributed load on the two adjoining bays. ' Thus the weif^ht at A (Fig. 39J must be =^ ^ that at C, before the load ban tie considered as upifqrmly distributed. If the uniformly rlistri- bnt'eii fooj} "were on tl)e "bottom Ijdom in the flg., the weiRhts concentrated at jD and I^jyoiild be equal. OIRSEKS ^IIH FASALLEL SliiAIGHT BOOMS. 97. Note. As the depth of the girder (a constant divisor in calculating the strains from moments) is uniiorm, the diagrams (4, 5) may be con- sidered to give strains instead of moments, if the vertical scale be multiplied by the nun^ber of unit^ of length in the depth of the girder. Thus, suppose the diagram bad been drawn to a vertical scale of six tons to the inch, and that the depth of the girder was two feet (the |^pt being the lineal unit used in the case), then the ordinates in the diagram may be considered to give actual strains instead of moments if they be rea4 off on a scale of three tons to the inch ; three tons to the inch being a sc^le twice as large as six tons to the inch. 98. ICoiE. The Uepth of the Girder is Ike disla^e between the centres of yravity of the cross sections of the booms, and must always be expressed in the same" units of measurement as the length of spap (S, 34). 99. Warren Clrder (single triangle), isosceles Bracine, Loaded oh Que Boom Only — for any case given under Moments of Rupture, pp. 2 to 15. BooMS.g— For the strain in any bay (84 — 87) of the unloaded booni,;^ Divide the moment of rupture (^x in the formula!, or the ordinate in the ^ Verticals from loaded bays to opposite apices may be considered to distri- bute the load between both the booms. ' ' "" t When two bars are said to form a pair, it is meant that the Mame amount o/ tJte vertical pressure of the load is transmitted through them both. X Any girder' whose web consists of more than one system of trian^ulation is considered a " latBce." § See also 91. 32 OALOuiiATioir or straisb. disgram, for the case T to 33) at the opposite apex, by the depth (98) of the girder. For any bay (84—87) of the loaded boom, take the arithmetical mean (half the sum) of the strains in the two opposite bays of the unloaded boom. Web. — For the strain on any pair (94) of diagonals forming an apex on the unloaded boom, — Multiply the shearing force (S Hx in the formula, or the ordinate in the diagram, for the case 34 to Sa), developed at the apex, by the secant of the angle which the brace makes with a vertical ; or increase the shearing force^ (as above) in the proportion of the inclined length of the br%ce to its vntical depth. For the counter strains (93) from jioving lo^ds, sf e (IflO). 100. The Connter 8t»lng (93) in girders not continnons will be given by the smaller value of the two shearing forces given in 42 or 4S, or by the ordinates (35) to the lines (AK^'K B. §^. 80,' or A E, £ B, fiig. 82) in the diagrams which accompany those ^verid cases ; ,tho values thns obtained being, of course, multiplied by the 8eet.'(n, ISft) of the angle between the brace and a vertical, and subject ta indeed all tiui other stipnlations made (under " Web ") for the strains normal. 101. In continuous girders with moving loads the connter strains are indefinite, but may be supposed to act equal in intensity to the values allowed for the regular strains, and for some distance on either side of the point of minimum shearing force * (51, SS). lOi. yranf^ Girder, Isoseeles or scalene BmcIiiK, wltb the I«ad bTonRbt danally upon Both pooms't*— for any case given under moments of rui^UT% pp. 2 to 1 6. Bpojw. J— Foir the strain in any bay (84 — 87) of either boom, — Divide the moment of ruptur^ ^Miin the fonr ite, or the ordinate in the diagram, for the ca8$ 3 tp 33) at the opposite apex, by the depth (98) of the girder. Wbb. — For the strain in any brace, multiply the shearing force (8 Hi in the formijls, or the ordinate in the diagram, for the case 34 to S9), developed at t^e mid'iength of the brace, by the secant (126) of the angle which the brace inaikes with a vertical ; or increase the shearing force (as above) in the proportion of the inclined length of the brace to its vertical depth. For the counter strains (93) from moving loads, see (100^, 101). 103, Gltrder with One Syst^jn of Vertical ^tinti gf^i Inclined Tlje», laajJ.ed either on the top or gig. 40. bottom, oi both — for any case given under moments of mptnre, pp. 2 to 15. BoouB.:^ — For the strain in any bay (C D for instance) of either boom (84—87), — Divide the moment of rupture (^x in the formnlee, or the ordinate in the diagram, for the cose 7 to * Actually there will be counter stains for a greater length of the several spans than there will be necessity practically to provide for them in, as their amount will always be exceedingly small near the piers compared with the other strains. f This is accomplished when there are verticals at the unloaded apices, X See also 97. SISAIGHT WARBES SIRSEHS. 33 33) at the opposite apex B, by the depth (98) of the girder. The strain in any bay of the upper boom in a whole girder (A B, fig. 40^ for instance,) will be equal to the strain m the bay (C D) pf the lower boom on the isiiiltspw ei^^ of t^e oppo- site apex ; for the moment of rapture will be taken at the seme p6m1.' (horizon^ tally considered) for both the bays. The bays G E and F G are unnecessary for an mdependent whole girder. Wkb. — Stntts. If the load he only on the upper boom, the compression on apy strut will equal the shearing force (S Hi iif the formute, or the ordinate in the diagram, for the case 34 to 62), developed at the middle of the adjoining bay on the far abutment side ; * if only on the lower, then that at the middle of the adjoining bay on the near dhvimeni side ; and if equally on the upper a/nd lower booms, then that dev0oped at the strut itself. Ties. For the tension on any inclined tie, — Multiply the shearing force (34) developed at its mid-length by the secant of the angle it wakes with a vertical, or increase that shearing force in the proportion of the length of the tie to the length of a strut. Counter Strains (93 — 101). In a girder vith the bracing disposed similar to that in the figure, a ni!iTin|,;leadj,- or a,.,3jp»J^^isering less than either half of the girder would "jfoifuop ten§op' p some of the verticals and compressiqri in some of the diagonals (to eatjmate which see 100). This can be obviSfed by introducing other ties (as H C ajid H F) wliioh will Buffer the tension otherwise bronght on the verticals. These ties would, however, be useless with a st^tipnary symmetrical load. 104. Warren Clrder witli scalene Bracing, Loaded on one Boom — for any case given under moment of rupture, pp. 2 to 15. Booms, t — iPor any'tftj {84 — 8 J) in the unloaded boom (CD,, fig. 41, for instance), — Divide the moiiient qf i;uptnre (Mz in the formulis, or the ordi- nate in the diagram, for the case T to 3S) at the"oppoaite apex (G), by the depth (98) of the girder. The strain on any hay of the loaded boom (as E G, fig. 41) will bear the same relation to the strains in the opposite bays (H C ^d C D) as its apex does to their apices horizontally considered (that is, as the point F does to the ppillts E and Ci). Therefore — F F Strain in E (J = Str. in H C -I- --- (str. in C D — str. in H C). (See 105). Jii ur Web. — For the strain in a pair (94) of liraees forming an apex on the unloaded boom, — Multiply the shearing force (SHa-inthe forraulaj, or the ordinate in the diagram, for the case 34 to S!!), developed at the middle of the loaded bay included^ between the bars, by the secants of the respective angles wfiirch they foi'm with, a vei-tical ; or increase the shearing force (as above) in the proportion of the inclined length of the respective braces to their vertical depth. For the counter strains (93) &om moving loads, see 100. 105. method of a|;plyine tbe Dlasramg for tbe Sloments of Rupture to the foregoing cases.t * Or ilie " no abutment" side in a semi-beam, t See also 97. S4 CALCVLATIpII Qf SIRAIS3, As before stated (96), leads w}iieh are in reality concentrated at the apices may be considered as unitonnly distributed if they be such as a uniformly distributed load would bring on the apices in question. Fig.«. Q Place the diagmiu of the moments of ruptoire (1) for the case of loadinpt, support, 4o, (T to 33) immediately above (or under), and dvawn to the same hori- zontal scale as the skeleton elevation (88, 50) of the girder. (See flg. 41, in which the load is supposed to be uniformly diaiributed (00, 19) nlong the upper boom.) Draw verticals right IhroORh the diagram at each of the Uiackd apices, whether on one or both booms{as IS Jl' and & G fig. 41.) If the lines to or from wluch the . A B C! D K F Q), bMt an;/ num- ber of intermediate systems losulcd miirorinl}- on on« bonni for an vtiiinl flistniiee on eitlivr side of flic mill span (see no); may be applied also to SM^s of continuous girders (30— 3.?). BsuWS.-'^Flsce the digram (5, 6, 97) for the monieptg of luptpH (1) for the case Qboye (or below) the outline sketch of the girder (59) »s in fig. it. Draw ebords mi t^ng^nts (as shown in fig. ) at points (0' E' Q') eowesponding with the diTinoQi made by the "primary triiiDgles." Tyro polygons will thqs be formed, one by (or including*) the Wbei)(?l) is applied the load being oentrally situated,. tjie chordg and tan. ifpnte will of coarse extend only along tbe parabola C B D (8g. 15). Fig. 43, STRAIQHT LATTICE SEMI-GIRUER. 35. ebards, another by (or including*) the tangents. For the strain in any- bay of the unloaded boom take the ordinate to the nrcumscribed polygon,* at the centre of the bay ; and for any bay (as E H) of the loaded boom take the ordinate to the in»criied polygon* at the centre of the bay. If the ordi- nates represent the moments of rapture (4, 5, 97), divide this value by the depth (98) of the girder. Web. — Divide the load per unit of length (34) by the number of sys- tems of triangles (^ the number of bays in Che base of one of the primary tri- angles) for a new unit of weight, w, ; thus- = Wi or - = »",. Then for the strain on any pair (95) of lattice bars, — Multiply the shearing force (S H^ in the formulse, or the ordinate in the diagram— .S4, 35,— calculated or constructed with the new unit u, , or tc'i ,) developed at their apex by the secant of the angle they make with a vertical, or increase that shearing force in the proportion of the inclined length of the bar to its vertical depth. 108. Any lattice Hemi-Kirder loaded on One only. If distributed, place the Pig. 43. Bogus. — if the loud be concentrated, see IIO. diagram for the moments of rupture (4, 5) immediately above (or below) the outline sketch of the girder (.19). Trace out the system of triangulation which terminates fairly, at the extremity of the girder (see the thick lines in the fig.) To the curve of moments (4, 5) draw chords and tangents at points ( E', D') corresponding to the divisions made '.l)y the triangles just traced out (D, E, fig. 43). If necessary for the con- struction, the diagram may be continued into the abutment. Two polygons will thus be formed, when, whatever be the number of systems of triangulation, — For the strain in apj bay (S^-r»j;i of the unloaded boom take tlie ordinate, at the centre of that bay, to the circumscribed polygon ; and for apy bay of the loaded boom, to the inscribed polygon. If the ordinates represent the moments of rupture (*, 5, 97), divide their value by the depth of the girder (98). Web. — If the load be concentrated, see (110). If distributed, divide the load per unit of length (w) by the numSer of systems of triangles (= the number of bays in the base of a primary triangle (4 in fig. 43) fbranew unit of load ( - = Wi ). Then for the strain on any pair (95) of braces forming an apex on the unloaded boom, — Multiply the shearing force (s Hx in the formulae, or the ordinate in the diagram for the case, calculated or constructed with the nf u' imit i;,— 34, 35), developed at their apex, by the secant of the angle which they (eaeh) make with a vertical, or increase the shearing force (as above) in the proportion of the inclined length of the brace to its vertical depth. ' See note, p. 34. D 2 36 CALOOLATIOS OF STRAINS. 109. Kattlce'Glrder, if whole, and of more than tvm aysiema of tri- angles, then not withoiil a etmiral apex on one or other of the booms (see 4g. 44), loailed nnlformly and equally on both the Boomn for an eaual Distance on either side of the Slid-span, may be applied to con- tinnoua girders (30 — 33); Boous. —Place the diagram (4, 4) of the moments of rupture (1) immeaiately above (or below) the outline sketch (6:t 39) Vig- 44. of the girder (as in the fig., where tlie load is 'supposed to extend over the whole length of the girder). Where the line to or from which the or- dinates are taken is curved, draw chords to it at points (E' P'G'C) correspond- ing to the several apices (on both booms) of tb% system of triangulation which has an apex at the mid span (E F G in fig.) For the strain in any bay (84—81) of either boom, take the ordinate to the polygon,*. thus formed, at the centre of that bay. if the ordinates represent the moments of rupture (4, 5, 9T), divide their value by the depth of the girder, WsB, — Divide the load per unit of length (w) by the number of systems of triangulation (= the number of bays in the base of one of the primary ('w \ - = till ). strain in any lattice bar, or pair t of bars. Multiply the shearing force (S Hi in the formulse, or the ordinate in the diagram for the case — 34, .15), developed at its or their mid-length, by the secant of the angle it or they (each) make with a vertical, or increase the shearing force (as above) in t]^ proportion of the inclined length of brace to the vertical depth. I09A. If there be only Two Systems of Triangvlation, Then for the strain in any bay (84 — 87) of either boom. Take the arithmetical mean between the moments of rupture at the two ends of the bay (p and ;), and divide it by the depth of the girder. „^ . . ■ , Mp + M, ' Stram in any bay = . ■ ■. 2u The ohords in the diagram irill, however, give the same result. 110. Concentrated loads on Ijtttice Girders. If a lattice girder bo subjected to theTaction of a concentrated load at any apex, as at D (fig. 42), the system of triangles uptn one apex of which the load is situated (ABODEFG, im.) should be oonsidered as constituting the sole web of the girder — ^that is, as far as the concentrated load is concerned, for there is also the distributed weight of the girder (61). The strains from the con- centrated load might be calculated as if the girder were a warren, and as if the intermediate braomg did not exist. Indeed, to introduce other systems of trian- gles for a stationaiy concentrated load would be an error in design, unless tihe boom upon which the load is placed be made sufficiently rigid to distribute part of the pressure upon adjacent intermediate apices, in wmch case the strains would be very indefinite. Then for the * If it be an application of case (SI) the lines A F and O B (fig. 16) must be con- sidered to form .part of the polygon, t There will be no pairs (95), cmless the number of systems of triangles be even ■ ODRVED OR OBHQUE OPEN-WBBBED GIBDEKS. 87 OIRSEBS WITH CDRVEO OB OBLIQUE BOOMS. 111. Any Carved of Abllaue wltole or Seml-elrder witli a Slncle- trlauKUlar IVeb, loaded on One Boom.* (This will include many roof principalB, bow-strinsf girders, bent cranes, die.) BoouB.— If alternate braces be Tertical, then for- the strain in an; bay of either boom; if not, then for the strain in any bay of the im- loaded ]Mom only, — Divide the moment of rupture (m< in the formule, or the ordinate in the diagram (4,5), for (be case 7 to 33) at the opposite apex by the length of the perpendicular let fall from that apex on to the bay. For the strains in the loaded bays, when all the braces are inclined, the best way is by the diagram as follows. Place the diagram of the moments of mpture, immediately above (or below) the outline sketch (53, 59) of the girde^^ If the lines in the diagram to or from which the ordinatea are directed to' be taken be carved, draw chords at points corresponding to the posi- tion of the apices in the loaded boom. Then for the strain in any bay of the loaded boom, divide the value of the ordinate to the just-formed polygon, taken at the opposite apex, by the length of a perpendicular let &U from that apex on to the bay (121). (See 154.) Web. — For any brace, first determine the shearing force (34), acting at the middle of the bay on the loaded boom, which forms part of the same triangle with the brace in qaestiou. Next, find the vertical components (80, 81) of the strains in those bays which are opposite sides of a quadrilateral figure, whose diagonal is the brace in questisnr. ' If either of these bays be part of a boom in tension, and sloped as a strat (91), or part of a boom in compreosion and sloped as a tie (91), add the vertical component' of the strain in it (alr^y found) to the shearing force first obtained. Again, if either of the two same bays be part of a boom in compression and sloped as a strut, or part of a boom in tension and sloped as a tie, tuibtraet its vertical component (already found) from the'i'esult of the last operation (the addition, if any). The total resulting quantity must then be multi- plied by the secant of the angle the brace makes with a vertical, or increased in the proportion of the inclined length of the brace to the ver- tical distance between its ends. If the sign of the reiuU be negaiwie (— ), it shows that the nature of the strain on the brace is apposite to that which its position would have indicated according to the general rules (91). If the brace be horizowliU, the shearing force must be disregarded, and the horizontal components of the bays substituted for their vertical components in the process detailed abotlt. The strain in a horizontal bay can have no vertical component. 112. Any Curved or Oblinne whole or Seml-Elrder with a slncle> triangmlar Web. loaded equally on Both Booms. Booms. — For the strain in any bay of either boom, — Divide the moment of rupture (Mz in the formulee, or the ordinate in the diagram (4, 5) for the oa£e 7 to 33), taken at the opposite apex, by the length of a perpendicular let fall from that apex on to the bay. (See 154.) Web. — As in the last case, excepting that the shearing force must be taken at the mid-length of a line joining the centres of the bays, which are two opposite sides of a quadrilateral whose diagonal is the brace. * SeefootnoteC"), p. 31. 09 CALOULATIOH OF STBIIHS. llSa For Continuons Girdera, and drdem flxed at the Ends, tlie foregoing methods may be used in connection with cases 24 to 33 and 4$ tii 53, provided that the conditions stated in the latter and belov (113A) are complied with. II3A. Wherever there is a negative moment of rupture at (or for safety in practice near) the abutments, the girder should be anchored down at its extremities. (See foot notes, pp. 13 and IS.) r 114. Flxlns the Points of Inflexion of Continuous Girders. — The points of inflexion (23) may be practically fixed at any part of a continuous girder subject to a movinjg load, by severing either boom at that part ; if the upper boom, the parts thus severed must be prevented ft'om coming in contact. The structure is thui resolved into a_feriea of independent girders, the strains upon which can then bi most T^ecSlMy calcvlated. Fig. 46. ExAHFliE. — In the accompanying fig.f4S), by aBvering(or really removmg) the bays of the tipper boom opposite and D, AC and D B become virtually senn- giiders, each having to sustain in addition to the distributed load upon its length, half the totta load on G D, suspended at the extremity (11, 39). C U is smiply an independent girder supported at both ends. 115. Thepointa of nation may ie conndered at fixed in those con- Fig. 46. on them may be obtained trom 155 or 118 tinuons girders, and girders fixed (or tied back) at the ends, whose depths vary as (or nearly as) the moments of mptnre. The strains in these also may be calculated as if the several divisions were independent girders. If theiteiision members running down from thS'fefrera (as in flg. 46) be made to act simplyw suspension chaiiiB, the strains CAlCUlATIOir BT THE COMPOSITION AND RESOLUTION OF FORCES. (applicable to all oases of OPEN-WBBBED INDBPBHDENT GIRDEB8.) 116. The following principles should be applied to the calculation of the strains on the various members of an open girder (discontinuous) by (1st) finding the reaction of the supports from any weight in the girder, and (2n ^ ; ...Horizontal length"*' 18 T. Straight 8emi-Blrder loaded In any manner. • Wbb. — Etery weight (125) on the girder it trantinittfd through ill own tyitem of trianglee to the abutment. The had on any brace is equal to tiie sum of all the weights upon its system, between it and the nnsupported end of the girder. For the tlrain on any brace multiply the load on it (aa above) by sec. 9. Books. — ^By (183), — ^The horizontal component of the strain in any brace = (strain in brace) x sin. 0. (1S«.) 128, — EzAXPLS. —Let fig. SO represent a parallel straight girder of otherwise irregular construction loaded at three points (a/c). The weight of the load at a isoonductedalong the braces (tied e,thatat(! along ode. If I ii! represent Uie STBAISHI WABREN SIBOERS. 43 weight Wi) ti a mil be the tension (01) on a b, and I a = increment (123) at a. If m n also equal Wi.ni will eqaal the compreasion in c &, and m b the increment at 6. Fif;. 60. The load on i: d is equal to Wi andWs together. IS op — this load, a e will equal the tension on d c, and p c the increment at c, and so on ; »t being equal to Wi, «/will equal the stntiu in fOt and £/the increment at/. Then (123) the tension (66) in bg = mb + la; that in <(0 = ii(r + (/+ strain in 6 ;; that in ii2=^gin any pair of diagonals forming an apex on the un- loaded boom. 2 = w « sec. 0, . . . (126,) where w = load per unit of length, and as = number of units between the apex and the unsupported end of the girder. BoouB. — ^The strain (S) in any bay of the loaded boom, S = I m(in— l) + i |wy tan. 9 . . . (126.) In sny bay of the unloaded boom, S = tn' w y tan. d, . . . (I26,) where m =■ number of tbe bay counted from the outer end of the girder, and y = length of bay. The fallowing cases (viz., 131 to 138 inclusive) do not apply to con- tinuous girders or whole girders fixed at the ends. 131. Stralsbt Warren Girder, Isosceles Braelns, loaded at Any Point. Wbb. — Strain (2) in any diagonal be- ween A and W, ^K- "• I 2 = W 5" sect. ». In any diagonal between B and W, 2 = Wjsec. «... (ia«.) BooxB. — strain in either end bay of tbe longer boom = (reaction of ^^ 0AL01TLAII01I or SIKAIIIB. Bnpport) X ten. (117, 186) ; and if this ralne be considered a unit, tlien the strains in the' bays of the longer boom will be proportional to the series !> 3> p, 7, &c. ; and in the shorter boom to the series 2, 4, 6, 8, &c., counting from tke supports. Thns the Btrain in tbe second bay of the longer boom, from pier B will equal (3 W -^ tan. < V Strain in the bay opposite the loaded apex= . . ; — d being the depth (98) of the gilder. ' " 138, If W Ibe In the centre of the clnler, the sttwn (2) on any diagonal, 2 = ^ see. 9 . . . (186.) And the strain (S) on the centre bay, 133. Btralcht Warren Girder, iMMweles Braelnc, with h cob- centmted HoTlnc load. WsB. — Every diagonal except the two end ones will be subject to counter strains (93). The maximum strain normal on any pair of braces forming an apex on the unloaded boom, will occur when IJie load is at the inner (mid- span) end of the inclosed bay. The maximum cqunter strain on any pair will obtain when the weight is at the outer (near abutment) end of the enclosed bay. The values can be obtained irom (131). Boons. — Tbe maximum strain on any bay (84-T) of the unloaded boom is when the weight (W) is at the opposite apex ; and on the loaded boom when the weight is at the next inner apex. The values can be obtained from (131). 134. Any Straight Warren or lattice Cirder (isosceles of scalene ^bracing), with any load gynunetrically disposed aboat the Centre, either on one or both booms. Suppose, the load collected at'the apices, according to (125). Web. — The load on any bar mil he equal to the nun of ^jj ^jjg ^^ ^jj]^ jqmi thg j,^ on j^ multi- plied by the secant of the angle it makes with a vertical (126). Let the aocompanyine fig. represent half a girder, with loads on both booms symmetiioally disposed atmut the centre ; AF C and its fellow, each \ ; and A D B, ' of the total load. The strains on the Tarious parts can then be obtained from (140). The compression along AB is uniform throughout 'the whole length. 48 OALOUIiATtON OF SIRAIKS. Aboh BnicaES. ABOHEB WITH SFAHDKIL BRAOIHO. 145. HVlth a Vnlform Horizontal toad. Let I = length of span, V = rise, or versine, y — horizontal distance of any point from the crown, to = load per unit of lengtli. Abched Bib. — Compression at the crown (C), C = ■ 8v Compression at any other point (C) "-VW^ The exprea^n for C is stricUy accurate only when the aroh* is a parabola,— the curve of equilibrium for the load is. question. It may, however, be safely used iu most cases of practice for arcs of dnsles. SpandbH. — If the arch* he parahoUc, the only strain on the spandril will be the vertical pressure of the load. (See foot note t, pi 49.) . The spandril may then condst of a number of pillars or struts (as in fig. 57), each sustaining at vertical pressure = nearly j N being the N — 1 number of spaces into which the pillars divide the span. If it consist of a continuofl^ web, the compression of it per unit of length will be equal to w. If the areh* be not a parabola, the strains in the spandril bracing may be obtained from 80, 81, (for continuous or "plate bracing"), or 111 (for diagonal open work), the lib being considered as the " compression boom." tot HoBlzoHTAL Mbhbbb. — WUh d parabolic (trek this member (B E fig. 67) simply acts as an immediate support for the load (87). With an ardi, not a parabola, there will be a strain on it acting in the direction of its length, the nature and amount of which may be determined from the Increments at the apices of the diagonals in the spandril, by the appli- cation of the law in 1S3 as illustrated in 134. 146. With a MovIdk load. For obbtlninf; the strains oil the various parts of a braced arch. Subject to a distributed moving load, the following method may be employed.t • More properly the neutral axis, or the line traced through the centres of gravity of the cross sections of the nb. t Given by B. StOney, B.A. (in " Theory of Strains ") j and others. BRACED AND IINEAB AKCHES. 49 * Conceive tbe load to be collected into weights at the several apices on the horizontal member, each apex sustain- ing half the load on the two adjacent ' ■ ^'K- *^' bays. The strains produced by each weight separately must then be found and tabulated, as in 135. Suppose the strains from weight W (fig. 58) are to be considered. At W draw the vertical J7 D. From the abutment A draw A C D through the crown of the arch (C) till it. intersects the perpendicular at D. Join D to B. If the weight W be resolved (118) in the directions A D and B S, the amounts and directions of the reactions of the two abutments will be found. This may be readily done by producing A D to E, and erecting a perpen- dicular at B. Then, if E B equal the weight W, D E will be the reaction of A, and D B the reaction of B. If the fbi'mer be traced up &om A towards the weight, and the latter from B, by the resolution of forces (118, 119) the strains on the various parts may be thus found ; and the same being done for the other weights in succession, the maximum strains produced by any position of the load may be derived from a table similar to that on page 4S. If the weight of the structure be small compared with that of the rolling load, it will be found that some of the end bays of the horizontal member, and of the middle bays in the ai-ch, will occasionally suffer tension. See also 137. UBBRAOED AKCHKS, Or arches whose stability depends upon the stiffness of the rib itself. 147. Tlie Neutral Snrfaee, or neutral curve of an arched rib, is a line traced through the centres of gravity (SSO) of the cross sections of the rib. 148. The Une of Pressures is a Une the ordinatet to which vary at iAe moments of rupture (1) for tlix load* The line of pressures is given at once in those diagrams (pp. 4 to 6), in which the ordinates for the moments of rupture are directed to be taken from and on the same side of one horizontal line. Where this is not the case (as in 20 or 21, for instance), the ordinates must be transferred to some new horizontal datum. The ordliiates may be taken to afiy scale for ready comparison with the neutral curve of the rib. 149. Wnenever in any arch the line of pressures eoineidei tott/t the neutral surface, the arch is in equilibrium, and the strain upon it ii everywhere compressive.f • For BIiuanr]ranclinanyottaerAreheBcasel4wUlbefoundexceedingIyoon- Tcnient. Tbe structure and its load should be conaideced as made up of several small portions, each collected at its centre of gravity'. The line of pressures can then be obtained by summing the ordinates for uiew outline, as there directed. t Strictly speaking, the pressure on an arch ofother than a thoroughly incom- pressible matsrial alters the form of the arch; and this alteration of form, or tmiUng action, induces strains similar to those found in beams (slee 1 and 93). The tension on equilibriatedribs (ISO, ISl) is in practice so small (even wl^ere it it developed,— for this does not occur till the tension induced by the bmdiag exceeds the direct compression on the rib) that it need not be regarded. •K 50 OALODLATIOS OP STKAISS, 150. The Stability of an Arch incawablc of Rcaistins Tension i|^ secure so long as the line of pressures does not at any point deviate from the neutral curre by an amount, the proportion of which to the depth of the rib is given by the values of q (153) for various forms of cross section, 151. Wherever the line of pressures deviates beyond this limit, there is a temion on the nb on the other, side of the neutral surface, incrSising with the deviation. 153. Pressure •along one edge of an arched rib invai'iablyaproduces tension on the other edge, whatever be the form of cross section. 153. Limits of the deviation of the line of pressures from the neutral surface consistent with there being no tension on ihe rib ^— = Form of Cross Sectiim, Value of q. * Rectangle . . ' -J- Ellipse and Circle ..'.... \ Hollow Rectangle (area = hh — b' h') \ also I formed section, i' being the ( , / b' h'^\ _ / b' h' \ sum of the breadths of the. lateral i ^ Kj; ~ b h? / "^ \ bh )' hollows. ) (V \ ^ + la /■ Ho'low Ellipse j(l.*_^)^(l_^'). Hollow Circle . i(>*F> b and V — external and internal breadths ; and h and A' — the external and internal heights or depths. I, section alike above and below . J- ( 1 + -77 — r~r )• A = area of each table or flange ;"' A' = area of the connecting web. 1 54. The above values of j should be applied as' a test for tension in cases of braced arches, and also where the spau'dril Fig, fig. consists of columns ; for the line of pressures* is th«n in reality !l polygon, with the angles at the apices on the rib. -(See fig. 59.) 154 A. When an Arched BrldBc consists of several unequal Spans, the neutral surfaces of aU*the ribs should be parts of the same figure-^ (generally arcs of the same circfe, or the sam« parabola). * These values or$ are from Professor Bankine'g " Civil Engineerinf;.' 8DSPBNSI0K BRIDSES. 51 SnsPEHSiON Bridois. ISS. ORDIHART SUSPENBION BSIDSB OP ONE BPAK, WITH A DNIPOBM HORIZONTAL LOAD. Maih CHilss.— The curve .which the main chains will as3«m« will be very nisarly a parabola, liet IS = load on each chain per nnit of length ; I = length of span ; V = Tersine, or degfession of the chain ; f = angle which a tangent to the cliain at any point makes with a horizontal ; y — horizontal distance of em; point from the mid-span. Tension at Centre, T = |^ ; ^.^ g„ At any ether point, «"=V(sy^""(^~^>=' SusPKUMNO Rods. — Let N =■ the number of spaces into which they divide the span, then, Tension an each^K Jf Towers and Cocnter-ohains. — The tension on the counter-chains, and the pressures on the towers may be readily found as follows : — Pro- duce the tangent to the main-chain at the tower (C) till its length (CD) on a scale of parts, equals the tension at that point (found from (F aboye). Through D draw D E parallel to the direction of the counter-chain (C F). Produce the centre line of the tower tillit inter- sects DE. Then (IIS) CE will give the pressure on the tower, and D E the tension on the counter-chain. 156. OBSINART SUSPENSION BBIDQE OF UORE THAN ONE SPAN, WITli A DNirOBM HORIZONTAL LOAD. When a suspenaioa bridge consists of several spans, the chains of all of them must form portions of one and the same parabola. The strains on the whole spans — as B C — will be the same as in (ISS). E 2 52 OALCULATION OV STRAINS. The Btrains on the outer spans (A B) are identical with those in B D, Tig. 81. as the parts A B ^d B D are equal. But where the abutment spans are not equal to half the inner ones, the first conditions made in this paragraph must be observed, and then T = ( — — j sec. (155), I bmg still the inner ipan. The compression on the towers will equal the disgonnl .of a parallelogram, whose sides oorrespond in direction and length to the tensions ip the obams on. either side (as at iT is &g. 61). 15T. SnSPENSIOH BKIDOB WITH SLOFIira BODS, WITH A VVIFOBK BOBI- ZOHTAL LOAD. MAm-0EAiR8.-^The curve which each half of the chains will assume will be a parabola with its axis parallel "^•*^' to the direction of the sloping rods. Tension at mid-span, T = -g^ } wnere (ml') is the total vertical load on the rods ; the other notations as in 155. Tension at any other pointy T sjs wy cosec. the angle which a tangent to the chain makes with the hccrizontaL^ak SLOPina KoDS.— Tension on a^rod, « = Wsec. /8; where W is the vertical load on the rod (half the load on the two adjacent bays) ; and J3 the an^ which the rods make with a vertical. HoBizoKTAii Meubeb. — ^The eompressioii (e) on the horizontal platform at any point distant (]f) from the mid-span, is e = wy tan. fi. For the towen, see 155 or 156. *' 158. SusrinaiOH BBinaEs with Moviko Loads are subject to mneh disfigurement, to prevent or modify which several means have been devised. 159. (1.) An auxiliary girder from pier to pier, anchored down to the abutments. If this girder be continuous for each span, its booms (fig. 38 a) for abont the middle half of its length must be able to resist a strain «'{' 1^2 = r-T-j, and the web a shearing force of about -r-; v/ bang the intensity of the moving load per unit of length ; I, the length of span, and d the depth (V8) of the girder. * ABUTMENTS AND PIERS. a3 IGO. (2.) When the brUge consists of several spans, — Fixing the chaim to the top of the, toweii, and considering the latter as semi-girder!:. each one loaded at the extremity with - — , notation as before (IS9). Wbll^ 8 V the tower is suffering the strains consequent on the application o'' .this force to its extremity, there is also a direct compression on it ~(w+ 7- )'• This latter, it must be remembered, modifies the ten - sional (4) strains produced by the former. 161. (3.) Inaerting diagonal bracing between the roadway and th^^ chains. The strains on the various parts may in this case be obtained aij iu 146 ; they will be altered in Icind only, not in intensity. 161 A. (4.) A pair of chains of identical curvature, placed one above the other, and having diagonal bracing between, the greatest shearing forcS on which would be — r- • 161 K. (5.) Counter chaim attached to the main chains at about | spa'i from the abutments or piers, and running down to the latter. They should be made to resist a strain of ( -^ x sec. 9); 9 being the angli? between the counter chains and a vertical. 16IC. (6.) Inclined straight chains, for oanying the platform ari-i the moving load. They extend from the towers, and meet or intersect each other. They are sustained in the required straight lines by rodS; which are connected to onrved chains, the latter having to carry th.-- weight of the straight chains only.* The tension on the latter may bs found from 155. The tension on the straight chains may be most readilv found by a parallelogram of forces (IIS). 162. Abntments and Picn. Girders, properly so called, viz., thosV. structures which simply rest upon the supports, bring upon those supporlt a vertical pressure equal to the shearing force developed there. (See last paragraph in 46. ) AliiUments of Arches. The thrust at the abutments of anarch is exactly equal to the compression in the arch rib at the springing, the valne of which may be determined from 145, 146. . ' For the towers and piers of suspension bridges, see 1S5, 156. Whenever the piers of a bridge consist of columns, their strength as such — their liability to flexure, &c. — must not be overlooked. (See 168.) SECTION m. DlSTRIBUIIOH Or'MATEBIAL 10 BliSIST THE CALCULATED SiRAIHS. (Embodying Frocesseses VI, VII, p. 24.) .. 163. Tlie Strenictli of a Stmctnre, or of any part of it, is its ability to resist the external forces tending to cause its rupture. * Mi. B. M, Ordisli's system. 54 DISTRIBUTION 07 TEE UATERUL. 164, Axiom. No whole is stronger than its wedkesl part. 105. Uniform Streiieth. AStructure is said to be nf nniform strength ■when no one part would yield bwi*"e another, supposing the structure to be subjected to the load, or a multiple of that load for which it was designed. In structures not of .uniform Btt^gth, all the material in excess of that neces- saryfor uniformity of strenghis redundant.* ' To secure wniformity of strength, a constant co efScient of safety ixii)- must be used for the same material strained in the same way. For beams of uniform strength, see SIS — 918. 16G. Vntts. It is necessary to adopt : I. Unit of Strain or Stress ; generally 1 lb. avoirdnpois.' 9. Unit of Sectional Area ; generally 1 superficial inch. 3. Compouvded Unit of Strain and Area ; 1 lb. per sq. inch. Let A = area of a section in units (sq. inches). S = calculated strain in units (pounds avoirdupois). n.,= ultimate strength, or breaking weight of the material — in -lbs. per sq. inch of section, (fm: numerical values, see 9Zl.) Co = coefSoient of safety (178). W S = working strain (119). FniNciPAL Strains to be met with in Bbidses, Girders, &c. IG7. Tension, causes or tends to cause the fracture of the material iipou which it acts by tearing asunder its particles. The resistance to Tension is directly as the area of the cross section of the material, takeik perpendicular to the direction of the strain (164). S X Co S Area necessary to safely resist a strain, A = — jt — = tF5' 168. Compression, causes or tends to cause the failure of the mate- rial, by crashing, buckling, or both combintfd. Crushvag. Materials in compression ( " struts ") can be considered liable to crushing alone, only when their least ^ameter (taken perpendicular to the direction of , the strain) is not more than about \ of their unsupported length,-)' The resistance is then directly as the sectional area. X Area necessary to safely resist a strain, A = = ■ f-r u ws Backling. When struts have an unsupported length equal to about 25 § times their least diameter, they may be considered to suffer almost en- * It cannot be said that the excess is entirely useless ; but more on this point would be out of place here. t Hodgkinson. t The resistance to crushing of a body whose diameter normal to the pres- sure far exceeds^ its dimensions* in a line with the pressure, is very great, but , equally indefinite. 9 For wi'ought-iron struts with riveted joints, from 40 to 50 times. BREAKING WEIOHT OP COHIMKS. 55 tirely ty being buckled, ». e., crumpled up.* Struts shorter than this fail, partly by crushing, and partly by .buckling. 169. bueakingI WEiaHT OF ooniHNS. — Prof. HodgkimotCs fotirmlce for cast-iron columns. When of more than from 25 to 80 diameters in length, Break, wt. = -^ for solid pillars ; and Break, wt. = 41-3 (i^ '» - d'^ ■ ) for hollow cylindrical pillars, flat, or firmly fixed at both ends, d being the external diameter, and d' the i/iternal diameter in inches, and I the length in feet. Columns rounded or moveable at both ends have but J the strength of those flat or fixed ; and the strength of those with one end fixed or flat, and the other rounded or moveable, is about an arithmetical mean between these two cases. When of less than from 25 to 30 diameters in length, let h be the value obtained by the above formulce, and c the crushing load of a short block (231) of the same sectional area as the column|iyuu the corrected breaking '*^^' " l + ic ■ OeneroU formula for the ireaMng meight of Cast onij Wrought Iron ■ Colamni. Let C= compressive resistance of a short block of the same sectional area. »• = T-= length of column divided by the greatest diameter. n For cast-iron, B. W. For wrought iron, B. W. = ^. 68 + -1 r C •85 + -Oir Breaking Weight in tons per sgwire inch of section. I ,. 13,600 I cast iron, „ ' — ;• T> J ) 330 + r' Bound ] •, ^ . 34,000 (wrought iron, ^^jjj,^ ciuare timber. 2000 + r" 850 350 + »■» Breaking Weight of timber pillars, taking the ttrength of a cube as unity. Value of r . Breaking weight . * Professor Hadgkinson. 1 1 12 i. 24 i 88 48 * 60 58 DISTBIBDIION OF HAIERIAL. 110. LOKO {IIBUI8 should b« made with a cross section, which will Figr. 63. insure a certain amount of rigidity or stiffnesSi 4Bd thus resist the tendency to bucl(ie : or they should be braced (either externally or internally), and thus divided into a number of shorter lengths, each of which (and, therefore, the braced strut as a whole) may be considered as suffering crushing alone. ITI. Staearlnc causes, or tends to cause, contiguous sections of the material to slide over each pig. ei. other (as at A B, sup? posing that D S is part of a and B C part of the load), is directly as the area that Area necessary ts safely A = loaded beam, A one support, The resistance to shearing wotdd be sheared. resist a shearing strain. S X Co U WS Shearing strains will b^aund to act on the vertical web * of continncjis webbed girders, and in{||p|ra generally, which see. For Bending, bending intenaitiet, and the resistance of materialt to them, see I, and 191 et eeq. » ITa.CoemclentB of Safety are numbers representing the proportions of the ultimate strength pf materials to the strains that can safely be brought upon them. Coefficients of safety may be variously estimated. The fol- lowing May,, however, be taken as a fair average of the factors at present in use where the materials employed and the workmanship are ordinarily good, Metals. Timber. Masonry. For a dead load . . 3 4 to 5 4 For a live load . . 5 8 to 10 8 Under dead load may be included all permanent or stationary loads, and loads very gradually applied. Under Hue load, all rapidly moving, and suddenly applied loads. ' Ultimate reaistance - Working strain. Coefficient of ufety 1T3. The Sfodnlns of Blasttelty (E) (in pounds per square inch as given in SSI) is> the weight (in pounds) required to elongate or shorten, by an amount equal to its original length, a bar of material (of one square inch of cross-section), and is on the supposition that the elasticity of the material would remain constant throughout the operation. strain (In Vn.) per gq. inch on a bar Mod.of elfti. (in lbs,} per iq. .Inch Increage or diminution of lenyth Original iengtli of tUe bar ' Joiiris. IT4. Jciintt thould always ie as strong as the porta they serve to join. • In the paragraphs 09 and 74. the flanges have not been supposed to ta,ke part in resistinfr the shearing farces (so called from the shearing strains above), any more than the web in taking its share of the horizont^ strains. (66.) IKON JOINTS AMD FASTEKINOS. 67 ITS. The variom partt of a joint, and the leveral pmrti in a fattening, thould be of uniform ttrengtk. IRON JOINTS AND VASIENINaS. 176. Rivets may fail in leveral different vays, depending on their office in a joint. The head may be shorn off (as at G D, j,. ^ EF) ; or the rivet nay be ruptured at any section (as A B) when the rivet is in tension. Let S = the tensional strain ; then d (diam. of tivet) should not be less than V 8 X Ux • ^0 7864 XT being the ultimate resistance of the material to tension (SSI) ; and C D or B F should not be less than f= — --.,„, (118), U being here the nltimate resistance U X 8'141o a ' to shearing (SSI). When the rivet has to resist a shearing strain (S) at AB, d must not be less than A/fr- — .-„-., U being the ultimate shearing resistance. Oenerdl Sulc—The height of the head (A) should tiever 6(! fc»» (htm half the diameter of the rivet. ITT. BoKs. — The diameter of a bolt liable to shearing at the fpiiidDe (as at A B, fig. 65) must be determi^^ from (ITS). If the Wt be in tension it will fail either^! st, by shearing 9ff the thr^ ; 2nd, or by shearing off the heftd i 9rd, qr ij tegsiffij^ Tt>p(an of the spindle. For a perfect thr«^d the height cf the qut and of tft« head should b? equal : but to alloir ^r inaccuracies of workinanship, the h^ht of the nut should b? about twice that of the head. The height sf &e put should not be less than the diameter of the spindle ; in praatiiN it is frequently made mnoh more than this, 178. The diameter of a nut or head ofaAoU, prqfihe head of a rtve^ ihffald he not maeh lem than tviee thSitifil^ ^indle. ITO. Plm Joints In Tension Bars. — (Such as in some suspension chuns, triangular girders, and trusses). Pin. — Let n = the least number of sections at which the pin must be divided before the joint can fail (4 in. fig. 66), <; = sectifmal area of pin, and S the tension on ike joint, then «=TmSSf^- TlietoefBcient(lT8)Bhoul!jbe larf^e, as any inaccuraoies in the workmanship will tend to concentrate the strain in certain parts. IMc. — The section of any link-head taken through the centre of the pin-hole (A B) should equal about half as muoh again as that b^ken through the body of the liuls (as ]>), in consequence of the inequality in the distribution of the strain. Fig, 86. JH 58 DISTBIBUTIOH OF UATEBIAL. Let I = the length of overlap, E F or G H (fig. 66) ; t= thiokneas of all the overlaps, in one series, taken together (2 or 3, fig. 66), then I = S X Co where S is the strain oi the joint, and U the ultimate resistance to shearing. General Sule. — Diameter of pin may be f of the width of the links. Riveted JoiiUa in Tension. 180. The Effective or Available Section of a plate with rivet hAposed to resist certain definite strains (68), and other parts, otber sirains (for instance, flanged girders '«^ith thin, contibnous webs, all open-webbed girden, trusses, ftCi) see 63 to Xi9i A inodd of. pro- cedure is there adopted which w6uld not be thoroughly appUcaUe to those beains in which evel-y fibr^ oi' ;^articl6 is cohlsidered io take part in r^isting the bending actiod of the moment of rupture (1), and where the whole secijon is liable to the action of the shearing force (1). 7» Designing a Beam — 198. Determine the nature of its cross section. If the exact proportions of the section are to he adhered to, and the area alone required, — express all the dimensions of the section in terms of one of them, that there may he but one unknown quantity. Thus— suppose a beam to support a given load is to be rectan- gular, with the deptti twice tue breadth .then let t = breadth, and 2 5 = depth. If all the dimetssions of the section, except one, he given, that Otte will of course be the unknown quantity. Then, — 193. Suhstitnte for M, in the equations given hereafter, its value as found from the span, manner of loading and supporting, £c.j pp. 2 to 15. The dimension, oi dimeiisions, reqtiired may then he obtained. Lastly, — 8TABIUIT or LOADBO BIA.U. 61 194. If at any Terdcal geetion, tbere be not sufficient material to resist the shearing force (1..34, et leq. and ITI), the necessary addition must be made to the section. This will seldom be required at other places tban near the snpporle in dts- continuoua beams, and near the points of contraiy flexure (193) m continuoas beams. 106. The vielght of the beam ittelf must alirays* be added to the estraneouB load, upon it ; and may be approximately estimated 1? a pro- cess similar to that in 60. *' 196. The Siabiuit of a Loaded Beau depends on the equation M = R. (1.) lOT. Abbreriations — Let M = moment of rupture (1, a), the values of which may be de- termined from the several oases 7 to 33. C I S == — = moment of resistance (1) of the section. I = moment 'of inertia of the section. t — distance of the neutral axis (198) from the farthest edge of the section. a — total area of the section. C = modulus of rupture (803). 198. TUe Keutral Axis (N — A in the sections, figs. 82—84) is a section of the neutral surface, — a layer in the beam (and the only one) which is neither extended nor shortened by the action of the load (4). 199. Provided that the limits of elasticity of the material of the beam be not exceeded, the neviral axa vnll ptut through the eetUre of gravity of the lectim (n^o). goo. There most be but one lineal unit need in obtaining the values of M, B, I, < ; and the superficial unit used for a must correspond to that lineal unit. 201. The section at which R and I (107) are taken may be made parallel to the reaction of the supports of the beam, i Fig. 76. S02. Wherever either the upper or lower surface of the beam is not perpendicular to the action of the load, then C must he modified to (C cos.' B), 6 being the inclination of the most inclined surface to that perpendicular. * Except in small girders, or beams of minor importance. t The section at which the moments of resistance should really be taken is a curved surface cutting the upper and lower edges of the beam, and the neutral surface all at rif^ht angles, whatever be the form of the beam ; and the moment of rupture to be equated with it should be taken at the intersection of this curved surface with the neutral surface of the beam. 62 BEAU8 or TARIODS SECTtOSS. 803, Hodnlm of RnptitFe.— The ilumttical-n,lne of C is thereeiatance of the material to direct compreBsion or teneion. But it is found from ekperimentB on cross breaking that this value is not sufficiently high. Amongst the reasons that have been assigned for this, are— 1st, that in addition to the resistances of the particles of the beam to a direct Strain, there is another resistance arising from the lateral adhesion of the fibres to each other, termed the " Resistance of Flexure." (See Barlow on the "Strength of Materials," eth edition.*) And 2nd, that in most metallic beams (especially when cast) the outer skin, which is strained more than any other part of the section, is very much stronger (from many well-known causes) than the nvernge section ; whereas if the direct tensile or compressive resistance of the same beam, in, the direction of ite length, were being experimentally ascertained, it would be i&t average section at least, and perhaps me centre (weaker) portion especially, from which the strength would be determined. However, there is evidently a necessity to employ a higher value than that for the direct resistance ; and Professor Hankine has adopted a modulus of rupture which is 18 times the load required to break a bar of 1 sq. inch section, supported on two pdnts one foot apart, and loaded in the middle between the -supports (831). MoMiKTS OP Inertia and Rksistanok of Beams of Vabiocs Seomohs. 804. Beam of a solid rectangnlar section. Fig. 76. 1 = Id' 12 ■ _ Cirfs Cad „ 20S. Beam of a hollow rectangnUir section. Fig. 77. I = bd^—b'd'" 12 - c(&d»-yd'3) „ K = — = M. 806. Beam of a solid circnlar section.. Fig. 78. I = -7854 r* = — '. 4 R = C -7854 r» = ^H = ji. i * Bdited by W.Hnmber. London : Lockwood & Co. K0XSHI8. or niBBIIA AHS BESISTAHCE. XOT. Beam of a hollow ciieular section. *I = -7854 (r* - j'*). _ •7854Cfr«- r'«) „ es Fig. 79. 20S. Beam of a solid elliptical section. I = -7854 b d\ K = -7854 C 6

) Fig. 81. %I0. Beam irith one flange. \ I = i { 5cP + i'd"- (y-6)d'" } K=2i=M. Fig. 82. 61 BEAKS or TABIOVS SEOTIOHS. 811, Beam witb tiro equal flanges. Fig. 83. I = 12 SIS. Beam vith two nnequid flanges. Fir. 84. I = J { 5 tP - (6 - *) (rf - c)» + 5' d" (l/-]c) id' - (/)' }• E = — = M. 813, To And tUe Homentg of Inertia and Rcslntsnce of anr CroHs Seetion made up of a number of simple figures. Find the moment of inertia of each of the simple figures about an axis traversing its centre of gravity parallel to the neutral axis of the complex figure. Multiply the area of each of the simple figures by the square of the distance between its centre of gravity and the nentnil axis of the whole figure. Add all the results together for the moment of inertia of the whole figure. Let I, = moment of inertia df one of the simple figures about its own neutral axis ; A its area ; v the distance &om its centre of gravity to that of the whole section (880) ; and I, moment of inertia of the whole section ; then, I = (Ii + v^ A) + &C. C I Moment of resistance, B = — (w.) 814. IWoiaeitts of Inertia dnd Resistance, Ultimate Strenattli, and Deflections of Similar Beams. Moments of inertia \ . .•-.•i»t ( *''* **'' power J Moments of resistance f ggij.l,- ) the 3rd power f of their linear Strengths I ) the 2nd power l dimensions. Deflections ) ^'J ** ( the 1st power ) nnsuAVtO BMAIts or vnivorh stkehqth. 65 BlAKB OF RKCTAttdtrtlA SKOTIOK AND OF UnIFOKH StBEKOTH (16R). SUvatiotu of Bmrta ^ GiHildent irioMh, and Plant of Smmt of Conttant l>epth. Fig. 85.— Elevation. 915. A tin!- beam (T) losbclgd Iritii a caiMKnir&tedl veigiit at iU iit' . tireMity. 21T. Beam sup- ported at both ends, loaded at any 'point (13) with a concentrated weight. SIB. SemUbeam (A) loaded uui' ibrmly over its en- tire length. ' A ^, aparabola, with its vertex at B. ^ A triangle. AB, a straight lUe. A'BandCB, a peir ofj^arabolaB, with their apices at B> AOandBC, pair of pELrabolag) with' . their vertices at A and B respec- tively.* A C D, B t), a pair of triangles, having a coimmon base, C D.» 318. Beam sup' ported at both ends, loaded uniformly . (ID) over its en tlfe length. 'A semi-ellipse, AB major ajuis.^ AOB. ADB, pairof parabolas, having their Vertices at mid'^pan.* * ¥he additional material dotted iA at the supports is necessary to resist the fehearmg force (104>. 66 SBTUEOTIOH. 919. How to cut the. (1.) Stroncu* and> (^<) BtilTut Reetansnlnr Beam firoin a Cylindrical loc Let the accompanying figs. 93, 94 represent sections of the log. - Pis. 93. I^fT. «*. strongest. Stlffest. Draw a diameter. For the beam whose ultimate strength will be the greatest^ trisect the diameter. For the beam which will deflect the least, divide the diameter into fonr equal parts. Draw perpendiculars to the diameter as shown, and their intersection with the circumference will determine the inscribed rectangle^ which is the section of the required beam. MO. To find the Centre of Gravity (199) of any Section. Let a, a', a", &o., represent the sectional area of the several elemen- tary parts into which the section may be decomposed ; g, g', gf', &c., the known distances of their respective centres of gravity from any fixed axis — say the lower edge of the beam — and S, the distance from the latter to the centre of gravity of the total section ; then, a _ "g + "'g' + <»"!/' + ^»- a + a! + a" + &0> DEFLECTION. 9X1. Deflection is the "displacement of any point of a loaded beam ftrom its position when the beam is unloaded." MS. Camber is an upward curvature, similar and equal to the maxi- mum calculated deflectiMi, given to a beam or girder or some line in it, in order to ensnre its hoiizontality when fiilly loaded. aiRDXM'¥aosB oaoss asoiibira are imiroiiM am> equal THBOUaHODT THKIK IiXireiB8. 3X3. The maximum deflections for several cases are given with the moments of rupture (7 to XS), fay the values for Def. ILAHOED OIBDEBS OP UHIFOBM SlBXMeTH. SS4. Girder sapported at both ends. Let D = central deflection. d s central depth. I = length of span. K = sum of the extension of one flange or boom, and the short- ening of the other by the strains upon them. (I05II1IV0V8 eiBDKBS. (S7 K may be found aa fultowa : Ii«t S = itrain in Iba. per sq. in. on either boom when the load imdniang the deflection is on the beam ; B = nlodulns of eUaUeity (»IS)j I = length of boom j then i-^I, + ^) = length of b|pm after the atrain ia on ; and if li coireapond aimilarly for the other boom, then k + Ji =li.. If the booms be of equal length and section, thenK = (2i). Let d = depth at support ; the other notations as before. The deflection at the unsupported extremity, — 2d tSt. ContiBnom Glrden, and einlers flxed »t One or Both Knds («4— 88). When any whole span ia aiulysed it will be seen (M to SO) that it ia equivalent to a whole giider supported at ue ends, and one or ma aemi-girdera, as the case may he. To UieBe the above fbrmnlee (MS— S) mn be applied, and the nuud- Fig. K. mum deflection obtained. Thus— Forasrtrder (aa AB);!^ at lalh ends {iA—M), the deflection of the semi-beama A C and D B at and D ( II or H5), added to the. deflection of C D below its ends, aa olibuned from (IS, 19, or »S4), will be the total maximum deflection of E, belnw A. B. Again, for eoMinuoiw girdtn with moving JoadttSX, SS), the maximum dtfitctian at the middle of a span (B C, flg. 26) vill aeeur AD8. , Then cuca-tain the load that would product that strain, and that will M the load requii«d. Remembering that if the itructwe he not of uniform itrengtA, the strength of the veakest part determines the strength of the whole (184). HM. BXAUTLS I.— What load, at th« centre of a wronght-iron, single-webbed or plate girder of uniform strength (165), and of the following dimensions, wonld cause the rupture of the girder,— Length of span, 20 ft. ; central depth (OTi IS), 1-6 it. ; effeouve section (180) of the lower flange at the centre, 4 sq. in. ; t|jp>n of average quality (SSI). ^ As the girder is of unifbrm strength, it matters not what part ia considered, for the same result would be obtained. Lower flange will be in tension (6S— 1). intimate strength of average wrought iron plate (>31), 66,000 lbs. per sq. in. iq.in. lbs, lbs. 4 X 66,000 = 220,000 = brealdng strength of lower flange. W i W X 20 Then from (19 and •!) strain on lower flange = — = 220,000 lb. = ^., ' Therftfore W = ^^^ ^ iXJ^ _ ^j^ y^ ^ ^.^ ^^^ 230. Bi-^MTLx n.— Required the greatest safe (172) load nniformly distributed on a rectangular beam of British oak projecting from a wall. Length of beams 6 ft. = 72 in. . . . (200.) Breadth „ =6 in. Depth „ s=9in. Taking coef, (1T2) as 6 ; O (203) as 10,000 (231.) M 8 B. . . . (1, 196.) "'• (a) E = ,£±L . . . (,01., V. 2 r(w(}38. 36 (w I) = 162,000. w I = 4,600 lbs. a 3 torn, 6 X Co 10,000 X 64X9 6x6 162,000 231. TaBL> of the SikKNOTH, &0., of MlTERIAtS IN FOUNDS AvOIBBVIOU fKB Square Incb of Seoiior, Jbtn-foii. Ifoihiluio/ JEliutieity. Aot •n lU.' '-"• '*x"- OuaHng. Cnu Bnating. MlTALS: Brass, Cast 18,000 49,000 60.000) 36,000 f 40,000 16,000 10,300 uOiOoo 110,000 27,700 40.000 9,000,000 14,230,000 17,000,000 18,000,000 1 487 ' to 624-4 633 1649 1666 444 „ Wire Copper, Rolled „ _ Bolts Iron, Cast „ American best „ „ average * The specific gravities may be readily found when it is known that a cubic foot of water weighs 62*6 lbs. BtnSKaiH, Ice, OF MATERIALS. 69 Tabli or THE Stmhgih, bto,, or MatsriaiiS — eontimied. mtlmateX'tManctto MaUriali. Xoduliu at OiuHeUy. Foot miti. T«uim. <*X"' Bhearim. Crou BnakinQ, XoMmof Suptw*. MOTAM:— «««. Iron, Wroaght .. ( 73,000 „ „ „ iverage 66,000 f 41,000 100 000> 36,000 60,000 43V60O) 29,000,000 480 „ .. Plates .... I 71,000 to 40,000 ' 16,000,000 » M •■ •▼wage cs.ooo „ ,, Jcunta— ■iDRle riveted . doubleriretecH- ■ 40,000 62,000 •■ It Wir© ■«•(■• 84,0001 88.000 r • •■ • •* *•• 16,000,000 .. „ ,.CBbI«.. 1 134,000 i ( 487 SteelBan 160,000 to 120,000 >■• 80,000,000 K «8 ;; hii^f:,.:. 9S«» ao,«o TiMBu : f 12,000) - Asb 16,H)0 9,000 1,400 \ to 1,600,000 47 ( 14.000) \ w,oooi Beech 11.600 9,000 i,3{o,oao Birch 15,000 6,600 >.* 11,700 1,600,000 44-4 Box 20,000 10,000 .., 1,000000 60 Cedm ll.MO 12,000 6,800 10,000 1,400 7^400 486.000 1,000,000 30-4 84 aam ^ ■" ••«•••....,•» ( 600 6,000 700,000 80 nr- 12,000 6,600 . to I 1,600 to 1,000,000 to 4« Uihogany f 8,000) to \ 20,000 ) 8,0fl0 f 7.«iii, sec.: Bride Fire „ Btnmg Ked „ Weak Bad Cbalk Gnalta f 276 i to • ( 300 ) CO ••• 1,700 i.ipo 6S0) to > I sooj 400 11,000 4,600 ».000 ■ to (6,600 ••* 1.100 ) to \ 2,360 1 2,600,000 115 ' to. I 136 rm '■•■to : I m PS I m no t 170 to ISO ISO to I 167 Mortar, ord. «■.. LlmeBtono *•■».. Saadatoqe S3!i. YaSIOUB VBtHODS OF DBAWIlta FAKABOLAa, SBB BaSIC AHD HuSBI BEINa aiVXN.* I. (Fhte ni., fig. 1.) Bt obdihatxs on oitseis niox a iakocdi (E 0) ID IBS PASABOLA AI ITS TIKIXX (S). Tbioagh h diav I> E parallel and equal to A C. The onlinatei or ofiets £rom any points in D E to the parabola will be proportional to the sqnarei of the distances of those points from S, Thus, if the ordinate at a be 1, then the ordinate at 6, tviee the distance of a bom. D.dnnst be 4 ; thSit at Ct three times the distance, most be 9 ; and so on. To proceed pra«tic^7 : DiTide E D into a number of equal parts (») as at a, i, ev he., fig, 1 ; then if B A be divided into (n') parts, each of these parts will' be' the required nnit, 1 of which is the o^t at a, 4 at &, 9 at e,. and so on. TiiTough the points «.' V. il, &o., thus detenuined, the required enrre can be' drawn. II. (Plate III.J fig. 3.) Bt OBDnrAncs fBOH ibb basi. Divide the base (half of which is repreeented by A 0) into an even number of equal parts ;> then if the haght or ordinate at centre D corre- qwnd to the square of. half the number-of those parts (8x8 = 64), the * The terms height (or ordinate at oentre) and beee have been need instead of absCiBBa for the former, and donble-ordinate for the latter, that tlie garabola might appear in a moie almple light than perbape it otherwiee would ave done. Any height con be adopted itar the parabola; oonvenience for scaling off the momenta (S), Ac., being alone studied. CONSTRUCTION OF PARABOLAS. Fiy 1. Fiy ;.'. / / ^ '-^ ~^' /' / ::* / ■ -, A C Fiy.3. ";- ': ' ^'' : ' - / ■ 1, / ■ ' c' ' / : / ■ ■ ,/■ / . ' / , Fit;- 4 E /■ \ /■■' a'" C 6 F 1 q ; C "H D K C /;, ,/,„„ „ r.,„.,,r,r ,..aP..,-,.M„:.r , „„y ynf„pr.,nt | />'"" 1 ^^" p/I Jc / 1 //'■' ! \ / \ P A M OONSTKCOIIOir OF PAKABOIiAS. 71 ordinate xt any other point (d for instance) will be represented by the product of the numbers of parts in the two segments into which d divides the base (4x12 = 48), The paraboia may then be drawn through the extremities of the ordinates. III. (Plate III., iig, 3.) Br the constki^toh of a duqbah. Draw D E parallel and equal to AG; divide D E and E A similarly ; the end B of E A corresponding to the end D of E D. Through a, b, &c., in £ D, draw a a, bb, &e„ parallel to D C Join D to the several points a' b', &c , in E A. The parabola will pass through the intersection of a (r with Va'ibb with D V, &c, ■ Note. If this mode of construction be adopted when the ordinates are I i at certain paints only (and this will generally be the case 4n piaciice), the actual curve need not be drawn, after the pointi it patati through have been determined. IV. (Plate III., fig. 4,) B7 the constritctiok of a siaorah. On the base AB describe an isosceles tiiangle, whose height C E is double that of the required parabola. • Divide the two sides A E, E B of the triangle into an even number of equal parts, and draw lines as in the figure. These lines will be tangents to the parabola, which may therefore be readily drawn. v. (Plate III., fig. S.) Bt means of a striso. Draw E D equal and parallel to AC. iToin C to F at the bisection of E D. Make G F 6 a right angle. Let F Q intersect the production of C D. MakeDK*= DG, parallel to F D ; H H is a "straightedge," against which slides the " set-square " S. A piece of thread or fine string equal in length to the distance A E is fixed, one end at E and the other at the point M, in the set square which will traverse the base A C as the set square slides along. A pencil, P, by which the string is kept tight, and close to the edge of the set square, will describe a true parabola. To DRAW A tAKOtST TO A FARABOIiA AT AKT POIKT P, (Plate III., fig. 6.) Draw P C perpendicular to the axis S G. Make D E = D C, Join E to P, and P B will be the required tangent. * K is the "focus "of the parabola. GG' the directrix. SCIEN-flFIC BOOKS , #t;te,isHXl} St D. VAN NOSTRAND, Publisher and Ittiporter. -o ■■ « ■■ » LOWELL HTDSAITLIC EZPEEIMEKTS— being a Selection from Experiments on Hydraulic Motors, Oh the Flow of Water over Weirs, and in Open Canals of Uniform Rectanguki" Section, made ii LoWisllj Mass. By J. B. Francis^ Civil Engififeeif. Secofid editiorij rfevised and . enlarged, including many New Experiments on Gauging Water in Open Canals, and on the Flow through Sub- merged OrificSs atid Diverging, Tubes. With 43 cappefptates, beautifully engaved, and about 100 new pages of text, i vol. 4tO, Cloth $1 5 . 00 ON THE STRENGTH OF CAST-IRON PI{.|.A^8. with Tables for the Use of Engineers, Architects, and Builders. By J. B. Francis, Civil Engineer. SvOt dloth 2 .00 ON THE STRENOf B OV MATSftlALS used in £n' gineering Construction. By J. K. Whilden. I vol. izmo, cloth 2 .00 3 TAN NOSHUIID'b BCIIENTIFIO FUBLICAII0II8. ON THE CONSTBXICTION OF IRON ROOFS. A Theoretical and Practical Treatise. By Francis Campin. With wood-cuts and plates of Roofs lately executed. . Large 8vo, cloth $3 . oo BROOKLYN WATER -WORKS AND SEWERS. Containing a Descriptive Account of the Con- struction of the Works, and also Reports on the Brooklyn, Hartford, Belleville, and Cambridge Pumping Engines. Prepared and printed by order of the Board of Water Commissioners. With illustrations, i vol. folio, cloth 1 5 . oo LONG AND SHORT SPAN RAILWAY BRIDGES. By John A. Roebling, C. E. Illustrated with large copperplate engravings of plans and views. In fress DESCRIPTION OF THE IRON RAILWAY BRIDGE ACROSS THE MISSISSIPPI RIVER AT aUINCY, ILLINOIS. By Thomas Curtis Clarke, Chief Engineer. Illustrated with numerous litho- ■ graphed plans, i vol. 8vo, cloth 7 • 50 SKELETON STRUCTURES, especially in their appli- cation to the Building of Steel and Iron Bridges. By Olaus Henrici. With folding plates and Dia- grams. 8vo, cloth 3 .00 TREATISE ON ROLL TURNING FOR THE MANU. FACTURE OF IRON. By Peter Tunner. Trans- lated and adapted by John B. Pearse, of the Penn- sylvania Steel Works. With numerous engravings and wood-cuts, tn press ON THE POWER OF WATER, as applied to drive Flour Mills, and to give motion to Turbines and other Hydrostatic Engines. By Joseph Glynn. Third edition, revised and enlarged, with numer- ous illustrations. 1 2mo, cloth 1.25 yAK KOBTBAKS'S BCIXIITZFIO FVSLlOtXlOSB. 3 UECHAHIC'S TOOL BOOK, with practical rules and suggestions, for the use of Machinists, Iron Work- ers, and others. By W. B. Harrison, associate editor of the "American Artisan." Illustrated with 44 engravings. 1 2mo, cloth $z ■ 50 MANUAL OF THE MECHANICS OF ENGINEER- ING and of the Construction of Machines. Part II. THEORETICAL MECHANICS, with an in- troduction to the Calculus. By Julius Wiesbach, P. H. D. 8vo, 1 100 pages, and upwards of 1000 illustrations. Translated frond the fourth aug- mented and improved German edition. By Eck- ley B. Coxe, A. M., Mining Engineer, la press. PEIRCE'S SYSTEM OF ANALYTIC MECHANICS. Physical and Celestial Mechanics. By Benjamin Peirce, PerkinsProfessorof Astronomy and Math- ematics in Harvard University, and Consulting Astronomer of the American Ephemeris and Nau- tical Almanac.- Developed in four systems of Analytic Mechanics, Celestial Mechanics, Potential Physics, and Analytic Morphology, i vol. 4to, cloth 10. bo OILLMORE. Practical Treatise on Liines, Hydraulic Cements, and Mortars. Papers on Practical En- gineering, U. S. Engineer Department No. 9, containing Reports of numerous fflcperiments con- ducted in New York City during the years 1858 to 1861, inclusive. By Q. A. Gillmore, Brig.- Geneial U. S. Volunteers, and Major U. S. Corps of Engineers. With numerous illustrations. One volume, octavo, cloth 4-00 CATALOGUE OF Scientific Books, PUBLISHED BT D. Van Nostrand, 23 MURRAY STREET and 27 WARREN STREET, PUBLISHER, IMPORTER and BOOKSELLEa NEW YORKt I 87 1. \* Copies of this Catalogue, and also Catalogue of Militaiy and Naval Books, will be sent post-paid to say address on uolicatioa. D. VAN NOSTRAND, 2$ MURRAY STREET AND 27 WARREN STREET, NEW YORK, PUBLISHES, IMPORTS, AND OFFERS FOR SALE, AT " WHOLESALE AND RETAIL, Works on Architecture, Carpentry, Building, Decorative Arts, Painting, Photography, Drawing, Cabinet Work, Astronomy Navigation, Meteorology, Naval Architecture, Mechanics, Engi neering. Chemistry, Physics, Microscopical Science, Brewing, Distilling, Wine Making, Iron and Steel, Coal, Coal Oil, Gas, Geology, Mineralogy, Mining, Metallurgy, Hydraulics, Hydro- statics, Electricity, Electric Telegraph, Magnetism, Natural Phi- losophy, Technology, Mathematics, Dyeing, Calico Printing, Leather Dressing, Sugar Manufactures, Cosmetics, Soap, Candles, Perfumery, Clock and Watch Making, Manu&ctures generally, and Works of Reference. %* General Catalogue of the above works, 64 paget 8p», tint tt any address on receipt of ten cents. S it. tAM H6stsA.m>'s tvajKUkTiom. UBOT (H. L.) Siege ArtiUety iti the Campaign against Richmond, with Notes on the 15-inch Gun, including an Algebraic Analysis of the Trajectory of a Shot in its ricochet upon smooth water. Illustrated with detailed drawings of the U S. and Confederate rifled projectiles. By Henry L. Abbot, Maj. of Engineers, and Bvt Maj.- General U. S. Vols., commanding Siege Artillery, Armies before Richmond. Paper No. 14, Professional Papers, Corps of Engineers, i vol. 8vo, cloth $3 • $<> AlEXANDER (J. H.) Universal Dictionary of Weights and Measures, Ancient and Modern, reduced to the standards of the United Sutes of America. By J. H. Alexander. New edition, enlarged, i vol. 8vo, doth. 3 . 50 BENET (a V.) Electro-Ballistic Machines, and the Schultz Chronoscope. By Brevet Lieut.-Colonel S. V. Benet. 1 vol. 4to, illustrated, cloth 4.O0 BROOKLYN WATER WORKS. Containing a Descrip- tive Account of the Construction of the Works, and also Reports on the Brooklyn, Hartford, Belleville, and Cambridge Pumping Engines. With illustrations. I vol. folio, cloth 15.00 B1TR0H (N. P.) Modern Marine Engineering, applied to Paddle and Screw Propulsion. Consisting of 36 colored plates, 259 Practical Woodcut Illustrations, and 403 pages of Descriptive Matter, the whole being an ex- position of the present practice of the following firms : Messrs. J. Penn & Sons ; Messrs. Maudslay, Sons 8e Field; Messrs. James Watt & Co.; Messrs. J. & G. Rennie ; Messrs. R. Napier & Sons ; Messrs. J. & W. Dudgeon ; Messrs. Ravenhill & Hodgson ; Messrs. Humphreys & Tenant ; Mr. J, T. Spencer, and Messrs. Forrester & Co. By N. P. Burgh, Engineer. In i thick ▼oL 4to, cloth 25.09 Do. do. half morocco. 30.ee ft. VAK sost&isb'a pvsiAok'nom. CAUPIir on the Construction of Iron Roofs. By Franda Campin. 8 vo, with plates, cloth $3 ■ oo CHAUVENET (Prof. Wm.) New Method of Correcting Lunar Distances, and Improved Method of Finding the Error and Rate ef a Chronometer, by equal altitudes. By Wm. Chauvenet, LL. D. i vol. 8vo, cloth 2 . 60 COLBUBN. The Gas Works of London. By Zerah Col- burn, C. E. 1 vol. i2mo, boards 75 CBAIQ (B. F.) Weights and Measures. An Account of the Decimal System, with Tables of Conversion for Commercial and Scientific Uses. By B. F. Craig, M. D. 1 vol. square 32mo, limp cloth 5x1 COFFIN. Navigation and Nautical Astronomy; pre- pared for the use of the U. S. Naval Academy. By Prof. J. H. C. Coffin. 4th edition enlarged, izmo, cloth 3 . 50 FRENCH'S ETHICS. Practical Ethics. By Rev. J. W. French, D. D., Professor of Ethics, U. S. Military Academy. Prepared for the Use of Students in the Military Academy, i vol. 8yo, cloth 4.50 FRENCH'S GRAMMAR. Part of a Course on Language. Prepared for Instruction in the U. S. Corps of Cadets. By Rev. J. W. French, D. D., Professor of Ethics and English Studies in the United States Military Academy, West Point, i vol. izmo, cloth 2 .50 FRANCIS. Lowell Hydraulic Experiments, being a selec- tion from Experiments on Hydraulic Motors, on the Flow of Water over Wiers, in Open Canals of Uniform Rectangular Section, and through submerged Orifices and diverging Tubes. Made at Lowell, Massachusetts. By James B. Francis, C. E. 3d edition, revised and enlarged, with many new experiments, and illustrated with twenty-three copperplate engravings, i vol. 410, doth. 15.00 4 Ik TAM MOOTBAini'B PVBIilOJlTIOlM. FEANCIS. On the Strength of Cast-iron Pillars, with Table* for the use of Engineers, Architects, and Builders. By James B. Francis, Civil Engineer, i vol. 8vo, cloth.. Sz.oo "A BCientiflc treatise ot ineetimable Tiilae to tbose for whom it is intended."— £oi(m Daily Mxertxur. GUIMOBE (Gen. Q;.A.) Treatise on Limes, Hydraulic Cements, and Mortars. Papers on Practical Engineer- ing, U. S. Engineer Department, No. 9, containing Reports of numerous Experiments conducted in New York City, during the years 1858 to 1861, inclusive. By Q^ A. Gillmore, Bvt. Maj.-Gen , U. S. A., Major, Corps U. S. Engineers. With numerous illustrations. t vol. 8vo, cloth 4-OA GOUGE. New System of Ventilation, which has been thoroughly tested under the patronage of many dis- tinguished persons. By Henry A. Gouge. Third edition, enlarged. With many illustrations. 8vo, doth 2 .00 HABBISON. The Mechanics' Tool Book, with Practical Rules and Suggestions for Use of Machinists, Iron Workers, and others. By W. B. Harrison, associate editor of the " American Artisan." Illustrated with 44 engravings. 1 2mo, cloth 2 . 50 EENBICI (Olaus). Skeleton Structures, especially in their application to the Building of Steel and Iron Bridges. By Olaus Henrici. With, folding plates and ' diagrams. I vol. 8vo, cloth 3 .00 EEWSON (Wm.) Principles and Practice of Embanking Lands from River Floods, as applied to the Levees of the Misassippi. By William Hewson, Civil Engineer. 1 vol. 8vo, cloth 2.00 ■< This Is a Tidtiable treatiae on the prlnclplea and practice of embanking lands flrom river floods, as applied to Lereea of the Mississippi, by a highly intelligent and expert- (need enghieer. The anthor says It is a llrst attempt to reduce to order and to rule tlu tailcn, ezeontion, and meaBorement of the Levees of the Mississippi. It is a most usefW «9d aswlwl snslrllratini to srientlflo litentnre,"-j%aa(ttl|ikta Stmimf JmntH. D. YAK KOBIBAKD B FDBUOATIOlnL EOLLET (A. L.) Railway Practice. American and Euro- pean Railway Practice, in the economical Generation of Steam, including the Materials and Constrdction of Coal- burning Boilers, Combustion, the Variable Blast, Va- pori^tion, Circulation, Superheating, Supplying and Heating Feed-water, &c., and the Adaptation of Wood and Coke-burning Engines to Coal-burning; and in Permanent Way, including Road-Hed, Sleepers, Rails, Joint-fastenings, Street Railways, &c., &c. By Alex- ander L. HoUey, B. P. With 77 lithographed plates. 1 vol. folio, cloth $12,00 • • • «jUI tbcse Babjecta are treated bj tba autbor in botb an Intelligent and Intel llgible manner. Tbe facts and ideas are well arranged, and preBented In a clear and Blm- pie style, accompanied by beautiful engravings, and we presume tbe work will be r» garded as Indispensable by all wbo are interested In a Icnowledge of tbe constmotion ol railroads and rolling stock, or the working of locomotivoB."— &fai({/Ic ^merfcan. HUNT (K. M.) Designs for the Gateways of the Southern Entrances to the Central Park. By Richard M. Hunt. With a description of the designs. I vol. 410, illus- trated, cloth 5 ofl KINO (W. H.) Lessons and Practical Notes on Steam, the Steam Engine, Propellers, &c., &c., for Young Ma- rine Engineers, Students, and others. By the late W. H. King, U. S. Navy. Revised by Chief Engineer J. W. King, U. S. Navy. Twelfth edition, enlarged. 8vo, cloth 2.f^ THE KANSAS CITY BRIDGE, with an account of the Regimen of the Missouri River, and a description of Methods used fot founding in that River. By O; Chanute, Chief Engineer, and George Morrison, Assistant Engineer. Illustrated with five lithographic views and twelve plates of plans. 4to, cloth 6.00 HeCOBMICK (R. C). Arizona : Its Resources and Pro»- pects. By Hon. R. C. McCormick. With map, 8vo, paper ,.,.»»» • J a. TAX HOSTSAITD'B FUXUOinOIM. miriFIE (Wm.) Mechanical Drawing. A Text-Book of Geometrical Drawing for the use of Mechanics and Schools, in which the Definitions and Rules of Geometry are familiarly explained ; the Practical Problems are ar- ranged, from the most simple to the more complex, and in their description technicalities are avoided as much as possible. With illustrations for Drawing Plans, Sec- tions, and Elevations of Buildings and Machinery; an Introduction to Isometrical Drawing, and an Essay on linear Perspective and Shadows. Illustrated with over 200 diagrams engraved on steel. By Wm. Minifie, Architect. Seventh Edition. With an Appendix on the Theory and Application of Colors. 1 vol. 8vo, cloth $4 00 WILLIAMSON. Practical Tables in Meteorology and Hypsometiy, in connection with the use of the Barometer. By Col. R. S. Williamson, U. S. A. I vol. 4to, flexible cloth 2 . 50 CULLEY. A Hand-Book of Practical Telegraphy. By R. S. CuUey, Engineer to the Electric and Interna- tional Telegraph Company. Fourth edition, revised and enlarged. 8vo. Illustrated. Cloth 5 .00 POPE. Modem Practice of the Electric Telegraph. For Electricians and Operators. By Frank L. Pope. Fourth edition, revised and illustrated. 8vo, cloth. . 2.00 HIHIiTE (Wm.) Geometrical Dravidng. Abridged from the Octavo edition, for the use of Schools. Illustrated with 48 steel plates. Fifth edition, i 'vol. i2mo, half roan 1.5^ * It ta won adapted ai a t«xt-book of drawing t* be ued In em High School! and icademies where thia tuehil branch of the fine arti has been hitherto too mnch aef- Wettd."— Arim JmrmiL 7 D. TiN K0SISAND*8 tVSUOkVtOim. PIERCE (Prof. Benj.) System of Analytical Mechanic*. Physical and Celestial Mechanics, by Benjamin Pierce, Perkins Professor of Astronomy and Mathematics in Harvard University, and Consulting Astronomer of the American Ephemeris and Nautical Almanac. Developed in four systems of Analytical Mechanics, Celestial Me- chanics, Potential Physics, and Analytic Morphology. 1 vol. 4to, cloth $10.00 "I bare rs-ezamlned tbe momoira of the great geometers, and lure strlnn to consoli- date their latest researches and their most exalted forms of thought into a consistent an4 uniform tceatisa. If I have hereby succeeded in opening to the students of my oonntry a readier access to these choice Jewels of intellect; if their brilliancy is not Impaired in this attempt to reset them; If, in their own constellation, they illustrate eaoh other, and con- centrate a stronger light upon the names of their discoverers; and, still more, if any gem wtaloh I may have presumed to add is not wholly lustreless in tbe ooUeotlony— I aiiaU feel itiai my work bos not been In Tain,"— £rirac win Rocks. By John G. Foster, Lieutenant-Colonel of Engineers, and Brevet Major-General, United States Army. With illustrations. 4to, cloth $3 . 50* SHAFFNER (T. P.) Telegraph Manual. A complete History and Uescripdon of the Semaphoric, Electric, and Magnetic Telegraphs of Europe, Asia, and Africa, with 625 illustrations. By Tal. P. Shaffher, of Ken- tucky. New edidon. 1 vol. 8vo, cloth, 850 pp.... 6.J0 SILVERSMITH (Julius). A Practical Hand-Book for Mi- ners, Metallurgists, and Assayers, comprising the most recent improvements in the disintegration, amalgama tion, smelting, and parting of the Precious Ores, with a Comprehensive Digest of the Mining Laws. Greatlt augumented, revised, and corrected. By Julius Silver- smith. Fourth edition. Profusely illustrated. I vol. I zmo, cloth 3 . 00 SIMM'S lEVELLINQ. A Treatise on the Principles and Practice of Levelling, showing its application to pur- poses of Railway Engineering and the Construction of Roads, &c. By Frederick W. Simms, C. E. From the fifth London edition, revised and corrected, with the addition of Mr. Law's Practical Examples for Setting Out Railway Curves. Illustrated with three lithographic plates and numerous wood-cuts. 8vo, cloth 2 . JO SUBMARINE WARFARE, Oifensive and Defensive, including a Discussioii of the Offensive Torpedo System. By Lieut -Cpftimander J. S. Barnes, U.S.N. With illustrations. 8vo, cloth 5 . 00 S. VAN NOSTRANDS PUBLICATIONS. PLATTNEE'S BLOW-PIPE ANALYSIS. A Complete Guide to Qualitative and Quantitative Examinations with the Blow-Pipe. Revised and enlarged by Prof. Richter, Freiberg. Translated from the latest Ger- man edition by Henry B. Cornwall, A. M., E. M 8vo, cloth $5-00 SFHEBICAL ASTBONOMT. By F. Brunnow, Ph. Dr. Translated by the Author from the Second German edition, i vol. 8vo, cloth 6. 5a STILLMAN (Paul). Steam Engine Indicator, and the Im- proved Manometer Steam and Vacuum Gauges — their Utility and Application. By Paul Stillman. New edition, i vol. i2mo, flexible cloth 1 .oo "Tbe imrpose of this naeflil volume la to bring to the notice of the muneroni dHB ol thoee interested in the bnildlng and the nse of steam engtaieg, the economy and safetf attend- ing the use of the instrument therein described. The Manometer has been long used— Iht InTentor is Watt in a cruder form; and the forms herein described are patented by tht author. The language of (he author, the diagrams, and Che scientific mode of treatmenti recommend the book to the careful consideration of all who hare engines in their cua,"» BmlonPotL SWEET (S. H.) Special Report on Coal; showing its Dis- tribution, Classification, and cost delivered over different toutet to various points in the State of New York, and the principal cities on the Atlantic Coast. By S. H. Sweet. With maps, 1 voL 8vo, cloth 3 .OC WILLIAMSOIf. Practical Tables in Meteorology and Hypsometiy, in connection with the use of the Barometer. By Col. R. S. Williamson, U. S. Army. 4to, flexible cloth 2.50 WALEEB (W. H.) Screw Propulsion. Notes on Screw Propulsion, its Rise and History. By Capt. W. H. Walker, U. S. Navy. 1 vol. 8vo, cloth 75 • "After thoroughly demonstratbig the efficiency of the screw, Ur. Walker proceeds to point out the various other pohits to be attended to in order to secure an effloient man-of- war, and eulogiiea throughout the readiness of the British Admiralty to test every nov- elty calculated to give satisfactory results. • • • • Commander Walker's book con- tains an immense amount of concise practical data, and every Item of Information ra- corded ftilly proves that the various points bearing upon It have bees well eoii8ld<>^ fnvlonsly to expressing an opinion."— Xondm Hining Jounul, D. VAN NOSTBAND S PTTBLIOAHONS. WEISBACH'S MECHANICS. New and revised edition. A Manual of the Mechanics of Engineering, and of the Construction of Machines. By Julius Weisbarh, Ph. D. Translated from the fourth augmented and improved German edition, by Eckley B. Coxe, A. M., Mining Engineer. Vol. I. — Theoretical Mechanics. I vol. 8vo, 1, 100 pages, and 902 wood-cut illustra- tions, printed from electrotype copies of those of the best German edition $io.oa Abstract of Contents. — Introduction to the Cal- culus — The General Principles of Mechanics — Pho- ronomics, or the Purely Mathematical Theory of Motion — Statics of Rigid Bodies — ^The Application of Statics to Elasticity and Strength — Dynamics of Rigid Bodies — ^Statics of Fluids — Dynamics of Fluids — The Theory of Oscillation, etc. " The present edition is an entirely new work, greatly extended and very mnoli Im- proved. It forms a text-book which must find its way into the hands, not only of every student, but of every engineer who desires to refresh his memory or acquire clear ideal on doubtful points." — The Sichnohgist. WARD (J. H.) Steam for the Million. A popular Trea- tise on Steam and its Application to the useful Arts, especially to Navigation. By J. H. Ward, Com- mander U. S. Navy. New and revised edition, i vol. 8vo, cloth 1 .00 WHILDEJI (J. K.) On the Strength of Materials uSed in Engineering Construction. By J. K. Whilden. I vol. i2mo, cloth 2 .00 WILLIAMSOar (R. S.) On the use of the Barometer on Surveys and Reconnaissances. Part I. Meteorology in its Connection with Hypsometry. Part II. Ba- rometric Hypsometry. By R. S. Williamson, Bvt. Lieut. -Col. U. S. A., Major Corps of Engineers. With Illustrative Tables and Engravings. Paper No. 15, .Professional Papers, Corps of Engineers. I vol. 4to, cloth 15 00 11 V. TAN HOSTBAND'S PCBLIdAXIOHB. EOEBLINQ (J. A.) Long and Short Span Railway Bridges. By John A. Roebling, C. E. Illustrated with large copperplate engravings of plans and views. Imperial folio, cloth $25 . oo CLARKE (T. C) Description of the Iron Railway Bridgef-over the Mississippi River, at'Quincy, Illi- nois. By Thomas Curtis Clarke, Chief Engineer. Illustrated with 27 lithographed plans, i vol. 8vo, cloth 7 50 TTHOTER (P.) A Treatise on Roll-Turning for the manufacture of Iron. By Peter Tunner. Trans- lated and adapted by John B. Pearse, of the Penn- sylvania Steel Works, with numerous engravings and wood-cuts, i vol. 8vo, text, and fol. vol. Plates, cloth 10.00 ISHEKWOOD (B. F.) Engineering Precedents for Steam Machinery. Arranged in the most practical and useful manner for Engineers, By B. F. Isher- wood. Civil Engineer, U. S. Navy. With illustra- tions. Two volumes in one. 8vo, cloth a . 5c JAUEEMAN. Treatise on the Metallurgy of Iron, con- taining outlines of the History of Iron Manufacture, methods of Assay, and analysis of Iron Ores, pro- cesses of manufacture of Iron and Steel, etc., etc. By H. Bauerman. First American edition. Re- vised and enlarged, with an appendix on the Martin Process for making Steel, from the report of Abram S. Hewitt Illustrated with numerous wood engra- vings, izmo, cloth 2 . 50 " This iB an Important addition to the stock of technical vorks pnhllBhed In thb country. It embodies the latest facts, diBCoverles, and processes connected with tba manofactore of Iron and steel, and should be In the hands of every person interested la the subject, as well as in all technical and scieutlllc libraries. "—&{m<(^ Aaurtean, PEET. Manual of Inorganic Chemistry for Students. By the late Dudley Peet, M. D. Revised and en- larged by Isaac Lewis Peet, A. M. i8mo, cloth. . . 75 12 D. Tin MOSTSAini'S FOBLIOUSMd. miQENT. Treatise on Optics : or, Light and Sight, the- oretically and practically treated ; with the applica- tion to Fine Art and Industrial Pursuits. By E. Nugent With one hundred and three illustrations. i2mo, cloth $2.00 " This book is of a practical rather than a theoretical kind, and is designed to aSbrd KCcnrate and complete Information to all interested in applications of the 8Cience.^/eou7id note. SABINE. HISTORY AND FBOGBESS OF THE ELEC- TRIC TELEGRAPH. By Robert Sabine, C.E. 2d edition, with additions. Fully illustrated. i2mo, clo. 1.75 OLYNN (J.) Treatise on the Power of Water, as applied to drive Flour Mills, and to give motion to Tur- bines and other Hydrostatic Engines.. By Joseph Glynn. Third edition, revised and enlarged, with numerous illustrations. 1 2 mo, cloth 1.25 PRIME. TREATISE ON ORE DEPOSITS. By Bern- hard Von Cotta. Translated from the Second Ger- man edition by Frederick Prime, Jr., Mining Engi- neer, and revised by the Author. With numerous illustrations. 8vo, cloth 4 ■ 00 HXJMBEB. A Handy Book for the Calculation of Strains in Girders and similar Structures, and their Strength, consisting of Formulae and corresponding Diagrams, with numerous details for practical application. By William Humber. i2mo, fully illustrated, cloth. . . 2.51 GIUiMORE. Engineer and Artillery Operations against Charleston, 1863. By Major-General Q. A. Gill- more. With 76 lithographic plates. 8vo, cloth... 10.00 Supplementaiy Report to the above, with 7 litho- graphed maps and views. 8to, cloth 5 .oe 18 D. TAN KOSTBAND'S'FUBUOATIOm AUCHIirCIOSS, Link and Valve Motions Simplified. Illustrated with 37 wood cuts, and 21 lithographic plates, together with a Travel Scale, and numerous useful Tables. By W. S. Auchincloss. 8vo., cloth, I3 oc JOYNSON. METALS USED IN CONSTRUCTIOH- Iron, Steel, Bessemer Metal, etc., etc. With illus- trations. i2mo, cloth 75 ELIOT AND STOEER. A compendious Manual of Qualitative Chemical Analysis. By Charles W. Eliot, and Frank H. Storer. Illustrated, i2mo., cloth,... i 50 VAN BUBEN. Investigations of Formulas, for the strength of the Iron parts of Steam Machinery. By J. D. Van Buren, Jr., C. E. Illustrated, 8vo., cloth, 2 00 JOYNSON. Designing and Construction of Machine Gearing. Illustrated, 8vo., cloth, '; 2 oc' HOW TO BECOME A SUCCESSFUL ENGINEER. Being hints to Youths intending to adopt the Pro- fession. By Bernhard Stuart. iSmo., cloth, 7;,' FREE HAND DRAWING, a Guide to Ornamental, Fig- ure and Landscape Drawing. By an Art Student. Profusely illustrated, iSmo., cloth, 75 THE EARTH'S CRUST. A handy Outlme of Geology. By David Page. Illustrated, 18 mo., cloth, 75 DICTIONARY of Manufactures, Mining, Machinery, and the Industrial Arts. By George Dodd. i2mo., cloth, ^ " og Bi Till kohbaxd'i puBUcaraoMk MAYER. Lecture Notes on Physics. By Alfred M. Mayer, Fh. D. 8vo, cloth $2 .00 A TREATISE OX THE RICHARDS STEAU-ENOINE INDICATOR, with Directions for its Use; By Chas. T. Porter. Revised with notes and large additions, as developed by American Practice, with an Appendix containing useful formulae and rules for Engineers. By F. W. Bacon, M. E., Member of the American Society of Civil Engineers. i8mo, illustraled. Cloth, i.oo ON THE FILTRATION OP RIVER WATERS, for the Supply of Cities, as practised in Europe, made to the Board of Water Commissioners of the City of St. Louis. By J. P. Kirkwood, Civil Engineer. Illus- trated by 30 engravings. 4to, cloth .- 15.00 THE PLANE-TABLE AND ITS USE IN TOFOQRAFH- ICAL SURVEYING. From the Papers of the U. S. Coast Survey. 8vo, illustrated. Cloth 2.00 REPORT on Machinery and Processes of the Industrial Arts and Apparatus of the fJxact Sciences. By F. A. 'P. Barnard, LL. D. Paris Universal Exposition, 1867. I vol. 8vo, cloth 5 .00 mON TRUSS BRIDGES FOR RAILROADS. The Method of Calculating Strains in Tn)ases, with a ICareful Comparison of the most Prominent Trusses in Reference to Economy in Combination, etc. By Brevet Colonel William E. Merrill, U. S. A. Illus- trated. 4to, cloth 5.00 USEFUL INFORMATION FOR RAILWAY MEN. By W. G. Hamilton, Engineer. Second edition, re- vised and enlarged. 600 pp. Morocco gilt For pocket i.c» Van Nostrand's eclectic Engineering Magazine. •»■ 112 Pages, lakoe'Sto, Monthlt. First Number was Issued Jan. i, 1869 Illustrated. PRICE Ss A YEAR IN ADVANCE. SINGLE COPIES 50 CTS. Vol. I. January to December, 1869. Cloth......... $5.cx3 " 2. " " June, 1870. " , 3.00 " 3. July to December, 1870. " 3.00 VA.N NOSTRAND'S MAGAZINE consists of Articles Sc- lected and Matter Condensed from all the Engineering Serial Publications of Europe and America. Few active engineers 01 artisans can take all or most of th« professional newspapers; none can afford the time to wade through the columns of the whole scientific press to get at the really important news, information, and opinions. The object of this Magazine is to present within limits of space and cost that all can afford, the cream of not less than fifly engi neering, mechanical, chemical, and metallurgical publications. The French and German Magazines will be largely translated Papers and discussions before Societies will be condensed. Professional newt from all sources will be compiled at length. IS ^ eale's Sej^y^^ OB SCIENTIFIC WORKS. %* Ibe entire Series is freel7 illustrated with wood and steel engrav- ings and littiographs where requisite. EBPOETED AND FOB SAT.Ti BT D. VAN NO ST RAND, 23 Murray and 37 "Warren St., New York. ♦ ■I RUDIMENTARY SERIES. 2. Natural Philosophy, by Charles TomliBBon 50 12. Pneumatics, " " 75 20: Perspective, by George Pyne $1 00 87. Painting; or, A . TAN KOSTBAND, 40. Glass Stainmg, by Dr. M. A. Gessert, with an Appen. dix on the Art of EuamelUug 50 41. Painting on Glass, from the German of Fromberg. . 50 50. La'w of Contracts for Work and Services, by David Gibbons 75 66. Clay Lands and Loamy Soils, by J. Donaldson. ... 60 69. Music, Treatise on, by C. C. Spencer $1 00 71. Piano-forte, Art of Playing, by C. C. Spencer 50 72. Recent and Fossil Shells (a Manual of the Mollusca), by S. P. "Woodward, F.G.S., A.L.S., etc. New edi- tion, with Appendix by Ealph Tate, F.G.S. Cloth extra, gilt top 3 75 79**. Photography, Popular Treatise on, from the French of Monckhoven, by W. H. Thomthwaite 75 86. Astronomy, by the Bev.E. Main. New and Enlarged edition ; with an appendix on "Spectrum Analysis." 113. Use of Pield Artillery on Service, by Jaubert, translated by Lieut. -Col. H. H. MaxweU 75 113*. Memoir on Svirords, by Marey, translated by Lieut. - Col. H. H. MaxweU 60 151, Handy Book on the Law of Friendly, Industrial and Provident, Building and Loan Societies. With Copious Notes. By Nathaniel White 50 163. The Law of Patents for Inventions, by F. W, Campin, Barrister 1 00 PHYSICAL SCIENCE. 1, Chemistry, by Prof. Fownes, including Agricultural Chemistry, for the use of Farmers 50 3. Geology, by Major-Gen. Portlock 75 4. Mineralogy, by Eamsay 1 50 7. Electricity, by Sir W. S. Harris 75 7*. Galvanism, Animal and Voltaic Electiicity, by Sir W. S. Harris 75 8. Magnetism, Exposition of, by Sic W. S. Harris 1 75 2 FUBLISHEB ANI) JMFOBHiB. 133. MeteiUurgy of Copper, by B. H. Lambom tl 00 134. Metallurgy of Silver and Lead, by K. H. Lambom. 1 00 135. Electro-Metallurgy, by A. Watt 1 00 138. Kaad-Book of the Telegraph, by E. Bond 50 143. Ezperimental Essays— On the Motion of Camphor and Modem Theory of Dew, by C. TomUnsou 60 li;l. Questions on Magnetism, Electricity, and Prac- tical Telegraphy, by W. McGregor 75 BUILDING AND ARCHITECTURE. 16. Architecture, Orders of, by W. H. Leeds 50 17. " Styles of, by T. Bury 75 18. " Principles of Design, by E.L.Garbett. 100 22. Building, the Art of, by E. Dobson 75 23. Brick and Tile Making, by Dobson and Mallett ..... 150 25. Masonry and Stone Catting, by E. Dobson 1 25 30. Draining and Sewage of Towns and Buildings, by G. D. Dempsey 1 00 With No. 29, DBAtN AGE OP Land, 2 vols, in 1 1 50 35. Blasting and Quarrying of Stone, and Blowing up of Bridges, by Lieut. -Gen. Sir J. Bnrgoyne. ... 75 00. Bictlpnciry of Technical Terms used by Architects, Builders, Engineers, Surveyors, etc 2 00 In cloth boards 2 50 Half morocco 3 00 i2. Cottage Building, by 0. B. Allen 50 44. Foundations and Concrete Works, by E. Dobson. 75 45. Limes, Cements, Mortars, Concrete, Mastics, etc., by G. R. Bumell 75 57. Warming and Ventilation, by C. Tomhnson 1 50 83**. Door Locks and Safes, by Tomlinson and Mallett. 1 25 HI. Arches, Piers and Buttresses, by W. Bland 75 116. Acoustics of Public Buildings, by T. B. Smith 75 123. Carpentry and Joinery, founded on Bobison and Tredgold 75 123*. Illustrative Plates to the preceding, fta 2 25 3 D. VAN NOSIBAMD, 12d. R00& for Public and Private Biiildiiigs, founded on Eobison, Price, and Tredgold 75 127. Architectural ModeUiug, Practical Instructions, by T. A. Kichardson w . . . . 75 128. Vitruvius's Architecture, translated by J. Gwilt, with Plates $2 50 130. Grecian Architecture, Principles of Beauty in, by the Earl of Aberdeen 50 132. Erection of Dwelling-Houses, with Specifications, Quantities of Materials, etc., by S. H. Brooks. 27 Plates 1 25 156. Quantities and Measurements ; How to Calculate and Take them in various kinds of Trades. By A. C. Beaton. New edition with additions 75 158. The Slide Rule, and How to Use it. With Slide Bule in Tuck of Cover 1 60 MACHINERY AND ENGINEERING. 33. Cranes and Machinery for Raising Heavy Bodies, the Art of Constructing, by J. Glynn 50 34. Steam Engine, by Dr. Lardner 50 43. Tubular and Iron Girder Bridges, including the Britannia and Conway Bridges, by G. D. Dempsey. . 75 47. Lighthouses, their Construction and Illuminatiou, by Allan Stevenson 1 50 59. Steam Boilers, their Construction and Management, by B. Armstrong 75 62. Railways, Construction, by Sir M. Stephenson 75 62*. Railway Capital and Dividends, with Statistics of Working, by E. D. Ohattaway 50 (Vols. 62 and 62* bound in one) -. 1 25 67. Clocks, Watches and Bells, by E. B. Deuison 1 75 Appendix (sold separately) 60 78. Steam and Locomoticn, on the Principle of connect- ing Science with Practice, by J. Sewell 1 00 78*. Locomotive Engines, by G. D. Dempsey 75 4 79*. niostrations to the above, 4to $2 2S 98. Mechanism and Constriiction of Machines, by T. Baker ; and Tools and Machines, by J. Nasmyth, ■with 220 Wood-outs 1 25 Hi. Machinery, Oonstruotion and Working, by 0. D. Abel 75 115. Plates to the above. 4to 3 75 139. SteamEngine,MathematioalTheoryof, by T.Baker. 50 155. Engineer's Guide to the Royal and Mercantile Navies, by a Practical Engineer 1 50 162. Brassfounder's Manual, by W. Graham 1 25 164. Modern Workshop Practice, as applied to Marine, Land, and Locomotive Engines, Floating Docks, Dredging Machines, Bridges, Cranes, etc., by J. G. Winton 1 50 CIVIL ENGINEERINC, ETC. 13. Civil Engineering, by H. Law and G. E. Bumell. . . 2 25 29. Draining Districts and Iiands, by G. D. Dempsey. . 75 (With No. 30, DEArNAOE amd Sewaoe of Towns, 2 vols, in one) 1 50 31. Well-Sinking, Boring and Pump Work, by J. G. Swindell, revised by G. B. Burnell 50 46. Road-Making and Maintenance of Macadamized Roads, by Gen. Sir J. Burgoyne 75 60. Land and Engineering Survejring, by T. Baker. . . 1 00 63. Agricultural Engineering, Buildings, Motive Powers, Field Engines, Machinery, and Imple- ments, by G. H. Andrews 1 50 77*. Economy of Fuel, by T. S. Prideaux 75 80*. Embanking Lands from the Sea, by J. Wiggins. , . 1 00 82**. Gas Works and Manufacturing Coal Gas, by S. Hughes 1 50 82***. Water-Works for Cities and Towns, by S. Hughes i 1 60 6 to. Vai» »os*6as*, 117 SubterrEtueons Surve3ring, and Ranging the Lind without the Magnet, by T. Penwiok, with Addi- tions by T. Baker $1 25 118. Civil Xlngineenng of Korth America, by D. Steven- son 1 60 120. Hydraulic Engineering, by G. E. Burnell 1 60 121. Kvers and Torrents, and a Treatise on Navigable Canals and Rivers that Carry Sand and Mud, from the Italian of Paul Frisi 1 25 125. Combustion of Coal, and the Prevention of Smoke; by 0. Wye Williams, M.IO.E. . . . .' 1.50 SHIP-BUILDING ANC^ NAVIGATION. 51. Naval Architecture, by J. Peake 1 60 63*. Ships for Ocean and River Service, Construction of, by Capt. H. A. SOmmerfeldt 50 53**. Atlas of 15 Plates to the Above, Drawn for Prac- tice. 4to 3 75 54. Masting, Mast-Making, and Rigging of Ships, by B. Kipping 75 54*. Iron Shipbuilding, by J. Grantham. Fifth edition with Supplement ' 2 00 54**. Atlas of 40 Plates to the preceding. Folio 19 00 55. Navigation ; the Sailor's Sea Book : How to Keep the Log and Work it off, etc. ; Law of Storms, and Ex- planation of Terms, by J. Greenwood 1 00 80. Marine Engines, and Steam Vessels, and the Screw, by E. Murray 1 25 83. bis. Ships and Boats, Forms of, by W. Bland 75 99. Nautical Astronomy and Navigation, by J. B. Young 1 00 100**. Navigation Tables, for Use with the above. .'. .... 75 306. Ships' Anchors for aU Services, by G. Ootsell 75 119. Sails and Sail-Mp king, by E. Kipping, N. A 1 25 6 RJSttsflEU Atto IJttOSIS*. ARITHMETIC AND MATHEMATICS. 6. Mechanics, by Charles Tomlinson 75 32. Mathematical Instruments, their Construction, Use, etc., by J. P. Heather 75 61*. Ready Reckoner for the Measnrement of Land, Tables of Work at from 2s. 6 A to £1 per acre, and valuation of Land from £1 to £1,000 per acre, by Arman 75 7$. Geometry, Descriptive, -with a Theory of Shadows and Ferspective, and a DescFiptio'n of the Principles and Practice of Isometrioal Projection, by J. F. Heather $1 00 .85. Eqnational Arithmetic : Tables for the Calculation of Simple Interest, with Logarithms for Compound Interest, and Annuities, by W. Hipsley 1 00 86. Algebra, by J. Haddon 1 00 86*. Key and Companion to the Above, by J. B. Young 75 88. Euclid's Geometry, with Essays on Logic, by H. Law 1 00 90. Geometry, Analytical and Conic Sections, by J. Hann. 50 91. Plane Trigonometry, by J. Hann 50 92. Spherical Trigonometry, by J. Hann 50 (The two volumes in one) 1 00 93. Mensuration, by T. Baker 75 94 Logarithms, Tables of ; with Tables of Natural Sines, Cosines, and Tangents, by H. Law 1 25 97. Statics and Dynamics, by T. Baker 50 101. Differential Calculus, by W. S. B. Woolhouse 50 101*. Weights and Measures of all Nations ; Weights of Coins, and Divisions of Time ; with the Prin- ciples which determine the Bate of Exchange, by W. S. B. Woolhouse 75 102. Integral Calculus, by H. Cox 60 103. Integral Calculus, Examples of, by J. Hann 50 101. Differential Calculus, Examples of, with Solutions, by J. Haddon 50 1 b. Van liosTbAKB, poBUsBXfi AitD i&t^oitieB. 105. Algebra, Geometsry, and Trigonometiy, First Mnemosical Lessons in, by the Bev. T. P. Eirkman. 75 131. Ready-Reckoner for Millers, Farmers and Mer- chants, showing the Value of any Quantity of Oom, with the approximate Value of MiU-stones, and Mm Work 60 %* Copies of any of the above wiU be sml by mail, post-paid, on receipt of price. WEISBACH'S MECHANICS. NEW AND REVISED EDITION. A MANUAL OF THE MECHANICS OF ENGINEERINa, AUD OF THE CONSTEUOTION OP MACHINES. Bt JULIUS WEISBACH, Ph. D. TBANSLATED FEOM THE FOUBTH ATJOMENiED AHD IMPE07BD GEEMAN EDITION, BtECKLEX B. COXE, a. M., Misxsa Enginees. Vol. I.-THEORETIOAL, MECHANICS. 1 Voii. 8yo, 1100 Pages, and 902 Wood-out IhiTjsteations, PBiwiy ed fbou elieotbotype cofies of those of the Best Gebman Edition. $10. Absteaot OF Contents.— Introduction to the Caloulna— The General Principles of Mechanics — Phoronomies, on the Purely Mathematical Theory of Motion — Mechanics, or the General Physical Theory of Motion — Statics of Eigid Bodies— The Appli- cation of Statics to Elasticity and Strength — Dynamics of Bigid Bodies — Statics of Fluids — Dynamics of Fluids — The Theory of Oscillation, etc. D. VAN NOSTRAND, Publisher, »3 Murray Street and g7 Warren Street, %• Copies sent free by mail on receipt of price. 8