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Thoaa too larga to ba ly includad in ona axpoaura ara filmad ling in tha uppar laft hand cornar, laft to md top to bottom, aa many framaa aa ad. Tha following diagrama illuatrata tha )d: Laa cartaa, planchaa, tablaaux, ate, pauvant Atra filmto k daa taux da reduction diff Aranta. Loraqua la documant aat trop grand pour Atra raproduit an un aaul clichA, II aat fiimA A partir da I'angia aupiriaur gaucha, da gaucha A droita. at da haut an baa, an pranant la nombra d'imagaa nAcaaaaira. Laa diagrammaa auivanta llluatrant la mAthoda. 1 2 3 1 2 3 4 6 6 (All Hghta reserved, J ADVANCE PROOF— (Siibject to revision). This Proof Is sent to y 'u for dlsou^on only, and en the express understanding that It is not to be used for any other purpose WhatSOever.~(>'^e« .See. so qf the ConsHtution.) INCORPORATED 1887. TRANSACTIONS. N.B.— This Society, as n body, does not Iiold itself responsible for the facts tncl opinions sta',ed in any of its piiblicntions. BRIDGE CALCULATIONS. By H. E. Vanvelet, M.Can.Soc.C.E. To be read lOth October. The opinion of many well known authorities is that it would be pre- ferable to use a distributed load, that would be safe for all existing types of locomotives in use on railways, and that would leave a margin for the probable increase of weight in the future. The wheel base of a loco- motive as well as the weight on each axle is limited by the radii of the curves and the section of the rail ; and although the weight of cars is constantly increasing, tiiurc exists a necessary relation between the en- gine and train weights. In general practice, the train weight is considered as distributed and the engine weight as concentrated. The author thinks that the weight of a wheel is always distributed by the rail and ties (more so in locomotives than in cars, owing to the lesser dis- tance between the wheels), and that both weights should be considered as distributed. It will always be necessary to use two different dis- tributed loads, and the equivalent distributed load will vary with the length of span. Furthermore the stresses, although calculated with the greatest care, are not the actual stresses in a .bridge, and frequently discrepancies, amounting to several thousand pounds, are shown during the erection. The general practice is to have the posts fastened by pins or rivets, al- lowing them to work as tension members, and the top chord is rigid, in- stead of having articulations at every panel point. It follows that the strains are not what they would be in an articulitcci system, where the posts could only take compression, the differences being more especially apparent in bridges with inclined top chords, which act partially as an arc, the posts acting as suspenders. Another cause of error is the use of stringers, rivetted to the floor beams, which act as pnrts of the top or bottom chords, as the case may be. It must not be supposed that the author is advocating free articula- tions with bolted stringers and slotted holes, as he believes that the actual practice increases the solidity of the bridges, and prefers to have stiffness in his work, at the cost of some uncertainty in his calculations. He would also follow in the lead of an eminent bridge engineer in the United States, whose trusses are rather light, a large quantity of ma- terial being used in the stiffening of the bridges, in the top and bottom laterals, sway bracing, and especially in the poitale, the last of which is certainly a very important part of a bridge. Most of the ac- tual specifications seem to be made for the perusal of outsiders more than for actual use, and it seems (to give one instance) that the rivetting foreman and inspector should know without being told, what the appear- ance of a rivet must be after it is driven. In treatises on bridges, written by French authors, it is always said that we must not rely too much on calculations, and the best that can be ."aid about their rules is that bridges built according to them and with a large factor of safety have withstood the to^t of time. Experience would sccui to show tliat, usually, the loiiiior tliospt^ciH. If instead of two weights V ami p a weight R is distributed The weights P and^ can then be replaced by a weight R such that orR = P-(P-^)(l-p^ The sliearing force immediately on the right of BD ^ YP(x + «1 -Y,) ^ MT _ Px' Nl 2N1 21 - rr.^ g'' - /^ g. ^ ^P + (»' - VP x ^^("1 - V) ^ (nl - V)^ 2N1 Nl Nl ^ 2N1 ^ Since M =• oj + n^ - V If the shearing force is to be a maximum with the condition ,._V(P -») + »!;) P»_VP + Mp jb — _ - or — ^— =— 1 Nl NP-p Y)[2NP7(l - yP(N- 1 )] + Nw^l''Pp 2Nl('NP-;j) For a uniformly distributed load the mix, shearin;; force = -"flL 2(N-1) Hence, that those two shearing; forces may be equal, NF-p \ ' '' \\-n\) NnT J By taking B=p-(p-i.)(i-]jy we have a close approximation on the safe side. The weights P and p can be replaced without material error by one weight R ; V being the length occupied by the distributed load P, and n the number of panels which must be fully loaded to give the maximum stress. 2° Graphic calculation of bending moments, taking every wheel into account. ' [ *» _ _• ^ ^ m m m k Jt. t^ 9m am ^ mm If a weight P is moving along AB, the bendin<^ moment at the point of application will be : V{l-x) h length occupied by the distributed load P, and n the number of panels which must be fully loaded to give the maximum stress. r 1 that -P rRl_ N-1) h s.nd P g the )aael8 1 into point To find the monieuts atD^ draw a Hcrlos of triangles ABC, A.Q.C, A.B^C^ etc., so tbiit AD = A,D,=A,D, = A^Do /».B = A , B , = A Ji^ -^ lenjrih of span I AA,, A, A.,. ..being the distances between the weights. As AAj A, = BBiB. =^ DD, D. it will be convenient to have those distances on a scale, and the whole series of triangles miiy be drown very quickly for every point at which the bending moment is required, and the sum MN + MN, f MN, will represent the moment at Do when the weight P. is ;it M. Now consider a motion of the wsights botw m two consecutive apices. Kach abscissa increas.;s or decreases in proportion to Jho distances moved, and it is nccciisary to reach an apex so that one of the increas- ing abscissa) may decrease. A ma.'cimuiu caa then oiily be reached at an apex, t. e., when one of the weights is applied at D^,. It will then only be necessary to consider the ordinates at the different apices, and a curve B:, b, b d;j ma. ai d a, A may be drawn whoso ordinates will repre- sent the bending moments at D from the timeP, enters the bridge until the I* leaves it. This solution brings forth a property of the parabola that the writer has never seen mentioned before, and whose limits can be enlarged by analytical demonstration. Draw the triangle AKB (fig. 6) KM = ?(^>x and VD= £(iz^)a = MN I Join (fig. 6) two points C and K of a parabola to the points of inter- section B and A, and of a perpendicular to the axis. The lengths VD and MN are equal. 3° Graphic calculation of stresses in the members of a truss, taking every wheel into account. This calculation is based on the two following theorems, for which the writer is indebted to Mr. Joseph Meyer, of the Union Bridge Co. To have the maximum stress in diagonal CB (fig. 7), the sum of the weights on the left of B including the weight at B must be larger than the sum of all the weights on the truss divided by the number of panels. To have the maximum stress in AB (fig. 7) the sum of the weights at the left of A, not including the weight at A, divided by the number of panel-: at the left, must be less than the sum of the weights on the bridge divided by the number of panels in the truss. A-^S>C+2)+E ^9 rij5 '»> >\+B+CtI> K M T The ordinates of the different points Bi C, etc., measure the moments in relation to B, C, etc., of all tlie weisrhts at the left, and if we want to find the values H x X / and Mi when M is at D and the ends of tlie truss at K and T|, Mi will bo measured by Di D^ and R X N ? by T, T, Remarking that for a distributed load the polygonal curve ABi CiDi becomes a parabola, we could find the demonstration of miiiiy interest- ing properties of tiic parabola. 4° Bending moments in continuous bridges of 2 spans. The max.imum bending moments at each point are given by consider- ing three cases of loading, viz., each span loaded and botit spans loaded. fi r- L.a -J A The maxima are then K'ven by the line and the two parabolas M,= ,VM?- (1) (2) (3) Km ? Let ij be the bending moment at r I'd" nliii d)i - ■'•(!' -7') ^(l^-/') (l-r)" ft ,X ^ Ij/ \i !j is to bj a luaxinium L. •'-'=; and • x — '> (\^\\^ "— TC-l)"(":")4] •"■^=[("-j)\'^)'']["H leiif^th oecupiuJ by the distributod loud P, and n tho number of panels which must be fully loaded to give tho uiaximuui stress. 2° Graphic calculation of beading nioin"iits, taking every wheel into account. . Ijfe-., f'"" If a woiirht V is moving along AB, tho bending moment at tho point of application will be ; m-x) Having drawn this parabola, the bending moment at a point D at a distance a from A will be : and CD being an ordinate of tho parabola for an abscissa a; = a CD = £(^«)« In the triangle ACB wc have : MN = CD?=iH =vtla=y l-a I 'IMu! bonding moment at D is then equal to tho ordinate of the triangle AC B at tho point of application of tho weight P. For anothev load Pi we should have another triangle AC, Bi :md if b is the distance between the two weights, MN + Mi Xj will represent liic bending moment at l> produced by tho weights Pand Pj. By sliding the triangle AC, Badis- tanee b to tlio loft, M i comes to M, and the bonding moment at D for any position of tho two weights is given by the ordinatcs MX + MX, of the two triangles. » The same reasoning will apply to any number of weights. To apply the method it is sufficient to draw the h parabola u = V ^ ' x to a convenient scale. ■^ I X A, i) i>/Ai M D This Formula has lieei, used, if the writer is not mistaken, by the Key- stone Bridge Co. To have the inaxiinuiii stress in diiigonal UB (%. 7), the sum of the weights on the left of B including the weight at B must he larger than the sum of ail the weights on the truss divided b)r the number of panels. To have tlie maximum stress in AB (fig. 7) the sum of the weights at the Id't of A, not including the ..u-Ight at A, divided by the numler of panels at the loft, must bo less than the sum of the weights on the bridge divided by the number of panels in the truss. MB->-C+J>*E h&<-D ^9 Fi' I ,' E/T^i^^- First draw the diagram shown in fig. 8, and let it be required to find the sum of weights on tru?s K T, and sum of weights at tlie left of M. The first sum be given at O b}- followinij^ the diagonnl E 0, ami the second sum at V by following the diagonal A V.- By moving the truss so that M occupies the different positions A B V, etc., it will be easy to find the worst situation of ^hc load, applying the theorems given tieforc. Now let Mj bo the bonding moment at M II the reaction at K Mi tlio lUDniont ill rolatinii to M of tho weights on the loft of M. Let I be tho panel length N '■ nuuiber of panels in truss loft of M MN: (RxNJ) ^ ""N -M. f as, to have the n)aximuni of .M N a part only ot M 8 is usually loaiK'd : we wart then only to know the quantities U ■< N /and M. To find those values wo will draw another diagram. 3 L.4 If a complete discussion were made, it would be found that for n length — from the centre, an hyperbola intervenes, increasing, the ne- 5 gative moments, and giviu<^ also positive moments as shown by double lines. But the results would not bo changed materially. If M^ is the bending moment produced by a distributed load p on » single span I we have a parabola MV= - pUl-z)-pJl-zY 2 2 /v. IX If An= UK the lines NH ami NK are M,= -P.^(Z-.)andM«=?g^(/-.) and it is easy to see that which gives an easy method to have the bending moments. The author thinks that the first formula given may have some practical value, and he would like to have thu opinion of bridge engineers about it, as well as the opinion of mechanical engineers as to the value to be given to the constants. The coefficients to be determined are Vp and V". lu the Canadian Pacific Ry. specification p is taken at 3000, and in calculations of many bridges the writer has taken P = 3730 and V= 105'0" and has found very little difference wiieii taking every wheel into account. For spans under 105 feel and over 21 feet he has taken P = 4600, iJ = 3240, V = 21'-0" and the formula becomes R = 3730 - 730 /I - 105y for spans over 21' R = 4600- 1360/ 1-^1 R.^4600 111 nl) '■ for spans over 21' for spacs under 21'