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TRANSACTIONS. N.B.— This SociPty, na a body, does not hold itself rPBpoiisible for the facts and opiiiioDS stated in any of its publications. CANTILEVER BRIDGES. -i (To he read mmSmmmktmm^ Jfumart/.^^ By C. F. Finuley, Can. A structure which is supported at one end of its length only, so that it reijuirea for equilibrium the application of other forces than the sup- port, is counnonly called a cantilever. If such a structure forms part of a bridge the bridge may be called a cantilever bridge. This descrip- tion agrees sufficiently well with the ordinary usage of these terms, and attempt at rigid definition would servo no good purpose. It is sometimes maiiitaiiu'd, that a cantilever bridge should be regard- ed and treated simply as a eoiititiunus-girder bridge, in which the points of contrary flexure are arbitrarily .uid peruiinently fixed by cutting the booms; but thisjioint of viev^ would exclude all cantilever bridges which are anchored at the abutments instead of being continuous with a neigli- boring span, and also it is both illogical and misleading to classify as a variety of continuous girders those in which the uiost prominent fea- ture is the breacli of contintiity at certain points. The cantilever is iu regard to the distribution of stress in it one of the simplest of framed structuies, and no one who is familiar with the analysis of stresses in ordinary bridge au 4 roof stresses, will find any difficulty in calculatitig the stresses in a cantilever bridge from given drawings and loads, difficulty, that is to say, inherent iu the cantilever system. There are, however, several features peculiar to the cantilever which deserve attention iu regaid to d'xlgn, and to these we shall confine our attention, taking it for granted that those points which the cantilever (las ill common with other bridges are familiar to our readers. The maimer in wliieli the cantilever is employed to form part of a bridge varies very much from one bridge to another. In some cases two similar cantilevers spring from neighbouring abutments in opposite directions, so tli.it one forms a counterbalance to the other except as regards moving loads. In otlier eases, a cantilever stands alone and is kept in eijiiilibrium by guys or backstays connected to an anchorage. Sonutinus an ojieniiig is spanned by two cantilevers with a central span resting on their ends, and sometimes by one cantilever with an independent span from its end to the other abutment. Examples oC tliise variotis types of cantilevers are shewn in the ac- companying diagrams, Plate... of which Fig. 1 represents the bridge now in course of construction over the Forth near Edinburgh in Scotland ; Fig. 2 represents a bridge being built over the Indus rivor in India ; Fig. ;^ represents the Jubilee ))ridgc crossing the Hooghly river near Cal- isiiiovi'i.: loiid.s. Ill otluT cases, ii cantilever stand.s aloiii! antl is kei)i in I'ljuilibriuin by fj;uys or backstays connected to uu ancliorago. So!m tinu'S an opening is spanned by two cantilevers with a central sjian restini:; on tlieir ends, and sometimes by one cantilever with an independent span from its end to the other abutment. Examples of th( so various types of cantilevers are shewn in the ac- companying diagrams, i'iato... of which Fig. 1 represents the bridge now in cours*! of construction over the Forth near Edinburgh in Scotland; Fig. 2 represents abridge being built over the Indus river in India; Fig. 3 represents the Jubilee bridge crossing the Hoo^"' ' nver near Cal- cutta in India ; Fig. 4 represents the bridge over the St. Lawrence below the Falls of Niagara. The great difference between the parts played by the cantilever in those different designs, is mainly accounted for by the character of the sites of the bridges. The reasons that may load to the adoption of the cantilever-bridge in preference of any other typo for certain situation are two : — First, where the span is very gi cat, the cantilever bridge generally requires less material than any other rigid structure of equal strength, even though anchorage may liave to be provided. Where two large spans have to be built, a double cantilever requiring no anchorage, uu»y effect a very considerable saving in material, thougli it must not l^e forgotten that in this case, u double pier of sufficient width for stability under all conditions of loading, is ncces.^ary. Secondly, where f dse works would be iuipossible or costly, the valu- able property of the cantilever — that it can be made to suppport itself during erection — gives it an immense advantage. If the design of the cantilever be such that it can be built out rapidly and cheaply, it will often be the most economical form in the end, even where its total quan- tity of material is nob so small as that required by some other type of bridge. In all ciigineering work quantity of material is only one of the elements, and it is of the greatest importance to bear this in mind in designing a cantilever bridge, because a want of regard to the method of erection, may easily add to its cost an amount much greater than can be Javed by economising material. The principal points which arise for consideration at the outset, in commencing to design a cantilever bridge are these : (1) The depth to be given to the catitilever at the abutment ; (2) The outline the structure is to liave in elevation, or in other words, the rate at which the depth is to vary in passing from the outer end of the cantilever; (3) The length of the panels into which the cantilever is to be divided by the bracing, and whether this length is to be uniform or not ; (4j The length of the cantilever in relation to the entire span. The first three of tliese points are intimately connected with one another, and none of them can be settled by a mathematical investigation applicable to all circumstances. The theoretical investigations which have been made on questions of this character are all vitiated (as regards their application to cantilevers) by the assumption that the load on the bridge is uniform throughout its length. This assumption is nearly enough correct for ordinary truss bridges, because the web is heavier where the booms are lighter, and so the weight uf the truss is dislriouted along its length with an approach to uniformity, and nlso because for all ordinary spans the weight of the girders, is considerably less than that of the floor and the moving load together, as that a small variation in the load per foot, due to the main girders is of less importance. With a cantilever, however, especially when used (as it generally is) for large spans the case is entirely diflFer- ent. The web and booms arc both heaviest at the abutment and dim- inished simultaneously on travilling away from the abutment. Also the weight per foot of the main framework of the cantilever, where greiitest, will in a large span be generally much greater than that from all other causes. When it is also considered that the cantilever as a rule, has to carry a heavy coiiconti ated load ou its end, it is obvious that all general conclusions founded on an assumption of uniform weight must be fallacious. As an illustration, it may be mentioned that the cantilevers of the Indus Bridge (Fig. 2) weigh about 1 ton per foot at their outer ends, and about 6^ tons per foot at the abutments (apart from the uniform load of the floor wiiich is about 4 tons per foot), and they are designed to carry a concentrated load of 300 tons at the end. Here we sue how misleading it is to regard the entire span as one con- tinuous girder bridge with fixed points of contrary flexure, because when the continuity is destroyed the central part is just as much an independent span as if it rested on stone piers. Whatever proportions would load to the greatest economy in an independent span of the same length should a fortiori be adopted for the central span of a cantilever bridge, because any extravagance in the central span has to be paid for over again in extra weight in the cantilevers. Jfalf the benefit of des- troying the continuity is lost, if we do not take advantage of it, to design each part of the bridge in the way most suitable for that part. In a comparison of the weights of girder bridges with those of cuntilever bridges, the concentration of weight near the abutments in the latter is an immense advantage on their side. The marked eft'eet of taking into account the varying weight of the cantilever is very well seen, if the theoretical case be considero''. in which the web mombe boom OX being horizontal^ let X y be the eooidlnates of any point P in the other boom. Let S be shearing force at 1' in tons. " jNF be bendino' luoincnt at V in foot-tons. irried by tlie cantilever, where C and D are arbitrary constants to be deternnned by the known conditions of the caso. Since x = o when y = o, we have 1) = o, dy T and since sheariii"- force at end of cantilever = L, we have II .= ^ " djr when X r= 0, and y := o, or C -= L, so tliat the eqnation becomes, lly-(m + 2KH) f + KIl ^^ + L x or x'' (m + 2 KII) + y^ KH + 2 U - 2 Hy r_-. This is an ctjuation to an ellipse, whose major axis is vertical, the ratio of the S(|uares oftlie axes bein!; 2 + i^„ The vortical tangent ' Kir to the ellipse at the end of its minor axis touches the curve at the point where y = j^^ so that the depth ol th(! longest cantilever that can be built with these loads, is \^ at the abutuunt, and is independent of m^ II and L. It will be noticed that the ftirm of the curve is independent of a, the length of the cantilever, when FT has been chosen. It is possiblv! to determine the value of H, which for any given length of cantilever of this elliptie form, will make the total weight a minimum, and this would be a method i f detennining the depth at tlie abutment, but such an investi nation would be men Iv a mathematical exercise- under hypotheses so remote from the facts as we have hitherto assumed, and results in a dei)th which is obviously too great for practical ccono my. in the first place diagonal bracing must be supplied to meet the case of moving loads, and as tlu; (h;ptli is increased, this bracing becomes heavier. Secondly, all secondary br:icing becomes lit;avier as the depth increases. Thirdly, all compression memliers in tlie web will have a smaller allowablo stress pei s(juare inch as the depth increases, owing to their greater length, and so tluir weight i>. increased. Fourthly, the cost of erection will in general be incroased by employing an excessive depth. Fifthly, a greater dcptli would properly be chosen for^a canti- lever where anchorage had to Ik; provided, than where two adjacent can- tilevers counterbahineed each other, owing to the greater depth reducing the material re(piired for guys and .un-liors. For all these reasons, it will be seen that it is useless to attempt to calculate the proper depth for a cantilever without a very extended investigation, for which the data are not at present available, hut both theory and experience point to tile advantage of choosing a greater depth than would be chosen lor I a girder bridge. An ordinary proportion for a large girder bridge would be jth to -ith of the sjian, and if for the girder were substituted two cantilevers im-eting in the middle of the >pan, the depth might witli a>e kept within such limits as will admit of reasonable proportions in Ocher respects, while tlie dia-jonal ties or struts may be allowed to vary in inclination to some extnt from one panel to another. ':ho eflfect must not be negleet'^d which the choice of panel length has upoa the weight of tlie floor system, sinoe it is just as necessary to observe economical proportions in that part of the bridge as in any other, and since it is obviously undesirable to attach cross girders to any part of the main trusses except where they are directly supported by the web members, if it can be .ivoided. Tlie principle to be observed here is as follows (and has ni.;s<) far as we know been stiited before), tliere is a minimum thickness for the wib of a plate girder, beyond which it can- not bo reduced however siu:iH be the stresses in it, Wliat this minimum should be is a matiei' of opinion, but havin- decided on it, tlic cross girder should (if otherwise piactieable) have a web of this thickness, and the spacing of the cross girders should be such as to take full advantage of the strength of the web, that is to say, the spacing should be such as to bring on to each cross girder 'i load that strains the metal of the web up to its limiting permissible stress in that position. The limit will generally bo reached in the bearing area ot the rivets through the web, before it is reached iu tliesliearing section olthcweb itself. If the spacing of the cross girders bo wider than this, tlie total weight of cross girders will remain unchanged, supposing them to be accu- rately proportioned to the stresses, because the load on eat-h cross girder increases directly as the space b^twefn tlum. and the number of cross girders diminishes as the inverse ol' the s;ime ratio. But while the total weight of cross gifder rcuiaiiis unchanged the weight of rail-bearing beams increases rapidly as their s})iiii increases, so that the total weight of the floor will be increased, it' the interval between cross girders be greater than that fi.Ked by the above principle. I Jut it should be observed that the proper spacing for cross girders thus arrived at is greater as the length of the cross girder is greater and in direct proportion to it. For the depth of the cross girder will be a constant fraction of its length, and therefore the safe load on a web of a given thickness will be proportional to the length of the cross girder. This is important for the reason, that it is not unusual to make can' ilevers of a tapering form in plan ; i.e., with width diminishing from the abutment cutwavds, to uive isreater stiff"ncss against wind. If this be done, the cross girder will vary in length and will be longer where the cantilever is deeper (if the depth vary), and we thus find that where wo wish to use longer panels in the main trusses, we shall promote economy in the floor system, by using them. The qualification must be observed, that when the cross girder reaches such a length as to make a framed truss more suitable for it than a plate girder, the principle above set forth ceases to hold good, because the wehM of the cross girder is then approximately propor- I 'I that of the cantiluvors, will j^oiiorally Ir^ voiy lar,u'ft ocunoinical, and if that proportion Ciinnot bi; usod, it is (ksirablo to know how the total (|uantity of material in the bridge wi I vary with the t'ifferent proportions that it may be open to \i< to select. Tlio umthematical analysis of this problem in general is sun oiiuded witli great difficulty, unless hypotheses arc made wiiich remove it out of the region of practical utility, but lor i.ny such type of cantilever as that of the St. I.awrence bridge, the following method will give all the desired information with sufficient accuracy for the pur- pose. The general features of the design are supposed to h:ive been provisionally settled, and it is only required to determine how the total weight will be iitfected by varying the length of the cantilever portion of the bridge. 'J he vertical loads oidy need be c of the weights of cantilevers of various lengths differing successively by ' i one panel. The labor in doing this is not excessive, because we do not ' require results correct in thcmselvts, but only results w hich are compar- ; able with each othev and relatively correct, and it is not necessary there- fore to have greal exactness in the permissible stresses per square inch, in compression members, or in the percentages to be added for covers and secondary braciuLrs and so on. A concentrated moving load is easily dealt ! with, as its effect on the who'e of the boom is greatest when it is at the , outer end of the cantilever, and on any member of the web when it is ' concentrated as nearly as possible above tlie outer end of that member. Thus the process of getting at the weights of cantilevers of various lengths loaded as in {a) and (b), is merely one of sucessive summation, one panel after another being adde 1 and no calculations repeated, as they would have to be m independent girders of different span. Next (c) let the loads be calculated which arc imposed on the end of the cantilever by central spans of different lengths. This load is made up ■ of half the dead weight of the central span and of more than half the Tiiovin'r load f i t^luMnouiu^ua^^ the reason, that it is not unusual to make; cantiltnor.s ol' a tapt riim toiin in plan; i.e., with width iliniini.shiiig from tho abutunsiit i utwaiils, to j^ive greater stiffncs.s afj;ainst winJ. If this be done, the ero.ss ^irdor will vary in length and will be longoi where the rantilever is deeper (if the depth vary), and we thus find tliat where we wisli to use longer [.aiiels in the main trusses, we shall proinoto eeonomy in the floor system, by using tliem. The qualifieation must be observed, that when the cross girder reaches such a length as to make a framed truss more suitable for it than a plate girder, the principle above set forth ceases to hold good, because the weight of the cross girder is then approximately propor- tioned to its load, and thercibre the same per foot run of the bridge, whatever the spacing may be. (We are here neglecting the consideration of the effect of the weights on separate axles of :i locomotive vhich would modily these conclusions where the cross girders were very close together). The panels should in no case be allowed to get too long, not only for tlic sake of beping down the weight of the floor system, but also because the difficulty of erection by building out will be vtistly in- creased, and also becausi; it will entail strengthening of tho booms to resist the local action of thoir own weight. Although it is impossible to lay down general rules, owing to the ex- tent to which the circumstances of diff"eront bridges vary, yet it may be said, that when the conditions admit of it, tho type of cantilever that will generally prove the cheapest, taking intoaccount facility of manufacture and erection, is the truss with vertical posts, with one horizontal boom and with one boom inclined and either straight throughout or at any rate with few imgles in it. The St. Laworence bridge (Fig. 4.) is a good example of this typo. If the lensional boom be horizontal, the diagonal ties are shorter, while if the cnmpressional boom be horizontal, the weight ot the two btonis together will be somewhut less. A Ibrm more pleasing to the eye is obtained by making the lower boom curved. By this means the wib stresses may be largely reduced, but the increased length of the web members and the additional weight in tlie lower boom, will usually mure than neutralize this advantage. The expense of manufacture will be also incrensed by a curved boom, and therefore it is not to be recommended except from asthetic considerationSj which are of course in certain cases of the highest importance. If a curved boom is used, it should be sufficiently flat that each length of boom from joint to joint can be made straight, without uiarring the out- line, and the central spiiU i>houl(J, if possible, have a curved boom of continuous curvature wiih il at ef the cantilcvir. Theles^tway of effecting this is by employing a hinged braced arch (as proposed by Mr. C. B. Bender^) instead of a truss i.t the center span. The hinged arch by means of a temporaiy connection with the tensional boom of the cantilever can be built out without false works in exactly tho same way as a trussed center span can be. The effect of substituting an arch for a truss in the center span is to increase largely the stress on the compressional boom of the cantilever, and at the same time to relieve tlie stress on the tensional boom and the anchorage. Care must be taken that under no circumstances c .n the stress on the tensional boom be reversed. The proportions which the length of the center spanf is to boar tl wlitii it is at the outer end of the cantiK:ver, aitd on any nu'whtr of the web when it is coiicentrig,c(l as nearly as possibli; above th*e outer end of that ujeniber. Thus the prdcess of getting at the weights of eantiU'vcrs of various lengths loailcd as in (), is merely one of sucessive summation, one panel after another boing ad' I and no calcul.'itions repeated, us they would have to be m independent girders of different span. Next (c) let tlie loads be calculated whieh arc imposed on the end of the cantilever by central spjms of different lengths. This load is made up of half the dead weight of tlie central span and of more than half the moving load (if the moving 'oad is not assumed as uniform). Finally make a list of the probable weights of material in tlie niiiin trusses of center spans of various lengths, taking care that the weights correspond to the kind of truss which it is actually intended to use, ttnd that they include only t-ueh parts as enter into the estimates previously nuide of weights of cantilevers. This list can be obtained from weights of e--:isting bridges, by estimating for one span in the same manner as the cantilever was estimated for, and thiiu comparing the result with the usual weight of bridges of the s;ime span and type, which will give a factor for reducing the usual weights of other spans so as to make them compaiable with the cantilever weights. We now have material for filling up a Table such as the foUowittg : 12 3 4 6 6 7 8 ■u § c .i 4. ci 1 •io^ .i a « if od 4) r. e o g* Cantilev ntral sp n it.inhu of tons. Can ry 111 load s. a Q a'-H bo biis B S tl - *- ^ <" ® a. " = -*.B o o O o a a B t; C P O ,n t; a to w " *j « 8J 0) OS Load from c resting dred Weight lever fc dred to it Weigh lever r central pB 'Z is Weigh lilever tral sp ft. ft. 300 290 20 280 40 270 60 260 80 250 100 240 120 230 140 &c. &c. 1 We have here supposed the Table to >e formed for a tot d span of 6C0 feet, which is to be made up of 2 cantilevers and a central span resting on them, and we liave supposed it to proceed by differences of 20 feet in the centre span. All the rows of figures in such a Table as this need not necessarily })e calculated. Some may be filled up by interpolation. Columns 4, 6 and 3 are the results of the preliminary calculations (a) (b) and (c) described above. Column 5 is obtained by multiplying the figures in column 3 by those in column 4. Column 8 is the desired total weight and will show the relative economy of different arrangements. The figures in column 8 will not of course, a<'-rec with those of an actual bridge, but the differences between them will, if the calculation is properly made, be the differences between the weights of actual bridges, because in the estimates, everything will have been considered which could cause a difference to arise. The Table given above by way of illustration, must of course be luodified to suit the circumstances of different sites. For instance, instead of two equal 5 cantilevers and a contral span, ii ( isc liko iIk; Hoonlily Bridge (Fig. 3) may nrise, wlu're tlion- is lur viwU opciiiiin one cantilever and a rthore 8pan. and in wliieh the antount of material between the piers, necessary to resist the hending nmnients at the i iers, must enter into the contpari- son. Attain, where iiiichoranv has to he provided, another column of weij;hts must In; added ff)r the weights of hackstays and anchors, and must l)c summed alonj^ with the weights of otlier parts of the brid'je. All this calculation may appear somewhat irksome, but in tlie case of a laru,e work, extensive e.ilculations can be afforded if their results air trustworthy; ami when the alternative is only to act on irrational annlooies fnmj other works, or on imperfect formulfo, a considerable amount of such preliminary labor should willingly be undergone . The weight of anchorage material necessary, is proportional to the binding moment nt the abutment and inversely as the depth of the cnntilever, other things being ei|ual. If the backstays are straight and terminate at ii given level, the most etononiieal nnglc for them is 45 '^ , regarding only the section necessary in them to meet the stress, hut by making the angle with the horizontal a little less tluui 45 ® , a certain amount of material may be saved in the joints of tlie backstays and also in the anchors, which more than cnnipensates the increased weight of the backstays themselves. Where other spans have to be built a'ijaoent to a largo cantilever span, it should not hastily be assumed tliat the proper plan is tieces- sarily to counteibalanee the cantilever by an adjueent cantilever in the opposite direction. Where good foundations exist at\d piers would not be expensive it may be cheaper to build a number of short independent side spans and anchor the cantilever independently of tliem. If this is done earc must bi' taken to provide in the abutment for the un- balancetl thrust in the lower boom of the cantilever. In estimating for a centre span, resting on two cantilevers, it must not be forgotten that every ton of weight in it has to be paid for again in extra weight in the cantile\er and, tlierelbrc, it is justifiable to incur greater expense than otherwise would be advisable, where weight can thereby be saved, as for instance by using tlie highe.-t class of material both for trusses and floor, and tlio liiilitest forms of eoaneetions. The joint between the central spin and the cantilever re(|uircs care- ful consideration, the coiniection should be such as to fulfil the following conditions : (1) Moth cantilevers must be free to expand and contract from change of temperature. (2) The wind pressure on the central span must bceijually borne by the two cantilevers. (3) There must be longitudintd support for the central span to resist the effect of a braking of a train on it or of a wind blowing diagonally. (4) The coiuH^etion at both ends nuist have sufficient lateral rigidity to check undue lateral viliratioii. Conditions (^1 j and (3) would be fulfill* d by su[)poiting the ecntnil sjian like ;iny ordinary bridge truss, on a rocker bolted down at on.' end and on a rocker resting on ((xpansion rollers at the other, but this would not satisfy the second of the condi- tions. A better plan would be to supjiort the span by rollers or links at both ends and to secure the centre s]ian to one cantilever only by a largo vertical pin on the centre line of the bridge adapted to tr.insfer all the lateral shearing force. A similar pin ;it the other end free to move in an elongated hole wonl'l :ilso lie the ninst satisfactory way of transmit- Ivb '•h [TUiH uietliod applies of oourso to otlier structures equally mth ean- tiloviiFH. Ill tliu cawii of iiidcpoiukiit missos the tlffonnution would he plottud coiunuiioiii^ from tho middle i.f tlio brid<,'c, wlu-rc uiidor a .syiumuti'ioal load tlio booms arc liorizoutal.] If thu aualytii-al nutliod Im" uwd, lot AIJVX ( I'ij,'. 7) bo ono paiiol of a canllluvcr whoso l)oouis AH mid X Y make rospcctively aiiulcs a® and.i® witli tho horizontal. Let 1,. 1^, 1,, bo the dirttancos rof^poc- tivoly of X, IJaiid Y from tho vortioal through thocstromity of the c:in- lilevcr. Tiicm if tholonj^'th AH of tho top boom, be expanded by an anjouut Kj x AB tho vortioal dofloction ut the end of i tho cantilever duo to this change, all the other uicuibors of the cantilever re- uiainin}; unaltered in length, is: — KrABl, Ki«. 7. (2) BX sin ABX K,XY Similarly for the length XY (»f bottom boom contracted by an amount (3) ■J^\ .^111 1'^*. i . (4) „^ = K4§^^;^^--l3(cotYnC-cotBXY)} K,.XY.l, BX.sin HXY : For the ^liagonal tie ]}Y : BYcos i ^ — —-^ t;^n MXY For the diagonal strut BX (5) ,,^K, ;:f-;;y-uMi,xv-c..ABX)} These expressions arc here given in tlieirmost general form. In any particular cnsc they will h'. Uiuoh sin.plitied. For instance, when the Looms are parallel (4) and (J)) consist of their first term only. If in addition tho strut BX is vortioal, sin ABX in ,5, and t liivi siirti('iriit luU'rul liu'ltlity to ciicuk iimliu' l:it( lid vilinitidii. ('iiiiililiHii> ( i / ,iiiil (li) woultl liu tultilliil liy su|)|iiii tiiip till' cciitcil ^]iitii likr .-niy (iiiliiinry liii(li;tt trusM, on a rocker iMtltcl liowii ;it nii rnd aiMlon a locki'i' resting; no cxpinisinti rolii Is iit tlitMitlnr. l»iir lliirt wmiM not >ati.'>ty tlit; sriroml of tlic coiidi- tioiiM. A bottiT |)l;m would Ik- to ,-u|i|iort tlic^-pan l»y rolli-rsor links at both cndH iiml to st'ciii*' tin- niitif >|iiin to one ciiniili'vcr only by a larj^o vertiuul pin on ihc centre line ol' tin bridijc^ adapted to tiMiister all the Jati>ral Hhcarinu; f'oice. A similar jtin ii the other end free to move in un elongated lioh; woiihl also ho the most satisfaetory way of transmit- tinj^ the Hhearini; I'oice at tli;it end also. 'Ihv points ot'eontrary flexnrc of the whole bridge umler wind pressun; would then he fixed, and all unoortaiuty as to wind storms removed, Tlu; plan adopted in the Indus Bridge, is to haveoxpinsion rollers at hot h ends, and to eonncot the span to both cantilevers by horizontal bolts alongthe centre line of the bridge by which an initial compn ssion is jiut on rubber s)trings enclosed in cylinders. These bolts do not transmit any shearing force, but only ferve to make the expansion e((nal at both ends, and remove theneee.ssity of bolting down the span to the cantilever at one end. The springs are not stift' enough to put any s(;rious additional strains on the bridge under the greatest clmnges of temperature. The study of the deflections and deformations of bridges has been too much passed over in works on the subject and ms it is one of great im. portancc in relation to cantilev«'is we will devote a little attention to it. It is advisable to ase(>rtain if possible the nature of the deformations which a bridge will undergo under its loads, before building it, and njt to depend simply on observjitions taken when the bridge is tested, because a study of the deformations may give valuable hints as to improvement in design. Whether the joints are hingod or rigid, it is permissible to treat the structure as a iraUiework with frictionless hinged joints in the first instance, as is universally done iti calculating the stresses in the various members. With this assumption, the deformation may be calculated either graphically or aiiiilytically. When; the angles between the web members and the booms iire not constant, the graphic method istlieniost rapid and in any case wo\ild be useful to check calculation and as giv- ing a general idea of the defoi inations at a glance. It rests on tlie fol- lowing propositions : — (1) *If a diagram be drawn representing a given hinged framework on a scale of ;};, and if starting from :i fixed point and a line of fixed direc- tion in the framework, the diajiram be redrawn with its lines increased or diminished in length, by the full-size elongation or compression of the various members of tlie framework, then in comparing the two diagrams thus obtained, tin' change nf position of any point in the dia- gram is the actual movenuMit \iiidor strain of the corresponding jioint in the atructurt!, and the change of angK' o<" any liiu^ in the diragram is m times the change of angle of ilio corresponding line in the structure. The only provist) is that the scale of the diagram must be sufficiently hirge thiit the elongation or cumiircssion of any niemb(>r may still be a small fraction oi' the line on the diagram affected by it : the fraction may be l-20th without seriously impairing the accuracy of the results. * This proposition ami the utiulytical expression iielow are given so far as we know, for the first time, though they are likely to have occurred to others. The geometrical proofs of them as also of the equations (2) (3) (4) and (5) are simple. i„„.ts^ivu. I.y tl..: •iii.^ii' r,l..|unualiou in u Inch iT.o joints wm^ ti-iatcl uH l.iii-.d. It will not U' accii lately so, boi-iiu»c at each joint the web iniiinbcrs and tlic boom act on cucb othor with niiial and opposite couples, ami also tlie .urvature of any web member, by Hhort«ning the aiHtan.J' between its en.ls, ten«ls to elevate the lower boom nn.l deproM the upper, but the ilVeet of these forces will not modify the form of the booui to any f-iuit extent ec.mpared to the whole defornmtion. It follows tlien from the similarity of the figures that iu the two can- tilevers compared, the radii of curvature of the booms at correHiHindin.u; points arc as m, : ui,. Let /, be the radius of curvature, r the distance from the neutral axis of the boom to its most strained edge, t the stress per Hq. inch, at that ed-e due to flexure, an- cot '0 . Suppose now o to hv, the angle between the booms (supposed to be straight). Then <■■ =: 3 -i O and a = 2 k cot (,^ + o) + cot ,5. If then ,'? and <5 be given A is a maximum when the booms are parallel and diminishes continually as o increases, thus showing that the distor- tion of the booms is less for a cantilever of varying depth than for one of unilorm deptli. For a given value of e, if J and 6 vary, it will be found that A is a 7 Fig. 8. I Fif,'. 9. uiaximnm when B is tho ))oiiit ol' contiict of a circle tlirouyh XV touch- ing tilt! line A 15 ^ vanislios wlion " = - — >', Till' W((li nioiiilKTs. liowt'ViT, Imve ouch a b»'n(liii<; moment at tho «"ud and tlusrt! muHt 1x3 a ciMiph; itctiin; on the boom, (((ual and oppo.sitn to the sum (or dift'oroiiee) (»!* the eonples aetiiij; on th<; ends of the two weh inonihers nieetiny at the point of the bomn inijuestion. Thus, when tho form of a frame with frictionKjis hiimes under strain would be th(! Figure as shi»wn in dotted lines (the booms being straiglit because cot II + cot (5 = ()) ~ then if the joints be rigid, the booms und webs would In; distorted as »hcwn in full lines, all an- gles at every joint being tlie same as those in the original figure. H', however (as is generally the ease in pin bridges), the booms (or oik^ of tluMn) be continuous and the web members be hinged, then neglecting friction of pins, in such a case when cot a \- cot d = o, the boom which would suffer distortion if cot ti + cot ('i were not zero will remain straight under strain. This haj)- ptiis in the finnextul fig- ure when ang. VVAI{ = ang. AXW. In practice only an approximation to nuh a condition could be realized because the panel c(»uld not be Mllowcd to vary so r.ipidly in length as constiint angles for bracing would necessitate. Hut if for ties AX and IJY, struts B\V and ('X were substituted, it »"ill be seen tli:it cot <, -\ cot <> becomes umeh larger at once, :md the strains on conipri'ssion boom, which is nccessirily continuous, will be much greater. If B li(! outside the two positions on At) given by n = - — ' , the cur- vature of the lower boom will be upwards because A becomes negative. Tako the case of a girder of uniform depth, n = /i and - = d Let d = depth a = length of panel a = d (cot j3 + cot (5) .-. A 2K a , A _ 2K Fig. 10. (1 01 — — — r a a Therefore the deviation of the boom per foot of length is constant and independent of the angles of tho br.icings. It is proportional to the stress per s(|. inch and inversely proportional to the depth. Attention should be directed to the deformation of the lower bonni of a cantilever or continuous girder at the abutment. Tf a diagram be drawn, as reconnncniled above, to a'^certain the general - tiiiK's sm-li a."< to 'iiv«' risr [o vrry m rinii- >fC(»iiilaiy strains in tin- li strain, an assump tion which is plausible enough for beams, but can have no relation whatever to the flexure of a biniit'd frauiework. We have seen how(!ver in e(|uations (4) and (5) how tlie deflexions are affectiid by the shearing stresses, and in that vi»^w it need not be surprising that u sudden an 1 great ehniigc in the shearing stress causes great local deformation. 'I'lie remarks apply e(ju;dly, of course, to con- tinuoua girders as to cantileveiH, and point to the desirability cd' hinging the lower boom at tlu; abutment when practicable. The calculation of drflcction in a cantilever may sometimes be facili- tated in the following manner. Suppose, by way of illustration, that the Mtresses on all the members of the bridge in tension are not far from 5 tons per sq. inch of gross section and that the compres- sional stress range from a to b tons. 'I'he form of thi; bridge would be un.'iltered by an ccjual compressionable stress p(!r square inch applied in all members alikt', for this would simply m;ike the bridge shrink in all directions alike, a similar result to that proilueed by I'all (d' temperature. Imagine then an initial eompressioii of live terns per s(juare inch applied to all members. This produces no angular deformation, but only a »hrinkage ecjuivalcnt to altering slightly the ;-e;dt; of the drawing, ^'ow combine this uniform stit^ss with the actual stresses in the bridge and we have the teiisinnal strcs-es so nearly neutralized, that tliey may be neglected, and the compressional stresses varying from a + 5 to b 1 f) tons per square inch. By this means we are saved the labour of calculating the ett'ect of ehangv of length in half the members of the bridge. If this nuiditication is applied to the graphic uu^thod, the resulting diagram will, in its angles, be unaffected by the moditica* tion. 'I'o liave the linear deformations correct, the whole diagram of deformation must bi; supposed expanded by the amount of the initial strain. If lor instance the ahutnientbc d ft below the end of the can- tilever the deflection given by diagram or eah;ulatiun will be in excess to the amount of K-d-t where k is the compression in a foot of length due to a .stress of 1 ton per sq. in., and t tons per sq. in. is tlie initial compresBiou suppo.scd to have boeu applied. 8