318 BRIDGES [FRAMES. Case 4. Warren girder; load on half the top joints, fig. 70. In designing a Warren girder it is necessary to provide for the advancing load. With a uniformly distributed load the GfJL i ij I. 70a. diagonal bracing consists of alternate struts and ties begin ning, if supported at the bottom, with a strut at each end, or with the tie at each end if the truss is hung from the top. At the centre there will be either two ties or two struts in juxtaposition; or, as in the case where there are an even number of loaded joints at the top, the central pair of diagonal members will be unstrained ; but an advancing load (neglecting the permanent load) will convert each diagonal member (except the end ones) alternately into a strut and a tie. It is necessary to provide for the maximum extension and maximum compression on each, taking into account the combined effects of the permanent and passing load ; the latter has generally sufficient effect to reverse the direction of the stress on the bracing near the centre, but not (in large bridges) towards the ends. Each member of the bracing towards the centre must, therefore, be so designed as to be capable of acting alternately as a strut and as a tie. The maximum stress on the top and bottom mem bers occurs when the bridge is fully loaded. This stress is easily calculated by the method of sections ; assume the girder cut by a vertical plane at the joint or pin opposite the member in question, or in other words, by a vertical plane pass ing through the vertex of the triangle of which the member in question is the base ; let d be the perpendicular distance between the member and the pin, t the thrust or tension on the member, and M the bending moment for the section calculated as for a girder loaded at the points corresponding to the joints. Then M = td. The stress on the diagonals corresponds to the shearing stress in the solid girder, and may consequently be calculated from the shearing diagrams, examples of which were given in 19. The loads may be referred to the joints before drawing the diagram, and the continuous curves of figures 18 to ISc will be replaced by lines consisting of a series of steps such as are shown in fig. 18a. We may then proceed to calculate the stress on any diagonal as fol lows : Take the shearing stress for the section at the joint where the diagonal in question abuts ; if there is a load on this joint, take the shearing stress for a section close to the joint, and on that side Fig. 71. of it which is next the diagonal in question ; call the-, shearing force thus found F ; let i be the slope of the diagonal or the angle which it makes with the horizon, then the tension or compression on the diagonal is F cosec. ? ; thus let fig. 71 represent a Warren girder of six equal bays, in which i zr60. Let the load on each top joint be 5 tons, the compressions and tensions on the diagonals are as follows : TABLE XT. Name of Brace. F x cosec. i. Compression. Tension 1 15 x 1-1547 17-32 2 10x1-1547 11-55
10x1-1547 11-55 4 5x1-1547 5-77 5 5x1-1547 577 6 Oxl-1547 These stresses, given simply as an example, .apply to the one special load, and are not to be confounded with the maximum stresses which an advancing load of the same intensity would produce. The stresses on the two halves of the girder are sym metrical. The arithmetical mode of computation is the simpler where the top and bottom members are parallel and the inclination of the diagonals constant; where these conditions are not fulfilled the method by reciprocal figures is preferable. In the actual design of any girder to suit various combinations of loads, care must be taken to design each member to suit the maximum stress which can arise from any combination. The maxi mum shearing stress is most easily selected by means of the dia grams 19. Care must be taken to meet both the maximum tension and maximum compression whenever the member is so placed that with some loads it is extended and with some compressed. We have just shown that this case arises in the diagonals near the centre of the girder. When frames are used as continuous girders, it is desirable to make the points of inflection coincide exactly with a joint. This may be done by cutting through or omitting the mem ber opposite the joint. This allows the reactions on each pier to be easily determined by the elementary principles of statics with any load, and without taking into account the form which the beam, assumes when deflected. Instead of a long continuous girder, we then have a series of girders, supported at or near the middle of their length by the piers, while a second series hang from the first by pins at determinate points. This arrangement greatly simplifies all the calculations without sensibly diminishing the advantage derived from the use of continuous girders. 55. Various Forms of Girder. The framed girder is sometimes made of the form in fig. 72, which has the Fig. 72. advantage of reducing the length of the end diagonals where the stress is heaviest. The reciprocal figure for this truss uniformly loaded on the bottom joints is shown in fig. 72. This girder is sometimes called a bowstring girder, though this name more properly belongs to an obsolete form with no diagonals. Z and X are the spaces between the end loads and the re actions which, for clearness, are shown as pulling up ; but the same reciprocal would result if the reaction were shown pushing up, and the letters X and Z were placed as dotted. This form has the advantage of reducing the compression on the diagonal struts to a comparatively small amount. A great part of the shearing stress is taken by the curved upper boom ; deeper girders can be profitably used of the bowstring than of the Warren type. The long and expen
sive struts in the latter form more than counterbalance