24. Counterbracing.—In the case of girders with braced webs, the tension bars of which are not adapted to resist a thrust, another circumstance due to the position of the live load must be considered. For a train advancing from the left, the travelling load shear in the left half of the span is of a different sign from that due to the dead load. Fig. 43 shows the maximum shear at vertical sections due to a dead and travelling load, the latter advancing (fig. 43, a) from the left and (fig. 43, b) from the right abutment. Comparing the figures it will be seen that over a distance x near the middle of the girder the shear changes sign, according as the load advances from the left or the right. The bracing bars, therefore, for this part of the girder must be adapted to resist either tension or thrust. Further, the range of stress to which they are subjected is the sum of the stresses due to the load advancing from the left or the right.
25. Greatest Shear when concentrated Loads travel over the Bridge.—To find the greatest shear with a set of concentrated loads at fixed distances, let the loads advance from the left abutment, and let C be the section at which the shear is required (fig. 44). The greatest shear at C may occur with W1 at C. If W1 passes beyond C, the shear at C will probably be greatest when W2 is at C. Let R be the resultant of the loads on the bridge when W1 is at C. Then the reaction at B and shear at C is Rn/l. Next let the loads advance a distance a so that W2 comes to C. Then the shear at C is R(n+a)/l-W1, plus any reaction d at B, due to any additional load which has come on the girder during the movement. The shear will therefore be increased by bringing W2 to C, if Ra/l+d > W1 and d is generally small and negligible. This result is modified if the action of the load near the section is distributed to the bracing intersections by rail and cross girders. In fig. 45 the action of W is distributed to A and B by the flooring. Then the loads at A and B are W(p-x)/p and Wx/p. Now let C (fig. 46) be the section at which the greatest shear is required, and let the loads advance from the left till W1 is at C. If R is the resultant of the loads then on the girder, the reaction at B and shear at C is Rn/l. But the shear may be greater when W2 is at C. In that case the shear at C becomes R(n+a)/l+d-W1, if a > p, and R(n+a)/l+d-W1a/p, if a < p. If we neglect d, then the shear increases by moving W2 to C, if Ra/l > W1 in the first case, and if Ra/l > W1a/p in the second case.
26. Greatest Bending Moment due to travelling concentrated Loads.—For the greatest bending moment due to a travelling live load, let a load of w per ft. run advance from the left abutment (fig. 47), and let its centre be at x from the left abutment. The reaction at B is 2wx²/l and the bending moment at any section C, at m from the left abutment, is 2wx²/(l-m)/l, which increases as x increases till the span is covered. Hence, for uniform travelling loads, the bending moments are greatest when the loading is complete. In that case the loads on either side of C are proportional to m and l-m. In the case of a series of travelling loads at fixed distances apart passing over the girder from the left, let W1, W2 (fig. 48), at distances x and x+a from the left abutment, be their resultants on either side of C. Then the reaction at B is W1x/l+W2(x+a)/l. The bending moment at C is
M = W1x(l-m)/l+W2m{1-(x+a)/l}.