Fig. 45.  Action distributed by flooring. Fig. 45. Fig. 44.  Shear when concentrated loads travel over the Bridge. Fig. 44.

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 W at C. If W passes beyond C, the shear at C will probably be greatest when W is at C. Let R be the resultant of the loads on the bridge when W 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 W comes to C. Then the shear at C is R(n+a)/l-W, 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 W to C, if Ra/l+d > W 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 W 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 W is at C. In that case the shear at C becomes R(n+a)/l+d-W, if a > p, and R(n+a)/l+d-Wa/p, if a < p. If we neglect d, then the shear increases by moving W to C, if Ra/l > W in the first case, and if Ra/l > Wa/p in the second case.

Fig. 48.  Series of travelling loads. Fig. 48. Fig. 47.  Travelling live load. Fig. 47.

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 2wx&SUP2;/l and the bending moment at any section C, at m from the left abutment, is 2wx&SUP2;/(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 W, W (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 Wx/l+W(x+a)/l. The bending moment at C is

M = Wx(l-m)/l+Wm{1-(x+a)/l}.

If the loads are moved a distance ∆x to the right, the bending moment becomes

M+∆M = W(x+∆x)(l-m)/l+Wm{1-(x+∆x+a)/l}

∆m = W∆x(l-m)/l-W∆xm/l,

and this is positive or the bending moment increases, if W(l-m) > Wm, or if W/m > W/(l-m). But these are the average loads per ft. run to the left and right of C. Hence, if the average load to the left of a section is greater than that to the right, the bending moment at the section will be increased by moving the loads to the right, and vice versa. Hence the maximum bending moment at C for a series of travelling loads will occur when the average load is the same on either side of C. If one of the loads is at C, spread over a very small distance in the neighbourhood of C, then a very small displacement of the loads will permit the fulfilment of the condition. Hence the criterion for the position of the loads which makes the moment at C greatest is this: one load must be at C, and the other loads must be distributed, so that the average loads per ft. on either side of C (the load at C being neglected) are nearly equal. If the loads are very unequal in magnitude or distance this condition may be satisfied for more than one position of the loads, but it is not difficult to ascertain which position gives the maximum moment.

Generally one of the largest of the loads must be at C with as many others to right and left as is consistent with that condition.

Fig. 49.  Beam with series of travelling loads. Fig. 49.

This criterion may be stated in another way. The greatest bending moment will occur with one of the greatest loads at the section, and when this further condition is satisfied. Let fig. 49 represent a beam with the series of loads travelling from the right. Let a b be the section considered, and let W be the load at a b when the bending moment there is greatest, and W the last load to the right then on the bridge. Then the position of the loads must be that which satisfies the condition

x
line
l
greater thanW+W+... W
line
W+W+... W
x
line
l
less thanW+W+... W
line
W+W+... W
Fig. 50.  Curve of bending moment. Fig. 50.

Fig. 50 shows the curve of bending moment under one of a series of travelling loads at fixed distances. Let W, W, W traverse the girder from the left at fixed distances a, b. For the position shown the distribution of bending moment due to W is given by ordinates of the triangle A′CB′; that due to W by ordinates of A′DB′; and that due to W by ordinates A′EB′. The total moment at W, due to three loads, is the sum mC+mn+mo of the intercepts which the triangle sides cut off from the vertical under W. As the loads move over the girder, the points C, D, E describe the parabolas M, M, M, the middle ordinates of which are &FRAC14;Wl, &FRAC14;Wl, and &FRAC14;Wl. If these are first drawn it is easy, for any position of the loads, to draw the lines B′C, B′D, B′E, and to find the sum of the intercepts which is the total bending moment under a load. The lower portion of the figure is the curve of bending moments under the leading load. Till W has advanced a distance a only one load is on the girder, and the curve A′F gives bending moments due to W only; as W advances to a distance a+b, two loads are on the girder, and the curve FG gives moments due to W and W. GB′ is the curve of moments for all three loads W+W+W.