BRIDGES

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Wj is at C. Then the reaction at B and shear at 0 is 'Rnjl. Next let the loads advance a distance a so that W2 comes to C. Then the shear at C is R(n + a)/7-Wj, plus any reaction d at B, due to any additional B load which has come on the girder during t the movement. The

- shear will therefore be

- increased by bringing

4 W2 to C, if Ra/Z + ^Wj F’S- 6. an(l ^ 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. 7 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 0 (Fig. 8) be the section at which the greatest shear is required, and let the loads advance from the left till Wj 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 he greater when W2 is at C. In that case the shear at C becomes R(rc + a)/7 +^-Wj, if a>p, and R(?i + a.)/7 ■-d— W^t/p, if a<p. If we neglect d, then the shear increases by moving W2 to C, if Ra/l>Wi in the first case, and if ~Ra/l>~W1a/p in the second case. For the greatest bending moment due to a travelling live load, let a load of w per foot run advance from the left abutment (Fig. 9), and let its centre be at x from the left abutment. The reaction at B is '2wx2/1 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 R C

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 W2 Wi that the average loads AG Q,„ per foot on either side of C (the load at C being neglected) are Ab nearly equal. If the h-x loads are very unequal 4,—1-m-—i km in magnitude or distance this condition '•1=2 C may be satisfied for more than one position Fig. 10. 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. 11 shows the curve of bending moment under one of a series of travelling loads at fixed distances. Let Wj, W2, W3 traverse the girder from the left at fixed distances a, b. For the position shown the distribution of bending moment due to AV1 is given by ordinates of the triangle A'CB'; that due to W2 by ordinates of A'DB'; and that due to AY3 by ordinates of A'EB'. The total moment at W1; due to three loads, is the sum rnG + mn + mo of the intercepts which the triangle sides cut off from the vertical under

Fig. 8. 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 Wi, W2 (Fig. 10), at distances x and x + u from the left abutment, be Wj. As the loads move over the girder, the points C, D, E describe their resultants on either side of C. Then the reaction at B is the parabolas Mj, M2, M3, the middle ordinates of which are JAV, W^ll + W2(a3+a)jl. The bending moment at C is £W2Z, and |W3Z. 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 a; . x+a' /7 M=W1y(Z-m) + W2m( 1 - —ysum of the intercepts which is the total bending moment under a load. The lower portion of the figure is the curve of bending moments If the loads are moved a distance kx to the right, the bending under the leading load. Till Wj has advanced a distance a only one moment becomes load is on the girder, and the curve A"F gives bending moments due ,, x + Axn . t x + a + x to Wi only ; as Wx advances to a distance a + b, two loads are on the M + AM = W1—j—+ 1 l girder, and the curve FG gives moments due to W2 and W2. GB" l is the curve of moments for all three loads Wq + W2 + AV3. Fig. 12 shows maximum bending moment curves for an extreme Am = WjW2~ case of a short bridge with very unequal loads. The three lightly and this is positive or the bending moment increases, if dotted parabolas are the curves of maximum moment for each of Wj( l-m)> W2m, or the loads taken separately. The three heavily dotted curves are if Wi/m > W2/(£ - m). curves of maximum moment under each of the loads, for the three But these are the loads passing over the bridge, at the given distances, from left to C b average loads per right. As might be expected, the moments are greatest in this foot run to the left case at the sections under the 15-ton load. The heavy continuous and right of C. line gives the last-mentioned curve for the reverse direction of Hence, if the aver- passage of the loads. sJ u x —1-x 1 age load to the left of With short bridges it is best to draw the curve of maximum a section is greater bending moments for some assumed typical set of loads in the way

- - 1-m—than that to the just described, and to design the girder accordingly. For longer

i right, the bending bridges the funicular polygon affords a method of determining (c. -1- 2 c n moment at the sec- maximum bending moments which is perhaps more convenient. tion will be increased But very great accuracy in drawing this curve is unnecessary, Fig- 9by moving the loads because the rolling stock of railways varies so much that the to the right, and vice precise magnitude and distribution of the loads which will pass over versa. Hence the maximum bending moment at C for a series of a bridge cannot be known. All that can be done is to assume a travelling loads will occur when the average load is the same on set of loads likely to produce somewhat severer straining than any either side of C. If one of the loads is at C, spread over a very probable actual rolling loads. Now, except for very short bridges