the section considered, and let Wx be the load at a b when the bending moment there is greatest, and Wn the last load to the right then on the bridge. Then the position of the loads must be that which satisfies the condition
xl greater than W1+W2+. . . Wx−1W1+W2+. . . Wn
xl less than W1+W2+. . . WxW1+W2+. . . Wn
Fig. 49.
Fig. 50.
Fig. 50 shows the curve of bending moment under one of a series of travelling loads at fixed distances. Let W1, W2, W3 traverse the girder from the left at fixed distances a, b. For the position shown the distribution of bending moment due to W1 is given by ordinates of the triangle A′CB′; that due to W2 by ordinates of A′DB′; and that due to W3 by ordinates A′EB′. The total moment at W1, due to three loads, is the sum mC+mn+mo of the intercepts which the triangle sides cut off from the vertical under W1. As the loads move over the girder, the points C, D, E describe the parabolas M1, M2, M3, the middle ordinates of which are 14W1l, 14W2l, and 14W3l. 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 W1 has advanced a distance a only one load is on the girder, and the curve A″F gives bending moments due to W1 only; as W1 advances to a distance a+b, two loads are on the girder, and the curve FG gives moments due to W1 and W2. GB″ is the curve of moments for all three loads W1+W2+W3.
Fig. 51.
Fig. 51 shows maximum bending moment curves for an extreme case of a short bridge with very unequal loads. The three lightly dotted parabolas are the curves of maximum moment for each of the loads taken separately. The three heavily dotted curves are curves of maximum moment under each of the loads, for the three loads passing over the bridge, at the given distances, from left to right. As might be expected, the moments are greatest in this case at the sections under the 15-ton load. The heavy continuous line gives the last-mentioned curve for the reverse direction of passage of the loads.
With short bridges it is best to draw the curve of maximum bending moments for some assumed typical set of loads in the way just described, and to design the girder accordingly. For longer bridges the funicular polygon affords a method of determining maximum bending moments which is perhaps more convenient. But very great accuracy in drawing this curve is unnecessary, because the rolling stock of railways varies so much that the precise magnitude and distribution of the loads which will pass over a bridge cannot be known. All that can be done is to assume a set of loads likely to produce somewhat severer straining than any probable actual rolling loads. Now, except for very short bridges and very unequal loads, a parabola can be found which includes the curve of maximum moments. This parabola is the curve of maximum moments for a travelling load uniform per ft. run. Let we be the load per ft. run which would produce the maximum moments represented by this parabola. Then we may be termed the uniform load per ft. equivalent to any assumed set of concentrated loads. Waddell has calculated tables of such equivalent uniform loads. But it is not difficult to find we, approximately enough for practical purposes, very simply. Experience shows that (a) a parabola having the same ordinate at the centre of the span, or (b) a parabola having the same ordinate at one-quarter span as the curve of maximum moments, agrees with it closely enough for practical designing. A criterion already given shows the position of any set of loads which will produce the greatest bending moment at the centre of the bridge, or at one-quarter span. Let Mc and Ma be those moments. At a section distant x from the centre of a girder of span 2c, the bending moment due to a uniform load we per ft. run is
M = 12we(c−x)(c+x).
Putting x = 0, for the centre section
Mc = 12wec2;
and putting x = 12c, for section at quarter span
Ma = 38wec2.
From these equations a value of we can be obtained. Then the bridge is designed, so far as the direct stresses are concerned, for bending moments due to a uniform dead load and the uniform equivalent load we.
Fig. 52.
27. Influence Lines.—In dealing with the action of travelling loads much assistance may be obtained by using a line termed an influence line. Such a line has for abscissa the distance of a load from one end of a girder, and for ordinate the bending moment or shear at any given section, or on any member, due to that load. Generally the influence line is drawn for unit load. In fig. 52 let A′B′ be a girder supported at the ends and let it be required to investigate the bending moment at C′ due to unit load in any position on the girder. When the load is at F′, the reaction at B′ is m/l and the moment at C′ is m(l−x)/l, which will be reckoned positive, when it resists a tendency of the right-hand part of the girder to turn counter-clockwise. Projecting A′F′C′B′ on to the horizontal AB, take Ff = m(l−x)/l, the moment at C of unit load at F. If this process is repeated for all positions of the load, we get the influence line AGB for the bending moment at C. The area AGB is termed the influence area. The greatest moment CG at C is x(l−x)/l. To use this line to investigate the maximum moment at C due to a series of travelling loads at fixed distances, let P1, P2, P3, . . . be the loads which at the moment considered are at distances m1, m2, . . . from the left abutment. Set off these distances along AB and let y1, y2, . . . be the corresponding ordinates of the influence curve (y = Ff ) on the verticals under the loads. Then the moment at C due to all the loads is
M = P1y1+P2y2+. . .
