If the loads are moved a distance [Delta]x to the right, the bending moment
becomes
M+[Delta]M = W_1(x+[Delta]x)(l-m)/l+W_2m{1-(x+[Delta]x+a)/l}
[Delta]m = W_1[Delta]x(l-m)/l-W_2[Delta]xm/l,
and this is positive or the bending moment increases, if W_1(l-m) > W_2m,
or if W_1/m > W_2/(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.
[Illustration: 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 [v.04 p.0552] the
section considered, and let W_x be the load at a b when the bending moment
there is greatest, and W_n the last load to the right then on the bridge.
Then the position of the loads must be that which satisfies the condition
x W_1+W_2+... W_{x-1}
--- greater than ------------------------
l W_1+W_2+... W_n
x W_1+W_2+... W_x
--- less than ------------------------
l W_1+W_2+... W_n
[Illustration: FIG. 50.]
Fig. 50 shows the curve of bending moment under one of a series of
travelling loads at fixed distances. Let W_1, W_2, W_3 traverse the girder
from the left at fixed distances a, b. For the position shown the
distribution of bending moment due to W_1 is given by ordinates of the
triangle A'CB'; that due to W_2 by ordinates of A'DB'; and that due to W_3
by ordinates A'EB'. The total moment at W_1, due to three loads, is the sum
mC+mn+mo of the intercepts which the triangle sides cut off from the
vertical under W_1. As the loads move over the girder, the points C, D, E
describe the parabolas M_1, M_2, M_3, the middle ordinates of which are
¼W_1l, ¼W_2l, and ¼W_3l. 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_1 has advanced a distance a only one load is on the girder,
and the curve A"F gives bending moments due to W_1 only; as W_1 advances to
a distance a+b, two loads are on the girder, and the curve FG gives moments
due to W_1 and W_2. GB" is the curve of moments for all three loads
W_1+W_2+W_3.
[Illustration: 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 w_e be the load per ft. run which would produce the maximum
moments represented by this parabola. Then w_e 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 w_e, 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 M_c and M_a 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 w_e per ft run is