tment, be their resultants on either
side of C. Then the reaction at B is W_1x/l+W_2(x+a)/l. The bending moment
at C is

M = W_1x(l-m)/l+W_2m{1-(x+a)/l}.

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.]

Fi