lling load has been given by
Prof. H.T. Eddy (_Trans. Am. Soc. C.E._ xxii., 1890). Let hk (fig. 56)
represent in magnitude and position a load W, at x from the left abutment,
on a girder AB of span l. Lay off kf, hg, horizontal and equal to l. Join f
and g to h and k. Draw verticals at A, B, and join no. Obviously no is
horizontal and equal to l. Also mn/mf = hk/kf or mn-W(l-x)/l, which is the
reaction at A due to the load at C, and is the shear at any point of AC.
Similarly, po is the reaction at B and shear at any point of CB. The shaded
rectangles represent the distribution of shear due to the load at C, while
no may be termed the datum line of shear. Let the load move to D, so that
its distance from the left abutment is x+a. Draw a vertical at D,
intersecting fh, kg, in s and q. Then qr/ro = hk/hg or ro = W(l-x-a)/l,
which is the reaction at A and shear at any point of AD, for the new
position of the load. Similarly, rs = W(x+a)/l is the shear on DB. The
distribution of shear is given by the partially shaded rectangles. For the
application of this method to a series of loads Prof. Eddy's paper must be
referred to.

29. _Economic Span._--In the case of a bridge of many spans, there is a
length of span which makes the cost of the bridge least. The cost of
abutments and bridge flooring is practically independent of the length of
span adopted. Let P be the cost of one pier; C the cost of the main girders
for one span, erected; n the number of spans; l the length of one span, and
L the length of the bridge between abutments. Then, n = L/l nearly. Cost of
piers (n-1)P. Cost of main girders nG. The cost of a pier will not vary
materially with the span adopted. It depends mainly on the character of the
foundations and height at which the bridge is carried. The cost of the main
girders for one span will vary nearly as the square of the span for any
given type of girder and intensity of live load. That is, G = al squared, where a
is a constant. Hence the total cost of that part of the bridge which varies
with the span adopted is--

C = (n-i)P+nal squared
= LP/l-P+Lal.

Differentiating and equating to zero, the cost is least when

dC LP
-- = - -- + La = 0,
dl l squared

/*
P = al squared = G;

that is, when the cost of one pier is equal to the cost erected of the main
girders of one span. Sir Guilford Molesworth puts this in a convenient but
less exact form. Let G be the cost of superstruct