ure of a 100-ft. span
erected, and P the cost of one pier with its protection. Then the economic
span is l = 100[root]P/[root]G.

30. _Limiting Span._--If the weight of the main girders of a bridge, per
ft. run in tons, is--

w_3 = (w_1+w_2)lr/(K-lr)

according to a formula already given, then w_3 becomes infinite if k-lr =
0, or if

l = K/r,

[v.04 p.0554] where l is the span in feet and r is the ratio of span to
depth of girder at centre. Taking K for steel girders as 7200 to 9000,

Limiting Span in Ft.
r = 12 l = 600 to 750
= 10 = 720 to 900
= 8 = 900 to 1120

[Illustration: FIG. 58.]

[Illustration: FIG. 59.]

[Illustration: FIG. 60.]

In a three-span bridge continuous girders are lighter than discontinuous
ones by about 45% for the dead load and 15% for the live load, if no
allowance is made for ambiguity due to uncertainty as to the level of the
supports. The cantilever and suspended girder types are as economical and
free from uncertainty as to the stresses. In long-span bridges the
cantilever system permits erection by building out, which is economical and
sometimes necessary. It is, however, unstable unless rigidly fixed at the
piers. In the Forth bridge stability is obtained partly by the great excess
of dead over live load, partly by the great width of the river piers. The
majority of bridges not of great span have girders with parallel booms.
This involves the fewest difficulties of workmanship and perhaps permits
the closest approximation of actual to theoretical dimensions of the parts.
In spans over 200 ft. it is economical to have one horizontal boom and one
polygonal (approximately parabolic) boom. The hog-backed girder is a
compromise between the two types, avoiding some difficulties of
construction near the ends of the girder.

[Illustration: FIG. 61.]

[Illustration: FIG. 62.]

Most braced girders may be considered as built up of two simple forms of
truss, the king-post truss (fig. 61, a), or the queen-post truss (fig. 61,
b). These may be used in either the upright or the inverted position. A
_multiple truss_ consists of a number of simple trusses, e.g. Bollman
truss. Some timber bridges consist of queen-post trusses in the upright
position, as shown diagrammatically in fig. 62, where the circles indicate
points at which the flooring girders transmit load to the main girders.
_Compound_ t