a bridge of
span l, both being considered for the present purpose to be uniform per ft.
run. Let k(W_l+W_f) be the weight of main girders designed to carry
W_l+W_f, but not their own weight in addition. Then

W_g = (W_l+W_f)(k+k^2+k^3 ...)

will be the weight of main girders to carry W_l+W_f and their own weight
(Buck, _Proc. Inst. C.E._ lxvii. p. 331). Hence,

W_g = (W_l+W_f)k/(1-k).

Since in designing a bridge W_l+W_f is known, k(W_l+W_f) can be found from
a provisional design in which the weight W_g is neglected. The actual
bridge must have the section of all members greater than those in the
provisional design in the ratio k/(1-k).

Waddell (_De Pontibus_) gives the following convenient empirical relations.
Let w_1, w_2 be the weights of main girders per ft. run for a live load p
per ft. run and spans l_1, l_2. Then

w_2/w_1 = 1/2 [l_2/l_1+(l_2/l_1)^2].

Now let w_1', w_2' be the girder weights per ft. run for spans l_1, l_2,
and live loads p' per ft. run. Then

w_2'/w_2 = 1/5(1+4p'/p)

w_2'/w_1 = 1/10[l_2/l_1+(l_2/l_1)^2](1+4p'/p)

A partially rational approximate formula for the weight of main girders is
the following (Unwin, _Wrought Iron Bridges and Roofs_, 1869, p. 40):--

Let w = total live load per ft. run of girder; w_2 the weight of platform
per ft. run; w_3 the weight of main girders per ft. run, all in tons; l =
span in ft.; s = average stress in tons per sq. in. on gross section of
metal; d = depth of girder at centre in ft.; r = ratio of span to depth of
girder so that r = l/d. Then

w_3 = (w_1+w_2)l^2/(Cds-l_2) = (w_1+w_2)lr/(Cs-lr),

where C is a constant for any type of girder. It is not easy to fix the
average stress s per sq. in. of gross section. Hence the formula is more
useful in the form

w = (w_1+w_2)l^2/(Kd-l^2) = (w_1+w_2)lr/(K-lr)

where K = (w_1+w_2+w_3)lr/w_3 is to be deduced from the data of some bridge
previously designed with the same working stresses. From some known
examples, C varies from 1500 to 1800 for iron braced parallel or bowstring
girders, and from 1200 to 1500 for similar girders of steel. K = 6000 to
7200 for iron and = 7200 to 9000 for steel bridges.

iv. _Wind Pressure._--Much attention has been given to wind action since
the disaster to the Tay bridge in 1879. As to the maximum wind pressure on
small plates normal to the wind, there is not much doubt. Anemometer
observations show that pressures of 30 lb per sq. ft. occur in storms
annu