will be the weight of main girders to carry Wl+Wf and their own weight (Buck, Proc. Inst. C.E. lxvii. p. 331). Hence,
Wg = (Wl+Wf)k/(1-k).
Since in designing a bridge Wl+Wf is known, k(Wl+Wf) can be found from a provisional design in which the weight Wg 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 w1, w2 be the weights of main girders per ft. run for a live load p per ft. run and spans l1, l2. Then
w2/w1 = ½ [l2/l1+(l2/l1)2].
Now let w1′, w2′ be the girder weights per ft. run for spans l1, l2, and live loads p′ per ft. run. Then
w2′/w2 = 1/5(1+4p′/p)
w2′/w1 = 1/10[l2/l1+(l2/l1)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; w2 the weight of platform per ft. run; w3 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