A purely empirical allowance for impact stresses has been proposed, amounting to 20% of the live load stresses for floor stringers; 15% for floor cross girders; and for main girders, 10% for 40-ft. spans, and 5% for 100-ft. spans. These percentages are added to the live load stresses.
iii. Dead Load. - The dead load consists of the weight of main girders, flooring and wind-bracing. It is generally reckoned to be uniformly distributed, but in large spans the distribution of weight in the main girders should be calculated and taken into account. The weight of the bridge flooring depends on the type adopted. Road bridges vary so much in the character of the flooring that no general rule can be given. In railway bridges the weight of sleepers, rails, etc., is 0.2 to 0.25 tons per ft. run for each line of way, while the rail girders, cross girders, etc., weigh 0.15 to 0.2 tons. If a footway is added about 0.4 ton per ft. run may be allowed for this. The weight of main girders increases with the span, and there is for any type of bridge a limiting span beyond which the dead load stresses exceed the assigned limit of working stress.
Let W be the total live load, W the total flooring load on a bridge of span l, both being considered for the present purpose to be uniform per ft. run. Let k(W+W) be the weight of main girders designed to carry W+W, but not their own weight in addition. Then
W = (W+W)(k+k2+k3 ...)
will be the weight of main girders to carry W+W and their own weight (Buck, Proc. Inst. C.E. lxvii. p. 331). Hence,
W = (W+W)k/(1-k).
Since in designing a bridge W+W is known, k(W+W) can be found from a provisional design in which the weight W 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, w be the weights of main girders per ft. run for a live load p per ft. run and spans l, l. Then
w/w = &FRAC12; [l/l+(l/l)2].
Now let w′, w′ be the girder weights per ft. run for spans l, l, and live loads p′ per ft. run. Then
w′/w = 1/5(1+4p′/p)
w′/w = 1/10[l/l+(l/l)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 the weight of platform per ft. run; w 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 = (w+w)l2/(Cds-l) = (w+w)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+w)l2/(Kd-l2) = (w+w)lr/(K-lr)
where K = (w+w+w)lr/w 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 annually in many localities, and that occasionally higher pressures are recorded in exposed positions. Thus at Bidstone, Liverpool, where the gauge has an exceptional exposure, a pressure of 80 lb per sq. ft. has been observed. In tornadoes, such as that at St Louis in 1896, it has been calculated, from the stability of structures overturned, that pressures of 45 to 90 lb per sq. ft. must have been reached. As to anemometer pressures, it should be observed that the recorded pressure is made up of a positive front and negative (vacuum) back pressure, but in structures the latter must be absent or only partially developed. Great difference of opinion exists as to whether on large surfaces the average pressure per sq. ft. is as great as on small surfaces, such as anemometer plates. The experiments of Sir B. Baker at the Forth bridge showed that on a surface 30 ft. × 15 ft. the intensity of pressure was less than on a similarly exposed anemometer plate.
In the case of bridges there is the further difficulty that some surfaces partially shield other surfaces; one girder, for instance, shields the girder behind it (see Brit. Assoc. Report, 1884). In 1881 a committee of the Board of Trade decided that the maximum wind pressure on a vertical surface in Great Britain should be assumed in designing structures to be 56 lb per sq. ft. For a plate girder bridge of less height than the train, the wind is to be taken to act on a surface equal to the projected area of one girder and the exposed part of a train covering the bridge. In the case of braced girder bridges, the wind pressure is taken as acting on a continuous surface extending from the rails to the top of the carriages, plus the vertical projected area of so much of one girder as is exposed above the train or below the rails. In addition, an allowance is made for pressure on the leeward girder according to a scale. The committee recommended that a factor of safety of 4 should be taken for wind stresses. For safety against overturning they considered a factor of 2 sufficient. In the case of bridges not subject to Board of Trade inspection, the allowance for wind pressure varies in different cases.
C. Shaler Smith allows 300 lb per ft. run for the pressure on the side of a train, and in addition 30 lb per sq. ft. on twice the vertical projected area of one girder, treating the pressure on the train as a travelling load. In the case of bridges of less than 50 ft. span he also provides strength to resist a pressure of 50 lb per sq. ft. on twice the vertical projection of one truss, no train being supposed to be on the bridge.