In Austria the official regulations require that railway bridges shall be designed for at least the following live loads per foot run and per track:—
| Span. | Live Load in Tons. | ||
| Metres. | Ft. | Per metre run. | Per ft. run. |
| 1 | 3.3 | 20 | 6.1 |
| 2 | 6.6 | 15 | 4.6 |
| 5 | 16.4 | 10 | 3.1 |
| 20 | 65.6 | 5 | 1.5 |
| 30 | 98.4 | 4 | 1.2 |
It would be simpler and more convenient in designing short bridges if, instead of assuming an equivalent uniform rolling load, agreement could be come to as to a typical heavy locomotive which would produce stresses as great as any existing locomotive on each class of railway. Bridges would then be designed for these selected loads, and the process would be safer in dealing with flooring girders and shearing forces than the assumption of a uniform load.
Some American locomotives are very heavy. Thus a consolidation engine may weigh 126 tons with a length over buffers of 57 ft., corresponding to an average load of 2.55 tons per ft. run. Also long ore wagons are used which weigh loaded two tons per ft. run. J. A. L. Waddell (De Pontibus, New York, 1898) proposes to arrange railways in seven classes, according to the live loads which may be expected from the character of their traffic, and to construct bridges in accordance with this classification. For the lightest class, he takes a locomotive and tender of 93.5 tons, 52 ft. between buffers (average load 1.8 tons per ft. run), and for the heaviest a locomotive and tender weighing 144.5 tons, 52 ft. between buffers (average load 2.77 tons per ft. run). Wagons he assumes to weigh for the lightest class 1.3 tons per ft. run and for the heaviest 1.9 tons. He takes as the live load for a bridge two such engines, followed by a train of wagons covering the span. Waddell’s tons are short tons of 2000 ℔.
ii. Impact.—If a vertical load is imposed suddenly, but without velocity, work is done during deflection, and the deformation and stress are momentarily double those due to the same load at rest on the structure. No load of exactly this kind is ever applied to a bridge. But if a load is so applied that the deflection increases with speed, the stress is greater than that due to a very gradually applied load, and vibrations about a mean position are set up. The rails not being absolutely straight and smooth, centrifugal and lurching actions occur which alter the distribution of the loading. Again, rapidly changing forces, due to the moving parts of the engine which are unbalanced vertically, act on the bridge; and, lastly, inequalities of level at the rail ends give rise to shocks. For all these reasons the stresses due to the live load are greater than those due to the same load resting quietly on the bridge. This increment is larger on the flooring girders than on the main ones, and on short main girders than on long ones. The impact stresses depend so much on local conditions that it is difficult to fix what allowance should be made. E. H. Stone (Trans. Am. Soc. of C.E. xli. p. 467) collated some measurements of deflection taken during official trials of Indian bridges, and found the increment of deflection due to impact to depend on the ratio of dead to live load. By plotting and averaging he obtained the following results:—
| Dead load in per cent of total load | 10 | 20 | 30 | 40 | 50 | 70 | 90 |
| Live load in per cent of total load | 90 | 80 | 70 | 60 | 50 | 30 | 10 |
| Ratio of live to dead load | 9 | 4 | 2.3 | 1.5 | 1.0 | 0.43 | 0.10 |
| Excess of deflection and stress due to moving load per cent | 23 | 13 | 8 | 5.5 | 4.0 | 1.6 | 0.3 |
These results are for the centre deflections of main girders, but Stone infers that the augmentation of stress for any member, due to causes included in impact allowance, will be the same percentage for the same ratios of live to dead load stresses. Valuable measurements of the deformations of girders and tension members due to moving trains have been made by S. W. Robinson (Trans. Am. Soc. C.E. xvi.) and by F. E. Turneaure (Trans. Am. Soc. C.E. xli.). The latter used a recording deflectometer and two recording extensometers. The observations are difficult, and the inertia of the instrument is liable to cause error, but much care was taken. The most striking conclusions from the results are that the locomotive balance weights have a large effect in causing vibration, and next, that in certain cases the vibrations are cumulative, reaching a value greater than that due to any single impact action. Generally: (1) At speeds less than 25 m. an hour there is not much vibration. (2) The increase of deflection due to impact at 40 or 50 m. an hour is likely to reach 40 to 50% for girder spans of less than 50 ft. (3) This percentage decreases rapidly for longer spans, becoming about 25% for 75-ft. spans. (4) The increase per cent of boom stresses due to impact is about the same as that of deflection; that in web bracing bars is rather greater. (5) Speed of train produces no effect on the mean deflection, but only on the magnitude of the vibrations.
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, &c., is 0.2 to 0.25 tons per ft. run for each line of way, while the rail girders, cross girders, &c., 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 Wl be the total live load, Wf 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(Wl+Wf) be the weight of main girders designed to carry Wl+Wf, but not their own weight in addition. Then
Wg = (Wl+Wf)(k+k2+k3 ...)
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 = 12 [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 = 15(1+4p′/p)
w2′/w1 = 110[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
w3 = (w1+w2)l2/(Cds−l2) = (w1+w2)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 = (w1+w2)l2/(Kd−l2) = (w1+w2)lr/(K−lr)
where K = (w1+w2+w3)lr/w3 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 ℔ 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 ℔ 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 ℔ 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
