Chapter 19

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:--

_Excess of Deflection and straining Action of a moving Load over that due to a resting Load._

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 W_l be the total live load, W_f 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_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 = ½ [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 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 [v.04 p.0549] 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.

19. _Stresses Permitted._--For a long time engineers held the convenient opinion that, if the total dead and live load stress on any section of a structure (of iron) did not exceed 5 tons per sq. in., ample safety was secured. It is no longer possible to design by so simple a rule. In an interesting address to the British Association in 1885, Sir B. Baker described the condition of opinion as to the safe limits of stress as chaotic. "The old foundations," he said, "are shaken, and engineers have not come to an agreement respecting the rebuilding of the structure. The variance in the strength of existing bridges is such as to be apparent to the educated eye without any calculation. In the present day engineers are in accord as to the principles of estimating the magnitude of the stresses on the members of a structure, but not so in proportioning the members to resist those stresses. The practical result is that a bridge which would be passed by the English Board of Trade would require to be strengthened 5% in some parts and 60% in others, before it would be accepted by the German government, or by any of the leading railway companies in America." Sir B. Baker then described the results of experiments on repetition of stress, and added that "hundreds of existing bridges which carry twenty trains a day with perfect safety would break down quickly under twenty trains an hour. This fact was forced on my attention nearly twenty-five years ago by the fracture of a number of girders of ordinary strength under a five-minutes' train service."

Practical experience taught engineers that though 5 tons per sq. in. for iron, or 6½ tons per sq. in. for steel, was safe or more than safe for long bridges with large ratio of dead to live load, it was not safe for short ones in which the stresses are mainly due to live load, the weight of the bridge being small. The experiments of A. Wöhler, repeated by Johann Bauschinger, Sir B. Baker and others, show that the breaking stress of a bar is not a fixed quantity, but depends on the range of variation of stress to which it is subjected, if that variation is repeated a very large number of times. Let K be the breaking strength of a bar per unit of section, when it is loaded once gradually to breaking. This may be termed the statical breaking strength. Let k_{max.} be the breaking strength of the same bar when subjected to stresses varying from k_{max.} to k_{min.} alternately and repeated an indefinitely great number of times; k_{min.} is to be reckoned + if of the same kind as k_{max.} and - if of the opposite kind (tension or thrust). The range of stress is therefore k_{max.}-k_{min.}, if the stresses are both of the same kind, and k_{max.}+k_{min.}, if they are of opposite kinds. Let [Delta] = k_{max.} ± k_{min.} = the range of stress, where [Delta] is always positive. Then Wöhler's results agree closely with the rule,

k_{max.} = ½[Delta]+[root](K²-n[Delta]K),

where n is a constant which varies from 1.3 to 2 in various qualities of iron and steel. For ductile iron or mild steel it may be taken as 1.5. For a statical load, range of stress nil, [Delta] = 0, k_{max.} = K, the statical breaking stress. For a bar so placed that it is alternately loaded and the load removed, [Delta] = k_{max.} and k_{max.} = 0.6 K. For a bar subjected to alternate tension and compression of equal amount, [Delta] = 2 f_{max.} and k_{max.} = 0.33 K. The safe working stress in these different cases is k_{max.} divided by the factor of safety. It is sometimes said that a bar is "fatigued" by repeated straining. The real nature of the action is not well understood, but the word fatigue may be used, if it is not considered to imply more than that the breaking stress under repetition of loading diminishes as the range of variation increases.

It was pointed out as early as 1869 (Unwin, _Wrought Iron Bridges and Roofs_) that a rational method of fixing the working stress, so far as knowledge went at that time, would be to make it depend on the ratio of live to dead load, and in such a way that the factor of safety for the live load stresses was double that for the dead load stresses. Let A be the dead load and B the live load, producing stress in a bar; [rho] = B/A the ratio of live to dead load; f_1 the safe working limit of stress for a bar subjected to a dead load only and f the safe working stress in any other case. Then

f_1 (A+B)/(A+2B) = f_1(1+[rho])/(1+2[rho]).

The following table gives values of f so computed on the assumption that f_1 = 7½ tons per sq. in. for iron and 9 tons per sq. in. for steel.

_Working Stress for combined Dead and Live Load. Factor of Safety twice as great for Live Load as for Dead Load._

+-------+----------+-----------------------------+ | Ratio | 1+[rho] |Values of f, tons per sq. in.| | [rho] | ------- +-----------------------------+ | | 1+2[rho] | Iron. | Mild Steel.| ----------------------+-------+----------+----------------+------------+ All dead load | 0 | 1.00 | 7.5 | 9.0 | | .25 | 0.83 | 6.2 | 7.5 | | .33 | 0.78 | 5.8 | 7.0 | | .50 | 0.75 | 5.6 | 6.8 | | .66 | 0.71 | 5.3 | 6.4 | Live load = Dead load | 1.00 | 0.66 | 4.9 | 5.9 | | 2.00 | 0.60 | 4.5 | 5.4 | | 4.00 | 0.56 | 4.2 | 5.0 | All live load | [inf] | 0.50 | 3.7 | 4.5 | ----------------------+-------+----------+----------------+------------+

Bridge sections designed by this rule differ little from those designed by formulae based directly on Wöhler's experiments. This rule has been revived in America, and appears to be increasingly relied on in bridge-designing. (See _Trans. Am. Soc. C.E._ xli. p. 156.)

The method of J.J. Weyrauch and W. Launhardt, based on an empirical expression for Wöhler's law, has been much used in bridge designing (see _Proc. Inst. C.E._ lxiii. p. 275). Let t be the _statical breaking strength_ of a bar, loaded once gradually up to fracture (t = breaking load divided by original area of section); u the breaking strength of a bar loaded and unloaded an indefinitely great number of times, the stress varying from u to 0 alternately (this is termed the _primitive strength_); and, lastly, let s be the breaking strength of a bar subjected to an indefinitely great number of repetitions of stresses equal and opposite in sign (tension and thrust), so that the stress ranges alternately from s to -s. This is termed the _vibration strength_. Wöhler's and Bauschinger's experiments give values of t, u, and s, for some materials. If a bar is subjected to alternations of stress having the range [Delta] = f_{max.}-f_{min.}, then, by Wöhler's law, the bar will ultimately break, if

f_{max.} = F[Delta], . . . (1)

where F is some unknown function. Launhardt found that, for stresses always of the same kind, F = (t-u)/(t-f_{max.}) approximately agreed with experiment. For stresses of different kinds Weyrauch found F = (u-s)/(2u-s-f_{max.}) to be similarly approximate. Now let f_{max.}/f_{min.} = [phi], where [phi] is + or - according as the stresses are of the same or opposite signs. Putting the values of F in (1) and solving for f_{max.}, we get for the breaking stress of a bar subjected to repetition of varying stress,

f_{max.} = u(1+(t-u)[phi]/u) [Stresses of same sign.] f_{max.} = u(1+(u-s)[phi]/u) [Stresses of opposite sign.]

The working stress in any case is f_{max.} divided by a factor of safety. Let that factor be 3. Then Wöhler's results for iron and Bauschinger's for steel give the following equations for tension or thrust:--

Iron, working stress, f = 4.4 (1+½[phi]) Steel, working stress, f = 5.87 (1+½[phi]).

In these equations [phi] is to have its + or - value according to the case considered. For shearing stresses the working stress may have 0.8 of its value for tension. The following table gives values of the working stress calculated by these equations:--

_Working Stress for Tension or Thrust by Launhardt and Weyrauch Formula._

+-------+-----------+--------------------+ | [phi] | [phi] | Working Stress f, | | | 1 + ----- | tons per sq. in. | | | 2 +--------------------+ | | | Iron. | Steel. | ------------------------+-------+-----------+--------------------+ All dead load | 1.0 | 1.5 | 6.60 | 8.80 | | 0.75 | 1.375 | 6.05 | 8.07 | | 0.50 | 1.25 | 5.50 | 7.34 | | 0.25 | 1.125 | 4.95 | 6.60 | All live load | 0.00 | 1.00 | 4.40 | 5.87 | | -0.25 | 0.875 | 3.85 | 5.14 | | -0.50 | 0.75 | 3.30 | 4.40 | | -0.75 | 0.625 | 2.75 | 3.67 | Equal stresses + and - | -1.00 | 0.500 | 2.20 | 2.93 | ------------------------+-------+-----------+--------------------+

[v.04 p.0550] To compare this with the previous table, [phi] = (A+B)/A = 1+[rho]. Except when the limiting stresses are of opposite sign, the two tables agree very well. In bridge work this occurs only in some of the bracing bars.

It is a matter of discussion whether, if fatigue is allowed for by the Weyrauch method, an additional allowance should be made for impact. There was no impact in Wöhler's experiments, and therefore it would seem rational to add the impact allowance to that for fatigue; but in that case the bridge sections become larger than experience shows to be necessary. Some engineers escape this difficulty by asserting that Wöhler's results are not applicable to bridge work. They reject the allowance for fatigue (that is, the effect of repetition) and design bridge members for the total dead and live load, plus a large allowance for impact varied according to some purely empirical rule. (See Waddell, _De Pontibus_, p.7.) Now in applying Wöhler's law, f_{max.} for any bridge member is found for the maximum possible live load, a live load which though it may sometimes come on the bridge and must therefore be provided for, is not the usual live load to which the bridge is subjected. Hence the range of stress, f_{max.}-f_{min.}, from which the working stress is deduced, is not the ordinary range of stress which is repeated a practically infinite number of times, but is a range of stress to which the bridge is subjected only at comparatively long intervals. Hence practically it appears probable that the allowance for fatigue made in either of the tables above is sufficient to cover the ordinary effects of impact also.

English bridge-builders are somewhat hampered in adopting rational limits of working stress by the rules of the Board of Trade. Nor do they all accept the guidance of Wöhler's law. The following are some examples of limits adopted. For the Dufferin bridge (steel) the working stress was taken at 6.5 tons per sq. in. in bottom booms and diagonals, 6.0 tons in top booms, 5.0 tons in verticals and long compression members. For the Stanley bridge at Brisbane the limits were 6.5 tons per sq. in. in compression boom, 7.0 tons in tension boom, 5.0 tons in vertical struts, 6.5 tons in diagonal ties, 8.0 tons in wind bracing, and 6.5 tons in cross and rail girders. In the new Tay bridge the limit of stress is generally 5 tons per sq. in., but in members in which the stress changes sign 4 tons per sq. in. In the Forth bridge for members in which the stress varied from 0 to a maximum frequently, the limit was 5.0 tons per sq. in., or if the stress varied rarely 5.6 tons per sq. in.; for members subjected to alternations of tension and thrust frequently 3.3 tons per sq. in. or 5 tons per sq. in. if the alternations were infrequent. The shearing area of rivets in tension members was made 1½ times the useful section of plate in tension. For compression members the shearing area of rivets in butt-joints was made half the useful section of plate in compression.

20. _Determination of Stresses in the Members of Bridges._--It is convenient to consider beam girder or truss bridges, and it is the stresses in the main girders which primarily require to be determined. A main girder consists of an upper and lower flange, boom or chord and a vertical web. The loading forces to be considered are vertical, the horizontal forces due to wind pressure are treated separately and provided for by a horizontal system of bracing. For practical purposes it is accurate enough to consider the booms or chords as carrying exclusively the horizontal tension and compression and the web as resisting the whole of the vertical and, in a plate web, the equal horizontal shearing forces. Let fig. 37 represent a beam with any system of loads W_1, W_2, ... W_n.

The reaction at the right abutment is

R_2 = W_1x_1/l+W_2x_2/l+...

That at the left abutment is

R_1 = W_1+W_2+...-R_2.

Consider any section a b. The total shear at a b is

S = R-[Sigma](W_1+W_2 ...)

where the summation extends to all the loads to the left of the section. Let p_1, p_2 ... be the distances of the loads from a b, and p the distance of R_1 from a b; then the bending moment at a b is

M = R_1p-[Sigma](W_1p_1+W_2p_2 ...)

where the summation extends to all the loads to the left of a b. If the loads on the right of the section are considered the expressions are similar and give the same results.