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 6&FRAC12; 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 be the breaking strength of the same bar when subjected to stresses varying from k to k alternately and repeated an indefinitely great number of times; k is to be reckoned + if of the same kind as k and - if of the opposite kind (tension or thrust). The range of stress is therefore k-k, if the stresses are both of the same kind, and k+k, if they are of opposite kinds. Let δ = k ± k = the range of stress, where δ is always positive.
Then Wöhler's results agree closely with the rule,
k = &FRAC12;δ+√(K&SUP2;-nδK),
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, δ = 0, k = K, the statical breaking stress. For a bar so placed that it is alternately loaded and the load removed, δ = k and k = 0.6 K. For a bar subjected to alternate tension and compression of equal amount, δ = 2 f and k = 0.33 K. The safe working stress in these different cases is k 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; ρ = B/A the ratio of live to dead load; f 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 (A+B)/(A+2B) = f(1+ρ)/(1+2ρ).
The following table gives values of f so computed on the assumption that f = 7&FRAC12; 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.
Values of f, tons per sq. in.
All dead load
Live load = Dead load
All live load
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 δ = f-f, then, by Wöhler's law, the bar will ultimately break, if
f = Fδ, . . . (1)