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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 kmax. be the breaking strength of the same bar when subjected to stresses varying from kmax. to kmin. alternately and repeated an indefinitely great number of times; kmin. is to be reckoned + if of the same kind as kmax. and - if of the opposite kind (tension or thrust). The range of stress is therefore kmax.-kmin., if the stresses are both of the same kind, and kmax.+kmin., if they are of opposite kinds. Let ? = kmax. ± kmin. = the range of stress, where ? is always positive. Then Wöhler's results agree closely with the rule,
kmax. = ½?+?(K²-n?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, ? = 0, kmax. = K, the statical breaking stress. For a bar so placed that it is alternately loaded and the load removed, ? = kmax. and kmax. = 0.6 K. For a bar subjected to alternate tension and compression of equal amount, ? = 2 fmax. and kmax. = 0.33 K. The safe working stress in these different cases is kmax. 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.
