Goodman Fatigue Criterion

Fatigue data is nearly always generated under fully reversed loading, where the stress swings symmetrically about zero. Real parts rarely do that — a spring is preloaded, a bolt is tightened, a rotating shaft carries steady bending — and a tensile mean stress is unambiguously harmful. The Goodman line is the correction: plot alternating stress on one axis and mean on the other, draw a straight line from the endurance limit $S_e$ to the ultimate strength $S_u$, and anything inside survives. With $\sigma_a = 100$ MPa against $S_e = 300$ and $\sigma_m = 150$ against $S_u = 600$, the two fractions are $1/3$ and $1/4$, summing to $7/12$, so $n = 12/7 = 1.71$.

John Goodman proposed the line in 1899 explicitly as a conservative simplification, not as a physical law. Gerber's parabola of 1874 fits scatter data better and Soderberg's line to yield is stricter still, but Goodman is what design codes and handbooks standardised on, precisely because it errs the safe way. The straight line is also easy to reason about: it says that a mean stress equal to half the ultimate uses up half your fatigue budget, whatever the material.

Two things to get right. $S_e$ must be the corrected endurance limit, not the polished-rotating-beam value — surface finish, size, load type, temperature and reliability factors typically knock the textbook $0.5\,S_u$ down to a third of that, and using the uncorrected figure is the single most dangerous shortcut in fatigue work. And a compressive mean stress does not fit this line at all; it is beneficial, which is the whole point of shot peening, cold-rolled threads and autofrettaged gun barrels. The Goodman check is also purely a fatigue check: you still have to confirm the peak $\sigma_m + \sigma_a$ stays below yield.