Mechanics of Materials · The Goodman line
Two stresses, one budget
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Two stresses, one budget

Almost nothing in a plant is loaded once. A stress that cycles between a maximum and a minimum is described by two numbers, and both matter. The alternating amplitude σa=σmaxσmin2\sigma_a = \dfrac{\sigma_{\max} - \sigma_{\min}}{2} is the size of the swing; the mean stress σm=σmax+σmin2\sigma_m = \dfrac{\sigma_{\max} + \sigma_{\min}}{2} is where the swing is centred. Both are stresses, both in MPa.

Each is charged against its own limit. The swing is charged against SeS_e, the corrected endurance limit — the fully reversed amplitude the part would survive forever. The mean is charged against SuS_u, the ultimate tensile strength. Goodman's line adds the two accounts: σaSe+σmSu=1n\dfrac{\sigma_a}{S_e} + \dfrac{\sigma_m}{S_u} = \dfrac{1}{n} — read aloud, sigma-a over S-e plus sigma-m over S-u equals one over n, where nn is the factor of safety against fatigue, a bare number with no units.

Read the sum as a budget. If the two fractions add to 1, the part sits exactly on the line and n=1n = 1: no margin. If they add to 0.5, half the budget is unspent and n=2n = 2. That reciprocal catches people out, so make it a habit — compute the sum, THEN flip it. And the reason the mean stress belongs in the account at all is the reason this line was drawn: a tensile mean holds a microcrack open, so the same swing does more damage riding high than it does about zero. A part can sit far under SeS_e on amplitude alone and still fail.