Goodman Mean-Stress Correction

Fold a nonzero mean stress into an equivalent fully-reversed stress and get the Goodman fatigue safety factor, with Gerber for comparison.

Inputσₐ/Sₑ + σₘ/Sᵤₜ = 1/n_f (modified Goodman), σₐᵣ = σₐ / (1 − σₘ/Sᵤₜ), Gerber: σₐ/Sₑ + (σₘ/Sᵤₜ)² = 1/n

Alternating stress σₐ (MPa)or enter ksi — converts to MPaMean stress σₘ (MPa)or enter ksi — converts to MPaUltimate strength Sᵤₜ (MPa)or enter ksi — converts to MPaEndurance limit Sₑ (fully reversed) (MPa)or enter ksi — converts to MPaYield strength S_y (optional, first-cycle check) (MPa) — optionalor enter ksi — converts to MPa

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The engineering

S-N data are almost always generated fully reversed (R = −1), but real parts carry a mean stress — bolt preload, pressure, centrifugal load — stacked under the vibration. The modified Goodman line trades alternating capacity linearly against mean stress up to Sᵤₜ, giving you an equivalent fully-reversed stress σₐᵣ you can take straight into an S-N curve, plus a load-line safety factor n_f (both σₐ and σₘ scaled together).

Goodman is deliberately conservative for ductile metals — test points cluster nearer the Gerber parabola, which this card prints alongside for comparison. Gotchas: a compressive mean stress helps fatigue life, but standard practice is to take no credit for it (σₐᵣ = σₐ, as done here); and a big Goodman factor means nothing if σ_max exceeds yield on the first cycle — enter S_y and the card runs the Langer check and tells you which line governs.

Sanity check: at σₘ = 0 the factor collapses to Sₑ/σₐ, and at σₐ = 0 it collapses to Sᵤₜ/σₘ. If your answer doesn't land between those two bounds, an input is wrong.

Where this math comes from

The mean-stress problem is as old as fatigue itself: August Wöhler's railway-axle experiments in the 1860s showed that cyclic amplitude, not just peak stress, kills parts, and Wilhelm Gerber fit a parabola to Wöhler's own data in 1874. John Goodman, a professor of engineering at Leeds, published the straight-line version in his 1899 textbook Mechanics Applied to Engineering — not because it fit the data better, but because it was simple, conservative, and needed only two numbers every materials handbook already listed.

B. P. Haigh gave the construction its graphical home in 1917 — the σₐ–σₘ diagram every fatigue engineer still sketches — and C. R. Soderberg offered a yield-based variant in 1930 for the truly cautious. The 'modified Goodman' pairing with the Langer first-cycle-yield line was canonized for generations of American engineers by Joseph Shigley's Mechanical Engineering Design, which is where the formulation on this card comes from.

1867August WöhlerRotating-bending axle tests establish the S-N curve and the role of stress amplitude.
1874Wilhelm GerberFits a parabolic mean-stress relation to Wöhler's data.
1899John GoodmanPublishes the linear σₐ–σₘ trade-off in Mechanics Applied to Engineering.
1917B. P. HaighIntroduces the constant-life (Haigh) diagram for plotting mean-stress criteria.
1963Joseph ShigleyMechanical Engineering Design standardizes the modified-Goodman + Langer workflow used here.

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