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Goodman diagram & mean-stress fatigue

A fully reversed endurance limit tells you nothing about a part that also carries a steady load. Four criteria, one Haigh diagram, your operating point plotted — including the yield check that all three fatigue criteria can quietly pass.

Stress state

At the critical location, after any stress concentration is applied.

Material

Worked examples

Safety factor

governing criterion

—
—
Modified Goodman—
Soderberg—
Gerber—
First-cycle yield (Langer)—
Stress ratio R = σmin/σmax—

Haigh diagram

Mean stress horizontally, alternating stress vertically. Inside a line is safe by that criterion.

Goodman Soderberg Gerber Yield Your point
Where does σe come from? That is usually the weakest number in the calculation. METALLAI predicts the endurance limit from composition, process route, surface condition and defect population — and states its uncertainty rather than assuming 0.5 × UTS. Open the app →

The four criteria

Goodman σa/σe + σm/UTS = 1/n Soderberg σa/σe + σm/σy = 1/n Gerber n·σa/σe + (n·σm/UTS)² = 1 Yield σa + σm = σy/n

The first three are the same idea drawn three ways: a locus from the endurance limit on the vertical axis to a strength on the horizontal one. They differ in which strength and in whether the path between them is straight.

CriterionEnds atShapeUse it when
Modified GoodmanUTSstraightDefault for steel. Conservative, and what most codes assume.
SoderbergYieldstraightMost conservative — also guarantees no yielding anywhere on the locus.
GerberUTSparabolicFits ductile steel data best. Closer to reality, less margin.
Langer yieldYieldstraight, slope −1Always. It catches the case all three above miss.
The yield check is not optional. Near the mean-stress axis, Goodman and Gerber both run out to UTS — but a part loaded to σm + σa above yield has already deformed on the first cycle. The governing safety factor is the smallest of the four, and this calculator reports that one as the headline.

Compressive mean stress

A negative mean stress genuinely helps — that is why shot peening and case hardening work. Design practice normally declines to claim the benefit: for σm < 0 the alternating stress alone is compared against the endurance limit. This tool does that, and tells you it has, rather than returning a flattering number off an extrapolated line.

Questions

Should I use Goodman, Soderberg or Gerber?
Goodman for most steel design work. Soderberg when you also need a guarantee against yielding. Gerber when you want the best fit to ductile-steel test data and are willing to carry less margin. Run all three — if they disagree sharply, your point is near the mean-stress axis and the yield check is what matters.
What endurance limit should I put in?
A measured one if you have it. The 0.5 × UTS rule of thumb is a rough starting point for steel below about 1400 MPa and does not apply to aluminium at all, which has no true endurance limit. Surface finish, size, notches and defect population all cut it, often by half or more.
Is this valid for a notched part?
Only if you have already applied the fatigue notch factor to σa. Practice differs on whether the mean stress gets the full factor too. Enter stresses at the critical location, not nominal section stresses.
What safety factor is enough?
That is a code and consequence question, not a physics one. What this tool can tell you is how much of your margin is being consumed by the mean stress rather than the alternating one — and if the four numbers sit far apart, that spread is itself information.

Get the endurance limit, not just the arithmetic

METALLAI predicts σe from composition and route, applies surface and defect knockdowns, screens with Goodman, and returns a P10 design life with its uncertainty.

Run a fatigue prediction — free Murakami √area calculator