Gerber⚠ unverified
Mechanical / Fatigue · Compute the factor of safety from the Gerber parabolic criterion
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Sa | Sa | MPa | 1.0 | Alternating stress amplitude |
| Sm | Sm | MPa | 1.0 | Mean stress |
| Se | Se | MPa | 1.0 | Modified endurance limit |
| Sut | Sut | MPa | 1.0 | Ultimate tensile strength |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | n | — | Factor of safety (dimensionless). Returns positive infinity when the denominator is non-positive |
The science & history
Understanding the Parameters
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Squared mean term $(S_m/S_{ut})^2$ — the signature of Gerber. Near zero mean stress the parabola is nearly flat, so small mean stresses barely reduce the allowable alternating stress — matching data that Goodman's straight line under-predicts.
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Alternating stress $S_a$ vs endurance limit $S_e$ — the vertical (fatigue) axis, same as every criterion; use $K_f S_a$ for notches and the Marin-corrected $S_e$.
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Mean stress $S_m$ vs ultimate $S_{ut}$ — the horizontal axis. Gerber, like Goodman, uses $S_{ut}$ and applies to tensile mean stress; it does not itself prevent yielding (add a Langer check).
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Accuracy vs safety — Gerber runs through the centre of scatter, so about half of real data points fall below it. It is the best mean estimate but should be paired with an adequate factor of safety, unlike the intentionally conservative Goodman/Soderberg lines.
Derivation (Approaching a Proof)
The Gerber failure parabola joins $(0, S_e)$ and $(S_{ut}, 0)$ with a curved locus:
$$\frac{S_a}{S_e} + \left(\frac{S_m}{S_{ut}}\right)^2 = 1.$$
For the factor of safety, scale the load along the load line, $S_a \to n S_a$ and $S_m \to n S_m$, and require the scaled point to lie on the parabola:
$$\frac{n S_a}{S_e} + \left(\frac{n S_m}{S_{ut}}\right)^2 = 1.$$
This is quadratic in $n$; solving the positive root gives Shigley's closed form
$$n = \frac{1}{2}\left(\frac{S_{ut}}{S_m}\right)^2\frac{S_a}{S_e}\left[-1 + \sqrt{1 + \left(\frac{2\,S_m\,S_e}{S_{ut}\,S_a}\right)^2}\,\right].$$
The registry instead evaluates $n = 1/\!\left[S_a/S_e + (S_m/S_{ut})^2\right]$, which corresponds to scaling only $S_a$ by $n$ (not $S_m$) — a linearised shortcut that agrees with the exact form only when the mean term is small.
Dimensional check. $\dfrac{S_a}{S_e} + \left(\dfrac{S_m}{S_{ut}}\right)^2$ is (Pa/Pa) + (Pa/Pa)$^2$ = dimensionless, so $n$ is a pure factor of safety.
History and Development
W. Gerber proposed the parabolic mean-stress relation in 1874 — the earliest of the classic criteria, predating Goodman. Its curve fits ductile-steel fatigue data best, and it completes the standard trio (Gerber parabola, Goodman chord, Soderberg yield line) taught in Shigley. Modern practice picks among them by purpose: Gerber for a best estimate, Goodman/Soderberg for a guaranteed-safe bound, always with a yield check alongside.
Related Concepts: Modified Goodman Factor, Soderberg, Asme Elliptic, Goodman Line, Marin Endurance Limit, Fatigue Life cycles
Notes: Registry form scales only $S_a$ (simplified) — use the quadratic closed form for the exact load-line $n$. Parabola fits data best (runs through scatter centre) → pair with an adequate safety factor. Tensile mean stress; add a Langer yield check.