Goodman Line⚠ unverified
Mechanical / Stress Analysis · Compute the modified Goodman interaction value for a fatigue stress state
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_a | σa | Pa | 1.0 | Alternating stress amplitude |
| sigma_m | σm | Pa | 1.0 | Mean stress |
| Se | Se | Pa | 1.0 | Endurance (fatigue) limit |
| Sut | Sut | Pa | 1.0 | Material ultimate tensile strength |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | GL | — | Dimensionless Goodman interaction value; failure is predicted when it reaches 1 |
The science & history
Understanding the Parameters
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Alternating stress $\sigma_a$ — the cyclic amplitude; the dominant fatigue driver, compared against $S_e$. Notched parts use $K_f\sigma_a$ (Fatigue Notch Factor).
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Mean stress $\sigma_m$ — the steady offset; a tensile mean is damaging and is penalised via $S_{ut}$. Goodman is for $\sigma_m \ge 0$ (compressive mean → take $\sigma_m = 0$).
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Endurance limit $S_e$ — the Marin-corrected part value (Marin Endurance Limit), anchoring the alternating axis.
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Ultimate strength $S_{ut}$ — anchors the mean axis.
- Utilisation $GL$ — how far along the failure line the operating point sits: $GL = 0.5$ means half the fatigue capacity is used, and the factor of safety is $n = 1/GL = 2$. This "usage ratio" form is convenient for reporting margins directly.
Derivation (Approaching a Proof)
The modified-Goodman failure line joins $(0, S_e)$ on the alternating axis to $(S_{ut}, 0)$ on the mean axis:
$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1.$$
Any point on the line is at the fatigue limit; the left-hand side, evaluated for the actual stresses, is the utilisation value
$$GL = \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}}.$$
$GL < 1$ places the point inside the safe region, $GL = 1$ on the line, $GL > 1$ outside (failure). Scaling the load until the point reaches the line shows the factor of safety is $n = 1/GL$ — the quantity Modified Goodman Factor returns.
Dimensional check. $GL = \dfrac{\sigma_a}{S_e} + \dfrac{\sigma_m}{S_{ut}} = \dfrac{\text{Pa}}{\text{Pa}} + \dfrac{\text{Pa}}{\text{Pa}}$ = dimensionless — a utilisation ratio, as required.
History and Development
John Goodman's 1899 mean-stress line, in its modified form (endurance limit on the alternating axis, ultimate on the mean axis), is the most-taught fatigue criterion. Expressing it as a utilisation value $GL$ rather than a factor of safety is common in code checks and automated design, where a single "usage ratio" against 1.0 is convenient. It is the linear, conservative middle ground between the Soderberg (yield) line and the Gerber parabola.
Related Concepts: Modified Goodman Factor, Marin Endurance Limit, Fatigue Notch Factor, Endurance Limit steel, Fatigue Stress Concentration, Factor of Safety yield
Notes: Returns the utilisation value $GL$ (fail at $GL=1$), the reciprocal of the factor of safety $n = 1/GL$ from Modified Goodman Factor. Use Marin-corrected $S_e$ and $K_f\sigma_a$. Tensile mean stress; add a Langer yield check separately.