Soderberg⚠ unverified
Mechanical / Fatigue · Compute the factor of safety from the Soderberg 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 |
| Sy | Sy | MPa | 1.0 | Yield 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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Yield strength $S_y$ on the mean axis — the defining choice. By pinning the line to $S_y$ (not $S_{ut}$), Soderberg forbids the mean stress alone from reaching yield, so any point below the line is safe against both fatigue and first-cycle yielding — no separate Langer check needed.
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Alternating stress $S_a$ — compared against the endurance limit $S_e$ as in every criterion; use $K_f S_a$ for notched parts.
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Endurance limit $S_e$ — the Marin-corrected value (Marin Endurance Limit).
- Conservatism — because $S_y$ can be well below $S_{ut}$ (e.g. $0.6$–$0.8\,S_{ut}$ for many steels), Soderberg predicts smaller $n$ than Goodman for the same stresses. It is the safe choice when mean stress is high or material data are uncertain, at the cost of heavier designs.
Derivation (Approaching a Proof)
The Soderberg failure line joins the endurance limit $(0, S_e)$ on the alternating axis to the yield strength $(S_y, 0)$ on the mean axis:
$$\frac{S_a}{S_e} + \frac{S_m}{S_y} = 1.$$
Scaling the load proportionally along the load line ($S_a, S_m \to n S_a, n S_m$) until it meets this line:
$$\frac{n S_a}{S_e} + \frac{n S_m}{S_y} = 1 \;\Longrightarrow\; \frac{1}{n} = \frac{S_a}{S_e} + \frac{S_m}{S_y}.$$
Because the intercept is $S_y$, the pure-mean-stress limit ($S_a = 0$) is $S_m = S_y$ — exactly the yield condition — so the line simultaneously enforces the yield constraint that Goodman leaves to a separate check.
Dimensional check. $\dfrac{S_a}{S_e} + \dfrac{S_m}{S_y}$ is (Pa/Pa)+(Pa/Pa) = dimensionless, so $n$ is a pure factor of safety.
History and Development
C.R. Soderberg proposed the yield-anchored line in 1930. It is the conservative bookend of the linear mean-stress family — Soderberg (yield) inside Goodman (ultimate) inside the Gerber parabola (best fit to data). Its guarantee against yielding made it popular for critical or uncertain applications, though modern practice more often uses Goodman or Gerber plus an explicit yield (Langer) check, which is less wasteful of material.
Related Concepts: Modified Goodman Factor, Gerber, Asme Elliptic, Goodman Line, Marin Endurance Limit, Static Failure Theories
Notes: Most conservative linear criterion (mean axis at $S_y$). Uniquely guards both fatigue and yielding — no separate Langer check needed. Use Marin-corrected $S_e$ and $K_f S_a$. For tensile mean stress.