Asme Elliptic⚠ unverified
Mechanical / Fatigue · Compute the factor of safety from the ASME-elliptic 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 radicand is non-positive |
The science & history
Understanding the Parameters
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Elliptical locus — the sum of squared ratios equals 1, so the failure boundary is an ellipse quadrant. It curves above the Soderberg chord (less conservative) but stays below the Gerber parabola near the yield intercept.
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Yield anchor $S_y$ — as in Soderberg, pinning the mean axis to $S_y$ means the pure-mean limit is the yield point, so the criterion inherently protects against yielding.
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Alternating stress $S_a$ vs $S_e$ — the fatigue axis, using Marin-corrected $S_e$ and notch-adjusted $K_f S_a$.
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ASME shaft context — this is the criterion embedded in the ANSI/ASME B106 shaft-design equation, which is why it appears alongside a combined-stress ($\sigma$, $\tau$) formulation in shaft work (Shaft Fatigue Factor).
Derivation (Approaching a Proof)
The ASME-elliptic failure locus is the quarter ellipse through the endurance limit on the alternating axis and the yield strength on the mean axis:
$$\left(\frac{S_a}{S_e}\right)^2 + \left(\frac{S_m}{S_y}\right)^2 = 1.$$
Scaling the load along the load line, $S_a \to n S_a$, $S_m \to n S_m$, and requiring the scaled point on the ellipse:
$$\left(\frac{n S_a}{S_e}\right)^2 + \left(\frac{n S_m}{S_y}\right)^2 = 1 \;\Longrightarrow\; n^2\!\left[\left(\frac{S_a}{S_e}\right)^2 + \left(\frac{S_m}{S_y}\right)^2\right] = 1.$$
Because both terms are squared, $n$ factors out cleanly as $n^2$ (unlike Gerber, where only the mean term is squared and the solve is a full quadratic). Taking the positive root:
$$n = \frac{1}{\sqrt{\left(S_a/S_e\right)^2 + \left(S_m/S_y\right)^2}}.$$
Dimensional check. Each ratio $S_a/S_e$, $S_m/S_y$ is dimensionless (Pa/Pa); their squares sum to a dimensionless radicand, so $n$ is a pure factor of safety.
History and Development
The elliptic criterion was adopted in the ASME shaft-design code as a compromise between the over-conservative Soderberg line and the sometimes-unsafe Gerber parabola, while retaining Soderberg's built-in yield protection. Its clean $n^2$-factoring (both terms squared) makes it algebraically convenient, and it underlies the widely-used ANSI/ASME B106.1M shaft equation that combines bending and torsion into an equivalent alternating and mean stress before applying the ellipse.
Related Concepts: Soderberg, Modified Goodman Factor, Gerber, Goodman Line, Shaft Fatigue Factor, Marin Endurance Limit
Notes: Elliptical locus, mean axis at $S_y$ → guards yielding (like Soderberg) but less conservative. Both terms squared → $n$ factors as $n^2$. Basis of the ANSI/ASME B106 shaft equation. Use Marin-corrected $S_e$ and $K_f S_a$.