Elliptical Interaction⚠ unverified
Mechanical / Stress Analysis · Compute the elliptical yield-locus interaction value for a combined stress state
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_x | σx | Pa | 1.0 | Normal stress along the x-axis |
| sigma_y | σy | Pa | 1.0 | Normal stress along the y-axis |
| tau | τ | Pa | 1.0 | Shear stress |
| Sy | Sy | Pa | 1.0 | Material yield strength |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | EI | — | Dimensionless elliptical interaction value; yielding is predicted when it reaches 1 |
The science & history
Understanding the Parameters
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Normalised normal stresses $\sigma_x/S_y$, $\sigma_y/S_y$ — each direct stress as a fraction of yield. Squaring and summing them makes the locus elliptical in the $(\sigma_x, \sigma_y)$ plane.
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Shear allowable $0.577\,S_y$ — the distortion-energy shear yield strength, $S_{sy} = S_y/\sqrt3 \approx 0.577\,S_y$. Normalising $\tau$ by it puts shear on the same footing as normal stress — the reason the $0.577$ appears.
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Interaction value $EI$ — a single utilisation ratio: $EI = 1$ is the yield boundary, $EI < 1$ safe, and the factor of safety is $n = 1/EI$.
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The missing cross term — true von Mises couples $\sigma_x$ and $\sigma_y$ through $-\sigma_x\sigma_y$, which reduces the equivalent stress when they have the same sign (biaxial tension is less severe than the sum of squares suggests) and increases it when opposite-signed. Dropping it makes this index inexact.
Derivation (Approaching a Proof)
The distortion-energy (von Mises) yield criterion in plane stress is
$$\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau^2 = S_y^2.$$
Dividing through by $S_y^2$ and taking the square root gives an interaction value that equals 1 at yield. The registry form drops the cross term and writes the remaining pieces with each component over its own allowable, using $S_{sy} = S_y/\sqrt3 = 0.577\,S_y$ for shear:
$$EI = \sqrt{\frac{\sigma_x^2}{S_y^2} + \frac{\sigma_y^2}{S_y^2} + \frac{\tau^2}{(0.577\,S_y)^2}} = \frac{1}{S_y}\sqrt{\sigma_x^2 + \sigma_y^2 + 3\tau^2}.$$
Comparing with the exact criterion, the two match only when $-\sigma_x\sigma_y$ is negligible; the index is a simplified, uncoupled approximation of the elliptical von Mises locus.
Dimensional check. Each ratio $\sigma/S_y$, $\tau/(0.577 S_y)$ is dimensionless (Pa/Pa), so the root-sum-of-squares $EI$ is dimensionless — a utilisation ratio, as required.
History and Development
The elliptical yield locus is the plane-stress signature of the von Mises / distortion-energy criterion (Huber 1904, von Mises 1913): plotting the yield boundary in the $(\sigma_x, \sigma_y)$ plane traces an ellipse, tilted because of the $-\sigma_x\sigma_y$ coupling. Interaction-ratio formulations like this one are common design shortcuts for combined loading; this particular version keeps the elliptical form but drops the tilt, trading rigour for a simpler expression.
Related Concepts: Von Mises Stress, Tresca Stress, Static Failure Theories, Factor of Safety yield, Principal Stresses, Octahedral Shear Stress
Notes: Omits the von Mises $-\sigma_x\sigma_y$ cross term — exact only when that term is small; non-conservative for like-signed biaxial tension. $0.577 = 1/\sqrt3$ (shear yield fraction). $EI = 1$ → yield; $n = 1/EI$. Use Von Mises Stress for rigour.