Tresca Stress⚠ unverified
Aerospace / Structures · Compute the Tresca equivalent stress for a plane-stress state
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_x | σx | Pa | 1.0 | Normal stress |
| sigma_y | σy | Pa | 1.0 | Normal stress |
| tau_xy | τxy | Pa | 1.0 | In-plane shear stress |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σtresca | Pa | Tresca equivalent stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Principal stresses $\sigma_1, \sigma_2, \sigma_3$ — the three normal stresses on the planes where shear vanishes (the eigenvalues of the stress tensor). Tresca uses only the extremes; the intermediate principal stress has no effect on the prediction.
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Maximum shear connection — the absolute maximum shear stress in 3D is $\tau_{max} = (\sigma_{max}-\sigma_{min})/2$, so $\sigma_{tresca} = 2\tau_{max}$. Comparing against $S_y$ is equivalent to comparing $\tau_{max}$ against $S_y/2$.
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Ignoring the middle principal — Tresca's blindness to $\sigma_2$ makes it conservative relative to von Mises (it predicts yield slightly earlier for most states), which is why codes that favour safety (e.g. ASME pressure-vessel design) adopt it.
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Equivalent stress — expressing a 3D state as one number lets you compare against a simple tension-test strength; $\sigma_{tresca}$ and $\sigma_{vm}$ (Von Mises Stress) are the two standard ductile-yield equivalents.
Derivation (Approaching a Proof)
Tresca's hypothesis: yielding begins when the maximum shear stress in the material equals the maximum shear at yield in a uniaxial tension test. In uniaxial tension at yield, $\sigma_1 = S_y$, $\sigma_2 = \sigma_3 = 0$, so the maximum shear is $\tau_y = S_y/2$ (Mohr's circle radius for that state).
For a general stress state, order the principals $\sigma_{max} \ge \sigma_{int} \ge \sigma_{min}$; the largest of the three Mohr circles has radius
$$\tau_{max} = \frac{\sigma_{max} - \sigma_{min}}{2}.$$
Setting $\tau_{max} = \tau_y = S_y/2$ gives the yield condition $\sigma_{max} - \sigma_{min} = S_y$. The left-hand side is defined as the Tresca equivalent stress:
$$\sigma_{tresca} = \sigma_{max} - \sigma_{min}, \qquad \text{yield when } \sigma_{tresca} = S_y.$$
Dimensional check. $\sigma_{tresca} = \sigma_{max} - \sigma_{min} = \text{Pa} - \text{Pa} = \text{Pa}$ — a stress, directly comparable to $S_y$.
History and Development
Henri Tresca proposed the maximum-shear-stress yield criterion in the 1860s from experiments on metal extrusion — the first quantitative theory of plastic yielding. It predates von Mises' distortion-energy criterion (1913) and remains widely used for its simplicity and conservatism, notably in the ASME Boiler & Pressure Vessel Code. The two criteria agree in uniaxial tension and differ by at most $\sim15\%$ elsewhere, with Tresca always the more cautious.
Related Concepts: Von Mises Stress, Principal Stresses, Max Principal Stress, Min Principal Stress, Mohr's Circle Radius, Static Failure Theories
Notes: $\sigma_{tresca} = 2\tau_{max}$; yield when it reaches $S_y$. Uses only the extreme principals (ignores the intermediate) — conservative vs Von Mises Stress. Basis of ASME pressure-vessel design.