Mohr's Circle Radius⚠ unverified
Mechanical / Stress Analysis · Radius of Mohr's circle (max shear stress)
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma_x | σx | Pa | 100000000.0 | Normal stress x |
| sigma_y | σy | Pa | 50000000.0 | Normal stress y |
| tau_xy | τxy | Pa | 30000000.0 | Shear stress |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| R | R | Pa | Mohr circle radius |
The science & history
Understanding the Parameters
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Stress difference $(\sigma_x-\sigma_y)/2$ — half the difference of the direct stresses. It is one leg of the right triangle whose hypotenuse is $R$; a large imbalance between $\sigma_x$ and $\sigma_y$ inflates the radius.
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Shear stress $\tau_{xy}$ — the other leg. Even with equal normal stresses, applied shear alone gives a finite radius $R = \tau_{xy}$.
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Radius = max in-plane shear — as the element rotates, the shear on its faces varies sinusoidally with amplitude $R$; the largest value is exactly $R$, occurring at $45^\circ$ to the principal directions. This is what feeds the maximum-shear-stress (Tresca) failure theory.
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Principal stresses — $\sigma_{1,2} = \sigma_{avg} \pm R$. The circle touches the normal-stress axis at these two points, where shear vanishes. A larger $R$ means principal stresses further apart — a more severe shear state.
Derivation (Approaching a Proof)
Rotating the axes by $\theta$, the transformed shear stress is
$$\tau_{x'y'} = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta,$$
and the transformed normal stress oscillates about $\sigma_{avg}$ with the complementary cosine terms. Treating $(\sigma_{x'}-\sigma_{avg})$ and $\tau_{x'y'}$ as coordinates and eliminating $2\theta$ via $\sin^2 + \cos^2 = 1$ gives the equation of a circle:
$$(\sigma_{x'} - \sigma_{avg})^2 + \tau_{x'y'}^2 = \left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2 = R^2.$$
The radius $R$ is the amplitude of the transformation sinusoids — the Pythagorean combination of the stress-difference and shear legs. Its maximum-shear interpretation follows because $\tau_{x'y'}$ ranges over $[-R, +R]$.
Dimensional check. $R = \sqrt{(\text{Pa})^2 + (\text{Pa})^2} = \text{Pa}$ — a stress, as required.
History and Development
Otto Mohr's 1882 circle recast Cauchy's stress-transformation equations as a single geometric object; the radius as maximum shear made the Tresca (maximum-shear-stress) yield criterion visually obvious — yield when the largest circle's radius reaches half the yield strength. Mohr's construction, and later his own Mohr–Coulomb failure envelope, remain foundational in mechanics of materials, soil mechanics, and plasticity.
Related Concepts: Mohr Circle Center, Principal Stresses, Max Principal Stress, Tresca Stress, Von Mises Stress, Shear Stress
Notes: $R$ = maximum in-plane shear (at $45^\circ$ to principal axes). Principal stresses $= \sigma_{avg} \pm R$. 2D (plane stress) — for 3D the absolute max shear may involve the out-of-plane principal (three Mohr circles). Feeds the Tresca criterion.