3D Mohr's Circle Calculator
Enter the full stress tensor to get the three principal stresses, the absolute maximum shear stress, and the Tresca and von Mises equivalent stresses — plotted live as three nested circles.
Unlike the 2D (plane stress) calculator, this one takes the full six-component stress tensor and solves for the three eigenvalues of the stress matrix. The big circle (between σ1 and σ3) sets the absolute maximum shear stress; the two smaller circles subdivide it. Every possible cutting plane through the point — pick one with the orientation sliders below — lands somewhere in the shaded region between them, never outside it.
Stress Tensor
Values in MPa. Tension positive, compression negative. τxy = τyx, τyz = τzy, τzx = τxz (symmetric tensor).
Glossary
- Principal stresses (3D)
- The three normal stresses σ1 ≥ σ2 ≥ σ3 acting on the three mutually perpendicular planes where shear stress is zero — the eigenvalues of the stress tensor.
- Absolute maximum shear stress
- τmax = (σ1 − σ3) / 2 — the radius of the largest of the three circles, and the true maximum shear stress at the point over every possible orientation (not just in-plane rotations).
- Tresca criterion
- Predicts yielding when the maximum shear stress reaches a critical value — equivalently, when σ1 − σ3 reaches the material's yield strength. Simple, conservative.
- von Mises criterion
- Predicts yielding based on distortion (shape-change) energy: σvM = √[½((σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²)]. Generally matches test data for ductile metals more closely than Tresca.
- Direction cosines
- The components (l, m, n) of a unit vector along the three principal axes — here, the orientation of an arbitrary cutting plane's normal, set via the zenith/azimuth sliders.
- Mean (hydrostatic) stress
- σmean = (σ1 + σ2 + σ3) / 3 — the volume-changing part of the stress state, as opposed to the distortional part that von Mises captures.
Frequently asked questions
Is the 3D Mohr's circle just a different way of drawing the same thing as the 2D one?
No — it solves a genuinely different problem. The 2D (plane stress) calculator takes 3 stress components (σx, σy, τxy) and has a closed-form solution. The 3D calculator takes the full 6-component stress tensor and requires finding the eigenvalues of a 3×3 symmetric matrix (the three principal stresses) — a materially bigger calculation, not just a different picture of the same numbers.
What do the three circles represent?
Each circle is the ordinary 2D Mohr's circle for a pair of principal stresses, drawn as if you were rotating the element about the third principal axis: the big circle (σ1, σ3) gives the absolute maximum shear stress, and the two smaller circles (σ1, σ2) and (σ2, σ3) bound it. Any plane through the point, at any orientation, has a stress state that lands in the shaded region between the big circle and the two small ones — never outside it.
What's the difference between the Tresca and von Mises criteria shown here?
Both are equivalent-stress measures used to predict yielding, computed directly from the three principal stresses. Tresca (maximum shear stress theory) uses σ1 − σ3 — twice the absolute maximum shear stress. Von Mises (distortion energy theory) uses √[½((σ1−σ2)² + (σ2−σ3)² + (σ3−σ1)²)]. Von Mises is generally a closer match to test data for ductile metals; Tresca is more conservative (predicts yielding at a lower load) and is simpler to apply by hand.
When do I actually need the 3D version instead of the 2D one?
Whenever the out-of-plane stress isn't negligible — thick sections, pressure vessels, contact stresses, anywhere the assumption behind plane stress (σz ≈ 0) doesn't hold. If you're working with a thin plate or shell loaded in its own plane, the 2D calculator is enough and simpler to use.
What are the "plane orientation" sliders for?
They set the orientation (zenith and azimuth angles, measured from the principal axes) of an arbitrary plane through the point, and the diagram plots that plane's actual (σn, τn) stress state as point P inside the shaded region — a live demonstration that no orientation can produce a stress state outside it.
References & further reading
- Mohr's circle — Wikipedia — covers the three-circle construction for the full 3D stress tensor, not just the 2D case.
- Tresca yield criterion — Wikipedia — the maximum shear stress failure theory tied to the largest of the three circles.
- Von Mises yield criterion — Wikipedia — the distortion-energy failure theory commonly compared against Tresca.
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