Mohr's Circle Calculator
Enter the stress state on an element (σx, σy, τxy) to get the principal stresses, the maximum in-plane shear stress and the rotated stress element — plotted live on an interactive Mohr's circle.
Drag the rotation angle below to see the stress element turn and Mohr's circle track it in real time: the two marked points (X and Y) always show the normal and shear stress on the element's two faces at the chosen orientation. Where the circle crosses the horizontal axis are the principal stresses (σ1, σ2); the top and bottom of the circle are the maximum in-plane shear stress (±τmax).
Stress State
Values in MPa. Tension positive, compression negative.
Mohr's Circle
Stress Element
- Tension (σ > 0)
- Compression (σ < 0)
- Shear stress (τ)
Glossary
- Plane stress
- A stress state where the out-of-plane normal and shear stresses are zero (or negligible) — the standard assumption for thin plates and shells, and the case this calculator covers.
- Principal stress
- The maximum (σ1) or minimum (σ2) normal stress at a point, occurring on a plane where the shear stress is exactly zero.
- Principal plane
- The orientation of the stress element at which a principal stress occurs, at angle θp from the reference (X-face) orientation.
- Maximum in-plane shear stress
- The largest shear stress achievable by rotating the element in the plane, equal to the radius of Mohr's circle and to (σ1 − σ2)/2. Occurs 45° from the principal planes.
- Average (hydrostatic-in-plane) normal stress
- σavg = (σx + σy)/2 — the center of Mohr's circle, and also the normal stress present on the planes of maximum shear.
Frequently asked questions
What is Mohr's circle used for?
Mohr's circle is a graphical method for finding the normal and shear stress on any plane through a point, given the stress state on two perpendicular planes (σx, σy, τxy). It's the fastest way to read off the principal stresses, the maximum in-plane shear stress, and the orientation at which they occur, without solving the transformation equations by hand for every angle.
What sign convention does this calculator use?
Normal stress is positive in tension, negative in compression. Shear stress is positive when it tends to rotate the element counter-clockwise. Positive rotation angles (θ) also turn the element counter-clockwise. This matches the convention used in most mechanics-of-materials textbooks (Hibbeler, Beer & Johnston, Gere).
Why is the angle on the circle double the angle on the element?
The stress transformation equations depend on 2θ, not θ — rotating the physical element by θ moves the corresponding point around Mohr's circle by 2θ. This is a direct consequence of the trigonometric identities used to derive the transformation (cos 2θ, sin 2θ), not an arbitrary drafting choice.
What are principal stresses and principal planes?
Principal stresses (σ1 and σ2) are the maximum and minimum normal stresses at a point, occurring on the planes where the shear stress is zero — the two points where Mohr's circle crosses the horizontal (σ) axis. Those orientations are called the principal planes, at angle θp from the reference X-face.
How is the maximum in-plane shear stress related to the principal stresses?
The maximum in-plane shear stress τmax equals the radius of the circle, R = √[((σx − σy)/2)² + τxy²], which also equals (σ1 − σ2)/2. It occurs 45° from the principal planes, on a plane where the normal stress equals the average stress σavg = (σx + σy)/2.
Does this handle 3D (triaxial) stress states?
No — this calculator covers 2D plane stress (σx, σy, τxy with the out-of-plane stress assumed to be zero), which is the standard case for thin plates, shells and most machine-design checks. A full 3D stress state needs three Mohr's circles, one per pair of principal stresses.
References & further reading
- Mohr's circle — Wikipedia — the graphical method itself: derivation, sign conventions and typical uses.
- Christian Otto Mohr — Wikipedia — the German civil engineer who developed the method in 1882.
- Stress (mechanics) — Wikipedia — background on the Cauchy stress tensor that Mohr's circle visualises.
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