Why does the point move by 2θ on Mohr’s circle?

The tensor transformation equations contain cos(2θ) and sin(2θ). A physical rotation of the stress element by θ therefore appears as a double-angle movement around the circle.

Does the simulator predict yielding or failure?

No. It transforms a local plane-stress state. To assess failure you would combine σ1, σ2, and τmax with a criterion such as maximum principal stress, Tresca, von Mises, or a material-specific model.

Mohr Circle & Stress Transformation

Plane stress σx, σy, τxy: Mohr circle, transformed stresses on a rotated element, principal stresses, τmax, and θp.

Mohr’s circle is a graphical form of the same tensor rotation equations. It is for a local plane-stress state; failure criteria, 3D stress, plasticity, and stress concentrations are outside this simple calculator.

Live graphs

About this model

Mohr’s circle is a graphical representation of the plane-stress tensor transformation. Starting from normal stresses σx and σy and engineering shear stress τxy, the circle center C = (σx + σy)/2 and radius R = sqrt(((σx − σy)/2)^2 + τxy^2) immediately give the principal stresses σ1, σ2 and maximum in-plane shear stress τmax. Rotating a physical stress element by θ corresponds to moving 2θ around the circle, so the simulator shows both the rotated element and the matching point on the σ–τ plot. This is a local elastic stress-transformation calculator: it does not choose a failure criterion, include 3D out-of-plane stresses, stress concentrations, plasticity, residual stress, or material anisotropy.

Who it's for: Mechanics of materials, machine design, civil/structural engineering, and finite-element stress post-processing introductions.

Key terms

Mohr circle
Stress transformation
Principal stress
Maximum shear stress
Plane stress

How it works

Plane-stress transformation calculator with Mohr’s circle: convert σx, σy, and τxy into principal stresses, maximum in-plane shear, and stresses on a rotated element.

Key equations

C = (σx + σy)/2, R = sqrt(((σx − σy)/2)^2 + τxy^2)

σ1,2 = C ± R; tan(2θp) = 2τxy/(σx − σy); τmax = R

Frequently asked questions

Why does the point move by 2θ on Mohr’s circle?: The tensor transformation equations contain cos(2θ) and sin(2θ). A physical rotation of the stress element by θ therefore appears as a double-angle movement around the circle.
Does the simulator predict yielding or failure?: No. It transforms a local plane-stress state. To assess failure you would combine σ1, σ2, and τmax with a criterion such as maximum principal stress, Tresca, von Mises, or a material-specific model.

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Mohr Circle & Stress Transformation

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About this model

How it works

Key equations

Frequently asked questions

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Live graphs

About this model

How it works

Key equations

Frequently asked questions