What do σ1 and σ2 mean?

σ1 and σ2 are the principal normal stresses in the 2D stress state. At the principal plane angles, shear stress is zero and the normal stresses reach their maximum and minimum in-plane values.

Is maximum shear stress always equal to the radius?

For the in-plane Mohr circle, yes. The maximum in-plane shear stress equals the radius R. For a true 3D stress state, the absolute maximum shear can involve the out-of-plane principal stress, so a 2D result may not be enough.

Why does the angle double in Mohr’s circle?

A physical rotation of the stress element by θ corresponds to a 2θ movement around Mohr’s circle. This is why the stress transformation equations use cos(2θ) and sin(2θ).

Can I use tensile stress as positive?

Yes, as long as you keep the sign convention consistent. This calculator treats the entered normal stresses directly and uses the listed transformation equations. If your textbook plots positive shear downward, the circle will be mirrored, but the principal stress magnitudes are unchanged.

Does Mohr’s circle predict failure?

Mohr’s circle gives the stress state. Failure prediction needs an appropriate criterion, such as maximum principal stress, Tresca, von Mises, Mohr-Coulomb, or another material-specific model.

Engineering

Mohr's Circle Calculator

Calculate principal stresses, maximum in-plane shear stress, Mohr circle radius, and transformed normal and shear stress for a 2D plane-stress state.

Enter stresses using one consistent unit such as MPa, ksi, psi, or N/mm². Positive shear sign conventions vary by textbook, so the result also shows the transformation equations used.

σx

σy

τxy

Plane angle θ (degrees)

Stress unit label

Principal stress result

—

Circle center and radius—

Maximum in-plane shear—

Principal plane angle—

Stress at θ—

—

How Mohr’s circle converts a stress element into principal stresses

Mohr’s circle plots the normal stress on the horizontal axis and shear stress on the vertical axis. The circle center is the average normal stress, and the radius combines the normal-stress difference and the shear stress. The right and left intersections with the normal-stress axis give the two in-plane principal stresses.

The DoITPoMS stress-analysis resource explains the same relationship by connecting principal stresses with the stress tensor and the rotated stress state in a Mohr’s circle stress analysis lesson.

Worked example

Given σx = 80 MPa, σy = -20 MPa, τxy = 30 MPa: C = (80 - 20) / 2 = 30 MPa R = √[((80 - (-20))/2)² + 30²] = 58.31 MPa σ1 = 88.31 MPa σ2 = -28.31 MPa

The maximum in-plane shear stress equals the circle radius. The normal stress on the maximum-shear planes equals the circle center, not zero unless the average stress is zero.

When should you not rely only on a 2D Mohr’s circle?

This calculator is for plane stress. Thick parts, pressure vessels, highly constrained bodies, and soil or rock stress states may require a full three-dimensional stress analysis. If out-of-plane stress is important, the third principal stress can change the absolute maximum shear stress.

Common questions

σ1 and σ2 are the principal normal stresses in the 2D stress state. At the principal plane angles, shear stress is zero and the normal stresses reach their maximum and minimum in-plane values.
For the in-plane Mohr circle, yes. The maximum in-plane shear stress equals the radius R. For a true 3D stress state, the absolute maximum shear can involve the out-of-plane principal stress, so a 2D result may not be enough.
A physical rotation of the stress element by θ corresponds to a 2θ movement around Mohr’s circle. This is why the stress transformation equations use cos(2θ) and sin(2θ).
Yes, as long as you keep the sign convention consistent. This calculator treats the entered normal stresses directly and uses the listed transformation equations. If your textbook plots positive shear downward, the circle will be mirrored, but the principal stress magnitudes are unchanged.
Mohr’s circle gives the stress state. Failure prediction needs an appropriate criterion, such as maximum principal stress, Tresca, von Mises, Mohr-Coulomb, or another material-specific model.