Mohr’s circle calculator

How Mohr’s circle works

At a point in a body under plane stress, the normal stress σ and shear stress τ on a plane depend on the angle of the plane. Plot σ against τ for every angle and the points form a circle. Mohr’s circle lets you read off the principal stresses, the maximum shear stress and the stresses on any rotated plane, without remembering the transformation equations.

1. Centre: σ_avg = (σ_x + σ_y) / 2.
2. Radius: R = √[((σ_x − σ_y) / 2)² + τ_xy²].
3. Principal stresses: σ_1,2 = σ_avg ± R, with tan 2θ_p = 2τ_xy / (σ_x − σ_y).
4. Maximum in-plane shear: τ_max = R, on planes 45° from the principal planes.

Sign convention

Tension is positive. A positive τ_xy acts in the +y direction on the +x face and in the −y direction on the −x face. Angles are anticlockwise from the x axis. In the diagram, positive shear is plotted downward so that the circle and the element turn the same way. Some courses plot shear upward with a clockwise-positive rule. The circle then looks mirrored, but the principal stresses and the maximum shear are the same.

Worked example

A point has σ_x = 80 MPa, σ_y = −40 MPa and τ_xy = 30 MPa. Find the principal stresses, the maximum shear stress, and the stresses on a plane turned 30° anticlockwise. The material yields at 250 MPa.

1. Sign convention
   Tension is positive and compression is negative. A positive shear stress τ_xy acts in the +y direction on the face whose outward normal is +x. Angles are measured anticlockwise from the x axis to the normal of a plane.
2. Centre of the circle
   σ_avg = (σ_x + σ_y) / 2 = (80 + (−40)) / 2 = 20.0 MPa
3. Radius of the circle
   R = √[ ((σ_x − σ_y) / 2)² + τ_xy² ] = √[ (60.0)² + (30)² ] = 67.1 MPa
4. Principal stresses
   σ₁ = σ_avg + R = 20.0 + 67.1 = 87.1 MPa
   σ₂ = σ_avg − R = 20.0 − 67.1 = −47.1 MPa
5. Principal directions
   tan 2θ_p = 2τ_xy / (σ_x − σ_y) = 2 × 30 / 120, so 2θ_p = 26.6° taken in the correct quadrant.
   The plane carrying σ₁ has its normal at θ_p1 = 13.3° from the x axis. The plane carrying σ₂ is at θ_p2 = −76.7°. There is no shear stress on either plane.
6. Maximum in-plane shear
   τ_max = R = 67.1 MPa, which equals (σ₁ − σ₂) / 2 = (87.1 − (−47.1)) / 2.
   It acts on planes at θ_s = θ_p1 − 45° = −31.7° (and 90° from there), where the normal stress on each plane is σ_avg = 20.0 MPa.
7. Absolute maximum shear
   With σ_z = 0 (plane stress) the three principal stresses are 87.1, 0 and −47.1 MPa. The absolute maximum shear stress is (σ_max − σ_min) / 2 = (87.1 − (−47.1)) / 2 = 67.1 MPa.
8. Element rotated by 30°
   σ_x′ = σ_avg + ((σ_x − σ_y) / 2) cos 2θ + τ_xy sin 2θ = 76.0 MPa
   σ_y′ = σ_avg − ((σ_x − σ_y) / 2) cos 2θ − τ_xy sin 2θ = −36.0 MPa
   τ_x′y′ = −((σ_x − σ_y) / 2) sin 2θ + τ_xy cos 2θ = −37.0 MPa
   Check: σ_x′ + σ_y′ = 40.0 and σ_x + σ_y = 40.0. The sum does not change when the element turns.
9. Yield check
   Von Mises: σ_vm = √(σ₁² − σ₁σ₂ + σ₂²) = 118 MPa
   Tresca: σ_T = largest difference between the principal stresses (including σ_z = 0) = 134 MPa
   With a yield strength of 250 MPa: factor of safety = σ_y / σ_eq = 2.12 (von Mises) and 1.86 (Tresca). A value below 1 means yielding is predicted.

Common stress states

Which formulas does this use?

Each step comes from a formula in the library: Mohr’s circle centre (mean in-plane stress), Principal stresses in plane stress, Principal angle in plane stress, Maximum shear stress (plane stress), Absolute maximum shear stress in plane stress, Von Mises equivalent stress, Tresca equivalent stress.

Common questions

What is Mohr’s circle?

Mohr’s circle is a graph of the normal stress σ and shear stress τ acting on every plane through a point. Its centre is the average normal stress, its radius is the maximum in-plane shear stress, and the points where it crosses the σ axis are the principal stresses. Turning the element by an angle θ moves a point round the circle by 2θ.

How do I find the principal stresses from σx, σy and τxy?

Find the centre σavg = (σx + σy) / 2 and the radius R = √[((σx − σy) / 2)² + τxy²]. Then σ₁ = σavg + R and σ₂ = σavg − R. The calculator above does this and shows each step.

What sign convention does the calculator use?

Tension is positive and compression is negative. A positive shear stress τxy acts in the +y direction on the face whose outward normal is +x. Angles are measured anticlockwise from the x axis. On the circle, positive shear is plotted downward so that a turn of the element and the matching turn on the circle go the same way. Check that this matches your textbook or lecture notes before comparing answers.

What is the difference between the maximum in-plane shear stress and the absolute maximum shear stress?

The in-plane maximum is the radius of the circle. The absolute maximum shear stress is half the difference between the largest and smallest of the three principal stresses, counting the out-of-plane stress as zero for plane stress. If σ₁ and σ₂ have the same sign, the absolute maximum is larger than the in-plane value.

How do I get the stresses on a plane at a given angle?

Use the angle field or the slider. The calculator applies σx′ = σavg + ((σx − σy) / 2) cos 2θ + τxy sin 2θ and τx′y′ = −((σx − σy) / 2) sin 2θ + τxy cos 2θ, and marks the rotated points on the circle.

Can I use it for strain, or for a strain gauge rosette?

Yes. Strain follows the same maths with the shear strain halved. The strain gauge rosette calculator turns three gauge readings into εx, εy and γxy and draws Mohr’s circle of strain.

Is it free, and does it keep my numbers?

It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.