Mohr's Circle Calculator — Principal Stresses & Stress Transformation

See Mohr's Circle Calculator — Principal Stresses & Stress Transformation in action

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1Enter σx, σy and τxy.

2Optionally enter a plane angle and a yield strength.

3Read the principal and shear stresses and the equivalent stresses.

Free Mohr's Circle & Plane Stress Calculator

Mohr's circle is the graphical way to understand how the stress at a point changes with the orientation of the plane you look at. Enter the normal stresses σx and σy and the shear stress τxy (in any consistent units, usually MPa) and this calculator gives the principal stresses σ₁ and σ₂, the angle of the principal planes, the maximum in-plane shear stress, the planes where it acts, the average stress (centre of the circle), and the stress components on any plane you choose at an angle θ — and draws the circle with points X and Y, the principal stresses, τmax and your chosen plane.

For failure checks it also reports the von Mises equivalent stress and the Tresca (maximum shear) equivalent for plane stress, the absolute maximum shear stress including the zero out-of-plane stress, and the safety factor against a yield strength you provide. A built-in check confirms σx + σy = σ₁ + σ₂, and the textbook example σx = 80, σy = −40, τxy = 30 MPa gives σ₁ = 87.08, σ₂ = −47.08, θp = 13.28° and τmax = 67.08 MPa.

Key features

How to use it

1. Enter σx, σy and τxy.
2. Optionally enter a plane angle and a yield strength.
3. Read the principal and shear stresses and the equivalent stresses.
4. Use the circle diagram to check signs and geometry.

Worked example

Example

With σx = 80 MPa, σy = −40 MPa and τxy = 30 MPa the circle has centre 20 MPa and radius 67.08 MPa; principal stresses are 87.08 and −47.08 MPa at 13.28°, von Mises is 117.9 MPa, and with a 250 MPa yield the safety factor is 2.12.

Who uses this tool

Tips for the best results

• Use tension positive, compression negative.
• Check the sign of τxy against your convention (here: upward on the +x face).
• Compare von Mises with yield for ductile metals.
• The principal angle is measured counter-clockwise from the x-axis.

Common mistakes to avoid

• Mixing units between σ and τ.
• Using Mohr's circle for 3-D stress states without adding σ₃.
• Confusing the angle on the circle (2θ) with the physical angle (θ).

Frequently asked questions