How to use the Mohr's Circle Calculator
- 1. Enter the stressesσx, σy (tension +) and the shear stress τxy in MPa.
- 2. Add a plane (optional)Give an angle θ to get normal and shear stress on that plane.
- 3. Read the circleSee σ1, σ2, τmax, θp and every step of the working.
Mohr's circle formulas
For plane stress the circle has its centre at C = (σx + σy)/2 on the σ axis and radius R = √[((σx − σy)/2)² + τxy²].
- Principal stresses: σ1 = C + R and σ2 = C − R. No shear acts on these planes.
- Principal angle: tan 2θp = 2τxy / (σx − σy).
- Maximum in-plane shear: τmax = R, on planes 45° from the principal planes, with normal stress C on them.
- On a plane at θ: σn = C + ((σx − σy)/2)cos 2θ + τxy sin 2θ and τ = −((σx − σy)/2)sin 2θ + τxy cos 2θ.
The tool plots point X = (σx, τxy) and Y = (σy, −τxy). The line XY is a diameter, and a rotation of θ on the element is a rotation of 2θ on the circle.
Worked example and exam tips
σx = 80 MPa, σy = −40 MPa, τxy = 30 MPa. C = 20 MPa and R = √(60² + 30²) = 67.08 MPa, so σ1 = 87.08 MPa, σ2 = −47.08 MPa and τmax = 67.08 MPa. tan 2θp = 60/120 gives θp = 13.28°.
- Check your answer: σ1 + σ2 must equal σx + σy.
- Books differ on whether τ is plotted up or down. The stresses do not change, only the picture flips. Follow your textbook's convention in the exam.
- For plane stress the third principal stress is zero, so the absolute maximum shear can be |σ1|/2 when both principal stresses have the same sign.
Frequently asked questions
What is Mohr's circle used for?
It is a graphical way to find principal stresses, maximum shear stress and the stress on any inclined plane from the known stresses on two perpendicular planes.
Why is the angle doubled on the circle?
The stress transformation equations use sin 2θ and cos 2θ, so a rotation θ of the element moves the point 2θ around the circle.
What sign convention does the tool use?
Tension is positive. τxy is positive when it acts upwards on the right-hand face. Angles are anticlockwise from the x-axis.
Is this calculator free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. Every calculation runs in your browser and nothing you type is sent anywhere.