Mohr’s Circle Calculator (Principal Stresses, Max Shear & Stress Transformation)
Plane-stress Mohr’s circle: principal stresses σ₁ and σ₂, principal angle, maximum shear stress, von Mises stress and the stresses on a plane rotated by any angle, from σx, σy and τxy.
How it works
Centre C = (σx + σy)/2 · radius R = √(((σx − σy)/2)² + τxy²) · σ₁,₂ = C ± R · τmax = R · tan 2θp = 2τxy/(σx − σy) · θs = θp ∓ 45° · σx′ = C + ((σx − σy)/2)cos 2θ + τxy sin 2θ, τx′y′ = −((σx − σy)/2) sin 2θ + τxy cos 2θ · von Mises σ_vm = √(σx² − σxσy + σy² + 3τxy²)
Mohr’s circle is the graphical form of the plane-stress transformation equations: every orientation of a small element maps to a point on a circle centred at the average normal stress, with a radius equal to the maximum in-plane shear. The two ends of the horizontal diameter are the principal stresses, which act on planes with no shear; the top and bottom of the circle give the maximum shear. Rotating the element by θ moves the point by 2θ on the circle. The von Mises stress combines the components for ductile-yield checks. Sign conventions and equations follow Gere & Goodno, Mechanics of Materials (ch. 7) and Hibbeler.
Worked example
With σx = 80 MPa, σy = −40 MPa and τxy = 30 MPa, the centre is (80 − 40)/2 = 20 MPa and the radius √(60² + 30²) = 67.08 MPa. So σ₁ = 20 + 67.08 = 87.08 MPa, σ₂ = −47.08 MPa and τmax = 67.08 MPa. The principal plane is at θp = ½ × atan(60/120) = 13.28°, and maximum shear at −31.72°. The von Mises stress is √(80² + 80 × 40 + 40² + 3 × 30²) = 117.90 MPa. Rotated by 30°: σx′ = 75.98 MPa, σy′ = −35.98 MPa, τx′y′ = −36.96 MPa.
Sources: Gere & Goodno — Mechanics of Materials, Ch. 7 (Analysis of Stress and Strain) · Wikipedia — Mohr’s circle
Frequently asked questions
What are principal stresses?
The maximum and minimum normal stresses at a point, acting on planes where the shear stress is zero. They are what failure theories and brittle-material checks use.
Why is the angle on Mohr’s circle doubled?
The transformation equations depend on 2θ, so turning the physical element by θ moves its point by 2θ around the circle. Principal planes 90° apart sit at opposite ends of a diameter.
What is the difference between in-plane and absolute maximum shear?
In-plane shear is the circle radius. If both principal stresses have the same sign, the true maximum shear involves the out-of-plane stress σ₃ = 0 and equals the larger of |σ₁|/2 and |σ₂|/2.
When should I use von Mises stress?
For ductile metals: yielding is predicted when the von Mises stress reaches the tensile yield strength. Brittle materials are better checked against the principal stresses.
Related calculators
Results assume the idealised textbook model stated above; real systems have losses and tolerances, so verify safety-critical values with measurements or a qualified engineer.