How Mohr's Circle Works for 2D Stress
Mohr's Circle is a graphical technique developed by Christian Otto Mohr for visualizing the 2D stress transformation equations. Given the normal stresses σx and σy acting on the faces of a small stress element, and the shear stress τxy acting on those same faces, every possible normal-shear stress pair (σ, τ) on a plane cut through that point traces out a circle in σ-τ space. This calculator plots that circle numerically and reads off its most useful landmarks: the principal stresses, the maximum in-plane shear stress, and the angle at which the principal planes occur.
Deriving the principal stresses and angle
The circle's center sits on the shear-free (τ = 0) axis at C = (σx + σy) / 2 — this is the average normal stress, which is invariant under rotation. Its radius is R = √[((σx − σy)/2)² + τxy²], derived directly from the stress transformation equations by treating σx, σy, and τxy as the legs of a right triangle in stress space. The circle crosses the τ = 0 axis at two points, the principal stresses: σ1 = C + R (the algebraically larger principal stress) and σ2 = C − R (the smaller one). The maximum in-plane shear stress τmax equals the radius R itself and occurs 90° around the circle from either principal stress — which corresponds to a physical rotation of only 45° from the principal planes, because Mohr's Circle doubles all angles. The angle from the x-axis to the σ1 plane is θp = ½ · atan2(2τxy, σx − σy).
Common mistakes
- Forgetting the angle-doubling rule: a 90° arc on Mohr's Circle corresponds to only 45° of physical rotation of the stress element — always divide the circle angle by 2.
- Mixing up σ1 and σ2 with σx and σy: the principal stresses are almost never equal to the input normal stresses unless τxy = 0, in which case the x-y axes already are the principal axes.
- Sign convention slips: tensile (positive) and compressive (negative) normal stresses must be entered with consistent signs, and shear stress sign follows the convention used to derive the transformation equations (clockwise shear on the positive x-face is typically taken as positive).
Real-world applications
- Mechanical and structural engineers use Mohr's Circle to check whether a shaft, beam, or pressure vessel wall reaches a critical combined stress state under combined bending, axial, and torsional loads.
- Geotechnical engineers use the same construction (with normal and shear stress on a soil or rock plane) to evaluate failure against the Mohr-Coulomb failure criterion.
- Machine design and fatigue analysis use the maximum shear stress and principal stresses from Mohr's Circle as inputs to failure theories such as maximum shear stress (Tresca) or distortion energy (von Mises).