Mohr's Circle Calculator — 2D Stress Transformation
Enter the normal stresses on the x- and y-faces and the shear stress to find the principal stresses (σ₁, σ₂), the maximum shear stress (τmax), and the angle of the principal planes — visualised as Mohr's circle on a σ-τ diagram.
MPa
MPa
MPa
Largest normal stress — shear stress is zero on this plane
- 1
Circle centre C = (σx + σy) ÷ 2
(80 + 20) ÷ 2 = 50 - 2
Half normal-stress difference
(80 − 20) ÷ 2 = 30 - 3
Radius R = √(half-diff² + τxy²)
√(30² + 30²) = 42.426The radius equals the maximum shear stress τmax. - 4
Maximum principal stress σ₁ = C + R
50 + 42.426 = 92.426
How does this calculator work?
For a 2D stress state (σx, σy, τxy): centre C = (σx+σy)/2, radius R = √[((σx−σy)/2)² + τxy²]. Principal stresses are σ₁ = C+R and σ₂ = C−R (zero shear). Maximum shear stress = R. The principal-plane angle is θp = ½·arctan(2τxy/(σx−σy)), which is half the arc angle on Mohr's circle.
Formula
How this is calculated
Mohr's circle is a graphical technique from mechanics of materials for transforming the state of stress at a point to any orientation. The x-face stress state (σx, τxy) and the y-face state (σy, −τxy) are plotted on a σ-τ diagram; they are always diametrically opposite on a circle whose centre lies at C = (σx + σy)/2 on the σ-axis and whose radius is R = √[((σx−σy)/2)² + τxy²].
The principal stresses σ₁ = C + R and σ₂ = C − R occur where the circle crosses the σ-axis — at those orientations the shear stress is zero. The maximum shear stress τmax = R occurs at the top and bottom of the circle, at planes rotated 45° from the principal planes. The physical angle θp to the first principal plane is half the angle 2θp = arctan(2τxy/(σx−σy)) read on the circle.
The calculator assumes a 2D (plane-stress or plane-strain) analysis in a homogeneous, isotropic, linear-elastic material. The third principal stress (out-of-plane, σz) is not computed here. Stresses are entered and shown in MPa; swap to any consistent unit (Pa, kPa, psi) — the geometry of the circle is unit-independent.
Frequently asked questions
Principal stresses are the normal stresses that act on planes where the shear stress is zero. Every stress state has such planes; σ₁ is the algebraically largest and σ₂ the smallest. They define the extremes of normal stress and are critical for failure analysis.
A rotation of the element by angle θ in physical space corresponds to a rotation of 2θ on Mohr's circle. This 2:1 mapping is a consequence of the stress-transformation equations and is the reason the principal-plane formula gives θp = (1/2)·arctan(2τxy/(σx−σy)).
A negative σ indicates compression (the material is being squeezed on that plane). If both σ₁ and σ₂ are negative the element is in biaxial compression. Tensile-dominated and compressive-dominated failures require different checks (von Mises yield vs. compressive crushing).
Also known as
TG we-Calculate Editorial Team. (2026). Mohr's Circle Calculator — 2D Stress Transformation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/mohr-circle-calculator
TG we-Calculate Editorial Team. "Mohr's Circle Calculator — 2D Stress Transformation." TG we-Calculate. 2026. https://we-calculate.com/calculator/mohr-circle-calculator.
TG we-Calculate Editorial Team, "Mohr's Circle Calculator — 2D Stress Transformation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/mohr-circle-calculator
@misc{wecalculate_mohr_circle_calculator, title = {Mohr's Circle Calculator — 2D Stress Transformation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/mohr-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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