Principal Stress Calculator — 2D Stress Transformation
Given normal stresses σx, σy and shear stress τxy on an element, find the principal stresses (σ₁, σ₂), the maximum shear stress, the angle of the principal plane, and the Von Mises equivalent stress — all from the 2D stress-transformation equations.
MPa
MPa
MPa
Largest normal stress — no shear acts on this plane
- 1
Average normal stress
(100 + 50) ÷ 2 = 75 - 2
Radius R = √(((σx−σy)÷2)² + τxy²)
√(((100 − 50) ÷ 2)² + 30²) = 39.05Radius of Mohr's circle — equal to the maximum shear stress. - 3
σ₁ = average + R
75 + 39.05 = 114.05
How does this calculator work?
Given σx, σy and τxy, the principal stresses are σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)²+τxy²] and the maximum shear stress is τmax = R = the square-root term. The Von Mises stress σ_VM = √(σ₁²−σ₁σ₂+σ₂²) is the ductile yield criterion. No shear stress acts on the principal planes, which are oriented at θp = ½ · atan2(2τxy, σx−σy) degrees.
Formula
How this is calculated
In 2D (plane) stress analysis, the stress state at a point is described by two normal stresses (σx, σy) and one shear stress (τxy). By rotating the coordinate axes to find the orientations at which the shear stress vanishes, we obtain the principal stresses — the maximum and minimum normal stresses that act on perpendicular planes called principal planes. The radius of Mohr's Circle, R = √[((σx−σy)/2)² + τxy²], gives both the maximum shear stress and the distance from the average normal stress to each principal stress.
The principal stress angle θp (relative to the x-axis) is found from θp = ½ · atan2(2τxy, σx − σy) and is measured in degrees. The two principal planes are mutually perpendicular, and the maximum shear stress planes are exactly 45° from them. No shear stress acts on the principal planes — this is by definition.
The Von Mises equivalent stress σ_VM = √(σ₁² − σ₁σ₂ + σ₂²) is the scalar failure criterion used in ductile materials (plane-stress assumption, σ₃ = 0). Material yields when σ_VM reaches the uniaxial yield strength. All inputs and outputs use MPa by convention, but the formulas are unit-agnostic — you can equally use psi or Pa as long as you are consistent.
Frequently asked questions
Principal stresses are the normal stresses on planes orientated so that shear stress is zero. Every 2D stress state has exactly two principal planes that are perpendicular to each other. The larger value σ₁ is the maximum principal stress; σ₂ is the minimum. They bound the range of normal stresses that act on any plane through the point.
Mohr's Circle is a graphical representation of 2D stress transformation. Its centre is at ((σx + σy)/2, 0) and its radius equals τmax = R. The principal stresses σ₁ and σ₂ are the rightmost and leftmost intercepts of the circle with the normal-stress axis. This calculator performs the same algebra numerically.
Von Mises stress is the most widely used failure criterion for ductile metals under complex loading. A component is considered safe when σ_VM < σ_yield / safety_factor. It accounts for the combined effect of all stress components, making it more realistic than comparing only the maximum principal stress to the yield strength (the maximum-normal-stress criterion, which is used instead for brittle materials).
Also known as
TG we-Calculate Editorial Team. (2026). Principal Stress Calculator — 2D Stress Transformation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/principal-stress-calculator
TG we-Calculate Editorial Team. "Principal Stress Calculator — 2D Stress Transformation." TG we-Calculate. 2026. https://we-calculate.com/calculator/principal-stress-calculator.
TG we-Calculate Editorial Team, "Principal Stress Calculator — 2D Stress Transformation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/principal-stress-calculator
@misc{wecalculate_principal_stress_calculator, title = {Principal Stress Calculator — 2D Stress Transformation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/principal-stress-calculator}}, year = {2026}, note = {TG we-Calculate} }
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