Principal Stress and Mohr's Circle
Principal stresses, maximum shear stress, and Mohr's circle relationships for 2D and 3D stress states.
Definition
At any point in a stressed body, there exist three mutually perpendicular planes on which the shear stress is zero. The normal stresses on these planes are the principal stresses σ₁ ≥ σ₂ ≥ σ₃. Maximum shear stress occurs on planes at 45° to the principal planes.
Equations (2D)
Principal stresses (plane stress): σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²] Maximum in-plane shear stress: τmax = √[((σx − σy)/2)² + τxy²] = (σ₁ − σ₂) / 2 Principal angle: tan 2θp = 2 τxy / (σx − σy) Mohr's circle centre: C = (σx + σy) / 2 Mohr's circle radius: R = τmax
Variables
| Symbol | Definition | Units |
|---|---|---|
| σx, σy | Normal stresses on x and y planes | MPa |
| τxy | Shear stress on the x-plane in y-direction | MPa |
| σ₁, σ₂ | Principal stresses (2D) | MPa |
| θp | Angle to principal plane from x-axis | degrees |
| τmax | Maximum in-plane shear stress | MPa |
Notes and Limitations
- 2D (plane stress) formulae give the in-plane principal stresses only
- For 3D stress states, the absolute maximum shear stress is (σ₁ − σ₃) / 2
- Mohr's circle sign convention: tension positive, shear positive if it causes clockwise rotation
- Principal stresses are independent of coordinate system orientation
Related Knowledge
See the Structural Analysis Knowledge category for stress transformation theory, tensor notation and 3D stress states.