Principal Stresses (Mohr's Circle)
Principal Stresses from Mohr’s Circle
A point in a loaded body generally carries combined normal and shear stresses. The principal stresses are the maximum and minimum normal stresses (on planes where shear vanishes) — the values compared against a material’s strength in a failure check. This worksheet computes them from a plane stress state.
With σ_avg = (σ_x + σ_y)/2 and the circle radius R = √[((σ_x − σ_y)/2)² + τ_xy²], the principals are σ_avg ± R.
Inputs
A plane stress state with two normal components and one shear component.
sigma_x := 80 MPa=80 MPa
sigma_y := 20 MPa=20 MPa
tau_xy := 30 MPa=30 MPa
Circle centre and radius
sigma_avg := (sigma_x + sigma_y)/2=50 MPa
R := sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2)=42.43 MPa
Principal and maximum shear stresses
sigma_1 := sigma_avg + R=92.43 MPa
sigma_2 := sigma_avg - R=7.57 MPa
tau_max := R=42.43 MPa
Results
σ₁ is the largest tensile principal stress and σ₂ the smallest (or most compressive); the maximum in-plane shear equals the circle radius R. Feed these into a yield criterion — maximum shear (Tresca) or distortion energy (von Mises) — to assess safety.
Assumptions & limitations
- Plane stress (the out-of-plane stress is zero).
- Linear-elastic, homogeneous, isotropic material.
- In-plane maximum shear only — the absolute maximum 3D shear may differ.