Stress state (plane stress)
Tension positive, compression negative. σz = τxz = τyz = 0.
Material
Results
Failure envelopes in the σ₁–σ₂ plane
The dashed load line runs from the origin through your point. Where it crosses an envelope is where that theory predicts failure; FoS = (distance to envelope) ÷ (distance to your point).
Mohr's circle PRO
Convention used: point X = (σx, τxy), point Y = (σy, −τxy); τ plotted positive upward.
Hand calculation — step by step PRO
Which theory should I use?
| Theory | Material | Notes |
|---|---|---|
| Distortion energy (von Mises) | Ductile | Best match to test data for ductile metals. This is the "equivalent stress" FEA tools show by default. |
| Max shear stress (Tresca) | Ductile | Slightly conservative (up to ~15% vs von Mises). Simple to calculate by hand. |
| Max normal stress (Rankine) | Brittle | Only safe for brittle materials when stresses have the same sign; unconservative in the mixed quadrant. |
| Coulomb–Mohr (brittle) | Brittle | Uses different tensile and compressive strengths. Recommended for cast iron in the mixed quadrant. |
References: Shigley's Mechanical Engineering Design, Ch. 5; Budynas & Nisbett. Plane stress assumed (third principal stress = 0).