Theories of Failure, Mohr's Circle & Fatigue
Principal stresses, which failure theory applies to which material, and S-N curves — the machine-design differentiator in interviews.
Key formulas
σ_vm = √(σ1² − σ1σ2 + σ2²); yield when σ_vm = Sy
Von Mises (plane stress): Distortion-energy theory — the default for ductile steel design and FEA stress plots.
τmax = (σ1 − σ3)/2; yield when τmax = Sy/2
Tresca (max shear): Conservative hexagon inside the von Mises ellipse; easy whiteboard check.
σa/Se + σm/Sut = 1/n
Goodman relation: Mean-stress correction for fatigue life; swap Sut for Sy (Soderberg) or use a parabola (Gerber).
Cheat sheet
- ▸Principal stresses σ1 ≥ σ2 ≥ σ3 are the extreme normal stresses where shear is zero; Mohr's circle radius = τmax = (σ1 − σ3)/2.
- ▸Ductile materials (yield criterion): maximum principal stress theory (Rankine) is unconservative for ductile — use Tresca (max shear) or von Mises (distortion energy, most accurate for ductile metals).
- ▸Brittle materials (fracture criterion): Rankine's maximum principal stress theory works; also Coulomb-Mohr for unequal tensile/compressive strength (e.g. cast iron).
- ▸Fatigue: below the endurance limit (steel ≈ 0.5 × UTS, at 10⁶–10⁷ cycles) infinite life is possible; non-ferrous metals have no true endurance limit.
- ▸S-N curve from rotating-bending (Wöhler) tests; design uses modified endurance limit with factors for surface finish, size, loading type and stress concentration (Kt, Kf).
- ▸Goodman / Soderberg / Gerber lines combine mean stress σm and alternating stress σa; Soderberg (through yield point) is the most conservative, Gerber the least.
Failure theories by material
| Theory | Criterion | Best for | Nature |
|---|---|---|---|
| Rankine (max principal) | σ1 = failure stress | Brittle (cast iron, ceramics) | Simple, conservative for brittle |
| Tresca (max shear) | τmax = Sy/2 | Ductile metals | Slightly conservative |
| Von Mises (distortion energy) | σ_vm = Sy | Ductile metals (steel, Al) | Most accurate, matches test data |
| Mohr-Coulomb | Envelope of Mohr circles | Brittle with σc ≠ σt | Handles unequal strengths |
Visual
Mohr's circle (plane stress, σx > σy > 0):
σ
σ1 ● centre C = (σx+σy)/2, radius R = √(((σx−σy)/2)² + τxy²)
/|\ circle from (σx, −τxy) to (σy, τxy)
----+--------→ σ σ3 ● = −R side; τmax = R at top of circleTop 4 campus interview questions
MCQ mastery quiz
1. The most accurate yield criterion for ductile steel is…
2. Maximum shear stress in plane stress equals…
3. The endurance limit of steel is roughly…
4. Which mean-stress line is the most conservative?