IDB-FAT-029
Fatigue · S-N · endurance limit · Goodman · stress concentration
Fatigue design primer
Designing against cyclic failure — the S-N curve and endurance limit, the Marin modifiers that knock it down, mean-stress (Goodman) correction, and the notches where cracks actually start.
Abstract
Most mechanical parts that break in service break by fatigue — cracking under cyclic stress well below the static strength. Fatigue is driven by the stress amplitude (and the mean stress), grows from stress raisers, and is hugely sensitive to surface finish. This primer is the high-cycle design workflow.
Section 1 covers what fatigue is. Section 2 is the S-N curve and endurance limit. Section 3 is the Marin modifiers that reduce the lab endurance limit to a real one. Section 4 is mean-stress correction (Goodman / Gerber / Soderberg). Section 5 is stress concentration and notch sensitivity. Section 6 is design rules and a checklist.
1.What fatigue is
A part loaded repeatedly can crack and fail at a stress far below its yield strength. A crack initiates at a stress raiser, grows a little with each cycle, and finally fractures the remaining section suddenly. The driver is the alternating stress, modified by the mean stress.
1.1Terms
High-cycle fatigue (>10³ cycles, elastic) is handled with the S-N / endurance approach below. Very-low-cycle, plastic fatigue needs strain-life methods (out of scope here).
2.The S-N curve and endurance limit
Plotting stress amplitude against cycles-to-failure gives the S-N (Wöhler) curve:
- Steels have an endurance limita knee near 10⁶ cycles below which life is effectively infinite. The lab estimate is
Se′ ≈ 0.5 · Sut(capped at ~700 MPa for Sut > ~1400 MPa). - Aluminium and most non-ferrous alloys have no true endurance limitthe curve keeps sloping, so they're rated at a finite life (e.g. fatigue strength at 5×10⁸ cycles).
- The finite-life region between ~10³ and 10⁶ cycles is roughly log-linear; interpolate there for a target life.
Se′ is a polished lab specimen. The real part is always weaker — that's what the Marin factors are for.
3.Endurance-limit modifiers (Marin factors)
Reduce the lab limit to a design value: Se = ka · kb · kc · kd · ke · Se′.
| Factor | Accounts for | Typical value |
|---|---|---|
| ka — surface | finish (cracks start at the surface) | ground ~0.9, machined ~0.7–0.8, hot-rolled ~0.5–0.7, as-forged ~0.3–0.5 |
| kb — size | larger sections fail sooner (bending/torsion) | ~1.0 (d ≤ 8 mm) down to ~0.75 (large); axial = 1.0 |
| kc — load type | bending vs axial vs torsion | bending 1.0, axial 0.85, torsion 0.59 |
| kd — temperature | strength change with temp | ~1.0 near room temperature |
| ke — reliability | scatter for higher survival % | 50% 1.00, 90% 0.90, 99% 0.81, 99.9% 0.75 |
Surface finish (ka) is usually the biggest single knock-down — a forged or corroded surface can halve the endurance limit. This is why fatigue-critical parts are ground, polished, or shot-peened. The standard surface factor is ka = a · Sut^b (a, b per finish; Sut in MPa).
4.Mean stress correction
A tensile mean stress lowers the allowable amplitude. Combine σₐ and σₘ with one of:
| Criterion | Equation | Use |
|---|---|---|
| Goodman (modified) | σₐ/Se + σₘ/Sut = 1/n | Default for brittle-ish / conservative design |
| Gerber | σₐ/Se + (σₘ/Sut)² = 1/n | Ductile steels, less conservative (fits data) |
| Soderberg | σₐ/Se + σₘ/Sy = 1/n | Most conservative; guards against yield too |
Here n is the fatigue factor of safety. Goodman is the usual default. Always also check static yield at the peak stress (σ_max ≤ Sy/n).
5.Stress concentration and notches
Fatigue cracks start at stress raisers — fillets, holes, keyways, threads, shoulders, tool marks. The geometric factor Kt multiplies nominal stress; the fatigue effect is Kf = 1 + q(Kt − 1), where q (notch sensitivity, 0–1) rises with material strength and notch radius.
- Apply Kf to the alternating stress (and often the mean) in the Goodman check.
- A generous fillet radius is the cheapest fatigue improvementsharp internal corners are crack factories.
- Threads, keyways and press-fit edges are classic initiation siteskeep them out of the highest-stress regions or detail them carefully (rolled threads, undercut radii).
6.Design rules and checklist
- Round every internal cornermaximise fillet radii at shoulders and section changes.
- Improve the surface where stress is highfine finish, and shot-peening / surface rolling to induce compressive residual stress (raises fatigue strength markedly).
- Avoid stacking stress raisersdon't put a hole in a fillet in a high-stress zone.
- Keep the mean stress downpreload, residual compression, and symmetric loading help.
- Use the right detail for weldsweld toes are severe notches; grind/peen toes and use fatigue-classified joint details.
- Design checklist: find σₐ and σₘ at the critical section → apply Kf → estimate Se = ka·kb·kc·kd·ke·Se′ → apply Goodman for fatigue n → check static yield at σ_max → if marginal, attack surface finish, fillets and residual stress first.
Pair this with the Ideambox spring and bolted-joint tools, whose fatigue factors use exactly this Goodman approach.