Fatigue Cycle Estimator
Safety margin
Acceptable
How it works
Fatigue is the progressive damage and eventual fracture of materials under repeated cyclic loading at stresses well below the material's ultimate tensile strength. The fatigue life is measured in cycles — the S-N (stress vs. number of cycles) curve defines this relationship.
**S-N curves and endurance limit** A Wöhler (S-N) curve plots stress amplitude vs. cycles to failure on log-log axes. Ferrous materials (steel, titanium) exhibit an endurance limit — a stress below which fatigue failure does not occur regardless of cycle count (typically ~50% of UTS for steel). Non-ferrous materials (aluminum, copper) have no endurance limit — the S-N curve continues declining with more cycles.
**Stress concentration and fatigue** Fatigue cracks almost always initiate at stress concentrations: surface defects, machining marks, holes, keyways, sharp fillets. Stress concentration factor (K_f) directly reduces fatigue strength. Shotpeening compresses the surface, closing crack initiation sites and extending fatigue life 20–200%. Polishing the surface of highly loaded shafts doubles fatigue strength.
**Goodman diagram: mean stress effect** Fatigue strength decreases as mean (tensile) stress increases. The modified Goodman diagram plots alternating stress vs. mean stress — the safe operating region lies below the Goodman line. Compressive mean stress actually improves fatigue life (extends the safe region).
**Variable amplitude loading (Palmgren-Miner rule)** For varying stress amplitudes, cumulative damage: D = Σ(n_i / N_i), where n_i is cycles at stress level i and N_i is cycles to failure at that stress level. Failure predicted when D = 1. This linear damage accumulation model is approximate but widely used in automotive, aerospace, and structural fatigue analysis.
Frequently Asked Questions
- The endurance limit (S_e) is the stress below which a ferrous material will not fail regardless of number of cycles. Approximate rule: S_e ≈ 0.5 × UTS for polished specimens in rotating bending (valid up to UTS = 1400 MPa for steel). For real components, apply modification factors: surface finish (rough machined: 0.7–0.9 of polished), size (>10mm diameter: 0.85–0.9), reliability (99.9%: 0.75), loading type (axial: 0.7, torsion: 0.577). Example: 1040 steel UTS = 620 MPa, S_e_base = 310 MPa, after modifications ≈ 310 × 0.85 × 0.88 × 0.70 = 162 MPa for a rough machined axial component.
- Modified Goodman criterion: σ_alt/S_e + σ_mean/UTS = 1 (safe if ≤1). For alternating stress σ_alt = 100 MPa and mean stress σ_mean = 200 MPa, S_e = 300 MPa, UTS = 600 MPa: 100/300 + 200/600 = 0.33 + 0.33 = 0.66 < 1 — safe. Goodman ratio = 0.66 means safety factor = 1/0.66 = 1.52. Tensile mean stress is harmful (reduces fatigue life). Compressive mean stress is beneficial — shot peening and pre-stressing (prestressed concrete, coil springs) introduce compressive residual stress to improve fatigue life. The Gerber parabola is less conservative; Soderberg is more conservative than Goodman.
- Theoretical stress concentration factor Kt = σ_max / σ_nominal (ratio of local peak stress to nominal stress). For a circular hole in a plate under tension: Kt = 3. For a sharp corner (notch radius → 0): Kt → infinity. The fatigue notch factor Kf = 1 + q × (Kt - 1), where q is the notch sensitivity (0–1, material and notch geometry dependent). High-strength steel: q ≈ 0.9. Low-strength steel: q ≈ 0.6. Kf directly reduces the fatigue strength: effective endurance limit = S_e / Kf. A shaft shoulder with Kt = 2.5, q = 0.9: Kf = 1 + 0.9 × 1.5 = 2.35 → fatigue strength is reduced to 1/2.35 = 43% of unnotched value.
- Real structures experience variable amplitude loading (random road inputs, wind loads, ocean waves). Rainflow counting extracts individual stress cycles from a complex loading history — counting each half-cycle by following the 'flow' down the peaks and valleys of the stress-time history like water flowing down a pagoda roof. Each identified cycle has an amplitude and mean stress. Apply Palmgren-Miner rule: D = Σ(n_i / N_i), where n_i is counted cycles at each stress level and N_i is cycles to failure from the S-N curve. Failure predicted when D = 1. Rainflow counting is implemented in fatigue analysis software (nCode, FE-Fatigue) and is the standard method per ASTM E1049.