Fatigue Life Estimator (S-N Curve)

What Is Metal Fatigue? Metal fatigue is the progressive cracking of a material under repeated cyclic loading, and it happens even when the peak stress sits well below the material’s tensile strength. Fatigue failure is responsible for 50 to 90% of all mechanical failures in structural components. Famous fatigue failures include the de Havilland Comet crashes of 1954 (caused by stress concentrations at square windows) and the collapse of the Silver Bridge in West Virginia in 1967.

The S-N Curve (Wöhler Curve) The S-N curve plots cyclic stress amplitude (S) vs. number of cycles to failure (N). It was developed by August Wöhler, a German engineer who conducted systematic fatigue tests on railway axles between 1858 and 1870 in Germany. Higher stress amplitude → fewer cycles to failure. Lower stress → more cycles.

Basquin’s Power Law The mathematical description of the S-N curve: σ_a = σ_f’ × (2N)^b Rearranged for life: N = ½ × (σ_a / σ_f’)^(1/b) Where: σ_a = stress amplitude (MPa), σ_f’ = fatigue strength coefficient (≈ UTS for steels), b = fatigue strength exponent (typically −0.05 to −0.12), N = cycles to failure.

Mind the direction of that ratio. It is stress divided by coefficient, and since b is negative and the ratio is less than one, raising it to 1/b turns a small number into a very large number of cycles. Flip the ratio by mistake and you get a fraction of a cycle, which reads as instant failure for a perfectly safe load.

Worked example. A36 steel at a 250 MPa amplitude, with σ_f’ = 460 MPa and b = −0.085. The ratio is 250/460 = 0.5435, and 1/b is −11.76, so 0.5435^−11.76 = 1,305 and N = 652 cycles. That is above the 200 MPa endurance limit, so a finite life is exactly what should come back.

Endurance Limit (Se) Many steels have an endurance limit, a stress level below which fatigue failure never occurs no matter how many cycles you run. For steels: Se ≈ 0.5 × UTS, for UTS below about 1400 MPa. Above that stress, the part will eventually fail. Aluminum alloys have no true endurance limit. They fail at any stress amplitude eventually, just after more cycles. The endurance limit applies to fully reversed bending (R = −1). Surface finish, size, reliability and mean stress all pull it down.

Stress Concentration Factor (Kt) Notches, holes, fillets and threads concentrate stress, which multiplies the local value the metal actually feels. Effective stress amplitude = Kt × nominal stress amplitude. A Kt of 2 does not halve the fatigue life, it cuts it by orders of magnitude, because life depends on stress raised to roughly the eleventh power. Smooth surfaces, compressive residual stresses from shot peening, and generous radii all push it back up.

Mean Stress Effects (Goodman Diagram) The S-N curve assumes fully reversed stress, meaning a mean of zero. A tensile mean stress reduces fatigue life, sometimes dramatically.

The Modified Goodman criterion converts a real load into the fully reversed amplitude that would do the same damage:

σ_ar = σ_a / (1 − σ_m / UTS)

Enter a mean stress below and the calculator applies exactly that correction before running Basquin’s law. A bolt preloaded to half its ultimate strength, cycling ±50 MPa, is not doing ±50 MPa of damage: it is doing the damage of a ±100 MPa fully reversed load.

The Gerber parabola, σ_a / Se + (σ_m / UTS)² = 1, is less conservative and often closer to test data, but Goodman is what design codes use. Compressive mean stress improves fatigue life, which is the whole point of shot peening and pre-stressing.

Common Material Fatigue Parameters Structural steel (A36): Se ≈ 200 MPa, UTS ≈ 400 MPa. High-strength steel (4340, heat treated): Se ≈ 500 to 700 MPa, UTS ≈ 1000 to 1400 MPa. That is a wide band because 4340 is defined by its chemistry, not its strength, and the tempering temperature moves it hundreds of megapascals. The preset below uses the softer end of the range. Aluminum 6061-T6: no endurance limit. This model puts its fatigue strength near 90 MPa at 10⁷ cycles. Titanium Ti-6Al-4V: Se ≈ 500 to 620 MPa, UTS ≈ 900 MPa. Gray cast iron: Se ≈ 100 to 170 MPa.

Safety Factor in Fatigue Design A safety factor of 1.5 to 2.0 on stress amplitude is typical for well-characterized loading. Apply it to the stress, not to the cycle count: because of that eleventh-power relationship, a factor of 2 on stress is worth a factor of several thousand on life, and a factor of 2 on life is worth almost nothing. Variable amplitude loading such as random vibration needs damage accumulation analysis instead, using Miner’s rule. Miner’s rule: Σ(ni/Ni) = 1 at failure. Conservative, widely used, and known to be imperfect.