Shaft Fatigue Safety Factor — Goodman Criterion and DIN 743 Explained

Before checking fatigue, the shaft must not yield under the maximum load combination. In the textbook route the bending normal stress, the axial direct stress and the torsional shear stress at each cross-section are combined into the von Mises equivalent stress, and the static safety factor is the yield strength divided by it. DIN 743 makes the same check its own way: each peak stress is divided by the component's own yield limit — which includes a static support factor and, at a circumferential notch, a modest yield increase — and the ratios are combined as S_F = 1/√((σ_zdmax/σ_zdFK + σ_bmax/σ_bFK)² + (τ_tmax/τ_tFK)²) ≥ 1.2.

For a solid circular section the bending section modulus is W = π d³ / 32 and the torsional one W_t = π d³ / 16, so σ_b = M / W and τ = T / W_t. Both routes use these nominal stresses; at a notch they are computed on the notch-root diameter.

Under purely reversed bending a shaft fails when the alternating stress amplitude reaches the endurance limit. With a mean stress on top, the modified-Goodman line joins the corrected endurance limit S_e on the amplitude axis to the ultimate strength S_u on the mean axis; the safety factor is the scale factor that puts the operating point on that line. The notch enters through Kf = 1 + q(Kt − 1), Peterson's notch-sensitivity relation, applied to the alternating stress.

DIN 743 reduces the smooth specimen's fatigue strength to the component's with four factors. K1, the technological size factor, lowers the strength of a bar that was heat-treated at a large diameter and so hardened less deeply. K2, the geometric size factor, reflects the gentler stress gradient in a thicker shaft. K_F penalises surface roughness, more so the stronger the steel. And β, the notch factor, is the form factor α divided by a support factor n that rewards a steep stress gradient at the notch root (β = α/n) — or, for keyways and retaining-ring grooves, a notch factor measured on test shafts and scaled to the actual diameter.

The mean stress then lowers the permissible amplitude through the mean-stress sensitivity ψ, computed from the component's fatigue limit and tensile strength. Load case 1 keeps the mean constant as the load grows; load case 2 lets mean and amplitude grow together, which is more conservative and what DIN 743 advises when in doubt. At high mean stress the yield limit takes over. The fatigue safety is S_D = 1/√((σ_zda/σ_zdADK + σ_ba/σ_bADK)² + (τ_ta/τ_tADK)²), and DIN 743 requires S_D ≥ 1.2.

DIN 743 sets S_min = 1.2 for both its proofs, presuming the loads are known accurately; where they are not, a larger margin is the designer's or the contract's call. The Goodman route has no standard minimum — practice for power-transmission shafts is often 1.5 to 2 on fatigue.

The two routes rarely agree to the second digit. DIN 743's notch factor through the support factor, its size factors K1 and K2, and ψ-based mean-stress effect are calibrated on shaft tests; the Goodman line is a general fatigue construction. On a shaft with a sharp shoulder and a large steady torque they can differ by tens of percent in either direction, which is why a German design review asks for the DIN 743 number.