Shaft Fatigue Safety Factor — Goodman Criterion and DIN 743 Explained

Before checking fatigue, the shaft must not yield statically under the maximum load combination. At each cross-section the bending normal stress and the torsional shear stress are combined into the von Mises equivalent stress, and the static safety factor is the yield strength divided by that equivalent stress. DIN 743-1 also adds the axial direct stress σ_ax = F_ax / A for shafts carrying a thrust load, so the full normal stress at the outer fibre is σ_b + σ_ax.

The section modulus for bending is W = π d³ / 32 and the polar section modulus for torsion is W_t = π d³ / 16, so for a solid circular section the torsional shear stress is τ = T / W_t and the bending stress is σ_b = M / W. These two quantities feed directly into the von Mises invariant, and the engine evaluates them at every segment boundary and at every load-application point to find the governing section.

Under purely reversed bending (R = −1) a shaft fails when the alternating stress amplitude equals the endurance limit. Under combined alternating and mean stress the Goodman criterion shows that the two terms are additive in the damage sense: the alternating part is normalised by the component endurance limit S_e and the mean part by the ultimate strength S_u. The safety factor SF_fat is the scalar by which both stresses must be scaled to land exactly on the failure line.

DIN 743 distinguishes the smooth-specimen material endurance limit from the component endurance limit S_e by introducing three reduction factors: the fatigue notch factor Kf (which accounts for the stress raiser geometry — a shoulder fillet, keyway or groove — from DIN 743-2), the size influence factor K_d (larger shafts are more likely to contain a critical defect, from DIN 743-3) and the surface factor K_R (a machined surface is weaker than a polished test specimen, from DIN 743-3). Because Kf is applied to the alternating stress component rather than to S_e directly, the alternating stress that enters the Goodman line already reflects the local notch severity — so a polished nominal stress never masquerades as infinite life on a notched shaft.

The fatigue notch factor Kf accounts for the stress concentration at a geometric feature. It is related to the theoretical stress-concentration factor Kt (from the DIN 743-2 charts for D/d and r/d at a shoulder, or from standardised tables for keyways and grooves) but is always ≤ Kt because the notch sensitivity q of the material means the crack-initiation stress is not as high as the elastic peak. Kf = 1 + q(Kt − 1). High-strength steels have high notch sensitivity (q → 1, Kf → Kt); mild steels and cast irons have lower q.

The size factor K_d captures the metallurgical size effect: a shaft of 100 mm diameter will have a lower effective endurance limit than a 10 mm test specimen turned from the same bar, because the larger volume has a higher probability of containing a fatigue-initiating defect. DIN 743-3 gives K_d as a function of diameter and heat-treatment condition. The surface factor K_R accounts for surface finish: a turned (Ra ≈ 1.6 µm) surface has a lower endurance limit than the polished specimens used in standard fatigue tests, and the correction grows with material strength. Both K_d and K_R are ≤ 1 and together with Kf they define how much of the laboratory endurance limit survives in the real component.

DIN 743-1 does not prescribe a single universal minimum — the required SF depends on load-data uncertainty, the consequence of failure and any additional dynamic factors not already captured in the load case. Industry practice for power-transmission shafts in gearboxes typically targets SF_fat ≥ 1.5 when load data are well known and rising to SF_fat ≥ 2.0 or higher where dynamic overloads are possible or failure is safety-critical. The static safety factor against yielding is usually required to be SF_stat ≥ 1.2 as a separate check.

MechanixCalc flags warning and fail states for the static safety factor and shows the raw margin at each segment so you can apply your own project requirement. The governing section — the notch with the lowest safety factor — is identified automatically across the full multi-segment geometry.