Goodman vs Soderberg vs Gerber: Mean-Stress Fatigue Correction Explained

The most useful way to visualise mean-stress effects is the Haigh (Goodman) diagram: mean stress σ_m is plotted on the horizontal axis and alternating stress amplitude σ_a on the vertical axis. The corrected endurance limit Se sits at (0, Se) and the relevant static limit (Sut or Sy) sits on the horizontal axis. A criterion draws a boundary between the safe region (below the line) and the failure region (above it). The distance from the operating point (σ_m, σ_a) to the boundary, measured along the ray from the origin, is the fatigue safety factor.

Plotting all three criteria on the same Haigh diagram makes their conservatism ordering immediately visible. The Soderberg line terminates at Sy — always to the left of Sut — so it is the most conservative at any non-zero mean stress. The Goodman line terminates at Sut and lies between the other two. The Gerber parabola curves outward and gives the highest allowable alternating stress at every mean stress level, so it is the least conservative; it is also the curve that best fits experimental data for ductile wrought steels.

A von Mises check against the static yield line (the Langer line: σ_a + σ_m = Sy) is always applied in parallel — the criterion with the lower resulting safety factor governs. Components whose operating point lies above the Langer line yield before they fail by fatigue.

All three criteria share the same corrected endurance limit Se (computed from the raw rotating-beam value Se_prime = 0.5·Sut, then multiplied by the Marin factors for surface finish, cross-section size, loading mode, temperature and reliability). They differ only in the static-limit term that governs the mean-stress penalty.

The modified Goodman criterion uses the ultimate tensile strength Sut as the static limit. It is the standard choice for most machine components because it is moderately conservative, it is linear (easy to apply), and it matches test data reasonably well for high-cycle fatigue. The Soderberg criterion replaces Sut with the yield strength Sy. Because Sy < Sut, the Soderberg line lies inside the Goodman line on the Haigh diagram and gives a smaller (more conservative) safety factor at any given mean stress — it never lets the design yield, even under static loads. It is sometimes preferred for critical brittle materials, but for ductile steels it is widely considered over-conservative. The Gerber parabola is the most accurate fit to test data for ductile steels but is non-linear; it gives a higher (less conservative) safety factor than Goodman, especially at moderate mean stresses, and is therefore preferred by some codes for wrought-steel shafts.

In practice neither σ_a nor σ_m is a simple nominal stress. Two adjustments are nearly always necessary before applying any of the three criteria.

First, stress concentration. At a notch (shoulder fillet, groove, keyway, transverse hole) the local stress is higher than the nominal value by a geometric factor Kt. For fatigue, the effective factor Kf is somewhat lower than Kt because small notches do not fully develop the theoretical concentration — the difference is captured by the Neuber notch sensitivity q (a function of the material's Sut and the notch root radius r): Kf = 1 + q·(Kt − 1). The engine applies Kf to both the alternating and the mean stress components before evaluating any criterion.

Second, combined normal and shear stress. When bending and torsion act simultaneously, the von Mises criterion converts them to a scalar equivalent: σ_a_eq = √(σ_a² + 3·τ_a²) and σ_m_eq = √(σ_m² + 3·τ_m²). These equivalents are then substituted into the Goodman, Soderberg or Gerber expressions directly. The √3 factor already captures the torsional endurance-limit reduction, so the torsional load factor kc is set to 1.0 (not 0.59) in the Se computation — applying both √3 and kc = 0.59 would double-count the same correction.

For most preliminary machine-design work, start with modified Goodman. It is the default in Shigley's Mechanical Engineering Design, gives a linear, easy-to-implement formula, and is moderately conservative for ductile steels. If the resulting safety factor is marginal, switch to Gerber to see whether the Goodman conservatism is hiding an otherwise adequate design; if the design must remain below yield under all circumstances (brittle material, safety-critical application), use Soderberg.

Goodman can be unconservative in two situations. First, at very high mean stresses (σ_m approaching Sut), where the Goodman line slightly over-predicts the experimental failure boundary for some steels — the Gerber parabola tends to be more accurate there. Second, in the low-cycle fatigue regime (fewer than about 10 000 cycles), where the Basquin S-N approach itself breaks down and the Coffin-Manson strain-life relation should be used instead. The calculator's Coffin-Manson panel handles this regime separately, and the mean-stress correction for LCF follows the Morrow or Smith-Watson-Topper correction rather than Goodman.

Always check the static yield safety factor in parallel: nY = Sy / (σ_a_eq + σ_m_eq). The lesser of nY and the fatigue safety factor is the governing result. A large fatigue safety factor accompanied by a small yield safety factor means the component yields before it fatigues — the design needs either higher-strength material or a larger cross-section.