Shaft Fatigue Factor⚠ unverified

Understanding the Parameters

• Alternating stress $\sigma_a$ — the amplitude of the stress cycle, $\sigma_a = (\sigma_{\max} - \sigma_{\min})/2$. For a rotating shaft, the bending stress is fully reversed, so bending is the main contributor to $\sigma_a$. Fatigue damage is driven primarily by this alternating component, which is why it is divided by the (smaller) endurance limit.

• Mean stress $\sigma_m$ — the steady offset, $\sigma_m = (\sigma_{\max} + \sigma_{\min})/2$. Steady torque and axial loads contribute here. A tensile mean stress reduces fatigue life (it holds cracks open), which the Goodman line accounts for by referencing $\sigma_m$ to the ultimate strength.

• Endurance limit $S_e$ — the fully-reversed stress the material survives indefinitely, after applying the Marin factors (Marin Modification Factors) for surface, size, loading, etc. It is much lower than $S_{ut}$, which is why the $\sigma_a/S_e$ term usually dominates.

• Ultimate strength $S_{ut}$ — anchors the mean-stress axis of the Goodman diagram.

Derivation (Approaching a Proof)

Fatigue under combined alternating and mean stress is described by a failure locus on a $\sigma_m$–$\sigma_a$ diagram. The modified Goodman line connects the endurance limit $S_e$ on the alternating axis to the ultimate strength $S_{ut}$ on the mean axis — a straight line whose equation is

$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1 \quad\text{(failure)}.$$

Any stress state $(\sigma_m, \sigma_a)$ on this line is at the fatigue limit. To express margin, scale both stress components by the factor of safety $n$ — i.e. ask by what factor the load could increase before the point reaches the line. Replacing $\sigma_a \to n\sigma_a$ and $\sigma_m \to n\sigma_m$ at the failure condition:

$$\frac{n\sigma_a}{S_e} + \frac{n\sigma_m}{S_{ut}} = 1 \;\Longrightarrow\; n\left(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}}\right) = 1 \;\Longrightarrow\; n = \frac{1}{\dfrac{\sigma_a}{S_e} + \dfrac{\sigma_m}{S_{ut}}}.$$

The reciprocal form is thus the distance-to-the-Goodman-line expressed as a load multiplier. Alternatives — the Soderberg line (uses $S_y$, more conservative) and the Gerber parabola (fits data better) — replace the straight Goodman locus but use the same factor-of-safety construction.

Dimensional check. Each ratio $\sigma/S$ is dimensionless, so $n$ is dimensionless. ✓

History and Development

The Goodman line dates to John Goodman (1899); the "modified" Goodman (endurance limit to ultimate strength) is the form standardised in Shigley and the ASME shaft code. It rests on Wöhler's 19th-century fatigue testing and the endurance-limit concept, combined with the Marin modification factors for real-part effects. It is the central design tool for infinite-life shaft and machine-element design.

Related Concepts: Goodman Line, Modified Goodman Factor, Marin Modification Factors, Fatigue Endurance Limit, Combined Stress Shaft, Factor of Safety

Notes: Modified Goodman (tensile mean stress). Use $S_e$ after Marin factors. Soderberg (uses $S_y$) is more conservative; Gerber fits scatter better. For combined loading, derive $\sigma_a$, $\sigma_m$ from the von Mises stresses of the alternating and mean load components.