Weld Fatigue Factor⚠ unverified

Understanding the Parameters

• Fatigue notch factor $K_f$ — how much the weld's geometry amplifies the alternating stress in fatigue. It is applied only to $\sigma_a$ because fatigue damage is driven by the cyclic component; mean stress is not notch-magnified the same way. For as-welded toes $K_f$ can be $2$–$4$; grinding or peening the toe lowers it markedly.

• Alternating stress $\sigma_a$ and mean stress $\sigma_m$ — decompose the load cycle: $\sigma_a = (\sigma_{\max}-\sigma_{\min})/2$, $\sigma_m = (\sigma_{\max}+\sigma_{\min})/2$. Goodman penalises both, but the alternating part (through $S_e$ and $K_f$) dominates.

• Endurance limit $S_e$ — the fully-corrected alternating-stress limit for infinite life; for welds it is heavily reduced by surface, size, and especially the weld condition. It anchors the alternating axis of the Goodman line.

• Ultimate strength $S_{ut}$ — anchors the mean-stress axis. Notably, welded-joint fatigue strength is far less sensitive to $S_{ut}$ than polished-specimen fatigue — a caution against reading too much into the $\sigma_m/S_{ut}$ term for welds.

• Factor $n$ — $n>1$ means the operating point lies inside the Goodman line (safe for the target life); $n<1$ predicts fatigue failure.

Derivation (Approaching a Proof)

The modified-Goodman line (Goodman Line) draws a safe boundary in the $\sigma_a$–$\sigma_m$ plane between the endurance limit $S_e$ on the alternating axis and the ultimate strength $S_{ut}$ on the mean axis. A stress state on the line satisfies

$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1.$$

Introduce the fatigue stress-concentration factor by magnifying the alternating stress, $\sigma_a \to K_f\sigma_a$ (the standard treatment for notched fatigue). To size a design, scale the load by a factor of safety $n$ — both stresses grow to $n K_f\sigma_a$ and $n\sigma_m$ — and place that scaled point exactly on the failure line:

$$\frac{n K_f\sigma_a}{S_e} + \frac{n\sigma_m}{S_{ut}} = 1.$$

Solving for $n$:

$$n = \frac{1}{K_f\,\dfrac{\sigma_a}{S_e} + \dfrac{\sigma_m}{S_{ut}}}.$$

This is the same construction as the Shaft Fatigue Factor and Bolt Fatigue Factor, specialised to a weld's $S_e$ and $K_f$.

Dimensional check. Each term $K_f\,\sigma_a/S_e$ and $\sigma_m/S_{ut}$ is (dimensionless)$\times$ $\text{Pa}/\text{Pa}$ = dimensionless, so the denominator is dimensionless and $n$ is dimensionless — a pure factor of safety.

History and Development

Goodman proposed his mean-stress line in 1899; the modified-Goodman form (endurance limit rather than a fraction of ultimate on the alternating axis) became the standard machine-design fatigue criterion in the 20th century (Soderberg, Gerber, and Goodman lines all appear in Shigley). For welds, however, laboratory-fatigue experience through the mid-20th century showed that joint geometry and defects dominate, and that weld fatigue strength is nearly independent of base-metal strength. Design practice therefore moved to detail categories with nominal-stress S-N curves (S-N Curve) — the AWS/AASHTO/Eurocode approach — which this Goodman formula approximates but does not replace.

Related Concepts: Goodman Line, Modified Goodman Factor, Shaft Fatigue Factor, Bolt Fatigue Factor, Fatigue Stress Concentration, S-N Curve, Endurance Limit steel

Notes: Machine-design Goodman method — real weld fatigue uses detail-category S-N curves (weld strength is nearly independent of $S_{ut}$). $K_f$ magnifies only $\sigma_a$; as-welded toes $K_f \approx 2$–$4$, reduced by toe grinding/peening. Returns $0$ for non-positive $S_e$ or $S_{ut}$.