Modified Goodman Factor⚠ unverified

Understanding the Parameters

• Alternating stress $S_a$ — the cyclic amplitude; it is the dominant fatigue driver and is compared against $S_e$. In a notched part, use $K_f S_a$ (Fatigue Stress Concentration).

• Mean stress $S_m$ — the steady offset. A tensile mean stress is damaging (it holds cracks open), which is why it is penalised via $S_{ut}$. Goodman is meant for $S_m \ge 0$; compressive mean stress is benign and usually taken as $S_m = 0$.

• Endurance limit $S_e$ — the Marin-corrected part endurance limit (Marin Endurance Limit), not the raw $S_e'$. Anchors the alternating axis.

• Ultimate strength $S_{ut}$ — anchors the mean axis. Using $S_{ut}$ (rather than yield $S_y$, as in Soderberg) makes modified Goodman less conservative than Soderberg but it does not itself guard against first-cycle yielding — a separate Langer yield check ($S_a + S_m \le S_y$) is needed.

• Factor $n$ — the scale factor by which the load can grow before the operating point reaches the Goodman line. $n>1$ is safe for infinite life.

Derivation (Approaching a Proof)

Plot the operating point $(S_m, S_a)$ in the mean–alternating plane. The modified Goodman failure line joins $(0, S_e)$ to $(S_{ut}, 0)$:

$$\frac{S_a}{S_e} + \frac{S_m}{S_{ut}} = 1.$$

To find the factor of safety, scale the load along the load line through the origin (both stresses grow in proportion, $S_a, S_m \to n S_a, n S_m$) until the scaled point lands on the failure line:

$$\frac{n S_a}{S_e} + \frac{n S_m}{S_{ut}} = 1 \;\Longrightarrow\; \frac{1}{n} = \frac{S_a}{S_e} + \frac{S_m}{S_{ut}}.$$

Solving, $n = \left(\dfrac{S_a}{S_e} + \dfrac{S_m}{S_{ut}}\right)^{-1}$. The line's straightness is the defining assumption — a deliberately conservative chord beneath the true (curved) mean-stress data.

Dimensional check. $\dfrac{S_a}{S_e} + \dfrac{S_m}{S_{ut}} = \dfrac{\text{Pa}}{\text{Pa}} + \dfrac{\text{Pa}}{\text{Pa}}$ = dimensionless, so $1/n$ is dimensionless and $n$ is a pure factor of safety.

History and Development

John Goodman proposed a mean-stress line in 1899. The modified Goodman (endurance limit on the alternating axis, ultimate strength on the mean axis) became the most-taught fatigue criterion of the 20th century (Shigley, Dowling). It sits between the conservative Soderberg (yield on the mean axis) and the data-fitting Gerber parabola: Goodman is the practical middle ground, its linearity trading a little accuracy for a transparent, always-safe estimate for ductile steels.

Related Concepts: Goodman Line, Soderberg, Gerber, Asme Elliptic, Marin Endurance Limit, Fatigue Life cycles

Notes: Linear, conservative; for tensile mean stress ($S_m \ge 0$). Use Marin-corrected $S_e$ and notch-adjusted $K_f S_a$. Does not guard yielding — add the Langer check $S_a + S_m \le S_y$. Compressive mean stress → take $S_m = 0$.