Fatigue Strength Coefficient⚠ unverified

Understanding the Parameters

• Ultimate strength $S_{ut}$ — the input; both the $10^3$-cycle strength and the true fatigue strength coefficient scale with it.

• The $0.9$ factor — Shigley's fatigue-strength fraction $f$ at $10^3$ cycles for steels ($f \approx 0.9$; slightly lower for very strong steels). Multiplying gives the high-stress anchor of the finite-life S-N line, whose low-stress anchor is the endurance limit $S_e$ at $10^6$ cycles.

• Versus the true $\sigma_f'$ — in strain-life (Coffin–Manson) fatigue, $\sigma_f'$ is the stress-amplitude intercept at one load reversal, typically $\approx \sigma_f$ (true fracture strength) $\approx S_{ut} + 345\ \text{MPa}$. That is *larger* than $0.9\,S_{ut}$ and serves a different (low-cycle, strain-based) purpose.

• Use — feed $0.9\,S_{ut}$ as $f S_{ut}$ into the finite-life fit constants $a,b$ (Fatigue Life cycles).

Derivation (Approaching a Proof)

The stress-life S-N line for steel is fitted between two anchors. At the high-stress end ($N = 10^3$ cycles) the fatigue strength is taken as a fraction of ultimate:

$$S_{f,\,10^3} = f\,S_{ut}, \qquad f \approx 0.9.$$

Hence the registry's $0.9\,S_{ut}$. The strain-life fatigue strength coefficient is a different construct: Basquin's law writes the elastic stress amplitude as $\sigma_a = \sigma_f'\,(2N)^{b}$, so $\sigma_f'$ is the amplitude extrapolated to a single reversal ($2N = 1$). Fitting strain-life data gives $\sigma_f' \approx \sigma_f$ (true fracture stress), well above $0.9\,S_{ut}$. The two coincide in symbol only; this card computes the $10^3$-cycle anchor.

Dimensional check. $\sigma_f' \approx 0.9\,S_{ut}$ carries the units of $S_{ut}$ ($\text{Pa} \to \text{Pa}$); $0.9$ is dimensionless.

History and Development

The two-anchor S-N construction (fatigue strength $f S_{ut}$ at $10^3$ cycles, endurance limit at $10^6$) is the Shigley stress-life recipe, with $f \approx 0.9$ the standard fraction for steel. The genuine fatigue strength coefficient $\sigma_f'$ belongs to the Coffin–Manson strain-life framework developed for low-cycle fatigue in the 1950s–60s. The overlapping symbol is a common source of confusion, which this card's naming reproduces.

Related Concepts: Fatigue Life cycles, Endurance Limit Unmodified, Endurance Limit steel, S-N Curve, Marin Endurance Limit, Cyclic Strain Hardening Exponent

Notes: Output $0.9\,S_{ut}$ is the $10^3$-cycle fatigue-strength anchor ($f S_{ut}$), not the strain-life coefficient $\sigma_f' \approx S_{ut}+345$ MPa (name/value mismatch). Feeds the finite-life fit (Fatigue Life cycles). $f \approx 0.9$ for steel (lower for very strong steels).