Endurance Limit Unmodified⚠ unverified
Mechanical / Materials · Estimate the unmodified rotating-beam endurance limit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Sut | Sut | Pa | 1.0 | Ultimate tensile strength of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Se' | Pa | Unmodified rotating-beam endurance limit, in pascals (Pa) |
The science & history
Understanding the Parameters
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Ultimate strength $S_{ut}$ — the single input. The empirical fact that a static property predicts the fatigue limit of steel to first order is what makes this one-line estimate possible.
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The 0.5 factor — the midline of the $0.4$–$0.6\,S_{ut}$ scatter band from rotating-beam (R.R. Moore) tests. It is a design estimate, not a per-heat value.
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The "unmodified" / prime — $S_e'$ is the polished, small, fully-reversed specimen value. A real part gets a lower $S_e$ after multiplying by the Marin surface, size, load, temperature, and reliability factors (Marin Endurance Limit).
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Missing ceiling — above $S_{ut} \approx 1400\ \text{MPa}$ the endurance limit plateaus near $700\ \text{MPa}$; this card's bare $0.5\,S_{ut}$ over-predicts for very high-strength steel.
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Steel only — the existence of a true endurance limit is a ferrous phenomenon; non-ferrous alloys use a fatigue strength at a specified life instead.
Derivation (Approaching a Proof)
The endurance limit is empirical, read from the S-N curve (S-N Curve): in a rotating-beam test a polished specimen sees fully-reversed bending, and for steels the S-N curve flattens to a horizontal asymptote near $10^6$ cycles — the amplitude of that asymptote is $S_e'$. Correlating $S_e'$ against $S_{ut}$ across many steels gives the band $S_e' \approx (0.4$–$0.6)\,S_{ut}$, whose midpoint is adopted:
$$S_e' = 0.5\,S_{ut}.$$
There is no mechanistic derivation of the $0.5$; the fatigue limit reflects the threshold below which microcracks do not propagate, which correlates with — but is not derivable from — the ultimate strength.
Dimensional check. $S_e' = 0.5\,S_{ut}$ carries the units of $S_{ut}$ ($\text{Pa} \to \text{Pa}$); $0.5$ is a dimensionless empirical factor.
History and Development
The $S_e' = 0.5\,S_{ut}$ estimate traces to August Wöhler's founding fatigue studies (1860s) and the R.R. Moore rotating-beam test that standardised endurance-limit measurement. It is the entry point of the Shigley stress-life method — every downstream step (Marin correction, Goodman/Gerber criteria, finite-life Basquin) builds on this one number. See Endurance Limit steel for the version with the high-strength ceiling.
Related Concepts: Endurance Limit steel, Marin Endurance Limit, Fatigue Strength Coefficient, Fatigue Life cycles, S-N Curve, Hardness to Tensile Strength
Notes: Duplicate of Endurance Limit steel ($0.5\,S_{ut}$); omits the $\approx 700\ \text{MPa}$ ceiling above $S_{ut} = 1400\ \text{MPa}$. Polished-specimen value — apply Marin factors (Marin Endurance Limit). Steels only.