Endurance Limit (steel)⚠ unverified
Mechanical / Fatigue · Rotating-beam endurance limit estimate for steel
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| ultimate_strength | Sut | Pa | 600000000.0 | Ultimate tensile strength |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| Se | Se' | Pa | Endurance limit |
The science & history
Understanding the Parameters
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Ultimate strength $S_{ut}$ — the single input. The remarkable empirical fact is that a static property predicts the fatigue limit of steel to first order: stronger steel endures higher cyclic stress — but only up to a point.
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The 0.5 factor — the mean of decades of R.R. Moore rotating-beam data; individual steels scatter roughly $0.4$–$0.6\,S_{ut}$. It is a design estimate, not a measured value for a specific heat.
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The 1400 MPa ceiling — above $\approx 1400\ \text{MPa}$ ($200$ ksi) the linear trend breaks: very high-strength steels are dominated by internal defects, so $S_e'$ flattens at $\sim 700\ \text{MPa}$. Pushing strength past this buys no fatigue benefit.
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"Rotating-beam" and the prime — $S_e'$ (with the prime) is the specimen value: polished, small, fully-reversed bending. Real parts get a de-primed $S_e$ after Marin correction.
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Steel only — the existence of a true endurance limit (a horizontal S-N asymptote) is a ferrous/BCC phenomenon. Aluminium and most non-ferrous alloys have no endurance limit; use a fatigue strength at a specified life instead.
Derivation (Approaching a Proof)
There is no first-principles derivation — the endurance limit is empirical, extracted from the S-N curve. In a rotating-beam (R.R. Moore) test a polished specimen is spun under a constant bending moment, so every surface fibre sees fully-reversed stress ($R = -1$). Testing many specimens at descending stress amplitudes and plotting stress versus cycles-to-failure produces the S-N curve (S-N Curve); for steels it develops a distinct knee near $10^6$ cycles below which specimens do not fail. The amplitude at that horizontal asymptote is $S_e'$.
Correlating $S_e'$ against $S_{ut}$ across hundreds of steels gives the band $S_e' \approx (0.4$–$0.6)\,S_{ut}$, whose midline $0.5\,S_{ut}$ is adopted as the estimate, with the $700\ \text{MPa}$ cap where the correlation saturates.
Dimensional check. $S_e' = 0.5\,S_{ut}$ carries the units of $S_{ut}$ directly ($\text{Pa} \to \text{Pa}$); $0.5$ is a dimensionless empirical factor.
History and Development
The rotating-beam fatigue test was standardised by R.R. Moore in the early 20th century, following August Wöhler's founding railway-axle fatigue studies (1860s) that first revealed the endurance limit. The $S_e' = 0.5\,S_{ut}$ rule with a $700\ \text{MPa}$ ceiling is the form given in Shigley and is the universal first step of the stress-life ("infinite-life") design method — everything downstream (Marin factors, Goodman/Gerber/Soderberg criteria) modifies this one number.
Related Concepts: Marin Endurance Limit, Endurance Limit Unmodified, S-N Curve, Fatigue Endurance Limit, Modified Goodman Factor, Fatigue Life cycles
Notes: Rotating-beam, polished, fully-reversed specimen value $S_e'$ — apply Marin factors for a real part (Marin Endurance Limit). $0.5\,S_{ut}$ up to $1400\ \text{MPa}$, then plateau $\approx 700\ \text{MPa}$. Steels only — non-ferrous alloys generally have no true endurance limit.