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In the high-cycle fatigue regime, the Basquin relation approximates the link between stress amplitude and cycles to failure.
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Formula
σa and σ′f use the same stress unit; Nf is cycles; b is dimensionless and typically negative on a descending S–N branch.
| Symbol | Meaning | Unit |
|---|---|---|
| Stress amplitude of a fully reversed loading cycle | Pa | |
| σ′f | Material coefficient of the stress–life curve in the chosen stress unit | Pa |
| Cycles to the defined fatigue failure criterion | load cycles | |
| Material exponent of the Basquin line | dimensionless |
\sigma_a=\sigma_f\prime(2N_f)^bThis is the stress–life form of an S–N curve and is often written using reversals 2Nf. It is an empirical material fit, not a universal fatigue-life guarantee. Here it assumes fully reversed loading and a range dominated by elastic strain. Mean stress, notches, surface condition, scatter, environment and load sequence can change life; suitable correction or fatigue models may be needed. Coefficients must match material, specimen condition, stress unit and test conditions.
Empirical S–N fit for the high-cycle, elastic-dominated regime and fully reversed loading. Material fit, specimen condition and stress unit must match; mean stress and scatter are not automatically included.
On a log–log plot, the power-law relation between stress amplitude and reversal count is a straight line.
Cycle count from stress amplitude
Only for positive amplitudes and a valid material fit; with b<0, Nf decreases as amplitude rises.
An explicitly assumed fit has σ′f=1000 MPa and b=−0.10. Estimate σa at Nf=106 cycles.
The reversal count is 2Nf=2·106.
Thus σa=1000·(2·106)−0·1 MPa≈234 MPa.
This value only demonstrates the calculation; it is not a property of a real material.
Using coefficients from another material or stress unit.
Use the fit, material condition, test conditions and unit together from the documented S–N evaluation.
Formula cluster
Basquin describes a material fit for constant amplitude; Miner approximately sums damage under varying loads.
A calculation example with explicitly assumed fit values only: σ′f=1000 MPa, b=−0.10 and Nf=106 cycles. Then 2Nf=2·106 reversals and σa=1000·(2·106)−0·1 MPa≈234 MPa. Check: because b is negative, predicted amplitude decreases as the cycle count rises; these assumed coefficients are not material data.
Represent S–N data in the high-cycle range and interpret stress amplitudes only with a fit appropriate to the material and test.
Here is how to work through a typical Basquin stress–life relation for fatigue strength (σₐ = σ′f (2Nf)ᵇ) task step by step:
Task
An explicitly assumed fit has σ′f=1000 MPa and b=−0.10. Estimate σa at Nf=106 cycles.
Solution path
The reversal count is 2Nf=2·106.
Thus σa=1000·(2·106)−0·1 MPa≈234 MPa.
This value only demonstrates the calculation; it is not a property of a real material.
The Basquin relation approximates the link between stress amplitude and cycle count in the high-cycle fatigue regime. On logarithmic axes, its power law becomes a straight line whose slope depends on the material fit. With the usual negative exponent, a greater supported cycle count corresponds to a smaller stress amplitude. This is an empirical description of test data, not a universal material constant or guaranteed life. Use coefficients matched to material condition, specimen geometry, loading type and stress unit. Mean stress or component notches may require additional models.
Nf counts complete loading cycles. A fully reversed cycle passes through two strain reversals: from tension to compression and back. Many sources therefore write the Basquin equation with 2Nf, the reversal count. Some references define the independent variable differently or absorb the factor into fitted coefficients. Compare equations only after checking the convention. Substituting Nf when the stated curve requires 2Nf shifts the result systematically and can lead to an incorrect life estimate. Record that convention alongside the fitted data.
Rearrange σa=σ′f(2Nf)^b to obtain Nf=½(σa/σ′f)^(1/b). The stress ratio is dimensionless, so its power is well-defined; halving the reversal count gives complete cycles. Use positive stress amplitudes and the exponent from a matching material fit. With negative b, predicted life increases as amplitude falls. Check that the result lies within the measured S–N data range. Extrapolation far beyond the tests is especially uncertain. Treat extrapolation beyond the test range with particular caution. Compare results only under the same fitted convention. within that test range.
Not by itself. The equation usually describes a fitted curve from fatigue tests under defined conditions. A real component may have different mean stress, notch effects, surface finish, size, corrosion, temperature and load sequence; fatigue data also scatter. Design needs suitable material and component rules, safety or reliability criteria, and documented loading assumptions. A Basquin curve can be an input or comparison, but it does not replace those steps. The numerical example here uses assumed fit values and must not be adopted as a design property.
σ′f is the fitted strength coefficient for the chosen Basquin representation. Since (2Nf)^b is dimensionless, σ′f carries the same stress unit as σa, such as MPa. Its numerical value therefore depends on the unit: a coefficient stated in MPa cannot be combined unchanged with stresses in pascals. It is also not a general tensile-strength value. Determine it with b from fatigue data for the stated material and test configuration, and document whether the curve uses cycles or reversals. A fit without these metadata is difficult to transfer.
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Which formula describes Basquin stress–life relation for fatigue strength?
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How do you rearrange Basquin stress–life relation for fatigue strength for Cycle count from stress amplitude?
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Which common mistake happens with Basquin stress–life relation for fatigue strength?
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