S/N Fatigue Curves: Constant vs Variable Amplitude Loading

Nearly every published S/N curve was measured under steady, repeatable loading. Real structures never see that.

Introduction

The S/N curve — stress amplitude versus cycles to failure — is the foundation of classical fatigue analysis. It is intuitive, experimentally well-established, and embedded in virtually every fatigue design standard. But there is a subtle and important limitation baked into almost every published S/N curve: the data were generated under constant amplitude, fully reversed loading. Real structures rarely see that.

This post examines how S/N curves are derived, what assumptions are embedded in them, and how fatigue behavior changes — sometimes dramatically — under variable amplitude loading.

How S/N Curves Are Derived

The Standard Test Protocol

A classical S/N curve is generated by testing a set of nominally identical specimens to failure, each at a different constant stress amplitude. The test machine applies a sinusoidal load at a fixed frequency, a fixed stress ratio R = σ_min / σ_max, and a fixed waveform shape — until the specimen fractures or a defined crack length is reached.

Key parameters held constant throughout each test:

Stress amplitude σ_a

Or stress range Δσ = 2σ_a

Stress ratio R

Commonly R = −1 (fully reversed) or R = 0 (zero-to-tension)

Frequency

Typically 10–100 Hz for metals in laboratory air

Environment

Temperature, humidity, and corrosive media held fixed

Results are plotted as stress amplitude versus cycles to failure (N_f) on log-log or semi-log axes. In the high-cycle regime a power law — the Basquin equation — is fit to the data:

where σ’_f is the fatigue strength coefficient and b is the Basquin exponent, typically −0.05 to −0.12 for structural metals.

The Endurance Limit

For ferrous metals — steels and cast irons — the S/N curve flattens at a stress amplitude below which fatigue failure does not occur within 10⁶–10⁷ cycles. This is the endurance limit S_e. Aluminum alloys, titanium, and most nonferrous metals do not exhibit a true endurance limit; their S/N curves continue to slope downward into the very high cycle fatigue (VHCF) regime beyond 10⁸ cycles.

Statistical Scatter

Fatigue data are inherently scattered. A published S/N curve is usually a mean fit, sometimes accompanied by −3σ or P50/P90/P99 scatter bands. Scatter arises from microstructural variability, surface finish differences, residual stresses, and test alignment errors. In design, a knockdown factor is applied below the mean curve to account for this uncertainty.

What Constant Amplitude Testing Ignores

The constant amplitude test is a controlled, repeatable measurement — but it embeds assumptions that do not hold in service:

Variable Amplitude Loading: What Changes

Miner’s Rule and Its Limitations

The standard method for applying S/N data to variable amplitude loading is Miner’s linear damage rule:

where n_i is the number of applied cycles at stress amplitude σ_i and N_i is the constant-amplitude life at that level from the S/N curve. Failure is predicted when D = 1.

Miner’s rule is appealingly simple and widely used, but it cannot capture sequence sensitivity — the critical shortcoming under variable amplitude loading.

Overload Retardation

A single tensile overload — a stress cycle significantly above the normal operating range — creates a large plastic zone ahead of the crack tip. Upon return to normal cycling, the compressive residual stress in this plastic zone retards crack growth. Variable amplitude loading with occasional tensile spikes can therefore give longer fatigue life than constant amplitude loading at the same RMS stress. Miner’s rule predicts the same life; the actual structure lives longer.

Underload Acceleration

Compressive underloads — stress cycles going significantly below the minimum of the normal range — can accelerate fatigue crack growth by damaging the crack closure mechanism. The crack closure stress is reduced, exposing the crack tip to more of the applied stress range. This is particularly important in variable amplitude spectra with high tensile mean stresses.

High-Low vs. Low-High Sequencing

Block loading tests have consistently shown:

High → Low sequence

Generally less damaging than Miner’s rule predicts — early overloads create beneficial compressive residual stresses at the crack tip.

Low → High sequence

Generally more damaging than Miner’s rule predicts — small cycles nucleate cracks that the subsequent large cycles then grow rapidly.

The unconservative error in Miner’s rule can be a factor of 2–3 in typical spectra; in aggressive two-block loading programs it can exceed a factor of 10.

Below-Endurance-Limit Cycles

For steels, the constant amplitude endurance limit implies that cycles below S_e cause zero damage. Under variable amplitude loading, this is not conservative. Small cycles below S_e, when interspersed with larger cycles that grow the crack to a critical size, do contribute to damage — the larger cycles raise the local stress intensity above threshold, and the smaller cycles then accumulate additional crack extension that would not occur in isolation.

The Effective S/N Curve Under Spectrum Loading

If one were to derive an S/N curve directly from variable amplitude fatigue tests — holding the shape of the load spectrum constant and scaling its amplitude — the resulting curve would differ from the constant amplitude curve in several ways:

Different slope

The effective Basquin exponent depends on spectral shape. Broad spectra with many small cycles tend to produce shallower effective slopes.

No endurance limit plateau

Because below-limit cycles contribute under spectrum loading, the effective curve continues to slope downward rather than flattening.

Shifted life predictions

Depending on the spectrum, effective fatigue strength at a given life can be higher or lower than constant amplitude data, for the sequence reasons described above.

This is the motivation behind fatigue damage spectrum (FDS) methods in vibration testing: the goal is to reproduce, in an accelerated test, the damage equivalent of the field spectrum rather than simply matching peak amplitudes.

Rainflow Counting: Bridging the Gap

The standard engineering approach to applying S/N data to variable amplitude loading is:

Step 1 — Rainflow count

Extract a distribution of stress cycles (range–mean pairs) from the variable amplitude time history.

Step 2 — Mean stress correction

Apply Goodman, Gerber, or Walker correction to convert each cycle to an equivalent fully reversed amplitude.

Step 3 — Miner’s rule

Sum damage contributions across all cycle bins using the S/N curve.

This is a reasonable engineering approximation embedded in most fatigue analysis codes. Its accuracy depends on how well Miner’s rule captures the actual sequence effects in the specific spectrum.

For random vibration environments where the power spectral density (PSD) is known, spectral fatigue methods — Dirlik, Narrowband, Tovo-Benassi — estimate the rainflow cycle distribution analytically from PSD moments, bypassing the need for time-domain simulation.

Practical Implications for Design

Use Spectrum-Appropriate Knockdown Factors

Design allowables derived from constant amplitude data should include a spectrum severity factor. A Miner’s rule damage sum at failure is commonly observed in the range of 0.3–0.7, not 1.0 — meaning the standard curve can be unconservative by up to a factor of 3.

Do Not Truncate High-Amplitude Cycles

In accelerated testing, it is tempting to remove rare high-amplitude cycles to reduce test time. This eliminates retardation effects and can make the test either more or less severe than the actual field environment — unpredictably.

Beware the Endurance Limit Under Spectrum Loading

For steel components in variable amplitude environments, design to a stress level below the spectrum-extended S/N curve — not the constant amplitude endurance limit.

Consider Dwell Effects for Titanium

Titanium alloys under sustained tensile stress (dwell fatigue) suffer additional damage not captured by rainflow cycle counting alone — another dimension in which constant amplitude S/N data is insufficient for life prediction.

Summary

S/N curves are derived from constant amplitude tests under carefully controlled, repeatable conditions. They are an essential tool, but they carry embedded assumptions — no sequence effects, no overload retardation, a well-defined endurance limit — that do not hold under the variable amplitude loading that structures actually experience.

The effective fatigue behavior under spectrum loading depends on spectral shape, load sequence, overload ratio, mean stress evolution, and environment. Miner’s linear damage rule bridges the gap with reasonable accuracy for many applications, but it can be unconservative by a factor of 2–10 in adverse loading sequences.

Understanding these differences is essential for engineers designing to fatigue-critical requirements under real-world loading conditions.