Probabilistic Fatigue Analysis

Why fatigue is intrinsically probabilistic

Fatigue life is one of the clearest examples of why a single deterministic margin can hide important engineering information. Nominally identical coupons tested at the same stress range do not fail at one cycle count; life can vary by orders of magnitude because crack initiation is sensitive to microstructure, surface condition, residual stress, manufacturing detail and environment. In a real structure there is additional uncertainty in applied load, load sequence, stress concentration, mean stress, temperature and the accuracy of the stress model. A probabilistic fatigue assessment treats these sources explicitly and asks a decision-focused question: what is the probability that fatigue failure, an unacceptable crack size or a specified damage state occurs before the required life? The purpose is not to replace sound deterministic fatigue practice, but to quantify the scatter that deterministic knock-down factors are intended to cover.

Define the fatigue limit state

Start by defining failure in engineering terms. For an S–N assessment the limit state may be life below the required cycles, Miner damage exceeding a specified threshold, or local stress exceeding the range for which the fatigue model is valid. For damage-tolerance work it may be crack size reaching an inspection or critical threshold. The limit-state definition must match the programme decision: safe-life, fail-safe, inspection interval or fleet risk. A common form is g(X)=N_f(X)-N_req, where the random input vector X contains uncertain stress, material fatigue parameters, geometry and environmental variables. Failure is g≤0. Treating the acceptance criterion explicitly prevents a vague distribution of predicted lives being mistaken for a reliability statement.

g(X) = N_f(X) - N_req
Failure when g(X) <= 0
Pf = P[g(X) <= 0]

Represent S–N scatter correctly

S–N data are usually modelled in logarithmic coordinates because fatigue life is positive and often approximately lognormal at a fixed stress amplitude. Regression uncertainty, specimen-to-specimen scatter and uncertainty in slope/intercept should be separated where possible. A fitted mean S–N curve is not itself a distribution of fatigue strength: the residual scatter about the curve matters, and confidence in the fitted curve depends on sample size. If design data already contain statistical tolerance factors or mandated scatter factors, do not add a second independent probabilistic scatter without understanding what is already embedded. For welded details, composite structures and additively manufactured parts, the form and variance of the fatigue model may differ materially from polished coupon data; the probability model must follow the applicable evidence.

Propagate load and stress uncertainty

Load uncertainty often dominates life because the S–N relationship is steep. A modest change in stress range can produce a large change in predicted life. The analyst should distinguish uncertainty in the external environment from uncertainty in the transfer from load to local stress. Structural dynamic amplification, contact state, bolt preload, weld geometry and stress-concentration modelling can all be uncertain. Where a finite-element model produces the stress, define which uncertain inputs genuinely change the response and preserve their dependence. It is usually inappropriate to assign an arbitrary independent percentage scatter directly to every stress result. A better model starts from physical uncertain quantities — loads, stiffnesses, dimensions and material properties — and propagates them to the stress history.

Variable-amplitude loading and cumulative damage

For variable-amplitude loading the damage model introduces another layer of uncertainty. Palmgren–Miner summation is convenient but does not capture sequence effects, overload retardation, load interaction or all endurance-limit behaviour. If Miner damage is used, uncertainty in cycle counting, spectrum severity and fatigue curve parameters should be propagated consistently. In simulation, each realisation should use a physically coherent load spectrum rather than independently randomising each bin unless that is genuinely how the environment varies. When the spectrum itself is measured from a limited population, separate unit-to-unit variability from uncertainty in the estimated population distribution. The result is a distribution of accumulated damage or fatigue life rather than one damage number.

D = sum(n_i / N_i)
Pf(t) = P[D(t) >= D_crit]

Monte Carlo, FORM and efficient assessment

Direct Monte Carlo is straightforward when the fatigue calculation is inexpensive: sample the uncertain inputs, calculate stress and fatigue life for each realisation, and estimate the fraction that fail the limit state. The challenge arises when every sample requires nonlinear FEA or a long transient. Then Latin hypercube sampling, response surfaces, polynomial chaos, Gaussian-process surrogates or FORM can reduce cost. Rare failure probabilities demand particular care; zero failures in a small Monte Carlo sample does not demonstrate zero risk. Use confidence bounds or a reliability method suited to the target probability. Surrogates must be validated around the failure boundary, not merely near the mean response, because errors near g=0 directly bias Pf.

Sensitivity, correlation and model validation

Sensitivity analysis should identify whether reliability is controlled by load scatter, fatigue strength, mean stress, stress concentration or another parameter. This directs useful engineering action: reducing uncertainty in a non-influential variable has little value. Correlation can be important — for example, manufacturing processes may simultaneously affect thickness, surface finish and residual stress. Validation should include deterministic fatigue checks, comparison with test data where available, unit checks, tail behaviour and convergence of the probability estimate. If several failure locations or modes exist, retain them individually before forming a system-level probability; taking only the most critical deterministic hot spot can miss a different location that becomes reliability-critical once scatter is included.

What to report

A defensible report states the fatigue model, failure criterion, source and statistical treatment of S–N data, load model, dependence assumptions, sampling method, convergence evidence and the resulting reliability metric. Report distributions or quantiles of life together with Pf at the required life, not only the mean life. Document model limitations such as Miner linearity, notch treatment, weld classification or unmodelled environment. Where the analysis supports inspection or maintenance, translate the result into an interval or decision criterion with the required confidence. The probability number is only meaningful when the chain from data to structural response to fatigue damage is traceable.

A mean fatigue life greater than the required life is not a reliability demonstration. Reliability depends on the lower tail of the life distribution and on how uncertainty in load and fatigue strength is represented.

Key takeaways

• Fatigue reliability is a limit-state problem, not a comparison of mean predicted life with required life.
• Load scatter and S–N scatter must be propagated from physically meaningful uncertain inputs and with dependence preserved.
• Rare-event probabilities require convergence evidence, validated surrogates or an appropriate reliability method.