Hybrid Reliability Analysis — Probabilistic and Non-Probabilistic Methods Combined

Aleatory uncertainty is irreducible natural variability — rolling a die, wind speed fluctuations, material strength scatter across a production batch. It is modelled with probability distributions. Epistemic uncertainty arises from lack of knowledge — a parameter not yet measured, a model whose form is disputed. More data or testing can in principle reduce it; until then it is expressed as an interval or fuzzy set. HRA keeps these two types separate to avoid the precision illusion created by assigning a made-up distribution to an epistemic variable.

The most common approach is a nested loop: the outer loop sweeps the epistemic interval variable across its range (using grid search, optimisation, or vertex methods for interval variables), and for each fixed value of the epistemic variable the inner loop computes the conditional failure probability using Monte Carlo simulation, FORM, or SORM on the remaining probabilistic variables. The minimum and maximum conditional failure probabilities over the sweep define the lower and upper bounds.

Yes, but the nested loop is then computationally expensive because each inner-loop FORM or Monte Carlo call requires many limit-state function evaluations. Surrogate models — response surfaces, polynomial chaos expansions, Gaussian process emulators — are typically trained on a small set of finite-element runs and substituted for the full model inside the loop to make the computation tractable.

A closed interval treats all values within [a, b] as equally possible and everything outside as impossible — a hard boundary. A fuzzy number lets you express that the central value is most plausible and possibility decreases gradually toward the edges. Use fuzzy representations when expert knowledge or partial data suggest a graded possibility, rather than a strict cut-off. If only upper and lower bounds are known with no preference information, a symmetric triangular or trapezoidal fuzzy number or a plain interval is appropriate.

They are related but distinct. HRA asks: given that some inputs are uncertain in a non-probabilistic sense, what is the range of failure probabilities? Robust reliability analysis or reliability-based robust design optimisation goes further and minimises the sensitivity of performance or failure probability to those uncertainties. HRA is a diagnostic and quantification tool; robust optimisation is a design decision tool that may use HRA as its inner evaluation engine.