Sensitivity and importance measures in structural reliability

TLDR

In multivariate reliability problems, the sensitivity factor αE[τ] is defined as the derivative of the equivalent reliability index βE(τ) with respect to the parameter τ. The authors derive an asymptotically exact approximation αE[τ]≈α[τ] for extreme reliability levels, provide simple formulas, and show that this approximation clarifies the use of alpha values as importance measures for stochastic variables.

Abstract

Abstract In multivariate reliability problems, which depend on one or more parameters τ, a sensitivity factor αE[τ] is defined as the derivative of the equivalent reliability index βE(τ)=−φ−1[P f (τ)] with P f (τ) the failure probability. αE[τ] expresses the change of αE(τ) due to small variations of τ. Since the numerical evaluation of αE[τ] is usually impractical, an approximation αE[τ]≈α[τ] is derived, which is asymptotically exact for extreme reliability levels. Simple formulae for α[τ] are given. The approximation αE[τ]:a[τ] also provides the basis for a better understanding of the commonly used alpha values αi=−1/βu i∗ as importance measures for stochastic variables.