Sensitivity and importance measures in structural reliability

Abstract

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