Sensitivity and importance measures in structural reliability

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.