Approximation and derivatives of probabilities of survival in structural analysis and design

Abstract

Yield stresses, allowable stresses, moment capacities (plastic moments), external loadings, manufacturing errors, etc., are not fixed quantities in practice, but must be modelled as random variables with a certain joint probability distribution. In reliability-oriented structural optimization the violation of the random behavioural constraints are evaluated by means of the corresponding probabilityp _s of survival. Hence, the approximative computation ofp _s and its sensitivities is of utmost importance. After the consideration of lower bounds ofp _s based on a selection of certain redundants in the vector of internal forces/bending moments, and the consideration of upper bounds ofp _s based on an optimizational representation of the yield or safety constraints by a pair of dual linear programs, a conical representation ofp _s is introduced based on a coneY _o of admissible pairs of external loads/strength increaments. Approximations ofp _s can be constructed then by replacing the (finitely generated) coneY _o by more simple ones, e.g. spherical or ellipsoidal cones. For the direct numerical computation of sensitivities ofp _s and its bounds or approximations by using e.g. sampling methods or asymptotic expansion techniques based on Laplace integral representation of multiple integrals, exact differentiation formulae — of arbitrary order — forp _s and its bounds or approximations with respect to deterministic input or design variables are obtained by applying the transformation method/stochastic completion techniques; the derivatives ofp _s are represented again by certain expectations or multiple integrals.