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

TLDR: In this article, a conical representation of yield stresses and their sensitivities is introduced based on a coneY¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ o                 of admissible pairs of external loads/strength increaments, 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 of the yield stresses are represented again by certain expectations or multiple integrals.

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.