Analytical derivation of the outcrossing rate in time-variant reliability problems

The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows one to compute the outcrossing rate by solving a two-component parallel system reliability problem using the First Order Reliability Method (FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then, it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load.