Convergence analysis of first order reliability method using chaos theory

First order reliability method (FORM) for the computation of reliability index has been used widely because of its advantages of the efficiency and effectiveness as well simplicity for many years. There exists the phenomenon of convergent failure in the FORM in calculating the reliability index iteratively for some limit state functions, for which the essential factor is investigated using chaotic dynamics theory in the present paper. The bifurcation plots of reliability index are presented for several typical limit state functions, and the computational results from those mapping functions due to FORM iterations show the complicated dynamics phenomena such as the periodic oscillation, bifurcation and chaos. Moreover, the Lyapunov exponents of non-linear map from FORM are calculated. From the numerical investigation of presented examples, it is concluded that the convergence of FORM does not depend on the curvature of design points of the limit state function, and the quantitative index for identifying the convergence of FORM iterative computation is the Lyapunov exponents of non-linear map corresponding to that limit state function.