Approximate Integral Method for Nonlinear Reliability Analysis

In the realm of reliability analysis methods, the first-order reliability method (FORM) exhibits excellent computational accuracy and efficiency in linear problems. However, it fails to deliver satisfactory performance in nonlinear ones. Therefore, this paper proposes an approximate integral method (AIM) to calculate the failure probability of nonlinear problems. First, based on the most probable point (MPP) of failure and the reliability index $β$ obtained from the FORM, the limit state function (LSF) can be equivalent to an approximate parabola (AP), which divides the hypersphere space into feasible and failure domains. Secondly, through the ratio of the approximate region occupied by a parabolic curve to the entire hypersphere region, the failure probability can be calculated by integration. To avoid the computational complexity in the parabolic approximate area due to high dimensionality, this paper employs a hyper-rectangle, constructed from chord lengths corresponding to different curvatures, as a substitute for the parabolic approximate area. Additionally, a function is utilized to adjust this substitution, ensuring accuracy in the calculation. Finally, compared with the calculated result of the Monte Carlo simulation (MCS) and the FORM, the feasibility of this method can be demonstrated through five numerical examples.