Inverse Reliability Analysis for Approximated Second-Order Reliability Method Using Hessian Update

According to order of approximation, there are two types of analytical reliability analysis methods; first-order reliability method and second-order reliability method. Even though FORM gives acceptable accuracy and good efficiency for mildly nonlinear performance functions, SORM is required in order to accurately estimate the probability of failure of highly nonlinear functions due to its large curvature. Despite its necessity, SORM is not commonly used because the calculation of the Hessian is required. To resolve the heavy computational cost in SORM due to the Hessian calculation, a quasi-Newton approach to approximate the Hessian is introduced in this study instead of calculating the Hessian directly. The proposed SORM with the approximated Hessian requires computations only used in FORM leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes the generalized chi-squared distribution in order to achieve better accuracy. Furthermore, an SORM-based inverse reliability method is proposed in this study as well. A reliability index corresponding to the target probability of failure is updated using the proposed SORM. Two approaches in terms of finding more accurate most probable point using the updated reliability index are proposed and compared with existing methods through numerical study. The numerical study results show that the proposed SORM achieves efficiency of FORM and accuracy of SORM.