A hyperspherical cap area integral method for reliability analysis

•The second-order reliability method and spherical coordinates are combined to accurately solve for the failure probability of an approximate parabolic failure surface.•The hyperspherical cap area is utilized to solve multidimensional reliability integral equations in reduced dimensions.•An equivalent failure probability expression is proposed for solving the general reliability problem.•The superiority of the proposed HCAIM is demonstrated through a large number of examples. In the second-order reliability method, the failure surface consisting of random variables is approximated as a paraboloid in standard normal space. The principal curvatures of the paraboloid are obtained by processing the Hessian matrix to compute the failure probability. However, Breitung's approximate formulation is not always accurate for the reliability problem with the highly nonlinear failure surface. In this paper, based on the approximated paraboloid, a hyperspherical cap area integral method (HCAIM) is presented to improve the accuracy with consistent efficiency. In HCAIM, the hyperspherical cap area expression is combined with the integral method, thus converting the multidimensional failure probability expression into a one-dimensional integral equation to solve for the failure probability of a paraboloid of revolution. An equivalent probability formula is proposed that replaces the failure probability of an elliptic paraboloid with the failure probabilities of multiple paraboloids of revolution. The performance of HCAIM is demonstrated by examples and compared with other methods. The results show that the proposed HCAIM is robust and accurate.