A new direct second-order reliability analysis method

In the second-order reliability method, the limit state function in arbitrarily distributed random variables is approximated by a quadratic polynomial of standard normal variables. The fitted quadratic polynomial is then used to calculate the probability of failure of the limit state. However, a closed-form solution for the probability of failure of a general quadratic polynomial surface is not available. As such, in this paper, a new second-order reliability method for reliability analysis is presented using saddlepoint approximation. The second-order approximation of the limit state function is obtained by the second-order Taylor series expansion at the most probable point. The cumulant generating function of the fitted quadratic polynomial of standard normal variables is derived analytically. The saddlepoint approximation is utilized to generate the probability density function, cumulative distribution function and probability of failure of the limit state. Finally, three numerical examples are used to compare the performance of the proposed second-order reliability with that of the first-order reliability method, the conventional second-order reliability method, and Monte Carlo simulation. The comparisons show that the proposed second-order reliability method gives accurate, convergent, and computationally efficient estimates of the probability of failure.