Second-Order Third-Moment Reliability Method

In the second-order reliability method (SORM), the failure probability is generally estimated using a parabolic approximation of a performance function. In the present paper, the moment properties of a second-order approximation of performance functions are investigated, and a moment approximation for a second-order reliability method and a simple second-order third-moment reliability index are proposed for the estimation of failure probability corresponding to both the simple and general parabolic approximations. Based on the property that the parabolic approximation approaches a unit normal random variable, the ranges of three parameters are investigated: the number of variables, the principal curvature, and the first-order reliability index. A simple analytical judgment formula is derived, which can help us judge when the first-order reliability method is sufficiently accurate and when the SORM is required. A simple second-order second-moment reliability index is also proposed for problems with relatively small principal curvatures. Through some numerical examples, the simplicity and accuracy of the second-order second- and third-moment reliability indices are demonstrated.