New approximations for SORM: Part 1

In the second-order reliability method the principal curvatures, which are defined as the eigenvalues of rotational transformed Hessian matrix, are used to construct a paraboloid approximation of the limit state surface and compute a second-order estimate of the failure probability. In this paper, the accuracy of the previous formulas of SORM are examined for a large range of not only curvatures but also number of random variables and first-order reliability indices. For easy practical application of SORM in engineering, a simple approximation of SORM is suggested and an empirical second-order reliability index is proposed. By the new approximations, SORM can be easily applied without rotational transformation and eigenvalue analysis of Hessian matrixes. The empirical reliability index proposed in this paper is shown to be simple and accurate among the existing SORM formulas with closed forms. The proposed empirical reliability index gives good approximations of exact results for a large range of curvature radii, the number of random variables, and the first-order reliability indices.