Second-order reliability method (SORM) is considered to be one of the most reliable computational methods for reliability. Its accuracy is generally dependent on three parameters, i.e. the curvature at the most probable point (MPP), the number of random variables and the first-order reliability index. Even though the curvature is usually defined by the eigenvalues of the rotational transformed Hessian matrix (curvature-fitting SORM), it can be obtained by an approximating paraboloid which is fitted to the failure surface at discrete points (point-fitting SORM) and by HL-RF algorithm (HL-RF based SORM). In this work, the accuracy of existing formulas for probability of failure by SORM is examined in terms of not only curvatures but also first-order reliability indices. In addition, advanced first order second moment (AFOSM) FORM method, curvature-fitting, point-fitting, and HLRF based SORM methods are studied and compared one another using several extreme case examples. Each method has its own advantages and disadvantages in the aspect of accuracy, computational complexity, efficiency, and application scope.
In the area of reliability-based design optimization (REDO), reliability analysis is needed to evaluate probabilistic constrains. It can be done in two different ways, the Reliability Index Approach (RIA) and Fixed Norm Formulation (FNF). Both of these involve a nonlinear programming problem with a single constraint and are numerically studied by using a sensitivity-based algorithm and a hybrid algorithm. FNF is shown to be more effective and robust in evaluating probabilistic failure modes in the RBDO process.
