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Two Second-Order Reliability Methods (SORM)SORM uses two different methods for limit-state function approximation:
An example of Curvature Fitting Method is shown in Figure 1. The main curvatures of the limit-state surface at the MPP in the standard normal space are positive for a convex failure set. There are n-1 main curvatures for a problem having n random variables in the standard normal space. These main curvatures are the Eigen values of the n-1 ´ n-1 second-order derivative matrix of the limit-state surface at MPP in the standard normal space. These main curvatures or the curvature matrix forms a paraboloid that is tangent to the first-order approximation surface at MPP and has the directional cosine of MPP as one of the principal axis in the standard normal space.
Figure 1 SORM, Curvature Fitting Method
For the Point Fitting SORM method , the fitted piecewise paraboloid are obtained by a set of discrete points on the limit state function around the MPP in the principal plane in the rotated standard normal space, . The rotated standard normal space can be generated by the Grand Smith orthogonal process with the directional cosine of MPP as one of the principle axes (see Figure 2).
Figure 2 SORM Point Fitting Method In SORM, the true limit-state functions are used only in the optimization procedure to identify the MPPs and in the calculation the curvatures of the limit-state functions at MPPs. Once MPPs and the corresponding curvatures of limit-state functions are found, the second-order approximate limit-state functions are used to calculate the probability. For component problems, analytical formulas can be used to compute the probability. However, for system problems, the probability can be only calculated by simulation methods and importance sampling methods (denoted as Second-Order Based Simulation, and Second-Order Based Importance Sampling, respectively).
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Last Updated 02/08/10
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