FORM (Rackwitz and Fiessler 1978), or by using second order correction in SORM
(Breitung 1984). Thus the safety index can be used directly as a measure of reliability.
One disadvantage of FORM and SORM methods is that there is no readily
available error estimate. The accuracy of FORM and SORM must be verified by other
methods, such as MCS. The errors of FORM and SORM may come from the errors
associated with MPP search and the nonlinearity of the limit state. The search of MPP
requires solving a nonlinear optimization problem, which is difficult to solve for some
problems. Wrong MPP usually leads to poor probability estimates, which is common
problem for MPP-based reliability analysis methods. FORM and SORM methods are also
not well suited for problems with many competing critical failure modes (i.e., multiple
MPPs). Due to the limitations of first-order and second-order approximations, FORM and
SORM methods do not perform well when the limit state surface is highly nonlinear
around MPP. This nonlinearity may come from the inherent nonlinearity of the problem
or may be induced by the transformation from X-space to U-space (Thacker et al. 2001).
For example, transforming a uniform random variable to a standard normal variable
usually increases the nonlinearity of the problem. When FORM and SORM methods
encounter difficulties, sampling methods with VRT such as Importance Sampling can be
employed to obtain/improve results with a reasonable amount of computational cost
compared to direct MCS.
Response Surface Approximations
Response surface approximations (RSA) (Khuri and Cornell 1996) can be used to
obtain a closed-form approximation to the limit state function to facilitate reliability
analysis. Response surface approximations usually fit low order polynomials to the
structural response in terms of random variables. The probability of failure can then be