The Second Order Reliability Method is used in the same context as the First Order Reliability: refer to for further details. The objective of SORM is to evaluate the probability content of the event :
where is the density function of the distribution in the standard space : that distribution is spherical (invariant by rotation by definition). That property implies that is a function of only. Furthermore, we suppose that outside the sphere which tangents the limit state surface in the standard space, is decreasing.
Breitung’s formula is an asymptotic results: the usual formula used in the normal standard space, has been generalized in [lebrun2009b] to standard spaces where the distribution is spherical, with the marginal cumulative density function of the spherical distributions in the standard space:
where is the cumulative distribution function of the standard 1D normal distribution.
Hohenbichler’s formula is an approximation of (3):
This formula is valid only in the normal standard space and if .
- Tvedt’s formula (Tvedt, 1988):
where is the real part of the complex number and the complex number such that and the cumulative distribution function of the standard 1D normal distribution. This formula is valid only in the normal standard space and if and .
Breitung K. a, “Asymptotic approximation for probability integral,” Probability Engineering Mechanics, 1989, Vol 4, No. 4.
Breitung K. b, 1984, “Asymptotic Approximation for multinormal Integrals,” Journal of Engineering Mechanics, ASCE, 110(3), 357-366.
Hohenbichler M., Rackwitz R., 1988, “Improvement of second order reliability estimates by importance sampling,” Journal of Engineering Mechanics, ASCE,114(12), pp 2195-2199.
Tvedt L. 1988, “Second order reliability by an exact integral,” proc. of the IFIP Working Conf. Reliability and Optimization of Structural Systems, Thoft-Christensen (Ed), pp377-384.
Zhao Y. G., Ono T., 1999, “New approximations for SORM : part 1”, Journal of Engineering Mechanics, ASCE,125(1), pp 79-85.
Zhao Y. G., Ono T., 1999, “New approximations for SORM : part 2”, Journal of Engineering Mechanics, ASCE,125(1), pp 86-93.
Adhikari S., 2004, “Reliability analysis using parabolic failure surface approximation”, Journal of Engineering Mechanics, ASCE,130(12), pp 1407-1427.