# SORM¶

The Second Order Reliability Method is used under the following context: let be a probabilistic input vector with joint density probability , let be a deterministic vector, let be the limit state function of the model and let be an event whose probability is defined as:

(1)¶

The principle is the same as for FORM: we map the physical space into the standard space through an isoprobabilistic transformation).

The integral (1) can be written as:

(2)¶

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.

The difference with FORM comes from the approximation of the limit state surface at the design point in the standard space: SORM approximates it by a quadratic surface that has the same main curvatures at the design point. Let be the dimension of the random vector and the main curvatures of the limit state function at the design point in the standard space. Several approximations are available, detailed here in the case where the origin of the standard space does not belong to the failure domain:

**Breitung’s formula** is an asymptotic result. 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:

(3)¶

where is the cumulative distribution function of the standard 1D normal distribution and the main curvatures of the homothetic of the failure domain at distance 1 from the origin.

**Hohenbichler’s formula** is an approximation of (3):

(4)¶

Recording to the Mill’s ratio, tends to 1 when tends
to .
This formula is valid **only** in the normal standard space and if:

for any .

**Tvedt’s formula** (Tvedt, 1988):

(5)¶

where , and are defined by:

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
for any .