SORM

The Second Order Reliability Method is used under the following context: let \vect{X} be a probabilistic input vector with joint density probability \pdf, let \vect{d} be a deterministic vector, let g(\vect{X}\,,\,\vect{d}) be the limit state function of the model and let \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} be an event whose probability P_f is defined as:

(1)P_f = \Prob{g(\vect{X}\,,\,\vect{d})\leq 0} = \int_{\cD_f} \pdf\, d\vect{x}

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)P_f = \Prob{h(\vect{U}\,,\,\vect{d})\leq 0} = \int_{\Rset^n} \boldsymbol{1}_{h(\vect{u}\,,\,\vect{d}) \leq 0} \,f_{\vect{U}}(\vect{u})\,d\vect{u}

where f_{\vect{U}} is the density function of the distribution in the standard space: that distribution is spherical (invariant by rotation by definition). That property implies that f_{\vect{U}} is a function of ||\vect{U}||^2 only.

Furthermore, we suppose that outside the sphere which tangents the limit state surface in the standard space, f_{\vect{U}} is decreasing.

The difference with FORM comes from the approximation of the limit state surface at the design point P^* in the standard space: SORM approximates it by a quadratic surface that has the same main curvatures at the design point. Let n be the dimension of the random vector \vect{X} and (\kappa_i)_{1 \leq i \leq n-1} the (n-1) 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 E the marginal cumulative density function of the spherical distributions in the standard space:

(3)P_{Breitung}^{generalized} \stackrel{\beta\rightarrow\infty}{=} \Phi(-\beta)\prod_{i=1}^{n-1}\frac{1} {\sqrt{1+\kappa_i^0}}

where \Phi is the cumulative distribution function of the standard 1D normal distribution and (\kappa_1^0, \dots, \kappa_d^0) 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)P_{Hohenbichler} = \Phi(-\beta) \prod_{i=1}^{n-1} \left( 1+\frac{\phi(-\beta)}{\beta \Phi(-\beta)}\kappa_i^0 \right) ^{-1/2}

Recording to the Mill’s ratio, \frac{\phi(-\beta)}{\beta \Phi(-\beta)} tends to 1 when \beta tends to +\infty. This formula is valid only in the normal standard space and if:

1+\frac{\phi(-\beta)}{\beta\Phi(-\beta)}\kappa_i^0 > 0

for any i.

Tvedt’s formula (Tvedt, 1988):

(5)P_{Tvedt} = A_1 + A_2 + A_3

where A_1, A_2 and A_3 are defined by:

A_1 & = \Phi(-\beta) \prod_{j=1}^{N-1} \left( 1+ \kappa_j^0 \right) ^{-1/2}\\ A_2 & = \left[ \beta \Phi(-\beta) - \phi(\beta)\right ] \left[ \prod_{j=1}^{N-1} \left( 1+\kappa_j^0 \right) ^{-1/2} - \prod_{j=1}^{N-1} \left( 1+(1 / \beta + 1) \kappa_j^0 \right) ^{-1/2} \right] \\ A_3 & = (1 + \beta) \left[ \beta \Phi(-\beta) - \phi(\beta)\right ] \\ & \quad \times \left[ \prod_{j=1}^{N-1} \left( 1+\kappa_j^0 \right)^{-1/2} - \operatorname{Re} \left(\prod_{j=1}^{N-1}\left( 1+(\imath / \beta + 1) \kappa_j^0 \right) ^{-1/2} \right) \right]

where {\cR}e(z) is the real part of the complex number z and \imath the complex number such that \imath^2 = -1 and \Phi the cumulative distribution function of the standard 1D normal distribution.

This formula is valid only in the normal standard space and if 1+\kappa_j^0 > 0 and 1+(1/\beta + 1) \kappa_j^0> 0 for any j.