SORM

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 \cD_f = \{\vect{X} \in \Rset^n \, / \, g(\vect{X}\,,\,\vect{d}) \le 0\} :

(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. After having mapped the physical space into the standard through an isoprobabilistic transformation (1) becomes:

(2)P_f = \Prob{G(\vect{U}\,,\,\vect{d})\leq 0} = \int_{\Rset^n} \boldsymbol{1}_{G(\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 \vect{P}^* in the standard space: SORM approximates it by a quadratic surface that has the same main curvatures at the design point.
Let us denote by n 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 results: 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}{=} E(-\beta)\prod_{i=1}^{n-1}\frac{1}{\sqrt{1+\beta\kappa_i}}

where \Phi is the cumulative distribution function of the standard 1D normal distribution.

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

    (4)\displaystyle P_{Hohenbichler} = \Phi(-\beta_{HL}) \prod_{i=1}^{n-1} \left( 1+\frac{\phi(-\beta_{HL})}{\Phi(-\beta_{HL})}\kappa_i \right) ^{-1/2}

    This formula is valid only in the normal standard space and if \boldsymbol{\forall i, 1+\frac{\phi(-\beta_{HL})}{\Phi(-\beta_{HL})}\kappa_i > 0}.

  • Tvedt’s formula (Tvedt, 1988):

    (5)\left\{ \begin{array}{lcl} \displaystyle P_{Tvedt} & = & A_1 + A_2 + A_3 \\ \displaystyle A_1 & = & \displaystyle \Phi(-\beta_{HL}) \prod_{i=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2}\\ \displaystyle A_2 & = & \displaystyle\left[ \beta_{HL} \Phi(-\beta_{HL}) - \phi(\beta_{HL})\right ] \left[ \prod_{j=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2} - \prod_{j=1}^{N-1} \left( 1+(1 + \beta_{HL}) \kappa_i \right) ^{-1/2} \right ] \\ \displaystyle A_3 & = & \displaystyle(1 + \beta_{HL}) \left[ \beta_{HL} \Phi(-\beta_{HL}) - \phi(\beta_{HL})\right ] \left[ \prod_{j=1}^{N-1} \left( 1+\beta_{HL} \kappa_i \right) ^{-1/2} \right.\\ & & \displaystyle\left. - {\cR}e \left( \prod_{j=1}^{N-1} \left( 1+(i + \beta_{HL}) \kappa_j \right) ^{-1/2} \right)\right ] \end{array} \right.

    where {\cR}e(z) is the real part of the complex number z and i the complex number such that i^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 \boldsymbol{\forall i, 1+\beta \kappa_i > 0} and \boldsymbol{\forall i, 1+(1 + \beta) \kappa_i> 0}.