Analytical derivation of the outcrossing rate in time-variant reliability problems

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1 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev Structures and Infrastructure Engineering, Vol. 00, No. 00, March 2006, 4 Analytical derivation of the outcrossing rate in time-variant reliability problems Bruno Sudret Electricité de France, R&D Division, Site des Renardières, F-7788 Moret-sur-Loing Revsion received 00 Month 200x) The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows to compute the outcrossing rate by solving a two-component parallel system reliability problem and its resolution by the First Order Reliability Method FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load. Keywords : Time-variant reliability - First passage problem - PHI2 method - outcrossing rate - FORM INTRODUCTION Structural reliability analysis aims at computing the probability of failure of a mechanical system with respect to a prescribed failure criterion by accounting for uncertainties arising in the model description geometry, material properties) or the environment loading). When the behavior of the system under consideration evolves in time, the reliability problem is referred to as time-variant. In these problems, the time dependency may be of two kinds : loading may be randomly varying in time : stochastic processes are introduced in the analysis. This allows to account for environmental loading such as wind velocity, temperature or wave height, occupancy loads, traffic loads, etc. material properties may be decaying in time. The degradation mechanisms usually present an initiation phase and a propagation phase. Both the initiation duration and the propagation kinetics may be considered as random in the analyses. A large amount of litterature on time-variant reliability is based on so-called asymptotic approaches, see Rackwitz 200) for a comprehensive review. However the mathematics of these approaches as well as their implementation reveal complex and can hardly be found in commercial softwares COMREL RCP Consult 998) being a remarkable exception). In this respect, an alternative class of methods called parallel system methods in the sequel) have been proposed. The original paper is due to Hagen and Tvedt 99). The idea has been further investigated by Li and Der Kiureghian 995), Der Kiureghian 2000), Koo and Der Kiureghian, A. 2003), Koo et al. 2005), Fujimura and Der Kiureghian, A. 2005) in the context of random vibrations, and by Der Kiureghian and Zhang 999) in the context of space-variant finite element reliability. A new formulation of the approach has been proposed recently by Sudret et al. 2002), Andrieu-Renaud et al. 2004) under the acronym PHI2 method. The present paper proposes new analytical derivations which allow to make the method more stable and easily applicable to time-variant reliability problems involving vector processes, using basic time-invariant reliability tools such as FORM/SORM methods. Structures and Infrastructure Engineering ISSN print / ISSN online c 2006 Taylor & Francis DOI: 0.080/573247YYxxxxxxxx

2 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 2 Analytical derivation of the outcrossing rate in time-variant reliability problems 2 BASICS OF TIME-VARIANT RELIABILITY ANALYSIS Let us denote by Xt,ω) the set of the random variables {X j ω),j =,...,p} and random processes {X k t,ω),k = p +,...,p + q} describing the randomness in the problem under consideration. In this notation, ω stands for the outcome in the space of outcomes Ω. Let us denote by g t,xt,ω)) the timedependent limit state function. As usual, the safe domain corresponds to positive values of g and the failure domain to negative values of g, whereas the limit state surface corresponds to zero values of g. The probability of failure of the system over the time interval [t,t 2 ] is defined as : P f t,t 2 ) = P ) t [t,t 2 ], g t,xt,ω)) 0 ) Denoting by N + t,t 2 ) the number of upcrossings of zero-value by the compound process g t,xt,ω)) from the safe domain to the failure domain within ]t,t 2 ], the probability of failure also reads : P f t,t 2 ) = P {g t,xt,ω)) 0} { N + t,t 2 ) > 0 }) 2) The following classical bounds on P f t,t 2 ) are available Rackwitz 200): max t t t 2 P f,i t) P f t,t 2 ) P f,i t ) + E [ N + t,t 2 ) ] 3) where P f,i t) is the so-called point-in-time or instantaneous probability of failure obtained by considering time as fixed. The mean number of outcrossings E[N + t,t 2 )] in Eq.3) can be evaluated from the outcrossing rate. The latter is defined as follows : ν + P N + t,t + t) = ) t) = lim t 0 t 4) The numerator of Eq.4) is nothing but the probability of having one outcrossing in the time interval [t,t + t]. Thus the alternative expression of ν + t) : ) P {g t,xt,ω)) > 0} {g t + t,xt + t,ω)) 0} ν + t) = lim t 0 t This expression is the starting point of the current communication. 5) 3 ANALYTICAL EXPRESSION OF THE OUTCROSSING RATE 3. Notation Let us introduce the following quantity : f t h) = P ) {g t,xt,ω)) > 0} {g t + h,xt + h,ω)) 0} 6) It is straightforward to see that f t 0) = 0 since the events in the intersection are disjoint. From this remark and Eq.5) it comes : ν + f t h) f t 0) t) = lim = f t 0) 7) h 0 h

3 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev Analytical derivation of the outcrossing rate in time-variant reliability problems 3 where f t.) is the derivative of f th) with respect to h. In the sequel, the notation will be used to denote the derivative of a function of one single variable with respect to this variable. When interpreting Eq.6) as a two-component parallel system, the use of the First Order Reliability Method FORM) leads to Ditlevsen and Madsen 996) : f t h) = Φ 2 βt), βt + h),ρt,h)) 8) where Φ 2 x,y,ρ) is the binormal CDF, βt) is the point-in-time reliability index at time instant t, βt+h) is the point-in-time reliability index at time instant t + h. Moreover the correlation coefficient ρt, h) between the limit state functions at t and t + h is defined as : ρt,h) = αt) αt + h) 9) i.e., it is the obtained by the dot product between the design point directions αt) = U t)/βt) and αt + h) = U t + h)/βt + h), U being the design point in the standard normal space. 3.2 Stationary case Let us suppose first that the point-in-time reliability problem is time-independent. This is the case when the limit state function does not depend explicitly on t, and when the random processes under consideration are stationary in nature. In this case, the instantaneous reliability index does not depend on time : βt) β. Thus : f t h) = Φ 2 β, β,ρt,h)) 0) f th) = Φ 2 ρt,h) β, β,ρt,h)) ρ h ) Reminding that Ditlevsen and Madsen 996) : Φ 2 x,y,ρ) ρ = 2 Φ 2 x,y,ρ) x y = ϕ 2 x,y,ρ) 2) one gets : f t h) = ϕ 2β, β,ρt,h)) ρt,h) h 3) The outcrossing rate ν + t) thus reads in the stationary case Eq.7)) : ν + = lim h 0 ϕ 2 β, β,ρt,h)) ρt,h) h 4) As the limit value of the above expression corresponding to h 0 is of interest, the following series expansion are used : αt + h) = αt) + hα t) + h2 2 α t) + oh 2 ) 5) α t + h) = α t) + hα t) + oh) 6)

4 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 4 Analytical derivation of the outcrossing rate in time-variant reliability problems where oh p ) means negligible with respect to h p. Reminding that α is a unit vector, one gets : αt) 2 = αt) 2 = αt) α t) = 0 7) [ αt) α t) ] = α t) 2 + αt) α t) = 0 8) Multiplying 5) by αt) yields : ρt,h) = αt) αt + h) = h2 2 αt) α t) + oh 2 ) = + h2 2 from which one gets : α t) 2 + oh 2 ) 9) ρt, h) h = αt) α t + h) = hαt) α t) + oh) = h α t) 2 + oh) 20) The binormal PDF can be classically expanded around a correlation coefficient ρ = + ε as follows see Appendix A) : ϕ 2 β, β, + ε) 2π 2ε e β2 /2 2) When using the latter approximation with ε = h2 2 α t) 2 see Eq.9)), one gets : ϕ 2 β, β,ρt,h)) 2πh α t) e β2 /2 22) Substituting for Eqs.20),22) in 4), one finally gets : ν + t) = ϕβ) 2π α t) 23) Note that ν + t) and α t) do not actually depend on t since the problem is stationary. 3.3 Non stationary case In the non stationary case, the second and third arguments of Φ 2 in Eq.8) depend on h. Thus the derivative : f th) = A t,h) {}}{ Φ 2 y βt), βt + h),ρt,h)) β t + h) + Let us focus now on the first term. First remind that see Appendix A): Φ 2 x,y,ρ) = ϕy)φ y A 2t,h) {}}{ Φ 2 ρt,h) βt), βt + h),ρt,h)) ρ h ) x ρy ρ 2 24) 25) Thus : A t,h) = ϕ βt + h))φ ) βt) + ρt,h)βt + h) β t + h) 26) ρ 2 t,h)

5 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev Analytical derivation of the outcrossing rate in time-variant reliability problems 5 Again, as the limit h 0 is of interest, the following series expansions are used : βt + h) = βt) + hβ t) + oh) 27) ρ 2 t,h) = αt) αt + h)) 2 = h 2 α t) 2 + oh 2 ) 28) βt) + ρt,h)βt + h) = hβ t) + oh) 29) Thus the limit : ) A t,0) = lim A t,h) = β t)ϕβt))φ β t) h 0 α t) 30) Let us consider now the second term of Eq.24). Using 2), one gets : where: ϕ 2 βt), βt + h),ρt,h)) = A 2 t,h) = ϕ 2 βt), βt + h),ρt,h)) ρt,h) h [ 2π ρ 2 t,h) exp 2 β 2 t) + 2ρt,h)βt)βt + h) + β 2 ] t + h) ρ 2 t,h) 3) 32) Using Eqs.9),27), the numerator in 32) can be rewritten after some basic algebra as: β 2 t) + 2ρt,h)βt)βt + h) + β 2 t + h) = h 2 β 2 t) α t) ) 2 + β 2 t) + oh 2 ) 33) whereas the denominator is given in Eq.28). Thus the series expansion of the binormal PDF for small values of h : [ ϕ 2 βt), βt + h),ρt,h)) 2πh α t) exp )] β 2 t) + β 2 t) 2 α t) 2 β ) 34) = h α t) ϕβt))ϕ t) α t) By substituting Eqs.20),34) in 3), one finally gets : Substituting 30),35) in 24) yields: A 2 t,0) = lim A 2 t,h) = α t) β ) t) ϕβt))ϕ h 0 α t) ν + t) = A t,0) + A 2 t,0) = α t) β ) ) t) ϕβt))ϕ α β t)ϕβt))φ β t) t) α t) 35) 36) Introducing the classical notation : Ψx) = ϕx) xφ x) 37) the outcrossing rate in the non stationary case eventually reads : ν + t) = α t) β ) t) ϕβt))ψ α t) 38)

6 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 6 Analytical derivation of the outcrossing rate in time-variant reliability problems Note that the latter equation contains in itself the stationary case see Eq.23)). Indeed, in this case, β t) = 0 and Ψ0) = / 2π. It is interesting to observe that the above equation looks like classical expressions of the outcrossing rate see e.g. Rackwitz 998)) obtained from generalizations of Rice s formula. 4 IMPLEMENTATION IN A TIME-INVARIANT RELIABILITY CODE The so-called PHI2 method which is described here has been presented by Sudret et al. 2002), Andrieu- Renaud et al. 2004) from an original idea by Hagen and Tvedt 99). It can be summarized as follows : The time-invariant reliability index βt) associated with the limit state {g t, Xt, ω)) 0} is computed after having frozen t in all functions of time and having replaced the random processes {X k t,ω),k = p +,...,p + q} by random variables {X ) k ω),k = p +,...,p + q}. FORM analysis corresponds to approximating the limit state surface by the hyperplane of equation βt) αt) u = 0 in the standard normal space. In this expression, αt) denotes the unit vector to the design point, which is the opposite of the normalized gradient of the limit state function at the design point. Consequently, the reliability index associated with {g t, Xt, ω)) > 0} is βt). The time-invariant reliability index βt+ t) associated with the limit state {g t + t, Xt + t,ω)) 0} is computed by another FORM analysis. It is important to notice that the random processes {X k t,ω),k = p +,...,p + q} are now replaced by another set of random variables {X 2) k ω),k = p +,...,p + q} that are different from, although correlated to X ) k ω). The coefficients of correlation between X ) k and X 2) l are computed from the autocorrelation coefficient matrix R of the vector random process : ρx ) k,x2) l ) = R k,l t,t + t) 39) The limit state surface is approximated by the hyperplane of equation βt + t) αt + t) u = 0 in the standard normal space. The outcrossing rate is evaluated by the finite difference version of Eq.38) : ) ν + αt + t) αt) βt + t) βt) PHI2 t) = ϕβt))ψ t αt + t) αt) 40) The latter equation is to be compared to the finite difference equation in the original PHI2 method: ν +,old PHI2 t) = Φ 2βt), βt + t),ρt, t)) t 4) The obtained outcrossing rate may be used to bound the probability of failure according to Eq.3). In case of stationary problems, it is constant, thus the expression P f t,t 2 ) ν + t t 2 ). In non stationary cases, two strategies may be adopted: when a quick approximate value of P f t,t 2 ) is looked after, asymptotic Laplace integration may be used see Rackwitz 998) for details). If the evolution in time of the probability of failure is of interest, then the outcrossing rate should be evaluated at various points in time and integrated using a trapezoidal quadrature scheme. The results in this case are usually more accurate and less computationally expensive than the repeated use of the asymptotic integration see Andrieu-Renaud et al. 2004)). Note that the major advantage of the PHI2 method as presented here is to decouple the computation of ν + t) and its time integration. This allows for a proper selection of the time integration scheme suitable to the problem under consideration.

7 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 5 Analytical application example Analytical derivation of the outcrossing rate in time-variant reliability problems 7 In order to validate the above mathematical derivations and to illustrate the PHI2 method step-by-step, let us consider in this section a simple non-stationary time-variant reliability problem. The limit state function reads : gt,r,st)) = R bt St) 42) where R is a Gaussian random variable mean value µ R, standard deviation σ R ) and St) is a stationary Gaussian random process mean value µ S, standard deviation σ S ). The autocorrelation coefficient function of the process is denoted by ρ S t). The term bt models a deterministic linear decrease in time of the resistance R. The random variables R and St) are independent. The outcrossing rate associated to the limit state surface can be computed analytically in this case Sudret et al. 2002). The conditional outcrossing rate obtained for a given realization r of R indeed reads Cramer and Leadbetter 967) : r bt ν + µs r,t) = ω 0 ϕ where Ψ.) is given in 37) and the cycle rate ω 0 is defined by : Thus the unconditional) outcrossing rate : ν + t) = σ S ) ) b Ψ ω 0 σ S 43) ω0 2 = d2 ρ S t) dt 2 = ρ S0) 44) t=0 This integral can be evaluated analytically Owen 980) : ) b ν + t) = ω 0 Ψ ω 0 σ S ν + r,t) ) r µr ϕ dr 45) σ R σ S ϕ σ R µ R bt µ S 46) As the limit state function Eq.42) is linear with respect to R and S, a method based on FORM should provide the exact result in terms of outcrossing rate. The PHI2 approach developed in Section 4 is now applied to confirm this statement. Time-invariant analysis at t. Let us consider the time-invariant limit state function : g t,r,s ) ) = R bt S ) 47) where S ) represents the Gaussian random variable St), with mean value µ S, standard deviation σ S. In order to transform the above limit state function in the standard normal space, let us introduce the standard normal random vector of components U,U 2 ) such that : R = µ R + σ R U S ) = µ S + σ S U 2 48) The limit state function 47) thus transforms into the standard normal space as : G t,u,u 2 ) = µ R bt µ S + σ R U σ S U 2 49)

8 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 8 Analytical derivation of the outcrossing rate in time-variant reliability problems It can by normalized as follows : G t,u,u 2 ) = µ R bt µ S σ R U + σ S U 2 ) 50) σr 2 + σ2 S The latter expression is obviously linear in the U-space. The reliability index β t) and related α-vector obtained by FORM which is exact due to linearity) read : βt) = µ R bt µ S σr 2 + σ2 S αt) = σr 2 + σ2 S ) σr σ S 5) 52) Time-invariant analysis at t + t. Let us consider now the second time-invariant FORM analysis at t + t. The limit state function reads : g 2 t,r,s 2) ) = R bt + t) S 2) 53) where S 2) represents the Gaussian random variable St + t), with mean value µ S, standard deviation σ S. Note that this random variable is correlated to S ) : ρ S ),S 2) = ρ S t) 54) The input random variables X = R,S ),S 2) ) are now transformed into standard normal variables U = U,U 2,U 3 ) using the Cholesky decomposition of the correlation matrix of X : R = µ R + σ R U S ) = µ S + σ S U 2 S 2) = µ S + σ S U U 3 ) 55) The normalized limit state function in the standard normal space now reads : G 2 t + t,u,u 2,U 3 ) = µ R bt + t) µ S [ σ R U + σ S U 2 + )] 2 U 3 σr 2 + σ2 S 56) Thus the FORM solution : βt + t) = µ R bt + t) µ S 57) σr 2 + σ2 S σ R αt + t) = σ S 58) σr 2 + σ2 S 2 σ S In order to evaluate the outcrossing rate by Eq.40), the increments of the reliability index and the α-vector between the situations should be computed. Note that the vector αt) should be given a third component

9 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev Analytical derivation of the outcrossing rate in time-variant reliability problems 9 corresponding to standard normal variable U 3 ) equal to zero in order to perform this calculation : From the latter equation one gets : βt) = b t 59) σr 2 + σ2 S 0 αt) = )σ S 60) σr 2 + σ2 S 2 σ S αt) = σ S 2 2 6) As we have posed = ρ S t), and t <<, the following approximation reads after Taylor expansion : 2 2 = 2 2ρS t) ω 0 t 62) Hence : αt) = t ω 0 σ S 63) Using Eqs.59),63), one gets : βt) αt) = b ω 0 σ S 64) Finally the outcrossing rate is obtained by substituting Eqs.5),6),64) in Eq.40) : ) b ν + t) = ω 0 Ψ ω 0 σ S σ S ϕ µ R bt µ S 65) which is exactly the analytical solution Eq.46). As a conclusion, this simple analytical example which has anyway some practical interpretation, as explained in the introduction of this section) shows that the proposed approach is exact when the problem under consideration is linear, provided that the time increment t tends to zero. 6 APPLICATION: DURABILITY OF A CORRODED BENDING BEAM 6. Problem statement Let us consider a steel bending beam. Its length is L = 5m, its cross-section is rectangular b 0 = 0.2 m, h 0 = 0.04 m. This beam is submitted to dead loads denoting by ρ st = 78.5 kn/m 3 the steel mass density, this load is equal to p = ρ st b 0 h 0 N/m)) as well as a pinpoint load F applied onto its middle point see Figure ). The bending moment is maximal in this point: M = Fl 4 + ρ stb 0 h 0 L )

10 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 0 Analytical derivation of the outcrossing rate in time-variant reliability problems Fω,t) d c t)=κ t corroded area h 0 b 0 sound steel Figure. Corroded beam under midspan load Assuming the steel has an elastic perfectly plastic constitutive law, and denoting by σ e the yield stress, the ultimate bending moment of the rectangular section is: M ult = bh2 σ e 4 67) It is now supposed that the steel beam corrodes in time. Precisely, the corrosion phenomenon is supposed to start at t = 0 and to be linear in time, meaning that the corrosion depth d c all around the section increases linearly in time d c = κt). Assuming that the corroded areas have lost all mechanical stiffness, the dimensions of the sound cross-section entering Eq.67) at time t writes: bt) = b 0 2κt ht) = h 0 2κt 68) Using the above notation, the beam fails at a given point in time if Mt) > M ult t) appearance of a plastic hinge in the middle of the span). The limit state function associated with the failure reads : gt,x) = M ult t) Mt) = bt)ht)2 σ e 4 Fl 4 + ρ stb 0 h 0 L 2 ) 8 where the dependency of the cross section dimensions to time have been specified in Eq.68). The random input parameters are gathered in Table. Table. Corroded bending beam - random variables and parameters Parameter Type of distribution Mean Coefficient of variation Load Gaussian 3500 N 20 % Steel yield stress Lognormal 240 MPa 0 % Beam breadth Lognormal 0.2 m 5 % Beam height Lognormal 0.04 m 0 % 69) The time interval under consideration is [0,20 years]. The corrosion kinetics is controlled by κ = 0.05 mm/year. The load is either modelled as a random variable see Table ) or as a Gaussian random process with the same statistics. In the latter case, the autocorrelation coefficient function is of exponential square type, with a correlation length l = day i.e year). It is emphasized that the time scale corresponding to the loading and to the corrosion are completely different, without introducing any difficulty in the PHI2 solving strategy. 6.2 Numerical results The initial probability of failure is computed by a time-invariant FORM analysis. It yields β = 4.53, i.e. P f,0 = The evolution in time of the outcrossing rate is plotted in Figure 2. Although the loading process is stationary, it appears that this outcrossing rate strongly evolves in time due to the change in the size of the beam section), since its value at t = 20 years is about 3 times that at t = 0. The evolution in time of the generalized reliability index β gen t) = Φ P f 0,t)) is represented in Figure 3. The first curve corresponds to values of the upper bound to P f 0,t) obtained from Eq.3), where

11 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev Analytical derivation of the outcrossing rate in time-variant reliability problems ν+t) t Figure 2. Reliability of a corroded beam - Outcrossing rate ν + t) PHI2 method Asymptotic time integration β gen t) t Figure 3. Reliability of a corroded beam - Generalized reliability index β gent) = Φ P f 0, t)) related to the upper bound of P f 0, t) the outcrossing rate has been computed by Eq.40). The second curve corresponds to values of the upper bound to P f 0,t) obtained by asymptotic Laplace integration in time domain see details in Rackwitz 200)). It appears that the reliability index decreases from 4.5 to.. The asymptotic approach provides slightly conservative results compared to PHI2. This result corroborates other benchmark studies presented earlier Sudret et al. 2002).

12 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 2 REFERENCES 6.3 Influence of the time increment t To show the improvement of the new implementation of the PHI2 method Eq.40)) over the original version Eq.4)), the outcrossing rate at initial instant t = 0 is computed by both formulae for various values of the time increment t/l where l = day). The results are plotted in Figure 4. In this figure, the ordinate is scaled, for each curve, by the value obtained for t/l = Old Version Eq.37) New Version Eq.36) ν + t=0, t/l) / ν + t=0,0.) Figure 4. Sensitivity of the computed outcrossing rate with respect to the time increment As observed earlier Andrieu-Renaud 2002), the first version of the PHI2 method Eq.4) and first curve in Figure 3) is quite sensitive to the choice of the time increment used in the finite difference computation: too large values make the finite difference inaccurate, whereas too small values lead to numerical instabilities. In contrary, the formula proposed in the present paper is totally insensitive to the time increment, provided it is sufficiently small to make sense in a finite difference context. We propose an increment such that t/l = % to get accurate results, l being the length in the autocorrelation function. t/l 7 CONCLUSION The present paper proposed an improvement of the so-called PHI2 method for solving time-variant reliability problems. The main contribution is the analytical derivation of the outcrossing rate for stationary and non stationary problems) in terms of the increments of the reliability index βt) and α-vectors αt). These formulae allow to evaluate the outcrossing rate ν + t) and the related probability of failure P f t,t 2 ) using solely time-invariant reliability tools. As the practical implementation is based on a finite difference scheme, the stability of the results with respect to the time increment t is studied. The formulae derived in the paper Eq.40)) reveal much more insensitive to t than the classical one Eq.4)). References Andrieu-Renaud, C. 2002). Fiabilité mécanique des structures soumises à des phénomènes physiques dépendant du temps. Ph. D. thesis, Université Blaise Pascal. Andrieu-Renaud, C., B. Sudret, and M. Lemaire 2004). The PHI2 method : a way to compute time-variant reliability. Rel. Eng. Sys. Safety 84,

13 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev REFERENCES 3 Cramer, H. and M. Leadbetter 967). Stationary and related processes. Wiley & Sons. Der Kiureghian, A. 2000). The geometry of random vibrations and solutions by form and sorm. Prob. Eng. Mech. 5), Der Kiureghian, A. and Y. Zhang 999). Space-variant finite element reliability analysis. Comp. Meth. Appl. Mech. Eng. 68, Ditlevsen, O. and H. Madsen 996). Structural reliability methods. J. Wiley and Sons, Chichester. Fujimura, K. and Der Kiureghian, A. 2005). FORM approximation of stationary first-passage probability of non linear systems. In G. Augusti, G. Schuller, and M. Ciampoli Eds.), Proc. 9th Int. Conf. Struc. Safe. Rel. ICOSSAR 2005), pp Millpress, Rotterdam. Hagen, O. and L. Tvedt 99). Vector process outcrossing as parallel system sensitivity measure. J. Eng. Mech. 70), Koo, H. and Der Kiureghian, A. 2003). FORM, SORM and simulation techniques for nonlinear random vibrations. Ph. D. thesis, University of California at Berkeley. 85 pages. Koo, H., Der Kiureghian, A., and K. Fujimura 2005). Design-point excitation for non-linear random vibrations. Prob. Eng. Mech. 202), Li, C. and A. Der Kiureghian 995). Mean out-crossing rate of nonlinear response to stochastic input. In M. Lemaire, J.-L. Favre and A. Mébarki Ed.), Proc. ICASP7 Applications of Statistics and Probability to Civil Engineering Reliability and Risk Analysis, pp Owen, D. 980). A table of normal integrals. Comm. Stat. Simul. Comp. B94), Rackwitz, R. 998). Computational techniques in stationary and non-stationary load combination - a review and some extensions. J. Struct. Eng. 25), 20. Rackwitz, R. 200). Reliability analysis - a review and some perspectives. Structural Safety 23, RCP Consult 998). COMREL user s manual. RCP Consult. Sudret, B., G. Defaux, C. Andrieu, and M. Lemaire 2002). Comparison of methods for computing the probability of failure in time-variant reliability using the outcrossing approach. In P. Spanos and G. Deodatis Eds.), Proc. 4th Int. Conf. on Comp. Stoch. Mech CSM4). Corfu. Appendix A: Results on the binormal PDF Lemma A. Proof of Eq.2) The binormal PDF classically reads : ϕ 2 x,y,ρ) = [ 2π ρ exp 2 2 Of interest is the quantity ϕ 2 β, β,ρ). From A) it comes : ϕ 2 β, β,ρ) = [ 2π ρ exp β2 2 2 x 2 2ρxy + y 2 ] ρ 2 )] 2 + 2ρ ρ 2 A) A2) Suppose ρ = + ε, where ε > 0, ε <<. The following series expansions hold : Substituting for Eqs.A3),A4) in Eq.A2), one gets Eq.2), i.e. : ρ 2 = 2ε + oε) A3) 2 + 2ρ = + o) A4) ρ2 ϕ 2 β, β, + ε) 2π 2ε exp β2 /2 A5)

14 November 4, 2006 :38 Structures and Infrastructure Engineering sudret-rev 4 REFERENCES Lemma A.2 Proof of Eq.25) The partial derivative of the binormal CDF reads: Φ 2 y x,y,ρ) = x [ 2π exp ρ 2 2 ξ 2 2ρξy + y 2 ] ρ 2 dξ A6) Straightforward algebra allows to write : 2 ξ 2 2ρξy + y 2 ξ ρy ρ 2 = + y ρ 2) 2 A7) Thus Eq.A6) rewrites : Φ 2 y x,y,ρ) = x 2π exp y2 /2) exp 2 ξ ρy ρ 2 ) 2 dξ ρ 2 A8) Introducing the mapping : u = ξ ρy ρ 2 A9) one gets : which finally reads : Φ 2 y x,y,ρ) = 2π exp y2 /2) Φ 2 x,y,ρ) = ϕy)φ y x ρy ρ 2 ) x ρy ρ 2 exp u 2 /2)du A0) A)