Reliability analysis examples

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1 Reliability analysis examples Engineeing Risk Analysis Goup, Technische Univesität München. Acisst., 80 Munich, Gemany. May 8, 08 Intoduction In the context of eliability analysis and ae event estimation, the pefomance of the system unde consideation can be descibed by a limit state function (L) g : Θ R. The failue hypesuface defined by g(θ) = 0 splits the space of uncetain vaiables Θ into the safe domain = {θ Θ : g(θ) > 0} and the failue domain = {θ Θ : g(θ) 0}. The L can include one o seveal distinct modes of failue. The pobability of occuence of Θ, called the pobability of failue, is defined by P() = I [θ]f(θ)dθ () Θ whee, f(θ) is the joint pobability density function (PD) of the model paametes and I[ ] is the indicato function, which takes the values I [θ] = when θ, and I [θ] = 0 othewise. A special challenge involves the analysis of ae events, that is, when Eq. () epesents the solution of a high dimensional integal fo which P() is vey small. In the following, we descibe 6 examples that ae commonly used as benchmaks in eliability analysis. These poblems ae solved in MATLAB and in python using the following appoaches: ist ode eliability method (ORM) with the Hasofe-Lind-Rackwitz-iessle (HLR) algoithm [, ]: unction ORM_HLR. ubset simulation (u) with adaptive conditional sampling (ac) [, ]: unction u. Coss entopy-based impotance sampling (I) using diffeent I densities (single Gaussian, Gaussian mixtue, von Mises-ishe-Nakagami mixtue) [6, 9]: unctions CEI_G, CEI_GM, and CEI_vMNM. equential impotance sampling (I) using diffeent MCMC algoithms (independent M- H with Gaussian mixtue poposal, adaptive C) [0]: unction I_GM, I_aC. o futhe questions o bugs in the codes please wite to: felipe.uibe@tum.de.

2 Example This poblem is poposed in []. Due to its simplicity and flexibility it can be used to investigate the behavio of the eliability methods in vaying dimensions. We conside a L expessed as a linea function of independent Gaussian andom vaiables: = d u i + β () whee, u is a d-dimensional standad Gaussian andom vecto. The pobability of failue is independent of the dimension d and can be deived analytically as P() = Φ( β), whee Φ( ) is the standad Gaussian cumulative distibution function (CD). igue shows the L in two dimensions; the pobability of failue in this case is P() 0 fo β =.090. i= u 0 0 u igue : Example : limit state function in Eq. () with d = and β =.090. Example This poblem is poposed in [7, 8]. We conside the L: g(x) = ± x i ± c () whee, x is a d-dimensional vecto of independent exponentially distibuted andom vaiables with paamete λ =. Afte tansfoming the vecto x to standad Gaussian vaiables u, the esulting L is highly nonlinea [], = ± λ i= ln(φ( u i )) ± c i= The pobability of failue can be computed analytically as P() = Γ (c; d, λ), whee c is a taget theshold level and Γ denotes the CD of the gamma distibution with shape d and scale λ. The negative sign in Eq. () coesponds to the lowe tail of Γ and the positive sign to the uppe tail of Γ, espectively. Example This poblem consists of a seies system involving a linea and a convex Ls in a two-dimensional standad Gaussian space [6]. This example is used fo investigating the pefomance stuctual

3 eliability methods in system poblems with limit state functions of diffeent type. The L is given by,. + (u + u ) = min 0.(u u ) (u + u ) () +. igue shows the L; the efeence pobability of failue in this case is P() = u u igue : Example : limit state function in Eq. (). Example This example is poposed in []. The L has a paabolic shape concave to the oigin of the two-dimensional standad Gaussian space and has two design points; it is given by = u 0.(u 0.) () igue shows the L; the efeence pobability of failue is P() =.0 0. u u igue : Example : limit state function in Eq. ().

4 6 Example This example is oiginally pesented in []. We define a seies system with fou banches such that the L in the two-dimensional Gaussian space is expessed as 0.(u u ) (u + u ) + = min 0.(u u ) + (u + u ) + u u + 7 u u + 7 Two of the components featue a paabolic limit state suface convex to the oigin, while the emaining two components ae of the linea type. We depict the L in igue. The efeence pobability of failue is estimated with Monte Calo simulation (N = 0 6 ) as P() =.6 0. (6) u u igue : Example : limit state function in Eq. (6). 7 Example 6 This example is pesented in []. Let u be a d-dimensional vecto of independent standad Gaussian andom vaiables. The failue domain is defined as all points u located outside of a hypesphee that has adius R >0, ( ) m = u u u ) m, fo m [0, ] the failue egion is independent of m. The coefficient m modifies the gadient of the limit state function in u -diection. The sum of squaed independent standad Gaussian andom vaiables follows a chi-squaed distibution. Theefoe, the analytical solution of the pobability of failue is given by [], P() = γ(d/, /d) Γ(d/) + ( u = Γ(d/, /d) Γ(d/) whee, γ(, ) is the lowe incomplete gamma function, and γ(, ) is the uppe incomplete gamma function. We depict the L in igue fo dimension d =, adius =.6, and m =. The efeence pobability of failue is P() = 0 6. (7)

5 u u igue : Example 6: limit state function in Eq. (7). Refeences [] Au,.-K., and J. L. Beck (00). Estimation of small failue pobabilities in high dimensions by subset simulation. In: Pobabilistic Engineeing Mechanics 6., pp [] Betz, W. (08). Bayesian infeence of engineeing models. Ph.D. Thesis, Ingenieufakultät Bau Geo Umwelt, Technische Univesität München. [] De Kiueghian, A., and T. Dakessian (998). Multiple design points in fist and second-ode eliability, tuctual afety 0., pp [] Engelund,., and R. Rackwitz (99). A benchmak study on impotance sampling techniques in stuctual eliability. In: tuctual afety., pp [] ujita, M., and R. Rackwitz (988). Updating fist- and second-ode estimates by impotance sampling. tuctual Engineeing/Eathquake Engineeing., pp. -9. [6] Geye,., I. Papaioannou, and D. taub (08). Coss entopy-based impotance sampling using Gaussian densities evisited. tuctual afety (in eview). [7] Hohenbichle, M., and R. Rackwitz (98). Non-nomal dependent vectos in stuctual safety. Jounal of the Engineeing Mechanics Division 07.6, pp [8] Hohenbichle, M., and R. Rackwitz (988). Impovement of second-ode eliability estimates by impotance sampling. Jounal of Engineeing Mechanics., pp. -9. [9] Papaioannou, I.,. Geye, and D. taub. A new flexible mixtue model fo coss entopy based impotance sampling. In pepaation. [0] Papaioannou, I., C. Papadimitiou, and D. taub (0). equential impotance sampling fo stuctual eliability analysis. In: tuctual afety 6, pp [] Papaioannou, I., W. Betz, K. Zwiglmaie, and D. taub (0). MCMC algoithms fo ubset imulation. In: Pobabilistic Engineeing Mechanics, pp [] Rackwitz, R. (976). Pactical pobabilistic appoach to design. ist ode eliability concepts fo design codes. Bulletin d Infomation du CEB, Vol.. [] Rackwitz, R., and B. iessle (979). tuctual eliability unde combined andom load sequences. Computes and tuctues, 9., pp [] Waats, P.-H. (000). tuctual eliability using finite element methods. An appaisal fo DAR: Diectional Adaptive Response suface ampling. Ph.D. thesis. Technical Univesity of Delft.