Abstract
The Nataf transformation has been proven very useful in reliability assessment when marginal distributions are statistically known and linear correlation is sufficient for modeling the dependence between random inputs. Under the assumption that the use of FORM is appropriate for the problem of interest, it is often of importance to quantify how the FORM solution is sensitive to the distribution parameters of the random inputs. Such information can be exploited in different contexts including optimal design under uncertainty. This chapter describes how sensitivities to marginal distribution parameters and linear correlation can be assessed numerically in the context of FORM based on the Nataf transformation. The emphasis is on the accuracy of such sensitivities with no other approximations than the one due to numerical integration. In the presented examples, the accuracy of these sensitivities is assessed w.r.t. reference solutions. The sensitivity to correlation brings useful information which are complementary to those w.r.t. marginal distribution parameters. High sensitivities may be detected such as illustrated in the context of stochastic crack growth based on the Virkler data set.
References
Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16(4):263–277. doi:10.1016/S0266-8920(01)00019-4
Blatman G, Sudret B (2010) An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab Eng Mech 25(2):183–197. doi:10.1016/j.probengmech.2009.10.003
Bourinet JM, Lemaire M (2008) FORM sensitivities to correlation: application to fatigue crack propagation based on Virkler data. In: Das PK (ed) Proceedings of 4th international ASRANet colloquium, Athens, Greece, June 25–27, 2008. http://maritime-conferences.com/asranet2010-conference/asranet2008/45%20Bourinet,%20Jean-Marc.pdf
Bourinet JM, Mattrand C, Dubourg V (2009) A review of recent features and improvements added to FERUM software. In: Furuta H, Frangopol DM, Shinozuka M (eds) Proceedings of 10th international conference on structural safety and reliability (ICOSSAR 2009), Osaka, Japan, September 13–17, 2009. CRC Press
Bucher CG, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7(1):57–66. doi:10.1016/0167-4730(90)90012-E
Der Kiureghian A, Lin HZ, Hwang SJ (1987) Second-order reliability approximations. J Eng Mech 113(8):1208–1225. doi:10.1061/(ASCE)0733-9399(1987) 113:8(1208)
Ditlevsen O, Madsen HO (2007) Structural reliability methods. Internet edition 2.3.7. http://od-website.dk/books/OD-HOM-StrucRelMeth-Ed2.3.7.pdf
Ditlevsen O, Olesen R (1986) Statistical analysis of the Virkler data on fatigue crack growth. Eng Fract Mech 25(2):177–195. doi:10.1016/0013-7944(86)90217-1
Dubourg V (2011) Adaptive surrogate models for reliability analysis and reliability-based design optimization. Phd thesis, Université Blaise Pascal, Clermont Ferrand, France. https://tel.archives-ouvertes.fr/tel-00697026v2
Gong JX, Yi P, Zhao N (2014) Non-gradient-based algorithm for structural reliability analysis. J Eng Mech 140(6):04014,029. doi:10.1061/(ASCE)EM.1943-7889.0000722
Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100(1):111–121
Hohenbichler M, Rackwitz R (1986) Sensitivity and importance measures in structural reliability. Civil Eng Syst 3(4):203–209. doi:10.1080/02630258608970445
Hohenbichler M, Rackwitz R (1988) Improvement of second-order reliability estimates by importance sampling. J Eng Mech 114(12):2195–2199. doi:10.1061/(ASCE)0733-9399(1988) 114:12(2195)
Kotulski ZA (1998) On efficiency of identification of a stochastic crack propagation. Arch Mech 50(5):829–847. http://am.ippt.pan.pl/am/article/viewFile/v50p829/621
Lebrun R, Dutfoy A (2009b) A generalization of the nataf transformation to distributions with elliptical copula. Probab Eng Mech 24(2):172–178. doi:10.1016/j.probengmech.2008.05.001
Lemaire M, Chateauneuf A, Mitteau JC (2010) Structural reliability. ISTE. doi:10.1002/9780470611708.fmatter
Liu PL, Der Kiureghian A (1986a) Multivariate distribution models with prescribed marginals and covariance. Probab Eng Mech 1(2):105–112. doi:10.1016/0266-8920(86)90033-0
Liu PL, Der Kiureghian A (1986b) Optimization algorithms for structural reliability analysis. Report no. UCB/SEMM-86/09, Department of civil and environmental engineering, University of California, Berkeley
Liu PL, Lin HZ, Der Kiureghian A (1989) CalREL user manual. Report no. UCB/SEMM-89/18, Department of civil and environmental engineering, University of California, Berkeley
Most T (2011) Efficient structural reliability methods considering incomplete knowledge of random variable distributions. Probab Eng Mech 26(2):380–386. doi:10.1016/j.probengmech.2010.09.003
Nataf A (1962) Détermination des distributions dont les marges sont données. Comptes Rendus de l’Académie des Sciences 225:42–43
Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494. doi:10.1016/0045-7949(78)90046-9
Rubinstein RY (1976) A monte carlo method for estimating the gradient in a stochastic network, technion, Haifa, Israel. Unpublished manuscript
Rubinstein RY (1986) The score function approach for sensitivity analysis of computer simulation models. Math Comput Simul 28(5):351–379. doi:10.1016/0378-4754(86)90072-8
Song S, Lu Z, Qiao H (2009) Subset simulation for structural reliability sensitivity analysis. Reliab Eng Syst Saf 94(2):658–665. doi:10.1016/j.ress.2008.07.006
Virkler DA, Hillberry BM, Goel PK (1979) The statistical nature of fatigue crack propagation. J Eng Mater Technol 101(2):148–153. doi:10.1115/1.3443666
Žanić V, Žiha K (1998) Sensitivity to correlation in multivariate models. Comput Assist Mech Eng Sci 5(1):75–84
Žanić V, Žiha K (2001) Sensitivity to correlations in structural problems. Transactions of FAMENA 25(2):1–26. https://bib.irb.hr/datoteka/87824.87824-FAMENA2001V25N2.pdf
Wei D, Rahman S (2007) Structural reliability analysis by univariate decomposition and numerical integration. Probab Eng Mech 22(1):27–38. doi:10.1016/j.probengmech.2006.05.004
Wu YT (1994) Computational methods for efficient structural reliability and reliability sensitivity analysis. AIAA J 32(8):1717–1723. doi:10.2514/3.12164
Zhang Y, Der Kiureghian A (1994) Two improved algorithms for reliability analysis. In: Rackwitz R, Augusti G, Borri A (eds) Proceedings of 6th IFIP WG 7.5 working conference on reliability and optimization of structural systems, Assisi, Italy, September 7–9, 1994. Springer, US, pp 297–304. doi:10.1007/978-0-387-34866-7_32
Acknowledgements
I wish to express my deep gratitude to Prof. Armen Der Kiureghian who has played an important role in the course of my career. His high-quality scientific work has been and still is a great source of inspiration in my research.
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Appendix: Determination of \(\mathbf{L}_0\) and \({\partial \mathbf{L}_0}/{\partial \rho _{ij}}\)
Appendix: Determination of \(\mathbf{L}_0\) and \({\partial \mathbf{L}_0}/{\partial \rho _{ij}}\)
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Bourinet, JM. (2017). FORM Sensitivities to Distribution Parameters with the Nataf Transformation. In: Gardoni, P. (eds) Risk and Reliability Analysis: Theory and Applications. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-52425-2_12
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