Structural and Multidisciplinary Optimization

, Volume 52, Issue 2, pp 251–268 | Cite as

The effect of ignoring dependence between failure modes on evaluating system reliability

  • Chanyoung Park
  • Nam H. KimEmail author
  • Raphael T. Haftka


Assuming independence between failure modes makes system reliability calculation simple but it adds approximation error. Interestingly, error due to ignoring dependence can be negligible for a highly reliable system. This paper investigates the reasons and the factors affecting the error. Error in system probability of failure (PF) is small for high reliability when tail-dependence is not very strong or the ratio between individual PFs is large. We created various conditions using copulas and observed the effect of ignoring dependence. Two reliability-based design optimization problems with a 2-bar and a 10-bar trusses are presented to show the effect of error on the optimum design and the system PF calculation. For the 10-bar truss, there were 5 % error in system PF and mass penalty less than 0.1 % in the optimum design for a target system PF of 10−7 even though five truss failures were strongly correlated.


Multiple failure modes Tail-dependence System reliability Dependence Correlation Reliability-based design optimization 



This work was supported by the National Science Foundation, under the grant CMMI-0856431 by the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378.


  1. Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insurance: Math Econ 44(2):182–198MathSciNetzbMATHGoogle Scholar
  2. Ang AH, Chaker AA, Abdelnour J (1975) Analysis of activity networks under uncertainty. J Eng Mech Div 101(4):373–387Google Scholar
  3. Ba-Abbad MA, Nikolaidis E, Kapania RK (2006) New approach for system reliability-based design optimization. AIAA J 44(5):1087–1096CrossRefGoogle Scholar
  4. Dey A, Mahadevan S (1998) Ductile structural system reliability analysis using adaptive importance sampling. Struct Saf 20(2):137–154CrossRefGoogle Scholar
  5. Ditlevsen O (1979) Narrow reliability bounds for structural systems. J Struct Mech 7(4):453–472CrossRefGoogle Scholar
  6. Eishakoff I (2004) Safety factors and reliability; friends or foes? Springer, New YorkCrossRefGoogle Scholar
  7. Frahm G, Junker M, Schmidt R (2005) Estimating the tail-dependence coefficient: properties and pitfalls. Insurance: Math Econ 37(1):80–100MathSciNetzbMATHGoogle Scholar
  8. Georges P, Lamy AG, Nicolas E, Quibel G, Roncalli T (2001) Multivariate survival modelling: a unified approach with copulas. Preprint from Groupe de Recherche Operationnelle, Credit LyonnaisGoogle Scholar
  9. Haldar A, Mahadevan S (2000) Probability, reliability, and statistical methods in engineering design. Wiley, New YorkGoogle Scholar
  10. Herencia JE, Haftka RT, Balabanov V (2013) Structural optimization of composite structures with limited number of element properties. Struct Multidiscip Optim 47(2):233–245MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hohenbichler M, Rackwitz R (1983) First-order concepts in system reliability. Struct Saf 1(3):177–188CrossRefGoogle Scholar
  12. Joe H (1997) Multivariate models and multivariate dependence concepts. CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  13. Kurowicka D, & Joe H (Eds.) (2011) Dependence Modeling: Vine Copula Handbook. World ScientificGoogle Scholar
  14. Li J, Chen JB, Fan WL (2007) The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 29(2):112–131CrossRefGoogle Scholar
  15. Lindskog F (2000) Linear correlation estimation. Preprint, ETH ZürichGoogle Scholar
  16. Melchers RE (1989) Importance sampling in structural systems. Struct Saf 6(1):3–10CrossRefGoogle Scholar
  17. Melchers RE (1999) Structural reliability analysis and prediction. Wiley, New YorkGoogle Scholar
  18. Neal DM, Matthews WT, Vangel MG, & Rudalevige T (1992) A Sensitivity Analysis on Component Reliability from Fatigue Life Computations, U.S. Army Materials Technology Laboratory MTL TR 92–5Google Scholar
  19. Nelsen RB (1999) An introduction to copulas. Springer, New YorkCrossRefzbMATHGoogle Scholar
  20. Noh Y (2009) Input model uncertainty and reliability-based design optimization with associated confidence level, Ph D. dissertation, Department of Mechanical Engineering, University of IowaGoogle Scholar
  21. Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidiscip Optim 40(1–6):35–51MathSciNetCrossRefzbMATHGoogle Scholar
  22. Schmidt R (2002) Tail dependence for elliptically contoured distributions. Math Methods Oper Res 55(2):301–327MathSciNetCrossRefzbMATHGoogle Scholar
  23. Sklar M (1959) Fonctions de répartition à n dimensions et leurs marges. Université Paris 8Google Scholar
  24. Vanmarcke EH (1973) Matrix formulation of reliability analysis and reliability-based design. Comput Struct 3(4):757–770CrossRefGoogle Scholar
  25. Venter GG (2002) Tails of copulas. Proc Casualty Actuarial Soc 89(171):68–113Google Scholar
  26. Zheng Y, Das PK (2000) Improved response surface method and its application to stiffened plate reliability analysis. Eng Struct 22(5):544–551CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Chanyoung Park
    • 1
  • Nam H. Kim
    • 1
    Email author
  • Raphael T. Haftka
    • 1
  1. 1.University of FloridaGainesvilleUSA

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