Reliability calculation method based on the Copula function for mechanical systems with dependent failure

  • Ying-Kui GuEmail author
  • Chao-Jun Fan
  • Ling-Qiang Liang
  • Jun Zhang
S.I.: Reliability Modeling with Applications Based on Big Data


In order to accurately calculate the reliability of mechanical components and systems with multiple correlated failure modes and to reduce the computational complexity of these calculations, the Copula function is used to represent related structures among failure modes. Based on a correlation analysis of the failure modes of parts of a system, a life distribution model of components is constructed using the Copula function. The type of Copula model was initially selected using a binary frequency histogram of the life empirical distribution between the two components. The unknown parameters in the Copula model were estimated using the maximum likelihood estimation method and the most suitable Copula model was determined by calculating the square Euclidean distance. The reliability of series, parallel, and series–parallel systems was analyzed based on the Copula function, where life was used as a variable to measure the correlation between components. Thus, a reliability model of a system with life correlations was established. Reliability calculation of a particular diesel crank and connecting rod mechanism was taken as a practical example to illustrate the feasibility of the proposed method.


Reliability Life distribution Copula function Multiple failure modes Parameter estimation Crank and connecting rod mechanism 



This research was partially supported by the National Natural Science Foundation of China under the Contract Number 61463021, the Natural Science Foundation of Jiangxi Province under the Contract Number 20181BAB202020, and the Young Scientists Object Program of Jiangxi Province, China under the Contract Number 20144BCB23037.


  1. Adduri, P. R., & Penmetsa, R. C. (2007). Bounds on structural system reliability in the presence of interval variables. Computers and Structures, 85(5–6), 320–329.CrossRefGoogle Scholar
  2. An, H., Yin, H., & He, F. K. (2016). Analysis and application of mechanical system reliability model based on Copula function. Polish Maritime Research, 23(s1), 187–191.CrossRefGoogle Scholar
  3. Barrera, J., Cancela, H., & Moreno, E. (2015). Topological optimization of reliable networks under dependent failures. Operations Research Letters, 43(2), 132–136.CrossRefGoogle Scholar
  4. Ditlevsen, O. (1982). System reliability bounding by conditioning. Journal of the Engineering Mechanics Division, 108(5), 708–716.Google Scholar
  5. Eryilmaz, S. (2014). Multivariate copula based dynamic reliability modeling with application to weighted k-out-of-n systems of dependent components. Structural Safety, 51, 23–28.CrossRefGoogle Scholar
  6. Guo, C., Wang, W., Guo, B., & Peng, R. (2013). Maintenance optimization for systems with dependent competing risks using a copula function. Eksploatacja I Niezawodnosc: Maintenance and Reliability, 15(1), 9–17.Google Scholar
  7. Jiang, C., Zhang, W., & Han, X. (2017). A Copula function based evidence theory model for correlation analysis and corresponding structural reliability method. Journal of Mechanical Engineering, 53(16), 199–209.CrossRefGoogle Scholar
  8. Li, C. P., & Hao, H. B. (2016). A copula-based degradation modeling and reliability assessment. Engineering Letters, 24(3), 295–300.Google Scholar
  9. Li, H., Huang, H. Z., Li, Y. F., Zhou, J., & Mi, J. (2018). Physics of failure-based reliability prediction of turbine blades using multi-source information fusion. Applied Soft Computing, 72, 624–635.CrossRefGoogle Scholar
  10. Li, X. Y., Liu, Y., Chen, C. J., & Jiang, T. (2015a). A copula-based reliability modeling for nonrepairable multi-state k-out-of-n systems with dependent components. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 230(2), 133–146.Google Scholar
  11. Li, Y. F., Mi, J., Liu, Y., Yang, Y. J., & Huang, H. Z. (2015b). Dynamic fault tree analysis based on continuous-time Bayesian networks under fuzzy numbers. Proceedings of the Institution of Mechanical Engineers, Part O, Journal of Risk and Reliability, 229(6), 530–541.CrossRefGoogle Scholar
  12. Lu, H., & Zhu, Z. C. (2018). A method for estimating the reliability of structural systems with moment-matching and copula concept. Mechanics Based Design of Structures and Machines, 46(2), 196–208.CrossRefGoogle Scholar
  13. Mi, J., Li, Y. F., Peng, W., & Huang, H. Z. (2018). Reliability analysis of complex multi-state system with common cause failure based on evidential networks. Reliability Engineering and System Safety, 174, 71–81.CrossRefGoogle Scholar
  14. Mi, J., Li, Y. F., Yang, Y. J., Peng, W., & Huang, H. Z. (2016). Reliability assessment of complex electromechanical systems under epistemic uncertainty. Reliability Engineering and System Safety, 152, 1–15.CrossRefGoogle Scholar
  15. Nelsen, R. B. (2006). An introduction to Copulas. New York: Springer.Google Scholar
  16. Park, C., Kim, N. H., & Haftka, R. T. (2015). The effect of ignoring dependence between failure modes on evaluating system reliability. Structural and Multidisciplinary Optimization, 52(2), 251–268.CrossRefGoogle Scholar
  17. Peng, W., Zhang, X. L., & Huang, H. Z. (2016). A failure rate interaction model for two-component systems based on copula function. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 230(3), 278–284.Google Scholar
  18. Qi, G., & Yang, G. (2015). Maintenance interval decision models for a system with failure interaction. Journal of Manufacturing Systems, 36, 109–114.CrossRefGoogle Scholar
  19. Stern, R. E., Song, J., & Work, D. B. (2017). Accelerated Monte Carlo system reliability analysis through machine-learning-based surrogate models of network connectivity. Reliability Engineering and System Safety, 164, 1–9.CrossRefGoogle Scholar
  20. Tang, X. S., Li, D. Q., & Zhou, C. B. (2015). Copula-based approaches for evaluating slope reliability under incomplete probability information. Structural Safety, 52, 90–99.CrossRefGoogle Scholar
  21. Wu, Z., Chen, J., & Wen, B. (2006). A new narrow-bound method for computing system failure probability. In Proceedings of geoshanghai international conference, pp. 98–102.Google Scholar
  22. Xiao, N. C., Huang, H. Z., Wang, Z., Li, Y., & Liu, Y. (2012). Reliability analysis of series systems with multiple failure modes under epistemic and aleatory uncertainties. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 226(3), 295–304.Google Scholar
  23. Yu, S., & Wang, Z. L. (2018). A novel time-variant reliability analysis method based on failure processes decomposition for dynamic uncertain structures. ASME Journal of Mechanical Design, 140(5), 051401.CrossRefGoogle Scholar
  24. Yu, S., Wang, Z. L., & Meng, D. B. (2018). Time-variant reliability assessment for multiple failure modes and temporal parameters. Structural and Multidisciplinary Optimization, 58(4), 1705–1717.CrossRefGoogle Scholar
  25. Zhang, X., & Alyson, W. (2017). System reliability and component importance under dependence: a Copula approach. Technometrics, 59(2), 215–224.CrossRefGoogle Scholar
  26. Zhang, X., Gao, H., Huang, H. Z., Li, Y. F., & Mi, J. (2018). Dynamic reliability modeling for system analysis under complex load. Reliability Engineering and System Safety, 180, 345–351.CrossRefGoogle Scholar
  27. Zhang, Y., & Lee, L. J. S. (2016). A copula approach in the point estimate method for reliability engineering. Quality and Reliability Engineering International, 32(4), 1501–1508.CrossRefGoogle Scholar
  28. Zhang, X. P., Shang, J. Z., Chen, X., Zhang, C. H., & Wang, Y. S. (2014). Statistical inference of accelerated life testing with dependent competing failures based on copula theory. IEEE Transactions on Reliability, 63(3), 764–780.CrossRefGoogle Scholar
  29. Zhou, J. Y., Xie, L. Y., & Qian, W. X. (2008). Reliability analysis on structural systems with load dependency. Chinese Journal of Mechanical Engineering, 44(5), 45–50.CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical and Electrical EngineeringJiangxi University of Science and TechnologyGanzhouPeople’s Republic of China

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