Journal of Mechanical Science and Technology

, Volume 32, Issue 11, pp 5063–5068 | Cite as

A direct-integration-based structural reliability analysis method using non-probabilistic convex model

  • Xiao-Bo Nie
  • Hai-Bin LiEmail author


In practical structural reliability analysis, there is not only random uncertainty but also fuzzy uncertainty. Aiming at the fuzzy reliability of structure, a novel fuzzy reliability method is proposed based on direct integration method and ellipsoidal convex model. Firstly, the decomposition of fuzzy mathematics principle is used to convert fuzzy reliability model into non-probabilistic reliability model, in which fuzzy variables are converted into interval variables. The upper and lower bounds of interval variables are determined by the possibility distribution function on the membership value. Secondly, multidimensional ellipsoid convex models are constructed to quantify the uncertainty because of the complexity of non-probabilistic reliability. Finally, sigmoid function with adjustable parameter is introduced to direct integration method for approximating the step function, and then direct integration method is used to solve the fuzzy reliability. Numerical examples are investigated to demonstrate the effectiveness of the present method, which provides a feasible way for the structural fuzzy reliability analysis.


Fuzzy variable Ellipsoid convex Interval variable Sigmoid function Direct integration method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. A. Zadeh, Fuzzy sets, Information & Control, 8 (3) (1965) 338–353.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems Man & Cybernetics smc, 3 (1) (1973) 28–44.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets & Systems, 1 (1) (1978) 3–28.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. A. Arbib, Introduction to the theory of fuzzy subsets, Vol. 1. by A. Kaufmann, Siam Review, 20 (2) (1978) 98–106.CrossRefGoogle Scholar
  5. [5]
    D. C. Komal and S. Y. Lee, Fuzzy reliability analysis of dual–fuel steam turbine propulsion system in LNG carriers considering data uncertainty, Journal of Natural Gas Science and Engineering, 23 (2015) 148–164.CrossRefGoogle Scholar
  6. [6]
    J. H. Purba, J. Lu, G. Q. Zhang and W. Pedrycz, A fuzzy reliability assessment of basic events of fault trees through qualitative data processing, Fuzzy Sets and Systems, 243 (2014) 50–69.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    C. Wang, Z. P. Qiu, M. H. Xu and H. C. Qiu, Novel fuzzy reliability analysis for heat transfer system based on interval ranking method, International Journal of Thermal Sciences, 116 (2017) 234–241.CrossRefGoogle Scholar
  8. [8]
    Y. G. Dong, X. Z. Chen, X. D. Zhao and Z. W. Quan, A general approach for fuzzy reliability analysis based on the fuzzy probability theory, Chinese Journal of Computational Mechanics, 22 (2013) 281–286.Google Scholar
  9. [9]
    H. Chen and X. Lou, A new fuzzy parameters reliability analysis model, International Conference on Broadcast Technology and Multimedia Communication (2010) 741–746.Google Scholar
  10. [10]
    C. F. Fuh, J. Rong and J. S. Su, Fuzzy system reliability analysis based on level (λ, 1) interval–valued fuzzy numbers, Information Sciences, 272 (2014) 185–197.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    H. Z. Huang, Y. Liu, Y. F. Li, L. H. Xue and Z. L. Wang, New evaluation methods for conceptual design selection using computational intelligence techniques, Journal of Mechanical Science and Technology, 27 (3) (2013) 733–746.CrossRefGoogle Scholar
  12. [12]
    A. Hai, W. An and Z. Ling, Fuzzy reliability analysis based on probability density function equivalent, Acta Aeronautica Et Astronautica Sinica, 30 (5) (2009) 886–894.Google Scholar
  13. [13]
    G. T. Zhao, Y. G. Dong and Z. Y. Song, Random reliability analysis based on the fuzzy theory, Journal of Hefei University of Technology, 33 (2) (2010) 249–253.Google Scholar
  14. [14]
    J. Song and Z. Z. Lu, Moment method for general failure probability with fuzzy failure state and fuzzy safety state, Engineering Mechanics, 25 (2) (2008) 71–77.MathSciNetGoogle Scholar
  15. [15]
    S. Guo, Z. Lv and L. Feng, A fuzzy reliability approach for structures in the possibility context, Chinese Journal of Computational Mechanics, 19 (1) (2002) 89–93.Google Scholar
  16. [16]
    Y. G. Dong, X. Z. Chen, H. D. Cho and J. W. Kwon, Simulation of fuzzy reliability indexes, KSME International Journal, 17 (4) (2003) 492–500.CrossRefGoogle Scholar
  17. [17]
    Y. Ben–Haim, A non–probabilistic concept of reliability, Structural Safety, 14 (4) (1994) 227–245.CrossRefGoogle Scholar
  18. [18]
    Y. Ben–Haim, A non–probabilistic measure of reliability of linear systems based on expansion of convex models, Structural Safety, 17 (2) (1995) 91–109.CrossRefGoogle Scholar
  19. [19]
    Y. Ben–Haim and I. Elishakoff, Discussion on: A nonprobabilistic concept of reliability, Structural Safety, 17 (3) (1995) 195–199.CrossRefGoogle Scholar
  20. [20]
    Y. I. Ping, Discussions on reliability measure for problems with bounded–but–unknown uncertainties, Chinese Journal of Computational Mechanics, 23 (2) (2006) 152–156.Google Scholar
  21. [21]
    X. F. Zhang, Y. Zhao and H. L. Shi, Study of structural nonprobabilistic measure based on convex model, Journal of Mechanical Strength, 29 (4) (2007) 589–592.Google Scholar
  22. [22]
    C. Jiang, X. Han, G. Y. Lu, J. Liu, Z. Zhang and Y. C. Bai, Correlation analysis of non–probabilistic convex model and corresponding structural reliability technique, Computer Methods in Applied Mechanics & Engineering, 200 (33) (2011) 2528–2546.CrossRefzbMATHGoogle Scholar
  23. [23]
    Y. C. Bai, X. Han, C. Jiang and R. G. Bi, A responsesurface–based structural reliability analysis method by using non–probability convex model, Applied Mathematical Modeling, 38 (15–16) (2014) 3834–3847.CrossRefGoogle Scholar
  24. [24]
    X. Liu and Z. Zhang, A hybrid reliability approach for structure optimization based on probability and ellipsoidal convex models, Journal of Engineering Design, 25 (4) (2014) 238–258.CrossRefGoogle Scholar
  25. [25]
    R. Rackwitz, Reliability analysis–a review and some perspectives, Structural Safety, 23 (4) (2001) 365–395.CrossRefGoogle Scholar
  26. [26]
    K. Y. Cai, C. Y. Wen and M. L. Zhang, Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets & Systems, 42 (2) (1991) 145–172.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    X. Chen and J. Fan, The application of convex models based on non–probabilistic concepts of reliability in bridge assessments, Journal of Huazhong University of Science & Technology, 37 (12) (2009) 120–123.zbMATHGoogle Scholar
  28. [28]
    S. X. Guo, L. Zhang and Y. Li, A solving method for nonprobabilistic reliability index of structures, Chinese Journal of Computational Mechanics, 22 (2) (2005) 227–231.Google Scholar
  29. [29]
    X. J. Wang, Z. P. Qiu and I. Elishakoff, Non–probabilistic set–theoretic model for structural safety measure, Acta Mechanica, 198 (1–2) (2008) 51–64.CrossRefzbMATHGoogle Scholar
  30. [30]
    L. P. Zhu, I. Elishakoff and J. H. Starnes, Derivation of multidimensional ellipsoidal convex model for experimental data, Mathematical and Computer Modeling, 24 (2) (1998) 103–114.CrossRefzbMATHGoogle Scholar
  31. [31]
    H. B. Li and X. B. Nie, Structural reliability analysis with fuzzy random variables using error principle, Engineering Applications of Artificial Intelligence, 67 (2018) 91–99.CrossRefGoogle Scholar
  32. [32]
    J. Mi, Y. F. Li, W. Peng and H. Z. Huang, Reliability analysis of complex multi–state system with common cause failure based on evidential networks, Reliability Engineering & System Safety, 174 (2018) 71–81.CrossRefGoogle Scholar
  33. [33]
    H. Z. Huang, C. G. Huang, Z. Peng, Y. F. Li and H. Yin, Fatigue life prediction of fan blade using nominal stress method and cumulative fatigue damage theory, International Journal of Turbo & Jet Engines, Doi:–2017–0015.Google Scholar
  34. [34]
    S. R. Yu, Y. H. Yin and B. Xu, Analysis of the randomfuzzy reliability based on the information entropy theory, Journal of Mechanical Strength, 28 (5) (2006) 695–698.Google Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceInner Mongolia University of TechnologyHohhot, Inner MongoliaChina

Personalised recommendations