Acta Mechanica Solida Sinica

, Volume 30, Issue 6, pp 638–646 | Cite as

Theoretical analysis of non-probabilistic reliability based on interval model

  • Xu-Yong Chen
  • Jian-Ping Fanb
  • Xiao-Ya Bian


The aim of this paper is to propose a theoretical approach for performing the non-probabilistic reliability analysis of structure. Due to a great deal of uncertainties and limited measured data in engineering practice, the structural uncertain parameters were described as interval variables. The theoretical analysis model was developed by starting from the 2-D plane and 3-D space. In order to avoid the loss of probable failure points, the 2-D plane and 3-D space were respectively divided into two parts and three parts for further analysis. The study pointed out that the probable failure points only existed among extreme points and root points of the limit state function. Furthermore, the low-dimensional analytical scheme was extended to the high-dimensional case. Using the proposed approach, it is easy to find the most probable failure point and to acquire the reliability index through simple comparison directly. A number of equations used for calculating the extreme points and root points were also evaluated. This result was useful to avoid the loss of probable failure points and meaningful for optimizing searches in the research field. Finally, two kinds of examples were presented and compared with the existing computation. The good agreements show that the proposed theoretical analysis approach in the paper is correct. The efforts were conducted to improve the optimization method, to indicate the search direction and path, and to avoid only searching the local optimal solution which would result in missed probable failure points.


Non-probabilistic Reliability Interval model Theoretical analysis Probable failure point 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Y. Ben-Haim, Convex models of uncertainty in radial pulse buckling of shells, J. Appl. Mech. 60 (3) (1993) 683–688.CrossRefGoogle Scholar
  2. 2.
    Y. Ben-Haim, A non-probabilistic concept of reliability, Struct. Saf. 14 (4) (1994) 227–245.CrossRefGoogle Scholar
  3. 3.
    Y. Ben-Haim, A non-probabilistic measure of reliability of linear systems based on expansion of convex models, Struct. Saf. 17 (2) (1995) 91–109.CrossRefGoogle Scholar
  4. 4.
    X.Y. Chen, C.Y. Tang, C.P. Tsui, J.P. Fan, Modified scheme based on semi-analytic approach for computing non-probabilistic reliability index, Acta Mech. Solida Sin. 23 (2) (2010) 115–123.CrossRefGoogle Scholar
  5. 5.
    I. Elishakoff, Discussion on: a non-probabilistic concept of reliability, Struct. Saf. 17 (3) (1995) 195–199.MathSciNetCrossRefGoogle Scholar
  6. 6.
    I. Elishakoff, Essay on uncertainties in elastic and viscoelastic structures: from A. M. Freudenthal’s criticisms to modern convex modeling, Comput. Struct. 56 (6) (1995) 871–895.CrossRefGoogle Scholar
  7. 7.
    J.P. Fan, S.J. Li, X.Y. Chen, Optimal searching algorithm for non-probabilistic reliability, Chin. J. Comput. Mech. 29 (6) (2012) 831–834 (in Chinese).zbMATHGoogle Scholar
  8. 8.
    J.P. Fan, S.J. Li, W. Qi, X.Y. Chen, Safety evaluation of non-probabilistic reliability model of structures, Chin. J. Solid Mech. 33 (3) (2012) 325–330 (in Chinese).Google Scholar
  9. 9.
    S.X. Guo, Z.Z. Lv, Comparison between the non-probabilistic and probabilistic reliability methods for uncertain structure design, Chin. J. Appl. Mech. 20 (3) (2003) 107–110.Google Scholar
  10. 10.
    S.X. Guo, Z.Z. Lv, Y.S. Feng, A non-probabilistic model of structural reliability based on interval analysis, Chin. J. Comput. Mech. 18 (1) (2001) 56–62 (in Chinese).Google Scholar
  11. 11.
    S.X. Guo, L. Zhang, Y. Li, Procedures for computing the non-probabilistic reliability index of uncertain structures, Chin. J. Comput. Mech. 22 (2) (2005) 227–231 (in Chinese).Google Scholar
  12. 12.
    J.E. Hurtado, D.A. Alvarez, The encounter of interval and probabilistic approaches to structural reliability at the design point, Comput. Methods Appl. Mech. Eng. 225–228 (2012) 74–94.MathSciNetCrossRefGoogle Scholar
  13. 13.
    T. Jiang, J.J. Chen, P.G. Jiang, Y.F. Tuo, A one-dimensional optimization algorithm for non-probabilistic reliability index, Eng. Mech. 24 (7) (2007) 23–27.Google Scholar
  14. 14.
    T. Jiang, J.J. Chen, Y.L. Xu, A semi-analytic method for calculating non-probabilistic reliability index based on interval models, Appl. Math. Model. 31 (7) (2007) 1362–1370.CrossRefGoogle Scholar
  15. 15.
    C. Jiang, Z. Zhang, X. Han, Y.C. Bai, An evidence-theory-based reliability analysis method for uncertain structures, Chin. J. Theor. Appl. Mech. 45 (1) (2013) 103–115.Google Scholar
  16. 16.
    C. Jiang, R.G. Bi, G.Y. Lu, X. Han, Structural reliability analysis using non-probabilistic convex model, Comput. Methods Appl. Mech. Eng. 254 (2013) 83–98.MathSciNetCrossRefGoogle Scholar
  17. 17.
    C. Jiang, Q.F. Zhang, X. Han, Y.H. Qian, A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model, Acta Mech. 225 (2014) 383–395.CrossRefGoogle Scholar
  18. 18.
    J.O. Lee, Y.S. Yang, W.S. Ruy, A comparative study on reliability-index and target-performance-based probabilistic structural design optimization, Comput. Struct. 80 (3–4) (2002) 257–269.CrossRefGoogle Scholar
  19. 19.
    S.J. Li, J.P. Fan, W. Qi, X.Y. Chen, The gradient projection method for non-probabilistic reliability index based on interval model, Chin. J. Comput. Mech. 30 (2) (2013) 192–197 (in Chinese).zbMATHGoogle Scholar
  20. 20.
    Z. Ni, Z.P. Qiu, Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability, Comput. Ind. Eng. 58 (3) (2010) 463–467.CrossRefGoogle Scholar
  21. 21.
    Z.P. Qiu, J. Wang, The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system, Eng. Fail. Anal. 17 (5) (2010) 1142–1154.CrossRefGoogle Scholar
  22. 22.
    L. Wang, X.J. Wang, Y. Xia, Hybrid reliability analysis of structures with multi-source uncertainties, Acta Mech. 225 (2) (2014) 413–430.CrossRefGoogle Scholar
  23. 23.
    L. Wang, X.J. Wang, R.X. Wang, X. Chen, Reliability-based design optimization under mixture of random, interval and convex uncertainties, Arch. Appl. Mech. 86 (7) (2016) 1341–1367.CrossRefGoogle Scholar
  24. 24.
    L. Wang, X.J. Wang, H. Su, G.P. Lin, Reliability estimation of fatigue crack growth prediction via limited measured data, Int. J. Mech. Sci. 121 (2017) 44–57.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2017

Authors and Affiliations

  1. 1.School of Resource and Civil EngineeringWuhan Institute of TechnologyWuhanChina
  2. 2.School of Civil Engineering & MechanicsHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations