Skip to main content
SpringerLink
Log in

A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Compared with a probability model, a non-probabilistic convex model only requires a small number of experimental samples to discern the uncertainty parameter bounds instead of the exact probability distribution. Therefore, it can be used for uncertainty analysis of many complex structures lacking experimental samples. Based on the multidimensional parallelepiped convex model, we propose a new method for non-probabilistic structural reliability analysis in which marginal intervals are used to express scattering levels for the parameters, and relevant angles are used to express the correlations between uncertain variables. Using an affine coordinate transformation, the multidimensional parallelepiped uncertainty domain and the limit-state function are transformed to a standard parameter space, and a non-probabilistic reliability index is used to measure the structural reliability. Finally, the method proposed herein was applied to several numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hasofer A.M., Lind N.C.: Exact and invariant second-moment code format. ASME J. Eng. Mech. Div. 100, 111–121 (1974)

    Google Scholar 

  2. Rackwitz R., Fiessler B.: Structural reliability under combined random load sequences. Comput. Struct. 9, 489–494 (1978)

    Article  MATH  Google Scholar 

  3. Hohenbichler M., Rackwitz R.: Non-normal dependent vectors in structural safety. ASME J. Eng. Mech. Div. 107, 1227–1238 (1981)

    Google Scholar 

  4. Breitung K.W.: Asymptotic approximation for multinormal integrals. ASCE J. Eng. Mech. 110, 357–366 (1984)

    Article  Google Scholar 

  5. Breitung K.W.: Asymptotic Approximations for Probability Integrals. Springer, Berlin (1994)

    MATH  Google Scholar 

  6. Polidori D.C., Beck J.L., Papadimitriou C.: New approximations for reliability integrals. ASCE J. Eng. Mech. 125, 466–475 (1994)

    Article  Google Scholar 

  7. Thoft-Christensen P., Murotsu Y.: Application of Structural Systems Reliability Theory. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  8. Ang A.H.S., Tang W.H.: Probability Concepts in Engineering Planning and Design. Volume II: Decision, Risk and Reliability. Wiley, New York (1984)

    Google Scholar 

  9. Ben-Haim Y., Elishakoff I.: Convex Models of Uncertainties in Applied Mechanics. Elsevier, Amsterdam (1990)

    Google Scholar 

  10. Ben-Haim Y.: Convex models of uncertainty in radial pulse buckling of shells. ASME J. Appl. Mech. 60, 683 (1993)

    Article  MATH  Google Scholar 

  11. Elishakoff I., Elisseeff P.: Non-probabilistic convex-theoretic modeling of scatter in material properties. AIAA J. 32, 843–849 (1994)

    Article  MATH  Google Scholar 

  12. Ben-Haim Y.: A non-probabilistic concept of reliability. Struct. Saf. 14, 227–245 (1994)

    Article  Google Scholar 

  13. Elishakoff I.: An idea on the uncertainty triangle. Editors rattle space. Shock Vib. 22, 1 (1995)

    Google Scholar 

  14. Qiu Z.P., Wang X.J.: Parameter perturbation method for dynamic responses of structures with uncertain-but-bounded parameters based on interval analysis. Int. J. Solids Struct. 42, 4958–4970 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wu J., Zhao Y.Q., Chen S.H.: An improved interval analysis method for uncertain structures. Struct. Eng. Mech. 20, 713–726 (2005)

    Article  Google Scholar 

  16. Zhou Y.T., Jiang C., Han X.: Interval and subinterval analysis methods of the structures and their error estimations. Int. J. Comput. Methods 3, 229–244 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhu, L.P., Elishakoff, I., Starnes, J.H. JR.: Derivation of multi-dimensional ellipsoidal convex model for experimental data. Math. Comput. Model. 24, 103–114 (1996)

    Google Scholar 

  18. Wang X.J., Elishakoff I., Qiu Z.P.: Experimental data have to decide which of the nonprobabilistic uncertainty descriptions—convex modeling or interval analysis—to utilize. J. Appl. Mech. 75, 041018 (2008)

    Article  Google Scholar 

  19. Jiang C., Han X., Lu G.Y., Liu J., Bai Y.C.: Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput. Method Appl. Mech. Eng. 200, 2528–2546 (2011)

    Article  MATH  Google Scholar 

  20. Elishakoff I.: Discussion on a non-probabilistic concept of reliability. Struct. Saf. 17, 195–199 (1995)

    Article  Google Scholar 

  21. Guo S.X., Lv Z.Z., Feng Y.S.: A non-probabilistic model of structural reliability based on interval analysis. Chin. J. Comput. Mech. 18, 56–60 (2001)

    Google Scholar 

  22. Guo S.X., Lv Z.Z.: Comparison between the non-probabilistic and probabilistic reliability methods for uncertain structure design. Chin. J. Appl. Mech. 20, 107–110 (2003)

    Google Scholar 

  23. Cao H.J., Duan B.Y.: An approach on the non-probabilistic reliability of structures based on uncertainty convex models. Chin. J. Comput. Mech. 22, 546–549 (2005)

    Google Scholar 

  24. Jiang T., Chen J.J.: A semi-analytic method for calculating non-probabilistic reliability index based on interval models. Appl. Math. Model. 56, 1362–1370 (2007)

    Google Scholar 

  25. Kang Z., Luo Y.J.: On structural optimization for non-probabilistic reliability based on convex models. Chin. J. Theor. Appl. Mech. 38, 807–815 (2006)

    Google Scholar 

  26. Luo Y.J., Kang Z., Luo Z., Alex L.: Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct. Multidiscip. Optim. 39, 297–310 (2008)

    Article  Google Scholar 

  27. Guo X., Bai W., Zhang W.S.: Extremal structural response analysis of truss structures under load uncertainty via SDP relaxation. Comput. Struct. 87, 246–253 (2009)

    Article  Google Scholar 

  28. Jiang C., Han X., Liu G.R.: Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Comput. Method Appl. Mech. Eng. 196, 4791–4890 (2007)

    Article  MATH  Google Scholar 

  29. Jiang C., Han X., Liu G.P.: A sequential nonlinear interval number programming method for uncertain structures. Comput. Method Appl. Mech. Eng. 197, 4250–4265 (2008)

    Article  MATH  Google Scholar 

  30. Du, X.P.: Interval reliability analysis. In: ASME Design Engineering Technical Conference & Computers and Information in Engineering Conference (DETC2007). Las Vegas, Nevada, USA (2007)

  31. Luo Y.J., Kang Z., Li A.: Structural reliability assessment based on probability and convex set mixed model. Comput. Struct. 87, 1408–1415 (2009)

    Article  Google Scholar 

  32. Jiang C., Li W.X., Han X., Liu L.X., Le P.H.: Structural reliability analysis based on random distributions with interval parameters. Comput. Struct. 89, 2292–2302 (2011)

    Article  Google Scholar 

  33. Jiang C., Han X., Li W.X., Liu J., Zhang Z.: A hybrid reliability approach based on probability and interval for uncertain structures. ASME J. Mech. Des. 134, 1–11 (2012)

    Article  Google Scholar 

  34. Jiang, C., Zhang, Q.F., Han, X.: Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty quantification. Int. J. Numer. Meth. Eng. (submitted)

  35. Nocedal J., Wright S.J.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  36. Elishakoff I., Haftka R.T., Fang J.: Structural design under bounded uncertainty-optimization with anti-optimization. Comput. Struct. 53, 1401–1405 (1994)

    Article  MATH  Google Scholar 

  37. Zhang W.G., He W., Zhong Z.H.: Techniques for Passenger Protection in Vehicle Crash. Hunan University Press, Changsha (2007)

    Google Scholar 

  38. Ma L.L., Yang N., Zhao G.F.: Structural crashworthiness analysis and improvement of the car. Mach. Des. Manuf. 6, 106–107 (2008)

    Google Scholar 

  39. Peng C.K., Wang G.Y., Wu Y.N.: Simulation study on crashworthiness of the car’s front side rail. Mech. Res. Appl. 20, 77–78 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha City, 410082, People’s Republic of China

    C. Jiang, Q. F. Zhang & X. Han

  2. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua City, 321004, People’s Republic of China

    Y. H. Qian

Authors
  1. C. Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Q. F. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. H. Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Jiang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, C., Zhang, Q.F., Han, X. et al. A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model. Acta Mech 225, 383–395 (2014). https://doi.org/10.1007/s00707-013-0975-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-013-0975-2

Keywords

Search

Navigation