Failure Probability of Structural Systems in the Presence of Imprecise Uncertainties

  • S. K. Spoorthi
  • A. S. BaluEmail author
Original Contribution


Structural reliability evaluation is considered to be the solution for modern complex engineering systems possessing uncertain parameters. Reliability estimation involves probabilistic theory when the uncertainties are defined as random variables, whereas with limited resources, it is strenuous to estimate precise parameters in the structural model. Therefore, for such cases, imprecise parameters should be treated appropriately in the design and analysis stage for the improvement of serviceability of the system. On the other side, analyses involving multi-dimensional, computationally expensive, and highly nonlinear structures are formidable in simulation-based methods in the presence of uncertainties. An efficient uncertainty analysis procedure is presented in this paper for analysing the systems with imprecise uncertainties defined as probability-box variables. The estimated bounds of failure probability for the numerical examples from structural mechanics are compared with the traditional approaches to demonstrate the efficiency of the methodology.


Failure probability HDMR Imprecise uncertainty Interval MCS Probability-box 



Not Applicable.


Not Applicable.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


  1. 1.
    H.R. Bae, R.V. Grandhi, R. Canfield, A Sensitivity analysis of structural response uncertainty propagation using evidence theory. Struct. Multidisc. Optim. 31, 270–279 (2006)CrossRefGoogle Scholar
  2. 2.
    D.A. Alvarez, J.E. Hurtado, I. Ramírez, Tighter bounds on the probability of failure than those provided by random set theory. Comput. Struct. 189, 101–113 (2017)CrossRefGoogle Scholar
  3. 3.
    T. Feng, S. Zhang, J. Mi, The reduction and fusion of fuzzy covering systems based on the evidence theory. Int. J. Approx. Reason. 53, 87–103 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. De, N. Dhang, A literature review on building typology and their failure occurrences. J. Inst. Eng. India Ser. A 100, 177–190 (2019)CrossRefGoogle Scholar
  5. 5.
    A.S. Balu, B.N. Rao, High dimensional model representation based formulations for fuzzy finite element analysis of structures. Finite Elem. Anal. Des. 50, 217–230 (2012)CrossRefGoogle Scholar
  6. 6.
    A.S. Balu, B.N. Rao, Efficient assessment of structural reliability in presence of random and fuzzy uncertainties. ASME J. Mech. Design 136, 1–11 (2014)CrossRefGoogle Scholar
  7. 7.
    S.X. Guo, Z.Z. Lu, A non-probabilistic robust reliability method for analysis and design optimization of structures with uncertain-but-bounded parameters. Appl. Math. Model. 39, 1985–2002 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Zhou, Y. Zhang, Explicit integration scheme for a non-isothermal elastoplastic model with convex and nonconvex subloading surfaces. Comput. Mech. 55, 943–961 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Echard, N. Gayton, M. Lemaire, N. Relun, A combined importance sampling and kriging reliability method for small failure probabilities with time-demanding numerical models. Reliab. Eng. Syst. Saf. 111, 232–240 (2013)CrossRefGoogle Scholar
  10. 10.
    Y. Luo, Z. Kang, Z. Luo, A. Li, Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct. Multidisc. Optim. 39, 297–310 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Z. Hu, S. Mahadevan, D. Ao, Uncertainty aggregation and reduction in structure-material performance prediction. Comput. Mech. 61, 237–257 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Zhu, D.M. Frangopol, Reliability assessment of ship structures using Bayesian updating. Eng. Struct. 56, 1836–1847 (2013)CrossRefGoogle Scholar
  13. 13.
    C. Simon, F. Bicking, Hybrid computation of uncertainty in reliability analysis with p-box and evidential networks. Reliab. Eng. Syst. Saf. 167, 629–638 (2017)CrossRefGoogle Scholar
  14. 14.
    S. Xie, B. Pan, X. Du, High dimensional model representation for hybrid reliability analysis with dependent interval variables constrained within ellipsoids. Struct. Multidisc. Optim. 56, 1493–1505 (2017)CrossRefGoogle Scholar
  15. 15.
    P.R. Adduri, R.C. Penmetsa, Bounds on structural system reliability in the presence of interval variables. Comput. Struct. 85, 320–329 (2007)CrossRefGoogle Scholar
  16. 16.
    O.G. Batarseh, Y. Wang, An interval-based approach to model input uncertainty in M/M/1 simulation. Int. J. Approx. Reason. 95, 46–61 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    H.B. Liu, C. Jiang, J. Liu, J.Z. Mao, Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation. Struct. Multidisc. Optim. 59, 61–74 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Schobi, B. Sudret, Structural reliability analysis for p-boxes using multi-level meta-models. Probab. Eng. Mech. 48, 27–38 (2017)CrossRefGoogle Scholar
  19. 19.
    N. Xiao, R.L. Mullen, R.L. Muhanna, Solution of uncertain linear systems of equations with probability-box parameters. Int. J. Reliab. Saf. 12, 147–165 (2016)CrossRefGoogle Scholar
  20. 20.
    X. Liu, Z. Kuang, L. Yin, L. Hu, Structural reliability analysis based on probability and probability box hybrid model. Struct. Saf. 68, 73–84 (2017)CrossRefGoogle Scholar
  21. 21.
    Q. Zhang, Z. Zeng, E. Zio, R. Kang, Probability box as a tool to model and control the effect of epistemic uncertainty in multiple dependent competing failure processes. Appl. Soft Comput. J. 56, 570–579 (2017)CrossRefGoogle Scholar
  22. 22.
    W. Gao, D. Wu, K. Gao, X. Chen, F. Tin-Loi, Structural reliability analysis with imprecise random and interval fields. Appl. Math. Model. 55, 49–67 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    A.S. Balu, B.N. Rao, Inverse structural reliability analysis under mixed uncertainties using high dimensional model representation and fast Fourier transform. Eng. Struct. 37, 224–234 (2012)CrossRefGoogle Scholar
  24. 24.
    J. Lim, B. Lee, I. Lee, Second-order reliability method-based inverse reliability analysis using Hessian update for accurate and efficient reliability-based design optimization. Int. J. Numer. Methods Eng. 100, 773–792 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    I. Lee, K.K. Choi, L. Du, D. Gorsich, Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Comput. Methods Appl. Mech. Engrg. 198, 14–27 (2008)CrossRefGoogle Scholar
  26. 26.
    H. Zhang, R.L. Mullen, R.L. Muhanna, Interval Monte Carlo methods for structural reliability. Struct. Saf. 32, 183–190 (2010)CrossRefGoogle Scholar
  27. 27.
    S. Biswal, A. Ramaswamy, Finite element model updating of concrete structures based on imprecise probability. Mech. Syst. Signal Process. 94, 165–179 (2017)CrossRefGoogle Scholar
  28. 28.
    G. Muscolino, A. Sofi, Analysis of structures with random axial stiffness described by imprecise probability density functions. Comput. Struct. 184, 1–13 (2017)CrossRefGoogle Scholar
  29. 29.
    A. Sil, T. Longmailai, Drift reliability assessment of a four storey frame residential building under seismic loading considering multiple factors. J. Inst. Eng. India Ser. A 98, 245–256 (2017)CrossRefGoogle Scholar
  30. 30.
    A. Guha Ray, D.K. Baidya, Reliability coupled sensitivity based design approach for gravity retaining walls. J. Inst. Eng. India Ser. A 93, 193–201 (2012)CrossRefGoogle Scholar
  31. 31.
    A. Guha Ray, S. Mondal, H.H. Mohiuddin, Reliability analysis of retaining walls subjected to blast loading by finite element approach. J. Inst. Eng. India Ser. A 99, 95–102 (2018)CrossRefGoogle Scholar
  32. 32.
    S.H. Kim, S.W. Na, Response surface method using vector projected sampling points. Struct. Saf. 19, 3–19 (1997)CrossRefGoogle Scholar
  33. 33.
    H. Rabitz, O.F. Aliş, J. Shorter, K. Shim, Efficient input–output model representations. Comput. Phys. Commun. 117, 11–20 (1999)CrossRefGoogle Scholar
  34. 34.
    G. Li, X. Xing, W. Welsh, H. Rabitz, High dimensional model representation constructed by support vector regression I. Independent variables with known probability distributions. J. Math. Chem. 55, 278–303 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    A.S. Balu, B.N. Rao, Confidence bounds on design variables using high-dimensional model representation-based inverse reliability analysis. ASCE J. Struct. Eng. 139, 985–996 (2013)CrossRefGoogle Scholar

Copyright information

© The Institution of Engineers (India) 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational Institute of Technology KarnatakaSurathkalIndia

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