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Construction of probability box model based on maximum entropy principle and corresponding hybrid reliability analysis approach

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Abstract

In this paper, a new method for constructing the probability box (p-box) model is developed based on maximum entropy principle. The distribution characteristics of probability box variable can be described by the nature of moments. The moment conditions are used to ensure the consistency of the cumulative distribution function (CDF), and the shape conditions are adopted to guarantee the validity of the cumulative distribution function. To ensure the uniqueness of the cumulative distribution function, simultaneously, the cumulative distribution function of the probability box variable is reconstructed based on maximum entropy principle. Then, considering that both aleatory and epistemic uncertainty exist in many engineering problems, a reliability analysis approach based on probability and probability box hybrid model is also developed for uncertain structures. Finally, four numerical examples and two engineering examples are investigated to demonstrate the effectiveness of the present method.

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References

  • Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. ASME J Eng Mech Div 100:111–121

    Google Scholar 

  • Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494

    MATH  Google Scholar 

  • Breitung K (1984) Asymptotic approximation for multi-normal integrals. ASCE J Eng Mech 110(3):357–366

    Google Scholar 

  • Liang JH, Mourelatos ZP, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. ASME J Mech Des 129:1215–1224

    Google Scholar 

  • Liu J, Liu H, Jiang C, Han X, Zhang DQ, Hu YF (2018) A new measurement for structural uncertainty propagation based on pseudo-probability distribution. Appl Math Model 63:744–760

    MathSciNet  Google Scholar 

  • Tsai YT, Lin KH, Hsu YY (2013) Reliability design optimisation for practical applications based on modelling processes. J Eng Design 24(12):849–863

    Google Scholar 

  • Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353

    MATH  Google Scholar 

  • Zadeh L (1978) Fuzzy set as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    MathSciNet  MATH  Google Scholar 

  • Klir GJ (2004) Generalized information theory: aims, results, and open problems. Reliab Eng Syst Saf 85(1–3):21–38

    Google Scholar 

  • Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainties in applied mechanics. Elsevier Science Publisher, Amsterdam

    MATH  Google Scholar 

  • Qiu ZP, Ma LH, Wang XJ (2006) Ellipsoidal-bound convex model for the non-linear buckling of a column with uncertain initial imperfection. Int J Nonlin Mech 41(8):919–925

    Google Scholar 

  • Jiang C, Han X, Lu GY, Liu J (2011) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Method Appl Mech 200(33–36):2528–2546

    MATH  Google Scholar 

  • Luo YJ, Kang Z, Li A (2009) Structural reliability assessment based on probability and convex set mixed model. Comput Struct 87(21–22):1408–1415

    Google Scholar 

  • Gao W, Song CM, Tin-Loi F (2010) Probabilistic interval analysis for structures with uncertainty. Struct Saf 32(3):191–199

    Google Scholar 

  • Ben-Haim Y (1993) Convex models of uncertainty in radial pulse buckling of shells. ASME J Appl Mech 60(3):683–688

    MATH  Google Scholar 

  • Cao HJ, Duan BY (2005) An approach on the non-probabilistic reliability of structures based on uncertainty convex models. Chinese J Comput Mech 22(5):546–549

    Google Scholar 

  • Du XP, Sudjianto A, Huang BQ (2005) Reliability-based design with the mixture of random and interval variables. ASME J Mech Des 127:1068–1076

    Google Scholar 

  • Adduri PR, Penmetsa RC (2007) Bounds on structural system reliability in the presence of interval variables. Comput Struct 85:320–329

    Google Scholar 

  • Williamson RC, Downs T (1990) Probabilistic arithmetic I: numerical methods for calculating convolutions and dependency bounds. Int J Approx Reason 4:89–158

    MathSciNet  MATH  Google Scholar 

  • Baudrit C, Dubois D (2006) Practical representations of incomplete probabilistic knowledge. Comput Stat Data An 51(1):86–108

    MathSciNet  MATH  Google Scholar 

  • Ferson S, Ginnzburg A (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54(1):133–144

    Google Scholar 

  • Ferson S, Nelsen R, Hajagos J, Berleant D, Zhang J, Tucker WT, Ginzburg L, Oberkampf WL. Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis, Sandia National Laboratories, SAND2004–3072, 2004

    Google Scholar 

  • Dubois D (2010) Representation, propagation, and decision issues in risk analysis under incomplete probabilistic information. Risk Anal 30(3):662–675

    Google Scholar 

  • Dutta P, Ali T (2012) A hybrid method to deal with aleatory and epistemic uncertainty in risk assessment. Int J Comput Appl 42(11):37–44

    Google Scholar 

  • Matthias T, Enrique M, Sebastien D (2013) On the connection between probability boxes and possibility measures. Inf Sci 224:88–108

    MathSciNet  MATH  Google Scholar 

  • Aughenbaugh JM, Paredis CJJ. Probability bounds analysis as a general approach to sensitivity analysis in decision making under uncertainty. in: SAE2007 transactions journal of passenger cars: mechanical systems, SAE, International, Warrendale, Pennsylvania, 2007, pp. 1325–1339

  • Du XP (2008) Unified uncertainty analysis by the first order reliability method. J Mech Des 130:091401–091410

    Google Scholar 

  • Oberguggenberger M (2015) Analysis and computation with hybrid random set stochastic models. Struct Saf 52:233–243

    Google Scholar 

  • Goldwasser L, Ginzburg L, Ferson S (2000) Quantitative methods for conservation biology. Springer-Berlag, New York

    Google Scholar 

  • Kriegler E, Held H (2005) Utilizing belief functions for the estimation of future climate change. Int J Approx Reason 39(2–3):185–209

    MathSciNet  MATH  Google Scholar 

  • Liu HB, Jiang C, Liu J, Mao JZ (2019) Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation. Struct Multidiscip Optim 59(1):61–74

    MathSciNet  Google Scholar 

  • Li YF, Ding Y, Zio E (2014) Random fuzzy extension of the universal generating function approach for the reliability assessment of multi-state systems under aleatory and epistemic uncertainties. IEEE T Reliab 63(1):13–25

    Google Scholar 

  • Liu X, Yin LR, Hu L, Zhang ZY (2017) An efficient reliability analysis approach for structure based on probability and probability box models. Struct Multidiscip Optim 56(1):167–181

    MathSciNet  Google Scholar 

  • Crespo LG, Kenny SP, Giesy DP (2013) Reliability analysis of polynomial systems subject to p-box uncertainties. Mech Syst Signal Process 37(1–2):121–136

    Google Scholar 

  • Chen N, Yu DJ, Xia BZ, Beer M (2016) Uncertainty analysis of a structural–acoustic problem using imprecise probabilities based on p-box representations. Mech Syst Signal Process 80:45–57

    Google Scholar 

  • Yang XF, Liu YS, Zhang YS, Yue ZF (2015) Hybrid reliability analysis with both random and probability-box variables. Acta Mech 226(5):1341–1357

    MathSciNet  MATH  Google Scholar 

  • Simon C, Bicking F (2017) Hybrid computation of uncertainty in reliability analysis with p-box and evidential networks. Reliab Eng Syst Saf 167:629–638

    Google Scholar 

  • Qiu Z, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. Int J Solids Struct 45:2850–2860

    MATH  Google Scholar 

  • Fetz T, Oberguggenberger M (2010) Multivariate models of uncertainty: a local random set approach. Struct Saf 32:417–424

    Google Scholar 

  • Xiao NC, Huang HZ, Wang Z, Pang Y, He L (2011) Reliability sensitivity analysis for structural systems in interval probability form. Struct Multidiscip Optim 44:691–705

    MathSciNet  MATH  Google Scholar 

  • Zhang H, Mullen RL, Muhanna RL (2010) Interval Monte Carlo methods for structural reliability. Struct Saf 32(3):183–190

    Google Scholar 

  • Oberguggenberger M, Fellin W (2008) Reliability bounds through random sets: nonparametric methods and geotechnical applications. Comput Struct 86(10):1093–1101

    Google Scholar 

  • Wang C, Zhang H, Beer M (2018) Computing tight bounds of structural reliability under imprecise probabilistic information. Comput Struct 208:92–104

    Google Scholar 

  • Ferson S, Kreinovich V, Ginzburg L, Myers DS, Sentz K. (2003) Constructing probability boxes and Dempster–Shafer structures, Technical Report SAND2002-4015, Sandia National Laboratories

  • Dempster AP (1967) Upper and lower probabilities induced by a multi-valued mapping. Ann Math Stat 38:325–339

    MATH  Google Scholar 

  • Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton, NJ

    MATH  Google Scholar 

  • Moore RE (1979) Methods and applications of interval analysis. Prentice-Hall Inc., London

    MATH  Google Scholar 

  • Xu J, Dang C, Kong F (2017) Efficient reliability analysis of structures with the rotational quasi-symmetric point-and the maximum entropy methods. Mech Syst Signal Process 95:58–76

    Google Scholar 

  • Jiang C, Han X, Liu GR (2008) a. A nonlinear interval number programming method for uncertain optimization problems, Eur. J Oper Res 188:1–13

    MathSciNet  MATH  Google Scholar 

  • Liu GR, Han X (2003) Computational inverse techniques in nondestructive evaluation. CRC Press, Florida

    MATH  Google Scholar 

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

    Google Scholar 

  • Polidori DC, Beck JL, Papadimitriou C (1994) New approximations for reliability integrals. ASCE J Eng Mech 125(4):466–475

    Google Scholar 

  • Au FTK, Cheng YS, Tham LG, Zeng GW (2003) Robust design of structures using convex models. Comput Struct 81:2611–2619

    Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 51775057, 51775414), the Scientific Research Fund of Hunan Provincial Education Department (No. 16B014), and Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education (kfj170401). The authors would also like to thank anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education, Changsha University of Science & Technology, Changsha, 410114, People’s Republic of China

    Xin Liu, Xinyu Wang & Jun Xie

  2. Key Laboratory of Education Ministry for Modern Design & Rotor-Bearing System, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Baotong Li

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  1. Xin Liu
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  2. Xinyu Wang
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  3. Jun Xie
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  4. Baotong Li
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Correspondence to Baotong Li.

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Liu, X., Wang, X., Xie, J. et al. Construction of probability box model based on maximum entropy principle and corresponding hybrid reliability analysis approach. Struct Multidisc Optim 61, 599–617 (2020). https://doi.org/10.1007/s00158-019-02382-9

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  • DOI: https://doi.org/10.1007/s00158-019-02382-9

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