Abstract
In this paper, a new reliability analysis method for engineering structures is developed based on probability and probability box (p-box) models. Random variable distributions are used to deal with the uncertain parameters with sufficient information, while the p-box models are employed to deal with the uncertain-but-bounded variables. Due to the existence of the p-box parameters, a limit-state band will result and a complex nesting optimization problem will be involved in this reliability analysis. To reduce the computational burden, an efficient decoupling strategy is developed to solve the nesting optimization problem. Through interval analysis for the probability transformation process, the complex nesting optimization problem can be transformed to a single-layer optimization model. Then, the optimum solution and corresponding reliability index can be obtained by introducing a sequential quadratic programming (SQP) method. Four engineering numerical examples are investigated to demonstrate the effectiveness of the present method.
Similar content being viewed by others
References
Angelis MD, Patelli E, Beer M (2015) Advanced line sampling for efficient robust reliability analysis. Struct Saf 52:170–182
Au FTK, Cheng YS, Tham LG, Zeng GW (2003) Robust design of structures using convex models. Comput Struct 81:2611–2619
Balu AS, Rao BN (2012) Multicut-high dimensional model representation for structural reliability bounds estimation under mixed uncertainties. Comput Aided Civ Inf Eng 27:419–438
Baudrit C, Dubois D (2006) Practical representations of incomplete probabilistic knowledge. Comput Stat Data Anal 51(1):86–108
Ben-Haim Y (1993) Convex models of uncertainty in radial pulse buckling of shells. ASME J Appl Mech 60(3):683–688
Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14(4):227–245
Breitung K (1984) Asymptotic approximation for multi-normal integrals. ASCE J Eng Mech 110(3):357–366
Cao HJ, Duan BY (2005) An approach on the non-probabilistic reliability of structures based on uncertainty convex models. Chin J Comput Mech 22(5):546–549
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
Dempster AP (1967) Upper and lower probabilities induced by a multi-valued mapping. Ann Math Stat 38:325–339
Du XP (2007) Interval reliability analysis. In: ASME 2007 Design Engineering Technical Conference & Computers and Information in Engineering Conference (DETC2007), Las Vegas, Nevada, USA
Du X (2008) Unified uncertainty analysis by the first order reliability method. J Mech Des 130:091401–091410
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
Dubois D (2010) Representation, propagation, and decision issues in risk analysis under incomplete probabilistic information. Risk Anal 30(3):662–675
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
Elishakoff I (1995) Discussion on a non-probabilistic concept of reliability. Struct Saf 17(3):195–9
Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54(2-3):133–144
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
Ferson S, Nelsen R, Hajagos J, Berleant D, Zhang J, Tucker WT, Ginzburg L, Oberkampf WL (2004) Dependence in probabilistic modeling, Dempster– Shafer theory, and probability bounds analysis, Sandia National Laboratories, SAND2004-3072
Fetz T, Oberguggenberger M (2010) Multivariate models of uncertainty: a local random set approach. Struct Saf 32:417–24
Guo X, Bai W, Zhang WS (2009) Confidence extremal structural response analysis of truss structures under static load uncertainty via SDP relaxation. Comput Struct 87:246–253
Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. ASME J Eng Mech Div 100:111–121
Hohenbichler M, Rackwitz R (1981) Non-normal dependent vectors in structural safety. ASME J Eng Mech Div 107:1227–1238
Jiang C, Han X, Liu GR (2007) Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Comput Methods Appl Mech Eng 196:4791–4800
Jiang C, Han X, Lu GY, Liu J, Zhang Z, Bai YC (2011a) Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput Methods Appl Mech Eng 200:2528–2546
Jiang C, Li WX, Han X, Liu LX, Le PH (2011b) Structural reliability analysis based on random distributions with interval parameters. Comput Struct 89:2292–2302
Li Y, Jiang P, Gao L, Shao XY (2013) Sequential optimisation and reliability assessment for multidisciplinary design optimisation under hybrid uncertainty of randomness and fuzziness. J Eng Des 24(5):363–382
Liang JH, Mourelatos ZP, Nikolaidis E (2007) A single-loop approach for system reliability-based design optimization. ASME J Mech Des 129:1215–1224
Liu X, Zhang ZY (2014) A hybrid reliability approach for structure optimization based on probability and ellipsoidal convex models. J Eng Des 25(4-6):238–258
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
Oberguggenberger M (2015) Analysis and computation with hybrid random set stochastic models. Struct Saf 52:233–243
Polidori DC, Beck JL, Papadimitriou C (1999) New approximations for reliability integrals. J Eng Mech 125(4):466–475
Qiu Z, Yang D, Elishakoff I (2008) Probabilistic interval reliability of structural systems. Int J Solids Struct 45:2850–2860
Rackwitz R, Fiessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494
Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton
Troffaes M, Destercke S (2011) Probability boxes on totally preordered spaces for multivariate modeling. Int J Approx Reason 52:767–791
Troffaes M, Miranda E, Destercke S (2013) On the connection between probability boxes and possibility measures. Inf Sci 224:88–108
Tsai YT, Lin KH, Hsu YY (2013) Reliability design optimisation for practical applications based on modelling processes. J Eng Des 24(12):849–863
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
Xiao N, Mullen R, Muhanna R (2016) Static analysis of structural systems with uncertain parameters using probability-box. In: The 7th International Workshop on Reliable Engineering Computing (REC2016), Bochum, Germany
Yang XF, Liu YS, Zhang YS, Yue ZF (2015) Hybrid reliability analysis with both random and probability-box variables. Acta Mech 226:1341–1357
Yao W, Chen X, Huang Y, Tooren M (2013) An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization. Reliab Eng Syst Saf 116:28–37
Youn BD, Wang P (2009) Complementary intersection method for system reliability analysis. ASME J Mech Des 131(4):041004–041018
Zhao ZH, Han X, Jiang C, Zhou XX (2010) A nonlinear interval-based optimization method with local-densifying approximation technique. Struct Multidiscip Optim 42:559–573
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No.51305047, 51475048), the Specialized Research Fund for the Doctoral Program of Higher Education (New Teachers: 20134316120003) and the Science Fundation of State Key Laboratory for Strength and Vibration of Mechanical Structures (No.SV2016-KF-09). The authors would also like to thank anonymous reviewers for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, X., Yin, L., Hu, L. et al. An efficient reliability analysis approach for structure based on probability and probability box models. Struct Multidisc Optim 56, 167–181 (2017). https://doi.org/10.1007/s00158-017-1659-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-017-1659-7