Due to epistemic uncertainty, precisely determining parameters of all distribution is impossible in engineering practice. In this article, a novel reliability analysis method based on the saddlepoint approximation is proposed for structural systems with parameter uncertainties. The proposed method includes four main steps: (1) sampling for random and probability-box variables, (2) approximating the cumulant generating functions for systems under the best and worst cases, (3) calculating saddlepoints for the best and worst cases, and (4) calculating the lower and upper bounds of the probability of failure. The proposed method is effective because it does not require a large sample size or solving complicated integrals. Furthermore, the proposed method provides results that have the same accuracy as the existing interval Monte Carlo simulation method, but with significantly reduced computational effort. The effectiveness of the proposed method is demonstrated with three examples that are compared against with the interval Monte Carlo simulation method.
KiureghianADDitlevsenO. Aleatory or epistemic? Does it matter?Struct Saf2009; 31(2): 105–112.
2.
HuangHZZhangXD. Design optimization with discrete and continuous variables of aleatory and epistemic uncertainties. J Mech Design2009; 131(3): 031006 (8 pp.).
3.
ZhangXDHuangHZ. Sequential optimization and reliability assessment for multidisciplinary design optimization under aleatory and epistemic uncertainties. Struct Multidiscip O2010; 40(1): 165–175.
4.
HuangHZZuoMJSunZQ. Bayesian reliability analysis for fuzzy lifetime data. Fuzzy Set Syst2006; 157(12): 1674–1686.
5.
DuX. Unified uncertainty analysis by the first order reliability method. J Mech Design2008; 130(9): 091401 (10 pp.).
6.
XiaoNCHuangHZWangZL. Reliability analysis of series systems with multiple failure modes under epistemic and aleatory uncertainties. Proc IMechE, Part O: J Risk and Reliability2012; 226(3): 295–304.
7.
SharmaRKKumarDKumarP. Fuzzy modeling of system behavior for risk and reliability analysis. Int J Syst Sci2008; 39(6): 563–581.
8.
FersonSRogerBNHajagosJ. Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis. Sandia National Laboratories report SAND2004-3072, October2004. Albuquerque, NM: Sandia National Laboratories.
DitlevsenOMadsenHO. Structural reliability methods. West Sussex: Wiley, 2007.
16.
HuangBQDuX. Probabilistic uncertainty analysis by mean value first order saddlepoint approximation. Reliab Eng Syst Safe2008; 93(2): 325–336.
17.
ZhaoYGOnoT. Moment methods for structural reliability. Struct Saf2001; 23(1): 47–75.
18.
KaymazI. Application of kriging method to structural reliability problems. Struct Saf2005; 27(2): 133–151.
19.
HenriquesAAVeigaJMCMatosJAC. Uncertainty analysis of structural systems by perturbation techniques. Struct Multidiscip O2008; 35(3): 201–212.
20.
OberkampfMLHeltonJCJoslynCA. Challenge problems: uncertainty in system response given uncertainty parameters. Reliab Eng Syst Safe2004; 85(1–3): 11–19.
21.
DuXSudjiantoAHuangBQ. Reliability-based design with the mixture of random and interval variables. J Mech Design2005; 127(6): 1068–1076.
22.
XiaoNCHuangHZWangZL. Unified uncertainty analysis by the mean value first order saddlepoint approximation. Struct Multidiscip O2012; 46(6): 803–812.
23.
ZamanKMcDonaldMMahadevanS. Probabilistic framework for uncertainty propagation with both probabilistic and interval variables. J Mech Design2011; 133(2): 021010 (14 pp.).
24.
MöllerBGrafWBeerM. Safety assessment of structures in view of fuzzy randomness. Comput Struct2003; 81(15): 1567–1582.
25.
WangZHuangH-ZLiY. An approach to system reliability analysis with fuzzy random variables. Mech Mach Theory2012; 52: 35–46.
26.
ZhouJMourelatosZP. A sequential algorithm for possibility-based design optimization. J Mech Design2008; 130(1): 011001 (10 pp.).
27.
FersonSKreinovichVHajagosJ. Experimental uncertainty estimation and statistics for data having interval uncertainty. Sandia National Laboratories report SAND2007-0939, May2007. Albuquerque, NM: Sandia National Laboratories.
28.
FersonSKreinovichVGinzburgL. Constructing probability boxes and Dempster-Shafer structures. Sandia National Laboratories report SAND2002-4015, January2003. Albuquerque, NM: Sandia National Laboratories.
29.
XiaoNCHuangHZWangZL. Reliability sensitivity analysis for structural systems in interval probability form. Struct Multidiscip O2011; 44(5): 691–705.
30.
ZhangH. Interval importance sampling method for finite element-based structural reliability assessment under parameter uncertainties. Struct Saf2012; 38: 1–10.
31.
ZhangHMullenRLMuhannaRL. Interval Monte Carlo methods for structural reliability. Struct Saf2010; 32(3): 183–190.
32.
ZhangHDaiHZBeerM. Structural reliability analysis on the basis of small samples: an interval quasi-Monte Carlo method. Mech Syst Signal Pr2013; 37(1–2): 137–151.
33.
FersonSTuckerWT. Sensitivity analysis using probability bounding. Reliab Eng Syst Safe2006; 91(10–11): 1435–1442.
34.
HuangBQDuXLakshminarayanaRE. A saddlepoint approximation based simulation method for uncertainty analysis. Int J Reliab Saf2006; 1(1/2): 206–224.
35.
DanielsHE. Saddlepoint approximations in statistics. Ann Math Stat1954; 25(4): 631–650.
36.
DuX. Saddlepoint approximation for sequential optimization and reliability analysis. J Mech Design2008; 130(1): 011011 (11 pp.).
37.
YuenKVWangJAuSK. Application of saddlepoint approximation in reliability analysis of dynamic systems. Earthq Eng Eng Vib2007; 6(45): 391–400.
38.
LugannaniRRiceSO. Saddlepoint approximation for the distribution of the sum of independent random variables. Adv Appl Probab1980; 12(2): 475–490.
39.
Barndorff-NielsenOE. Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika1986; 73(2): 307–322.
40.
GillespieCSRenshawE. An improved saddlepoint approximation. Math Biosci2007; 208(2): 359–374.
41.
ReidN. Saddlepoint methods and statistical inference. Stat Sci1988; 3(2): 213–227.
42.
AlvarezDA. On the calculation of the bounds of probability of events using infinite random sets. Int J Approx Reason2006; 43(3): 241–267.
43.
GuoSXZhangLLiY. Procedures for computing the non-probabilistic reliability index of uncertain structures. Chinese J Comput Mech2005; 22(2): 227–232.
44.
GuoSX. Non-stochastic reliability and optimization of uncertain structural systems. Xi’an: Northwestern Polytechnic University, 2005.
