A modified method to calculate reliability index using maximum entropy principle

Abstract

Routine reliability index method, first order second moment (FOSM), may not ensure convergence of iteration when the performance function is strongly nonlinear. A modified method was proposed to calculate reliability index based on maximum entropy (MaxEnt) principle. To achieve this goal, the complicated iteration of first order second moment (FOSM) method was replaced by the calculation of entropy density function. Local convergence of Newton iteration method utilized to calculate entropy density function was proved, which ensured the convergence of iteration when calculating reliability index. To promote calculation efficiency, Newton down-hill algorithm was incorporated into calculating entropy density function and Monte Carlo simulations (MCS) were performed to assess the efficiency of the presented method. Two numerical examples were presented to verify the validation of the presented method. Moreover, the execution and advantages of the presented method were explained. From Example 1, after seven times iteration, the proposed method is capable of calculating the reliability index when the performance function is strongly nonlinear and at the same time the proposed method can preserve the calculation accuracy; From Example 2, the reliability indices calculated using the proposed method, FOSM and MCS are 3.823 9, 3.813 0 and 3.827 6, respectively, and the according iteration times are 5, 36 and 10⁶, which shows that the presented method can improve calculation accuracy without increasing computational cost for the performance function of which the reliability index can be calculated using first order second moment (FOSM) method.

Keywords

reliability index / maximum entropy principle / first order second moment / Newton iteration / Monte Carlo simulation

References

[1] PhoonK KReliability-based design in geotechnical engineering: Computations and applications [M], 2008London and New YorkTaylor & Francis372-375

[2] JuangC H, FangS Y, KhorE H. First-order reliability method for probabilistic liquefaction triggering analysis using CPT [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2006, 132(3): 337-350

[3] HuangJ S, GriffithsD V. Observations on FORM in a simple geomechanics example [J]. Structural Safety, 2011, 33(1): 115-119

[4] XiangY B, LiuY M. Inverse first-order reliability method for probabilistic fatigue life prediction of composite laminates under multiaxial loading [J]. Journal of Aerospace Engineering, 2011, 24(2): 189-198

[5] RackwitzR. Reliability analysis-A review and some perspectives [J]. Structural Safety, 2001, 23(4): 365-395

[6] ZhaoY G, TetsuroO. A general procedure for first/secondorder reliability method (FORM/SORM) [J]. Structural Safety, 1999, 21(2): 95-112

[7] AngH S, TangW HProbability concepts in engineering planning and design, 20074nd editionNew YorkJohn Wiley133-134

[8] ZhaoY G, TetsuroO. Moment methods for structural reliability [J]. Structural Safety, 2001, 23(1): 47-75

[9] LowB K, TangW H. Reliability analysis using object oriented constrained optimization [J]. Structural Safety, 2004, 26(1): 69-89

[10] LowB K, TangW H. Efficient spreadsheet algorithm for first-order reliability Method [J]. Journal of Engineering Mechanics, ASCE, 2007, 133(12): 1378-1387

[11] ZhengJ J, LiuY, GuoJ. Equal step iteration method for reliability index and its proof of convergence [J]. Journal of Huazhong University of Science and Technology: Natural Science Edition, 2008, 36(8): 129-132

[12] ZhangJ, ZhangL M, TangW H, DistM. Reliability-based optimization of geotechnical systems [J]. Journal of Geotechnical and Geoenvironmental Engineering, 2011, 137(12): 1211-1221

[13] JaynesE T. Information theory and statistical mechanics [J]. Physic Review, 1957, 106(4): 620-630

[14] PhillipsS J, AndersonR P, SchapireR E. Maximum entropy modeling of species geographic distributions [J]. Ecological Modeling, 2006, 190(3/4): 231-259

[15] HyonY, CarrilloJ A, DuQ, KineticC L, ModelsR. A maximum entropy principle based closure method for macro-micro models of polymeric metals [J]. Kinetic and Related Models, 2008, 1(2): 1-14

[16] ZhaoW Y, XuZ J, ZhengJ J. Reliability analysis of vertical bearing capacity of pile using random-fuzzy entropy principle [C]. Sixth International Conference on Natural Computation (ICNC 2010), 20104198-4193

[17] BanayarJ R, MaritanA, VolkovI. Applications of the principle of maximum entropy: from physics to ecology [J]. Journal of Physics: Condensed Mater, 2010, 22(6): 1-13

[18] ZhengJ J, XuZ J, LiuY, MaQ. Reliability analysis for vertical bearing capacity of pile based on the maximum entropy principle [J]. Chinese Journal of Geotechnical Engineering, 2010, 32(11): 1643-1647

[19] TaylorM EPartial differential equation III: nonlinear equations [M], 2011New YorkSpringer Science & Business Media, LLC176-182