An efficient method for the estimation of structural reliability intervals with random sets, dependence modeling and uncertain inputs

A general method for estimating the bounds of the reliability of a system in which the input variables are described by random sets (probability distributions, probability boxes, or possibility distributions), with dependence modeling is proposed. The method is based on an analytical property of the so-called design point vector; this property is exploited by constructing a nonlinear projection of Monte Carlo samples of the input variables in a two-dimensional diagram from which the analyst can easily extract the relevant samples for computing both the lower and upper bounds of the failure probability using random set theory. The method, which is illustrated with some examples, represents a dramatic reduction in the number of focal element evaluations performed when applying the Monte Carlo method to random set theory.

[1]  R. Nelsen.  An Introduction to Copulas (Springer Series in Statistics) , 2006 .

[2]  M. Beer,et al.  Structural reliability analysis on the basis of small samples: An interval quasi-Monte Carlo method , 2013 .

[3]  Hung T. Nguyen,et al.  An Introduction to Random Sets , 2006 .

[4]  R. Rackwitz,et al.  Quadratic Limit States in Structural Reliability , 1979 .

[5]  R. Nelsen.  An Introduction to Copulas , 1998 .

[6]  Makoto Ohsaki,et al.  Optimization And Anti-Optimization Of Structures Under Uncertainty , 2010 .

[7]  Xiaoping Du,et al.  Sensitivity Analysis with Mixture of Epistemic and Aleatory Uncertainties , 2007 .

[8]  J. Lawry,et al.  Fuzzy label methods for constructing imprecise limit state functions , 2003 .

[9]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[10]  M. Valdebenito,et al.  The role of the design point for calculating failure probabilities in view of dimensionality and structural nonlinearities , 2010 .

[11]  A. Kiureghian,et al.  Second-Order Reliability Approximations , 1987 .

[12]  Xiaojun Wang,et al.  Probability and convexity concepts are not antagonistic , 2011 .

[14]  I. Elishakoff.  Essay on uncertainties in elastic and viscoelastic structures: From A. M. Freudenthal's criticisms to modern convex modeling , 1995 .

[15]  Yan-Gang Zhao,et al.  New Approximations for SORM: Part 2 , 1999 .

[16]  Xiaoping Du.  Unified Uncertainty Analysis by the First Order Reliability Method , 2008 .

[17]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[18]  Fulvio Tonon.  Using random set theory to propagate epistemic uncertainty through a mechanical system , 2004, Reliab. Eng. Syst. Saf..

[19]  C. Jiang,et al.  Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique , 2011 .

[20]  L. Tvedt.  Distribution of quadratic forms in normal space-application to structural reliability , 1990 .

[21]  Y. Ben-Haim.  Robust reliability in the mechanical sciences , 1996 .

[22]  G. Schuëller,et al.  A critical appraisal of methods to determine failure probabilities , 1987 .

[23]  Christopher D. Eamon,et al.  Evaluation of accuracy and efficiency of some simulation and sampling methods in structural reliability analysis , 2005 .

[24]  Diego A. Alvarez,et al.  A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type , 2009, Fuzzy Sets Syst..

[25]  Pekka Neittaanmki,et al.  Structural Optimization with Uncertainties , 2009 .

[26]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[27]  Kiyohiro Ikeda,et al.  Stability and optimization of structures , 2007 .

[28]  Scott Ferson,et al.  Constructing Probability Boxes and Dempster-Shafer Structures , 2003 .

[29]  A. Sklar,et al.  Random variables, distribution functions, and copulas---a personal look backward and forward , 1996 .

[30]  Sankaran Mahadevan,et al.  Multiple Linearization Method for Nonlinear Reliability Analysis , 2001 .

[31]  I. Elishakoff,et al.  Are Probabilistic and Anti-Optimization Approaches Compatible? , 1999 .

[32]  Jon C. Helton,et al.  Evidence Theory for Engineering Applications , 2004 .

[33]  Jorge E. Hurtado,et al.  Dimensionality reduction and visualization of structural reliability problems using polar features , 2012 .

[34]  Robert E. Melchers,et al.  Structural Reliability: Analysis and Prediction , 1987 .

[35]  Robert G. Sexsmith,et al.  Probability-based safety analysis — value and drawbacks , 1999 .

[36]  Jorge E. Hurtado,et al.  The encounter of interval and probabilistic approaches to structural reliability at the design point , 2012 .

[37]  Shigeru Nakagiri,et al.  Finite element interval analysis of external loads identified by displacement input with uncertainty , 1999 .

[38]  Diego A. Alvarez,et al.  On the calculation of the bounds of probability of events using infinite random sets , 2006, Int. J. Approx. Reason..

[39]  Wolfgang Fellin,et al.  The fuzziness and sensitivity of failure probabilities , 2005 .

[40]  Lambros S. Katafygiotis,et al.  Geometric insight into the challenges of solving high-dimensional reliability problems , 2008 .

[41]  Yan-Gang Zhao,et al.  A general procedure for first/second-order reliabilitymethod (FORM/SORM) , 1999 .

[42]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[43]  Christian Cremona,et al.  The possibilistic reliability theory: theoretical aspects and applications , 1997 .

[44]  Jorge E. Hurtado,et al.  Fuzzy structural analysis based on fundamental reliability concepts , 2012 .

[45]  M. Beer,et al.  Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics , 2004 .

[46]  R. Rackwitz,et al.  Structural reliability under combined random load sequences , 1978 .

[47]  G. Muscolino,et al.  Interval analysis of structures with uncertain-but-bounded axial stiffness , 2011 .

[48]  F. Thouverez,et al.  ANALYSIS OF MECHANICAL SYSTEMS USING INTERVAL COMPUTATIONS APPLIED TO FINITE ELEMENT METHODS , 2001 .

[49]  A. M. Hasofer,et al.  Exact and Invariant Second-Moment Code Format , 1974 .

[50]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[51]  G. Schuëller,et al.  Chair of Engineering Mechanics Ifm-publication 2-374 a Critical Appraisal of Reliability Estimation Procedures for High Dimensions , 2022 .

[52]  F. L. Chernousko.  What is Ellipsoidal Modelling and How to Use It for Control and State Estimation , 1999 .

[53]  G. Roeck,et al.  Improving interval analysis in finite element calculations by means of affine arithmetic , 2010 .

[54]  Diego A. Alvarez,et al.  A method for enhancing computational efficiency in Monte Carlo calculation of failure probabilities by exploiting FORM results , 2013 .

[55]  S. Mcwilliam.  Anti-optimisation of uncertain structures using interval analysis , 2001 .

[56]  I. Hlavácek.  Uncertain input data problems and the worst scenario method , 2011 .

[57]  Z. Qiu,et al.  Comparison of dynamic response of structures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach , 2003 .

[58]  Fulvio Tonon,et al.  Hybrid analysis of uncertainty: probability, fuzziness and anti-optimization , 2001 .

[59]  Xiaoping Du,et al.  Reliability sensitivity analysis with random and interval variables , 2009 .

[60]  Søren Nielsen,et al.  Closure of "Interval Algebra to Deal with Pattern Loading and Structural Uncertainties" , 1995 .

[61]  Isaac Elishakoff,et al.  Refined second-order reliability analysis☆ , 1994 .

[62]  Zhiping Qiu.  Convex models and interval analysis method to predict the effect of uncertain-but-bounded parameters on the buckling of composite structures , 2005 .

[63]  A. Kiureghian,et al.  Multivariate distribution models with prescribed marginals and covariances , 1986 .

[64]  M Cliff Joslyn and Jane Booker,et al.  Generalized Information Theory for Engineering Modeling and Simulation , 2004 .

[65]  I. Elishakoff.  Essay on Reliability Index, Probabilistic Interpretation of Safety Factor, and Convex Models of Uncertainty , 1991 .

[66]  R. Mullen,et al.  Interval Monte Carlo methods for structural reliability , 2010 .

[67]  I. Molchanov.  Theory of Random Sets , 2005 .

[68]  I. Elishakoff,et al.  Convex models of uncertainty in applied mechanics , 1990 .