Structural analysis with probability-boxes

Probability-box (p-box) is a rigorous and practical way to represent epistemic sources of uncertainty where the available knowledge is insufficient to construct the required probability distributions. In this paper, interval finite element (FE) methods are combined with the concept of p-box to analyse structures subjected to uncertain loads modelled by p-boxes. Two methods, namely the discrete p-box convolution and interval Monte Carlo methods, are presented along with example problems. The computational efficiency of the p-box FE method is also presented.

[1]  Robert L. Mullen,et al.  Bounds of Structural Response for All Possible Loading Combinations , 1999 .

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

[3]  Daniel Berleant,et al.  Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals , 1998, Reliab. Comput..

[4]  Robert L. Mullen,et al.  Combined Axial and Bending Stiffness in Interval Finite-Element Methods , 2007 .

[5]  Chris P. Pantelides,et al.  Load and resistance convex models for optimum design , 1999 .

[6]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[7]  Alfredo Ang H.-S.,et al.  Probability concepts in engineering planning and design, vol i : basic principles , 1979 .

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

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

[10]  Jon C. Helton,et al.  Representation of analysis results involving aleatory and epistemic uncertainty , 2010, Int. J. Gen. Syst..

[11]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[12]  Ramana V. Grandhi,et al.  Using random set theory to calculate reliability bounds for a wing structure , 2006 .

[13]  M. Elisabeth Paté-Cornell,et al.  Uncertainties in risk analysis: Six levels of treatment , 1996 .

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

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

[16]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[17]  S. Ferson,et al.  Different methods are needed to propagate ignorance and variability , 1996 .

[18]  Pol D. Spanos,et al.  Computational Stochastic Mechanics , 2011 .

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

[20]  Robert L. Mullen,et al.  Penalty-Based Solution for the Interval Finite-Element Methods , 2005 .

[21]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[22]  David Moens,et al.  A survey of non-probabilistic uncertainty treatment in finite element analysis , 2005 .

[23]  Hao Zhang,et al.  Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics , 2007, Reliab. Comput..

[24]  Isaac Elishakoff,et al.  A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties , 1998 .

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

[26]  Helmut Schweiger,et al.  Reliability Analysis in Geotechnics with the Random Set Finite Element Method , 2005 .

[27]  Didier Dubois,et al.  Representing parametric probabilistic models tainted with imprecision , 2008, Fuzzy Sets Syst..

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

[29]  Hao Zhang,et al.  Nondeterministic Linear Static Finite Element Analysis: An Interval Approach , 2005 .

[30]  Wolfgang Fellin,et al.  Reliability bounds through random sets , 2008 .

[31]  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.

[32]  Philipp Limbourg,et al.  Uncertainty analysis using evidence theory - confronting level-1 and level-2 approaches with data availability and computational constraints , 2010, Reliab. Eng. Syst. Saf..

[33]  Bruce R. Ellingwood,et al.  Quantifying and communicating uncertainty in seismic risk assessment , 2009 .

[34]  R. Mullen,et al.  Uncertainty in mechanics problems-interval-based approach , 2001 .

[35]  Didier Dubois,et al.  Random sets and fuzzy interval analysis , 1991 .