Comparison between non‐probabilistic interval analysis method and probabilistic approach in static response problem of structures with uncertain‐but‐bounded parameters

The uncertainty present in many practical engineering analysis and design problems can be modelled using probabilistic or non-probabilistic interval analysis methods. One motivation for using non-probabilistic interval analysis models rather than probabilistic models for uncertain variables is the general dearth of information in characterizing the uncertainties. Non-probabilistic interval analysis methods are less information-intensive than probabilistic models, since no density information is required. Instead of conventional optimization studies, where the minimum possible response is sought, here an uncertainty model is developed as an anti-optimization problem of finding the least favourable response and the most favourable response under the constraints within the set-theoretical description. Non-probabilistic interval analysis methods have been used for dealing with uncertain phenomena in a wide range of engineering applications. This paper is concerned with the problem of comparison between the non-probabilistic interval analysis method and the probabilistic approach in the static response problem of structures with uncertain-but-bounded parameters from mathematical proofs and numerical calculations. The results show that under the condition of the interval vector of the uncertain parameters determined from the probabilistic and statistical information, the width of the static displacement obtained by the non-probabilistic interval analysis method is larger than that by the probabilistic approach for structures with uncertain-but-bounded structural parameters. This is just the result that we expect, since according to the definition of probabilistic theory and interval mathematics, the region determined by the non-probabilistic interval analysis method should contain one predicted by the probabilistic approach. Copyright © 2004 John Wiley & Sons, Ltd.

[1]  F. Schweppe.  Recursive state estimation: Unknown but bounded errors and system inputs , 1967 .

[2]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[3]  S. Bialas,et al.  Necessary and Sufficient Conditions for the Stability of Interval Matrices , 1983 .

[4]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[5]  I. Elishakoff,et al.  Probabilistic Methods in the Theory of Structures , 1984 .

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

[7]  Yakov Ben-Haim,et al.  Convex models of vehicle response to unknown but bounded terrain , 1991 .

[8]  S. H. A. Chen,et al.  Natural frequencies of structures with uncertain but nonrandom parameters , 1995 .

[9]  J. H. Starnes,et al.  The bound set of possible eigenvalues of structures with uncertain but non-random parameters , 1996 .

[10]  Chris P. Pantelides,et al.  Convex models for impulsive response of structures , 1996 .

[11]  Yakov Ben-Haim,et al.  Maximum Structural Response Using Convex Models , 1996 .

[12]  Chris P. Pantelides,et al.  Design of Trusses under Uncertain Loads Using Convex Models , 1998 .

[13]  I. Elishakoff,et al.  Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis , 1998 .

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

[15]  C. Pantelides,et al.  Optimum structural design via convex model superposition , 2000 .

[16]  Singiresu S. Rao,et al.  Fuzzy boundary element method for the analysis of imprecisely defined systems , 2001 .

[17]  Chris P. Pantelides,et al.  COMPARISON OF FUZZY SET AND CONVEX MODEL THEORIES IN STRUCTURAL DESIGN , 2001 .

[18]  Z. Qiu.  Comparison of static response of structures using convex models and interval analysis method , 2003 .

[19]  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 .