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Research Papers

# Comparisons of Probabilistic and Two Nonprobabilistic Methods for Uncertain Imperfection Sensitivity of a Column on a Nonlinear Mixed Quadratic-Cubic Foundation

[+] Author and Article Information
Xiaojun Wang

Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R.C.xjwang@buaa.edu.cn

Isaac Elishakoff

Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991elishako@fau.edu

Zhiping Qiu

Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R.C.zpqiu@buaa.edu.cn

Lihong Ma

Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R.C.lhma@ase.buaa.edu.cn

J. Appl. Mech 76(1), 011007 (Oct 31, 2008) (8 pages) doi:10.1115/1.2998763 History: Received January 10, 2008; Revised June 20, 2008; Published October 31, 2008

## Abstract

Two nonprobabilistic set-theoretical treatments of the initial imperfection sensitive structure—a finite column on a nonlinear mixed quadratic-cubic elastic foundation—are presented. The minimum buckling load is determined as a function of the parameters, which describe the range of possible initial imperfection profiles of the column. The two set-theoretical models are “interval analysis” and “convex modeling.” The first model represents the range of variation of the most significant $N$ Fourier coefficients by a hypercuboid set. In the second model, the uncertainty in the initial imperfection profile is expressed by an ellipsoidal set in $N$-dimensional Euclidean space. The minimum buckling load is then evaluated in both the hypercuboid and the ellipsoid. A comparison between these methods and the probabilistic method are performed, where the probabilistic results at different reliability levels are taken as the benchmarks of accuracy for judgment. It is demonstrated that a nonprobabilistic model of uncertainty may be an alternative method for buckling analysis of a column on a nonlinear mixed quadratic-cubic elastic foundation under limited information on initial imperfection.

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Topics: Stress , Buckling , Modeling

## Figures

Figure 4

Buckling load α∗ from interval analysis

Figure 5

The ellipse enclosed by a rectangle

Figure 6

Buckling load α computed from convex modeling

Figure 7

Probability density function for a truncated normally distributed random variable

Figure 1

A column on a nonlinear mixed quadratic-cubic elastic foundation

Figure 2

Figure 3

Buckling load α∗ versus the initial imperfection amplitude

Figure 8

Comparison of the buckling load computed from probabilistic and nonprobabilistic methods for cases of b=0.1: (a) ξ¯c=0.0, and (b) ξ¯c=0.01

Figure 9

Comparison of the buckling loads computed from probabilistic and nonprobabilistic methods for case of b=1.0: (a) ξ¯c=0.0, and (b) ξ¯c=0.01

Figure 10

Comparison of the buckling loads computed from probabilistic and nonprobabilistic methods for case of b=0.1: (a) ξ¯c=0.0, and (b) ξ¯c=0.02

Figure 11

Comparison of the buckling loads computed from probabilistic and nonprobabilistic methods for case of b=1.0: (a) ξ¯c=0.0, and (b) ξ¯c=0.02

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