The Perturbation Method for Stability Analysis of Uncertain Structures

Abstract:

This paper is concerned with the structural stability problem involving uncertain-but-bounded parameters, specified as bounds on these parameters. This produces interval stand and geometry stiffness matrices, and the problem is transformed into a interval buckling eigenvalue problem in interval mathematics. The perturbation method is proposed to determine the lower and upper bounds on the buckling eigenvalues and due to uncertain-but-bounded parameters. Moreover, the critical load of the uncertain structure can be obtained. The effectiveness of the presented method was demonstrated by comparison with conventional stability theory, using a typical numerical example.

• Periodical: Applied Mechanics and Materials (Volume 367)
• Pages: 302-307
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.367.302
• Online since: August 2013
• Authors: Chang Rui Ji, Zao Ni
• Keywords: Critical Buckling Load, Interval Sets, Perturbation Method, Stability Analysis, Uncertain Variables

References

[1] S. J. Britvec. The Stability of Elastic Systems, Pergamon Press: New York, (1973).

[2] A. Chajes. Principles of Structural Stability Theory, Prentice-Hall, Inc, Englewood Cliffs: New Jersey, (1974).

[3] Y. Ben-Haim, I. Elishakoff. Convex Models of Uncertainty in Applied Mechanics, Elsevier: Amsterdam, (1990).

[4] I. Elishakoff, Y. W. Li, J. H. Starnes. A deterministic method to predict the effect of unknown-but-bounded elastic moduli on the buckling of composite structures. Computer Methods in Applied Mechanics and Engineering 1994; 111: 155-167.

DOI: 10.1016/0045-7825(94)90043-4

[5] I. Elishakoff, P. Eliseeff, S. Glegg. Convex modelling of material uncertainty in vibrations of a viscoelastic structure. AIAA Journal 1994; 32: 843-849.

[6] Z. P. Qiu, S. H. Chen, Elishakoff I. Natural frequencies of structures with uncertain but non-random parameters. Journal of Optimization Theory and Applications 1995; 86: 669-683.

DOI: 10.1007/bf02192164

[7] Z. P. Qiu, I. Elishakoff. Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis. Computer Methods in Applied Mechanics and Engineering 1998; 152: 361-372.

DOI: 10.1016/s0045-7825(96)01211-x

[8] Z. P. Qiu, P. C. Mueller, A. Frommer. Stability robustness bounds for linear state-space models with structured uncertainty based on ellipsoidal set-theoretic approach. Mathematics and Computers in Simulation 2001; 56: 35-53.

DOI: 10.1016/s0378-4754(00)00200-7

[9] Z. P. Qiu, P. C. Mueller, A. Frommer. An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices. Communications in Numerical Methods in Engineering 2001; 17: 239-251.

DOI: 10.1002/cnm.401

[10] Z. P. Qiu, I. Elishakoff, J. H. Strarnes. The bound set of possible eigenvalues of structures with uncertain but non-random parameters. Chaos, Solitons and Fractals 1996; 7: 1845-1857.

DOI: 10.1016/s0960-0779(96)00041-0

[11] S. H. Chen, X. W. Yang. Interval finite element method for beam structures. Finite Element in Analysis and Design 2000; 34: 75-88.

DOI: 10.1016/s0168-874x(99)00029-3

[12] R. L. Mullen, R. L. Muhanna. Bounds of structural response for all possible loading combinations. Structural Engineering (ASCE) 1999; 125: 98-106.

DOI: 10.1061/(asce)0733-9445(1999)125:1(98)

[13] C. P. Pantelides, S. Ganerli. Comparison of fuzzy set and convex model theories in structural design. Mechanical Systems and Processing 2001; 15(3): 499 -511.

DOI: 10.1006/mssp.2000.1379

[14] S. Ganzerli, C. P. Pantelides. Optimum structural design via convex model superposition. Computers and Structures 2000; 74(6): 639-647.

DOI: 10.1016/s0045-7949(99)00077-2

[15] S. Ganzerli, C. P. Pantelides. Load and resistance convex models for optimum design. Structural Optimization 1999; 17(4): 259 -268.

DOI: 10.1007/bf01207002

[16] R. E. Moore. Methods and Applications of Interval Analysis, SIAM: Philadephia, (1979).

[17] G. Alefeld, Herzberger J. Introduction to Interval Computations, Academic Press: New York, (1983).

[18] G. Genta. Vibration of Structures and Machines: Practical Aspects. 3rd edition, Springer-Verlag: New York, (1999).

[19] H. C. Hu. The Theory of Natural Vibration of MDOF Structures, Science Press of China, 1987 (in Chinese).

[20] P. Lancaster. Theory of Matrices, Academic Press: New York, (1985).

[21] J. N. Franklin. Matrix Theory. Prentice-Hall, Inc, Englewood Cliffs: New Jersey, (1968).