Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-03T06:27:05.717Z Has data issue: false hasContentIssue false
  • English
  • Français

A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam

Published online by Cambridge University Press:  10 December 2010

Carlo Lovadina ,
David Mora  and
Rodolfo Rodríguez
Carlo Lovadina
Affiliation:
Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy. [email protected]
David Mora
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Casilla 5-C, Concepción, Chile. [email protected]
Rodolfo Rodríguez
Affiliation:
CIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. [email protected]
Get access

Abstract

The aim of this paper is to develop a finite element method which allows computing the buckling coefficients and modes of a non-homogeneous Timoshenko beam. Studying the spectral properties of a non-compact operator, we show that the relevant buckling coefficients correspond to isolated eigenvalues of finite multiplicity. Optimal order error estimates are proved for the eigenfunctions as well as a double order of convergence for the eigenvalues using classical abstract spectral approximation theory for non-compact operators. These estimates are valid independently of the thickness of the beam, which leads to the conclusion that the method is locking-free. Numerical tests are reported in order to assess the performance of the method.

Keywords

Finite element approximationeigenvalue problemsTimoshenko beams
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 603 - 626
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N., Discretization by finite elements of a model parameter dependent problem. Numer. Math. 37 (1981) 405–421. CrossRef
I. Babuška and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis II, P.G. Ciarlet and J.L. Lions Eds., North-Holland, Amsterdam (1991) 641–787.
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).
Dauge, M. and Suri, M., Numerical approximation of the spectra of non-compact operators arising in buckling problems. J. Numer. Math. 10 (2002) 193–219.
Descloux, J., Nassif, N. and Rappaz, J., On spectral approximation. Part 1: The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97–112. CrossRef
Descloux, J., Nassif, N. and Rappaz, J., On spectral approximation. Part 2: Error estimates for the Galerkin method. RAIRO Anal. Numér. 12 (1978) 113–119. CrossRef
R.S. Falk, Finite Elements for the Reissner-Mindlin Plate, in Mixed Finite Elements, Compatibility Conditions, and Applications, D. Boffi and L. Gastaldi Eds., Springer-Verlag, Berlin (2008) 195–230.
Hernández, E., Otárola, E., Rodríguez, R. and Sanhueza, F., Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry. IMA J. Numer. Anal. 29 (2009) 180–207. CrossRef
T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966).
Lovadina, C., Mora, D. and Rodríguez, R., Approximation of the buckling problem for Reissner-Mindlin plates. SIAM J. Numer. Anal. 48 (2010) 603–632. CrossRef
J.N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York (1993).
Szabó, B. and Királyfalvi, G., Linear models of buckling and stress-stiffening. Comput. Methods Appl. Mech. Eng. 171 (1999) 43–59. CrossRef