A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, p. 603-626

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.

DOI : https://doi.org/10.1051/m2an/2010071
Classification:  65N25,  65N30,  74S05,  74K10
Keywords: finite element approximation, eigenvalue problems, Timoshenko beams
@article{M2AN_2011__45_4_603_0,
author = {Lovadina, Carlo and Mora, David and Rodr\'\i guez, Rodolfo},
title = {A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {4},
year = {2011},
pages = {603-626},
doi = {10.1051/m2an/2010071},
zbl = {1267.74049},
mrnumber = {2804652},
language = {en},
url = {http://www.numdam.org/item/M2AN_2011__45_4_603_0}
}

Lovadina, Carlo; Mora, David; Rodríguez, Rodolfo. A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, pp. 603-626. doi : 10.1051/m2an/2010071. http://www.numdam.org/item/M2AN_2011__45_4_603_0/

[1] D.N. Arnold, Discretization by finite elements of a model parameter dependent problem. Numer. Math. 37 (1981) 405-421. | MR 627113 | Zbl 0446.73066

[2] 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. | MR 1115240 | Zbl 0875.65087

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[4] M. Dauge and M. Suri, Numerical approximation of the spectra of non-compact operators arising in buckling problems. J. Numer. Math. 10 (2002) 193-219. | MR 1935966 | Zbl 1099.74545

[5] J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97-112. | Numdam | MR 483400 | Zbl 0393.65024

[6] J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 2: Error estimates for the Galerkin method. RAIRO Anal. Numér. 12 (1978) 113-119. | Numdam | MR 483401 | Zbl 0393.65025

[7] 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. | Zbl 1167.74041

[8] E. Hernández, E. Otárola, R. Rodríguez and F. Sanhueza, Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry. IMA J. Numer. Anal. 29 (2009) 180-207. | MR 2470946 | Zbl 1155.74042

[9] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966). | MR 203473 | Zbl 0836.47009

[10] C. Lovadina, D. Mora and R. Rodríguez, Approximation of the buckling problem for Reissner-Mindlin plates. SIAM J. Numer. Anal. 48 (2010) 603-632. | MR 2669998 | Zbl pre05866183

[11] J.N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York (1993). | Zbl 0561.65079

[12] B. Szabó and G. Királyfalvi, Linear models of buckling and stress-stiffening. Comput. Methods Appl. Mech. Eng. 171 (1999) 43-59. | Zbl 0944.74028