## Preprint 2010-07

## Carlo Lovadina, David Mora, Rodolfo Rodríguez:

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

### Abstract:

The aim of this paper is to develop a finite element method which allows omputing 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.

This preprint gave rise to the following definitive publication(s):

**Carlo LOVADINA, David MORA, Rodolfo RODRíGUEZ: ***A locking-free finite element method for the buckling problem of a non-homogeneous Timoshenko beam*. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 45, 4, pp. 603-626, (2011).