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

¹ Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy. carlo.lovadina@unipv.it
² Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Casilla 5-C, Concepción, Chile. dmora@ubiobio.cl
³ CIMA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rodolfo@ing-mat.udec.cl

Received: 24 March 2010
Revised: 15 July 2010

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.