Locking-free finite element method for a bending moment formulation of Timoshenko beams

Abstract

In this paper we study a finite element formulation for Timoshenko beams. It is known that standard finite elements applied to this model lead to wrong results when the thickness of the beam is small. Here, we consider a mixed formulation in terms of the transverse displacement, rotation, shear stress and bending moment. By using the classical Babuška–Brezzi theory, it is proved that the resulting variational formulation is well posed. We discretize it by continuous piecewise linear finite elements for the shear stress and bending moment, and discontinuous piecewise constant finite elements for the displacement and rotation. We prove an optimal (linear) order of convergence in terms of the mesh size for the natural norms and a double order (quadratic) in L2-norms for the shear stress and the bending moment. These estimates involve constants and norms of the solution that are proved to be bounded independently of the beam thickness, which ensures the locking-free character of the method. Numerical tests are reported in order to support our theoretical results.