Finite Element Analysis of a Bending Moment Formulation for the Vibration Problem of a Non-homogeneous Timoshenko Beam

In this paper we analyze a low-order finite element method for approximating the vibration frequencies and modes of a non-homogeneous Timoshenko beam. We consider a formulation in which the bending moment is introduced as an additional unknown. Optimal order error estimates are proved for displacements, rotations, shear stress and bending moment of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are independent of the beam thickness, which leads to the conclusion that the method is locking free. For its implementation, displacements and rotations can be eliminated leading to a well posed generalized matrix eigenvalue problem for which the computer cost of its solution is similar to that of other classical formulations. We report numerical experiments which allow us to assess the performance of the method.