Shear locking in one-dimensional finite element methods

The accuracy of the results obtained with Finite Element Methods depends on the basis functions employed to approximate the displacement fields. If the beam is very thin, there can be shear locking and the numerical approximation leads to erroneous solutions. In this work, shear locking is analyzed by calculating expressly the curves of the transverse deflection, the cross-section rotation, and the shear strain in different approaches. The influence of the ratio between the shear stiffness and the bending stiffness is explicitly shown when force and moment loads are applied. It is concluded that the locking behavior at nodes is not the decisive factor to assess the quality of the solution. The important point is to analyze the entire curves when the ratio between the stiffnesses is varied. It is verified that the mixed interpolation and the discrete shear gap approaches are superior to the pure displacement and to the field consistency approaches.