Analysis of delayed convergence in the three-noded Timoshenko beam element using the function space approach

Abstract

Despite satisfying only completeness and continuity requirements, elements often perform erroneously in a certain class of problems, called the locking situations, where they display spurious stress oscillations and enhanced stiffness properties. The function space approach that effectively substantiates the postulates of the field consistency paradigm is an efficient tool to reveal the fundamental cause of locking phenomena, and propose methods to eliminate this pathological problem. In this paper, we review the delayed convergence behaviour of three-noded Timoshenko beam elements using the rigorous function space approach. Explicit, closed form algebraic results for the element strains, stresses and errors have been derived using this method. The performance of the field-inconsistent three-noded Timoshenko beam element is compared with that of the field-inconsistent two-noded beam element. It is demonstrated that while the field-inconsistent two-noded linear element is prone to shear locking, the field-inconsistent three-noded element is not very vulnerable to this pathological problem, despite the resulting shear oscillations.