Unified and integrated approach in a new Timoshenko beam element

This paper presents a new unified and integrated approach to construct locking-free finite elements for bending of shear deformable beam element. The new UI (Unified and Integrated) element, with two nodes and three degrees of freedom (d.o.f.) per node, is formulated based on a pure displacement formulation and utilises vertical displacement, rotational and curvature as three d.o.f. at the nodes. A continuity of C² Hermite shape functions for bending deflection v_b is used in which rotation function of θ and curvature of χ are dependently expressed as the first and second derivatives of bending deflection. The formulation of element takes account of the effect of shear transversal forces in order to behave appropriately in the analysis of thin and thick beams. A shear influence factor of ϕ is expressed explicitly, which is a function of length thickness ratio (L/h), as a control for shear deformation. The resulting UI element is absolutely free from locking and preserves the high accuracy of the standard locking-free finite elements and classical Bernoulli Euler element. Finally, several numerical tests are presented to confirm the performance of the proposed formulations.