An Innovative Timoshenko Beam Element

Timoshenko's beam theory is an extension of the Euler-Bernoulli beam theory to allow for the effects of transverse shear deformations which are often significant in the vertical displacements of short beams. For statically indeterminate beams and rigid frames, the inclusion of shear deformation effects will cause some changes in the magnitudes of external reactions, which in turn would affect the magnitudes of all internal forces and joint displacements. Flexibility methods are used for analysis of indeterminate structures. In this paper, a new flexibility based Timoshenko beam element is presented. First, shear deformation effects are included to kinematic equations. Then, equilibrium and constitutive equations are added in kinematic equations. Next, the basic linear formulation and stiffness matrix of the beam element are obtained which include the shear deformation effects without shear locking. Material and geometric nonlinearity can be considered easily in this approach with some changes in linear formulation. A '2-node' element is used in this method. It should also be noted that the main characteristics of this formulation are the exact fulfilment of equilibrium of forces at any interior point, with no additional hypotheses about the variation of displacements, strains and stresses.