Abstract
The stiffness, mass, and consistent force matrices for a simple two-node Timoshenko beam element are developed based upon Hamilton's principle. Cubic and quadratic Lagrangian polynomials are used for the transverse and rotational displacements, respectively, where the polynomials are made interdependent by requiring them to satisfy the two homogeneous differential equations associated with Timoshenko's beam theory. The resulting stiffness matrix, which can be exactly integrated and is free of ‘shear-locking’, is in agreement with the exact Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element exactly predicts the displacement of a short beam subjected to complex distributed loadings using only one element, and the current element predicts shear and moment resultants and natural frequencies better than existing Timoshenko beam elements.
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