Bounds for Stiffness of Prismatic Beams

A linearly elastic beam finite element, dual to the Timoshenko beam model, is developed. Shear effects are included, but unlike the Timoshenko element, which is essentially an equilibrium model (stress compatibility enforced), the proposed element is a displacement model (strain and displacement compatibility enforced). Shear distorsions are accounted for by defining a set of internal degrees of freedom, which are subsequently eliminated through use of a condensation procedure to yield the element stiffness matrix in terms of nodal degrees of freedom only. The stiffnesses thus obtained are shown to be upper bound estimates of the corresponding exact values, whereas the theory of the Timoshenko beam provides the associated lower bound estimates. A comparison of the predictions of the two models shows a good agreement over a wide range of aspect ratios.