Accuracy of Co-rotational formulation for 3-D Timoshenko's beam

An accuracy of finite element solutions for 3-D Timoshenko's beams, obtained using a co-rotational formulation, is discussed. The co-rotational formulation has often been used with an assumption that the relative deformations are small. A fundamental question, therefore, has been raised as to whether or not the numerical solutions obtained approach the solutions of the exact theory. In this paper, from theoretical point of view, we investigate the accuracy of the co-rotational formulation for 3-D Timoshenko's beam undergoing finite strains and finite rotations. It is shown that the use of the conventional secant coordinates fails to give satisfactory numerical solutions. We introduce a new local coordinate system in which a linear beam theory is used to construct the strain energy function. It is shown that the finite element solutions obtained converge to those of the exact beam theory as the number of element increases.