Simple Degenerate Formulation for Large Displacement Analysis of Beams

A finite-element formulation for the large displacement analysis of beams is proposed. It is based on the degeneration approach: The governing equations for a general solid are directly discretized. The assumptions of the Timoshenko beam theory are implemented in the discretization process by devising beam elements and utilizing the penalty method. The formulation for 2D beam analysis is first presented and the 3D formulation follows. Characteristically, the proposed beam elements possess relative nodes and rotations are excluded from nodal variables. The beam formulations thus developed are quite simple and straightforward. It is noteworthy that unlike conventional formulations, the present formulation for 3D beam analysis is just a simple extension of the 2D case, which can be attributed mainly to the avoidance of rotations in nodal variables. In numerical examples, the approximate penalty number is investigated first by analyzing a cantilever beam, and it turns out to be 103 times Young's modulus. With this value, example problems are solved and excellent agreement with the existing solutions is observed, confirming the validity of the present formulation.