Three-dimensional non-linear Timoshenko beam based on the core-congruential formulation

Abstract

A three-dimensional, geometrically non-linear, two-node Timoshenko beam element based on the total Lagrangian description is derived. The element behaviour is assumed to be linear elastic, but no restrictions are placed on the magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for bending-stretching and bending-torsional-coupling effects without special provisions. The core-congruential formulation (CCF) is used to derive the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples are presented to test and validate the present element.