A 3-DIMENSIONAL NONLINEAR TIMOSHENKO BEAM BASED ON THE CORE-CONGRUENTIAL FORMULATION

A three-dimensional, geometrically non-linear, two-node Timoshenko bea m element based on the total Lagrangian description is derived. The el ement behaviour is assumed to be linear elastic, but no restrictions a re placed on the magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six ro tational-vector components. The formulation uses the Green-Lagrange st rains and second Piola-Kirchhoff stresses as energy-conjugate variable s and accounts for bending-stretching and bending-torsional-coupling e ffects 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 m atrix are developed at the particle level. A sequence of matrix transf ormations carries these equations to beam cross-sections and finally t o the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choi ce of over-the-element interpolation in the last one. The tangent stif fness matrix is found to retain symmetry if the rotational vector is c hosen to measure finite rotations. An extensive set of numerical examp les are presented to test and validate the present element.