Dynamics of corotational beam elements in large displacements and rotations - some aspects on the kinetic energy and the integration of the equations of motions

Classical structural theories able to deal with beams undergoing large displacements and rotations, with small strains, are that of 'geometrically exact' elements and the corotational approach. In this work an attempt is made to combine geometrically exact and corotational approaches, adopting the typical viewpoint of geometrically exact models in order to evaluate the inertial forces of a beam element. A new procedure for the evaluation of the inertial forces of the beam is proposed. This is based on an exact description of the kinematics of the element and on a spatial discretization which employs only nodal variables defined in the global frame of reference of the structural problem, so avoiding approximations typically introduced by the classic corotational formulations. The issue of integration of the equations of motion is addressed taking into account the non-linear character of the element configuration space, which is due to the presence of large cross-sectional rotations. It's also shown that the proposed procedure allows for a standard time stepping scheme in integrating the equations of motions, with practical advantages on the reuse of standard numerical algorithms.