Co-rotational 3D beam element using quaternion algebra to account for large rotations: Formulation theory and static applications

This paper presents a co-rotational beam element based on quaternion algebra as a means of parameterizing large rotations. This co-rotational framework is based on a decomposition of beam kinematics into a rigid element frame, which follows the element and its pure deformation. Once the decomposition between rigid body motion and deformation is obtained, the principle of virtual work allows calculating the element response projected onto large displacements and rotations. This 3D co-rotational element lies within the framework of incremental formulations. The special feature of this formulation pertains to the decomposition of kinematics to extract the rigid body motion and pure deformations through use of quaternion algebra by solving an internal nonlinear kinematic equation. Thus, from the 2 quaternions defining global rotations and the 2 displacement vectors at the ends of the beam, this method is able to extract: 1 quaternion and 1 displacement vector parameterizing the rigid body motion, plus 2 other quaternions defining the internal rotations and beam elongation. The proposed formulation based on quaternion algebra therefore constitutes an alternative to the literature's treatment of large-rotation kinematics using a quaternion algebra. The operators are also linearized in order to obtain the algorithmic stiffness operator. Eleven distinct static numerical applications are presented, along with comparisons from the literature in the aim of demonstrating the efficiency of this proposed element.