Nonlinear three-dimensional beam theory for flexible multibody dynamics

In flexible multibody systems, it is convenient to approximate many structural components as beams or shells. Classical beam theories, such as Euler–Bernoulli beam theory, often form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the beam’s section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions. Kinematically, the problem is decomposed into an arbitrarily large rigid-section motion and a warping field. The sectional strains associated with the rigid-section motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the equations describing geometrically exact beams and those describing local deformations. The governing equations for geometrically exact beams are nonlinear, one-dimensional equations, whereas a linear, two-dimensional analysis provides the detailed distribution of three-dimensional stress and strain fields. Within the stated assumptions, the solutions presented here are the exact solution of three-dimensional elasticity for beams undergoing arbitrarily large motions.