Coupled Deformation Modes in the Large Deformation Finite-Element Analysis: Problem Definition

In the classical formulations of beam problems, the beam cross section is assumed to remain rigid when the beam deforms. In Euler–Bernoulli beam theory, the rigid cross section remains perpendicular to the beam centerline; while in the more general Timoshenko beam theory the rigid cross section is permitted to rotate due to the shear deformation, and as a result, the cross section can have an arbitrary rotation with respect to the beam centerline. In more general beam models as the ones based on the absolute nodal coordinate formulation (ANCF), the cross section is allowed to deform and it is no longer treated as a rigid surface. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross section and other modes of deformations such as bending and they lead to a new set of modes referred to in this paper as the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. In this investigation, three different large deformation dynamic beam models are discussed and compared in order to investigate the effect of the ANCF-coupled deformation modes. The three methods differ in the way the beam elastic forces are calculated. The first method is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second method is based on the elastic line approach that systematically eliminates these modes. The ANCF-coupled deformation modes eliminated in the elastic line approach are identified and the effect of such deformation modes on the efficiency and accuracy of the numerical solution is discussed. The third large deformation beam model discussed in this investigation is based on the Hellinger–Reissner principle that can be used to eliminate the shear locking encountered in some beam models. Numerical examples are presented in order to demonstrate the use and compare the results of the three different beam formulations. It is shown that while the effect of the ANCF-coupled deformation modes is not significant in very stiff and moderately stiff structures, the effect of these modes can not be neglected in the case of very flexible structures.