There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.
The absolute nodal coordinate formulation has been recently extended to shear deformable beam or plate elements. This has been accomplished, in practice, by parameterizing the complete volume of the elements instead of a line or surface in the element kinematics description. In the absolute nodal coordinate formulation, the position of any point of the element volume is defined employing independent slope coordinates. The use of a large number of slope coordinates leads to unusual kinematic features that must be accounted for in order to avoid the element locking. This study demonstrates that the shear deformable element based on the absolute nodal coordinate formulation suffers from curvature thickness locking and shear locking in addition to the previously reported Poisson's locking. Due to the tendency of locking, the use of the absolute nodal coordinate formulation can lead to elements with weak performance. In order to eliminate locking problems, this study introduces a new absolute nodal coordinate-based finite element. The introduced element uses redefined polynomial expansion together with a reduced integration procedure. The performance of the introduced element is studied by means of certain dynamic problems. The element exhibits a competent convergence rate and it does not suffer from the previously mentioned locking effects.
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