Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation

In the finite element literature, beams and plates are not treated as isoparametric elements because the displacement gradients at the nodes are used to describe infinitesimal rotations. Such a linearization of the slopes leads to incorrect rigid body equations of motion when the beams and plates rotate as rigid bodies. A simple and efficient absolute nodal coordinate formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, can be systematically developed and efficiently used in many large deformation applications as well as in the analysis of curved structures. This formulation leads to a constant mass matrix, and as a consequence, an efficient procedure can be used for solving for the nodal accelerations. In the absolute nodal coordinate formulation, the displacement gradients at the nodes are determined in the undeformed reference configuration using simple rigid body kinematics. The problems that arise from the linearization of the slopes and the use of the finite rotations as nodal coordinates are discussed in this paper.