Three-dimensional absolute nodal co-ordinate formulation: plate problem

In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no inÞnitesimal or Þnite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate co-ordinates. Using this interpretation of the plate co-ordinates the new method does not require the use of co-ordinate transformation to deÞne the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The sti¤ness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental Þnite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain