Three-dimensional finite element analysis in cylindrical coordinates for nonlinear solid mechanics problems

Abstract

A formulation for three-dimensional nonlinear finite element analysis in cylindrical coordinates in presented. The elements are isoparametric with the same interpolation functions used to represent the geometry and the physical displacement components. The elements can be used for general three-dimensional analysis, but they are most effective for cases when the geometry and the response are best described in cylindrical coordinates. In contrast to formulations in Cartesian coordinates, the foregoing formulation allows the exact representation of a circular shape. For structures with circular geometries, the improved accuracy of the elements can provide better finite element predictions and reduce the number of elements needed in the circumferential direction. The reduction in the number of elements can result in a significant recuction in computer resources needed for large three-dimensional analyses, particularly in the presence of nonlinearities. Whereas the elements may, for some problems, be less efficient than semianalytic elements, the generality of the interpolation provides greater flexibility in modeling different shapes and responses. The mathematical formulation of the three-dimensional finite element equations in cylindrical coordinates is first described for finite strain and deformation. The advantages and additional complexities associated with the cylindrical coordinate formulation are discussed, and the implementation into finite element codes is described. Numerical studies are then presented for two static nonlinear three-dimensional problems: a pneumatic tire in contact with a rigid pavement and the skew bending of a stiffened annular plate. The finite element predictions of the cylindrical coordinate formulation are compared to experimental data and to the predictions from a commercial code using isoparametric finite elements developed in Cartesian coordinates.