A curved finite element for general thin shell structures

Abstract

This work describes the development of a curved quadrilateral shell finite element which demonstrates very good convergence properties. A general description is used in deriving the element so that it may be applied to any thin shell problem. The element is shown to be very efficient. It has a total of 36 degrees-of-freedom with 9 at each of the corners of the element. There are several distinct advantages that the element offers for practical applications. Most of the shell elements that have been presented in the past are limited to problems in which the coordinates on the shell surface are orthogonal. The element that is described in the paper is derived using a general description so that it may be applied to any thin shell problem including those in which the shell coordinates are not orthogonal. The degrees-of-freedom at each of the four nodes are the three Cartesian displacements and their first derivatives with respect to the two surface coordinates. The imposition of boundary conditions is simplified since each of the degrees-of-freedom can be associated with a quantity which has a simple physical meaning. During the course of the derivation of the element, the strain displacement relationships are derived in a very simple manner consistent with Love's first approximation for thin shells. The derivation in the paper starts from basic principles and should help to shed some light on the proper form for the bending strain. Two primary contributions are presented in this work. The first is the presentation of a procedure for the development of a general quadrilateral shell element. The second is the simple derivation of the bending strain for the thin shells which apparently has not been presented previously.