A COMPUTERIZED SYMBOLIC INTEGRATION TECHNIQUE FOR DEVELOPMENT OF TRIANGULAR AND QUADRILATERAL COMPOSITE SHALLOW-SHELL FINITE ELEMENTS

摘要

Computerized symbolic integration is used in conjunction with group-theoretic techniques to obtain analytic expressions for the stiffness, geometric stiffness, consistent mass, and con¬sistent load matrices of composite shallow shell structural elements. The elements are shear flexible and have variable curvature. A stiffness (displacement) formulation is used with the fundamental unknowns consisting of both the displacement and rotation components of the refer¬ence surface of the shell. Both triangular and quadrilateral elements are developed. The tri¬angular elements have six and ten nodes. The quadrilateral elements have four and eight nodes and can have internal degrees of freedom associated with displacement modes which vanish along the edges of the element (bubble modes).nThe stiffness, geometric stiffness, consistent mass, and consistent load coefficients are expressed as linear combinations of integrals (over the element domain) whose integrands are products of shape functions and their derivatives. The evaluation of the elemental matrices is thus divided into two separate problems - determination of the coefficients in the linear com¬bination and evaluation of the integrals. The latter problem can be computationally the more time consuming and the present study focuses on simplifications and means of reducing the effort involved in this task.nThe integrals are performed symbolically by using the symbolic-and-algebraic-manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floating-point arithmetic operations required in this approach with those required by a commonly used numerical quadrature technique.