ABSTRACT
Formulations for one and two-dimensional elements were developed in Chapters 3 and 4. The equations for the shape functions were specific to the elements because the terms in them were dependent on the geometrical co-ordinates of the nodes (see Equations (3.5), (4.14) and (4.20)). The development of an appropriate stiffness matrix for the bar element and for the constant strain triangle was not particularly onerous. The development of an appropriate stiffness matrix for the simplest rectangular element was shown, in contrast, to be relatively laborious, and it was suggested that a numerical integration technique should be employed to calculate each entry in the stiffness matrix. Further, the formulation was developed only for the rectangular element, and that for a general quadrilateral element, or even for a rectangular element skewed relative to the global axes, would be more difficult still. In order to overcome these difficulties, the parametric element formulations have been developed, in which the shape functions are defined in terms of natural co-ordinate systems (common to all elements of a particular type) and the elements are mapped onto real space using transformation matrices. This proves to be a very powerful technique, which permits the study of complicated geometries using relatively simple element formulations combined with numerical integration to calculate the stiffness matrices.
