Flat plate structures which vibrate in their plane, such as shear wall buildings, can be analysed by dividing the plate up into an assemblage of two-dimensional finite elements, called membrane elements. The most common shapes of element used are triangles, rectangles and quadrilaterals. These elements can also be used to analyse the low frequency vibrations of complex shell-type structures such as aircraft and ships. In these cases the membrane action of the walls of the structures are more predominant than the bending action.

In Chapter 3 it is shown that in order to satisfy the convergence criteria, the element displacement functions should be derived from complete polynomials. In one dimension the polynomial terms are 1, x, x2, x3,…, etc. Complete polynomials in two variables, x and y, can be generated using Pascal's triangle, as shown in Figure 4.1. Node points are normally situated at the vertices of the element, although additional ones are sometimes situated along the sides of the element in order to increase accuracy. (This technique is analogous to having additional node points along the length of a one-dimensional element, as described in Section 3.8.) When two adjacent elements are joined together, they are attached at their node points. The nodal degrees of freedom and element displacement functions should be chosen to ensure that the elements are conforming, that is, the displacement functions and their derivatives up to order (p − 1), are continuous at every point on the common boundary (see Section 3.2). In some cases it is not possible to achieve the necessary continuity using complete polynomials [4.1, 4.2]. This is overcome by using some additional terms of higher degree. When selecting these terms care should be taken to ensure that the displacement pattern is independent of the direction of the coordinate axes. This property is known as geometric invariance. For the two-dimensional case, the additional terms should be chosen in pairs, one from either side of the axis of symmetry in Figure 4.1. As an example, consider the derivation of a quadratic model with eight terms. Selecting all the constant, linear and quadratic terms plus the x2y and xy2 terms, produces a function which is quadratic in x along y = constant and quadratic in y along x = constant. Thus the deformation pattern will be the same whatever the orientation of the axes. This would not be true if the terms x3 and x2y had been selected. In this case the function is cubic in x along y = constant and quadratic in y along x = constant. Therefore, the deformation pattern depends upon the orientation of the axes. Note that complete polynomials are invariant.