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12.8.2 Stiffness matrix for a triangular finite element

Triangular finite elements are used in the solution of plane stress and plane strain problems. Their advantage over other shaped elements lies in their ability to represent irregular shapes and boundaries with relative simplicity.

In the derivation of the stiffness matrix we shall adopt the step by step procedure of the previous example. Initially, therefore, we choose a suitable coordinate and node numbering system for the element and define its nodal displacement and nodal force vectors. Figure 12.13 shows a triangular element referred to axes Oxy and having

F*J'Ui

Fig. 12.13 Triangular element for plane elasticity problems.

nodes i, j and k lettered anticlockwise. It may be shown that the inverse of the [A] matrix for a triangular element contains terms giving the actual area of the element; this area is positive if the above node lettering or numbering system is adopted. The element is to be used for plane elasticity problems and has therefore two degrees of freedom per node, giving a total of six degrees of freedom for the element, which will result in a 6 x 6 element stiffness matrix [A^]. The nodal forces and displacements are shown and the complete displacement and force vectors are m = {

FXJc I Fy,k

We now select a displacement function which must satisfy the boundary conditions of the element, i.e. the condition that each node possesses two degrees of freedom. Generally, for computational purposes, a polynomial is preferable to, say, a trigonometric series since the terms in a polynomial can be calculated much more rapidly by a digital computer. Furthermore, the total number of degrees of freedom is six, so that only six coefficients in the polynomial can be obtained. Suppose that the displacement function is u(x, y) = on + a2x + Ï v{x,y) = a4 + a5x + a6y J

The constant terms, a^ and q4, are required to represent any in-plane rigid body motion, i.e. motion without strain, while the linear terms enable states of constant strain to be specified; Eqs (12.82) ensure compatibility of displacement along the edges of adjacent elements. Writing Eqs (12.82) in matrix form gives ai v(x,y) f

0 0