## 3. A simple element analysis (cont'd)

The natural-coördinate method (Desai & Abel, 1972, pp. 88-91) expresses the location of any point in the triangular element by the area coördinates $$(\zeta_1, \zeta_2, \zeta_3)$$, where $$\zeta_i = A_i / A$$; $$A$$ is the total area of the triangle and the $$A_i$$ are as shown in the Figure. (Note that the three $$\zeta_i$$ are not independent, since they add up to 1.) We can use the $$\zeta_i$$ as the set of basis functions. It can be shown that in this case the coefficients $$c_i$$ become the nodal displacements $$w_i$$. Following the Rayleigh-Ritz procedure again leads to equation 10.

R. Funnell