The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 385

FIGURE 6.31
Subdivision of the distance between side 1 and node 1.

FIGURE 6.32
Labeling of a triangle, m = 3.

and the other coefficients α

2 ,

β

2 ,

... , γ

3 are defined by cyclic interchange of the subscripts

in the order 1, 2, 3, e.g., x 1 y 3 .

The location of a point in the triangle will be identified using a special construction. As
illustrated in Fig. 6.31, draw dashed lines parallel to side 1, dividing the distance between
side 1 and node 1 into m equal segments. Label one of these lines as p . Draw similar sets
of lines parallel to side 2, and label one of these lines as q . Similarly, label one of a set
of lines parallel to side 3 as r . A point at the intersection of lines p, q, and r can now be
designated by the three digits p, q, and r . Also, the vertices and points inside a triangle can
be assigned pqr labels. A typical labeled triangle is shown in Fig. 6.32. Note that for any
point, p

α = x 3 y 1 −

2

+ + =

m .
Assume node point displacements are given the same pqr designation as the nodal points
themselves. The triangular coordinate approach has permitted a triangle to be “subdivided”
with uniquely labeled interior nodes. Trial functions of the form

q r

1
2 (

m + )

N pqr v pqr

1 )( m + 2

u = Nv = (6.76)