Page 203 - Rock Mechanics For Underground Mining

THE FINITE ELEMENT METHOD

The components of [N], i.e. the terms N i , are prescribed functions of position, and
e
[u ] is a column vector listing the nodal displacements u xi , u yi , u xj . . . etc.
The interpolation functions which constitute the elements of [N] must be chosen
to return the nodal displacements at each of the nodes. This requires that

[N i ] xi,yi = [I]
[N i ] xi,yj = [0], etc.
where [I] and [0] are the identity and null matrices respectively. Also, since both
components of displacement at a point are to be interpolated in the same way, it is
clear that
[N i ] = N i [I]

where N i is a scalar function of position within the element.
A simple development of a linear interpolation function is demonstrated by repre-
senting the displacements in terms of linear functions of position, i.e.

u x = 1 + 2 x + 3 y
(6.37)
u y = 4 + 5 x + 6 y
The six interpolation constants are determined by ensuring that the displacements
u x , u y assume the nodal values when nodal co-ordinates are inserted in equation
6.37. Thus 1 , 2 , 3 are determined by solving the simultaneous equations

u xi = 1 + 2 x i + 3 y i
u xj = 1 + 2 x j + 3 y j
u xk = 1 + 2 x k + 3 y k

Solution for 1 , 2 , 3 and some rearrangement, produces
1
u x = [(a i + b i x + c i y)u xi + (a j + b j x + c j y)u xj + (a k + b k x + c k y)u xk ]
2
(6.38)

where

a i = x j y k − x k y j
b i = y j − y k
c i = x k − x j

with cyclic permutation of i, j, k to obtain a j , etc., and

2 = 2 × area of the triangular element
⎡ ⎤
1 x i y i
= 2 1 x j y j ⎦
⎣
1 x k y k
185