Page 199 - Rock Mechanics For Underground Mining

THE BOUNDARY ELEMENT METHOD

nodes 1, 2, 3 of the element, suppose that a set of functions is defined by

1
N 1 =− (1 − )
2
2
N 2 = (1 − ) (6.25)
1
N 3 = (1 + )
2
The property of these functions is that each takes the value of unity at a particular
node, and zero at the other two nodes. They are therefore useful for interpolation of
element geometry from the nodal co-ordinates in the form

x( ) = x 1 N 1 + x 2 N 2 + x 3 N 3 = x N
(6.26)

y( ) = y N
It is seen that the properties of the interpolation functions 6.25 ensure that equations
6.26returnthepositionco-ordinatesofthenodes,1,2,3,for =−1, 0, 1respectively.
Also, equations 6.25 and 6.26 can be interpreted to define a transformation from the
element local co-ordinate to the global x, y system.
In seeking a solution to the boundary value problem posed in Figure 6.6a, it is
known that the stress and displacement distribution in the medium exterior to S is
uniquely determined by the conditions on the surface S (Love, 1944). Thus if some
method can be established for inducing a traction distribution on S identical to the
known, imposed distribution, the problem is effectively solved. Suppose, for example,
continuous distributions q x (S), q y (S), of x- and y-directed line load singularities
are disposed over the surface S in the continuum. Using the solutions for stress
components due to unit line loads (Appendix B), and the known tangent to S at
y
x
any point i, the x and y component tractions T , T x and T , T y induced by the
xi yi xi yi
distributions of x- and y-directed line loads, can be determined. When point i is a
node of the surface, the condition to be achieved to realise the known condition on S
is
%
x y
q x (S)T + q y (S)T
xi xi dS = t xi
s
(6.27)
%
x y
q x (S)T + q y (S)T
yi yi dS = t yi
s
Discretisation of equation 6.27 requires that the surface distributions of fictitious load,
q x (S) and q y (S), be expressed in terms of the nodal values of these quantities. Suppose
that, for any element, the interpolation functions 6.25 are also used to define fictitious
load distributions with respect to the element intrinsic co-ordinate , i.e.

q x ( ) = q x1 N 1 + q x2 N 2 + q x3 N 3 = q x N
(6.28)

q y ( ) = q y N
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