Page 200 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

The discretised form of equation 6.27 can be written

n %

x y
q x (S)T + q y (S)T xi dS = t xi (6.29)
xi
j=1 s e
n %

x y
q x (S)T + q y (S)T (6.30)
yi yi dS = t yi
j=1 s e
where n is the number of boundary elements, and each surface integral is evaluated
j
over the range S e of each boundary element j. Considering a particular element, one
of the surface integrals can be expressed by
%
x y
q x (S)T + q y (S)T xi dS =
xi
s e
% 1 % 1
x dS y dS
q x N ( )T ( ) d + q y N ( )T ( ) d (6.31)
xi xi
−1 d −1 d
Theintegralsoftheinterpolationfunction(N)–kernel(T )productsdefinedinequation
6.31 can be evaluated readily by standard Gaussian quadrature methods. When all
components of equations 6.29 and 6.30 have been calculated using the procedure
defined in equation 6.31, it is found that for the m boundary nodes
m

q xj T x∗ + q yj T y∗
x x = t xi
j=1
(6.32)
m

q xj T x∗ + q yj T y∗
x y = t yi
j=1
where T ,etc.,aretheresultsofthevariousinterpolationfunction–kernelintegrations
x∗
x
and, for the end nodes of each element, a summation with the appropriate integral for
the adjacent element. When equations similar to 6.32 have been established for each
of the m boundary nodes, they may be recast in the form
∗
[T ][q] = [t] (6.33)
Equation 6.33 represents a set of 2m simultaneous equations in 2m unknowns, which
are the nodal values of fictitious boundary load intensity.
Once equation 6.33 has been solved for the vector [q] of nodal load intensities,
all other problem unknowns can be calculated readily. For example, nodal displace-
ments, or displacements at an internal point i in the medium, can be determined
from

%
x y
u xi = q x (S)U + q y (S)U xi dS
xi
s
(6.34)
%
x y
u yi = q x (S)U + q y (S)U yi dS
yi
s
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