Page 201 - Rock Mechanics For Underground Mining

THE FINITE ELEMENT METHOD

Similarly, stress components at an internal point i in the medium are given by

%
x y
xxi = q x (S) + q y (S) dS
xxi xxi
s
%
x y
yyi = q x (S) yyi + q y (S) yyi dS (6.35)
s
%
x y
xyi = q x (S) + q y (S) dS
xyi xyi
s
y
y
x
x
In equations 6.34 and 6.35, U , xxi , U , xxi , etc., are displacements and stresses
xi
xi
induced by x- and y-directed unit line loads, given by the expressions in Appendix B.
Equations 6.34 and 6.35 may be discretised using the methods defined by equations
6.29–6.31, and all the resulting integrals can be evaluated using standard quadrature
formulae.
In setting up equations 6.27, 6.34 and 6.35, the principle of superposition is ex-
ploited implicitly. Thus the method is applicable to linear elastic, or at least piece-wise
linear elastic, behaviour of the medium. Also, since both element geometry and ficti-
tious load variation are described in terms of quadratic interpolation functions (equa-
tions 6.25), the method may be described as a quadratic, isoparametric formulation
of the boundary element method.
The introduction of fictitious load distributions, q x (S), q y (S), in the solution proce-
dure to satisfy the imposed boundary conditions results in this approach being called
an indirect formulation of the boundary element method. In the alternative direct
formulation, the algorithm is developed from a relation between nodal displacements
[u] and tractions [t], based on the Betti Reciprocal Work Theorem (Love, 1944).
These formulations are also isoparametric, with element geometry, surface tractions
and displacements following imposed quadratic variation with respect to the element
intrinsic co-ordinate.

6.6 The finite element method

The basis of the finite element method is the definition of a problem domain surround-
ing an excavation, and division of the domain into an assembly of discrete, interacting
elements. Figure 6.7a illustrates the cross section of an underground opening gener-
ated in an infinite body subject to initial stresses p xx , p yy , p xy . In Figure 6.7b, the
selected boundary of the problem domain is indicated, and appropriate supports and
conditions are prescribed at the arbitrary outer boundary to render the problem stati-
cally determinate. The domain has been divided into a set of triangular elements. A
representative element of the set is illustrated in Figure 6.7c, with the points i, j, k
defining the nodes of the element. The problem is to determine the state of total
stress, and the excavation-induced displacements, throughout the assembly of finite
elements. The following description of the solution procedure is based on that by
Zienkiewicz (1977).
In the displacement formulation of the finite element method considered here, the
initial step is to choose a set of functions which define the displacement components
at any point within a finite element, in terms of the nodal displacements. The various
steps of the solution procedure then develop from the imposed displacement field.
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