Page 197 - Rock Mechanics For Underground Mining

THE BOUNDARY ELEMENT METHOD

Solutions to the more complex excavation design problems may usually be obtained
by use of computational procedures. The use of these techniques is now firmly em-
bedded in rock mechanics practice. The following discussion is intended to indicate
the potential of various computational methods of analysis in excavation design. The
description is limited to the formulation of solution procedures for plane geometric
problems, with the implication that, conceptually at least, there is no difficulty in
extending a particular procedure to three-dimensional geometry.
Computational methods of stress analysis fall into two categories – differential
methods and integral methods. In differential methods, the problem domain is di-
vided (discretised) into a set of subdomains or elements. A solution procedure may
then be based on numerical approximations of the governing equations, i.e. the differ-
ential equations of equilibrium, the strain–displacement relations and the stress–strain
equations, as in classical finite difference methods. Alternatively, the procedure may
exploit approximations to the connectivity of elements, and continuity of displace-
ments and stresses between elements, as in the finite element method.
The characteristic of integral methods of stress analysis is that a problem is spec-
ified and solved in terms of surface values of the field variables of traction and
displacement. Since the problem boundary only is defined and discretised, the so-
called boundary element methods of analysis effectively provide a unit reduction in
the dimensional order of a problem. The implication is a significant advantage in
computational efficiency, compared with the differential methods.
The differences in problem formulation between differential and integral methods
of analysis lead to various fundamental and operational advantages and disadvantages
for each. For a method such as the finite element method, nonlinear and heteroge-
neous material properties may be readily accommodated, but the outer boundary of
the problem domain is defined arbitrarily, and discretisation errors occur throughout
the domain. On the other hand, boundary element methods model far-field boundary
conditions correctly, restrict discretisation errors to the problem boundary, and en-
sure fully continuous variation of stress and displacement throughout the medium.
However, these methods are best suited to linear material behaviour and homoge-
neous material properties; non-linear behaviour and medium heterogeneity negate
the intrinsic simplicity of a boundary element solution procedure.
In describing various computational procedures, the intention is not to provide a
comprehensive account of the methods. Instead, the aim is to identify the essential
principles of each method.

6.5 The boundary element method

Attention in the following discussion is confined to the case of a long excavation of
uniform cross section, developed in an infinite elastic body subject to initial stress. By
way of introduction, Figure 6.5a illustrates the trace of the surface S of an excavation
to be generated in a medium subject to uniaxial stress, p xx ,inthe x direction. At
any point on the surface, the pre-excavation load condition is defined by a traction

t (S). After excavation of the material within S, the surface of the opening is to be
x
traction free, as shown in Figure 6.5b. This condition is achieved if a distribution
of surface traction, t x (S), equal in magnitude, but opposite in sense to that shown
in Figure 6.5a, is induced in a medium that is stress free at infinity. The required
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