374 Design and analysis of underground excavations
shown in this diagram, and are mathematically defined as
U1 n L2 = JP
U, n U, = EPand, from this diagram because JP and EP have no common sector,
JP n EP = 0 and hence the block is removable. By an extension of this
procedure, the removability of all blocks with respect to all potential excava-
tion planes can be established. The power of the method lies in its ability
to convert three-dimensional polyhedra (i.e. the blocks of rock) into
mathematically defined sets, and to use mathematics to establish
kinematic feasibility.
The mathematics of block theory is beyond the scope of this work but is
well presented in the seminal book by Goodman and Shi (1985).
20.2 Design against stress-controlled
instability
In the introduction to the chapter, we mentioned that rock instability
around an excavation can occur due to block movement, stress
effects or sometimes both mechanisms can occur concurrently. In this
section, we describe design against stress-controlled instability through
an understanding of the stress field around excavations, and how one
can defend against the development of high stresses on the boundary
of an excavation. Also described are the effect of rock bolting on the
stress field and the use of the ground response curve to understand both
the rock response to excavation and the potential need for installed
support.20.2. I Zone of influence
When studying elastic stress distributions around underground openings,
as described in Section 19.2.1, we note that the excavation affects the
stresses and displacements for an infinite distance away from the opening.
This is because, in the mathematical derivation of the various equations,
the assumption has been made that material surrounding the opening
extends to infinity. As engineers, we are only interested in signifcant
changes to the stress field and displacements: below a certain level, it can
be assumed that the changes have no significant engineering influence.
This leads to the concept of the zone of influence, which is the zone around
the excavation in which the stresses are perturbed from their in situ values
by more than a defined amount.
For example, we could define the zone of influence around the excava-
tion as the zone within which at least one component of the stress tensor
is perturbed by greater than, say, 5% of its in situ value, expressed
mathematically as