Engineering Rock Mechanics

348 Underground excavation instabilijl mechanisms

nature of stress has to be considered, but in CHILE materials there is a
relative simplicity in the associated stress-controlled instability mechanisms.
The analysis of stress-controlled instability must begin with a knowledge
of the magnitudes and directions of the in situ stresses in the region of the
excavation. The induced stresses can then be determined, i.e. the in situ
stresses after perturbation by engineering. There exist closed form solutions
for the induced stresses around circular and elliptical openings (and
complex variable techniques extend these to many smooth, symmetrical
geometries), and with numerical analysis techniques the values of the
induced stresses can be determined accurately for any three-dimensional
excavation geometry. Finally, a rock failure criterion expressed in terms of
stresses is required; failure has already been discussed in Chapter 6 for
intact rock, in Chapter 7 for discontinuities and in Chapter 8 for rock
masses.
It is now appropriate to consider stress distributions around under-
ground openings in order to determine the extent of stress-controlled
instability mechanisms. In the text that follows a series of elastic solutions
for various geometries will be presented, but the derivation of each solution
is not included.
All analytical closed form solutions must satisfy the following criteria.
(a) Equations of equilibrium-three equations of the form

az,, ar a0

ax ay aZ

-+A +I-+z=o.

@) Strain compatibility equations-three equations of the form

and three of the form

where the symbols are as defined in Chapters 3 and 5.

(c) Boundary conditions-e.g. zero traction or uniform pressure on the

(d) Conditions at infinity-e.g. field stresses.

It is from these conditions that the solutions for the circular and elliptical

openings that follow have been derived. As the conditions require that the

derivatives of various functions exist, openings with sharp corners cannot

be exactly modelled, although solutions for openings with small radii

comers have been developed using the theory of complex variables. The

solutions can be cumbersome, inhibiting simple analysis of parameter

variation, and it is for this reason that we do not present them here-

concentrating instead on the simpler instructive solutions, of which the

Kirsch equations are perhaps the paradigm.

excavation boundary.