Stress-controlled instability mechanisms 34979.2.7 Stresses and displucements around a circular
excavation
In rock mechanics and rock engineering, the Kirsch equations are the most
widely used suite of equations from the theory of elasticity. They allow
determination of the stresses and displacements around a circular excava-
tion, and are given in Fig. 19.10. The pre-eminent nature of these equations
is due to the requirements of stress determination techniques in circular
boreholes and consideration of the stability of circular tunnels. The
equations apply to openings made in previously stressed CHILE materials,
rather than the case of openings made in unstressed materials. The authors
had considerable difficulty in reconciling the many different expressions
given in the literature for u, and u8, but are confident that the expressions
given in Fig. 19.10 are correct. The angle 8 is measured anticlockwise
positive from the horizontal axis in the figure.
Some special cases of interest are now given in which the Kirsch
equations are used to demonstrate a number of important points. These
occur at specific locations (i.e. the boundary of the excavation) and with
specific stress fields (i.e. uniaxial and hydrostatic).
Stresses at the boundary of a circular opening. We see from Fig. 19.10 that
the stresses on the boundary (i.e. when r = a) are given by
0, = 0
08 = ~~((1 + k) + 2(1- k)c0~28}
and Z,O = 0.Note that the first of the stresses is zero because there is no internal
pressure, and the last of the stresses must be zero at a traction-free
boundary (the excavation boundary is a principal stress plane). The
variation in boundary tangential stress at the end points of horizontal and
vertical diameters for 0 I k I 1 is shown in Fig. 19.11.
Figure 19.10 Stresses and displacements induced around a circular excavation in
plane strain (for a CHILE material).