Engineering Rock Mechanics

Design against stress-controlled instability 375

where qinduced represents any component of induced stress, and

the number 0.05 represents 5%-but may be any other percent-

age value relevant to the engineering objective.

Considering the stresses around a circular excavation (see Figs 19.10 and

19.11), the perturbation to the tangential stress component on the horizon-

tal plane through the centre of the excavation can be calculated from the

second of the equations shown in Fig. 19.10. For the example of k = 1, the

equation reduces to

and, by substituting this expression into the equation above, the 5% zone
of influence is given by r5% = ahO. Thus, the 5% zone of influence is 4.47~
(as measured from the centre of the excavation) or 3.47~ measured from
the wall of the excavation. In this case, the vertical and horizontal in situ
stress components are equal, i.e. k = 1, and so this distance to the boundary
of the zone of influence applies in all directions.
For other components of the stress field, and for other values of k, similar
calculations can be made. For example, when r = 5a and k = 1, or = 0.96~~
and o, = 1.04pZ, indicating that the 4% zone of influence (based on these
components) then extends to Y = 5a.
This principle of the zone of influence and the method of establishing
its extent is directly applicable to any stress distribution, whether obtained
by a closed form solution or numerically.

Elliptical approximation. In the case of a circular excavation when k # 1,

the zone of influence is not circular in shape. The shapes of the zones of

influence associated with each component of induced stress may be very

different, as demonstrated by Fig. 20.17, but an approximation to the overall

zone of influence may be found by drawing a circumscribed ellipse to the

various perturbation contours. For the example shown in the figure, where

k = 0.5, the major and minor axes of the ellipse are 11.76~ and 7.98a,

respectively. The circumscribed ellipse in Fig. 20.17 does not indicate the

magnitude of the stresses per se, but rather the magnitude of the

perturbations to the in situ stress components. Thus, although the induced

stresses would be expected to be greatest along the horizontal axis (for this

value of k), this is not the case for the perturbations.

Similarly the value of the concept of the zone of influence is not in

assessing the likelihood of inducing stresses which will lead to failure of

the rock, but in determining-for the purposes of design-at which

locations the induced stress field may be regarded as being unperturbed

from the in situ stress field, and hence at what separations proximate

excavations can be positioned. This is our next subject.

Multiple openings. In the case of adjacent circular openings, the stress dis-

tribution due to the two excavations can be approximated by summing the

distributions due to the two single excavations. This provides two items of

information for design: the stresses induced by multiple excavations; and the

locations where the individual zones of influence overlap (or, are distinct).