Design against stress-controlled instability 379in this figure and that shown in Fig. 20.17 is clear, except that now we are
considering the excavation shape itself to be an ellipse, rather than a circle.
The principle is that a percentage value, c, is chosen for the zone of
influence, and then the width and height of the circumscribing zone of
influence ellipse are determined from the equations in Fig. 20.21.
The value of c provides the value of A, and then the values of Wi and Hi
can be directly evaluated from the equations in Fig. 20.21, using the values
of k, q and a and the criteria given. Although in Fig. 20.21 the elliptical
approximation to the zone of influence is indicated with its major axis in
the vertical direction, this will not always be the case because the aspect
ratio of this ellipse will depend on the parameters just described.
In Figure 20.22, two examples of this zone of influence are presented,
both having a W/H value of 2, but with differing stress ratios k. The two
cases have been chosen for comparison because they illustrate the use of
the criteria presented in Fig. 20.21. In the left-hand diagram, the limits of
the zone of influence are determined by the 5% contours-given by 1.05
and 0.95-associated with the vertical stress component. In the right-hand
diagram, the limits of the zone of influence are determined by the 0.95
contours (associated with the vertical stress component) and, now, the 0.15
contour (associated with the horizontal stress component).
The 5% zone of influence produces the 0.95 and 1.05 contours for the
perturbation to the vertical stress in both diagrams in Fig. 20.12. In the
case of the horizontal stress component, we consider the criterion
1 0, - pmin I > 0.05pm,, and so, because pmin = kp,,,, the required contours
are for
03 > k + 0.05 pmaxWhichever
is greaterWhichever
is greaterc% roneot
influenceFigure 20.21 Elliptical approximation to the zone of influence around an elliptical
excdvdtion (from chapter by J. W. Bray in Brown, 1987).