Engineering Rock Mechanics

320 Design and analysis of surface excavations

Basal planes:

analysed using poles

Dip direction of slope&

Oblique toppling:

critically increases

intersections appro

Block edees:

Figure 18.7 Illustration of the direct toppling instability modes.

An interesting aspect of this analysis is that the slope angle is not
explicitly used. The direction of the slope serves to indicate in which area
of the projection one needs to look for poles and intersections as potential
candidates for inducing instability. Similarly, although not shown in
Fig. 18.8, it must be remembered that those basal planes whose dips lie
outside the main region of instability must also be considered as candidates
for defining toppling blocks. It is quite likely that oblique blocks formed by
basal planes dipping into the slope will topple.

Flexural toppling instability. In Figure 17.11, the analysis of flexural
toppling was illustrated using the $j theory. The geometrical analysis
and the associated criteria for inter-layer slip can be used to form the basis
of a kinematic feasibility analysis for flexural toppling using the overlay
method. In fact, the angles and the criterion are clarified by the use of
such a method. From Section 17.1.4, the geometrical criterion for inter-layer
slip to occur is t,u2 90 + $ - /3, where Vis the dip of the slope, $ is the friction
angle associated with the discontinuities and /3 is the dip of these
discontinuities.
The criterion was expressed in this way because we were interested in
the slope angle at which inter-layer slip could occur. In terms of the
projection overlay, we wish to know the positions of the discontinuity poles
on the projection which would indicate the potential for inter-layer slip.
Thus, the criterion above can be rewritten as p^2 $ + (90 - w). This allows
not only the creation of the instability overlay for flexural toppling, but
also identification of the various components of the criterion on the
projection.

(a) There is one set of discontinuity planes dipping into the slope, at a
sufficiently high angle to generate inter-layer slip, following the crite-
rion above.

The criteria are as follows.