3 1 8 Design and analysis of surface excavations
From criterion (a) an overlay for an intersection plot is required; from
criterion @) an overlay for a pole plot is required. For this the intersection
and pole plots are superimposed and a composite overlay is used.
In an analogous fashion to the previous analyses of plane and wedge
instability, in Figs 18.6(a) and (b) the generation of the hemispherical
projection instability overlays based on the criteria above is shown.
In Fig. 18.6(a), the radial solid line pointing to the left is again taken to
be the slope direction, Because the interest is in the angles between the
vertical and both the plunge of the lines of intersection (criterion (a) above)
and the dip of the basal plane (criterion @) above), the overlay will consist
only of concentric circles. The concentric circles are numbered from the
perimeter inwards for the intersections, and from the centre outwards for
the poles. (Because the intersection lines are dipping into the slope, whereas the
basal planes are dipping out of the slope, the overlay criteria are both on the same
side of the composite generic overlay shown in Fig. 18.6(a)--and on the opposite
side to the direction of the slope dip.) The two dashed radial lines represent a
'sub-criterion', in that observations have indicated that toppling tends to
occur within a k20" sector of the slope dip, except for very steep slopes
where the sector can be considerably enlarged.
Given the criteria, the necessary bounds can be drawn and the overlay
produced. Figure 18.6@) shows the overlay for this example. There are
many instability regions associated with a direct toppling overlay,
depending on the combinations of the occurrences of overlaid poles and
intersections. Figure 18.7 clarifies these possibilities. The upper suite of
sketches refers to the basal plane occurrences; the lower suite of sketches
refers to the intersection occurrences. In this sense, the occurrence of direct
toppling instability is not so sharply focused as with the previous two
overlays, but again illustrates the value of this approach.
The many modes of toppling instability can be established from the
sketches in Fig. 18.7 and any specific example can be interpreted with the
aid of the overlay technique. Moreover, once a potential mode has been
established from the analysis, the engineer can return to the field and
consider the mechanism in situ. This provides a powerful technique for
establishing the real likelihood of instability: attempting to establish the
direct toppling modes without such a visual and integrating analysis would
be most unsatisfactory.
To assess the kinematic feasibility for direct toppling instability, the spe-
cific overlay (in this case, Fig. 18.6@)) is superimposed onto a composite
projection representing all the inter-set intersections and all the poles for
the rock mass discontinuity data (in this case, the data shown in Fig. 18.1).
This is shown in Fig. 18.8.
It can be seen that the potential for toppling is not high. The main
possibility is for set F to form the basal plane, and the block edges to be
formed by intersection IAE-a typical example of the need to return to the
field and assess the mechanism visually. The circumstances are akin to a
combination of the two left-hand sketches in Fig. 18.7, with oblique
toppling occurring due to the intersections not falling within the main
region of instability. In the specific example shown in Fig. 18.8, the potential
toppling direction is diagonally southwards across the slope.