344 Underground excavation instability mechanisms
on the inclined projection by the dashed great circle. The relative positions
of the inclined horizontal plane and NhI are used to distinguish between
’up’ and ‘down’: any line which appears on the &-side of the inclined
horizontal plane is directed downwards (because we started with a lower-
hemispherical projection and Nh was initially coincident with the down-
wards directed vertical line). The requirement to be able to distinguish
between up and down is essential in interpreting potential gravitationally
induced instability!
The elegance of this technique lies in the simple graphical transformation
(illustrated by the curved lines representing, for example, the Nl to Nl~
inclination) which is the representation of an equivalent 3 x 3 matrix
multiplication. Also, considering the associated D1 to DII inclination, we see
immediately that DII does not correspond to the mid-point of the great
circle, which is expected, in terms of a lower-hemispherical projection, to
be the line of maximum dip. Relative to the global frame of reference, DII
remains the line of maximum dip. Relative to the local frame of reference
(the inclined projection), the mid-point of the great circle is no more than
the line on the plane which makes the maximum angle to the plane of the
projection (which is the rock surface) and has no general engineering
utility.
Thus, the inclined hemispherical projection technique retains the inter-
pretive character of representing the three-dimensional rock structure
geometry, whilst enabling rapid study equivalent to lengthy mathematical
operations.
In the following paragraphs, we demonstrate the method of identifying
falling, sliding and stable blocks utilizing the inclined hemispherical
projection technique.
Identification of falling blocks. In Section 19.1.1, procedures were presented
for identifylng the kinematic feasibility of a falling block using the
lower-hemispherical projection to represent a horizontal roof. These
same basic procedures, shown in Fig. 19.1, can be used but with the inclined
hemispherical projection accounting for excavation surfaces at any
orientation.
Figure 19.6 illustrates the identification of a block falling from an inclined
surface. The various great circles and poles on this diagram have been
constructed using the procedures shown in Fig. 19.5. Note particularly the
great circle, H, representing the horizontal plane and the associated pole,
AIhI, representing the vertical. This vertical line is also shown in the
accompanying sketch of such a block.
By comparison with Fig. 19.1, the highlighted spherical triangle in both
cases contains the pole representing the vertical direction-and hence the
block will fall from an overhanging surface, because the spherical triangle
surrounds the downward directed vertical. The latter point is related to the
following discussion on stable blocks, which cannot fall from non-
overhanging surfaces because they have upward directed verticals.Identification of sliding blocks. By comparison with Fig. 19.2, a similar
procedure can be used for inclined projections to identify blocks which can