342 Underground excavation instability mechanismsFigure 19.3 Kinematic identification of stable blocks.surfaces which are not horizontal-the concept of inclined hemispherical
projection can be used.
The purpose of inclining the projection is to be able to identify tetra-
hedral blocks formed by three discontinuities and the excavation surface,
the latter being retained as the perimeter of the projection. This is achieved
by ensuring that the plane of projection is coincident with the plane of the
excavation surface. Naturally, the position of the discontinuity planes and
the point representing the vertical direction on the projection will change,
and friction circles will no longer be concentric with the perimeter of the
net but must be constructed as circles representing friction cones around
the vertical.
With the extended technique of the inclined projection, the block iden-
tification procedures can be retained and similar techniques for establishing
kinematic feasibility used as for horizontal roofs. The details of the
procedure required to construct an inclined hemispherical projection are
given in the second Appendix, and so here we concentrate on the required
angle of inclination and interpretation of the resulting diagram.
Angle of inclination of the hemispherical projection. In Fig. 19.4, there is a
generic excavation with five main excavation surfaces: the roof, the shoul-
der, the sidewall, the knee and the floor. The inclined hemisphere is shown
as it relates to each of the excavation surfaces. The lower-hemispherical
projection at the roof is inverted to an upper-hemispherical projection
at the floor: between these extremes the hemisphere has been inclined
by 45" at the shoulder, 90" at the vertical sidewall and 135" at the knee.
The hemisphere can be inclined in this way to accommodate any required
excavation surface.
The key to the procedure is to incline the projection such that it becomes
coincident with the outward directed normal to the excavation surface. This
is achieved by performing the following steps.
First, plot on an overlay, the normal to the excavation surface in question
(Nf in Fig. 19.5), the normals to the various discontinuity surfaces
(Nl, N2 in Fig. 19.5) and the normal to the horizontal plane (Nh in Fig.
19.5, which is coincident with the centre of the projection, i.e. vertical).
Then rotate the overlay such that Nf lies on the E-W line.