Design against structurally-controlled instabilijl 37 1Block falling
from roofBOAB = plane 1, OBC = plane 2, OCA = plane 3
ABC = excavation roof
vertical force, F~ = 2 Vertical components of S,,S, and S,The direction cosines of a line trend a and plunge p,
using right hand axes, are
u, = cos a cos p, a, = sin (Y cos p, a- = sin p
Thus, assuming a friction-only material, on any
face in contact with the rock there is a normal force N
and a shear force S, which can be expressed as N tan 4.
If we can examine the vertical components of these forces
for all three faces of the block, we have
FT = &‘V& + b- tan 4i)
where bz, are the vertical direction cosines of the
bisector; of the apical angle on each face in contact
with the rock, nz- is the vertical component of the
normal to the ith’face. For wedge stability, FL + W < 0,
otherwise the block will fall under the action of gravity.( Vertical components of N,,N, and N, 1
3
,=I I -1Figure 20.13 Analysis of a tetrahedral block subject to in situ stresses and the action
of gravity.sophisticated analytical techniques based on computer methods and to
incorporate all of the analyses we have discussed so far in one integrated
approach.
The underlying principle of block theory is the recognition that blocks
are formed from the intersection of a number of non-parallel and non-
coincident planes. Any particular plane can be regarded as dividing the
space occupied by the rock into two half-spaces: for the sake of simplicity
these are called the ‘upper half-space’ and the ’lower half-space’. Thus, any
great circle on a hemispherical projection, e.g. one of those in Fig. 20.8, also
may be regarded as dividing space into these two half-spaces, and by
convention they are coded with a numerical value of 0 for the upper half-
space and 1 for the lower half-space. This idea stimulates the concept of
extending the hemispherical projection beyond the customary boundary
(which represents a horizontal plane) such that the upper and lower half-