Design against sfrucfura//y-contro//ed instabilify 369Under the circumstances of a compressive rock stress, it is noted in the
figure that there will only be restraint, i.e. P > 0, if a < Cp. This accords with
the Cpj theory which was discussed in Section 16.5, for very similar circum-
stances, i.e. slip around the boundary of an excavation. Also, the k, and k,
referred to in Fig. 20.10 are the same as the k,, and k,, in Section 7.3.1, except
that the second subscript has been omitted because there are no cross-
stiffnesses in this case. In Fig. 20.11, we show a variation in P/2H0, as a
function of the semi-apical angle for different ratios of normal to shear
discontinuity stiffness.
(a) regardless of the ratio of discontinuity stiffness, P is always zero when
a= Cp;
(b) there is a tendency for the prism to be expelled from the surface when
the semi-apical angle exceeds the angle of friction, which is exacerbated
for high values of kdks; and,
(c) the relation between P/2H0 and a becomes close to linear in the semi-
apical angle range of 20-60".
In the case of asymmetric triangular roof prisms, the analysis presented
above has to be extended to include two different semi-apical angles, as
shown in Fig. 20.12. A similar sensitivity analysis to the one illustrated in
Fig. 20.11 can be generated to show how the components of the apical angle
effect the stability of the roof prism.
There are three interesting characterisitics indicated in Fig. 20.11:Tetrahedral blocks. To consider the stability of a general tetrahedral block,
through a three-dimensional analysis and taking into account Coulomb
friction on the three faces of the block that are in contact with the rock mass,
the earlier simple sliding analysis illustrated in Fig. 20.9 can be extended.
This extended analysis is shown in Fig. 20.13.
The normal forces on each of the block faces shown in Fig. 20.13 can be
obtained by transforming the stress state in the rock to obtain the normal
Figure 20.11 Effect of stress in the surrounding rock and discontinuity stiffness on
the generation of constraining force for a symmetric triangular roof prism.