Engineering Rock Mechanics

292 Surkrce excavation instabiliv mechanisms

sliding directly, and by making suitable assumptions to render the problem
two dimensional, the solution is straightforward.
The right-hand side of Fig. 17.5 shows an idealized form of the plane
instability condition. This demonstrates two of the underlying assumptions
in the analysis: the strikes of the plane of sliding and the slope face are
parallel, and there are no end restraints caused by adjacent blocks of rock.
The free body diagram shows the forces acting on the unstable block of
rock. In the case shown, a partially water-filled tension crack has been
included, with the result that there are water pressure distributions along
the tension crack and the plane of sliding. The usual assumption for these
distributions is that they are linear, and the water pressure on the plane of
sliding is zero at the plane’s intersection with the slope face.
Making the assumptions that the rock mass is impermeable, the sliding
block is rigid, the strength of the sliding plane is given by the
Mohr-Coulomb criterion and that all forces pass through the centroid of
the sliding block (so that moment equilibrium is automatically maintained),
then by defining the factor of safety as the ratio between the forces resisting
sliding and the forces driving the sliding, we have

c’(H - z)cosecy, + (Wcos yp - U - Vsiny,)tan@’

Vcosy,+Wsiny,

F=

Similar formulations can be derived for other cases, such as a horizontal
sliding plane, no tension crack, a sloping upper surface or dry conditions.
The last of these cases can over-estimate the stability of the slope and
should only be used when there is confidence in the knowledge of the
hydraulic regime.
The effective stress parameters C and @’ have been used in the analysis
above. It is by no means clear without further information whether, in fact,
the most appropriate parameters are the traditionally used total stress
parameters c and @ of rock engineering which imply drained conditions,
or the traditionally used effective stress parameters c’ and 4‘ of soil
engineering which incorporate the effect of water pressure resulting from
undrained conditions. This is a complex subject, and a thorough knowl-

Tension crack

• f

U

y, = unit weight of water

Y y = unit weight of rock

Geometry Free body diagram

Figure 17.5 Geometry of static analysis of plane instability.