Engineering Rock Mechanics

Slope instabiliv 291

number of unknowns. The usual method of doing this is by sub-dividing
the mass under consideration into 'slices', and analysing each slice on the
basis that it is in limiting equilibrium, i.e. each N and S is linked through
the strength criterion of the slip surface.

to it, and equilbrium analysis leads to

0%

4-

N2 Q.3

The margin sketch shows a typical slice with the various forces applied s "2

[(W - ub) tan @ + bc] sec a

F[1+ (tana tan@) / F]

S=

which enables S to be expressed in terms of the other slice parameters. The
analysis of the factor of safety, F, of the entire mass then depends on whether
the slip surface is generally non-circular, or specifically truly circular.
In the former case, resolving horizontally and vertically for all of the slices
leads to

Typical slice

C (FSsec a)

C (W tana) + C (Ssec a - w tana)

F=

In the latter case, the equation is simpler, and reduces to 'N

where H is the hydrostatic thrust from the tension crack and the other
parameters are illustrated in the margin sketch.
The anticipated location of the slip surface can now be found from analysis
of the whole range of possible surfaces, and taking the actual surface to be
that which gives the lowest factor of safety. Curvilinear slips are, in general,
truly three-dimensional in that they resemble the bowl of a spoon, and hence
the analysis here is an approximation. The seminal references for this type
of two-dimensional analysis are Bishop (1955) for the circular slip surface and
Janbu (1954) for the non-circular slip surface, with further explanation
specifically related to rock slopes in Hoek and Bray (1977).

77.7.2 Plane sliding

In Fig. 17.4, we showed the variety of curvilinear slips that can occur for

different geological circumstances. In this section, we concentrate on the

type of failure illustrated in the central diagram of Fig. 17.4, where major

structural features are present which are much weaker than the rock on

either side. Because the slip generally occurs on a major discontinuity, it

will usually have a planar form-owing to the planar nature of the pre-

exisiting discontinuity. In fact, when the instability is dictated by the

presence of pre-exisiting discontinuities, the instability takes the form of

plane sliding, wedge sliding or toppling, as illustrated in elements (b)-(d)

of both Figs 17.2 and 17.3. In this and the two following sub-sections, we

deal with these in turn.

The case of plane sliding is unlike that of curvilinear slip, in that it is

statistically determinate. We can calculate the factor of safety for plane