Question and answers: foundation and slope instability mechanisms 293from which we see that the three angles of /3, e and $r are required
to describe the problem. These are established using the hemispherical
projection, as shown below. In constructing this projection, it is import-
ant to realize that the angles /3 and ( are measured in the plane that is
perpendicular to the line of intersection that forms the wedge. The great
circle representing this plane is shown dashed in the projection, and has
as its normal the line of intersection of planes 1 and 2.Substituting the various numerical values into the equation for the
wedge factor of safety then gives
sinj3 tan@ sin 101 tan29 0.982 0.554
sin $6 tan@ sin36 tan32 0.588 0.625Fw = - x-- - x-- -- x-- - 1.48.
Notice that the magnitude of kw is 0.982/0.588 = 1.67, which shows
that the confinement offered by the fractures forming the sides of the
wedge contributes an additional 67% to the factor of safety, over and
above that obtained from plane sliding. In this case, the plane case would
be unstable.
The calculation presented here is approximate, because the various
angles have been measured on the hemispherical projection. The equa-
tion for the factor of safety is, however, exact. To demonstrate this, if
the various angles are computed using vector analysis, we find that
= 100.9", 6 = 72.0" and pb = 32.59 and from this we obtain Fw = 1.454.
The answer computed using a wedge analysis algorithm, such as those
presented by Hoek and Bray (1977), is 1.453.
Finally, we see that neither the orientation of the slope face nor the
unit weight of the rock material enters into the calculation. This is
because the stability of the wedge does not depend on its size, as it is a
friction-only case. If either cohesion or water pressures are present, then
the size of the wedge has to be taken into account. However, the slope
face orientation should be such that a wedge is actually formed, and