Civil Engineering Reference
In-Depth Information
where the angles
ξ
and
β
are defined in
Figure 7.4(a). Angles
ξ
and
β
are measured on
the great circle containing the pole to the line
of intersection and the poles of the two slide
planes. In order to meet the conditions for equi-
librium, the normal components of the reactions
are equal (equation (7.4)), and the sum of the
parallel components equals the component of
the weight acting down the line of intersection
(equation (7.5)).
The values of
R
A
and
R
B
are found from equa-
tions (7.4) and (7.5) by solving and adding as
follows:
of a wedge with cohesion and friction acting on
the slide planes and water pressure. The complete
set of equations for stability analysis of a wedge
is shown in Appendix III; this analysis includes
parameters to define the shape and dimensions of
the wedge, different shear strengths on each slide
surface, water pressures and two external loads.
This section shows the important influence of
the wedging action as the included angle of the
wedge decreases below 90
◦
. The increase by a
factor of 2 or 3 on the factor of safety determ-
ined by plane failure analysis is of great practical
importance because, as shown in Figure 7.5,
the factor of safety of a wedge can be signific-
antly greater than that of a plane failure. There-
fore, where the structural features that are likely
to control the stability of a rock slope do not
strike parallel to the slope face, the stability
analysis should be carried out by means of three-
dimensional methods discussed in this chapter.
W
cos
ψ
i
sin
β
sin
(ξ/
2
)
R
A
+
R
B
=
(7.6)
Hence
sin
β
sin
(ξ/
2
)
·
tan
φ
tan
ψ
i
FS
=
(7.7)
In other words,
7.4 Wedge analysis including cohesion,
friction and water pressure
Section 7.3 discussed the geometric conditions
that could result in a wedge failure, but this
kinematic analysis provides limited information
of the factor of safety because the dimensions
of the wedge were not considered. This section
describes a method to calculate the factor of safety
of a wedge that incorporates the slope geometry,
different shear strengths of the two slide planes
and ground water (Hoek
et al
., 1973). However,
the limitations of this analysis are that there is
no tension crack, and no external forces such as
bolting can be included.
Figure 7.6(a) shows the geometry and dimen-
sions of the wedge that will be considered in
the following analysis. Note that the upper slope
surface in this analysis can be obliquely inclined
with respect to the slope face, thereby removing
a restriction that has been present in the stabil-
ity analyses that have been discussed so far in the
topic. The total height of the slope
H
is the differ-
ence in vertical elevation between the upper and
lower extremities of the line of intersection along
which sliding is assumed to occur. The water
FS
W
=
K
FS
P
(7.8)
where FS
W
is the factor of safety of a wedge sup-
ported by friction only, and FS
P
is the factor of
safety of a plane failure in which the slide plane,
with friction angle
φ
, dips at the same angle as the
line of intersection
ψ
i
.
K
is the wedge factor that, as shown by
equation (7.7), depends upon the included angle
of the wedge
ξ
and the angle of tilt
β
of the wedge.
Values for the wedge factor
K
, for a range of
values of
ξ
and
β
are plotted in Figure 7.5.
The method of calculating the factor of safety
of wedges as discussed in this section is, of course,
simplistic because it does not incorporate differ-
ent friction angles and cohesions on the two slide
planes, or ground water pressures. When these
factors are included in the analysis, the equations
become more complex. Rather than develop these
equations in terms of the angles
ξ
and
β
, which
cannot be measured directly in the field, the more
complete analysis is presented in terms of directly
measurable dips and dip directions. The follow-
ing section gives equations for the factor of safety
