Civil Engineering Reference
In-Depth Information
γ
w
is the unit weight of the water,
H
is the total
height of the wedge. The dimensionless factors
X
,
Y
,
A
and
B
depend upon the geometry of the
wedge.
The values of parameters
X
,
Y
,
A
and
B
are
given in equations (7.10)-(7.13):
(a)
Upper slope surface,
which can be obliquely
inclined with respect
to the face
4
Plane A
Plane B
Face
5
3
sin
θ
24
sin
θ
45
cos
θ
2.na
X
=
(7.10)
2
sin
θ
13
sin
θ
35
cos
θ
1.
nb
1
Y
=
(7.11)
cos
ψ
a
−
cos
ψ
b
cos
θ
na
.
nb
sin
ψ
5
sin
2
θ
na
.
nb
A
=
(7.12)
cos
ψ
b
−
cos
ψ
a
cos
θ
na
.
nb
sin
ψ
5
sin
2
θ
na
.
nb
B
=
(7.13)
(b)
where
ψ
a
and
ψ
b
are the dips of planes A and
B respectively and
ψ
5
is the dip of the line
of intersection, line 5. The angles required for
the solution of these equations can be measured
most conveniently on a stereoplot that defines the
geometry of the wedge and the slope (Figure 7.7).
The application of the equations discussed in
this section is illustrated in the following example,
using the parameters shown in Table 7.1.
The total height of the wedge
H
is 40 m, the
unit weight of the rock is 25 kN
/
m
3
, and the unit
weight of the water 9.81 kN
/
m
3
.
The stereoplot of the great circles represent-
ing the four planes involved in this example is
presented in Figure 7.7, and all the angles required
for the solution of equations (7.10)-(7.13) are
marked in this figure.
Determination of the factor of safety is most
conveniently carried out on a calculation sheet
such as that presented on Table 7.2. Setting
the calculations out in this manner not only
enables the user to check all the data, but it also
shows how each variable contributes to the over-
all factor of safety. Hence, if it is required to check
the influence of the cohesion on both planes fall-
ing to zero, this can be done by setting the two
groups containing the cohesion values
c
A
and
c
B
to zero, giving a factor of safety of 0.62. Altern-
atively, the effect of drainage can be checked by
H
½
H
Assumed water pressure
distribution
Figure 7.6
Geometry of wedge used for stability
analysis including the influence of friction and
cohesion, and of water pressure on the slide surfaces:
(a) pictorial view of wedge showing the numbering of
the intersection lines and planes; (b) view normal to
the line of intersection (5) showing wedge height and
water pressure distribution.
(Hoek
et al
., 1973):
A
2
γ
r
X
tan
φ
A
3
γ
r
H
(c
A
X
γ
w
FS
=
+
c
B
Y)
+
−
B
2
γ
r
Y
tan
φ
B
γ
w
+
−
(7.9)
where
c
A
and
c
B
are the cohesive strengths, and
φ
A
and
φ
B
are the angles of friction respectively on
planes A and B,
γ
r
is the unit weight of the rock,
