Civil Engineering Reference
In-Depth Information
Tension crack
Ground water
surface
Typical slice
z
w
z
2
z
/3
½
H
Failure through
toe of slope
L
Water force =
w
h
w
(
∆
x
/cos
b
)
d
∆
x
Factor of safety:
FS =
f
0
∑
X
/(1 +
Y
/FS
)
∑
(8.10)
Z
+
Q
where
X
=[
c
+(
] (1 + tan
2
r
h
-
w
h
w
) tan
b
)
∆
x
(8.11)
(8.12)
(8.13)
(8.14)
Y
= tan
b
tan
Z
=
r
h
x
tan
b
∆
h
w
z
2
Q
=½
Figure 8.17
Janbu's
modified method of
slices for the analysis of
non-circular failure in
slopes cut into
materials in which
failure is defined by the
Mohr-Coulomb failure
criterion.
h
w
Slice weight
=
r
h
∆
x
Note: angle
b
is negative when sliding uphill
Approximate correction factor
f
0
f
0
=1+
K
(
d
/
L
- 1.4(
d
/
L
)
2
)
b
(8.15)
for
c
=0;
K
= 0.31
c
> 0,
>0;
K
= 0.50
used, and a new factor of safety is calculated from
equations (8.3) and (8.10) given in Figures 8.16
and 8.17, respectively. If the difference between
the calculated and the assumed factors of safety is
greater than 0.001, the calculated factor of safety
is used as a second estimate of FS for a new factor
of safety calculation. This process is repeated
until the difference between successive factors of
safety is less than 0.001. For both the Bishop and
the Janbu methods, approximately seven iteration
cycles will be required to achieve this result for
most slope and slide surface geometries.
Step 5: Conditions and corrections.
Figure
8.16 lists two conditions (equations (8.8)
and (8.9)) that must be satisfied for each slice in
the Bishop analysis. The first condition ensures
that the effective normal stress on the base
of each slice is always positive. If this con-
dition is not met for any slice, the inclusion
of a tension crack into the analysis should be
considered. If it is impossible to satisfy this
condition by readjustment of the ground water
conditions or the introduction of a tension
crack, the analysis as presented in Figure 8.16
should be abandoned and a more elaborate form
of analysis,
to be described later,
should be
adopted.
Condition 2 in Figure 8.16 was suggested by
Whitman and Bailey (1967) and it ensures that
the analysis is not invalidated by conditions which
can sometimes occur near the toe of a slope in
which a deep slide surface has been assumed. If
this condition is not satisfied by all slices, the
slice dimensions should be changed and, if this
