Civil Engineering Reference
In-Depth Information
in the slope and the uplift pressure
U
can
exceed that shown in Figure 6.3. For the
idealized rectangular pressure distribution
shown in Figure 6.5(a), the uplift force
U
is
given by
of soil above the slope crest, or an assumed loc-
ation may be required for design. Under these
circumstances, it becomes necessary to con-
sider the most probable position of a tension
crack.
When the slope is dry or nearly dry
(z
w
/z
0
)
,
equation (6.4) for the factor of safety can be
modified as follows:
=
U
=
Ap
(6.10)
where
A
is the area of the sliding plane given
by equation (6.5) and
p
is the pressure in the
plane (and at the base of the tension crack)
given by
A
W
sin
ψ
p
+
c
·
FS
=
cot
ψ
p
tan
φ
(6.13)
The critical tension crack depth
z
c
for a dry slope
can be found by minimizing the right-hand side
of equation (6.13) with respect to
z/H
. This gives
the critical tension crack depth as
p
=
γ
w
z
w
(6.11)
The condition shown in Figure 6.5(a) may
only occur rarely, but could result in a low
factor of safety; a system of horizontal drains
may help to limit the water pressure in the
slope.
−
cot
ψ
f
tan
ψ
p
z
c
H
=
1
(6.14)
and the corresponding position of the critical
tension crack
b
c
behind the crest is
(d)
Ground water level in the slope is below the
base of the tension crack so water pressure
acts only on the sliding plane (Figure 6.5(b)).
If the water discharges to the atmosphere
where the sliding plane daylights on the face,
then the water pressure can be approximated
by a triangular distribution, from which the
uplift force
U
is given by
H
=
cot
ψ
f
cot
ψ
p
−
b
c
cot
ψ
f
(6.15)
Critical tension crack depths and locations for a
range of dimensions for dry slopes are plotted in
Figure 6.6(a) and (b). However, if the tension
crack forms during heavy rain or if it is located
on a pre-existing geological feature such as a ver-
tical joint, equations (6.14) and (6.15) no longer
apply.
1
2
z
w
sin
ψ
p
h
w
γ
w
U
=
(6.12)
where
h
w
is the estimated depth of water at
the mid-point of the saturated portion of the
sliding plane.
6.3.3 The tension crack as an indicator of
instability
Anyone who has examined excavated rock slopes
cannot have failed to notice the occasional ten-
sion cracks behind the crest (Figure 6.7). Some
of these cracks have been visible for tens of years
and, in many cases, do not appear to have had any
adverse influence upon the stability of the slope.
It is interesting, therefore, to consider how such
cracks are formed and whether they can give any
indication of slope instability.
In a series of very detailed model studies on the
failure of slopes in jointed rocks, Barton (1971)
6.3.2 Critical tension crack depth
and location
In the analysis, it has been assumed that the
position of the tension crack is known from its
visible trace on the upper surface or on the face
of the slope, and that its depth can be established
by constructing an accurate cross-section of the
slope. However, the tension crack position may
be unknown, due for example, to the presence
