Geoscience Reference
In-Depth Information
Note, that the forces parallel to the slope,
E
1
and
E
2
, are in balance, and cancel
out in the equilibrium. The same holds for the shear forces on the vertical sides.
Therefore, (10.2) represents a lower limit. Usually, with depth undrained strength
and weight increase. Formula (10.2) is valid for shallow slip surfaces. A
submerged slope under still water has the same result, except for the fact that in
stead of
w
should be used.
Let the soil be fully drained and the shear strength expressed by |
the submerged weight
' =
| < c +
'
tan
. The cohesion may allow for a steeper slope than the natural slope angle (
)
>
), as shown by the Mohr circle in Fig 10.1b. The stress state is at limit state,
expressed by point T
s
, representing the normal and shear stress at the slip line.
Obviously,
W =
Hb
cos
)
=
s
b/
cos
)
= |
T
s
O
|b
, where |T
s
O
|
is indicated in the
Mohr diagram. Hence,
|
T
s
O
| =
H
cos
)
. From the Mohr diagram it can be shown
that
|
T
s
O
| = c
cos
/
sin
(
)
)
, and a critical layer thickness is then
H = c
cos
/(
cos
)
sin(
)
))
(10.3a)
The corresponding stability factor is expressed by
cos
c
(
)
F =
with
)
>
(10.3b)
cos
)
sin(
)
)
H
, at which a soil layer of
thickness
H
is just stable. When there is no cohesion
c
= 0, obviously
For
F = 1
, formula (10.3b) indicates the slope angle
)
)
=
satisfies
F
= 1.
;
water film
x
p
0
/
w
z
'
'
)
?
q
slip plane
)
(a) (b)
Figure 10.2
For a saturated drained slope, the pore pressures induced by local groundwater
flow has a pronounced effect on the stability. Let the shear strength of the soil be
defined by the Mohr-Coulomb criterion: |
. Note, that cohesion is
disregarded. Assume a local groundwater flow, with a specific discharge
q
inclined
at an angle
| <
'
tan
(see Fig 10.2a).
First, the corresponding pore pressure
u
is determined. Assume a (locally)
uniform flow field, where the pore pressure is linear: or
u = A
;
+ B
+ C
. The
condition that
u = p
0
along the surface
=
0 yields
u = B
+ p
0
. The
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