Civil Engineering Reference
In-Depth Information
we have
T
=
W
sin
α
N
=
W
cos
α
(20.20)
where
is the average inclination of the slip surface at the base of the slice. Hence we
may calculate
T
and
N
for each slice and, for equilibrium, making use of Eq. (20.19),
W
sin
α
(
W
cos
φ
α
=
α
−
ul
) tan
(20.21)
where
u
is the average pore pressure over the length
l
of the base of each slice. Instead
of making use of Eq. (20.20) we may calculate
T
and
N
for each slice from force
polygons like those shown in Fig. 20.10(b). The calculations are assisted by the use of
a table such as that shown in Fig. 20.19(c) in Example 20.4. As before, it is necessary
to examine a number of different mechanisms to locate the critical slip circle; the slope
is taken to be in a state of collapse if Eq. (20.21) is satisfied for any mechanism.
(b) The Bishop routine method (Bishop, 1955)
Here it is assumed that the resultant of the interslice forces is horizontal. Hence
0
as shown in Fig. 20.12 and each slice is statically determinate. After resolving, taking
moments and summing over the whole mechanism, the solution comes out in the form
θ
=
W
sin
(
W
φ
−
ub
)sec
α
tan
α
=
(20.22)
φ
1
+
tan
α
tan
where
b
is the width of each slice. In practice, evaluation of Eq. (20.22) is simplified
if use is made of a table similar to that in Fig. 20.19(c). As before, it is necessary to
examine a number of different mechanisms to locate the critical slip circle; the slope is
then taken to be in a state of collapse if Eq. (20.22) is satisfied for any mechanism.
20.6 Other limit equilibrium methods
So far we have considered mechanisms consisting either of a single straight slip surface
or a circular arc. The limit equilibrium method is not restricted to these geometries
and there are two other commonly used arrangements of slip surfaces.
Figure 20.13 shows a mechanism consisting of several straight slip surfaces forming
two triangular wedges and a block; this mechanism is appropriate where a layer of
relatively weak soil occurs within the slope as shown. The shear and normal forces
across each slip surface are marked. In this case, unlike the method of slices, the soil
in the vertical slip surfaces is at failure and so the shear stresses can be determined
from either Eq. (20.1) or (20.2) and the lengths of the slip surfaces. Working from the
left-hand wedge towards the right, the forces on each block are statically determinate.
Figure 20.14 shows a mechanism in which there is a single continuous slip surface of
general shape. The solution is found using the method of slices, as described above, for
which at least one simplifying assumption is required. Thus the Swedish method (
X
and
E
0) can be applied to general slip surfaces.
Other solutions were developed by Janbu (1973) and byMorgenstern and Price (1965).
=
0) or the Bishop routine method (
X
=
