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considering the fact that the angle between
s-
direction and
r-
direction (radial) is
constant, i.e. ½
+
.
$
c/ta
n
q
s
q
0
c
A
s
@A
@$
*
3
*
1
Figure 12.2b Stress state
Kötter's equation for drained soil along the
s
-directions, according to equation
(11.4a), becomes (the soil weight of the plastic zone has been disregarded)
0
,s
+
2
0
,s
tan
=
0
or
(ln
0
+
2
tan
)
,s
= 0
(12.4)
Integration yields
ln
0
+
2
tan
= C
s
in the s-direction
(12.5)
From zone 1 to zone 3 this gives
03
= e
2(
3
1
)tan
ln
01
+
2
1
tan
=
ln
03
+
2
3
tan
or
01
/
(12.6)
With
3
1
=
/2,
q + c
/tan
=
*
1
(1+sin
) and
q
0
+
c/
tan
=
*
3
(1
sin
),
see Fig 12.2b, keeping in mind that
01
/
03
=
*
1
/
*
3
, this provides the ultimate
bearing capacity
q = N
q
q
0
+ N
c
c
(12.7a)
1
sin
tan
with
N
q
=
e
and
N
c
= (N
q
- 1)/tan
(12.7b)
1
sin
=
30
o
, the bearing capacity factors become
N
q
=
18.4 and
N
c
=
30.1. In reality, the specific soil weight
For the present case, with
affects the pattern of slip
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