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at the sides. The corresponding dissipation work rate is elaborated and results are
presented in Table 9.1 for the 10 subzones. Hence, with
k
z
the apparent cohesion in
the Pleistocene sand layer, the dissipation work rate
W
B,t
in zone B becomes for
Case II
W
B,t
=
?
,t
kh
(½
k'/k
+
3
/
2
k
z
/k
+ 6.0 + 1.875) =
?
,t
kh
(7.875 +
(k'+
3
k
z
)/2
k
)
(9.45)
TABLE 9.1 DISSIPATION WORK IN ZONE B DUE TO THE DOWEL
subdomain
B1
B2
B3
B4
B5 + B6
B7 + B8
B9
B10
¾ k
z
/k
¾ k
z
/k
½+¼ k'/k
½+¼ k'/k
2
2
½
½
W
B,t
/ ?
,t
kh
remark
plastic
plastic
rigid
rigid
plastic
plastic
rigid
rigid
Finally the total dissipation work becomes (
s
A
and
s
C
are the lengths of the
circular slip surface of zone A and zone C, respectively)
I
Without dowel W
,t
=
?
,t
kh
(
s
A
k
A
/kh +
1.875 +
k'
/
k + s
C
k
C
/kh
)
(9.46a)
II
With dowel W
,t
=
?
,t
kh
(
s
A
k
A
/kh +
7.875 +
(
k'+
3
k
z
)/2
k + s
C
k
C
/kh
)
(9.46b)
Here,
k
A
and
k
C
represent an apparent dynamic cohesion. Adopting
k
A
=
8
k
(much higher stress level and stiffness in the dike),
k'=
0 (uplift: no effective
stress),
k
C
= k
, and
k
z
=
2
k
(relatively low because of high pore pressures)
,
yields
with
s
A
=
1
h
and
s
C
=
½
h
I
Without dowel W
,t
/
?
,t
kh =
(13.8
+ 1.875) = 45.2
(9.47a)
II
With dowel W
,t
/
?
,t
kh =
(13.8
+ 10.875) = 54.2
(9.47b)
If the situation without dowel corresponds to an ultimate limit state, then the
effect of the dowel is an improvement of the stability factor to 54.2/45.2 = 1.20.
The stiff dowel method has been checked with centrifuge tests and FEM
simulations (stability factor improvement from 1.00 to 1.10 - 1.15). At present dike
dowelling is accepted as an official improvement method for dike stability in cases
where space for conventional dike enlargement is limited.
The method of rigid-viscoplastic analytic elements can be used by hand and
represents an elegant manner to check failure states. That it works only in two-
dimensions, is a limitation.
application 9.1
The stability and accuracy of numerical calculations depends on the
discretisation in space and time. For the storage equation (consolidation), which is
of the type
f
,
t
=
c
v
f
,
xx
, using forward time approximation and central spatial
difference, the truncation error shows that the solution converges, if
x
)
2
2
c
v
t <
(
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