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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 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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