156 Ground improvement by deep vibratory methods
∆
=
−
(
) (
)
A A
A A
/
/
c
c
1
1
1
(4.55)
with µ S = 0.33
A
K
K
K
c
a c
m ax
ac
ac
max
A





 = −
⋅
⋅
− +
⋅ ⋅
−
± ⋅
⋅
⋅
− +
1
4
2 5
2 4
1
1
2
4
2
(
)
(
)
(
)
β
β
5 5
4
1
16
1
4
1
2
⋅
−





 +
⋅
⋅
−
⋅
−
K
K
K
ac
ac
max
ac
(
)
β
(4.56)
On the other hand, the improvement factor β 1  so calculated, which is
mainly based on the assumption of a lateral support of the column by the
surrounding soil, does not take into account the increasing support of the
soil with depth t. Therefore, the improvement factor is increased by introducing the depth factor f t to
β
β
2
t
1
= ⋅
f
(4.57)
where
f
K
tp
t
0c
0c
s
c
/
/
= +
−
⋅ ∑ ⋅ ∆
1
1
1
[(
)
] (
)
K
γ
(4.58)
p
p
A A p p
c
c
c
c
s
/
/ / /
=
+ −
A A [(
) (
)]
1
(4.59)
p
p
f
A A
K f
A A
c
s
s
c
ac
s
c
=
+
⋅
1 2
/
( , / )
( , / )
µ
µ
(4.60)
K 0c
c
= −
1 sin ϕ
(4.61)
The factor f t is limited by
f t ≤
E E
p p
c
s
c
s
/
/
(4.62)
and the improvement factor β must not rise above β lim given by
β β
2
1
1
≤
= +
⋅
−






lim
A
E
E
c
c
s
A
(4.63)
The above equations are best programmed, for example, in a spreadsheet,
together with the equations for classical settlement calculation under load.
The results of the Priebe analysis are presented in Table 4.6. Ground
pressure in the middle of the embankment is calculated by using
p =
⋅
=
14
21 8
305
m
kN/m
k N/m
3
2
.
(4.64)