Geoscience Reference
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D = c
/2(
kM
)
½
=
c
/2
M
2
1
(12.51)
/k
is constant (residual
deformations under cyclic loading are independent of the frequency), which
renders the damping ratio into a constant
Hysteretic damping assumes that the factor
c
2
c
c
2
k
c
2
k
c
2
1
1
1
D
h
(12.51)
2
k
2
k
2
k
2
kM
2
kM
k
/
M
kM
1
Hence, the damping ratio is constant, independent of the speed or frequency of
the loading. There is no critical damping in soils with hysteretic damping.
application 12.1
A steel pipe pile with length
H =
30
m, radius
R =
0.7
m and wall thickness
d =
4
cm is driven through a thick clay layer with
c
u
=
50 kPa down to a rigid sand bed
with CPT value of
q
c
=
20 MPa. An estimate of the elasticity modulus
E
can be
obtained from the CPT value by the formula
E
an empirical constant.
The bearing capacity
Q
c
consists of the shaft resistance
Q
s
and the point resistance
Q
e
. The shaft friction includes the inner and outer side and using (12.19) one finds
q
c
/
)
with
)
Q
s
=
2
H
(
R + R
d
)
s
c
u
4
RH
x 0.7 x 50 = 9.24 MN
The point bearing
Q
e
is due to the point cross-section, i.e. a cylindrical strip.
Thus
Q
e
Rdq
e
, with
q
e
the local uniform pressure. The maximum value of
q
e
can be related to the CPT value by
q
e
=
1.3
q
c
, where the coefficient 1.3 is a shape
factor (strip load versus point load). Hence, the end-bearing capacity becomes
2
Q
e
=
2
Rdq
e
=
2.6
Rdq
c
=
2.6
0.7 x 0.04 x 20 = 4.67 MN
The total bearing capacity becomes
Q
c
= Q
s
+ Q
e
=
13.9
MN with the main
contribution from the shaft friction.
Using solution (12.24) the displacement can be approximated by subtracting an
opposite cylindrical load with radius
R-d
. Under ultimate loading the vertical
displacement
w
can be approximated by
2
)
/E =
2.6
q
c
d
(1
2
)
/
(
q
c
/
2
)
d
w =
2
q
e
d
(1
)
)
=
2.6
)
(1
For the empirical constant
)
0.1
(for sand) is chosen and for the Poisson ratio
the value
=
0.35 is adopted. Then, the vertical displacement becomes
w =
0.23
d
=
0.009 m, i.e. 9 mm, which is quite acceptable. In compacting soils, the side
friction may give rise to additional settlement. Often a plug of compacted soil may
stick firmly in the pile toe, which changes the situation drastically.
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