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# Nonlinear Characteristics of Floating Piles Under Rotating Machine Induced Vertical Vibration

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This study presents a finite element based approach to evaluate the linear and nonlinear frequency–amplitude response of floating piles subject to rotating machine induced vertical vibrations. A Matlab program is developed to compute the response of a single pile with floating tip condition for a linear and two nonlinear soil models. The variation of complex soil stiffness parameters with frequency has also been presented using different boundary zone parameters (shear modulus reduction ratio, thickness ratio and damping ratio). A detailed investigation of soil–pile system stiffness and damping parameters has been done considering different values of soil–pile separation lengths and boundary zone parameters. To verify the effectiveness of this proposed approach vertical vibration tests were conducted in the field on a single pile of diameter 0.114 m and length of 2.85 m by constructing floating tip condition. The frequency–amplitude response obtained from field vibration test has been compared with the theoretical results for all the soil models. From the comparison results it is observed that the proposed theory can predict the nonlinear response of floating piles very efficiently with proper inclusion of boundary zone parameters and soil–pile separation lengths.
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1 23
Geotechnical and Geological
Engineering
An International Journal
ISSN 0960-3182
Geotech Geol Eng
DOI 10.1007/s10706-015-9885-5
Nonlinear Characteristics of Floating Piles
Under Rotating Machine Induced Vertical
Vibration
Sumeet Kumar Sinha, Sanjit Biswas &
1 23
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ORIGINAL PAPER
Nonlinear Characteristics of Floating Piles Under Rotating
Machine Induced Vertical Vibration
Sumeet Kumar Sinha .Sanjit Biswas .
Received: 23 August 2014 / Accepted: 18 April 2015
ÓSpringer International Publishing Switzerland 2015
Abstract This study presents a ﬁnite element based
approach to evaluate the linear and nonlinear frequen-
cy–amplitude response of ﬂoating piles subject to
rotating machine induced vertical vibrations. A Matlab
program is developed to compute the response of a
single pile with ﬂoating tip condition for a linear and
two nonlinear soil models. The variation of complex
soil stiffness parameters with frequency has also been
presented using different boundary zone parameters
(shear modulus reduction ratio, thickness ratio and
damping ratio). A detailed investigation of soil–pile
system stiffness and damping parameters has been
done considering different values of soil–pile separa-
tion lengths and boundary zone parameters. To verify
the effectiveness of this proposed approach vertical
vibration tests were conducted in the ﬁeld on a single
pile of diameter 0.114 m and length of 2.85 m by
constructing ﬂoating tip condition. The frequency–
amplitude response obtained from ﬁeld vibration test
has been compared with the theoretical results for all
the soil models. From the comparison results it is
observed that the proposed theory can predict the
nonlinear response of ﬂoating piles very efﬁciently
with proper inclusion of boundary zone parameters
and soil–pile separation lengths.
Keywords Floating pile Vertical vibration Soil–
pile interaction Nonlinear dynamic response Soil–
pile separation length Complex soil stiffness
1 Introduction
Floating piles carry superstructure load by developing
shear strength through friction between pile and
surrounding soil. Most often during installation of
bored cast in situ piles, some amount of bentonite
slurry (polymud) remain at the bottom of the boreholes
even after construction of piles. This situation leads to
improper contact between the pile tip and soil, which
results the reduction in soil–pile stiffness under
dynamic loading. Therefore there is a need to establish
a method for calculating accurate dynamic response of
ﬂoating piles for safe and economic design.
Precise evaluation of the dynamic response of a pile
foundation involves detailed analysis of stiffness and
damping parameters of soil–pile system considering
actual ﬁeld conditions. In past years several re-
searchers (Kobori et al. 1975; Nogami and Novak
1976; Novak 1974; Tajimi 1966) contributed to the
formulation and development of the continuum ap-
proach to analyse the dynamic response of pile
S. K. Sinha S. Biswas (&)B. Manna
Department of Civil Engineering, IIT Delhi,
New Delhi 110016, India
e-mail: sanjit.jal@gmail.com
S. K. Sinha
e-mail: sumeet.kumar507@gmail.com
B. Manna
e-mail: bmanna@civil.iitd.ac.in
123
Geotech Geol Eng
DOI 10.1007/s10706-015-9885-5
Author's personal copy
foundations. The continuum approach was mainly
described by Novak and Aboul-Ella (1978) to derive
the impedance function of single piles in layered soil
medium considering linear elastic (Novak et al. 1978)
behaviour of soil. Later Novak and Sheta (1980)
included a weak cylindrical zone around the pile with
reduced soil shear modulus than outer zone to
incorporate an approximate nonlinear phenomenon
of soil–pile system subjected to rotating machine
induced vibrations. One of the assumptions of this
method was the negligence of the inner weak zone soil
mass to avoid the complex calculations involving
wave reﬂections between interface of inner zone and
outer zone. To overcome this assumption Veletsos and
Dotson (1988) evaluated a process to include inner
zone soil mass. Han and Sabin (1995) proposed a soil
model with parabolic variation of soil shear modulus
in the inner weak zone considering the mass of the soil
to provide a non-reﬂective surface at inner soil
boundary. Based on this model Han (1997) performed
a parametric study on boundary zone parameters under
vertical vibration. Manna and Baidya (2010) used the
Novak’s continuum approach to evaluate the nonlin-
ear response of model piles and compared the results
with vertical vibration ﬁeld test response.
To test the efﬁciency of the proposed analytical
methods, experimental study is essential. Among
different type of forced vibration testing, small scale
pile tests are very popular because it is inexpensive,
relatively easy to perform and less time consuming.
Novak and Grigg (1976) conducted dynamic tests on
small piles in the ﬁeld and compared the test results
with the numerical solution of Novak’s plain strain
model (1974). Novak (1977) presented an ap-
proximate analytical solution for the vertical response
of ﬂoating piles. Using this method the impedance of
the soil–pile system were obtained and compared with
the dynamic ﬁeld test results. The comparison showed
that the motion of the tip could not be neglected unless
it was an extremely long pile. Han and Novak (1988)
conducted dynamic tests on large scale model piles in
the ﬁeld subjected to strong vertical excitation. The
measured response curves were compared with the
theoretical curves calculated using continuum ap-
proach. Burr et al. (1997) performed the dynamic tests
on model pile groups embedded in different types of
soil.
From the literature review it is understandable that
both the analytical and experimental study on ﬂoating
piles under rotating machine induced vertical vibra-
tion are rarely performed. Hence, in this paper a
Matlab program SPVVA 1.0 (Single-Pile Vertical
Vibration Analyzer) has been developed based on the
continuum approach of Novak and Aboul-Ella (1978)
to evaluate the dynamic response of ﬂoating piles
considering different types of soil stiffness models
(Han and Sabin 1995; Novak et al. 1978; Novak and
Sheta 1980). To verify the applicability of the
proposed model, dynamic forced vibration tests were
performed in the ﬁeld on a hollow steel pipe
(L/D
p
=25) by constructing ﬂoating tip condition. A
parametric study has been also performed to monitor
the effect of the inﬂuencing parameters on the stiffness
and damping of pile foundations under vertical
vibration.
2 Characteristics of the Soil–Pile System
The dynamic nonlinear vertical response of pile
foundation is governed by complicated interaction
phenomenon between soil and pile. The interaction
between the pile and the surrounding soil develops
stiffness of the soil–pile system and generate
damping through radiation and material (hysteresis)
damping. This phenomenon can be modeled by the
complex impedance functions where the real and
imaginary parts represent the stiffness and damping
of the soil–pile system respectively. This section of
the paper presents a model for evaluating the
dynamic response of ﬂoating piles which derives
the soil–pile complex stiffness based on the con-
tinuum approach and the ﬁnite element method. The
variation of complex stiffness parameters with
frequency obtained using different soil models are
also presented in this section.
2.1 Stiffness of the Soil Medium
The variation of soil properties along depth and radial
direction plays a signiﬁcant role in determining the
stiffness of the soil medium. These factors along with
the application of large strain on the pile adjacent to
soil during vibration lead to the consideration of
complex soil behavior for the evaluation of complex
soil stiffness. In this section the complex stiffness of
one linear and two nonlinear soil models are derived
considering small defection of plane strain models.
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The derivation assumes the soil medium to be
composed of horizontal layers which are homogenous,
isotropic, linear viscoelastic having frequency inde-
pendent material damping. However the pile is
considered to be rigid, circular, mass less inﬁnitely
long cylinder with the boundary between the two
medium. The schematic diagram of the soil models are
shown in Fig. 1.
2.1.1 Linear Model
In this model the dynamic complex stiffness function
of soil is calculated considering no slippage between
the pile and soil. The soil properties are taken as
constant for each layer but may differ for individual
layers. The solution of the equation of motion of this
viscoelastic medium (Fig. 1a) with hysteresis type
damping is obtained from the solution of wave
propagation equations of elastic medium (Novak
et al. 1978). The complex soil stiffness in vertical
direction is derived as
KL¼2pGsa
0ð1þDsÞI1a
0

I0a
0
 ð1Þ
a
0¼a0i.ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þiDs
pð2Þ
Finally the vertical soil stiffness can be expressed as
KL¼GsSv1a0;Ds
ðÞþiSv2a0;Ds
ðÞ½ð3Þ
where S
v1
and S
v2
are the real and imaginary part of the
soil stiffness (K
L
) respectively in vertical direction
which depends upon shear modulus of soil (G
s
), soil
damping (D
s
) and dimensionless frequency parameter
a
0
. Here a
0is the complex dimensionless frequency
and I
0
and I
1
are the modiﬁed Bessel function of
second kind with order 0 and 1 respectively.
2.1.2 Nonlinear Model I
In this approximate nonlinear model, the soil stiffness
is formulated based on the model given by Novak and
Sheta (1980). To account for the soil nonlinearity and
slippage between the pile and soil, a cylindrical
Fig. 1 Schematic diagram of soil models with boundary zone: aLinear Model, bNonlinear Model I, cNonlinear Model II
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annulus of softer soil (Fig. 1b) is taken around the pile
which has less soil modulus and more damping than
outer zone. The inner zone is considered as mass less
(q=0) to avoid wave reﬂection from the interface of
the inner and outer zone. The vertical soil stiffness is
deduced by
KNS ¼2pGws=Gsð1þiDws Þ
ln Ri
RiþTws þGwsð1þiDws Þ
Gsð1þiDsÞ
I0a
0
ðÞ
a
0I1a
0
ðÞ
ð4Þ
After separating the real and imaginary part the
equation can be expressed as
KNS ¼GsSv1a0;Gws
Gs
;Tws
Ri
;Ds

þiSv2a0;Gws
Gs
;Tws
Ri
;Ds
ð5Þ
where S
v1
and S
v2
are the real and imaginary part of the
complex soil stiffness (K
NS
) respectively in vertical
direction and R
i
is the radius of the pile. The soil
stiffness and damping depends on shear modulus
reduction ratio (G
ws
/G
s
), thickness ratio (T
ws
/R
i
), inner
weak zone damping (D
ws
) and dimensionless frequen-
cy parameter (a
0
) where G
ws
and T
ws
are the shear
modulus and thickness of the inner zone (weak soil
medium) respectively.
2.1.3 Nonlinear Model II
A more realistic model with smooth parabolic
variation of the soil shear modulus from inner weak
zone to outer zone has been proposed by Han and
Sabin (1995) which provides non-reﬂective wave
surface at the interface of the inner and outer soil zone
(Fig. 1c) by considering the soil mass of the cylindri-
cal boundary zone. The complex stiffness of the soil in
vertical direction is given by
KH¼2pa Gws
Gs
ð1þi2DwsÞdw
dxx¼x1
ð6Þ
a¼x1Ri=Tws ð7Þ
x1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1Gwsð1þi2Dws Þ=Gsð1þi2DsÞ
pð8Þ
wðxÞ¼X
n¼1
n¼0
Anxnð9Þ
A0¼dI0ðhR0=RiÞ;A1¼dI0ðhR0=RiÞ;
A2¼ðabA0þA1Þ=2að10a;b;cÞ
h¼a0i.ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þi2Ds
p;d¼1=wðx1Þð11a;bÞ
An¼nðn1Þ2An1þa½bþðn2Þðn1ÞAn2
½bþðn3Þðn1ÞAn3o=nðn1Það12Þ
a¼aR0=Ri;b¼ðh=aÞ2ð13a;bÞ
After simplifying the vertical stiffness, it can be
expressed as
KH¼GsSv1a0;Gws
Gs
;Tws
Ri
;Ds

þiSv2a0;Gws
Gs
;Tws
Ri
;Ds

ð14Þ
where S
v1
and S
v2
are the real and imaginary part of
soil stiffness (K
H
) respectively in vertical direction, h
is the complex dimensionless frequency parameter, a
is a constant and a,b,x
1
,dare the numerical notations.
2.1.4 Comparison of Soil Complex Stiffness
Parameters
The soil stiffness (S
v1
) and damping (S
v2
) parameters
obtained from Linear Model, Nonlinear Model I and
Nonlinear Model II are compared for different
variations of boundary zone parameters (G
ws
/G
s
,T
ws
/
R
i
and D
ws
). In Nonlinear Model I the inner medium is
considered as mass less whereas in Nonlinear Model II
the mass density of inner medium is taken equal to the
mass density of the outer medium.
The comparison of soil stiffness (S
v1
) and damping
(S
v2
) parameters for different values of shear modulus
reduction ratio (G
ws
/G
s
=0.2, 0.5, 0.8) is shown in
Fig. 2. From the ﬁgure it can be noted that the stiffness
parameter (S
v1
) and damping parameter (S
v2
) are
increased with the increase in shear modulus reduction
ratio (G
ws
/G
s
) for Nonlinear Model I whereas for
Nonlinear Model II the trend is quite opposite. It is
also well observed that the soil stiffness and damping
parameters are very sensitive to shear modulus
reduction ratio (G
ws
/G
s
) for both nonlinear models
though the parameters show large difference in
magnitudes with the change in shear modulus
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reduction ratio (G
ws
/G
s
) for the case of Nonlinear
Model I than Nonlinear Model II. The change in
magnitudes of the parameters is more prominent with
the reduction in shear modulus ratio. The larger value
of stiffness and damping parameters of Nonlinear
Model II is observed than Nonlinear Model I because
of the consideration of inner weak zone soil mass. The
comparison curves of soil stiffness (S
v1
) and damping
(S
v2
) parameters for different values of thickness ratio
(T
ws
/R
i
=0.5, 0.75, 1.0) and inner zone damping
(D
ws
=0.2, 0.3, 0.4) are also presented in Figs. 3and
4respectively. It is found from Fig. 3that the
magnitude of stiffness parameter (S
v1
) and damping
parameter (S
v2
) are decreased as the value of thickness
ratio (T
ws
/R
i
) increases. However in Fig. 4the stiffness
parameter (S
v1
) is decreased and damping parameter
(S
v2
) is increased with the increase in inner zone
damping ratio (D
ws
). Both the comparison curves
show no signiﬁcant change in magnitude with the
change in thickness ratio (T
ws
/R
i
) and inner zone
damping ratio (D
ws
). In all the comparison curves as
the frequency increases the stiffness parameter (S
v1
)
and damping parameter (S
v2
) are increased but in the
case of Nonlinear Model II the damping parameter
(S
v2
) is decreased and shows very high value at low
frequency level. The Linear Model shows com-
paratively larger stiffness (S
v1
) and damping pa-
rameter (S
v2
) values and remains constant for a
particular damping value (D
s
). It is observed from
the all the ﬁgures that the Nonlinear Model II shows
lower stiffness and higher damping parametric values
than the other two soil models due the consideration of
the weak zone soil mass in Nonlinear Model II. On the
other hand, the Nonlinear Model I produces lower
parametric values in general than Linear Model and
this is because of the introduction of weak boundary
zone in Nonlinear Model I. However, in the case of
Nonlinear Model I the rate of change in stiffness
parameter values are more than Linear Model and as a
result the higher stiffness parameter values are
obtained by Nonlinear Model I in higher frequency
level.
Fig. 2 Comparison of
complex stiffness
parameters of soil obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, D
ws
=0.3, T
ws
/
R
i
=0.75): asoil stiffness
parameter (S
v1
), bsoil
damping parameter (S
v2
)
Fig. 3 Comparison of
complex stiffness
parameters of soil obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, D
ws
=0.3, G
ws
/
G
s
=0.5): asoil stiffness
parameter (S
v1
), bsoil
damping parameter (S
v2
)
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2.2 Impedance of the Soil–Pile System in Layered
Medium
The stiffness of the soil–pile system is complex, with
real (in phase) part describing the true stiffness and
imaginary (out of phase) part as the total damping
have been proposed to model the complex soil–pile
behaviour, in which ﬁnite element approach offers
simplicity with great versatility. The pile foundation
resists the load by developing the soil–pile interaction
phenomenon which produces the reaction from the
surrounding soil to the pile elements. The stiffness and
damping of the soil–pile system is calculated by
dividing the pile into two nodded one dimensional
elements and solving simultaneous equations of
motions derived from each of the pile elements
(Novak and Aboul-Ella 1978).
2.2.1 Elemental Stiffness Matrix
The impedance function of the pile foundation in the
composite medium is computed from the combination
of elemental stiffness matrixes. The elemental stiff-
ness matrix is derived for each pile element by
considering homogeneous, vertical prismatic elements
extending between the interfaces of each layer as
shown in Fig. 5. The properties of each element are
fully described by its complex stiffness matrix which
includes the properties of both pile and soil. The
embedded elemental reactions can be derived by the
differential equation of motion in the vertical direction
considering the vertical amplitude of defection as
v(z,t).
l
vþc_
dz2þGs½Sv1þiSv2v¼0ð15Þ
where l=mass of the pile per unit length; _
v=time
derivative; c=coefﬁcient of pile internal damping;
E
p
=Young’s modulus of pile material and
A=cross-sectional area of the pile. Assuming the
Fig. 4 Comparison of
complex stiffness
parameters of soil obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, G
ws
/G
s
=0.5,
T
ws
/R
i
=0.75): asoil
stiffness parameter (S
v1
),
bsoil damping parameter
(S
v2
)
Fig. 5 Pile embedded in layered soil medium with node
numbering of elements
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harmonic motion v(z,t)=v(z)e
-ixt
having complex
amplitude v(z), the above equation can be simpliﬁed to
d2v
dz2þk
h

2
v¼0ð16Þ
where, the value kis the equivalent complex frequen-
cy parameter.
k¼hﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
lx
2GsSv1icxþGsSv2
ðÞðÞ
EpA
sð17Þ
The complex amplitude can be written as the
following general equation in which B&Care
integration constants
vðzÞ¼Bcos k
h
þCsin k
h
 ð18Þ
The dynamic stiffness of the pile is derived using
two boundary conditions. Let’s consider a general pile
element inside the soil having a length of h
i
and area of
A
i
(diameter of D
i
) with Young’s modulus of E
p
.As
shown in the Fig. 5, the nodes have been numbered as
iand i?1 along the depth of the element i. The
amplitude of the axial force Fi
vðzÞcan be represented
by the following equation up to the elastic range of pile
material.
Fi
vðzÞ¼EpAi
dvðzÞ
dz ð19Þ
Considering the ﬁrst boundary condition as
v(0) =1 and v(h
i
)=0 i.e. unit displacement at the
node iwith other node i?1 ﬁxed, gives the value of
B=1 and C=-cot(k).
ki;i¼Fi
vð0Þ¼EpAi
hi
kcotðkÞð20Þ
ki;iþ1¼Fi
vðhÞ¼EpAi
hi
kcosecðkÞð21Þ
Similarly considering v(0) =0 and v(h
i
)=1 i.e.
unit displacement at the node i?1 with other node i
ﬁxed, gives the value of B=0 and C=cosec(k).
kiþ1;iþ1¼Fiþ1
vð0Þ¼EpAi
hi
kcotðkÞð22Þ
kiþ1;i¼Fiþ1
vðhÞ¼EpAi
hi
kcosecðkÞð23Þ
Therefore the overall stiffness of an individual pile
member can be formulated in the matrix form as
ki
v¼EpAi
hi
kcotðkÞcosecðkÞ
cosecðkÞcotðkÞ

ð24Þ
The end forces F
v
i
and F
v
i?1
corresponding to end
displacements v
i
and v
i?1
of element iare expressed as
Fi
v
Fiþ1
v

¼ki
v
vi
viþ1
 ð25Þ
2.2.2 Dynamic Vertical Stiffness of Pile
The total stiffness matrix {K
TV
} of the pile can be
integrated by diagonally aligning the elemental stiff-
ness matrix as shown in Eq. 26. The bandwidth of this
stiffness matrix is 2.
KTV ¼
k11 k12
k21 k22 þk22 k23
k32 k33 þk33
..
.
..
.kn1n
knn þknn knnþ1
knþ1nknþ1nþ1
2
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
5
ð26Þ
The complex vertical stiffness of the pile at
different nodes can be derived by
K1
v
K2
v
.
.
.
.
.
.
Knþ1
v
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
;
¼KTV
½
v1
v2
.
.
.
.
.
.
vnþ1
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
ð27Þ
where v
n
is the displacement of the node nand Kn
vis
the corresponding force at node n. The amplitude of
vertical displacement of the pile foundation can be
found directly by the vertical displacement of the pile
head. Hence, the interest is to ﬁnd out the vertical
stiffness of the pile at the topmost pile node. The
stiffness of the soil–pile system (K1
v) is derived as
described below
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K1
v
0
.
.
.
.
.
.
0
8
>
>
>
>
>
<
>
>
>
>
>
:
9
>
>
>
>
>
=
>
>
>
>
>
;
¼KTV
½
1
v2
.
.
.
.
.
.
vnþ1
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð28Þ
K1
v¼1
KTV
½
1
hi
1;1
ð29Þ
The total vertical stiffness of the pile is complex
where, the real part corresponds to the true stiffness k1
s
and the imaginary part corresponds to the equivalent
viscous damping k1
d. Thus the complex vertical
stiffness of single ﬂoating piles can be expressed as
K1
v¼k1
sþixk1
dð30Þ
The dimensionless stiffness and damping pa-
rameters F
v1
and F
v2
are calculated as
Fv1¼real K1
v

R1
EpA1
ð31Þ
Fv2¼img K1
v

Vn
s=x
EpA1
ð32Þ
where A
1
and R
1
are the area and radius of the topmost
pile element respectively and Vn
sis the shear wave
velocity of the soil at the bottommost layer.
2.2.3 Comparison of Impedance Parameters
of the Soil–Pile System
The dimensionless soil–pile system stiffness parameter
(F
v1
) and damping parameters (F
v2
) has been compared
for Linear Model, Nonlinear Model I and Nonlinear
Model II for different values of shear modulus reduction
ratio (G
ws
/G
s
), separation length ratio (S
L
/D
P
)and
thickness ratio (T
ws
/R
i
). A model concrete pile of 2 m
length is used for the comparison assuming constant
boundary zone parameters along the pile depth. The
effect of inner zone damping (D
ws
) is comparatively
ignorable in nonlinear models (Fig. 4) and thus the soil
damping values (D
s
=0.1, D
ws
=0.3) are kept con-
stants. To incorporate the separation length in the
analysis the shear modulus reduction ratio is taken as
zero (G
ws
/G
s
=0) in the top most layer division.
The comparison of the dimensionless soil–pile
system stiffness parameter (F
v1
) and damping pa-
rameters (F
v2
) for different values of shear modulus
reduction ratio (G
ws
/G
s
=0.2, 0.5, 0.8) is shown in
Fig. 6. It is found from the curves that the soil–pile
system stiffness (F
v1
) and damping (F
v2
) parameter are
increased for Nonlinear Model I with the increase in
shear modulus reduction ratio (G
ws
/G
s
)butthepa-
rameters are decreased for Nonlinear Model II. It can
also be observed that the stiffness (F
v1
) and damping
(F
v2
) parameter obtained from Nonlinear Model I are
more sensitive than Nonlinear Model II with the change
of shear modulus reduction ratio (G
ws
/G
s
). The com-
parison of the dimensionless stiffness(F
v1
) and damping
(F
v2
) parameters for different values of separation
length ratio (S
L
/D
p
=1.0, 2.0, 3.0) and thickness ratio
(T
ws
/R
i
=0.5, 0.75, 1.0) are presented in Figs. 7and 8
respectively. From the curves it is observed that the
stiffness (F
v1
) and damping (F
v2
) parameters are
decreased with the increase in separation length ratio
(S
L
/D
p
) and thickness ratio (T
ws
/R
i
) though the change in
magnitude is negligible. From all the comparison curves
it is noted that the soil–pile system stiffness (F
v1
)and
damping (F
v2
) parameters are decreased with the
increase in frequency but at higher frequency level it
starts increasing. In addition to that the damping
Fig. 6 Comparison of
impedance parameters of
soil–pile system obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, D
ws
=0.3, G
ws
/
G
s
=0.50, T
ws
/R
i
=0.75):
astiffness parameter (F
v1
),
bdamping parameter (F
v2
)
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parameter (F
v2
) reverses its sequence with the change of
boundary zone parameters (G
ws
/G
s
,S
L
/D
P
,T
ws
/R
i
)after
a certain level of frequency.
The comparison of the soil–pile system stiffness
(F
v1
) and damping (F
v2
) parameters obtained from
nonlinearmodels for a value of dimensionless frequency
(a
0
=0.30) is listed in Table 1. The value of soil–pile
system stiffness (F
v1
) and damping (F
v2
) parameters are
0.0132 and 0.0517 respectively for Linear Model at
a
0
=0.30. From the curves and the table it is observed
that the Nonlinear Model I and Linear Model show
lower and higher value of soil–pile system stiffness
parameter (F
v1
) respectively. However the Nonlinear
Model I and Nonlinear Model II indicate lower and
higher damping parameter (F
v2
) values invariably.
3 Experimental Investigation
3.1 Site Characterization
To verify the efﬁciency of the developed theory forced
vibration tests were performed in the ﬁeld. The test site
was located in between block 2 and 3 at Indian Institute of
Technology Delhi, Hauz Khas, New Delhi, India. To
characterize the site soil condition both in situ test and
laboratory tests were conducted. The standard penetra-
3 m at different interval (Bureau of Indian Standards
1981) and simultaneously soil samples (disturbed and
undisturbed) were also collected from the borehole for
laboratory testing. The laboratory experiments such as
bulk density determination (Bureau of Indian Standards
1970), natural water content determination (Bureau of
Indian Standards 1973), particle size distribution test
(Bureau of Indian Standards 1985a), Atterberg’s limits
tests (Bureau of Indian Standards 1985b) and uncon-
solidated undrained triaxial test (Bureau of Indian
Standards 1993) were carried out to determine the
relevant properties of soil for theoretical analysis. Based
on all the ﬁeld and laboratory test results, the whole
vertical soil proﬁle has been classiﬁed as inorganic clayey
silt with low plasticity (ML-CL) as per Uniﬁed Soil
Classiﬁcation System. The approximate value of Pois-
son’s ratio is considered 0.3. The Young’s modulus (E)of
soil is found out from the correlation (Bowles 1996)
Fig. 7 Comparison of
impedance parameters of
soil–pile system obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, D
ws
=0.3, S
L
/
D
P
=0, T
ws
/R
i
=0.75):
astiffness parameter (F
v1
),
bdamping parameter (F
v2
)
Fig. 8 Comparison of
impedance parameters of
soil–pile system obtained
from Linear Model,
Nonlinear Model I and
Nonlinear Model II
(D
s
=0.1, D
ws
=0.3, S
L
/
D
P
=0, G
ws
/G
s
=0.50):
astiffness parameter (F
v1
),
bdamping parameter (F
v2
)
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E¼400Suð33Þ
where S
u
is the undrained shear strength of soil which
is equal to the cohesion value obtained from uncon-
solidated undrained triaxial test. Form this value of
Young’s modulus ﬁnally the value of shear modulus is
obtained. All the soil properties obtained from the ﬁeld
and laboratory tests are listed in Table 2.
3.2 Dynamic Vertical Response of Test Pile
Vertical vibration tests were performed in the ﬁeld on
a single pile by constructing ﬂoating pile condition.
The mechanical oscillator having two counter rotating
(Lazan type) eccentric masses was used to produce the
harmonic excitation force on the pile. The magnitude
of the exciting force was controlled by adjusting the
eccentric distance between the two masses. When the
mechanical oscillator is driven by a motor, the
oscillator develops unidirectional sinusoidal vibratory
force passing through the centre of the oscillator. The
dynamic force (P) can be expressed as
P¼ðW:eÞ=gx2sin xtð34Þ
where Wis the total weight of eccentric rotating
masses, eis the eccentric distance between the rotating
masses, gis the acceleration due to gravity, xis the
circular frequency of vibration and tis the time.
A hollow steel pipe of length (L) 2.85 m and outer
diameter (D
P
) 0.114 m (thickness =3 mm) has been
used as a pile for dynamic ﬁeld testing. After closing
the bottom of the pipe with a circular plate the pipe
was driven with a SPT hammer in a undersize borehole
which was made by a hand auger of diameter =0.1 m
to ensure a good contact between the pile and soil.
Table 1 Comparison of
impedance parameters of
the soil–pile system for
Nonlinear Model I and
Nonlinear Model II
(a
0
=0.3, D
ws
=0.3 and
D
s
=0.1)
a
F
v1
=stiffness parameter
of the soil–pile system
b
F
v2
=damping
parameter of the soil–pile
system
Variation of
parameters
Nonlinear Model I Nonlinear Model II
F
v1
a
F
v2
b
F
v1
a
F
v2
b
S
L
/D
p
G
ws
/G
s
=0.5 and T
ws
/R
i
=0.75
1.00 0.0103 0.0332 0.0122 0.0704
2.00 0.0096 0.0314 0.0124 0.0661
3.00 0.0089 0.0297 0.0126 0.0619
G
ws
/G
s
S
L
/D
P
=0 and T
ws
/R
i
=0.75
0.20 0.0062 0.0212 0.0164 0.0774
0.50 0.0110 0.0351 0.0118 0.0749
0.80 0.0128 0.0426 0.0079 0.0694
T
ws
/R
i
S
L
/D
P
=0 and G
ws
/G
s
=0.5
0.50 0.0115 0.0383 0.0123 0.0759
0.75 0.0110 0.0351 0.0118 0.0748
1.00 0.0106 0.0325 0.0114 0.0739
Table 2 Properties of soil
proﬁle depth wise Properties of soil Depth
0.0–1.0 m 1.0–2.0 m 2.0–3.0 m
Moisture content (%) 11.33 13.27 13.39
Bulk density (kN/m
3
) 17.05 18.03 18.85
Liquid limit (%) 25.09
Plastic limit (%) 19.02
Grain size distribution (%) Sand =30.30
Silt =58.80
Clay =10.90
Cohesion (kPa) 81.00
SPT-N11 11 16
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Before the installation of pile sufﬁcient amount of
bentonite slurry was put into the borehole to simulate
the ﬂoating tip condition. After installation the pile
was kept in rest for a period of 2 months to overcome
the disturbances caused by the pile installation
process. A pile cap of dimension 0.5 m 90.5 m 9
0.05 m (weight =1 kN) was connected to the pile by
a specially fabricated pile cap connector after reaching
the soil–pile system in its natural condition. Then the
mechanical oscillator assembly (weight =4 kN) was
mounted on the pile cap and followed by 10 steel
plates (weight =0.5 kN 910 =5 kN) on the top of
the mechanical oscillator. The steel plates are mounted
on the top of the pile cap to get well pronounced
resonant peaks within the maximum running limit of
the AC motor. The whole assemble was properly
tightened with the pile cap by four long bolts in such a
way that it could act as a single unit. The oscillator was
connected with a 10 hp AC motor by means of a
ﬂexible shaft. The speed of AC motor was controlled
by a speed control unit for running the oscillator at
different frequencies. To measure the amplitude of
vibration a piezoelectric acceleration pickup was
connected to the centre of the top most plate and it
was further connected to a compatible vibration meter
to display the steady-state amplitude of vibration for a
certain frequency. The complete experimental setup
has been shown in Fig. 9.
Steady-state vertical response of the soil–pile
system was measured for different eccentric moments
(W.e =0.369, 0.735, 1.100, and 1.448 N m) under
the static load (W
s
) of 10 kN. The oscillator was run in
a controlled manner through the motor using the speed
control unit to avoid sudden application of high
magnitude dynamic load within a frequency range of
5.0–45.0 Hz. The frequency of the oscillator was
recorded directly by the frequency indicator of speed
control unit which was connected with the motor and
the corresponding vertical amplitude of vibration was
measured by the vibration meter. The frequency
versus amplitude response curves of the ﬂoating pile
has been presented in Fig. 10 for different eccentric
Fig. 9 Schematic diagram
of vertical vibration test
setup
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moments. From Fig. 10, a single resonant peak is
observed which indicates the response of a single
degree of freedom system. The response curves also
display nonlinearity as the resonant frequencies are
decreased with the increasing excitation intensity and
also the resonant amplitudes are not proportional with
the excitation intensity. However, the differences
between the consecutive resonant frequency values
are found less because of the small differences of the
consecutive eccentric moments.
4 Comparison Between the Experiments
and Numerical Analysis
The test pile has been analyzed with all the soil models
i.e. Linear Model, Nonlinear Model I and Nonlinear
Model II to get the frequency–amplitude response of
the soil–pile system. The value of soil material
damping is assumed to be 0.10 for analysis. The
results obtained from the vertical vibration test and the
numerical analysis is summarized in Table 3for
4.1 Linear Model
The comparison of frequency–amplitude response
obtained from linear analysis and vertical vibration
test results are presented in Fig. 11. It can be well
concluded from the ﬁgure that the theoretical resonant
amplitudes are lower and the resonant frequencies are
higher than the dynamic test results. This can be
explained by the assumption of no weak inner zone
which results in larger stiffness of soil–pile system
which produces larger resonant frequencies and lower
amplitude values. The Linear Model fails to predict
the nonlinear response of the soil–pile systems and
shows same resonant frequency and proportional
amplitude values for all excitation levels.
4.2 Boundary Zone Parameters
To get the nonlinear response of the pile foundation,
the boundary zone parameters like modulus reduction
ratio (G
ws
/G
s
), thickness ratio (T
ws
/R
i
), weak zone
soil damping (D
ws
) and separation length (S
L
) are
considered in nonlinear analysis. The boundary zone
parameters of the weakened zone are adjusted based
on the literature (Elkasabgy and El Naggar 2013;
Han and Novak 1988; Manna and Baidya 2009;
Vaziri and Han 1992) so that theoretical response
curves approach towards dynamic test curves. How-
ever, there is no speciﬁc guideline available in the
literature to estimate the actual value of nonlinear
parameters for different levels of eccentric moment
with depth and soil conditions. The depth of the
separation length has been presumed as 0.15–0.24 m
from ground level for excitation intensity of
0.369–1.448 N m respectively. Figure 12 shows the
variation of boundary zone parameters (G
ws
/G
s
,T
ws
/
R
i
and D
ws
) with depth for different excitation levels.
It is observed from the curves that the shear modulus
reduction ratio (G
ws
/G
s
) is reduced with the increase
in excitation intensity whereas the thickness ratio
(T
ws
/R
i
) and weak zone soil damping (D
ws
) are
increased. As the depth increases the values of shear
modulus reduction ratio (G
ws
/G
s
) are increased but
the thickness ratio (T
ws
/R
i
) and weak zone soil
damping (D
ws
) are decreased for all excitation level.
In this study, the modulus reduction ratio (G
ws
/G
s
)
varies from 0.15 to 0.51, the thickness ratio (T
ws
/R
i
)
varies from 0.53 to 0.90 and the weak zone soil
damping (D
ws
) varies from 0.13 to 0.35 for different
eccentric moments (0.369–1.448 N m) and for dif-
ferent depths (0–3.0 m). It is also found that the
nonlinear response of pile foundations is mainly
inﬂuenced by the geometric nonlinearity (soil–pile
separation) and material nonlinearity (shear modulus
reduction).
Fig. 10 Frequency versus amplitude response curves obtained
from vertical vibration test
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4.3 Nonlinear Model I
After incorporating the boundary zone parameters, the
nonlinear analysis is performed with the Nonlinear
Model I. The frequency–amplitude response of the
pile foundation obtained from the analysis is com-
pared with the dynamic test result and presented in
Fig. 13. From the comparison it is observed that the
Nonlinear Model I shows a very well deﬁned match
with the test results. The nonlinearity is also exhibited
from the ﬁgure by increasing disproportional ampli-
tudes and decreasing resonant frequency with the
increasing exaction levels.
4.4 Nonlinear Model II
The nonlinear analysis of the ﬂoating pile is also
performed with the Nonlinear Model II using same
boundary zone parameters. The obtained frequency–
amplitude response is compared with the dynamic test
results and shown in Fig. 14. It is found from the ﬁgure
that the Nonlinear Model II produces very high
resonant frequency and less amplitude values as
compared to dynamic test results. As it is found from
Figs. 6,7and 8that the stiffness (F
v1
) and damping
(F
v2
) parameters of Nonlinear Model II show relative-
ly higher value than other soil models at lower
frequency level, the analysis produces large resonant
frequency and lower amplitude values.
Table 3 Comparison of
results obtained from
vertical vibration test,
Linear Model, Nonlinear
Model I and Nonlinear
Model II
Eccentricity (W.e) Resonant frequency (Hz) Vertical amplitude (mm)
Test results
0.369 33.80 0.203
0.735 32.37 0.356
1.110 31.58 0.512
1.448 30.33 0.675
Linear Model analysis
0.369 37.25 0.1624
0.735 37.25 0.3234
1.110 37.25 0.4840
1.448 37.25 0.6372
Nonlinear Model I analysis
0.369 33.00 0.1769
0.735 32.25 0.3516
1.110 31.25 0.5205
1.448 30.25 0.6795
Nonlinear Model II analysis
0.369 67.75 0.1141
0.735 67.55 0.235
1.110 67.30 0.3611
1.448 67.00 0.4875
Fig. 11 Comparison of frequency versus amplitude response
curves obtained from vertical vibration test and Linear Model
analysis
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4.5 Stiffness and Damping of the Floating Pile
Since Nonlinear Model I predicts the dynamic
response of the ﬂoating pile more accurately than
other two models, the stiffness and damping of soil–
pile system obtained from Nonlinear Model I for
different excitation levels are presented in Fig. 15.
From the ﬁgure it is noted that the stiffness of the soil–
pile system is increased with the increase in frequency
but in the case of damping, it is decreased rapidly as
Fig. 12 Variation of
boundary zone parameters
with depth for different
excitation levels
Fig. 13 Comparison of frequency versus amplitude response
curves obtained from vertical vibration test and Nonlinear
Model I analysis
Fig. 14 Comparison of frequency versus amplitude response
curves obtained from vertical vibration test and Nonlinear
Model II analysis
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the frequency increases for all excitation intensities. It
can also be observed that both the stiffness and
damping are decreased with the increase in excitation
levels. This is because of the growth of the inner weak
soil zone around the pile with the increasing excitation
level. It can be seen from the ﬁgure that the soil–pile
system stiffness is frequency independent at lower
frequency range. This is because at lower frequency
level the dynamic stiffness is close to the static
stiffness of the pile.
The analytical soil–pile system stiffness and damp-
ing for linear model and nonlinear models are
compared in Fig. 16 for higher and lower loading
eccentricities. The Linear Model over estimates the
stiffness and damping values than the Nonlinear
Model I at all frequencies. However the Nonlinear
Model II shows very large stiffness and damping value
than all the other models at low frequencies and it
starts converging with other models at high frequency
level. As the frequency increases the damping values
are rapidly decreased because of the conversion of
circular frequency (x) and frequency independent soil
material damping (b) to the equivalent viscous damp-
ing coefﬁcient (c)as
c¼2b=xð35Þ
The stiffness values obtained from the Linear
Model and Nonlinear Model I are increased as the
frequency increase whereas it is decreased for the
Nonlinear Model II with the increase in frequency.
5 Conclusions
The principle objective of this present study is to
propose a ﬁnite element model to evaluate the
frequency–amplitude response of the ﬂoating piles
subject to rotating machine induced vertical vibra-
tions. The study includes the characterization of a
linear and two nonlinear soil models for different
variations of boundary zone parameters and soil–pile
separation lengths. A vertical vibration ﬁeld test was
also conducted on a single ﬂoating pile and based on
that results the effectiveness of the proposed model
has been monitored for different types of soil model.
The major conclusions of this present study are listed
below.
1. From both the dynamic test and analysis results it
is observed that the response of the single ﬂoating
pile under rotating machine induced vertical
vibration is nonlinear.
2. Prediction of a well deﬁned combination of
separation length and boundary zone parameters
for different eccentric levels is the key aspect to
Fig. 15 Stiffness and damping of soil–pile system obtained
from Nonlinear Model I
Fig. 16 Comparison of
impedance of soil–pile
system obtained from Linear
Model, Nonlinear Model I
and Nonlinear Model II:
astiffness, bdamping
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perform the theoretical nonlinear analysis
accurately.
3. The ﬁndings of this study prove the acceptance of
the Nonlinear Model I over Linear and Nonlinear
Model II for modeling single ﬂoating piles under
vertical vibration.
4. As the eccentric moment increases, the stiffness
and damping of the soil–pile system decreases due
of the extent of weakened soil zone around the
pile.
5. For Nonlinear Model II, the stiffness values are
higher than Linear and Nonlinear Model I and
they decrease with the increase in frequency in
contrast to the other soil models.
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... Yang et al. (2009) obtained analytical and semi-analytical solutions for single piles under vertical harmonic load. Sinha, Biswas, and Manna (2015) conducted dynamic forced vertical vibration tests on hollow steel pipe pile constructed in the field with floating tip condition. Bhowmik, Baidya, and Dasgupta (2016) investigated the dynamic vertical response of three single vertical piles with different lengths both numerically and experimentally. ...
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... However, the estimation of precise boundary zone parametric values is difficult without the help of experimental results due the lack of guidelines in the literature. Therefore, the values of boundary zone parameters are achieved by trial and error method based on the ranges reported in various literatures (Han and Novak 1988;Vaziri and Han 1992; Elkasabgy and Naggar 2013; Sinha et al. 2015). For different eccentric moments, the values of soil parameters of the weakened zone are varied so that the nonlinear analytical response curves approach towards the coupled vibration test responses. ...
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... Later, Han and Sabin [6] proposed a soil model with parabolic variation of boundary zone shear modulus of soil to provide non-reflective wave interface. Sinha et al. [7] performed parametric study of different soil models using continuum approach of Novak [2] and found that the soil model of Novak and Sheta [5] was very efficient to determine the nonlinear responses of single piles under machine induced vibrations. ...
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