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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 &

Bappaditya Manna

1 23

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ORIGINAL PAPER

Nonlinear Characteristics of Floating Piles Under Rotating

Machine Induced Vertical Vibration

Sumeet Kumar Sinha .Sanjit Biswas .

Bappaditya Manna

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.ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þiDs

pð2Þ

Finally the vertical soil stiffness can be expressed as

KL¼GsSv1a0;Ds

ðÞþiSv2a0;Ds

ðÞ½ð3Þ

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

x1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1Gwsð1þi2Dws Þ=Gsð1þi2DsÞ

pð8Þ

wðxÞ¼X

n¼1

n¼0

Anxnð9Þ

A0¼dI0ðhR0=RiÞ;A1¼dI0ðhR0=RiÞ;

A2¼ðabA0þA1Þ=2að10a;b;cÞ

h¼a0i.ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þi2Ds

p;d¼1=wðx1Þð11a;bÞ

An¼nðn1Þ2An1þa½bþðn2Þðn1ÞAn2

½bþðn3Þðn1ÞAn3o=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

under dynamic loading. Various approximate methods

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_

vEpAd2v

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.

k¼hﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

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

vðzÞ¼EpAi

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

vðhÞ¼EpAi

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

vðhÞ¼EpAi

hi

kcosecðkÞð23Þ

Therefore the overall stiffness of an individual pile

member can be formulated in the matrix form as

ki

v¼EpAi

hi

kcotðkÞcosecðkÞ

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-

tiontests(SPT)wasperformedintheﬁelduptoadepthof

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

different loading eccentricities.

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.

References

Bowles JE (1996) Foundation analysis and design, 5th edn. The

McGraw-Hill Companies Inc, New York

Bureau of Indian Standards (1970) Classiﬁcation and identiﬁ-

cation of soils for general engineering purposes. IS 1498,

Manak Bhavan, New Delhi, India

Bureau of Indian Standards (1973) Determination of water

content. IS 2720 (Part 2), Manak Bhavan, New Delhi, India

Bureau of Indian Standards (1981) Method for standard

penetration test of soils. IS 2131, Manak Bhavan, New

Delhi, India

Bureau of Indian Standards (1985a) Grain size analysis. IS

2720(Part 4), Manak Bhavan, New Delhi, India

Bureau of Indian Standards (1985b) Determination of liquid and

plastic limit. IS 2720 (Part 5), Manak Bhavan, New Delhi,

India

Bureau of Indian Standards (1993) Determination of the shear

strength parameters of a specimen tested in unconsolidated

undrained triaxial compression without the measurement

of pore water pressure. IS 2720 (Part 11), Manak Bhavan,

New Delhi, India

Burr JP, Pender MJ, Larkin TJ (1997) Dynamic response of

laterally excited pile groups. J Geotech Geoenviron Eng

ASCE 123(1):1–8

Elkasabgy M, El Naggar MH (2013) Dynamic response of

vertically loaded helical and driven steel piles. Can Geo-

tech J 50:521–535

Han YC (1997) Dynamic vertical response of piles in nonlinear

soil. J Geotech Geoenviron Eng ASCE 123(8):710–716

Han Y, Novak M (1988) Dynamic behavior of single piles under

strong harmonic excitation. Can Geotech J 25:523–534

Han YC, Sabin GCW (1995) Impedances for radially inhomo-

geneous viscoelastic soil media. J Eng Mech Div ASCE

121(9):939–947

Kobori T, Minai R, Baba K (1975) Dynamic interaction between

an elastic cylinder and its surrounding visco-elastic half-

space. In: Proceedings of the 25th Japan national congress

of applied mech, University of Tokyo Press, Tokyo, Japan,

pp 215–226

Manna B, Baidya DK (2009) Vertical vibration of full-scale

pile—analytical and experimental study. J Geotech

Geoenviron Eng ASCE 135(10):1452–1461

Manna B, Baidya DK (2010) Dynamic nonlinear response of

pile foundations under vertical vibration-theory versus

experiment. Soil Dyn Earthq Eng 30:456–469

Nogami T, Novak M (1976) Soil–pile interaction in vertical

vibration. Int J Earthq Eng Struct Dyn 5(3):277–293

Novak M (1974) Dynamic stiffness and damping of piles. Can

Geotech J 11:574–598

Novak M (1977) Vertical vibration of ﬂoating piles. J Eng Mech

Div 103(1):153–168

Novak M, Aboul-Ella F (1978) Impedance functions of piles in

layered media. J Eng Mech Div 104(3):643–661

Novak M, Grigg RF (1976) Dynamic experiments with small

pile foundations. Can Geotech J 13(4):372–385

Novak M, Sheta M (1980) Approximate approach to contact

problems of piles. In: O’Neill M (ed) Proceedings of the

dynamic response of pile foundations: analytical aspects,

New York, ASCE, pp 53–79

Novak M, Nagomi T, Aboul-Ella F (1978) Dynamic soil reac-

tion for plain strain case. J Eng Mech Div ASCE

104(EM4):953–959

Tajimi H (1966) Earthquake or foundation structures. Report of

the Faculty of Science and Engineering, Nihon University,

Tokyo City, Japan, pp 1.1–3.5

Vaziri H, Han Y (1992) Nonlinear vibration of pile groups under

lateral loading. Can Geotech J 29:702–710

Veletsos AS, Dotson KW (1988) Vertical and torsional vibra-

tion of foundations in inhomogeneous media. J Geotech

Geoenviron Eng ASCE 114(9):1002–1021

Geotech Geol Eng

123

Author's personal copy