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Nonlinear dynamic response of floating piles under vertical vibration

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This paper presents the influence of non-linearity on the dynamic response of floating pile foundation subject to vertical vibration. A detailed theoretical investigation of stiffness and damping parameters is made for floating piles with different shear modulus reduction ratio of the weak zone soil. The accuracy of the non-linear analysis for predicting the dynamic response depends upon the choice of boundary zone parameters and the pile separation length. The tip resistance of pile is neglected and comparison of the theoretical curves with the experimental results has been made. In this study it is shown that close agreement between theoretical and measured response curves of floating piles can be achieved by considering precise boundary zone parameter values and separation between pile and soil.
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Nonlinear dynamic response of floating piles under vertical vibration
S. Kumar, S. Biswas & B. Manna
Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi, India
ABSTRACT: This paper presents the influence of non-linearity on the dynamic response of floating pile
foundation subject to vertical vibration. A detailed theoretical investigation of stiffness and damping parameters
is made for floating piles with different shear modulus reduction ratio of the weak zone soil.The accuracy of the
non-linear analysis for predicting the dynamic response depends upon the choice of boundary zone parameters
and the pile separation length. The tip resistance of pile is neglected and comparison of the theoretical curves
with the experimental results has been made. In this study it is shown that close agreement between theoretical
and measured response curves of floating piles can be achieved by considering precise boundary zone parameter
values and separation between pile and soil.
1 INTRODUCTION
Floating Piles are Friction piles which transfer their
load to ground through skin friction. Most often due
to the improper cleaning of boreholes some amount
of bentonite slurry (polymud) remains present at the
bottom of the boreholes even after the installation of
bored cast in situ concrete piles. Due to the improper
contact between pile tip and soil, the end bearing
resistance is not fully developed which is a common
phenomenon in clayey soils. This results in higher
amplitude of vibration under dynamic loading which
can cause massive destruction to the machine founda-
tions. Therefore there is a need to establish a method
to calculate the dynamic nonlinear response of pile
foundations under induced forces of rotation machines
without considering the end bearing resistance of pile
foundation.
The interaction between the pile and soil determines
the load resistance and serviceability of the structure.
This interaction results in development of stiffness
and damping of the pile which can be expressed by
impedance functions (Novak 1974). The total stiff-
ness of the pile is complex where the real (in phase)
part describes the true stiffness and complex (out of
phase) part signifies total damping. Various approx-
imate linear methods have been proposed to model
the non-linear behavior of pile in which finite ele-
ment approach offers accuracy with great versatility.
Matlock et al. (1978) introduced lumped mass models
with nonlinear discrete springs, dashpot, and fric-
tion elements to simulate the nonlinear behaviour
of pile foundation. Novak and Grigg (1978) per-
formed dynamic experiments and investigated fre-
quency response of piles. Novak et al. (1978) proposed
a linear soil model with a constant shear modulus
to predict the dynamic behavior of piles. Later weak
cylindrical zone around the pile with step variation
of shear modulus was proposed by Novak and Sheta
(1980) to get the nonlinear effect of pile soil system.
Han (1997) introduced a parabolic variation of shear
modulus for the weak inner zone of the pile to sim-
ulate a better of pile estimation of nonlinear pile soil
response.
The main objective of the present study is to monitor
the nonlinear dynamic response of floating piles under
vertical vibration. To fulfill this objective the contin-
uum approach analysis described by Novak and Aboul
Ella (1978) has been used to model the response of
single floating piles under vertical vibration of rotat-
ing machines ignoring the end bearing resistance of
pile. To incorporate the nonlinear effect the soil model
describe by Novak and Sheta (1980) assuming a cylin-
drical zone around the pile with less inner zone soil
modulus than outer zone. The variation of dimension-
less stiffness parameter fw1and damping parameter fw2
with dimensionless frequency in floating piles with
varying shear modulus reduction ratio has been inves-
tigated and presented. The effectiveness of the model is
also monitored by comparing the theoretical response
curve with experimental results (Manna and Baidya,
2010) of two different soil-pile separation conditions
ignoring the end bearing resistance of the pile.
2 THEORETICAL STUDY
Among all the methods described by many researchers
for predicting the response of pile foundations under
dynamic load, the continuum approach is most promis-
ing and widely used. In Continuum Approach the
stiffness and damping are calculated by solving the
equation of motion which is derived by dividing
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the whole pile into two nodded one dimensional pile
elements. This method is first presented by Novak and
Aboul Ella (1978). Based on their assumptions and the
model, a computer code SPVVA (Single – Pile Verti-
cal Vibration Analyzer) has been developed in Matlab
(R2012b) to calculate the nonlinear response of single
piles without considering the pile tip reaction.
2.1 Soil stiffness model
The pile foundation resists the load by developing
the soil-pile interaction phenomenon between the sur-
rounding soils and the pile elements. The method
of determine the dynamic soil stiffness has been
described by many researchers. Here the general com-
plex soil stiffness is derived for infinite long rigid
embedded piles under uniform harmonic motions
in vertical direction. The soil stiffness in vertical
direction can be defined by
where, dimensionless frequency a0=r0ω/V 0;r0=
equivalent radius of cross-section of pile; V0=shear
velocity; ω=frequency; i=(1)0.5;Sw1are Sw2real
and imaginary parts of the dimensionless complex soil
stiffness.
The magnitude and variation of soil properties along
depth and radial direction plays a significant role in
determining the dynamic soil stiffness and damping.
These factors along with slippage, lack of bond are
accounted for the inclusion of cylindrical zones around
the pile whose shear modulus and material damping
differ from those of the surround medium. Many mod-
els have been developed in an attempt to simulate the
nonlinear soil behavior.
In this study the soil model describe by Novak and
Sheta (1980) is used to approximate the soil nonlin-
ear behavior. The soil is assumed to be composed of
horizontal layers that are homogenous, isotropic, lin-
ear viscoelastic with frequency independent material
damping. The pile is taken as rigid, circular, mass
less and infinitely long cylinder with the boundary
between the two media.A cylindrical annulus of mass
less (ρ=0) softer soil (an inner weakened zone) is
considered around the pile than the outer medium.
The soil stiffness and damping depends on inner weak
zone damping (Dws); shear modulus reduction ratio
(Gws/Gs); dimensionless frequency parameter (a0)
and thickness ratio (Tws/Dp) and is shown in Fig-
ure 1. Weakened bond and slippage are accounted by
reduced shear modulus and increased damping in the
inner layer. Hence the vertical soil stiffness can be
expressed as
The variation of the dimensionless soil stiffness and
damping parameters with dimensionless frequency for
Figure 1. Schematic representation of cylindrical boundary
zone around the pile and the variation of shear modulus of
soil in the layered soil media.
different shear modulus reduction ratio and assuming
other parameter constant is shown in Figure 2. It is
observed from the curves that the value of Sw1and Sw2
increases with the increase of shear modulus reduction
ratio and frequency. At higher range of shear modu-
lus reduction ratio the dimensionless soil stiffness and
damping parameters are not increasing proportionally.
2.2 Analytical model
The impedance function of the pile in the composite
medium is derived from the combination of element
stiffness matrixes. The element stiffness matrix is
derived for each pile element by considering homoge-
neous, vertical prismatic elements extending between
the interfaces of each layer (Figure 3). The properties
of each element are fully described by its complex
stiffness matrix which includes the properties of both
pile and soil. The embedded element reactions can be
described by the differential equation of motion in the
vertical direction w(z,t)as
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Figure 2. The variation of dimensionless soil stiffness
and damping parameters with dimensionless frequency for
different shear modulus reduction ratio.
Figure 3. Pile embedded in soil strata with nodes
numbering.
where, μ=mass of the pile per unit length; ˙w=time
derivative; c=coefficient of pile internal damping;
Ep=Young’s modulus and A=the cross-sectional area
of the pile respectively.
Assuming the harmonic motion w(z,t)=w(z,t)eiωt
having complex amplitude w(z), the above equation
can be simplified to
where, the value λis the equivalent complex frequency
parameter. Thus the complex amplitude can be writ-
ten as the following general equation in which B,
C=integration constants
2.2.1 Dynamic stiffness matrix
The dynamic stiffness of the pile can be defined by
w(0) =1, w(h)=0 and w(0) =0, w(h)=1 boundary
conditions. Let consider a general element just beneath
the soil layer as shown in the Figure 1 of length h. In the
figure, the nodes have been numbered as iand i+1
down the depth of the element i. The amplitude of the
axial force N(z) can be represented by the following
equation up to the elastic range of pile material.
Considering first as w(0) =1 and w(h)=0 i.e. unit
displacement at the node iwith other node i+1 fixed
give the value of B=1 and C=−cot(λ).
Similarly when w(0) =0 and w(h)=1 i.e. unit dis-
placement at the node i+1 with other node ifixed
give the value of B=0 and C=cosec(λ)
Therefore the overall stiffness of the individual
member can be formulated in the matrix form as
Then the end forces N1and N2corresponding to end
displacements w1and w2are expressed as
The total stiffness of the pile can be integrated from
the elemental matrix diagonally. The bandwidth of the
stiffness matrix is 2.
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2.2.2 Vertical stiffness of the pile
For the vertical direction, the stiffness matrix [Kw]is
assembled and using this stiffness matrix, the complex
vertical stiffness of the pile at different nodes can be
derived by
where, Kn
wis the corresponding force at node nand
wnis the displacement of node nand [Kw] can be
represented by a band matrix of 2 as
The settlement of any structure can be found directly
by the settlement of pile head, so it is of interest to f ind
the stiffness of the pile at tip. Hence the stiffness K1
w
is derived as described below
The stiffness of the pile head is complex where,
the real part corresponds to the true stiffness k1
wand
imaginary part corresponds to the equivalent viscous
damping C1
w. Thus the complex vertical stiffness of
single pile is expressed as
The dimensionless stiffness and damping parame-
ters fw1and fw2can be calculated as
where r,Ep,Aare of the top-most layer and Vsis of the
bottom most layer.
Figure 4. The variation of dimensionless stiffness and
damping parameters with dimensionless frequency for dif-
ferent shear modulus reduction ratio.
3 STIFFNESS AND DAMPING PARAMETERS
The behavior of the pile in vertical vibration can be
characterized by the dimensionless Stiffness parame-
ter fw1and damping parameter fw2. These parameters
depends upon the boundary zone parameters like shear
modulus reduction ratio (Gws/Gs), inner weak zone
damping (Dws), thickness ratio (Tws /Dp) and also on
separation length at pile soil interface (SL).
In this present study a detailed comparison of
dimensionless stiffness and damping parameters (fw1
and fw2) are presented for different values of shear
modulus reduction ratio (Gws/Gs)=0.3, 0.5, 0.8 and
1. Other boundary zone parameters are kept constant
which are assumed to be as 0.25 for thickness ratio;
0.1 for outer soil damping and 0.3 for inner weak zone
damping. The length of the pile is taken as 2m which
is divided into 10 elements.
The variation of fw1and fw2with dimensionless fre-
quency for different shear modulus reduction ratio
under vertical vibration is presented in Figure 4. It
is observed from the figure that the dimensionless
stiffness and damping parameters increases with the
increase of shear modulus reduction ratio. However,
the increment is not very significant for higher range
of shear modulus reduction ratio values. It is also noted
that at lower frequency range, the value of damping
parameter is decreasing rapidly because the viscous
damping of the system is inversely proportional with
frequency. Whereas the stiffness parameter is almost
remain constant with frequency.
4 COMPARATIVE STUDY
Based on Novak’s continuum approach (Novak and
Aboul-Ella, 1978) SPVVA1 (Single – Pile Vertical
Vibration Analyzer Program) is developed in Matlab
954
Figure 5. Comparison of experimental and theoretical
response of single pile without pile-soil separation.
(R2012b) to determine the impedance function of the
single pile subject to vertical vibration.
The results obtained from this compute program
are compared with the experimental results which are
reported by Manna and Baidya (2010) using same soil
as well as boundary zone properties presented in that
paper. In that study the forced vibration tests of sin-
gle piles were conducted at a site located adjacent to
Hangar, at Indian Institute of Technology, Kharagpur
Campus, India. The soil consisted of a 1.2 m layered
stratum of yellow organic silty clay with low plastic-
ity, resting on a thick 1.1 m stratum of brown medium
stiff inorganic claywhich itself rested on gravel mixed-
stiff inorganic clay at a depth of 2.3m. The variation
of shear wave velocity Vsof soil layers was obtained
using established empirical equations.
Here, the frequency-amplitude response single pile
obtained from the developed non-linear computer
program is compared with the experimental results
presented in Manna and Baidya (2010). A single pile
results of l/d=20 and static load (Ws)=10 kN are
used for comparison with and without introducing the
length of separation between pile and soil from the
ground level (SL). The pile tip resistance is assumed to
be zero to stimulate the floating piles condition.
As previously discussed two soil-pile systems are
modeled here, case I – without considering pile-soil
separation length and case II – with consideration of
pile-soil separation length (SL). In total four eccen-
tricities (W.e=0.187, 0.278, 0.366 and 0.450 Nm,
where W=weight of two counter rotating masses of
the oscillator and e=eccentric distance between the
masses) are considered for both the cases and the sep-
aration lengths (SL=0.18Dpfor W.e=0.187 Nm to
2.4Dpfor W.e=0.450 Nm) are taken as described in
the Manna and Baidya (2010) for pile with diame-
ter (Dp) of 0.1 m. The nonlinear dynamic responses
are calculated by introducing a cylindrical weak zone
Figure 6. Comparison of experimental and theoretical
response of single pile considering pile-soil separation.
around the pile and the nonlinear boundary zone
parameters are also considered according to that study.
The comparison curves considering the without and
with separation condition are shown in Figure 5 and
Figure 6 respectively. It can be noted from both the
comparison curves that the theoretical curves are quite
well match with the experimental results though in
higher eccentric moment it differs a little. The non-
linearity is also observed with the phenomenon of
decreasing resonant frequencies with the increasing
excitation intensity and also the amplitudes are not pro-
portional to the excitation intensity. For the modeled
graphs with and without separation length resemble
each other except at higher excitation intensities where
increase in pile separation decreases stiffness of the
pile foundation which results the decrease in the res-
onant frequency of the system. The little discrepancy
in amplitude is may be due to the development of end
bearing in model pile but not considered in the case of
analysis.
5 CONCLUSIONS
The principle objective of the present study is to inves-
tigate the non-linear frequency amplitude response of
floating piles under vertical vibration and to determine
the variation of stiffness and damping parameters fw1
and fw2with frequency. A comparative study between
the theoretical and experimental response curves con-
sidering with and without pile separation is also been
presented. The findings of this study have provided
a clear insight about the soil-pile interaction phe-
nomenon on dynamic responses of floating piles under
vertical vibration. Some major conclusions that can
be made from the theoretical study are summarized
below.
1. The soil-pile interaction has a significant effect
on the dynamics of pile which is itself largely
955
dependent upon the boundary zone parameters and
pile separation length. However the introduction of
separation length leads to an overall reduction in
pile-soil system stiffness.
2. The dimensionless stiffness and damping param-
eters fw1and fw2also depend upon the boundary
zone parameters and soil-pile separation length. In
this study the predicted dimensionless stiffness and
damping parameters follow a very evident trend
under vertical vibration.
3. It is found from the frequency-amplitude compari-
son curves that the theoretical curves are quite well
match with the experimental results. The resonant
frequencies of both theoretical and experimental
curves are almost matches whereas the amplitude
results are little bit higher because of the assump-
tion of lower damping value of soil for the analysis.
This is may be due to the development of end bear-
ing in model pile which has not been considered for
analysis.
Finally it can be concluded that the present model
can predict the dynamic nonlinear response of sin-
gle floating piles reasonably well with the accurate
assumption of nonlinear boundary zone parameters
and separation length. Though considering other tran-
sition function between week soil zone and outer soil
layer may produce more realistic results for predicting
the nonlinear response of pile foundation.
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Han,Y.C. 1997. Dynamic vertical response of piles in nonlin-
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Engineering, ASCE, 123(8): 710–716.
Manna, B., and Baidya, D. K. 2010. Dynamic nonlinear
response of pile foundations under vertical vibration-
theory versus experiment, Soil Dynamics and Earthquake
Engineering, 30: 456–469.
Matlab R2012b, 64bit under Windows 8.1 64bit.
Matlock, H., Foo, St. H.C. and Bryant, L.M. 1978. Simu-
lation of lateral pile behaviour under earthquake motion.
Proceedings Conference ASCE Earthquake Engineering
and Soil Dynamics; Pasadena, pp: 600–618.
Novak, M. 1974. Dynamic stiffness and dampingof piles,
Canadian Geotechnical Journal, 11: 574–598.
Novak, M. and Grigg, R.F. 1976. Dynamic experiments with
small pile foundations. Canadian Geotechnical Journal,
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Novak, M., andAboul-Ella, F. 1978. Impedance functions for
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