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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2009; 38:16651685Published online 26 May 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.918
Simple formulas for the dynamic stiffness of pile groups
Reza Taherzadeh1, Didier Clouteau1,, and Regis Cottereau1,2
1Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande Voie des Vignes,
92295 ChatenayMalabry, France2 International Center for Numerical Methods in Engineering (CIMNE), Universitat Politecnica de Catalunya,
Jordi Girona 13, 08034 Barcelona, Spain
SUMMARY
Simple formulas are derived for the dynamic stiffness of pile group foundations subjected to horizontal androcking dynamic loads. The formulations are based on the construction of a general model of impedancematrices as the condensation of matrices of mass, damping, and stiffness, and on the identification ofthe values of these matrices on an extensive database of numerical experiments computed using coupledfinite elementboundary element models. The formulations obtained can be readily used for the design ofboth floating piles on homogeneous halfspace and endbearing piles and are applicable for a wide rangeof mechanical and geometrical parameters of the soil and piles, in particular for large pile groups. Forthe seismic design of a building, the use of the simple formulas rather than a full computational modelis shown to induce little error on the evaluation of the response spectra and time histories. Copyright q2009 John Wiley & Sons, Ltd.
Received 19 August 2008; Revised 24 December 2008; Accepted 9 March 2009
KEY WORDS: soil impedance matrix; pile group foundation; design formulas; lumpedparameter models;hidden variables models
1. INTRODUCTION
Whatever the mode of vibration, the dynamic stiffness of a pile group cannot be computed by
simply adding the stiffnesses of the individual piles. Depending on the mechanical and geometrical
parameters of the soil and piles, the dynamical behavior of each pile can be heavily influenced by
that of its neighbors [1]. Among other phenomena, it is clear that the dynamic resonance of thesoil constrained within a cluster of piles cannot be modeled when the complex dynamic interaction
between these piles is neglected.
The main approach to solve this strongly coupled problem is the use of full numerical models,
taking into account the soil and the piles with equal rigor. This is however a computationally very
Correspondence to: Didier Clouteau, Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande Voiedes Vignes, 92295 ChatenayMalabry, France.
Email: [email protected]
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1666 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
demanding approach, in particular for large number of piles, and has only been attempted, to the
knowledge of the authors, by Kaynia [2], using the boundary element (BE) method. All othernumerical methods in the literature seem to include some simplifying assumptions. For example,
the axisymetrical finite element (FE) model
[3
], the ringpile model
[4
], or the closely spaced plates
model [5] can be used, when the geometrical layout of the pile group allows for it. The latter twoapproaches consist in grouping the piles in concentric circles or soilpilestripped upright plates,
respectively, both allowing for an easier evaluation of the interaction effects. Another approach
consists in replacing the pile group with a single equivalent upright beam [6]. In any case, theseapproaches are not adapted to the needs of civil and structural engineers, who need to design pile
foundations with little recourse to computational tools.
A more interesting approach for the design purposes consists in providing analytical formulas,
whose structure is usually derived from physical considerations, and with tabulized parameters,
depending on the geometrical and mechanical parameters of the soil and piles. The simplest type
of such approaches is based on Winklers spring model for the soil, for which radiation damping
and inertial effects are neglected [7 9]. A relatively simple method was proposed by Gazetas andDobry
[10
]for estimating the damping characteristics of horizontally loaded single pile in layered
soil. Following Wolfs approach [11] for the modeling of soilstructure interaction (SSI), otherresearchers [1217] have replaced the soilpile system by a one degreeoffreedom (DOF) mass,with a damper and a spring. Inertial effects and radiation damping are therefore taken into account
to some extent, but the general dynamical behavior, and in particular the interaction between the
different piles, is heavily simplified. To improve these models, Dobry and Gazetas [18] proposedan approximate formulation accounting for the interaction between the piles by modeling the
waves emanating from each excited pile. The additional term is therefore based on the computation
of the propagation of a wave, supposed to be cylindrical, emanating from a single excited pile
in a homogeneous domain. The method was further refined by Gazetas and coworkers [1921]to attempt to model multiple reflections within the pile group in layered soil. However, a few
attempts have been made at accurately modeling the large pile group foundation, in particular for
the complex frequency dependance of endbearing pile foundations. (Konagai et al. [6] provideformulas valid only for sway, Mylonakis and Gazetas [21] provide formulas valid for all movementsbut only for a group of nine piles, and Nikolaou et al. [22] provide for kinematic pile bending fora group of 20 piles).
Despite the significant progress in pile dynamics [23], there is however still a need for simpleengineering procedures for their design, following the example of the code provisions developed
for the seismic design on spread footings [24, 25]. This paper aims at providing such formulas to beused for both small and large pile groups, as well as for both floating pile groups on homogeneous
halfspace and endbearing pile groups. The novelty of this paper is that the formulas are valid
over a range of parameters larger than the formulas previously available in the literature (see above
references). In particular, they can be used for large numbers of piles. This is made possible by the
use of a very general dynamic model for the representation of stiffness impedance matrices, the
hidden state variable model (Section 3.1). The parameters appearing in this model are then fittedusing an extensive database of full coupled FEBE computations of soilpile systems (Section 3.2).
The sway and rocking of the foundation are accounted for, in a large range of parameters of the
soil and piles, and the formulas are given independently for floating (Section 4.1) and endbearing
piles (Section 4.2). Further, for the seismic design of a building, the use of the simple formulas
rather than a full computational model is shown to induce little error on the evaluation of the
response spectra and time histories (Section 5).
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1667
The reader interested in a fast use of the formulas can refer directly for a floating pile group
on homogeneous halfspace (respectively, an endbearing pile group) to Table II (respectively,
Table IV) the coefficients of which are to be used in Equations (9) and (8) (respectively, Equations
(14) and (10)). In these equations, both the dynamic stiffness and the frequency are normalized,
as described in Section 2.2.
2. THE IMPEDANCE MATRIX OF A PILE GROUP: DEFINITION AND NOTATIONS
In this section, we introduce the main notations and define the impedance matrix of a pile group.
The normalizations, which will be used throughout the paper, of both the impedance and the
frequency are also introduced.
2.1. Notations
In all formulas, the indices s and p will refer to the soil and the piles, respectively. When
considering two layers of soil, the top layer will still be denoted as s, whereas the bedrock will be
denoted as b. Es, s, Gs, and s (respectively, Ep, p, Gp, p, and Eb, b, Gb, b) hence denote
Youngs modulus, Poissons ratio, the shear modulus, and the unit mass of the soil (respectively,
of the piles, and of the bedrock). Vs and s (resp., Vb and b) denote the shear wave velocity and
the hysteretic damping of the soil (resp., of the bedrock). All the piles in a group are supposed
to be identical with a diameter d, a length lp, an inertial moment Ip, and they are separated from
each other by a distance s (see Figure 1). They are rigidly attached to a massless square cap
with a halfwidth Bf, which is supposed to have no contact with the soil. Further we define
L0 = (EpIp/Es)0.25, closely related to the critical pile length defined by several authors [26, 27]and an equivalent radius for the cap Rf =2Bf/
.
Two cases will be considered in this paper: (1) floating pile groups on homogeneous halfspace
and (2) endbearing pile groups. In the former case, the soil is a homogeneous halfspace, whereas
in the latter case, the soil is composed of a layer of thickness H, resting over a bedrock, in whichthe tips of the piles are embedded (lp>H).
2.2. Impedance matrix
The impedance matrix or dynamic stiffness matrix Z() of a pile group relates the vector of forces
and moments applied on the rigid cap at the top of the piles to the resulting vector of displace
ments and rotations at the same point. Since the rigid cap is supposed to be square, symmetry
Figure 1. Definition of pile group foundation.
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1669
of the parameters of the final formulas is then performed by regression from a database of FEBE
computations.
Note that the hidden state variable model that is chosen here might appear more mathematical
than the previous formulations found in the literature. However, remember that this structure is
just an intermediate state that allows to find the final formulas that will eventually be used bythe engineers. In Section 4, we will see some analogies between the formulas proposed, and the
mechanical systems created as sets of springs, dampers, and masses. The main difference with
lumpedparameters models is that, in the case of the hidden state variables model, the equivalent
mechanical model comes out naturally as a consequence of the regression rather than being chosen
a priori.
3.1. The hidden variables model
The construction of the hidden variables model of an impedance matrix is based on the supposition
that, besides the n physical DOFs on which the impedance is defined (typically the rigidbody
modes of the cap of the pile group), there exist nI additional DOFs that represent some internal
resonance phenomena inside the soil and the pile group. The resonance modes corresponding tothese DOFs cannot be physically identified, as only their influence on the impedance matrix is
observable, thus the DOFs are referred to as hidden or inner.
With respect to the n =n+nI DOFs, matrices of mass M, damping D, and stiffness K canthen be identified, and the dynamic stiffness matrix S(a0) is defined as
S(a0)= (Ka20 M)+ia0C (4)The impedance matrix corresponding to the hidden variables model is then the condensation on
the n physical DOFs of the stiffness matrix S(a0). More specifically, introducing the block
decomposition of Equation (4),
S(a0) Sc(a0)STc (a0) SI(a0)
=K Kc
KTc KIa20
M Mc
MTc MI+ ia0
C Cc
CTc CI (5)
the impedance matrix is defined as
Z(a0)=S(a0)Sc(a0)S1I (a0)STc (a0) (6)As the hidden variables are not necessarily physical DOFs, but rather state variables in the
background of the physical model, the matrices M, D, and K are really generalized mass, damping
and stiffness matrices and do not correspond a priori with the classical mass, damping and stiffness
matrices, or to those obtained through the application of some modal reduction technique. Another
equivalent form of the hidden variables model can be derived [30], where the hidden parts of thematrices are diagonal, and with no coupling in mass. In that case, the impedance can be written as
Z(a0)= (Ka20 M)+ ia0Cn
h=1
(ia0Cc +K
c)(ia0C
c +K
c)
T
(kIa20 mI)+ia0cI(7)
where Cc and Kc are the th columns of Cc and Kc, and m
I, c
I, and k
I are the diagonal elements
of MI, CI, and KI.
The main interest of this hidden variables model is its generality. Its structure makes it suitable
for the representation of any type of impedance matrix, provided that an appropriate number of
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1670 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
0 2 4 6 8 10
0
2
4
6
2x 109
Frequency [Hz]
Dynamicstiffness
[N/m]
0 2 4 6 8 100
2
4
6
8x 109
Frequency [Hz]
Damping[N/m]
Figure 2. FE model (left) of the 4 9 pile group within a block of soil and comparison of the real (right,up) and imaginary (right, down) parts of the horizontal impedance computed using the FEBE model
(solid line) and the BE model (dashed line) [36].
hidden variables is used. Note that the numerical identification of the matrices M, C, and K is
entirely performed only from the knowledge of the impedance matrix, and that the number of
hidden variables can be automatically chosen based on a precision criteria for the approximation
of the impedance matrix [3032].Contrarily to the lumpedparameter models of the impedance matrix [33], in which the iden
tification of the mechanical elements may yield negative values of the springs, dashpots, and/or
masses in the hidden state variable model, the causality and stability of the soil impedance matrixare directly related to the positivity of M, K, and C. In other words, in comparison with lumped
parameter models, the diagnosis of unphysical models is very natural.
In the next section, the numerical method that is used to derive the reference impedance matrices,
and to identify the parameters of the formulas, is described. The methodology for the identification
of the hidden variables model of a given impedance matrix is also described in Appendix A.
3.2. The reference FEBE model
We suppose, for the reference computations, that both the soil and the piles behave linearly and
that the contact between the piles and the soil is continuous in all directions without any slippage
or gap. The elastodynamic equations are therefore linear. The numerical approach used to derive
the reference results for the calibration of the simple formulations is based on an efficient FEBEcoupling technique that is described in detail in [34, 35] and is briefly recalled below.
The soil is separated into two blocks: one, bounded and containing the piles, which is modeled
by the FE method, and the other, surrounding the previous one, which is modeled by the BE
method (see Figure 2). Within the FE block, the piles are modeled as Bernoulli beam elements.
The two blocks are then assembled using the CraigBampton coupling technique [37], so as tolower the computational cost, which may reach high levels for large pile groups. This numerical
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0 1 2 3 4 5 6
0
10
20
30
0 1 2 3 4 5 60
10
20
30
40
50
60
Figure 3. Real (left) and imaginary (right) parts of the normalized horizontal impedance matrix for floating(solid line) and endbearing (dashed line) pile groups.
model was already validated for stiffness problems taken from the literature (in particular [2]) andthe results are given in [35]. However, these validation results only concerned floating pile groupson homogeneous halfspaces so that we present here a comparison, on a particular example, of
the FEBE model with the BE approach described in [36].We therefore consider a 49 pile group embedded in a soil with two layers (see Figure 2). The
piles have Youngs modulus of 25 GPa, a diameter of d=1.3m and are separated by s =2.6m. Thefirst layer of soil is H=9.5 m thick, and is formed of a very soft saturated organic clay with Swavevelocity Vs =80 m/s, unit mass s =1.5 Mg/m3, and Poissons ratio s =0.49. The lower layer ofsoil is a stiff sand with Swave velocity Vd=300m/s, unit mass d=2 Mg/m3, and Poissonsratio d=0.4, in which the piles penetrate 6 m. In both layers the hysteretic damping is takenas s =d=0.05. As seen in Figure 2, the agreement between the results in the two numericalapproaches is very good.
It should be noted that the frequency dependance of pile groups is particularly sensitive to the
number of piles and to its character of floating or end bearing. The dynamic stiffness of single
piles and pile groups with a small number of piles is nearly independent of frequency [38], whilethat of larger pile groups may show large variations with frequency. Likewise, the behavior of
endbearing pile groups is much more erratic with frequency than that of floating pile groups on
homogeneous halfspace. These physical results are retrieved with the FEBE approach and an
example of such a comparison is shown in Figure 3. These results were obtained considering the
sample number 4 in Tables I and III.
It is also interesting to note, in Figure 3 for the endbearing pile group, that the imaginary
part of the impedance (it is also true for the rocking term, not shown here) presents a small andalmost constant value below some cut off frequency, which is the resonance frequency of the top
layer of soil. Indeed, for very low frequencies, surface waves cannot build up in that top layer
and take energy away from the foundation, so that the radiation damping is very low. Above
that cut off frequency, a large peak can be observed on the imaginary part (with the real part
almost cancelling), indicating a resonance within the soil that tends to soak energy away from the
foundation.
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4. COMPUTATION OF SIMPLE FORMULAS FOR PILE GROUPS
In this section, we present the derivation of the simple formulas in the cases of the floating pile
groups on homogeneous halfspace and endbearing pile groups and using the ideas discussed
above. Depending on the type of pile group and on the type of element of the impedance matrix,more or less hidden variables are necessary to describe its behavior, and, correspondingly, more
or less parameters are needed in the formulas.
4.1. Floating pile groups
We first consider the floating pile groups embedded in a homogeneous halfspace. In that case,
the variation of the dynamic stiffness with the frequency is rather smooth, as seen in Figures 4
and 5. More precisely, the dynamic stiffness always has a parabolic variation, while the damping
coefficient is approximately linear. The parabolic decrease of the real part seems to indicate that
a mass remains entrapped between the piles and vibrates inphase with the cap.
The hidden variables model predicts in all cases in the database (described in Table I) a two
DOFs system, one for the sway and one for the rocking, and with no hidden variables. Note thatthe coupling term is negligible. In Figure 6, a schematic drawing of a system corresponding to
such impedance is presented. The superstructure is subjected to the seismic horizontal force fs.
The elements of the normalized impedance can therefore be written as
Zh(a0) = (kh a20 mh)+ia0chZr(a0) = (kr a20 mr)+ ia0crZsr(a0) = 0
(8)
where the values of kh, ch, mh, kr, cr, and mr depend on the case considered. Remember that the
definition of the normalized frequency a0 is given in Section 2.2 and that the normalized values in
these formulas (8) must be scaled by the static stiffness to yield the actual value of the impedance
matrix, as described in Section 2.2.
0 1 2 3 4 5 6
0
5
10
0 1 2 3 4 5 60
10
20
30
40
50
Figure 4. Real (left) and imaginary (right) parts of the horizontal impedance matrix for different pileseparations: s/d=2 (solid line), s/d=2.5 (dashed line), and s/d=3.5 (soliddashed lines). The figures
correspond to a 1515 pile group with Ep/Es =300 and Rf/ lp =1.1.
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0 1 2 3 4 5 6
0
10
20
0 1 2 3 4 5 60
10
20
30
40
50
60
Figure 5. Real (left) and imaginary (right) parts of the rocking term of the dynamic stiffness matrixfor different pile separations: Rf/ lp
=0.7 (solid line), Rf/ lp
=0.65 (dashed line), and Rf/ lp
=0.55
(soliddashed lines). The figures correspond to a 14 14 pile group with Ep/Es =375 and s/d=2.
Table I. The database of soilpile group systems used to derive the simple formulations for floating pilegroups on homogeneous halfspace.
Piles Ep d Es Bf lp s Vs
Sample (Dimensionless) (GPa) (m) (GPa) (m) (m) (m) (m/s)
1 88 30 1 0.1 15 18 3.3 1402 1111 30 1 0.1 20 18 3.3 1403 1616 30 1 0.1 30 18 3.3 1404 13
13 40 1 0.08 20 24 2.8 130
5 1313 30 1 0.08 20 24 2.8 1306 1313 20 1 0.08 20 24 2.8 1307 1515 25 1.3 0.08 20 18 2.5 1308 1515 25 1 0.08 20 18 2.5 1309 1515 25 0.7 0.08 20 18 2.5 13010 1818 30 1 0.02 25 14 2.6 6011 1818 30 1 0.08 25 14 2.6 13012 1818 30 1 0.1 25 14 2.6 14013 1616 30 1 0.08 15 28 2.0 13014 1616 30 1 0.08 15 20 2.0 13015 1616 30 1 0.08 15 14 2.0 130
The range of parameters is 250Ep/Es1500, 2s/dp3.6, and 0.55Rf/ lp2 and constant hystericdamping s =0.05.
In previous works, the leading parameters for this type of pile groups were identified to be the
ratio of Youngs moduli Ep/Es and the normalized separation of the piles s/d [2, 39], or the factorL0 = (EpIp/Es)0.25 related to the active pile length [26, 27, 40]. We decide here to use as leading
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1674 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
Figure 6. A schematic drawing of a simple model for floating pile group.
Table II. Coefficients for the horizontal and rocking elements of theimpedance of a floating pile groups on homogeneous halfspace.
=0
Rflp
1 L0s
20 1 2 R (%)
kh 6.8 0 0.3 80ch 5 0 0.5 90mh 0.4 0 1.6 86
kr 8 0.6 0.4 83cr 5 0.5 0.2 72mr = mhl2eq/4 0.7 1 0.4 96
parameters the normalized radius of the foundation Rf/ lp and a normalized active pile length ratio
L0/s. We therefore provide equations of the parameters
{k
h,c
h,
m
h,k
r,c
r,
m
r}in the form
=0
Rf
lp
1L 0s
2(9)
with the values of0, 1, and 2 being provided for each of the parameters. A multiple regression
analysis was then conducted with respect to the two quantities Rf/ lp and L 0/s, and lead to the
values described in Table II. The regression coefficient R is also indicated in the same table to
provide an indicator of the accuracy of the regression analysis.
In general terms, the formulas in Table II corroborate the observed results that, for short
separations between the piles and for weak soils, both normalized dynamic stiffness and damping
increase. Note that, as indicated by the zeros in Table II, the influence of the ratio R f/lp on the
horizontal impedance is negligible, while it is rather important for the rocking term. Besides the
uniform presentation in Table II, the reader may also find an expanded, nonnormalized version
of the same formulas in Appendix B for easier reading.
4.2. Endbearing pile groups
We then consider endbearing pile groups. As stated earlier, their dynamical behavior is much
more complicated than that of floating pile groups on homogeneous halfspace. The structure of
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1675
Table III. The database of soilpile group systems used to derive the simple formulationsfor endbearing pile groups.
Piles Ep d Es Bf lp H Vs Vb s
Samples (Dimensionless) (GPa) (m) (GPa) (m) (m) (m) (m/s) (m/s) (m)
1 88 30 1 0.1 15 16 14 140 830 3.32 1111 30 1 0.1 20 16 14 140 830 3.33 1616 30 1 0.1 30 16 14 140 830 3.34 1313 40 1 0.08 20 22 20 130 620 2.85 1313 30 1 0.08 20 22 20 130 620 2.86 1313 20 1 0.08 20 22 20 130 620 2.87 1515 25 1.3 0.2 37 20 18 200 780 4.38 1818 30 1 0.04 32 20 18 90 440 3.19 1818 30 1 0.1 32 20 18 140 700 3.110 1414 30 1 0.08 26 26 24 130 1000 3.311 1414 30 1 0.08 26 20 18 130 1000 3.312 1414 30 1 0.08 26 16 14 130 1000 3.313 13
13 30 1 0.06 26 18 16 110 620 3.6
14 1313 30 1 0.06 26 18 16 110 620 3.615 1313 30 1 0.06 26 18 16 110 620 3.6The range of parameters is 125Ep/Es750, 2.8s/dp4.4, 1Rf/H2.1, and 3Vb/Vs8 and constant
hysteretic damping s =0.05.
the approximation for the impedance matrix is therefore difficult to guess a priori and we use the
hidden variables model in a very general setting. Note that, as the coupling term is negligible in
the cases considered, the hidden variables model was identified independently on the horizontal
and rocking terms of the impedance matrix.
The identification of the hidden variables model for all the cases in the database described
in Table IV suggests the consideration of three hidden variables for the sway and none for the
rocking. Besides, no coupling in the stiffness for the first hidden variable and no coupling in thedamping for the two others seemed to be necessary. The chosen structure for the endbearing pile
groups is therefore written as a special case of Equation (7) for the hidden variables model
Zh(a0) = (kh a20 mh)+ia0ch +a20 c
21
(k1 a20 m1)+ia0c1 k
22
(k2 a20 m2)+ ia0c2
k23
(k3 a20 m3)+ ia0c3Zr(a0) = (kr a20 mr)+ ia0cr
Zsr(a0) = 0
(10)
and represented as a set of masses, springs, and dampers in Figure 7. In these formulas, the static
stiffness coefficient is k0 = kh k2 k3. The previous observation for the coupling with the hiddenvariables can be translated in Figure 7 by the fact that the mass m1 is linked to the foundation by
a dashpot, whereas the masses m2 and m3 are linked to it through springs. This fact arises from
the presence of the cut off frequency of the top layer of soil, which was discussed in Section 3.2.
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Table IV. Coefficients for the horizontal and rocking elements of the impedanceof an endbearing pile group.
=0
RfH
1
L 0s 2
bVbs Vs 3
0 1 2 3 R (%)
k0 = kh k2 k3 10 0.5 0.35 0 93ch = c0 + c1 1 0.5 0.5 0.5 60mh = m0 0.5 1 0 0 82k1 2.6 1 0 0 65c1 1.9 1.5 0 0 75m1 1.4 1 0 0 80k2 1.25 0.35 1 0.5 60c2 0.04 0 1 1 62m2 0.08 1 1 0.5 80k3 16.1 3 3 0.5 75c3 3 2 3 1.5 70m3 0.6 1 3 0.5 75
kr 15 0.5 1 1 98cr 17 0.5 2 0.5 95mr = m0l2eq/4 1.6 1.5 1 0.5 90
Figure 7. A schematic drawing of a simple model for endbearing pile group.
More physical remarks can be made in the different frequency ranges defined by the resonance
frequencies a0 of the masses m representing the hidden variables. In the lowfrequency range
(a0
a01), a firstorder expansion gives
Zh(a0)= k0 +ia0(ch + c2 + c3) (11)
It is worth noticing that the slope of the imaginary part c0+ c1 + c2+ c3 is not small since it allowsto quickly reach the level of the hysteretical damping. In the range of resonance of mass m1(a0 a01 21a01, with =c/(2
km)), and supposing that all the resonance frequencies are
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far enough from each other (a0 a02 22a02 and a0 a03 23a03), one has
Zh(a0) = kh + k1 k2a202
a2
02a2
0
k3a203
a2
03 a2
0
+ia0
ch c1 + c2
a202
a202 a20
2+ c3
a203
a203a20
2 (12)which means that around a01, the mass m1 has the same displacement as the foundation, so that
there is no damping contribution from c1. For a0 a02, masses m2 and m3 are also linked to thefoundation, but the dashpots c2 and c3 introduce some damping. The equivalent slope around a01tends to c0 + c2 + c3 =eq/a01, which is actually small as expected to model the sole hystereticaldamping eq. Usually, c1 a01eq. In the range of resonance of the mass m2 (an equivalent formulacan be derived for mass m3), a large imaginary part is brought on by k/2, which corresponds
to the peaks observed in Figure 3. Finally, at high frequency (a0
a03), one has
Zh(a0)=
kh c21m1
a20 mh
+ ia0ch (13)
which classically corresponds to all the masses m1, m2, and m3 being fixed. One can see c1 as
the radiative damping that occurs only above a01 since for this frequency we have shown that
the damping is only eq/a01. Thus, this model reproduces the cut off frequency at the resonance
frequency of the layer.
Once the structure of the approximation has been decided, a multiple regression analysis is
performed on the same leading parameters as before, plus the ratio (bVb)/(sVs) to yield the
formulas presented in Table IV. Note that several coefficients appear as zeros in the table, which
means that the parameters modeled do not have any influence on the formula. Note also that, as
before, the formulas are presented in a nonnormalized manner in Appendix B for easier reading.The general formulas for the parameters are
=0
Rf
H
1L 0s
2 bVbs Vs
3(14)
It is particularly interesting to note that although the formulas were derived from rather mathe
matical considerations (the hidden variables model and a regression analysis), they yield a very good
evaluation of the resonance frequencies of the soil layer. Indeed, the first fundamental frequency
of the soil layer s01 =2Vs/(4H) and
k1/m1 =1.4Vs/H (see Appendix B for nonnormalizedformulas) coincides. Likewise, the second fundamental frequency of the soil layer s02 =6Vs/(4H)is very well approximated by the third resonance of the simple model
k3/m3 =5.1Vs/H.
5. IMPACT OF THE FORMULAS ON THE EVALUATION OF DESIGN QUANTITIES
In this last section, we discuss the accuracy of the proposed formulas on two practical cases.
More particularly, the accuracy of the predicted transfer functions, spectral acceleration on top
of a building, and relative displacement between the top and bottom of the building, using the
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1678 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
0 1 2 3 4 5
0
x 1010
x 1010 x 1013
x 1013
1
Frequency [Hz]
Realpart[N/m]
0 1 2 3 4 50
1
2
Frequency [Hz]
Imaginarypart
[N.m
]
Imaginarypart
[N.m
]
0 1 2 3 4 50
1
2
2
Frequency [Hz]
Realpart[N/m]
0 1 2 3 4 50
1
Frequency [Hz]
Figure 8. Comparison between the real (up) and imaginary (down) parts of the horizontal (left) androcking (right) elements of the impedance matrix for a 10 10 endbearing pile group computed using
the simplified formulas (10) (dashed line) and the FEBE model (solid line).
proposed formulas, is demonstrated. In a second test, we compare the accuracy of our proposed
formula with another one from the literature.
5.1. Case 1
For this validation, a 10
10 endbearing pile group is used, with piles with dp
=1 m, lp
=22m,
s =5m, and connected by a 1.1m thick, rigid, cap with Bf =25m. The mechanical properties ofthe piles are Ep =30GPa, p =0.25, and p =2500kg/m3. This pile group stands in H=20m thicksoil layer with properties Es =60MPa, s =0.4, and s =1750kg/m3. The mechanical propertiesof the underlying halfspace are Eb =1.5GPa and b =0.3 and b =2000kg/m3. The real andimaginary parts of the impedance are shown on Figure 8, both as computed using the numerical
FEBE model, and using the simple formulas of Equation (10). The agreement between the two
approaches is good, in particular for the shaking term, considering the important variability in the
frequency. Note that the pile group considered here was not used for the regression analysis that
determined the parameters in Table IV.
We now turn to the observation of the accuracy of the proposed formulations for the estimation of
engineering quantities of interest. We therefore consider a 60 m high building (20 floors), with floors
of 22.5 m22.5m, and 6 columns6 columns. The slab weight per unit area is 500kg/m2
and thecharacteristics of the beams and columns are, respectively, E I=5.1MNm2 and E I=1MNm2.
We first consider the estimation of transfer functions in two different cases: (1) using the entire,
66, impedance matrix computed from the FEBE model, and considering both the kinematicand inertial interaction and (2) using only the horizontal and rocking elements of the impedance
matrix computed with the proposed formula (10) and neglecting the kinematic interaction. For
both cases, the displacement field is decomposed on a basis, which contains the rigidbody modes
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1679
of the building (lm ), which coincide with those of the foundation and the flexible modes of the
building on a rigid basis (/n):
u(,
x)=m cm()lm(x)+n n()/n(x)=[c a]
L
U
(15)
where L is the matrix of the rigidbody modes of the structure and U is the matrix of the eigenmodes
of the structure clamped at its base. The response of the structure, taking into account SSI, is then
computed using the following formula:Z() 0
0 0
+(1+2i)
0 0
0 K
2
M M
M I
c()
a()
=
Z()c0()
0
(16)
where the diagonal matrix K contains the squares of the lowest circular frequencies of the structure
on fixed base and I is the identity matrix arising from the orthogonality of the eigenmodes with
respect to the mass matrix. stands for the rigidbody modes and for the eigenmodes on fixed
base, while c0 is the kinematic interaction. The differences between the two models with respectto this formulation are the impedance matrix Z() and the kinematic interaction factor takes equal
to Dui () with D having null components but a unitary for the sway term. Besides, it is worth
noticing that the simplified model does not correspond to the physical model sketched on Figure 7
subjected to an uniform acceleration ai . Indeed, inertial forces are not applied on mass m1, m2,
and m3 since these masses are in the soil and have their inertial forces already balanced in the soil.
The resonance frequencies of the soil are computed at fs01 =1.55Hz and fs02 =4.6Hz. Assuminga horizontal harmonic base motion at the bedrock, the horizontal transfer function at the free
surface and at the top of the building are represented in Figure 9. It clearly shows the effect of the
0 1 2 3 4 50
2
4
6
8
10
12
14
Frequency [Hz]
Figure 9. Transfer function at ground surface free field (solid line) of the structure without SSI (dashedline), and of the structure with SSI (dasheddotted line), all computed with the FEBE approach, andtransfer function of the structure with SSI computed with the proposed formulas (dotted line). The figures
correspond to a structure resting on a 10 10 endbearing pile group.
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1680 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
0 4 8 12 16
0
3
Time [s]
Acceleration
[m/s2]
0 4 8 12 16
0
3
Time [s]
Acceleration
[m/s2]
0 4 8 12 160
2
4
6
8
10
Frequncy [Hz]
Spectralacceleration[
m/s2]
Figure 10. Ground acceleration (left) and 5%damped response spectra (right) recorded in Aegion (Greece)in 1995 (top and solid line), and in Friuli (Italy) in 1976 (bottom and dashed line).
0 2 4 6 8 100
2.5
5
7.5
10
Frequency [Hz]
Spectralacceleration[m/s2]
0 2 4 6 8 10
0
2
4
6
8
Frequency [Hz]
Spectralacceleration[m/s2]
Figure 11. Comparison of the acceleration response spectra at the top of the building for theFriuli earthquake (left) and the Aegion earthquake (right) using the complete FEBE model(solid line) and the simple formulation (dashed line). The figures correspond to a structure
resting on a 1010 endbearing pile group.
SSI, as well as the ability of the formulas (10) to compute the resonance frequency of the coupledsystem (the peaks of the dotted and dashdotted lines in Figure 9 coincide almost exactly).
We then consider two real recordings of earthquakes, with different frequency contents (see
Figure 10) and peak ground accelerations at about 0.3g. In Figure 11 a comparison is given in the
spectral acceleration on top of the building computed in the two cases considered earlier of the
FEBE model supposing inertial and kinematic interaction and the simple formulas (10) neglecting
the kinematic interaction. In Figure (12), a comparison is given in the time histories of the relative
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1681
0 6 12 18 24 30
0
0.04
0.08
0.12
Relativedisplacem
ent[m]
Time [s]
0 6 12 18 24 30
0
0.01
0.02
0.03
Relativedisplacem
ent[m]
Time [s]
Figure 12. Comparison of the relative displacements between the top and the base of thebuilding for the Friuli earthquake (left) and the Aegion earthquake (right), using the completeFEBE model (solid line) and the simple formulation (dashed line). The figures correspond to
a structure resting on a 1010 endbearing pile group.
0 1 2 3 4 5 6
0
1
2
3x 109
Frequency [Hz]
Dy
namicstiffness[N/m]
0 1 2 3 4 5 60
0.5
1
1.5
2
2.5x 109
Frequency [Hz]
Damping[N/m]
Figure 13. Real (left) and imaginary (right) parts of the horizontal impedance matrix for a 36 endbearingpile group computed using the simplified formulas (10) (solid line) and BE solution (dashed line) and
simplified analytical solution of [21, 36] (dotted line).
displacements between the top and the base of the building. In both figures, the agreement between
the two approaches is very good.
5.2. Case 2
In this example, we compare our simplified formulations (10) for the impedance of the end
bearing pile group introduced in Section 3.2 with the simplified formulas proposed in [21, 36].We use an equivalent of Rf =8m, because the formulas in Table IV are proposed for a circularor square foundation. Figure 13 shows this comparison along with the value of the BE solution.
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1682 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
Our formulas seem to behave at least as well as the previously available one. Remember that its
range of application, in particular in terms of the numbers of piles, is much larger.
6. CONCLUSION
Simple formulations have been derived for the dynamic stiffness matrices of pile group foundations
subjected to horizontal and rocking dynamic loads. These formulations were found using a large
database of impedance matrices computed using a FEBE model. They can be readily employed for
the design of large foundations on piles and are shown to yield very accurate values of the estimated
quantities of interest for building design. The formulations have been derived both for floating
pile groups on homogeneous halfspace and endbearing pile groups in a homogenous stratum.
They can be used for large pile groups (n50), as well as for a large range of mechanical and
geometrical parameters of the soil and the piles. They provide a first step toward code provisions
specifically focused on pile footings.
APPENDIX A
In this appendix, the practical methodology for the construction of the reduced matrix S(a0)=Ka20 M+ ia0C is introduced. Three main steps are identified:
The impedance of the FEBE model is computed. More specially a set of values {Z(a0l)} ofthe impedance matrix at a finite number of frequencies (a0l)1lL is computed.
The set of values {Z(a0l)} is interpolated to yield a matrixvalued rational function in theform a0 N(a0)/q(a0), which approximates the behavior of the impedance matrix {Z(a0)} ofthe model. The function a0 N(a0) is a matrixvalued polynomial in (ia0), and the functiona0 q(a0) is a scalar polynomial in (ia0). Many methods can be used to achieve that goal.
The identification of the matrices K, C, and M from the polynomials a0
N(a0) and a0
q(a0) is then performed. This step does not involve any approximation and is further detailedin [32].
APPENDIX B
In this appendix, we present an extended version of the formulas presented in Tables II and IV in
a nonnormalized form.
For the case of the floating pile groups on homogeneous halfspace the coefficients appearing
in the nonnormalized version of Equation (8) are
kh =
6.8GsR
fL0
s0.3
ch = 5GsR
2f
Vs
L 0
s(B1)
mh = 0.4sR3f
L 0
s
1.6
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SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1683
and
kr = 8GsR3flp
Rf
0.6
L0
s 0.4
cr = 5GsR
3f
Vs
R flp
L0
s
0.2
mr = 0.7GsR4flp
L 0
s
0.4(B2)
The notations are defined in Section 2.1, and the range of parameters is 250Ep/Es1500,
2s/dp3.6, 0.55Rf/ lp2, 0.55L 0/s1.05 and constant hysteric damping s =0.05.For the case of endbearing pile groups, the coefficients appearing in the nonnormalized version
of Equation (10) are
k0 =khk2 k3 = 10GsR f
Rf
H
0.5L 0
s
0.35
ch = c0+c1 =GsR
2f
Vs
H
R f
L 0
s
b Vb
s Vs(B3)
mh = 12sH R2f
k1 = 2.6Gs R2f
H
c1 = 1.9GsR
2f
Vs
H Rf
1.5(B4)
m1 = 1.4sR2f H
k2 = 1.25GsRf
R f
H
0.35s
L0
bVb
sVs
c2 = 0.04GsR2fbVb
sV2
s
s
L 0
(B5)
m2 = 0.08sR2f Hs
L 0
bVb
sVs
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1684 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU
k3 = 16.1GsR4f
H3
L0
s
3s Vs
b Vb
c3 = 3 GsVs
R4f
H2
L0s
3sVsbVb
1.5
m3 = 0.6sR4f
H1
L 0
s
3sVs
bVb
(B6)
and
kr = 15GsR3f
Rf
H
L0
s
bVb
s Vs
cr=
17GsR
4f
VsL 0
s
2
R fH
sVs
bVb
mr = 1.6sR4fL0
s
Rf H
s Vs
b Vb
(B7)
The notations are defined in Section 2.1, and the range of parameters is 125Ep/Es750,
2.8s/dp4.4, 1Rf/H2.1, 4.1bVb/(sVs)11, 0.45L 0/s0.8 and constant hysteretic
damping s =0.05.
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