Taherzadeh et al. (2009)

download Taherzadeh et al. (2009)

of 21

Embed Size (px)

Transcript of Taherzadeh et al. (2009)

  • 8/2/2019 Taherzadeh et al. (2009)

    1/21

    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 Chatenay-Malabry, France2 International Center for Numerical Methods in Engineering (CIMNE), Universitat Politecnica de Catalunya,

    Jordi Girona 1-3, 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 half-space and end-bearing 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; lumped-parameter 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 Chatenay-Malabry, France.

    E-mail: [email protected]

    Copyright q 2009 John Wiley & Sons, Ltd.

  • 8/2/2019 Taherzadeh et al. (2009)

    2/21

    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 ring-pile 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 soilpile-stripped 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 degree-of-freedom (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 co-workers [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 end-bearing 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

    half-space and end-bearing 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 end-bearing

    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).

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    3/21

    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 half-space (respectively, an end-bearing 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 mass-less square cap

    with a half-width 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 half-space

    and (2) end-bearing pile groups. In the former case, the soil is a homogeneous half-space, 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.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    4/21

  • 8/2/2019 Taherzadeh et al. (2009)

    5/21

    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

    lumped-parameters 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 rigid-body

    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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    6/21

    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 lumped-parameter 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    7/21

    SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1671

    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 end-bearing (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 half-spaces 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 S-wavevelocity 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 S-wave 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

    end-bearing pile groups is much more erratic with frequency than that of floating pile groups on

    homogeneous half-space. 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 end-bearing 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.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    8/21

    1672 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU

    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 half-space and end-bearing 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 half-space. 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 in-phase 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.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    9/21

    SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1673

    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 half-space.

    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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    10/21

    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 half-space.

    =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, non-normalized version

    of the same formulas in Appendix B for easier reading.

    4.2. End-bearing pile groups

    We then consider end-bearing pile groups. As stated earlier, their dynamical behavior is much

    more complicated than that of floating pile groups on homogeneous half-space. The structure of

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    11/21

    SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1675

    Table III. The database of soilpile group systems used to derive the simple formulationsfor end-bearing 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 end-bearing 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.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    12/21

    1676 R. TAHERZADEH, D. CLOUTEAU AND R. COTTEREAU

    Table IV. Coefficients for the horizontal and rocking elements of the impedanceof an end-bearing 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 end-bearing 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 low-frequency range

    (a0

    a01), a first-order 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    13/21

    SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1677

    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 non-normalized 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 non-normalizedformulas) 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    14/21

    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 end-bearing 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 end-bearing 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 half-space 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 rigid-body modes

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    15/21

    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 rigid-body 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 rigid-body 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 end-bearing pile group.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    16/21

    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 end-bearing 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    17/21

    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 end-bearing 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 end-bearingpile 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.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    18/21

    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 half-space and end-bearing 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 matrix-valued 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 matrix-valued 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 non-normalized form.

    For the case of the floating pile groups on homogeneous half-space the coefficients appearing

    in the non-normalized 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    19/21

    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 end-bearing pile groups, the coefficients appearing in the non-normalized 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

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    20/21

    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.

    REFERENCES

    1. Wolf JP. Foundation Analysis using Simple Physical Model. Prentice-Hall: Englewood Cliffs, NJ, 1994.

    2. Kaynia AM. Dynamic stiffness and seismic response of pile groups. Research Report R82-03, MasshachussettsInstitute of Technology, 1982.

    3. Waas G, Hartmann HG. Seismic analysis of pile foundations including soilpilesoil interaction. Proceedings of

    the 8th World Conference on Earthquake Engineering, San Francisco, July 1984; 555562.

    4. Takemiya H. Ring-pile analysis for a grouped pile foundation subjected to base motion. Structural

    Engineering/Earthquake Engineering 1986; 3(1):195202.

    5. Ohira A, Tazoh T, Dewa T, Shimizu K, Shimada M. Observation of earthquake response behaviours of foundations

    piles for road bridge. Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, vol. 3,

    July 1984; 577584.

    6. Konagai K, Ahsan R, Maruyama D. Simple expression of the dynamic stiffness of grouped piles in sway motion.

    Journal of Earthquake Engineering 2000; 4(3):355376.

    7. Crouse CB, Cheang L. Dynamic testing and analysis of pile group foundation. In Dynamic Response of Pile

    FoundationsExperiment, Analysis and Observation, Nogami T (ed.). ASCE: New York, 1987; 7998.

    8. Mylonakis G, Nikolaou A, Gazetas G. SoilPile-Bridge seismic interaction: kinematic and inertial effects. Part

    I: soft soil. Earthquake Engineering and Structural Dynamics 1997; 26(3):337359.9. Hutchinson TC, Chai YH, Boulanger RW, Idriss IM. Inelastic seismic response of extended pile-shaft-supported

    bridge structures. Earthquake Spectra 2004; 20(4):10571080.

    10. Gazetas G, Dobry R. Horizontal response of piles in layered soil. Journal of Geotechnical Engineering 1984;

    110(1):2034.

    11. Wolf JP. SoilStructure Interaction Analysis in Time Domain. Prentice-Hall: Englewood Cliffs, NJ, 1988.

    12. Levine MB, Scott RF. Dynamic response verification of simplified bridge-foundation model. Journal of

    Geotechnical Engineering 1989; 115(2):12461260.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe

  • 8/2/2019 Taherzadeh et al. (2009)

    21/21

    SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS 1685

    13. Spyrakos CC. Assessment of SSI on the longitudinal seismic response of short span bridges. Engineering

    Structures 1990; 12(1):6066.14. Harada T, Yamashita N, Sakanashi K. Theoretical study on fundamental period and damping ratio of bridge

    pier-foundation system. Proceedings of the Japan Society of Civil Engineers 1994; 489(127):227234.15. Chaudhary MS, Parakash S. Dynamic soil structure interaction for bridge abutment on piles. Geotechnical Special

    Publication 1998; 2:12471258.16. Spyrakos CC, Loannidis G. Seismic behavior of a post-tensioned integral bridge including soilstructure interaction

    (SSI). Soil Dynamics and Earthquake Engineering 2003; 23(1):5363.17. Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soilstructure interaction. Soil Dynamics

    and Earthquake Engineering 2003; 23(4):287302.18. Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechnique

    1988; 38(4):557574.19. Gazetas G, Makris N. Dynamic pilesoilpile interaction. Part I: analysis of axial vibration. Earthquake

    Engineering and Structural Dynamics 1991; 20(2):115132.20. Makris N, Gazetas G. Dynamic pilesoilpile interaction. Part II: lateral and seismic response. Earthquake

    Engineering and Structural Dynamics 1992; 21(2):145162.21. Mylonakis G, Gazetas G. Lateral vibration and internal forces of grouped piles in layered soil. Journal of

    Geotechnical and Geoenvironmental Engineering (ASCE) 1999; 125(1):1625.22. Nikolaou S, Mylonakis G, Gazetas G, Tazoh T. Kinematic pile bending during earthquake: analysis and field

    measurement. Geotechnique 2001; 51(5):425440.23. Pender M. Aseismic pile foundation design analysis. Bulletin of the New Zealand National Society for Earthquake

    Engineering 1993; 26(1):4960.24. ATC. Tentative Provisions for the Development of Seismic Regulations of Buildings : Cooperative Effort with the

    Design Profession, Building Code Interests and the Research Community, Washington, DC, 1978.25. NEHRP, National Earthquake Hazards Blackuction Program. Recommended Provisions for the Development of

    Seismic Regulations for New Buildings, Washington, DC, NEHRP, 1997.26. Randolph M. Response of flexible piles to lateral loading. Geotechnique 1981; 31(2):247259.27. Poulos HG, Hull TS. The role of analytical geomechanics in foundation engineering. Foundation Engineering:

    Current Principles and Practices (ASCE) 1989; 2:15781606.28. Gazetas G. Analysis of machine foundation vibrations: state of the art. Soil Dynamics and Earthquake Engineering

    1983; 2(1):242.29. Sieffert JG, Cevaer F. Handbook of Impedance Functions. Surface Foundations Editions Ouest-France: France,

    1992.

    30. Cottereau R, Clouteau D, Soize C. Construction of a probabilistic model for impedance matrices. ComputerMethods in Applied Mechanics and Engineering 2007; 196(1720):22522268.31. Cottereau R, Clouteau D, Soize C. Probabilistic impedance of foundation: impact of the seismic design on

    uncertain soils. Earthquake Engineering and Structural Dynamics 2008; 37(6):899918.32. Cottereau R. Probabilistic models of impedance matrices. Ph.D. Thesis, Ecole Centrale Paris, France, 2006.

    Available from: http://tel.archives-ouvertes.fr/tel-00132950/en/.33. Wolf JP. Consistent lumped-parameter models for unbounded soil: physical representation. Earthquake Engineering

    and Structural Dynamics 1991; 20(1):1232.34. Clouteau D, Taherzadeh R. Soil, pile group and building interactions under seismic loading. Proceedings of the

    1st European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland, September 2006. In

    CDROM.35. Taherzadeh R, Clouteau D, Cottereau R. Identification of the essential parameters for the lateral impedance of

    large pile groups. Proceedings of the 4th International Conference on Geotechnical Earthquake Engineering,

    Thessaloniki, Greece, June 2007. In CDROM.36. Gazetas G, Hess P, Zinn R, Mylonakis G, Nikolaou A. Seismic response of a large pile group. Proceedings of

    the 11th European Conference on Earthquake Engineering, Paris, September 1998. In CDROM.37. Craig RJ, Bampton M. Coupling of substructures for dynamic analyses. AIAA Journal 1968; 6(7):13131319.38. Miura K, Kaynia AM, Masuda K, Kitamura E, Seto Y. Dynamic behaviour of pile foundations in homogeneous

    and non-homogeneous media. Earthquake Engineering and Structural Dynamics 1994; 23(2):183192.39. Gazetas G. Seismic response of end-bearing single piles. International Journal of Soil Dynamics and Earthquake

    Engineering 1984; 3(2):8293.40. Poulos HG. Behavior of laterally loaded piles: part IIgroup piles. Journal of the Soil Mechanics and Foundations

    Division (ASCE) 1971; 97(SM5):733751.

    Copyright q 2009 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:16651685

    DOI: 10.1002/eqe