Taherzadeh et al. (2009)

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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 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.


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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 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

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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 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.


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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


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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


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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 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.


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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 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.


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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 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


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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


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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.


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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


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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 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


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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


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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.


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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


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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.


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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


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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


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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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