Simple formulas for the dynamic stiﬀness of pile … · SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2002; 00:1–6 Prepared using eqeauth.cls [Version: 2002/11/11 v1.00]

Simple formulas for the dynamic stiffness of pile groups

Reza Taherzadeh1, Didier Clouteau1,∗, Regis Cottereau2

1 Laboratoire MSSMat, Ecole Centrale Paris, CNRS UMR 8579, Grande Voie des Vignes, 92295Chatenay-Malabry, France

2 International Center for Numerical Methods in Engineering (CIMNE), Universitat Politecnica deCatalunya, Jordi Girona 1-3, 08034 Barcelona, Spain

SUMMARY

Simple formulas are derived for the dynamic stiffness of pile group foundations subjected to horizontaland rocking dynamic loads. The formulations are based on the construction of a general model ofimpedance matrices as the condensation of matrices of mass, damping and stiffness, and on theidentification of the values of these matrices on an extensive database of numerical experimentscomputed using coupled Finite Element-Boundary Element (FE-BE) models. The formulationsobtained can be readily used for design of both floating piles on homogeneous half-space and end-bearing piles, and are applicable for a wide range of mechanical and geometrical parameters of the soiland piles, in particular for large pile groups. For the seismic design of a building, the use of the simpleformulas rather than a full computational model is shown to induce little error on the evaluation ofthe response spectra and time histories. Copyright

2002 John Wiley & Sons, Ltd.

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 computedby simply adding the stiffnesses of the individual piles. Depending on the mechanical andgeometrical parameters of the soil and piles, the dynamical behavior of each pile can be heavilyinfluenced by that of its neighbors [1]. Among other phenomena, it is clear that the dynamicresonance of the soil constrained within a cluster of piles cannot be modeled when the complexdynamic 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 computationallyvery demanding approach, in particular for large number of piles, and has only been attemptedat, to the knowledge of the authors, by Kaynia [2], using the Boundary Element (BE) method.All other numerical methods in the literature seem to include some simplifying assumptions.

∗Correspondence to: [email protected]

Copyright

2002 John Wiley & Sons, Ltd.

2 R. TAHERZADEH, D. CLOUTEAU & R. COTTEREAU

For example, the axisymetrical Finite Element (FE) model [3], the Ring-Pile model [4], or theclosely-spaced plates model [5] can be used, when the geometrical layout of the pile group allowsfor it. The latter two approaches consist in grouping the piles in concentric circles or soil-pile-stripped upright plates, respectively, both allowing for an easier evaluation of the interactioneffects. Another approach consists in replacing the pile group by a single equivalent uprightbeam [6]. In any case, these approaches are not adapted to the needs of civil and structuralengineers, who need to design pile foundations with little recourse to computational tools.

A more interesting approach for 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 simplesttype of such approaches is based on Winkler’s spring model for the soil, for which radiationdamping and inertial effects are neglected [7, 8, 9]. A relatively simple method was proposed byGazetas and Dobry [18] for estimating the damping characteristics of horizontally loaded singlepile in layered soil. Following Wolf’s approach [10] for the modeling of soil-structure interaction(SSI), other researchers [11, 12, 13, 14, 15, 16] have replaced the soil-pile system by a onedegree-of-freedom (DOF) mass, with a damper and a spring. Inertial effects and radiationdamping 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 improvethese models, Dobry and Gazetas [17] proposed an approximate formulation accounting forthe 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 [19, 20, 21] to attempt to modelmultiple reflections within the pile group in layered soil. However, a few attempts have beenmade at accurately modeling the large pile group foundation, in particular for the complexfrequency-dependance of end-bearing pile foundations. (Konagai [6] provides formulas validonly for sway, Mylonakis and Gazetas [21] provide formulas valid for all movements but onlyfor a group of nine piles and Nikolaou et al. [40] provide for kinematic pile bending for a groupof twenty piles).

Despite the significant progress in pile dynamics [22], there is therefore still a need for simpleengineering procedures for their design, following the example of the code provisions developedfor the seismic design on spread footings [23, 24]. This paper aims at providing such formulas,to be used for both small and large pile groups, as well as for both floating pile groups onhomogeneous half-space and end-bearing pile groups. The main novelty of this paper is thatthe formulas are valid over a range of parameters larger than formulas previously availablein the literature (see above references). They can be used for large numbers of piles. This ismade possible by the use of a very general dynamic model for the representation of stiffnessimpedance matrices, the hidden state variable model (Sec. 3.1). The parameters appearing inthis model are then fitted using an extensive database of full coupled FE-BE computationsof soil-pile systems (Sec. 3.2). The sway and rocking of the foundation are accounted for, ina large range of parameters of the soil and piles, and the formulas are given independentlyfor floating (Sec. 4.1) and end-bearing piles (Sec. 4.2). Further, for the seismic design of abuilding, the use of the simple formulas rather than a full computational model is shown toinduce little error on the evaluation of the response spectra and time histories (Sec. 5).

The reader interested in a fast use of the formulas can refer directly, for a floating pilegroup on homogeneous half-space (respectively, an end-bearing pile group) to Table II (resp.,

Copyright2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 00:1–6Prepared using eqeauth.cls

SIMPLE FORMULAS FOR THE DYNAMIC STIFFNESS OF PILE GROUPS 3

Figure 1. Definition of pile group foundation.

Table IV), the coefficients of which are to be used in Eq. (9) and (8) (resp., Eq. (14) and (10)).In these equations, both the dynamic stiffness and the frequency are normalized, as describedin Sec. 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 pilegroup. The normalizations, that will be used throughout the paper, of both the impedanceand 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. Whenconsidering two layers of soil, the top layer will still be denoted “s” while the bedrock willbe denoted “b”. Es, νs, Gs, and ρs (respectively, Ep, νp, Gp, ρp, and Eb, νb, Gb, ρb) hencedenote Young’s modulus, Poisson’s ratio, the shear modulus, and the unit mass of the soil(respectively, of the piles, and of the bedrock). Vs and βs (resp., Vb and βb) denote the shearwave velocity and the hysteretic damping of the soil (resp., of the bedrock). All the piles ina group are supposed identical, with a diameter d, a length lp, an inertial moment Ip, andthey are separated from each other by a distance s (see Fig. 1). They are rigidly attached toa mass-less square cap with a half-width Bf , which is supposed to have no contact with thesoil. Further we define L0 = (EpIp/Es)

0.25, closely related to the critical pile length definedby several authors [25, 26] and an equivalent radius for the cap Rf = 2Bf/

√π.

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,while, in the latter case, the soil is composed of a layer of thickness H, resting over a bedrock,in which the tips of the piles are embedded (lp > H).

2.2. Impedance matrix

The impedance matrix or dynamic stiffness matrix Z(ω) of a pile group relates the vector offorces and moments applied on the rigid cap at the top of the piles to the resulting vector of



displacements and rotations at the same point. Since the rigid cap is supposed to be square,symmetry considerations ensure that the impedance matrix, in the basis of the rigid bodymovements of the pile cap, is not full. More precisely, only the diagonal terms, and the couplingterms between the sway along one of the axes of symmetry and the rotation around the otherone, are non-null. Besides, the two horizontal axes of symmetry are equivalent, so that thecorresponding terms in the impedance matrix are equal. Hence, only five elements should beconsidered: horizontal, rocking, pumping, torsion, and horizontal-rocking coupling. Finally, asis usually done in earthquake engineering, we disregard the pumping and torsion terms as lesssignificant, and therefore write the impedance as

Z(ω) =

[

Zh(ω) Zhr(ω)Zhr(ω) Zr(ω)

]

, (1)

where Zh(ω), Zr(ω) and Zhr(ω) are, respectively, the horizontal, rocking, and coupling elements.To simplify the comparisons between different soils and/or foundation sizes, it is customary

in foundation design [27, 29] to normalize the impedance matrix by a constant static stiffnessmatrix K0, given here by

K0 =

[

GsRf 00 GsR

3f

]

. (2)

It is also classical to use a dimensionless frequency a0 = ωRf/Vs, where ω is the circularfrequency. The impedance matrix is finally written

Z(a0) = K1/20 Z(a0)K

1/20 , (3)

where the tilde symbol¯refers to normalised quantities.Note that the imaginary part of the impedance matrix is sometimes plotted in the literature

after a division by ω. While this is interesting in the simple cases where that imaginary partremains more or less linear, such as for circular rigid foundations over homogeneous soils, it isnot relevant here, so that that custom will not be followed in the plots of this paper.

3. PRINCIPLE OF THE COMPUTATION OF SIMPLE FORMULAS

As described in the introduction, there is an engineering need for simple formulations of theimpedance matrix of large pile groups. The goal is to avoid using complicated and time-consuming computational models, while retaining their accuracy. The usual approach to thederivation of these simple formulations consists in two steps:

1. choose a structure for the formulation, usually based on physical considerations andsimplifying assumptions

2. set the values of the parameters for a range of soil and pile group characteristics, usuallyprovided in a tabulized form.

In the literature (see the introduction for references), the first step is basically a decision of thedesigner, which, however, conditions in a large part the quality of the formulation. We thereforechoose here a more rational approach, in which no specific structure is chosen a priori, andmake use of the so-called ”hidden variables model” of the impedance matrix [35, 36]. Theidentification of the parameters of the final formulas is then performed by regression from a



database of FE-BE computations. Note that the hidden state variable model that is chosenhere might appear more mathematically than previous formulations found in the literature.However, remember that this structure is just an intermediate state that allows to find thefinal formulas that will eventually be used by the engineers. In section 4, we will see someanalogies between the formulas proposed, and mechanical systems created as sets of spring,dampers and masses. The differences with lumped parameters is that with the hidden statevariables model, the equivalent mechanical model comes out naturally as a consequence of theregression, 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 thesupposition that, besides the nΓ physical DOFs on which the impedance is defined (typicallythe rigid-body modes of the cap of the pile group), there exists nI additional DOFs thatrepresent some internal resonance phenomena inside the soil and the pile group. The resonancemodes corresponding to these DOFs cannot be physically identified, as only their influence onthe impedance matrix is observable, so the DOFs are referred to as “hidden”, or ”inner”.

With respect to the n = nΓ + nI DOFs, matrices of mass M, damping D, and stiffness K

can then be identified, and the dynamic stiffness matrix S(a0) is defined as

S(a0) = (K − a20M) + ia0C. (4)

The impedance matrix corresponding to the hidden variables model is then the condensationon the nΓ physical DOFs of the stiffness matrix S(a0). More specifically, introducing the blockdecomposition of Eq. (4),

[

SΓ(a0) Sc(a0)S

Tc (a0) SI(a0)

]

=

([

KΓ Kc

KTc KI

]

− a20

[

MΓ Mc

MTc MI

])

+ ia0

[

CΓ Cc

CTc CI

]

, (5)

the impedance matrix is defined as

Z(a0) = SΓ(a0) − Sc(a0)S−1I (a0)S

Tc (a0). (6)

As the hidden variables are not necessarily physical DOFs, but rather state variables inthe 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, dampingand stiffness matrices, or to those obtained through the application of some modal reductiontechnique. Another equivalent form of the hidden variables model can be derived [35], wherethe hidden parts of the matrices are diagonal, and with no coupling in mass. In that case, theimpedance can be written

Z(a0) = (KΓ − a20MΓ) + ia0CΓ −

nh∑

`=1

(ia0C`c + K

`c)(ia0C

`c + K

`c)

T

(kÌ − a2

0mÌ) + ia0c`

I

(7)

where C`c and K

`c are the `th columns of Cc and Kc, and m`

I , cÌ and k`

I are the diagonalelements of MI , CI and KI .

The main interest of this hidden variables model is its generality. Its structure makes itsuitable for the representation of any type of impedance matrix, provided that an appropriatenumber of hidden variables is used. Note that the numerical identification of the matrices M,



C, and K is entirely performed from the knowledge only of the impedance matrix, and thatthe number of hidden variables can be automatically chosen based on a precision criteria forthe approximation of the impedance matrix [39, 35, 36].

Contrary to the lumped-parameter models of the impedance matrix [28], in which theidentification of the mechanical elements yields negative values of the springs, dashpots, and/ormasses, in the hidden state variable model, the causality and stability of the soil impedancematrix are directly related to the positivity of M, K and C. In other words, in comparisonwith 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 impedancematrices, and to identify the parameters of the formulas, is described. The methodology forthe identification of the hidden variables model of a given impedance matrix is also describedin App. 6.

3.2. The reference FE-BE model

We suppose, for the reference computations, that both the soil and the piles behave linearlyand that the contact between the piles and the soil is continuous in all directions, without anyslippage or gap. The elastodynamic equations are therefore linear. The numerical approachused to derive the reference results for the calibration of the simple formulations is basedon an efficient FE-BE coupling technique that is described in detail in [30, 31] and is brieflyrecalled below.

The soil is separated into two blocks: one, bounded and containing the piles, which is modeledby the FE method, and the other, surrounding the previous one, which is modeled by the BEmethod (see Fig. 2). Within the FE block, the piles are modeled as Bernouilli beam elements.The two blocks are then assembled using the Craig-Bampton coupling technique [32], so as tolower the computational cost, which may reach high levels for large pile groups. This numericalmodel was already validated for stiffness problems taken from the literature (in particular [2])and the results are given in [31]. However, these validation results only concerned floatingpile groups on homogeneous half-spaces so that we present here a comparison, on a particularexample, of the FE-BE model with the BE approach described in [34].

We therefore consider a 4 × 9 pile group embedded in a soil with two layers (see Fig. 2).The piles have a Young’s modulus of 25 GPa, a diameter of d = 1.3 m and are separated bys = 2.6 m. The first layer of soil is H = 9.5 m-thick, and is formed of a very soft saturatedorganic clay with S-wave velocity Vs = 80 m/s, unit mass ρs = 1.5 Mg/m3 and Poisson’s ratioνs = 0.49. The lower layer of soil is a stiff sand with S-wave velocity Vd = 300 m/s, unit massρd = 2 Mg/m3 and Poisson’s ratio νd = 0.4, in which the piles penetrate 6 m. In both layersthe hysteretic damping is taken as βs = βd = 0.05. As seen on Fig. 2, the agreement betweenthe results in the two numerical approaches is very good.

It should be noted that the frequency-dependance of pile groups is particularly sensitive tothe number of piles and to its character of floating or end-bearing. The dynamic stiffness ofsingle piles and pile groups with a small number of piles is nearly independent of frequency [33],while that of larger pile groups may show large variations with frequency. Likewise, the behaviorof end-bearing pile groups is much more erratic with frequency than that of floating pile groupson homogeneous half-space. These physical results are retrieved with the FE-BE approach andan example of such comparison is shown in Fig. 3. These results were obtained considering thesample number 4 in Tables I and III.



0 2 4 6 8 10−6

−4

−2

0

2x 10

9

Frequency [Hz]

Dyn

amic

stif

fnes

s [N

/m]

0 2 4 6 8 100

2

4

6

8x 10

9

Frequency [Hz]

Dam

ping

[N/m

]

Figure 2. FE model (left) of the 4 × 9 pile group within a block of soil andcomparison of the real (right, up) and imaginary (right, down) parts of thehorizontal impedance computed using the FE-BE model (solid line) and the BE

model (dashed line) [34].

0 1 2 3 4 5 6−30

−20

−10

0

10

20

30

Dimensionless Frequency [−]

Nor

mal

ized

Rea

l Par

t [−

]

0 1 2 3 4 5 60

10

20

30

40

50

60


Nor

mal

ized

Imag

inar

y P

art [

−]

Figure 3. Real (left) and imaginary (right) parts of the normalized horizontalimpedance matrix, for floating (solid line) and end-bearing (dashed line) pile

groups.

It is also interesting to note, on Fig. 3, for the end-bearing pile group, that the imaginarypart of the impedance (it is also true for the rocking term, not shown here) present a small andalmost constant value below some cut-off frequency, which is the resonance frequency of thetop layer of soil. Indeed, for very low frequencies, surface waves cannot build up in that toplayer 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 thereal part almost cancelling), indicating a resonance within the soil, that tends to soak energyaway from the foundation.

4. COMPUTATION OF SIMPLE FORMULAS FOR PILE GROUPS

In this section, we present the derivation of the simple formulas in the cases of the floatingpile groups on homogeneous half-space and end-bearing pile groups, and using the ideas



0 1 2 3 4 5 6−15

−10

−5

0

5

10


Nor

mal

ized

Rea

l Par

t [−

]

0 1 2 3 4 5 60

10

20

30

40

50


Nor

mal

ized

Imag

inar

y P

art [

−]

Figure 4. Real (left) and imaginary (right) parts of the horizontal impedancematrix for different pile separations: s/d = 2 (solid line), s/d = 2.5 (dashed line)and s/d = 3.5 (solid-dashed line). The figures correspond to a 15 × 15 pile group

with Ep/Es = 300 and Rf/lp = 1.1.

0 1 2 3 4 5 6−40

−30

−20

−10

0

10

20


Nor

mal

ized

Rea

l Par

t [−

]

0 1 2 3 4 5 60

10

20

30

40

50

60


Nor

mal

ized

Imag

inar

y P

art [

−]

Figure 5. Real (left) and imaginary (right) parts of the rocking term of thedynamic stiffness matrix for different pile separations: Rf/lp = 0.7 (solid line),Rf/lp = 0.65 (dashed line) and Rf/lp = 0.55 (solid-dashed line). The figures

correspond to a 14 × 14 pile group with Ep/Es = 375 and s/d = 2.

discussed above. Depending on the type of pile group, and on the type of element of theimpedance 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 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 Fig. 4and 5. More precisely, the dynamic stiffness always has a parabolic variation while the dampingcoefficient is approximately linear. The parabolic decrease of the real part seems to indicatethat 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) atwo-DOFs system, one for the sway and one for the rocking, and with no hidden variables. Note



Table I. The database of soil-pile group systems used to derive the simpleformulations for floating pile groups on homogeneous half-space. The range ofparameters is 250 ≤ Ep/Es ≤ 1500, 2 ≤ s/dp ≤ 3.6 and 0.55 ≤ Rf/lp ≤ 2 and

constant hysteric damping βs = 0.05.

Sample Piles Ep d Es Bf lp s Vs

[-] [GPa] [m] [GPa] [m] [m] [m] [m/s]

1 8 × 8 30 1 0.1 15 18 3.3 140

2 11 × 11 30 1 0.1 20 18 3.3 140

3 16 × 16 30 1 0.1 30 18 3.3 140

4 13 × 13 40 1 0.08 20 24 2.8 130

5 13 × 13 30 1 0.08 20 24 2.8 130

6 13 × 13 20 1 0.08 20 24 2.8 130

7 15 × 15 25 1.3 0.08 20 18 2.5 130

8 15 × 15 25 1 0.08 20 18 2.5 130

9 15 × 15 25 0.7 0.08 20 18 2.5 130

10 18 × 18 30 1 0.02 25 14 2.6 60

11 18 × 18 30 1 0.08 25 14 2.6 130

12 18 × 18 30 1 0.1 25 14 2.6 140

13 16 × 16 30 1 0.08 15 28 2.0 130

14 16 × 16 30 1 0.08 15 20 2.0 130

15 16 × 16 30 1 0.08 15 14 2.0 130

that the coupling term is negligible. On Fig. 6, a schematic drawing of a system correspondingto such impedance is presented. The superstructure is subjected to the seismic horizontal forcefs. The elements of the normalized impedance can therefore be written

Zh(a0) = (kh − a20mh) + ia0ch

Zr(a0) = (kr − a20mr) + ia0cr

Zsr(a0) = 0

, (8)

where the values of kh, ch, mh, kr, cr and mr depend on the case considered. Remember thatthe definition of the normalized frequency a0 is given in Sec. 2.2 and that the normalizedvalues in these formulas (8) must be scaled by the static stiffness to yield the actual value ofthe impedance matrix, as described in Sec. 2.2.

In previous works, the leading parameters for this type of pile groups were identified to bethe ratio of Young’s moduli Ep/Es and the normalized separation of the piles s/d [2, 37], or thefactor L0 = (EpIp/Es)

0.25 related to the active pile length [38, 25, 26]. We decide here to use asleading 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 χ ∈ kh, ch, mh, kr, cr, mr



Figure 6. A schematic drawing of a simple model for floating pile group.

in the form

χ = λ0

(

Rf

lp

)λ1(

L0

s

)λ2

, (9)

with the values of λ0, λ1 and λ2 being provided for each of the parameters. A multiple regressionanalysis was then conducted with respect to the two quantities Rf/lp and L0/s, and lead tothe values described in Table II. The regression coefficient R is also indicated in the same tableto provide an indicator of the accuracy of the regression analysis.

In general terms, the formulas in Table II corroborate the observed results that, for shortseparations between the piles, and for weak soils, both normalized dynamic stiffness anddamping increase. Note that, as indicated by the zeros in Table II, the influence of the ratioRf/lp on the horizontal impedance is negligible, while it is rather important for the rockingterm. Besides the uniform presentation in Table II, the reader may also find an expanded,non-normalized, version of the same formulas in App. 6, for easier reading.

4.2. End-bearing pile groups

We then consider end-bearing pile groups. As stated earlier, their dynamical behavior is muchmore complicated than that of floating pile groups on homogeneous half-space. The structureof the approximation for the impedance matrix is therefore difficult to guess a priori and weuse the hidden variables model in a very general setting. Note that, as the coupling term isnegligible in the cases considered, the hidden variables model was identified independently onthe horizontal and rocking terms of the impedance matrix.

The identification of the hidden variables model for all the cases in the database describedin Table IV suggest the consideration of three hidden variables for the sway and none for therocking. 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



Table II. Coefficients for the horizontal and rocking elements of the impedance ofa floating pile groups on homogeneous half-space.

χ = λ0

(

Rf

lp

)λ1 (

L0

s

)λ2

λ0 λ1 λ2 R [%]

kh 6.8 0 0.3 80

ch 5 0 0.5 90

mh 0.4 0 1.6 86

kr 8 -0.6 0.4 83

cr 5 -0.5 0.2 72

mr = mhl2eq/4 0.7 -1 0.4 96

pile groups is therefore written, as a special case of Eq. (7) for the hidden variables model,

Zh(a0) = (kh − a20mh) + ia0ch +

a2

0c2

1

(k1−a2

0m1)+ia0c1

− k2

2

(k2−a2

0m2)+ia0c2

− k2

3

(k3−a2

0m3)+ia0c3

Zr(a0) = (kr − a20mr) + ia0cr

Zsr(a0) = 0

,

(10)and represented as a set of mass, springs, and dampers in Fig. 7. The previous observation forthe coupling with the hidden variables can be translated in Fig. 7 by the fact the mass m1 islinked to the foundation by a dashpot while the masses m2 and m3 are linked to it throughsprings. This fact arises from the presence of the cut-off frequency of the top layer of soil thatwas discussed in Sec. 3.2.

More physical remarks can be made in the different frequency ranges defined by theresonance frequencies a0α of the masses mα representing the hidden variables. In the lowfrequency 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 sinceit allows to quickly reach the level of the hysteretical damping. In the range of resonance ofmass m1 (a0 − a01 2ζ1a01, with ζα = cα/(2

√kαmα)), and supposing that all the resonance

frequencies are far enough from each other (a0 − a02 2ζ2a02 and a0 − a03 2ζ3a03), onehas

Zh(a0) =

[

kh + k1 − k2a202

a202 − a2

0

− k3a203

a203 − a2

0

]

+ ia0

[

ch − c1 + c2

(

a202

a202 − a2

0

)2

+ c3

(

a203

a203 − a2

0

)2]

(12)

which means that around a01, the mass m1 has the same displacement as the foundation, sothat there is no damping contribution from c1. For a0 a02, masses m2 and m3 are also linkedto the foundation but the dashpots c2 and c3 introduce some damping. The equivalent slope



Figure 7. A schematic drawing of a simple model for end bearing pile group.

around a01 tends to c0 + c2 + c3 = βeq/a01 which is actually small as expected to model thesole hysteretical damping βeq. Usually c1 a01βeq. In the range of resonance of the mass m2

(an equivalent formula can be derived for mass m3), a large imaginary part is brought on bykα

2ζα, which corresponds to the peaks observed in Fig. 3. Finally, at high frequency (a0 a03),

one has

Zh(a0) =

(

kh − c21

m1− a2

0mh

)

+ ia0ch (13)

which classically corresponds to all the masses m1, m2 and m3 being fixed. One can see c1

as the radiative damping which occurs only above a01 since for this frequency we have shownthat the damping is only βeq/a01. Thus this model reproduces the cut-off frequency at theresonance frequency of the layer.

Once the structure of the approximation has been decided, a multiple regression analysisis performed on the same leading parameters as before, plus the ratio (ρbVb)/(ρsVs) to yieldthe 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 alsothat, as before, the formulas are presented in a non-normalized manner in App. 6 for easierreading. The general formulas for the parameters are

χ = λ0

(

Rf

H

)λ1(

L0

s

)λ2(

ρbVb

ρsVs

)λ3

. (14)

It is particularly interesting to note that, although the formulas were derived from rathermathematical considerations (the hidden variables model and a regression analysis), theyyield a very good evaluation of the resonance frequencies of the soil layer. Indeed, the firstfundamental frequency of the soil layer ωs

01 = 2πVs/(4H) and√

k1/m1 = 1.4Vs/H (see App. 6for non-normalized formulas) coincide. Likewise, the second fundamental frequency of the soil



Table III. The database of soil-pile group systems used to derive the simpleformulations for end-bearing pile groups. The range of parameters is 125 ≤

Ep/Es ≤ 750, 2.8 ≤ s/dp ≤ 4.4, 1 ≤ Rf/H ≤ 2.1 and 3 ≤ Vb/Vs ≤ 8 andconstant hysteretic damping βs = 0.05.

Samples Piles Ep d Es Bf lp H Vs Vb s

[-] [GPa] [m] [GPa] [m] [m] [m] [m/s] [m/s] [m]

1 8 × 8 30 1 0.1 15 16 14 140 830 3.3

2 11 × 11 30 1 0.1 20 16 14 140 830 3.3

3 16 × 16 30 1 0.1 30 16 14 140 830 3.3

4 13 × 13 40 1 0.08 20 22 20 130 620 2.8

5 13 × 13 30 1 0.08 20 22 20 130 620 2.8

6 13 × 13 20 1 0.08 20 22 20 130 620 2.8

7 15 × 15 25 1.3 0.2 37 20 18 200 780 4.3

8 18 × 18 30 1 0.04 32 20 18 90 440 3.1

9 18 × 18 30 1 0.1 32 20 18 140 700 3.1

10 14 × 14 30 1 0.08 26 26 24 130 1000 3.3

11 14 × 14 30 1 0.08 26 20 18 130 1000 3.3

12 14 × 14 30 1 0.08 26 16 14 130 1000 3.3

13 13 × 13 30 1 0.06 26 18 16 110 620 3.6

14 13 × 13 30 1 0.06 26 18 16 110 620 3.6

15 13 × 13 30 1 0.06 26 18 16 110 620 3.6

layer ωs02 = 6πVs/(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 ontop of a building and relative displacement between top and bottom of the building, usingthe proposed formulas, is demonstrated. In a second test, we compare the accuracy of ourproposed 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 = 5 m, and connected by a 1.1 m-thick, rigid, cap with Bf = 25 m. The mechanicalproperties of the piles are Ep = 30 GPa, νp = 0.25, and ρp = 2500 kg/m3. This pile groupstands in H = 20 m-thick soil layer, with properties Es = 60 MPa, νs = 0.4 and ρs = 1750



Table IV. Coefficients for the horizontal and rocking elements of the impedanceof an end-bearing pile group.

χ = λ0

(

Rf

H

)λ1 (

L0

s

)λ2

(

ρbVb

ρsVs

)λ3

λ0 λ1 λ2 λ3 R [%]

k0 = kh − k2 − k3 10 0.5 0.35 0 93

ch = c0 + c1 1 -0.5 0.5 0.5 60

mh = m0 0.5 -1 0 0 82

k1 2.6 1 0 0 65

c1 1.9 -1.5 0 0 75

m1 1.4 -1 0 0 80

k2 1.25 0.35 -1 0.5 60

c2 0.04 0 -1 1 62

m2 0.08 -1 -1 0.5 80

k3 16.1 3 3 -0.5 75

c3 3 2 3 -1.5 70

m3 0.6 1 3 -0.5 75

kr 15 0.5 1 1 98

cr 17 0.5 2 -0.5 95

mr = m0l2eq/4 1.6 -1.5 1 -0.5 90

kg/m3. The mechanical properties of the underlying half-space are Eb = 1.5 GPa and νb = 0.3and ρb = 2000 kg/m3. The real and imaginary parts of the impedance are shown on Fig. 8, bothas computed using the numerical FE-BE model, and using the simple formulas of Eq. (10). Theagreement between the two approaches is good, in particular for the shaking term, consideringthe important variability in frequency. Note that the pile group considered here was not usedfor the regression analysis that determined the parameters in Table IV

We now turn to the observation of the accuracy of the proposed formulations for theestimation of engineering quantities of interest. We therefore consider a 60 m high-building(20 floors), with floors of 22.5 m ×22.5 m, and 6 columns ×6 columns. The slab weight perunit area is 500 kg/m2 and the characteristics of the beams and columns are, respectively,EI = 5.1 MN.m2 and EI = 1 MN.m2.

We first consider the estimation of transfer functions in two different cases: (1) using theentire, 6 × 6, impedance matrix computed from the FE-BE model, and considering both thekinematic and inertial interaction, and (2) using only the horizontal and rocking elements ofthe impedance matrix computed with the proposed formula (10) and neglecting the kinematicinteraction. For both cases, the displacement field is decomposed on a basis that contains therigid body modes of the building (lm), which coincide with those of the foundation, and theflexible modes of the building on a rigid basis (φn):

u(ω,x) =∑

m

cm(ω)lm(x) +∑

n

αn(ω)φn(x) =[

c α]

L

Φ

(15)



0 1 2 3 4 5−1

0

1x 10

10

Frequency [Hz]

Rea

l par

t [N

/m]

0 1 2 3 4 50

1

2x 10

10

Frequency [Hz]

Imag

inar

y pa

rt [N

.m]

0 1 2 3 4 50

1

2x 10

13

Frequency [Hz]

Rea

l par

t [N

/m]

0 1 2 3 4 50

1

2x 10

13

Frequency [Hz]

Imag

inar

y pa

rt [N

.m]

Figure 8. Comparison between the real (up) and imaginary (down) parts of thehorizontal (left) and rocking (right) elements of the impedance matrix for a 10×10end-bearing pile group computed using the simplified formulas (10) (dashed line)

and the FE-BE model (solid line).

where L is the matrix of the rigid body modes of the structure and Φ is the matrix of theeigenmodes of the structure clamped at its base. The response of the structure, taking intoaccount soil-structure interaction, is then computed using the following formula

([

Z(ω) 0

0 0

]

+ (1 + 2iβ)

[

0 0

0 Λ

]

− ω2

[

MΓ MΓΩ

MΩΓ I

])

c(ω)α(ω)

=

Z(ω)c0(ω)0

(16)

where the diagonal matrix Λ contains the squares of the lowest circular frequencies of thestructure on fixed base and I is the identity matrix arising from the orthogonality of theeigenmodes with respect to the mass matrix. Γ stands for the rigid body modes and Ω for theeigenmodes on fixed base, while c0 is the kinematic interaction. The differences between thetwo models with respect to this formulation are the impedance matrix Z(ω) and the kinematicinteraction factor takes equal to ∆ui(ω) with ∆ having null components but a unitary forthe sway term. Besides, it is worth noticing that the simplified model does not correspond tothe physical model sketched on Fig. 7 subjected to an uniform acceleration ai. Indeed, inertialforces are not applied on mass m1, m2 and m3 since these masses are in the soil and have theirinertial forces already balanced in the soil.

The resonance frequencies of the soil are computed at fs01 = 1.55 Hz and fs

02 = 4.6 Hz.Assuming a horizontal harmonic base motion at the bedrock, the horizontal transfer functionat the free surface and at the top of the building are represented on Fig. 9. It clearly shows theeffect of the soil-structure interaction, as well as the ability of the formulas (10) to computethe resonance frequency of the coupled system (the peaks of the dotted and dash-dotted lineson Fig. 9 coincide almost exactly).

We then consider two real recordings of earthquakes, with different frequency contents (seeFig. 10) and peak ground accelerations (PGA) at about 0.3 g. On Fig. 11 a comparison is givenof the spectral acceleration on top of the building computed in the two cases considered earlierof the FE-BE model supposing inertial and kinematic interaction and the simple formulas (10)neglecting the kinematic interaction. On Fig. (12), a comparison is given of the time historiesof the relative displacements between the top and the base of the building. On both figures,the agreement between the two approaches is very good.



0 1 2 3 4 50

2

4

6

8

10

12

14

Frequency [Hz]

Tra

nsfe

r F

unct

ion

[−]

Figure 9. Transfer function at ground surface free field (solid line), of the structurewithout SSI (dashed line), and of the structure with SSI (dashed-dotted line), allcomputed with the FE-BE approach, and transfer function of the structure withSSI computed with the proposed formulas (dotted line). The figures corresponds

to a structure resting on a 10 × 10 end-bearing pile group.

0 4 8 12 16−3

0

3

Time [s]

Acc

eler

atio

n [m

/s2 ]

0 4 8 12 16−3

0

3

Time [s]

Acc

eler

atio

n [m

/s2 ]

0 4 8 12 160

2

4

6

8

10

Frequncy [Hz]

Spe

ctra

l acc

eler

atio

n [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).

5.2. Case 2

In this example, we compare our simplified formulations 10 for the impedance of the end-bearing pile group introduced in Sec. 3.2 with the simplified formulas proposed in [34, 21]. Weuse an equivalent with of Rf = 8 m, because the formulas in Table IV are proposed for circularor square foundation. Fig. 13 shows this comparison, along with the value of the BE solution.Our formulas seems to behave at least as well as the previously available one. Remember thatits range of application, in in particular in terms of the numbers of piles is much larger.



0 2 4 6 8 100

2.5

5

7.5

10

Frequency [Hz]

Spe

ctra

l acc

eler

atio

n [m

/s2 ]

0 2 4 6 8 100

2

4

6

8

Frequency [Hz]

Spe

ctra

l acc

eler

atio

n [m

/s2 ]

Figure 11. Comparison of the acceleration response spectra at the top of thebuilding for the Friuli earthquake (left) and the Aegion earthquake (right), usingthe complete FE-BE model (solid line) and the simple formulation (dashed line).The figures corresponds to a structure resting on a 10×10 end-bearing pile group.

0 6 12 18 24 30−0.12

−0.08

−0.04

0

0.04

0.08

0.12

Time [s]

Rel

ativ

e di

spla

cem

ent [

m]

0 6 12 18 24 30−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time [s]

Rel

ativ

e di

spla

cem

ent [

m]

Figure 12. Comparison of the relative displacements between the top and the baseof the building for the Friuli earthquake (left) and the Aegion earthquake (right),using the complete FE-BE model (solid line) and the simple formulation (dashedline). The figures corresponds to a structure resting on a 10× 10 end-bearing pile

group.

6. CONCLUSIONSimple formulations have been derived for the dynamic stiffness matrices of pile groupfoundations subjected to horizontal and rocking dynamic loads. These formulations were foundusing a large database of impedance matrices computed using a FE-BE model. They can bereadily employed for design of large foundations on piles and are shown to yield very accuratevalues of the estimated quantities of interest for building design. The formulations have beenderived both for floating pile groups on homogeneous half-space and end-bearing pile groupsin a homogenous stratum. They can be used for large pile groups (n ≥ 50), as well as for alarge range of mechanical and geometrical parameters of the soil and the piles. They providea first step towards code provisions specifically focused on pile footings.



0 1 2 3 4 5 6−2

−1

0

1

2

3x 10

9

Frequency [Hz]

Dyn

amic

stif

fnes

s [N

/m]

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5x 10

9

Frequency [Hz]

Dam

ping

[N/m

]

Figure 13. Real (left) and imaginary (right) parts of the horizontal impedancematrix, for a 36 end-bearing pile group computed using the simplifiedformulas (10) (solid line) and BE solution (dashed line) and simplified analytical

solution of [8] (dotted line).

REFERENCES

1. Wolf JP. Foundation analysis using simple physical model. Prentice-Hall: Englewood Cliffs, New-Jersey,1994.

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

3. Waas G, Hartmann HG. Seismic analysis of pile foundations including soil-pile-soil interaction. Proceedingsof the 8th World Conference on Earthquake Engineering San Francisco, July 1984; 555–62.

4. Takemiya H. Ring-Pile analysis for a grouped pile foundation subjected to base motion. StructuralEngineering/Earthquake Engineering 1986; 3(1):195–202.

5. Ohira A, Tazoh T, Dewa T, Shimizu K, Shimada M. Observation of earthquake response behaviours offoundations piles for road bridge. Proceedings of the 8th World Conference on Earthquake Engineering,San Francisco, July 1984; 3:577–584.

6. Konagai K, Ahsan R, Maruyama D. Simple expression of the dynamic stiffness of grouped piles in swaymotion. Journal of Earthquake Engineering 2000; 4(3):355–376.

7. Crouse CB, Cheang L. Dynamic testing and analysis of pile group foundation. In Dynamic Response ofPile Foundations-Experiment, Analysis and Observation, Nogami T. (ed). ASCE: New-York, 1987; 79–98.

8. Mylonakis G, Nikolaou A, Gazetas G. Soil-Pile-Bridge seismic interaction: kinematic and inertial effects.Part I: soft soil. Earthquake Engineering and Structural Dynamics 1997; 26(3):337–359.

9. Hutchinson TC, Chai YH, Boulanger RW, Idriss IM. Inelastic seismic response of extended pile-shaft-supported bridge structures. Earthquake Spectra 2004; 20(4):1057–1080.

10. Wolf JP. Soil-structure-interaction analysis in time domain. Prentice-Hall: Englewood Cliffs, New-Jersey,1988.

11. Levine MB, Scott RF. Dynamic response verification of simplified bridge-foundation model. Journal ofGeotechnical Engineering 1989; 115(2):1246–1260.

12. Spyrakos CC. Assessment of SSI on the longitudinal seismic response of short span bridges. EngineeringStructures 1990; 12(1):60–66.

13. Harada T, Yamashita N, Sakanashi K. Theoretical study on fundamental period and damping ratio of bridgepier-foubdation system. In Proceedings of the Japan Society of Civil Engineers 1994; 489(1-27):227–234.

14. Chaudhary MS, Parakash S. Dynamic soil structure interaction for bridge abutment on piles. InGeotechnical Special Publication 1998;2:1247–1258.

15. Spyrakos CC, Loannidis G. Seismic behavior of a post-tensioned integral bridge including soil-structureinteraction (SSI). Soil Dynamics and Earthquake Engineering 2003; 23(1):53–63.

16. Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soil-structure interaction. SoilDynamics and Earthquake Engineering 2003; 23(4):287–302.

17. Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechnique1988; 38(4):557–574.



18. Gazetas G., Dobry R. Horizontal response of piles in layered soil. Journal of Geotechnical Engineering1984; 110(1):20–34.

19. Gazetas G, Makris N. Dynamic pile-soil-pile interaction. Part I: Analysis of axial vibration. EarthquakeEngineering and Structural Dynamics 1991; 20(2):115–132.

20. Makris N, Gazetas G. Dynamic pile-soil-pile interaction. Part II: Lateral and seismic response. EarthquakeEngineering and Structural Dynamics 1992; 21(2):145–162.

21. Mylonakis G, Gazetas G. Lateral vibration and internal forces of grouped piles in layered soil. Journal ofGeotechnical and Geoenvironmental Engineering, ASCE 1999; 125(1):16–25.

22. Pender M. Aseismic pile foundation design analysis. Bulletin of the New Zealand National Society forEarthquake Engineering 1993; 26(1):49-160.

23. ATC: Tentative provisions for the development of seismic regulations of buildings: cooperative effort withthe design profession, building code interests and the research community. Washington, D.C, 1978.

24. NEHRP, National Earthquake Hazards blackuction Program. Recommended provisions for the developmentof seismic regulations for new buildings. Washington, D.C, 1997.

25. Randolph M. Response of Flexible Piles to Lateral Loading. In Geotechnique, 1981; 31(2):247–259.26. Poulos HG, Hull TS. The role of analytical geomechanics in foundation engineering. In Foundation

Engineering: Current Principles and Practices, ASCE, 1989; 2:1578–1606.27. Gazetas G. Analysis of machine foundation vibrations: state of the art. In Soil Dynamics and Earthquake

Engineering, 1983; 2(1):2–42.28. Wolf JP. Consistent lumped-parameter models for unbounded soil: physical representation. In Earthquake

Engineering and Structural Dynamics, 1991; 20(1):12–32.

29. Sieffert JG, Cevaer F. Handbook of impedance functions. Surface foundations. Ouest Editions, 1992.30. Clouteau D, Taherzadeh R. Soil, pile group and building interactions under seismic loading. In Proceedings

of the First European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland,September 2006, in CDROM.

31. Taherzadeh R, Clouteau D, Cottereau R. Identification of the essential parameters for the lateral impedanceof large pile groups. In Proceedings of the 4th International Conference on Geotechnical EarthquakeEngineering, Thessaloniki, Greece, June 2007, in CDROM.

32. Craig RJ, Bampton M. Coupling of substructures for dynamic analyses. AIAA Journal 1968; 6(7):1313–1319.

33. Miura K, Kaynia AM, Masuda K, Kitamura E, Seto Y. Dynamic behaviour of pile foundations inhomogeneous and non-homogeneous media. Earthquake Engineering and Structural Dynamics 1994;23(2):183–192.

34. Gazetas G, Hess P, Zinn R, Mylonakis G, Nikolaou A. Seismic response of a large pile group. In Proceedingsof the 11th European Conference on Earthquake Engineering, Paris, September 1998, in CDROM.

35. Cottereau R, Clouteau D, Soize C. Construction of a probabilistic model for impedance matrices. ComputerMethods in Applied Mechanics and Engineering 2007; 196(17-20):2252–2268.

36. Cottereau R, Clouteau D, Soize C. Probabilistic impedance of foundation: Impact of the seismic design onuncertain soils. Earthquake Engineering and Structural Dynamics 2008; 37(6):899-918.

37. Gazetas G. Seismic response of end-bearing single piles. International Journal of Soil Dynamics andEarthquake Engineering 1984; 3(2):82–93.

38. Poulos HG. Behavior of laterally loaded piles: PART II - group piles. Journal of the Soil Mechanics andFoundations Division, ASCE, 1971;733–751.

39. Cottereau R. Probabilistic models of impedance matrices. PhD thesis, Ecole Centrale Paris, France,(http://tel.archives-ouvertes.fr/tel-00132950/en/); 2006.

40. Nikolaou S., Mylonakis G., Gazetas G. and Tazoh T. Kinematic pile bending during earthquake: analysisand field measurement. Geotechnique 2001; 51(5):425-440.

APPENDIX A

In this appendix, the practical methodology for the construction of the reduced matrixS(a0) = K − a2

0M + ia0C is introduced. Three main steps are identified:

The impedance of the FE-BE model is computed. More specially a set of values Z(a0l)of the impedance matrix at a finite number of frequencies (a0l)1≤l≤L is computed.The set of values Z(a0l) is interpolated to yield a matrix-valued rational function in

the form a0 → N(a0)/q(a0), which approximates the behavior of the impedance matrixZ(a0) of the model. The function a0 → N(a0) is a matrix-valued polynomial in (ia0),



and the function a0 → q(a0) is a scalar polynomial in (ia0). Many methods can be usedto 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 isfurther detailed in [39].

APPENDIX B

In this appendix, we present an extended version of the formulas presented in Tables IIand IV, in a non-normalized form. For the case of the floating pile groups on homogeneoushalf-space the coefficients appearing in the non-normalized version of Eq. (8) are

kh = 6.8GsRf

(

L0

s

)0.3

ch = 5GsR2

f

Vs

√

L0

s

mh = 0.4ρsR3f

(

L0

s

)1.6

(17)

and

kr = 8GsR3f

(

lpRf

)0.6(

L0

s

)0.4

cr = 5GsR3

f

Vs

√

Rf lp(

L0

s

)0.2

mr = 0.7GsR4f lp

(

L0

s

)0.4

(18)

The range of parameters is 250 ≤ Ep/Es ≤ 1500, 2 ≤ s/dp ≤ 3.6 and 0.55 ≤ Rf/lp ≤ 2 andconstant hysteric damping βs = 0.05.

For the case of end-bearing pile groups, the coefficients appearing in the non-normalizedversion of Eq. (10) are

k0 = kh − k2 − k3 = 10GsRf

(

Rf

H

)0.5(

L0

s

)0.35

ch = c0 + c1 =GsR2

f

Vs

√

HRf

L0

sρbVb

ρsVs

mh = 12ρsHR2

f

, (19)

k1 = 2.6GsR2

f

H

c1 = 1.9GsR2

f

Vs(HRf )

1.5

m1 = 1.4ρsR2fH

, (20)

k2 = 1.25GsRf

(

Rf

H

)0.35s

L0

√

ρbVb

ρsVs

c2 = 0.04GsR2f

ρbVb

ρsV 2s

(

sL0

)

m2 = 0.08ρsR2fH s

L0

√

ρbVb

ρsVs

, (21)

k3 = 16.1GsR4

f

H3

(

L0

s

)3√

ρsVs

ρbVb

c3 = 3Gs

Vs

R4

f

H2

(

L0

s

)3(

ρsVs

ρbVb

)1.5

m3 = 0.6ρsR4

f

H1

(

L0

s

)3√

ρsVs

ρbVb

, (22)



and

kr = 15GsR3f

√

Rf

HL0

sρbVb

ρsVs

cr = 17GsR4

f

Vs

(

L0

s

)2√

Rf

HρsVs

ρbVb

mr = 1.6ρsR4f

L0

s

√

RfH ρsVs

ρbVb

. (23)

The range of parameters is 125 ≤ Ep/Es ≤ 750, 2.8 ≤ s/dp ≤ 4.4, 1 ≤ Rf/H ≤ 2.1 and3 ≤ Vb/Vs ≤ 8 and constant hysteretic damping βs = 0.05.


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