Vibration Control of Structures through Structure-Soil ...paginas.fe.up.pt/~eurodyn2014/CD/papers/077_MS02_ABS_1619.pdf · vibration control. In this paper it is proposed for the

ABSTRACT: Devices such as isolators, dampers and tuned mass dampers are now widely used in the construction industry for earthquake engineering to reduce vibration in new and, in a few cases, existing buildings. The application in to existing building is in general limited by the costs and in the case of historical building by local regulations. Aiming at the vibration control of existing structures in this paper it is proposed for the first time to exploit the structure-soil-structure mechanism as a vehicle to reduce the vibrations of structures due to seismic action. Specifically a novel device, herein called Vibrating Barrier (ViBa) hosted in the soil and detached from the structure is designed to protect a given structure. The ViBa is a massive structure whose structural parameters are calibrated to absorb portion of the ground motion input energy so to reduce the vibrations of the structure. Modelling the ground motion as zero-mean quasi-stationary response-spectrum-compatible Gaussian stochastic process, the soil as visco-elastic medium and linear behaving structures various examples are produced to show the effectiveness of the novel approach. Reduction up to 44% of the response has been achieved.

KEY WORDS: Structure-Soil-Structure-Interaction, vibration control, Ground motion excitation, vibrating barrier.

1 INTRODUCTION

The problem of reducing vibrations in structures, generally known as vibration control, arises in various branches of engineering: civil, aeronautical and mechanical. Unpredicted vibrations can lead to the deterioration or collapse of structures. In the field of earthquake engineering, modern strategies of seismic design aim to reduce structural vibrations by: i) increasing the dissipative properties of the structure; ii) moving the natural frequencies of the structure away from the frequencies in which the seismic action possess the highest energy; iii) modifying the energy transferred from the earthquake to the structure. The control of vibrations of structures is currently performed using passive, active or hybrid strategies. In the framework of passive control systems is it possible to categorize the following three general devices: i) tuned mass dampers, manufactured by adding one or more oscillators to the structure; ii) dampers able to transform the seismic energy in heat which is then diffused in the environment; iii) isolation systems, used mainly for earthquake applications, based on the idea of shift the fundamental frequency of the superstructure far from the main frequency of the earthquake. Apart from few attempts to protect existing structures the use of vibration control devices is still restricted to new buildings and/or constructions. One main reason is that the introduction of control devices in existing structures is too invasive, costly and requires the demolishing of some structural and/or non-structural components. This is clearly prohibitive for developing countries and for historical buildings. An alternative solution is to protect the structures introducing trenches or sheet-pile walls in the soil. However this approach seems to be more effective for surface waves coming from railways rather than seismic waves.

There are, therefore, up-to-date no currently feasible solutions to protect cities from seismic ground vibrations. Bearing in mind the global necessity to protect existing structures from earthquakes and the limitation of current technologies a novel control strategy is proposed in this paper. The concept is based on the generally known structure-soil-structure interaction (SSSI) and on the findings in the pioneristic works of Warburton et al. [1] and Luco and Contesse [2].

The dynamic structure-soil-structure interaction among the structures occurs through the radiation energy emitted from a vibrating structure to the other structure. As a consequence, the dynamic response of one structure cannot be studied independently from the other one. Warburton et al. [1] studied the dynamic response of two rigid masses in an elastic subspace showing the influence of one mass respect to the other. Luco and Contesse [2] studied the dynamic interaction between two parallel infinite shear walls placed on rigid foundations and forced by vertically incident SH wave. They showed the interaction effects are especially important for a small shear wall located close to a larger structure. Kobori and Kusakabe [3] extended the structure-soil-structure interaction study to flexible structures and pointed out that the response of a structure might be sensibly smaller due the presence and interaction of another structure. A recent review of the structure-soil-structure interaction problem can be found in Menglin et al.[4]. An extension to the traditional structure-soil-structure interaction problem where only two structures are considered in the study is performed through the site-city interaction problem. Due to the difficulties involved in modelling the multiple interactions and the sustained progress in computational mechanics numerical approaches based on wave propagation and finite or boundary element analysis are usually adopted for the study of site city interaction [5-7].

Vibration Control of Structures through Structure-Soil-Structure-Interaction

Pierfrancesco Cacciola1 and Alessandro Tombari1 1School of Environment and Technology, University of Brighton, Cockcroft Building, Lewes Road, BN24GJ Brighton, UK

email: [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

559

Interestingly in [7] it has been shown as the energy of ground motion at the free field in the city might be reduced by around 50% due to the perturbation induced by resonant buildings. Analytical studies on site-city interaction have also been proposed in the literature by Gueguen et al. [8] and Boutin and Roussillon [9]. In [8] the effect of the city is accounted for by modelling the structures as a simple oscillator, while in [9] the multiple interactions between buildings are studied through homogenization methods.

The SSSI study is generally dealt with through a deterministic approach and few attempts have been made to consider seismic action as a random process. In the framework of the stochastic seismic analysis of the structure-soil-structure interaction problem Behnamfar and Sugimura [10] performed a parametric study of a 2D system consisting of two structures connected to a homogeneous, visco-elastic medium resting over a half-space by considering P-, SV-and Rayleigh wave propagation modelled as a stationary process. Alexander et al. [11] conducted the study of the SSSI problem using modelling through a discrete system forced by stationary filtered white noise excitation.

Even if it has been observed in the past by several authors the beneficial effect of the presence of a structure to reduce the vibrations of another this interaction has not yet used to the best knowledge of the authors as a tool for the seismic vibration control. In this paper it is proposed for the first time to design a structural system, herein called Vibrating Barrier (ViBa), to reduce the vibration of a neighborough structure. To this aim the stochastic analysis of the ViBa-Soil-Structure Interaction problem is performed. The structure and the ViBa are both modelled as linear behaving systems, the soil as a visco-elastic medium and the ground motion excitation as a zero-mean response-spectrum compatible Gaussian stationary/quasi stationary process. According to this hypothesis the structural parameter of the ViBa are optimized so to reduce the 50% fractile of the maximum response of the target structure. Numerical results showed a remarkable reduction of the response up to 40% in terms of maximum displacement.

2 PROBLEM FORMULATION

Consider the ViBa-soil-structure system illustrated in Figure 1. The system consists of a cantilevered structure to be protected and the ViBa both assumed linear behaving and supported by monopile foundations. The soil medium is assumed visco-elastic.

In order to analyse the mutual interaction between ViBa and structure, the general problem is discretized in n degrees of freedoms according to the Finite Element Method (FEM), as depicted in Figure 2. Due to the visco-elastic model adopted for the soil a frequency domain approach is followed to cast the equations of motion of the ViBa-soil-structure interaction model in terms of absolute displacements, that is

(1)

where M is the real [n x n] global mass matrix, is the complex [n x n] global stiffness matrix, is the [n x 1] vector of the frequency Fourier transform of the nodal absolute displacements and is the [n x 1] vector of the frequency Fourier transform of the nodal forces

Figure 1. ViBa-soil-structure system

. For clarity’s sake Equation 1 is also written in extended form as follows:

SS,1 SF,1

FS,1 FF,1

0 0

FF,c 0

0 FF,c

0 0FF,2 FS,2

SF,2 SS,2

2

SS,1 00 FF,1

0 00 0

0 00 0

FF,2 00 SS,2

,1

,1

,2

,2

0

,1

,2

0

(2)

Figure 2. Discrete ViBa-soil-structure interaction model

ViBa

Structure to beprotected

Visco-elastic soil

FF,2

ukS,1

uhS,2

urF,1 ul

F,2

FF,1

SS,1

SS,2

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

560

where the masses are assumed lumped for each node, ∙ , and ∙ , indicates the sub matrices related to the superstructure and the soil-foundation sub system, respectively, ∙ , and ∙ , refers to the coupling between the superstructure and the foundation (for i =1,2). Furthermore ∙ , is referred to the structure-soil-structure interaction

effects, that is it indicates the coupling between the foundations of the two considered structures. Hysteretic damping [12] is used in the analysis to model the dissipation of energy in both the soil and the structures, therefore all the stiffness sub-matrices in Equation 2 can be written as

K , K , 1 jη 3

where is the loss factor and √ 1. In Equation 2,

, is the stiffness matrix of the pile and it is given by the sum of the stiffness matrix of the pile structure

, and the diagonal matrix of the visco-elastic soil

, , that is

, , , (4)

where , lists the contribution of both soil-structure-interaction (SSI) and coupling between the two structures, i.e. the structure-soil-structure-interaction (SSSI) defined through the Winkler type model as shown in Figure 2. Note that in Equation 4 the dependency by the circular frequency is due to the additional geometrical dissipation of energy called radiation damping.

To determine the values of both k and the k , in this paper, the solution obtained by Novak [13], under the hypothesis of plane strain conditions and determined for a rigid, massless, circular cylinder of radius r and height dz, is used. That is,

k

2

1∙

(5)

where: 1 2 /2 1 , , is the Hankel function of the second kind of order n, / , the shear modulus, is the Poisson’s ratio and is the shear wave velocity of the soil deposit. Note that the dependency from in Equation 4 is omitted for simplicity’s sake. The stiffness k represents the coupling interaction due to the propagation of the vibrations arising from each structure and traveling through the soil. The procedure proposed by Dobry and Gazetas [14] to determine the attenuation function necessary to obtain the displacement field around a vibrating foundation is herein used to analyse the SSSI problem. From the acoustic theory, Morse and Ingard [15], obtained the following relation for asymptotic cylindrical waves:

, ≅√

exp ∙ exp (6)

where S is the spacing between source and receiver pile, is the amplitude and is the damping ratio of the soil,

3.4 1⁄ is the Lysmer’s analogue’ velocity.

Therefore, the displacement of the pile under its own dynamic load is given by:

, ≅√

∙ exp (7)

By dividing Equations 6 by Equation 7, the attenuation , yields

,,,

exp ∙ exp

(8)

As the two springs k and the k are in parallel, therefore the total stiffness is determined through the inverse of the superposition of the two compliance contributions given by k and α , k . It yields

k k 1 α , (9) and

k kα ,

1 α ,

(10)

Note that it has been assumed the two piles possess the same radius r. Therefore, the matrix FF,c lists the coupling terms given by Equation 10. Moreover, the nodal forces

are determined through the seismic wave propagation of shear waves from the bedrock through the free field as

U (11)

where is the 2 1 vector of the transfer functions of the soil deposit, U is the Fourier transform of the ground motion displacement at the bedrock and is the 2 matrix defined as:

0 0, 0

0 ,

0 0

(12)

where , is a diagonal matrix listing the stiffness k connected to the joints of the piles.

3 STOCHASTIC RESPONSE

After determining the equation of motion in the frequency domain the stochastic structural response to a stationary Gaussian zero-mean ground motion process is determined in this section. From Equation 1, by denoting

, the solution can be rewritten as:

(13) And using Equation 11

∙ U (14) The power density spectrum matrix of the displacement is obtained from Equation 13 by multiplication by its complex conjugate, as:

∗

∙∗

∙ U

(15)


561

where ∗ is the complex conjugate transpose, therefore after applying the expectation ∙ to Equations 15, and defining

, it yields

∙ G (16)

where is the power spectral density matrix of the response in terms of absolute displacements and G is the power spectral density function of the ground motion at the bedrock. Note that the response power spectral density matrix lists both the response of the ViBa and of the Structure to be protected. Therefore it can be used as vehicle to minimize the maximum response statistics by calibrating the ViBa structural parameters as shown in the next section.

4 ViBa DESIGN PROCEDURE

The aim of the paper is to determine the optimal dynamic characteristics of the ViBa in order to attenuate the vibration of adjacent structures. Therefore the optimization problem can be set to find the structural parameters SS,1, SS,1, , so to minimize the median value of the peak of the relative horizontal displacements of the structure to be protected; at a given not-exceeding probability p. The fractile of order p of the distribution of maxima is determined through the first crossing problem:

X , , , (17)

Where is the is the time observing window; is the peak factor; , is the zero-order response spectral moment. The peak factor determined by Vanmarcke [16] is used:

,

2 ln 2 1 . ln 2

(18)

with

2πln,

,

(19)

and

1,

, ,

(20)

where the response spectral moments , are given by the following equation:

, G (21)

G is the power spectral density function of the horizontal relative displacements between top of the superstructure and of the foundation; it can be obtained by the following relations:

G G G GG

(22)

where the subscripts U and UF are related to the absolute displacement of the top of structure and of the foundation, respectively.

5 NUMERICAL ANALYSES

5.1 Case studies

The previous procedure is applied to obtain the optimal ViBa parameters in order to minimize the relative displacement response of the system depicted in Figure 1. The structure to be protected is idealized as single degree of freedom system cantilevered structure made in reinforced concrete with circular section of 1 meter radius. The height of the pier is 2.5 meter. A point mass of 500.000 kg is placed on the top of the pier. The structure is founded on a 10 meter monopile of same diameter. The foundation of the ViBa is also a monopile 10 meter depth and with radius 1meter. The mass, the height and the hysteretic damping factor of the ViBa will be the design parameters calibrated to optimize the response of the target structure. Hysteretic damping for both structure and soil is fixed at 0.1.

The structures are founded on a 30 meter thickness deposit. Several shear wave velocities, as reported in Table 1, are considered in the analysis in order to study the response for the soil types described by seismic design code [9].

Table 1. Soil properties

Soil Type [m/s] [kg/m3] A 800 2000 0.45 B 400 2000 0.45 C 200 2000 0.45 D 100 2000 0.45

5.2 Determination of the input PSD

The stationary power spectral density function od the ground displacement G used in this analysis is determined from the response-spectrum-compatible model obtained for the ground motion acceleration by Cacciola et al. [17]. Specifically,

44

,̅ ,

Δ

(23)

where is the peak factor given in Equation 18 with and approximated as follows

2πln

(24)

and

11

11

2 0

1

(25)

Moreover, 0 where ≅ 1 / is the lowest bound of the existence domain of ̅ . Figure 3 shows the input PSD used in the analysis for damping ratio 0.05 and 20 . It is consistent with the assigned elastic response spectrum defined at the outcropping bedrock, classified as soil type A according to Eurocode 8. Therefore G G / . Finally, a linear site response analysis is performed to achieve the PSD of the free field motion at each depth in which the pile is discretized.


562

Figure 3. Input PSD defined at bedrock outcropping

5.3 Numerical results

This section presents the numerical results of the case study previously described in section 5.1. A parametric study is initially conducted varying the mass of the ViBa between 0.25 to 2 times the mass of the structure to be protected. Moreover, optimal parameters (i.e. the height of the ViBa and the hysteretic damping) are evaluated for each of the several pile-pile spacing considered in the analysis. Lower bound of the hysteretic damping, has been set equal to 0.02. The analyses are carried out for each of the soil deposit reported in Table 1.

Preliminary eigenvalues analyses are carried out for the structure before being protected. Table 2 lists the first fundamental frequencies (in Hz) of the soil deposit and the uncoupled structure for each soil type.

Numerical results are illustrated in terms of reduction factor RF defined as:

RF (26)

where and are the median values of the

peak obtained by Equation 17 of the relative displacements of the structure after and before being protected, respectively.

Figure 4 shows the reduction factor curves for the several soil types obtained for assigned mass ratios by varying the spacing between the structure and the ViBa. Each marker is determined by independent optimization of the ViBa. Clearly, The higher the RF value the lower is the efficiency of the ViBa. Also the RF values will be higher as the distance between the ViBa and structure increases. Nevertheless, a reduction of around 20% , RF= 0.83, is achieved for the case related to the soil type C at long distance (50 m) when the mass of the ViBa is high (i.e. 2 times the mass of the structure).

It is worth noting that the RF assumes small values at spacing less than 10 meters; it proves the high efficacy of the ViBa for engineering practical distances. The best case is achieved for the case of soil type C, with the higher mass and closest spacing in which the obtained RF is 0.56, that is a 44% of reduction compared to the uncoupled structure.

Table 2. Natural frequencies

Soil Type [Hz] [Hz] diff. % A 6.67 2.67 150 B 3.33 2.08 60.1 C 1.67 1.57 6.4 D 0.83 1.15 -28.0

Figure 4 Reduction factor curve for soil type a) A, b) B, c) C, and d) D

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4x 10

-3

Circular Frequency [rad/s]

Gag

()

[m2 s-3

]

0 10 20 30 40 500.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Pile-Pile Spacing [m]

RF

0 10 20 30 40 500.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


RF

0 10 20 30 40 500.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


RF

0 10 20 30 40 500.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1


RF

mViBa/mstr = 0.25

mViBa/mstr = 0.75

mViBa/mstr = 0.1

mViBa/mstr = 2

a)

c)

b)

d)


563

By focusing on the case related to the spacing = 1 m, Figure 5 shows the RF curves for each soil type on varying the mass ratio. The curves show that the efficiency of the ViBa is different according to the soil type. The smallest RFs are obtained for the soil type C. One cause can be elicited from Table 2; the smaller is the difference between the natural frequency of the soil and of the structure the higher is the efficiency of the ViBa, namely the smaller is the RF. Indeed, when the two natural frequencies are closer each other, the structure is in the situation referred to as double resonance and the displacements of the uncoupled structure are high; the protection provided from the ViBa sensibly decreases the structural displacement by absorbing a significant part of the seismic input energy.

Figure 6 shows the results of the optimal ViBa height for the same cases previously described in Figure 5. The optimal value decreases with the increase of the mass ratio. The results related to the softer soil type D are higher than the other cases because the maximum reduction has been obtained by absorbing the input signal close to the frequency of the soil deposit (0.83 Hz) whereas in the other soil types, the smaller RF has been related to the frequency of the structure. This aspect will be later analysed in Figure 8.

Figure 7 shows the effects of the ViBa with the distance. The ViBa has been optimized for the spacing = 1 and its effects are sensible as far as around 120m where the curves become asymptotic to the unity where the structure to be protected is no more affected by the coupling interaction with the ViBa.

Figure 8 shows the PSD functions of the response acceleration G G for the several soil types adopted in this paper obtained for the case of mass ratio = 0.75 and pile-pile spacing = 1m. The continuous line is related to the PSD in term of absolute acceleration of the response of the structure after being protected by the ViBa; the dotted line is related to the same response before being protected by ViBa (uncoupled case). The higher reduction is obtained for the case related to the soil type C where the structure is in resonance with the ground motion excitation due to the natural frequency of the soil deposit; in this case the ViBa is able to reduce of about 40% the maximum absolute acceleration.

A relevant reduction is achieved also for the case in the soil type D, where the difference between the natural frequency of the soil and of the structure is moderate. Moreover, the higher peak is related to the frequency of the soil deposit (0.83 Hz) whereas in the other cases the higher peaks are consistent with the frequency of the structure to be protected.

It is worth noting that in every case the PSD of the coupled case shows a further peak with regards to the uncoupled case; this is due to the influence of the ViBa that produce the typical shape of the Tuned Mass Damper. A similar behaviour has been observed by Alexander et al. [11] in the case of the study of cross dynamic interaction of structures founded on surface foundations.

The trajectories of the relative response of the structure protect (mass ratio = 0.75 and spacing = 1m) and non-protected to a randomly selected ground motion time history are depicted in Figure 9 for illustrative purpose.

Figure 5. RF curves for spacing = 1 m

Figure 6. Optimal ViBa heights for spacing = 1 m

Figure 7. RF curves for spacing = 1 m

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

mViBa/mstr

RF

Soil Type ASoil Type BSoil Type CSoil Type D

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

mViBa/mstr

HV

iBa [

m]

Soil Type ASoil Type BSoil Type CSoil Type D

50 100 150 200 250 3000.4

0.5

0.6

0.7

0.8

0.9

1

1.1


RF

Soil Type A

Soil Type B

Soil Type C

Soil Type D


564

Figure 8. Acceleration PSD functions for soil type a) A, b) B, c) C, and d) D obtained for mass ratio = 0.75 and spacing = 1m

Figure 9 Structural relative displacements for uncoupled and coupled case for soil type a) A, b) B, c) C and d) D obtained for mass ratio = 0.75 and spacing = 1m for one sample.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25


Ga [

m2s-3

]

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5


Ga [

m2s-3

]

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10


Ga

[m2 s-3

]

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Ga [

m2 s-3

]

coupled structure

uncoupled structure

c)

b)

d)

a)

RF = 0.82HViBa=3.84m

ViBa=0.02

RF = 0.85H

ViBa=3.69m

ViBa=0.02


ViBa=0.02


ViBa=0.02

0 5 10 15 20 25 30-0.01

-0.005

0

0.005

0.01

t [s]

U [

m]

0 5 10 15 20 25 30-0.02

-0.01

0

0.01

0.02

U [

m]

t [s]

0 5 10 15 20 25 30-0.06

-0.03

0

0.03

0.06

U [

m]

t [s]0 5 10 15 20 25 30

-0.015

0

0.015

U [

m]

t [s]

b)

d)

a)

c)


565

6 CONCLUSIONS

A novel vibration control strategy has been proposed in this paper to reduce the vibrations of structure due to seismic waves. The new strategy exploits the structure-soil-structure interaction mechanism to develop a novel device herein called Vibrating Barrier (ViBa). This barrier is hosted in the soil nearby the structure to be protected and will absorb part of the seismic input. Remarkably the device is not in contact with the structure, therefore can be used to protect existing buildings. The case of a simplified structural model has been scrutinized. Specifically a cantilevered structure founded on monopile has been selected as the structure to be protected, therefore the ViBa has been modelled as well as a cantilevered structure founded on monopile. Various soil condition according to the Eurocode 8 classification has been considered showing the effectiveness of the ViBa in different scenarios. Interestingly a reduction of up to 44% has been achieved. It is worth noting that as the ViBa is hosted in the soil it can be able protect more than one structure according to the Poulos’s superposition procedure [19]. Specifically, the response of a pile groups can be obtained from the study of only two piles at a time by assuming “transparent” the other piles. As proven by several authors Roesset [20], Kaynia and Kausel [21], the results of this approximation are in very good agreement with more rigorous dynamic solutions since the pile diameter is small with respect to the wavelength.

ACKNOWLEDGMENTS

This research was supported by the EPSRC First Grant EP/I033924/1 “Vibrating Barriers for the control of seismic waves (ViBa)”, EP/K004867/1.

REFERENCES [1] G.B. Warburton, J.D. Richardson, J.J. Webster, Forced vibrations of two

masses on an elastic half space. Journal of Applied Mechanics-Transactions, ASME 1971;38(1):148–56.

[2] J.E. Luco, L. Contesse, Dynamic structure–soil–structure interaction. Bulletin of the Seismological Society of America 1973;63(4):1289–303.

[3] T .Kobori, K. Kusakabe, Cross-interaction between two embedded structures in earthquakes. In: Proceedings of the seventh world conference on earthquake engineering. Istanbul, Turkey; 1980. p. 65–72.

[4] L. Menglin, W. Huaifeng, Xi C., Z. Yongmei, Structure–soil–structure interaction: Literature review, Soil Dynamics and Earthquake Engineering, 31 1724–1731, 2011.

[5] D. Clouteau and D. Aubry, Modifications of the ground motion in dense urban areas, Journal of Computational Acoustics, 9, 1659-1675, 2001.

[6] F.J. Chávez-García and M. Cárdenas-Soto, The contribution of the built environment to the free-field ground motion in Mexico city, Soil Dyn. Earthq. Eng. 22, 773-780, 2002.

[7] M. Kham, J-F. Semblat, P-Y. Bard, P. Dangla, Seismic site-city interaction: main governing phenomena through simplified numerical models, Bull. Seism. Soc. Am. 96, no.5, 1934-1951, 2006.

[8] P. Guégue, P.-Y. Bard, F.J. Chavez-Garcia, Site-City Interaction in Mexico City-Like environments : An Analytical Study, Bull. Seism. Soc. Am., 92,2, 794-811, 2002.

[9] C. Boutin and P. Roussillon, Assessment of the urbanization effect on seismic response, Bull. Seism. Soc. Am. 94(1), 251-268. 2004.

[10] F. Behnamfar and Y. Sugimura, Dynamic response of adjacent structures under spatially variable seismic waves, Probabilistic Engineering Mechanics 14 33-44, 1999.

[11] N.A. Alexander, E. Ibraim, H. Aldaikh, Exploration of Structure-Soil-Structure Interaction Dynamics, in Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping and Y. Tsompanakis,

(Editors), Civil-Comp Press, Stirlingshire, United Kingdom, paper 216, 2011.

[12] A.L. Kimball and D.F. Lovell, Internal friction in solids, Physics Review 30 948, 1927.

[13] M. Novak, Dynamic Stiffness and Damping of Piles, Canadian Geotechnical Journal, 11, 574-598, 1974.

[14] R. Dobry and G. Gazetas, Simple method for dynamic stiffness and damping of floating pile groups, Geotechnique 38, 24, pp.554-574, 1988.

[15] P. M. Morse, K. U. Ingard, Theoretical Acoustics. Princeton University Press, Princeton, New Jersey, 1968..

[16] E. H. Vanmarcke, Properties of spectral moments with applications to random vibration, J. Eng. Mech., 98(2), 425–446, 1972.

[17] Commission of European Communities, Eurocode 8, earthquake resistant design of structures, Brussels, Belgium, 1998.

[18] P. Cacciola ,P. Colajanni ,G. Muscolino, Combination of modal responses consistent with seismic input representation,, Journal of Structural Engineering (ASCE130(1), 47-55,), 2004.,

[19] H. G. Poulos, Behavior of laterally-loaded piles II: pile groups. Journal of Soil Mechanics and Foundations Division, ASCE, 91, SM5,733-751, 1971.

[20] J.M. Roesset, Dynamic stiffness of pile groups, Analysis and Design of Pile Foundations, ASCE, 263-286, 1984.

[21] A. M. Kaynia and E. Kausel, Dynamic stiffness and seismic response of pile groups. Research Report R82-03. Massachusset Institute of Technology, 1982.


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Vibration Control of Structures through Structure-Soil ...paginas.fe.up.pt/~eurodyn2014/CD/papers/077_MS02_ABS_1619.pdf · vibration control. In this paper it is proposed for the