Seismic Isolation Foundations

Seismic isolation foundations with effective attenuation zones

Zhifei Shi n, Zhibao Cheng, Hongjun XiangSchool of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China

a r t i c l e i n f o

Article history:Received 12 August 2013Received in revised form20 November 2013Accepted 22 November 2013Available online 14 December 2013

Keywords:Band of frequency gapAttenuation zonesPeriodic foundationSeismic isolation

a b s t r a c t

In this paper, a new conguration of seismic isolation foundation containing several concrete layers andsome rubber blocks is proposed. The concrete layers and the rubber blocks are placed periodically toform a periodic foundation. To study the isolation ability of this new conguration of periodic foundation,an equivalent analytical model is established. For practical applications, two very useful formulas areobtained. Using these formulas, the low bound frequency and the width of the rst attenuation zone canbe directly approximated without the calculation of dispersion structure. This new conguration ofseismic isolation foundation enjoys the rst attenuation zone between 2.15 Hz and 15.01 Hz, whichmeans that the components of seismic waves with frequencies from 2.15 Hz to 15.01 Hz cannotpropagate upward in the foundation. To illustrate the efciency of this seismic isolation foundation,the seismic responses of a 6-story frame with three different foundations are simulated. Numericalsimulations show that the seismic responses of the structure with the periodic foundation are greatlyattenuated as compared with those of the structure with no isolation base or with traditional rubberbearings.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Base isolation is an effective way to improve structural seismicresponse and reduce damages that may be caused by earthquakes.In the past several decades, a number of important achievementsrelating to base isolation have been achieved. Kelly conductedtheoretical and experimental work with ber-reinforced bearingsas elastomeric isolators aimed at addressing the shortcomings ofconventional isolators [1]. Tsai and Kelly analyzed the bucklingload of isolators by solving a cubic equation established usingbeam theory [2]. Jangid and Datta studied the response of atorsionally coupled base isolated building for two-componentrandom ground motions [3,4]. Kikuchi et al. [5] and Yamamotoet al. [6] studied the response of yielding in seismically isolatedstructures suggesting that ductility in a seismically isolated struc-ture should be limited contrary to current seismic design philo-sophies. Warn and his co-workers conducted both experimentaland numerical investigations on the critical load capacities ofelastomeric and lead-rubber seismic isolation bearings [7,8].In addition, Warn and Whittaker also investigated the inuenceof vertical earthquake excitation on the response of a bridgeisolated with low-damping rubber and lead-rubber bearingsthrough earthquake simulation testing [9].

By developing the resettable variable stiffness damper and thevariable friction damper as well as the leverage-type stiffnesscontrollable damper, Lu et al. studied the performances of thesemi-active isolation systems equipped with different dampers[1012]. The simulated results demonstrated that the slidingisolation system equipped with the resettable variable stiffnessdamper was able to attenuate the low-frequency resonancebehavior of the seismic isolation system induced by long-periodground motions [10]. In order to overcome the limitation of thetraditional friction pendulum isolators, Krishnamoorthy developeda variable curvature pendulum isolator and a variable frictionpendulum isolator, and studied the effectiveness of these isolationsystems through a three-span continuous bridge [13,14]. Replacingthe conventional columns by seismic isolation columns, Ribakovand his co-workers developed a hybrid seismic isolation systemfor protection of structures against near fault earthquakes [1517].In order to limit the displacements in the isolating columns,variable friction dampers were added. By mixing shredded rubbertire particles with soil materials and placing the mixtures aroundbuilding foundations, Tsang et al. proposed a potential seismicisolation method for the protection of low-to-medium-rise build-ings [18,19]. Considering the dynamic interaction between soil andstructure, Spyrakos et al. investigated the seismic responses ofbase-isolated structures by formulating the equations of motion inthe frequency domain and assuming frequency-independent soilstiffness and damping constants [20,21]. Based on nonlinearregression analysis, Ryan et al. developed the equations to esti-mate the lateral force distribution in the superstructure and

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0267-7261/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.soildyn.2013.11.009

n Corresponding author. Tel.: 86 10 51688367.E-mail address: [email protected] (Z. Shi).

Soil Dynamics and Earthquake Engineering 57 (2014) 143151

evaluated an alternative normalized strength characterization againstthe equivalent linear characterization [2224]. Their works for evalua-tion include the ability to effectively account for variations in groundmotion intensity and the ability to effectively describe the energydissipation capacity of the isolation system [24].

Aforementioned seismic isolation systems include elastomericbearings, frictional/sliding bearings, roller bearings and so on. Themechanisms of these traditional base isolation systems can beascribed to two points. One is to shift the fundamental frequencyof a structure away from the dominant frequencies of earthquakeground motion and fundamental frequency of the xed base super-structure. Another is to add some preliminary elements to undergoenergy dissipation. Different from the above traditional base isolationmethods, the method discussed in the present paper is the so-calledperiodic foundations which are studied in the frequency domain.Periodic foundations have a special dynamic property, namedattenuation zones in which waves/vibrations are blocked. It is hopedthat the frequency attenuation zone can cover the main frequencyregion of the seismic vibration and the characteristic frequency of thesuperstructure. Therefore, it can reduce the seismic energy input anddecrease the seismic response of superstructure. Both numerical andexperimental investigations show that periodic structures made ofcommon construction materials can greatly reduce the responses ofisolated structures [2530].

The content of this paper is organized as follows. In Section 2,the background as well as the aims and the scope of this paper arepresented. The governing equation for shear waves in an innitelayered periodic structure is given in Section 3. The dispersionequation is obtained and two very useful formulas to approximatethe low bound frequency and the width of the rst attenuationzone are found. A new conguration of layered periodic structurescontaining several concrete layers and some rubber blocks isproposed and the analytical model is established in Section 4.In Section 5, the seismic responses of a 6-story frame with threedifferent foundations are simulated in order to illustrate theefciency of this isolation foundation. The present numericalsimulations show that the periodic foundation can greatly attenu-ate the seismic responses of the structure compared with the onewith traditional rubber bearings, which indicates that the newconguration of layered periodic structures has a bright future inengineering applications. In addition, detailed discussions aboutthe advantages and disadvantages of the proposed congurationare presented at the end of Section 5. Finally, some conclusions aregiven in Section 6.

2. Aims and scope

Recently, investigations in the eld of solid-state physics haveshown that phononic crystals, one kind of periodic materials orperiodic structures, enjoy bands of frequency gaps. If the excitationfrequencies fall within the range of the gaps, waves cannotpropagate in or through the material. This kind of periodicmaterials has found wide applications in engineering such asfrequency lter, noise control, structure isolation, vibrationattenuation and so on. Enlightened by the concept of frequencygap existing in phononic crystals, a novel isolation system calledperiodic foundation was proposed to attenuate seismic waves [25].Theoretically speaking, periodic foundations can prevent seismicwaves coming from all directions if the foundations have periodicityin three dimensions. However, the present paper concentrates on thelayered periodic foundations with periodicity only in the verticaldirection.

Similar to the property of phononic crystal, different kinds ofperiodic foundations can be developed according to the periodicity offoundations. In our previous works, foundations with periodicity in

one dimension [2628], two dimensions [25,29] and three dimen-sions [30] have been proposed and the feasibility study on seismicisolation of these foundations has been conducted. Different fromphononic crystal structures considered in the solid-state-physics, theaforementioned periodic foundations are made of common construc-tion materials such as concrete, rubber, steel and so on. In addition,the size of periodic foundations is much larger than that consideredin the solid-state-physics.

Focused on the layered periodic foundation, previous investi-gations on the dynamic property of the attenuation zone wereconducted. Based on the theoretical analysis and numericalsimulation, Xiang et al. studied the feasibility of the layeredperiodic foundation composited of concrete layers and rubberlayers [26]. In the work [27,28], a layered periodic foundation wasfabricated and shake table tests were performed, in which greatattenuations were found when the exciting frequencies fell intothe band of frequency gaps.

Though it is demonstrated that layered periodic foundationscan greatly reduce seismic responses of isolated structures, furtherinvestigations should be conducted to make periodic foundationsmore applicable in civil engineering. First, originated in the solid-state-physics, it is a hard work for civil engineers to understandthe periodic theory, and it wastes a lot of time in determining theregion of attenuation zones of periodic foundations. Hence, it isessential to nd a simple formula to determine the rst attenua-tion zone, which is one of the major aims of this paper. Given thatearthquake is one kind of low-frequency broadband stochasticvibrations, periodic foundations should be designed with low andwide attenuation zones in order to effectively isolate the super-structure from the seismic energy. The second objective of thispaper is to develop a new conguration of layered periodicstructures in order to produce lower and wider attenuation zones.The present investigation shows that lower and wider attenuationzones could be obtained by replacing rubber layers in the layeredperiodic foundations [2628] by rubber blocks. In addition, seismicrecords with the main frequency falling in the attenuation zoneswere chosen in our previous studies. In order to consider differentsite conditions, more seismic records are used to verify theefciency of the new conguration of layered periodic structuresproposed in the present paper.

3. Basic theory

3.1. Dispersion structure

An innite layered periodic structure as shown in Fig. 1 isconsidered. Material A and material B are arranged alternatively toform this kind of periodic structure. Due to periodicity, a typicalcell as shown in Fig. 1 can be drawn to study the property of thisperiodic structure. The thickness of every layer of material A andmaterial B is denoted by hA and hB, respectively. Therefore, thethickness of the typical cell is H hAhB.

Let uz; t be the component of displacement in the x direction.Under the assumption of continuous, isotropic, perfectly elasticand small deformation as well as without consideration of damp-ing, the governing equation for shear wave (Sx) can be given as:

G2uz2

2ut2

1

where G is the shear modulus and is the mass density. Thesolution of Eq. (1) is assumed as:

uz; t Uzeit 2

Z. Shi et al. / Soil Dynamics and Earthquake Engineering 57 (2014) 143151144

where is the angular frequency and U(z) is the steady-statedisplacement component. Furthermore, the steady-state displace-ment component and shear stress component can be expressed as:

Uz sin z=c cos z=c 3a

z sin z=c cos z=c 3b

where and are constants to be determined, cG=

pis the

shear wave velocity and G=c is a wave impedance parameter.In the local coordinate system, Eq. (3a) and (3b) is valid for

every layer of the periodic structure. To express the displacementeld in detail, four unknown constants (A,B, A, B) should beintroduced, which can be determined after considering theboundary conditions at the interfaces.

Assuming that the layers are bonded perfectly at the interfaces,the inner boundary conditions for the typical cell can be given as:

UBhB UA0; BhB A0 4

Here, for the considered unit cell as shown in Fig. 1, UBhB andBhB are the displacement and shear stress at the top interface ofmaterial B, respectively; similarly, UA0 and A0 are the dis-placement and shear stress at the bottom interface of material A,respectively.

According to the BlochFloquet theory, the periodic boundaryconditions of the typical cell can be given as:

UB0eikH UAhA; B0eikH AhA 5

where k is the wave number. UB0 and B0 are the displacementand shear stress at the bottom interface of material B, respectively;UAhA and AhA are the displacement and shear stress at the topinterface of material A, respectively.

Substituting Eqs. (4) and (5) into Eq. (3a) and (3b), thefollowing set of four homogeneous equations are obtained:

sin AhA cos AhA 0 1A cos AhA A sin AhA B 00 eikH sin BhB cos BhBAeikH 0 B cos BhB B sin BhB

266664

377775

A AB B

266664

377775 0 6

where 1=c.To nd the non-trivial solution of Eq. (6), the determinant of

the matrix of coefcients must vanish, which results in thedispersion equation. By scanning the wave number along therst Billouin zone, the dispersion relationship can beobtained. The dispersion equation can be given in the followingform:

cos kH cos hAcA

cos

hBcB

0:5 A

BBA

sin

hAcA

sin

hBcB

7

Usually, the wave number k a ib is a complex number andrepresents the wave model. The symbols a and b represent thephase constant and the attenuation constant, respectively. If b 0,the corresponding wave can propagate in the periodic structurewithout any attenuation, which corresponds to the pass band.If a n=H n 1; 2; and ba0, the corresponding wave isevanescent and cannot pass through the periodic structure, whichis the so-called stop band in the eld of solid-state physics. In thefollowing discussion, the terms stop band and band of frequencygap will be replaced by another term attenuation zone in order tobe easily understood by many civil engineering readers.

3.2. Attenuation zones

From the view of potential application, the layered periodicstructure made of common construction materials such as con-crete and rubber is considered. The material parameters are givenin Table 1. Taking the thicknesses of concrete layer as hA 0:2 m,Fig. 2 presents the dispersion structure of innite layered periodic

Table 1Material parameters.

Materials Young's modulus E (Pa) Poisson ratio Density (kg/m3)

Concrete 2.50E10 0.2 2300Rubber 1.37E05 0.463 1300

0 5 10 15 20 250

-1

-2

-3

-4

-5

-6

-7hB=0.1mhB=0.2mhB=0.25m

Atte

nuat

ion

cons

tant

Frequency(Hz)

Fig. 2. Dispersion structure of layered periodic structures with hA0.2 m.

HhA

hB

x

Sx wave

oBzB

xB

zAxAoA

z

Material A

Material B

Typical cell

Fig. 1. Schematic of a layered periodic structure.

Z. Shi et al. / Soil Dynamics and Earthquake Engineering 57 (2014) 143151 145

structures with three different thicknesses of rubber layer.Obviously, the attenuation zones are located in low frequencyregion (o25 Hz), which corresponds well with the main fre-quency region of engineering vibration.

To block seismic waves or vibrations effectively, a wider rstattenuation zone and lower bound frequency of the rst attenua-tion zone are always desired. Therefore, the lower bound of therst attenuation zone (LBFAZ) and the width of the rst attenua-tion zone (WFAZ) are two very important parameters of a periodicstructure. In addition, an explicit formula for calculating LBFAZ(WFAZ) is very useful for researchers and engineers. To nd theseformulae, the following parameters are introduced:

finpf o

; finp 2

; fo 0:5cBhB

; S hAhB

; AB

; 0:5S

8

where finp is the frequency of an input wave, fo is the naturalfrequency of a layer of material B, is the ratio of material density

and S is the ratio of the thickness of the two different layers. Theparameter denotes the material mismatch. The dispersionequation Eq. (7) can then be simplied as:

cos kH cos sin 9

where is a relative parameter of frequency.Given that the rst attenuation zone is produced by

cos kH 1, the relative LBFAZ can be obtained by the rst rootof Eq. (9). Sackman et al. [31] gave two approximations for therelative LBFAZ (i.e., L):

L 12 for a small

2=p

for a large

(10

It can be examined that the above approximation is valid for alarge . However, it is not exact for a medium or a small . On theother hand, intermediate values of must be involved in practicaldesign for a lter [31]. Besides, Eq. (10) is applicable only forqualitative analysis, it is not suitable for practical design. To get abetter approximation, Eq. (10) is modied by a piecewise functionas follows:

L 233 r0:041:087 lg 1:36 0:04oo42=

pZ4

8>>>: 11

The curves of Eq. (11) are shown in Fig. 3. It can be found thatthe relative LBFAZ (L) tends to lower with the increase of thematerial mismatch, especially for 44.

Similarly, the upper bound of the rst attenuation zone(UBFAZ) corresponds to the case U . In this case, the frequencyof the input wave should be equal to the natural frequency of alayer of material B. Thus, the WFAZ can be given as:

233 r0:041:087 lg 1:36 0:04oo4

2=

pZ4

8>>>: 12

For comparison purposes, four models including the modelgiven later in the present paper are used to examine the correct-ness of the approximation equations [32], which are plotted inFig. 3. It is clear that the LBFAZ and WFAZ can be obtainedprecisely.

Eqs. (11) and (12) are very useful in practical engineering becausethe LBFAZ and WFAZ of a layered periodic structure can be obtaineddirectly without the calculation of the dispersion structure. Once atypical cell is given (i.e., the parameter is given), the LBFAZ andWFAZ can easily be found. Inversely, the material properties and thethicknesses of the layers can be optimized in order to effectivelyblock waves with certain frequencies ().

10-4 10-3 10-2 10-1 100 101 102 103 104

WFAZ LBFAZ Cheng( = 0.88) [28] Sackman( =1.11)[31] Cao( = 3.34)[32] Present model ( = 9.83)

Rel

ativ

e fr

eque

ncy

para

met

er (

)

0

Fig. 3. Approximation for the LBFAZ and WFAZ.

Rubber block

Concrete layer

L

L

h A

x

yz

l l

hB

Fig. 4. Schematic of a new conguration of periodic foundation.

Equivalent layered cellTypical cell

x

z

x

z

x

z

Equivalent mass-spring cell

2N

Fig. 5. Analytical models.


4. A new conguration of seismic isolation foundationand its equivalent analytical models

As a kind of random vibrations, seismic waves usually transmita large amount of energy by its low frequency components.Therefore, to block seismic waves effectively, the periodic founda-tion should be designed with a lower LBFAZ and a wider WFAZsimultaneously. The LBFAZ below 2 Hz or 3 Hz is always longed for.To meet this requirement, a new conguration of seismic isolationfoundation is proposed in this section based on the equivalentanalytical models.

In order to design a periodic foundation with a lower LBFAZand a wider WFAZ, engineering materials or congurations withbig relative ratios for density and shear modulus should beconsidered. In the present paper, a new conguration of seismicisolation foundation containing several concrete layers and somerubber blocks is proposed as shown in Fig. 4. The sizes of theconcrete layer and the rubber block are L L hA and l l hB,respectively. The concrete layers and the rubber blocks are placedperiodically to form a periodic foundation. Due to symmetry, atypical cell as shown in Fig. 5(a) is chosen in the following analysis

and the Sx wave is considered only in this present paper. To studythe dispersion properties of this new periodic foundation, therubber block layer is treated as a continuous material layer basedon the equivalent principle, which results in a layered typical cellas shown in Fig. 5(b). The reasonability of this equivalent layeredmodel can be veried by comparing the dispersion structure ofthis new conguration of periodic foundation with that based onthe mass-spring model, which can be found in Appendix A.

The shear modulus G and the mass density of the equivalentcontinuous material layer can be determined as follows:

L2 l2

GL2 Gl2(

13

Here, means to sum up all rubber blocks in a rubber layer.Eq. (13) means that both the mass and the shear stiffness of theequivalent continuous material layer are equivalent to those of therubber block layers.

Taking the thickness of the concrete layer as 0.2 m, Fig. 6 givesthe rst attenuation zone versus the thickness of the rubberblocks. It is found that the LBFAZ and the UBFAZ decrease rapidlywith the increase of the thickness of the rubber blocks.

5. Vibration attenuation simulations

5.1. Models and inputs

For comparison purposes, a 6-story frame with three differentfoundations as shown in Fig. 7 is considered. One is with noisolation foundation, i.e., the upper structure is xed on theground. Another is with traditional rubber bearings (rubberbearings provided between the upper structure and the base).And the third one is with a periodic isolation foundation. Oursimulations are conducted by using the commercial softwareANSYS 10.0. The size of the upper structure is 9 m9 m3.3 m.The rectangular cross sections of the concrete columns and thebeams of the frame are 0.5 m0.5 m and 0.5 m0.3 m, respec-tively. Beams and columns are simulated by using the elementBEAM188. The oor thickness is 0.1 m. Floors are modeled by theelement Shell63. By performing the model analysis, the rst fourcharacteristic frequencies of the upper structure with no isolationfoundation can easily be found as 1.739 Hz, 5.537 Hz, 10.181 Hz,and 15.38 Hz.

0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

50

60

Freq

uenc

y (H

z)

Thickness of the rubber block (m)

LBFAZ UBFAZ WFAZ

Fig. 6. The rst attenuation zone versus the thickness of the rubber block as takinghA0.2 m.

A A A

z

x

z

x

z

x

Rubberbearings

Periodicfoundation

Fig. 7. A 6-story frame with three different foundations. (a) No isolation, (b) rubber bearing isolation and (c) periodic foundation isolation.


For the traditional rubber bearings, the type GZP500-V6A isadopted in the present simulation. The bearings are added atthe bottom of each column and total nine bearings are used.The height and the horizontal stiffness of each GZP-500-V6A [33]is 194 mm and 1.22 kN/mm, respectively. To simplify the analysis,in numerical simulation, rubber bearings are replaced by concretecolumns with the same height based on the equivalence ofhorizontal stiffness. The rst four characteristic frequencies of

the structure with traditional rubber bearings are 0.7859 Hz,3.281 Hz, 6.732 Hz, and 10.351 Hz.

The periodic foundation contains three unit cells. The periodicconstant is taken as H0.4 m. The unit cell consists of a concretelayer and nine rubber blocks as shown in Fig. 5(a). The sizes of theconcrete layer and the rubber block are 10 m10 m0.2 m and1 m1 m0.2 m, respectively. Both the concrete layers and therubber blocks are simulated by using the element SOLID45. Concrete

Table 2Earthquake acceleration records [34].

Site type Earthquake (record place) Magnitude (time) Acceleration peak (gal)

Hard site Anza (Anza Fire Station) 4.7 (1980.02.25) 64.764Medium site Imperial Valley (Superstition Mtn Camera) 7.3 (1999.11.12) 108.891Soft site Loma Prieta (Alameda Naval Air Stn Hanger) 7.1 (1989.10.18) 261.908

-100

-50

0

50

100

Acc

eler

atio

n(ga

l)

Time(s)

Anza Fire Station

0

1

2

3

Four

ier a

mpl

itude

(gal

)Frequency (Hz)

Anza Fire Station

-200

-100

0

100

200

Acc

eler

atio

n(ga

l)

Time(s)

Superstition Mtn Camera

0

1

2

3

4

5

Four

ier a

mpl

itude

(gal

)

Time(s)

Superstition Mtn Camera

-300

-200

-100

0

100

200

Acc

eler

atio

n (g

al)

Time(s)

Alameda Naval Air Stn Hanger

0 2 4 6 8 10 12 0 5 10 15 20 25

0 5 10 15 20 25 30 0 5 10 15 20 25

0 5 10 15 20 25 30 0 5 10 15 20 25

0

2

4

6

8

10

Four

ier a

mpl

itude

(gal

)

Frequency(Hz)

Alameda Naval Air Stn Hanger

Fig. 8. Acceleration records and Fourier spectra of three seismic waves: (a) Anza, (b) Imperial Valley and (c) Loma Prieta.


layers and rubber blocks are bonded together perfectly using theVGLUE command. To x the frame on the foundation, the elementBEAM188 and the element SOLID45 are connected by using a set ofconstraint equations. The rst attenuation zone of the periodicfoundation can be found between 2.15 Hz and 15.01 Hz.

In addition, three ground motions as shown in Table 2 areconsidered in the present simulations, i.e., Anza 1980 (Anza FireStation), Imperial Valley 1999 (Superstition Mtn Camera) andLoma Prieta 1989 (Alameda Naval Air Stn Hanger). All the groundmotions can be obtained from the PEER Ground Database [34]. Theselected seismic waves correspond to three different types of soilsite conditions, i.e., hard site, medium site and soft site. Theacceleration records of these three seismic waves and their Fourierspectra are shown in Fig. 8.

5.2. Results and discussions

In numerical simulation, the acceleration records are applied toall the nodes in the x direction on the bottom of the structure by

using the Big-Mass method. The corresponding accelerationresponses in the x direction of node A of the frame are drawn inFig. 9. Obviously, responses for the system with periodic founda-tion are much lower than those for other cases. Fig. 10 gives themaximum relative displacement in the x direction of upperstructure with different isolation bases under the above seismicwaves. It can be seen that the seismic responses of the structurewith the periodic foundation are greatly attenuated as comparedwith those of the structure with no isolation base or withtraditional rubber bearings.

To show the ltering effect of the periodic foundation clearly,the Fourier spectra of the acceleration responses of node A are alsogiven in Fig. 9. The gray shadow part shows the rst attenuationzone of the periodic foundation. It is obvious that the second andthe third characteristic frequencies of the structure with noisolation base are located in the rst attenuation zone of theperiodic foundation. The second, third and fourth characteristicfrequencies of the structure with traditional rubber bearings arealso located in the rst attenuation zone of the periodic

0 2 4 6 8 10-300

-150

0

150

300

Acc

eler

atio

n re

spon

se (g

al)

Time (s)

No isolation Rubber bearing isolation Periodic foundation isolation

0 5 10 15 20 2510-4

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

103

10-4

10-3

10-2

10-1

100

101

102

103

Four

ier a

mpl

itude

(gal

)

Frequency(Hz)


0 5 10 15 20 25 30-900

-600

-300

0

300

600

900

Acc

elea

rtion

resp

onse

(gal

)A

ccel

earti

on re

spon

se (g

al)

Time (s)

No isolaiton Rubber bearing isolation Periodic foundation isolation

0 5 10 15 20 25

Four

ier a

mpl

itude

(gal

)

Frequency (Hz)


-1500

-1000

-500

0

500

1000

1500

Time (s)


0 5 10 15 20 25 30 0 5 10 15 20 25

Four

ier a

mpl

itude

(gal

)

Frequency(Hz)


Fig. 9. Acceleration in the x direction of node A and their Fourier spectra when the frame is under different seismic inputs: (a) Anza, (b) Imperial Valley and (c) Loma Prieta.


foundation. So, it is easily understood that the new congurationof seismic isolation foundation proposed in this paper can greatlyblock seismic waves compared with the structures with noisolation base or with traditional rubber bearings.

Good performance of the periodic foundations can be easily under-stood. To some extent, the layered periodic foundations considered inour series investigations can be viewed as an enlarged model ofthe traditional laminated rubber bearings. Having lower horizontalrigidness, the periodic foundations can reduce the responses forboth acceleration and relative displacement between oors ofupper structures. However, as aforementioned, the mechanisms

between the periodic foundations and the traditional laminatedrubber bearings are different. That is why the periodic foundationshaving the same horizontal rigidness as the traditional laminatedrubber bearings work more effectively.

On the other hand, good performances do not mean that theproposed isolation method can replace the well-developed isola-tion systems. Because of the small horizontal rigidness, thedisplacement responses of the upper structures with the periodicfoundations may be enlarged under seismic loads or wind loads.Therefore, further studies should focus on the inuences of thedamping of the periodic foundations and the additional measuresto decrease the displacement response of upper structures.

In addition, it is well known that most of the conventionalseismic isolation devices are effective for new structures as well asfor retrotting of existing ones. Comparatively, the size of theproposed isolation system is relatively large, it may not beconvenient for retrotting of existing structures. Nevertheless,the relative displacements between different oors of upperstructures with periodic foundations are much smaller, which isbenecial for civil engineering applications.

6. Conclusions

Based on the periodic theory usually used in solid-state physics,a new conguration of seismic isolation foundation is proposedand its isolation ability is numerically simulated. Some conclusionscan be drawn as follows:

(1) Periodic foundations can be constructed by using severalconcrete layers and some rubber blocks. The analytical modelcan be established based on the equivalent principle on both

1

2

3

4

5

6

Maximum relative displacement (mm)

Floo

r


1

2

3

4

5

6

Maximum relative displacement (mm)

Floo

r


1

2

3

4

5

6

0 1 2 3 0 2 4 6 8 10

0 10 20 30Maximum relative displacement (mm)

Floo

r


Fig. 10. Maximum relative displacement of upper structure under different seismic waves: (a) Anza, (b) Imperial Valley and (c) Loma Prieta.

0

5

10

15

20

25 Equivalent layered cell model Mass-spring cell model(N=1) Mass-spring cell model(N=2) Mass-spring cell model(N=3)

2

Freq

uenc

y (H

z)

Normalzied wave vector0

Fig. 11. Dispersion curve of the proposed periodic structure obtained by using theequivalent layered cell model and the mass-spring cell model.


mass and shear stiffness. This new conguration of seismicisolation foundation enjoys the rst attenuation zone between2.15 Hz and 15.01 Hz.

(2) Two very useful formulas in practical engineering areobtained. These formulas can be used directly to approximatethe low bound frequency and the width of the rst attenuationzone without the calculation of the dispersion structure.

(3) Numerical simulations show that the seismic responses of thestructure with the periodic foundation are greatly attenuatedas compared with those of the structure with no isolation baseor with traditional rubber bearings.

(4) Given that this new conguration of seismic isolation founda-tion can produce very low bound frequency and much widerattenuation zones, it is more suitable for civil engineeringapplications compared with the periodic foundations pro-posed in our previous works.

(5) The applicability of the proposed foundation to retrot theexisting structure is not very easy. Some additional measuresshould also be applied to reduce the displacement of the upperstructures.

Acknowledgment

This work is supported by the National Natural Science Foun-dation of China (51178036) and the Fundamental Research Fundsfor the Central Universities (No. 2013JBM010).

Appendix A

For comparison purposes, the proposed periodic structure canalso be simplied as a mass-spring model as shown in Fig. 5(c).The concrete layer is replaced by a mass M. If we divide everyrubber block layer into 2N parts in the layer direction (z-direction),each part can be considered as a mass-spring system. Theconcentrated mass m is connected by two linear springs. The in-plane stiffness of the linear spring is denoted by k. The parametersM, m and k of this mass-spring model can be determined by theparameters of concrete layer and rubber block as follows:

M L2hA Concrete layer A 1

2Nml2hBk2N

Gl2

hB

8>: Rubber block A 2Therefore, the Lumped Mass method can then be used to calculatethe dispersion relationship.

Taking the thickness of both the concrete layer and the rubberblock layer as 0.2 m, Fig. 11 presents the dispersion curves of theproposed periodic structure with the size length of the concretelayer L 10 m and the side length of the rubber block l 1 m.Results show that the mass-spring model converges very quickly.And, the dispersion relationship obtained by the equivalentlayered periodic structure model is in line with those obtainedby the mass-spring model.

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Seismic isolation foundations with effective attenuation zonesIntroductionAims and scopeBasic theoryDispersion structureAttenuation zones

A new configuration of seismic isolation foundation and its equivalent analytical modelsVibration attenuation simulationsModels and inputsResults and discussions

ConclusionsAcknowledgmentAppendix AReferences

Documents

Seismic Isolation Foundations