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Sustainable Cities and Society 1 (2011) 315

Contents lists available at ScienceDirect

Sustainable Cities and Society

j o u rn a l hom ep age : www.e l sev i e r. com/ loca t e / s c s

Smart passive damper control for greater building earthquake resilience insustainable cities

I. Takewaki a,, K. Fujita b , K. Yamamoto b , H. Takabatake ca Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8540, Japanb Department of Urban and Environmental Engineering, Graduate School of Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto 615-8540, Japanc Department of Architecture, Kanazawa Institute of Technology, 7-1 Ohgigaoka, Nonoichi, Ishikawa 921-8501, Japan

a r t i c l e i n f o

Keywords:Sustainable buildingsRehabilitation of buildingsPassive dampers

a b s t r a c t

Passive dampers are used recently in many mid and high-rise buildings. This trend is accelerated by theincreased demand and desire for safer, more reliable and more comfortable buildings under uncertainexternal loading and environment. Viscous, visco-elastic, hysteretic and friction dampers are represen-tatives of passive dampers. Such passive dampers also play a keyrole in theimplementationof structuralrehabilitationwhichis essential for the realizationand promotionof sustainable buildings.The techniqueof structural health monitoring is inevitable for the reliable and effective installation of passive dampersduring the structural rehabilitation or retrot.

Thedesign earthquake groundmotions changefrom time to time when a newclass of groundmotions(e.g. long-period groundmotions dueto surfacewaves) is observed or a newtype of damage appears dur-ing severe earthquakes. The concept of critical excitation is useful in responding to this change togetherwith the usage of passive dampers from the viewpoint of sustainable buildings and cities.

In this paper, a historical review is made on the development of smart or optimal building structuralcontrol withpassivedampersandsome possibilities of structuralrehabilitationby useof passive dampersare discussed.

2010 Elsevier B.V. All rights reserved.

1. Introduction

The structural rehabilitation or retrot of buildings has beenconductedfor a long time allover the world andthe structural con-trol using passive dampers plays a key role in the implementationof the structural rehabilitation or retrot which is essential for therealization of sustainable buildings and cities.

The structural control has a long and successful history inmechanical and aerospace engineering. This is because these eldsusually deal with controllable external loading and environmentwith little uncertainty. However, in the eld of civil engineer-ing, it has a different background ( Casciati, 2002; Christopoulos &Filiatrault,2006;de Silva,2007;Housner et al., 1994;Housner et al.,1997; Johnson & Smyth,2006;Kobori et al., 1998; Soong & Dargush,1997; Srinivasan & McFarland, 2000 ). Building and civil structuresare often subjected to severe earthquake ground motions, winddisturbances and other external loading with large uncertainties.It is therefore inevitable to take into account these uncertainties inthe theory of structural control and its application to actual struc-tures.There areve importantareasimpacted by structural control

Corresponding author.E-mail address: [email protected] (I. Takewaki).

(Soong, 1998 ), i.e. (a) systems approach, (b) deepening effect, (c)broadening effect, (d) experimental research and (e) creative engi-neering. Among these ve areas, the broadening effect includes thesmart use of passive dampers in building structures.

In the early stage of development in passive structural control,the installation itself of supplemental dampers in ordinary build-ings was the central objective and many successful applicationshave been made. It seems natural that, after extensive develop-ments of various damper systems, another objective and targetwere aimed at accelerating the development of smart and effectiveinstallation of supplemental passive dampers. This trend corre-sponds well to the promotion of new design methods for buildingstructures from the viewpoint of sustainability and efcient use of materials.

Although themotivation wasinspired anddirectedto smart andeffective installation of supplemental passive dampers, researchon optimal passive damper placement has still been very limited.The following studies may be relevant to this subject. Gurgoze andMuller (1992) presented a numerical method for nding the opti-mal placement and the optimal damping coefcient for a singleviscous damper in a given linear multi-degree-of-freedom system.Zhang and Soong (1992) proposed a seismic design method to ndthe optimal location of viscous dampers for a building with speci-edstorystiffnesses. While their methodis based upon an intuitive

2210-6707/$ see front matter 2010 Elsevier B.V. All rights reserved.

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4 I. Takewaki et al. / Sustainable Cities and Society 1 (2011) 315

Table 1Pros and cons of passive dampers.

Pros Cons

Viscous damper Do not introduce excessive additionalforce into structural frames (Phasedelay and relief mechanism)

Cannot respond to impulsive loading

Visco-elastic damper Cost effective Introduce excessive additional force into structural frames;Temperature, frequency, amplitude-dependence

Hysteretic damper (shear, buckling-restrained brace) Cost effective Introduce excessive additional force into structural frames

Friction Easy control of slippage force Reliability of control of slippage force

criterion that an additional damper should be placed sequentiallyon the story with the maximum interstory drift, it is pioneering.Hahn andSathiavageeswaran (1992) performedseveral parametricinvestigations on the effects of damper distribution on the earth-quake response of shear buildings, and showed that, for a buildingwith uniform story stiffnesses dampers should be added to thelower half oors of the building. Tsuji and Nakamura (1996) p ro-posed an algorithm to nd both the optimal story stiffness anddamper distributions for a shear building model subjected to a setof spectrum-compatible earthquakes.

Rather recently Takewaki (1997, 1999a,b) opened another doorof smart passive damper placement with the help of the conceptsof inverse problem approaches and optimal criteria-based designapproaches. He solved a problem of optimal passive damper place-ment by deriving the optimality criteria and then by developingan incremental inverse problem approach. For many years, thisresearch played a role as a pioneering work in this area and manyresearchers referred to this article and compared the results bytheir methods with the result by Takewaki (1997) . Subsequently,Takewaki and Yoshitomi (1998) , Takewaki, Yoshitomi, Uetani, andTsuji(1999) and Takewaki (2000) introduceda new approach basedon the concept of optimal sensitivity. The optimal quantity of pas-sive dampers is obtained automatically together with the optimalplacement through this new method. The detailed explanation of this approach is made in Takewaki (2009, 2010) .

After these researches, many related works have been devel-

oped ( Aydin et al., 2007; Cimellaro, 2007; Fujita, Moustafa, &Takewaki,2010;Fujita,Yamamoto, & Takewaki,2010;Garcia,2001;Lavan & Levy, 2006; Silvestri et al., 2003; Silvestri & Trombetti,2007; Singh & Moreschi, 2001; Trombetti & Silvestri, 2004; Uetaniet al., 2003; Yamamoto et al., 2010 ). Although most of theresearches are based on gradient-based approaches, a GA-basedapproach is also investigated ( Lavan & Dargush, 2009 ). Most of them investigated new optimal design methods of supplementaldampers and proposed effective and useful methods.

There are several textbooks dealing with the design of passivedampers. Connor andKlink(1996) introduceda concept of motion-baseddesignand providedversatileexplanationon various passiveand active control systems, i.e. visco-elastic, viscous and tuned-mass dampers, base-isolation systems and active control systems.

Soong and Dargush (1997) explain the fundamental mechanicalaspects of passive dampers andpresent many practicalexamplesof application to realistic buildings. Hanson and Soong (2001) beginwith basic concepts of passive dampers and present a few exam-ples of application. Christopoulos and Filiatrault (2006) deal withpassive energy dissipation systems and base-isolated buildings.They treat several different systems of supplemental dampers, i.e.metallic and friction dampers, viscous and visco-elastic dampers,self-centering characteristic dampers, tuned-mass dampers, etc.They also explain the energy principle and performance-baseddesign principle. de Silva (2007) collects many useful chapters forpassive damper systems and gives an up-to-date review. Takewaki(2009) provided several gradient-based approaches.

The design earthquake ground motions change from time to

time when a new class of ground motions (e.g. long-period ground

motions due to surface waves) is observedor a newtype of damageappears during severe earthquakes. The concept of critical exci-tation is useful in responding to this change and should be usedas a next-generation paradigm for unpredictable design groundmotions together with theusageof passive dampers from theview-point of sustainable buildings and cities.

2. Realization of sustainable buildings from the viewpointof structural engineering

The structural rehabilitation is essential for the cost-effectiverealization of sustainable buildings and many useful methods havebeen proposed. Passive dampers enable the structural rehabilita-tion inevitable for the realization and promotion of sustainablebuildings.The mostadvantageousaspects of usingpassive dampersare to be able to use various types of passive dampers, i.e. vis-cous, visco-elastic, hysteretic and friction dampers, depending onthe type of buildings to be rehabilitated. Each type of damper hasits features and simultaneous usage of different dampers helps thecompensation of demerits.

Table 1 shows thepros andcons of various passive dampers. Themost important aspect is to prevent introducing excessive addi-tional forces in the original frames to be rehabilitated. The phasedelay andreliefmechanism in viscous (oil) dampers andthe series-type allocation of multiple passive dampers are regarded as keymechanisms.

In the structuralrehabilitation, the structuralhealth monitoringplays a signicant role so as to maintainthe effectiveness andrelia-bility of rehabilitation. Many useful system identication methodshave been proposed so far. Interested readers should read Boller etal. (2009) . There are two types of system identication techniques,i.e. modal parameter system identication and physical parametersystem identication ( Takewaki & Nakamura, 2000, 2005 ).

The design earthquake ground motions change from time totime when a new class of ground motions (e.g. long-period groundmotions due to surface waves) is observedor a newtype of damageappears during severe earthquakes. Passive dampers are useful torespond to this change regardless of whether the object building isa newly constructed one or one to be rehabilitated. The concept of critical excitation is also useful to respond to this change together

with the usage of passive dampers from the viewpoint of sustain-able buildings and cities. The critical excitation method plays animportant role in the point that it can incorporate inexperienced,undesirable inputs in the design stage (see Fig. 1).

There are various buildings in a city. Each building has its natu-ral period and original structural properties. When an earthquakeoccurs, ground motionsof various properties areinducedin thecity.The combination of the building natural period with the predomi-nant period of the induced ground motion may lead to disastrousphenomena in the city. Many past earthquake ground motionsexhibited such phenomena. To the authors knowledge, the con-cept of critical excitation and the structural design based uponthis concept can become one of such new paradigms.

It may be natural to assume that earthquake has a bound on its

magnitude. In other words, the earthquake energy radiated from

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I. Takewaki et al. / Sustainable Cities and Society 1 (2011) 315 5

Fig. 1. Various critical excitations for different types of buildings.

the fault has a bound. The problem is to nd the most unfavorableground motion for a building or a group of buildings (see Fig. 1,Takewaki, 2007, 2008a ).

A displacement spectrum or acceleration spectrum of groundmotions has been proposed at the rock surface depending on theseismic moment,distancefrom thefault, etc.( Fig.2 ). Suchspectrummay have uncertainties. One possibility or approach is to specifythe acceleration or velocity power and allow the variability of thespectrum.

The problem of ground motion variability is very importantand tough. Code-specied design ground motions are usuallyconstructed by taking into account the knowledge from past obser-vation and the probabilistic insights. However, uncertainties inthe occurrence of earthquakes (or ground motions), the fault rup-ture mechanisms, the wave propagation mechanisms, the groundproperties, etc. cause much difculty in dening reasonable designground motions especially for important buildings in which severedamage or collapse has to be avoided absolutely.

The signicance of critical excitation is supported by its broadperspective. There are two classes of buildings in a city (see Fig. 3).One is the important buildings which play an important role dur-ing disastrous earthquakes. The otherone is ordinarybuildings.Theformerone shouldnot have damage duringearthquake andthe lat-ter one may be damaged partially especially for critical excitation

larger than code-specied design earthquakes. The concept of crit-ical excitation may enable structural designers to make ordinarybuildings more seismic-resistant.

3. Fundamentals of passive damper installation

Three principal types of passive control systems installed inbuilding structures are (i) story-installation typesupplementalpas-sive dampers (viscous damper, visco-elastic damper, hystereticdamper), (ii) tuned-mass dampers (TMD) and (iii) base-isolationsystems as shown in Fig. 4.

Story-installation type supplemental passive dampers are prin-cipally treated in this section. As for tuned-mass dampers andbase-isolated buildings (see Fig. 4), refer to Takewaki (2009) . Inorder to present fundamental basics for mechanical modeling of these story-installation type supplemental passive dampers, vis-cous and visco-elastic dampers are taken as examples.[Viscous dampers]

The passive damper system, as shown in Fig. 5, including a vis-cous damper can be modeled into two models. One is the dashpotandthe other is thedashpot supportedby a spring. Thelatter modelis a Maxwell-type model. Let c denote thedamping coefcientof thedashpot and ks denote the stiffness of the supporting spring. This

Fig. 2. Earthquake ground motion depending on fault rupture mechanism, wave propagation and surface ground amplication, etc.

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Fig. 3. Relation of critical excitation with code-specied ground motion in public building and ordinary building.

Fig. 4. Three principal installation types of passive dampers.

Fig. 5. Passive damper system including a viscous damper and its modeling intodashpot model and Maxwell model.

Fig. 6. Passive damper system including a visco-elastic damper and its modelinginto KelvinVoigt model and KelvinVoigt model with support.

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Fig. 7. A representative schematic diagram of optimization procedures.

supporting spring represents the stiffness of the viscous damperdevice itself, e.g. an oil damper, or the stiffness of the surroundingsupporting system.

As for the Maxwell-type model, let us assume that the force-displacement relation in the frequency domain can be describedby

F () = (K R + iK I )U () = (kV + ic V )U ( ). (1)

In Eq. (1) , kV and c V denotethe stiffness of the spring andthe damp-ing coefcient of the dashpot of the pseudo KelvinVoigt modeltransformed from the Maxwell-type model.

The complex stiffness in Eq. (1) may be derived as follows fromtheformulationof theseries model of thedashpot andthe support-

Fig. 9. Correspondence of time and frequency-domain dual formulations.

ing spring.

K R + iK I =1

(1 /ic ) + (1 /k S) (2)

This is the standard treatment of series models. After some manip-ulation, the real and imaginary parts of the complex stiffness in Eq.(2) m ay be expressed by

K R( ) = kV ( ) =kSc 2 2

kS2 + c 2 2

(3)

K I () = c V () = kS2c

kS2 + c 2 2

. (4)

Fig. 8. Theoretical basis of effectiveness of supplemental dampers.

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Fig. 10. Shear building model.

It can be understood from Eqs. (3) and (4) that kV and c V are func-tions of the excitation frequency . This property may be used inthe representationof frequency-dependent characteristicvia seriesmodels.

[Visco-elastic dampers]The passive damper system, as shown in Fig. 6, including a

visco-elastic damper can be modeled into two models. One is theKelvinVoigt model and the other is the KelvinVoigt model witha support. Let k denote the stiffness of the visco-elastic damperitself and c denote the damping coefcient of the visco-elasticdamper itself. On the other hand, kS denotes the stiffness of thesupporting spring. This supporting spring represents the stiffnessof thevisco-elastic damperdevice itself, e.g. steel attachment of thevisco-elastic material, or the stiffness of the surrounding support-

Fig. 11. Schematic diagram of the relation of the gradient vector of the objectivefunction with the constraint on damper quantity.

Fig. 13. Multi-criteria plot by using two-step procedure for deformation-acceleration simultaneous control.

ing system. It is well known that k and c of most of visco-elasticmaterials depend on frequency, vibration amplitude and tempera-

ture, etc. Therefore the treatment of visco-elastic damper devicesis more difcult than viscous dampers in general.

As for the KelvinVoigt model with a support, assume thatthe forcedisplacement relation in the frequency domain can bedescribed by

F ( ) = (K R + iK I )U ( ) = (kE + ic E )U ( ). (5)

In Eq. (5) , kE and c E denotethe stiffness of the spring andthe damp-ing coefcient of the dashpot of the pseudo KelvinVoigt modeltransformed from the KelvinVoigt model with a support.

Thecomplex stiffnessof this pseudoKelvinVoigt model maybederived as follows from the normalformulation of the series modelof the KelvinVoigt model and the supporting spring.

K R + iK I =1

(1 / (k + ic )) + (1 /k S) (6)

After some manipulation, the real and imaginary parts of the com-plex stiffness in Eq. (6) may be expressed by

K R() = kE () =kS(k2 + kkS + c 2 2 )

(k + kS)2 + c 2 2

(7)

K I ( ) = c E ( ) =kS

2 c (k + kS)

2 + c 2 2. (8)

It can be understood from Eqs. (7) and (8) that kE and c E are func-tions of the excitation frequency as in Eqs. (3) and (4) .

Fig. 12. Multi-criteria plot with respect to sum of mean-square deformations and mean-square acceleration; (a) deformation minimization, (b) acceleration maximization.

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Fig. 14. Two-dimensional planar building frame.

Fig. 15. Algorithm for optimal damper placement; (a) fundamental algorithm, (b) augmented algorithm.

Fig. 7 shows a representative schematic diagram for thegradient-based optimization procedures. The optimization proce-dures are based on the optimality criteria and related performancesensitivities. The damper placement criteria are derived fromtheseoptimality criteriaand performance sensitivities. In Fig.7 , the initialdesign is a bare frame without supplemental dampers. Sensitivityanalysis of the objective function with respect to a design variable(damping coefcient of supplemental damper) is performed rstforthisbare frameand thehighestperformancesensitivityis found.Then the additional supplemental damper is added to this story.This implies that the supplemental damper with the highest per-formance sensitivitycan decrease theperformance mosteffectivelyandthe damping coefcientshould be increased in thissupplemen-tal damper. Again sensitivity analysis is performed for the framewith a supplemental damper and the highest performance sensi-tivityis found sequentially. If the multiple stories show the highestperformance sensitivity, then the damping coefcients of the cor- Fig. 16. Three-dimensional shear building model.

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Fig. 17. Shear building-ground system with a tuned-mass damper.

responding supplemental dampers are increased. Again sensitivityanalysis is performed for the frame with supplemental dampersand repeat the procedure explained above until the required totalquantity of supplemental dampers is reached.

4. Theoretical basis of effectiveness of supplementaldampers

If the earthquake input energy criterion holds even approxi-mately regardless of the existence of supplemental dampers andthesupplemental passive dampers canabsorb theearthquakeinputenergy as greatly as possible, the input energy to the frame can bereduced drastically (see Fig. 8). Although main frames are usuallydesigned so as to remain elastic in the case of using passive energydissipation systems, inelastic dynamic responses of building struc-tures with viscous or hysteretic dampers are also discussed fromthe viewpoint of effectiveness of viscous and hysteretic dampers.

5. Constant earthquake input energy criterion to MDOFmodel in frequency domain

Consider a proportionally damped multi-degree-of-freedom(MDOF) structurewith themass matrix [ M ].Let { x} denotethe hor-izontal nodal displacements of masses relative to the ground andlet {1} denote the inuence coefcient vector. The input energy tothis MDOF structure may be described as

E I =

{ x}T [M ]{1}u g dt. (9a)

Application of the inverse Fourier transformation { x}T =

{ X }

T eit d/ 2 to Eq. (9a) leads to

E I =

12

{ X }T

eit d [M ]{1}u g dt (9b)

Let {H V ( ; i, h i)}, i, h i and [ ] denotethe velocity transfer func-tion, the ith undamped natural circular frequency, the ith dampingratio and the modal matrix. Substitution of the relations { X ()} =[ ]{H V ( ; i, h i)}U g ( )and U g ( ) =

u g (t )eit d t intoEq. (9b)

provides

E I =

0U g ( )

2F MP () d (10)

Fig. 18. Shear-exural building model supported by ground.

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Fig. 19. Comparison of the proposed steepest direction search method with the conventional steepest descent method.

In Eq. (10) , F MP ( ) may be dened by

F MP () = 1

{Re[H V ( ; i, h i)]}T [ ]T [M ]{1} (11)

If U g () is constant ( = A) with respect to frequency, the inputenergy may be expressed as

E I =

A2

0 F MP ( ) d. (12)

Substitution of Eq. (11) into Eq. (12) leads to

E I = A2

0Re[H V ( ; i, h i)]d

T

[ ]T [M ]{1}. (13)

With the help of the residue theorem in each mode, the inputenergy to the proportionally damped MDOF structure may resultin

E I =12

A2 {1}T [ ]T [M ]{1} =12

A2N

j= 1m j. (14)

N denotes the number of masses and m j is the mass correspondingtothe jth horizontal nodaldisplacement.Eq. (14) implies that, if the

Fourier amplitude is constant with respect to frequency, the inputenergyto the proportionallydampedMDOFstructuredepends onlyon the total mass of the model.

Therelationof Eq. (14) canalso bederivedby the ideain the timedomain because the initial velocity A is given simultaneously atall the masses by an ideal input with a constant Fourier amplitudespectrum (see Fig. 9, Takewaki & Fujita, 2009 ).

6. Structural building models for damper design

The problems of optimal viscous damper placement have beentreated in various types of structures or structural systems. Someof them will be shown here.

A shear building model as shown in Fig. 10 is a simplest

model and is being used as a benchmark model for demon-stration of damper effectiveness. Takewaki (1997) presented asimple gradient-based and optimality criteria-based approach tothe smart installation problem of dampers in this model. Fig. 11shows a schematic diagram of the relation of the gradient vec-tor of the objective function (performance) V = 1 (c ) + 2 (c ) ( i(c ):amplitude of interstory-drift transfer function at the undampedfundamental natural frequency) with the constraint c 1 + c 2 = W on total damper quantity.

Takewaki (1999a) introduced a new concept of displacement-acceleration simultaneous control via stiffness-damping collabora-tion. Fig. 12 illustrates the multi-criteria plot with respect to sumof mean-square deformations and mean-square acceleration: (a)Plot for deformation minimization, (b) Plot for acceleration maxi-

mization.For more exible control, Fig. 13 shows the multi-criteria

plot by using the two-step procedure for deformation-accelerationsimultaneous control.

A shear-exural building model is an extended model of theshear building model and can take into account the overall exuralbehavior of high-rise buildings. The number of degrees-of-freedomof theshear-exuralbuildingmodelis twotimes thenumber of sto-ries. A planar frame model as shown in Fig. 14 is the most detailedtwo-dimensional model which can consider the member responseof framed buildings. Takewaki (2000) and Takewaki et al. (1999)proposeda new gradient-basedapproachto the problemof optimaldamper placement for smart design. Fig. 15(a ) presents the funda-mental algorithm and Fig. 15(b) shows the augmented algorithmwhich is needed for avoid of negative damper installation.

A 3-D shear building model as shown in Fig. 16 is another exten-sion of the shear building model which can deal with the torsionalresponse of 3-D buildings. Other extensions of the shear buildingmodel can be explained by the building-ground model as shown inFigs. 17 and 18 , building-pile-soil model and building-TMD modelas shown in Fig. 17 , etc. These models can take into account theinteraction of the building with surrounding mediums or devices.

Fig. 19 explains the comparison of the proposed steepest direc-tion search method ( Fujita, Moustafa, & Takewaki, 2010; Fujita,Yamamoto, & Takewaki, 2010; Takewaki, 2000; Takewaki et al.,1999 ) with the conventional steepest descent method. It should benoted that the ratio of the stiffness of members or elements sup-porting the dampers to the story stiffness of the structural frame isa key parameter in the effective use of passive dampers ( Takewaki& Yoshitomi, 1998; Fujita, Moustafa, & Takewaki, 2010 ).

7. Objective functions for optimal damper design

The examples of the objective function used in the problems of optimal damper placement are as follows:

Fig.20. Comparison of objectivefunctions for random damper placementwith that

for the optimal placement.

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Fig. 21. Design-dependent critical excitation and effective damper placement forcritical excitation.

Fig. 22. Info-gap robustness function with respect to level of design requirement f C.

Fig.23. Six-storyshearbuilding model:(a) bareframe,(b) frame withaddeddamperinthe rststory, (c)framewith addeddamperinthe thirdstory, (d)framewith addeddamper in the sixth story.

Fig. 24. Supplemental viscous damping coefcient; rst-story allocation, third-story allocation, sixth-story allocation from the top.

The sum of the interstory-drift transfer function amplitudesat the fundamental natural frequency through stories is a funda-mental exibility measure of the shear building model ( Takewaki,1997 ). This quantity is a measure of the building itself and doesnot depend on the input of earthquake ground motions. Otherexamples of the objective function are a probabilistic response

quantity, an earthquake response quantity, the maximum value of interstory-drift transfer function ( Fujita, Yamamoto, & Takewaki,2010 ) and an H norm ( Yamamoto et al., 2010 ).

Theexplainedgradient-basedapproach is basedon theLagrangemultiplier method for optimization. It is useful to investigatethe global optimality of the solution. Fig. 20 shows the com-

Fig. 25. Plot of the info-gap robustness function m with respect to the level of theload spectral uncertainty s for various requirements of earthquake input energies

E I =4.0 106

, 6.0 106

, 8.0 106

(Nm) (rst-story damping model).

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Fig. 26. H norm of the transfer function matrix of interstory drifts.

parison of the variation of the objective function, i.e., the sumof the mean-squares of the interstory drifts, for various struc-tural models with randomly placed VD under a constant totaldamper capacity. The stiffnesses of damper-supporting membersare treated here. From Fig. 20 , the optimal damper placementderived by the Lagrange multiplier method decreases the objec-tive function most effectively andgives the globaloptimal solution.This gure veries the validity of the proposed optimizationmethodology.

Fig.27. Optimal viscous damping coefcientdistribution for minimum H norm of the transfer function matrix of interstory drifts.

8. Optimal damper design under uncertain conditions

Sources of uncertainties in structural engineering usually comefrominput earthquake ground motionsandparameter variability instructures( Takewaki & Ben-Haim,2005;Takewaki, 2007,2008a,b ).It is well accepted in earthquake prone countries that the formeruncertainties govern the principaldesign stage. However, the latteruncertainties are also important in the stage of design decision.Fig. 21 showsthe design-dependent critical excitationand effectivedamper placement for critical excitation. It is very important in

Fig. 28. Maximum interstory drift of the models with various damper placements under recorded ground motions by time history response analysis. (a) El Centro NS 1940,

(b) Hachinohe NS 1968, (c) Taft EW 1952, (d) JMA Kobe NS 1995, (e) four recorded ground motions for a model without passive dampers.

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this situationto introducea frameworkto describe theuncertainty.Takewaki andBen-Haim(2005, 2008, chap. 19) developed a uniedtheory for taking into account of both uncertainties stated aboveby introducing an info-gap robustness function. Fig. 22 shows theinfo-gap robustness function with respect to design requirement f C.

Consider a six-story shear building model: (a) bare frame, (b)frame with added damper in the rst story, (c) frame with addeddamperin thethirdstory, (d)framewith added damperin thesixthstory as shown in Fig. 23 . The supplemental damper has an uncer-tainty in its damping coefcient. The description of the uncertaintyin terms of the parameter m is illustrated in Fig. 24 . The plot of the info-gap robustness function m with respect tothe level of theloadspectraluncertainty s for various requirements of earthquakeinput energies E I =4.0 10 6 , 6.0 10 6 , 8.0 10 6 (N m) (rst-storydamping model) is shown in Fig. 25. It can be observed that theinfo-gap robustness function m and the level of the load spectraluncertainty s introduce a new trade-off relationship.

Another approach to the smart passive damper design underuncertainties of ground motions is the method for the minimumH norm. Yamamoto et al.(2010) developed an optimal placementmethod of passive viscous dampers in a shear building model. TheH norm of the transfer function matrix of interstory drifts hasbeen adopted as theobjective function.Sinceit represents themag-nitude of vibration transfer ( Fig. 26 ), one can attenuate the gainof the frequency response by minimizing this norm. The sequen-tial quadratic programming (SQP) method has been employed foroptimization.

To demonstrate the practical applicability of the present designmethod, a ve-story damped shear building model has been con-sidered. The oor masses and story stiffnesses are prescribedas mi = 5.12 10 5 [kg] and ki = 6 .02 10

8 [N/m ] (i = 1 , , 5),respectively. The structural damping ratio of the main frame isassumed to be 0.02 (stiffness-proportional damping).

Fig. 27 provides the distribution of damping coefcientsobtained by this optimization method. Fig. 28(a)(d) presents themaximum interstory drifts of the models with various damper

placements (1st story alone, uniform placement, optimal place-ment) under recorded ground motions. In addition, Fig. 28(e)shows the maximum interstory drifts of the model without pas-sive dampers under four recorded ground motions. These valuesare obtained by time history response analysis. As seen in these g-ures, theproposed methodcan achieve a remarkable improvementfor various earthquake ground motions.

9. Future directions

Passive dampers possess various mechanical properties withfairly large degree of uncertainty. Dependency of their propertieson frequency, temperature, loading-history and surrounding stiff-

ness etc. is crucial for the development of reliable design methods.A unied treatment of these dependencies may be difcult andindividual inclusion of these effects may be possible. For example,dependency on frequency may be overcome by the combinationof springs and dashpots explained in Figs. 5 and 6 . For nonlineardampers, an optimization method using numerical design sensi-tivities and reduced models ( Fujita, Moustafa, & Takewaki, 2010;Suzuki et al., 2009 ) will be useful.

As the amount of dampers increases, the force in the damper-supporting member becomes larger ( Fujita, Moustafa, & Takewaki,2010 ). Introduction of excessive forces into the frame maybe unfa-vorable in the structural rehabilitation aimed for weak frames.Devices or mechanisms to avoid the introduction of such excessiveforces may be desired. Gradient-based approaches including the

stress distributions in the original frame and the supporting mem-

bers ( Fujita, Moustafa, & Takewaki, 2010 ) can be used as promisingones.

Since a nite number of damper systems can be installedin actual buildings, discrete optimal design/placement may beanother topic of great interest. Combination of gradient-basedapproaches and discrete design approaches can be one possibilitywith high efciency and reliability.

10. Conclusions

The following conclusions have been obtained.

(1) Passive dampers areeffective in reducing thebuilding responseunder earthquake ground motions. The ratio of the stiffness of supporting members to the frame stiffness is very important inthe effective use of passive dampers.

(2) Theoptimal damper placementdependsstronglyon themodelsof building structures.

(3) Gradient-based optimization methods are useful in the opti-mal damper placement and can be applied to various models of building structures.

(4) Passive dampers play a key role in the implementation of structural rehabilitation or retrot which is essential for therealization and promotion of sustainable buildings. Structuralhealth monitoring should be incorporated in the process of structural rehabilitation from the viewpoint of its reliability.

(5) The design earthquake ground motions change from time totime when a new class of ground motions (e.g. long-periodground motions due to surface waves) is observed or a newtype of damage appears during severe earthquakes. The con-cept of critical excitation is useful in responding to this changetogether with the usage of passive dampers from theviewpointof sustainable buildings and cities.

Acknowledgements

Part of the present work is supported by the Grant-in-Aid forScientic Research of Japan Society for the Promotion of Science(nos. 20656086, 21360267). This support is greatly appreciated.

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