Rehabilitation of existing structures by optimal placement of viscous dampers

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1 Lfe-Cycle and Sustanablty of Cvl Infrastructure Systems Strauss, Frangopol & Bergmester (Eds) 3 Taylor & Francs Group, London, ISBN Rehabltaton of exstng structures by optmal placement of vscous dampers A. Gh. Prcope & D. Cretu Techncal Unversty of Cvl Engneerng Bucharest, Bucharest, Romana ABSTRACT: The am of ths paper s to study the optmal placement of vscous dampers n exstng structures, n order to mnmze the dsplacement response, for pulse lke sesmc condtons of Romana. The optmal behavor s determned usng the dynamc equaton of moton, wrtten n the frequency doman. The optmzaton process s based on mnmzng the sum of the mean square response of nterstorey dsplacement. Usng the senstvtes of the transfer functon, the optmal poston of the vscous dampers wll be evaluated for a structural confguraton and ncreased n an teratve process. A case study on an exstng concrete frame structure s presented. The structure s modeled wth fnte elements, and the optmal vscous damper placement s determned. The sesmc responses of the exstng structure and of the structure rehabltated wth optmal vscous dampers are compared. Also the nfluence of the dampers s assessed on the adjacent connectng elements. INTRODUCTION The current approach to sesmc desgn of buldngs supposes three dfferent levels of performance. The structure needs to behave dfferently for dfferent earthquake ntenstes. In the event of a mnor earthquake the structure must not sustan damage, whle n the event of a moderate earthquake the structure can sustan damage n certan parts of the structure. In the event of a severe earthquake, buldngs must not collapse, however consderable damage s allowed. Ths prncple requres buldngs to be repared after each moderate or severe earthquake. A modern approach to rehabltaton s to ncrease the ablty of the structure to absorb energy through passve devces. The vscous damper s a passve devce, whch has been shown to sgnfcantly mprove the response of a structure durng an earthquake. References to these devces have been mplemented n desgn codes lke the FEMA 356 (). It s shown that, n general, ths type of devce mproves the sesmc behavor of structures. Whle there are some studes on the effect of passve dampng devces n the structure, ther optmal placement s not as well researched. Most studes on the optmal dstrbuton use teratve nonlnear tral and error analyses to obtan an optmal damper dstrbuton. In ther study, Martnez and Romero (3) dstrbute the hghest dampng capacty n the stores wth the maxmum relatve velocty. The evaluaton of ths parameter s, however, dependent on a certan sesmc acton, thus a more theoretcal approach needs to be studed. Such an approach has been developed by Takewak (9) usng optmal desgn theory, and wll be used n the followng study. In addton, the study ams to address optmal damper placement for the partcular sesmc condtons of pulse lke, long perod earthquakes produced by the Vrancea source. Currently, desgn codes n Romana want to ncrease the level of sesmc hazard, to a more sutable mean return perod of 475 years. It s argued that for these types of earthquakes the forces n the vscous dampers need to be exceedngly hgh, ntroducng addtonal nternal forces nto the adjacent member. The man objectves of the study are the followng:. Test the Optmal Damper Placement method developed by Takewak (9).. Determne the opportunty of usng vscous dampers for the rehabltaton of structures under the partculartes of Vrancea earthquakes and the new desgn provsons. VISCOUS DAMPERS Vscous dampers are passve dsspaton devces. Ths type of damper s very robust and has been 53

2 B35x6 B35x6 C8x8 C6x6 B35x6 B35x6 B35x6 B35x6 Lnear vscous damper Fgure. Elevaton of orgnal structure. used n both new and exstng projects. The vscous damper s bult lke a pston wth two chambers, one of whch s flled wth vscous flud. When the pston moves, t forces the lqud through an orfce generatng a resstng force. The value of the force developed n the damper F vs s: Fvs sgn() vcv () where v s the relatve speed between the ends of the damper, C s the damper constant and s a power exponent of relatve speed, between The artcle wll refer only to the lnear vscous damper, for whch =. 3 NUMERICAL PROCEDURES The study ams to present and apply an optmal damper placement strategy and test t to the specfc, pulse lke, Vrancea earthquake. Frst of all, the dynamc analyss of the structure s computed wthout dampers. The code requrements are assessed usng the performance levels expressed n FEMA 356 (), whch wll also be adapted to the Romanan code P (6). Two sets of constrants on the vscous dampers are mposed. Frstly, usng equvalent vscous dampng, an overall sum of dampng coeffcents (C tot ) s determned for the structure. Secondly, for techncal and economc reasons a lmt value s mposed on the dampng constant, CC lm. From the full fnte element model of the structure a shear buldng model s constructed, n order to smplfy the amount of calculaton n the optmzaton process. Usng the orgnal structure and the constrants determned for the dampers, the optmal damper dstrbuton s determned, consderng the March 4 th, 977 Vrancea earthquake. Once the optmal dstrbuton s obtaned, the model wth a unform damper dstrbuton and the optmal damper dstrbuton are subjected to an Fgure. Elevaton of Retroftted Structure. ncremental dynamc analyss (IDA). For the ncremental dynamc analyss (IDA) 4 accelerograms are generated usng Vanmarke (967) method. Each of these accelerograms PGA s scaled usng 4 levels (Sf={.6,,.5, }). For each one of the cases, a nonlnear dynamc analyss s run. A total of 3 damper dstrbutons (no dampers, unform dampers dstrbuton, optmal damper dstrbuton) are tested aganst 5 accelerograms, each wth 4 scalng coeffcents, resultng n 6 nonlnear dynamc analyses. Each dynamc nonlnear analyss s performed usng SAP v4 software. The results of the analyses are compared and the performance levels are assessed for the 3 dstrbutons. 4 THE TEST STRUCTURE For the numerc experments, a symmetrc concrete structure s used. Only one of the central frames of the structure s studed and ts elevaton s shown n Fgure. The structure has 6 stores of 3m each and 4 spans of 6m each. The buldng s checked and comples wth current desgn specfcatons correspondng to a year mean return perod PGA. The rebar grade consdered s S35 and the concrete class s C/5. The desgn ncludes dead loads (5 kpa), whch comprse slab fnshng layers, parttons, and lve loads ( kpa). The nelastc response of the structure s modeled usng plastc hnges whch can form n both ends of each bar element. The plastc hnges are assgned a Takeda type hysteretc behavor. For the beam plastc hnges only the moment curvature relaton s consdered and dealzed for blnear behavor. For column plastc hnges, the nteracton between axal force and moment s consdered. The relaton moment curvature for each level of axal force s consdered blnear. Each of the moment curvature relatons are deduced usng the average strength of the concrete and steel used. Addtonally, for the concrete strength 54

3 and stran, the effect of confnement s consdered usng formulas gven by the P (6) desgn standard. The acceptance crtera for the plastc hnge rotaton are extracted from FEMA 356 and are: Table. Acceptance crtera for plastc rotaton angle (rad) Performance Level IO LS CP Beams...5 Columns.5.5. All of the elements have adequate transverse renforcement. Also, t s consdered that the beam column connecton s strong enough to avod any shear deformaton or yeldng. The model s consdered fxed to the ground at the base of the bottom storey. In order to determne the optmal dstrbuton of the vscous dampers, the current frame s further smplfed to a shear frame model. 5 DESIGN METHOD In the followng chapter, the proposed desgn method s developed usng theory by Takewak (9). Frstly, some theoretcal consderatons are presented and explaned. In the second part of the chapter the logcal steps for programmng are presented and n the last part of the chapter the used accelerograms are dscussed. 5. Theoretcal consderatons The problem whch needs to be solved can be formulated n the followng manner. Gven a structure, ts dynamc characterstcs and the power spectral densty (PSD) of the nput accelerogram, the optmal poston of the vscous dampers needs to be evaluated, so that t mnmzes the sum of mean squares of nterstorey dsplacements. The problem needs to be solved whle accountng for two constrants. Frstly, the sum of the damper coeffcents (C ) s equal to a set value (C tot ). Secondly, each of the damper coeffcents wll be smaller than a certan value (C lm ). 5 C C tot () C C (3) lm 5 d d (4) Ths problem has been studed by Takewak (9). The method uses optmal desgn theory, supposng the structure remans elastc. The artcle wll apply the method to the present problem and extrapolate the results for the nonlnear response of the structure. The problem can be formulated usng generalzed Lagrange formulaton and Lagrange multplers (,, ): LC (,,, ) ( C Ctot) ( C) d 6 ( C Clm) (5) For the reduced model the equaton of moton s wrtten n the frequency doman: g K C M v Mrv (6) where M s the mass matrx, C s the dampng matrx, K s the stffness matrx, r s a column vector wth on every poston, v() s the Fourer transform of the dsplacement vector and v g () s the Fourer transform of the ground acceleraton. In order to smplfy the statement the followng notatons are made: AK C M (7) The equaton of moton s wrtten: Av( ) Mrv g ( ) (8) The relaton between dsplacement and nterstorey dsplacement s expressed usng a transformaton matrx(t): d v d v d3 v3 d4 v4 d 5 v 5 d6 v6 (9) d ( ) TA Mrv ( ) () g or wrtng Hd( ) TA rm as the transfer functons for each of the nterstorey dsplacements. Usng random vbraton theory, the mean square response of the nterstorey dsplacement d can be expressed: Hd ( ) P ( ) d g d () Where, P g represents the power spectral densty functon of the nput acceleraton. The next step s to 55

4 assess the frst order senstvty of the mean square response of the nterstory dsplacement to each damper (C j ): d H ( ) ( ) d H d Hd ( ) P ( ) ( ) ( ) g d Hd P g d C C C j j j And the second order senstvty: d H ( ) ( ) d H d Pg ( ) d Cj Cl C j Cl Hd ( ) H ( ) d Pg ( ) d C C l j Hd ( ) H ( ) d Hd ( ) P ( ) ( ) ( ) g d Hd P g d C C C C j l j l () (3) After determnng the frst and second order senstvtes, the followng algorthm s used to solve the optmal dstrbuton problem. 5. Optmal dstrbuton algorthm A routne has been programmed nto MATLAB to solve the optmal lnear vscous damper placement. It s shown that the followng algorthm solves the Lagrange problem. Step. Intalze all dampng constants C j =; Step. Defne Dynamc Characterstcs (M,K,C) and constrants (C tot,c lm ), PSD functon (P g ) and number of steps (n); Step 3. Fnd l damper so that D / Cl s mnmum and ncrease the dampng constant of damper l Cl W / n Step 4. Update the objectve functon usng a lnear approxmaton D CD/ Cl and ts senstvty D/ C D/ C Cl; Step 5. If there s another damper m such that D / Cm D/ Cl, compute the ncrement Cm Step 6. Update C matrx and contnue from step 3 for the remanng number of steps. If n step 3 there are multple dampers l lk wth the same senstvty, all of ther dampng coeffcents are ncreased usng the followng relaton: lk lk D D D D Cl... Clk Cl C l l Cl Clk C l lk Cl (4) 5.3 Input accelerograms The most mportant accelerogram to be used s the record of the March 4 th, 977 Vrancea Earthquake PSDMarch 4 4th977 3 PSDCrtcal 5 5 Power Spectrum(cm /s 3 ) Angular Frequency(rad/s) Fgure 3. PSD of March 4 th, 977 recorded accelerogram and the Crtcal PSD. (N-S prncpal component, recorded at INCERC). For ths accelerogram the PSD s plotted n Fgure 3. Ths record, beng one of the only strong motons recorded n Romana, wll consttute the bass for the optmal desgn procedure. In order to account for the varablty of the PSD functon, a crtcal approach method s used, also proposed by Takewak(7). It s consdered that the crtcal exctaton PSD has the same total power as of the March 4 th, 977 accelerogram and the ampltude s equal to the peak value of the recorded accelerogram PSD. The crtcal PSD s centered on the frst angular frequency of the structure. Thus, for the proposed crtcal exctaton, the response s almost resonant. The peak value of the PSD s s=756 cm /s 3 and the area of the PSD S=75cm /s 4. The crtcal exctaton PSD used for the optmal desgn process has a constant ampltude of 756 cm /s 3 on a.7 rad/s nterval centered on the frst perod of the studed frame =.4 rad/s (T =.6s). Fgure 3 plots the crtcal exctaton PSD and the PSD of the March 4 th, 977 earthquake. In order to confrm the obtaned results through the nonlnear tme hstory analyss, another seres of 4 accelerograms are generated. The accelerograms are spectrum compatble and have been generated usng Vanmarke (976) algorthm. In Fgure 4 the target Acceleraton(g) Perod (s) Generated Generated Generated 3 Generated 4 Target Spectrum Fgure 4. Target Spectrum and Generated Accelerogram Spectra. 56

5 spectrum s presented, along wth the spectra of the generated accelerograms. It must be noted that a seres of 5 accelerograms were generated out of whch 4 have been chosen to resemble the March 4 th, 977 accelerogram n terms of PGA and Aras ntensty. All the accelerograms used, have a PGA of.4g, whch corresponds to current Romanan desgn codes for condtons n Bucharest. Ths PGA level corresponds to a mean return perod of years. The next generaton of Romanan codes ams to rase the mean return perod of the desgn earthquake to 475 years. Table presents the levels of sesmc hazard and the acceptance crtera whch need to be fulflled for each hazard level. Table. Levels of consdered sesmc hazard Mean Return Perod (years) Scalng Factor (Sf).6.5 Performance Level IO LS LS CP 6 RESULTS OF OPTIMAL DISTRIBUTION ALGORITHM After the complete frame s modeled usng fnte elements, a reduced shear buldng model s produced. The characterstcs of the reduced shear frame model are presented n table 3. Table 3. Characterstcs of the shear buldng model Settng m k c 3 kg kn /m kn s/m Storey Storey Storey Storey Storey Storey The mass matrx s determned usng the fnte element model. The structural dampng s assumed to be Raylegh proportonal to the stffness of the structure. The dampng matrx results consderng 5% fracton of crtcal dampng for the frst perod of the structure. A certan level of dampng s mposed. Because ths study ams to prove the opportunty of usng vscous dampers for partcular sesmc condtons, an average level of equvalent vscous dampng s chosen for structures outftted wth vscous dampers. By choosng an equvalent vscous dampng level of d =5% of crtcal dampng the followng formula can be used to compute an unform dstrbuton of lnear vscous dampers (C unf ): 4d m Cunf 4 kns/ m (5) T m cos r 9 C 8 7 C 6 C3 5 C4 4 C5 3 C6 3 4 Algorthm Step Number Fgure 5. Evoluton of dampng coeffcents wth respect to desgn step. Dampng Coeffcent kn s/m Where d s the equvalent vscous dampng ntroduced by the dampers, m r,,t are the storey mass, normalzed dsplacement n the frst mode, relatve normalzed dsplacement n the frst mode, the angle between the damper and horzontal, respectvely the frst perod of the structure. Thus, the unform dstrbuton uses 6 lnear vscous dampers, each one wth a dampng constant (C unf =4 kns/m). For the optmal dstrbuton of the dampers the followng constrants are employed. Frstly, the sum of the dampng coeffcents for the whole structure wll be the same as n the unform dstrbuton case (C s =4kNs/m). The second constrant of the algorthm needs to be chosen consderng two aspects. Frstly, a very hgh dampng constant results n large forces whch can cause local structural falure, for the elements wth whch the damper s connected. Secondly, techncal aspects need to be taken nto account as producers usually have a lmt force for ther dampers. In the case of ths study the lmt on the dampng constant consdered s C lm =8 kns/m). For these constrants, consderng the crtcal PSD, shown n Fgure 3, and choosng a number of steps (n=4), the damper dstrbuton presented n Fgure 5 s obtaned. Interstorey Dsplacement Senstvty.E+ -.E- -.E- -3.E- -4.E- Algorthm Step Number 3 4 C C C3 C4 C5 C6-5.E- Fgure 6. Evoluton of Interstorey Dsplacement Senstvty wth respect to desgn step. 57

6 Objectve Functon E4 3 4 AlgorthmStepNumber Optmal Fgure 7. Evoluton of objectve functons wth desgn step. The optmal damper dstrbuton uses 3 dampers on stores,3 and 4 each wth a dampng coeffcent C lm =8 kns/m. The results ndcate that the 4 th storey damper s the most useful for the structure. As ts value ncreases the senstvty of the 4 th storey damper (Fgure 6) reaches the senstvty of the nd storey damper and the ncrease n dampng coeffcent s dstrbuted between the two dampers. In the last steps of the process both the nd and the 3 rd storey dampers reach the mposed constrant of the algorthm and ther ncrease s transferred to the 4 th storey damper whch also reaches the constrant n the last step of the algorthm. Usng equaton (5), the equvalent vscous dampng can be assessed for both dstrbutons. The equvalent vscous dampng for the unform dstrbuton s equal to 6%, whle the equvalent vscous dampng for the optmum dstrbuton s almost.5 tmes hgher (38%). In Fgure 7 the objectve functon, the sum of nterstorey dsplacement s plotted through the algorthm steps for the optmal dstrbuton and for the unform dstrbuton. It s clear that through all the desgn steps, the optmal dstrbuton obtaned provdes lower nterstorey dsplacement than the unform dstrbuton. PGA Scalng Factor.5.5 NoDampers Optmum Drft Fgure 8. IDA-maxmum drft for March 4 th, 977 accelerogram. The gap between the two curves starts to decrease towards the end as the constrants n the algorthm lmt the values of the most useful dampng coeffcents. The dfference between the two objectve functons (sum of the nterstorey dsplacements) at the end of the algorthm s 3%. 7 RESULTS OF NONLINEAR DYNAMIC ANALYSIS The nonlnear dynamc analyss needs to establsh the opportunty of usng the lnear vscous dampers for specfed sesmc condtons. The structure s analyzed wthout vscous dampers and wth two dstrbutons of vscous dampers (unform dstrbuton and optmal dstrbuton). In Fgure 8, the results for the nonlnear analyss are dsplayed for the structure n the condtons of the March 4 th 977 accelerogram and all of the three damper dstrbutons. The postve effect of the dampers s evdent on the maxmum drfts of the structure. From the Fgure t results that the dampers are even more effectve n reducng dsplacements as the level of the PGA ncreases. If the decrease from the no damper dstrbuton to the optmal dstrbuton s % for Sf=.6, t ncreases to 5% for Sf=. The dfferences n drft between the unform dstrbuton and the optmal dstrbuton range between 8% and 6%, approxmately the same as the elastc structure. In Fgure 9 the maxmum results from the 4 generated accelerograms are presented. It s evdent that the trends mantan. The dfferences n drft between the bare structure and the optmum dstrbuton of dampers range from 5% to 5%, whle the dfferences between the unform dstrbuton and the optmal dstrbuton are between 7% and %. The maxmum results of the generated accelerograms vary on average by only 7% wth regard to the results obtaned usng the recorded accelerogram, whch means that the generated accelerograms smulate wth a satsfactory degree the behavor of PGA Scalng Factor Maxmum Drft NoDampers Optmal Fgure 9. IDA Maxmum drft for generated accelerograms. 58

7 PGA Scale Factor.5 Optmal.5 6 Maxmum Damper Force(kN) Fgure. IDA Maxmum Damper Force. the recorded accelerogram. In Fgure the IDA results for the maxmum damper force are presented. The damper forces are much hgher (5%) for the optmal dstrbuton than for the unform dstrbuton, llustratng the effectveness of the optmal dstrbuton. Another nterestng aspect s the evoluton of nternal forces n the central column to whch the damper s connected. In Fgures and the envelopes of the axal force and moment for the central column and the three dstrbutons are shown. The mpact of the damper force on the axal force of the column has an nfluence proportonal to the scale of the PGA. The axal force vares from the axal force n the structure wthout dampers to the optmal dstrbuton by 5% to a maxmum of 6%, for the hghest PGA levels. Wth respect to the unform damper dstrbuton the ntal axal force n the column vares from 4% to 47% from lowest to hghest PGA levels. The level of moment n the frame also changes but much less than the axal force. The envelope of the moment s presented n Fgure. The moments decrease from the structure wthout dampers to the structure wth the optmal dstrbuton, however the maxmum dfferences are only % and an average would be 8%. PGA Scale Factor.5 Optmal Envelope Moment(kNm) Fgure. IDA Envelope of central column moment. Thus, t s safe to assume that the change n the maxmum values of the moment s mnmal. The decrease n axal force can cause damage to the column; however ths was not the case for the current study. Although the axal force decreases and the moments reman constant, the value of the moment assocated wth the low axal force s not hgh enough to cause the appearance of plastc hnges. Rather the number of plastc hnges and ther development s reduced from the model wthout vscous dampers to the model wth vscous dampers. Fnally, the response of the structure s studed wth respect to the plastc rotaton acceptance crtera. The results ndcate that the hgher the PGA of the earthquake, the more the dampers mpact the formaton of plastc hnges. For the lowest scalng factor of the PGA, the number of plastc hnges s almost the same n the model wthout dampers as n the model wth dampers. However, for the hgher levels of scalng of the PGA, the effect s more pronounced. For a Sf=, the structure wth no dampers develops a storey mechansm wth plastc rotatons above the lmt of collapse preventon. Ths does not happen once the unform damper dstrbuton s nstalled. The dfference s even more obvous for the optmal dstrbuton. In the case of ths dstrbuton, although the number of plastc hnges s the same as PGA Scale Factor.5 Envelope Axal Force Optmal Number of Plastc Hnges Sf=.6 Sf= Sf=.5 Sf= Optmal Optmal Optmal Collapse Optmal <CP <LS <IO Fgure. IDA Envelope of central column axal force. Fgure 3. Number of plastc hnges and correspondng performance crtera for each Sf. 59

8 the unform dstrbuton the degree of ther rotaton ensures the lfe safety acceptance crtera. In the event of an ncrease of the sesmc hazard n Romanan codes (from Sf= to Sf=.5), the structure wthout dampers would be n need of rehabltaton. Both cases of dstrbuton meet the desgn objectves whch s lfe safety for Sf=.5 and collapse preventon for Sf=. The optmal dstrbuton performs even better ensurng the lfe safety objectve. 8 CONCLUSIONS The sesmc performance of a sx-storey concrete frame to partcular sesmc condtons gven by the Vrancea source s analysed n vew of upcomng change of desgn codes. A comparatve study s performed n order to study the opportunty of usng lnear vscous dampers to dmnsh the sesmc response of the structure. At the same tme the method developed by Takewak (9) to assess optmal damper dstrbuton s tested. The proposed structure s outftted wth two damper dstrbutons, a unform dstrbuton and an optmal dstrbuton. The two confguratons are studed usng the only strong moton recorded earthquake from the Vrancea source (March 4 th, 977). Moreover a set of 4 spectrum compatble accelerograms are generated n such a manner as to resemble as much as possble the recorded accelerogram. Four degrees of the PGA are consdered correspondng to mportant mean return perods, amng to prove that an eventual ncrease of sesmc hazard can be compensated by the ntroducton of the lnear vscous dampers. The results show that the lnear vscous dampers reduce the dsplacement of the structure n the event of a pulse lke earthquake produced by the Vrancea source. It s shown that the ntroducton of a usual level of dampers reduces dsplacements from % up to 5% dependng on the PGA of the earthquake. The tested optmal dstrbuton algorthm provdes a decrease n dsplacements from the unform dstrbutons of about 5%. Another nterestng aspect s the change n nternal forces for the members to whch the dampers are connected. The results show that although the moment n the column does not decrease, the axal force vares extensvely, by up to 6%. Although there s a change n axal force, the nonlnear analyses do not show the formaton of addtonal plastc hnges on the columns adjacent to the dampers, whch means that the decrease n axal force s not n phase wth the maxmum moments. One must note that for the current case study the change n nternal forces does not affect the structural response, however the changes, beng extensve, could have a negatve effect on a dfferent structure. The eventual ncrease of the mean return perod for the desgn earthquake n Romanan codes could render a lot of structures n need of rehabltaton. For the case study the structure does not meet the requred acceptance crtera for the ncreased hazard, unless outftted wth vscous dampers. The vscous dampers prove ther effectveness n reducng dsplacements and plastc rotatons. Furthermore, wth the use of the optmal dstrbuton algorthm, damper dstrbutons can be mproved. For the case study, the optmal dstrbuton uses half of the number of dampers the unform dstrbuton does, lmtng both costs and extent of nterventon. It s nterestng to note that although lnear vscous dampers provde a better structural response, the nonlnear vscous dampers are generally preferred because they lmt the force whch they develop. Ths study shows that the lnear vscous dampers can successfully be employed, but a study of the nonlnear vscous dampers s already beng developed and the results are promsng. REFERENCES Gasparn D.A., Vanmarke EH 976. Smulated earthquake motons compatble wth prescrbed response spectra. MIT cvl engneerng reaserch report R76-4. Martnez-Rodrgo M., Romero M.L. 3. An optmum retroft strategy for moment resstng frames wth nonlnear vscous dampers for sesmc applcatons. Engneerng Structures 5: Symans M.D., Constantnou M.C Sem-actve control for sesmc protecton of structures: a state-of-the-art revew. Engneerng Structures (6): ; Takewak I. 9, Buldng Control wth Passve Dampers Optmal Performance-based Desgn for Earthquakes, Kyoto, John Wley & Sons (Asa):5-3; Takewak I.. Optmal damper placemnt for crtcal extaton, Probablstc Engneerng Mechancs, no. 4:37-35,Elsever B.V. P-/6. Sesmc Desgn Code for Buldngs (Romanan Code); FEMA 356/. Prestandard and Commentary for the Sesmc Rehabltaton of Buldngs 6