Design Method for RC Building Structure Controlled by Elasto-Plastic Dampers Using Performance Curve

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1 Design Metho or RC Biling Strctre Controlle by Elasto-Plastic Dampers Using Perormance Crve W. P Whan University o Technology, China K. Kasai Tokyo Institte o Technology, Japan SUMMARY: This paper proposes a esign metho o elasto-plastic ampers or seismic control o reinorce concrete biling strctres. The ynamic property parameters o the single egree o reeom passive control system are escribe, an the parametric sty is carrie ot. The perormance crves which escribe the relationship between ampers, strctral component, an seismic inpt intitively are evelope; the perormance crve can be se to etermine the necessary amper qantity or given target strctral perormance. A amper istribtion metho or MDOF strctres is also propose, which is base on the trning o the eqivalent stiness o strctre. The time history analysis on a 7-story RC biling with elasto-plastic ampers esigne by the propose metho veriie its accracy an applicability. Keywors: Elasto-plastic amper, RC strctre, seismic control, passive control esign. ITRODUCTIO Energy issipation technology or seismic protection o biling strctres has got active practice an signiicant avance in recent years. As one o major types o energy issipation evice, elasto-plastic ampers incling steel amper an riction amper are being wiely se in both new constrction an retroitting o varios types o strctres, incling reinorce concrete (RC biling strctres. Compare with other types o ampers, elasto-plastic amper has relatively high energy issipation capability an less epenence on the sage environment. The elasto-plastic amper has become one kin o most poplar energy issipation evices in strctral engineering. On the other han, e to the complex hysteretic characteristics o RC strctral members, there is still no applicable metho to make eicient amper esign with comprehensive consieration o hysteresis o both main strctre an amper. The time history analysis is able to accomplish this work an proces accrate reslt, bt repeate analysis or etermining amper qantity is time-consming. It is highly esirable to evelop a comprehensive esign metho that clearly inicates eects as well as necessary capacities o the ampers or the target perormance o RC strctres. Prior to this research, the writers have extene esign metho or elastic strctres (JSSI 005, 007 as well as elasto-plastic strctres incling steel strctre an timber strctre (Kasai an P 00, Kasai et al. 00. An so ar, only the visco-elastic amper was consiere in these methos or elasto-plastic strctres. By extening the previos methos, this sty proposes a amper esign metho or RC strctre ae with elasto-plastic ampers. The passively controlle strctre esigne by present metho evelops ientical story rit in each story as prescribe. The perormance crves or single egree o reeom system an a amper istribtion rle or mlti-egree o reeom system are propose.

2 . DYAMIC PROPERTIES OF SYSTEM.. Stiness egraing hysteretic moel or RC rame The stiness egraing Takea moel (Takea et al. 97 is employe to reproce the hysteretic behavior o RC strctres. As shown in Figre, Takea moel has tri-linear skeleton crve. Displacement c, eines happening o cracking o RC, an isplacement y, eines yieling o reinorce bar. The strctre behaviors elastically, i max < c. Figre (a shows the hysteresis an skeleton crve or c < max < y ; Figre (b shows the hysteresis or y < max. α K Skeleton Crve F α K Stationary hysteresis E p K K l c y β c/ y F K K / F Damper Frame System F (a Hysteresis beore yieling F α K α K K l K c y E p F K K/ (a Beore yieling o rame F F Damper Frame System (b Hysteresis ater yieling (b Ater yieling o rame Figre. Stiness egraing moel Figre. Stationary state hysteresis o system The elastic stiness, post-crack stiness an post-yiel stiness are represente by K, α K, an α K, respectively. Hereinater, ctility ratios an c are se respectively to enote the maximm an cracking isplacements normalize by y, as given in Eq. (.a, b. max / y, c c / y (.a,b The stationary state hysteresis crves o Takea moel are governe by its loaing an nloaing rles which are epenent on maximm isplacement. Unloaing stiness K is given by Eq. (.a an (.b, epening on the ctility ratio. K c + α c K ( c < (.a + c α( K c + c K < (.b 0.4 ( + c Strctral loaing stiness is maximm eormation-oriente. The reloaing crve can be constrcte by connecting the point with zero orce an the point reache in the previos cycle, i that point lies on the skeleton crve or on a line aime at a point on the skeleton crve. The more etails abot this moel can be on in Takea et al. (970. The amper consiere in this paper is a combination o an elasto-plastic energy issipating element an an elastic spporting sch as brace. Becase both elasto-plastic energy issipating element an brace are isplacement-epenent, or prpose o convenience, this combination assembly is calle as elasto-plastic amper in this paper. Obviosly, as long as the elastic stiness o amper K an ctility ratio o amper are etermine, or given stiness o brace, the necessary stiness an

3 ctility ratio or energy issipating element can be calclate reaily. Becase energy issipating element an brace are connecte in series, the yieling orces or amper an energy issipating element are eqal, an the elastic stiness an ctility ratios or them become large or small simltaneosly. Figre shows the hysteresis o strctral system an its elements.. Dynamic properties o system The ratio o amper stiness to the elastic stiness o RC rame is eine as amper stiness ratio r. The elastic stiness o system is enote by K, the ratio o K to K can be calclate by Eq. (.3 K / K ( K + K / K + r (.3 Use T, h to enote the natral perio, amping ratio o RC rame, the natral perio T 0 an amping ratio h 0 o the elastic system with ae elasto-plastic amper can be expresse as: T 0 / T / + r, h 0 / h / + r (.4 The eqivalent stiness o RC rame K eq, is eine as the secant stiness o RC rame at the maximm eormation, an expresse as Eq. (.5, K eq, pk (.5 Where p is the ratio o the eqivalent stiness to the initial elastic stiness o RC rame, an expresse as the nction o ctility ratio, as given in Eq. (.6. c + α( c p, c < (.6a + α( + α( p c c, > (.6b The eqivalent stiness K eq an eqivalent perio T eq o the system are given as ollows. K eq Keq, + Keq, Keq, K / T eq (.7 (.8 / T r / + p (.9 Dissipate energy E p in one stationary state cycle is given by Eq. (.0a an Eq. (.0b, respectively. E E p p K K y y pc( p, c < (.0a + p c λ p ( + c p, > (.0b c + α( c The energy issipate by amper in one stationary state cycle, E, is easily obtaine as Eq. (.. E 4K ( y (.

4 The strain energy possesse by system can be calclate base on the eqivalent stiness o the system, an it incles strain energy o the rame E s an o amper E s. Es + Es K ( p + r / y (. Thereore, the amping ratio contribte by plastic eormation o RC rame is obtaine as ollows. E p p ( p h c p, c < (.3a 4π Es + Es π ( p + r / ( c + p λ p ( + c hp p, > (.3b π ( p + r / c + α( c The amping ratio contribte by amper is, h E r ( (.4 4π E s + Es π ( p + r / The above escribes the eqations o energy issipation an the amping ratios or system an its constittion elements in stationary state. Figre 3 plots the crves or T eq /T verss amping ratio h eq, where the ctility ratio o amper is set to be,, 3, 4, an 5 an amper stiness ratio set to be, 0., 0., 0.35, an 0.5. Generally, as K /K increases, the T eq ecreases; as ctility ratio o amper becomes larger, h eq increases signiicantly. On the other han, or small ctility ratio o amper, as the qantity o amper is increase, the ecrease o T eq is more signiicant than the increase o h eq. h eq 0.5 (5 ( h eq 0.5 (5 ( h eq (4(5 K /K h eq (4 (5 K /K (3 ( ( K /K T eq /T (3 ( ( 0. K /K 0. T eq /T (3 ( ( T eq /T (3 ( ( T eq /T (a 0.7 (b (c.5 (.0 Figre 3. Damping ratio verss eqivalent perio In orer to se the elastic response spectrm, the elasto-plastic amping system is converte to an eqivalent elastic system. The eqivalent stiness has been eine in Eq. (.7. The amping ratio or system in stationary state has been eine; however, the response o system sbjecte to earthqake excitation is o great ranomness, the irect se o the previosly eine amping ratio lacks o reasonability. Hereinater, the average amping propose by ewmark an Rosenbleth (97 is introce. Average amping ratio is the average o the amping ratios o the stationary state cycles with amplite ranging rom 0 to the maximm. With the initial amping ratio incle, the total eqivalent amping ratio is calclate by Eq. (.5. h + eq h + 0 ( hp( h ( (.5

5 3. DAMPER DESIG BASED O PERFORMACE CURVE FOR SDOF SYSTEM 3. Response rection o passive system The evalation metho o response rection o passive control biling has been propose base on the elastic response spectrm previosly. In this section, the esign metho or RC strctre ae with elasto-plastic amper is constrcte by extening the previos metho. The isplacement spectrm, pseo-velocity spectrm, pseo-acceleration spectrm are eine as S, S pv an S pa, respectively. From any one o the three spectra, the other two spectra can be calclate in terms o Eq. (3.. T T S ( T, h S pv( T, h S pa( T, h π π (3. The response rection is the combination eect o ecrease vibration perio an increase amping eect by aing amper into strctre. We eine the ratios o the isplacement, orce F o passive system to the isplacement an orce o original elastic rame el, F el as isplacement rection ratio R, pseo-acceleration rection ratio R pa, respectively. S ( Teq, heq Teq S pv( Teq, h R Dh (3.a S ( T, h T S pv( T, h S pa( Teq, heq T S pv( Teq, h R pa Dh (3.b S pa( T, h Teq S pv( T, h where S pv( Teq, h is the average pseo-velocity spectrm over perio range rom T 0 to T eq (Kasai et al. 005, 009. The reason to se average pseo-velocity spectrm is in that the spectrm is smoothe an the possible ierence between the spectrm vale at T an T eq can be rece in sch a way. Assme the pseo-acceleration spectrm in short-perio range an pseo-velocity spectrm in meim- to long-perio range are ientical, respectively, the response rection o system can be rewritten as R Teq( T + T0 Dh, Teq R Dh, T eq T T R pa R (S pa Const. (3.3a,b Teq T R pa R (S pv Const. (3.4a,b Teq Given parameters o main rame h, c, α, α,, an parameters o amper, K /K, calclate T 0 /T, T eq /T, an h eq, an then obtain R an R pa by sing Eqs. (3.3 an (3.4. The plot or R -R pa estimate or varie parameters is eine as perormance crve. Assme c 0., α 0., α an h 0.0, the perormance crves or 0.7 an are plotte in Figre 4, where constant pseo-acceleration spectrm S pa, an constant pseo- acceleration spectrm S pa are consiere separately. The ctility ratio or amper ~8 an amper stiness ratio K /K 0~ are consiere. The constant a in D h is 75 Figre 4 show the ollowing tenency. The se o large stiness o amper an low yiel strength can make the amper yiel early, an proce large ctility ratio o amper. Sch a esign can rece the isplacement as well as shear orce. This means the riction amper with high initial stiness is very

6 eective. On the other han, i amper has high yiel strength, the ctility ratio becomes small. In sch a case, increasing stiness ratio has little eect to rece the isplacement, an increase o shear orce shol be pay attention to. Comparatively, the response rection eect in more signiicant in S pa constant than that or S pv constant R pa K/K R K/K R Figre 4 Passive control perormance crves 3. Passive control esign base on perormance crve The alreay-known incle the initial perio T, initial amping ratio h, all strctral parameters or RC rame, brace stiness ratio K b /K, target ctility ratio. Base on the perormance crve, the passive control esign to etermine the qantity o amper can be perorme as ollowing. (. Make perormance crves or target ctility ratio. (. Calclate responses o elastic rame el, F el. (3. Calclate the response rections R, R pa. (4. Determine the amper stiness ratio K /K, with an assme amper ctility ratio. (5. Get the pseo-acceleration rection R pa, an sbseqently get maximm shear orce o system. I both the target maximm orce an maximm isplacement are satisie, the esign is inishe. 4. DAMPER DISTRIBUTIO RULE FOR MDOF SYSTEM 4. Design procere The writers have evelope a series o esign methos or elastic an elasto-plastic rames with varios types o ampers, an valiate them by extensive nmerical tests consiering arbitrary parametric istribtions. The present sty extens the previos methos to RC biling strctres which have more complex hysteretic properties. The general esign procere is shown as ollows. Step. Establish the reqire basic esign ata, incling mass m i an story height h i o each story o biling, an etermine the target rit angle, esign spectrm an earthqake motions or ynamic analysis. Step. Establish the constittive moel or each story base on the pshover analysis, an constrct a MDOF shear beam moel or biling. Step 3. Rece the egree o reeoms to get an eqivalent SDOF moel at target rit level. Step 4. Get the necessary amper qantity or SDOF system throgh perormance crves. Step 5. Distribte amper into each story o MDOF system (Section 5.3. Step 6. Examine the perormance o passive control system by time history analysis. 4. Conversion o MDOF rame to SDOF rame R pa (a 0.7, SpaConstant (b 0.7, SpvConstant (c.0, SpaConstant (.0, SpvConstant The conversion o MDOF strctral rame into an eqivalent SDOF rame is ivie into three cases accoring to the target rit angle θ t. Firstly, the elastic stiness o the eqivalent SDOF system is R pa R R pa R

7 estimate by, / K K i hi H e, H e mi H i / mi H i (4.a,b i i i where H e is the eective height o the RC biling, an straight-line moe giving niorm story rit angle is assme, as oten consiere by the writers (Kasai et al. 006, 00. The storage an issipate energies are respectively assme eqal between eqivalent SDOF system an MDOF system, namely, pk ( H e θ t Esi, qk ( H e θt E pi (4.a,b i i In case o i >, namely, yieling happens in one or more stories at target rit angle θ, it is assme similarly that > hols tre or SDOF system. In aition to the previos two constraint conitions, c an α o SDOF system are assme to be the average vale o ci, α i o each story level o MDOF rame (Eq. (4.3. α α i, c ci (4.3a,b i i In this way, an eqivalent SDOF system is etermine. It is note that the eqivalent SDOF system epens on the target isplacement level, becase the eqivalent SDOF is mae base on the eqivalence o amping an perio SDOF an MDOF systems at speciie perormance level. 4.3 Damper istribtion rle or MDOF system The amper qantity obtaine in eqivalent SDOF system is istribte to each story o MDOF system by sing the ollowing two constraint conitions. (a The eqivalent amping ratios o SDOF an MDOF passive control systems are eqal. (b The eqivalent stiness o passive control MDOF system at target rit angle is proportional to the story shear orce Q i acting on each story. Eqations (4.4 an (4.5 are constrcte base on the above conitions. E + p E i ( E pi + Ei h 4π ( E + s Es 4π i ( Esi + Esi (4.4 Q i Qi hi θ hi ( pik i + Ki / i hi ( pik i + Ki / (4.5 The stiness o elasto-plastic amper istribte to into story K i is obtaine as Eq. (4.6. Ki θ Qi hi 8( Esi E pi ( πh Qihi K i pi ( DESIG EXAMPLE Time history analysis is perorme to veriy the accracy o present esign metho. The x-irection o a 7-story RC biling consiere in Kasai et al. (0 is se as example. The total height o the

8 biling is 8.8m, an the namental perio in x-irection is.43s. In Kasai et al. (0, the pshover crves have been converte into the tri-linear skeleton crves, o which parameters are shown in Table. Table. Parameters o Takea moel or or 7 story RC biling Story o. W i (k h i (cm K i (k/cm a i K i (k/cm a i K i (k/cm ci (cm yi (cm F ci (k F yi (k Table. Parameters o Takea moel or eqivalent SDOF rame H e (cm K (k/cm a a c h p θ/ θ/ For artiicial earthqake waves incling BCJ-L, Hachinone, JMA Kobe, Tohok University are se in esign. These waves show almost constant pseo-acceleration response spectrm between s, an constant pseo-velocity spectrm over 0.64s. The response spectrm o % amping have constant spectrm o S pa 434cm/s an S pv 43cm/s in corresponing range (Figre 5. The constant a in D h is set to be 5 or JMA Kobe wave, an 75 or other 3 waves. Throgh time history analysis, it is on that the 7-story RC biling proces very large story rit o abot /4 in n an 3 r story when sbjecte to the esign earthqake waves. In the passive control esign, the target rit angle is set to be /75 an /00, which lea to ctility ratio o main rame o abot.5 an.0 in each story. The parameters or the eqivalent SDOF rame are shown in Table. The target ctility ratio or amper is consiere to be 5. In terms o the propose metho, the amper stiness ratios are esigne as shown in Table 3 an Table 4. Figres 6(a an 6(b show the maximm story rit o RC biling or θ/75 esign an θ/00 esign, respectively. The response or biling with ampers is shown by soli line; the response or biling withot ampers is shown by broken line. Beore ampers are ae, the biling proces irreglar story rit over the height, an very large story rit in n an 3 r story. By aing the amper esign by the propose metho inproportionally, the story rit angles are almost ientical an satisy the target vale. Table 3. Damper esign reslts or MDOF system (θ/00 θ/00 a75 a5 Story o. K i (k/cm K i /K i F i (k K i (k/cm K i /K i F i (k

9 Table 4. Damper esign reslts or MDOF system (θ/75 θ/75 a75 (k/cm a5 Story o. K i (k/cm K i /K i F i (k K i (k/cm K i /K i F i (k Spv (cm/s Spv43(cm/s Spa (cm/s (a (b Withot amper Spectrm or esign Spectrm o each motion Spa434(cm/s 50 Average 500 h0.0 h (a Pseo-velocity (b Pseo-acceleration /00 /75 Figre 5. Response spectra o gron motions Figre 6. Comparison o maximm story rit angle o example biling with an withot ae ampers 6. COCLUSIOS This paper propose a passive control esign metho or RC strctre ae with elasto-plastic amper. The ynamic properties o the system constitte by the main rame an the ae component in parallel are iscsse, the variation o eqivalent perio an amping were escribe. The ampers are esigne by tning the eqivalent stiness o the system so that an optimm eqivalent stiness istribtion can be achieve at target rit by aing elasto-plastic ampers, while the earthqake energies are issipate by plastic eormation o ampers as well. The time history analysis on strctres with irreglar istribtion o strctral parameters was perorme, an goo accracy o the propose metho was veriie. REFERECE: JSSI Manal n Eition (005, 007, Design an Constrction Manal or Passively Controlle Bilings, Japan Society o Seismic Isolation (JSSI, Tokyo, JAPA, st E., 405 pages, 003, n E., 55 pages, 005, n E. an n Print, 509 pages, 007 (in Japanese Kasai K., P W. Passive control esign metho or MDOF slip-hysteretic strctre ae with visco-elastic amper, Jornal o Strctre an Constrction Engineering, o.650, pp.78~790, Kasai K., Ogawa R., P W. et al. Passive control esign metho or elasto-plastic rame ae with visco-elastic Dampers, Jornal o Strctre an Constrction Engineering, o.655, pp.65~634, Takea T., Sozen M.A. an ielsen.. Reinorce Concrete Response to Simlate Earthqakes, Jornal o the Strctral Division, ASCE, Vol. 96, o.st, ewmark,.m. an Rosenbleth, E. Fnamentals o Earthqake Engineering. Prentice-Hall Inc., 97.

10 Kasai K., Ito H., an Watanabe A., Peak Response Preiction Rle or a SDOF Elasto-Plastic System Base on Eqivalent Linearization Techniqe, Jornal o Strctre an Constrction Engineering, o.57, pp.53~6, Kasai K., ishihara K., P W. et al. Peak Response Evalation Metho or Slip-Hysteretic Strctre Ae with Visco-elastic Damper, Jornal o Strctre an Constrction Engineering, o.646, pp.7~36, Kasai K., Kawanabe Y., Eqivalent Linearization to Preict Dynamic Properties an Seismic Peak Responses o a Strctral System with High Viscos Damping an Hysteretic Damping, Jornal o Strctre an Constrction Engineering, o. 59,pp.43~5, Kasai K., Minato., an Kawanabe Y., Passive Control Design Metho Base on Tning o Eqivalent Stiness o Visco-Elastic Damper, Jornal o Strctre an Constrction, o. 60, pp , Kasai K., L X.L., P W.C., Weng D.G., Waa A., Zho Y.: China-Japan (SFC-JST research on se o ampers or repair o RC biling amage ring 008 Wenchan earthqake (Part : Design metho or RC biling repair sing ampers, Proceeing o 8 th International Conerence on Urban Earthqake Engineering, Tokyo, 0.

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