EFFICIENT ANALYSIS OF THE NONLINEAR DYNAMIC RESPONSE OF A BUILDING WITH A FRICTION-BASED SEISMIC BASE ISOLATION SYSTEM

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1 ECCOMAS hematc Confeence on Computatonal Methods n Stuctual Dynamcs and Eathquake Engneeng M. Papadakaks, D.C. Champs, N.D. Lagaos, Y. sompanaks (eds.) Rethymno, Cete, Geece, June 7 EFFICIEN ANALYSIS OF HE NONLINEAR DYNAMIC RESPONSE OF A BUILDING WIH A FRICION-BASED SEISMIC BASE ISOLAION SYSEM Rob H.B. Fey 1, Hlco M.R. Suy, Flavo M.B. Galant 3, and Henk Njmeje 1 1 Endhoven Unvesty of echnology, Depatment of Mechancal Engneeng P.O. Box 513, 56 MB Endhoven, the Nethelands R.H.B.Fey@tue.nl, H.Njmeje@tue.nl NXP Semconductos Reseach Hgh ech Campus 5, 5656 AE Endhoven, the Nethelands Hlco.Suy@NXP.com 3 NO Bult Envonment and Geoscences, Cvl Infastuctue Depatment P.O. Box 49, 6 AA Delft, the Nethelands Flavo.Galant@tno.nl Keywods: Nonlnea dynamcs, Sesmc base solaton, Fcton, Peodc solutons, Stckslp moton. Abstact. Many dynamc cvl stuctues ae subject to some fom of non-smooth o dscontnuous nonlneaty. One emnent example of such nonlneaty s fcton. hs s caused by the fact that fcton always opposes the decton of movement, thus changng sgn when the sldng velocty changes sgn. In ths pape, a stuctue wth fcton-based sesmc base solaton s egaded. Sesmc base solaton can be employed to decouple a supestuctue fom the potentally hazadous suoundng gound moton. As a esult, the sesmc esstance of the supestuctue can be mpoved. In ths case study, a lnea fnte element model of the supestuctue s dynamcally educed enablng effcent nonlnea dynamc analyss afte couplng wth the nonlnea base solaton system, whch s composed of lnea lamnated ubbe beangs, lnea vscous dampes, and nonlnea fcton elements. he effects of vaous modelng appoaches on the total system's dynamc esponse ae consdeed. Futhemoe, the dynamc pefomance of the nonlnea system s studed by both tansent and steady-state analyses. It s shown that, n addton to standad tansent analyses, steady-state analyses can povde valuable nsght n detaled undestandng of the mpoved sesmc esstance of a buldng wth a fcton-based base solaton system.

2 1 INRODUCION One way of potectng stuctues aganst eathquakes s by addng mechancal devces to educe the sesmc esponse [1, ]. In ths pape a base-solaton system consstng of spng, dampe, and fcton elements s appled. Fcton elements, whch have a nonlnea (even dscontnuous) natue, ae capable of decouplng stuctue moton fom gound moton. In ecent yeas, many papes and books appeaed focused on effcent modelng and analyss of dscontnuous nonlnea dynamc systems (see e.g. [3]). he behavo of these systems s not fully undestood yet and numecal analyss of these systems s often cumbesome, especally fo systems wth many degees of feedom. In ths pape, an effcent modelng and analyss methodology s pesented fo cvl stuctues wth nonlnea (dscontnuous) elements. he (lnea) model fo the supestuctue wll be dynamcally educed. wo dffeent Coulomb fcton models wll be employed fo the fcton elements n the base solaton system: a dscontnuous swtch model [3] and a smoothed model. he equatons of moton of the total system (supestuctue plus base solaton system) wll be deved usng Lagange s equatons fo systems wth constants. ansent analyses wll be pefomed usng measuement data of the 1995 Kobe eathquake [4]. he man novelty of the pape les n the study of steady-state behavo of cvl stuctues wth fcton elements, see also [5, 6], usng new tools developed fo dscontnuous dynamc systems. It wll be shown that ths study nceases the nsght n the (standad) tansent esponse analyss, whch s common pactce n eathquake engneeng. he pape s oganzed as follows. he modelng of the supestuctue and the base solaton system wll be dscussed n sectons and 3 espectvely. In secton 4 the equatons of moton of the total stuctue wll be deved. hs wll be followed by tansent analyses (secton 5) and steady-state analyses (secton 6). Fnally, n secton 7 conclusons wll be dawn. DYNAMIC MODELING OF HE SUPERSRUCURE An exstng unbaced hgh-se steel buldng of 11 constucton levels (1 stoes) wth a total heght of 53 m, measued fom gound-level, s chosen as the supestuctue, because ts dynamc popetes wee known to the authos. he buldng s located n a egon wthout sesmc actvty and actually has no base solaton system. hs s of no mpotance hee, snce t s meely ou ntenton to show a methodology fo modelng and analyzng the nonlnea dynamcs of supestuctues wth fcton-based base solaton systems, and not to study o mpove the sesmc esstance of ths specfc supestuctue. So, n ths study, the model of the supestuctue wll be coupled to a base solaton system. he buldng has a steel Moment Resstng Fame wthout dagonal bacng membes and wthout concete stuctual membes. he concete floos (sufaces: 15 by 8 m) and concete wall elements contbute to the stuctual mass, but not to the stuctual stffness. It s assumed that the stuctue wll always eman n ts elastc ange (wth o wthout base solaton). he valdty of ths assumpton s vefed n the numecal smulatons, as pesented late. Fnte Element (FE) modelng s appled to deve the netal and stffness chaactestcs of the supestuctue. he two components of the hozontal gound moton ae often not coelated and have the maxmum values at dffeent nstants. he effect of the vecto sum of the two components on the magntude of the total hozontal moton, can theefoe be dsegaded. Moeove, otatonal exctaton can be neglected fo a stuctue of ths sze and wth ths symmetc layout. heefoe, only lateal exctaton of the buldng s egaded, as ths s the most vulneable exctaton decton. In tansent analyss (secton 5), both hozontal and vetcal exctaton wll be taken nto consdeaton.

3 Fg. 1(a) shows the lnea D FE-model based on Benoull-Eule beam elements. Each node has thee dof's (hozontal and vetcal dsplacement and otaton). Fg. 1(b) shows the lowest fou undamped egenmodes and egenfequences of the stuctue. In ths fxed-base stuaton, the sx degees of feedom (dof's) of the two nodes that ae located at gound-level, ae suppessed. hese sx dof's ae called bounday o nteface dof's and ae, n a late stage, needed fo the couplng of the supestuctue to the base solaton system. he emanng dof's of the stuctue ae called ntenal dof's. he dampng atos of the fst fou (fxed-base) egenmodes, as pesented n Fg. 1(b), ae chosen equal to 3, 4, 5 and 6%, espectvely. he fst dampng ato has been measued expementally. he othe values ae based on the fact that hghe modes usually exhbt hghe dampng, due to lage flexual and shea defomaton of the stuctue. (b) 4 4 elevaton [m] 1 1 f 1 =.46 Hz f = 1.7 Hz (a) span [m] f 3 = 1.98 Hz 1 1 f 4 =.98 Hz Fgue 1: (a) FEM-model of the supestuctue; (b) Fst fou egenmodes and egenfequences of the fxed-base supestuctue. A lage numbe of dof's may lead to computatonally neffcent nonlnea dynamc analyses n sectons 5 and 6. heefoe, model educton s appled to the supestuctue usng the method of Cag-Bampton [7]. In ths case, the ognal dsplacement feld s appoxmated by a lnea combnaton of 6 constant modes, whch ae defned fo the 6 bounday dof s, and 7 fxed-nteface egenmodes, based on a cut-off fequency of 5 Hz. hs fequency s chosen such that (the lagest pat of) the domnant fequency ange of most eathquakes (between.1 and 1 Hz [1, 8]) s coveed. Use of a cut-off fequency of Hz dd not sgnfcantly nfluence the analyss esults, as pesented late n the pape, because the egenmodes between 5 and Hz manly consst of modes wth only vetcal floo esonance. Howeve, the numbe of dof's would be unnecessaly nceased. Of couse, to cuent standads, the FE model used hee s athe smplstc. Howeve, ths s elevant, snce ths pape focuses on the nonlnea dynamc modelng and analyss methodology, whch wll be elaboated late on. Obvously, a detaled 3D model of the supestuctue could be educed analogously. It s noted that the set of n g = 13 dof's of the educed model contans n b = 6 physcal bounday dof's, to enable easy couplng of the supestuctue to the solaton system, and n = 7 genealzed dof s coespondng to the 7 fxed-nteface egenmodes. In the emande of the pape, the educed 13 dof supestuctue model wll be used. 3 SEISMIC BASE ISOLAION A sesmc base solaton system may pevent damage to the supestuctue n case of an eathquake. In ths type of solaton, mechancal devces ae placed undeneath the supestuc- 3

4 tue [1, ]. o ensue that these mechancal devces ae defomed unfomly, the supestuctue s equpped wth a elatvely gd daphagm at the solaton level. he base solaton system should comply wth vaous chaactestcs, such as a lage degee of lateal flexblty fo decouplng of the supestuctue fom the suoundng gound, vetcal load cayng capacty, gdty to lateal nonsesmc sevce loads, and enegy dsspaton. heefoe, multple devces ae often combned nto one system. In ths eseach, a base solaton system s chosen that conssts of lnea lamnated ubbe beangs, lnea vscous dampes, and nonlnea fcton elements wth ectlnea sldng sufaces [1, ]. In ths pape, the focus wll be on the effect of the latte elements on the system's esponse. he dy fcton element conssts of two paallel hozontal sufaces, whch slde elatve to each othe. As the fcton foce always opposes the elatve decton of moton, ts behavo s nonlnea and even dscontnuous. In lteatue, vaous fcton models exst [3]. Hee, t s assumed that the fcton foce can be epesented by the Coulomb fcton model, see Fg... M. M = N N. M = N Fgue : Schematc epesentaton of the Coulomb fcton model. In ths model, dung stck, n whch case the elatve sldng velocty x& =, the fcton foce s bounded by [ F w, max, Fw,max ], counteactng all othe extenal foces, esultng n zeo acceleaton. When the extenal foces exceed the maxmum fcton foce, slp occus ( x& ) and the fcton foce s ±, the sgn dependng on the sgn of x&. F w, max 4 DERIVAION OF HE EQUAIONS OF MOION In ode to deve the dynamc model of the base-solated supestuctue ncludng Coulomb fcton elements, fst, the knematcs of the model ae elaboated, see Fg. 3. G C I K F A HI JHK? JK HA / K D H K L A H ; : N EI HA = JEL A O HEC E= F D H= C I JHK? JK HA > = I A EI = JE I O I JA I K HH E C C H Fgue 3: Knematcs of the model of the base-solated stuctue. 4

5 he two bounday nodes of the supestuctue ae located at gound-level (GL) and ae labeled 1 and. In Fg. 3, the column of dof's q of the supestuctue s measued elatve to the ght-handed coodnate system (X,Y,Z) wth ogn O, whch s fxed to the suoundng gound. hs dof column s gven by: [ q q ] = [ x y θ x y q ] q b θ = (1) wth x, y and θ the hozontal dsplacement, vetcal dsplacement, and otaton of bounday nodes = 1,, espectvely. he n b = 6 bounday dof's ae collected n q b. he n = 7 genealzed dof s ae collected n q. A base solaton system s attached to each of the bounday nodes 1 and. he two (dentcal) solaton systems consst of a lnea spng, a lnea dampe, and a nonlnea fcton element. he hozontal dsplacement of the solaton system measued elatve to (X,Y,Z) equals x so = x 1 x (due to the vey stff daphagm). he hozontal foce due to the lnea elements of base solaton system ( = 1,) s: F k x cx& lnea, = lam + () whee k lam s the lateal stffness of the lamnated ubbe beang and c s the dampng constant. he stffness and dampng contbutons of the two base solaton systems ae added to the supestuctue's educed stffness matx K and dampng matx C, espectvely: K q = K q + k [ ( )] lam x1 x zeos 1, n C q& = Cq& + c[ x& x zeos( 1, n )] 1 & Hee, ( 1 ) epesents a ow of zeos wth dmensons ( ) zeos,n 1,n. Usng Lagange's equatons, the followng equatons of moton of the base-solated supestuctue wthout fcton elements can be deved: M q& + C q& + K q = MΓu& whee M s the (educed) mass matx of the supestuctue. he hozontal and vetcal gound acceleaton, elatve to netal space, and the gavtatonal acceleaton g, ae collected n the exctaton column u & = [ u&& ( )] ho u& ve + g Γ = [ ] ( = [ 1 1 Γ ] (3) (4) (5). he exctaton nfluence matx Γ, [ ] Γ ho Γ ve ho, ho Γ ve = 1 1 Γ, ve ) has also undegone the Cag-Bampton educton fom the uneduced set of dof's to the new educed set of dof's q, esultng n the nfluence columns Γ, ho and Γ, ve wth sze ( n,1). Next, the two Coulomb fcton elements ae ncluded n the equatons of moton. A possble mathematcal fomulaton of the Coulomb fcton foce model of Fg. s [3]: F w, = F w,,max Sgn μfn,, f x& >, μ n, = μfn,, f x& <, (6) [ μfn,, μfn, ], f x& =. ( x& ) = F Sgn( x& ) 5

6 whee μ [-] denotes the fcton coeffcent, and F n, [N] s equal to the nomal foce between the two fcton sufaces of fcton element ( = 1, ). o numecally tackle the change of state between stck and slp (o vce vesa), a tanston phase can be ntoduced. hs so-called swtch-model [3] captues the actual non-smooth dynamcs of the Coulomb fcton model. In the swtch-model, the tanston state s composed of a naow band o bounday laye aound the hypesuface of the stck phase n Ω = { x R g n x& = x& = }, whee R g epesent the ng dmensonal state space [ q q& ] 1 x =. he swtch band x& = 1, η foms a subspace, n whch the vecto feld s such that the soluton s foced towads ts mddle, the hypesuface. Hee, the wdth of the swtch band s chosen η = 1 5 [m/s], small enough to yeld a good appoxmaton. Altenatvely, the Coulomb fcton foce can be appoxmated by the smoothed model: whee ε [-] stongly nfluences the slope of F π ( εx& ) w, = μfn, actan (7) w F, nea x& =. he value of ε = 75 s chosen as a compomse between pesevaton of the non-smooth chaacte of the Coulomb fcton model and the stffness of the equatons of moton, whch may esult n ntegaton poblems. Only neglgble dffeences wee found fo the tansent and steady-state esponses of the base-solated buldng between the two dffeent fcton models (6) and (7). he equatons of moton ncludng the Coulomb fcton elements ae deved now usng Lagange s Equatons fo Systems wth Constants. Assumng the constants & y 1 = & y = (no uplft), the vetcal nomal foces F n, 1 and F n, n the nodes 1 and of Fg. 3, collected n λ = [ ] Fn,1 F n,, ae teated as constant foces leadng to the followng equatons of moton: ( q q& u& ) + Sλ R λ M q& h,, = (8) R q& = (9) whee: h ( q q&, u&& ) = C q& K q MΓu&, (1) the constant equatons (9) and the constant foces 1 R = 1 and the fcton foces ae ntoduced va the column: F Sλ = w,1 F n,1 F w, 6 F n, R λ ae ntoduced va the matx: zeos zeos ( 1, n ) ( ) 1, n zeos zeos ( 1, n ) ( 1, n ) It s emphaszed that fo the swtch-model mplementaton, the expessons fo R, S and λ depend on the system s state (stck, slp o tanston) and ae unequal to the ones mentoned above. Fo nstance, f a fcton element s n stck, the coespondng fcton foce s also a constant foce, togethe wth the nomal foces. λ (11) (1)

7 hee ae two ways to numecally solve the constaned equatons of moton (8)-(9): ethe by stablzaton [5, 6] of the magnally stable constant equaton (9) o by tansfomng the constaned equatons of moton to unconstaned equatons of moton. Hee, only the latte appoach wll be pesented. he tansfomaton: s substtuted n (8), followed by pemultplcaton wth Hee, matx Q ( n g, n mn ) of ank mn q = Qz (13) Q, esultng n: ( Qz, Qz&, u& ) + Q Sλ = Q R λ = Q MQz & Q h (14) n s chosen such that Q R = O. Fo the smoothed fcton model (and fo the swtch model n the state that both fcton elements slp) z = [ x x q ] 1 θ1 θ and the numbe of ndependent dof s s n = 11 mn. he nomal (constant) foces can now be obtaned by combnng (8), (9), and (13): [ RM ( R S )] RM h( Qz, Qz&, u&& ) λ = (15) he nomal foces λ depend on the dynamc esponse of the total system. Hee, n all dynamc analyses, the nomal foces appea to be compessve foces. In othe wods, equaton (9) ndeed s vald and the elatve vetcal dsplacements y 1 and y ae always equal to zeo. 5 RANSIEN ANALYSIS In ths secton, tme smulatons wll be caed out fo both the fxed-base and solatedbase stuctue unde eathquake gound moton exctaton. hs exctaton s assumed to act smultaneously at both suppot ponts of the system. Acceleatons ecoded at fou dffeent sesmc statons dung the 1995 Kobe eathquake [4] have been used as exctaton sgnals to ensue that a boad banded exctaton spectum s coveed. All ecodngs have been scaled to a hozontal Peak Gound Acceleaton (PGA) of 3 m/s so that also the fxed-base stuctue emans n ts elastc ange fo each ecod. hs s done to enable a compason between the fxed-base and solated-base stuaton, late n ths secton. Befoe tme smulatons can be caed out, sutable values of the desgn vaables of the two dentcal solaton systems ( k lam, c, and μ ) must be obtaned. hese values ae found n an teatve paamete lne-seach [6], n whch the Intestoy Dft Rato ( IDR ) and the absolute hozontal floo acceleatons a ae smultaneously mnmzed. hs mnmzaton s subjected to a sesmc gap constant of x so. 4 m. he IDR s defned as the quotent of the elatve hozontal dsplacement between two constucton levels, and the constucton level heght. he IDR and the absolute hozontal floo acceleaton a can be elated to the amount of (non)stuctual damage [9]. he optmal values of the desgn vaables esultng fom the paamete lne-seach ae: k lam = 5 kn/m, c = 1 kns/m, and μ =. 3 (whch n appoxmaton s the statc and dynamc fcton coeffcent fo eflon on steel). hese values ae used n all comng analyses. Subsequently, a compason s made between the smulaton esults of the fxed-base stuaton and the solated-base stuaton. he IDR (defned as the maxmum absolute value ove the ente tme esponse) as a functon of the constucton level s gven n Fg. 4 fo the ecods of the Shn-Osaka staton and the KJMA staton, espectvely. Clealy, sesmc base solaton benefts the supestuctue's sesmc pefomance. 7

8 .1.1 Fxed base Isolated base.1.1 Fxed base Isolated base.8.8 IDR [ ].6 IDR [ ] (a) constucton level [ ]. (b) constucton level [ ] Fgue 4: IDR, fo the fxed-base and solated-base system, as a functon of the constucton level, fo the acceleatons of the Shn-Osaka (a) and KJMA (b) staton. he amount of beneft of usng a base solaton system depends on the auto powe spectum of the exctaton sgnal. Fg. 4 shows that the beneft of base solaton wth espect to the IDR s clealy less fo the KJMA nput ecod than fo the Shn-Osaka ecod. he same concluson can be dawn wth espect to the absolute hozontal floo acceleaton a. Wth the ad of steady-state nonlnea dynamc analyses, whch wll be pesented n secton 6, ths dependency on the nput exctaton spectum wll be qualtatvely clafed. Vaous modelng appoaches ae used n engneeng pactce, as pescbed n ecent buldng codes [1]. o study the effect of some of these appoaches, eathquake tme smulatons ae agan employed. Fst of all, the effect of takng vetcal gound exctaton nto account s consdeed. Next, the effect of the vayng nomal (constant) foce s nvestgated. hs s done by compang the esults of the ognal model, whee the nomal foce s calculated as a constant foce to detemne the fcton foce, to the esults of a modfed model. In ths modfed model, the nomal foce n the Coulomb fcton model s fxed at ts statc equvalent, caused by gavtatonal loadng. hs mples that, n the case of the modfed model, the fcton foce does not depend on the dynamc esponse of the supestuctue, but s a functon of sldng velocty only. Fnally, the effect of the dynamcs of the supestuctue s studed. hs s done by compang the esults of the ognal model wth the esults of a model whee the supestuctue s egaded as a gd body. Fom these analyses, whch ae dscussed n detal n [6], t can be concluded that the supestuctue flexblty geatly affects the total system's esponse, as t nfluences the shea foces that ae exeted on the solaton system. he othe two effects (vetcal exctaton and vayng nomal foce) have elatvely lttle nfluence on the esponse of the solaton system and supestuctue. Vetcal exctaton wll be neglected n the next secton. 6 PERIODIC SOLUIONS ANALYSIS he steady-state behavo of a peodcally excted nonlnea dynamcal system s often studed to detect esonance fequences o bfucaton ponts whch may gve se to undesed dynamc behavo. he esultng vbatons may have a peodc, quas-peodc o chaotc natue [11]. he system s excted by the followng hozontal, hamonc acceleaton of the gound: u& = A sn πf t, & = (16) ho ho ( ) whee A ho and f ho ae the hozontal acceleaton ampltude and fequency, espectvely. As mentoned befoe, the fequency ange.1 to 1 Hz coves the most mpotant fequences n ho u& ve 8

9 an eathquake exctaton sgnal [1, 8]. In ths pape, the focus s on fndng stable peodc solutons (wth a peod equal to the peod of exctaton: f s) of the system fo exctaton fequences between.1 and 1 Hz. hs somewhat lage fequency ange s chosen because the fst esonance fequency of the base-solated buldng, whch should hadly be excted dung eatquakes, wll appea to be.6 Hz, whch s vey close to: whee bs klam ms ( π ) 1 ho f = Hz (17) m s s the mass of the supestuctue. In (17), f bs s the egenfequency of the gd supestuctue suppoted by an undamped base solaton system wthout fcton elements. Vaous numecal algothms exst fo effcent calculaton of peodc solutons of (14) and the local stablty. In ths eseach, the shootng method s used [11]. Fo a dscontnuous model specal measues must be taken to apply ths method [3]. Path-followng based on the shootng method s appled to calculate banches of peodc solutons wth f ho as contnuaton paamete. In some fequency egons the convegence of the shootng method s poo. In these egons stepped fequency sweepng based on standad numecal ntegaton s appled. In the latte case the ntegaton tme s chosen long enough to ensue that tansents have ded out and a steady-state soluton s eached. Fg. 5 shows the IDR, the absolute hozontal floo acceleaton a, and solato dsplacement x so (all defned as the maxmum absolute value ove the ente esponse peod and, fo the IDR and a, ove all constucton levels) as a functon of f ho fo vaous acceleaton exctaton ampltudes A ho, based on the magntudes of the PGA of most eathquakes [4] IDR [ ] (a) 1 1 a [m/s ] (b) 1 1 x so [m] A =.1 ho A =.5 ho A ho =1. A =. ho f ho [Hz] f ho [Hz] f [Hz] ho Fgue 5: Fequency ampltude dagams fo vaous exctaton ampltudes A ho (n m/s ) he acceleaton ampltude A ho s the same fo all fequences. A moe ealstc exctaton sgnal s typcally based on a constant exctaton dsplacement ampltude n the low fequency ange, a constant exctaton velocty ampltude n the mddle fequency ange, and a constant exctaton acceleaton ampltude n the hgh fequency ange. Howeve, an mpotant objectve of ths pape s meely to show that steady-state analyss may gve valuable addtonal nsght n the nonlnea dynamc behavo of a fcton-based base-solated supestuctue, next to the standad tansent analyss. Fg. 5(c) shows fequency anges whee x so = (ths value s not ndcated n the doublelogathmc gaph) and anges whee the solato exhbts stck-slp moton ( x so > ). In the fequency ange above.5 Hz, fo all values of A ho consdeed, the system s n stck dung the ente peod. Note that stck s only appoxmated by the smoothed Coulomb fcton model (7). Howeve, these esults ae all vefed wth the swtch-model (6). (c) 9

10 When A ho s nceased, the shea foces exeted by the supestuctue on the solaton system, ncease as well. heefoe, the lage A ho, the lage the fequency egons wth solato stck-slp moton. In addton, the shea foces that ae exeted on the solaton system ae the lagest nea the esonance fequences of the supestuctue. hs s the eason why n fgue 5(c), fo A ho = m/s, solato stck-slp moton s nduced aound the nd and 3d fxedbase egenfequency at 1.7 and 1.98 Hz, espectvely (Fg. 1(b)). Lage esonances can be seen nea f bs. 6 Hz (fo A ho. 5 m/s ), because nea ths fequency, stck s elatvely nsgnfcant compaed to slp. Actually, ths esonance should not occu n pactce, because k lam s chosen such that f bs <. 1 Hz (ecall that.1 Hz s consdeed to be the lowe bound of the fequency content of most eathquakes [1, 8]). In fact, fo A ho.5 m/s, a sngle-dof model based on a gd supestuctue shows a esponse fo x so, whch s n vey good ageement wth Fg. 5(c) fo f ho. 1 Hz [5, 6]. Fo vey low A ho (e.g. A ho =. 1 m/s ), the system acts as a fxed-base supestuctue ove almost the ente fequency ange, except fo a small fequency band nea the fst fxed-base egenfequency at.46 Hz. Above.5 Hz, the system behaves as a fxed-base supestuctue and fxed-base egenfequences (e.g. at.98 Hz, Fg. 1(b)) can be dstngushed n the esponses of IDR and a. In Fg. 6 a compason s made between the steady-state peodc behavo of the supestuctue n the fxed-base and solated-base stuaton fo A =. 5 m/s. ho 1 1 (a) Fxed base Isolated base 1 (b) Fxed base Isolated base IDR [ ] 1 3 a [m/s ] f ho [Hz] f [Hz] ho Fgue 6: Fequency ampltude dagams n the fxed-base and solated-base stuaton fo A ho =.5 m/s In the fxed-base stuaton, the system s lnea and, theefoe, the fequency ampltude dagams can be egaded as (lnea) Fequency Response Functons (FRF's). Fg. 6 shows that the sesmc esstance s stongly mpoved n the fequency ange nea.46 Hz. In ths fequency ange, maxmum values of IDR and a ae stongly deceased due to decouplng of the gound moton and the fst fxed-base egenmode wth egenfequency at.46 Hz (Fg. 1(c)) ealzed by slppng of the base-solaton system, see also Fg. 5(c). Fo hghe fequences, and fo A ho =. 5 m/s, the system's dynamc behavo s hadly affected by the basesolaton system, because t emans n stck fo almost evey exctaton fequency. A deteoated pefomance s found aound f bs.6 Hz. Howeve, as stated befoe, ths s below the domnant exctaton fequences of most eathquakes. 1

11 1 (a) KJMA Shn Osaka 15 KJMA Shn Osaka (b) PSD [m /s 4 /Hz] 1 1 PSD [m /s 4 /Hz] f [Hz] f [Hz] Fgue 7: Auto powe specta of the Shn-Osaka and KJMA hozontal gound acceleatons (Δf=.1 Hz, PGA=3 m/s ) on double log scale (a) and double lnea scale (b) Now, the meanng of studyng steady-state esponses of stuctues potected aganst sesmc exctaton va nonlnea devces can be llustated. Fg. 4 shows that the base-solaton system yelds a lage mpovement of the tansent sesmc esponse fo the Shn-Osaka exctaton sgnal than fo the KJMA exctaton sgnal. A qualtatve explanaton fo ths fact can be gven by studyng Fgs. 5(c) (fo A ho =. 5 m/s ), 6 and 7. Fg. 7 shows that n the fequency ange.1-1. Hz the powe spectum of the Shn-Osaka sgnal mostly domnates the KJMA spectum (at.46 Hz t s a facto 15 lage). Fgs. 5(c) and 6 show that fo A ho =. 5 m/s n a fequency band aound.46 Hz due to the fcton elements decouplng between the hozontal gound moton and the supestuctue can be ealzed ( x so =. 5 m) esultng n deceased IDR and a values. It s less lkely that the lage value of the Shn Osaka powe spectum nea.8 Hz wll stongly contbute to the slp moton because nea that fequency ant-esonances occu n Fgs. 5(c) (fo A ho =. 5 m/s ) and 6. Of couse, the exctaton level dependency of the esponse complcates the ntepetaton. hee s no doubt, howeve, that a bette nsght n the tansent esponse s obtaned by also studyng the steady-state esponse. 7 CONCLUSIONS hs study foms an example of an emegng applcaton of new modelng and analyss technques fo non-smooth dynamcal systems n the feld of eathquake engneeng. In ths pape, the non-smooth dynamc modelng and analyss of stuctue wth a fcton-based base solaton system have been dscussed. A compact dynamc model of the system has been deved by couplng a dynamcally educed fnte element model of the buldng to the fcton-based base solaton system, usng Lagange's method fo systems wth constants. hs compact model allows fo both effcent and accuate nonlnea dynamc analyss n the fequency ange of nteest. Responses obtaned usng a smooth appoxmaton of the dscontnuous fcton foce coesponded vey well wth esponses based on a non-smooth swtch-band fcton model. In tansent tme smulatons, the beneft of the base solaton system has been clealy demonstated usng fou ecodngs of the Kobe eathquake as gound exctaton sgnals. he dynamcs of the (flexble) supestuctue lagely nfluence the total system's dynamc esponse. Vetcal gound acceleaton and a vayng nomal foce on the fcton elements, have elatvely lttle nfluence on the solato dsplacement and the supestuctue esponse. Steady-state analyss wth the ad of peodc soluton solves povdes valuable addtonal nsght n the nonlnea dynamc behavo of the base-solated stuctue next to standad tansent analyss. he obtaned fequency ampltude dagams evealed that, fo an acceleaton 11

12 exctaton level of.5 m/s, the fcton-based base solaton system s manly actve n the fequency exctaton ange nea.46 Hz. Fo hghe fequences, the solaton system emans n stck dung the ente peod of the peodc soluton and the system acts as a fxed-base stuctue. Futhemoe, a base solaton esonance fequency s found nea.6 Hz, below the domnant fequency ange (.1-1 Hz) of most eathquakes. Hee, the supestuctue appoxmately behaves as a gd body. he extent of beneft of the base solaton system n the tansent esponse may be qualtatvely explaned by examnng the nput exctaton spectum and the nonlnea steady-state dynamcs. Howeve, the exctaton level dependency of the tansent and steady-state esponse does complcate the ntepetaton. Summazng, t may be expected that theoy and numecal tools fo non-smooth dynamcal systems may pove to be vey useful n the futue desgn of eathquake esstant stuctues. REFERENCES [1] R. Sknne, W. Robnson, G. McVey, An ntoducton to sesmc solaton. John Wley & Sons, Inc., New Yok, [] F. Naem, J. Kelly, Desgn of sesmc solated stuctues: fom theoy to pactce. John Wley & Sons, Inc., New Yok, [3] R.I. Lene and H. Njmeje, Dynamcs and bfucatons of non-smooth mechancal systems. Lectue notes n appled and computatonal mechancs, Vol. I8. Spnge, Beln, 4. [4] Pacfc Eathquake Engneeng Reseach Cente (PEER), Stong moton database, eteved fom 8 June 4. [5] H.M.R. Suy, R.H.B. Fey, F.M.B. Galant, H. Njmeje, Nonlnea dynamc analyss of a stuctue wth a fcton-based sesmc base solaton system, Nonlnea Dynamcs, specal ssue on Dscontnuous Dynamcal Systems, accepted fo publcaton, 7. [6] H.M.R. Suy, Nonlnea dynamc analyss of a stuctue wth a fcton based sesmc base solaton system, epot DC 5-5, Endhoven Unvesty of echnology, 5. [7] R.R. Cag J. and A.J. Kudla, Fundamentals of stuctual dynamcs, nd Edton. John Wley & Sons, Inc., Hoboken, New Jesey, 6. [8] Clough, R., and Penzen, J., Dynamcs of stuctues, nd Edton. McGaw-Hll Inc., New Yok, [9] K. Pote, A. Kemdjan, J. Legue, S. Kng, A buldng damage estmaton method fo busness ecovey. 1 th Wold Confeence on Eathquake Engneeng, Auckland, New Zealand, Januay 3- Febuay 4,. [1] ECS EN 1998, Euocode 8: Desgn of stuctues fo eathquake esstance, Euopean Commttee fo Standadzaton, 5-6. [11].S. Pake and L.O. Chua, Pactcal numecal algothms fo chaotc systems. Spnge-Velag Inc., New Yok,