THE INFLUENCE OF AMPLITUDE AND PHASE DIFFERENCES IN BI-DIRECTIONAL GROUND MOTION ON THE BEHAVIOUR OF IRREGULAR STRUCTURES.

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1 th World Conference on Earthquake Enneern Vancouver, B.C., Canada Auust -6, Paper No. 7 THE INFLUENCE OF AMPLITUE AN PHASE IFFERENCES IN BI-IRECTIONAL GROUN MOTION ON THE BEHAVIOUR OF IRREGULAR STRUCTURES. Ncholas A ALEXANER SUMMARY The eneraton of an appled torque spectrum for asmmetrc buldn structures s dscussed. The crtcal condtons surroundn peaks n ths torque spectrum are noted. The are found to be a functon of phase tunn of component frequences. A parameter termed the deree of phase tunn s proposed. Larer values of the deree of phase tunn produce larer total power of the appled torque. The deree of phase tunn s nvestated for actual acceleroram records and found to be dfferent. The nfluence of ths appled torque on the structural response of the buldn sstem s nvestated n both lnear and nonlnear structural sstems. Larer derees of phase tunn produce larer power of the torsonal structural response. The mportance of the deree of phase tunn s underlned wth reference to selecton of a statstcall unbased set of accelerorams selected for nonlnear tmehstor analss of asmmetrc structures. INTROUCTION In recent ears there has been a move towards nonlnear tme-hstor analss of buldn structures subject to sesmc actons. Ths has manl been due to ncreased computatonal power and better more useable nonlnear fnte element codes. The feasblt of performn such analses on comple, materall nonlnear buldn structures has come nto the realm of the enneern desner not just the researcher. Thouh, perhaps, ths s true onl for unusual, epensve or safet crtcal structures at present. The most wdel used alternatve s, of course, classcal lnear modal analss whch emplos a pseudo-nonlnear and hhl smoothed desn spectrum. The dsadvantaes of usn modal analss are well known; however the desn spectrum contans a useful summar and, n some crude sense, a probablstc nterpretaton of the characterstcs of a spectrum of severe sesmc events. Thus the abandonment of modal analss n favor of a nonlnear tme-hstor approach rases the queston about how to use the desn spectrum. Man researchers have advocated usn the desn spectrum to enerate a seres of artfcal accelerorams. Tpcall, the ampltude spectrum of an artfcal and random snal are adjusted such that the response spectrum of a snle deree of freedom sstem matches appromatel the desn spectrum. The phase content of the artfcal record s tpcall mantaned as random. An alternatve s to use recorded accelerorams for actual sesmc events. In both cases the queston remans Lecturer n Structural Enneern, epartment of Cvl Enneern, Unverst of Brstol, UK.

2 about the selecton of accelerorams and how man to use for the purposes of a relable estmate of the performance of the buldn structure when subject to the unknown future sesmc event. Ths paper consders the analss of asmmetrc buldn structures. Partcularl t attempts to determne f certan crtcal phase and ampltude combnatons n orthoonal acceleroram pars can produce more or less torsonal moton n such structures. The reason wh ths ma be mportant s that t s currentl tpcal to use onl a small sample of real accelerorams, perhaps onl, [], n a nonlnear tmehstor analss. A small sample ma or ma not accuratel descrbe an unknown populaton statstc, and often t doesn t; unless more nformaton s known about the characterstcs of the unknown populaton. The unknown populaton here s the probablt of a certan unknown future sesmc event producn a partcular crtcal structural response. In the case of artfcal accelerorams a larer number of records are often used. However, wthout understandn of the effect of the round moton phase components on the structural performance, t s stll possble that mss-represent the unknown populaton statstcs. FREQUENCY COMPONENTS OF APPLIE INERTIAL TORQUE, a mult-store buldn structure can be dealzed b the follown snle store dealzaton, shown n Fure. The lumped buldn mass m s eccentrcall supported b a hpothetcal column. The equaton of moton, under lnearl elastc condtons, of such a structure s well documented [], [], etc. ; and t can, n an case, be derved b emplon Euler-Laranan equatons derved from the varatonal prncple of least acton. Thus equaton of moton, defned wth a coordnate orn at the elastc centre of stffness, s as follows n equaton () B a process of sub-structurn that adopts three master derees of freedom (,,θ ) ε ε ε ε && && && rθ + [ C] & & & rθ + θ rθ && = && ε && ε && () [ M ] u& + [ C] u& + [ K] u = u& where r s the radus of raton of the buldn mass defned at the orn of the buldn. Eccentrct ratos ε = e r, ε e r Frequenc parameters = K m = K m, = K mr C s a =,. [ ] dampn matr. &, & are the orthoonal, horzontal, round moton components that ecte the structure. The eternal actons have a torque component τ () t whch can be obtan for the last row of the RHS of equaton () hence equaton () B emplon a Fourer transform of () t & & τ () t & () t ε & () t ε θ = () &,, the nfluence of a frequenc component ( ) of the torque τ s ven b equaton (). Now the modulus of ths epresson s ven b equaton (5) where the phase dfference spectra, equaton (). & () t = A ( ) ep( φ ( ) ) ep( t) d, & () t A ( ) ep φ ( ) π π θ ( ) ep( t) = d

3 τ () t ( ) ep( t) π = d θ ε m ε Fure, dealzed eccentrc snle store structure { ( ) A ( ) ε sn( φ ( ) )} ( ) = ep( φ ( ) ) A ( ) ε A ( ) ε cos( φ ( ) ) () ( ) = ( ) φ ( ) φ ( ) φ ( ) = () {( ε ) + ( A ( ) ε ) } { ( A ( ) ε )( A ( ) ε ) cos( φ ( ) )} = µ ( ) σ ( ) A (5) Phase tunn µ represents an ampltude dependent term of torque power,.e. the part that s not affected The term ( ) b phase tunn. σ ( ) s an ampltude and phase dfference dependent term, of the torque power,.e. the part that s affected b phase tunn. The terms µ ( ) and σ ( ) can onl be evaluated numercall for partcular acceleroram pars. In order to provde a meannful comparson the follown accelerorams are scaled usn ther Aras ntenstes []. The mean Aras ntenst I a of & & and & & s compute, equaton (6) and then the &, & accelerorams b dvdn b I a. Thus the normalzed, scaled, acceleroram par has a new mean Aras ntenst of m/s. Ths also mples that, for the condton of equal eccentrct ratos, the total power of the fed ampltude dependent term σ ( ) n equaton (5) s constant, shown n (6), for an normalzed acceleroram par. I a π = && () t dt + () t dt &&, µ ( ) d = ε π (6) Fure, raphcall dsplas equaton (5) for a two acceleroram pars. For ths fure the eccentrct ratos are ε =. σ. It s ε. Fure also dsplas the relatve mantudes of terms µ ( ) and ( ) = clear from vsual nspecton that torque power for the Kobe event s larer that for the Northrde event.

4 Remember the mean total power of & & and & & the same for both events, due to scaln. Ths dfference n mantude s not dependant on scaln, or s t dependant on the structural confuraton; thouh larer eccentrctes wll enerall ncrease the torque power. The dfference s entrel due to the phase tunn term σ ( ). Both raphs are dsplaed from to 5 Hz as almost all the power s lmted to ths narrow rane. The frequences at whch peaks n the torque power occur are often caused b phase tunn as well as larer ampltudes. In the Northrde eample plot the peak at.8hz s amplfed b phase tunn n ths wa. However ths s not a unversal result. Consder the peak at about.hz n ths plot, the phase dfference s such that t has almost no effect torque power. At.Hz there are larer ampltude than at.8hz but the phase tunn s such that the are not utlzed because σ ( π.). Ths s mportant, for knowlede of ampltude content s necessar but not suffcent to quantf torque power. The selecton of acceleroram pars for tme-hstor analss based solel on total power such as Aras ntenstes etc. wll mss some mportant nformaton n the case of asmmetrc buldns. ) f r T( w e P o e T orqu ) f r T( w e P o e T orqu Northrde, USA, 7//9 - : PST frequenc f [Hz] ( ) µ ( ) σ ( ) Kobe, Japan, 7//95 - : µ ( ) σ ( ) ( ) η =.76 η = frequenc f [Hz] Fure, Comparson of two appled torque power for two events, ε ε =. = The condtons for the mamum of (5) are overned b σ ( ) and are ven n equaton (7). Ampltude terms A ( ), A ( ) are postve, b defnton.

5 ,, ( ) occurs at φ ( ) =, ( ) occurs at φ ( ) ma π ε ε > ε ε <, ε ε > = mn (7) π, ε < ε The relatve mantudes of terms µ ( ) and σ ( ) depend on the ampltude eccentrct terms A ( ) ε and A ( ) ε. If ether s zero the phase tunn term σ ( ) s zero,.e. phase components have no nfluence on the mantude of ( ) n ths case. Structures wth onl one as of asmmetr [],.e. ε = orε =, wll not observe the nfluence of phase tunn even f two orthoonal accelerorams are emploed. If one of the ampltude eccentrct terms s lare relatve to the other then the mantude of µ. As the ampltude eccentrct terms tend to smlar values,.e. ( ) s overned b ( ) A ( ) ε A ( ) ε = equaton (5) smplfes to ( ) = A ( ) ( ε ) cos φ ( ) { ( )} (8) In ths case the nfluence of phase tunn s most snfcant. At crtcal phase dfferences t can double the ampltude of ths torque component or reduce t to zero. Total power of appled nertal torque The mamum and mnmum mantudes of Torque ( ), wth respect to phase dfference, are ven as follows n equatons (9) and (). ue to the nature of (7) the equaton (9) and () are dependant on the sn eccentrct rato product; where sn ( ) s the snum functon. The provde bounds to the total power of the Torque can be epressed n (). These bounds mark the nfluence of the phase dfference content of the orthoonal acceleroram par. * σ ( ) ( A ( ) ε )( A ( ) ε ) * ( ) = µ ( ) σ ( ).sn( ε ) mn ε * ( ) = µ ( ) + σ ( ).sn( ε ε ) ma = (9) () ( ) d ( ) d ( ) d mn ma () The deree of phase tunn Ths leads to an assessment of the deree of phase tunn η n a par of orthoonal accelerorams, ven b equaton () and ths η s propert of the acceleroram par and almost ndependent of the structural eccentrct ratos; onl the sn of the eccentrct rato product s mportant. When η equals the phase dfference of the acceleroram par s such that t leads to a mnmum power of the appled nertal torque. When η equals the phase dfference of the acceleroram par s such that t leads to a mamum power of the appled nertal torque. The plus or mnus sn s a consequence of the condtons (9) and () and whch follow drectl from (7). Wthout numercall evaluatn ρ n equaton (), for a partcular acceleroram par, t s not possble to ascertan whch condton ε ε > or ε ε < wll result n the larer η and hence evaluate the larer appled torque from a partcular acceleroram par.

6 ( ) d ( ) ( ) d ( ) ( ε ε ) ( ) A ( ) cos( φ ( )) A ( ) A ( ) d d A η = mn d = sn = ( ± ρ) () d ma mn There s propert about equaton () such that the follown statements () are also vald. The bound on larer η s such that t s alwas between.5 and. Let η = ( ρ), η = ( + ρ) then η +η =. () In Fure, the value of η, deree of phase tunn, s ven of each acceleroram par. The Kobe earthquake has the larer η. Ths mples that f both records where used n a tme-hstor stud, and are scaled to comparable ampltude levels the Kobe par would nduce larer torsonal nertal actons on the buldn. The usefulness of the η s that t s predomnantl a characterstc of the round moton not the structure,.e. t appears self-evdent that some events produce more torque power than others. However, t has et to be shown that ths ncrease n torque power, the sstem nput, results n larer structural torsonal responses, the sstem output. 5 5 γ = ) γ =.5 ( z ln frequenc rato Ω = / Fure, Torsonal response transfer functon: ε ε =. = LINEAR STRUCTURAL RESPONSE TO APPLIE INERTIAL TORQUE B takn the Fourer transform of equaton (), the frequenc doman representaton of () s ven n equaton (). u & {[( ) + ( [ C] )] } u&& ( ) = H ( ) ( ) = [ K ] [ M ] [ ] u& ( ) ()

7 Whle t s, currentl, feasble to evaluate ths transfer matr [ ( ) ] H usn a computer alebra packae [5]; the eneral, alebrac, result s dense, comple and rather un-elucdatn. In order to eamne some of H a few condtons are assumed. Frst dampn n nored, and structural frequenc the features of [ ( )] parameters are assumed = ; ths condton should place the structure near the crtcal condton = θ of coupled swa and torsonal resonance. The response rotatonal acceleraton of the structure s ven b the last row of (); thus t can be shown that the buldn rotatonal acceleraton s the product of the real. functon z ( ) and the nertal torque ( ) & ( ) = z( ) ( ), z ( ) r θ & Ω =, ( Ω + Ω ( ε ε ) Ω = (5) The form of ths transfer functon z ( ) can be vsualzed n Fure. In the more eneral case, when dampn s present, z ( ) s a comple functon ven n equaton (6) and ts form s also dsplaed n γ Fure ; where γ s the rato of crtcal dampn and dampn matr[ C] [ K] s an een-frequenc. =. In ths sstem z ( + Ωγ ) ( + Ω Ωγ) Ω Ω 6 Ωγ Ω Ω γ Ω γ Ω z( )=:= ( ) ( + + ε Ω 6 6 Ω 5 γ Ω γ Ω 6 ε 8 Ω γ Ω 5 γε Ω ε Ω 6 ε Ω ε γ ) (6) ) ( ) T n d a ) ( z )Ṭ ( ( z c e l. A n al R otato Frequenc [Hz] Frequenc [Hz] - z ( ) r & θ ( ) ( ) rθ & () t Tme [s] Fure, Kobe acceleroram par, ε = ε =., = = θ = π, γ =. 5 In Fure an eample of the frequenc doman approach s vsualzed, torque power and torque transfer functons, top raph, are multpled to produce torsonal response power, centre raph, and tme doman torsonal response acceleraton, bottom raph. From the tme doman responses t s possble to produce torsonal acceleraton response spectra, shown n Fure 5. The torsonal acceleraton response of the

8 Kobe event s enerall larer than the Northrde event. The peak torsonal acceleratons are of the order of % larer for the Kobe event. Notce, however, at the hh frequenc rane the Northrde spectrum eceeds the Kobe. If Fure s consdered, there s some torque power n ths hh frequenc rane for the Northrde event whle the Kobe event has almost no power at ths frequenc rane. Thus the nfluence of the deree of phase tunn on the torsonal response s dependant on structural confuraton. 6 5 Kobe n ra t o le A cce n T orso Northrde - Frequenc /π [Hz] Fure 5, Torsonal Acceleraton Response Spectra, ε ε =., = = θ, γ =. 5 = It has now been mathematcall establshed that the mantude of the response torsonal acceleraton s nfluenced drectl b the phase tunn term σ ( ) n the case of a lnear structural sstem (). There s an mportant corollar of (6); the nfluence of the phase tunn term s mtated b the transfer functon z ( ). The reatest nfluence of the phase tunn on the rotatonal acceleraton s when frequences at whch lare torque power concde appromatel wth frequenc band around the structural frequenc parameter. NONLINEAR STRUCTURAL RESPONSE TO APPLIE INERTIAL TORQUE Lnear behavor s sesmc structural dnamcs s prosac, and there s a need to carr forward deas n the prevous secton to ncludn the nfluence of nelastc behavor. Because of the complet of the dfferental sstem t s not feasble to derve analtcal epressons for responses (solutons) n terms of appled nertal actons. Numercal and computatons soluton schemes, n the tme doman, take over from the analtc frequenc doman approach n the prevous secton. Ths allows the sstem nonlneart to be nvestated. The nonlnear buldn model emploed s taken from [6], [7]. Nonlnear Buldn Model An mult-store structure, wth a lare number of derees of freedom, can be analsed b consdern a reduced or sub-structured sstem. In ths paper the notaton [ u ] s a vector of elements matr of elements γ, j u and [ γ ]. The prncpal or master derees of freedom of the eneral buldn are to be two horzontal swa ordnates, and an anular dsplacement ordnate ϕ, three n total: [ ] [ ϕ] T, j s a u =.

9 The coordnate orn s the buldn centre of mass (CM). The anular dsplacement s defned ϕ = r mθ where r m s the radus of raton of the buldn mass about the CM and θ s the rotaton of the structural mass about the CM. In eneral the sstem s defned b equaton (7). Ths dffers from equaton () n that t has a dfference orn. [ u & ] + [ ][ u& ] + [ f ] = [ & ] γ (7), j Under lnearl elastc normalsed stffness acton vector [ f ] s defned n [] and equaton (7) [ ] [ k ][ u ] = f =, j λ εm λ λ λ εm εm λ εm λ t ϕ where ( λ = λ =, λt = t ) acceleraton vector s [ ] [ ] T & = && &&. The non-dmensonal sstem stffness matr s [ k, j ] orthoonal dampn matr s [ γ ], are the frequenc rato parameters. The round (8) and the, j. Thus n ths snle store buldn dealsaton the elastc stffness matr resulted n a concse parametrc formulaton n terms of s ke parameters: () whch can be thouht as a pro for the buldn fundamental frequenc, () ε m () ε m whch epress the buldns plan eccentrct (v) λ the rato of buldn stffness n the to drectons (v) λ s parameter that s zero for structures that have no rreulart n elevaton and have resstn elements alned to the lobal aes. (v) λt the rato of buldn stffness n the θ to drectons. The advantae of ths lnear parametrc formulaton s that the actual confuraton of a partcular buldn need not be stated. In the case of nonlnear buldn behavour [ f ] needs to be consderable more elaborate. [7] descrbes the detals of the alorthm emploed. The eneral form of the lnear epresson n (7) s etended to nclude some smple nonlnear character. The am s ntroduce the mnmum number of etra, nonlnear, parameters. The formulaton s based on a multvarate Talor seres epanson up to and ncludn quadratc terms. Cross quadratc terms u u j are nelected. ue the nature of the problem an antsmmetrc functon shown n equaton (9) s conjectured. [ f ] [ k ][ u ] c( ) [ k ][ u u ] = (9), j, j [ f ] [ k ][ u ] + [ h ] =, () j At the orn ths functonal relatonshp (9) s dentcal to equaton (8). In order to ncorporate buldn nelastct there s a need for a dfferent set of functons, equaton (), for the undeformn paths. These alternate functons are parallel to equaton (8) thus the mpose a coordnate shft [ h ] of orn [7]. There s need to defne an alorthm for controlln the use of equatons (9) and (). The smplct of such an approach, from a parametrc pont of vew, s that all lnearl elastc parameters are mantaned and one etra nonlnear parameter s added.e. sstem strenth c. A comparson of ths nonlnear model wth other approaches s ven n [6]

10 Numercal Results The nonlnear tmehstor analss s performed on the two earthquakes used prevousl. The phase dfference content s modfed for both records such that, n effect, each event produces a set of three pars of records: () the ornal unmodfed acceleroram par, equaton () () An artfcal acceleroram par that have dentcal ampltude spectra as the ornal par () but have no phase dfference content; see equaton () () An artfcal acceleroram par that have dentcal ampltude spectra to () but have mamal phase dfference content, see equaton () () t && () t, && () t && () t & = = () t = && () t & () t = A ( ) ep( φ ( )) ep( t) d & (), () t = && () t & () t = A ( ) ep( ( φ ( ) + π )) ep( t) d & (), In Fure 6, the nfluence of phase content on the acceleraton response spectra s noted. The total acceleratons n the and drecton are onl marnall nfluenced b varatons n phase content. However t s clear that the torsonal acceleraton s snfcantl nfluenced b phase content. Ths shows the aruments developed for lnear sstems are stll vald, n some measure, n the case of nonlnear hsteretc buldn sstems. () ( & ) & φ ( ) = φ ( ) φ ( ) φ ( ) = π φ ( ) = - ( & ) & φ ( ) = φ ( ) φ ( ) φ ( ) = π φ ( ) = - ϕ& & 6 5 φ ( ) = φ ( ) φ ( ) φ ( ) = π ( ) = φ - Frequenc Parameter /π [Hz] Fure 6, Northrde, nonlnear response spectra, c =., λ =, λ =, ε =., ε =., γ =.5 mt m m

11 In Fure 7 there s a comparson of the nonlnear torsonal acceleraton spectra for the Northrde and Kobe records. Both records underlne the nfluence of phase content on torsonal response. The Kobe record stll has larer torsonal acceleratons. The dfferences between the Kobe and Northrde plots for equaton () and () ndcate that the dffern ampltude contents also has an effect here; remember the σ n equaton (5). formulaton of the phase tunn term ( ).5 Northrde c e l. A n al T orso.5.5 φ ( ) = φ ( ) = φ ( ) φ ( ) φ ( ) = π Kobe φ ( ) = π c e l. A n al T orso φ ( ) = φ ( ) φ ( ) φ ( ) = - Frequenc Parameter /π [Hz] Fure 7, Comparson on Nonlnear Torsonal Responses c =., λ =, λ =, ε =., ε =., γ =.5 mt m The results n Fure 7 seem to ndcate lttle overall dfference between the lnear and nonlnear sstems. The reason for ths s that the eccentrct ratos are small and hence the torsonal moton s enerall small. The structural ductle behavor s nduced, predomnantl, b swa motons and not torsonal motons. In order to nvestate the dfferences here another combnaton of sstem parameters s emploed n Fure 8. The larer eccentrct ratos cause larer appled torque power whch nduces ductle behavor caused b buldn torson. As a consequence, n ths nonlnear model, the torsonal acceleratons are bounded b the structure s torsonal strenth. Thus the nfluence of Kobe s larer deree of phase tunn on the torsonal acceleraton s markedl reduced; both Kobe and Northrde records produce more comparable levels of torsonal acceleraton. However, when consdern the torsonal ductlt demand spectra, t s clear that the larer appled torque power, nduced b the Kobe par of records, manfests tself n the nonlnear structural sstem b larer ductlt demands. Thus the eneral concluson s that larer derees of phase tunn n an acceleroram par, can n nonlnear structural confuratons, lead to ether larer torsonal acceleraton or larer ductlt demands. m

12 6 c e l. A n al T orso 5 Northrde Kobe - d n m a e u ctlt n al Northrde Kobe T orso - Frequenc Parameter /π [Hz] Fure 8, Comparson of uctlt demand and Torsonal acceleraton c =., λ =, λ =, ε =., ε =., γ =.5 mt m CONCLUSIONS In ths paper, nonlnear analses of asmmetrc buldns subjected to sesmc actons s nvestated. The nfluence of the round moton, ampltude and phase dfferences frequenc components s dscussed. Epressons for the frequenc contents of the appled torque are proposed. Ampltude and phase dfference terms are dentfed. The condtons for mamal torque power are eplored. These mama occur when the ampltude content at a certan frequenc s lare and of smlar mantude n both the and drecton and ths stuaton s combned wth the condton of crtcal phase dfferences between these components. Under these condtons of crtcal phase tunn, the and round moton can amplf the appled torque power b a factor of two or reduce t to zero. A concept, the deree of phase tunn between a par of orthoonal accelerorams s ntroduced. Ths quantfes the promt to crtcal phase tunn of a par of records. The deree of phase tunn s dfferent for dfferent records. What s also clear s that some pars of acceleroram records produce more appled torque power than other due to ther lare deree of phase tunn. Usn a classcal lnear frequenc doman analss, the nfluence of the appled torque power s mapped throuh the structural sstem s comple transfer matr to elct the structural response. It has been mathematcall demonstrated that larer deree of phase tunn results n larer appled torque power and subsequentl larer structural torsonal acceleratons. Lnear response spectra for events presented n ths paper show as much as % ncrease n peak acceleratons due to the phase tunn effect. Usn a nonlnear, hsteretc, buldn model a more detaled analss s performed. For nonlnear structural sstems that ehbt ductle behavor, predomnantl due to swa moton, the ncrease n torsonal acceleratons due to phase tunn s also observed. In the case of nonlnear structural sstems m

13 that ehbt ductle behavor due to torsonal moton there s no snfcant ncrease n torsonal acceleratons due to phase tunn. However there s a snfcant ncrease n torsonall nduced ductlt demand. It s clear that knowlede of the ampltude-frequenc content, of the round acceleraton, s necessar but not suffcent to quantf the power of the appled torque or the torsonal response of asmmetrc structures. Larer derees of phase tunn enerate, n nonlnear structural sstems, ether larer torsonal acceleratons or larer torsonal ductlt demands. Thus, when selectn a set of accelerorams for use n nonlnear tmehstor studes, of asmmetrc buldns, t s necessar to reconze the nfluence of the deree of phase tunn. Otherwse the statstcal predctons from small samples of analses ma be based and unrepresentatve of the structures performance n the case of the unknown future sesmc event. REFERENCES CEN European Standard. Eurocode 8: esn of structures for earthquake resstance. Part General rules, sesmc actons and rules for buldns. CEN PrRAFT No., Ma. Secton..., pp. Chandler, A.M.,Correnza J.C. & Hutchnson G.L, 995 Ultmate lmt state Sesmc Torsonal provsons n Eurocode 8. Proc. Instm Cv. Enrs Structs & Blds., Feb., pp Hejal, R. & Chopra, A.K Earthquake response of torsonall coupled buldns Earthq. En. Research Center, Un. of Calforna at Berkele UCB/EERC-87/. Aras, A 97, A measure of Earthquake ntenst R.J ed. Sesmc esn for Nuclear Power Plants, MIT press, Cambrde, Massachusetts, pp Maple 7., Ma 8,.Waterloo Maple Inc. 6. Aleander, N.A. & Goorvadoo N. Inelastc models and ductlt demand of asmmetrc buldns, th European Conference on Earthquake Enneern,, Aleander N.A., Goorvadoo N., Noor F.A. and Chanerle A.A. A new parametrc store nonlneart ncludn hsteretc behavour n the dnamc analss of buldns under b-drectonal sesmc acton. The 5 th Internatonal Conference on Computatonal Cvl & Structural Enneern, Belum, ; :