VALIDATION OF SINGLE STOREY MODELS FOR THE EVALUATION OF THE SEISMIC PERFORMANCE OF MULTI-STOREY ASYMMETRIC BUILDINGS

Transcription

1 13 th Wold Confeence on Eathquake Engineeing Vancouve, B.C., Canada August 1-6, 4 Pape No. 13 VALIDATION OF SINGLE STOREY MODELS FOR THE EVALUATION OF THE SEISMIC PERFORMANCE OF MULTI-STOREY ASYMMETRIC BUILDINGS Gonzalo ZÁRATE 1 and A. Gustavo AYALA 1 SUMMARY Of all the models used in the past fo the development and evaluation of seismic design citeia fo asmmetic stuctues, the most common has been the single-stoe model with the same numbe of esisting planes as the stuctue it epesents, same uncoupled dnamic popeties and thee degees of feedom. When consideing the coupled dnamic popeties of these models, the ae diffeent to those of the stuctue the aim to epesent. The onl model that allows a coespondence of esults with the stuctues epesented is the paametic model poposed b Kan and Chopa. Based on this model, and with the pupose of evaluating and tansfoming to eal buildings the massive amount of available esults geneated with single-stoe models and investigating new cases of inteest, this pape pesents an appoimate pocedue that allows the definition of a 3-degee of feedom simplified stuctue epesenting the eal stuctue and satisfing the equisites of the paametic model of Kan and Chopa. The non-linea foce-displacement elationships fo this model ae obtained fom the capacit cuves of the eal stuctue. Once the pefomance point of the appoimate model is obtained, the seismic pefomance of the eal building is obtained using the coespondence elationships to tansfom this pefomance point to the esponse of the eal stuctue. Finall the foces and displacements coesponding to this pefomance point ae ecoveed fom the esults of a pushove analsis. INTRODUCTION Most studies elated to the tosional behaviou of buildings have been based on simplified single-stoe models with thee degees of feedom (3DOF), Figs.1 and. The assumption on which these studies ae based has been that these models epesent, in the linea an non-linea ange of behaviou, the esponse of multi-stoe tosionall coupled buildings subjected to seismic foces, e.g., Bustamante and Rosenblueth [1] fo linea models and Aala and Gacía [] fo non-linea models The infomation deived fom these models is vast and has been etapolated diectl to multi-stoe buildings. In fact, the validit of singlestoe models to appoimatel epesent multi-stoe buildings esponse is questionable. Seveal authos have concluded that the single stoe analog to multi-stoe buildings can be used onl with shea buildings with constant stiffness atios between lateal load esistant elements on each stoe (in these 1 Instituto de Ingenieía UNAM, Méico D.F., Méico, gaala@dali.fi-p.unam.m

2 stuctues the stiffness centes of all floos ae located along a vetical ais), Yoon and Staffod-Smith [3]. Figue 1. Multi-stoe asmmetic building. Isometic view Plan view Figue. Equivalent single stoe asmmetic model. 3. SX CM 3. esx=. CM= 3. esx=.1 CM= DMAX/DSYM DMAX/DSYM DMAX/DSYM XR XR XR 3. esx= CM= 3. esx=.1 CM=.1 3. esx=. CM=. DMAX/DSYM DMAX/DSYM DMAX/DSYM XR XR XR DMAX/DSYM esx=.3 CM= XR. Left element "" Cente element "" Right element "" BIRECTIONAL EARTHQUAKE Design 1, Mass at (.,.) es = = T = T =1.5 seg, h = b R = 1.5 Rn, Q = Q = 4. Figue 3. Illustation of the seismic pefomance of a single-stoe 3DOF model. With the aim to adequatel intepet data fom eseach of single-stoe models, this pape pesents a coespondence law between the esults of an equivalent model; e.g. Fig.3, and the eal building. Equations that allow the definition of single-stoe models with 3DOF and natual fequencies appoimating the fequencies of the coesponding coupled modes of the eal building ae pesented. The same equations ma be used to find the seismic esponse of eal

3 stuctues fom the esponses of thei equivalent linea elastic models. The assumptions used to fomulate this equivalent model estict its application to a class of buildings with cetain geometic and asmmet chaacteistics. Consideing that the esponse of asmmetic buildings subjected to design demands beond sevice conditions incusion in the non-linea ange of behaviou, this pape establishes the concepts and steps equied to fomulate a non-linea single-stoe 3DOF model simila to the elastic model defined analticall. This model is simple than those based on conventional pushove analses which associate the inelastic seismic esponse of a eal stuctue to that of a simplified single degee of feedom model. DEFINITION OF THE EQUIVALENT ELASTIC SINGLE-STOREY MODEL In 1971 Kan and Chopa [4] developed a fomulation to define a single-stoe 3DOF sstem with eigenvalues that appoimate the fist thee coupled fequencies of a eal building. The pesented a method to constuct an appoimation of the modal shapes of a tosionall coupled multi-stoe building fom a linea combination of the uncoupled modal shapes, obtaining an equation that allows the estimation of the maimum esponse fom a modal spectal analsis. Fo a building subjected to eathquake-like ecitations, the dnamic equilibium ma be witten as: [ ] [ ] [ ] { d } K K m && θ { d} && ug[ m]{} 1 [ m] {&& d } K K + θ { d} = && ug [ m]{} 1 (1) [ mθ ] {&& d } [ K ] [ ] { d } θ θ K θ K θθ θ [m]: Tanslational mass mati (diagonal). m : Rotational mass mati (diagonal). [m θ ]=[ ][ ] [ ]: Mati of the squaed gation adii (Diagonal). [K ]: Stiffness mati patition coesponding to the tanslational stiffness in X diection. [K ]: Stiffness mati patition coesponding to the tanslational stiffness in Y diection. [K θθ] Stiffness mati patition coesponding to the tosional stiffness. [K θ ]: Stiffness mati patition coesponding to the coupled Xθ stiffness. [K θ ]: Stiffness mati patition coesponding to the coupled Yθ stiffness. ü g and ü g : Gound acceleation in X and Y diections, espectivel. {1}: Unit vecto. {d }: Displacement vecto in X diection. {d }: Displacement vecto in Y diection. {d θ }: Rotation vecto. In geneal, Eq. 1 can be eaanged and modified accodingl to the tansfomation poposed in this wok: [ ] { } [ K ] [ K ] m && u t { u } && ug [ m]{} 1 [ m] { u } [ Kt ] [ Ktt ] K && θ + t { uθ} = [ m] { u } { } ug[ m]{} 1 K u && t K && ()

4 [ K ] [ ] [ K ][ ] tt 1 1 θθ = (3) [ K ] [ K ][ ] 1 t = (4) θ [ ] 1 K = K t θ (5) { u } { d } 1 u = { uθ} = [ R] { dθ} { u} { d} {} [ ] [ I ] 1 R = [ ], [ ] = [ I ] 1 (6) (7) [R]: Tansfomation mati. []: Radius of gation mati (diagonal). [I]: Identit mati. In the paticula case of the single-stoe 3DOF stuctue used to epesent a multi-stoe building, Eq. becomes: ek k m && u u && u g ek k ek m θ && uθ + u m θ = m u u u && ek && g k (8) m: Tanslational mass fo the 3DOF model. : Radius of gation of the 3DOF model. k : Tanslational stiffness in X diection. k : Tanslational stiffness in Y diection. k θ : Rotational stiffness. e : Eccenticit in X diection. e : Eccenticit in Y diection. u i : Displacement of the i=, and θ degee of feedom.

5 With this equation it is possible to detemine fo a eal building an equied esponse vaiable, howeve in this wok onl the time histo of displacements will be consideed as vaiable of stud, because, having detemined the histo of displacements of the equivalent model, the methods to obtain othe vaiables ae standad and can be consulted in the specialized liteatue, Chopa [5]. Equivalent model obtained fom the Petubation Theo The eigenvalue poblem of the single-stoe equivalent model can be appoimated using a second ode petubation of the multi-stoe eigenvalue poblem of the oiginal model, Wilkinson [6] and Kan and Chopa [7]. The application of a petubation to an eigenvalue poblem epessed in its canonical fom consists in appoimating the mati of coefficients as the sum of two matices and its eigenvalues and eigenvectos b powe seies, Wilkinson [6]. In this wa, the eigenvalue poblem elated with the oiginal model can be epessed as: [ K ] [ E] ω [ M] { } { } ( + ) Φ = (9) [ ] [ K ] [ ] [ ] Kt m : [ ] [ ] [ ] [ ] [ ] [ ] tt t t [ K ] [ K ] [ m] t K K, E : K K, M : m (1) ω: Fequenc. [K ]: Uncoupled stiffness mati (unpetubed sstem). [E]: Petubation mati (petubed sstem). [M]: Tanslational mass mati. {Φ}: Modal shape. K = & K + E [ ] [ ] [ ] The paametes of the equivalent model deived fom an application of the Petubation Theo coespond to a second ode appoimation in the powe seies and ae epessed in tems of the patitions, eigenvalues and eigenvectos of the uncoupled mati. Using the Petubation Theo the modal shapes of the oiginal model can be epessed as a linea combination of those coesponding to the uncoupled sstem, Wilkinson [6] and Kan and Chopa [7]: { Φ } = [ Ψ]{ α} & (11) [ Ψ ] { Ψ } {} {} = {} { Ψθ } {} {} {} { Ψ } (1)

6 { α} α = αθ α (13) T { α} { α } = 1 (14) [Ψ]: Modal shapes of the unpetubed sstem ([K ]). {Ψ }: Fist modal shape coesponding to the [K] patition. {Ψ }: Fist modal shape coesponding to the [K] patition. {Ψ θ }: Fist modal shape coesponding to the [Ktt] patition. {α}: Modal shape of the equivalent model. In Eq. (11) the {α} vecto ma be intepeted as the vecto of coupling coefficients (linea combination factos) needed to obtain the coupled modal shapes of the iegula building fom its uncoupled modal shapes. Eq. (11) states the coespondence elationship between the esponse paametes of the single stoe model and those of the eal building. Using the above appoach, the eigenvalue poblem associated to the oiginal model ma be epessed as: t [ ] ω M [ ] { α} { } t ( Ψ [ K] Ψ Ψ [ ] Ψ ) = (15) Developing T { }[ K ]{ θ } ω ω Ψ t Ψ α T T { }[ t]{ } t { } K θ Kt { θ} Ψ Ψ ω ω Ψ Ψ αθ = T { Ψ } { } K α t Ψθ ω ω (16) { Ψ T }[ ]{ } { T }[ ]{ } { T m Ψ Ψθ m Ψθ Ψ }[ m]{ Ψ } 1 = = = (17) ω, ω, and ω t : Fequencies of the fist modes of the uncoupled sstem ([K ], [K ] and [K tt ] patitions espectivel).

7 Establishing the eigenvalue poblem of the single-stoe 3DOF model: e k ω m k α e e k kt ω m k αθ = α e k k ω m (18) Compaing tem b tem Eqs. 16 and 18: Kθ k = ω m, kt = ωθ m, k = ω m, kt = (19) e e T { Ψ } K t { Ψ θ } = () ω m T { Ψ }[ K t]{ Ψ θ } = (1) ω m Solving the eigenvalue poblem of the equivalent stuctue, the fequencies obtained ae an appoimation to the fist thee natual fequencies of the oiginal model, and the coesponding modal shapes ae the coefficients which allow the modal shapes of the oiginal model to be epessed as a linea combination of the modal shapes of the uncoupled sstem. One featue of the model poposed b Kan and Chopa [7] is that it is paametic in natue. i.e., the mass and stiffness matices ae modified in such wa that the ae factoized b the adius of gation, not allowing the eplicit definition of geomet, stiffness and mass. Poposed tansfomation In this wok a linea tansfomation elating the mass and stiffness matices to the Kan and Chopa [7] model is poposed. Fo the paticula case of a building with equal mass at all levels and mass centes located along a vetical ais b assigning the mass and adius of gation of an level of the oiginal model to the equivalent model, it is possible to define the mass and tansfomation matices as: [ m ] m = m m ()

8 [ ] [ I ] 1 R = [ ], [ ] = [ I ] 1 (3) [m ]: Tanslational masses mati fo the 3DOF model (containing the tanslational mass of an levels in oiginal stuctue). m : Tanslational mass of the equivalent model. []: Radius of gation mati of an floo of the oiginal stuctue (diagonal). [R]: Tansfomation mati. The eigenvalue poblem fo the equivalent sstem is: e e W m = k k k ω m α t θ = ([ ] ω o [ ]){ α} e k k m α m α e k k (4) [W o ]: Equivalent stiffness mati. The canonical fom of the eigenvalue poblem associated to the equivalent model, illustated in Fig 4, is: k e k m m 1 α e k k k e t Π ω α = ω 1 αθ = m m m e k k m m ([ ] [ I ]){ } 1 α (5) [Π] Coefficient mati of the canonical fom of the eigenvalue poblem. Multipling Eq. 5 b mati [m ], Eq., leads to: 1 ([ Wo ] ω [ m ])[ R] [ R]{ α} = (6)

9 Pe-multipling this equation b [R] -1 : 1 1 [ R] ([ Wo ] ω [ m ])[ R] [ R]{ α} epanding: = (7) ([ ] 1 [ ][ ] 1 [ ] 1 R W [ ][ ] 1 o R ω R m R )[ R]{ α} = (8) the following mati equation is obtained: ([ KC] ω m p ){ β} = (9) [ K ] [ R] [ m ] [ R] C 1 1 = [ ] Π (3) [ ] [ ][ ] 1 1 mp = R m R (31) { β} [ R]{ α} = (3) [m p ]:Acceleation coefficients mati (Diagonal). [K C ]:Displacement coefficients mati (Diagonal). As a esult of the pevious fomulation, the dnamic equilibium equation fo a single stoe 3DOF stuctue in its conventional fom is: { } + [ ]{ } = { } m d && K d m A (33) p C p V V M t u u u θ Figue 4. Equivalent 3DOF model.

10 EXAMPLE Building desciption To illustate the application of the above pocedue, a einfoced concete mass asmmetic building is used. The building is egula in plan and in elevation with a ectangula plan, thee bas in the tansvese diection and fou in the longitudinal diection (7 and 8 m long espectivel) with stoe heights of 3.3 m, beams sections of.4.8 m and column sections of.8.8 m. The consideed elastic modulus is 136. kg/cm and the masses in eve floo located along a vetical ais, ae ton*sec /m, Figs. 5 and 6. The building was designed b Chípol [8] in accodance with the cuent Meico Cit code, DDF [9]. 7 m C.M. (1.8,1.6) 7 m 7 m 8m 8m 8m 8m Figue 5. Plan view of the building. 7 m 7 m 7 m 8 m 8 m 8 m 8 m Figue 6. Lateal views of the building. Equivalent model In this wok, the pogam to calculate the equied paametes fo the paticula case of equal floo masses cented along a vetical ais was developed in Fotan language. With this pogam mati [Π] defined b Eq. is evaluated as: Π = [ ] (34) The vibation peiods of the equivalent model and the coesponding peiods obtained fom the oiginal model ae pesented in Table 1:

11 The modal shapes associated to the equivalent model ae: [ ] α o = (35) Table 1. Compaison of peiods of the equivalent model vs. the peiods fom the oiginal model. T 3 T T 1 Oiginal Mode l Equivale nt M odel Eo (%) Modal shapes fom the eigenvalue poblem of the oiginal model (complete stiffness and mass matices), and the modal shapes deived fom Petubation Theo and the equivalent model ae shown in Table. Relative eo is calculated between modal shapes fom the oiginal model and the built modal shapes. Futhemoe, the maimum eo between the modal shapes is also pesented. Table. Modal shapes obtained fom the Petubation Theo and fom the oiginal model. Φ 3 Φ Φ 1 Φ R3 Φ R Φ R1 e 3 (%) e (%) e 1 (%) MAXIMUM ERROR (%)

12 Φ 1, Φ and Φ 3 : Modal shapes of the fist, second and thid modes, espectivel, fom the eigenvalue poblem associated to the oiginal poblem. Φ R1, Φ R and Φ R3 : Modal shapes built fom the equivalent model and the modal shapes of the unpetubed sstem. ( Φ Φ i Ri e ) i = 1% : Eo between Φ i and Φ Ri, (i=1,,3). Φ i Consideing the floo mass of the oiginal building: [ m ] = (36) Fo the tansfomation mati: [ ] R P =.. 1. (37) Fom Eqs. 5 and 6 it is possible to obtain the matices of displacement coefficients and acceleation coefficients, espectivel: [ ] K C = (38) m p = (39) DEFINITION OF THE EQUIVALENT INELASTIC SINGLE-STOREY MODEL To obtain the inelastic esponse of a multi-stoe iegula building with less effot than that involved in detemining its inelastic esponse fom a conventional non-linea dnamic analsis, the capacit spectum concept is suitable. Howeve, thee ae some concepts that must be consideed. One is the fact that in a non-linea stuctual model it is onl possible to obtain the lateal stiffness (uncoupled stiffness) in both othogonal diections and the otational stiffness fom pushove analses with the distibutions of equivalent static lateal foces and floo moments applied at all levels of the stuctue, Fig. 7, in ode to obtain thee capacit cuves epessed as oof displacement (otation) vs. base shea (moment), Fig 8. Fom these capacit cuves it is possible to deive the lateal and tosional stiffnesses needed in Eq. 8. The seismic esponse of this non-linea single-stoe model ma be obtained b an of the alead available

13 step b step integation pocedues. It is impotant to point out that the appoimation of the seismic pefomance of eal asmmetic buildings calculated with this method is highl dependent the distibution of the applied lateal loads and floo moments and must be pefomed caefull in ode to obtain ational esults, Aala and Tavea [1]. CM Z Y X Figue 7. Diection of foces fo the thee pushove analses of an asmmetic building. VBX (ton) DX (m)..4.6 VBY (ton) DY (m)..4.6 MB (ton-m) Rt (ad).1..3 a) Diection X. b) Diection Y. c) Diection θ Figue 8. Illustation of base shea vs. oof displacement and base moment vs. oof otation cuves. The pocedue poposed is valid onl fo a famil of multi-stoe buildings with the same estictions as s fo the elastic cases. The fact that the matices obtained do not allow the identification of its esistant planes of the pototpe o its eccenticities makes it impossible to model the stength distibution and its chaacteistics in ode to elate the inelastic esponse of the simplified model with the eal model. These topics equie futhe eseach to modif the pocedue pesented in this pape to fomulate a single stoe inelastic model and fom its seismic esponse obtain the inelastic esponse of a eal building.

14 CONCLUSIONS The fomulation of simplified single-stoe 3DOF models equivalent to multi-stoe asmmetic building models pesented in this pape leads to dnamic sstems with equal uncoupled and appoimate tosionall coupled fequencies guaanteeing that the seismic esponses of the simplified models also epesent those of the pototpes. This chaacteistic opens up also the possibilit of tansfoming the seismic pefomance esults fom these models to eal asmmetic buildings. This esult is of paamount impotant in the intepetation of esults deived simplified single-stoe models The model fomulated in this wok is deived fom an invese tansfomation oiginall poposed b Kan and Chopa [4], with dnamic equilibium equations with stiffness and mass matices coesponding to those of an asmmetic multi-stoe building subjected to bi-diectional seismic demands. The matices obtained epesent the global values of mass and lateal and tosional stiffness of the asmmetic multistoe building ignoing the in-plan distibution of esisting elements. This chaacteistic makes the esults of numeous studies on single-stoe models, with emphasis on the behaviou of thei diffeent esisting planes, patl useful as thee is not possible to etapolate esults on the pefomance of these esisting planes to the coesponding planes in the actual asmmetic stuctue. Regading the poposed model, it is ecommended to continue with the eseach to develop a pocedue to define simplified models fom which the pefomance of eal multi-stoe asmmetic buildings could be obtained. These models, if single-stoe 3DOF, could be used to validate and eassess the esults obtained since the 198 s, to investigate the seismic behaviou of mass and stiffness asmmetic buildings. These models should be capable of coectl epesenting the tosional coupling with consistent equations to allow the tansfomation of the stuctual popeties of a eal building to those of a coesponding simplified model and the seismic pefomance of this simplified model to the pefomance of the eal stuctue. ACKNOWLEDGEMENTS The sponsoship of the poject Development of Seismic Design Citeia fo Tosion and the patial scholaship ganted to the fist autho to complete his gaduate studies b the Geneal Diectoate fo Affais of the Academic Pesonnel of UNAM, DGAPA ae acknowledged. The eview and citical comments of Mauo Niño ae appeciated. REFERENCES 1. Bustamante JI, Rosenblueth E. Building code povisions on tosional oscillations. Poceedings of the nd Wold Confeence on Eathquake Engineeing, Toko and Koto Japan, 196; Aala AG, Gacía O. Seismic behaviou of buildings design in accodance with a nom fo tosional design. (in Spanish), Poc. of the IX National Congess on Eathquake Engineeing and IX National Congess on Stuctual Engineeing, Manzanillo, Col., Meico, 1991; Vol. II. 3. Yoon S, Staffod-Smith B. (1995) "Estimating peiod atio fo pedicting tosional coupling" J. Engng. Stuct, 1995; 17(1): Kan CL, Chopa AK. Coupled lateal tosional esponse of buildings to gound shaking. Repot no , UCB/EERC, Univesit of Califonia; Bekele, CA, Chopa AK. Dnamics of stuctues. Theo and applications to eathquake engineeing. Pentice Hall, Englewood Cliffs, N.J., Wilkinson JH. Petubation theo, in The algebaic eigenvalue poblem. Claendon Pess, Ofod, 1965; 6 71.

15 7. Kan CL, Chopa AK. Elastic eathquake analsis of tosionall coupled multisto buildings. Eathquake Engineeing and Stuctual Dnamics, 1977; Vol. 5: Chípol A. Stud of the seismic esponse of 3D models of tosionall coupled buildings. (in Spanish), Maste of Engineeing Thesis. Gaduate Pogam in Engineeing. UNAM. Meico, DDF. "Constuction Code fo the Fedeal Distict and Complementa Technical Noms." (RCDF and NTC), (In Spanish), Official Gazette of the Fedeation, Meico, Aala AG, Tavea EA. A New Appoach fo the Evaluation of the Seismic Pefomance of Asmmetic Buildings. Poceedings of the 7 th National Congess on Eathquake Engineeing, Pape no. 137, EERI, CD-R,.