AN ENERGY BASED METHOD FOR SEISMIC EVALUATION OF STRUCTURES

Transcription

1 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia AN NRGY BASD MTHOD FOR SISMIC VALUATION OF STRUCTURS Sutat Leelataviwat, Wiai Saewo, Subhash C. Goel Assistat Professor, Dept. of Civil gieerig, Kig Mogkut s Uiversity of Techology Thoburi, Thailad Structural gieer, Bouygues Thai Ltd., Bagkok, Thailad meritus Professor, Dept. of Civil ad virometal gieerig, Uiversity of Michiga, A Arbor, USA mail: sutat.lee@kmutt.ac.th ABSTRACT : This paper presets a seismic evaluatio procedure based o a eergy cocept that has bee recetly developed ad successfully used for desig purposes called Performace-Based Plastic Desig (PBPD). The uderlyig theory ad the framework for carryig out the aalysis are first preseted. The skeleto forcedisplacemet (capacity) curve of the structure is coverted ito eergy capacity plot which is superimposed over the correspodig eergy demad plot for the give hazard level to determie the expected peak respose. The method is applied to a umber of example sigle-degree-of-freedom (SDOF) ad multi-degree-of-freedom (MDOF) structural systems with excellet results. The results are compared with those obtaied from oliear dyamic aalyses as well as those from methods proposed by other ivestigators icludig the Modal Pushover Aalysis Method ad the FMA Displacemet Coefficiet Method. For SDOF systems, the results idicate that the proposed method provides respose values that are idetical to those obtaied from a well-established procedure usig ielastic desig spectrum. For MDOF systems also, the proposed method provides respose values that are reliable whe compared to the results from o-liear dyamic aalysis ad other wellestablished oliear static procedures. KYWORDS: ergy Method, ergy Balace Cocept, Noliear Static Aalysis, Seismic valuatio, Performace-Based Plastic Desig INTRODUCTION Noliear Static Procedure (NSP) has bee widely accepted as a useful tool for performace-based seismic desig ad evaluatio of structures (FMA 6). I the short time that has elapsed sice its itroductio to the egieerig commuity, the method has bee a subject of extesive research ad several ew aalysis approaches have bee proposed. I most cases, the behavior of the structure is characterized by the capacity curve which is represeted by a plot of the base shear versus the roof displacemet. The capacity curve is used to establish a equivalet sigle-degree-of-freedom (SDOF) system. The target displacemet ca the be predicted usig oe of the methods such as oliear dyamic aalysis, capacity spectrum approach, the modificatio coefficiet approach, or the direct use of ielastic costat ductility spectra. Oce determied, the target displacemet ca the be projected back to the roof displacemet from which the story drifts ad compoet forces ca be obtaied for compariso with available capacities. The absorbed eergy (or work doe by the exteral forces) has also bee suggested as a alterative idex to establish the capacity curve (Herádez-Motes et al. 4; Mezzi et al. 6). The objective of this study is to preset a seismic evaluatio procedure based o a eergy cocept that has bee recetly developed ad successfully used for desig purposes called Performace-Based Plastic Desig (PBPD) method. The uderlyig theory ad the framework for carryig out the aalysis are first preseted. The aalysis procedure is the applied to SDOF ad multi-degree-of-freedom (MDOF) structures to estimate the displacemet demads. The results obtaied from the proposed procedure are the evaluated ad compared with the results from o-liear dyamic procedure (NDP) as well as those from other well established NSPs.

2 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia. NRGY BALANC CONCPT IN PRFORMANC-BASD PLASTIC DSIGN Recetly, a iovative performace-based desig procedure which directly accouts for structural ielastic behavior has bee developed (Leelataviwat et al. 999; Lee ad Goel ; Dasgupta et al. 4; Chao ad Goel 6a; Chao ad Goel 6b). The desig base shear for a selected hazard level ad yield mechaism is calculated by equatig the work eeded to push the structure mootoically up to the target drift to that required by a equivalet elastic-plastic sigle degree of freedom system to achieve the same state. The method has bee called Performace-Based Plastic Desig (PBPD). At the heart of PBPD methodology is the eergy balace cocept. This cocept has bee used by early researchers, i a desig cotext, to obtai the amout of plastic eergy to be absorbed by the structure (Houser 956) as well as to derive costat ductility ielastic desig spectra (Newmark ad Hall 98). The eergy balace cocept is based o the assumptio that the eergy computed from the mootoic loaddeformatio respose of the ielastic system ad the oe computed from the correspodig elastic system are the same. For a SDOF system with elastic-plastic load-deformatio characteristic, the eergy balace equatio is give by: = MSv = Vy Dy + Vy( Dm Dy ) () where M is the mass of the system, S v is the pseudo velocity, V y is the yield stregth, D y is the yield displacemet, D m is the maximum ielastic displacemet. As recogized by Newmark, the above eergy balace relatioship is valid oly for systems with period i the acceleratio sesitive regios of the spectrum. I the PBPD the eergy balace equatio is modified so that it ca be used for all periods. Lee ad Goel () itroduced the eergy factor, γ, ad modified the eergy balace equatio as follows: γ = γ MSv = Vy Dy + Vy( Dm Dy ) () i which γ is the eergy factor ad is defied as the ratio of the eergy absorbed by the ielastic system to that of the equivalet elastic system. The eergy factor ca be computed by rearragig quatio : VyDy + Vy(Dm Dy ) VyDy + Vy(Dm Dy ) μ γ = = = () MS Ry v VyDyRy where μ is the displacemet ductility factor (D m /D y ), ad R y is the yield stregth reductio factor. The yield stregth reductio factor, R y, is defied as the ratio of the stregth required for the system to remai elastic to the yield stregth of the system. The modified eergy balace cocept is illustrated i Figure a. I order to compute the eergy factor i quatio, a relatioship betwee Ry-μ-T is eeded. Usig Ry-μ-T equatios such as the oe developed by Newmark ad Hall (98), the factor γ ca be determied for a give ductility level as show i Figure b. For seismic desig purposes, a target ductility level ca be selected ad, quatio ca the be used to obtai the required yield stregth of the system The eergy balace equatio ca be exteded to MDOF systems by usig the cocept of equivalet simple oscillator. The equivalet simple oscillator of a MDOF system i the th mode is defied as a SDOF system with the same period ad carries a mass equal to: T ( φ ) m M = φ m φ T (4)

3 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia where M is geerally called the effective modal mass for the th mode of the MDOF system, m is the mass matrix, φ is the th mode shape, ad is a uit vector. The sum of all effective modal masses for the MDOF system is equal to the total mass of the MDOF system. Further, for this SDOF system carryig a mass equal to M, the SDOF becomes equivalet to the th mode of the MDOF system i terms of base shear. Ielastic System Correspodig lastic System. Force Ve Vy MS v γ MS v γ μ = μ = 5. μ = Dy De Dm Displacemet (a) (b) Figure. (a) Modified ergy Balace Cocept (Lee ad Goel, ); (b) ergy Factor as a Fuctio of Period ad Ductility Ratio based o Newmark ad Hall s R y -μ-t quatios. By assumig that the couplig betwee the modes which arises due to yieldig of the system ca be eglected (Chopra ad Goel ) ad that the mode shapes remai costat eve after yieldig, it ca be show (Leelataviwat et al. 7) that the equivalet simple oscillator ca be used to calculate the eergy demad of the th mode of the MDOF system resultig i the eergy balace equatio: γ M Sv = Vbyuy + Vby ( um uy ) (5) where V by is the yield base shear of the MDOF i the th mode ad u y ad u m are the yield ad maximum displacemets at the ceter of the applied forces, respectively. The left had side of quatio 5 is the eergy demad ad the right had side of the equatio is the work doe by the applied forces or the absorbed eergy. The eergy balace cocept for MDOF systems ad the cocept of equivalet simple oscillator are illustrated i Figure T (sec) V b u V b V by γ M S v V b D M V b u y u m (a) (b) Figure. (a) ergy balace cocept for MDOF system i the th mode; (b) quivalet simple oscillator. SISMIC VALUATION BASD ON NRGY BALANC CONCPT The above eergy balace cocept ca be applied i seismic evaluatio of structures by evaluatig the terms o both sides of quatio 5. The eergy capacity plot for the system is first obtaied. The covetioal push-over curve (geerally represeted by the base shear versus roof displacemet plot) ca be geerated ad the coverted ito the eergy capacity curve. The absorbed eergy ca be determied from the total absorbed eergy due to idividual lateral forces or from the sum of the areas uder the lateral load-deflectio curves at all u V b

4 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia floor levels. This eergy capacity curve represets the right-had side of quatio 5. For the demad side, a eergy demad curve ca be geerated usig the term o the left-had side of quatio 5. As ca be see, the eergy demad is a fuctio of the eergy factor, γ, which i tur is a fuctio of ductility factor, μ, ad the yield stregth reductio factor, R y. I dealig with MDOF systems, the cocept of displacemet ductility may be somewhat difficult to defie. Hece the eergy factor ad the yield stregth reductio factor ca be more appropriately writte i terms of eergy ductility, μ, which is defied as the ratio of the maximum absorbed eergy to the absorbed eergy at yield ad is related to the displacemet ductility by: VyDm VyDy μ = = μ VyDy The eergy factor i terms of the eergy ductility is give by substitutig quatio 6 ito quatio : μ γ = (7) R y where γ is the eergy factor i terms of eergy ductility. The yield stregth reductio factor, R y, ca also be cast i terms of eergy ductility. The eergy ductility ca be approximately take as the ratio of the total absorbed eergy to the absorbed eergy whe system yieldig occurs. This ca be determied based o the ratio of the total absorbed eergy to the absorbed eergy at the yield poit obtaied from a bi-liear approximatio of the pushover curve of the system. The itersectig poit of the eergy demad ad capacity curves determies the expected level of the respose of the system. The respose parameters of iterest, correspodig to the displacemet level obtaied from the itersectio of the eergy demad ad capacity curves ca the be extracted. The proposed eergy balace cocept ca be applied i the first or multiple mode aalysis. However, combiig idepedet modal resposes is a complex problem. Future research is eeded to explore the applicability of the eergy-based multiple mode aalysis. The cocept of eergy-based demad-capacity diagram is illustrated i Figure. It should be oted that the roof displacemet used i the push-over curve serves oly as a tool for visualizatio purposes. The estimate of the respose does ot deped o the choice of the locatio of the referece displacemet. It is also oteworthy that there is o eed to covert the structure ito a equivalet SDOF system eve though the cocept of equivalet simple oscillator plays a importat role i the uderlyig process. The respose ca be computed directly from the MDOF model. (6) f = T α ( φ m ) u r V Push-Over Curve γ = M S v V (a) u r (b) u r (c) u r (d) u max u r Figure. Proposed ergy-based valuatio Method: (a) Push-Over Curve, (b) ergy-displacemet Capacity Diagram, (c) ergy Demad Diagram, ad (d) Determiatio of Displacemet Demad. 4. SDOF SYSTM XAMPLS I order to validate the proposed eergy-based evaluatio method, the procedure was implemeted to determie maximum ielastic displacemets of six elastic-plastic SDOF systems. These systems were first used i aother study by Chopra ad Goel (999). I their study, Chopra ad Goel used a capacity-demad diagram method that

5 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia was based o Newmark ad Hall s ielastic desig spectrum to obtai the maximum displacemet of those systems. The procedure used by Chopra ad Goel was a improved aalysis procedure over the covetioal Capacity Spectrum Method to determie peak deformatio of ielastic systems. The properties of the elasticplastic systems are summarized i Table. The peak displacemets as determied by Chopra ad Goel alog with the peak displacemets obtaied by the proposed method discussed earlier are also show i Table. For brevity, the resultig eergy demad-capacity diagram of oly oe of the systems (System 6) is show here i Figure 4. Table. Properties of Six lastoplastic Systems Used to valuate the Proposed Method. System T V e /m V y /m D y R y Peak Respose (s) (g) (g) (cm) (V e /Vy) Chopra & Goel This Study μ D m (cm) μ D m (cm) / y Absorbed ergy Demad D (cm) Figure. 4. ergy Capacity-Demad Diagram of xample System 6. As ca be see, the results obtaied by the two methods are essetially idetical. This is because the same ielastic respose spectrum was used i both methods. Hece, for SDOF systems, it ca be cocluded that the proposed method gives idetical results to those obtaied from a well-established procedure based o ielastic desig spectrum. 5. MDOF SYSTM XAMPLS The procedure preseted herei was applied to determie the respose of example three-story ad ie-story steel frames show i Figure 5. These structures have bee used i various studies ad detailed iformatio about these structures ca be foud elsewhere (Herádez-Motes et al. 4, Goel ad Chopra ). For this study, the buildigs were modeled usig the ceterlie dimesios without cosiderig the pael zoe deformatios. All members were assumed to have elastic-plastic behavior with % strai hardeig. Dampig of 5% critical was assiged for the first ad secod modes. Gravity load ad P-delta effects were ot icluded i the aalysis for the three-story frame. For the ie-story frame, the P-delta effect was cosidered usig dummy colum represetig the gravity colums i the buildig. The aalyses were carried out usig SAPNL (Computer ad Structures ). The example frames were aalyzed to determie the story displacemet ad drift demads for the 94 NS l Cetro groud motio scaled i such a way that the roof displacemets were equal to % ad % i oliear dyamic aalysis. This results i groud motio multipliers (GMMs) of.88 (% drift) ad.9 (% drift) for

6 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia the three-story frame ad GMMs of.5 (% drift) ad.8 (% drift) for the ie-story frame. W4x68 W4x W4x57 W4x8 W4x7 W4x7 W4x57 W4x8 W4x7 W4x455 W4x5 W7x84 Wx99 W6x5 W6x5 W6x5 W6x5 W6x6 W6x6 W6x6 ft ft (Basemet) W4x57 W4x57 W4x57 W4x W4x W4x W4x68 Wx6 Wx8 ft 9 ft 5 ft Figure 5. Nie-story ad three-story example frames For each frame, the capacity curve was first obtaied by calculatig the work doe by the applied forces i the pushover aalysis. Lateral forces were applied i proportio to the first mode shape. The absorbed eergy was calculated by umerically itegratig the lateral load-deflectio curves at the floor levels. The demad curve was obtaied based o the effective modal mass of the first mode ad the pseudo-velocity value determied from the 5% damped respose spectrum of the l Cetro groud motio with correspodig groud motio multiplier. The eergy factor, γ, was determied based o Newmark ad Hall s R y -μ-t relatioship. The eergy ductility used to evaluate the eergy factor, γ, was based o the ratio of the total absorbed eergy to the absorbed eergy at system yield poit determied from bi-liear approximatio of the pushover curve. The eergy capacity ad demad curves ormalized by the absorbed eergy at yield of the two frames are show i Figure 6. Usig the itersectio poits of the demad ad capacity curves as the maximum displacemet values, the story displacemet ad drift demads could be determied Demad (GMM =.9) Demad (GMM=.88) Capacity (-Story) Demad (GMM =.8) Demad (GMM =.5) Capacity (9-Story) / y 4.. / y Roof Drift (%) -Story Frame. 4 5 Roof Drift (%) 9-Story Frame Figure 6. ergy demad ad capacity curves of the three- ad ie-story example frames The accuracy of the method was assessed by comparig the results with those from the NDP ad the results from other well-established NSPs icludig the modal pushover aalysis (MPA) by Goel ad Chopra () ad the displacemet coefficiet method accordig to FMA (). For the MPA method, the first three modes were employed ad, for each mode, modal pushover, i cojuctio with dyamic aalysis, was performed to determie the modal displacemet demads. The displacemets obtaied from modal respose of the first three modes were the combied. For the FMA method, pushover aalysis was performed for each

7 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia frame usig the target displacemet specified by FMA. Figure 7 shows floor displacemet ad story drift demad estimates from differet aalysis procedures for the three- ad ie-story example frames. It ca be see that, for the % roof drift level (GMMs of.88 ad.5), where the respose remaied essetially i the elastic rage, all of the NSPs provided very close estimates of the respose for both frames. For the % roof drift level (GMMs of.9 ad.8) where ielastic respose became sigificat, the floor displacemet ad story drift estimates were less accurate but were withi acceptable limit cosiderig the approximate ature of NSPs. As expected, the story drift estimates were less accurate tha the floor displacemet estimates for most the cases especially for the ie-story frame. For the case of the iestory frame, the MPA provided closer drift results i the upper stories where the ifluece of the higher modes may be sigificat. Overall, the proposed eergy method provides results that are comparable to those from well-established procedures. It should be oted that i this example, the proposed eergy method was carried out usig Newmark ad Hall s Ry-μ-T Relatioship which was developed to represet respose for a wide rage of groud motios. The accuracy of the proposed eergy method could be improved by usig a more specific Ry-μ-T relatioship developed for the groud motio uder cosideratio. NDP This Study MPA FMA Floor No. -Story GMM =.88 4 Floor No Story 9-Story 9-Story GMM =.9 GMM =.5 GMM = Floor Displacemet / Total Height (%) Story Story -Story -Story 9-Story 9-Story GMM =.88 GMM =.9 GMM =.5 GMM = Peak Story Drift (%) Figure 7. Floor displacemets ad story drifts from differet aalysis procedures 7. SUMMARY AND CONCLUSIONS I this study, a applicatio of the eergy balace cocept i seismic evaluatio of structures is preseted. The mai fidigs of this study are:. ergy balace cocept ca be exteded to MDOF systems ad ca be used i seismic evaluatio of structures. The seismic demad curve is first established for varyig displacemet levels. The capacity curve of the structure is the preseted i terms of the eergy absorbed by the structure at the correspodig displacemet levels. The target displacemet is determied based o the itersectio of eergy demad ad

8 The 4 th World Coferece o arthquake gieerig October -7, 8, Beijig, Chia capacity curves. The mai feature of the proposed method is that the demad ad capacity curves are produced i terms of eergy. Hece, it avoids the arbitrary use of the base shear versus roof displacemet represetatio of the capacity curve. The respose ca also be computed directly from the MDOF, eve though the cocept of equivalet simple oscillator plays a importat role i the uderlyig process.. The accuracy of the proposed method was ivestigated for SDOF ad MDOF systems. The results are compared with those from differet aalysis procedures. For these example systems ad groud motios, the proposed method provided results that are accurate ad comparable with established procedures. RFRNCS. Chao, S.-H., ad Goel, S. C. (6a), "Performace-Based Desig of ccetrically Braced Frames Usig Target Drift ad Yield Mechaism," gieerig Joural, America Istitute of Steel Costructio, Third Quarter, 6, PP Chao, S.-H., ad Goel, S. C. (6b). Performace-Based Plastic Desig of Seismic Resistat Special Truss Momet Frames, Report No. UMC 6-, Dept. of Civil ad vr. g., Uiversity of Michiga, A Arbor, MI.. Chopra, A.K., ad Goel, R.K. (999). Capacity-Demad-Diagram Methods for stimatig Deformatio of Ielastic Structures: SDF Systems, Report No. PR-999/, Pacific arthquake Research Ceter, Uiversity of Califoria, Berkley. 4. Computer ad Structures (). SAP Noliear Versio 8, Berkeley, USA. 5. Dasgupta, P., Goel, S.C., ad Parra-Motesios, G. (4). Performace-Based Seismic Desig ad Behavior of a Composite Bucklig Restraied Braced Frame (BRBF). Paper No. 497, Proc., World Coferece o arthquake gieerig, Vacouver, BC, August FMA (). Prestadard ad Commetary for Seismic Rehabilitatio of Buildigs, FMA 56, Federal mergecy Maagemet Agecy, Washigto D.C. 7. FMA (6). Improvemet of Noliear Static Seismic Aalysis Procedures, FMA 44, Federal mergecy Maagemet Agecy, Washigto D.C. 8. Goel, R.K., ad Chopra, A.K. (). A Modal Pushover Aalysis Procedure to stimate Seismic Demads for Buildigs: Theory ad Implemetatio, Report No. PR-/, Pacific arthquake Research Ceter, Uiversity of Califoria, Berkley. 9. Herádez-Motes,., Kwo, O-S, ad Aschheim, M.A. (4). A ergy-based Formulatio for Firstad Multiple-Mode Noliear Static (Pushover) Aalyses. Joural of arthquake gieerig, 8(), Houser, G.W. (956). Limit Desig of Structures to Resist arthquakes. Proc.,World Coferece o arthquake gieerig, 5,-.. Lee, Soo-Sik, ad Goel, S.C. (). Performace-Based Desig of Steel Momet Frames Usig Target Drift ad Yield Mechaism, Research Report No. UMC -7, Dept. of Civil ad vr. g., Uiversity of Michiga, A Arbor, MI.. Leelataviwat, S., Goel, S.C., ad Stojadiović, B. (999), Toward Performace-Based Seismic Desig of Structures. arthquake Spectra, 5(), Leelataviwat, S., Saewo, W., ad Goel, S.C (7) "A ergy Based Method for Seismic valuatio of Structures," Proceedigs of the Structural gieers Associatio of Califoria Covetio SAOC 7, Lake Tahoe, Califoria, Sep Mezzi, M., Comodii, F., Lucarelli, M., Parducci, A., ad Tomassoli,. (6) Pseudo-ergy Respose Spectra for the valuatio of the Seismic Respose from Pushover Aalysis. Proc., First uropea Coferece o arthquake gieerig ad Seismology, Switzerlad. 5. Newmark, N.M., ad Hall, W.J. (98). arthquake Spectra ad Desig, RI, Berkley, CA., USA.