INFLUENCE OF HYSTERETIC BEHAVIOR ON THE NONLINEAR RESPONSE OF FRAME STRUCTURES

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1 th World Conference on Erthquke Engineering Vncouver, B.C., Cnd August -, Pper No. 9 INFLUENCE OF HYSTERETIC BEHAVIOR ON THE NONLINEAR RESPONSE OF FRAME STRUCTURES Ricrdo A. MEDINA nd Helmut KRAWINKLER SUMMARY This pper focuses on understnding nd ssessing the effect of hysteretic behvior (i.e., biliner, pekoriented nd pinching) in the evlution of pek deformtion demnds nd their distribution over the height for regulr frme structures over wide rnge of stories (from to ) nd fundmentl periods (from. s. to. s.). The ground motions used re those with frequency content chrcteristic of ordinry ground motions (no ner-fult or soft soil effects). The hysteretic models utilized in this study do not exhibit monotonic or cyclic deteriortion; thus, the discussion is most relevnt for performnce levels relted to dmge nd loss of functionlity. Results suggest tht the degree of stiffness degrdtion is importnt for the seismic performnce evlution of regulr frmes becuse systems with lrge degree of stiffness degrdtion tend to exhibit lrger pek drift demnds nd less uniform distribution of pek drifts over the height. The type of hysteretic behvior lso hs significnt influence on the dynmic response of long, flexible frmes tht re prone to globl dynmic instbility due to P-Delt effects. This study lso demonstrtes the need to develop relible procedures to estimte the properties of equivlent or reference SDOF systems when they re used to evlute the response of complex MDOF structures with vrious hysteretic responses t the component level. INTRODUCTION Performnce evlution of moment resisting frme structures subject to severe ground shking requires nlyticl models ble to resonbly represent the cyclic nonliner behvior of structurl components. Numerous studies hve delt with the seismic demnd evlution of systems with vrious types of hysteretic behvior, e.g., systems with rther stble hysteresis loops nd systems tht include stiffness degrdtion typicl of reinforced-concrete nd timber components. However, most efforts hve focused on stiffness degrding SDOF systems, e.g., Rhmn nd Krwinkler [], Oh et. l [], Song nd Pincheir [], Chung nd Loh [], Mirnd nd Grci [5], Frrow nd Kurm [], nd there is no consensus regrding the effect of the type of hysteretic response t the component level on the behvior of MDOF systems with nd without stiffness degrdtion. Assistnt Professor, Dept. of Civil nd Environmentl Engineering, University of Mrylnd, College Prk. Emil: rmedin@umd.edu. Professor, Dept. of Civil nd Environmentl Engineering, Stnford University, Stnford, CA. Emil: krwinkler@stnford.edu.

2 The objective of this pper is to provide comprehensive informtion on the influence of hysteretic behvior on the nonliner response of regulr frme structures over wide rnge of stories (from to ) nd fundmentl periods (from. sec. to. sec.). The response prmeters used to quntify behvior re the mximum roof drift nd the mximum story drift over the height. The ground motions utilized re those with frequency content chrcteristics of ordinry ground motions. The hysteretic models do not exhibit monotonic or cyclic deteriortion; thus, results re most relevnt for performnce levels relted to dmge nd loss of functionlity. A comprison between the nonliner response of MDOF nd SDOF systems with vrious hysteretic models is presented s well s n evlution of the effect of the hysteretic model in the probbilistic seismic performnce ssessment of regulr frmes. FRAME MODELS USED IN THIS STUDY Generl Description This study is bsed on nonliner time history nlyses using twelve two-dimensionl, single-by generic frme models. The generic frme models do not exhibit mss, strength or stiffness irregulrities. Results presented by Medin nd Krwinkler [7] demonstrte tht generic frme models, such s the ones used in this study, re dequte to represent the globl dynmic response of more complex regulr multi-by frmes. Frme models with number of stories, N, equl to,, 9,, 5, nd, nd fundmentl period, T, of.n nd.n re utilized. These T vlues re deemed to represent lower nd upper bound estimtes of the fundmentl period of regulr moment-resisting frmes with totl heights consistent with the ones used in this reserch. Frmes with period of.n re denoted s stiff frmes, wheres frmes with period of.n re denoted s flexible frmes. Properties of Generic Frme Models The min chrcteristics of the generic frme models re s follows:. The sme mss is used t ll floor levels. Reltive stiffnesses re tuned so tht the first mode is stright line (K stiffness pttern). Bem-hinge (BH) models re utilized, i.e., plstifiction only occurs t the end of the bems nd the bottom of the first story columns [Fig. ]. Frmes re designed so tht simultneous yielding t ll plstic hinge loctions is ttined under prbolic (NEHRP, k = ) lod pttern (S strength pttern) 5. The bse sher strength is defined by the prmeter γ = V y /W (V y = bse sher yield strength, W = seismiclly effective weight). Concentrted plsticity is modeled by utilizing nonliner rottionl springs 7. Grvity lod moments nd the effects of xil column forces on bending strength re not considered. Globl (structure) P-Delt is included (member P-delt is ignored). The grvity lod cusing P- Delt effects is tken s the ded lod plus live lod equl to % of the ded lod. The reference vlue for the P-Delt effect is the elstic first story stbility coefficient, θ = P δ s /V h, where P is the first story P-Delt grvity lod, δ s nd V re the first story drift nd sher force, respectively, nd h is the first story height 9. For the nonliner time history nlyses, 5% Ryleigh dmping is ssigned to the first mode nd the mode t which the cumultive mss prticiption exceeds 95%. HYSTERETIC BEHAVIOR AT PLASTIC HINGE LOCATIONS Nonliner behvior t the component level is modeled with concentrted plsticity pproch in which non-deteriorting rottionl springs re used to represent the member cyclic response. In this study, three different hysteretic models re used: pek-oriented, biliner nd pinching (Fig. ). The pinching model corresponds to severely pinched cse in which the coefficients κ f nd κ d in Fig. (d) re equl to.5.

3 The effect of these prmeters on the pinching response of the system is illustrted in Fig., which depicts the response of two pinching models with κ f nd κ d equl to.5 nd.5, respectively. All three models hve % strin hrdening (the post-yield stiffness is % of the initil stiffness) nd the unloding stiffness is equl to the initil stiffness, i.e., unloding stiffness degrdtion is not considered. F y + F,7 K rel K e Envelope Curve 5 δ y - δ y + K rel - F y 9 δ Envelope Curve 5 + F y - δ y (c) F F y - K e δ y +, δ Envelope Curve 7 5 F mx + F y K rel, F p =κ f.f y - F K e - F y F min (d) K rel, δ per K rel,b κ d.δ per F p =κ f.f mx δ Fig. : Bem-hinge mechnism, pek-oriented hysteretic behvior,(c) biliner hysteretic behvior, (d) pinching hysteretic behvior.5 RESPONSE OF PINCHING MODEL Strin hrdening =., κ f = κ d =.5.5 RESPONSE OF PINCHING MODEL Strin hrdening =., κ f = κ d = M/M y M/M y θ/θ y θ/θ y Fig. : Vrious degrees of pinching in the moment-rottion reltionship, κ f = κ d =.5,.5

4 ORDINARY GROUND MOTIONS A set of recorded ordinry ground motions from Clifornin erthqukes with moment mgnitude between.5 nd.9 nd closest distnce to the fult rupture between km nd km is utilized. In this context, ordinry ground motions refer to those tht do not exhibit ner-fult, forwrd-directivity, soft-soil effects, nd (c) long-durtion effects. This set of ground motions is referred to s LMSR-N in subsequent plots. All ground motions were recorded on NEHRP site clss D (FEMA [9]). A rndom horizontl component of ground motion is selected t ech sttion to void bis in the selection process. A comprehensive description of the properties of these ground motion records is presented in Ref. [7]. ANALYSIS METHOD Engineering Demnd Prmeters The Engineering Demnd Prmeters (EDPs) of primry interest re roof nd story drifts. As used in this context, the term drift refers to the rtio of reltive displcement to the corresponding height, i.e., it defines the tngent of the drift ngle, which is equl to the drift ngle for the rnge of interest. The roof drift is considered mesure of the globl response (nd globl dmge) of the system while the story drift is ssumed to be relevnt for structurl nd non-structurl dmge ssessment s well s globl collpse ssessment due to dynmic instbility cused by P-Delt effects. Nonliner Time History Anlyses The bsic nlysis pproch consists of performing nonliner time history nlysis for given structure nd ground motion, using the DRAIN-DX computer progrm []. The hysteretic models under considertion were incorported into DRAIN-DX s new subroutines to crry out this study. For the dynmic nlyses, the ground motion intensity is relted to the structure strength by the reltive intensity prmeter, [S (T )/g]/γ, where S (T ) is the 5% dmped spectrl ccelertion t the fundmentl period of the structure, nd γ is the bse sher coefficient, i.e., γ = V y /W. The reltive intensity represents the ductility dependent response modifiction fctor (often denoted s R µ ), which is equivlent to the conventionl R-fctor if no overstrength is present. The use of [S (T )/g]/γ s reltive intensity mesure permits n ssessment of the dynmic response of frmes bsed on two different pproches: decresing the bse sher strength of the structure while keeping the ground motion intensity constnt (the R-fctor perspective) nd incresing the intensity of the ground motion while keeping the bse sher strength constnt [the Incrementl Dynmic Anlysis (IDA) perspective, Vmvtsikos nd Cornell [], see Fig. ]. Grphicl Representtion of Results Two bsic grphicl communiction schemes for given structure nd ground motion re presented. First, grphs of the type shown in Fig. re used. In this cse, the reltive intensity is plotted on the verticl xis, nd the mximum drift is plotted on the horizontl xis. The mximum drift is normlized by the spectrl displcement t the first mode period of the system, S d (T ), divided by structure height H. Second, normlized story drift profile curves re lso generted for discrete vlues of reltive intensity (Fig. ). In this representtion, for ech structure nd ground motion, the number of stories is plotted on the verticl xis nd the normlized drift is plotted on the horizontl xis. Dt of the type presented in Fig. permits n evlution of the distribution of story drift over the height of the frmes.

5 [S (T )/g]/ NORMALIZED MAXIMUM STORY DRIFT N=9, T =.9, ξ=.5, model, θ=.5, BH, K, S, LMSR-N Individul responses Medin th percentile th percentile 5 Norm. Mx. Story Drift Over Height, θ s,mx /[S d (T )/H] Fig. : Normlized mx. story drift over the height N = 9, T =.9 s., pek-oriented model Story Number MAX. STORY DRIFT PROFILES-[S (T )/g]/γ=. N=9, T=.9, ξ=.5, model, θ=.5, BH, K, S, LMSR-N Individul responses th percentile Medin th percentile 5 Normlized Mximum Story Drifts, θ si,mx /[S d (T )/H] Fig. : Normlized mx. story drift profiles N = 9, T =.9 s., pek-oriented model If bsolute vlues re of interest, given the ground motion hzrd S (T )/g, n pproprite bse sher strength cn be selected nd both the verticl nd horizontl xis of Fig. cn be de-normlized to obtin their corresponding IDAs. This de-normliztion process is illustrted with the dt from Fig. 7 nd Fig., which present the response of 9-story frme with T =. seconds in the normlized nd IDA domin, respectively. Similrly, given the ground motion hzrd nd the bse sher strength of the structure, grphs of the type presented in Fig. cn be used to generte bsolute vlues of the distribution of story drifts over the height. The second line of the title of the grphs shown in Figs. nd corresponds to prmeters tht hve been identified in previous sections. These prmeters describe the bsic properties of the structurl model. Sttisticl Representtion of Results The sttisticl evlution of results is performed using counted sttistics, in which vlues re sorted from smllest to lrgest, nd percentiles re counted rther thn computed bsed on specific distribution. Counted sttistics is needed to evlute those cses in which dt points re lost due to dynmic instbility cused by P-Delt effects (i.e., globl collpse cses). For consistency, counted sttistics is utilized t ll reltive intensity levels even if the full set of dt points is vilble. Thus, for set of dt points, the medin is the verge between the th nd st sorted vlues, the th percentile is the verge between the th nd 7 th sorted vlues, nd the th percentile is the verge between the rd nd th sorted vlues. The stndrd devition of the nturl logrithm of the dt (from here on referred to s dispersion) is estimted by using the counted th percentile vlue, x. If the counted medin is denoted s x 5, the dispersion of the dt is given by the nturl logrithm of the rtio (x 5 / x ). EFFECT OF HYSTERETIC BEHAVIOR ON THE NONLINEAR RESPONSE OF FRAMES The sensitivity of mximum roof nd story drifts to the type of hysteretic model is evluted for the fmily of twelve generic frme models exposed to the LMSR-N set of ordinry ground motions. In ll figures, unless otherwise specified, pinching model with κ d = κ f =.5 is utilized nd is denoted s pinching (the prmeters κ d nd κ f, which control the mount of stiffness degrdtion in the pinching model, re defined in Fig. (d)). A vlue of.5 is chosen becuse it is representtive of severe stiffness degrdtion in the response. Mximum Roof nd Story Drift Demnds In generl, frmes with severe pinching (κ d = κ f =.5) exhibit normlized mximum roof drift demnds lrger thn those observed for the cse of frmes with pek-oriented nd biliner hysteretic behvior, s

6 illustrted in Figs. 5 to 7 (which present representtive results for the N =, 9, nd generic frmes). Severe stiffness degrdtion cuses the system to become softer, nd hence, experience lrger deformtion demnds. It is importnt to note tht models with pek-oriented hysteretic behvior, which lso experience stiffness degrdtion, exhibit mximum roof drift demnds comprble to (nd in some cses smller thn) those observed for the biliner model (except for the frme with T =. s. in which demnds for the pek-oriented model re consistently lrger thn those experienced by the biliner model). These observtions indicte tht for medium to long-period structures, limited stiffness degrdtion (i.e., pek oriented cse) cn in some cses improve the seismic behvior of regulr frme structures. However, when the mount of stiffness degrdtion is lrge, it becomes detrimentl to the behvior of the system. This pttern is lso vlid for roof nd mximum story drift demnds, s the rtio of the two EDPs is essentilly independent of the hysteretic model s shown in Fig.. This figure depicts the vrition of the medin rtio of the mximum story drift over the height to the mximum roof drift with the fundmentl period of frmes, for reltive intensity [S (T )/g]/γ equl to.. NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=, T =., ξ=.5, Diff. hysteretic models, θ=., BH, K, S, LMSR-N NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=, T =., ξ=.5, Diff. hysteretic models, θ=.7, BH, K, S, LMSR-N [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) Fig. 5: Medin normlized mx. roof drift, N =, vrious hysteretic models, T =. s. nd. s. NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=9, T =.9, ξ=.5, Diff. hysteretic models, θ=.5, BH, K, S, LMSR-N NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=9, T =., ξ=.5, Diff. hysteretic models, θ=., BH, K, S, LMSR-N [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) Fig. : Medin normlized mx. roof drift, N = 9, vrious hysteretic models, T =.9 s. nd. s.

7 NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=, T =., ξ=.5, Diff. hysteretic models, θ=., BH, K, S, LMSR-N NORMALIZED MAXIMUM ROOF DRIFT-MEDIANS N=, T =., ξ=.5, Diff. hysteretic models, θ=., BH, K, S, LMSR-N [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) [S (T )/g]/γ 5 Normlized Mximum Roof Drift, θ r,mx /(S d (T )/H) Fig. 7: Medin normlized mx. roof drift, N =, vrious hysteretic models, T =. s. nd. s. MAX. STORY DRIFT/MAX. ROOF DRIFT-T =.N [S (T )/g]/γ=., Medin vlues, ξ=.5, BH, K, S, LMSR-N MAX. STORY DRIFT/MAX. ROOF DRIFT-T =.N [S (T )/g]/γ=., Medin vlues, ξ=.5, BH, K, S, LMSR-N θs,mx/θr,mx θs,mx/θr,mx Fundmentl Period, T (s)..... Fundmentl Period, T (s) Fig. : Medin rtio of mx. story drift to mx. roof drift, [S (T )/g]/γ =. stiff nd flexible frmes P-Delt Effects The pttern of behvior discussed in the previous prgrph is reversed in the cse of flexible structures tht re sensitive to P-Delt effects. P-Delt sensitive cses re defined s those in which the effect of grvity lods on the deformed configurtion of the system cuses lrge second order effects nd, sooner or lter, dynmic instbility in the response. For these cses [e.g., N =, T =. s., Fig. 7], it is the biliner model tht cuses the lrgest EDPs becuse the response of the system with biliner hysteretic behvior spends more time on the envelope of the moment-rottion reltionship of its components, which leds to rtcheting of the response nd potentil dynmic instbility problems if P-Delt effects produce negtive post-yield tngent stiffness. Figure 9 presents globl pushover curve for the -story, T =. s. frme, which represents the reltionship between globl deformtion prmeter (e.g., roof displcement) nd globl strength prmeter (e.g., bse sher) obtined by subjecting the structure to predetermined lterl lod pttern (in this cse prbolic, NEHRP k = pttern). This pushover curve is common to ll hysteretic models since the envelope of the moment-rottion reltionship t plstic hinge loctions is the sme regrdless of the type of hysteretic model. Note tht for this cse the post-yielding tngent stiffness hs lrge negtive vlue. It is this negtive post-yielding tngent stiffness tht cuses rtcheting of the dynmic response (increse in drift in subsequent cycles) nd ultimtely collpse in sideswy mode. Such collpse is

8 observed only if the post-yielding tngent stiffness is negtive, nd it occurs t reltive intensity tht decreses rpidly with n increse in the negtive slope of the post-yielding tngent stiffness. Normlized Strength, V/V y..... GLOBAL PUSHOVER CURVES T =. s., N =.9K i K i.k i Without P-Delt effects With P-Delt effects -.K i Normlized Roof Displcement, δ r /δ yr Floor Level DEFLECTED SHAPE FROM PUSHOVER ANALYSIS T =. s., N = Normlized Roof Displcement, δ r /δ yr Fig. 9: Pushover nlysis, N =, T =. s., globl pushover curve, deflected shpes t vrious levels of roof displcement Historiclly, the elstic story stbility coefficient, Pδ/(Vh), hs been used to quntify the importnce of the P-Delt effect. This study nd studies performed by others (e.g., Bernl [], Gupt nd Krwinkler [], nd Aydinoglu []) hve demonstrted tht this elstic story stbility coefficient my be poor mesure of the importnce of P-Delt effects in structures tht respond inelsticlly. A much better mesure cn be obtined from the globl pushover curve. Studies reported in [] nd [5] hve shown tht globl inelstic stbility coefficient, defined s the difference between the post-yielding tngent stiffnesses without nd with P-Delt effects of the globl pushover curve, nd normlized by the elstic stiffness, is good mesure of the importnce of P-Delt effect. For the -story frme with T =. s. the elstic globl stbility coefficient is.9, nd the inelstic globl stbility coefficient is.7. Thus, the inelstic stbility coefficient is more thn four times s lrge s the elstic one. The reson is evident from Fig. 9, which shows pushover deflection profiles for this frme. The elstic deflected shpe is close to stright line; however, once the structure yields, there is concentrtion of lrge story drifts in the bottom stories due to the presence of P-Delt effects. As the roof displcement increses, the bottom story drift vlues increse t rpid rte until dynmic instbility is pproched [see curve for δ r /δ yr =. in Fig. 9]. Mximum Story Drift Profiles A more comprehensive picture of the effect of the hysteretic model on the nonliner response of regulr frmes cn be obtined by studying the distribution of mximum story drifts over the height. Regulr frmes tht exhibit pinching hysteretic behvior t the component level experience less uniform distribution of mximum story drifts over the height s compred to frmes with biliner nd pekoriented behvior. Differences in the distribution of mximum story drifts over the height re more pronounced t the top nd t the bottom of the structure. These observtions re illustrted in Fig., which presents medin mximum story drift profiles of the 9-story structure with fundmentl period of.9 seconds for vrious vlues of reltive intensity. The generl conclusion is tht regulr frmes tht hve severe pinching in the hysteresis response not only experience the lrgest roof nd story drifts over the height (s presented in previous sections) but lso experience the lrgest mount of totl dmge (ssuming tht the mximum story drift is good indictor of structurl nd non-structurl dmge).

9 These ptterns of behvior cn hve significnt (or t lest noteworthy) influence on the seismic performnce of existing buildings tht will experience significnt stiffness degrdtion when exposed to strong ground shking, e.g., old reinforced-concrete construction. Story Number MAX. STORY DRIFT PROFILES-MEDIANS N=9, T =.9, ξ=.5, model, θ=.5, BH, K, S, LMSR-N [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ = Normlized Mximum Story Drifts, θ si,mx /[S d (T )/H] Story Number MAX. STORY DRIFT PROFILES-MEDIANS N=9, T =.9, ξ=.5, model, θ=.5, BH, K, S, LMSR-N [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ = Normlized Mximum Story Drifts, θ si,mx /[S d (T )/H] Story Number MAX. STORY DRIFT PROFILES-MEDIANS N=9, T =.9, ξ=.5, model, θ=.5, BH, K, S, LMSR-N [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ =. [S(T)/g]/ γ = Normlized Mximum Story Drifts, θ si,mx /[S d (T )/H] (c) Fig. : Medin normlized mx. story drift profiles, N = 9, T =.9 s., vrious hysteretic models pek-oriented, biliner, nd (c) pinching (κ d = κ f =.5) Sensitivity of the Response to the Degree of Stiffness Degrdtion The results presented so fr suggest tht the mximum roof nd story drift responses re lrger for systems with hysteretic behvior tht exhibits severe stiffness degrdtion such s the pinching model with κ d = κ f =.5. Unless such severe stiffness degrdtion is present, the differences between the pinching nd pek oriented models re not importnt s shown in Fig.. This figure presents mximum roof drifts for reltive intensity of [S (T )/g]/γ =. for both stiff nd flexible frmes with hysteretic behvior represented by biliner, pek-oriented nd pinching (κ d = κ f =.5) models s well s by pinching model with κ d = κ f =.5 [denoted in the grphs s pinching () ]. The mximum roof drift demnds re similr between the pinching () (represented by the filled squres in the plot) nd pekoriented models.

10 .5 NORMALIZED MAXIMUM ROOF DRIFTS-T =.N [S (T )/g]/γ=., Medin vlues, ξ=.5, BH, K, S, LMSR-N ().5 NORMALIZED MAXIMUM ROOF DRIFTS-T =.N [S (T )/g]/γ=., Medin vlues, ξ=.5, BH, K, S, LMSR-N () θr,mx/(s d (T )/H).5 θr,mx/(s d (T )/H) Fundmentl Period, T (s)..... Fundmentl Period, T (s) Fig. : Medin normlized mx. roof drift demnds, [S (T )/g]/γ =., stiff nd flexible frmes. Comprison to SDOF Systems Fig. presents medin rtios of mximum inelstic to elstic displcement for SDOF systems with strength reduction fctor, R = [S (T )/g]/η equl to., % strin hrdening, nd biliner, pek-oriented nd pinching (κ d = κ f =.5) hysteretic behvior (the prmeter η represents the strength of the SDOF system, which is nlogous the MDOF bse sher strength prmeter, γ). The effect of P-Delt t the SDOF level is represented by rotting the hysteresis digrm by n ngle equl to the elstic first story stbility coefficient of the.n nd.n frme structures. A comprison of Figs. nd shows tht except for T =. s., nd except for long period P-Delt sensitive systems, the difference in inelstic displcement demnds between the pinching model nd the other two models re more pronounced in the MDOF domin. This observtion hs implictions in seismic design nd evlution pproches in which the response of n MDOF system is represented by n equivlent SDOF model..5 RATIO OF INELASTIC TO ELASTIC DISP.-.N R=[S (T )/g]/η=., Medin vlues, ξ=.5, LMSR-N.5 RATIO OF INELASTIC TO ELASTIC DISP.-.N R=[S (T )/g]/η=., Medin vlues, ξ=.5, LMSR-N δinelstic/s d (T ).5 δinelstic/s d (T ) Period, T (s)..... Period, T (s) Fig. : Medin rtio of inelstic to elstic displcement, SDOF systems, vrious hysteretic models, R = [S (T )/g]/η =. P-Delt slope corresponding to.n nd P-Delt slope corresponding to.n For long period SDOF systems (T >. s.), the response of the biliner model becomes P-Delt sensitive while the response of the pek-oriented nd pinching models is stble. In this rnge, the differences between SDOF nd MDOF system responses become very lrge, for resons discussed in Ref. [5]. Adm et l. [5] present comprehensive discussion on the use of reference SDOF models to predict globl dynmic instbility in the response of flexible MDOF frmes.

11 EFFECT OF HYSTERETIC BEHAVIOR ON PROBABILISTIC SEISMIC PERFORMANCE ASSESSMENT In this section, the implictions of differences in the hysteretic behvior on the probbilistic seismic performnce ssessment of regulr frmes re illustrted. In this context, performnce ssessment implies tht the structure is given (i.e., γ = V y /W, T, nd other properties re known), nd decision vribles, such s losses, hve to be evluted (Krwinkler nd Mirnd []). Prt of this process is probbilistic seismic demnd nlysis tht includes quntifiction of engineering demnd prmeters (EDPs) nd their ssocited uncertinties. This quntifiction cn be crried out by computing the men nnul frequency of exceednce of n EDP, given by [ EDP y IM x] dλ ( ) λ ( y) = P = x () EDP IM where λ EDP (y) P[EDP y IM = x] λ IM (x) = men nnul frequency of EDP exceeding the vlue y = probbility of EDP exceeding y given tht IM equls x = men nnul frequency of IM exceeding x (ground motion hzrd) A prerequisite to the implementtion of Eq. () is hzrd nlysis for ground motion intensity mesure. In this study, the spectrl ccelertion t the first mode period of the structure, S (T ), is used s the intensity mesure. Similr to the IM hzrd, the men nnul frequency of the EDP exceeding certin vlue lso cn be represented in hzrd curve. Two EDPs re evluted: the mximum roof drift nd the mximum story drift over the height. Mximum Drift Hzrd Curves Evlution of Eq. () to compute the the mximum drift hzrd (men nnul frequency of drift exceeding y, given tht S (T ) equls S ) requires hzrd nlysis informtion on S (T ), which represents the term λ IM (x) in Eq. (), s well s probbilistic dt on mximum drift demnds. Since this evlution is performed for structures of given strength (γ), it is convenient to represent the mximum drift dt in the conventionl IDA form of S (T ) versus mximum drift, given the strength γ. Representtive IDA curves (medins nd dispersions) using the mximum roof drift ngle, θ r,mx, nd the mximum story drift ngle over the height, θ s,mx, for frme with N = 9, T =. seconds nd γ =., nd vrious hysteretic models re shown in Fig.. Spectrl Accelertion, S (T )/g.... MEDIAN IDA CURVES N=9, T =., γ=., ξ=.5, model, BH, K, S, LMSR-N Pek oriented-roof Drift -Roof Drift -Roof Drift Pek oriented-story Drift -Story Drift -Story Drift Mximum Drift Angle, θ mx Spectrl Accelertion, S (T )/g.... STD. DEV. OF LN(θ mx ) GIVEN S (T )/g N=9, T =., γ=., ξ=.5, model, BH, K, S, LMSR-N Pek oriented-roof Drift -Roof Drift -Roof Drift Pek oriented-story Drift -Story Drift -Story Drift.... Stndrd devition of ln(θ mx ) Given S (T )/g Fig. : Incrementl dynmic nlyses, N = 9, T =. s., γ =., vrious hysteretic models, medins nd dispersions

12 A rigorous evlution of Eq. () cn be done by crrying out numericl integrtion ssuming lognorml distribution of the EDP bsed on the medin nd dispersion presented in Fig.. However, informtion on the S (T ) hzrd is required to crry out such n evlution. Bsed on the work of Cornell et l. [7] the S (T ) hzrd cn be represented by curve of the type: S ( T ) ( S ) P S k [ ( T S ] = k S λ = ) () where the exponent k pproximtes the locl slope of the hzrd curve (in the log domin) round the return period of primry interest. For illustrtion of drift hzrd curves, n S (T ) hzrd curve is utilized tht is estimted from the equlhzrd response spectr vlues clculted for Vn Nuys, CA, site (site clss NEHRP D) s prt of the PEER reserch effort to develop Performnce-Bsed Erthquke Engineering Methodologies (Somerville nd Collins []). At the period of. sec., nd round the hzrd level of primry interest (the /5 hzrd), the following expression for the S (T ) hzrd is derived:. [ (.) S ] =. S o λ (.) ( S ) P S 7 () S = Mximum roof nd story drift hzrd curves computed from numericl integrtion of Eq. () re shown in Fig.. The drift hzrd curves permit probbilistic ssessment of mximum roof nd story drifts, in which the record-to-record vribility is considered. For given men nnul frequency of exceednce of mximum drift vlue, the pinching model clerly exhibits the lrgest mximum roof nd story drifts, which is consistent with the generl pttern of behvior evluted in previous sections of this pper. Men Annul Freq. of Exceednce, λ θ.e-.e-.e- MAXIMUM DRIFT HAZARD N=9, T =., γ=., ξ=.5, model, BH, K, S, LMSR-N Pek oriented-roof Drift -Roof Drift -Roof Drift Pek oriented-story Drift -Story Drift -Story Drift Mximum Drift Angle, θ mx Fig. : Mximum drift hzrd curves, N = 9, T =. s., γ =., vrious hysteretic models The conclusion is tht lrge degree of stiffness degrdtion does hve noticeble effect on the behvior nd the probbilistic seismic performnce ssessment of regulr frmes. Thus, n pproprite modeling of the hysteretic behvior t the component level is importnt in order to crry out relible seismic performnce ssessment of existing buildings.

13 CONCLUSIONS This pper evlutes the sensitivity of importnt drift prmeters (roof nd mximum story drifts) of regulr frmes to the hysteretic chrcteristics of the components tht control the inelstic response. Three types of hysteretic chrcteristics t the component level re considered: pek-oriented, biliner nd pinching. Monotonic nd cyclic deteriortion re not considered; therefore, results re representtive of levels of inelstic behvior t with monotonic nd cyclic deteriortion re not expected to be criticl. Moreover, only ordinry ground motions re utilized. Conclusions re to be interpreted within the conditions previously specified. Results demonstrte tht, except for long, flexible frmes tht re sensitive to P-Delt effects, both the biliner nd the pek-oriented models exhibit similr pek roof nd story drift demnds regrdless of the level of inelstic behvior. In some cses, the pek-oriented model experiences pek drift demnds smller thn those experienced by systems with biliner hysteretic chrcteristics. However, once significnt stiffness degrdtion is present, e.g., hysteretic behvior represented by the pinching model with smll vlues of κ d nd κ f, pek roof nd story drift demnds re clerly lrger thn those exhibited by the pek-oriented nd biliner models. Moreover, these ptterns re more pronounced for MDOF structures thn for SDOF systems. For inelstic systems, story drift demnds of frmes with pinched hysteretic chrcteristics re prticulrly lrger t the top nd t the bottom stories. These ptterns of behvior re reflected in the probbilistic seismic demnd evlution of frmes s illustrted in the lst section of this pper. The forementioned ptterns re reversed in the cse of flexible structures tht re sensitive to P-Delt effects. P-Delt sensitive cses re defined s those in which the effect of grvity lods on the deformed configurtion of the system cuses lrge displcement mplifiction in the dynmic response nd, sooner or lter, dynmic instbility. For these cses (e.g., N =, T =. s.), the biliner model predicts the lrgest EDPs becuse the response of the system with biliner hysteretic behvior spends more time on the envelope of the moment-rottion reltionship of its components, which leds to rtcheting of the response nd potentil dynmic instbility problems if P-Delt effects produce negtive post-yield tngent stiffness. The conclusions obtined in this study highlight the importnce of: A resonble representtion of the hysteretic response of structurl elements in MDOF models tht re used for seismic response nlyses nd probbilistic seismic performnce ssessment. The degree of stiffness degrdtion in the seismic response of regulr frmes, which is prticulrly importnt for the evlution nd rehbilittion of old reinforced-concrete construction. The effect of the hysteretic behvior when globl collpse due to dynmic instbility is of concern. Relible procedures for the identifiction of equivlent or reference SDOF systems when they re used to represent the behvior of MDOF structures. ACKNOWLEDGEMENTS This reserch ws crried out s prt of comprehensive effort t Stnford's John A. Blume Erthquke Engineering Center to ssess seismic demnds for frme structures. This effort is supported by the NSF sponsored Pcific Erthquke Engineering Reserch (PEER) Center. Additionl funding ws provided by The University of Mrylnd s prt of the fculty strt-up pckge of the first uthor of this pper. Both sources of support re grtefully cknowledged.

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