On Modeling Viscous Damping in Nonlinear Dynamic Analysis of Base-Isolated Reinforced Concrete Buildings

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1 O Modelig Viscous Dampig i Noliear Dyamic Aalysis of Base-Isolated Reiforced Cocrete Buildigs D.R. Pat & A.C. Wijeyewickrema Tokyo Istitute of Techology, Japa SUMMARY: The effects of modelig viscous dampig o the respose idicators of base-isolated reiforced cocrete buildigs subjected to earthquake groud motios are ivestigated usig a three-story buildig previously tested o a shakig table. Three-dimesioal fiite elemet simulatios are carried out, usig may differet approaches of applyig viscous dampig to the superstructure of the buildig. The approaches cosidered here are developed withi the framework of Rayleigh dampig. Noliear behavior of the superstructure as well as the isolatio system is cosidered i the aalyses. It is recommeded to use Rayleigh dampig where the dampig ratio is carefully selected ad the coefficiets multiplyig the mass ad stiffess matrices are calculated from the frequecies of the buildig based o the post-elastic stiffess of the isolatio system. Alteratively, stiffess-proportioal dampig where the coefficiet multiplyig the stiffess matrix is calculated from the frequecy of the superstructure for a fixed-base coditio ca be used. Keywords: Base isolatio, oliear aalysis, Rayleigh dampig, shakig table test. INTRODUCTION Modelig viscous dampig is a challegig task for base-isolated buildigs which cosist of two subsystems viz. the isolatio system ad the superstructure, with substatially differet eergy dissipatio properties. It is logical to prescribe viscous dampig separately for the isolatio system ad the superstructure, where the use of viscous dampig i the isolatio system ca be avoided by usig hysteretic models of bearigs to accout for all the eergy dissipatio. O the other had, the applicatio of viscous dampig to the superstructure aloe ca be doe usig various approaches withi the Rayleigh dampig framework, where dampig matrix c is give as, c= a m+ k (.) a, where m is the mass matrix, k is the stiffess matrix, ad a ad a are the dampig coefficiets. th The dampig ratio for the mode of a structure ξ is defied as, ξ a aω o = +, (.2) 2ω 2 th where ω is the frequecy of the mode. The coefficiets a ad a are determied assumig the same dampig ratio ξ for two selected modes as, a 2ωω 2,, i j = ξ a = ξ ωi + ωj ωi + ωj (.3)

2 where ω i ad ω j are the frequecies of mode i ad mode j, respectively. The two modes i ad j are chose to esure early the same amout of dampig for all the modes sigificatly cotributig to the respose of the structure (Chopra 27). Typically, ω i is selected to be the frequecy of the first mode ad ω j correspods to a higher mode. Mass-proportioal dampig where c= am ad a = 2 ξω i ; ad stiffess-proportioal dampig where c= ak ad a = 2 ξ ω i, ca be viewed as special cases of Rayleigh dampig. Covetioally, the dampig matrix c is computed usig Rayleigh dampig based o iitial elastic properties of the system. This practice is based o the otio that eergy dissipatio due to viscous dampig is egligible, compared with the much higher eergy dissipatio due to material oliearity i the ielastic rage. However, covetioal applicatio of Rayleigh dampig could lead to urealistically large dampig forces, resultig i a ucoservative desig (Hall 26). While the issue of modellig viscous dampig has bee explored extesively for fixed-base buildigs (see for example Léger ad Dussault 992, Charey 28, ad Erdura 22), it is oly recetly that the baseisolated buildigs have bee cosidered (see Hall 26 ad Rya ad Polaco 28). Hall (26) studied the effects of mass-proportioal dampig ad stiffess-proportioal dampig usig a rigid superstructure model. Rya ad Polaco (28) recommeded the applicatio of stiffessproportioal dampig istead of Rayleigh dampig to the superstructure of the buildig but their study was based o two-dimesioal models of the buildigs assumig liear elastic behavior of the superstructure. Stiffess-proportioal dampig was also used by Pat ad Wijeyewickrema (22) to evaluate the performace of a base-isolated reiforced cocrete (RC) buildig subjected to seismic poudig usig three-dimesioal oliear fiite elemet (FE) models. However, the effects of modelig viscous dampig o the structural respose of the buildig were ot discussed i that study. Therefore, a systematic study ivestigatig the effects of modelig viscous dampig o the importat respose idicators of base-isolated buildigs such as floor displacemets, floor acceleratios ad story shear forces, usig rigorous three-dimesioal FE models, is eeded. I additio, oe of these previous studies have compared umerical aalysis results with the results from experimets or istrumeted buildigs. I the preset study, the cosequeces of modelig viscous dampig i time-history aalysis of baseisolated RC buildigs by usig various approaches based o the Rayleigh dampig framework, are ivestigated. Existig shakig table test results of a three-story base-isolated buildig, where the superstructure was shake strogly but did ot udergo sigificat ielastic excursios, are compared with the umerical aalysis results to arrive at the appropriate techique for modelig viscous dampig. I these umerical simulatios, eergy dissipatio due to ielastic material behavior of the superstructure was simulated usig well-established material models, while the viscous dampig was applied to the superstructure usig may differet approaches. All the eergy dissipatio i the isolatio system was explicitly modeled usig biliear hysteretic models of bearigs. 2. BENCHMARK BUILDING AND NUMERICAL MODELING The base-isolated bechmark buildig used to compare umerical aalysis results is a.4 scale model of a three-story two-bay by oe-bay RC structure built i Japa (Fig. 2.). To maitai proper scalig, packets of lead billets were tied dow to each slab ad the total weight of the model was 4 kn. Three phases of shakig table tests were carried out followig iitial system idetificatio tests icludig the static pull-back tests. However, due to improper calibratio of loadcells durig the first static pull-back test, localized crackig was observed especially i first-floor beam-colum joits (Clark et al. 997). I the first phase of the shakig table tests, the structure was braced i the trasverse directio as well as the logitudial directio to eable compariso of various isolatio systems by avoidig damage to the superstructure. Desig-level ad desig-level 2 earthquake excitatios, accordig to Japaese practice were used as iput i this stage. I the secod phase of the

3 tests, the model was braced oly i the trasverse directio ad the two desig-level earthquakes were ru agai, followed by very high-itesity motios iteded to cause damage to the structure. I the fial phase, the model was repaired ad moderate-itesity motios were applied. It is oted that i all the tests, the earthquake excitatio was oly i the logitudial directio. This study focuses o the secod phase of the tests where desig-level earthquakes were used. I this phase, idetical 76 mm diameter high dampig rubber bearigs with twety 2.2 mm thick rubber layers alteratig with mm thick steel shims were used uder each of the six colum bases. Desig-level 2 tests which led to yieldig i the RC frame durig the tests are cosidered i the preset study. I these tests, recordigs of the 94 El Cetro, 968 Tokachi-oki, ad 979 Miyagike-oki earthquakes were ru at a itesity correspodig to the full-scale peak velocity of 5 cm/sec (correspodig to the desig-level 2 earthquake) ad are referred to as ELC-5, HACH-5, ad MIYA-5, respectively. Oe additioal excitatio that represeted the desig forces typically specified by the US codes was also used i these tests. The excitatio referred to as ELC-S, is a sythetic record geerated from the 94 El Cetro earthquake recordig. The sequece of excitatio was i the followig order: ELC-S, ELC-5, HACH-5, ad MIYA-5. 2@.87m 2.4m A B C 2 2@.87m 2.4m.25m 2@.2m excitatio shakig table shakig table (a) (b) (c) Fig. 2.. Geometry of the three-story buildig tested o shakig table: (a) pla; (b) elevatio alog logitudial directio; ad (c) elevatio alog trasverse directio (ote that o braces were used alog grid lie B). 2.. Modelig of structural elemets Three-dimesioal FE model of the buildig was developed i OpeSees (2). Beams ad colums were modeled usig force-based, Euler-Beroulli fiber beam-colum elemets that accout for the spread of ielasticity alog the legth of the elemet. For cocrete the modified Ket ad Park model (Park et al. 982) was used i compressio ad a iitial liear elastic brach together with a liear softeig brach up to zero stress was used i tesio. The model proposed by Yassi (994) was used to accout for cocrete damage ad hysteresis. For reiforcig steel, the costitutive model of Meegotto ad Pito (973), which icludes strai hardeig ad the Bauschiger effect, was used. I this study, the strai-hardeig ratio is take as %. Bearigs were modeled usig elastomeric bearig elemets with a biliear hysteretic model used to represet the lateral force-deformatio relatioship of each elemet ad to simulate all the eergy dissipatio i the isolatio system. The parameters of the biliear hysteretic model for each elemet as determied from the compoet test results of a bearig that was similar to the oes used i the buildig, are iitial stiffess K =,67.5 kn/m, postelastic stiffess K 2 = 3 kn/m, ad yield force F y = 3.94 kn. The vertical force-deformatio relatioship of each elemet was simulated usig a liear elastic compressio-oly sprig with a stiffess K v = 338, kn/m. The stadard horizotal characteristic test of the bearig at % shear strai level ad a axial load of 78.5 kn was reproduced quite well usig these properties as show i Fig. 2.2.

4 Force (kn) Experimet Aalysis Displacemet (mm) Fig Hysteretic loop of the bearig for oe cycle of displacemet at % shear strai level. Experimetal curve is reproduced after Clark et al. (997) Modelig of viscous dampig Viscous dampig was applied oly to the superstructure of the buildig usig may differet approaches of computig the dampig matrix c that cover a wide rage of optios available i may existig FE programs (Table 2.). The relevat frequecies as obtaied from modal aalyses are also show i Table 2.. Sice the modal dampig values could ot be estimated reasoably well durig the test (Clark et al. 997), umerical simulatios were performed usig dampig ratio ξ = %, 2%, 3%, 4%, ad 5%. This also eables the ivestigatio of the effects of dampig ratio o the aalysis results. The approaches cosidered here deped o: () Dampig: Rayleigh, mass-proportioal or stiffess-proportioal dampig. (2) Dampig coefficiets a, a : The dampig coefficiets a ad a are costat throughout the aalysis (based o the iitial stiffess) or updated i each aalysis step (based o the taget stiffess). (3) Basis for computig a, a : Modes of deformatio chose ad the structural model cosidered for modal aalysis to calculate frequecies ecessary for the evaluatio of dampig coefficiets. (4) Stiffess matrix: Iitial or taget stiffess matrix used i Eq. (.). The approaches are grouped as follows: () Group A: Rayleigh dampig; dampig coefficiets are costat ad calculated from modal aalysis of the superstructure for a fixed-base coditio (es 4). (2) Group B: Rayleigh dampig; dampig coefficiets are costat ad calculated from modal aalysis of the base-isolated buildig with iitial stiffess of the isolatio system (es 5 8). (3) Group C: Rayleigh dampig; dampig coefficiets are costat ad calculated from modal aalysis of the base-isolated buildig with post-elastic stiffess of the isolatio system (es 9 2). (4) Group D: Rayleigh dampig; dampig coefficiets are updated ad calculated from modal aalysis of the base-isolated buildig (es 3 6). (5) Group E: Mass-proportioal dampig (es 7 2). (6) Group F: Stiffess-proportioal dampig; dampig coefficiet is costat ad calculated from modal aalysis of the superstructure for a fixed-base coditio (es 2 22). (7) Group G: Stiffess-proportioal dampig; dampig coefficiet is calculated from modal aalysis of the base-isolated buildig (es 23 28).

5 Table 2.. Differet approaches of modelig viscous dampig. Basis for Dampig computig a Group Dampig coefficiets, a Stiffess ω ω i j a, a Modes Structural matrix (rad/s) (rad/s) i, j model A Rayleigh Costat, 3 SS Iitial A Rayleigh Costat, 3 SS Taget A Rayleigh Costat, 4 SS Iitial A Rayleigh Costat, 4 SS Taget B Rayleigh Costat, 3 BIB-I Iitial B Rayleigh Costat, 3 BIB-I Taget B Rayleigh Costat, 4 BIB-I Iitial B Rayleigh Costat, 4 BIB-I Taget C Rayleigh Costat, 3 BIB-P Iitial C Rayleigh Costat, 3 BIB-P Taget C Rayleigh Costat, 4 BIB-P Iitial C Rayleigh Costat, 4 BIB-P Taget D Rayleigh Updated, 3 BIB Iitial 4 D Rayleigh Updated, 3 BIB Taget 5 D Rayleigh Updated, 4 BIB Iitial 6 D Rayleigh Updated, 4 BIB Taget 7 E Mass prop. Costat SS E Mass prop. Costat BIB-I E Mass prop. Costat BIB-P E Mass prop. Updated BIB 2 F Stiffess prop. Costat SS Iitial F Stiffess prop. Costat SS Taget G Stiffess prop. Costat BIB-I Iitial G Stiffess prop. Costat BIB-I Taget G Stiffess prop. Costat BIB-P Iitial G Stiffess prop. Costat BIB-P Taget G Stiffess prop. Updated BIB Iitial 28 G Stiffess prop. Updated BIB Taget 3. COMPARISON OF TEST AND NUMERICAL ANALYSIS RESULTS Noliear time-history aalysis of the buildig was carried out usig OpeSees (2) for a sigle cotiuous sequece of cocateated records ELC-S, ELC-5, HACH-5, ad MIYA-5 usig the 28 approaches of modelig viscous dampig. The P effect for the superstructure as well as the isolatio system was icluded i the aalysis to cosider geometric oliearity effects. Numerical aalysis results oly for the weakest (i.e., ELC-S) ad the strogest (i.e., MIYA-5) excitatios (Fig. 3.) are discussed i this paper, for brevity. The selected respose idicators are relative floor displacemets, absolute floor acceleratios, ad story shear forces. To eable the compariso of various approaches of modelig dampig, the error E of the peak value of the respose idicator represetig displacemet, acceleratio or shear force at th story was calculated as,

6 E N T =, = (base),, 2, 3 (roof), (3.) T where N ad T are the peak values of umerical aalysis ad test results, respectively of the respose idicator at th story. Maximum of E values at all the stories for various dampig ratios ad approaches of modelig viscous dampig are show i Fig. 3.2 ad Fig. 3.3 for the ELC-S excitatio ad the MIYA-5 excitatio, respectively. Maximum errors whe the viscous dampig was ot cosidered i the aalysis are also show for compariso purposes. Figures 3.2 ad 3.3 show that the error treds usig various approaches are early the same for the ELC-S ad MIYA-5 excitatios with a few exceptios, especially i Group E (mass-proportioal dampig). I geeral, the errors uder the stroger excitatio are smaller tha those uder the weaker excitatio. It is oted that the peak iter-story drift ratios were foud to be relatively small at.2% ad.2% for the ELC-S ad MIYA-5 excitatios, respectively. Although there are differeces betwee errors uder the two excitatios, the superstructure does ot experiece eough damage to sigificatly affect the dyamic characteristics of the buildig uder both the excitatios. Therefore, uless specified otherwise, the discussio from here owards will be focused o the respose uder the MIYA-5 excitatio. Displacemet (mm) Time (s) (a) Displacemet (mm) Time (s) (b) Fig. 3.. Displacemet time histories of the iput motios: (a) ELC-S ad (b) MIYA-5 Clark et al. (997). It is clearly see from Fig. 3.3 that errors for a particular dampig ratio usig various approaches withi ay group other tha Group E, do ot show sigificat differeces for all the respose idicators. This implies that the choice of (a) higher mode (mode 3 or mode 4) frequecy ad (b) stiffess matrix (iitial or taget), i the applicatio of viscous dampig has egligible ifluece o the respose idicators. The choice of stiffess matrix does ot have a sigificat ifluece because stiffess-proportioal dampig costitutes a very small fractio of viscous dampig, compared with mass-proportioal dampig. I additio, the fact that the stiffess of the superstructure is ot chaged sigificatly durig the excitatios also implies that the choice of stiffess matrix i the applicatio of dampig has egligible ifluece. This is differet from fixed-base or base-isolated structures subjected to strog groud motios which result i sigificat stiffess degradatio. The errors for a particular dampig ratio usig the approaches withi Group E are sigificatly differet. I additio, the error treds with respect to dampig ratio are chaged for es 8 2 compared with other approaches. Although the related results are ot show here for brevity, it was foud from the umerical aalysis results that es 8 2 uderestimate the floor displacemet respose but overestimate the floor acceleratio ad story shear force resposes for almost all the dampig ratios. The floor displacemet respose usig es 8 2 were foud to be slightly smaller tha the correspodig Rayleigh dampig couterparts, while the floor acceleratios ad story shear forces were sigificatly larger for all the dampig ratios. This is because mass-proportioal dampig, which ca be visualized as dampers coectig each mass to a exteral support, does ot correspod to a realistic physical pheomeo (Pg. 455 Chopra 27). Therefore, the applicatio of oly massproportioal dampig is ot desirable. Here it is oted that although the mass-proportioal dampig domiates, the errors obtaied usig oly the mass-proportioal dampig part of es 6, are ot the same errors whe oly mass-proportioal dampig is cosidered i es 7 2, as the

7 Floor displacemet error (%) Group A Group B o dampig Group C Group D Group E Group F % dampig 2% dampig 3% dampig 4% dampig 5% dampig Group G (a) Floor acceleratio error (%) (b) Story shear force error (%) (c) Fig Maximum errors i peak values of respose idicators for all the stories for the ELC-S excitatio: (a) relative floor displacemet; (b) absolute floor acceleratio; ad (c) story shear force. The dashed lie idicates the maximum error whe viscous dampig was ot cosidered.

8 Floor displacemet error (%) (a) Group A Group B o dampig Group C Group D Group E Group F % dampig 2% dampig 3% dampig 4% dampig 5% dampig Group G Floor acceleratio error (%) (b) Story shear force error (%) (c) Fig Maximum errors i peak values of respose idicators for all the stories for the MIYA-5 excitatio: (a) relative floor displacemet; (b) absolute floor acceleratio; ad (c) story shear force. The dashed lie idicates the maximum error whe viscous dampig was ot cosidered.

9 dampig coefficiet a is computed usig differet equatios. For example, for ω = 37.4 rad/s ad ω 3 = rad/s (ω 4 = 3.62 rad/s), a = 47.7ξ (a = 57.8 ξ ) for the mass-proportioal part of Rayleigh dampig ad a = 74.3ξ for oly mass-proportioal dampig. Therefore, the dampig coefficiet for oly mass-proportioal dampig is larger tha that for the mass-proportioal part of Rayleigh dampig, which essetially results i higher degree of dampig. Similar results were observed by Erdura (22) for fixed-base steel buildigs. Aother importat observatio from the results is that for the floor displacemets, smallest errors occur whe viscous dampig was eglected i the model (Figs. 3.2(a) ad 3.3(a)). However, Figs. 3.2(b), 3.2(c), ad 3.3(b) suggest that eglectig such dampig may lead to higher errors i terms of the floor acceleratios ad story shear forces. I certai scearios, viscous dampig is also desirable by aalysis programs for purposes of umerical stability. I additio, the higher error due to a particular approach of modelig dampig, results maily because of the iclusio of higher degree of dampig. It is observed that the errors usig the updated dampig coefficiets a ad a are smaller compared with usig costat a ad a, ad the differeces are more obvious for higher dampig ratios. For example, cosider the errors usig approaches i Group B (costat a ad a ) ad Group D (updated a ad a ), where i the latter group the errors are smaller (Figs. 3.2 ad 3.3). It is iterestig to cosider the results obtaied usig the approaches i Group C, where dampig coefficiets were computed based o the modal aalysis of the base-isolated buildig with post-elastic stiffess of the isolatio system. The errors usig the approaches i Group C are either smaller tha or comparable with the errors i Group D for all the dampig ratios (Figs. 3.2 ad 3.3). I additio, the aalysis time with the approaches i Group C was foud to be about oe fourth of the aalysis time usig approaches i Group D, where dampig coefficiets were updated i each aalysis step. Therefore, approaches i Group C are more attractive from the view poit of computatioal cost tha approaches i Group D. The aalysis results idicate that there is o uique best approach for the modelig of viscous dampig suitable for all situatios. The followig two methods seem to yield the least errors: Method : Group C type dampig, i.e., Rayleigh dampig, with the dampig coefficiets a ad a computed from the frequecies of the base-isolated buildig with the post-elastic stiffess of the isolatio system. However, it should be oted that with this type of dampig, the errors deped greatly o the selected dampig ratio ξ. Method 2: Group F type dampig, i.e., stiffess-proportioal dampig with dampig coefficiet a computed from the frequecy of the superstructure for a fixed-base coditio. It should be oted that with this type of dampig, the errors do ot vary sigificatly depedig o the selected dampig ratio ξ, compared with the Group C type dampig, idicatig less ucertaity i the predicated respose with the choice of ξ. Group F type dampig also icludes the approaches recommeded by Rya ad Polaco (28). 4. CONCLUDING REMARKS The cosequeces of modelig viscous dampig i base-isolated RC buildigs are ivestigated, by usig a three-story buildig tested o a shakig table. The study has led to the followig coclusios ad recommedatios: () Dampig ratio as well as the approach adopted to model dampig is crucial because the floor displacemets, floor acceleratios, ad story shear forces are affected sigificatly by it.

10 (2) Applicatio of Rayleigh dampig where the dampig coefficiets a ad a are calculated from the frequecies of the superstructure for a fixed-base coditio (Group A) ad massproportioal dampig (Group E) should be avoided, as these approaches could produce large errors i respose predictios. (3) Rayleigh dampig where dampig coefficiets a ad a are calculated from the frequecies of the base-isolated buildig with the post-elastic stiffess of the isolatio system istead of the iitial stiffess of the isolatio system (Group C), could be used. However, the dampig ratio should be carefully selected, as the buildig respose depeds greatly o it. As a alterative, stiffess-proportioal dampig where the dampig coefficiet a is computed from the frequecy of the superstructure for a fixed-base coditio (Group F) could be used; with this approach the buildig respose does ot deped sigificatly o the selected dampig ratio. ACKNOWLEDGEMENTS The first author is pleased to ackowledge a Mobukagakusho (Miistry of Educatio, Culture, Sports, Sciece ad Techology, Japa) scholarship for graduate studets. Fiacial support from Ceter for Urba Earthquake Egieerig (CUEE), Tokyo Istitute of Techology is gratefully ackowledged. REFERENCES Charey, F.A. (28). Uiteded cosequeces of modelig dampig i structures. Joural of Structural Egieerig (ASCE) 34:4, Chopra, A.K. (27). Dyamics of Structures: Theory ad Applicatios to Earthquake Egieerig, Pearso Educatio: New Jersey. Clark, P.W., Aike, I.D., ad Kelly, J.M. (997). Experimetal studies of the ultimate behavior of seismicallyisolated structures: Techical Report, UCB/EERC-97/8, Earthquake Egieerig Research Ceter, Uiversity of Califoria, Berkeley. Erdura, E. (22). Evaluatio of Rayleigh dampig ad its ifluece o egieerig demad parameter estimates. Earthquake Egieerig ad Structural Dyamics (i press). Hall, J.F. (26). Problems ecoutered from the use (or misuse) of Rayleigh dampig. Earthquake Egieerig ad Structural Dyamics 35, Léger, P. ad Dussault, S. (992). Seismic-eergy dissipatio i MDOF structures. Joural of Structural Egieerig (ASCE) 8:5, Meegotto, M. ad Pito, P. (973). Methods of aalysis for cyclically loaded R/C frames. Symposium of Resistace ad Ultimate Deformability of Structure Acted by Well Defied Repeated Load, Lisbo, Portugal. OpeSees (2). Ope system for earthquake egieerig simulatio. Computer Program, Uiversity of Califoria, Berkeley. Available from: Pat, D.R. ad Wijeyewickrema, A.C. (22). Structural performace of a base-isolated reiforced cocrete buildig subjected to seismic poudig. Earthquake Egieerig ad Structural Dyamics (i press). Park, R., Priestley, M.J.N., ad Gill, W.D. (982). Ductility of square-cofied cocrete colums. Joural of Structural Egieerig (ASCE) 8, Rya, K.L. ad Polaco, J. (28). Problems with Rayleigh dampig i base-isolated buildigs. Joural of Structural Egieerig (ASCE) 34:, Yassi, M.H.M. (994). Noliear aalysis of prestressed cocrete structures uder mootoic ad cyclic loads. Ph.D. Dissertatio, Uiversity of Califoria, Berkeley.