Elastic and Inelastic Seismic Response of Buildings with Damping Systems

Elastic and Inelastic Seismic Responseof Buildings with Damping Systems

Oscar M. Ramirez,a) Michael C. Constantinou,b) M.EERI,Andrew S. Whittaker,c) M.EERI, Charles A. Kircher,d) M.EERI,and Christis Z. Chrysostomoue)

The effect of damping on the response of elastic and inelastic single-degree-of-freedom systems was studied by nonlinear response-history analy-sis using earthquake histories that matched on average a 2000 NEHRP spec-trum on a stiff soil site for a region of high seismic risk. New displacementreduction factors for levels of damping greater than 5% of critical are pre-sented. New equations to relate inelastic and elastic displacements in theshort-period range, for levels of damping greater than 5% of critical, are pre-sented. The technical basis for reducing the minimum design base shear indamped buildings by a maximum of 25%, from that required for the corre-sponding undamped building, is derived based on comparable levels of dam-age in both the damped and undamped buildings. [DOI: 10.1193/1.1509762]

INTRODUCTION

Conventionally constructed earthquake-resistant buildings rely on significant inelas-tic action (energy dissipation) in selected components of the framing system for designand maximum earthquake shaking. For the commonly used special moment-resistingframe, inelastic action should occur in the beams near the columns and in the beam-column panel joint: both zones form part of the gravity-load-resisting system. Inelasticaction results in damage, which is often substantial in scope and difficult to repair. Dam-age to the gravity-load-resisting system can result in significant direct and indirectlosses.

The desire to avoid damage to components of gravity-load-resisting frames in build-ings following the 1989 Loma Prieta and 1994 Northridge earthquakes spurred the de-velopment of passive energy dissipation systems. Passive metallic yielding, viscoelastic,and viscous damping devices are now available in the marketplace, both in the UnitedStates and abroad. Soong and Dargush (1997), Constantinou et al. (1998), and Hansonand Soong (2001) describe these and other types of passive dampers. The primary ob-jective of adding energy dissipation systems to building frames has been to focus the

a) Prof., Director, Centro Experimental de Ingenieria, Universidad Technologica de Panama, El Dorado Panama,Rep. de Panama

b) Prof. and Chmn., Dept. of Civ., Struct. and Envir. Engrg., Univ. at Buffalo, State Univ. of New York, Buffalo,NY 14260

c) Associate Prof., Dept. of Civ., Struct. and Envir. Engrg., Univ. at Buffalo, State Univ. of New York, Buffalo,NY 14260

d) Principal, C. A. Kircher and Associates, Palo Alto, CA 94303e) Lecturer, Dept. of Civ. Engrg., Higher Technical Institute, 2152 Nicosia, Cyprus

531Earthquake Spectra, Volume 18, No. 3, pages 531–547, August 2002; © 2002, Earthquake Engineering Research Institute

532 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

energy dissipation during an earthquake into disposable elements specifically designedfor the purpose of dissipating energy, and to substantially reduce or eliminate energydissipation in the gravity-load-resisting frame. Since energy dissipation or damping de-vices do not form part of the gravity-load-resisting system they can be replaced after anearthquake without compromising the structural integrity of the frame.

One impediment to the widespread use of passive energy dissipation systems hasbeen the lack of robust and validated guidelines for the modeling, analysis and design ofenergy dissipation systems, and testing of damping devices. Considerable research effortin the 1990s resulted in the development of at least five code-oriented procedures relatedto the implementation of passive energy dissipation systems. The Structural EngineersAssociation of Northern California (SEAONC) published the first procedures in 1992(Whittaker et al. 1993). The Federal Emergency Management Agency (FEMA) includeddraft guidelines for the implementation of passive energy dissipation devices in newbuildings in the 1994 edition of the NEHRP Recommended Provisions for the SeismicRegulations for New Buildings (BSSC 1994). Guidelines for the implementation of pas-sive energy dissipation devices in retrofit construction were published in 1997 in theFEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings (ATC 1997).In 1999, the SEAOC Ad Hoc Committee on Energy Dissipation published guidelines forimplementing energy dissipation devices in new buildings in the SEAOC Blue Book(SEAOC 1999) in a format consistent with that of the 1997 Uniform Building Code(ICBO 1997).

In 2001, FEMA published the 2000 edition of the NEHRP Recommended Provisionsfor Seismic Regulations for New Buildings and Other Structures (BSSC 2001), in whichcompletely revised procedures for implementing passive energy dissipation devices innew buildings are outlined and robust linear procedures (equivalent lateral force andresponse-spectrum methods) for analysis are described. The development and verifica-tion of the analysis methods for buildings with damping systems in the 2000 NEHRPRecommended Provisions (hereafter termed the 2000 NEHRP Provisions) are the resultof the collective efforts of members of Technical Subcommittee 12 of the Building Seis-mic Safety Council and researchers at the University at Buffalo. These efforts are de-scribed in Ramirez et al. (2000).

The 2000 NEHRP Provisions analysis methods for buildings with damping systemswere written around a number of significant simplifications and limits, some of whichare outlined below:

1. A multi-degree-of-freedom (MDOF) building with a damping system can betransformed into equivalent single-degree-of-freedom (SDOF) systems usingmodal decomposition procedures. Such procedures do not strictly apply to ei-ther yielding buildings or buildings that are non-proportionally damped.

2. The response of an inelastic single-degree-of-freedom system can be estimatedusing equivalent linear properties and a 5% damped response spectrum. Spectrafor damping greater than 5% can be established using damping coefficients, andvelocity-dependent forces can be established using either pseudo-velocity andmodal information or by applying correction factors to the pseudo velocity.

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 533

3. The minimum design base shear for buildings with damping systems is lessthan that for conventional buildings without damping systems, based on limit-ing the deformations and ductility demands in the damped building to those as-sumed for undamped (conventional) buildings.

This paper is the first of two presenting the development and verification of the 2000NEHRP Provisions procedures for buildings with damping systems. This paper presents(a) the studies that produced the damping coefficients listed in the 2000 NEHRP Provi-sions to modify the 5% damped response spectrum for the effects of higher damping, (b)a study of the relation between elastic and inelastic displacements in viscously dampedbuildings, and (c) a comparison of ductility demands in structures without and withdamping systems, where the damped buildings are designed for a smaller base shearthan conventional buildings.

The companion paper (Ramirez et al. 2002) describes the simplified method ofanalysis for single-degree-of-freedom structures with linear viscous, nonlinear viscousand hysteretic damping systems, presents methods to calculate maximum velocity andmaximum acceleration using pseudo-velocity and pseudo-acceleration data, and summa-rizes the results of a comprehensive study of the simplified methods of analysis.

MODIFICATION OF RESPONSE SPECTRUM FOR HIGH DAMPING

Traditionally, 5% damping has been assumed for the construction of elastic responsespectra that are used for design of earthquake-resistant structures. Spectra for higher lev-els of damping must be constructed for the application of simplified methods of analysisof structures with damping systems. Elastic spectra constructed for levels of viscousdamping greater than 5% are used for the analysis of linearly elastic structures with lin-ear viscous damping systems. Moreover, such spectra are used for the nonlinear analysisof yielding structures because these methods facilitate the direct evaluation of inelasticresponse using demand spectra, which are established using a 5%-damped pseudo-acceleration response spectrum and adjustment factors for the increased effective damp-ing in the structure. The typical approach to construct an elastic spectrum for dampingother than 5%, Sa(T,b), is to divide the 5%-damped spectral acceleration by a dampingcoefficient B that is a function of the damping ratio, b, namely,

Sa~T,b!5Sa~T,5%!

B~b!(1)

where T is the elastic period.

The values of the damping coefficient that appeared in the 1994 NEHRP Recom-mended Provisions (BSSC 1995) were based on the study of Wu and Hanson (1989).The FEMA 273 guidelines (ATC 1997) were developed using damping coefficients thatwere based on the work of Newmark and Hall (1982) but were extended to higher valuesof the damping ratio. The extension of the work by Newmark and Hall to higher valuesof the damping ratio was necessary for two reasons. First, the simplified methods ofanalysis in FEMA 273 could result in high effective damping due to the combined ef-fects of yielding of the building frame and added viscous damping. (For information, thevalues assigned to B in FEMA 273 are presented in Table 1.) Second, under certain con-

534 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

ditions the damping ratio in higher modes may be very large and could reach critical orovercritical values in buildings having a complete vertical distribution of viscous energydissipation devices. Three of the shortcomings of the values assigned to B in FEMA 273are as follows:

1. The values of B in the constant acceleration region of the spectrum (T<Ts inFigure 1) are larger than those in the constant velocity region of the spectrum.This contradicts the fact that there is only a modest reduction of displacementwith increased damping in very stiff structures, and leads the user to the erro-neous conclusion that damping systems are most effective when used in stiffstructures.

2. The effect of damping beyond 50% of critical is ignored leading to conservativeestimates of displacement in highly damped buildings, which may be the casefor yielding frames equipped with supplemental viscous damping systems.

3. Limiting the damping coefficient to 2.0 for T>Ts and b.50% results in a con-servative estimate of the maximum velocity, which is of great significance indetermining forces in viscous dampers.

In the study reported in this paper, values for the damping coefficient for damping

Table 1. Values of damping coefficient B

EffectiveDamping

b

FEMA 273 Ramirez al. (2000)2000 NEHRP

Provisions

Bs1 B1

2 Bs3 B1

4 B5

,0.02 0.8 0.8 0.80 0.80 0.80.05 1.0 1.0 1.00 1.00 1.00.10 1.3 1.2 1.20 1.20 1.20.20 1.8 1.5 1.50 1.50 1.50.30 2.3 1.7 1.70 1.70 1.80.40 2.7 1.9 1.90 1.90 2.10.50 3.0 2.0 2.20 2.20 2.40.60 3.0 2.0 2.30 2.60 2.70.70 3.0 2.0 2.35 2.90 3.00.80 3.0 2.0 2.40 3.30 3.30.90 3.0 2.0 2.45 3.70 3.61.00 3.0 2.0 2.50 4.00 4.0

1. For T<TsBs /B1 ; Ts is the corner point in the spectrum per Figure 1; see the 2000NEHRP Provisions for definition of terms.2. For T>TsBs /B1

3. Valid at T50.2Ts ; for 0.2Ts,T,Ts , B is determined by linear interpolation betweenvalues Bs and B1 ; for T,0.2Ts , B is determined by linear interpolation between valuesof 1.0 (at T50.0) and Bs (valid at T50.2Ts).4. For T>Ts

5. For T>0.2Ts ; B51.0 at T50.0; values of B for 0,T,0.2Ts can be obtained by lin-ear interpolation.

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 535

ratios up to 100% of critical are calculated and compared with the values presented inFEMA 273. The procedure followed for obtaining these coefficients is described below.

METHODOLOGY FOR ESTABLISHING VALUES OF THE DAMPINGCOEFFICIENT

Values of the damping coefficient, B, for a particular period can be obtained as theratio of the 5%-damped design spectral acceleration to the average spectral accelerationfor a different damping ratio, b, by re-organizing Equation 1:

B5Sa~T,5%!

Sa~T,b!(2)

Linear response-history analysis was used to obtain the spectral accelerations,Sa(T,b). Twenty horizontal components of ten earthquake history sets were selected forthe analysis. Each of these sets were associated with earthquakes with a magnitudelarger than 6.5, an epicentral distance between 10 and 20 km, and site conditions char-acterized by Site Class C to D in accordance with the 2000 NEHRP Provisions. The ap-plicable design response spectrum, which represents the target design-response spec-trum, had parameters SDS51.0, SD150.6, and Ts50.6 second. The 20 horizontalcomponents were amplitude scaled using a process that preserved the frequency contentof the histories and ensured an equal contribution of each record to the average responsespectrum (Tsopelas et al. 1997). The 10 sets of earthquake histories are presented inTable 2 together with their scale factors. No near-field or soft-soil histories were in-cluded in the set of 20 records and as such the results presented below may not be validfor such histories.

Figure 1 shows that the average response spectrum of the 20 scaled motions repre-sents well the target NEHRP design-response spectrum. The maximum and minimum

Figure 1. Maximum, average, and minimum spectral acceleration values of scaled motions.

536 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

spectral acceleration values of the 20 scaled motions that are also shown in Figure 1demonstrate the variability in the characteristics of the scaled motions, which is implicitin the definition of seismic hazard.

Average relationships between the damping coefficient, B, and the period, were es-tablished using the results of the response-history analysis, and are shown in Figure 2.Using these data, the trilinear relationship of Figure 3 is proposed, for which the damp-ing coefficient is constant in the constant velocity segment of the spectrum and reducesin value to 1.0 at zero period. The proposed model requires three parameters for eachdamping value, Bs , B1 , and Ts .

Best-fit values of the damping coefficient for damping ratios in the range of 2 to 100percent were obtained. The values of Ramirez et al. for Bs and B1 based on the best fit ofthe calculated damping coefficient are generally smaller and larger, respectively, than thevalues given in FEMA 273. Accordingly, the values for Bs recommended by Ramirez

Figure 2. Calculated damping coefficient as a function of period.

Table 2. Earthquake histories used in the analysis and scale factors

Year Earthquake Station Components Scale Factor

1949 Washington 325 (USGS) N04W, N86E 2.741954 Eureka 022 (USGS) N11W, N79E 1.741971 San Fernando 241 (USGS) N00W, S90W 1.961971 San Fernando 458 (USGS) S00W, S90W 2.221989 Loma Prieta Gilroy 2 (CDMG) 90,0 1.461989 Loma Prieta Hollister (CDMG) 90,0 1.071992 Landers Yermo (CDMG) 360,270 1.281992 Landers Joshua (CDMG) 90,0 1.481994 Northridge Moorpark (CDMG) 180,90 2.611994 Northridge Century (CDMG) 90,360 2.27

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 537

et al. that are presented in column 4 of Table 1 are those of the study (and significantlysmaller than those of FEMA 273 in column 2). The values of B1 recommended byRamirez et al. that are presented in column 5 of Table 1 are slightly smaller than thevalues established in the study but were set equal to the values of FEMA 273 to elimi-nate confusion with only little conservatism. For inclusion in the 2000 NEHRP Provi-sions, the presentation of the damping coefficient was further simplified with little lossof accuracy to a two-parameter model as characterized using the data in the last columnof Table 1.

EFFECT OF DAMPING ON INELASTIC DISPLACEMENTS

The ratio of maximum inelastic displacement to maximum elastic displacement cal-culated was introduced in FEMA 273 as coefficient C1 to facilitate the calculation ofdisplacements in short-period yielding structures.

Mander et al. (1984), Riddell et al. (1989), Nassar and Krawinkler (1991), Vidicet al. (1992), and Miranda (1993, 2001) considered the problem of deriving simple ex-pressions for coefficient C1 . Typically, these studies proposed expressions relating eitherthe coefficient C1 to the ductility-based portion of the R factor and the elastic period, orthe ductility-based portion of the R factor to the ductility ratio and the elastic period.Key parameters such as the ratio of post-elastic stiffness to elastic stiffness and period Ts

also appear in these expressions. With the exceptions of Newmark and Hall (1973) andRiddell and Newmark (1979), none of the expressions considered the effect of viscousdamping. Unfortunately, the relationships of Newmark and Riddell are too complex tobe inverted to obtain expressions for C1 .

Miranda and Bertero (1994) presented a comprehensive evaluation of the studies onthe ductility-based portion of the R factor. The data of Miranda and Bertero were used inpart to establish expressions for coefficient C1 in FEMA 273. Equation 3 below presentsthe expression for C1 that is used in FEMA 273. Mander et al. (1984) proposed a rela-tionship of this form with the only difference being a different definition for Ts .

Figure 3. Proposed relationship between damping coefficient and period.

538 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

C151

RmF11~Rm21!STs

T DG>1.0 for Te<Ts

(3)51.0 for Te>Ts

Consider now Figure 4 that illustrates the idealized behavior of a single-degree-of-freedom oscillator. Under elastic conditions, the seismic demand consists of peak forceVe and peak displacement De . Under inelastic conditions, the peak displacement is Di .By definition

C15Di

De5

m•Dy

De5

m

Rm(4)

where

m5Di

Dy(5)

is the displacement ductility ratio, and

Rm5Ve

Vy5

De

Dy(6)

is the ratio of the required elastic strength, Ve , to the yield strength, Vy , and representsthe ductility-based portion of the R factor or the ductility factor per ATC 19 (ATC 1995).For this study, Te is the elastic period (based on stiffness Ke), a is the ratio of post-elasticstiffness to elastic stiffness and Dy is the yield displacement.

Of use in the analysis of structures with viscous damping systems is a calibrated re-lationship between the coefficient C1 and the post-elastic to elastic stiffness ratio, a, inthe practical range of 0 to 0.5, the elastic period, Te , the viscous damping ratio underelastic conditions, bv , in the range of 0.05 to 0.30, and the period Ts that in part char-acterizes the design response spectrum. Such a relation does not exist in the literature

Figure 4. Elastic and idealized inelastic behavior of a single-degree-of-freedom structure.

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 539

but has been established (Ramirez et al. 2000, 2002) using the exhaustive database ofnonlinear response-history analysis results for yielding structures with damping systems.

DEVELOPMENT OF EQUATION FOR COEFFICIENT C1

To establish the relationship for C1 as a function of the variables identified above,nonlinear analysis was performed on bilinear hysteretic systems with linear viscousdamping. The ranges of the variables studied were as follows:

1. 0.2<Te<3.0 seconds in steps of 0.1 second

2. a50.05, 0.15, 0.25, 0.50, 1.0 (where 1.0 is elastic behavior)

3. Rm52.0, 3.33, 5.0

4. Linear viscous damping under elastic conditions bv50.0, 0.15, 0.25, whenadded to inherent damping of 0.05 gave a total viscous damping ratio underelastic conditions of 0.05, 0.20, and 0.30.

The ductility factor of Equation 6 was calculated as

Rm5mSa~Te ,b50.05!

VyB(7)

where m is the mass of the system, Sa is the spectral acceleration for 5% damping, andB is the damping coefficient for the total damping ratio. The trilinear model for thedamping coefficient (columns 4 and 5 of Table 1) was used to calculate the yieldstrength of the analyzed models.

Values for the coefficient C1 were obtained from response-history analysis (using thehistories of Table 2) as the ratio of the average peak inelastic displacement to the averagepeak elastic displacement. Plots of this coefficient versus period Te revealed the basicnature of the relation. Moreover, since C1 should converge to unity when either Rm or ais equal to 1.0, the following relation was proposed:

C1511~12a!~Rm21!

RmSa Ts

TeDb

(8)

where parameters a and b were obtained by calibration of the model using the results ofdynamic analysis. The simplest form of these parameters was found to be

a5a0~210.45Rm! (9)

Table 3. Values of parameter a0

a

0.05 0.15 0.25 0.50

bv50.05 0.116 0.100 0.093 0.071bv50.30 0.195 0.160 0.143 0.111

540 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

b53.2420.10Rm24.5bv (10)

where values of a0 are given in Table 3 below. Linear interpolation can be used to esti-mate intermediate values of a0 .

Figures 5 through 7 compare values of coefficient C1 obtained by nonlinearresponse-history analysis to the predictions of the model given by Equation 8. Theresponse-history results presented in Figures 5 through 7 are the average of the results ofanalyses using the 20 earthquake histories of Table 2. The relation of Equation 8 de-

Figure 5. Values of C1 for a total viscous damping ratio of 0.05.

Figure 6. Values of C1 for a total viscous damping ratio of 0.20.

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 541

scribes well the calculated values of the coefficient and it follows the desired behaviorfor large values of period Te , where Equation 8 predicts a value near 1.0 for Te>Ts . Thisrelation can be used to estimate displacement demands in structures with damping sys-tems without the need to use iterative methods of analysis. One interesting observationfrom Figures 5 through 7 is that the effect of viscous damping on C1 in the range of 5 to30 percent is not significant. Similar results were observed for the other values of bv .

DUCTILITY DEMANDS IN STRUCTURESWITH VISCOUS DAMPING SYSTEMS

A building frame without a damping system would typically be designed for code-prescribed lateral loads equal to the elastic inertia forces divided by a response modifi-cation factor that is denoted as R in U.S. practice. Such a frame, if assumed to be elas-toplastic will have a yield strength Vy given by

Vy5Ve~b50.05!

Rm(11)

where Rm is the ductility-based portion of the R factor. In Equation 11, the required elas-tic strength Ve is calculated assuming a damping ratio of 5%. A frame designed usingEquation 11 will undergo inelastic deformations when subjected to a design earthquake.

Consider now a frame with a viscous damping system that is designed using a simi-lar approach. The frame is designed to have a yield strength Vyd given by

Vyd5Ve~b50.05!

RmB(12)

where B is the damping coefficient for the viscous damping ratio of the structure under

Figure 7. Values of C1 for a total viscous damping ratio of 0.30.

542 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

elastic conditions (bv). Note that the ratio Ve /B in Equation 12 is the base shear of thedamped frame calculated assuming that the frame is elastic with damping ratio equal tobv . If bv is equal to 0.20 (0.05 inherent plus 0.15 added viscous damping), then B isequal to 1.5 (see the last column of Table 1). Accordingly, for the same value of Rm , thestrength of the damped frame, Vyd , will be substantially less than that of the undampedframe, Vy , or for this example, Vyd50.67Vy . This observation raises one important ques-tion, namely, whether the two frames will have comparable displacement ductility de-mands, considering that the damped frame is less stiff than the undamped frame. Thenext section of the paper presents the results of a systematic study that answered thisquestion.

EVALUATION OF DISPLACEMENT DUCTILITY DEMANDS

Bilinear hysteretic single-degree-of-freedom (SDOF) systems without and withsupplemental linear viscous damping were considered in the study. Each system withoutsupplemental damping was characterized by the elastic period Te , ductility factor Rm ,ratio of post-elastic stiffness to elastic stiffness, a, and an inherent viscous damping ra-tio, bI , of 0.05. Values of a50.05, 0.15, 0.25, 0.50, Rm52.0, 3.33, 5.0, and Te between0.2 and 2.0 seconds were considered in the study. Such a range encompasses all practicalvalues for ductile framing systems.

Each bilinear system with supplemental damping was characterized by the same pa-rameters noted above, a value of the elastic period Ted that was larger than Te , andsupplemental linear viscous damping bv50.15, 0.25 under elastic conditions. Accord-ingly, the total damping ratio under elastic conditions was either 0.20 or 0.30. Thedamped SDOF systems (that is, those with supplemental damping) had lower yieldstrengths than the corresponding undamped systems (that is, those with no supplementaldamping) as indicated by Equations 11 and 12.

The elastic period Ted of the damped system was related to the period Te of the cor-responding undamped system as follows:

Ted

Te5Bh (13)

where h is a parameter that is dependent on the period range under consideration and thegeometry of the beam and column sections in the frame. Note again that the strength(and thus stiffness) of the damped system is substantially smaller than that of the un-damped system per Equation 12. Different values of h were established for cases whereboth Te and Ted fell in either (a) the constant acceleration segment of the spectrum, thatis, T<Ts , or (b) the constant velocity segment of the spectrum, that is, T>Ts . For case(a), Vy /Vyd5B and h was approximately 0.75 and 0.55 for rectangular and wide-flangecross section shapes, respectively. For case (b), Vy /Vyd5B11h and h was of the order of1 for both rectangular and wide-flange cross section shapes.

Accordingly, analyses were performed using Equation 13 to calculate the period ofthe damped structure with h50.5 when Te<Ts and h51.0 when Te.Ts . The frameswere designed to have a yield strength given by Equations 11 through 13:

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 543

Vyd

Vy5

1

Bfor Te ,Ted,Ts

(14)5

1

B2 for Te ,Ted>Ts

Figure 8. Comparison of average displacement ductility ratio for 5%- and 20%-dampedsystems.

Figure 9. Comparison of average displacement ductility ratio for 5%- and 30%-damped sys-tems.

544 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

that is, the yield strength of the damped systems varied between 0.35 and 0.67 times thestrength of the corresponding undamped systems.

Response-history analyses were performed using the 20 scaled earthquake historieslisted in Table 2 to compare the displacement ductility demands in the undamped anddamped frames. Figures 8 and 9 compare the calculated average displacement ductilityratio for the undamped and the damped systems, where the average is that of the 20calculated values for each combination of parameters. The ductility demand in the twosystems is nearly identical. Figure 10 presents a comparison of the displacement ductil-ity ratio of the undamped and the 20%-damped systems for a50.05. In this figure, theaverage, maximum and minimum displacement ductility ratios for the 20 calculated val-ues are presented; the values are nearly identical for the same Rm and a. The data ofFigure 10 also reveal the possible scatter in the ductility demand in a design earthquake.Interestingly, the scatter in the ductility demand in the damped systems is similar to thatin the corresponding undamped systems, further supporting the approach of Equation12.

On the basis of the results presented above and those of Wu and Hanson (1989), the2000 NEHRP Provisions permit the design of building structures with damping systemsfor a lower minimum base shear than that for the corresponding undamped structure.The minimum base shear for damped structures is the greater of V/B or 0.75V, where Vis the minimum seismic base shear for the design of the undamped structure and B is thedamping coefficient for the combined inherent and viscous damping under elastic con-ditions.

Figure 10. Comparison of maximum, average and minimum displacement ductility ratios of5%- and 20%-damped systems with a50.05.

ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 545

SUMMARY AND CONCLUSIONS

Studies were undertaken to support the development of the 2000 NEHRP Provisionsfor the design of buildings with energy dissipation systems. The numerical simulationsmade use of 20 earthquake histories that were scaled to match on average the 2000NEHRP spectrum for SDS51.0, S150.6, and Ts50.6 second. Near-field and soft-soilearthquake histories were not included in the set of 20 records and as such the conclu-sions presented below should not necessarily be directly applied to such conditions. Thekey products and conclusions of this study are as follows.

1. New values for the damping coefficient that are used to calculate spectral ac-celerations for damping different from 5% have been established. The values arevalid for viscous damping ratio in the range of 2% to 100% of critical. The val-ues for B derived in this study were adopted, after simplification, in the 2000NEHRP Provisions.

2. A new relationship describing the ratio of peak inelastic displacement to thepeak elastic displacement for systems with supplemental viscous damping hasbeen presented. This relationship can be used to directly estimate displacementdemands in short-period yielding structures with damping systems. The effectof viscous damping in the range of 5% to 30% on the value of C1 is not sig-nificant.

3. The design base shear for damped buildings can be reduced from that for theundamped frame, without increasing the displacement ductility demand on theframe. This conclusion is now reflected in the 2000 NEHRP Provisions where adamped frame can be designed for a base shear strength that is up to 25% lessthan the corresponding undamped frame. This result is a paradigm shift fromprevious guidelines that required the damped frame to be designed for the samebase shear as the corresponding undamped frame.

ACKNOWLEDGMENTS

Financial support for this project was provided by the Multidisciplinary Center forEarthquake Engineering Research, Task on Rehabilitation Strategies for Buildings(Projects No. 982403, No. 992403, and No. 00-2042). The work was performed underthe direction of Technical Subcommittee 12, Base Isolation and Energy Dissipation, ofthe Building Seismic Safety Council, which was tasked with developing analysis anddesign procedures for inclusion in the 2000 edition of the NEHRP Recommended Pro-visions for Seismic Regulations for New Buildings and Other Structures. The work ofTechnical Subcommittee 12 was supported by the Federal Emergency ManagementAgency.

REFERENCES

Applied Technology Council (ATC), 1995. Structural Response Modification Factors, ReportNo. ATC 19, Redwood City, CA.

ATC, 1997. NEHRP Guidelines for the Seismic Rehabilitation of Buildings and NEHRP Com-mentary on the Guidelines for the Seismic Rehabilitation of Buildings, Report Nos. FEMA

546 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

273 and 274, prepared for the Building Seismic Safety Council, published by the FederalEmergency Management Agency, Washington, DC.

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(Received 28 November 2001; accepted 28 May 2002)