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Elastic and Inelastic Seismic Response of 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. Chrysostomou e) 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 displacement reduction factors for levels of damping greater than 5% of critical are pre- sented. New equations to relate inelastic and elastic displacements in the short-period range, for levels of damping greater than 5% of critical, are pre- sented. The technical basis for reducing the minimum design base shear in damped 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 design and maximum earthquake shaking. For the commonly used special moment-resisting frame, 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. Inelastic action 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 indirect losses. 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 United States and abroad. Soong and Dargush (1997), Constantinou et al. (1998), and Hanson and 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 94303 e) Lecturer, Dept. of Civ. Engrg., Higher Technical Institute, 2152 Nicosia, Cyprus 531 Earthquake Spectra , Volume 18, No. 3, pages 531547, August 2002; 2002, Earthquake Engineering Research Institute

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energy dissipation during an earthquake into disposable elements specifically designed for the purpose of dissipating energy, and to substantially reduce or eliminate energy dissipation 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 an earthquake without compromising the structural integrity of the frame. One impediment to the widespread use of passive energy dissipation systems has been the lack of robust and validated guidelines for the modeling, analysis and design of energy dissipation systems, and testing of damping devices. Considerable research effort in the 1990s resulted in the development of at least five code-oriented procedures related to the implementation of passive energy dissipation systems. The Structural Engineers Association of Northern California (SEAONC) published the first procedures in 1992 (Whittaker et al. 1993). The Federal Emergency Management Agency (FEMA) included draft guidelines for the implementation of passive energy dissipation devices in new buildings in the 1994 edition of the NEHRP Recommended Provisions for the Seismic Regulations for New Buildings (BSSC 1994). Guidelines for the implementation of pas- sive energy dissipation devices in retrofit construction were published in 1997 in the FEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings (ATC 1997). In 1999, the SEAOC Ad Hoc Committee on Energy Dissipation published guidelines for implementing 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 Provisions for Seismic Regulations for New Buildings and Other Structures (BSSC 2001), in which completely revised procedures for implementing passive energy dissipation devices in new buildings are outlined and robust linear procedures (equivalent lateral force and response-spectrum methods) for analysis are described. The development and verifica- tion of the analysis methods for buildings with damping systems in the 2000 NEHRP Recommended Provisions (hereafter termed the 2000 NEHRP Provisions ) are the result of 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 systems were written around a number of significant simplifications and limits, some of which are outlined below: 1. A multi-degree-of-freedom (MDOF) building with a damping system can be transformed into equivalent single-degree-of-freedom (SDOF) systems using modal 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 estimated using equivalent linear properties and a 5% damped response spectrum. Spectra for damping greater than 5% can be established using damping coefficients, and velocity-dependent forces can be established using either pseudo-velocity and modal information or by applying correction factors to the pseudo velocity. 532 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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3. The minimum design base shear for buildings with damping systems is less than 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 2000 NEHRP 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 damped buildings, and (c) a comparison of ductility demands in structures without and with damping systems, where the damped buildings are designed for a smaller base shear than conventional buildings. The companion paper (Ramirez et al. 2002) describes the simplified method of analysis for single-degree-of-freedom structures with linear viscous, nonlinear viscous and hysteretic damping systems, presents methods to calculate maximum velocity and maximum 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 response spectra 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 analysis of structures with damping systems. Elastic spectra constructed for levels of viscous damping 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 analysis of yielding structures because these methods facilitate the direct evaluation of inelastic response 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 damping other than 5%, ), is to divide the 5%-damped spectral acceleration by a damping coefficient that is a function of the damping ratio, , namely, ,5% (1) where 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 that were based on the work of Newmark and Hall (1982) but were extended to higher values of the damping ratio. The extension of the work by Newmark and Hall to higher values of the damping ratio was necessary for two reasons. First, the simplified methods of analysis 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, the values assigned to in FEMA 273 are presented in Table 1.) Second, under certain con- ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 533

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ditions the damping ratio in higher modes may be very large and could reach critical or overcritical values in buildings having a complete vertical distribution of viscous energy dissipation devices. Three of the shortcomings of the values assigned to in FEMA 273 are as follows: 1. The values of in the constant acceleration region of the spectrum in Figure 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 displacement with increased damping in very stiff structures, and leads the user to the erro- neous conclusion that damping systems are most effective when used in stiff structures. 2. The effect of damping beyond 50% of critical is ignored leading to conservative estimates of displacement in highly damped buildings, which may be the case for yielding frames equipped with supplemental viscous damping systems. 3. Limiting the damping coefficient to 2.0 for and 50% results in a con- servative estimate of the maximum velocity, which is of great significance in determining forces in viscous dampers. In the study reported in this paper, values for the damping coefficient for damping Table 1. Values of damping coefficient Effective Damping FEMA 273 Ramirez al. (2000) 2000 NEHRP Provisions 0.02 0.8 0.8 0.80 0.80 0.8 0.05 1.0 1.0 1.00 1.00 1.0 0.10 1.3 1.2 1.20 1.20 1.2 0.20 1.8 1.5 1.50 1.50 1.5 0.30 2.3 1.7 1.70 1.70 1.8 0.40 2.7 1.9 1.90 1.90 2.1 0.50 3.0 2.0 2.20 2.20 2.4 0.60 3.0 2.0 2.30 2.60 2.7 0.70 3.0 2.0 2.35 2.90 3.0 0.80 3.0 2.0 2.40 3.30 3.3 0.90 3.0 2.0 2.45 3.70 3.6 1.00 3.0 2.0 2.50 4.00 4.0 1. For is the corner point in the spectrum per Figure 1; see the 2000 NEHRP Provisions for definition of terms. 2. For 3. Valid at 0.2 ; for 0.2 is determined by linear interpolation between values and ; for 0.2 is determined by linear interpolation between values of 1.0 (at 0.0) and (valid at 0.2 ). 4. For 5. For 0.2 1.0 at 0.0; values of for 0 0.2 can be obtained by lin- ear interpolation. 534 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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ratios up to 100% of critical are calculated and compared with the values presented in FEMA 273 . The procedure followed for obtaining these coefficients is described below. METHODOLOGY FOR ESTABLISHING VALUES OF THE DAMPING COEFFICIENT Values of the damping coefficient, for a particular period can be obtained as the ratio of the 5%-damped design spectral acceleration to the average spectral acceleration for a different damping ratio, , by re-organizing Equation 1: ,5% (2) Linear response-history analysis was used to obtain the spectral accelerations, ). Twenty horizontal components of ten earthquake history sets were selected for the analysis. Each of these sets were associated with earthquakes with a magnitude larger 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 DS 1.0, 0.6, and 0.6 second. The 20 horizontal components were amplitude scaled using a process that preserved the frequency content of the histories and ensured an equal contribution of each record to the average response spectrum (Tsopelas et al. 1997). The 10 sets of earthquake histories are presented in Table 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 valid for 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. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 535

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spectral acceleration values of the 20 scaled motions that are also shown in Figure 1 demonstrate the variability in the characteristics of the scaled motions, which is implicit in the definition of seismic hazard. Average relationships between the damping coefficient, 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 reduces in value to 1.0 at zero period. The proposed model requires three parameters for each damping value, and Best-fit values of the damping coefficient for damping ratios in the range of 2 to 100 percent were obtained. The values of Ramirez et al. for and based on the best fit of the calculated damping coefficient are generally smaller and larger, respectively, than the values given in FEMA 273 . Accordingly, the values for 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 2.74 1954 Eureka 022 (USGS) N11W, N79E 1.74 1971 San Fernando 241 (USGS) N00W, S90W 1.96 1971 San Fernando 458 (USGS) S00W, S90W 2.22 1989 Loma Prieta Gilroy 2 (CDMG) 90,0 1.46 1989 Loma Prieta Hollister (CDMG) 90,0 1.07 1992 Landers Yermo (CDMG) 360,270 1.28 1992 Landers Joshua (CDMG) 90,0 1.48 1994 Northridge Moorpark (CDMG) 180,90 2.61 1994 Northridge Century (CDMG) 90,360 2.27 536 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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et al. that are presented in column 4 of Table 1 are those of the study (and significantly smaller than those of FEMA 273 in column 2). The values of recommended by Ramirez et al. that are presented in column 5 of Table 1 are slightly smaller than the values 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 loss of accuracy to a two-parameter model as characterized using the data in the last column of 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 to facilitate the calculation of displacements in short-period yielding structures. Mander et al. (1984), Riddell et al. (1989), Nassar and Krawinkler (1991), Vidic et al. (1992), and Miranda (1993, 2001) considered the problem of deriving simple ex- pressions for coefficient Typically, these studies proposed expressions relating either the coefficient to the ductility-based portion of the factor and the elastic period, or the ductility-based portion of the factor to the ductility ratio and the elastic period. Key parameters such as the ratio of post-elastic stiffness to elastic stiffness and period also appear in these expressions. With the exceptions of Newmark and Hall (1973) and Riddell and Newmark (1979), none of the expressions considered the effect of viscous damping. Unfortunately, the relationships of Newmark and Riddell are too complex to be inverted to obtain expressions for Miranda and Bertero (1994) presented a comprehensive evaluation of the studies on the ductility-based portion of the factor. The data of Miranda and Bertero were used in part to establish expressions for coefficient in FEMA 273 . Equation 3 below presents the expression for 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 Figure 3. Proposed relationship between damping coefficient and period. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 537

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1.0 for (3) 1.0 for 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 force and peak displacement Under inelastic conditions, the peak displacement is By definition (4) where (5) is the displacement ductility ratio, and (6) is the ratio of the required elastic strength, to the yield strength, and represents the ductility-based portion of the factor or the ductility factor per ATC 19 (ATC 1995). For this study, is the elastic period (based on stiffness is the ratio of post-elastic stiffness to elastic stiffness and is the yield displacement. Of use in the analysis of structures with viscous damping systems is a calibrated re- lationship between the coefficient and the post-elastic to elastic stiffness ratio, ,in the practical range of 0 to 0.5, the elastic period, the viscous damping ratio under elastic conditions, in the range of 0.05 to 0.30, and the period 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. 538 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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but has been established (Ramirez et al. 2000, 2002) using the exhaustive database of nonlinear response-history analysis results for yielding structures with damping systems. DEVELOPMENT OF EQUATION FOR COEFFICIENT To establish the relationship for as a function of the variables identified above, nonlinear analysis was performed on bilinear hysteretic systems with linear viscous damping. The ranges of the variables studied were as follows: 1. 0.2 3.0 seconds in steps of 0.1 second 2. 0.05, 0.15, 0.25, 0.50, 1.0 (where 1.0 is elastic behavior) 3. 2.0, 3.33, 5.0 4. Linear viscous damping under elastic conditions 0.0, 0.15, 0.25, when added to inherent damping of 0.05 gave a total viscous damping ratio under elastic conditions of 0.05, 0.20, and 0.30. The ductility factor of Equation 6 was calculated as mS 0.05 (7) where is the mass of the system, is the spectral acceleration for 5% damping, and is the damping coefficient for the total damping ratio. The trilinear model for the damping coefficient (columns 4 and 5 of Table 1) was used to calculate the yield strength of the analyzed models. Values for the coefficient were obtained from response-history analysis (using the histories of Table 2) as the ratio of the average peak inelastic displacement to the average peak elastic displacement. Plots of this coefficient versus period revealed the basic nature of the relation. Moreover, since should converge to unity when either or is equal to 1.0, the following relation was proposed: (8) where parameters and were obtained by calibration of the model using the results of dynamic analysis. The simplest form of these parameters was found to be 0.45 (9) Table 3. Values of parameter 0.05 0.15 0.25 0.50 0.05 0.116 0.100 0.093 0.071 0.30 0.195 0.160 0.143 0.111 ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 539

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3.24 0.10 4.5 (10) where values of are given in Table 3 below. Linear interpolation can be used to esti- mate intermediate values of Figures 5 through 7 compare values of coefficient obtained by nonlinear response-history analysis to the predictions of the model given by Equation 8. The response-history results presented in Figures 5 through 7 are the average of the results of analyses using the 20 earthquake histories of Table 2. The relation of Equation 8 de- Figure 5. Values of for a total viscous damping ratio of 0.05. Figure 6. Values of for a total viscous damping ratio of 0.20. 540 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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scribes well the calculated values of the coefficient and it follows the desired behavior for large values of period where Equation 8 predicts a value near 1.0 for This relation can be used to estimate displacement demands in structures with damping sys- tems without the need to use iterative methods of analysis. One interesting observation from Figures 5 through 7 is that the effect of viscous damping on in the range of 5 to 30 percent is not significant. Similar results were observed for the other values of DUCTILITY DEMANDS IN STRUCTURES WITH 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 in U.S. practice. Such a frame, if assumed to be elas- toplastic will have a yield strength given by 0.05 (11) where is the ductility-based portion of the factor. In Equation 11, the required elas- tic strength is calculated assuming a damping ratio of 5%. A frame designed using Equation 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 given by 0.05 (12) where is the damping coefficient for the viscous damping ratio of the structure under Figure 7. Values of for a total viscous damping ratio of 0.30. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 541

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elastic conditions ). Note that the ratio in Equation 12 is the base shear of the damped frame calculated assuming that the frame is elastic with damping ratio equal to If is equal to 0.20 (0.05 inherent plus 0.15 added viscous damping), then is equal to 1.5 (see the last column of Table 1). Accordingly, for the same value of the strength of the damped frame, will be substantially less than that of the undamped frame, or for this example, 0.67 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. The next section of the paper presents the results of a systematic study that answered this question. EVALUATION OF DISPLACEMENT DUCTILITY DEMANDS Bilinear hysteretic single-degree-of-freedom (SDOF) systems without and with supplemental linear viscous damping were considered in the study. Each system without supplemental damping was characterized by the elastic period ductility factor ratio of post-elastic stiffness to elastic stiffness, , and an inherent viscous damping ra- tio, of 0.05. Values of 0.05, 0.15, 0.25, 0.50, 2.0, 3.33, 5.0, and between 0.2 and 2.0 seconds were considered in the study. Such a range encompasses all practical values 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 ed that was larger than and supplemental linear viscous damping 0.15, 0.25 under elastic conditions. Accord- ingly, the total damping ratio under elastic conditions was either 0.20 or 0.30. The damped SDOF systems (that is, those with supplemental damping) had lower yield strengths than the corresponding undamped systems (that is, those with no supplemental damping) as indicated by Equations 11 and 12. The elastic period ed of the damped system was related to the period of the cor- responding undamped system as follows: ed (13) where is a parameter that is dependent on the period range under consideration and the geometry 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 were established for cases where both and ed fell in either (a) the constant acceleration segment of the spectrum, that is, or (b) the constant velocity segment of the spectrum, that is, For case (a), and was approximately 0.75 and 0.55 for rectangular and wide-flange cross section shapes, respectively. For case (b), and was of the order of 1 for both rectangular and wide-flange cross section shapes. Accordingly, analyses were performed using Equation 13 to calculate the period of the damped structure with 0.5 when and 1.0 when The frames were designed to have a yield strength given by Equations 11 through 13: 542 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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for ed (14) for ed Figure 8. Comparison of average displacement ductility ratio for 5%- and 20%-damped systems. Figure 9. Comparison of average displacement ductility ratio for 5%- and 30%-damped sys- tems. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 543

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that is, the yield strength of the damped systems varied between 0.35 and 0.67 times the strength of the corresponding undamped systems. Response-history analyses were performed using the 20 scaled earthquake histories listed in Table 2 to compare the displacement ductility demands in the undamped and damped frames. Figures 8 and 9 compare the calculated average displacement ductility ratio for the undamped and the damped systems, where the average is that of the 20 calculated values for each combination of parameters. The ductility demand in the two systems is nearly identical. Figure 10 presents a comparison of the displacement ductil- ity ratio of the undamped and the 20%-damped systems for 0.05. In this figure, the average, maximum and minimum displacement ductility ratios for the 20 calculated val- ues are presented; the values are nearly identical for the same and . The data of Figure 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 that in the corresponding undamped systems, further supporting the approach of Equation 12. On the basis of the results presented above and those of Wu and Hanson (1989), the 2000 NEHRP Provisions permit the design of building structures with damping systems for a lower minimum base shear than that for the corresponding undamped structure. The minimum base shear for damped structures is the greater of or 0.75 where is the minimum seismic base shear for the design of the undamped structure and is the damping coefficient for the combined inherent and viscous damping under elastic con- ditions. Figure 10. Comparison of maximum, average and minimum displacement ductility ratios of 5%- and 20%-damped systems with 0.05. 544 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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SUMMARY AND CONCLUSIONS Studies were undertaken to support the development of the 2000 NEHRP Provisions for the design of buildings with energy dissipation systems. The numerical simulations made use of 20 earthquake histories that were scaled to match on average the 2000 NEHRP spectrum for DS 1.0, 0.6, and 0.6 second. Near-field and soft-soil earthquake 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. The key 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 are valid for viscous damping ratio in the range of 2% to 100% of critical. The val- ues for derived in this study were adopted, after simplification, in the 2000 NEHRP Provisions 2. A new relationship describing the ratio of peak inelastic displacement to the peak elastic displacement for systems with supplemental viscous damping has been presented. This relationship can be used to directly estimate displacement demands in short-period yielding structures with damping systems. The effect of viscous damping in the range of 5% to 30% on the value of is not sig- nificant. 3. The design base shear for damped buildings can be reduced from that for the undamped frame, without increasing the displacement ductility demand on the frame. This conclusion is now reflected in the 2000 NEHRP Provisions where a damped frame can be designed for a base shear strength that is up to 25% less than the corresponding undamped frame. This result is a paradigm shift from previous guidelines that required the damped frame to be designed for the same base shear as the corresponding undamped frame. ACKNOWLEDGMENTS Financial support for this project was provided by the Multidisciplinary Center for Earthquake Engineering Research, Task on Rehabilitation Strategies for Buildings (Projects No. 982403, No. 992403, and No. 00-2042). The work was performed under the direction of Technical Subcommittee 12, Base Isolation and Energy Dissipation, of the Building Seismic Safety Council, which was tasked with developing analysis and design procedures for inclusion in the 2000 edition of the NEHRP Recommended Pro- visions for Seismic Regulations for New Buildings and Other Structures . The work of Technical Subcommittee 12 was supported by the Federal Emergency Management Agency. REFERENCES Applied Technology Council (ATC), 1995. Structural Response Modification Factors , Report No. 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 ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 545

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273 and 274 , prepared for the Building Seismic Safety Council, published by the Federal Emergency Management Agency, Washington, DC. Constantinou, M. C., Soong, T. T., and Dargush, G. F., 1998. Passive Energy Dissipation Sys- tems for Structural Design and Retrofit , Monograph Series No. 1, Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, State University of New York at Buffalo, Buffalo, NY. Building Seismic Safety Council (BSSC), 1995. NEHRP Recommended Provisions for the Seis- mic Regulations for New Buildings , 1994 Edition, Report No. FEMA 222A , developed for the Federal Emergency Management Agency, Washington, DC. BSSC, 2001. NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures , 2000 Edition, Report Nos. FEMA 368 and 369 , Federal Emergency Man- agement Agency, Washington, DC. Hanson, R. D., and Soong, T. T., 2001. Seismic Design with Supplemental Energy Dissipation Devices , Earthquake Engineering Research Institute, Oakland, CA, 135 pp. International Conference of Building Officials (ICBO), 1997. Uniform Building Code , Vol. 2Structural Engineering Design Provisions, Whittier, CA. Mander, J. B., Priestley, M. J. N., and Park, R., 1984, Seismic Design of Bridge Piers , Report 84-2, Department of Civil Engineering, University of Canterbury, New Zealand. Miranda, E., 1993. Site-dependent strength reduction factors, J. Struct. Eng. 119 (12), 3503 3519. Miranda, E., 2001. Estimation of inelastic deformation demands of SDOF systems, J. Struct. Eng. 127 (9), 10051020. Miranda, E., and Bertero, V. V., 1994. Evaluation of strength reduction factors for earthquake- resistant design, Earthquake Spectra 10 (2), 357379. Nassar, A. A., and Krawinkler, H., 1991. Seismic Demands for SDOF and MDOF Systems , Re- port No. 95, John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA, 204 pp. Newmark, N. M., and Hall, W. J., 1973. Seismic Design Criteria for Nuclear Reactor Facilities Report No. 46, Building Practices for Disaster Mitigation, National Bureau of Standards, U.S. Department of Commerce, pp. 209236. Newmark, N. M., and Hall, W. J., 1982. Earthquake Spectra and Design , Earthquake Engineer- ing Research Institute, Berkeley, CA, 103 pp. Ramirez, O. M., Constantinou, M. C., Kircher, C. A., Whittaker, A. S., Johnson, M. W., Gomez, J. D., and Chrysostomou, C. Z., 2000. Development and evaluation of simplified procedures for analysis and design of buildings with passive energy dissipation systems, Report No. MCEER 00-0010 , Revision 1, Multidisciplinary Center for Earthquake Engineering Re- search, University at Buffalo, State University of New York, Buffalo, NY, 470 pp. Ramirez, O. M., Constantinou, M. C., Gomez, J. D., Whittaker, A. S., and Chrysostomou, C. Z., 2002. Evaluation of simplified methods of analysis of yielding structures with damping sys- tems, Earthquake Spectra 18 (3), 501530 (this issue). Riddell, R., Hidalgo, P., and Cruz, E., 1989. Response modification factors for earthquake re- sistant design of short period structures, Earthquake Spectra (3), 571590. Riddell, R., and Newmark, N. M., 1979. Statistical Analysis of the Response of Nonlinear Sys- tems Subjected to Earthquakes , Structural Research Series No. 468, Department of Civil En- gineering, University of Illinois at Urbana-Champaign, Urbana, Il, 291 pp. 546 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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Structural Engineers Association of California (SEAOC), 1999. Recommended Lateral Force Requirements and Commentary, 7th Edition , Sacramento, CA. Soong, T. T., and Dargush, G. F., 1997. Passive Energy Dissipation Systems in Structural En- gineering , Wiley, & Sons Ltd., London (UK) and New York (USA). Tsopelas, P., Constantinou, M. C., Kircher, C. A., and Whittaker, A. S., 1997. Evaluation of simplified methods of analysis for yielding structures, Report No. NCEER 97-0012 , National Center for Earthquake Engineering Research, University at Buffalo, State University of New York, Buffalo, NY. Vidic, T., Fajfar, P. and Fischinger, M., 1992. A procedure for determining consistent inelastic design spectra, Workshop on Nonlinear Seismic Analysis of RC Structures, Proceedings Bled, Slovenia. Whittaker, A. S., Aiken, I. D., Bergman, D., Clark, P. W., Cohen, J. M., Kelly, J. M., and Scholl, R. E., 1993. Code requirements for the design and implementation of passive energy dissi- pation systems, Proceedings, Seminar on Seismic Isolation, Passive Energy Dissipation, and Active Control, Report No. ATC-17-1 , Applied Technology Council, Redwood City, CA. Wu, J., and Hanson, R. D., 1989. Study of inelastic spectra with high damping, J. Struct. Eng. 115 (6), 14121431. (Received 28 November 2001; accepted 28 May 2002) ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 547

Ramirez a Michael C Constantinou b MEERI Andrew S Whittaker c MEERI Charles A Kircher d MEERI and Christis Z Chrysostomou e The effect of damping on the response of elastic and inelastic single degreeoffreedom systems was studied by nonlinear respo ID: 23380

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Elastic and Inelastic Seismic Response of 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. Chrysostomou e) 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 displacement reduction factors for levels of damping greater than 5% of critical are pre- sented. New equations to relate inelastic and elastic displacements in the short-period range, for levels of damping greater than 5% of critical, are pre- sented. The technical basis for reducing the minimum design base shear in damped 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 design and maximum earthquake shaking. For the commonly used special moment-resisting frame, 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. Inelastic action 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 indirect losses. 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 United States and abroad. Soong and Dargush (1997), Constantinou et al. (1998), and Hanson and 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 94303 e) Lecturer, Dept. of Civ. Engrg., Higher Technical Institute, 2152 Nicosia, Cyprus 531 Earthquake Spectra , Volume 18, No. 3, pages 531547, August 2002; 2002, Earthquake Engineering Research Institute

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energy dissipation during an earthquake into disposable elements specifically designed for the purpose of dissipating energy, and to substantially reduce or eliminate energy dissipation 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 an earthquake without compromising the structural integrity of the frame. One impediment to the widespread use of passive energy dissipation systems has been the lack of robust and validated guidelines for the modeling, analysis and design of energy dissipation systems, and testing of damping devices. Considerable research effort in the 1990s resulted in the development of at least five code-oriented procedures related to the implementation of passive energy dissipation systems. The Structural Engineers Association of Northern California (SEAONC) published the first procedures in 1992 (Whittaker et al. 1993). The Federal Emergency Management Agency (FEMA) included draft guidelines for the implementation of passive energy dissipation devices in new buildings in the 1994 edition of the NEHRP Recommended Provisions for the Seismic Regulations for New Buildings (BSSC 1994). Guidelines for the implementation of pas- sive energy dissipation devices in retrofit construction were published in 1997 in the FEMA 273 NEHRP Guidelines for the Seismic Rehabilitation of Buildings (ATC 1997). In 1999, the SEAOC Ad Hoc Committee on Energy Dissipation published guidelines for implementing 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 Provisions for Seismic Regulations for New Buildings and Other Structures (BSSC 2001), in which completely revised procedures for implementing passive energy dissipation devices in new buildings are outlined and robust linear procedures (equivalent lateral force and response-spectrum methods) for analysis are described. The development and verifica- tion of the analysis methods for buildings with damping systems in the 2000 NEHRP Recommended Provisions (hereafter termed the 2000 NEHRP Provisions ) are the result of 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 systems were written around a number of significant simplifications and limits, some of which are outlined below: 1. A multi-degree-of-freedom (MDOF) building with a damping system can be transformed into equivalent single-degree-of-freedom (SDOF) systems using modal 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 estimated using equivalent linear properties and a 5% damped response spectrum. Spectra for damping greater than 5% can be established using damping coefficients, and velocity-dependent forces can be established using either pseudo-velocity and modal information or by applying correction factors to the pseudo velocity. 532 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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3. The minimum design base shear for buildings with damping systems is less than 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 2000 NEHRP 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 damped buildings, and (c) a comparison of ductility demands in structures without and with damping systems, where the damped buildings are designed for a smaller base shear than conventional buildings. The companion paper (Ramirez et al. 2002) describes the simplified method of analysis for single-degree-of-freedom structures with linear viscous, nonlinear viscous and hysteretic damping systems, presents methods to calculate maximum velocity and maximum 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 response spectra 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 analysis of structures with damping systems. Elastic spectra constructed for levels of viscous damping 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 analysis of yielding structures because these methods facilitate the direct evaluation of inelastic response 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 damping other than 5%, ), is to divide the 5%-damped spectral acceleration by a damping coefficient that is a function of the damping ratio, , namely, ,5% (1) where 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 that were based on the work of Newmark and Hall (1982) but were extended to higher values of the damping ratio. The extension of the work by Newmark and Hall to higher values of the damping ratio was necessary for two reasons. First, the simplified methods of analysis 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, the values assigned to in FEMA 273 are presented in Table 1.) Second, under certain con- ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 533

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ditions the damping ratio in higher modes may be very large and could reach critical or overcritical values in buildings having a complete vertical distribution of viscous energy dissipation devices. Three of the shortcomings of the values assigned to in FEMA 273 are as follows: 1. The values of in the constant acceleration region of the spectrum in Figure 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 displacement with increased damping in very stiff structures, and leads the user to the erro- neous conclusion that damping systems are most effective when used in stiff structures. 2. The effect of damping beyond 50% of critical is ignored leading to conservative estimates of displacement in highly damped buildings, which may be the case for yielding frames equipped with supplemental viscous damping systems. 3. Limiting the damping coefficient to 2.0 for and 50% results in a con- servative estimate of the maximum velocity, which is of great significance in determining forces in viscous dampers. In the study reported in this paper, values for the damping coefficient for damping Table 1. Values of damping coefficient Effective Damping FEMA 273 Ramirez al. (2000) 2000 NEHRP Provisions 0.02 0.8 0.8 0.80 0.80 0.8 0.05 1.0 1.0 1.00 1.00 1.0 0.10 1.3 1.2 1.20 1.20 1.2 0.20 1.8 1.5 1.50 1.50 1.5 0.30 2.3 1.7 1.70 1.70 1.8 0.40 2.7 1.9 1.90 1.90 2.1 0.50 3.0 2.0 2.20 2.20 2.4 0.60 3.0 2.0 2.30 2.60 2.7 0.70 3.0 2.0 2.35 2.90 3.0 0.80 3.0 2.0 2.40 3.30 3.3 0.90 3.0 2.0 2.45 3.70 3.6 1.00 3.0 2.0 2.50 4.00 4.0 1. For is the corner point in the spectrum per Figure 1; see the 2000 NEHRP Provisions for definition of terms. 2. For 3. Valid at 0.2 ; for 0.2 is determined by linear interpolation between values and ; for 0.2 is determined by linear interpolation between values of 1.0 (at 0.0) and (valid at 0.2 ). 4. For 5. For 0.2 1.0 at 0.0; values of for 0 0.2 can be obtained by lin- ear interpolation. 534 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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ratios up to 100% of critical are calculated and compared with the values presented in FEMA 273 . The procedure followed for obtaining these coefficients is described below. METHODOLOGY FOR ESTABLISHING VALUES OF THE DAMPING COEFFICIENT Values of the damping coefficient, for a particular period can be obtained as the ratio of the 5%-damped design spectral acceleration to the average spectral acceleration for a different damping ratio, , by re-organizing Equation 1: ,5% (2) Linear response-history analysis was used to obtain the spectral accelerations, ). Twenty horizontal components of ten earthquake history sets were selected for the analysis. Each of these sets were associated with earthquakes with a magnitude larger 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 DS 1.0, 0.6, and 0.6 second. The 20 horizontal components were amplitude scaled using a process that preserved the frequency content of the histories and ensured an equal contribution of each record to the average response spectrum (Tsopelas et al. 1997). The 10 sets of earthquake histories are presented in Table 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 valid for 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. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 535

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spectral acceleration values of the 20 scaled motions that are also shown in Figure 1 demonstrate the variability in the characteristics of the scaled motions, which is implicit in the definition of seismic hazard. Average relationships between the damping coefficient, 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 reduces in value to 1.0 at zero period. The proposed model requires three parameters for each damping value, and Best-fit values of the damping coefficient for damping ratios in the range of 2 to 100 percent were obtained. The values of Ramirez et al. for and based on the best fit of the calculated damping coefficient are generally smaller and larger, respectively, than the values given in FEMA 273 . Accordingly, the values for 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 2.74 1954 Eureka 022 (USGS) N11W, N79E 1.74 1971 San Fernando 241 (USGS) N00W, S90W 1.96 1971 San Fernando 458 (USGS) S00W, S90W 2.22 1989 Loma Prieta Gilroy 2 (CDMG) 90,0 1.46 1989 Loma Prieta Hollister (CDMG) 90,0 1.07 1992 Landers Yermo (CDMG) 360,270 1.28 1992 Landers Joshua (CDMG) 90,0 1.48 1994 Northridge Moorpark (CDMG) 180,90 2.61 1994 Northridge Century (CDMG) 90,360 2.27 536 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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et al. that are presented in column 4 of Table 1 are those of the study (and significantly smaller than those of FEMA 273 in column 2). The values of recommended by Ramirez et al. that are presented in column 5 of Table 1 are slightly smaller than the values 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 loss of accuracy to a two-parameter model as characterized using the data in the last column of 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 to facilitate the calculation of displacements in short-period yielding structures. Mander et al. (1984), Riddell et al. (1989), Nassar and Krawinkler (1991), Vidic et al. (1992), and Miranda (1993, 2001) considered the problem of deriving simple ex- pressions for coefficient Typically, these studies proposed expressions relating either the coefficient to the ductility-based portion of the factor and the elastic period, or the ductility-based portion of the factor to the ductility ratio and the elastic period. Key parameters such as the ratio of post-elastic stiffness to elastic stiffness and period also appear in these expressions. With the exceptions of Newmark and Hall (1973) and Riddell and Newmark (1979), none of the expressions considered the effect of viscous damping. Unfortunately, the relationships of Newmark and Riddell are too complex to be inverted to obtain expressions for Miranda and Bertero (1994) presented a comprehensive evaluation of the studies on the ductility-based portion of the factor. The data of Miranda and Bertero were used in part to establish expressions for coefficient in FEMA 273 . Equation 3 below presents the expression for 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 Figure 3. Proposed relationship between damping coefficient and period. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 537

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1.0 for (3) 1.0 for 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 force and peak displacement Under inelastic conditions, the peak displacement is By definition (4) where (5) is the displacement ductility ratio, and (6) is the ratio of the required elastic strength, to the yield strength, and represents the ductility-based portion of the factor or the ductility factor per ATC 19 (ATC 1995). For this study, is the elastic period (based on stiffness is the ratio of post-elastic stiffness to elastic stiffness and is the yield displacement. Of use in the analysis of structures with viscous damping systems is a calibrated re- lationship between the coefficient and the post-elastic to elastic stiffness ratio, ,in the practical range of 0 to 0.5, the elastic period, the viscous damping ratio under elastic conditions, in the range of 0.05 to 0.30, and the period 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. 538 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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but has been established (Ramirez et al. 2000, 2002) using the exhaustive database of nonlinear response-history analysis results for yielding structures with damping systems. DEVELOPMENT OF EQUATION FOR COEFFICIENT To establish the relationship for as a function of the variables identified above, nonlinear analysis was performed on bilinear hysteretic systems with linear viscous damping. The ranges of the variables studied were as follows: 1. 0.2 3.0 seconds in steps of 0.1 second 2. 0.05, 0.15, 0.25, 0.50, 1.0 (where 1.0 is elastic behavior) 3. 2.0, 3.33, 5.0 4. Linear viscous damping under elastic conditions 0.0, 0.15, 0.25, when added to inherent damping of 0.05 gave a total viscous damping ratio under elastic conditions of 0.05, 0.20, and 0.30. The ductility factor of Equation 6 was calculated as mS 0.05 (7) where is the mass of the system, is the spectral acceleration for 5% damping, and is the damping coefficient for the total damping ratio. The trilinear model for the damping coefficient (columns 4 and 5 of Table 1) was used to calculate the yield strength of the analyzed models. Values for the coefficient were obtained from response-history analysis (using the histories of Table 2) as the ratio of the average peak inelastic displacement to the average peak elastic displacement. Plots of this coefficient versus period revealed the basic nature of the relation. Moreover, since should converge to unity when either or is equal to 1.0, the following relation was proposed: (8) where parameters and were obtained by calibration of the model using the results of dynamic analysis. The simplest form of these parameters was found to be 0.45 (9) Table 3. Values of parameter 0.05 0.15 0.25 0.50 0.05 0.116 0.100 0.093 0.071 0.30 0.195 0.160 0.143 0.111 ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 539

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3.24 0.10 4.5 (10) where values of are given in Table 3 below. Linear interpolation can be used to esti- mate intermediate values of Figures 5 through 7 compare values of coefficient obtained by nonlinear response-history analysis to the predictions of the model given by Equation 8. The response-history results presented in Figures 5 through 7 are the average of the results of analyses using the 20 earthquake histories of Table 2. The relation of Equation 8 de- Figure 5. Values of for a total viscous damping ratio of 0.05. Figure 6. Values of for a total viscous damping ratio of 0.20. 540 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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scribes well the calculated values of the coefficient and it follows the desired behavior for large values of period where Equation 8 predicts a value near 1.0 for This relation can be used to estimate displacement demands in structures with damping sys- tems without the need to use iterative methods of analysis. One interesting observation from Figures 5 through 7 is that the effect of viscous damping on in the range of 5 to 30 percent is not significant. Similar results were observed for the other values of DUCTILITY DEMANDS IN STRUCTURES WITH 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 in U.S. practice. Such a frame, if assumed to be elas- toplastic will have a yield strength given by 0.05 (11) where is the ductility-based portion of the factor. In Equation 11, the required elas- tic strength is calculated assuming a damping ratio of 5%. A frame designed using Equation 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 given by 0.05 (12) where is the damping coefficient for the viscous damping ratio of the structure under Figure 7. Values of for a total viscous damping ratio of 0.30. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 541

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elastic conditions ). Note that the ratio in Equation 12 is the base shear of the damped frame calculated assuming that the frame is elastic with damping ratio equal to If is equal to 0.20 (0.05 inherent plus 0.15 added viscous damping), then is equal to 1.5 (see the last column of Table 1). Accordingly, for the same value of the strength of the damped frame, will be substantially less than that of the undamped frame, or for this example, 0.67 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. The next section of the paper presents the results of a systematic study that answered this question. EVALUATION OF DISPLACEMENT DUCTILITY DEMANDS Bilinear hysteretic single-degree-of-freedom (SDOF) systems without and with supplemental linear viscous damping were considered in the study. Each system without supplemental damping was characterized by the elastic period ductility factor ratio of post-elastic stiffness to elastic stiffness, , and an inherent viscous damping ra- tio, of 0.05. Values of 0.05, 0.15, 0.25, 0.50, 2.0, 3.33, 5.0, and between 0.2 and 2.0 seconds were considered in the study. Such a range encompasses all practical values 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 ed that was larger than and supplemental linear viscous damping 0.15, 0.25 under elastic conditions. Accord- ingly, the total damping ratio under elastic conditions was either 0.20 or 0.30. The damped SDOF systems (that is, those with supplemental damping) had lower yield strengths than the corresponding undamped systems (that is, those with no supplemental damping) as indicated by Equations 11 and 12. The elastic period ed of the damped system was related to the period of the cor- responding undamped system as follows: ed (13) where is a parameter that is dependent on the period range under consideration and the geometry 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 were established for cases where both and ed fell in either (a) the constant acceleration segment of the spectrum, that is, or (b) the constant velocity segment of the spectrum, that is, For case (a), and was approximately 0.75 and 0.55 for rectangular and wide-flange cross section shapes, respectively. For case (b), and was of the order of 1 for both rectangular and wide-flange cross section shapes. Accordingly, analyses were performed using Equation 13 to calculate the period of the damped structure with 0.5 when and 1.0 when The frames were designed to have a yield strength given by Equations 11 through 13: 542 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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for ed (14) for ed Figure 8. Comparison of average displacement ductility ratio for 5%- and 20%-damped systems. Figure 9. Comparison of average displacement ductility ratio for 5%- and 30%-damped sys- tems. ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 543

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that is, the yield strength of the damped systems varied between 0.35 and 0.67 times the strength of the corresponding undamped systems. Response-history analyses were performed using the 20 scaled earthquake histories listed in Table 2 to compare the displacement ductility demands in the undamped and damped frames. Figures 8 and 9 compare the calculated average displacement ductility ratio for the undamped and the damped systems, where the average is that of the 20 calculated values for each combination of parameters. The ductility demand in the two systems is nearly identical. Figure 10 presents a comparison of the displacement ductil- ity ratio of the undamped and the 20%-damped systems for 0.05. In this figure, the average, maximum and minimum displacement ductility ratios for the 20 calculated val- ues are presented; the values are nearly identical for the same and . The data of Figure 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 that in the corresponding undamped systems, further supporting the approach of Equation 12. On the basis of the results presented above and those of Wu and Hanson (1989), the 2000 NEHRP Provisions permit the design of building structures with damping systems for a lower minimum base shear than that for the corresponding undamped structure. The minimum base shear for damped structures is the greater of or 0.75 where is the minimum seismic base shear for the design of the undamped structure and is the damping coefficient for the combined inherent and viscous damping under elastic con- ditions. Figure 10. Comparison of maximum, average and minimum displacement ductility ratios of 5%- and 20%-damped systems with 0.05. 544 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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SUMMARY AND CONCLUSIONS Studies were undertaken to support the development of the 2000 NEHRP Provisions for the design of buildings with energy dissipation systems. The numerical simulations made use of 20 earthquake histories that were scaled to match on average the 2000 NEHRP spectrum for DS 1.0, 0.6, and 0.6 second. Near-field and soft-soil earthquake 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. The key 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 are valid for viscous damping ratio in the range of 2% to 100% of critical. The val- ues for derived in this study were adopted, after simplification, in the 2000 NEHRP Provisions 2. A new relationship describing the ratio of peak inelastic displacement to the peak elastic displacement for systems with supplemental viscous damping has been presented. This relationship can be used to directly estimate displacement demands in short-period yielding structures with damping systems. The effect of viscous damping in the range of 5% to 30% on the value of is not sig- nificant. 3. The design base shear for damped buildings can be reduced from that for the undamped frame, without increasing the displacement ductility demand on the frame. This conclusion is now reflected in the 2000 NEHRP Provisions where a damped frame can be designed for a base shear strength that is up to 25% less than the corresponding undamped frame. This result is a paradigm shift from previous guidelines that required the damped frame to be designed for the same base shear as the corresponding undamped frame. ACKNOWLEDGMENTS Financial support for this project was provided by the Multidisciplinary Center for Earthquake Engineering Research, Task on Rehabilitation Strategies for Buildings (Projects No. 982403, No. 992403, and No. 00-2042). The work was performed under the direction of Technical Subcommittee 12, Base Isolation and Energy Dissipation, of the Building Seismic Safety Council, which was tasked with developing analysis and design procedures for inclusion in the 2000 edition of the NEHRP Recommended Pro- visions for Seismic Regulations for New Buildings and Other Structures . The work of Technical Subcommittee 12 was supported by the Federal Emergency Management Agency. REFERENCES Applied Technology Council (ATC), 1995. Structural Response Modification Factors , Report No. 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 ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 545

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273 and 274 , prepared for the Building Seismic Safety Council, published by the Federal Emergency Management Agency, Washington, DC. Constantinou, M. C., Soong, T. T., and Dargush, G. F., 1998. Passive Energy Dissipation Sys- tems for Structural Design and Retrofit , Monograph Series No. 1, Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, State University of New York at Buffalo, Buffalo, NY. Building Seismic Safety Council (BSSC), 1995. NEHRP Recommended Provisions for the Seis- mic Regulations for New Buildings , 1994 Edition, Report No. FEMA 222A , developed for the Federal Emergency Management Agency, Washington, DC. BSSC, 2001. NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures , 2000 Edition, Report Nos. FEMA 368 and 369 , Federal Emergency Man- agement Agency, Washington, DC. Hanson, R. D., and Soong, T. T., 2001. Seismic Design with Supplemental Energy Dissipation Devices , Earthquake Engineering Research Institute, Oakland, CA, 135 pp. International Conference of Building Officials (ICBO), 1997. Uniform Building Code , Vol. 2Structural Engineering Design Provisions, Whittier, CA. Mander, J. B., Priestley, M. J. N., and Park, R., 1984, Seismic Design of Bridge Piers , Report 84-2, Department of Civil Engineering, University of Canterbury, New Zealand. Miranda, E., 1993. Site-dependent strength reduction factors, J. Struct. Eng. 119 (12), 3503 3519. Miranda, E., 2001. Estimation of inelastic deformation demands of SDOF systems, J. Struct. Eng. 127 (9), 10051020. Miranda, E., and Bertero, V. V., 1994. Evaluation of strength reduction factors for earthquake- resistant design, Earthquake Spectra 10 (2), 357379. Nassar, A. A., and Krawinkler, H., 1991. Seismic Demands for SDOF and MDOF Systems , Re- port No. 95, John A. Blume Earthquake Engineering Research Center, Department of Civil Engineering, Stanford University, Stanford, CA, 204 pp. Newmark, N. M., and Hall, W. J., 1973. Seismic Design Criteria for Nuclear Reactor Facilities Report No. 46, Building Practices for Disaster Mitigation, National Bureau of Standards, U.S. Department of Commerce, pp. 209236. Newmark, N. M., and Hall, W. J., 1982. Earthquake Spectra and Design , Earthquake Engineer- ing Research Institute, Berkeley, CA, 103 pp. Ramirez, O. M., Constantinou, M. C., Kircher, C. A., Whittaker, A. S., Johnson, M. W., Gomez, J. D., and Chrysostomou, C. Z., 2000. Development and evaluation of simplified procedures for analysis and design of buildings with passive energy dissipation systems, Report No. MCEER 00-0010 , Revision 1, Multidisciplinary Center for Earthquake Engineering Re- search, University at Buffalo, State University of New York, Buffalo, NY, 470 pp. Ramirez, O. M., Constantinou, M. C., Gomez, J. D., Whittaker, A. S., and Chrysostomou, C. Z., 2002. Evaluation of simplified methods of analysis of yielding structures with damping sys- tems, Earthquake Spectra 18 (3), 501530 (this issue). Riddell, R., Hidalgo, P., and Cruz, E., 1989. Response modification factors for earthquake re- sistant design of short period structures, Earthquake Spectra (3), 571590. Riddell, R., and Newmark, N. M., 1979. Statistical Analysis of the Response of Nonlinear Sys- tems Subjected to Earthquakes , Structural Research Series No. 468, Department of Civil En- gineering, University of Illinois at Urbana-Champaign, Urbana, Il, 291 pp. 546 O. M. RAMIREZ, M. C. CONSTANTINOU, A. S.WHITTAKER, C. A. KIRCHER, AND C. Z. CHRYSOSTOMOU

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Structural Engineers Association of California (SEAOC), 1999. Recommended Lateral Force Requirements and Commentary, 7th Edition , Sacramento, CA. Soong, T. T., and Dargush, G. F., 1997. Passive Energy Dissipation Systems in Structural En- gineering , Wiley, & Sons Ltd., London (UK) and New York (USA). Tsopelas, P., Constantinou, M. C., Kircher, C. A., and Whittaker, A. S., 1997. Evaluation of simplified methods of analysis for yielding structures, Report No. NCEER 97-0012 , National Center for Earthquake Engineering Research, University at Buffalo, State University of New York, Buffalo, NY. Vidic, T., Fajfar, P. and Fischinger, M., 1992. A procedure for determining consistent inelastic design spectra, Workshop on Nonlinear Seismic Analysis of RC Structures, Proceedings Bled, Slovenia. Whittaker, A. S., Aiken, I. D., Bergman, D., Clark, P. W., Cohen, J. M., Kelly, J. M., and Scholl, R. E., 1993. Code requirements for the design and implementation of passive energy dissi- pation systems, Proceedings, Seminar on Seismic Isolation, Passive Energy Dissipation, and Active Control, Report No. ATC-17-1 , Applied Technology Council, Redwood City, CA. Wu, J., and Hanson, R. D., 1989. Study of inelastic spectra with high damping, J. Struct. Eng. 115 (6), 14121431. (Received 28 November 2001; accepted 28 May 2002) ELASTIC AND INELASTIC SEISMIC RESPONSE OF BUILDINGS WITH DAMPING SYSTEMS 547

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