Inelastic Deformation Ratios for Design and Evaluation of Structures: Single-Degree-of-Freedom Bilinear Systems

Transcription

1 Inelastic Deformation Ratios for Design and Evaluation of Structures: Single-Degree-of-Freedom Bilinear Systems Anil K. Chopra, M.ASCE, 1 and Chatpan Chintanapakdee, A.M.ASCE 2 Abstract: The relationship between the peak deformations of inelastic and corresponding linear single-degree-of-freedom SDF systems is investigated. Presented are the median of the inelastic deformation ratio for 214 ground motions organized into 11 ensembles of ground motions, representing large or small earthquake magnitude and distance, and National Earthquake Hazards Reduction Program NEHRP site classes B, C, and D; near-fault ground motions are also included. Two sets of results are presented for bilinear nondegrading systems over the complete range of elastic vibration period, T n : C for systems with known ductility factor,, and C R for systems with known yield-strength reduction factor, R y. The influence of postyield stiffness on the inelastic deformation ratios C and C R is investigated comprehensively. All data are interpreted in the context of acceleration-sensitive, velocity-sensitive, and displacement-sensitive regions of the spectrum for broad applications. The median C versus T n and C R versus T n plots are demonstrated to be essentially independent of the earthquake magnitude and distance over their ranges considered, and of site class. In the acceleration-sensitive spectral region, the median inelastic deformation ratio for near-fault ground motions is systematically different when plotted against T n ; however, when plotted against normalized period T n /T c where T c is the period separating the acceleration- and velocity-sensitive regions they become very similar in all spectral regions. Determined by regression analysis of the data, two equations one for C and the other for C R have been developed as a function of T n /T c, and or R y, respectively, and are valid for all ground motion ensembles considered. These equations for C and C R should be useful in estimating the inelastic deformation of new or rehabilitated structures where the global ductility capacity can be estimated and existing structures with known lateral strength. DOI: / ASCE : CE Database subject headings: Deformation; Inelastic action; Ground motion; Structural analysis. Introduction Current structural engineering practice usually estimates seismic demands by applying the nonlinear static procedure or pushover analysis presented in FEMA273 FEMA 1997, FEMA356 FEMA 2000, or ATC-40 guidelines ATC Seismic demands are computed by nonlinear static analysis of the structure subjected to monotonically increasing forces until a predetermined target displacement is reached. More recently, a modal pushover analysis procedure based on structural dynamics theory has been developed Chopra and Goel 2002, where seismic demands due to individual terms in the modal expansion of the effective earthquake forces are determined by a pushover analysis using the inertia force distributions associated with each mode up to a modal target displacement. The target roof displacement in all of these pushover procedures is determined from the peak deformation of an inelastic single-degree-of-freedom SDF system with its force-deformation relation defined from the pushover curve. This has led to renewed interest in the relationship between 1 Johnson Professor, Dept. of Civil and Environmental Engineering, Univ. of California, Berkeley, CA Lecturer, Dept. of Civil Engineering, Chulalongkorn Univ., Bangkok, Thailand. Note. Associate Editor: Gregory A. MacRae. Discussion open until February 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on October 10, 2002; approved on March 27, This paper is part of the Journal of Structural Engineering, Vol. 130, No. 9, September 1, ASCE, ISSN /2004/ /$ the peak deformations of inelastic and corresponding linear SDF systems, u m and u o, respectively, a problem first studied by Veletsos and Newmark The inelastic deformation ratio, if expressed as a function of elastic vibration period T n and ductility factor, can be used to determine the inelastic deformation of a new or rehabilitated structure where global ductility capacity can be estimated. Veletsos and Newmark 1960 proposed and evaluated two approaches: equal-deformation rule, i.e., u m /u o 1; and equating the strain energy for elastic and inelastic systems, leading to u m /u o / 2 1. This ratio was shown to differ significantly in different spectral or frequency regions, with the equal deformation rule being valid only in the displacement-sensitive spectral region Veletsos et al The equal-deformation rule was later recommended also for the velocity-sensitive spectral region Veletsos and Vann With the accumulation of strong motion records and ease of computation, it became possible to investigate the statistics of the inelastic deformation ratio over large ensembles of ground motions. Response data for 124 excitations and bilinear systems with 3% postyield stiffness ratio demonstrated that the inelastic deformation ratio was strongly dependent on the elastic vibration period in the acceleration-sensitive spectral region and that the period beyond which the equal-deformation rule is valid depends on the site conditions rock, alluvium, or soft soil and on the ductility factor Miranda 1991, A more recent investigation based on 264 records concluded that the earthquake magnitude and rupture distance have little influence on the inelastic deformation ratio Miranda 2000 ; however, at short periods s, this ratio for fault-normal near-fault records with forward di- JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004 / 1309

2 rectivity effects was shown to be much larger compared to farfault records Báez and Miranda Regression analyses of these data led to an equation, which is restricted to elastoplastic systems, for the inelastic deformation ratio as a function of elastic vibration period and ductility factor Miranda The inelastic deformation ratio, if expressed as a function of elastic vibration period and yield-strength reduction factor R y strength required for the structure to remain elastic yield strength of the structure, can be used to determine the inelastic deformation of an existing structure with known lateral strength. For reinforced concrete structures, researchers have developed an equation that specifies the range of R y and T n where the equaldeformation rule is valid Shimazaki and Sozen 1984, established the short-period range where the inelastic deformation ratio is sensitive to both strength and stiffness of the system, and developed an upper bound equation for the deformation Qi and Moehle An evaluation of the FEMA-273 procedure for estimating deformation demonstrated that the ratio of the mean inelastic and elastic deformations exceeds unity if R y exceeds 5, regardless of T n Whittaker et al Response data for 216 ground motions recorded on NEHRP site classes B, C, and D demonstrated that the mean inelastic deformation ratio is influenced little by soil condition, by magnitude if R y 4 but significantly for larger R y ), or by rupture distance so long as it exceeds 10 km Ruiz-Garcia and Miranda Regression analysis of these data led to an equation for the inelastic deformation ratio as a function of T n and R y ; this equation is restricted to elastoplastic systems. Researchers have demonstrated that the inelastic deformation ratio in the acceleration-sensitive spectral region is reduced because of postyield stiffness Veletsos 1969; Qi and Moehle 1991, and increased due to stiffness degradation Clough 1966; Qi and Moehle 1991; Song and Pincheira 2000 and pinching Gupta and Krawinkler 1998; Gupta and Kunnath 1998; Song and Pincheira 2000 of the hysteresis loop. However, others have concluded that the influence of postyield stiffness on the ductility demand for constant-strength systems is not significant, especially at longer periods Clough 1966 and that mean responses of constantductility systems can be conservatively estimated using the elastoplastic model Riddell et al ; it is noticeable on the order of 10 20% but not predominant on inelastic strength demands Nassar and Krawinkler As an alternative to the empirical equations for the inelastic deformation ratio, R y T n relations can be used to estimate the peak deformation of an inelastic SDF system Chopra and Goel 1999; Fajfar 2000, but this indirect method is slightly biased towards underestimating the deformation Miranda The objective of this paper is to present the median and dispersion of the inelastic deformation ratio of SDF systems for several ensembles of ground motions, representing large or small earthquake magnitude and small or large distance, NEHRP site classes B, C, and D all firm sites ; near-fault ground motions are also included. Two sets of results are presented: one for systems with known ductility factor and the other for systems with known yield-strength reduction factor. The earthquake responses of bilinear systems are investigated comprehensively, but degradation of stiffness or strength, or pinching of the hysteresis loop is not considered. Investigated is the influence of earthquake magnitude and distance, site class, and near-fault ground motions on the inelastic deformation ratio. Presented are two equations in terms of or R y for the inelastic deformation ratio, valid for all ground motions ensembles considered, in a form convenient for application to design or evaluation, respectively, of structures. Fig. 1. Bilinear force deformation relationship of inelastic singledegree-of-freedom system and corresponding elastic system Inelastic Deformation Ratio: Theory Bilinear Systems The initial loading curve for a system with bilinear force deformation relation f s (u,sgn u ) is shown in Fig. 1. The elastic stiffness is k and the postyield stiffness is k, where is the postyield stiffness ratio; elastoplastic systems 0 are included but not systems with negative postyield stiffness 0. The yield strength is f y and the yield deformation u y. Unloading and reloading of the hysteretic system occurs without any deterioration of stiffness or strength. Within the linearly elastic range the system has a natural vibration period T n frequency n 2 /T n ) and damping ratio. The yield strength reduction factor R y is defined by R y f o u o (1) f y u y where f o and u o minimum yield strength and yield deformation required for the structure to remain elastic during the ground motion, or the peak response values for the corresponding linear system. The peak force in the inelastic system is f m Fig. 1. The peak or absolute without regard to algebraic sign maximum deformation of the bilinear system is denoted by u m and its ductility factor by u m (2) u y It can be shown that the inelastic deformation ratio, defined as the ratio of deformations of inelastic and corresponding linear systems, is u m (3) u o R y Computing Inelastic Deformation Ratio The equation governing the deformation u(t) of an SDF system of mass m due to earthquake acceleration ü g (t) is written in two alternative forms ü 2 n u f s /m ü g t (4a) or ü t 2 n u f s /m 0 (4b) where u t u u g. The deformation responses of an inelastic system defined by period T n, postyield stiffness ratio, and damping ratio and its corresponding linear system are deter / JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004

3 mined by numerical solution of Eq. 4a to obtain the peak deformations u m and u o, respectively, leading to the inelastic deformation ratio. This ratio will be denoted by C u m u o (5a) or C R u m (5b) u o Consistent with Miranda s notation Miranda 2001, the subscripts and R y in Eq. 5 represent systems with known ductility capacity or known yield strength defined by the reduction factor R y, respectively. The inelastic deformation ratio C R for a system of known yield strength can be determined directly from the computed peak deformations u m and u o of inelastic and elastic systems; the corresponding ductility demand is given by Eq. 3, wherein R y is readily available from Eq. 1. An iterative procedure is necessary to determine the inelastic deformation ratio C for a specified ductility factor because the yield strength corresponding to a selected cannot be determined directly Veletsos and Newmark 1964; Chopra 2001, Sec Because C is not unique, as more than one value of f y may yield the same, following convention the smallest f y is chosen that gives the largest C Veletsos and Newmark 1964; Chopra 2001, Sec The median value and dispersion of C and C R, defined, for example, in Chopra et al. 2003, is then computed for each of the nine ensembles of excitations described later. Limiting Values of Inelastic Deformation Ratio This ratio can be determined analytically for two limiting cases: T n tends to zero and T n tends to infinity. The first implies a very-short-period system that, for a fixed mass, is essentially rigid; thus u(t) 0, u (t) 0, and ü t (t) ü g (t). Eq. 4b then gives f s (t) mü g (t) and its peak value is f so mü go where the subscript o denotes the peak value, which is independent of the force deformation relation and hence valid for bilinear and corresponding linear systems. Thus for the limiting case of T n 0, f m f o, an equal-force rule based on Fig. 1 can be expressed as Dividing by f y gives f y 1 f y f o 1 1 R y (6) Substituting Eq. 6 in Eq. 3, the deformation ratio u m /u o for T n 0 systems can be expressed as a function of or of R y L 1 1 (7a) Fig. 2. Influence of postyield stiffness ratio on limiting values of inelastic deformation ratio as elastic vibration period T n tends to zero: a L and b L R or L R 1 R y 1 R y 1 (7b) Eqs. 7a and 7b can be interpreted as the limiting values of u m /u o as T n tends to zero for systems with constant or constant R y, respectively, independent of damping. For elastoplastic systems 0, L and L R ; thus, the limiting value of u m /u o for systems with T n 0is for constant- systems but is unbounded for constant-r y systems. In passing, note that the prevailing view that R y 1atT n 0, independent of, is valid only for elastoplastic systems; Eq. 6 indicates that R y is slightly larger for bilinear systems. Figs. 2 a and b plot Eqs. 7a and 7b for various values of and R y, respectively. For the limiting case of T n 0, u m is greater than u o, and u m /u o increases with Fig. 2 a and with R y Fig. 2 b i.e., decreasing yield strength. This ratio rapidly increases with R y near R y 1, implying that the deformation of the inelastic system is much larger even if its strength is only slightly below that required for the structure to remain elastic, an observation first made by Veletsos et al Postyield stiffness reduces the u m /u o ratio below its value for elastoplastic systems, which as mentioned earlier is for constant- systems but is unbounded for constant-r y systems; in the latter case the ratio is bounded if the system has the slightest postyield stiffness. Postyield stiffness is much more influential in reducing u m /u o for constant-r y systems compared to constant- systems. Thus, much difference between the C and C R would be expected in the short-period range. The ratio of deformations of inelastic and corresponding linear systems with very long period T n can be determined by physical reasoning. The mass of such an extremely flexible system would be expected to stay essentially stationary while the ground below moves. Thus, its peak deformation would tend to the peak ground displacement, u go, as T n tends to infinity, independent of the force deformation relation, and, therefore, valid for bilinear and corresponding linear systems. Thus for the limiting case of T n, u m u o u go, and C C R 1 (8) which is the well-known equal-displacement rule Veletsos and Newmark Eqs. 7 and 8 are independent of the excitation. Ground Motions and Elastic Response Spectra Seven ensembles of far-fault ground motions, each with 20 records, were included in this investigation. The first group of ensembles, denoted by LMSR, LMLR, SMSR, and SMLR represent four combinations of large (M ) or small (M ) magnitude and small (R km) or large (R km) distance. Record-to-source distance is defined as the closest distance to the fault rupture zone except for two records: Point Mugu-Port Hueneme PM73phn in the SMSR ensemble and Borrego-El Centro Array No. 9 BO42elc in the SMLR ensemble where the hypocentral distance is reported as 25 and 49 km, respectively. These ground motions and their parameters are tabulated in Chopra and Chintanapakdee All JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004 / 1311

4 Fig. 3. Median elastic response spectrum for LMSR ensemble of far-fault earthquake ground motions, shown by a solid line, together with idealized version in dashed line and spectral regions; 5% these records except one correspond to NEHRP site class D; the Morgan Hill San Juan Bautista MH84sjb record in the SMLR ensemble corresponds to NEHRP site class C. The second group of three ensembles is categorized by NEHRP site classes B, C, or D. These ground motions were recorded during earthquakes with magnitudes ranging from 6.0 to 7.4 at distances ranging from 11 to 118 km horizontal distance from the edge of horizontal projection of the fault rupture area to the site. Four ensembles of near-fault NF ground motions are considered in this paper. The first two ensembles, denoted by NF-fault normal FN and NF-fault parallel FP, are the two horizontal components of 15 near-fault ground motions, recorded during earthquakes of magnitudes ranging from 6.2 to 6.9 and at distances ranging from 0 to 9 km. These ground motions were all recorded on firm soil NEHRP site class D or rock; the rock motions have been modified to reproduce soil site conditions Somerville The other two ensembles, denoted by NF-FN soil33 and NF-FP soil33, are the fault-normal and fault-parallel components of 33 near-fault ground motions, all recorded on soil during earthquakes of magnitudes ranging from 6.0 to 7.6 and at distances ranging from 0.2 to 16.4 km. However, each of these two ensembles includes 11 ground motions from the above set of 15 near-fault records. All of the ground motion ensembles mentioned above are tabulated in Chopra and Chintanapakdee The median response spectrum for each ensemble of ground motions was determined. The one for the LMSR ensemble is presented in Fig. 3 as a four-way logarithmic plot. The idealized version of the response spectrum shown in dashed line was constructed according to the procedure described in Riddell and Newmark 1979, where the spectrum is divided logically into three period ranges Chopra 2001, Section 6.8. The long-period region to the right of point d, T n T d, is called the displacementsensitive region; the short-period region to the left of point c, T n T c, is called the acceleration-sensitive region; and the intermediate-period region between points c and d, T c T n T d, is called the velocity-sensitive region. Similar plots for the other ground motion ensembles and a table for T a, T b, T c, T d, T e, and T f are available in Chopra and Chintanapakdee The median pseudoacceleration spectra normalized to the peak ground acceleration of the ensemble are presented in Fig. 4. Clearly, the spectral shapes for the LMSR, LMLR, SMSR, and Fig. 4. Comparison of median pseudoacceleration response spectra among: a LMSR, LMLR, SMSR, and SMLR ensembles of ground motions; b LMSR and site class B, C, and D ensembles; and c far-fault LMSR, and near-fault fault-normal and near-fault faultparallel ensembles; 5% SMLR ensembles are very similar Fig. 4 a, and for the site class B, C, and D ensembles are similar, but differ from the LMSR shape in the period range 0.3 to 0.8 s Fig. 4 b. However, the shapes of median response spectra from the NF-FN and NF-FP ensembles are significantly different than the LMSR ensemble Fig. 4 c. Deformation of Systems with Known Ductility Fig. 5 presents the median value of C for the LMSR ensemble as a function of T n for fixed damping ratio 5%; all results presented in this paper are for this damping ratio. The results for fixed postyield stiffness ratio Fig. 5 a permits the following observations on how the degree of inelastic action indicated by influences the relationship between u m and u o in the various Fig. 5. Median of inelastic deformation ratio C for bilinear systems subjected to LMSR ensemble of ground motions: a influence of ductility factor and b influence of postyield stiffness ratio 1312 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004

5 Fig. 6. Ratio of median deformations u m for bilinear and elastoplastic systems: a influence of ductility factor at postyield stiffness ratio 3%; and b influence of at 4 Fig. 7. Dispersion of inelastic deformation ratio C for bilinear systems subjected to LMSR ensemble of ground motions: a influence of ductility factor at postyield stiffness ratio 3% and b influence of at 4 spectral regions. In the acceleration-sensitive region, u m u o at T n T c but u m exceeds u o increasingly for shorter periods and larger, indicating greater inelastic action, and approaches the limit given by Eq. 7a as T n tends to zero. In the velocitysensitive region, u m u o and is affected very little by. Inthe displacement-sensitive region, u m u o for systems in the period range T d to T f, where u m decreases for increasing ; however, for systems with periods longer than T f, u m u o essentially independent of and u m approaches u o Eq. 8 as T n tends to infinity, independent of. The results for fixed ductility factor Fig. 5 b indicate that postyield stiffness reduces the deformation of bilinear systems relative to elastoplastic systems over the entire period range except for T n T f where u m u o. While this observation is valid over the range of 0 to 10%, it may not be valid for larger values of the postyield stiffness ratio, e.g., in the period range T d T f, u m will eventually increase for large enough values of to u o at 1. Computed from the data of Fig. 5, the ratio of median u m for bilinear and elastoplastic systems is plotted against T n for fixed and four values of Fig. 6 a and fixed and three values of Fig. 6 b. Fig. 6 indicates that the percentage reduction in deformation due to postyield stiffness is roughly constant over a wide period range, increases slightly for very short periods (T n T a ), but disappears for very long periods (T n T f ). The reduction in deformation is roughly independent of period because the degree of inelastic action, as indicated by the value, is kept constant; consistent with intuition, greater reduction is achieved at larger ductility factors. The reduction of deformation due to postyield stiffness is modest for realistic values of and ; e.g., for 4 and 3% the deformation is reduced by less than 15% over the entire period range Fig. 6 a. These results support the view that for systems with known and small ductility, elastoplastic models provide a usefully conservative estimate of deformation Riddell and Newmark 1979; Riddell et al and is consistent with the earlier result that R y is affected little by postyield stiffness Nassar and Krawinkler The dispersion of C for the LMSR ensemble is plotted against T n for fixed and four values of Fig. 7 a and for fixed with four values of Fig. 7 b. Here the dispersion is 1 zero as T n tends to zero or infinity because the limiting value of C Eqs. 7a and 8 is independent of the ground motion; 2 roughly similar over a wide range of periods from T b to T e ; 3 increases with Fig. 7 a consistent with intuition; and 4 affected very little by postyield stiffness over the range of considered Fig. 7 b, contrary to intuition. Deformation of Systems with Known R y Fig. 8 presents the median value of C R for the LMSR ensemble as a function of T n for 5%. The results for fixed postyield stiffness ratio Fig. 8 a permit the following observations on how the yield strength influences the relationship between the deformations u m and u o of inelastic and elastic systems in the various spectral regions. In the acceleration-sensitive region, starting from u m u o at T n T c, u m exceeds u o increasingly for shorter periods where u m is very sensitive to the yield strength, increasing as the yield strength is reduced; u m approaches the limit given by Eq. 7b as T n tends to zero. Very short period systems (T n T a ), even systems with strength only slightly smaller than the minimum strength required for the systems to remain elastic e.g., R y 1.5), experience deformations much larger than the elastic deformation. Just as in the case of constant- systems, u m u o in the velocity-sensitive region and is essentially independent of the yield strength. In the displacement-sensitive region, the relationship between u m and u o is similar to that observed for constant- systems: u m u o for systems in the period range T d T f where u m decreases as strength is reduced; u m u o, essentially independent of strength, for periods longer than T f ; and u m approaches u o Eq. 8 as T n tends to infinity, independent of yield strength. The results for fixed or constant R y presented in Fig. 8 b indicates that postyield stiffness reduces the deformation of a bilinear system relative to its value for elastoplastic systems. Although the deformation is reduced over the entire period range, the ratio of median u m for bilinear and elastoplastic systems demonstrates that postyield stiffness reduces the deformation only moderately in the velocity and displacement-sensitive regions; e.g., for R y 4 and T n T c this reduction is less than 13, Fig. 8. Median of inelastic deformation ratio C R for bilinear systems subjected to LMSR ensemble of ground motions: a influence of strength reduction factor R y and b influence of postyield stiffness ratio JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004 / 1313

6 Fig. 9. Influence of postyield stiffness ratio on: a ratio of median deformations u m for bilinear and elastoplastic systems and b ductility demand; R y 4 Fig. 11. Comparison of median inelastic deformation ratio for LMSR, LMLR, SMSR, and SMLR ensembles of far-fault ground motions: a C for 4 and 3%; b C R for R y 4 and 3% 17, and 22% for 3, 5, and 10%, respectively Fig. 9 a. In these regions, the percentage reduction in deformation due to postyield stiffness is insensitive to the period because the median ductility demand is roughly constant over this period range Fig. 9 b. However, the deformation is reduced considerably in the acceleration-sensitive region, especially for periods shorter than T b. The constant-r y plots in Fig. 9 a demonstrate that postyield stiffness is much more effective in reducing deformation in the acceleration-sensitive region than observed from the constant- plots of Fig. 6 a. Such is the case because ductility demands for systems with a selected R y say, R y 4) are much larger than the R y say, 4 value in the acceleration-sensitive spectral region Fig. 9 b ; and larger implies larger reduction in deformation due to postyield stiffness Fig. 6 b. This observation implies that ignoring postyield stiffness in estimating deformation is too conservative for seismic evaluation of existing structures with periods in the acceleration-sensitive region. While factual and seemingly significant, this observation may not be important for real structures because they are usually strong enough to remain elastic for most ground motions. The dispersion of C R for the LMSR ensemble is plotted against T n for fixed and four values of R y Fig. 10 a and for fixed R y with four values of Fig. 10 b. Here the dispersion is: 1 zero as T n tends to zero except for elastoplastic systems, 0, Fig. 10 b or as T n tends to infinity because the limiting values of C R Eqs. 7b and 8 are independent of the ground motion; 2 larger for weaker larger R y ) systems over a wide range of periods except for T n T b where this trend is reversed; 3 essentially independent of postyield stiffness ratio Fig. 10 b for periods in the velocity and displacement-sensitive regions; and 4 reduced significantly with the slightest postyield stiffness in the very short-period (T n T b ) portion of the accelerationsensitive region. Influence of Earthquake Magnitude and Distance The computations that led to the results of the preceding section were repeated for the LMSR, LMLR, SMSR, and SMLR ground motion ensembles. As observed earlier, the shapes of the four median response spectra are very similar Fig. 4 a and their T c values 0.43, 0.46, 0.41, and 0.44 s, respectively are also close. This similarity suggests that the median C versus T n and C R versus T n functions should be quite similar. This expectation is confirmed by the deformation ratio-period plots for the four ensembles shown in Figs. 11 a and b. Influence of Firm Site Classes Similar computations were repeated for three ensembles of ground motions recorded on firm sites: NEHRP site classes B, C, and D. As observed earlier, the shapes of the three median response spectra are similar Fig. 4 b and their T c values 0.33, 0.33, and 0.41 s, respectively are also similar. Although the T c values for the three site classes vary over a wider range compared to when earthquake magnitude and recording distance were varied, the median C versus T n curves Fig. 12 a and C R versus T n curves Fig. 12 b for the three site classes are very similar; they are close to the LMSR result in spite of the difference in their spectra Fig. 4 b. Fig. 10. Dispersion of inelastic deformation ratio C R for bilinear systems subjected to LMSR ensemble of ground motions: a influence of strength reduction factor R y at postyield stiffness ratio 3% and b influence of at fixed R y 4 Fig. 12. Comparison of median inelastic deformation ratio for LMSR and NEHRP site class B, C, and D ensembles of far-fault ground motions: a C for 4 and 3%; b C R for R y 4 and 3% 1314 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004

7 Fig. 13. Comparison of median inelastic deformation ratio C for far-fault LMSR and near-fault ground motions plotted versus: a elastic vibration period T n and b normalized period T n /T c ; both plots are for ductility factor 4 and postyield stiffness ratio 3% Fig. 15. Comparison of mean inelastic deformation ratio C estimated by equations developed by various researchers and LMSR data for ductility factor 6: a elastoplastic systems and b postyield stiffness systems 10% Near-Fault Ground Motions As mentioned earlier, the median response spectra for the faultnormal and fault-parallel components of near-fault ground motions display a significantly different shape with much different T c 0.85 and 0.61 s compared to far-fault ground motions, e.g., the LMSR ensemble Fig. 4 c. Because of these differences, the inelastic deformation ratios C and C R for near-fault ground motions is significantly different than far-fault motions Figs. 13 a and 14 a, which is consistent with earlier work reported by Báez and Miranda This systematic difference between the values of C R for near-fault and far-fault ground motions in the acceleration-sensitive spectral region is primarily due to the difference between the period T c values for the two sets of excitations, as demonstrated by plotting the ensemble median of the individual ground motion data for C and C R ) expressed as a function of the normalized vibration period T n /T c Figs. 13 b and 14 b, first introduced by Chopra and Chintanapakdee Now the C values for the far-fault ground motions and both fault-normal and fault-parallel components of near-fault ground motions are close in the acceleration-sensitive (T n T c ) region of the spectrum, and also in the velocity and displacementsensitive spectral regions, T n T c Fig. 13 b ; the same observation applies also to C R Fig. 14 b. Estimating Deformations of Inelastic Systems Simplified equations for inelastic deformation ratios C and C R would obviously facilitate estimation of the deformation of inelastic SDF system because the deformation of the corresponding linear system is readily known from the elastic design spectrum. Fig. 14. Comparison of median inelastic deformation ratio C R for far-fault LMSR and near-fault ground motions plotted versus: a elastic vibration period T n and b normalized period T n /T c ; both plots are for R y 4 and postyield stiffness ratio 3% Such an equation for C could be used to determine the inelastic deformation of a new or rehabilitated structure where the global displacement ductility capacity can be estimated. Similarly an equation for C R could be used to determine the inelastic global and local deformations of existing structures with known lateral strength. Such equations are developed in this section. Equation for C Miranda 2000 developed the following equation for the mean inelastic deformation ratio for elastoplastic systems: C exp 12T 0.8 Alternatively, the deformation of inelastic systems can be determined from Eq. 3 using R y T n relations, which have been developed by several researchers. The earliest of these relations for elastoplastic systems Veletsos and Newmark 1960, and consistent with Newmark Hall inelastic design spectra Newmark and Hall 1982, is available in Fig of Chopra Substituting this relation in Eq. 3 gives (9) T n T a C / 2 1 T b T n T c (10) 1 T n T c wherein T c T c 2 1/ and equations for the transitional period regions are not included. Substituting the relation developed by Krawinkler and Nassar 1992, based on earthquake response of bilinear systems, in Eq. 3 gives C c 1 1 1/c (11a) where T a n c T n, a 1 T b T n n (11b) and the numerical coefficients depend on the postyield stiffness ratio, : a 1 and b 0.42 for 0; a 1 and b 0.37 for 2%; and a 0.8 and b 0.29 for 10%. Fig. 15 a compares Eqs. 9, 10, and 11 for 0 with the mean value of C determined earlier Fig. 5 for elastoplastic systems subjected to the LMSR ensemble of ground motions. Only results for ductility factor of 6 the largest value considered are presented because it is the severest test of the C equations. Even for this extreme case, Eqs. 9, 10, and 11 provide excellent estimates of the inelastic deformation for elastoplastic systems, irrespective of the fact that they were developed for ground JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004 / 1315

8 Fig. 16. Comparison of median inelastic deformation ratio C estimated by proposed equation with computed data for LMSR far-fault and near-fault fault-normal component ground motions for elastoplastic 0 and bilinear 10% systems; and 6 motion ensembles different than the LMSR ensemble; the latter two equations are based on an indirect method, with the bias identified by Miranda A similar comparison for 10% Fig. 15 b demonstrates that Eq. 11 specialized for 10% is satisfactory over a wide range of periods but overestimates C at very short periods because as T n tends to zero, Eq. 11 approaches a limiting value that is valid only for elastoplastic systems instead of Eq. 7a derived earlier. Because Eqs. 9 and 10 were developed from response data for elastoplastic systems, as expected they overestimate C for bilinear systems; note that Eq. 9 is based on mean value of C, which is larger than the median value for probability density functions that are skewed to the right. Presented next is an improved equation that fits the median C data for any ensemble of ground motions, but ignores the data showing C 1 over the period range T d to T f Fig. 5 a and satisfies the limiting values of C at T n 0 and T n Eqs. 7a and 8. Such a function for C has been derived Chopra and Chintanapakdee 2003 in terms of ductility factor and the normalized period T n /T c C 1 L 1 1 a c T d n b T c 1 (12) Fig. 17. Comparison of median deformation ratio C estimated by proposed equation with computed data for a b LMSR, LMLR, SMSR, and SMLR far-fault ground motions; c d NEHRP site class B, C, and D ensembles; and e f near-fault fault-normal and fault-parallel ensembles of 33 records on soil where L is given by Eq. 7a. The numerical parameters a, b, c, and d were determined from the response data by the algorithm fmins.m in MATLAB 1997 to minimize the residual error, defined as the sum of the squares of differences between the actual data and the regression equation, but with a penalty for underestimation: a 72, b 2.2, c 1.6, and d 1.5 using the LMSR data and a 105, b 2.3, c 1.9, d 1.7 using the data for four LMSR, LMLR, SMSR, and SMLR ensembles; these parameter values are independent of postyield stiffness ratio. Further information is available in Chopra and Chintanapakdee Fig. 16 shows that Eq. 12 with the first set of parameter values closely matches the median C computed for the LMSR and near-fault ground motions. The excellent agreement with the LMSR data is expected because the parameters were determined from those data. Most impressive is that the same parameters provide a good match with the near-fault data, which has been achieved because Eq. 12 is developed as a function of T n /T c, the normalized period, instead of T n. Fig. 17 shows that Eq. 12, which uses the second set of values for a, b, c, and d, provides a satisfactory fit to the median C computed for LMSR, LMLR, SMSR, and SMLR ground motion ensembles Figs. 17 a and b ; NEHRP site class B, C, and D ensembles Figs. 17 c and d ; and near-fault NF-FN soil33 and NF-FP soil33 ensembles Figs. 17 e and f. Thus, these parameter values are generally applicable for a wide range of conditions, except for soft soil sites, and even for a large ensemble of near-fault ground motions. A more complicated equation would be necessary to match the C 1 data in the T d -to-t f period range; however, these data should be re-examined with P effects included before developing such an equation. Equation for C R Ruiz-Garcia and Miranda 2003 developed the following equation for the mean inelastic deformation ratio for elastoplastic systems: 1 C R 1 a T/T s 1 (13) c R 1 b where a 50; b 1.8; c 55; and T s 0.75, 0.85, or 1.05 for NEHRP site class B, C, or D, respectively. Alternatively, the deformation of inelastic systems can be determined from Eq. 3 using R y T n relations, as demonstrated earlier for constant-ductility systems. Substituting the R y T n relations, for elastoplastic systems, consistent with Newmark and Hall 1982 in Eq. 3, gives T n T a C R R 2 y 1 /2R y T b T n T c (14) 1 T n T c and using the relation developed by Krawinkler and Nassar 1992 for bilinear systems gives C R 1 R y 1 1 c R c y 1 (15) where c is given by Eq. 11b / JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004

9 Fig. 18. Comparison of mean inelastic deformation ratio C R estimated by equations developed by various researchers and LMSR data for R y 4: a elastoplastic systems and b bilinear systems 10% Fig. 18 a compares Eqs. 13, 14, and 15 for 0 with the mean value of C R determined earlier Fig. 8 for elastoplastic systems subjected to the LMSR ensemble of ground motions; results are presented only for R y 4. Eqs. 13, 14, and 15 provide good estimates of the inelastic deformation ratio for elastoplastic systems with initial period T n longer than T b, although they were developed for ground motion ensembles different than the LMSR ensemble. A similar comparison for bilinear systems with 10% Fig. 18 b demonstrates that surprisingly Eq. 15 specialized for 10% is not much better than Eqs. 13 and 14, which were developed from response data of elastoplastic systems. All the relations overestimate C R at very short periods because they approach infinity as T n tends to zero, instead of the bounded value given by Eq. 7b. Because Eqs. 13 and 14 were developed from response data for elastoplastic systems, as expected they overestimate C R for bilinear systems. Presented finally is an equation that fits the median C R data for any ensemble of ground motions, but ignores the data showing C R 1 over the period range T d T f e.g., Fig. 8 a and satisfies the limiting values of C R at T n 0 and T n Eqs. 7b and 8. Such a function for C R has been derived Chopra and Chintanapakdee 2003 in terms of the yield-strength reduction factor R y and the normalized period T n /T c C R 1 L R 1 1 a R y b c T n d T c 1 (16) Observe that the form of this equation is the same as Eq. 12 for C where L is replaced by L R and by R y. Using the same procedure and criteria, nonlinear regression analysis of the LMSR data led to a 63, b 2.3, c 1.7, and d 2.3, and of the data for four ensembles of ground motion led to a 61, b 2.4, c 1.5, and d 2.4. Fig. 19 shows that Eq. 16 using the first set of parameter values agrees well with the median C R computed for the LMSR and near-fault ground motions, which is expected because the parameters were determined from those data. Most impressive is that the same parameters provide a good match with the near-fault data, which has been achieved because Eq. 16 is developed as a function of T n /T c, the normalized period, instead of T n. Fig. 20 shows that Eq. 16 with the second set of parameter values provides a good, generally modestly conservative estimate of the median C R for all LMSR, LMLR, SMSR, and SMLR ground motion ensembles Figs. 20 a and b ; NEHRP site class B, C, and D ensembles Figs. 20 c and d ; and near-fault NF-FN soil33 and NF-FP soil33 ensembles Figs. 20 e and f. Thus, these parameter values are generally applicable for a wide range of conditions, except for soft soil sites, and even for a larger ensemble of near-fault motions. Fig. 19. Comparison of median inelastic deformation ratio C R estimated by proposed equation with computed data for LMSR far-fault and near-fault fault-normal component ground motions for elastoplastic 0 and bilinear 10% systems; and R y 6 Conclusions This investigation of the median and dispersion of the ratio of deformations denoted by C for system with known ductility factor and by C R for systems with known yield strength reduction factor R y of bilinearly inelastic and corresponding linear systems over eight ensembles of ground motions, has led to the following conclusions: 1. Equations have been derived for L and L R, the limiting values of C and C R as the elastic vibration period T n of the system tends to zero. L depends on and, the postyield stiffness ratio; L R depends on R y and. For the special case of elastoplastic systems, L and L R. Postyield stiffness is much more influential in reducing L R from an unbounded value to a finite value than L. For the limiting case of T n, physical reasoning leads to C C R 1, the well-known equal deformation rule Veletsos and Newmark Fig. 20. Comparison of median inelastic deformation ratio C R estimated by proposed equation with computed data for a b LMSR, LMLR, SMSR, and SMLR far-fault ground motions; c d NEHRP site class B, C, and D ensembles; and e f near-fault faultnormal and fault-parallel ensembles of 33 records on soil JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004 / 1317

10 2. The median values of C and C R ) for the seven ensembles of far-fault ground motions confirm known results that C and C R ) exceed 1 in the acceleration-sensitive spectral region, increasing with and R y ); 1 in the velocitysensitive and displacement-sensitive regions, essentially independent of and R y ); except that they fall below 1 in the period range T d T f of the latter region, decreasing for increasing and R y ). For very short-period systems (T n T a ), C R is very sensitive to the yield strength and can be very large even if the strength of the system is only slightly smaller than that required for it to remain elastic. 3. In contrast to the prevailing view, based largely on earthquake response of SDF systems for fixed ductility factor, ignoring postyield stiffness in estimating deformation is too conservative for seismic evaluation of existing structures with known R y in the acceleration-sensitive region. 4. The dispersion of C is zero as T n tends to zero or infinity; roughly similar over a wide period range, T b T f ; increasing with ; and affected little by postyield stiffness. 5. The dispersion of C R is zero as T n tends to zero except for elastoplastic systems or T n tends to infinity; is larger for weaker systems over a wide range of periods except for T n T b where this trend is reversed; and is essentially independent of postyield stiffness ratio for the range of considered in the velocity- and displacement-sensitive regions, but is reduced significantly with the slightest postyield stiffness in the short-period portion of the acceleration-sensitive region. 6. The median C versus T n plots for four ground motion ensembles denoted by LMSR, LMLR, SMSR, and SMLR representing four combinations of large (M ) or small (M ) magnitude and small (R km) or large (R km) distance are very similar; the C R T n plots of the four ensembles are also very close. 7. The median C T n and C R T n plots are essentially independent of NEHRP site classes B, C, and D, all of which are firm soil sites. 8. In the acceleration-sensitive spectral region, the median C T n and C R T n plots for near-fault ground motions are systematically different than far-fault ground motions; however, when plotted against normalized period T n /T c where T c is the period separating the acceleration- and velocitysensitive regions, they become very similar in all spectral regions. 9. Simplified equations for inelastic deformation ratios of C Eq. 12 and C R Eq. 16 for elastoplastic and bilinear systems have been developed as functions of T n /T c, and or R y, respectively. These equations satisfy the limiting values: L and L R at T n 0, and 1 at T n. Determined by regression analysis of the C and C R data for four ensembles LMSR, LMLR, SMSR, and SMLR of far-fault ground motions, the numerical values for the parameters in these equations are as follows: a 105, b 2.3, c 1.9, and d 1.7 in Eq. 12 ; and a 61, b 2.4, c 1.5, and d 2.4 in Eq. 16. These equations and parameters are also valid for near-fault ground motions recorded on soil and rock Chopra and Chintanapakdee 2003 and far-fault ground motions recorded on NEHRP site classes B, C, and D. Thus the equations presented for C and C R should be valid for almost any ensemble of ground motions recorded at firm sites. The equation for C should be useful in estimating the inelastic deformation of a new or rehabilitated structure where the global ductility capacity can be estimated. Similarly, the equation for C R should be useful for existing structures with known lateral strength. Acknowledgments This research investigation is funded by the National Science Foundation under Grant No. CMS , a part of the United States Japan Cooperative Research in Urban Earthquake Disaster Mitigation. This financial support is gratefully acknowledged. Our research has benefited from discussions with Professor Helmut Krawinkler and Professor Eduardo Miranda of Stanford Univ., who also provided some of the ground motion ensembles, and reviewed the final draft of this manuscript. This paper was prepared during the first writer s appointment in the Miller Institute for Basic Research in Science at the Univ. of California, Berkeley, Calif. Notations The following symbols are used in this paper: a parameter in empirical formula for C R, C,orR y ; b parameter in empirical formula for C R, C,or R y ; C R ratio of peak deformations of inelastic and corresponding elastic single-degree-of- freedom systems for systems with known yield-strength reduction factor; C ratio of peak deformations of inelastic and corresponding elastic single-degree-of- freedom systems for systems with known ductility factor; c parameter in empirical formula for C R, C,or R y ; d parameter in empirical formula for C R, C ; f m peak resisting force of inelastic system; f o peak resisting force of elastic system; f s resisting force of system; f y yield strength of inelastic system; k elastic initial stiffness of system; deformation ratio C R for zero-period system; L R L deformation ratio C for zero-period system; M earthquake magnitude; m mass of system; R record-to-source distance defined as closest distance to fault rupture zone; R y T a yield-strength reduction factor; period defined in Newmark Hall smooth design spectrum Fig. 3 ; T b period defined in Newmark Hall smooth design spectrum Fig. 3 ; T c period separating acceleration- and velocity-sensitive regions; T c transition period in Newmark Hall R y T n relations; T d period separating velocity- and displacement-sensitive regions; T e period defined in Newmark Hall smooth design spectrum Fig. 3 ; T f period defined in Newmark Hall smooth design spectrum Fig. 3 ; T n elastic natural vibration period; T s characteristic period in empirical formula by Miranda; u t total displacement of mass of system; u(t) deformation of single-degree-of-freedom system; ü g (t) earthquake ground acceleration; 1318 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / SEPTEMBER 2004

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