EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2001; 30:1791–1816 (DOI: 10.1002/eqe.93)
Hysteretic energy spectrum and damage control Rafael Riddell1;∗;† and Jaime E. Garcia2 1 Department
of Structural and Geotechnical Engineering; Universidad Cat olica de Chile; Santiago; Chile 2 Department of Civil Engineering; Universidad de Cuenca; Azuay; Ecuador
SUMMARY The inelastic response of single-degree-of-freedom (SDOF) systems subjected to earthquake motions is studied and a method to derive hysteretic energy dissipation spectra is proposed. The amount of energy dissipated through inelastic deformation combined with other response parameters allow the estimation of the required deformation capacity to avoid collapse for a given design earthquake. In the =rst part of the study, a detailed analysis of correlation between energy and ground motion intensity indices is carried out to identify the indices to be used as scaling parameters and base line of the energy dissipation spectrum. The response of elastoplastic, bilinear, and sti?ness degrading systems with 5 per cent damping, subjected to a world-wide ensemble of 52 earthquake records is considered. The statistical analysis of the response data provides the factors for constructing the energy dissipation spectrum as well as the Newmark–Hall inelastic spectra. The combination of these spectra allows the estimation of the ultimate deformation capacity required to survive the design earthquake, capacity that can also be presented in spectral form as an example shows. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS:
hysteretic energy; intensity index; energy spectrum; non-linear response; ultimate deformation; structural damage
INTRODUCTION Over the last 10 or 15 years the concern in seismic design has been progressively shifting to performance. Damage observed during earthquakes seems to have called the attention of the earthquake design community, including developed countries, in the sense that a code designed building may not necessarily ful=ll the earthquake design philosophy [1] that “if an unusual earthquake, somewhat greater than the most severe probable earthquake that is likely within the expected life of the building, should occur, the structure may undergo larger deformations and have serious permanent displacements and possibly require major repair, but it ∗
Correspondence to: Rafael Riddell, Department of Structural and Geotechnical Engineering, Universidad CatGolica de Chile, Casilla 306-Correo 22, Santiago, Chile. † E-mail:
[email protected] Contract=grant sponsor: National Science and Technology Foundation of Chile (FONDECYT); contract=grant number: 1990112
Copyright ? 2001 John Wiley & Sons, Ltd.
Received 18 May 2000 Revised 2 November 2000 and 12 February 2001 Accepted 14 March 2001
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will not collapse”. The reason is that the seismic code emphasis is on strength, while toughness should result from compliance with the material design code, but no accurate veri=cation of the seismic performance of the designed structure is ever made. Although there is possible agreement that non-linear 3-D history analyses for veri=cation ground motions is the answer, signi=cant improvement and standardization of these procedures is still necessary for general use in the profession. But the problem is not only one of veri=cation against collapse. The number of deaths and the important economic losses induced by recent earthquakes suggest that the acceptable level of damage also needs to be revised. It is therefore expected that damage assessment will become a central issue in the years to come. Until the aforementioned sophisticated analysis procedures become generally available, simpler approaches are necessary. The development of simple techniques also permits one to gain insight into the fundamental principles governing a problem. With the previous objectives in mind, hysteretic energy dissipation was studied, starting from a thorough consideration of the correlation between energy and intensity indices and ending with the rules to construct energy spectra. The energy spectrum is necessary to asses seismic structural damage, since recent approaches consider structural damage as a combination of maximum deformation (or ductility) and the e?ect of repeated cyclic response in the inelastic range or cumulative damage. Available damage models can be directly applied from the energy spectrum and standard Newmark–Hall spectra, establishing a direct relationship between strength, ductility, deformation, energy dissipation and damage. STRUCTURAL MODELS AND GROUND MOTIONS CONSIDERED A simple SDOF system was used in this study, with the force–displacement relationship given by three non-linear models: elastoplastic, bilinear, and sti?ness degrading (Figure 1). These
Figure 1. Single-degree-of-freedom system and non-linear load–deformation models used. Copyright ? 2001 John Wiley & Sons, Ltd.
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models cover a broad range of structural behaviour; they are intended to represent overall generic behaviour, rather than speci=c characteristics of individual systems [2; 3]. Strength deterioration was not considered, mainly because in a well-detailed structure it should only occur at extreme deformations near the failure state. A damping factor = 0:05 (5 per cent of critical) was used. Fifty-two earthquake records were used as input ground motion (Table I). These records represent moderate-to-large intensities of motion. Damage was observed near the recording site of all these motions. Most of them satisfy the following intensity condition: peak ground acceleration larger than 0:25 g and=or peak ground velocity larger than 25 cm=sec. It was not attempted to group the records according to similar characteristics regarding soil conditions, tectonic environment, Mercalli Intensity, distance to fault, or others; the reason was in part the lack of data—on geotechnical information for example—and the diOculty to form groups of statistical signi=cance. Indeed, if families of similar characteristics could be arranged in future studies, the scatter of results should decrease so that energy estimates could be made more accurately. It must be pointed out, however, that the =ndings of the study may not be directly extrapolated to soft soils since most of the ground motion data considered is on =rm ground. EQUATION OF MOTION AND ENERGY EXPRESSIONS The equation of motion of the system shown in Figure 1 can be written as u(t) Q + 2!u(t) ˙ +
R(u) = −y(t) Q m
(1)
where u is the relative displacement of mass m with respect to its base, ! = k=m is the undamped elastic circular frequency, R(u) is the hysteretic restoring force with sti?ness parameter k (Figure 1), = c=2!m is the damping factor as a fraction of the critical value, and y(t) Q is the base acceleration. Integrating Equation (1) with respect to u leads to the well-known energy balance equation [4; 5] which must hold at any time during motion:
E K + E D + E H + ES = EI where, using du = u˙ dt,
EK =
0
t
1 ˙ 2 − u(0) u(t) Q u(t) ˙ dt = [u(t) ˙ 2] 2
(2)
(3)
represents the kinetic energy per unit of mass, which becomes null if the initial velocity is zero and the integration is carried out long enough until the system comes to rest; t u˙ 2 (t) dt (4) ED = 2! 0
is the energy per unit of mass dissipated by the viscous damper; 1 t E H + ES = R(u)u(t) ˙ dt m 0 Copyright ? 2001 John Wiley & Sons, Ltd.
(5)
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Table I. Earthquakes records used in this study. Station, component, date
CMD Vernon, U.S.A. S08W (10=3=1933) El Centro, U.S.A. S00E (18=5=1940) Olympia, U.S.A. N86E (13=4=1949) Eureka, U.S.A. N79E (21=12=1954) Ferndale, U.S.A. N44E (21=12=1954) Kushiro Kisyo-Dai, Japan, N90E (23=4=1962) Ochiai Bridge, Japan, N00E (5=4=1966) Temblor, U.S.A. S25W (27=6=1966) Cholame 2, U.S.A. N65E (27=6=1966) Cholame 5, U.S.A. N85E (27=6=1966) Lima, Peru, N08E (17=10=1966) El Centro, U.S.A. S00W (8=4=1968) Hachinohe, Japan, N00E (16=5=1968) Aomori, Japan, N00E (16=5=1968) Muroran, Japan, N00E (16=5=1968) Itajima Bridge, Japan, Long. (6=8=1968) Itajima Bridge, Japan, Long. (21=9=1968) Toyohama Bridge, Japan, Long. (5=1=1971) Pacoima, U.S.A. S16E (9=2=1971) Orion LA, U.S.A. N00W (9=2=1971) Castaic, U.S.A. N21E (9=2=1971) Bucarest, Romania, S00E (4=3=1977) San Juan, Argentina, S90E (23=11=1977) Ventanas, Chile, Trans. (7=11=1981) Papudo, Chile, Long. (7=11=1981) La Ligua, Chile, Long. (7=11=1981) Rapel, Chile, N00E (3=3=1985) Zapallar, Chile, N90E (3=3=1985) Llo-Lleo, Chile, N10E (3=3=1985) Vina del Mar, Chile, S20W (3=3=1985) UTFSM, Chile, N70E (3=3=1985) Papudo, Chile, S40E (3=3=1985) Llay Llay, Chile, S10W (3=3=1985) San Felipe, Chile, N80E (3=3=1985) El Almendral, Chile, N50E (3=3=1985) Melipilla, Chile, N00E (3=3=1985) Pichilemu, Chile, N00E (3=3=1985) Iloca, Chile, N90E (3=3=1985) SCT, Mexico, N90E (19=9=1985) Corralitos, U.S.A. N00E (18=10=1985) KSR Kushiro, Japan, N63E (15=1=1993) Pacoima DAM, U.S.A. S05E (17=1=1994) Newhall, U.S.A. N00E (17=1=1994) Pacoima-Kagel, U.S.A. N00E (17=1=1994) Sylmar, U.S.A. N00E (17=1=1994) Santa Monica, U.S.A. N90E (17=1=1994)
Maximum acceleration (g)
Maximum velocity (cm=sec)
Maximum displacement (cm)
−0:133 −0:348
−29:03 −33:45
−19:50 −12:36 −9:38 −12:55
0.280 0.258 −0:159 0.478 −0:276 0.348 0.489 0.434 0.409 0.130 0.269 −0:257 −0:220 0.612 −0:261 0.450 1.171 0.255 0.316 0.206 0.193 0.268 −0:603 −0:469 0.467 0.304 −0:712 0.363 0.176 0.231 −0:352 0.434 0.297 −0:686 0.259 0.278 −0:171 0.630 0.725 −0:415 0.591 0.433 0.843 −0:883
17.09
−29:38 −35:65 −20:01
23.66
−22:52
78.08 25.44 −15:20 −25:81 −35:43 −39:12 30.28 −22:56 −12:93 15.90 113.23 30.00 17.16 75.12 −20:60 −17:87 −18:93 −18:83 −21:64 13.46 −40:29 30.74 14.60 12.41 −41:79 −17:77 −28:58 34.25 −11:68 15.09 −60:61 −55:20 33.59 44.68 −94:73 −50:88 −128:88 41.75
14.72 5.22 8.36 −5:55 −26:27 −6:89 −11:67 12.96 −9:68 −19:97 7.90 −4:59 −2:80 3.38 −41:92 16.53 −5:05 −19:93 6.33 −8:04 −7:43 4.49 −6:54 −1:69 −10:49 −5:42 3.11 1.60 8.43 −3:50 −5:78 12.02 3.73 1.39 21.16 12.03 4.73 4.65 28.81 −6:64 −30:67 −15:09 Continued
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Table I. Continued Station, component, date
Maximum acceleration (g)
Moorpark, U.S.A. S00E (17=1=1994) Castaic, U.S.A. N90E (17=1=1994) Arleta, U.S.A. N90E (17=1=1994) Century City-LA, U.S.A. N90E (17=1=1994) Obregon Park-LA, U.S.A. N00E (17=1=1994) Hollywood-LA, U.S.A. N00E (17=1=1994)
0.292 0.568 0.344 0.256 −0:408 −0:389
Maximum velocity (cm=sec) 20.28
−51:51 −40:37
21.36 −30:86 22.26
Maximum displacement (cm) 4.67
−9:19
8.36
−6:51 −2:65
4.27
is a term that composes the hysteretic energy EH , or energy dissipated per unit of mass by inelastic behaviour, and the stored elastic-strain energy per unit mass ES , which also vanishes when the system comes to rest; and u t y(t) Q du = − y(t) Q u(t) ˙ dt (6) EI = − 0
0
is the energy input per unit of mass, or energy supplied to the system by the moving base. Then, at the end of the motion, Equation (2) becomes E H + ED = EI
(7)
i.e. the total energy imparted to the structure by the earthquake must be dissipated by damping and inelastic deformations. CORRELATION BETWEEN ENERGY AND GROUND MOTION INTENSITY INDICES The correlation between EI and EH and various indices that have been proposed to characterize the intensity of earthquake motions was studied with the purpose of identifying appropriate normalization or scaling parameters to derive energy spectra. It is well known that the intensity of motion cannot be satisfactorily characterized by a single parameter. Consequently, as in the case of response spectra, it was expected that di?erent intensity measures would be suitable in the three characteristic spectral regions: short period (acceleration sensitive systems), intermediate period (velocity controlled responses) and long period (displacement sensitive systems). Thus, a number of indices were considered as described next. Arias [6; 7] proposed a measure of earthquake intensity that relates to the sum of the energies dissipated, per unit of mass, by a population of damped oscillators of all natural frequencies (0¡!¡∞): tf cos−1 IA () = yQ 2 (t) dt (8) g 1 − 2 0 where tf is the total duration of the ground motion and g is the acceleration of gravity. Housner [8] argued that a measure of seismic destructiveness could be given by the average rate of buildup of the total energy per unit mass input to structures; considering that the integral of Copyright ? 2001 John Wiley & Sons, Ltd.
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the squared ground acceleration was proportional to the total input energy, he proposed the earthquake power index t2 1 yQ 2 (t) dt (9) P= t1 − t2 t1 where t1 and t2 are the limits of the strong portion of motion. Mathematically, Equation (9) is the average value of the squared acceleration over the interval between t1 and t2 . The popular de=nition of signi=cant duration of motion after Trifunac and Brady [9] was adopted in this study, i.e. the interval between instants t5 and t95 at which 5 and 95 per cent of the integral in Equation (8) is attained, respectively. Then the earthquake power, or mean-square acceleration, is given by Equation (10), and similarly the indices mean-square velocity Pv and mean-square displacement Pd can be de=ned as given by Equations (11) and (12): t95 1 Pa = yQ 2 (t) dt (10) t95 − t5 t5 t95 1 y˙ 2 (t) dt (11) Pv = t95 − t5 t5 t95 1 y2 (t) dt (12) Pd = t95 − t5 t5 where y(t) ˙ and y(t) are the ground velocity and displacement histories, respectively. Hereafter, the signi=cant duration will be designated as td = t95 − t5 . For simplicity, without regard to duration or to the constants involved in Equation (8), the integrals of the squared ground motions have been used [10] as indices in the form: tf Ea = yQ 2 (t) dt (13)
Ev =
0
tf
0
Ed =
y˙ 2 (t) dt
(14)
y2 (t) dt
(15)
tf
0
√ √ The root-mean-square values of the ground motions, or e?ective values arms = Pa , vrms = Pv , √ [11–13] and drms = Pd have also been considered as potential measures of earthquake intensity √ √ as well as√the square root of the integral of the squared ground motions: ars = Ea , vrs = Ev , and drs = Ed . Other indices, which are based on the previously discussed quantities and include new parameters, have been proposed as descriptors of earthquake intensity. Araya and Saragoni [14] de=ned the ‘potential destructiveness’ of an earthquake as
PD =
IA 02
(16)
where IA is given by Equation (8) and 0 is the number of zero-crossings per unit of time of the accelerogram; the signi=cance of this parameter is the incorporation of the frequency Copyright ? 2001 John Wiley & Sons, Ltd.
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content of the ground motion through 0 . Park et al. [15] found that the ‘characteristic intensity’ 0:5 IC = a1:5 rms td
(17)
was a reasonable representation of the destructiveness of ground motions because it correlated well with structural damage expressed in terms of their damage index (Equation (37)). Fajfar et al. [16] proposed the expression IF = vmax td0:25
(18)
as a measure of the ground motion capacity to damage structures with fundamental periods in the intermediate period range, wherein vmax is the peak ground velocity. All the above indices depend only on the ground motions. They were used together with the peak values of ground acceleration amax , ground velocity vmax , and ground displacement dmax to test their correlation with the input and hysteretic energies. Only one response-related parameter, Housner’s spectral intensity, was used as well. Since the pseudo-velocity response Sv and the maximum strain energy stored in a linearly elastic system are related by ES max = mSv2 =2, Housner [17] argued that the spectrum itself was a measure of the severity of the earthquake, and de=ned the spectral intensity 2:5 Sv (; T ) dT (19) SI () = 0:1
Systems associated to three control frequencies 0.2, 1 and 5 cps were chosen as representative of the three characteristic spectral regions. Two types of energy were computed for each control frequency: input energy for an elastic system, or energy dissipated by damping (EI = ED ), and hysteretic energy EH dissipated by an inelastic system for a response associated to a ductility factor = 3. Use of other values of led to the same conclusions. To visualize the correlation among energy and the intensity indices, plots like Figure 2 were made for all the indices. In particular, Figure 2 shows the relation between index Ev (Equation (14)) and EH , where each asterisk corresponds to each of the 52 earthquake records; it can be seen that Ev and EH correlate well for intermediate and low-frequency systems (1 and 0:2 cps), but they show no relation at all for 5 cps. To have an objective measure of the correlation, a curve of the form E = Q
(20)
was =tted to the data, where E is the energy, Q is the intensity index and and are the non-linear regression parameters (or linear regression between the logarithms of the variables); the goodness of the =t is quanti=ed by the correlation coeOcient given by n (‘n Q ‘n E) − ‘n Q ‘n E (21) = (n (‘n Q)2 − ( ‘n Q)2 )(n (‘n E)2 − ( ‘n E)2 ) The correlation coeOcients for energy vs intensity, for all the indices, and for the three control frequencies, are summarized in Tables II and III; the indices ranked top-=ve for each frequency are noted. The correlation coeOcient is the same for indices that di?er only by a constant or by the exponent. Several observations can be made from the results presented in these tables: Copyright ? 2001 John Wiley & Sons, Ltd.
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Figure 2. Hysteretic energy per unit of mass EH vs ground motion intensity index Ev for control frequencies of 0.2, 1 and 5 cps.
(a) as expected, no index shows satisfactory correlation with energy in the three spectral regions simultaneously, indeed, acceleration related indices (amax ; ars ; arms ; IC ) are better for rigid systems (5 cps), velocity related indices (vmax ; vrs ; vrms ; IF ; SI ) are better for intermediate frequency systems (1 cps), and displacement related indices are better for Yexible systems (dmax ; drs ) although some velocity related indices also do well in the displacement region; (b) the peak ground motion parameters (amax ; vmax ; dmax ) show good correlation, specially in the displacement and acceleration regions where dmax and amax are the best, or nearly the best indices; (c) considering the previous observation, and recalling that Nau and Hall [10] tested the same indices used herein (except PD ; IC and IF ) and found that none of them Copyright ? 2001 John Wiley & Sons, Ltd.
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Table II. Correlation coeOcient between input energy EI for elastic systems and various intensity indices. Index dmax vmax amax Ed and drs Ev and vrs IA and Ea and ars Pd and drms Pv and vrms Pa and arms PD IC IF SI td
f = 0:2 cps
f = 1 cps
Rank
0.862 0.736 0.127 0.811 0.905 0.341 0.748 0.761 0.139 0.685 0.289 0.817 0.842 0.201
2
0.469 0.657 0.353 0.403 0.785 0.612 0.323 0.574 0.294 0.553 0.536 0.772 0.792 0.301
5 1
4 3
f = 5 cps
Rank 4 2 5
3 1
Rank
0.244 0.083 0.664 0.216 0.029 0.713 0.309 0.091 0.514 0.156 0.693 0.039 0.012 0.122
3 1 4 2
Table III. Correlation coeOcient between hysteretic energy EH for elastoplastic systems with response ductility = 3 and various intensity indices. Index dmax vmax amax Ed and drs Ev and vrs IA and Ea and ars Pd and drms Pv and vrms Pa and arms PD IC IF SI td
f = 0:2 cps
f = 1 cps
Rank
0.918 0.750 0.027 0.886 0.871 0.175 0.839 0.766 0.052 0.669 0.140 0.804 0.826 0.129
1
0.629 0.781 0.276 0.531 0.901 0.549 0.478 0.723 0.281 0.478 0.488 0.878 0.917 0.245
2 3 4
6 5
f = 5 cps
Rank 4 2 5
3 1
0.163 0.108 0.817 0.249 0.050 0.786 0.201 0.140 0.751 0.044 0.839 0.069 0.133 0.110
Rank
2 3 4 1
provided noteworthy advantage over the peak ground motions to predict elastic and inelastic spectral ordinates, amax ; vmax and dmax must be regarded as signi=cant intensity parameters to characterize the earthquake demand, and especially because they can be estimated for future earthquakes with relative ease; (d) Housner’s intensity is the best index for f = 1 cps, ranks well for f = 0:2 cps, but does poorly for rigid systems; it should also be noted that SI is a response variable, hence, it is less desirable as a predictor variable; (e) similarly, Nau and Hall [10] found that using Housner’s intensity as scaling parameter, but computed over three di?erent ranges of frequency, provided less dispersion in the ordinates of normalized elastic Copyright ? 2001 John Wiley & Sons, Ltd.
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spectra than that which resulted from normalization to the peak ground motion parameters; however, the advantage faded away for inelastic systems with response ductilities larger than 3; (f) although the duration of motion itself does not correlate well with input energy, nor dissipated energy, it provides a signi=cant improvement of correlation when combined with other indices (Park’s index presents better correlation with energy than arms , and Fajfar’s index improves the correlation of vmax ). The previously discussed study permitted the narrowing down of the possibilities to a few indices. Then two further analyses were carried out. First, examining the scatter of energy spectra (EH ) computed for numerous frequencies, it was con=rmed that the above trends were not limited to the particular control frequencies used, but actually extended to the entire frequency range they were meant to represent. And second, considering the convenience of incorporating td , new compound intensity indices of the form I = Q 1 td 2
(22)
were evaluated. The exponents 1 and 2 were determined by means of an optimization scheme. The objective was to minimize, over the three relevant spectral regions, the average coeOcient of variation COV of energy spectra (EH ) for the 52 records normalized using I as scaling parameter. The de=nition of COV will be presented in the next section. Exponents were calculated for numerous cases: using the most promising intensity indices Q in the three spectral regions, for the three types of force–deformation relationships, and for =ve levels of the response ductility factor (1.5, 2, 3, 5 and 10). For each case, the optimization procedure consisted of varying 1 and 2 in 0.1 increments starting from 1 = 2 = 0, computing COV for each pair ( 1 ; 2 ) and plotting contour curves of COV. A typical example of such plots is presented in Figure 3, where the optimum pair is approximately 1 = 0:75 and 2 = 0:35, for a minimum COV = 0:43. It can be seen in Figure 3 that COV is not very sensitive to 1 and 2 since the surface COV( 1 ; 2 ) has small curvature in the vicinity of its minimum value. It was found that the indices Q in Equation (22) that led to smaller COVs were dmax in the displacement region, vrms in the velocity region and amax and arms in the acceleration region; however, using vrms instead of vmax produced less than 9 per cent reduction on COV, while using arms instead of amax resulted in negligible variation of COV for elastoplastic and bilinear systems and 13.6 per cent reduction for sti?ness degrading systems. Considering that recommendations to estimate the root-mean-square values of future ground motions are not available, compound intensity indices including only the peak ground motion parameters dmax ; vmax and amax are proposed. On the other hand, since COV is not very sensitive to 1 and 2 , and the optimum pairs ( 1 ; 2 ) are not signi=cantly di?erent when di?erent ductility levels and di?erent load–deformation relationships are considered, values 1 and 2 that can be approximately applied for all cases were selected. Thus, the following compound intensity indices are recommended to normalize ground motions to predict energy dissipation during earthquakes: Id = dmax td1=3
(23)
2=3 1=3 td Iv = vmax
(24)
Ia = Copyright ? 2001 John Wiley & Sons, Ltd.
amax td1=3
(25a)
amax
(25b) Earthquake Engng Struct. Dyn. 2001; 30:1791–1816
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Figure 3. Contours of the average coeOcient of variation in the velocity region for 1 energy spectra normalized to vmax td 2 . Sti?ness degrading systems with 5 per cent damping and response ductility = 3.
where Id applies in the displacement region of the spectrum for any load–deformation model, Iv applies in the velocity region for any model too, and Ia applies in the acceleration region, with Equation (25a) being suitable for sti?ness degrading models and Equation (25b) for elastoplastic or bilinear systems.
STATISTICAL ANALYSIS OF EH SPECTRA Spectra of energy dissipated by inelastic behaviour EH were computed for the 52 records listed in Table I. An example is √shown in Figure 4. It was found convenient to present the energy spectrum in terms of EH , because this quantity is linearly proportional to the ground motion amplitude, i.e. if the ground acceleration is ampli=ed by a factor ! the energy spectrum is ampli=ed by the same factor. At the same time, since the yield levels of the inelastic systems are taken as a fraction of the elastic response displacement, when a record is ampli=ed by a factor ! and the yield level is ampli=ed by the same factor, the response ductility factor is the same as that of a system with the non-ampli=ed load–deformation relationship subjected to √ the non-ampli=ed motion. In turn, since EH corresponds to dissipated energy per unit of mass, EH has velocity units and the three axes of the tri-partite logarithmic plot have the same units of the conventional response spectrum. Therefore, it is appropriate to Copyright ? 2001 John Wiley & Sons, Ltd.
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Figure 4. Dissipated energy spectrum for the Sylmar N00E record of 17 January 1994. Sti?ness degrading systems with 5 per cent damping.
refer to the three regions of the energy spectrum as displacement, velocity, and acceleration regions. As a =rst step, average spectra are computed. Designating the energy spectrum as (26) SH = EH = SH (f; ; ; R(u)) the normalized average spectrum is given by 1 SHi (f) SZH (f) = n i=1 Ii n
(27)
where f is the frequency (f = !=2#), Ii is the normalization factor for the ith record, n is the number of records and SHi is the ith spectrum for given values of and and for a given restoring-force model. Average energy spectra for sti?ness degrading systems are 2=3 and amax , presented in Figures 5, 6 and 7, for spectra normalized to the indices dmax ; vmax respectively; each =gure is pertinent only in the frequency range that corresponds to the scaling index used. The shapes of the average spectra for elastoplastic and bilinear systems are similar to those shown in√these =gures. It can be observed that: (a) for low frequencies, and for a given value of , EH =! slightly increases with !, it does not vary signi=cantly with , and it is approximately equal to the peak ground displacement dmax (Figure 5); (b) Copyright ? 2001 John Wiley & Sons, Ltd.
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Figure 5. Average energy spectra for records normalized to peak ground displacement (dmax ). Sti?ness degrading systems with 5 per cent damping.
√ for intermediate frequencies (Figure 6), EH does not vary signi=cantly √ with either, but increases as ! increases; and (c) for high frequencies (Figure 7), ! EH decreases as ! increases, but increases with . The average spectra suggest an energy spectrum shaped as shown in Figure 8. Note that the trilinear spectrum is not parallel to the three axes of the tripartite logarithmic plot. The next step is to compute the corner frequencies fdv and fva (Figure 8) that de=ne the three spectral regions. Lower and upper limits of 0.05 and 20 cps were arbitrarily chosen as boundaries of the frequency range of practical interest. The trilinear spectrum is de=ned in the logarithmic space by exponential curves
$ d = % d f &d
(28a)
$ v = % v f &v
(28b)
&a
(28c)
$a = % a f
where the six regression coeOcients % and & are determined by minimizing the square error between the trilinear spectrum $ and the average energy spectrum SZH (Equation (27)) in each spectral region: [2 = Copyright ? 2001 John Wiley & Sons, Ltd.
nf j=1
wj [SZH (fj ) − $(fj )]2
(29)
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2=3 Figure 6. Average energy spectra for records normalized to vmax . Sti?ness degrading systems with 5 per cent damping.
where nf is the number of frequencies, wj = 0:5(fj+1 − fj−1 )=(fu − f‘ ) is a weight factor to account for unequally spaced frequencies, and f‘ and fu are the lower and upper frequencies of the corresponding spectral region. The iterative procedure begins with assumed values of the corner frequencies, and new values are computed in each cycle. The conditions $v = !dv $d and $a = !va $v hold at the corner frequencies fdv and fva ; respectively (with !dv = 2#fdv and !va = 2#fva ). Thus, using Equations (28), the corner frequencies are
fdv(i) =
fva(i) =
%(i) v 2#%d(i) %(i) a 2#%(i) v
1
1+&d(i) −&v(i)
1
1+&v(i) −&a(i)
(30a)
(30b)
where i denotes the ith cycle. The procedure converges rapidly until f(i+1) is as close to f(i) as desired at each corner. The =nal step is to compute statistics in each spectral region, for each ductility factor, and for each restoring force model. The variance and the standard Copyright ? 2001 John Wiley & Sons, Ltd.
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Figure 7. Average energy spectra for records normalized to peak ground acceleration (amax ). Sti?ness degrading systems with 5 per cent damping.
Figure 8. Shape of the smoothed spectrum of energy dissipation.
Copyright ? 2001 John Wiley & Sons, Ltd.
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deviation are de=ned as n 1 VAR($) = n i=1
)($) =
fu f‘
[SHi (f)=Ii − $(f)]2 df fu − f‘
VAR($)
(31) (32)
where i denotes the ith record, and n is the number of records. In turn, as mentioned in the previous section, for the evaluation of the compound intensity indices, the average coeOcient of variation over a spectral region was computed as COV($) =
j
wj COV[$(fj )]
(33)
where COV[$(fj )] is the discrete coeOcient of variation for each frequency in the region given by
1 1 [SHi (fj )=Ii − $(fj )]2 (34) COV[$(fj )] = $(fj ) n i The =nal results of the method outlined above are summarized in Tables IV, V and VI for the elastoplastic, bilinear and sti?ness degrading models, respectively. These tables give the coeOcients % and & (Equation (28)) and statistics according to Equations (32) and (33); in addition to the results associated with normalization to the indices given by Equations (23) –(25), which produce the least dispersion, the results related to normalization by the peak ground motion parameters with no regard to td are also provided. The calculated statistics are√based on the assumption that the square root of the energy dissipated per unit of mass ( EH ) is normally distributed. The assumption is sound if the derived probability distribution for EH ; the variable of interest, presents good =t with the actual response data. The =tness was checked applying the Kolmogorov–Smirnov test at three control frequencies that presented the most scatter; it was found that the test satis=ed a signi=cance level of 5 per cent by an ample margin. As a part of the study, factors for constructing elastic and inelastic design spectra were also obtained. The procedure due to Veletsos–Newmark–Hall-Mohraz [18–23] and later revised by Riddell–Newmark [3; 24] is well known. Indeed, the method is simpler than the extension to energy spectra presented above, because the ordinates of the trilinear design spectrum, in the region response ampli=cation, are parallel to the corresponding axes of the tripartite grid. Factors for elastoplastic systems with 5 per cent damping are presented in Table VII. These factors can be conservatively used for bilinear and sti?ness degrading systems, since on the average the ductility demand on them is less than that imposed on elastoplastic systems, as earlier reported [3; 25] and con=rmed in this work. The ordinates of the design spectrum S for given ductility factor are obtained applying the ampli=cation factors (Table VII) to the corresponding peak ground motion parameters pg : S = Copyright ? 2001 John Wiley & Sons, Ltd.
pg
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Table IV. Factors for constructing energy dissipation spectra (EH ) for elastoplastic systems with 5 per cent damping. Spectral region and normalization index
$ = % f&
Ductility
Standard deviation
COV
%
&
)
1.5 2 3 5 10
0.58 0.60 0.59 0.53 0.44
0.18 0.11 0.04 −0:14 −0:05
0.17 0.18 0.17 0.17 0.16
0.38 0.34 0.31 0.31 0.34
Displacement dmax
1.5 2 3 5 10
1.49 1.58 1.53 1.38 1.15
0.20 0.13 0.06 −0:01 −0:04
0.44 0.50 0.55 0.59 0.57
0.38 0.37 0.38 0.42 0.48
2=3 1=3 td Velocity vmax
1.5 2 3 5 10
1.28 1.55 1.75 1.85 1.88
0.15 0.17 0.18 0.17 0.14
0.82 0.94 0.99 0.99 0.93
0.57 0.53 0.48 0.45 0.42
2:3 Velocity vmax
1.5 2 3 5 10
3.20 3.88 4.42 4.74 4.86
0.13 0.15 0.15 0.15 0.13
2.13 2.50 2.78 2.90 2.83
0.60 0.57 0.55 0.53 0.51
Acceleration amax
1.5 2 3 5 10
3.56 4.40 5.16 5.50 5.73
−0:48 −0:44 −0:39 −0:29 −0:18
0.49 0.65 0.84 1.06 1.31
0.38 0.38 0.38 0.38 0.35
Displacement
dmax td1=3
where pg represents dmax , vmax , or amax depending on the spectral region of interest. The elastic spectrum Se is given by Equation (35) for the particular case = 1; wherefrom the inelastic spectrum can be alternatively obtained as S = , Se
(36)
where , is also given in Table VII. It is worth commenting here that Equation (36) is often misunderstood as equivalent to ‘deriving inelastic spectra from elastic response analyses’. This is certainly not the case because actual inelastic responses directly lead to the factors (average ampli=cation with respect to the peak ground motion parameters). Simple approximations for , are well known: the ratios 1= for the displacement and velocity regions, and 1= 2 − 1 for the acceleration region (which were shown [3] to be unconservative for high ductility and high damping, as also apparent in this study). Table VII also provides the standard deviation ) and the coeOcient of variation COV = ) = calculated over the corresponding spectral regions. Copyright ? 2001 John Wiley & Sons, Ltd.
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Table V. Factors for constructing energy dissipation spectra (EH ) for bilinear systems with 5 per cent damping. Spectral region and normalization index
$ = % f&
Ductility
Standard deviation
COV
%
&
)
Displacement dmax td1=3
1.5 2 3 5 10
0.59 0.62 0.59 0.51 0.41
0.19 0.12 0.04 −0:03 −0:08
0.17 0.18 0.17 0.17 0.14
0.38 0.33 0.31 0.31 0.34
Displacement dmax
1.5 2 3 5 10
1.52 1.61 1.53 1.33 1.05
0.21 0.14 0.06 −0:02 −0:08
0.44 0.50 0.56 0.58 0.53
0.38 0.37 0.39 0.43 0.48
2=3 1=3 td Velocity vmax
1.5 2 3 5 10
1.29 1.56 1.77 1.89 1.88
0.16 0.18 0.18 0.15 0.14
0.84 0.96 1.01 0.98 0.89
0.58 0.54 0.48 0.44 0.40
2=3 Velocity vmax
1.5 2 3 5 10
3.22 3.93 4.51 4.85 4.86
0.14 0.16 0.16 0.13 0.12
2.22 2.63 2.88 2.91 2.76
0.62 0.59 0.56 0.53 0.50
Acceleration amax
1.5 2 3 5 10
3.64 4.67 5.39 5.85 5.95
−0:49 −0:46 −0:37 −0:27 −0:13
0.51 0.71 0.95 1.24 1.59
0.39 0.40 0.41 0.40 0.37
The COV values in Table VII are consistent with previous studies. Riddell and Newmark [3] found COVs in the range 0.18–0.22 in the acceleration region of the spectrum, 0.31–0.39 in the velocity region and 0.41–0.49 in the displacement region, for between 1 and 10, while Riddell [25] obtained COVs between 0.19–0.31, 0.25–0.4, and 0.33–0.44 in the mentioned regions respectively, for the same damping and range of . Miranda [26] and Riddell [25] have reported COV values for the response modi=cation factor R (R , the ratio between elastic and inelastic response, is a close relative of , , the former being calculated for individual frequencies while the latter is a frequency-band ratio). Miranda [26] found COV(R ) varying between about 0.25 and 0.45 for groups of records on rock and alluvial soils, for between 2 to 6, with COV increasing as ductility increased; Riddell [25] obtained practically the same COVs for R . It is worth recalling that Miranda [26] and Riddell [25] considered earthquake records grouped according to soil conditions and thus the dispersion should decrease due to the similar frequency content of the motions, whereas in this study a wide variety of ground motions regarding soil conditions and tectonic settings were used. In a recent study, Ordaz Copyright ? 2001 John Wiley & Sons, Ltd.
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Table VI. Factors for constructing energy dissipation spectra (EH ) for sti?ness degrading systems with 5 per cent damping. Spectral region and normalization index
$ = % f&
Ductility
Standard deviation
COV
%
&
)
1.5 2 3 5 10
0.69 0.66 0.56 0.46 0.34
0.21 0.14 0.06 0.01 −0:04
0.18 0.17 0.16 0.13 0.10
0.34 0.30 0.29 0.29 0.30
Displacement dmax
1.5 2 3 5 10
1.78 1.69 1.44 1.17 0.88
0.22 0.15 0.07 0.01 −0:03
0.50 0.52 0.50 0.45 0.36
0.37 0.36 0.37 0.39 0.40
2=3 1=3 td Velocity vmax
1.5 2 3 5 10
1.67 1.85 1.91 1.87 1.75
0.17 0.18 0.18 0.18 0.18
1.00 1.04 1.00 0.92 0.83
0.52 0.48 0.44 0.40 0.37
2=3 Velocity vmax
1.5 2 3 5 10
4.24 4.70 4.88 4.81 4.54
0.15 0.16 0.16 0.16 0.17
2.84 3.02 2.97 2.83 2.61
0.59 0.55 0.52 0.49 0.47
Acceleration amax td1=3
1.5 2 3 5 10
2.18 2.39 2.37 2.27 2.12
−0:41 −0:33 −0:21 −0:08 −0:06
0.41 0.50 0.60 0.71 0.83
0.49 0.46 0.42 0.38 0.34
Acceleration amax
1.5 2 3 5 10
5.49 6.05 6.10 5.88 5.54
−0:41 −0:33 −0:22 −0:09
1.10 1.39 1.71 2.10 2.62
0.53 0.50 0.47 0.45 0.42
Displacement
dmax td1=3
0.05
and Perez [27] proposed rules to predict R that featured better accuracy than other available relationships; it should be noted, though, that they predicted R on the basis of response quantities: the relative velocity spectrum and=or the displacement response spectrum. The factors in Tables IV–VII, however, are predicted on the basis of a priori parameters: the peak ground motion parameters or other ground motion intensity indices. COV (average COV) of hysteretic energy spectra in the displacement and acceleration regions can be held in the range 0.3–0.5 if the appropriate normalization index is used (indices including td ). The larger COV in the velocity region could be reduced if vrs o vrms were used instead of vmax ; as mentioned above; however, estimates of the former indices for future earthquakes are not available and Copyright ? 2001 John Wiley & Sons, Ltd.
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Table VII. Factors for constructing elastic and inelastic design spectra for systems with 5 per cent damping. Spectral region and normalization index
Ductility
Displacement dmax
1 1.5 2 3 5 10
Velocity vmax
Acceleration amax
)
COV
,
1.76 1.07 0.76 0.50 0.30 0.15
0.77 0.45 0.32 0.23 0.14 0.07
0.44 0.42 0.42 0.45 0.46 0.48
1.00 0.61 0.43 0.29 0.17 0.08
1 1.5 2 3 5 10
1.67 1.05 0.78 0.54 0.36 0.22
0.78 0.42 0.31 0.21 0.14 0.08
0.46 0.40 0.40 0.39 0.37 0.36
1.00 0.62 0.47 0.32 0.22 0.13
1 1.5 2 3 5 10
2.09 1.46 1.23 1.02 0.84 0.67
0.71 0.38 0.28 0.22 0.18 0.16
0.34 0.26 0.23 0.21 0.21 0.24
1.00 0.70 0.59 0.49 0.40 0.32
2=3 1=3 so the latter was preferred. If vmax td is used as recommended (Equation (24)), COV in the velocity region ranges between 0.37 and 0.58. Such a range denotes large uncertainty but it is not extraordinarily larger than the above-commented values.
CONSTRUCTION OF DISSIPATED ENERGY SPECTRA AND APPLICATION TO DAMAGE CONTROL The =rst step in the construction of energy and design spectra involves the de=nition of the earthquake hazard in terms of estimates of the expected peak ground motion parameters at the site under consideration. A discussion of this subject is beyond the scope of this paper. The spectra presented in this section will be anchored to a peak ground acceleration of 1g; which is only a referential value selected for illustrative purposes and has no e?ect on the observations to be made. The design peak ground velocity and displacement will be de=ned 2 . In this study, mean on the basis of average values of the ratios vmax =amax and amax dmax =vmax values of 98:5cm=sec=g and 4 were obtained for the previously mentioned ratios, while Riddell and Newmark [3] found averages of 89 and 6, respectively. Assuming vmax =amax = 85 and 2 = 6; vmax = 85 cm=sec and dmax = 44 cm are obtained. Next, energy dissipation amax dmax =vmax spectra and inelastic spectra required for damage assessment will be constructed. Spectra for elastoplastic systems with 5 per cent damping and response ductility = 5 will be considered. Copyright ? 2001 John Wiley & Sons, Ltd.
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Energy spectrum Assuming that no estimate of the ground motion duration td is available, the design ground motion (Figure 8) for constructing the energy spectrum (ES) is simply taken as dmax = 44 cm, 2=3 vmax = 852=3 = 19 cm=sec; and amax = 1g. The factors to determine the spectral ordinates are given in Table IV. Supposing that some degree of conservatism is desired, factors associated with the mean plus one standard deviation level will be used (15.9 per cent probability of exceedance). Thus, the spectral ampli=cation factors for = 5 are $d = (1:38 + 0:59)f−0:01 = 1:97f−0:01 in the displacement region, $v = (4:74+2:9)f0:15 = 7:64f0:15 in the velocity region, and $a = (5:5 + 1:06)f−0:29 = 6:56f−0:29 in the acceleration region. For f = 0:05; the lowest frequency of the spectrum (Figure 8), the ampli=cation factor $d = 2:03 is ob√ tained; then EH =! = dmax $d = 89 cm√determines point J of the spectrum (Figure 8). Similarly, for f = 20; $a = 2:75 gives ! EH = 2:75amax = 2:75g which corresponds to point M 2=3 (Figure 8). The corner frequency fdv is calculated from the condition dmax $d ! = vmax $v i.e., √ 2=3 −0:01 0:15 ! = 145:2f 86:7f which yields fdv = 0:207 and EH = vmax $v = 145:2(0:207)0:15 = 2=3 $v ! = amax $a ; 115 cm=sec (point K in Figure 8). The corner frequency fva results from vmax √ 2=3 0:15 −0:29 i.e. 145:2f ! = 6433f ; wherefrom fva = 3:882cps and EH = vmax $v = 145:2(3:882)0:15 = 178 cm=sec (point L in Figure 8). The completed trilinear energy spectrum is plotted in √ Fig√ √ ure 9, for which the relevant labels of the tripartite grid axes are EH =!, EH ; and ! EH in the displacement, velocity, and acceleration regions, respectively. It can be seen in Figure 9 that the energy dissipation demand varies considerably with frequency. At the peak of the smoothed energy spectrum (f = 3:88 cps) EH = 31684 cm2 =sec2 while at the ends of the spectrum EH is 780 and 460 cm2 =sec2 ; respectively, i.e. ratios of the order of 40 and 70, respectively. Inelastic spectra As selected above, the design ground motion parameters are dmax = 44 cm; vmax = 85 cm=sec; and amax = 1g. In this case, the spectrum to be constructed =rst is the inelastic yield spectrum [3] (IYS), also known as constant-ductility spectrum [28], which corresponds to a plot of the yield deformation uy necessary to limit the maximum deformation of the system to a speci=ed multiple of the yield deformation itself (umax = uy ). Since the factors given in Table VII synthesize the characteristics of a family of 52 earthquake records, the spectrum corresponds to a smoothed design spectrum, in opposition to a response spectrum that refers to the response to one speci=c excitation. In this case, the spectral quantities of interest are uy in the displacement axis and !2 uy in the acceleration axis; the latter multiplied by the mass gives the yield resistance Fy of the system, which in the case of an elastoplastic system is also its maximum strength. In the ampli=ed region of the spectrum, between 0.15 and 10 cps; the spectral ordinates are obtained using the mean-plus-one-standard-deviation ( +) ) factors given in Table VII for = 5: 0.44, 0.5 and 1.02 for the displacement, velocity, and acceleration regions, respectively. Thus, the spectral ordinates are: 44 × 0:44 = 19 cm; 85 × 0:5 = 43 cm=sec and 1g × 1:02 = 1:02g. The spectrum is completed with transition zones: in the lowest frequency (0:05cps) the spectral ordinate is dmax = = 44=5 = 8:8cm; in the highest frequency (33cps) the spectral ordinate is [3] amax −0:07 = 0:89g. The complete inelastic design spectrum is presented in Figure 9. The second spectrum of interest is the total deformation Copyright ? 2001 John Wiley & Sons, Ltd.
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Figure 9. Inelastic design spectrum, dissipated energy spectrum, and required deformation capacity spectrum, for design ductility = 5 and design ground motion speci=ed by amax = 1g; vmax = 85 cm=sec, and dmax = 44 cm.
spectrum (TDS), which simply gives the maximum deformation of the system umax = uy . It is easily obtained by multiplying the IYS or constant ductility spectrum by . In this case the relevant axis of the tripartite logarithmic plot is the displacement axis which corresponds to umax . Although this type of presentation has been available for around 40 years, it is less known; it should become more popular as the need to estimate maximum deformations arises in connection with displacement based design. This spectrum is not included in Figure 9, to avoid confusion. Damage considerations A great deal of e?ort has been taken to gain insight into the factors controlling damage and collapse during earthquakes. A detailed presentation and discussion of this subject exceeds the scope of this paper, instead several key references which contain numerous entries to the literature on the subject are given [29–36]. There seems to be agreement on the fact that earthquake damage occurs not only due to maximum deformation or maximum ductility attained, but is associated with the hysteretic energy dissipated by the structure as well. Park and Ang [30] proposed a simple index for seismic damage assessment of reinforced concrete Copyright ? 2001 John Wiley & Sons, Ltd.
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structures: DPA =
umax mEH + uu Fy uu
(37)
where umax is the maximum deformation under earthquake excitation (as de=ned above), uu is the ultimate deformation capacity under monotonic loading, Fy is yield resistance (as de=ned above), mEH is the total hysteretic energy dissipation, is a parameter that weights the e?ect of cyclic loading on structural damage and DPA ¿1 means complete collapse or total damage. Note that umax =uu would be equal to 1 if umax was measured in a monotonic loading test, in turn mEH =Fy uu would also equal 1 under such a test. The clear implication of Equation (37) is that, under earthquake loading, when energy dissipation takes place, umax cannot reach uu . Values of based on experimental data [30] varied between −0:3 and 1.2, with a median [35] of 0.15. Since the latter value has been also used by other authors [30; 34–36], = 0:15 will be taken for the following example. Further elaboration on appropriate values of for di?erent structural materials and con=gurations is probably needed. Using the energy spectra presented herein combined with Newmark–Hall spectra, a simple estimation of the required deformation capacity of a structure can be made. Indeed, taking DPA = 1; the ultimate deformation capacity supplied to the structure must comply with uu ¿umax +
mEH Fy
(38)
recalling that uu is the design capacity based on monotonic testing data and monotonic behaviour knowledge, while the second member of Equation (38) corresponds to earthquake response quantities. The latter quantities are directly read from the spectra presented above; in fact, umax is the TDS, Fy =m is the IYS but read in the acceleration axis (or IYS · !2 if read in the displacement axis), and EH is the ES squared. In symbolic form, for each frequency, Equation (38) can be written as UDCS¿TDS +
ES2 IYS
(39)
where UDCS is the ultimate deformation capacity spectrum. In other words, UDCS gives the required monotonic deformation capacity for the structure to survive the design earthquake without collapse. The UDCS is plotted in Figure 9 according to Equation (39); naturally it is read in the displacement axis. Note also that di?erent levels of acceptable damage, i.e. performance, may be established by taking di?erent values of DPA ; for example taking DPA = 0:5 the design condition given by Equation (38) becomes mEH uu ¿2 umax + (40) Fy The appeal of this expression is that the quantity in parenthesis in the second member does not change, i.e. it is obtained from the same design spectra based on the same design ground motion, but certainly the structure should be provided more displacement capacity if better performance is desired. The implication is that performance based design need not be speci=ed through a set of ground motions of di?erent intensities, but through one design motion with performance controlled by the design parameters selected (Fy or ) and deformation capacity Copyright ? 2001 John Wiley & Sons, Ltd.
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supplied (uu ). Finally, it is worth mentioning that an alternative approach to damage control has been proposed [33] in terms of ‘equivalent ductility’, which corresponds to a ductility limit that cannot be exceeded in order to satisfy a given performance level (permissible damage). As a consequence of the deterioration resulting from cyclic inelastic behaviour, the equivalent ductility max is smaller than the ultimate ductility capacity u determined under monotonic loading. The method requires estimation of a parameter that depends on EH and umax ; which can be readily determined with the energy spectra and total deformation spectra presented in this paper. SUMMARY AND CONCLUSIONS This study has attempted to contribute to a better understanding of hysteretic energy dissipation in single-degree-of-freedom systems, since it has been recognized for some time that energy dissipation is a form of structural damage, and thereby plays an important role in the assessment of seismic performance. A basic investigation was carried out to identify ground motion intensity indices that correlated well with input and dissipated energy. It was found that: (a) no index shows satisfactory correlation with energy in the three spectral regions simultaneously; (b) peak ground motions parameters present good correlation with energy, specially in the displacement and acceleration regions, while Housner’s intensity is the best in the intermediate region; and (c) compound indices of the form I = Q 1 td2 ; where Q is a peak ground motion parameter and td is the signi=cant duration of motion, are recommended as most appropriate to normalize hysteretic energy spectra, since incorporating td provides less dispersion of the spectral ordinates, and because design ground motion parameters can be selected with relative ease. The next step was the statistical analysis of hysteretic energy spectra computed for 52 earthquake records, to produce rules for constructing energy dissipation spectra applying ampli=cation factors to a selected design ground motion, in much the similar fashion as the Newmark–Hall method, although somewhat more elaborated in this case because the spectral ordinates are not parallel to the axes of the tripartite logarithmic plot. Factors to construct energy spectra for elastoplastic, bilinear and sti?ness degrading systems are presented, including statistics to account for spectral ordinates associated to di?erent probabilities of exceedance. Finally, on the basis of the damage model due to Park and Ang, and using the proposed hysteretic energy spectra combined with Newmark–Hall spectra, estimates of the required deformation capacity of a structure associated with di?erent performance levels can be made. With the energy spectrum the response information for single-degree-of-freedom systems is complete, and relationships between strength, deformation, ductility, energy dissipation, and damage become established. Now, the reliability of the damage prediction or assessment will mostly depend on the adequacy of the damage model.
ACKNOWLEDGEMENTS
This study was carried out in the Department of Structural and Geotechnical Engineering at the Universidad CatGolica de Chile with =nancial assistance from the National Science and Technology Foundation of Chile (FONDECYT) under grant No. 1990112. Copyright ? 2001 John Wiley & Sons, Ltd.
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