Arroyo y Ordaz 2007 Energía Histerética EESD

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2007; 36:23652382Published online 27 July 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.736

On the estimation of hysteretic energy demandsfor SDOF systems

Danny Arroyo1,2,, and Mario Ordaz2

1Departamento de Materiales, Universidad Autonoma Metropolitana, Azcapotzalco Avenida San Pablo 180,Colonia Reynosa Tamaulipas, Mexico DF 02200, Mexico

2Instituto de Ingeniera, Universidad Nacional Autonoma de Mexico, Avenida Universidad 300, CiudadUniversitaria, Coyoacan, Mexico DF 04510, Mexico

SUMMARY

Based on a statistical study of the dynamic response of single degree of freedom (SDOF) systems subjectedto earthquake ground motions, a rule to estimate hysteretic energy demands is proposed. Expressions forelasticperfectly plastic, stiffness degrading and pinching SDOF systems were obtained. The proposedrule does not explicitly depend on the period of the system; instead, it depends on the elastic pseudo-acceleration and elastic velocity spectra. It is shown that the proposed rule can be applied to computehysteretic energy demands for systems located at different soil conditions. In addition, information aboutscatter and bias of the proposed rule is also presented. Copyright q 2007 John Wiley & Sons, Ltd.

Received 8 November 2006; Revised 12 April 2007; Accepted 17 June 2007

KEY WORDS: hysteretic energy; strong ground motion duration; constant ductility spectra; stiffnessdegrading system

INTRODUCTION

Several researchers have included the effect of ground motion duration on the seismic design ofstructures through the use of the energy approach [115]. Most of these studies have consideredthat the damage on a structure subjected to earthquake ground motions can be related to the peakdisplacement that the system undergoes and to the amount of hysteretic energy that the systemdissipates during a seismic event. Usually, these seismic demands are compared with the structuralsupplies through the use of seismic damage indexes to assess the seismic behavior of structures.In order to apply these methods in the seismic design of structures, it is necessary to have accurateestimations of seismic demands and also of structural supplies.

Correspondence to: Danny Arroyo, Departamento de Materiales, Universidad Autonoma Metropolitana, AzcapotzalcoAvenida San Pablo 180, Colonia Reynosa Tamaulipas, Mexico DF 02200, Mexico.

E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

2366 D. ARROYO AND M. ORDAZ

Abundant research has been aimed at developing expressions to compute hysteretic energydemands. In the following paragraphs, we discuss some of the related previous studies.

Fajfar and VidicFajfar and Vidic [10] proposed the following expression to compute hysteretic energy demandsper unit of mass (EH) for bilinear single degree of freedom (SDOF) systems with 10% of strainhardening:

EH =(SA(T )R(T )

)2(1)

where SA(T ) is the pseudo-acceleration spectrum valued at the period of the system, T ; is themaximum ductility demand; is the circular frequency of the oscillator; R(T ) is the correspondingstrength-reduction factor and is given in the following equation:

= zT zzg (2)where

zT =

0.9 if TT1

0.9 0.25 T T1T2 T if T1TT2

0.65 if TT2

(3)

T1 = 2 cv PGVca PGA

(4)

T2 = 2 cd PGDcv PGV

(5)

z = ( 1)c

(6)

zg =(

a2 dtPGA PGV

)cg(7)

where PGA, PGV and PGD are the peak ground acceleration, velocity and displacement, respec-tively. T1 and T2 are the limits between the short and medium period range and between the mediumand large period range of the NewmarkHall spectrum, respectively. For bilinear non-degrading sys-tems with damping proportional to the instantaneous stiffness, c = 0.67, cg=0.4, ca = 2.4, cv=1.9and cd = 1.7.

Fajfar and Vidic [10] stated that Equation (1) could not be applied in the case of narrow-bandground motions, such as the motions recorded at very soft soil sites.

The basic shape of Equation (1) is given by the pseudo-velocity spectrum (SV(T )= SA(T )/)normalized by the R spectrum. Hence, the effect of ground motion duration comes indirectlythrough SV(T ) and the parameter zg . In addition, note that Equation (1) depends on response

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2007; 36:23652382DOI: 10.1002/eqe

ESTIMATION OF HYSTERETIC ENERGY DEMANDS FOR SDOF SYSTEMS 2367

parameters (SV(T ) and R(T )) and explicitly (although weakly) depends on the period of thesystem, through the parameter zT .

ManfrediManfredi [13] proposed the following equation to compute EH for elasticperfectly plastic SDOFsystems:

EH = (c 1) neq(

SA(T )

)2 ( 1R(T )

)2(8)

where c is the maximum cyclic ductility demand and neq is the number of equivalent cycles,given by

neq = 1 + 0.18(R(T ) 1)3/5 ID1/61/2 (9)

ID = tE

0 a(t)2 dt

PGA PGV(10)

={

T/T1 if TT11 if T>T1

(11)

={/0 if 01 if = 0

(12)

In Equations (10)(12), tE is the earthquake duration, 0 = 0.05, and T1 is the limit between theshort and medium period range of the NewmarkHall spectrum for each record, computed withEquation (4), according to Fajfar and Vidic [10].

As in Fajfar and Vidics rule, the basic shape of Equation (8) is given by the SV(T ) spectrumnormalized by the R spectrum. The effect of ground motion duration comes indirectly throughSV(T ) and the parameter ID . Note that Equation (8) also depends on response parameters (SV(T )and R(T )) and explicitly, but weakly, depends on the period of the system, through the parameter .

Manfredi stated that since a large number of records were used, Equation (8) could be appliedto compute hysteretic energy demands for different types of ground motions. However, he acknow-ledged that Equation (8) could overestimate hysteretic energy demands. In addition, he observedthat Equation (8) was not able to fit the shift of the period of the peak value between the SV(T )spectrum and the hysteretic energy spectrum, shown by some earthquakes.

Riddell and Garcia

Riddell and Garcia [14] proposed the following expression to construct tri-linear logarithmicspectra of the square root of EH for elasticperfectly plastic, bilinear and stiffness degradingSDOF systems:

v = f (13)where v is the square root of EH normalized by certain ground motion parameters dependingon the spectral region considered (i.e. acceleration, velocity or displacement region), f is the



frequency of the system and and are the coefficients that depend on the type of hystereticmodel, maximum ductility demand and on the spectral region considered.

Conversely to the rules listed before, the equations proposed by Riddell and Garcia do notdepend on response parameters. They depend on PGA, PGV, PGD, T and ground motion duration.They used a set of 52 ground motions, most of them recorded at firm soil sites. They reportedcoefficients of variation of the square root of EH in the range of 0.30.49.

Kunnath and Chai

Kunnath and Chai [15] proposed to compute hysteretic energy demands as a fraction of the elasticinput energy demand per unit of mass (EI), as follows:

EH = cEI (14)with

c = 1.13 ( 1)0.82

(15)

EI = 0.5v2e = 0.5(v PGV)2 (16)

v ={v(2T /Tc (T/Tc)2) if TTc

(17)

v =0.25 PGA

PGV

td Tc

+ 0.52 + 2 (18)

where Tc is equal to T1 in Fajfar and Vidics model (Equation (4)), td is the ground motion durationaccording to Trifunac and Brady [16] and is a parameter that depends on the type of groundmotion. They used = 0.5 for the type of ground motions considered in their study (they mainlyused ground motions recorded at firm soil sites). They stated that Equation (15) was obtained fromstiffness degrading SDOF systems considering only four ground motions. As Riddell and Garciasrule, this method does not depend on the response parameters, but explicitly depends on T .

As it can be observed, most of the methods available to compute EH were developed for firmsoil ground motions. Only the model proposed by Manfredi was developed to be applied on systemslocated at different soil conditions. Nevertheless, this model is useful only for elasticperfectlyplastic systems, although it has been demonstrated [11] that the type of hysteretic behavior canhave an important effect on the hysteretic energy demands, especially for ground motions withlarge duration, as those recorded at very soft soils.

In view of this situation, the objective of this study was to obtain empirical rules to computehysteretic energy demands for SDOF systems with different hysteretic behavior. The presentedmodel was developed to be applied to systems located at various soil conditions. In addition,measures of the bias and variability of the model are presented.

HYSTERETIC MODELS CONSIDERED

Four hysteretic models were used: elasticperfectly plastic plus three strength- and stiffness-degrading models. The degrading models were obtained from the model proposed by Kunnath



Figure 1. Hysteretic models considered.

et al. [17], which requires the definition of three parameters that control the level of stiffnessdegradation (k), strength degradation (k) and pinching (k). Although Kunnaths model canconsider a tri-linear envelope of the forcedisplacement relationship, in this paper we decided touse a bi-linear envelope with zero strain hardening.

Figure 1 shows a sketch of the different models considered and in the following paragraphs, thethree degrading hysteretic models are described.

The first degrading model utilized is the modified non-degrading Clough model. Accordingto Kunnath et al. [17] setting k = k = and k = 0.0, their model reduces to a modi-fied non-degrading Clough model. For the numerical analyses, the following values wereadopted: k = 20, k = 0.0 and k = 2.0. As it can be observed in Figure 1(b), this modelresembles the behavior of well-detailed reinforced concrete structures, where the responseis primarily flexural. Note that in this model, only the stiffness of reloading branches isdegraded.

The second degrading model used is Takedas model [18]. According to Kunnath et al. [17]setting k = 2, k = 0.1 and k = 2, their model reduces to Takedas. Note that in this model,the stiffness of the unloading and reloading branches is degraded each time a maximum dis-placement occurs. Also, the strength deteriorates depending on the hysteretic energy dissipatedfor the system. As it can be observed in Figure 1(c), this model resembles the behavior ofreinforced concrete structures with strength and stiffness degradation, but without significantdegradation of energy capacity dissipation.

The third degrading model used is the modified Takedas model. According to Kunnathet al. [17] setting k = 2, k = 0.1 and k = 0.5, their model reduces to modified Takedasmodel. This model is similar to Takedas model, but the energy capacity dissipation is highlydeteriorated due to pinching of hysteretic loops. As it can be observed in Figure 1(d), this



model resembles the behavior of reinforced concrete structures with degradation caused byhigh shear stresses and slippage of steel bars.

PROPOSED ESTIMATION RULE

The form of the proposed rule is defined in Equation (19). It depends on two response parametersand on one duration parameter. In early studies it was considered that under certain hypothesis,hysteretic energy demands could be estimated from elastic spectral velocity [1]; hence, the elasticvelocity spectrum normalized by PGV was adopted as the basic shape of Equation (19). For longperiod systems the ratio between elastic spectral velocity and PGV tend to unity; however, forthis range of periods, EH must tend to zero. Then, the term SA/PGA was included in order toattain the correct limit for long period systems. Finally, the effect of duration on EH was includedthrough the parameter . Thus, the proposed rule is the following:

EHPGV

=(

SA(T )PGA

)a0 (V (T )PGV

)a1a2 sf (19)

where SA(T ) and V (T ) are the pseudo-acceleration spectrum and the elastic velocity spectrumvalued at a period equal to T , respectively; is the normalized duration of the ground motiondefined in Equation (20); sf is a scale factor and a0, a1 and a2 are the free parameters that accountfor the effect of on EH. In turn,

= td Vmax2SDmax

(20)

where Vmax and SDmax are the maxima of the elastic velocity and displacement spectra, respectively,and td is the ground motion duration defined according to Trifunac and Brady [16].

In Equation (20) the ground motion duration is normalized by a measure of its predominantperiod; hence, is a measure of the number of cycles of ground motion. Several ways to normalizetd to obtain the number of cycles of ground motion were studied; the approach presented inEquation (20) was selected because it yielded the best fit in the regression analysis. However, itmust be acknowledged that for each spectral region, a different normalization of td produces thebest correlation with EH, as Riddell and Garcia [14] have observed.

One special difference between Equation (19) and the available rules is the parameter , whichtakes into account the shift of the period of the peak value between the elastic spectra and thehysteretic energy spectrum, shown by some earthquakes. In order to exemplify this observation,V and constant ductility EH spectra for El Centro 1940 N-S record and SCT 1985 E-W recordare shown in Figure 2. The former accelerogram was recorded at a firm soil station in California,while the latter was recorded at a soft soil station in Mexico City during the great 1985 Michoacanearthquake. As it can be observed, especially for the recording at soft soil, the shapes of the EHspectra tend to depart from that of the velocity spectrum. For SCT record, the peak of V spectrumis observed at a period of 2.4 s, while the peak of EH spectra is observed at periods rangingbetween 2 s for elasticperfectly plastic systems and 1.6 s for the modified Takeda systems. ForEl Centro record, the peak value of the V spectrum is observed at a period of 5 s, while the peakof EH spectra is observed at periods ranging between 3.8 s for elasticperfectly plastic systemsand 3.2 s for the modified Takeda systems.



Figure 2. V and EH spectra for El Centro 1940 N-S record and SCT 1985E-W record, = 4 and = 0.05.

Another difference between the rule proposed and the existing ones is that the formulationpresented does not depend explicitly on the value of T . As shown in Figure 2, the relationshipbetween EH and T depends on the soil type. For soft soil ground motions, which in general arenarrow banded, greater values of EH are observed for systems with values of T near to resonance.On the other hand, for firm soil ground motions, which in general are wide-band signals, sucheffect is not observed. Therefore, it was decided to select a model that does not depend explicitlyon T . Hence, it could be applied for systems located at different soil conditions.

Finally, note that Equation (19) has theoretically correct limits: as T approaches zero, Equation(19) approaches zero, while as T approaches infinity, Equation (19) approaches zero at a rate thatdepends on SA(T ). Finally, for = 1, Equation (19) yields zero.

REGRESSION ANALYSIS

A set of 47 strong ground motion records was used in the regression analysis. The records consid-ered are listed in Table I. The set comprises accelerograms recorded at different soil conditions.PGA is in the range of 2.61470 cm/s2, PGV is in the range of 0.1125 cm/s, td is in the rangeof 1.1192 s and is in the range of 2.162.9. As it can be observed, the set is composed of 24accelerograms recorded at rock and firm soil sites and 23 accelerograms recorded at very soft soilsites, in order to attain the same level of accuracy at different soil conditions.



Table I. Strong ground motions records utilized in the regression analysis.

Station Date PGA (cm/s)2 PGV (cm/s) td (s) SoilCaleta 09/19/85 127.0 12.5 27.5 13.0 RockCaleta 02/08/99 3.5 0.1 7.0 24.2 RockSCT 09/19/85 160.0 58.0 36.8 15.0 Very soft soilSCT 09/30/99 20.6 6.4 79.6 35.4 Very soft soilCerro de piedra 09/19/85 15.1 2.8 17.2 5.2 RockCerro de piedra 01/13/90 138.0 3.4 1.1 3.4 RockSector popular 09/14/95 40.4 13.8 92.5 46.0 Very soft soilVilla del mar 09/30/99 38.3 12.5 101.0 38.5 Very soft soilAeropuerto 09/30/99 24.6 10.6 147.0 32.1 Very soft soilCU 09/19/85 32.5 8.3 56.5 23.3 Firm soilCaleta 12/29/99 4.9 0.5 22.0 16.3 RockAeropuerto 09/14/95 30.9 13.8 192.0 39.7 Very soft soilTlahuac 09/19/85 117.0 34.2 57.7 23.5 Soft soilTacubaya 09/19/85 31.8 8.6 61.8 19.2 Firm soilFilo de Caballo 09/19/85 64.0 8.3 26.6 11.2 RockCentral del Abastos 09/19/85 76.5 38.5 128.0 32.3 Very soft soilCentral del Abastos 09/30/99 29.7 16.7 122.0 40.6 Very soft soilCentral del Abastos 09/14/95 32.2 12.8 111.0 36.2 Very soft soilAeropuerto 01/11/97 20.4 11.7 122.0 25.2 Very soft soilXochimilco 09/30/99 38.9 16.7 150.0 42.8 Very soft soilXochimilco 12/10/94 11.8 4.7 128.0 39.9 Very soft soilViveros 09/19/85 40.1 10.7 46.2 16.4 Firm soilRoma A 12/10/94 19.4 6.3 80.2 36.1 Very soft soilEl Centro 05/18/40 350.0 67.6 14.2 3.1 RockCordoba 09/30/99 30.5 9.7 116.0 52.7 Very soft soilIbero 09/30/99 30.4 7.7 76.6 49.1 Very soft soilEl Centro 12/16/55 11.6 1.8 22.9 4.6 RockApeel2 Redwood City 17/10/89 268.8 53.7 9.3 8.4 Soft soilFoster City 17/10/89 277.0 46.1 14.5 5.8 Soft soilTreasure Island 17/10/89 155.0 34.0 4.5 2.1 Soft soilSan Francisco Airport 17/10/89 328.0 29.1 11.3 11.9 Soft soilNorthwestern Balkan/P 13/08/81 2.6 0.1 5.4 5.3 Soft soilCibeles 04/25/89 54.5 19.6 66.2 29.2 Very soft soilCenapred 01/11/97 5.6 1.8 66.6 14.4 Firm soilAeropuerto Zihuatanejo 09/19/85 155.0 17.7 19.1 10.9 RockAcapulco 10/24/93 58.3 2.1 9.5 12.6 RockCayaco 09/30/99 4.8 0.5 55.3 12.4 RockFilo de Caballo 04/25/89 15.3 2.5 19.5 8.0 RockAutodromo 04/25/89 19.2 9.1 141.0 31.3 Very soft soilTCU046 09/20/99 105.0 11.0 16.1 5.7 RockCape Mendocino 04/25/92 1470.0 125.0 6.2 2.2 RockLlolleo 03/03/85 698.0 41.5 35.7 25.5 Firm soilTecamachalco 09/14/95 8.3 2.5 168.0 62.9 Firm soilCampo Libre 09/14/95 13.8 4.5 63.8 21.0 Firm soilFremont mission 17/10/89 121.6 11.5 33.1 9.9 Firm soilParkfield, Stone corral 05/02/83 148.1 8.7 10.7 5.4 RockSan Bernardino 06/28/92 90.4 13.7 26.2 14.9 Firm soil



The size of the set used in the regression analysis is quite arbitrary. However, as it will be shownlater in the paper, when the proposed rule was applied to a second set of ground motions, themeasures of accuracy of the proposed rule were very similar than those observed in the first set.Hence, it was considered that the size of the set of ground motions used in the regression analysiswas acceptable.

In order to fit Equation (19) to EH spectra obtained from SDOF systems subjected to the setof accelerograms considered, the following modified version of Equation (19) was utilized:

y = a0x1 + a1x2 + a2x3 + b (21)where y, x1, x2, x3 and x4 are the natural logarithms of

EH/PGV, SV(T )/PGA, V (T )/PGV,

and sf, respectively.Thus, for a given value of , the coefficients a0, a1, a2 and b can be obtained by multiple

linear regression analysis. The values of = 1.5, 2, 3, 4 and 6 were considered and the values ofT between 0.1 and 5 s were used, while a damping value of = 0.05 was considered.

Then, for each ductility demand and each hysteretic model, the value of that yielded theminimum error was computed by an iterative procedure. The measure of error used is defined inthe following equation

= 1

N

Ni=1

(ln(

EHiEHi

))2(22)

where EH is the observed value of the hysteretic energy and EH is the estimated value of EH,while N is the number of data points considered in the analysis; for each value of , N is equalto the product between the number of periods and the number of records considered.

Since the values of a0, a1, a2 and b are fitted by multiple linear regression analysis, the esti-mations obtained with the proposed model, on average, tend to be unbiased. However, in orderto show this feature, the bias of the estimation was also computed, according to the followingequation:

E = 1N

Ni=1

ln(

EHiEHi

)(23)

Note that the positive values of E imply that on average the model underestimates the observedvalues, while the negative values of E mean that on average the observed values are overestimated.

The obtained results are summarized in Table II. As expected, increases with ductility andwith the level of degradation of the hysteretic behavior. Also, for a given hysteretic model, a0tends to be fairly independent of , while a1 tends to decrease as increases. The parameter a2has more influence on elasticperfectly plastic and Clough systems than on Takedas and modifiedTakedas models; for the former models, a2 increases as ductility increases. Finally, sf = exp(b)increases as ductility increases and is in the range of 0.6873 to nearly 1.

For given T and , the proposed rule can be applied by the interpolation of values given inTable II, which might be impractical. Therefore, it was decided to fit some mathematical functionsto the coefficients of Table II. In all the cases, the simplest function was used. In addition, a0, a1and a2 were constrained to positive values in order to avoid problems with the limit of the modelas T approaches to infinity.



Table II. Coefficients obtained from regression analysis, = 0.05. a0 a1 a2 exp(b) Hysteretic model

1.5 1.0200 0.3328 0.7450 0.0229 0.6873 EPP2.0 1.0333 0.3413 0.7016 0.0700 0.7589 EPP3.0 1.0733 0.3423 0.6462 0.1185 0.7788 EPP4.0 1.0967 0.3463 0.6006 0.1427 0.7767 EPP6.0 1.1400 0.3518 0.5378 0.1676 0.7591 EPP

1.5 1.0845 0.3693 0.6320 0.0895 0.8354 Clough2.0 1.1575 0.3690 0.5773 0.1174 0.8822 Clough3.0 1.3100 0.3640 0.5222 0.1416 0.8865 Clough4.0 1.4333 0.3663 0.4833 0.1446 0.8994 Clough6.0 1.7000 0.3593 0.4573 0.1594 0.8791 Clough

1.5 1.1125 0.3714 0.6370 0.0836 0.7919 Takeda2.0 1.2000 0.3712 0.5811 0.0965 0.8628 Takeda3.0 1.3900 0.3645 0.5367 0.1014 0.9056 Takeda4.0 1.5500 0.3645 0.5112 0.0966 0.9331 Takeda6.0 1.8500 0.3607 0.4903 0.0987 0.9356 Takeda

1.5 1.1426 0.3624 0.6375 0.0840 0.7872 mod-Takeda2.0 1.2394 0.3635 0.5669 0.0958 0.8623 mod-Takeda3.0 1.4353 0.3612 0.5084 0.0901 0.9262 mod-Takeda4.0 1.6348 0.3564 0.4888 0.0885 0.9411 mod-Takeda6.0 1.9000 0.3645 0.4468 0.0779 0.9684 mod-Takeda

mod-Takeda, modified Takeda model.

Based on these considerations, the following expressions to compute a0, a1, a2 and sf = exp(b)are proposed:

= 1 + ( 1)m (24)a0 = k0 (25)

a1 = 1k1 + k2 ln() (26)

a2 = k3 k4/ (27)sf = F0 (1 e(1)) (28)

The values of parameters m, k0, k1, k2, k3, k4, F0 and are given in Table III. Although some dif-ferences were observed, it was considered that Equations (24)(28) yielded acceptable estimationsof , a0, a1, a2 and sf.

The values of obtained with Equation (21) and Equations (24)(28), for a given ductilitydemand, are listed in Table IV. We observe a slight increment in as increases. The values inthe range between 0.526 and 0.627 were obtained.

The value of is comparable to the coefficient of variation of EH; thus, the results presented inTable IV can be compared with the coefficients of variation presented by Riddell and Garcia [14].



Table III. Parameters of equations (24)(28).Hysteretic model m k0 k1 k2 k3 k4 F0

EPP 0.03 0.3428 1.1716 0.3685 0.2152 0.2892 0.7708 4.4301Clough 0.14 0.3655 1.4195 0.4441 0.1825 0.1356 0.8876 5.6537Takeda 0.17 0.3664 1.4678 0.3368 0.1058 0.0273 0.9167 3.8306mod-Takeda 0.19 0.3616 1.4143 0.4637 0.0872 0.0 0.9355 3.4811mod-Takeda, modified Takeda model.

Table IV. The values of and E obtained with the proposed rule.

E

Hysteretic model = 1.5 = 2 = 3 = 4 = 6 = 1.5 = 2 = 3 = 4 = 6EPP 0.583 0.567 0.577 0.593 0.627 0.004 0.010 0.020 0.011 0.035Clough 0.525 0.543 0.569 0.580 0.590 0.021 0.004 0.005 0.005 0.007Takeda 0.534 0.554 0.569 0.571 0.582 0.007 0.078 0.027 0.008 0.070mod-Takeda 0.526 0.555 0.574 0.576 0.583 0.004 0.093 0.045 0.009 0.056mod-Takeda, modified Takeda model.

They obtained values of the coefficient of variation of the square root of EH in the range between0.31 and 0.6, that is, between 0.62 and 1.2 for the coefficient of variation of EH. Note that the peakcoefficient of variation obtained with the proposed method is almost half of the peak coefficientof variation reported by Riddell and Garcia [13]. In addition, Fajfar and Vidic [10] identified thatcoefficients of variation of energy are larger than the coefficients of variation of displacementparameters. Therefore, the accuracy of the proposed method can be considered acceptable.

The values of E obtained with the proposed model are also given in Table IV. As expected, theestimations are almost unbiased. The maximum and minimum values of 0.07 and 0.093 wereobtained, respectively. This means that, on average, the largest bias of the models is about 9%.It is important to recall that none of the rules previously derived to compute EH demands havediscussed the size of their corresponding biases.

The information presented in Table IV can be considered as a global measure of the accuracy ofthe proposed model. However, it is also important to study the accuracy of the model for given T combinations. Figures 3 and 4 show and E spectra for estimations obtained with the proposedmodel.

In general, all hysteretic models have very similar values of . The largest values of areobserved for systems with T0.4 s, tends tobe fairly independent of T .

As in the case of the spectra, the largest values of bias E are observed at the short period region.Here, the negative values of E are observed, except for the case of elasticperfectly plastic systemswith = 1.5 and 2. Thus, at this period range, the proposed model systematically overestimatesthe observed values of EH, except for elasticperfectly plastic systems undergoing small ductility



Figure 3. spectra for the proposed rule.

demands, where underestimations were systematically observed. For systems with T>1 s, the Espectra fluctuate around zero. That is, the estimations tend to be unbiased. At this period range,E spectra are fairly independent from T .

From the results presented in this section it can be concluded that the accuracy of the proposedmodels is satisfactory.

In order to qualitatively illustrate the ability of the proposed rule to estimate EH demands forground motions recorded at different soil conditions, Figure 5 compares the estimated values of

EH/PGV obtained with the proposed rule for the Cape Mendocino and Cordoba records. TheCape Mendocino record is a very impulsive ground motion recorded at a rock soil site; on theother hand, the Cordoba record is a very long, nearly harmonic ground motion recorded atthe lake bed zone of Mexico City. As it can be observed, the proposed rule is able to capturethe differences between spectral shapes of

EH/PGV spectra observed in ground motions

recorded at firm and very soft soil sites.

COMPARISON WITH OTHER FORMULATIONS

In this section, the accuracy of the proposed method is compared with the accuracy of the oneproposed by Manfredi. Only this rule was considered because it was the only method that can beapplied at various soil conditions [13].



Figure 4. E spectra for the proposed rule.

Figure 5. Estimations of

EH/PGV obtained with the proposed rule for ground motions recorded atrock sites and very soft soil sites, = 0.05.

The comparisons shown in this section were obtained from elasticperfectly plastic SDOFsystems subjected to a second set of ground motions, comprising 46 accelerograms recorded atdifferent soil conditions during several seismic events, which were not used in the regressionanalysis of this paper, nor in Manfredis study. These ground motions are listed in Table V;as shown, there are accelerograms recorded at four different soil conditions. According to theUSGS site classification, Type A ground motions were recorded at sites with shear wave velocities



Table V. Second set of ground motions utilized.

Station Date Comp PGA (cm/s2) PGV (cm/s) td (s) SoilLamont 11/12/99 N-S 27.8 11.0 19.1 4.0 ALamont 11/12/99 E-W 52.2 5.3 18.5 3.8 AGilroy array no. 1 10/18/89 0 403.0 31.6 6.5 4.7 AGilroy array no. 1 10/18/89 90 464.0 33.9 3.7 5.8 ALucerne 6/28/92 275 707.0 97.7 13.1 2.9 ALucerne 6/28/92 0 770.0 31.9 13.8 4.8 AMT Wilson-CIT 1/17/94 0 229.0 7.3 9.9 19.7 AMT Wilson-CIT 1/17/94 90 132.0 5.7 13.8 17.6 ALake Hughes no. 9 02/09/71 21 154.0 4.5 9.4 10.5 ALake Hughes no. 9 02/09/71 291 131.0 3.9 11.8 7.3 ALA-Wonderland Ave 10/01/87 75 38.4 1.7 6.7 8.6 ALA-Wonderland Ave 10/01/87 165 46.5 1.5 7.1 14.0 ADuzce 11/12/99 180 341.0 60.0 10.9 2.3 CDuzce 11/12/99 270 525.0 83.5 10.8 2.2 CErzikan 03/13/92 N-S 505.0 84.0 7.5 3.3 CErzikan 03/13/92 E-W 486.0 64.3 7.4 2.6 CHollywood Storage 10/01/87 0 217.0 9.0 11.5 12.3 CHollywood Storage 10/01/87 90 122.0 6.9 13.8 12.8 CPetrolia 04/25/92 0 578.0 48.1 17.7 14.2 CPetrolia 04/25/92 90 650.0 89.7 16.1 9.9 CSan Bernardino 06/28/92 90 90.4 13.8 23.2 13.1 CSan Bernardino 06/28/92 180 98.8 11.8 23.6 14.1 CTakatori 01/16/95 0 600.0 127.0 11.4 6.1 ETakatori 01/16/95 90 604.0 121.0 9.9 6.2 ETakarazu 01/16/95 0 680.0 68.3 4.6 2.7 ETakarazu 01/16/95 90 680.0 85.3 3.7 2.7 ESalton Sea 04/26/81 225 195.0 16.3 8.4 5.3 ESalton Sea 04/26/81 315 173.0 12.3 9.2 6.2 EApeel 1 Redwood City 04/24/84 40 44.9 3.4 21.3 18.5 EApeel 1 Redwood City 04/24/84 310 66.6 4.0 19.0 21.8 EMontebello, Bluff 01/17/94 206 176.0 9.4 10.6 13.9 EMontebello, Bluff 01/17/94 296 126.0 5.9 13.8 7.7 EAmbarli 08/17/99 N-S 244.0 40.0 36.6 8.8 EAmbarli 08/17/99 E-W 181.0 33.2 37.2 13.7 EAlameda 09/30/99 N-S 26.8 8.0 99.6 46.3 Very soft soilAlameda 09/30/99 E-W 22.6 4.8 91.9 49.8 Very soft soilBuenos Aires 04/25/89 N-S 54.4 23.5 107.0 36.0 Very soft soilBuenos Aires 04/25/89 E-W 58.9 22.5 87.7 28.7 Very soft soilGaribaldi 09/30/99 N-S 27.2 8.3 92.5 43.5 Very soft soilGaribaldi 09/30/99 E-W 18.3 5.9 108.0 54.4 Very soft soilHospital Juarez 12/29/99 N-S 4.9 1.5 72.5 34.1 Very soft soilHospital Juarez 12/29/99 E-W 6.1 1.4 58.9 26.6 Very soft soilLiverpool 09/30/99 N-S 31.1 11.1 109.0 50.1 Very soft soilLiverpool 09/30/99 E-W 26.1 7.7 99.6 46.1 Very soft soilTlatelolco 04/25/89 N-S 44.8 12.0 49.7 24.7 Very soft soilTlatelolco 04/25/89 E-W 30.7 9.6 64.2 30.4 Very soft soil



Table VI. The values of and E for the second set of ground motions.

E

= 1.5 = 2 = 3 = 4 = 6 = 1.5 = 2 = 3 = 4 = 6This study 0.561 0.521 0.514 0.515 0.530 0.008 0.012 0.042 0.045 0.029Manfredi 1.335 0.942 0.694 0.611 0.557 1.222 0.757 0.415 0.292 0.196

greater than 750 m/s, Type C ground motions were recorded at sites with shear wave velocitiesbetween 180 and 360 m/s and Type D ground motions were recorded at sites with shear wavevelocities lower than 180 m/s. The very soft soil ground motions were recorded at sites withshear wave velocities lower than 100 m/s. PGA is in the range of 4.9770 cm/s2, PGV is inthe range of 1.489.7 cm/s, td is in the range of 3.7108 s and is in the range of 2.254.4.As it can be observed, almost the same number of accelerograms for each type of ground mo-tion was included in order to assess the accuracy of the rules considered under different soilconditions.

The values of and E obtained with the proposed model and Manfredis rule are given inTable VI. Although the set ground motions listed in Table V were not used in the regressionanalysis, the values of given in Table VI related to the proposed rule are very similar tothe values presented in Table IV. Hence, it can be considered that the regression analysis wassatisfactory.

As it can be observed, the proposed method yields smaller values of than Manfredis rule,especially for


Figure 6. Comparison between and and E spectra for different estimations of EH.

the Manfredis rule, the greater differences between are about 25% for = 3, with the smallestvalues observed also for Type C ground motions; larger errors are observed for very soft soilground motions.

For the proposed model, the values of related to peak energy demand are similar to the valuesof considering all period range, except for very soft ground motions where larger errors areobserved in peak energy demand for values = 4 and 6. For the values of = 1.5, 2, the smallestvalues of are observed for very soft soil ground motions, while the largest values are observed forthe Type A ground motions. On the other hand, for = 36 the smallest values of are observedfor Type E ground motions, while the largest values are observed for very soft soil ground motions.For a given , the largest difference between values related to peak energy demand is about 40%for = 1.5. In the case of Manfredis rule, it can be observed that the values of related to peakenergy demand are smaller than those observed considering the whole period range. The smallestvalues of are observed for very soft soil ground motions, while the largest values are observedfor the Type A ground motions. For a given , the largest difference between values related topeak energy demand is about 45% for = 2.

In this section it has been shown that, in general, the estimations obtained with the proposedrule are more accurate than the estimations obtained with the other rule available to compute EHdemands for systems located at various soil conditions. The proposed model has the advantagethat it can be applied for SDOF systems with different hysteretic models. One special feature ofthe proposed model is that it tends to yield unbiased estimations of EH, a feature that can beimportant particularly in the context of probabilistic seismic hazard analysis, where estimations ofmedian values of response parameters are often needed.



Figure 7. Comparison between values for different soil types.

CONCLUSIONS

Expressions to compute hysteretic energy demands for SDOF systems have been presented. Theproposed equations were developed for four types of hysteretic behavior: elasticperfectly plastic,Clough, Takedas and modified Takedas models. The proposed method depends on elastic pseudo-acceleration and velocity spectra and on a measure of the number of cycles of ground motion. Itaccounts for the shift of the period of the peak value between elastic spectra and the hystereticenergy spectrum.

It has been shown that the proposed rules can be applied to compute hysteretic energy demandsfor systems located at different soil conditions and that the accuracy of the proposed model isbetter than that of the other comparable available rule. In addition, it is shown that the proposedmethod tends to yield unbiased estimations of hysteretic energy demands.

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