Development of Earthquake Energy Demand Spectraweb.mit.edu/liss/papers/ES2015-31-1667-1689.pdf · 2016-06-02 · Development of Earthquake Energy Demand Spectra Ahmet Anıl Dindar,a)

Development of Earthquake EnergyDemand Spectra

Ahmet Anıl Dindar,a) Cem Yalçın,b) Ercan Yüksel,c) Hasan Özkaynak,d)

and Oral Büyüköztürke)

Current seismic codes are generally based on the use of response spectra inthe computation of the seismic demand of structures. This study evaluates the useof energy concept in the determination of the seismic demand due to its potentialto overcome the shortcomings found in the current response spectra–based meth-ods. The emphasis of this study is placed on the computation of the input andplastic energy demand spectra directly derived from the energy-balance equationwith respect to selected far-field ground motion obtained from Pacific EarthquakeEngineering Research (PEER) database, soil classification according to NationalEarthquake Hazards Reduction Program (NEHRP) and characteristics of thestructural behavior. The concept and methodology are described through exten-sive nonlinear time history analyses of single-degree-of-freedom (SDOF) sys-tems. The proposed input and plastic energy demand spectra incorporatedifferent soil types, elastic perfectly plastic constitutive model, 5% viscousdamping ratio, different ductility levels, and varying seismic intensities. [DOI:10.1193/011212EQS010M]

INTRODUCTION

Current procedures for the design of earthquake-resistant structures generally considerlateral forces induced by the strong ground motions as equivalent loads. Current seismicanalysis methods and design codes primarily rely on the strength and displacement capabil-ities of the structural members, such as ASCE/SEI 7-10 (2010), FEMA 356 (2000), Eurocode8 (2004), and Turkish Earthquake Code (2007). However, another important parameter,energy, has not been explicitly specified in the determination of the earthquake affect.With improved knowledge on the characteristics of the strong ground motions and the avail-ability of efficient tools for calculating the structural response, more rational approaches cannow be implemented for the prediction of seismic effects as a basis for the development ofappropriate seismic design provisions. Use of the energy concept appears to have a greatpotential in the analysis of seismic demands and design of the structural members sincesuch an approach includes both strength and displacement characteristics of the structureas well as hysteretic behavior of the structural members. This subject was initially discussedby Housner during the First World Conference on Earthquake Engineering (1956). Since

a) Department of Civil Engineering, Istanbul Kultur University, Bakirkoy, 34156 Istanbul, Turkeyb) Department of Civil Engineering, Bogazici University, Bebek, 34342 Istanbul, Turkeyc) Faculty of Civil Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkeyd) Department of Civil Engineering, Beykent University, Sisli-Ayazaga, 34396 Istanbul, Turkeye) Department of Civil and Environmental Engineering, MIT, Cambridge, MA 02139

1667Earthquake Spectra, Volume 31, No. 3, pages 1667–1689, August 2015; © 2015, Earthquake Engineering Research Institute

then, various researchers have applied the energy principles to the seismic analysis anddesign of structural members. Particularly in the 1980s, the use of energy-based algorithmsgained popularity among the investigators (Zahrah and Hall 1984, Akiyama 1985, Berteroand Uang 1988, Kuwamura and Galambos 1989). The use of the energy-based analysis anddesign of the structures was discussed as a part of a conference in Bled, Slovenia (Fajfar andKrawinkler 1992), where the researchers agreed that the next generation building codeswould consider this approach.

The use of energy spectra offers a great potential to identify the structural resistanceagainst earthquake-induced effects since they cover a wide range of structural and groundmotion characteristics related to the seismic analysis (Fajfar and Vidic 1994a). This advan-tage comes with a question as how to apply the energy demand spectra in the determination ofthe seismic demand on the structures and, consequently, design of the structural members(Fajfar and Vidic 1994b). Some researchers have already proposed several indices for thecalculation of the earthquake-induced energy and its dissipation for elastic (Decanini andMollaioli 1998) and inelastic systems (Decanini and Mollaioli 2001). Due to the randomnature of the ground motion records during the earthquakes, the proposed indices had tobe based on simplifying assumptions (Manfredi 2001). However, since the structuralresponse characteristics and the ground motion records strongly affect the seismic demand,an energy demand spectrum should be derived carefully by considering the various propertiesof the structure and earthquake records (Chai 2004). Recently, some researchers have focusedon the relationship between the seismic demand and the structural response by using energyprinciples such as capacity curve of the structures (Leelataviwat et al. 2009).

The current building codes employ the elastic response spectra, which are constructed bytaking the extreme values of the time history analysis for a single-degree-of-freedom (SDOF)system excited by a set of motion records, to define the seismic demand (Chopra 2006). Theresponse spectrum approach only indicates the peak responses and disregards the loss ofsome valuable information such as the total frequency content and duration of the earthquakerecords which are important in seismic design (Gupta 1990). Expressing earthquake-induceddemand in terms of energy includes explicitly the level of damage since the frequency contentand the duration of the motion, while these features are not covered in the response spectrasince it accounts for only the maximum response values throughout the excitation history(Bertero and Uang 1988).

The current study aims to define the seismic demand on structures in the form of inputand plastic energy demand spectra. This is accomplished by conducting extensive timehistory analyses applied to SDOF systems. Throughout the analyses, near-fault (NF) andfar-field (FF) types of earthquakes, seismic intensities, site classes, and constitutive behaviormodels of SDOF systems coupled with ductility levels are taken into consideration, whileutilizing the energy-balance equation in order to derive the seismic demand for imparted anddissipated energies. Current force-based design practice uses load reduction factor (R) inorder to define the inelastic response of the structure provided there is sufficient deformationcapability exist. However, in the energy-based approach, the displacement ductility (μ) isselected for the evaluation of the inelastic performance of the structure since it includesthe displacement-based and also the performance-based design methodologies (Benavent-Climent et al. 2010). The site classes and the epicentral distances were explicitly included

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into the computation of the energy-based spectra by using carefully selected earthquakerecords from the PEER database with various intensities. A further filtering process wascarried out in order to select the ground motions that were used in the analyses. Nonlineartime history analyses of the SDOF systems having different constitutive behavior modelswere performed for a range of periods. The outcomes of the nonlinear time history analyseswere then used in the energy-balance equation in order to obtain the spectral values of inputand plastic energies. A comprehensive evaluation was performed in terms of the input andplastic energies and their ratios. The smoothed mass-normalized input and plastic energydemand spectra parameters are proposed based on the most suitable structural behaviormodel, ductility level, site class, and seismic intensity with a corresponding scalingrelationship.

ENERGY BALANCE EQUATION

The energy-balance equation is derived by directly integrating the equation of motionwith respect to the relative displacement response of the system illustrated in Figure 1(Uang and Bertero 1990).

EQ-TARGET;temp:intralink-;e1;62;441

ðmuðtÞduþ

ðc_uðtÞduþ

ðf sdu ¼ �

ðmugðtÞdu (1)

where m is the mass of the structure, u is the relative acceleration, c is the damping coeffi-cient, u is the relative displacement, _u is the relative velocity, f s is the resisting force and ug isthe ground acceleration. Response of the system is taken relative to the ground level (fixed-based system) instead of absolute response for simple representation of energy terms(Figure 1).

The terms on the left hand side of Equation 1 represent the energy components of thestructure, namely, kinetic (EK), damping (ED), and absorbed (EA) energies. The right-handside of the equation represents the total input energy (EI) that is imposed to the structure.Thus:

EQ-TARGET;temp:intralink-;e2;62;287EK þ ED þ EA ¼ EI (2)

Figure 1. Fixed-based SDOF system.

DEVELOPMENTOF EARTHQUAKE ENERGY DEMAND SPECTRA 1669

The absorbed energy (EA) term also includes the strain (ES) and plastic (EP) energiescaused by elastic and inelastic response of the system, respectively, and it is expressed as

EQ-TARGET;temp:intralink-;e3;41;615EA ¼ ES þ EP (3)

Therefore, the final form of the energy balance equation becomes

EQ-TARGET;temp:intralink-;e4;41;572EK þ ED þ ðES þ EPÞ ¼ EI (4)

The elastic strain energy (ES) is calculated by using the elastic stiffness (k) of the systemas follows:

EQ-TARGET;temp:intralink-;e5;41;517ES ¼u2ðtÞ2k

(5)

The energy dissipated by the inelastic displacements could be determined by calculatingthe area of the enclosed loop of the force-displacement relationship (Wong and Yang 2002).Analytically, the plastic energy (EP) could also be calculated by subtracting the kinetic,damping and elastic energies from the input energy, as expressed in Equation 6:

EQ-TARGET;temp:intralink-;e6;41;425EP ¼ EI � ðEK þ ED þ ESÞ (6)

The input energy (EI) imparted into the structure throughout the earthquake duration isstored and dissipated by these four components as shown in Figure 2. These componentscould be separated into two groups: stored and dissipated energies. The elastic strain andkinetic energies are stored components that diminish at the end of the motion when no excita-tion exists, whereas the damping and plastic energies are dissipated throughout the motion.

The dissipated energies are critical in the evaluation and design of the structures. Somestudies (Decanini and Mollaioli 2001, Manfredi 2001, Lopez-Almansa et al. 2013) suggestedthat there is a consistent relation between the input and plastic energies. Therefore, instead of

Figure 2. Energy components.

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using cumbersome relations in obtaining the plastic energies from input energies for thedesign purpose (Wong 2004), the energy demand may be derived from the direct proceduresthrough the evaluation of wide range of nonlinear time history analyses results (Fajfar andVidic 1994b).

METHODOLOGY

The methodology for determining energy demand spectra as proposed herein relies on thedirect derivation of the energy terms through the energy balance equation. A lumped-masscantilever column with potential plastic hinging section near its fixed support is used to com-pute the spectral values in the nonlinear response and the energy time history analyses. Sixdifferent constitutive models, as shown in Figure 4, have been examined in the computationof the spectral values.

By evaluating various characteristics such as mass, ductility ratio, constitutive model,seismic intensity and soil condition, a complete set of equations describing the smoothedenergy demand spectra is proposed with a numerical example. Smoothing of the spectrais conducted by applying linear and nonlinear regression analyses using the mean-plus-one standard deviation values of the numerous spectral waveforms of the filtered energytime history analyses. Also, a quadratic scaling procedure is introduced for the computationof the spectral energy values at different seismic intensities.

COMPUTATION OF THE ENERGY TERMS

A computer algorithm using MATLAB (Release 2010a; The Mathworks, Inc. 2011) wasdeveloped in order to determine the energy time histories for computing the seismic input andplastic energy demand spectra. The structural behavior models including stiffness and duc-tility properties and the selected earthquake records are the main input parameters of thedeveloped algorithm as explained in four successive steps, as follows:

Step 1: Earthquake time histories, range of ductility levels, initial stiffness values and vary-ing mass conditions, structural behavior models, and constant damping ratio of 5%were used as the input parameters to carry out the analyses defined in subse-quent steps.

Step 2: Linear and nonlinear time history analyses were performed by using the computerprogram IDARC2D (Reinhorn et al. 2009, version 7.0) that is capable of applyingseveral constitutive models by control parameters, as shown in Figure 3.

Figure 3. Control parameters used in IDARC2D (Reinhorn and Sivaselvan 1999).


In Figure 3, α, β, and γ are the values that describe the change of the stiffness, strengthand pinching of the hysteresis curves, respectively. The intervals of these modification para-meters have certain limits in IDARC2D program. Based on these intervals, the values inTable 1 represent the constitutive models evaluated in this study.

By using these control parameters utilized in six different constitutive models namelyelastic perfectly plastic (EPP), bilinear, Takeda and Clough models with and without pinch-ing (mild and severe), as shown in Figure 4. Behavior models were chosen in order to repre-sent a wide range of structures such as reinforced concrete, steel and masonry.

The structures responding within the inelastic response range are assumed to have theability to deform up to an ultimate displacement beyond the yield level defined by a ductilityfactor (μ), as illustrated in Figure 5.

During any credible earthquake event, the structure experiences many cyclic reversalloadings and numerous random amplitude cyclic displacements. In order to avoid any con-fusion, the constant-ductility approach is favored in the computations to be consistent withthe expected nonlinear behavior level of the systems under different earthquake records. Theconstant-ductility approach (Chopra 2006) is achieved by modifying the yield strength of thesystem along with the initial stiffness (k1), iteratively, until the target ductility is reached. The

Figure 4. The constitutive behavior models and hysteretic responses.

Table 1. Parameters chosen for the constitutive models

Models Behavior HC HBD HBE HS IBIL EI3

Elastic-perfectly plastic Flexural 200 0.01 0.01 1.00 1 0%Bilinear Flexural 200 0.01 0.01 1.00 1 10%Clough Flexural 200 0.01 0.01 1.00 0 10%Takeda Flexural 4 0.01 0.01 0.01 0 10%Pinching (mild) Slip 4 0.01 0.01 0.05 0 10%Pinching (severe) Slip 4 0.60 0.60 0.50 0 10%

HC, HBD, HBE, HS, IBIL, and EI3 refer to stiffness degrading, ductility-based strength degrading, energy-controlledstrength degrading, pinching, idealization, and post-yielding stiffness ratio to initial stiffness parameters, respectively.

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iteration procedure of the constant-ductility approach is shown in Figure 6. The computationto reach the target ductility is performed by an algorithm referred by Kunnath and Hu (2004)is employed. The convergence of the iteration is checked by the predefined control parameterof ductility when it reaches a level that is within �1% of the target ductility.

At first glance, this assumption may seem misleading with respect to the relation betweenthe reinforcement configuration and the yield strength of the reinforced concrete (RC) sec-tions (Priestley et al. 2007), however, it is assumed that the initial stiffness is not affectedfrom the modification of the yield level of the system (Kunnath and Hu 2004). Hence, theconstruction of an energy spectrum through the use of constant ductility could be used in thedetermination of the seismic demand for various inelastic states.

Figure 5. Definition of the ductility (μ) in inelastic behavior.

Figure 6. Iteration procedure in constant ductility analysis.


Step 3: Energy terms are determined from the results of nonlinear time history analysesaccomplished in the previous step, and spectral values are kept for the next step.This process is repeated within a range of SDOF periods and given earthquaketime histories.

Step 4: Input energy (EI) and plastic energy (EP) demand spectra are plotted with respect tothe given period range. To generalize the energy demand spectra, the linear and non-linear regression analyses are performed at the end of each ductility level. The mean-plus-one-standard deviation curve is plotted using the calculated energy demandspectra values. Finally, the smoothing process is applied to this obtained averagetrend line.

A detailed description of these steps and, accordingly, the developed algorithm can befound in Dindar (2009). The relations between the steps are summarized in the flowchartgiven in Figure 7.

VERIFICATION OF THE DEVELOPED COMPUTER ALGORITHM

The developed algorithm was verified with the existing literature for two cases. Theenergy term values defined in Wong and Yang (2002) for elastic and inelastic caseswere evaluated, and perfect agreement was found, as shown in Figure 8.

After the energy time histories were computed successfully, the construction of the inputenergy (EI) demand spectra was also verified (Figure 9) by using an example given byBertero and Uang (1992) in which 1985 Chile Earthquake recorded at Llolleo Station, com-ponent 10° and 1986 San Salvador Earthquake recorded at Geotech Investigation CenterStation, 90° component time histories were used in the computations. Similarly, perfectagreement was also achieved. Therefore, this validates the computer algorithm developedin this study.

SELECTION OF THE GROUND MOTIONS

The construction of a spectrum requires carefully selected ground motion records toreflect the characteristics of the earthquakes. Two horizontal components of the 114 realearthquake records, in total 228 time histories, those including NF and FF types, fromfree-fault stations (on surface or single story light buildings) were taken from PEER databaseby considering their PGA values of 0.15–0.80 g and moment magnitudes of 5.0 or higher.

Near-fault ground motions often contain significant waves like pulses with single ordouble-sided amplitudes (Bray and Marek 2004). Pulse-type excitations could induce drastichigh response in fixed-based buildings (Bertero et al. 1978). Because the relative energyterms were evaluated in this study, it was needed to examine the effects of the NF groundmotions on the energy demand spectra. It is well known that the NF ground motions withdirectivity effects tend to have high PGV/PGA ratios, which dramatically influence theresponse characteristics of the structures (Malhotra 1999). Based on this fact, some of therecords with greater than the mean-plus-one standard deviation of PGV/PGA values wereexcluded due to their pulse-type NF characteristics (Changhai et al. 2007). Consequently,the number of the earthquake records dropped from 228 to 145 after the NF type recordsexcluded. The number of the earthquake time histories with respect to the soil conditions

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Figure 7. Flowchart for the determination of the energy spectra.


and the epicentral distances are given in Tables 2 and 3, respectively. The detailed informa-tion on the selected earthquake time histories could be found in Dindar (2009).

The effect of the NF ground motions on energy demand spectra with μ ¼ 2 and 5% vis-cous damping ratio was apparent after filtering process, as seen in Figures 10 and 11.

Figure 8. Verification of the energy time history calculations for the case of 1994 NorthridgeEarthquake recorded at Sylmar Hospital 360° component for 5% viscous damping ratio and nat-ural period of 1.5 s.

Figure 9. Verification of mass-normalized input energy (EI) demand spectra calculations for twoearthquakes while μ ¼ 1 and 5% viscous damping ratio.

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It is clearly observed from Figure 12 that the coefficient of variance (COV) values remainthe same for NFþ FF and FF records for both input energy (EI) and plastic energy (EP)within a 1 s period range. As the period increases beyond 1 s, the scatterness is more apparent.Thus, their COV values are different from each other. This variation of COV beyond 1speriod indicates the behavior difference of the energy values. In the case of NFþ FF, con-stant plateau is observed, whereas in the FF case, a descending energy values are moreapparent.

Table 2. Distribution of records with respect to site classes (BSSC 1995)

Site class VS30 (m∕s)Near-fault and

far-field (NFþ FF)Far-field(FF)

A (Rock) >760 6 5B (Stiff soil) 360–760 82 66C (Soft soil) 180–360 106 53D (Very soft soil) <180 34 21Total 228 145

Table 3. Distribution of records with respect to epicentral distance in km

Epicentral distance (km)Near-fault and

far-field (NFþ FF)Far-field(FF)

0.0 < Df < 5.0 36 05.0 < Df < 12.0 70 6512.0 < Df < 30.0 92 5930.0 < Df 30 21Total 228 145

Figure 10. Mass-normalized input energy (EI) demand spectra for Soil B with and without NFtime histories for ductility level 2 and 5% viscous damping ratio.


In order to derive the most appropriate input and plastic energy spectra that adequatelyreflect the remaining 145 earthquake records, a probabilistic approach was employed. Thisprobabilistic analysis is essential in smoothing the average response (here, energy) spectra.The smoothing of the demand energy spectra were utilized by estimation of the amplificationfactors obtained for mean-plus-one standard deviation spectrum (84.1% probability level)assuming a lognormal distribution (Clough and Penzien 2003, Chopra 2006).

THE INFLUENCE OF BEHAVIOR MODELS ON ENERGY DEMAND SPECTRA

A comprehensive comparative analysis was conducted on the influence of behaviormodels on the energy spectra values computed for selected 145 FF time histories on differentsoils. As an example, input energy (EI) and plastic energy (EP) spectral values(mean-plus-one standard deviation) from 66 time histories recorded on Soil B are plottedin Figure 13.

Figure 11. Mass-normalized plastic energy (EP) demand spectra for Soil B with and without NFtime histories for ductility level 2 and 5% viscous damping ratio.

Figure 12. COV of input and plastic energy demand spectra for before and after filtering the timehistories for Soil B, ductility level 2 and 5% viscous damping ratio.

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Figure 13. Input and plastic energy demand spectra for six different constitutive models withvarious ductility levels and 5% viscous damping ratio.


The influence of the constitutive model on the imparted energy values is almost identicalat short periods, however post peak response periods show different energies. On the otherhand, the plastic energies for different constitutive models reveal that smaller the area enclosedin the hysteresis, the lesser the dissipated energy demand. Due to the fact that the plastic energyis related to the damage occurrence in the structural elements, the ratio of the plastic to the inputenergy demand spectra gives a better understanding in the influence of the variation of theconstitutive models on energy dissipation by plastic energy as seen in Figure 14.

The dissipated energy is defined by the area of the force-displacement hysteresis loops.Therefore, the larger the loop, the greater the energy dissipated, and vice versa. As in the caseof the pinching, the area of the hysteresis loops is considerably less than that of the well-confined flexural cases. Even though at first sight, the EP∕EI ratios may seem deceptive inrealizing the difference between a flexural and pinching behavior, that is highly probable dueto the demand and capacity consideration resulting from the nonlinear time history analysis.The lower level of EP∕EI ratio of the slip models compared to the flexural models indicatesthat plastic energy demand is low due to the inadequate resistance, whereas the flexural mod-els have more plastic energy demand. Therefore, the input and plastic energy demand spectrashould be interpreted as an indicator to the demand of the ground motion, while their ratiowould mean the capacity of the given structure.

Figure 14. EP∕EI ratio of constitutive models for different ductility levels (μ ¼ 2, 4, 6) and 5%viscous damping ratio.

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Even though Fajfar et al. (1992) recommended to use EP∕EI ¼ 0.70 constant value forthe entire period range in the spectra for the structures having 5% viscous damping ratio, itwas found in the study, and as seen in Figure 14, that this constant value is conservative onlyfor ductility level of 2 at low period ranges, but different at other ductility levels and longperiod ranges. In this study, the EPP model was chosen since it encompasses other behaviormodels (as seen in Figure 14), and thus, the plastic energy (EP) demand spectra becomesmore conservative.

Many researchers recommended some indices in order to describe the destructiveness ofthe earthquakes in terms of energy related parameters. Akiyama (1985), Kuwamura andGalambos (1989), Benavent-Climent et al. (2002), Ghodrati et al. (2010), Tselentis andDanciu (2010) introduced the design energy input spectra in terms of velocity that was com-puted from the terms in the above-mentioned energy balance equation (Figure 15) and pro-posed several formulations on the relation of EP∕EI to determine the energy demand valuerelated to the damage on the structure.

Decanini and Mollaioli (1998) developed elastic input energy (EI) spectra formulationconsidering several structural and ground motion parameters such as hysteretic models, duc-tility, focal distance and site class. Later, Decanini and Mollaioli (2001) introduced a meth-odology that computes not only input energy but also plastic energy under certain structuraland ground motion parameters. Even if their study and this study examines the similar struc-tural and ground motion parameters, their spectral energy values were based on EP∕EI rela-tions as well as several additional variables as function of the geometrical properties of theelastic input energy (EI) demand spectra (Figure 16).

Figure 15. Examples of design energy input spectra in terms of velocity (Benavent-Climent et al.2010).


Contrary to the studies suggesting that the ratio EP∕EI is a stable parameter in the damagedefinition on the members, the authors believe that this ratio may lead to inaccurate results inthe performance-based design which uses the capacity curve of a structural component com-puted by the push-over analysis. The capacity curve of a system is actually the backboneenvelope of the force-displacement hysteresis of a system under reverse cyclic action.Even if the performance-based design methodology accounts for the inelastic response inits procedure, it does not reflect the hysteresis behavior. This exclusion may lead the engineerto not realize the distinctive difference between the energy dissipation capacities from fullyflexural to pinching-dominant systems, which is particularly crucial in reinforced concretemembers (Yalcin et al. 2008). The reverse-cyclic action and cumulative damage occurrenceon the members due to the earthquake motion is not included in the current design codes.

THE EFFECT OF SEISMIC INTENSITY ON ENERGY DEMAND SPECTRA

A set of nonlinear analyses were carried out on a SDOF system (T ¼ 1.0 s), having EPPbehavior, using the 1989 Loma Prieta earthquake recorded at Gilroy Array 0° componenttime history for various ductility and increased PGA levels. As the intensity of the groundmotion increases, the level of maximum displacement also increases. Since the input energyis a coupled term of the ground motion and the relative displacement response, the amount ofthe input energy is proportional to the severity of the ground motion (seismic activity), sinceit is the integration of the ground acceleration with respect to the displacement. The incrementof the input and plastic energy time series’ values with increasing ground motion intensity areplotted for scaled earthquake records in Figure 17.

Comparing the input and plastic energy time histories in Figure 17, the relation betweenthe values of PGA ¼ 0.2 g, 0.3 g, 0.4 g, 0.6 g with respect to PGA ¼ 0.1 g is in quadratictrend. For example, the input and plastic energy values of the time series cases of 0.2 g, 0.3 g,0.4 g, and 0.6 g are found as 4, 9, 16, and 36 times of the values of 0.1 g case, respectively. Inaddition to energy time histories, the spectral values derived from the nonlinear analysis forthe all 145 time histories also possess the similar relation for both input and plastic energyvalues as seen in Figure 18.

Based on this scaling relation, the energy demand spectra values for higher seismic intensityareas could be scaled with respect to certain PGA level, 0.1 g-based as proposed in this study.

Figure 16. Spectral shapes proposed by Decanini and Mollaioli (2001).

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PROPOSED INPUT AND PLASTIC ENERGY DEMAND SPECTRA

The mean-plus-one standard deviation average of the time histories with respect to thedifferent soil and ductility levels were employed in linear and nonlinear regression analysis inorder to have a smoothed energy demand spectra. The resulting views of the input and plasticenergy demand spectra are plotted in Figure 19.

After the regression analysis, the general formulation of the input and plastic energydemand spectra become as follows:

EQ-TARGET;temp:intralink-;e7;62;160EPGAI ¼

�PGA0.1 g

�2

� E0.1 gI � m (7)

where PGA is the peak ground acceleration of the seismic intensity and EI0.1 g is the mass-

normalized input energy for the base seismic intensity (PGA ¼ 0.1 g) in Equation 7.

Figure 18. The quadratic ratio relation of input energy (EI) and plastic energy (EP) spectralvalues from 145 time histories having 4 different PGA values of SDOF (T ¼ 0.05 s, 0.2 s,0.5 s, 1 s, 2 s, and 3 s), ductility level (μ ¼ 2) and 5% viscous damping ratio.

Figure 17. The variation of input and plastic energy time histories of SDOF (T ¼ 1.0 s) systemunder the scaled ground motions.


Figure 19. Proposed input and plastic energy demand spectra for different site classes con-sidering 5% viscous damping ratio and EPP constitutive model with 84.1% confidencelevel.

Table 4. The parameters of the proposed input energy (EI ) and plastic energy (EP) demandspectra

Parameters

Soil A Soil B Soil C Soil D

EI∕m EP∕m EI∕m EP∕m EI∕m EP∕m EI∕m EP∕m

μ ¼ 1A (m2∕s2) 0.0059 0.0000 0.0050 0.0000 0.0040 0.0000 0.0045 0.0000B (m2∕s2) 0.0650 0.0000 0.0705 0.0000 0.0770 0.0000 0.0850 0.0000TC (s) 0.50 0.50 0.65 0.65 0.90 0.90 1.05 1.05k 0.910 0.000 1.095 0.000 1.456 0.000 1.658 0.000

μ ¼ 2A (m2∕s2) 0.0051 0.0027 0.0046 0.0022 0.0038 0.0019 0.0038 0.0019B (m2∕s2) 0.0551 0.0321 0.0624 0.0340 0.0650 0.0380 0.0700 0.0400TC (s) 0.45 0.45 0.60 0.60 0.80 0.80 0.95 0.95k 0.848 0.828 1.071 0.984 1.320 1.270 1.554 1.500

μ ¼ 4A (m2∕s2) 0.0044 0.0030 0.0042 0.0028 0.0037 0.0024 0.0037 0.0024B (m2∕s2) 0.0444 0.0306 0.0500 0.0320 0.0530 0.0350 0.0580 0.0380TC (s) 0.40 0.40 0.55 0.55 0.70 0.70 0.85 0.85k 0.889 0.954 1.107 1.127 1.260 1.310 1.450 1.520

μ ¼ 6A (m2∕s2) 0.0040 0.0032 0.0035 0.0039 0.0036 0.0026 0.0036 0.0027B (m2∕s2) 0.0350 0.0291 0.0410 0.0300 0.0440 0.0330 0.0480 0.0360TC (s) 0.35 0.35 0.50 0.50 0.65 0.65 0.80 0.80k 0.861 0.950 1.110 1.150 1.272 1.280 1.414 1.469

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EQ-TARGET;temp:intralink-;e8;62;640EI0.1 g ¼

8<:

Aþ ðB� AÞ∕ðT � 0.05ÞBðTC∕TÞk 0.05 s ≤ T ≤ TC

TC ≤ T ≤ 3.0 s

9=; (8)

where values of A, B, k, and TC in Equation 8 are given in Table 4.

The plastic energy formulation is same with the formulation of input energy as given inEquations 7–8, even though the values of A, B, k are different from input energy values asfollows:

EQ-TARGET;temp:intralink-;e9;62;532EPGAP ¼

�PGA0.1 g

�2

� E0.1 gP � m (9)

EQ-TARGET;temp:intralink-;e10;62;486E0.1 gP ¼

8<:

Aþ ðB� AÞ∕ðT � 0.05ÞBðTC∕TÞk 0.05 s ≤ T ≤ TC

TC ≤ T ≤ 3.0 s

9=; (10)

where the terms of A, B, k and TC are given in Table 4.

NUMERICAL EXAMPLE

An indicative example is given here to demonstrate the implementation of the energy-based seismic energy demand. The parameters required in the computation of the input energy(EI) and plastic energy (EP) spectral values are the initial period (T), the mass (m) and the targetductility level (μ) of the structure as well as the seismic intensity of the site.

The proposed energy demand spectrum is compared with the input energy spectrumgiven in Decanini and Mollaioli (2001) which was computed using Kobe-JMA 180° com-ponent (PGA ¼ 0.8 g) recorded 1995 Great Hanshin earthquake on stiff soil (360m∕s <VS30 < 760m∕s), viscous damping ratio 5% and ductility level of μ ¼ 4. The plastic energy

Figure 20. Comparison of the mass-normalized input energy (EI) and plastic energy (EP) spectracomputed in Decanini and Mollaioli (2001) and this study.


(EP) spectra constructed by using the formulation proposed in this study and the plasticenergy spectra given in Decanini and Mollaioli (2001) are plotted together in Figure 20.

The scaling factor used in the construction of the proposed spectra is taken as 64 since thenatural accelerogram of Kobe-JMA 180° component has PGA of 0.8 g. The proposed spectraare in good agreement with Decanini and Mollaioli’s (2001) study.

If the energy imparted (EI) into the structure and the amount of the energy dissipated (EP)by the structure through the plastic rotations are sought then the spectral values of theseenergies could be directly obtained from the spectra computed in Figure 20 correspondingto the structure’s initial period by multiplying the mass of the structure.

CONCLUSIONS

The estimation of seismic demand in earthquake resistant-design of structures is an essen-tial step. As presented in this study, the existing methodologies to estimate the seismicdemand in terms of strength and displacement may contain some shortcomings. The useof energy concept in the seismic demand analysis is a fundamental alternative to overcomethese shortcomings.

This paper aims developing an earthquake energy demand spectra given the structuralproperties and site conditions including earthquake intensity, soil class and a target structuralductility. The use of the spectra together with period and mass of the structure will allow theengineer to determine the earthquake input energy and structural plastic energy demandrequirement of the structure. Plastic energy is considered to be a measure of energy dissipa-tion and hence it can be attributed to the damage development in the structure under earth-quake. For a given system, the required plastic energy value determined from the demandspectra, and thus the energy dissipation capacity of the structure represents an essential para-meter towards design on the basis of which detailing of the structure could be possible.

The energy balance equation is applied through a developed computer algorithm to com-pute the nonlinear energy time history of the structures at a given ductility level. For a givenearthquake record, the algorithm iterates the yield level of the SDOF system so that the ulti-mate drift ductility reaches the target level with presumed tolerance of�1%. Properties of thestructure such as ductility, constitutive model and the characteristics of the ground motionsuch as seismic intensity, soil conditions were thoroughly evaluated in the energy-balanceequation to understand their influence on the seismic energy demand.

The following conclusions have been drawn from extensive case studies that have beenperformed as part of this work:

• When comparing the cases of NF+FF and FF-type ground motions for the deter-mination of energy spectra, it is clearly seen from the COV diagrams that NFþ FFhas more scatterness than that of FF, indicating that the shape of the resulting energyspectrum should have a descending branch rather than a constant plateau.

• As the ductility increases, the amplitude of the mass-normalized input energy spectrafor various constitutive models decreases, whereas the mass-normalized plastic energyspectra slightly increases. For a specific period, the ratio of EP∕EI increases withincrement of ductility level, since more energy is dissipated at higher ductility levels.

1686 DINDAR ET AL.

• The EP∕EI ratio of 0.70 proposed by Fajfar et al. (1992) is conservative only forductility level of 2 at low period ranges, but different at other ductility levels andlong period ranges.

• Out of six constitutive models used in the study, EP∕EI diagram of EPP modelencompasses all other constitutive models. Therefore, the corresponding energydemand spectra should be used since this yields a higher plastic energy demandthan those of the others.

• It was concluded that in order to determine the input and plastic energies for a spe-cific PGA value, the 0.1 g-based energy spectra is multiplied by the square of theratio of actual PGA to the base PGA of 0.1 g.

• The smoothed input energy (EI) and plastic energy (EP) demand spectra were for-mulated in accordance with seismic intensity, site class and the ductility level of thesystem.

The use of the energy demand spectra recommended in this study for the assessment ofexisting structures requires redevelopment of all the spectra consistent with the material beha-vior characteristics of the existing structure or to convert the currently developed spectra tothe desired spectra by using suitable conversion factors.

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(Received 12 January 2012; accepted 2 November 2013)


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Development of Earthquake Energy Demand Spectraweb.mit.edu/liss/papers/ES2015-31-1667-1689.pdf · 2016-06-02 · Development of Earthquake Energy Demand Spectra Ahmet Anıl Dindar,a)