Development of Earthquake Energy Demand Spectra

Transcription

1 Development of Earthquake Energy Demand Spectra Ahmet Anıl Dindar, a) Cem Yalçın, b) Ercan Yüksel, c) Hasan Özkaynak, d) and Oral Büyüköztürk e) Current seismic codes are generally based on the use of response spectra in the computation of the seismic demand of structures. This study evaluates the use of energy concept in the determination of the seismic demand due to its potential to overcome the shortcomings found in the current response spectra based methods. The emphasis of this study is placed on the computation of the input and plastic energy demand spectra directly derived from the energy-balance equation with respect to selected far-field ground motion obtained from Pacific Earthquake Engineering Research (PEER) database, soil classification according to National Earthquake Hazards Reduction Program (NEHRP) and characteristics of the structural behavior. The concept and methodology are described through extensive nonlinear time history analyses of single-degree-of-freedom (SDOF) systems. The proposed input and plastic energy demand spectra incorporate different soil types, elastic perfectly plastic constitutive model, 5% viscous damping ratio, different ductility levels, and varying seismic intensities. [DOI: /011212EQS010M] INTRODUCTION Current procedures for the design of earthquake-resistant structures generally consider lateral forces induced by the strong ground motions as equivalent loads. Current seismic analysis methods and design codes primarily rely on the strength and displacement capabilities of the structural members, such as ASCE/SEI 7-10 (2010), FEMA 356 (2000), Eurocode 8 (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 availability of efficient tools for calculating the structural response, more rational approaches can now be implemented for the prediction of seismic effects as a basis for the development of appropriate seismic design provisions. Use of the energy concept appears to have a great potential in the analysis of seismic demands and design of the structural members since such an approach includes both strength and displacement characteristics of the structure as well as hysteretic behavior of the structural members. This subject was initially discussed by Housner during the First World Conference on Earthquake Engineering (1956). Since a) Department of Civil Engineering, Istanbul Kultur University, Bakirkoy, Istanbul, Turkey b) Department of Civil Engineering, Bogazici University, Bebek, Istanbul, Turkey c) Faculty of Civil Engineering, Istanbul Technical University, Maslak, Istanbul, Turkey d) Department of Civil Engineering, Beykent University, Sisli-Ayazaga, Istanbul, Turkey e) Department of Civil and Environmental Engineering, MIT, Cambridge, MA Earthquake Spectra, Volume 31, No. 3, pages , August 2015; 2015, Earthquake Engineering Research Institute

2 1668 DINDAR ET AL. then, various researchers have applied the energy principles to the seismic analysis and design of structural members. Particularly in the 1980s, the use of energy-based algorithms gained popularity among the investigators (Zahrah and Hall 1984, Akiyama 1985, Bertero and Uang 1988, Kuwamura and Galambos 1989). The use of the energy-based analysis and design of the structures was discussed as a part of a conference in Bled, Slovenia (Fajfar and Krawinkler 1992), where the researchers agreed that the next generation building codes would consider this approach. The use of energy spectra offers a great potential to identify the structural resistance against earthquake-induced effects since they cover a wide range of structural and ground motion characteristics related to the seismic analysis (Fajfar and Vidic 1994a). This advantage comes with a question as how to apply the energy demand spectra in the determination of the seismic demand on the structures and, consequently, design of the structural members (Fajfar and Vidic 1994b). Some researchers have already proposed several indices for the calculation of the earthquake-induced energy and its dissipation for elastic (Decanini and Mollaioli 1998) and inelastic systems (Decanini and Mollaioli 2001). Due to the random nature of the ground motion records during the earthquakes, the proposed indices had to be based on simplifying assumptions (Manfredi 2001). However, since the structural response characteristics and the ground motion records strongly affect the seismic demand, an energy demand spectrum should be derived carefully by considering the various properties of the structure and earthquake records (Chai 2004). Recently, some researchers have focused on the relationship between the seismic demand and the structural response by using energy principles such as capacity curve of the structures (Leelataviwat et al. 2009). The current building codes employ the elastic response spectra, which are constructed by taking 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). The response spectrum approach only indicates the peak responses and disregards the loss of some valuable information such as the total frequency content and duration of the earthquake records which are important in seismic design (Gupta 1990). Expressing earthquake-induced demand in terms of energy includes explicitly the level of damage since the frequency content and the duration of the motion, while these features are not covered in the response spectra since 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 input and plastic energy demand spectra. This is accomplished by conducting extensive time history analyses applied to SDOF systems. Throughout the analyses, near-fault (NF) and far-field (FF) types of earthquakes, seismic intensities, site classes, and constitutive behavior models of SDOF systems coupled with ductility levels are taken into consideration, while utilizing the energy-balance equation in order to derive the seismic demand for imparted and dissipated energies. Current force-based design practice uses load reduction factor (R) in order to define the inelastic response of the structure provided there is sufficient deformation capability exist. However, in the energy-based approach, the displacement ductility (μ) is selected for the evaluation of the inelastic performance of the structure since it includes the displacement-based and also the performance-based design methodologies (Benavent- Climent et al. 2010). The site classes and the epicentral distances were explicitly included

3 EQ-TARGET;temp:intralink-;e1;62;441 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1669 into the computation of the energy-based spectra by using carefully selected earthquake records from the PEER database with various intensities. A further filtering process was carried out in order to select the ground motions that were used in the analyses. Nonlinear time history analyses of the SDOF systems having different constitutive behavior models were performed for a range of periods. The outcomes of the nonlinear time history analyses were then used in the energy-balance equation in order to obtain the spectral values of input and plastic energies. A comprehensive evaluation was performed in terms of the input and plastic energies and their ratios. The smoothed mass-normalized input and plastic energy demand spectra parameters are proposed based on the most suitable structural behavior model, ductility level, site class, and seismic intensity with a corresponding scaling relationship. ENERGY BALANCE EQUATION The energy-balance equation is derived by directly integrating the equation of motion with respect to the relative displacement response of the system illustrated in Figure 1 (Uang and Bertero 1990). ð ð ð ð müðtþdu þ c_uðtþdu þ f s du ¼ mü g ðtþdu (1) where m is the mass of the structure, ü is the relative acceleration, c is the damping coefficient, u is the relative displacement, _u is the relative velocity, f s is the resisting force and ü g is the ground acceleration. Response of the system is taken relative to the ground level (fixedbased 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 the structure, namely, kinetic (E K ), damping (E D ), and absorbed (E A ) energies. The right-hand side of the equation represents the total input energy (E I ) that is imposed to the structure. Thus: EQ-TARGET;temp:intralink-;e2;62;287E K þ E D þ E A ¼ E I (2) Figure 1. Fixed-based SDOF system.

4 1670 DINDAR ET AL. The absorbed energy (E A ) term also includes the strain (E S ) and plastic (E P ) energies caused by elastic and inelastic response of the system, respectively, and it is expressed as EQ-TARGET;temp:intralink-;e3;41;615E A ¼ E S þ E P (3) Therefore, the final form of the energy balance equation becomes EQ-TARGET;temp:intralink-;e4;41;572E K þ E D þðe S þ E P Þ¼E I (4) The elastic strain energy (E S ) is calculated by using the elastic stiffness (k) of the system as follows: EQ-TARGET;temp:intralink-;e5;41;517E S ¼ u2 ðtþ 2k (5) The energy dissipated by the inelastic displacements could be determined by calculating the area of the enclosed loop of the force-displacement relationship (Wong and Yang 2002). Analytically, the plastic energy (E P ) 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;425E P ¼ E I ðe K þ E D þ E S Þ (6) The input energy (E I ) imparted into the structure throughout the earthquake duration is stored and dissipated by these four components as shown in Figure 2. These components could be separated into two groups: stored and dissipated energies. The elastic strain and kinetic energies are stored components that diminish at the end of the motion when no excitation 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. Some studies (Decanini and Mollaioli 2001, Manfredi 2001, Lopez-Almansa et al. 2013) suggested that there is a consistent relation between the input and plastic energies. Therefore, instead of Figure 2. Energy components.

5 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1671 using cumbersome relations in obtaining the plastic energies from input energies for the design purpose (Wong 2004), the energy demand may be derived from the direct procedures through the evaluation of wide range of nonlinear time history analyses results (Fajfar and Vidic 1994b). METHODOLOGY The methodology for determining energy demand spectra as proposed herein relies on the direct derivation of the energy terms through the energy balance equation. A lumped-mass cantilever column with potential plastic hinging section near its fixed support is used to compute the spectral values in the nonlinear response and the energy time history analyses. Six different constitutive models, as shown in Figure 4, have been examined in the computation of 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 smoothed energy demand spectra is proposed with a numerical example. Smoothing of the spectra is conducted by applying linear and nonlinear regression analyses using the mean-plusone standard deviation values of the numerous spectral waveforms of the filtered energy time history analyses. Also, a quadratic scaling procedure is introduced for the computation of the spectral energy values at different seismic intensities. COMPUTATION OF THE ENERGY TERMS A computer algorithm using MATLAB (Release 2010a; The Mathworks, Inc. 2011) was developed in order to determine the energy time histories for computing the seismic input and plastic energy demand spectra. The structural behavior models including stiffness and ductility properties and the selected earthquake records are the main input parameters of the developed algorithm as explained in four successive steps, as follows: Step 1: Earthquake time histories, range of ductility levels, initial stiffness values and varying mass conditions, structural behavior models, and constant damping ratio of 5% were used as the input parameters to carry out the analyses defined in subsequent steps. Step 2: Linear and nonlinear time history analyses were performed by using the computer program IDARC2D (Reinhorn et al. 2009, version 7.0) that is capable of applying several constitutive models by control parameters, as shown in Figure 3. Figure 3. Control parameters used in IDARC2D (Reinhorn and Sivaselvan 1999).

6 1672 DINDAR ET AL. Table 1. Parameters chosen for the constitutive models Models Behavior HC HBD HBE HS IBIL EI3 Elastic-perfectly plastic Flexural % Bilinear Flexural % Clough Flexural % Takeda Flexural % Pinching (mild) Slip % Pinching (severe) Slip % HC, HBD, HBE, HS, IBIL, and EI3 refer to stiffness degrading, ductility-based strength degrading, energy-controlled strength degrading, pinching, idealization, and post-yielding stiffness ratio to initial stiffness parameters, respectively. In Figure 3, α, β, and γ are the values that describe the change of the stiffness, strength and pinching of the hysteresis curves, respectively. The intervals of these modification parameters have certain limits in IDARC2D program. Based on these intervals, the values in Table 1 represent the constitutive models evaluated in this study. By using these control parameters utilized in six different constitutive models namely elastic perfectly plastic (EPP), bilinear, Takeda and Clough models with and without pinching (mild and severe), as shown in Figure 4. Behavior models were chosen in order to represent a wide range of structures such as reinforced concrete, steel and masonry. The structures responding within the inelastic response range are assumed to have the ability to deform up to an ultimate displacement beyond the yield level defined by a ductility factor (μ), as illustrated in Figure 5. During any credible earthquake event, the structure experiences many cyclic reversal loadings and numerous random amplitude cyclic displacements. In order to avoid any confusion, the constant-ductility approach is favored in the computations to be consistent with the expected nonlinear behavior level of the systems under different earthquake records. The constant-ductility approach (Chopra 2006) is achieved by modifying the yield strength of the system along with the initial stiffness (k 1 ), iteratively, until the target ductility is reached. The Figure 4. The constitutive behavior models and hysteretic responses.

7 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1673 Figure 5. Definition of the ductility (μ) in inelastic behavior. iteration procedure of the constant-ductility approach is shown in Figure 6. The computation to 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 parameter of 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 between the reinforcement configuration and the yield strength of the reinforced concrete (RC) sections (Priestley et al. 2007), however, it is assumed that the initial stiffness is not affected from the modification of the yield level of the system (Kunnath and Hu 2004). Hence, the construction of an energy spectrum through the use of constant ductility could be used in the determination of the seismic demand for various inelastic states. Figure 6. Iteration procedure in constant ductility analysis.

8 1674 DINDAR ET AL. Step 3: Energy terms are determined from the results of nonlinear time history analyses accomplished 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 earthquake time histories. Step 4: Input energy (E I ) and plastic energy (E P ) demand spectra are plotted with respect to the given period range. To generalize the energy demand spectra, the linear and nonlinear regression analyses are performed at the end of each ductility level. The meanplus-one-standard deviation curve is plotted using the calculated energy demand spectra values. Finally, the smoothing process is applied to this obtained average trend line. A detailed description of these steps and, accordingly, the developed algorithm can be found in Dindar (2009). The relations between the steps are summarized in the flowchart given in Figure 7. VERIFICATION OF THE DEVELOPED COMPUTER ALGORITHM The developed algorithm was verified with the existing literature for two cases. The energy term values defined in Wong and Yang (2002) for elastic and inelastic cases were evaluated, and perfect agreement was found, as shown in Figure 8. After the energy time histories were computed successfully, the construction of the input energy (E I ) demand spectra was also verified (Figure 9) by using an example given by Bertero and Uang (1992) in which 1985 Chile Earthquake recorded at Llolleo Station, component 10 and 1986 San Salvador Earthquake recorded at Geotech Investigation Center Station, 90 component time histories were used in the computations. Similarly, perfect agreement was also achieved. Therefore, this validates the computer algorithm developed in this study. SELECTION OF THE GROUND MOTIONS The construction of a spectrum requires carefully selected ground motion records to reflect the characteristics of the earthquakes. Two horizontal components of the 114 real earthquake records, in total 228 time histories, those including NF and FF types, from free-fault stations (on surface or single story light buildings) were taken from PEER database by considering their PGA values of g and moment magnitudes of 5.0 or higher. Near-fault ground motions often contain significant waves like pulses with single or double-sided amplitudes (Bray and Marek 2004). Pulse-type excitations could induce drastic high response in fixed-based buildings (Bertero et al. 1978). Because the relative energy terms were evaluated in this study, it was needed to examine the effects of the NF ground motions on the energy demand spectra. It is well known that the NF ground motions with directivity effects tend to have high PGV/PGA ratios, which dramatically influence the response characteristics of the structures (Malhotra 1999). Based on this fact, some of the records with greater than the mean-plus-one standard deviation of PGV/PGA values were excluded 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 records excluded. The number of the earthquake time histories with respect to the soil conditions

9 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1675 Figure 7. Flowchart for the determination of the energy spectra.

10 1676 DINDAR ET AL. Figure 8. Verification of the energy time history calculations for the case of 1994 Northridge Earthquake recorded at Sylmar Hospital 360 component for 5% viscous damping ratio and natural period of 1.5 s. Figure 9. Verification of mass-normalized input energy (E I ) demand spectra calculations for two earthquakes while μ ¼ 1 and 5% viscous damping ratio. and the epicentral distances are given in Tables 2 and 3, respectively. The detailed information 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% viscous damping ratio was apparent after filtering process, as seen in Figures 10 and 11.

11 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1677 Table 2. Distribution of records with respect to site classes (BSSC 1995) Site class V S30 (m s) Near-fault and far-field (NF þ FF) Far-field (FF) A (Rock) > B (Stiff soil) C (Soft soil) D (Very soft soil) < Total Table 3. Distribution of records with respect to epicentral distance in km Epicentral distance (km) Near-fault and far-field (NF þ FF) Far-field (FF) 0.0 <D f < <D f < <D f < <D f Total It is clearly observed from Figure 12 that the coefficient of variance (COV) values remain the same for NF þ FF and FF records for both input energy (E I ) and plastic energy (E P ) 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 1s period indicates the behavior difference of the energy values. In the case of NF þ FF, constant plateau is observed, whereas in the FF case, a descending energy values are more apparent. Figure 10. Mass-normalized input energy (E I ) demand spectra for Soil B with and without NF time histories for ductility level 2 and 5% viscous damping ratio.

12 1678 DINDAR ET AL. Figure 11. Mass-normalized plastic energy (E P ) demand spectra for Soil B with and without NF time 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 time histories for Soil B, ductility level 2 and 5% viscous damping ratio. In order to derive the most appropriate input and plastic energy spectra that adequately reflect the remaining 145 earthquake records, a probabilistic approach was employed. This probabilistic analysis is essential in smoothing the average response (here, energy) spectra. The smoothing of the demand energy spectra were utilized by estimation of the amplification factors 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 behavior models on the energy spectra values computed for selected 145 FF time histories on different soils. As an example, input energy (E I ) and plastic energy (E P ) spectral values (mean-plus-one standard deviation) from 66 time histories recorded on Soil B are plotted in Figure 13.

13 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1679 Figure 13. Input and plastic energy demand spectra for six different constitutive models with various ductility levels and 5% viscous damping ratio.

14 1680 DINDAR ET AL. The influence of the constitutive model on the imparted energy values is almost identical at short periods, however post peak response periods show different energies. On the other hand, the plastic energies for different constitutive models reveal that smaller the area enclosed in the hysteresis, the lesser the dissipated energy demand. Due to the fact that the plastic energy is related to the damage occurrence in the structural elements, the ratio of the plastic to the input energy demand spectra gives a better understanding in the influence of the variation of the constitutive 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 case of the pinching, the area of the hysteresis loops is considerably less than that of the wellconfined flexural cases. Even though at first sight, the E P E I ratios may seem deceptive in realizing the difference between a flexural and pinching behavior, that is highly probable due to the demand and capacity consideration resulting from the nonlinear time history analysis. The lower level of E P E I ratio of the slip models compared to the flexural models indicates that plastic energy demand is low due to the inadequate resistance, whereas the flexural models have more plastic energy demand. Therefore, the input and plastic energy demand spectra should be interpreted as an indicator to the demand of the ground motion, while their ratio would mean the capacity of the given structure. Figure 14. E P E I ratio of constitutive models for different ductility levels (μ ¼ 2, 4, 6) and 5% viscous damping ratio.

15 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1681 Even though Fajfar et al. (1992) recommended to use E P E I ¼ 0.70 constant value for the entire period range in the spectra for the structures having 5% viscous damping ratio, it was found in the study, and as seen in Figure 14, that this constant value is conservative only for ductility level of 2 at low period ranges, but different at other ductility levels and long period ranges. In this study, the EPP model was chosen since it encompasses other behavior models (as seen in Figure 14), and thus, the plastic energy (E P ) demand spectra becomes more conservative. Many researchers recommended some indices in order to describe the destructiveness of the earthquakes in terms of energy related parameters. Akiyama (1985), Kuwamura and Galambos (1989), Benavent-Climent et al. (2002), Ghodrati et al. (2010), Tselentis and Danciu (2010) introduced the design energy input spectra in terms of velocity that was computed from the terms in the above-mentioned energy balance equation (Figure 15) and proposed several formulations on the relation of E P E I to determine the energy demand value related to the damage on the structure. Decanini and Mollaioli (1998) developed elastic input energy (E I ) spectra formulation considering several structural and ground motion parameters such as hysteretic models, ductility, focal distance and site class. Later, Decanini and Mollaioli (2001) introduced a methodology that computes not only input energy but also plastic energy under certain structural and ground motion parameters. Even if their study and this study examines the similar structural and ground motion parameters, their spectral energy values were based on E P E I relations as well as several additional variables as function of the geometrical properties of the elastic input energy (E I ) demand spectra (Figure 16). Figure 15. Examples of design energy input spectra in terms of velocity (Benavent-Climent et al. 2010).

16 1682 DINDAR ET AL. Figure 16. Spectral shapes proposed by Decanini and Mollaioli (2001). Contrary to the studies suggesting that the ratio E P E I is a stable parameter in the damage definition on the members, the authors believe that this ratio may lead to inaccurate results in the performance-based design which uses the capacity curve of a structural component computed by the push-over analysis. The capacity curve of a system is actually the backbone envelope of the force-displacement hysteresis of a system under reverse cyclic action. Even if the performance-based design methodology accounts for the inelastic response in its procedure, it does not reflect the hysteresis behavior. This exclusion may lead the engineer to not realize the distinctive difference between the energy dissipation capacities from fully flexural to pinching-dominant systems, which is particularly crucial in reinforced concrete members (Yalcin et al. 2008). The reverse-cyclic action and cumulative damage occurrence on 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 EPP behavior, using the 1989 Loma Prieta earthquake recorded at Gilroy Array 0 component time history for various ductility and increased PGA levels. As the intensity of the ground motion increases, the level of maximum displacement also increases. Since the input energy is a coupled term of the ground motion and the relative displacement response, the amount of the input energy is proportional to the severity of the ground motion (seismic activity), since it is the integration of the ground acceleration with respect to the displacement. The increment of the input and plastic energy time series values with increasing ground motion intensity are plotted for scaled earthquake records in Figure 17. Comparing the input and plastic energy time histories in Figure 17, the relation between the 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 quadratic trend. 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. In addition to energy time histories, the spectral values derived from the nonlinear analysis for the all 145 time histories also possess the similar relation for both input and plastic energy values as seen in Figure 18. Based on this scaling relation, the energy demand spectra values for higher seismic intensity areas could be scaled with respect to certain PGA level, 0.1 g-based as proposed in this study.

17 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1683 Figure 17. The variation of input and plastic energy time histories of SDOF (T ¼ 1.0 s) system under the scaled ground motions. Figure 18. The quadratic ratio relation of input energy (E I ) and plastic energy (E P ) spectral values 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. PROPOSED INPUT AND PLASTIC ENERGY DEMAND SPECTRA The mean-plus-one standard deviation average of the time histories with respect to the different soil and ductility levels were employed in linear and nonlinear regression analysis in order to have a smoothed energy demand spectra. The resulting views of the input and plastic energy demand spectra are plotted in Figure 19. After the regression analysis, the general formulation of the input and plastic energy demand spectra become as follows: EQ-TARGET;temp:intralink-;e7;62;160E PGA I ¼ PGA 2 E 0.1 g I m (7) 0.1 g where PGA is the peak ground acceleration of the seismic intensity and E I 0.1 g is the massnormalized input energy for the base seismic intensity (PGA ¼ 0.1 g) in Equation 7.

18 1684 DINDAR ET AL. Figure 19. Proposed input and plastic energy demand spectra for different site classes considering 5% viscous damping ratio and EPP constitutive model with 84.1% confidence level. Table 4. spectra The parameters of the proposed input energy (E I ) and plastic energy (E P ) demand Soil A Soil B Soil C Soil D Parameters E I m E P m E I m E P m E I m E P m E I m E P m μ ¼ 1 A (m 2 s 2 ) B (m 2 s 2 ) T C (s) k μ ¼ 2 A (m 2 s 2 ) B (m 2 s 2 ) T C (s) k μ ¼ 4 A (m 2 s 2 ) B (m 2 s 2 ) T C (s) k μ ¼ 6 A (m 2 s 2 ) B (m 2 s 2 ) T C (s) k

19 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1685 EQ-TARGET;temp:intralink-;e8;62;640E I 0.1 g ¼ 8 < : A þðb AÞ ðt 0.05Þ BðT C TÞ k 0.05 s T T C T C T 3.0 s 9 = ; (8) where values of A, B, k, and T C in Equation 8 are given in Table 4. The plastic energy formulation is same with the formulation of input energy as given in Equations 7 8, even though the values of A, B, k are different from input energy values as follows: EQ-TARGET;temp:intralink-;e9;62;532E PGA P ¼ PGA 2 E 0.1 g P m (9) 0.1 g EQ-TARGET;temp:intralink-;e10;62;486E 0.1 g P ¼ 8 < : A þðb AÞ ðt 0.05Þ BðT C TÞ k 0.05 s T T C T C T 3.0 s 9 = ; (10) where the terms of A, B, k and T C are given in Table 4. NUMERICAL EXAMPLE An indicative example is given here to demonstrate the implementation of the energybased seismic energy demand. The parameters required in the computation of the input energy (E I ) and plastic energy (E P ) spectral values are the initial period (T), the mass (m) and the target ductility 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 spectrum given in Decanini and Mollaioli (2001) which was computed using Kobe-JMA 180 component (PGA ¼ 0.8 g) recorded 1995 Great Hanshin earthquake on stiff soil (360 m s < V S30 < 760 m s), viscous damping ratio 5% and ductility level of μ ¼ 4. The plastic energy Figure 20. Comparison of the mass-normalized input energy (E I ) and plastic energy (E P ) spectra computed in Decanini and Mollaioli (2001) and this study.

20 1686 DINDAR ET AL. (E P ) spectra constructed by using the formulation proposed in this study and the plastic energy 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 the natural accelerogram of Kobe-JMA 180 component has PGA of 0.8 g. The proposed spectra are in good agreement with Decanini and Mollaioli s (2001) study. If the energy imparted (E I ) into the structure and the amount of the energy dissipated (E P ) by the structure through the plastic rotations are sought then the spectral values of these energies could be directly obtained from the spectra computed in Figure 20 corresponding to 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 essential step. As presented in this study, the existing methodologies to estimate the seismic demand in terms of strength and displacement may contain some shortcomings. The use of energy concept in the seismic demand analysis is a fundamental alternative to overcome these shortcomings. This paper aims developing an earthquake energy demand spectra given the structural properties and site conditions including earthquake intensity, soil class and a target structural ductility. The use of the spectra together with period and mass of the structure will allow the engineer to determine the earthquake input energy and structural plastic energy demand requirement of the structure. Plastic energy is considered to be a measure of energy dissipation and hence it can be attributed to the damage development in the structure under earthquake. For a given system, the required plastic energy value determined from the demand spectra, and thus the energy dissipation capacity of the structure represents an essential parameter 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 compute the nonlinear energy time history of the structures at a given ductility level. For a given earthquake record, the algorithm iterates the yield level of the SDOF system so that the ultimate drift ductility reaches the target level with presumed tolerance of 1%. Properties of the structure such as ductility, constitutive model and the characteristics of the ground motion such as seismic intensity, soil conditions were thoroughly evaluated in the energy-balance equation to understand their influence on the seismic energy demand. The following conclusions have been drawn from extensive case studies that have been performed as part of this work: When comparing the cases of NF+FF and FF-type ground motions for the determination of energy spectra, it is clearly seen from the COV diagrams that NF þ FF has more scatterness than that of FF, indicating that the shape of the resulting energy spectrum should have a descending branch rather than a constant plateau. As the ductility increases, the amplitude of the mass-normalized input energy spectra for various constitutive models decreases, whereas the mass-normalized plastic energy spectra slightly increases. For a specific period, the ratio of E P E I increases with increment of ductility level, since more energy is dissipated at higher ductility levels.

21 DEVELOPMENT OF EARTHQUAKE ENERGY DEMAND SPECTRA 1687 The E P E I ratio of 0.70 proposed by Fajfar et al. (1992) is conservative only for ductility level of 2 at low period ranges, but different at other ductility levels and long period ranges. Out of six constitutive models used in the study, E P E I diagram of EPP model encompasses all other constitutive models. Therefore, the corresponding energy demand spectra should be used since this yields a higher plastic energy demand than those of the others. It was concluded that in order to determine the input and plastic energies for a specific PGA value, the 0.1 g-based energy spectra is multiplied by the square of the ratio of actual PGA to the base PGA of 0.1 g. The smoothed input energy (E I ) and plastic energy (E P ) demand spectra were formulated in accordance with seismic intensity, site class and the ductility level of the system. The use of the energy demand spectra recommended in this study for the assessment of existing structures requires redevelopment of all the spectra consistent with the material behavior characteristics of the existing structure or to convert the currently developed spectra to the desired spectra by using suitable conversion factors. REFERENCES Akiyama, H., Earthquake-Resistant Limit-State Design for Buildings, University of Tokyo Press. American Society of Civil Engineers (ASCE), Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-10 Standard, Reston, VA. Bertero, V. V., Mahin, S. A., and Herrera, R. A., Aseismic design implications of San Fernando earthquake records, Earthquake Engineering and Structural Dynamics 6, Bertero, V. V., and Uang, C. M., Implications of Recorded Earthquake Ground Motions on Seismic Design of Building Structures, Research Report, UCB/EERC-88/13, University of California at Berkeley, Los Angeles, CA. Bertero, V. V., and Uang, C. M., Issues and future directions in the use of an energy approach for seismic-resistance of design structures, in Proceedings, Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings, July, Bled, Slovenia, Benavent-Climent, A., Pujades, L. G., and Lopez-Almansa, F., Design energy input spectra for moderate-seismicity regions, Earthquake Engineering and Structural Dynamics, 31, Benavent-Climent, A., Lopez-Almansa, F., and Bravo-Gonzalez, D. A., Design energy input spectra for moderate-to-high seismicity regions based on Colombian earthquakes, Soil Dynamics and Earthquake Engineering 30, Bray, J. D., and Rodriguez-Marek, A., Characterization of forward-directivity ground motions in the near-fault region, Soil Dynamics and Earthquake Engineering 24, Building Seismic Safety Council (BSSC), Edition, NEHRP Recommended Provisions for Seismic Regulations for New Buildings, FEMA 222A/223A, Vol. 1 (Provisions) and Vol. 2 (Commentary), Federal Emergency Management Agency, Washington, D.C. Chai, Y. H., Incorporating low-cycle fatigue model into duration-dependent inelastic design spectra, Earthquake Engineering and Structural Dynamics 34,

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