Hysteretic Energy Demand in SDOF Structures Subjected to an Earthquake Excitation : Analytical and Empirical Results

In energy-based seismic design approach, earthquake ground motion is considered as an energy input to structures. The earthquake input energy is the total of energy components such as kinetic energy, damping energy, elastic strain energy and hysteretic energy, which contributes the most to structural damage. In literature, there are many empirical formulas based on the hysteretic model, damping ratio and ductility in order to estimate hysteretic energy, whereas they do not directly consider the ground motion characteristics. This paper uses nonlinear time history (NLTH) analysis for energy calculations and presents the distribution of earthquake input energy and hysteretic energy of single-degree-of-freedom (SDOF) systems over the ground motion duration. Seven real earthquakes recorded on the same soil profile and three different bilinear SDOF systems having constant ductility ratio and different natural periods are selected to perform NLTH analyses. As results of nonlinear dynamic analyses, input and hysteretic energies per unit masses are graphically obtained. The hysteretic energy to input energy ratio ( E H / E I ) is investigated, as well as the ratio of other energy components to energy input. E H / E I ratios of NLTH analysis are compared to the results of empirical approximations related E H / E I ratio and a reasonable agreement is observed. The average of E H / E I ratio is found to be between 0.468 and 0.488 meaning nearly half of the earthquake energy input is dissipated through the hysteretic behavior.


Introduction
Conventional design and analysis methods mainly focus on establishing a particular peak demand parameter such as member force, maximum displacement and displacement ductility.In these design procedures, capacity of structural components is taken to be independent of the earthquake excitation and the cumulative damage associated with ground motion duration and numerous inelastic deformation cycles that the structure might experiences is not accounted for.Although displacement-based seismic design methods correlating the imposed displacement to the structure and the structural damage have recently been developed, the problem related to the cumulative damage has not been overcome.Accordingly, neither force-based methods nor displacement-based methods provide the whole necessary information to quantify the level of structural damage.
More recently, it has been widely recognized that the energy is one of the key factors related to the structural damage in strong ground motions since the level of damage depends on both maximum deformations and response history characteristics.Accordingly, more rational seismic design methods based on energy criterions incorporating forces and displacements have been developed where the loading effect of earthquake is interpreted in terms of input energy (EI) [1][2][3][4][5][6][7][8][9][10][11][12][13][14].These pioneer studies have provided new insights into the field of modern earthquake engineering and the energy concept is an important topic of current interest [15][16][17][18][19][20][21][22][23][24][25][26][27].Since the earthquake input energy is estimated as the integral of velocity response of inelastic system with respect to earthquake duration, all of the inelastic deformation cycles are considered in energy-based methods.Another major advantage of energy-based approach is that the structural resistance and the earthquake effect in terms of energy are basically uncoupled since input energy is a quite stable response parameter and hardly depends on hysteretic characteristics of the structure.Accordingly, there have been extensive attempts for estimating earthquake input energy [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42].
Usually, the design criterion in energy-based methods is satisfied by providing adequate capacity to dissipate the seismic energy imposed on structures by earthquake ground motions.Therefore, in order to achieve energy-based design criterions, it seems quite imperative to have the accurate evaluation of both seismic energy demands and energy dissipation capacity of structural components which strongly depends on the loading history.Meanwhile, quantification of the demand in terms of energy is the preliminary task.When the structure comes to rest at the end of the ground motion, the kinetic energy (EK) and the elastic strain energy (ES) of the system essentially vanishes and the total energy imposed by the earthquake excitation is dissipated in part by the damping energy (ED) and the hysteretic energy (EH) components.Meanwhile, estimation of hysteretic energy demands imparted to structures by earthquake ground motion is a crucial issue since it is associated with the damage potential of structures.In this regard accurate estimation of the portion of cumulative hysteretic energy to energy input is of great importance.Moreover, it would be quite easy to compute the hysteretic energy demand from input energy spectra if the ratio of hysteretic energy to energy input was known.Accordingly, several approximations have been proposed in order to estimate hysteretic energy demand and the hysteretic energy to input energy ratio (EH/EI) in structures [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57].
The main objective of the paper is to obtain input and hysteretic energy time history curves of several inelastic SDOF systems by means of NLTH analysis and to investigate the effectiveness of some approximate methods related to hysteretic and input energy.Accordingly, three SDOF systems having natural periods as 0.2 s, 0.6 s and 1.0 s are selected to perform energy analyses.Firstly, the hysteretic energy to energy input ratio versus time (EH/EI-t) graphs of bilinear SDOF systems with a damping ratio of 5% and a ductility ratio of 2 are obtained from NLTH analyses.Subsequently, EH/EI ratios obtained from reference approximations are indicated in these graphs with the result graphs of nonlinear dynamic analyses.Finally, EH/EI ratios from time history analyses are compared with EH/EI values of approximate estimations.The results and findings of the study are presented by graphs and tables.

Seismic Energy Components
The general energy balance of structural systems can be derived by integrating the governing equation of motion with respect to relative displacement of the mass.Energy components which are defined in energy-based seismic design and evaluation rise from the integral terms of this equation.Consequently, the energy balance equation of an SDOF system can be written as in the form of its well-known expression from dynamics of structures as given in Eq. ( 1) [2,4,5,13,15,21,25,26,32,37,40,41,46].
In the basic energy balance equation; m is the seismic mass of the structure, c is the damping coefficient, u is the relative displacement of the system, fs is the restoring force,   () is the horizontal ground acceleration,  () is the relative velocity and  () is the relative acceleration of the system.The righthand side of Eq. ( 1) expresses the total energy input to the system (EI) with the strong ground acceleration.The energy input (EI) points out the energy demand of an earthquake and imposes the seismic energy to the structure.The first term on the left-hand side of Eq. ( 1) represents the kinetic energy of the mass (EK), the second term represents the damping energy (ED) and the last term indicates the total absorbed energy by the structure with both linear-elastic and nonlinear behavior (Ea).The summation of these energy components constitutes the total energy input and Eq. ( 1) can be rewritten by using the symbolic seismic energy terms as: The absorbed energy (Ea) term also includes both the elastic strain energy component (ES) and the hysteretic energy component (EH).Hysteretic energy is generally considered to be the most important energy component contributing to structural damage [13,21,46,56].It may be thought that the hysteretic energy composing the very significant portion of the absorbed energy is directly related with inelastic response of the system.Besides Eq. ( 2) can be rewritten by expanding the total absorbed energy (Ea) in terms of the elastic strain energy (ES) and the hysteretic energy (EH) as: Eq. ( 3) yields the energy response parameters of SDOF systems subjected to earthquake excitation.ES has the significant portion in the elastic response of the system whereas it approaches nearly to zero at the end of the ground motion.In inelastic behavior of the system, the components EK and ES are negligible compared to EH and almost at the end of the earthquake ground motion, Eq. ( 3) may be practically expressed as:

Reference Approximations for Hysteretic to Input Energy Ratio
In scientific literature there exist many previous studies related to the hysteretic energy to input energy ratio (EH/EI) of SDOF systems.Researchers generally defined EH/EI ratio as functions of viscous damping ratio (), ductility factor () and the hysteretic behavior [2,3,43,44,56,57].Akiyama proposed a relationship between the input and hysteretic energies in terms of equivalent velocities [3].Based on analysis of SDOF systems having elasticperfectly plastic restoring force characteristics, Akiyama expressed the ratio of damage velocity (as a function of the energy contributing to damage) to the equivalent velocity (as a function of EI) in terms of the viscous damping ratio ().At the end of the earthquake ground motion duration since EK and ES components are almost zero, the energy which contributes to structural damage can be approximately taken equal to the hysteretic energy EH [40].Accordingly, Akiyama's empirical expression is given below [3]: Fajfar and Vidic [43] proposed an expression based on the results of some parametric studies considering elastic-perfectly plastic SDOF systems.They expressed EH/EI ratio as a function of viscous damping (), ductility factor () and hysteretic behavior [43,56].Their simple formula to describe EH/EI ratio is as follows: where cE and cH coefficients depending on the type of hysteretic model and damping model are taken to be 1.05 and 0.95 for 5% damping, respectively [43].These coefficients used within the study are for bilinear hysteretic model and instantaneousstiffness-proportional damping.
Manfredi [44] used many earthquake acceleration records and carried out statistical analyses on the used ground motion records.The following expression was given by Manfredi for the damping ratio equal to 0.05: (7) where c is the cyclic ductility ratio [44].
Khashaee [57] eliminated the cyclic ductility c in Manfredi's formula and applied regression analysis on hysteretic and input energy data obtained from 160 accelerograms and proposed an expression for EH/EI ratio for systems having =2, 3, 4 and 5 as: In addition to the above empirical formulas, an extensive research has been devoted to estimate the ratio of EH/EI.Kuwamura and Galambos [4], Akbaş et al. [15], Decanini and Mollaioli [16] and Benavent et al. [32,37] are some of the leading researchers who made studies to estimate the ratio of EH/EI in SDOF systems [40].It was clearly seen from these researchers' studies that proposed expressions to estimate the ratio of EH/EI are overly conservative and generally overestimate the EH/EI ratio [25].

Ground Motion Database
A total of seven real accelerograms are assembled considering the magnitude, distance, fault type, and soil profile type information.The accelerograms are compiled from the strong ground motion database of Pacific Earthquake Engineering Research Center [58].
The accelerograms have a magnitude range of 6.5 to 7.5 and a source-to-site distances less than 80 km.The peak ground acceleration (PGA) value of the earthquake ground motions is larger than 0.1g, where g is the gravitational acceleration.The site conditions of the assembled accelerograms represent the features of NEHRP site class D (stiff soil) according the available average shear-wave velocity to 30 m depth of subsoil (VS30).The selected ground motions have strike-slip focal mechanism.It should also be noted that the selected ground motion records are identified as no pulse-like records in the Pacific Earthquake Engineering Research Center (PEER) ground motion database.The overall characteristics of the collected strong ground motion records are presented in Table 1, where Mw is the moment magnitude of earthquake, RJB is the Joyner-Boore distance, PGV and PGD are the peak values of ground velocity and ground displacement, respectively.Meanwhile, the acceleration time histories of the assembled ground motions are demonstrated in Fig. 1.
Horizontal components of actual accelerograms are considered in NLTH analysis for energy calculations and the structural response to one horizontal component is evaluated.The action effects due to the combination of the horizontal components of ground motion is not taken into consideration in order to obtain input (EI) and hysteretic (EH) energy time history curves, as well as EH/EI variations.
Plotted in Fig. 2 is the non-scaled inelastic acceleration response spectra of individual records developed for a damping ratio of 5%.Response spectra are constructed using PRISM software [59].
The ground motion records are not scaled since the paper does not focus on estimating of input and hysteretic energy time history curves of inelastic SDOF systems subjected to earthquake ground motions compatible with elastic design acceleration spectra of any earthquake design code.The study only evaluates EH/EI ratios the selected SFOF systems under a set of non-scaled recorded earthquake under ground motions.

Characteristics of SDOF Systems and Energy Graphs
Three SDOF systems having various natural periods of Tn=0.2 s, 0.6 s and 1.0 s are selected as shown in Fig. 3.A constant ductility demand of =2 is taken into consideration.Pre-yield damping ratio is taken as =5%.The non-linear material behavior is modeled as a bilinear non-degrading hysteretic model with postyield strain-hardening ratio of 10% (Fig. 4).Strength degradation and pinching effects are neglected within the study.Many structures subjected to reverse cyclic loading exhibit some level of stiffness and strength degradation [59].However, the implemented hysteretic model does not incorporate any level of strength or stiffness degradation.The motivation of a simpler bilinear hysteretic model relies on the fact that the approximate formulas considered in the study are mainly based on elasto-plastic (a bilinear strength hardening model with post-yield stiffness equal to zero) or bilinear non-degrading hysteretic model.

Figure 4. Bilinear non-degrading hysteretic model with 10% strain hardening
Both the input energy and hysteretic energy graphs of SDOF systems in Fig. 3 are obtained by using the relevant expressions in Eq. ( 1).Earthquake input energies are computed from the right-hand side of general energy balance equation and hysteretic energies are determined from the last term (inelastic part) of the left-hand side of the same equation.Velocity time histories of inelastic SDOF systems subjected to horizontal component of earthquake ground motions in Fig. 1 are computed by using PRISM Software [60].Then, hysteretic energies and input energies are computed by using the Excel programming written by the authors.Hysteretic energy tends to be constant over very large duration of earthquake ground motion and is considered to be the main design parameter in energy based seismic design of structures [15,25].Nonetheless, hysteretic to input energy (EH/EI) ratio generally tends to be constant over the certain duration of ground motion.

Figure 5.
Input and hysteretic energy graphs of an SDOF system under an earthquake effect

Hysteretic to Input Energy Ratios of SDOF Systems
Nonlinear response parameters (i.e.nonlinear displacements and velocities) of the SDOF systems subjected to assembled ground motions are obtained through NLTH analysis.Firstly, seismic energy input to the systems is determined by integrating the product of ground motion accelerations and nonlinear velocities over the entire duration of the earthquake.In this way time histories of seismic input energy per unit mass are assessed.Then, hysteretic energy demands for SDOF systems are obtained by integrating the resulting nonlinear forces of dynamic analysis over the earthquake duration.It is quite significant to estimate the hysteretic energy demand since in energy based design procedures the structural damage is limited by providing adequate dissipated energy capacity of the structural system.Accordingly, shown in Fig. 6 are the input and hysteretic energy time histories of inelastic SDOF systems subjected to assembled ground motions.
Fig. 6 indicates that the maximum input and hysteretic energies occur at the end of the earthquake excitation.Therefore, the duration of earthquake ground motion affects these energy components.The dissipated hysteretic energy increases as SDOF systems experience inelastic deformations.However, it is almost zero at the linear elastic response of the systems.Similarly, elastic input energy is very small.The maximum EI/m values obtained for the considered SDOF systems are listed in Table 2.After estimating the seismic energy demand imposed by the earthquake, the percentage of the input energy to be dissipated through inelastic hysteretic behavior is calculated and the variation of hysteretic energy to input energy (EH/EI) ratio is obtained (Fig. 7).The results of the time history analyses show that this ratio is a stable quantity.Additionally, the constant EH/EI ratios obtained from the reference empirical approximations are also plotted in Fig. 7.
Approximate EH/EI ratios are quite close to each other and it is found that the reference approximate formulas estimate EH/EI ratios within reasonable limits.The portion of the earthquake input energy distributed among the total of the kinetic energy of the mass, the damping energy and the elastic strain energy (EK+ED+ES) and the hysteretic energy (EH) is listed in Table 4 in percent.For all period values, the summation of these values corresponds to the total of the input energy (EI).All earthquake ground motions reflect their own characteristics to results.It should also be noted that more realistic results may be calculated using ground motion records containing accelerograms of two horizontal components.However, considering the limits of the software used for seismic response analysis of SDOF systems in the study, the seismic action is described by one horizontal component.This point should be taken into consideration in order to evaluate the results of the input and hysteretic energy computations of this study.

Conclusions and Recommendations
Earthquake input energy can be used as a measure of the intensity of ground motion and besides it accounts for the duration of ground motion.Having estimated the energy input to the system (e.g. from energy input spectrum), one can easily determine the portion of the input energy converted to hysteretic energy since EH/EI is generally supposed to be a stable quantity.In this study, input and hysteretic energy time histories of SDOF structures having bilinear behavior under the effect of selected earthquakes are investigated.EH/EI ratio graphs are obtained from nonlinear time history analyses and compared with some approximate formulas given by prior researchers.
Earlier studies indicated that structural properties such as ductility, damping ratio and the shape of hysteresis loop do have a significant influence on earthquake input energy and hysteretic energy time histories of structures.It is found that the characteristics of the employed earthquake ground motions significantly affect the energy time histories.Maximum input energies have tendency to increase from Tn=0.2 s to Tn=0.6 s for almost all selected earthquakes while there is an increasing or decreasing in the energies from Tn=0.6 s to Tn=1.0 s.Erzincan Earthquake gives the maximum input and hysteretic energies among all selected earthquakes, for SDOF systems having =2, =5% and =0.10.
Considering the results of the presented study, it can be concluded that the input energy and especially the hysteretic energy tend to be constant over a wide duration range.In respect to this, EH/EI ratios are generally obtained almost constant at the same duration range.Nonlinear time history results are compared to the approximate results given in Eqs.
(5)- (8).Fajfar and Vidic's estimation about EH/EI ratio gives the maximum value as 0.525, the second maximum is Akiyama's approximation as 0.497, then Manfredi's formula gives EH/EI=0.480 and finally the value of 0.443 is obtained from Khashaee's equation as the minimum among all considered estimations in the study.Time history results for EH/EI ratios give very compatible results with selected researchers' approximate estimations.For EH/EI ratios; the mean result of time history analyses for Tn=0.2 s and Tn=0.6 s is obtained as 0.468 and the mean result for Tn=1.0 s is obtained as 0.488.The mean value of selected researchers' formulas is calculated as 0.486.The mean results of EH/EI ratios are obtained too close within this study when the results of time history analyses and approximate equations are compared.
Further research may be conducted to obtain more detailed conclusion about the dissipation of hysteretic energy in SDOF systems and the variation of EH/EI ratios.More ground motion records can be used to generalize the energy results.Using a wide range of ground motion set may lead to more sensitive and more reliable generalizations about the EH/EI ratios.Analyses may be performed to determine the input energy and hysteretic energy dissipation of multi-degree-of-freedom (MDOF) systems.Moreover, different hysteretic models, ductility ratios, damping ratios and characteristics of earthquake ground motion records used in the analyses can change the energy results.The other researchers' estimations may be investigated to compare the degree of approximation of EH/EI ratios with the results of dynamic analyses.
In this study, ground motion records are selected based on magnitude, distance, and focal mechanism and it is only aimed to obtain EH/EI ratios from time history analyses and from approximate formulas.If seismic assessment is to be performed, the rigorous selection of ground motions will be an important consideration and holistic ground motion selection methods should be used for realistic structural responses.

Figure 1 .
Figure 1.Acceleration time histories of the assembled ground motions In the presented study, both the input energy and hysteretic energy demands of SDOF structures and EH/EI ratios are obtained by means of NLTH analysis using the assembled ground motions and the results of EH/EI ratios are compared with those of Eqs.(5)-(8).

Figure 2 .
Figure 2. Inelastic acceleration spectra of the records

Fig. 5
Fig. 5 is a representative figure which explains the graphs of input and hysteretic energy time histories.Hysteretic energy tends to be constant over very large duration of earthquake ground motion and is considered to be the main design parameter in energy based seismic design of structures[15,25].Nonetheless, hysteretic to input energy (EH/EI) ratio generally tends to be constant over the certain duration of ground motion.

Table 1 .
Major seismological parameters of the assembled ground motions

Table 2 .
Maximum input energy per unit mass

Table 3
summarizes the maximum EH/EI ratios of the individual ground motions as well as the approximate estimations of different researchers.It is observed that the average of EH/EI ratios is rather constant and the reference empirical formulas estimate this value quite reasonable rather than the EH/EI ratios of individual earthquakes.

Table 4 .
Energy ratios of bilinear SDOF systems near the end of the earthquake duration