Advances in Civil Engineering

Advances in Civil Engineering / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 1548319 |

Mebrahtom Gebrekirstos Mezgebo, Eric M. Lui, "Hysteresis and Soil Site Dependent Input and Hysteretic Energy Spectra for Far-Source Ground Motions", Advances in Civil Engineering, vol. 2016, Article ID 1548319, 29 pages, 2016.

Hysteresis and Soil Site Dependent Input and Hysteretic Energy Spectra for Far-Source Ground Motions

Academic Editor: Lucio Nobile
Received30 Jun 2016
Revised14 Sep 2016
Accepted26 Sep 2016
Published06 Dec 2016


Earthquake input energy spectra for four soil site classes, four hysteresis models, and five ductility levels are developed for far-source ground motion effect. These energy spectra are normalized by a quantity called velocity index (VI). The use of VI allows for the creation of dimensionless spectra and results in smaller coefficients of variation. Hysteretic energy spectra are then developed to address the demand aspect of an energy-based seismic design of structures with 5% critical damping and ductility that ranges from 2 to 5. The proposed input and hysteretic energy spectra are then compared with response spectra generated using nonlinear time history analyses of real ground motions and are found to produce reasonably good results over a relatively large period range.

1. Introduction

Performance-based design is a design philosophy that aims at designing structures to resist seismic forces with acceptable damage. The two most widely used performance-based design procedures are the force-based design and displacement-based design methods. In the force-based design (FBD) method, a design seismic force for a target structure is specified on the basis of an elastic acceleration response spectrum. This seismic design force is called the design base shear. To account for inelasticity (ductility effect), the design force of the target structure obtained from the elastic acceleration response spectrum is divided by a force-reduction factor. The structure is then designed for the reduced force, and the displacement can be checked so that the code-specified serviceability limits are met. The FBD method is not without limitations and drawbacks. Smith and Tso [1] through their study on a large class of reinforced concrete members such as piers, flexural walls, and ductile moment resisting fames claimed that force-based design procedure was inconsistent. They argued the assumption that the stiffness of the lateral force resisting elements was essentially independent of their strength was inconsistent as strength and stiffness are usually related. Moreover, the problems associated with this method, as pointed out by Priestley et al. [2] are as follows. The elastic stiffness is not known at the start of the design process, and very approximate values have to be used. Foundation effects are generally ignored in FBD and are difficult to incorporate in the design process as they affect both the elastic period and displacement ductility demand. Even though the design force is calculated from an allowable displacement ductility factor, it does not properly address the force-displacement relationship of the structure.

The displacement-based design (DBD) method takes displacement as a design parameter as opposed to using base shear as in the case for FBD. As a result, the important task in a displacement-based design approach is to estimate the maximum displacement demand in a structure with reasonable simplicity and accuracy as a function of its local mechanical characteristics, such as member strain and deformation limits. FEMA 440 [3] displacement-based coefficient method is one of the currently available displacement-based seismic design methods. The coefficient method modifies the linear elastic response of an equivalent single degree of freedom (SDOF) system by multiplying it by a series of coefficients to estimate a global displacement, commonly termed as the target displacement. This method uses an idealized force-displacement curve (pushover curve), which is a plot (for a given damping coefficient) of base shear versus roof displacement developed for a multiple degree of freedom (MDOF) structure. A corresponding spectral value for an effective period, , of an equivalent SDOF system is then obtained from an elastic response spectrum corresponding to a design ground motion. The target displacement is then calculated using an empirical formula that involves modifying coefficients and the spectral value for the corresponding effective period. The effective period is obtained from an initial period of the structure and accounts for the loss of stiffness in the transition from elastic to inelastic behavior.

The accuracy of the DBD method is highly dependent on how closely the equivalent SDOF system and its corresponding MDOF system are related through the idealized pushover curve. Several researchers have identified glitches in the use of roof displacement-based pushover curve. Hernandez-Montes et al. [4] noted that the use of roof displacement in generating the capacity curve could be misleading because the capacity curve so obtained sometimes tends to show the structure as a source of energy rather than an entity for absorbing energy. They suggested that an energy-based pushover analysis be used instead whereby the lateral force is plotted against a displacement which is a function of energy. Manoukas et al. [5] also developed an energy-based pushover procedure for estimating structural performance under strong earthquakes. They showed through numerical examples that their procedure provided better results compared to those produced by other similar methods.

It should be noted that neither the FBD method that uses base shear as a design parameter nor the DBD method that uses displacement as a design parameter can directly consider the cumulative damage effect that result from numerous inelastic cycles of vibration due to deterioration of the structure’s hysteretic behavior. Moreover, the effect of earthquakes on structures should be interpreted not just as a force or displacement quantity, but as a product of both, that is, in terms of energy. This is the underlying concept for the energy-based seismic design (EBSD) method. In EBSD, a structure is designed so its energy absorbing capacity exceeds the energy demand from earthquakes. To perform EBSD, one needs to address both the capacity and demand sides of the problem. This paper is focused on the demand aspect of EBSD, while the capacity aspect will be discussed in a separate paper. In what follows, the basic concept of EBSD is reviewed, and input and hysteretic energy spectra equations for four soil site classes, four hysteresis models, and five ductility levels will be developed and proposed for use in EBSD.

2. Energy-Based Seismic Design (EBSD)

The equation of motion for a single degree of freedom (SDOF) inelastic system under ground motion excitation is given by where is mass of the system; is damping coefficient; is restoring force; is ground acceleration; and , , and are the relative acceleration, velocity, and displacement of the system with respect to the ground, respectively.

The energy balance equation for this SDOF structure based on relative motion can be written asEquation (2) can be rewritten as where in (3) vanishes when vibration of the structure ceases. is related to the inherent viscous damping of the structure and any supplemental damping devices provided to the structure. consists of two different types of energy: elastic strain energy, , and hysteretic energy, HE. Elastic strain energy does not cause permanent deformation, as it occurs as a result of elastic deformation of the structure, is recoverable, and vanishes when vibration ceases. Hysteretic energy is related to the inelastic deformation of the structure and is the energy that is expended through inelastic hysteretic response. It is obtained by subtracting the strain energy recovered during the unloading process from the strain energy accumulated during the loading process [6]. For a given hysteresis model, hysteretic energy is represented by the area enclosed by the hysteresis loops.

The philosophy of EBSD primarily focuses on ensuring that structures are designed to meet the energy demand of an earthquake. In EBSD, if the energy demand of a structure due to an earthquake can be dissipated through damping or a controlled inelastic deformation mechanism in the structure, the design is said to be satisfactory. Therefore, energy is considered to be the main design parameter in any energy-based seismic design procedure. EBSD is believed to be a rational design approach for seismic design because it takes into account the accumulated earthquake induced damage in the design procedure. However, the viability of EBSD depends on the accuracy in developing inelastic design spectra for SDOF as well as the quality of the equations that relate the input energy to the hysteretic energy. Furthermore, when applying to MDOF structures, a reliable and rational procedure for distributing the hysteretic energy over the height of the structure or among the various energy dissipating mechanisms needs to be developed.

3. Input and Hysteretic Energies

3.1. Input Energy

Earthquake input energy is the amount of energy an earthquake imparts to a structure. If relative velocity of the structure is used in the computation, the resulting input energy as shown in (7) is called the relative input energy. On the other hand, if the relative velocity in the equation is replaced by the total velocity of the structural system , the resulting input energy is referred to as the absolute or total input energy.

In a study by Bruneau and Wang [7] who used closed-form energy expression for a SDOF system subjected to rectangular and harmonic base excitation, a close relationship between relative input energy and relative displacement was reported. As a result, they recommended that a relative input energy formulation be used for assessing earthquake damage on structures. Henceforth, the relative input energy IE is used in the present study to quantify the energy content of an earthquake. In another study, Fabrizio et al. [8] performed correlation study between input and hysteretic energies with roof and maximum interstory displacements using six hysteresis models and 900 recorded ground motions. Two pairs of indices were proposed to quantify the correlation for both single degree and multiple degrees of freedom systems.

Over the years a number of researchers have recommended empirical formulae to estimate earthquake input energy. Housner [9] computed the input energy per unit mass of a SDOF system as proportional to the square of the pseudospectral velocity. Akiyama [10], using Japanese design earthquakes, proposed the input energy per unit mass of an elastic SDOF structure due to a given earthquake as proportional to the square of the natural period of vibration of the structure or the predominant period of ground motion, whichever is smaller. Kuwamura and Galambos [11] extended Akiyama’s work and included the effects of severity and duration of the earthquake in their energy expression. Fajfar et al. [12], using 40 accelerograms and examining structures that fall within the constant velocity region of the response spectra, proposed an input energy expression as a function of the duration of strong motion as defined by Trifunac and Brady [13] and the peak ground velocity of the ground motion. Uang and Bertero [14], using five accelerograms and the absolute input energy, concluded that Housner’s (1956) expression for input energy reflected the maximum elastic energy stored in the structures but did not include the damping energy. Like Fajfar et al., they proposed an expression for the absolute input energy per unit mass as a function of the duration of strong motion and the peak ground velocity of the ground motion. Manfredi [15], using 244 accelerograms, proposed an input energy expression that was applicable for structures with vibration periods that fell in the constant velocity region of the spectra. Manfredi’s input energy expression includes parameters such as pseudospectral velocity, earthquake intensity, peak ground acceleration, peak ground velocity, and a term referred to as cyclic ductility , which is expressed as a function of the maximum cyclic plastic deformation and the yield deformation of the structure. Khashaee [16] proposed an equation for estimating seismic input energy expressed in terms of an earthquake intensity index proposed by Park and Ang [17], as well as the natural period and ductility of the structure. Other researchers who have proposed input energy spectra are Decanini and Mollaioli [18], Benavent-Climent et al. [19, 20], Amiri et al. [21], López-Almansa et al. [22], and Cheng et al. [23, 24].

Because of the different assumptions and methodologies used, all the aforementioned approaches that have been proposed to quantify the amount of earthquake input energy can show large variations and deviations. In addition, because most researchers used elastoplastic or bilinear hysteresis models in their computations, the accuracy of the estimates could deteriorate for structures that exhibit different kinds of hysteretic behavior. In the present study, earthquake input energy expressions that take into account ground motion characteristics, hysteretic behavior, ductility level, and soil site conditions are proposed and compared with nonlinear time history analysis results. The four hysteresis models used are shown in Figure 1. The site soil classes considered are labeled B, C, D, and E in accordance with ASCE/SEI 7-10 [25] as shown in Table 1.

Soil site classSoil description (m/s)

CVery dense soil and soft rock
DStiff soil
ESoft soil

: average shear wave velocity of the upper 30 m depth of soil profile of a site.
3.2. Hysteretic Energy

Hysteretic energy is used as a parameter to quantify the needed energy dissipation capacity of structures subject to earthquake excitation. This is because hysteretic energy can be used to measure the inelastic cumulative damage of structures during an earthquake. Knowing the amount of energy that needs to be dissipated allows a designer to determine if a structure possesses sufficient energy dissipation capacity to avoid collapse. Hysteretic energy is the energy associated with the hysteretic behavior of a structural system and can be computed numerically by integrating the inelastic portion of the hysteretic force-displacement relationship of the structure under an earthquake excitation. This is equal to the area bounded by the hysteresis loops when the structure undergoes vibration under the excitation force. For a given earthquake and a SDOF system, hysteretic energy is related to input energy and can be estimated from the corresponding input energy. Its accuracy primarily depends on the accuracy of the expressions used for estimating the input energy. Unlike input energy, hysteretic energy is independent of whether relative or absolute energy is used in the energy balance equation.

One of the first researchers to propose equations for estimating hysteretic energy of SDOF systems from the corresponding input energy was Housner (1956), who defined the energy that contributed to the damage of a structure as the total seismic input energy, IE, minus the energy dissipated through inherent damping, . According to this definition, the damage energy is the sum of the absorbed and kinetic energy. However, at the end of ground motion duration, the kinetic energy becomes small, and since the elastic energy contributes only to recoverable elastic deformation, only the hysteretic energy is assumed to cause damage to the structure. Housner proposed that the ratio of damage energy per unit mass to input energy per unit mass (both have unit of velocity square) be equal to the square root of the respective energies. Akiyama (1985), based on analyses of SDOF systems with elastic-perfectly-plastic restoring force characteristics, proposed the damage to input energy per unit mass ratio as a function of the structure’s damping ratio . Fajfar and Vidic [26] performed a parametric study on nonlinear elastoplastic SDOF systems subjected to five different ground motions from different countries and for proposed an expression for the energy ratio as a function of the ductility ratio of the structure. For a structure with damping ratio , Manfredi (2001) recommended the hysteretic energy to input energy ratio be expressed as a function of the earthquake intensity, peak ground acceleration, peak ground velocity, and cyclic ductility . Khashaee (2004), with the intention of eliminating the cyclic ductility variable used by Manfredi, applied regression analysis on input and hysteretic energies data obtained from 160 accelerograms and proposed an expression for the energy ratio which is dependent only on the ductility ratio . López-Almansa et al. (2013), using 149 ground motion records extracted from the Turkish dataset registers classified into eight groups by soil type, surface magnitude, and source distance effects, proposed empirical criteria for estimating hysteretic energy from an input energy expressed in terms of equivalent velocity. Their study concluded that the influence of period on the hysteretic to input energy ratio was noticeable, whereas the effect of soil and ground motion parameters was observed to be negligible.

In the present work, equations that relate hysteretic to input energy that are both period and ductility dependent taking into consideration the effects of system hysteretic behavior and soil class will be presented.

4. Ground Motion Ensembles, Target Spectra, and Hysteresis Models

To develop the input energy spectra, four ground motion ensembles based on ASCE/SEI 7-10 (2010) site soil classes B, C, D, and E as shown in Table 1 are used. The numbers of earthquake records used for the four site classes are 38, 42, 38, and 26, respectively. All these ground motion ensembles were extracted from the Pacific Earthquake Engineering Research Center (PEER) ground motion record database. They are summarized in Tables 25. Note that, for each selected earthquake, both the fault normal and fault parallel components were used in the study. In Tables 25, the next-generation attenuation (NGA) number, event, year, station, moment magnitude , closest distance to the rupture plane , average shear wave velocity of the upper 30 m depth of soil profile , and scale factor (discussed in the section on Target Spectra) are given. Additional information on these ground motion records can be found in Some general comments in regard to earthquake magnitudes, site-to-source distances, target spectra, and damping used in the present study are given below.

NGA #EventYearStation (km) (m/s)Scale factor

2107Denali-Alaska2002Carlo (temp)7.950.9963.90.8985
946Northridge-011994Antelope Buttes6.6946.9821.71.7258
804Loma Prieta1989So. San Francisco-Sierra Pt.6.9363.11020.60.9561
1033Northridge-011994Littlerock-Brainard Can6.6946.6821.71.2158
2111Denali-Alaska2002R109 (temp)7.943963.90.8937
788Loma Prieta1989Piedmont Jr High6.9373895.41.0313
797Loma Prieta1989SF-Rincon Hill6.9374.1873.10.8949
1021Northridge-011994Lake Hughes #4-Camp Mend6.6931.7821.71.2027
925Big Bear-011992Rancho Cucamonga-Deer Can6.4659.4821.72.0055
1074Northridge-011994Sandberg-Bald Mtn6.6941.6821.70.8161
1060Northridge-011994Rancho Cucamonga-Deer Can6.6980821.71.4735
1096Northridge-011994Wrightwood-Jackson Flat6.6964.7821.71.5746
943Northridge-011994Anacapa Island6.6968.9821.72.4764
795Loma Prieta1989SF-Pacific Heights6.93761249.91.1371
1041Northridge-011994Mt Wilson-CIT Seis Sta6.6935.9821.71.0261

NGA #EventYearStation (km) (m/s)Scale factor

353Coalinga-011983Parkfield-Gold Hill 4W6.3641.1438.31.4575
762Loma Prieta1989Fremont-Mission San Jose6.9339.5367.60.9005
798Loma Prieta1989SF-Telegraph Hill6.9376.5712.82.182
980Northridge-011994Huntington Beach-Lake St6.6977.5370.81.6481
1015Northridge-011994LB-Rancho Los Cerritos6.6951.9405.21.3589
1026Northridge-011994Lawndale-Osage Ave6.6939.9361.20.9721
1027Northridge-011994Leona Valley #16.6937.2684.91.5469
1028Northridge-011994Leona Valley #26.6937.24461.4716
1029Northridge-011994Leona Valley #36.6937.3684.91.3395

NGA #EventYearStation (km) (m/s)Scale factor

1816Hector Mine1999North Palm Springs Fire Sta #367.1361.8345.41.9373
862Landers1992Indio-Coachella Canal7.2854.2345.41.3058
941Big Bear-011992Yermo Fire Station6.46[71.0]353.62.7963
907Big Bear-011992Hesperia-4th & Palm6.46[44.8]345.42.2741
1762Hector Mine1999Amboy7.1343271.40.7422
1791Hector Mine1999Indio-Coachella Canal7.1373.5345.41.6048
1776Hector Mine1999Desert Hot Springs7.1356.4345.42.1505

NGA #EventYearStation (km) (m/s)Scale factor

3319Chi-Chi- Taiwan-061999CHY1076.379.8175.71.8138
962Northridge-011994Carson-Water St6.6949.8160.61.903
808Loma Prieta1989Treasure Island6.9377.4155.11.0323

4.1. Earthquake Magnitudes

The ground motion records have moment magnitudes that vary from 6.2 to 7.9. This range covers ground motions referred to as strong to major earthquakes.

4.2. Site-to-Source Distances (km)

The effect of near-source ground motions on structures is more pronounced compared to that of far-source ground motions. Hall et al. [27], Campbell [28], and Bazzurro and Luco [29] have suggested that pronounced effects of near-source ground motions need to be addressed differently during design. In other words, the result of a study based on a mix of near-source and far-source ground motion records is likely to give misleading conclusions. As a result, this study has excluded near-source ground motion records in the selected ground motion ensemble and is primarily focused on far-source ground motion effects. Near-source and far-source ground motion records are considered to have an epicentral distance of less than 15 km and greater than 30 km, respectively, to the site of the structure or recording station. In this study, the epicentral distance of the selected ground motions spans the range of approximately 30 to 80 km.

4.3. Target Spectra

In developing design spectra, it is desirable to have the group of selected ground motion records exhibit similar spectral characteristics. The resulting design spectra developed using such similar spectra should yield smaller standard deviation. However, because ground motions that occur at different sites have different magnitudes, peak values, and durations and when averaged, the resulting standard deviations could still be high. To minimize the anticipated high discrepancies and variations, ground motions are often scaled to match some target spectra before they are used in time history analysis. In this study, ground motion records were selected and scaled to match soil site-based response spectra generated by geometrically combining the Pacific Earthquake Engineering Research Center’s Next Generation Attenuation (PEER-NGA) ground motion models developed by Abrahamson and Silva [30], Campbell and Bozorgnia [31], Boore and Atkinson [32], and Chiou and Youngs [33] so they represent an 84th percentile (i.e., one standard deviation above the mean values) of the geometric mean of these four attenuation models. The target spectra for the four soil site classes are shown in Figure 2. Also shown in this figure are the geometric and arithmetic means of the ensembles of earthquake spectra generated from the selected earthquake records for each respective soil site class. The parity between the mean and target spectra indicates that the selected ground motion records satisfactorily represent ground motions recorded at each specific soil site class. A statistical analysis yields a correlation coefficient of 0.99, which signifies that each selected ground motion ensemble is a good representative of its respective soil site class. The selection and scaling of all the ground motions have been made in such a way that the resulting ground motion spectra match the target spectra in the period range 0.1 to 3.5 s with equal weights. This range of vibration periods was chosen because most medium- to high-rise buildings have fundamental periods of vibrations of 0.3 to 3.0 s according to the PEER manual, and most bridges have fundamental periods of vibrations in the 0.2 to 1.2 s range according to Kunde and Jangid [34].

4.4. Damping

In the USA, the design of earthquake-resistant structures is often based on a damping ratio of 5% as seismic design maps contained in ASCE/SEI 7-10 (2010) and IBC [35] for building design and AASHTO [36] for bridge design were all developed based on 5% of critical damping. It is also common to use 5% damping to capture the damping behavior of structures that exhibit inelastic behavior when subjected to dynamic loading. Thus, a damping value of 5% was used in this study to develop the site class and hysteretic specific input energy spectra.

4.5. Ductility

Ductility plays an important role in the response of structures to dynamic loading. The more ductile the structure is, the less likely it will experience catastrophic damage during a major seismic event. Constant ductility spectra developed by Peng and Conte [37], Inel et al. [38], and Zhai and Xie [39] supported the same hypothesis. According to Chopra and Goel [40], constant ductility spectra are being used more often for seismic design. In view of this, the input energy spectra developed in this study are chosen to be constant ductility spectra with ductility levels of and 5. Note that means the structure remains elastic.

4.6. Hysteresis Models

The force-displacement relationship or hysteretic behavior of a structure affects its seismic response and hence the energy spectra. Thus, it is important that the hysteretic behavior of a structure being designed is known and considered in the determination of the input energy. In this study, four different hysteresis models, namely, bilinear plastic (BP), stiffness degradation or modified clough (SD), bilinear flag (BF), and bilinear slip (BS) as shown in Figure 1, are considered. The BP model is often used to model the cyclic behavior of steel structures with strain hardening effect, while the SD model is used to model reinforced concrete structures that exhibit stiffness degradation under repeated loading and unloading [41]. The BF model is generally used for modeling structures that are capable of recentering after an earthquake [42, 43], and the BS model is used to model structures that experience joint or bond slippage [44]. In the figure, is the initial stiffness, which changes in accordance with the period of the SDOF system, but a constant post- to preyield stiffness ratio of is used because real structures do not undergo complete plastification after first yield. The effect of the value of on the shape or size of the input energy spectra is not very significant. For instance, Nakashima et al. [45] studied bilinear SDOF and MDOF structures with post- to preyield stiffness ratio up to 0.75 and concluded the effects of only have a minor effect on the input energy.

5. Energy Spectra and Normalization Parameter

Using the scaled ground motion records and hysteresis models, nonlinear time history analyses were performed and (7) was numerically integrated to generate constant ductility spectra with damping ratio . In addition to accounting for structural properties (hysteresis model, period, ductility, and damping) and soil site effects, it is desirable to consider the effect of earthquake intensity in the development of the proposed energy spectra. One way to incorporate the earthquake intensity effect is to normalize the energy spectra using some seismic damage indices or quantities that are used to measure the intensity of seismic events. In the present study, the following ground motion indices have been used to determine this normalization parameter.(i)Cumulative absolute velocity (CAV) (EPRC [46]):(ii)Arias intensity () (Arias [47]):(iii)Seismic damage index () (Manfredi (2001)):(iv)Velocity index (VI) (proposed in the present study): where is ground motion acceleration, is acceleration due to gravity 9.81 m/s2 (32.2 ft/s2), PGA is peak ground acceleration, PGV is peak ground velocity, and is duration of ground motion.

Applying the above indices as normalization parameters, the coefficients of variation for various periods and ductility ratios calculated from the suite of earthquake records for soil site class B and bilinear plastic (BP) hysteresis model are plotted in Figure 3. As can be seen, the coefficients of variation for the input energy per unit mass normalized by the proposed velocity index are the lowest for the range of periods and ductility used. Similar observations are made for other soil site classes and hysteresis models. It should be noted that a key advantage of CAV over other peak ground motion and response-spectral parameters is that through integration of the absolute value of the ground acceleration, it can account for the cumulative effect of the ground motion duration. According to EPRC (1998), CAV is also a ground motion parameter that best correlated with structural damage out of the various ground motion parameters that it investigated. One shortcoming of CAV is that although it has unit of velocity, it is not directly related to the ground motion velocity, so some characteristics of the ground motion are lost. By combining both CAV and PGV into a single parameter VI, we are able to take advantage of the desirable characteristics of both parameters. Hence, the proposed velocity index will be used as the normalization parameter for all subsequent computations. Note that the unit of VI is (distance/time)2, which is the same as that of energy per unit mass. Thus, normalization of the input energy per unit mass by VI results in an energy spectrum that is nondimensional. Nondimensional spectra have an added advantage that they allow designers to use any units of their choice. Moreover, for a given seismic site and soil class, the two quantities PGV and CAV required to calculate VI can readily be obtained from the literature. Publications by Power et al. [48], Campbell and Bozorgnia [49], and Bradley [50] are among some research findings that give CAV and PGV prediction equations.

6. Proposed Normalized Energy Spectra

Nonlinear analysis software called BISPEC [51] was used to obtain input energy (IE) and hysteretic energy (HE) spectral values of SDOF systems. The spectral values were obtained from nonlinear time history analysis of an SDOF system subjected to earthquake ground motion records by using the widely used Newmark integration method. In this study, values of beta and gamma (i.e., the average acceleration method) were used. Time steps for all the analyses were taken as the minimum of three quantities: the recorded earthquake time step, the period of the SDOF divided by 25, and 0.01 s. Once the input energy spectral values were obtained, they were normalized as discussed below.

The normalized energy (NE) proposed in this study is defined as the square root of the input energy per unit mass, , divided by the velocity index, VI; that is,The use of (12) as a normalization parameter not only converts the input energy per unit mass to a nondimensional quantity but also makes the energy spectra more narrowly banded (i.e., relatively smaller peaks and valleys) when compared with the actual input energy spectra. This is demonstrated in Figure 4 where the actual input energy spectra, , (Figure 4(a)) are compared with the VI normalized input energy spectra (NE) (Figure 4(b)) for SDOF structures with hysteresis model BP, ductility ratios of , and soil site class B. The figure clearly shows that, upon normalization by VI, the peaks and valleys of the input energy spectra are less pronounced, thus reducing the dispersion of the spectral values. Also shown in Figure 4 are the and spectra, which can be seen as enveloping a rather high percentage of the input energy spectra for the selected ground motion records, particularly in the range of periods from 0.5 to 5 s. Similar observations have been made for the other three hysteresis models and soil site classes investigated in this study. It should be noted that the use of the or spectra as opposed to the mean spectra is recommended for use in the proposed EBSD not only because they envelope a relatively high percentage of the input energy spectra but also because the potential damage caused by the kinetic energy component of the input energy is ignored and so a more conservative approach is warranted.

The equations used to represent the and VI normalized input energy spectra are to be derived so they will conform to the general shape of design spectra often used in seismic codes. That is, they will consist of three regions: a straight line for structures in the short period range, a constant (plateau) for structures in the intermediate period range, and a power curve for structures in the long period range. Thus, the proposed VI normalized input energy spectra, NE, in the period range  s will have the following form. where, with reference to Figure 5, the constants and define the linear part of the spectra, is the maximum value of the NE spectra, and and are the periods that separate the short and long period regions from the intermediate region of the spectra.

Values of , , , , , , and for the SDOF normalized energy (NE) spectra are obtained for the four site classes and four hysteresis models using a three-part nonlinear regression analysis: one each for the short, intermediate, and long period regions. They are then expressed as a function of ductility as shown in Tables 69.

Mean + Mean +

Hysteresis model BP

Hysteresis model SD

Hysteresis model BF

Hysteresis model BS

Mean + σMean +

Hysteresis model BP

Hysteresis model SD

Hysteresis model BF