Advances in Civil Engineering

Volume 2016 (2016), Article ID 1548319, 29 pages

http://dx.doi.org/10.1155/2016/1548319

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

^{1}Michael Baker International, Hamilton, NJ 08619, USA^{2}Department of Civil and Environmental Engineering, Syracuse University, Syracuse, NY 13244-1240, USA

Received 30 June 2016; Revised 14 September 2016; Accepted 26 September 2016

Academic Editor: Lucio Nobile

Copyright © 2016 Mebrahtom Gebrekirstos Mezgebo and Eric M. Lui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.