Estimation of Floor Response Spectra for Self-Centering Structural Systems with Flag-Shaped Hysteretic Behavior

: Spectral floor-acceleration demands are necessary for the seismic design of acceleration-sensitive non-structural components (NSCs). Existing studies to estimate floor response spectra (FRS) using empirical equations are based on elasto-plastic and stiffness degrading hysteretic behavior of the primary structure. In the present study, the FRS for self-centering (SC) structural systems with flag-shaped hysteretic behavior under far-fault ground motions are investigated. The FRS and dynamic amplification factor (DAF) are obtained from nonlinear response history analysis (NLRHA) of SC structural systems with flag-shaped hysteretic behavior. An equation to estimate the FRS is proposed and verified by carrying out NLRHA using a different set of far-fault ground motions. The equation to estimate FRS is shown to predict floor acceleration demands with very good accuracy.


Introduction
The adoption of seismic design has greatly reduced the damage to the structural elements of buildings. However, damage to secondary structural elements and non-structural components (NSCs) can lead to huge economic losses, due to associated loss of functionality of important facilities and business downtime after a major earthquake (see for eg., EERI [1], Villaverde [2], Dhakal et al. [3], Devin and Fanning [4], Wang et al. [5]). Although existing building codes such as Eurocode 8 [6] and ASCE 7-16 [7] provide expressions to determine the acceleration demand for the estimation of seismic design force for acceleration-sensitive NSCs, several studies have shown that the peak floor acceleration and floor response spectra (FRS) estimated from such codes are not accurate (see for e.g., Sullivan et al. [8], Vukobratovic and Fajfar [9], Aragaw and Calvi [10], Kazantzi et al. [11], Vukobratovic et al. [12]).
Estimation of FRS using empirical equations based on a single-degree-of-freedom (SDOF) system for the primary structure have been reported in previous studies. Oropeza et al. [13] considered primary structures with an elasto-plastic and modified Takeda hysteretic model, while Sullivan et al. [8] also considered primary structures using a modified Takeda hysteretic model. Vukobratovic and Fajfar [14] used the equation of Yasui et al. [15] in the pre-and post-resonance region and an empirically developed expression in the resonance region for primary structures with an elasto-plastic and stiffness degrading model to estimate FRS. Welch and Sullivan [16] modified the equation developed by Sullivan et al. [8] to estimate the FRS.
Several studies have also extended the approach to estimate the FRS using empirical equations based on SDOF systems to multi-degree-of-freedom (MDOF) systems. For instance, the estimation of FRS for reinforced concrete (RC) frame systems by Calvi and Sullivan [17], Vukobratovic and Fajfar [18], and Merino et al. [19], for RC walls by Welch and Sullivan [16] and Vukobratovic and Ruggieri [20], and for base rocking wall buildings by Aragaw and Calvi [10]. The excellent seismic performance of self-centering (SC) structural systems with flagshaped hysteretic behavior have been reported previously (see for e.g., Kurama [21], Smith et al. [22], Belleri et al. [23], Buddika and Wijeyewickrema [24], Shrestha et al. [25]). In the present study, the FRS for SC structural systems with flag-shaped hysteretic behavior under far-fault ground motions are investigated. The FRS and dynamic amplification factor (DAF) are obtained from nonlinear response history analysis (NLRHA) of SC structural systems with flag-shaped hysteretic behavior. An equation to estimate the FRS is proposed and verified by carrying out NLRHA using a different set of far-fault ground motions.

Self-Centering (SC) SDOF System with Flag-Shaped Hysteretic Behavior, Ground Motion Records, and Numerical Modeling
The force-displacement relationship of the self-centering (SC) SDOF system with flag-shaped hysteretic behavior, which is the primary structure is shown in Figure 1. The nonlinear response history analysis (NLRHA) of the SC flag-shaped SDOF system is carried out with the following parameters: initial vibration period 0.1s, 0.3s, 0.5s, 0.75s,   (Table A-4A, FEMA [31]) and given in Table 1. In the present study, the NLRHA is carried out using OpenSees [32]. The SC flagshaped SDOF system (primary structure) is modeled using a zero-length element with the SelfCentering material model. The NSC (secondary structure) is represented by an elastic SDOF system and modeled using a zero-length element. The numerical integration of the equations of motion is accomplished using the Newmark constant average acceleration method ( 0.25, 0.5)

Floor Response Spectra and Dynamic Amplification Factor from Nonlinear
Response History Analysis (NLRHA)

Floor Response Spectra
The floor response spectra for self-centering (SC) structural systems with flag-shaped hysteretic behavior are determined using the following procedure: (a) The NLRHA of the prescribed self-centering (SC) SDOF system with flag-shaped hysteretic behavior (primary structure), for the set of ground motion records and determination of the total floor acceleration response history for each ground motion record. Here, the total floor acceleration response history is the sum of the floor acceleration response history relative to the ground and the ground acceleration response history. (b) Linear RHA of the elastic SDOF system (secondary structure), using the set of total floor acceleration response histories determined from step (a), to generate floor response spectra. observed that when the primary structure is elastic, i.e. 1 R = , a single peak of the mean normalized FRS at / 1 s p T T = can be observed. When 1 R > , the primary structure behaves inelastically and rather than a single peak, the maximum value of the mean normalized FRS remains nearly constant over a wide period range and forms a spectral plateau. Moreover, the width of the spectral plateau increases with increase in R and decrease in β , which is due to the higher ductility demand on the primary structure. In most cases, the peak value of mean normalized FRS increases when R changes from 1 to 2 and then decreases for 2 R > .
To investigate the effect of β more clearly, the mean normalized FRS are shown for Figure 4. The results show that the mean normalized FRS decreases when β increases. A wider spectral plateau is observed with decrease in β .

Maximum Dynamic Amplification
Factor max DAF Figure 5 shows where , , a b and c are constant coefficients determined from nonlinear regression analysis, separately for each R and given in Table 2. Note that in the present study, the viscous damping ratio of the secondary structure  Table 3

Post-Resonance Dynamic Amplification Factor
where the effective period e T associated with the secant stiffness at peak response of the primary structure is given by, (see Welch and Sullivan [16]), where α = post-yield stiffness ratio (taken as 5% in this study), and the ductility demand μ is computed using the equation for constantstrength inelastic displacement ratio R C given by Zhang et al. [29] for SC systems as,

Equation to Estimate FRS
In the present study, the following equation is proposed to estimate the FRS for the primary structure as SC systems,

T DAF T T T A T R T DAF T T T A
Note that an equation of similar form was used by Sullivan et al. [8] for the estimation of FRS for well detailed RC structures.
The acceleration demand of the secondary structure increases linearly in the pre-resonance region 0 ≤ <

Verification of Proposed Equation to Estimate FRS
For the verification, a primary structure which is a SC structural system with flagshaped hysteretic behavior, with initial vibration period 0. Here, 11 far-fault ground motions are considered from the 22 far-fault ground motion set of FEMA P695 (Table A-4A, FEMA [31]) and given in Table 3. Note that these 11 far-fault ground motions are not the same ground motions used in Sec. 2. The mean floor response spectra are determined from NLRHA using the procedure given in Section 3.1. The mean peak acceleration demand of the secondary structure s A obtained from Equation (5) (5) slightly underestimates the peak acceleration demand of the secondary structure in the post-resonance region but is mostly conservative in the other regions. It is also observed that the width of the spectral plateau with varying R and β can be well estimated from Equation (5). In general it can be concluded that the s A estimated from Equation (5) shows good accuracy when compared with NLRHA results.

Comparison of the Proposed FRS with Existing Direct Methods
The mean peak acceleration demand of the secondary structure s A obtained from Equation (5) of the present analysis and existing direct methods of Sullivan et al. [8] and Vukobratovic and Fajfar [14] are compared with the NLRHA results in Figure 10. The equations to estimate the FRS using existing direct methods are given in Appendix B. For the comparison, a primary structure with flag-shaped hysteretic behavior, with initial vibration period 1.0 s, p T = response reduction factor 2; R = post-yield stiffness ratio 5%; α = energy dissipation parameter 0%, 20%, ,100% β =  , and viscous damping ratio 5% p ξ = ; and a secondary structure with viscous damping ratio 5% s ξ = is considered. Figure 10. Comparison of the mean floor response spectra from Equation (5) of the present analysis, Sullivan et al. [8], Vukobratovic and Fajfar [14], and NLRHA for a primary structure with initial vibration period 1.0 s ; and (f) . Note: The methods of Sullivan et al. [8] and Vukobratovic and Fajfar [14] are for primary structures with modified Takeda and elasto-plastic hysteretic behavior, respectively.
It is observed that the s A estimated from Equation (5) of present analysis provides a good estimate in the resonance region and slightly overestimates in the other regions (see Figure 10). The method of Sullivan et al. [8] slightly overestimates in pre-resonance region but underestimates in the resonance region. Moreover, in the post-resonance region, s A is underestimated for  (5) of the present analysis shows good accuracy than the existing direct methods when compared with NLRHA results.

Conclusions
In this study, the floor response spectra (FRS) for self-centering (SC) structural systems with flag-shaped hysteretic behavior is investigated using nonlinear response history analysis (NLRHA), and an equation to estimate the FRS is proposed. In particular, a primary structure with post-yield stiffness ratio 5%; = α viscous damping ratio 5% = p ξ ; and a secondary structure with viscous damping ratio 5% = s ξ is considered. The proposed equation is then verified using a different set of far-fault ground motions. Based on the study, the following conclusions can be drawn: 1. The effect of the primary structure initial vibration period p T , response reduction factor R , and energy dissipation parameter β on the FRS is studied. A single peak was observed on the mean normalized FRS for 1 = R , but for 1 > R , the maximum value of the mean normalized FRS is nearly constant over a wide period range and forms a spectral plateau. The width of the spectral plateau increases with increase in R and decrease in β , which is due to the higher ductility demand on the primary structure. In addition, the peak value of mean normalized FRS increases when R changes from 1 to 2 and then decreases for  Acknowledgments: The authors B.K.S. and N.M. gratefully acknowledge Monbukagakusho (Ministry of Education, Culture, Sports, Science, and Technology, Japan) scholarships for graduate students.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Comparison of the Normalized Floor Response Spectra
The normalized FRS from Equation (5)