Constant ductility inelastic displacement ratio spectra of shape memory alloy-friction self-centering structures based on a hybrid model

Previous researches on the constant ductility inelastic displacement ratio spectra (Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document}) of self-centering structures have been conducted based on the typical flag-shaped self-centering (FS) model. However, the arising shape-memory-alloy-friction self-centering structures (SMAFSs) own their specialized force–displacement relationship different from the typical FS model. In this research, a hybrid force–displacement model composed of the typical FS model and Coulomb friction model is employed and an efficient calculating procedure is proposed to statistically investigate the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of SMAFSs. Comparison with the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of the typical FS system with similar self-centering capacity shows that the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of the SMAFS is significantly different from that of the typical FS system, which explains the necessity of employing the hybrid model for SMAFSs and the effects of the friction. The effects of seismic parameters and structural parameters on the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of SMAFSs are investigated. Furthermore, the formula for estimating the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of SMAFSs is proposed through statistical regression. This proposed formula could estimate the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of SMAFSs and describe the effects of structural parameters more accurately. The research could provide a basis of estimating the Cμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}_{\upmu }$$\end{document} of SMAFSs to obtain reliable seismic design results of the structures.


Introduction
The traditional seismic design of structures mainly focuses on achieving the target of no collapse under strong earthquakes based on the ductility principle. However, previous strong earthquakes have proved that the action of strong earthquakes would lead to significant plastic deformation and residual deformation of structures. Excessive residual deformation will cause structures to lose their function and make it difficult to restore their function (Dong et al. 2022;Chen et al. 2016). At present, residual deformation has been used as 1 3 an important indicator to measure the post-earthquake resilience and seismic performance level of structures in performance-based seismic design (Zhong et al. 2022). In existing researches, various self-centering structures that seek to minimize the residual deformation have been proposed and rapidly developed (Lee et al. 2011;Qiu et al. 2017;Hu et al. 2021a, b).
Self-centering structures include but not limited to rocking self-centering structures (Rahgozar et al. 2019;Eatherton et al. 2014;Hu et al. 2021a, b), frame structures with self-centering beam-column connections (Priestley et al. 1993;Eldin et al. 2019) or with self-centering energy dissipation dampers or braces Xu et al. 2020). No matter what kind of self-centering structures, they can be regarded as being functionally composed of self-centering elements and energy dissipation elements. In current research, the commonly used self-centering elements include pre-stressed tendons (Priestley et al. 1993;Eldin et al. 2019;Li et al. 2021), pre-pressed disc spring components (Xu et al. 2020;Yousef-Beik et al. 2020;Dong et al. 2019;Xue et al.2021), and shape memory alloys (SMAs) (Ozbulut et al. 2010;Gur et al. 2013;Qzbulut et al. 2016;Xue et al. 2020;Wang et al. 2020;Shi et al. 2020;Qiu et al. 2022;Zhang et al. 2021;Shams et al. 2021;Mirzai et al. 2020;Wang et al.2019). The common energy dissipation elements mainly include steel-based dampers (Eatherton et al. 2014;Li et al. 2021;Dong et al. 2019), friction-based dampers (Eldin et al. 2019;Xu et al. 2020;Xue et al. 2021;Ozbulut et al. 2010;Gur et al. 2013;Ozbulut et al. 2016;Zhang et al. 2021;Shams et al. 2021;Mirzai et al. 2020;Wang et al. 2019;Westenenk et al. 2019), and viscous fluid dampers (Kam et al. 2010). Among these self-centering elements, SMAs have specific superelasticity characterized with excellent self-centering function and certain energy-dissipating capacity, and SMAs are thus ideal materials for constructing self-centering dampers. In practical application, SMAs still need to be compounded with high energy dissipation elements such as buckling-restrained braces (BRBs) or friction elements to make up their insufficient energy dissipation capacity. For example, Qiu et al. (2020) studied the seismic design method and seismic performance of SMA-BRB hybrid bracing self-centering structures, while Gur et al. (2013), Ozbulut et al. (2016), Soul et al. (2017), Qian et al. (2016) and Zhang et al. (2020a, b, c) proposed various SMA-friction self-centering structures or dampers composed of SMA elements and friction elements. Friction elements are characterized with small yielding deformation and zero post-yielding stiffness. These characteristics make friction elements have lower equivalent stiffness and higher equivalent damping ratio compared to other displacement-dependent dampers. Therefore, combining friction elements with SMA-based dampers could significantly enhance their hysteretic energy dissipation performance. Noted that, the force-deformation relationships of SMA-friction hybrid structures are no longer the typical flag-shaped self-centering (FS) relationship as SMAs. Therefore, it should employ a factual hybrid model rather than the typical FS model to simulate the mechanical behavior of SMA-friction hybrid structures in static and dynamic analysis.
The self-centering structures are developed based on the concept of performance-based design. The rapid development of self-centering structures promotes research on corresponding performance-based seismic design theories and methods (Zhang et al. 2020a, b, c;Christopoulos et al. 2003). The displacement-based design (DBD) method is one of the most commonly used performance-based seismic design methods. Estimating the maximum displacement of self-centering structures is a key step in DBD methods (Liu et al. 2018;Qiu et al. 2021). An important and convenient method for estimating displacement is the coefficient method based on the inelastic displacement ratio spectra (Priestley et al. 2008;ASCE 41-17 2017;Konstandakopoulou et al. 2020;Yang et al. 2023). In 1960, Veletsos and Newmark (1960) studied and defined the inelastic displacement ratio spectra for the first time. Since then, scholars have performed plenty of statistical analysis on the inelastic displacement ratio spectra of different nonlinear structures through time-history analysis, and proposed different prediction formulas (e.g., Miranda 2001;Wen et al. 2014;Yaghmaei-Sabegh et al. 2017;Hassani et al. 2018;Wu et al. 2019;Konstandakopoulou et al. 2020). Currently, research on the inelastic ratio spectra mainly focuses on common elastic-plastic structures, and the constitutive models of these structures mainly include the bilinear model, elastic-plastic model, and degradation model. However, the research on the inelastic displacement ratio spectra of self-centering structures is still limited.
In recent years, several researches on the inelastic displacement ratio spectra of selfcentering structures have been conducted. Rahgozar et al. (2016) used the typical FS model shown in Fig. 2a to study the constant-strength inelastic displacement ratio spectra ( C R ) of the self-centering rocking system under near-field earthquakes, and determined the formula for estimating the C R by two-stage regression method. Zhang et al. (2018) studied the peak displacements of self-centering systems based on C R , which were calculated from more than five million nonlinear time history analyses of single-degree-of-freedom (SDOF) systems using ground motions representing a site with stiff soil conditions, and developed a simple equation that can be used during design to estimate the displacement demands on self-centering systems. Liu et al. (2019) used a hybrid mechanical model composed of bilinear elastic constitutive and elastic-plastic hysteretic constitutive to represent the force-displacement relationship of the self-centering structure composed of the prestressed tendons and metal energy dissipation dampers, and proposed the formula of the C R through statistical regression. Recently, Francesco et al. (2019) studied the constant ductility inelastic displacement ratio spectra (C ) of self-centering structures by employing the typical FS model shown in Fig. 2a, analyzed the effects of the seismic site types, duration and peak acceleration as well as the post-yielding stiffness ratios and ductility levels of the structures on C , and finally proposed the estimated formula of the C for the self-centering structures. Dong et al. (2020) studied the C of the self-centering system under near-field seismic motions based on the typical FS model. The results show that the C of self-centering structures under near-field seismic motions is significantly greater than that under far-field seismic motions in the acceleration sensitive region of the spectra, and finally proposed the formula for estimating the C of the self-centering structure under near-field earthquakes.
Above research shows that the typical FS model was commonly employed to simulate the mechanical behavior of self-centering structures in calculating the C (Francesco et al. 2019;Dong et al. 2020). However, the force-displacement relationship of SMAFSs is no longer a typical FS model due to the combination of friction devices and SMA devices. Therefore, it should employ a factual hybrid model to simulate force-displacement relationship of SMAFSs in calculating the C of SMAFSs. However, there is not the research on the C of SMAFSs based on the hybrid model composed of SMA model and friction model at present.
In this paper, a hybrid model composed of the typical FS model and coulomb friction model is employed to simulate the mechanical behavior of SMAFSs, and an efficient searching method is proposed to calculate the C of SMAFSs. For comparison, the C μ of the typical FS system is also calculated and discussed. The comparison results show that the C μ of the SMAFS is significantly different from that of the typical FS system when they have similar self-centering capacity, which explains the necessary of employing the hybrid model for SMAFSs. The effects of seismic motion characteristics and structural parameters on C are statistically investigated. Through statistical regression, a concise formula for 1 3 estimating the C of SMAFS is proposed, which can accurately predict the C of SMAFSs with different parameter combinations. The research results can provide a basis for estimating the maximum displacement of SMAFSs for seismic design.

Mechanical model
As an example of SMAFSs, Fig. 1a presents a structure isolated with shape memory alloysliding friction isolators, which have been proved to be effective in significantly reducing the seismic force of building (Gur et al. 2013). SMAFSs can be equivalent to a single-degree-offreedom (SDOF) system as shown in Fig. 1b. Unlike the SMA self-centering (SMAS) structure, the resisting force in SMAFS is provided by SMA devices and friction devices in parallel. The constitutive relationship of the SMA is a typical flag-shaped self-centering model as  Fig. 2b, and the hybrid hysteresis model of the SMAFS is thus derived as shown in Fig. 2c. The dynamic equation of the equivalent SDOF system shown in Fig. 1b under seismic motions can be written as follows: where u(t) represents the displacement of the isolated structure, represents the structural damping ratio,m represents the mass of the isolated structure, T represents the natural period of the corresponding linear system of SMAFS, ü g (t) represents the acceleration of seismic motion, and F SF represents the resisting force provided by the SMA device and the pure friction bearings in parallel. Figure 2 presents the schematic diagram of the force-displacement relationship of SMA, friction and SMAFS respectively, where F SMA ,F f , and F SF represent the forces of the SMA element, friction element, and SMA-friction hybrid self-centering structure, respectively. It is easy to know that in which, F SMA is determined by the Graesser-Cozzarelli model (Graesser et al. 1991) as Eqs. (3) and (4), in which, F sy , k 1 , and s represent the yield force, initial stiffness, and post-yield stiffness ratio of SMA respectively, a is one-dimensional back stress, f T , c and a are the parameters for controlling the hysteretic property of force-displacement relationship of SMA. The notations erf() and [U()] are used to represent the error function and unit step function, respectively. The functions are expressed as follow: In addition, the Coulomb friction force F f can be determined by the Bouc-Wen model as Eqs. (7) and (8) (1) where F fy and u fy denote the sliding-started friction force and displacement respectively, z is the evolutionary variable to account for the hysteresis property of friction, , and are the parameters for controlling the shape of the hysteretic curve of F f . According to the constitutive model described by Eqs.
(2)-(8) and Fig. 2. It can be easily known that Let where r s is defined as the SMA resisting force ratio.
According to Eqs. (9)-(11), it can be obtained that Figure 2c suggests that SMAFS has self-centering capability as < 1 . According to Eq. (12), the condition that SMAFS has self-centering capability can be obtained as follows, For verification, the above model is used to simulate the force-displacement relationship of the displacement-amplified SMA-friction damper (DASMAFD) in the literature (Zhang et al. 2020a, b, c). The DASMAFD can be regarded as a SMAFS due to its configuration composed of SMA and friction devices in parallel. In the specimen of DASMAFD, SMA device contains 16 NiTi wires with working length 100 mm and diameter 1 mm, and the friction force is supplied by the contact surfaces of the two steel components named guiding pad and actuator lever, in which the friction coefficient is 0.19 and the normal pressure on the friction surface is 8.46 kN. The parameters of SMA and friction devices adopted in the analytical model are tabulated in Table 1. The simulation results are shown in Fig. 3, which verifies that the above models can accurately simulate the force-displacement relationship of SMAFS. Note that, the special attention should be provided on the effects of scale of the tested specimens in the practical engineering, since scaled testing protocol may lead to discrepancies with respect to the full-scale response, just as the effects of the scale of the friction-based isolator on its friction coefficient (Quaglin et al. 2022). In this paper, the topic focuses on the constant ductility inelastic displacement ratio spectra of SMAFS, while the scaled specimen of DASMAFD is adopted to mainly show the feasibility of the analysis model. Therefore, the effects of the scale of the specimen are not further discussed.

Calculation method of C
The ductility factor refers to the ratio of the maximum inelastic displacement response ( u in ) to the yield displacement ( u y ) of inelastic structures (Francesco et al. 2019;Dong et al. 2020), namely The yield deformation u y of SMAFS is defined in Fig. 2c, and initial stiffness k e of SMAFS is calculated as k e = F y ∕u y , and the initial period ( T ) of the linearized system corresponding to SMAFS is calculated as The ratio of the maximum inelastic displacement response ( u in ) of the inelastic structure with given to the maximum elastic displacement response ( u e ) of the linearized elastic structure with the same initial T as the inelastic structure is defined as constant ductility inelastic displacement ratio C (Dong et al. 2020), i.e., According to the definition of C , a new method is proposed to calculate the C of SMAFSs, in which the factual hybrid force-displacement model shown in Fig. 2c is employed to simulate the mechanical behavior of SMAFSs. Figure 4 shows the calculation procedure of C of the SMAFS . In this calculating method, a half-interval searching Verification of the hybrid model method is employed to determine the suitable yield force of the inelastic structure with given . In detail, the calculating steps of C are narrated as follows: i. Given the mass m , the natural period T and the structural damping ratio of the linearized elastic system, calculate the stiffness k e = m(2π∕T) 2 . ii. Calculate the seismic responses of the linearized elastic system by employing Newton's iterative method, find the maximum elastic displacement u e . Let F yu = k e u e and F yl = 0. iii. Given the ductility factor , the post-yield stiffness ratio s , the hysteresis coefficient s , and the resisting force ratio r s of SMA. Assume that the yield force of the SMAFS is F y = (F yu + F yl )∕2 , and the yield displacement is u y = F y ∕k e . iv. Calculate the yield force ( F sy = r s F y ), the initial stiffness ( k 1 = F sy ∕u y ) of SMA and the sliding friction force of the friction element ( F fy = 1 − r s F y ). v. Calculate the seismic responses of the SMAFS through solving Eq. (1) by direct integration method, find the maximum inelastic displacement u in , and calculate the ductility factor i = u in ∕u y . vi. If ( i − )∕ ≤ , calculate C = u in ∕u e , and end the procedure. Otherwise,F yu and F yl will need to be reassigned: If i > , let F yl = F y ; if i < , let F yu = F y , and continue to iterate the steps iii-v.

Selection of seismic motions
A total of 200 ground motion records were selected for calculating and investigating the C of SMAFSs. These ground motion records were selected from earthquakes with magnitudes ranging from 6.0 to 7.9 to represent the moderate and larger earthquakes, the epicentral distances of the sites cover a wide range, and the V s,30 is bigger than 180 m/s. According to the ground soil conditions classified by the United States Geological Survey (Surhone et al. 1998), the ground motions can be subdivided into two groups, one of which is the ground motions on soft site where V s,30 is less than 360 m/s, and another of which is the ground motions on rock site where V s,30 is larger than 360 m/s. The two groups of ground motions are tabulated in Tables 4 and 5 in the Appendix respectively, which are all selected from the PEER Ground Motion Database. The acceleration spectrum of considered ground motions are plotted as shown in Fig. 5. The figure shows that the acceleration spectrum for considered ground motions cover different seismic characteristics, which suggests the selected ground motions could represent uncertainty of earthquakes whether on soft site or rock site. Therefore, enough number of ground motions are needed to obtain reliable results for C in a statistical sense. The results also show that the predominant period corresponding to the peak of acceleration spectrum for ground motions on soft site cover a wider range compared to that on rock site, which suggests that the ground motions on soft site could include more comprehensive seismic characteristics. Therefore, the ground motions on soft site were adopted to investigate the effects of structural parameters and the PGV/PGA of ground motions on C .

Calculation examples for verification
SMAFS can realize the idealized self-centering function and seismic mitigation performance by combining the self-centering capacity of SMAs and the high energy dissipation efficiency of fiction elements. This section mainly calculates and compares the C of SMAFS and SMAS to verify the proposed calculating procedure shown in Fig. 4 and explain the necessity of employing the hybrid force-displacement relationship for SMAFSs. In order to make the two structures have similar self-centering performance, let the hysteresis coefficient s of SMAS be equal to the hysteresis coefficient of SMAFS. In SMAS, the post-yield stiffness ratio of SMAS s = 0.1, and the energy dissipation Elastic acceleration spectra for considered ground motions: a soft site, b rock site coefficient s f= 0.8. In SMAFS, the SMA resisting force ratio r s = 0.7 , the post-yield stiffness s = 0.1, and the energy dissipation coefficient s = 0.29. According to Eq. (12), it can be obtained that = 0.8 . The comparison between the hysteretic loops of SMAFS and SMAS for a given displacement demand is conducted in Fig. 6 according to parameters of the two structures assumed above, on the condition that the two structures have same yield force F sy and ductility coefficient . The figure shows that two structures have same height of the hysteretic loops (i.e. s = ), which assures that the two structures have similar selfcentering performance except a small possible residual displacement of SMAFS due to the combination of SMA and friction elements. Meanwhile, the difference between the two hysteretic loops is obvious, i.e. the hysteretic loop of the SMAFS is fuller than SMAS, which indicates that SMAFS has advantage in term of energy dissipation. Therefore, the SMAFS could have idealized self-centering performance and more significant seismic mitigation at the same time, compared to SMAS. The C of SMAS and SMAFS with different under the seismic motions of soft sites are calculated and compared in Fig. 7, where C _SMAS represents the C of SMAS, and C _SMAFS represents the C of SMAFSs. Figure 7a shows that C _SMAS gradually decreases sharply with the increase of T and increases obviously with increasing as 0.1s ≤ T < 0.6s . However, the C _SMAS become insensitive to T and as T ≥ 0.6s . The C _SMAS is similar as the C of the self-centering structures with typical FS force-displacement relationship in the literature (Dong et al. 2020), which suggests that the proposed calculating procedure of C is credible. Figure 7b shows that C _SMAFS has the similar change law on T and asC _SMAS ,while the C _SMAFS is smaller than C _SMAS as is small. In addition, the C _ SMAFS is more sensitive to than C _SMAFS in long natural period region and can be smaller than 1 as is 2 or 3. In order to clearly show the difference between C _SMAFS and C _SMAS , Fig. 7c presents the ratio of C _ SMAS to C _ SMAFS as T increases from 0.1 s to 6 s for different . The results show that C _SMAS ∕C _SMAFS is significantly greater than 1 as 0.2s < T < 2s and ≤ 4 . The above results indicate that addition of friction can improve the control performance of the inelastic displacement of the SMA-based self-centering structure when is smaller than 4. In other words, employing the hybrid model shown in Fig. 2c is necessary for SMAFSs to obtain accurateC . The later part of this paper will mainly study the effects of properties of seismic motion and structural parameters on the C _ SMAFS of SMAFSs. For convenience, C is used hereinafter to replaceC _SMAFS .

Effects of structural parameters on C
The mechanical properties of SMAFSs are the intrinsic factors that determine their seismic performance. Based on the force-displacement relationship of SMAFS in Sect. 2, the main parameters affecting the mechanical properties of SMAFS include: the SMA post-yield stiffness ratio s , hysteresis coefficient s and SMA resisting force ratio r s . In this section, the different parameters combinations in Table 2 are used to study the effects of parameters on C .  3.1 SMA resisting force ratio r As described in Sect. 2, SMA resisting ratio r s refers to the ratio of the yield force of SMA element ( F sy ) to the yield force of SMAFS ( F y ). When r s = 1, the system is transformed into a pure SMA self-centering structure; when r s = 0, the system is transformed into a pure friction isolation structure (e.g. a sliding friction isolation structure). In order to make that the structures have reliable self-centering capability, r s = 0.5-1 is considered in this section. Taking the seismic motions in Table 4 in the Appendix as the excitation, the C of SMAFS with different r s are calculated according to calculating procedure discussed in Sect. 2.2, and results are shown in Fig. 8a. The results show that the C increases with the increase of r s as is given, and the effect of r s on C is more significant as 0.3s < T < 2s . This is because, the resisting force ratio supplied by friction element in SMAFS decreases with the increase of r s to result in the decrease of energy dissipation capacity of SMAFS, and finally result in an increase of the inelastic displacement demand of the structure. When r s = 1, the system degenerates into a pure SMA self-centering system, and its C is the greatest and tends to 1 as T is long enough, which is consistent with the analysis results on conventional self-centering system in the literature (Dong et al. 2020). The above results show that appropriately increasing the ratio of friction elements in SMAFS can enhance the control effect on inelastic displacement of the structure. Figures 8b and c respectively show the force-displacement curves and time-history displacement curves of SMAFS with different r s under Imperial Valley-02 seismic motions when T is given. The results show that the hysteresis curve of the SMAFS is fuller as r s is small, which means that the energy dissipation capacity increases, thereby the structure obtains a smaller structural displacement response. Although Fig. 8b shows that the self-centering capacity of the SMAFS becomes weaker with r s decreasing, the residual deformation of the structure obtained from displacement time-history curves shown in Fig. 8c is still ignorable even if r s = 0.5 . This phenomenon can be explained by the research results in literature (Eatherton et al. 2011). The research results of this literature, obtained through the statistical analysis of time-history responses of a self-centering system, show that the self-centering system with hysteretic coefficient up to 1.5 could still eliminate residual drifts reliably in probability. In Fig. 8, the can be calculated to be 1.25 according to Eq. (12) as r s = 0.5 , which is smaller than 1.5. Therefore, the ignorable residual drift of SMAFS with r s = 0.5 in Fig. 8c is reasonable. The research results indicate that the increase of friction in selfcentering structures can reduce significantly the inelastic displacement response under the premise of the self-centering capacity of SMAFS.

Energy dissipation coefficient ˇ
As shown in Fig. 2a, the energy dissipation coefficient s is an indicator describing the energy dissipation capacity of SMA elements. Increasing s could enhance the energy dissipation capacity of the SMAFS, but weaken the self-centering capability. When = 2, 4, and 6, the C of SMAFS with different s are shown in Fig. 9a. The results show that C slightly decreases with s increasing. However, the effects of s on C is not obvious, even if is bigger. This is because that SMA mainly plays a self-centering role in SMAFS based on the fact that the energy dissipation capacity of SMA element is far weaker than that of the friction element. Therefore, increasing s has inapparent effect on the hysteretic energy dissipation capacity of the whole SMAFS, which in turn leads to that the effect of s on C is not significant. In order to more clearly show the effect of s on seismic responses of SMAFS, Fig. 9b and c present respectively the force-displacement curves and displacement time-history curves of SMAFS with different s under Imperial Valley-02 seismic motion. The results show that the effect of s on the maximum displacement response of the SMAFS is not significant, although it has a significant effect on the shape of the force-displacement curve of the SMAFS. Meanwhile, increasing s will significantly reduce the self-centering capability of the SMAFS as shown in Fig. 9b. The above results show that SMA in SMAFS mainly plays the role of self-centering rather than energy dissipation. Therefore, the s should be limited to assure the self-centering capacity of SMAFSs, without significant effects on the damping performance of SMAFSs. Noted that s = 0 is a hypothetical extreme situation for SMAs rather than a reality due to the hysteresis loops of SMAs. Considering that the hysteresis loops of some SMAs are insignificant, e.g. the SMAs in the research (Konopatsky et al. 2021;Yaacoub et al. 2020), s = 0 is included in this research to study the effect of s in a wider range.

Post-yield stiffness ratio ˛
The post-yield stiffness ratio s of SMA is major parameter to indicate the nonlinear characteristics of SMA element. According to the present literature (Zhang et al. 2020a, b, c;Xu et al. 2018;Wang et al. 2012), the s for SMA wires, springs or rods could be up to 20%. Therefore, the s = 0 − 0.2 is considered to cover the possible cases. The effects of s on C are shown in Fig. 10. Figure 10a shows that C decreases obviously with the increase of s as 0.1s < T ≤ 0.5s , and this phenomenon becomes more pronounced with the increase of . As0.5s < T < 6s , the C become insensitive to s , and the C with different s tends to be similar as is given. These results can be explained as follows. The Fig. 9 Effects of s on C : a C of SMAFS ( r s = 0.7, s = 0.1) , b Force-displacement curves and c displacement time-history curves of SMAFS subjected to Imperial Valley-02 seismic motion equivalent stiffness of the SMAFS significantly increases with the increase of s when T is short, thus the inelastic displacement demand of the SMAFS will decrease obviously.

(a) (b) (c)
With T increasing, the equivalent stiffness of the SMAFS become insensitive to s , and the increase of the displacement response caused by the increase of s is thus no longer significant, thereby the effect of s on C becomes weaker. In order to intuitively show the effect of s on the seismic response of SMAFS, the force-displacement curves and displacement time-history curves of SMAFS with different s under Imperial Valley-02 seismic motions are presented in Fig. 10b and c, respectively. Figure 10b and c show that the displacement response of SMAFS slightly decreases with the increase of s , but the maximum resisting force of SMAFS increases significantly with the increase of s .

Site types
This part mainly studies the effects of site types on C . According to Sect. 2.3, the sites are divided into two major types: soft site (SS) and rock site (RS). The seismic records of the corresponding sites are as shown in Tables 4 and 5 in Appendix, respectively. Let the postyield stiffness ratio s = 0.1, the energy dissipation coefficient s = 0.5, and the SMA resisting force ratio r s = 0.7 in SMAFS. SMAFS with this group of parameters has full hysteresis curves and self-centering capacity according to the self-centering condition as Eq. (13), and this group of parameters will be adopted in the following research on the effects of seismic motion characteristics. The C of the SMAFS under seismic motions on soft and rock sites are respectively calculated as varies from 2 to 10 according to the calculation procedure as shown in Fig. 4. The calculation results are shown in Fig. 11, in which the C _SS is the C for soft site and the C _RS is the C for rock site. Figure 11a and b show that C _SS and C _RS all increase with the increase of and decreases with the increase of T . However, C _SS is significantly bigger than C _RS as 0.1 < T < 6 . In order to clearly show this difference, Fig. 11c presents the ratio of C _SS to C _RS for different ductility levels. The results show that C _SS is significantly greater than C _ RS , and the ratio of C _SS to C _RS is bigger than 1 and increases with the increase of . When T = 0.15-0.4 s, C _ SS ∕C _ RS can reach 1.2-1.5 for different ductility levels. The above results show that the C of SMAFS under the soft site is greater than that under the rock site, and this phenomenon is significant when T = 0.15 s − 0.4 s.  LOH et al. (2002) indicated that the ratio of peak ground velocity (PGV) to peak ground acceleration (PGA) is a main indicator to identify near-fault seismic effects, and the seismic motions with a PGV/PGA value greater than 0.2 has a significant damage effect on building structures. Based on the range of the ratio of PGV/PGA, the seismic motions in Table 4 can be divided into three groups, namely 0 < PGV PGA ≤ 0.12 , 0.12 < PGV PGA ≤ 0.2 , and 0.2 < PGV PGA ≤ 0.7 . The corresponding calculation results of C are shown in Fig. 12a-c, and are respectively marked as C _0< PGV PGA ≤0.12 , C _0.12< PGV PGA ≤0.2 , and C _0.2< PGV PGA ≤0.7. The results show that the C of SMAFS increases significantly with the increase of PGV/PGA. In order to quantitatively study the difference between the three, the C of different groups was normalized based on C _0< PGV PGA ≤0.12 , the normalized results are shown in Fig. 12d. The results show that C _0.12< PGV PGA ≤0.7 /C _0< PGV PGA ≤0.12 increases with the increase of and are all greater than 1 for 0.1s < T ≤ 1s , and this ratio for different is approximately equal to 1 as 1s < T ≤ 6s . At the same time, C _0.12< PGV PGA ≤0.7 /C _0< PGV PGA ≤0.12 are greater than 1 for whole T region, and the ratio also increases with the increases of and is more sensitive to as 0.2s < T ≤ 6s than that as 0.1s < T ≤ 0.2s . This result is attributed to that the bigger PGV/ PGA means more long-period components in ground motions, which could result in that SMAFS with a long natural period T easily obtain more intensive seismic responses.

Formula for estimating the C µ of SMAFS
Estimating the inelastic displacement demand is critical for direct-displacement-based seismic design. In order to facilitate the seismic design of the structure, it is necessary to establish the formula for estimating the C of SMAFS. Currently, researchers have proposed the formulas for estimating the C of traditional structures. For example, Miranda et al. (2001) proposed the formula for predicting the C of the ideal elastoplastic system, and Chopra and Chintanapakdee (2004) proposed the C formula of the bilinear system based on the Miranda's formula.
Based the above analysis results, the C of SMAFS is affected by structural characteristics. According to the analysis results in Sect. 4, the post-yield stiffness ratio s , SMA resisting force ratio r s and ductility factor have significant effects on the C of SMAFSs, so the C can be regarded as the function of , s , r s and T , i.e. C = C , T, s , r s . Based on the parametric analysis results, the expression of the C of SMAFSs is assumed as in which, where,a ,b ,c ,d , are the regression parameters.
According to formula (17), it is easy to known that C → C _ hT when T → ∞ (in the long period region).
The Eq. (17) is used to fit the C shown in Fig. 11a. Figure 11a shows that C basically keep constant when T is in the region of 2-6 s, thus average value of C in this natural period region can be identified as C _hT in Eq. (17). The C _hT for different is determined as above method, and the relationship between C _hT and μ is nearly linear as shown the black square point in Fig. 13a. Equation (19) is employed to fit the C _hT -μ relationship to obtain the red solid fitted curves shown in Fig. 13a and the regression parameters a = 0.08, b = 1.03, m = 1.40. Substituting the parameters into Eq. (19) and Eq. (17), and then Eq. (17) is employed to fit the C − T relationship. In Fig. 13b, the fitted curves for C − T relationship with different are compared with the statistical C − T curves obtained by the above calculation procedure, and the regression Table 3. The results show that the predictive results by Eq. (17) fit the statistical C − T relationship of SMAFS with different accurately. Table 3 shows that the c first decreases significantly with the increase of and then tends to be constant, while d almost keep a constant. Therefore, d can be represented by the average value for different , namely d = 1.05 . Using the piecewise linear relationship to regress the relationship between the parameters c and , it can be obtained that Using the regressed C − T relationship obtained by Eqs. (17)-(20), the effects of r s and s on C spectra can be described. The regressed C spectra and numerical statistical C spectra of SMAFS with different r s and s are compared in Fig. 14a and d, respectively. The results show that the regressed C spectra by the proposed formulas can well describe the effects of , r s and s on C .

Conclusion
The purpose of this paper is to investigate the constant ductility inelastic displacement ratio spectra C of shape memory alloy-friction self-centering structures (SMAFSs) for their performance-based seismic design. Based on a hybrid force-displacement model constituting the typical flag-shaped self-centering (FS) model and the Coulomb friction model, a new procedure was proposed to calculate C of SMAFSs, when a halfinterval searching method was employed to determine the yield force of the inelastic structure with given ductility factor. The proposed calculating procedure for the C of SMAFSs was examined by a calculating example. Furthermore, the effects of seismic motion parameters and structural parameters on the C were investigated, and a concise estimating formula of the C of SMAFSs was developed through statistical regression.
The main conclusions are drawn as follows: (1). Compared to the typical FS system with similar self-centering capacity, the C μ of the SMAFS is significantly different from that of the typical FS system, which explains that it is the necessary to employ the hybrid model rather than the typical FS model for calculating the C μ of SMAFSs. Meanwhile, the C of SMAFS is significantly smaller than that of the typical FS system when the natural period is among the region 0.2-3 s and ductility factor is smaller than 4, which suggests that addition of friction could improve the seismic performance of the SMA-based self-centering structures.
(2). The C of SMAFS is insensitive to the hysteretic energy dissipation coefficient of SMA, due to the weaker energy dissipation efficiency of SMAs compared to friction elements. Increasing hysteretic energy dissipation coefficient of SMA will slightly decrease the C of SMAFS, but it will significantly weaken the self-centering capability of the structure. In other words, SMAs mainly play a role of self-centering function rather than energy dissipation in SMAFSs. Meanwhile, increasing the proportion of the restoring force supplied by SMA in SMAFS can significantly increase the C . Therefore, it is recommended that increasing the proportion of friction force in SMAFS and reducing the hysteretic energy dissipation coefficient of SMA to get significant inelastic displacement mitigation under the premise that the SMAFS has certain self-centering capacity. In addition, increasing the post-yield stiffness ratio s of SMA will slightly reduce the C of SMAFS, but will significantly increase the restoring force demand of SMAFSs. (3). The C of SMAFS on the soft site is greater than that on the rock site, especially for short-medium natural period and high ductility level. With the increase of PGV/PGA, the C increases obviously and becomes more sensitive to ductility factor. (4). A concise formula for estimating the C of SMAFS is proposed, which could estimate the C of SMAFS and describe the effects of the parameters, including ductility factor, SMA post-yield stiffness ratio and the proportion of the restoring force supplied by SMAs, on C of SMAFSs more accurately. The proposed prediction formula for the C of SMAFSs supplies a basis of estimating the inelastic displacement for performancebased seismic design of the structures.