*3.2. Bond Curves*

The bond curves in Figure 8 were obtained from the results in Figure 7. The bond curves in the initial slip phase have a smaller initial stiffness, and the bearing capacity and initial stiffness gradually increased in the increasing phase. Finally, the bearing capacity of the specimens increased slowly and reached the maximum axial force, and the slope of the bond curve began to decrease in the decreasing phase. All of the bond curves are origin-symmetric under axial tension and axial compression. In addition, as shown in Figure 8a,b, the bearing capacity of specimens SCB-12-0, SCB-18-0, SCB-24-0, SCB-36-0 increased as the loading rate increased, and the same conclusion can be obtained for the initial strain rates for specimens SCB-12-0, SCB-12-25, SCB-12-50, and SCB-12-100.

**Figure 7.** (**a**) Loading rate; (**b**) initial strain. Hysteresis curves of self-centering SMA brace specimens.

**Figure 8.** (**a**) Loading rate; (**b**) initial strain. Bond curves of self-centering SMA brace specimens.

#### *3.3. Secant Stiffness*

In this paper, the stiffness degradation of the self-centering SMA brace is presented by the secant stiffness coefficient, *Ksi*, which can be calculated by:

$$K\_{\rm si} = \frac{|F\_{\rm i,max}| + |-F\_{\rm i,min}|}{|D\_{\rm i,max}| + |-D\_{\rm i,min}|} \tag{1}$$

where *F<sup>i</sup>* ,max, *F<sup>i</sup>* ,min, *D<sup>i</sup>* ,max, and *D<sup>i</sup>* ,min represent the maximum axial force, minimum axial compression, maximum displacement, and minimum displacement at the *i*-th hysteretic cycle under the load displacement of *i*.

Figure 9 shows the secant stiffness curves of all of the test specimens. The secant stiffness coefficient decreases after reaching the maximum value as the applied displacement increases and the reduction rate gradually decreases. The initial increase in the secant stiffness coefficient is mainly because of the slip friction of the slip components. In addition, as the loading rate and initial strain increase, it can be seen that the secant stiffness coefficients of the self-centering SMA braces increase gradually.

**Figure 9.** Secant stiffness of self-centering SMA brace specimens.

#### *3.4. Energy Dissipation Coefficient*

The different energy dissipation coefficients, *E<sup>i</sup>* , which are calculated by the enclosed area of the hysteresis curve for each specimen, are shown in Figure 10. The maximum *E<sup>i</sup>* of the specimens SCB-12-0, SCB-18-0, SCB-24-0, and SCB-36-0 are 317.37 J, 320.80 J, 310.73 J, and 291.13 J, respectively. An increase in the *E<sup>i</sup>* in each test case can be observed when the loading rates are increased from 0.0012 s−<sup>1</sup> to 0.0018 s−<sup>1</sup> , an a gradual decrease occurs as the loading rate increases to 0.0036 s−<sup>1</sup> , which is influenced by the pinch phenomenon in the hysteresis curves [22]. In addition, an increase in the *E<sup>i</sup>* for specimens SCB-12- 0, SCB-12-25, SCB-12-50, and SCB-12-100 can be clearly seen under different test cases where there is an increase of initial strain from 0 to 0.01, which was mainly caused by the increase in the hysteresis areas. Therefore, the energy dissipation capacity can be effectively increased by increasing the initial strain, but the influence law of the loading rate for the self-centering SMA braces is uncertain, meaning that further research should be on the equivalent damping coefficient.

**Figure 10.** Energy dissipation coefficient of specimens.

## *3.5. Equivalent Damping Coefficient*

The equivalent damping coefficient, *ξeq*, is also an important parameter that can be used to evaluate the energy dissipation capacity, which can be calculated as follows [23]:

$$\mathfrak{f}\_{eq} = \frac{1}{2\pi} \frac{\mathbb{S}\_{EBG} + \mathbb{S}\_{DEG}}{\mathbb{S}\_{OBA} + \mathbb{S}\_{ODE}} \tag{2}$$

where *SEBG* and *SDEG* represent the areas of the closed geometrical figures EBG and DEG, which are enclosed by the vertical axial force and horizontal displacement of the coordinates, and *SOBA* and *SODC* represent the areas of the triangles OBA and ODC in Figure 11.

**Figure 11.** Calculation of the energy dissipation coefficient.

For all of the test specimens, the equivalent damping coefficient during the loadingunloading process is presented in Figure 12. The equivalent damping coefficient of all of the specimens increases as the applied displacement increases. For the specimens with different loading rates, the equivalent damping coefficient of specimen SCB-18-0 is higher than that of specimens SCB-12-0, SCB-24-0, and SCB-36-0; this is mainly caused by the hysteresis area and energy dissipation value. Compared to specimen SCB-12-0, SCB-12-25, SCB-12-50, and SCB-12-100 have a larger maximum axial force *F* and nearly the same energy dissipation value *E<sup>i</sup>* , resulting in a lower equivalent damping coefficient as the initial strain increases.

**Figure 12.** Equivalent damping coefficient of self-centering SMA brace specimens.

#### *3.6. Self-Centering Capacity Ratio*

The proposed self-centering SMA brace is mainly composed of a slip component and re-centering component, and consequently, the axial force, *FSCB*, can be written by:

$$F\_{\rm SCB} = F\_{\rm SMA} + F\_{\rm slip} \tag{3}$$

where *FSMA* and *Fslip* are the forces of the SMA wires and slip, as shown in Figure 13.

**Figure 13.** (**a**) SMA, (**b**) slip component, and (**c**) self-centering SMA brace force–displacement curve of self-centering SMA brace.

The self-centering capacity ratio, *δ*, is an important parameter that can be used to evaluate the re-centering capacity of self-centering SMA braces and can be calculated as follows:

$$
\delta = \frac{D - D\_1}{D} \tag{4}
$$

where *D* is the maximum applied displacement, and *D*<sup>1</sup> is the residual displacement.

Based on the test results in Section 3.1, the maximum *Fu,SCB*, *Fslip*, *FSMA*, *D*, *D*1, and *δ* of each specimen can be concluded and are shown in Table 5. The *Fslip* was obtained by the vertical force of the rectangular loops around the origin point in Figure 7, and the *FSMA* is equal to the *Fu,SCB* minus the *Fslip*. By increasing the loading rate, the *Fu,SCB* and *δ* increased, while the *Fslip* and *D*<sup>1</sup> decreased. In addition, a larger initial strain also resulted in a greater *Fu,SCB* and larger *δ* at the maximum displacement, and yet both the *Fslip* and *D*<sup>1</sup> show opposite trends. Moreover, it should be noted that the maximum self-centering capacity ratio is 89.38%, showing that the self-centering SMA brace has an excellent re-centering capacity.


**Table 5.** Performance indices of self-centering SMA braces.

#### **4. Numerical Results**

The SIMULINK toolbox from MATLAB was used to simulate the seismic performance of the self-centering SMA brace, and the numerical and test results will be compared with the same conditions.

## *4.1. Numerical Model of Self-Centering SMA Brace*

In Equation (3), the force of the SMA wires can be expressed by:

$$F\_{SMA} = \sigma\_{SMA} A\_{\rm s} \tag{5}$$

where *σSMA* and *A<sup>s</sup>* are the stress and cross-sectional area of the SMA wires, respectively.

According to the improved Graesser and Cozarelli model by Graesser [24] and Qin [21], the *σSMA* can be calculated as follows:

$$\dot{\sigma} = E \left[ \dot{\varepsilon} - |\dot{\varepsilon}| \left( \frac{\sigma - \beta}{Y} \right)^{n-1} \left( \frac{\sigma - \beta}{Y} \right) \right] \tag{6}$$

$$\mathcal{J} = \text{Eq}\left\{ \varepsilon\_{\text{in}} - \frac{\sigma}{E} + f\_T |\varepsilon|^c \text{erf}(a\varepsilon) [u(-\varepsilon \dot{\varepsilon})] + f\_M [\varepsilon - \varepsilon\_{M\_f} \text{sgn}(\varepsilon)]^m [u(\varepsilon \dot{\varepsilon})] [u(|\varepsilon| - \varepsilon\_{M\_f})] \right\} \tag{7}$$

where *ε*, *E*, and *Y* are the strain, elastic modulus, and yield stress of the SMA wire, respectively; *n* is a constant controlling the sharpness of the transition from the elastic state to the phase transformation; *β* is the one-dimensional back stress; *α* is equal to *Ey*/(*E*−*Ey*); *E<sup>y</sup>* is the slope of stress–strain curve in the plastic range; *εin* is the inelastic strain; *fT*, *a,* and *c* are the material constant controlling the type and size of the hysteresis, the amount of elastic recovery, and the slope of the unloading stress plateau, respectively; *εMf* is the Martensite finish transformation strain; and *f<sup>M</sup>* and *m* are the constants controlling the Martensite hardening curve.

The error function *erf* (x), Heaviside function *u* (x), and sigh function sign (*x*) can be expressed as [21]:

$$\operatorname{erf}(\mathbf{x}) = \frac{2}{\sqrt{\mathbf{x}}} \int\_0^\mathbf{x} e^{-t^2} dt \tag{8}$$

$$u(\mathbf{x}) = \begin{cases} 1 & (\mathbf{x} \ge \mathbf{0}) \\ 0 & (\mathbf{x} < \mathbf{0}) \end{cases} \tag{9}$$

$$\text{sgn}(\mathbf{x}) = \begin{cases} +1 & \mathbf{x} > \mathbf{0} \\ \mathbf{0} & \mathbf{x} = \mathbf{0} \\ -1 & \mathbf{x} < \mathbf{0} \end{cases} \tag{10}$$

Therefore, the numerical model of the SMA wires will be revealed in detail using Equations (7) and (8) during the numerical analysis.

To accurately simulate the mechanical properties of the slip component, the Bouc-Wen model [25] is presented and described by:

$$F\_{slip} = \lambda kd + (1 - \lambda)kD\_yZ \tag{11}$$

where *d*, *k*, *Dy*, *λ,* and *Z* are the deflection, initial stiffness, yield displacement, ratio of plastic and elastic stiffness, and non-dimensional displacement, respectively.

The first-order non-dimensional displacement equation yields can be expressed by:

$$\dot{Z}D\_y = -\gamma \left| \dot{d} \right| Z |Z|^{\eta - 1} - \beta \dot{d} |Z|^{\eta} + \theta \dot{d} \tag{12}$$

where *γ*, *β*, and *θ* are the parameters to control the shape and size of the hysteresis curve, and *η* is a scalar value to govern the smoothness of the transition from the elastic stage to the plastic stage.

In addition, according to the material properties of the SMA wires in Section 2.2.1 and the test results in Section 3, the model parameters of the SMA wire and slip model can be determined and are listed in Table 6.

**Table 6.** Determined the parameters of the SMA wire and slip model.

