*4.2. Comparison of Test and Numerical Results* 4.2.1. Loading Rate

The hysteresis curves of the numerical results were simulated using the improved Graesser model and Bouc–Wen model, which need to combine with the main parameters in Tables 2 and 6. Figure 14 displays the comparison of the test and numerical results under different loading rates. At the applied displacements of 2.40 mm and 4.80 mm for specimens SCB-12-0, SCB-18-0, SCB-24-0, and SCB-36-0, a great difference in the hysteresis curves between the test and numerical results can be observed, which can be explained by the existing errors between the slip bolt and slip hole in the test model. However, both the test and numerical hysteresis curves have almost the same initial stiffness for all of the specimens at the different loading rates. As the applied displacement increases, the hysteresis curves between the test and numerical results for each specimen are very close, and only a small error exists near σ *MA s* , which is the same as the error for the SMA wire at the different loading rates [22].

**Figure 14.** (**a**) SCB-12-0; (**b**) SCB-18-0; (**c**) SCB-24-0; (**d**) SCB-36-0. Comparison between test and numerical hysteresis curves under different loading rates.

For the different loading rates, the secant stiffness and energy dissipation coefficients of the specimens obtained from the test and numerical results with the applied displacement values ranging from 2.40 mm to 14.40 mm are shown in Tables 7 and 8. At the displacements of 2.40 mm and 4.80 mm, the maximum error between the test and numerical secant stiffness is 31.76%. Meanwhile, the maximum error is also 56.63% for the energy dissipation coefficient. The main reason for the error may be due to the difference between the slip bolt and hole in the test brace. As the displacement increases from 7.20 mm to 14.40 mm, the maximum errors in the secant stiffness and energy dissipation coefficient between the test and numerical results are 5.68% and 8.02%, respectively, showing good accuracy. Based on the above analysis, the presented numerical model determined from Equations (3)–(12) can be used to simulate the seismic performance of self-centering SMA braces with different loading rates.


**Table 7.** Comparison of secant stiffness under different loading rates.

**Table 8.** Comparison of energy dissipation coefficient under different loading rates.


Note. Error = (Numerical − Test)/Test: 'Tes.' and 'Num' denote the test and numerical results, respectively.

#### 4.2.2. Initial Strain

Figure 15 shows the comparison between the test and numerical results under different initial strains at the applied displacements ranging from 2.40 mm to 14.40 mm. At the displacements of 2.40 mm and 4.80 mm for specimens SCB-12-0, SCB-12-25, SCB-12-50, and SCB-12-100, an obvious contrast between the test and numerical results can be observed, and the cause of the contrast is the same as it is with the loading rates. As the displacement increased from 7.20 mm to 14.40 mm, the hysteresis curves of the specimens obtained by the test results all show close agreement with the numerical results.

The secant stiffness and energy dissipation coefficient of the specimens calculated from test and numerical results are shown for the different initial strains in Tables 9 and 10. At the displacement of 2.40 mm, the maximum errors of the secant stiffness and energy dissipation coefficient between the test and numerical results are 54.95% and 31.76%, respectively. As the strain amplitude increases, the numerical results of the secant stiffness and energy dissipation coefficient for all of the specimens become gradually closer to the test results, which have a maximum error of 7.64%, showing great agreement. Therefore, the proposed numerical model can also be used to analyze the seismic performance of the self-centering SMA braces with different initial strains.

**Figure 15.** (**a**) SCB-12-0; (**b**) SCB-12-25; (**c**) SCB-12-50; (**d**) SCB-12-100. Comparison between test and numerical hysteresis curves under different initial strains.


**Table 9.** Comparison of energy dissipation capacity under different initial strains.


**Table 10.** Comparison of secant stiffness under different initial strains.

## **5. Conclusions**

In this paper, the cyclic loading test and numerical analysis were studied to carry out an innovative self-centering brace with the effect of the loading rate and initial strain, and the seismic performance of the brace was investigated. The following conclusions can be obtained:


**Author Contributions:** Conceptualization, Y.J., S.H. and S.Z.; methodology, B.Z., S.H. and S.Z.; software, F.T. and S.H.; validation, B.Z., S.Z. and S.H.; formal analysis, Y.J., B.Z., F.T. and S.H.; writing—original draft preparation, B.Z., S.Z. and F.T.; writing—review and editing, Y.J. and S.H.; supervision, Y.J. and S.H.; project administration, Y.J., S.Z. and S.H.; funding acquisition, Y.J., S.Z. and S.H. and W.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors gratefully acknowledge the research grant provided by the National Nature Science Foundation of China (No. 51908268, 51268044, 51878108).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used to support the findings of this study are available from the corresponding author upon request.

**Acknowledgments:** The authors gratefully acknowledge the support by Guoyang Guan from Engineering Mechanics Experiment Center, Nanchang University during the tests conducted for this study.

**Conflicts of Interest:** The authors confirm that there are no conflicts of interested associated with this article.
