# Piecewise Function Hysteretic Model for Cold-Formed Steel Shear Walls with Reinforced End Studs
## 4 Summary and Conclusions  
A theoretical hysteretic model, which is concise enough to be mathematically described and can reflect the shear behavior of the wall with reinforced end studs, was proposed. A degradation coefficient of secant stiffness was first introduced, and a discrete coordinate method was proposed to establish the load-displacement skeleton curve of the wall; on this basis, a piecewise function was  
Figure 12: Cumulative energy dissipation of test and the proposed hysteretic model. (**a**) 89-type walls; (**b**) 140-type walls; (**c**) Relative errors \(\varepsilon\) between calculated and test results.  
adopted to capture the hysteretic loop relative to each governing deformation, the hysteretic model of the wall was further established, and the criteria for the dominant parameters of the model was stated. Finally, the hysteretic model was validated by the experimental results of [PERSON] [22]. The conclusions drawn are as follows:  
1. Elastic lateral stiffness \(K_{\mathrm{e}}\) and shear capacity \(F_{\mathrm{P}}\) are key factors determining the load-displacement skeleton curve of a CFS shear wall; where \(K_{\mathrm{e}}\) and \(F_{\mathrm{P}}\) can be predicted by the equivalent-bracing model and the proposed simplified calculation model, respectively, and the relative errors between their predicted and test results are within 11% and 15%, respectively.
2. The load-displacement skeleton curves determined by the discrete coordinate method agree well with the test results, which fully reflect the shear behavior (e.g., nonlinearity, stiffness, and strength deterioration) of the walls with reinforced end studs l.
3. During cyclic loading, test hysteretic curves and the calculated results that were determined by the proposed hysteretic model are of high consistency in both the pinching characteristic and energy dissipating level of the walls. Due to a larger stiffness of the reinforced end studs, the relative error of the total energy dissipation between calculated and test results for 140-type walls is within 10%; whereas, for 89-type walls with reinforced end studs having lower stiffness, both steel frame buckling and a supplementary torsional deformation in screw connections were involved in energy dissipation, resulting in the calculated results being lower than the test results during the later stage of loading.
4. Several hysteretic models (e.g., Pinching4 material in OpenSees, BWBN model, EPHM, Pivot model, as well as the three-segment nonlinear pinching hysteretic model), were able to reproduce the behavior of the traditional shear wall with coupled C section end studs; in contrast, the proposed piecewise function hysteretic model is suitable for the wall with reinforced end studs (i.e., continuous concrete-filled rectangular steel tube columns), which is more in line with the requirements of mid-rise CFS structures, and the model has intuitional expressions with clear physical interpretations for each parameter.
5. Due to the significant reduction in cumulative energy dissipation that is caused by insufficient deformation of screw connections, the energy dissipating capacity of a CFS shear wall with reinforced end studs should not be considered when the wall's aspect ratio is larger (in this study, \(H/L\) = 2.5, where \(H\) and \(L\) are wall height and wall length, respectively).  
The work described in this paper was fully supported by the National Key Program Foundation of China (51538002).  
[PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest.