# Piecewise Function Hysteretic Model for Cold-Formed Steel Shear Walls with Reinforced End Studs
## 3 Hysteretic Model of CFS Shear Walls with Reinforced End Studs
### Hysteretic Model of Load-Displacement Curve
#### 3.2.2 Modeling of Hysteretic Loop  
It is difficult to describe the _f-\(\delta\)_ curve with turning points as a single equation with clear physical interpretations. Therefore, \(f\) = 0 is taken as the first demarcation point to divide the ascending branch of the curve into unloading segment AB and loading segment BCD. Furthermore, \(\delta\) = 0 is taken as the second demarcation point to divide the loading segment BCD into loading segment I and loading segment II (see Figure 8b). Consequently, the ascending branch of the typical hysteretic loop consists of three segments with monotonic stiffness development, namely, the unloading segment, loading segment I, and loading segment II, where loading segment I (i.e., the slipping segment) is described  
Figure 8: Hysteretic model of a wall with reinforced end studs. (**a**) Hysteretic laws; (**b**) calculation model for the ascending branch of the curve. Note: The symbols’ meanings are identical to those in Equations (8)–(16).  
Figure 7: Typical hysteretic loops.  
linearly, because its stiffness can be seen as a constant, and the unloading segment and loading segment II are described using Richard-Abbott expressions [26]. In summary, the ascending branch of a typical hysteretic loop can be captured using the following piecewise function:  
\[\text{Unloading segment}:\ f=\frac{(k_{\text{a}}-k_{\text{b}})(\delta-\delta_{ \text{n}})}{\left[1+\left|\frac{(k_{\text{b}}-k_{\text{b}})(\delta-\delta_{ \text{n}})}{f_{\text{mu}}}\right|^{n_{\text{u}}}\right]^{\frac{1}{n_{\text{u}} }}+k_{\text{b}}(\delta-\delta_{\text{n}}),\ f\leq 0 \tag{8}\]  
\[\text{Loading segment I}:\ f=\ k_{\text{b}}\delta+f_{0},\ 0{<}f\leq f_{0} \tag{9}\]  
\[\text{Loading segment II}:\ f=f_{0}+\frac{(k_{\text{b}}-k_{\text{c}})\delta}{ \left[1+\left|\frac{(k_{\text{b}}-k_{\text{c}})\delta}{f_{\text{ml}}}\right|^ {n_{\text{l}}}\right]^{\frac{1}{n_{\text{l}}}}}+k_{\text{c}}\delta,\ f>f_{0} \tag{10}\]  
where \(k_{\text{a}}\) is the initial unloading stiffness at point A; \(k_{\text{b}}\) is the asymptote slope of the unloading segment in the vicinity of point C; \(k_{\text{c}}\) is the asymptote slope of the loading segment at point D; \(\delta_{\text{n}}\) (including \(\delta_{\text{n}+}\) and \(\delta_{\text{n}-}\) in opposite directions) is the governing deformation of the unloading segment at its initial point; \(f_{\text{mu}}\) is the reference load of the unloading segment, which is the difference between \(f_{\text{n}-}\) and the load of point \(P_{1},f_{\text{ml}}\) is the reference load of loading segment, which is the difference between the pinching load \(f_{0}\) at zero displacement point and the load of point \(P_{2}\), points \(P_{1}\) and \(P_{2}\) locate on the asymptotes of \(k_{\text{b}}\) and \(k_{\text{c}}\), respectively; \(n_{\text{u}}\) and \(n_{\text{l}}\) are the shape parameters of the unloading segment and loading segment II, respectively.  
Equations (8)-(10) show that the dominant parameters of the ascending branch of a typical hysteretic loop include \(k_{\text{a}}\), \(k_{\text{b}}\), \(k_{\text{c}},f_{\text{mu}},f_{\text{ml}}\), \(n_{\text{u}}\), \(n_{\text{l}}\), and \(f_{0}\).