# 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
#### 3.2.3 Criteria for the Dominate Parameters  
1 Stiffness parameters \(k_{\text{a}}\), \(k_{\text{b}}\) and \(k_{\text{c}}\)  
\(k_{\text{a}}\) and \(k_{\text{c}}\) are the stiffness of the \(f\)-\(\delta\) curve in two opposite directions at each governing deformation (see Figure 8b), respectively, whose values under different load levels can be determined according to the test hysteretic loops measurement. The formulas of \(k_{\text{a}}\) and \(k_{\text{c}}\) can be obtained through the statistical analysis of the experimental results in the study by [PERSON] and [PERSON] [21], given by:  
\[k_{\text{a}}=\ R_{\text{a}}\delta_{\text{n}}+k_{\text{a}0} \tag{11}\]  
\[k_{\text{c}}=\ R_{\text{c}1}e^{R_{\text{c}2}\delta_{\text{n}}}+R_{\text{c}3}e ^{R_{\text{c}4}\delta_{\text{n}}} \tag{12}\]  
where \(R_{\text{a}}\) is the gradient of \(k_{\text{a}}\) with governing deformation \(\delta_{\text{n}};k_{\text{a}0}\) is the value of \(k_{\text{a}}\) at \(\delta_{\text{n}}=0\); \(R_{\text{c}1}\) and \(R_{\text{c}3}\) are the controlling coefficients of \(k_{\text{c}}\); \(R_{\text{c}2}\) and \(R_{\text{c}4}\) are the index gradients of \(k_{\text{c}}\) with \(\delta_{\text{n}}\). It can be observed form Figure 8b that \(k_{\text{b}}\) approximates the slipping segment slope of the hysteresis loop. Therefore, the development process of those hysteretic loops obtained from previous experiments is listed in Figure 9 in order to investigate the developing rule of the slipping segment slope. It is indicated that during the initiation of the loading process, the wall is in elasticity, the curve is linear, and the slipping segment slope is equal to the elastic lateral stiffness of the wall; when the load increased, screw connections fail progressively, pinching appears in the curve because of the opening and closing of the screw holes, and the slipping segment slope decreases continuously; after an even larger amount of wall displacement, an aggravated extrusion deformation of screw connections appears, the loose screw holes cause the appearance of some no-load slipping of the screw connections, which leads the slipping axis to have a tendency towards a horizontal configuration. It can be concluded that the developing rule of the slipping segment slope is governed by a process whereby the elastic lateral stiffness of the wall decreases to a minimum value. Based on the assumption of a linear relationship between the slipping segment slope and load level, \(k_{\text{b}}\) can be determined:  
\[k_{\text{b}} = (1-\chi)k_{1} \tag{13}\] \[k_{1} = (1-\chi)k_{0} \tag{14}\]  
where \(k_{1}\) is the difference of the slipping segment slopes between elastic and destruction phrases; \(k_{0}\) is the slipping segment slope in elastic phrase, which is assigned with the elastic stiffness of \(f\)-\(\delta\) curve (i.e., the ratio of \(f_{\text{e}}\) to \(\delta_{\text{e}}\) that corresponds to the elastic strength limit \(F_{\text{e}}\), see Figure 8b); \(\lambda\) is the coefficient relative to the slipping segment slope in the destruction phrase, which approximates to 0.1, according to the statistical analysis of experimental results; \(\chi=C_{i}/C\), in which \(C_{i}=(\delta_{n}/\delta_{\text{min}}-1)\), \(C=(\delta_{n\text{max}}/\delta_{\text{min}}-1)\), and \(\delta_{\text{max}}\) and \(\delta_{\text{min}}\) are the maximum governing deformation and the minimum load level difference, respectively. 2 Pinching load\(f_{0}\)  
As shown in Figure 2, for each hysteretic loop, the reverse loading paths approximately direct to certain points, like pinching points C and F in Figure 8a. Screws are in the no-load slipping phrase before those pinching points are reached, at that moment, the wallboard has the least restraint on the steel frame, and the shear capacity of the frame can thereby be taken as the pinching load (approximately). Pinching load \(f_{0}\) is a direct reflection of the pinching characteristics of the curve.