# 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
### Modeling of Backbone Curve
#### 3.1.2 Discrete Coordinate Method
where, \(K_{\rm e}\) is the elastic lateral stiffness of the wall; \(P\) is lateral load; \(\Delta\) is lateral displacement of the wall; \(i_{\rm c}\) is line rigidity of end studs; \(H\) is wall height; \(L\) is wall length; \(E_{\rm bi}A_{\rm bi}\) is axial stiffness of the brace, which is simplified from the \(i\)-layer wallboards on one side of the wall and can be determined by Equation (4); \(G_{i}\) and \(t_{i}\) are shear modulus and thickness of the \(i\)-layer wallboard, respectively; \(f_{i}\) and \(d_{\rm ei}\) are shear strength and elastic deformation of the screw connection corresponding to the \(i\)-layer wallboard, respectively; during the calculation of \(E_{\rm bi}A_{\rm bi}\), due to the arrangement of double-row screws for the end stud and single-line screws for the track, as shown in Figure 1, in which \(n_{\rm s}\) and \(n_{\rm e}\) are screw numbers along the vertical and horizontal edges on one side of the wall, respectively, an average space \(S_{\rm a}\) of those screw connections in the perimeter of the wall was calculated first by Equation (5), and then an equivalent number \(n_{\rm se}\) of those screws on either the left or right side of the wall was determined according to Equation (6). 2. Determination of shear capacity \(F_{\rm p}\)  
Numerous experimental results show that shear capacity of CFS shear walls depends on the shear strength of the screw connections in the perimeter of the wall. The elastic model, which is usually used to determine the shear capacity of CFS shear wall [24], is not suitable for a wall with reinforced end studs. It can be explained in that the elastic model takes the load that screw connections on four corners of the wall experience as they reach their ultimate shear strength as the wall's shear capacity, the rest of the screw connections on the end studs do not reach their ultimate strength because their horizontal force components are defined by linear conversion according to the four screw connections' shear strength, and the failure mode of all screws on the end studs being sheared off thereby cannot be reflected during cyclic loading, resulting in lower calculated results. It also can be concluded from Table 1 that the calculated results based on the elastic model are generally 20%-35% lower than the corresponding test results. Compared to the strip model [25] for steel plate shear walls, it can be concluded that screw connections in the perimeter of CFS shear walls can be taken as the end fields of the equivalent strips. Consequently, in considering the strip model, a simplified calculation model for the wall \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen Number [21]** & **Test Results/kN** & **Calculated Results Based on Elastic Model/kN** & **Relative Errors** \\ \hline WA1 & 100.0 & 79.7 & \(-\)20.3\% \\ WA2 & 101.5 & 81.4 & \(-\)19.8\% \\ WA3 & 109.0 & 81.4 & \(-\)25.3\% \\ WB1 & 126.5 & 101.5 & \(-\)19.8\% \\ WB2 & 60.3 & 45.3 & \(-\)24.9\% \\ WC1 & 109.8 & 70.6 & \(-\)35.7\% \\ WC2 & 95.4 & 70.6 & \(-\)26.0\% \\ WC3 & 67.0 & 45.3 & \(-\)32.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the shear capacity \(F_{\rm p}\) of the wall with reinforced end studs. Figure 5: Equivalent-bracing model. (**a**) Simplification model; (**b**) calculation model. Reprinted from [20]. Copyright 2015, with permission from Elsevier.