# 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
### Model Verification
Taking specimen W89-1, for example, details are as follows: (1) dimensionless treatment was performed on the load-displacement skeleton curve that was determined by the discrete coordinate method, \(\delta_{\text{n}}\) and its corresponding load \(f_{\text{n}}\) of the hysteretic loop were determined based on the load case relative to each governing deformation, and the slipping segment slope \(k_{0}\) in elastic phrase was calculated (see Figure 8b); (2) on the basis of \(k_{0}\), \(\delta_{\text{n}}\), and \(f_{\text{n}}\), the dominant parameters of the hysteretic model were obtained by Equations (11)-(16), as listed in Table 4, the controlling factors of parameters \(k_{\text{a}}\), \(k_{\text{c}}\), and \(n_{\text{u}}\), \(n_{\text{l}}\) are listed in Table 5; (3) after substituting those dominant parameters that are listed in Table 4 into Equations (8)-(10), the \(f\)-\(\delta\) hysteretic loops corresponding to different governing deformations were established, and the hysteretic model of the wall was further obtained through substituting wall height \(H\) and the calculated result of shear capacity \(F_{\text{PC}}\) into those \(f\)-\(\delta\) loops. Comparisons of the hysteretic curves between the calculated results and the test results are shown in Figure 11. Good agreement is observed between the test results and the calculated results, which fully reflect the hysteretic characteristics of the walls during cyclic loading. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Load** & \(\delta_{\text{n}}\) & \(f_{\text{n}}\) & \(k_{0}\) & \(k_{\text{a}}\) & \(k_{\text{b}}\) & \(k_{\text{c}}\) & \(f_{\text{mu}}\) & \(f_{\text{ml}}\) & \(n_{\text{u}}\) & \(n_{\text{l}}\) & \(f_{\text{0}}\) \\ \hline
10 mm & 0.0033 & 0.847 & 125.0 & 788.4 & 102.3 & 502.4 & 1.074 & 1.258 & 1.45 & 1.35 & 0.11 \\
15 mm & 0.005 & 0.948 & 125.0 & 827.3 & 92.0 & 285.6 & 1.298 & 1.050 & 1.40 & 1.40 & 0.11 \\
20 mm & 0.0067 & 0.998 & 125.0 & 866.2 & 81.8 & 206.2 & 1.436 & 1.041 & 1.35 & 1.46 & 0.11 \\
30 mm & 0.01 & 1.0 & 125.0 & 941.8 & 61.4 & 148.8 & 1.504 & 1.212 & 1.25 & 1.57 & 0.11 \\
40 mm & 0.0133 & 0.916 & 125.0 & 1017.3 & 40.9 & 118.7 & 1.350 & 1.317 & 1.15 & 1.68 & 0.11 \\
50 mm & 0.0167 & 0.791 & 125.0 & 1095.1 & 20.4 & 95.4 & 1.022 & 1.252 & 1.04 & 1.78 & 0.11 \\
60 mm & 0.02 & 0.657 & 125.0 & 1170.7 & 12.5 & 77.2 & 0.797 & 1.247 & 0.94 & 1.89 & 0.11 \\ \hline \hline \end{tabular}  
* \(k_{0}\) is the slipping segment slope of the wall in elasticity (see Figure 8b); the rest of the symbols’ meanings are identical to those in Equations (8)–(10).  
\end{table} 
Table 4: The dominant parameters used to the hysteretic model of specimen W89-1 1. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Specimen** & \multirow{2}{*}{**Wall Size/m**} & \multicolumn{3}{c}{**Elastic Lateral Stiffness \(K_{\text{e}}\)**} & \multicolumn{3}{c}{**Total Energy Dissipation**} \\
**Number** & & \multicolumn{3}{c}{**(\(L\times H\))**} & \multicolumn{3}{c}{**(N/mm)**} & \multicolumn{3}{c}{**Shear Capacity \(F_{\text{p}}\)/kN**} & \multicolumn{3}{c}{**E/KJ**} \\ \cline{3-10}
**[22]** & & \(K_{\text{et}}\) & \(K_{\text{ec}}\) & \(\kappa_{\text{1}}\) & \(F_{\text{pt}}\) & \(F_{\text{pc}}\) & \(\kappa_{\text{2}}\) & \(E_{\text{t}}\) & \(E_{\text{c}}\) & \(\kappa_{\text{3}}\) \\ \hline W89-1 & 3.6 \(\times\) 3.0 & 9670 & 10177 & 0.05 & 110  
Energy dissipation is an important indicator to measure the seismic performance of a structure or a component. The proposed hysteretic model was thereby validated by comparing the energy dissipation of the walls. Only the energy dissipation relative to the first cycle of each load level was considered, because strength deterioration in successive cycles of the same amplitude was not involved in the model.