## 4 Structural Vulnerability Analyses on Two-Story CFS Shear Wall Systems
### Traditional Shear Wall System \(W_{1}\)  
For a traditional shear wall, the construction of a beam-column joint is depicted in Figure 1a, and the deformation diagram is shown in Figure 2a. There are not only ductile CFS members but also brittle sheathing boards in the traditional shear wall system \(W_{1}\). According to the clustering criteria of the structural vulnerability theory (obtained in Section 3.2), the hierarchical model of the shear wall system is established in Figure 7. In addition, possible failure modes (FM) of the shear wall system were obtained based on the unzipping criteria, as described in Section 3.3, and the unzipping process was depicted in Figure 7. Only the collapse modes are summarized in Table 2 and Figure 8.  
Figure 9 shows the experimental behavior of specimen \(W_{1}\). It appears that the total fracture of the sheathing boards occurred on one side of the specimen, and the boards were separated from the end studs. On the opposite side of the wall, the boards essentially were detached from the end studs due to the screws between the sheathings and end studs failing in shear or being pulled out of the boards. The connections between the end studs and their bases could not effectively restrain the studs' rotation. To protect the experimental equipment and ensure the security of the test, the loading procedure was terminated before the specimen attained its total collapse mode. However, the failure modes (Figure 9) reveal that when subjected to a horizontal load, the specimen has a total collapse tendency as a result of the separation between the boards and end studs. Such failure modes are resulting in a geometric variable system of the wall and further overturning collapse. This collapse tendency coincides with that shown in Figure 8a, meaning fractures of the equivalent braces. This type of collapse mode is the most incidental because of the highest vulnerability index amongst all collapse modes. Thus, the applicability of the structural vulnerability theory to the whole structure of such a traditional CFS shear wall system is verified.  
It can be derived from Figure 8 and Table 2 that there is a notable difference in the vulnerability results between the lower and upper wall segments of the traditional CFS shear wall system, which means that a key wall element needs to be identified because the failure of the key element at different locations will induce various damage degrees to the structure. In the case of a severe earthquake, where the sheathings are the primary lateral load-bearing structural components, local collapse or even total collapse may take place as soon as they are damaged. That is because the structure has no redundant loading path if the sheathings are damaged.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Collapse Mode** & **Failure Mode** & **Separateness \(\boldsymbol{\gamma}\)** & **Relative Damage** & **Vulnerability** \\  & & & **Demand \(D_{\mathbf{r}}\)** & **Index \(\boldsymbol{\phi}\)** \\ \hline \multirow{3}{*}{Total collapse mode} & **Figure 8a** & **1.0** & **0.1581** & **6.325** \\  & Figure 8b & 1.0 & 0.1949 & 5.131 \\  & Figure 8c & 1.0 & 0.2614 & 3.826 \\ \hline \multirow{3}{*}{Local collapse mode} & **Figure 8d** & 0.355 & 0.1581 & 2.246 \\ \cline{1-1}  & & & & \\ \hline \hline \end{tabular} Note: Bold indicates the same failure mode as in the test results.  
\end{table}
Table 2: Vulnerability results of the traditional shear wall system \(W_{1}\).