## 4 Case Study of Masonry Wall Building Reinforced with X-Bracing Concrete Frames  
A reinforced rigid frame model is shown in Figure 5A. The columns were reinforced with steel bars and the x-bracing concrete frame technique, because the shear strength acts on columns in buildings. The beams were reinforced with steel bars, as shown in Table 8. This is equivalent to using plastic hinges, which may be placed at the end of beams. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Damage Characteristics**} & \multicolumn{3}{c}{**Damage Index (DI)**} \\ \cline{2-4}  & **WS01** & **WS02** & **WS03** \\ \hline Total collapse of infill & 1.57 & 1.52 & 1.16 \\ Extensive large cracks & 1.07 & 1.03 & 0.84 \\ Corner crushing & 0.77 & 0.74 & 0.52 \\ First diagonal crack & 0.48 & 0.47 & 0.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Damage index at each damage level. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample Name**} & \multicolumn{5}{c}{**\(\beta\)**} \\ \cline{2-6}  & **0.10** & **0.15** & **0.20** & **0.25** & **0.30** & **0.35** \\ \hline No. 4 & 0.818 & 0.837 & 0.846 & 0.846 & 0.855 & 0.864 \\ WS01 & 0.818 & 0.836 & 0.845 & 0.845 & 0.854 & 0.863 \\ WS02 & 0.810 & 0.819 & 0.824 & 0.824 & 0.829 & 0.834 \\ W503 & 0.810 & 0.819 & 0.824 & 0.824 & 0.829 & 0.833 \\ AFKJ1 & 0.822 & 0.843 & 0.854 & 0.854 & 0.865 & 0.876 \\ TA2 & 0.833 & 0.867 & 0.883 & 0.883 & 0.900 & 0.917 \\ IF-1 & 0.808 & 0.817 & 0.821 & 0.821 & 0.825 & 0.829 \\ IF-2 & 0.806 & 0.812 & 0.815 & 0.815 & 0.818 & 0.821 \\ Average & 0.816 & 0.831 & 0.839 & 0.839 & 0.847 & 0.855 \\ SD. & 0.009 & 0.018 & 0.023 & 0.023 & 0.027 & 0.032 \\ Avg. + SD. & 0.825 & 0.849 & 0.862 & 0.862 & 0.874 & 0.887 \\ Avg. – SD. & 0.806 & 0.813 & 0.816 & 0.816 & 0.819 & 0.823 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Damage index (DI) using various coefficients (\(\beta\)) of the model samples. Figure 5: Part of the wall frame model. (**A**) An analytical model of the masonry walls according to the equivalent compressive x-brace concrete frame method. (**B**) Relation of the lateral force to the displacement value. \begin{table}
\begin{tabular}{c c c c} \hline
**Building** & **Size (mm.)** & **Main Reinforcement (mm.)** & **Stirup Reinforcement (mm.)** \\ \hline C1 & 200 \(\times\) 200 & 8 DB16 & RB6@150 \\ B2 & 200 \(\times\) 400 & 5 DB16 & RB6@150 \\ RB & 200 \(\times\) 400 & 5 DB16 & RB6@150 \\ W1 & 125 \(\times\) 200 & 2 RB9 & RB6@150 \\ \hline \end{tabular}
\end{table} 
Table 8: The details of the reinforced steel bars. The column's moment resistance (\(M_{S}\)) can be calculated from the sum of the original column's moment resistance (\(M_{c}\)) and the x-bracing concrete frames' moment resistance (\(M_{B}\)) as follows:  
\[M_{S}=\ M_{C}+M_{B} \tag{7}\]  
The x-bracing concrete frames' moment resistance (\(M_{B}\)) can be calculated from the sum of the concrete brace moment resistance and the steel bar moment resistance as follows:  
\[M_{B}=\sum\
olimits_{i=1}^{n}\left(f_{ys}(S_{s})i_{i}\right)+\sum\
olimits_{i= 1}^{n}(f_{tc}(S_{c})i_{i}) \tag{8}\]  
where  
\(f_{ys}\) and \(f^{\prime}c\) = yield the strength of the steel bar and compressive strength of the concrete brace;  
\(Ss\) and \(Sc\) = cross-section of the steel bar and concrete brace;  
\(V_{BF}\) = lateral load resistance of the reinforced rigid frame. \[V_{BF}=2(Mpj+Ms)/h=2(Mpj+Mc+Mb)/h \tag{9}\]  
where \(M_{pj}\) = the joint connecting plastic moment, which is considered to include the smallest moment of the column \(M_{pc}\) plastic moment, the beam \(M_{pb}\) plastic moment, and the joint connecting moment of the column-beam.