## 5 Reinforced Building Concrete Frame Analysis
The original wall has a stiffness value of 1.94 kN/mm and a strengthened value of 71.96 kN, as shown in Figure 11. Figures 9-11 summarize the stiffness values, the strengthened values, and the ductility values of the three-story concrete building structures shown in Table 5. The original wall has a natural frequency period of 1.04 s. The reinforced wall has an increased stiffness value that can lead to a decrease in the natural frequency period, which is 0.36 s. On the other hand, calculating the shear force requirement using the equivalent static force method  
Figure 11: Comparison of the original walls and reinforced walls. Figure 12: Comparison of the original concrete building structure and reinforced concrete building structure. (pushover analysis), we obtain \(T=0.22\) s. Hence, the shear force required for the reinforced structure is the same. Reinforcement resulted in an increase in the structure's strength by 4.23 times. The reinforced column had a strength increase of 5.54 times, and the reinforced wall had a strength increase of 2.10 times. However, the total strengths of the reinforced column and reinforced wall are 81% and 19% of that of the entire three-story concrete building structure, respectively. When comparing the ductility of the reinforced frame structure, which has a ductility value of 1.73, we found that it was decreased compared to the original frame structure, which has a ductility value of 1.86. In addition, the reinforced column structure has a ductility value of 1.64, showing a decrease from the original column structure, which has a ductility value of 1.89. In contrast, the wall reinforced by ferrocement and expanded metal has a ductility value of 1.63, which is increased compared to the ductility value of the original wall structure. The original wall has a ductility value of 1.38. The damage levels of the original structure and the reinforced structure were determined using the damage index, as follows in Equation (11) [15]. Figure 6:0  
where  
\(S_{m}\) is the maximum deflection value of the system due to seismic force;  
\(S_{u}\) is the maximum deflection value under monotonic loading;  
\(F_{y}\) is the yield point of the structural system;  
\(E_{h}\) is the hysteretic energy value of the structural system;  
\(\beta\) is a constant which indicates the scale of damage by the response modal shape. The damage index values of the original building structure and reinforced building structure are shown in Figures 13 and 14 and were obtained using criteria for measuring the degree of structural damage, defined based on the total damage index (DI) of the structural parts at the plastic hinge, which is the sum of the damage derived from the story drift structural value and the hysteretic energy value in response to the seismic forces. The definitions used to measure the results are as follows: a damage index DI which has a value less than 0.4 indicates a low-damage level, a DI value between 0.4 and 0.6 indicates repairable damage, a DI value more than 0.6 but not over 1.0 indicates severe damage which cannot be repaired, and a DI value more than 1.0 indicates the erosion of the structure. The analysis results showed that the original building structure had lower-level damage at both ends of the column and erosion, with a DI value over 1.0, when it was reinforced. Moreover, the damage level was decreased to a level not exceeding 0.6, which indicated a repairable level of damage. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Structures** & **Stiffness** & **Seismic Resistance** & \\  & **(kN/mm)** & **(kN)** & **Ductility** \\ \hline The original frame & 2.29 & 127.91 & 1.86 \\ The reinforced frame & 26.15 & 702.35 & 1.73 \\ The original column & 1.07 & 55.95 & 1.89 \\ The reinforced column & 21.07 & 551.35 & 1.64 \\ The original wall & 1.94 & 71.96 & 1.38 \\ The reinforced wall & 6.56 & 151.04 & 1.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Stiffness, seismic resistance, and ductility of the structures.