## 4 Numerical Simulation
### Damage Identification Based on the Proposed Method
#### 4.6.4 Damage Case 4  
In damage case 4, the damage of a single beam or column is studied. The beams and columns in the model are numbered for ease of illustration. For columns, they are numbered from left to right and then from bottom to top. A total of 48 columns are numbered 1-48. Then the beams are consecutively numbered in accordance with the same rule. There are 40 beams numbered 49-88. The damage of a single beam or column is studied under four separate damage cases, which are shown in Table 6: The 10 th column with a damage factor of 0.5 and 0.8; and the 55 th beam with a damage factor of 0.5 and 0.8. The 10 th column and the 55 th beam belong to the 2 nd substructure, as shown in Figure 7. Under the above four damage cases, the numerical simulation is carried out and the damage of eight substructures is identified. The damage factors of other substructures are close to the theoretical value of 1, so only the results of the second substructure are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Damage Case** & \begin{tabular}{c} **10 th Column with** \\ **Damage Factor 0.5** \\ \end{tabular} & \begin{tabular}{c} **10 th Column with** \\ **Damage Factor 0.8** \\ \end{tabular} & \
\begin{tabular}{c} **55 th Beam with** \\ **Damage Factor 0.5** \\ \end{tabular} &
\begin{tabular}{c} **55 th Beam with** \\ **Damage Factor 0.8** \\ \end{tabular} \\ \hline The improved method & 0.9205 & 0.9802 & 0.9642 & 0.9908 \\ The traditional method & 0.9246 & 0.9824 & 0.9538 & 0.9875 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Damage identification results of 2 nd substructure with single column/beam damage. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No** & **Theory** & **The Improved Method** & **The Traditional Method** \\ \hline
1 & 1.0000 & 0.9999 & 1.0000 \\
2 & 0.9000 & 0.8836 & 0.8926 \\
3 & 1.0000 & 0.9767 & 0.9933 \\
4 & 0.9000 & 0.9042 & 0.8897 \\
5 & 0.9400 & 0.9376 & 0.9339 \\
6 & 1.0000 & 0.9999 & 0.9952 \\
7 & 0.9600 & 0.9754 & 0.9752 \\
8 & 1.0000 & 1.0000 & 0.9998 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Damage identification results with multiple smaller damages. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No** & **Theory** & **The Improved Method** & **The Traditional Method** \\ \hline
1 & 1.0000 & 1.0000 & 1.0000 \\
2 & 1.0000 & 0.9999 & 1.0000 \\
3 & 1.0000 & 1.0000 & 1.0000 \\
4 & 1.0000 & 1.0000 & 1.0000 \\
5 & 1.0000 & 0.9998 & 1.0000 \\
6 & 1.0000 & 1.0000 & 1.0000 \\
7 & 0.9600 & 0.9633 & 0.9653 \\
8 & 1.0000 & 0.9911 & 0.9900 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Damage identification results with a single smaller damage. As shown in Table 6, when the damage factor of the 10 th column is 0.5, it can be detected that the 2 nd substructure has been damaged. However, it is difficult to detect the structural damage when the damage factor is smaller than 0.8. Compared with columns, the contribution of beams to structural lateral stiffness is smaller, so the damage is more difficult to detect under the same damage factor. When the damage factor of the 10 th column is 0.5, it can be detected that the 2 nd substructure has been damaged. In order to further identify damage, each of the beams and columns is treated as a separate substructure. Therefore, there are 11 substructures and 11 damage factors to be identified in the second floor. The damage identification results are shown in Table 7. From Table 7, it can be roughly judged that a column has been damaged, but it is almost impossible to identify which column has been damaged. It is because the identification used the structural frequency and mode shape, which conveys global information about the structure and are insensitive to local damage. If local damage is to be identified, a method suitable for local damage identification should be used. Such a method has been proposed by the author and it can be found in Reference [29] that is referenced in the introduction.