# A Novel Hybrid Whale-Chimp Optimization Algorithm for Structural Damage Detection
## 4 Numerical Simulations  
### Structural Damage Detection of Two-Story Rigid Frame  
A two-story rigid frame structure is divided into 18 elements, as shown in Figure 4. The numbers in the circles represent the element number and the other numbers represent the node number. To prevent the interference of other factors, the nodes of the two-story rigid frame are all set as steel nodes. The overall degree of freedom of the structure is 16 and the first type of constraint is used to prevent rigid body displacement. Physical parameters of the two-story rigid frame structure are listed in Table 3 as follows:To simplify the comparison process, both the ChOA and the W-ChOA use the same parameters. The parameters are set as: population size \(N=50\) and maximum iterations \(T_{\text{max}}=200\). Modal order is set to five and search dimension is set to 18. The optimal value is selected by running five times for each case. Excluding the set of real damage elements, damage detection values greater than \(10^{-3}\) are considered misjudgments. In structures, the failure of the frame structure often occurs in the mid-span where it is subjected to a big bending moment. Therefore, the damage simulation is set in the middle element of the beam. The damage degree is realized by reducing the stiffness of the element. As listed in Table 4, the stiffness of the elements is reduced by 10% to simulate different damage situations of the frame structure. In addition, to accurately simulate the effects of measurement errors on natural frequency and mode shape, random noise is added to the measured modal parameter, as shown:  
\[S_{\text{noise}}=S_{\text{original}}\big{(}1+R_{p}N_{\text{variable}}\big{)} \tag{19}\]  
where \(S_{\text{noise}}\) is the modal parameter with noise variable. \(S_{\text{original}}\) is the original modal parameter of the rigid frame. \(R_{p}\), \(N_{\text{variable}}\) represents the noise level and the type of noise subject to the standard normal distribution, respectively. Figure 5 shows the single damage results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 5a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 17 without misjudgment. In contrast, the identification results of the ChOA have a misjudgment at element 2. As shown in Figure 5b, if the damage percent is 10% and the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Damages** & **Damage Elements** & **Degree of Damage** \\ \hline
1 & Single damage & 17 & 10\% \\
2 & Double damages & 8, 17 & 10\%, 10\% \\
3 & Triple damages & 5, 8, 17 & 10\%, 10\%, 10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Different damage cases of two-story rigid frame. Figure 4: Finite element model of rigid frame. noise level is 10%, both the ChOA and the W-ChOA have a misjudgment at element 2 and the misjudgment value of the ChOA is greater than the W-ChOA. Figure 6 shows the double damage detection results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 6a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 8 and element 17 without misjudgment. In contrast, the identification results of the ChOA have a misjudgment at element 3. As shown in Figure 6b, if the damage percent is 10% and the noise level is 10%, the ChOA results have three misjudgments in element 2, element 10, and element 11. The identification results of the W-ChOA are much better with only one misjudgment at element 11. Figure 7 shows the triple damage detection results by the ChOA and the W-ChOA under different noise levels. As shown in Figure 7a, if the damage percent is 10% and the noise level is 5%, the W-ChOA can accurately identify damage location and damage degree at element 5, element 8, and element 17 with only one misjudgment at element 11.