## 3 Results and Discussion
### Damage Severity Identification of Hinge Joint Based on Artificial Bee Colony
#### Numerical Simulations
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & \multirow{2}{*}{**Slab No.**} & \multirow{2}{*}{**Algorithms**} & \multicolumn{4}{c}{**Hinge Joint No.**} \\ \cline{4-7}  & & & **1** & **2** & **3** \\ \hline \multirow{8}{*}{1} & & ABC & 0.18 & \(5.643\times 10^{-10}\) & 0 \\  & & ABCG & 0.18 & \(5.646\times 10^{-10}\) & 0 \\  & 1 & ABCLGII & 0.18 & \(5.641\times 10^{-10}\) & 0 \\  & 1 & HABCDE & 0.18 & 0 & 0 \\  & 1 & GA & 0.1802 & \(7.630\times 10^{-6}\) & \(4.252\times 10^{-9}\) \\  & 1 & PSO & 0.18 & 0 & 0 \\ \cline{2-7}  & & ABC & 0.18 & \(3.512\times 10^{-10}\) & \(1.724\times 10^{-9}\) \\  & 1 & ABCG & 0.18 & \(3.504\times 10^{-10}\) & \(1.725\times 10^{-9}\) \\  & 3 & ABCLGII & 0.18 & \(3.505\times 10^{-10}\) & \(1.725\times 10^{-9}\) \\  & 1 & HABCDE & 0.18 & \(3.506\times 10^{-10}\) & \(1.725\times 10^{-9}\) \\  & 1 & GA & 0.1799 & \(3.064\times 10^{-5}\) & \(1.526\times 10^{-5}\) \\  & 1 & PSO & 0.18 & 0 & 0 \\ \hline \multirow{8}{*}{2} & & ABC & \(4.140\times 10^{-10}\) & 0.33 & 0 \\  & 1 & ABCG & \(4.142\times 10^{-10}\) & 0.33 & 0 \\  & 2 & ABCLGII & \(4.139\times 10^{-10}\) & 0.33 & 0 \\  & 1 & HABCDE & \(4.141\times 10^{-10}\) & 0.33 & 0 \\  & 1 & GA & \(2.998\times 10^{-8}\) & 0.33 & \(6.199\times 10^{-6}\) \\  & 1 & PSO & 0 & 0.33 & 0 \\ \cline{2-7}  & & ABC & \(1.753\times 10^{-9}\) & 0.33 & 0 \\  & 1 & ABCG & \(1.753\times 10^{-9}\) & 0.33 & \(0\) \\  & 1 & ABCGII & \(1.753\times 10^{-9}\) & 0.33 & \(1.998\times 10^{-15}\) \\  & 1 & HABCDE & \(1.754\times 10^{-9}\) & 0.33 & \(2.998\times 10^{-15}\) \\  & 1 & GA & 0 & 0.3457 & 0 \\  & 1 & PSO & 0 & 0.33 & 0 \\ \hline \multirow{8}{*}{3} & & ABC & 0.05 & 0.15 & 0 \\  & 1 & ABCG & 0.05 & 0.15 & 0 \\  & 1 & HABCDE & 0.05 & 0.15 & 0 \\  & 1 & HABCDE & 0.05 & 0.15 & 0 \\  & 1 & HABCDE & 0.05 & 0.15 & 0 \\  & 1 & GA & 0.0489 & 0.1485 & 0.0039 \\  & 1 & PSO & 0 & 0.1377 & 0 \\ \cline{2-7}  & & ABC & 0.05 & 0.15 & 0 \\  & 1 & ABCG & 0.0499 & 0.15 & 0 \\  & 1 & ABCGII & 0.05 & 0.15 & 0 \\  & 1 & HABCDE & 0.05 & 0.15 & 0 \\  & 1 & GA & 0.0547 & 0.1484 & 0 \\  & 1 & PSO & 0 & 0.159 & 0 \\ \hline \multirow{8}{*}{4} & & ABC & 0.22 & 0.1 & 0.43 \\  & 1 & ABCG & 0.22 & 0.1 & 0.43 \\  & 1 & HABCDE & 0.22 & 0.1 & 0.43 \\  & 1 & HABCDE & 0.22 & 0.1 & 0.43 \\  & 1 & HABCDE & 0.22 & 0.1 & 0.43 \\  & 1 & HABCDE & 0.22 & 0.1 & 0.43 \\  & 1 & HABCDE & 0.22 & 0.1 & 0.43 \\  & 1 & PSO & 0.22 & 0.1 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Damage identification results for cases 1 to 4. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Case No.**} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**3**} & \multicolumn{2}{c}{**4**} \\ \hline \multicolumn{2}{c}{**Slab No.**} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**3**} & \multicolumn{2}{c}{**4**} \\ \hline \multirow{8}{*}{Algorithms} & ABC & 1 & **3** & **2** & **4** & **2** & **3** & **1** & **2** \\ \cline{3-8}  & ABC & 432 & 429 & 434 & 435 & 449 & 454 & 476 & 471 \\  & ABCG & 435 & 430 & 432 & 431 & 453 & 451 & 475 & 473 \\  & ABCLGII & 53 & 49 & 51 & 54 & 52 & 51 & 54 & 56 \\  & HABCDE & 56 & 55 & 57 & 56 & 59 & 63 & 62 & 65 \\  & GA & 23 & 22 & 19 & 21 & 25 & 23 & 28 & 29 \\  & PSO & 20 & 21 & 20 & 22 & 21 & 22 & 26 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Time costs for cases 1 to 4 (unit: s). As can be seen from Table 9, the relative error of a single hinge joint damage level was no more than \(2\times 10^{-5}\), while those of case 1 to case 3 were much less than this value. It demonstrates that the damage identification method based on MHPM and original or improved ABC algorithms can successfully identify the location and degree of hinge joint damage, which has favorable applicability and accuracy. Taking into consideration the convergence rate, identification accuracy, and time cost, the ABCLGII algorithm is more suitable for the damage identification method proposed in this paper. Comparations with PSO and GA were also conducted.