## 3 Results and Discussion
### Damage Severity Identification of Hinge Joint Based on Artificial Bee Colony
#### Numerical Simulations
Particle swarm optimization presented favorable convergence speed and identification accuracy for most of the damage cases except case 4 (multiple damages for joints 1, 2, 4, and 5). In this case, the identification results of PSO were unacceptable. As for GA, it presented the most unsatisfactory results among these methods. In order to verify the applicability of the proposed damage identification method in this article, we compared the damage identification results for several cases of Tables 3 and 4 (as shown in Table 11), which were calculated by the algorithms in this article and References [19; 20], respectively. The error in Table 11 was computed by Equation (19). As can be seen from Table 11, the proposed algorithm in this paper has zero error, which means it has the best performance for damage identification. Although some undamaged hinge joints were identified as damage through the method of this paper; the identified damage was less than \(3.035\times 10^{-9}\) and we have reasons to confirm they are intact. The identification errors of other methods in the literature [19; 20] revealed that these two methods cannot identify the damage location and extent exactly, and the errors rise with the actual damage degree increase. For cases 2 and 7, both of the methods even have wrong diagnoses. Through comparative analysis, it verifies the feasibility and correctness of the proposed method once more. In this paper, the damage identification method based on ABCLGII has better performance than other ABC algorithms, but the time cost (from 50 to 70 s for the damage cases) is still high for the damage cases with 500 iterations. And from Tables 7 and 10, we can obtain that the more complicated the damage condition is, the higher the time cost will be. The hinge joint damage condition is usually more complicated for real engineering problems. As a result, time cost could be a barrier for the proposed method. It should be noticed that the damage identification algorithm is based on LLD influence lines. If we want to analyze hinge joint damage conditions of simply supported hinged-slab bridges, the LLD influence line measured and calculated by field experiment must be acquired first. Although the principle of field experiment is simple, it is time-consuming and is costly, which is a disadvantage for the proposed algorithm in this article. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & \multirow{2}{*}{**Slab No.**} & \multicolumn{6}{c}{**LLD Influence**} & \multicolumn{1}{c}{**Line Vertical Value**} \\ \cline{3-8}  & & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \multirow{2}{*}{5} & 1 & 0.26629 & 0.18381 & 0.14782 & 0.12150 & 0.10314 & 0.09153 & 0.08590 \\  & 4 & 0.11814 & 0.13872 & 0.15553 & 0.16616 & 0.15493 & 0.13749 & 0.12904 \\ \hline \multirow{2}{*}{6} & 2 & 0.29513 & 0.29808 & 0.28782 & 0.03595 & 0.03052 & 0.02708 & 0.02542 \\  & 5 & 0.02767 & 0.02948 & 0.03322 & 0.23341 & 0.23824 & 0.22593 & 0.21205 \\ \hline \multirow{2}{*}{7} & 1 & 0.34244 & 0.21413 & 0.18510 & 0.16820 & 0.03313 & 0.02940 & 0.02760 \\  & 3 & 0.17655 & 0.22862 & 0.23877 & 0.23183 & 0.04567 & 0.04053 & 0.03804 \\ \hline \multirow{2}{*}{8} & 4 & 0.12022 & 0.15568 & 0.25309 & 0.27135 & 0.07970 & 0.06187 & 0.05807 \\  & 7 & 0.02463 & 0.03190 & 0.05186 & 0.05895 & 0.22806 & 0.28812 & 0.31649 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: LLD influence lines for damage cases listed in Table 4.  
Figure 13: Iteration process of damage identification for Case 5 to Case 8: (**a**) case 5: slab 1; (**b**) case 5: slab 4; (**c**) case 6: slab 2; (**d**) case 6: slab 5; (**e**) case 7: slab 1; (**f**) case 7: slab 3; (**g**) case 4: slab 4; (**h**) case 4: slab 7.