## 3 Results and Discussion
### Damage Severity Identification of Hinge Joint Based on Artificial Bee Colony
#### Numerical Simulations  
For damage severity assessment of hinge joint damage in simply supported bridges with four slabs, LLD influence lines for damage cases listed in Table 3 were first obtained and listed in Table 5. Iteration processes for damage identification using ABC algorithms, GA, and PSO were illustrated in Figure 12. Corresponding identification results and time costs are listed in Tables 6 and 7. As can be seen from the results, ABC, ABCG, ABCLGII, and HABCDE presented favorable identification accuracies for all damage cases of hinge joints. Particle swarm optimization also possessed good precision except in case 3 (multiple damages for joints 1 and 2). The relative errors for slabs 2 and 3 in case 3 were 8.2% and 6.0%, respectively. The GA presented the worst identification results among these methods. As for calculation speed, PSO possessed the lowest time cost. Among the ABC algorithms, ABCLGII presented the most favorable efficiency. A simply supported hinged-slab bridge composed of seven slabs was also adopted as the numerical analysis example. Under the damage cases of Table 4, the LLD influence lines calculated by MHPM were shown in Table 8. After the damage identification calculated by the proposed method, the iteration process and damage condition results based on ABC algorithms, GA, and PSO are described in Figure 13 and Tables 9 and 10. As shown in Figure 13, damage identification based on ABCLGII had the fastest convergence rate among the ABC algorithms. It could converge after 100 iterations from case 1 to case 3, while the convergence rate decreased to 180 cycles for case 4, which is a more complicated damage condition. The HABCDE algorithm ranked only second to the ABCLGII in terms of convergence rate. With the complexity of damage conditions increased, its convergence rate decreased from 200 to 400 iterations. As for the other two algorithms, the convergence rate of ABCG was unstable. The original ABC algorithm had the worst performance, which converged after almost 500 iterations in all damage cases. In the aspect of calculation error, the three improved ABC algorithms had similar performances, of which the error was approximately 5 \(\times\)\(10^{-10}\). But in the last case, ABCG had a maximum error of \(4.72\times 10^{-6}\). The original ABC algorithm error fluctuated within a range of 1.79 \(\times\)\(10^{-9}\) and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case No.**} & \multirow{2}{*}{**Slab No.**} & \multicolumn{4}{c}{**LLD Influence Line Vertical Value**} \\ \cline{3-6}  & & **1** & **2** & **3** & **4** \\ \hline \multirow{2}{*}{1} & 1 & 0.41844 & 0.19806 & 0.17576 & 0.16496 \\  & 3 & 0.18972 & 0.26694 & 0.27956 & 0.27774 \\ \hline \multirow{2}{*}{2} & 2 & 0.35677 & 0.36376 & 0.14416 & 0.13531 \\  & 4 & 0.12700 & 0.13531 & 0.35677 & 0.38093 \\ \hline \multirow{2}{*}{3} & 2 & 0.29225 & 0.32085 & 0.20157 & 0.18919 \\  & 3 & 0.17522 & 0.20219 & 0.31539 & 0.31137 \\ \hline \multirow{2}{*}{4} & 1 & 0.49264 & 0.23551 & 0.17375 & 0.07614 \\  & 2 & 0.25219 & 0.37177 & 0.28635 & 0.12548 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: LLD influence lines for damage cases listed in Table 3.  
\(9.68\times 10^{-7}\). The time cost of the damage identification algorithm based on ABCLGII and HABCDE did not exceed 70 s for 500 iterations, while that of the others needed 10 to 15 min. Figure 12: Iteration process of damage identification for cases 1 to 4: (a) case 1: slab 1; (b) case 1: slab 3; (c) case 2: slab 2; (d) case 2: slab 4; (e) case 3: slab 2; (f) case 3: slab 3; (g) case 4: slab 1; (h) case 4: slab 2.