## 4 Conclusions  
In this study, a damage identification algorithm based on a modified hinged plate method and improved artificial bee colony algorithms was proposed to identify the hinged joint damage of simply supported hinged-slab bridges. The following conclusions were obtained:  
1. The damage factor through substitution of a relative displacement into the canonical equations can realize the simulation of hinge joint damage. The lateral load distribution influence line calculated by modified hinge plate method coincided with the result computed by the finite element method. The maximum error of damage cases in this study by modified hinge plate method was less than 1.9%. 2. Hinge joint damage can lead to cross phenomenon of lateral load distribution influence lines, which is suitable for the damage localization of hinged-slab bridges with single hinge damage. Moreover, the offset degree of lateral load distribution influence line is proportional to damage degrees, which can realize the qualitative assessment of hinge damage. However, cross phenomenon is not effective to identify the damage location with multiple hinge damages. 3. Original and improved artificial bee colony algorithms successfully identified the location and degree of hinge joint damages, of which the maximum error did not exceed 4.72 \(\times\) 10\({}^{-6}\). Based on ABCLGII and HABCDE, the algorithms had the lowest time cost (less than 70 s). Moreover, ABCLGII converged after 100 iterations approximately, while the others did not. So ABCLGII is the most suitable for the proposed damage identification algorithm among artificial bee colony algorithms in this work. 4. The results of comparison with particle swarm optimization and genetic algorithm revealed that both PSO and GA converged after 100 iterations at most and the time costs of them were no more than 30 s, which presented satisfactory convergence speed and time cost. However, the accuracy of damage identification algorithm based on PSO was not stable; namely, the minimum and maximum errors were 1 \(\times\) 10\({}^{-9}\) for single damage condition and 0.028 for multiple hinge damages, respectively. As for GA, its error fluctuated between 0.0005 and 0.022, which demonstrated it had the most unsatisfactory identification results among these methods. 5. It demonstrated again that the proposed algorithm was accurate through comparison with methods in the literature [19; 20]. The former algorithm had zero error while the latter ones had larger errors ranging from 0.003 to 0.164. Even the latter algorithms identified the damage degree and location improperly. The proposed algorithm presents good performances for hinge joint damage identification, but there are still some barriers for it, such as the computation time of the algorithm and the cost and efficiency of field experiment. In the future, the damage identification algorithm should be improved \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case No.** & **Slab No.** & **Algorithms** & **Damage Identification Results** & **Error** \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{2} & In this article & [4.139 \(\times\) 10\({}^{-10}\), 0.33, 0] & 0 \\  & & [19] & [0.04, 0.41, 0] & 0.079 \\  & & [20] & [0, 0.38, 0.03] & 0.063 \\ \hline \multirow{3}{*}{3} & \multirow{3}{*}{3} & In this article & [0.05, 0.15, 0] & 0 \\  & & [19] & [0.033, 0.14, 0] & 0.017 \\  & & [20] & [0.046, 0.155, 0] & 0.014 \\ \hline \multirow{3}{*}{5} & \multirow{3}{*}{1} & In this article & [0.05, 5.275 \(\times\) 10\({}^{-10}\), 0, 0, 0, 3.035 \(\times\) 10\({}^{-9}\)] & 0 \\  & & [19] & [0.057, 0, 0, 0, 0] & 0.005 \\  & & [20] & [0.048, 0, 0, 0, 0] & 0.003 \\ \hline \multirow{3}{*}{7} & \multirow{3}{*}{3} & In this article & [0.1, 3.608 \(\times\) 10\({}^{-10}\), 7.175 \(\times\) 10\({}^{-10}\), 0.5, 0, 0] & 0 \\  & & [19] & [0.15, 0.03, 0.12, 0.41, 0.07, 0] & 0.164 \\ \cline{1-1}  & & [20] & [0.13, 0, 0.09, 0.54, 0.049, 0] & 0.118 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Hinge joint damage identification results comparison.