## 3 Results and Discussion  
### Lateral Load Distribution Evaluation Based on Modified Hinge Plate Method  
Simply supported slab bridges composed of four and seven slabs were selected as the research objects in this study, whose cross-section form of a single slab was rectangular. The sizes of width and height were 1 m and 0.6 m, respectively. The span of the bridge was 12 m, and the material properties of elements were elastic modulus \(E=3.5\times 10^{10}\) Pa, Poisson's ratio \(\mu=0.2\), density \(\rho=2500\) kg/m\({}^{3}\). The finite element model was established by ANSYS 14.0 and composed through SOLID 65 element. A general hinge joint mode is shown in Figure 5a. For the convenience of analysis, the hinge joint was simplified as shown Figure 5b. The spring element COMBIN 7 was selected to simulate the damage through reduction of spring stiffness, which is depicted in Figure 5c. Here, an undamaged hinge joint was considered to have infinite stiffness, while the spring stiffness with damage extent \(\mu_{i}\) could be calculated by Equation (17). It should be noted that all numerical computations and algorithm programs were implemented in MATLAB 2014b and run on a Core i5-6500 personal computer with Windows 7 system, 3.20 GHz CPU and 8 GB RAM Memory. \[k_{i}=\frac{g_{i}}{\mu_{i}\big{(}\delta_{i,p}+\delta_{i,j}g_{j}\big{)}}\times 0.5, \tag{17}\]  
where \(k_{i}\) is the spring stiffness. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **The Parameter Values** \\ \hline ABCG 1  & \(G=0.1\); \(\varepsilon=2^{-52}\); \(\alpha=0.95\); \(\beta=0.15\); \(\delta=0.95\); \(\xi=0.98\) \\ ABCLGII 2  & \(r=2\); \(q=0.2\); \(P_{str}=0.5\) \\ HABCDE 3  & \(C=1\); \(CR=0.6\); \(F=0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization parameters values. The deflections of each slab at mid-span cross section can be calculated based on finite element simulation. According to the statics calculating procedure of LLD, it can be obtained by:  
\[\eta_{ki}=\frac{\omega_{ki}}{\sum\limits_{j=1}^{n}\omega_{ji}}, \tag{18}\]  
where \(\omega_{ki}\) is deflection of the \(k\)th slab at mid-span cross-section when the external load is located at mid-span of the \(i\)th slab. The efficiency and accuracy of MHPM were verified by comparing its results with those calculated through FEM. Damage cases of hinge joints for simply supported bridges with four and seven slabs (as shown in Figure 6) are listed in Tables 3 and 4, respectively. Corresponding LLD influence lines of overall cases were shown in Figures 7 and 8. In the figures, MHPM and FEM represent two methods, while D and U mean the hinge joint was damaged and undamaged, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Case No.** & **Damage Location and Extent** & **Slab Nos. for LLD** \\ \hline
5 & [0.05, 0, 0, 0, 0, 0] & 1, 4 \\
6 & [0, 0, 0.6, 0, 0, 0] & 2, 5 \\
7 & [0.1, 0, 0, 0.5, 0, 0] & 1, 3 \\
8 & [0.1, 0.15, 0, 0.4, 0.05, 0] & 4, 7 \\ \hline \end{tabular}
\end{table} 
Table 4: Hinge joint damage cases for simply supported bridge with 7 slabs. Figure 7: LLD influence lines under damage and intact cases of 4-slab bridge: (**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. As can be seen from Figures 7 and 8, LLD influence lines computed by MHPM were consistent with the results obtained through FEM. Compared with FEM, which was taken as the benchmark method, the relative error of LLD for a single slab calculated by MHPM did not exceed 1.9%. It demonstrates the correctness of MHPM proposed in this paper. In the case of single hinge joint damage, namely cases 1, 2, 5, and 6, cross phenomena of LLD influence lines between damaged and undamaged cases coincided with results found in the literature [18].