## 2 Discussion on the Working Conditions of a Bridge Restrainer
### Outline of Dynamic Response Analysis  
#### 2.1.1 Analysis Model  
Figure 3 shows a general drawing of the simple span steel I-girder bridge considered in this study, which has typical specifications for the road width, spans and bridge types in Japanese urban areas. The bridge specifications were defined considering actual bridges: span length, 35 m; height of the girder, 2 m; width of the road, 10.5 m with two lanes each way; height of the pier, 12 m; and a soil type of soft ground. Figure 4 shows the analysis model. In this figure, the points basically represent  
the nodes, and the lines connected between the two nodes represent the elements. Regarding the mesh division, the superstructure divided one span into four sections and the pier was divided into three. The abutment was a rigid element, and a spring element capable of expressing bounding was placed either between the pier or between the pier and the abutment. As a boundary condition, the base of the pier was fixed, with a spring representing the ground. Tables 1-4 list the parameters used in the analysis. In this analysis model, a structure-simplified frame finite element (FE) model was employed because it is commonly used for bridge design, and it can reduce the calculation cost. The superstructure, constituting a slab and girder, was modeled using the linear beam model, and the pier was modeled as a nonlinear beam model, which is defined based on the perfect elastic-plasticity M-\(\varphi\) model shown in Figure 5. A rigid beam element was used to match the gap between the centroid axis of the superstructure and the top of the substructure. Figure 3: General drawing of a target bridge. (**a**) Side view; (**b**) cross-section view; and (**c**) pier view. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Part**} & \multicolumn{2}{c}{**Cross-Section A (m2)**} & \multicolumn{2}{c}{**Sectional Secondary Moment I (m4)**} & \multirow{2}{*}{**Weight (kN)**} & \multicolumn{2}{c}{**Elastic Coefficient (kN/mm2)**} & \multicolumn{2}{c}{**Elastic Shear Modulus (N/mm2)**} \\ \hline \multicolumn{2}{c}{**Girder**} & 0.60 & 0.121 & 4600 & 200,000 & 77,000 \\ \hline \multirow{2}{*}{Pier} & Upper & 8.55 & 2.005 & & \\ \cline{2-5}  & Middle & 8.55 & 2.420 & 2000 & 26,500 & 11,500 \\ \cline{2-5}  & Low & 8.55 & 2.977 & & \\ \hline Rigid Member & 1000 & 1000 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of beam elements. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Ground**} & \multicolumn{2}{c}{**Girder-Collision**} & \multirow{2}{*}{**Friction Force**} \\ \multicolumn{2}{c}{**Horizontal-Direction**} & \multicolumn{2}{c}{**Rotation-Direction Spring Constant (kN/m)**} & \multicolumn{2}{c}{**Spring Constant (kN/m)**} \\ \hline
3,300,000 & 75,000,000 & 13,714,286 & 115 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of spring elements. Figure 4: Analysis model. (**a**) Overall model diagram and (**b**) detail of each part. The mass of the structure was distributed among the nodes, and the ground spring comprised of four spring elements (a horizontal spring, vertical spring, rotation spring and coupled spring). Furthermore, considering the bounding within the superstructure, a collision spring was defined as shown in Figure 6 (compression side). With this model assumed as the elastic collision spring model, based on the specifications [2], this spring could be obtained using Equation (1):  
\[k_{I}=n\times E_{s}\times A_{s}/L_{s} \tag{1}\]  
where \(k_{I}\) is the collision spring constant, \(n\) is the partition number of the beam element, \(E_{s}\times A_{s}\) is the axial direction stiffness of the superstructure and \(L_{s}\) is the length of the girder. Furthermore, the friction between the superstructure and substructure after breaking the fixed support is expressed, as shown in Figure 7. The coefficient of dynamic friction was assumed as \(0.05\), based on a previous study [4].