# Seismic Effectiveness of Multiple Seismic Measures on a Continuous Gider Bridge
## 3 Establishment of a Finite Element Model of a Bridge  
### Description of the Bridge  
A four-span continuous girder bridge is studied to evaluate the effect of seismic measures. The bridge has a single span of 30 m and a total span of 120 m, as shown in Figure 1a. Figure 1b shows the section of the girder, which consists of six small box girders with either reinforced or pre-stressed concrete, with a width of 17 m and a height of 1.6 m. The compressive strength of concrete used in the girder is 50 MPa, the elastic modulus is 34,500 MPa, and the bulk density is 25 kN/m\({}^{3}\)[21]. The piers of the bridge have a square cylindrical section, with a height of 15 m and a side length of 1.8 m, as shown in Figure 1c. The compressive strength of concrete used in piers is 30 MPa, the elastic modulus is 30,00 MPa, and the bulk density is 25 kN/m\({}^{3}\). The total weight of the bridge is 8137 kN [25]. The girder and pier of the bridge are modelled using the elastic beam-column elements. The pier foundation is assumed to be fixed and the soil structure interaction is neglected.  
Figure 1: Finite-element model of bridge: (**a**) detail of bridge; (**b**) section of girder; (**c**) section of pier.  
The bearings adopted on #1 and #5 piers and #2, #3, and #4 piers are different. Slide rubber bearings are adopted on #1 and #5 piers, in order to satisfy the deformation requirements of the structure under a static load, whereas laminated rubber bearings are adopted on #2, #3, and #4 piers. The bearings consisting of an elastomeric rubber pad are modelled by the elastic, perfectly plastic element to represent the frictional force developed at the elastomeric pad. The parameters of the bearings are shown in Table 1[36].  
The initial gap of expansion device is 160 mm, as shown in Figure 2. The steel members of the expansion device have a yield strength of 345 MPa and an ultimate strength range of 490-675 MPa. The lower limit of tensile strength of rubber shear springs is 20 MPa [32]. The expansion devices are modelled by the clearance element: the clearance quantity is the same as initial gap, 160 mm, the effective stiffness is 5320 kN/mm, and the damping \(c\) is 5.16 kN\(\cdot\)s/m, with the ultimate impact as 15,000 kN [37].  
High-strength steel bars or cables are desirable options for unseating prevention restrainers, which can generally be implemented in the longitudinal direction of bridges, unlike concrete shear keys. They can be installed at either end abutments, intermediate piers, or in-span hinges, and such applications have proved effective at reducing the likelihood of bridge unseating [27]. As steel cable/bar restrainers are the least expensive option, they are extensively used as bridge unseating prevention restrainers.  
Therefore, this study selects high-strength steel bars as restrainers and steel cables as unseating prevention devices.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters of Bearings** & **Slide Rubber Bearings** & **Laminated Rubber Bearings** \\ \hline Plane size of bearings (mm) & 400 \(\times\) 450 & 400 \(\times\) 450 \\ \hline Maximum bearing pressure/(kN) & 1716 & 3481 \\ \hline Height of bearings (mm) & 116 & 130 \\ \hline Total thickness of rubber layer (mm) & 82 & 95 \\ \hline Longitudinal displacement of bearing without & / & 45 \\ braking force (mm) & 90 & 63 \\ \hline Allowable shear deformation (mm) & 81.2 & 91 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of bearings.  
Figure 2: Finite element model of expansion devices.