## 4 Girder End Reliability under the Relative Transverse Displacements
### Transverse Behavior of the Girder End Calculated by the Longitudinal Displacement Data  
Based on the longitudinal displacement data of the bearings and expansion joints on the bridge girder end, the cross-section rotation of the girder and the relative transverse displacement of the expansion joint can be calculated using the same principle (as shown in Figure 10). \(L\) can be the transverse length of a girder or an expansion joint, and \(H\) can be the distance between two sensors. The \(L\)/cos\(\theta\) in Step 4 is the length of the triangular hypotenuse in Step 3. It should be noted that because the calculated \(\theta\) is small (less than \(6.5\times 10^{-5}\) rad), \(L\)/cos\(\theta-L\) (the relative transverse displacement between two measured points, i.e., the transverse deformation of the girder cross-section) must be chosen, and not \(\theta\) (cross-section rotation), to characterize the transverse behavior of the girder. According to the calculation presented in Figure 10, Figure 11 shows the transverse deformation of the girders and the relative transverse displacement of the expansion joints at the bridge girder ends in 2017. Figure 11a is the transverse deformation of the girder (30 m apart, calculated from B-7-11 and B-7-12) at Pier 7 from June to December; Figure 11b is the relative transverse displacement of \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Location of Bearings** & **Pier 1** & **Pier 2** & **Pier 3** & **Pier 5** & **Pier 6** & **Pier 7** \\ \hline
**Cumulation Displacement in** & & & & & & \\
**One Year (unit: km)** & 0.6165 & 1.0910 & 1.1018 & 1.0489 & 0.9456 & 0.6154 \\ \hline
**Wear Rate in One Year** & & & & & & \\
**(Compared with 50 km)** & 0.0123 & 0.0218 & 0.0220 & 0.0210 & 0.0189 & 0.0123 \\ \hline
**Wear Time Every 50 km (unit: Years)** & 81.10 & 45.83 & 45.38 & 47.67 & 52.88 & 81.24 \\ \hline \end{tabular}
\end{table} 
Table 1: Cumulative longitudinal displacement and wear details of the bridge bearings. the downstream expansion joint (15 m apart, calculated from EJ-7-1 and EJ-7-2) at Pier 7 from June to December; Figure 11c is the relative transverse displacement of the expansion joint (30 m apart, calculated from EJ-7-1 and EJ-7-4) at Pier 7 from June to December; and Figure 11d is the relative transverse displacement of the expansion joint (30 m apart, calculated from EJ-1-1 and EJ-1-3) at Pier 1 from June to July. Figure 11: Relative transverse deformation/displacement at the girder ends: (**a**) main bridge girder (from B-7-11 and B-7-12); (**b**) expansion joint (from EJ-7-1 and EJ-7-2); (**c**) expansion joint (from EJ-7-1 and EJ-7-4); (**d**) expansion joint (from EJ-1-1 and EJ-1-3). Figure 10: Process used to calculate the relative transverse displacement based on longitudinal displacement data. As shown in Figure 11, the relative transverse displacement of the downstream expansion joint (15 m, including two expansion devices) is not significantly smaller than that of the whole expansion joint (30 m, including four expansion devices) due to the relative transverse displacement of the expansion joint, which depends not only on the main bridge deformation but also the ramp bridge deformation. Specifically, the 30-m-wide ramp bridge on Pier 7 includes two independent 15-m-wide girders. The maximum transverse deformation of the girder is nearly 2 mm, and the maximum relative transverse displacement of the expansion joint is nearly 1 mm. The transverse deformation (2 mm) is a low level of deformation for a steel girder with a transverse length of 30 m. However, for the expansion joint (and the expansion devices in the expansion joint), which does not have any transverse stiffness, the effect of 1 mm of relative transverse displacement between the main bridge and the ramp bridge must be investigated further. Furthermore, according to the study of [PERSON] et al.