## 4 Girder End Reliability under the Relative Transverse Displacements
### Reliability and Early Warning of the Girder End of the Train-Bridge System
codes of Europe and United States [34; 35], the expansion device will not reach the cut-off limit of fatigue (the lowest stress amplitude that will lead to fatigue in the steel; the lowest cut-off limit of Europe code [34] is 15 MPa, and the lowest cut-off limit of US code [35] is 10 MPa) in most cases. The fatigue reliability of the expansion device under the diurnal relative transverse displacements is sufficient The fatigue of the girder end may depend more on the vertical train loads during bridge operation. For the train crossings, the transverse deflection of the rail is used to analyze the girder end reliability. Figure 14 gives the transverse rail deflections under the load necessary to achieve \(-1\) mm of relative transverse displacement on the fixed side. The rail deflects at approximately 1 mm in the longitudinal distance of 2000 mm. For a China Railway High-speed 3 (CRH3) type electric multiple unit (EMU) train with cars that are 25 m in length, the rail transverse deflection will give an impulse to each crossing train wheel. The transverse stability of the crossing high-speed trains will be significantly weakened. Considering the cumulative effect of some other factors (e.g., the transverse track irregularity), the transverse instability of the train running will be further weakened. The transverse deflections of the rail should be monitored carefully during diurnal bridge operation. Next, assuming that \(\mathbf{Z}\) is the discrete data of rail transverse deflections, the probability (CDF value) that the element of \(\mathbf{Z}\) is less than \(\alpha r\) can be expressed as  
\[P=P(z\leq\alpha r)=\int_{-\infty}^{\alpha r}f_{\mathbf{Z}}(z)\mathrm{d}z \tag{12}\]  
Figure 14: Transverse deflections of the rail under \(-1\) mm relative transverse displacement of the fixed side. Figure 13: Kernel probability density function (PDF) of the von Mises stresses belonging to the calculated maximum points of the expansion device members: (a) data-driven calculated results; (b) rainflow equivalent amplitude. where \(f_{\mathbf{Z}}(z)\) is the PDF of \(\mathbf{Z}\); \(r\) is the resistance of the transverse rail deflections at the expansion device, which is determined by the code limit or engineering experience; and \(\alpha\) is the statistical redundancy of the early warning. Figure 15 shows the kernel PDF and kernel CDF of transverse rail deflections based on the data of EJ-1-1 to EJ-1-3 from June through July 2017. As shown in Figure 15b, the value of \(\alpha r\) is approximately 0.6 mm at the probability of 99.995%. The train running stability limit of transverse rail deflections for the expansion device (\(r\)) is 1 mm, and the statistical redundancy of the early warning (\(\alpha\)) can be set as 0.6. If the calculated transverse rail deflections based on the real-time monitoring data exceed 0.6 mm, a deceleration instruction should be sent to the crossing high-speed trains until the rails at the girder end can be inspected and repaired. The determination of the reliable probability in the diurnal early warning depends on the maintenance experience of the bridge engineers. The original intention of bridge health monitoring is to detect bridge damage as early as possible, to facilitate the rapid repair of a bridge subject to diurnal operation. With the accumulation of monitoring data, the reliable probability and the statistical redundancy of the early warning system can be updated for more informed long-term bridge operation. This approach promotes smart maintenance of the girder end of the long-span high-speed railway bridge.