## 4 Girder End Reliability under the Relative Transverse Displacements
### Reliability and Early Warning of the Girder End of the Train-Bridge System  
Adding the calculated relative transverse displacements to the fixed side (the main bridge side) of the expansion device FE model, the mechanical behaviors of the expansion device during the operating period can be calculated. Then, the reliability analysis of the girder end train-bridge system can be conducted based on the monitoring data. Considering the missing data of sensor EJ-1-3 (valid only in June and July, 2017), and the maximum relative transverse displacements that occur at EJ-1-1 to EJ-1-3 in July 2017, the relative transverse displacements used for reliability analysis come from the data of EJ-1-1 to EJ-1-3 for June through July 2017. Most of the long-term responses will not obey a single common distribution (e.g., normal distribution). The relative transverse displacement data-driven calculated responses of expansion device FE model do not obey a simple probability distribution too. Hence, a non-parametric estimation should be used in the description of their statistical features [31; 32; 33]. The kernel density estimation is used to calculate the statistical features of the mechanical behaviors of the expansion device. A kernel distribution is defined by a smoothing function and a bandwidth value, which control the smoothness of the resulting density curve. The kernel density estimator is the estimated probability density function (PDF) of a random variable. For any discrete variables of \(x\), the kernel density estimator's formula is given by  
\[f_{h}(x)=\frac{1}{nh}\sum_{i=1}^{n}K\Big{(}\frac{x-x_{i}}{h}\Big{)} \tag{10}\]  
where \(x_{1},x_{2},\dots,x_{n}\) are samples from an unknown distribution; \(n\) is the sample size; \(K(\cdot)\) is the kernel smoothing function, which uses a normal distribution function in this article; and \(h\) is the bandwidth. The kernel estimator for the cumulative distribution function (CDF), for any real values of \(x\), is given by  
\[F_{h}(x)=\int_{-\infty}^{x}f_{h}(t)\mathrm{d}t=\frac{1}{n}\sum_{i=1}^{n}G\Big{(} \frac{x-x_{i}}{h}\Big{)} \tag{11}\]  
where \(G(x)=\int_{-\infty}^{x}K(t)\mathrm{d}t\). In this article, the smoothing function uses _normal_, \(h\) equals (data maximum - data minimum)/1000. For the fatigue reliability of the girder end (expansion device), the kernel PDF of the data-driven calculated von Mises stresses and their rainflow equivalent amplitudes belonging to the calculated maximum points of the sliding supports, guide rails, rails, and scissor-like connecting rods are shown in Figure 13. Under the diurnal relative transverse displacements, the von Mises stress amplitudes are mainly in the range of 0-3 MPa. The maximum von Mises stress of the sliding support is 100.71 MPa, but only 214 amplitudes exceed 10 MPa in the 2-month study period. According to the fatigue design \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mechanical Responses** & **Maximum Value** & **Maximum Value Location** \\ \hline Transverse deflection of rails & \(-\)1.00069 mm & Rails fastened on the steel sleepers at the fixed side. \\ \hline [PERSON] stress of sliding supports & 107.60 MPa & Contact areas of the sliding support and the fixed box mouth. \\ \hline [PERSON] stress of guide rails & 48.05 MPa & Contact areas of the guide rails and the hanging steel \\  & & \
\begin{tabular}{c} seeper (including the fasteners). \\ \end{tabular} \\ \hline von Mises stress of rails & 88.44 MPa & Contact areas of the rails and the hanging steel sleeper (including the fasteners). \\ \hline von Mises stress of scissor-like connecting rods & 42.72 MPa &
\begin{tabular}{c} Middle rotating shafts of the scissor-like connecting rods connected with the steel sleeper. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Device mechanical responses under \(-\)1 mm of transverse displacement on the fixed side.