## 4 Girder End Reliability under the Relative Transverse Displacements
### Mechanical Characteristics of the Expansion Device under Relative Transverse Displacements  
A three-dimensional FE model of the expansion device is established using ANSYS to analyze the mechanical behavior of the expansion device under relative transverse displacements. The rails, the steel sleepers, the expansion boxes and the sliding support in the expansion boxes, the guide-rails, and the scissor-like connecting rod are established using the SOLID187 element with the corresponding parameters for steel, while the concrete sleepers are established using the SOLID187 element with the parameters for concrete. According to the design documents and the engineering experience, the tensile yield strength and tensile strength of the steel used on the expansion device are 370 and 510 MPa (Chinese Q370 qE steel), and the tensile yield strength and tensile strength of the steel rail are 460 and 880 MPa (Chinese U71 Mn steel); the elastic modulus of all steel materials is set to \(2.06\times 10^{5}\) N/mm\({}^{2}\) before yielding, and is set to 0 N/mm\({}^{2}\) after yielding; the elastic modulus of concrete sleeper is set to \(3.6\times 10^{4}\) N/mm\({}^{2}\); the damping ratio of the steel material is set to 0.02, while the damping ratio of the concrete material is set to 0.05. The dynamic effects of the operating loads are small for the expansion device, such that the stiffness and damping of all of the fasteners and the sliding plates are neglected. All of the fasteners are modeled by coupling the degrees of freedom (DOF) in the corresponding directions, while all of the sliding plates are modeled by releasing the DOF in the corresponding directions. The expansion box and sliding support, the sleeper and rail, and the sleeper and guide-rail at the fixed side are coupled in the longitudinal (X), transverse (Y), and vertical (Z) directions. The expansion box and sliding support, the sleeper and rail, and the sleeper and guide-rail at the free side are coupled in the Y and Z directions. The scissor-like connecting rods are coupled with the steel sleepers in the X, Y, and Z directions at the connected nodes, allowing the connecting rods to rotate around the Y axis. Figure 12 shows the scalar displacement contours and the von Mises stress contours of the expansion device under 1 mm relative transverse displacement using three types of loading methods. Figure 12a presents the results under \(-0.5\) mm of displacement on the fixed side and \(+0.5\) mm of displacement on the free side; Figure 12b displays the results under \(1\) mm of displacement on the free side; and Figure 12c,d show the results under \(-1\) mm of displacement on the fixed side. In the FE model of the expansion device (Figure 12), the left side is fixed (main bridge side), and the right side is free (ramp bridge side). The displacement loads are applied at the nodes of the expansion box bottom and the concrete sleeper bottom. The FE model of the expansion device is symmetric along the upstream and downstream directions (the transverse direction is the Y direction), and all three  
loading methods can represent the typical situation of the expansion device under relative transverse displacements. The three loading methods used to generate the 1-mm relative transverse displacement generate different displacement contours but nearly the same stress contours. As shown in Figure 12, the displacement maximum under 1 mm of displacement on the free side is 0.58% larger than that under \(-1\) mm of displacement on the fixed side; when subjected to 1 mm of relative transverse displacement, the rails, guide-rails, and scissor-like connecting rods, which are connected to the steel sleepers, exhibit relatively higher displacements, and the contact areas of the hanging steel sleepers with the rails, guide rails, sliding supports, and scissor-like connecting rods will have relatively higher stresses. However, the stress of the entire structure remains at a low level comparing with the yield strength of steel materials.