# Track-Bridge Interaction of CWR on Chinese Large-Span Bridge of High-Speed Railway
## 2 Research Model of Track-Bridge Interaction
### Research Model Proposed by Scholars  
In 1962, in the Soviet Union, [PERSON] [9] regarded the track as an elastic rod, and 25 m rails were built in each subgrade section, and the longitudinal displacement was restrained at the end of the track. Assuming that the longitudinal resistance of the track is linear, the longitudinal spring is used to simulate the pier stiffness, and the train braking force is loaded on the track. The second-order differential equation (Equation (1)) about the relative displacement of the beam and rail is obtained, and then the longitudinal displacement and internal force of the rail are solved:  
\[-\text{EA}\frac{d^{2}u(x)}{dx^{2}}+ku(x)=q \tag{1}\]  
where \(k\) is the longitudinal resistance coefficient, \(u(x)\) is the longitudinal displacement of the rail at \(x\), \(q\) is the train braking force concentration, and \(EA\) is the product of the rail's Young's modulus and cross-sectional area. In 1974, [PERSON], a Czechoslovak, proposed the quasi-static force analysis method to calculate the distribution of train braking force and traction force in rail and bridge. Based on the linear line resistance, the balance differential equation of track-bridge rail integration was established [10]. After laying CWR on Liuli river bridge on the Beijing-Guangzhou railway for the first time in China, the Research Institute of Ministry of railways, Southwest Jiaotong University, Beijing Jiaotong University, Central South University (Changsha Railway College), and other units have conducted systematic theoretical and experimental studies on the track-bridge interaction [11]. In 1984, [PERSON] simplified the calculation method of rail expansion force and displacement by the virtual beam method according to the principle of structural mechanics [12]. In 1998, [PERSON] collected a large number of research and test data on CWR at home and abroad and introduced a variety of methods for determining the longitudinal resistance of CWR. When calculating the braking force, the maximum braking force ratio is taken as the braking adhesion coefficient (the value is 0.2\(\sim\)0.3, summarized from the domestic and foreign track braking force test data). In addition to full-span loading, it should also load 40 m outside the bridge, and the maximum loading length should not exceed 400 m. In the calculation of temperature force: For the fixed area of the line, it is recommended to refer to the German code and calculate the annual temperature difference relative to the locked rail temperature, and the temperature change of the bridge can be \(\pm\)30 \({}^{\circ}\)C. For the case of setting the rail expansion regulator on the bridge or beam end, the daily temperature difference can be 30 \({}^{\circ}\)C for a concrete bridge and 15 \({}^{\circ}\)C for a steel bridge. When calculating the bending force (longitudinal force generated by the relative displacement of the bridge and the long rail caused by the deflection of the bridge under train load), the vertical live load can be 0.8 UIC, and the loading position is the same as the braking load position. In the calculation model, the rigid arm element is added to the strut and tie model recommended by German code to simulate the height of the beam body, and the vertical bar is used to transfer the vertical force. On this basis, the influence of the simply supported beam span number, substructure stiffness, and line resistance coefficient on the force of the rail and pier on the simply supported beam is analyzed. In addition, the longitudinal force of the long-span continuous steel derivative bridge, steel analysis arch bridge, concrete continuous beam bridge, and steel cable-stayed bridge are calculated [13]. In 2000, in his doctoral dissertation, [PERSON] compiled the static nonlinear finite element software with the tangent stiffness method and the detail increment method and performed parametric analysis on the rail-rail interaction problem of a multi-span, simply supported beam bridge.