# A New Top-Mounted Shear-Hinge Structure Based on Modal Theory and Rubber-Pad Damping Theory
## 3 Finite-Element Modeling
### TMSH Strength-Check Calculation  
To verify the load-carrying capacity of the TMSH structure and ensure that it conforms to the ultimate load range under the most unfavorable operating conditions, the Mises stress criterion [32] of the fourth strength theory, i.e., the distortional energy density theory, is introduced to calibrate the shear-hinge strength, which is mainly calculated from the principal stresses in three directions using Equation (6). \[\sigma=\sqrt{\frac{1}{2}\Big{[}(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma _{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\Big{]}} \tag{6}\]  
In the above formula, \(\sigma\) represents the Mises stress, and \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) represent the first, second, and third principal stresses, respectively. Strength theory suggests that if the distortion energy density reaches a certain limit related to the material properties, the material will yield. The establishment of the floating-slab shear-hinge refinement model is shown in Figure 7. To save computing resources and ensure the authenticity of the calculated structure as far as possible, the validation model establishes a three-link refined floating-slab structure, with TMSHs and SMSHs using refined models, and rails simulated using Timoshenko Beam B21. In order to prevent rail deformation from influencing the results, rail  
Figure 6: TMSH constraints and loads during modal analysis. stiffness is calculated according to the material properties, and fasteners and steel-spring isolators are simulated using Connector, as shown in Table 4.  
When a train runs on a floating slab, the internal forces on the track components caused by wheel loads can be equated to those caused by static loads. Taking Metro A as the standard, each bogie is equipped with two-wheel pairs for a total of four wheels, with a wheel spacing of 2.5 m and a single axle weight of 16 t. According to the description given in Appendix F (Track-Strength Verification) in the Code for the Design of Railway Tracks [33], the dynamic wheel load is expressed as the maximum possible value of the equivalent static load. The equivalent vertical static load is calculated as follows:  
\[\left\{ \begin{array}{cc}\upsilon\leq\frac{120\;\mathrm{km}}{h}&P_{d}=P_{o}( 1+\alpha)\\ \frac{120\;\mathrm{km}}{h}<\upsilon\leq\frac{160\;\mathrm{km}}{h}&P_{d}=P_{o}( 1+\alpha)(1+\alpha_{1})\end{array} \right. \tag{7}\]  
where \(P_{d}\) is the vertical equivalent static load of the wheel acting on the rail, \(P_{o}\) is the static wheel weight, \(V\) is the travel speed, and \(\alpha,\;\alpha_{1}\) is the speed coefficient. \[\alpha=0.6\upsilon/100 \tag{8}\]  
\[\alpha_{1}=0.3\Delta\upsilon_{1}/100 \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline
**Component** & **Parameter** & **Value** \\ \hline Prefabricated floating slab & Floating-slab size & 4.776 m \(\times\) 2.7 m \(\times\) 0.35 m \\  & Young’s modulus & 34,500 MPa \\  & Density & 2500 kg/m\({}^{3}\) \\  & Poisson’s ratio & 0.2 \\ Steel-spring isolator & Vertical stiffness & \(60\times 10^{6}\) N/m \\  & Lateral stiffness & \(7\times 10^{8}\) N/m \\  & Vertical damping & \(12,000\) N \(\times\) s/m \\  & Layout distance & 1.2 m \\ Fastener & Vertical stiffness & \(3\times 10^{7}\) N/m \\  & Lateral stiffness & \(1\times 10^{7}\) N/m \\  & Vertical damping & \(10,000\) N \(\times\) s/m \\ Rail & Mass & 60.64 kg/m \\  & [PERSON]’s modulus & \(2.01\times 10^{9}\) Pa \\ \hline \end{tabular}
\end{table} 
Table 4: Floating-slab model parameters. Figure 7: Strength-verification model of TMSH. The average speed of operation on the straight section is 80 km/h, and the axle weight of the train is multiplied by 1.48 coefficients. When the bearing capacity of the floating slab is checked, the axle weight of the train is also multiplied by 1.4 sub-coefficients. The gravity coefficient is taken as 9.8 m/s\({}^{2}\) and substituted into Equations (7)-(9) to calculate \(P_{d}=162.4\) (kN).