## 3 Proposed Calculation Method of the Guardrail Interval Limit Based on the Vehicle Running Attitude  
The guardrail interval limit ensures that the wheels of a vehicle do not hit the opening section when it enters the guardrail section. Actually, the existing calculation method is a one-dimensional model, which does not consider the influence of the vehicle running attitude and wheelset angle of attack, and thus cannot account for the worst-case conditions. The actual worst-case conditions are as follows: when the bogie with the worst-case parameters passes the turnout in the side-facing direction, it enters the opening section at the toe end of the guardrail in the free inscribing state. In view of this, a two dimensional model is developed in this section to calculate the guardrail interval limit, which can consider the worst-case conditions. First, the most adverse bogie parameters were studied. The existing calculation method for the guardrail interval limit indicates that a small wheel flange thickness and wheelset inner distance increase the likelihood that the wheel hits the guardrail. Therefore, the wheel flange thickness d and wheelset inner distance T were still set to the minimum values of 22 and 1350 mm, respectively. The possible reduction of the wheelset inner distance due to the vehicle load \(\varepsilon\) was considered, and it was set to a value of 2 mm. When a vehicle is running in the lead curve section, decreasing the bogie axis distance increases the angle of twisting when it reaches the free inscribing state, and the wheel becomes more likely to hit the guardrail. Therefore, the bogie axis distance Q was set to the minimum value of 1.8 m, which matches the actual dimensions of railway tracks. In the free inscribing state, the front wheelset flange of the bogie is in contact with the upper rail of the curve, while the rear wheelset flange is in contact with the lower rail of the curve. The radius of the wheel flange rolling circle \(R_{f}\) and the height difference between the top of the wheel flange and top surface of the rail \(\Delta h\) determine the twisting angle of the bogie in the free inscribing state, as shown in Figure 5. A small \(R_{f}\) and \(\Delta h\) decrease the length of the secant line between the flange rolling circle and the top surface of the rail \(L_{r}\), which weakens the ability of the rail to limit the twisting of the wheelset and increases the twisting angle of the bogie in the free inscribing state. Therefore, the minimum value of \(R_{f}\) can be calculated using Equation (2):  
\[R_{f\text{min}}=R_{\text{min}}+H_{\text{min}}, \tag{2}\]  
where \(R_{f\text{min}}\) is the minimum value of \(R_{f}\), \(R_{\text{min}}\) is the minimum value of the nominal rolling circle radius of the wheel (i.e., 420 mm), and \(H_{\text{min}}\) is the minimum value of the wheel flange height (i.e., 27 mm). Therefore, \(R_{f\text{min}}\) is 447 mm. When a vehicle passes through a small-radius curve at a high speed, the bogie may experience a slight flange climb due to centrifugal forces after reaching the free inscribing state, as shown in Figure 6. This will further reduce the height difference between the top of the wheel flange and the top surface of the rail \(\Delta h\) and further increase the twisting angle of the bogie, which is defined as the extreme inscribing state. In this case, the minimum value of \(\Delta h\) should be calculated using Equation (3) to consider the slight flange climb of the wheels in the extreme inscribing state. \[\Delta h_{\text{min}}=H_{\text{min}}-\sigma, \tag{3}\]  
where \(\Delta h_{\text{min}}\) is the minimum value of \(\Delta h\) and \(\sigma\) is the flange climb of the wheel in the extreme inscribing state, which can be set to 12 mm. Then, \(\Delta h_{\text{min}}\) is 15 mm.