## 3 Proposed Calculation Method of the Guardrail Interval Limit Based on the Vehicle Running Attitude
According to the above parameters, when the bogie reaches the maximum twisting angle, the probability of the flange of the lower wheel of the curve hitting the guardrail is closely related to the height difference between the top of the flange and the top surface of the guardrail \(\Delta h^{\prime}\), as shown in Figure 7. A larger value of \(\Delta h^{\prime}\) will cause a longer length of the secant line between the lower wheel flange rolling circle and the top surface of the guardrail \(L_{g}\) and reduce the distance from the endpoint of the secant line to the working edge of the upper rail of the curve, which increases the probability that the flange hits the guardrail. In this case, the maximum value of \(\Delta h^{\prime}\) is calculated using Equation (4). \[\Delta h^{\prime}_{\text{max}}=H_{\text{min}}+\delta_{0}, \tag{4}\]  
where \(\Delta h^{\prime}_{\text{max}}\) is the maximum value of \(\Delta h^{\prime}\) and \(\delta_{0}\) is the height difference from the top surface of guardrail to the top surface of stock rail, which can be set to 12 mm based on the main types of turnouts in China. Then, \(\Delta h^{\prime}_{\text{max}}\) is 39 mm. Based on the abovementioned parameters, the length of the secant line between the wheel flange rolling circle and top surfaces of the rail and guardrail under the worst-case conditions can be calculated using Equation (5). \[\left\{ \begin{array}{l}L_{\text{rmin}}=2\sqrt{{R_{f\text{min}}}^{2}-\left({ R_{f\text{min}}}-\Delta h_{\text{min}}\right)^{2}}\\ L_{\text{gmax}}=2\sqrt{{R_{f\text{min}}}^{2}-\left({R_{f\text{min}}}-\Delta h ^{\prime}_{\text{max}}\right)^{2}}\end{array} \right., \tag{5}\]  
where \(L_{\text{rmin}}\) is the minimum length of the secant line between the wheel flange rolling circle and the top surface of the rail (i.e., 230 mm) and \(L_{\text{gmax}}\) is the maximum length of the  
Figure 5: Relationship between the height difference between the top of the wheel flange to the top surface of the rail \(\Delta h\) and the twisting of the wheelset. Figure 6: Scenario of slight climb of the wheel in the extreme inscribing state. secant line between the wheel flange rolling circle and the top surface of the guardrail (i.e., 365 mm). Based on the above analysis, Figure 8 shows a bogie model under the worst-case conditions. Because the radius of the lead curve, gage-widening segment, and other alignment parameters vary among different types of turnouts, the worst-case attitude also differs for a bogie entering the guardrail opening section, which affects the guardrail interval limit. Based on the established worst-case bogie model and specific parameters for the turnout alignment, the inscribing state can be constructed based on the length of the secant line \(L_{\text{rmin}}\) of flange rolling circles of the upper wheel of the front wheelset and the lower wheel of the rear wheelset. In the inscribing state, when the front wheelset of the bogie is at the end of the buffer segment at the toe end of the guardrail, then the vertical distance from the end of the secant line \(L_{\text{gmax}}\) of the lower wheel flange rolling circle to the working edge of the upper rail of the curve is the necessary guardrail interval limit. This is shown in Figure 9.  
Figure 8: Most adverse bogie model. Figure 7: Relationship between the height difference between the top of the wheel flange to the top surface of the guardrail \(\Delta h^{\prime}\) and the flange hitting the guardrail.