# Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons
## 2 Method for Assessment of Theoretical Safety against Freight Wagon Derailment
The reduced gauge is determined via Equation (4):  
\[\sigma=2s-2d= \Delta+\delta \tag{4}\]  
Figure 3: Representation of bogie moving in curved section of the track. Figure 2: Movement of bogie in curved section of the track. where \(\Delta\) is the total clearance between the flange and the rail, equal to 0.01 m; and \(\delta\) is an additional expansion of the rail track in a curved section depending on the radius of the calculation curve, as determined with data from Table 1[27]. In this way, the calculated value represents the maximum total clearance between the rails and the axle in a curved section of the rail track. In Figure 3, the attacking axle (point A) always contacts the outer rail. Depending on the movement speed and the radius of the curve, the second axle (point B or B\({}^{\prime}\) or B\({}^{\prime\prime}\)) can take one of the following positions:  
* AB--maximum crossing (\(\sigma_{b}=\sigma=\Delta+\delta\));
* AB\({}^{\prime}\)--free settling (\(0\leq\sigma_{b}\leq\delta\));
* AB\({}^{\prime\prime}\)--maximum displacement (\(\sigma_{b}=0\)). When moving in a curve, the following forces act on the bogie (Figure 4):  
1. The transverse force, \(H\), is induced by the centrifugal (\(H_{c}\)) and wind (\(H_{w}\)) forces. It is applied at the mass center of the wagon and is determined via Equation (5): \[H=H_{c}+\ H_{w}\] (5)
2. The centrifugal force is defined using Equation (6): \[H_{c}=P.(\frac{v^{2}}{R.g}-\frac{h}{2.s})\] (6) where \(P\) (in [N]) is the weight of the wagon, \(v\) (in [m/s]) is the movement speed, \(g\) (in [m/s\({}^{2}\)]) is the gravitational acceleration, \(R\) (in [m]) is the curve radius, \(2.s\) (in [mm]) is the \begin{table}
\begin{tabular}{c c} \hline
**Radius \(R\) (m)** & \(\delta\) **(mm)** \\ \hline
125–150 & 20 \\
150–180 & 15 \\
180–250 & 10 \\
250–300 & 5 \\ Over 300 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Additional expansion of the rail track in a curved section depends on the radius of the calculation curve [13]. Figure 4: Forces acting on the bogie while moving in a curve. distance between wheels rolling circles (normal track width 2\(s\) = 1500 mm), and \(h\) (in [mm]) is the overhang of the outer rail determined from the table in Figure 5[27]. 3. The wind force is determined via Equation (7): \[H_{w}=F.W\] (7) where \(F\) is the surface of the wagon on which the wind is acting (in [m\({}^{2}\)]), and \(W\) is the wind pressure (in [N/m\({}^{2}\)]). 4. The frictional forces \(\Phi\) obtained because of the rotation around the pole \(M\) are determined using Equation (8): \[\Phi\ =\mu.N_{st}\] (8) where \(\mu\) is the coefficient of friction between the wheel and the rail, and \(N_{st}\) is the static vertical load on one wheel, as determined using Equation (9): \[N_{st}=\frac{P}{N}\] (9) where \(N\) is the number of wheels. For compliance with European standards, it should be noted that \(N_{st}=Q_{nom}\). 5. The total reaction \(Y_{i}\) from rails on the wheelset \(i\) are obtained from the equilibrium conditions \(\Sigma Y=0\) and \(\Sigma M_{M}=0\), according to Equation (10): \[\left| \begin{array}{l}Y_{1}-Y_{2}-H-2.\Phi.cos\alpha_{1}+2\Phi cos\alpha_{2}=0\\ Y_{1}x+Y_{2}(2l-x)-H(x-l)-2\Phi.r_{1}-2\Phi.r_{2}=0\end{array} \right.\] (10) where \(\Phi_{yi}\) is the component of force \(\Phi\) along the \(y\)-axis, and \(r_{i}\) is the distance from pole \(M\) to the corresponding contact point between the rail and wheel of the \(i\)-th wheel axle. Figure 5: Overhang of the outer rail, depending on radius of curve and movement speed [27]. In Equation (10), there are four unknown terms implicitly set by the centrifugal forces: \(Y_{1}\), \(Y_{2}\), \(x\), and speed \(v\). Therefore, the total reactions, \(Y_{1}\) and \(Y_{2}\), are determined according to the following methodology, and the graphical representation is shown in Figure 6.  
Step 1. It is assumed that the bogie is in the limit state between \"maximum overshoot\" and \"free settling\". It is possible at a precisely determined but unknown speed, \(v_{1}\).