# Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons
## 2 Method for Assessment of Theoretical Safety against Freight Wagon Derailment  
In accordance with EN 14363:2019 [9], Section 6.1, Method 2 (paragraph 6.1.5.2), when analyzing the safety criterion against derailment, Equation (1) must be considered:  
\[\left(\frac{Y}{Q}\right)_{ji}=\frac{Y_{ja}}{Q_{jk,\min}+\Delta Q_{jH}};\qquad \frac{Y_{ja}}{Q_{jk,\min}+\Delta Q_{jH}}\leq\left(\frac{Y}{Q}\right)_{\lim} \tag{1}\]  
where \(Y_{ja}\) is the reaction of the rail at its contact with the attacking wheel, \(Q_{jk,\min}\) is the lowest value of the vertical reaction of the wheel calculated when the frame of the wagon is twisted, and \(\Delta Q_{jH}\) is a load on the wheel from the moment of the forces acting on the 2 wheels of the examined wheel axle (Figure 1). The value of \((Y/Q)_{\lim}\) is equal to 1.2, and the railway vehicle is safe against derailment if \((Y/Q)_{ja}\leq 1.2\). The component \(\Delta Q_{jH}\) is determined using Equation (2):  
\[\Delta Q_{jH}=(Y_{ja}+Y_{ji}).\frac{h}{2b_{0}} \tag{2}\]  
where \(Y_{ja}\) is the total reaction of the rail at its contact with the attacking (outer) wheel; \(h\) is the effective height above the rail of the journal box suspension (for the most commonly used bogie, Y25, \(h=365\) mm is assumed); \(Y_{ji}\) is horizontal load force between the inner (non-attacking) wheel of the examined axle and the inner rail (Figure 1); \(2b_{0}\) is the nominal transverse distance between the contact points of the wheels (\(2b_{0}=1500\) mm is assumed); \(j\) is index (number) of the examined axle; \(a\) is index (number) of the outer wheel; and \(i\) is index of the inner wheel. For the theoretical assessment of the criterion from Equation (1), it is necessary to apply appropriate methods for the theoretical determination of the following parameters:  
* \(Y_{ji}\)--the total reaction of the rail in contact with the attacking (outer) wheel. The parameter is involved in Equations (1) and (2). * \(Y_{ji}\)--the horizontal load force between the inner (non-attacking) wheel of the examined track axle and the inner rail. The parameter is involved in Equation (2). * \(Q_{jk,\min}\)--the lowest value of the vertical reaction of the wheel calculated when the frame of the wagon is twisted. The parameter is involved in Equation (1). The methods for determining these three important parameters are given in the next three subsections and represent the main goal of our paper. Figure 1: Forces acting in wheel–rail contact. Methodology for Theoretical Determination of Leading Forces, \(Y_{a}\), on Axles of Railway Vehicles with Bogies  
In [27], a theoretical method for determining the total reaction of the rail on the wheels of a bogie is proposed. Briefly, the method consists of the following: when moving on a curved section of the rail track, the wagon performs two movements--translational and rotational. The rotation occurs around the center of rotation, \(M\) (Figure 2), characterized by the pole distance, \(x\), which can be determined using Equation (3):  
\[x=\;l+\frac{R.\sigma_{b}}{2.l} \tag{3}\]  
where \(2.l\) is the distance between wheelsets of the bogie (for compliance with European standards, it should be noted that \(2.l=2a^{+}\)), \(R\) is the radius of the calculation curve, and \(\sigma_{b}\) is the current coordinate. The current coordinate, \(\sigma_{b}\), depends on the position of the bogie when passing a curved section of the track. In Figure 3, the bogie is represented by section AB (AB\({}^{\prime}\) or AB\({}^{\prime\prime}\)). For this purpose, the transverse dimensions of the track with gauge \(2s\) and of the bogie are reduced by the constant amount, \(2d\), defining the transverse distance between the bases of the wheel flanges of the same axle. The bases of the flanges of the two wheels merge and are represented in Figure 3 as points A and B (B\({}^{\prime}\) and B\({}^{\prime\prime}\)). The same points depict the attacking (A) and the non-attacking (B) axle, respectively.