# Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method for Assessment of Theoretical Safety against Freight Wagon Derailment

_{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 ΔQ

_{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}≤ 1.2.

_{jH}is determined using Equation (2):

_{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.

- Y
_{ja}—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).

#### 2.1. Methodology for Theoretical Determination of Leading Forces, Y_{a}, on Axles of Railway Vehicles with Bogies

^{+}), R is the radius of the calculation curve, and σ

_{b}is the current coordinate.

_{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′ or AB″). 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.

- AB—maximum crossing (σ
_{b}= σ = Δ + δ); - AB′—free settling (0 ≤ σ
_{b}≤ δ); - AB″—maximum displacement (σ
_{b}= 0).

- 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):

- 2.
- The centrifugal force is defined using Equation (6):

^{2}]) is the gravitational acceleration, R (in [m]) is the curve radius, 2.s (in [mm]) is the 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):

^{2}]), and W is the wind pressure (in [N/m

^{2}]).

- 4.
- The frictional forces Φ obtained because of the rotation around the pole M are determined using Equation (8):

_{st}is the static vertical load on one wheel, as determined using Equation (9):

_{st}= Q

_{nom}.

- 5.
- The total reaction Y
_{i}from rails on the wheelset i are obtained from the equilibrium conditions ΣY = 0 and ΣM_{M}= 0, according to Equation (10):

_{yi}is the component of force Φ 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.

_{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.

_{1}. From the condition for the considered boundary condition, it follows that the distance from pole M, according to Equation (11), is

_{1}and v

_{1}, which are typical for the limited state. When solving Equation (10), it is possible for Y

_{1}or for v

_{1}to obtain negative values. This indicates that the boundary condition is not valid for the specified track and bogie parameters. In this case, it is necessary to go to Step 3 of the current methodology.

_{1}, and the pole distance will be x = x

_{max}. Therefore, the system of Equation (10) can be solved concerning Y

_{1}and Y

_{2}by setting discrete movement speed values in the specified interval.

_{2}and force Y

_{1}, in this case, can be found by solving Equation (10) under the conditions of Equation (13).

_{2}= 0), then, in Equation (10), there are three unknowns: Y

_{1}, v, and x. In this case, the condition in Equation (14) is relevant:

_{k}, is higher than v

_{2}, it is necessary to build the third zone of the horizontal dynamic calculations, i.e., the zone of maximum displacement. In this case, the condition in Equation (15) is valid:

_{2}and v

_{k}, the full reaction forces, Y

_{1}and Y

_{2}, can be determined.

_{1}and Y

_{2}, of the first- and second-wheel axles of each bogie at different speeds, curve radii, specific track parameters, different wheel loads, different bogie wheel axle distances, and other parameters.

#### 2.2. Methodology for Theoretical Determination of the Horizontal Load Force between the Inner (Non-Attacking) Wheel Y_{ji} of the Investigated Wheel Axle and the Inner Rail

_{ji}is the vertical load force of the inner wheel (index i) on axle j, Q

_{nom}is the nominal vertical load force of the wagon wheels, and Q

_{ja,min}is the minimum vertical force acting on the outer (attacking) wheel of axle j. It is determined in accordance with the methodology given in Section 2.3. of this paper. Q

_{nom}is determined using the ratio of the force from the weight of the wagon Q and the number of wheels of the vehicle N, as given in Equation (17):

#### 2.3. Methodology for Theoretical Determination of the Smallest Value of the Vertical Reaction of the Wheel, Q_{jk,,min}, Calculated during Torsion of the Wagon Frame

_{jk,,min}, allows us to obtain the corresponding maximum value of this parameter, Q

_{jk,,max}. The calculations are carried out in the following sequence:

- 2.
- In accordance with EN 14363 [9], the minimum deflection of the frame Δz* is determined, which should be reached during the real (in situ) testing of the wagon. It is determined via Equation (18), subject to the requirement in Equation (19). In this case, 2a* is valid for wagon frames with pivot distances between 4 and 30 m.

- 3.
- Recalculation of the force ΔF
_{p}from step 1 for loading the wagon frame to achieve the minimum deflection Δz* according to Equation (20):

_{z*}significantly loads the two unilaterally located wheels and significantly less for the other two.

- 4.
- The force ΔF
_{z*}is then transmitted from the lateral support to the side beams of the bogie with a value of ΔF′_{z*max}and ΔF′_{z*min}according to Equations (21) and (22). The corresponding distances, b_{1F}and b_{s}, are shown in Figure 9.

_{z*max}and ΔF′

_{z*min}are distributed between the two axle journals of the overloaded and the two axle journals of the unloaded wheels, with the forces ΔF′

_{z*max}and ΔF′

_{z*min}acting on the first (attacking) wheel axle, defined using Equations (23) and (24):

- 5.
- The minimum value of the wheel reaction, Q
_{jk,,min}, is determined using Equation (27), and the maximum value is evaluated using Equation (28), respectively:

_{nom}is determined via Equation (17).

## 3. Results from the Theoretical Derailment Safety Assessment

- Tare weight of the wagon, 27.5 t;
- Curve radius, R = 150 m;
- Clearance between flanges and rail threads in a straight section of the track, equal to δ = 0.01 m;
- Additional tracks widening in a curved section, δ = 0.002 m (in accordance with the test data of the wagon [30]);
- Coefficient of friction between the rail and the wheel, μ = 0.4;
- Wheel axle distance, a
^{+}= 1.8 m; - Pivot distance (for one wagon section only), a* = 11.995 m;
- Speed of passing through the curve, v = 7 km/h (in accordance with the test data of the wagon [30]);
- Distance between the rolling circles of the two wheels of the same axle, 2b
_{0}= 1.5 m; - Transverse distance between the axle journals, 2b
_{jF}= 2.0 m; - Distance between the side supports on the bogie, 2b
_{s}= 1.7 m; - Overhang of the outer rail, h = 0.15 m;
- Earth acceleration, g = 9.81 m/s
^{2}.

## 4. Discussion

_{1i}, Δz*, Q

_{nom}, and Q

_{jk,,min}. A comparison of the results from tests and calculations is given in Table 3.

_{1i}, were measured in the test on the flat track with a radius of 150 m, while forces Q

_{nom}and Q

_{jk,min}were measured on the twist test rig, as well as Δz* [11,30]. It should be noted that all values of parameters measured in the tests are the average values from different numbers of tests (seven tests on twist rig and three on flat track) conducted. In [30], the measurement uncertainty was determined at 1.4% for vertical forces and 1.2% for displacements and twists. This, along with wagon imperfections caused by production, welding, and other influential factors, explains the deviation from the theoretical assumptions.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Overhang of the outer rail, depending on radius of curve and movement speed [27].

**Table 1.**Additional expansion of the rail track in a curved section depends on the radius of the calculation curve [13].

Radius R (m) | δ (mm) |
---|---|

125–150 | 20 |

150–180 | 15 |

180–250 | 10 |

250–300 | 5 |

Over 300 | 0 |

Parameter | Value | Remark |
---|---|---|

v_{1} | 58.3 km/h | Methodology from Section 2.1. |

Y_{1} = Y_{1a} | 24.718 kN | Methodology from Section 2.1. |

Y_{1i} | −14.024 kN | Methodology from Section 2.2. |

g* | 3.251‰ | Equation (19) |

Δz* | 39 mm | Equation (18) |

ΔF_{p} | 50 kN | The selected load value for the torsional stiffness test [30] |

Δz_{p} | 0.08265 mm | Deflection of the frame under the Load, ΔF _{p}, determined in [31] |

ΔF_{z*} | 23.59 kN | Equation (20) |

ΔF’_{z*}_{,max} | 21.82 kN | Equation (21) |

ΔF’_{z*}_{,min} | 1.769 kN | Equation (22) |

ΔF’_{1z*,max} | 10.909 kN | Equation (23) |

ΔF’_{1z*,min} | 0.885 kN | Equation (24) |

ΔQ_{1,max} | 12.58 kN | Equation (25) |

ΔQ_{1,min} | −0.7862 kN | Equation (26) |

Q_{nom} | 22.48 kN | Equation (17) |

Q_{jk,min} | 21.695 kN | Equation (27) |

Q_{jk,max} | 35.061 kN | Equation (28) |

Parameter | Value from Calculation | Value from Test |
---|---|---|

Y_{1i} | −14.024 kN | −13.5 kN |

Δz* | 39 mm | 40.1 mm |

Q_{nom} | 22.48 kN | 22.10 kN |

Q_{jk,min} | 21.695 kN | 19.81 kN |

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**MDPI and ACS Style**

Stoilov, V.; Slavchev, S.; Maznichki, V.; Purgic, S.
Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons. *Appl. Sci.* **2023**, *13*, 12698.
https://doi.org/10.3390/app132312698

**AMA Style**

Stoilov V, Slavchev S, Maznichki V, Purgic S.
Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons. *Applied Sciences*. 2023; 13(23):12698.
https://doi.org/10.3390/app132312698

**Chicago/Turabian Style**

Stoilov, Valeri, Svetoslav Slavchev, Vladislav Maznichki, and Sanel Purgic.
2023. "Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons" *Applied Sciences* 13, no. 23: 12698.
https://doi.org/10.3390/app132312698