# Analysis of Lateral Forces for Assessment of Safety against Derailment of the Specialized Train Composition for the Transportation of Long Rails

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## Abstract

**:**

## 1. Introduction

- Reducing the share of road transport, with 50% of all medium-distance transport being carried out through rail and water transport;
- By 2030, 30% of freight transport over 300 km should be carried out through rail or water transport, and by 2050, over 50%;
- By 2050, most of the passenger transport over more than 300 km should be carried out through rail transport.

- The derailment phenomenon has a distinctly stochastic character. It depends on many factors, which at a certain moment in a complex combination can lead to its occurrence. Current research addresses the issue of derailment from the point of view of theory enrichment by accounting for one or more additional factors that could influence the occurrence of the adverse event. This effort to clarify the mechanism of the process is undeniably positive for two main reasons: first, although the wagons commissioned into operation meet the requirements of the regulatory documents, in practice, railway accidents caused by derailments often occur; and second, it could lead to an adjustment of national or international standards, which would reduce the likelihood of derailment.
- Although the proposed criteria are generally valid for the cases of examination of railway vehicles in a loaded or empty state, the safety assessment against derailment is carried out on an unloaded wagon. The reasons for this are logical—with a full wagon, the vertical force Q increases significantly (especially for freight wagons up to 3–4 times), while the lateral force Y increases less or in the same order under the influence of centrifugal forces.
- Currently, there are regulatory documents that, regardless of their shortcomings, should be respected to reduce the likelihood of derailment.
- In most analyzed publications and in all normative documents related to the derailment process, the evaluation is reduced to the application of Nadal’s criterion [21]. The limit value of the ratio (Y/Q) at a coefficient of friction equal to 0.36 was determined via Nadal’s criterion using Equation (1):

- Determination of the lateral forces acting on the central bearings of the bogies, obtained because of the elastic deformations of the long rails transported.
- Application of the method for theoretical assessment of safety against derailment, considering the permissible vertical load and the total values of lateral forces resulting from the movement of the train composition in a curved section of the track and from the elastic deformation of the transported rails.

- Development of a computational model for the research object;
- Numerical solution for determining the interaction forces between the deformed rails and the vertical columns of the wagon;
- Determination of the additional forces acting on the central bearings of the bogies.

## 2. Computational Model

- The composition consists of nine wagons; the rails are located on support frames (Figure 2b) and are grouped in separate sections of five rails each.
- It is assumed that the first four wagons have entered the curve.

_{0}defines the position of the fixing block in the middle wagon, and the remaining elements (A

_{1}to A

_{12}) define the position of the support frames. The front wagon is equipped with two support frames (A

_{11}and A

_{12}) in accordance with the requirements of [29]. The locomotive is not illustrated in Figure 3.

- The friction between the rails is neglected due to the peculiarities of the investigated structure.
- In the wagon with the fixing blocks, it is assumed that the rails are immovably fixed, i.e., in the fixing block, it is assumed that we have a fixed beam (Figure 5a);
- At the locations of the supports A
_{i}, it is assumed that unknown forces caused by the deformation of the beam act on the lateral direction (y axis) as shown in Figure 5b. The forces are directed perpendicular to the beam axis (x axis); - Because of the large dimensions and deformations of the system and considering that the task is statically indeterminate, it is solved with a linear analysis method for the approximate determination of the forces in the supports. The obtained results contain some inaccuracies, which can be estimated by the studies presented in [25,27];
- When moving in a curve with radius R = 150 m, the speed of the train composition is small; therefore, the inertial forces are neglected.

**Figure 5.**Calculation model of the fixed beam and lateral forces acting on it: (

**a**) fixed beam in fixing block; (

**b**) unknown forces acting on the lateral direction.

_{i}(Figure 5). The type of finite element is shown in Figure 6 [32].

_{i}is the length of the element and E is the modulus of elasticity.

_{i}and correspondingly different stiffness matrices. Its position is described by 2n + 2 global coordinates (Figure 7a). The forces act on the nodes, and the moment acts in node A

_{0}(Figure 7b).

_{i}along the y-axis direction) and elements of the matrix [K]. It is determined using Equation (9):

_{A0}and F

_{A0}can be determined from Equation (6) or from the two equilibrium conditions of the entire beam.

## 3. Results from the Numerical Calculations

#### 3.1. Results from Calculation of Lateral Forces between Rails and Vertical Columns of the Wagon

_{i}(and the displacements ${u}_{iy}$, respectively) along the y-axis, it is assumed that the composition is in a curve with radius R (Figure 2a). It is assumed that the points A

_{1}, A

_{2}, A

_{4}, A

_{5}, A

_{7}, A

_{8}, A

_{10}and A

_{11}lie on the theoretical arc of a circle because that is where the central bearings of the wagons are positioned. The remaining points A

_{3}, A

_{6}, A

_{9}and A

_{12}are located on the chords between the above points (Figure 3). Their position is determined by the line connecting the central bearings of the wagon. Point A

_{1}is located at the beginning of the curve.

_{0}–A

_{1}, and its direction relative to the curve is determined by the two central bearings of the wagon with the fixing blocks (Figure 2a).

_{Ai}and the moment M

_{A0}is carried out in the MATLAB program environment using the displacements from Table 1 (third column). Table 1 (column 5) also contains the obtained values for force reactions in points A

_{i}. It should be noted that the values are obtained under the condition that there is no horizontal clearance between the rails and the support stanchions, i.e., ∆h = 0 (Figure 2d).

**Table 1.**Coordinates of points A

_{i}along the bending line and values of reaction forces (and moment M

_{A0}) in points A

_{i}without horizontal clearance between the rails and the support stanchions (∆h = 0).

Point | Coordinate x (m) | Coordinate y = u_{iy} (m) | Reaction Identifier (Unit) | Value |
---|---|---|---|---|

A_{0} | 0 | 0 | M_{A0} (kN·m) | −20.9627 |

A_{0} | 0 | 0 | F_{A0} (kN) | −20.9627 |

A_{1} | 3.000 | 0 | F_{A1} (kN) | 15.4067 |

A_{2} | 8.190 | −0.168 | F_{A2} (kN) | 28.7388 |

A_{3} | 12.690 | −0.525 | F_{A3} (kN) | −45.0839 |

A_{4} | 17.190 | −0.883 | F_{A4} (kN) | 22.1165 |

A_{5} | 22.380 | −1.540 | F_{A5} (kN) | 21.0936 |

A_{6} | 26.880 | −2.320 | F_{A6} (kN) | −42.6837 |

A_{7} | 31.380 | −3.100 | F_{A7} (kN) | 21.6934 |

A_{8} | 36.570 | −4.241 | F_{A8} (kN) | 20.7446 |

A_{9} | 41.070 | −5.436 | F_{A9} (kN) | −43.0170 |

A_{10} | 45.570 | −6.631 | F_{A10} (kN) | 25.6134 |

A_{11} | 50.760 | −8.245 | F_{A11} (kN) | 4.9837 |

A_{12} | 56.260 | −10.200 | F_{A12} (kN) | −8.6432 |

_{3}, A

_{6}and A

_{9}). This is logical and is dictated by the fact that the clearance ∆h is not provided in the model, i.e., ∆h = 0 (Figure 2d). To increase the adequacy of the model, a limited horizontal displacement of the points was allowed due to the available constructively provided clearances between the rails and the support frames. With successive iterations based on a possible displacement of the bending line in the support points, new results for the y-coordinates of points A

_{i}and values for the force reactions in points A

_{i}were obtained and are presented in Table 2.

_{0}and A

_{1}of the wagon with fixing block and at points A

_{11}and A

_{12}of the front (closest to the locomotive) wagon.

**Table 2.**Adjusted coordinates of points A

_{i}along the bending line and values of reaction forces (and moment M

_{A0}) in points A

_{i}with horizontal clearance between the rails and the support stanchions (∆h = 90 mm).

Point | Coordinate x (m) | Coordinate y = u_{iy} (m) | Reaction Identifier (Unit) | Value |
---|---|---|---|---|

A_{0} | 0 | 0 | M_{A0} (kN·m) | −21.8257 |

A_{0} | 0 | 0 | F_{A0} (kN) | −21.8257 |

A_{1} | 3.000 | 0 | F_{A1} (kN) | 22.8997 |

A_{2} | 8.190 | −0.1436 | F_{A2} (kN) | −1.0583 |

A_{3} | 12.690 | −0.4350 | F_{A3} (kN) | 1.3180 |

A_{4} | 17.190 | −0.8733 | F_{A4} (kN) | −1.0934 |

A_{5} | 22.380 | −1.5400 | F_{A5} (kN) | −1.1026 |

A_{6} | 26.880 | −2.2530 | F_{A6} (kN) | 1.1627 |

A_{7} | 31.380 | −3.1000 | F_{A7} (kN) | −0.0952 |

A_{8} | 36.570 | −4.2410 | F_{A8} (kN) | −0.2744 |

A_{9} | 41.070 | −5.3690 | F_{A9} (kN) | −0.4341 |

A_{10} | 45.570 | −6.6274 | F_{A10} (kN) | 1.2717 |

A_{11} | 50.760 | −8.2450 | F_{A11} (kN) | 4.8339 |

A_{12} | 56.260 | −10.1100 | F_{A12} (kN) | −5.6022 |

#### 3.2. Results from Calculation of Additional Lateral Forces in Wheel–Rail Contact

_{0}and A

_{1}of the wagon with fixing block with their respective values from Table 2 are applied to the wagon, as shown in Figure 9.

_{11}and A

_{12}of the front wagon with their respective values from Table 2 are applied to the wagon, as shown in Figure 10.

_{ri}is trivial, and their values are given in Table 3.

## 4. Assessment of Safety against Derailment

- The vertical load from the weight of 60 (4 rows × 15 rails per row) rails of type 60E1 with a length of 120 m (on the entire train composition) was applied;
- The horizontal forces from Table 3 are applied in the central bearings of the wagon.

- The tare weight of each wagon is 21 tons;
- The distance between pivots is 9000 mm;
- The maximum load capacity is 69 tons;
- The maximum permissible load capacity when transporting rails is 0.8 × 69 tons = 49.6 tons according to [29];
- The wagon gauge is G1;
- The weight of two fixing blocks at the middle wagon is 2 × 2.5 tons = 5 tons;
- The weight of one support frame is 1 ton.

- The wagon equipped with fixing block to limit the longitudinal movements of the rails during transportation—the reason for this is that this is the wagon with the maximum tare weight (taking into account the weight of the installed frames) and in it, the lateral forces F
_{A0}and F_{A1}(Table 2), respectively, and the lateral force H_{ri}have a maximum values; - Front wagon with protective walls. It has a reduced weight of the payload, which is placed asymmetrically on the wagon;
- Intermediate wagon (heaviest loaded wagon of all the wagons discussed above).

Criterion | Wagon with Fixing Block | Front Wagon | Intermediate Wagon | Empty Wagon |
---|---|---|---|---|

${\left(\frac{Y}{Q}\right)}_{ja}\text{\u2264}1.08$ | 1.0698 | 0.8024 | 0.8295 | 0.8268 |

## 5. Conclusions

- Four different types of train composition wagons were considered: a wagon with fixing blocks, front wagon, intermediate wagon, and an empty wagon. The analysis of the results shows that the wagon with fixing blocks is most at risk of derailment. The obtained value (Table 4) fully corresponds to the requirements of the normative documents not only for real tests (<1.2), but also for theoretical studies (<1.08).
- The front and intermediate wagons have criterion values very close to that of the empty wagon. This shows that the emerging horizontal elastic forces do not significantly influence the derailment process.
- The developed model allows us to evaluate the influence of individual factors like pivot distance, own weight, number of rails, movement speed, overhang of the outer rail of the track, friction coefficient, etc., on the safety against derailment criterion.
- The transportation of long rails can be carried out with wagons that have different parameters (length, base, width, own weight, load capacity, etc.). After analyzing the mentioned parameters, the authors propose the most efficient version of the specialized train composition for 120 m long rails. When using another modification in the wagons or another rail length, the mentioned efficiency cannot be guaranteed.
- The obtained results show that the transportation of long rails with specialized train composition can be realized on four levels. This will significantly increase the efficiency of operators when delivering long new rails.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Train composition for the transport of long rails in a curve: (

**a**) middle view; (

**b**) front view (source: personal archive).

**Figure 2.**Train composition for transportation of long rails: (

**a**) composition in the curve; (

**b**) middle wagon with frames; (

**c**) load/pressure forces in the fixing block; (

**d**) movement limitations in support frames.

**Figure 6.**Finite beam element [28].

**Figure 7.**Discretized model of the beam: (

**a**) with defined nodes coordinates; (

**b**) with defined loads in nodes.

Wagon with Fixing Block | Front Wagon | Unit | Value |
---|---|---|---|

H_{r1} | - | (kN) | 4.84 |

H_{r2} | - | (kN) | 5.94 |

- | H_{r3} | (kN) | 2.62 |

- | H_{r4} | (kN) | 3.42 |

**Table 4.**Results from calculation of parameters used for assessment of safety against derailment of wagon with fixing block.

Parameter Description | Symbol/Equation | Value |
---|---|---|

Total reaction force of the attacking wheel of wheelset ^{1} | Y_{1a} ^{1} | 122.814 kN |

Total reaction force of the non-attacking wheel of wheelset ^{1} | Y_{1i} ^{1} | −70.732 kN |

Minimum deflection of the bogie frame to be reached during tests of the wagon [28] | $\mathsf{\Delta}{z}_{\mathrm{jk}}^{+}={\mathrm{g}}_{\mathrm{lim}}^{+}.2{a}^{+}$ | 0.0076 m |

Twist of the bogie frame during test (2a^{+} is distance between wheelsets) [28] | ${g}_{\mathrm{lim}}^{+}=7-5/2{a}^{+}$ | 4.2222‰ |

Minimum deflection of the wagon frame to be reached during tests of the wagon [28] | $\mathsf{\Delta}z*={g}_{\mathrm{lim}}^{*}.2a$ | 33.2 mm |

Twist of the wagon frame during test (2a is the pivot distance) [28] | ${g}_{\mathrm{lim}}^{*}=\frac{15}{2a}+2$ | 3.6483‰ |

Load force during torsion test | $\mathsf{\Delta}{F}_{\mathrm{p}}$ | 10 kN |

Deflection of the wagon frame under ΔF_{p} | $\mathsf{\Delta}{z}_{\mathrm{p}}$ | 88.22 mm |

Force needed to achieve Δz* | $\text{\u2206}{F}_{\mathrm{z}*}=\frac{\text{\u2206}{F}_{\mathrm{p}}.\text{\u2206}{z}^{*}}{\text{\u2206}{z}_{\mathrm{p}}}$ | 3.763 kN |

Maximum force additionally loading bogie frame under ΔF_{z*} | $\text{\u2206}{{F}^{\text{\u2032}}}_{\mathrm{z}*,\mathrm{max}}=\frac{\text{\u2206}{F}_{\mathrm{z}*}.({b}_{1\mathrm{F}}+{b}_{\mathrm{s}})}{2{b}_{1\mathrm{F}}}$ | 3.481 kN |

Minimum force additionally loading bogie frame under ΔF_{z*} | $\text{\u2206}{{F}^{\text{\u2032}}}_{\mathrm{z}*,\mathrm{min}}=\text{\u2206}{F}_{\mathrm{z}*}-\text{\u2206}{F\text{\u2019}}_{\mathrm{z}*,\mathrm{max}}$ | 0.282 kN |

Maximum force overloading first wheel | $\text{\u2206}{{F}^{\text{\u2032}}}_{1\mathrm{z}*,\mathrm{max}}=\frac{\text{\u2206}{{F}^{\text{\u2032}}}_{\mathrm{z}*,\mathrm{max}}}{2}$ | 1.7405 kN |

Minimum force overloading first wheel | $\text{\u2206}{{F}^{\text{\u2032}}}_{1\mathrm{z}*,\mathrm{min}}=\frac{\text{\u2206}{{F}^{\text{\u2032}}}_{\mathrm{z}*,\mathrm{min}}}{2}$ | 0.141 kN |

Additional maximum vertical reaction force in wheels (Equation (25) of [36]) | $\text{\u2206}{Q}_{1,\mathrm{max}}$ | 2.007 kN |

Additional minimum vertical reaction force in wheels (Equation (25) of [36]) | $\text{\u2206}{Q}_{1,\mathrm{min}}$ | −0.125 kN |

Nominal vertical force loading the wheels | ${Q}_{\mathrm{nom}}=Q/N$ | 93.401 kN |

Minimum vertical wheel reaction force | ${Q}_{\mathrm{jk},\mathrm{min}}={Q}_{\mathrm{nom}}+\text{\u2206}{Q}_{1,\mathrm{min}}$ | 93.276 kN |

Minimum vertical wheel reaction force | ${Q}_{\mathrm{jk},\mathrm{max}}={Q}_{\mathrm{nom}}+\text{\u2206}{Q}_{1,\mathrm{max}}$ | 95.408 kN |

^{1}Calculated according to methodology described in [36].

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**MDPI and ACS Style**

Stoilov, V.; Sinapov, P.; Slavchev, S.; Maznichki, V.; Purgic, S.
Analysis of Lateral Forces for Assessment of Safety against Derailment of the Specialized Train Composition for the Transportation of Long Rails. *Appl. Sci.* **2024**, *14*, 860.
https://doi.org/10.3390/app14020860

**AMA Style**

Stoilov V, Sinapov P, Slavchev S, Maznichki V, Purgic S.
Analysis of Lateral Forces for Assessment of Safety against Derailment of the Specialized Train Composition for the Transportation of Long Rails. *Applied Sciences*. 2024; 14(2):860.
https://doi.org/10.3390/app14020860

**Chicago/Turabian Style**

Stoilov, Valeri, Petko Sinapov, Svetoslav Slavchev, Vladislav Maznichki, and Sanel Purgic.
2024. "Analysis of Lateral Forces for Assessment of Safety against Derailment of the Specialized Train Composition for the Transportation of Long Rails" *Applied Sciences* 14, no. 2: 860.
https://doi.org/10.3390/app14020860