# Mechanism of Sleeper–Ballast Dynamic Impact and Residual Settlements Accumulation in Zones with Unsupported Sleepers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of the Literature

^{2}when the rail roughness was approximately 2 mm or more. The effect of unsupported sleepers on dynamic characteristics of ballast beds was studied using the DEM of a ballast bed in [38]. The influence of unsupported sleepers and their number on the dynamic response of the ballast bed was analysed. The results show that the unsupported sleeper will redistribute the contact forces of ballast particles so that the number of strong force chains in the ballast will decrease, and the number of strong force chains in the adjacent ballast sleepers will increase.

## 3. Experimental Measurements and Analysis of the Void Dynamic Behaviour in a Void Zone

## 4. Numerical Simulation of the Dynamic Impact

^{2}, fastening foundation coefficient 200 MN/m

^{2}, fastening damping 185 kN·s/m

^{2}, sleeper meter mass 300 kg/m, sleeper layer bending stiffness 0.01 MN·m

^{2}, sleeper–ballast foundation coefficient 600 MN/m

^{2}, sleeper–ballast damping 75 kN·s/m

^{2}, ballast meter mass 500 kg/m, ballast–subgrade foundation coefficient 200 MN/m

^{2}and ballast–subgrade damping 680 kN·s/m

^{2}. The vehicle parameters are as follows: car body mass for 1 wheel 7000 kg, unsprung wheel mass 400 kg, contact stiffness 2400 MN/m, contact damping 155 kNs/m, suspension stiffness ${k}_{v}$ = 0.6 MN/m and suspension damping 25 kNs/m.

^{2}, the maximal wheel acceleration is about 10 m/s

^{2}. Two acceleration zones are visible; one represents the void closing, and the other represents the void opening. The maximal wheel acceleration is shifted in time relative to the maximal acceleration of the sleepers.

## 5. Analysis of the Influence of the Sleeper and Void Size on Impact Loading, Acceleration and Velocity

## 6. Analytic Explanation of the Void Impact Mechanism

## 7. Estimation of Residual Settlement Intensity of the Ballast Layer in the Void Zone

- The well-supported mean part of the sleeper with the mean loading of about 205 N/particle under the sleeper and about 127 N/particle over the subgrade;
- The sleeper sides 0.25–0.5 m with the particle loading breakdown to 145–162 N/particle under the sleeper and about 94–101 N/particle over the subgrade;
- Zones outside the sleeper that carry not more than 10% of the loading.

## 8. Discussion

## 9. Conclusions

- The void zones are characterised by impact interactions that appear even for low-weight trains and velocities.
- The sleeper–ballast impact appears due to void closing with a time shift before the wheel reaches the void zone.
- The impact accelerations of the sleeper at a void zone are more than 2 times higher than the corresponding wheel and rail accelerations for the considered lightweight loading and low-velocity case.
- The ballast loading in the void zone consists of different loading patterns of impact and quasistatic loading that depend on the ballast support position along the void zone.
- The ballast impact increases with the increase in the void depth up to a certain depth value that depends on the void zone length. The further void depth increase causes the reduction in the ballast impact.
- The vertical impact velocity of sleepers depends linearly on the longitudinal wheel velocity.
- The theoretical estimation of the impact loading influence on the settlement accumulation shows that it causes high acceleration of the settlement intensity. The ballast impact loading of about 63 kN that appears for the case of the void depth of about 3 mm and the length of about 4 m causes up to 8 times higher settlement intensity than the quasistatic loading of 96 kN for the normal-weight loading without void.
- The pre-stressed zones neighbouring the void zone have about as low settlement intensity as the track without void despite having about 30% higher maximal loading.
- The high settlement accumulation under the impact loading with the following full unloading in the void zone accompanies the reduction in the sleeper support under the sleeper ends.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic explanation of the phenomenon of sleeper–ballast dynamic impact in the void zone.

**Figure 3.**Experimental measurement of trackside rail deflections (

**top, right**) in the void zone for a slow lightweight post train (

**top, left**) and the measurement results (

**bottom**).

**Figure 5.**The mechanical model of track and vehicle interaction in the void zone [44].

**Figure 6.**The simulated rail deflections for the slow lightweight post train in the rectangular form void.

**Figure 7.**Track loadings (bottom), accelerations (center) and deflections (top) along the track in the impact moment.

**Figure 13.**Relation of the quasistatic loading to the void depth inside the void zone (

**top**) and outside the void zone (

**bottom**).

**Figure 15.**Rail deflection line for wheel point loading case with the distributed own weight loading and the maximal deflection using exact and approximate solutions.

**Figure 20.**The distribution of the normal force over the particles after impact loading series (top) and after the quasistatic cyclic loading (bottom).

**Figure 21.**The mean normal loading distribution in the particles under the sleeper (

**top**) and under the ballast bed (

**bottom**) after the quasistatic loading cycles.

**Figure 22.**The mean normal loading distribution in the particles under the sleeper (

**top**) and under the ballast bed (

**bottom**) after the impact loading cycles.

Case | Sleeper | Impact Shift, m | Wheel Acceleration, m/s^{2} | Ballast | Rail Deflection | Void | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

Impact Velocity, cm/s | Max Acceleration, m/s^{2} | Impact Loading, kN | QS Loading in Void Zone, kN | QS Loading Outside Void Zone, kN | Negative, mm | Positive, mm | Void Length, m | Void Depth, mm | |||

L2V1 | 1.4 | 4.9 | 0.24 | 1.1 | 5.44 | 6.21 | 52.1 | 1.01 | 0.1 | 1.4 | 0.85 |

L2V2 | 1.4 | 0.86 | 0.82 | 0 | 0 | 59.4 | 1.23 | 0.1 | 1.49 | 1.84 | |

L3V1 | 1.6 | 2.64 | 0.56 | 0.36 | 3 | 23 | 51.6 | 1.1 | 0.1 | 1.93 | 0.75 |

L3V2 | 2.3 | 6.65 | 0.28 | 3.02 | 2.94 | 10.82 | 61.13 | 1.83 | 0.1 | 2.05 | 1.671 |

L3V3 | 2.0 | 1.24 | 1.24 | 0 | 0 | 65.8 | 2.16 | 0.13 | 2.2 | 2.65 | |

L4V1 | 3.4 | 4.26 | 0.12 | 1 | 27.88 | 47.36 | 0.97 | 0.15 | 2.4 | 0.53 | |

L4V2 | 1.6 | 12.51 | 0.63 | 2.6 | 10.01 | 22.45 | 58.78 | 1.77 | 0.16 | 2.58 | 1.42 |

L4V3 | 2.7 | 15 | 0.35 | 5.52 | 16.5 | 14.65 | 64.18 | 2.6 | 0.16 | 2.77 | 2.36 |

L4V4 | 3.2 | 3.52 | 2.73 | 0 | 0 | 67.54 | 3.29 | 0.16 | 2.94 | 3.31 | |

L5V1 | 3.1 | 5.86 | 1.25 | 0.034 | 4.55 | 28.63 | 43.77 | 0.88 | 0.12 | 2.78 | 0.26 |

L5V2 | 1.3 | 13.16 | 0.87 | 1.5 | 11.77 | 24.6 | 54.49 | 1.66 | 0.19 | 3.07 | 1.08 |

L5V3 | 2.7 | 23.24 | 0.64 | 6.02 | 22.04 | 29.8 | 60.8 | 2.38 | 0.24 | 3.27 | 1.97 |

L5V4 | 3.9 | 19.66 | 0.38 | 7.9 | 20.56 | 25.77 | 64.64 | 3.14 | 0.38 | 3.44 | 2.87 |

L5V5 | 4.5 | 5.23 | 0.41 | 11.04 | 4.49 | 17.38 | 67.52 | 4.31 | 0.37 | 3.5 | 4.26 |

L6V1 | 4.0 | 4.42 | 1 | 0.13 | 4.35 | 33.23 | 40.83 | 0.61 | 0.15 | 3.23 | 0.25 |

L6V2 | 0.9 | 9.96 | 1.02 | 1.09 | 9.33 | 33.74 | 49.86 | 1.07 | 0.39 | 3.57 | 0.8 |

L6V3 | 2.3 | 19.79 | 1.24 | 3.9 | 16.08 | 33.21 | 56.39 | 1.87 | 0.35 | 3.74 | 1.46 |

L6V4 | 3.7 | 27.15 | 1.08 | 7.6 | 20.71 | 31.76 | 60.33 | 2.76 | 0.29 | 3.87 | 2.31 |

L6V5 | 4.9 | 29.72 | 0.87 | 9.95 | 21.8 | 29.72 | 63.14 | 3.69 | 0.23 | 4.04 | 3.2 |

L6V6 | 5.8 | 31.58 | 0.76 | 11.02 | 28.54 | 26.25 | 64.86 | 4.6 | 0.17 | 4.11 | 4.18 |

L6V7 | 6.0 | 30.95 | 0.56 | 11.15 | 29.1 | 22.61 | 65.79 | 5.49 | 0.1 | 4.18 | 4.99 |

L7V2 | 5.7 | 10.28 | 1.3 | 0.72 | 9.67 | 24.78 | 45.41 | 0.74 | 0.34 | 3.95 | 0.14 |

L7V3 | 1.7 | 19.2 | 1.46 | 2.36 | 18.14 | 33.35 | 52.43 | 1.64 | 0.1 | 4.16 | 0.86 |

L7V4 | 3.1 | 23.07 | 1.25 | 4.93 | 23.22 | 37.12 | 56.17 | 2.54 | 0.1 | 4.34 | 1.63 |

L7V5 | 4.5 | 32.1 | 1.55 | 9.42 | 30.27 | 36.71 | 59.87 | 3.82 | 0.1 | 4.51 | 2.86 |

L7V6 | 6.1 | 38.14 | 0.99 | 10.65 | 37.75 | 36.4 | 60.84 | 4.24 | 0.1 | 4.6 | 3.29 |

L7V7 | 6.6 | 48.1 | 0.76 | 12.81 | 46.05 | 35.74 | 62.36 | 5.08 | 0.1 | 4.69 | 4.15 |

L7V8 | 7.3 | 45.82 | 1.29 | 13.34 | 43.86 | 34.27 | 63.16 | 5.89 | 0.1 | 4.7 | 5.03 |

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Sysyn, M.; Przybylowicz, M.; Nabochenko, O.; Liu, J.
Mechanism of Sleeper–Ballast Dynamic Impact and Residual Settlements Accumulation in Zones with Unsupported Sleepers. *Sustainability* **2021**, *13*, 7740.
https://doi.org/10.3390/su13147740

**AMA Style**

Sysyn M, Przybylowicz M, Nabochenko O, Liu J.
Mechanism of Sleeper–Ballast Dynamic Impact and Residual Settlements Accumulation in Zones with Unsupported Sleepers. *Sustainability*. 2021; 13(14):7740.
https://doi.org/10.3390/su13147740

**Chicago/Turabian Style**

Sysyn, Mykola, Michal Przybylowicz, Olga Nabochenko, and Jianxing Liu.
2021. "Mechanism of Sleeper–Ballast Dynamic Impact and Residual Settlements Accumulation in Zones with Unsupported Sleepers" *Sustainability* 13, no. 14: 7740.
https://doi.org/10.3390/su13147740