# Effect of Random Lateral Ballast Resistance on Force-Deformation Characteristics of CWR with a Small-Radius Curve

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## Abstract

**:**

## 1. Introduction

## 2. Field Tests

#### 2.1. Testing Method

#### 2.2. Testing Results

#### 2.3. SW Test for Normality of Lateral Ballast Resistance

_{0}, is that the overall distribution of the lateral ballast resistance samples follows a normal distribution. The test results, given a 5% level of significance, are shown in Table 1.

## 3. Numerical Study

#### 3.1. Finite Element (FE) Model of CWR Track

#### 3.2. Force-Deformation Characteristic Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Nomenclature | Description |
---|---|

CWR | Continuous welded rail |

FE | Finite element |

$A$ | Cross-sectional area of rail |

$E$ | Elastic modulus of rail |

$N$ | Compressive force of rail |

${d}_{lo}$ | Longitudinal deflection |

${r}_{lo}$ | Longitudinal ballast resistance |

${d}_{la}$ | Lateral deflection |

${r}_{la}$ | Lateral ballast resistance |

$n$ | Number of samples |

$\alpha $ | Thermal expansion coefficient |

${x}_{\left(i\right)}$ | Sample observations |

$\overline{x}$ | Mean value of sample observations |

${(\cdot )}^{T}$ | Transpose of a matrix |

${(\cdot )}^{-1}$ | Inverse of a matrix |

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**Figure 4.**Field tests of lateral ballast resistance. (

**a**) Forcing device; (

**b**) Displacement measured by a dial gauge.

**Figure 9.**Generated normally distributed lateral ballast resistance. (

**a**) Histogram of generated normally distributed lateral ballast resistance; (

**b**) Randomly distributed lateral ballast resistance along the CWR track.

**Figure 17.**The maximum, minimum, mean, and standard deviation of lateral deflection under different mean values.

**Figure 19.**The maximum, minimum, mean, and standard deviations of lateral deflection under different standard deviations.

Data | ${\mathit{x}}_{\left(\mathit{i}\right)}$ | ${\mathit{x}}_{\left(\mathit{n}-\mathit{i}+1\right)}$ | ${\mathit{d}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{a}}_{\mathit{i}}{\mathit{d}}_{\mathit{i}}$ | ${\left({\mathit{x}}_{\mathit{i}}-\overline{\mathit{x}}\right)}^{2}$ |
---|---|---|---|---|---|---|

6.025 | 6.025 | 10.71958 | 4.694577 | 0.515 | 2.417707 | 5.701912 |

6.877692 | 6.877692 | 10 | 3.122308 | 0.3306 | 1.032235 | 2.356763 |

6.933333 | 6.933333 | 9.689655 | 2.756322 | 0.2495 | 0.687702 | 2.189022 |

7.036061 | 7.036061 | 9.576923 | 2.540862 | 0.1878 | 0.477174 | 1.895597 |

7.334783 | 7.334783 | 9.420492 | 2.085709 | 0.1353 | 0.282196 | 1.162267 |

7.46 | 7.46 | 9.354516 | 1.894516 | 0.088 | 0.166717 | 0.907957 |

7.544828 | 7.544828 | 9.296154 | 1.751326 | 0.0433 | 0.075832 | 0.753493 |

8.924 | 0.261256 | |||||

9.296154 | 0.780195 | |||||

9.354516 | 0.886702 | |||||

9.420492 | 1.015307 | |||||

9.576923 | 1.355025 | |||||

9.689655 | 1.630187 | |||||

10 | 2.518989 | |||||

10.71958 | 5.32091 |

Rail | Height | 176 mm |

Cross-sectional area | 77.45 cm^{2} | |

Moment of inertia about the lateral axis | 3217 cm^{4} | |

Moment of inertia about the vertical axis | 524 cm^{4} | |

Elastic modulus | 2.06 × 10^{11} N/m^{2} | |

Poisson’s ratio | 0.3 | |

Fastener | Lateral stiffness | 9 × 10^{6} N/m |

Longitudinal stiffness | 5 × 10^{6} N/m | |

Horizontal torsional stiffness | 207 N·m/rad | |

Torsion moment | 150N·m | |

Sleeper | Length | 2.6 m |

Elastic modulus | 3.6 × 10^{10} N/m^{2} | |

Sleeper spacing | 0.6 m |

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**MDPI and ACS Style**

Wang, H.; Xing, C.; Deng, X.
Effect of Random Lateral Ballast Resistance on Force-Deformation Characteristics of CWR with a Small-Radius Curve. *Materials* **2023**, *16*, 2876.
https://doi.org/10.3390/ma16072876

**AMA Style**

Wang H, Xing C, Deng X.
Effect of Random Lateral Ballast Resistance on Force-Deformation Characteristics of CWR with a Small-Radius Curve. *Materials*. 2023; 16(7):2876.
https://doi.org/10.3390/ma16072876

**Chicago/Turabian Style**

Wang, Huan, Chengwei Xing, and Xiaohui Deng.
2023. "Effect of Random Lateral Ballast Resistance on Force-Deformation Characteristics of CWR with a Small-Radius Curve" *Materials* 16, no. 7: 2876.
https://doi.org/10.3390/ma16072876