# Influence of Variable Height of Piers on the Dynamic Characteristics of High-Speed Train–Track–Bridge Coupled Systems in Mountainous Areas

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## Abstract

**:**

## 1. Introduction

## 2. Train–Track–Bridge Coupled Vibration Model

#### 2.1. Train Model

#### 2.2. Track and Bridge Model

#### 2.3. Rail Irregularity

#### 2.4. Train–Track–Bridge Coupled Vibration

#### 2.5. Equation Solving of System Coupled Vibration

## 3. Analysis and Calculation Parameters

#### 3.1. Research Scenario

#### 3.2. Pier Height Working Condition

#### 3.3. Earthquake Excitation Model

- (1)
- Class I site soil: rocky, compact gravelly soil.
- (2)
- Class Ⅱ site soil: medium-dense, loose gravel soil, dense, medium-dense gravel, coarse and medium sand; clayey soil with foundation soil with a permissible bearing capacity $\left[{\sigma}_{0}\right]$ > 150 kPa.
- (3)
- Class Ⅲ site soil: loose gravel, coarse and medium sand, dense and medium-dense fine and silty sand, clayey soil with a permissible bearing capacity [σ0] ≤ 150 kPa and fill soil with $\left[{\sigma}_{0}\right]$ ≥ 130 kPa.
- (4)
- Class Ⅳ site soil: silty soil, loose fine and chalky sand, recently deposited clayey soil; fill with foundation soil with an allowable bearing capacity $\left[{\sigma}_{0}\right]$ < 130 kPa.

## 4. Dynamic Response of the System

#### 4.1. Validation of the System

#### 4.2. Train Running Safety Indicators

#### 4.2.1. Derailment Coefficient

#### 4.2.2. Rate of Wheel Load Reduction

#### 4.3. The Influence of Train Speed on System Dynamic Response

#### 4.4. The Influence of Pier Height on System Dynamic Response

## 5. Conclusions

- (1)
- The dynamic response of the system and the safety index of the train generally increase with the increase in the train running speed.
- (2)
- Under seismic excitation, the dynamic response of the system is significantly increased, and the lateral dynamic response of the system is more affected by seismic excitation than the vertical response.
- (3)
- Compared to equal-height piers, the peak lateral dynamic response of the system with unequal-height piers (gradually increasing) decreases, which is beneficial for stabilizing the vehicle body. The peak lateral dynamic response of the system with unequal-height piers (steep increase in pier height) increases sharply, seriously reducing passenger comfort.
- (4)
- The vertical dynamic response of trains under two unequal-height pier conditions increases, and the safety indicators of equal-height piers are significantly better than the two unequal-height pier conditions. The recommended design is the optimal choice for equal-height bridge piers, followed by gradually increasing pier heights, and avoiding steep increases in pier heights.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Model validation with the example in Ref. [38].

Vertical | Longitudinal | Lateral | Yaw | Roll | Pitch | |
---|---|---|---|---|---|---|

Car body | ${z}_{c}$ | ${x}_{c}$ | ${y}_{c}$ | ${\theta}_{zc}$ | ${\theta}_{xc}$ | ${\theta}_{yc}$ |

Front bogie | ${z}_{tf}$ | ${x}_{tf}$ | ${y}_{tf}$ | ${\theta}_{ztf}$ | ${\theta}_{xtf}$ | ${\theta}_{ytf}$ |

Rear bogie | ${z}_{tb}$ | ${x}_{tb}$ | ${y}_{tb}$ | ${\theta}_{ztb}$ | ${\theta}_{xtb}$ | ${\theta}_{ytb}$ |

First wheel set | ${z}_{w1}$ | ${x}_{w1}$ | ${y}_{w1}$ | ${\theta}_{zw1}$ | ${\theta}_{xw1}$ | - |

Second wheel set | ${z}_{w2}$ | ${x}_{w2}$ | ${y}_{w2}$ | ${\theta}_{zw2}$ | ${\theta}_{xw2}$ | - |

Third wheel set | ${z}_{w3}$ | ${x}_{w3}$ | ${y}_{w3}$ | ${\theta}_{zw3}$ | ${\theta}_{xw3}$ | - |

Fourth wheel-set | ${z}_{w4}$ | ${x}_{w4}$ | ${y}_{w4}$ | ${\theta}_{zw4}$ | ${\theta}_{xw4}$ | - |

${\mathsf{\Omega}}_{c}$ | ${\mathsf{\Omega}}_{r}$ | ${\mathsf{\Omega}}_{s}$ | ${A}_{g}$ | ${A}_{v}$ | ${A}_{a}$ |

0.8264 | 0.0206 | 0.438 | $5.32\times {10}^{-8}$ | $2.119\times {10}^{-7}$ | $4.032\times {10}^{-7}$ |

Parameters | Definitions | Values | Units |
---|---|---|---|

${E}_{b}$ | Elastic modulus | 3.45 × 10^{10} | ${\mathrm{N}/\mathrm{m}}^{2}$ |

$I$ | Mass moment of inertia of cross section | 12.744 | ${\mathrm{m}}^{4}$ |

$\mu $ | Poisson’s ratio | 0.2 | / |

${\overline{m}}_{b}$ | Mass per unit length | 2.972 × 10^{4} | $\mathrm{kg}/\mathrm{m}$ |

${L}_{e}$ | Length of element | 3.2 | $m$ |

$\zeta $ | Damping ratio | 0.03 | / |

Working Condition Type | ${\mathit{H}}_{1}$ | ${\mathit{H}}_{2}$ | ${\mathit{H}}_{3}$ | ${\mathit{H}}_{4}$ |
---|---|---|---|---|

Equal-height pier (m) | 8 | 8 | 8 | 8 |

Gradually increasing (m) | 8 | 10 | 12 | 14 |

Sharply increasing (m) | 8 | 8 | 14 | 14 |

Parameters | Site Classification | |||
---|---|---|---|---|

I | II | III | IV | |

${\zeta}_{g}$ | 0.728 | 0.775 | 0.822 | 0.868 |

${\zeta}_{f}$ | 0.411 | 0.557 | 1.140 | 2.208 |

${\omega}_{g}(\mathrm{rad}\cdot {\mathrm{s}}^{-1})$ | 24.763 | 18.656 | 13.491 | 9.848 |

${\omega}_{f}(\mathrm{rad}\cdot {\mathrm{s}}^{-1})$ | 0.453 | 0.355 | 0.154 | 0.082 |

${S}_{0}({\mathrm{cm}}^{2}{/\mathrm{s}}^{2})$ (PGA = 0.1 g) | 11.241 | 15.546 | 22.370 | 31.201 |

${S}_{0}({\mathrm{cm}}^{2}{/\mathrm{s}}^{2})$ (PGA = 0.2 g) | 43.028 | 59.512 | 85.598 | 119.304 |

${S}_{0}({\mathrm{cm}}^{2}{/\mathrm{s}}^{2})$ (PGA = 0.4 g) | 172.103 | 238.028 | 342.905 | 478.009 |

Working Condition Type | 1st Modulus | 2nd Modulus | 3rd Modulus | 4th Modulus |
---|---|---|---|---|

Equal-height pier | 13.187 | 13.302 | 15.806 | 21.443 |

Gradually increasing | 10.452 | 10.518 | 12.998 | 15.538 |

Sharply increasing | 9.630 | 10.832 | 13.964 | 15.595 |

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## Share and Cite

**MDPI and ACS Style**

Zeng, Y.; Jiang, L.; Zhang, Z.; Zhao, H.; Hu, H.; Zhang, P.; Tang, F.; Xiang, P.
Influence of Variable Height of Piers on the Dynamic Characteristics of High-Speed Train–Track–Bridge Coupled Systems in Mountainous Areas. *Appl. Sci.* **2023**, *13*, 10271.
https://doi.org/10.3390/app131810271

**AMA Style**

Zeng Y, Jiang L, Zhang Z, Zhao H, Hu H, Zhang P, Tang F, Xiang P.
Influence of Variable Height of Piers on the Dynamic Characteristics of High-Speed Train–Track–Bridge Coupled Systems in Mountainous Areas. *Applied Sciences*. 2023; 13(18):10271.
https://doi.org/10.3390/app131810271

**Chicago/Turabian Style**

Zeng, Yingying, Lizhong Jiang, Zhixiong Zhang, Han Zhao, Huifang Hu, Peng Zhang, Fang Tang, and Ping Xiang.
2023. "Influence of Variable Height of Piers on the Dynamic Characteristics of High-Speed Train–Track–Bridge Coupled Systems in Mountainous Areas" *Applied Sciences* 13, no. 18: 10271.
https://doi.org/10.3390/app131810271