# Effect of Groundwater Level Rise on the Critical Velocity of High-Speed Railway

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Solution Method

#### 2.1. Biot’s Porous Media Theory

_{i}and w

_{i}represent soil skeleton displacement and pore water displacement relative to soil skeleton, respectively; ρ

_{b}= nρ

_{f}+ (1 − n)ρ

_{s}is the density of saturated soil, where ρ

_{s}and ρ

_{f}are the density of soil particles and the density of pore fluid, respectively; n is the porosity; m is the effective density, m = a

_{∞}ρ

_{f}/n, where a

_{∞}= $1/\sqrt{n}$ is a measure of soil pore curvature; b = ρ

_{f}g/k

_{D}, b represents the viscous coupling between pore fluid and soil particles, k

_{D}is the Darcy permeability coefficient of saturated soil in m/s; g is gravitational; α, M is the Biot constant, where $\alpha =K/{K}_{s}$, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$M$}\right.=(n/{K}_{f})+(\alpha -n/{K}_{s})$, K, K

_{s}, and K

_{f}are the bulk modulus of the soil skeleton, soil particles, and pore fluid, respectively; subscripts i, j = x, y, z are tensor notations; λ is the first parameter of Lame of the soil skeleton, μ is the second parameter of Lame of the soil skeleton; superscripts ‘∙’, ‘∙∙’, represent the first- and second-order derivatives with respect to time, respectively.

#### 2.2. 2.5D Finite Element Solution

_{x}represents the wave number along the x direction; ω represents the circular corner frequency. $h$ is a variable in the space and time domain and $\tilde{\overline{h}}$ is a variable in the wave number and frequency domain.

**M**represents the mass matrix;

**K**and

**L**represent the stiffness matrices;

**U**and

**W**represent the soil skeleton displacement matrix and the pore fluid relative displacement matrix, respectively;

**F**

^{s}and

**F**

^{f}represent the external load vectors.

#### 2.3. Train-Track-Embankment Coupling

**K**and

_{T}**M**represent the stiffness matrix and mass matrix of the rail; ${\tilde{\overline{\mathit{F}}}}_{IT}$ is the supporting force at the embankment surface; ${\tilde{\overline{\mathit{P}}}}_{k}$ represents the force vector of the train load on the surface of the rail, which is usually calculated by the quarter car model [21]. Figure 1 shows the geometric profile of train wheel loads of k carriages.

_{T}_{ni}is the axle load for ith wheelsets of the nth carriage. As indicated in Figure 1, L

_{n}is the length of the nth carriage, a

_{n}is the length of the bogie wheelbase, and b

_{n}is the distance from the second to third axles of the carriage of the nth carriage. ${x}_{n1}={\sum}_{n=1}^{k}{L}_{n}+{x}_{0}$; ${x}_{n2}={a}_{n}+{\sum}_{n=1}^{k}{L}_{n}+{x}_{0}$; ${x}_{n3}={a}_{n}+{b}_{n}+{\sum}_{n=1}^{k}{L}_{n}+{x}_{0}$; ${x}_{n4}=2{a}_{n}+{b}_{n}+{\sum}_{n=1}^{k}{L}_{n}+{x}_{0}$, among which x

_{0}is the distance to the first axle load position.

#### 2.4. Model Validation

## 3. Numerical Modelling

#### 3.1. Introduction of the Model

#### 3.2. Calculated Cases

## 4. Numerical Analysis

#### 4.1. Critical Velocity

_{s}). On the other hand, point B reflects the intensity of the surface vibration of the foundation, and its critical velocity is referred to as the foundation critical velocity (V

_{g}).

_{s}) being higher than the critical velocity of the foundation (V

_{g}) under the same groundwater levels. By comparing the system critical velocities under different groundwater levels in Figure 6a, it can be seen that V

_{s}decreases as the groundwater level rises. Specifically, for Case 1, V

_{s}is 145 m/s, while for Case 2 (groundwater level rising to the surface of the foundation), V

_{s}decreases by 13.8% to 125 m/s. When the groundwater level is further raised to the surface of the subgrade (Case 3), V

_{s}reduces by 22.8% to 112 m/s as compared to Case 1. These findings indicate that the increase in groundwater level in the embankment has a more significant impact on V

_{s}than that on the foundation. While the foundation critical velocity V

_{g}is slightly less affected by the groundwater level rise, it still decreased by 10.7% and 22.8% when the groundwater level was raised from Case 1 to Case 2 and Case 3, respectively. Even though the vibration intensity of Case 3 is lighter than Case 2, the elevation of the groundwater level in the embankment still further reduces V

_{g}.

_{s}) and foundation critical velocity (V

_{g}) for each Case, respectively.

#### 4.2. Dynamic Response in the Foundation

#### 4.3. Displacement Response Spectrum

_{s}, 1 V

_{s,}and 1.2 V

_{s}. In this context, the value of V

_{s}in Case 3 is 112 m/s.

_{s}, the high-frequency response in the foundation attenuates rapidly, while the high-spectrum lines at the other two speeds remain in a higher position. This suggests that the high-frequency response at high speed attenuates slowly in the foundation, leading to a larger high-frequency response area and a slower attenuation rate in the surrounding foundation as the train speed increases.

## 5. Summary and Conclusions

- (1)
- The critical velocity of the high-speed railway consistently decreases with the groundwater level rise. Moreover, the rise of the groundwater level within the embankment exerts a more pronounced influence on the system’s critical velocity compared to the rise in groundwater level within the foundation. This underscores the significance of effective embankment waterproofing in controlling track vibrations;
- (2)
- Train operations can induce deformation in both the embankment and foundation, with deformation significantly increasing as the groundwater level rises. In particular, when the groundwater level ascends from the foundation bottom to the subgrade surface, the deformation of the subgrade surface escalates by approximately 55%;
- (3)
- The frequency spectrum of ground vibration increases significantly in the high-frequency region with the rising groundwater levels, and this increase affects a wider frequency range as the water level rises;
- (4)
- This study indicates that the increase in groundwater level not only amplifies vibrations but also contributes to the extended propagation of high-frequency vibrations. Consequently, a more comprehensive analysis of the correlation between vibration propagation mechanisms and rising groundwater levels is imperative for future research;
- (5)
- A limitation of this study is that the materials in the model are simulated using isotropic linear elastic properties. Future research could explore the anisotropic nature of materials and the polyphase composition of the media for a more comprehensive understanding.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Development of the maximum vertical displacements versus train speed at different depths under different cases. (

**a**) Point A; (

**b**) Point B.

**Figure 8.**Three-dimensional displacement contours at the foundation surface. (

**a**) Case 1; (

**b**) Case 2; (

**c**) Case 3.

**Figure 10.**Frequency spectrum of vertical displacement for different train speeds at different observation points. (

**a**) Point A; (

**b**) Point C.; (

**c**) Point D.

**Figure 11.**Frequency spectrum of vertical displacement for different groundwater levels of different points. (

**a**) Point A; (

**b**) Point C.; (

**c**) Point D.

Soil Layer | Biot’s Constant α | Biot’s Constant M (MPa) | Young’s Modulus E (MPa) | Poisson’s Ratio v | Density of Soil Particles (kg/m^{3}) | Liquid Density (kg/m ^{3}) | Soil Damping D_{0} | Porosity n | Permeability Coefficient k_{D} (m/s) |
---|---|---|---|---|---|---|---|---|---|

Roadbed | 0.001 | 0.001 | 240 | 0.25 | 2500 | 0.001 | 0.05 | 0.001 | 10^{−20} |

Subgrade | 0.001 | 0.001 | 140 | 0.3 | 2200 | 0.001 | 0.05 | 0.001 | 10^{−20} |

Soil layer 1 | 0.001 | 0.001 | 113 | 0.35 | 2700 | 0.001 | 0.05 | 0.001 | 10^{−20} |

Soil layer 2 | 0.001 | 0.001 | 113 | 0.35 | 2700 | 0.001 | 0.05 | 0.001 | 10^{−20} |

Soil layer 3 | 0.001 | 0.001 | 135 | 0.35 | 2700 | 0.001 | 0.05 | 0.001 | 10^{−20} |

Soil Layer | Biot’s Constant α | Biot’s Constant M (MPa) | Young’s Modulus E (MPa) | Poisson’s Ratio v | Density of Soil Particles (kg/m^{3}) | Liquid Density (kg/m ^{3}) | Soil Damping D_{0} | Porosity n | Permeability Coefficient k_{D} (m/s) |
---|---|---|---|---|---|---|---|---|---|

Roadbed | 0.001 | 0.001 | 240 | 0.25 | 2500 | 1000 | 0.05 | 0.001 | 1 |

Subgrade | 1.000 | 6400 | 80 | 0.3 | 2700 | 1000 | 0.05 | 0.3 | 10^{−6} |

Soil layer 1 | 1.000 | 3520 | 45 | 0.35 | 2700 | 1000 | 0.05 | 0.6 | 10^{−6} |

Soil layer 2 | 1.000 | 3520 | 45 | 0.35 | 2700 | 1000 | 0.05 | 0.6 | 10^{−8} |

Soil layer 3 | 1.000 | 3520 | 60 | 0.35 | 2700 | 1000 | 0.05 | 0.6 | 10^{−6} |

Rail Mass per Linear Meter (kg/m) | Rail Bending Stiffness (MNm^{2}) | Slab Bending Stiffness (MNm ^{2}) | Mass per Linear Meter of Slab (kg/m) | Stiffness of CA Mortar Layer (MN/m/m) |
---|---|---|---|---|

60.64 | 6.625 | 40 | 950 | 100 |

Damping of CA mortar layer (Ns/m/m) | Bending stiffness of the concrete base (MNm ^{2}) | Mass per linear meter of the concrete base (kg/m) | Fastener stiffness (MN/m/m) | Fastener damping (Ns/m/m) |

2 × 10^{5} | 190 | 1800 | 28.5 | 5 × 10^{4} |

Parameter Name | Value |
---|---|

Carriage mass/kg | 45,000 |

Bogie mass/kg | 3600 |

Wheelset quality/kg | 1700 |

Carriage length/m | 24.8 |

Centre-to-centre distance of adjacent bogies/m | 14.9 |

Bogie length/m | 2.5 |

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**MDPI and ACS Style**

Hu, J.; Jin, L.; Wu, S.; Zheng, B.; Tang, Y.; Wu, X.
Effect of Groundwater Level Rise on the Critical Velocity of High-Speed Railway. *Water* **2023**, *15*, 3764.
https://doi.org/10.3390/w15213764

**AMA Style**

Hu J, Jin L, Wu S, Zheng B, Tang Y, Wu X.
Effect of Groundwater Level Rise on the Critical Velocity of High-Speed Railway. *Water*. 2023; 15(21):3764.
https://doi.org/10.3390/w15213764

**Chicago/Turabian Style**

Hu, Jing, Linlian Jin, Shujing Wu, Bin Zheng, Yue Tang, and Xuezheng Wu.
2023. "Effect of Groundwater Level Rise on the Critical Velocity of High-Speed Railway" *Water* 15, no. 21: 3764.
https://doi.org/10.3390/w15213764