# Numerical Simulation Study Considering Discontinuous Longitudinal Joints in Soft Soil under Symmetric Loading

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## Abstract

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## 1. Introduction

## 2. Modelling Strategy

- Based on the Winkler method, radial ground contact springs ${k}_{s}$ and tangential ground contact springs ${k}_{z}$ are introduced to simulate the contact between the segment structure and the strata (see Figure 2);
- According to the mechanical properties of the soil, the tensile-compressive springs can only withstand compression but not tension, i.e., ${k}_{t}=0$;
- At the joints of the segments, three sets of mechanical springs with discontinuous deformations are used to simulate tensile-compressive springs (${k}_{n}$), shear springs (${k}_{r}$), and rotational springs (${k}_{\u03f4}$); it is assumed that the node displacements at the joint positions of adjacent segments are different in the calculation (see Figure 2); among them, the influence of tensile-compressive springs and shear springs on the internal forces and deformations of the structure is relatively small [23]; this paper only considers the influence of rotational springs on the structure, and sets the tensile-compressive springs and shear springs to relatively large values;
- In the calculation of the segment structure, we focus more on the bending behavior of the structure, ignoring shear deformations, and use the Euler–Bernoulli beam model to simulate the segment structure.

## 3. Subgrade Reaction Coefficient and Lateral Pressure Coefficient

#### 3.1. Subgrade Reaction Coefficient

#### 3.2. Lateral Pressure Coefficient

## 4. Analysis of Individual Joint Bending Stiffness

## 5. Conclusions

- (1)
- Based on the conditions of the soft soil layers in Shanghai, numerical simulation results show that the tunnel experiences negative bending moments at the crown and bottom, while positive bending moments occur along the sides. The axial force is negative, peaking at the tunnel sidewalls. Tunnel deformation takes on an elliptical shape. Additionally, under the influence of bending moments, joints exhibit angular distortion.
- (2)
- When the subgrade reaction coefficient decreases, the absolute values of bending moments, axial forces, and deformations at the tunnel crown and bottom generally increase. A discontinuity in axial forces occurs at the joints. The effectiveness of the model was verified by comparing the deformation of 3D laser scanning point cloud sections. When the foundation resistance coefficient is 5000 $\mathrm{k}\mathrm{N}/{\mathrm{m}}^{3}$, the deformation is closer to the actual values, with the maximum error occurring at the left sidewall of the tunnel (6.5 mm), followed by the tunnel crown (3.1 mm). When the lateral pressure coefficient decreases, the absolute values of bending moments and deformations at the tunnel crown, bottom, and sidewalls generally increase, while the absolute values of axial forces at the top and bottom decrease.
- (3)
- The bending stiffness of individual joints of single-ring segments in soft soil layers was reduced. When the bending moment at the joint is negative, a decrease in joint bending stiffness leads to an increase in nearby bending moments, while a decrease in joint bending stiffness leads to a decrease in nearby bending moments when the bending moment at the joint is positive. As the bending stiffness decreases, the joint opening angle increases, affecting the entire ring through beam transfer. The extent and magnitude of this effect depend on the bending moment at the joint and the joint’s position, providing reference for the calculation of bending moments in traditional indirect methods for ring segments and joints.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The earth pressure applied on the tunnel lining under semi-infinite conditions with consideration of ground surcharge.

**Figure 4.**Flowchart of the FEM process considering joint discontinuity and nonlinear ground springs.

**Figure 5.**The illustration of results with respect to the reference angle when the subgrade reaction coefficient is altered. (

**a**) Bending moment; (

**b**) axial force; (

**c**) radial deformation; (

**d**) rotational deformation.

**Figure 6.**The illustration of results with respect to the reference angle when lateral pressure coefficient is altered. (

**a**) Bending moment; (

**b**) axial force; (

**c**) radial deformation; (

**d**) rotational deformation.

**Figure 7.**The illustration of results with respect to the reference angle when joint B1-D is altered. (

**a**) Bending moments; (

**b**) axial force; (

**c**) radial deformations, and (

**d**) rotational deformations.

**Figure 8.**The illustration of results with respect to the reference angle when joint L1-B1 is altered. (

**a**) Bending moments; (

**b**) axial force; (

**c**) radial deformations, and (

**d**) rotational deformations.

**Figure 9.**The illustration of results with respect to the reference angle when joint F-L1 is altered. (

**a**) Bending moments; (

**b**) axial force; (

**c**) radial deformations, and (

**d**) rotational deformations.

Symbol | Properties | Unit | Value | |
---|---|---|---|---|

Material parameters of soil | ${\rho}_{s}$ | Density of soil | ${\mathrm{k}\mathrm{N}/\mathrm{m}}^{3}$ | 18 |

${K}_{s}$ | Subgrade reaction coefficient(compression) | ${\mathrm{k}\mathrm{N}/\mathrm{m}}^{3}$ | 3000~15,000 | |

${K}_{t}$ | Subgrade reaction coefficient(tension) | ${\mathrm{k}\mathrm{N}/\mathrm{m}}^{3}$ | 0 | |

${K}_{0}$ | Coefficient of lateral pressure | 0.5~0.7 | ||

${\nu}_{s}$ | $\mathrm{Poisson}\u2019\mathrm{s}\mathrm{ratio}$ of soil | 0.35 | ||

${E}_{s}$ | Young’s modulus of soil | $\mathrm{k}\mathrm{P}\mathrm{a}$ | 1.68 × 10^{4} | |

$H$ | Embedded depth | $\mathrm{m}$ | 14 |

Parameter Classification | Symbol | Properties | Unit | Value |
---|---|---|---|---|

Geometrical parameters of tunnel lining | $R$ | Radius | $\mathrm{m}$ | 2.925 |

$b$ | Width | $\mathrm{m}$ | 1 | |

$h$ | Thickness | $\mathrm{m}$ | 0.35 | |

Material parameters of tunnel lining | ${\rho}_{c}$ | Density of segment | ${\mathrm{k}\mathrm{N}/\mathrm{m}}^{3}$ | 2.5 |

${E}_{c}$ | Young’s modulus of segment | $\mathrm{k}\mathrm{P}\mathrm{a}$ | 3.45 × 10^{7} | |

${\nu}_{c}$ | Poisson’s ratio of segment | / | 0.2 | |

Stiffness parameters of soil spring | ${k}_{sn}$ | Radial | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 450 |

${k}_{zn}$ | Tangential | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 150 | |

Stiffness parameters of joints | ${k}_{n}$ | Compress spring | $\mathrm{k}\mathrm{N}/\mathrm{m}$ | 1 × 10^{7} |

${k}_{r}$ | Shear spring | $\mathrm{k}\mathrm{N}/\mathrm{m}$ | 1 × 10^{10} | |

${k}_{\u03f4}$ | Moment spring | $\mathrm{k}\mathrm{N}\xb7\mathrm{m}/\mathrm{r}\mathrm{a}\mathrm{d}$ | 1.54 × 10^{5} | |

Load | ${P}_{0}$ | ground overload | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 20 |

${P}_{g}$ | Self weight of the tunnel lining | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 317 | |

${P}_{1}$ | Vertical overburden earth pressure at the tunnel crown | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 270 | |

${P}_{2}$ | Vertical overburden earth pressure at the tunnel invert | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 317 | |

${P}_{3}$ | Lateral earth pressure at the tunnel crown | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 175 | |

${P}_{4}$ | Additional earth pressure at the tunnel invert | $\mathrm{k}\mathrm{P}\mathrm{a}/\mathrm{m}$ | 245 |

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**MDPI and ACS Style**

He, X.; Xu, X.; Yang, H.
Numerical Simulation Study Considering Discontinuous Longitudinal Joints in Soft Soil under Symmetric Loading. *Symmetry* **2024**, *16*, 650.
https://doi.org/10.3390/sym16060650

**AMA Style**

He X, Xu X, Yang H.
Numerical Simulation Study Considering Discontinuous Longitudinal Joints in Soft Soil under Symmetric Loading. *Symmetry*. 2024; 16(6):650.
https://doi.org/10.3390/sym16060650

**Chicago/Turabian Style**

He, Xianwei, Xiangyang Xu, and Hao Yang.
2024. "Numerical Simulation Study Considering Discontinuous Longitudinal Joints in Soft Soil under Symmetric Loading" *Symmetry* 16, no. 6: 650.
https://doi.org/10.3390/sym16060650