# On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulations

_{1}is the thickness of soil layer 1, h

_{2}is the thickness of soil layer 2, D is the relative distance between the wave crest and the slope top of the left slope, H is the height of the solitary wave, W is the width of the trench, B is the height of the trench, d is the water depth, and U

_{0}is the initial current velocity.

#### 2.1. Wave–Current Sub-Model

#### 2.1.1. Continuity Equations and Momentum Equations

#### 2.1.2. Turbulence Models

#### 2.1.3. Boundary Conditions for Solitary Wave Generation

#### 2.2. Seabed Sub-Model

#### 2.2.1. Seepage Pressure

#### 2.2.2. Strength Reduction Method for the Seabed

_{1}is the first invariant stress tensor, J

_{2}is the second invariant deviatoric stress tensor. m, ${\alpha}_{0}$ and ${k}_{0}$ are the parameters related to soil material parameters:

#### 2.3. Boundary Conditions

#### 2.4. Integration of Sub-Models

#### 2.5. Convergence of the FEM Meshes

_{ec}| is the maximum pore pressure, and σ′

_{0}is the initial effective stress at point A. To achieve the computational accuracy, a mesh number with the smallest standard deviation was selected. The FEM mesh adopted in the computation for the seabed in the vicinity of the foundation trench is shown in Figure 3. The mesh refinement near the foundation trench was adopted to achieve satisfactory calculation.

## 3. Model Validation

## 4. Results and Discussion

#### 4.1. Consolidation of the Seabed

#### 4.2. Stability Index for the One-Stage Slope under Solitary Wave Loading

_{min}) was regarded as the stability index for the slope with the corresponding slope ratio. With the decrease of slope ratio, the FOS

_{min}increased as expected. It was observed that FOS

_{min}was bigger than 1 when the slope ratio was 1:2.5 in this case. Therefore, when the slope ratio was smaller than 1:2.5, the slope was stable under the combined actions of self-weight, wave pressure, and induced seepage force in the seabed in this study. With the decrease of slope ratio, the FOS

_{min}increased as expected.

#### 4.3. Influence of Soil Strength Parameters on the Slope Stability

_{min}, are also shown in Figure 7. As expected, the stability of slope increases with the increase of soil strength.

#### 4.4. Influence of the Slope Ratio on Slope Stability

#### 4.5. Influence of Current Direction on Slope Stability

_{min}in Figure 10a,d were smaller than those in Figure 10b,c. This phenomenon may be attributed to the fact that, when the currents propagated away from the slope surface, the current-induced pressure acting on the slope surface may have reduced the stability of slope.

#### 4.6. Influence of Slope Ratio on Two-Stage Slope

_{min}with the slope ratio of upper slope. It could be noted that FOS

_{min}increased slightly as the upper slope ratio increased. Thus, it was concluded that the slope ratio of the lower slope had more significant influence on the stability of the whole slope compared with the upper slope ratio.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Variation of the wave induced maximum excess pore pressure (|u

_{ec}|/σ′

_{0}) (at point A in Figure 1) with the mesh number N.

**Figure 4.**Comparison of wave surface profile between the proposed model and the experimental data [23]: (

**a**) t = 10 s, (

**b**) t = 20 s, (

**c**) t = 25 s, (

**d**) t = 30 s.

**Figure 5.**Initial state of the trench after excavation: (

**a**) horizontal displacement, (

**b**) vertical displacement, (

**c**) stress state.

**Figure 7.**Destruction areas of the slope with different soil parameters: (

**a**) c′ = 15 kPa, 20 kPa, 25 kPa, 30 kPa; (

**b**) φ′ = 12°, 16°, 20°, 24°.

**Figure 8.**Plastic penetration zones in the slope with different slope ratios: (

**a**) slope ratio 1:2.5, (

**b**) slope ratio 1:3, (

**c**) slope ratio 1:3.5, (

**d**) slope ratio 1:4.

**Figure 9.**Sliding displacements in the slope with different slope ratios (scale factor is 20): (

**a**) slope ratio 1:2.5, (

**b**) slope ratio 1:3, (

**c**) slope ratio 1:3.5, (

**d**) slope ratio 1:4.

**Figure 10.**Deformation of the trench slopes under the currents in different directions: (

**a**) left slope, 1 m/s; (

**b**) right slope, 1 m/s; (

**c**) left slope, −1 m/s; (

**d**) right slope, −1 m/s.

**Figure 11.**Variation of FOS

_{min}with the slope ratio of upper slope for various lower slope ratios.

Parameters | Characteristics | Value | Unit |
---|---|---|---|

Wave Parameters | Wave height (H) | 3 | m |

Water depth (d) | 10 | m | |

Soil Parameters | Seabed thickness (h) | 40.5 | m |

Shear modulus (G) | 6.56 × 10^{6} | Pa | |

Soil porosity (n) | 0.41 | - | |

Poison’s Ratio (μ) | 0.35 | - | |

Elastic modulus (E) | 1.77 × 10^{7} | Pa | |

Soil permeability (k) | 8 × 10^{−6} | m/s | |

Density of soil grain (ρ_{s}) | 2.71 × 10^{3} | kg/m^{3} | |

Effective cohesion (c′) | 15 | kPa | |

Effective internal friction angle (φ′) | 20 | ° | |

Trench width (W) | 34 | m | |

Trench height (B) | 18 | m | |

Water parameters | Bulk modulus (K_{w}) | 2 × 10^{9} | Pa |

Density of water (ρ_{w}) | 1000 | kg/m^{3} |

Parameters | Characteristics | Value | Unit |
---|---|---|---|

Soil Parameters in Upper Slope | Seabed thickness (h_{1}) | 8 | m |

Shear modulus (G_{1}) | 4.33 × 10^{6} | Pa | |

Soil porosity (n_{1}) | 0.56 | - | |

Poison’s ratio (μ_{1}) | 0.35 | - | |

Elastic modulus (E_{1}) | 1.17 × 10^{7} | Pa | |

Soil permeability (k_{1}) | 1 × 10^{−9} | m/s | |

Density of soil grain (ρ_{s1}) | 2.75 × 10^{3} | kg/m^{3} | |

Effective cohesion (c_{1}′) | 12 | kPa | |

Effective internal friction angle (φ_{1}′) | 13 | ||

Soil Parameters in Lower Slope | Seabed thickness (h_{2}) | 32.5 | m |

Shear modulus (G_{2}) | 6.56 × 10^{6} | Pa | |

Soil porosity (n_{2}) | 0.41 | - | |

Poison’s ratio (μ_{2}) | 0.35 | - | |

Elastic modulus (E_{2}) | 1.77 × 10^{7} | Pa | |

Soil permeability (k_{2}) | 8 × 10^{−6} | m/s | |

Density of soil grain (ρ_{s2}) | 2.71 × 10^{3} | kg/m^{3} | |

Effective cohesion(c_{2}′) | 15 | kPa | |

Effective internal friction angle (φ_{2}′) | 20 |

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**MDPI and ACS Style**

Chen, W.; Wang, D.; Xu, L.; Lv, Z.; Wang, Z.; Gao, H. On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave. *J. Mar. Sci. Eng.* **2021**, *9*, 526.
https://doi.org/10.3390/jmse9050526

**AMA Style**

Chen W, Wang D, Xu L, Lv Z, Wang Z, Gao H. On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave. *Journal of Marine Science and Engineering*. 2021; 9(5):526.
https://doi.org/10.3390/jmse9050526

**Chicago/Turabian Style**

Chen, Weiyun, Dan Wang, Lingyu Xu, Zhenyu Lv, Zhihua Wang, and Hongmei Gao. 2021. "On the Slope Stability of the Submerged Trench of the Immersed Tunnel Subjected to Solitary Wave" *Journal of Marine Science and Engineering* 9, no. 5: 526.
https://doi.org/10.3390/jmse9050526