# Numerical Limit Analysis of the Stability of Reinforced Retaining Walls with the Strength Reduction Method

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Limit Analysis

#### 2.1. Theory

#### 2.2. Lower-Bound Principle

#### 2.3. Upper-Bound Principle

#### 2.4. Mesh Detail

## 3. Verification of the Numerical Model

#### 3.1. Case 3-1

#### 3.2. Case 3-2

## 4. Parametric Study and Results

#### 4.1. Numerical Model

#### 4.1.1. Parameter Value Ranges and Baseline Case

#### 4.1.2. Geometry and Boundary Conditions

#### 4.1.3. Soil Constitutive Models and Properties

#### 4.1.4. Reinforcement Properties

#### 4.1.5. Interface Properties

#### 4.1.6. Critical Reinforcement Lengths

#### 4.2. Wall Height

#### 4.3. Reinforcement Spacing

#### 4.4. Horizontal Seismic Load Originating from Earthquake

_{h}/g

_{v}, with g

_{v}= 9.8 m/s

^{2}, were considered. The effect of the seismic load, ranging from 0.05 g to 0.2 g, on the critical reinforcement length is shown in Figure 11. An increase in seismic coefficient value required a larger critical length of the reinforcement to satisfy the stability requirements of the sliding and global failure modes.

## 5. Recommendations for Design

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the different numbers of elements. (

**a**) Safety factor = 0.8752 (elements = 10,000). (

**b**) Safety factor = 0.8781 (elements = 20,000).

Input Data | Case 3-1 | Case 3-2 |
---|---|---|

Wall height | 8.2 | 12.1 |

Reinforcement spacing (m) | 0.4 | 1.34 |

Reinforcement length (m) | 1.5 | 7.5 |

Reinforcement soil unit weight (kN/m^{3}) | 22 | 15.64 |

Reinforcement soil angle of friction (°) | 45 | 39.5 |

Retaining soil unit weight (kN/m^{3}) | 22 | 15.64 |

Retaining soil angle of friction (°) | 45 | 39.5 |

Foundation soil unit weight (kN/m^{3}) | 22 | 15.64 |

Foundation soil angle of friction (°) | 45 | 39.5 |

Ultimate strength of geogrid reinforcement (kN/m) | 9.0 | 10.0 |

Soil–geogrid angle of friction (°) | 35 | 39 |

Safety Factor | Lower Bound | Upper Bound | Average Value | FLAC Analysis |
---|---|---|---|---|

Case 3-1 | 1.039 | 1.110 | 1.075 | 1.09 (1) |

Material | Reinforced Soil | Retained Soil | Foundation Soil | Block Facing |
---|---|---|---|---|

Constitutive mode | Mohr-Coulomb | Mohr-Coulomb | Linearly elastic | Linearly elastic |

Unit weight (kN/m^{3}) | 18 | 18 | 18 | 23 |

Young’s modulus (MPa) | 20 | 20 | 2000 | - |

Poisson’s ratio | 0.3 | 0.3 | 0.3 | - |

Cohesion (kPa) | 0 | 0 | 0 | - |

Friction angle (°) | 35 | 35 | 35 | 35 |

Dilation angle (°) | 5 | 5 | 5 | - |

Materials | Secant Stiffness at 2%, J _{2%} (kN/m^{2}) | Tensile Strength (kN/m^{2}) |
---|---|---|

geogrid | 400 | 20 |

Interface | Friction Angle (°) | Dilation Angle (°) | Cohesion (kPa) | Normal | Shear Stiffness |
---|---|---|---|---|---|

Backfill-reinforcement | 35 | 5 | 0 | - | - |

Block-reinforcement | 25 | 0 | 0 | - | - |

Case | Parameter in the Numerical Model | Failure Mechanism | |||
---|---|---|---|---|---|

Wall Height (m) | Reinforcement Length (m) | Reinforcement Space (m) | K (g) | ||

1 | 6 | 2 | 0.6 | 0 | |

2 | 10 | 3 | 0.6 | 0 | |

3 | 13 | 4 | 0.6 | 0 | |

4 | 6 | 9 | 0.6 | 0 | |

5 | 10 | 10 | 0.6 | 0 | |

6 | 13 | 12 | 0.6 | 0 | |

7 | 10 | 2 | 0.4 | 0 | |

8 | 10 | 3 | 0.6 | 0 | |

9 | 10 | 5 | 0.9 | 0 | |

10 | 10 | 8 | 0.4 | 0 | |

11 | 10 | 8 | 0.6 | 0 | |

12 | 10 | 9 | 0.9 | 0 | |

13 | 10 | 5 | 0.6 | 0.1 | |

14 | 10 | 7 | 0.6 | 0.2 | |

15 | 10 | 12 | 0.6 | 0.1 | |

16 | 10 | 15 | 0.6 | 0.2 |

Case | L/H | Length (m) |
---|---|---|

Base conditions | 0.8 | 12 |

Seismic loading | 0.9 | 15 |

Case | Spacing (m) |
---|---|

Base conditions | 0.7 |

Seismic loading | 0.5 |

Limited reinforcement length | 0.5 |

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**MDPI and ACS Style**

Li, J.; Li, X.; Jing, M.; Pang, R. Numerical Limit Analysis of the Stability of Reinforced Retaining Walls with the Strength Reduction Method. *Water* **2022**, *14*, 2319.
https://doi.org/10.3390/w14152319

**AMA Style**

Li J, Li X, Jing M, Pang R. Numerical Limit Analysis of the Stability of Reinforced Retaining Walls with the Strength Reduction Method. *Water*. 2022; 14(15):2319.
https://doi.org/10.3390/w14152319

**Chicago/Turabian Style**

Li, Jinsheng, Xueqi Li, Mingyuan Jing, and Rui Pang. 2022. "Numerical Limit Analysis of the Stability of Reinforced Retaining Walls with the Strength Reduction Method" *Water* 14, no. 15: 2319.
https://doi.org/10.3390/w14152319