# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of Failure Model

#### 2.1. Failure Model of the Soil Region at the Front Surface of the Anchorage Segment

- (1)
- Failure mechanism

- (2)
- Velocity field

- (3)
- Geometrical construction of the 3D velocity discontinuity surface in logarithmic spiral region

#### 2.2. The Failure Model of the Anchorage Segment Region

- (1)
- Failure mechanism

- (2)
- Velocity field

## 3. Upper-Bound Solution of Ultimate Pullout Capacity of Expanded Anchor Cable

#### 3.1. Calculation Theory of Upper-Bound Theorem of Limit Analysis

#### 3.2. Ultimate End Resistance Calculation

#### 3.2.1. Power Calculation of Logarithmic Spiral Region

#### 3.2.2. Determination of Minimum Upper-Bound Solution of End Resistance

#### 3.3. Ultimate Lateral Resistance Calculation

- (1)
- Rate of internal energy dissipation of the anchorage segment region

- (2)
- Rate of work of anchorage segment weight

## 4. Example Analysis and Solution Validation

#### 4.1. Theoretical Solution

^{3}, the elastic modulus was 13.5 MPa, the shear modulus was 5 MPa, the friction angle was 10.6°, and the cohesion was 8 kPa.

#### 4.2. Numerical Solution

#### 4.3. Solution Validation

## 5. Model Application

#### 5.1. Effect of Anchorage Segment Diameter

#### 5.2. Effect of Anchorage Segment Length

#### 5.3. Effect of the Inclination Angle of the Anchor Cable

#### 5.4. Effect of the Buried Depth of the Anchor Cable

## 6. Conclusions

- (1)
- A failure model of an expanded anchor cable located in a homogeneous stratum was constructed. The logarithmic spiral failure model was used as the failure model of the soil region at the front surface of the anchorage segment, representing the first time that this model has been implemented to calculate the end resistance of an expanded anchor cable. The failure mechanism of the anchorage side surface was assumed to satisfy the slippage model. The expressions of the ultimate lateral resistance and ultimate end resistance could be derived, respectively, from the power calculations for the two models and several algebraic operations. Due to the particularity of the boundary conditions in this paper, the rotation center points of the logarithmic spiral lines were uncertain. Therefore, the particle swarm optimization algorithm was used to determine the optimal solution of the end resistance.
- (2)
- The theoretical calculation results were compared with the numerical simulation in three cases. The results showed that the error between the numerical solution and the theoretical solution for the lateral resistance in the three cases was only 6.0%; however, the error for the end resistance was relatively large, though still within 30%, and so the maximum error for the ultimate pullout capacity was 7.7%. These errors were much smaller than the errors between the numerical simulation and the theoretical calculation method proposed in [18] for the same cases. This indicated that the proposed theoretical model had high reliability and superiority in the analysis of the ultimate pullout capacity of the expanded anchor cable.
- (3)
- The ultimate lateral resistance and total ultimate pullout capacity increased significantly with the increase in the anchorage segment diameter, anchorage segment length, and buried depth of the expanded anchor cable. The ultimate end resistance increased significantly with the buried depth, increased slightly with the anchorage segment diameter, and was almost unaffected by the anchorage segment length.
- (4)
- With the increase in the inclination angle of the anchor cable, the end resistance and the ultimate pullout capacity gradually decreased, while the lateral resistance increased first and then decreased. However, in general, the change in the inclination angle of the anchor cable had a relatively small effect on the ultimate lateral resistance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Spatial discretization technique for the generation of the failure model: (

**a**) profile diagram, (

**b**) front-end surface of the anchorage segment.

**Figure 7.**The velocity field of the anchorage segment region: (

**a**) profile diagram, (

**b**) cross-sectional diagram.

**Figure 9.**Cross-sections of the failure mechanisms for different coordinates of point O: (

**a**) intersection with the ground surface, (

**b**) all intersection with the potential slip surface.

**Figure 10.**Integral diagram of vertical and horizontal soil pressure: (

**a**) model diagram; (

**b**) vertical component of soil pressure; (

**c**) horizontal component of soil pressure.

**Figure 16.**The ultimate force variation curves according to changes in the anchorage segment diameter.

**Figure 17.**The ultimate force variation curves according to an increase in the anchorage segment length.

**Figure 18.**The ultimate force variation curves according to an increase in the inclination angle of the anchor cable.

**Figure 19.**The ultimate force variation curves according to the change in the buried depth of anchor cable.

Case | Anchorage Segment Length (m) | Free Segment Length (m) | Anchorage Segment Diameter (m) | Inclination Angle of Anchor Cable (°) | Stratum Friction Angle (°) | Buried Depth (m) |
---|---|---|---|---|---|---|

M1 | 10 | 9 | 0.6 | 30 | 10.6 | 7 |

M2 | 12 | 9 | 0.6 | 30 | 10.6 | 7 |

M3 | 10 | 9 | 0.6 | 45 | 10.6 | 7 |

Case | Lateral Resistance P_{1} (kN) | End Resistance P_{2} (kN) | Ultimate Pullout Force P (kN) |
---|---|---|---|

M1 | 610.1 | 217.4 | 861.7 |

M2 | 761.5 | 217.4 | 1019.3 |

M3 | 580.8 | 144.9 | 782.9 |

Material | Density (kg/m^{3}) | Elastic Modulus (MPa) | Shear Modulus (MPa) | Friction Angle (°) | Cohesion (kPa) |
---|---|---|---|---|---|

Steel strand | 7800 | 200,000 | 77,000 | - | - |

Soil | 1770 | 13.5 | 5 | 10.6 | 8.0 |

Bearing body | 7000 | 22,000 | 9361.70 | - | - |

Grouting body | 2200 | 21,897 | 9317.87 | 50.2 | 1960 |

**Table 4.**Comparison between ultimate pullout capacity according to numerical and theoretical solutions.

Case | Lateral Resistance P_{1} (kN) | End Resistance P_{2} (kN) | Ultimate Pullout Capacity P (kN) | ||||||
---|---|---|---|---|---|---|---|---|---|

Numerical | Theoretical | Error (%) | Numerical | Theoretical | Error (%) | Numerical | Theoretical | Error (%) | |

M1 | 575.5 | 610.1 | 6.0 | 176.5 | 217.4 | 23.2 | 800 | 861.7 | 7.71 |

M2 | 724.6 | 761.5 | 5.1 | 170.8 | 217.4 | 27.3 | 950 | 1019.3 | 7.29 |

M3 | 608.9 | 580.8 | 4.6 | 146.7 | 144.9 | 1.2 | 830 | 782.9 | 5.67 |

**Table 5.**Comparison between numerical solutions and theoretical solutions based on [18].

Case | Lateral Resistance P_{1} (kN) | End Resistance P_{2} (kN) | Ultimate Pullout Capacity P (kN) | ||||||
---|---|---|---|---|---|---|---|---|---|

Numerical | Zeng [18] | Error (%) | Numerical | Zeng [18] | Error (%) | Numerical | Zeng [18] | Error (%) | |

M1 | 575.5 | 587.9 | 2.15 | 176.5 | 54.8 | 68.95 | 800 | 642.7 | 19.67 |

M2 | 724.6 | 705.4 | 2.64 | 170.8 | 54.8 | 67.92 | 950 | 760.2 | 19.97 |

M3 | 608.9 | 587.9 | 3.45 | 146.7 | 54.8 | 62.65 | 830 | 642.7 | 22.57 |

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**MDPI and ACS Style**

Cheng, X.; Wang, B.; Ma, L.; Xue, C.; Yu, Y.
Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable. *Appl. Sci.* **2023**, *13*, 2357.
https://doi.org/10.3390/app13042357

**AMA Style**

Cheng X, Wang B, Ma L, Xue C, Yu Y.
Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable. *Applied Sciences*. 2023; 13(4):2357.
https://doi.org/10.3390/app13042357

**Chicago/Turabian Style**

Cheng, Xingyuan, Bo Wang, Longxiang Ma, Chenxi Xue, and Yunxiang Yu.
2023. "Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable" *Applied Sciences* 13, no. 4: 2357.
https://doi.org/10.3390/app13042357