## 4.1 Introduction

## 4.2 Analytical Modelling

### 4.2.1 Bond-Slip Model

_{p}, the shear stress increases linearly from zero to the shear strength of the bolt/grout interface. Then, with δ increasing to δ

_{r}, the shear stress decreases linearly to the residual shear strength of the bolt/grout interface. After δ

_{r}, the shear stress keeps constant and equals τ

_{r}.

_{p}: shear strength of the bolt/grout interface; τ

_{r}: residual shear strength of the bolt/grout interface; and δ

_{r}: shear slippage when the residual shear strength of the bolt/grout interface reaches.

### 4.2.2 Governing Equation

_{b}(x): rock bolt displacement at the position of x; and u

_{m}(x): confining medium displacement at the position of x.

_{b}(x): axial stress in the rock bolt at the position of x; E

_{m}: elastic modulus of the confining medium; and σ

_{m}(x): axial stress applied on the confining medium.

_{m}: cross-section area of the confining medium; and r

_{b}: rock bolt radius.

### 4.2.3 Pull-Out Stages of Rock Bolts

#### 4.2.3.1 Elastic Stage

_{p}into Eq. (4.24) leads to:

_{sof}: pull-out load when the bolt/grout interface starts softening.

#### 4.2.3.2 Elastic-Softening Stage

_{s}is the softening length. In the elastic-softening stage, the interface in the domain [0, L − a

_{s}] is still elastic. A boundary condition is Eq. (4.20) and the other boundary condition is:

_{s}into Eq. (4.29), the axial stress can be acquired:

_{s}, L], the bolt/grout interface softens. Substituting Eq. (4.1b) into Eq. (4.12) leads to:

_{r}. Therefore, substituting τ(x = L) = τ

_{r}into Eq. (4.37) leads to:

_{1}(L − a

_{s})) is equal to 1. Then, the softening length when the elastic-softening stage ends can be acquired:

_{s1}: the softening length when the elastic-softening stage ends.

#### 4.2.3.3 Elastic-Softening-Debonding Stage

_{r}, the bolt/grout interface enters the elastic-softening-debonding stage. In this stage, the full bolt/grout interface is composed of elastic section, softening section and debonding section, as shown in Fig. 4.5.

_{d}, the bolt/grout interface in the domain [0, L − a

_{d}− a

_{s}] is still elastic. Therefore, Eqs. (4.28), (4.29) and (4.30) are still valid if L is replaced with L − a

_{d}:

_{d}− a

_{s}, L − a

_{d}], the bolt/grout interface softens. Therefore, Eqs. (4.35), (4.36) and (4.37) are still valid if L is replaced with (L − a

_{d}):

_{d}) = τ

_{r}into Eq. (4.47) leads to:

_{d}and a

_{s}. It shows that in this stage, the softening length keeps varying with the debonding length increasing. Combining Eqs. (4.45) and (4.48) together, the shear slippage of the bolt/grout interface at the position of x = L − a

_{d}can be acquired:

_{d}into Eq. (4.46), the axial stress can be acquired:

_{d}, L], the bolt/grout interface debonds and the shear stress equals τ

_{r}:

_{d}= a

_{s}. Substituting it into Eq. (4.48) leads to:

_{s2}: softening length when the elastic-softening-debonding stage ends.

#### 4.2.3.4 Softening-Debonding Stage

_{d}], the bolt/grout interface softens. Substituting Eqs. (4.9), (4.19) and (4.20) and τ(x = L − a

_{d}) = τ

_{r}into Eq. (4.34) leads to:

_{d}into Eqs. (4.58) and (4.59) leads to:

_{d}, L], the bolt/grout interface debonds. Substituting Eqs. (4.19), (4.61) and (4.62) into Eq. (4.52) leads to:

_{d}= L into Eqs. (4.65) and (4.66), the pull-out displacement and load of the rock bolt at the end of the softening-debonding stage can be acquired:

_{b0}: pull-out displacement of the rock bolt at the end of the softening-debonding stage; and F

_{0}: pull-out load of the rock bolt at the end of the softening-debonding stage.

#### 4.2.3.5 Debonding Stage

## 4.3 Calibration of the Input Parameters

_{b}, E

_{b}and E

_{m}, can be acquired based on the rock bolt pull-out scenario. Specifically, r

_{b}and E

_{b}can be acquired from the rock bolt specification provided by the bolt manufacturer. E

_{m}can be acquired by conducting uniaxial compressive strength tests on samples cored from the confining medium.

_{p}equals the pull-out displacement at Point 1. Also, F

_{sof}equals the pull-out load at Point 1. Substituting F

_{sof}and δ

_{p}into Eq. (4.26), τ

_{p}can be acquired.

_{r}can be selected in the range between the pull-out displacement at Point 1 and the pull-out displacement at Point 2. Then, τ

_{r}can be calibrated until the maximum analytical pull-out load fits well with the maximum experimental pull-out load.

## 4.4 Validation of the Analytical Model

### 4.4.1 Validation with a Pull-Out Test

r _{b} (mm) | E _{b} (GPa) | L (m) | E _{m} (GPa) | A _{m} (m^{2}) |
---|---|---|---|---|

14 | 51 | 3 | 6 | 1 |

τ _{p} (MPa) | δ _{p} (mm) | τ _{r} (MPa) | δ _{r} (mm) |
---|---|---|---|

2.2 | 3.57 | 1 | 8.91 |

### 4.4.2 Validation with the Other Pull-Out Test

r _{b} (mm) | E _{b} (GPa) | L (m) | E _{m} (GPa) | A _{m} (m^{2}) |
---|---|---|---|---|

19 | 83 | 10 | 57 | 1 |

τ _{p} (MPa) | δ _{p} (mm) | τ _{r} (MPa) | δ _{r} (mm) |
---|---|---|---|

1.34 | 10.37 | 0.47 | 35.02 |

## 4.5 Parametric Study

### 4.5.1 Elastic Modulus of the Confining Medium

r _{b} (mm) | E _{b} (GPa) | L (m) | E _{m} (MPa) | A _{m} (m^{2}) |
---|---|---|---|---|

10 | 196 | 1.5 | 10 | 1 |

50 | ||||

90 |

τ _{p} (MPa) | δ _{p} (mm) | τ _{r} (MPa) | δ _{r} (mm) |
---|---|---|---|

2 | 1.5 | 0.8 | 3.5 |

### 4.5.2 Shear Strength of the Bolt/Grout Interface

r _{b} (mm) | E _{b} (GPa) | L (m) | E _{m} (GPa) | A _{m} (m^{2}) |
---|---|---|---|---|

10 | 196 | 1.5 | 1 | 1 |

τ _{p} (MPa) | δ _{p} (mm) | τ _{r} (MPa) | δ _{r} (mm) |
---|---|---|---|

2 | 1.5 | 0.5 | 3.5 |

4 | |||

6 |

### 4.5.3 Residual Shear Strength of the Bolt/Grout Interface

r _{b} (mm) | E _{b} (GPa) | L (m) | E _{m} (GPa) | A _{m} (m^{2}) |
---|---|---|---|---|

10 | 196 | 1.5 | 1 | 1 |

τ _{p} (MPa) | δ _{p} (mm) | τ _{r} (MPa) | δ _{r} (mm) |
---|---|---|---|

4 | 1.5 | 0.5 | 3.5 |

1 | |||

1.5 |