## 1 Introduction

## 2 Background

### 2.1 Instrumented Bolts

### 2.2 Numerical Modelling

## 3 Discontinuity-Bolt Angle

### 3.1 Calculating the Discontinuity-Bolt Angle

_{0}and N

_{0}are the shear load and axial load at point O. A is the point of maximum bending strain and p

_{u}is the rock/grout reaction. According to the analytical model, the elastic axial strain in the rock bolt is dependent on the angle between the rock bolt and the discontinuity and the shear displacement of the bolt. The shear displacement of the bolt can be calculated from using the bending strain and La (active length) obtained from the instrumented rock bolt strain data. Once the shear displacement is known the angle between the rock bolt and the discontinuity can be back calculated from the axial strain obtained from the instrumented rock bolt strain data.

_{o}can be calculated as:

_{z}is the area moment of inertia along the z-axis of the bolt and D

_{b}is the bolt diameter.

_{y}is calculated as:

_{A}is the length of section OA (measured from the instrumented rock bolt data), p

_{u}is the bearing capacity per unit length of the soil.

_{u}before reaching maximum value is calculated as:

_{u}is given by:

_{c}is the unconfined compressive strength (UCS) of rock.

_{1}is the stiffness of the bond-slip model in the elastic stage (Ma, Zhao et al. 2019), θ is the bolt deflection angle, A is the area of the bolt cross-section, α is the angle between bolt and discontinuity, and Δ

_{ext}is the axial extension in the bolt.

### 3.2 Validation of Discontinuity-Bolt Angle Calculation

#### 3.2.1 Double Shear Test

Test | Concrete | UCS (MPa) |
---|---|---|

1 | Low strength | 20.5 |

2 | Low strength | 18.5 |

3 | Low strength | 22.8 |

Average UCS | 20.6 | |

4 | Medium strength | 28.1 |

5 | Medium strength | 33.5 |

6 | Medium strength | 27 |

Average UCS | 29.5 |

Elastic modulus (GPa) | 200 |

Diameter (mm) | 22 |

Yield strength (MPa) | 340 |

Ultimate strength (MPa) | 490 |

#### 3.2.2 Angled Shear Test

Bolt | |||||
---|---|---|---|---|---|

Bolt diameter (mm) | Bolt grade | Bolt length (mm) | Number of slots | Rebar design | |

19 | 60 | 1198.5 ± 7.5 | 2 | Double threaded ends | |

Concrete | |||||

Average compressive strength (MPa) | Average modulus of elasticity (GPa) | ||||

31 | 28 |

#### 3.2.3 Numerical Modelling

Elastic modulus (MPa) | UCS (MPa) | UTS (MPa) |
---|---|---|

Concrete | ||

26,000 | 30 | 3 |

Grout | ||

3,000 | 184 | 13.8 |

Interface | ||

80,000 | 0.57 | 0 |

#### 3.2.4 Validation with Laboratory and Numerical Modelling Results

Test | E _{b} (bending strain at A) | E _{a} (bending strain at A) | L _{A} (length of section OA) |
---|---|---|---|

DST 1 | 1772 | 246 | 0.0174 |

DST 2 | 3692 | 327.5 | 0.0312 |

DST 3 | 1635 | 168.5 | 0.0247 |

DST 4 | 717.5 | 103.5 | 0.0191 |

DST 5 | 1042 | 82.2 | 0.0386 |

DST 6 | 1110 | 266 | 0.0300 |

DST (Kostecki 2019) | 970 | 80 | 0.0313 |

DST 80° (Jessu et al. 2016) | 272 | 325 | 0.0300 |

60° Numerical model | 446 | 813 | 0.0160 |

45° Numerical model | 140 | 620 | 0.02 |

## 4 Calibration of Pile Element Parameters

### 4.1 Load vs Displacement Plot

_{0}can be calculated using Eq. (1). The axial load N

_{0}can be calculated as:

### 4.2 Validation with Laboratory Double Shear Test

## 5 In-Situ Modelling

### 5.1 In-Situ Instrumented Rock Bolt Study

Length (m) | 2.4 |

Diameter (mm) | 22 |

Young’s modulus (GPa) | 200 |

Yield strength(kN) | 270 |

Failure strain | 0.15 |

### 5.2 Calibration of the Numerical Model with In-Situ Data

UCS (MPa) | 181 |

Young’s modulus (GPa) | 64.2 |

Poisson’s ratio | 0.324 |

Cohesion (MPa) | 27.5 |

Friction angle | 35 ^{o} |

Tensile strength (MPa) | 8.1 |

Defect set | Dip/Dip direction (°) | Type | Trace length (m) | Planarity | Roughness |
---|---|---|---|---|---|

1 | 64° → 110 | Joints and quartz veins | 1–7 | Planar and undulating | Rough and smooth |

2 | 51° → 259 | Foliation and joints | 1–10 | Planar and undulating | Rough and smooth |

3 | 59° → 020 | Joints and quartz veins | 1–5 | Planar | Rough |

4 | 85° → 179 | Joints | > 5 | Planar and Stepped | Rough to slightly rough |

Depth 627 m | Major (MPa) | 54.3 | 13 | 313 |
---|---|---|---|---|

Intermediate (MPa) | 24.1 | 54 | 204 | |

Minor (MPa) | 8.0 | 32 | 52 |

Bolt 7 | Bolt 10 | |
---|---|---|

Diameter (m) | 0.022 | 0.022 |

L _{A} (m) | 0.019 | 0.0127 |

E _{b} (Bending strain) | 370e−6 | 1529e–6 |

E _{a} (Axial strain) | 226e−6 | 1467e–6 |

Bolt-Discontinuity angle (°) | 81.2 | 70 |

Bolt angle with horizontal (°) | 50 | 60 |

Discontinuity orientation with rock (°) | 48.8 | 50 |

Normal stiffness (MPa/m) | 50,000 |

Shear stiffness (MPa/m) | 17,000 |

Friction angle (°) | 26.5 |

Length (m) | 2.4 |

Diameter (mm) | 22 |

Young’s Modulus (GPa) | 200 |

Yield Strength(kN) | 270 |

Failure strain | 0.15 |

Normal spring stiffness (GPa) | 20 |

Normal spring cohesion (Nm) | 2.75e6 |

Normal spring friction angle | 25 |

Shear spring stiffness (GPa) | 20 |

Shear spring cohesion (Nm) | 8e5 |

Shear spring friction angle | 23 |