## 2 Theoretical Analysis
### Failure Mode and Force Analysis  
The degree of influence of factors on the mechanical properties, which range from most significant to least significant, are as follows: bolt yield stress, diameter and surface shape. The failure modes of the specimens can be divided into three main types:  
1. Bolt pull-out failure. This failure mode occurs when the tensile stress of the bolt is higher than the bond strength between the bolt and the CBGM, resulting in the bolt pulling out of the CBGM before reaching tensile stress. The mode of destruction is shown in Figure 1a.  
Using the dx micro-terminus, if the bolt pulls out from the CBGM or splitting/shear failure occurs, the force of the bolt appears as shown in Figure 2a, and the force relationship is written as follows:  
\[F>\int_{0}^{l_{\text{a}}}\pi d\tau\text{d}x \tag{1}\]  
where \(F\) is the bolt pulling force (kN), \(d\) is the bolt diameter (mm), \(\tau\) is the bond stress between the bolt and CBGM (MPa) and \(l_{\text{a}}\) is the anchorage length of the bolt (mm).  
Assuming that the bond stress is evenly distributed along the bolt, Equation (1) can be expressed as follows:  
\[F>\pi dl_{\text{a}}\tau \tag{2}\]  
Bolt fracture failure. The bond strength between the bolt and the CBGM is higher than the tensile stress of the bolt when the bolt stress grade is lower or the stress grade of the CBGM is higher. Therefore, the bolt stress reaches tensile strength before pulling out, which results in the bolt fracturing and being destroyed. The mode of destruction is shown in Figure 1b.  
If the bolt fractures, the force of the bolt appears as shown in Figure 2b, and the force relationship can be calculated as follows:  
\[F>\int_{0}^{S}\sigma\text{d}\text{A} \tag{3}\]  
where \(S\) is the area of the bolt section (mm\({}^{2}\)), while \(\sigma\) is the stress from the bolt section (MPa).  
Figure 1: Schematic diagram of failure modes between bolt and Cement-based grouting materials (CBGM): (**a**) bolt pull-out failure; (**b**) bolt fracture failure; (**c**) CBGM splitting failure.  
Figure 2: Schematic diagram of bolt damage force. (**a**) Schematic diagram of bolt pulls out/splitting failure force. (**b**) Schematic diagram of bolt fracture/failure force.  
These values can be obtained via the following formula:  
\[F>\sigma S \tag{4}\]
3. Specimen splitting failure or shear failure. The protruding transverse ribs on the bolt surface produce an oblique extrusion force on the CBGM, which can be divided into the axial component force along the bolt surface and the radial component force. The axial component force makes the CBGM subject to circumferential tensile force, and the radial component force causes longitudinal cracks in the CBGM with the increase in load. The CBGM will be split and broken if the CBGM is not restrained by concrete and circumferential hoops. Conversely, the protruding transverse ribs on the bolt surface will crush and shear the CBGM. The mode of destruction is shown in Figure 1c.  
If the specimen splitting failure or shear failure occur, the bolt pulling force will be the same as bond strength between the bolt and the CBGM. The force relationship is written as follows:  
\[F=\pi\mathrm{d}l_{\mathrm{a}}\tau \tag{5}\]