## 4 Results and Discussion
### Validity Study of Equation (4)  
This section further discusses the validity of Equation (4). In the equation, parameter \(A\) is the theoretical peak anchorage force when the radial stiffness of the confinement \(S\)\(\rightarrow\)+\(\infty\), i.e., the peak axial force of a bolt anchored within an extremely rigid confining rock mass. A previous study [41] identified that rockbolting failure while subjected to the axial load can be classified into two modes: parallel shear failure and expansive slippage failure, as shown in Figure 9.  
Figure 8: Relationship between tube radial stiffness and measured peak anchorage force. (**a**) Bold with rib spacing of 12 mm; (**b**) bolt with rib spacing of 48 mm.  
Parallel shear failure occurs along a cylindrical interface between bolt and resin, whereas expansive slippage failure takes place within the resin material under the transverse rib of the bolt and then slips along the axial direction while expanding in the surrounding material. Under the condition of the infinite radial stiffness of the confining tube, expansion slip failure cannot occur. In other words, anchorage failure can merely exhibit parallel shear failure. In such a situation, the axial force of the bolt equals the shear stress along the failure interface [41]:  
\[F_{T}=\tau_{bg}A_{bg}+\tau_{g}A_{g} \tag{7}\]  
where  
\(F_{T}\) is the axial load (N);  
\(\tau_{bg}\) is the shear stress along the resin/bolt interface (MPa);  
\(A_{bg}\) is the area of the resin/bolt interface (mm\({}^{2}\));  
\(\tau_{g}\) is the shear strength of the resin of the parallel shear failure interface (MPa); and  
\(A_{g}\) is the area of the parallel shear failure interface (mm\({}^{2}\)).  
In critical state,  
\[\tau_{bg}=c_{bg}+p\tan\phi_{bg} \tag{8}\]  
\[\tau_{g}=c_{g}+ptan\phi \tag{9}\]  
where  
\(c_{bg}\) is the cohesion of the resin-bolt interface (MPa);  
\(p\) is the confining pressure when failure occurs (MPa);  
\(\phi_{bg}\) is the internal friction angle of the resin-bolt interface (\({}^{\circ}\));  
\(c_{g}\) is the resin cohesion (MPa); and  
\(\phi\) is the internal friction angle of resin (\({}^{\circ}\)).  
According to the specimen size and dextral-bolt dimensions (Table 1 and Figure 9), ratios _l/L_ were 64% and 91% for the original and modified dextral-threaded bolt, respectively. The outer diameter of the bolt \(R\) = 11.8 mm under both conditions. Therefore, for the bolt with a rib spacing of 12 mm, we have \(A_{bg}=2657\) mm\({}^{2}\) and \(A_{g}=4757\) mm\({}^{2}\). For the bolt  
Figure 9: Rockbolting failure modes. Red line: parallel shear failure; blue line: expansive slippage failure [41].  
with the rib spacing of 48 mm, we have \(A_{bg}=664\) mm\({}^{2}\) and \(A_{g}=6750\) mm\({}^{2}\). For resin mechanical properties, \(c_{g}=19.1\) MPa and \(\phi=\)32.3\({}^{\circ}\); for the mechanical properties resin-steel interface, we assumed \(c_{bg}=c_{g}/2\) and \(\phi_{bg}=\phi/2\). Confining pressure \(p\) is mainly caused by the shrinkage of the resin, thus, we assumed \(p=5.0\) MPa [41]. Substituting these parameters into Equations (7)-(9), the theoretical \(F_{712}=135.1\) kN, and \(F_{748}=157.6\) kN. Compared with the experimental data of \(F_{p12}=146.0\) kN and \(F_{p48}=174.8\) kN, two theoretical values agreed well.  
According to Equation (3), \((A-B)\) is the anchorage force when radial stiffness S\(\rightarrow\)0. At present, there is no commonly accepted approach to estimate the peak anchorage force in such situations. The minimal peak anchorage force should be related to bolt type and grout strength but is independent of the surrounding rock conditions. In this study, the theoretical peak anchorage force without the radial stiffness of confining material for the original dextral bolt was 58.9 kN, which can be used as a reference for subsequent study.