# The Bolt Anchorage Performance of Fractured Rock under a Freeze-Thaw Cycle Load
## 4 Analysis and Discussion  
### The Influence of Fracture Dip Angle on Anchor Specim Strength  
Through a multifactorial analysis of variance and visual examination of the experimental results, it has been discerned that the fracture dip angle stands as the primary controlling factor governing the strength of specimens. As shown in Figure 4, under varied conditions, the strength of specimens with a 45\({}^{\circ}\) fracture dip angle consistently falls below that of specimens without fractures and those with a 90\({}^{\circ}\) fracture dip angle (see Figure 3). With equivalent anchor restraint conditions and freeze-thaw cycles, the following relationships in terms of strength among the specimens emerge: 45\({}^{\circ}\) fracture dip angle < 90\({}^{\circ}\) fracture dip angle < no fractures. Under conditions devoid of anchor restraints, the tangent modulus of specimens exhibits the following relationship: 45\({}^{\circ}\) fracture dip angle < 90\({}^{\circ}\) fracture dip angle \(\approx\) no fractures. For instance, considering specimens subjected to ten freeze-thaw cycles without anchor restraints, the average strengths are 18.3 MPa, 5.2 MPa, and 13.4 MPa, respectively, for specimens with no fractures, a 45\({}^{\circ}\) fracture dip angle, and a 90\({}^{\circ}\) fracture dip angle. Notably, the strength of specimens with a 45\({}^{\circ}\) fracture dip angle and 90\({}^{\circ}\) fracture dip angle accounts for 28.4% and 73.4%, respectively, of that of specimens without fractures. The average tangent moduli for specimens with no fractures, a 45\({}^{\circ}\) fracture dip angle, and a 90\({}^{\circ}\) fracture dip angle are 3.26 GPa, 1.38 GPa, and 2.12 GPa, respectively. This indicates that the strength of specimens with a 45\({}^{\circ}\) fracture dip angle and 90\({}^{\circ}\) fracture dip angle corresponds to 42.3% and 65.2%, respectively, of that of specimens without fractures, as depicted in Table 6.  
Figure 3: Example stress–strain curve of strain-softening/hardening model. (\(e^{e}\): elastic strain, \(e^{p}\): plastic strain). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Density (kg/m\({}^{3}\))**} & **Young’s Modulus** & **Poisson’s Ratio** & **Yield Strength** & ** Elastoplastic Tangent** \\  & & **(MPa)** & **(MPa)** & **Modulus (GPa)** \\ \hline Value & 7900 & 200 & 0.25 & 450 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of bolt mechanical model. The presence of fracture angles induces shear failure along the predetermined planes of weakness in the specimens, with inclined sliding planes, typified by a 45\({}^{\circ}\) fracture dip angle, proving less advantageous than vertical sliding planes. Consequently, the stress conditions become more intricate, resulting in heightened shear stress on structural planes under identical loads, thereby rendering them more susceptible to failure. Similarly, variations in the tangent moduli follow suit, with the deformation of specimens with a 45\({}^{\circ}\) fracture dip angle primarily governed by the shear modulus on weak structural planes, manifesting in their lower values and diminished overall stiffness compared to specimens without anchor restraints. This discrepancy in the deformation behaviour between specimens with and without anchor restraints signifies a distinct phenomenon warranting further investigation. Combining specimens subjected to varying freeze-thaw cycles for statistical analysis, as depicted in Figure 5, reveals that specimens with a 45\({}^{\circ}\) fracture dip angle exhibit the lowest degree of dispersion in their strength, followed by those with a 90\({}^{\circ}\) fracture dip angle. This variability in dispersion data underscores the distinct characteristics of the reliability of strength for inclined fractures approximating 45\({}^{\circ}\), compared to those nearing a vertical orientation.