# A Modified Bearing Capacity Model for Inclined Shallow Anchor Cable with Experimental Verification
## 3 Validation by Model Experiment
### Experimental Result  
Figure 5 illustrates the load-displacement curves of anchor cables at different anchor inclinations. Notably, each anchor inclination underwent three repeated tests, demonstrating that the results within the same group are consistent, thereby confirming the reliability of the experimental findings. Furthermore, the load-displacement curves for different anchor inclinations exhibit similar characteristics. Initially, during the elastic phase, the load increased almost linearly. As the system entered the elastoplastic phase, the rate of increase in the load-displacement curve accelerated, indicating a nonlinear relationship between load and displacement. In the plastic failure phase, the rate of displacement increase further intensified. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elastic Modulus (MPa)** & **Poisson’s Ratio** & **Cohesion (kPa)** & **Internal Friction Angle (\({}^{\circ}\))** & **Density (g/cm\({}^{3}\))** \\ \hline
124.5 & 0.37 & 9.58 & 39 & 1.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Ground parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Samples**} & \multicolumn{2}{c}{**Elastic Modulus (MPa)**} & \multicolumn{2}{c}{**Poisson’s Ratio**} \\ \cline{2-5}  & **Experimental Value** & **Error** & **Experimental Value** & **Error** \\ \hline
1 & 136.4 & 10\% & 0.35 & 5.7\% \\
2 & 122.1 & 2\% & 0.39 & 5.1\% \\
3 & 114.9 & 8\% & 0.37 & 0.3\% \\ Average & 124.5 & / & 0.37 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Elastic modulus and Poisson’s ratio. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Samples** & **Quality (g)** & **Density (g/cm\({}^{3}\))** & **Error** \\ \hline
1 & 356 & 1.81 & 2.3\% \\
2 & 342 & 1.74 & 1.7\% \\
3 & 349 & 1.78 & 0.6\% \\
4 & 339 & 1.73 & 2.3\% \\
5 & 353 & 1.8 & 1.7\% \\
6 & 345 & 1.76 & 0.6\% \\
7 & 333 & 1.70 & 4.0\% \\ Average & 345 & 1.77 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Density. Figure 5: The load–displacement curves for the tests: (**a**) 30\({}^{\circ}\); (**b**) 40\({}^{\circ}\); (**c**) 50\({}^{\circ}\); (**d**); 60\({}^{\circ}\); (**e**) 70\({}^{\circ}\); (**f**) anchor cables at different inclination angles. For analysis, the median curve from the three repeated tests within the same group was selected, as shown in Figure 5f. It was determined that when the displacement increment of the anchor cable exceeds 2 mm during a specified load-holding duration, it can be considered a sign of anchor failure. Regarding the influence of the anchor inclination, it was observed that the ultimate bearing capacity increased as the anchor inclination decreased, which is consistent with the findings of [PERSON] [39]. In the initial phase, the mechanical response was predominantly governed by the elastic deformation of the anchor cable and the rock mass system, with the effect of the inclination being minimal. As loading increased, the nonlinear effects at the interface between the rock mass and the anchor cable began to manifest, leading to a gradual separation of the curves. To further investigate the effects of anchor cable inclination on the failure mechanism, gypsum slurry was poured into the failure pit formed after the anchor cable was fully extracted under laboratory conditions. This approach allows for clearer and more visual observation of the failure surfaces through the hardened gypsum. Figure 6 presents the hardened gypsum at various anchoring forces. As shown in Figure 6, the failure models of the anchor cables, pulled out at different angles, exhibit an asymmetric \"inverted trumpet\" shape, caused by the varying stress distributions around the anchor cable during pull-out. This observation suggests that the failure surfaces at different anchoring points can be approximated as inverted conical.