## 4 Example Verification and Engineering Application  
### Example Verification and Analysis  
In a field pullout experiment conducted by [PERSON] et al. [26] in a mountainous area, as depicted in Figure 6, the anchoring bar had a diameter (\(D\)) of 42 mm, a Young's modulus (\(E\)) of 210 GPa, and an anchoring length (\(L\)) of 3 m. The grouting body was composed of fine gravel concrete with a uniaxial compressive strength of 27.10 MPa and a deformation modulus of 26 GPa. The diameter of the drilled holes was 150 mm. Strain gauges were strategically placed at 150 mm intervals along the anchored section, capturing changes in displacement and axial force at the top of the anchor during the upward pull-out process. The interfacial mechanical parameters obtained were as follows: \(S_{1}\) = 0.21 mm, \(S_{2}\) = 0.38 mm, \(\tau_{1}\) = 3.84 MPa, \(\tau_{2}\) = 1.15 MPa. Using Equations (14) and (15), the ultimate length of the elastic state of the anchor (\(L_{t}\)) is determined to be 1.045 m, with an ultimate elastic load (\(P_{ik}\)) of 175 kN. Building on the research outlined in Section 3.2, the ultimate length of the plastic state (\(L_{8}\)) is calculated as 207 mm, and the ultimate plastic load (\(P_{sk}\)) is found to be 271 kN. Incorporating field-measured data into Equations (11), (12), (16), (17), (21), and (24), the theoretical curve depicting the relationship between pull-out load and anchor displacement is obtained, as illustrated in Figure 7. The figure reveals distinct stages as the pull-out load increases, namely the elastic stage (OA), plastic stage (AB), and debonding stage (BC). The OA stage represents the elastic phase, with point A acting as the boundary between the elastic and plastic stages. Section AB signifies the plastic stage, with point B marking the transition to the plastic-debonding stage. Section BC corresponds to the debonding stage. Observing Figure 7, the load-displacement relationship obtained through the theoretical calculation aligns well with the measured data. The variance between the measured and calculated values falls within the range of 1.3% to 8.8%, affirming the validity of the theoretical derivation. Figure 5: The flow chart of the calculation method of the anchoring system at each stage. In Figure 8, a comparison of the pull-out load-displacement prediction curve generated by the model presented in this paper is shown alongside other published mathematical models, including the trilinear bond-slip model, double exponential function model, and exponential-hyperbolic composite model. The results of the comparison indicate that, during the elastic state of the anchorage body, the calculation outcomes of each model are relatively close. However, as the anchorage body transitions into the plastic and debonding states, differences emerge in the calculated results and the curve trends. This discrepancy may arise from varying assumptions each model makes for the plastic stage, resulting in a more significant difference in debonding state results compared to the elastic state. It is worth noting that the length of the plastic state of the anchorage body in this project is relatively short, which could contribute to a less pronounced difference in the results during the plastic state. Figure 6: Diagrammatic drawing of anchor pull-out tests. Figure 7: Load–axial displacement curves of the anchor. Utilizing Equations (12), (17), and (22), and field-measured results, the axial force distribution curves of the anchor along the anchoring depth under pull-out loads of \(P\) = 160 kN, 250 kN, and 300 kN are calculated, as depicted in Figure 9. Notably, the calculated theoretical curve of axial force distribution aligns well with the measured curve, effectively reflecting the axial force distribution in the anchoring section of the anchor under the influence of pulling forces. In Figure 9, as the pull-out load increases, the axial force of the anchor does not distribute evenly and exhibits a nonlinear pattern.