# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 4 Example Analysis and Solution Validation
### Numerical Solution  
In order to verify the accuracy of the theoretical solution, the finite difference software FLAC3D was adopted to calculate the magnitude of the ultimate pullout capacity of an expanded anchor cable, and the results were compared with the theoretical upper-bound solution.  
Corresponding to the theoretical analysis in the above sections, only the ultimate pullout capacity of a single expanded anchor cable arranged in a homogeneous soil stratum was analyzed in the numerical simulations. The three-dimensional numerical model was established according to the conditions of the expanded anchor cable considered in this paper (Figure 1), as shown in Figure 13. In this Figure, the parameters of the anchor cable accorded with those of case M1 listed in Table 1. In addition, the two bearing bodies had the same length of 0.5 m (Figure 14). In order to ensure the simula  
Figure 11: Iteration process of optimal solution (case M3).  
Figure 12: Velocity discontinuity surface: (**a**) 3D model diagram; (**b**) cross-sectional diagram.  
width, and height of the numerical model were 22.4 m, 20 m, and 17.0 m, respectively (as shown in Figure 13). The boundary conditions were set to normal constraints for all four vertical model planes, and the displacements at the bottom of the model were fixed. On the excavation side of the foundation pit, normal constraints were applied below the bottom of the pit. In the model, 3D grids were used to model the soil, the steel strand, the bearing body, and the grouting body of the anchorage segment. The steel strand and bearing body adopted an elastic constitutive model, and the other 3D grids adopted the Mohr-Coulomb constitutive model. The calculation parameters of the soil and anchor cable are shown in Table 3.  
the effect of isolating pipes outside the steel strand. The pullout capacity provided by the free segment of the anchor cable was much smaller than that of the anchorage segment; hence, it was neglected in the model. Thus, the cohesion and friction angle of the contact surface between the grouting body and the surrounding soil were also taken as 0 in the free segment.  
In the numerical simulation, the pullout process of the expanded anchor cable was simulated by applying the pullout force gradually from a small magnitude to a large magnitude that was uniformly taken on by the four steel strands. The ultimate pullout capacity was determined according to the following computation convergence characteristics: the pullout stage at which the computation stopped converging was first determined, and then the pullout force in the previous pullout stage could be considered as the ultimate pullout capacity.  
Figure 15 depicts the relationship between the pullout force \(P\) and the slipping displacement \(S\) of the anchorage segment in case M1. As can be seen in Figure 15, when the pullout force applied to the steel strands was small, the displacement of the anchorage segment was small; when the applied pullout force was close to the ultimate pullout capacity, the anchorage segment presented significant slip; indeed, when the pullout force exceeded the ultimate pullout capacity, the slipping displacement of the anchorage segment could not achieve a convergence state. To be specific, when the pullout force was less than 750 kN, the slipping displacement was small; when reaching 750 kN, the slipping displacement of the anchorage segment was only 42.9 mm; when the pulling force continued to increase and reached the ultimate pullout capacity of 800 kN, a large slip with a magnitude of 570 mm occurred in the anchorage segment, but the anchor-cable-soil system could still approach an equilibrium state; when the pullout force reached 850 kN, the computation could not approach a convergence state. Thus, it could be concluded that the ultimate pullout capacity of the expanded anchor cable was 800 kN.