# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 6 Conclusions  
A failure model based on the upper-bound theorem of limit analysis was presented in this paper. The expressions of the upper-bound solution for the ultimate lateral resistance and the ultimate end resistance were derived according to the principle of virtual power, and the final solution for the ultimate end resistance was determined by the particle swarm optimization algorithm. The upper-bound solution for the ultimate pullout capacity was obtained by summing the ultimate lateral resistance and ultimate end resistance. The accuracy of the proposed analytical solution was verified through a comparison with three-dimensional numerical simulations, which were calculated by the finite difference software FLAC3D. In addition, the superiority of the proposed analytical solution was well demonstrated through a comparison with an existing theoretical method. Regarding the application of the failure model, the influence of the anchor cable's key parameters on the ultimate lateral resistance, end resistance, and ultimate pullout capacity was discussed in detail. The following conclusions could be drawn:  
1. A failure model of an expanded anchor cable located in a homogeneous stratum was constructed. The logarithmic spiral failure model was used as the failure model of the soil region at the front surface of the anchorage segment, representing the first time that this model has been implemented to calculate the end resistance of an expanded anchor cable. The failure mechanism of the anchorage side surface was assumed to satisfy the slippage model. The expressions of the ultimate lateral resistance and ultimate end resistance could be derived, respectively, from the power calculations for the two models and several algebraic operations. Due to the particularity of the boundary conditions in this paper, the rotation center points of the logarithmic spiral lines were uncertain. Therefore, the particle swarm optimization algorithm was used to determine the optimal solution of the end resistance. 2. The theoretical calculation results were compared with the numerical simulation in three cases. The results showed that the error between the numerical solution and the theoretical solution for the lateral resistance in the three cases was only 6.0%; however, the error for the end resistance was relatively large, though still within 30%, and so the maximum error for the ultimate pullout capacity was 7.7%. These errors were much smaller than the errors between the numerical simulation and the theoretical calculation method proposed in [18] for the same cases. This indicated that  
Figure 19: The ultimate force variation curves according to the change in the buried depth of anchor cable. the proposed theoretical model had high reliability and superiority in the analysis of the ultimate pullout capacity of the expanded anchor cable. 3. The ultimate lateral resistance and total ultimate pullout capacity increased significantly with the increase in the anchorage segment diameter, anchorage segment length, and buried depth of the expanded anchor cable. The ultimate end resistance increased significantly with the buried depth, increased slightly with the anchorage segment diameter, and was almost unaffected by the anchorage segment length. 4. With the increase in the inclination angle of the anchor cable, the end resistance and the ultimate pullout capacity gradually decreased, while the lateral resistance increased first and then decreased. However, in general, the change in the inclination angle of the anchor cable had a relatively small effect on the ultimate lateral resistance. Conceptualization, B.W. and L.M. ; methodology, B.W. and L.M. ; software, X.C. and Y.Y. ; validation, X.C., C.X. and Y.Y. ; formal analysis, Y.Y. ; investigation, C.X. and Y.Y; resources, L.M. ; data curation, X.C. ; writing--original draft preparation, X.C. and Y.Y. ; writing--review and editing, X.C., B.W. and L.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding.