# Upper-Bound Limit Analysis of Ultimate Pullout Capacity of Expanded Anchor Cable
## 3 Upper-Bound Solution of Ultimate Pullout Capacity of Expanded Anchor Cable
### Ultimate End Resistance Calculation
#### 3.2.2 Determination of Minimum Upper-Bound Solution of End Resistance  
The upper-bound solution of the ultimate end resistance was determined by the coordinates of the rotation center point O. It was necessary to decide the coordinates of point O that could minimize the possible upper-bound solution in order to obtain the real upper-bound solution for the ultimate end resistance. In the present study, the particle swarm optimization (PSO) algorithm [28; 29] was adopted to identify the optimal upper-bound solution for the ultimate end resistance of the expanded anchor cable and determine the corresponding failure model under the determined ultimate end resistance.  
The application of the particle swarm optimization algorithm was as follows: (1) each particle represented the coordinates of the rotation center O, denoted \(\mathbf{x}_{i}^{j}(Y_{i}^{j},Z_{i}^{j})\), with index \(i\) representing the number of particles and index \(j\) representing the number of iterations; (2) each particle's fitness, which was represented by the ultimate end resistance corresponding to the rotation center \(\mathbf{x}_{i}^{j}(Y_{i}^{j},Z_{i}^{j})\), could be calculated according to Equation (9). For the initial population number \(N\), each particle had an initial position \(\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\cdots,\mathbf{x}_{N}^{0}\) and an initial velocity \(\mathbf{v}_{1}^{0},\mathbf{v}_{2}^{0},\cdots,\mathbf{v}_{N}^{0}\). The particle fitness values calculated from all the initial positions were regarded as the individual optimal solutions \(\mathbf{p}_{1}^{0},\mathbf{p}_{2}^{0},\cdots,\mathbf{p}_{N}^{0}\), and the minimum value of all the individual optimal solutions was taken as the group optimal solution \(\mathbf{p}_{\mathbf{g}}^{0}\). In the second step of the iteration, the particle velocity and position were updated according to Equations (13) and (14), respectively, and then the individual optimal solutions \(\mathbf{p}_{1}^{1},\mathbf{p}_{2}^{1},\cdots,\mathbf{p}_{N}^{1}\) and the group optimal solution \(\mathbf{p}_{\mathbf{g}}^{1}\) could be obtained following the same method. This process was repeated until \(G\) iterations had been completed, and the individual optimal solutions \(\mathbf{p}_{1}^{G},\mathbf{p}_{2}^{G},\cdots,\mathbf{p}_{N}^{G}\) and group optimal solution \(\mathbf{p}_{\mathbf{g}}^{G}\) could be obtained. In each iteration, the optimal solution (i.e., minimum upper-bound solution for the end resistance) needed to be compared with the group optimal solution. If the optimal solution was greater than the group optimal solution, the optimal solution was updated with the current group optimal solution. To be specific, in the first iteration, the optimal solution was equal to \(\mathbf{p}_{\mathbf{g}}^{0}\); after G iterations, the optimal solution was the minimum of \(\mathbf{p}_{g}^{1},\mathbf{p}_{g}^{2},\cdots,\mathbf{p}_{g}^{G}\). In order to ensure the accuracy of the calculation results, the population number and the number of iterations were determined through a corresponding convergence analysis:  
\[\mathbf{v}_{i}^{j}=w\mathbf{v}_{i}^{j-1}+c_{1}r_{1}(\mathbf{p}_{i}^{j-1}- \mathbf{x}_{i}^{j-1})+c_{2}r_{2}(\mathbf{pg}^{j-1}-\mathbf{x}_{i}^{j-1}) \tag{13}\]  
\[\mathbf{x}_{i}^{j}=\mathbf{x}_{i}^{j-1}+\mathbf{v}_{i}^{j} \tag{14}\]  
where \(w\) is the weighting factor; \(c_{1}\) and \(c_{2}\) are the individual learning factor and group learning factor, respectively; and \(r_{1}\) and \(r_{2}\) are two random numbers with a value in the range [0, 1], which were introduced to increase the computation randomness.