## 4 Example Calculation and Comparative Analysis
### Example 1  
The strip foundation of example 1 is located at a two-layered soil with an upper sand soil layer and a lower clay layer. The foundation width of the two-layered strip foundation is 2 m, regardless of the ground overload. The soil parameters are listed in Table 1. The angle of internal friction is not considered for the clay layer, and the clay weight is taken as 19 kN/m\({}^{3}\). The parameters of the IRMO algorithm are \(N=50\), \(M=51\), and \(G=500\).  
The ultimate bearing capacity of the foundation calculated by the IRMO algorithm in this paper is compared with the corresponding calculation results of predecessors ([PERSON] and [PERSON] [24] and [PERSON] [25]), as shown in Table 2. The solution in this paper is higher than [PERSON]'s, [PERSON]'s, and [PERSON]'s solutions. The reason for this kind of deviation may be that although the three solutions are the same upper bound solution of limit analysis, the specific optimization algorithms implemented are different. For example, the IRMO algorithm is used in this paper to search and solve, and there are 25 rigid blocks for calculation. [PERSON] uses the Monte Carlo optimization method to solve, and the rigid block division is not as detailed as in this paper, resulting in different calculation results. Therefore, the comparative analysis in this example verifies the feasibility of the IRMO algorithm in calculating the ultimate bearing capacity of a two-layered foundation.  
After repeated calculation 20 times in this example, the search results of the critical sliding surface of the two-layered foundation were obtained by running the IRMO algorithm 20 times, as shown in Figure 11. The corresponding upper limit solution of the ultimate bearing capacity of the two-layered foundation is shown in Figure 12.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Soil Layer Classification** & \(H\) **(m)** & \(\varphi\) **(\({}^{\circ}\))** & \(c\) **(kPa)** & \(\gamma\) **(kN/m\({}^{3}\))** \\ \hline Upper soil & 4 & 35 & 0 & 17 \\ Lower soil & – & – & 50 & 19 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soil layer parameter information of calculation example.  
Figure 11: Search results of the critical sliding surface of the two–layered foundation after 20 times IRMO algorithm operation.  
Figure 11 shows that the results of 20 searches of the IRMO algorithm almost overlap, the relative error control between the calculation results of each generation is quite small. Figure 12 shows that the standard deviation is 0.2599, and the average is 810.7433 (the position of the red line in the figure). Therefore, it demonstrates the great stability of the algorithm in determining the critical sliding surface of the two-layered strip foundation and calculating the corresponding ultimate bearing capacity.  
Figure 13 shows the convergence of the upper limit solution of the ultimate bearing capacity of the two-layered strip foundation with the number of iterations after the algorithm has been run three times under the same conditions. The IRMO algorithm has gradually become stable after 250 generations. The convergence rate of the proposed algorithm is compared with the convergence generations of [PERSON] [26] which shows that the algorithm in this paper is efficient and has an ideal convergence efficiency.