## 3 Verification with Numerical Solutions
### Comparison of Upper Bound Solutions with Numerical Analysis Result  
Figure 7 shows the soil thrust (both upper bound and numerical solutions) with respect to \(c_{u}\) with the identical \(\sigma_{c}\), \(\beta\), and dimensions \(H\), \(L\), and \(D\), as cited by [PERSON] et al. [22]. In the numerical analysis results, the soil thrust is calculated by doubling the sum of the reaction force at all nodes on the front panel of the track, where velocity was constrained. Once the soil thrust converged in the numerical analyses, the final soil thrust was estimated by averaging soil thrust values for 1.5 s.  
In Figure 7, solid, dashed, and dotted lines represent \(F_{b}\) (from the block failure mode), \(F_{t}\) (from the triangle wedge failure mode), and \(F_{p}\) (from the trapezoidal failure mode), while the thick gray solid line represents the upper bound solution \(F_{x}\), which is the minimum  
Figure 6: Mesh configurations applied to the numerical analysis: (**a**) lowest mesh density; (**b**) intermediate mesh density; and (**c**) highest mesh density.  
Figure 7: Upper bound solutions (\(F_{b}\), \(F_{t}\), \(F_{p}\)) and minimum upper bound solution \(F_{x}\) overlapped with numerical solutions with different mesh densities.  
between \(F_{b}\), \(F_{t}\), and \(F_{p}\). In Figure 7, when \(c_{u}\) is less than about 8 kPa, \(F_{b}\) is less than \(F_{t}\) and \(F_{p}\); thus, in this case, \(F_{x}=F_{b}\). If \(c_{u}\) is greater than 8 kPa, \(F_{t}\) is smaller than \(F_{b}\) and \(F_{p}\); thus, in this case, \(F_{x}=F_{t}\). Figure 7 also plots the numerical solutions overlapped with the upper bound solutions. In Figure 7, the green, blue, and red symbols correspond to the numerical solutions with the lowest, intermediate, and highest mesh densities. As mesh density increases, the numerical solution more closely approximates the analytical solution proposed in this study; the general rule in the numerical simulation is that the numerical analysis with the finer mesh approaches the exact solution; thus, the analytical solution in this study is very close to the exact solution.  
Figure 8 shows the distribution of equivalent plastic strain in the ground from the results of the numerical simulation with the highest mesh density. When \(c_{u}=4\) kPa (Figure 8a), the numerical solution clearly shows the block failure, which is identical to the failure mode determined by the upper bound solution. When \(c_{u}=\)15, 20, and 25 kPa (Figure 8d-f), the numerical solutions definitely depict the triangular wedge failure mode, which is identical to the failure mode determined by the upper bound solution; in Figure 8d-f, as \(c_{u}\) increases, the length of the triangular wedge decreases. When \(c_{u}\) is 7.5 and 10 kPa, as shown in Figure 8b,c, as the failure mode transitions from the block to the triangular wedge modes, numerical analyses simulated both failure modes concurrently. This implies that, as \(c_{u}\) increases, rather than an abrupt change in the failure mode, both failure modes occur simultaneously. As a result, following the comparison with the numerical solution, the proposed upper bound solution can accurately predict the soil thrust value and the corresponding failure mode.