## 4 Parametric Study  
This study analyzed the influence of parameters (geometry, vehicle design, and ground strength) on the soil thrust over clay slope using the proposed upper bound theoretical solution. Table 2 lists the parameters considered; this study considered the ground slope as a geometry parameter and undrained shear strength \(c_{u}\) as a ground strength parameter. For the vehicle parameters, this study included ground contact pressure \(\sigma_{c}\), track height \(H\), track length \(L\), and track width \(D\). Table 2 also presents the values and ranges of the considered parameters. Figure 8: Equivalent plastic strain distribution when undrained shear strength \(c_{u}\) is (**a**) 5 kPa, (**b**) 7.5 kPa, (**c**) 10 kPa, (**d**) 15 kPa, (**e**) 20 kPa, and (**f**) 25 kPa. According to [PERSON] et al. [30], 95.5% of the seabed has slopes of 6\({}^{\circ}\) or less, while only 4.5% exceed this angle; consequently, the slope angle (\(\beta\)) was set within a range of 0\({}^{\circ}\) (which represented the flat surface) to 10\({}^{\circ}\) (which corresponded to the extreme slope). For the ground contact pressure (\(\sigma_{c}\)), this study established a range of 5 to 50 kPa to cover the diverse types of underwater heavy vehicles. For the undrained shear strength \(c_{u}\), following [PERSON], [PERSON], [PERSON] et al. [27], this study set a range of \(c_{u}\) from 2 to 40 kPa. Following [PERSON] et al. [22], the height, length, and width of a track system were set to 0.035, 0.124, and 0.124 m, respectively. Additionally, to assess the impact of these aspect ratios, the upper bound solution was calculated and compared while varying the dimensions of height, length, and width. This study parametrically studied the responses of the stationary upper bound solution \(F_{x}\) (=min(\(F_{b}\), \(F_{t}\), \(F_{p}\))) with respect to the considered parameters (\(\beta\), \(c_{u}\), \(\sigma_{c}\), \(H\), \(L\), and \(D\)). Figure 9 illustrates how soil thrust responds to the changes in \(\beta\) and \(\sigma_{c}\), while \(c_{u}\) and \(H\), \(L\), and \(D\) remain constant (dimensions cited from [PERSON] et al. [22]). Figure 9a plots the soil thrust versus the angle \(\beta\), where the block failure predominates due to low undrained shear strength (\(c_{u}\) = 2 kPa), while Figure 9b illustrates the soil thrust against the angle \(\beta\), where the triangle wedge failure predominates because of high undrained shear strength (\(c_{u}\) = 40 kPa). As the angle \(\beta\) increases for all failure modes, there is a noticeable decrease in soil thrust; this decrease is further amplified as \(\sigma_{c}\) increases. In very soft ground conditions (Figure 9a), soil thrust becomes negative with higher \(\beta\) and \(\sigma_{c}\), indicating that the tracked vehicle cannot climb the slope and begins to slip downward. On the other hand, in stiff ground conditions (Figure 9b), where the triangular wedge failure is dominant, the soil thrust remains positive, enabling the vehicle to climb the slope. Figure 10 displays how soil thrust changes with \(c_{u}\) and \(\sigma_{c}\) while \(\beta\) is constant, and dimensions \(H\), \(L\), \(D\) from [PERSON] et al. [22] are also fixed. As \(\sigma_{c}\) increases, the kink point, where the failure mode transitions from the block failure mode to triangle wedge failure mode, moves to the upper left side. This implies that the tendency of the block failure mode increases with the vehicle load. In Figure 10 (and Figure 9a), when \(\sigma_{c}\) increases with low \(c_{u}\)