## 4 Parametric Study
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameters** & & **Values** \\ \hline Geometry & Slope angle & \(\beta\) (\({}^{\circ}\)) & 0–10 \\ \hline Ground Strength & Undrained shear strength & \(c_{u}\) (kPa) & 2–40 \\ \hline \multirow{4}{*}{Vehicle} & Ground contact pressure & \(\sigma_{c}\) (kPa) & 5–50 \\  & Height of the track & \(H\) (m) & 0.035, 0.105 \\ \cline{1-1}  & Length of the track & \(L\) (m) & 0.124, 0.372 \\ \cline{1-1}  & Width of the track & \(D\) (m) & 0.124, 0.372 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and their ranges in the parametric study. Figure 9: Soil thrust versus ground slope angle with varying ground contact pressure \(\sigma_{c}\) with (**a**) very soft clay ground (\(c_{u}\) = 2 kPa) and (**b**) stiff clay ground (\(c_{u}\) = 40 kPa). negative soil thrust occurs, causing the vehicle to slip down. Before the kink points (where block failure mode happens), with fixed \(c_{u}\), higher soil thrust is generated as \(\sigma_{c}\) is lower (\(c_{u}\) = 2 kPa in Figure 10); however, after the kink points (where triangular wedge failure mode happens), with fixed \(c_{u}\), the higher soil thrust is generated as \(\sigma_{c}\) is higher (\(c_{u}\) = 35 kPa in Figure 10). Subsequently, this study examined the effect of dimensions (\(H\), \(L\), and \(D\)) of the track system on vehicle slippage on a clay slope. As shown in Figure 9, the most critical case is that the heavy vehicle climbs up a slope consisting of very soft clay, where the block failure mode is dominant; thus, the present analysis only considered the block failure mode with fixed slope (\(\beta\) = 8\({}^{\circ}\)) and \(\sigma_{c}\) (=40 kPa) and with varied undrained shear strength \(c_{u}\) and dimensions (\(H\), \(L\), and \(D\)) of the track. Figure 11 illustrates soil thrust versus \(c_{u}\) with the different dimensions of the track system. The dashed line represents soil thrust for the default dimensions (\(H\) = 0.035 m, \(L\) = 0.124 m, and \(D\) = 0.124 m, cited from [PERSON] et al. (2020)), while the solid colored lines indicate soil thrust against \(c_{u}\) for modified dimensions ((3\(H\), \(L\), \(D\)), (\(H\), 3\(L\), \(D\)), and (\(H\), \(L\), 3\(D\))). As illustrated in Figure 11, increasing \(H\) (solid blue line) leads to a rise in both the intercept and slope of the soil thrust curve with respect to \(c_{u}\); however, increasing \(L\) or \(D\) (solid green or orange line) increases the slope of the soil thrust curve while reducing its intercept, which does not cut negative soil thrust region along the horizontal axis (or \(c_{u}\) axis). With the definition of \(\sigma_{c}\) (=\(W_{g}/(LD)\)), Equation (7) can be rewritten as  
\[F_{b}=c_{u}L(D+2H)-\sigma_{c}LD\text{sin}\,\beta. \tag{24}\]  
In Equation (24), an increase in \(H\) causes the first term \(c_{u}L(D+2H)\) to increase, while the second term \(\sigma_{c}LD\text{sin}\beta\) remains unchanged; accordingly, the soil thrust \(F_{b}\) increases. As \(L\) and \(D\) increase, however, the first and second terms increase together, and the second term \(\sigma_{c}LD\text{sin}\beta\) is also proportional to \(\sigma_{c}\). Since soil thrust is calculated by subtracting these terms, increases in \(L\) and \(D\) do not significantly boost soil thrust; in fact, they sometimes result in a reduction as the second term \(\sigma_{c}LD\text{sin}\beta\) increases. Therefore, when operating underwater heavy tracked vehicles on very soft ground where negative soil thrust is a concern, it is necessary to select a more effective strategy adjusted to site conditions: an increase in the track height or a reduction in the vehicle weight. Figure 10: Soil thrust versus undrained shear strength \(c_{u}\) with different ground contact pressure \(\sigma_{c}\).