## 3 Upper-Bound Solution for Different Potential Failure Modes on a Clay Slope
### Triangular-Wedge Failure  
A potential triangular-wedge failure of a clay slope associated with the movement of a track system is presented in Figure 5a. In this failure mode, the failure line initiated from the toe of the track system may propagate toward the top of the clay block; thus, the triangular-wedge failure can be defined by \(\theta\), with \(0<\theta\leq\tan^{-1}(L/H)\). Given the coordinate system \(x\)-\(y\), the velocity for the motion of the failure wedge can be expressed as:  
\[\mathbf{V}=(V\cos\theta)\mathbf{e}_{x}+(V\sin\theta)\mathbf{e}_{y}. \tag{6}\]  
Accordingly, the total external power (\(P\)) and the dissipation rate (\(D\)) can be expressed as:  
\[ \begin{array}{l}P=[F_{x}+F_{v}\sin\psi,-F_{v}\cos\psi]\cdot[V\sin\theta,V\cos \theta]\\ P=\left\{F_{x}\sin\theta+F_{v}(\sin\psi\sin\theta-\cos\psi\cos\theta)\right\}V,\\ D=\frac{c_{u}HV}{\cos\theta}.\end{array}  \tag{7}\]  
Figure 4: Variation in the soil thrust \(F_{x}\) normalized by \(c_{u}\) under the block failure mode with different slope angles (\(\psi\)) and weight of vehicle \((F_{v})\). The geometry of a soil block with \(H=\) 0.05 m and \(L=\) 0.15 m was adopted for the parametric study. Equating the total external power to the internal energy dissipation rate yields the equation of the soil thrust under the triangular-wedge failure as follows:  
\[F_{x}=F_{v}\bigg{(}\frac{\cos\psi\!\cos\theta}{\sin\theta}-\sin\psi\bigg{)}+ \frac{c_{u}H}{\sin\theta\!\cos\theta}. \tag{9}\]  
We note that \(c_{u}\), \(H\), \(F_{v}\), and \(\psi\) are the given constants from the soil-track system. Since the failure wedge under the triangular failure mode is defined by \(\theta\), we can obtain the critical value of the soil thrust by the minimization of \(F_{x}\) with respect to \(\theta\) based on the upper-bound limit analysis, that is:  
\[\frac{\partial F_{x}}{\partial\theta}=0\Rightarrow\tan^{2}\theta=\frac{c_{u }H+F_{v}\!\cos\psi}{c_{u}H}, \tag{10}\]  
\[\therefore\tan^{2}\theta=\bigg{(}1+\frac{F_{v}\!\cos\psi}{c_{u}H}\bigg{)}. \tag{11}\]  
Since \(\theta>0\) under the triangular failure mode, we can express the condition of \(\theta\) that minimizes the soil thrust \(F_{x}\) as follows:  
\[\theta=\tan^{-1}\sqrt{1+\frac{F_{v}\!\cos\psi}{c_{u}H}}. \tag{12}\]  
Since \(\psi\) denotes the inclination angle of the clay slope and \(F_{v}\) indicates the magnitude of the overburden force associated with the weight of the vehicle, the minimum value of \(\theta\) is achieved either when \(F_{y}=0\) or when \(\psi\) approaches \(\pi/2\). In the case of \(F_{y}=0\), the minimum value of \(\theta\) is \(\pi/4\), which indicates that the upper bound solution of \(F_{x}\) becomes:  
\[F_{x}=2c_{u}H, \tag{13}\]  
Figure 5: Triangular- and trapezoidal-wedge failure modes of a single-track system on a clay slope. \(F_{x}\), \(F_{v}\), \(V\), and \(\psi\) denote the soil thrust, overburden force transmitted from the weight of the vehicle, velocity, and slope angle, respectively. (**a**) Triangular-wedge failure mode: \(0<\theta\leq\tan^{-1}(L/H)\); (**b**) Trapezoidal-wedge failure mode: \(\tan^{-1}(L/H)<\theta\leq\pi/2\). which is independent of the slope angle \(\psi\). When \(\psi\) approaches \(\pi/2\), \(\theta\) becomes close to \(\pi/4\), and then the soil thrust is approached as follows. The parametric changes in \(F_{x}\) in Equation (9) associated with the variation in \(\theta\) and \(\psi\) under different values of \(F_{v}\) are depicted in Figure 6. For this analysis, the geometry of the soil block was assumed as \(H=0.05\) m and \(L=0.15\)\(m\), which leads the maximum value of \(\theta\) to be approximately \(71.65^{\circ}\). Different \(F_{v}\) values were also selected as 0, 5, 10, and 15 kN/m, and their resultant \(F_{x}\) values correspond to Figure 6a-d, respectively. An undrained shear strength \(c_{u}\) of 10 kPa was considered as an example, and the normalized soil thrust values \(F_{x}/c_{u}\) are demonstrated to focus on the influence of \(\psi\) and \(\theta\) on \(F_{v}\) in this study.