## 3 Theoretical Analysis of the Grouser Effect
### Calculation and Analysis of the Traction Force under the Grouser Effect  
The traction force produced by the track shoe during the process of movement, that is, the reaction force received by the track shoe during the shearing process, can usually be divided into the vertical thrust \(F_{1}\) acting on the surface of the track shoe, the force \(F_{2}\) acting on the bottom of the grouser, the force \(F_{3}\) acting on both sides of the track shoe and the grouser, and the force \(F_{4}\) acting on the grouser. The shear stress of a single-track shoe on the ground is  
\[F_{0}=F_{1}+F_{2}+F_{3}+F_{4} \tag{10}\]  
According to the Coulomb formula:  
\[F_{1}=B_{1}(c+p_{1}\tan\varphi)\Big{(}1-e^{-i/k}\Big{)} \tag{11}\]  
\[F_{2}=B_{2}(c+p_{2}\tan\varphi)\Big{(}1-e^{-i/k}\Big{)} \tag{12}\]  
In Figure 6, the force acting on both sides of the track shoe can be regarded as a strip load on a semi-infinite body. Under the action of the strip load, \(p\), the stress at any point in the semi-infinite body in the \(x\) direction is:  
\[\delta_{x}=-\frac{p}{2\pi}\int_{\theta_{1}}^{\theta_{2}}\sin^{2}\theta d\theta\]  
When \(\theta_{1}=0\),  
\[\delta_{x}=-\frac{p}{2\pi}(\pi-2\theta-\sin 2\theta)\]  
Figure 5: Schematic diagram of the interaction force between a K-shaped track shoe and the soil. (**a**) Side view of the force diagram of a K-shaped track shoe. (**b**) Dimension diagram of a K-shaped track shoe.  
Moreover, because  
\[\tan\theta=\frac{h}{b}\]  
\[p=\frac{W\prime}{bl}\]  
The force \(F_{3}\) acting on both sides of the track shoe and the grouser is  
\[F_{3}=\frac{W^{\prime}h}{b\pi}\arctan\left(\frac{h}{b}\right) \tag{13}\]  
The force \(F_{4}\) acting on the spine was analyzed using the principle of a retaining wall, as shown in Figure 7. According to [PERSON]'s earth pressure theory, under the action of a uniformly distributed load \(p\) on the track shoe, the passive ground pressure on the horizontal grouser is  
\[\sigma=\gamma zN_{\varphi}+pN_{\varphi}+2c\sqrt{N_{\varphi}}\]  
where \(\gamma\) is the weight of the soil unit and \(N_{\varphi}\) is the flow value.  
\[F_{41}=\int_{0}^{h}\sigma bdz=\frac{1}{2}\gamma h^{2}bN_{\varphi}+phbN_{ \varphi}+2 chb\sqrt{N_{\varphi}}\]  
In the K-shaped track shoe shown in Figure 5b, the passive ground pressure on a single inclined grouser is  
\[F_{42}=\frac{1}{2}\gamma h^{2}w\sin\alpha N_{\varphi}+pltw\sin\alpha N_{ \varphi}+2 chw\sin\alpha\sqrt{N_{\varphi}}+chw\cos\alpha\]  
If we consider the four common track shoes in Figure 3 and compare their structural features comprehensively, the force \(F_{4}\) acting on the grouser can be obtained as follows  
\[F_{4}=n_{1}F_{41}+n_{2}F_{42}\]  
Figure 6: Stress distribution of the semi-infinite body under a band load.  
Figure 7: Schematic diagram of the force on the grouser.  
where \(n_{1}\) is the number of horizontal grousers, \(n_{1}\) can be 0 or 1; and \(n_{2}\) is the number of inclined grousers, either 0 or 2. When \(n_{1}=1\) and \(n_{2}=0\), the force on the grouser of a T-shaped track, a \(\pi\)-shaped track, and a V-shaped track can be solved; when \(n_{1}=1\) and \(n_{2}=2\), the force on the grouser of a K-shaped track can be solved; and when \(n_{1}=0\) and \(n_{2}=2\), the force on the grouser of a figure-eight track can be solved.  
From Equations (9) and (10), it can be seen that the traction force on a single-track shoe is  
\[F=F_{0}-R \tag{14}\]