## 3 Soil Thrust Assessment Method for Continuous Tracks
### Proposed Method for Evaluating the Soil Thrust of a Continuous Track  
As shown in Figure 3, at the soil-track interface, rectangular soil blocks are formed between adjacent grousers and each of them is independently sheared [5; 6; 7; 8; 9]. This means that the shear failure plane at the soil-track interface is not continuous but divided into separate sections (i.e., separate soil blocks). The soil block located at the front in the traveling direction of the off-road tracked vehicle experiences little slip displacement and soil thrust without much shearing, and the slip displacement and soil thrust increase as shearing progresses toward the rear.  
As shown in Figure 2, when the off-road tracked vehicle travels with a slip ratio \(i\), the slip displacement (\(j_{n}\)) occurring in the grouser (i.e., soil block) that is separated by a specific distance (\(x_{n}\)) from the first point of the soil-track interface can be expressed as the product of the slip ratio and the distance (\(j_{n}=ix_{n}\)). If the grouser spacing is \(l_{t}\), the total length \(l\) of the soil-track interface is divided into \(N=l/l_{t}\) soil blocks and each block is separately sheared. The distance between the first block of the soil-track interface and the soil block at the \(n\)th position is \(x_{n}=l_{t}\ (n-1)=l(n-1)/N\). Therefore, the slip displacement occurring in the soil block at the \(n\)th position from the first block of the soil-track interface can be expressed as follows.  
\[j_{n}=ix_{n}=il_{t}(n-1)=il(n-1)/N \tag{8}\]  
Additionally, in Equation (1), if the length \(l\) of the soil-track interface is replaced with \(l_{t}\), the maximum soil thrust \(F_{max}\) developed on each soil block can be obtained; it is given in Equation (9).  
\[F_{max(s)}=wl_{t}\big{(}c+\sigma_{v}^{\prime}\tan\phi\big{)}+l_{t}h(2c+0.64 \cot^{-1}\frac{h}{w}\sigma_{v}^{\prime}\tan\phi\big{)} \tag{9}\]  
By substituting the slip displacement from Equation (8) and maximum soil thrusts from Equation (9) in Equations (2)-(4), the soil thrust (\(F_{n}\)) generated by the soil block at the \(n\)th position from the first block of the soil-track interface can be calculated. \(F_{n}\) under hardening, softening, and hump behaviors under the slip ratio \(i\) are given in Equations (10)-(12), respectively.  
\[F_{n}=F_{max(s)}\{1-\exp(-il(n-1)/NK)\} \tag{10}\]  
\[F_{n}=F_{max(s)}K_{r}[1+\{1/K_{r}(1-1/e)-1\}\exp(1-il(n-1)/NK_{w})][1-\exp(-il (n-1)/NK_{w})] \tag{11}\]  
\[F_{n}=F_{max(s)}(il(n-1)/NK_{h})\exp(1-il(n-1)/NK_{h}) \tag{12}\]  
The total soil thrust of the continuous track can be evaluated as the sum of the soil thrusts developed in each soil block. Therefore, the total soil thrusts on the off-road vehicle under hardening, softening, and hump behaviors are as shown in Equations (13)-(15), respectively.  
\[F_{t}=F_{max(s)}\sum_{n=1}^{N}\{1-\exp(-il(n-1)/NK)\} \tag{13}\]  
\[F_{t}=F_{max(s)}K_{r}\sum_{n=1}^{N}[1+\{1/K_{r}(1-1/e)-1\}\exp(1-il(n-1)/NK_{w })][1-\exp(-il(n-1)/NK_{w})] \tag{14}\]  
\[F_{t}=F_{max(s)}\sum_{n=1}^{N}(il(n-1)/NK_{h})\exp(1-il(n-1)/NK_{h}) \tag{15}\]  
The driving force and driving efficiency can be predicted by evaluating the total soil thrust according to the slip ratio of the off-road tracked vehicle to which the grouser is attached. Unlike the previous studies, Equations (13)-(15) were derived after incorporating the differences in soil thrusts according to the grouser spacing (\(l_{t}\)). In other words, obtaining the optimal grouser shape ratio is possible by analyzing the change in the total soil thrust according to the shape ratio (\(l_{t}/h\)), which has a significant impact on the soil thrust of an off-road tracked vehicle.