# Shear Strength Analysis and Slope Stability Study of Straight Root Herbaceous Root Soil Composite
## 2 Methods and Materials
### Shear Strength Expression of the Soil-Root Composite  
The soil shear strength is not constant; rather, it increases with normal stress \(\sigma\) on the shear moving surface. [PERSON] summarized the failure phenomena and the influencing factors of soil and proposed the shear strength calculation formula as  
\[\tau_{f}=c+\sigma\tan\ \varphi \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Root** & **Depth/cm** & **Moisture** & **Dry Density/** & **Average Root** & **Main Root System** \\
**Content/\%** & **Content/\%** & **Content/\%** & **(g/cm\({}^{3}\))** & **Diameter/mm** & **Length/cm** \\ \hline
0 & & & & / & / \\
0.12 & 0-20 & 16.7 & 1.58 & 2.7 & 11.4 \\
0.31 & & & & 2.1 & 13.2 \\
0 & & & & 2.4 & 14.6 \\
0.11 & & & & / & / \\
0.15 & 20–40 & 23.4 & 1.42 & 1.1 & 3.6 \\
0.18 & & & & 0.7 & 3.9 \\
0 & & & & / & / \\
0.06 & 40–60 & 28.6 & 1.33 & 0.3 & 0.9 \\
0.07 & & & & 0.4 & 0.7 \\
0.09 & & & & 0.3 & 0.8 \\ \hline \end{tabular}
\end{table}
Table 2: Sample group parameters for roots and soil.  
where \(\tau_{f}\) is the shear stress on the shear sliding plane (kPa), \(\sigma\) tan \(\varphi\) is the frictional strength (kPa), \(\varphi\) is the internal friction angle (\({}^{\circ}\)), and \(c\) is the cohesion (kPa).  
Based on reinforced soil principles, scholars have analyzed the stressed state of the herbaceous plant soil-root composite by treating the plant root distribution in the soil as a three-dimensional distribution of reinforcing fibers. The reinforcement process provides additional cohesion \(\Delta c\) to the soil. Furthermore, the wrapping effect of the root system restricts the lateral deformation of the soil, thereby effectively improving the bearing capacity of the slope soil [16; 17; 30]. Here, the soil shear strength of the soil is expressed as  
\[\tau_{f}=c+\sigma\tan\ \varphi+\Delta c \tag{2}\]  
\[\Delta c=\frac{T_{N}}{A}\cos\ \psi+\frac{T_{N}}{A}\sin\ \psi\tan\ \varphi \tag{3}\]  
\[\psi=\arctan\!\frac{1}{k}\ \text{or}\ \psi=\left|\arctan\!\frac{1}{k+\cot i}\right| \tag{4}\]  
where \(\Delta c\) is the additional cohesion (kPa), \(T_{N}\) is the tensile stress on roots (\(N\)), \(A\) is the contact area between a single root and the surrounding soil (m\({}^{2}\)), \(\psi\) is the angle between the shearing direction and the root (\({}^{\circ}\)), \(i\) is the original angle (\({}^{\circ}\)) between the shearing plane and the root (\({}^{\circ}\)), and \(k\) is the shear deformation ratio.  
In contrast, other scholars believe that the cohesion of the soil-root composite is collectively borne by the internal particles of the soil and the constraints between the root and the soil. The shear strength index of the soil-root composite is defined as the combination of the comprehensive \(c\) and the comprehensive \(\varphi\). The total cohesion is the sum of the cohesion due to roots and the soil's own cohesion [24; 33; 37]:  
\[c=c_{s}+c_{r} \tag{5}\]  
where \(c\) is the comprehensive cohesion (kPa), \(c_{r}\) is the cohesion due to roots (kPa), and \(c_{s}\) is the soil cohesion (kPa).