## 2 Unsaturated Subgrade Filler Triaxial Testing System
### Analysis of Experimental Results
#### 3.4.3 Analysis of Shear Strength  
Recent studies have consistently suggested that the distance from the center of the Mohr circle to the failure point is commonly considered the radius of the Mohr circle, \(R_{d}\)[44; 49; 50; 51].  
\[R_{d}=\frac{\sigma_{1d}-\sigma_{3d}}{2}\]  
However, due to the nonlinear mechanical behavior of unsaturated soils, discrepancies between experimental results and theoretical values, \(d_{0}\), exist.  
\[d_{0}=R_{d}-\frac{\sigma_{1d}-\sigma_{3d}}{2}\]  
Consequently, this study reformulates the determination of shear strength parameters as a least squares optimization issue, with the aim of minimizing the error between theoretical and experimental data through parameter optimization. The specific objective is to minimize the function \(G(X)\).  
\[G(X)=\sum_{i=1}^{n}(d_{0})^{2}=\left(R_{d}-\frac{\sigma_{1d}-\sigma_{3d}}{2} \right)^{2}=\sum_{i=1}^{n}\left\{\frac{c_{0}+\left(\frac{\sigma_{1d}+\sigma_{3 d}}{2}-u_{a}\right)\tan\varphi_{0}+s_{d}\tan\varphi_{b}}{\left(\tan\varphi_{0} \right)^{2}+\left(\tan\varphi_{b}\right)^{2}+1}-\frac{\sigma_{1d}+\sigma_{3d} }{2}\right\}\]  
Here, \(C_{0}\) is cohesive strength under saturated conditions, \(\sigma_{1d}\) is the maximum principal stress under shear failure conditions, \(\sigma_{3d}\) is the minimum principal stress under shear failure conditions, \(u_{a}\) is the pore air pressure, \(\varphi_{0}\) is the frictional angle under saturated conditions, and \(\varphi_{b}\) is the angle that indicates how shear strength changes in response to variations in matric suction.  
Substituting the data from Table 4 into Equation (3) yields the best fit, with \(C_{0}\), \(\varphi_{0}\), and \(\varphi_{b}\) values of 18.74 kPa, 30.2\({}^{\circ}\), and 14.09\({}^{\circ}\), respectively. The coefficient of determination, \(R^{2}\), of the regression equation is 0.98, with a standard deviation of 2.88 kPa, indicating a strong correlation between the independent variables (\(C_{0}\), \(\varphi_{0}\), and \(\varphi_{b}\)) and the dependent variables (\(s_{d}\), \(\sigma_{1d}-u_{a}\), and \(\sigma_{3d}-u_{a}\)). Figure 14 illustrates the Mohr-Coulomb circles and the failure envelope at the shear failure point. The predicted results closely align with the experimental stress range, demonstrating an ideal fit.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Serial Number** & \(s_{d}\) **(kPa)** & \(\sigma_{1d}\)–\(u_{a}\) **(kPa)** & \(\sigma_{3d}\)–\(u_{a}\) **(kPa)** \\ \hline \multirow{4}{*}{Set 1} & 1 & 35.7 & 95.6 & 5 \\  & 2 & 63.9 & 127.3 & 5 \\  & 3 & 127.9 & 197.8 & 5 \\  & 4 & 164.9 & 249.4 & 5 \\  & 5 & 256.8 & 290.7 & 5 \\  & 6 & 322.9 & 336.3 & 5 \\ \hline \multirow{4}{*}{Set 2} & 1 & - & - & 200 \\  & 2 & - & - & 200 \\  & 3 & 121.7 & 748.9 & 200 \\  & 4 & 171.9 & 786.9 & 200 \\  & 5 & 196.9 & 830.9 & 200 \\ \hline \multirow{4}{*}{Set 3} & 1 & - & - & 200 \\  & 2 & - & - & 200 \\  & 3 & - & - & 200 \\  & 4 & - & - & 200 \\  & 5 & - & - & 200 \\  & 6 & - & - & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Soil sample coefficients at the failure point under shear loading conditions.