## 3 Results
### Shear Strength Prediction of Loess and Paleosol Samples  
The calculation of the shear strength for unsaturated soil, generally, consists of two parts: the saturated shear strength and the shear strength contributed by suction. A shear strength prediction model (Equation (2)) was proposed by [PERSON] et al. [43] based on Canadian glacial till. However, [PERSON] et al. [40] found that Equation (2) was not suitable for direct use on loess because its shear strength was greatly underestimated. So, they added a correction coefficient (g) to Equation (2) and thereby produced a shear strength expression (Equation (3)) applicable to loess.  
\[\tau_{f}=c^{\prime}+(\sigma-u_{a})\tan(\varphi^{\prime})+(u_{a}-u_{w})\Theta^{ \kappa}\tan(\varphi^{\prime}) \tag{2}\]  
\[\tau_{f}=c^{\prime}+(\sigma-u_{a})\tan(\varphi^{\prime})+(u_{a}-u_{w})(\texttt{ g}\cdot\Theta^{\kappa})\tan(\varphi^{\prime}) \tag{3}\]  
where \(c^{\prime}\) and \(\varphi^{\prime}\) are strength parameters of saturated soil; \((\sigma-u_{a})\) is normal stress; for a conventional direct shear test under atmospheric pressure, \(u_{a}=0\); \(u_{w}\) is the pore-water pressure; \((u_{a}-u_{w})\) is matric suction; \(\Theta\) is the relative volume moisture content, \(\Theta=\theta_{w}/\theta_{si}\)\(\kappa\) is the fitting parameter; and g is the correction coefficient.  
The unsaturated shear strength formulas of loess and paleosol were established by fitting Equation (3) with the experimental data at a normal stress of 200 kPa. The shear strength parameters and fitting parameters in Equation (3) are shown in Table 5. Finally, their unsaturated shear strength formulas are  
\[\tau_{f,loess}=9.1+\sigma\tan 26.1^{\circ}+1.29(u_{a}-u_{w})\Theta^{0.72}\tan 2 6.1^{\circ} \tag{4}\]  
\[\tau_{f,paleosol}=17.6+\sigma\tan 28.3^{\circ}+2.96(u_{a}-u_{w})\Theta^{1.39} \tan 28.3^{\circ} \tag{5}\]  
Figure 7 manifests the relationship between shear strength from the conventional direct shear test and matrix suction after shearing. The excellent fitting degrees (0.991 and 0.960, respectively) verified their accuracy and the suitability of Equations (4) and (5) in describing the relationship between shear strength and matrix suction for unsaturated loess and paleosol.  
Figure 7: Relationship between shear strength and matrix suction.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample** & \(\boldsymbol{c^{\prime}/\kappa}\)**Pa** & \(\boldsymbol{\varphi^{\prime}/\sigma}\) & \(\mathbf{g}\) & \(\boldsymbol{\kappa}\) & \(\boldsymbol{R^{2}}\) \\ \hline Loess & 9.1 & 26.1 & 1.29 & 0.72 & 0.991 \\ Paleosol & 17.6 & 28.3 & 2.96 & 1.39 & 0.960 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters in Equation (3) for loess and paleosol samples.  
To verify the applicability of Equations (4) and (5) and the accuracy of the parameters in the prediction models, the unsaturated shear strength models were verified for various shear strengths from conventional direct shear tests with different normal stresses and initial moisture contents. A comparison of tested and predictive values is depicted in Figure 8, showing that the results are well-distributed near the standard line. Furthermore, SPSS version 19 software was used for significance analysis using the paired \(t\)-test with a critical level of \(\alpha=0.05\). The significance evaluation of the predictive models for loess and paleosol produced results of 0.061 and 0.088, respectively (both greater than 0.05). These can be regarded as demonstrating that there was no significant difference between predictive values and test values. So, we have reason to believe that Equations (4) and (5), determined based on the Vanapalli model, can well predict the unsaturated shear strength of undisturbed loess and paleosol.