# Modified Equation of Shear Strength with Respect to Saturation
## 4 Results and Discussion
### Comparison with Literature Data from Clayed-Silty Sand and Ankara Clay  
In order to further examine the predictive values of Equation (5), we refer to the laboratory test reported in [PERSON] (2016) and [PERSON] (2017) in the case of the clayed-silty sand and Ankara clay, respectively. The four optimal parameters in Equation (5) proposed in this paper were also computed by employing the nonlinear least squares method (Table 3).  
**Clayed-Silty sand**. The clayed-silty sand (sand 0%, silt 92%, clay 8%) has a relatively uniform grain-size distribution with a median grain size D50 of approximately 0.1 mm. [PERSON] (2016) also adopted a pressure-plate system to test the soil-water characteristic curve, a method similar to this research. The difference was that he used the direct shear apparatus to obtain the shear strength. The relationship between mattic suction, effective stress parameter, and saturation of various compaction pressures for the modeled data and measured data are presented in Figures 18 and 19, respectively. Figures 20 and 21 show the prediction of the proposed model for the clayed-silty sand for various compaction pressures. The computed results were compared with the reported data in that study. The correlation coefficients between the experimental data and fitting data were 0.95 and 0.98 at a compaction pressure of 100 kPa and 600 kPa, respectively.  
Figure 17: The comparison between the predicted and measured shear strength.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\xi_{2}\) & \(\xi_{3}\) & \(\xi_{4}=\xi_{1}\cdot\mathbf{P_{0}}\) & **m** \\ \hline Clayed-silty sand (e = 1.31) & 2.63 & \(-\)1.42 & 6.52 & 0.82 \\ Clayed-silty sand (e = 1.08) & 89.07 & \(-\)1.79 & 551.05 & 0.58 \\ Ankara clay & 52.59 & \(-\)3.42 & 160.82 & 0.38 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of the model parameter fitting for the literature data.  
Figure 19: Effective stress parameter \(\chi\) of the calculated points and fitted curve at different compaction pressures.  
Figure 20: Calculated results on the shear strength of the compaction pressure for 100 kPa.  
**Clayey soil**. The clayey soil (clay fraction 67.9%) researched by [PERSON] [27] was obtained from METU (Middle East Technical University). The specific gravity was 2.73, the liquid limit was 48%, the plastic limit was 21%, and the plasticity index was 27%. Figures 22 and 23 show the modeled and experimental data for the relationship between matric suction and saturation as well as the effective stress parameter, respectively, with good agreement. The relationship between the shear strength and saturation under various net normal stresses is shown in Figure 24. The calculated values were close to the experimental points in [PERSON]'s paper. All correlation coefficients were larger than 0.98.  
Figure 21: Calculated results on the shear strength of the compaction pressure for 600 kPa.  
Figure 23: Effective stress parameter \(\chi\) based on the calculated data [27] and the fitted curve.  
Figure 22: Soil–water characteristic curve based on experimental points [27] and the fitted curve.  
In geotechnical engineering, after obtaining the soil sample using a cutting ring, it is easy for engineers to obtain the degree of saturation through a simple laboratory test. The measurement of the matric suction is expensive, time-consuming, and difficult. Therefore, knowing the degree of saturation, the shear strength can be determined using Equation (5) without knowledge of the matric suction. This could also guide practical engineering applications such as slope stability.