# Evaluation of the Strength Characteristics of Sility-Clayey Soils during Freezing-Thawing Cycles
## 4 Discussion  
Based on the results of laboratory research, a dependence was derived that allows to determine the angle of internal friction or cohesion on the given data of void ratio, initial soil moisture, and number of freeze-thaw cycles. A program based on the least-squares method has the following form:  
\[y=e^{a_{0}+a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}+b_{1}X_{1}{}^{2}+b_{2}X_{2}{}^{2}+ b_{3}X_{3}{}^{2}+c_{1}X_{1}X_{2}+c_{2}X_{1}X_{3}+c_{3}X_{2}X_{3} \tag{1}\]  
where \(\mathbf{a}_{i},\ \mathbf{b}_{i},\ \text{and}\ \mathbf{c}_{i}\) is the approximation coefficient;  
\(\mathbf{X_{i}}\) is the value of the factor under study. The developed program for calculating the dependence of the desired parameter on the given values allows us to process the results of a three-factor experiment. The general view of the program is presented in Figure 7. The figure shows the results of calculating the approximation coefficients for the angle of internal friction. For the angle of internal Friction (2) and Cohesion (3), the mathematical dependence can be represented as follows:  
\[y=e^{5.093-3.328\cdot X_{1}-0.162\cdot X_{2}+1.797\cdot X_{3}-1.628\cdot X_{1} {}^{2}-0.002\cdot X_{2}{}^{2}-0.188\cdot X_{3}{}^{2}+0.332\cdot X_{1}X_{2}+0.005 \cdot X_{2}X_{3}} \tag{2}\]  
\[y=e^{2.696+10.277\cdot X_{1}-0.129\cdot X_{2}-4.049\cdot X_{3}+5.206\cdot X_{1} {}^{2}+0.019\cdot X_{2}{}^{2}+0.436\cdot X_{3}{}^{2}-1.057\cdot X_{1}X_{2}-0.19 \cdot 2\cdot X_{1}X_{3}-0.012\cdot X_{2}X_{3}} \tag{3}\]  
where \(X_{i}\) is the value of the factor under study (\(X_{1}\) is the soil void ratio; \(X_{2}\) is the initial soil moisture content; \(X_{3}\) is the number of freezing-thawing cycles). Figure 8 shows a graph of the correspondence of the results of laboratory and theoretical values of strength characteristics. The axis of the abscissa indicates the value of the characteristic calculated according to Equations (2) and (3). The axis of the ordinate represents the value of the characteristic obtained as a result of laboratory studies. With perfect correspondence, the points should be located on the line \(Y=X\). Figure 8 shows that the closest correspondence is observed for the angle of internal friction. Figure 7: General view of the program. Thus, by setting the values of void ratio, initial soil moisture, and number of freeze-thaw cycles, it becomes possible to calculate the angle of internal friction and/or cohesion for each type of soil. However, the presented dependencies work correctly only near the points of interest. Therefore, additional tests are required inside the cube (Figure 1) to clarify these mathematical dependences. 5 Example Calculation of Engineering Characteristics of Soil Bases by the Proposed Analytical Method  
In order to verify the proposed method, in this subsection, the following example was considered to determine the angle of internal friction and cohesion and compare it with the calculated values; the bearing capacity of the soil was determined according to the standard [18] of the Russian Federation. According to the design standards, when calculating the strains of the soil base of building foundations, the average pressure under the foundation footing should not exceed the design soil resistance R, which is calculated by Equation (4) in accordance with SP 22.13330.2016 \"Soil bases of buildings and structures\" [18] which allows the determination of the linear load in the soil base,  
\[R=\frac{\gamma_{c1}\gamma_{c2}}{k}\left[M_{\gamma}k_{z}b\gamma_{III}+M_{q}d_{l} \gamma_{II}^{\prime}+\left(M_{q}-1\right)d_{b}\gamma_{II}^{\prime}+M_{c}c_{II}\right] \tag{4}\]  
As an example, below is the calculation of the general case of strength characteristics with the following initial data:  
1. The foundation footing width \(b=10\) m, therefore \(k_{z}=\frac{z_{0}}{b}+0.2=1\). 2. The foundation depth \(d_{l}=5\) m.
3. The basement depth \(d_{b}\) is assumed equal to 2 m in accordance with SP 22.13330.2016 [18]. 4.