# Predictive Models for Correlation of Compaction Characteristics of Weakly Cohesive Soils
## 5 Results and Discussions  
### The Law of Dependence Between Static Longitudinal Modulus and Dry Density  
For the type of foundation soil (weakly cohesive soil-clay, with moisture content w = 12..15%), based on experimental determinations and laboratory investigations, correlations were established between the values of the static modulus of longitudinal deformation \(\rm{E_{st}}\) of the soil and its dry density \(\rm{\varrho_{d}}\), using different approximation functions (linear, polynomial, exponential, logarithmic, power) [6]. Initially, the authors set out to find specific approximation laws for the compaction of layers of different thicknesses, but the general tendencies of the dependence for \(\rm{E_{st}=f(\rm{\varrho_{d}})}\) are very similar (Figure 4), and therefore a valid law will be proposed for layers with thicknesses between 24.5 and 29.5 cm. The centralization of all these laws, obtained for the data of the four types of weakly cohesive soils (tested in measurement points F, G, H, I) is given in Table 1.  
By comparing the mathematical expressions for each previously listed law, of the form \(\rm{E_{st}=f(\rm{\varrho_{d}})}\), within the actual physical domain of existence (Figure 4), it was straightforward to select an optimal variant based on minimizing error (i.e., mean square deviation \(\rm{R^{2}}\)) across the considered domain. A comparative analysis of the data in Table 1 reveals that the second-degree polynomial provides the best approximation of the relationship between the two parameters (\(\rm{E_{st}}\) and \(\rm{\varrho_{d}}\)) with the smallest error. However, the linear model also performs very well, with a difference of less than 1% from the best fit. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Measurement Points**} & \multicolumn{1}{c}{**Layer Thicknesses [cm]**} & \multicolumn{1}{c}{**Approximative Laws**} & \multicolumn{1}{c}{**RMSE**} \\ \hline \multicolumn{3}{c}{E\({}_{\rm st}=1000.6\rm{\varrho_{d}^{2}-2230.8}\rm{\varrho_{d}}+1235.1-0.720\)} & \(\rm{R^{2}=0.9927}\) \\ \multicolumn{3}{c}{E\({}_{\rm st}=3.579\rm{\varrho_{d}^{8}}\rm{\varrho_{d}^{6}}\rm{\varrho_{d}^{ 8}}\rm{\varrho_{d}^{8}}\rm{\varrho_{d}^{8}}\rm{R^{2}}=0.9852}\) \\ \multicolumn{3}{c}{F, G, H, I} \\ \multicolumn{3}{c}{24.5–29.5} & \(\rm{E_{st}=814.25}\rm{\varrho_{d}}-1056.9\) & \(\rm{R^{2}=0.9842}\) \\ \multicolumn{3}{c}{E\({}_{\rm st}=0.0224\rm{e^{5.68}}\rm{\varrho_{d}}\rm{\varrho_{d}^{8}}\rm{R^{2}}=0.9802}\) \\ \multicolumn{3}{c}{E\({}_{\rm st}=1229.6 ln(\rm{\varrho_{d}})-327.5\)} & \(\rm{R^{2}=0.9787}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Laws for \(\rm{E_{st}=f(\rm{\varrho_{d}})}\) dependence. For implementing an algorithm in a computational program based on these parameters--particularly for a specific phase of the technological process or in an evolutionary manner during the compaction process after a certain number of passes--the \(\mathrm{E_{st}}\) law, expressed as a function of \(\mathrm{E_{st}}=\mathrm{f}(\varrho_{\mathrm{d}})\) and applicable to weakly cohesive soils, can be represented with a convenient analytical expression.