# 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
This expression largely accounts for the magnitude of the applied load as well as the soil's condition, particularly its degree of compaction, as follows:  
\[\mathrm{E_{st}}=1000\left[\rho_{\mathrm{d}}^{2}-\mathrm{P_{c}}\varrho_{\mathrm{ d}}+\left(\mathrm{P_{c}}-1\right)\left[\mathrm{daN/cm^{2}}\right],\right. \tag{4}\]  
where \(\mathrm{E_{st}}\) is the static modulus of linear deformation of the soil in [\(\mathrm{daN/cm^{2}}\)], \(\varrho_{\mathrm{d}}\) is the dry density of the soil, achieved at a given time, in [g/cm\({}^{3}\)], and \(\mathrm{P_{c}}\) represents the technological factor of compaction depending on the characteristics of the compactor and the soil, calculated with the following relationship:  
\[\mathrm{P_{c}}=\frac{\mathrm{Q}}{\mathrm{b^{*}}\cdot\mathrm{D_{r}}}\left[ \mathrm{daN/cm^{2}}\right], \tag{5}\]  
Figure 4: Dependence of the static modulus of linear deformation on the dry density for weakly cohesive soils (experiments under static conditions). where \(Q\) represents the roll weight [daN], \(b^{*}\) is the footprint or contact patch of the machine [cm], and \(D_{r}\) is the roll diameter [cm]. Experimental results have shown that the width \(b^{*}\) of the roller's footprint at the interface with the ground typically falls within the range \(b^{*}=(0.1\ldots 0.2)D_{r}\). The size of the contact patch depends on the condition of the terrain, specifically its compaction level, as well as the weight of the roller. This value fluctuates within narrow limits, reflecting the compaction efficiency. The authors suggest using an intermediate value of \(b^{*}=0.18D_{r}\), and after substituting this into Equation (5), the pressure factor is expressed as follows:  
\[P_{c}=\frac{Q}{0.18D_{r}^{2}}\left[daN/cm^{2}\right]. \tag{6}\]  
For the VV170 compactor operating in static mode, and for the tested soil, a value of \(P_{c}=2.22\) was obtained. The proposed relationship \(E_{st}=f(\varrho_{d})\), which applies to the static compaction of weakly cohesive soils with layer thicknesses between 24.5 and 29.5 cm, is expressed as follows:  
\[E_{st}=1000\left(p_{d}^{2}-2.22\varrho_{d}+1.22\right)\left[\frac{daN}{cm^{2} }\right]. \tag{7}\]  
The comparative representation of two approximate polynomial laws is shown in Figure 5. The first law is derived from experimental data (Table 1), while the second is proposed by the authors, as expressed in Equation (4). In conclusion, the results presented in Figure 5 demonstrate a strong similarity in the curve's shape, thereby confirming the validity of the polynomial law proposed by the authors in Equation (7). If we examine the experimental results obtained at each of the four measurement points (see Figure 6), we can refer to the summarized data presented in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Measurement Points**} & **Layer** & **Initial Density** & **Initial Moisture** & **Approximative** \\  & **Thicknesses** & **[cm]** & **[g/cm3]** & **Content [\%]** & **Laws** \\ \hline F & 26.5 & 0.745\(\varrho_{\text{dmax}}\) & 0.60\(w_{\text{opt}}\) & \(E_{st}=714.28\varrho_{d}-903.71\) \\ G & 24.5 & 0.797\(\varrho_{\text{dmax}}\) & 0.85\(w_{\text{opt}}\) & \(E_{st}=833.33\varrho_{d}-1091.33\) \\ H & 28.0 & 0.785\(\varrho_{\text{dmax}}\) & 0.66\(w_{\text{opt}}\) & \(E_{st}=833.33\varrho_{d}-1079.25\) \\ I & 29.5 & 0.817\(\varrho_{\text{dmax}}\) & 1.04\(w_{\text{opt}}\) & \(E_{st}=1000\varrho_{d}-1352\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Centralization of variation laws \(E_{st}=\varrho_{d}\) for each type of soil analyzed. Figure 5: Comparative representation of the law \(E_{st}=f(\varrho_{d})\): experimental results (in blue) versus the proposed analytical model (in red).