# A Modified Pressure-Sinkage Model for Studying the Effect of a Hard Layer in Sandy Loam Soil
## 4 New Pressure-Sinkage Relationship Equation/Modified Sinkage Modulus (k)  
The conventional pressure-sinkage models can be used for homogenous, infinite soil thickness when an examination of the impact of the hard layer is not required. It was intended that certain proposals would be made for models that would consider the effect of the hard layer, but the models were too complicated and could not be applied. Hence, the present study's main aim is to improve a simple pressure-sinkage model (shown in Equation (1)) to assess the affection of the rigid layer. Most proposed pressure-sinkage models have tried to develop the sinkage modulus (k) as functions concerning the plate dimensions or wheel dimensions. These models have also regarded the sinkage exponent as a constant soil parameter that is not impacted by other factors. The exponent (n), as well as the sinkage modulus (k), remain constant in the present study for the first part of the pressure-sinkage curve. These soil parameters, however, do not remain constant for the second part, because the bottom of the soil bin impacts the pressure distribution while increasing compaction. The compaction following the breaking point becomes a more exponential function, and hence, the exponent (n) is not constant. It is possible to improve Equation (1) by altering the two parameters so that the second part of the pressure-sinkage curve can be more accurately characterized. As modifying both parameters is not effective, the sinkage modulus (k) can be modified using a simpler method. The exponent (n) characterizes the soil's deformation and compaction behavior under vertical loading. On the other hand, k indicates the soil's rigidity and is mainly influenced by the soil's properties. The experimental results concerning the two soil thickness levels at varied initial bulk densities show that a pressure-sinkage model is developed by presenting a new sinkage modulus concerning the affection of the soil's hard layer (Equations (3) and (4)):  
\[\mathrm{p}=\mathrm{k}_{\mathrm{app}}\Big{(}\frac{\mathrm{z}}{\mathrm{D}} \Big{)}^{\mathrm{n}} \tag{3}\]  
\[\mathrm{k}_{\mathrm{app}}=\mathrm{k}+\mathrm{B}\Big{(}\mathrm{e}^{\mathrm{c}( \mathrm{\hat{b}}-\frac{\mathrm{z_{0}}}{\mathrm{D}})}-1\Big{)} \tag{4}\]  
where \(\mathrm{k}_{\mathrm{app}}\) is the apparent sinkage modulus in unit N/cm\({}^{2}\), and B and c are constant. z/D > z\({}_{\mathrm{o}}\)/D (for the second part of the pressure-sinkage curve). The apparent sinkage modulus is equal to the sinkage modulus at z/D < z\({}_{\mathrm{o}}\)/D (for the first part of the pressure-sinkage curve), as seen in Equation (5). \[\mathrm{k}_{\mathrm{app}}=\mathrm{k} \tag{5}\]  
\(\mathrm{\Delta k}=\mathrm{k}_{\mathrm{app}}-\mathrm{k}\) and \(\mathrm{\Delta z/D}=\mathrm{z/D-z}_{\mathrm{o}}\)/D for z/D > z\({}_{\mathrm{o}}\)/D, by substituting these two terms in Equation (4), the following equation becomes:  
\[\mathrm{\Delta k}=\mathrm{B}\Big{(}\mathrm{e}^{\mathrm{c}\mathrm{\Delta z}}-1 \Big{)} \tag{6}\]  
Considering the logarithm on both sides of Equation (6) leads to the new equation given below:  
\[\mathrm{In}\Big{(}\frac{\mathrm{\Delta k}}{\mathrm{B}}+1\Big{)}=\mathrm{c} \cdot\mathrm{\Delta}\Big{(}\frac{\mathrm{z}}{\mathrm{D}}\Big{)} \tag{7}\]  
To determine the constants B and c, two measured points on the pressure-sinkage curve can be used. The example given below presents the calculation involved. Table 4 presents the data of the example. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**z\({}_{\mathrm{o}}\)/D**} & **Measured** & \multirow{2}{*}{\(\mathrm{\Delta z}\)/D = z/D-z\({}_{\mathrm{o}}\)/D} & **k** & **Measured k** & **\(\mathrm{\Delta k}\) = k\({}_{\mathrm{app}}-\mathrm{k}\)** \\  & **z/D** & & [N/cm\({}^{2}\)] & **or k\({}_{\mathrm{app}}\)** & **\(\mathrm{\Delta k}\) = k\({}_{\mathrm{app}}-\mathrm{k}\)** \\ \hline \multirow{2}{*}{0.2} & 0.3 & 0.1 & \multirow{2}{*}{9} & 11 & 2 \\  & 0.6 & 0.4 & & 24.5 & 15.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The data of the example.