# 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)
First, in the case of the 30 cm soil thickness and the 1.3 g/cm\({}^{3}\) density, \(\Delta\)k and k\({}_{\rm app}\) should be below 17.2 N/cm\({}^{2}\) and 30.76 N/cm\({}^{2}\), respectively, which are the corresponding values of the 1.4 g/cm\({}^{3}\) density for the same thickness. However, the obtained results are 46.1 N/cm\({}^{2}\) and 76.4 N/cm\({}^{2}\), consecutively. Second, with the 40 cm soil thickness and the 1.4 g/cm\({}^{3}\) density, \(\Delta\)k and k\({}_{\rm app}\) should be below 37.9 and 50.82 N/cm\({}^{2}\), respectively, which are the corresponding values of the 1.5 g/cm\({}^{3}\) density for the same thickness. Nevertheless, the acquired values are 40.6 N/cm\({}^{2}\) and 70.38 N/cm\({}^{2}\), consecutively. This irregularity occurred because the soil filled up as layers in the soil bin. Therefore, it was difficult to ensure that all layers were in the same compressed level and soil distribution. To present the relationship between \(\Delta\)k and \(\Delta\)(z/D), Equation (6) used the values of B and c shown in Table 5. Figures 10 and 11 illustrate this relationship, along with the value of \(\Delta\)k at \(\Delta\)(z/D) = 0.5. Most of the pressure-sinkage models in the literature have been applied to specify the behavior of homogeneous soils (without a hard layer). However, in the field, a vehicle usually operates in soils with a hard layer located at a certain depth. There have been some attempts to study the behavior of soils with a hard layer. [PERSON] et al. [30] tested the hard layer's effect on sandy loam soil and noted that the hard layer raised the pressure and compaction, thus altering the soil behavior; however, this study did not improve a new pressure-sinkage model for studying the affection of a hard layer. [PERSON] [25] developed an analytical pressure-sinkage model by examining invariant soil parameters that are not impacted by other factors, including the plates' dimensions. Nevertheless, this model is difficult to apply while studying the hard layer. [PERSON] and [PERSON] [35] modified the well-known Terzaghi bearing capacity equation by adjusting the shape factors first. Then, they improved the bearing capacity factor to account for the differences in the bearing capacity of the finite layer. However, this model is applicable for the foundation, not for the vehicle operating in a field. In contrast, the present study proposed a new pressure-sinkage relationship that can describe the soil's hard layer in a simple and general manner with meaningful parameters. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Soil Thickness** **[cm]** & **Density** **[g/cm\({}^{3}\)]** & **\(\Delta\)k (0.5)** **[N/cm\({}^{2}\)]** & **k\({}_{\rm app}\) (0.5)** **[N/cm\({}^{2}\)]** & **c** & **B** \\ \hline \multirow{4}{*}{30} & 1.3 & 46.1 & 76.4 & 2.5 & 18.53 \\  & 1.4 & 17.2 & 30.76 & 2 & 10.33 \\  & 1.5 & 31.1 & 49.1 & 2.5 & 12.5 \\  & 1.3 & 19.27 & 29.70 & 1.5 & 12.2 \\
40 & 1.4 & 40.6 & 70.38 & 2.5 & 16.3 \\  & 1.5 & 37.9 & 50.82 &Figure 10: The relationship of \(\Delta\)k and \(\Delta\)(z/D) for a soil thickness level of 30 cm at bulk densities of (**a**) 13 g/cm\({}^{3}\), (**b**) 1.4 g/cm\({}^{3}\), and (**c**) 1.5 g/cm\({}^{3}\). Figure 11: The relationship of \(\Delta\)k and \(\Delta\)(z/D) for a soil thickness level of 40 cm at bulk densities of **(a)** 1.3 g/cm\({}^{3}\), **(b)** 1.4 g/cm\({}^{3}\), and **(c)** 1.5 g/cm\({}^{3}\).