# 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)
By substituting Table 4\({}^{\prime}\)s data in Equation (7), the following two equations are provided:  
\[\text{In}\left(\frac{2}{\text{B}}+1\right)=0.1\text{-c}\]  
\[\text{In}\left(\frac{15.5}{\text{B}}+1\right)=0.4\text{-c}\]  
To determine the values of B and c, the two equations form an implicit equation that can be solved using an implicit equation solver program. This program identified the constants as B = 4.306 and c = 3.815. Figure 6 presents a model of the relationship between \(\Delta\)k and \(\Delta\)(z/D) that was suggested after several calculations of Equation (6) were completed to remove the calculation errors. It is seen that Equation (6) is flexible at different values of B and c, but converging values were noticed at \(\Delta\)(z/D) = 0.5, where \(\Delta\)k = 30 [N/cm\({}^{2}\)]. The range of B is 2-50, while the range of c is 1-5; hence, \(\Delta\)(z/D) = 0.5 is regarded as a fixed point to identify the optimum range for B and c. Figure 7 shows the relationship between B and c that can help determine their range. Hence, Equation (6) was used to calculate the constant B with a \(\Delta\)k range between 10 and 50 and a c range between 1 and 5. The B and c values are coherent for \(\Delta\)k (0.5) of 50, 30, and 20 (N/cm\({}^{2}\)). In the present study, the optimum range for the constant B is between 10 and 20, and between 1 and 3 for constant c.  
Figure 6: Model of the relationship of \(\Delta\)k versus \(\Delta\)(z/D). Experimental verification of the proposed sinkage modulus (k\({}_{\rm app}\)) was conducted to determine the affection of a hard layer in the soil. Figures 8 and 9 show the pressure-sinkage curve, load-bearing capacity factor (k), and k\({}_{\rm app}\) curve results for the sinkage plate diameter of 20 cm at 30 cm and 40 cm soil thickness levels, respectively, with varied densities. The first part of the pressure-sinkage curve involved subjecting the build-up zone to Equation (1). To identify the equation's parameters, a regression analysis (Table 3) was used. In the first part of the curve (prior to the breaking point), where k\({}_{\rm app}\) = k at z/D \(<\) z\({}_{\rm o}\)/D, no changes were seen in the sinkage exponent (n) or the sinkage modulus (k). There is an increase in pressure as a function of the relative sinkage as the logarithmic scale for showing the relation, and hence, the pressure and relative sinkage have a linear relationship. The load-carrying capacity factor (k) depends on density. Accordingly, its value increases with the rise in the soil density, with some abnormalities. These abnormalities occur because of the soil filling up in the bin in layers, as it is not easy to control all of the layers and ensure they have the same compaction and pressure distribution. The curve's second part shows the compact zone's interaction with the rigid layer. With increased deformation, the pressure also increases with relative sinkage. For this portion, the proposed Equation (4) was used, where z/D \(>\) z\({}_{\rm o}\)/D. Moreover, the figures show that the values of k\({}_{\rm app}\) and the applied pressure are coherent. Hence, with increasing pressure, there is an increase in the sinkage modulus (k\({}_{\rm app}\)), which leads to an increase in the soil's load-bearing capacity. Figure 7: The relationship of constants B and c in Equation (6). Figure 8: Pressure–sinkage curves with kapp for a soil thickness level of 30 cm at bulk densities of (**a**) 1.3 g/cm\({}^{3}\), (**b**) 1.4 g/cm\({}^{3}\), and (**c**) 1.5 g/cm\({}^{3}\). Figure 9: Pressure–sinkage curves with kapp 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\({}^{3}\). Table 5 presents the values of B, c, \(\Delta\)k, and k\({}_{\rm app}\) at \(\Delta\)(z/D) = 0.5, in terms of two soil thickness levels at diverse densities. The table shows that there are anomalies in the outcomes of two of the tested conditions.