# A Modified Pressure-Sinkage Model for Studying the Effect of a Hard Layer in Sandy Loam Soil
In this model, the dimensions of the parameters concerning the soil modulus rely on a sinkage exponent. [PERSON]'s model was further improved upon in 1965 by [PERSON], who used two dimensionless parameters [23]. In 1959, [PERSON] suggested a pressure-sinkage equation that was based on the Boussinesq theory regarding the elastic half-space indicated in Equation (1) [24]. \[\mathrm{p}=\mathrm{k}\left(\frac{\mathrm{z}}{\mathrm{D}}\right)^{\mathrm{n}} \tag{1}\]  
where p is the average pressure under an indenter, D is the diameter of the indenter, z is the vertical soil deformation (sinkage), and k is the sinkage modulus (load-carrying capacity factor). It is noteworthy that Equation (1) applies to homogeneous soils that do not have a hard layer; that is, there is a decrease in monotonic sinkage when ground pressure increases. In reality, vehicles tend to work in soils that have a hard layer, or hardpan, at a depth [25]. Because of the tillage operations and the resulting settling, varied local load-carrying capacities are generated at diverse depths. While the upper, loose layer's load-carrying capacity is low, the deeper layer's load-carrying capacity is based on the tillage operation, although it increases with depth. [PERSON]'s theory states that there is a gradual reduction in the exerted surface pressure as a function of depth. At a given depth, if there is a rigid layer, it alters the pressure distribution as a function of relative depth [26]. An analytical pressure-sinkage model was presented by [PERSON] that considered the soil hardpan effect [27; 28]. In this model, the equation for assessing the soil's bearing capacities in terms of infinite hardpan does not apply to some types of soil. To address the [PERSON] model's limitations, in 2010, [PERSON] developed an analytical model by considering invariant soil parameters that are not impacted by other factors, including the plates' dimensions [25]. These are two complicated models that are difficult to apply. [PERSON]'s model focused on the homogenous soil's stress and displacement of finite thickness foundation. This model can be applied for evaluating the deformation of a soil layer that is underlain by a hard layer with no compaction. According to this model, deformations under foundations, even with no underlying hard layer, focused on the massif's upper part and are not deeply entrenched. This leads to an \"active zone\", where the base can be considered deformed. It is crucial to consider the core depth problem when calculating the finite thickness base. This can only be experimentally addressed, as per the measurements of the building soil's layer-by-layer deformations [29]. [PERSON] et al. examined the hard layer's affection on the sandy loam soil behavior and noted that the hard layer increased the pressure and compaction, thus altering the soil behavior. Moreover, it increased deformation and changed the pressure-sinkage equation parameters [30]. The present study aims to characterize the pressure-sinkage relationship that can indicate the soil's hard layer simply and in a general manner with meaningful parameters. The study also intends to specify the zone (breaking point) in which the rigid layer impacts the compaction. Further, the study presents a new way to characterize the pressure-sinkage relationship by altering the sinkage modulus (k) in Equation (1). However, there are certain problems when determining the sinkage modulus k, and these problems can be caused by extreme variability in the soil textures, the distribution of the moisture content and its impact on the mechanical properties, and the hardness distribution in the hardpan at finite depth. Hence, large circular plates were used to ensure that the loading conditions were similar to those occurring in a real tire.