### Lateral Stress Theories in Literature
In this equation, \(\psi\) is the wall rigidity factor (1 for flexible walls and 2 for rigid walls), \(x\), \(y\), and \(z\) are the Cartesian coordinates (m), and \(q\) is the applied area load at the soil surface (kPa). However, since the properties of the soils are not taken into account (such as relative density and internal friction angle), it is known that similar results can be obtained for different soil types. However, when parameters other than the Poisson's ratio change, it is inevitable to obtain different lateral stress values. \[\sigma_{x}(y,z)=\frac{\psi q}{2\pi}[f(b,d)-f(b,c)-f(a,d)+f(a,c)] \tag{3}\]  
\[f(\beta,\delta)=tan^{-1}\left[\frac{\beta(\delta-y)}{z\sqrt{\beta^{2}+\left( \delta-y\right)^{2}+z^{2}}}\right]-\frac{\beta z(\delta-y)}{(\beta^{2}+z^{2}) \sqrt{\beta^{2}+\left(\delta-y\right)^{2}+z^{2}}} \tag{4}\]  
Analytical research was conducted, and an equation estimating the lateral stresses in the soil was presented. [PERSON] [45] calculated the settlement of a rectangular foundation with a linearly distributed load on a homogeneous soil. Then, the author determined the settlement of a heterogeneous soil and calculated the lateral increase in stress under a linearly distributed rectangular loaded area, taking into account the Poisson's ratio (\(\upsilon\)) of the soil. \[\sigma_{x}=P\left(K_{2}-(1-2\upsilon)K_{2}^{\prime}\right) \tag{5}\]  
In Equation (5), variables \(K_{2}\) and \(K_{2}^{\prime}\), which depends on the size of the foundation and the depth of the soil, are used. The \(\sigma_{x}\) value is the lateral stress value occurring under a rectangular loaded foundation. \(P\) is the maximum load before the failure. One study stated that the soil pressure at rest is related to the relative density. It is observed that there are very few studies measuring lateral stresses, and almost all of these studies are aimed at determining the \(K_{0}\) value. However, vertical stresses are mainly determined in theoretical studies, and there are very few recognized studies on lateral stresses. In most studies investigating the lateral behavior of soil, lateral deformations are measured and modeled using numerical methods. Lateral stresses are crucial design parameters in many engineering projects, such as retaining walls, silos, and tunnel designs. Therefore,  
Figure 1: Lateral stress due to finite area load [44]. this study measured the lateral stresses occurring in sandy soils with different internal frictions. Furthermore, the experimental results obtained were compared with the results of Equation (5). Therefore, the section comparing the experimental and theoretical results contributes significantly to this literature. Then, nonlinear regression analysis was performed based on the experimental results and geometric parameters. An empirical equation that can be used to predict additional lateral stress increases was developed. This study explored the impacts of relative density and foundation size on the bearing capacity, lateral stresses, and deformation behavior of rigid shallow foundations on sandy soil. Model experiments were used to determine the load-settlement relationships and lateral stress distributions that varied with depth. The theoretical calculation method was also used to assess the ultimate bearing capacity of the foundations. The results calculated using the equation presented by [PERSON] [45], one of the rare studies on this subject, were compared with the experimental lateral stress values. Classical elasticity theory-based relations, such as Boussinesq, can estimate the stress distribution in the subsoil in a general framework; however, they are limited in directly estimating lateral stresses. These theories mainly focus on the vertical stress distribution under vertical loading and calculate the horizontal stress values assuming an ideal elastic medium. Therefore, the use of these theories for lateral stress estimation may not provide sufficient accuracy, especially in cases where plastic deformations or changes in soil density need to be considered.