### Lateral Stress Theories in Literature  
The accurate determination of stress distribution in soils is of great importance in geotechnical engineering in terms of foundation design, stability of soil structures, and safety of underground structures. The widely used Boussinesq theory can only calculate the vertical stress distribution in isotropic and homogeneous soils within the framework of elasticity theory [32]. This theory, solved the problem of stress distribution in an elastic, isotropic, infinite half-space by taking into account a point load on its planar surface. However, since this theory is based on certain assumptions, it may not always accurately reflect the actual soil conditions (such as soil type, relative density, internal friction angle, cohesion, and void ratio). Although the Boussinesq theory is an essential tool in soil mechanics, it may not be sufficient in all cases due to the complexity of real soil conditions. In particular, in cases such as elastoplastic behavior, layered soil structure, and stress increases in the soil, more accurate analyses should be conducted through alternative empirical studies. Since then, numerous studies have been conducted to investigate the applicability of the Boussinesq theory in determining the stress distribution in soils resulting from surface loading or foundation pressures [33; 34; 35; 36; 37; 38; 39]. It has been realized that the soil, being a non-elastic and anisotropic material, does not behave as perfectly as the Boussinesq theory assumes; therefore, experimentalists have proposed several modifications to the basic formulations [36; 37; 38; 40]. Certain formulations are available in the literature for estimating stress in soil mechanics, particularly for infinite loading. [PERSON] and [PERSON] (1974) performed stress calculations under various loading and geometric conditions to determine the stress distributions in soil, particularly under surface and rigid loading [41]. These calculations were performed using the basic equations of elastic theory. The lateral stress distribution in the soil under a rectangular area with a uniform load applied to the surface;  
\[\sigma_{x}=\frac{2q}{\pi}\frac{x^{2}z}{\left(x^{2}+z^{2}\right)^{2}} \tag{2}\]  
where \(q\) is the uniformly distributed load, \(z\) is the depth, and \(x\) is the distance parallel to the wall surface, as expressed in Equation (2). It should also be noted that finite-area loading solutions are available, but they have limitations. [PERSON] (1935) and [PERSON] (1936) provided an equation and chart for calculating the vertical stresses under the corner of a finite rectangular area, from which extrapolation can be made to find the loading at any point under the rectangle, considering rectangular areas of different sizes [35; 42]. However, these methods are time-consuming and are used to control vertical stress in the soil. The literature review also shows that Poisson's ratio is generally assumed to be equal to 0.50 while developing formulations for lateral stress. As stated by [PERSON] (2017) [43], Poisson's ratio for soil can vary from about 0.25 (granular and hard cohesive soils) to 0.49 (soft cohesive soils); therefore, load reductions can be made if the soil properties are known. In appropriate cases, these lateral stress solutions can be represented by Poisson's ratio of the soil considered in the formulations. For this purpose, a study was conducted that presented closed-form solutions for the stresses in the soil due to finite areas and loads [44]. In this analytical study, considering the three-dimensional Cartesian coordinate system, including finite line and finite area loads, the aim is to calculate the vertical and lateral stress values directly at any point. In this study, formulations are presented that calculate lateral stress estimates by considering the geometric dimensions and the effect of the soil's Poisson ratio (Figure 1). The lateral stress calculations for a uniformly loaded finite area are presented in Equations (3) and (4).