# Experimental and Numerical Studies of Vertical Stresses
## 1 Introduction  
The self-weight of the soil layers and the applied external loads cause stresses in the soil. Estimating the distribution and amount of these stresses in the soil is important to solve many problems in geotechnical engineering. These stresses depend on the soil properties, the intensity of the applied load, and the dimensions of the loaded area. The estimation of the stresses within a soil mass due to applied loads has concerned researchers for many years. Many approaches have been developed using the theory of elasticity to determine the stress distribution in soil. These approaches are similar, and the differences between them are the assumptions made to determine the elastic conditions of the soil. Due to the complex structure of the soil, it is very difficult to make realistic stress-strain analyses. For this reason, analyses are usually performed with the assumption that the soil is isotropic and linearly elastic. Therefore, soil properties are defined by two parameters, elasticity modulus, E, and Poisson's ratio, \(\mu\). [PERSON]'s [1] equation is one of the most frequently used and practical equations, which is used to obtain the stress values in the soil due to external loads. This equation was originally developed to calculate the vertical stresses for a point load in a homogeneous, isotropic, weightless, linearly elastic half-space, and it is integrated over the footing area to calculate the stresses under a distributed load. It is often used when stresses and deformations in the soil due to loads applied from circular areas on the soil surface are applied by circular storage silos, foundations, and wheels. [PERSON] and [PERSON] [2] and [PERSON] and [PERSON] [3] carried out studies on the calculation of stress and deformations in a homogeneous and isotropic medium due to loads applied to circular areas. In these studies, simple equations required for calculations and influence diagrams and coefficient tables for use in equations are presented. For the solution of foundation engineering problems, it is very important to know the vertical stresses that occur in the soil due to external loads. The variation of vertical stresses under a uniform load applied to a circular area with respect to the radius of the loaded area and the depth from which the vertical stress value will be calculated is shown in Figure 1.  
In [PERSON]'s [1] solution, vertical stress distributions are independent of the material properties. In addition, parameters such as the type of soil, density and consistency cannot be taken into account, and there is no difference between the stress distributions in soil types with different properties. [PERSON] [4] presented an equation for a vertical point load at the surface and discussed the stress functions for plane strain problems. Unlike the Boussinesq solution, the Poisson ratio of the soil medium is required as the material parameter in the Westergaard equations for stress calculations. Although the soil medium is considered isotropic and homogeneous in the above-mentioned elastic solutions, most soils are neither isotropic nor homogeneous. Previous studies including laboratory experiments and numerical analyses on the subject have shown that the stress distribution in soils varies significantly depending on the material parameters of the soil. For that reason, it is required to find the vertical stress values experimentally and to compare these with numerical and theoretical solutions. When the previous studies on the vertical stress distribution in soils due to applied loads are examined, it is seen that not enough experimental studies have been performed. The vertical stress value at any depth of z under the centerline for a uniformly distributed load applied to a circular area is:  
\[\sigma_{z}=q\left[1-\frac{z^{3}}{\left({R_{0}}^{2}+{z}^{2}\right)^{3/2}}\right] \tag{1}\]  
The study of [PERSON] and [PERSON] [5] is considered to be one of the first experimental studies to determine the stress values in soils.