# Experimental and Numerical Studies of Vertical Stresses
## 1 Introduction
In the study, laboratory model tests were carried out to obtain the stress distribution in the sand soil. For this purpose, a sand fill  
Figure 1: Vertical stress under a uniformly loaded circular area. was loaded, and stresses at particular depths of horizontal planes were measured with instruments located at measurement depths. [PERSON] and [PERSON] [5] compared the results obtained from the experimental study with the theoretical solutions and indicated that vertical stress distributions obtained from the experimental study beneath the loaded areas were in good agreement with the predictions from the Boussinesq equations [6]. In the experimental study carried out by [PERSON] and [PERSON] [7], vertical stresses were measured under model foundations resting on the sand, and it was observed that there was a general agreement with the stress values obtained by elastic solutions. [PERSON] et al. [8] examined factors such as non-homogeneity, non-linearity, and anisotropy that affect the vertical stress values and concluded that the Boussinesq solution gives a reasonably accurate vertical stress distribution under a uniformly loaded area for different soil conditions. In the study by [PERSON] et al. [9], dimensionless equations are derived to calculate the vertical stress distribution under rigid circular foundations on sands using the elastoplastic approach. The resulting distribution was used to calculate stresses at various points using elastic theory. In the study, the obtained results are presented graphically and are compared with the distribution of Boussinesq. [PERSON] [10] algebraically formulated the vertical stress distribution below the origin of the linearly distributed Boussinesq loading case. In the study, the equation was evaluated to establish that the currently used equation of the uniformly loaded rectangular loading case is valid. This paper concludes that the formula that was developed is valid. [PERSON] [11] carried out a series of laboratory model tests to investigate [PERSON]'s theory on the vertical stress increment in the sandy soil mass caused by surface loading. The test results were compared with the theoretical values of Boussinesq. The theoretical stress values obtained by the Boussinesq method were always smaller than the test results under the footing, regardless of the depth. Outside of the footing, the values were greater than the stress measured only at the depth of the foundation width. It has been seen that the theoretical and experimental results are close to each other when the depth reaches two and three times the width of the footing. In the study, the measured vertical stress decreased as the applied load increased. These trends were confirmed to be valid for the considered range of the relative density of sand and/or width of the footing. According to the results obtained from the study, it has been seen that if the Boussinesq theory is used, more accurate values can be obtained by correcting the theoretical values using these results. [PERSON] et al. [12] carried out an experimental investigation and compared the results with the numerical and theoretical results. They indicated that vertical stress distributions beneath the uniformly loaded rectangular area show the same characteristics in all methods. By using Boussinesq displacement potential functions, [PERSON] et al. [13] determined the vertical stress distribution and vertical displacements under the influence of a uniformly distributed load on a rectangular area, assuming that the soil medium is homogeneous, elastic, and semi-infinite. The results show that the strain-displacement relations for small displacement assumptions and [PERSON]'s generalized stress-strain laws for linear elastic, isotropic, homogeneous bodies are applied simultaneously to derive the vertical fields from the displacement fields. [PERSON] et al. [14] carried out a study to obtain the vertical stress values beneath the center of continuous foundations resting on geotextile-reinforced soil.