# Experimental and Numerical Studies of Vertical Stresses
## 5 Conclusions  
The vertical stress distribution beneath the circular footings on sand were investigated experimentally and numerically. The obtained results were compared with [PERSON]'s solution. Based on the results of this research, the following main conclusions can be drawn:  
Figure 23: Vertical stress distribution (D = 200 mm).  
Figure 22: Vertical stresses beneath the 150 mm circular footing: (a) 2 DM; (b) 3 DM.  
* The vertical stress distribution in horizontal planes at depth z from the footing base shows the same characteristic form in all the methods presented in this study, and the vertical stresses beneath the centerline of the model footing decrease as the depth increases.
* Experimental results obtained from the laboratory tests show that the zone where the measured vertical stress value is 10% of the applied load is approximately 2.0 D (two times the footing diameter), and this is consistent with numerical and theoretical solutions.
* Test results show that at the same ratio of z/D, vertical stress values increase with increasing the diameter of the footing under the same plate load, especially at shallow depths. It can be concluded that if two footings of various sizes subjected to the constant stress level are placed on the same sand with the same density, the vertical stresses beneath these two footings will be different. In other words, the larger footing produces a greater stress bulb than the smaller footing. The bigger footing produces a larger stress bulb than the smaller footing does. This behavior is observed in numerical analyses of both 2 DM and 3 DM but not in [PERSON]'s solution because it does not take into account the footing size.
* According to the results of the model tests performed at different footing diameters for the same applied stress, beneath the center line of the 150 mm diameter circular footing, compared to the 85 mm diameter circular footing, higher vertical stress values were obtained at the rate of 20% at 1.0 D, 12.5% at 1.5 D and 1% at 2.0 D depth.
* Considering the results of the numerical analysis performed at different footing diameters for the same applied stress, the vertical stress values were obtained beneath the centerline of the 150 mm diameter circular footing, up to 2.4% higher in the 2D model and up to 12% in the 3D model, compared to the 85 mm diameter circular footing.
* Differences of up to 13% were observed between the vertical stress values obtained by the experimental and Boussinesq method.
* Based on the numerical results from the parametric study, it appears that the 2 DM results are greater than the 3 DM results at all depths for three circular footings of different diameters.
* According to the results obtained from the parametric study, Boussinesq and numerical results give relatively close values up to 2.0 D depth, while the Boussinesq method gives lower vertical stress values than the numerical results at depths greater than 2.0 D.
* The vertical stress values obtained from 2 DM and 3 DM analyses are tolerably different from each other, and they both give conservative results. Because of the difficulty of the 3D modeling and because it normally takes a longer time to solve, it can be concluded that the 2D axisymmetric model can be suggested to be sufficiently appropriate in the solution of vertical stress increment problems.
* As a result of the analyses carried out to determine the effect of the material model on the vertical stresses, it was determined that the two-dimensional HS model results were quite close to the three-dimensional DP results and were approximately 5-17% lower than the two-dimensional DP model results depending on the footing diameter and measurement depth.  
Conceptualization, project administration, supervision: M.S.K.; writing--review and editing, and writing--original draft: S.B.; methodology, data curation, software, validation, and formal analysis: M.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.