# Experimental and Numerical Studies of Vertical Stresses
## 4 Results and Discussions
### Effect of Footing Size  
Two models of circular footings with different diameters were tested to evaluate whether the vertical stresses occurring below the centerline of the circular footing depend on the footing width, D. Vertical stress values obtained from the tests are compared in Figure 10. Figure 10 shows that at the same ratio of z/D, vertical stress values increase with increasing the diameter of the footing under the same plate load. As seen in Figure 10, measured stresses beneath the 150 mm circular footing are considerably higher than the stresses beneath the 85 mm circular footing, especially at shallow depths. According to the results of the model tests performed at different footing diameters, beneath the centerline 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 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.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Depth, z**} & \multicolumn{4}{c}{**Vertical Stresses, \(\Delta\sigma_{\mathbf{z}}\) (kN/m\({}^{2}\))**} \\ \cline{2-5}  & **Experimental** & **2 DM** & **3 DM** & **Boussinesq** \\ \hline
1.0 D & 3.18 & 3.22 & 2.88 & 2.85 \\
1.5 D & 1.62 & 1.69 & 1.46 & 1.46 \\
2.0 D & 0.99 & 1.00 & 0.91 & 0.87 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Vertical stress values beneath the centerline of the 150 mm circular footing.  
Figure 9: Vertical stress distribution beneath the 150 mm circular footing.  
Figures 11-13 present the vertical stress values obtained from 2 DM, 3 DM and theoretical analyses, respectively. As seen from Figure 11, although the vertical stresses under the 150 mm diameter footing are higher in the 2 DM analysis, the amount of increase is quite low. There is a maximum difference of 2.4% between the vertical stress values obtained for 85 and 150 mm diameter footings.  
Figure 12 shows that vertical stress values below 150 mm circular footing in 3 DM are up to 12% greater than stresses below 85 mm circular footing. As can be clearly seen in Figure 13, the size of the footing has no effect on the vertical stresses calculated with the Boussinesq method, as the formula does not consider the size of the footing.  
Figure 11: Vertical stress values obtained from 2 DM analysis.  
Figure 10: Measured vertical stress values for circular footings.  
The effect of the foundation diameter on the vertical stresses can be explained as follows: As the foundation diameter increases, the area on which the load acts becomes larger, and this causes a greater number of sand grains to transfer the vertical stresses to the deeper soil layers under the foundation, especially in dense sands. This result is similar to the results obtained from the study by [PERSON] et al. (2014). As a result, a greater stress bulb is formed under the foundation, and at the same depth/foundation diameter ratios, greater stress values occur under the large foundation diameter. This also confirms the stress-dependent behavior of sand soils.