# Experimental and Numerical Studies of Vertical Stresses
## 4 Results and Discussions
### Effect of Material Model  
A series of two-dimensional finite element analyses was carried out in order to examine the effect of the material model used in the numerical analysis on the vertical stress values due to external loads under circular footings on sand. For this purpose, vertical stress values under circular footings with 85 and 150 mm diameters were obtained with the PLAXIS  
Figure 12: Vertical stress values obtained from 3 DM analysis.  
Figure 13: Vertical stress values obtained from Boussinesq theory.  
program using the experimental setup dimensions. In the analyses, the elastoplastic hyperbolic model described as the hardening soil model (HS) was used to describe the non-linear sand behavior. When subjected to primary deviatoric loading, cohesionless soil shows a decreasing stiffness, and simultaneously, irreversible plastic strains develop. The observed relationship between the pressure and axial strain can be well approximated by a hyperbola as used in the variable elastic, hyperbolic model [28]. However, the HS model is far superior to the hyperbolic model, being capable of simulating non-linear, inelastic, stress-dependent material behavior. Limiting states of stress are described by means of the friction angle (\(\mathcal{O}\)), the cohesion (c), and the dilatancy angle (\(\psi\)). In addition, the increase in soil stiffness with pressure is accounted for in all three stiffnesses used, i.e., the triaxial loading stiffness \(\mathrm{E}_{50}\), the triaxial unloading/reloading stiffness \(\mathrm{E}_{\mathrm{ur}}\) and the oedometer loading stiffness \(\mathrm{E}_{\mathrm{oed}}\)[29]. Soil parameters used in the numerical analysis obtained by consolidated-drained triaxial compression tests on sand are shown in Table 5.  
The stress values obtained as a result of the analyses carried out to examine the effect of the material model on the vertical stresses are shown in Figures 14 and 15. Figure 16 shows the vertical stress distribution beneath the footings. As seen from the figures, the vertical stress values obtained using the hardening soil material model in two-dimensional conditions are closer to the three-dimensional Drucker-Prager results than the two-dimensional Drucker-Prager results. When comparing the vertical stress values obtained with the HS and DP models in two-dimensional conditions for the 85 mm diameter footing, it is seen that the two-dimensional DP model results are approximately 17% greater at 1.0 D depth, 11% at 1.5 D depth, and 10% at 2.0 D depth than the HS model results. For the 150 mm diameter footing, it is seen that the two-dimensional DP model results are approximately 12%, 10% and 5% larger, respectively, at the same depths.  
It can be said that the difference between the vertical stress values obtained with different material models may be due to the assumptions made in the soil models in question and the difference in the finite element mesh between the two computer programs. As it is known, sand soils are stress-dependent soils. Considering this feature of sand soils, it can be said that the hardening soil model can more realistically model the stress-dependent behavior of sand soils.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Symbol** & **Unit** & **Value** \\ \hline Unit weight & V & \(\mathrm{kN/m^{3}}\) & 17.1 \\ Primary loading stiffness & \(\mathrm{E}_{\mathrm{ret}}^{50}\) & \(\mathrm{kN/m^{2}}\) & 28,000 \\ Initial stiffness & \(\mathrm{E}_{\mathrm{oed}}\) & \(\mathrm{kN/m^{2}}\) & 28,000 \\ Unloading/reloading stiffness & \(\mathrm{E}_{\mathrm{ur}}\) & \(\mathrm{kN/m^{2}}\) & 84,000 \\ Cohesion & c & \(\mathrm{kN/m^{2}}\) & 0.10 \\ Internal friction angle & \(\mathcal{O}\) & (\({}^{\circ}\)) & 41 \\ Dilatancy angle & \(\psi\) & (\({}^{\circ}\)) & 11 \\ Poisson’s ratio & \(\mu\) & - & 0.20 \\ \hline \end{tabular}
\end{table}
Table 5: Parameters of HS model.  
Figure 14: Vertical stresses with HS and DP models (85 mm footing).  
Figure 15: Vertical stresses obtained with HS and DP models (150 mm footing).