# Numerical Parametric Studies on the Stress Distribution in Rocks around Underground Silo
## 5 Numerical Results  
### Displacements  
Exaggerated deformed shapes resulting from the excavation of the underground silo are shown in Figure 11 for three different cases of Ko values. For the case of Ko = 0.5, the maximum displacement (0.85 mm) occurred in the vertical direction at the central point of the silo's bottom surface (Location F) since the in situ vertical stresses were twice as high as the in situ horizontal stresses. For the case of Ko = 2.0, on the other hand, the maximum displacement (1.32 mm) occurred in the radial direction at mid-point of the cylindrical wall (Location D) since the in situ horizontal stresses were twice as high as the in situ vertical stresses. For the case of Ko = 1.0, where the in-situ stresses were hydrostatic, the maximum vertical displacement (0.786 mm) at central point of the silo's bottom surface (Location F) were somewhat higher than the radial displacement (0.652 mm) at the mid-point of the cylindrical wall (Location D).  
The displacements at key locations obtained through 2D axial symmetric analysis based on change in the ratio of in situ horizontal stress to vertical stress (Ko) are presented in Table 4. The displacements at key locations obtained through 3D analysis based on change in the ratio of in situ horizontal stress to vertical stress (Ko) are presented in Table 5. It can be seen that numerical results of the 3D analysis were almost identical everywhere to those of the 2D axial symmetric analysis. Therefore, the reliability of the 2D finite element model was confirmed.  
Figure 10: Key locations of the finite element model after excavation of the underground silo.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Locations** & **Direction** & **Ko = 0.5** & **Ko = 1.0** & **Ko = 2.0** \\ \hline \multirow{2}{*}{A} & Vertical & \(-0.485\) & \(-0.375\) & \(-0.153\) \\ \cline{2-5}  & Radial & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \multirow{2}{*}{B} & Vertical & \(-0.318\) & \(-0.234\) & \(-0.064\) \\ \cline{2-5}  & Radial & \(-0.076\) & \(-0.230\) & \(-0.538\) \\ \hline \multirow{2}{*}{C} & Vertical & \(0.284\) & \(0.284\) & \(0.283\) \\ \cline{2-5}  & Radial & \(-0.208\) & \(-0.540\) & \(-1.205\) \\ \hline \multirow{2}{*}{D} & Vertical & \(0.087\) & \(0.070\) & \(0.038\) \\ \cline{2-5}  & Radial & \(-0.318\) & \(-0.651\) & \(-1.316\) (max) \\ \hline \multirow{2}{*}{E} & Vertical & \(0.281\) & \(0.206\) & \(0.055\) \\ \cline{2-5}  & Radial & \(-0.067\) & \(-0.191\) & \(-0.438\) \\ \hline \multirow{2}{*}{F} & Vertical & \(0.848\) & \(0.785\) & \(0.658\) \\ \cline{2-5}  & Radial & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Displacements at key locations obtained through 2D analysis (unit: mm).  
Figure 11: Deformed shapes of the underground silo.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Locations** & **Direction** & **Ko = 0.5** & **Ko = 1.0** & **Ko = 2.0** \\ \hline \multirow{2}{*}{A} & Vertical & \(-0.489\) & \(-0.379\) & \(-0.158\) \\ \cline{2-5}  & Radial & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \multirow{2}{*}{B} & Vertical & \(-0.321\) & \(-0.235\) & \(-0.064\) \\ \cline{2-5}  & Radial & \(-0.078\) & \(-0.234\) & \(-0.547\) \\ \hline \multirow{2}{*}{C} & Vertical & \(0.286\) & \(0.285\) & \(0.283\) \\ \cline{2-5}  & Radial & \(-0.209\) & \(-0.542\) & \(-1.208\) \\ \hline \multirow{2}{*}{D} & Vertical & \(0.087\) & \(0.703\) & \(0.037\) \\ \cline{2-5}  & Radial & \(-0.319\) & \(-0.652\) & \(-1.320\) (Max) \\ \hline \multirow{2}{*}{E} & Vertical & \(0.282\) & \(0.206\) & \(0.054\) \\ \cline{2-5}  & Radial & \(-0.067\) & \(-0.191\) & \(-0.439\) \\ \hline \multirow{2}{*}{F} & Vertical & \(0.850\) (Max) & \(0.786\) (Max) & \(0.659\) \\ \cline{2-5}  & Radial & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Displacements at key locations obtained through 3D analysis (unit: mm).