# Numerical Parametric Studies on the Stress Distribution in Rocks around Underground Silo
## 5 Numerical Results
### Stresses  
This section presents the numerical results for stresses obtained by 2D axial symmetric analysis. The results of computed stresses examined at the key locations are presented in Figure 10. Nine different lines were selected around the silo: a vertical upward line, A1, above dome crown; a vertical upward line, B2; a diagonal line, B3; a horizontal line, B4; a horizontal line, C5, at the dome's bottom; a horizontal line, D6, at the middle of the storage wall; a horizontal line, E7; a vertical downward line, E8, at the silo's bottom corner; and a vertical downward line, F9, at the bottom center of the silo. Among all these locations, it was found that deviatoric stresses were most concentrated around two locations: C and D. Therefore, in the following analysis, stress results will be interpreted along the two corresponding lines, C5 and D6. In Figures 12-14, the stress state in the p-q plane is summarized for Ko values of 0.5, 1.0, and 2.0, along with the strength envelopes corresponding to the triaxial compression and extension modes. Using the given material properties (\(\varnothing=43\) and \(\mathrm{c}=2.26\) MPa, these two upper- and lower-bound strength envelopes were computed based on the following equations:  
\[\mathrm{q}_{\mathrm{TXC}}=[6\sin\varnothing/(3-\sin\varnothing)]\cdot\mathrm{P }+[6\cos\varnothing/(3-\sin\varnothing)]\cdot\mathrm{c}=1.765\;\mathrm{P}+4.27\]  
\[\mathrm{q}_{\mathrm{TXC}}=[(3-\sin\varnothing)/(3+\sin\varnothing)]\cdot \mathrm{q}_{\mathrm{TXC}}=0.63\;\mathrm{q}_{\mathrm{TXC}}\]  
Generally, the deviatoric shear stresses increased with the distance from the storage wall, but the mean pressures remained more-or-less constant, except for the first three points along line C5 for Ko = 0.5. For Ko = 0.5, all deviatoric stresses were somewhat below the lower-bound strength envelope in the triaxial extension mode. For Ko = 1.0, deviatoric stress in the first element in line C5 reached the lower-bound strength in the triaxial extension mode. For Ko = 2.0, the deviatoric stresses in the first two elements in lines C5 and D6 were above the lower-bound strength envelope and the first element in line C5 almost reached the upper-bound shear strength in the triaxial compression mode. The plots in the p-q plane shown in Figures 11-13 provide a good evaluation of the stress states along lines C5 and D6 by comparing the upper- and lower-bound strength envelopes. However, these plots do not show the ultimate strength corresponding to each stress point since the ultimate strengths are dependent on both mean pressure and Lode angle, as explained in the material model in Section 3[34; 35]. Therefore, in order to evaluate more accurately, an additional plot showing the stress state in the octahedral plane is necessary. The stress states in the p-q and t-q planes are summarized in Figures 15-17 for Ko values of 0.5, 1.0 and 2.0, respectively. The t-q plane means a \(\emptyset\)-q plane failure surface. The t-q plot, as introduced in this study, is essentially the simpler form of deviatoric stress plot in the octahedral plane shown in Figures 6 and 7. For Ko = 0.5, the stress state was somewhat close to the triaxial compression mode at Location C, but was closer still to the triaxial extension mode at Location D. The safety factors, as defined in Figure 17 in the next section, were 2.61 and 1.58 at Locations C and D, respectively. For Ko = 1.0, the stress state was very close to the triaxial compression mode at Location C, but was very close to the deviatoric pure shear mode at Location D, with a Lode angel of \(-1.9^{\circ}\). The safety factors were 1.47 and 1.34 at Locations C and D, respectively. For Ko = 2.0, the stress state at Location C was almost in the triaxial compression mode, reaching ultimate strength with a safety factor of 1.02. The stress state at Location D was also close to the triaxial compression mode, with a Lode angle of \(-20.9^{\circ}\), reaching ultimate strength with a safety factor of 1.08.