## 4 Results
### Dynamic Simulations  
Table 4 summarizes the results of the dynamic simulations. The maximum von Mises equivalent stresses and plastic strains of the cylindrical shell, roof dome, ring and roof stiffeners and rafters (beam) and anchors are evaluated. In addition, the deformations of the cylindrical shell and roof as well as the maximum uplift height are analyzed. Furthermore, Figure 13 displays the stress distribution of the overall tank while Figure 14 depicts the displacement figures of the cylindrical tank shell. The description of the results uses the following location descriptions along the tank circumference:  
* Frontal tank wall: the tank wall in the direction of the explosion source. * Tank wall sides: the tank wall parallel to the direction of the explosion source. * Tank rear wall: the tank wall on the side facing away from the explosion pressure wave. Figure 12: Results of implicit, static analyses; stress results in (Pa). Figure 13: von Mises stress of tank elements as a result of the various simulations, in Pa. Note: Simulations 1.2a and 1.2b as well as 2.2a and 2.2b result in the same stress distribution. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{**Maximal von Mises Stress (MPa)**} & \multicolumn{4}{c}{**Maximal Deformation (mm)**} & \multicolumn{4}{c}{**Maximal Plastic Strain (\%)**} \\ \hline
**Sim** & **Shell** & **Roof** & **Beam** & **Anchor** & **Shell** & **Roof** & **Uplift 1 ** & **Shell** & **Roof** & **Beam** & **Anchor** \\ \hline
1.1a & 352 & 346 & 356 & 362 & 1089 b  & 109 & 79 & 1.444 & 0.249 & 1.510 & 1.919 \\
1.1b & 493 & 431 & 503 & 505 & 1044 b  & 106 & 61 & 0.904 & 0.156 & 1.160 & 1.160 \\
1.2a & 115 & 233 & 124 & 121 & 11 & 57 & 6 & 0 & 0 & 0 \\
1.2b & 115 & 233 & 124 & 121 & 11 & 57 & 6 & 0 & 0 & 0 \\
1.3a & 175 & 345 & 345 & 345 & 34 & 82 & 11 & 0 & 0.067 & 0.114 & 0.049 \\
1.3b & 173 & 402 & 386 & 395 & 36 & 82 & 11 & 0 & 0.028 & 0.003 & 0.019 \\ \hline
2.1a & 350 & 228 & 358 & 360 & 1004 b  & 50 & 63 & 1.129 & 0 & 1.555 & 1.650 \\
2.1b & 476 & 227 & 494 & 501 & 949 b  & 48 & 51 & 0.535 & 0 & 0.934 & 1.032 \\
2.2a & 149 & 158 & 110 & 345 & 17 & 34 & 9 & 0 & 0 & 0 & 0.005 \\
2.2b & 149 & 158 & 110 & 353 & 17 & 34 & 9 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}  
* Maximal uplift height of annular plate. \({}^{\text{b}}\) Timestep of simulation at 1022 ms, when the pressure wave leaves the last elements of the simulation model.  
\end{table} 
Table 4: Summary of results of tank structure simulations. Figure 13 shows the von Mises equivalent stress for the various configurations of the dynamic calculation. For all figures, the color scale is selected so that the maximum value is at the nominal yield point of 345 MPa. This means that areas showing stress values of this order of magnitude or higher are displayed in red and are easily comparable. As expected, the different load profiles and material formulations used in the analyses show different results. Applying a uniform surface pressure to half of the cylinder and the roof (Simulation 1.1a/-b) results in high stresses in the order of magnitude and above the yield on the tank wall sides, where the loaded tank shell presses against the unloaded shell. This results in plastic strain (Table 4). Maximal lifting heights of 79 mm in Simulation 1.1a and 61 mm in Simulation 1.1b and plastic strain of the anchorage are also observed. When using the Johnson-Cook (JC) formulation with the more accurate modeling of the material at the transition to the plastic region, strain rate effects are taken into account, which lead to smaller plastic deformations overall; for instance, in the tank shell, the plastic strains are only 63% compared to bilinear (BL) material formulation (\(\epsilon_{\text{pl,shell,JC}}\) = 0.904%,  
Figure 14: Displacement of tank shell elements as a result of the various simulations, in m. Note: Simulations 1.2a and 1.2b as well as 2.2a and 2.2b result in the same shell deformation figure.