## 6 Result of Case Studies
### Model 3--LNG Cargo Containment Boxes with Hull Structure
Tables 12 and 13 show the ultimate dynamic shear and bending capacities of Model 3, i.e., the LNG cargo containment system, including the hull structure, which were calculated using Equation (10) and DIR, including the nonlinear boundary conditions. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Model 2**} & \multicolumn{4}{c}{**Model 3**} \\ \cline{2-10}  & **Median** & **Max.** & **Mean** & **SD.** & **Mean** \(\pm\) **2 \(\times\) SD** & **Median** & **Max.** & **Mean** & **SD.** & **Mean** \(\pm\) **2 \(\times\) SD** \\ \hline Shear & \(-\)2.3\% & 5.2\% & \(-\)2.7\% & 3.4\% & \(\begin{array}{c}4.2,\\ -9.5\%\end{array}\) & 2.5\% & 6.1\% & 1.4\% & 4.5\% & \(\begin{array}{c}10.3,\\ -7.5\%\end{array}\) \\ \hline Bending & 7.6\% & 11.4\% & 6.6\% & 3.5\% & \(\begin{array}{c}13.6,\\ -0.4\%\end{array}\) & \(-\)1.0\% & 20.9\% & \(-\)2.0\% & 11.9\% & \(\begin{array}{c}21.9,\\ -25.8\%\end{array}\) \\ \hline \hline \end{tabular} Note: SD = Standard deviation.  
\end{table} 
Table 11: Statistical data of difference in the shear/bending stresses calculated using TIRF with linear boundary conditions and DIR with nonlinear boundary conditions for Model 2 and Model 3. In this study, series FE assessments using DIR and TIRF were carried out on the ultimate shear and bending capacities of LNG cargo containment boxes for 12 load cases using load areas, as defined in Figure 5. Because the total amount of external energy is proportional to the load area, the 12 load areas can be simplified as 1 \(\times\) 1 (= 0.13 m\({}^{2}\)), 3 \(\times\) 1 (= 0.40 m\({}^{2}\)), 2 \(\times\) 2 (= 0.53 m\({}^{2}\)), 3 \(\times\) 2 (= 0.80 m\({}^{2}\)), and 3 \(\times\) 3 (= 1.20 m\({}^{2}\)) with respect to the load area size. The minimum shear and bending strengths of the LNG cargo containment boxes for each load area are summarized in Tables 12 and 13, respectively. The results of Tables 12 and 13 are shown in Figure 23a,b using normalized values, i.e., DIR/TIRF, for comparison purposes. All the ultimate shear and bending capacities calculated using Equation (10) show practically conservative results against the DIR. There have been several studies investigating the use of TIRF approaches for dynamic transient simulation purposes. However, it should be noted that the TIRF approach is theoretically applicable only to linear systems, such as steel structures. In contrast, the current study focuses on nonlinear systems, including nonlinear geometry, nonlinear material properties, and nonlinear boundary conditions, to evaluate whether the linear \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Rise Time** [msec]} & \multicolumn{2}{c}{**Area = 0.13 m\({}^{2}\)**} & \multicolumn{2}{c}{**Area = 0.40 m\({}^{2}\)**} & \multicolumn{2}{c}{**Area = 0.53 m\({}^{2}\)**} & \multicolumn{2}{c}{**Area = 0.80 m\({}^{2}\)**} & \multicolumn{2}{c}{**Area = 1.20 m\({}^{2}\)**} \\ \cline{2-10}  & **TIRF** & **DIR** & **TIRF** & **DIR** & ** TIRF** & **DIR** & ** TIRF** & **DIR** & ** TIRF** & **DIR** \\ \hline
1 & 15.2 & 21.3 & 21.9 & 21.8 & 22.4 & 28.1 & 18.8 & 26.0 & 22.3 & 25.9 \\
2 & 15.1 & 21.8 & 21.9 & 23.4 & 22.5 & 27.6 & 19.7 & 27.4 & 22.3 & 27.3 \\
5 & 15.1 & 22.8 & 21.9 & 23.6 & 22.5 & 28.2 & 20.3 & 28.1 & 22.3 & 28.6 \\
10 & 15.1 & 23.2 & 21.8 & 23.8 & 22.6 & 28.5 & 20.5 & 28.2 & 22.3 & 28.3 \\
20 & 15.1 & 23.3 & 21.7 & 23.7 & 22.6 & 28.4 & 20.6 & 28.1 & 22.3 & 28.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Ultimate dynamic bending capacities of Model 3 using TIRF with linear boundary condition and DIR with nonlinear boundary condition (unit: bar). Figure 23: Normalized ultimate dynamic capacities using TIRF and DIR: (**a**) Shear capacity (**b**) Bending capacity.