## 5 Results and Discussion
### Midship Section  
For each of the versions, the midship section is designed considering the minimum requirements by the classification society in order to obtain a comparison criterion between them. Figure 19 shows the midship section of each version.  
Figure 19: _Cont._  
Figure 18: Comparison in the bending moment value between container ship versions and loading conditions.  
\(V_{2}\) contains the largest quantity of steel, due to a reduced moment of inertia in comparison with the other models. Conversely, the remaining versions exhibit a comparable amount of steel, with \(V_{1}\) having the least. Specifically, the ratio of A235 steel to AH315 steel stands at 69% to 31% for V\({}_{0}\), 67% to 33% for \(V_{1}\), 61% to 39% for \(V_{2}\), and 77% to 23% for \(V_{3}\).  
Figure 20 shows the distribution of shear forces in the midship section (red lines, numbers and arrows refer to the modeling process, not numerical values) for the \(V_{2}\) caused by vertical shear force associated with a vertical bending moment in still water  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(V_{1}\) & \(V_{2}\) & \(V_{3}\) \\ \hline Area of cross-section of ordinary & Steel (A235) & 2.272 & 2.561 & 2.202 \\ stiffeners and plates (m\({}^{2}\)) & Steel (AH315) & 0.573 & 0.611 & 0.663 \\ Moment of inertia, \(y\)-transverse axis (m\({}^{4}\)) & 98.147 & 84.343 & 103.489 \\ Position of neutral axis above baseline (m) & 5.99 & 5.522 & 5.633 \\ Modulus at deck (m\({}^{3}\)) & 10.218 & 11.31 & 8.222 \\ Modulus at bottom (m\({}^{3}\)) & 16.37 & 15.27 & 18.369 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Section properties of the different versions.  
Figure 19: Midship section scantling. (**a**) \(V_{1}\), (**b**) \(V_{2}\), (**c**) \(V_{3}\).  
(\(Q_{s}\)), the vertical shear force associated with vertical bending moment due to wave (\(Q_{W}\)), the horizontal shear force associated with horizontal bending moment due to wave (\(Q_{lic}\)), Saint-Venant component of hydrodynamic and static torsional moment (\(M_{wtc}\), \(M_{slc}\)), and warping torsional moment of hydrodynamic and static torsional moment (\(M_{wtc}\), \(M_{stc}\)).  
Vertical shear forces are most pronounced along the shell side and double hull side near the neutral axis, particularly where the double bottom connects to the inner side. The region with the highest shear stress is within the double bottom cell, but its thickness balances the stresses, making them similar to areas with lower shear stress and thinner sections in the double bottom. Maximum shear stress distribution (\(Q_{hc}\)) is found in the bottom and double bottom strikes near the neutral axis and smaller scantlings. For \(M_{wtc}\) and \(M_{stc}\) (Saint-Venant), the highest stresses occur where multiple shear stresses converge at intersections. In contrast, \(M_{wtc}\) and \(M_{stc}\) (warping) stresses are notably lower, peaking near the neutral axis along the double side and at the centerline.  
Figure 21 shows the von Mises stress distribution and the deformed structure of the beam model representing the cross-section of the midship section of the \(V_{1}\) for standard upright and heel conditions. The values in red represent the highest values expressed in stresses (permissible values lower than the yield stress), while the values in blue represent the opposite, the intermediate colours refer to intermediate stresses located between the maximum and minimum stresses.