## 3 FE Analysis
### FE Modeling  
The commercial software Ansys Mechanical APDL (Version 2022 R2) was used to conduct nonlinear FE analysis using an elastic-perfectly plastic (EPP) material curve [34]. This study employed the FE modeling technique validated through numerical benchmark studies [35; 36].  
The ISSC and [PERSON] et al. [36; 37] recommended the use of 10 elements in the plate breadth direction for ultimate compressive strength calculation. However, a finer mesh  
Figure 2: Effect of element size around crack zone.  
with more elements is necessary to achieve a more detailed stress distribution and accurately capture the ultimate strength behavior. In this study, quadrilateral shell elements with a size of 21.5 mm were assigned as follows, considering the aspect ratio of the plate, web, and flange: 40 elements in the plate breadth (b), 200 elements in the plate length (L), 20 elements in the web height (h\({}_{w}\)), and 6 elements in the flange breadth (b\({}_{l}\)).  
The structures in this study were assumed to be simply supported because they were surrounded by longitudinal stiffeners (or girders) or transverse frames, which are strong structural members. Figure 3 illustrates the boundary conditions applied to the stiffened panels.  
For ultimate strength calculation, applied loads must be sufficiently high to induce structural collapse. In this study, the applied line pressure was at 1.1 times the yield strength multiplied by thickness (plate, web and flange, respectively), and five loading ratios (\(\sigma_{y}\):\(\sigma_{x}\) = 1:0, 3:1, 1:1, 1:3, 0:1) derived from experience [36] were considered to calculate the strength under uniaxial and biaxial compressive loads.  
Due to welding, initial deflection and residual stress are inevitable initial imperfections in ship structures. This study applied buckling initial deflection, column-type initial deflection, and sideways distortion to the plates, stiffened panels, and stiffeners, respectively, as shown in Figure 4 and determined using Equations (1)-(3) [38]. However, residual stress from welding was not considered. This is because the stress may be relieved after cracking and depends on crack size and type.  
\[w_{opl}=A_{o}sin\Big{(}\frac{m\pi x}{L}\Big{)}sin\Big{(}\frac{\pi y}{b}\Big{)} \ \ \text{for plate}, \tag{1}\]  
\[w_{oc}=B_{o}sin\Big{(}\frac{\pi x}{L}\Big{)}sin\Big{(}\frac{\pi y}{B}\Big{)} \ \ \text{for entire stiffened panel}, \tag{2}\]  
\[w_{os}=C_{o}\frac{z}{h_{w}}sin\Big{(}\frac{\pi x}{L}\Big{)}\ \ \text{for stiffener ( sideways)}, \tag{3}\]  
where \(A_{o}=0.1\beta^{2}t_{p}\), \(B_{o}=0.0015L\), \(C_{o}=0.0015L\), \(\beta\) is the plate slenderness ratio, \(t_{p}\) is the plate thickness, \(L\) is the plate length, b is the plate breadth, B is the panel breadth, and \(m\) is the buckling half-wave number in the length direction, which is the minimum integer satisfying Equation (4) [38].  
\[\frac{\big{(}m^{2}/L^{2}{+}1/b^{2}\big{)}^{2}}{m^{2}/L^{2}{+}c/b^{2}}\leq\frac {\big{[}(m+1)^{2}/L^{2}{+}1/b^{2}\big{]}^{2}}{(m+1)^{2}/L^{2}{+}c/b^{2}} \tag{4}\]  
where \(c\) is the loading ratio (\(\sigma_{y}/\sigma_{x}\)).  
Figure 3: Applied boundary conditions considering simply supported structures.