## 3 Results and Discussion
### Verification and Validation of Computational Modeling of Buckling
#### 3.1.1 Verification of Elastoplastic Buckling Model Under Combined Loading on Unstiffened Plates  
The first verification was carried out considering [PERSON] and [PERSON] [23]'s study, which applied longitudinal compressive loading \(\sigma_{X}\) and transverse compressive \(\sigma_{Y}\) loading in different ratios, and a constant lateral pressure \(P_{z}=0.16\) MPa to a simply supported unstiffened plate, with \(a=4300\) mm, \(b=815\) mm, \(t=19\) mm, \(E=208.5\) GPa, \(\sigma_{y}\ =\)315 MPa, and \(\
u\ =\ 0.3\). An initial imperfection of plate as \(w_{0}=b/200\), while the proposed model considered \(w_{0}=b/2000\) (Equation (8)). The geometry of the unstiffened plate with boundary conditions of the proposed model is shown in Figure 8.  
Figure 7: Search space sequence for exhaustive search technique application (adapted from [PERSON] et al. [13]).  
The finite element size used in the proposed computational modeling was defined according to the mesh convergence test considering the loading condition \(\sigma_{X}\ :\sigma_{Y}\) = 0.8:0.2, as shown in Table 1.  
Considering the element size with 25 mm in Table 1, the proposed model resulted in \(\sigma_{uX}\) = 225.22 MPa and \(\sigma_{uY}\) = 56.30 MPa. When compared to the values obtained by [PERSON] and [PERSON] [23], which were \(\sigma_{uX}\) = 222.96 MPa and \(\sigma_{uY}\) = 55.72 MPa, the differences are 1.01% and 1.04% higher, respectively. For the ratio \(\sigma_{X}\ :\sigma_{Y}\) = 0.7:0.3, the proposed model resulted in \(\sigma_{uX}\) = 159.07 MPa and \(\sigma_{uY}\) = 68.17 MPa. When compared to the values obtained by [PERSON] and [PERSON] [23] which were \(\sigma_{uX}\) = 153.97 MPa and \(\sigma_{uY}\) = 65.99 MPa, the differences are 3.31% and 3.30% higher, respectively. Despite accounting for different boundary conditions and values of initial imperfections, the ultimate stress values confirm that the proposed model is verified for use in the elastoplastic analysis of unstiffened plates.  
The second verification was performed considering [PERSON] et al.'s [11] work, which applied biaxial compressive loading and a constant lateral pressure \(P_{z}\) = 0.0507 MPa to a plate made of AH-36 steel, with \(a\) = 2000 mm, \(b\) = 1000 mm, \(t\) = 12 mm, \(E\) = 210 GPa, \(\sigma_{y}\) =355 MPa, and \(\
u\) = 0.3, with a rectangular perforation in the center, with length \(a_{0}\) =450 mm and width \(b_{0}\) = 667 mm, as shown in Figure 9.  
[PERSON] et al. [11] used the SHELL281 finite element with 30 mm and a quadrilateral form. The \(\sigma_{u}\) obtained with the proposed computational model was achieved according to the mesh convergence test shown in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline
**Element Length** & **Number of** & \(\sigma_{uX}\) \\
**(mm)** & **Elements** & **(MPa)** \\ \hline
100 & 387 & 225.22 \\
75 & 638 & 225.22 \\
50 & 1462 & 225.22 \\
25 & 5676 & 225.22 \\ \hline \end{tabular}
\end{table}
Table 1: Mesh convergence test for verification of elastoplastic buckling model under combined loading on unstiffened plate.  
Figure 8: Unstiffened plate geometry and boundary conditions of the proposed model, with the blue symbols representing the displacement constraints according to the axis direction, the red arrows representing the biaxial compressive loading, and the red lines on the plate representing the applied lateral pressure.  
In the present study, a \(\sigma_{u}~{}=~{}28.40\) MPa was obtained with 200 sub-steps of loading. When compared to \(\sigma_{u}=~{}28.76\) MPa with 1000 sub-steps of loading by [PERSON] et al. [11], the result is 1.25% lower. Despite the different reference loading sub-steps considered in each model, the ultimate stress values also confirm that the proposed model is verified for use in the elastoplastic analysis of unstiffened plates.