## 2 Materials and Methods
### Computational Modeling of Buckling
#### 2.1.2 Computational Modeling of Elastoplastic Buckling  
Although the elastoplastic buckling was defined with bilinear isotropic hardening, the material was considered as a linear elastic-perfectly plastic model, without strain-hardening, resulting in more conservative results. This regime involves the presence of geometric and material nonlinearities, causing \([K_{G}]\) and \([U]\) to no longer be proportional to the stresses. The initial imperfections of the elastic buckling were considered with the maximum amplitude value of initial imperfections [19]:  
\[w_{b}\;=\;\frac{b}{2000} \tag{8}\]  
where \(w_{b}\) is the maximum amplitude value of initial imperfections and \(b\) is the width of the plate. This maximum amplitude value multiplied to the first buckling mode displacements results in the initial shape of plate in the elastoplastic buckling, and it is considered in ANSYS, as described by [PERSON] [20].  
To determine the resultant displacements, two reference loads were defined as corresponding to the material's yield load for the plate and for the stiffeners:  
\[P_{Rp}\;=\;\sigma_{y}t_{p}, \tag{9}\]  
\[P_{Rs}=\sigma_{y}t_{s}, \tag{10}\]  
where \(P_{Rp}\) represents the reference load of the plate, \(\sigma_{y}\) represents the yield stress of the material, \(t_{p}\) represents the plate thickness (or only \(t\) for an untiffened plate), \(P_{Rs}\) represents the reference load of the stiffener, and \(t_{s}\) represents the stiffener thickness. The application of these reference loads ensures the application of the \(\sigma_{y}\) over the stiffened plate. The reference loads are divided into sub-steps and incremented with the Newton-Raphson iterative method, where the unbalanced load vector represents the difference between the external loads and the nonlinear internal forces at each iteration. At each iteration, the displacement vector was recalculated:  
\[\{\varphi\}_{r+1}\;= \{P\}_{i+1}-\{N_{NL}\}_{rr}, \tag{11}\] \[\{\varphi\}_{r+1}\;= \{[K_{T}\}_{r}][\Delta U]_{r+1},\] (12) \[\{\Delta U\}_{r+1}\;=\;\{U\}_{r}\;+\;\Delta U, \tag{13}\]  
where \(\{\varphi\}\) represents the unbalanced load vector, \(\{P\}\) represents the vector of applied external forces, \(\{N_{NL}\}\) represents the vector of nonlinear internal forces, \(\{K_{T}\}\) represents the tangent stiffness matrix of the material, and \(\Delta U\) represents the displacement increment vector needed to reach the equilibrium configuration.  
The ultimate buckling load \(P_{u}\) and ultimate buckling stress \(\sigma_{u}\) were determined when, as the load sub-steps increments were applied, the material deformations reached magnitudes so large that the internal loads could no longer balance or meet the equilibrium tolerance, indicating that the material had reached its maximum stress [13].  
For the case study, due to the higher number of simulations, the number of sub-steps was standardized according to the sub-steps convergence test realized before the simulations. To verify and validate the computational model of buckling, the number of sub-steps was set to 200, with a maximum of 400 and a minimum of 100. The solution convergence criteria were based on ANSYS's standards for force, moment, and displacements, i.e., a nonlinear solution is considered converged when the normalized residual of any key variable (force, moment, or displacement) falls within an acceptable threshold. This occurs when the sum of the absolute values of the unbalanced residuals, whether in terms of force, moment, or displacement, is at most 0.5% of the corresponding applied or incremental quantity. In this context, the unbalanced residual represents the portion of the force, moment, or displacement that has not yet been balanced in the iterative process, while the reference value corresponds to the total applied or incremental quantity used for comparison.