## 2 Materials and Methods
### Constructal Design Method and Exhaustive Search Technique  
As in [PERSON] et al. [13], the application of constructal design starts from an unstiffened plate that is considered as a reference, and its ultimate buckling strength \(\sigma_{uk}\) is adopted as a comparison parameter. The \(a\) and \(b\) were maintained constantly, and \(t\) was reduced to \(t_{p}\) to allocate volume for the stiffeners according to a defined volume fraction \(\phi\), as follows:  
\[\phi=\frac{V_{s}}{V_{p}}=\frac{N_{ls}(ah_{s}t_{s})+N_{ls}(b\ -N_{ls}t_{s})h_{s}t_{s})}{ abt}, \tag{14}\]  
where \(\phi\) represents the fraction of the total volume of the reference plate that is transformed into stiffeners, \(V_{s}\) is the total volume of stiffeners, and \(V_{p}\) is the total volume of the reference plate. The number of stiffeners \(N_{ls}\) and \(N_{ls}\) and their respective \(h_{s}/t_{s}\) are degrees of freedom (DOF) necessary for the application of constructal design, generating new geometric configurations of stiffened plates. The \(\sigma_{u}\) results of these proposed geometries were obtained and evaluated numerically. It is important to emphasize that the \(h_{s}\) and \(t_{s}\) are variables, but in each new configuration generated, they had the same values for both longitudinal and transverse stiffeners. How \(N_{ls}\) and \(N_{ls}\) are degrees of freedom, to respect the symmetrical distribution of the stiffeners on the plate, the spacings between stiffeners were defined as:  
\[S_{ls}=\frac{b}{N_{ls}+1}, \tag{15}\]  
\[S_{ls}=\frac{a}{N_{ls}+1}, \tag{16}\]  
where \(S_{ls}\) is the spacing between longitudinal stiffeners and \(S_{ls}\) is the spacing between transverse stiffeners. To understand how the \(P_{z}\) and \(h_{s}/t_{s}\) impacts on \(\sigma_{u}\), three conditions of lateral pressure were considered: an application of lateral pressure in a positive direction of the \(z\)-axis, in a  
Figure 3: Boundary conditions of a simply supported stiffened plate, with displacement restrictions represented in blue on the stiffened plate boards, according to the respective axis directions. negative direction of the \(z\)-axis, and null pressure. However, the optimized stiffened plate configurations were investigated only for the application of positive lateral pressure and biaxial compressive loading, being the unstiffened plate under this condition defined as the reference plate. This was adopted due to the desirable positive displacement over \(z\)-axis, similarly to sagging behavior (Figure 3). Because of this, its respective ultimate buckling stress reference \(\sigma_{uR}\) was considered for comparisons with the other plate geometries. To compare these results, the \(\sigma_{u}\) resulted from geometric configurations proposed by constructal design were dimensionless as normalized ultimate buckling strength \(\sigma_{uN}\). This serves as the performance indicator, and is calculated as:  
\[\sigma_{uN}\ =\ \frac{\sigma_{u}}{\sigma_{uR}}, \tag{17}\]  
where \(\sigma_{uN}\) is the normalized ultimate buckling stress, \(\sigma_{u}\) is the ultimate buckling stress obtained from the varying DOF, and \(\sigma_{uR}\) is the ultimate buckling stress of the reference plate. In turn, the constructal design method is applied according to the following steps, shown in Figure 4.  
The fraction \(\phi\) was set as 0.3, identified by [PERSON] et al. [13] as the best ratio between \(V_{s}\) and \(V_{p}\) to improve \(\sigma_{u}\) in elastoplastic buckling due to uniaxial compression, for a plate with the same dimensions considered in this study, with \(N_{ls}\) and \(N_{ls}\) values ranging from 2 to 5, resulting in sixteen plate configurations. While the \(t_{s}\) was defined as a minimum of 5 mm, set according to commercial dimensions, and varied incrementally up to 45 mm, and the \(h_{s}\) values range depending on the available volume, respecting the volume conservation.