## 1 Introduction  
In recent decades, the size of vessels has significantly increased to accommodate more containers, thereby reducing operational costs and enhancing the sustainability of the shipping process [1, 2]. This growth in vessel dimensions has profound implications for the ship's structural components, particularly the hull's bottom panels, which are primarily composed of thin plates. These thin plates are subjected to various stress types, includingtension, compression, bending, and torsion, due to their weight, the weight of the cargo, lateral water pressure, navigation forces, and sea conditions [3]. Due to their slenderness, thin plates inherently have low resistance to compressive loads. At the bottom of the hull, these plates are subjected to biaxial compressive stress and lateral water pressure [4]. This combined loading can lead to a phenomenon known as buckling, which initially occurs in the elastic regime, where the panel experiences instability and reversible deformation under a critical load. The buckling can transition to the elastoplastic regime, causing stress redistribution and developing a transverse tensile membrane, allowing the structures to support stress beyond critical strength without failure [5]. To mitigate this issue of buckling in thin plates, adding longitudinal and transverse stiffeners can significantly enhance their mechanical resistance. These stiffeners can support the weight of the cargo, in-plane loadings, and lateral water pressure [6]. In plates under longitudinal loading, the longitudinal stiffeners are responsible for carrying a portion of the longitudinal force applied to the plate, while the transverse stiffeners are used to subdivide the plate into smaller units [7]. The behavior is the opposite for plates under transverse loading, with the transverse stiffeners supporting the plate, and the perpendicular stiffeners subdividing the plate. For plates under biaxial loading, both longitudinal and transverse stiffeners perform both functions due to the application of longitudinal and transverse loadings. However, randomly adding stiffeners to non-stiffened plates does not efficiently improve resistance to buckling. Therefore, it is essential to understand how the geometric configuration of stiffened plates influences displacement and stress distribution, thus affecting their buckling resistance under different loading conditions. In this context, the application of the constructal design method, developed by [PERSON] in 1996, emerges as an innovative solution. [PERSON], when studying heat flow in cooling problems for electronic components, realized that the ideal solution resembled the structure of trees. This insight led him to formulate the constructal law, which states that for a finite system (natural or not) to persist, its configuration must evolve to facilitate flow [7,8]. This approach can also be applied to the buckling problem in plates, where evolving the system and facilitating flow means redistributing the displacement and stress to increase buckling resistance [9]. Recent studies have successfully applied the constructal design method to plates under different types of loading by reducing thickness and converting the corresponding volume into stiffeners, without adding materials, thus keeping the total material volume constant. In addition, several researchers have investigated the influence of stiffened plate geometry in different loading scenarios, aiming to understand the mechanical behavior and improve its resistance. [PERSON] et al. [10] developed a novel methodology based on numerical buckling analyses to formulate empirical equations and design curves. This approach allows for predicting the maximum lateral pressure that stiffened panels under combined loading can withstand, considering various load scenarios, stiffener characteristics, and initial imperfections.