## 1 Introduction  
The ultimate strength of plates and stiffened panels with cracks under compressive loads has been extensively studied by researchers due to its critical significance in various engineering applications, particularly in civil, mechanical, and marine/ocean engineering. Over the years, numerous experimental [1, 2, 3, 4], analytical [5, 6], and numerical [2, 7, 8] investigations have been conducted to understand the complex behavior of these structural components and enhance their performance. The effect of cracks in plates and stiffened panels on the ultimate strength behavior has also been widely explored in the fields of ships and offshore engineering. Cracks progressively weaken structural integrity, with their severity increasing over time. Ships and ship-shaped offshore structures are continuously exposed to vertical bending moments during their operation. In particular, cracks may propagate over time due to the cyclic tensile and compressive loads induced by vertical bending moments, such as hogging and sagging during operation [1, 7, 9]. Various studies have been conducted to obtain the ultimate compressive strength or residual ultimate compressive strength of plates [9; 10; 11; 12; 13] and stiffened panels [4; 8; 10; 14; 15; 16; 17]. Recently, the ultimate limit state of a cracked ship hull girder was examined by [PERSON] and [PERSON] [18]. The residual ultimate strength characteristics of plates and stiffened panels with cracks may be influenced by the following key parameters: (1) material properties, (2) crack geometry and location, (3) loading conditions, (4) stiffening techniques, and (5) analysis methods. Material properties, such as yield strength, elastic modulus, and ductility, significantly affect the ultimate strength of plates and stiffened panels. Researchers have explored different materials, including steel, aluminum, and advanced composites, to assess the influence of material properties on panels' performance under compressive loads. [PERSON] et al. [19] recently conducted axial compression tests on cracked aluminum stiffened panels and found that material selection significantly influences a panel's ultimate strength, resistance to crack propagation, and overall structural stability. The geometry and location of cracks in plates and stiffened panels are critical factors influencing their ultimate strength [10]. The crack length, orientation, and position can significantly impact a panel's load-carrying capacity and failure mechanisms. For instance, cracks near a panel's edges or stiffeners often lead to high-stress concentrations, reducing ultimate strength. The effect of cracks is highly dependent on loading conditions. Ships experience cyclic loading due to waves, repeatedly undergoing hogging and sagging moments. While tensile loads promote crack propagation, compressive loads, on the other hand, can suppress crack growth by closing the crack surfaces. Researchers have investigated the effects of various loads on panels' performance, including compression [10; 16] and shear [1; 7; 20; 21]. Recently, [PERSON] et al. [22; 23; 24] numerically predicted the residual ultimate strength of plates, stiffened panels, and box girders under cyclic loading conditions. [PERSON] et al. [25] conducted a sensitivity analysis on a cracked plate under axial compression via elastic buckling analysis. While crack closure can mitigate local crack-tip stresses under cyclic or bending-dominated conditions, its role under monotonic compressive loading remains limited. Several studies [26; 27; 28] have shown that in thin-walled structures under dominant compression, crack-face contact may occur but contributes little to global load transfer due to slippage or partial interaction. Moreover, stress redistribution around closed cracks can still trigger local buckling or failure, meaning that closure does not guarantee structural integrity.