1. Introduction

In recent years, the cruise ship industry has experienced a continuous increase in vessel size and architectural complexity, driven by market demands for enhanced onboard amenities, passenger comfort, and operational efficiency. As a result, cruise ship superstructures have evolved into large-scale, multi-deck assemblies extending significantly in the longitudinal direction. To control structural weight while maintaining sufficient global and local strength, thin-walled plate structures stiffened by longitudinal and transverse members are extensively adopted in these superstructures. Although such structural configurations provide considerable advantages in terms of weight efficiency and constructability, their slender geometry inherently increases vulnerability to buckling under in-plane compressive, transverse, and shear loads. Buckling is widely recognized as a governing failure mode for thin-walled plated structures, particularly when subjected to combined loading conditions. In cruise ship superstructures, these loads arise from a complex interaction of global hull girder bending, local deckhouse loads, thermal effects, and operational conditions. Consequently, ensuring adequate buckling strength is a critical requirement in the structural design and safety assessment of thin-walled stiffened plates. An insufficient consideration of buckling behavior may lead not only to local structural damage but also to progressive failure mechanisms that compromise overall structural integrity. Traditionally, the buckling strength of stiffened plates has been evaluated using detailed finite element analysis (FEA), which allows accurate representation of geometric nonlinearity, material yielding, and initial imperfections. While such numerical approaches provide high fidelity results, they require substantial modeling effort, computational cost, and engineering time. These limitations make them less suitable for rapid design iterations or extensive parametric studies, particularly during early design stages when multiple structural configurations must be evaluated efficiently.

To address these challenges, classification societies have developed simplified yet robust assessment tools based on established buckling theories. Among them, the Panel Ultimate Limit State (PULS) method developed by DNV has gained wide acceptance for the evaluation of stiffened plate structures. The PULS method enables rapid estimation of ultimate and buckling strength by incorporating analytical formulations, empirically calibrated interaction equations, and rational consideration of initial imperfections. Owing to these characteristics, PULS provides a practical balance between computational efficiency and engineering reliability, making it particularly suitable for comparative studies and preliminary design assessments. Despite the widespread application of the PULS method, there remains a need for systematic investigation into the combined effects of key design parameters on buckling performance, especially for thin-walled plates used in cruise ship superstructures. In particular, the interaction between plate thickness and stiffener configuration including the presence of secondary stiffeners has not been sufficiently quantified from a parametric perspective. In practical design, increasing plate thickness or adding secondary stiffeners is often assumed to monotonically improve buckling strength; however, such assumptions may overlook adverse effects such as local stiffener buckling or unfavorable load redistribution.

The primary objective of this study is to conduct a comprehensive parametric buckling strength assessment of thin-walled stiffened plates representative of cruise ship superstructures using the DNV-PULS method. The study focuses on evaluating the influence of plate thickness variation and secondary stiffener installation under combined in-plane loading conditions, while explicitly accounting for initial geometric imperfections in accordance with classification society requirements. Through systematic comparison of multiple structural configurations, this research aims to identify dominant buckling modes, clarify non-linear trends in buckling resistance, and establish practical insights for optimal structural design.

The outcomes of this thesis are expected to contribute to improved understanding of buckling behavior in thin-walled stiffened plates and to provide designers with rational guidance for selecting appropriate combinations of plate thickness and stiffener arrangements. Ultimately, this work seeks to support safer and more efficient structural design practices for cruise ship superstructures by bridging the gap between detailed numerical analysis and practical engineering assessment methods.

Hendrik Naar(2006) investigates the ultimate longitudinal strength of hull girders in large passenger ships, where the combined behavior of the hull and multi-deck superstructure (with many openings) makes collapse mechanisms more complex than in conventional single-deck ships. It develops a nonlinear Coupled Beams (CB) method that models decks and hull components as coupled thin-walled beams connected by nonlinear vertical and shear members, enabling efficient prediction of the ship’s nonlinear response up to collapse. The method is validated and benchmarked against nonlinear 3D finite element analyses, including mesh sensitivity work, using a post-Panamax passenger ship case study under both hogging and sagging bending. Results show good agreement between CB and FE up to the post-peak region, and indicate that failure initiates by shear collapse in longitudinal bulkheads, highlighting the critical role of shear strength and shear stiffness of bulkheads/side structures in passenger-ship ultimate strength.

Chenfeng Li et al.(2019) conducts a comprehensive nonlinear finite element investigation into the ultimate compressive strength of welded stiffened plates fabricated from common shipbuilding steel grades(S235∼S390), explicitly accounting for initial geometric imperfections and welding-induced residual stresses. The results demonstrate that welding residual stress reduces ultimate strength by up to approximately 10%, while increasing steel yield strength effectively enhances ultimate capacity only for plates with low to moderate column slenderness ratios, where elasto-plastic buckling governs collapse behavior. For highly slender stiffened plates dominated by elastic buckling, the benefit of higher-strength steel is shown to be limited, highlighting the necessity of integrated design consideration of material grade, slenderness, and fabrication effects in ship structural design.

Alexander G. Jerne(2021) investigates how elastic buckling at subcritical load levels affects the global load-carrying mechanism of a modern passenger ship, with particular emphasis on stiffness reduction and load redistribution rather than ultimate collapse. Using a nonlinear finite element global model, the results demonstrate that elastic buckling of thin deck plates can occur well below the ultimate limit state and can significantly alter load paths and force distribution among decks, stiffeners, and girders effects that are not captured by conventional linear static analysis. The study highlights the necessity of nonlinear analysis in the early design and assessment of passenger-ship superstructures, as ignoring subcritical buckling-induced stiffness degradation may lead to inaccurate evaluation of structural behavior and safety margins.

Gan Jin et al.(2021) performs a nonlinear finite element analysis to evaluate the ultimate strength of typical perforated high web-frame (plate–frame) structures widely used in the superstructures of large cruise ships under longitudinal compressive loading. The results show that thin plate frames are highly sensitive to initial geometric imperfections, while the contribution of longitudinal girders to ultimate capacity is significant, and the ultimate strength is relatively insensitive to hole shape and hole ratio within a reasonable range; instead, the location of openings governs the position of the buckling and collapse bands. The study concludes that properly designed web openings can be adopted without substantial loss of load-carrying capacity, providing practical guidance for lightweight design and safety assessment of cruise ship superstructures.

Yu Yang et al.(2023) numerically investigates the ultimate bearing capacity of an unconventional stepped deck structure used in modern cruise ships by performing nonlinear quasi-static finite element analyses and comparing its collapse behavior with that of a conventional flat deck. The results reveal that the stepped deck exhibits a significantly reduced ultimate strength (about 20% of a conventional deck), with collapse initiating at the junction of steps with the largest height difference, and that the ultimate capacity is strongly influenced by the thickness of structural components— particularly the longitudinal girder web. Based on the identified weak regions, the study proposes pillar reinforcement and longitudinal girder strengthening schemes, demonstrating that targeted local reinforcement can substantially improve ultimate bearing capacity with only a minor increase in structural weight, providing practical guidance for cruise ship deck design and optimization.

Lei Ao et al.(2024) investigates the ultimate strength and collapse behavior of a large passenger ship’s funnel structure with openings under extreme wind pressure through combined scaled model experiments and nonlinear finite element analysis. The results demonstrate that structural failure is governed by local buckling and plastic collapse concentrated at the corner regions of the openings, where stress concentration at stiffener ends initiates progressive collapse, and the experimental observations show strong agreement with numerical predictions. Based on validated nonlinear similarity laws, the study quantifies the reduction in ultimate strength due to openings and provides practical design guidance for improving the structural safety of passenger ship funnel structures subjected to severe wind loads.

Across the six reference studies, the shared technical premise is that passenger-ship and cruise-ship superstructures are governed by instability-driven strength limits, but each paper addresses this premise at a different structural scale and with different dominant drivers: Naar (2006) and Jerne (2021) focus on global load-carrying mechanisms (hull-girder collapse and subcritical stiffness degradation, respectively), whereas Ao et al. (2024), Gan et al. (2021), and Yu et al. (2023) quantify how geometric discontinuities and atypical arrangements (openings in funnel structures, perforated web-frame plate–frame systems, and stepped decks) trigger localized buckling/plastic collapse and thereby reduce capacity, and Li et al. (2019) further shows that even for “conventional” stiffened plates the ultimate compressive strength is strongly conditioned by as-built effects (imperfections/residual stress) and slenderness-dependent collapse modes. In contrast to these works most of which rely on nonlinear global FE, component-level nonlinear FE + tests, or fabrication-sensitive FE parametrics to explain collapse your current study is positioned as a design-stage decision framework: it uses the DNV-PULS method to perform a systematic parametric screening of thin-walled stiffened plates representative of cruise-ship superstructures under combined in-plane loads, explicitly including initial imperfections, and it demonstrates that “thickness increase” or “secondary stiffener addition” does not guarantee monotonic improvement because the governing limit state can shift toward local stiffener buckling and stiffness-interaction effects. Therefore, the logical contribution of the current thesis relative to the six prior studies is not to replace high-fidelity collapse simulations, but to bridge them: it translates the instability mechanisms highlighted in global-collapse, opening-sensitive, configuration-sensitive, and fabrication-sensitive studies into a rapid, classification-aligned parametric methodology that produces actionable design insight (dominant modes, non-linear trends, and robust parameter combinations) suitable for early-stage optimization of cruise-ship superstructure plate panels.

2. Superstructure Design

2.1 Main components and specifications

This chapter outlines the structural configuration, analytical framework, and assessment methodology employed in this study.

It begins by describing the representative thin-walled stiffened plate model, including its principal dimensions, material properties, and stiffener arrangements, which collectively form the basis for the parametric investigation. The rationale for selecting specific geometric parameters and load conditions is explained in the context of typical cruise ship superstructure design requirements. Subsequently, the chapter introduces the core analytical tool—the DNV Panel Ultimate Limit State (PULS) method. A detailed explanation of its theoretical foundation is provided, focusing on the formulation of the usage factor as a scalar safety measure under combined in-plane stresses, and its integration of initial imperfections consistent with classification society standards. Finally, the numerical modeling strategy and the definition of the parametric design variations are presented, establishing a clear and systematic workflow that links design inputs, analytical procedures, and the evaluation of buckling performance.

Figure 1 illustrates a representative cruise-ship superstructure and its detailed structural components, highlighting the extensive use of thin-walled stiffened plate systems in multi-deck arrangements to achieve weight efficiency while maintaining global continuity. From a theoretical perspective, these components operate in a high-slenderness regime, where the load-carrying capacity is governed not by material yielding alone but by elastic and elastoplastic buckling phenomena arising under combined in-plane compression, transverse load, and shear induced by hull-girder bending and local deckhouse actions. The figure establishes the structural context of the study by demonstrating that superstructure panels form an integral part of the ship’s longitudinal strength system, making their buckling resistance a critical design concern rather than a purely local issue.

2.2 Buckling Strength Assessment for PULS

DNV-PULS(Panel Ultimate Limit State) is a specialized computational program developed by Det Norske Veritas (DNV) for the assessment of buckling and ultimate strength of stiffened plate and shell structures commonly used in ship and offshore structural applications. The program is widely adopted in classification society rule development, design verification, and direct strength assessment (DSA) due to its balance between computational efficiency and engineering accuracy.

The primary objective of DNV-PULS is to evaluate elastic buckling strength, post-buckling behavior, and ultimate load-carrying capacity of plate-stiffener assemblies subjected to various combinations of compressive, shear, and lateral pressure loads. Unlike full nonlinear finite element analysis (NLFEA), PULS employs advanced semi-analytical formulations calibrated against extensive numerical simulations and experimental databases, allowing rapid yet reliable structural assessment. The theoretical foundation of DNV-PULS is based on classical plate and shell buckling theory, extended through empirical and semi-empirical formulations to account for real structural behavior. Key theoretical elements include:

In addition to its theoretical foundation, the trustworthiness of the DNV-PULS program is strongly supported by its extensive validation history and long-term application in classification practice. The PULS formulations have been systematically calibrated against a large body of nonlinear finite element analyses and experimental test results covering a wide range of plate slenderness ratios, stiffener configurations, material grades, and imperfection patterns. This calibration ensures that the predicted buckling and ultimate strength values are neither purely analytical nor empirical, but represent a rational synthesis of theory and verified structural behavior. From a regulatory perspective, DNV-PULS constitutes a core assessment engine embedded within DNV’s rule development framework for ship and offshore structures. The method has been continuously refined through its use in rule checks, direct strength assessments, and approval-in-principle (AiP) evaluations for commercial vessels and offshore units. As a result, the safety margins produced by PULS are implicitly aligned with the reliability targets adopted by classification societies, making its results directly interpretable within a limit-state design philosophy.

The influence of DNV-PULS extends beyond a single classification society, as its assessment concepts—such as imperfection-sensitive buckling reduction factors, interaction equations for combined in-plane loads, and unified usage-factor criteria—are consistent with, and have informed, similar formulations adopted by other major classification societies and international guidelines. Consequently, PULS is widely regarded not merely as a software tool, but as a standardized engineering methodology for preliminary and comparative buckling strength evaluation. Owing to its balance between computational efficiency and structural fidelity, DNV-PULS has become particularly influential in early-stage design, parametric studies, and optimization workflows, where large numbers of structural variants must be assessed within practical time constraints. While it does not replace high-fidelity nonlinear finite element analysis for final verification, its demonstrated consistency with detailed numerical and experimental results establishes it as a reliable screening and decision-support tool for identifying dominant buckling modes, non-linear strength trends, and rational design directions.

3. FE-analysis and results

This chapter presents a systematic analysis of the buckling strength and safety margins for the parametric stiffened plate models under combined in-plane loading. The results, computed using the DNV-PULS method, are evaluated primarily through the usage factor (η), which quantifies the proximity to the ultimate limit state. The discussion focuses on elucidating the influence of two key design variables plate thickness and the presence of secondary stiffeners on the buckling performance. By comparing multiple configurations and their corresponding failure modes, this section identifies critical trends, including the non-linear relationship between plate thickness and buckling margin, the conditional effectiveness of secondary stiffeners, and the transition of governing failure mechanisms from global panel buckling to local stiffener instability. The findings provide actionable insights into the integrated optimization of plate and stiffener parameters for enhanced buckling resistance in thin-walled cruise ship superstructures.

3.1 FE-modeling and constraint condition

Figure 2 presents the finite-element (FE) idealization adopted to evaluate the buckling behavior of these stiffened plates, where boundary conditions and load applications are defined to reproduce the theoretically idealized panel behavior assumed in classification-based buckling formulations.

From an engineering standpoint, the FE model bridges classical plate-and-stiffener buckling theory and practical ship design by ensuring compatibility with the assumptions embedded in the DNV-PULS method, including representative support conditions, effective panel spans, and imperfection sensitivity. Consequently, Fig. 2 provides the analytical foundation that enables a rational comparison between theoretical buckling capacity, numerical response, and classification-oriented safety margins, thereby justifying the parametric assessment framework employed in this study.

The boundary conditions were implemented to model a representative interior segment of a continuous stiffened panel field, employing simply-supported constraints on all four edges. This approach effectively represents the restraint provided by surrounding structural members, such as transverse frames and adjacent panels. The in-plane loading applied to the model consisted of combined longitudinal compressive stress (50MPa), transverse compressive stress (15MPa), and in-plane shear stress (5 MPa). These specific stress magnitudes were selected to reflect a realistic and critical load case derived from the operational stress levels typical in the superstructure of a reference passenger ship, thereby ensuring the buckling assessment is grounded in practical engineering conditions.

Tables 1 and 2 collectively define the foundational parametric framework for the buckling strength assessment. Table 1 specifies the geometric dimensions of the representative thin-walled stiffened plate panel, establishing a baseline configuration typical of cruise ship superstructures. The panel measures 2400 mm in length with a full breadth of 1200 mm, analyzed in a half-breadth model of 600mm to leverage symmetry. The parametric variation is introduced through two plate thicknesses (6.0mm and 6.5mm) and the presence or absence of secondary stiffeners. The primary longitudinal stiffeners are standardized with angle-bar profile (web height 100mm, flange width 75mm, thickness 7mm), while the secondary stiffeners are configured as flat bars (75mm height, 6 mm thickness).

Table 2 complements this by detailing the material properties, assuming a common shipbuilding steel with an elastic modulus of 210,000MPa, a yield strength of 235MPa, and a Poisson's ratio of 0.3. Together, these tables provide the essential and consistent input parameters encompassing geometry, configuration, and material behavior necessary for the subsequent application of the DNV-PULS method and the interpretation of resulting buckling strengths and safety margins.

3.2 Methodology and imperfection

DNV-PULS(DNV, 2006) formalizes the safety assessment procedure by defining the usage factor (η) within the context of multi-axial load interaction at the ultimate limit state (ULS). The factor is calculated as the ratio of two radius vectors in a three-dimensional load space defined by the principal in-plane stress components: longitudinal compressive stress (σ_lx ), transverse stress (σ_ly), and shear stress (σ_q ). The applied load vector, L_q , is the Euclidean norm of the design load components as noted equation 3.1:

This scalar value represents the magnitude of the combined stress state acting on the panel. The corresponding ultimate strength vector, L_u, is defined similarly using the panel's ultimate capacities under each independent load component as indicated equation 3.2:

Each ultimate strength component (σ_lx,u , σ_ly,u, σ_q,u) is derived from the DNV-PULS formulation(DNV, 2006), which accounts for elasto-plastic buckling interaction, initial imperfections, and stiffener effectiveness.

The usage factor is then equation 3.3:

This formulation is grounded in the principle of interaction for combined stress states at collapse. It extends the classical von- Mises yield criterion to the realm of stability failure. The underlying theory posits that failure (buckling or yielding) under combined loading occurs when a scalar function of the applied stresses reaches a critical value defined by the material and geometric properties. By normalizing the load vector by the ultimate strength vector, the method creates a dimensionless measure of proximity to the ULS(ultimate limit states) envelope. A value of η(usage factor)=1.0 signifies that the combined stress state lies precisely on the interaction failure surface. The formulation inherently satisfies the requirement that the effects of axial compression, transverse compression, and shear are not additive linearly but interact synergistically to reduce the structural reserve, a phenomenon well-established in plate buckling theory. This approach provides a rational, mechanics-based safety metric that is consistent with limit state design philosophy and allows for direct comparison across diverse loading scenarios and structural configurations.

Figure 3 schematically outlines the systematic workflow employed for the parametric buckling assessment of thin-walled stiffened plates, integrating geometric modeling, classification society methodology, and performance evaluation. The procedure begins with the definition of a parameterized baseline finite element model, which encapsulates the key geometric variables, plate thickness, primary stiffener spacing, and the presence or absence of secondary stiffeners. This model is subjected to a combined in-plane load case (longitudinal compression, transverse compression, and shear) representative of critical operational conditions in a cruise ship superstructure. The core of the workflow is the application of the DNV Panel Ultimate Limit State (PULS) method. This semi-analytical, rule-based tool calculates the usage factor (η) for each design variant. The calculation explicitly incorporates initial geometric imperfections (both plate and stiffener distortions) as per classification society standards, ensuring the assessment reflects fabrication-induced sensitivities.

The usage factor, derived from a multi-axial load interaction formulation, serves as a scalar safety metric quantifying the proximity to the ultimate limit state. The final stage involves the systematic comparison and interpretation of results. The calculated usage factors and derived buckling margins are analyzed across the parameter space to identify dominant trends, non-linear interactions, and shifts in the governing failure mode. This structured workflow bridges detailed geometric parameterization with a standardized, code-compliant assessment, providing a rapid and reliable methodology for comparative buckling strength evaluation during preliminary design.

Figure 4 illustrates the assumed initial geometric imperfections applied to the stiffened plate models, including both local plate out-of-plane deformation and global stiffener-induced distortion(IACS, 2012), which are introduced to reflect realistic fabrication tolerances and welding-induced imperfections.

From an engineering and theoretical standpoint, the inclusion of these imperfection shapes is essential because buckling and ultimate strength of thin-walled structures are highly imperfection-sensitive, and their presence ensures that the numerical and PULS-based evaluations(DNV, 2006) represent conservative and physically meaningful collapse behavior consistent with classification-society safety assumptions.

Figure 5 illustrates the three distinct stiffened plate configurations used for the parametric buckling analysis, designated as Model (a), (b), and (c) to isolate the influence of stiffener arrangement. Model (a) represents the base configuration, consisting of a thin-walled plate reinforced solely by primary longitudinal stiffeners. This model serves as the reference case to evaluate the fundamental buckling strength of a conventionally stiffened panel, where instability is primarily governed by the interaction between plate slenderness and the bending rigidity of the primary stiffeners under combined in-plane loads. Model (b) introduces a secondary stiffener, positioned between the primary longitudinal members. The purpose of this configuration is to investigate the effect of reducing the unsupported plate panel size. The secondary stiffener aims to suppress global plate buckling by subdividing the wider plate field, thereby potentially shifting the governing failure mode and enhancing the ultimate capacity through a reduction in the effective buckling length of the plate. Model (c) depicts a configuration where the secondary stiffener is removed, but the plate thickness is increased relative to Model (a).This model is designed for a direct comparative assessment, isolating the effect of material addition (via plate thickening) against the effect of geometric reinforcement (via secondary stiffening in Model (b)). It allows for the evaluation of whether simply increasing plate mass is more or less effective than strategic structural detailing in improving buckling resistance.

3.3 Strength analysis results

Table 3 summarizes the usage factors (UF) obtained from the DNV-PULS assessment for six stiffened plate configurations subjected to combined in-plane loading. The usage factor, defined as the ratio between the applied load vector and the ultimate strength vector, represents the proximity of each configuration to the ultimate limit state (ULS). A value approaching unity indicates that the structural capacity is nearly exhausted, whereas lower values imply a larger safety margin. For Models 1 to 4, which are equipped with primary longitudinal stiffeners, the calculated usage factors range from 0.48 to 0.74, indicating a sufficient but non-negligible safety margin against buckling failure. Within this group, a clear and consistent trend is observed with respect to both plate thickness and the presence of secondary stiffeners. A direct comparison between Model 1 (6.0 mm plate without secondary stiffeners, UF = 0.74) and Model 2 (6.0 mm plate with secondary stiffeners, UF = 0.53) demonstrates that the introduction of a secondary stiffener leads to a substantial reduction in the usage factor, corresponding to an improvement in buckling resistance of approximately 28%. This result confirms that, for relatively slender plates, the subdivision of the plate field by a secondary stiffener effectively reduces the buckling length and enhances the overall stability of the panel. A similar but less pronounced effect is observed for the thicker plate configurations. Model 3 (6.5 mm plate without secondary stiffeners) exhibits a usage factor of 0.65, while Model 4 (6.5 mm plate with secondary stiffeners) shows a reduced value of 0.48. Although the secondary stiffener still provides a measurable improvement, the relative gain is smaller than that observed for the thinner plate. This indicates that, as plate thickness increases, the marginal benefit of additional stiffening diminishes due to a shift in the governing instability mechanism. In contrast, Models 5 and 6, which do not include any longitudinal stiffeners, exhibit significantly higher usage factors of 0.94 and 0.80 for plate thicknesses of 6.0 mm and 6.5 mm, respectively. These values indicate that unstiffened plates operate very close to the ultimate limit state under the same loading conditions. The results clearly demonstrate that primary longitudinal stiffeners play a dominant role in resisting combined in-plane loads, and that plate thickening alone is insufficient to ensure adequate buckling safety when stiffeners are absent. The reduction in usage factor from 0.94 (Model 5) to 0.80 (Model 6) with an increase in plate thickness confirms that plate slenderness remains a critical parameter for unstiffened configurations. However, even with increased thickness, the safety margin remains inferior to that of stiffened panels, underscoring the structural inefficiency of relying solely on material addition without appropriate stiffening.

Figure 6 plots the relationship between buckling factor and plate thickness for the analyzed stiffened plate configurations. The results demonstrate a clear divergence in structural performance based on the presence of secondary stiffeners. For models without secondary stiffeners, buckling factor shows a steady, near-linear increase with plate thickness. This trend aligns with classical plate theory, where thicker plates directly enhance flexural rigidity and critical buckling stress. In contrast, for models with secondary stiffeners, the strength curve exhibits a distinct non-linear behavior: rapid initial improvement at lower thicknesses, followed by a significant reduction in the rate of gain, leading to a plateau. This saturation occurs because, beyond a critical thickness, the governing failure mode shifts from global plate buckling to local stiffener instability (e.g., stiffener tripping). At this point, the system's ultimate capacity becomes controlled by the stiffener's cross-sectional properties, which are not improved by further plate thickening. Therefore, Figure 6 provides a key optimization insight: while secondary stiffeners are highly effective for slender plates, their benefit diminishes once the plate is sufficiently thick, guiding designers toward a balanced system-level approach rather than unilateral plate thickening.

Figure 7 illustrates the non-linear relationship between the buckling margin (defined as the inverse of the usage factor), and plate thickness for thin-walled stiffened plates under combined in-plane loading. The plotted curves distinctly separate the structural response based on the presence of secondary stiffeners, providing critical insight into the efficiency of design strategies aimed at enhancing buckling resistance. Theoretically, the buckling margin represents a normalized measure of the safety reserve against the ultimate limit state. For configurations without secondary stiffeners, the margin increases in a near-linear manner with plate thickness. This trend aligns with classical plate buckling theory, where the critical buckling stress for a simply supported plate is proportional to the square of the thickness-to-width ratio. Therefore, increasing plate thickness directly enhances the plate's flexural rigidity (D), thereby elevating the global elastic buckling load of the panel. In contrast, for panels equipped with secondary stiffeners, the margin exhibits a markedly different behavior: a steep initial improvement at lower thicknesses, followed by a distinct plateau as thickness increases.

This saturation effect signifies a fundamental shift in the governing failure mechanism. Initially, the secondary stiffener effectively subdivides the plate field, reducing the unsupported panel width and shifting the buckling mode from a global panel instability to a more localized one. However, beyond a critical plate thickness (approximately 8.5 mm in this study), the system's ultimate capacity becomes governed not by plate buckling, but by the local instability of the stiffeners themselves—specifically, stiffener tripping (torsional-flexural buckling) or web buckling.

This transition is consistent with the principles of interactive buckling. As the attached plate thickens, its stability increases, transferring a higher proportion of the load to the stiffeners. The stiffener's torsional rigidity, however, does not scale with plate thickness. Consequently, the limiting failure mode shifts from plate-dominated buckling to stiffener-dominated instability. The plateau in Figure 7 quantitatively captures this phenomenon, indicating that further material addition via plate thickening yields diminishing returns on the overall system strength once stiffener local buckling becomes critical.

Figure 8 provides a direct visual comparison of the buckling mode shapes for stiffened plate panels with two distinct plate thicknesses: 9 mm and 9.5 mm. This visualization serves as crucial physical evidence for the failure mode transition discussed in the parametric trend analysis. For the panel with a 9 mm plate thickness, the buckling deformation exhibits a coupled, global pattern. Both the plate field and the secondary stiffener undergo coordinated out-of-plane displacement, indicating an interactive buckling mode where the stiffener participates in the overall panel instability while simultaneously providing restraint to the plate. This mode is characteristic of a system where the plate slenderness is still a dominant factor. In contrast, the buckling mode for the 9.5 mm panel shows a pronounced localization of deformation. The failure concentrates predominantly in the web region or at the flange-web junction of the secondary stiffener, with the adjacent plate remaining relatively stable. This shift to a localized stiffener tripping or web buckling mode signifies that increasing the plate thickness has sufficiently stabilized the plate field, thereby transferring the governing limit state to the cross-sectional stability of the stiffener itself. The comparison in Figure 8 thus anchors the analytical findings in a clear mechanistic reality. It visually validates the parametric trend where, beyond a critical thickness, the buckling margin saturates (as seen in Figures 7 and 9). This transition from a global, plate-stiffener interactive mode to a local, stiffener-dominated mode explains why simply adding plate material becomes ineffective. The figure underscores that optimizing design requires anticipating and controlling this failure hierarchy, as material addition beyond a certain point merely shifts the structural weak link without significantly enhancing the system's ultimate capacity.

Figure 9 presents the relationship between buckling margin and plate thickness across an extended parametric range, elucidating the complex, non-linear interaction that dictates the safety performance of thin-walled stiffened plates. The plotted curves distinctly delineate the behavioral divergence between configurations with and without secondary stiffeners, revealing a critical design inflection point that challenges conventional, monotonic assumptions about structural reinforcement. Theoretically, the initial positive correlation observed across all configurations where buckling margin increases with plate thickness is governed by the enhanced flexural rigidity of the plate element. According to classical orthotropic plate theory, the buckling resistance of a stiffened panel is a function of both plate slenderness and stiffener slenderness. At lower plate thicknesses, the system's ultimate capacity is primarily controlled by meaning that increasing plate thickness directly mitigates global plate buckling, resulting in a near-linear improvement in the margin. However, the curves diverge significantly beyond a plate thickness of approximately 8.5 mm. For panels without secondary stiffeners, the buckling margin continues on a steady, linear upward trajectory. This indicates that the failure mechanism remains dominated by global plate buckling, and the benefit of added material through plate thickening is consistently realized. In stark contrast, for panels equipped with secondary stiffeners, the curve exhibits a pronounced reduction in slope, asymptotically tending toward saturation. This plateau represents a fundamental shift in the structural system's sensitivity and governing limit state. As the plate thickness increases, the attached plate becomes sufficiently rigid, thereby stabilizing the plate field. Consequently, the system's ultimate capacity becomes increasingly governed by the local stability of the stiffeners themselves specifically, their torsional (tripping) buckling strength or web buckling strength. These stiffener-dominated failure modes are inherently less sensitive to further increases in the attached plate thickness. The saturation point, therefore, marks the transition where the "weakest link" in the structural system shifts from the plate to the stiffener.

Thus, Figure 9 provides quantitative and graphical evidence for a key engineering insight: the strategic benefit of adding secondary stiffeners to enhance buckling margin is most potent within the low-to-mid range of plate slenderness. Beyond a critical threshold, the return on investment diminishes due to a failure mode transition. This graph effectively guides designers away from unilateral plate thickening and toward a balanced, system-level optimization where plate dimensions and stiffener properties are harmonized to achieve weight-efficient buckling resistance without triggering premature local stiffener instability.

Figure 10 presents a critical comparative visualization of the buckling mode shapes for stiffened panels with 9 mm and 9.5 mm plate thicknesses, corresponding to the parametric region where the buckling margin exhibits saturation, as identified in Figure 9. This visual evidence serves as the definitive mechanistic validation for the non-linear strength trends and the governing limit state transition. The buckling deformation pattern for the 9 mm panel demonstrates a coupled interactive buckling mode. In this state, the plate field and the secondary stiffener undergo synchronized, global out-of-plane displacement. This indicates that the structural system is failing as an integrated unit, where the stiffener provides restraint but also actively participates in the overall instability. The mode is governed by the interactive slenderness of the plate-stiffener combination. In stark contrast, the buckling mode for the 9.5 mm panel exhibits a highly localized failure mechanism. Deformation is concentrated almost exclusively in the web of the secondary stiffener or at its junction with the flange, characteristic of stiffener tripping (torsional buckling) or localized web crippling. The attached plate remains notably stable, indicating that the increased plate thickness has effectively eliminated plate buckling as the primary failure path. Consequently, the system's "weakest link" has shifted decisively from the plate to the cross-sectional stability of the stiffener itself.

Therefore, Figure 10 does more than display two deformation shapes; it provides physical proof of the failure mode hierarchy shift. This visual confirmation directly explains the saturation plateau seen in the buckling margin curves (Figures 7 & 9): once the plate is stabilized, further thickening does not prevent failure because the critical load is now determined by the stiffener's torsional rigidity, which is independent of plate thickness. This underscores a fundamental design principle: optimizing for buckling resistance requires a holistic system analysis to ensure that strengthening one component (the plate) does not inadvertently trigger a premature and inefficient failure in another (the stiffener).

4. Conclusions and future works

This study employed the DNV Panel Ultimate Limit State (PULS) method to investigate the buckling behavior of thin-walled stiffened plates representative of cruise ship superstructures under combined in-plane loading. The results confirm that the PULS method offers a practical and efficient tool for preliminary design by capturing the non-linear interaction between plate slenderness and stiffener rigidity while maintaining consistency with classification-society design philosophy.

The parametric analyses show that secondary stiffeners can significantly enhance buckling resistance by reducing the effective buckling length of slender plates. However, the benefit of plate thickening is not monotonic. Beyond a critical thickness, the governing failure mechanism shifts from global panel buckling to local stiffener instability, such as stiffener tripping or web buckling, resulting in a saturation of the buckling margin. This transition indicates that the ultimate capacity of the structural system becomes increasingly controlled by stiffener stability rather than by plate strength alone.

These findings highlight the necessity of an integrated design strategy in which plate thickness and stiffener configuration are optimized simultaneously. Simply increasing plate thickness without corresponding consideration of stiffener behavior may lead to inefficient designs and premature local failure. The observed change in buckling modes provides physical justification for the non-linear strength trends and reinforces the importance of harmonizing plate and stiffener parameters to achieve rational and weight-efficient buckling performance in cruise ship superstructure design.

While this study provides valuable insights for the preliminary design phase, further investigations are recommended to expand its applicability and depth: