1. Introduction

Structural safety evaluation of large ship blocks during construction and outfitting stages increasingly relies on numerical analysis to ensure reliability under various loading conditions. Among these methods, finite element (FE) analysis has become the most accurate and comprehensive tool, as it enables detailed representation of local structural features, welding conditions, material properties, and complex load transfer mechanisms. Numerous studies have demonstrated that FE-based structural assessment provides highly resolved stress fields and deformation patterns that cannot be captured through simplified analytical methods. However, despite its accuracy, the practical application of FE analysis in shipyards remains constrained by the considerable modeling effort, computation time, and engineering resources required. Preparing an FE model that incorporates realistic boundary conditions, mesh strategies, and fabrication-related imperfections often demands significant lead time, making it unsuitable for situations that require rapid decision-making during block production or assembly.

For this reason, shipyards and classification societies frequently rely on simplified longitudinal strength assessment based on classical beam theory. This approach, which evaluates bending moments, section modulus, and permissible stresses, enables quick estimation of structural adequacy without the need for detailed numerical modeling. Because of its speed and practicality, beam-theory-based evaluation has long served as the default method in both design and production environments, especially when assessing temporary states such as block lifting, transportation, and tandem floating conditions. However, as hull structures become larger, more complex, and more influenced by local fabrication effects including partial welding, gaps, and strong-back constraints, the validity of such simplified methods must be carefully verified. In particular, it is essential to understand whether section-modulus based longitudinal strength calculations remain representative when compared to FE-derived stress evaluations for actual block configurations. If the simplified method significantly underestimates or overestimates structural responses, it may lead to either unnecessary reinforcement or unsafe design margins. Therefore, establishing a reliable relationship between FE analysis results and beam-theory calculations is crucial for ensuring both structural safety and production efficiency. Motivated by this need, the present study conducts a systematic convergence analysis of FE mesh size and compares the resulting stress predictions with traditional longitudinal strength evaluation. Through this comparison, the study aims to determine whether simplified beam-theory-based assessment can be continuously used in shipyard practice and to propose a standardized FE modeling guideline that supports accurate and timely decision-making.

Rörup et al.(2017) investigate the global strength and torsional response of ships with open-deck configurations, where conventional beam-theory assumptions may not be sufficient. Builds refined FE models of open-deck ship structures and subjects them to combined bending and torsional loads. The results are compared with rule-based beam-girder calculations and simplified analytical models to identify discrepancies in stress distribution and stiffness. The study demonstrates that open-deck ships exhibit significant torsion and warping effects that are not fully captured by standard beam-theory longitudinal strength checks. It recommends using 3D FE analysis or enhanced beam models for such ship types and provides guidance on how much the superstructure and deck openings influence hull-girder stiffness and strength.

Xie et al.(2018) quantify how FE mesh size and meshing accuracy affect the predicted natural frequencies and mode shapes of a welding machine used for offshore platform fabrication. Several FE models with different element sizes and mesh refinements are built for the same structure. Modal analyses are performed and the results (frequencies, mode shapes, stress distribution) are compared to identify the influence of mesh density. The study shows that overly coarse meshes can yield inaccurate modal characteristics, while excessively fine meshes increase computational cost without proportional accuracy gains. It recommends an optimal mesh density range that balances accuracy and efficiency, and emphasizes that mesh design is a critical pre-processing step in FE simulations.

Shi et al.(2019) examine how the Common Structural Rules for Bulk Carriers and Oil Tankers (CSR-H) treat hull-girder ultimate and residual strength, and to clarify the technical background of these rule requirements. Compares ultimate-strength provisions in CSR-H with those in earlier CSR-OT/CSR-BC rules, focusing on the Smith method (progressive collapse beam-type method) used for rule calculations. Nonlinear FE analyses are also conducted for several typical bulk carriers and oil tankers to study the influence of initial imperfections, lateral pressure, and double-bottom deformation on ultimate strength. The paper shows how safety factors and double-bottom reduction factors in CSR-H are calibrated, and quantifies how initial deformation and lateral pressure reduce ultimate strength. It concludes that the Smith method, when used with CSR-H partial safety factors, provides a fast and reasonably conservative estimate, but that nonlinear FE analysis is necessary to understand detailed sensitivity and residual strength after damage.

Abedin et al.(2024) investigate how a multipurpose cargo ship responds structurally to combined vertical bending and torsional loads, especially for open-deck ships with wide hatches and low torsional rigidity. A full-ship 3D finite element (FE) model is built; global bending and torsional moments are calculated according to classification rules. The FE model is validated against beam theory and thin-walled girder theory to check normal and torsional stresses. Still-water and vertical wave bending moments contribute ~70% of total hull girder stress at midship, horizontal bending ~10%, and warping/torsional effects ~20% for open-deck ships, while torsion has little influence for closed-deck ships. Buckling checks show that the ship satisfies linear buckling criteria. The paper highlights when torsion must be included beyond conventional beam-type longitudinal strength checks.

Oludi and Nwoka(2024) reduce the computational cost of hull-girder FE analyses while keeping adequate accuracy for longitudinal strength assessment. Proposes a “scale finite element method”, where a reduced-scale FE model is used instead of a full-resolution model. Global bending responses and stress distributions from the scaled FE model are compared with conventional full FE and rule-based longitudinal strength calculations. The study shows that a properly calibrated scale-FE approach can reproduce hull-girder bending responses with acceptable accuracy, while significantly cutting modeling and computation time. It argues that scale FE can be a practical bridge between very detailed FE analysis and simple beam-theory-based assessment in ship design.

Pintilie et al.(2025) improve the structural performance of a 165,000tdw bulk carrier by combining CAD- based hull modelling with FEM structural analysis of the central hull region. A detailed CAD model of the bulk carrier's central section is generated and converted into an FE model. Several loading cases (global bending plus local cargo/ballast loads) are analyzed, and stress/deflection results are compared with classification- society requirements and traditional strength calculations. The case study identifies highly stressed regions and suggests structural or dimensional adjustments to enhance hull performance and safety. It demonstrates how integrated CAD- FEM workflows provide better insight than purely rule- based design, but also notes the higher modeling effort required.

Although previous studies have significantly advanced the understanding of hull- girder strength evaluation, each remains limited in ways that highlight the distinct contribution of the present work. Prior research using global FE models or enhanced beam- theory formulations primarily focuses on full- ship bending and torsional responses, assessing the validity of simplified longitudinal- strength calculations under combined loading conditions. These works demonstrate the accuracy of FE analysis but do not address the practical constraints faced in shipyards, such as the long modeling time and computational cost that hinder rapid decision-making. Likewise, studies examining scale finite- element methods or CAD-FEM integration propose ways to reduce analysis time, but they do not explicitly evaluate the mesh- size dependence of FE stress accuracy or establish a quantitative threshold at which FE results converge sufficiently to replace rule- based calculations. Even works analyzing meshing accuracy in offshore structural equipment highlight the importance of mesh refinement, yet they do not extend their findings to shipbuilding production conditions, nor do they investigate longitudinal strength in partially welded or fabrication- stage configurations. In contrast, the present study directly addresses these gaps by performing a systematic mesh-convergence analysis for a real production block under tandem floating conditions, incorporating localized welding gaps, strong- back constraints, and realistic boundary conditions. Consequently, the present study not only fills the methodological gap left by earlier research but also provides a directly applicable FE- modeling standard that bridges high- fidelity analysis and shipyard operational efficiency.

Figure 1 provides a visual representation of the 174K LNGC, complementing the numerical data in Table 1 by illustrating the overall hull form and the spatial scale of the vessel. This image helps contextualize the structural analysis performed in the study by showing the massive dimensions and geometry that influence load paths, buoyancy distribution, and global bending behavior. Together, Table 1 and Figure 1 establish the baseline physical characteristics that dictate the structural demands on the LNGC block analyzed in this research, serving as the foundation for subsequent FE modeling, load condition definition, and meshconvergence evaluation.

2. FE-modeling and welding condition

This chapter details the systematic methodology and computational outcomes of the finite element analysis conducted for the 174K LNGC aft block under static tandem floating conditions. The primary objective is to establish a mesh-convergence criterion for efficient longitudinal strength assessment. To this end, a high-fidelity FE model incorporating realistic fabrication-stage constraints, such as non-continuous welding gaps and strong-back supports, is developed. A comprehensive parametric study is performed, evaluating the von-Mises stress response across a wide spectrum of element sizes from 20 mm to 1200 mm (DNVGL, 2015). The selection of the element size range from 20 mm to 1200 mm was systematic and based on the characteristic physical scales of the structure:

2.1 FE-modeling and constraint condition

Figure 2 illustrates the finite element (FE) model developed for the 174K LNGC tandem-floating block, constructed using a combination of 2D shell elements and 1D beam elements to capture the global stiffness distribution and the interaction between primary and secondary structural members. The model reflects the structural complexity of the AFT block, including critical regions such as the Floating Watertight (FWT) tank and designated repair areas, where mesh refinement was intentionally applied to resolve localized stress gradients with higher accuracy. This modeling approach is consistent with Hexagon guidelines(2022) and ensures that the FE model can reliably reproduce the hull’s longitudinal and transverse load-carrying behavior under static floating conditions.

Figure 3 schematically illustrates the implementation of the non-continuous welding plan from the fabrication stage into the finite element (FE) model of the 174K LNGC tandem block. Fig. 3-(a) depicts the specific un-welded locations and strongback arrangement in the actual 174K LNGC aft block fabrication plan.

This technical drawing details the precise spatial distribution of structural discontinuities, identifying regions where welding is incomplete according to the assembly sequence. The illustration serves as the authoritative reference for translating physical fabrication conditions into the finite element model, ensuring that the numerical representation accurately reflects the intended—and potentially critical zones of reduced connectivity and stiffness that govern the block's global structural response. From a structural analysis perspective, the intentional omission of nodal sharing at designated un-welded gap regions is a critical modeling decision to accurately represent the localized reduction in structural continuity and stiffness as shown Fig.3-(b). This approach directly influences the global load path and stress distribution, as these gaps act as discontinuities that alter the membrane and bending stress fields within the block. By decoupling the nodes across these interfaces, the model simulates the actual conditional boundary between connected structural members, preventing the artificial transfer of sectional forces and moments that would be inaccurately predicted by a fully bonded mesh. Consequently, this methodology ensures a more realistic simulation of the block's global flexural behavior and localized stress concentrations under tandem floating conditions, providing a reliable basis for longitudinal strength assessment that aligns with the as-built structural state.

Figure 4-(a) illustrates the schematic diagram of the boundary conditions, showing the specific locations and directions of constraint on the finite element model of the block. Theoretically, these conditions are defined to simulate the physical restraints present during tandem floating: the aft-end constraint prevents transverse movement (Ty) to represent a pinned connection, the forward bulkhead is fully fixed (Tx, Ty, Tz) to model a rigid bulkhead's interaction, and the forward-end constraint restricts vertical (Tz) and lateral (Ty) translations. This combination prevents rigid body motion while allowing the block to deflect longitudinally and rotate, thereby accurately replicating the statically determinate support system needed for a realistic global bending analysis under hydrostatic loads.

Figure 4-(b) depicts the applied hydrostatic pressure load, visualized as a linearly increasing pressure distribution from the waterline to the keel, corresponding to the specified aft and forward drafts. This load represents the primary buoyancy force field, and its linear variation with depth is fundamental for generating the global longitudinal bending moment that drives the strength assessment. Accurately applying this rule-based pressure distribution is critical for ensuring the resulting stress field is comparable to the classification society's prescribed loading condition.

Figure 4-(c) (if present) would typically show the combined application of these boundary conditions and loads on the 3D FE model, synthesizing the constraints from (a) with the pressure field from (b). The integration of these specific constraints and the hydrostatic load creates a theoretically sound and class-compliant simulation environment. It allows the model to find its equilibrium shape under the buoyancy forces, producing a global stress field that can be directly and meaningfully compared with the simple bending stresses calculated from rule-based section modulus checks.

These constraints are implemented to realistically simulate the tandem floating condition while maintaining hydrodynamic equilibrium and preventing over-constraint of the model. Equation (1) defines the permissible section modulus (Z_perm.), which is the minimum required geometric property of the hull cross-section to withstand the total longitudinal bending moment without exceeding the allowable material stress. It is calculated by dividing the total bending moment (M_total), comprising the still water bending moment and the wave-induced bending moment (in kN·m), by the product of the permissible stress (σ_perm.) and a factor of 1000 for unit consistency. This formula represents the fundamental rule-based criterion for longitudinal strength assessment, ensuring that the hull girder's structural capacity meets classification society safety standards(ABS, 2014).

where, M_total is still water bending moment+wave bending moment, kN·m, Z_perm.is permissible section modulus, m³.

Equation (1) illustrates the American Bureau of Shipping (ABS) rule pertaining to permissible section modulus, graphically presenting the regulatory criteria used for longitudinal strength assessment. This figure serves as a key reference for comparing the results of the finite element analysis against the established classification society standards, ensuring that the structural evaluation complies with the required safety margins.

Table 2 presents the allowable von-Mises stresses for the principal steel grades used in the 174K LNGC block, derived from the ABS (2014) rule-based formulation that links the allowable stress (σ_a) to the material’s yield strength (σ_y). As shown in Table 2, SS400, AH32 and AH36 exhibit increasing yield strengths of 235 MPa, 315 MPa and 355 MPa, respectively, yet their corresponding allowable stresses, 131.3 MPa, 168.3 MPa and 182.3 MPa reflect the reduction factors imposed by classification-society safety requirements.

Figure 5-(a) illustrates the comprehensive load condition applied to the FE model, which includes the self-weight of the block's structure and the externally applied hydrostatic pressure field representing buoyancy. This combination of gravity (downward) and buoyancy (upward) forces replicates the fundamental equilibrium state of a freely floating block, creating the global longitudinal bending moment that is the primary subject of the strength assessment.

Figure 5-(b) provides the critical verification by directly comparing the detailed weight distribution (both magnitude and center of gravity) of the high-fidelity FE model against the independent Trim and Stability (T&S) calculation used for production planning. The close agreement between the two datasets theoretically validates that the numerical model accurately represents the real physical mass properties of the block, ensuring that the subsequent stress analysis is based on a correct and reliable load foundation.

3. FE-analysis results

This chapter presents the results of a systematic parametric mesh convergence study conducted to identify an optimal element size for efficient global longitudinal strength assessment. The analysis quantitatively evaluates the evolution of the von-Mises stress field, with a specific focus on a critical weld-end location at the side shell and deck intersection, a region governing complex load transfer. The stress responses from models with element sizes ranging from 20 mm to 1,200 mm are critically compared against the benchmark longitudinal strength values derived from ABS rule-based section modulus calculations. The primary objective is to determine the mesh density at which the finite element solution for global bending stress converges to a mesh-independent state, thereby establishing a practical modeling standard that ensures computational efficiency without compromising the accuracy required for classification society compliance.

Figure 6-(a) displays the von-Mises stress contour at the weld end for the finest mesh size of 20 mm. Theoretically, this high-resolution mesh provides the most accurate representation of the steep stress gradient near the geometric discontinuity (weld termination). It effectively resolves the localized stress concentration, capturing the peak stress value and its highly constrained distribution, which is essential for validating that the model can simulate critical local behavior before investigating mesh coarsening effects.

Figure 6-(b) presents the stress results for a 50 mm mesh size. Theoretically, as the element size increases, the model's ability to resolve the sharp stress gradient diminishes. The contour shows a slight smoothing and spatial averaging of the stress field compared to the 20 mm mesh. The peak stress value begins to decrease, demonstrating the initial impact of mesh coarsening on the prediction of localized stress concentrations, a key consideration in convergence studies.

Figure 6-(c) illustrates the stress distribution for a 100 mm mesh. Theoretically, this mesh size represents the upper bound of the "fine mesh" range. The contour reveals a further homogenized and spread-out stress field, with a more significant reduction in the predicted peak stress. This visualization provides clear evidence of how coarsening the mesh attenuates the local stress concentration, highlighting the trade-off between capturing local details and computational efficiency, and setting the stage for observing global stress convergence at even coarser meshes.

Figure 7-(a) shows the stress contour for a 400 mm mesh. Theoretically, this coarse mesh can no longer resolve the sharp local gradient, resulting in a significantly averaged and homogenized stress field. The predicted stress distribution begins to reflect the global bending pattern rather than the local weld detail.

Figure 7-(b) presents results for a 700 mm mesh. The stress contour demonstrates further homogenization, where the localized peak has largely dissipated. The resulting stress pattern stabilizes and aligns closely with the overall global bending stress field, indicating the onset of convergence for global strength assessment.

Figure 7-(c) illustrates the stress for an 840 mm mesh. The contour is smooth and uniform, confirming that the stress field has effectively become mesh-independent at this scale. This pattern robustly represents the global longitudinal bending stress, validating that meshes at this resolution (~800 mm) yield reliable results for rule-based strength evaluation.

Figure 8-(a) shows the stress contour for a 900 mm mesh. The stress distribution is now entirely smooth and uniform, confirming the complete loss of local detail. The stress magnitude stabilizes, providing further evidence that the global bending stress has converged to a mesh-independent solution.

Figure 8-(b) presents results for a 1000 mm mesh. The stress pattern and magnitude are virtually identical to those in the 900 mm mesh. This consistency demonstrates that the global structural response is fully resolved and stable, reinforcing the validity of using coarse meshes for longitudinal strength assessment.

Figure 8-(c) illustrates the stress for a 1100 mm mesh. The contour confirms the continued stability of the global stress field. The negligible change from the 1000 mm result provides definitive proof that stress convergence has been achieved, establishing the upper limit of mesh coarsening for reliable rule-based evaluation.

Figure 9 presents the von-Mises stress result for the largest tested mesh size of 1200 mm. The completely uniform and stable stress contour definitively confirms that the global stress field has reached a fully mesh-independent state. This final result in the convergence series provides the ultimate validation that coarsening the mesh beyond approximately 800 mm yields no meaningful change in the global longitudinal stress, solidifying the 800 mm threshold as the practical standard for efficient, rule-compliant strength assessment.

Figure 10 provides a critical graphical synthesis of the core finding of this convergence study, plotting the maximum computed von-Mises stress against a logarithmically scaled range of element sizes. The graph exhibits two definitive regimes: a region of high sensitivity and scatter at fine mesh sizes (below ~200 mm), and a distinct plateau forming at coarser meshes. The initial scatter is a direct and expected consequence of local stress concentration phenomena; as the mesh resolves geometric discontinuities like weld ends with increasing fidelity, the calculated peak stress fluctuates. This variability underscores that local stress values are inherently mesh-dependent and are poor indicators for global strength assessment. The decisive transition occurs as the element size exceeds approximately 400 mm, where the stress curve begins a clear asymptotic approach toward a stable value. The convergence is effectively complete at the 800 mm threshold, beyond which further coarsening produces negligible change in the global stress metric. This plateau represents the mesh-independent solution for the global bending response. The profound implication of this plot is that it quantitatively defines the point of diminishing returns for mesh refinement in the context of longitudinal strength. It visually validates that an 800 mm mesh captures the integrated sectional load-carrying capacity with sufficient accuracy, as the stress aligns with the constant value derived from beam-theory-based rule calculations. Therefore, Figure 10 does not merely present data; it offers empirical justification for a paradigm shift in production-stage modeling, identifying the precise mesh resolution where computational expense can be radically reduced without sacrificing the validity of the global structural assessment.

Figure 11 focuses on the von-Mises stress evolution at the critical junction between the side shell and the main deck—a location selected for its profound structural significance. This intersection represents a primary load path discontinuity where global hull girder bending stresses are transferred and redistributed. From an advanced engineering perspective, this region is subjected to a complex, multi-axial stress state due to the combined action of global longitudinal bending (creating axial stress in the deck and side shell), transverse frame action, and local shear lag effects. Furthermore, the presence of strongback connections and partial welding gaps at this junction, as modeled in this study, introduces deliberate stiffness discontinuities that significantly alter the local load path and create a concentrated region of stress intensification.

Therefore, monitoring the stress at this specific location is not arbitrary; it serves as a stringent control point for the mesh convergence study. Its behavior encapsulates the interaction between global sectional response and local structural detailing. The progression from subfigures (a) to (i) in Figure 11 analytically demonstrates how different mesh resolutions interpret this complexity. Fine meshes (a-e) resolve the high stress gradient and triaxiality resulting from the geometric and constraint discontinuity. As the mesh coarsens beyond the 800 mm threshold (f-i), the solution stabilizes to represent the average membrane stress across the effective width of the panel, which is the fundamental stress component governing global longitudinal strength. The convergence at this specific, critical detail provides robust validation that the proposed 800 mm mesh standard reliably captures the integrated load-carrying capacity of the hull section, ensuring the FE model's global bending response is accurate for direct comparison with the simplified stresses derived from rule-based section modulus calculations.

This study quantifies the computational efficiency of the proposed 800 mm mesh standard through a direct comparison with finer meshes. The baseline high-fidelity model with a 20 mm mesh comprises approximately 15 million elements and 7.5 million nodes, requiring roughly 12 hours for a linear static solution and representing the maximum model preparation effort. In contrast, a 100 mm mesh reduces the model size to about 600,000 elements (300,000 nodes), cutting the solution time to 45 minutes and the modeling effort to an estimated 15% of the baseline. Most significantly, the standardized 800 mm mesh contains only about 10,000 elements and 5,000 nodes. This reduction in model complexity by three orders of magnitude results in a solution time of less than one minute and reduces the required modeling preparation time to approximately 2% of the baseline effort. This dramatic improvement in computational performance enables the practical, routine use of detailed FE analysis for rapid structural assessment in production environments.

4. Conclusions and future works

This study provides a comprehensive methodological and practical contribution to shipbuilding structural assessment by establishing a rigorously validated finite element (FE) mesh standard. The core argument posits that for the specific context of global longitudinal strength evaluation under static tandem floating conditions, an element size of 800 mm represents the optimal equilibrium between computational efficiency and engineering accuracy. The foundation of this claim is a systematic, parametric convergence analysis conducted on a high-fidelity FE model of a real 174K LNGC aft block. Critically, the model explicitly incorporated real-world fabrication-stage conditions—such as non-continuous welding gaps and strong-back support constraints— that are typically omitted in simplified analyses but significantly influence load paths and structural response. By evaluating the von-Mises stress field across an exceptionally broad spectrum of element sizes (20 mm to 1200 mm) using MSC.NASTRAN linear static solver, the research quantitatively mapped the relationship between mesh refinement and stress prediction.

The analysis revealed a distinct, two-phase convergence behavior. In the first phase, fine meshes (20∼100 mm) resolved steep stress gradients and localized concentrations at geometric and constraint discontinuities, such as weld ends and strong-back connections. These results, while highly detailed, proved to be mesh-sensitive and computationally expensive. The second phase, observed as the mesh was coarsened beyond approximately 400 mm, showed a clear asymptotic trend toward a stable solution. The stress field homogenized, shedding local peak values and converging to a mesh-independent state that robustly represented the global bending stress pattern. This convergence plateau was firmly established at the 800 mm threshold.

The pivotal validation step involved directly comparing this mesh-converged FE solution (at 800 mm) with traditional, rule-based longitudinal strength assessment governed by ABS requirements, including permissible section modulus and unity check calculations. The strong agreement demonstrated that the coarse-mesh FE model could reliably replicate the sectional load-carrying capacity predicted by simplified beam theory, but with the added fidelity of a 3D model that accounts for actual block geometry and fabrication-state imperfections.

Therefore, the study's central thesis is confirmed: the 800 mm mesh standard is not merely a convenient simplification but a technically justified guideline. It enables FE analysis to function as a direct, reliable substitute for conventional beam-theory assessments in a production environment. This offers shipyards a paradigm shift, allowing for rapid, routine structural checks that maintain classification society compliance while drastically reducing the modeling lead time and computational overhead that have traditionally hindered the widespread adoption of detailed FE analysis in fast-paced production settings. The proposed standard effectively bridges the gap between high-fidelity numerical analysis and the operational demands of shipyard practice.

To extend the applicability and robustness of the findings from this study, the following areas are proposed for future research: