Single-Objective PSA Framework for Longitudinal Girder Scantlings with Full-Ship and Ultimate Strength Verification for a 500-Tonne Catamara

3.1. General procedure

It is critical in ship design to ensure the structural safety of the hull under all operating sea states. At a minimum, this is commonly secured by designing in accordance with established standards, regulations, and the rules of classification societies, which have evolved largely from accumulated experience. These rule-based approaches are simplified and formalised to facilitate practical use in determining the scantlings of principal members and arranging the primary structural layout In general, there are two approaches for structural design: the comparative method and the direct method, as shown in Fig. 1. The former is a common approach for designing hull scantlings based on a previous ship design. It can provide a sound basis for standard or conventional designs that is similar to a previous proven design with a good track record. However, it may not be reliable when a major design parameter changes significantly. Under these conditions, the latter is preferred, particularly for a novel design that goes beyond the proven design. It is based on finite-element modelling to secure hull structural safety beyond what existing standards, regulations, or class rules explicitly cover. It is now common practice to evaluate, as accurately

as practicable, the full set of loads arising from the actual operating sea states and to apply these loads to the structural safety assessment, thereby achieving a more realistic analysis. Even with these two design approaches, there are still some structural members that are overly heavy while others barely meet the strength criteria, and errors may remain hidden until construction. These tendencies are even worse in the design of small special ships. Therefore, to prevent such issues at the initial design stage of these vessels, scantlings should be optimised and reviewed numerically. Accordingly, the present study proposes a new procedure (Fig. 2) that adds an explicit optimisation stage to the conventional direct structural analysis. The procedure first verifies safety by direct analysis under the load cases. It then runs an optimisation loop that adjusts scantlings within rule constraints to meet strength and buckling criteria with minimum weight. The result is a design that remains rule-compliant and structurally safe, while improving structural efficiency and design transparency. As indicated in Figs. 1 and 2, the proposed framework extends the conventional direct approach by inserting an explicit scantling optimisation stage between the initial direct strength assessment and the final verification analyses.

3.2. Optimisation-based direct analysis procedure

3.2.1. Extreme design wave Primary member scantlings must be set against the most severe loading condition that a ship may

Fig. 2. Proposed design procedure for scantling optimisation.

experience during its lifetime. This condition is determined using probabilistic–statistical methods. Under the common assumption that the lifetime irregular sea can be represented as a superposition of all possible regular (sinusoidal) waves, one first computes structural responses to regular waves and then performs statistical analysis based on a wave spectrum (short-term sea) and long-term wave data / scatter diagrams to obtain the most probable extreme value. For the location of interest, compute loads for all regular waves, estimate the extreme load by statistical analysis, and select the single equivalent regular wave—the design wave—that produces that extreme. Three-dimensional structural analysis under this wave then yields the stress distribution under the extreme load, which is checked against allowable stress for final acceptance. Because analysis is required only for a few governing load cases and their design waves, time and effort are reduced.

The present study adopts design waves determined by such a procedure in a separate analysis; here, attention is focused on the structural responses under the selected design waves rather than on the detailed derivation steps. 3.2.2. Scantling optimisation In this study, the optimisation is applied to the longitudinal girders, which are selected as the primary design variables, while other members remain as in the rule-based baseline design. More generally, such a framework can be extended to treat the scantlings of longitudinal plate panels, longitudinal stiffeners and primary support members, as well as selected spacing parameters, as design variables within prescribed practical limits. As the spacing of major supporting members is usually constrained by cargo arrangement and construction requirements, such extended applications would mainly target plate thicknesses and the

sizes of small or highly stressed stiffeners and girders. These variables are adjusted in a single-objective framework in which structural weight is minimised while at least maintaining, or where possible increasing, structural safety within the limits prescribed by the classification rules. The optimisation is implemented using a heuristic Pareto-based simulated annealing algorithm in MAESTRO, which is employed here as a constrained single-objective search scheme. For each candidate design, the three-hold model is re-analysed, and the resulting strength, buckling and weight measures are used to guide the search towards designs that minimise weight while satisfying all prescribed criteria. Upon convergence, the optimised scantlings are verified by finite element analyses using both a fullship model and a one-slice model representing the critical transverse cross-section(s) under the governing load combinations. Global and local responses are checked against the prescribed acceptance criteria. If all checks are satisfied with adequate margins, the optimised scantlings are accepted as ensuring structural integrity under the specified design loads; otherwise, the optimisation–analysis cycle is repeated with adjusted constraints or design variables until a compliant solution is obtained. The full-ship finite element model is employed to establish the global load environment and to verify the final structural response, whereas the three-hold model is adopted for the optimisation loop in order to achieve computational efficiency while preserving sufficient fidelity in representing the governing vertical and transverse bending boundary conditions. The three-hold model inherits its loading conditions directly from the full-ship analysis, ensuring consistency between global response and local optimisation.

4.2. Mesh models

In this study, finite element analyses are carried out for three models. Fig. 4 describes the three FE mesh models for the present study. The primary structural members in all three models are represented by four-node quadrilateral and three-node triangular shell elements, while local stiffeners and girders are modelled using equivalent shell strips consistent with MAESTRO practice. 4.2.1. Full-ship model The full-ship model is used to evaluate hydrostatic and hydrodynamic loads under the governing full-load condition and the associated design waves specified in Table 3, which represent the most probable extreme vertical and transverse bending responses for the selected trading route. The hydrostatic pressure is determined such that buoyancy and weight are in equilibrium and no net unbalanced force is generated. In this study, wave-induced vertical and transverse bending moments at midship, as defined in Table 3, are considered as the governing global loading conditions. 4.2.2. Three-hold model In the three-hold finite element model, the structural strength is evaluated under the governing sagging and hogging vertical bending moments (122.2 and −163 MNm), which are determined in accordance with the Korean Register (KR, 2024) wave load criteria and are consistent with the design-wavebased global analysis. This model is employed as the basis for the scantling optimisation.

4.1. Target vessel

The target vessel, a 500-tonne steel catamaran, was illustrated in Fig. 3. The principal dimensions are summarized in Table 1. It is considered that various vehicle loading conditions for the target vessel are applied, including passenger cars, cargo trucks, large buses and heavy trucks. A ferry with a breadth of 19.8m is selected as the final design, and the optimal vehicle arrangement on the car deck is determined accordingly.

4.2.3. One-slice model To evaluate and compare the variation in ultimate strength of the steel catamaran resulting from the optimisation, a nonlinear structural analysis is performed using a one-frame slice at the midship region representing the critical transverse section. The full-ship model consists of 36,823 nodes and 151,609 elements, the three-hold model comprises 6,661 nodes and 21,600 elements, and the one-slice model comprises 642 nodes and 1,472 elements. In all models, only the principal structural members of the steel catamaran are represented, while outfit items such as seats, stairways and other minor fittings are excluded from the modelling.

Fig. 4. Finite element models: full-ship, three-hold and one-slice.

4.3. Boundary conditions and loading

In the present study, three boundary points (A-C) are introduced, as shown in Fig. 5. For full-ship mesh model, three nodal points were selected for the boundaries. The details of them were summarised in Table 2. The loading conditions applied in the full-ship analysis are defined with the same draft for all cases, as follows:

• Full-load still-water condition • Wave-induced vertical bending moments in sagging and hogging • Wave-induced transverse bending moments in sagging and hogging For the three-hold model, the structural analysis is performed under the following wave-induced vertical bending moments:

Table 2. Applied boundary conditions for the full-ship mesh model. Points

• Sagging: 122.2 MNm • Hogging: −163 MNm For the one-slice model, the ultimate strength of the midship transverse section is evaluated by applying displacement-controlled loading to both sides of the section until the maximum load-carrying capacity is reached.

4.4. Weight distribution

Using the generated full-ship mesh model, the distribution of weights acting on the hull was obtained by applying the light weight and the specified distributed loads. For calculation of light weight, mild steel was applied (yield stress σ y =235MPa, elastic modulus E =204GPa, density ρ =7850kg/m3 , Poisson’s ratio υ =0.3) Figs. 6 and 7 show the weight and buoyancy distribution of the full-ship model in full load condition. It is noted that the maximum weight occurs 20m away from stern. The maximum buoyancy appears between 15-20m away from the stern. In particular, the internal structural responses under the full-load condition are examined in Fig. 8.

These results are used to verify that a physically consistent longitudinal and transverse load environment has been established prior to detailed strength assessment and optimisation. In Fig. 8(a), the vertical shear force varies smoothly along the length, with negative values developing in the midship region and recovering towards the bow. The zero-crossings indicate locations where the cumulative weight and buoyancy are in local equilibrium. The absence of abrupt changes implies that the assumed weight–buoyancy distribution does not introduce unrealistic local load concentrations, and that the longitudinal load balance is reasonable. Fig. 8(b) shows the vertical bending moment distribution, which exhibits a typical sagging-dominant pattern, reaching its maximum magnitude in the midship region and decreasing towards the ends. This confirms that the central part of the catamaran is subjected to the most severe global vertical bending, consistent with conventional ship-type behaviour, and provides the governing condition for the longitudinal strength assessment and subsequent scantling optimisation. The transverse bending moment distribution in Fig. 8(c), plotted with respect to the distance from the centreline, indicates significant transverse bending across the bridge deck between the twin hulls. Positive moments occur near the centreline, while negative moments develop towards the outer sides, reflecting the load transfer mechanism between the demi-hulls and the cross-deck structure. The magnitude and shape of this curve highlight that the bridge

Table 3. Full-load conditions and design waves. Design Waves Loading Conditions

deck, transverse frames and cross-deck girders must be designed to resist transverse bending as one of the key governing actions. In Fig. 8(d), the horizontal bending moment along the ship length remains comparatively small and varies smoothly, indicating that lateral global bending is not a dominant design driver for this vessel under the considered loading condition. Consequently, the structural design is primarily governed by the vertical bending and the transverse bending associated

with the catamaran configuration, while the horizontal bending moment is confirmed to be of secondary importance. Therefore, these shear force and bending moment distributions demonstrate that the adopted weight and buoyancy model produces a consistent and realistic global load environment for the 500-tonne steel catamaran and provides a clear basis for the subsequent full-ship, three-hold and one-slice finite element analyses.

5.1. Optimisation results

The optimisation technique applied to the hull structure is the Pareto simulated annealing method implemented in MAESTRO. In the present study, only longitudinal girders were treated as design variables; plating and transverse members remain as in the rulebased baseline. The design variables are constrained as follows: • web height, Hw , 300 ≤ Hw ≤ 750 mm, • web thickness, Tw , 8 ≤ Tw ≤ 12 mm, • flange breadth, Bf , 100 ≤ Bf ≤ 200 mm, • flange thickness, Tf , 8 ≤ Tf ≤ 14 mm. There are 17 types of girders in the target vessel and their initial dimensions and their optimisation results are listed in the Table 4. In this optimised configuration, the resulting scantlings provide a mod-

est but meaningful reduction in hull girder weight while satisfying all prescribed strength criteria. For the three-hold cargo model, the steel weight decreases from 121.3 tonne in the initial design to 115.8 tonne after optimisation, corresponding to a reduction of 4.53%. When the optimised scantlings are consistently extended to the full ship model, the total hull weight is reduced from 695.7 tonne to 669.2 tonne, i.e. by 3.81%. These results confirm that the conservative nature of the initial scantlings can be relaxed towards the lower bounds of the admissible design space without compromising rule compliance, thereby achieving a more weight-efficient structural configuration. In Fig. 9, the global stress distributions of the three-hold FE model under the governing sagging and hogging bending moments specified in Table 3 are compared between the original and optimised scantlings. In the original configuration, relatively high von Mises stresses are observed in the upper deck

Fig. 8. Internal structural response under the full load condition.

Table 4. Comparison of girder section dimensions before and after optimisation. Initial (Hw × Tw × Bf × Tf )

longitudinal and around the opening corners, forming localised hot spots under both sagging and hogging conditions.

After applying the proposed optimisation, the stress fields become more uniform and the peak von Mises stresses are slightly reduced for both sagging and hogging load cases. High-stress regions at the deck and longitudinal girder connections are mitigated, and the overall stress levels in the primary longitudinal members remain below the allowable limit with similar or slightly reduced peak values while the global deformation pattern remains essentially unchanged. This demonstrates that the optimised longitudinal girder scantlings not only achieve weight reduction but also improve the structural efficiency by alleviating local stress concentrations, without introducing any adverse effect on the global strength response. In Fig. 10, the von Mises stress distributions on the 1st , 2nd and 3rd decks are compared between the original and optimised scantlings. The comparison shows that, with the proposed single-objective optimisation, all decks retain acceptable stress levels, and in several regions the stress field becomes more uniform, demonstrating that the achieved weight reduction is obtained without compromising deck structural integrity.

Fig. 9. Comparison of von Mises stress distribution for three-hold FE model under the sagging and hogging bending moment.

Fig. 10. Comparison of von Mises stress distribution for three-hold FE model under the sagging bending moment: (a) 1st deck, (b) 2nd deck, (c) 3rd deck.

In Fig. 11, the von Mises stress distributions of the three-hold FE model under the rule-based hogging bending moment are compared between the original and optimised scantlings for (a) the 4th deck, (b) the hull centre region, and (c) the hull side shell.

In both the original and optimised configurations shown in Fig. 11(a), higher stresses are confined to the regions adjacent to the openings and the side shell, while the remaining deck area remains in a low-to-moderate stress range. After optimisation, the

Fig. 11. Comparison of von Mises stress distribution for three-hold FE model under the rule-based hogging bending moment: (a) 4th deck, (b) hull centre, (c) hull side.

overall stress pattern is essentially unchanged; local peak stresses vary only marginally and remain within the allowable limit. This indicates that the weight reduction of the longitudinal girders does not impair

the load-carrying behaviour or safety of the 4th deck, and no new critical hot spots are introduced. For the hull centre region illustrated in Fig. 11(b), including the connections between longitudinals,

decks and side structure, the stress distributions before and after optimisation exhibit similar smooth gradients with only limited localised peaks. The optimised scantlings result in stress levels and band shapes comparable to the original design, without any excessive expansion of high-stress zones. This confirms that the optimised configuration maintains adequate global stiffness and strength in the central part of the hull under hogging conditions. Along the side shell shown in Fig. 11(c), the characteristic tension–compression band induced by hogging is observed in both cases, with maximum stresses remaining within the permissible range. After optimisation, the stress pattern is preserved and the stress levels are similar or slightly more uniform in some regions. There is no evidence of increased stress concentration or reduced margin against buckling. Therefore, the comparisons for the 4th deck, hull centre and hull side demonstrate that the proposed single-objective PSA-based optimisation achieves longitudinal girder weight reduction without degrading the global or local strength performance of the hull, as all critical regions retain sufficient safety margins under the hogging design load. Fig. 12 compares the von Mises stress distributions of the transverse frames in the three-hold FE model under the rule-based hogging bending moment, before and after optimisation, for (a) between bottom and 1st deck, (b) between 1st and 2nd deck, (c) between 2nd and 3rd deck, and (d) between 3rd and 4th deck. For the transverse frames between the bottom and 1st deck, both the original and optimised configurations show higher stresses mainly in the centre regions, while the overall stress level remains within the allowable range. After optimisation, the stress pattern is essentially preserved, and although some local peaks change slightly, no new critical hot spots appear. This indicates that the weight reduction of the longitudinal girders does not adversely affect the structural behaviour of the lower transverse framing. Between the 1st and 2nd decks, band-shaped stress regions occur along the upper parts of the frames and deck connections in both cases. The optimised scantlings lead to stress levels and distributions similar to those of the original design, with only minor increases in some members that still remain below the allowable limit. This confirms that the load-carrying capacity of the transverse system in this region is maintained. For the transverse frames between the 2nd and rd 3 decks, the original design exhibits somewhat higher stresses at selected transverse girders and longitudinal–frame intersections. In the optimised case, certain upper members experience slightly increased stresses, but the overall pattern remains

comparable and all stresses are within acceptable limits. The optimisation thus causes some redistribution without compromising structural safety. Between the 3rd and 4th decks, both designs are dominated by low stress levels, with only small localised peaks. The optimised scantlings do not introduce any noticeable increase in stress concentration, and sufficient margins are preserved. Therefore, the comparison demonstrates that the proposed single-objective PSA-based optimisation of longitudinal girders achieves weight reduction without degrading the strength of the transverse framing system; the transverse members across all deck levels retain adequate safety margins under the hogging design condition.

5.2. Full-ship response

As noted earlier, after completion of the optimisation, global finite element analyses were performed using the full-ship model with the optimised scantlings for the five loading conditions defined in Table 3, in order to compare and assess the structural responses before and after optimisation. Fig. 13 compares the von Mises stress distributions of the full-ship FE model between the original and optimised scantlings for (a) full-load still water, (b) vertical bending moment in sagging, (c) vertical bending moment in hogging, (d) transverse bending moment in sagging, (e) transverse bending moment in hogging. Across all load cases, the optimised configuration exhibits stress patterns that are very similar to those of the original design, and in many regions the stress levels are slightly reduced or remain essentially unchanged: In full load-still water condition, the overall hull girder stresses remain low in both designs. The optimised scantlings do not introduce any additional high-stress zones, indicating that the still-water strength is fully preserved. For vertical bending, sagging (b) and hogging (c), the characteristic global bending stress distributions along the hull are maintained after optimisation. Peak von Mises stresses in the deck and bottom regions are comparable to, or marginally lower than, the original values and remain below the allowable limit. No new critical hot spots appear, confirming that the longitudinal strength under primary vertical bending is not compromised.

5.3. Ultimate strength

In this section, a one-slice hull girder model is employed to evaluate the change in ultimate strength before and after optimisation, and to assess its adequacy against the relevant classification society criteria.

Fig. 12. Comparison of von Mises stress distribution for transverse frame in three-hold FE model under the rule-based hogging bending moment: (a) between bottom and 1st deck, (b) between 1st and 2nd , (c) between 2nd and 3rd , (d) between 3rd and 4th .

Fig. 13. Comparison of pressure and von Mises stress distribution for the full-ship FE model between the original and optimised scantlings for (a) full load–still water, (b) vertical bending moment in sagging, (c) vertical bending moment in hogging, (d) transverse bending moment in sagging, (e) transverse bending moment in hogging.

Fig. 14 presents the von Mises stress distributions obtained from the one-slice FE model for the sagging and hogging vertical bending moments, comparing the original and optimised scantlings. For both loading directions, the overall stress patterns of the optimised section closely follow those of the original design. The stress is smoothly distributed along the decks, side shell and inner structures, and the

local high-stress regions remain confined to the same limited connections as in the baseline case. No new hotspots appear and no significant increase in peak stress is observed after optimisation. This indicates that the longitudinal girder weight reduction does not adversely affect the stress state used as input for the ultimate strength assessment of the hull girder cross-section.

Fig. 14. Comparison of von Mises stress distribution for one-slice FE model: (a) sagging, (b) hogging.

Fig. 15 shows the vertical bending moment– curvature relationships and the corresponding ultimate hull girder strengths for the one-slice model, compared with the KR requirement (KR, 2024). One-slice hull girder ultimate strength assessment indicates that the optimisation leads to only a modest reduction in the hull girder capacity. In hogging, the ultimate bending moment decreases from 0.723 GNm to 0.679 GNm, corresponding to a 6.1% reduction, while in sagging it decreases from 0.614 GNm to 0.582 GNm, corresponding to a 5.2% reduction. Despite these reductions, the optimised scantlings still provide safety margins of approximately 10.9% in hogging and 13.5% in sagging relative to the KR

requirements, thereby confirming that the proposed optimisation maintains an adequate ultimate strength capacity of the hull girder. The observed reductions in ultimate bending capacity after optimisation are primarily attributed to the reduction of sectional area and moment of inertia of the longitudinal girders near the design lower bounds, which leads to earlier onset of yielding and local buckling in critical deck and girder components. Nevertheless, the progressive collapse mechanism and overall load-carrying behaviour remain unchanged, and sufficient safety margins relative to the KR requirements are preserved.

Fig. 15. Comparison of vertical bending moment–curvature relationships and ultimate hull girder strength for the one-slice model: KR requirement, original design, and optimised design (hogging and sagging).

5.4. Discussion

In the present study, the PSA scheme in MAESTRO is employed as a constrained single-objective search method. The scantling optimisation results in optimised longitudinal members that are concentrated near the lower bounds of the design domain, such as 300.0×8.0×100.0×8.0 mm. This tendency is not attributed to a deficiency of the optimisation algorithm, but is a direct consequence of the adopted problem formulation. The design variables, namely web height Hw , web thickness Tw , flange breadth Bf and flange thickness Tf are bounded within practically acceptable limits. The single-objective framework is constructed such that the primary objective is the reduction of structural weight, whereas strength and buckling are required to remain below the prescribed allowable limits and are thus treated effectively as constraints. Under this configuration, the sectional properties increase monotonically with each design variable, so that any increase in Hw , Tw , Bf or Tf leads to a heavier structure without providing an explicit benefit in the objective space, as long as all candidate sections already satisfy the rule-based criteria. As a result, designs located away from the lower bounds are dom-

inated by lighter designs that still meet all constraints, and are therefore discarded by the PSA mechanism. The solution archive consequently collapses towards corner solutions at the minimum permissible scantlings, which explains the apparent clustering of the optimised sections at one boundary. This outcome indicates that, within the given bounds and constraint set, the initial scantlings were conservative and that the lower-bound configurations ensure sufficient safety. If additional performance measures such as deformation limits, local stiffness requirements, redundancy or fabrication preferences were to be included explicitly as separate objectives or tighter constraints, a more distributed set of non-dominated candidates and a wider variety of optimal scantling combinations would be expected. The optimisation does not compromise structural safety: rule-based strength, buckling and serviceability checks remain satisfied; full-ship global responses under the governing vertical and transverse bending load cases are essentially unchanged compared with the baseline design; transverse hotspots occur at the same locations with similar or lower peak von Mises stresses, indicating that no new critical regions are introduced; and the one-slice ultimate strength assessment confirms that the optimised section still exceeds

the KR hull-girder requirements by approximately 10.9% in hogging and 13.5% in sagging. The PSA algorithm employed in this study follows the principles of stochastic global search, in which controlled random perturbations and probabilistic acceptance criteria enable the exploration of the discrete design space beyond local minima. When applied as a constrained single-objective scheme, the PSA mechanism effectively drives the solution towards the lightest feasible designs located near the admissible lower bounds while maintaining all structural constraints. Hence, the proposed optimisation-based framework reduces weight without degrading the ultimate or rule-based safety of the hull structure.