Nomenclature
e:
Nominal strain
E:
Elastic modulus
K:
Strength coefficient in Swift hardening law
n:
Hardening exponent in Swift hardening law
Q:
Material constant of Voce hardening law
s:
Nominal stress
T:
Simplified lashing force period
tp:
Deck plate thickness
α:
Weighting factor of combined Swift-Voce combined hardening law
β:
Material constant of Voce hardening law
ε:
True strain
εo:
Initial yield strain
εp:
True plastic strain
θh:
Horizontal lashing angle
θυ:
Vertical lashing angle
ν:
Poisson’ ratio
ρ:
Material density
σ:
True stress
σ0:
Initial yield strength
σs:
Swift constitutive equation
συ:
Voce constitutive equation
1. Introduction
Accidents during the operation of roll-on/roll-off (RO-RO) ships can cause large-scale casualties, and their risk has been continuously raised. Accordingly, related safety regulations (International Maritime Organization, 2011) have been strengthened, and studies on the structural strength of RO-RO ships have been conducted (Alie & Latumahina, 2019; Korkut et al., 2005; Szymonski, 2024; Wang et al., 2018; Windyandari et al., 2022).
The accident of Ferry MV SEWOL, in which 304 fatalities occurred, is known to have been caused by the failure to use a lashing device for securing cargo and vehicles. Cargo that is not properly secured on a ship can significantly affect the ship’s restoring stability. Jia (2007) confirmed the possibility of vehicle securing (free lashing) using only the friction force between the deck and the tire, rather than the conventional lashing system generally used to secure vehicles on RO-RO ships, in order to reduce labor and time during the lashing process, but still emphasized that vehicles should be secured in preparation for rapidly changing sea conditions. The lashing force acting on RO-RO ships originates from the motion of the hull. Kreuzer et al. (2007) developed cargo load optimization software that considers ship motion, the dynamic behavior of trailers, and the characteristics of lashing devices by utilizing the principles of dynamics acting on multibody systems. Wang et al. (2023) determined the motion period of RO-RO ships considering extreme sea conditions. The load acting with that period was used in dynamic simulations for the lanyard connected to the vehicle, and the maximum stress acting on the lanyard was analyzed. Choung and Jeong (2019) derived the acceleration distribution through seakeeping analysis for a car ferry and predicted the lashing force acting on vehicle wheels based on it. Sasa et al. (2023) developed a motion model of trailers in rough seas through numerical analysis and emphasized the importance of lashing by simulating the Ferry Ariake accident, which is known to have been caused by the loss of restoring stability due to the failure of a lashing device. Yulianto et al. (2018) evaluated the structural strength of a turnbuckle according to the inclination of a container ship.
However, the scope of guidelines and standards such as ISO 23575 and the IMO CSS Code mainly focuses on the specifications and performance of the lashing device itself or on the general principles and operational procedures of cargo securing, and it is difficult to say that they directly provide procedures for quantitatively evaluating the local detailed structure when the lashing point is integrated into the deck.
Recently, the use of the integrated lashing pot, which has been increasingly introduced to improve shipyard productivity and reduce construction costs, is composed by machining a lashing hole in the deck and attaching a lashing cup underneath. Such a structure increases geometric discontinuities and therefore has a high possibility of causing stress concentration and large plastic deformation around the lashing hole. In other words, apart from whether the individual device satisfies the load requirement, a structural evaluation criterion is required to determine whether the integrated deck structure satisfies functional requirements such as the prevention of hook detachment and the allowable deformation at the serviceability limit state (SLS). Accordingly, this study presents a rational strength evaluation methodology based on the SLS for a deck-integrated structure including an integrated lashing pot, and calibrates the combined Swift–Voce constitutive equation as a flow stress model to reliably reproduce the behavior in the local large-deformation region, and applies it to nonlinear finite element analysis.
2. Lashing Pot Structures and Lashing Hooks
2.1 Lashing Pot Structures
A lashing pot is an anchor point used to secure vehicles or cargo, such as automobile trailers, on RO-RO ships. It consists of a lashing hole, where a lashing hook is attached, and a lashing cup, which reinforces the structural strength of the lashing pot weakened by the presence of the hole. The most widely used shapes of lashing holes are the cloverleaf type, as shown in Fig. 1(a), and the circular type, as shown in Fig. 1(b). In this study, the cloverleaf type was selected for analysis because its more complex geometric shape makes it more susceptible to local stress concentration.
Traditionally, lashing pots have been installed after the block assembly stage by cutting a hole in the deck and welding the structure around the opening, as illustrated in Fig. 2(a). However, to improve construction productivity compared with the conventional approach, shipyards have increasingly adopted a deck-integrated lashing pot installation method. In this approach, the lashing hole is directly machined into the deck during the early block fabrication stage, and the lashing cup is welded directly beneath the deck, thereby simplifying the installation process, as shown in Fig. 2(b).
As shown in Fig. 3(a), a lashing pot is structurally most vulnerable when it is located at the center of both the web frames and the longitudinal stiffeners. One possible reinforcement method is to weld a transverse carling, as illustrated in Fig. 3(b). Kim (2019) demonstrated through finite element analysis (FEA) that carling reinforcement beneath the deck can effectively distribute the applied load. In some cases, the lashing pot is located directly above a longitudinal stiffener, as shown in Fig. 3(c). There are also many instances where the lashing pot is positioned above a web frame, as illustrated in Fig. 3(d).
2.2 Lashing Hook
Among the reference hooks specified in ISO 23575 (International Organization for Standardization, 2022), the chain lashing hook shown in Fig. 4(a) is also widely used. However, considering the actual operational practice and compatibility with cloverleaf-type holes in large RO-RO ships, the elephant foot hook type lashing hook was adopted in this study, as shown in Fig. 4(b).
3. Material Constant Calibration
To accurately evaluate the plastic deformation behavior of the lashing pot structure, it is necessary to employ a modeling approach that can precisely describe the stress–strain relationship of the material from the onset of yielding to the large-deformation regime. Since the reliability of FEA largely depends on the accuracy of the input material properties, it is essential to define a reasonable flow stress model that can properly reflect the hardening characteristics of the material even at high levels of plastic strain.
3.1 Theoretical Backgrounds for Flow Stress
From the relationship between tensile force and gauge displacement obtained from a tensile test, the relationship between nominal stress and nominal strain can be derived. The nominal stress–nominal strain curve can be converted into the uniform true stress–uniform true strain (σ – ε) relationship using Eqs. (1)–(2), assuming volume conservation during plastic deformation. After necking occurs, the σ – ε relationship is no longer uniform within the gauge length, and the stress state changes from uniaxial stress to triaxial stress. In general, the necking strain of structural steels is known to be approximately 0.1–0.2, which is considerably smaller than the plastic strain at fracture. Therefore, various material constitutive laws have been proposed to estimate the post-necking σ – ε relationship, i.e., the true stress–true plastic strain relation, based on the σ–ɛ relation before necking. For this reason, several methods have also been proposed to correct the flow stress after necking.
A theoretical expression for converting the triaxial stress after necking in a round-bar specimen into an equivalent stress was proposed by Bridgman (1964). Choung and Cho (2008) also proposed an empirical formula for correcting the post-necking flow stress based on numerical analysis of square cross-section specimens.
Recently, more advanced models have been developed. The Swift constitutive equation in Eq. (3) is known to describe the flow stress more accurately at low plastic strains compared with Hollomon’s power law. The Voce model in Eq. (4) (Voce, 1948) is known to be suitable for materials in which strain hardening can be neglected. Because of the different characteristics of these two hardening models, the Swift constitutive equation shows a tendency for the stress to increase continuously as deformation progresses, whereas the Voce constitutive equation exhibits behavior in which the stress approaches a certain saturation value despite increasing strain. In other words, in the large-deformation regime, the Swift constitutive equation tends to overestimate the stress and predicts continuous hardening, whereas the Voce constitutive equation tends to underestimate the stress because no further hardening is considered. In reality, the large-deformation behavior of materials lies between these two models. Therefore, the combined Swift–Voce constitutive equation, shown in Eq. (5), was proposed to derive a flow stress that more closely represents the actual material behavior by combining the two models (Pack & Mohr, 2017; Pack et al., 2018, 2019). The validity of this material hardening model has been demonstrated by Cerik et al. (2019a, 2019b, 2020) and Sung et al. (2010).
As the weighting factor α in Eq. (5) approaches 1, the model becomes closer to the Swift constitutive law, implying that the flow stress is less corrected for the effect of triaxial stress. Conversely, as α approaches 0, the model becomes closer to the Voce model, and the flow stress tends to be interpreted as a saturated flow stress.
3.2 Tensile Test
In this study, AH36 steel plates with a thickness of 15.0 mm were obtained, and smooth-bar specimens (FB) and notched specimens (NT) were fabricated to determine the material constants of the flow stress, as illustrated in Fig. 5. Tensile tests were then conducted using a 50-ton MTS universal testing machine to obtain the relationship between tensile force and gauge displacement. During the tests, the load was applied at a rate of 1.0 mm/min, and an extensometer with a gauge length of 50 mm was used to measure the gauge displacement.
3.3 Flow Stress Calibration
From the tensile test of the smooth specimen, the engineering stress–engineering strain curve (s – e) shown in Fig. 6(a) was obtained. Using Eqs. (1)–(2), the curve was converted into the uniform true stress–uniform true strain curve (σ – ε), which is presented in Fig. 6(b). Based on the uniform true stress–uniform true plastic strain curve prior to necking, the material constants of the Swift and Voce models were determined, and the results are summarized in Table 1.
Necking in the uniform true stress–uniform true strain curve occurred at a strain of 0.177, and therefore the flow stress corresponding to larger plastic strains was required. To determine this, FEA was repeatedly performed while updating the weighting factor α of the combined Swift–Voce model until the FEA results matched the tensile force–elongation relationship obtained from the notched specimen tests (Cerik & Choung, 2020). The FEA model of the notched tensile specimen is shown in Fig. 7. Based on previous studies, an element size of 0.1 mm was applied in the gauge section of the specimen (Cerik et al., 2019a; Cerik et al., 2019b; Park et al., 2019). The material properties of the specimen are summarized in Table 2, and the boundary and loading conditions applied to the specimen are presented in Fig. 7 and Table 3.
As shown in Fig. 8, the results of the tensile test and the FEA showed the best agreement when the weighting factor α was set to 0.8. The combined Swift–Voce flow stress corresponding to α = 0.8 is presented in Fig. 9.
In this study, the material used was assumed to be AH36 steel with a yield strength of 355 MPa. However, the yield strength obtained from the previously conducted smooth tensile test was 397.74 MPa. After taking the logarithm of both sides of the combined Swift–Voce data shown in Fig. 9, the stress was shifted in parallel so that the yield strength became 355 MPa, as illustrated in Fig. 10(a). In this study, the flow stress with a yield strength of 355 MPa and the same Swift material constants as those obtained from the experimental data is referred to as the design flow stress, and it is presented in Fig. 10(b).
4. Finite Element Analysis
4.1 Evaluation Criterion
ISO 23575 provides structural strength criteria for lashing devices, including lashing pot structures, used on RO-RO ships. According to this standard, when a load equal to 1.25 times the minimum securing load (MSL) of the lashing device is applied and then unloaded, the residual deformation at the lashing hole must not exceed 3 mm. This requirement is intended to prevent the lashing hook from disengaging from the lashing hole. In this study, not only the residual deformation criterion but also the condition that the lashing hook should not be released from the lashing pot, verified through simulation, was adopted as the SLS evaluation criterion.
4.2 Geometric Modeling
The lashing pot configurations shown in Fig. 2 were constructed as FEA models, as illustrated in Fig. 11. As shown in Fig. 11(a), the lashing pot is located at the mid-span between the web frames and also at the mid-span between the longitudinal stiffeners, without any additional reinforcement. Therefore, this configuration was designated as no reinforcement (NR). To minimize the influence of boundary conditions, the longitudinal and transverse extents of the NR model include one web frame spacing and three longitudinal stiffener spacings, respectively.
The configuration shown in Fig. 11(b) was designated as CL, since one transverse carling was installed as reinforcement in the NR configuration. The modeling extent of the CL model is the same as that of the NR model. In Fig. 11(c), the lashing pot is located directly on a longitudinal stiffener, and thus this configuration was named on longitudinal stiffener (OL). The longitudinal and transverse extents of the OL model include one web frame spacing and four longitudinal stiffener spacings, respectively. The configuration shown in Fig. 11(d) places the lashing pot directly on a web frame, and therefore it was designated as on web frame (OW). The longitudinal and transverse extents of the OW model include two half web frame spacings and four longitudinal stiffener spacings, respectively.
Considering that the upper thickness of the lashing cup used in RO-RO ships is generally 16.0 mm, the minimum deck thickness was also set to 16.0 mm. The detailed dimensions of the FEA models are summarized in Table 4. The elements used in the four models were reduced-integration shell elements (S4R and S3R). As shown in Fig. 12(a) and Fig. 12(b), the element size used in all four models was set to 1.5 mm, which was chosen to adequately represent the curvature radius of the cloverleaf hole. In addition, since the element size corresponds to only approximately 9% of the minimum deck thickness, it was considered sufficiently small for structural strength evaluation; therefore, a separate mesh convergence study was not performed. It was assumed that the deck, longitudinal stiffeners, web frames, and lashing cups used in the NR, CL, OL, and OW models were made of the same material. In other words, all components were assumed to be uniformly made of AH36 steel.
Based on the elephant foot hook drawing shown in Fig. 4(b), an FEA model of the lashing hook was generated with emphasis on the regions expected to come into contact with the lashing pot structure, as illustrated in Fig. 13. Since the lashing hook has sufficiently higher stiffness than the lashing pot structure, the hook in the FEA model was treated as a rigid body and modeled using rigid elements (R3D4 and R3D3). The number of elements used in the lashing pot structure and the lashing hook is summarized in Table 5.
4.3 Load and Boundary Conditions
The loading angle acting on the lashing hook should be set so that it can induce significant out-of-plane deformation of the deck. According to ISO 23575 (International Organization for Standardization, 2022), the lashing angle is defined by two components: the vertical lashing angle (θυ) and the horizontal lashing angle (θh ), as illustrated in Fig. 14(a). In this reference, the specified values of the vertical lashing angle θυ are 10°, 45°, and 80°, while the horizontal lashing angle θh values are 0°, 45°, and 90°. As shown in Fig. 14(b), the lashing angle θυ = 80° and θh = 0°, which is considered to have the most critical effect on the lashing pot structure, was applied in this study.
As shown in Fig. 15, a local coordinate system inclined by the lashing angle was defined. Assuming that the distance between the lashing hole (anchor point) and the vehicle securing point is sufficiently long, the lashing angle can be assumed to remain constant during loading. Accordingly, the load was applied in the y-direction, while the transverse movements of the inclined hook, namely the x- and z-direction displacements, were constrained. In addition, the rotations of the hook about the x- and y-axes were restricted so that the hook could move only along a single direction. Meanwhile, since the FEA model of the hull structure was sufficiently large compared with the lashing pot structure, the outer boundaries of the four models were assumed to be fully constrained in all six degrees of freedom. The boundary conditions applied to the FEA models including the hook and the lashing pot are summarized in Table 6.
The load corresponding to the SLS criterion defined in this study is 1.25 MSL. This load consists of the static load due to the initial securing force and the dynamic load induced by the motion of the ship. The irregular dynamic load caused by ship motion is expected to exhibit a pattern similar to that shown in Fig. 16(a). In this study, the resultant of the initial static securing force and the dynamic securing force induced by ship motion was assumed to be 1.25 MSL, and the loading cycle was simplified as having a period T, as illustrated in Fig. 16(b). In other words, after applying and then unloading the 1.25 MSL securing force, the satisfaction of the SLS criterion was evaluated. In this study, the minimum breaking load (MBL) of the assumed lashing rod was 30 t. Therefore, according to DNV (2024) and ISO 23575, the minimum securing load (MSL) is 15 t (MSL = 0.5 MBL).
The contact between the lashing pot and the lashing hook was defined using a surface-to-surface contact condition. A friction coefficient of 0.7 was applied to account for metal-to-metal friction (Sullivan, 1988). In this study, ABAQUS/Explicit (Simulia, 2024) was employed to perform the nonlinear structural analysis.
4.4 Simulation Time
Although the securing force induced by ship motion can be classified as a dynamic load, it may be treated as a quasi-static load because the motion period of the ship is sufficiently long. To exclude the effects of dynamic load amplification, a sufficiently long loading duration is required in the analysis. To determine this loading duration, a convergence study was conducted using the NR model, which was expected to have the lowest structural stiffness.
A review of the literature (Choung et al., 2016) indicated that the roll period of a 1,633-ton car ferry built in Korea was approximately 6.3 s. Therefore, the reference loading period (T) was assumed to be 6 s. Based on this reference value, a convergence study was performed for eight cases at 6 s intervals. It should be noted that T represents the time required for the load to reach 1.25 MSL, excluding the unloading phase. The value of T determined from this convergence study effectively corresponds to 2T in the actual loading cycle, and thus it was expected to sufficiently suppress dynamic effects. As shown in Fig. 17, the applied load was assumed to increase linearly with time.
The displacement–tensile force relationship of the hook after it comes into contact with the lashing pot is presented in Fig. 18(a). Immediately after the hook contacts the lashing pot, the curves show a converging trend as the simulation period increases. To closely examine the dynamic behavior of the hook, the results corresponding to the initial contact stage are shown in Fig. 18(b).
The vertical axis intercept of each curve represents the initial dynamic load (impact load) generated at the moment when the hook first contacts the lashing pot. As observed in the dynamic response during the initial contact stage, when the analysis time is short, the rapid load application causes significant inertial effects, resulting in a large fluctuation in the dynamic load. However, as the analysis time increases, the influence of inertia gradually decreases, and the load–displacement curves exhibit a converging tendency.
In particular, when comparing the results obtained for 30 s, 36 s, 42 s, and 48 s, almost no difference in the initial dynamic load is observed, and the overall load–displacement curves are nearly identical. Therefore, in this study, a simulation time of 30 s was selected as the final analysis duration, as it ensures stable quasi-static behavior.
5. Result
As shown in Table 4, a load corresponding to 1.25 MSL was applied to the lashing pot structure with an initial deck thickness tp of 16.0 mm, and the plastic deformation behavior was analyzed for each reinforcement configuration. After unloading, the residual displacement of the lashing pot is presented in Fig. 19, where clear differences in residual deformation at the lashing hole can be observed depending on the type of lower reinforcement structure.
As summarized in Table 7, the NR model, which has no additional reinforcement, exhibited a residual deformation of 7.61 mm after unloading under the 16.0 mm thickness condition. This value significantly exceeds the allowable deformation limit of 3 mm specified in ISO 23575, indicating that the unreinforced structure cannot provide sufficient stiffness against the applied external load, resulting in excessive plastic deformation.
In contrast, the models with reinforcements showed smaller residual deformations compared with the NR model. Under the 16.0 mm thickness condition, the CL, OL, and OW models exhibited residual deformations of 1.33 mm, 1.01 mm, and 0.38 mm, respectively. In particular, the OW model, where the lashing pot is installed directly above the web frame, demonstrated the highest structural stiffness and thus the best performance in suppressing deformation. The CL and OL models, reinforced using carling and longitudinal stiffeners, respectively, were also found to effectively distribute the applied load and control plastic deformation.
To evaluate whether the SLS criterion, which aims to prevent hook disengagement and excessive residual deformation of the structure, is satisfied, additional analyses were conducted while increasing the deck thickness, as summarized in Table 7. For the reinforced models (CL, OL, and OW), the residual deformation remained below 3.0 mm even at the initial design thickness of 16.0 mm, thereby fully satisfying the SLS criterion. This indicates that when an appropriate lower reinforcement design is provided, sufficient serviceability can be achieved even with a deck thickness smaller than the initial design thickness.
In contrast, the NR model, which has no reinforcement, did not satisfy the SLS criterion at 16.0 mm. Therefore, additional analyses were performed by increasing the deck thickness in increments of 0.5 mm. As the thickness increased, the residual deformation gradually decreased to 5.16 mm (tp = 16.5 mm), 4.06 mm (tp = 17.0 mm), and 3.31 mm (tp = 17.5 mm). When the deck thickness reached 18.0 mm, the residual deformation further decreased to 2.67 mm, thereby satisfying the SLS criterion.
Consequently, when applying an integrated lashing pot design aimed at improving production efficiency, the use of lower reinforcement members effectively increases structural stiffness and suppresses plastic deformation. This demonstrates that structural safety can be ensured without increasing the deck thickness.
6. Conclusion
In this study, a deck-integrated lashing pot structure, which has been introduced to improve the construction productivity and reduce the cost of RO-RO ships, was investigated to establish a rational structural strength evaluation criterion and to quantitatively assess its safety under various reinforcement conditions. To accurately reproduce the large-deformation behavior occurring around the lashing hole, the combined Swift–Voce constitutive equation, derived through cross-validation between tensile tests and finite element analysis, was applied. Through this nonlinear material model, an analysis framework capable of predicting complex stress states and nonlinear plastic deformation behavior with high reliability was established, which are difficult to capture using simple linear analysis or conventional empirical approaches.
The structural analysis results confirmed that the residual deformation of the structure varies depending on the arrangement of the reinforcement members beneath the lashing pot. In the NR model, where no additional reinforcement was applied, excessive plastic deformation was concentrated around the lashing hole under loading, and significant residual deformation remained after unloading, indicating insufficient structural stiffness. In contrast, the CL, OL, and OW models, in which reinforcement members such as carling, longitudinal stiffeners, and web frames were effectively integrated, were able to redistribute the locally concentrated loads to surrounding structural members. As a result, the extent of plastic deformation was minimized, and the structural integrity was maintained more stably.
The SLS evaluation based on ISO 23575 and related regulations also clearly demonstrated the difference in design efficiency depending on the presence of reinforcement. The NR model, without reinforcement, required the deck thickness to be increased from the initial design value of 16.0 mm to 18.0 mm in order to satisfy the allowable deformation criterion. However, the reinforced models were able to fully satisfy the SLS criterion even with the initial deck thickness of 16.0 mm, indicating that structural safety can be ensured without additional steel material.
In conclusion, the nonlinear analysis method based on the SLS evaluation procedure established in this study is considered to be an effective methodology for quantitatively verifying the structural safety of integrated lashing pot structures. The rational structural strength evaluation criterion proposed in this study is expected to serve as a practical design guideline for shipyards when applying various types of integrated structures aimed at improving production efficiency, enabling them to secure structural reliability while avoiding unnecessary overdesign.
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