Nomenclature
SRF: Springing reduction factor
Dr-HydroE: Rigid body fatigue damage index without springing from the hydroelastic analysis
Dt-HydroE: Total fatigue damage index including springing from the hydroelastic analysis
DQS: Quasi-static fatigue damage
D(QS+Dynamic): Total fatigue damage including Quasi-static and dynamic response
ωcut-off: Cut-off frequency
ω1: The first major contributing natural frequency of the hull girder
ξ: Modal damping coefficient of the first vibration mode of the hull girder
[RR]: Rigid body part of the matrices
[RE]: Rigid-Elastic interaction part
[ER]: Elastic-Rigid interaction part
[EE]: Elastic part
[K]: Structural (modal) stiffness
[FR]: Rigid body excitation force
[FE]: Elastic body excitation force
[ζ]: Total response
[ζR]: Total rigid body response
[ζE]: Total elastic modes response
Vs: Service speed
Vd: Design speed
ηi: the modal damping coefficient for the ith hull girder natural frequency
&omegasi: the frequency of the ith hull girder natural vibration mode, in rad/s
g: Gravitational acceleration
L: Length between perpendiculars
1. Introduction
With the expanding global economy, the size of merchant ships used for maritime transportation has increased in response to market demand. These ships are primarily used to transport resources, natural energy, and products. Container ships, which play a crucial role in maritime logistics, have hatch openings to allow containers to be loaded in cargo holds. Although their draft is restricted to pass through canals, they are designed with a wide beam to carry large quantities of cargo. Due to their long length and low section modulus, these ships have low natural frequencies and exhibit elastic behavior along with six-degree-of-freedom rigid body motion when subjected to dynamic wave loads. This behavior is known as the hydroelastic effect, which can be further classified into whipping and springing. Whipping is occurred by the impact load when a ship with a large flare angle at the bow and stern encounters high waves, increasing the vertical bending moment at the center of the hull. In contrast, springing refers to the continuous vibration of the hull caused by wave loads. Whipping is primarily associated with ultimate strength and can lead to hull failure, whereas springing is related to the fatigue strength of the hull. Notable marine accidents caused by the hydroelastic effect include the MSC Napoli container ship accident in 2007, linked to whipping, and the Great Lake bulk carrier accident in 1950, associated with springing.
Various studies on the hydroelastic effect of container ships have been conducted worldwide, with a focus on full-scale measurements, model tests, and numerical simulations. In full-scale measurements, Kim et al. (2018a) analyzed that vibration contributed 41% to 45% of the total fatigue damage when two-node and three-node vertical bending moments were considered for a 9,400 TEU container ship. Kim et al. (2018b) reported that vibration accounted for 40% to 50% of the total fatigue damage when the two-node vertical bending moment was considered for a 13,000 TEU container ship. Barhoumi and Storhaug (2014) found that vibration contributed 57% of fatigue damage for an 8,600 TEU container ship and 46% for a 9,400 TEU container ship. Storhaug (2014) further analyzed that vibration contributed 55% to 60% of fatigue damage for the same 8,600 TEU container ship. Kahl et al. (2014) observed that vibration accounted for 37% of fatigue damage for a 4,600 TEU container ship and 57% for a 14,000 TEU container ship. Similarly, Renaud et al. (2013) found that vibration contributed 45% of fatigue damage for a 9,400 TEU container ship. The results from these full-scale measurements indicate that vibration contributes significantly to fatigue damage, although it is difficult to establish a clear trend in the contribution of fatigue damage based on ship size. In model tests, Drummen et al. (2008) showed that vibration contributed 40% of fatigue damage when only the two-node vertical bending vibration mode was considered for the bow wave in a North Atlantic environment. Moe (2005) analyzed that vibration contributed 50% of fatigue damage under the Pacific Ocean environment. Storhaug (2014) found that vibration contributed 86% to 87% of fatigue damage for an 8,600 TEU container ship and 65% to 78% for a 13,000 TEU container ship. Storhaug et al. (2010) further reported that vibration contributed 65% of fatigue damage for a 13,000 TEU container ship, with significant fatigue damage occurring at the center of the ship, while the contribution of vibration to fatigue damage was highest at the stern. The results of model tests indicate that relatively greater fatigue damage occurs in the midship region compared to the bow and stern regions. However, as with full-scale measurements, it remains difficult to establish a clear trend in the contribution of fatigue damage based on ship size. In numerical simulations, many researchers have conducted hydroelastic analyses using either time-domain or frequency-domain approaches, employing one-dimensional or three-dimensional hull structure models (Hirdaris et al., 2003; Jensen and Dogliani, 1996; Kim et al., 2009; Kim et al., 2013; Malenica et al., 2003; Price & Temarel, 1982; Wu and Moan, 1996). Drummen et al. (2008) reported that linear motion analysis produced more accurate results than time-domain nonlinear motion analysis when compared with full-scale measurement results under bow wave conditions. From the research trends on the hydroelastic effect of ships, no clear pattern of fatigue damage contribution has emerged. However, it has been consistently observed that the total fatigue damage is highest in the midship region and decreases toward the bow and stern. Additionally, linear numerical analysis results align more closely with full-scale measurement results.
This study aims to perform hydroelastic fatigue analysis by applying Lloyd’s Register (LR) and Bureau Veritas (BV) procedures based on linear hydroelastic motion analysis. In accordance with the procedures of these two classification societies, hydroelastic fatigue analysis was conducted in the after end (AFT), engine room (ER), midship (MID), and forward end (FWD) regions at the hatch coaming top corners, which are fatigue-sensitive areas of a container ship. For quasi-static fatigue damage, LR’s full spectral fatigue analysis technique was employed. Hydroelastic motion analysis was carried out under the identical condition to compare the hydroelastic stress response amplitude operator (RAO) and the fatigue damage increase ratio for each classification society.
2. Hydroelastic Fatigue Analysis Procedures by Classification Societies
Classification societies have presented procedures for hydroelastic fatigue analysis, and the characteristics of these procedures are summarized in Table 1.
LR (2022) presented a procedure in which the hydroelastic stress RAO and hydroelastic fatigue damage are calculated by multiplying the mode RAO for each vibration mode, obtained from frequency-domain linear hydroelastic motion analysis, by the hot-spot stress of each vibration mode. Rigid body motion fatigue damage is calculated from the hydroelastic stress RAO below the cut-off frequency. The springing reduction factor, a fatigue damage ratio, is determined by dividing the rigid body motion fatigue damage by the hydroelastic fatigue damage. BV (2015) proposed a procedure where the dynamic elastic mode response is calculated using the hydrodynamic coefficients obtained from frequency-domain linear hydroelastic motion analysis, and dynamic stress RAO is derived by multiplying the hot-spot stress of each vibration mode. The final hydroelastic stress RAO is obtained by adding the dynamic stress RAO and quasi-static stress RAO to calculate the hydroelastic fatigue damage. This quasi-static stress RAO is calculated from full spectral fatigue analysis. The American Bureau of Shipping (2017) introduced a procedure in which the first-order and second-order springings in the frequency domain are calculated, considering only two-node vertical bending vibration, and the hydroelastic stress RAO is determined by multiplying the hot-spot stress of each vibration mode. The hydroelastic stress RAO is divided into rigid body motion RAO by waves and springing RAO based on 2.0 [rad/s]. The springing fatigue damage contribution is calculated by dividing the hydroelastic fatigue damage, obtained from the hydroelastic stress RAO, by the rigid body motion fatigue damage derived from the rigid body motion RAO. In contrast, Det Norske Veritas (2021a) requires the fatigue damage increase coefficient due to springing to be calculated using a specific formula. The Korean Register (2018) presents various methodologies based on time-domain hydroelastic motion analysis. It outlines a procedure in which linear springing analysis is conducted using time-domain hydroelastic motion analysis, and fatigue damage is either directly obtained by acquiring the hydroelastic stress RAO, or the linear springing correlation coefficient is calculated by dividing the fatigue damage obtained from the stress RAO in linear rigid body motion analysis. Alternatively, nonlinear springing is analyzed, and long-term fatigue damage is directly obtained by acquiring the hydroelastic stress time series. This value is then divided by the long-term fatigue damage obtained from the rigid body stress time series in nonlinear rigid body motion analysis to calculate the nonlinear springing correlation coefficient. In the procedures of classification societies other than BV and KR, hydroelastic fatigue damage is obtained by multiplying or dividing the quasi-static fatigue damage, derived from full spectral fatigue analysis, by the ratio specified by each classification society.
Full spectral fatigue analysis is a technique used for assessing hull fatigue strength, and classification societies provide their own procedures for conducting full spectral fatigue analysis (American Bureau of Shipping, 2004; Det Norske Veritas, 2021b; LR, 2002). In this method, the quasi-static stress RAO is obtained by performing frequency-domain ship rigid body motion analysis and hull structure analysis. Quasi-static fatigue damage is then calculated by applying the S-N curve that corresponds to the sea state and local structural conditions.
3. Analysis Conditions
3.1 Ship Modeling
The full ship finite element (FE) model of the target container ship is shown in Fig. 1, and its principal dimensions are listed in Table 2. Fatigue analysis regions were defined as AFT, ER, MID, and FWD at the hatch coaming top corners, in the longitudinal direction from the stern of the ship. As shown in Fig. 2, each position was modeled as t×t, corresponding to the plate thickness (t), and a dummy rod element was included at the hatch coaming top corners to capture hot-spot stress.
HydroE-FD 1.2.2.0, distributed by LR, was used for hydroelastic motion analysis. HydroE-FD is software developed based on 3D velocity potential theory, which can perform frequency-domain hydroelastic motion analysis for elastic vessels to predict rigid-body ship motion and elastic deformation. Shin et al. (2015) validated HydroE-FD and WISHFLEX-beam, a time-domain hydroelastic software program, as the section force RAO results obtained using both HydroE-FD and WISHFLEX-beam were similar for a 16,000 TEU container ship. MSC Nastran 2023.3 was employed for the normal mode analysis required for hydroelastic motion analysis and for linear elastic analysis in the full spectral fatigue analysis.
The full ship FE model must accurately represent the container modeling and weight distribution of the ship. For container modeling, LR (2022) provided a method that models the inside of the hatch using 20 ft (6.096 m) containers connected to the inner bottom, and 40 ft (12.192 m) containers connected to the top of the deck, as shown in Fig. 3(a). BV (2015) offered a different modeling approach, where the inside of the hatch is modeled with 20 ft (6.096 m) containers connected to all positions in contact with the container corners, and 20 ft (6.096 m) containers at the top of the deck, as illustrated in Fig. 3(b). It is estimated that for LR’s modeling method, the container load in the cargo hold is transmitted only to the inner bottom, which supports the containers, whereas for BV’s modeling method, the load is transmitted to the transverse and longitudinal bulkheads via the cell guide in the cargo hold, in addition to the inner bottom. In this study, the LR method was employed. Containers were modeled as mass elements that reflect both the mass and mass moment of inertia, with the assumption that cargo had been loaded up to 90% of the container height (H) for each container, as shown in Fig. 4. This assumption aligns with the 90% container height generally used in loading manuals, as cargo does not typically occupy 100% of the internal space of a container. The RBE3 interpolation element, which is Nastran rigid body element (RBE), was used to connect the mass elements to the hull. The RBE3 element consists of a master node, slave nodes, and the degrees of freedom and weights assigned to each slave node. The load acting on the master node is distributed to the slave nodes according to their respective degrees of freedom and weights. Therefore, when force is applied by installing the mass elements that represent the container load at the master node and applying acceleration, the slave nodes connected to the hull can properly distribute the container load from the master node to the hull based on the predefined degrees of freedom and weights. For the containers placed on the hatch cover of the upper deck, the RBE3 element was modeled such that 15% of the weight is distributed in the longitudinal direction of the hatch coaming and 85% in the transverse direction, as shown in Fig. 5 (LR, 2021). A software program was developed to identify container positions and automatically generate mass and interpolation elements by inputting the information from the full ship FE model and loading manual, as shown in Fig. 6.
The mass elements, excluding the containers, were used for the hull weight. The length between perpendiculars (LBP) was equally divided into 20 s, creating a total of 22 blocks, including the bow and stern parts outside the LBP. The ship’s mass was then distributed across these blocks. The center of gravity (COG) for each block was determined in the longitudinal, transverse, and vertical directions, and the overall COG in the full ship FE model was obtained by distributing the mass elements to match the COG of each block. A mass generation software program was developed to perform this iterative task. Table 3 shows the actual weight and COG of the container ship, including containers, as well as the weight and COG generated in the full ship FE model. The results demonstrate consistency.
3.2 Fatigue Damage Calculation by Classification Societies
LR (2022) proposed that the springing reduction factor, SRF, be calculated for hot-spot stress, as shown in Eq. (1), and that hydroelastic fatigue life be obtained by multiplying the fatigue life obtained from full spectral fatigue analysis by the springing reduction factor.
As shown in Fig. 7, Dr-HydroE represents the fatigue damage obtained using the hydroelastic stress RAO below the cut-off frequency, and Dt-HydroE represents the fatigue damage obtained using the hydroelastic stress RAO. The cut-off frequency, ωcut-off, is calculated using Eq. (2).
ξ is given by Eq. (3).
BV (2015) presented Eqs. (4) and (5) using the hydrodynamic coefficients obtained from hydroelastic motion analysis. These are expressed as follows:
The quasi-static response, [ζ], from the total hydroelastic response is defined in Eq. (6). Based on this, Eqs. (7) and (8) can be derived as follows:
Eq. (9) can be derived from Eqs. (4) to (8), allowing for the extraction of the dynamic elastic modes response,
[ ζ E D ]
The final hydroelastic stress RAO can be obtained by adding the dynamic elastic mode stress RAO, which multiplies the hot-spot stress of each vibration mode by the dynamic elastic modes response
[ ζ E D ]
Table 4 summarizes the hydroelastic motion analysis conditions presented by LR and BV, along with the values used in this study.
Here, the structural damping of LR, ηi, is given by Eq. (10).
4. Hydroelastic Fatigue Analysis
4.1 Analysis Procedure
Fig. 9 illustrates the hydroelastic fatigue analysis procedure used in this study. Normal mode analysis is performed on the full ship FE model, which includes a fine mesh model that accounts for the weight of the containers and the hull. Hot-spot stress and modal displacement are obtained for each mode. The side shell model is extracted from the FE model, and the hydro panel model for hydroelastic motion analysis is generated. The deformation of the hydro panel caused by hydroelastic behavior is determined by mapping the modal displacement of the side shell model onto the hydro panel. Hydroelastic motion analysis is conducted using HydroE-FD, from which mode RAO and hydrodynamic coefficients are extracted. Mode RAO is then multiplied by the hot-spot stress for each mode to obtain the mode stress RAO, and the cut-off frequency and fatigue damage increase ratio are derived following the LR procedure. Dynamic elastic modes response is determined using the hydrodynamic coefficients extracted according to the BV procedure, and the dynamic elastic mode stress RAO is calculated by multiplying the hot-spot stress for each mode. Quasi-static stress RAO is obtained through full spectral fatigue analysis. Specifically, the hot-spot stress for each unit load is determined by applying the unit load to each side shell element of the full ship FE model and performing linear structural analysis. The rigid body pressure RAO of the hydro panel, obtained using HydroE-FD, is mapped onto the side shell model to acquire the rigid body pressure RAO. Quasi-static stress RAO is then obtained by multiplying the hot-spot stress for each unit load by the rigid body pressure RAO. Finally, the fatigue damage increase ratio is calculated using the dynamic elastic mode stress RAO and the quasi-static stress RAO.
Before conducting the hydroelastic fatigue analysis, a full spectral fatigue analysis was performed for the AFT, ER, MID, and FWD regions of the container ship to identify the hot spot elements in each region. The procedure for the full spectral fatigue analysis is illustrated in Fig. 10. The target fatigue life was set at 20 years. Since the full ship FE model consists of mass elements, linear structural analysis was carried out using the inertia relief method to achieve force balance. The worldwide sea state was applied (Det Norske Veritas, 2021c), and a short-crest wave with a was used. For the S-N curve of the hatch coaming top corners, the B-curve presented by Det Norske Veritas (2021b) was utilized. Typically, the hydroelastic response spectrum forms a broadband spectrum. Therefore, LR (2022) proposed the Park model (Park et al., 2014) as a method for calculating fatigue damage in broadband spectra, and this model was applied in the current study.
4.2 Normal Mode and Hydroelastic Motion Analyses
A total of 60 modes were calculated in the normal mode analysis, with 16 global vibration modes below 3 Hz identified. Table 5 shows the natural frequency and symmetry for each mode. The size of the hydro panel used in the hydroelastic motion analysis ranged from 0.8 to 0.95 m. The displacement of the hydro panel for each mode, required for conducting the hydroelastic motion analysis, was mapped from the full ship FE model, as shown in Fig. 11.
The rigid body pressure RAO of the hydro panel, obtained from the hydroelastic motion analysis, was mapped onto each side shell element of the full ship FE model to calculate the rigid body pressure RAO. Fig. 12 shows the results of pressure mapping in both the low-frequency and high-frequency regions that maximize the vertical bending moment RAO at the midship, with the ship traveling at a speed of 10.803 m/s and a heading angle of 135°. Two pressure mapping methods were developed. The first method involves identifying the hydro panel whose center is closest to the center of the side shell element and assigning the pressure RAO of this hydro panel to the corresponding side shell element. The second method identifies four hydro panels closest to the center of the side shell element and assigns weights to the pressure RAO values in inverse proportion to the distance of each hydro panel, thus calculating the pressure RAO for the side shell element. In this study, the first method was used.
5. Hydroelastic Fatigue Analysis Results and Analysis
Fig. 13 shows the cut-off frequency results by heading angle for the LR procedure. It is evident that most cut-off frequencies are determined by mode 8 (the vertical bending mode), while mode 7 (the torsional mode) determines the cut-off frequency in the beam sea direction. No occurrence of ω1 the stress RAO was observed around the beam sea.
Fig. 14 compares the stress RAO results for the AFT region obtained by each classification society. The LR method calculated stress RAO using the sum of vibration modes, while the BV method combined quasi-static and dynamic stress RAO for a wave amplitude of 1 m. The responses of LR and BV differ in the low-frequency region. This discrepancy appears to result from the modes below 3 Hz being insufficient to represent the behavior of the AFT region in the low-frequency region, as the distance from the deck house to the stern is shorter than the cargo area. Therefore, when modes above 3 Hz are included, the LR response is expected to align more closely with that of BV in the low-frequency region.
Figs. 15–17 compare the stress RAO results of each classification society for the ER, MID, and FWD regions. The LR method derived stress RAO from normal mode and hydroelastic motion analyses below the waterline, while the BV method combined quasi-static and dynamic elastic mode stress RAO for a wave amplitude of 1 m using full spectral fatigue analysis. Despite the differences in methodology, the results show that sufficiently similar stress RAO values can be obtained at the hatch coaming top, even when using vibration mode analysis below 3 Hz.
Table 6 and Fig. 18 summarize the fatigue damage increase ratio according to each classification society. The fatigue damage increase ratio was highest at MID for the LR method and at ER for the BV method. The contribution of vibration to fatigue damage was generally consistent with full-scale measurement research results, as shown in Table 7 and Fig. 19, with the highest values observed at MID for the LR method and at ER for the BV method. Storhaug et al. (2010) found that the contribution of vibration to fatigue damage was high at AFT, but the results of this study displayed different tendencies.
In this study, the number of modes was selected based on the recommendations provided by the classification societies. However, it appears necessary to include additional modes above 3 Hz when calculating the fatigue damage increase ratio for the AFT region in terms of the LR method. For the BV method, the dynamic stress RAO in a specific amount exists in the low-frequency region, far from the resonance frequency between the following sea and beam sea, as shown in Fig. 20. This response seems unreasonable and requires further investigation.
6. Conclusion
In this study, hydroelastic fatigue analysis was performed according to the procedures outlined by LR and BV. The weight distribution of the containers and the hull was matched by applying mass elements and a RBE in the full ship FE model. Hot-spot stress and modal displacement for each mode were obtained through normal mode analysis. The modal displacement was then mapped to the hydro panel for hydroelastic motion analysis, utilizing the frequency-domain-based HydroE-FD method. The fatigue damage increase ratio was calculated using the mode RAO and hydrodynamic coefficients derived from HydroE-FD, the hot-spot stress from normal mode analysis, and the quasi-static stress RAO obtained from full spectral fatigue analysis, following the procedures of each classification society.
Hydroelastic fatigue analysis was conducted in the AFT, ER, MID, and FWD regions at the hatch coaming top corners. The results showed that the fatigue damage increase ratio was consistent with full-scale measurement research. The highest fatigue damage increase ratio was observed at MID for the LR method and at ER for the BV method.
The LR and BV methods produced similar stress RAO values in the ER, MID, and FWD regions. However, for the LR method, the stress RAO differed significantly from the BV results in the AFT region, due to an insufficient number of vibration modes being included. It is necessary to incorporate modes above 3 Hz for accurate hydroelastic fatigue damage calculations in the AFT region. For the BV method, the dynamic stress RAO exhibited an unreasonable behavior in the low-frequency range between the following sea and beam sea. Further research is required to minimize such responses.
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