A broad CFD-based study of the non-linear effects on wave-induced bending moments and shear forces on monohull ships

1 A broad CFD-based study of the non-linear effects on wave-induced bending moments and shear forces on monohull ships Tormod Landet1, Marine Lasbleis2, Kei Sugimoto3, Guillaume de Hauteclocque2, Min Xu4, and Håvard Austefjord1 1 IACS – DNV, Oslo, Norway 2 IACS – Bureau Veritas, Paris, France 3 IACS – Nippon Kaiji Kyokai (ClassNK), Japan 4 IACS – China Class Society, Shanghai, China Abstract. The International Association of Classification Societies (IACS) is about to update its design requirements for steel ships longer than 90 m, namely Unified Requirement (UR) S11 and S11A, applying to the majority of the worlds shipping. As a part of this work an extensive CFD-based study of wave loads on 45 monohull ships of a variety of ship types (oil/ore/gas/container carriers, Ro-Ro, Cruise) has been conducted and is presented in this paper. The goal of the study is to provide the technical background for updating the rules for the non-linear effects on the extreme global wave-induced hull-girder loads, in particular the midship vertical wave-induced bending moment (VBM) and the quarter-length wave-induced vertical shear forces (VSF) since these are among the most important input parameters for structural design of large sea-going ships. We start by showing that by use of modern CFD-based fully non-linear methods, multiple participants in the project are able to show good agreement between simulations and model tests of a rigid container ship in irregular waves. We then present a large database of ships of various relevant ship types and show the results of extensive CFD calculations to find trends in how the non-linear factors change with the main parameters of the ships in the database. Regression fits based on ship main parameters are suggested for predicting non-linear wave loads. Keywords: Wave Loads; CFD; Non-linear factor; VBM; VSF; IACS; CSR; UR S11; UR S11A. 1. Introduction The International Association of Classification Societies (IACS) has established a project team to update the hydrodynamic wave loads applied by the member societies in rule checks for class approval. Among these prescriptive rules are the requirements to longitudinal strength for all steel ships longer than 90m. We will here present a summary of the work performed by this project team which will result in an update to IACS Unified Requirements (UR) S11 and S11A. The previous update to the unified requirements for longitudinal strength was performed in 2015 and was limited to container ships only [1]. That work, which lead to UR S11A, was based on a database of non-linear hydrodynamic calculations made available by the participating class societies. At that time the members made use of semi-non-linear potential-flow-based simulation tools [2]. As noted in the presentations from that project team there were quite large discrepancies between the results from the various tools and statistical methodologies employed. The resulting UR S11A formulas are used today in IACS for container vessels and the older UR S11 formulas are used for all other ship types. The IACS rule formulas for motions, accelerations, and wave loads such as bending moments, shear forces and wave-frequent external pressures are all formulated as products, 𝑋 = 𝐶𝑊,𝑋 𝑋𝑙𝑖𝑛 𝑓𝑛𝑙−𝑋, where 𝐶𝑊,𝑋 is the wave parameter, 𝑋𝑙𝑖𝑛 is the dimensional linear formulation, and 𝑓𝑛𝑙−𝑋 is the non-linear amplification or reduction factor. For motions and accelerations, the non-linear factors are all 𝑓𝑛𝑙−𝑋 = 1.0, while for some global loads and pressures it may be higher or lower than unity depending on the load mechanism. The development of the new wave parameters and the linear formulations are documented separately [3], we will here present only the work performed to update the formulations of the non-linear factors for the wave-induced vertical bending moment, VBM, vertical shear force, VSF, and local external wave-frequent pressures under the equivalent design waves maximizing moment. To avoid the large spread in results found in the development of UR S11A, it was decided to establish new and higher standards for the applied numerical and statistical methods, made possible by both the increase in available computational power, and the increased attention to and development of fully non-linear methods for sea-keeping simulations. All participants were now required to use fully non-linear CFD-based methods where the two-phase 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 Navier-Stokes equations are solved with physically correct handling of dry and wet areas with associated external pressures. The International Ship and Offshore Structures Congress, ISSC 2021 Load Committee have created a benchmark including experimental model tests of a 6750 TEU container vessel [4]. The experimental study evaluates bending moments in selected extreme sea states with a very rigid ship model, allowing the extraction of the rigid body bending moment without the effect of Whipping. Container vessels are known to have large nonlinear contribution to hull girder loads. It is assumed that if participants in the current study can reproduce the ISSC model test by combining CFD with appropriate statistical methods, then the same CFD software, mesh, timestep, and statistical procedure may be used to compute non-linear factors for the vessels covered by UR S11 and S11A. Details on the statistical and numerical methods applied are shown in section 2. The results of the validation done for the ISSC experimental benchmark container carrier are shown in section 3. Figure 1. The global hull-girder vertical bending moment, VBM, and shear force, VSF, with positive sign directions indicated. A large number of existing ocean-going ships were selected to form the basis for the new CFD-based study. IACS Common Structural Rules (CSR) cover Oil tankers and Bulk carriers and the CSR has inherited the VBM and VSF requirements from UR S11. These ship types are well represented in our database of non-linear calculations along with the Container carriers (UR S11A) and the ship types covered by UR S11 that are assumed to have significant non-linear effects or a tendency to be dimensioned by global bending. The included ship types outside of Oil tankers, Bulk carriers, and Container vessels, are Ro-Ro carriers, Passenger/Cruise vessels and Gas carriers (LNG/LPG). The vessel database and the resulting non-linear factors for vertical bending moments from CFD are shown in section 4 along with the resulting regression formulas. Shear forces associated with the maximum moments are described in section 5, and the external sea pressure associated with the extreme moments are shown in section 6. The wave-induced non-linear sea pressure factors are a part of CSR and not UR S11/S11A. 2. Numerical and statistical methods One of the participating class societies is using the commercial CFD solver Simcenter STAR-CCM+ from Siemens with the HRIC method [5] for wave-interface capturing. The two others are using in-house versions of the open-source interFoam solver from OpenFOAM with the MULES method for interface capturing [6]. The incoming waves are enforced by Dirichlet boundary conditions and by forcing-zone methods to dampen and stop any disturbances to the incoming wave field created by the vessel from getting close to the boundaries and creating unwanted reflections. Such forcing zones can significantly reduce the computational domain, see e,g. [7] for an introduction. The incoming waves are represented by irregular long-crested waves described either by linear wave kinematics, or by use of a higher-order spectral representation of the wave field [8] [9] (HOSM). Two of the participants are performing the statistical assessment using Response Conditioned Waves (RCW, see e.g. [10] [11]) and one is using a full Monte-Carlo approach with long simulations in Random Irregular Waves, RIW. The Response Conditioned Wave method, in brief, involves combining the linear Response Amplitude Operator, RAO, and the sea-state wave spectrum to compute the linear response spectrum and then compute the wave event that is the most-likely irregular-wave time history to produce a target linear response level given the wave-frequency distribution of the response spectrum [10]. In this case the target response is the 25-year linear long-term extreme from a 3D potential theory panel method. One participant is extending the standard RCW theory by computing the most-likely non-linear HOSM wave profile by use of a linearized response model based on the RAO combined with the FFT of the wave elevation time series in a non-linear optimization loop, see [8] for details. VBM (Hogging positive) VSF Aft Fore 3 The Monte-Carlo RIW approach is simply to simulate sufficiently long time series of the vessel subjected to irregular non-linear HOSM waves in CFD. As a rough rule of thumb, when the expected return period of the target response in the sea state is simulated twenty times, then the statistics for the corresponding response level can be considered converged. This was done by adding random seed simulations of 20 minutes each until the 20-timesreturn-period criterion was met, most often ending up at about nine hours of CFD simulations per vessel and loading condition. In some cases, extrapolation of the tail of the resulting empirical probability distribution using a Weibull assumption was needed towards the bow and stern of the vessel where the sea state selected as worst for the midship bending moment may not be the optimal choice. The Weibull-assumption was not needed in the midship region where the maximum non-linear VBM is found, and hence this is only relevant for the distribution of VBM along the vessel, not for the overall non-linear factor which is found from the maximums of the linear and non-linear distributions. For both the selected statistical methods, RCW and RIW, the first step is to select one or more sea states where the non-linear response should be estimated. The extreme load is expected to occur in a sea state close to the steepness limit for rigid-body VBM at five knots. It is assumed that the 𝑓𝑛𝑙 changes with the target load level, but not very much with the sea state parameters 𝐻𝑆 and 𝑇𝑍 within the group of sea states that are likely to cause the linear maximum load. In this work the following procedure is used: the linear 25-year long-term CoC (Coefficient of Contribution, see e.g. [12]) is computed for each sea state. For each considered wave direction the sea state period 𝑇𝑍 with the maximum sum of CoC in that heading is used as the basis for the calculations, and the wave height 𝐻𝑆 is taken as the maximum in the scatter diagram for this 𝑇𝑍, which results in a very steep sea state where the 25-year extreme is likely to occur relatively frequently. The selected sea state is a good representative for the sea states likely to give the extreme response since the non-linear factor can be assumed to be independent of the significant wave height, though it is strongly dependent on the target linear response level. I.e., the 25-year extreme VBM will occur more often in our simulations by selecting the maximum realistic 𝐻𝑆, but the resulting non-linear factor should be the same as what would be found for a lower 𝐻𝑆 for the same 𝑇𝑍, see e.g. [13] and [14]. Note that the incident wave physics would typically not include wave breaking in the legacy semi-non-linear potential theory programs, while the 𝐻𝑆 in the CFD simulations should not be increased much beyond the steepness limit as the waves are here much more accurately modelled. If most of the steep waves end up breaking in the CFD wave simulation then the actual waves encountered by the vessels will be flatter and longer than the intention, and the increased-𝐻𝑆 method will not give the desired accuracy. We rely on the fact that increasing the 𝐻𝑆 is just a scaling of the response spectral amplitudes, while wave breaking will also cause a shift in the overall shape of the response spectrum since the wave spectrum energy content changes. Figure 2. Illustration of the wave scatter diagram with sea-state probabilities (blue dots), the linear 25-year VBM CoC (yellow/red), an empirical sea-state steepness limit (black), and the short-term linear iso-response curves (green). The blue column has the maximum integrated Coefficient of Contribution in head seas for this 280m Bulk carrier in full load. Selected sea state 4 The procedure applied to find the sea-state to simulate in CFD is illustrated in Figure 2. The VBM RAO is taken from a 280m long Bulk carrier in the homogenous fully loaded condition. The scatter diagram applied is the North-Atlantic from IACS Recommendation 34 (Rev. 2), but it should be noted that the CoC distribution and choice of the 𝑇𝑍 to apply in the calculation is not very sensitive to the exact details of the scatter diagram. The selected sea state is always picked to be near the steepness limit, so the energy content in another scatter diagram (in terms of CoC as a function of 𝑇𝑍) must be very different for the sea state that is picked to be very different from the one we have selected here. The linear target VBM response is on the other hand quite important, as the nonlinear factor changes with the target response level in the sea state, so a milder scatter diagram causes non-linear factors that are closer to unity, while a harsher scatter diagram (higher 25-year linear extreme) will cause the opposite. The applied statistical and CFD methods used by the participating class societies in the IACS working group for wave loads are summarized in Table 1. The CFD solvers are run without any turbulence modelling. This simplification gives equivalent results in the benchmark to using either an Euler solver or a full URANS solver with turbulence modelling, confirming the negligible influence of viscosity on the vertical global loads. All simulations are performed at full scale. Table 1. Summary of numerical and statistical methods employed to compute non-linear factors for VBM and VSF and typical mesh size. For two of the participating class societies, one sea state requires, at minimum, two CFD simulations of Response Conditioned Waves (RCW), one for sagging and one for hogging VBM at the linear 25-year long-term response level. The last participant uses a Monte Carlo approach with Random Irregular Waves (RIW) with a duration of 20 times the linear 25-year long-term response return period in the selected sea state. The listed duration is per sea state (and vessel and loading condition). CLASS SOCIETY CFD SOLVER WAVE KINEMATICS STATISTICAL METHOD MESH SIZE DURATION A OpenFOAM HOSM RCW 1.2 M cells ~ 2 x 60 sec. B OpenFOAM HOSM RIW 0.3 M cells ~ 9 hours C Star CCM+ Linear RCW 1.2 M cells ~ 2 x 60 sec. 3. Benchmarking of the CFD and statistical methods Due to the large discrepancies found between different numerical and statistical methods in the work leading up to the publication of IACS UR S11A [2], the current rule-update CFD project started with a qualification step to ensure the methods employed by the participating class societies could accurately predict the results from an existing model test of rigid-body VBM. The ISSC loads committee benchmark of global loads on a 6750 TEU container vessel [4] was selected. Model tests were performed at Ecole Centrale Nante, ECN, both for regular [15] and irregular [14] waves. The participants CFD codes were first benchmarked by use of the regular wave experiments. Here we focus on the irregular sea-state loads prediction, as this is what will be done for a large set of vessels later to establish the rules-background database. Figure 3 shows the results provided by the participating class societies in an irregular sea state with 𝐻𝑆=10𝑚 and 𝑇𝑃=14𝑠. ECN has performed a ~30-hour realization in the model basin in this sea state and provided time-series of the resulting midship vertical bending moment for 10 seeds (about 25hours of usable data). Each blue dot in the figure represents one positive or negative moment extrema from the model test. Class society B has replicated the model test by running 30-hours coarse-mesh CFD in the same sea state, but with their own HOSM waves that are not replicating the waves measured in the model basin. Class societies A and C have used irregular conditioned extreme waves which can be seen at some selected probabilities of exceedance. 5 Figure 3. Wave-induced dynamic VBM vs the probability of exceedance, Q, for the ISSC loads benchmark in irregular waves. The 20 most extreme points are semi-transparent as the empirical value of Q will be less accurate for these. The discrepancies in the calculation of VBM using each class-society’s linear potential flow tool can be seen to be of the same magnitude as the discrepancies between the CFD results. It hence seems, from these results, that a properly executed CFD analysis should not introduce significantly more errors than a linear potential flow seakeeping simulation. It should be noted, however, that this is probably not true for an inexperienced user whose chances of obtaining erroneous results is still much higher with a complex non-linear CFD solver. The discretized mesh applied by society B is optimized to capture the largest waves causing the largest moments. It was confirmed separately that the CFD sagging moment in medium-height waves, where there are some differences between the orange and blue points in Figure 3, does approach the model test if the mesh density is increased. Since the objective of the study is to look at the extremes it was decided that the mesh density was adequate. As a point of comparison with the study performed in 2015, class society B has provided results using a quasinon-linear potential flow solver of the same type as the ones used in [2]. The only main difference is that HOSM waves have been applied at this time instead of linear irregular waves to enable direct comparison with CFD. As can be seen this type of method tends to overpredict the VBM. This is a known issue across many such tools, and we can hence expect the results from the CFD-based study documented below to be closer to reality both for the container vessels currently using the UR S11A, and probably also for the other ship types using UR S11, an older ruleset which is not based on a database of direct load calculations. 4. Vertical wave-induced bending moment The study initially contained 103 vessels in homogenous fully loaded and ballast conditions. Passenger and Cruise vessels are only included in the fully loaded condition since the ballast and fully loaded conditions are typically very close in terms of displacement and trim for such vessels. The other vessel types are included with two loading conditions meant to span the full range of possible operational draughts. The initial CFD simulations for all vessels were performed in head sea waves for sea states near the breaking limit. The sea states were selected as explained in section 2. From the resulting time series and videos it was found that several of the vessels got significant amounts of green water on the foredeck during the maximum sagging wave encounters. The wave-induced sagging moment is in part due to the pressure from a wave crest at the bow, pushing the bow upwards. The pressure from the green water on the foredeck is hence counteracting the wave crest pressure and reducing the sagging moment. One alternative for how to deal with this problem is to continue the bulwark geometry vertically until there is no green water observed (or lift the whole foredeck or breakwater depending on the ship type and design). For some vessels this was found to work without resulting in designs that are obviously different from real ships in operation [16], but for others the vessel drawings indicated that the freeboard in the CFD models was reasonable and that the waves would realistically break over the bow in the 25year extreme wave encounters. As seen in Figure 2 the encountered waves in the worst storms in the North-Atlantic Ocean can be very large, especially considering that the maximum wave height encountered can be roughly two times the significant wave height, HS. 6 It should be noted that the RIW simulations also showed that the most extreme sagging moments had water on the foredeck. It is hence not the case that slightly lower waves would be more critical since no green water would be present for those. In the long irregular wave simulations those slightly lower waves are present in the encountered irregular seas, and they were not giving VBM exceeding the wave encounters that gave water on deck so we can conclude they were not more critical in terms of vertical wave-induced bending moments. After running some initial tests with following sea waves in CFD it was decided that all vessels included in our CFD database, created explicitly for the purpose of fitting new rule formulas, should have the non-linear bending moment calculated also in following sea waves by the exact same procedure as was used for head sea waves. For many of the included vessels the linear long-term statistics indicate a very slight reduction in VBM when switching from head to following sea waves, the marginal CoC in following sea waves is just below the marginal CoC for head sea waves (for some vessels it is even slightly higher). This is due to the low forward speed used as rule basis in extreme conditions, a maximum of 5 knots is assumed in the most extreme sea states in the North-Atlantic [17]. Our final database hence consists of 45 vessels where head and following sea results are available. This is unfortunately smaller than the ~100 vessels we aimed for but is still a sizable database covering a large range of relevant vessel designs and sizes. A very rough estimate is that 40 000 CPU-days across three different modern high-performance compute clusters were used to generate the presented results. The main parameters of the vessels in the CFD database are listed in Table 2 and shown in Figure 4. Table 2 Main parameters of the vessels included in the CFD database with results for headand following-sea waves. Cargo Type Count Length [m] Breadth [m] CB [-] CWP [-] Container 11 120 – 380 20 – 51 0.54 – 0.72 0.66 – 0.91 Bulk 8 140 – 280 24 – 45 0.74 – 0.88 0.81 – 0.97 Gas 8 110 – 270 19 – 49 0.65 – 0.78 0.74 – 0.89 Oil 8 160 – 260 24 – 49 0.69 – 0.86 0.78 – 0.94 Passenger 6 120 – 330 22 – 47 0.59 – 0.72 0.82 – 0.93 Ro–Ro 4 160 – 250 28 – 32 0.57 – 0.69 0.78 – 0.92 Total 45 110 – 380 19 – 51 0.54 – 0.88 0.66 – 0.97 Figure 4. Main parameters of the vessels included in the CFD database with results for headand following-sea waves. Ship dimensions: L=length [m], T=draught [m], B=breadth [m], CB=block coefficient [-], CWP=water-plane coefficient [-] The resulting non-linear factors for VBM in the IACS Head Sea Moment (HSM) wave conditions can be seen in Figure 5, and the following-sea wave results (FSM) can be seen in Figure 6. The formulas used for the x-axis are explained further below. The non-linear factor is computed as the maximum VBM from CFD along the length of the vessel divided by the maximum VBM from linear potential flow along the length of the vessel, see Equation 1. These longitudinal VBM distributions are extracted at the same 25-year-extreme long-term probability level. It is important to note that the CFD and linear moment distributions may not have their maximums at the same longitudinal location, so the reported non-linear factors for VBM are global parameters, independent of location. 7 𝒇𝒏𝒍,𝑽𝑩𝑴 =𝐦𝐚𝐱 𝒙 ∈ [𝟎,𝑳] 𝑽𝑩𝑴𝐂𝐅𝐃(𝒙) 𝐦𝐚𝐱 𝒙 ∈ [𝟎,𝑳] 𝑽𝑩𝑴𝐥𝐢𝐧.(𝒙) (1) Figure 5. The non-linear factors for VBM from the head-sea wave simulations. The 𝑓𝑛𝑙 from CFD is computed according to equation 1. Figure 6. The non-linear factors for VBM from the following-sea wave simulations. The 𝑓𝑛𝑙 from CFD is computed according to equation 1. A comparison between the resulting non-linear factors from the HSM and FSM simulations can be seen in Figure 7. In general, the following-sea wave simulations gives higher sagging moments for the fully loaded condition, while it is the head-sea waves that give the highest sagging moments in ballast. This seems reasonable since green water should be much less likely for vessels in ballast condition, when they typically have a quite high freeboard at the bow. For hogging we see that the occurrence of the most extreme non-linearities (𝑓𝑛𝑙 below 0.7) are reduced when following-sea waves are included. For vessels that are approximately equally likely to see the highest vertical wave hogging moment in head or following seas at low forward speed (based on linear theory), it may hence be advisable to include following-sea wave simulations when running CFD since the non-linear factor may then be conservative compared to the non-linear factor in head-sea waves. Please note that when including fluid-structure interaction effects such as Whipping it is not expected that the same is true since the vertical whipping bending moment is typically highest at higher forward speed in more benign sea states when significant bow slamming occurs. Please also note that the effect of whipping and springing are not included in the IACS 𝑓𝑛𝑙 factor, it covers only rigid-body non-linearities. Hull flexibility is not a part of the CFD-based rule update study documented herein. 8 Figure 7. Comparison of resulting 𝑓𝑛𝑙 for VBM from head and following sea waves. The 𝑓𝑛𝑙 from CFD is computed according to equation 1. 4.1. Result envelopes for VBM and regression formulas The maximum non-linear factor found between the head-sea and following-sea wave simulations is used to define an envelope non-linear factor which is to be applied to the bending moment in both the HSM and FSM equivalent design waves. The maximum of the two is shown in Figure 8. Here the statistical method is highlighted instead of ship type, and we remind the reader that the results from the RCW and RIW methods are computed separately by non-collaborating CFD experts each in their separate class societies. The vessel geometry and mass models are picked from that class society’s database of real vessel designs. It is hence a good quality assurance to see that the results fall roughly in line. Figure 8. The non-linear factors for VBM, taking the maximum from head-sea and following-sea waves. The RMSRE regression fitting errors are 5.6% (hogging) and 7.4% (sagging). The 𝑓𝑛𝑙 from CFD is computed according to equation 1. Our suggested formulation for the non-linear factors for VBM in head and following sea waves have been used as the x-axis values in the above figures. For the vertical wave-induced hogging moment the proposed non-linear factor found from a regression fit is 𝒇𝒏𝒍−𝒗𝒉 = 𝟏.𝟎𝟔(𝑩𝑻 𝑳)𝟎.𝟏𝟓 𝑪𝑩 𝟎.𝟔, (2) 9 and the proposed non-linear factor regression fit for vertical wave-induced sagging moment is 𝒇𝒏𝒍−𝒗𝒔 = 𝟎.𝟗𝟖 (𝑪𝑩𝑪𝑾𝑷)−𝟎.𝟔. (3) The parameters T (draught amidships), 𝐶𝐵 (block coefficient), and 𝐶𝑊𝑃 (water-plane coefficient) are taken from the actual loading conditions in the above plots (where these formulas are used on the x-axis) and in the regression fitting process used to find the proposed prescriptive rule formulas. The IACS rule formulas may in some cases use the values from scantling draught instead when computing the non-linear factor to apply. The ship length is L, and breadth is B. The length parameters (L, B, T) are to be taken in meters, and the other coefficients are dimensionless. 5. Vertical wave-induced shear force The wave-induced hull-girder Vertical Shear Force, VSF, has been extracted from the same CFD wave simulations as were used for the VBM. The vertical shear forces in head and following seas are assumed to occur simultaneously with the maximum vertical bending moments in the Equivalent Design Wave approach used by the current IACS rules. We do not have reasons to modify this, so the Head and Following Sea Moment design waves, HSM and FSM, will be used also to produce the non-linear factor for VSF. This factor must be found both aft and forward of midship due to the two-peaked VSF distribution along the hull. For each of the vessel halves, the shear force during both the extreme wave-induced sagging and hogging bending moments need to be established, leading to four formulations for the non-linear factor in VSF. This is equivalent to the current IACS procedure. Also equivalent is our suggested approach of using the non-linear factor for VBM to estimate the nonlinear factors for VSF. The results from the CFD analyses are shown in Figure 9. The x-axes in the sub-figures show the calculated rule value from the suggested regression fits, which are shown in the axis labels. The y-axes show the result from the CFD calculations. Just like for VBM, the CFD non-linear factors are calculated from the extremum of the nonlinear distributions on each half of the vessel divided by the extremum of the linear distributions on the same halves of the vessel. The non-linear and linear distributions may not have their extremum at the exact same longitudinal position.