Update of the Rule Load Formulations in the IACS Common Structural Rules

1 Update of the Rule Load Formulations in the IACS Common Structural Rules Quentin Derbanne1, * , Håvard Austefjord2, Kei Sugimoto3 and Xu Min4 1 Bureau Veritas, Nantes, France 2 DNV, Oslo, Norway 3 Nippon Kaiji Kyokai (ClassNK), Tokyo, Japan 4 China Classification Society, Shanghai, China Abstract. Following the update of the Recommendation n°34, which describes the sea state statistics encountered by ships operating in the North Atlantic, the International Association of Classification Societies has undertaken a comprehensive revision of the load formulations in the Common Structural Rules for Oil Tankers and Bulk Carriers (CSR). The objective was to achieve a good consistency between the rule loads and the loads estimated using a direct approach from the scatter diagram. A large database of more than 250 vessels, covering major ship types (oil tankers, bulk carriers, containerships, gas carriers, ore carriers, cruise ships…), in both full and ballast loading conditions, was built. For each vessel a long-term statistical approach was used to compute the extreme loads, corresponding to a 25-year return period, as well as the fatigue loads from the 3D Boundary Elements Method transfer functions. Approximately 200 different loads quantities (motions, accelerations, hull girder loads and pressure along different sections) have been estimated for every vessel across 12 Equivalent Design Waves (EDW). This extensive database served as the reference results to calibrate the rule formulations for loads envelopes and for pressures and Load Combination Factors across all EDWs. The difference between the rule formulations, which only require a few key ship parameters, and the reference computations were minimized and thoroughly documented. This has resulted in a significantly more consistent set of rule loads that will be used in the future revision of the CSR. Keywords: Common Structural Rules, Equivalent Design Waves, Hydrodynamic loads 1. Introduction In 2006, the International Association of Classification Societies (IACS) issued two different set of rules, for Oil Tankers [1] and for Bulk Carriers [2]. A few years later, these two sets of rules were harmonized and merged into a single set, the Common Structural Rules for Bulk Carriers and Oil Tankers (CSR) [3]. This set of rules was submitted to an audit by the International Maritime Organization (IMO) to check the compliance with the Goal Based Standard (GBS) [4]. According to the GBS, “ships shall be designed in accordance with North Atlantic environmental conditions and relevant long-term sea state scatter diagrams”. IACS explained the North Atlantic environmental conditions are described by a scatter diagram in the Recommendation n°34 [5], and that this recommendation has been used in the rule development. In 2016, the GBS audit report within MSC 96/5 includes the observation IACS/2015/FR1-8/OB/02, saying that “IACS' Rec. n°34 that is based on old wave statistics was last revised in 2000/2001 and there is no evidence of monitoring since its adoption”. The audit team concluded that they had “not found sufficient justification that the wave data used in the rules properly represent North Atlantic conditions.” Following this observation IACS has put in place a dedicated project team to update the scatter diagram. The work was done between 2017 and 2022. By combining modern hindcast data with ship AIS positions, a new scatter diagram has been issued [6]. More details can be found in [7]. Starting from this updated scatter diagram, another project team worked from 2022 to 2025 to update all the load formulations from the CSR. The present paper shows the methodology used to derive new load formulations from direct analysis results using the updated scatter diagram. Only the linear loads formulations are shown in this paper. Further non-linear corrections based on CFD calculations, for vertical bending moment and shear force, are presented in [8]. * Correspondence to: [email protected] 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 2. Rule update principles The updates follow the main principles of current CSR. For extreme loads and fatigue loads, the rules give envelope values for motions, accelerations and hull girder loads. The new envelope formulations are detailed in 3. The Equivalent Design Wave (EDW) concept with regular design waves and load combination factors (LCF) remain. For each EDW, the instantaneous values of motions, accelerations and hull girder loads are given by a LCF multiplied with the envelope value (see 4), while the wave pressure is directly given by rule formulations (see 5). All these rule formulations are calibrated versus direct computations. 2.1. Evaluated fleet 2.1.1. Type of vessels A very large hydrodynamic database of more than 250 vessels has been used to compute extreme and fatigue loads, and to fit the rule formulations. In principle the update of the CSR requires computing all types of loads (motions, accelerations, hull girder loads, pressure) for only bulk carriers and oil tankers. It was however decided to have a broader database, including more ship types. It has been checked that including non-CSR vessels in the regressions of rule formulations do not significantly impact the accuracy for CSR vessels. On the opposite, it prevents us from doing over-fitting (as some parameters, such as 𝐶𝐵 and 𝐶𝑊, are very correlated for CSR vessels), and it brings more robustness for the application of the rules to vessels with abnormal shapes. Table 1 shows the number of vessels in the database, per ship type and loading condition. Table 1. Number of vessels by ship type in the database Ship type Full load Ballast CSR vessels Bulk carrier 47 43 Oil tanker 48 43 Non-CSR vessels Containership 52 41 Cargo carrier 30 30 Gas carrier 30 20 Ro-ro ship 18 13 Passenger ship 11 6 Offshore vessel 21 21 Other 7 8 Total 264 225 In the following graphs and tables, the vessels will be split into four groups: CSR vessels in full or in ballast loading condition, non-CSR vessels in full or ballast loading condition. 2.1.2. Main characteristics Figure 1. Main vessels characteristics. 3 The different vessels of the database have been selected to cover a wide range of dimensions and shapes. A single vessel (and loading condition) is characterized by the following quantities: rule length 𝐿 (m), breadth 𝐵 (m), average draught at the corresponding loading condition 𝑇 (m), block coefficient at the corresponding loading condition 𝐶𝐵=∆/𝐿𝐵𝑇 (∆ being the displacement), waterplane coefficient at the corresponding loading condition 𝐶𝑊𝑃=𝐴𝑊/𝐿𝐵 (𝐴𝑊 being the waterplane area), and roll period 𝑇𝜃 (s). Figure 1 shows the values of these quantities for all the vessels of the database. 2.2. Computation of the linear long-term hydrodynamic loads 2.2.1. Loads transfer functions For each vessel and each loading condition in the database, loads transfer functions (RAOs) are computed using linear potential flow solvers Hydrostar, Wasim, 3-DPM.L or WALCS. RAOs are computed for motions (roll and pitch), accelerations at the center of gravity (surge, sway, heave, roll and pitch), hull girder loads at 21 sections along the ship (vertical and horizontal bending moment, vertical shear force, torsion) and pressures at 105 different points along the wetted hull. RAOs are computed for two speeds: 5 knots and 75% of the design speed. This is in total about 400 RAOs per vessel and loading condition. 2.2.2. Long-term computation The loads RAOs are combined with the scatter diagram to compute the extreme and fatigue loads. The assumptions of this computation are the ones given in Rec.34 rev.2. The sea states statistics are given by the scatter diagram. The heading distribution is uniform. The sea states are modeled by a JONSWAP spectrum with 𝛾=1.5 and a directional spreading in cos3. Extreme loads are computed at 5 knots and correspond to a 25-year return period. They are defined by the following equation, where 𝑝𝑖 is the probability to encounter a given sea state and heading, 𝑇𝑧𝑖 is the mean response period on this sea state (in seconds), and 𝑃𝑖(𝑥) is the short-term non-exceedance probability given by a Rayleigh distribution: ∑𝑝𝑖 𝑇𝑧𝑖(1−𝑃𝑖(𝑋25)) 𝑖=1 25∗365.25∗24∗3600 (1) Fatigue loads are computed at 75% of design speed and correspond to a 10-2 probability of exceedance. They are defined by the following equation: ∑𝑝𝑖 𝑇𝑧𝑖(1−𝑃𝑖(𝑋10−2)) 𝑖=10−2∑𝑝𝑖 𝑇𝑧𝑖 𝑖 (2) These extreme and fatigue loads are called ‘envelope’ loads. They are not concomitant. To obtain a realistic combination of loads, we must use the EDWs. 2.2.3. Equivalent design waves The EDWs are the same as the ones already used in the CSR. We have 7 EDWs for extreme loads and 5 for fatigue. Each EDW is a regular wave that is tailored to reach the envelope value of its governing load. For instance, HSM is maximizing the vertical bending moment at midship in head sea and BSP is maximizing the wave pressure at midship waterline in beam sea. The exact definition of all the EDWs can be found directly in the CSR, but a summary is given in Table 2. Table 2. Equivalent Design Waves definition Name Heading Governing load HSM 180° Vertical Bending Moment at midship Extreme and Fatigue HSA 180° Vertical acceleration at Fore Perpendicular Extreme only FSM 0° Vertical Bending Moment at midship Extreme and Fatigue BSR 90° / 270° Roll angle Extreme and Fatigue BSP 90° / 270° Pressure at midship waterline Extreme and Fatigue OST 60° / 300° Torsion at 0.25L and baseline Extreme and Fatigue OSA 120° / 240° Pitch acceleration Extreme only 4 Using the calculated envelope value for the governing loads, and using the RAOs for all the other loads, we were able to compute the instantaneous value of all the loads (motions, accelerations, hull girder loads and pressure) under each EDW. It means that we end up with a huge database of results where for each ship and loading condition, we have the value of 200 loads components under the 7 extreme EDWs and 5 fatigue EDWs. 2.3. Methodology for developing the rule formulations 2.3.1. Error quantification For each vessel, and each loading condition, the rule value is compared to the computed value. The error is defined as: 𝜀=𝐶𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒−𝑅𝑢𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑖𝑛𝑔 𝑡𝑒𝑟𝑚 (3) The normalizing term can be defined in different ways, depending on the type of load we want to compare: • For envelope values of motions and accelerations which are, by definition, always strictly positive, the normalizing term is chosen to be the rule value. Hence the error is expressed as a fraction of the rule value. • For envelope values of hull girder loads, the normalizing term is chosen to be the maximum value along the ship of the rule envelope. Indeed, at fore and aft end of the vessel, where the hull girder loads are negligible, it makes no sense to express the error as a fraction of a very small quantity. Hence the error, at any location, is expressed as a fraction of the maximum rule envelope value. • For instantaneous loads under a given EDW, the normalizing term is chosen to be the envelope rule value for motions and accelerations, and the maximum value along the ship of the envelope rule value for hull girder loads. Hence the error is expressed as a fraction of the maximum envelope value. • For pressure, the normalizing term is equal to 1. The error is then the difference of pressure (in kN/m²). In any case, a positive error means that the computed value is bigger than the rule value, and a negative error means that the computed value is smaller than the rule value. Once the error is computed for each vessel, each loading condition (and each longitudinal location for hull girder loads and pressures), the following global quantities are defined: Mean Error (ME) 𝑀𝐸=1𝑁∑𝜀𝑖 𝑁 𝑖=1 (4) Root Mean Squared Error (RMSE) 𝑅𝑀𝑆𝐸=√1𝑁∑𝜀𝑖2 𝑁 𝑖=1 (5) 2.3.2. Rule formulations Rule formulations should give a simple but accurate estimation of the computed hydrodynamic loads, using only a few parameters: 𝐿, 𝐵, 𝑇, 𝐶𝐵, 𝐶𝑊, 𝑇𝜃 and 𝑉. They should, as much as possible, be consistent from a unit point of view. For each formulation, the error is computed for every vessel and loading condition, and the ME and RMSE are computed. A good formulation should have 𝑀𝐸=0, and a RMSE as small as possible. 5 3. Envelopes for extreme loads and fatigue loads 3.1. Rule formulation template 3.1.1. Generalized wave parameter A generalized wave parameter, 𝐶𝑤, is used in the definition of all the loads. This wave parameter takes a similar form as the one already defined in UR S11A [9]. It is computed from the rule length 𝐿 and a reference length 𝐿𝑟𝑒𝑓 which is defined for each load. 𝐶𝑤= { 1−1.3(1−√𝐿 𝐿𝑟𝑒𝑓)1.8𝑖𝑓 𝐿<𝐿𝑟𝑒𝑓 1−0.9(√𝐿 𝐿𝑟𝑒𝑓−1)1.8𝑖𝑓 𝐿≥𝐿𝑟𝑒𝑓 (6) The wave parameter is considered to have a unit in meter. A detailed definition of what the wave parameter represents can be found in [10]. This methodology has already been used to define the wave parameter in UR S11A, as explained in [11]. In a nutshell, it represents the interaction between the wave scatter diagram and the load RAO of a given vessel. As illustrated in Figure 2, the RAOs of longer vessels have their peak at longer periods than shorter vessels. Hence the worst sea state in the scatter diagram for longer vessels is not the same as the worst sea state for shorter vessels. The wave parameter is somehow proportional to this worst wave height for a given vessel. The quantity 𝐿𝑟𝑒𝑓, defined in meters, gives the ship length for which the wave parameter reaches its maximum value. When the ship length 𝐿 is close to 𝐿𝑟𝑒𝑓, it means that the highest sea states in the scatter diagram are the worst sea state for this vessel and this load parameter. 𝐿𝑟𝑒𝑓 is defined for each load (motions, accelerations, hull girder loads). It reflects the fact that for the same vessel, the peak of the different loads RAOs are at different frequencies, and that the worst sea state for vertical bending moment is not the worst sea state for horizontal bending moment or pressure. It is important to understand that, up to a multiplicative constant, the wave parameter can be computed for a given vessel and a given load, from the corresponding RAO and the scatter diagram. Figure 2. Illustration of the interaction between the scatter diagram and the vessel RAOs 3.1.2. Probability factor As in the current CSR the fatigue loads envelopes are defined from the extreme load envelopes multiplied by a probability factor 𝑓𝑝. Hence this coefficient considers the speed effect (from 5 knots to 75% of design speed) and the probability level (from 25 years to 10-2). For extreme loads we have 𝑓𝑝=1. 6 3.1.3. Envelope template and fitting methodology The rule formulations for envelopes values, which by definition are always strictly positive, are made using the following template: { 𝑋=𝑎0 𝑓𝑝𝐶𝑤 𝐿𝑛 ∏𝑃𝑖𝑎𝑖𝑓(𝑥′) 𝑓𝑝=𝑏0 𝐿𝑏7 ∏𝑃𝑖𝑏𝑖 𝑥′=𝑥𝐿 ⁄−𝑥𝑎 𝑥𝑓−𝑥𝑎 𝑥𝑎=𝑐0+∑𝑐𝑖𝑃𝑖 𝑥𝑓=𝑑0+∑𝑑𝑖𝑃𝑖 𝑃𝑖∈[𝐵𝐿,𝑇𝐿,𝐶𝐵,𝐶𝑊𝑃,𝐹𝑟=𝑉 √𝑔𝐿,𝑇𝑅=𝑇𝜃√𝑔𝐿] (7) 𝑛 is the Froude dimension of the transfer function of the considered load: -1 for linear acceleration and angular motions, -2 for angular accelerations, +2 for shear forces, and +3 for moments. 𝑃𝑖 are a set of dimensionless parameters, 𝑎𝑖, 𝑏𝑖,𝑐𝑖 and 𝑑𝑖 are numbers to be tuned. 𝑓(𝑥′) is the distribution function (used only for hull girder loads). To consider the different behavior of deep and shallow draft vessels, CSR and non-CSR vessels, the distribution functions are defined as a function of 𝑥′. It means that the distribution function will start at 𝑥𝑎𝐿 (not necessary 𝑥=0) and end at 𝑥𝑓𝐿 (not necessary 𝑥=𝐿). The best formulation is found by minimizing the RMSE, ensuring a ME equal to zero, and trying to use the smallest number of parameters 𝑃𝑖: no need to have a very complex formulation for a small gain of RMSE. If needed, physical constant such as 𝜌 and 𝑔 are included in the parameter 𝑎0 to have a consistent formulation in terms of units, assuming the wave parameter is in meter. For instance, linear accelerations will be proportional to 𝑔𝐶𝑤𝐿−1, and moments will be proportional to 𝜌𝑔𝐶𝑤𝐿3. The first step is to fit the extreme envelope, by tuning the parameters 𝑎𝑖. For hull girder loads, this fit is done on the maximum bending moment along the vessel, and the maximum shear force and torsion moment in the aft part or in the fore part. In a second step the formulations for fatigue loads are adjusted by tuning the parameters 𝑏𝑖 of the probability factor. In a last step, for hull girder loads only, the distribution functions are fitted by tuning the coefficients 𝑐𝑖 and 𝑑𝑖 and adjusting the distribution function. 3.2. Motions and accelerations The formulations in Table 3 are proposed for the motions and accelerations extreme values. The draft 𝑇, the block coefficient 𝐶𝐵, the waterplane coefficient 𝐶𝑊𝑃 and the roll period 𝑇𝜃 are depending on the loading condition (full or ballast). The Froude number 𝐹𝑟 should be computed at a speed of 5 knots. Table 3. Ship motions and accelerations: new formulations for extreme and fatigue envelopes, mean and root mean squared errors Load Rule formulation 𝐿𝑟𝑒𝑓 𝑓𝑝 Extreme Fatigue ME RMSE ME RMSE Surge acc. 12.1 𝑔𝑓𝑝𝐶𝑤𝐿−0.85𝑇−0.15𝐶𝐵−0.6 203/𝐶𝑊𝑃 0.9 1.03 𝐿−0.2𝐵−0.15𝑇−0.1 0.4% 2.4% 0.7% 10.3% Sway acc. 11.1 𝑔𝑓𝑝𝐶𝑤𝐿−0.35𝐵−0.5𝑇−0.15 132(𝐿𝐵)0.65 0.55 𝐿−0.08𝐵−0.15𝐶𝐵−0.1 -0.1% 3.1% -0.5% 4.7% Heave acc. 56 𝑔𝑓𝑝𝐶𝑤𝐿−0.8𝐵−0.2𝐶𝑊𝑃 −0.2 192(𝐿𝑇)0.35 1.87 𝐿−0.4𝐶𝐵−0.5 -0.4% 3.6% 0.1% 9.7% Roll motion 𝑓𝑝3050(1−𝑇𝜃/54) 𝐵+51 2530 𝑇𝑅2 0.805 𝐿−0.2𝑇𝑅−0.2𝐶𝐵−1.2 0.6% 31.2% 0.1% 23.0% Roll acc. 𝑓𝑝136(1−𝑇𝜃/53) 𝑇𝜃1.2𝐶𝑊𝑃1.3(𝐵+16) 2530 𝑇𝑅2 5.45 𝐿−0.45𝑇𝑅−0.45 -0.1% 20.8% 1.9% 27.0% Pitch motion 3650 𝑓𝑝𝐶𝑤𝐿−1𝐶𝑊𝑃 −0.75𝐹𝑟025 309 0.86 𝐿−0.15𝐵−0.2𝐶𝑊𝑃 −0.2 0.0% 4.7% -0.5% 6.0% Pitch acc. 182 𝑓𝑝𝑔𝐶𝑤𝐿−1.4𝐵−0.6𝐶𝑊𝑃 1.4𝐹𝑟0.3 132(𝐿𝐵)0.8 0.72 𝐵−0.3𝐶𝑊𝑃 −0.6 0.2% 4.9% -0.2% 9.1% 7 Figure 3 shows the error for every vessel. All the dots represent individual vessels, split into 4 groups (CSR vessels in full load or in ballast condition, non-CSR vessels in full load or ballast condition). Large diamonds with the horizontal error bar represent the mean error of each group, plus or minus the standard deviation of the error. The agreement is very good: the mean error is close to zero for all the loads (small differences are due to the rounding of the numerical coefficient 𝑎0 in the formulation), and the variability is quite low: RMSE is less than 5% for extreme loads and less than 10% for fatigue loads (except for roll motions and acceleration). Furthermore, the variability is the same for all the 4 categories, showing that the effects of loading condition and ship shape have been properly considered in the rule formulation. A significant scatter exists however for the roll motion and the roll acceleration. Roll motion is indeed very difficult to compute, and even direct computation may not represent accurately the reality. Furthermore, it was difficult to fit a good formulation to the computed data. It was decided to keep a formulation like the current roll formulation in the CSR and not to use the wave parameter. Figure 3. Accuracy of the new load formulations (New rules) compared to linear direct computations (Rec.34 Rev.2) 3.3. Hull girder loads The hull girder loads are defined by one or more characteristic loads using the template of equation (7) and by one or more distribution functions. The longitudinal distributions are described by sinusoidal functions, to fit as much as possible to the calculations. The parameters 𝑥𝑎 and 𝑥𝑓 are fitted to minimize the error in all sections. The formulations presented in equations (8) to (12) are fitted to the linear calculations. In a second step, CFD computations are used to correct for non-linear effects for vertical bending moment and shear force (see [8]). Nonlinear effects will affect the amplitude of the loads using the non-linear factor 𝑓𝑛𝑙 and the longitudinal distribution by adjusting the parameters 𝑥𝑎 and 𝑥𝑓. 3.3.1. Vertical wave bending moment The formulation for vertical bending moment (VBM) is given in equation (8). It is built from the maximum value along the ship, 𝑀𝑤𝑣−𝑚𝑎𝑥, and a distribution function 𝑓𝑚(𝑥′). 𝑀𝑤𝑣=𝑀𝑤𝑣−𝑚𝑎𝑥 𝑓𝑚(𝑥′) 𝑀𝑤𝑣−𝑚𝑎𝑥=0.118 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿2.3𝐵0.7𝐶𝑊𝑃 1.4 𝑓𝑚(𝑥′)=𝑠𝑖𝑛2(𝜋𝑥′) 𝐿𝑟𝑒𝑓=358𝐶𝑊𝑃 −1.3 𝑓𝑝=1.56 𝐿−0.4𝐶𝑤𝑝 −1.25 𝑥𝑎=0.34+0.3𝐶𝐵−0.6𝐶𝑊𝑃−𝑇/𝐿 𝑥𝑓=0.95+0.5𝐶𝐵−0.4𝐶𝑊𝑃+0.1𝑇/𝐿 (8) 8 This can be compared to the original CSR [3] formulation given in equation (9): 𝑀𝑤𝑣=𝑀𝑤𝑣−𝑚𝑎𝑥 𝑓𝑚(𝑥) 𝑀𝑤𝑣−𝑚𝑎𝑥=0.19𝐶𝑤−𝐶𝑆𝑅𝑓𝑝𝑓𝑛𝑙𝐿2𝐵𝐶𝐵 𝑓𝑚(𝑥)= { 𝑥 0.4𝐿 𝑓𝑜𝑟 0≤𝑥<0.4𝐿 1 𝑓𝑜𝑟 0.4𝐿≤𝑥<0.65𝐿 1−𝑥/𝐿 0.35 𝑓𝑜𝑟 𝑥≤0.65𝐿<𝐿 (9) Figure 4 shows the different steps of the rule optimization process. Figure 4 (a) shows the comparison of the computed vertical bending moment and the linear CSR bending moment (without 𝑓𝑛𝑙), and the corresponding error (expressed as a percentage of 𝑀𝑤𝑣−𝑚𝑎𝑥). There are obviously a lot of errors, not only around the midship area, but also in the fore and aft part. Figure 4 (b) shows the same comparison where the new formulation is used for 𝑀𝑤𝑣−𝑚𝑎𝑥 but the CSR distribution function is kept. While the error is largely reduced around midship, this rule formulation is largely over-conservative in the fore and aft area. The computed bending moment distributions are smooth and do not match at all the CSR distribution. This is why we have tried to use a sinusoidal distribution. Figure 4 (c) shows this comparison where the distribution function is 𝑓𝑚(𝑥)=𝑠𝑖𝑛2(𝜋𝑥/𝐿). The fit is much better. However, it is now quite clear that, in the aft part, the rule formulation underestimates the bending moment of the full loading conditions, while it over-estimates the bending moment of ballast loading conditions. In the fore part the rule formulation underestimates the bending moment of CSR ships, while it overestimates the bending moment of non-CSR ships. To correct for the effect of loading condition and ship type, the final distribution function is now defined as function of 𝑥′. The effect of 𝑥′ is to stretch the sinusoidal function, depending on the ship parameters (𝐶𝐵, 𝐶𝑊𝑃 or 𝑇/𝐿). The final comparison is shown in Figure 4 (d): The accuracy of the rule formulation is much better. The error is now similar at all longitudinal sections, with no significant effect of ship type or loading condition. (a) (b) (c) (d) Figure 4. Extreme vertical bending moment envelope compared to different rule formulations: (a) is original CSR, (b) is the new formulation for 𝑀𝑤𝑣−𝑚𝑎𝑥 combined with the CSR distribution function, (c) is the new formulation for 𝑀𝑤𝑣−𝑚𝑎𝑥 combined with the sinusoidal distribution function 𝑓𝑚(𝑥)=𝑠𝑖𝑛2(𝜋𝑥/𝐿), (d) is the new formulation for 𝑀𝑤𝑣−𝑚𝑎𝑥 and the new distribution function 𝑓𝑚(𝑥′). 3.3.2. Vertical wave shear force The formulation for vertical shear force (VSF) is given in equation (10). It is built from three characteristic values (maximum VSF in the aft part 𝑄𝑤𝑣−𝑎𝑓𝑡, minimum VSF in the midship region 𝑄𝑤𝑣−𝑚𝑖𝑑, and maximum VSF in the fore part 𝑄𝑤𝑣−𝑓𝑜𝑟𝑒) and three distribution functions 𝑓𝑎(𝑥′), 𝑓𝑚(𝑥′) and 𝑓𝑓(𝑥′). 𝑀𝑤ℎ=𝑄𝑤𝑣−𝑎𝑓𝑡 𝑓𝑎(𝑥′)+𝑄𝑤𝑣−𝑚𝑖𝑑 𝑓𝑚(𝑥′)+𝑄𝑤𝑣−𝑓𝑜𝑟𝑒 𝑓𝑓(𝑥′) 𝑄𝑤𝑣−𝑎𝑓𝑡=0.395 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿1.3𝐵0.7𝐶𝑊𝑃 1.1 𝑓𝑎(𝑥′)={𝑠𝑖𝑛2(2𝜋𝑥′) 𝑓𝑜𝑟 0<𝑥′<0.5 0 𝑓𝑜𝑟 𝑥′≤0 𝑜𝑟 𝑥′≥0.5 (10) 9 𝑄𝑤𝑣−𝑚𝑖𝑑=0.09 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿1.7𝐵0.3𝐶𝑊𝑃 1.2 𝑓𝑚(𝑥′)={𝑠𝑖𝑛2(2𝜋(𝑥′−0.25)) 𝑓𝑜𝑟 0.25<𝑥′<0.75 0 𝑓𝑜𝑟 𝑥′≤0.25 𝑜𝑟 𝑥′≥0.75 𝑄𝑤𝑣−𝑓𝑜𝑟𝑒=0.718 𝜌𝑔𝐶𝑤𝑓𝑝𝑓𝑛𝑙𝐿𝐵𝐶𝑊𝑃 0.9 𝑓𝑓(𝑥′)={𝑠𝑖𝑛2(2𝜋𝑥′) 𝑓𝑜𝑟 0.5<𝑥′<1.0 0 𝑓𝑜𝑟 𝑥′≤0.5 𝑜𝑟 𝑥′≥1.0 𝐿𝑟𝑒𝑓=375𝐶𝑊𝑃 −1.3 𝑓𝑝=1.37 𝐿−0.35𝐶𝑊𝑃 −0.6 𝑥𝑎=0.39+0.4𝐶𝐵−0.8𝐶𝑊𝑃−1.0𝑇/𝐿 𝑥𝑓=0.89+0.4𝐶𝐵−0.2𝐶𝑊𝑃+0.3𝑇/𝐿 3.3.3. Horizontal wave bending moment The formulation for horizontal bending moment (HBM) is given in equation (11). It is built from the maximum value along the ship, 𝑀𝑤ℎ−𝑚𝑎𝑥, and a distribution function 𝑓𝑚(𝑥′). 𝑀𝑤ℎ=𝑀𝑤ℎ−𝑚𝑎𝑥 𝑓𝑚(𝑥′) 𝑀𝑤𝑣−𝑚𝑎𝑥=0.24 𝜌𝑔𝐶𝑤𝑓𝑝𝐿2.2𝐵−0.1𝑇0.9 𝑓𝑚(𝑥′)=𝑠𝑖𝑛2(𝜋𝑥′) 𝐿𝑟𝑒𝑓=358(𝐿/𝑇)0.3 𝑓𝑝=2.44 𝐿−0.55𝐵0.3𝐶𝐵1.6 𝑥𝑎=−0.31+0.4𝐶𝐵 𝑥𝑓=1.0 (11) 3.3.4. Torsion moment The formulation for torsion moment is given in equation (12). It is built from the maximum value along the ship, 𝑀𝑤𝑡−𝑚𝑎𝑥, and three distribution functions 𝑓𝑎(𝑥′), 𝑓𝑚(𝑥′) and 𝑓𝑓(𝑥′). 𝑀𝑤𝑡=𝑀𝑤𝑡−𝑚𝑎𝑥(𝑓𝑎(𝑥′)+0.25𝑓𝑚(𝑥′)+0.76𝑓𝑓(𝑥′)) 𝑀𝑤𝑡−𝑚𝑎𝑥=1.43 𝜌𝑔𝐶𝑤𝑓𝑝𝐿0.6𝐵1.4𝑇𝐶𝑊𝑃 𝑓𝑎(𝑥′)={𝑠𝑖𝑛2(𝜋𝑥′/0.6) 𝑓𝑜𝑟 0<𝑥′<0.6 0 𝑓𝑜𝑟 𝑥′≤0 𝑜𝑟 𝑥′≥0.6 𝑓𝑚(𝑥′)={𝑠𝑖𝑛2(𝜋(𝑥′−0.3)/0.4) 𝑓𝑜𝑟 0.3<𝑥′<0.7 0 𝑓𝑜𝑟 𝑥′≤0.3 𝑜𝑟 𝑥′≥0.7 𝑓𝑓(𝑥′)={𝑠𝑖𝑛2(𝜋(𝑥′−0.4)/0.6) 𝑓𝑜𝑟 0.4<𝑥′<1.0 0 𝑓𝑜𝑟 𝑥′≤0.4 𝑜𝑟 𝑥′≥1.0 𝐿𝑟𝑒𝑓=375(𝐿/𝑇)0.3 𝑓𝑝=0.303 𝐿0.2𝑇−0.45 𝑥𝑎=−0.05 𝑥𝑓=1.07 (12) 3.3.5. Comparisons with computed values Figure 5 shows the comparison of these linear rule formulations with the linear computations. The errors are also included in the titles of each graph. We can see the very good agreement for VSF and VBM for all types of vessels, with RMSE in the order of 10%. Comparisons of the non-linear coefficients are reported in [8]. For torsion and HBM it was found difficult to fit the formula for all ship types and loading conditions. This is again because of roll motion that could increase significantly these loads, especially for ballast cases. The rule update is therefore based on CSR vessels and full load condition as the moment was found to increase with draft: the rule formulations are approximately proportional to the draft 𝑇, it has been checked that these loads at full draft are always higher than the value at ballast draft, despite the effect of roll. When considering only the CSR vessels in full load, the RMSE for HBM is 8.6% for extreme loads and 6.0% for fatigue loads, while the RMSE for torsion is 9.9% for extreme loads and 11.8% for fatigue loads. 16 6. Fatigue period 6.1. Methodology In the current CSR, the fatigue damage is computed from the 10-2 stress range and the stress period 𝑇= 4log10𝐿. This stress period only depends on the ship length and is the same for all the structural details onboard the ship. When fatigue damage is computed using a spectral approach, the mean cycle period might be different for each detail onboard the ship. It is computed from the stress RAO of this particular detail. In order to derive a rule formulation, we should compute stress RAOs on multiple details onboard many different ships. However, the 102 stress range being computed from the stress ranges computed on the 5 fatigue Equivalent Design Waves, we can assume that the mean stress period is equal to the mean period of the governing load (the governing parameter of the design wave maximizing the stress range). Hence it makes sense to compare the mean cycle period of the 4 governing load parameters (Vertical bending moment, torsion moment, roll motion and waterline pressure at midship). Figure 10 shows this comparison. The mean cycle periods of the governing loads are very different from the rule period 4log10𝐿. The mean roll period is usually higher than the rule period, but the mean periods of the other loads are always smaller, which mean that the rules are unconservative. 6.2. Proposed formulations 4 formulations are proposed for the mean cycle period: one for each governing load. The mean period to be used for each EDW is given in the table below, together with the quantification of the error. Table 8. Fatigue period formulations EDW Rule formulation RMSE HSM and FSM 0.8 √𝐿−𝐿/50+1.84 𝐶𝐵 2.9% OST 3.41𝑙𝑜𝑔10(𝐿) 5.9% BSP 5.71 𝐵0.07𝐶𝐵0.15 4.2% BSR 𝑚𝑖𝑛(0.7 𝑇𝑟𝑜𝑙𝑙+2.8 ;6 𝐿0.13) 5.4% Figure 10. Comparison of the fatigue rule period and the governing loads mean period 6.3. Selection of the fatigue period In principle we could compute the fatigue damage for each of the 5 design waves, using for each of them the 10-2 stress range and the corresponding mean period, and take the highest fatigue damage. It is however simpler to select the governing design wave as being the one maximizing the quantity Δ𝜎𝑖3 𝑇𝑖. 17 7. Conclusion Following the update of the North Atlantic scatter diagram on the Recommendation n°34, this paper explains how the CSR load formulations have been updated. The objective was to achieve a good consistency between the rule loads and the loads estimated using a direct approach from the scatter diagram. The first step was to build a huge hydrodynamic database of loads RAOs for more than 250 vessels of major ship types (oil tankers, bulk carriers, containerships, gas carriers, ore carriers, cruise ships…), in both full load and ballast conditions. By combining RAOs and the scatter diagram, extreme and fatigue loads were then computed. The second step was to fit the rule formulations for loads envelopes for motions accelerations and hull girder loads, and the corresponding LCF and sea pressure under the different EDWs. Each time the procedure was similar: trying to minimize the error between the rule formulation and the computed value, while keeping relatively simple formulations. The accuracy of the rule formulation is, for each formulation, quantified by the RMSE. The result is a set of formulation which is consistent with the recommendation n°34 assumptions on the environmental (encountered sea states) and operating (heading and speed) conditions. At the time of the finalization of this paper, the consequence assessment of these new loads on the ship scantlings is still ongoing. The plan is to adopt the Rule Change Proposal in July 2027, for a tentative date of entry into force in July 2029. Acknowledgments All work performed in this study was sponsored by IACS through the Hull Panel PT PH49 project grant to update the prescriptive wave loads in CSR and UR S11/S11A. References [1] IACS, Common Structural Rules for Double Hull Oil Tankers, 2006. [2] IACS, Common Structural Rules for Bulk Carriers, 2006. [3] IACS, Common Structural Rules for Bulk Carriers and Oil Tankers, 2014. [4] IMO, "Goal-based ship construction standards for bulk carriers and oil tankers," in Resolution MSC.287(87). [5] IACS, Recommendation 34 Standard Wave Data - Rev.1, 2000. [6] IACS, Recommendation 34 Standard Wave Data - Rev.2, 2022. [7] H. N. Austefjord, G. de Hauteclocque, M. C. Johnson and T. Y. Zhu, "Updat of wave statistics standards for classification rules," in MARSTRUCT, Gothenburg, 2023. [8] T. Landet, G. de Hauteclocque, K. Sugimoto, Q. Derbanne and H. Austefjord, "A broad CFD-based study of the nonlinear effects on wave-induced bending moments and shear forces on monohull ships," in PRADS, Ann Arbor, 2025. [9] IACS, UR S11A: Longitudinal Strength Standard for Container Ships, 2015. [10] G. de Hauteclocque and Q. Derbanne, "Generalized wave parameter for rules formulae," in PRADS, Copenhagen, 2016. [11] Q. Derbanne, G. Storhaug, V. Shigunov, G. Xie and G. Zheng, "Rule formulation of vertical hull girder wave loads based on direct computation," in PRADS, Copenhagen, 2016.