ABS TECHNICAL PAPERS 2008 PROBABILISTIC PRESENTATION OF THE STILL WATER LOADS. WHICH WAY AHEAD?

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1 Proceedings of the 7th International Conference on Offshore Mechanics and Arctic Engineering OMAE008 June 15-0, 008, Estoril, Portugal OMAE PROBABILISTIC PRESENTATION OF THE STILL WATER LOADS. WHICH WAY AHEAD? Lyuben D Ivanov Life Cycle Support Technology, Operational Safety and Evaluation, American Bureau of Shipping, Houston, USA Ge Wang Life Cycle Support ABSTRACT This paper analyzes the state-of-the-art concerning this subject. The objectives are: 1) to provide data of still water bending moments and shear forces that were collected from loading manuals of dozens of tankers ) conduct statistical analysis of the loading cases and 3) discuss possible ways for further improvement of knowledge about the variability of still water loads during real ships operation. 1 INTRODUCTION In almost all publications in the public domain, the probabilistic distribution functions of still water loads were determined based on statistical analysis of still water loads collected from voyage data of real ships. These studies were performed at a time when onboard computers were used in a limited number of ships. The rapid development of computer technology has caused many changes in ships operation. One result is the widespread use of computers for controlling and monitoring the stability and longitudinal strength of ships. It has become the routine to use a computer in planning cargo loading and monitoring the resulting stresses on the hull structure. The randomness of the still water loads does not disappear, but likely has changed. However, the impact of this change has not been investigated yet. Since the introduction of probabilistic methods in assessment of the wave-induced loads (St. Denis and Pierson, 1953), hundreds of papers have been published, which reflect the research interest all over the world. Probabilistic treatment of the still water loads, however, started much later and only around a dozen publications exist. A major reason is that the wave-induced loads are often greater than the still water loads and people tend to address first the loads of higher magnitude. Another reason is the premises that still water loads can be efficiently controlled by ship s operator, and as such, the still water loads do not exceed given permissible values if the loading manual prepared by the ship s designer was followed. The first attempt to present the still water loads in probabilistic terms for sea-going ships was done by W. Trafalski (1967). Later, Abrahamsen et al (1970) and Truhin (1970) published their results of probabilistic still water loads. Several publications on the same subject followed during the 1970s for different ship types (Lewis, 1973; Ivanov, 1973, 1975; Maximadji, 1973; Król, 1974; Mano et al, 1977; Sőding, 1979). All these publications were based on the analysis of real cargo plans of different ship types. They provided information about the type of probabilistic distribution of maximum still water loads to be expected during ships operation (in almost all cases this was Gaussian distribution). During the 1980 s and 1990 s, major contributions to the subject came from numerous works done by Guedes Soares (198, 1984, 1988, 1990, 199, 1996, 000). One should also note the interesting analytical work on the probabilistic treatment of still water loads done by Gran (1991 and 199) and Hansen and Ditlevsen (001). The objectives of this paper are: To analyze the loading manuals for new tanker designs and the derivation of the probabilistic distributions for full, partial, and ballast load conditions depending on the ships size. Although the loading cases in loading manuals are considered by designers as equally likely to occur, the information can be used as a starting point in the probabilistic assessment of hull girder ultimate and fatigue strength. To discuss the obtained results and propose further refinement of the probabilistic still water and total loads, which can be used in a reliability-based analysis. Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 97

2 PROBABILISTIC DISTRIBUTION FUNCTIONS OF STILL WATER LOADS.1 Methods to Determine the Probabilistic Distribution of the Still Water Loads: In general, there are three ways to determine the probabilistic distribution of the still water loads: Statistical analysis of real cargo plans for ships with given type, size, cargo, operational area, etc. The so obtained probabilistic distributions are valid for the past and can be used for new vessels if they operate in the same environment. Statistical analysis of the loading manuals. They include a few cargo loading cases, for which ship s trim, stability and hull girder strength are checked. Often, the loading cases in these manuals are assumed as equally likely to occur. Therefore, the probabilistic distribution of the still water loads derived from the manuals is not real. In some sense, the probabilistic distribution of the still water loads derived from the loading manuals is the input. One should have an operator (as an analog to the statistical dynamics) to transform the input into an output (i.e., obtaining more realistic probabilistic distributions). This operator depends on future routes, type of cargo, ship s size, age, etc. Once the operator is determined (at least approximately), the frequency of each loading case in the corresponding loading manual can be obtained based on which more realistic probabilistic distributions can be derived. Thus, a step forward will be done toward application of the time-variant reliability to ship structures design, repair and maintenance. During ship s operation, the number of cargo loading cases may well exceed the number of loading cases in the loading manuals. In all cases, the calculated still water shear forces and bending moments should be smaller than the permissible corresponding values specified by the classification societies rules. Because of that, there are many more variations of the still water shear forces and bending moments than in the loading manuals. This second approach is the approach the authors plan to follow in their work. Qualitative approach. Based on experience and engineering intuition, some types of probabilistic distributions are assumed. Application of simulation methods such as, e.g., Monte Carlo simulation should be involved. Especially for completely new design, this approach may be the best available.. Investigated Data This paper concentrates on the analysis of loading manuals in order to derive the probabilistic distribution functions of still water loads. The loading manuals of double hull tankers and 1 single hull tankers were analyzed. Their principal dimensions are given in Table 1. One should emphasize again that the derived probabilistic distributions may not reflect the variability of real ships operation. Table 1 Principal dimensions of tankers No. Type Year built DWT Length L BP Draft Width B [m] Depth D [m] [m] T design [m] 1 DH , DH , DH , DH , DH , DH , DH , DH 00 77, DH , DH , DH , DH , DH , DH , DH , DH , DH , DH , DH , DH , DH , DH , SH , SH , Probabilistic Presentation Of The Still Water Loads. Which Way Ahead?

3 No. Type Year built DWT Length L BP Draft Width B [m] Depth D [m] [m] T design [m] 5 SH , SH , SH SH , SH , SH , SH , SH , SH , SH , Note: DH = double hull, SH = single hull Further, these tankers were divided into three groups (see Table ) to reflect the fact that tankers with deadweight greater than 150,000 DWT cannot pass through the Suez Canal. Table ing of the tankers Range of deadweight [tonnes] Double hull tankers Single hull tankers I < 150K 13 8 II 150K-300K 6 4 III > 300K Candidate probabilistic distribution functions Some probabilistic distribution functions were selected as candidate functions. Under certain conditions, some exotic probabilistic distribution may appear to be the best fit following Kolmogorov-Smirnov criterion. It was decided to ignore these non-typical or exotic functions and concentrate only on the candidate distribution functions: Gaussian, two-parameter Weibull and lognormal, and one-parameter Rayleigh functions. a) Gaussian (Normal) probability density function: 1 1 x μ f ( x) = exp (1) σ π σ where μ and σ are, correspondingly, the mean and standard deviation of x. b) Two-parameter Weibull probability density function: λ 1 λ λ x x f ( x) = exp () α λ α where α and λ are, correspondingly, scale and shape parameters c) Two-parameter Lognormal probability density function: 1 1 lnx m f( x) = exp (3) xs π s where m and s are, correspondingly, the mean and standard deviation of lnx. d) One-parameter Rayleigh probability density function: ( ) x 1 f x exp x = σ σ where σ is the standard deviation of x (4) The results of the calculations are given in Appendix I for tankers with deadweight smaller than 150,000 DWT, in Appendix II for tankers with deadweight between 150,000 DWT and 300,000 DWT, and in Appendix III for tankers with deadweight greater than 300,000 DWT. Truncation of the probabilistic distribution is also considered due to the existing limits of the still water loads. Because the area below the probability density function (p.d.f.) should always be equal to unity, one should increase the ordinates of the p.d.f. by a constant C that is be determined in the following way: u u f C ( x) dx= Cf ( x) dx= CFu ( ) Fb ( ) = 1 (5) b b ( ) ( ) (6) C= 1/ F u F b where u = upper limit, b = lower limit (they are determined based on the permissible values given in loading manuals), f C (x) = truncated p.d.f. f(x) = p.d.f. without limits, F(u) = cumulative distribution function (c.d.f.) at u, F(b) = c.d.f. at b, C = constant EasyFit computer program was used to derive probabilistic distributions that best fit the available data for still water shear forces and bending moments using Kolmogorov-Smirnov criterion. There are cases when the best fit is not shown in the graphs. It means that this is one of those exotic types of probabilistic distribution as mentioned above. The selected probability distribution types are given in all the graphs. If some of them are missing, it means that they do not meet the Kolmogorov- Smirnov criterion. All SWBM are expressed as a ratio over the permissible values approved by the classification societies rules. Therefore, they are dimensionless and less than unity. 3 ANALYSIS RESULTS A summary of the results for the probabilistic distributions of the still water loads (see Appendices I, II and III) is shown in Table 3. Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 99

4 Table 3 Types of probabilistic distributions Load case +SF -SF +BM -BM FL W W W W I B W W N N PL W W W W FL LN W R W II B LN W N N PL W W N W FL LN LN No data LN III B W R LN No data PL W LN LN LN Notes: +SF = positive shear force, -SF = negative shear force, +BM = positive bending moment (hogging), -BM = negative bending moment (sagging), W = Weibull distribution, N = normal distribution, LN = lognormal distribution, R = Rayleigh distribution, FL = full load, PL = partial load, B = ballast (the parameters of the probabilistic distributions are given in Tables 4 7). When the numeric characteristics of the two-parameter Lognormal distribution are known, the mean μ and variance σ of the random variable can be calculated by the formulae: s ( ) ( μ = exp m + σ = exp m + s. exp s ) 1 (7) where m and s are the mean and variance of the logarithm of x (see Eq. (3) and Table 8). Table 4 II III Numeric characteristics of two-parameter Lognormal distribution st.dev. Load SW load mean s of pattern type m of Lnx Lnx FL +SF B +SF SF FL -SF BM B +BM SF PL +BM BM Table 5 Numeric characteristics of Normal distribution Load pattern SW load type mean μ of x st. dev. σ of x COV +BM B I -BM BM B -BM II PL +BM COV = σ /μ coefficient of variance Table 6 I II III Numeric characteristics of two-parameter Weibull distribution Load SW load pattern type α λ +SF FL -SF BM BM B +SF SF SF 7.70 PL -SF BM BM FL -SF BM B -SF SF PL -SF BM B +SF PL +SF When the numeric characteristics of the two-parameter Weibull distribution are known, the mean and variance of the random parameter can be calculated by the formulae (see Table 9): 1 mx =αγ 1+ λ 1 Dx =α Γ 1+ Γ 1+ λ λ (8) Table 7 Numeric characteristic of one-parameter Rayleigh distribution Load pattern SW load st.dev. type σ of x II FL +BM III B -SF Table 8 Means, st. dev. and COV of the SWSF and SWBM when the two-parameter Lognormal distribution is followed II III Load pattern SW load type mean μ of x st.dev. σ of x COV FL +SF B +SF SF FL -SF BM B +BM SF PL +BM BM Probabilistic Presentation Of The Still Water Loads. Which Way Ahead?

5 When the standard deviation of one-parameter Rayleigh distribution is known, the mean of the random parameter can be calculated by the formula (see Table 10): μ = 1.53σ (9) Further analysis is made for the average mean values and standard deviations when all results are regrouped on the basis of load pattern. The results are shown in Table 11 where μ is the mean value of SWSF or SWBM and σ is the standard deviation of SWSF or SWBM. As shown in Table 3, the Normal distribution appears to be a good fit for only two situations. The Weibull distribution fits well with many in I tankers. More diverse are the probabilistic distributions for the tankers in the second group. In group III, the Lognormal distribution is more frequently met. Another fact is that the spread (represented by the COV) of the still water loads for tankers in the first group is greater than for the tankers in the other two groups (see Tables 8, 9, 10). This may reflect the peculiar operations of I tankers, many of which are not built to operate between specified ports as in liner shipping. Table 9 Means, st. deviations and COV of the SWSF and SWBM when the two-parameter Weibull distribution is followed I II III Load patte rn SW load type mean st. dev. COV +SF SF FL +BM BM SF B -SF SF SF PL +BM BM FL -SF BM B -SF SF PL -SF BM B +SF PL +SF Table 10 Means, st. dev. and COV of the SWSF and SWBM when the one-parameter Rayleigh distribution is followed Load pattern SW load type mean μ of x st.dev. σ of x COV II FL +BM III B -SF The tankers in the second and third group (i.e., VLCC and ULCC) operate, in general, in a manner closer to liner shipping because most of the them are used to carry crude oil between two major ports. One can also observe the smaller spread of SWBM in ballast conditions for all sizes of tankers (the average mean is 0.756, the average standard deviation is 0.173, and the COV is 0.9) than in full or partial load conditions (the average mean is 93, the average standard deviation is 0.458, and the COV is 0.77). This is understandable because the variants for ships ballast conditions envisaged in design are not as numerous as in comparison with the variations in cargo distribution. Table 11 Mean values, st. deviations and COV of SWSF and SWBM vs. load case Load case Full load Ballast Partial load μ, σ, COV SF (+) SF (-) BM (+) BM (-) μ σ COV μ σ COV μ σ COV WHICH WAY AHEAD? Any analysis of past experience provides real information for a given time period in the past. This is the so-called aposteriori information. However, the real challenge and the most difficult task is to provide apriori information to be used in ships design and operation. The difficulty stems from the unavoidable uncertainties in ships operation due to many reasons. One could point to a few of them: change in the political situation worldwide, discovery of new reservoirs of crude oil or exhausting of the old ones, technological progress, etc. The latter is characterized mostly by the dynamic Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 101

6 development of computer technology and weather prediction. Almost all modern tankers are equipped with onboard computers, which are used to calculate the ship s trim, stability, strength, etc. for any cargo loading pattern. The ship s operator relies heavily on these loading computers to predict and control the hull stresses that a cargo loading pattern can impose on his/her ship. The satellite technology and the international cooperation provides another valuable source of information that can be used by ships operators to predict the ship s behavior during the voyage between any port A and port B depending on the weather during the voyage. Although this service is not 100% accurate yet, its quality increases constantly and it gradually becomes one of the basic factors influencing the shipping industry. This reasoning leads to the conclusion that the prediction of the still water loads should reflect these new developments. A preferable way is to derive the functions by simulating ship operation, where the data from the loading manuals is used as inputs. Such a simulation may be able to consider the effect of weather forecasting. This is not an easy task and requires involvement of all parties in ships operation. Another issue that deserves to be re-visited is the existing design practice of treating the still water loads and the wave-induced loads as independent parameters and summing them up to derive the total loads. The latter are calculated by different methods, all of which are based on assumptions. This practice is inherited from the past when the weather forecast was not well developed and computers were not available onboard ships. As mentioned above, the times have changed and the ships operation also changes in pace with the new developments. We now check the ultimate hull girder strength using the total loads and the fatigue strength of structural components and joints using the information for the wave-induced loads causing cyclic loading on these components and joints around some mean loads, which also change over time. The total load should be the basis for assessing the hullgirder ultimate strength, which is the most important measure of structural strength. Therefore, the target should be the probabilistic distribution of the total load. Properly defined probabilistic distribution of the still water loads is important but insufficient to solve this problem. Data is necessary to derive the probabilistic distribution of the total load. It can be used as a calibration criterion of the traditional practice of summing up the probabilistic distributions for still water and wave-induced loads. This seems to be one of the few fields where a large gap in knowledge exists. The calculation of the fatigue strength of structural components and joints requires another approach that is closer to the existing practice but on a higher level. This higher level should include presentation of the still water loads (the mean loads around which the cyclic loads are applied) and wave-induced loads (the cyclic loads) as random functions changing over time. 5 CONCLUSIONS The paper presents a study on the probabilistic distributions of the still water loads using the loading manuals of tankers. The results revealed the effect of the ships size: For tankers with deadweight smaller than 150K tonnes, the dominant probabilistic distribution is the Weibull distribution. For tankers with deadweight greater than 150K but smaller than 300K tonnes (i.e., VLCC), there is no dominant probabilistic distribution. For tankers with deadweight greater than 300K tonnes (i.e., ULCC), the dominant probabilistic distribution is the lognormal distribution. Although the loading cases in loading manuals are treated as equally likely to occur, the information can be used as a starting point in the probabilistic assessment of hull girder ultimate and fatigue strength using the numeric characteristics of the corresponding probabilistic distributions given in the paper. For obtaining the probabilistic distribution of total SWBM and SWSF for modern ships operation we find necessary considering the effect of onboard computers used for checking the ships strength under any loading pattern and the effect of the weather forecast. Only then could one have greater confidence in the calculation of the hull girder ultimate strength where these total loads are used. 6 ACKNOWLEDGEMENT The authors thank the students Thomas Lynch, Deidre Norman, Steve Tuttle, Jason Searle from the Memorial University in Newfoundland, Canada and Dr. Abulbashar Alam from ABS for their help in extracting the raw data. 7 REFERENCES Alexander I Maximadji Estimation of the still water bending moments for tankers (in Russian), Proceedings of the Central Scientific Research Institute of Merchant Fleet, Leningrad, 1973, vol. 169, pp Carlos Guedes Soares Combination of primary load effects in ship structures, Probabilistic Engineering Mechanics, vol. 7, 199, pp Carlos Guedes Soares - Influence of human control on the probability distribution of maximum still water load effects in ships, Marine Structures, vol. 3, No. 4, 1990, pp Carlos Guedes Soares Probabilistic models for load effects in ship structures, Dissertation submitted in partial fulfillment of the requirements for the degree of Doktor IngeniÖr at the Norwegian Institute of Technology, Trondheim, June 1984, 386 pages 10 Probabilistic Presentation Of The Still Water Loads. 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7 Carlos Guedes Soares Stochastic Modeling of Maximum Still-Water Load Effects in Ship Structures, Journal of Ship Research, September 1990, vol. 34, No. 3, pp Carlos Guedes Soares, Mario Dogliani Probabilistic modeling of time-varying still-water load effects in tankers, Marine Structures, vol. 13, 000, pp Carlos Guedes Soares, Surgina Dias - Probabilistic Models of Still-Water Load Effects in Containers, Marine Structures, vol. 9, 1996, pp Carlos Guedes Soares, Torgeir Moan - Statistical analysis of still water bending moments and shear forces in tankers, ore and bulk carriers, Norwegian Maritime Research, No.3 198, pp Carlos Guedes Soares, Torgeir Moan - Statistical Analysis of still-water bending moments, shear forces in tankers, ore, and bulk carriers, III Iberoamerican Congress of Naval Engineering, Madrid 31 May - 5 June 198, 15 pages Carlos Guedes Soares, Torgeir Moan Statistical analysis of still-water load effects on ship structures, Transactions of the Society of Naval Architects and Marine Engineers, vol. 96, 1988, pp Edward V Lewis Load Criteria for Ship Structural Design, Report No. SSC-40, Ship Structure Committee, Washington, D.C., 1973 Egil Abrahamsen, Nils Nordenstrøm, Eivald M. Q. Røren Design and Reliability of Ship Structures, Det Norske Veritas Publication No. 73, November 1970 Hajimu Mano, Hiroshi Kawabe, Kunio Iwakawa, Nobuyki Mitsumune - Statistical character of the demand on longitudinal strength (Second Report) Long Term Distribution of Still Water Bending Moment (in Japanese), Journal of the Society of Naval Architects of Japan, 1977, vol. 14, pp Heinrich SÖding The Prediction of Still-Water Bending Moments in Containerships, Schiffstechnik, vol. 6, 1979, pp Lyuben D Ivanov Statistical estimation of still water bending moments for cargo ships - Fifth International Ship Structures Congress, Hamburg, September 1973, Proceedings, vol., pp Lyuben D Ivanov, Hristo Madjarov - The statistical estimation of still water bending moments for cargo ships, Shipping World & Shipbuilder, 1975, No. 3908, pp Manley St. Denis, Willard J Pierson, Jr. - On the motions of ships in confused seas, Transactions of SNAME, 1953, pp Michel Ochi Applied Probability and Stochastic Processes in Engineering and Physical Sciences, John Wiley & Sons, 1989 Peter Friis-Hansen, Ove Ditlevsen - A Stochastic Still Water Response Model, Technical Univ. of Denmark, DK-800 Kgs. Lyngby, Department of Mechanical Engineering, Section for Maritime Engineering, November 13, 001, 5 pages Sverre Gran Short-Term Still-Water Load Statistics, DnV Technical Report No , 199, 35 pages Sverre Gran Still-Water Load Statistics, DnV Technical Report No , December 1991, 6 pages Tadeusz Król Analysis of the statistical information for the load distribution onboard ships (in Polish), Politechnika Gdanska, Institute of Shipbuilding, Gdansk, Poland, 1974, 63 pages Truhin B V Determination of the Margin of Safety of Ships (in Russian), Proceedings of the Gorky Institute for Water Transport Engineers, vol. 88, 1970, pp , Gorky, USSR Wlodzimerz Trafalski Cargo Loads of General Cargo Ships (Preliminary study), Ship Design and Research Center, Strength Analysis Division, Proceedings, vol. No. 3/67, March 1967, Gdansk, Poland, 30 pages (in Polish) Appendix I: Tankers with DW < 150K DWT Weibull_p [Rank-] Rayleigh_1p [Rank-5] Lognormal [Rank-7] Normal [Rank-1] Fig. 1 Positive shear forces for 11 full load conditions of tankers with DW < 150K DWT Lognormal [Rank-4] Rayleigh_1p [Rank-8] Normal [Rank-11] Fig. Negative shear forces for 114 full load conditions of tankers with DW < 150K DWT Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 103

8 Normal [Rank-13] Logrnormal [Rank-9] Rayleigh_1p [Rank-] Fig. 3 Hogging bending moments for 4 full load conditions of tankers with DW < 150K DWT Normal [Rank-14] Lognormal [Rank-9] frequency n / N [-] Weibull_p [Rank-5] Rayleigh_1p [Rank-7] Fig. 6 Negative shear forces for 41 ballast conditions of tankers with DW < 150K DWT Normal [Rank-6] Logrnormal [Rank-9] Rayleigh_1p [Rank-11] Fig. 4 Sagging bending moments for 105 full load conditions of tankers with DW < 150K DWT Normal [Rank-10] Lognormal [Rank-8] frequency n / N [-] Rayleigh_1p [Rank-4] Fig. 5 Positive shear forces for 145 ballast conditions of tankers with DW < 150K DWT Histogrm Lognormal[R-11] Normal[R-3] 5 0 Rayleigh_1p[R-1] Fig. 7 Hogging bending moments for 39 ballast conditions of tankers with DW < 150K DWT Histogrm Normal [Rank-3] Lognormal [Rank-8] Rayleigh_1p [Rank-13] Fig. 8 Sagging bending moments for 8 ballast conditions of tankers with DW < 150K DWT 104 Probabilistic Presentation Of The Still Water Loads. Which Way Ahead?

9 Lognormal [Rank-] Rayleigh_1p [Rank-5] Normal [Rank-14] Fig. 9 Positive shear forces for 91 partial load conditions of tankers with DW < 150K DWT Lognormal [Rank-3] Normal [Rank-11] Rayleigh_1p [Rank-13] Fig. 10 Negative shear forces for 31 partial load conditions of tankers with DW < 150K DWT Lognormal [Rank-3] Normal [Rank-9] Rayleigh_1p [Rank-16] Fig. 11 Hogging bending moments for 3 partial load conditions of tankers with DW < 150K DWT Lognormal [Rank-4] Normal [Rank-8] Rayleigh_1p [Rank-1] Fig. 1 Sagging bending moments for 19 partial load conditions of tankers with DW < 150K DWT Appendix II: Tankers with deadweight 150K DWT 300K DWT Lognormal [Rank-] Weibull_p [Rank-3] Rayleigh_1p [Rank-9] Normal [Rank-11] Fig. 13 Positive shear forces for 89 full load conditions of tankers with DW = 150K Lognormal [Rank-4] Rayleigh_1p [Rank-11] Normal [Rank-13] Fig. 14 Negative shear forces for 89 full load conditions of tankers with DW = 150K Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 105

10 Lognormal [Rank-9] Normal [Rank-7] Rayleigh_1p [Rank-4] Fig. 15 Hogging bending moments for 35 full load conditions of tankers with DW = 150K DWT - 300K DWT Lognormal [Rank-3] Rayleigh_1p [Rank-1] Normal [Rank-10] Fig. 16 Sagging bending moments for 56 full load Rayleigh_1p [Rank-6] Lognormal [Rank-3] Normal [Rank-1] Weibull_p [Rank-4] Fig. 17 Positive shear forces for 8 ballast conditions of tankers with deadweight 150K DWT - 300K DWT Lognormal [Rank-] Rayleigh_1p [Rank-6] Normal [Rank-1] Fig. 18 Negative shear forces for 8 ballast conditions of tankers with deadweight 150K DWT - 300K DWT Lognormal [Rank-4] Normal [Rank-] Rayleigh_1p [Rank-13] Fig. 19 Hogging bending moments for 19 ballast Normal [Rank-3] Lognormal [Rank-6] Rayleigh_1p [Rank-13] Fig. 0 Sagging bending moments for 13 ballast 106 Probabilistic Presentation Of The Still Water Loads. Which Way Ahead?

11 Normal [Rank-6] Rayleigh_1p [Rank-16] Lognormal [Rank-10] Fig. 1 Positive shear forces for 14 partial load Weibull [Rank-1] Lognormal [Rank-5] Normal [Rank-6] Rayleigh_1p [Rank-15] Fig. Negative shear forces for 14 partial load Lognormal [Rank-5] Normal [Rank-] Rayleigh_1p [Rank-1] Fig. 3 Hogging bending moments for 101 partial load p. d. f. [-] Lognormal [Rank-7] Rayleigh_1p [Rank-13] Fig. 4 Sagging bending moments for 61 partial load Appendix III: Tankers with DW > 300K DWT Rayleigh_1p [Rank-9] Lognormal [Rank-1] Normal [Rank-11] Fig. 5 Positive shear forces for 3 full load conditions of tankers with DW > 300K DWT Normal [Rank-8] 5 Lognormal [Rank-] Rayleigh_1p [Rank-13] Fig. 6 Negative shear forces for 3 full load conditions of tankers with DW > 300K DWT Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 107

12 Normal [Rank-8] Lognormal [Rank-] Rayleigh_1p [Rank-1] Fig. 7 Sagging bending moments for 3 full load conditions of tankers with DW > 300K DWT Rayleigh_1p [Rank-9] 5 Lognormal [Rank-3] Normal [Rank-11] Fig. 30 Hogging bending moments for 13 ballast conditions of tankers with DW > 300K DWT Normal [Rank-17] Rayleigh_1p [Rank-6] Lognormal [Rank-9] Fig. 8 Positive shear forces for 13 ballast conditions of tankers with DW > 300K DWT Normal [Rank-10] Weibull_p [Rank-3] Rayleigh_p [Rank-15] Fig. 31 Positive shear forces for 101 partial load conditions of tankers with DW > 300K DWT Rayleigh_1p [Rank-] Lognormal [Rank-9] Normal [Rank-1] Weibull_p [Rank-14] Fig. 9 Negative shear forces for 13 ballast conditions of tankers with DW > 300K DWT Lognormal [Rank-] Normal [Rank_5] Rayleigh_1p [Rank-14] Fig. 3 Negative shear forces for 10 partial load conditions of tankers with DW > 300K DWT 108 Probabilistic Presentation Of The Still Water Loads. Which Way Ahead?

13 Normal [Rank-8] Lognormal [Rank-4] Rayleigh_1p [Rank-11] Fig. 33 Hogging bending moments for 58 partial load conditions of tankers with DW > 300K DWT Normal [Rank-7] 0 Lognormal [Rank-4] Rayleigh_1p [Rank-13] Fig. 34 Sagging bending moments for 44 partial load conditions of tankers with DW > 300K DWT Probabilistic Presentation Of The Still Water Loads. Which Way Ahead? 109