Department of Naval Architecture Faculty of Engineering Kyushu University

Transcription

1 Contribution to 3rd 1.S.S.C., Oslo, 1967 THEORETICAL DETERMINATION OF DESIGN WAVE BENDING MOMENTS by Jun-ichi Fukuda (Reprinted from Japan Shipbuilding & Marine Engineering, Vol. 2, No. 3, May 1967) Department of Naval Architecture Faculty of Engineering Kyushu University Hakozaki, Fukuoka, Japan

2 Theoretical Determination Of Design Wave Bending Moments By Jun-ichi Fukuda Department of Naval Architecture Kyushu University Introduction In 1935, an experimental study on ship hull stresses in waves was planned by the Naval Technical Research Institute of Japan, following the accidents of two destroyers breaking up in a severe typhoon during their battle practice. In 1936, the first towing model test was carried out in an experimental tank by SatO'>, which heralded the beginning of experimental study on wave bending moments in the world, and was followed afterwards by many researches in this field. In this writing, we are to review the past history of study on wave induced loads on ships in Japan and to observe the present situation. After the preliminary test in l936 Sâto tried a series of model experiments in 1939 and later on, but he had to give up his work at the end of the Second World War in 1945 without completing it. Since Sato s experiments there was nothing notice able in that sort of research except the full scale measurements on the M/S "San Francisco" and S/S "Ocean Vulcan", until Lewis' experiments on a T2 tanker model2' were reported in Following Lewis work experimental studies on wave loads on ship hulls were forwarded in a great number over the world. In Japan also, a number of experimental reports by Akita3>,. Ochi4> and others appeared in. succession. In 1961, there were contributed three experimental works on T-2 tanker models (by Taniguchi, Akita and Fukuda) to the Committee on Wave Loads on Ships 1st International Ship Struc tures Congress, Glasgow5>. And, in 1964,:threë experimental reports on two destroyer models and a T-2 tanker model (by. Fukuda, Nozaki and Yamanouchi) were presented to the 2nd. International Ship Structures Congress, Delft6>. While those experimental studies were carriçd Out, several theoretical researches on wave bending moments were performed by Watanabe7' 8) and Hanaoka9'. Besides, Fukuda' > tried a series of theoretical calculations.. on ship motions. and wave bending moments, based upon the linear strip theory given by Watanabe8>, which was similar in principle to Korviii-Kroukovsky'S method11>. Fukuda's calculations were made on Taniguchi's experiments with a T-2 tanker model5', and the results showed a fair1 good agreement between theory and experiment. Fukudat2> developed his method and completed the computer program for response operators of ship motions and wave bending moment in regular oblique waves. Besides those experimental and. theoretical studies on the response to regular waves, statistical studies On the response to irregular sea waves were forwarded by a number of authors. Akita and others13' 14) tried, short-term predictions Of wave bending moments induced on ship hulls in rough seas, based upon the linear superposition method proposed by St. Denis and Pierson15>, and utilizing response operators approximately estimated. Fukuda and others'6' 17, 18) made similar shortterm predictions on a destroyer ship form and a super tanker, where they used the response operators calculated by a linear Strip method. Masuda'9' 2) investigated longterm distributions of the so-called V'E-value (root mean square of all amplitudes) of wave bending moments on ships and estimated approximate design wave bending moments for ships, which were taken into account for the Construction Rules of Nippon Kaiji Kyokai. 'Studies on wave induced loads on gigantic tankers were recently informed by several authors. Shimada and others21> made short-term predictions of wave induced shearing forces and bending moments distributed along the hull length of a gigantic tanker in irregular head seas by means of theoretical calculations, And, Fukuda and others22' 23) carried out short- and long-term predictions of wave bending moments for gigantic tankers. Their work is entirely based upon theqti cal methods by iicing...calculated response operats in reu1ar ob1iqtiwaves. and availing lone-term frequencies of observed ocean waves. For the purpose of investigating short- and longterm distributions of hull stresses on ships in operation, full-scale measurements on actual ships have been continued for many years since 1954 by the Shipbuilding Research Association of Japan, Ship May

3 Research Institute and others, and statistical analyses of those have been forwarded24' 25, 26) Some improved proposals were made for the analysis of fullscale tests by Sakao27), Wätanabe28) and others The above is a survey of the. history of study on wave induced loads in Japan. At present, research works are forwarded mostly in order to determine the design wave bending moments by means of theoretical methods and full-scale tests. There will be described, in the following sections, an outline of recent researches for the theoretical deter-. mination of design wave bending moments, and also discussed on the longitudinal distribution of wave bending moments along a ship hull. Response Operator of Wave Bending Moment According to experimental works on T-2 tanker models carried out by Taniguchi, Akita and Fukuda5, and the theoretical analysis on those made by Fukuda1), the characters of vertical wave bending moments on a ship in regular head waves were gradually revealed, and the following facts were confirmed. The effect of ship speed 7n midship wave bending moment is generally not so significant at, a range of usual ship speed. Though the wave length that gives the maxi mum amplitude of midship wave bending moment varies with ship speed, the wave length nearly equal to the ship length gives generally the maximum amplitude of midship wave bending moment. A linear relation holds fairly good between wave height 'apd- amplitude of midship wave bending moment. - - (d) The weight.distrihution that gives a hogging midship bending moment in still water causes generally smaller amplitudes of midship wave bending moment than those caused by the weight distribution resulting in a sagging midship bending moment in still watcr. (C) The longitudinal distribution of amplitudes of wave bending moment along a ship hull is approximately symmetric about midship and the maximum amplitude takes place very closely amidships at a range of low ship speed. At high speed, however, the longitudinal location where the maximum, amplitude occurs shifts rather forwards. - Those findings are valuable but not sufficient to be availed for the determination of design wave bending moments. As a step for that purpose, is necessary to obtain" the respànse operators of wave bending moment in regular waves. -Theoretical methods based upon a linear 'strip theory are quite useful for evaluating such response operators. The author tried previously theoretical calculations of wave bending moments on a T-2 tanker model and a destroyer model, and, found a practically satisfactory agreement between calculated arid experimental results' 29) Some examples of those are shown in Figs. 1 and. 2, where are given the results on a T-2 tanker model in regular head waves. The 'following notations are employed in the diagrams..,k/l.75 :XpRlN -: CALCULATION A/L=L25 T - :EXPERIMENT CALCULATION.. A/L=I yr ' ' -. a.2 A/L= '12 3 o Fr..' - Fr. Fig. 1- Comparison of Calculated and Experimental Results of Wave Bending Moment Amplitudes at Midhip On A T-2 Tanker Model in Regular Head Waves - Japan Shipbuilding & Marine Engineering

4 o : EXPERIMENT. - CALCULATION A/L., Fr..15 A/I.., Fr..2 a.. C.=.83 L/B6., L./d= 7.5 Bid -2.9)7 V'-O (HEAD WAVES) -' _.._ ---- Fr.- " -.5 : " =.2.) // "S" 1 I S 2. Fig. 2 Comparison of Calculated and Experimental Results of Wave Bending Moment Amplitudes Distributed over the Length of A T-2 Tanker Model in Regular Head WaveS M: amplitude of vertical wave bending moment p density of water acceleration of gravity L: ship length between perpendiculars B: ship breadth h wave amplitude V A: wave length Fr.: Froude number The calculation method is based on Watanabe's linear strip theory8), taking into account effects of ship speed and wave orbital motion, and using Tasai's cross sectional values for damping and added mass'. The author developed his method and extensively applied to the case in regular oblique waves, and completed the computer program for response operators of ship motions and wave bending moment in regular oblique waves'21. Examples of the computer program results for a gigantic tanker ship form are shown in Figs. 3 and 4, where the dimensionless amplitudes of midship wave bending moment are given as functions of v'ship length/wave length with the parameters of Froude number and heading angle to waves. The following notation is used in the diagrams besides those described above. 4: heading angle to waves (i = : head waves) Main particulars of the ship form are given in Table 1, and the weight distribution data, in Table 2. Thus, we can evaluate the response operators of wave bending moment on a ship in regular May 1967 Fig. Amplitudes of Midship Wave Bending Moment on A Gigantic Tanker in Regular Head Waves ) C.=o.83 L/B6., Lid = 7.5 B/d2.917 Fr..15 :. - 3 _-_ :I2 ----, V4U VI,./, /\j \\ Fig. 4 Amplitudes of Midship Wave Bending Moment on A Gigantic Tanker in Regular Oblique Waves Table 1 Main Particulars of A Gigantic Tanker Ship Form Length/Breadth (L/B) 6. Length/Draught (L/d) 173 Breadth/Draught (Bid) Block Coefficent (cb).83 Water Plane Area Coefficient (C,,,).891 Midship Coefficient (Cm).993 Center. of BUoyancy from Midship (forwards).336l oblique waves by the aid of linear strip theory. Therefore, it is possible to predict short- and longterm distributions of wave bending moment on a ship in ocean waves by the theoretical methods availing the obtained response operators and the long-term wave statistics. 15

5 Thble 2 WeIght Distribution for A Gigantic Tanker Ship Form Weight 1st. Mt. about Midship C. G. from Midship 2nd. Mt. about Midship Longitudinal Gyradius Still Water Midship Bending Moment After Body Fore Body.4825W.5175W WL.12124WL.1816L O.2343L.2225WL2 Short-Term Parameter of Wave Bending Moment O.389WL2 Total In the statistical methods for analysing a ship response to sea waves such as the wave bending moment, the basic assumption is that the elevation of sea waves and the response of a ship to waves are stochastic processes, which follows approximately the "Gaussian" law of statistics, and the peak values of those (usually called amplitudes) are distributed practically according to the "Rayleigh" law if taken over a sufficiently short period. The applicability of this assumption has been verified by a number of investigators withship data of full-scale measurements. The Rayleigh distribution has been called so often the "short-term. distribution", because it can practically express the short-term distribution 5U the ship response to waves as_well as that (if the elevation of sea waves.. There has been used the standard deviation "R" or the so-called /-value (which is the root mean square of all amplitudes and equal to R) as a convenient parameter of the Rayleigh distribution. If we know the value of R, we can statistically estimate the expected maximum of a certain number of amplitudes, tlie probabilitythat the amplitude takes on values larger than a constant value, etc.,availing the R-ve. Thus the characters of a short'term distribution of the ship response to waves can be represented by the parameter R. Therefore, let us have considerations on the R-value. The standard deviation of wave bending moment in a short period can be evaluated theoretically by the calculations on energy spectrum of wave bending moment, based on the linear superposition method proposed by St. Denis and Pierson15, as follows: Supposing that a ship operates among a seaway with a constant speed and a constant heading angle to waves, 16 W.OSS6IWL.3361L.6115WL2.245L -.3OSWL (Sag) where R2 : variance of wave bending moment R : standard deviation of wave bending moment [f(w,x)]2 : spectral density of directional component wave [M(o,9 x)1 : response amplitude for heading angle (9x) w : circular frequency of component wave x : angle between a component wave direction and the average wave direction : heading angle of ship to the average wave direction A number of theoretical wave spectra have been propose4 by ocean-graphers. Here, let us take up the modified Pierson-Moskowitz wave spectrum (I.S.S.C. spectrum)3i), assuming that the short crested: ness of sea waves can be simly introduced ly, (cosine)2 distribution for the directional. spectral density. Then, [f(cv, x]2 in the formula (1) may be given by where R2= (w,x)]2[mo(o,o )]2dwdX (1) [f(w,x)]2 = [f()2 cos2 x: Tr/2 x r/2 (2) =: elsewhere [f(w)j2.11 H2wo1(w/wo)6 exp ( O.44(w/w)) ( 3) wo=27r/t H : significant wave height T : average wave period Formulas (2) and (3) represent the short crested irregular waves having the significant wave height H and the average wave period T. When the long crested irregular waves are assumed, only the formula (3) should be taken, and the following formula may be used instead of the formula (1). R2= [f(w)]2[mo(w,9)j2dw...(4) There are shown, in Figs. 5 and 6, examples of the calculated results of standard deviation of midship wave bending moment for gigantic tanker ship forms, whose main particulars and weight distribu tion data are given in Tables 1 and 2. Calculations were carried out by using such response operators as shown in Fig. 4, and applying formulas (1), (2) and (3) on the assumption of short crested irregular waves. In Fig. 5, the dimensionless standard deviations of midship wave bending moment on a 3-meter long tanker are given as functions of average wave period with the parameters of Japan Shipbuilding & Marine Engineering

6 .4 - C =.83. L3M L/B6., L/dl7.5. Bid ' (HEAD WAV5) C L/8-G., Lid = 17.5, Bid d'(head WAVES), Fr I Fr. =.15 e-o' Fr.= -'--''j '5' T(sec) Fig. 5 Standard Deviations of Midship Wave Bending Moment on A Gigantic Tanker n Irregular Waves Froude number and heading angle to waves. And, in Fig. 6, those for the geometrically similar tankers having different lengths, which operate in head seas, are shown as functions of ship length with the parameter of average wave period. As mentioned above, the value of R that represents a short-term distribution of wave bending moment can be evaluated for a given ship as a function of sea state (which is defined by H and T), heading angle to waves and ship speed by the theoretical methods. In the following section, there will be discussed on the resulting long-term dis. tribution of wave bending moment predicted by availing the obtained response operators and longterm frequencies of observed ocean waves. Long-Term Distribution of Wave Bending Moment 9d'1 When the short-term parameter of wave bending moment "R" is known, the probability q(m>m1), which is the probability that the variable M (peak value of wave bending moment) takes on values larger than M1, is given by q(m>m1)=exp(m12/2r2) (5) Accordingly, the total exceeding probability' Q(M>M1) that M exceeds M1 during a very long period such, as a life time of a ship can be obtained by integrating q(m>m1) multiplied by p(r), which is the distribution function of R, over all values of R. Namely,.2-.I - L Cm) Fig. 6 Standard Deviations of Midship Wave Bending MOment on Geometrically Similar Tankers in Irregular Head Waves Q(M> M1) = exp (M12/2R2).p(R)dR...(6) The integral may be also evaluated numerically by the summation as follpws; Q(M>M1)= exp(.m12/2r2).p,, (7) where p is the long-term probability that R yields among seaways on which a ship operates. Some' authors considered theoretical distribution functions, such as "log normal" and "Weibull" dis. tributions, to be fitted to the long-term distribution of R-value, and estimated the resulting long-term distributions of.peak value of wave bending moment. It is, however, not always necessary to theoretical distribution of R-value in order to ohtai,. the long-term distribution of peak value of wave bending moment, if we have sufficient data on the g-term distribution of. sea waves by sea and route to be availed for evaluation of a large number of R-values. The final object is not the long-term pi'ediction of R-value but that of peak value of wave bending moment. Since the short. term parameter of wave bending moment can be evaluated for a given ship as a function of sea state (which is defined by the significant wave height H and the average wave period T), heading angle to waves and ship speed by the theoretical methods, the long-term distribution of peak value of wave bending moment will be obtained by the method as described below. Statistics of the significant wave height H and the average wave period T for sea areas were given by the Committee on Environmental Conditions, I.S.S.C., 196c'>. Those on the North Atlantic were reported by Roll32> and recently by Walden33> in full detail, and those on the North Pacific were also May

7 given in detail by Yamanouchi and others34. In those reports, the long-term distributions of ocean waves were given as functions of the significant wave height H and the average wave period T which were classified into -a number of divisions respectively. rr* Supposing that H denotes the significant wave height of the interval No. i, and T, the average wave period of the interval No. j, which are the mean values of lower and upper limits of the intervals No. i and No. j respectively, let S denote a seaway having the significant wave height H. and the average wave period T,. Then, the probability p that denotes the long-term frequency of the seaway S,, can be known for sea areas and routes from those wave statistics mentioned above. In the second place, supposing that the whole range of heading angle to waves (which is equal to 2ir) is divided into a number of equal intervals and e, denotes the interval No. k, which is the mean values of lower and upper limits of the interval No. k, let RJk denote the short-term parameter of wave bending moment (standard deviation of wave bending moment) on a ship operating among a seaway S,,, with a constant heading angle 9k and a constant speed. Then, R12,, can be evaluated theoretically by the method described in the preceding section, and the probability of occurrence of R,fr is equal to p, which is the long-term frequency of Se,, if Th1j operates among seaways always with a constant heading angle k and a constant speed. Accordingly, the long-term distribution of peak value of wave bending moment can be Obtained by using the formula (7), which may be written in this case as follows; Qk(M>Mi)= E E exp(_ml2/2r2jk)pl1...(8) where Qk(M>Mi) denotes the total, exceeding probability that the peak value of wave bending moment M takes on values larger than M1 when a ship operates among seaways always with a constant heading angle e, and. a constant speed. If the probability P'ijk is known, which denotes the probability that a ship takes a constant heading angle k in a seaway S,,, keeping a constant speed, the long-term distribution of peak value of wave bending moment will be evaluated by taking all heading angles into consideration, as follows; Q(M> M1) = exp (Ml2/2R2jjk)p'JkpJ...( 9) where Q(M>Mi) denotes the total exceeding probability that the peak value of wave bending moment M takes on values larger than M1 when all heading angles are taken into account. Assuming that P'ijk is given by the following, independent on a seaway Si,, a heading angle ek and ship speed, 18 PUk 1/N (1) the formula (9) may be written simply as f11ows: Q(M>M1)=(l/N) E Qk (11) Ic where N is the total number of intervals of heading angle. Main results of the long-term predictions of midship wave bending moment on gigantic tankers, performed by the author2! recently, are shown in Figs. 7ll. Those predictions have been made on tankers, operating among the North Atlantic by availing Walden's wave statistics33, where sea states are most severe 'and wave data are collected most plentifully, though the sea area is not considered as a typical tanker route. e -9'( BEAM) IN THE NORTH ATLANTIC C. O.83. L3OOu L/B-6.O, Lid BId =2.917 Fr. =.15 ALL. HEADINGS B O(HEAD) 8-i 8'(FOLLOwING) FIg. 7 Long-Term Distributions of Midship Wave Bending Moment on A Gigantic Tanker IS XIS4) C IN THE NORTH ATLANTIC LJB6.O ALL HEADINGS Lid = I 7.5 Q=Io' Fr,=.15 B/d =29I M/pgL'B H, BY N.K. RULES L Im) Q IO 4 5 Fig. 8 Midship Wave Bending Moments and Effective Wave Heights Predicted with Different Exceeding Probabilities for Geometrically Similar Tankers Japan Shipbuilding & Marine Engineering 5

8 C, =.83 LIB =6. Lid = I 7.5 ' Bid J C,=O.83 L/B-5.O, B/d I Lid , (2 M -7., =2.5 SPEED 1 CT. 1 OO SPEED I'<r. 8-6 '( (2 SPEED: I 8icr IO C LiB -6. Lid - I 7.5 Bid -2.9I7.. IN THE NORTH ATLANTIC ALL HEADINGS, Q -I ( 2 3, 4 5 L Cm) Fig. 1 Effective Wave Heights for Tankers Having Different Length/Breadth Ratios May = =18. :SPEED= OKT L (ml Fig. 9 Midship Wave Bending Moments and Effective Wave. Heights Predicted with Exceeding Probability of 1-8 for Geometrically Similar Tankers The following notations are used in the diagrams: M : value of wave bending moment expected to be exceeded with a certain probability He: effective wave height, corresponding to M Q : exceeding probability There are shown, in Fig. 7, the longterm distributions of midship wave bending moment on a 3-meter long tanker evaluated by using such R- values as shown in Fig. 5, and applying formulas (8) and (11). Similar calculations have been made on the geometrically similar tanker ship forms having different lengths and different speeds. Results from those are given in Figs. 8 and 9. The former shows the dimensionless values of midship wave bending moment and the corresponding effective wave heights, which are predicted with different exceeding' probabilities, as functions of ship lcngth, and the latter shows those predicted with the exceeding probability of Q=18 at different ship speeds. There are given in Fig. 8 also the design values of effective wave heights recommended formerly by Nippon Kaiji Kyokai; as compared with N 'a JO SC a SPEED: 4 r. I 8 c =.83 1 LIB 6. DO SC 2 SPEED: 18KT. 1 8 SPEED: KT. 6 I 2 Lid = 12., Bid -2. (5., =a.5 =(7.5,- = L (ml 5 Fig. 11 Effective Wave Heights for Tankers Having Different Length/Draught Ratios the predicted results. It seems to.,be reasonable that the dimensionless values of midship wave bending moment or the corresponding effective wave heights predicted with the exceeding probability of l8 or I among the North Atlantic will be taken as the design values for large tankers, arid the design values of, effective wave, height may be constant for large tankers longer than 35 meters independent on their lengths. 19

9 Similar statistical predictions have been tried on a series of tanker ship forms having a constant length/draught ratio and different length/breadth ratios, and another series of those having a constant length/breadth ratio and different length/ draught ratios, in order to investigate the effects of principal dimension ratios on the long-term trends of midship wave bending moment. Results are shown in Figs. 1 and 11, where the effective wave heights predicted with the exceeding probability of QlO8 among the North Atlantic are given for different ship speeds as functions of ship lengtli. The effect of length/breadth ratio is shown in Fig. 1, and that of length/draught ratio in Fig. 11. As found in those diagrams, the larger length/breadth ratio gives generally a little larger effective wave height, but the trend of the effect of length/draught ratio is not uniform. Those results obtained by the theoretical evaluations explained above will give answers to the question how io certify the same level of risk probability for all ships, so far as the wave bending fnoment is concerned, even though the absolutely definite design values of wave bending moment cannot be determined by those. It is, however, of minor importance what probability level is adopted, as long as the same- probability of the risk is used for all ships. It is the fact that ships can never be designed without any risk probability at all, and the most important. task is to standardize the risk probability based on the theoretical methods and the empirical facts. Such methods as described above will be adequate to that purpose. Longitudlinal Distribution of Wave Bending Moment along Ship Length Much have been discussed on the short- and long-term distributions of midship wave bending moment in the preceding sections. Besides those, however, it is necessary to investigate the longitudinal distribution of wave bending moments along a ship hull, in order to achieve the practical design for the longitudinal strength. Such investigations have been reported recently by Shimada and others21 They have investigated the short-term wave bending moments induced along the hull length of a gigantic tanker in long crested. irregular seas. by means. of.theoretical methods, Their work is based upon the method similar to that described in the section on.shortterm parameter of wave bending moment, but they tised the modified Neumann spectrum instead.of the modified Pierson-Moskowitz spectrum in order to formulate the long crested irregular waves. There are assumed the long crested irregular seas corresponding to the average- sea states of the North Atlantic as functions of wind velocity based on the wave, data given 2 a AP Fig. 12 Longitudinal Distributions of Amplitudes of Wave. Bendmg Moment along A Gigantic Tanker's Hull in Regular Head Waves a 3 tn REGULAR HEAD WAVES (AlL = I CX lo) 3.. IN IRREGULAR HEAD SEAS 2.. AP Fr.= - = :., =.2 WIND VEL. 35m/sec H- 8.52m T sec Fig. 13 Longitudinal Distributions of Amplitudes of Wave Bending Moment along A Gigantic Tanker's Hull in Irregular Head Waves by Ro1132, for the purpose of estimating the average values of 1/1 highest of wave bending moments induced along the hull length of a gigantic tanker in those sea waves. Examples of the results of their calculations are shown in Figs. 12 and 13. Thosç diagrams show the results for a gigantic tanker in full loaded con dition, whose main particulars are given in Table 3 There are shown in Fig 12 the longitudinal dis tributions of amplitudes over the hull length of the. ship in regular head waves, and, in Fig. 18, the average values of 1/1 highest of those in shortterm irregular head waves where is assumed a sea way corresponding to the average sea state of the North Atlantic when the wind of 35 rn/sec blows. In Fig. 13, the following notation is used: M(l/lo) : average value of 1/1 highest of wave bending moment amplitudes Japan Shipbuilding & Marine. Engineering.. FP FP

10 Table 3 Main Particulars of A Gigantic Tanker Length between Perpendiculars (L) Breadth (B) As found in Fig. 12, amplitudes of wave bending moment on the ship in regular head waves are distributed approximately symmetric about midship and the maximum amplitude occurs nearly amidships at low speed, but the distribution of amplitudes of wave bending moment is no more symmetric and the maximum amplitude occurs rather in front of midship at high speed. However, as shown in Fig. 13, those trends at high speed are not so significant in irregular head waves as compared with the case in regular head waves, and the longitudinal 'distribution of amplitudes of wave bending moment 'on the ship in irregular head waves may be looked upon practically symmetric at a range of usual ship speed. From those results, we can assume approximately that the maximum amplitude of wave bending moment occurs at midship in a short-term irregular seaway, and consequently the long-term midship wave bending moments may be accepted as the design values of wave bending momeñt for ships. Shimada and others have made the similar calculations on the ship in ballast loaded condition, and obtained the results of similar trends to those in full loaded condition. It is noticed, however, that the amplitudes of midship wave bending moment are generally a little larger in ballasted condition as compared with those in full loaded condition. Conclusion 81. m 48.4 m Depth (D) 23.6m Draught (d) 17.8 m Block Coefficient (Cb).88 Center of Buoyancy from Midship (forwards).826l Longitudinal Gyradius.2837L An outline of recent research works on the theoretical determination of design wave bending moments in Japan is as above-mentioned, and a promise for solving the problem is given. However, in order to determine the sufficiently reasonable design values of wave bending moment for ships, further investigations will be necessary on the effects of ship size, hull form, loading condition, weight distribution, ship speed, etc., upon the long-term trends of wave bending moments. For that purpose, the theoretical methods described in this paper may be quite adequate, if more accurate wave data will be obtained over the world sea areas and routes in the near future. Reference 1) M. Sato: "Model Experiments on the Longitudinal Strength of. Ships Running among Waves", Journal of the Society of Naval Architects of Japan, VoL 9 (1956).2) E. V. Lewis: "Ship Model Tests to Determine Bending Moments, in Waves", Trans. S.N.A.M.E., Vol. 62 (1954) Y. Akita and K. Ochi: "Model Experiments on the Strength of Ships in Waves", Trans. S.N.A.M.E., Vol. 63 (1955) K. Ochi: "Model Experiments on Ship Strength and Slamming in Regular Waves", Trans. S.NAM.E., Vol. 66 (1958) "Report of Committee on Wave Loads", International Ship Structures Congress, Glasgow (1961) "Report of Committee 2b-I on Wave Bending, Shear and Torsion, Model and Theory", Proceedings of International Ship Structures Congress, Delft (1964) Y. Watanabe: "The Strength of Ship Going among Waves" Memoirs of the Faculty of Engineering, Kyushu University, Vol. 16, No. 4 (1957) Y. Watanabe: "On the Theory of Heaving and Pitching Motions", Technology Reports of the Faculty of Engineering, Kyushu University, Vol. 31, No. 1 (1958) T. Hanaoka: 'On the Calculation of Motion and Bending Moment of a Ship among Waves", journal of the Society of Naval Architects of Japan, Vol. 11' (1957) J. Fukuda: "On the Bending Moments of a Ship in Regular Waves", Journal of the Society of Naval Architects of Japan, Vol. 11 (1961) and Vol. 111(1962) B. V Korvin-Kroukovsky and W. R. Jacobs: "Pitching and Heaving Motions of a Ship in Regular Waves", Trans. S.N.A.M.E. Vol. 65 (1957) J. Fukuda: "Computer Program Results for Response Operators of Wave Bending Moment in Regular Oblique Waves", Memoirs of the Faculty of Engineering, Kyushu University, Vol. 26, No. 2 (1966) Y. Akita: "Bending Moments of Ships in a Seaway", Journal of the Society of Naval Architects of Japan, Vol. 12 (1958) W. Akita S. Tashiro and K. Goda: "A Method of Predicting Stresses of Ships in Rough Seas", Journal of the Society of Naval Architects of Japan, Vol. 16 (196)' M. St. Denis and W. J. Pierson, Jr.: "On the Motions of Ships in Confused Seas", Trans. S.N.A.M.E., Vol. 61 (1953) J. Fukuda, J. Shibata and H. Toyota: "On the Longitudinal Strength of a Super Tanker in Regular Head Waves", Journal of the So- May

11 Japan ciety of Naval Architects of West Japan, No. 26 (1963) J. Fukuda, J. Shibata and H. Toyota: "Midship Bending Moments Acting on a Destroyer in Irregular Seas", Journal of the Society of Naval Architects of Japan, Vol. 114 (1963) J. Fukuda and J. Shibata: "The Effects of Ship Length, Speed and Course on Midship Bending Moment, Slamming and Bow Submergence in Rough Seas", Memoirs of the Faculty of Engineering, Kyushu University, Vol. 25, No. 2 (1966) Y. Masuda: "Stat stical Amidship Bending Moment for Ships", Journal of the Society of Naval Architects of Japan1 Vol. 111 (1962) Y. Masuda: "Proposed Rules for the Longitudinal Strength of Cargo Ships", Journal of thea Society of Naval Architects of Japan, Vol. 113 (1963) H. Shimada; M. Ogata and M. Konuma: "Longitudinal Distribution of Wave Bending Moments and Shearing Forces of a Gigantic Tanker in Regular and Irregular Head Waves", read at the Spring Meeting of the Society of Naval Architects of Japan in May J. Fukuda: "A Method for Predicting Long- Term Wave Loads on Ship Hulls", Bulletin of the Society of Naval Architects of Japan, NO. 448 (1966) J Fukuda, M. lizuka, M. Ogata and M. Konuma "Long-Term Prediction of Wave Bending Moments on Gigantic Tanker Hulls", read at the Spring Meeting of the Society of Naval Architects of West Japan in May Y. Akita and I. Ishiyama: "Statistical Measurement of Ship Stress on M/S Hodakasan Maru", Journal of the Society of Naval Architects of Japan, Vol. 15 (1959) "Experiments on the Stress Frequency and Deck Wave Load Acting on the High Speed Ships in Rough Seas", Report of the Shipbuilding Research Association of Japan, No. 49 (1965) N. Ando, H. Nagasawa and N. Shimada: "Experiments on the Strength of Liner Cargo Boat in Service", Journal of the Society of Naval Architects of Japan, Vol. 119 (1966) M. Sakao: "On the Long-Term Statistical Distribution of Random Variable in Ships", Journal of the Society of Naval Architects of Japan, Vol. ill (1962) Y. Watanabe: "On the Statistical Method of Analysis of Bending Stresses of Ship at Sea", Bulletin of the Society of Naval Architects of Japan, No. 429 (1965) J. Fuküda, J. Shibata, H. Toyota and A. Hoshi kuma: "Theoretical Evaluation of Bending Moments of a Destroyer in Regular Waves", Journal of the Society of Naval Architects of Japan, Vol. 112 (1962) F. Tasai "On the Damping and Adde4 Mass of Ships Heaving and Pitching", Reports of Research Institute for Applied Mechanics, Kyushu University, Vol. 7, No. 26 (1959) and Vol. 8, No. 31 (196) "Report of the Committee on Environmental Conditions", Proceedings of LS.S.ç., Delft (1964) H. U. Roll: "Height, Length and Steepness of Seawaves in the North Atlantic and Dimensions of Seawaves as Functions of Wind Forces", Technical and. Research Bulletin No. 1-19, S.N.A.M.E. (1958) H. Walden: "Die Eigenschaften der Meerswellen im Nordatlantischen Ozean", Deutscher Wetterdienst, Seewetteramt, Einzelveröffentlichungen Nr. 41, Humburg (1964) Y. Yamanouchi, S. Unoki and T. Kanda: "On the Winds and Waves on the Northern North Pacific Ocean and South Adjacent Seas. of Japan as the Environmental Condition for the Ship", Papers of Ship Research Institute, Tokyo, No. 5 (1965). 22. Shipbuilding & Marine Engineering