Draft

l4th International Ship Structure Congress

_1970.

Report of Committee

2.

(Draft)

Membership of Committee:

Prof.Ir. J. Gerritsma (Chairman) Ir. F.X.P. Soejadi (Secretary) Prof. J. Fukuda

R.A. Goodman

Prof. G.J. Goodrich Dr. P. Kaplan

Prof. E.V. Lewis Ir. J.H.G. Verhagen

1.Mandate of the Comittee.

To survey and summarize current activities in the field of model

experiments and theoretical investigations to determine wave loads on ship such as bending, shear, torsion and local loads, including hydrodynamic aspects of slamming

2.Work of the Committee.

Since the 3rd International Ship Structure Congress in Oslo the Com-mittee has met at Feltham

₍₁₉₆₈₎

_{and in Rome}

_(1968).

_{In addition a meeting} of the Chairmen of Committees 1, 2 and 3 was held at Feitham

_{(1968) to}

discuss matters of common interest. It was agreed that the function of Committee 2 _{should be to provide the information required by Committee 3} for the estimation of wave loads.

In accordance with the

₁₉₆₇

_{ISSC recommendations and the mandate of this} Committee, attention has been directed to matters such as bending moments and torsion in oblique waves, local pressure loads and the hydrodynamics of slamming. As far as information was available, these subjects are reviewed in the Report; two Appendices, written _{by Committee members concei-riing the} calculation of wave bending moments in oblique waves and the hydrodynamic aspects of slamming, are added.

-2-Ó

3.Wave bending moments and torsion.

3.1.Model experiments and calculations.

In the 1967 report of Committee 2b' (wave loads, model and theory), it was stated that the computation of the wave bending moments in

longitudinal waves by using the strip theory is sufficiently accurate for practical purposes. Model experiments have confirmed this statement to some extent. As a result an increasing use of the computer for wave bending moment calculations could be observed. In this way,a number of authors have studied the influence of systematic variation of the main dimensions and weight distribution of the ship on the wave bending moment and pressure distributions. One illustrative example of such a parameter study is given by Marsich and Merega (i). They studied the hull forms which are used for tankers and bulk-carriers. The most probable peak value of the vertical wave bending moment over 1O

cycles has been considered in a long crested

60

knot Scott sea spectrum, as a function of the length-breadth (L/B), the length-draft (L/T), the Froude number and the weight distribution. A very large number of combinations of these parameters were analysed and the results lead to some general conclusions. From the analysis o the total set of results a formula was produced which seems to cover well all possible values within the considered range of parameters:

Cm = - 21.38Fn)p +

- (1.91-12.-

8.l5Fn)]

iO3Li

where:

c = M X* _{maximum energy wave length}

m

3'

pgEL

DAVXAV+DPJXJJ DL

D = displacement

DAy displacement forward of the ship's centre of gravity

XAV = distance between ship's centre of gravity and DAy's centre of gravity

D and are the corresponding values for the displacement aft.

-3-weight distribution. It was found that for constant Fn (up to Fn = 0.10) Cm is a linear function of p, although the loading conditions were

different.

Also Teraawa published results of wave bending moment response operators for cargo shipforms and tankers up to a blockcoefficient of CB = 0,85 and with L/B ratio's as low as 5. (2).

The methods of calculation are invariably based on the strip theory and thus two dimensional solutions for the hydrodynamic forces on oscillating cylinders in the surface of a fluid are of prime interest, especially

because unconventional cross sectional forms,( such as extreme bulbous bows) are now used in modern ships. The classic Lewis transformation which has been used. with success in many cases is not sufficient for arbitrary cross

sections, so multi-coefficient mapping formulas are now in use with

considerable success(,1i,.Also other methods, for instance those employing a distribution of singularities on the cylinder surface, as reported by Frank (5), have proved to be useful in some cases.

For oblique headings the linearization and the use of strip theory methods yields reasonable results for ship motions in longcrested waves and,

consequently, in this case it may also be expected that wave bending moments and perhaps other wave loadings may be determined with a fair degree of accuracy.

The most simple solution, where swaying, yawing and rolling are neglected is treated by Fukuda in Appendix 1: a practical method of calculating

vertical ship motions and wave loads in regular oblique waves. The calculated midship wave bending moments in oblique waves agree satisfactory with model

experiments, carried out with a Series Sixty block .70 model and a T2 tanker model. Fukuda also published the results of systematic calculations for two

Series Sixty models in oblique waves (6).

In the more general case swaying, yawing and rolling motions are to be included in the analysis of ship motions and wave loading.

Added mass and damping for swaying and rolling cylinders were calculated by Tasai

₍₇₎

by extending Ursell's method for the considered lnode of motions.

Vugts compared the results of swaying, rolling and heaving cylinders with similar calculations, based on the Jong's work (3,1f). In figure:1,a comparison of computed and measured added mass and damping for a swaying cylinder with a ship-like cross section is shown as an example.

Research of this kind will lead to a more general description of the motions and wave loading of a ship in oblique waves.

wave loads in oblique waves is given by Kaplan as a basis for computer simulation of ship structural response (8).

Kaplan summarizes the pro5ect as follows (9):

The ultimate purpose of this program is to develop a mathematical model that will allow evaluation of ship dynamic response under all environmental conditions of ship wave interaction The existance of such a model would allow simple solution of various structural and ship motion problems that presently require expensive and laborious full scale trials, model tests, etc. A number of different phases were considered in this work, and a description of the objectives

and achievements obtained to date in this work are described. The first task assembled various mathematical models that could describe ship-wave interactions, with particular emphasis on the bending moment and slamming responses for a conventional ohipform

in a seaway. The results of that investigation are described in (8), which considers the use of linearized motion theories applied to this general problem area, together with the possible approaches for carrying out computations.

This studr produced equations that represent the vertical and lateral bending moments due to waves for a ship moving at any direction to

such long-crested waves, where these results are a generalization, for the case of vertical bending moments, of the treatment of

motions in head seas, while the lateral bending moments represent a basic development. The treatment of lateral bending moments in

(8)

includes three degrees of freedom; yaw, sway and roll. Numerical results obtained by simple "hand" calculations showed good agreement with experiment, thereby providing a basis for extending the te techniques to other cases. In figures 2, 3, 14 and ₅ this comparison

is shown for the "Wolverine State" with regard to the vertical wave bending moment. For the case of lateral bending moments the condition corresponding to headings. of 600 and 1200 were chosen, because

these result in large lateral bending moment values In figures 6 and 7 the effect of roll motion on the calculated lateral bending moment is shown. Figure 6 indicates the inability of the theory to properly account for short waves

_(XIL<o,6).

That particular condition is quite close to roll resonance and deviations of this nature might be expected in that region. A comparison with

experiments is shown in figures 8 and

_9.

It can be seen that

figure

8.

The problem of determiningslaiu-induced bending moments was also considered in (8), and various mathematical models for this effect were described. The particular type of computer that could be used

for this problem, as well as for the slowly varying wave-induced bending moments, was dependent upon the nature of the form of output desired, i.e.whether in spectral form or in time history form.

Some limited considerations for determining the computer system necessary for obtaining the various results was presented, but the

final choice is dependent upon the results of computer experiments. The inclusion of roll effects in determining lateral bending moments allows extension to predict torsional moments on ships in oblique waves, as long as additional information on ship inertial properties

is available. This would extend the mathematical model in

(8)

to obtain all of the slowly varying wave-induced structural responses by means of computer evaluation.

In order to evaluate the effectiveness of the mathematical model, a study was carried out to convert the equations in (8) to a computer program and for analog system, and obtain results that could be cómpared with available experimental data. The results of that study are described in (io), where consideration was given to the use of analog, digital and hybrid computer methods. Computations were made with a digital computer to obtain bending moment responses in regular oblique waves for two ships: viz, a cargo ship and an aircraft

carrier. Good agreement was obtained with the experimental data for the cases investigated, and the digital program has sufficient flexibility that would allow variations which would encompass other ships as well.

The work in (io) considers the computer system hardware requirements and time requirements for determining the various bending moment quantities, and the work provides a basis for further computer

evaluations to produce additional verification and/or predictions for a larger class of ships.

The work in

₍₉₎

was devoted to vertical and lateral wave induced bending moments in oblique seas, as well as slamming responses due to bow flare. The mathematical models required for the slamming responses are described, including two computational approaches to solve

the partial differential equations that represent the elastic response of the ship. Outputs in the form of spectra for wave-induced motions, as well as time histories of these motions

-6-of the slamming responses due to bow flare.

As a result of the demonstrated capacity of the equation system, as described in (9), further extensions of this work are presently being carried out.

These extensions are the inclusion of roll effects applied to evaluating torsional moments in oblique waves; the development of a computer technique to obtain "springing" response in time

history and spectral forms, thereby contributing to the problem of wave induced vibratory stresses; and the prediction of slamming-induced bending moments due to bottom impact with various models of hydrodynamic force input. In addition a complete documentation of the final computer programs will be prepared so that evaluation of these various phenomena can be obtained for arbitrary ships. This last stage of work is aimed at the verification of the entire procedure by comparing the results with available model tests and for full scale data, and was started in June 1969 with expectation of completion within 18 months thereafter.

Amidships lateral and vertical wave bending moments as well as shear forces were measured and calculated in oblique waves by Wahab (ii) on a Series Sixty block 0.80 ship model.

The response curves of the vertical and horizontal bending moments and of the shear forces have a very fluctuating character as can be seen from Figure 10. This implies that extensive testing is neded. for an accurate determination of the response operators.

The correlation with calculations is rather good as seen in Figure 11. By using a cos2 wave energy spreading function it was concluded that short crested waves in head and following seas have little effect on the wave bending moments.

3.2.Influence of spectrum shape on wave bending moments.

In predicting the long term trends of wave bending moments an important problem is the effect of spectrum shape (with area constant) on wave

bending moment. The known sea spectrum formulations (according to

Darbyshire, Pierson-Moskovitch, Neumann and B.T.T.P.), when reduced to the same significant wave height show quite small variations. Ewing and

Goodrich have shown that the resulting ship motions for ship lengths greater than 300 ft. are not affected by these variations (12). The same applies for vertical wave bending moments. However, actual measured sea spectra

have a much larger variety of form. It has been shown by Lewis (13) that if a group of spectra are randomly selected - all having the

same significant height or area - one can determine not only the average bending moment expected, but also the standard deviation of bending moment. In the case of the "Wolverine State", for example,

in waves of significant height of 30 ft. the ratio of standard deviation to mean bending moment was found to be

0012/

₀₀₅₉

0.20. Such an

estimate of standard deviation is a necessary part of the prediction of long-term trends of wave bending moment from model tests.

14.Wave induced vibrations.

Wave induced vibrations are reported in a number of cases where the frequency of wave encounter equals the natural frequency of the 2 node

vertical vibration of the ship

_{(114, 15,}

16, iT). In (ii) Maximadji describes these phenomena for the tanker "Sophia". In the ballast condition a

noticable 2 node vertical vibration was excited by bow waves of relatively small length (12 to 15 meters). Similar effects occurred in the loaded condit3on and were also reported from the tanker "Oley Koshevoi". In all of these cases a small change of speed or heading eliminated these vibrations

completely.

With increasing ship-length the 2 node natural frequency is reduced and resonance conditions occur at wave conditions having a corresponding

increased wave length and wave energy. The wave induced vibrations increase the midship bending moments and are a matter of concern for those who are

responsible for the design of very large tankers, bulkcarriers and Great Lakes bulkcarriers.

The analysis of wave induced vibrations requites the knowledge of the wave excitation forces along the length of the ship for relatively short wave lengths and of the elastic properties of the ship's hull including the structural damping.

The vibration amplitude at any point x along the ship length due to a sinus-oidal wave force F' acting at x is given by the following expression (18):

F' X (x)_{x (}

x)

+ c e\

12

(

w 2

JÇX2(x)x

'I

-8-We frequency of wave encounter.

X(x) = normalised 2-node vibration profile.

e = structural damping

P = apparent sectional mass.

In this expression only the 2 node vibration is considered because higher nodes of vibration are not important for "springing" effects.

The vibration profile y(x) is found from a vectorial summation of the sectional wave forces according to this expression. The corresponding stress follows from the known structural ship response.

However there is almost no experimental evidence to show that the cross-. sectional wave excitation F', as calculated by using strip theory, is sufficiently accurate for very small wave lengths.

An experiment with a model divided in several sections, as done earlier for ship motion work (lOE), should be carried out for these extreme

conditions to check the theory. Correlation with full scale measurements is considered as an essential extension of the research in this field.

In the case of very full ships flow visualization tests have shown the presence of large bow vortices. The shedding of these vortices may also be a cause of hull vibrations, similar to the effect of "springing".

5. Local hydrodynamic loading.

5.1.Local transverse loading.

Recent experience has shown that the transverse strength of large bulk carriers and tankers is a matter which requires very close study. Many computer programs are now available to handle the computational work involved in ship transverse strength analysis, but little accurate

information about the local, or transverse, loading which is needed for input to these programs is available. It is clear, therefore, that if

further significant advances towards a more rational analysis of transverse strength are to be made in the near future, further research into local hydrodynamic loading is necessary. A certain amount of theoretical and experimental work on pressures on transverse ship sections located in the mid-portion of the vessel has already been carried out by several authors.

(20-28). A short summary of some of the results is included in the second part of this chapter.

oscillating cylinder in the free surface of a fluid is easily found from p = - p, where the velocity potential can be determined for instance by tJrsell's method and a conformal mapping method to take into account the actual shape of the cross section.

For ship forms where the strip theory yields a good correlation with experimental motions and wave bending moments, also the measured pressures agree satisfactory with calculations, at least for the mid-portion of the ship. So far as the extreme ends of the vessel are concerned, there is very little information available, except for flat bottom presstires due to slamming. It is suggested, therefore that

consideration should be given to methods of estimating local pressures in this very important region where the ship structure, in all types of vessel, often sustains serious damage.

5.2.Results of model experiments and calculations.

Hoffman measured ship motions, bending moments, shearing forces and water pressures on a 4.26 ni. T2 SEAl tanker model in longitudinal regular waves (20). To allow the measurement of vertical bending moments and shearing forces, the model was cut into three parts at the quarter lengths and was rejoined by means of flexure beams. In the

final stage of the testing the water pressures were measured with fifteen transducers: six in station 1, six in station

8,

one in the forefoot and two on the wall sided part of the midship section. Motion phase angles appeared to be important because of critical

dependency of the theoretical pressure distribution on these quantities. Measurements were taken both for head and following seas.

The correlation of measured and calculated pressures according to the striptheory is satisfactory as shown in Table 1.

-10-.

Table

Comparison of experimental and. calculated pressure at the keel at station 8 of a T2 tanker model.

From

₍₂₀₎

Sponsored by the American Bureau of Shipping this work is extended at Webb Institute of Naval Architecture as part of a more extensjve project related to the longitudinal, transverse and local strength of ships.

Pressure measurements were carried out at Webb with a 5 foot Mariner model

at

0.2

L aft of the F.P. in a range of regular waves and in an irregular seaway corresponding to a LW knot Pierson-Moskovitch spectrum. The

measured pressure spectrum was compared with results predicted from the regular wave response amplitude operators and showed excellent agreement,

Experimental Theoretical Fn A/L Head Seas 0 .10 .15

.20

.25

.30

.5

5.1

3.0

2.6

2.1

2.6

1.6

14.3

_2.6

_2.0

_2.0

_1.8

_1.9

.75

5.1

10.3

6.8

14.9

3.9

14.9

6.8

_9.9

6.6

5.1

14.14

5.1

1.0

6.9

13.1

114.1

_13.6

_11.3

7.0

.6

12.1

13.14 _114.5

_11.5

_8.8

1.25

5.1

7.14

_10.6

_13.8

_15.9

_17.6

8.1

8.3

11.1

_11.9

16.0

1.5

14.14

5.8

8.0

_10.7

_11.8

_12.5

6.0

5.2

6.6

10.7

11.3

10.6

1.75

14.1

5.5

7.1

8.5

9.1

9.7

5.6

6.0

7.1

8.2

9.14

_9.5

2.0

14.8

5.3

6.5

7.0

7.2

7.14

5.3

5.5

14.14

5.2

6.2

6.14

which is another check of the usual assumption of linearity.

These results indicated the possibility of the use of small models for pressure measurements and a more systematic series of tests was then

performed using a 5 feet model of the "Wolverine State", for which full-scale data are presently available.

Pressure measurements were recorded at the following five locations aft of F.P: 0,1 L on the keel, 0,2 L on the keel, bilge and side and at O,I5 L on the keel. The test program included regular wave and irregular wave conditions. The theoretical pressure will be calculated, using essentially the method given in (20), with some more recent refinements incorporated into the mapping method and in the coefficients of equations of motion. The above studies will be associated with future work at Webb concerning transverse structural strenght of large tankers .Preliminay calculations of pressure distribution over the shell of a 200.000 tontanker have been

carried out and compared with static values.

Earlier tests by Hou-Wen 1-luang (29), who measured the pressure

distribution on a model at nine cross sections, showed a good correlation with two dimensional calculations at the midship section; but differences due to three dimensional effects were found at the fore and aft parts of the hull. Tasai's analysis (21) indicated that in order to calculate the pressure distribution at the midship section one has to consider not only the heaving but also the pitching motion of the ship, because terms

containing vó and V should be present in the expression for the pressure. A method for calculating the hydrodynamic pressure distribution along the contour of a transverse section in beam waves was proposed by Tasai (22).

A summary of Goda's work up to 1961 was given at the 3rd ISSC (25). An extension of this work is given in a note to this Committee (26). Again a T2-SE-A1 tanker model (length 14,5 in) was used to compare theory and experiment. The self propelled model was tested in regular head waves of constant height and the hydrodynamic pressures on the hull at midship, at O,25L and at 0,15L from F.P. were measured in addition to the

shipmotions. Tasai's and Fukuda's methods were used to calculate the corresponding pressures according to the strip-theory (20,29). Examples of the comparison between calculated and measured pressure amplitudes are shown in Figure 12. The agreement is satisfactory in several cases.

Further investigations will include the analysis of hydrodynamic

pressures of the midship section in regular beam seas, and the consideration of long term distributions of hyth'odynamic pressures to arrive at standards for the transverse wave loads on ship structures.

Terazawa and Nakamura have reported on systematic computer calculations

of response operators of hydrodynainic pressures acting on the section contour

-12-of a heaving and pitching ship in regular waves as a project -12-of the 90th Research Committee of the Shipbuilding Research Association of Japan (28).

Three Series Sixty models (CB =

0,55, 0,65, 0,75)

and a tanker model (CB = 0,80) were selected with weight distributions as given in Table 2 of (31). The calculation is in general based on Tasai'C theory (21). The present report extends his method to calculate the hydrodynarnic pressure distribution on various sections considering the effects of heaving and pitching motions. The response operators of the hydrodynamic pressure at ten points along the half girth Öf the section contours were calculated as a function of ship speed and wave length at the midship section and the Quarter length sections. One example of the calculated results is shown in Figure 13.

5.3.Dynamic loading due to the motion of liquid cargo in tanks.

In the design of tankers and other vessels carrying liquid cargo, the dimensions of the tanks are important with respect to the dynamic forces on the bulkheads and other structural members. An extensive experimental research ori the motions of cargo oil and the resulting pressure and loading of bulkheads was carried out by Akita with the aid of a model tank in

transient and regular pitching and surging motion (32, 33). The dynamic forces effected on the bulkheads by the motion of the liquid cargo depends on the type of motion of the tank, the length of the tank and the level of the liquid.

The maximum torces are approximately proportional to the length of the oil tank. In the case of transient motion, where the pitch angle changes from zero to a certain constant value, the forces due to dynamic pressures are 60% of those resulting from a regular pitching motion. From tests with a self propelled ship model equiped with a tank, it was concluded that for actual ships in irregular waves an intermediate mode of motion is experienced which lies between these two extremes.

It vas also concluded that the regulations of the Classification Societies for the design of watertight bulkheads are reasonable, but for swash bulkheads this is not so.

Swash bulkheads have little merits in view of hull weight reduction resulting from reduced dynamic force by the motion of liquid cargo.

However they remain necessary from the point of view of transverse strength of the hull structures Two results of the model experiments are shown in

Figures 114 and 15.

A number of empirical formulas for the dynamic forces on watertight bulk-heads and awash bulkbulk-heads result from the experiments. They may be used for design purposes.

5.+.Impact pressure and whipping stresses due to shipping green water. Serious damage on various parts of ship structures due to the impact of shipping green water has been reported in many cases. The damage concerns the fore castle deck, forwards parts of upper decks, pillars between decks, hatch covers, bridge fronts etc.

To analyse the mechanism of generation and the peak pressures of shipping wave impacts with respect to ship motions and ship speed, and to estimate the transient whipping phenomena due to these impacts, an experimental investigation was carried out by Kawakami (31) with a 2,83 meter wooden model (CB 0,68) in waves. The model consisted of six separate sections which were connected by bending moment-measuring apparatus. The natural 2 node-vibration period was 0,095 sec. Simultaneous observations were made of wave height at the bow, pitching and heaving motions, shipping wavepressures on the forward deck, vertical wave bending moment and vibratory moments due to whipping. A range of wave

lengths, wave heights and ship speeds were considered.

The heavy impact pressures of shipping green seas on the forward deck of the ship in rough seas are generated by dropping of piled-up spray, swelled-up waves or the scooping of green water at heavy pitching conditions. The measured. pressures are complicated functions of ship speed and wave length ratio. In the severest conditions impact pressures on the forecastle deck were as high as 30-50 ton/rn

An important conclusion is that whipping bending moment stresses may reach 140_70% of the sagging bending moment in regular waves, as shown in Figure 16.

6.Hydrodynamic aspects of slamming.

The prediction of slamming and related phenomena (such as whipping stresses, local high pressures and decelerations) should be based on fundamental knowledge of the hydrodynamic phenomena of shiplike bodies entering the free surface of a fluid. At present solutions for simplified

cases are available, for instance a flat plate feeling on an undisturbed water surface. For practical purposes the general three dimensional problem of an arbitrary shaped fore body of a ship entering the wavy water surface as a result of heaving, pitching and other motions, has not been solved to

its full extent.

Empirical approaches are used for the prediction of the occurrence of slamming in a seaway. To this end, the conditions leading to slamming have been studied by means of model tests and full scale experiments. In this

respect the definition of slamming has to be stated; slams have been detected and more or less defined by pressure measurements in the bottom of the ship or by the resulting whipping stresses or by deceleration peaks in the motion. For instance, in the analysis of seakeeping trials Aertssen defines the occurrence of a slam when the whipping stress exceeds

60 kg/cm2 (35). Other definitions of such an arbitrary nature are possible. Tick considered three conditions leading to slamming, namely: bow emergence, certain values of the relative velocity between ship and water, and the angle between keel and wave surface at the instant of impact (36),

Ochi reduced these conditions to -bow emergence and a threshold relative velocity, below which slamming does not occur. From various sources Ochi has evaluated this threshold velocity as

_3,5

rn/s for a ship length of

150 m. The impact load will be greater the more the threshold value is exceeded.

Assuming that bow emergence and relative bow velocity in a random sea are statistically independent events the frequency of occurrence per unit time N ma be estimated from:

i

'm

.2

N =

_2um

exp(-T

_{/2m )}

I f os * os

\ os,

where: m is the variance of relative bow motion,

os

m . is the variance of relative bow velocity

os

Tf is the draught forward of the ship.

is the threshold value of the relative velocity.

A reasonable agreement between theory and model experiments was shown by

Ochi (37).

From experimental data it was concluded that the impact pressure associated with slamming is approximately proportional to the square of the relative velocity. The probability distribution of the relative velocity follows a truncated Rayleigh distribution. From these two conditions the probability density function of the impact pressure can be derived and the probability of the impact pressure exceeding a certain magnitude per cycle of wave encounter can be obtained.

Also in this particular case, where the impact pressures at 0,1L aft of the F.P. were considered, a good correlation with "Mariner" ship model experiments in a severe sea state condition was found.

Ferdinande studied the influence of ship speed on the threshold

velocity with model tests for one particular cargo ship form. He concluded that the threshold velocity, which has to be introduced in order to

have a strict minimum value; it has to be regarded as a rather artificiel concept (38).

Not only bottoni impact forces, but also hydrodynamic impact loads on the immersed bow due to flare, have to be considered because of resulting whipping stresses. The mathematical model for this case is described by Kaplan as part of the general wave load problem in (8).

A "state of the art" report "On the hydrodynamic impact problem for flat-bottomed bodies" by Verhagen is given as Appendix 2.

-16-References.

(i) Marsich, S., Merega, F.

"Strip theory and power spectral density function application to the study of ship geometry and weight distribution influence on wave bending moment".

7th Symposium on Naval Hydrodynamics

_1968.

Terazawa, K.

"Response operators of ship motions and midship bending moments in regular waves calculated by the Research Committee no.

90

of the Shipbuilding Research Association of Japan".

Eleventh International Towing Tank Conference

1966.

Vugts, J.H.

"The hydrodynatnic coefficients for swaying, heaving and rolling cylinders in a free surface".

Report no. 191. Deift Shipbuilding Laboratory,

_1968.

()

de Jong, B.

"Berekening van de hydrodynamische coefficienten van oscillerende cylinders"

Report No. Deift Shipbuilding Laboratory,

_1967.

(14a) Hoffman, D.

"Lecture notes on conformal mapping techniques in ship hydrodynamics" draft

_1969.

Frank, W.

"Oscillation of Cylinders in or below the free surface of deep fluids". Report 237S,

_1967.

Fukuda, J.

"Computer Program results for ship behaviour n regular oblique waves". Eleventh International Towing Tank Conference

_1966.

Tasai, F.

"Hydrodynamic force and moment produced by swaying and rolling oscillation of cylinders on the free surface".

Report of Research Institute for Applied Mechanics, Japans Volume

_9,

no.

_35,1961.

Kaplan,P.

"Development of mathematical models for describing ship structural response in waves".

Progress Report on Project 1714 "Ship Computer Response" to the Ship Structure Committee

_1969.

(9) Kaplan, P.

"Computer simulation of ship structural response". Note prepared for Committee 2, I.S.S.C. 1969.

(io) Kaplan, P., Sargent,T.P., Raff, A.I.

"An investigation of the utility of Computer Simulation to predict ship structural response in waves".

Oceanics, Techn. Report No. 68-57, 1968.

(To be published as a Ship Structures Committee Report)

(ii) Wahab, R.

"Amidship forces and moments on a CB = 0,80 Series Sixty model in waves from various directions".

International Shipbuilding Progress 1967.

Ewing, J.A. and Goodrich, G.J.

"The influence on ship motions of different wave spectra and of ship length".

Transactions Royal Institution of Naval Architects, 1967.

Lewis, E.V.,

"Predicting long-term distributions of wave induced bending moment on ship hulls".

Proceedings of Spring meeting SNAME 1967.

(iii) Taylor, K.V. and Bell, A.0.

"Wave-excited hull vibration stresses measurements on a 147000-ton deadweight tanker".

BSRA report no. 115, 1966.

Mathews, S.T.

"Main hull girder loads on a Great Lakes bulk carrier". SNAME 1968.

Belgova, M.A.

"Determination of overall bending moments caused by elastic vibrations of ships".

Transactions of the Leningrad Institute of Water transport 1962. B.S.R.A. translation.

(i7)

Maximadji, A.I.

"On the problem of norms for longitudinal rigidity of ship's hulls made from high tensile steel".

Proceedings Third International Ship Structures Congress(discussiofl to the report of committee 3d).

-18-Mc Goidrick, R.T. Gleyzal, A.N., Hess, R.L., Hess, G.K. "Recent developments in the theory of ship vibration"

Report 739, 1953, David Taylor Model Basin.

Gerritsma, J. and Beukelman, W.

"Analysis of the modified theory for the calculation of ship motions and wave bending moments".

International Shipbuilding Progress 1967.

Hoffman, D.

"Distribution of wave caused hydrodynamic pressures and forces on a ship hull in waves".

Norwegian Ship Model Tank Publication 9)4, 1966.

Tasai, F.

"An approximate calculation of hydrodynamic pressure on the midship section contour of a ship heaving and pitching in regular head waves". Reports of Research Institute for Applied Mechanics.

Kyushu University, Vol. 1)4, no.)48, 1966.

Tasai, F.

"Pressure fluctuation on the ship hull oscillating in beam seas". Journal of the Society of Naval Architects of West Japan.

No. 35, 1968.

Goda, K.

"Transverse wave loads on a ship in waves".

Journal of the Society of Naval Architects of Japan. Vol.121, 1967.

(2)4)

Goda, K.

"Hydrodynainic pressure on a midship in waves"

Journal of the Society of Naval Architects of West Japan, no.35, 1968.

Goda, K.

"Transverse wave loads on a ship in waves" (2nd report).

Journal of the Society of Naval Architects of Japan. Vol.123, 1968.

Goda, K.

"Transverse wave loads on a ship in waves". Contribution to 3rd ISSC. Oslo 1967.

Goda, K.

"Transverse wave loads on a ship in waves"

A note prepared for the Rome meeting of Committee 2, ISSC, 1968.

Terazawa, K. and Nakwnura, S.

"Calculation of hydrodynamic pressure on the section contour of a

ship in regular head waves"

A note prepared for the Rome meeting of Committee 2, ISSC, 1968.

Hou-Wen Huang,

"Measurement of pressures and hydrodynamic forces on a shiplike model oscillating in a free surface".

I.E.R. Report 1965.

Fukuda, J.

"On the midship bending moments of a ship in regular waves".

Journal of the Society of Naval Architects of Japan. Vol.110, Dec.1961.

Terazawa, K.

"Response Operators of ship motions and midship bending moments in regular waves, calculated by the Research Committee no.90 of the Shipbuilding Research Association of Japan".

Proceedings 11th ITTC 1966.

Akita, L

"Dynamic Pressure of cargo oil due to pitching and effectiviness of swash bulkhead in long tanks".

Japan Shipbuilding & Marine Rngirieering 1967.

(3)

Akita, Y, Maeda, T, Furuta, K.

"On the motion of cargo oil in long tanks".

Journal of the Society of Naval Architects of Japan. Vol.123, 1968.

(3I) Kawakami, M.

"On the impact strength of ships due to shipping green seas - Towing experiments of a ship model in regular waves".

Society of Naval Architects of Japan, 1969.

(35) Aertssen; G.

"Service-performace and sea-keeping trials of M.V. Jordaens". Transactions oyal Institution of Naval Architects, 1966.

-20-Tick, L.J.

"Certain probabilities associated with bow submergence and ship slamming in irregular seas".

Journal of Ship Research 1958.

Ochi, M.K.

"Prediction of occurrence and severity of ship slamming at sea". Fifth Symposium on Naval Hydrodynamics ONR 196k.

Ferdinande, V.

"Analysis of slamming phenomena on a model of a cargo ship in irregular waves".

Appendix I.

A practical Method of Calculating Vertical Ship Motions and Wave Loads in Regular Oblique Waves.

*

by Jun-ichi Fukuda

Summary.

A practical method of calculating response operators of vertical ship motions, shearing force and bending moment in regular oblique waves is

described in this note, along with results of comparison between calculation and experiment. The calculation method is based on the linear strip theory, which is originally established for the case in longitudinal waves, and. extensively applied to the case in oblique waves. Primarily, heave and

pitch of a ship in regular oblique waves are solved. Then, vertical motions, shearing force and bending moment at a certain longitudinal location of the hull will be calculated by using the solutions of heave and pitch.

Contents.

i Introduction

2 Modified Strip Theory

3 Heave and Pitch

Vertical Motions

5 Vertical Shearing Force and Bending Momènt

6 Comparisons between Calculation and Experiment References

i .Introduction.

About a decade ago, a method of calculating heaving and pitching motions of a ship in regular longitudinal waves was proposed by Korvin-Kroukowsky {i}, jihich was based on the linear strip theory with application of Munk's method for a problem of airship fluid-dynamics. This method

was immediately followed by Watanabe's modified one [2] and widely

developed for calculating vertical ship motions and wave loads in longitudinal waves with the aid of calculation methods of two dimensional added mass

*

Professor of Naval Architecture, Kyushu University.

-22-and damping for shiplike cross sections according to Grim [3], Tasai [ii] and Porter

[5].

Subsequently, a method of calculating response operators of vertical motions, shearing force and bending moment in regular oblique waves was given by the author [6, 7, 8], which was based on Watanabe's

theory (originally established for the case in longitudina1 waves) and extensively applied to the case in oblique waves. At present, the similar methods for calculating response operators in regular oblique waves are adopted in Japan

[9,

10, 11] , Netherlands [12] and Norway- [13]. Since details of the author's calculation method are described in his published papers

[8, th]

, its outline is shown here along with results of

comparison between calculation and exeriment performed by other authors.

2.Modified Strip Theory.

The application of Watanabe's method [2] was attempted to the case in regular oblique waves, though the original theory was established for the case in regular head or following waves. In the method described below, only the heaving and pitching motions are taken into consideratioñ and the other motions; surge, sway, yaw, roll and drift are all ignored.

Consider the case when a ship goes forward with a constant speed in regular oblique waves. As shown in Figures Al-1 and A2-2, the coordinate system O-XYZ is fixed in space such that the XY-plane lies in the

undisturbed water surface, and the coordinate system o-xyz is fixed to the ship such that the origin is situated at the midship. The ship goes straight on with a heading angle 4i to the wave coming from the positive X-direction to the negative direction. Then, the surface elevation of

regular oblique wave encountered with the ship will be expressed as follows in the vertical plane including the ship centre line:

h = h cos(k*x + tüt) (i)

where

h: elevation of surface wave h :amplitude of surface wave

o

k*:kcosp, _{k = 2,T/A, A: wave length, heading angle}

+ k*V : circular frequency of wave encounter w = : circular frequency of wave

g : aceleration of gravity, V: velocity- of ship, t: time

When the ship is heaving and pitching in regular waves, the vertical oscillatory force per unit length acting on a section distant x from the

and

dF d dF w

-(ps),

p : density of water

y : half breadth of water line at x

w

pN : sectional damping coefficient at x

pa : sectional added mass at x

w : weight of hull and load per unit length at x

Z : vertical displacement of hull at x

Z : relative vertical displacement between hull and effective

re

wave at x

The values of sectional damping coefficient and added mass for cross sections can be evaluated according to Tasai's method { i«j

When the ship is heaving and pitching in regular oblique waves, Z and Zre will be given by:

Z = ; + (x.xG)+,

Ze

_{Ç + (x_xG) - he} (11)

where

Ç : heaving displacement

pitching angle

XG : x-coordinate of the centre of gravity of ship h effective wave elevation

e

and it follows that:

= Ç + (X - xG)4, _re + (x_xG) - Vc

-=+(xxG),

ZreG2h1eJ

(5)

As shown in Figure 1, the surface elevation of regular oblique wave encountered with the ship can be described as:

h(x,y,t) h0cos(kXcosiP - kysin _{+ met)}

midship is given by follows according to Watanabe's strip theory [2:

dF 1 2

= -+ - +

+ dx dx dx dx dx where: =

2PgyZ

- pN re

-1--at p(x,y)in a transverse section of the ship, And, the average wave elevation in the transverse section of the ship will be:

Yw sin(ky sirup) h(x,y,t)dy =

kysinP

h0cos(k*x +wet) = Ceh Ce = sin(kysinip)/kysinip

The effect of decreasing orbital motion of wave for the wave induced force on the strip is introduced with:

*

exp(-k d ) = exp(-lcad)

where:

O: Sectional area coefficient at x

d: draught at x

Ce is the coefficient which represents the average wave elevation

for the wave induced force on the strip and comes to a unity in longitudinal waves and in long waves. And exp(-k d*) represents the so-called "Smith

effect" for the wave induced force. By introducing Ce and exp(-k d*), the effective wave elevation for evaluating the wave induced force on the section may be given by:

* *

-kd -kd

h=Ce

h=Ce

hcos'kx+wt

e e e o e

By substituting the formulae (iL) and

₍₅₎

into the formula (3) and taking into account of Cx = -V), the vertical forces on the section are found as follows: dF

= -2PY

(x_xG)

- h)

dF -pN {ç + (x_xG) - V -

h)

-

+ (xxYq - 2V

_-

h)

} G e - - !. { + (x_xG) }

The equations of heaving and pitching motions are obtained by integrating the force of the formula (2) and the moment with respect to the centre of gravity produced by the force from the after end to the fore end of the ship and by putting them equal to zero. Namely,

dF3 dx dF14 dx

(9)

j

X1 FP *

/ dF

(dF

F

Cx)

-j

dx

j

dx AP X1 FP X dF dF /

- x-x )dx

M*(x1)

= J

(x-x1)dx AP

=-/

dx where: F*(x1)

oscillatory shearing force at x1 (upward force on the forward side of section or downward force on the after side of section is defined as positive)

oscillatory bending moment at x1 (hogging moment is defined as positive)

Strictly speaking, the method described above is not exact for the three dimensional problem. And it has weak points in evaluating hydrodynamic force and moment induced by the regular oblique waves even though the strip theory could be admitted. At present, however, when the more elegant one such as a method of singularity has less applicability to the actual ship forms, the method described here will be adequate for the practical purpose.

3.Heave and Pitch.

Performing the integral operation of the left sides of equations

(io)

and neglecting the secondary terms, we obtain the equations of heaving and pitching motions in the form of:

a

+c+d+e,c+g4,=Fcos(wt

(I) ço e A 4, + B 4) + C + D + E _c + G t = M Cos(w t + (I) 4) 4) 4)0 e

M

where: ac + Pfsdx. W: weight of ship b = pfNdx Cc =

2pgJydx

= Pfs(x_xG)cix FP FP

( dF

I

dx=O,

j

(x_xG)axO

"AP AP

When the equations of heaving and pitching motions have been solved, the shearing force and bending moment at a section distant x1 from the midship can be calculated by follows:

(io)

(12)

e ₌

PfN(x_xG)dx - VpJsdx

=

2Pgfy(x_x)dx

- Vb

(i1)

I y"s(xx) dx,

I: longitudinal moment of inertia of ship

B _{= PJN(X_XG)2dX} C = 2Pgfy

_{(x_xG)}

dx - VE =

_{N(xxG)ax +}

_VpJsdx =

2PgfY(x_x)dx

and: F (COSOE (f1 + f2 + f3

FJ Ft.f= htft

+ +

cM

M

=h

1+m2+m3)

M I _cosß

_m

M

o{S±fl8}

o{;

+ m + =

Ce

_J dx : }

2PfY

_kd* [cos k X j

we

i

*

sin k x J * f2 ₁ (-sin k xf f ) =

wPjCee_kd

cc:s * " clx k xJ * eQs k xl f3 * kd c1x f } =

-

ePJe

sin kx) * (cos k

xl

2pg c

e_*

*

(x_x)x

mJ

Jwe

_tsinkxJ

* -sifl k*x ) m2 } = wpTNC e_kd *

(x_x)dx

m

J

e

jcoskxj

* reos k*x m3

fsc

*

(x_xGdx

Ì

= e

e

m3

_tinkxJ

*

+ wvf

_kd* (-sin k x

sCe

dx

I

cos k

e

*

The numerical integrations in (114) and (16) should be carried out from the after end to the fore end of the ship.

The solutions of equations (13) are found in the form of:

I

J

1

(15)

= C cosw t -

8mw t

C cos(w t+a

c e s e o e C

4) =

s8±et = 4)cos(wt+ß4))

where:

C0: heaving amplitude

phase angle between the heaving motion and the wave elevation at midship.

4): pitching amplitude

phase angle between the pitching motion and the wave elevation at midship

C C cosc , C C Bina

C O C S O C

4)=

4)cos4,

4)s = (posinß4)

)4.Vertical Motions.

Various vertical motions of a longitudinal location distant x1 from the midship can be calculated by using the solutions of heaving and pitching motions.

The vertical motion at X1 is derived from (i') as follows:

Z Z coBw t - Z sinwt = Z cos(w t + e

c e s e o e z

where:

Z0 amplitude of vertical motion at X1.

e phase angle between the vertical motion at and the wave

elevation at midship.

Z = Z cose _{= C +}

C O Z C

Z = Z sine _{= C + (xl-.xG)4)}

s o z s

The vertical velocity and acceleration at x1 can be easily obtained from (18) and (19).

The relative vertical motion between the hull and the undisturbed wave surface can be obtained from (i), (18) and (19) as follows:

Z = Z cosw t - Z sinw t = Z cos(w t+ ) (20)

r rc e rs e ro e zr

where:

Z : amplitude of relative vertical motion at X

ro 1

czr: phase angle between the relative vertical motion at x1 and the wave elevation at midship.

and -28-and Z rc Z rs

=Z cose

ro zr = Z sine ro zr C e = C s + + (xl_xG)4)c 1_XG)4)s -

hcosk*x1

hsink*x1

(21)

}

(17)

and

V

=V Sifl

rs ro vr

= + (x1_xG)CJ - V -wh cos k*x1

The results of (18)-(23) will be available for the purpose of pursueing the seakeeping problems such as ship acceleration, deck wetness, propeller racing and slamming in rough seas.

5.Vertical Shearing Force and Bending Moment.

The oscillatory shearing force and bending moment at a section distant from the midship can be calculated according to (li) and (12) by using the solutions of heaving and pitching motions.

By performing the integral operations of the right side of (ii) and neglecting the secondary terms, the shearing force at x1 will be obtained in the form of

*

*

*.

*

F (x ) = F cos t - F sinw t = F cos(w t+1) (2h)

e e s e o e

where:

amplitude of shearing force at X1

y : phase angle between the shearing torce at and the wave

elevation at midship

and, F and F: are given by follows:

F*)

r; F _J

_t

_i

k

j

1

ORtR,+R,

s s e s +

f

+

Q+C

_{+ Q2[} +h

1123

(25) where:

₂₁

P1 =

- wt_

wfdx + pfs dxl +

2pgJydx

P2 ₌

WPfN dx

+ w Vps e xl

And, the relative vertical velocity between the hull and the surface wave will be given by:

y

v coswt-v sinwt=v cos(wt+c

) (22)

r re e rs e ro e vr

where

V:

amplitude of relative vertical velocity at

e: phase angle between the relative vertical velocity at X1 and the wave elevation at midship.

V

=V

cose

re ro vr

=

e(s

+ (xl_xG)S

J -

V4, +h sinC o i

Ql = _we1Jw _xG)dX

+ pfs(x_xG)dx] +

2Pgfy

_{(x_xG)dx + v2ps}

1

-VpJNdx

Q2 = _WePf X_XG)dX

+ we

PJ&dX +P3l(xl_XG)}

_2pg

C

e_*

Jcosk*x1

Jwe

_sinkxJ

::}

wf

NC e"1«1

*

-sink x J

dx

cosk xj

e

*1

*,

dx

: } = WU) p

"SC

e_t*

cosk X

ej

e

s1

xJ

kd*

(_sink*x

J

+wVps Ce

_xle

_*

cosk X1)

By performing the similar integral operations in the right side of (12)

and neglecting the secondary terms, the berding moment at x1 will be

obtained in the forni of

* *

*.

*

M (x

)

= M cosw t - M sinw t = M cos(w t+tS)

(21)

i e

e

s

e

o e

where:

*

M :

amplitude of bending moment at x1

ô

: phase angle between the bending moment at x1 and the wave

elevation at midship

and,

M*

_{and M: are given by follows:}

*

Ml

cl

Ici

t;

s

T

2{1

h

[ri+r2+r3t

*=pl

+p2

+

q1j

+ q

/

M I _{t; 4)} 4)

r

where:

e(g

x-x1)dx+ PJs(x_x1)dxj_ 2Pgf3r(x_x1)dx

p2 = WePt xxl)dx + WVPfs dx

2'l r

q1 = w

_{e1 g)}

_{- Iw(x_xG)(x_xl)dx + pfs(x_xG)(x_xl)dxì}

_2PgfY(x_x)(x_x1)dx

+

V2pJsdX

+

VPfN(x_x1)dx

q2 = wePfN(X_X )(x-.x )dx + w V(x -x

_{G 1}

_e

1

J

(26)

-30-cos

yCe

* (x-x1)dx

r11

2pg

I

_kd*

rJ

J

we

(sinkxJ

*

k xi = e r2 wpfNe _kd*

f_sin

(x-x1

rJ

cos

kx

r3}

r' WWp SCee

[

_* ( (x-x1)dx -

f

_kd*

cos kx

3

1.sinkxj

+

w\TPJ

kd* 1-sin kx

sCe

dx cos k x e *

In the formulae (26) and (29), numerical integrations should be carried out from the after end to x1 or from the fore end to x1. And

5xi

in (26) denotes the sectional added mass at x1.

When calculating the shearing force and bending moment at midship, the formulae (26) and (29) will be more simplified by putting X1 = 0.

6.Comparisons between Calculation and Experiment.

A number of model experiments have been carried out for the purpose of pursueing the response characteristics in regular head waves at

experimental tanks in the world. And some of those results vere compared with calculated results based on the strip theory. Such investigations

[9, 14,

15,

16]led

us to the general conclusion that the calculated

response operators of vertical motions, shearing force and bending moment in head waves were sufficiently valid at least for the practical purpose.

As to the case in oblique waves, however, we had not sufficient

informations on the validity of the strip theory intil the calculations by Joosen [12} and Nordenström [13] were recently reported, where the

calculated heave, pitch, relative bow motion and midship bending moment in oblique waves were compared with the results of experiments carried out at the Netherlands Ship Model Basin. In Japan, meanwhile, comparisons between calculation and experiment for the vertical ship motions and midship bending moment in oblique waves were tried by several authors [iî, 18, 19], where the calculation method described above is adopted.

Examples of the comparison between calculation and experiment performed by Japanese authors are shown below.

Fig. Al-3 shows results of calculation and experiment for pitch and heave of a Series 60, 0.70 block coefficient ship model in regular oblique

waves carried out by Yamanouchi and Ando [17] at the Seakeeping Model Basin in the Ship Research Institute, Tokyo.

Konuma

[19]

_{tried the calculation of midship bending moments on a} T2-tanker model in regular oblique waves in order to compare with the

experiment carried out by Yamanouchi, Goda and Ogawa [20] at the Ship Research Institute. The results are given in Fig.A1-l4.

According to those calculations compared with the experiments at the N.S.M.B. and the S.R.l. in Japan, the applicability of the strip theory

in oblique waves seems to be unexpectedly larger than the limitation of the theory. And it may be said that the strip theory is valid for

evaluating the response operators of vertical motions and wave loads in oblique waves though further investigations are necessary for the

applicability of the theory.

Acknowledgement.

The author wishes to express his deep thanks to Dr. Y. Yaxnanouchi and his staff in the Ship Research Institute and to Mr. M. Konuina for

their sincere cooperations on the occasion of this works.

References.

[i] B.V. Korvin-Kroukowsky and W.R. Jacobs: "Pitching and Heaving Motions of a Ship in Regular Waves" TSNPIME,

vol.65, 1957.

Y. Watanabe: "On the Theory of Heaving and Pitching Motions"

Technology Report of the Faculty of Engineering, Kyushu University,

Vol.31,

No. 1,

_1958.

0. Grim: "A Method for a More Precise Computation of Heaving and Pitching Motions in SmoOth Water and in Waves" 3rd Symposium of Naval Hydrodynamics, Scheveningen,

_1960.

[]

F. Tasai: "On the damping Force and Added 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, 1960.

[51

W.R. Porter: "Pressure Distribution, Added Mass and Damping Coefficient for Cylinders Oscillating in a Free Surface" University of California, Institute of Engineering Research, Series No. 82,

_1960.

[6]

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. 1i1,

_1963.

-32-{ 7]

J. Fukuda and J. Shibata: "The effects of Ship Length, Speed and Course on Midship Bending Moment, Slamming and Bow Submergence in Rough Seas" The Memoirs of the Faculty of Engineering, Kyushu University,

Vol.25,

No. 2, 1966.

J. Fukuda: "Computer Program Results for Response Operators of Wave Bending Moment in Regular Oblique Waves" The Memoirs of the Faculty of Engineering, Kyushu University, Vol.

26, No. 2, 1966.

J. Fukuda: "Theoretical Determination of Design Wave Bending Moments" Japan Shipbuilding & Marine Engineering,

Vol.2, No. 3, 1961.

[io]

J. Fukuda: "Trends of Extreme Wave Bending Moment According to Long-Term Predictions" Journal of the Society of Naval Architects of Japan,

Vol.

123, 1968.

[11] J. Fukuda: "Predicting Long-Term Trends of Deck Wetness of Ships in Ocean Waves" To Be Read at the Autumn Meeting of the Society of Naval Architects of Japan and Published at the End of

_1968.

[121 W.P.A. Joosen, R. Wahab and J.J. Woortman "Vertical Motions and Bending Moments in Regular Waves. A Comparison between Calculation and Experiment"

ISP, Vol. 15,

No.

161, 1968.

[13]

N. Nordenström and B. Pederson: "Calculations of Wave Induced Motions and Loads. Progress Report No.

6,

Comparisons with Results from Model

Experiments and Full Scale Measurements" Det Norske Ventas Research Department, Report No.

68-12-S, 1968.

[114] J. Fukuda: "On the Midship Bending Moment of a Ship in Regular Waves"

Journal of the Society of Naval Architects of Japan, Vol.110, ₁₉₆₁ and

Vol. 111, 1962.

J. Gerritsma and W. Beukelman: "Analysis of the Modified Strip Theory for the Calculation of Ship Motions and Wave Bending Moments" ISP, Vol. 114,

No.

156, 1967.

M. Lötveit and K. Raslum: "Comparison between Calculated and Measured Wave Bending Moments, Shearing Forces and Pitching Motion for a T2 Tanker Model in Regular Waves" Det Norske \Ieritas Research Department, Report

No. 614-314-S,

19614.

Y. Yamanouchi and S. Ando: "Experiments on a Series 60, CB 0.10 Ship Model in Oblique Regular Waves" Proceedings of 11th ITTC, Tokyo,

_1966.

[181 Y. Ymanouchi, H. Oi, Y Takaishi, H. Kihara, T. Yoshino and M. lizuka:

"On the Ship Motions and Accelerations of a Nuclear Ship in Waves" Journal of the Society of Naval Architects of Japan, Vol.

123, 1968.

[191 M. Konujua: "Comparisons between Calculation and Experiment for Vertical Motions and Bending Moments in Regular Oblique Waves" Technical Report of Nagasaki Shipyard, Mitsubishi Heavy Industries Ltd., ₁₉₆₆

(Unpublished).

{ 2O Y. Ye.manouchi, K. Goda and A. Ogàwa: "Bending and Torsional Moments

and Motions of a T2-.SE-A1 Tanker Model in Oblique Regular Waves" Note Presented to 2nd ISSC, Deift, 1961 (Unpublished).

-31f-Appendix I.

On the hydrodynamic impact problem for flat-bottomed bodies. by

J.H.G. Verhagen. Introduction.

The research on ship slamming is concerned with the following areas of investigation:

The study of ship motions in waves.

The study of the mechanism of slamming and the prediction of slamming loads.

The response of the local structure as well as the hull girder response. The complete solution of the slamming problem needs a combination of these studies.

However, progress hae been made by neglecting the interaction effects upon the load generated, and considering each of these aspects as distinct and separable parts of the total problem. The principal problem is the fundamental research on the mechanism of hydrodynamic impact. Together with model

experiments conducted in a Seakeeping Tank it gives the starting point ori which the statistical studies on the frequency of occurrence of impact and its magnitude can be based.

A survey of the impact problem is given by Szebehely [i], Szebehely and Ochi [2], Chu and Abramson {3, Moran {hand recently by Meyerhoff [5].

In the following we will concern ourselves to the impact problem for flat-bottomed bodies which has received considerable attention during the last few years.

The conventional approach.

The hydrodynwnic impact theory based on the fundamental work of van Krmn and Wagner usually deals with an incompressible fluid and is essentially involved in the wedge entry problem.

The results of these theories become increasingly inaccurate as the dead-rise angle of the wedge decreases. In the extreme case of a flat bottom the theory predicts infinitely high pressures at the instant of impact. Taking the compressibility of the water into account (von Kg.rmn [6], Egorov

_[TI,

Ogilvie 8]) results into an initial pressures equal to the acoustic pressure in water. The high pressure predicted by these theories could not be verified experimentally. Measurements show a much lower impact pressure over a longer time period. Moreover, the impact pressure is

approximately proportional to the square of the impact velocity instead of linearly proportional. Theories which include the cushioning effect of the air layer entrapped between body and water surface at the moment of

impact are developed by several investigators. By this approach the essential features of the impact phenomena obtained from two dimensional drop tests can be largely explained.

Studies on the effect of entrapped air.

Fujita

_[9]

considers the impact mechanism as due to the pressure increase caused by the rapid compression. of a fixed amount of air

entrapped between the body and the water. Also the first calculation of Lewison and Maclean [io] is based on this assumption.

The amount of air must be chosen to fit the experimental results, Egorov

_Lii]

calculated the amount of air from the deformation of the vater surface at the moment of impact. An improved theory considering the motion in the air layer as a one dimensional compressible ideal gas flow is developed by Lewison [io] , [12] . Compressible and incompressible

movement of the water is assumed to take place. The method of solution is somewhat crude. Arbitrary assumtions are made and factors introduced, which have to be deterininated experimentally. One of the difficulties lay in the initial conditions that were assumed to prevail when calculations of the air pressure began. The outflow of the air at the edge of the

model desires a more careful study. To prevent the horizontal air velocity in the calculation from increasing to unreasonably high values (several times sonic speed) Lewison introduced into the computer program an upper bound to the outflow of air equal to sonic speed. To his opinion the chocking proceBs of the air flow will be a gradual one instead of an arbitrary discontinuity. The theoretical curves do not reproduce the experimental ones very well although their general characters are much similar.

Further refinement of the theory will be needed.

Experiments were designed to check the theoretical approach. The air velocity under the falling flat bottomed model was measured by means of high-speed motion photography of styrofoam balls carried along by the air flow.

Simultaneously the motion of the water surface was recorded as well as

the pressure at the center and near the edge. The results indicated that the air velocities were less than expected but the experimental technique

could have been improved. No significant downward motion of the water surface was observed before impact.

These tests were performed on the ship dynamic test machine at the University of California [13]. This machine provides a facility to study

on a relatively large scale, two-dimensional impact phenomena and local structural response to impact load. The preliminary tests results

-36-obtained by Maclean [i on this facility show that the maximum pea} pressure increases linearly with drop height and. therefore with the square of the impact velocity. The peak pressure diminished towards the free edges of the model. The peak pressure at the edges time lead that on the centerline. The maximum pressure at the centerline occurred when the model has just passed the original water level. The increasn peak pres-sure with the model loading as meapres-sured was much less than predicted by Lewison's theory. An important result obtained from the work of Lewison and Maclean is that the peak pressure can be drastically reduced if flanges are fitted to the flat bottom to entrap air.

In order to clarify the relationship between the maximum pressure and the impact velocity, Nagai carried out a systematic test program on a dropping test tower, which is a small-scale version of the California ship dynamic test machine [22]. From the obtained experimental data, he summarized the following results:

The relationship between maximum pressure and impact velocity depends mainly upon the weight and the angle between both bottom and water surfaces, rather than upon the bottom width.

2e

Within five degrees angle between bottom and water surface the relation-ship turned out to be p where 1.l-t < n < 2.2, but no clear

max

quantitative relation was found between the weight and the maximum pressure.

The two-dimensional problem of the air flow between a rigid flat-bottomed body falling towards a rigid plane is solved by Johnson [15]. From his careful numerical study it is concluded that the deformation of the water surface must be taken into account to obtain pressures that agree with experimental data. The behaviour of the free surface near the edge of the body requires a careful study, because small changes in the exit area

will significantly affect the amount of entrapped air and hence the pressure beneath the body.

Verhagen

(16]

treated the flow in the air-layer as a one-dimensional compressible flow of a perfect gas in the same way as Lewison. The

disturbance of the water surface due to the imposed air pressure is calculated neglecting the water compressibility. The initial conditions at the starting point of the numerical calculation are derived under the assumptin that initially the air flow can be considered as incompressible and the water surface undisturbed. From the theoretical treatment it appears that the air-layer is entrapped as soon as the local velocity of the air at the edges reaches the velocity of sound. Afterwards an adiabatic

assumed, The solution of that part of the problem is similar to the approach of Fujita. The numerical results gave an excellent agreement with the experimental data obtained from his fairly lightly loaded two-dimensional test body. The increase in peak pressure with the model loading according to this theory is also overestimated. It is concluded that compressibility effects of the water cannot be neglected when the mass of the body is large compared to the added mass. An interesting experimental study on the importance of air density and fluid properties to water impact pressures is given by Gerlach [17]. By reducing the environmental pressure, he found the peak pressure increasing with

diminishing gas density, up till quite closely the acoustic liquid pressure. The peak pressure for a given gas fluid and body appeared to be a function of the variable h/pe1 where h is the drop height and

e the environmental

pressure. The presence of' waves on the liquid surface tends to reduce the peak pressure. A theoretical study on the influence of air on the

impact pressure of blunt bodies is given by Greenberg [18]. He considered the air as incompressible because the air velocities are anticipated in his case as sufficiently subsonic. Also the water is considered as incompressible. The non-linear free surface conditions are retained. The analysis will be

coded for solution.

From model experiments on slwnniing impact conducted in regular and irregular waves Ochi [19] obtained a relationship between maximum pressure and

relative velocity between bottom and wave at various locations along the ship's bottom. A meanline representing the pressure on the keel plate for a 13 feet Mariner model at station 2 is drawn for comparison with results of various two-dimensional drop-tests in Figure A2-1.

The reason for the large discrepancy is not clear.

Conclus Ion.

The theoretical treatment of the impact problem which include the effect of entrapped air yield results which are in reasonable agreement with results obtained from two-dimensional drop-tests. However, the results obtained from model experiments in waves cannot be satisfactory explained by theory. Further research is recommended.