Ship bending moments in irregular seas predicted from model tests

WEBB INSTITUTE OF

NAVAL

ARCHITECTURE

GLEN COVE

LONG ISLAND

NEW YORK

December 1963

SHIP BENDING MOMENTS

IN IRREGULAR SEAS

PREDICTED FROM MODEL TESTS

by

Robert B. Zubaly

and

Edward V. Lewis

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DCCUMENTAT DATUM:

Final Report on Phase I of Research Project

Sponsored by American Bureau of Shipping

New York City

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SHIP BENDING MOMENTS

tN IRREGULAR SEAS

PREDICTED FROM MODEL TESTS

By

Robert B. Zubaly

and

Edward V. Lewis

Final Report on Phase I of Research Project Sponsored by American Bureau of Shipping

New York City

ABSTRACT

Results are presented of a study of comparative trends of ship hull bending moments in realistic short-crested irregular seas. The work involved the prediction of the wave-induced bending moments

to be expected at sea, by means of the method of superposition, making use of available results of systematic model tests in regular waves and available ocean wave spectra obtained from records taken at sea. Variations in non-dimensional wave bending moment coefficients are presented graphically as a function of ship size, fullness, speed

and heading -- and to a limited extent of severity of the sea. The approximate effect of lateral bending is indicated.

It is shown that the calculation procedure followed can pro-duce useful information regarding trends of wave-inpro-duced bending

moments under realistic sea conditions. Certain specific conclusions are drawn regarding the relative importance of the factors mentioned

INTRODUCTION

Background

Research on the problem of wave induced bending moments on ships' hulls has made considerable progress in recent years. Full-scale

strain measurements on ships have been made in various countries, and statistical methods of analyzing them are being developed. At the same time, model test techniques are being used to determine wave bending moments at various headings to regular waves of different lengths and

heights. The development of theoretical methods of calculating wave bending moments, taking into account ship and wave motions, is making

good progress.

Furthermore, new theories for describing the actual irregular wave patterns at sea in precise statistical terms have been developed by oceanographers, and sea surface records are becoming more plentiful. The superposition technique has proved to be of real value in predicting ship performance in these irregular seas on the basis of experimental and theoretical data on bending moments in regular waves.

However, little attempt has yet been made to apply this accu-mulated research knowledge to the urgent practical problems with which

ship designers and classification societies are faced. Instead, the

basis for design remains a standard.bending moment calculation with the

ship

poised statically on a wave of its own length. Wave height and allowable stress are determined on an empirical basis to allow approxi-mately for the effects of ship size and other characteristics. Such a

semi-empirical design procedure would undoubtedly be satisfactory if

there were no changes in the size and speed of ships. But there has

been a remarkable trend toward larger tankers and bulk cargo carriers,

and the end is not in sight. Not only are waves of the length of these large ships less frequent, but their heights are known to be generally

not so great as those of shorter waves. Some reduction in design wave height therefore seems permissible, but the question is, how much? As

-5-stated by Mr. David P. Brown, Chairman of the Board, American Bureau of Shipping, in his half-yearly statement of July, 1961, "There still remains the very important question as to the degree to which recogni-tian may be given to what might be termed the diminishing relative severity of storm conditions as the sizes of ships increase."

At the same time, general cargo ships and naval vessels have shown a continuing upward trend in speed, without the spectacular size

increases noted for the bulk cargo ships. Questions have been raised as to the effect of speed itself on the wave bending moments, as well as to the influence of the finer form coefficients which accompany higher

design speeds. Some of the classification societies take block

co-efficient into account in their strength standards.

tn Mr. Brown's report, he added: "The development of new

facilities for model testing in wavemaking tanks and of instrumentation to be applied to ships at sea is providing increasing opportunities to obtain information on these vexatious problems." Accordingly a project was undertaken at Webb Institute of Naval Architecture, under the

sponsorship of the American Bureau of Shipping, to attempt to answer

some of the above questions.

Objectire

It is the purpose of this report to describe the results of the first phase of a continuing investigation to determine possible applications of bending moment research results

to design

problems. The

objective of the overall study, as stated in the Proposal for Research

(l)*, is:

To apply presently available knowledge of ocean wave patterns and hull bending moments in waves to the prediction of the relationship between wave bending moments and ship

character-istics under realistic sea conditions. The variables of

particular interest are severity of sea, ship size, fullness,

speed and heading.

*Numbers in parentheses refer to references listed at the end of this

The first phase discussed herein was directed toward

obtain-ing comparative trends of bendobtain-ing moment in realistic irregular seas,

computed by the superposition method, using model test results in com-bination with recently acquired data on ocean wave spectra. Results can be interpreted ìn terms of effective wave heights in a conventional

bending moment calculation. An iuuuediate application of such

qualita-tive results is to obtain a more rational basis for determining the

variations in standards of hull strength with ship size, fullness and

speed in rough seas.

The second phase involves the use of probability techniques to analyze and interpret the results of accumulated service data on ship

hull stresses at sea. The goal is to develop procedures whereby stress data taken over a period of two or three years can be used to predict the highest wave bending moment to be expected in the life of a ship. Ultimately these two approaches will be brought together so that with the use of model tests and probability theory valid predictions can be

made for any ship in the design stage. The work of phase 2 will be the

subject of a later report.

Procedure

The prediction of wave-induced bending moments in ships opera-ting in realistic short-crested irregular seas is accomplished by

application of the principle of superposition, in which systematic model test results are combined with sea spectra obtained from actual

wave records. The result of this combination is a statistical predic-tion of wave bending moment for each model and spectrum assumed (2).

By applying this technique to different ship forms and char-acteristics, comparative results can be obtained which demonstrate the variation of wave bending moments with ship size, fullness, heading and

speed. The trends of predicted bending moment thus obtained are

pre-sented here in terms of effective wave heights.

In the following sections, the Sea Spectra, Model Data, Response Spectra, and Results of Calculations under Phase i will be

BASIS FOR CALCULATIONS

Sea Spectra

At the outset of this project, very few authentic sea spectra were available, but some records taken during an extra-tropical cyclone

in 1953 and analyzed by Pierson (3) were selected as the best available

storm sea data. The particular spectrum chosen for most of the early calculations was the one taken at the height of the storm after the wind had blown at 62 knots for 4 hours. The waves reached a significant height of 42 feet, and it was believed at the time of the analysis in

1959 that they were "probably among the highest significant waves yet recorded by means of an instrument." This storm sea spectrum is referred to in this report as the 62-knot spectrum. Two less severe spectra were also used in the bending moment calculations. The three spectra,

repre-senting three stages in the development of the same storm in the North Atlantic between Iceland and the British Isles, are compared in Figure 1

and have the following characteristics:

Time of Observation Wind Speed Significant Height

Date Hour Knots Feet

The first calculations were based on the above spectra. During the course of the work, about four hundred additional sea spectra became available in reports published by Moskowitz, Pierson and Mehr (4). Wave records were obtained by means of Tucker shipborne wave recorders on weather ships in the North Atlantic at or near positions I and J shown

in the map in Figure 2. The records were analyzed by electronic computer

using the procedures given by Blackman and Tukey (5).

The most critical calculations were then repeated using a mean

spectrum based on the new information. This spectrum was the average of

the thirteen most severe seas among all the spectra in refererLces (3) and (4), including the 62-knot spectrum used originally. All spectra having

11/15/53 1153 37 16.6

11/15/53 2355 52 31.4

significant heights of more than 37 feet were included. The resulting mean spectrum is compared with the 62-knot spectrum in Figure 3. In this

plot the spectra have been transformed into the non-dimensional slope

fornì found to be most suitable for carrying out the bending moment cal-culations and interpreting the results. In this form, the ordinates are a function of wave slope squared (rather than wave height) arid are plotted

against the natural logarithm of the wave length (rather than frequency)

for convenience in comparing the behavior of ships of different size. This form of presentation is explained in Appendix A and Reference 6.

It will be noted that the 62-knot spectrum has a very different

appearance when transformed in this manner. The high frequency (short wave length) end, which appeared insignificant in Figure 1 now appears

much more important, and the peak which is so prominent in the

conven-tional plot almost disappears in Figure 3. There are strong theoretical grounds for using this slope-log method of plotting for the present

purpose.

Comparing the two spectra in Figure 3 reveals considerable divergence in the short wave length portions. It is in this area that

the wave records and the analysis method used are somewhat doubtful in

accuracy. But in the region of most importance (wave lengths greater than 4U0 ft.) the two spectra are very similar.

A composite plot of the thirteen spectra used to obtain the mean curve in Figure 3 is shown in Figure 4. It was impossible to select

a single worst spectrum from the group, because different ones are highest in different ranges of wave lengths. An envelope curve through

the highest point at each wave length produced an unreasonably severe

spectrum. Hence it was decided to use the average of these thirteen

severe seas. Spectral values at wave lengths less than 250 feet

(logA= 5.5) are so uncertain that they have not been shown in the plot. The probability of a vessel encountering a worse sea than that represented by the average spectrum is very remote, especially on typical tanker

routes. Therefore, predicted bending moment trends in this sea should

be of considerable significance in design.

The sea spectra shown in Figure 4 are point spectra,

-9-ing irregular seas as observed at a fixed point with no indication of the

direction of the component waves. To represent a short-crested sea in which wave components come from many directions at the same time, a

family of wave spectra is required, each member of the family indicating

the energy in different angular bands of wave components. This is accomplished by multiplying the stationary spectrum by 2/if

_cos2/,

the angular spreading function suggested in Pierson, Neumann and James (7), where the angle » gives the direction of a wave component relative

to the direction of the wind (assumed to be constant). The spectrum can then be represented by a family of curves, each curve representing the total energy in the components coming from a particular direction. Figure 5 shows such a plot for the 62-knot spectrum, at angular incre-ments of 22½0, the two plots being the "wave height" and "wave slope"

forms of the directional spectrum, respectively. It is the latter form

which has been used in the calculations.

A two-dimensional, or directional, spectrum, can also be repre-sented by a Contour plot as in Figure 6 which shows the 62-knot spectrum with the above spreading function applied in the conventional form of

plotting. It is of interest to compare the two-dimensional spectrum of Figure 6 with an actual spectrum obtained by means of a special wave buoy which measures wave slopes. Canham, Cartwright, et al (8), give contour

plots of several directional spectra, the most severe of which (Figure 7) is strikingly similar to the 62-knot spectrum using the cos2 spreading function. On the basis of this comparison the choice of spreading

function appears reasonable. However, for use in bending moment calcula-tians it is more convenient to use the spectrum in the form of a family of curves, as mentioned above (Figure 5).

Model Data

In order to make comparative predictions of ship bending moments at all headings to the short-crested irregular seas described above, it

is necessary to know the bending moment response of different ships to

regular waves of a wide range of lengths and directions. Fortunately an

unusually complete program of model tests has been carried out by Vossers, Swaan, and Rijken at the Netherlands Ship Model Basin and the results

published (9). The models were derived from Series 60 lines and covered a wide range of block coefficient, hull proportions, and speeds, as well as different wave lengths and directions. A full and a fine model were chosen for the present calculations, as described below.

The full ship form chosen for the study was that which most

nearly represented recent large tankers. The following form

character-istics apply:

Block coeff., CB = 0.80 Prismatic coeff., C = 0.805 L/H = 17.50 Waterplane coeff., Cw = 0.871 LIB = 7.00 Midship coeff., Cm = 0.994

C.B. fw'd. .025 L

Here L is length, H is draft, B is breadth, and C.B. is center of buoyancy.

That these characteristics are typical of large tankers is evident from Figure 8, which shows beam, draft and block coeffIcient

against length for the above form (straight lines) compared with those

for representative tankers (data spots). (10; il; current periodical literature).

There is considerable variation in the characteristics of finer cargo ships, but the Vossers model based on Series 60, block coefficient

0.60,was used as a typical form quite similar to the Mariner class. The characteristics follow: Cß = 0.60 C = 0.614 p L/H = 17.50 C = 0.706 w LIB = 7.00 C = 0.977 in C.B. aft = .015L

The models had been tested in regular waves, with five wave lengths varying from 0.6 to 1.8 times model length, and five headings (ship to wave angles) varying from 170° to l0. The range of Fraude

numbers,

v/1/jF

, covered in the tests was from 0.10 to 0.25.

in the present project, measured bending moments from the tests described above were put into the form of nondimensional bending moment

coefficients, h/L, where Lhe effective wave height, h, is defined as

static bending momeìt calculation (Smith effect excluded) gives the same wave bending moment as that measured in the model tests. The response amplitude operator is [(h/L)

/U'0/A) ]

, where the ratio

C0/A

, wave amplitude over length, is proportional to wave slope. The

derivation of the wave bending moment coefficient and the response ampli-tude operator is detailed in Appendices B and C, respectively.

After conversion to this form, the operators can be plotted

against the logarithm of wave length. Since the longest wave in which the models were tested was l.8L, the data had to be extrapolated to longer waves. The extrapolated portions of the curves were faired so as to approach the static wave bending moment curves (with Smith effect) as a

limit. A sample plot is given in Figure 9, showing how the fairing was

carried out.

Figure lO has been prepared to show graphically the calcula-tion for the case of the 0.80 block ship heading into short-crested

irregular seas. The upper portion of the figure shows the different directional components of the spectrum based on the average of the 13

worst records (Figure 3). The second part of the figure shows the family of curves representing the response amplitude operators derived from the model test results, each curve for the 600 ft. length defining the

response of the model to the waves coming from a particular angle. The

curves are labeled with the angles 1Q indicating the responses to the

same angular wave components as those shown in the sea spectrum. Each of these component response curves was derived from the model tests at different angles to the waves by picking off the results at the

appro-priate angles.

Also shown in this plot are the head sea response operators

expanded to lengths of 300, 900, and 1200 feet. The other angular

com-ponents for these lengths have been omitted from the figure for clarity.

A comparison of the operator curves for different ship lengths demon-strates the advantage of the form of presentation used in these calcula-tions -- the response operators for any series of geometrically similar

ships plot as a set of identically shaped curves, shifted on the logÀ axis according to the absolute sizes of the ships. Portions of the

curves shown by broken lines are extrapolated beyond the measured data, as explained above.

The bottom portion of Figure 10 shows the bending moment

response spectra, which will be discussed later.

Ship Response Spectra

The product of a sea spectrum component for a certain angle and the response amplitude operator component associated with that wave direction gives a response spectrum component curve. The family of curves obtained in this way (one curve for each wave component) is then

integrated over direction (angle) to obtain a single response curve.

Four such integrated response curves for the four ship lengths are shown

in ttie lower plot of Figure 10. The angular components of the response spectra have not been plotted.

The double Integration of the family of response spectrum com-ponent curves over both angle and wave length gives the cumulative

energy density, R, for the bending moment coefficient. Since the

response curves shown in Figure lO are already integrated over angle, the cumulative density Is equal to the integral of each of these curves

over wave length only. From values of R for each ship size statistical parameters, such as the average value of the highest expected wave bend-ing moment coefficient out of a total of N oscillations, may be

calcu-lated from the expression, h /L = C

uT,

e

where the multiplier C takes different values depending on the number of

oscillations considered. For example:

Average h/L = 0.866 Ti

Average of 1/10 highest h/L = 1.80

VT

Highest expected he/L in 100 oscillations = 2.28

Vi

1,000 'I = 2.73

\fi

n _{" 10,000 = 3.145}

ifi

The statistical parameter used in these calculations was the

highest expected bending moment in 10,000 cycles. This criterion is

-13-roughly equivalent to the highest expected bending moment in about 20 years

of operation, since it represents 20 hours of operation in severe seas which

might be expected to occur for perhaps 20 hours In a 20 year period.

The

use of the same criterion for different ships implies that all ships are

in the same service.

This criterion is used here for qualitative

compari-sons only, however.

The question of a suitable criterion to use in a

quantitative prediction of design bending moment is not considered.

RESULTS

Effect of Ship Speed

The variation of wave bending moment with ship speed is shown

in Figure 11 for the 62-knot spectrum and head seas.

It is evident that

the speed of a ship has little effect on wave bending moments.

Decreas-ing speed can, in fact, increase the wave bendDecreas-ing moil]ents slightly (see

CB = .60 curves).

No consideration is given here to two other effects

of speed, namely the increase in the bending moment caused by ship-produced

waves as speed increases, and the effect of speed on slauatiing which may

increase midship hull stresses.

The former causes a shift of the mean

value discussed later; the effect of slallffL!ing requires further detailed

study.

Effect of Ship Heading

The vertical wave bending moment is influenced by the direction

of the ship's travel relative to the waves.

In a short-crested sea the

wave components come from various directions simultaneously, so that

re-gardless of the heading the ship reacts to waves coming from many angles.

The heading of a ship is defined here as the angle between the direction

of ship's motion and that of the dominant waves, i.e. of the wind.

The

calculated bending moments are the result of superimposing the ship's

response to all wave components present for each heading.

The effect on wave bending moment of ship heading is shown in

Figure 12 for ships of 600 foot length in both short and long-crested

maximum bending moments are reached in head seas, as expected, and are then less in realistic short-crested than in hypothetical long-crested

seas. It also shows the reduction in bending moments in beam seas is

comparatively small when the waves are short-crested, especially for

fine ships.

The comparatively high values of bending moment calculated in beam seas seems reasonable on the basis of the principle of superposition. However, it should be noted that the application of this principle to

ship behavior in short-crested seas has not yet been confirmed through

model tests. It is to be hoped that facilities for generating realistic short-crested seas in a model tank will be developed by some laboratory

in order to check and confirm the superposition principle.

Effect of Ship Size and Fullness

The results of the calculations for tanker type vessels with Cß = 0.80 in the average severe spectrum are shown in Figure 13, which

gives effective wave height as a function of length. A low ship speed of Froude number = 0.10 (8.25 knots for a 600 foot ship) was considered

to be a reasonable maximum speed in an extremely rough sea. The curve

0.6

crosses the L/20 line at L = 500 feet, and coincides with the 0.6 L

wave from about 500 feet to 650 feet. The matching of the calculated trend with these other criteria thus provides a sound basis for the

comparison of the larger ships with those of 500 to 650 feet, even if

the absolute significance of the statistical parameter is doubtful.

The calculated trend indicates that at lengths greater than 600 feet the increase ìn effective wave height with length is less rapid _than

is shown by the other criteria.

The results for the finer ships are shown in Figure 14, where

the curve is compared to the CB 0.80 curve. A somewhat higher speed

(Fr = 0.15; 12.4 knots for a 600 foot ship) _{was used since the finer}

ships could be expected to make better speed in rough _{seas. Possible}

increased stresses caused by slamming were not inc1.uded. The trend with length is similar to that for the fuller ships, and from 15 to 207. lower.

Thus the bending moment coefficient is not quite proportional to block

coefficient, since in that case the reduction would have been 257e.

How-ever, it should be noted that fullness is already taken into account in the bending moment coefficient h/L which includes the waterplane

co-efficient.

All figures are for mean wave bending moments (hog and sag) not corrected for the shift of the mean value measured in waves from the initial zero corresponding to the model at rest in still water. This shift of mean value causes an increase of 11 to l27 in hogging and

de-crease in sagging bending moment for the full ships, and a very small increase in sagging moment for the finer ships, when compared to the mean

values. The method of dealing with this effect is described in Appendix

D. Also not included here is the effect of lateral bending on deck edge

stress.

Another factor not taken into account is the indication from other research (12) that the highest wave bending moment occurs somewhat

forward of midships. For ships of similar form and speed, such as the 0.80 block series of different sizes, this should not affect the

compara-tive results shown. However, it might affect the comparison between 0.80 and 0.60 block coefficient forms.

Combined Vertical and Lateral Bending

The stresses associated with vertical bending moments are the mean of those measured in the deck on the port and starboard sides of the

ship. A ship operating in oblique seas is subjected to unsylLIIetrical

bending, so that the stresses measured at one deck edge will usually

exceed the mean value. This diagonal bending in an oblique sea can be

dealt with as the combination of vertical and lateral bending components. Since a limited amount of data on bending in both directions, and their phase relationships, were measured in the model tests, it was possible to

calculate roughly the combined effect.

To calculate the combined effect of lateral and vertical bending on deck edge stress, an effective bending moment for maximum combined stress was defined as follows:

where:

M = effective bending moment e

= amplitude of deck edge stress due to combined vertical

and lateral bending, at deck edge where combined stress

is maximum.

= section modulus of hull amidships with respect to

vertical bending.

The derivation of this expression for effective bending moment and its response amplitude operator are given in Appendix E.

The use of the section modulus with respect to vertical bending in the definition of Me makes possible a direct comparison of these effect-ive bending moments with the vertical bending moments reported previously. The deck edge stress, and therefore the effective bending moment, are functions of the vertical and lateral bending moments, the section moduli with respect to vertical and lateral bending (i and ZL), and the phase

angle between lateral and vertical bending moments. Results for the full

(CB = .80) ships with several values of ZV/ZL are plotted in Figure 15

in the form of effective wave height, as a function cE ship heading, in

the 62-knot spectrum. Bending moments for other ratios may be interpo.la-ted from the graphs. Calculation of some typical section moduli showed

that values of ZV/ZL 0.60 to 0.70 are coumon for large tankers.

Com-parison of the curves in Figure 15 with the curve for Z./ZL = O from Figure 12 shows a generally similar picture to that for pure vertical bending, but a steady increase in h/L as the ratio ZV/ZL increases.

Curves for ZV/ZL 1.00, 0.60 and 0.30 recalculated for the

average severe spectrum are compared in head seas with pure vertical

bending (Zv/ZL O ) in figure 16. The ZV/ZL = 0.60 curve would be

re-presentative of typical large tankers, and the effective wave heights

for this case run from 157. (at L = 300 ft.) to 257. (at L = 1,200 ft.) above the h for vertical bending only. These results are approximate to the extent that model phase angle data were not as complete as they should

Effect of Sea Severity

Limited computations of vertical bending moment were carried out for the 37 and 52 knot spectra previously described (Figure i) for the case of head seas. Results are tabulated below:

These results show a definite reduction in bending moment with

sea severity. However, these spectra are not necessarily typical, and further data on sea spectra must be analyzed before realistic trends of bending moment with sea severity can be firmly established.

The table also shows clearly the advantage of increased length of ship in all sea conditions shown, not only the most severe case. Note

that for the ship with CB = . 80 the coefficient decreases with increas-ing length of ship for any wind speed. It also shows that in this case a

fine hull (Cß .60) has a less rapid increase of bending moment with wind speed than a full hull (CB = .80). This leads to the curious situation that a fine 450 foot ship in moderately rough seas has a higher bending moment coefficient than a full 600 foot tanker; but in very rough seas

the reverse is the case. This apparent anomaly appears to be generally

in accordance with results obtained from full-scale ship stress observa-tions (13).

In order to explain the above findings Figure 17 has been

pre-pared. The plot shows the relationship between the sea spectra and

ship response spectra for the two ships in both the moderate and very

severe seas. Here it may readily be seen why in head seas the fine 450 foot ship is better off in a very rough sea, but inferior to the 600 foot full ship in a moderate sea.

Ship

CB _Fr

he/L in given sea spectra

Length 37 knot 52 knot 62 knot

450 .80 .10 .038 .053 .057

600 .80 .10 .022 .043 .048

900 .80 .10 .009 .027 .036

SUNMARY

Conclusions

Various conclusions drawn herein may be sunnnarized as follows:

The calculated results confirm the generally accepted idea that effective wave height (or wave bending moment coefficient)

in extremely rough seas does not increase In proportion to

ship length.

In fact, effective wave height increases less rapidly with ship length, for the particular severe sea spectrum studied,

than is assumed by various commonly used wave height functions. Wave bending moment coefficients in head seas are 15 to 207.

greater for full ships (CB = 0.80) at a reasonably slow speed (Fr = 0.10) than for fine ships (CB = 0.60) at a suitable

higher speed (Fr = 0.15).

Ship speed has little effect in general on extreme wave bend-ing moments, although high speeds do tend to increase them

slightly.

Head seas produce the maximum wave bending moments, but the

reduction in beam seas is not great (about 30 to 407. reduction) with short-crested waves.

The effect of lateral bending is to produce effective vertical bending moments which are greater than those caused by pure

vertical bending. The increase becomes larger with increas-ing ratio of vertical to lateral section modulus, ZV/ZL In

a typical large tanker (ZV/ZL = 0.60) the increase caused by lateral bending may be about 207,.

Hogging bending moments are 11-127. greater than the mean wave

bending moments in the case of 0.80 block coefficient. Sagging moments are very slightly greater than the mean

for the 0.60 block coefficient.

The comparative performance of different sizes of ships in different sea conditions can readily be evaluated by the

methods used in this study. It is possible, for example, to show that one ship may be superior in one sea condition and

inferior in another.

-19-Re coiiniiendat ions

Further research is needed in the following areas:

Collection and spectrum analysis of wave records on important

trade routes.

Carrying out additional systematic model tests to extend the

scope of presently available results, including more complete

data on phase angles between lateral and vertical bending.

Experimental model studies of bending moments in short-crested

seas, in order to check the applicability of the superposition

principle.

Investigation of the effect of slaiuuiing on midship hull

stresses and the influence of hull fullness and other

character-istics thereon.

Extension of the present work into quantitative predictions of

highest bending moments to be expected in the lifetime of any

ship in any service.

REFERENCE S

"Proposal for Research: Trends of Wave Bending Moments on Ship Hulls for American Bureau of Shipping, New York." Webb Institute

of Naval Architecture, 27 October 1961.

E. V. Lewis, "A Study of Midship Bending Moments in Irregular

Head Seas," Journal of Ship Research, Vol. 1, No. 1, April 1957.

W. J. Pierson, "A Study of Wave Forecasting Methods and of the Height of a Fully Developed Sea on the Basis of Some Wave Records Obtained by the OW.S. Weather Explorer During a Storm at Sea," Deutsche Hydrographische Zeitschrift, Band 12 Heft 6, 1959.

L. Noskowitz, W. J. Pierson Jr., and E. Mehr, "Wave Spectra Estimated from Wave Records Obtained by the OWS WEATHER EXPLORER and the OWS WEATHER REPORTER," New York University Research Division Report, Part I, November 1962 and Part II, March 1963.

R. B. Blackman and J. W. Tukey, "The Measurement of Power Spectra from the Point of View of Communications Engineering," Bell

System Technical Journal, 1958.

E. V. Lewis, "The Superposition Principle Applied to the Prediction of Ship Behavior," from Lecture Notes on Ship Motions in Irregular

Seas. Webb Institute of Naval Architecture Report, October 1963.

W. J. Pierson, G. Neumann and R. W. James, "Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectra and Statistics," H. O. Pub. 603, U. S. Navy Hydrographic Office,

1955.

H. J. S. Canham, D. E. Cartwright, G. J. Goodrich, and N. Hogben,

"Seakeeping Trials on O.W.S. Weather Reporter," Trans. Royal

Institution of Naval Architects, 1962.

G. Vossers, W. A. Swaan, and H. Rijken, "Vertical and Lateral Bending Moment Measurements on Series 60 Models." International Shipbuilding Progress, Vol. 8, No. 83, July 1961.

W. O. Nichols, M. L. Rubin and R. V. Danielson, "Some Aspects of Large Tanker Design," Trans. SNAME, Vol. 68, 1960.

C. L. Long, J. L. Stevens, Jr. and J. T. Tompkins, Jr., "Modern High Speed Tankers," Trans. SNAME, Vol. 68, 1960.

Z. .eorge Wachnik and Frank M. Schwartz, "Experimental Letermination of Bending Moments and Shear orces in a Multi-Segmented Ship Model

Moving in Waves," International Shipbuilding Progress, Vol. 10,

No. 101,

Jan. 1963.

-21-R. Bennet, A. Ivarson and N. Nordenstrom, "Results from Full Scale Measurements and Predictions of Wave Bending Moments Acting on

Ships," Report No. 32 of the Swedish Shipbuilding Research

Foundation, October 1962.

W. A. Swaan, "Ainidship Bending Moments for Ships in Waves," International ShipbuIlding Progress, Vol. 6, 1959, p. 398-408.

o

C,

cg

Q

r

09 /.0

//

/2

W, 5EC Fig. i

Sea spectra obtained froni wave records taken aboard a Weather Ship In the North Atlantic Ocean.

80

70

0

50

70 J

\J

70

20

/0

J

-f-Nil

+

Fig. 2

Weather ship stations in the North Atlantic.

All sea spectra given

in this report were obtained at stations I or J.

/0 20 70

tvtAP OF 77-/E

wEATHE,

SHIP

S74T/OVS IN THE

NOeTH

ATLANTIC

cl

30 '0

o

/0 20

LÀ]

4

5

_/

i ¡ I Ï I I I

II

I I

Il

I I /5° 2O0

3ti

0 X 900

/0a0

,'90

222

3000

I I I

Ill

i J i I I I I I

¿I

1.0

9

.8

.7

.,3

.1

.3

Fig. 3

Average of thirteen severe North Atlantic

sea spectra (4) compared to

62-knot spectrum (Fig. 1), plotted in slope-log form.

\\

\

\

\

X

.3

'çi

cii CULA 7Eo

_/2O/A/r

/

G2-AJOT 5PCT,M

/

4PtE SP-cTJ/q

X

\

7

cf

logeA

T:

4LL

A

///C,VT

iPECTt74

4ì

çeE1r

11

uiwr

"y,

7

a

X

Fig. 4

Composite plot of the thirteen severe sea spectra averaged in Fig. 3.

s

L°

X)4

Fig. 5 Directional sea spectrum obtained by applying an ideal spreading function

to the 62-knot spectrum of Fig. 1.

0o

k

r._ _QQ

3

2

çIj

0.Z O. 0. 1.0 1.2

W, 5EC.

I I i

Iiii

i i I i

3cwo0

/O

200

V'/AVE LEAIGT/-/) fEET

-29-7

I I. i i I I I I t i i I

I huit

I _I

ZOO

300

4Û0 (OO 9OO

/000

/500 ¿7OO

3000

VVAVE LE.k/6rH FEET

Fig. 7 _{Directional sea spectrum components, 62-knot spectrum of Fig. 5, plotted} in conventional form and in slope-log forni.

N

N

\

\\

_\

= o°

.80

.75

.70

/4O

/20

L

K

80

-5O

-ç

+

-++

+

++

+t

±

C88O

+

A

IiF

PIP

f + + +

i

700

800

9O0

/000

LE,&/r/i' 'v /r

R'e/L12

RIA]

2.0

/0

o

5

RANGE 0F WAVE

m L EA/G TH5 COVE?EO

-lAi MOLI'EL TESTS

SHIP PA TA

L

00'

C

F,- -0./O

II2°

/57.!

\ \

5 7A TIC Vt/A Vi 8FA/D/,V MOMEAI T

Ct},'VE

(5 WAA Ai 4PPk'QKI/vf,4 T/OAJ)

REE /4)

/350

_1/2/

/350

Fig. 9

Typical response amplitude operators, showing

extent of model data and

extrapolation to long wave lengths.

3

7

z

(4e/4)

JA)

£2F

¿

Ii'

67/2

,It4, - O°

(o

\\

_\\

\\ \\

_{\\ \\}

N N

LJûO

z9/etc T/OAJ/7

\SfI

JC7tJ/1

/3 JEV,

Nok'r4' 4r24VT/c

ST,,t1

7

8

/000

-ecx

>,

/

7_ I

III

F I I

III

v'

' iea' -

't77i'

s/I/p eLs/7w35

-

.-

-

-.--- -.---

.-- .--r.--

-

1

-7

6

9

,

\S/-//P 1fes1oNs

'a-',&C/(.'

6L9û'

û0

_A/orE:

_{/iìL/ES r S Mr COEFF}

LxperED /ìV /0, ODO CyciE3,

J ,fOT'T/QN4L

7) 72it

!Q(/,,çE 1('Ûor & 7

2<' 7

/.zo' 72,

/f?ePoìv5E

('L/I?V.

8

9

JILA

Fig. 10

Graphical summary of bending moment predictions for

ships of 0.80 block

coefficient in severe short-crested irregular head seas

(Froude No. = 0.10)

A

Oc

.20 /0

o

/0

/5

1.Sii 'S'PEO -%-Fig. 11

Variation of effective wave height with ship speed in short-crested irregular head seas (62-knot spectrum).

T

L9O0

C...

L=E0o'

I L I i i t I i t

1CQLL O W/ì4f'

SEAS

35

3'O

¿'s

20

/5

/0

5

o

Fig. 12 Variation of effective wave height and bending morn it coefficient with

heading in short-crested irregular seas (62-knot spectrum).

Vertical Bending Only.

7

.05

.04-.

.O3

.02

0/

o

5H/P Lg,&/GTH

//

/

/

Fr

Vk = 1G. so

= 0.80

J

al

e

' V/G3O

/

A

0.60

J

¡(leA

(/X LßC<,

L

öi

7E1

SE4

c

.60

= ì

I

-i

o

/35_

//E4 Q/AJG,

SEA frl

JEAS

/4540

SE4

§0

20

/0

o

lE

o

he

300

400

500

00

700

500

00

/000

1/00 /200

L5AV6TH,

Fig. 13

Trend of effective wave height with ship length computed for ships of 0.80 block coefficient in severe short-crested irregular head seas, in comparison with other wave height formulations.

40

¿PECT/

4, 'r

C4LCUL47-EO 4

e

cpLßÇ.

1±'T/C1L

(4'r

h4 v

a

4

_t'd7//V

/Y/4

I

,k,,wE,vr

/,v /4

"ECTLD

/,,

OOC

ao,,fû

26O,

6

.5

C

.3Oo

4?o

0Ö 7Oo 900 1000 //O O / 200

L(NTN, F7

Fig. 14

20

/5-IC

5-o

ç

-37-.06

.05 .Q-Q /

o

R

_//

//

) -.--= C- 7

.oc

NL

/

Z

5//P LrAi 7/Í = 600'

=

V

=rc2S

=./o

C

/g

eo

.4'eA

&/k11G,

¿EGkEES

Fig. 15 Variation of effective wave height and bending moment coefficient with heading in short-crested irregular seas (62-knot spectrum).

Lateral Bending Included.

FQLLOVI/,JG'

8 EA. A-1

/-/EAO

&rEcr/

v

VE 141çwr,

4e'

,4Çf/-L300

400

500 60e 7oo

LLNcTW, IEEr

C

80,

/0

Fig. 16

Trend of effective wave height with ship length computed for ships

of 0.80

block coefficient in severe short-crested irregular head seas,

taking

into account the effect of lateral bending.

Zy/z .3 =0 _L

CCJL.1TE.

4

'-fxp4rc7D

fT/C ¿

/,v JEv'E

4q

D 5E s 4/ /4

000

C,VCLE5

(4&ûìr

¿0

/hips)

90o

/OO

120o

fhe/L

L

IA

FLINCTI 01/ 0F

LLi2

2

N

N

N

IN 37-kNOT SEA

L

450'

CL3

L=

oo

0.80

6 -I ¿ _COO'

C: O. SO

END/ÑG

M/v1E,V7

SPE YPA

-39-SF4 SPEC TRA

2- KA'OT

37-Kwor

5/7//P

RP0NS

OPERA 7i9S

FR. M7.r 0./O

N

N

L -I 7

8

/

2-kor

SEA:

LOo'

\ç-

c. go

L= 45o'

c= 2.6O

Fig. 17 _{Graphical sunary of bending moment predictions for two}

different ships in two different sea conditions (head sea

components only).

6

7

8

Appendix A

WAVE SLOPE SPECTRUM

The wave bending moment calculations illustrated in Figure 10 re-quire the sea spectrum to be represented in wave slope form as shown in

the upper plot of that figure. The transfoLination of a sea spectrum

from the conventional energy form as a function of W to wave slope

form as a function of log)\ is given below.

W = circular wave frequency =

A

= wave length

z

[r(w)] = ordinate of conventional energy spectrum

r

Jr(Io.X)

= ordinate of wave slope spectrum

LÀ]

Here the term "wave slope" refers to wave amplitude/length ratio. Strictly speaking, maximum wave slope is

IL

where is wave amplitude and is wave length.

Considering an increment

6ú)

in the conventional system, the area of

the rectangle represents the square of the amplitude of a component wave. Similarly in the slope form of the spectrum the area of

[r'loge A)/AJZ

(IO

A)

represents the square

of the amplitude/length ratio of a component wave. Writing the square of the amplitude of a component wave in both systems establishes the

following relationship between the systems:

Since

and

therefore

r(log

_]

(Io

)=

[rw]2

w

(i)

(2)

loge A -

10e

(Zirg) -

()_) (3)

Substituting (4) into (i), we get

z z

[r(loge A)J (-

¿'w)

₌

_f

r(w)]

cW

(5)

and the relationship between the ordinates of the spectrum in the two

systems is:

FrJog A)

-W5()j2

L

À

J

-[]2

_8zz

(6)

Both sides of equation (6) represent the average squared value of the

ratio (amplitude/length). The function

r(IOíe A)

is

A

Appendix B

WAVE BENDING MOMENT COEFFICIENT

The wave-induced bending moments calculated by the methods described in this report are all expressed in terms of either effective

wave height, h, or wave bending moment coefficient, h/L. The deriva-tian of the wave bending moment coefficient follows:

M Static wave bending moment = cpgL2BhC

st w

where

c _{a coefficient defined by the above equation}

p mass density of water g = acceleration of gravity

L = ship length

B

ship

beam

h = wave height used in static calculation

C waterplane coefficient

w Similarly,

= wave bending moment obtained in an irregular sea

= cpgL2 B h C

where

he = effective wave height

and other quantities are as defined above. Solving for the effective wave height,

h-

M

e

The effective wave height may then be non-dimensionalized to the wave

bending moment coefficient by dividing by L,

jj_

M

Appendix C

RESPONSE ANPLITUDE OPERATORS

The response amplitude operators were derived from the model

test results (2) as given below.

Wave bending moments measured in the model tests were plotted in the following form originally:

Mw g h BL

where

= bending moment coefficient (Vossers)

M = measured wave bending moment = wave amplitude used in the tests g = weight density of water

B = beani of model

L Length of model

After plotting and fairing, the response amplitude operators were obtained by the following conversion:

r

Z. z z

Jhe/L

M

(A

- cgLBÇj

ZIA

1Z1__

Appendix D

Still Water Bending Moment

The bending moments reported herein are wave induced bending

moments only. The defined zero value (mean value) of bending moment is that present when the model is at rest in still water with the same weight distribution as at the time of the tests. Thus, the still water zero speed bending moment for any condition of loading must be

obtained by static calculation, and it can be added to the wave bending

moments.

Shift of Mean Value

Model tests reveal a shift in the mean value of bending moment measured in waves from the initial still water bending moment. One

cause of this shift is the bending moment produced by the waves formed by the forward motion of the ship, whether or not sea waves are present. Another factor is that there may be non-linear hydrodynamic forces caused

by the sloping sides of the ship.

The first of these effects is evaluated by running the model

in still

water and assuming the effect to be the sanie in regular or

irregular waves. It is dependent only on the ship form and speed, and produces a hogging bending moment for full forms and a sagging bending moment for fine forms, the magnitude of the bending moments increasing

with increased speed.

The second effect is evaluated by comparing the mean value of bending moment when running in waves with that when running at the same

speed in still water. It Is a function of wave height and heading as

well as ship form and speed. The correction can be either hogging or

sagging.

CORRECTION FOR SHIFT OF MEAN VALUE OF BENDING MOMENT MEASURED IN WAVES

Both of the above corrections can be determined in the model tests and applied to the expanded results tor a ship in irregular wave The method of applying the zero shift correction is illustrated in the figure, which shows a case in which hogging is greater than sagging. The assumption on which the correction is based is that the response amplitude operators at each wave frequency are increased by the amount of the zero shift (as in hogging amplitude for the figure shown). Using the fictitious hogging amplitude should lead to an approximately

correct prediction of hogging moment in irregular waves. Similarly,

a reduced fictitious amplitude is necessary to predict sagging moment

in irregular waves.

t

TRUE ZERO

AMPLITUDE

SHIFT

UMt 5!NUSOWAL

BENDIF'4G MOMEi.4T VAR,ATIc»I

IN

CIULAR WAVES

411

FC.TTIOUS AMPLITUDE To

ACOUN1 FOR ZF.RO SHIF-r

IN HOGCING =

TRUE AMPL. + ZERO SfflFr

Method of correcting for shift of base line in bending moment records

obtained in regular waves.

45-f

TRUE ZERO

.1

5AG

TRUE ZERO

Aooendix E

EFFECTIVE VERTICAL BENDING MOMENT FOR DECK-EDGE STRESS

When lateral and vertical bending are combined, the effective bending moment for the combined (deck edge) stress is derived as given

below.

°=

OL

f

'

V

where

0

, O are vertical and lateral wave bending stresses, respectively.

, MWL are vertical and lateral wave

bending moments.

Z,

,

ZL are vertical and lateral section moduli.

The combined stress at the deck edge due to vertical and

lateral bending in regular waves is:

=

_{± 2cç,o}

_{CO5 (2)}

where

8

= phase angle between vertical and lateral stress

or bending moment. Substituting,

i

Z1

ZVZL

2 M/VMWL cos

(3)

The effective bending moment for maximum combined stress is defined as

follows (see text of report):

Thus

M

M

+

ML(-)

1-

2MwvMwL(

i CO5

s

Z%

MeZv

(4)

where M_WV, M and cos are obtainable from model test results. Each

WL

of the wave bending moments in the above equation (M , M and M ) can

e wv WL

be expressed as a response amplitude operator as defined in Appendix C. The above relationship can then be written in terms of _response

ampli-tude operators as follows:

/ 2

/he/L\

(he/L

/he/L2 /Zv\Z

_{¿(he/L/he/L\}

+

CO5 'V °' 'L

c0/A

1 Ç0/

-47-=

()