Determination of wave bending moments for ship design

ARCHIEF

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Deift

DETERNATION

WAVE BENDING MONTS FOR SHIP DESIGN

by Rutger Bennet

August 1963

(Paper to be presented to the Scandinavian Ship Technical Conference, Abo, Finland, Lth October, 1963)

INTRODUCTION

According to the rules of the Classification Societies,

the longitudinal bending moment is decisive for the midship

section modulus, and hence also to a great extent for the

steel weight This bending moment consists of several rather

independent parts, only one of which - the still water moment

-can be calculated exactly. However, this part often has only

a minor influence on the design. The remaining parts are

often given the common name wave bending moment, which

con-tains both the moment really caused by the waves, and the

added effect of local loads of various kind: slamming,

vibra-tions, temperature variations and others.

For design, part of this moment is calculated according

to a method which in principle has been standardised for a

long time, and the rest of the load is taken care of by a

nominal permitted stress. The notion about what is covered by respectively the bending moment and the stress is vague, but the ratio between them is given a value, the section

modulus, which has been found adequate from experience. It is probable that the scantlings, determined in this way, are

on the whole suitable for ships of those types and sizes

where we have sufficient experience. The difficulties have

started now we begin to build ships outside this experience,

2

In order to learn the real loads on a ship hull in waves,

a great number of measurements have been carried out, both

full scale and on models. For model tests to have any meaning,

it must be possible to convert their results to the

corre-sponding ship. The greatest problems in the comparison between

model and ship are the ratio of bending moment to stress, and

the generation of a realistic wave system in the tank, which

can be directly compared with the conditions during the full

scale tests.

In the ships the stress variation is usually measured

at a point of the hull, where there are no stress

concentra-tions or other disturbances. Calibration of the hull at the

gage 10-cation, i.e. recording of the stress caused by a known

change of static bending moment, has been made in several

cases, and generally the agreement with theory has been

satisfactory. Even if considerable doubts have been expressed

by some investigators, it is believed that the so called

simple beam theory will give a sufficiently accurate value

of the moment from the measured stress. At least this is so

as long as the stress is caused by a static or semistatic

moment, varying only at the slow rate of the waves. The

action of high frequency, dynamic loads is more doubtful,

which also depends on the fact that the instruments used often are too slow to record such variations. At the Swedish

tests this has been considered rather an advantage, as the

primary purpose has been to investigate the 7pui'e,

seniista-tic wave bending moment. The statisseniista-tical pattern which has

been suggested for the variation cannot be expected to be

valid for the vibrations, caused by slamming _{or 7whipping.}

These stresses must be treated separately and added to the design afterwrds. They are not mentioned in this paper.

u

3

The next problem is the wave system. The statistical wave theories have advanced very fast, and many equations have been proposed for wave spectra. During the second half

of 1962 a project was initiated by W. Pierson at New York

University, which is aimed at creating order in this confusion.

A great number of wave records, gathered during several _years

on the British weather ships by means of the Tucker Wave

Re-corder, are treated in a computer. Two reports _{are already}

published, containing about 2+00 such spectra, together with

information on corresponding wind speed, /1/e It is thus

possible to find out what wind generated spectra really look

like, and a comparison with one of the available theoretica.1

equations, which was recently performed at Webb Institute of

Naval Architecture, N.Y., has given interesting results which

will be mentioned later in this paper, /2/.

With a great many results of full scale stress

measure-ments available, several model test series, and a number of

recorded wave spectra, it is now essential to find methods of

using all this knowledge0 The difficulty of getting agreement

between model and full scale tests is mainly caused by the

problem of creating comparable _{wave conditions. Even if the}

superposition principle is valid, and typical spectra are known, such a comparison is difficult because _{so few full}

scale tests have been accompanied by wave measurements. For

the model both the moment and the _{wave system are known, for}

the ship only the moment and quite generally which wave

spectra may be expected, but not simultaneous values of these two variables. The only possibility must therefore be _a

sta-tistical treatment, similar for ship and model, and then the results can be compared on equal probability levels in both

-

2-cases, Such calculations have been performed independently

at the Swedish Shipbuilding Research Foundation _{- Chalmers}

Technical University, Göteborg, and at Webb _{Institute of}

Naval Architecture, Glen Cove,

N.Y.

attempts cannot be regarded as any final answer to all

questions, but the results are certainly promising. It _ought

to be possible very soon to predict from model tests the

wave bending moments acting on a ship during its service

life. This may in the first place have _{importance for ships}

of unusual form or size, where experience is now lacking. It may, however, later become a normal procedure for every new ship, just as it is to determine the required horse po-wer from model tests. A similar analysis _{can also be applied}

to systematic investigations of the _{influence of various}

hull parameters on bending moment, motions or speed in waves. Regarding the design of the structure, _{it iaust of course}

be pointed out that everything is far _{from finished with the}

still water and wave bending moment. First, a number of other loads have to be added, and further it is not clear what is really decisive for the structural strength. On the other hand

the wave bending moment is certainly _{a major part of the}

va-rying load, and the question of the _{importance of fatigue can}

never be solved unless the long term distribution _{of this} moment is known /3/.,

ANALYSIS OF FULL SCALE TESTS

Stress measurements at sea have been statistically

ana-lysed since Jasperas classical paper to SNAME in 1956. As in

so many other cases, Jasper had a predecessor in Ltcmdr Roop, USN, who more or less intuitively _{suggested a statistical}

5

approach already in l9LO /L/. Roop found empirically _a

distri-bution of wave stresses, which later proved to be _{very close}

to the Rayleigh distribution. This has _{now been found as a}

result of the assumption that the _{waves can be regarded as}

a normal, stochastic process, and it is the generally

accep-ted basis of all statistical analysis of ship _{response to}

waves . Incidentally, Roop even went a step further and indi-cated what long term distribution might be expected, if a

great number of short term distributions were added.

The agreement between different investigators _{in this}

area seems to end with the Rayleigh distribution, and there is great uncertainty as to how to continue the _{analysis of}

data. One reason for this is uncertainty _{as to what is the}

actual purpose. Many authors consider it _{necessary to find}

the greatest value of wave stress, or bending moment, which can ever occur. Attempts have been made to define _{the most}

severe condition _{for a certain ship length and form, then}

to extrapolate test results to this condition. _{It is, however,}

not certain this is possible, _{or even desirable. The wave}

moment is depending on so many different _{factors, all of}

which have a wide statistical scatter, that it is practically impossible to find the combination giving the _absolute

ex-treme value. An important factor, for _{instance, is the total}

time the ship will encounter _{every such combination, In}

gene-ral it must be assumed that this time is _{shorter, the more}

severe the wave condition becomes, which means a lower

proba-bility of high bending moments. These _{are two factors of}

directly opposite influence. Further, there is the very little

known influence of how the ship is handled, what somebody has called _{the captain's spectrum. It is undoubtedly a fact,}

6

that the greatest wave stress recorded in _{many ships has}

occurred in 7- Beau'ort, which is not so very extreme0

If the wave system is represented by _{a spectrum, this}

has three important parameters: general shape, _area f'o-quency range, especielly the fref'o-quency of the peak0 At least

for higher wind spueds, the shape does not _{seem to vary too}

much. The area: which determines the wave heights2 may, how-ever, vary between very _{de limits, even for constant wind}

speed, and this is also the case for the peak frequency. The

result is that the wave bending moment _{may be of widely}

different order of magnitude at different occasions in wave

systems, which are visually judged to _{be identical, either}

they are defined by wind speed or by sorne observed wave

height0 To be sure of having measured the _{highest possible}

value ±n each condition so defined, _{it may be necessary to}

carry cn the tests for the enti:'e serioo life of the _ship0

The only possibility of getting _{any information regarding}

extreme values within a reasonable length of time, is by

statistical analysis _{It is then of no importance how the}

wave systems are defined, onJy a statistical distribution of the moment variation is determined _{for each such s:rstem}

according to the chosen definition.

There are many advantages in _{expressing the wave}

con-ditions by the Beaufort number0 Ships' _{officers have much}

experience of this procedure, and there is generally a fairly good agreei'ient with corresponding _{wind speeds. The}

available statistics on wind speeds is _{still much better}

than on wave heights, and it is therefore easier to

deter-mine the probability of _{a certain wind speed than of a}

7

of a given bending moment it is essential to know the

per-centage of the total time at sea every wave condition is

expected to prevail.

In reference /5/ was shown, how a long term

distribu-tion of bending moments in each eaufort number, or small

group of such numbers, can be calculated by convolution of

two separate distributions: the Rayleigh distribution _in

every single wave spectrum, and the distribution of its

parameter r (root mean spuare) in the different spectra with-in each group, A similar c1cu.lation hs b2en made for an

Ameriáan C 4-ship, fig. l If then the probability of each

group is known, which is the same as the time the ship will meet corresponding wave conditions during a specified period,

the expected ma:circium value is given. In fig. 1 is

illustra-ted, how the maximum for this ship _{is very nearly constant}

in all conditions more _{severe than about 7 aft. It is}

per-haps more correct to _{y, that the probability of a certain}

high value of bending moment is nearly _{constant in all wave}

conditions above a certain limit. This also _{agrees with the}

experience that the highest value sometimes _{has been measured}

in rather moderate weather. Above that limit it is quite

fortuitous in which of all possible conditions _{the highest}

value happens to be measured. This limit _{may not be the}

same for all ships, among other things it is probably

de-pending on ship length. The point _{on the weather scale}

where the curve flattens out is moving towards more severe

conditions with increasing ship length. This _{can be} calcu-lated from model tests

As was pointed out above, not only the _{extreme values} are of interest, but all amplitudes which are expected to occur at _{least up to io6 times during a normal service}

period. All the weather groups must then be added, each

weighted according to its own probability of _occurrence,

fig. 2. The principle of this calculation was demonstrated

in ref. /5/.

This method of regarding the final long term

distribu-tion as a sum of a. great number of short term distribudistribu-tions,

added after being weighted according to weather _statistics,

leads naturally to a similar method for analysis of model test results4 A direct comparison between model and ship is

then possible, and consequently also general conclusions can

be drawn from model tests0

ANALYSIS OF MODEL TESTS

Model tests in regular waves have no meaning if they

cannot be evaluated in a way to make them valid in real,

irregular wave systems0 The only possibility for this is given by the principle of superposition, which assumes that

the response to each frequency in the wave spectrum can be

determined by a test in a regular wave with the same

fre-quency, or wave length.

Rather different results have been obtained regarding the validity of this principle0 Most available test results

show that the linear approach is goDd enough for practical

use, but there are exceptions and the question may not be

quite satisfactorily solved.

Model tests may also be carried out in irregular _waves,

either with a spectrum directly programmed to _{agree with an}

actually recorded one, or with a quite arbitrary spectrum.

As it is practically impossible to run tests in all the

superposi-

-9-tion is still necessary. If the model spectrum is

approxi-mately of the same shape as the real _{one, the influence of}

any possible unlinearity will be small, and that may be a

reason for trying to find good approximations to wave

spec-tra.

Under a cont.ìct with the American Bureau of Shiping,

an investigation is being made at Webb Institute of Naval

Architecture by EV. Lewis and others, about how model

tests might be made to agree with full scale tests, _{and how}

such results are to be interpreted. VosserTh tests _in

regu-lar waves with models of Series 60 /6/ were used as a basis

for calculations. A certain number of the _{spectra published}

in ref. /1/ were also investigated, in order to find ave-rages for their shape and area at various wind speeds, i'2/.

At first, bending moment spectra were calculated in three

typical wave spectra, recorded in raspectively 37, 52 and 62 knots winde The results indicate the influence of length,

block coefficient and speed, fig. _{3. Preliminary studies}

showed the necessity of taking the short-crestedness _{of the}

waves into account, which is an effect of the angular spread of the wave energy. A good general agreement with measured three dimensional spectra /7/ was obtained by letting the

energy vary as cos2a, where a is the deviation from the

main direction. The ordinate of the unidirectional spectrum

was thus multiplied by the factor ? cos2a, _{which makes the}

integral between plus and minus 900 equal to unity.

If the angular spread of the energy is to be _taken

into account, model tests in _{many different headings must}

be available. For these calculations _{were used 0, 20, 2+0,} 60, 0 and 90°. It must be emphasized that such _{a calculation}

10

-The trends with length and block, shown in fig. _{3, have}

also been found at full scale tests. The nondimensional

mo-ment is decreasing with increasing length, and increasing with the block coefficient. Comparing L _{= 15O ft, C = .60}

with L = 600 ft, _{0B = the short, slender ship has a}

greater relative moment in wind speeds below about 50 knots, while the longer and fuller ship has the greatest moment in more severe weather. The curve for the former ship is

consi-derably more flat in its upper part, and the moment is only

slightly increasing above 50 knots. This is a result of the

frequency ranges of the spectra, and especially _{of the}

fre-quency for the maximum energy, in relation to the ship

length.

For a better view of the extreme values, the _results

in the 62 knots spectrum are shown in fig. L1., as function

of ship length. The scale of moments here _{represent the}

highest expected value of 10,000, which _{corresponds to}

about 20 hours in this spectrum. In the fig. _{are also shown}

estimated maximum values from full scale tests _{according to}

ref. /5/.

There is certainly at least a qualitative agreement

between models and ships in fig. L1., but the comparison is

still not satisfactory. One single spectrum _{cannot represent}

the most severe conditions during the _{entire service life}

of a ship. For a true comparison, the model _{tests must be}

analysed statistically with the _{same method as the full}

scale tests. It is then _{necessary to have a statistical}

distribution of spectra.

An investigation of a great number of the _spectra

suggested by Darbyshire in ref. // was found to agree very

well with most of the records as to shape and peak frequencies,

but not so well as to area, i.e. wave heights. To find an expressionf' the variation of the heights, the measured

spectra were grouped according to wind speed, and the

distri-bution of the root mean square heights was determined in

each group. The mean values were compared with observations,

given by Roll /9/, with the surprising result that the wave height, visually observed on the weather ships is on the average equal to the root mean square of all heights in the

spectrum, the r-value. This indicates that spectra are

about L.O% higher than previously assumed, when the observed

height was taken to be the significant value. An average

spectrum at a certain wind speed consequently seems to have

Darbyshire's shape and frequencies, and area according to

visual observations.

On the basis of this investigation a series of average

spectra in various wind speeds were calculated, which may be

assumed to represent typical North Atlantic conditions. By an approximate method the scatter of the measured spectra about this average was included in the calculation, and some account was also taken of the fact that the relative heading to the dominant wave direction is varying. For each spectrum the corresponding moment spectrum was calculated, which

directly gave the parameter r of the Rayleigh distribution.

Finally all the Rayleigh short term distributions were added, with regard to the probability of each spectrum from weather statistics /9/. The resulting long term distribution

is directly comparable to the one calculated from full scale tests according to the method given in ref. /5/. The model

12

-At the same time _{a similar investigation was conducted}

at SSF-CTH in Gothenbu.rg, with somewhat different assumptions regarding the wave spectra /13/. _{The same Darbyshire equation}

was used, but it was modified to make both _{heights and}

fre-quencies agree with visual observations In refG /9/ are

given distributions of _{simultaneously observed heights and}

wave lengths. A spectrum was fitted to each of those combi-nations, and it was assumed _{to be valid during a length of}

time corresponding to the _{relative number of observations.}

The observed height was taken to be the significant _height,

and the observed _{wave length to be the significant}

apparent

period. According to the _{Webb investigation the visual}

height seems to _{agree better with the root}

mean square than with the significant value, and this method _{therefore gives}

too low bending moments. On the other hand _{no account was}

taken of the

energy spread, nor of the fact that the ship

has various headings _{relative to the dominant wave direction.} These two factors both _{tend to make the calculated moments}

too high. Results for two ships are given in fig. 6.

DETERMINATION OF WAVE BENDING MOMENT FOR A NEW SHIP A calculation of expected wave bending moments in a

ship of a certain length and hull form requires _the follow-ing conditions to be fulfilled:

The amplitude _{operator is completely determined.} The superposition method _{of calculating the moment}

spectrum from amplitude operator and wave spectrum is valid.

Every wave spectrum the _{ship will ever encounter} is

13

-L) The Rayleigh distribution is valid for the morent

amplitudes in each of these spectra0

These four points obviously contain all _necessary

in-formation for a complete calculation of the probability of any arbitrary bending moment, and such a calculation would

be exact in a statistical sense0

The amplitude operator, or the wave moment _{as a}

func-tion of wave lengths, can be determined in regular _{or in}

irregular waves. An irregular system with a spectrum

approxi-mately similar to the real sea spectrum in shape and fre-quency range is bound to give the quickest result, as the complete function is obtained by a small number of runs. A transformation from one spectrum to another one, even

with very different degree of severity, has been shown _to

agree very well at tests in the Davidson Laboratory /10/.

If the operator is obtained in regular or irregular

waves, it is in either case necessary to take into account the effect of the angular spread of wave energy, usually

called the short-crestedness of the sea. Results from test

runs in different headings must then be available, and the

energy spreading function must be known. After a sufficient

number of such complete tests, it might be possible to find general correction factors, by means of which results in

only one direction could be made valid in _{a three dimensional}

spectrum. It is, however, evident that the spread of _energy

has an essential influence on both moment and other _responses.

During 25 years a ship may encounter about 100,000 dif-ferent spectra, and it is hardly conceivable _{even with} elec-trafic computers to calculate that number of Rayleigh

to one single long term distribution. Moreover, _{it is not}

possible to know all the spectra, because each _{ship will}

only meet a random sample of the infinite _{number of spectra}

that may occur at sea. A statistical approximation is thus simply necessary.

In the chapter on Analysis of Model Tests, two methods

were described, which have been used to obtain _{a long term}

distribution of spectra. A continued analysis of the lOO

spectra published in ref. /1/ (which probably will be

fol-lowed by more) will undoubtedly give a much better basis for this type of calculations. It might, for instance, be

possi-ble directly to determine distributions _{for the ordinates of}

all spectra at every frequency _{(= wave lengths), and in that}

way obtain a statistical _{mean spectrum covering a long}

period of time, together with the actual _{scatter about this} mean.

The method of superposition has _{in many cases been}

shown to be applicable in practice, _{even if it is not}

strict-ly proved, and the Rayleigh short _{term distribution has been}

ccnfirmed by so many investigations that _{it may be adopted}

with great confidence. By means of an amplitude operator

and a distribution of spectra, a long term distribution of

bending moment amplitudes _{can consequently be calculated.}

Incidentally, this method is not only _{applicable to the}

bending moments, but to all ship _{responses to waves: motions,}

accelerations, speed, or power. It should be possible to

find by integration an _{average speed during, say ten years}

from such results as are given by Gerritsma and others in

15

-A true wave statistics may be obtained in two ways:

continued work on actual wave records, and analysis accord-ing to the proposed method of all model test results now

available. By comparison with the full scale tests, also

available to a great number, empirical spectrum fami1ies

can be obtained, which give the best result for the greatest

number of models, within the confidence limits that are calculated /1L1/. For normal ships, results for the most similar model in a test series may be used as an approximate

basis for design; for ships of unusual size _{or form a}

speci-al model test has to be done. It is probable that it will

not be long before it becomes part of normal routine to test

models of all new ships also in waves, with measurements

including bending moment and shear force.

This paper is to a great extent based _{on work done}

during a visit to Webb Institute of Naval Architecture,

N.Y., as part of a contract with American _{Bureau of Shipping.}

The author is indebted to these institutions _{and to Professor}

E.V. Lewis for this opportunity, and for _{permission to}

pub-lish the results. A travel grant from OECD, _{allocated through}

the Swedish State Technical Research _{Council is also}

REFERENCES

/i/

_{Moskowitz, L., Pierson, W.J,, Mehr, E.}

Wave Spectra Estimated

_{from Wave Records Obtained}

_{by the}

OWS WEATHER EXPLORER and the OWS WEATHER REPORTERO

New York University.

Department of Meteorology

_{and Oceanography.}

Part I, Nov. 1962, Part

_{II, March}

/2/

_{Goodrich, G,H., Bonnet, R0, Lewis, E.V.}

The Use of Model Test

_{Data to Predict Statistical}

_Trends

of Wave Bending Marnent0

Prcgress

Report No0

of Project for American

Bureau of

Shipping0 (Unpublished)

Webb Institute of Naval Architecture.

_April

1963,

/3/

Nibbering, J.JOW.

Vermoeiing van scheepsconstructies.

Schip en Werf

30(1963) 10, s. 275-25.

/L./

Roop, W.P,

Service Strain Tests,

_{Technique and Procedure0}

US Experimental Model Basin.

Report No,

/5/

Bennet, R., Ivarson, A.,

_{Nordenström, N,}

Results from Full Scale

_{Measurements and Predictions of}

Wave Bending Moments

_{Acting on Ships0}

Stiftelsen för Skeppsbyggnadsteknisk

_{For skning, Got eborg0}

Report No, 32, 1962,

/6/

_{Vossers, G,, Swaan, W.A0,}

_{Rijken, H.}

Experiments with Series 60

_{Models in Waves.}

Trans, Soc. Nay0 Arch,

_{Mar, Eng.}

/7/

Canham, HOJOSO,

_{Cartwright, D.E., Goodrich,}

_{G., Hogben, N,}

Seakeeping Trials

Trans0 Royal Inst. Nay,

Arch0

1962,

//

Darbyshire, J0

A Further Investigation

_{of Wind Generated Waves.}

Deutsche Hydrogra.phische

_Zeitschrift

12(1959) 1.

/9/

Roll, H,U.

Height, Length and Steepness of Seawaves in the North

_Atlantic0

Soc0 Naii-, Arch, Mar.

Eng,

Technical and Research

_{Bulletin 1-19,}

_l95.

/10/ Dalzell, J0F.

Trends with Speed, Ship Length and Sea Severity

_of

Midship Bending Moment of

_{a Destroyer Type Ship in Head}

Seas,

Davidson Lab, Stevens

_{Inst. of Technology.}

Report

R-92, 1962.

/11/

Bennet, R

A Comparison of Measured and Statistically Calculated

Wave Stress Distributions,

Progress Report No0

_{L- of Project for AmerIcan Bureau of}

Shipping0 (Unpublished)

Webb Institute of Naval

_{Architecture0}

Juno

/12/

Zubaly, R.B,

Trends of Bending Moments in

_{Irregular Seas.}

Progress Report No, 2 of Project

_{for American Bureau of}

Shipping, (Unpublished)

Webb Institute of Naval

_{Architecture0}

January

/13/

Ivarson, A,

Ore Carrier Model Tests,

Stiftolson för Skeppsbyggnadsteknisk

_{Forskning. Göteborg}

Report No

/14/

Nordonström, N,

On Estimation of Long Term

_{Distributions of Wave Induced}

Midship Bending Moments in Ships,

Chalmers Tekniska 1-lögakola,

Institutionen för Skeppsbyggnadsteknik.

Aug,

/15/

Gorritsma, J,, van den Bosch,

_{J.J,, Beukelrnan, W.}

Propulsion in Regular and Irregular

_Waves.

mt, Shipbuilding Progress,

(196l)

2,

(June), s

A6/

Swaan,

_{:J.A0, Rijken,}

Speed Loss at Sea as a Function of Longitudinal Weight

Distribution0

North East Coast Inst. Eng.

Shipb,

z

FIG. 2. LONG TERM

_{DISTRIBUTION 0F EFFLCTIVE}

WAVE PLIGHT

SAIE SHIP AS FIG. 1.

'DN

(SERIES 60)

/

L

N

r'

FIG. I. EFFECTIVE WAVE HEIGHT EXPECTED

WITH P% PROBABILITY IN VARIOUS _{BEAUFORT NUI'4BERS}

SHIP: S.S. WOLVERINE STATE, L

¿96'3

_{0B = 0,65}

m-WIND SPEED

KNOTS

FIG. 3. VARIATION OF W.B.M. COEFFICIENT WITH WIND

ROOT MEAN SQUARE VALUE OF ni.

_{(REF. /12/)}

[F. [5]. CB 0W - 066

o PEF[53, C.O-l5-D.BO

SHIP LENGTH

I I I I I

FEE T

FIG.

Li.. VARIATION OF W.B.I. COEFFICIE

_{WITH SHIP LENGTH.}

EXPECTED MAX. OF 10,000 IN 62 KNOTS

500

ft.. SEC.

-50 KNOTS (Jo)

IjO KNOTS (s-9)

OKNOTS (7)

ZOKNOTS (5)

FIG. 5. MODIFIED DARBYSHIRE

4

-3

-2

-I _f

6 a. BULK CARRIER. L

? 8 7 i$ 5 4 3 2 I 0 72 -IO o 6 z ß(/L/( CAìi!/Ei #100(6 TíT PgZß/Cf(ON f.YE4It CANADA (iN/P C)

6 b. DRY CARGO SHIP. L

FIG.

6.

AND FULL SCALE TESTS.