Statistics and wave loads

comments on the report:

ON ESTIMATION OF LONG TERM

DISTRIBUTIONS OF WAVE INDUCED MIDSHIP

BENDING MOMENTS IN SHIPS

with applications of proposed methods

by

NILS NORDENSTROM

Repon from the Division of Ship Design Chalmers University of Technology The measurements have been made ih cooperation with. the Swedish Shipbuilding

Research Foundation

The analysis is supported by a grant from the Swedish Technical Research Council

Gothenburg May 1964 Anders Svennerud

Preface

These comments tO my earlier report "On _{stimatiOn of} Long Term Distributions of Wave Induóed Midship Bending

Moments in Ships" have two purposes.

Firstly I want to point out to those less familiar with statistics, the central role of statistIcs ii all

na-tural science research, _{I do not expect that my very short} introductIon, will make the statistical _{concepts clear, but}

I hope to awaken _{an interest in these important fundamentals.} Secondly I want to give a more useful and understand-able description of the methods _{presented iii the earlier}

report. _{Some views and conclusions regarding methods and} results are given.

I especially thank Professor Anders Svemienid whose understanding and interest in the subject has made this work

possible and Mr. Ru-bger Bennet for _{our interest±aag and} st1mulating discussions.

Gothenburg, May 1964

Appendix I, _{TIathema±ica1 treatment of the}

còmection between _{error of measurement and}

transformation function

0e0.0...0. ....ó...,.e.0.

49c

Appendix II A note on the choise of

"representative r value"

_{000,00 0000OflØ0fl.o5.00.. o,.}

₅

j-Appendix III, _{Sorne simple ways to estimate} the parameters of the Weibull _distribution

55, Appendix IV, _{Statistical anaiysis of the}

data presented by Göransson and Olofsson _in

reference

/6/

_{. co,: 000000000,o 900000C}

- 58

IntroductjÓn

i

Summary of reference /13/

_{0000000e..oe.00}

Other methods cáo OSI 00.0. o

.10

A practical application ₁₁

The type of the long term peak

value distribution 0 . _{0 0 0 0 i o e o o o} e. t 15. Thê importance of accuracy of the

method used to estimate r values _{00 00 00 00 00 00 00.4,002 17.}

"The lagest possible value'! _{C 0} 0 3

'.0 -Moient factors 27. Summary

_{00100.500 0000000000*01St 00000 006 0.0.00}

r O 43 Reference s 0 2 0 0 0 o e n i o o o o o o e o e o a o s o i _{o o s o o o s o o I I * O 0.0 0 0} j 07 r I

Ïntxoduction.

Two probiem concerning wave bending thoments aro discussed in the report /13/.

Estimation of wave loads from full scale rneäsremen-ts at sea.

2. Predict5ion bf long téith t.butions oí wave loads

for the entire life of ä ship.

The results can be used. f o.r the analysis of other

vari-ables e.g. the motions of a ship izia seaway.

A shit :j subjected to a very Jarge number (about 108)

of stress cycles during the.lifetime. The amplitude of the loads vary at random and. statistical distrihutiòn functions are tie only means of analysis of this variation.

Let us bgin with a brief diseuCsion of some important

fundamentals for all scientific work, _{Carnap /3 4/ and.}

Nagel /12/have treated this subject in considérable d.tai10

There are two kinds of reasoning and. conclus±ons,

Analytic or d.eductive.0

This way of reasoning cannot increase ou imowledge of

other facts than those which are contain od. in the pre-mises. E.g. mathematicai proofs are ana1ytic

Synthetic or inductie«,

This way o± reasoning is the only way to increase our

lmowledge of facts that are nQt:»ontain.ed. in the pro-mises. Then you Imow some events that already have

taken place, and want to predic.b the outcome of the same kind of events in the future you have to use

ductivereasoniiig. The best way to handle thiC prolem

is to use statistical method.

All statements that can be made about facts

must be

based on statistics, and the introduction of statistical

rnethod does not lead to a provisory solution, which

s

actu-ally said sometimes, but is on the contrary the only sien.

tif±c approach.

The terra Drobability is used under many different

cir--cumsta.ces an.d has many different meanings.

As

exarpie

probability is used in colloquial language when we want to

express a vague beliéf in something we cannot verify

This

is probably oie of the reasöns why the term often is

mis-understood when it appears with a quite different meaning in

scientific language.

Here we use probability defined in the following two

ways.

See Carnap /4/.

1

"Probability is the degree of cofiratio. of a hypo-.

thesis h with resDect to an eidence statement e egc

an observational reporte

This is a logical, semantical

concept.

A sentence about this cOncept is based, not on obser-..

vation of facts but on logical.

analysis.---(änalytic)"

2

"Probability is the relative frequency (in the long run)

of one property of eènts or things with respect to

another,

A sentence about this concept is factual

empiricaL"

(synthetic)

One usually speaks about the probability of an event

This makes it easy to misunderstand the öoncepts

Probabil-ity is not a property of events or things

ut of 3entonoe

about events or ±hiigs.

One sentence must be either false

or true.

A sentence can, however, be probable with respect

to other sentencies because one sentence can be true more

Often than another sentence in the long run,

If for instance you say "The probability to ge six in

this bast with this dice is 1/6", you can not verify or falsi-fy the statement by casting the dice. once. Therefore the

stateent is meaningless. If on the other hand you say "It

will be six" you can immediatelyprove if the statement is

false or true by casting the dice once and therefore the i.at-ter statement lias a meaning. Theh you say "The probability to get six Is 1/6" aid more or less unconsciously think of

some kind of Ititherentu probability you should think "The statement 'it i1 be si±' is true with the relative

fre-quency 1/6 in thé lOng rt.u"ò

If you accept a cëtain hypothesis aftr a statitical

test of say 5 percents significance level, _{this will mean}

nothing (inherent) but making a decision in a way that 95

percents of all similar decisions will be correct in. the long run if the hypothesis is true.

A statistical distribution function is a tool deter-mining the probability that a variable takes on values smal-1er than or equal to a certain given value.

The distribution function gives _{a clear and useful}

de-scription of events that already have taken place. _Prom

such an estimated (measured) distribution ftmction we can predict how a variable under unchanged _{conditions with a}

certain calculable. degree of confidence will behave in the

future. _{The distribution function thus gives us a rational}

expectation about the future behaviour of _{a variable.} The mentioned degree of confidence cannot of course

become solid )mowledge as analytical deduction _{is not possible.} We cannot be completely sure that the sun will rise to-morrow but it is rational to act .as ifit would do so. This is self-evident for most people. _{It is therefore remarkable that}

these. very persons often. _{are suspicious of statistics.}

Summary of reference /13/.

A large number of full scale measurements _{during the}

past thirty years have shòwii that the short term (about half

an hour) distribution of peak stresses can be welL described by a certain kind of distribution function Which is called

the Rayleigh distribution0 _{This distribution holds for the}

discrete peak values and not for the entire _continous

vari-ation of the amplitude. _{The latter variable is approximately} normally distributed, _{The Rayleigh distribution is of a}

simple mathematical form and it is fully _{charachterized by}

on _{single parameter which we denote r. This parameter is}

estimated from a record and _{can be interpreted as a measure}

of the severity of the load situa-tian. _{The parameter r is}

the root mean square of the _{peak values. Piire 1 illustrates}

th _{cÓrnection be-ween a record9 corresponding Rayleigh}

distri--bu-tian and the predicted future behaviour of the variable0 óertain Rayleigh d±strïbu-tion, correporidiìig _to

certain value of r, cives the distrbution _{of peak values} which you expect Lf yóu cöntinüé _{a rëóord under.tuadhanged}

eiterna _{conditions (wêather speed, öburse etöo) ò}

on the whole if you take _{a recOrd under eondit±ons whiöh cn be}

chatäcterjzed by ±he same value ö,f r _{Thëlàttercan oCcur}

in an infini-te numbor o± combina-±ons _{of óxtornal cönditions0}

The entire "load life" of a ship is imagined to consist

of a sum of a very large number of short term distributions

The lifetime al' _{a ship is thus divided into short periods}

where the loads in. each period is expected to follow a Ray-leigh distribution which is _{charaCterized by a certain value}

of r. _{Itis then iìediately obvious that if you lmow the}

long ter _{distribution of the r values9 you can predict an}

expected long term distribution of peak values by summing contributions from each short term _{distribution of peak}

alues. _{The total probability for the stress to exceed a}

certain stres level is obtained as a. sum of the

robabii-ties to exceed the _{saIne level within each short term} distri-bution. _{By repeating this procedure at}

some different stress

of peak values.

If we accept the above mentioned procedure of suation we can make the following statement of the probleme

To get a representative sample öf r values,

To estima-te long term distributions of r values. To transform _{a. long term distribution of r values to} a long term, distribution of peic values.

To estimate the confidence oÍ' the resulting long term distribution of peak values.

i Then we. make a nwnber of measurements on boárd a ship

e.g. during one vOyage across the North Atlantic we get a

sample of' r values. _{Such a sample is seriously affected by}

the fact that consecutive measurements are made under similar

conditions. This is due to the slow change in external

con-ditions. One voyage may contain measurements only in heavy

weather and head seas. Even if we ma1e rneasurements during the entire life of a ship we may get a sarnple that is not at

all representative för the trad _{in question or on the whole}

for the type of ship.

In order to get a r distribution which is more

repre-sentative than the one we happen to get in a limited time, we use a weighting procedure. - We choose one Or more

vari-ables (weather, wave direction etO.) that strongly affect

the wave loads and estimate an expected long term distri-buton of these variables by using Weather statistics, _We then put larger (smaller) weight to measurements made under conditions that have occurred more seldom (often) than _we

expect them to do. _{Note that there isno need. to Imow}

the loads are affected by the chosei variable. _{it is,}

how-evêr, desirable that the loads are affected by the chosen

Variable.

The data used in this report are Weighted with respect

to the sea state defined by-the Beaufort _{scale. In this way}

we elim.nate a great deal of uncertainty and it becomes

possible to compare different ships as if they. were sub-jected to the saine weather conditions.

Two earlier proposed types of long term distributions of r, viz, the normal one and the log-normal _{one are}

ana-lysed in the report /13/. The transformed variable log

(r + e) is found to be approximately normally distributed,'

This new distribution describes thê long term r

value.distri-bution much better than does the above mentioned ones, It

lies in between the normal one and the log-normal one, The

latter distributions are obtained _{as extreme cases when the}

constant c approaches eternity and zero. _{A comparison of}

the different r distributions is given in the next chapter.

Results from the above mentioned _{'summätion of short}

term distributions are presented in /13/. The three above

mentioned long term distributions of _{r are used0 The sum is} obtained by numerical integration _{of the corresponding}

func-tions in an electronic computer1' _{The result is presented in}

dimensionless graphs which directly _{give the resulting long} term distributions of peak values as function of the

para-meters of the long -term distributions _{of r, The graphs give} sufficient accuracy for practical use.

Analysis of the integration shows _{that the upper part} of the resulting long term _{distribution of peak values is}

almost solely dependent _{on extremely large r values which} are exceeded with very small probability. _{It is shown that} it is necessary -to carry _{on the integration up to five times}

the standard deviation of r above the mean of r, ±n order to

'ge± _{reliable inforrnation about the in-eresting largest loads}

that occur only a ±ew times _{during the lifetime of a ship0} You cn object that such _{a high level in the r value}

distribution is not realistic because, _{it corresponds to a}

larer number of r values than _{a ship is ever subjected to}

during the lifetime. _{This objection is, however,} insignifi-cant because even i _{you are interested in the commonly}

largest r values, you have to _{consider the risk of obtaining}

i) This work was made at Faci-t _{lectronics AB i-n othenburg.}

7. even larger values, The distribution of r values does of

course 'not stop at the expected largest value0 _{The largest}

value takes on different values in every sample and is in itself a statistical variable with a distribution of its own.

The largest value (second largest etc.) will thus be both larger and smaller than the expected largest value (second largest etc,) and the distribution above the expected largest

value is of great importance. _{It cannot be insisted 'upon}

too strongly that you must not let the most important part of the distïbution slip out of your hands when you carry out the integration0

4 _{A sample of r values cannot for natural reasons cover}

the entire life of a ship. _{An available long term}

distri-bution of r values will therefore be a random sample from

the total number of r values a ship is subjected to during the lifetime0 If you take a number of such samples from the

same ship and estimate a number of long term peak value

distributions with aid of the graphs mentioned _{above (3),}

you will get the same number of different results. _The

weighting procedure mentioned above (1) does not exclude this random variation but only reduces the number of measutements necessary to get a representative sampled

To be able to draw any conclusions from _{such a random} result it is necessary to know the magnitude _{of the expected}

deviation from the "truth" _{The "truth" here means the}

re-sult you would obtain in the 'ong _{run, on the same ship using}

the same method of measurement0 _{his kind of reasonable}

truth must not be mixed up with _{any kind of metaphysic} abso-lute truth that has nothing to do with _{natural science0}

The report /13/ presents methods to _{compute the}

devi-ations as confidence limits0 _{These are fuictions of the}

total number of measrements, the number f _{r values within} different groups9 e.g. weather _{groups and the values of the}

pa'ameters of the r distributions0 _{The confidence limits}

gives an interval on both sides of _{a computed value. This} interval contains the "truc" value _{With, a óertain}

pröba-bility. _{1Tote that the confidence limits contain the "tri.ie"}

value with a certain probability, and that the estimated

value is distfibuted around. the !ttfl.jeTf value, _{not vice vérsa..}

The confidence limits give a

easue

Of the reliability

öf a result, and conversely they give the _{number of} measure-ments necessary to get a certain desired _{accuracy. t is}

also possible to estimate how to distribute measurements among different groups (e.g. weather groups) ±n order _{to get}

optimum accuracy, and thus get _{as much informtion as possible} out of a ertain number of measurements.

Other methods0

The methods described in the report /13/ presupposes that you use the Rayleigh distribution to describe the short term distrIbutions, and then use estimated parameters for

.further calculations. One could ask why it is not better tO use strain cycle counters and thus directly obtain an 'Texact

distribution. Some aspects to this question is of öurrent

interest.

The factual wave loads do of ourse not change if you

change the recording method so the analysis will be based on the same random sample of wave loads whichever method

you may use..

Knowledge of the types of distribution functions, which by experience have well described a great many measurements

from many ships makes it possible to use parameters to de-scribe results from measurements. Such parameter values

give more exhaustive and accessible information about a

result, than does a detailed desciption of a random sample in an individual case0 _{This applies for instance to long} term distributions from strain cycle counters. In order to use such distributions you have -to fi-t hypothetical distri-bution functions to the data and -test the goodness of fit0

In other case there is no possibility to judge the relïa-bility of a result, and draw any conclusions of general

applicability.

Consequently, values from strain cycle counters can

give reliable results if they are handled according to the

same principles as used when evaluating our records with

the aid of r values. _{The d±fference being that in the former}

case we do not make use of the knowledge about the type of

the short terni distribution. _{This can be considered as an}

advantage only if we do not rely on thirty years experie.ee of short term distributions. _{There is, however, a need for}

control.

The estimated long term distributions that _{are obtained} from any method can be controlled by _{comparing the largest}

(second largest etc.) value recorded with the largest (second largest etc.) value predicted for the sarne period.

Conse-quently the same possibility of control applies to results from strain cycle counters as well _{as results from r values}

Short term distributions characterized by r values form at present the most simple and effective basis for a weigh-ting procedure. They are also the best basis for a detailed

analysis of the influence of different variables (weather,

wave direction, speed, loading conditions etc.) on the _wave

loads. _{Weighting of values from strain cycle coünters is}

practicable only if you use comparatively expensive and

complicated measuring equipment on board, e.g. tape punch or tape recorder, or if you read off the counters at short

intervals and punch the data on cards or tape for use in a

computer.

It has been said that it would be less expensive and

better to measure only the extreme values during rather long

periods with a simple device and use the data for

extreme-value analysis. That is certainly of considerable value,

but in that case you have to give up all interesting in-formation about other values than the extreme ones, and the

advantage of that method lies in the possibility _of

mea-suring _{inexpensively only the extreme values on many ships.} The external conditions _{vary cQnsiderably during these long} periods. _{This makes weighting ineffective and you have to}

carry on the measurements for a very long time in order to

get sufficient accuracy.

All different methods that _{can be used for the analysis}

of distributions of wave loads are, however, of great in-terest as they make possible comparisons between results

from different methods, especially now when we have rather little experience in this complicated field.

At present we cbisider r values to be the best basis

k practical application.

Pigure 2 shows a comparison between different _methods to analyse a total number of 1577 short term _{recörds of}

mid-ship bending stresses in 7 shïps. The distribution of r

values for each of these ships is shown in the _{report /13/.} Some relevant characteristics of the _{ships are given in table}

II.

The following three methods have been used for the

estimation of the long term peak stress distributions.

The sample of r values from each ship _{is divided into} groups defined by the weather at the period of

measure-ment. _{The Beaufort scale is used, the groups thus being} 0-3, 4-5, 6-7, 8-9 and 10-12, _{We assume that the r}

values are normally distributed within _{each group and.} compute the parameters of the normal _{r distributions} from the corresponding values of r. The parameters are

used to estimate the long term peak stress _distribution

within each group. _{The total long term peak stress}

distribution is obtained _{as a weighted sum of the} distri-butions within the groups.

This method was proposed in reference /1/.

The entire long -term distribution of r values from each ship is assumed -to be normal. _{The parameters of this}

distribution is obtained by plotting _{weighted r values}

on normal probability paper aaad fit the straight _line to the data. _{The fit to the upper part of the}

distri-bution should be considered most _{important. This is of} course not in order to pay special _{consideration to any}

random scatter of the largest _{r values, but depends on}

the fact that the data when plotted in this way will

get a curvature in its lower part. _{The lower values}

have very little influence _{on the result and should not} have any influence on _{a line which fits to the most}

im-portant upper part of the distribution. _{The parameters} obtained from the normal probability, paper are used to

estimate the long term peak value distribution from the graphs in report /13/.

The variable log (r + e) is assumed to be normal. See

further method 2. It is shown in the report /13/ that we can use e = 2 kg/mm2.

The comparison of methods refers to the point of peak stress distributions corresponding to the lifetime of a ship.,

i.e. the maximum stress we expeöt the ship to be subjected to

only once in its iifetime _{As we see in figure 2 methods 2} and 3 give similar results, the difference being that method

3 gives a little larger values. _{This depends on the fact that}

the distribution fanction for _{r that we use in method 3 gives} greater probability tò get large r values than does the normal r distribution in method 2. It is shown in the report /13/ that method 2 gives an underestimation and method _{3 possibly}

an _{overestimation o± the stress. It is therefore probable}

that these two methods give a lower and an upper limit för

the stresse

The r distribution proposed in method _{3 fits the data} much better than does the normal distribution _{of method 2 so}

the former method is to be preferred. Method 2, however,

sometimes gives a rather similar result fitting the data of measurements fairly weil so this simpler methòd _{can be used}

in some cases. _{These cases are determined by plotting the}

data on normal probability _paper.

Method 3 gives a result that in three _{cases diverges}

from the result of the other two methods. _{The shortest and} the two longest ships have got _{the largest divergencies.}

This is probably due to the fact that we have got very few

records in heavy weather from these ships. _{This is of great}

importance when we use method 1,but is _{relatively unimportant}

for the other two methods. _{Methods 1 and 3 give good} agree-ment for the remaining ships.

Tue confidence limits (see table I) explain to a. great

13. methods. Tie result from method 1 is, however, even more

dislocated because the two longest ships have got no records at all in Beaufôrt 10-12. _{This points out a serious weabiess} of methöd. 1, for the. result should not be affected by such a

random eent Method i _{also gives considerably widei}

confi-dence imits than the other methöds, _{especially in thos} cases where the upper Beaufort groups contain fe

measure-ments. _{Phe confidence limits also varies with the total}

num-ber of measurements but it is shown in the _{report /13/ that}

the confidence of method 1 solel de.ends on the u.'er Beau-fort groups. _{It is possible to increase the reliability of} method 1 by uziiîying at least 3 of the upper Beaufort groups, but then it is necessary to do _{an internal weighting in tuis}

larger group and thus wo approach _{methods 2 and 3.}

As a matter of fact it is self-evident _{that the most}

important upper Beaufort _{groups should not have least} reli-ability. _{Comparing method 1 with the other two methods we}

find the most striking difference being that method 1 handles a number of separate groups but the other _{two methods so to}

say connect the groups so that they cELn help each other to tell what the. distribution looks like in its most important but troublesome upper part. _{If one happens to get very} un-expected reóords or no _{ecords at all in heavy weather,}

methods 2 and 3 do not lead to any catastophe. _{This is,} however, the óase if _{we use method 1. It is easy to sâtisfy} oneself about this fact by adding a large r value in heavy weather and observe the jump _{of the predicted values. This}

is the fact shown by the _{colifidence limits.}

The free choice of weighting _{procedure is another} ad-vantage of _{ethods 2 and 3. Using these methods it is}

easy

to extend the weighting to _{more variables (e.g. weather,} wave direction, speed) because then the _{weighting is made}

first and does not make the integration more laborious. _Use of method 1, however, makes _{extended weighting very laborious,} or at least expensive in a computer, _{because each group is}

integrated separately and the number _{of groups incréases}

It is also possible to make _{a l?ss laborious weighting}

accor-ding to method 1 _{by first weighting wit}-iin groups and. then}

flake the final weighting afte' -the integration of eabh groupc This method is, however, less efficient.

Method i was the best one as long as we did no-t have any good. approximation for the entire long term r

distribu-tiön, but it does no longer give the bèst reliability when

we ow such a long term r distribution. However, method. i

is still of great interest because it visualizes the

influ-ence of the weather.

-Reliability and. simplicity is of vital importance in

choice o-f method. There is no doubt thatmethod 2 änd

The type of the long terni peak value distribution.

Figure 3 shows a graph that directly gives the para-meters of a long term peak value distribution as functions

of the parameters of a long term normal r distribution. The

long term peak value distribution is described by the Weibull

distribution (see Appendix III and figure 2). This

distri-bution fits well to the result from the integration that is

presented in the report /13/.

It is interesting to note that the parameter k, that

solely determines the curvature of the WeibUll distribution,

directly can be estimated from the parameters of a normal r

distribution. This is interesting because the value of k has a great influence on the risk of fatigue failure as

com-pared to the risk of failure at one single large load. n-creasing k values increases the risk of fatigue failure be-cause then the maguitude of loads occurring òften increase as compared to loads occurring seldom, and conversely de-creasing k values do increase the risk of failure at one

single large load as compared to the risk of fatigue failure.

Figure 2 shows k values for our ships as a function of

ship length. The k values were estimated. from figure 3 (method 2) and Appendix III (method 3). The figure shows

that k has a minimum at a ship length of about 170 meters. The bloók coefficient seems to hava little influence on k be-cause ships No. 2 and 3 have quite different bloòk coeffi-cients but similar k values and about the same length. If

k varies in this manner with the length of ship then hips

of the given length are. subjected to one single comparatively

large load with larger probability than longer aid shorter

ships.

Extended measurements can give a better estimate of k

as a function of variables of current interest. This is of

great importance as k undoubtedly is a very suitable

para-meter in the analysis of wave loads. It should be noted that a large k value does not necessarily give _{more fatigue} failures because those ships may well be of a typethat usually is better constructed in this respect.

It is not surprising that k increases with a large length of ship because of the smaller probability for a ve long ship to meet waves f a Oharacter Ìiecessary to cause a load. öorresponding to the greatest load, experienced by a

shorter ship j This depends on the fact that the wave steep-nss decreases as the wave length increases. Very long waves

are also less freq,uent than sOmewhat shorter ones. See Roll /14/. It should. alCo be noted that k does not tell aiything

about the absqlute values of wave loads but oily characterizes the relation between latge and. smil wave loads on the ae

ship.

Wave statistics and model experiments in 'aes can give many interesting Contributions to the solution of these

17.

The importance of accuracy of the method used to estimate r

values.

The uncertainty of the methOd of measurement has two

main influences on the estimated long term peak stress

distri-bution.

Larger standard error of measurement leads to wider

confidence limits around the predicted stress and it also

leads to a larger overestimation of the stress.

We begin to investigate how the uncertainty of the

method of measurement influences the long term r distribution.

We later use the uncertainty of the r distribution to estimate

the uncertainty of the peak stress distribution.

The methods used to estimate r introduces a standard

error of the method of measurement that is proportionì to

the r value.

_{This gives larger absolute errors to the larger}

r values than to the smaller ones.

Therefore an estimated

distribution of r v1ues always has larger skewness i.e.

longer "right tail" than te distribution we intend to

esti-mate.

If for example r is exactly normally distributed, an

estimated distrilution of r values would not be iiormal, but

the largest estimated r values would be larger than the

corresponding normal. values.

Larger uncertainty gives larger skewness.

If the

stan-dard deviation of the error of measurement is great as

com-pared to the standard deviation of r, then the upper part of

the estimated r distribution becomes lognormal independent

of the type of the "true" distribulion of r.

This depends on

the fact that the staildard error of measurement is

a linear

function of the r value.

See Appendix I.

This is one

expla-nation to the fact that the lognormal distribution has been

successfully fitted to r value data, in spite of the fact

that a lognormal r distribution leads to a preposterous peak

stress distribution (see /13/).

It also points out what

erroneous results it is possible to get if one does not take

great care of the error of measurement.

Consequently, uncertainty of the method of measurement

leads to the use of r distribution functions with too large

skewness, and thus to overestimation of the risk of getting

large r values. -- .

Overestimation of the standard deviation of r is an

other conequenbe. of the introduced error of measurement.

This depends on the fact that the estimated standard

devi-ation of r is composed of two components.. One compoflent

being the spread of r values depending ori varying external

conditions (weather etc.), and the other component being the inescapable error of measurement. The ferner component can only be estimated from data, but the latter component can be estimated analytically and. is dependent on the method

ofmea-surement. _{It is therefore possible to estimate th influence}

of the error of measurement on the estimated standard

devi-ation of r. Figure 4 shows ti4s influence..

The. influence of the error of measurement increases as

the r value increases. Thereföre a grouped sample of r values

wi];l get larger estmated standard deviation of the groups

with larger r values, even if the t?teIt variable would have the same standard deviation of all _{groups independent of the}

size of the r values. . This is connected With the above

men-tioned skewness. See Appendix I.

In this cOnnection we remind of the properties of the

distribution functions mentioned on page 11. The normal distribution is symmetrical. The normally distr-ibuted

vari-able log (r + c) gives little skewness of the r istribution when e is large. _{The skewness increases when c dçcreases.}

The skewness is largest in the log-normal distribution when

c equals zero. _{The skewness is,however, not solely} depen-dent on C, but is also dependent on the other parameters of

the distribution..

Persons unaccustomed to statistical .thinkin, too _often feel strongly attempted to accept the disastrous

rnisuder-standing that a little systematic _{error can be jt.ggled away} in a magic maimer by making a lat'ge number of measurements. This is of course wrong. _{In our case it is not possible to}

measure-19.

ment on the r distribution by increasing the number of

mea-surements. An increased number of mèasurements can reduce the scatter of the predictions but this is of little or no interest if the points are scattering around the wrong value.

It is necessary to do a very complicated analysis o the problem if we want to lmow the exact magnitude of the

discussed deviations from the distributions we intend to

esti-mate. We will here only.point out some tendencies and do a simple estimate of the magnitude of the deviation caused by

a certain error of measurement. We know from the report /13/ that a peak value distribution estimated from a normal r

distribution is an approximately linear function of the stan-dard deviation of the r distribution, and that the width of the confidence interval is proportional to this standard

devi-ation. Let us assumO that the standard deviation of r is

small as compared to the mean of r. This is the case in a Beaufort grouped sample We can then disregard the skewness

within a group and directly read off in figure 4 the influeflce

of the error Of measurement on an estimated stress value and

the corresponding confidence interval. If for instance you set the upper limit of the allowed systematic deviation to 5 % you can read off in. figure 4 the corresponding desired accuracy f the measurements. It is also possible to use figure 4 to correct far the error of measurement.

It is evident from figure 4 that the error of measure-ment has great influence on groups with large r values but

has little influence on groups with small r values. If you unite all groups to one single long term r

distribution you will find that this will get a considerably larger standard deviation and therefore becomes less _sensitive

to the error of measurement, especially as compared to those

groups containing the largest r values. _{From this point of}

view one single r distrihütion is to be preferred, but the skewness can become important when we use one single r

distri-bution and the skewness also introduces an overestimation. The influence of the error of measurement _{on the confidence} limits is, however, always favourable for the single

diCtri-but ion.

Prom the above it is evident that it is important to handle evex sIngle measurement in the same careful maimet in order not.to introduce any unimowi ui.certaintiésinthe

caidulatiöns. It is also evident that one single long term r distribution is less sensitive to the error of measurement

"The largest possible value'1.

Many authors have discussed a concept that is denominated. in a.y different ways, and which we cali "the largest posihle

value". It has been said that a physical upper limit'exists

and that statistical distribution funct±ons w! thout upper lim:Lt for this reason cannot be valid above this magic point. It ha

for instance been proposed that we should determine an "upper limit for wave bending thoments" and that we should use distri-bution functions which stop at this limit0 Trials have also

been made to compute "the largest possible wave load' in Waves of unrealistic steepness in order to fix an uupper limit" and concepts as "largest r value" and "most severe sea state" have

been discussed.

This is only a sample of a great number of snailar

íoru-lations. It is evident that these terms sometimes really are interpreted literally and that an absOlutely largest value is

considered to exist. We start with a discussion of this intor-pretation,

The basis for reasoning about "largest possible values" is probably the fact that a sea wave for instance cannot be

higher than the diameter of our planets PrOm this one has drawn the conclusion that there must be a certain limit to

natures own achievements, It seems like "the largest possible

value" is treated as a physical constant.

Probably nobody would believe that "the largest possible value" is common, but on the contrary very uncommon0 "The

largest possible value" is also usually said to be much larger than common values. _{By so to speak pushing the problem} suffi-ciently for away in this manner it is easy to maintain coníused.

conceptions because one would not usually experience any value

that is larger than "the largest possible value",

As the term "the largest possible value" in many cases

is based on misinterpretations of basic concepts there seems

to be a need for an analysis of these concepts _{even. ifas a} matter of fact it is self-evident that any absolutely largest

value doe not exist.

To be of any interest "the largest possible value" will

have to comply with at least the following two demands.

The hypothesis "the largest possible value exists" must

have a meaning.

A hypothesis has a meaning scientifically, if any one logical consequence of the hypothesis leads tö sentencies

that can be.empirically verified.

"The largest possible value" must be applicable.

This demand is in fact covered by demand No. i. Here we

only mean that the concept should be useful for

construc-tion purposes.

"The largest possible value" does not comply with any

of these demands and. is therefore of no interest. Let us

ex-plain this statement. We refer to the introduction of this

report.

If "the largest possible value" exists it is a factual

empirical thing. The magnitude of this valu e cannot be deter-mined analytically. Thus any sentence about the magnitude of

this value will have to be empirically verified. Then we have

to use inductive reasoning that cannot give full certainty. It is therefore evident in this case that it does not make

sense to define a value that can be reaóhed but (with probabil-ity i) cannot be exceeded. _{However large a value observed}

there is always a probability (larger than zero) to exceed

that value. _{It is therefore impossible to verify a statethent} about the magnitude of "the largest possible value.".

Consequently demand No. i cannot be complied with.

Assume for a moment that "the largest possible value"

does exist and that we biow its magnitude. What use could we

make of this lmowledge?

It would certainly not be wise to determine the dimen-sions of the ship structure to sustain "the largst possible

23. load", vthieh seems to be very unlikely to occur, if at the

same time we did not know its probability of occurrence or,

correctly seaking, did not Imow the probability to exceed a

value, close below "the largest possible value". It is obvIous that this probability is not. constant under all circumstances,

so the interesting thing would not be the magnitude of "the largest possible value" but the probability to exceed values

below "the largest possible value".

Consequently demand. No. 2 cannot be complied with.

We can thus state that "the largest possible value" is of no sense and of no interest if it is not defined as a value

that is exceeded with a certain small probability.

Pormula-tions similar to "the largest possible value"

e.g. "most severe sea state" havé not always meant values defined in our manner, but seem _tO result from the difficulty

to accept.. statistical thinking.

The whole problem of strength of structures is basïcally h statistical problem.

This has to be accepted.

"Thé largest possible value" interpreted as absolutely largest is a metaphysical concept _{- a late remembrance of the}

"Ideas" of Plato - and has nothing to do in natural science.

Summing up we recapitulate the following points of the

reasoning.

Wave loads are facts.

Sentencies about facts have no meaning if they cannot be

empirically verified.

Predictions regarding the magnitude 'of wave loads cannot be solid knowledge.

Wave loads are not interesting unless they are associnted

with probabilities.

Let us finally accept that any absolutely larget load

does not exist. Then below we speak about the larges-t value we mean a value that is exceeded with a certain small proba.-bility. This is obviously the only interpretation of the ter

tithe largest possible value" that does make sense.

Our discussion has showed that it makes no sense to look

for any upper limit for wave loads. We now proceed to discuss

the following question. Is it possible to use "the largest r value" or "the most severe sea state" that is exceeded with a

small probability for any conclusions about dimensioning?

It is not possible to draw any such conclusions. This will be evident in the discuss-ion below. As each sea state

corresponds to a certain r value it is sufficient to. discuss r.

The conclusions in question are usually drawn by using

a "'epresentative large r value". The stress (load) is com-puted from the corresponding Rayleigh distribution One

usual-ly states that the probability to exceed the computed stress equals the product of the probability to exceed r in the r

distribution and the probability to exceed the stress in

question in the Rayleigh distributiOn. This conclusion is completely wrong and very misleading. It is therefore easy to understand that it leads to the senséless consequencies

that have been pointed out iii. several cases. The fact that a ship has to be subjected to a "representative large r value" during an absurd length of time to reach a stress value

predic-ted by the methods handled in the report /13/ does not tell

anything about the reliability of those methods.

The total probability to exceed a certain stress is a sum of probabilities from each of all r values a ship is

sub-jected to during the lifetime. If you choose one "represen-tative r value" (sea state) you do not consider the larger or smaller probability to exceed a certain stress level at every

other r value. TIiis is the reason why a ship cs to live

through one r value during a very long time before the largest

stress computed from the "reDresentative large r value" eauals

the correctly estimated largest stresse, Even if you choose a very large r value the time necessary may well be longer than

the ordinary lifetime 0±' a ship, because the most important part of the r distribution lies at a very high level. Canse-quently even a very large r value can give a sc !Laafl

contribu-tion to the total probability of exceeding the expected lar

gest stress value, that the ship can live through th:Ls r value

for a practically unlimited time without considerable risk of

exceeding the expected largest stress value0 See Appendix IL

See also pages ami 7 in ai reporte

Finally we can state that it is not interesting to look

for "the largest r value", "the most severe sea state", "the

most severe wave- or response-spectrum" or any similar concept even if it is defined as a large quantity which is exceeded

with little probability0 _{The only way to u r vaJ.ues (or}

similar) is to use the entire distribution

of r

_{values that} we expect during the lifetime of a

ship0

One may

ask wh

we do not simply investigate an example

where wo calculate a "true'1 long term peak stress distrihation

by integration of

a r

dllstribution We could then choose a

r

value"

in such a

_manner

that the stress

com-puted from the corresponding

Rayleigh distibution

_{equals the}

"true" stress0 This can 'e obt-ai:aecl. by ohoàsing suitable

values for the probability to exceed "the _{representative r.} value" and the probability to exceed the stress _{in question}

in the Rayleigh distribution, _{We then get two fictive}

proba-bilities that in certain case give ihe amo result as a

cor-rect öalculationc,

This "practical method io howeve:c, not useful because the fictive prObabilities vary from one case another0

Short and long term distributions _{can not be described by the} same function with the exception of -the completely uiirealistic case when r is constant throughout the 11f etime _Therefore the proposed "practical"

method will

_{give a comp1etelr wrong}

slope to the long te stress distributioi which obviously is not Rarleigh distributed. Moreover the siöpe of the r distrI-bution varies from one case to another Por this reason two

r distributions can give completely differing

bug

term stress

distributions even if the r distributions happen to coincide

in one "representative" point. Consequently "the most severe

sea state" is not interesting because it means quite different

things on different ships r

We are thus again baôk to the. demand of using the entire long term distribution of r values, sea states, spectra etc. tö get the possibility of using these concepts in. the solution of the wave load problem.

Moment factors.

The variable we primarily measure on board the ships is

the extension in. fore and aft direction between two points in deck ör side close, to the stringer corner nrnidships. From

this extension we simply compute the stress with Hookes law.

Athwartships stresses that may occur are not taken into

con-sideration. Estimated r values are stresses in kg/mm2.

The stresses are computed as half the sum of the hogging

and sagging stresses. These parts are later determined as

percentage parts of the sum. This method is chosen because it

gives better statistical accuracy of the estimate of both the sum and the parts as compared to estimates of each part

sepa-rately. Here we first discuss the sum.

In order to compare data from different ships and be able to draw general conclusions we have to use nondimensional

mornent factors. Different types of moment factors wilJ. be discussed in this chapter. The problem of estimating wave

loads is divided into an "internal" and an "external" part.

The "internal" problem is constituted by-the fact that the section modulus of the transverse section amidships _varies

considerably between different ships even if they have similar principal dimensions. _{Therefore we compute t midship bending} moment from -the stress recorded and the cornputed section modu-lus of the measuring point. It has been

said,

by other authors

that a certain inaccuracy is introduced in the _{estimate of the}

section modulus because the elements of a transverse section do not have the same effectiveness in different _{types of}

trans-verse sections. This inaócuracy is, however, certainly much smaller than the error introduced by neglecting _{the difference} between ships and determine the stress _{as function of the}

principal dimensions as some authors _{propose. It has been} shom experimentally that the difference between the "tnie"

and the estimated section modulus is much smaller than.. the

di±'ferencjes between section moduli of different ships. A

misestimate of -the section modulus will be of little importance

if the section modulus is computed in the _{same manner at}

surement and at design. This holds, however, only for ships of traditional design.

When choosing the required section modulus we meet

another side of the "internal" problem. This depends on the

different ability of different designs to withstand a certain

load without failures. This problem will not be discussçd in

detail here.

The "external" problem is to pay proper consideration,

to varying external dimensions (including the distribution of weight in fore and aft direction) as well as other external

conditions (weather etc.) -that varies from one case to another

and have influence on the wave loads This problem wille

discussed here0

The problem of determining the requiréd section modulus

has been discussed for a considerable length of -time and many

ways of approach have been proposed However, it seems im-possible to find any general methods because in each way of approach we haxe to consider the very special pro-requisite

conditions and take into consideration which way we want to use a result. It is evident that if e try to determine the required section modulus from the result of full scale tests,

we cannot use the same methods as used when determining the

section modulus 'from model tests in waves, taking standard

series of model tests in waves into consideration (analogous

to ordinary tank tests).

Therefore, it seems better to accept the d.ifferencies in pro-requisite conditions and define those quanti'bies best

suited to each way of approach. There is, however, one wa to compare results from different ways of approach by

com-paring long term distributioiis of wave loads.

Here we define a moment factor as a function of ship parameters that has constant value at the same probability level in long term distributions from full scale measurements

on different ships. It is evident that a function of this

kind, gives the best description of the connection between

- 'computations or model tests - have to be compared with re-sults from full scale measurements.

The entire pròblem is thus divided into two comptely

separate parts. On one hand the "internal" ability of the

ship to Withstand. certain loads, on the other hand the in-fluence of principal dimensions and. other "external" quanti-ties on the wave loads. _{This division is the basis of the} rules of the classification societies where the problems, however, have been coupled together and, the permitted stress

is not only dependent on the "internal" ability of the ship to sustain a certain load but is also dependent _{on variations}

in "external" conditions. This is due to the fäct that earlier

it was ot possible to observe anything but the total result

i.e. the failures, and there were. no possibilities to separa'e the causes. _{The mentioned. division is necessary for an} ana-lysis of the causes of failures. _{Wave load statistics is} necessary for the division.

It is also necessary in this connection to _{d,vide the}

total wave bending moment into _{one still water part and one}

wave part as we here only discuss the part that varies in

waves. _{Here below wave bending moment only means the varying}

part of the moment.

In order to compare our results from full _scale measure-ments on different ships we.first form a moment factor m0 in

traditional manner (Murray). _{Then we discuss different types} of moment factors and _{we determine a moment factor that}

ac-cording to our above mentioned definition _{gives the best fit}

to the data.

Nowadays it is commonly accepted that the magnitude of the wave bending moment is better described by ship length and breadth than by length and _{displacement formerly used.} This depends on the fact that the draught _{has little influence}

on the wave bending moment which _{primarily is affected by the} hull form in the vicinity of the water linee _{See Swaan /16/} and Vossers /17/. _{We use the largest stress X8 we expect a}

M8 = S

M8

y B L

In the first trial to determine a moment factor fitting the

data we also use section modulus at measi.ring point S, speci

fic weight of sea water , mouldêd breadth B and length bet-ween perpendiculars L. For the present we put tIe fictivel) wave height H.proportional to and later we discuss H as a functiOn of L. Particulars of the ships are given in table

II. In the following we use meter and toi = 1000kg.

Figure 5 shows m as a function of L. The same figure also shows the water plane area coefficient and the pris-matic coefficient OP as functions of, L. The curves describing

m0, O and. O shows a striking resemblance and the fullness of the ship is no doubt one variable that is strongly connec-ted to the difference between the two groups of ships separa-ted in the figure.

The moment factor m1 was proposed in reference /1/.

M na

-1

yBL3C

Reference was then given to Swaan /16/. 11e shows that

a statical calculation (without Smith's efeet) for ships

with vertical sides gives approximately. a linear relation

between M and _{Det Norske Ventas applies a dependence} of the fullness that is similar to the dependence in _{m1 (the} basis being a statically calculated sagging moment giving a

coiservativè prediction for fine ships). See equatiOn (2). ccórding to Swaan, however, M is not prOportional to but the relation is of the type

M = const. - a) (i)

where a 0.4 when 08 < _{< 0.9 arid when the wave length}

Consequently, reference to Swaan /16/ gives the following

moment factor.

31.

is similar to the ship length. Waves of this length give the largest wave loads, It is therefore explicable that m1 (see figu.re .5) can not describe the influence of the fullness on the recorded wave loads.

II' the statically calculated moment varies in a certain

way with certain parameters this is of course no reason to

believe that the dynamical moment has to vary in the same way. The hydrody-namic moment is different from the hydrostatic one

and aòceleration forces from the weight of ship and surrounding

water are added in the dynathical case. The relation, if any,

betweènCtatical and dynamical moments has to be empirically

verified. Statical moments are, however, interesting because they can lead us to try reasonable functions. Model

experi-ments (see Vossers /17/) show that the influence of fullness on wave bendìng moment in some cases are similar to relations

statically calculateda However, one has to be very cautious

about conclusions from statical calculations.

Prom reference /6/ we get (see Appendix IV) the following

relation between C _andC,.

0WL = 0.68 (C + 048) (2)

Vossers /17/ gives the formula

3 0 = 2 + i

which also can be written

0.67 (C + 0.50)

The latter formula is very similar to (2). This gives

an indication that the data of reference /6/, which we will

use below, are representative for a large number of ships. We put a = 0.4 and get from (i) and (2)

M8

yBL C

This moment factor has approximately the same.dependence

of fullness as the formula used by Lloyd's Register0 They assume the wave bending moment to be proportional to aBel)

In this connection the fore and aft distribution of

weight and buoyancy is of particular interest and C is

there-fore more adequate than CBO The latter is affected by irrele--vant factors as bilge radious and rise o the bottom0 If we

hold desplacement, length9 breadth and draught constant and

increase the bilge radious we increase the wave bending moment0

At the same time C increases but C remains constant6 We therefore propose C to be used in this connectioli.

It is evident froh figure 5 that m2 is better than m but that m2 still does not.describe the influence of the

full-ness in a way that fits our data. The wave bending moment recorded. thus shows a stron dependence of the fulJ,.ness than

is given by m1 and m2.

Many other investigations support this conclusion0 Johnson /9/ presents results from the very extensive

full scale measurements carried out by the British Ship

Research Association (B S R A). Pigure 15. in /9/ gives

ap-proximately the following relation which refers to the rnoent

a ship is expected to meet 1 time in 100 days This is

ap-proximately equivalent to the probability level io_6 M = const. (CB - 0.5)

It should be noted that the influence of _CB this

great in spite of the fact that the ine ships as an average have met more heavy weather aiid have spent longer time on the

North Atlantic than have the full ones

i) C denotes block coefficient.

Bermet /2/ has analysed B S R A's measurements and he

gives effective wave height in Beaufort 8-9 asa function of'

CB. Figure 5 in /2/ gives approximately the following rda.

tion.

M = const. (CB - 0.16)

It should be noted that this holds only for Beaufort

8-9.

Zubaly /18/ presents results from calculations based on

model exeriments and wave statistics. The moment factor

is given for two ships (L = 137 in; _{0B = 0.60, C =}

_{071 and}

0B = 0.80, C = 0.87) as a function of wind, speed W in knots The following relations are derived from /18/.

M = const. _{0WL (CB -} 0.11)

M = const. C, (C - 0.21)

By inserting equation (2) we get

M = const. (CB - 0.33) IV = const. (CD - 0.38) W 60 W _{= 35} W = 60 W _{= 35}

Göransson änd Olofeson /6/ present statically

calcu-lated _{wave bending moments of 30 ships having 0.63}

Op

0.82. They g±ve the moment factors

h (hogging) and (sagging). M M s 2 9 1k = 2

BL H0

_yBL

H

H0 is the wave height. H0 _{L/20 is used in /6/.}

Statistical analysis shows (see Appendix IV) that the

three moment factors

1h' and h + = 1kh + are very

strongly correlated to C ,O _{and OBI No one of the latter}

parameters is significantly superior to _{the others. Prom}

Appendix IV we get

+ = conet. (c - 0.13).

Neither 0B' a nor O can be expected to give a complete

description of the influence of the fullness on the statically calculated moment. One reason being that the wave bending moment varies with

0E

(c) when

0WL (cr) is constant. This

is due to the fact that the wave bending moment _{is dependent} on the slope of the cross sections in the vicinity of the

water linea _{Vertical sides give the smallest momeat (u shape).}

Increasing angles to the vertical plane _{increases the moment}

(y shape) _{The degree of u shape can for a bonstant value of}

or C be described by C.

_{C/C. This is}

_{a kind of}

vertical prismatic doefficient. _{Statical calculations for 4} fine ships (C _{= 0.686, reference /6/) and 3 full ships (C =}

0804, Ivarseon at the Swedish Shipbuilding Research Fornida-tion) show that the moment is approximately a linear decreasing function (not proportional) of _{CV when the shape of the} sec-tians vary from y shape to u shapes The moment of the fine ship is more sensitive to changes of than is the moment of the full ship. _{This depends on the fact that a smaller part} of the full ship is affected by a change of shapes The stati-cally calculated moment of the fine _{ship is about 15 larger}

at y shape (Cv. _{= 0.855) than at u shape (C = 0.928).}

According to Swaan /16/ C _{is the parameter best suited} to describe the influence of the fullness on the wave bending

moment. He considers the influence of _{the shape of the}

sec-tions to be negligible as compared to the great degree of

ap-proximation introduced by the statical _{approach. However,}

Appendix IV, based on reference /6/, _{shows that in statical}

calculations for realistic hull forms C _{is not superior to}

C.

C and C are slightly superior to _{0B which, for above} mentioned reasons can be expected. _{Concerning the shape of}

the sections it is pointed _{out above that y shape gives}

con-siderably higher moments than gives u shape and one can no

doubt assume that differencies in statically calculated moments correspond to differencies in dynamical moments although the absolute values of neither the _{moments nor the differencjes}

35.

Thereforé it seems reasonable to expect the shape o± _the sec-tions to become of increased importance when _{our mowledge}

about wave loads increases.

The longitudinal moment of inertia is another variable having influence on the wave loads, and _{even if we hold the}

lQflgitudinal moment of inertia òonstant changes in the

lOngi-tudinal weight distribution will change the wave bonding mo-ments. _{Unfortunately wedo not have the longitudinal weight}

distribution at each period of measurement easily _accessible,

but _{does to a great extent give a measure of the average}

longitudinal weight distribution. _{The hydrodynaimical part of} the moment is closely related to Ci,.

For simplicity here we consider to give a usefil

de-scription of the longitudinal distribution _{of weight and}

buoy-ancy. C _{is at any rate one single parameter giving a good}

description of the influence of these _{quantities. The next}

step will be to introduce additional parameters describing the influenóe of the shape of the sections _{and the longitudinal} weight distribution.

The data presented in referencies /2, 6, _{9, 18/,} refer-red to above, gives reason to expect the following moment fac-tors m3 and in4 to fit our data _{better than do m1 and}

M3 _M8

BL3(C-O.2)

in432

const. (c - 0.35)

It is evident from figure 6 that _{Is better than in1}

and m2, but that m3 still is _{not satisfactory, On the}

con-trary m4 gives a good. fit to our data. Note that in4 closely resembles the mentioned, result from Ïnodel _{tests presented by} Zubaly /18/. _{In the future we will possibly get a function} of the type (c - a)2 + b. _{This is indicated by model tests}

and by statical calculations.

We now try to fit a function to the results from methods 2 and 3 (see page ii) that is _{obtained by using m4 (figure 6),}

In4 = a e

The result from method 1 seems to be. doibtful because the

right

part of the curve obviously has got to large steepness0

The moment factor m is proportional to H/LI Here it is not necessary to make any distinction between H and He

be-cause these two quantities are related to L in the same way0 It is well Imown from statical calculations (Swaan /16/) and. from model tests (Dalzell

/5/,

Ivarsson /8/, Vosers /17/)

that waves of' about the same length as the ship give th lar-gest moments, and

that

the moment

is

approximately proportional

to the wave height. Consequently we expect the mOment factor

(at _{a certain probability level) as a function of ship length}

to be proportional to the wave steepness (at a certain

proba-bili-ty level) as a function of wave length. The wave steepness

has a finite maximum for small wave lengths, it decreases as

the wave length increases and it approaches zero when the wave length approaches eternity (see Roll /14/). This sim le

rela-tion between wave steepness and moment factor is no doubt fundamental for the understanding of the wave load

problem.

Many model tests (e.g. Zubaly /18/) supports

this

statement.

We therefore try the function

which has a finite maximum for L = O aid approaches zero when L approaches eternity. _{Figure 6 shOws that the following}

fuñe-tions give a good fit to the results of _{methods 2 and 3}

L 2m4 = 0.00270 e 250

L 100 L 240

3 = 0.00275 e

270

The largest moment we expect a ship to meet during the

lifetime

is

thus probably contained

_within

the limits (see

page 12) given by the following formulas.

= const. L e

100 L 240

270

It

is interestin.g to compare the following three ways of describing fictive wave height.

L.

i. H1 = const. L 260

₍₃₎

const. L°'5 Lloyd's Register

113 const. L°'3 Dot Norske Ventas

Figure 7 shows

a qualitative comparison between the three functions. The constants are given values so that the functions coincide at the ship length 200 meters. It is

evident from

figure

7 that the function H. is of a quite different type than 112 and H3 The function

Hi reaches a

maximum, in this case at 260 meters,

and

it decreases for larger L.

Puidàmentally the same type of function has been ob-tained from model tests (Dalell /5/, Lewis /10/9 Ivarssoi

at the Swedish Shipbuilding Research Foundatiön).

Murray /.11/ has presented statistics of failures from a very large number of ships. It shows that

within the region

(medium size ships) where H1 is relatively larger than 112 a

37.

L

0.00270

y B L3 e - 250 100 L 240 (4) 3 =

0.00275

y B L3 e 270 C 0.65 _0p 0.85

Prom these

formulas

we get fictive wave height as a

function of

ship

length.

L