Stress and motion measurements on ships at sea. Part 1: Theoretical basis for statistical analysis Part 2: Results of statistical analysis. Part 3: Test results from ms Canada and ms Minnesota

Report No. 13

STIFTELSEN FOR

ij

STRESS AND MOTION MEASUREMENTS

ON SHIPS AT SEA

(Project S-1)

/ I. Theoretical basis for statistical analysis

II. Results of statistical analysis

by

Rutger Bennet

Measurements at sea carried out by:

Ingenjör Sten Brämberg, AB Götaverken, Göteborg°

Mr. John T. Birmingham, David Taylor Model Basin, Washington D.C.° Civilingenjör Tord Byquist, Tekniska Högskolan, Stockholm

Civilingenjör Ove Heliman, AB Lindholmens Vary, Göteborg Civilingenjör Erik So'rensen, Kockums Mek. Verkstad, Malmö

Civilingenjör Rutger Bennet, Stiftelsen för Skeppsbyggn.-tekn. Forskning

* installation of instruments f

Statistical investigations carried out by:

Professor Ulf Grenander, Stockholms Högskola Fil. lic. Uno Zackrisson, Göteborgs Universitet Amanuens Esbjörn Cariström, Göteborgs Universitet

Energy spectra and amplitude distributions have been worked out on the electronic computer Aiwac III E at the ADB Institute, Göteborg. The pro-gramme for the spectrum calculations was prepared by Mr. Sture Laryd.

Measurement data have been transferred to punched cards at SAAB, Linköping. Figures and diagrams have been drawn by ingenjör Sven Niisson,

AB Götaverken.

The report has been translated from Swedish by Mr. Eric Elliot, Göteborg.

The Stress Measurement Committe

of the Swedish Shipbuilding Research Foundation: Professor Anders Svennerud, Chairman

Civilingenjör Bengt Bengtsson Civilingenjör Rutger Bennet Ingenjör Sten Brämberg Professor Curt Falkemo Tekn. lic. Erik Hultmark Tekn. lic. Gunnar Nilsson Professor Erik Steneroth

FOREWORD

Developments in shipbuilding and seafaring during recent years have been towards even larger and faster vessels. Swedish shipyards have made their contribution in this respect by designing and building types of ships that are advanced in many

respects. In this connection there has been an increased demand for information

based on a more thorough knowledge concerning the behaviour and the occurrence of stresses in ships at sea. As early as 1954 a committee within the Ingenjörsveten-skapsakademien took the initiative in making certain stress measurements in a fast cargo vessel. After the establishment of the Swedish Shipbuilding Research Founda-tion in 1955, research work was taken over by this body.

With the passage of time it has become clearer than ever that the only possibility of obtaining, from the extensive measurements available, some idea of the relationship between the bending moments and other factors such as course, speed, wave

condi-tions, displacement, cargo distribution, etc., is by means of statistical analysis

with the application of the statistical wave theories. Since these theories are completely new and have, as yet, not been fully verified, it has been considered justifiable to review them in Part I of this report. Part H which follows, deals with the statistical results of the analyses. The aim of the work here has been, through various ways, to clarify the results of the statistical methods. Only a limited part of the measurement material has been analysed here. The intention is to deal with the remaining measurement values using the experiences gained here, and present the results of this work in a later supplementary report.

This research work has been financed by means of grants from:

Statens Tekniska Forskningsräd (The Swedish Technical Research Council)

Hugo Hammars fond för sjöfartsteknisk forskning (Hugo Hammars Foundation for Maritime Research) and

Stiftelsen för Skeppsbyggnadsteknisk Forskning (The Swedish Shipbuilding Research

Foundation).

Valuable co-operation has been established with the David Taylor Model Basin,

Washington D. C., U. S. A., from where a set of instruments has been made available and from where significant impulses concerning the actual measuring procedures

have originated.

The Johnson Line Shipping Company and the Transatlantic Shipping Company have

been kind enough to collaborate with us by allowing the measurements to be carried

out on board M/S "CANADA" and M/S "MINNESOTA" respectively, the officers and

crews in both cases showing great interest and being of valuable help.

Many people from shipyards and various other institutions have taken part in the measuring work and have contributed with various investigations. The measurements have also been discussed by the SSF Stress Measurement Committee.

1. THEORETICAL BASIS FOR STATISTICAL ANALYSIS

INTRODUCTION

For some considerable period of time it has been general practice for naval architects to define the surface of sea waves by using the trochoid, i. e. the curve described by a point on the radius of a circle rolling on the underside of a horizontal line. A wave obtained in this way has been considered to conform closest to observations made concerning wave surface configuration, the circular motion of the particles, pressure distribution within the wave, etc. This has, however, only applied to each individual wave considered separately and it has not been possible to use the trochoid to describe the whole surface of the sea where each wave has a different height, length, and, to a certain extent, direction. Neither has it been possible, with the help of the trochoidal

wave theory, to establish any relationship between wind and waves, even though attempts

have been made in this direction. Sea wave observations have been made for many years but, for a long time, there was no possibility of arriving at any definite system in the apparently complete irregularity, particularly concerning wave height.

In the absence of any possibility of devising any mathematical model of the configura-tion of the sea surface, i. e. the appearance of the "path" followed by a ship, it was also impossible to produce any proper mathematical picture of the behaviour of a ship travelling along this surface, the motions it described and the stresses to which it was

subjected. As a base for material strength calculations, one has been obliged to use idealized wave forms such as the trochoid and assume the ship to be in a state of

static equilibrium relative to this wave.

During the last twenty years, the oceanographers conception of wind-generated seas has been basically revised. Theories exist today which, though far from complete, make possible a mathematical representation of the irregularity of waves. These advances have been made possible through the development of mathematical statistics, a branch of mathematics which is being used more and more to define natural phenomena.

MATHEMATICAL REPRESENTATION OF SEA WAVES

The trochoid has characteristics which do not satisfy all hydrodynamic demands. The

current classical theory of gravitation waves led by LAMB [1], amongst others,

builds on the sine-formed wave which is completely regular and moves forwards at constant speed. This wave alone cannot, however, describe the irregular configura-tion of the sea surface. If, however, a large (theoretically infinite) number of sine waves are superimposed on each other, each with its own frequency and amplitude and with a different phase angle relative to the others, a function can be obtained,

the appearance of which depends on the amplitudes and phase angles of the

compo-nents. (Fig. 1). This function of time is represented symbolically as follows:

r(')

=/

s/c t #

(w)J

where:

r(t)

= the function ordinate at time t

E (cs>) = phase displacement for component wave with frequency w

If the phase displacement were constant in time, the resulting wave would be periodic, that is to say, a certain wave sequence would occur at regular intervals. This does

not coincide with the observations made. It is therefore assumed that e (w) is completely random, that is to say it can, for every .' and at every instant, assume

any value between O and 2 lT. In this way a constantly varying function is obtained

with the same basic components, which never repeats itself even if t tends towards

infinity.

Component Waves

(Length of each. 5(2 T . 2OO/u' Height ./5' (u5 Spa ct r um

Wen. El.nøten r(t).f W50 [utn.n(u.)J

'ji

Ilk

ahi

iIiIIi'ìì..

%%W7%%,,W1aj

Frequency wn2/T Time

Fig. 1. Typical energy spectrum approximated with a finite number of

sine-waves [14].

The elementary wave amplitude is denoted by Vír ( )]2

d. A curve of [r (c

as a function of the frequency (Fig. 1) will then have the property that the area of an element having width d represents the square of the mean amplitude within the

frequency range t dc./2. Since the energy of a wave motion is proportional to

(amplitude)2, the above-mentioned area is proportional to the energy, and the whole area under the curve will represent the total energy of the wave system per unit length.

Since the function thus represents the contribution of the different frequencies to the total energy, it is usually called the energy spectrum.

The statistical branch of mathematics has been widely employed on such so-called time series. TUKEY, WIENER and in this country, CRAMR, can be mentioned. It

has been shown that a function such as equation (1) with continuous spectrum and

random phase angle has certain statistical properties. It has thus been possible to

apply a fixed statistical pattern to the irregu1arity itself which enables predictions

to be made regarding probable wave heights, etc [11].

The primary of these properties is that the wave ordinate at a certain point during different times is distributed according to Gauss - distribution function, that is to say, normally distributed. The same applies to the wave ordinate at a certain time at different points in a locality where wind conditions are the same. We thus have:

k2

p(b< r(L)<k) =

P

(a).

i. e. the probability that the wave ordinate at the time t1 selected at random lies between k1 and k2 is represented by the right-hand side in (2).

+ POINv$ OSTAIN(D WoM AN A LV SiS

SCALE OF DEVIATIONS OF POINTS 0M RECO*D FROM MC*W

Fig. 2. Distribution of wave ordinates from measurements at sea. (Reproduced

from [15]).

+ + NORMAL PI5TRIUTION u _CURVi Q I., O O u + z Ii a w k 4.

According to (2) the distribution has a mean value of O and variance (12 _{= R/2. It can}

be shown that R = the total area under the energy spectrum which, according to

definition, is proportional to the total energy in the wave system and in this way a

relationship between the spectrum and wave surface configuration has been obtained.

I addition, the wave heights, that is to say, the vertical distance from wave crest to

the adjacent wave trough, have a certain distribution which can be calculated, given

the extent of the area under the spectrum. If 2 h represents wave height, we have:

Ì2A

R

p(4<h(h2)

=J

-e

SA

A,

that is to say, the probability of h assuming a value between h1 and h2 is equal to the integral in the right-hand side of the equation (3).

xz

e

Fig. 3. Rayleighs frequency function

The distribution function(3), which in this connection is usually known as "Rayleighs

function", (Fig. 3), has only one parameter, R. Given this value, it is thus possible

statistically fully to plot the waves which at a given instant exist within an area where the wind blows at near enough constant strength and direction, as well as the wave

crests which pass a given point during an interval of time with constant wind.

(3)

There are important reservations concerning the validity of these theories. The wave heights are assumed to be small relative to wave length, which enables linear condi-tions to be accepted as a basis. Observacondi-tions have shown, however, that waves can occur which are higher than would be expected according to equation (3) and the first

reaction has been to explain this by arguing that the linear assumption no longer applies above some certain limit. Another condition is that the energy spectrum should be "narrow" for (3) to be valid. The greater part of the total energy should thus be concentrated within a small frequency interval. Some authors have considered this

condition to be decisive and CARTWRIGHT has taken the amplitude distribution for a

spectrum of random width and has shown that (3) is a special case which only applies when the width of the spectrum tends to O. This is discussed more fully later on.

THE RESPONSE OF SHIPS TO CONFUSED SEAS

When the oceanographers began to bring some order into the chaos which a confused sea gives the impression of being, the shipbuilding technicians soon followed suit.

Studies of the motions of ships have been made more difficult by the lack of knowledge

regarding the variations of the outside forces, which caused a wide gulf between theory and actuality. The first work published, where the new conception of waves has been seriously applied, was a lecture before SNAME in 1953 by St. Denis and Pierson [2j. For the first time in the long history of shipbuilding, an oceanographer and a naval architect cooperated in a mutual attempt to solve the problem of the motion of a ship in really confused seas.

The basis for the hypotheses set up regarding the three most important motions, pitching, rolling and heaving, is that the ship s response in these three degrees of freedom is linearly dependent on wave height, and that a super-positioning law can be applied so that the sum of the responses to a number of elementary waves is equal to the total response to one wave which is composed of the same elements. The variation in, for example, the pitching angle, will then assume the same character as that of the wave surface, that is to .say, may be expressed thus:

s(é) ¿(4))] V[S())Jéc

(4)

where s(t) = the value of the ordinate for the observed motion at time t, = a phase angle selected at random for every frequency,

Y[sca32

2

= the amplitude of the components in question,

and thus [«41)) = the ordinate in the "energy spectrum" of the motion.

The consequence of this hypothesis is that the amplitudes for these motions have the same statistical distribution according to "Rayleigh s function" as applied to wave

The bending moment in the hull of a ship is also vitally affected by the waves which

the ship encounters and it is reasonable to investigate whether the same argument can be applied to the variations in the moment and, consequently, in the stresses arising as a result thereof. JASPER has investigated the statistical distribution of stress amplitudes in full scale ships and has found good agreement with the Rayleigh distribution [3], and LEWIS has carried out a large number of measurements of

midship bending moments on models and has found not only that the amplitudes have

this distribution but also that the variation can be expressed by equation (4) [4].

Static calculations of the bending moment in waves of different heights show that it is reasonable to extend the assumption of linear dependence on wave height to the bending

moment [5, 6]. Application of the above-mentioned super-position law then leads to the following conclusions as regards rolling, pitching and heaving as well as bending

moment, in future grouped under the term, the ships "response".

In a simple formed wave with amplitude H, these responses will vary in a sine-formed sense with an amplitude S, which for moderate wave heights is directly pro-portional to H. This can be expressed:

5= AH

A (e), which is a function of the frequency of encounter, is defined as the amplitude of the respective responses per unit wave height and is called "the response amplitude operator". If this function is known, the ordinate for the response in question can be

calculated from the following equation:

s(z') E(Lv)JI4r(a,)J2[A(,)J2 J

(5)

D

where [r(ø)]2 = the ordinate in the wave spectrum.

The response energy spectrum can be calculated according to:

fs4)/2

= [r/

2fA(/2

(6)

The response amplitude operator for different frequencies is dependent on the natural

frequency of undamped motion in each degree of freedom. The function has a peak

near resonance with the frequency of encounter and falls off on both sides of this. The appearance of the curve decides to a great extent the influence both of the ships speed and its course relative to the dominating wave direction, and it must therefore be of the greatest significance to obtain an idea of how the amplitude operator varies with different factors.

In order to calculate a response spectrum with the aid of the amplitude operator from

the prevailing wave spectrum, the latter must be transformed to the "encounter

MODEL SPEED 2.4 FT/SEC-SHIP SPEED 4.5 KNOTS

RESPONSE AMPLITUDE OPERATOR FROM REGULAR WAVE TESTS

2

OBSERVED WAVE SPECTRUM

FROM MOVING PICK-UP (AVG. OF 3 RUNS)

W,. 21'/T,

Fig. 4. Amplitude operator from tests in regular waves.

According to Lewis, ref. [4J.

0 2 4 6 8

We 27r/Te

Fig. 5. Bending moment spectra. Calculated with amplitude operator in fig. 4 and observed in wave spectrum

according to fig. 4.

1f the speed y = O and regular waves come from dead ahead with frequency o, the

frequency of encounter is _e = ci

At all other speeds and at other angles between the ship and wave direction, the frequency of encounter will differ from ci. The relationship between them can be expressed thus:

G

G)(/-OC)

G) Y

where =

cas X

= the angle between the ship s course and the direction of advance of the waves. 300-w X w > d loo z UI 2 14

(7)

OBSERVED' SPECTRUM OF ENDING MOMENT (AVG. OF 3 RUNSS' 400

-OO s PREDICTED BENDING MOMENT SPECTRUM OF OO

fr

'loo 'III IO 12 14 e lo 'I I-U. )-_u z w D o w I-X w X w

Note that in this definition of the relative direction of approach _e = 1800 with _{a sea}

from dead ahead and 00 _{with a following sea. When the waves meet the ship from}

directions forward of abeam, o. will therefore be negative and c.Je will always be greater than W.

(FoLLov/vG SEA)

4 RAD.

The total energy in the wave system is not altered in the relative co-ordinate system. The area under the spectrum must thus remain unaltered after transformation and a

factor can be determined from this condition, which is a function of ec and by which

the ordinate in the original spectrum should be multiplied to give a correct encounter spectrum.

Fig. 6 shows an ideal, unidirectional Neumann spectrum [13] at different speeds in a head and following sea respectively. With increased speed in a head sea, the

spectrum widens and at the same time moves to higher frequencies. In a following sea, it becomes narrower whilst the energy is concentrated to lower frequencies. The position of the spectrum maximum, together with the amplitude operator, has the

greatest significance in explaining how an alteration in the ships course or speed may influence the motions or bending moment. Assume that the amplitude operator has a maximum at almost the same frequency as the peak in the wave spectrum at relatively low speed. The peak of the motion spectrum will then be at this frequency and the

result will be a large amplitude of the motion. If the speed is increased, a smaller

part of the wave energy will influence the ship at this resonance frequency, whilst the

greater part of the energy operates at a higher frequency. The result must be a

decrease in the motion amplitudes. If, however, the amplitude operator had its peak at a higher frequency, the maximum motions would only occur when the speed had become so high that the encounter spectrum maximum coincided with the resonance frequency. With further increase of speed, the amplitudes would decrease again.

The amplitude operator for pitching depends on the natural frequency of the unrestricted

pitching motion and on the relationship between the wave length and length of the ship. The results from the OCEAN VULCAN measurements show that pitching generally

occurs in the lowest of natural frequency or encounter frequency. At such speeds and

in wave systems where the natural frequency is lower than the dominating encounter frequency, pitching will thus occur in the former, with ever decreasing amplitudes the more the speed is increased. If, however, the dominating encounter frequency is very low, as is the case with, for example, a following sea, pitching takes place in this low frequency and the amplitudes can reach high values.

As far as the bending moment is concerned, it is difficult to describe any natural frequency with which is would be possible to obtain resonance with the operative forces

(the waves). Experiments with models in regular waves [4, 6] have shown that the bending moment has a maximum at a resonance factor approximately equal to 0. 9 between the natural pitching frequency and the encounter frequency. Full scale measurements show that the moment varies with frequencies about equal to the

encounter frequency; in analyses of the OCEAN VULCAN experiments, this relation-ship has been used to calculate the wave lengths with a good degree of conformity.

No actual calculations of the moment amplitude operator for full-scale ships appear

The amplitude operator has, perhaps, its greatest significance for the interpretation of model experiments in regular waves.

If the super-position law mentioned on page 8 can be applied, the amplitude operator can be determined through such experiments, after which the ships responses to any wave spectrum can be calculated. The response amplitude is determined at anumber

of different wave lengths ( = encounter frequencies) and a curve through the points so obtained gives a continuous transfer function (Figs. 4 and 5). If this is multiplied by the mutually connected values on a wave spectrum, a response spectrum is obtained from which probable resulting amplitudes can be calculated. This procedure has, as far as is known, only been applied to models [4, 12], so that the question of scale

factors in transferring the results to ships is still unanswered, but there is scarcely

any reason to assume any other scale factors than those which can be obtained through dimension studies.

AMPLITUDE DISTRIBUTION

It can be seen from the foregoing that much valuable information about the variations of the bending moment would be obtainable if its energy spectrum in different external

conditions were known, that is to say, different speeds and courses relative to the waves, different wind and wave conditions, different displacements and load distribu-tions. A more direct picture showing what values the total amplitudes attain is obtained by studying their statistical distribution. As briefly mentioned above, it can be shown

that it nearly coincides with the Rayleigh distribution if certain conditions regarding, amongst others, width of energy spectrum, are fulfilled.

The amplitudes can be studied in two ways (Fig. 8). They can be measured as the distance at right-angles to the time-axis between a maximum and the next preceding

or following minimum without regard to the double-amplitude mean level. Rayleigh

showed (1880), that for a time function f (t) with a narrow spectrum, the distance 2 a

has the following statistical frequency function:

tu(o) =

2G_M,

o

02

e

where p (a) = probability for a value a m0 = mean square value of f (t)

It has hitherto appeared impossible to deduce the distribution for a in cases where f (t) has a spectrum of different width.

A large number of measurements of wave heights have been analysed in this way 19, 10, 11] and generally speaking a good measure of agreement has been found

between equation (8) and histograms produced from the measurements. In addition,

¡e

t

where e2'

L constitutes a measurement of the relative width of the spectrum. The integral in (10) is the Gaussian function, and the following is found for the two extreme cases E = O (infinitely narrow spectrum) and ¿= i (infinitely wide spectrum):

£2

2

=

,. e

e = o

(/2)

and p(7)

e E

/

(/3)

(12) is the Rayleigh distribution and (13) the normal distribution. Fig. 7 shows

equation (10) for different values of E.

a

the rolling, heaving and pitching amplitudes and bending moment together with the

stresses caused thereby, have been shown with great probability to follow the same

distribution [3, 4, 9].

Another way of viewing the amplitudes is to measure the distance from the mean level of f (t) to the respective peaks, when the maxima and minima are dealt with

individually. The distribution of "wave height" so defined has been deduced by RICE [8] and applied by CARTWRIGHT and LONGUET-HIGGINS [9] in connection

with waves and ship motions. This method is particularly valuable where bending moment is concerned since the hogging and sagging moments are of individual interest. Let f (t) be a time function according to equation (1) and let E (L)) be the ordinate

in its energy spectrum. The nth moment of the spectrum about the origin is designated m and is expressed as

F?7

G) (.

(9)

o

(as will be seen, m0 = R = the area under the spectrum) II! = the height of a peak above the mean level of f (t) and 7

the following statistical frequency function for _'7.is obtained:

11

Fig. 7. Distribution of heights of maxima for different values of width ¿ of the energy spectrum [9].

Since the ordinates of the peaks are counted from the mean level of the function,

negative maxima (and positive minima) can be found. In fig. 8

A, C, G, J, L are positive maxima B, D, F, M, O are negative minima

E, N. negative maxima H, K. are positive minima.

Fig. 8

There is a direct relationship between the proportion negative maxima and the

spectrum width . If r represents the ratio between the number of negative

maxima and the total number of maxima, the following will apply according to

[9]:

frequency function and the probability for a maximum of given magnitude is the same

as for a minimum having the same absolute value. This is apparent directly from the

fact that phase displacement in equation (1) assumes random values between O and 2 T . There is thus the same probability for a value E as for (E t ir), that is to say

the same probability for + f (t) as for - f (t). (Note. L. here is the phase angle in equation (1).) This agrees neither in theory nor with measurements actually carried out as regards variations in bending moments. The sagging moment is usually greater

than the hogging moment and the relationship between them depends, amongst other

things, on hull shape and block coefficient and is thus not constant for all ships. In addition, there is a not inconsiderable sagging moment arising from the pressure distribution in the ship s own wave system and which is dependent on speed. This must further indicate that the moment variation is not in reality of the type which can be represented by equation (1), but it does not necessarily mean that theories

deduced from equation (1) cannot be applied. Fig. 7 denotes that only when E. =

0. 2 - 0. 3, the difference between actual amplitude distribution and the Rayleigh distribution becomes so great as to imply any practical significance for probable maximum values. The same limit presumably applies for the total variations to conform to the Rayleigh distribution with sufficient exactitude. Spectra of bending moments or stresses hitherto calculated, show that E. very rarely exceeds 0. 3.

It should, therefore, be possible to assume good statistical agreement between the Rayleigh distribution and distribution of the total variations. This will be discussed in more detail in connection with spectra calculated from measurements. As far as the individual hogging and sagging moments are concerned, their values counted from the calm water level and not from the mean level, are of interest. Because of the greater values of the sagging moments, the mean level is displaced in the sagging direction. If the direct influence of speed is excluded and the zero level is defined as the mean level for all the small variations which usually lie symmetrically about a given line, it is reasonable to assume that the amplitudes from this line in the

respective directions do not deviate too much from the theoretical distribution. Such a hypothesis would, mathematically speaking, presumably be considered quite

arbitrary, but if it can be determined empirically that it applies with sufficient

statistical certainty, possibly modified by certain empirically determined correc-tions, it must be a useful basis for evaluating the magnitude of the amplitudes which probably can be expected to appear.

ESTIMATION OF THE PARAMETER R

The hypothesis that hogging and sagging moments respectively conform to the Rayleigh

distribution is thus strictly speaking based on false assumptions. The amplitudes of these moments defined as above must, however, have some statistical distribution and when it is a matter of analysing measured values, it must lie nearest to hand first to test the Rayleigh distribution. Such a test first and foremost implies to estimate the parameter in the function, R, from the measured values, so that the observed distribu-tion conforms in the best possible way to the theoretical curve. Estimadistribu-tion of R can

09

1A&A

,___

_._

.,aa.rau SU'arra.

---

-

_----i.

_{__I}

-o o Fig. 9

be made in many ways and in the first place is dependent on how the data have been

measured. Only a few of these methods are run through here without proof.

Bending moments, or the stresses directly caused thereby, are measured with

recording instruments and the result is a diagram similar to that shown on fig. 9. The variable consists of the amplitudes measured from the zero level O - O, designated x. Thus all the values in the sample are known here, the total number being

design-ated N. It can then be shown that the best evaluation of the parameter is expressed by:

Zx

R

_N

that is to say, the mean value of the squares of all values ofx1. For different reasons it may be desired to reject all X1 values below a certain limit, which can be called x1. All values x1 , x1, where x1 O, are thus known.

A truncated sample is then obtained where the parameter is evaluated as follows:

Exit

N

If the values above a limit x2 are also rejected, that is to say, all the values

(x1 x x2) are known, the parameter for a double-faced truncated sample is

obtained:

8 Z

i

x,e

2 R

-

a

X2

N

1

Besides with recording instruments, observations can also be taken with statistical counters, which count the number of values between previously determined limits. The

magnitude of the individual values is thus not known but only how many lie within a

number of definite intervals. The parameter can also be estimated in this way.

2.

X,

o

Let ni be the number of values between the limits x1 and x1 ₊

and _k be the number of values greater than the upper limit xk.

The best estimate of R is then obtained from the equation:

k

k-i

i.,. x1

a a

fl

(,Z

3) ₌ ne..

';:,-/ I

_/

£

R

-J

If the intervals are so small that (x _{+ i}

- x) . O,

it can be shown that

a

t

-a

t

,

X. 4;.,

-e

R

and the following simpler expression is obtained:

in,.

x,.3

iVn

_k

which is analogous with equation (16) above.

II, with only two counters, the number of values exceeding two fixed limits has

been obtained, the following is thus known:

the number n1 which exceeds the value of x1 and

the number n2 which exceeds the value of x2

then R can be evaluated from the following expression:

a t

xix,

I,, ", - In n3

If the function studied is a time series according to equation (1) and the Rayleigh distribution is regarded as the distribution of the maxima of such a time function, the parameter R is equal to double the variance of the function ordinates taken at random values of the time t.

Let y be the ordinate at time t, counted from an arbitrary level.

-The mean level ordinate will then be

_N

(zi c)

(

-

_(a/B)

for large values of N.

Whereafter is obtained R

2 $

(2/)

The variance c2

/

1V

-

N-(20)

(is)

According to the definition of the energy spectrum, R is also equal to the total area

under this.

Since R is the only parameter in the assumed distribution, the latter becomes fully

known when R is determined.

EXTREME VALUES

Of special interest are the greatest amplitude values which with a certain probability occur within a limited space of time. The probability of a valueXm being exceeded is, for the Rayleigh function

P(x,

For large values of the number in a sample N, the probability of a value _{Xm just} being exceeded within this sample, can be estimated as:

P(xx) =

f IV

e

4.2

dx

2

-=

e

(23)

(2 2)

From (22) and (23) is then obtained:

X=

V1RInN

4)

Equation (24) does not express any exact value forXm which always must be obtained

for given R and N, not even for very large values of N and not even if it is known that the sample is taken from a population conforming exactly to the Rayleigh distribution. It must be borne in mind the whole time that all expressions in this connection are only true statistically speaking and as a result come under the laws of chance. The greatest value of xm thus has its own probability distribution which is of vital significance for correct conclusions from a limited number of measured values.

Longuet-Higgins is one of those who has been engaged with this extreme value

distribution [lOJ. With some revision, Longuet-Higgins' result can be expressed

as follows:

R(/n N

=

I - e

(2c)

As shown by Fig. 10, the frequency function has a maximum for A = O, which implies

that equation (24) expresses the value of Xm which most often comes nearest if several samples are drawn from the same population. It is further shown, which is particularly important, that ¿1 can more often be expected to be positive than negative.

J-0 a 8 60 5 4 30 20 /0

Fig. 10. Extreme value distribution according to Longuet-Higgins [10].

96 ea

6-c

ft75r )E'-UE/V7

Fig. 11. Confidence limits for y_max = x_max

/ '/

/0 20 30 40 50 o 70 0 o g0

t.

/00 600 700 800 900 f0o 'V

From equation (25) can thus be calculated the limits between which Xm can be

expected to lie with certain given probability, for a certain value of R. Different methods have previously been shown for estimating this parameter from observed

values. These estimates, in their turn, are obviously fraught with errors. In

order to arrive at an exact calculation of R it would be necessary to know all individuals in the whole population. It can, however, be assumed with sufficient

exactitude, that with the relatively large samples generally available, the common estimates of R are normally distributed. From the expressions of their variances it is thus possible to calculate confidence limits also for R, that is to say, the limits between which the true value of R should lie with desired probability. By taking into consideration both the variance in the estimated R value and the random

variation of the extreme value, it is possible finally to calculate the limits between which the largest of N values can be expected. These limits are found in Fig. 11. These curves should be interpreted in the following manner. If x in equation (22) is replaced by the standardized variable

X

(22 a)

the distribution function assumes the following form:

2

P(v

.i) =

e (22

)

The method for evaluating R which gives the least variance, must be equation (15) since, amongst other things, the greatest possible amount of information is available (all values in the sample). Provided that this estimate is normally distributed, those limits are obtained between which vm can be expected to lie with given probability

indicated in Fig. 11. If R is estimated according to any other method which presumes less information of the real values, the estimate will be less efficient and the variance will increase correspondingly. The probable limits will then be further widened.

CALCULATION OF THE ENERGY SPECTRUM

As previously mentioned, the assumed time function can be expressed as follows:

s()

+ e(a))] p2.ç(a)?la J() (ao)

[ (

))J2

here represents the ordinate in the energy spectrum at frequency C-) and

has dimensions [amplitude]2 per unit frequency, generally radians/sec. The integral

cannot be solved as such but must be regarded solely as a mathematical symbol for

finite sum if the spectrum is known. The function value at time t can then be

expressed:

2

.s(é)

=JCO51V

¿

e(c

)7/'s(cJ

,,

/a)

-

)

(27)

L 2n#i a,,..,#J 2n. ¿,

,, o

If, on the other hand, s (t) is known, for example in the form of a measurement diagram, the energy spectrum can be calculated in several, in principle completely different, ways. The method used in the following description of measurement

analysis is indicated by, amongst others, Pierson-St. Denis /2] and is based on

calculation of the auto-correlation function.

The non-normalized auto-correlation factor is defined as:

Qi%)

L,m

'p

2

t

_'p

where to = time of origin

t

= total time for the analysed function

h = one time interval

From the auto-correlation function a spectrum is obtained as follows:

= COS

A.

(29)

o

With numerical calculation, these integrals are approximated by sums. The function is divided in intervals of equal time periods, LI t. The length of the interval must be chosen bearing in mind that the lowest time period occurring in the spectrum with this method, is equal to 2 ¿1 t, so that one must be aware of where the spectrum probably dies out. The ordinate is then read off at every dividing point from an

arbitrary base line. The mean value of the ordinates is calculated according to

equation (21 a), page 18 and if the deviation of an arbitrary ordinate from the mean value is represented by s (tq) the auto-correlation factor is obtained as:

n-A 2

4's(é)

$(_lI J

Q(h)=6 ¿__

_f

...ni

Forh=O, Q(o)

=

I

[s(Ç1J2

=

z

R

£4.' £4

$ (t .' A). cié

(30)

The function will swing round the zero value with decreasing amplitudes for

increasing h. In order to secure the essential part of the spectrum in the calculation, m must have a sufficiently high value. At the same time, a large value of m means

that the calculations become very tedious so that the value chosen must not be unnecessarily high. The product of m 4 t is the longest period which will occur in

the calculation and a suitable value for m can thus be judged.

When all Q (h) values have been calculated, a first rough value of the energy is

obtained by means of the formula:

-,

7TÁb

=

1-/Q(o)

2XQ64)

cas 4. 7Y;/7 (3/) 4=,

*

0,1, 2....

m

in the frequency interval

71(4 Vi)

_{< G) <}

4m

These rough values must be corrected for the errors arising from the fact that they are calculated from discreet values taken from a finite diagram. This correction gives the energy coefficients

=

c234,

o,c4L4 #

Finally the ordinate in the spectrum is obtained:

z

¿441m

fs(&JJ= 77-at frequency 6) =

4m

74

w 71«'k# Ya)

4E in

(sa)

(33)

H. RESULTS OF STATISTICAL ANALYSIS

Measurements of stresses and motions have been carried out in M/S CANADA,

belonging to the Johnson Line, during the years 1956- 1958. The instrument equipment was made available by the David Taylor Model Basin, Washington, D. C., USA. This

consisted of a five-channel registering oscillograph and a counter unit. The latter contained 6 counters for each one of the five channels and by means of a contact system in the oscillograph, the number of double amplitudes exceeding six previously

determined levels could be counted.

The following was registered on the five channels of the recorder:

Longitudinal normal stress in the shelter deck amidships as a result of bending

parallel to the centre line plane ("vertical" stress).

Longitudinal normal stress in the shelter deck amidships as a result of bending parallel to the deck plane ("horizontal" stress).

Vertical acceleration amidships.

Rolling angle. Pitching angle.

MEASURING STRESSES

The stresses were measured with wire strain gauges attached directly on the under-side of the deck plating at a distance of 100 mm from the stringer corner between frames 100 and 101 (see Fig. 12). Two units each containing 8 gauges were connected up on either side of the ship as shown in the wiring diagram in Fig. 13

[17]. With

this connection an electrical analogy to the stress-strain equation was obtained:

G ₌

[E

. )J. E

- (i -

y)

T]

(31)

where

Ç

= stress in x-axis (fore and aft)

= elongation in x-axis (fore and aft) = elongation in y-axis (athwartships)

y = Poissons constant

= temperature coefficient of steel

T

= temperature difference

The relationship between the resistances of the gauges positioned longitudinally and athwartships was chosen on the assumption that Poisson s constant is equal to 1/3. In this way the direct variation in longitudinal stress was registered and the influence of

transverse stress on elongation was eliminated. When the term "stress" is used in

this report, this is defined as the result of measurement made according to this

ANENOIIE TER

70 80 9

A OCELEROtIE TER STRAIN GA 6ES

d = 268" = 8.13 m

corresponding deadweight = 9085 tons Machinery output = 2 x 7000 BHP

speed = 19.5 knots

Fig. 12. M/S CANADA. Location of gauges.

Oscillator Input B Signal O ut put -4-z Athwart ships Direction

R1, R2, R3 and R4 are gauges with the same resistance, attached directly to the deck plating in the directions shown. B1 and B2 are temperature compensating gauges having twice as great resistance as the others. They are attached to a plate which is

welded to the deck at one point only.

Fig. 13. Wiring-up of strain gauges.

110 120 130 140 150 160 170 F80 0 Longitudinal Direction Loa = 500'2" = 152.46 m = 46411" = 141.70 m BM =

640"

= 19. 50 m DM = 3910" = 12.14 m

Measuring control centre

Wire strain gauges on underside of

shelter deck.

The bending of a ship can be considered to be made up of two components, one in the

centre line plane and one parallel to the deck plane. The first of these gives rise to

a stress ,,

the same both starboard and port. The other gives a stress in the same

direction, G , which because of the symmetry about the ct,, has the same numerical

value but different signs starboard and port. The gauge bridges were connected

together so that channel i on the oscillograph relayed the sum of the signals from the port and starboard sides whilst channel 2 relayed the difference between them. The

reading on channel i was therefore proportional to Ç, and on channel 2 to G _{. In}

future these stresses will be called the "vertical" and "horizontal" stresses respec-tively.

MEASURING THE MOTIONS

Pitching and rolling angles were measured with the aid of a stabilized gyroscope located in the measuring control centre. The gyroscope could be turned through known angles in the fore and aft and athwartships planes, so that precise calibration

could be carried out. An accelerometer in the gyro room at the after end of the

engine room registered the vertical acceleration.

OTHER OBSERVATIONS

Wind strength and direction were read off from a cup anemometer placed on top of the chartroom sufficiently high up so that interference from the superstructure and funnel could be considered eliminated.

It is extremely difficult to judge wave heights with any exactitude. In order to relate stresses and motions to the prevailing wave conditions, the wave system must be characterized in some way. It is then possible to use the statistical distribution of the wave heights as a basis and to select a definite point on the distribution curve as the "significant wave height". For this, it is usual to employ the average of the highest third of all waves. For example, the 33 highest waves from 100 are picked out and their average height called the "significant height". If the wave heights are presumed to be distributed according to the frequency function given in Chapter 1, this is fully

determined by a significant height so defined. If this is represented by ¡i , it will

mean, amongst other things, that the highest wave of 1000, with 90 % probability will not exceed 2.2 . nor be less than 1.7

When observing the waves it is naturally impossible to determine the significant height with any degree of exactness. The only possibility is to try to judge the height of as many as possible of the highest, most easily distinguished waves and then take an

average figure for these heights. Lower waves and secondary crests are ignored. The wave heights estimated during the measurements have been compared with the theore-tical values which, according to Pierson, Neumann and James [111 should exist at the prevailing wind speed and generally speaking, conformity has been rather good.

In addition, course, speed and propeller revolutions were recorded with the ordinary

ships instruments. A special recorder was connected to the SAL log so that

varia-tions in speed could be studied more closely.

ANALYSIS OF MEASURED RESULTS

The analysis has hitherto included stresses and pitching angles. From a large number of measurements made, 12 have been selected for careful evaluation. Table I gives a picture of these 12 measurement periods. In order to render possible the

use of an electronic calculating machine, the measurement diagrams from these 12 periods were transferred to punched cards with the help of a semi-automatic curve reading apparatus. In such a way, the ordinates for positive and negativepeaks, and ordinates with a constant time difference of 1.1 secs. were read off from afixed

base line. The amplitude distributions were calculated from the former and energy spectra from the latter. The results are shown in figs. I - XVII and the Tables II - VI.

AMPLITUDE DISTRIBUTIONS

In order to examine the conformity of the registered amplitudes to the Rayleigh

function according to equation (3), the mean value of the squares of all values in

each sample comprising the best estimation of the parameter R, were calculated. Histograms of the measured values have been prepared and compared with the function calculated with this R value. The only hitherto analysed measurement

period showing complete deviation from the theoretical function is No. 4 (Figs. I d,

II b, Ill b) and the reason for this will be discussed further on.

Figs. IV - VI show the conformity to the distribution function, that is to say, the

integral curve of the frequency function. A point on this curve indicates the probability of the variable being less than or at most, equal to, this value. The confidence limits

are calculated on the assumption that the fractiles are normally distributed with variance

The 90 % limits imply that, in a sample of given magnitude, there will be 90 %

probability of the observed values lying within these bounds. In one case out of ten, the f ractile value can by chance lie outside these limits, even if the sample istaken

from a population exactly conforming to the theoretical distribution.

2

,P(/-P)

5=

_1u2.w

where p = frequency function value P = distribution function value N = number of observations

STATISTICAL TEST

The tables II - W show the results of different methods of testing conformity. For

the f-test is calculated as follows:

2

2 fr,.

-

tV1P(x)J

=

where ni = observed number in the class i

and

Np()

= theoretical " " 't t? _j

The class divisions represented on the histograms have been used without any

com-bining of the higher groups with only i or 2 values. In some cases, therefore, X

has become very large and does not always give a fair reading. Uncertainty as regards the measured values and above all, reading off the diagrams, means that a very small value for P ( j2) should be used in order to reject the test hypothesis

(viz, that the observations belong to a Rayleigh-distributed population). It is probably not unreasonable to set the limit at P ( Z2) = 0.1 %.

In Chapter I it was mentioned that the greatest of the N values can be expressed:

2

x =

Rc'(/7NL1)

mo. X

and that ¿1 has a distribution which, according to Longuet-Higgins, is represented

by Fig. 10. The values of Li, with the probability of obtaining a numerically

equally large or larger Li than that observed, are also given in Tables II - W. The R value derived as above has been used in calculating the observed value of d In the relatively large sample involved here, this estimate can be assumed to be normally distributed with variance R2/N and the mean deviation R/V'N. A confidence

interval for R is thus obtained, so that

R(/)<R<

/

_vT

(37)

(3e)

(.38)

where R = actual value

R1 = estimated value

and u = the normal distribution value which corresponds to the chosen

probability oc.

If the following is assumed

that is to say, the limits which with probability «. contain the actual values of R and K respectively. Then it is possible to write:

P(R)

= P(K)

(p4/)

P(R«/c)

_{- P(Rd)}

P(A)

2

(42)

since R and K are independent of each other. Confidence intervals for the maximum

value can then be obtained.

a

)'

X

<R

_/

with a statistical certainty of ¿X2. If the standardized variable

X

-is introduced, the following can be expressed:

<

_max2

<(I..

)nN-l..)

o

(It should be noted that since the -distribution is skew, the upper and lower limits for á are not numerically equal).

These confidence limits for Vmax are shown in Fig. 11 and the probabilities P (y) indicated in Tables H - W are taken from this.

It is clear from the tables that the observed maximum value in most cases conforms more closely to the theoretical than does the whole distribution according to the

f-test. This can certainly partially be explained by the unfavourable manner in

which X2 has been calculated. In several cases, however, a large Z2_{was obtained}

in spite of the fact that the higher values, including the highest, showed very good conformity, and a joining of groups had not helped here. This circumstance, that the

maximum value conforms better to the theory than the curve as a whole is, indeed, favourable, as it is the magnitude of the extreme values which one is most interested in predicting with the aid of the parameter R.

GRAPHICAL DETERMINATION OF R

A Rayleigh distribution having a lower truncating point x1 can be expressed:

x

sr

e

(4f)

Ñ(X) =

R (43) (44)

The probability of a value being greater than x is:

z a

P(x .>,t) =

fpEx

-

(4e)

X. L

From the observed values can be estimated:

where n = number of observed values greater than x1

and n1= xl

If both these probabilities are equalized, the following is obtained:

2 *

X. - X

_{4 V}

¿n,'nn.

¿

(47)

(48)

This relationship applies for aU Ys. In a diagram with x2 along one axis and in ni along the other, the distribution will clearly be represented as a straight line with

slope - 1/R.

The counters coupled to the measuring instrument register directly n for six values of x1. The counter values have been plotted in such diagrams and a straight line has been adapted to the points, after which R has been determined from the slope of this line. Figs. XIV - XVI show some examples of this and table V - VI show the result of a comparison between the R values determined in this manner and those calculated

analytically.

This estimate obviously has a lower efficiency than the maximum likelihood estimate. If an extreme case is assumed where there are only two counters which count n1 variations with a greater amplitude than x1 and n2 greater than x2 respectively, the

following is obtained:

a *

The variance for this estimate can be calculated from the following expression:

R2

'

ì0

ftn

,D - ¿fr7,10

ja

N

2

wherep = e

_IV

If the efficiency of the estimate is defined as the relationship between the variance for the best estimate, which can be calculated as R2/N and the variance for the estimate according to equation (49), the efficiency S (R1) is obtained thus:

= fi.. ,P&

[t'?,A.

-

¿7/C7

J

a

(.c/)

-The magnitude of S is shown in Fig. XVII for different values of p1 and p2. An efficiency of S % can be interpreted so, that one obtains the same variance as if

one used the maximum likelihood estimate on only S % of the total number of values

in the sample. An evaluation of R by adapting a straight line to several counter values gives an average figure for several such estimates according to equation (49) and the variance ought therefore to become lower than when there are only two points.

An advantage with the graphical method is that it is possible to truncate the

distribu-tion arbitrarily. Especially when measuring stresses a large number of small

variations emanating from vibrations in the hull are received. These cannot be re-garded as belonging to the distribution of bending moment stresses and should be excluded since they decrease the mean value. In the same way, individual high values have too great an influence on R, if this is calculated from all the values. When adapting the straight line, more consideration has generally been paid to

counters Nos. 3, 4 and 5 than Nos. 1, 2 and 6. A subjective factor is certainly introduced here but it is probably not unjustified.

The result of the comparison shown in Tables V and VI is considered to be so good

that in future the graphical method will be used on all measurements carried out,

in order to obtain a long-time distribution of stresses arising. An important purpose with measurements of this kind is to examine the correlation between stresses and

motions on the one hand and different prevailing outside circumstances such as course, speed, wave heights, displacement etc., on the other. This demands a very large volume of measurement data in order to achieve statistical security in the

result. Only with the aid of a simple, quick method of analysis is it possible to

obtain the required data for this and the graphical method described can be considered

to fulfil these conditions.

According to fig. XVII the highest efficiency that can possibly be reached with only

two counters is about 65%. One of the counters then must count all variations greater

than zero. This is very difficult to achieve, and, as was pointed out before, the

distribution ought to be truncated to eliminate vibrations. The value of p1 must then be assumed to be less than unity. It should, however, be possible to choose x1 and

x2 so that an efficiency of about 50 % is maintained within the normal limits of the

R-value. The smallest number of variations can then be computed, on which the estimation of R can be based, to give a reasonable variance. The necessary time interval between reading off the counters can also be fixed from the same condition. It might even be possible to design an instrument, which automatically makes a calculation of R according to ekv. (49) at certain intervals of time, and thus gives a direct indication of the risk for a given stress.

ENERGY SPECTRA

A total of 16 energy spectra has been calculated from some of the measuring periods in Table 1. For this purpose the method described on pages 21 - 23 has been used. The result is shown in Figs. VII - XIII. Measurements i - 4 were made immediately after one another over a period of about two hours and the only factor which varies to any extent is the course relative to the dominant wave direction. These results have therefore been inserted in the same diagram, showing how the spectrum changes

according to the angle of the waves. The angle Xeis measured in accordance with the definition on page 10. Spectra i - 3 are concentrated at a period of encounter of about 6 seconds, this being probably almost the same as the pitching period of the ship. Since no wave measurements have been made, the wave spectrum is,

unfor-tunately, not known.

The stress and pitching spectra are lowest in a head sea and highest in quarter head sea. The ship s amplitude operator should have its peak at a period slightly over the natural period, 6 seconds, and the encounter spectrum of the waves during measure-ment No. 3 has also probably had its maximum here. During measuremeasure-ment No. 1, with a head sea, this maximum has been at a lower period and also had a lower value (compare Fig. 6), so that the response spectra have also been considerably

lower. The fact that such high spectra were obtained during measurement No. 2, with the sea on the beam, shows that the waves had a very broad directional spectrum so that it is very difficult to draw any definite conclusions.

Measurement No. 4, with quarter following sea, shows a completely different picture from the others. The amplitude distributions were also completely different

from the others. As opposed to Nos. i - 3 these spectra, in accordance with Cartwright s definition, are very "wide" and the amplitudes should then, in

accordance with Fig. 7, be normally distributed. Since the negative maxima have

not been attained, this is not unreasonable (Fig. i d). It is interesting to note

wind strength, then a spectrum transformed to 450 will have a maximum at

almost exactly the same frequency as both pitching and stresses. Fig. XIII shows the remarkable agreement between the three calculated spectra.

It is obvious that with a following sea, where the encounter spectrum is concentrated to a very low frequency, there are also variations in the reponse of the ship at this

slow tempo. Since the wave energy is so extremely concentrated, the responses have very large amplitudes even if the amplitude operator does not have so high a value here. It is also probable that waves with frequencies, which normally are so high

that they have no significance, now have a so much lower relative frequency and greater energy, that they can be responsible for noticeable responses. As is the case with the spectra Nos. 2 and 3 calculated here, the wave spectra have often a small peak at high frequency [18], which normally does not exert any influence. Through

the transformation of the encounter spectrum this peak will, in the case of travel in following sea, move to lower frequencies and, at the same time, become higher and narrower. It then gives rise to the other peak in spectrum No. 4, with a subsequently large number of small variations. The distribution then has the appearance shown in the figure.

The calculated energy spectra show a clear connection between pitching and vertical bending moment. The same applies to the R values calculated froni the resulting amplitudes. It can be generally maintained that when the ship has an accentuated pitching motion then the bending moment is also large and that if, by altering course

or speed, this motion is reduced it is reasonable to assume that the stresses will

also be lower. One should possibly exercise care when drawing this conclusion in the case of a following sea where pitching is reduced more than the stresses. It is

possible that the stress amplitude operator has, at low frequencies, a relatively

higher value than that of the pitching. This can only be clarified by, simultaneously with measurements of this type, carrying out wave measurements, enabling wave and response spectra to be inter-related. Model experiments being carried out at various towing tanks, may also give answers to these questions.

CONCLUSION

It is very obvious that the only possibility of judging the results of measurements

of this type is by the use of statistical analysis. The greatest stresses measured

during half an hour, a voyage or a year have, as such, no signigicance. The average value of stresses during similar periods have no significance either unless the relationship of other values to the average stress value is known. The statistical distribution function must, in other words, be known.

Such statistical loading analysis has long been used in other technical branches, for example road and air transport technique. Since the theories concerning the statis-tical nature of sea waves have now provided a basis for a similar analysis of the motion and moments caused by the waves, it is only natural that loading statistics have also made advances in shipbuilding technique. The aim must naturally be to obtain such knowledge of externally-operating forces that the design of a ship can

also be based on actual loading values and not only on experience and comparison

LITERATURE

LAMB, H. : Hydrodynamics. Dover Publications, New York, 1945.

St. DENTS, M. & PIERSON, W. : On the motions of ships in confused seas. Trans. S.N.A.M.E. Vol. 61(1953), s. 280 - 332.

JASPER, N. : Statistical distribution patterns of ocean waves and of wave-induced ship stresses and motions, with engineering applications.

Trans. S.N.A.M.E. Vol. 64 (1956), s. 375 - 432.

LEWIS, E. V. : A study of midship bending moments in irregular head seas,

T2-SE-A1 tanker model. Journal of Ship Research, Vol. 1(1957) nr. 1, s. 43-54.

ABRAHAMSEN, E. & VEDELER, G.: The strength of large tankers. Eur. Shipbuilding Vol. 6 (1957) : 6 & Vol. 7(1958): 1.

CHRISTENSEN, Hj., LÖTVEIT, M. & MURER, Chr.: Modellförsök for â be-stemme skjrkrefter og böyemomenter i et skip i regelmessige böiger.

3. Nordiska skeppstekniska ârsmötet, Göteborg 1958.

CARTWRIGHT, D. E. & RYDILL, L. J.: The rolling and pitching of a ship at sea.

A direct comparison between calculated and recorded motions of a ship in sea

waves. Trans. I.N.A. 1957, s. 100 - 135.

RICE, S. O.: Mathematical analysis of random noise. The Bell System Technical

Journal, Vol. 23 (1944) & Vol. 24 (1945).

CARTWRIGHT, D. E. & LONGUET-HIGGINS, M. S.: The statistical distribution of

the maxima of a random moving surface. Proc. Royal Society, Ser. A 1956, s. 212. LONGUET-HIGGINS, M. S.: On the statistical distribution of the heights of sea waves. Journal of Marine Research, Vol. XI, 1952.

PIERSON, W. J., NEUMANN, G. & JAMES, R. W.: Practical methods for

observing and forecasting ocean waves by means of wave spectra and statistics.

Hydrographic Office Pubi. No. 603, New York 1955.

LEWIS, E.V. & DALZELL, J. F.: Motion, bending moment and shear measure-ments on a destroyer model in waves. Stevens ETT Report No. 656, April 1958.

NEUMANN, G.: On ocean wave spectra and a new method of forecasting

wind-generated sea. Beach Erosion Board, Technical Memorandum No. 43, 1953.

LEWIS, E.V.: Ship speeds in irregular seas. Trans. S.N.A.M.E. Vol. 63

(1955), s. 134 - 202.

LEWIS, E.V.: Summer seminar on ship behavior at sea. Stevens ETT Report

No. 619, 1956.

PIERSON, W. J.: Interpretation of the observable properties of "sea" waves in terms of the energy spectrum of the Gaussian record. Trans. American

Geophysical Union, Vol. 35:5 (1954).

JASPER, N. H.: Service stresses and motions of the "Esso Asheville", a T-2 tanker, including a statistical analysis of experimental data. D. Taylor Model Basin Report No. 960, 1955.

PIERSON, W. J. & MARKS, W.: The power spectrum analysis of ocean wave

records. Trans. American Geophysical Union, Vol. 33:6 (1952).

de LEIRIS, H.: La détermination statistique des contraintes, subies par le

Note.

X

V max

max

_v

P(v) = probability for Vmax observed in accordance with Fig. 11

P(s) =

It

10

i a t

P ( fl

i

i -distribution

Table I. Analysed measuring periods

Table Il a. Sagging stresses

lest. no. Speed knots R.P.M. Wind Waves Dead-weight

Beaufort _directionRelative Height metres Relative direction Xe 1 14 1/2 94 6 00 5 1800 9000 2 16 1/2 94 6 900 5 900 9000 3 15 1/2 94 6 45° 5 135° 9000 4 17 1/2 94 6 135° 5 450 9000 5 17 90 5 20° 1,5 160° 3300 6 16 90 6 30° 2 150° 3300 7 15 90 6 00 4,5 1800 6700 8 5 30 6 00 4,5 180° 6700 9 15 90 6 00 4,5 180° 6700 10 15 1/2 88 8 30° 5 150° 8800 11 12 1/2 73 9 30° 5 150° 8800 12 10 65 9 30° 5 150° 8800 Test no. N VT kp/mm2 y max calcu-lated vmax observed P(v) approx

4

P(d) approx

e

P(12) approx % 1 321 1,30 2,40 3,73 0,2 8,1 0 21,8 14 10 2 272 1,63 2,37 2,95 8 3,1 5 162,8 14 0 3 279 1,92 2,37 2,75 15 1,9 12 52,4 15 0 4 168 1,31 2,26 2,26 50 0 64 58,9 12 0 5 384 0,53 2,44 2,29 30 -0,7 13 26,0 9 0,1 6 290 1,11 2,38 2,72 20 1,7 16 22,4 13 5 7 309 1,58 2,39 2,86 13 2,44 8 58,9 14 0 8 129 1,02 2,20 2,34 40 0,61 40 28,0 10 0,2 9 125 1,01 2,20 2,50 25 1,41 21 14,4 10 15 10 499 1,42 2,49 3,66 0,5 7,1 0 585,0 17 0 11 389 1,66 2,44 2,78 20 1,8 14 54,9 15 0 12 380 1,39 2,44 3,09 5 3,6 2 18,4 14 20

Table II b. Hogging stresses

Table li c. Total vertkal stress variations Test no. N kp/mm 2 V max calcu-lated y max observed p (y) approx % ¿3 P (6 ) approx % p ( Z) approx % 330 1,07 2,41 2,16 15 -1,14 5 12,6 9 20 2 259 1,14 2,36 2,44 40 0,40 49 18,7 12 10 3 274 1,22 2,37 2,07 5 -1,34 3 13,4 10 20 4 186 1,03 2,29 2,56 25 1,34 23 61,5 10 0 5 380 0,53 2,43 2,39 40 -0,23 28 14,7 10 15 6 291 1,12 2,39 2,09 10 -1,31 3 8,7 9 50 7 306 1,32 2,39 1,91 1 -2,06 0,5 28,0 10 0,2 8 131 0,98 2,21 2,24 45 0,15 57 4,9 9 80 9 126 1,10 2,20 2,14 40 -0,27 27 18,0 15 25 10 484 0,97 2,49 2,53 50 0,24 55 34,8 10 0,01 11 369 1,03 2,43 2,87 15 2,29 10 31,9 12 0,2 12 363 1,03 2,43 2,34 35 -0,43 22 20,0 10 2,5 Test no. N kp/mm2 V max calcu-lated y max observed P (y) approx % ¿ P (â) approx % 12 _f P ( V)_approx % 1 646 2,29 2,54 2,92 15 2,03 12 21,2 14 10 2 530 2,66 2,51 2,56 50 0,26 54 24,4 14 5 3 552 3,04 2,51 2,49 50 -0,11 32 18,8 16 30 4 352 2,07 2,42 2,35 40 -0,32 26 346,0 10 0 5 763 1,04 2,57 2,29 5 -1,39 2 43,5 15 0,01 6 580 2,20 2,52 2,44 30 -0,40 23 28,8 11 0,25 7 614 2,83 2,53 2,22 5 -1,47 1 47,0 13 0 8 259 1,95 2,36 2,35 50 -0,05 35 15,8 15 30 9 250 2,06 2,35 2,16 20 -0,86 10 18,2 14 20 10 978 2,33 2,62 3,01 15 2,19 10 15,3 15 30 11 755 2,62 2,57 2,65 40 0,41 49 15,4 15 30 12 741 2,35 2,57 2,77 30 1,07 30 17,1 14 25

Table Ill. Total horizontal stress variations

Table IV. Pitching angle Test no. N kp/mm2 V calcu-lated -max observed P(v) approx P(12) approx Z2 f approx 647 0,91 2,54 2,73 30 0,98 31 7,1 10 70 2 583 0,92 2,52 3,17 5 3,69 2 9,9 12 60 3 626 1,03 2,54 2,99 12 2,52 7 11,0 13 60 4 504 0,80 2,49 3,31 3 4,76 1 321,9 11 0 Test no. N degrees y max -calcu lated y max observed P (y) approx %

p ()

approx % p

(Z)

approx % 1 503 3,62 2,49 2,72 30 1,15 27 22,4 11 2 2 475 5,14 2,48 2,57 40 0,z16 46 19,6 15 20 3 444 5,88 2,47 2,27 20 -0,97 8 24,7 12 2 4 245 2,67 2,35 2,46 40 0,52 45 25,6 7 0,05 7 567 4,30 2,52 2,59 45 0,35 50 16,1 12 20 8 232 2,99 2,33 2,39 50 0,28 52 22,9 10 1 9 195 3,36 2,30 2,06 15 -1,04 6 18,7 16 30

Table V. Comparison between analytical and graphic calculations of R. Vertical stress variation

Table VI. Pitching angle Test no.

Analytically

Graphically Observed max value R kp/mm2 90% confidence limits for R Most probable max value kp/mm2 90 o confidence limits for max value Max value 1 5,26 2,29 2,20 - 2,38 5,83 5,58 - 6,06 2,3 5,8 6,69 2 7,07 2,66 2,54 - 2,77 6,66 6,37 - 6,94 2,8 6,7 6,80 3 9,27 3,04 2,91 - 3,17 7,65 7,33 - 7,96 2,8 7,1 7,58 4 4,27 2,07 1,95-2,17 5,00 4,73-5,26 2,9 6,3 4,86 5 1,09 1,04 1,00 - 1,08 2,69 2,59 - 2,79 1,0 2,6 2,39 6 4,82 2,20 2,10 - 2,28 5,54 5,31 - 5,76 2,2 5,5 5,36 7 8,03 2,83 2,72 - 2,94 7,18 6,89 - 7,46 2,4 6,3 6,25 8 3,81 1,95 1,83 - 2,07 4,60 4,31 - 4,87 1,9 4,5 4,58 9 4,22 2,06 1,92 - 2,18 4,83 4,52 - 5,12 2,0 4,7 4,44 10 5,42 2,33 2,26 -2,40 6,11 5,92 -6,29 2,6 6,8 7,01 lI 6,84 2,62 2,52 - 2,71 6,74 6,49 - 6,97 2,9 7,5 6,94 12 5,52 2,35 2,26 - 2,43 6,04 5,82 - 6,25 2,4 6,2 6,51 Test no.

Analytically

Graphically Observed max value R degrees 90 % confidence limits for R Most probable max value 90 % confidence limits for max value Max value 1 13,12 3,62 3,45 - 3,78 9,01 8,59 - 9,41 3,0 7,4 9,84 2 26,47 5,14 4,91 - 5,37 12,75 12,18 - 13,32 5,0 11,9 13,20 3 34,57 5,88 5,60 - 6,15 14,52 13,83 - 15,19 5,6 13,3 13,30 4 7,14 2,67 2,50 -2,83 6,28 5,88 - 6,65 3,7 8,0 6,60 7 18,48 4,30 4,11 -4,47 10,84 10,36 - 11,26 4,3 10,6 11,20 8 8,96 2,99 2,80 - 3,18 6,98 6,52 - 7,41 3,7 8,1 7,16 9 11,30 3,36 3,12 - 3,59 7,72 7,18 - 8,26 - - 6,91

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Fig. i b. Histogram of hogging stresses

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