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HYDROMECHANICS STATISTICAL DISTRIBUTION PATTERNS OF OCEAN WAVES AND OF WAVE-INDUCED SHIP STRESSES AND MOTIONS,

WITH ENGINEERING APPLICATIONS

o

AERODYNAMICS

o

FRUCTURAL MECHANICS

o

APPLIED MAThEMATICS

Nw'Y DEPTMEW

by Norman H. Jasper

RESEARCH AND DEVELOPMENT REPORT

1b. v

Scheebot

c.

:'

STRUCTURAL MECHANICS LABORATORY

October 1957 Report 921

S

)

(2)

STATISTICAL DISTRIBUTION PATTERNS OF OCEAN WAVES AND OF WAVE-INDUCED SHIP STRESSES AND MOTIONS,

WITH ENGINEERING APPLICATIONS

by

Norman H. Jasper

Reprint of Paper Published in

The Society of Naval Architects and Marine Engineers Transactions, Vol. 64, 1956.

(3)

SOURCES OF DATA

Stresses

Motions Wave Data TABLE OF CONTENTS

Page

INTRODUCTION

i

STATISTICAL BACKGROUND 3

ANALYSIS OF SRORT-TERM DISTRIBUTION FUNGFIONS

ANALYSIS OF LONG-TLRM DISTRIBUTION FUNCTIONS 15

Background 15

Heights and Lengths of Ocean Waves 15

Stress Variations

22

Esso Asheville

22 USS Fessenden 29 SS Ocean Vulcan 29 SSGopherMariner 29 Destroyer 29 Aircraft Carrier 31 USCOCUnimak 31 Ship Motions 31

APPLICATIONS OF STATISTICAL DISTRIBUTION PATTERNS 33

Specification of Engineering Requirements 33

Prediction of Life Expectancy of Structures

36

Prediction of Optimum Operating Conditions 37

DISCUSSION 38 SUMMARY 39 ACKNOWLEDGMENTS 40 BIBLIOGRAPHY 40 DISCUSSION 41 11 8 8 8 10 10

(4)

Statistical Distribution Patterns of:

Ocean Waves and of Wave-Induced Ship

Stresses and Motions, with

Engin ring Applications1

Little is known about the frequency of

occurrence of the various magnitudes of

ocean waves.

Even less is known about

the severity of the wave-induced motions

and stresses which ships experience in

service. The intent of this research effort

is to show that, by utilization of statistical

methods, it

is possible. to describe and

predict service conditions for ships in an orderly and relatively simple manner

de-spite the general complexities of the

prob-lem. Wave observations taken

continu-ously over a period of 6 years at several

weather stations in the Atlantic Ocean

were studied. Wave-induced motions and

stresses in ships obtained under a wide

range of operating conditions were studied

INTRODUCTION

It is probably correct to state that the

ship-building industry knows less about the service

conditions under which its product must operate than does any other major construction industry.

In contrast to the aircraft builders, shipbuilders

have made only a small effort to establish actual service stresses and motions or in incorporating

i The paper given here was originally submitted to the School of Engineering and Architectui of the Catholic University of America Washington. D. C., in partial fulfillment of the requirements for the Degree of Doctor of Engineering.

2 Engineer and Deputy Division Head, Vibrations Division, David Taylor Model Basin. Navy Department, Washington. D. C.

Presented at the Annual Meeting, New York, N. Y., November 15-16, 1956, of SOcIETY OF NAVAL ARCHiTECTS AND MraE

EÑGINEERS.

B N. H. JASPER,2 MEMBER

i

for seven different ships. On the basis of

an analysis Of voluminous experimental data

it is concluded that the probability dis

tributions of wave height, wave length,

wave-induced pitch, roll, and heave

mo-tions of ships, and wave-induced ship

stresses may all be approximated by a óne-parameter-type distribution function when

the environmental conditions are steady,

whereas these variables will tend to follow

the two-parameter logarithmically normal distribution when the environmental

con-ditions are allowed to vary over a wide

range. Applications of the results of this study to the solution of engineering prob-lems are. described with the help of

perti-nent illustrations.

the results of such studies into methods of ship design. A deterrent to progress in this direction has been the complexity of the problém' as

evi-denced by the extensive variability of the sea

and of the response of the ship to the sea. The

intent of this paper is to show that, by utilizing

statistical methods, it is possible to describe and predict service conditions for ships in an orderly

and relatively simple manner, even though the

problem appears formidable. .

The experimental approach taken in this pape/

should complement the theoretical work, of St Denis and Pierson although the present study

(5)

given variate E P(x) p(z) population sample c.d.f. d.f. V mean value of z z e variance of z. stanard dOviation quantile or fractile coñfidénce bands random

TABLE i STATISTICAL TEis AND DEFINITIONS

It should be noted that in the present work only estimates of the statistical parameters can be

Property under Study, for example, magnitude of stress variation, height of

waves.

N

.z2

The mean square value of z, E -_____

Probability of the event x X0 with respect to the random experiment e. The probability P mean8 that in a long series of repetitions of it is practically certain that the relative frequency of occurrence of x X0 is equal to P where

O < P < I A value of P = I indicates that the event will occur every

expenment e P =O indicates that the event will never occur

= P (z)

Aggregate of all the members of the distribution subject to a statistical study. The members may be individual measurements or the results of

indi-vidual experiments.

Number of N members drawn (usually at random) from the population. Here

N la the "sample Size."

Abbreviation of the term "cumulative distribution function" P (z) which denotes the probability that the variate z will take on a value less than or

equalto z, i;e., P(z0)=P[z z0:1.

Abbreviation of the term "distributiön function" p(z) which is the derivative

of P(z) with respect to z, i.e.,

P(x)=P(x)andfls(x)dx=1

E"4

- '

where z, is the individual member of the population or sample and N

is the number of members of the population or sample.

Mean value of a sample of values of

Mean value of the population.

(z)2

. .

-

N

..The

variance is a measure of the dispersion of the values of

z. The greater the variañce the greater the dispersion. Variance of the population.

Square root Of the variance

Corresponding to a given value of the c.df. P(z), there is a value of the variate below which a fraction P(z) of all the members of the population lie; The value of this variate is the quantile or fractile corresponding to P(z). Interval within which the "true" distribution will fall with a certain probability.

A method of drawing a sample when each member of the population has an equal

(6)

limited by the assumptions - required by their

mathematical analysis.

The purpose of this paper is to present evidence

that the wave-induced pitching,

rolling, and

heaving motion of ships, as well as the associated hull-girder stresses follow simple, mathematically defined statistical frequency or probability dis-tributions. It will be shown that the distribution functions of motiöns and stresses are of the same

type as those applicable to the height of ocean

waves.3 In particular it will be shown that the

distribution function approximates the single

parameter "Rayleigh" type for a given state of

the sea, ship speed, and course. On the other

hand, when distributions of wave height, ship motion or stress experienced over a wide range

of conditions are considered, the two-parameter

"logarithmically normal" distiibution is shown

to be applicable over the tange of conditions

observed thus far. The latter distribution is the result of a summation of many distributions of

the first type.

Finally, the extreme values of

ship motions and stresses will be examined to evaluate the feasibility of the prediction of

ex-treme values.

For the sake of brevity, the distribution ap-plicable to uniform conditions of the sea, ship

speed, and course will be called "short-term"

distribution, whereas the function which repre-sents the distribution obtained when the seas,

ship speeds, and courses are allowed to vary over

a range of conditions, will be designated as a

'long-term'S distribution.

The conclusions arrived at in this paper are

based on numerous observations of wave height, hull-girder stress4

(for seven ships) and ship

motions (for five ships). The experimental data

were obtained over a period of years and cover

a wide range of service conditions.

There are many applications in which a

knowl-edge of the frequency distribution of hull

mo-tions, stresses, and the heights of ocean waves can

be used to advantage. For example:

i

Prediction of the most probable amplitudes

of roll and pitch motion of ships under given

environmental conditions.

2 Estimation of the extreme values of ship

response or of wave heights encountered over

given periods of time.

3 Statistical estimation of the capacity. for

which shipboard stabilization equipment must be

designed.

'The study of ocean waves originated as a by-product of the basic investigation of the ship's response to the sea.

4Wherever the term stress is used in this paper, a stress computed from the measured strain is to be inferred.

3

4 Estimation of the endurance strength of the

ship structuxe

The experimental work carried out in

connec-tion with this problem has been a co-operative

effort of many people and several organizations.

In particular the U. S. Navy, The Society of

Naval Architects and Marine Engineers, the

U. S. Coast Guard, the U. S. Weather Bureau, and the Esso Company hase contributed much

to make this research possible.

The special

instrumentation and methods used to collect the data, as well.as some of the sea tests are described in references (1-4) This paper will not be con-. cerned with the methods with which the data were

obtained except in so far as they are pertinent

to the statistical analysis.

The general plan of presentation will be as

follows: Some pertinent statistical background

information will be given first, followed by a short

outline of the sources of the data employed in this study.

The analyses of short-term and

long-term distributions will be given separately. Each analysis applies statistical tests of significance to

the hypothesis that the experimental data are

samples from a specified distribution. The re-sults of the analyses are discussed and

applica-tions of the results to practical problems are

illustrated, followed by a summary of the major

conclusions.

STATISTICAL BACKGROUND

The wave heights, the variations in ship

mo-tions, and hull-girder stresses experienced under

a given set of conditions will be described in terms

of their distribution functions. The statistical

methods applied in the present study comprise techniques which were used to obtain, classify,

analyze, and present large masses of experimental data. In particular they were used to:

i

Collect the data by sampling procedures.

2

Classify and condens the data. The data

were classified according to the magnitude and

frequency of the measured variations. Each vari-ation is understood tO mean a fluctuvari-ation in the magnitude of the variable from the largest

posi-tive value to the succeeding largest negaposi-tive

value; for example in the case of a rolliñg motion

the variation would denote a motion of the ship from the extreme starboard position to the

ex-treme port position.

3 Present the data.

The data are given in

tabular form as well as in the form of statistical

distribution patterns, such as histograms and dis-tribution fúnctions.

4 Analyze the data. With the data presented

'Numbers in parentheses refer to the Bibliography at the end of the paper.

(7)

LBP 528' Beam 76' Depth 44½' Heave Acceleration at C.G. LBP 300' Beam 41'3/4" Depth 24' 7½" Gage Locations Loaded Displacement, 21,050 Tons

SS GOPHER MARINER, C-4,. Dry Cargo

Gage Locations

Loaded Displacement, Gages; on main deck,

21,880 Tons 9½"inside rail, at

Frame 57. SS ESSO ASHEVILLE, T-2, Tanker

Loaded Displacement, 13,000 Tons SS OCEAN VIJLCAN

Gage Location

Gage Location

Gages; 6" below main deck on side plating; 8"fwd. Frame 106.

Gage; approx, upper

deck, 12'stbd. of

centerline, Frame 92.

Gage; on centerline, main deck, 18"fwd. Frame 90. 10.71' above neutral axis.

Loaded Displacement, 2,500 Tons Main Deck LBP 416' Beam 56'10½" Depth 37'4" LBP 503' Beam 68' Depth 39½' 125 100 75 50 25 5:p AP USCGC UNIMAK

FIG. 1 LOCATIoN OF STRAIN GAGES (Circles indicategúgelocations.)

LBP 306' Loadéd Displacement

Beam 36' 1630 Tons

Depth 20.67'

LJSS FESSENDEN, DER-142

Longit. Hull GirderStress at Amidships

(8)

in a suitable manner, analytic functions are fitted

to the data, and tests of significance are applied to determine whether certain statistical predic-tiôns and conclusions can be reached with

con-fidence on the basis of the available data.

Some basic terms used in statistical work are

défined in Table 1. A distribution function indi-cates the relative frequency or probability of oc-currence of a particular eveÚt. The best known type of distribution is the "normal" distribution.

It is theoretically possible, but not often prac-ticable, to convert distributiOns 'which are not

"normal" into a normal distribution by a change of variables. One such conversion which will be utilized here is to take the logarithm of the

vari-able. Thus the "10g-normai" distribution is

ob-tained, which signifies that the logarithm of the

variable is normally distributed. Commercial

chart paper (probability chart) is available which

is designed so that the 'integral of the normal or

log-normal distribution will plot as a straight line

on the chart.

The log-normal law has been

found applicable in various diversified fields.

See, for example, the extensive bibliography given

by V. T. Chow (5).

The probability P(x) takes on definite values for a given population. An estimate of P(x) is often

desired on the basis of samples of the members

of the population. Statistics provide the tools for makiùg such estimates. It will be assumed that

the sample is taken from a population for which

the quantity under discussion is defined by a

certain analytical expression.

The reliability

of this hypothesis is evaluated by the application of statistical tests of significance which enable one to estimate the expected variation of the measured

data from the analytically defined values, if the

hypothesis is true.

In order to determine, whether the variation of the sample data from the assumed distribution is consistent with the hypothesis, use will be made

of a theorem in statistics (6) which states that

the sample quantile6 Xp is asymptotically normal

with mean value E and standard deviation

i

¡P(iF)

P

N

where N is the size of the sample, p the probability

density, P the. cumulative probability at x =

and E is the corresponding quantile of the popu-lation from which the sample is drawn.

Other-wise statéd: If a sample of N-values is chosen

at random and if these values are numbered in the order of increasing magnitude, theñ the

magni-6A quantile denotes the variate corresponding to a given fraction;

i.e., the 0.40 quantile is that ialüeof the variate below which 40 per cent of all values lie.

5

tude of any particular value, say the rth cannot be predicted in advance of the drawing of the

sample; there exists a certain probability function.

If N is large, and if neither rIN nor (N - r)/N is

too small, then the probability function forthe rth value is approximately normal in form.

If N is

increased and if r is also increased so as to keep rIN as nearly constant as possible (as fOr a fixed quan tile of the sample) then the approximation

to normality increases, becoming perfect in the

limit as N - ; the width of the probability

spread simultaneously decreases, becoming zero

in the limit.

If a curve is plotted on each side of the analytic

cumulative distribution curve at a distance of

1.65 standard deviation of the distribution of the corresponding quantiles, then on the average nine

tenths of the observed quantiles would be'

ex-pected to fall between these limits and 10 per cent

of the measured quantiles would be expected to

fall outside these limits This control or

confi-dence curve is therefore designated as the 90

per cent significance level7 and indicates the

probability of a random value falling within the

limits when the hypothesis is true. In order for

the deviation between the hypothetical

distribu-tion and the sample values to be considered not

significant 'at this level, it would be necessary to

have less than 10 per cent of a large number of

observed values fall outside these limits.

In

practice it has been found that wrong conclusions

will seldom be drawn if the' level of significance is

set at 90 per cent or less; that is, the hypothesis is probably true if the deviations are no greater than could be expected, on the- average, in one out of

10 tests. On the other hand if the deviations lie

outside the 99 per cent limits, that is, if they are larger than would be-expected once in a hundred

similar tests, then the hypothesis should be

re-jected confidently. A distribution may be of

practical significance even though the statistical

tests indicate a significant deviation between it

and the measured data.

In many cases a visual inspection of the scatter

is sufficient to indicate that the assumed

distri-bution is an acceptable representation of the actual distribution.

The types of basic distributions

that will be considered here are the normal, log-normal, and the Rayleigh distributions.

The normal distribution of x is defined as

fol-lows:

1

(xf)'

p(x) -

1.e, 2«2

- <x < ±

o y 2ir

Some statisticians prefer to label the limits according to the probability with which a random measuiement would fall without the limits; i.e., the 90 per cent significance limit would then be re-ferred to as the 10 per cent level.

(9)

P(x)

=1

p(x)dx

where

p(x) is the probability density of x x is the variate

is the mean value of the variate

is the variance of the population

Thus the two parameters and define the

dis-tribution completely.

The logarithmically normal or log-normal dis. tri bution is a normal distribution of the logarithm

ofx,and

1 -(Iogx - u)2

p(x) -

1-e

2e'

Nov

where u is the mean value of log x and u is the

standard déviation of log x.

The Rayleigh distribution is defined as follows

p(x) =eIE,x5O

where E is the mean value of the squares of x; thus

(5

5:

o

=

then -../E p(x) = 2ve'h"

Test Conditions: Sea State 5, Head Seas, Ship Speed 7 knots

Significant lave Height 21 It.

The ship pitched at a rate of about 330 variatioos per hour. The test sample coosisled of 236 variations.in pitch angle.

TABLE 2 USCGC UNIMAK-COMPTJTATION OF

CONFI-DENCE Lmeus FOR DATA SHOWN ¡N FIGS. 2

(Pitch angle, sea state 5. head seas, ship speed71/,knots.)

4 6 8 10 12 14 16 10 20

Variation in Pitch Angle - degrees

FIG. 2 USCGC UNnLAX, DzsmrnurION OF VARIATION IN PITcH

ANGLE-SAioPLE i

(This sample i, representative of 129 samples tht were analyzed.)

the single parameter E defines the distribution. Integrating to obtain the cuniulative probability

It is possible to represent the distributicOns cor-

,.-

. 2

r-

i Ç' r

responding to different values of E by a single

j,

px1

-

VE.UX4J -

vE LI-e

distribution, employing a change of 'variables, or

Thuslet 2

P(x4)

=

[1 - e

J

Thenretical Rayleigh Distribution

E 41.3 (degree)2 Eoperimental Histogram z Variation Ia P11thAn0le dug p Prnb40lIIty 99osIty (u,i dupe,) 2.z P Probability P_fee o -e Standard Deviation nl Fiantiln C ayant Limit ute a-n Lomar 1.init nl.

!I/!ii

e

p N 6Cnofid.oce Llmil3. dna

0.5 0.0210 0.050 0.2186 0.12 0.29 1 0.0413 0.0206 0.2236 u o.iu 2 0.0715 0.0809 0.2290 2.23 1.77 3 0.9046 0.1729 0.2353 3.24 .2.76 4 0.1193 02923 02482 4.25 3.75 5 0.1244. 0.4097 0.2572 5.26 4.74 6 0.1190 n.5294 0.2730 6.27 5.73 7 0.1051 0.6441 0.2965 7.30 6.70 r 0.0875 0.7107 0.3200 6.33 7.67 9 0.0608 0.8198 0.3644 0.36 6.02 10 0.0511 .0.8796 0.4160 10.42 9.59 11 0.0302 0.9221 0.4000 11.48 00.52 12 0.0243 0.9519 . 04732 02.51 11.53 14 0.0118 0.9784 0.8019 14.80 1320 16 0.0052 0.9911 1.1756, 17.18 14.82

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-i - F(x) =

Thçrefore if the variáble. under study is èxpressed

in terms of the standardized variate r then the

function [1 e-J will represent the cumulative

dlstnbutlofi of all Rayleigh distributions This

cumuläthre-diStributiön ftmction will plot ás a

straight line if the logarithm of [1

- P(x1)],

that is, the probability of exceeding x, is plotted against r2 as has been done in Fig 3.

It should be mited thát the methOds of

sta-tistics are strictly valid only if the measured

val-ues are independent samples.

The physical

20

o15

a-25

data are not, in general, independent of one

an-other However if the tests of significance indi cate, on the basis of assumed independence of all

observations, that these. data reasonably can be assumed to derive from the distribution speci-fied by tIte hypothesis, theñ this conclusion is even more sound than the tests of sigmficance

indicate.

SouÌ.cÈs 0! DATA

Siresses

Service stresses were Obtained fòr the follöiving ships:

i

The'SS Esso Asheville, a T-2 Maritime

Coin-mission design tanker, which operated m

coast-wise service, between the. Unitéd StateS, Central

and South America.

2 The SS Gopher Mariner, a C 4 Maritime

Commission design drr-cargo vessel, which oper-ated in the transatlantic service.

3 The SS Ocean Vulcan (7), a dry-cargo vessel. No Specific data are available to the author as to,

its area of 9peration.

4 The USS Fessendên, a destroyer escort, oper-ating in coastal waters off the U.S. East Coast.

5 The USCGC Unimak, a fòrmer tJ.S. Navy AVP class vessel, now 'fitted as,a weather ship,

operating' 'at weather station B in'

the' Nofth

Atlantic Ocean. Some data also were obtained

for a sister

ship, the USCGC Casco

These

vessels are approximately 300 ft long and displace

about 2500 tons.

6 A destroyer approximately 380 ft long

oper-ating in the Atlantic Ocean.

7 An aircraft carrier ,approximately 900 ft

long operating in the Atlantic Ocean.

The actual measurements of strain obtained

Test Cooditions: Sea State 5, Quarter Following Seas,

Ship Speed 7 knots, Significant

Vlave Height 2i ft.

The ship pitched at a rate of about 370 variatinos per hour. The test oample consisted of 175 variations in pitchangle.

O i 2 3 4 '5 6 '7' 8 9 10 II' 12

Vàriatios in Pitch Angle in degrees

FIG. 4 JSCGC TJNIMAX, DISTRIBUTION or VAxIATI0N IN PITCH ANGLESAIOLE 2 (This sample represents the poorest fit to the Rayleigh distribution of 129 samples that were

an-alyzed.)

8

Thenreticnl Rayleigh Distribution

-E 20.2 (degreè)2

_Eoperiirrental Histogram

for these ships, excepting the Ocean Vulcan (7),

were made by the U.S. Navy's 'David Taylor

Model Basin. In many cases these

measure-ments were obtained by means of automatic

instrumentation which did not require the

pres-ence of an Operator' aboard the ship., The strain

gages were installed on the strength deck, near

aniidships, see Fig. 1. The tests of six of these

ships (Gopher Mariner, Esso Asheville, Unimak, Fessenden, Destroyer, Ca±rier) covered a period

'of ábòut 3 to 8 months, predominantly in the

winter season; the' Oceañ Vulcan data 'however cover a period of 23 years.

Motiòns

Ship motions were obtained by Taylor Model

Basin personnel on the Esso Asheville, Casco,

'Unirnak, a destroyer, and an aircraft carrier.

The motions were recorded by means of an

auto-matic recorder which took a record at preset

intervals of time for a. predetermined dation of

time. The complete tests on the Esso A sheville;

Casco and Unimak are described in references

(12)

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(13)

Wave Data

Observation of sig!iificant wave height8 and wave period, obtained over a period of 6 years, have been macle available by the U.S. Wèather

Bureau. These, observations, were made every 3 hr by trained weather ôbservers in accordance with instructions prescribed by the U.S. Weather

Bureau (8).

Only one quantitative measure

ment was reported each time the sea was

ob-served, Darbyshire (9) obtained measurements

of wave heights, at Ocean Stations I and J by

means of a wave meter which has been developed

in England. Thus one set of measurements is

available which may be used as a check ofwave

height observations reported by weather ships

ANALYSIS OF SHORT-TERM DISTRiBUTION

FÜ IoNs

It is the purpose of this section to show that,

on the basis of the experimental evid4ce, wave

induced motions andi stresses of ships are dis tributed according to, the Rayleigh distributioi

with a high degree of statistical significance pro vided the sea conditions, ships speed, .and head ing remain uíiiform' throughout the period uncle:

consideratibn.

it has been accepted generally that the height of ocean wavés passing any given point follow th

Rayleigh distribution, provided the associatec

power spectrum has most of the energy con

centrated ia a narrow frequency band. This dis tribution is valid if measurements of wave heighi

are taken over a relatively short period of time

of the order of i hr, during which interval the se

conditions do not change appreciably.

It car

be shown that this distribution is the same a

that representing all :wave heights, in the arer under consideration, at one instant of time. Thi

statistics of the Rayleigh distribution were applied

by Longuet-Higgins (10) to the analysis' of the distribution of wave heights in a sea

character-ized by a narrow spectrum. This work was

ex-tensively applied by Pierson, Neumann, and

James (11) to the study of ocean waves.

Inasmuch as hull stresseS and motióñs are

caused by the passage,of the waves past the ship

it will be appropriate to test the hypothésis that

the responses of the ship to the waves, Le., stresses

and motions, may similarly be Statistically répre-sented by a Rayleigh distribution.

For this purpose a number of tests9 were run

0Significant wave height is the average height of the upper third

highest waves.

'Each test yielded a continùotxs '1,-hr oscillogeam record of ship motions and stresses applicable to a specific combiflation of condi-tions.

10

TABLE 3 Cm-SQu TEST APPUED TO TEST GOODNESS

OF FIT OF RAYLEIGH DISTRIBUTION TO TEST RRsui.rs

i

r

s

- on the USCGC Unimak under a wide variety of combinations of sea, speed and ships heading to the waves. The hull stresses, pitching, rolling,

L

and heaving motions were measured for 39 sets of L these combinatioñs.

The parameter E which

defines the Rayleigh distribution

p(x) =

was comptted for each of the frequency

distri-butions of motions and stresses measured under the various combinations of environmental con

ditidns. The individual histograms were plotted

and compared with the Rayleigh distributions

corresponding to the computed value of E. A

typical. distribution, thus determined,, is shown

in Figs. 2 and 3. The pattern which evidenced

the poorest'. agreenient' between the

experi-mental and the Rayleigh distribution is shown

in Figs. 4 and 5

It has been shown in the section "Statistical

Background," that all Rayleigh distributions may be represented by a single analytic expression if a

new variable y is used rather than x directly. The

cumulative distribution function then plots as

Sample Tested -Clans Linrits Numbqr xl Variations Measured Expected Number al 4tons -VatiS f Ch, Square Remark,

-ANIMAR 0-. 3 dil 40 48 i.3ä I.2 6.4

Pitcir Angie 3- 6 64 77 0.64 Good Fit See Figuies 2,3 6- 8 49 50 0.02 P 0.15 8-10 33 33 0 10-14 27 26 0.04 04-89 5 2 4.5

UNIMAK 0- 2 IJ8deg 46 35 3.46 I:ii.2 - OLI

Pitch Angle 2 1/8- 31/8 25 33 1.93 Fair Fit See Figuren 4, 31/8- 40/8 41/6- 5016 31 23 36 28 0.26 0.89 P - 0.92 51/8- 61/8 14 20 8.80 6018- 81/8 27 19 3.36 81/8-Ill/B 6 6 0 DESTROYFR 0- 1 dog 14 16 0.25 £.2 .4.4 Pitch Angie

1- 0 34 41 1.20 Viry Good Fit

See Fignren 6,7 2- 3 54 0.02 °- 0.30 3-4 4- 5 56 45 54 43 0.07, 0.09 5: 6 37 30 ' 1.02 6-7 , 1-10 1g II 17 15 0.06 1.07

AIRCRAFT CARRIER h 5 dog 43 45.96 0.19 Z2 2.8

Pitch Aogle

5- 9 76 69.59 0.59 Very Good Fit

See Figures 8.9 P-0.30

9-12 36 43.69 1.35

12-15 32 27.93 0.59

15-18 II 16.03 9.04

- 18-23 8 7.42 0.05

AIRCRAVT CARRIER 0- 750 psi 35 28.75 ' 1.36 I? -6.8

Etici, in Muir Reck 750-tobo ' 68 66.47 004 Good FiL

See Figuie 10' 10 P. 0.75 1500-2250 2250-3000 61 31 66.40 41.29 0.44 2.58 3000-3730 25 18.97 0.92 3750-5500 10 8.13, 0.(3

(14)

TABLE 4 USCGC UNIMAK-PITCH ANGLE-COMPARISON OF PREDICTED

MEASURED MAXIMUM VALUES

-25

a straight line on the type of' probability chart that was devised in the section mentioned (see

Fig.3).

A statistical test of significance was pplied

to the hypothesis: "The variátions of the

meas-ured sample from the assumed population,

specified by E, are no larger than may be

ex-pected according to the laws' of chance."

Aside from the deviations attributable to chance

it was necessary to consider the expected

vari-Procedure for Prediction st Probable Maaimuo Value Orne a Rayleigh Distribution

Let o mio - most probable mavimum value oto tabee trum a sample cnntaining-tl values at o Then according to Loveuet-Higginm

ornai _t16314

where '6. lOBat4 - toge - .! (o_e.6)]

Sample Size IO 20 50 loo 200 -5go 1000 2000 5000 10.000 20,000 50.000-100,000 °mao l4 0.707 l.0'30 1.366 1.583 1.770 2.000 2.172 -2.323 2.509 - 2.642 2.769 2.929 3.044 3.155 3.296 3.430

Theship pitched it a rate of about 400 variations per hour. The teat sample consisted of 270 variations in pitch angle.

Test Conditions

Significant Wave Height, 14 ft

Head Seas

Ship Speed,15knots

Theeretical Rayleigh 0stribotion

-- E 17.1(degree)

-Experimental Histogram

11

ations due to the inaccuracies- of- the measurement

process. The variances of-. the- fratiles- for the

sample under consideration were computed

ac-cording to the theorem stated in the section cited.

Fig. 3 which represents the results for a typical

test indicates that the measured data fall well

within the 90 per cent limits; a second set of curves,

the 67 per cent limits alsó was computed and the

data satisfy this more restrictive test. Thus a

high degree of confidence can be placed in the

by-sea State - Sigo Baue Height -Heading at Waves Relatime lt 56,9 'Ship -towed knots Sampling Tiere -min litai Neat Vaniationi in sample. N Cuerputnd Value at E, (deg)2 Ueamured Oie Value' deg Predicted Mae Value - deg 2 6 - Otead Seas 10 30 240 - 4.00 4.0 - 4.7 -- 14 32 296 4.46 5.3 - 5.1 Quarter 10 31 249 1.97 2.9 - 3.3 Heed Seas 14 32 '214 1.06 3.3 3.2 - Beam Seas lt - 2966 276 5.40 0.3 5.5 II 32 343 9.25 5.1 4.7 Qaarter Follow ng 14 - 29 262 1.68 3.9 3.1 ' Ftllowmrg sean 14 32 - 308 1.06 3.6-- 3.3 -3 7.9 - lead Seas -10 26 253 10.35 7.3 7.6 - 14 - 32 321 9.62 7.3 1.5 07 27 277 9.59 -6.3 7.3 Quartmr IO 213 249 1.79 3.1 3.2 Head Seam 17 37 345 2.02 5.9' - 3.5 Beam Seis 10 29 262 3.52 - 4.3 -4.4 - Following 17 31 210 1.85 3.9 3.2 Seas Following Seas-17 -23 193 1.20 2.4 -- 2.6 -4 16 -Head Seas 14 34 289 20.12 10.3 12.2 5 21 Read Seas 79 276 206 47.32 10.0 16.0 18 24 235 56.00 15.5 17.5 Quarter 716 - 3616- 347 35.10 13.1 14.3 Head Seas 14 2016 274 - 47.14 . -16.5 16.6 940810 714 30 102 16.76 -0.8 9.3 - Quarter 714 2866 175 20.24 9.5 10.4 Falloreurg Seam 10 29 143 2e.00 11.5 11.8 14 27 136 17.2 8.3 9.2 Following 714 33 160 31.12 12.5 12.6 Sean 14 31 040 20.54 - 10.5 10.1 4 5 6 7 10 11

Variation ia Pitch Angle--in degrees

FoG. 6 DISTRIBUTION os' VARIATION IN Pucia ANGLE. DESTROYER

'5.

(15)

s-a, -C)

PP P

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(16)

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FIG. 8 DISTRIBUTION OF VARIATION m PitcH ANGLE, AIRCRAFT CARRIER

TABLE 5 STATISTICS op LOGNORMAL DIsmmurxoN OF WAVE-HEIGHT OBSERVATIONS IN THE NORTE AThANTIC OCEAN

pothesis that this sample was taken from a Ray-leigh distribution., Consideration of the vari-ances due to instrument errors would have

re-sulted in wider confidence limits, Table 2;

As a further check on the validity of the

hy-pothesis, the chi-suare test was applied to the

grouped data, in accordance with the method of

chapter 23, reference (12).

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rather low value of chi-square, Table 3, thus indi-cating that there are no inconsistencies between the test data and. the hypothesis.

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poor-est set of data obtained in the sea tpoor-ests of the

Unitnak. In this case also there are no signifi.

Test Conditions: inificant Wave Height. 16 lt

Head Seap

Ship Speed,About 6 knots

The ship pitched at a rate of about -320 variations r hour.

Thé test sample consisted of 210 variations in pitch angle.

Experimental Histogram

2 3- 4

Variation in Pitch Angle in degrees

13

Theeretical Rayleigh Distribution

E 6.3 (degree)2

cant differences between

test data and -

the

hypothetical Ray1eih distribution on the basis

of either the chi-square test or the variation in the

distribution of quantiles., - -. - -

-- Inasmuch as none of the more than one hunthed

distributions of ship motion or htill stresses

(longi-tdinrsl,' hull-girder bending moments) indicated

significant deviations from the hypothesis, this

hypothesis may be accepted confidently for ships

of-the type tested.

-Thus fär only- the results of the Unitnak sea trials have been discussed. A similaz analysis

also has been made of the results óbtained from

sea tests (in the Atlantic Ocean) on a destroyer

-:::

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(18)

r-and a large aircraft carrier. Figs. 6 through 11 show typical distributions obtained from the sea

trials of the destroyer and carrier.

It is agâin

evident that there is no significant difference be-tWeen the measured data and the hypothesis that

the data are samples taken from a Rayleigh dis-tribution..

It will be of interest to compare the largest

measured value out of a given number of

meas-urements with the most probable largest vàlue

predicted by the statistical theory.

Longuet-Higgins shows that the largést probable value

x. out of N-measurements is /E times a

con-stant if the population is of the Rayleigh type

where the constant is a function Of N oniy For

N large this constant becomes approximately

equal to /iN. Table. 4 gives the comparison

of predlicted and measured values for a number of tests. There appears to be satisfactory

agree-ment between rneasureagree-ment..and predicted values.

On the basis of these data, applicable to a wide

variety of sea conditiOns and ship types, it is

concluded that the wave-induced 'motions and

stresses of ships are distributed according to the Rayleigh distribution, for any given set of steady conditions of the seai ship speed and heading.

ANALYSIS OF LONG-TERM DISTRIBUTION

FUNCTIONS

Background

It is the main purpose of this section to show that wave-induced hull stresses and, motions of

ships may be approximated by a logarithmically normal distribution if the environmental condi-tions under which the data are secured include a

wide variety of sea conditións, ship's headings

and speeds, such as a ship would be likely to

encounter over à period of weeks, months or years.

It is a secondary Objective to show that te

heights of ocean waves may be repres''ted by a

log-normal distribution.

When this study was initiated several years

ago, there were few data available on the motions

and hull-girder bending moments which ships

experience in service. It was realized that large quantities of data would have to be collected in order to provide a reasonably flrth basis for the

objective of this program which is the'

specifica-tion and predicspecifica-tion of the service stresses and

motions of ships Statistical methods were to be

utilied in the planning of the tests and the

col-lection, presentation, and analysis of test results. It was felt désirable, early in the planning stage, to have a preliminary idea of the general type of frequency distributions which might be applicable

as well as of the length of time over which

meas-15

urements would have to be made in order to

estab-lish the desired distribution patterns

Because 'tbe tresses and motions of ships are

caused by the passage of the waves past the ship,

it was thought desirable to study the extensive

wave-height observations that had been collected over a period of many years both in the Atlantic and Pacific Oceans with the. expeètation that the frequency-distribution patterns of ship response would, to some degree, be similar to those appli

cable to the wave data. Inasmi.ich as thè

wave-height data comprise visual observations, the dis

tributions will be evaluated in a qualitative

sense only; that is, it will not be possible to assign

statistical significance levels. The wave

observa-tions will be analyzed first, followed by the hull-girder stress and ship-motion data. Most of the

analysis will be concentrated on the hull stresses because much more stress than ship-motion data

have been measured to date.

Heights and Lengths of Ocean Waves

Fig., 12 shows a chart on which the several weather stations, manned by weather ships, are

marked.

For each of these stations the U.S.

Weather Bureau has collected observations of the

significant wave heightio and wave period,

ré-ported every 3 hr, over a period of about 6 years

in accordance with instructions of the U.S.

Weather Bureau(8).

Detailed study of these

observations mdicate that if the data are plotted on log-normal probability paper and if a log-'

normal distribution is fitted to the plotted data," very good agreement is obtained. Fig. 13 shows

the distribution for Station C

This curve is

typical. of the distributions obtained from the

U.S. Weather Bureau data., Station C, is the one

at which the sea trials of the USCGC Casco (i) were held.. The essential statistical data for sig

nificant wave height are given in Table' 5.

Com-parison of the distributions obtained for each

year with the sum of the observation overa period of about 6 years indicates that one year's observa tion gives a good approximation to the distribu

tion obtained over the lOnger period of time.

The détails of the comparison will not be given

here.

Recently J. Darbyshire (9) has obtained

meas-urements of wave heights by means of a wave

meter installed on a weather ship. The

measfire-ments were made over a period of about a

year-February 1953 to January Ï954at the North

lO The significant wave height is the meán height of the upper third highest waies.

Il The fitted log-normal distribution corresponds to the value of the mean and viriancé computed directly from the wave data.

(19)

a-45 40 35 g 30 25

l

20 15 10 , (Kipo \

Theoretical Rayleigh Distribution, E = 4.21I-2-)

Experimental Histogram

The ship experieoced about 460 variationS per hour.

-The test sample consisted of 230 stress variations.

Atlantic weather stations I and J, see Fig. 12.

Darbyshire reported the maximum wave height measured each time observations were made at

3-hr intervals, while the ship was at sea. The

visual observations made by weather observers, on the other hand, are reported as the significant wave height. It will be of interest to compare

the visual observations with the measurements that have been obtained with the wave meter. If the hypothesis is accepted that the short-term

distribution of wave height follows the Rayleigh distribution, then the maximum wave height and the significant wave height are related by a

con-stant factor.

Thus the distributions of

maxi-mum and significant wave heights should both be

of the same type, log normai in this case, and

differ only in their mean values.

The U.S.

Weather Bureau data indicate that the standard

deviation'2 of the Log6 (significant wave height)

is 0.62 at Station J and 0.61 at Station I as

compared to a value of 0.57 for the Log6

(maxi-mum wave height) for the

measurements at

Stations I and J reported by Darbysbire, Fig. 15.

The wave-meter data bave been fitted with a

log-normal distribution on the assumption that

the distribution of the maximum wave heightsis

log normal, Fig.

15.

The experimental data

indicate excellent agreement with the fitted

Test Conditions: Significant Wave Height, 16 it Head Seas

Ship Speed, About 6 knots

2 3 4

Stress in Main Deck a,midship, Kips per square inch

(1 Kip 1000 pounds)

Fic. lo DIsrRiiTIoN OF VAR-cATION IN THE LoNGrruDIN STRESS, MAIN DECK. AMIDSHIP, Antcso.AR-r CARIuER

12 The numerical values given here refer to wave heights measured 1* The numerical valueofthe constant (5.12) does not affect the

in feet. typeof distribution.

16

distribution, well within the accuracy of the wave measurements. The latter fact, together with the good agreement between the standard deviations

of the distribution of significant and maximum

wave heights, support the hypothesis that the

distribution of wave heights may be

approxi-mated by Rayleigh and log-normal distributions

for the short and long term respectively. It is

realized that the visual estimates of significant

wave heights obtained by the U.S. Weather

Bureau may not be accurate estimates of the

mean value of the third highest waves. But as

long as these estimates are proportional to the actual significant wave height, the form of the cumulative distribution (log-normal type) and its variance remain unchanged; only the mean

value would change since log (constant) x is equal to log (constant) plus log x.

Fig. 14 shows the distribution of the predomi-nant wave periods observed at Ocean Station B.

Here again the log-normal distribution appears

to provide a reasonably good fit. If these wave

periods are converted to wave lengths by means

of the formula,1° L = 5.12 T2, where L is the

length of the wave in feet and T is its period in seconds, which is applicable to gravity waves

in deep water, it may be seen that the wave

length will follow the log-normal distribution

(20)

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(21)

FIG. 12

LocA'rIoN oi WEATHERSHIP STATIONS IN NORTH AT1ANTIC OCEAN

80 70 60 50 40 20 10 o io 60

-.

'

CANADI IRELAND

-..

-r r 50

____

50 K M.., g 40 g -, 70 60 50 40 30 20 1 0 10 20

(22)

volving only a change in scale from the

distribu-tion of periods, Fig. 14.

Observations of significant wave heights in

the Pacific Ocean made by merchant vessels

have also been studied (13).

Although these

data are not as extensive or systematic as the

Weather Bureau data for the North Atlantic

Ocean, the indication is that the cumulative

distribution functions of significant wave heights

in the

Pacific Ocean are approximately log

normal.

The log-normal distribution of "significant"

19 99.99 99.9 99.8 99.5 99 70 60 50 40 30 20 10 5

wave heights may be thought of as a distribution of E where each value of E corresponds to a

defi-nite sea state which may be alternatively speci-fied by a significant wave height, if it may be

assumed that the Rayleigh distribution holds for

the distribution of individual wave heights in a

given sea state. This assumption appears to be

justified (11). On the basis of this assumption it

is possible to arrive at the long-term distribution of wave heights (as apart from significant wave

heights) in accordance with the following

ap-proach.

__i_t H

'

:i

Wave Heht) 'I

L

, . I I

':1 }i

. I .

. Experimental Data (Significant

X Computed Fractiles of Distribution o Individual Wave Heights

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¡ji ::: :: : : L ne Line ® represents observed were of 4 from ® represents derived by line the obtained years. from A. the values observed the the

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of significant wave height which 11l

at intervals of 3 hours over a period jj

The position of the line was computed Ii" data.

distribution of individual wave heights

¡

significant wave heights represented

:=4ï , !W11h13' JVi Ag i

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Wave Height Crest to Trough, ft

FIG. 13 DxsrituTIoN ø ii HEIGHTS OF OCEAN WAvEs AT WEATHER STATION C, 52° N 37° W,NORTH ATLANTIC

OCEAN

(This distribution is based on 12365 observations made over a period of 4'!, yr by U.S. Weather Bureau personnel.)

0.01 0.05 0.1 0.2 0.5 1 2 = w e Q-10 Do w = a 20 Do = 30 u 'C w ' 40 50 60 70 80 90 95 98 95 90

(23)

Mathematical Synthésis, of Cumulative. Long-Term Distribution of Wave Heights Based on

As-sumption of a Log-Normal Distribution of

Sig-nificant Wave Height. Let the instantaneous

dis-tribution of the wave heights pertaining to a

particular sea, be defined by the Rayleigh

dis-tribution, thus

p(x)

where14 E1 is the mean square value of 1l the

individual wave heights comprising the ith sea.

Let E

= cEV be the significant wave height

where c is a constant. The distribution of E is

log normal, thus

TABLE 6 SS Esso ASHEVILLE-BASIC DATA ON WAVE-INDUCED STRESSES FOR THE PiuoD SEPTEM-BER 1, 1953 TO APRIL 3, 1954

p (log E)

i

_(logÊ_u)2

--e

2e1

u/2ir

where CT2 is the variance, and u is the mean vahe

of log E. Therefore

ilE is also eqúal to the total area under the power spectrum. See

references (14, 15) tor a discussion of the power-spectrum eoñcept.

20

p(Ê)=p(iog)

and the probability P that x take on a value x is E=r P(x1) (logÊ

--e

2'

CTE V'2ir X = Xi

f2

2 -C2X1 - e dx dE x=O

The second integral is integrable, thus Ê=oo Ê=O -(logÊ - u)2

f

£

e

-[i - e

Ê' ] dE

The integral may be evaluated numerically by

a summation, thus j=N P(x1)tr

E

f.[1-e

xsl/EI] J=1 -.f!.

-

. St

. Numberof Stress Cycles Falling between Stated Limltsf (lar a 30.Dly Period at Sea)

For a Sample of2 min Taken Every Hour FaraSarople of2min Taken Every Fourth Hourtt

'' q

w E ,

!..

!IIL b7 !,

!,

E

L

§-

L

§ ER

L

O w .. - o w

-

,,,, .,, Stress A L 15 59 79 2.832 1:208 221 31 3 0 0 - -. -Stbi E L 15.25 18.422 3370 473 46 0 0 A L 68 40.9 60 2,136 - Ï,226 206

'

4 E L ,, 80.89 99,171 51,315 I6,'6 .7,685 .*8 324 Stress A B 69 55 94 3 120 1690 142 14 0 0 0 Stbd E B 13.85 23,406 1,967 194 0 0 0 -A E 8 B 61.9 6L9 100 3,304 '. - 52.3 1,409 73,691 1,336 59,873 427 22,332 115 6,05 17, 889 5 262 Stress ' A L 75 65 87 3,120 ' 1,767 343 55 8 4 0 Port E L 13.85 24,473 4,751 762 111 55 0 A L 68 34.8 51 1,808 - - 987 706 222 71 0 1 E L 9L53 90,340 64,620 20,320 7,048 0 92 Stress Port A, E B B 69 58: 84 2,784 15.52 1,442 22,380 280 4,345 38 590 2 31 0 0 0 0 -A 8 51.9 35.2 57 1,922 ' - 785 685 280 125 48 lt 'E B 89.90 79,752 61,582 25,172 11,238 4,315 1,618

The factors K mrd K2 are thefacbrs by which the cycles measured on the oscillograni samples are multiplied ¡À oTder to estimate the number ofotress cycles that would1have been obtained if continuous measurements had beso tateb over a period of 30 days at ses.

jPeat-to-pet variation.

tiEver, fourth sample recorded was utilized in this wialysis.

P(x)

=

f

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