HYDROMECHANICS STATISTICAL DISTRIBUTION PATTERNS OF OCEAN WAVES AND OF WAVE-INDUCED SHIP STRESSES AND MOTIONS,
WITH ENGINEERING APPLICATIONS
o
AERODYNAMICSo
FRUCTURAL MECHANICSo
APPLIED MAThEMATICSNw'Y DEPTMEW
by Norman H. JasperRESEARCH AND DEVELOPMENT REPORT
1b. v
Scheebot
c.
:'
STRUCTURAL MECHANICS LABORATORY
October 1957 Report 921
S
)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.
SOURCES OF DATA
Stresses
Motions Wave Data TABLE OF CONTENTSPage
INTRODUCTIONi
STATISTICAL BACKGROUND 3ANALYSIS OF SRORT-TERM DISTRIBUTION FUNGFIONS
ANALYSIS OF LONG-TLRM DISTRIBUTION FUNCTIONS 15
Background 15
Heights and Lengths of Ocean Waves 15
Stress Variations
22Esso Asheville
22 USS Fessenden 29 SS Ocean Vulcan 29 SSGopherMariner 29 Destroyer 29 Aircraft Carrier 31 USCOCUnimak 31 Ship Motions 31APPLICATIONS OF STATISTICAL DISTRIBUTION PATTERNS 33
Specification of Engineering Requirements 33
Prediction of Life Expectancy of Structures
36Prediction of Optimum Operating Conditions 37
DISCUSSION 38 SUMMARY 39 ACKNOWLEDGMENTS 40 BIBLIOGRAPHY 40 DISCUSSION 41 11 8 8 8 10 10
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 motionsand 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 andstresses in ships obtained under a wide
range of operating conditions were studiedINTRODUCTION
It is probably correct to state that the
ship-building industry knows less about the serviceconditions 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 ofperti-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
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 Nis 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 ofz. 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
limited by the assumptions - required by their
mathematical analysis.
The purpose of this paper is to present evidence
that the wave-induced pitching,
rolling, andheaving 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 ofthe 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 ofthe first type.
Finally, the extreme values ofship 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 datawere 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 amplitudesof 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 muchto make this research possible.
The specialinstrumentation 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 tothe hypothesis that the experimental data are
samples from a specified distribution. The re-sults of the analyses are discussed andapplica-tions of the results to practical problems are
illustrated, followed by a summary of the majorconclusions.
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 motionthe 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.
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
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 measureddata 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 approximationto 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 ninetenths 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 thelimits 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.
Inpractice 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 statisticaltests 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.
P(x)
=1
p(x)dxwhere
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-
,.-
. 2r-
i Ç' rresponding 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
JThenretical 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
ep 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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The physical
20
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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 MaritimeCoin-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' NofthAtlantic Ocean. Some data also were obtained
for a sister
ship, the USCGC Casco
Thesevessels 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 thesemeasure-ments were obtained by means of automatic
instrumentation which did not require thepres-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 anauto-matic recorder which took a record at preset
intervals of time for a. predetermined dation oftime. The complete tests on the Esso A sheville;
Casco and Unimak are described in references
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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 developedin England. Thus one set of measurements is
available which may be used as a check ofwaveheight 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 heighiare 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. Thistatistics 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 shipit 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 shownin 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
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 beex-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.
s-a, -C)
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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 ofchapter 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.
Figs. 4 and 5 show the distribution for the
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 -
thehypothetical 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
-:::
SigLwnveHigh! Mill, s"ee°uho i,2)ut Sgi,WaveH:ight o/io. 6/54 A 6'05N woo'v izati 6.34 1.0472 0.4524-1/49-12/54 8 56'391 Si'OO 15,587 6.50 '1.8858 0.4434 0/49-12/54 C 5r4019 ss'ao'e 16,857 6.75 1.9100 0.3763 0/49-12/54 D 44'0O.M 4I'00'V 16,804 6.26 0.8338 0.3643 -1/49-12/54 E - 35'OO'N 48'O0 16,777 6.56 0.5164 0.3765 1/47. W53 I 1'O0il 15'208 11.276 7.36 L9066 0.3767 0147 6/53 J 52°308 2000M 12.016 - 7.40 2.0021 6.3853 0/49-12/53 8 45'0FM 16'Dfl 11082 6.20 1.6240 0.3033 0149-12/53 U 600011 02'O(E 14,324 4.00 1.0076 0.2344 0/49- 6/54 H 36'ODhI 20'0V 01,007 - 5.08 1.6250 0.4237 -. ellacb .bat,.!!.. Cbr.CbT!n.. -. s!.t.. All lc(rftb re SUn! !0U1a
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IIIlIllIIlllllIlII lu lUll IlIllIllIllIll w Ill IUIIIIUlIIIIlIIIIII cv C.J.1 = 0I øI C Oai ai ai ø ci
c C.J CVi -co ue3ied d - I uo3J3d dco r co ai
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co u, -cv c-J=
=
co co u, ai C, o, a, C, o o. 0. cor-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 ofmeas-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 ForN 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 satisfactoryagree-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 andstresses 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 alog-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 assignstatistical 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 theseobservations 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 distribution obtained over the lOnger period of time.
The détails of the comparison will not be givenhere.
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.
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 at3-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 atStations I and J reported by Darbysbire, Fig. 15.
The wave-meter data bave been fitted with a
log-normal distribution on the assumption thatthe 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 distributionsfor the short and long term respectively. It is
realized that the visual estimates of significantwave 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 wavesin deep water, it may be seen that the wave
length will follow the log-normal distributionin-,
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IUa3Jad d r-. T -==
d ¿dd
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c.1 U, (D r-. 0)-4FIG. 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 20volving 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 thesedata 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 heightsin 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
fljIJ
I1"'
.
..uut '
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, . o percen in e process o integration ll plotted error numerical H: Z -. .::
j
.. :: -'r
::.vj:;, ::::
:::: :::: :::. . . ...--
___
____u1u ....::.nui.
. ... ::. -t :-r...
JIjj1JtI
¡ji ::: :: : : L ne Line ® represents observed were of 4 from ® represents derived by line the obtained years. from A. the values observed the thelog normal distribution fitted to he ¡II
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
t
r i
d!I1IIIUII
unii w
ii u nr nIl
Ir
IJHj: *.. :111.111.
= w t, wQ-n
e u, Do 0 80 u,wu M w o o .0a .0o Q-2 4 6 8 10 12 14 161820 30 40 60Wave 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
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 Rayleighdis-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 heightwhere 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
2e1u/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 = Xif2
2 -C2X1 - e dx dE x=OThe 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 !,
!,
EL
§-
L
§ ERL
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 1,618The 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)
=