Midship wave bending moments in a model of the mariner class cargo ship CALIFORNIA BEAR running at oblique headings in regular waves

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ARCHIEF

_

;Tr '

SSC-202

SHIP

DEL OF THE MARINERCL

HIP "CALIFORNIA BEAR"

LI

UE HEADINGS IN R

This document has been approved

for public release and sale; its

distribution is unlimited:

SHIP STRUCTURE

COMMITTEE

NOVEMBER 1969

Lab1 v. Schee -s

Technische-. . .

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SHIP STRUCTURE COMMITTEE

November 1969

Dear Sir:

The Ship Structure Committee has sponsored a

series of studies relating bending moments and strains

on towing-tank models to those on actual ships. The

enclosed report presents towing-tank data on the

Cali-fbrnia Bear

for which full-scale data has been

previ-ously obtained.

This report is being distributed to persons interested in the Ship Structure Committee's work.

Your comments would be appreciated. Sincerely,

C. P. Murp y

Rear Admiral, U.S. Coast Guard Chairman, Ship Structure Committee

MEMBER AGENCIES: ADDRESS CORRESPONDENCE TO:

UNITED STATES COAST GUARD SECRETARY

NAVAL SHIP SYSTEMS COMMAND SHIP STRUCTURE COMMITTEE MILITARY SEA TRANSPORTATION SERVICE U.S. COAST GUARD HEADQUARTERS

MARITIME ADMINISTRATION WASHINGTON, D.C. 20591

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SSC-202

Technical Report

'on

Project SR-165

"Bending Moment Determination"

to the

Ship Structure Committee .

MIDSHIP WAVE BENDING MOMENTS IN A MODEL OF THE _{MARINER-CLASS}

- CARGO SHIP

CALIFORNIA BEAR

RUNNING AT OBLIQUE HEADINGS IN REGULAR WAVES

by

E. Numata and W. F. Yonkers Stevens Institute of Technology

Hoboken, New Jersey

under

- Department of the Navy

NAVSEC Contract N00024-67-C-5218

This

_.document

_{has been approved for public}

reiedseand

ite

distribution is unlimited. U. S. Coast Guard Headquarters

Washington, D. C.

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-Vertical and lateral wave bending moments were measured at

the midship section of a 1/96-scale model of the C4-S-1A MARINER-class cargo ship CALIFORNIA BEAR.

The model was self propelled

through a ship speed-range of 10 to 22 knots at seven headings to

regular waves of lengths between 0.2 and 2.0 times the length

be-tween perpendiculars; moderate wave heights not exceeding 1/50 of

the model length were used Results 6re presented in charts of bending-moment-amplitude/wave-amplitude versus ship speed, with

wave length as the parameter

l'tlm ship loading conditions, rep-resentative

of actual westbound and eastbound trans-Pacific

voy-ages are covetsed.

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APPENDI X

f-,

INTRODUCTIONF

-DESCRIPTION OF THE _EXPERIMENT"

DISCUSSION . . .

-CONCLUOI,N:G.;.REMAR'KS . , .

.:ACKNOWLEDGEKtNTs.

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The SHIP STRUCTURE COMMITTEE is constituted to prosecute a research to

improve the hull structures of ships by an extension of knowledge pertaining to

design, materials and methods of fabrication.

RADM C. P. Murphy, USCG - Chairman Chief, Office of Merchant Marine Safety

U. S. Coast Guard Headquarters

Captain W. R. Riblett, USN Head, Ship Engineering Division Naval Ship Engineering Center

Captain T. J. Banvard, USN Maintenance and Repair Officer Military Sea Transportation Service

SHIP STRUCTURE COMMITTEE

SHIP STRUCTURE SUBCOMMITTEE

The SHIP STRUCTURE SUBCOMMITTEE acts for the Ship Structure Committee

on technical matters by providing technical coordination for the determination

of goals and objectives of the program, and by evaluating and interpreting the

results in terms of ship structural design, construction and operation.

NAVAL SHIP ENGINEERING CENTER U. S. COAST GUARD

Mr. E. S. Dillon

Chief, Division of Ship Design Office of Ship Construction Maritime Administration Mr. D. B. Bannerman, Jr. Vice President - Technical American Bureau of Shipping

Mr. J. J. Nachtsheim - Chairman

Mr. J. B. O'Brien - Contract Administrator Mr. G. Sorkin - Member Mr. H. S. Sayre - Alternate Mr. I. Fioriti - Alternate MARITIME ADMINISTRATION Mr. F. Dashnaw - Member Mr. A. Maillar - Member Mr. R. Falls - Alternate Mr. W. G. Frederick - Alternate AMERICAN BUREAU OF SHIPPING

Mr. G. F. Casey - Member Mr. F. J. Crum - Member

OFFICE OF NAVAL RESEARCH

Mr. J. M. Crowley - Member Dr. W. G. Rauch - Alternate

MILITARY SEA TRANSPORTATION SERVICE

Mr. R. R. Askren - Member

Lt. J. G. T. E. Koster, USN - Member

iv

CDR C. R. Thompson, USCG - Member LCDR C. S. Loosemore, USCG - Member LCDR L. C. Melberg, USCG - Alternate CDR L. A. Colucciello, USCG - Alternate

NAVAL SHIP RESEARCH & DEVELOPMENT CENTER Mr. A. B. Stavovy - Alternate

NATIONAL ACADEMY OF SCIENCES

Mr. A. R. Lytle, Liaison Mr. R. W. Rumke, Liaison Mr. M. L. Sellers, Liaison

AMERICAN IRON AND STEEL INSTITUTE Mr. J. R. LeCron, Liaison

BRITISH NAVY STAFF

Mr. H. E. Hogben, Liaison CDR D. Faulkner, RCNC, Liaison

WELDING RESEARCH COUNCIL

Mr. K. H. Koopman, Liaison Mr. C. Larson, Liaison

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2.

'

.LIST OF FIGURES

Body Plan of _CALIFORNIA

REAR-VertiCal and Lateral Bending Moment' Balance. 3. _{PrOfile-pf -CALIFORNIA BEAR}

(Photdgra0),, and

Berldihg.:MOmentiBal-ande

in Position

_Across

Model

Gut

:(Photograph) . . . .

. .

, .

- f

Weight Distribution 'Diagram (Displacement 16,840 Tns,

Drafts 23.70 ' Fwd, 25.55 ' Aft)

.

Weight Distribution Diagraffi (Displacement _{13,990 Tans;}

.

Drafts 19.25 ' Fwd, 22.55 ' Aft)

-Vertical Wave Bending Moments (Drafts: _19.25'

_rwO,

Aft; 20 Knots) . . .

. ... :

. .

Vertical Wave Betiding Moments (Drafts: 19.25' 'Fwd, 22.55'

Aft; '20 Knots)". . .

. ...

". . : .

-.Lateral Wave Bending Moments (Drafts 19.25' Fwd,,-.22.55!

Aft;. 20 Knotsr . . . " . . . .

Lateral 'Wave Bending Moments (Drafts: 19.25' Fwci_;,22,.S5

, Aft; 20 Knots) .. . .

..

10. _{' Vertical 'Wave ^Bending .Moments (Drafts:. 23.70'. Tw4i-;;2,$,,5}

'Aft.; 20 .Knots)

. . . .

Vertical Wave 'Bench twMoments (Drafts:. 23.7

Aft; . . . "

' 12, Lateral ::,Wave. Be-riding Moments (Drafts: 23.76'

rya,

25 55'

Aft; 20 Knot)

. . .

.

13. _{Lateral .Wave Bending Moments (Drafts':}

'23.701-Aft; 20 Knots)

14, _{Vertical WaVe.Bending'MOMenfs -''Oraffs: 19.25'}

Aft; 20 Knots)

... .

. . .

....

.

Lateral Wave Bending Moments (Drafts.: f9-.25' Fwd, 22L551

Aft; 20 Knots) . .

. .... .

. .

:

. .

Vertical and Lateral- Wave Bending. _moine'ili (Drafts : _23

Fwd, 25.55' Aft; 20- Knots)

17. _{Vertical and Lateral Wave Bending Moments (Drafts:}

23.7 ' _ Fwd, 25.55' 'Aft; 20 Knots) _13 12 12,_ 13

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-vi PAGE 14 14 17 17 17 17 18 18 18 18 19 19 19 19 20 20

Wave Bending Moments (Drafts: 19.25' Fwd, 22.55' Aft; 20 Knots)

Wave Bending Moments (Drafts: 23.70' Fwd, 25.55' Aft; 20 Knots)

APPENDIX FIGURES

Vertical Wave Bending Moments (Drafts: 19.25' Fwd, 22.55' Aft; 180° Heading)

Vertical Wave Bending Moments (Drafts: 19.25' Fwd, 22.55' Aft; 900 Heading)

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Lateral Wave Bending Moments (Drafts: 19.25' Fwd, 22.55'

Aft;. '

...

.

20

Lateral Wave Bending Moments (Drafts: 19.25' Fwd, 22.55' Aft; 150° Heading). . . .

Lateral _Wave Bending Moments (Drafts.; Fwd, 22.55

Aft; 120° Heading)

-Lateral Wave Bending Moments (Drafts:.19.25' Fwd, 22.55'

Aft; 120°- Heading)'. . .

' Lateral Wave Beni:hi-1g Moments (DraftS: .19.25' Fwd, 22.55'

Aft; 900 Heading) . . . .

lateral Wave Bending Moments (DraftS:' 1972

Aft;:'90°.Headin4), . .

.A-21. _{Lateral Wave .Bending Moments (Drafts: 19-.2} Aftl-- 606- Headi 6)- . . .

Lateral Wave Bending-: Moments., (Drafts: 19.25' TWd,

Aft; 60°.-.Heading) . .

Lateral Wave .Bending_ Moments (Drafts: 19.25! Fwd, 2255'

Aft; 30° Heading)- . . . -22

Lateral Wave Bending Moments (Drafts: 19.25' Fwd, 22.55' Aft; 30° Heading) . -.i: . -.- . . .

. ...

. _ . 22

_

Lateral Wave 'Bending _Moments (Drafts: 19.25' Fwd, 22.55'

Aft;- 00 %Headi ng): . '. . .

-23

Vertical Wave Bending Moments (Drafts: Aft:;1.180°..fleading).

..

Vertical Wave Bending Moments _{23.70t..Fwd, 25.55'}

Aft;"-180°-.Heading).':-. .

A-28. _{Vertical Wave BerldinT-Momerits.-(Drafts: 23.70' Fwd,}

Aft;, 1500 .Headif10:

A-29._ Vertical- Wave Bending Moments_ (Drafts: _23.70!

'Aft; 150° -Heading)'._{_}

- ,

Fw , 5.55'

Fwd., ,25..55':

Vertical Wave Bending Moments (Drafts : _{3.70" Fwd,'25.55!}

Aft; 120°

Headfmr.-.'

_ . . . .

' _A-31. _{Vertical Wave Bending' Moments (Drafts:}

23.70' Fwd, Aft;.120° Heading)... . .

- ... .

. . . . vii PAGE 20 . 21 22 23 23 23

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Vertical Wave Bending Moments (Drafts- '23'

Aft

90.0 .Heading):

. .... .

. . . -. .

A-33. ,yerl:'Wave Ben'ding Moiments (Drafts: .23.,7-0!-',Tyd',25;5

Aff, 600 Heiding): .

. . .

. ....

:

. .

-%Vertical - WiVe:Bendih§ .komens (Drafts

23 7O'Fqd, 25:55)

:"-Aft;' 60° Heading).

. . . .

Veftfcal

ertiC41.-WaveBeiidingM6Menis-Cbrafts::237

't;-30°'--Headiii-gf.:?-''

_ _,. _

. ,.

...

.

647tT.-17,'Wave Bending Moments (D:rafft,:. 2,

-..7,

ervOieadihy .-.:::-..---'

_.. _. 1- -...,ri..,.J,...-.:-.

',,

-.

:ii-'..., , ...,:4,,.::.,,,,

etfearWaVe-BendingrMoMent's (Drafts: 23.,

_, ,W4---k..':'

Aft, 0° Heading)

Lateral Wave Bending7Moments (Drafts: 23.70',.E,44,

'-'Aft;'18.0° Heading)'.,-.

. .

. ... .

-.- Lateral Wave Bending, Moments (Drafts: 23.70,', FWelc?:25,

'

Aft; 150°-Heading)

. .

. ... .

. .

-,

A741.

Lateral Wave Bending 'Moments (Drafts:

23.70'.Fw0;--Aft; 120°- HeaaingT.

.

...

. . .

.

,

_ .

Lateral Wave Bending Moments (Drafts;

Aft90°,

Heading)-Laterai-WaVe Bending Moments (Drafts.: 0:7pe,_

Aft; 60°. Heading'),;

. . . .

Lateral : Waiei.pendihg:M0ents

ft, 30° H0846,4).-

. .

,

.

LaterarNave.Bending=ftments (Drafts : 23,1

0° Heading)

'

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INTRODUCTION

In 1960 the Ship Structure Committee authorized Davidson Laboratory to initiate a research program entitled "Bending Moment Determination," using ship-model tests

to investigate hull bending moments in regular and irregular waves.

The initial phase of this program, Project SR-157, covered investigations of trends of midship bending moment as a function of wave steepness in models of

a MARINER-class cargo ship with variations in freeboard and weight distribution,

a destroyer, and (3) a tanker. _{Dalzelli'2 concluded that, within practical}

operational and design limits, no dramatic upper limit of wave bending moments at

amidships is to be expected as the ratio of wave height to wave length increases to

a value of about 1:9.

Since this conclusion was limited to midship bending moments, and it was known

that maximum moments under certain circumstances could _{occur elsewhere, the next}

phase of the study, Project SR-165, examined the longitudinal distribution of

bend-ing moments in a MARINER-class cargo-ship model _{in regular waves of extreme steepness.}

Maniar2'4 concluded that, within _{practical operational limits of speed for the MARINER,}

maximum wave bending moments would occur in the region from amidships to 0.125L aft

of amidships. _{Thus the practice of concentrating on midship bending moments both in}

design studies and in full-scale measurements _{appears to be justified.}

Another part of the investigation involved the testing of the MARINER model in

high irregular waves to obtain time history records of wave bending moment and wave

elevation. _{Wave and bending moment energy-spectra were computed and used to derive}

equivalent regular-wave bending-moment "response operators" which were shown to be in reasonable agreement with response results obtained from model tests in regular waves.

Such favorable agreement inspires confidence in the alternate procedure of

using a "response operator" from tests in regular waves to predict the energy spectrum

of bending-moment response of either a ship model in a known wave-spectrum or a ship

in a real seaway the energy spectrum of which can be determined. Before taking this

latter step, however, it is necessary to demonstrate satisfactory correlation

be-tween model and ship bending-moment responses.

Over a period of years, full-scale statistical data on midship bending-moment responses have beenicollected by the Teledyne Materials Research Company (under Project SR-153), on several vessels including (1) the MARINER-class, cargo ship

CALIFORNIA BEAR operated by the Pacific Far East Line in trans-Pacific service

between San Francisco and Yokohama and (2) the cargo ship 'WOLVERINE STATE _operating

on the United States-Northern Europe route in the North Atlantic Ocean. Davidson

Laboratory proposed the conduct of scale-model tests of these vessels at various

speeds and headings in regular waves, to obtain bending-moment _{response operators.}

' Webb Institute of Naval Architecture, in a parallel effort, proposed an analysis

of statistical data for the ships and the prediction of ship bending-moment

statistics from Davidson Laboratory's model-test results. The tests of the

WOLVERINE STATE model have been reported in Reference 5.

This report describes the tests of the CALIFORNIA BEAR model at Davidson Lab-oratory, under Project SR-165, and presents the wave-bending-moment results which have been used by Webb Institute in their model-ship correlation program under Project SR-177.

Two ship loading conditions were used in this experiment. Both were taken

from the reported cargo and tankage distributions for actual trans-Pacific _voyages

of the CALIFORNIA BEAR, on which the Teledyne Research Corporation had installed instruments to collect strain data on ship bending moments.

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-2-DESCRIPTION OF THE EXPERIMENT

Model

The tests were conducted on a 65-inch (1/96-scale) fiberglass model of a

stand-ard MARINER-class cargo vessel. This model was originally built for the tests

reported in Reference 3. Figure 1 shows the body plan. The model was cut at a

point corresponding to 260 feet aft of the ship's fore perpendicular. The two halves

of the model were connected by the standard Davidson Laboratory two-component

bend-ing-moment balance, illustrated in Fig. 2. This balance consists of two pedestals

with an integral flexure beam of cruciform cross-section milled from a single block

of aluminum alloy. The componental deflections of the beam in its vertical and

lateral planes of symmetry are mechanically amplified by linkages and sensed by

linear variable differential transformers. Figure 3 is a photograph of the model

in profile and a photograph of the setup of the bending-moment balance across the

model cut. The midships cut was sealed with a thin sheet of rubber fixed to the

hull with vinyl-plastic electrical tape.

An electric propulsion motor was installed to turn a stock four-bladed

pro-peller of approximately the desired scale diameter. The rudder was built to scale

and was operated by a servomotor which formed part of an automatically controlled steering system.

Loading Conditions

Pacific Far East Line made available complete records of cargo and tankage distributions for nine voyages covering two years of instrumented operation between

January 1966 and December 1967.

Two representative loading conditions were used for ballasting the model. For the first condition, representing a full load of cargo typical of a westbound voyage (San Francisco to Yokohama) of the CALIFORNIA BEAR, the loading condition

chosen was that of westbound voyage number 33 of October 1967. The mean draft

on this voyage was nearly identical to the average of the mean drafts of nine

westbound voyages, numbers 25 through 33. Figure 4 is a weight distribution diagram

for westbound voyage 33, and shows a displacement of 16,840 tons at a mean draft, at mid-voyage, of 24.6 feet with a trim of 1.8 feet by the stern.

The second condition was meant to represent a low-density, dry-cargo loading

typical of an eastbound voyage from Yokohama to San Francisco. The average of the

displacements of the nine sample eastbound voyages was 12,000 tons. This value

was too low to be simulated on the model. To facilitate proper ballasting of

the model, the cargo-loading of eastbound voyage 25 of March 1966 was chosen. Figure 5 is the weight distribution diagram for this voyage and shows a displace-ment of 13,900 tons at a mean draft, at mid-voyage, of 20.9 feet with a trim of 3.29 feet by the stern.

Reported cargo and tankage distributions were used to simulate each loading

condition on model scale. Each half of the model was ballasted to the desired

scaled values of weight, LCG, and pitch gyradius. Gyradius was checked by

sus-pending each model-half from a knife edge, oscillating it as a compound pendulum,

and measuring its natural period. The gyradius was calculated from this period

and the distance from the knife-edge to the CG. Tare weights were substituted for

the bending-moment balance to permit separate ballasting of the model-halves. The vertical center of gravity of each half was adjusted to a common value based

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-Profile of CALIFORNIA BEAR

Fig. 5. Weight Distribution Diagram Displacement: 13,900 Tons, Drafts: 19.25' 'Fwd, 22.55' Aft Loaded Ship L Light Ship. 71_ 500 A P1

Fig1:11:3.;'.Bending Moment Balance in

Position Across Model

, 4.. Weight Distribution Diagram Displacement: 16,840 . Tons -Drafts: 23.70' Fwd, '25.55'1-Aft , ' 200 Tans Foot 760 Ao 30 f .P. 0 10

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On the reported load ing.

.

Ballast was adjusted,,)aterally to obtain the scaled:-

.

value of tkenat'uralrolling period oft he complete model

_, _. _. .

,

-

.

- :The ,'ship

:characteristics-for the ,iwo ' load

.- .

conditions 6re:tabulated be low. 1

DisplaCernent,,..long tons

Draft

-- Fore ,pgrpendicOlar, ft

-Aft' Perpendicular,'-ft

Mean, ft

LOG .aft.,tf-amidshlP, ft

,

Pitch radius of 'gYrai ion, ft

NatUra I

_{rolling period, sec}

Forebody

b iSplacement, ,long tons,

'.CG forward tf

0";

Pitch gyradiU

ft

V CG above baseline, ft

Afterbody

D isplacement,.:Jong

LOG aft_of amidships,

Pi tch. gyred i

ft

, ₇

VCG.:6,13oVe baseline, ft

6,129

- 8,-079

105.0 ,98.2

66.5

-

_24:0

:: _27.O -,r--, ,- ----:. -',..

,-7,771

8,767

107.2

--

_ r-170.37,, -

71.4

S f,".:-._

,'

27.0

'0 `,.,C'r,7 , ,

LP44ULEE:.

-11:=,

TABLE OF SHIP CHARACTERISTICS

Length Between Perpendiculars.., ft.

.

528.0

Eastbound

13,900

' I

-. 14-.4 Westbound

-:16,840

23.70

25.5 .5

24.62

8.8

123.6

16.7

The e)cperimerit=iWas. CorlduCied

_{Labora'tory!s Tank 2 (75' x' 75! x 4,5").}

This facility ihbliYijet a Wiavernaker:a long

_{one 75-foot- side of the tank, a wave}

ab-sorber alOhb' the opposite side,and a:thbVab le bridge "spanhir6 the 'tank.' the...bridge

supports a' monorail-carriage - dr iven by 6'

_{servo-controlled -Motor.'}

_"

. _-W.

Suspended from the carriage is

_{a six-degree-of-freedom -,rflot ions apparatus which}

is servo-dr iYen,to .follow a. self-propelled,

_{automatically steered model in waves'.}

A 'Vertical heave. rod 'rides in. bear ings on the apparatus' and is attached to the

model through a threedegree-of=freeddm gimbal.

_{Power and ContrOl, wieis, for the}

rudder and propulsion mdfOrs

,

_{as well as signal cab les from the,:,bending-moment}

.

ba 1 ante

6 re led upward to the

_{carriage and thence to a recording and control station}

at -tankside.

_{The moment-balance output signals were fed through}

_{a Sanborn 350-1100}

Series carrier amplifier system, and then through

_{Krohn-Hite Model. 35 low-pass act ive}

filters to minimize high-frequency noise in the

records..

_{The signals were then}

dis-played by a v is i corder Model 1108

_{as time histories on osci 1 lograph chart Paper.}

-- r

,ine heights of -a 1 1

regular waves were calibrated before model

_{tests, by?}

traversing the "reading" section.. of carriage travel with

a resistance-type wave

probe at the model location./.'IThg,:resulting records, of 'WaveferevatiOn time history

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-6-were reduced to obtain the average wave height for each wave length at each bridge

heading. During each test of the model, the wavemaker speed-control was adjusted

to maintain the wave period used during the wave calibrations. This procedure was

preferred to the measuring of the wave elevation during each model run, because it is known that model-generated waves can influence wave-probe readings.

Test Procedure

The bridge was positioned for model headings of 180 degrees (head seas),

150 degrees, 120 degrees, 60 degrees, 30 degrees, and zero degrees. At each

head-ing angle, at least eight wave lengths were tested and a minimum of three speeds

in the range of 10-22 knots, full scale. A nominal wave height of 1/50 of the

model length was used for all waves except the relatively short ones, for which a

reduced height was substituted. The mean speed of the model was averaged over a

distance of about four model lengths (20 feet).

The bending moment balance was calibrated periodically by applying known moments to the model while it was afloat.

Data Reduction and Presentation

Time histories of vertical and lateral bending moments were reduced to obtain

the average range of moment during the 20-foot interval of model travel. The

phase between the two moments was also determined.

The moment data are presented in dimensional form (bending-moment-amplitude/ wave-amplitude in ft-tons/ft) as trends versus ship speed: for all wave-length/

ship-length ratios at a given heading angle; separate chart S are presented for

vertical and lateral moments. Figures A-1 through A-25 of the Appendix 'contain

the results for the 20,9-ft mean draft condition; Figures A-.26 through A-45 present

the results at 24.6-ft 'mean draft. The reported bending moments are due Solely to

wave-induced loads and are measured with respect: to a still-water datum.

Still-water hogging moments were 86,000 ft-tons at '20.9-ft mean draft and 70,000 ft-tons

at 24.6-ft mean draft; these are calculated values.

Phase is presented as the lag of lateral moment behind vertical moment for a

given wave-length/ship-length ratio. The, phase results are consistent with a

headinl-angle convention of representing head seas by 180 degrees and following seas

by' zero degrees. At intermediate heading angles, the waves approach the port 'bow

(150 degrees, 120 degrees), the port beam (90 degrees), or the port quarter (60

degrees, 30 degrees). Each phase angle is the lag of starboard lateral moment

behind hogging vertical moment, with a starboard lateral moment corresponding to

a hull-deflection curve which. is concave to starboard. The phase results are

presented' tabular form

in

Tables A-1 and A-Z of the Appendix; a single phase value iS given fOr'each combination of heading angle and wave-length/ship-length

ratio.

' DISCUSSION

The object of this'investigation was to obtain wave-bending-moment response

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-7-prediction-and-correlation project. The moment curves and phase results in the

Appendix are convenient and practicable for the purpose. Sample response-curves

for the average Pacific crossing-speed of 20 knots have been constructed from the charts in the Appendix and are shown in Figs. 6-13.

The double peaks in some of the vertical-moment curves, Figs. 6, 7, 10, and

11, have been documented by investigators at the St. Albans Tank in England (most

recently by Murdey ), but only for head seas. Murdey, with the help of

analy-tical predictions of the inertial and hydrodynamic components of veranaly-tical moment,

explains why the peaks occur. He shows that each component contributes one peak

to the total vertical moment. Fukuda,7 using an analytical prediction technique,

has found double peaks in vertical-moment response curves for other heading

angles.

A typical set of vertical-moment response curves, Figs. 6 and 7 for example, shows a consistent tendency for the moment peaks to shift to shorter wave lengths

as the heading angle changes from 0 or 180 degrees to 90 degrees. In 180-deg

head seas the peak moment occurs in a wave length about equal to the ship length. Thus it may be visualized that maximum sagging moment occurs when the trough of

such a wave is at amidships with wave crests near each perpendicular. To obtain

a similar wave-ship geometry, at a 120-deg heading, the wave length should be

about equal to the effective ship length (i.e. , L cos pw) where L is ship length

and

pw is heading angle.

Figures 14-17 show trends of wave bending moments versus wave length/L cosilw

for the various heading angles at a ship speed of 20 knots. This method of

plotting shows that the moment peaks generally occur at a constant value of wave length / L cos pw of about 0.9.

Figures 18 and 19 show cross-plots of moments versus heading angle at a value of wave length / L cos pw of 0.9; with certain exceptions, noted by

asterisks, these are curves of peak values. Data were taken at too few headings

to permit definition of the angles at which maximum moments occur. However, the

general headings at which highest values occur are head seas for vertical moment

and bow seas (about 130 degrees) for lateral moment. Both moments exhibit

secondary peaks in quartering seas (about 50 degrees). These trends are in

general agreement with analytical predictions for a ship of similar fineness, by Fukuda.7

CONCLUDING REMARKS

The primary objective of this experimental model investigation was to obtain wave-bending-moment data and to reduce them to a form usable by Webb Institute

for correlation with full-scale measurements. _{This has been accomplished and the}

general trends of the results appear reasonable when compared with published data. Analysis of the results shows that peak moments tend to

occur--At a constant value of wave length/effective ship length, and

In head seas for vertical moment and in bow seas for lateral _moment.

In view of the self-sufficient nature of these results, it is recommended

that no further model testing be conducted in connection with the CALIFORNIA BEAR project.

ACKNOWLEDGEMENTS

The authors wish to thank two persons who made significant contributions

to this investigation: _{Mr. V. J. Bahorich, Superintending Engineer, Pacific}

Far East Line, for furnishing an extensive amount of information on the loading statistics of the CALIFORNIA BEAR; and Mr. M. J. Chiocco, for conducting the model investigation of the westbound loading condition.

(18)

'Vertical Plomeist' AMPI I iude

Sove Amp I tude "."

10.0007 rr.'n 00 lieed ind = 0 , 5.000 . 5 2.0Wave 0. Length Ship Length Z

Vertical Whe pending.-Moments

prafts...;,..; 19.25' Fwd, -Aft Speed:

20 Knots

'Vert loll' riMiehi-!A'MP. fi tudre.

Wave: 0.5 . . ' TS' Ship Length :.-- ' =

Fig . 7. Vertical Wave Bendin4.:Mortient...

Drafts 19 25' Fwd,'-22 55' Aft- Speed

(19)

Lateral Moment Amplitude liave.Amolitude 5,000

-Nom

i3O00

Lateral 'HOmeni AMOiituda

- Wave Amplitude., Ft-Tons -Ft Ft-Tons Ft -30 Reading 0! Heading 5 1.0 1.5 2:0 Sh1p Length

Fig.. 8. Lateral Wave Bending Moments .

.Drafts : 19.25' Fwd, 22.55' ,Aft Speed: 20 Knots

no° Heading:,

150! Heading

Wave Bending Moments

Drafts : Fwd, 22.55' Aft Speed:

(20)

:riot)

Vert tail MomentAMPlitUde Amplitude:Jr Ft-Tons Ft Vertical Momeni-Amelltude

"

Amplitude.: -1.0 Wave , Ship Length

."c131-;if '"<"

'

.Wive .Bendfqg Som`rerits: .

firiftS:

58!".Aft., Speed:'

20 Knots

ding.

..-.,1800 Heading :I50°.lineding . i.0 Lnqth-Ship

Length--

sNertidal WaverBendin6' .Moments

Drafts

23 70' Fwd, 25:55'. Aft ::,Sfreed:

(21)

Lateral Moment 'Antal tude 'Wave Amplitude 5;000 Ft-Toni' Ft Ft _Toni:. 60'2" Heading _ 1500. Heidin

-.lie

Head ing

. Fig.

13.

_Lateral Wave Bending MpmeRts

D-Tftrs:.'2J.16'

.Fwd, 25.55';Aft Speed: Knots

"..

0° Head in

ad ing

Wave Length,it; Ship Length

Fig. 12. Lateral Wave Bending Mothents,,..:

Drafts : 23.70' Fwd, 25..551:Aft, SP-e0:

(22)

Wive Lek-U. Ship Length cos is

' HEADING

13!

0.5

Wes4 Lenetn

Ship Length cos ow

Drafts: 19.25.,FueS 22.55._- Aft SPoed: 20 knots HEADING CI.30° 1.5 1 2.0

, Fig. 15. ,La.teral. Wave Bending- Moments

Drafts: 19.25' Fwd, 2:551Aft

:Speed:-, 20 KnOtt, 'Lateral: -tiow_eitiv_pn Wave Areplittele .Lt roris ' t 8.0 6,0-00 4.000 '6,000 4,00 2.000

(23)

8,000 6,000- 4,000-2,000r La'teral MoaeniAmolitude' Wave Amplitude 6.060 ' 4,0 J13-6,000 8,000 2,000 , NeriiCal,AOMeni Amplitude Wav.o.Amplitude

Fig-16,.

Vertical-

and Lateral Wave

'Bending_MOifieritt Drafts: Fwd,

25.55: Aft . Speed:, 20 .Knots

Ft-Tons Lateral Moment-AnialitUde. Wave Amplitude, COO j54T4 Wave ''Lenciih" ViRTI-GAL. Ship Leng.th,cos,u.' -LATERAL 0 ,

Vertical 'moment Amislitude.

- Wave:Amplitude-

:-Ft-Tons

. 2,000:

Fig. 17. Vertical and Lateral. Wave Bending Moment t Draft: 23.-7_0' 'Fwd;,

25.55' Aft Speed: 20 Knots.'

HEADING '0 180° .13 Ise fie -` 2.0

(24)

Moment Amp1 tude

-Wave. AmplItinio.:

5,..cto0 Tt =Toni

Wave Length _0.9

Ship Length cos law

.

10,000 -

-LATLRAL' BENDINGI'MOMENT

VEAT I "CALBENDING'KOME.NT

*

150

Fig. 19... Wave .Bending'Moments,

Drafts: 23.70L-Fw0;,25.55!. _

20 Knot

180

Fig.1

-Wave Bending Moments Drafts: ,Fwd,,22.55' Aft Speed: 20 Knots

Moment AmplIttiie Wave Amplitude...

15,000 _Tt

.1.

VERTICAL BEND 1 gG MOMENT

- 'LATERAL BENDING MOMENT

(25)

-15-REFERENCES

-I. DALZELL, J. F., An Investigation of Midship Bending Moments Experienced in

Extreme Regular Waves by Models of the MARINER Type Ship and Three Variants.

Ship Structure Committee, Serial

SSC-I55,

January 1964.

DALZELL, J. F., An Investigation of Midship Bending Moments Experienced in

Extreme Regular Waves by Models :of a Tanker and a Destroyer. Ship Structure

Committee, Serial SSC-156, February 1964.

MANIAR, N. M., Investigation of Bending Moments Within the Midship Half Length

of a MARINER Model in Extreme Waves. Ship Structure Committee, Serial SSC-163,

June 1964.

MANIAR, N. M. and NUMATA, E., "Bending Moment Distribution in a Mariner Cargo Ship Model in Regular and Irregular Waves of Extreme Steepness," Davidson

Laboratory Report 1094, Stevens Institute of Technology, March 1968.

CHIOCCO, M. J. and NUMATA, E., "Midship Wave Bending Moments in a Model of the Cargo Ship WOLVERINE STATE Running at Oblique Headings in Regular Waves,"

Davidson Laboratory Report 1299, Stevens Institute of Technology, October 1968.

MURDEY, D. C., "On the Double Peaks in Wave Bending Moment Response Curves,"

Trans., Royal Institution of Naval Architects, 1968.

FUKUDA, JUN-ICHI, "Computer Program Results for Response Operators of Wave

Bending Moment in Regular Oblique Waves." Reprint from Memoirs of the Faculty

of Engineering, Kyushu University, Vol. XXVI, No. 2 (contribution to 3d

(26)

_Wavi,_Lendth Snip' Length

020

0.30 ,0.40 -v_{_} ,0.60 01.70 0.80 0.90 1.00 1.10 1.25

1

:.50 -1.45 2.00. .7 600 200 -140 - 175 -175 80 160.' .150 -. . _.3-35

146

In) 135 80 155 -180 90 '175, 190 165 ' .205 -65 190. 226 140

-Lateral moment after vertical moment.

--Phases constant with speed from,10,to 20 knots)"

150 145 145 4o 200 145 200. 120 150

(27)

.030'

0.'50 0.60' 1.'10 1.29 -1.90 -1.79 2'. 60 .;

TABLE A.'

pmASE-ANGLES

Drafts:

23;.,40, FW-d.';,:,25.:.'55' Aft

189

200.

4:;1,atera 1 moment after vertical moment,

.13h a'ies . -cons tant with .speed: 'from .1,0 to20knots.,

-Head ings '90° 1.49 )40. 135, 135.-140 149 145 ,

(28)

Vertical heventhealline.

- Wave Ampl

'2t-tans '

15.000

5.000

Vert tealN;Iimm;ivA..0 I iiude'

Wan,. ft" ,1;.-4Ta Ft. Speed, Anais. 5 itz L - Is I -Speed. 141-ots

Fig: A'41% -Vertical Wave Bending Moments Vertical Wave Bending Moments

Drafts: 19.25' Fwd 22'85!,,Aft-' 180° .,, Drafts.; 9.25' Fwd, 22..55' Aft 180°

Heading ' -' Heading, = _ 0.50 0.40 Wave _th Ship Length C BO C.70 -0,0 Aio i tude -4 'Wave:Amplitude 15.000 0.40 10' . Length .Ship Length . 'Ship-Length 1.00 0.50 %SO ,,,121pme Ship Lengtn 20 :\

Speed, Wren Speed. Snots,

AL3:::-)YgtiCal Wa4 Bending Moments .F.1".g,._A7.4r.- YeetiOal Wave Bending_Moments

Drafts:, 19451-:-FWd, 22.55! Aft 150°:' : Drafi's-,:_19:Wryd,, 24551.Aft 150°

(29)

_

-Vert 1 cal Moment Aoof I tude

Move, Amol.1 tudo :S.A.IpiLpngth

;

.. _

tient I te1,41cenent 'Aelo I I tude

WeeeAntplI tune

to.ixisT

'Vert 1.1- &sent' Amp I I thee Worrre 'kw I I t , . 0601.Di ; 777-. , 0 S Feed,- Knot.. Speed,. Knots 5;00.0 is= sed..:;K4e; ,0.60 0.90 1.00

Fig . ,Vetica1Wael.:5endin4-,2MOmehts

Fig A6

Verti cal Wave Bening 'Moments

". Drafts: 19'. 25 -FW:d; Drafts: =19.25V--.-Fwd, 22.55 Aft 1200

Heading Head

ing-vert Ica I-Moment Amp) I tudo

--Wave Amp I Ituda

-;aye Lenot4 ' Ship Length 0.40 1.50 '0.20 - _ ' --- '

-Fig . : Vertical _{_} Momints' Fig

A-

:.Vertjcal:;WaveyBending Moments

_ Drafts 19 25' Fwd, ,22:551-Af,t-900 : Drafts : 19.25! .--FWd 2255' Aft 90°.

(30)

'Vertical Moment Analitede

.121JtV96

ooá

10.000

0.30

vavtraet moment Analltvaa --- Wave Amplituass ic --5ined. Knots , , 0.90 1.10 s040 -1.00 Vertlialleinant amelltuar WeveAteptitede 15400, -.77.3.t IC , ;715 -...

Speed ,-Knots ' : Speed. Knots

% ,,c4.:-.

Fig. A7,9....

Vertical Wave B000191Moiridhts-

Fig. A-10.,...,

Vertical Wave Bending Moments

Drafts: 19.26): FWd,.22-.551 Mt.

609 .::-..."

Draftt.:14:5:: fwd, 22,65'

Aft

60°-Heading, -..,-' - --t, Heading'. . ., ' ifertIcel:Mdment AnsIltuda, Ft-Tans 15000

7--. Ft 7154W61. 0;80 - '

Fig,

A-11. -Vertvcal.Wave,-.Bendlngl4Omentt Vertical Wave Bending Moments Drafts 19.25',f-Wd22,55-j:Aft 309 Fwd, - 22.55' Aft 309

HeadiDg.. 140qflilei 0:50 0.60 0.70 1.00 1.50 's1RT; 20 '5 10

(31)

Wave Ampl 1 tude.

15.009 _Ltko_ns

1.et erel -Women t Amp 1 It ode

---Weve'Ampiltude 000 fIL-1:1L-"' . Ft, -3.000 . -3,000 0.. 10 15 Spend, Knots 15

,fl

0 /21. F1L4' n ci Cu. A . 10 20 20

WøneL.ngth. ' _ Vert 1 ta Ampl 1 tude

Shl p 'Length Amp! I t ode

Wave Length' 911 p Length' 0.70, 0.80 1.25 0.40 0.50 1.75 0.60 0.90 100

Fig. A-15:. Lateral Wave Bending Moments Drafts:- 19.25'.Fwd,:22.55' Aft 1509,,,,

Heading,

15,000Ft-Tons_Ft,

.5.000

,

Fig. Vertical .Wave Bending MoMents Fig, A-14. Ve'rtical Wave Bending -Moments

Drafts .: Fwd, 22.55' Aft 0°- Drafts: 19.25'. Fwd, 22.55' Aft 0!'

Heading Heading 3.600a'J'aa 0,90 -10 .20 300_ 0.80 0.60 1.25 10 2 0.80 0.70 u) -1 0 - 2.00 ' 15 Spend, idiots 1, 15 20 ,111 1 p Length - 1

-Fig. A,16.,0..dterai Wave Bending Moments_{_}

_

Drafts: 19,25' Fwd,-22.55' Aft 150?

- Heading

}agora/ Moment Amyl 1 tude Seen Length

Wave Amp 1 1 tude ShIp;Langth

0.70

1:50

(32)

.:Lite.e I1400rent- Ae,i !tide

-0 0 Arep.I I:t utt

5. tc.; --20 ' Ship Length - 20 20 46.0-Length Ship.Length

o050

0.40 0.80

Fig. -Lateral WaVe.'Beriding -Moments Fig.: :A=1:8; ILateral -Wave- Beng Moments

Drafts:-19-.251. -Fwd, Aft 1200 Drafts:-19.:25' Fwd, 22-.55'- Aft 1200

Headingi,! - - Heading

Fig. -A=19;

Lateral:WaVeIgenting:;MoMents

Fi:g.

A-26.

-Lateral, Wave Bending Moments

Drafts- 19':25,'2".-FWid; :Aft. 90°- Draftt: 19.25' Fwa,,22..55',. Aft 90°

Headingc- Heading

(33)

lateral Moment Anelltude Wave Amplitude

5.000

io

lateral Moment Amplitude Way* Amplitude it Sewed, Knots 15 Speed,, 5h;S, 20 Wave L.vvth Ship Length 0.30 0.20 M.013LIMSh Ship Length

Fig. A-23. Lateral Wave Bending Moments Drafts: 19.25' Fwd, 22.55' Aft 300

Heading

-23-}stem] Mommit Amplitude Wave Amplitude

-5.000

lateral Moment amplitude Wave Amplitude 5,000NM= 0.90 o 5,000 2.0 Wave Length Ship Length 0.70 1.00 Speed, Knots

Fig. A-21. Lateral Wave Bending Moments Fig. A-22. Lateral Wave Bending Moments

Drafts: 19.25' Fwd, 22.55' Aft 60° Drafts: 19.25' Fwd, 22.55' Aft 60°

Heading Heading

11121-1d5lEst

Ship Length

0.60

2.00

Fig. A-24. Lateral Wave Bending Moments Drafts: 19.25' Fwd, 22.55' Aft 30° Heading 10 IS Speed,'Itnets 15 20 10 15 20

(34)

34tOra I lletret-ei Andiltude-blare Amplitude_ 4,000 0.50 0.70 0.80 0A 0.90 so o oo ot

Ira

or,- I 00 - 15 k..12

Vert ic1 Komerrtkoplitude lieve-koplitude_

,

0.80.

1.25

10 15 20

Fig. A-27. Veriial''.2161aCie-,Bending

Moments Drafts:- 23;70LFwd, 26:55' Aft 1800 Heading- ' ' 1.10 1.25 1.50, 1.75-2.00 10,000 10 s7d Knot

Fig. A-25.,. Lateral WiVeiBending Moments Fig. A-26. Vertical Wave Bending Moments

- .-lia: !r . r

!,-Drafts: 19.25' Fwd, 22.55 Aft 0° Drafts: 23.70' Fwd, 25.55' Aft 180°

Heading Heading

Speed, Knots

Fig"::1A-128.'Vertical Wave Bending

Moments Drafts: 23 70' Fwd, 25.55' Aft 1500 Heading 1.00 0.60 1.50 I-.75 -5.000 - _ 20 5;000 o I. n 0 0.40 ;!. fons _Length -Ship Lgngth

(35)

-.Vertical Aament-Amplitude Wave 'collude

Ilertical We/Mirk Amplitude

Wave Amplitude is Ft.,..7t&ns 15.000 10,000 5:000 10 10.000 4 _ 3.70.. SpeZd.'1(mstc

ji Fig. A-29.- Vertical Wave- Bending

';Moments Drafts : 23.70! ;Fwd;;`25.55.',. ; Aft 1500 Heading AMva.Length 'ShipLength Wave Leugih , Ship Length 1.1C ! :

;-Fig. A-31. Vertical _Wave Bending

! Moments Drafts.: 23.70' Fwd,25.55'. ; Aft 120° Heading - .---;'.);.--,---:, -_ 500

Fig. VerticaL Wave Bending

Moments ,Drafts:.23.70' Fwd, 25.55' Aft 120°- Heading - Vertical.htcmnt Amplitude. - Wave Maplltude 2.000 2.000. 1Lits -Wcie Length :Ship Length '

Fig. A-32. `.-Vertical Wave Belding Moments Drafts: _23.70'. Fwd, 25.55' Aft 90° Heading'. -' 7:2-5 wove Length Ship Length 2.00 c . 0 2.42 Tab i 75 75 20 Spc07 Knott ' 5.000 .25 2,000 1.50 2,000

(36)

Vertical velment. Pap 1 itude Wane M,plltce EC+7 _ Wave Length Ship Length

5.-

I . Spied. Knott {7'

eal Waver Ben`ding,

MoMents--.DraftS':

Aft 660::.14sidii-ig

, -0.50 Wave Largth Ship Length

0.70-

-26-Fig

A-34.

Vertical 'Wave Bending

Moments Drafts:

-.23.70' Frid,-, 26.

55'

Aft 60° Heading.

-. , Kart iotl...vh.caren,mpti

15 Oct Sh.1 p. length

Fig.

A-35.!:.

Vertical Wave Bei-idt_rig;:. r

,

Vertitil Wave Bending

.

Moments Draftt,:-:-.23i.3

OnleiTtS7Dfafts: 23.;7o' Fwd, 25.

55"-Aft 30?,-,H4ittri6'.

- .Heading;

ed:- Knots:

1.75

-:._

(37)

5 '.000

.

, verif eat *914weie

j .- Wave Mali twigs

-sum Ft- -imoo 5,000 - Ft s 2;000 ' 15 qL60 0.50 Soentii, Knots

Fig.

A=37...

Veetical Wave Bending Moments

Fig.

A-3 ..,..1-

Vertical:Aave Bending Moments

Drafts :

23.70'-_Fwd, 25.~.551..:Aft.-,0°- -':-...-; "

:DraftS: 23:i7dVh4-4;

2-5'.(:5-5',,Aft: o:--.

Heading -

'

--.:.-

'

" = 2. .::' - . :Heading '

- '''--

-

'

'' -"' '

.

r.!''

,

'1'.

Speed,. Knots

Fig:

A=39. ,l_ateral Wave Berid_ing,,Moments

Drafts:

2.70'

Fwd,

Aft i80°

Heading

Winie Length' Ship Length'

1.75

' sprd. 41`AP1

Fig. A-40-.

Lateral.,

Wave-Bending .11o,

ments Drafts: '23.701.-Fwd

25.55'

-Aft

150°

Headiii4

, 0,90 1.00 .1.25 `' 1.50 1.10 .000

00

7.50 - 0 AO .°°!?n- Ship Length 1L 1:50

-tairal.lioment Ampl i ,Letera 1 Mcmenthot 1Mule

(38)

-Mai-met. Iftado

Fig:. A4.1:-.'Latei!alx",:voVel-eild:rikg',_

Moments Drafts

237O'

25.-..55!)

-Aft- 120°'.1-.1eading 15 Soe'sel. Knott W.V. Length . Selp-Length Later*1 Wave...Bend:trig

Moments:--briFfit:,'23.17wl'Fild; 2555'

-Aft- 60°-,Tleading

--- -al nereist:Amplis-de - Psnol toile 2,0005' -- 1.75 :25 k:,: 41,_ 50 0 _0:70 - 2C Wave Length :Ship Length , .

- Lat-ea1", W Ale" Bending

Mo-ments Drafts :

Fwd, 25.55' Aft

90-0

Heading,:-0:70 0.00 1.00 i.2i

::o

175

Loterol .eOnnt AMP I Wave heel tudeL"

5.000IZt_goise-Ship Length

0.40

1.50

Lateraj Waye Bending

Mo-.thents".0;:afti-,2.70,'- NO, 25.,55' Aft

30°' Head:it-le,.

-

₄₄₀

(39)

, . .

:.Late%n,Montan, ?XI .tuele

+4

smba I 1.1arth_{Thip Length} 0.70

0.90

(.75

1.25 1.50

Fig. A-45. Lateral Wave Bending

Mo=-me-iits Drafts: 23.70' Fwd, 25.55' Aft '0° Heading 15 20" 9peed, Knot 0 ' 0.50 244. _0.80 040 x:

(40)

UNCLASSIFIED

Security Classification

FORM

1 JAN 64 _UNCLASSIFIED

Security Classification

DOCUMENT CONTROL DATA - R&D

(Security claseification of title, body of abstract and indexing annotation must be entered when the overall report is classified)

I. ORIGINATING ACTIVITY(Corporate author)

Davidson Laboratory

Stevens Institute of Technology

2a. REPORT SECURITY C LAssiFICATION

Unclassified

2b. GROUP

3. REPORT TITLE

.

MIDSHIP WAVE BENDING MOMENTS IN A MODEL OF THE MARINER-CLASS CARGO SHIP "CALIFORNIA BEAR" RUNNING AT OBLIQUE HEADINGS IN REGULAR WAVES

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

Technical Report

5. AUTHOR(S)(Last name, first name, initial)

Edward Numata William F. Yonkers

6. REPORT DATE

,

Ncvember 1969

Te. TOTAL NO. OF PAGES

29

76. NO. OF REFS 7

13a. CONTRACT OR GRANT NO.

N00024-67-C-5218

b. PROJECT NO.

SF013-03-04, Task 2022

C.

d.

SA. ORIGINATOR'S REPORT NUMISER(S)

Report 1344

9b. _{this report)}OTHER REPORT No(S) (Any other numbers that may be assigned

SSC-202

10. AVAILABILITY/LIMITATION NOTICES

Distribution of this document is unlimited.

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Ship Systems Command

13. ABSTRACT

Vertical and lateral wave bending moments were measured at the midship section of a 1/96-scale model of the C4-S-1A MARINER-class cargo ship

CALIFORNIA BEAR. _{The model was self-propelled through a ship speed-range}

of 10 to 22 knots at seven headings to regular waves of lengths between 0.2 and 2.0 times the length between perpendiculars; moderate wave heights

rot exceeding 1/50 of the model length were used. _{Results are presented in}

charts of bending-moment-amplitude/wave-amplitude versus ship speed, with

wave length as the parameter. Two ship loading conditions, representative

(41)

HIslCIASUFTFD Security Classification 14. KEY WORDS Cargo Ship Ship Hulls

Ship Bending Moment Model Tests Water Waves LINK A ROLE WT LINK 8 ROLE WT LINK C ROLE WT INSTRUCTIONS

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UNCLASSIFIED