Basic Ship Theory Vol.1 and Vol.2 BasicShipTheoryK.J. RawsonMSc,DEng, FEng, RCNC, FRINA, WhSchE.C. TupperBSc, CEng, RCNC, FRINA, WhSchFifth editionVolume 1Chapters1 to 9Hydrostatics and StrengthOXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEWDELHIButterworth-HeinemannLinacreHouse, Jordan Hill,Oxford OX28DP225 Wildwood Avenue, Woburn, MA 01801-2041A division ofReedEducationaland ProfessionalPublishing LtdA member oftheReed Elsevier plcgroupFirst published by Longman Group Limited 1968Second edition 1976(in twovolumes)Third edition 1983Fourth edition 1994Fifth edition 2001#K.J. Rawson and E.C. Tupper 2001All rights reserved. Nopart ofthis publicationmay bereproduced inany material form (including photocopying orstoring inany medium byelectronic means andwhetherornottransiently orincidentallyto someother use ofthis publication) without thewritten permission ofthecopyright holderexcept inaccordance with theprovisions oftheCopyright,Designs and Patents Act 1988orunder the termsofalicence issuedby theCopyrightLicensing Agency Ltd, 90 Tottenham Court Road, London,England W1P0LP. Applications forthecopyright holder's writtenpermission toreproduce any part ofthis publication should beaddressedto thepublishersBritish Library Cataloguing in Publication DataRawson, K.J. (Kenneth John), 1926Basic shiptheory. 5thed.Vol. 1, ch.19: Hydrostatics and strength K.J. Rawson,E.C. Tupper1. Naval architecture 2. ShipbuildingI. Title II. Tupper,E.C. (Eric Charles), 1928623.8/1Library of Congress Cataloguing in Publication DataRawson, K.J.Basic shiptheory/K.J. Rawson, E.C. Tupper. 5th ed.p. cm.Contents: v.1. Hydrostatics and strength v.2.Ship dynamics and design.Includes bibliographicalreferences and index.ISBN 0-7506-5396-5 (v.1:alk.paper) ISBN0-7506-5397-3 (v.2: alk.paper)1. Naval architecture I.Tupper, E.C. II.Title.VM156 .R37 2001623.8/1dc21 2001037513ISBN 0 75065396 5Forinformationon allButterworth-Heinemannpublications visitour website atwww.bh.comTypesetinIndia at Integra Software Services Pvt Ltd,Pondicherry,India 605005; www.integra-india.comIntroduction ...............................Symbols and nomenclature ..............................1 Art or science? .......................................1.1 Authorities .........................................2 Some tools ................................2.1 Basic geometric concepts .......................................2.2 Properties of irregular shapes .................................2.3 Approximate integration ..........................................2.4 Computers ..........................................2.5 Appriximate formulae and rules ..............................2.6 Statistics .......................................2.7 Worked examples ...................................................2.8 Problems .......................................3 Flotation and trim ...........................................3.1 Flotation ......................................3.2 Hydrostatic data ..................................................3.3 Worked examples ...................................................3.4 Problems .......................................4 Stability ...........................4.1 Initial stability .............................................4.2 Complete stability ...................................................4.3 Dynamical stability ..................................................4.4 Stability assessment ...............................................4.5 Problems .......................................5 Hazards and protection ..................................5.1 Flooding and collision .............................................5.2 Safety of life at sea .................................................5.3 Other hazards ...............................................5.4 Abnormal waves ...................................................5.5 Environmental pollution ..........................................5.6 Problems .......................................6 The ship girder .......................................6.1 The standard calculation .........................................6.2 Material considerations ...........................................6.3 Conclusions ............................................6.4 Problems .......................................7 Structural design and analysis ......................7.1 Stiffened plating ..................................................7.2 Panels of plating ...................................................7.3 Frameworks ............................................7.4 Finite element techniques .......................................7.5 Realistic assessment of structral elements .............7.6 Fittings ....................................7.7 Problems .......................................8 Launching and docking .................................8.1 Launching .........................................8.2 Docking .....................................8.3 Problems .......................................9 The ship environment and human factors ...9.1 The external environment. The sea ........................9.2 Waves ...................................9.3 Climate ....................................9.4 Physical limitations .................................................9.5 The internal environment ........................................9.6 Motions .....................................9.7 The air ...................................9.8 Lighting .....................................9.9 Vibration and noise .................................................9.10 Human factors .................................................9.11 Problems .........................................Bibliography ................................Answers to problems ........................................Index ...................ForewordtotheftheditionOver the last quarter of the last century there were many changes inthemaritime scene. Ships maynowbe muchlarger; their speeds are generallyhigher; thecrews havebecomedrasticallyreduced; therearemanydierenttypes(includinghovercraft, multi-hull designsandsoon); muchquickerandmoreaccurateassessments of stability, strength, manoeuvring, motions andpoweringarepossibleusingcomplexcomputerprograms;on-boardcomputersystemshelptheoperators; ferriescarrymanymorevehiclesandpassengers;andsothelistgoeson.However,thefundamentalconceptsofnavalarchitec-ture, whichtheauthorsset out whenBasicShipTheorywasrst published,remainas valid as ever.As with many other branches of engineering, quite rapid advances have beenmadeinshipdesign, productionandoperation. Manyadvancesrelatetotheeectiveness (in terms of money, manpower and time) with which older proced-ures or methods can be accomplished. This is largely due to the greatereciency and lower cost of modern computers and proliferation of informationavailable. Other advances are relatedtoour fundamental understandingofnavalarchitectureandtheenvironmentinwhichshipsoperate.Thesetendtobeassociatedwiththemoreadvancedaspectsof thesubject: morecomplexprograms for analysing structures, for example, which are not appropriate to abasic textbook.The naval architect is aected not only by changes in technology but also bychanges in society itself. Fashions change as do the concerns of the public, oftenstimulated by the press. Some tragic losses in the last few years of the twentiethcenturybrought increasedpublic concernfor the safetyof ships andthosesailinginthem, bothpassengersandcrew. Itmustberecognized, ofcourse,that increased safety usually means more cost so that a conict between moneyand safety is to be expected. In spite of steps taken as a result of theseexperiences, thereare, sadly, still manylossesofships, somequitelargeandsome involving signicant loss of life. It remains important, therefore, to strivetoimprovestillfurtherthesafetyofshipsandprotectionoftheenvironment.Steady, if somewhat slow, progress is being made by the national and interna-tional bodies concerned. Public concern for the environment impacts upon shipdesignand operation. Thus,tankers must be designed to reducethe risk of oilspillageandmoredangerouscargoesmustreceivespecialattentiontoprotectthepublicandnature. Respectfortheenvironmentincludingdischargesintothe sea is an important aspect of dening risk through accident or irresponsibleusage.Alot ofinformationisnowavailableontheInternet, includingresultsofmuch research. Taking the Royal Institution of Naval Architects as an examplexiof a learned society, its website makes available summaries of technical papersandenablesmemberstojoininthediscussionsofitstechnical groups. Otherdata is available in a compact form on CD-rom. Clearly anything that improvesthe amount and/or quality of information available to the naval architect is tobewelcomed.However,itisconsideredthat,forthepresentat anyrate,thereremains aneedfor basic text books. The twoare complementary. Abasicunderstandingof thesubject isneededbeforeinformationfromtheInternetcanbeusedintelligently. Inthiseditionwehavemaintainedtheobjectiveofconveyingprinciples andunderstandingtohelpstudent andpractitioner intheir work.The authors have again been in a slight dilemma in deciding just how far togo in the subjects of each chapter. It is tempting to load the books with theorieswhichhave become more andmore advanced. What has beendone is toprovideaglimpseintodevelopmentsandadvancedworkwithwhichstudentsandpractitionersmust becomefamiliar. Towardstheendof eachchapterasectiongivinganoutline of howmatters are developinghas beenincludedwhichwill helptoleadstudents, withtheaidof theInternet, toall relevantreferences.Someweb site addresses have also been given.Itmustbe appreciatedthatstandards changecontinually,as dothe titlesoforganizations. Every attempt has been made to include the latest at the time ofwritingbutthereadershouldalwayschecksourcedocumentstoseewhetherthey still apply in detail at the time they are to be used. What the reader can relyon is that the principles underlyingsuch standards will still be relevant.2001 K J R E C Txii ForewordtothefiftheditionAcknowledgementsTheauthorshavedeliberately refrainedfromquotingalarge number of refer-ences. However, wewishtoacknowledgethecontributions of manypracti-tioners and research workers to our understanding of naval architecture, uponwhose workwe have drawn. Manywill be well knowntoany student ofengineering. Thoseearlyengineers intheeldwhoset thefundamentals ofthesubject, suchasBernoulli, Reynolds, theFroudes, Taylor, Timoshenko,Southwell andSimpson, arementionedinthetext becausetheir names aresynonymous with sections of navalarchitecture.Othershavedevelopedourunderstanding, withmorepreciseandcompre-hensivemethodsandtheoriesastechnologyhasadvancedandtheabilitytocarry out complex computations improved. Some notable workers are notquoted as their workhas been too advanced for a bookof this nature.We are indebted to a number of organizations which have allowed us to drawupontheir publications, transactions, journals andconference proceedings.This has enabledus toillustrateandquantifysomeof thephenomenadis-cussed. Theseincludethelearnedsocieties, suchas theRoyal InstitutionofNaval ArchitectsandtheSocietyofNaval ArchitectsandMarineEngineers;research establishments, such as the Defence Evaluation and Research Agency,the Taylor Model Basin, British Maritime Technology and MARIN; theclassicationsocieties; andGovernmentdepartmentssuchastheMinistryofDefence and the Department of the Environment, Transport and the Regions;publications such as those of the International Maritime Organisation and theInternational TowingTank Conferences.xiiiIntroductionIntheir youngdays theauthors performedthecalculations outlinedinthisworkmanually aidedonly by slide rule and, luxuriously, calculators. Thearduous natureof suchendeavours detractedfromthecreativeaspects andaectedtheenjoymentofdesigningships. Today, whileitwouldbepossible,such prolonged calculation is unthinkable because the chores have beenremovedtothecareof thecomputer, whichhasgreatlyenrichedthedesignprocess by giving time for reection, trial and innovation, allowing the eects ofchangesto be examined rapidly.It would be equally nonsensical to plunge into computer manipulation with-out knowledge of the basic theories, their strengths and limitations, which allowjudgement tobe quantiedandinteractions tobe acknowledged. Asimplechange indimensions of anembryoship, for example, will aect otation,stability, protection, powering, strength, manoeuvringandmanysub-systemswithin,thataect alandarchitect tomuchlessanextent. Forthisreason, theauthors havedecidedtoleavecomputer systemdesigntothosequaliedtoprovidesuchimportant tools andtoensurethat thestudent recognizes thefundamental theory on which they are based so that he or she may understandwhat consequences the designer's actions will have, as they feel their waytowardsthe bestsolutionto an owner's economic aims or military demands.Manipulation of the elements of a ship is greatly strengthened by such a `feel'andexperienceprovidedbypersonalinvolvement.Virtuallyeveryship'schar-acteristic andsystemaects everyother shipsothat some formof holisticapproachis essential.Acruderepresentationoftheprocessofcreatingashipisoutlinedinthegure.xivEconomics of tradeorMilitary objectiveVolumeHull shapeWeightResistance & PropulsionDimensionsSafetyArchitectureStructureProductionManoeuvringFlotation & stabilityChoice of machineryDesignThisis,ofcourse,onlyabeginning.Moreover,thearrowsshouldreallybepointing in both directions; for example, the choice of machinery to serve speedand endurance reects back on the volume required and the architecture of theship which aects safety and structure. And so on. Quantication of thechangesiseectedbythechoiceofsuitablecomputerprograms.Downstreamof this processlies designof systems to supporteach function but this, for themoment, is enough to distinguish betweenknowledge and application.The authors have had to limit their work to presentation of the fundamentalsof naval architectureandwouldexpect readerstoadopt whatevercomputersystems are available tothemwitha soundknowledge of their basis andfrailties.Thesequenceofthechapterswhichfollowhasbeenchosentobuildknowledge in a logical progression. The rst thirteen chapters address elementsof ship response to the environments likely to be met; Chapter 14 adds some ofthe major systems needed within the ship and Chapter 15 provides somediscipline to the design process. The nal chapter reects upon some particularship types showing how the application of the same general principles can leadto signicantly dierent responses to an owner's needs. A few worked examplesare included to demonstrate that there is real purpose in understanding theoret-ical navalarchitecture.Theopportunity, aordedbythepublicationof afthedition, has beentaken to extend the use of SI units throughout. The relationships between themand the old Imperial units, however, have been retained in the Introduction toassistthosewhohavetodealwitholdershipswhoseparticularsremainintheold units.Carehasbeentakentoavoidduplicating, asfarasispossible, workthatstudents will cover in other parts of the course; indeed, it is necessary to assumethatknowledgeinallsubjectsadvanceswithprogressthroughthebook. Theauthors havetriedtostimulateandholdtheinterest of students bycarefularrangement of subject matter.Chapter 1 and the opening paragraphs of eachsucceedingchapterhavebeenpresentedinsomewhatlyricaltermsinthehopethattheyconveytostudentssomeoftheenthusiasmwhichtheauthorsthem-selves feel for this fascinating subject. Naval architects need never fear that theywill,duringtheir careers,have to facethe same problems,day after day.Theywill experience as wide a variety of sciences as are toucheduponby anyprofession.Beforeembarkingonthebookproper, it isnecessarytocomment ontheunits employed.UNITSIn May 1965, the UKGovernment, in common with other governments,announcedthatIndustryshouldmovetotheuseofthemetricsystem.Atthesametime,arationalizedsetofmetricunitshasbeenadoptedinternationally,followingendorsementbytheInternationalOrganizationforStandardizationusingthe Syste meInternationald'Unite s (SI).The adoption of SI units has been patchy in many countries while some haveyet to change fromtheir traditional positions.Introduction xvInthefollowingnotes, theSIsystemof unitsispresentedbriey; afullertreatment appears in British Standard 5555. This book is written using SI units.The SI is a rationalized selection of units in the metric system. It is a coherentsystem,i.e.theproductorquotientofanytwounitquantitiesinthesystemisthe unitof the resultant quantity. The basic units are as follows:Quantity Nameofunit UnitsymbolLength metre mMass kilogramme kgTime second sElectriccurrent ampere AThermodynamictemperature kelvin KLuminousintensity candela cdAmountofsubstance mole molPlaneangle radian radSolidangle steradian srSpecial names have been adopted for some of the derived SI units and theseare listedbelow together withtheir unit symbols:Physicalquantity SIunit UnitsymbolForce newton N = kg m,s2Work,energy joule J = NmPower watt W= J,sElectriccharge coulomb C = AsElectricpotential volt V = W,AElectriccapacitance farad F = As,VElectricresistance ohm= V,AFrequency hertz Hz = s1Illuminance lux lx = lm,m2Selfinductance henry H = Vs,ALuminousux lumen lm = cdsrPressure,stress pascal Pa = N,m2megapascal MPa = N,mm2Electricalconductance siemens S = 1,Magneticux weber Wb = VsMagneticuxdensity tesla T = Wb,m2The following two tables list other derived units and the equivalent values ofsome UKunits,respectively:Physicalquantity SIunit UnitsymbolArea squaremetre m2Volume cubicmetre m3Density kilogrammepercubicmetre kg,m3Velocity metrepersecond m,sAngularvelocity radianpersecond rad,sAcceleration metrepersecondsquared m,s2xvi Introduction xvi IntroductionAngularacceleration radianpersecondsquared rad,s2Pressure,stress newtonpersquaremetre N,m2Surfacetension newtonpermetre N,mDynamicviscosity newtonsecondpermetresquared Ns,m2Kinematicviscosity metresquaredpersecond m2,sThermalconductivity wattpermetrekelvin W,(mK)Quantity Imperialunit EquivalentSIunitsLength 1 yd 0.9144 m1 ft 0.3048 m1 in 0.0254 m1mile 1609.344 m1nauticalmile(UK)1853.18 m1nauticalmile(International)1852 mArea 1 in2645.16 106m21 ft20.092903 m21 yd20.836127 m21mile22.58999 106m2Volume 1 in316.3871 106m31 ft30.0283168 m31UKgal 0.004546092 m3= 4.546092litresVelocity 1 ft,s 0.3048 m,s1 mile,hr 0.44704 m,s ; 1.60934 km,hr1 knot(UK) 0.51477 m,s ; 1.85318 km,hr1 knot(International) 0.51444 m,s ; 1.852 km,hrStandardacceleration,g 32.174 ft,s29.80665 m,s2Mass 1 lb 0.45359237 kg1 ton 1016.05 kg = 1.01605tonnesMassdensity 1 lb,in327.6799 103kg,m31 lb,ft316.0185 kg,m3Force 1 pdl 0.138255 N1 lbf 4.44822 NPressure 1 lbf,in26894.76 N,m20.0689476barsStress 1 tonf,in215.4443 106N,m215.443 MPaorN,mm2Energy 1 ft pdl 0.0421401 J1 ft lbf 1.35582 J1 cal 4.1868 J1 Btu 1055.06 JPower 1 hp 745.700 WTemperature 1Rankineunit 5,9Kelvinunit1Fahrenheitunit 5,9CelsiusunitNotethat,whilemultiplesofthedenominatorsarepreferred,theengineeringindustry has generally adopted N,mm2for stress instead of MN,m2which has,of course, the same numericalvalue and are the same as MPa.Introduction xviiPrexestodenotemultiplesandsub-multiplestobeaxedtothenamesofunits are:Factorbywhichtheunitismultiplied Prefix Symbol1 000 000 000 000=1012tera T1 000 000 000=109giga G1 000 000=106mega M1 000=103kilo k100=102hecto h10=101deca da0.1=101deci d0.01=102centi c0.001=103milli m0.000 001=106micro j0.000 000 001=109nano n0.000 000 000 001=1012pico p0.000 000 000 000 001=1015femto f0.000 000 000 000 000 001=1018atto aWe list, nally, some preferred metric values (values preferred for density offresh and salt waterare based on a temperature of 15C(59F)).Item AcceptedImperialfigureDirectmetricequivalentPreferredSIvalueGravity,g 32.17 ft,s29.80665 m,s29.807 m,s2Massdensity 64 lb,ft31.0252 tonne,m31.025 tonne,m3saltwater 35 ft3,ton 0.9754 m3,tonne 0.975 m3,tonneMassdensity 62.2 lb,ft30.9964 tonne,m31.0 tonne,m3freshwater 36 ft3,ton 1.0033 m3,tonne 1.0 m3,tonneYoung'smodulusE(steel) 13,500 tonf,in22.0855 107N,cm2209 GN,m2orGPaAtmosphericpressure 14.7 lbf,in2101,353 N,m2105N,m2orPa10.1353 N,cm2or1.0barTPI(saltwater)______Aw420tonf,in 1.025 Aw(tonnef,m) 1.025 Awtonnef,mAw(ft2) Aw(m2)NPC 100.52 Aw(N,cm)NPM Aw(m2) 10,052 Aw(N,m) 104Aw(N,m)MCT1//(saltwater)GML12Ltonfftin(Unitsoftonfandfeet)Onemetretrimmoment,(inMNortonnef mm,intonnef )GMLLMNmm_ _GMLLMNmm_ _Forcedisplacement 1 tonf 1.01605tonnef 1.016 tonnef9964.02N 9964 NMassdisplacement 1 ton 1.01605 tonne 1.016 tonneWeightdensity:Saltwater 0.01 MN,m3Freshwater 0.0098 MN,m3Specicvolume:Saltwater 99.5 m3,MNFreshwater 102.0 m3,MNxviii Introduction xviii IntroductionOf particularsignicancetothenaval architect aretheunitsusedfordis-placement,densityandstress.Theforcedisplacement,undertheSIschememust be expressed in terms of newtons. In practice the meganewton (MN) is amore convenient unit and 1 MN is approximately equivalent to 100 tonf (100.44more exactly). The authors have additionally introduced the tonnef (and,correspondingly,thetonneformassmeasurement)asexplainedmorefullyinChapter 3.EXAMPLESA number of worked examples has been included in the text of most chapters toillustratetheapplicationoftheprinciplesenunciatedtherein. Somearerela-tivelyshortbutothersinvolvelengthycomputations.Theyhavebeendeliber-ately chosen to help educate the student in the subject of naval architecture, andtheauthorshavenotbeenundulyinuencedbythethoughtthatexaminationquestions often involve about 30 minutes'work.In the problems set at the end of each chapter, the aim has been adequately tocover the subjectmatter,avoiding, as far as possible, examples involving merearithmetic substitutionin standard formulae.REFERENCESANDTHEINTERNETReferences for eachchapter aregiveninaBibliographyat theendof eachvolume with a list of works for general reading. Because a lot of usefulinformationistobefoundthesedaysontheInternet,somerelevantwebsitesare quoted at the end of the Bibliography.Introduction xixSymbolsandnomenclatureGENERALa linear accelerationA area in generalB breadth in generalD, d diameter in generalE energy in generalF force in generalg acceleration due to gravityh depth or pressure head in generalhw, wheight of wave, crest to troughH total head, BernoulliL length in generalLw, ` wave-lengthm massn rate of revolutionp pressure intensitypvvapour pressure of waterpambient pressure at innityP power in generalq stagnation pressureQ rate of owr, R radius in generals length along patht time in generalttemperature in generalT period of time for a complete cycleu reciprocal weight density, specic volume,u, v, w velocity components in direction of x-, y-, z-axesU, V linear velocityw weight densityW weight in generalx, y, z body axes and Cartesian co-ordinatesRight-hand system xed in the body, z-axis vertically down,x-axis forward.Origin at c.g.x0, y0, z0xed axesRight-hand orthogonal system nominally xed in space,z0-axis vertically down, x0-axis in the general direction of the initial motion.c angular acceleration specic gravity circulationc thickness of boundary layer in general0 angle of pitchj coecient of dynamic viscosityi coecient of kinematic viscosity, mass densityc angle of roll, heel or list angle of yaw. angular velocity or circular frequency\ volume in generalxxGEOMETRYOFSHIPAMmidship section areaAWwaterplane areaAxmaximum transverse section areaB beam or moulded breadthBM metacentre above centre of buoyancyCBblock coecientCMmidship section coecientCPlongitudinal prismatic coecientCVPvertical prismatic coecientCWPcoecient of neness of waterplaneD depth of shipF freeboardGM transverse metacentric heightGMLlongitudinal metacentric heightILlongitudinal moment of inertia of waterplane about CFIPpolar moment of inertiaITtransverse moment of inertiaL length of shipgenerally between perpsLOAlength overallLPPlength between perpsLWLlength of waterline in generalS wetted surfaceT draught displacement force` scale ratioship/model dimension\ displacement volume displacement massPROPELLERGEOMETRYADdeveloped blade areaAEexpanded areaAOdisc areaAPprojected blade areab span of aerofoil or hydrofoilc chord lengthd boss or hub diameterD diameter of propellerfMcamberP propeller pitch in generalR propeller radiust thickness of aerofoilZ number of blades of propellerc angle of attackc pitch angle of screw propellerRESISTANCEANDPROPULSIONa resistance augment fractionCDdrag coe.CLlift coe.CTspecic total resistance coe.CWspecic wave-making resistance coe.D drag forceFnFroude numberI idle resistanceJ advance number of propellerKQtorque coe.KTthrust coe.L lift forceSymbolsandnomenclature xxiPDdelivered power at propellerPEeective powerPIindicated powerPSshaft powerPTthrust powerQ torqueR resistance in generalRnReynolds numberRFfrictional resistanceRRresiduary resistanceRTtotal resistanceRWwave-making resistancesAapparent slip ratiot thrust deduction fractionT thrustU velocity of a uidUvelocity of an undisturbed owV speed of shipVAspeed of advance of propellerw Taylor wake fraction in generalwFFroude wake fractionWnWeber numberu appendage scale eect factoru advance angle of a propeller blade sectionc Taylor's advance coe.j eciency in generaljBpropeller eciency behind shipjDquasi propulsive coecientjHhull e.jOpropeller e. in open waterjRrelative rotative eciencyo cavitation numberSEAKEEPINGc wave velocityf frequencyfEfrequency of encounterIxx, Iyy, Izzreal moments of inertiaIxy, Ixz, Iyzreal products of inertiak radius of gyrationmnspectrum moment where n is an integerMLhorizontal wave bending momentMTtorsional wave bending momentMVvertical wave bending moments relative vertical motion of bow with respect to wave surfaceS(.), S0(.), etc. one-dimensional spectral densityS(.,j), S0(.,j), two-dimensional spectraletc. densityT wave periodTEperiod of encounterTznatural period in smooth water for heavingT0natural period in smooth water for pitchingTcnatural period in smooth water for rollingY0(.) response amplitude operatorpitchYc(.) response amplitude operatorrollY(.) response amplitude operatoryawu leeway or drift anglecRrudder angle phase angle between any two harmonic motions instantaneous wave elevationxxii SymbolsandnomenclatureAwave amplitudewwave height, crest to trough0 pitch angle0Apitch amplitudei wave number.Efrequency of encounter tuning factorMANOEUVRABILITYACarea under cut-upARarea of rudderb span of hydrofoilc chord of hydrofoilK, M, N moment components on body relative to body axesO origin of body axesp, q, r components of angular velocity relative to body axesX, Y, Z force components on bodyc angle of attacku drift anglecRrudder angle heading angle.Csteady rate of turnSTRENGTHa length of plateb breadth of plateC modulus of rigidity linear strainE modulus of elasticity, Young's moduluso direct stressoyyield stressg acceleration due to gravityI planar second moment of areaJ polar second moment of areaj stress concentration factork radius of gyrationK bulk modulusl length of memberL lengthM bending momentMpplastic momentMABbending moment at A in member ABm massP direct load, externally appliedPEEuler collapse loadp distributed direct load (area distribution), pressurep/distributed direct load (line distribution)t shear stressr radiusS internal shear forces distance along a curveT applied torquet thickness,timeU strain energyW weight, external loady lever in bendingc deection, permanent set, elemental (when associated with element of breadth, e.g. cb), mass densityv Poisson's ratio0 slopeSymbolsandnomenclature xxiiiNOTES(a) A distance between two points is represented by a bar over the letters defining the two points,e.g. GM is the distance between G and M.(b) Whenaquantityistobeexpressedinnon-dimensional formitisdenotedbytheuseoftheprime /. Unless otherwise specified, the non-dimensionalizing factor is a function of p, L and V,e.g. m/ = m,12,L3, x/ = x,12,L2V2, L/ = L,12,L3V2.(c) Alower case subscript is used to denote the denominator of a partial derivative, e.g.Yu = 0Y,0u.(d) For derivatives with respect to time the dot notation is used, e.g. _ x = dx,dt.xxiv Symbolsandnomenclature1 Artorscience?Many thousands of years ago when people became intelligent and adventurous,those tribes who lived near the sea ventured on to it. They built rafts or hollowedout treetrunksandsoonexperiencedthethrill of movingacross thewater,propelled by tide or wind or device. They experienced, too, the rst sea disasters;their boats sankor broke, capsizedor rottedandlives were lost. It was natural thatthose builders of boats whichwere adjudgedmore successful thanothers, receivedtheacclaimoftheirfellowsandweresoonregardedascraftsmen. Theintel-ligent craftsman observed perhaps, that capsizing was less frequent when usingtwotrunksjoinedtogetherorwhenanoutriggerwasxed,orthatitcouldbemanoeuvred better with a rudder in a suitable position. The tools were trial anderror and the stimulus was pride. He was the rst naval architect.Thecraftsmen'sexpertisedevelopedasitwaspasseddownthegenerations:the Greeks built their triremes andthe Romans their galleys; the Vikingsproducedtheirbeautiful crafttocarrysoldiersthroughheavyseasandontothebeaches. Several hundredyears later, thecraftsmenweredesigningandbuilding great square rigged ships for trade and war and relying still on know-ledgepasseddownthroughthegenerationsandguardedbyextremesecrecy.Still, they learned by trial and error because they had as yet no other tools andthe disastersat sea persisted.Theneedforascienticapproachmust havebeenfelt manyhundredsofyearsbeforeit waspossibleandit wasnot possibleuntil relativelyrecently,despite the corner stone laid by Archimedes two thousand years ago. Until themiddle of the eighteenth century the design and building of ships was wholly acraft and it was not, until the second half of the nineteenth century that scienceaectedshipsappreciably.Isaac Newton and other great mathematicians of the seventeenth century laidthe foundations for so many sciences and naval architecture was no exception.Without any doubt, however, the father of naval architecture was PierreBouguerwhopublishedin1746, Traite duNavire. Inhisbook, Bouguerlaidthe foundations of many aspects of naval architecture which were developed laterin the eighteenth century by Bernoulli, Euler and Santacilla. Lagrange and manyothers made contributions but the other outstanding gure of that century wastheSwede,FrederickChapmanwhopioneeredworkonshipresistancewhichledupto the greatworkof WilliamFroude ahundredyearslater.A scienticapproachto navalarchitecture wasencouragedmoreon the continentthaninBritainwhereit remaineduntil the1850s, acraft surroundedbyprideandsecrecy. On 19 May 1666, Samuel Pepys wrote of a Mr Deane:Andthenhe fell toexplaintome his manner of castingthe draught ofwaterwhichashipwill drawbefore-hand; whichisasecret theKingand1all admire in him, and he is the rst that hath come to any certainty before-hand of foretellingthe draught of waterof a shipbeforeshe be launched.The second half of the nineteenth century, however, produced Scott Russell,Rankine and Froude and the development of the science, and dissemination ofknowledge in Britain was rapid.NAVALARCHITECTURETODAYIt wouldbe quite wrongtosaythat the art andcraft built upover manythousands of years has beenwhollyreplacedbyascience. The needfor ascienticapproachwas felt, rst, becausetheart hadprovedinadequatetohalt the disasters at sea or to guarantee the merchant that he or she was gettingthe best value for their money. Science has contributed much to alleviate theseshortcomings but it continuesto require the injection of experience of success-ful practice. Science produces the correct basis for comparison of ships but theexact value of the criteria which determine their performances must, as in otherbranches of engineering, continue to be dictated by previous successful practice,i.e. like most engineering, this is largelyacomparative science. Where thescientictool islessprecisethanonecouldwish, itmustbeheavilyoverlaidwithcraft; where aprecise tool is developed, the craft must be discarded.Becausecomplexproblems encouragedogma, thishas not alwaysbeen easy.Thequestion, `Art or Science?' is thereforeloadedsinceit presupposes achoice. Naval architecture is art and science.Basically, naval architecture is concerned with ship safety, ship performanceand ship geometry,although theseare not exclusivedivisions.With ship safety, the naval architect is concerned that the ship does not cap-size in a seaway, or when damaged or even when maltreated. It is necessary toensure that the ship is suciently strong so that it does not break up or fracturelocallytolet thewaterin. Thecrewmust beassuredthat theyhaveagoodchance of survival if the ship does let water in through accident or enemy action.Theperformanceoftheshipisdictatedbytheneedsoftradeorwar. Therequiredamount of cargomust be carriedtothe places whichthe ownerspecies in the right condition and in the most economical manner; the warshipmust carry the maximum hitting power of the right sort and an ecient crew tothe remote parts of the world. Size, tonnage, deadweight, endurance, speed, life,resistance, methodsofpropulsion, manoeuvrabilityandmanyotherfeaturesmustbematchedtoprovidetherightprimaryperformanceattherightcost.Over 90 per cent of the world's tradeis still carried by sea.Ship geometry concerns the correct interrelation of compartments which thearchitect of a house considers on a smaller scale. In an aircraft carrier, the navalarchitecthas2000roomstorelate,onewithanother,andmustprovideuptofty dierent piping and ducting systems to all parts of the ship. It is necessaryto provide comfort for the crew and facilities to enable each member to performhis or her correct function. The ship must load and unload in harbour with theutmost speed and perhaps replenish at sea. The architecture of the ship must besuchthat it canbeeconomicallybuilt, andaestheticallypleasing. Thenaval2 Basicshiptheoryarchitectisbeingheldincreasinglyresponsibleforensuringthattheenviron-mental impact of the product is minimal both in normal operation and follow-ing any foreseeable accident. There is a duty to the public at large for the safetyof marine transport. In common with other professionals the naval architect isexpected to abide by a stringent code of conduct.It must beclearthat naval architectureinvolvescomplexcompromisesofmanyof these features. The art is, perhaps, the blendinginthe right pro-portions. Therecanbefewother pursuits whichdrawonsuchavarietyofsciencestoblendthemintoanacceptablewhole.Therecanbefewpursuitsasfascinating.SHIPSShips aredesignedtomeet therequirements of owners or of war andtheirfeatures are dictated by these requirements. The purpose of a merchant ship hasbeendescribedasconveyingpassengersorcargofromoneporttoanotherinthe most ecient manner. This was interpreted by the owners of Cutty Sark asthe conveyance of relatively small quantities of tea in the shortest possible time,because this was what the tea market demanded at that time. The market mightwell have required twice the quantity of tea per voyage in a voyage of twice thelengthof time, whenafundamentallydierent designof shipwouldhaveresulted. Theeconomicsofanyparticularmarket haveaprofoundeect onmerchant ship design.Thus, the change in the oil market following the secondworld war resulted in the disappearance of the 12,000 tonf deadweight tankersand the appearance of the 400,000 tonf deadweight supertankers. The econom-icsofthetradingoftheshipitselfhaveaneectonitsdesign;thedesire,forexample, for small tonnage (andtherefore small harbour dues) withlargecargo-carryingcapacitybroughtaboutthethreeislandandshelterdeckshipswherecargocouldbestowedinspacesnotcountedtowardsthetonnageonwhichinsurance rates andharbour dues were based. Suchtrends have notalways been compatible with safety and requirements of safety now also vitallyinuenceshipdesign. Specializeddemandsoftradehaveproducedthegreatpassengerlinersandbulkcarriers, thenatural-gascarriers, thetrawlersandmanyother interestingships. Indeed, the trendis towards more andmorespecializationin merchant shipdesign (seeChapter16).Specialization applies equally to warships. Basically, the warship is designedto meet a country's defence policy. Because the design and building of warshipstakes several years, it is an advantage if a particular defence policy persists forat least ten years and the task of long term defence planning is an onerous andresponsibleone. TheDefenceStainterpretsthegeneral Governmentpolicyinto the needs for meeting particular threats in particular parts of the world andthescientistsandtechnologistsproduceweaponsfordefensiveandoensiveuse. The naval architect then designs ships to carry the weapons and the men tousethemtothecorrectpartoftheworld. Thus, nationslikeBritainandtheUSAwithcommitments the other sideof theworld, wouldbeexpectedtoexpendmoreoftheavailablespaceintheirshipsonfacilitiesforgettingtheweaponsandcrewinasatisfactorystatetoaremote, perhapshot,areathanArtorscience? 3a nation which expects to make short harrying excursions from its home ports.It is important, therefore, to regard the ship as a complete weapon system andweapon and ship designersmust workin the closest possible contact.Nowhere, probably, was this more important than in the aircraft carrier. Thetype of aircraft carriedsovitallyaects anaircraft carrier that the shipisvirtuallydesignedaroundit; onlybyexceedingall theminimumdemandsofanaircraft andproducing monster carriers, cananyappreciable degree ofexibilitybe introduced. Theguidedmissile destroyer results directly from theDefenceSta'sassessmentoflikelyenemyaircraftandguidedweaponsandtheir concept of howandwhere they wouldbe used; submarine designisprofoundly aected by diving depth and weapon systems which are determinedbyoensiveanddefensiveconsiderations.Theinventionofthetorpedoledtothemotortorpedoboatwhichinturnledtothetorpedoboatdestroyer; thesubmarine, as an alternative carrier of the torpedo, led to the design of the anti-submarinefrigate; themissile carryingnuclear submarine ledtothe hunterkillernuclearsubmarine. Thus, theparticulardemandof war, asisnatural,producesa particularwarship.Particulardemandsoftheseahaveresultedinmanyotherinterestingandimportant ships: the self-righting lifeboats, surface eect vessels, containerships,cargodrones,hydrofoilcraftandahostofothers.Allaregovernedbythe basic rules and tools of naval architecture which this book seeks to explore.Precision in the use of these tools must continue to be inspired by knowledge ofseadisasters; Libertyships of the secondworldwar, the loss of the RoyalGeorge, the loss of HMSCaptain, and the lossof theVasa:In1628,theVasasetoutonamaidenvoyagewhichlastedlittlemorethantwo hours. She sank in good weather through capsizing while still in view ofthe people of Stockholm.That disasters remainaninuence upondesignandoperationhas beentragically illustrated bythe losses of theHeraldof Free Enterprise and Estoniainthe1990s, whileferrylossescontinueatanalarmingrate, ofteninnationswhich cannot aord the level of safetythat theywould like.AuthoritiesCLASSIFICATIONSOCIETIESTheauthoritieswiththemost profoundinuenceonshipbuilding, merchantshipdesignandshipsafetyaretheclassicationsocieties. AmongthemostdominantareLloyd'sRegisterofShipping,detNorskeVeritas,theAmericanBureau of Shipping, Bureau Veritas, Registro Italiano, Germanische Lloyd andNipponKaiji Kyokai. Thesemeettodiscussstandardsundertheauspicesofthe International Associationof Classication Societies (IACS).It is odd that the two most inuential bodies in the shipbuilding and shippingindustries should both derive their names from the same owner of a coee shop,Edward Lloyd, at the end of the seventeenth century. Yet the two organizations4 Basicshiptheoryareentirelyindependentwithquiteseparatehistories.Lloyd'sInsuranceCor-poration is concerned with mercantile and other insurance. Lloyd's Register ofShippingisconcernedwiththemaintenanceofpropertechnical standardsinshipconstructionandtheclassicationofships,i.e.therecordofallrelevanttechnical details andthe assurance that these meet the requiredstandards.Vessels soregisteredwithLloyd's Register are saidtobe classedwiththeSocietyandmaybeawardedtheclassication 100A1. ThecrossdenotesthattheshiphasbeenbuiltunderthesupervisionofsurveyorsfromLloyd'sRegister while 100Ashows that the vessel is built inaccordance withtherecommended standards. The nal 1 indicates that the safety equipment,anchors andcables are as required. Other provisos totheclassicationareoften added.The maintenance of these standards is animportant functionof Lloyd'sRegister whorequiresurveys of aspeciedthoroughness at statedintervalsby the Society's surveyors. Failure to conform may result in removal of the shipfrom class and a consequent reduction in its value. The total impartiality of theSociety is its great strength. It is also empowered to allot load line (see Chapter5)certicatestoships, toensurethattheyareadheredtoand, asagentsforcertainforeigngovernments, toassess tonnage measurement andtoensurecompliance withsafetyregulations. Over 1000surveyors, scatteredall overthe world, carry out the required surveys, reporting to headquarters in Londonor other national centres where the classicationof the ships are considered.The standards to which the ships must be built and maintained are laid downin the rst of the two major publications of Lloyd's Register, Rules andRegulationsfortheClassicationofShips.Thisisissuedannuallyandkeptupto date to meet new demands. The other major publication is the Register Bookinseveral volumes, whichlists everyknownship, whether classedwiththeSociety or not, together with all of its important technical particulars. Separatebooksappearforthebuildingandclassicationofyachtsandtherearemanyotherpublications to assist surveyors.Anumberofclassicationsocieties, includingLR, DNVandABS, oeraserviceforclassifyingnavalcraft. Typically, suchrulescovertheshipanditssystems including those that support the ghting capability of the craft. They donotcoverthemilitarysensors,weaponsorcommandandcontrolsystems,sothat the classicationsocietyconcentrates onthe shipas at for purposeweaponplatform.Thenavyconcernedactsasbuyerandownerandcancon-tinuetospecifyanyspecialmilitaryrequirements.Thetechnicalrequirementsthat make a ship t for naval service, and which would be dened by the navyconcerned, and make the ship dierent from a typical merchant ship, are:1. dierent strength requirementsto give a designable to acceptdamage;2. weaponandsensor supports taking account of possible deformationofstructure;3. the ability to withstand enemy action, including appropriate strength, stabil-ity, shock and redundancy features;4. allowancefor the eectsof impacting weapons.Artorscience? 5Themainelementsoftheclassrulesarecommonfornavalandciviliancraft.This ensures compliance with international regulations such as those of SOLASandMARPOL. Thewarshipisissuedwiththesamerangeof technical andoperational certicates as would be the case fora merchant ship.One advantage is that the navy, through its chosen shipbuilder, has access tothe world wide organization of the classication society in relation to materialand equipmentacceptance.GOVERNMENTBODIESThe statutory authority in the United Kingdomfordeclaring the standards ofsafety formerchant ships,relatedto damage, collision,subdivision,life savingequipment, loading, stability, re protection, navigation, carriage of dangerousgoods,loadlinestandardsandmanyalliedsubjects,istheDepartmentoftheEnvironment, Transport and the Regions (DETR). This department is also theauthorityontonnagemeasurementstandards.Itisresponsibleforseeingthatsafety standards, many of which are governed by international agreements, aremaintained.Executiveauthorityformarinesafetywasinvestedin1994intheMarine SafetyAgency(MSA) createdfromthe former Surveyor General'sorganization.Thenin1998,anexecutiveagencyofDETR,theMaritimeandCoastguardAgency(MCA),wasformedbymerging theMSAand theCoast-guard Agency (CA). The MCA provides three functions, survey and inspectionof vessels, co-ordinationof searchandrescue, andmarinepollutioncontrolresponse. DETRis responsiblefor enquiringintoseadisasters throughtheMarine Accident Investigation Branch. Responsibility for the safety of oshorestructures was transferredin1994 fromthe Department of Energy totheHealthand Safety Executive following the Piper Alphadisaster.Ship surveyors in the Marine Division and similar national authorities in othercountries, like the US Coastguard, carry, thus, an enormous responsibility.INTERNATIONALBODIESTheInternational MaritimeOrganization(IMO), representsover150of themaritime nations of the world. The organization sponsors international actionwith a view to improving and standardizing questions relating to ship safety andmeasurement. It sponsors the International Conventions on Safety of Life at Seawhich agree to the application of new standards of safety. The same organiza-tion sponsors, also, international conferences on the load line and standardizingaction on tonnage measurement and many other maritime problems.LEARNEDSOCIETIESTheInstitutionofNaval Architectswasformedin1860wheninterestinthesubject in Britain quickened and it has contributed much to the development ofnaval architecture. It became the Royal Institution of Naval Architects in 1960.Abroad, amongthe manysocieties worthyof mentionare the AssociationTechnique Maritime et Ae ronautique in France, the Society of Naval Architectsand Marine Engineers in the USA and the Society of Naval Architects of Japan.6 Basicshiptheory2 SometoolsNo occupation can be properly developed without tools, whether it be garden-ing or naval architecture or astro-navigation. Many tools needed for the studyofnaval architecturearealreadytohand, providedbymathematics, appliedmechanics and physics and it will be necessary to assume as the book progressesthat knowledge in all allied subjects has also progressed. Knowledge, forexample, of elementary dierential andintegral calculus is assumedtobedevelopingconcurrentlywiththis chapter. Moreover, the tools needtobesharp; denitions must be precise, while the devices adopted from mathematicsmust be pointed in such a way as to bear directly on ship shapes and problems.As a meansof examining this science, theseare the tools.It isconvenient too, toadopt aterminologyorparticularlanguageandashorthand for many of the devices to be used. This chapter lays a rm founda-tionfromwhichtobuildupthe subject. Finally, there are short notes onstatisticsand approximate formulae.Basic geometricconceptsThe main parts of a typical ship together with the terms applied to the principalpartsareillustratedinFig. 2.1. Because, atrst, theyareoflittleinterestorinuence, superstructures and deckhouses are ignored and the hull of the ship isconsideredasahollowbodycurvedinalldirections,surmountedbyawater-tight deck. Most ships have only one plane of symmetry, called the middle lineplane which becomes the principal plane of reference. The shape of the ship cutbythisplaneisknownasthesheerplanorprole.Thedesignwaterplaneisaplaneperpendicular to the middle lineplane,chosenas a planeof referenceator near the horizontal: it may or may not be parallel tothe keel. Planesperpendicular toboththe middle line plane andthe designwaterplane areFig. 2.17calledtransverseplanes andatransversesectionof theshipdoes, normally,exhibit symmetry about the middle line. Planes at right angles to the middle lineplane, andparallel tothedesignwaterplanearecalledwaterplanes, whetherthey are in the water or not, and they are usually symmetrical about the middleline. Waterplanes are not necessarily parallel to the keel. Thus, the curved shapeof a ship is best conveyed to our minds by its sections cut by orthogonal planes.Figure 2.2illustrates these planes.Transverse sections laidone on top of the otherform a body plan which, byconvention, whenthesectionsaresymmetrical, showsonlyhalfsections, theforward half sections on the right-hand side of the middle line and the after halfsections on the left. Half waterplanes placed one on top of the other form a halfbreadth plan. Waterplanes looked at edge on in the sheer or body plan are calledwaterlines. The sheer, the body plan and the half breadth collectively are calledthe lines plan or sheer drawing and the three constituents are clearly related (seeFig. 2.3).It is convenient if the waterplanes andthe transverse planes are equallyspacedanddatumpointsareneededtostartfrom.ThatwaterplanetowhichFig. 2.2Fig. 2.3 Lines plan8 Basicshiptheorythe shipisbeingdesignediscalledtheloadwaterplane (LWP) or designwater-planeandadditional waterplanes for examiningtheship's shapearedrawnaboveitandbelowit,equallyspaced,usuallyleavinganunevenslicenearthekeelwhich is bestexamined separately.A reference point at the fore end of the ship is provided by the intersection ofthe load waterline and the stem contour and the line perpendicular to the LWPthrough this point is called the fore perpendicular (FP). It does not matter wheretheperpendicularsare,providedthattheyarepreciseandxedfortheship'slife, thattheyembracemostoftheunderwaterportionandthattherearenoserious discontinuities between them. The after perpendicular (AP) is frequentlytaken through the axis of the rudder stock or the intersection of the LWL andtransom prole. If the point is sharp enough, it is sometimes better taken at theaftercutuporataplaceinthevicinitywherethereisadiscontinuityintheship's shape. The distance between these two convenient reference lines is calledthe length between perpendiculars (LBP or LPP). Two other lengths which will bereferred to and which need no further explanation are the length overall and thelength on the waterline.The distance between perpendiculars is divided into a convenient number ofequalspaces, oftentwenty, togive,includingtheFPandtheAP, twenty-oneevenly spaced ordinates. These ordinates are, of course, the edges of transverseplanes looked at in the sheer or half breadth and have the shapes half shown inthe body plan. Ordinates can also dene any set of evenly spaced reference linesdrawn on an irregular shape. The distance from the middle line plane along anordinate in the half breadth is called an oset and this distance appears again inthebodyplanwhereitisviewedfromadierentdirection.Allsuchdistancesforall waterplanesandall ordinatesformatableofosetswhichdenestheshape of the hull and from which a lines plan can be drawn. A simple tableofosetsis used in Fig. 3.30to calculate the geometric particulars of the form.Areferenceplaneisneededabout mid-lengthof theshipand, not unnat-urally, the transverse plane midway between the perpendiculars is chosen. It iscalledamidshipsormidshipsandthesectionof theshipbythisplaneisthemidshipsection. It may not be the largest sectionandit shouldhave nosignicance other thanits positionhalfwaybetweenthe perpendiculars. Itsposition isusually dened by the symbol .Theshape,lines,osets anddimensionsof primaryinterest to the theory ofnaval architecture are those which are wetted by the sea and are called displace-ment lines, ordinates, osets, etc. Unless otherwise stated, this bookrefersnormallytodisplacementdimensions.Thosewhichareofinteresttotheship-builderarethelinesoftheframeswhichdierfromthedisplacementlinesbythe thickness of hull plating or more, according to how the ship is built. Theseare calledmoulded dimensions. Denitions of displacement dimensions aresimilar to thosewhich follow but will dierby platingthicknesses.The moulded draught is the perpendicular distance in a transverse plane fromthe topof the at keel tothe designwaterline. If unspecied, it refers toamidships. The draught amidships is the mean draught unless the meandraughtis referred directly to draughtmarkreadings.Sometools 9Themouldeddepthistheperpendiculardistanceinatransverseplanefromthetopoftheatkeel totheundersideofdeckplatingattheship'sside. Ifunspecied,it refersto this dimension amidships.Freeboard is the dierence between the depth at side and the draught. It is theperpendicular distance in a transverse plane from the waterline to the uppersideof the deck platingat side.Themouldedbreadthextremeis the maximumhorizontal breadthof anyframesection.The terms breadth and beam are synonymous.Certain other geometric concepts of varying precision will be found useful indeningtheshapeofthehull.Riseofooristhedistanceabovethekeelthatatangent tothebottomatornearthekeel cutsthelineofmaximumbeamamidships(see Fig.2.6).Fig. 2.4 Moulded and displacement linesFig. 2.510 BasicshiptheoryTumblehomeisthetendencyofasectiontofallintowardsthemiddlelineplane from the vertical as it approaches the deck edge. The opposite tendency iscalled are (see Fig. 2.6).Deckcamberorrounddownisthecurveappliedtoadecktransversely.Itisnormally concave downwards, a parabolic or circular curve, and measured as xcentimetresin y metres.Sheeris the tendencyof a deck to rise above the horizontalin prole.Rake is the departure from the vertical of any conspicuous line in prole suchas a funnel, mast,stem contour,superstructure,etc. (Fig. 2.7).There are special words applied to the angular movements of the whole shipfromequilibriumconditions.Angularbodilymovementfromtheverticalinatransverseplaneiscalledheel. Angular bodilymovement inthemiddlelineplane is called trim. Angular disturbance from the mean course of a ship in thehorizontal planeiscalledyawordrift.Notethattheseareall anglesandnotrates,which are considered in later chapters.There are two curves which can be derived from the osets which dene theshapeofthehullbyareasinsteadofdistanceswhichwilllaterproveofgreatvalue. By erecting a height proportional to the area of each ordinate up to theLWPat each ordinate station on a horizontalaxis,a curveisobtained knownasthecurveofareas.Figure2.8showssuchacurvewithnumber4ordinate,takenasanexample. Theheightofthecurveofareasatnumber4ordinaterepresentstheareaof number4ordinatesection; theheight at number5isproportional to the area of number 5 section and so on. A second type of areacurvecanbeobtainedbyexaminingeachordinatesection. Figure2.8againFig. 2.6Fig. 2.7Sometools 11takes 4 ordinate section as an example. Plotting outwards from a vertical axis,distancescorresponding to the areasof a section up to each waterline, a curveknown as a Bonjean curve is obtained. Thus, the distance outwards at the LWLis proportional to the area of the section up to the LWL, the distance outwardsat 1WL is proportional to the area of section up to 1WL and so on. Clearly, aBonjeancurvecan be drawn foreach sectionand a set produced.Thevolumeofdisplacement, \,isthetotalvolumeofuiddisplacedbytheship. It is best conceived by imagining the uid to be wax and the ship removedfrom it; it is then the volume of the impression left by the hull. For convenienceof calculation,it is the addition of the volumes of the main bodyand append-ages such as the slices at the keel, abaft the AP, rudder, bilge keels, propellers,etc., with subtractions forcondensor inlets and other holes.Finally, in the denition of hull geometry there are certain coecients whichwill laterprove of value as guidesto the fatnessor slimnessof the hull.Thecoecientofnenessofwaterplane,CWP,istheratiooftheareaofthewaterplane to the area of its circumscribing rectangle. It varies from about 0.70forshipswithunusuallyneendstoabout0.90forshipswithmuchparallelmiddlebody.CWP =AWLWLBThe midship section coecient, CM, is the ratio of the midship section area totheareaofarectanglewhosesidesareequal tothedraughtandthebreadthextremeamidships. Its valueusually exceeds0.85for shipsother than yachts.CM =AMBTTheblockcoecient, CB, istheratioofthevolumeofdisplacementtothevolume of a rectangular block whose sides are equal to the breadth extreme, themean draught and the length betweenperpendiculars.CB =\BTLPPFig. 2.812 BasicshiptheoryMean values of block coecient might be 0.88 for a large oil tanker, 0.60 for anaircraftcarrierand 0.50 fora yacht form.The longitudinal prismatic coecient, CP, or simply prismatic coecient is theratioofthevolumeofdisplacementtothevolumeofaprismhavingalengthequalto the length between perpendiculars and a cross-sectional areaequaltothe midshipsectional area.Expectedvalues generallyexceed0.55.CP =\AMLPPFig. 2.9 Waterplane coefficientFig. 2.10 Midship coefficientFig. 2.11 Block coefficientSometools 13Theverticalprismaticcoecient,CVPistheratioofthevolumeofdisplace-ment tothevolumeof aprismhavingalengthequal tothedraught andacross-sectional areaequalto the waterplane area.CVP =\AWTBefore leaving these coecients for the time being, it should be observed thatthe denitions above have useddisplacement andnot mouldeddimensionsbecause it is generally in the very early stages of design that these are of interest.Practice in this respect varies a good deal. Where the dierence is signicant, asfor example in the structural design of tankers by Lloyd's Rules, care should betakentocheckthedenitioninuse.Itshouldalsobenotedthatthevaluesofthe various coecients depend on the positions adopted for the perpendiculars.Properties ofirregular shapesNow that the geometry of the ship has been dened, it is necessary to anticipatewhat propertiesof theseshapesaregoingtobeuseful andndout howtocalculatethem.PLANESHAPESWaterplanes,transversesections,at decks,bulkheads, the curve of areasandexpansionsofcurvedsurfacesaresomeoftheplaneshapeswhosepropertiesareofinterest.TheareaofasurfaceintheplaneofOxydenedinCartesianco-ordinates,isA =_y dxin which all strips of length y and width cx are summed over the total extent ofx. Because y is rarely, with ship shapes, a precise mathematical function of x theintegration must be carried out by an approximate method which will presentlybe deduced.Fig. 2.12 Longitudinal prismatic coefficient14 BasicshiptheoryTherearerstmomentsofareaabouteachaxis.(FortheguresshowninFig. 2.14,x1and y1are lengths and x and y are co-ordinates.)Myy =_xy1 dx and Mxx =_x1y dyFig. 2.14Dividingeachexpressionbytheareagivestheco-ordinatesof thecentreofarea, ( x,y): x =1A_xy1 dx andy =1A_x1y dyFor the particularcase of a gure bounded on one edge by thex-axisM+y =_12y2dx andy =12A_y21 dxFor a plane gure placed symmetrically about the x-axis such as a waterplane,Mxx =_x1y dy = 0 and the distance of the centre of area, called in the particularcase of a waterplane, the centre of otation (CF), from the y-axis is given by x =MyyA=_xy1 dx_y1 dxFig. 2.15*Notethat My = Mxx.Fig. 2.13Sometools 15It is convenient to examine such a symmetrical gure in relation to the secondmoment of area, since it is normally possible to simplify work by choosing onesymmetrical axisforshipshapes.Thesecondmoments ofareaormomentsofinertia about the two axesfor the waterplane shownin Fig. 2.15 are givenbyIT =13_y31 dx aboutOxforeachhalfIyy =_x2y1 dx aboutOyforeachhalfTheparallel axistheoremshowsthatthesecondmoment ofareaofaplanegure about any axis,Q, of a setof parallel axes isleastwhen thataxis passesthrough the centre of area and that the second moment of area about any otheraxis,R, parallelto Q and at a distance h fromit is givenby (Fig. 2.16)IR = IQAh2Fromthis, itfollowsthattheleast longitudinal secondmoment ofareaofawaterplaneisthataboutanaxisthroughthecentreofotationandgivenby(Fig. 2.17)IL = IyyA x2i.e.IL =_x2y1 dx A x2Fig. 2.16Fig. 2.1716 BasicshiptheoryTHREE- DIMENSIONALSHAPESIt has already been shown how to represent the three-dimensional shape of theship by a plane shape, the curve of areas, by representing each section area by alength (Fig. 2.8). This is one convenient way to represent the three-dimensionalshape of the main underwater form (less appendages). The volume of displace-ment isgiven by\ =_x2x1Adxi.e. it is the sum of all such slices of cross-sectional area A over the total extentof x (Fig. 2.18).Theshapeoftheshipcanequallyberepresentedbyacurveofwaterplaneareasonavertical axis(Fig. 2.19), thebreadthofthecurveatanyheight, z,abovethekeel representingtheareaof thewaterplaneat that draught. Thevolume of displacement is again the sum of all such slices of cross-sectional areaAw, over the total extent of z fromzeroto draught T,\ =_T0Aw dzFig. 2.18Fig. 2.19Sometools 17TherstmomentsofvolumeinthelongitudinaldirectionaboutOzandinthe vertical directionabout the keelare givenbyML =_Ax dx and MV =_T0AWz dzDividing bythe volume ineachcase gives the co-ordinates of the centre of volume.Thecentreof volumeof uiddisplacedbyashipisknownasthecentreofbuoyancy; itsprojectionsintheplanandinsectionareknownasthelongi-tudinal centre of buoyancy (LCB) and the vertical centre of buoyancy (VCB)LCBfromOy =1\_Ax dxVCBabovekeel =1\_AWz dzFig. 2.20 Centre of buoyancy projectionsShould the ship not be symmetrical below the waterline, the centre of buoyancywill not lie in the middle line plane. Its projection in plan may then be referred toasthetransversecentreofbuoyancy(TCB).Hadzbeentakenasthedistancebelow the waterline, the second expression would, of course, represent the pos-ition of the VCBbelowthe waterline. Dening it formally, the centre of buoyancyof a oating body is the centre of volume of the displaceduidinwhichthe body isoating. The rst moment of volume about the centre of volume is zero.The weight of a body is the total of the weights of all of its constituent parts.First moments of the weights about particular axes divided by the total weight,dene the co-ordinates of the centre of weight or centre of gravity (CG) relativeto those axes. Projections of the centre of gravity of a ship in plan and in sectionareknownasthelongitudinal centreofgravity(LCG)andvertical centreofgravity(VCG)and transverse centreof gravity (TCG).LCGfromOy =1W_xdWVCGabovekeel =1W_z dWTCGfrommiddlelineplane =1W_y dW18 BasicshiptheoryDening it formally, the centre of gravity of a body is that point through which,for statical considerations, the whole weight of the body may be assumed to act.The rstmomentof weightabout the centre of gravityis zero.Fig. 2.21 Centre of gravity projectionsMETACENTREConsider any body oating upright and freely at waterline WL, whose centre ofbuoyancy is at B. Let the body now be rotated through a small angle in the planeof the paper without altering the volume of displacement (it is more convenienttodrawifthebodyisassumedxedandthewaterlinerotatedtoW1L1).Thecentre of buoyancy for this new immersed shape is at B1. Lines through Band B1Fig. 2.22Sometools 19normal totheir respective waterlines intersect at Mwhichis knownas themetacentresinceitappearsasifthebodyrotatesaboutitforsmallanglesofrotation. The metacentre is the point of intersection of the normal to a slightlyinclined waterplane of a body, rotated without change of displacement,throughthecentreofbuoyancypertainingtothatwaterplaneandtheverticalplane through the centre of buoyancy pertaining to the upright condition. Thetermmetacentreisreservedforsmall inclinationsfromanuprightcondition.The pointof intersectionof normals throughthe centres of buoyancypertain-ingtosuccessivewaterplanesofabodyrotatedinnitesimallyatanyangleofinclination without change of displacement, iscalled thepro-metacentre.If thebodyisrotatedwithout changeof displacement, thevolumeof theimmersedwedgemustbeequaltothevolumeoftheemergedwedge.Further-more, the transfer of this volume from the emerged to the immersedside mustberesponsibleforthemovementofthecentreofbuoyancyofthewholebodyfrom B to B1; from this weconclude:(a) that the volumes of the two wedges mustbe equal(b) thattherstmomentsofthetwowedgesabouttheirlineofintersectionmust, for equilibrium, be equal and(c) that the transfer of rst moment of the wedges must equal the change in rstmomentof the whole body.Writingdown theseobservationsin mathematicalsymbols,Volumeof immersedwedge =_y 12ycdx= Volumeof emergedwedge1stmomentof immersedwedge =_12y2c_ _23y dx= 1stmomentof emergedwedgeTransferof 1stmomentof wedges = 2 _13y3cdx =23c_y3dxTransferof 1stmomentof wholebody = \BB/ = \ BM c{\ BM c =23c_y3dxButwehavealreadyseenthatI =23_y3dxabouttheaxisofinclinationforboth half waterplanes{BM=I\Thisisanimportantgeometricpropertyofaoatingbody. IftheoatingbodyisashiptherearetwoBMsofparticularinterest,thetransverseBMforrotation about a fore-and-aft axis and the longitudinal BM for rotation about atransverseaxis, thetwoaxes passingthroughthecentreof otationof thewaterplane.20 BasicshiptheoryHOLLOWSHAPESThehull ofashipis,ofcourse,ahollowbodyenclosedbyplating. Itwillbenecessary to nd the weight and positions of the centre of gravity of such shapes.A pseudo-expansionof theshapeisrstobtainedbya methoddescribedfullyinatextbookonlayingo. Briey, thegirthsofsectionareplottedat eachordinate and increased in height by a factor to allow for the dierence betweenprojectedand slant distances inplan. Ameanvalue of this factor is foundfor eachordinate.It is now necessary to apply to each ordinate a mean plating thickness whichmustbefoundbyexaminingtheplatingthicknesses(orweightsperunitarea,sometimescalledpoundages)alongthegirthateachordinate(Fig.2.23).Thevariation is usually not great in girth and an arithmetic mean t/ will be given bydividing the sum of each plate width plate thickness by the girth. If the weightdensityofthematerialisw,theweightofthebottomplatingisthusgivenbyW = w_g/t/ dx and the positionof the LCGis givenby x =wW_g/t/x dxTo nd the position of the VCG, it is necessary to return to the sections andto nd the position by drawing. Each section is divided by trial and error, intofour lengths of equal weight. The mid-points of two adjacent sections are joinedand the mid-points of these lines are joined. The mid-point of the resulting lineFig. 2.23 Bottom platingSometools 21is the required position of the c.g. and its height h above the keel measured. Forthe whole body, then, the positionof the VCG above the keelis givenby

h =wW_g/t/h dxVariousfactorscanbeappliedtotheweightdensityguretoaccountfordierentmethodsofconstruction. Anallowancefortheadditional weightoflaps, if the plating is raised and sunken or clinker, can be made; an addition canbe made for rivet heads. It is unwise to apply any general rule and these factorsmust be calculated foreach case.SYMBOLSANDCONVENTIONSIt wouldbesimplerif everyoneusedthesamesymbolsforthesamethings.Variousinternationalbodiesattempttopromotethisandthesymbolsusedinthisbook,listedat thebeginning,followthegeneralagreements.Thesymbolsand units associated with hydrodynamics are those agreed by the InternationalTowing TankConference.ApproximateintegrationAnumberofdierent propertiesof particularinterest tothenaval architecthavebeenexpressedassimpleintegralsbecausethisisaconvenient formofshorthand. It is not necessary to be familiar with the integral calculus, however,beyondunderstandingthattheelongatedSsign,_,meansthesumofallsuchtypicalpartsthatfollowthesignovertheextentofwhateverfollowsd.Sometextbooks at this stage would use the symbol which is simply the Greek letterS. It is now necessary to adopt various rules for calculating these integrals. Theobvious way to calculate A =_y dx is to plot the curve on squared paper andthen count up all the small squares. This could be extended to calculate the rstmoment, M =_xy dx, inwhichthe number of squares inacolumn, y, ismultipliedbythenumberofsquaresfromtheorigin,x,andthisaddedforallFig. 2.24 VCG of bottom plating22 Basicshiptheoryof the columns that gotomake upthe shape. Clearly, this soonbecomeslaborious andother means of determiningthevalueof anintegral must befound. The integral calculus will be used to deduce some of the rules but thosewhoarenotyetsucientlyfamiliarwiththatsubjectandindeed, bythosewhoaretheyshouldberegardedmerelyastoolsforcalculatingthevariousexpressions shown above in mathematicalshorthand.TRAPEZOIDALRULEA trapezoid is a plane four-sided gure having two sides parallel. If the lengthsofthesesidesarey1andy2andtheyarehapart,theareaofthetrapezoidisgivenbyA =12h(y1y2)Fig. 2.26 A trapezoidA curvilinear gure can be divided into a number of approximate trapezoidsbycoveringit withnequallyspacedordinates, hapart, thebreadths at theordinates in orderbeingy1, y2, y3, . . . , yn.Fig. 2.25 The squared paper approachSometools 23Commencingwiththeleft-handtrapezoid, theareasofeachtrapezoidaregiven by12h(y1y2)12h(y2y3)12h(y3y4) . . .By addition, the total area Aof the gure is givenbyA =12h(y12y22y3 yn)= h12y1y2y3 12yn_ _This is termed the Trapezoidal Rule. Clearly, the more numerous theordinates,themoreaccuratewillbetheanswer.Thus,toevaluatetheexpres-sionA =_y dxtheshapeisdividedintoevenlyspacedsectionshapart, theordinatesmeasuredandsubstitutedintherulegivenabove. If theordinatesrepresent cross-sectional areas of a solid, then the integration gives the volumeof that solid, \ =_Adx.Expressions can be deduced for moments, but these are not as convenient touse as thosethat follow.SIMPSON' SRULESGenerallyknownasSimpson'srules, theserulesforapproximateintegrationwere, infact, deducedbyothermathematiciansmanyyearspreviously. Theyare a special case of the NewtonCotes' rules. Let us deduce a rule for integrat-ing a curve y over the extent of x. It will be convenient to choose the origin to bein the middle of the base 2h long, having ordinates y1, y2and y3. The choice oforigin in noway aectsthe results as the student should verifyforhimself.Assume that the curve can be represented by an equation of the third order,y = a0a1x a2x2a3x3Fig. 2.27 Curvilinear figure represented by trapezoids24 BasicshiptheoryThe areaunder the curveis givenbyA =_hhy dx =_hh(a0a1x a2x2a3x3) dx= a0x a1x22 a2x33 a3x44_ _hh= 2a0h 23a2h3(1)Assume nowthat the areacan be givenby the expressionA = Ly1My2Ny3(2)Nowy1 = a0a1h a2h2a3h3y2 = a0y3 = a0a1h a2h2a3h3Substituting in Equation(2)A = (L MN)a0(L N)a1h (L N)a2h2(L N)a3h3(3)Equatingthe coecients of a in equations (1) and (3)L MN = 2hL N = 0L N =23h___{L = N =13h and M =43hTheareacanberepresentedbyEquation(2), therefore, providedthat thecoecients are thosededuced andA =13hy143hy213hy3 =13h(y14y2y3)Fig. 2.28Sometools 25This is known as Simpson's First Rule or 3 Ordinate Rule. A curved gure canbe divided by any uneven number of equally spaced ordinates h apart. The areawithin ordinatesnumbers1 and 3 isA1 =13h(y14y2y3)within 3 and 5 ordinatesA2 =13h(y34y4y5)within 5 and 7 ordinatesA3 =13h(y54y6y7)and so on.The total areais thereforegivenbyA =13h(y14y22y34y42y54y62y7 yn)i.e.A =23h12y12y2y32y4y52y6y7 12yn_ _Thisisthegeneralizedformoftherstruleappliedtoareas.Thecommonmultiplier is13 the common interval h and the individual multipliers are 1, 4, 2,4, 2, 4, . . . , 2, 4, 1.The rule is one for evaluating an integral. While it has been deduced using areaas an example, it is equally applicable to any integration using Cartesian or polarco-ordinates. Toevaluatetheintegral Mx =_xy dx, forexample, insteadofmultiplying thevalueof yat eachordinatebythe appropriate Simpsonmulti-plier, the value of xy is so treated. Similarly, to evaluate Ix =_x2y dx, the valueof x2yat eachordinateismultipliedbytheappropriateSimpsonmultiplier.Fig. 2.2926 BasicshiptheoryAll of these operations are best performed in a methodical fashion as shown inthe following example and more fully in worked examples later in the chapter.Students should develop a facility in the use of Simpson's rules by practice.EXAMPLE EXAMPLE1.Calculatethevalueoftheintegral_Pdvbetweenthevaluesofvequalto 7 and 15.The valuesof P at equalintervalsof v are as follows:v 7 9 11 13 15P 9 27 36 39 37Solution: The common interval, i.e. the distance between successive values of v,is 2. Settingout Simpson's rulein tabular form,v P Simpson'smultipliersFunctionsof_Pdv7 9 1 99 27 4 10811 36 2 7213 39 4 15615 37 1 37382_Pdv =132 382 = 254.7The unitsare, of course, thoseappropriateto P v.Toapplythetrapezoidalruletoacurvilinearshape,wehadtoassumethatthe relationship between successive ordinates was linear. In applying Simpson'srst rule, we have assumed the relationship to be bounded by an expression ofthe third order. Much greater accuracy is therefore to be expected and for mostfunctions innaval architecture the accuracysoobtainedis quite sucient.Wherethereisknowntobearapidchangeof form, it iswisetoput inanintermediate ordinate and the rule can be adapted to do this. Suppose that therapidchange is knownto be betweenordinates3 and 4 (Fig.2.30).Fig. 2.30Sometools 27Areabetween1and3ords. =13h(y14y2y3)Areabetween3and4ords. =13h2(y34y312y4) =13h12y32y31212y4_ _Areabetween4 and6ords. =13h(y44y5y6)Totalarea =13h y14y2112y32y312112y4_4y52y6 yn_Note that, unlessa second half ordinate is inserted,n must nowbe even.Rules can be deduced, in a similar manner, for gures bounded by unevenlyspacedordinates or dierent numbers of evenlyspacedordinates. For fourevenly spacedordinatesthe rule becomesA =38h(y13y23y3y4)Fig. 2.31This is knownas Simpson's SecondRule. Extendedfor alarge number ofordinates,it becomesA =38h(y13y23y32y43y53y62y7 yn)Thus, the common multiplier in this case is38 times the common interval and theindividual multipliers, 1, 3, 3, 2, 3, 3, 2, 3, 3, 2, . . . , 3, 3, 1. It is suitable for 4, 7,10, 13,16, etc.,ordinates. It can be proved in a mannerexactly similar to thatemployed for the rst rule, assuming a third order curve, and it can be used likethe rst rule to integrateany continuousfunction.Another particular Simpson's rule which will be useful is that which gives thearea between two ordinates when three are known. The area between ordinates1 and 2 is givenbyA1 =112h(5y18y2y3)28 BasicshiptheoryThis rule cannot be used for moments. The rst moment of area of that portionbetweenordinates1 and 2 about number1 ordinate isMx =124h2(3y110y2y3)These two rules are known loosely as the 5,8 minus one Rule and the 3,10 minusone Rule. They are somewhat less accurate than the rst two rules. Incidentally,applying the 5,8minus onerule backwards, the unshaded area of Fig.2.32 isA2 =112h(y18y25y3)and adding this to the expressionforA1, the total area isA =13h(y14y2y3)If the common multiplier has been forgotten, the student can quickly deduce itby applying the particular ruleto a rectangle.Rulescanbecombinedonewithanotherjust astheunit foreachruleiscombined in series to deal with many ordinates. It is important that any discon-tinuity in a curve falls at the end of a unit, e.g. on the 2 multiplier for the rst andsecond rules; if this is so, the rules can be used on curves with discontinuities. Ingeneral, because of dierences in the common interval necessary each side of adiscontinuity, it will be convenient to deal with the two parts separately.The rst rule deals with functions bounded by 3, 5, 7, 9, 11, 13, etc., ordinatesand the second rule with 4, 7, 10, 13, etc., ordinates. Let us deduce a rule for sixordinatesFig. 2.33Fig. 2.32Sometools 29Areabetween1and2ords. =112h(5y18y2y3)= h512y1812y2112y3_ _Areabetween2and5ords. =38h(y23y33y4y5)= h38y298y398y438y5_ _Areabetween5and6ords. =112h(y48y55y6)= h 112y4812y5512y6_ _Total areaA = h512y12524y22524y32524y42524y5512y6_ _=2524h(0.4y1y2y3y4y50.4y6)Table2.1NewtonCotes'rulesNumberof Multipliersforordinatesnumbersordinates 1 2 3 4 5212123164616418383818579032901290329079061928875288502885028875288

7418402168402784027284027840

875117280357717280132317280298917280298917280

998928350588828350928283501049628350454028350

Area = L (Multiplier ordinate)Ordinatesareequallyspacedwithendordinatescoincidingwithends of curve. Multipliers are always symmetrical and areindicatedbydots.30 BasicshiptheoryThesefewSimpson'srules,appliedinarepetitivemanner,havebeenfoundsatisfactory for hand computation for many years. The digital computer makessomewhatdierentdemandsandthemoregeneralizedNewtonCotes' rules,summarized in Table2.1, maybe found moresuitablefor somepurposes.TCHEBYCHEFF' SRULESReturning to Equation (2) under Simpson's rules, the rule required was forcedtotaketheformofthesumofequallyspacedordinates,eachmultipliedbyacoecient. The rule could have been forced to take many forms, most of theminconvenient.One formwhich does yield a convenient rule results fromassuming that the areacanberepresentedbythesumofordinatesplacedaspecialdistancex(whichmay be zero) from the origin, all multiplied by the same coecient, i.e. insteadof assumingthe form as before,assume that the area can be representedbyA = p(y1y2y3). y2beingat the originNowy1 = a0a1x a2x2a3x3y2 = a0y3 = a0a1x a2x2a3x3adding:A = p(3a02a2x2)equating coecients of a above withthose of Equation(1) on p. 253p = 2h {p =23hand2px2=23h3= ph2{x =1

2_h = 0.7071 hThe total shadedareainFig. 2.34cantherefore be calculatedbyerectingordinates equal to0.7071of thehalf lengthfromthemid-point, measuringtheirheightsandthatofthemid-ordinate, addingthethreeheightstogetherand multiplyingthe total by two-thirdsof the half length.A =23h(y1y2y3)where y1and y3are 0.7071of the halflength h, from the mid-point.ThisisTchebyche'sruleforthreeordinates.Similarrulescanbededucedfor 2, 3, 4, 5, 6, 7 and 9 ordinates when spacings become as shown in Table 2.2.Theeight andtenordinaterulespacings havebeendeducedbyapplyingthefourandveordinateruleseachsideofthemid-pointofthehalflength.Sometools 31Table2.2Tchebyche'srulespacingsNumberof Spacingeachsideofmid-ordinateasafactorofthehalflengthh Degreeofordinates,n curve2 0.57735 33 0 0.70711 34 0.18759 0.79465 55 0 0.37454 0.83250 56 0.26664 0.42252 0.86625 77 0 0.32391 0.52966 0.88386 78 0.10268 0.40620 0.59380 0.89733 59 0 0.16791 0.52876 0.60102 0.91159 910 0.08375 0.31273 0.50000 0.68727 0.91625 5Thecommonmultiplierforall rulesisthewholelength2hdividedbyn, thenumberof ordinates used, 2h,n.Tchebyche'srulesareusednotinfrequently, particularlythetenordinaterule,forcalculatingdisplacementfroma`Tchebychebodyplan',i.e.abodyplan drawn with ordinate positions to correspond to the Tchebyche spacings.Areas of the sections are calculatedby Simpson's rules or by otherconvenientmeans, merely added together and multiplied by 2h,n to give volume ofdisplacement. Linesare, infact, oftenfairedonaTchebychebodyplantoavoidthemoreprolongedcalculationbySimpson'sruleswitheachiteration.Since fairing is basically to a curve or areas, this assumes the use of Tchebycheordinatesto dene the body plan.GAUSSRULESIt has been seen that the Simpson rules and NewtonCotes' rules employ equallyspaced ordinates with unequal multipliers and the Tchebyche rules use constantFig. 2.3432 Basicshiptheorymultipliers with unequal ordinate spacing. A third set of rules, the Gauss rules,uses unequal spacing of ordinates and unequal multipliers as shown in Table 2.3.Table2.3Gauss'rulesNumberofordinatesSpacingeachsideofmid-ordinateasafactorofthehalflength.Multiplier2 Spacing 0.57735Multiplier 0.500003 Position 0 0.77460Multiplier 0.44444 0.277784 Position 0.33998 0.86114Multiplier 0.32607 0.173935 Position 0 0.53847 0.90618Multiplier 0.28445 0.23931 0.11846Integral = Sumof products wholebaseThe Gauss rules have the merit of being more accurate thaneither theSimpsonor Tchebycherules, but their applicationinvolves more tediouscalculationwhenmanual methodsareused. ByusingGaussrules, thenavalarchitect caneither obtaingreater accuracy by using the same number ofordinates or obtainthe same accuracywithfewerordinates and in less time.It can be shownthat:(a) aSimpsonrulewithanevennumberofordinatesisonlymarginallymoreaccurate than the next lower odd ordinate rule; odd ordinate Simpson rulesare therefore to be preferred,(b) a Tchebycherule with an even number of ordinates gives the sameaccuracy as the nexthighest odd ordinate rule. Even ordinate Tchebycherulesare therefore to be preferred,(c) aTchebycherulewithanevennumber of ordinates gives anaccuracyrather better thanthe next highest oddordinate Simpsonrule, i.e. thetwo-ordinate Tchebycherule is more accurate thanthe three-ordinateSimpson rule,(d) The ve-ordinate Gauss rule gives an accuracy comparable with thatachieved with nine-ordinateSimpson or Tchebyche rules.EXAMPLE EXAMPLE2. Integratey = tan xfromx = 0tox = :,3bytheveordinaterules of (a) Simpson, (b) NewtonCotes, (c) Tchebyche, (d ) Gauss.Solution: The precise solutionis_:,30tan x = [log cos x[:,30= 0.69315= 0.22064:Sometools 33(a) Simpsonx y SM f (A)0 0120:,12 0.26795 2 0.53590:,6 0.57735 1 0.57735:,4 1.00000 2 2.00000:,3 1.73205120.866033.97928Area =23:123.97928 = 0.22107:(b) NewtonCotesx y M f (A)0 0 7 0:,12 0.26795 32 8.57440:,6 0.57735 12 6.92820:,4 1.00000 32 32.00000:,3 1.73205 7 12.1243559.62695Area =190:359.62695 = 0.22084:(c) Tchebychex y:,6(1 0.83250) = 5.0250.08793:,6(1 0.37454) = 18.7640.33972:,6 = 300.57735:,6(1 0.37454) = 41.2360.87655:,6(1 0.83250) = 54.9751.426743.30829Area =:153.30829 = 0.22055:(d ) Gaussx y M f (A):,6(1 0.90618) = 2.8150.04918 0.11846 0.00583:,6(1 0.53847) = 13.8460.24647 0.23931 0.05898:,6 = 300.57735 0.28445 0.16423:,6(1 0.53847) = 46.1541.04114 0.23931 0.24916:,6(1 0.90618) = 57.1851.55090 0.11846 0.183720.66192Area =:30.66192 = 0.22064:34 BasicshiptheoryComputersDIGITALCOMPUTERSA digital computer is an electronic device capable of holding large amounts ofdata andof carrying out arithmetical computations andgroups of logicalprocessesathighspeed.Assuchitisasmuchatoolforthenavalarchitecttouse, asthesliderule, ortheruleforcalculatingareasandvolumes. It is, ofcourse, muchmore powerful but that makesit even more important to under-stand. It isnot necessarytoknowindetail howit worksbut itsbasicchar-acteristics,strengthsand weaknesses,should be understood.The system includes input units which accept information in a suitably codedform(CD-romor disk readers, keyboards, optical readers or light pens);storageormemoryunitsforholdinginstructions;acalculationunitbywhichdataismanipulated; acontrol unit whichcallsupdataandprogramsfromstorageinthecorrectsequenceforusebythecalculationunitandthenpassestheresultstooutputunits; outputunitsforpresentingresults(visual displayunits, printers, or plotters);and a power unit.The immediate output may be a magnetic tape or disk which can be decodedlateronseparateprint-out devices. Sometimestheoutput isuseddirectlytocontrolamachinetoolorautomaticdraughtingequipment.Inputandoutputunits mayberemotefromthecomputer itself, providinganumber of out-stationswithaccesstoalargecentralcomputer,orprovidinganetworkwiththe ability to interactwithother users.Aswithanyotherformofcommunication, thatbetweenthedesignerandthe computermust be conducted in a language understood by both. Therearemany such languages suitable for scientic, engineering and commercial work.Thecomputeritself usesacompilertotranslatetheinput languageintothemorecomplex machine languageit needs forecient working.Input systems are usuallyinteractive, enablingadesigner toengage inadialogue withthe computer or, more accurately, withthe software inthecomputer. Software is of twomaintypes; that whichcontrols the generalactivitieswithinthecomputer(e.g.howdataisstoredandaccessed)andthatwhichdirectshowaparticularproblemistobetackled. Thecomputermayprompt the operator by asking for more data or for a decision between possibleoptionsitpresentstohim. Thesoftwarecanincludedecisionaids, i.e. itcanrecommendaparticularoptionasthebestinthecircumstancesandgiveitsreasons if requested. If the operator makes, or appears to make, a mistake themachine can challenge the input.Displays can be in colour and present data in graphical form. Colour can beuseful in dierentiating between dierent elements of the total display. Red canbeusedtohighlighthazardoussituationsbecausehumansassociateredwithdanger. However, forsomeapplicationsmonochromeissuperior. Shadesofonecolourcanmorereadilyindicatetherelativemagnitudeofasinglepara-meter, e.g. shades of blueindicatingwaterdepth.Graphical displays are oftenmoremeaningful tohumansthanlongtabulationsofgures. Thusaplotofpoints which should lie on smooth curves will quickly highlight a rogue reading.Sometools 35Thisfacilityisusedasaninputcheckinlargeniteelementcalculations.Thecomputer can cause the display to rotate so that a complex shape, a ship's hullfor instance, can be viewed from a number of directions. The designer can viewaspaceor equipment fromanychosenposition. Inthis waychecks canbemade, as the design progresses, on the acceptability of various sight lines. Canoperators see all the displays in, say, the Operations Room they need to in orderto carryout their tasks? Equally, maintainers can check thatthere isadequatespace around an equipmentfor opening it upand workingon it.Takingthis onestagefurther, thecomputer cangeneratewhat is termedvirtual reality(VR). TheuserofVRiseectivelyimmersedin, andinteractswith, acomputer generatedenvironment. Typicallyahelmet, or headset, isworn which has a stereoscopic screen for each eye. Users are tted with sensorswhich pick up their movements and the computer translates these into changingpictures on the screens. Thus an owner could be taken on a `walk through' of aplanned vessel, or those responsible for layouts can be given a tour of the space.Colours, surface textures and lighting can all be represented. Such methods arecapableofreplacingthetraditional mock-upsandthe3-Dand2-Dlineoutsusedduringconstruction. All this canbe done before anysteel is cut. Toenhancethesenseofrealismgloves,orsuits,withforcefeedbackdevicescanbe worn to provide a sense of touch. Objects can be `picked up' and `manipul-ated' in the virtual environment.It does not follow that because a computer can be used to provide a service itshouldbesoused. Itcanbeexpensivebothinmoneyandtime. Thusintheexampleabove, thecomputermaybecheaperthanafull scalemock-upbutwould a small scale model suce? A computer is likely to be most cost eectiveaspartofacomprehensivesystem.Thatis,aspartofacomputeraideddesignand manufacture system(CAD/CAM). Insuchsystems the designer uses aterminal toaccess data anda complete suite of design programs. Severalsystems have been developed for ship design, some concentrating on the initialdesign phase and others on the detailed design process and its interaction withproduction.Thusoncethecomputerholdsadenitionofthehullshapethatshapecanbecalledupforsubsequentmanipulationandforall calculations,layouts, etc. forwhichitisneeded. Thisclearlyreducesthechanceoferrors.Again, once the structure has been designed the computer can be programmedtogenerateamaterialsorderinglist. Thengivensuitableinputs it cankeeptrackofthematerialthroughthestoresandworkshops.Thecomplexityofaship, andthemanyinter-relationships betweenits component elements, aresuchthat it is anideal candidatefor computerization. Thechallengelies inestablishing allthe interactions.In the case of the design and build of a Landing Platform Dock (LPD) for theUK MOD a system was used involving 250 work stations for creating 2-D and3-D geometry and data, 125 work stations for viewing the data and 140 PCs foraccessingandcreatingtheproduct denitionmodel data. Thesystemenabled`virtual prototyping' andearlycustomer approval of subjectiveareas of theship. Amongother uses of thecomputer whichareof interest tothenavalarchitect, are:36 BasicshiptheorySimulation modelling. Provided that the factors governing a real life situationareunderstood, it maybepossibletorepresent it byaset of mathematicalrelationships. In other words it can be modelled and the model used to study theeectsofchangingsomeofthefactorsmuchquicker,cheaperandsaferthancouldbeachievedwithfull scaleexperimentation. Considertankersarrivingataterminal.Factorsinuencingthesmoothoperationarenumbersofships,arrivalintervals,shipsize,dischargerateandstoragecapacities.Asimulationmodel could be produced to study the problem, reduce the queuing time and toseetheeectsofadditionalberthsanddierentdischargeratesandshipsize.The economics of such a procedure is also conducive to this type of modelling.Expert systems and decision aiding. Humans canreasonandlearnfromprevious experience. Articial Intelligence(AI) isconcernedwithdevelopingcomputer-based systems endowed with such higher intellectual processes. It hasbeen possible to program them to carry out fairly complex tasks such as playingchess. Thecomputerusesitshighspeedtoconsiderall possibleoptionsandtheir consequences and to develop a winning strategy. Such programs are calledexpert systems. These systems can make use of human inputs to help decide onthe signicance of certain situations and what action is advisable. These know-ledge-based expert systems have been used as decision aids. An early applicationof such techniques was in medicine. Medical ocers could not be provided forall ballistic missile submarines, but they did carry a qualied sick berth attend-ant(SBA).TheSBAwouldexamineasickcrewmember,takingtemperatureandotherreadingstofeedintoacomputerprogramcontainingcontributionsfromdistinguisheddoctors.Thecomputerthenanalyzedthedataitreceived,decided what might be wrong with the patient and asked for additional facts tonarrow down the possibilities. The end result was a print out of the most likelycomplaints and their probability. This enabled the command to decide whether,or not, to abortthe mission.In the sameway, the navalarchitectcan developdecisionaidsfor problemswhere a number of options are available and experience is useful. Such aids canenlist thehelpof leadingdesigners, makingtheirexpertiseavailabletoeveninexperiencedsta.SIMULATORSAnticipatingsomeof theworkof laterchapters, thebehaviourof ashipinresponse to applied forces and moments can be represented by a mathematicalequation.Theappliedforcesmayarisefromthedeliberateactionofthoseonboardinmovingacontrol surfacesuchas arudder or fromsomeexternalagencysuch as the seaway in which the shipis operating.In itssimplest form,the equationmaybe alinear dierential equationinvolvingone degree offreedombut, toachievegreateraccuracy, mayincludenon-linearandcross-coupling terms.The same formof equation canbe represented by a suitably contrivedelectricalcircuit.Inthecaseofashipturningundertheactionoftherudder,ifthecomponentsoftheelectricalcircuitarecorrectlychosen, byvaryinganinputsignalinconformitywiththeruddermovements,thevoltageacrosstwoSometools 37points of the circuit canbe measuredtorepresent the ship's heading. Byextending the circuitry, more variables can be studied such as the angle of heel,drift angle andadvance. T