Ship Hydrostatics and Stability - A B Biran PDF

Ship Hydrostatics and Stability Ship Hydrostatics and Stability A.B. Biran Technion – Faculty of Mechanical Engineering AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann An imprint of Elsevier Linacre House, Jordan Hil...

Ship Hydrostatics and Stability Ship Hydrostatics and Stability A.B. Biran Technion – Faculty of Mechanical Engineering AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann An imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803 First published 2003 Copyright
c 2003, A.B. Biran. All rights reserved The right of A.B. Biran to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK. Phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 4988 7 For information on all Butterworth-Heinemann publications visit our website at www.bh.com Typeset by Integra Software Services Pvt. Ltd, Pondicherry, India www.integra-india.com Printed and bound in Great Britain To my wife Suzi Contents Preface xiii Acknowledgements xvii 1 Definitions, principal dimensions 1 1.1 Introduction........................... 1 1.2 Marine terminology....................... 2 1.3 The principal dimensions of a ship............... 3 1.4 The definition of the hull surface................ 9 1.4.1 Coordinate systems................... 9 1.4.2 Graphic description................... 11 1.4.3 Fairing.......................... 13 1.4.4 Table of offsets..................... 15 1.5 Coefficients of form....................... 15 1.6 Summary............................. 19 1.7 Example............................. 20 1.8 Exercises............................. 21 2 Basic ship hydrostatics 23 2.1 Introduction........................... 23 2.2 Archimedes’ principle...................... 24 2.2.1 A body with simple geometrical form......... 24 2.2.2 The general case..................... 29 2.3 The conditions of equilibrium of a floating body........ 32 2.3.1 Forces.......................... 33 2.3.2 Moments........................ 34 2.4 A definition of stability..................... 36 2.5 Initial stability.......................... 37 2.6 Metacentric height........................ 39 2.7 A lemma on moving volumes or masses............ 40 2.8 Small angles of inclination................... 41 2.8.1 A theorem on the axis of inclination.......... 41 2.8.2 Metacentric radius.................... 44 2.9 The curve of centres of buoyancy................ 45 2.10 The metacentric evolute..................... 47 2.11 Metacentres for various axes of inclination........... 47 viii Contents 2.12 Summary............................. 48 2.13 Examples............................. 50 2.14 Exercises............................. 67 2.15 Appendix – Water densities................... 70 3 Numerical integration in naval architecture 71 3.1 Introduction........................... 71 3.2 The trapezoidal rule....................... 72 3.2.1 Error of integration by the trapezoidal rule....... 75 3.3 Simpson’s rule.......................... 77 3.3.1 Error of integration by Simpson’s rule......... 79 3.4 Calculating points on the integral curve............. 80 3.5 Intermediate ordinates...................... 83 3.6 Reduced ordinates........................ 84 3.7 Other procedures of numerical integration........... 85 3.8 Summary............................. 86 3.9 Examples............................. 87 3.10 Exercises............................. 90 4 Hydrostatic curves 91 4.1 Introduction........................... 91 4.2 The calculation of hydrostatic data............... 92 4.2.1 Waterline properties................... 92 4.2.2 Volume properties.................... 95 4.2.3 Derived data....................... 96 4.2.4 Wetted surface area................... 98 4.3 Hydrostatic curves........................ 99 4.4 Bonjean curves and their use.................. 101 4.5 Some properties of hydrostatic curves.............. 104 4.6 Hydrostatic properties of affine hulls.............. 107 4.7 Summary............................. 108 4.8 Example............................. 109 4.9 Exercises............................. 109 5 Statical stability at large angles of heel 111 5.1 Introduction........................... 111 5.2 The righting arm......................... 111 5.3 The curve of statical stability.................. 114 5.4 The influence of trim and waves................. 116 5.5 Summary............................. 117 5.6 Example............................. 119 5.7 Exercises............................. 119 6 Simple models of stability 121 6.1 Introduction........................... 121  Contents ix 6.2 Angles of statical equilibrium.................. 124 6.3 The wind heeling arm...................... 124 6.4 Heeling arm in turning...................... 126 6.5 Other heeling arms........................ 127 6.6 Dynamical stability....................... 128 6.7 Stability conditions – a more rigorous derivation........ 131 6.8 Roll period............................ 133 6.9 Loads that adversely affect stability............... 135 6.9.1 Loads displaced transversely.............. 135 6.9.2 Hanging loads...................... 136 6.9.3 Free surfaces of liquids................. 137 6.9.4 Shifting loads...................... 141 6.9.5 Moving loads as a case of positive feedback...... 142 6.10 The stability of grounded or docked ships............ 144 6.10.1 Grounding on the whole length of the keel....... 144 6.10.2 Grounding on one point of the keel........... 145 6.11 Negative metacentric height................... 146 6.12 The limitations of simple models................ 150 6.13 Other modes of capsizing.................... 151 6.14 Summary............................. 152 6.15 Examples............................. 154 6.16 Exercises............................. 155 7 Weight and trim calculations 159 7.1 Introduction........................... 159 7.2 Weight calculations....................... 160 7.2.1 Weight groups...................... 160 7.2.2 Weight calculations................... 161 7.3 Trim............................... 164 7.3.1 Finding the trim and the draughts at perpendiculars.. 164 7.3.2 Equilibrium at large angles of trim........... 165 7.4 The inclining experiment.................... 166 7.5 Summary............................. 171 7.6 Examples............................. 172 7.7 Exercises............................. 174 8 Intact stability regulations I 177 8.1 Introduction........................... 177 8.2 The IMO code on intact stability................ 178 8.2.1 Passenger and cargo ships................ 178 8.2.2 Cargo ships carrying timber deck cargoes....... 182 8.2.3 Fishing vessels..................... 182 8.2.4 Mobile offshore drilling units.............. 183 8.2.5 Dynamically supported craft.............. 183 8.2.6 Container ships greater than 100 m........... 185 x Contents 8.2.7 Icing........................... 185 8.2.8 Inclining and rolling tests................ 185 8.3 The regulations of the US Navy................. 185 8.4 The regulations of the UK Navy................. 190 8.5 A criterion for sail vessels.................... 192 8.6 A code of practice for small workboats and pilot boats..... 194 8.7 Regulations for internal-water vessels.............. 196 8.7.1 EC regulations...................... 196 8.7.2 Swiss regulations.................... 196 8.8 Summary............................. 197 8.9 Examples............................. 198 8.10 Exercises............................. 201 9 Parametric resonance 203 9.1 Introduction........................... 203 9.2 The influence of waves on ship stability............. 204 9.3 The Mathieu effect – parametric resonance........... 207 9.3.1 The Mathieu equation – stability............ 207 9.3.2 The Mathieu equation – simulations.......... 211 9.3.3 Frequency of encounter................. 215 9.4 Summary............................. 216 9.5 Examples............................. 217 9.6 Exercise............................. 219 10 Intact stability regulations II 221 10.1 Introduction........................... 221 10.2 The regulations of the German Navy.............. 221 10.2.1 Categories of service.................. 222 10.2.2 Loading conditions................... 222 10.2.3 Trochoidal waves.................... 223 10.2.4 Righting arms...................... 227 10.2.5 Free liquid surfaces................... 227 10.2.6 Wind heeling arm.................... 228 10.2.7 The wind criterion.................... 229 10.2.8 Stability in turning................... 230 10.2.9 Other heeling arms................... 231 10.3 Summary............................. 231 10.4 Examples............................. 232 10.5 Exercises............................. 236 11 Flooding and damage condition 239 11.1 Introduction........................... 239 11.2 A few definitions......................... 241 11.3 Two methods for finding the ship condition after flooding... 243 11.3.1 Lost buoyancy...................... 246  Contents xi 11.3.2 Added weight...................... 248 11.3.3 The comparison..................... 250 11.4 Details of the flooding process................. 251 11.5 Damage stability regulations.................. 252 11.5.1 SOLAS......................... 252 11.5.2 Probabilistic regulations................. 254 11.5.3 The US Navy...................... 256 11.5.4 The UK Navy...................... 257 11.5.5 The German Navy.................... 258 11.5.6 A code for large commercial sailing or motor vessels. 259 11.5.7 A code for small workboats and pilot boats....... 259 11.5.8 EC regulations for internal-water vessels........ 260 11.5.9 Swiss regulations for internal-water vessels...... 260 11.6 The curve of floodable lengths.................. 261 11.7 Summary............................. 263 11.8 Examples............................. 265 11.9 Exercise............................. 268 12 Linear ship response in waves 269 12.1 Introduction........................... 269 12.2 Linear wave theory....................... 270 12.3 Modelling real seas....................... 273 12.4 Wave induced forces and motions................ 277 12.5 A note on natural periods.................... 281 12.6 Roll stabilizers.......................... 283 12.7 Summary............................. 286 12.8 Examples............................. 287 12.9 Exercises............................. 290 12.10 Appendix – The relationship between curl and rotation..... 290 13 Computer methods 293 13.1 Introduction........................... 293 13.2 Geometric introduction..................... 294 13.2.1 Parametric curves.................... 294 13.2.2 Curvature........................ 295 13.2.3 Splines.......................... 296 13.2.4 Bézier curves...................... 298 13.2.5 B-splines........................ 302 13.2.6 Parametric surfaces................... 303 13.2.7 Ruled surfaces...................... 305 13.2.8 Surface curvatures.................... 305 13.3 Hull modelling.......................... 308 13.3.1 Mathematical ship lines................. 308 13.3.2 Fairing.......................... 308 13.3.3 Modelling with MultiSurf and SurfaceWorks...... 308 xii Contents 13.4 Calculations without and with the computer.......... 316 13.4.1 Hydrostatic calculations................. 317 13.5 Simulations........................... 319 13.5.1 A simple example of roll simulation.......... 322 13.6 Summary............................. 324 13.7 Examples............................. 326 13.8 Exercises............................. 326 Bibliography 327 Index 337 Preface This book is based on a course of Ship Hydrostatics delivered during a quarter of a century at the Faculty of Mechanical Engineering of the Technion–Israel Institute of Technology. The book reflects the author’s own experience in design and R&D and incorporates improvements based on feedback received from students. The book is addressed in the first place to undergraduate students for whom it is a first course in Naval Architecture or Ocean Engineering. Many sections can be also read by technicians and ship officers. Selected sections can be used as reference text by practising Naval Architects. Naval Architecture is an age-old field of human activity and as such it is much affected by tradition. This background is part of the beauty of the profession. The book is based on this tradition but, at the same time, the author tried to write a modern text that considers more recent developments, among them the theory of parametric resonance, also known as Mathieu effect, the use of personal computers, and new regulations for intact and damage stability. The Mathieu effect is believed to be the cause of many marine disasters. German researchers were the first to study this hypothesis. Unfortunately, in the first years of their research they published their results in German only. The German Federal Navy – Bundesmarine – elaborated stability regulations that allow for the Mathieu effect. These regulations were subsequently adopted by a few additional navies. Proposals have been made to consider the effect of waves for merchant vessels too. Very powerful personal computers are available today; their utility is enhanced by many versatile, user-friendly software packages. PC programmes for hydro- static calculations are commercially available and their prices vary from several hundred dollars, for the simplest, to many thousands for the more powerful. Programmes for particular tasks can be written by a user familiar with a good software package. To show how to do it, this book is illustrated with a few examples calculated in Excel and with many examples written in MATLAB. MATLAB is an increasingly popular, comprehensive computing environment characterized by an interactive mode of work, many built-in functions, imme- diate graphing facilities and easy programming paradigms. Readers who have access to MATLAB, even to the Students’ Edition, can readily use those exam- ples. Readers who do not work in MATLAB can convert the examples to other programming languages. Several new stability regulations are briefly reviewed in this book. Students and practising Naval Architects will certainly welcome the description of such rules and examples of how to apply them. xiv Preface This book is accompanied by a selection of freely downloadable MATLAB files for hydrostatic and stability calculations. In order to access this mate- rial please visit www.bh.com/companions/ and follow the instructions on the screen. About this book Theoretical developments require an understanding of basic calculus and analytic geometry. A few sections employ basic vector calculus, differential geometry or ordinary differential equations. Students able to read them will gain more insight into matters explained in the book. Other readers can skip those sections without impairing their understanding of practical calculations and regulations described in the text. Chapter 1 introduces the reader to basic terminology and to the subject of hull definition. The definitions follow new ISO and ISO-based standards. Trans- lations into French, German and Italian are provided for the most important terms. The basic concepts of hydrostatics of floating bodies are described in Chap- ter 2; they include the conditions of equilibrium and initial stability. By the end of this chapter, the reader knows that hydrostatic calculations require many inte- grations. Methods for performing such integrations in Naval Architecture are developed in Chapter 3. Chapter 4 shows how to apply the procedures of numerical integration to the calculation of actual hydrostatic properties. Other matters covered in the same chapter are a few simple checks of the resulting plots, and an analysis of how the properties change when a given hull is subjected to a particular class of transformations, namely the properties of affine hulls. Chapter 5 discusses the statical stability at large angles of heel and the curve of statical stability. Simple models for assessing the ship stability in the presence of various heel- ing moments are developed in Chapter 6. Both static and dynamic effects are considered, as well as the influence of factors and situations that negatively affect stability. Examples of the latter are displaced loads, hanging loads, free liquid sur- faces, shifting loads, and grounding and docking. Three subjects closely related to practical stability calculations are described in Chapter 7: Weight and trim calculations and the inclining experiment. Ships and other floating structures are approved for use only if they comply with pertinent regulations. Regulations applicable to merchant ships, ships of the US Navy and UK Navy, and small sail or motor craft are summarily described in Chapter 8. The phenomenon of parametric resonance, or Mathieu effect, is briefly descri- bed in Chapter 9. The chapter includes a simple criterion of distinguishing between stable and unstable solutions and examples of simple simulations in MATLAB.  Preface xv Ships of the German Federal Navy are designed according to criteria that take into account the Mathieu effect: they are introduced in Chapter 10. Chapters 8 and 10 deal with intact ships. Ships and some other floating struc- tures are also required to survive after a limited amount of flooding. Chapter 11 shows how to achieve this goal by subdividing the hull by means of watertight bulkheads. There are two methods of calculating the ship condition after dam- age, namely the method of lost buoyancy and the method of added weight. The difference between the two methods is explained by means of a simple example. The chapter also contains short descriptions of several regulations for merchant and for naval ships. Chapters 8, 10 and 11 inform the reader about the existence of requirements issued by bodies that approve the design and the use of ships and other floating bodies, and show how simple models developed in previous chapters are applied in engineering calculations. Not all the details of those regulations are included in this book, neither all regulations issued all over the world. If the reader has to perform calculations that must be submitted for approval, it is highly recom- mended to find out which are the relevant regulations and to consult the complete, most recent edition of them. Chapter 12 goes beyond the traditional scope of Ship Hydrostatics and pro- vides a bridge towards more advanced and realistic models. The theory of linear waves is briefly introduced and it is shown how real seas can be described by the superposition of linear waves and by the concept of spectrum. Floating bodies move in six degrees of freedom and the spectrum of those motions is related to the sea spectrum. Another subject introduced in this chapter is that of tank stabilizers, a case in which surfaces of free liquids can help in reducing the roll amplitude. Chapter 13 is about the use of modern computers in hull definition, hydro- static calculations and simulations of motions. The chapter introduces the basic concepts of computer graphics and illustrates their application to hull defini- tion by means of the MultiSurf and SurfaceWorks packages. A roll simulation in SIMULINK, a toolbox of MATLAB, exemplifies the possibilities of modern simulation software. Using this book Boldface words indicate a key term used for the first time in the text, for instance length between perpendiculars. Italics are used to emphasize, for example equilibrium of moments. Vectors are written with a line over their name: KB, GM. Listings of MATLAB programmes, functions and file names are written in typewriter characters, for instance mathisim.m. Basic ideas are exemplified on simple geometric forms for which analytic solutions can be readily found. After mastering these ideas, the students should practise on real ship data provided in examples and exercises, at the end of each chapter. The data of an existing vessel, called Lido 9, are used throughout the xvi Preface book to illustrate the main concepts. Data of a few other real-world vessels are given in additional examples and exercises. I am closing this preface by paying a tribute to the memory of those who taught me the profession, Dinu Ilie and Nicolae Pârâianu, and of my colleague in teaching, Pinkhas Milkh. Acknowledgements The first acknowledgements should certainly go to the many students who took the course from which emerged this book. Their reactions helped in identifying the topics that need more explanations. Naming a few of those students would imply the risk of being unfair to others. Many numerical examples were calculated with the aid of the programme system ARCHIMEDES. The TECHNION obtained this software by the courtesy of Heinrich Söding, then at the Technical University of Hannover, now at the Technical University of Hamburg. Included with the programme source there was a set of test data that describe a vessel identified as Ship No. 83074. Some examples in this book are based on that data. Sol Bodner, coordinator of the Ship Engineering Program of the Technion, provided essential support for the course of Ship Hydrostatics. Itzhak Shaham and Jack Yanai contributed to the success of the programme. Paul Münch provided data of actual vessels and Lido Kineret, Ltd and the Özdeniz Group, Inc. allowed us to use them in numerical examples. Eliezer Kantorowitz read initial drafts of the book proposal. Yeshayahu Hershkowitz, of Lloyd’s Register, and Arnon Nitzan, then student in the last graduate year, read the final draft and returned helpful comments. Reinhard Siegel, of AeroHydro, provided the drawing on which the cover of the book is based, and helped in the application of MultiSurf and SurfaceWorks. Antonio Tiano, of the University of Pavia, gave advice on a few specialized items. Dan Livneh, of the Israeli Administration of Shipping and Ports, provided updating on international codes of practice. C.B. Barrass reviewed the first eleven chapters and provided helpful comments. Richard Barker drew the attention of the author to the first uses of the term Naval Architecture. The common love for the history of the profession enabled a pleasant and interesting dialogue. Naomi Fernandes of MathWorks, Baruch Pekelman, their agent in Israel, and his assistants enabled the author to use the latest MATLAB developments. The author thanks Addison-Wesley Longman, especially Karen Mosman and Pauline Gillet, for permission to use material from the book MATLAB for Engi- neers written by him and Moshe Breiner. The author thanks the editors of Elsevier, Rebecca Hamersley, Rebecca Rue, Sallyann Deans and Nishma Shah for their cooperation and continuous help. It was the task of Nishma Shah to bring the project into production. Finally, the author appreciates the way Padma Narayanan, of Integra Software Services, managed the production process of this book. 1 Definitions, principal dimensions 1.1 Introduction The subjects treated in this book are the basis of the profession called Naval Architecture. The term Naval Architecture comes from the titles of books pub- lished in the seventeenth century. For a long time, the oldest such book we were aware of was Joseph Furttenbach’s Architectura Navalis published in Frankfurt in 1629. The bibliographical data of a beautiful reproduction are included in the references listed at the end of this book. Close to 1965 an older Portuguese manuscript was rediscovered in Madrid, in the Library of the Royal Academy of History. The work is due to João Baptista Lavanha and is known as Livro Primeiro da Architectura Naval, that is ‘First book on Naval Architecture’. The traditional dating of the manuscript is 1614. The following is a quotation from a translation due to Richard Barker: Architecture consists in building, which is the permanent construc- tion of any thing. This is done either for defence or for religion, and utility, or for navigation. And from this partition is born the division of Architecture into three parts, which are Military, Civil and Naval Architecture. And Naval Architecture is that which with certain rules teaches the building of ships, in which one can navigate well and conveniently. The term may be still older. Thomas Digges (English, 1546–1595) published in 1579 an Arithmeticall Militarie Treatise, named Stratioticos in which he promised to write a book on ‘Architecture Nautical’. He did not do so. Both the British Royal Institution of Naval Architects – RINA – and the American Society of Naval Architects and Marine Engineers – SNAME – opened their websites for public debates on a modern definition of Naval Architecture. Out of the many proposals appearing there, that provided by A. Blyth, FRINA, looked to us both concise and comprehensive: Naval Architecture is that branch of engineering which embraces all aspects of design, research, developments, construction, trials 2 Ship Hydrostatics and Stability and effectiveness of all forms of man-made vehicles which operate either in or below the surface of any body of water. If Naval Architecture is a branch of Engineering, what is Engineering? In the New Encyclopedia Britannica (1989) we find: Engineering is the professional art of applying science to the optimum conversion of the resources of nature to the uses of mankind. Engineering has been defined by the Engineers Council for Professional Development, in the United States, as the creative application of “scientific principles to design or develop structures, machines... ” This book deals with the scientific principles of Hydrostatics and Stability. These subjects are treated in other languages in books bearing titles such as Ship theory (for example Doyère, 1927) or Ship statics (for example Hervieu, 1985). Further scientific principles to be learned by the NavalArchitect include Hydrodynamics, Strength, Motions on Waves and more. The ‘art of applying’ these principles belongs to courses in Ship Design. 1.2 Marine terminology Like any other field of engineering, Naval Architecture has its own vocabulary composed of technical terms. While a word may have several meanings in com- mon language, when used as a technical term, in a given field of technology, it has one meaning only. This enables unambigous communication within the profession, hence the importance of clear definitions. The technical vocabulary of people with long maritime tradition has peculiar- ities of origins and usage. As a first important example in English let us consider the word ship; it is of Germanic origin. Indeed, to this day the equivalent Dan- ish word is skib, the Dutch, schep, the German, Schiff (pronounce ‘shif’), the Norwegian skip (pronounce ‘ship’), and the Swedish, skepp. For mariners and Naval Architects a ship has a soul; when speaking about a ship they use the pronoun ‘she’. Another interesting term is starboard; it means the right-hand side of a ship when looking forward. This term has nothing to do with stars. Pictures of Viking vessels (see especially the Bayeux Tapestry) show that they had a steering board (paddle) on their right-hand side. In Norwegian a ‘steering board’ is called ‘styri bord’. In old English the Nordic term became ‘steorbord’ to be later distorted to the present-day ‘starboard’. The correct term should have been ‘steeringboard’. German uses the exact translation of this word, ‘Steuerbord’. The left-hand side of a vessel was called larboard. Hendrickson (1997) traces this term to ‘lureboard’, from the Anglo-Saxon word ‘laere’ that meant empty, because the steersman stood on the other side. The term became ‘lade-board’ and  Definitions, principal dimensions 3 ‘larboard’because the ship could be loaded from this side only. Larboard sounded too much like starboard and could be confounded with this. Therefore, more than 200 years ago the term was changed to port. In fact, a ship with a steering board on the right-hand side can approach to port only with her left-hand side. 1.3 The principal dimensions of a ship In this chapter we introduce the principal dimensions of a ship, as defined in the international standard ISO 7462 (1985). The terminology in this document was adopted by some national standards, for example the German standard DIN 81209-1. We extract from the latter publication the symbols to be used in draw- ings and equations, and the symbols recommended for use in computer programs. Basically, the notation agrees with that used by SNAME and with the ITTC Dictionary of Ship Hydrodynamics (RINA, 1978). Much of this notation has been used for a long time in English-speaking countries. Beyond this chapter, many definitions and symbols appearing in this book are derived from the above-mentioned sources. Different symbols have been in use in continental Europe, in countries with a long maritime tradition. Hervieu (1985), for example, opposes the introduction of Anglo-Saxon notation and justifies his attitude in the Introduction of his book. If we stick in this book to a certain notation, it is not only because the book is published in the UK, but also because English is presently recognized as the world’s lingua franca and the notation is adopted in more and more national standards. As to spelling, we use the British one. For example, in this book we write ‘centre’, rather than ‘center’ as in the American spelling, ‘draught’ and not ‘draft’, and ‘moulded’ instead of ‘molded’. To enable the reader to consult technical literature using other symbols, we shall mention the most important of them. For ship dimensions we do this in Table 1.1, where we shall give also translations into French and German of the most important terms, following mainly ISO 7462 and DIN 81209-1. In addition, Italian terms will be inserted and they conform to Italian technical literature, for example Costaguta (1981). The translations will be marked by ‘Fr’ for French, ‘G’for German and ‘I’for Italian.Almost all ship hulls are symmetric with respect with a longitudinal plane (plane xz in Figure 1.6). In other words, ships present a ‘port-to-starboard’ symmetry. The definitions take this fact into account. Those definitions are explained in Figures 1.1 to 1.4. The outer surface of a steel or aluminium ship is usually not smooth because not all plates have the same thickness. Therefore, it is convenient to define the hull surface of such a ship on the inner surface of the plating. This is the Moulded sur- face of the hull. Dimensions measured to this surface are qualified as Moulded. By contrast, dimensions measured to the outer surface of the hull or of an appendage are qualified as extreme. The moulded surface is used in the first stages of ship design, before designing the plating, and also in test-basin studies. 4 Ship Hydrostatics and Stability Table 1.1 Principal ship dimensions and related terminology English term Symbol Computer Translations notation After (aft) perpendicular AP Fr perpendiculaire arrière, G hinteres Lot, I perpendicolare addietro Baseline BL Fr ligne de base, G Basis, I linea base Bow Fr proue, l’avant, G Bug, I prora, prua Breadth B B Fr largeur, G Breite, I larghezza Camber Fr bouge, G Balkenbucht, I bolzone Centreline plane CL Fr plan longitudinal de symétrie, G Mittschiffsebene, I Piano di simmetria, piano diametrale Depth D DEP Fr creux, G Seitenhöhe, I altezza Depth, moulded Fr creux sur quille, G Seitenhöhe, I altezza di costruzione (puntale) Design waterline DWL DWL Fr flottaison normale, G Konstruktionswasserlinie (KWL), I linea d’acqua del piano di costruzione Draught T T Fr tirant d’eau, G Tiefgang, I immersione Draught, aft TA TA Fr tirant d’eau arrière, G Hinterer Tiefgang, I immersiona a poppa Draught, amidships TM Fr tirant d’eau milieu, G mittleres Tiefgang, I immersione media Draught, extreme Fr profondeur de carène hors tout, G größter Tiefgang, I pescaggio Draught, forward TF TF Fr tirant d’eau avant, G Vorderer Tiefgang, I immersione a prora Draught, moulded Fr profondeur de carène hors membres, Forward perpendicular FP Fr perpendiculaire avant, G vorderes Lot, I perpendicolare avanti  Definitions, principal dimensions 5 Table 1.1 Cont. English term Symbol Computer Translations notation Freeboard f FREP Fr franc-bord, G Freibord, I franco bordo Heel angle φs HEELANG Fr bande, gı̂te, Krängungswinkel I angolo d’inclinazione trasversale Length between Lpp LPP Fr longueur entre perpendiculars perpendiculaires, G Länge zwischen den Loten, I lunghezza tra le perpendicolari Length of waterline LWL LWL Fr longueur à la flottaison, G Wasserlinielänge, I lunghezza al galleggiamento Length overall LOA Fr longueur hors tout, G Länge über allen, I lunghezza fuori tutto Length overall LOS Fr longueur hors tout immergé, submerged G Länge über allen unter Wasser, I lunghezza massima opera viva Lines plan Fr plan des formes, G Linienriß, I piano di costruzione, piano delle linee Load waterline DWL DWL Fr ligne de flottaison en charge, G Konstruktionswasserlinie, I linea d’acqua a pieno carico Midships Fr couple milieu, G Hauptspant, I sezione maestra Moulded Fr hors membres, G auf Spanten, I fuori ossatura Port P Fr bâbord, G Backbord, I sinistra Sheer Fr tonture, G Decksprung, I insellatura Starboard S Fr tribord, G Steuerbord, I dritta Station Fr couple, G Spante, I ordinata Stern, poop Fr arrière, poupe, G Hinterschiff, I poppa Trim Fr assiette, G Trimm, I differenza d’immersione Waterline WL WL Fr ligne d’eau, G Wasserlinie, I linea d’acqua 6 Ship Hydrostatics and Stability LOA ✛ ✲ Midships Sheer at FP Sheer at AP ❅ ❘ ❅ ❄ ❄ Deck ✁ ❇ ✁ ❇✻ ✻✁ ❇❇ ✁ PP ✁ ✏ P P ❇ ✁✁ ❆ ❇ ❆ Baseline ✑ AP FP ✛ = ✲✛ = ✲ LPP ✛ ✲ LWL ✛ ✲ LOS ✛ ✲ Figure 1.1 Length dimensions Steel plating ❆❆ ❅ ✁✁ ❆ ❆✠ ❅ ❘✁ ✁ ❅ ❆❆ ✁✁ ❆❆ ✁✁ ❆❆ ✁✁ ✛ LPP ✲ AP FP Figure 1.2 How to measure the length between perpendiculars  ✞ ❥ ✆ ✝ PP ❄ PP ✻ ❅ TF ❅❄ ✥ ✥✥ TA ✥ ✥✥✥ ❅ ✥✥ ✻ ✟ ✥ ✟✟ ✟ ✟ ✥✥✥ ✥ ✥ ✥✥✥ ❅ TM ❄ ✥✥ ❅ ❅ AP LPP ✻ FP ✛ ✲ Figure 1.3 The case of a keel not parallel to the load line  Definitions, principal dimensions 7 ✛ B ✲ Camber ❄ ✻ ✻ f ❄ ✻ D T ✫ ✪❄ ❄ Figure 1.4 Breadth, depth, draught and camber The baseline, shortly BL, is a line lying in the longitudinal plane of symmetry and parallel to the designed summer load waterline (see next paragraph for a definition). It appears as a horizontal in the lateral and transverse views of the hull surface. The baseline is used as the longitudinal axis, that is the x-axis of the system of coordinates in which hull points are defined. Therefore, it is recommended to place this line so that it passes through the lowest point of the hull surface. Then, all z-coordinates will be positive. Before defining the dimensions of a ship we must choose a reference waterline. ISO 7462 recommends that this load waterline be the designed summer load line, that is the waterline up to which the ship can be loaded, in sea water, during summer when waves are lower than in winter. The qualifier ‘designed’means that this line was established in some design stage. In later design stages, or during operation, the load line may change. It would be very inconvenient to update this reference and change dimensions and coordinates; therefore, the ‘designed’ datum line is kept even if no more exact. A notation older than ISO 7462 is DWL, an abbreviation for ‘Design Waterline’. The after perpendicular, or aft perpendicular, noted AP , is a line drawn perpendicularly to the load line through the after side of the rudder post or through the axis of the rudder stock. The latter case is shown in Figures 1.1 and 1.3. For naval vessels, and today for some merchant vessels ships, it is usual to place the AP at the intersection of the aftermost part of the moulded surface and the load line, as shown in Figure 1.2. The forward perpendicular, F P , is drawn per- pendicularly to the load line through the intersection of the fore side of the stem with the load waterline. Mind the slight lack of consistency: while all moulded dimensions are measured to the moulded surface, the F P is drawn on the outer side of the stem. The distance between the after and the forward perpendicular, measured parallel to the load line, is called length between perpendiculars and its notation is Lpp. An older notation was LBP. We call length overall, LOA , 8 Ship Hydrostatics and Stability the length between the ship extremities. The length overall submerged, LOS , is the maximum length of the submerged hull measured parallel to the designed load line. We call station a point on the baseline, and the transverse section of the hull surface passing through that point. The station placed at half Lpp is called midships. It is usual to note the midship section by means of the symbol shown in Figure 1.5 (a). In German literature we usually find the simplified form shown in Figure 1.5 (b). The moulded depth, D, is the height above baseline of the intersection of the underside of the deck plate with the ship side (see Figure 1.4). When there are several decks, it is necessary to specify to which one refers the depth. The moulded draught, T , is the vertical distance between the top of the keel to the designed summer load line, usually measured in the midships plane (see Figure 1.4). Even when the keel is parallel to the load waterline, there may be appendages protruding below the keel, for example the sonar dome of a warship. Then, it is necessary to define an extreme draught that is the distance between the lowest point of the hull or of an appendage and the designed load line. Certain ships are designed with a keel that is not parallel to the load line. Some tugs and fishing vessels display this feature. To define the draughts associated with such a situation let us refer to Figure 1.3. We draw an auxiliary line that extends the keel afterwards and forwards. The distance between the intersection of this auxiliary line with the aft perpendicular and the load line is called aft draught and is noted with TA. Similarly, the distance between the load line and the intersection of the auxiliary line with the forward perpendicular is called forward draught and is noted with TF. Then, the draught measured in the midship section is known as midships draught and its symbol is TM. The difference between depth and draft is called freeboard; in DIN 81209-1 it is noted by f. The moulded volume of displacement is the volume enclosed between the submerged, moulded hull and the horizontal waterplane defined by a given draught. This volume is noted by ∇, a symbol known in English-language litera- ture as del, and in European literature as nabla. In English we must use two words, ‘submerged hull’, to identify the part of the hull below the waterline. Romance languages use for the same notion only one word derived from the Latin ‘carina’. Thus, in French it is ‘carène’, while in Catalan, Italian, Portuguese, Romanian, and Spanish it is called ‘carena’. In many ships the deck has a transverse curvature that facilitates the drainage of water. The vertical distance between the lowest and the highest points of the ✗✔ ✏ ✗✔ (a) (b) ❅ ✑ ✒ ✖✕ ✖✕❅ ❅ Figure 1.5 (a) Midships symbol in English literature, (b) Midships symbol in German literature  Definitions, principal dimensions 9 deck, in a given transverse section, is called camber (see Figure 1.4). According to ISO 7460 the camber is measured in mm, while all other ship dimensions are given in m. A common practice is to fix the camber amidships as 1/50 of the breadth in that section and to fair the deck towards its extremities (for the term ‘fair’ see Subsection 1.4.3). In most ships, the intersection of the deck surface and the plane of symmetry is a curved line with the concavity upwards. Usually, that line is tangent to a horizontal passing at a height equal to the ship depth, D, in the midship section, and runs upwards towards the ship extremities. It is higher at the bow. This longitudinal curvature is called sheer and is illustrated in Figure 1.1. The deck sheer helps in preventing the entrance of waves and is taken into account when establishing the load line in accordance with international conventions. 1.4 The definition of the hull surface 1.4.1 Coordinate systems The DIN 81209-1 standard recommends the system of coordinates shown in Figure 1.6. The x-axis runs along the ship and is positive forwards, the y-axis is transversal and positive to port, and the z-axis is vertical and positive upwards. The origin of coordinates lies at the intersection of the centreline plane with the transversal plane that contains the aft perpendicular. The international standards ISO 7460 and 7463 recommend the same positive senses as DIN 81209-1 but do not specify a definite origin. Other systems of coordinates are possible. For example, a system defined as above, but having its origin in the midship sec- tion, has some advantages in the display of certain hydrostatic data. Computer programmes written in the USA use a system of coordinates with the origin of coordinates in the plane of the forward perpendicular, F P , the x-axis positive Bow, Prow Port x z Starboard Stern y AP Figure 1.6 System of coordinates recommended by DIN 81209-1 10 Ship Hydrostatics and Stability afterwards, the y-axis positive to starboard, and the z-axis positive upwards. For dynamic applications, taking the origin in the centre of gravity simplifies the equations. However, it should be clear that to each loading condition corresponds one centre of gravity, while a point like the intersection of the aft perpendicular with the base line is independent of the ship loading. The system of coordinates used for the hull surface can be also employed for the location of weights. By its very nature, the system in which the hull is defined is fixed in the ship and moves with her. To define the various floating conditions, that is the positions that the vessel can assume, we use another system, fixed in space, that is defined in ISO 7463 as x0 , y0 , z0. Let this system initially coincide with the system x, y, z. A vertical translation of the system x, y, z with respect to the space-fixed system x0 , y0 , z0 produces a draught change. If the ship-fixed z-axis is vertical, we say that the ship floats in an upright condition. A rotation of the ship-fixed system around an axis parallel to the x-axis is called heel (Figure 1.7) if it is temporary, and list if it is permanent. The heel can be produced by lateral wind, by the centrifugal force developed in turning, or by the temporary, transverse displacement of weights. The list can result from incorrect loading or from flooding. If the transverse inclination is the result of ship motions, it is time-varying and we call it roll. When the ship-fixed x-axis is parallel to the space-fixed x0 -axis, we say that the ship floats on even keel. A static inclination of the ship-fixed system around an axis parallel to the ship-fixed y-axis is called trim. If the inclination is dynamic, that is a function of time resulting from ship motions, it is called pitch. A graphic explanation of the term trim is given in Figure 1.7. The trim is measured as the difference between the forward and the aft draught. Then, trim is positive if the ship is trimmed by the head. As defined here the trim is measured in metres. z ✻ z0 z z0 ❖❈ ✻ ❑ ❆ ❈ φ ✛ ✿❆ s ✘ ✘ ❈ ❆ ❆ ❈ ✂✂ ❈ ✂ ❆ ✟❆✟ ✟ ✟✟ ❈ ❇ N ✘✂✘ ✿ ❈ ✘✘✘ ❆✟✟ ✘✘✘✘ θs ✲x y0 ✛ ✟ ❈ ✘✘✘✘✘ ✟ ❅ ✘✘✘✘✘ ❅❈ ✘✘✘✘ ✻ x0 ✟✟ ✙ ✟ ❅ ✘✘ ❈✘✘ ✘✘ y (a) heel (b) trim Figure 1.7 Heel and trim  Definitions, principal dimensions 11 1.4.2 Graphic description In most cases the hull surface has double curvature and cannot be defined by simple analytical equations. To cope with the problem, Naval Architects have drawn lines obtained by cutting the hull surface with sets of parallel planes. Readers may find an analogy with the definition of the earth surface in topography by contour lines. Each contour line connects points of constant height above sea level. Similarly, we represent the hull surface by means of lines of constant x, constant y, and constant z. Thus, cutting the hull surface by planes parallel to the yOz plane we obtain the transverse sections noted in Figure 1.8 as St0 to St10, that is Station 0, Station 1,... Station 10. Cutting the same hull by horizontal planes (planes parallel to the base plane xOy), we obtain the waterlines marked in Figure 1.9 as WL0 to WL5. Finally, by cutting the same hull with longitudinal planes parallel to the xOz plane, we draw the buttocks shown in Figure 1.10. The most important buttock is the line y = 0 known as centreline; for almost all ship hulls it is a plane of symmetry. Stations, waterlines and buttocks are drawn together in the lines drawing. Figure 1.11 shows one of the possible arrangements, probably the most common one. As stations and waterlines are symmetric for almost all ships, it is sufficient to draw only a half of each one. Let us take a look to the right of our drawing; we see the set of stations represented together in the body plan. The left half of the body plan contains stations 0 to 4, that is the stations of the afterbody, while the right half is composed of stations 5 to 10, that is the forebody. The set of buttocks, known as sheer plan, is placed at the left of the body plan. Beneath is the set of waterlines. Looking with more attention to the lines drawing we find out that each line appears as curved in one projection, and as straight lines in St 0 St 1 St 2 St 3 St 4 St 5 St 6 St 7 St 8 St 9 St 10 Figure 1.8 Stations 12 Ship Hydrostatics and Stability WL 5 WL 4 WL 3 WL 2 WL 1 WL 0 Figure 1.9 Waterlines the other two. For example, stations appear as curved lines in the body plan, as straight lines in the sheer and in the waterlines plans. The station segments having the highest curvature are those in the bilge region, that is between the bottom and the ship side. Often no buttock or waterlines cuts them. To check what happens there it is usual to draw one or more additional lines by cutting the hull surface with one or more planes parallel to the baseline Buttock 1 Buttock 2 Buttock 3 Centreline Figure 1.10 Buttocks  Definitions, principal dimensions 13 Sheer plan Body plan Buttock 3 Buttock 2 Buttock 1 Afterbody Forebody 10 WL 5 0 9 1 2 8 7 WL 0 3 6 4 5 St 0 St 1 St 2 St 3 St 4 St 5 St 6 St 7 St 8 St 9 St 10 Waterlines plan Figure 1.11 The lines drawing but making an angle with the horizontal. A good practice is to incline the plane so that it will be approximately normal to the station lines in the region of highest curvature. The intersection of such a plane with the hull surface is appropriately called diagonal. Figure 1.11 was produced by modifying under MultiSurf a model provided with that software. The resulting surface model was exported as a DXF file to TurboCad where it was completed with text and exported as an EPS (Encapsu- lated PostScript) file. Figures 1.8 to 1.10 were obtained from the same model as MultiSurf contour curves and similarly post-processed under TurboCad. 1.4.3 Fairing The curves appearing in the lines drawing must fulfill two kinds of conditions: they must be coordinated and they must be ‘smooth’, except where functionality requires for abrupt changes. Lines that fulfill these conditions are said to be fair. We are going to be more specific. In the preceding section we have used three projections to define the ship hull. From descriptive geometry we may know that two projections are sufficient to define a point in three-dimensional space. It follows that the three projections in the lines drawing must be coordinated, otherwise one of them may be false. Let us explain this idea by means of Fig- ure 1.12. In the body plan, at the intersection of Station 8 with Waterline 4, we measure that half-breadth y(WL4, St8). We must find exactly the same dimen- sion between the centreline and the intersection of Waterline 4 and Station 8 in the waterlines plan. The same intersection appears as a point, marked by a circle, 14 Ship Hydrostatics and Stability y(WL4, St8) WL4 10 8 z (Buttock1, St10) A St 8 y(WL4, St8) B Figure 1.12 Fairing in the sheer plan. Next, we measure in the body plan the distance z(Buttock1, St10) between the base plane and the intersection of Station 10 with the longi- tudinal plane that defines Buttock 1. We must find exactly the same distance in the sheer plan. As a third example, the intersection of Buttock 1 and Waterline 1 in the sheer plan and in the waterlines plan must lie on the same vertical, as shown by the segment AB. The concept of smooth lines is not easy to explain in words, although lines that are not smooth can be easily recognized in the drawing. The manual of the surface modelling program MultiSurf rightly relates fairing to the concepts of beauty and simplicity and adds: A curve should not be more complex than it needs to be to serve its function. It should be free of unnecessary inflection points (reversals of curvature), rapid turns (local high curvature), flat spots (local low curvature), or abrupt changes of curvature... With other words, a ‘curve should be pleasing to the eye’ as one famous Naval Architect was fond of saying. For a formal definition of the concept of curvature see Chapter 13, Computer methods. The fairing process cannot be satisfactorily completed in the lines drawing. Let us suppose that the lines are drawn at the scale 1:200. A good, young eye can identify errors of 0.1 mm. At the ship scale this becomes an error of 20 mm that cannot be accepted. Therefore, for many years it was usual to redraw the lines at the scale 1:1 in the moulding loft and the fairing process was completed there. Some time after 1950, both in East Germany (the former DDR) and in Sweden, an optical method was introduced. The lines were drawn in the design office at the scale 1:20, under a magnifying glass. The drawing was photographed on glass plates and brought to a projector situated above the workshop. From there  Definitions, principal dimensions 15 Table 1.2 Table of offsets St 0 1 2 3 4 5 6 7 8 9 10 x 0.000 0.893 1.786 2.678 3.571 4.464 5.357 6.249 7.142 8.035 8.928 WL z Half breadths 0 0.360 0.900 1.189 1.325 1.377 1.335 1.219 1.024 0.749 0.389 1 0.512 0.894 1.167 1.341 1.440 1.463 1.417 1.300 1.109 0.842 0.496 0.067 2 0.665 1.014 1.240 1.397 1.482 1.501 1.455 1.340 1.156 0.898 0.564 0.149 3 0.817 1.055 1.270 1.414 1.495 1.514 1.470 1.361 1.184 0.936 0.614 0.214 4 0.969 1.070 1.273 1.412 1.491 1.511 1.471 1.369 1.201 0.962 0.648 0.257 5 1.122 1.069 1.260 1.395 1.474 1.496 1.461 1.363 1.201 0.972 0.671 0.295 the drawing was projected on plates so that it appeared at the 1:1 scale to enable cutting by optically guided, automatic burners. The development of hardware and software in the second half of the twentieth century allowed the introduction of computer-fairing methods. Historical high- lights can be found in Kuo (1971) and other references cited in Chapter 13. When the hull surface is defined by algebraic curves, as explained in Chapter 13, the lines are smooth by construction. Recent computer programmes include tools that help in completing the fairing process and checking it, mainly the calcu- lation of curvatures and rendering. A rendered view is one in which the hull surface appears in perspective, shaded and lighted so that surface smoothness can be summarily checked. For more details see Chapter 13. 1.4.4 Table of offsets In shipyard practice it has been usual to derive from the lines plan a digi- tal description of the hull known as table of offsets. Today, programs used to design hull surface produce automatically this document. An example is shown in Table 1.2. The numbers correspond to Figure 1.11. The table of offsets contains half-breadths measured at the stations and on the waterlines appearing in the lines plan. The result is a table with two entries in which the offsets (half-breadths) are grouped into columns, each column corresponding to a station, and in rows, each row corresponding to a waterline. Table 1.2 was produced in MultiSurf. 1.5 Coefficients of form In ship design it is often necessary to classify the hulls and to find relationships between forms and their properties, especially the hydrodynamic properties. The coefficients of form are the most important means of achieving this. By their definition, the coefficients of form are non-dimensional numbers. 16 Ship Hydrostatics and Stability ← DWL ← Waterplane ← Submerged hull Figure 1.13 The submerged hull The block coefficient is the ratio of the moulded displacement volume, ∇, to the volume of the parallelepiped (rectangular block) with the dimensions L, B and T : ∇ CB = (1.1) LBT In Figure 1.14 we see that CB indicates how much of the enclosing parallelepiped is filled by the hull. The midship coefficient, CM , is defined as the ratio of the midship-section area, AM , to the product of the breadth and the draught, BT , AM CM = (1.2) BT Figure 1.15 enables a graphical interpretation of CM. T L B Figure 1.14 The definition of the block coefficient, C B  Definitions, principal dimensions 17 Midship area T B Figure 1.15 The definition of the midship-section coefficient, C M The prismatic coefficient, CP , is the ratio of the moulded displacement vol- ume, ∇, to the product of the midship-section area, AM , and the length, L: ∇ CB LBT CB CP = = = (1.3) AM L CM BT L CM In Figure 1.16 we can see that CP is an indicator of how much of a cylinder with constant section AM and length L is filled by the submerged hull. Let us note the waterplane area by AW. Then, we define the waterplane-area coefficient by AW CWL = (1.4) LB L AM Figure 1.16 The definition of the prismatic coefficient, C P 18 Ship Hydrostatics and Stability AW B L Figure 1.17 The definition of the waterplane coefficient, C WL A graphic interpretation of the waterplane coefficient can be deduced from Figure 1.17. The vertical prismatic coefficient is calculated as ∇ CVP = (1.5) AW T For a geometric interpretation see Figure 1.18. Other coefficients are defined as ratios of dimensions, for instance L/B, known as length–breadth ratio, and B/T known as ‘B over T’. The length coefficient of Froude, or length–displacement ratio is L  m= (1.6) ∇1/3 and, similarly, the volumetric coefficient, ∇/L3. Table 1.3 shows the symbols, the computer notations, the translations of the terms related to the coefficients of form, and the symbols that have been used in continental Europe. T AW Figure 1.18 The definition of the vertical prismatic coefficient, C VP  Definitions, principal dimensions 19 Table 1.3 Coefficients of form and related terminology English term Symbol Computer Translations notation European symbol Block coefficient CB CB Fr coefficient de block, δ, G Blockcoeffizient, I coefficiente di finezza (bloc) Coefficient of form Fr coefficient de remplissage, G Völligkeitsgrad, I coefficiente di carena Displacement ∆ Fr déplacement, G Verdrängung, I dislocamento Displacement mass ∆ DISPM Fr déplacement, masse, G Verdrängungsmasse Displacement ∇ DISPV Fr Volume de la carène, volume G Verdrängungs Volumen, I volume di carena Midship CM CMS Fr coefficient de remplissage au coefficient maı̂tre couple, β, G Völligkeitsgrad der Hauptspantfläche, I coefficiente della sezione maestra Midship-section AM Fr aire du couple milieu, G Spantfläche, area I area della sezione maestra Prismatic CP CPL Fr coefficient prismatique, φ, coefficient G Schärfegrad, I coefficiente prismatico o longitudinale Vertical prismatic CVP CVP Fr coefficient de remplissage vertical ψ, coefficient I coefficiente di finezza prismatico verticale Waterplane area AW AW Fr aire de la surface de la flottaison, G Wasserlinienfläche, I area del galleggiamento Waterplane-area CWL Fr coefficient de remplissage coefficient de la flottaison, α, G Völligkeitsgrad der Wasserlinienfläche, I coefficiente del piano di galleggiamento 1.6 Summary The material treated in this book belongs to the field of Naval Architecture. The terminology is specific to this branch of Engineering and is based on a long maritime tradition. The terms and symbols introduced in the book comply with recent international and corresponding national standards. So do the definitions of the main dimensions of a ship. Familiarity with the terminology and the cor- responding symbols enables good communication between specialists all over 20 Ship Hydrostatics and Stability the world and correct understanding and application of international conventions and regulations. In general, the hull surface defies a simple mathematical definition. Therefore, the usual way of defining this surface is by cutting it with sets of planes parallel to the planes of coordinates. Let the x-axis run along the ship, the y-axis be transversal, and the z-axis, vertical. The sections of constant x are called sta- tions, those of constant z, waterlines, and the contours of constant y, buttocks. The three sets must be coordinated and the curves be fair, a concept related to simplicity, curvature and beauty. Sections, waterlines and buttocks are represented together in the lines plan. Line plans are drawn at a reducing scale; therefore, an accurate fairing process cannot be carried out on the drawing board. In the past it was usual to redraw the lines on the moulding loft, at the 1:1 scale. In the second half of the twenti- eth century the introduction of digital computers and the progress of software, especially computer graphics, made possible new methods that will be briefly discussed in Chapter 13. In early ship design it is necessary to choose an appropriate hull form and estimate its hydrodynamic properties. These tasks are facilitated by character- izing and classifying the ship forms by means of non-dimensional coefficients of form and ratios of dimensions. The most important coefficient of form is the block coefficient defined as the ratio of the displacement volume (volume of the submerged hull) to the product of ship length, breadth and draught. An example of ratio of dimensions is the length–breadth ratio. 1.7 Example Example 1.1 – Coefficients of a fishing vessel In INSEAN (1962) we find the test data of a fishing-vessel hull called C.484 and whose principal characteristics are: LWL 14.251 m B 4.52 m TM 1.908 m ∇ 58.536 m3 AM 6.855 m2 AW 47.595 m2 We calculate the coefficients of form as follows: ∇ 58.536 CB = = = 0.476 Lpp BTM 14.251 × 4.52 × 1.908 AW 47.595 CWL = = = 0.739 LWL B 14.251 × 4.52  Definitions, principal dimensions 21 AM 6.855 CM = = = 0.795 BT 4.52 × 1.908 ∇ 58.536 CP = = = 0.599 AM LWL 6.855 × 14.251 and we can verify that CB 0.476 CP = = = 0.599 CM 0.795 1.8 Exercises Exercise 1.1 – Vertical prismatic coefficient Find the relationship between the vertical prismatic coefficient, CVP , the waterplane-area coefficient, CWL , and the block coefficient, CB. Exercise 1.2 – Coefficients of Ship 83074 Table 1.4 contains data belonging to the hull we called Ship 83074. The length between perpendiculars, Lpp , is 205.74 m, and the breadth, B, 28.955 m. Com- plete the table and plot the coefficients of form against the draught, T. In Naval Architecture it is usual to measure the draught along the vertical axis, and other data – in our case the coefficients of form – along the horizontal axis (see Chapter 4). Exercise 1.3 – Coefficients of hull C.786 Table 1.5 contains data taken from INSEAN (1963) and referring to a tanker hull identified as C.786. Table 1.4 Coefficients of form of Ship 83074 Draught Displacement Waterplane volume area T ∇ AWL CB CWL CM CP m m3 m2 3 9029 3540.8 0.505 0.594 0.890 0.568 4 12632 3694.2 0.915 5 16404 3805.2 0.931 6 20257 3898.7 0.943 7 24199 3988.6 0.951 8 28270 4095.8 0.957 9 32404 4240.4 0.962 22 Ship Hydrostatics and Stability Table 1.5 Data of tanker hull C.786 LWL 205.468 m B 27.432 m TM 10.750 m ∇ 46341 m3 AM 0.220 AWL 3.648 Calculate the coefficients of form and check that CB = CP CM 2 Basic ship hydrostatics 2.1 Introduction This chapter deals with the conditions of equilibrium and initial stability of floating bodies. We begin with a derivation of Archimedes’ principle and the definitions of the notions of centre of buoyancy and displacement. Archimedes’principle provides a particular formulation of the law of equilibrium of forces for floating bodies. The law of equilibrium of moments is formulated as Stevin’s law and it expresses the relationship between the centre of gravity and the centre of buoyancy of the floating body. The study of initial stability is the study of the behaviour in the neighbourhood of the position of equilibrium. To derive the condition of initial stability we introduce Bouguer’s concept of metacentre. To each position of a floating body correspond one centre of buoyancy and one metacentre. Each position of the floating body is defined by three parameters, for instance the triple {displacement, angle of heel, angle of trim}; we call them the parameters of the floating condition. If we keep two parameters constant and let one vary, the centre of buoyancy travels along a curve and the metacentre along another. If only one parameter is kept constant and two vary, the centre of buoyancy and the metacentre generate two surfaces. In this chapter we shall briefly show what happens when the displacement is constant. The discussion of the case in which only one angle (that is, either heel or trim) varies leads to the concept of metacentric evolute. The treatment of the above problems is based on the following assumptions: 1. the water is incompressible; 2. viscosity plays no role; 3. surface tension plays no role; 4. the water surface is plane. The first assumption is practically exact in the range of water depths we are interested in. The second assumption is exact in static conditions (that is without motion) and a good approximation at the very slow rates of motion discussed in ship hydrostatics. In Chapter 12 we shall point out to the few cases in which vis- cosity should be considered. The third assumption is true for the sizes of floating bodies and the wave heights we are dealing with. The fourth assumption is never 24 Ship Hydrostatics and Stability true, not even in the sheltered waters of a harbour. However, this hypothesis allows us to derive very useful, general results, and calculate essential properties of ships and other floating bodies. It is only in Chapter 9 that we shall leave the assumption of a plane water surface and see what happens in waves. In fact, the theory of ship hydrostatics was developed during 200 years under the hypothesis of a plane water surface and only in the middle of the twentieth century it was recognized that this assumption cannot explain the capsizing of a few ships that were considered stable by that time. The results derived in this chapter are general in the sense that they do not assume particular body shapes. Thus, no symmetry must be assumed such as it usually exists in ships (port-to-starboard symmetry) and still less symmetry about two axes, as encountered, for instance, in Viking ships, some ferries, some offshore platforms and most buoys. The results hold the same for single-hull ships as for catamarans and trimarans. The only problem is that the treatment of the problems for general-form floating bodies requires ‘more’ mathematics than the calculations for certain simple or symmetric solids. To make this chapter accessible to a larger audience, although we derive the results for body shapes without any form restrictions, we also exemplify them on parallelepipedic and other simply defined floating body forms. Reading only those examples is suf- ficient to understand the ideas involved and the results obtained in this chapter. However, only the general derivations can provide the feeling of generality and a good insight into the problems discussed here. 2.2 Archimedes’ principle 2.2.1 A body with simple geometrical form A body immersed in a fluid is subjected to an upwards force equal to the weight of the fluid displaced. The above statement is known as Archimedes’ principle. One legend has it that Archimedes (Greek, lived in Syracuse – Sicily – between 287 and 212 bc) discovered this law while taking a bath and that he was so happy that he ran naked in the streets shouting ‘I have found’ (in Greek Heureka, see entry ‘eureka’ in Merriam-Webster, 1991). The legend may be nice, but it is most probably not true. What is certain is that Archimedes used his principle to assess the amount of gold in gold–silver alloys. Archimedes’ principle can be derived mathematically if we know another law of general hydrostatics. Most textbooks contain only a brief, unconvincing proof based on intuitive considerations of equilibrium. A more elaborate proof is given here and we prefer it because only thus it is possible to decide whether Archimedes’principle applies or not in a given case. Let us consider a fluid whose specific gravity is γ. Then, at a depth z the pressure in the fluid equals γz. This is the weight of the fluid column of height z and unit area cross section. The  Basic ship hydrostatics 25 pressure at a point is the same in all directions and this statement is known as Pascal’s principle. The proof of this statement can be found in many textbooks on fluid mechanics, such as Douglas, Gasiorek and Swaffiled (1979: 24), or Pnueli and Gutfinger (1992: 30–1). In this section we calculate the hydrostatic forces acting on a body having a simple geometric form. The general derivation is contained in the next section. In this section we consider a simple-form solid as shown in Figure 2.1; it is a parallelepipedic body whose horizontal, rectangular cross-section has the sides B and L. We consider the body immersed to the draught T. Let us call the top face 1, the bottom face 2, and number the vertical faces with 3 to 6. Figure 2.1(b) shows the diagrams of the liquid pressures acting on faces 4 and 6. To obtain the absolute pressure we must add the force due to the atmospheric pressure p0. Those who like mathematics will say that the hydrostatic force on face 4 is the integral of the pressures on that face. Assuming that forces are positive in a rightwards direction, and adding the force due to the atmospheric pressure, we obtain
T 1 F4 = L γzdz + p0 LT = γLT 2 + p0 LT (2.1) 0 2 ✛ B ✲ 1 (b) 1 6 3 ✻ ✲ L ✛ ✻ L ✲ ✛L z Waterplane T ✲ p = γz ✛ L ❄ ✲ ✛ L ❄ ✲ ✛ L ✻✻✻✻✻ p = γT (a) 5 ✻ L 4 6 ❄ 3 (c) Figure 2.1 Hydrostatic forces on a body with simple geometrical form 26 Ship Hydrostatics and Stability Similarly, the force on face 6 is
T 1 F6 = −L γzdz − p0 LT = − γLT 2 − p0 LT (2.2) 0 2 As the force on face 6 is equal and opposed to that on face 4 we conclude that the two forces cancel each other. The reader who does not like integrals can reason in one of the following two ways. 1. The force per unit length of face 4, due to liquid pressure, equals the area of the triangle of pressures. As the pressure at depth T is γT , the area of the triangle equals 1 1 T × γT = γT 2 2 2 Then, the force on the total length L of face 4 is 1 F4 = L × γT 2 + p0 LT (2.3) 2 Similarly, the force on face 6 is 1 F6 = −L × γT 2 − p0 LT (2.4) 2 The sum of the two forces F4 , F6 is zero. 2. As the pressure varies linearly with depth, we calculate the force on unit length of the face 4 as equal to the depth T times the mean pressure γT /2. To get the force on the total length L of face 4 we multiply the above result by L and adding the force due to atmospheric pressure we obtain 1 F4 = γLT 2 + p0 LT 2 Proceeding in the same way we find that the force on face 6, F6 , is equal and opposed to the force on face 4. The sum of the two forces is zero. In continuation we find that the forces on faces 3 and 5 cancel one another. The only forces that remain are those on the bottom and on the top face, that is faces 2 and 1. The force on the top face is due only to atmospheric pressure and equals F1 = −p0 LB (2.5) and the force on the bottom, F2 = p0 LB + γLBT (2.6) The resultant of F1 and F2 is an upwards force given by F = F2 + F1 = γLBT + p0 LB − p0 LB = γLBT (2.7)  Basic ship hydrostatics 27 The product LBT is actually the volume of the immersed body. Then, the force F given by Eq. (2.7) is the weight of the volume of liquid displaced by the immersed body. This verifies Archimedes’ principle for the solid considered in this section. We saw above that the atmospheric pressure does not play a role in the derivation of Archimedes’ principle. Neither does it play any role in most other problems we are going to treat in this book; therefore, we shall ignore it in future. Let us consider in Figure 2.2 a ‘zoom’ of Figure 2.1. It is natural to consider that the resultant of the forces is applied at the point P situated in the centroid of face 2. The meaning of this sentence is that, for any coordinate planes, the moment of the force γLBT applied at the point P equals the integral of the moments of pressures. In the same figure, the point B is the centre of volume of the solid. If our solid would be made of a homogeneous material, the point B would be its centre of gravity. We see that P is situated exactly under B, but at double draught. As a vector can be moved along its line of action, without changing its moments, it is commonly admitted that the force γLBT is applied in the point B. A frequent statement is: the force exercised by the liquid is applied in the centre of the displaced volume. The correct statement should be: ‘We can consider that the force exercised by the liquid is applied in the centre of the displaced volume’. The force γLBT is called buoyancy force. We have analyzed above the case of a solid that protrudes the surface of the liquid. Two other cases may occur; they are shown in Figure 2.3. We study again the same body as before. In Figure 2.3(a) the body is situated somewhere between the free surface and the bottom. Pressures are now higher; on the vertical faces their distribution follows a trapezoidal pattern. We can still show that the sum of p0 ✲ ✻ LL ✛ ✻ ✻ ✲ ✛L z ✲ T/2 ✛L ❄ ✲ ✛ L p = γz ✲ ❄ tB ✛ L T ✲ ✛ L L ✲✛ B/2 ✲ ✛ L ✲ ✛ L ✲ ✛ L ✲ tP ✛ L ❄ ✻✻✻✻✻✻✻✻✻ p= γT Figure 2.2 Zoom of Figure 2.1 28 Ship Hydrostatics and Stability ✻ ✻ ✻ (a) (b) z p = γz ❄ ✲❄ ❄ ❄ ❄ ✛ ❄ ✄✲ ✛❈ z1 ✄ ✲ ✛ ❈ p= γ z1 z+H ✄ ✲ ✛ ❈ ✄ ✲ ✛ ❈ ✄ ❈ ✄ ✲ ✛ ❈ ✲❄ ❄ ❄ ❄ ✛ ❄ ❄ ✄ ✲ ✛ ❈  ✲ ✛ ❉ ✄ ✲ ✛ ❈  ✲ ✛ ❉ ✄ ✲ ✛ ❈ ❄ ✲ ✛ ❉ ✻✻✻✻✻  ✲ ✛ ❉  ❉ p = γ (z +H ) ✲ ✛  ❉  ✲ ✛ ❉  ✲ ✻ R ✛ ❉  ✲ ✛ ❉ Figure 2.3 Two positions of submergence the forces on faces 3 to 6 is zero. It remains to sum the forces on faces 1 and 2, that is on the top and the bottom of the solid. The result is γ(z + H)LB − γzLB = γLBH (2.8) As γLBH is the weight of the liquid displaced by the submerged body, this is the same result as that obtained for the situation in Figures 2.1 and 2.2, that is Archimedes’ principle holds in this case too. In Figure 2.3(b) we consider the solid lying on the sea bottom (or lake, river, basin bottom) and assume that no liquid infiltrates under the body. Then no liquid pressure is exercised on face 2. The net hydrostatic force on the body is γz1 LB and it is directed downwards. Archimedes’ principle does not hold in this case. For equilibrium we must introduce a sea-bottom reaction, R, equal to the weight of the body plus the pressure force γz1 LB. The force necessary to lift the body from the bottom is equal to that reaction. However, immediately that the water can exercise its pressure on face 2, a buoyancy force is developed and the body seems lighter. It is as if when on the bottom the body is ‘sucked’ with a force γz1 LB. Figure 2.3(b) shows a particular case. Upwards hydrostatic forces can develop in different situations, for example: if the submerged body has such a shape that the liquid can enter under part of its surface. This is the case of most ships; the bottom is not compact and liquid pressures can act through it. This phe- nomenon is taken into account in the design of dams and breakwaters where it is called uplift.  Basic ship hydrostatics 29 In the two cases mentioned above, the upwards force can be less than the weight of the displaced liquid. A designer should always assume the worst situation. Thus, to be on the safe side, when calculating the force necessary to bring a weight to the surface one should not count on the existence of the uplift. On the other hand, when calculating a deadweight – such as a concrete block – for an anchoring system, the existence of uplift forces should be taken into account because they can reduce the friction forces (between deadweight and bottom) that oppose horizontal pulls. 2.2.2 The general case In Figure 2.4 we consider a submerged body and a system of cartesian coordi- nates, x, y, z, where z is measured vertically and downwards. The only condition we impose at this stage is that no straight line parallel to one of the coordinate axes pierces the body more than twice. We shall give later a hint on how to relax this condition, generalizing thus the conclusions to any body form. Let the surface of the body be S, and let P be the horizontal plane that cuts in S the largest contour. The plane P divides the surface S into two surfaces, S1 situated above P , and S2 under P. We assume that S1 is defined by z = f1 (x, y) x dy dx pdA y Z = f1(x, y) Z = f2(x, y) z Figure 2.4 Archimedes’ principle – vertical force 30 Ship Hydrostatics and Stability and S2 by z = f2 (x, y) The hydrostatic force on an element dA of S1 is pdA. This force is directed along the normal, n, to S1 in the element of area. If the cosine of the angle between n and the vertical axis is cos(n, z), the vertical component of the pressure force on dA equals γf1 (x, y) cos(n, z)dA. As cos(n, z)dA is the projection of dA on a horizontal plane, that is dxdy, we conclude that the vertical hydrostatic force on S1 is

γ f1 (x, y)dxdy (2.9) S1 Let us consider now an element of S2 ‘opposed’ to the one we considered on S1. We reason as above, taking care to change signs. We conclude that the hydrostatic force on S2 is

−γ f2 (x, y)dxdy (2.10) S2 and the total force on S,

F =γ [f1 (x, y) − f2 (x, y)]dxdy (2.11) S The integral in Eq. (2.11) yields the volume of the submerged body. Thus, F equals the weight of the liquid displaced by the submerged body. It remains to show that the horizontal components of the resultant of hydrostatic pressures are equal to zero. We use Figure 2.5 to prove this for the component parallel to the x-axis. The force component parallel to the x-axis acting on the element of area dA is p cos(n, x)dA = γz dydz On the other side of the surface, at the same depth z, there is an element of area such that the hydrostatic force on it equal