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  1. 1. Dynamics ofOffshoreStructuresJames F. Wilson, EditorJohn Wiley & Sons, Inc.
  2. 2. Dynamics of Offshore Structures
  3. 3. Cover photograph. This historical Argus Island Tower was a U.S. Navy facility, locat-ed 39 km off the southwest coast of Bermuda in a water depth of 58 m. Built in 1960,the Tower was used for about 10 years for underwater acoustic research and for sub-marine detection. The two enclosed levels on top of this four legged jacket structurehad space for diesel generators, living quarters, and laboratories. During the first fewyears of the Tower’s existence, it was subjected to storm-generated waves approaching21 m, which was also the wave height upon which the Tower design was based. The1969 inspections of the Tower revealed storm damage to many of its subsurface weld-ed brace connections, damage that was deemed too closely to repair and subsequentlymaintain. Thus, demolition using shaped charges toppled the Tower in 1976, and itsremains now rest on the coral floor of the sea.
  4. 4. Dynamics ofOffshoreStructuresJames F. Wilson, EditorJohn Wiley & Sons, Inc.
  5. 5. This book is printed on acid-free paper. ᭺ ϱCopyright © 2003 by John Wiley & Sons, Inc. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in CanadaNo part of this publication may be reproduced, stored in a retrieval system ortransmitted in any form or by any means, electronic, mechanical, photocopying,recording, scanning, or otherwise, except as permitted under Section 107 or 108 ofthe 1976 United States Copyright Act, without either the prior written permissionof the Publisher, or authorization through payment of the appropriate per-copy feeto the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923,(978) 750-8400, fax (978) 750-4470, or on the web at Requeststo the Publisher for permission should be addressed to the PermissionsDepartment, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201)748-6011, fax (201) 748-6008, e-mail: of Liability/Disclaimer of Warranty: While the publisher and author have usedtheir best efforts in preparing this book, they make no representations or warrantieswith respect to the accuracy or completeness of the contents of this book and specifi-cally disclaim any implied warranties of merchantability or fitness for a particularpurpose. No warranty may be created or extended by sales representatives or writtensales materials. The advice and strategies contained herein may not be suitable foryour situation. You should consult with a professional where appropriate. Neither thepublisher nor author shall be liable for any loss of profit or any other commercialdamages, including but not limited to special, incidental, consequential, or other dam-ages.For general information on our other products and services or for technical sup-port, please contact our Customer Care Department within the United States at(800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.Wiley also publishes its books in a variety of electronic formats. Some content thatappears in print may not be available in electronic books.Library of Congress Cataloging-in-Publication Data:Wilson, James F. (James Franklin), 1933– Dynamics of offshore structures / James F. Wilson, Bruce J. Muga, Lymon C. Reese.—2nd ed. p. cm. Includes index. New ed. of: Dynamics of offshore structures / James F. Wilson, editor. 1984. ISBN 0-471-26467-9 (cloth: alk. paper) 1. Offshore structures. I. Muga, Bruce J. (Bruce Jennings) II. Reese, Lymon C., 1917– III. Dynamics of offshore structures. IV. Title. TC1665.W55 2002 627Ј.98—dc21 2002028858Printed in the United States of America.10 9 8 7 6 5 4 3 2 1
  6. 6. Contents Preface Xl Contributors Xlll Acknowledgments xv1 Structures in the Offshore Environment 1 James F. Wilson1.1 Historical Perspective, 21.2 Platforms, 4 Fixed-Bottom Platforms, 4 Compliant Platforms, 7 Buoyant Platforms, 71.3 Moorings, 9 Temporary Anchor Moorings, 9 Platform Pile and Single-Point Moorings, 101.4 Pipelines, 11 Sea Floor Pipelines, 11 Vertical Pipelines, 121.5 Challenges, 14 Environmental Forces, 14 Structural Materials, 14 Modeling and Analysis, 15 Experimental Evaluations, 15 References, 152 Structure-Environmental Force Interactions 17 James F. Wilson2.1 Single Degree of Freedom Structures, 172.2 Fluid-Induced Structural Forces, 21 Classical Inviscid Fluid Flow, 21 Real Viscous Fluid Flow, 23 Conservation of Linear Momentum and Flow Superposition, 25 Buoyancy and Gravity, 28 Winds and Currents, 332.3 Earthquakes, Ice Impact, and Wave Slamming, 36 Earthquake Forces, 36 Ice Impact Forces, 38 Wave Slamming Forces, 392.4 Structural Mass, Damping, and Restraint, 40 Structural Mass and Stiffness, 40 v
  7. 7. VI CONTENTS Cable Restraints, 43 Soil Foundation Restraints, 52 Problems, 54 References, 56 3 Deterministic Descriptions of Offshore Waves 61 Bruce 1. Muga 3.1 Description of Plane Waves, 62 3.2 Linear Plane Waves, 65 3.3 Nonlinear Waves, 68 Trochoidal Theory, 69 Cnoidal Theory, 69 Stokes Theory, 69 Solitary Theory, 72 Numerical Theory, 73 3.4 Domains of Validity for Wave Theories, 75 Problems, 80 References, 81 4 Wave Forces on Structures 84 James F. Wilson 4.1 Wave Loading of Flexible Cylinders, 84 4.2 Classification of Fluid Load Regimes, 87 4.3 Flow Regimes for Offshore Structures, 91 4.4 Summary of Flow Coefficients CD and C M , 93 4.5 Transfer Functions for Wave Loading, 94 Problems, 96 References, 98 5 Deterministic Responses for Single Degree of Freedom Structures 100 James F. Wilson 5.1 Natural Frequencies of Linear Systems, 101 Direct Method, 101 The Rayleigh Method, 105 5.2 Frequencies for Nonlinear Structures, 111 5.3 Response Functions for Linear Structures, 115 Harmonic Response Function, 116 Impulse Response Function, 118 Convolution Integral, 119 5.4 Response of Linear Structures to Earthquake Loading, 120 5.5 Response Characteristics of Nonlinear Structures, 123 Responses to Harmonic Excitation, 123
  8. 8. CONTENTS vii Jumps, 126 Subharmonics, 1295.6 Nonlinear Responses for a SALM Buoy, 131 Physical Description and Dynamic Model, 132 Numerical Results and Conclusions, 136 Problems, 138 References, 1426 Statistical Descriptions of Offshore Waves 144 Bruce J. Muga6.1 Introduction to Wave Spectra, 1446.2 Concept of the Significant Wave, 147 Wave Height Distributions, 148 Wave Height-Wave Spectrum Relationships, 1506.3 Descriptions of Wave Energy Spectra, 152 Alternative Definitions, 152 Empirical Forms, 1546.4 Selection of Design Wave Spectra, 1566.5 Synthesis of Time Histories from Spectra, 159 References, 1627 Statistical Responses for Single Degree of Freedom Linear Structures 165 James F. Wilson7.1 Averages and Probabilities, 1667.2 Stationary and Ergodic Hypotheses, 1697.3 Autocorrelation and Spectral Density, 1727.4 Structural Response Statistics: Part I, 173 The Fourier Transform, 174 The Autocorrelation Functions, 175 Response Parameters, 176 Approximate Responses, 180 Extensions, 182 7.5 Structural Response Statistics: Part II, 183 State Variable Form, 184 Covariance Propagation Equation: Derivation, 185 Steady State Solutions, 187 A Closed Form Solution, 188 Problems, 190 References, 193 8 Multi-Degree of Freedom Linear Structures 195 James F. Wilson 8.1 Equations of Motion: General Form, 196
  9. 9. Vlll CONTENTS The Coordinate Vector, ~, 196 The Loading Vector, p, 198 The Mass Matrix, M, 199 The Stiffness Matrix, K, 201 The Damping Matrix, C, 203 8.2 Equations of Motion: Newtons Method, 204 8.3 Equations of Motion: Lagranges Formulation, 208 System Energies, 208 Hamiltons Principle, 209 Derivation for a Simple System, 209 Comments, 212 8.4 Free, Undamped Motion, 214 Frequencies, 214 Modal Vectors and Normalization, 214 Orthogonality of the Modal Vectors, 215 8.5 Forced, Damped Motion, 216 The Mode Shape Matrix, X, 217 Modal Damping Matrix, C, 217 Uncoupling the Equations of Motion, 218 Steady State Solutions, 219 8.6 Summary of the Normal Mode Method, 219 Problems, 220 References, 221 9 Applications of Multi-Degree of Freedom Analysis 223 James F. Wilson 9.1 A Fixed Leg Platform: Time Domain Responses, 224 Mathematical Model, 224 Frequencies, 225 Modal Vectors and Normalization, 226 Response to a Harmonic Wave, 227 Response to Earthquake Excitation, 231 9.2 A Monopod Gravity Platform: Free Vibration and Stability, 233 Mathematical Model, 233 Frequencies and Mode Shapes, 236 Dynamic Stability, 238 9.3 Structural Response Statistics for Wave Loading, 239 9.4 A Fixed Leg Platform: Statistical Responses, 242 Problems, 246 References, 246 10 Continuous Systems 248 James F. Wilson 10.1 Modeling Beams and Cables, 250 Governing Equations, 250
  10. 10. CONTENTS ix Cable Frequencies and Mode Shapes, 252 Beam Frequencies and Mode Shapes, 25410.2 Cable Responses, 259 Transverse End Excitation, 259 Parametric Excitation, 26110.3 Beam Responses, 265 Transverse Excitation, 266 Parametric Excitation, 269 Statistical Wave Excitation, 27210.4 Deployment of an OTEC Pipeline, 275 Mathematical Model, 276 Pipe Excitations by Barge and Waves, 277 Numerical Results, 280 Problems, 284 References, 28711 Behavior of Piles Supporting Offshore Structures 289 Lyman C. Reese11.1 Characteristics of Soil and Response to External Loading, 29011.2 Design of Single Piles under Axial Loading, 29111.3 Design of Single Piles under Lateral Loading, 296 Pile Model and Method of Solution, 297 Response of Soil, 298 Short-Term Static Loading, 299 Cyclic Loading, 299 Scour, 300 Solutions Using Nondimensional Parameters, 301 Computer-Aided Solutions, 30611.4 An Example: Lateral Pile Loading, 307 Problem Description, 308 Solutions Using Nondimensional Parameters, 310 Computer-Aided Solutions, 313 Conclusions, 31611.5 Response of Piles to Dynamic Loading, 316 Earthquakes, 316 Time-Dependent Loading Above the Mudline, 317 Problems, 318 References, 319 Conversion Table 321 Index 323
  11. 11. PrefaceThis book is intended for three groups: (1) students and professors of structuraland ocean engineering; (2) engineers and scientists in academic institutions,government laboratories, and industries involved in research on offshore installa-tions, especially fluid-structure-soil interactions; and (3) practicing professionalengineers who consider conceptual designs and need to employ dynamic analy-sis to evaluate facilities constructed offshore. The material herein was originallyprepared by the three contributors for short courses attended by engineeringpractitioners, and for university courses taken by engineering seniors and grad-uate students. Compared to the first edition, this second edition includes more exampleproblems to illustrate the dynamic modeling, analysis, and solution of deter-ministic and stochastic responses for a wide variety of structures offshore, whichinclude buoys, moored ships, and platforms of the fixed-bottom, cable-stayed,and gravity-type designs. Also, the extensive references of the first edition areupdated, especially source material involving offshore waves, structural modaldamping, and fluid-structure-soil interactions. As in the first edition, this second edition addresses the basic physical ideas,structural modeling, and mathematical methods needed to analyze the dynamicbehavior of structures offshore. Chapter 1 summarizes existing installationsand points out future challenges. In subsequent chapters, careful attention isgiven to the many and sometimes subtle assumptions involved in formulatingboth the structural model and the natural forces imposed by the often hostileenvironment. The analyses in these chapters focus on plane motions of elasticstructures with linear and nonlinear restraints, motions induced by the forces ofcurrents, winds, waves, and earthquakes. Chapters 2 through 5 address singledegree of freedom structural models that, together with plane wave loadingtheories, lead to time history predictions of structural responses. Chapters 6and 7 extend these analyses to statistical descriptions of both wave loading andstructural motion. Chapters 8 and 9 include the analysis and examples of multi-degree of freedom linear structures. Chapter 10 deals with continuous systemanalysis, including the motion of cables and pipelines. Chapter 11 addressescurrent practice related to submerged pile design for structures offshore. I sincerely hope that this book will be useful and serve as an inspirationto engineers and researchers who design and analyze structures for the offshoreenvironment. JAMES F. WILSON Chapel Hill, North Carolina Xl
  12. 12. Contributors Bruce J. Muga, Professor Emeritus of Civil and Environmental Engineer-ing at Duke University, received his B.S. in Civil Engineering from the Universityof Texas, and his M.S. and Ph.D. degrees in Civil Engineering (Hydrodynamics)from the University of Illinois. From 1961 to 1967 he was employed as a Project Engineer in the Port andHarbor Division of the U.S. Naval Civil Engineering Laboratory, Port Huen-eme, California. In 1964, he was assigned as Consultant to the U.S. MilitaryAssistance Command, Vietnam, to advise on coastal and harbor engineeringprojects. In 1967, Dr. Muga accepted a position in teaching and research at DukeUniversity and was Chairman of the Department of Civil Engineering in 1974.He has served as a consultant to many international corporations engaged in off-shore and deep ocean engineering activities. He has written numerous technicalpapers and for seventeen years served on the North Carolina Marine SciencesCouncil. Prior to retirement, Dr. Muga was a Registered Professional Engineerin California, a member of the American Society of Civil Engineers and theMarine Technology Society. He is a life member of the Permanent InternationalAssociation of Navigational Congresses. Lymon C. Reese is the Nasser I. AI-Rashid Chair Emeritus and Professor,Department of Civil Engineering, The University of Texas, Austin, Texas, andis principal at Ensoft, Inc., a distributor of engineering software. Some of hisconsulting activities are carried out through Lymon C. Reese & Associates, asubsidiary of Ensoft. He received his Bachelors and Masters degrees in Civil Engineering fromThe University of Texas at Austin and his Ph.D. from The University of Cali-fornia at Berkeley. Dr. Reese has had several years of industrial experience andhas been a consultant to a number of companies and governmental agencies.He was formerly Assistant Professor of Civil Engineering at Mississippi StateUniversity. Dr. Reese has done extensive research in the field of geotechnical engineer-ing, principally concerning the behavior of deep foundations. He has pioneeredin performing field studies of instrumented piles and has developed analyticalmethods now widely used in the design of major structures. He has authoredover 400 technical papers and reports and presented a number of invited lecturesand talks in North and South America, Australia, Africa, Asia, and Europe. XUl
  13. 13. xiv CONTRIBUTORS Dr. Reese is an Honorary Member of the American Society of Civil Engineersand was selected as Terzaghi Lecturer in 1976; he received the Terzaghi Awardin 1983. He received the Distinguished Achievement Award for Individuals fromthe Offshore Technology Conference in 1985 and was elected to membership inthe National Academy of Engineering in 1975. He is a registered professionalengineer in Texas and Louisiana. James F. Wilson earned an A.B. degree from the College of Wooster, aB.S. degree in Mechanical Engineering from MIT in 1956, and a Ph.D. degreein applied mechanics from The Ohio State University, where he was a FordFoundation Fellow and a Freeman Scholar. He worked in research and develop-ment for several companies and government agencies before joining the facultyat Duke University in 1967. During his academic career, Dr. Wilson was a NASA-ASEE Faculty Fellow,a lecturer at three NATO Advanced Study Institutes, and a Visiting Scholarat Colorado State University and the University of Melbourne, Australia. Hehas been active in national committees for the American Society of MechanicalEngineers (ASME) and the American Society of Civil Engineers (ASCE), andreceived national awards for innovative experimental research (ASME, 1977),and the years best state-of-the-art civil engineering journal publication (ASCE,1987). He is a Life Fellow in ASME and a retired Fellow of the National Academyof Forensic Engineers. As a registered professional engineer, he regularly servesas an expert witness, testifying on structural failures, product performance, andvehicle accident reconstruction. He is author or coauthor of over 200 works, which include technical reports onforensic engineering, refereed symposium papers and journal articles, two bookson structural dynamics, a three-volume work on experiments in engineering,and two U.S. patents. His experimental research on robotics was highlighted inthe 1989 BBC documentary, Natures Technology. During Dr. Wilsons career at Duke University, he has taught courses inapplied mechanics, structural dynamics, and experimental systems, and wasthe major research advisor for over 35 graduate students, including postdoctoralfellows. He also served as the Director of Graduate Studies for the Departmentof Civil and Environmental Engineering. As Professor Emeritus since 1998, Dr.Wilson continues to pursue his research and writing interests and consultingpractice.
  14. 14. Acknowledgments I am grateful to those who have made this edition possible: to Robert E.Sandstrom for his contributions to Chapter 1; to Academic Press, Inc., forpermission to use parts of Chapter 1, first published in the Encyclopedia ofPhysical Science and Technology, III, 2001; and to Gary Orgill and David M.Wilson who performed many of the calculations for the sample problems. I especially acknowledge the patience and careful scholarship of the twoguest contributors: Bruce J. Muga for Chapters 3 and 6 on fluid mechanics,and Lymon C. Reese for Chapter 11 on soil mechanics. Only by including themechanics of fluids and soils can realistic analyses of offshore structures be made. JAMES F. WILSON Chapel Hill, North Carolina xv
  15. 15. 1Structures in theOffshore EnvironmentJames F. Wilson
  16. 16. 2 STRUCTURES IN THE OFFSHORE ENVIRONMENT In perspective, offshore structures include a great deal more than the towersand platforms. They include moored or mobile ships whose positions may beprecisely controlled. They include the guy lines for compliant towers, the cablesfor buoys and for tension-leg platforms, and the associated pipelines withoutwhich the platforms and submerged oil production systems would be useless.Detailed descriptions of such installations may be found in the references at theend of this chapter. Of particular note is the review article on compliant offshorestructures by Adrezin et al.(1996), with its 130 citations to the world literatureon the subject up to the mid-1990s. For descriptions of current practice in alltypes of offshore installations, the reader is referred to the yearly conferenceproceedings such as found in the References at the end of this chapter. This chapter begins with a short history of offshore structures, describes typ-ical state-of-the-art installations, and concludes with a discussion of engineeringchallenges for future designs. Subsequent chapters address in some detail boththe mathematical modeling and the environmental loading of offshore struc-tures, together with ways to predict their dynamic responses and structuralintegrity, from both the deterministic and the statistical viewpoints.1.1 HISTORICAL PERSPECTIVEThe earliest offshore structure for oil drilling was built about 1887 off the coastof southern California near Santa Barbara. This was simply a wooden wharfoutfitted with a rig for drilling vertical wells into the sea floor. More elaborateplatforms supported by timber piers were then built for oil drilling, includinginstallations for the mile-deep well in Caddo Lake, Louisiana (1911) and the plat-form in Lake Maracaibo, Venezuela (1927). Soon after these early pier systemswere built, it became apparent that the lifetime of timber structures erectedin lakes or oceans is severely limited because of attacks by marine organisms.For this reason, reinforced concrete replaced timber as the supporting structurefor many offshore platforms up to the late 1940s. Over the next 50 years about12,000 platform structures were built offshore, usually of steel but more recentlyof precast concrete. The chief features of these structures, together with theirsupporting components such as mooring systems and pipelines, are discussed inthis chapter. See also Gerwick (1999) and Will (1982). Offshore mooring systems have a variety of configurations. All have anchorsor groups of piles in the seabed with flexible lines (cables, ropes, chains) leadingfrom them to buoys, ships, or platform structures. The function of a mooringsystem is to keep the buoy, ship, or platform structure at a relatively fixedlocation during engineering operations. Engineering efforts in mooring systemshave focused in recent years on the development of new anchor configurationswith higher pullout loads, larger capacity and lower cost of installation for deeperwater applications. When pipelines were first laid offshore, no extraordinary analyses or deploy-ment techniques were needed since they were in shallow water and were of smalldiameter, somewhat flexible, and made of relatively ductile steel. As platformswere built in deeper and deeper water with multiple well slots, larger diameter
  17. 17. HISTORICAL PERSPECTIVE 3pipelines of higher strength were required. During the 1960s, engineers metthis challenge with new designs and with refined methods of analysis and de-ployment. Pipeline systems evolved into two main types: sea floor and verticalconfigurations. Both are used to transport gas and oil, but the vertical sys-tems also include risers to carry drilling tools, electric power lines, dredge pipesfor deep sea mining, and cold water pipes (CWP) for ocean thermal energyconversion (OTEC). Throughout the world there are at present about 80,000 km of marinepipelines. Since 1986, the rate of building new marine pipelines has been about1000 km per year. Individual pipelines on the sea floor vary in length from 1 to1000 km and in diameter from 7 to 152 em. For instance, a Norwegian projectfeatures a 1000 km line extending from the Troll field to Belgium, which wascompleted in 1992. At present, Kuwait has the loading line of largest diameter,152 em. The pipelines of smaller diameter are used to transport oil and gasfrom wellheads, and those of larger diameter are used to load and unload oilfrom tankers moored at offshore terminals. The deepest sea floor pipelines atpresent are the 46 em diameter gas lines in the Gulf of Mexico, for which the e 500 ~ ~ 1000 a:: i:! ~ 1500 2000 (f) Figure 1.1 Six offshore platforms at their maximum depths: (a) jackup rig; (b)gravity platform; (c) jacket structure; (d) compliant tower; (e) tension leg platform; (f) semisubmersible.
  18. 18. 4 STRUCTURES IN THE OFFSHORE ENVIRONMENTmaximum depth is 1400 m. Sea floor pipelines are often anchored to the seabedor buried in trenches for protection from erosion and the undermining effects ofcurrents. Some seabed pipelines have a coating of concrete to add protectionand to reduce buoyancy. Figure 1.2 A mat-type jackup rig at a Louisiana dock.1.2 PLATFORMSSix general types of offshore platforms are depicted in Figure 1.1. The first threeare designed for depths up to about 500 m, and the last three are for depths to2000 m. Not shown are subsea production platforms, which are presently ratedfor 3000 m depths.Fixed-Bottom Platforms A mobile structure often used for exploratory oil-drilling operations is theself-elevating platform commonly called a jackup or mat-supported rig. A con-structed version of this platform, depicted schematically in Figure, is shownin Figure 1.2. Typically, such a platform is supported by three to six legs thatare attached to a steel mat resting on the sea floor. In soft soils, the legs passthrough the mat and may penetrate the soil to depths of up to 70 m. To thebottom of each leg is attached a steel saucer or spud can to help stabilize the
  19. 19. PLATFORMS 5structure and to minimize leg penetration into the soil. The height of theplatform above the seafloor, up to 100 m, may be adjusted by using motordrives attached to each leg. Figure 1.3 A jacket-template platform (courtesy of IHI Co. Ltd., Japan). A platform designed to be used in a fixed location as a production unitis shown in Figure 1.1b. Such a unit, called a gravity platform, consists of acluster of concrete oil-storage tanks surrounding hollow, tapered concrete legsthat extend above the water line to support a steel deck. See Graff and Chen(1981). A typical unit, of which there were 28 operating in the North Sea in1999, has one to four legs and rests directly on a concrete mat on the sea floor.
  20. 20. 6 STRUCTURES IN THE OFFSHORE ENVIRONMENTWith ballast consisting of sand in the bottom of the tanks and seawater inthe legs, these structures depend on self-weight alone to maintain an uprightposition when subjected to the highest waves that are expected to occur in a100 year time period. A realistic 100 year wave that may occur in the northernNorth Sea is 27.8 m. At present, the largest concrete gravity platform is theTroll structure, and one of moderate size is the Statfjord-A Condeep structure,both located in the North Sea. The latter structure is 250 m high and hasthree legs. Located off the coast of Norway, the Statfjord-A Condeep unit hasslots for 42 oil wells that reach to depths of 2800 m. When in operation, itaccommodates a crew of 200 people who live and work on this structure. COMPLIANT ~ JACKET TOWER : : TEMPLATE i,,,~ Idi ~ ,i i I ~ I J ~~ " ... ~ Jt ········~~:-.-:".:."tw.. o 0.1 0.2 WAVE FREQUENCY, HzFigure 1.4 Two storm wave height spectra in the Gulf of Mexico, showing two offshorestructures with natural frequencies beyond the frequencies of the highest energy waves. Found more frequently among the permanent, fixed-bottom structures, how-ever, is the steel truss or jacket template structure shown schematically in Figure1.1c, where an installed structure is depicted in Figure 1.3. As for the gravityplatform, each steel jacket unit is designed for a fixed location and a fixed waterdepth. The first such structure was operational in 1955 in water 30 m deep. By1999 there were approximately 6500 jacket structures, the tallest of which wasthe Bullwinkle unit located in the Gulf of Mexico. The common characteristicsof these jacket structures are their tubular legs, somewhat inclined to the verti-cal, and reinforced with tubular braces in K or X patterns. Piles driven throughthese legs into the sea floor and clusters of piles around some of the legs main-tain structural stability in adverse weather. One of the largest jacket structuresis the 380 m high Cognac unit, which has 10 legs with 24 piles extending 140 minto the soft clay of the Gulf of Mexico. As with all jacket template structures,its natural or fundamental bending frequency of 0.17 Hz is above the 0.11 Hzfrequency of the highest energy sea waves in the Gulf of Mexico during stormconditions, as depicted in Figure 1.4.
  21. 21. PLATFORMS 7Compliant Platforms An alternative class of offshore structures meant for depths from 300 to 800m is the compliant tower such as that shown in Figure 1.1d. Such a towermayor may not have mooring lines. It is a pile-supported steel truss structuredesigned to comply or flex with the waves and has considerably less structuralmaterial per unit height when compared with a common jacket template tower. The first compliant tower was the Lena, which was installed in the early1980s in the Gulf of Mexico. Including its three-level drilling and productiondeck and its drilling rigs, this tower reaches a total height of 400 m. Eachof the 20 stabilizing cables, attached 25 m below the water line and arrangedsymmetrically about the structure, extends a horizontal distance of about 1000m to a line of clumped weights that rest on the sea floor, to an anchor cableand an anchor pile. Under normal weather or small storm conditions, the cablesact as hard springs, but with severe storms or hurricanes, the cable restraintsbecome softer or compliant. That is, the amplitude of tower rotation increasesat a rate greater than that of the loading, since the clumped weights lift off thesea floor to accommodate the increased storm loads on the tower. When stormsor hurricane conditions are anticipated, operations on compliant towers ceaseand the crew is evacuated. Installation of the Lena cables was more difficult and costly than anticipated.Subsequently, compliant towers without cables have been designed by Exxon,and two such designs were installed in 1999 in the Gulf of Mexico. Unlikethe jacket-template structures, the compliant towers have natural frequenciesin bending or sway near 0.03 Hz, or well below the 0.05 Hz frequency of thehighest energy sea waves in the Gulf of Mexico during storm conditions. Thus,an important feature of such structures is that they are designed to have naturalsway frequencies well removed from the frequency range of the highest energywaves for normal seas (0.1 to 0.15 Hz) and for storm seas (0.05 to 0.1 Hz). Thisfrequency spread is necessary to avoid platform resonance, which can lead tofailure. The sway frequencies of two platforms in comparison to the frequencyrange for the spectrum of the highest energy storm waves in the Gulf of Mexicoare depicted graphically in Figure 1.4. The measurement and meaning of thiswave height spectra, which is highly site-dependent, will be discussed in detailin subsequent chapters.Buoyant Platforms The tension leg platform (TLP) can be economically competitive with com-pliant towers for water depths between 300 m and 1200 m. The schematic designof the TLP is depicted in Figure 1.1e. In such designs, the total buoyant forceof the submerged pontoons exceeds the structures total gravity or deadweightloading. Taut, vertical tethers extending from the columns and moored to thefoundation templates on the ocean floor keep the structure in position duringall weather conditions. The heave, pitch, and roll motion are well restrained bythe tethers; but the motions in the horizontal plane, or surge, sway, and yaw,are quite compliant with the motion of the waves. The first production TLP
  22. 22. 8 STRUCTURES IN THE OFFSHORE ENVIRONMENTwas built 150 km off the coast of Scotland in the mid-1980s. Conoco installedthe Julliet in 1989, and Saga Petroleum installed the Snone near Norway in1991. The tethers for the Snone are 137 cm in diameter. By the late 1990s,a total of eleven TLPs were installed, three in the North Sea and eight in theGulf of Mexico. For water depths of about 1500 m, a subsea production system provides anexcellent alternative to a fixed surface facility. Much of a subsea system restson the ocean floor, and its production of oil and gas is controlled by computerfrom a ship or other buoyant structure above the subsea unit. The buoyantstructure and the subsea unit are often connected by a marine riser, which willbe discussed presently. Figure 1.5 A semisubmersible platform (courtesy of the builder, Mitsubishi, Ltd., Tokyo and the owner, Japan Drilling Co.). A popular buoyant structure is the floating production system. Such a struc-ture is practical for water depths up to 3000 m, and also at lesser depths wherethe field life of the structure is to be relatively short. An example of a buoyantstructure is the semisubmersible with fully submerged hulls, shown schemat-ically in Figure 1.1f , with an installed design shown in Figure 1.5. Other
  23. 23. AJOORlNGS 9examples include ships converted to floating production systems. In the late1960s, companies initiated research and design for these semisubmersible, multi-hull tubular structures and ships that would remain relatively stable in roughseas. In the late 1990s, the first three draft caisson vessels, or spars, wereinstalled for use in 180 m water depths. Spars are floating vertical cylinders thatsupport production decks above storm waves. These structures are controlledto remain essentially still in stormy seas. Some need to be towed from place toplace; others are self-propelled. During drilling and production operations, thesestructures are kept in place with mooring lines and thrusters. The computer-controlled thrusters monitor the mooring line forces and accurately positionthe structure over the wellhead. One of the first semisubmersible structureswas the Sedco 709 with a water depth rating of 1800 m. By the year 2000,semisubmersibles using dynamic positioning were designed for 3000 m waterdepths.1.3 MOORINGSTemporary Anchor Moorings A classical example of temporary offshore mooring is the spread mooringconfiguration for a ship in relatively shallow water. Six to eight cables of wirerope or chain are unreeled from onboard winches symmetrically placed aroundthe perimeter of the ship. Tug boats aid in spread mooring installations. Inplace, each cable hangs as a catenary curve and is attached either directly toa drag embedment anchor in the seabed or to a buoy that is anchored. Anexample of a spread mooring configuration is shown in Figure 1.6. Particularmooring configurations were reported by Baar et al. (2000) and OBrian andMuga (1964). SWAY PLAN VIEW t~~Q ~ . Figure 1.6 A spread mooring configuration. In a typical spread mooring operation for a semisubmersible in deep water,a work vessel transports each anchor while pulling out its cable attached to thesemisubmersible. The vessel lowers the anchor and installs a locating surfacebuoy just above it. At present, temporary systems for semisubmersible drilling
  24. 24. 10 STRUCTURES IN THE OFFSHORE ENVIRONMENTrigs and construction barges are used in water depths of up to 2000 m. Anexample of an early and successful drilling rig is the Ocean Victory installed ata water depth of 450 m. This rig employed 12 anchors, each with a holding powerof 200,000 newton (N) or about 45,000 lb. Each catenary line was about 2500m long and consisted of two equal segments: one of 8.9 cm diameter steel wirecable and the other of 8.3 cm diameter chain. Newer generation rigs designedfor deep water have ten times the anchor-holding capacity of the Ocean Victory.An anchor design based on suction is shown schematically in Figure 1.7. LOWERING CABLES MOORING CHAIN Figure 1.7Platform Pile and Single-Point Moorings For installing piles for platform moorings, specially fitted derrick bargesmay use hydraulic hammers, drilling equipment, or possibly a jetting system.In jetting, seawater is forced around the base of a pile, blasting away the soilto make way for pile embedment. When installing mooring piles for tension legplatforms, template structures are carefully positioned at the site, and piles arehammered through the template that serves as a pile guide. Suction piles areemployed in deep water where the use of hydraulic hammers is impractical ortoo costly. Suction piles employ hydrostatic pressure to push the piles to fullpenetration. Single-point mooring (SPM) systems are designed to accommodate deep-draft tankers while they transfer crude oil and fuel oil to and from shore. Twotypical designs are shown in Figure 1.8: the single anchor leg mooring (SALM),and the catenary anchor leg mooring (CALM). By the year 2000, there wereabout 50 SALM systems and 150 CALM systems in operation throughout theworld. Their common features are the rotating head on the buoy and the ver-tical chain that anchors the buoy to the sea floor. While a few SALM systemsmay have taut mooring lines for added buoy stability, all CALM systems havemultiple catenary lines (anchor legs). A third type of SPM system, the ar-ticulated column, has been designed and laboratory-tested but has yet to beinstalled offshore.
  25. 25. PIPELINES 11 MOORING nJRNTABLE BUOY HAUSERS MOORING BUOY LINE FLOATING I 1 HOSE I I (a) (b) I " MOORING CHAINS I (60R8) , I lWlNHOSES CHAIN , ,. I SWIVEL .- PIPELINE ", MANIFOLD RISER HOSES Figure 1.8 Two single-point mooring systems: (a) a single anchor leg mooring (SALM); (b) a catenary anchor leg mooring (CALM). Successful SPMs are found at the worlds largest terminals. One of these isSaudi Arabias Juaymah exporting terminal, which has two SALM buoys andfour CALM systems. For this SPM system, crude oil and fuel oil are loadedsimultaneously to a moored tanker through the swivel assembly on the seabed.A second example is the CALM system for service vessels associated with theCognac platform in the Gulf of Mexico. This SPM buoy has 12 catenary lines,each anchored to 0.76 m diameter piles. The water depth here is 275 m.1.4 PIPELINESSea Floor Pipelines Most of the 80,000 km of offshore pipelines have been installed by one ofthe following three methods. In these methods, the deployed pipeline forms anS-shape between the vessel and the sea floor. TENSIONER SEAFLOOR Figure 1.9 The laybarge method of pipeline construction offshore.
  26. 26. 12 STRUCTURES IN THE OFFSHORE ENVIRONMENT 1. Laybarge method. Pipe sections, which sometimes have been coated pre-viously with concrete for protection, are welded together on the deck of a bargeand deployed on rollers over the stern. Near the stern, the pipeline passesover pontoons called stingers that relieve excess bending in the pipeline as it isdeployed to the seabed. See Figure 1.9. 2. Reel barge method. Small to medium diameter pipe sections (up to 41 cmin diameter) are prewelded and coiled onto a reel mounted to the deploymentvessel. As the pipeline is unreeled at sea, it passes through straightening rollersand then deployed as in the laybarge method. 3. Bottom pull method. Pipe sections are assembled on shore, and the pipestring is towed into the sea by a barge. During the launch to its place onthe seabed, pontoons are often used under the string to avoid excess pipelinebending. The following two methods of pipeline deployment have been studied exten-sively but have yet to be used. 1. J-Iay method. A dynamically positioned vessel such as a drill ship, aconverted pipelaying vessel, or a semisubmersible may be used in this operation.On board the vessel a derrick is used to hold the pipeline vertically as it islowered, and the pipeline forms a J-shape between the vessel and the seafloor.An efficient single station pipe welding procedure has to be developed beforethis method can be adapted to common practice. 2. Floating string method. According to this concept, the pipeline is floatedon pontoons at the water surface. Then the pontoons are released successivelyso that the pipe string gradually sinks to the sea floor.Vertical Pipelines One type of vertical offshore pipeline is the marine riser, which is shownfor several of the structures in Figure 1.1. Although marine risers make upa fraction of the 80,000 km network, they are nonetheless key components ofoffshore structures and serve a variety of functions. For example, a riser maycontain a bundle of smaller pipelines connecting a wellhead to its platform, orit may transport oil directly from its platform to a sea floor pipeline. A risermay act as a drilling sleeve, or it may contain electric power lines for operatingseafloor mining vehicles or other subsea facilities. A typical drilling riser, with aball joint at the bottom and a telescoping joint at the top, is maintained undertension to ensure stability and is kept to within 8 degrees of the vertical bycomputer-controlled positioning of its parent drilling vessel. In the mid-1980s,the longest riser was operating off the east coast of the United States, wheredepths of 2100 m were reached. At present, risers for depths approaching 3000m are becoming a reality. Vertical pipelines are used in deep-sea dredging operations. A most chal-lenging engineering problem, which began to receive serious attention in thelate 1960s, was that of designing dredge pipes to suck up and transport man-ganese nodules from depths of 3000 to 5500 m to the ship. See Lecourt andWilliams (1971).These nodules, about the size of a mans fist and found in a
  27. 27. PIPELINES 13monolayer on the sea floor in many parts of the world, contain other mineralsas well, including cobalt, copper, and nickel with traces of molybdenum, vana-dium, and titanium. By the mid-1980s, remote-controlled mining of the noduleswas achieved with a 5500 m long dredge pipe. In this design, air bubbles arepumped into the string at various points along its length to aid in pumpingthe nodules to the surface. The dredge pipes position is computer-controlled,synchronized to the movement of both the self-propelled mining scoop at oneend and the ship at the other end. A schematic diagram of this system is shownin Figure 1.10a. (a) (b) MANGANESE NODULESFigure 1.10 Two vertical pipeline systems: (a) a dredge pipe for mining manganese nodules; (b) a cold water pipe for ocean thermal energy conversion. Vertical cold water pipe systems for ocean thermal energy conversion (OTEC)have been studied extensively. See Wilson et al. (1982). Calculations show thatpractical full-scale pipe designs are 10-20 m in diameter, are about 1000 m long,and may be used to raise the cooler water in the deep ocean to the warmersurface water. With the differential temperature between the ends of the pipe(10-20°C), it is possible to produce net power through heat exchange. Such asystem is depicted in Figure 1. lOb. In 1979, a small-scale OTEC power plantoperated successfully on a barge 2.3 km offshore of Kona, Hawaii. This pipewas 61 em in diameter, 660 m long, and made of polyurethane. The CWP wassupported by a surface buoy at the barge stern and was tension-moored with
  28. 28. 14 STRUCTURES IN THE OFFSHORE ENVIRONMENTa smaller diameter polyurethane pipeline, a seafloor wire rope to shore, andconcrete blocks on the sloping sea floor. This was the worlds first and so farthe only OTEC system at sea to generate net useful power. In the year 2001,further development work on OTEC systems was underway in India.1.5 CHALLENGESTo meet the need for new sources of energy and minerals, marine engineers mustwork at the frontiers of known technology. Their main challenges are to design,deploy, and operate facilities and equipment in environments where none havebefore existed-in deeper and deeper water, on the slopes of the continentalshelves, and in the hostile Arctic seas. To meet the challenges, marine engineerscontinue their research and development efforts in the following four broad andnecessarily overlapping topics of concern.Environmental Forces Every offshore facility is subjected to several types of environmental loadsduring its lifetime. Thus, site-dependent databases are being developed to char-acterize the time-varying fluid-induced loads of winds, currents, and waves.Such loads occur both on a short time scale of seconds and minutes, inducedby periodic vortex shedding, wind gusts, and wave slamming; and over longerperiods of hours, days, and even years, where the loads are induced by steadywaves, tides, and hurricanes. At some sites, data are also needed on subseaearthquake intensity, on the scouring of sea floor foundations by currents, andon the total and differential settlements of the sea floor due to the withdrawalof hydrocarbons during the lifetime of the structure. At present, there is theparticular challenge of minimizing differential settlements for gravity platforms.For the Ekofisk structural complex in the North Sea, however, this challengeappears to have been met by employing water injection. In Arctic regions, dataare needed on the rates of ice accretion and on the velocity, yield strength,and mass of floating ice. Reliable deterministic and statistical methods are be-ing developed to measure and interpret time-dependent field data suitable forpredicting structural loadings.Structural Materials An offshore structure should have a high ratio of strength to self-weight.For instance, for each added unit of deck weight for a tension leg platform,an additional 1.3 units of hull weight are required for buoyancy support, andan additional 0.65 unit of mooring pretension force is needed. In this case,high-strength steels with high fracture toughness are being investigated for thepurpose of reducing hull weight. Hollow cylindrical steel link chains or syntheticmooring line materials, such as Kevlar with an abrasion-resistant polyethylenecover, are being developed to increase mooring capacity. To determine thesuitability of new, high-strength steels and composite materials in the offshoreenvironment, test data are being generated. These data involve measures of cor-rosion fatigue, fracture toughness, stress corrosion cracking, and weldability for
  29. 29. REFERENCES 15steels and the reliability of several types of high-strength, light-weight syntheticrope mooring lines.Modeling and Analysis Once the site, the environmental load conditions, and the preliminary struc-tural design are determined, a mathematical model of the structure is formu-lated, and computer-aided analyses are performed to determine the overall mo-tion, the critical stresses, and the reliability of the design. The fundamentalmechanics of fatigue and fracture were reviewed by Petroski (1984), who dis-cussed fatigue failures that have occurred at the welded tubular joints, whichcan be subjected to millions of loading cycles during a structures lifetime. Itis particularly important to include fatigue in the structural reliability analysis.Novel and improved methodology for reliability assessment of offshore structuralwelded joints, with applications to jackup rigs, was discussed by Etube (2001).Computer graphics and finite element methods of structural analysis that aidin this design process are continually being improved. Structural optimization,least weight criteria, nonlinear dynamics, fluid-structure-soil dynamic interac-tions, and statistical methods are being added and refined in the mathematicalmodels. The aims are to achieve more accurate representations for structuraldynamic behavior and improve the predictions of structural lifetimes. See Buch-holt (1997) and Jin and Bea (2000).Experimental Evaluations Before a structure is installed offshore, extensive tests are made on itscomponent materials and also on its overall dynamic behavior using scaled-down laboratory models with simulated environmental load conditions. Presentlaboratory-scale efforts are focused on two issues: fluid-structural load inter-actions, especially the effects of fluid vortices on structural motion; and themechanics of structural-soil foundation interactions. Once installed in the field, the structure may be instrumented for a period oftime to determine whether its behavior under a variety of natural forces has beenaccurately predicted by the mathematical models and analyses used in its design.Such data are useful for future similar designs. In addition, for platforms, buoys,and pipeline systems nearing the end of their lifetimes, frequent inspection andmaintenance are required to assure continuing performance. Improvements arebeing made in the computer systems used for the retrieval, storage, and analysisof large amounts of data from both laboratory scale and field tests of offshorestructural systems.REFERENCES Adrezin, R., Bar-Avi, P. and Benaroya, H., Dynamic Response of Compliant Offshore Structures-Review, Journal of Aerospace Engineering, October, 114-131, 1996. Baar, J. J. M., Heyl, C. N., and Rodenbusch, G., Extreme Responses of Turret Moored Tankers, Proceedings of the Offshore Technology Conference, OTC- 12147, 2000.
  30. 30. 16 STRUCTURES IN THE OFFSHORE ENVIRONMENTBuchholt, H. A., Structural Dynamics for Engineers, Thomas Telford Publications, London, 1997.Etube, L. S., Fatigue and Fracture Mechanics of Offshore Structures, Professional Engineering Publishing Limited, London and Bury St. Edmunds, Suffolk, UK, 2001. Gerwick, B. C., Construction of Marine and Offshore Structures, second ed., CRC Press, Boca Raton, FL, 1999. Graff, W. J., and Chen, W. F., Bottom-Supported Concrete Platforms: Overview, Journal of the Structural Division, ASCE l07(ST6), 1059-1081, June 1981. Jin, Z., and Bea, R., Enhancements of TOPCAT-3 Dimensional Loadings, Relia- bility, and Deck Structure Capabilities, Proceedings of the Offshore Technology Conference, OTC-11939, 2000. Lecourt, E. J., and Williams, D. W., Deep Ocean Mining-New Applications for Oil and Field Marine Equipment, Proceedings of the Offshore Technology Confer- ence, OTC-1412, 1971. Mulcahy, M., Fixed Offshore Structure Design Responds to Tougher Operating Re- quirements, Sea Technology, 10-12, June 1979. OBrian, J. T., and Muga, B. J., Sea Tests on a Spread-Moored Landing Craft, Proceedings of the Eighth Conference on Coastal Engineering, Lisbon, Portugal, 1964. Petroski, H.J., Fracture Mechanics and Fatigue in Offshore Structures, chapter 13 of Dynamics of Offshore Structures, J. F. Wilson, editor, Wiley, New York 1984. Will, S. A., Conventional and Deep Water Offshore Platforms, Civil Engineering 58, February 1982. Wilson, J. F., Muga, B. J., and Pandey, P., Dynamics During Deployment of Pipes for Ocean Thermal Energy Conversion, Proceedings of the First Offshore M echan- ics/Arctic Engineering/ Deepsea Systems Symposium, ASME, New Orleans, LA, 53-58, March 1982.
  31. 31. 2Structure-EnvironmentalForce InteractionsJames F. Wilson
  32. 32. 18 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONSfor multi-degree of freedom problems in three-dimensional physical space, themathematical models in this text are limited to motion in the vertical plane, arepresentation that is generally adequate for a preliminary dynamic design ofoffshore structures. The simplest mathematical model has just one equation of motion, writtenas a differential equation in terms of only one time-dependent scalar coordinatewhich uniquely describes the structures position. Such a model defines a singledegree of freedom system. Assume that a structure or portion of a structureis rigid, or nearly so, and has a virtual mass m. Because of structure-fluidinteractions, discussed later in this chapter, virtual mass includes all or part ofthe structural mass, together with some water that the structure drags with itduring motion. Assume further that m is sufficiently higher than the mass ofthe system restraints (guy lines or soil foundation) that limit its motion. Letthe motion of m be restricted to a plane on which the absolute displacementcoordinate of its mass center G is v = v(t). Let £Fv denote the sum of thefour types of external force components on m, in line with v and positive in thepositive v direction. In these terms, Newtons second law states ,LFv = mii (2.1)where ii is the absolute acceleration of G. This equation is particularly useful insingle degree of freedom models involving rigid body translation only in whichthe flexible supports restraining the motion of m are in line with v. w tv ~ flv) . G m G FOR SHIP BUOYANT AT REST FORCE = W Figure 2.1 Free body sketch for the spread moored ship of Figure 1.6. Example Problem 2.1. Consider the surge motion for the spread mooredship shown in Figure 1.6. This ship is asumed to be a rigid structure for whichthe motion is described solely by its horizontal displacement coordinate v = v(t)of its mass center at G. Sway, pitch, heave, and yaw motions are neglected. Thefree body sketch of the ship in the vertical plane, shown in Figure 2.1, depicts thefour types of externally applied loads: self-weight or ship displacement W, whichis balanced by its buoyant force equal in magnitude to W; the equivalent of all v-directed, time-dependent environmental forces PI (t); the equivalent v-directedmooring line restraint force of the general form q( v); and the velocity-dependent
  33. 33. SINGLE DEGREE OF FREEDOM STRUCTURES 19damping force of the general form f(v). When equation (2.1) is applied in thev direction to this free body sketch, the resulting equation for surge motionbecomes mv + f(v) + q(v) = Pl(t) (2.2)In the first term of equation (2.2), the virtual mass m for surge motion is about15 percent higher than the actual ships mass. Quantitative values for theremaining terms of this equation are discussed later in this chapter. Other examples of single degree of freedom offshore structures include thepure, plane rotational motion of buoys such as shown in Figures 1.8 and ofgravity platforms rocking in the vertical plane, such as depicted in Figure 2.2.With all out of plane motion and translational motion supressed, the angulardisplacement of these structures, modeled as rigid bodies of virtual mass m, theangular displacement is uniquely described by the single coordinate B = B(t).Suppose that such a structure rotates about a fixed point 0 in the plane. Let£1vlo denote the sum of all external moments in the plane of motion, acting onm, positive in the positive direction of B. These moments, which are due to thefour types of external forces discussed above, are all expressed with respect tothe same fixed point O. In this case, the equation of motion has the generalform (2.3)in which Bis the absolute angular acceleration of the rigid body and Jo is thevirtual mass moment of inertia of this body with respect to the reference axisthrough point 0 and perpendicular to the plane of motion. The value of J o isdefined as o Lr J = 2 dm (2.4)where r is the distance from that reference axis to the virtual mass elementdm, and the integration is over the whole rigid body. In applications it is oftenconvenient to express Jo in terms of J e , or the value of J o when the point 0coincides with the mass center G. The connection is through the parallel axistheorem, or Jo = Je +mhb (2.5)where he is the distance between 0 and G. Values of Je for a variety of solidsof uniform density are listed in most elementary texts on rigid body dynamics.For a relatively rigid structure composed of such shapes, the structures total J ovalue can be estimated by calculating J o for each elementary component usingequation (2.5) and then superimposing the results.
  34. 34. 20 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS ~DECK I I I _LEG -,.... 1_ B h G ~CAISSONS hG hb ~SKIRT 0 III ,,1 EQUILIBRIUM STATE FREE BODY SKETCH Figure 2.2 Monopod gravity platform in pure rotation. Example Problem 2.2. Consider the rotational motion of a rigid gravityplatform shown in Figure 2.2. This structure has an actual mass of rno, abuoyant mass of rnb, and is supported at the sea floor by a soil foundation. Letthe motion be limited to rotations 0 about the base pivot point O. Define M pcas the net moment about point 0 due to the pressure differences across the topof the caissons, and let F(t) represent the net horizontal load due to currents,winds, and waves, located at height h o above O. Let f(B) and q(O) be therespective foundation reaction moments for damping and rotational restraint.In these terms, the application of equation (2.3) to the free body sketch of thegravity platform of Figure 2.2 leads to the following equation of motion: JoB + f(B) + q(O) - (rnoghc - rnbghb) sinO = -F(t) ho - Mpc (2.6)In equation (2.6), Jo is based on the virtual mass of the submerged portion of thestructure. For instance, for a submerged cylinder, virtual mass is approximatelythe cylinder mass plus the mass of the water displaced by the cylinder. Quan-titative calculations for Jo and the other terms of equation (2.6) are illustratedin Example Problem 5.2. In the last example problem, the rigid body assumption may not always bea realistic one. The rigid body model would be entirely useless, for instance,if the dynamic flexural stress were needed in the legs of the gravity platform.
  35. 35. FLUID-INDUCED STRUCTURAL FORCES 21Nonetheless, the rigid body assumption may be warranted if only an estimate ofthe overall dynamic stability of the platform-soil foundation system is needed.Thus the choice of the mathematical model is strongly tempered by the partic-ular goals of the analysis.2.2 FLUID-INDUCED STRUCTURAL FORCESThere is a wealth of literature on the theory and measurement of forces on solidbodies moving or at rest in dynamic fluid fields. An excellent critique of thisliterature that extends back to the early nineteenth century and up to 1981 isgiven by Sarpkaya and Isaacson (1981). The essential physical ideas and gov-erning nondimensionalloading parameters presently perceived as characterizingthese dynamic forces are now highlighted. Most of the following discussion islimited to an isolated, fully submerged, right circular cylindrical solid for whichthe incident fluid velocity is perpendicular to its longitudinal axis. In the planeflow cases considered, shown in Figures 2.3 through 2.6, the fluid or cylindermotion is in line with its net force per unit length, q. The fluid is assumedto be incompressible, and for the present, effects of nearby objects and solidboundaries are not included. As restrictive as these assumptions may seem, theresults none the less demonstrate the basic ideas of fluid loading for a majorportion of offshore structures and their components, including pipelines, cables,tubular structural members, and many types of submerged tanks and caissons.Classical Inviscid Fluid Flow One classical loading parameter is the inertia coefficient, eM, alternativelydenoted as CJ or Gi , a parameter that originated with hydrodynamics or thetheory of ideal, inviscid flow, formulated during the nineteenth and early twen-tieth centuries (Batchelor, 2000; Lamb, 1945). This coefficient relates the forceper unit length qJ that is required to hold a rigid cylinder stationary in a fluidof uniform, constant free stream acceleration of magnitude u. That is, (2.7)where p is the fluid density and D is the cylinder diameter. Shown in Figure2.3 is this case of unseparated, unsteady, ideal flow, together with values of C Mfor several ratios of cylinder length to diameter, €/ D. These results, reportedby Wendel (1956), are based on theoretical values of another nondimensionalparameter, the added mass coefficient CA, defined by (2.8)It is observed from the theoretical data in Figure 2.3 that as the cylinder lengthbecomes much larger than its diameter, the value of GM approaches the limitof 2, for which C A approaches unity by equation (2.8).
  36. 36. 22 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS l/I2~ 1.2 1.62 2.5 1.78 5.0 1.90 9.0 1.96 00 2 Figure 2.3 A rigid, stationary cylinder in an ideal, accelerating fluid. Consider now the case of Figure 2.4, a rigid cylinder with a mass per unitlength mo, immersed in a fluid. This cylinder has an absolute translationaldisplacement v = v(t) in the fluid medium that would normally be at rest,except for the presence of the cylinder. Based on Newtons second law of motionand classical hydrodynamic theory, the force per unit length required to achievean acceleration v for the cylinder is iiI = (m o + CAP7r~2) = mv V (2.9)Equation (2.9) defines the cylinders virtual mass per unit length, m. Thus, mis the sum of mo in vacuo and the added or apparent mass per unit length,CAP7r D 2/4, resulting from those fluid particles that are pulled along by theintruding cylinder. v Figure 2.4 A rigid cylinder accelerating in an ideal fluid.
  37. 37. FLUID-INDUCED STRUCTURAL FORCES 23Real Viscous Fluid Flow Measurements show that CM and CA are time-dependent, which is due tofluid viscosity. There is flow separation behind the cylinder, accompanied bydifferential pressure forces opposing cylinder motion. Such forces are referredto as form drag. In applications, a root-mean-square (rms) measured average ofeach coefficient C M and C A is used. If such data are lacking, it is appropriateto choose C A = 1 for design purposes, provided that the geometric ratio f! / D ismuch greater than one. Figure 2.5 Viscous drag on a rigid, stationary cylinder. Another classical loading parameter is the viscous, frictional drag coefficient,CD. Define qD as the force per unit length necessary to hold a fully immersedcylinder stationary as it is subjected to a constant free stream fluid velocity, u.In these terms, measurements show that (2.10)The use of the absolute value sign on one of the velocity terms guarantees thatqD will always oppose the direction of u, as shown in Figure 2.5. For thisflow case, the experimental relationships of CD to two nondimensional parame-ters, cylinder roughness, and the Reynolds number, are well known (Schlichting,1968). Here the Reynolds number is defined by Re = puD (2.11) JLwhere JL is the absolute viscosity of the fluid. For a smooth cylinder subjectedto this constant, uniform, free stream flow, the value of CD is approximatelyunity for Re in the range of about 1000 to 200,000. If the cylinder were rotating about its longitudinal axis or if it were notcircular, or if other solid elements or rigid boundaries were nearby, one wouldneed an additional loading parameter, the lift coefficient CL . In such cases, thelift force per unit length, which is perpendicular to u and qD, has the same form
  38. 38. 24 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONSas equation (2.10), where CL depends on cylinder roughness, Reynolds number,and the proximity of objects nearby. In a uniform flow field, both separation and periodic wakes or vortices mayform behind the stationary cylinder. This phenomenon, discussed extensively byBlevins (1977), is depicted in Figure 2.6. The vortices behind the cylinder detachalternately. Accompanying this is a periodic pressure fluctuation, top to bottom,at a characteristic frequency of fs, typically expressed in units of Hz (cycles persecond). Periodic vortices or vortex sheets occur for Reynolds numbers in therange of 60 to 10,000 and sometimes even higher. (Swimmers can observe thisphenomenon, for instance, by moving a hand downward through the water withfingers spread and feeling a tendency for the fingers to vibrate horizontallyor side to side).The nondimensional parameter that correlates vortex-sheddingdata for the flow of Figure 2.6 is the Strouhal number, defined by S = fsD (2.12) u pattern times: t = 11f;, 21f;, ... nlf; pattern times: t = 1.51f;, 2.51f;, ... (n+O.5)1f; (n = 1, 2, .. .) (n = 1, 2, ...) Figure 2.6 Periodic vortices trailing behind a rigid, stationary cylinder. Generally, S correlates well with the Reynolds number. For instance, corre-lations of S with Re showing the apparent effects on vortex-shedding frequencyof a cylinder in proximity to a twin cylinder and a ground plane are reported byWilson and Caldwell (1971). In some instances, vortex-induced pressure forceson a cylindrical structure could be large enough to destroy the structure. Forinstance, periodic vortices behind the cylindrical piles supporting an offshorepier lead to a complete destruction of that pier in a tidal current of two knots.Since vortex-induced loading is unpredictable, one generally makes provisions toavoid periodicity of the vortices. Although many methods have been proposedto suppress vortices (Hafen et al., 1976), a particularly practical one is the ad-dition of helical strakes around the cylinder, as shown in Figure 2.7. Optimalstrake geometries for the least vortex-induced loads on circular cylinders werestudied by Wilson and Tinsley (1989).
  39. 39. FLUID-INDUCED STRUCTURAL FORCES 25 rJTJ[11J Figure 2.7 A cylinder with a helical strake to negate periodic vortices. The last nondimensional fluid-loading parameter highlighted herein is theKeulegan-Carpenter number, Kc, which arises when a free stream, plane, peri-odic flow is imposed on a stationary cylinder. If neither u or u is constant, butboth are described by a single, simple, plane wave of time period T, then Kccorrelates well with the force data on the cylinder. This flow parameter is Kc = uoT (2.13) Dwhere Uo is the amplitude of the wave velocity (Keulegan and Carpenter, 1958).Conservation of Linear Momentum and Flow Superposition For a stationary cylinder in a plane flow field with the free stream velocityu = u(t), the total time--varying load per unit length on the cylinder may beexpressed by superimposing the two flow models described by Figures 2.3 and2.5. Thus, by adding their respective drag and inertial loadings expressed byequations (2.10) and (2.7), the result becomes (2.14)Equation (2.14) was first proposed by Morison and his colleagues (1950) as anempirical result, and that equation now bears his name. :e- : -- z r-D-j DISC WAVE I DISC I--Qi1Z LOADING--"~~~~ I I L __ J RIGID I CYLINDER {CONTROL VOLUME Figure 2.8 A rigid, circular cylinder in a flow field.
  40. 40. 26 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS There is a theoretical basis for Morisons equation, the general form forwhich can be deduced by applying the principle of conservation of linear fluidmomentum. This conservation principle is based on Newtons second law asapplied to the fluid occupying a fixed control volume V at any instant of time.In this case, V is chosen as the imaginary rectangular box surrounding a discelement of a submerged circular cylinder, as shown in Figure 2.8. Here, V =D2 6.z, where D is the disc diameter and 6.z is the disc height. For water ofdensity p and with a horizontal velocity u along x, the net horizontal shear load2;Fx on the disc is given by LFx = ~ r PUdV+j pu - udAo (2.15) ut lv Aowhere dV is an element of the control volume and dAo is the element of areaon the surface of the control volume perpendicular to the flow. The reader isreferred to a standard text such as Munson et al. (1998) for a general derivationof equation (2.15). The terms of equation (2.15) are now evaluated. First, 2;Fx = -q6.z whereq is the loading per unit length. The negative sign is chosen since the net shearreaction load must oppose the direction of flow. The first integral on the rightis approximated by ~ 8t r pu dV ~ - pD lv 2 6.z it (2.16a)The negative sign is chosen since the fluid decelerates inside the control volumesurrounding the rigid disc. The remaining integral represents the net momentumflux along x, or the difference between the out-flowing momentum and the in-flowing momentum through area Ao = D 6.z. Thus 1 net pu-udAo =1 out pu·udAo - r lin pu·udAo =O-pu·l u ID6.z (2.16b)where lui is the absolute value of u and is used to preserve the sign of q. Whenu reverses direction, so does q. With equations (2.16), equation (2.15) becomes (2.17)To correlate equation (2.17) with experimental results, the coefficients CD/2and C M /4 are inserted as multiples of the two respective terms on the rightside of equation (2.17), a result that then agrees with Morisons form, equation(2.14). According to the database summarized in Chapter 4, CD and C M both havea range from 0.4 to 2.0. However, based on a multitude of experiments compiledby the British Ship Research Association (1976) and by Sarpkaya and Isaacson(1981), it is quite apparent that CD and CM are not simple constants. In thecase of an imposed periodic plane flow such as offshore waves of period T, if theroot-mean-square (rms) value is chosen for each over a time that is sufficiently
  41. 41. FLUID-INDUCED STRUCTURAL FORCES 27large compared to T, then data show that CD and C M are functions of threeparameters, expressed as CD = CD(Re, Kc, cylinder roughness) (2.18a) C M = CM(Re, Kc, cylinder roughness) (2.18b)Further, Sarpkaya (1976) proposed that a frequency parameter (3 replace Re inthose relationships, where 2 (3 = Re = pD (2.19) Kc JLT CD = C D ((3, Kc, cylinder roughness) (2.20a) C M = C M ((3, Kc, cylinder roughness) (2.20b)The main advantage of Sarpkayas forms is that Uo, the free stream amplitude ofthe periodic velocity, then appears only once in each function of equations (2.20),instead of twice in each function of equations (2.18). For periodic flows, thisalternative gives efficient correlations of measured cylinder forces with CD andCM . Among the phenomena neglected in both of these functional relationshipsare cavitation, fluid compressibility, three-dimensional flow, proximity effects,and all movement of the cylinder. In the following example problem, Morisonsequation is modified to account for one of these neglected factors: the lateralvibrations of the cylinder. - ... v, v -- ev lev U, u FREE BODY SKETCH Figure 2.9 Cylinder model used to characterize fluid-structure interactions. Example Problem 2.3. Consider fluid-solid interactions for the vibratingcylinder based on the single degree of freedom model shown in Figure 2.9. Thecylinder is modeled as a rigid body with an elastic restraint of stiffness k perunit length and with a linear viscous structural damping constant of c per unitlength. This cylinder is an approximate model of a flexible, tubular cross-member of an offshore platform whose legs provide the cylinders end restraint.
  42. 42. 28 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONSThe free stream horizontal flow velocity u = u(t) is in line with the cylinderstranslational motion v = v(t). As shown on its free body sketch, the restraintand damping forces per unit length that oppose the cylinder motion are kv andelJ, respectively. The virtual mass per unit length, in, is deduced from equation(2.9). The fluid loading per unit length is given by equation (2.14), modified sothat the drag force is based on the relative velocity (u - lJ) between the fluidand the cylinder. When Newtons second law, equation (2.1), is applied to thiscylinder, the equation of motion becomes ( - + e A P1r D 2 ).. + cv + k- v = e DP D Iu - v. (u - v.) + CM P1r D 2 U• mo _. (2.21) 4 v 2 4The term involving eM results from fluid motion only, such as wave action,where it is the absolute acceleration of the fluid. Equation (2.21) is nonlinear as a consequence of the drag force term. Bergeand Penzien (1974) linearized this equation for small motion, or Cb = CD lu - vi r:::: constant (2.22)With this assumption, equation (2.21) becomesEquation (2.23) clearly shows that the damping of a moving cylinder is increaseddue to the fluid drag force, a force that in general overwhelms the internalstructural damping e. The conclusion holds true for its nonlinear counterpartalso, equation (2.21). Thus, to solve equation (2.21) or (2.23) for the structural displacementv = v(t), one needs to know four structural parameters: mo, D, e, and k;the fluid density Pi the free stream flow field u = U(t)i and the three empiricalconstants C A, CD and eM. In subsequent examples, k and e will be estimated forparticular cases, and the dependency of the latter three empirical constants onoffshore waves will be discussed in greater detail. Other environmental factorsaffecting the motion of offshore structures are now quantified.Buoyancy and Gravity It is well known that a solid object can be lifted much more easily whenin water than in air. This is because the water pressure exerts an upward orbuoyant force on the submerged solid. The Greek mathematician Archimedes(287-212 B.C.) stated this principle in precise terms: A solid body partially submerged in a fluid is buoyed up by a force mbg equal to the weight of the fluid displaced. (mb is the mass of the fluid displaced.)
  43. 43. FLUID-INDUCED STRUCTURAL FORCES 29 Figure 2.10 Cross section of a ship showing buoyant and gravitational forces. For example, consider the ship shown in Figure 2.10 which has an actualweight (in air at sea level) of mag. The volume of water that the ship displacesis V and the weight density of the water is denoted by Tw The result of applyingArchimedes principle to this system is (2.24)This buoyant force Tw V acts at point B on the ship, in a vertical directionopposite to that of the resultant gravitational force mag. The latter force actsat the ships center of gravity. Thus, a ships actual weight is generally referredto as its displacement. In general, point B is not coincident with point G of the solid body. Inprecise terms: If the displaced fluid is of constant density, B is located at the cen- troid of the displaced fluid volume V.This statement can be proved in general for solid body of arbitrary shape (Batch-elor, 2000). Its validity can be simply illustrated for the following special caseof a solid, prismatic block partially submerged in water. Refer now to the se-quence of illustrations (a) through (f) in Figure 2.11. Suppose that the solidblock (a) is removed from the water and then replaced by pressure forces aroundthe void of volume V so that the water remains undisturbed (b). The pressuredistribution, shown as "gage" pressure, is constant on any horizontal plane andvaries linearly with depth. The pressure distribution on the solid block (c) isidentical to that on the void. In fact this identical pressure distribution wouldalso occur on a water block (d) of volume V which would just fill the same void.
  44. 44. 30 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS AIR ~ ~ I!;". .-.... r.. ~~oln-TT1"TTT"TTTT"TT11Tf" WATER . . (a) FLOATING SOLID (b) WATER CAVITY tV (c) ISOLATED SOLID ~ fllll 1Ilf!1lI~ If If (d) ISOLATED WATER BLOCK m"g YwV G B m,g m"g (e) STATICALLY EQUIVALENT (f) STATICALLY EQUIVALENT SOLID BLOCK WATER BLOCK Figure 2.11 Illustration of Archimedess principle. To maintain static equilibrium of this water block, two conditions must bemet. First, the force of the water block in (d), or mbg which acts at B, thecentroid of V, must be balanced by its net pressure forces along its horizontalboundaries. Since the pressure forces on the vertical boundaries balance, theyare of no consequence in this problem. Second, the resultant of the boundarypressure forces must pass through B to avoid rotation of the water block. Sincethe solid block (e) and the water block (f) have identical boundary pressuredistributions, the buoyant force mbg acts at B on the solid block also. Example Problem 2.4. Consider a gravity platform partially submerged insoft mud, as shown in Figure 2.12. Here the buoyant force of air may be ne-glected because the gravitational force mag is defined as the structures weightin air. Assume that the water-saturated mud layer behaves as a liquid of approx-imately constant density "(Tn and thus contributes to the structures buoyancy.Liquefaction of the mud foundations of gravity platforms can occur during astorm due to caisson vibrations and repeated shear stress reversals at the soil-caisson interface (Graff and Chen, 1981). This leads to a gradual increase inpore water pressure which reduces the shear strength of the mud foundation,causing the foundation to behave as a liquid.
  45. 45. FLUID-INDUCED STRUCTURAL FORCES 31 WATER Y.v :. .. .. (a) Figure 2.12 A gravity platform partially submerged in a mud layer or in liquefied soil. The three buoyant forces on this platform are: (1) a buoyant force due tothe mud of Ym Vm , which is located at a distance of hm from the base and actsupward at the centroid of the displaced mud volume Vm ; (2) a buoyant forcefor the portion of the caisson in water of Y w V e , which is located at he from thebase and acts upward at the centroid of its displaced water volume Ve ; and (3)a buoyant force on the submerged portion of each leg of Y w Ve, which is locatedat he from the base and acts at the centroid of its displaced water volume ofVe. If the structure has N identical legs, the total buoyant force mbg and itsstatically equivalent location h b on the structure are, respectively (2.25) (2.26)Here, h b is located on the vertical centerline of the upright structure, wherethe identical legs are symmetrically placed with respect to this axis. However,that if the structure is tipped at angle e, the resultant buoyant force retainsits same magnitude for all practical purposes, but its location shifts to an off-center position, from B to B. The location of B relative to the mass center Gdetermines the static stability of the whole structure, as the following exampleproblem demonstrates.
  46. 46. 32 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS (a) STABLE (b) STABLE (c) UNSTABLEFigure 2.13 Static stability of three gravity platforms supported by a liquefied soil foundation. Example Problem 2.5. Consider now the stability of the monopod gravityplatforms shown in Figure 2.13, as they rock in plane motion on the liquifiedmud foundation. Assume that the rocking motions are very slow so that thestructures inertias can be neglected, for which mbg :::= mag. If B and also Bremain always above G as a monopod tips to a small angle 8, the structurewill return to vertical equilibrium (8 = 0). This configuration, shown in Figure2.13(a), is stable because the resultant buoyant force and gravity force producea couple opposite to the direction of rotation. The restoring moment imposedby the liquified mud foundation would be relatively insignificant in this case.If the design is such that B and B remain below G, the structure may still bestable, but only if the metacenter M is above G. The metacenter is the point ofintersection of the vertical line through B with the original centerline, as shownin Figure 2.13(b). When M is above G, the floating structure is dynamicallystable because the restoring, gravity-produced couple is in a direction that willreduce 8. If, however, M is below G, as shown in Figure 2.13(c), then the coupledue to the buoyancy and gravity forces will increase 8, and the structure willtopple in the absence of the restoring forces of the foundation. The qualitative analysis presented in the last example problem is analogousto the elementary static stability analysis of ships and other floating objects aspresented, for instance, by Munson et al. (1998). In practice, a more thoroughanalysis of structural stability is needed, which is accomplished by studying thedifferential equations des.cribing structural motion. Such dynamic analyses willbe illustrated in the forthcoming chapters.
  47. 47. FLUID-INDUCED STRUCTURAL FORCES 33Winds and Currents For a typical fixed-bottom offshore platform, the static drag force due towind on the superstructure amounts to about 15 percent of the total force onthe structure (Muga and Wilson, 1970), and often accounts for about 25 percentof the total overturning moment (Graff, 1981). The wind-induced overturningmoment increases linearly with the height of the structure, and thus, as thesestructures are built in deeper and deeper water, the effects of wind drag thenbecome increasingly significant in design. A measure of wind velocity is needed to predict both wind loading on thesuperstructure and to predict the magnitude of the wind-generated wave forceson the submerged portion of the structure. A windstorm is often described asairflow with a mean or steady velocity u(z), with a superimposed fluctuatingvelocity. Here z is the height above the still water level. Gould and Abu-Sitta(1980) point out that the averaging period of one hour has been used in Europeand Canada in presenting data for u(z) = u(h), for a reference height of eitherz = h= 30 ft or 10 m. Gaythwaite (1981) states that in Great Britain valuesof u( h) chosen for structural design have traditionally been averaged over onlyone minute. In the United States, however, the concept of the fastest mile ofwind speed is used to define u(h). That is, measures are made of wind velocityduring the time it takes for a mile of air to pass a fixed point, and the annualextreme condition is used as the reference value. Sachs (1972) and Simiu (1976)discuss methods of converting such data to a mean velocity. Gaythwaite (1981)suggests that all but temporary marine structures should be designed using themean, fastest mile of wind speed associated with return periods of 50 to 100years. Once u(h) is established for a particular offshore site, the mean horizontalwind velocity at height z above the sea surface is u(z) = z) (h lin u(h) (2.27)The exponent n depends on many factors. For instance, n = 3 fits data forrough coastal areas; n = 7 to 8 for sustained winds over an unobstructed sea;and n = 12 to 13 for gusts. At heights of 100 ft or more above the surfaces,the vertical gust-velocity becomes about the same as its horizontal value. Winddata u(h) and further discussions of equation (2.27) are given by Muga andWilson (1970), Sherlock (1953), Simiu and Scanlan (1978), and Vellozzi andCohen (1968). On a member of the superstructure which is nine or more diameters removedfrom neighboring structural elements, u = u(z) calculated from equation (2.27)can be used with equation (2.10) to estimate wind drag force, where CD is basedboth on Reynolds number and on the cross-sectional shape of the member.Values of CD for common shapes are readily available (Hoerner, 1965, andPattison et al., 1977). For a truss structure in wind, the sum of the drag forces on each individualmember may give a low estimate of the total drag force. This effect, called
  48. 48. 34 STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONSsolidification, modifies the drag coefficient as follows: CD = 1.8¢, for 0 < ¢ < 0.6 (2.28a) CD = 2, for ¢ 2: 0.6 (2.28b)where the solidity ratio ¢ is given by ¢ = projected area of truss members only (2.29) projected area of the enclosed solidHere the projected area is in a plane normal to the prevailing direction of thewind. For members of the superstructure closer than nine diameters, shielding ef-fects (sometimes referred to as sheltering) are apparent. For instance, a cylinderjust behind a lead cylinder facing the wind experiences a considerable drop indrag force, and sometimes a negative drag force at close spacings. Wind tunneldata for the shielding of cylinders are reported by Pagon (1934) and Wilson andCaldwell (1971). Numerous references to calculated results based on classicalhydrodynamics are discussed by Muga and Wilson (1970). Values of shieldingfactors varying from 0 to 1.0 and useful for design purposes are recommended byGraff (1981). To obtain the drag force for a shielded component, the shieldingfactor is multiplied by the static drag force of its unshielded counterpart. In addition to these static loads, the dynamic effects of wind on offshorestructures should be considered also. For instance, for a moored structure whosefundamental period in free oscillation is close to the period of wind gusts, thedynamic deflection of the structure could become significant. However, for atruss structure with fixed legs, the overall dynamic response due to gusts aloneare generally insignificant. This is because the time lag between gust arrivalat the leading edges and its arrival on the downstream portions of the openstructure tends to minimize the overall loading. Static loading of individualmembers to gusts could be significant since wind gust speeds may be quitehigh. For instance, a value of 120 knots is often used for the design of structuresin the North Sea, a value about 1.4 times the steady or sustained wind speedu(z) at a height of 30 m (Graff, 1981). On the other hand, adverse vibrations of a structural component may arise insteady winds due to vortex shedding, if these vortex frequencies are in tune or inresonance with a free vibration frequency of a structural member. This may leadto large displacements or flutter of platelike members and to galloping beam andcable components. As discussed earlier in this chapter, helical strakes or otherspoilers can be used to eliminate periodic vortices on tubular members. Vortex-induced resonance is further discussed by Blevins (1977), Gould and Abu-Sitta(1980), and Simiu and Scanlan (1978). Extended discussions concerning thephysical basis for both wind and ocean currents are given in the classical treatiseof Neumann and Pierson (1966). In his summary of ocean currents, Gaythwaite (1981) indicates that tidal cur-rents and wind-stress currents are the two most relevant ones in the structural