Hydromechanically Loaded Shells: Proceedings of the 1971 Symposium of the International Association for Shell Structures, Part I 9780824895426

Table of contents :
CONTENTS
FOREWORD
PREFACE
PRESIDENT'S ADDRESS
Session I DESIGN CRITERIA AND CONCEPTUAL DESIGN, Part I
General Report
NUMERICAL SOLUTION TO THE DYNAMICS OF VISCOUS FLUID FLOW BY A FINITE-ELEMENT ALGORITHM. A FIRST STEP TOWARDS COMPUTATIONAL CONTINUUM MECHANICS
THE INFLUENCE OF A HIGH VELOCITY FLUID ENVIRONMENT ON THE STATIC AND DYNAMIC STABILITY OF THIN CYLINDRICAL SHELL STRUCTURES
DESIGN CRITERIA FOR SUBMARINE SHELLS IN DEEP ONE-ATMOSPHERE SYSTEMS
DISCUSSION
Session II DESIGN CRITERIA AND CONCEPTUAL DESIGN, Part II
General Report
THE CONCEPTUAL DESIGN OF AN ADVANCED UNDERSEA HABITAT
STRUCTURAL CONCEPT OF THE FLOATING CITY
THE CITY AND THE SEA
A SYSTEMS APPROACH TO DESIGN OF AN OFFSHORE OIL STORAGE RESERVOIR
DIRECT SOLUTION OF DROP-SHAPED TANKS
INFLATABLE SHELLS IN THE FLUID ENVIRONMENT
THE USE OF ROTATION CYCLIDES IN UNDERWATER STRUCTURES
UNDERSEA OIL STORAGE SYSTEM MODELS I AND II
"AD HOC" CONTRIBUTION
DISCUSSION
Session III SURFACE AND SHALLOW WATER SHELL STRUCTURES
General Report
OPTIMUM DESIGN OF REINFORCED CONCRETE SHELLS AND SLABS
THE THEORY AND SYNTHESIS OF THIN-SHELL SHIP STRUCTURES
A PROPOSAL FOR OFFSHORE BUILDING STANDARDS
THE BUCKLING OF CONCRETE COOLING-TOWER SHELLS
SEMI-SPHERICAL SHELLS AS "FLOATING FOUNDATIONS
and G. Verhamme
DISCUSSION
Session IV SUBMERGED SHELLS
General Report
FAILURE MODES OF SPHERICAL ACRYLIC SHELLS UNDER EXTERNAL HYDROSTATIC LOADING
ANALYSIS OF BEHAVIOR OF UNSTIFFENED TOROIDAL SHELLS
DISCONTINUITY STRESSES IN AN UNDERSEAS VEHICLE COMPOSED OF TWO DISSIMILAR SHELLS
ANALYSIS FOR DESIGN OF ELLIPTIC UNDERWATER VESSELS
SUBMERGED CYLINDRICAL SHELLS SUBJECTED TO MOVING LOADS
RELIABILITY CONCEPTS AND SAFETY FACTORS IN STRUCTURAL DESIGN OF PRESSURE HULLS
DISCUSSION
Session V STATIC AND STABILITY ANALYSIS OF HYDROMECHANICALLY LOADED SHELLS
General Report
ANALYSIS OF TOTALLY SUBMERGED THIN SHELLS BY FINITE ELEMENT TECHNIQUE
STABILITY OF RING-STIFFENED CYLINDRICAL SHELLS WITH CLOSED ENDS UNDER EXTERNAL PRESSURE
STABILITY OF CYLINDRICAL SHELLS BY THE USE OF RECTANGULAR BAR CELLS
THE BUCKLING OF PIPELINES UNDER NONUNIFORM EXTERNAL PRESSURE
EFFECT OF OUT-OF-ROUNDNESS ON THE ELASTIC INSTABILITY OF THIN CIRCULAR CYLINDRICAL SHELLS
NONLINEAR ANALYSIS OF DEEP OCEAN STRUCTURES
BUCKLING OF THIN SHALLOW ANISOTROPIC SPHERICAL SHELLS WITH A CONSIDERATION OF TRANSVERSE SHEAR
DISCUSSION
Session VI HYDRODYNAMICALLY LOADED SHELLS
General Report
THE EFFECTS OF CONTAINED LIQUID ON THE DYNAMICS OF ELASTIC SHELLS
RESPONSE OF VISCOELASTIC SHALLOW S
STATICS AND DYNAMICS OF HYDROSTATICALLY LOADED SHELLS BY FINITE ELEMENT METHOD
NONLINEAR DYNAMIC ANALYSIS OF MODERATELY THICK SHELLS
THE FINITE ELEMENT DYNAMIC STABILITY ANALYSIS OF THIN SHELLS SUBMERGED IN FLUIDS
NUMERICAL ANALYSIS OF VISCOELASTOPLASTIC TRANSIENT RESPONSE OF SUBMERGED SHELLS
FREE VIBRATIONS OF A PARTIALLY SUBMERGED CYLINDRICAL SHELL
INELASTIC RESPONSE OF SUBMERGED SPHERICAL STRUCTURES
AN INTEGRAL EQUATION FORMULATION OF ACOUSTIC FLUID - ELASTIC SHELL DYNAMIC INTERACTION PROBLEMS
ACOUSTIC RADIATION FROM A RANDOMLY EXCITED THIN VISCOELASTIC SPHERICAL SHELL IMMERSED IN A FLUID
VIBRATION FREQUENCIES OF A FREE-FREE CYLINDRICAL BEAM OF FINITE LENGTH IMMERSED IN AN INVISCID FLUID
DISCUSSION
Session VII PERTINENT SHELL THEORIES AND METHODS FOR ANALYSIS
General Report
ON THEORIES OF SHELLS
CREEP IN MULTI-LAYERED SHRINK FITTED CYLINDRICAL PRESSURE VESSELS
OPTIMUM DESIGN OF LIQUID STORAGE TANKS
DERIVATION OF THE COMPLETE DIFFERENTIAL EQUATION OF THE TOROIDAL SHELL UNDER UNIFORM LOADING AND ITS SOLUTION BY MEANS OF DIGITAL COMPUTER
STRESS ANALYSIS OF A CIRCULAR CYLINDRICAL SHELL, REINFORCED BY EQUALLY SPACED RING FRAMES UNDER UNIFORM PRESSURE
CREEP OF LAMINATED ANISOTROPIC CYLINDRICAL SHELLS
EXTENDED FINITE STRIP METHOD FOR PRISMATIC PLATE AND SHELL STRUCTURES
A SOLUTION OF THE PLASTIC DEFORMATION OF CLAMPED RECTANGULAR MEMBRANES SUBJECTED TO DYNAMIC LOADING
DISCRETIZATION OF STRUCTURAL PROBLEMS BY A GENERALIZED VARIATIONAL APPROACH
MEMBRANE SHELLS WITH POINT SUPPORTS
DISCUSSION
Session VIII MODEL TESTS, CONSTRUCTION METHODS AND RELATED FIELDS
General Report
DESIGN OF RING-STIFFENED CYLINDRICAL SHELLS ENCASED IN CONCRETE
DESIGN OF MODELS OF HYDRODYNAMICALLY LOADED SHELLS
PHOTOELASTIC STRESS ANALYSIS OF A RING STIFFENED SHELL
CERTIFICATION FOR MATERIAL SAFETY OF HYPERBARIC FACILITIES
CONSTRUCTION OF LARGE CONCRETE SHELLS FOR OCEAN STRUCTURES: TECHNIQUES AND METHODS
THE COLLAPSE STRENGTH OF CYLINDRICAL SHELLS STIFFENED BY HAT-SHAPED STIFFENERS
EXPERIMENTAL STUDIES ON CONCRETE
CONCRETE HULLS FOR OCEAN TANKERS
DISCUSSION
CLOSING REMARKS
LIST OF PARTICIPANTS
AUTHOR INDEX

Citation preview

HYDROMECHANICALLY LOADED SHELLS

HYDROMECHANICALLY LOADED SHELLS Proceedings of the 1971 Symposium of the International Association for Shell Structures Pacific Symposium Part I

edited by Rudolph Szilard

The University Press of Hawaii Honolulu

Information on the Proceedings for PART II "Tension Structures and Space Frames" can be obtained by writing IASS Organizing Committee Architectural Institute of Japan 2-19, Ginza 3-chome Tokyo, 104 JAPAN

Copyright © 1973 by The University Press of Hawaii All rights reserved Library of Congress Catalog Card Number 72-93559 ISBN 0-8248-0264-0 Manufactured in the United States of America

CONTENTS

Page Foreword

XI

Xlll

Preface Opening Remarks

XV

Session I: Design Criteria and Conceptual Design, Part I 3

General Report John P. Craven Numerical Solution to the Dynamics of Viscous Fluid Flow by a Finite-Element Algorithm; A First Step Towards Computational Continuum Mechanics Allen J. Baker

5

The Influence of a High Velocity Fluid Environment on the Static and Dynamic Stability of Thin Cylindrical Shell Structures Walter Horn, Gerald Barr, and Ronald Stearman

24

Design Criteria for Submarine Shells in Deep One-Atmosphere Systems Angelo Di Tommaso and Federico M. Mazzolani

48

Discussion

66

Session II: Design Criteria and Conceptual Design, Part II

69

General Report John P. Craven

71

The Conceptual Design of an Advanced Undersea Habitat E. Eugene Allmendinger

73

Structural Concept of the Floating City Kiyonori Kikutake

88

The City and the Sea John P. Craven

92

A Systems Approach to Design of an Offshore Oil Storage Reservoir Charles M. Hix, Jr.

100

Direct Solution of Drop-Shaped Tanks Celai N. Kostem

115

Inflatable Shells in the Fluid Environment John W. Leonard

124

The Use of Rotation Cyclides in Underwater Structures Jack R. Maison and Jack R. Vinson

140

V

Page Undersea Oil Storage System, Models I and II H. Itokawa, T. Ohira, T. Hirose and M. Sakuta

150

'Ad Hoc'Contribution Paul Conil

180

Discussion

183

Session III: Surface and Shallow Water Shell Structures

185

General Report Jerry D. Stachiw

187

Optimum Design of Reinforced Concrete Shells and Slabs Troels Brndum-Nielsen

190

The Theory and Synthesis of Thin-Shell Ship Structures J.B. Caldwell

201

A Proposal for Offshore Building Standards Masakazu Ozaki

222

The Buckling of Concrete Cooling-Tower Shells David P. Billington

239

Semi-Spherical Shells as 'Floating Foundations' M. Van Laethem, J. de Coen, D. Jiang, J. Rogier, and G. Verhamme

248

An Analysis of Ferro-Cement Boat Hulls J.C. Scrivener and A.J. Carr

268

Discussion

277

Session IV: Submerged Shells

283

General Report Y. Tsuboi

285

Failure Modes of Spherical Acrylic Shells Under External Hydrostatic Loading Jerry D. Stachiw

288

Analysis of Behavior of Unstiffened Toroidal Shells William J. Nordell and John E. Crawford

304

Discontinuity Stresses in an Underseas Vehicle Composed of Two Dissimilar Shells S.T. Gulati

315

Analysis for Design of Elliptic Underwater Vessels James Ting-Shun Wang

328

Submerged Cylindrical Shells Subjected to Moving Loads J.E. Russell and G. Herrmann

341

vi

Page Reliability Concepts and Safety Factors in Structural Design of Pressure Hulls H. Ebner and K. Kokkinowrachos

351

Discussion

370

Session V: Static and Stability Analysis of Hydromechanically Loaded Shells

373

General Report F. W. Bornscheuer

375

Analysis of Totally Submerged Thin Shells by Finite Element Technique Oktay Ural

386

Stability of Ring-Stiffened Cylindrical Shells with Closed Ends under External Pressure J.E. Goldberg, A. V. Setlur, and D. V. Pathak

400

Stability of Cylindrical Shells by the Use of Rectangular Bar Cells A. Hrennikoff, C.I. Mathew and R. Sen

412

The Buckling of Pipelines under Non-Uniform External Pressure P.J. Moss

427

Effect of Out-of-Roundness on the Elastic Instability of Thin Circular Cylindrical Shells James R. McDonald and Kenneth R. White

442

Nonlinear Analysis of Deep Ocean Structures E.L. Wilson, T.M. Hsueh and L.R. Jones

457

Buckling of Thin Shallow Anisotropic Spherical Shells with a Consideration of Transverse Shear

475

William A. Nash and P.K. Hsueh Discussion

489

Session VI: Hydrodynamically Loaded Shells

495

General Report Rudolph Szilard

497

The Effects of Contained Liquid on the Dynamics of Elastic Shells H. Norman Abramson Response of Viscoelastic Shallow Spherical Shells to Dynamic Pressures Megumi Sunakawa and Nori Kumai Statics and Dynamics of Hydrostatically Loaded Shells by Finite Element Method C. Brebbia, S. Sabanathan, U. Tabhildar and H. Tottenham Nonlinear Dynamic Analysis of Moderately Thick Shells Robert R. Archer and MuktiL. Das

vii

Page The Finite Element Dynamic Stability Analysis of Thin Shells Submerged in Fluids T.J. Chung and J.F. Jenkins

572

Numerical Analysis of Viscoelastoplastic Transient Response of Submerged Shells T.J. Chung, R.L. Eidson and J.F. Jenkins

581

Free Vibrations of a Partially Submerged Cylindrical Shell Victor I. Weingarten, Sami F. Masri and Mehran Lashkari

591

Inelastic Response of Submerged Spherical Structures Bijoy K. Bhattacharya and John F. Carney III

602

An Integral Equation Formulation of Acoustic Fluid — Elastic Shell Dynamic Interaction Problems Richard P. Shaw

615

Acoustic Radiation from a Randomly Excited Thin Viscoelastic Spherical Shell Immersed in a Fluid Edward B. Magrab and Robert T. Menton

633

Vibration Frequencies of a Free-Free Cylindrical Beam of Finite Length Immersed in an Inviscid Fluid Antonio Pita

647

Discussion

665

Session VII: Pertinent Shell Theories and Methods for Analysis

667

General Report Troels Brndum-Nielsen

669

On Theories of Shells W. Zerna

673

Creep in Multi-Layered Shrink Fitted Cylindrical Pressure Vessels Mehdy Sabbaghian

687

Optimum Design of Liquid Storage Tanks J.C. Shang

697

Derivation of the Complete Differential Equation of the Toroidal Shell under Uniform Loading and its Solution by Means of Digital Computer Ivo Senjanovic

711

Stress Analysis of a Circular Cylindrical Shell Reinforced by Equally Spaced Ring Frames under Uniform Pressure Dragan Stulhofer

727

Creep of Laminated Anisotropic Cylindrical Shells Sam Tang

742

viii

Page Extended Finite Strip Method for Prismatic Plate and Shell Structures Ghulam Husain Siddiqi and C. V. Girija Vallabhan A Solution of the Plastic Deformation of Clamped Rectangular Membranes Subjected to Dynamic Loading Franklin J. Kay and Gerald D. Whitehouse

753

766

Discretization of Structural Problems by a Generalized Variational Approach Walter Wunderlich

779

Membrane Shells with Point Supports P. Csonka

794

Discussion

800

Session VIII:

Model Tests, Construction Methods and Related Fields

General Report Ben C. Gerwick,

803 805

Jr.

Design of Ring-Stiffened Cylindrical Shells Encased in Concrete R.K. Jain, A.R. Griffith and E. W. Kiesling

809

Design of Models of Hydrodynamically Loaded Shells Glenn Murphy

823

Photoelastic Stress Analysis of a Ring Stiffened Shell Philip M. Hoyt and Bin Chang

833

Certification for Material Safety of Hyperbaric Facilities William J. Bobisch and Michael Yachnis

841

Construction of Large Concrete Shells for Ocean Structures: Techniques and Methods Ben C. Gerwick, Jr.

850

The Collapse Strength of Cylindrical Shells Stiffened by Hat-Shaped Stiffeners L.F. Greimann and R.C. DeHart

856

Experimental Studies on Concrete Spherical Shells under Hydrostatic Loading Harvey H. Haynes and Lawrence F. Kahn

874

Concrete Hulls for Ocean Tankers Rowland G. Morgan

898

Discussion

913

Closing Remarks

919

List of Participants

920

Author Index

927

ix

FOREWORD The world has come to that point in time where it must of necessity develop much closer working relationships with its oceans. Engineers and architects, interacting with other scientific specialists, are concerned with creative application of their knowledge and skills for the benefit of man. This conference is dedicated to the realization that the oceans offer great promise in the era ahead in terms of contribution toward solutions to problems of society. Certainly, problems of pollution, energy production, transportation, food supplies, and recreation will require a more knowledgeable and professional use of the almost infinite ocean resource; if the optimum, most beneficial, solutions are to be developed. Part I of the IASS Pacific Symposium "Hydromechanically Loaded Shells" was designed to provide a forum for many of the world's leading scientists, engineers, and architects; a time and place where they could meet and discuss the current state-of-the-art in that branch of knowledge related to the structural mechanics of shell type enclosures and surfaces which are subjected to fluid pressure loadings. Knowledge of this subject, of course, is a vital prerequisite to any concerted effort man may make in the direction of a more intense use of the seas which, for practical purposes, surround all the land masses of his world. With this purpose in mind, the University of Hawaii proposed to the U.S. National Science Foundation, the Advanced Research Projects Agency of the U.S. Department of Defense, the U.S. Office of Naval Research, the State of Hawaii, and to the International Association for Shell Structures, the sponsorship of a symposium to speak to this subject. Response to the request was gratifying. Papers submitted, and presented, were outstanding. Dr. Rudolph Szilard, who assisted in this effort, served as chairman of the technical program committee. These proceedings represent a series of relevant papers on this most essential subject. As man moves into the sea, surely he will use this volume as one "stepping stone."

Howard Harrenstien Conference Chairman Currently Dean of Engineering and Environmental Design University of Miami Miami, Florida, U.S.A.

xi

PREFACE Presently, we notice a growing interest in scientific and technical exploration of the sea with its multifold applications of shell structures. Part I of the IASS Pacific Symposium was organized as a response to this growing interest in ocean engineering. Thus, the primary objectives of the symposium was to bring together the research and design engineers, the ocean scientists, the naval architects, and the construction specialists from all over the world; and to provide an international forum, where their existing knowledge could be reviewed and discussed, and where the future research areas which need urgent attention could be outlined. During the 1967 IASS annual conference held in Mexico City, Dr. Howard Harrenstien's suggestion for arranging a symposium on "shell structures in ocean environment" met with great encouragement by Prof. Dr. Ir. A. M. Hass, President of IASS. The Executive Council of the International Association for Shell Structures gave its final approval to the proposal during the Association's meeting, held in Vienna, in 1970. In the meantime, a group of Japanese scientists had submitted a separate proposal for an equally challenging symposium on "tension structures and space frames". Upon President Haas' recommendation the representatives of the two organizing committees met in Tokyo and Kyoto (1971) to discuss the apparently conflicting time schedules of the two conference proposals. Although the considerable national prestige involved in being the host-country for an IASS conference might have created friction, the complete agreement and cooperation achieved between the two organizing committees produced the concept of a joint PACIFIC SYMPOSIUM with Part I in Hawaii and Part II in Japan. It is hoped that the international cooperation shown by this group of scientist-engineers will be an example of good will for all to follow in settling what would appear to be a conflict of national interests. In spite of the papers, eight general papers, all have been the various authors symposium.

highly specialized theme of the symposium, the final program was comprised of sixty-one reports, and numerous ad hoc discussions from the floor. Disregarding the length of some published unabridged, with only minor editorial corrections, to retain the individual styles of representing some fifteen countries, which reflects the true international flavor of this

The symposium was attended by 140 participants, 46 of which came from outside of the United States. The list of the participants can be found at the end of this volume. It has been the aim to publish the Proceedings of both parts of the IASS Pacific Symposium as soon as possible, in order to disseminate the timely information generated during the meetings and to provide a valuable source of reference for experienced and inexperienced engineers who are interested in these specialized fields. This Symposium, and hence this Proceedings volume, would not have been possible without the cooperation of many people. We wish to thank all of those who contributed. We feel, however, particularly indebted to Mrs. Sue Perry, conference coordinator; Mrs. Mary Kamiya, research associate; and Mr. George Sheets, engineering editor (all of them from the Center for Engineering Research, University of Hawaii) whose devoted work was certainly beyond the call of duty.

xiii

The Pacific Symposium, as mentioned above, was organized under the auspices of the International Association for Shell Structures. Sponsoring organizations were the University of Hawaii, the State of Hawaii, the National Science Foundation, the Office of Naval Research, and the Advanced Research Projects Agency. In conclusion, we would like to express our sincere gratitude to these sponsoring agencies for their support, as well as to acknowledge the financial assistance of the following Hawaiian organizations and firms: Austin, Smith and Associates, Inc. Honolulu, Hawaii

Continental Drilling Company Waipahu, Hawaii

Kutaka, Portugal, Ibara, Inc. Lihue, Kauai, Hawaii

Marine Technology Society Hawaii Section

Alfred A. Yee and Associates, Inc. Honolulu, Hawaii

Rudolph Szilard, Dr.-Ing., Editor Chairman of Technical Program Committee

xiv

PRESIDENT'S ADDRESS

A. M. Haas President International Association for Shell Structures

Ladies and gentlemen, I feel honored to be called upon to speak to you. First of all, I would like to express our appreciation for the invitation to have the IASS come to Hawaii. In the month of September 1967 during an IASS Symposium held at Mexico City, Professor Harrenstien and I had a brief encounter. He proposed to have a symposium on shell structures with special reference to the ones designed and built in Hawaii. To explain his proposal he gave a short exposition of what then was called "Shell Structures in an Ocean Environment." He told me that at the University of Hawaii there exists a center for deep ocean laboratory projects. The research has many aspects: to explore the deeper layers of the ocean and of its soil for food production, to investigate the possibilities of shelters for various purposes in deep water, etc. In addition, a percentage of the ocean floor could be made accessible through the use of concrete hulls. Continental shelves and submarine banks to a depth of 3500 feet are available. They are in general flat to gently sloping, and are located in the vicinity of land, thus offering ideal construction sites for concrete hulls.

xv

Moreover, this invitation offered the opportunity to combine the results of scientific research and knowledge together with the pleasures to be enjoyed at a prominent beach resort. I consider it a sign of intelligence that the IASS grasped this opportunity. Also, I realized that it would be an extension of the work of our Association into a realm not yet mastered: the design and calculation of thick shells. So far, in building construction, we occupy ourselves chiefly with thin shells, a type of shell that is largely used. Through several approximations and simplifications we have in this field arrived at tractable methods of calculation. When we came to an agreement, and a call for papers had been sent out, it resulted in many contributions. Among them are excellent ones, coming from all parts of the world, covering a broad field of investigation, design and calculations. To us it meant that the initiative of Professor Harrenstien and his co-investigators apparently had been well chosen. It meant a Symposium having eight half-day sessions; moreover the title of the whole was transformed into "Hydromechanically Loaded Shells". The IASS was founded in 1959 by Professor Torroja of Madrid. It brought together all shell-minded people: architects, structural engineers, general contractors. Thus, we constitute a group of shell people, theoretical as well as practical. We lay stress on knowledge, creativeness, and execution (building). We have members all over the world from every country, from Argentina to Continental China, from South Africa to Honolulu. We have less than a thousand members; we do not go by the usual urge to become larger and bigger. We aim towards quality. The name IASS stands for International Association for Shell Structures. It recently has been extended to spatial structures. In it we try to be really international, not only in writing, and in speech, but also in behavior. We try to approach each other without national resentment, without bias or prejudice. This is, I know, no easy matter; nevertheless we keep trying. Sometimes I try to explain this attitude in the following manner. The tiny shell cannot help to be conceived and born, viz., to be designed and calculated, in some country or another. The engineer and the builder do not occupy themselves with boundaries. However, the latter are man-made and are established on principles which were formulated in former times. Therefore, let us be aware of the mental pattern we put on shells, and on each other. Let us at this moment realize that we stand at the opening of a Pacific Symposium, a fine piece of practical international work, effectuated as a result of the collaboration of architects and engineers in the United States, as well as in Japan, on both sides of the Pacific Ocean.

Session I DESIGN CRITERIA AND CONCEPTUAL DESIGN, Part I

Session Chairman:

David P. Billington (USA)

General Reporter:

John P. Craven (USA)

General Report Session I DESIGN CRITERIA AND CONCEPTUAL DESIGN, PART I Dr. John P. Craven Dean of Marine Programs University of Hawaii Honolulu, Hawaii, U.S.A.

Within the present state-of-the art, any particular geometric configuration for shell structures which we desire to utilize will probably be solvable by somebody's routine on somebody's computer without the elegance of the classical solutions for which researchers in this field have become famous. In spite of this, I consider that a very significant (if not landmark) paper is presented here by Dr. Baker as he takes the governing equations of fluid mechanics and replaces them with a finite element numerical-solution algorithm in a manner similar to the finite element approach used in solid mechanics, recognizing that, after all, continuum mechanics is continuum mechanics, and hydrodynamics, plasticity and elasticity are just specific forms of more general field equations. From an analytical standpoint, I note that Dr. Baker has not gone beyond the two-dimensional and asymmetric case and while I do not hesitate to ask this question, because I am sure it will lead to a great many more insights, I would be interested to know how he would approach the more generalized case of three-dimensional flow. The paper presented by Dr. Baker is of specific importance, since without a complete incorporation of the hydromechanical effects into the matrix-displacement equations of shell structures, we will be left with very weak approximations on the important forcing functions and the dynamic interaction of solids and fluids. The importance of the Baker paper has been highlighted by the paper presented this morning on the influence of high velocity fluid environment on the static and dynamic stability of thin cylindrical shell structures by Stearman, Horn and Barr. The paper by Professors Di Tommaso and Mazzolani deals with the design of submarines subjected to static loadings. At present there is a significant difference in the design approach of aerospace structures versus ship and submerged shell structures, namely, in the first case we are dealing with very sophisticated techniques, while in the second case we generally use less sophisticated approaches. Indeed, those who have been involved in the submarine underwater technology business believe that we have, from a practical application standpoint, tremendous strides to be made immediately. The solutions for practical, feasible under-the-water structures are in fact in great demand, alongside the aerospace structures. In both cases we are dealing with structures which are extremely difficult to design. We must realize, however, that there is a definite need for generating some rather straightforward, if not elementary, design considerations for the submarine environment, to be used in preliminary design and to check the results of the computerized approach. As we look at the aerospace structures, we see here another example of designing very close to the allowable limits. It is evident that the slightest deviation from the theoretical shape is going to introduce problems that we have not heretofore anticipated and that a good part of this ability to not anticipate derives from our lack of understanding of the flow field and how the flow field has, in fact, been modified by the changes in shape. The paper written by Stearman et al., which was both informative and instructional, is yet another in a series of excellent papers we have come to see, namely the possible applications of aerodynamic solutions to their hydrodynamic counterparts. As we come down to the submarine department we note a very interesting item. As Professors Mazzolani and Di Tommaso have indicated, the rather archaic ring-stiffened submarines of military design of the past are now to be replaced by essentially monocoque structures of both spheres and ring-stiffened cylinders. For these types of shells even the elementary design tables and charts, which are required to do the first-cut design, have not been adequately developed nor are they available to the designers. I think that this paper does provide the required design criteria for submarine shells which will permit first-cut determinations of the sizes of the above vehicles. I have been promoting for the last five years an item which the curves developed by Di Tommaso and Mazzolani eloquently demonstrate,

3

that is, if we build submersibles out of glass or ceramics, a large portion of our problems are essentially solved. These are, indeed, wonderful materials. The problems we presently face in submarine design are similar to the problems we would face in aerospace design if we still insisted that such structures be made of heavy ferrous materials, rather than materials of modern space technology. I would like to point out that there is again a danger involved in blindly utilizing the design criteria developed by Mazzolani and Di Tommaso because inevitably if one designs a submersible structure, one must deviate from the simple spheres and cylinders. Most deep submersibles use these well-tested forms, those having intersections and connections, longitudinal and transverse ribs and stiffeners. As a result, inevitably the buckling modes will not be the simple modes as indicated. There are "accordian" modes, "pleated" modes, and many other kinds of buckling modes. Therefore, the submarine designer must consider these buckling shapes in the final design, although he can (with appropriate factors of safety) utilize the criteria generated in this paper for the badly needed preliminary design of these structures. In summary these papers say that in general the solutions of aerodynamic problems are in a much more sophisticated state of development than their hydromechanic counterparts, but even the aerodynamic problems require a better approach, i.e., a better generation of the fluid field in which they are going to operate. The Baker paper states that these problems can be solved through the use of computerized solutions. Certainly this hope is completely realizable in the case of the two-dimensional or asymmetric flow situation, particularly in the critical areas of transient. The paper by Di Tommaso and Mazzolani tells us that we in the underwater technology field can afford to wait for some of these sophisticated solutions because we have a great deal of progress yet to be made in this area. In the meantime, solutions of rather elementary simplicity, but of great value to the designers, must be used.

4

NUMERICAL SOLUTION TO THE DYNAMICS OF VISCOUS FLUID FLOW BY A FINITE-ELEMENT ALGORITHM. A FIRST STEP TOWARDS COMPUTATIONAL CONTINUUM MECHANICS Allen J. Baker Principal Scientist Continuum Mechanics Research Textron's Bell Aerospace Company Buffalo, New York, U.S.A.

ABSTRACT A finite-element numerical-solution algorithm is derived for the transient swirling flow of an incompressible viscous fluid. Dependent-variable transformations are utilized to render the governing system of partial differential equations mathematically well-posed, as an initial-boundary-value problem, for each computed dependent variable. The numerical algorithm, derived for this common equation system, transforms it to a system of first-order, ordinary differential equations, written on the finite-element discretization of the dependent variable. The algorithm is completely compatible with a finite-element representation of a concomitant (structural) boundary; it thus represents a step towards computational continuum mechanics. The procedure accommodates arbitrarily mixed specification of fixed and/or gradient boundary conditions for each physical and computational variable. The finite-element idealization allows accurate modeling of irregular geometrical shapes through use of a completely arbitrary computational mesh. Non-negligible boundary deformation is allowed by recomputation of finite-element geometrical characteristics, in the affected regions only, while enforcing stress-tensor continuity between fluid and boundary. Efficient solution of the equation systems is achieved through use of an optimally stable, predictor-corrector numerical integration algorithm. Numerical results are presented for an illustrative flow problem.

THE GOVERNING DIFFERENTIAL EQUATIONS FOR VISCOUS FLUID DYNAMICS The complete description of a fluid dynamical state is contained within the solution of the system of coupled, nonlinear, second-order partial differential equations describing local conservation of mass, m o m e n t u m and energy, in conjunction with appropriate specifications of constitutive behavior and applicable initial and boundary conditions. These equations for a single-component, viscous, incompressible fluid are 0 = - [uj] ; i

(1) (2)

v - h v ^ v x ] ; ; (

H

"p),t

=

"[

U i I I

"Vj-p

T

J

; i

The dependent variables in Eqs. (1) through (3) have their usual fluid dynamic interpretation with Uj the vector velocity, p the pressure, p the (constant) density, r-- the stress tensor, H . the stagnation enthalpy, k the thermal conductivity, and T the local temperature. Cartesian tensor notation is used [9] ; the semicolon denotes the vector derivative of a Cartesian tensor, while the comma denotes the usual gradient operator. Summation convention is employed for repeated subscripts. The solution of Eqs. (1) through (3) requires specification of constitutive relationships between appropriate dependent variables. For laminar flow of a Newtonian fluid, the dynamic relations are contained within Stokes viscosity law, which becomes, for an incompressible fluid

5

Tjj = M u i ; j + u j ; i ]

(4)

An appropriate equation of state relates thermodynamic variables. Before transforming Eqs. (1) through (3) to the desired form, it is convenient to nondimensionalize all variables to extract the useful fluid dynamic parameters. Selecting a characteristic length (L), velocity ( U ^ ) , and uniform density (p) as the reference parameters, Eqs. (1) through (3) become 0 = - [14] ; i \ t

= - [

U

i V

P

(5)

V ^

T

i j ] ; j

(6)

( H - E c f e ) ) ^ - ^ . ; I v j - R ^ i ] , The nondimensional groupings of fluid parameters are defined as UocL

Reynolds Number:

Re =

Prandtl Number:

Pr

Eckert Number:

Ec =—=— Cpioo

=

^oo

(8) (9)

X

(10)

DIFFERENTIAL EQUATION SYSTEM DEVELOPMENT FOR THE DESIRED CLASS OF PROBLEMS IN INCOMPRESSIBLE VISCOUS FLUID DYNAMICS The solution of Eqs. (5) through (7) is a formidable task. It is desired to restrict the generality of the description to the degree that the concomitant mathematical advantages are applicable to nontrivial problems. The essence of this development is restriction to two independent space dimensions, but in which three-dimensional flow fields may exist. The derivations will be outlined; the interested reader is referred to [4], Baker (1971), for details. Continuity Equation From mathematical physics it is known that a vector field is defined when its divergence and curl are known. The vector field of fluid mechanics is velocity, and Eq. (5) defines its vanishing divergence. Hence, it is possible to completely specify this field in terms of the curl of some vector potential, Little analytical benefit accrues from this specification (as does occur, for example, in Maxwell's equations) in fluid dynamics, due primarily to the nonlinearity of Eq. (6). Significant numerical benefit can occur, however, if either a) the velocity vector Uj is planar, or b) one scalar component, say u 3 , of Uj, is independent of the corresponding independent variable, x^. In these cases, the continuity equation can be identically satisfied in terms of the x^ scalar component of the vector potential and u 3 , as u

isTe3ij*3a

+ u

35i3

0 0

In Eq. (11), e 3 i j is the Cartesian alternating tensor and J is the determinant of the space metric. Restricting attention to spaces spanned by rectangular, cylindrical and/or spherical Cartesian coordinate systems, Eq. (11) can be advantageously written in terms of a scalar function , and the gradient operator as Uj

= — 1 — e3ifP • + u3ôi3 r sin 0 J ' J

6

(12)

In Eq. (12), a is nonzero only for spherical coordinates, when it is unity, r is set equal to unity (and 0) for rectangular coordinates, and x^ corresponds to the azimuthal angle (0) in both cylindrical and spherical coordinates with range 0 < 277. It is the reasonable assumption that the space of fluid dynamical problems to be considered can be conveniently spanned by at least one of these Cartesian bases. It is readily shown [4], that Eq. (12) identically satisfies Eq. (5) in these bases. Compatibility Equation With the zero divergence of the flow field ensured by identification of this scalar function (streamfunction), it remains to obtain the curl of the velocity uj. Identify the vorticity vector as =

"i

(13)

J eijkuk;j

The X j scalar component (oj) of £2-, in the three Cartesian bases, is

Cl> =e

(14)

¿ 3ijuj;i

with a interpreted as before. In the derived solution algorithm, this scalar component will be employed as an auxiliary dependent variable. Substituting Eq. (12) into Eq. (14), and using the skew-symmetric properties of the alternating tensor contraction, the following compatibility equation is obtained 1

cj =

1

(15)

•4>> k

Lrsin a 4> '

;k

In rectangular coordinates, Eq. (15) is the familiar Poisson equation involving the Laplacian operator. In the other coordinate systems, it lacks certain metric terms of being Laplacian. Equation (15) ensures compatibility between the (to be) computed vorticity and the defined streamfunction. The Curl of the Navier-Stokes Equation The curl of the Navier-Stokes equation ensures the existence of the streamfunction. Since the problem class is restricted from full dimensional generality, determination of the Xj scalar component of the curl, coupled with the X3 scalar component, is adequate to determine Uj. From Eq. (7), the desired curl equation is 3ki

u

i,t+

[uiVièrìj]

; j

(16)

= 0 ;k

Advantage has been taken in Eq. (16), of the symmetry properties of the alternating tensor to eliminate the explicit appearance of pressure — a prime computational feature of the dependent-variable transformation. Equation (16) is rearranged to explicit appearance of the computational variables, using Eqs. (12) and (14) to replace the velocity and velocity derivative. Omitting details (see [4]), Eq. (16) written in terms of vorticity, streamfunction and x^ component of velocity U j becomes raco,

-e 3ki

>*

a

r sin 0 / ;k

+ u e

3 eiC

i>a

/

a

r sin 0/;3/'Jc

* k£ \

Re

(17)

(",ke3kiu3;i);3

7

While clarity results from identifying velocity in Eq. (17), the indicated form is desired for implementation in the computer code. Selection of a particular coordinate system allows specification of the second term in brackets (modified by the alternating tensor) in terms of the u^ velocity component. For example, in cylindrical coordinates, with the space defined as

(18)

jz,r,0 j \

the last term becomes " e 3 k i u 3 e 3il

(19)

2r / z

rsina0/;3k

Note that U3 may be a function of independent variables other than X3. The X3 Component of the Navier-Stokes Equation Determination of U3, in terms of co and l

!q!m

^

Equation (51) is valid in each m subdomain, and, hence, in all M finite-elements. Since nodes are located on 9 R m , they are selectively common to neighboring finite elements. Since all Q* converge to unique values at these nodes, irrespective of which particular element formulation generated them, and is outwards pointing, the contributions of the integrals over 9 R m in Eq. (51) vanish in pairs along all internal boundaries. On all external boundaries Q* (and \p*) take on known boundary values, Eq. (47), and are evaluated accordingly. Before assembly of Eq. (51) into a global system, it is useful to identify several finite-element matrices. Mm

3 5

MHTdr

r i m /

[Hm

[r^ki' H ^ H 1 - ^ ™ r^'

i

L

f m "

lrlm

f

M

"

f

d

*

'

'

r sin 0

'

Rm 9Rm )

Y

( m - [ n l ^ S

K-jxj

r sin 0 Q

m

-kda

(52)

In Eq. (52), advantage has been taken of the functional form of the various matrices to extract those terms independent of the integration variables. Using Eq. (52), (51) can be compactly written as [ClmWm

= -C[K1] - [K2]

+

[Zl]m) jq(t)(m + j L f m + ] Y L

(53)

The superscript ' denotes ordinary differentiation of the unknown node-point variable vector with respect to time. The Boolean assembly of the M N Eq. (53), where N is the number of nodes per finite element, produces the deterministic ordinary differential equation system written on the global unknown vector j q(t) j . Nonzero occur only for those finite elements located on 9R. Formally premultiplying contributions to [Zl] and both sides of the global equation (which is identical to Eq. (53) with subscripts m removed) by [C] renders it in form for integration, as an initial value problem, by any consistent numerical integration algorithm. For the steady-state solution, the left side of Eq. (53) (without subscripts m ) vanishes, and solution involves formal premultiplication by the inverse of the matrix modifying the unknown vector. Since generating the inverse of large matrices is usually uneconomical, it is preferential to establish solution to Eq. (53) by use of a matrix equation solver. Many have been devised, e.g., Gauss—Seidel, Choleskey, Newton-Raphson, over-relaxation, etc., and the features of each are well known. Each is potentially applicable to Eq. (53) for either the steady-state or transient solution.

14

Economical Finite-Element-Solution Algorithm for the Inhomogeneous Elliptic Compatibility Equation The compatibility equation relating streamfunction to vorticity, Eq. (15), is a particularly simple example of Eq. (45), since for Q interpreted as 1//, the function f cancels the convective acceleration term. A potential solution procedure, suggested by Baker [2], involves decomposition of the streamfunction in R m into its complementary and particular solutions ^m(XiJt) = V ^ O + ^ i V )

(54)

The complementary solution, satisfies a homogeneous Laplace equation, a well-posed boundary-value problem with boundary conditions expressed by Eq. (47). The particular solution, is a function satisfying Eq. (55). ( 1 P A ) =_rawm \rsina0/;k

(55)

The solution to Eq. (55), in the m ^ subdomain R m , may be achieved analytically since co m is discretized by the finite-element approximation. Expanding the particular solution into the same series of functions, as the finite-element approximation for Q, Eq. (49), the undetermined expansion coefficients are algebraically related to the node-point behavior of vorticity, co m . Hence, the particular solution in R m is expressible as ^(xi)t)=

jx|T[r]mji2(t)|m

(56)

The subscript bar in the spatial vector indicates that its members include higher-order polynomials than those appearing in Eq. (49). With the analytic determination of the particular streamfunction solution possible, it remains to obtain the finite-element solution for the boundary-value problem written on complementary streamfunction. Expanding the complementary solution as in Eq. (49), forming the weighted residual, Eq. (44), and identifying the matrices

[Kilm-tr]£/

jx[)kjxjTkr-^

M m - P i J /

!

9Rm

*

[r]m

K

r sin 0

>

c


m , Eq. (49). A

degenerate-table look-up (only one entry) accounts for its allowed constancy. Numerical Results The numerical algorithm has been exercised against a representative sample problem of isothermal internal f l o w to f o r m a proof o f concept and evaluate solution accuracy and computational speed. For this purpose, a 2-ft by 25-ft long duct was selected, wherein fluid, initially at rest, was impulsively accelerated to a velocity o f 10 ft/sec at a Reynolds Number o f 200. The transient solution was followed until steady-state conditions were approached. From symmetry considerations, only one half-plane o f the problem domain need be considered. As shown in Fig. 2, the region above the duct centerline was discretized into 48 finite elements, with the smallest elements located where the maximum vorticity gradients were expected. Re

- 200

Complementary

48

Finite E l e m e n t s

Total

33 Nodes

L o n g i t u d i n a l D i s p l a c e m e n t , ft

Fig. 2 Steady-State Streamfunction Distribution

19

Solid Lines

Dashed Lines

The numerical integration of the equation system was performed using an optimally stable, one-step predictor-corrector integration algorithm derived by Nigro [11]. Internal, propagated error control is provided by specification of a user-supplied convergence parameter. The value selected for these studies (0.1) has provided adequate numerical accuracy in previous numerical experimentation. The integration algorithm maximizes step size (At) at each point in the solution by comparison between the convergence parameter and a weighted, local truncation-error coefficient. The transient solution was pursued through sufficient time to allow the fluid, originally contained in the duct, to be completely replaced by new fluid entering from the left end. For the sample Reynolds Number and inlet velocity, this would occur after about 2.0 seconds. At this time, the steady-state solution should be approached in the region 1 6 < x < 2 5 ft. Table II contains selected sample points in the computed time-de pendent vorticity solution. The final data point of 2.2 sec compares favorably with the steady-state vorticity distribution, obtained from a 400 cell finite-difference solution, using the program of Gosman, et. al [8]. The computed time history for vorticity is stable, with a tendency to selectively overshoot the steady-state value by a small amount, and to then approach steady-state from above.

TABLE II Time-Dependent Nodal Vorticity Distribution j i 2 ( t ) j Nodal Coordinates y

X

Solution Time (sec) 0

0.5

L0

1.6

2.2

OO

0.5

0.5 1.0 2.0 3.5 5.0 7.5 10.0 16.0 25.0

0 0 0 0 0 0 0 0 0

0 0.67 0.79 0.56 0.31 0.21 0.16 0.13 0.13

0 0.22 0.53 1.13 1.14 0.82 0.52 0.37 0.36

0 0.50 0.66 0.69 1.11 1.46 1.13 0.70 0.60

0 0.22 0.36 0.88 0.95 1.24 1.53 1.10 0.92

0 0.26 0.62 0.90 1.10 1.24 1.40 1.50 1.50

0.9

0.5 1.0 2.0 3.5 5.0 7.5 10.0 16.0 25.0

0 0 0 0 0 0 0 0 0

0 3.83 3.29 2.47 2.14 1.95 1.75 1.66 2.00

0 3.89 3.66 3.48 3.07 2.71 2.39 2.20 2.35

0 3.76 3.87 3.32 3.17 3.13 2.77 2.46 2.48

0 3.91 3.67 3.54 3.04 3.02 2.96 2.61 2.56

0 4.26 3.57 3.20 3.04 2.90 2.80 2.70 2.70

The total computer time to generate the transient solution was 320 CPU seconds on an IBM 360/65, with all algebra performed in single precision. This time includes the set-up of the entire problem geometry and generation of all element matrix coefficients, as well as that spent performing the differential equation system integration. The initial time step was set at At = 0.001 sec; it was automatically incremented a total of 224 times such that an integration step of 0.224 sec was taken at the final stage. A steady-state solution was also obtained directly, using the same integration package, by inputting a vorticity and streamfunction distribution close to the steady-state values. Integration was then carried forward until the

20

maximum node-point vorticity-time derivative fell below about 0.05. The calculated steady-state complementary streamfunction and total streamfunction are plotted in Fig. 2. They correctly predict fully developed flow at the duct exit. The calculated vorticity distribution was essentially identical, to within ±0.1, to the finite-difference solution of Table II, except at the two nodes located at x,y values of (1.0,0.9) and (2.0,0.9), where they were respectively under- and over-predicted by 0.4 and 0.2. These deviations result from the rather large discretization used in this region, where both the vorticity and vorticity gradient are large. Total CPU time for this steady-state solution was 80 seconds.

CONCLUSIONS A finite-element numerical-solution algorithm has been derived for the transient flow of an incompressible, viscous fluid. It transforms the governing partial differential equation system into a system of first-order, ordinary differential equations written on the arbitrarily discretized equivalent of the dependent variables. Using simplex-element representations, the theory has been exercised against a problem of transient flow in a two-dimensional duct. The computed results compare favorably with the results of other available analyses, and were economically obtained through use of an optimally-stable integration algorithm. Continuing research with application to problems having irregularly shaped and deforming walls is being undertaken, to more fully explore this new diagnostic tool.

NOTATION a

boundary condition coefficient

c

expansion coefficient; specific heat

E

Young's modulus

Ec f,g

Eckert number

H

stagnation enthalphy

».J J

rectangular Cartesian unit vector

k

function of known argument; thermal conductivity

L

differential operator; characteristic length

function of known argument

determinant of Cartesian space metric

unit outward pointing vector N

nodes per finite-element

P

pressure

Pr

Prandtl number

jr , < M j

spherical coordinate basis

Q,q

generalized dependent variable

R

residual

R

domain of independent space variables

Re

Reynolds number

t

time

T

temperature; total stress tensor

u, U

velocity; general dependent variable

w

weighting function

21

I x, y, z I

rectangular coordinate basis

Xj

independent space variables

I z, r, 61

cylindrical coordinate basis

a

(0,1) coefficient for spherical coordinates

T

finite-element

local coefficient matrix

Sjj

Kronecker delta

3R

bounding surface of domain R

e

alternating tensor; strain

K

functional

v

kinematic viscosity ; unit normal vector

p

density

a

Poisson ratio

T-j

stress tensor

tyj

vector streamfunction scalar component of streamfunction

Î2

discretized vorticity

co

X3 scalar component of vorticity

j I

column vector

[ ]

matrix

*

approximate solution

'

temporal derivative

T

matrix transpose

i,j,k

tensor indices

m

pertaining to the m

00

reference condition

tVi

22

subdomain (finite-element)

BIBLIOGRAPHY 1. Baker, A.J., "A Numerical Solution Technique for a Class of Two-Dimensional Problems in Fluid Dynamics Formulated With the Use of Discrete Elements," Report TCTN-1005,1969, Bell Aerospace Company. 2. Baker, A.J., "Finite-Element Theory for Viscous Fluid Dynamics," Research Report No. 9500-920189, Aug. 1970, Bell Aerospace Company. 3.

Baker, A.J., "Finite-Element Solution Algorithm for Viscous Incompressible Fluid Dynamics," Int. J. Num. Mtd. Engr., to appear.

4.

Baker, A.J., "Finite-Element Theory for the Mechanics and Thermodynamics of a Viscous, Compressible Multi-Species Fluid," Research Report No. 9500-920200, June 1971, Bell Aerospace Company.

5. Finlayson, B.A., and Scriven, L.E., "The Method of Weighted Residuals and Its Relation to Certain Variational Principles for the Analysis of Transport Processes," Chem. Engr. Sci., 20,1965, pp. 3 9 5 4 0 4 . 6. Finlayson, B.A., and Scriven, L.E., "On the Search for Variational Principles," Int. Journal of Heat Mass Transfer, Vol. 10, No. 6 , 1 9 6 7 , pp. 799-821. 7.

Finlayson, B.A., and Scriven, L.E., "The Method of Weighted Residuals - A Review," Appl. Mech. Rev., Vol. 19, No. 9, 1966, pp. 735-748.

8. Gosman, A.D., et al., Heat and Mass Transfer in Recirculating Flows, Academic, London, 1969. 9. Jaunzemis, W., Continuum Mechanics, Ch. 1, MacMillan, New York, 1967. 10. Kantorovich, L.V., and Krylov, V.I., Approximate Inter-Science, New York, 1958.

Methods of Higher Analysis, translated by C.D. Benster,

11. Nigro, B.J., "The Derivation of Optimally Stable, K-Stage, One Step, Explicit Numerical Integration Methods," Technical Note TCTN-1008, Apr. 1970, Bell Aerospace Company.

23

THE INFLUENCE OF A HIGH VELOCITY FLUID ENVIRONMENT ON THE STATIC AND DYNAMIC STABILITY OF THIN CYLINDRICAL SHELL STRUCTURES Walter Horn Graduate Student Aerospace and Engineering Mechanics Department The University of Texas at Austin Austin, Texas, U.S.A. Gerald Barr Staff Member Dynamics Analysis Research Division Sandia Laboratories Albuquerque, New Mexico, U.S.A. Ronald Stearman Associate Professor Aerospace and Engineering Mechanics Department The University of Texas at Austin Austin, Texas, U.S.A.

ABSTRACT A combined theoretical and experimental research program has been carried out to investigate the stability characteristics of thin-walled circular cylindrical shell structures exposed to a high velocity fluid dynamic (aerodynamic) environment. As a result of this study, several basic observations were made concerning the degree of sophistication required in the analytical modeling of this problem. It was found, for example, that small details in the description of the structural boundary conditions can strongly influence the stability of the shell. One of the more significant structural boundary condition effects was observed when the shell geometry and loading conditions were such that the edge bending disturbances were propagated well into the interior of the shell. On the other hand, when conditions were such that these edge bending disturbances were confined to a small boundary layer region near the ends of the shell no significant edge effects due to bending were noticed on the overall shell stability. Small initial deviations of the shell's surface from its idealized shape were also shown to drastically reduce its resistance to panel flutter, a dynamic instability, even though the deviations were only on the order of one shell thickness or less. Panel flutter instabilities in the presence of a laminar fluid boundary layer profile were found to be much less destructive to the shell than those originating in the presence of a turbulent profile. Furthermore, the laminar profile was also found to induce panel flutter at much lower levels of free stream energy. The highly divergent panel flutter, occurring in the presence of a turbulent profile, appeared to have a characteristic wavelength that was small compared to the radius and length of the shell. In contrast, panel flutter, occurring in the presence of a laminar profile, had a characteristic wavelength that was on the order of the radius or length of the shell. The experimentally determined panel flutter boundaries were also found to be in poor agreement with all existing theoretical predictions. At supersonic Mach numbers no significant air stream influence was noticed on the shell buckling loads. The wealth of available still air buckling data can consequently be employed to determine the buckling loads of cylindrical shells exposed to a supersonic air stream. Although most of the study was conducted on the isotropic cylindrical shell, analyses have been carried out illustrating how these results could be extended to certain types of ring and longeron stiffened cylindrical shells. When the rings and longerons divide the shell surface into identical panel elements the analyses can be reduced in a rigorous manner to that of an equivalent panel element of the system due to the circulant form of the equations of motion. This reduction procedure allows for all types of interelement (panel) coupling and is subject to the sole restriction that the dynamic phenomena be satisfactorily described by linear theory. The present study shows that a supersonic fluid dynamic environment can significantly complicate the dynamic stability characteristics of thin cylindrical shell structures while introducing no significant complications to its static stability features.

24

INTRODUCTION This research embraces a combined experimental and analytical program to investigate the stability characteristics of thin-walled cylindrical shell structures exposed to a high velocity fluid environment. It includes the basic evaluation of the past and currently employed analytical modeling of the problem and suggests areas where the modeling needs further refinements. The general problem becomes one of practical consideration in the design of skin panels on space shuttle vehicles, reusable launch boosters, and high performance supersonic aircraft. Although preliminary design criteria are evolving for flat panel elements, very little design information is currently available for thin-walled shell-type structures. Further research is necessary to obtain a better understanding of the stability and also the response characteristics of thin shell structures subjected to a high velocity fluid environment which may include aerodynamic noise, boundary layer turbulence, and buffeting conditions or large scale turbulence. The present investigation is concerned only with the question as to how the stability characteristics of such structures are influenced by a high velocity external flow environment parallel to the shell axis. The experimental phase of the research was carried out in the AEDC propulsion wind tunnel facility of the Arnold Engineering Development Center over the Mach number range 1.2 to 3.5. Fourteen different cylindrical shell configurations were studied under different internal stress levels and supersonic flow conditions. All of the shells had a common length to radius ratio of two and radius to thickness ratios from 2000 to 4000. The experimental data reduction and analytical portion of the study was carried out on the University of Texas CDC 6600 digital computer employing a FORTRAN IV computer program. As a result of the theoretical studies several observations were made concerning the degree of sophistication required in the analytical modeling of the problem. It was found, for example, that small changes in the description of the structural boundary conditions can strongly influence the stability characteristics of the shell. One of the more significant structural boundary condition effects was observed for shells with small to moderate radius to thickness ratios preloaded under combined internal pressure and axial compressive end loads. On the other hand, when the combined geometry and loading conditions on the shell were such that they caused the shell to respond more like a membrane, the induced bending disturbances from the edge constraints were confined to a small boundary layer region near the ends of the shell and no significant edge effects due to bending were noticed on the overall shell stability. Small deviations of the shell's surface from its idealized shape were also shown to drastically reduce its resistance to panel flutter, a dynamic instability of the shell, even though the deviations were only on the order of one shell thickness or less. Even the best manufacturing methods admit this magnitude of imperfection in the fabricated shell geometries. During the wind tunnel experiments, panel flutter instabilities in the presence of a laminar boundary layer profile were found to be much less destructive to the shell than those originating in the presence of a turbulent profile. On the other hand, the laminar or nearly laminar boundary layer profile will induce a limited amplitude panel flutter at much lower levels of free stream energy than will a turbulent profile. When panel flutter does occur in the presence of a turbulent profile, however, it was always found to be catastrophic. The visible form of the flutter mode for the highly divergent panel flutter instability appeared to have a characteristic wavelength that was small compared to the radius and length of the shell. For the more mild limited amplitude flutter, however, this characteristic wavelength appeared to be on the order of the radius and length of the shell. The experimentally determined flutter boundaries were found to be in poor agreement, however, with the existing theoretical predictions employing both short wavelength and long wavelength approximations for the flutter mode. A comparison was also made with two traveling wave analyses for an infinite length cylindrical shell with the highly divergent flutter. This comparison was justified on the basis that the characteristic lengths of the initial unstable wave forms were small compared with the shell radius and length. The flutter boundaries predicted by this analysis occurred at a much lower level of free stream energy than the experimental boundaries. All of the theoretical predictions concerning panel flutter were thus found to be in poor agreement with the experimental observations. Both experimental and analytical results from this investigation demonstrate that the still-air buckling characteristics of thin cylindrical shells were not significantly influenced by the supersonic air stream. The wealth of available still-air buckling data can consequently be employed to determine the buckling loads of cylindrical shells exposed to a supersonic flow field.

25

Although most o f the analytical studies were conducted on the isotropic cylindrical shell, analyses have been carried out illustrating how these results could be extended to certain types of ring and longeron stiffened cylindrical shells. When the rings and longerons divide the shell surface into identical panel elements the analysis can be reduced in a rigorous manner to that o f an equivalent panel element of the system due to the circulant form of the equations o f motion. This reduction procedure allows for all types o f interelement (panel) coupling and is subject to the sole restriction that the dynamic phenomena be satisfactorily described by linear theory. The influence o f a high velocity fluid environment on the stability features of thin cylindrical shell structures involves many parameters which significantly complicate the problem. The basic development and evaluation of suitable methods o f analysis for this problem must, by necessity, involve both theoretical and experimental investigations. This approach has been followed in the present program. The analytical definition and modeling of the problem are presented first along with the theoretical results obtained from the problem solutions. Experimental observations made during this study are then presented. Finally, evaluation o f existing theoretical models of this problem are made based upon the available experimental data, and recommended refinements for future analytical modeling are presented.

THEORETICAL STUDY Due to the expense and difficulty involved in conducting an experimental study of this problem the shell aeroelastic stability was investigated initially from an analytical point of view to gain further insight into the parameters that may significantly influence the experiment. The model under study, f l o w conditions, and

S P E E D OF SOUND

static loading conditions are shown in Fig. 1. This model

STATIC

PRESSURE

M A S S DENSITY

represents a thin-walled, finite length, isotropic cylindri-

VELOCITY

^

U

cal shell structure with a large radius to thickness ratio and small initial deviations of its median surface from that of a perfect circular cylindrical shell. The initial imperfections are axisymmetric having a harmonic wave form in the axial direction. The outer surface of the shell is exposed to a supersonic potential f l o w parallel to the shell axis. The static loading on the shell consists o f an

— INTERNAL P R E S S U R E

axially compressive end loading, an internal pressure

p(p»lg)

Fig. 1

loading, and a hydrostatic pressure loading on the outer surface due to the f l o w field. This model was thought to

Shell Geometry and Flow Conditions

include most o f the significant parameters that influence the shell's aeroelastic stability. That is, the influence of such different parameters as initial geometric imperfections, structural boundary conditions, different potential f l o w approximations, and prestress levels' could all be readily investigated with this model. FORMULATION * I

Problem Formulation The

PERFECT

PATH »1

SMELL

FREELY-EXPANDED

EDGE

STATIC PRESTABILITY

problem formulation is sepa-

RUTH » 2

rated into two major categories as illus-

• I 0

trated in Fig. 2 [3]. The first formulation

BC'i: •„• 0. F

»

-0,

R^T

RESPONSE 0 i

DYNAMIC STABILITY SOLUTION

„ " O

.11

CONSTRAINED

EDGE

is based on a perfect circular cylindrical shell, whereas

the

second

formulation

centers around a shell with axisymmetric initial imperfections o f harmonic

wave

FORMULATION* 2

form. Both formulations contain the so-

N o . 2 and N o . 4, as special cases. In the

FREELY-EXPANDED

PATH « 4

V

M > 1.0

classical analysis, a kinetic stability solu-

EDGE

STATIC PRESTABILITY

(9.)

called classical analysis, P A T H S N o . 1 and N o . 3, and a nonclassical analysis, P A T H S

PATH » 3

IMPERFECT S H E L L

BC't:

DYNAMIC STABILITY SOLUTION

RESPONSE

•O.

•Sr> CONSTRAINED

E DOC

Fig. 2

tion is considered by investigating the

Formulation o f Aeroelastic Stability Problem

26

behavior in time of small displacement perturbations with respect to the middle surface of the original shell geometry which is either a perfect shell, PATH No. 1, or an imperfect shell, PATH No. 3. Any change in the shell static equilibrium shape due to axially compressive end loading, internal pressure loading, or fluid dynamic loading prior to the instability is ignored. The stress state in the shell is determined from membrane theory and the static or dynamic stability of the shell is investigated about its originally specified geometric shape. In contrast, the nonclassical analysis determines the statically deformed middle surface (predeformation shape) of the shell due to initial loading conditions prior to an instability and then investigates the stability of the shell about this newly calculated equilibrium surface employing a kinetic stability approach. In essence, the nonclassical approach attempts to more realistically model the actual structural boundary conditions in the problem. An additional formulation of the problem was also carried out employing the so-called "freely supported" boundary conditions in contrast to those shown in Fig. 2. These basic formulations lead to refinements in the structural modeling of the problem in at least two areas. They can be classified as a predeformation effect and a direct structural boundary condition effect. The latter results from specifying different structural boundary conditions on the ends of the shell whereas the former results from an interaction of the initial prestress, in an otherwise perfect or imperfect shell, with the structural boundary constraints. A set of nonlinear Donnell type shell equations coupled with a linear potential flow theory was employed to describe the motion of the shell and the radial surface pressure loading respectively in the presence of a supersonic flow field. Geometric nonlinearities along with the initial imperfections are introduced into the shell theory through the strain-displacement relations. In-plane inertia of the shell has been neglected in the equations of motion due to the anticipated predominately radial motion of the shell during an instability. The governing equations of motion of a cylindrical shell with small initial deviations of the median surface are written in terms of the radial displacement w , and the total stress function F , as ^

D v 4 w

V

+

S

4

F

F

=

W^y - W ^ W

yy t

^ W^

- W ^ W ^ y

,xx - F,yy(w,xx+^r,xx) " F ^ w , y y + w r , y y )

- W ^ W ^

+ 2F

t

,xy(w,xy+^Vy) = P i * ^ )

(1)

(2)

where the commas denote partial differentiation. The associated boundary conditions of a shell with simply supported "zero tangential shear stress" edges at x = 0 , L are w(x,y,t) = w C x . y . t ) ^ = F i x . y . t ) ^ = 0

^

F(x,y,t)yy = Nx while the classical "freely supported" boundary conditions are u i x . y . t ) ^ = v(x,y,t) = w ( x , y , t ) = w ( x , y , t ) j X X = 0

(4)

The surface loading on the shell may be defined as p(x,y,t) = p - p s h w u + p a

(5)

where the first term represents the static pressure differential across the skin of the shell and the second term is the inertial loading resulting from the motion of the shell surface. The radial aerodynamic pressure is obtained from limiting cases of the exact supersonic potential flow solution over a finite length shell with a harmonically oscillating surface. The approximations are valid for the so-called long and short wavelength modes of shell instability. They are obtained from the solution of the following boundary value problem.

27

J_

+

r2 302 30' 3r L

+

3r 2

r 9r

2

3x 2

~

3w a ï

u

-

2 920 ^

I 30

+

M 320 a

dx9t

+

1 320 a2 3t2

1 3w _ , . û 3 T - ^ ^

x< 0 r r

P

= 9 (d± \3x

(6)

oo

x> 0

+1 U 3t /

where 0 is a perturbation velocity potential induced by the shell radial deformation w . The total velocity potential being 4> = U(x+0). In addition, r , 9 , and x are polar coordinates as indicated in Fig. 1 while U is the magnitude of the free stream velocity, M the free stream Mach number, and a the speed of sound in the undisturbed flow. The pressure coefficient Cp is defined in terms of the perturbation pressure, p , and the free stream static pressure, Poo, and the dynamic pressure, q ^ , as P-Poo

The systems of Eqs. (1), (2), (3) or (4), (5), and (6) are solved subject to the restriction of simple harmonic motion and the stability analysis reduced to the solution of a complex eigenvalue problem. More specifically, the combination of physical parameters (such as flow velocity, fluid density, shell geometry, etc.) that lead to one or more real valued eigenvalues represents the desired solutions of interest. Method of Solution A complete consistent solution to the stability problem is accomplished in two steps. The first step determines the prestability deformation of the middle surface due to the application of initial preloads and radial pressure from the air stream. This deformation is determined in closed form from a set of steady-state response equations obtained by separating Eqs. (1) and (2) into their static and dynamic components after making the following substitutions w(x,y,t) = w s (x) + w°(x,y,t) F(x,y,t) = F s ( x , y ) + F ° ( x , y , t )

(7)

p(x,y,t) = p s (x) + p°(x,y,t) This is possible because of the axisymmetric property of the initial imperfections, the surface loading, and the resulting prestability deformations. A set of equations governing the static prestability deformations (subscript s) is obtained by virtue of the equilibrium state existing prior to the instability. A second set of linearized equations in terms of the dynamic components (superscript o) govern the dynamic or static stability of the shell about its deformed middle surface. The two systems of equations are coupled through the induced static deformation and stress terms. The deformed middle surface of the shell (predeformation state) is determined from the following set of steady state response equations Dw

and

s,xxxx

+

0 / R ) F s p o c - F s,yy w spcx

(l/Eh)V4Fs = (1/R)wsxx

28

=

P S ( X ) + F s,yy W r,xx

(8) (9)

and the associated boundary conditions at x=0,L of w s (x) = w ^ x ) ^ = F s (x,y) > x y = 0;

Fs(x,y)yy = Nx

(10)

or in the case of freely-supported edges u s ( x ) ^ = v s (x) = w s (x) = w s ( x ) ; X X = 0

(11)

The initial geometric imperfection is defined as w = M sin (rîrx/L)

(12)

The general analytic solution is obtained by solving Eq. (9) for the appropriate derivatives of the stress function which reduce Eq. (8) to the form of an ordinary fourth order linear inhomogeneous differential equation with constant coefficients. This is then readily solved in closed form by standard methods. After this first step of the solution is completed, the second procedure involves obtaining nontrivial solutions to the following linearized dynamic set of linear partial differential equations possessing variable coefficients °V4w° ^

" F s,xx w ?yy " F ? y y K

- Fs ^ x

+

+ 2 F ^ y w ^ y = p° ( x,y ; t)

(13)

and =

(14)

The primes denote ordinary differentiation with respect to x. The associated boundary conditions are w°(x,y,t) = w°(x,y,t) ) X X = F

0

= F0(x,y,t)yy = 0

^ ) ^

(15)

or for the freely-supported case u°(x,y . t ) ^ = v°(x,y,t) = w°(x,y ,t) = w

0

^ ) ^

= 0

(16)

The specific form of the variable coefficients are obtained from a solution of the static response problem and the form of initial imperfection. Modal solutions of the dynamic equations are sought in the form

w°(a,e ,T) = f(a) cos n 5 e i k r F ° ( a , M = g( a ) cos nfle 0 "" .,« Ut with: r = k K

coR ( n = -¡-¡- (k real) U

(17) _x a== K

where the axial deformation function and axial stress function satisfying the boundary conditions are denoted by

f(a)

=

N .. X sinZ ma m -\ l m

m

N ~ g(a) = 2 Ym_ sin Z1, a sin Zmm a m=l

(18) Z

iy , _ = mr/(L/R)

or for freely-supported edges by N f(a)

=

g(a)

=

m m?-il

~ X

msinZma

N ~ m-l

Y

(19) msinZma

29

Galerkin's approximate solution method was then applied simultaneously to Eqs. (13) and (14) reducing the equations to the matrix form of a nonlinear eigenvalue problem with eigenvalue k. [k^Q-kBj -B2]X = 0

(20)

The system was then transformed to the form of an equivalent linear eigenvalue problem, det [A - kl] = 0

(21)

Having the same eigenvalues as Eq. (20) [7]. The stability criteria employed in the analysis observes the behavior of the eigenvalue k as a function of the free stream static pressure or applied axial load. The time dependent part of the dynamic solution is of the form w°(a,0 ,r) ~

e

ikr

.

The onset of a dynamic instability occurs when the imaginary part of k changes sign from a positive to negative value while the real part remains finite but nonvanishing. At the critical stability boundary the shell motion becomes undamped simple harmonic. In contrast, a static instability occurs at zero frequency. The real part of k vanishes as the imaginary part of k changes sign from a positive to negative value. The latter form of instability is frequently classified as a buckling or divergence phenomena while the former represents a panel flutter instability. In the above method of solution, the specific form of the aerodynamic pressure approximations was not mentioned and a few comments on this are in order. In keeping with the general form of solution, the perturbation velocity potential, radial displacement and down wash components were expressed as 0(x,r,0,t) = 0 n (x,r) cos n0 e l c d t w(x,0,t) = w n (x) cos n0 e i w t

(22)

W(x,0,t) = W n ( x ) c o s n 0 e i o j t Upon substituting this into Eq. (6) and applying the Laplace transformation 0 n (p,t) = 7 e - P x 0 n ( x , r ) d x o

(23)

To the resulting equations one obtains the following solution for the transformed perturbation velocity potential 0 n ( P , t ) = RW n (p) ^

(24)

where q2

and as

= (^)

2

p

2

+

2 M

2ia;(R)p.aj2M2

is a modified Bessel Function of the second kind. Upon inversion the velocity potential may be expressed 0 n (x,r,0 ,t) = R cos n0

f W n (z) U n (x-z) dz o

30

(25)

Where the aerodynamic Kernel function is defined in terms of the inverse Laplace transform K(qr) eP x " W S

T 2m Joo

= _1

U (XJ) n

d

"

R

*

{

y

^

-

f

g

/

V

1

a+i°° e? x K n (?r)

f

©

^

.

^

.

^

»

^

}

C27,

with =

a

L

.{oo

5

The pressure loading on the shell surface associated with the n

=

M

X

circumferential deformation mode is then given by

which after an appropriate integration by parts may be expressed as

. OJL M 2

Now for short wavelength instabilities occurring at high Mach numbers the integral correction term becomes quite small and can be neglected [5,9]. This resulting piston theory expression with a first-order curvature correction term was employed to estimate the unsteady air loads on the shell in the so-called short wavelength approximations. The long wavelength approximation is based upon the following asymptotic approximation to the aerodynamic Kernel function U n (x) [5,6,10]. T T ^ - T T « n(a> (n+a)

ln2+a2|3/2>n2+l

aK

The above approximation is exact as a tends to either 0 or °° . That is, as the shell surface approaches a slender or planar configuration to the air stream. In the limiting case a-K) the aerodynamic Kernel function takes on the characteristic of a delta function, i.e., Va)

j

%(a)

n

Un(a)^-^S(x) and the pressure coefficient can be approximated as

31

The pressure singularity at the leading edge of the shell will give rise to a concentrated generalized aerodynamic force acting at this edge when it is not restrained against radial deflections. This is a consequence of applying slender body theory to a configuration possessing a discontinuity in cross section. In essence the pressure singularity need be retained only in those flutter analyses where the shell leading edge is not completely restrained against radial movement. Slender body theory and piston theory are a plausible approximation to the aerodynamic pressures for predicting a low supersonic Mach number long wavelength instability and a high supersonic Mach number short wavelength instability, respectively. They were the two potential flow approximations employed in the present study to estimate the steady and unsteady pressure loadings on the shell surface due to the supersonic flow field. Theoretical Results The theoretical investigations carried out to date have looked into the influence of initial geometric imperfections, refinements in modeling structural boundary conditions, and different aerodynamic approximations on the accuracy in estimating the aeroelastic stability characteristics of thin cylindrical shell structures. Several definitive observations have resulted from the theoretical study which are detailed in the following. a)

Influence of Structural Boundary Conditions on the Shell's Aeroelastic Stability

Refinements in modeling the structural boundary conditions were carried out in two basic areas of interest. These were classified as a predeformation effect and a direct structural boundary condition effect. The latter results from specifying different structural conditions on the ends of the shell whereas the former results from an interaction of the initial prestress, in an otherwise perfect or imperfect shell, with the structural boundary constraints. That is, when predeformations are considered, stability is investigated about the deformed state of the shell instead of its freely expanded or undeformed state. This deformed state results from preloading the shell with a combined internal pressure and/or uniform axial loading in the presence of an air stream and realistic boundary constraints. The modified piston theory approximation, given by Eq. (29), with the integral term neglected, was employed to estimate the aerodynamic pressure loadings. The basic problem under consideration here was the formulation no. 1 illustrated in Fig. 2. When predeformation effects were considered path no. 2 was followed in this problem formulation and the shell's aeroelastic stability investigated about its predeformation state. When path no. 1 was followed in the problem formulation the preloading on the shell did not change the basic shell geometry and stability was investigated , about the shell's initial or undeformed state.This slight i refinement in structural modeling implied by path no. 2 ] can have an influence on the stability characteristics of < the shell as illustrated in Figs. 3 and 4. In Fig. 3 it is j shown that when the combined loading and geometry were such that the predeformation effects were propai gated from the ends of the shell well into its interior a , 15 to 20 percent shift was observed in the flutter ! boundary of highly stressed shells. This predeformation influence was found to stabilize the shell. On the other

UNSTABLE

—CT

LEGEND PATH # I - - P A T H # 2 o 2 4 TERMS A 28 TERMS • 3 0 TERMS

STABLE

DATA E - 1 4 . 1 0 6 psi R/h « 5 0 0 L/R • 727 N , ' - 375 • • 3

Prtttobiltly Dtformotions Prof'let

4 6 8 10 INTERNAL PRESSURE PARAMETER %

Fig. 3 Aeroelastic Stability Boundaries for Copper Shell at Mach 5.0 and R/h=500

SHELL

INTERNAL PRESSURE

p(piig)

Fig. 4 Aeroelastic Stability Boundaries at N x =0 Path #1 and # 2 Solutions

32

hand, for certain shell geometries and loading conditions the predeformation influences were restricted to a small region or boundary layer adjacent to the ends of the shell and no significant influences were obtained as illustrated in Fig. 4. The influence of applying different structural boundary conditions on the ends of the shell was investigated over a range of parameters where predeformation effects were not found to be significant. Small changes in the structural description of the boundary constraints can produce significant shifts in the aeroelastic stability of the shell. This is illustrated in Fig. 5 where changes only in the in-plane boundary conditions produced significant shifts in the aeroelastic stability boundary at the higher levels of internal stress or pressurization. Although these refinements in the modeling of the structural boundary conditions did not explain the existing discrepancy between theory and the earlier experiments, information was obtained on the accuracy required in the modeling of the structural boundary conditions for a realistic analysis. In summary it was found that the correct structural modeling of the shell must include the option of employing different structural boundary conditions as well as predeformation effects. b)

Fig. 5 Cylindrical Shell Flutter Boundaries for Two Different Structural Boundary Conditions

Influence of Initial Geometric Imperfections on the Shell's Aeroelastic Stability

This study represented an additional structural refinement introduced in the aeroelastic modeling of cylindrical shell structures. It represents an extension of the study conducted under part (a) and was again undertaken to help clarify the reason for the large discrepancy that existed between experimental and theoretical observations on cylindrical shell panel flutter. More specifically the existing analytical models were extended to determine the influence that small initial geometric imperfections, due to fabrication techniques, have on the aeroelastic stability behavior of cylindrical shells. The modified piston theory aerodynamic approximation was again used and the basic problem under consideration here was the formulation no. 2 illustrated in Fig. 2. When the predeformation influence of part (a) is also considered, path no. 4 is followed in the problem formulation. When the predeformation effects are neglected path no. 3 represents the appropriate formulation. The present PATH # 2 study demonstrates how an apparently better correlaUNSTABLE tion between the theory and earlier experiments can be achieved when initial geometric imperfections are considered in the analysis. The nature of the imperfections were estimated from experimental measurements on fabricated shells. This apparent improved PATH # 4 , r - 3 , J •! 0 correlation with experiments is illustrated in Fig. 6, where the results of formulation no. 1 path no. 2 are compared with the results of formulation no. 2 path no. 4. It is evident that initial geometric imper.5 10 fections drastically reduced the shell's ability to resist SHELL INTERNAL PRESSURE p (psig) panel flutter even when the imperfection amplitudes Fig. 6 were only on the order of one shell thickness. The Aeroelastic Stability Boundaries; Demonstration of the results of the present analysis extended our concept Effect of Initial Imperfections concerning the correct structural modeling for studies n

related to the aeroelastic stability of thin cylindrical shells. In essence, the correct structural modeling of the shell must include the option of employing different structural boundary conditions, predeformation effects, as well as initial geometric imperfections that may be present in the shell surface. 33

It is of interest to note that a limit check on the analysis of parts (a) and (b), concerning conclusions on structural modeling, was obtained by setting the air speed equal to zero. This stability study then reduces to a classical shell buckling analysis under the influence of predeformation effects and initial geometric imperfections. This has been extensively studied in the literature and much data is readily available for comparison. The limit checks used in the present study agreed with the published results in all cases. This is illustrated in Fig. 7 [1]. This was assumed to be a sufficient verification of the analyses and computer codes developed during these studies. c)

£ > I 4 I I 0

STATIC

SOLUTION

-1.0

-.5 IMPERFECTION

pii

R / h » 100 L/R • 3 2 p • 0 . 0 pti

UNSTABLE

AMPLITUDE

RATIO

Fig. 7 Numerical Solution Limit Check

Influence of Supersonic Flow Field on Critical Buckling Loads of Cylindrical Shells

The structural loads imposed on many aerospace vehicles passing through the region of maximum dynamic pressure during the boost or reentry phase of a trajectory are quite severe and may result in the structure becoming aeroelastically unstable in a static mode. That is, the structure may encounter a buckling type collapse rather than a panel fluttering instability due to the combined air and inertia loads imposed upon it. Since the primary structural member is usually the thin walled cylindrical shell the problem becomes one of investigating its stability against buckling when its outer surface is exposed to a high velocity air stream. The initial preload was a combined internal pressure end axial compressive and loading and the modified piston theory approximation was again used. Primary findings in this area indicate that if the critical loading conditions occur at supersonic air speeds then the existing still air buckling data should be beneficial in the design of such shell structures. The influence of the supersonic air stream has no significant effect on the shell's ability to resist a buckling collapse. These conclusions have been confirmed on both a theoretical and experimental basis. The theoretical study employed the computer code developed under parts (a) and (b) above. Some of the n — 1 — I — • — r E • 14 x 10 psi theoretical data substantiating these conclusions are L/R • 2.0 R/h • 2000. illustrated in Fig. 8. In this illustration three different I M-3.0 WH • 5 shell internal pressures p were examined and the J 24 TERMS | corresponding stability boundaries plotted in Fig. 8. It is seen that for each shell internal pressure level, the £ BUCKLING AND DIVERGENCE LOAD stability behavior exhibits^ a similar trend as a function y p • 0.0 psi, N • - .695, n • 29 of the applied axial load N x . As the applied axial load is £ p • 0.5 psi, Î Î • -.95 . n • 24 increased, the ability of the shell to resist a dynamic 1 p - l . O p s i , N • -1.02, n - 0 (fluttering) type instability decreases by as much as & 85-90 percent of the zero axial load case. This pro- if nounced destabilizing trend occurs at axial loads less than 40 percent of the static still air buckling load of the -.4 -.6 -.8 -1.0 shell. For axial loads above approximately 40 percent of CRITICAL BUCKLING LOAD RATIO N_ the critical buckling load, the freestream static pressure Fig. 8 required to cause an aeroelastic instability remains fairly Influence of Axial Compressive Loading on the constant but the critical circumferential wave number Aeroelastic Stability Boundary decreases until a critical axial load is reached. At this axial load, the shell diverges into a static buckled shape with a corresponding circumferential wave number equivalent to that predicted from still air buckling studies. Also, the critical divergence axial load obtained in the presence of a supersonic flow field is essentially the same as the critical buckling load predicted by the still air buckling study at corresponding shell internal pressures. x

It by the below. certain

is apparent from this study that the axial load carrying capability of thin cylindrical shells is not influenced existence of an external supersonic flow field. This was also confirmed in the experimental study discussed However, this does not preclude the necessity of investigating the dynamic stability of the shell since for flow and stress conditions the shell may be statically stable but dynamically unstable.

34

d)

Evaluation of Different Potential Flow Approximations

Although the aerodynamic theory most commonly employed in the reported theoretical results was the modified simple piston theory approximation, several studies have been carried out using the slender body approximation valid for long wavelength instabilities occurring at low supersonic Mach numbers [5,2]. Unfortunately, after extensive study it was found that this simple aerodynamic theory could not be expected to predict even the qualitative features of the shell dynamic instabilities that were observed experimentally. Comparisons with a good number of experimental observations indicated that slender body theory does not predict a dynamic instability (panel flutter) over the range of experimental parameters where a dynamic instability was observed. Instead slender body theory always predicted a static or divergence instability over the complete range of parameters investigated. In addition, this aerodynamic approximation indicated a significant influence of the supersonic flow field on the buckling characteristics of thin cylindrical shells that was not predicted by the modified piston theory approximation or observed experimentally. In view of this, the theory was considered to be inadequate for the present problem. The modified piston theory approximation showed reasonably close correlation with all of the earlier experimental observations on a mild limited amplitude panel flutter occurring on initially unstressed shells. It became increasingly unconservative, however, as the initial prestress in the shell was increased. In addition, it did not compare very well with a more highly divergent flutter reported for the first time in the present experimental study. Experimental observations discussed below cast further doubt on even the validity of employing the modified piston theory aerodynamic model for predicting the shell dynamic stability. In essence, all potential flow models appear to be on questionable grounds for predicting the dynamic stability characteristics of thin shell structures.

EXPERIMENTAL OBSERVATIONS To further evaluate the accuracy of the analytical modeling in the above analysis and in the analyses carried out to date on this problem, a wind tunnel experiment was designed and conducted on 14 different cylindrical shell configurations under different internal stress levels and supersonic flow conditions. All of the shells had a common length to radius ratio of two and radius to thickness ratios from 2000 to 4000. These shell geometries were taken to be the same as those from an earlier program to facilitate a comparison of the present results with these earlier experiments [11,12], The facilities of the 16-ft supersonic and transonic Propulsion Wind Tunnel of the Arnold Engineering Development Center was employed for the study. Full details of the wind tunnel facility are presented in [16]. Description of Flutter Model and Instrumentation The experimental flutter model was a ogive cylinder configuration cantilevered at its base from the wind tunnel sting. The thin shells under study were isotropic circular cylindrical shells fabricated from copper by an electroforming process. The shells were bonded to two heavy copper end rings and mounted near the base of the ogive cylinder model. The electroforming process provided shell models with a high degree of uniformity in both material and geometric properties while minimizing geometric imperfections and initial fabrication stresses. In addition, extremely thin shells could be easily fabricated in this manner. A photograph of a typical test shell without end rings, and its installation in the wind tunnel is shown in Fig. 9. Several types of instrumentation were employed on the model to observe and control the shell instabilities. Transducers were employed to monitor both static and dynamic displacements of the shell skin. Loading mechanisms were also employed that could simulate a variety of stress states in the shell by employing axial compressive end loads and/or internal pressurization. A boundary layer control slot was located in the model nose cone to artificially trip an existing laminar boundary layer profile or thicken an already fully developed turbulent profile. A variety of boundary layer rakes and static surface pressure probes were also employed to determine the nature of the local flow field over the shell. Finally, visual monitoring through high speed photography and on-line television cameras provided an additional mode of instrumentation for observing the shell instabilities. Further detailed descriptions on the model and instrumentation can be found in [8].

35

Observed Shell Instabilities Two basic types of shell instabilities were observed during the course of the experiments. This included a divergence or buckling instability and a dynamic or panel fluttering instability of the shell skin in the presence of a supersonic air stream. This latter instability is similar to flag or sail flutter, while the former appears identical to the classical static buckling instability of thin shells under combined axial compressive end loading and/or internal pressure. The basic experimental procedure followed during the course of the study was to maintain wind tunnel conditions fixed and change the shell model internal stress state or boundary layer features to initiate or suppress a shell instability. Buckling or Divergence Instability Buckling or divergence studies were conducted to determine the influence of the supersonic air stream on the shell classical still air buckling loads, and to establish a safe shell loading limit for avoiding a divergence or static mode of shell instabilities over a range of supersonic flow conditions. Although the onset of a static divergence or buckling collapse was quite evident due to its catastrophic nature, several experimental indicators were employed to distinguish its characteristics [8]. The influence of the supersonic air stream on the classical buckling loads of thin cylindrical shells under combined axial end compressive loading and internal pressurization was investigated by correlating data from at least a dozen different test conditions. These included shell buckling studies both with and without the influence of an external supersonic flow. The data are presented in Table I and on the buckling interaction curve illustrated in Fig. 10. The data Fig. 9 represented by the shaded points in Fig. 10 were taken from Cylindrical Shell Flutter Model and Wind Tunnel still air buckling tests, while the open symbols represent wind Installation tunnel test points. The flow conditions and radius-to-thickness ratios of each shell are indicated beside each test point. The triangular open symbols represent dynamic instabilities (panel flutter), while the remaining open symbols represent static buckling in the presence of an air stream. The data on this interaction curve indicate that no significant shift occurs in the wind tunnel buckling load points (open non-triangular symbols) when compared to their still air buckling counter parts (shaded symbols). The primary findings here indicate that if the critical loading conditions on the shell cause it to become unstable in a divergence or buckling mode at supersonic air speeds then the wealth of existing still air buckling data should be beneficial in estimating the buckling loads of such structures. The interaction design curve suggested in [17] which is

36

4> >
>

V >

>>

>

3

•e

u "3 •S Ih

u 3 -IH S

I

I

iE

_1

a .5 E 3

a .5 Ë 3

I

I

I

S e

>>

e«

S •o c

a .5 E m -i

3

3 •8

•8

H

H

H

H

vq ri es

p

VO

eS

n es o o d

es o p d

es en O O d

•si

o o o\

O

o f-H OS o

o 00

o es »o O

¿

Ti-

O o

o VO p d

es

fi o

o res o

V)

o o

o IO o 9

^

o d

en

o o

o IO en

o o Tf

o

o

o en »—i

O

o 1 es

o

o

o

o o es

I

I

o o VO es

o o so eS

o o o eS

o o O eS

o o vo CS

o o o

. o es o eS^

o o o

Ov en »o

Ov en •o

en O

es o Tt

O

r-

3 d o

^ '""'

p O

p o

p o

o en es o

00

io o es o

o W) O es

o >o Ov

o o o eS

o o d

0\ en o p o

T O O d

o o o

t— en o o

en o\ en

o o

o OV

O O c-

o o> en

V5

O

_

(S o

o

o es 00

o

o

i-H o

z z

a

= z(

b

= z

>a)

(^b)

119

(26)

Application of the Bernoulli-Euler-Lagrange equations (Eq. (21)), by taking r' and 0' (derivatives with respect to z) will give the new optimization equations,

2 7 rr + X

^

9 9

i csc^ 0 + X2 2 ( r c o s 0 " ^

s i n 0 ^ - ^ - ^ dz N Q dz

s n

' ^ ~r ^

cos

0)+

1

cos

= 0

0 "r

(27)

fa^

si n 0

=

0

(28)

In order to perform the integration after the variable transformation, it is required to define the initial values for new X2 1 and X2 2 • This is accomplished by the re-utilization of the natural boundary conditions. The new set of natural boundary conditions will be 8F2

= 0

(29)

z=z„ , where

, y

j

=

dr d0 di'dT

Similarly, after a certain range of integration, when the tangent of the generator loses its steepness, a new variable transformation is undertaken. The net set of governing differential equations will be expressed in terms of independent variable r . Because of the new variable transformation, the natural boundary conditions need to be reapplied (Eq. (25)).

NUMERICAL SOLUTION The numerical integration of the governing differential equations, and required variable transformations require extensive but repetitive numerical computations. Therefore, it is prohibitively complex and prone to error for hand computations. Consequently the use of digital computers is essential. The numerical integration of the differential equations is attempted after the rearrangement of the equations to permit the use of integration schemes. Depending upon the independent variable range, the equations to be integrated are: 1.

Equations (3), (14), (23), (24) and the parametric expression of stability (Eqs. (3), (17) and (18)).

2.

Equations (4), (15), (27), (28) and the parametric expression of stability (Eqs. (9), (17) and (18)).

Numerical integration is carried out by the use of 4 order Runge-Kutta numerical integration scheme [7]. The same integration was also performed by using the Euler integration scheme, even though the Euler integration scheme required somehow less computational effort, no discernible numerical variations were observed. Nevertheless, in view of possible addition of more constraint equations, it is believed that the use of Runge-Kutta scheme should be preferable. The only aspect of the numerical integration scheme which requires attention is the definition of the points where the variable transformation will be performed. Some values for these respective transformation points are given as [4]: the first transformation to be made when the azimuthal angle reaches the value of about 50° and the second transformation is at about 140°. This study has used a different approach. In each integration step there was an increase in the rate of change in the azimuthal angle. The first transformation was done when the rate of change was ten times the rate of change at the origin. Similarly the second transformation was done when the rate of change became about 1/10 of the rate of change observed when the azimuthal angle reaches the value of 90°.

120

Byproduct of this approach is the numerical values for the thickness of the shell (Fig. 3). Along the integration path, some regions are observed where there is a pronounced increase in the shell thickness as compared to the other regions. Since it may be preferable to maintain a constant shell thickness, these regions should be considered as those which may require rib stiffeners to maintain a uniform factor of safety when the tank is drained and subjected to predefined external loads.

0.5

T

1.0 15

Results obtained from a numerical example are shown in Fig. 3. The numerical values used for this problem are given as V = 1000 gallons (water)

H = 2 ft

Allowable stress = 36 ksi

E = 25,000 ksi

f = 20

a = 0.20

Fig. 3 Nondimensionalized Generator Geometry and Thickness Variation (dashed line corresponds to the membrane thickness)

The solution by the prescribed method gives » 75°

110°

T = 1.90 in.

For clearer visualization of the problem shell geometry and thickness variation are given in nondimensionalized form. The regions where the rib stiffeners are recommended is shown on the thickness variation curve.

CONCLUSIONS The study provides a direct variational method in the analysis of drop-shaped tanks. The constraints, geometrical, stress, stability, and other forms can be taken into account by the addition of more Lagrangian multipliers. Due to the incremental nature of the integration, inclusion of external modification parameters is also possible. The compact nature of the computational algorithm enables the use of small digital computers in the analysis. The stiffener requirements for the drained drop-shaped tanks can be accomplished either by using variable thickness, which leads to a not-too-practical solution, or by rib stiffeners of equivalent area.

121

NOTATION

ds

differential arc length of the shell

dr

differential radius of the shell

dz

differential depth of the shell

E

Young's Modulus for the shell material

f

factor of safety against buckling

F*

generalized optimization functional

F* j

optimization functional, r - independent variable

F*2

optimization functional, z - independent variable

I*

optimization integrand

N^

meridional stress

Ng

tangential stress

N0

constant surface stress

Pg

tangential external load

Pr

radial external load

qcr

critical buckling load

r

r-ordinate, cylindrical coordinates

rj

principal radius of curvature, perpendicular to the generator plane principal radius of curvature, in-plane

ra

radius at the first variable transformation, z a =z(0 a )

r^

radius at the second variable transformation, rj ) =r(i^ ) )

rc

bottom ordinate, rc=(0=18O°)

t

shell thickness

V

reservoir volume

z

z-abcissa, cylindrical coordinates

za

depth at the first variable transformation, z a =z(0 a )

z^

depth at the second variable transformation, z^)=z(0j5)

zc

total shell depth, zc=z(v?=180°)

a

buckling parameter

7

unit weight of the liquid

122

BIBLIOGRAPHY 1. Bouman, C.A., "Strength Tests on a Spheroid Tank," De Ingenieur, Vol. 53,1938, pp. 3 9 4 6 . 2. Codegone, C., "Serbatoi a Involucro Uniformamente Teso," Ann. Lavori Publ. No. 79,1941, pp. 179-183. 3. Elsgolc, L.E., Calculus of Variations, Addison-Wesley Pub. Co., Reading, Mass., 1962, pp. 127-146. 4. Flügge, W., Stresses in Shells, 2nd ed., Springer-Verlag, Vienna, 1962, pp. 376-378. 5. Girkmann, KFlaechentragwerke, Sechste Auflage, Springer-Verlag, Vienna, 1963, pp. 376-378. 6. Haas, A.M., Design of Thin Concrete Shells, Vol. 2, Wiley & Sons, New York, 1967, pp. 204-230. 7. Ketter, R.L., and Prawel, SP., Jr., Modern Methods of Engineering Computation, 1969, pp. 4 2 7 4 5 5 .

McGraw-Hill, New York,

8. Kostem, C.N., "A Bibliography on Discretized Systems in Structural Mechanics," Report No. 237.52, Feb. 1969, Fritz Engineering Laboratory, Lehigh University Office of Research, Bethlehem, Pennsylvania. 9. Kostem, C.N., "Synthetic Design of Minimum Weight Membranes," International Association for Shell Structures, Madrid Colloquium, Vol. Ill, Sep. 1969. 10. Kostem, C.N., "Optimum Weight FoldedPlate Systems," International Association for Shell Structures, Vienna Symposium, Sep. 1970, (to be published). 11. Novozhilov, V.V., The Theory of Thin Shells, P. Noordhoff Ltd., Groeningern, The Netherlands, 1959, pp. 124-130. 12. Pelikan, J., "Form Determination of Shell Structures," Bulletin of the International Association for Shell Structures, No. 43, Sep. 1970, pp. 3-8. 13. Runge, C., and König, H., Vorlesungen ueber Numerisches Rechnen, Springer-Verlag, Berlin, 1924. 14. Szmodits, K., "Water Tower with Drop-Shaped Tank," Proceedings of the Symposium on Tower-Shaped Steel and Reinforced Concrete Structures, International Association for Shell Structures, Bratislava, June 1966, pp. 289-290.

123

I N F L A T A B L E SHELLS IN THE F L U I D E N V I R O N M E N T John W. Leonard Associate Professor Illinois Institute o f Technology Chicago, Illinois, U.S.A.

ABSTRACT The use o f prefabricated inflated membrane shells as temporary protection against the underwater environment is a promising technique for the construction o f interconnected domed villages on the ocean floor and of decompression stations anchored at various depths below the surface. Inflatable shells have not been used extensively because it has not heretofore been possible to predict their complex behavior. Solution methods are now available for the pressurization and in-service isotropic behavior o f an inflatable shell o f revolution. These methods have been incorporated into a single computer program which is described herein. It is possible to determine the initial shape o f the deflated shell required to obtain the desired configuration of the meridian for the final shell. The static and dynamic response o f the shell to in-service loads such as concrete dead load and pressure transients can be analyzed. The choice o f meridional shape is arbitrary and can be described either by equations, by pointwise values o f the radii o f curvature, or by Chebyshev interpolation of pointwise values of the radius normal to the axis o f revolution. The solution methods and the capabilities o f the computer program are demonstrated by means of static and dynamic example problems for a drop-shaped shell, which is perhaps ideally suited to carry hydrostatic loads. Some observations are made on the effects o f approximated geometry and of initial pressurization stress on the in-service dynamic behavior.

INTRODUCTION There

has

been

considerable

attention

given

of

late

to

the

nonlinear

behavior

of

extremely

thin

pressure-stabilized shell structures, commonly called pneumatic or inflatable shells. The distinguishing characteristic o f such structural components is that broadly distributed environmental loads are supported by an initially-stressed membrane, the initial stress being provided by internal pressure. One potential area of application for inflatable shells is as protection against the underwater environment. It is the purpose of this paper to discuss these potential applications and to present solution methods adequate for the proper analysis o f the nonlinear behavior of such shells. Some o f the advantages o f inflatable shells for use in the fluid environment are that 1) they are light in weight and collapsible implying ease in transportation and erection of components, 2) the environmental loads are efficiently carried by direct tensile stress without bending, 3 ) the leakage of gases through punctures provides early warning o f collapse and repairs can be easily made by patching, and 4 ) the primary load carrying mechanism is part o f the habitable environment itself, i.e., a pressurized mixture o f gases. On the other hand, a disadvantage o f inflatable shells over other structural systems is their susceptibility to large displacements and stress concentrations due to concentrated loads and dynamic overpressures. Since the primary environmental loads are hydrostatic in nature and since sensitive pressure control devices can be included to stiffen the shell against vibrations, these disadvantages can be overcome. However, analytic capability to treat dynamic and static in-service loads should be available in order to predict the shell behavior under such conditions. Inflatable shells made o f a thin metallic film or vinyl-coated fabric could be used to provide temporary or semi-permanent roofing for interconnected domed villages on the ocean floor or for enclosures anchored at various

124

depths below the surface to be used as decompression stations. If a permanent structure was desired, concrete could be sprayed on the interior of the inflated shell using modern shotcreting techniques. The inflated shell in this instance would act as temporary formwork to support the concrete during the curing process. If the inflated shell were used as formwork for the casting of concrete, two alternative methods of construction seem feasible. In one method, the formwork would be submerged in a packaged configuration and inflated to the desired configuration. Then, a two or three man crew on the ocean floor could attach steel reinforcement to the form and spray concrete on the interior with shotcreting equipment. The other alternative would be to submerge a double-walled form and to inflate the interior compartment of the form. Then concrete, possibly fiber reinforced, could be pumped from the surface to displace the air or water used to inflate the compartment. Only recently has nonlinear membrane theory been studied [2,11]. A specialized theory for inflatable shells has been reported [4,5], and solution methods for pressurized shells of revolution based on this theory have been presented for the static and dynamic behavior during the pressurization and in-service phases [4,6,7]. The basis of derivation of that theory is that the known geometry coincides with the fully inflated shape. There have also been conducted studies of inflatables assuming a given initial state and nonlinear material behavior [3,9]. In the present study, solution methods for the pressurization and in-service phases of an inflatable shell of revolution in the fluid environment are presented in a unified context. A single master computer program is described which, depending on the choice of input parameters, can determine either the initial shape required for a given final shape after pressurization, the static response to in-service loads, the frequencies and mode shapes of free vibration, or the forced dynamic response to in-service loads. The choice of meridional contour is arbitrary and the computer program allows for description of the shell geometry either by equations using an angular coordinate system, by pointwise values of the radii of curvature, or by Chebyshev interpolation of pointwise values of the radius normal to the axis of revolution using an axial coordinate system. Several example problems are given to demonstrate the various solution techniques and associated subprograms of the system, and to illustrate some interesting effects of the initial stress due to pressurization.

MATHEMATICAL MODEL For convenience in the mathematical analysis of inflatable shells their behavior can be considered in three stages: the unfolding phase in which the shell is inflated from its compact packaged configuration into its initial unstrained shape; the pressurization phase wherein the shell undergoes further nonlinear displacements from its unstrained shape into the desired final configuration; and the in-service phase in which the fully pressurized shell is subjected to various external loads or vibrations. The unfolding phase is primarily a problem in the mathematical topology of isometric surfaces and will not be considered here. The pressurization phase is a large displacement problem for which a consistent nonlinear theory has been developed [4]. The fundamental assumptions are: the reference surface is the deformed middle surface of the desired shape; the shell is extremely thin; and the material is elastic and isotropic. Fig. 1 Geometry of Shell of Revolution

125

For the special case of a hydrostatically loaded, axially-symmetric shell of revolution (see Fig. 1) the Poincaré perturbation technique [10] can be used to determine in a recursive fashion the stress resultants and displacements as follows R

Nj j =

R

2Pi

-

2

f 7(H+z)r r'dx

(la)

r R n

u

P

=

i

R 2 [Pi-T(H+z>] ~

w

!,ȓp r

u

=

22

r

(

N

P

=

Ì!wìp R

?\/a77

(lb)

n

?

(lc)

R9

}

=

(ld)

(le)

= N n /Eh

(10

A j = N22/Eh 1

"

u-

A

w-

u- •

w- •

w'

[(

u-

w' •

1

VAlr 1

(lg)

(lh>

foM

2(l-u2)

j=l

VA

n

r

R2

VA

n

r

u- •

R2

where f ' = 9f/9x , P j = internal pressure, H = depth of apex below surface, z = axial distance from apex, R] ,R 2 = radii of curvature, r = normal distance from axis of revolution, E = Young's modulus, v = Poisson's ratio, h = thickness of shell, V A j j = meridional component of metric tensor, x = meridional coordinate, N j j = meridional stress resultant, N 2 2 = hoop stress resultant, Up = tangential pressurization displacement, Wp = normal pressurization displacement, and Cj are constants of integration to be determined by the base boundary conditions. The in-service phase can be considered as the superposition of a linear problem, either static or dynamic, onto the previous nonlinear problem posed by the pressurization phase — if one makes the additional assumption that the initial pressurization has stiffened the shell sufficiently that any additional deformations are small. For this case, a linearized theory has been developed [5,6,7] which, unlike classical linear theory, incorporates the effect of initial pressurization stress in the governing equations of equilibrium. For the special case of a shell of revolution with nonsymmetric superposed loads, the equations admit solutions of the form u = 2 u n (x,t) cos n0

(2a)

v = 2 v n (x,t) sin nQ

(2b)

w = 2 w n (x,t) cos n9

(2c)

P j = 2 P j n ( x , t ) cos nd

(3a)

P 2 = 2 P 2 n ( x , t ) sin n0

(3b)

P 3 = 2 P 3 n ( x , t ) cos n0

(3c)

if the applied loads can be written as

126

where P j = load in meridional direction, P 2 = load in hoop direction, P3 = load normal to shell. For this assumed solution, the equations of motion for the in-service phase behavior of a shell of revolution are given by 82U "at

+ n v

PinAllO*2) Eh

=

u

n(1+Kl)

+ u

n[gl(1+K2)-g9(1+Kl)l - w ^ O * ^ ) ^ - ^ ) ]

n ^ T 83 + u n [ ( 8 2 - 8 1 8 9 X ^ 2 ) - 8 ? ( 1 - K 2 ) - n 2 g | ( ^ + K 2 ) ] - n v ^ g ^

+ 2K 2 )

+ W n t g j g ^ l + K j ) + g 6 g 8 (i>-K 2 ) + 2 g l ( g 6 - g 5 ) ( i ^ + K 2 )]

P(l-^ 2 ) E

A

a2y

n

P^lli1^2) Eh

_

(4a)

,r ,, . 1+P v n ( x + K 1 ) - v n [ g 9 ( 2 - + K 1 ) - g 1 ( - T + K 2 ) ] - n u n ^ - g3

- n u n g l g 3 ( ^ + 2K 2 ) - v n [ ( g 2 - g 1 g 9 X ~ + K 1 ) + g 2 ( ^ + K 2 ) + n 2 g 2 ( l + K 2 ) ]

+ nwng3 [ g ^ l + K ^ g ^ - K j ) ]

ofl„2x A

E

32wn PsnAnil-p2) 11 - J - - " ^ E h dt

(4b)

=


)/E and the mixed initial-value and boundary-value problem posed by Eqs. (4) is converted into a static boundary-value problem of the eigenvalue type. TABLE I GEOMETRIC FUNCTIONS FOR SHELL OF REVOLUTION definition

ip as coord.

1

r'/r

g 6 cot 1p

r

'/Sio

2

r"/r

g7

r

"/8io

3

VÄ^/r

84/810

84/glO

4

VA„

R

d+(r')2)H

5

VÄ^/Rj

1

•82/(8384)

6

VÄH/R2

g4/R2

!/8l0

7

r

2^r2

RI/g4

•3g1g5g1o-r"7(g2gio)

8

R

2^R2

gj-cotip

8l+89

9

A'n/(2An)

87

8181/83 2

10

r

R2 sin 1p

r

8i

grg6

1

Note: s i n ^ = g 6 / g 3 and cos ^ = g i / g 3

128

z as coord.

The method used to determine the natural frequencies and mode shapes for this eigenvalue problem is to select a trial value of to and to check if it provides a unique solution to the equivalent static problem. If to were a natural frequency, a nontrivial family of solutions differing only by a multiplicative constant would have been found. Using the static solution procedure described above, the partial solutions are propagated along the meridian. For the free-vibration equations, the total solution at the boundary point is given by

{

true

homogeneous boundary conditions

{0}

(7a)

bound. ' where a.j are the constantsconds.J of combination for which a nontrivial solution exists if

det

homogeneous boundary conditions

= 0

(7b)

Since the determinant is a continuous function of GO, the trial and error search for the natural frequencies consists of computing the determinant for each trial to and comparing it to zero. If the test determinant is zero, a natural frequency has been found. A change in sign in the determinant for two successive trials implies the existence of a natural frequency between those two trials. A binary search is then instituted to converge on the natural frequency. The natural mode shape is then given to within a multiplicative constant by true Ì mode > shape J

homogeneous solutions

(7c)

It should be noted here that the presence of the initial pressurization stress in the equations of motion has a great effect on the values of the natural frequencies and mode shapes. For lower levels of the initial stress (of the order N j j / E h < 0.05) it has been found [6] that an approximate set of equations in which the initial stress effect is neglected in the first two of Eqs. (4) yields excellent results in comparison to the exact equations. However, for some materials and for stress levels encountered in the fluid environment, the initial stress ratio N j j / E h is not small and significant errors occur if the above approximation is made. Moreover, contrary to results reported previously [ 7 ] , for higher levels of the initial stress ratio the mode shape as well as the natural frequency is affected by the value of the initial stress. This is discussed subsequently when the example problems are considered. Once the modes of free vibration have been determined, the forced response of the inflated shell to time-dependent in-service loads Pj(t) can be determined by modal analysis. Assume that the displacements are an infinite sum of the natural modes, each mode being multiplied by an unknown participation factor u(x,0,t) = 2 2 u*j(x) cos n0 F n j ( t ) i n

(8a)

v(x,0,t) = 2 2 v*i(x) sin n9 F .(t) l n

(8b)

w(x,0,t) = 2 2 w*j(x) cos nd F n i ( t )

(8c)

where u * j , v*j, wjijj denote the displacements of the i ^ ordered mode of the n1*1 harmonic. If Eqs. (8) are substituted into Eqs. (4) (with damping terms added), and the free vibration equations and orthogonality properties of the natural modes are used to combine and eliminate terms, then the following equation is obtained for the participation factor of each mode [7]

129

a2F„; at2

+

9t

ni

, = Q(t) m

(9a)

where to ^ is the natural frequency of the i^ 1 ordered mode for the n^ 1 harmonic, (3 = ratio of the coefficient of damping to the coefficient of critical damping, and 1 Q

ni

=

2tr base I

ap{x

. [piufn c o s

n0 + P

2vfn sin

nd + P

3wfn

cos

dx

(9b)

where K f i j = modal normalization constant

K

ni

=

JducX ' Kn

+ v

fn

+ w

fn 1 ^

dx

(9c)

and Pj(x,0 ,t) = specified in-service dynamic loads. Equations (9) constitute a classical initial-value problem for which a multitude of solution techniques are available. The generalized Holzer method was used in this study.

COMPUTER PROGRAM The solution methods described in the preceding section have been incorporated into a single computer program capable of providing solutions in tabular and graphical form to any combination of static and dynamic problems for either the pressurization or in-service phases of an arbitrary shell of revolution. The program is designed to be general and flexible in nature. All special operations dependent on the problem considered have been segregated from the main flow of the program so as to readily accept changes. For example, the user can choose from a variety of subprograms to model the particular boundary conditions, load configurations, and input modes he requires. The flow of command in the program to solve particular aspects of inflatable shell behavior is controlled by certain input parameters generated by the user. The block diagram of the main program controlling the flow of information and calculation is shown in Fig. 2. The results of the calculation in each block are stored off-line and can be recalled by different blocks. The final step in the flow for each problem is the processing of the solutions generated. The input information, and displacement and stress solutions are recalled from off-line and displayed in

Fig. 2 Flow Diagram of Computer Program

130

tabular form and graphically using a CALCOMP plotter. The graphical results are plotted in SVL x 11 blocks suitable for direct conversion to figures. For example, the figures accompanying the example problems considered in the succeeding section were obtained directly from the CALCOMP plotter under the control o f this program. The separate calculation blocks in the main block diagram will now be discussed. The S H E L L block is used to calculate the geometric quantities gj, i = l , 1 0 (given in Table I) from which all other pertinent properties can be easily derived. Subroutine S H E L L was designed to be flexible in regards to the nature of the shell geometry, the chosen coordinate system, and mode o f input desired. The geometric quantities can be calculated directly from formulae provided by the user, from inputted pointwise values o f the radii of curvature given in an angular coordinate system, or from interpolation formulae based on inputted pointwise values of r , the distance of the meridian from the axis of revolution. This last capability given above is o f value when an arbitrary shape for the meridian is desired and no exact equation for the meridional curve is available. The method of interpolation used is based on Chebyshev polynomials since they, amongst the various alternatives tested, provided the most accurate approximation to the third derivative o f the meridional curve required to calculate g^ = R j / R j . The effect of this approximation on the results for in-service shell behavior is considered in the following section on example problems. Special calculations are required in cases where the shell is continuous at the apex, i.e., an umbilical point; and an axial coordinate system is adopted. Since at an umbilical point, the axial coordinate system is singular, an angular spherical coordinate system was imposed near the apex for an arbitrarily chosen small axial distance. Within this " c a p " region the meridian was subdivided into ten angular increments and gj was calculated using spherical formulae. At the transition point between the angular and axial coordinate system transformation formulae are required to propagate derivatives past that point. These transformations are df dz

1

ctf

(10a) INPUT LOADS

d2f d2f H = — ^ ^ ,2 dz r 2

(*)

1_ ch ctf d I dip

TYPE ä.C.'s NEAR"" ^ ^ A P E X /

(10b)

Although the displacements and stress resultants o f the pressurization phase have been written in closed-form by Eqs. ( 1 ) , the integrals required in Eqs. ( 1 ) can only be evaluated if the specific problem dependent geometric function gj can also be expressed in closed-form. For the case o f a general shell of revolution with an arbitrary meridional contour such is not the case. Therefore, a numerical integration procedure was adopted in subroutine P R E S S U R to calculate N J J , N 2 2 , Up, and w . Special care had to be taken in the integration process to propagate the solution past the transition point if a mixed angular and axial coordinate system was adopted as discussed above.

CONTINUOUS

SET (U) ,

APPLY FROBENIUS

FOR EACH PARTIAL SOLUTION

TO D AT APEX FOR

luì- + < A U )

1 C A L C U L A T EE R R O R A (U>(u) = N t w -(U) 0 L D F R O M D.E s .'

c

(SID) = |U. U. U I T KOR ITH P A R T I A L S O L U T I O N

1

SUPPRESS A N D I N I T I A L I Z E FOR N E X T R E G I O N

)

U P D A T E L O C A T I O N OF S U P P R E S S I O N POINTS

T

Fig. 4 Flow Diagram of MARCH

EXAMPLE PROBLEMS In order to demonstrate the solution methods and computer program described in the preceding sections several particular problems were considered for inflatable shells in the fluid environment. Results have been reported elsewhere [4,5,6,7] for example static and dynamic problems involving spheres, paraboloids, and toroidal sections. The in-service static and dynamic behavior of hemispherical shells under constant internal pressure and externa] hydrostatic pressure have been considered.

Fig.5 Flow Diagram of Suppression Routine

In this section the pressurization phase and the static and dynamic in-service phase of a drop-shaped tank will be considered in detail. Such a tank, as described by Fliigge [1] in section 2.3.1 of the text Stresses in Shells, is perhaps ideally suited for subsurface applications. The tank geometry coincides with the shape that a drop of water could take resting on a plane surface. Fliigge considered the case of a tank filled with liquid and which therefore is subjected to constant tension in the skin. This corresponds to surface tension in the equivalent drop of water. If

132

this concept is reversed, one obtains the shape o f a shell surrounded by liquid with constant compression in the skin. This implies an optimal shape for a concrete tank on the ocean floor. Another reason for considering such a geometry, besides its optimal character for hydrostatic applications, is that this shape for a shell o f revolution involves a meridian contour in an axial coordinate system for which the closed-form equation o f the contour is unknown. Thus, it provides an excellent test of the capabilities of the computer program to treat arbitrary meridian contours. The geometry o f the drop-shaped tank is governed by two differential equations obtained from the equilibrium equations o f the shell. These equations are [ 1 ] d sin tp dr

sin ip _ 7 ( H + z ) r

sin ip 9 1 - sin

(11a)

CTT

dz

(lib)

dF

where 7 = unit weight of water,

.50 Q .70 0 90 I

•

MEMBRANE THICKNESS VARIATION

Z* = . 4 9 0

R* = . 5 2

H* AT 0 0ECREES - e.0514 H* AT 0 DEGREES - 1.4604 H * m O OECREES - l.ie4S

R O *

• .39A .41 O .43 © .45 • .47 Z

Z* = . 4 9 0

H* AT 0 H» fll D H* AT 0 H* AT 0 H* AT 0

DECREES DECREES DECREES DECREES OECREES

• •

E-66SE E.5C97 E-40E6 e.eS41 E.1910

Fig. 12

Fig. 11

From Fig. 11 it is apparent that the greater the value of R* the smaller the value of H*, and subsequently, from Eq. (25) the smaller the stress at zero degrees. Furthermore, since the variation of h / h m a x is smaller for larger R*, also noted from Fig. 11, the two-fold benefit of lower stress and reduced thickness variation is realized. Figure 12 confirms the indications of the first chart, and now the designer has the choice of five configurations of approximately the same W*/D, H* and h / h m a x variation. The process can be repeated for new values of Z* until the designer is satisfied. It must be noted that a comparison between the Lockheed Deep Quest and an equivalent rotation cyclide shell is somewhat artificial. The design of the submersible was based upon a buckling mode of failure whereas the preceding analysis postulates a strength failure. Valuable insight can be gained however. A structure similar to the Deep Quest but subject to internal pressurization would have thinner walls. The W*/D for such a shell is found to be 1.69. Comparison with the equivalent rotation cyclide which has a W*/D of 1.58 indicates that a reduction in weight of approximately 6 percent can be made by using the rotation cyclide geometry. Examination of Fig. 12 provides an explanation of the weight savings. The submersible has a step increase in thickness at the waist, but the constant stress rotation cyclide shows a steep but finite thickness gradient. The weight savings is achieved by adhering to the thickness variation of the equivalent rotation cyclide.

147

CONCLUSIONS Rotation cyclide shells have been shown to offer a degree of geometric flexibility previously unknown in underwater shell design. The membrane analysis of this new class of axisymmetric monocoque shells was presented in a form useful to designers of underwater pressure resistant structures. The next step in the development of rotation cyclides for undersea use is to evaluate their buckling resistance.

NOTATION a

shell size parameter

cn

Jacobi elliptic cosine function

dn

Jacobi elliptic delta function

D

volumetric displacement

h

shell wall thickness

h

/ h max H*

shell thickness variation

le

elliptic modulii describing shell shape

fl'V

v H'v

dimensionless constant relating shell thickness at apex to shell descriptors quarter period of elliptic functions integrated moment resultants

MrMe

integrated stress resultants uniform pressure loading

P

o rQ o

integrated shear resultant radial shell coordinate

max

maximum radius, located where r'=0

o r

radius at shell waist

r r

(1-R*2)^

R*

r

R*o sn

r 0 / r m a x , shell descriptor Jacobi elliptic sine function

W

shell weight

W/D

weight-to-displacement ratio

W*/D

material- and pressure-independent weight-to-displacement ratio

z

axial shell coordinate

z' || x„ _=n0 z| max Z*

axial length at shell apex; often maximum axial dimension axial coordinate of maximum radius

a

(r'2+z'2)*

f

shell thickness coordinate

e

shell circumferential coordinate

M

argument of elliptic function describing shell shape

v

argument of elliptic function; independent variable

m a x / z | x = 0 > s h e 1 1 descriptor

(lrmax)/(Z|x=0)

148

>sheUdescriPtor

Pm "w a

all X

density of shell wall material density of water allowable material stress transformed independent variable; shell meridional coordinate

0

inclination of tangent to shell meridian

a

l-sn 2 (fjt)dn 2 0xjc) prime denotes differentiation with respect to x

% vol

ratio of rotation cyclide shell volume to cylinder volume

BIBLIOGRAPHY 1•

Hagan, R.H., "Design and Fabrication of Pressure Hulls for Deep Diving Submersibles," presented at joint meeting SNAME and AWS, Philadelphia, Pennsylvania, Feb. 21, 1969.

2.

Maison, Jack R., Ph.D. Thesis, 1970, University of Delaware.

3.

Moon, P., and Spencer, D.E., Field Theory Handbook, Springer-Verlag, Berlin, 1961.

4.

Reissner, E., "On the Theory of Thin Elastic Shells," Reissner, U., Anniversary Volume, Edwards, Ann Arbor, Michigan, 1949.

5.

Wizansky, G., "High Pressure Technology in the Deep Quest Research Submersible," Preprint from Symposium on Advances in High Pressure Technology, AIChE, New Orleans, Mar. 16-20,1969.

149

UNDERSEA OIL STORAGE SYSTEM MODELS I AND II T. Hirose Koyo Iron Works and Construction Co., Ltd. Japan

H. Itokawa President Association of Marine Industry and Engineering Japan T. Ohira Executive Secretary Association of Marine Industry and Engineering Japan

M. Sakuta Fuyo Ocean Development and Engineering Co., Ltd. Japan

MODEL I ABSTRACT In the absence of any suitable on-shore site, the utilization of off-shore space as a storage base of imported crude oils presents one of the most promising alternative plans. We have designed a series of flexible undersea storage tanks all of flying saucer type. The main feature of this flexible type is its ability to prevent oil pollution because the tank deforms in accordance as it is being charged or discharged. This flexibility is achieved by the membrane structure in the upper half of the tank. We have carried out extensive experiments in Tokyo Bay using a one-tenth size model with major and minor axis diameters of 5 and 3 m, respectively. No problem was found in either charging or discharging operations. No oil leakage was found from either the connections of oil supply hose or the joint of tank membrane and steel structures. The tank deformation actually observed was in accordance with the design. The whole system performed satisfactorily.

INTRODUCTION Demand for petroleum in Japan has been showing remarkable growth almost every year. The present annual consumption of petroleum now exceeds a 200 million level. The rate of this growth shows no sign of reaching a peak level as yet. The storage site of imported crude oils needs to be expanded accordingly. Under the present domestic situations prevailing in Japan, it is almost impossible to find a huge storage site which includes a harbor large enough for the entering or leaving of a large tanker and at the same time is sufficiently close to the site of petroleum consumption. It is exactly for this reason that much attention is now being paid to the utilization of off-shore space as a storage base of crude oil. As part of the project for the construction of a storage tank to be installed on the sea floor, we have carried out the basic research and the design of the system USOS Model I and also its model test in Tokyo Bay.

STRUCTURE OF TANK The diameter of a one-tenth size model tank constructed was chosen as 5 m, assuming that the full scale tank of 50 m in diameter will be installed at a depth of 50-100 m. In order to reduce the horizontal component of flow resistance due to the horizontal currents such as ocean current, tidal current and tsunami, etc., we have chosen a flying saucer-like flattened spherical shape tank having minor axis diameter of 3 m. When the tank is installed within the ocean and the crude oil is either charged into or discharged from, the tank, an equivalent amount of sea water equivalent to the displaced oil must necessarily be replaced. If this is replaced directly, oily sea water will be discharged from the tank polluting the sea surface. To resolve this problem,

150

we have designed a flexible tank so that we can replace the sea water rather indirectly by changing the tank volume itself. The lower half of the tank below the major axis is made of steel shell plate but the upper half has a membrane structure of flexible material. This structure is quite effective for the stability of the flexible region for almost a complete range of tank charge from full to empty levels through an intermediate level. We installed a ring-shaped pontoon around the joint of the upper and lower sections which is capable of adjusting the buoyancy force of the tank as well as of reinforcing the tank structure against external pressure. The radius of a flexible membrane surrounding this joint allowing for some clearance is 25 m with the tank height of 4 m. Design Condition of Tank

Depth: - 8.5 m at the major axis of tank Content: Kerosene having specific gravity P2 = 0.78 Sea water: Specific gravity p j = 1.03 Dynamic pressure on tank: ignored in this experiment Pj = PjH + pjh

?2

=

P j H + pfi.

The external pressure on the tank P3 is given as P3 = Pj - P 2 = (pj - p 2 ) h Based on this information, the following material has been chosen for the tank. Lower half: Upper half:

3/16 in. steel plate 1.7 mmt membrane (Vinylon fabric coated with Nitrile rubber)

Mooring of tank The tank has a buoyancy force even in the absence of oil inside. Therefore, it is moored to the concrete anchor block by wire rope.

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Tank Nozzle Four inches o f tank nozzles are used for both charging and discharging purposes. The charge nozzle is installed near the major axis o f the tank in a downward position so that enough stirring is provided particularly near the tank bottom as it is charged. The discharge nozzle is located on the tank bottom.

PURPOSE OF E X P E R I M E N T 1.

T o obtain sufficient basic data for the design of an undersea oil storage system.

2.

To

examine methods o f measuring varying tank volumes under sea level and other various measurement

methods. 3.

Investigation

o f stress distribution on the tank due to the density difference o f sea water and crude oil

4.

Investigation of tension on anchor rope due to buoyancy force and tidal current

5.

Study o f anchor foundation

6.

Study

7.

Study o f placing methods o f tank

o f the joint structure between steel shell (lower half) and flexible membrane (upper half)

ITEMS OF E X P E R I M E N T 1. 2.

On-site tank assembly Installation of anchor block into ocean floor

3.

STRAIN GAUGE (FOR STEEL)

SWITCH BOX

RECORDER

Installation of tank into ocean floor

4.

STRAIN GAUGE / (FOR MEMBRANE) /

Ni

Undersea coupling to wire rope

PRESSURE GAUGES

o f anchor 5.

\

Undersea coupling of oil supply rubber hose and tank nozzle

6.

LOAD CELL

Coupling between pumping ship and rubber hose

7.

Charging or discharging opera-

RUBBER HOSE

tion of crude oil 8.

T

Observation o f flexible mem-

. O .1

>7i • • '

» • i 77777777)

/ n h

)~J "

7

r r \ r'\> ^TTYtttt'

WIRE ROPE rrnn7j7TTrn~

brane deformation during oil charging or discharging 9.

Various measurements ACTIVE GAUGE

Instruments used: Strain gauge for membrane (upper part)

DUMMY

Strain gauge for membrane (lower part) Buoyancy meter (load cell) Pressure gauge

152

GAUGE

er u *

z

et

o

3

000*21

n

io

o

o

? in O i/> D

t t/1 0 ui 3

153

RESULTS Because it is not permitted to carry out the experiment using petroleum in Tokyo Bay due to an actual danger of oil pollution, we have used instead colored sea water. Because of this, we were not able to measure the effect of the density difference of sea water and oil on the tank as we originally set out to do. Except for this inevitable omission, we obtained useful results. Leakage

EMPTY

No leakage was observed on the tank joint between upper and lower tank nor on any coupling joints of hoses, etc. Deformation of Upper Membrane

Records of Buoyancy Meter

FULL

yywoo FULL

The deformation of membrane in charging or discharging tank turned out to be as we expected.

•

DISCHARGE

Oòó

•

EMPTY

WW

The records of buoyancy meter did show fluctuations in buoyancy forces. We believe that these are due to the waves. The following factors cannot be completely ruled out for this fluctuation at this stage however. 1.

Difference of resistance coefficients at full and empty tank levels.

2.

The relation of wave effect and the depth

3.

Correlation of amplitude and phase between wave and buoyancy fluctuations.

FUTURE STUDY Many problems remain unresolved in the development work of undersea oil storage systems including measures against marine life or marine mosses, method of corrosion proof, measures for sludges and drain, detection method of oil leakage, development of volume meter, undersea construction method and control of oil charging or discharging operations, etc. If we plan to build a large capacity oil storage base on a larger scale, countermeasures for oil leakage and the design of a much larger tank become even more urgent. Model USOS 2 is the offspring of such need and will be introduced next.

NOTATION h

height of the tank in meters

H

height above the tank in meters

Pj

pressure in the sea water in kg/cm 2

P2

pressure within the tank in kg/cm 2

P3

external pressure on the tank in kg/cm 2

P\

specific gravity of sea water, gr/cm^

P2

specific gravity of kerosene, gr/cm^

154

MODEL II In Japan we have the direct crude oil sea transportation systems to the consumer from the oil production field, and the refinery plants system are close to the consumers, generally. The crude oil consumption which is mostly imported in Japan increased rapidly in recent years to 2 x 10^ K£ in 1970 and will increase to 3 x 1 0 ^ KC, and 7 x 10^ K2 in 1975 and 1985, respectively. We have 39 refinery plants with capacities of 510,000 K£/day and each plant has small storage tank facilities and each loading and unloading ones to and from oil tankers (2-5 x 10 5 D.W.T., 2-5 x 10 2 D.W.T.). Because tanker sizes become larger, and deeper depths are required in berths (20M-50M depth),because refinery plants become the center of oil chemical "KOMBINAT" and of consumers, and because safety of water crafts course, efficient loading, and balanced stock holding for national economy and national energy sources are required, we have to construct large volume storage tank yards (tanks group), called CTS (Central Terminal Station) in Japan today. These CTS sites on seashores have open ocean bay, and have deep depths, as mentioned before (Fig. 3). The preliminary and feasibility studies are being done now in Japan by many firms.

Association of Marine Industry and Engineering of Japan Undersea (Underwater) O i l Storage System Study Team Marubeni-lida C o . , L t d . Shumitomo Shoji C o . , Ltd. Taiyo Kogyo C o . , Ltd. Koyo Iron Works and Construction C o . , L t d . Hiraoka Shokusen C o . , Ltd. Taisei Construction C o . , Ltd. Asahi Glass C o . , L t d . Tokyo Kaiki C o . , L t d . F u j i O i l C o . , Ltd. Nippon Kokan C o . , L t d . Tokyo Electric C o . , Ltd. Kansai Electric C o . , L t d . Chubu Electric C o . , L t d . Systems Engineering Institute

Crude oil storage systems are, by positioning or leveling, classified: (1) on land, (2) in land, (3) on sea, (4) in sea, and (5) in sea bed. USOS Model II is type (3), on sea, and Model I is type (4), in sea.

PLANNING AND LAYOUT Our USOS study team began some preliminary and feasibility studies for these CTS systems in 1967 and ended with USOS Model I in 1968. We are now studying USOS Models II and III on shallow sea bed, with fixed rigid cofferdams and storage in these enclosed dams with steel tanks skin floating on the closed sea water. Model II design conditions are shown in Table I and Fig. 6 and we check some feasibilities on Model II in Shibushi Bay in Kyushu Island (south) in Japan. A flowchart of the total system is shown in Fig. 7.

STRUCTURES This type of crude oil storage tanks are set in enclosed cofferdams on sea and these cofferdam structures are used as protectors against both wave and oil, completely shut and separate from open sea influence. The tanks, made 155

Fig. 1 Refinery Plants Sites and Capacities (Unit: KL/D)

156

X 108KL

^ ^

S I ORAGE TANK V O L U M E S / / / / / / /

77777777/T////////////// V////////////////////, 1968

1970 Fig. 2

1975

Estimation of Oil Consumption in the Nations (Imported Oil Volumes in Japan)

157

1980

© Fig. 3 CTS Project Sites

158

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oo CO

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o

C/3 1-. « ra >/")

SP iS

160

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