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Computational Tensor Analysis of Shell Structures PDF

320 Pages·1990·14.572 MB·English
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Lecture Notes in Engineering The Springer-Verlag Lecture Notes provide rapid (approximately six months), refereed publication of topical items, longer than ordinary journal articles but shorter and less formal than most monographs and textbooks. They are published in an attractive yet economical format; authors or editors provide manuscripts typed to specifications, ready for photo-reproduction. The Editorial Board Managing Editors C. A Brebbia S.AOrszag Wessex Institute of Technology Applied and Computational Mathematics Ashurst Lodge, Ashurst 218 Fine Hall Southampton S04 2AA (UK) Princeton, NJ 08544 (USA) Consulting Editors Materials Science and Computer Simulation: S. Yip Chemical Engineering: Dept of Nuclear Engg., MIT J. H. Seinfeld Cambridge, MA 02139 (USA) Dept. of Chemical Engg., Spaulding Bldg. Calif. Inst. ofTechnology Mechanics of Materials: Pasadena, CA 91125 (USA) FA Leckie Dept. of Mechanical Engineering Dynamics and Vibrations: Univ. of California P'Spanos Santa Barbara. Department of Mechanical and CA 93106 (USA) Civil Engineering, Rice University A R. S. Ponter P. O. Box 1892 Dept. of Engineering, The University Houston, Texas 77251 (USA) Leicester LE1 7RH (UK) Earthquake Engineering: Fluid Mechanics: AS. Cakmak K.-P' Holz Dept. of Civil Engineering, Princeton University Inst. fUr Stromungsmechanik, Princeton, NJ 08544 (USA) Universitat Hannover, Callinstr. 32 0-3000 Hannover 1 (FRG) Electrical Engineering: P. Silvester Nonlinear Mechanics: Dept. of Electrical Engg., McGill University K.-J. Bathe 3480 University Street Dept. of Mechanical Engg., MIT Montreal, PO H3A 2A7 (Canada) Cambridge, MA 02139 (USA) Geotechnical Engineering and Geomechanics: Structural Engineering: C.S. Desai J. Connor College of Engineering Dept of Civil Engineering, MIT Dept. of Civil Engg. and Engg. Mechanics Cambridge, MA 02139 (USA) The University of Arizona W. Wunderlich Tucson, AZ 85721 (USA) Inst fUr Konstruktiven Ingenieurbau Ruhr-Universitat Bochum Hydrology: Universitatsstr.150, G.Pinder 0-4639 Bochum-Ouerenburg (FRG) School of Engineering, Dept. of Civil Engg. Princeton University Structural Engineering, Fluids and Princeton, NJ 08544 (USA) Thermodynamics: J. Argyris Laser Fusion - Plasma: Inst fUr· Statik und Dynamik der R. McCrory Luft-und Raumfahrtkonstruktion Lab. for Laser Energetics, University of Rochester Pfaffenwaldring 27 Rochester, NY 14627 (USA) 0-7000 Stuttgart 80 (FRG) Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag 58 S. Naomis, P. C. M. Lau Computational Tensor Analysis of Shell Structures Springer-Verlag Berlin Heidelberg New York London '--...~ ""-----' Paris Tokyo Hong Kong Barcelona Series Editors C. A. Brebbia . S. A. Orszag Consulting Editors J. Argyris . K.-J. Bathe· A. S. Cakmak . J. Connor· R. McCrory C. S. Desai· K. -Po Holz . F. A. Leckie· G. Pinder· A. R. S. Pont J. H. Seinfeld . P. Silvester· P. Spanos· W. Wunderlich· S. Yip Authors Dr. Steve Naomis Dr. Paul C. M. Lau Civil Engineering Department University of Western Australia Nedlands, WA 6009 Australia ISBN-13:978-3-540-52852-4 e-ISBN-13:978-3-642-84243-6 001: 10.1007/978-3-642-84243-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1990 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 2161/3020-543210 Printed on acid-free paper. PREFACE This book presents a method which is capable of evaluating the deformation characteristics of thin shell structures A free vibration analysis is chosen as a convenient means of studying the displacement behaviour of the shell, enabling it to deform naturally without imposing any particular loading conditions. The strain-displacement equations for thin shells of arbitrary geometry are developed. These relationships are expressed in general curvilinear coordinates and are formulated entirely in the framework of tensor calculus. The resulting theory is not restricted to shell structures characterized by any particular geometric form, loading or boundary conditions. The complete displacement and strain equations developed by Flugge are approximated by the curvilinear finite difference method and are applied to computing the natural frequencies and mode shapes of general thin shells. This approach enables both the displacement components and geometric properties of the shell to be approximated numerically and accurately. The selection of an appropriate displacement field to approximate the deformation of the shell within each finite difference mesh is discussed in detail. In addition, comparisons are made between the use of second and third-order finite difference interpolation meshes. The method is applied to a series of flat plate, cylindrical, spherical and conical shell structures to verify its accuracy. The computed natural frequencies and their associated mode shapes are documented and compared with published results. IV The authors wish to acknowledge the supports provided by the Civil Engineering Department, University of Western Australia to the development of the computational tensor analysis of shell structures. TABLE OF CONTENTS NOTATION IX 1 . INTRODUCTION 1 2. GENERAL THEORY 9 2.1 A Summary of the Tensorial Quantities Required in the Formulation of a Shell Theory 11 . 2.1.1 Base Vectors 11 2.1.2 Metric Tensors 12 2.1.3 Coordinate Transfonnations 15 2.1.4 Christoffel Symbols 18 2.1.5 Covariant Derivatives 19 2.2 Surface Geometry 22 2.2.1 Curvilinear Coordinates on a Surface 22 2.2.2 Geometry of a Curved Surface 22 2.3 The Strain Tensor 30 2.4 The Stress Tensor 33 2.5 The Constitutive Equations 37 2.6 The Theory of Shells 40 2.6.1 Shell Geometry 40 2.6.2 Deformation Characteristics 43 VI 2.6.3 The Change in Curvature Tensor 47 2.6.4 The Strain-Displacement Equations 50 2.6.5 Interpretation and Discussion of dIe Strain- Displacement Equations 56 3. NUMERICAL FUNDAMENTALS 60 3.1 The Curvilinear Finite Difference Method 62 3.2 The Numerical Computation or the Surface Geometric Quantities 69 3.2.1 Base Vectors and Metric Tensors 70 3.2.2 The Christoffel Symbol 73 3.2.3 The Curvatufe Tensor 74 3.2.4 Covariant Derivative of the Curvature Tensor 75 3.3 The Principle or Virtual Displacements 77 3.4 Discretization and Displacement Fields 78 3.5 The Numerical Implementation or the General Surface Stress, Strain and Displacement Components 85 3.5.1 The General Surface Displacement Components 86 3.5.2 The General Strain Tensor 87 3.5.3 The General Stress Tensor 89 3.6 Boundary Conditions 91 3.7 The Numerical Solution of the Eigenvalue Problem 98 VII 4. NUMERICAL IMPLEMENTATION 101 4.1 A Second Order Implementation 103 4.1.1 Second Order CFD Approximation 103 4.1.2 Numerical Integration Scheme 108 4.1.3 Numerical Examples 116 4.2 A Third Order Implementation 146 4.2.1 Third Order CFD Approximation 146 4.2.2 Boundary Conditions 151 5. NUMERICAL APPLICATIONS 157 5.1 Simply Supported Plate 158 5.2 Cantilever Plate 161 5.3 Spherical Cap on a Square Base 164 5.4 Cylindrical Panel 170 5.5 Curved Fan Blade 173 5.6 Conical Shell Panel 176 5.7 Cylindrical Tank 179 6. SUMMARY 182 REFERENCES 184 APPENDIX A: Displacement Transformations 196 APPENDIX B: 'Finite Difference Expressions 201 VIII APPENDIX C: Numerical Integration of the Stiffness Matrix 233 APPENDIX D: Application of the CFD method to the Analysis of Beam Bending Problems with Fixed Edges 240 APPENDIX E: Transformation of the Generalized Eigenvalue Problem to Standard Form 250 APPENDIX F: Numerical Results 252 Simply supported plate 253 Cantilever plate 253 Curved fan blade 267 Spherical shell 274 Cylindrical panel 281 Cylindrical tank 288 Conoidal shell 295 SUBJECT INDEX 302 NOTATION General [ ] Matrix notation. { } Vector notation. Partial differentiation with respect to the surface coordinates. i Covariant differentiation with respect to the coordinates Oil Covariant differentiation with respect to the surface u coordinates xu. Time derivative. Transpose of a matrix. Inverse of a matrix. Structure's stiffness matrix Structure's mass matrix. Two dimensional region. Boundary curve. r. Integration point for a Gauss-Legendre numerical integration I scheme. W. Weight factors for a Gauss-Legendre numerical integration I scheme. Coordinates of a general three-dimensional curvilinear coordinate system. Coordinates of a general two-dimensional curvilinear coordinate system on the shell's middle surface. Coordinate axis normal to xu. &i Kronecker delta. I Eigenvalue.

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