J._N._Reddy-Mechanics_of_Laminated_Composite_Plates_and_Shells__Theory_and_Analysis__Second_Edition-CRC_Press_(2003).pdf PDF

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ME[HANI[S of LAMINATED [OMPOSITE PLATES and SHELLS Theory and Analysis SECOND EDITION -.1.N. REDDY CRC PRESS Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Pnblication Data Reddy, J, N. (Junuthula Narasimha), 1945- Mechanics of laminated composite plates and shells: t...

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ME[HANI[S of LAMINATED [OMPOSITE PLATES and SHELLS Theory and Analysis SECOND

EDITION

-.1.N. REDDY

CRC PRESS Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Pnblication Data Reddy, J, N. (Junuthula Narasimha), 1945Mechanics of laminated composite plates and shells: theory and analysis I J.N. Reddy.2nd ed. p. cm. Rev. ed. of: Mechanics of laminated composite plates. c 1997. Includes bibliographical references and index. ISBN 0-8493-1592-1 (alk. paper) I. Plates (Engineering) - Mathematical models. 2. Shells (Engineering) - Mathematical models. 3. Laminated materials-Mechanical properties-Mathematical models. 4. Composite materials-Mechanical properties-Mathematical models. I. Reddy, J. N. (Junuthula Narasimha), 1945-. Mechanics of laminated composite plates. II. Title. TA660.P6R42 2003 624.1'7765 -dc22

2003061067

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Visit the CRC Press Web site at www.crcpress.com © 2004 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-1592-1 Library of Congress Card Number 2003061067 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

To the Memory of

My parents, My brother, My brother in-law, My father in-law, Hans Eggers, Kalpana Chawla, ...

About the Author J. N. Reddy is a Distinguished Professor and the inaugural holder of the Oscar S. Wyatt Endowed Chair in the Department of Mechanical Engineering at Texas A&M University, College Station, Texas. Prior to his current position, he worked as a postdoctoral fellow at the University of Texas at Austin (1973-74), as a research scientist for Lockheed Missiles and Space Company (1974), and taught at the University of Oklahoma (1975-1980) and Virginia Polytechnic Institute and State University (1980-1992), where he was the inaugural holder of the Clifton C. Garvin Endowed Professorship. Professor Reddy is the author of over 300 journal papers and 13 text books on theoretical formulations and finite-element analysis of problems in solid and structural mechanics (plates and shells), composite materials, computational fluid dynamics and heat transfer, and applied mathematics. His contributions to mechanics of composite materials and structures are well known through his research on refined plate and shell theories and their finite element models. Professor Reddy is the first recipient of the University of Oklahoma College of Engineering's Award for Outstanding Faculty Achievement in Research, the 1984 Walter L. Huber Civil Engineering Research Prize of the American Society of Civil Engineers (ASCE), the 1985 Alumni Research Award at Virginia Polytechnic Institute, and 1992 Worcester Reed Warner Medal and 1995 Charles Russ Richards Memorial Award of the American Society of Mechanical Engineers (ASME). He received German Academic Exchange (DAAD) and von Humboldt Foundation (Germany) research awards. Recently, he received the 1997 Melvin R. Lohmann Medal from Oklahoma State University's College of Engineering, Architecture and Technology, the 1997 Archie Higdon Distinguished Educator Award from the Mechanics Division of the American Society of Engineering Education, the 1998 Nathan M. Newmark Medal from the American Society of Civil Engineers, the 2000 Excellence in the Field of Composites Award from the American Society of Composite Materials, the 2000 Faculty Distinguished Achievement Award for Research, the 2003 Bush Excellence Award for Faculty in International Research award from Texas A&M University, and 2003 Computational Structural Mechanics Award from the U.S. Association for Computational Mechanics. Professor Reddy is a fellow of the American Academy of Mechanics (AAM), the American Society of Civil Engineers (ASCE), the American Society of Mechanical Engineers (ASME), the American Society of Composites (ASC), International Association of Computational Mechanics (IACM), U.S. Association of Computational Mechanics (USACM), the Aeronautical Society of India (ASI), and the American Society of Composite Materials. Dr. Reddy is the Editor-in-Chief of the journals Mechanics of Advanced Materials and Structures (Taylor and Francis), International Journal of Computational Engineering Science and International Journal Structural Stability and Dynamics (both from World Scientific), and he serves on the editorial boards of over two dozen other journals.

Contents

Preface to the Second Edition ............................................ xix Preface to the First Edition ............................................... xxi 1 Equations of Anisotropic Elasticity, Virtual Work Principles, and Variational Methods ................................................. 1 l.1 Fiber-Reinforced Composite Materials .................................. 1 l.2 Mathematical Preliminaries ............................................. 3 l.2.1 General Comments .................... '" ........................ 3 1.2.2 Vectors and Tensors .............................................. 3 l.3 Equations of Anisotropic Entropy ...................................... 12 l.3.1 Introduction .................................................... 12 1.3.2 Strain-Displacement Equations .................................. 13 l.3.3 Strain Compatibility Equations .................................. 18 1.3.4 Stress Measures ................................................. 18 l.3.5 Equations of Motion ............................................ 19 1.3.6 Generalized Hooke's Law ........................................ 22 l.3.7 Thermodynamic Principles ...................................... 34 1.4 Virtual Work Principles ............................................... 38 1.4.1 Introduction .................................................... 38 1.4.2 Virtual Displacements and Virtual Work ........................ 38 l.4.3 Variational Operator and Euler Equations ....................... 40 l.4.4 Principle of Virtual Displacements ............................... 44 1.5 Variational Methods ................................................... 58 l.5.1 Introduction .................................................... 58 l.5.2 The Ritz Method ............................................... 58 1.5.3 Weighted-Residual Methods ..................................... 64 1.6 Summary ............................................................. 71 Problems ................................................................. 72 References for Additional Reading ......................................... 78 2 Introduction to Composite Materials .................................. 81 2.1 Basic Concepts and Terminology ...................................... 81 2.l.1 Fibers and Matrix ........... '" ...................... , ......... 81 2.l.2 Laminae and Laminates ......................................... 83 2.2 Constitutive Equations of a Lamina ................................... 85 2.2.1 Generalized Hooke's Law ........................................ 85 2.2.2 Characteristics of a Unidirectional Lamina ....................... 86

x

CONTENTS

2.3 Transformation of Stresses and Strains ................................. 89 2.3.1 Coordinate Transformations ..................................... 89 2.3.2 Transformation of Stress Components ........................... 90 2.3.3 Transformation of Strain Components ........................... 93 2.3.4 Transformation of Material Coefficients .......................... 96 2.4 Plan Stress Constitutive Relations ..................................... 99 Problems ................................................................ 103 References for Additional Reading ........................................ 106

3 Classical and First-Order Theories of Laminated Composite Plates ....................................................... 109 3.1 Introduction ......................................................... 109 3.1.1 Preliminary Comments ......................................... 109 3.1.2 Classification of Structural Theories ............................ 109 3.2 An Overview of Laminated Plate Theories ............................ 110 3.3 The Classical Laminated Plate Theory ................................ 112 3.3.1 Assumptions ................................................... 112 3.3.2 Displacements and Strains ..................................... 113 3.3.3 Lamina Constitutive Relations ................................. 117 3.3.4 Equations of Motion ........................................... 119 3.3.5 Laminate Constitutive Equations ............................... 127 3.3.6 Equations of Motion in Terms of Displacements ................ 129 3.4 The First-Order Laminated Plate Theory ............................. 132 3.4.1 Displacements and Strains ..................................... 132 3.4.2 Equations of Motion ........................................... 134 3.4.3 Laminate Constitutive Equations ............................... 137 3.4.4 Equations of Motion in Terms of Displacements ................ 139 3.5 Laminate Stiffnesses for Selected Laminates ........................... 142 3.5.1 General Discussion ............................................. 142 3.5.2 Single-Layer Plates ............................................. 144 3.5.3 Symmetric Laminates .......................................... 148 3.5.4 Antisymmetric Laminates ...................................... 152 3.5.5 Balanced and Quasi-Isotropic Laminates ....................... 156 Problems ................................................................ 157 References for Additional Reading ........................................ 161 4 One-Dimensional Analysis of Laminated Composite Plates ......... 165 4.1 Introduction ......................................................... 165 4.2 Analysis of Laminated Beams Using CLPT ........................... 167 4.2.1 Governing Equations ........................................... 167 4.2.2 Bending ....................................................... 169 4.2.3 Buckling ....................................................... 176 4.2.4 Vibration ...................................................... 182

CONTENTS

Xl

4.3 Analysis of Laminated Beams Using FSDT ........................... 187 4.3.1 Governing Equations ........................................... 187 4.3.2 Bending ....................................................... 188 4.3.3 Buckling ....................................................... 192 4.3.4 Vibration ...................................................... 197 4.4 Cylindrical Bending Using CLPT ..................................... 200 4.4.1 Governing Equations ........................................... 200 4.4.2 Bending ....................................................... 203 4.4.3 Buckling ....................................................... 208 4.4.4 Vibration ...................................................... 209 4.5 Cylindrical Bending Using FSDT ..................................... 214 4.5.1 Governing Equations ........................................... 214 4.5.2 Bending ....................................................... 215 4.5.3 Buckling ....................................................... 216 4.5.4 Vibration ...................................................... 219 4.6 Vibration Suppression in Beams ...................................... 222 4.6.1 Introduction ................................................... 222 4.6.2 Theoretical Formulation ........................................ 222 4.6.3 Analytical Solution ............................................ 227 4.6.4 Numerical Results ............................................. 230 4.7 Closing Remarks ..................................................... 232 Problems ................................................................ 232 References for Additional Reading ........................................ 242

5 Analysis of Specially Orthotropic Laminates Using CLPT .......... 245 5.1 Introduction ......................................................... 245 5.2 Bending of Simply Supported Rectangular Plates ..................... 246 5.2.1 Governing Equations ........................................... 246 5.2.2 The Navier Solution ........................................... 247 5.3 Bending of Plates with Two Opposite Edges Simply Supported ....... 255 5.3.1 The Levy Solution Procedure .................................. 255 5.3.2 Analytical Solutions ............................................ 257 5.3.3 Ritz Solution .................................................. 262 5.4 Bending of Rectangular Plates with Various Boundary Conditions .... 265 5.4.1 Virtual Work Statements ....................................... 265 5.4.2 Clamped Plates ................................................ 266 5.4.3 Approximation Functions for Other Boundary Conditions ....... 269 5.5 Buckling of Simply Supported Plates Under Compressive Loads ....... 271 5.5.1 Governing Equations ........................................... 271 5.5.2 The Navier Solution ........................................... 272 5.5.3 Biaxial Compression of a Square Laminate (k = 1) .... '" ...... 273 5.5.4 Biaxial Loading of a Square Laminate .......................... 274 5.5.5 Uniaxial Compression of a Rectangular Laminate (k = 0) ....... 274

xii

CONTENTS

5.6 Buckling of Rectangular Plates Under In-Plane Shear Load ........... 278 5.6.1 Governing Equation ............................................ 278 5.6.2 Simply Supported Plates ....................................... 278 5.6.3 Clamped Plates ................................................ 280 5.7 Vibration of Simply Supported Plates ................................ 282 5.7.1 Governing Equations ........................................... 282 5.7.2 Solution ....................................................... 282 5.8 Buckling and Vibration of Plates with Two Parallel Edges Simply Supported ................................................... 285 5.8.1 Introduction ................................................... 285 5.8.2 Buckling by Direct Integration ................................. 287 5.8.3 Vibration by Direct Integration ................................ 288 5.8.4 Buckling and Vibration by the State-Space Approach ........... 288 5.9 Transient Analysis ................................................... 290 5.9.1 Preliminary Comments ......................................... 290 5.9.2 Spatial Variation of the Solution ............................... 290 5.9.3 Time Integration ............................................... 292 5.10 Closure ............................................................. 293 Problems ................................................................ 293 References for Additional Reading ........................................ 296 6 Analytical Solutions of Rectangular Laminated Plates Using CLPT ............................................................. 297 6.1 Governing Equations in Terms of Displacements ...................... 297 6.2 Admissible Boundary Conditions for the Navier Solutions ............. 299 6.3 Navier Solutions of Antisymmetric Cross-Ply Laminates .............. 301 6.3.1 Boundary Conditions .......................................... 301 6.3.2 Solution ....................................................... 304 6.3.3 Bending ....................................................... 308 6.3.4 Determination of Stresses ...................................... 309 6.3.5 Buckling ....................................................... 317 6.3.6 Vibration ...................................................... 323 6.4 Navier Solutions of Antisymmetric Angle-Ply Laminates .............. 326 6.4.1 Boundary Conditions .......................................... 326 6.4.2 Solution ....................................................... 328 6.4.3 Bending ....................................................... 329 6.4.4 Determination of Stresses ...................................... 330 6.4.5 Buckling ....................................................... 335 6.4.6 Vibration ...................................................... 337 6.5 The Levy Solutions ................................................... 339 6.5.1 Introduction ................................................... 339 6.5.2 Solution Procedure ............................................. 342 6.5.3 Antisymmetric Cross-Ply Laminates ............................ 348 6.5.4 Antisymmetric Angle-Ply Laminates ........................... 353

CONTENTS

xiii

6.6 Analysis of Midplane Symmetric Laminates ........................... 356 6.6.1 Introduction ................................................... 356 6.6.2 Governing Equations ........................................... 356 6.6.3 Weak Forms ................................................... 357 6.6.4 The Ritz Solution .............................................. 358 6.6.5 Simply Supported Plates ....................................... 358 6.6.6 Other Boundary Conditions .................................... 360 6.7 Transient Analysis ................................................... 361 6.7.1 Preliminary Comments ......................................... 361 6.7.2 Equations of Motion ........................................... 361 6.7.3 Numerical Time Integration .................................... 362 6.7.4 Numerical Results ............................................. 364 6.8 Summary ............................................................ 371 Problems ................................................................ 371 References for Additional Reading ........................................ 375

7 Analytical Solutions of Rectangular Laminated Plates Using FSDT ............................................................. 377 7.1 Introduction ......................................................... 377 7.2 Simply Supported Antisymmetric Cross-Ply Laminated Plates ........ 379 7.2.1 Solution for the General Case .................................. 379 7.2.2 Bending ....................................................... 381 7.2.3 Buckling ....................................................... 388 7.2.4 Vibration ...................................................... 394 7.3 Simply Supported Antisymmetric Angle-Ply Laminated Plates ........ 400 7.3.1 Boundary Conditions .......................................... 400 7.3.2 The Navier Solution ........................................... 402 7.3.3 Bending ....................................................... 404 7.3.4 Buckling ....................................................... 405 7.3.5 Vibration ...................................................... 406 7.4 Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported ............................................ .412 7.4.1 Introduction ................................................... 412 7.4.2 The Levy Type Solution ....................................... 413 7.4.3 Num...