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Mechanics of Laminated Composite Plates and Shells Theory and Analysis J. N. Reddy Department of Mechanical Engineering Texas A&M University College Station, Texas, USA 77843 Second Edition CRC Press Boca Raton New York London Tokyo

Mechanics of Laminated Composite Plates and Shells · 2017-05-28 · Mechanics of Laminated Composite Plates and Shells Theory and Analysis J. N. Reddy Department of Mechanical Engineering

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Mechanics of LaminatedComposite Plates and Shells

Theory and Analysis

J. N. ReddyDepartment of Mechanical Engineering

Texas A&M UniversityCollege Station, Texas, USA 77843

Second Edition

CRC Press

Boca Raton � New York � London � Tokyo

insert

Library of Congress Data

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 OscarS. Wyatt Endowed Chair in the Department of Mechanical Engineering at TexasA&M University, College Station, Texas. Prior to his current position, he workedas a postdoctoral fellow at the University of Texas at Austin (1973�74), as aresearch scientist for Lockheed Missiles and Space Company (l974), and taughtat the University of Oklahoma (1975�1980) and Virginia Polytechnic Institute andState 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 Þnite-element analysis of problems in solid andstructural mechanics (plates and shells), composite materials, computational ßuiddynamics and heat transfer, and applied mathematics. His contributions tomechanics of composite materials and structures are well known through his researchon reÞned plate and shell theories and their Þnite element models.Professor Reddy is the Þrst recipient of the University of Oklahoma College

of Engineering�s Award for Outstanding Faculty Achievement in Research, the1984 Walter L. Huber Civil Engineering Research Prize of the American Society ofCivil Engineers (ASCE), the 1985 Alumni Research Award at Virginia PolytechnicInstitute, and 1992 Worcester Reed Warner Medal and 1995 Charles Russ RichardsMemorial Award of the American Society of Mechanical Engineers (ASME). Hereceived German Academic Exchange (DAAD) and von Humboldt Foundation(Germany) research awards. Recently, he received the 1997 Melvin R. LohmannMedal from Oklahoma State University�s College of Engineering, Architectureand Technology, the 1997 Archie Higdon Distinguished Educator Award from theMechanics Division of the American Society of Engineering Education, the 1998Nathan M. Newmark Medal from the American Society of Civil Engineers, the2000 Excellence in the Field of Composites Award from the American Societyof Composite Materials, the 2000 Faculty Distinguished Achievement Award forResearch, the 2003 Bush Excellence Award for Faculty in International Researchaward from Texas A&M University, and 2003 Computational Structural MechanicsAward 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 ofMechanical Engineers (ASME), the American Society of Composites (ASC),International Association of Computational Mechanics (IACM), U.S. Association ofComputational Mechanics (USACM), the Aeronautical Society of India (ASI), andthe American Society of Composite Materials. Dr. Reddy is the Editor-in-Chief ofthe journals Mechanics of Advanced Materials and Structures (Taylor and Francis),International Journal of Computational Engineering Science and InternationalJournal Structural Stability and Dynamics (both from World ScientiÞc), and heserves 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

1.1 Fiber-Reinforced Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.2.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31.2.2 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

1.3 Equations of Anisotropic Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Strain-Displacement Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.3 Strain Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181.3.4 Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.6 Generalized Hooke�s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221.3.7 Thermodynamic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 Virtual Work Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4.2 Virtual Displacements and Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . 381.4.3 Variational Operator and Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 401.4.4 Principle of Virtual Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

1.5 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .581.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.5.2 The Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.5.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.1.1 Fibers and Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.1.2 Laminae and Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.2 Constitutive Equations of a Lamina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.2.1 Generalized Hooke�s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .852.2.2 Characteristics of a Unidirectional Lamina . . . . . . . . . . . . . . . . . . . . . . .86

x CONTENTS

2.3 Transformation of Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .892.3.1 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .892.3.2 Transformation of Stress Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.3.3 Transformation of Strain Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.3.4 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 LaminatedComposite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093.1.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1093.1.2 ClassiÞcation of Structural Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.2 An Overview of Laminated Plate Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.3 The Classical Laminated Plate Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1123.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1123.3.2 Displacements and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.3.3 Lamina Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.3.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.3.5 Laminate Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1273.3.6 Equations of Motion in Terms of Displacements . . . . . . . . . . . . . . . . 129

3.4 The First-Order Laminated Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1323.4.1 Displacements and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.4.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.4.3 Laminate Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1373.4.4 Equations of Motion in Terms of Displacements . . . . . . . . . . . . . . . . 139

3.5 Laminate Stiffnesses for Selected Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . .1423.5.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1423.5.2 Single-Layer Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1443.5.3 Symmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483.5.4 Antisymmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523.5.5 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1674.2.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.2.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1764.2.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

CONTENTS xi

4.3 Analysis of Laminated Beams Using FSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1874.3.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.3.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1924.3.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

4.4 Cylindrical Bending Using CLPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2004.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2004.4.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.4.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2084.4.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

4.5 Cylindrical Bending Using FSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2144.5.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.5.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2164.5.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.6 Vibration Suppression in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2224.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2224.6.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2224.6.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.6.4 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 . . . . . . . . . . . . . . . . . . . . . 2465.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2465.2.2 The Navier Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

5.3 Bending of Plates with Two Opposite Edges Simply Supported . . . . . . . 2555.3.1 The Levy Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.3.2 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2575.3.3 Ritz Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5.4 Bending of Rectangular Plates with Various Boundary Conditions . . . . 2655.4.1 Virtual Work Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2655.4.2 Clamped Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2665.4.3 Approximation Functions for Other Boundary Conditions . . . . . . .269

5.5 Buckling of Simply Supported Plates Under Compressive Loads . . . . . . . 2715.5.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2715.5.2 The Navier Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.5.3 Biaxial Compression of a Square Laminate (k = 1) . . . . . . . . . . . . . 2735.5.4 Biaxial Loading of a Square Laminate . . . . . . . . . . . . . . . . . . . . . . . . . . 2745.5.5 Uniaxial Compression of a Rectangular Laminate (k = 0) . . . . . . . 274

xii CONTENTS

5.6 Buckling of Rectangular Plates Under In-Plane Shear Load . . . . . . . . . . . 2785.6.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2785.6.2 Simply Supported Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.6.3 Clamped Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280

5.7 Vibration of Simply Supported Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2825.7.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2825.7.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

5.8 Buckling and Vibration of Plates with Two Parallel EdgesSimply Supported . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2855.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2855.8.2 Buckling by Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2875.8.3 Vibration by Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2885.8.4 Buckling and Vibration by the State-Space Approach . . . . . . . . . . .288

5.9 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2905.9.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2905.9.2 Spatial Variation of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2905.9.3 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292

5.10 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .296

6 Analytical Solutions of Rectangular Laminated PlatesUsing 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 . . . . . . . . . . . . . . 3016.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.3.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3046.3.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3086.3.4 Determination of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.3.5 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3176.3.6 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.4 Navier Solutions of Antisymmetric Angle-Ply Laminates . . . . . . . . . . . . . . 3266.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.4.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3286.4.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3296.4.4 Determination of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3306.4.5 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3356.4.6 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

6.5 The Levy Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3396.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.5.2 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3426.5.3 Antisymmetric Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . .3486.5.4 Antisymmetric Angle-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

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6.6 Analysis of Midplane Symmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . .3566.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3566.6.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3566.6.3 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.6.4 The Ritz Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3586.6.5 Simply Supported Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3586.6.6 Other Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

6.7 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.7.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3616.7.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.7.3 Numerical Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3626.7.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .375

7 Analytical Solutions of Rectangular Laminated PlatesUsing FSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

7.2 Simply Supported Antisymmetric Cross-Ply Laminated Plates . . . . . . . . 3797.2.1 Solution for the General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3797.2.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.2.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3887.2.4 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

7.3 Simply Supported Antisymmetric Angle-Ply Laminated Plates . . . . . . . . 4007.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4007.3.2 The Navier Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4027.3.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4047.3.4 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4057.3.5 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

7.4 Antisymmetric Cross-Ply Laminates with Two OppositeEdges Simply Supported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4127.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4127.4.2 The Levy Type Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4137.4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

7.5 Antisymmetric Angle-Ply Laminates with Two OppositeEdges Simply Supported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4217.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4217.5.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4217.5.3 The Levy Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4237.5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

7.6 Transient Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .430

7.7 Vibration Control of Laminated Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4377.7.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4377.7.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .438

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7.7.3 Velocity Feedback Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4387.7.4 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4397.7.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .445

8 Theory and Analysis of Laminated Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

8.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4508.2.1 Geometric Properties of the Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4508.2.2 Kinetics of the Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4548.2.3 Kinematics of the Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4558.2.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4578.2.5 Laminate Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

8.3 Theory of Doubly-Curved Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4628.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4628.3.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

8.4 Vibration and Buckling of Cross-Ply LaminatedCircular Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4738.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4738.4.2 Analytical Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4758.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4798.4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483

9 Linear Finite Element Analysis of Composite Plates and Shells . . . .487

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

9.2 Finite Element Models of the Classical Plate Theory (CLPT) . . . . . . . . . 4889.2.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4889.2.2 Spatial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4909.2.3 Semidiscrete Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4999.2.4 Fully Discretized Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . .5009.2.5 Quadrilateral Elements and Numerical Integration . . . . . . . . . . . . . .5039.2.6 Post-Computation of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5109.2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

9.3 Finite Element Models of Shear Deformation Plate Theory (FSDT) . . . 5159.3.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5159.3.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5169.3.3 Penalty Function Formulation and Shear Locking . . . . . . . . . . . . . . .5209.3.4 Post-Computation of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5249.3.5 Bending Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5259.3.6 Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5409.3.7 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

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9.4 Finite Element Analysis of Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5439.4.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5439.4.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5469.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .560

10 Nonlinear Analysis of Composite Plates and Shells . . . . . . . . . . . . . . . . 567

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

10.2 Classical Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56810.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56810.2.2 Virtual Work Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56910.2.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .572

10.3 First-Order Shear Deformation Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . 57510.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57510.3.2 Virtual Work Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57610.3.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .578

10.4 Time Approximation and the Newton-Raphson Method . . . . . . . . . . . . . .58310.4.1 Time Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58310.4.2 The Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58410.4.3 Tangent Stiffness Coefficients for CLPT . . . . . . . . . . . . . . . . . . . . . . . 58610.4.4 Tangent Stiffness Coefficients for FSDT . . . . . . . . . . . . . . . . . . . . . . . 59010.4.5 Membrane Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

10.5 Numerical Examples of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59610.5.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59610.5.2 Isotropic and Orthotropic Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59610.5.3 Laminated Composite Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60110.5.4 Effect of Symmetry Boundary Conditions on Nonlinear

Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60410.5.5 Nonlinear Response Under In-Plane Compressive Loads . . . . . . . 60810.5.6 Nonlinear Response of Antisymmetric Cross-Ply Laminated

Plate Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60810.5.7 Transient Analysis of Composite Plates . . . . . . . . . . . . . . . . . . . . . . . 612

10.6 Functionally Graded Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61310.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61310.6.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61510.6.3 Thermomechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61610.6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

10.7 Finite Element Models of Laminated Shell Theory . . . . . . . . . . . . . . . . . . . 62010.7.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62010.7.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62210.7.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

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10.8 Continuum Shell Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62710.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62710.8.2 Incremental Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62810.8.3 Continuum Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63110.8.4 Shell Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63310.8.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63810.8.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

10.9 Postbuckling Response and Progressive Failure of CompositePanels in Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64510.9.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64510.9.2 Experimental Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64510.9.3 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64710.9.4 Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64810.9.5 Results for Panel C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65010.9.6 Results for Panel H4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

10.10 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .664

11 Third-Order Theory of Laminated Composite Plates and Shells . .671

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

11.2 A Third-Order Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67111.2.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67111.2.2 Strains and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67411.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674

11.3 Higher-Order Laminate Stiffness Characteristics . . . . . . . . . . . . . . . . . . . . . 67711.3.1 Single-Layer Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67811.3.2 Symmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68011.3.3 Antisymmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .681

11.4 The Navier Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68211.4.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68211.4.2 Antisymmetric Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . .68411.4.3 Antisymmetric Angle-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . 68711.4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

11.5 Levy Solutions of Cross-Ply Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69911.5.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69911.5.2 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70111.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

11.6 Finite Element Model of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70611.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70611.6.2 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70711.6.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71211.6.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

CONTENTS xvii

11.7 Equations of Motion of the Third-Order Theory of Doubly-CurvedShells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .721

12 Layerwise Theory and Variable Kinematic Models . . . . . . . . . . . . . . . . .725

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72512.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72512.1.2 An Overview of Layerwise Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

12.2 Development of the Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73012.2.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73012.2.2 Strains and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73312.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73412.2.4 Laminate Constitutive Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .736

12.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73812.3.1 Layerwise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73812.3.2 Full Layerwise Model Versus 3-D Finite Element Model . . . . . . . 73912.3.3 Considerations for Modeling Relatively Thin Laminates . . . . . . . 74212.3.4 Bending of a Simply Supported (0/90/0) Laminate . . . . . . . . . . . . 74612.3.5 Free Edge Stresses in a (45/-45)s Laminate . . . . . . . . . . . . . . . . . . . .753

12.4 Variable Kinematic Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75912.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75912.4.2 Multiple Assumed Displacement Fields . . . . . . . . . . . . . . . . . . . . . . . . 76212.4.3 Incorporation of Delamination Kinematics . . . . . . . . . . . . . . . . . . . . .76412.4.4 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76612.4.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769

12.5 Application to Adaptive Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78012.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78012.5.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78312.5.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78512.5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787

12.6 Layerwise Theory of Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79412.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79412.6.2 Unstiffened Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79412.6.3 Stiffened Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79812.6.4 Postbuckling of Laminated Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . .806

12.7 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

References for Additional Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .816

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

PREFACE TO THE SECOND EDITION xix

Preface to the Second Edition

In the seven years since the Þrst edition of this book appeared some signiÞcantdevelopments have taken place in the area of materials modeling in generaland in composite materials and structures in particular. Foremost among thesedevelopments have been the smart materials and structures, functionally gradedmaterials (FGMs), and nanoscience and technology � each topic deserves to betreated in a separate monograph. While the author�s expertise and contributionsin these areas are limited, it is felt that the reader should be made aware ofthe developments in the analysis of smart and FGM structures. The subject ofnanoscience and technology, of course, is outside the scope of the present study.Also, the Þrst edition of this book did not contain any material on the theory andanalysis of laminated shells. It should be an integral part of any study on laminatedcomposite structures.The focus for the present edition of this book remains the same � the education of

the individual who is interested in gaining a good understanding of the mechanicstheories and associated Þnite element models of laminated composite structures.Very little material has been deleted. New material has been added in mostchapters along with some rearrangement of topics to improve the clarity of theoverall presentation. In particular, the material from the Þrst three chapters iscondensed into a single chapter (Chapter 1) in this second edition to make room forthe new material. Thus Chapter 1 contains certain mathematical preliminaries, astudy of the equations of anisotropic elasticity, and an introduction to the principleof virtual displacements and classical variational methods (the Ritz and Galerkinmethods). Chapters 2 through 7 correspond to Chapters 4 through 9, respectively,from the Þrst edition, and they have been revised to include smart structures andfunctionally graded materials. A completely new chapter, Chapter 8, on theoryand analysis of laminated shells is added to overcome the glaring omission in theÞrst edition of this book. Chapters 9 and 10 (corresponding to Chapters 10 and13 in the Þrst edition) are devoted to linear and nonlinear Þnite element analysis,respectively, of laminated plates and shells. These chapters are extensively revised toinclude more details on the derivation of tangent stiffness matrices and Þnite elementmodels of shells with numerical examples. Chapters 11 and 12 in the present editioncorrespond to Chapters 11 and 12 of the Þrst edition, which underwent signiÞcantrevisions to include laminated shells. The problem sets essentially remained thesame with the addition of a few problems here and there.

xx PREFACE TO THE SECOND EDITION

The acknowledgments and sincere thanks and feelings expressed in the prefaceto the Þrst edition still hold but they are not repeated here. It is a pleasure toacknowledge the help of my colleagues, especially Dr. Zhen-Qiang Cheng, for theirhelp with the proofreading of the manuscript. Thanks are also due to Mr. RomanArciniega for providing the numerical results of some examples on shells included inChapter 9.

J. N. Reddy

College Station, Texas

PREFACE TO THE FIRST EDITION xxi

Preface to the First Edition

The dramatic increase in the use of composite materials in all types of engineeringstructures (e.g., aerospace, automotive, and underwater structures, as well as inmedical prosthetic devices, electronic circuit boards, and sports equipment) and thenumber of journals and research papers published in the last two decades attest tothe fact that there has been a major effort to develop composite material systems,and to analyze and design structural components made from composite materials.The subject of composite materials is truly an interdisciplinary area where

chemists, material scientists, chemical engineers, mechanical engineers, andstructural engineers contribute to the overall product. The number of studentstaking courses in composite materials and structures has steadily increased in recentyears, and the students are drawn to these courses from a variety of disciplines. Thecourses offered at universities and the books published on composite materials areof three types: material science, mechanics, and design. The present book belongsto the mechanics category.The motivation for the present book has come from many years of the author�s

research and teaching in laminated composite structures and from the fact theredoes not exist a book that contains a detailed coverage of various laminate theories,analytical solutions, and Þnite element models. The book is largely based on theauthor�s original work on reÞned theories of laminated composite plates and shells,and analytical and Þnite element solutions he and his collaborators have developedover the last two decades.Some mathematical preliminaries, equations of anisotropic elasticity, and virtual

work principles and variational methods are reviewed in Chapters 1 through 3. Areader who has had a course in elasticity or energy and variational principles ofmechanics may skip these chapters and go directly to Chapter 4, where certainterminology common to composite materials is introduced, followed by a discussionof the constitutive equations of a lamina and transformation of stresses and strains.Readers who have had a basic course in composites may skip Chapter 4 also.The major journey of the book begins with Chapter 5, where a complete

derivation of the equations of motion of the classical and Þrst-order sheardeformation laminated plate theories is presented, and laminate stiffnesscharacteristics of selected laminates are discussed. Chapter 6 includes applicationsof the classical and Þrst-order shear deformation theories to laminated beamsand plate strips in cylindrical bending. Here analytical solutions are developedfor bending, buckling, natural vibration, and transient response of simple beamand plate structures. Chapter 7 deals with the analysis of specially orthotropicrectangular laminates using the classical laminated plate theory (CLPT). Here,the parametric effects of material anisotropy, lamination scheme, and plate aspectratio on bending deßections and stresses, buckling loads, vibration frequencies, andtransient response are discussed.

xxii PREFACE TO THE FIRST EDITION

Analytical solutions for bending, buckling, natural vibration, and transientresponse of rectangular laminates based on the Navier and Levy solution approachesare presented in Chapters 8 and 9 for the classical and Þrst-order shear deformationplate theories (FSDT), respectively. The Rayleigh-Ritz solutions are also discussedfor laminates that do not admit the Navier solutions. Chapter 10 deals with Þniteelement analysis of composite laminates. One-dimensional (for beams and platestrips) as well as two-dimensional (plates) Þnite element models based on CLPTand FSDT are discussed and numerical examples are presented.Chapters 11 and 12 are devoted to higher-order (third-order) laminate theories

and layerwise theories, respectively. Analytical as well as Þnite element models arediscussed. The material included in these chapters is up to date at the time of thiswriting. Finally, Chapter 13 is concerned about the geometrically nonlinear analysisof composite laminates. Displacement Þnite element models of laminated plates withthe von Karman nonlinearity are derived, and numerical results are presented forsome typical problems.The book is suitable as a reference for engineers and scientists working in industry

and academia, and it can be used as a textbook in a graduate course on theoryand/or analysis of composite laminates. It can also be used for a course on stressanalysis of laminated composite plates. An introductory course on mechanics ofcomposite materials may prove to be helpful but not necessary because a review ofthe basics is included in the Þrst four chapters of this book. The Þrst course maycover Chapters 1 through 8 or 9, and a second course may cover Chapters 8 through13.The author wishes to thank all his former doctoral students for their research

collaboration on the subject. In particular, Chapters 7 through 13 contain results ofthe research conducted by Drs. Ahmed Khdeir, Stephen Engelstad, Asghar Nosier,and Donald Robbins, Jr. on the development of theories, analytical solutions, andÞnite element analysis of equivalent single-layer and layerwise theories of compositelaminates. The research of the author in composite materials was inßuenced by manyresearchers. The author wishes to thank Professor Charles W. Bert of the Universityof Oklahoma, Professor Robert M. Jones of the Virginia Polytechnic Institute andState University, Professor A. V. Krishna Murty of the Indian Institute of Science,and Dr. Nicholas J. Pagano of Wright-Patterson Air Force Base. It is also theauthor�s pleasure to acknowledge the help of Mr. Praveen Grama, Mr. DakshinaMoorthy, and Mr. Govind Rengarajan for their help with the proofreading of themanuscript. The author is indebted to Dr. Filis Kokkinos for his dedication andinnovative and creative production of the Þnal artwork in this book. Indeed, withouthis imagination and hundreds of hours of effort the artwork would not have lookedas beautiful, professional, and technical as it does.The author gratefully acknowledges the support of his research in composite

materials in the last two decades by the Office of Naval Research (ONR), the AirForce Office of ScientiÞc Research (AFOSR), the U.S. Army Research Office (ARO),the National Aeronautics and Space Administration (NASA Lewis and NASALangley), the U.S. National Science Foundation (NSF), and the Oscar S. WyattChair in the Department of Mechanical Engineering at Texas A&M University.Without this support, it would not have been possible to contribute to the subjectof this book. The author is also grateful to Professor G. P. Peterson, a colleague

PREFACE TO THE FIRST EDITION xxiii

and friend, for his encouragement and support of the author�s professional activitiesat Texas A&M University.The writing of this book took thousands of hours over the last ten years. Most

of these hours came from evenings and holidays that could have been devoted tofamily matters. While no words of gratitude can replace the time lost with family,it should be recorded that the author is grateful to his wife Aruna for her care,devotion, and love, and to his daughter Anita and son Anil for their understandingand support.During the long period of writing this book, the author has lost his father,

brother, brother in-law, father in-law, and a friend (Hans Eggers) - all suddenly.While death is imminent, the suddenness makes it more difficult to accept. Thisbook is dedicated to the memory of these individuals.

J. N. ReddyCollege Station, Texas

All that is not given is lost