J. N. Reddy - Mechanics of Laminated Composite Plates and Shells.pdf

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_J_N. REDDY

SECOND EDITION

ME[HANl[S of LAMINATED COMPOSITE PLATES and SHELLS Theory and Analysis

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2003061067 TA660.P6R42 2003 624. l '7765-dc22

Reddy, J. N. (Junuthula Narasimha), 1945- Mechanics of laminated composite plates and shells : theory and analysis I 1.N. Reddy.- 2nd ed.

p. cm. Rev, ed. of: Mechanics of laminated composite plates. cl997. lncludes 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. Il. Title.

Library of Congress Cataloglng-In-Publ³cat³on Data

My parenis, My brother,

My brother in-law, My father in-law, H ans Eggers, Kalpana Chawla, ...

To the Memory of

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 Memoria[ 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 N athan M. N ewmark M edal 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.

About the Author

2 Introduction to Composite Materials 81

2.1 Basic Concepts and Terminology 81 2.1.1 Fibers and Matrix 81 2.1.2 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 U nidirectional Lamina 86

1.1 Fiber-Reinforced Composite Materials 1 1.2 Mathematical Preliminaries 3

1.2.1 General Comments 3 1.2.2 Vectors and Tensors 3

1.3 Equations of Anisotropie Entropy 12 1.3.1 Introduction 12 1.3.2 Stra³n-Displacement Equations 13 1.3.3 Strain Compatibility Equations 18 1.3.4 Stress Measures 18 1.3.5 Equations of Motion 19 1.3.6 Generalized Hooke's Law 22 1.3. 7 Thermodynamic Principles 34

1.4 Virtual Work Principles 38 1.4.1 Introduction 38 1.4.2 Virtual Displacements and Virtual Work 38 1.4.3 Variational Operator and Euler Equations 40 1.4.4 Principle of Virtual Displacements .44

1.5 Variational Methods 58 1.5. l Introduction 58 1.5.2 The Ritz Method 58 1.5.3 Weighted-Residual Methods 64

1.6 Summary 71 Problems 72 References for Additional Reading 78

1 Equations of Anisotropie Elasticity, Virtual Work Principles, and Variational Methods 1

Preface to the Second Edition xix

Preface to the First Edition xxi

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 Classica! 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.l Generai 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-Isotropie 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

X CONTENTS

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 Numerica! 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. l Governing Equations 246 5.2.2 The Navier Solution 247

5.3 Bending of Plates with Two Opposite Edges Simply Supported 255 5.3. l The L®vy 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. l 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 27 4 5.5.5 Uniaxial Compression of a Rectangular Laminate (k =O) 274

CONTENTS Xl

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 Laminateci Plates U sing 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 L®vy 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

xii CONTENTS

7.1 Introduction 377 7.2 Simply Supported Antisymmetric Cross-Ply Laminated Plates 379

7.2.l Solution for the Generai 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 L®vy Type Solution 413 7.4.3 Numerica! Examples 415

7.5 Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported .421 7.5.1 Introduction 421 7.5.2 Governing Equations 421 7.5.3 The L®vy Solution 423 7.5.4 Numerica! Examples 425

7.6 Transient Solutions 430 7.7 Vibration Contro! of Laminated Plates 437

7.7.1 Preliminary Comments 437 7. 7.2 Theoretical Formulation 438

7 Analytical Solutions of Rectangular Laminateci Plates Using FSDT 377

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 Numerica! Time Integration 362 6.7.4 Numerica! Results 364


CONTENTS xiii

7.7.3 Velocity Feedback Control. 438 7.7.4 Analytical Solution 439 7.7.5 Numerica! Results and Discussion 441

7.8 Summary 442 Problems 444 References far Additional Reading 445

8 Theory and Analysis of Laminated Shells 449

8.1 Introduction 449 8.2 Governing Equations 450

8.2.1 Geometrie Properties of the Shell 450 8.2.2 Kinetics of the Shell 454 8.2.3 Kinematics of the Shell 455 8.2.4 Equations of Motion 457 8.2.5 Laminate Constitutive Relations 461

8.3 Theory of Doubly-Curved Shells 462 8.3.1 Equations of Motion 462 8.3.2 Analytical Solution 463

8.4 Vibration and Buckling of Cross-Ply Laminated Circular Cylindrical Shells 473 8.4.1 Equations of Motion 473 8.4.2 Analytical Solution Procedure 475 8.4.3 Boundary Conditions 4 79 8.4.4 Numerica! Results 480

Problems 483 References far 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 Classica! Plate Theory (CLPT) 488

9.2.l Weak Forms 488 9.2.2 Spatial Approximations 490 9.2.3 Semidiscrete Finite Element Model 499 9.2.4 Fully Discretized Finite Element Models 500 9.2.5 Quadrilatera! Elements and Numerica! Integration 503 9.2.6 Post-Computation of Stresses 510 9.2.7 Numerica! Results 510

9.3 Finite Element Models of Shear Defarmation Plate Theory (FSDT) 515 9.3.l Weak Forms 515 9.3.2 Finite Element Model 516 9.3.3 Penalty Function Formulation and Shear Locking 520 9.3.4 Post-Computation of Stresses 524 9.3.5 Bending Analysis 525 9.3.6 Vibration Analysis 540 9.3.7 Transient Analysis 542

xiv CONTENTS

9.4 Finite Element Analysis of Shells 543 9.4.1 Weak Forms 543 9.4.2 Finite Element Model 546 9.4.3 Numerical Results 549


10 Nonlinear Analysis of Composite Plates and Shells 567

10.1 Introduction 567 10.2 Classical Plate Theory 568

10.2.1 Governing Equations 568 10.2.2 Virtual Work Statement 569 10.2.3 Finite Element Model. 572

10.3 First-Order Shear Deformation Plate Theory 575 10.3.1 Governing Equations 575 10.3.2 Virtual Work Statements 576 10.3.3 Finite Element Model 578

10.4 Time Approximation and the Newton-Raphson Method 583 10.4.1 Time Approximations 583 10.4.2 The Newton-Raphson Method 584 10.4.3 Tangent Stiffness Coefficients for CLPT 586 10.4.4 Tangent Stiffness Coefficients for FSDT 590 10.4.5 Membrane Locking 594

10.5 Numerical Examples of Plates 596 10.5.l Preliminary Comments 596 10.5.2 Isotropie and Orthotropic Plates 596 10.5.3 Laminated Composite Plates 601 10.5.4 Effect of Symmetry Boundary Conditions on Nonlinear

Response 604 10.5.5 Nonlinear Response Under In-Plane Compressive Loads 608 10.5.6 Nonlinear Response of Antisymmetric Cross-Ply Laminated

Plate Strips 608 10.5. 7 Transient Analysis of Composite Plates 612

10.6 Functionally Graded Plates 613 10.6.1 Background 613 10.6.2 Theoretical Formulation 615 10.6.3 Thermomechanical Coupling 616 10.6.4 Numerical R.esults 617

10. 7 Finite Element Models of Laminated Shell Theory 621 10. 7.1 Governing Equations 621 10.7.2 Finite Element Model. 622 10.7.3 Numerical Examples 625

CONTENTS xv

10.8 Continuum Shell Finite Element 627 10.8.l Introduction 627 10.8.2 Incremental Equations of Motion 628 10.8.3 Continuum Finite Element Mode 631 10.8.4 Shell Finite Element 633 10.8.5 Numerica! Examples 638 10.8.6 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . 644

10.9 Postbuckling Response and Progressive Failure of Composite Panels in Compression 645 10.9.1 Preliminary Comments 645 10.9.2 Experimental Study 645 10.9.3 Finite Element Models 647 10.9.4 Failure Analysis 648 10.9.5 Results for Panel C4.... . . . . . . . . . 650 10.9.6 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 671

11.2.1 Displacement Field 671 11.2.2 Strains and Stresses 674 11.2.3 Equations of Motion 674

11.3 Higher-Order Laminate Stiffness Characteristics 677 11.3.1 Single-Layer Plates 678 11.3.2 Symmetric Laminates 680 11.3.3 Antisymmetric Laminates 681

11.4 The N avier Solutions. . . . . . . . . . . . . . . . . . . 682 11.4.l Preliminary Comments 682 11.4.2 Antisymmetric Cross-Ply Laminates 684 11.4.3 Antisymmetric Angle-Ply Laminates 687 11.4.4 Numerical Results 689

11.5 L®vy Solutions of Cross-Ply Laminates 699 11.5.l Preliminary Comments 699 11.5.2 Solution Procedure 701 11.5.3 Numerical Results 704

11.6 Finite Element Model of Plates. . . . . . . . . 706 11.6.1 Introduction 706 11.6.2 Finite Element Model. 707 11.6.3 Numerical Results 712 11.6.4 Closure 714

xvi CONTENTS

11. 7 Equations of Motion of the Third-Order Theory of Doubly-Curved Shells 718

Problems 720 References for Additional Reading 721

12 Layerwise Theory and Variable Kinematic Models 725

12.1 Introduction 725 12.1.1 Motivation 725 12.1.2 An Overview of Layerwise Theories 726

12.2 Development of the Theory 730 12.2.1 Displacement Field 730 12.2.2 Strains and Stresses 733 12.2.3 Equations of Motion 734 12.2.4 Laminate Constitutive Equations 736

12.3 Finite Element Model 738 12.3.1 Layerwise Model 738 12.3.2 Full Layerwise Model Versus 3-D Finite Element Model 739 12.3.3 Considerations for Modeling Relatively Thin Laminates 7 42 12.3.4 Bending of a Simply Supported (0/90/0) Laminate 746 12.3.5 Free Edge Stresses in a ( 45/-45)8 Laminate 753

12.4 Variable Kinematic Formulations 759 12.4.1 Introduction 759 12.4.2 Multiple Assumed Displacement Fields 762 12.4.3 Incorporation of Delamination Kinematics 764 12.4.4 Finite Element Model 766 12.4.5 Illustrative Examples 769

12.5 Application to Adaptive Structures 780 12.5.1 Introduction 780 12.5.2 Governing Equations 783 12.5.3 Finite Element Model 785 12.5.4 An Example 787

12.6 Layerwise Theory of Cylindrical Shells 794 12.6.1 Introduction 794 12.6.2 Unstiffened Shells 794 12.6.3 Stiffened Shells 798 12.6.4 Postbuckling of Laminated Cylinders 806

12.7 Closure 812 Ref erences for Addi tional Reading 816

Subject Index 821

CONTENTS XVll

In the seven years since the first edition of this book appeared some significant developments have taken place in the area of materials modeling in general and in composite materials and structures in particular. Foremost among these developments have been the smart materials and structures, functionally graded materials (FGMs), and nanoscience and technology each topic deserves to be treated in a separate monograph. While the author's expertise and contributions in these areas are limited, it is felt that the reader should be made aware of the developments in the analysis of smart and FGM structures. The subject of nanoscience and technology, of course, is outside the scope of the present study. Also, the first edition of this book did not contain any materiai on the theory and analysis of laminated shells. It should be an integrai part of any study on laminated composite structures. The focus for the present edition of this book remains the sarne ~ the education of

the individuai who is interested in gaining a good understanding of the mechanics theories and associated finite element models of laminated composite structures. Very little materiai has been deleted. New materiai has been added in most chapters along with some rearrangement of topics to improve the clarity of the overall presentation. In particular, the materia! from the first three chapters is condensed into a single chapter ( Chapter 1) in this second edition to make room for the new materiai. Thus Chapter 1 contains certain mathematical preliminaries, a study of the equations of anisotropie elasticity, and an introduction to the principle of virtual displacements and classical variational methods (the Ritz and Galerkin methods). Chapters 2 through 7 correspond to Chapters 4 through 9, respectively, from the first edition, and they have been revised to include smart structures and functionally graded materials. A completely new chapter, Chapter 8, on theory and analysis of laminated shells is added to overcome the glaring omission in the first edition of this book. Chapters 9 and 10 ( corresponding to Chapters 10 and 13 in the first edition) are devoted to linear and nonlinear finite element analysis, respectively, of laminated plates and shells. These chapters are extensively revised to include more details on the derivation of tangent stiffness matrices and finite element models of shells with numerica! examples. Chapters 11 and 12 in the present edition correspond to Chapters 11 and 12 of the first edition, which underwent significant revisions to include laminated shells. The problem sets essentially remained the sarne with the addition of a few problems here and there. The acknowledgments and sincere thanks and feelings expressed in the preface

to the first edition still hold but they are not repeated here. It is a pleasure to acknowledge the help of my colleagues, especially Dr. Zhen-Qiang Cheng, for their help with the proofreading of the manuscript. Thanks are also due to Mr. Roman

Preface to the Second Edition

PREFACE TO THE SECOND EDITION xix

J. N. Reddy College Station, Texas

Arciniega for providing the numerica! results of some examples on shells included in Chapter 9.

XX PREFACE TO THE SECOND EDITION

The dramatic increase in the use of composite materials in all types of engineering structures (e.g., aerospace, automotive, and underwater structures, as well as in medical prosthetic devices, electronic circuit boards, and sports equipment) and the number of journals and research papers published in the last two decades attest to the 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, and structural engineers contribute to the overall product. The number of students taking courses in composite materials and structures has steadily increased in recent years, and the students are drawn to these courses from a variety of disciplines. The courses offered at universities and the books published on composite materials are of three types: material science, mechanics, and design. The present book belongs to the mechanics category. The motivation far the present book has come from many years of the author's

research and teaching in laminated composite structures and from the fact there does not exist a book that contains a detailed coverage of various laminate theories, analytical solutions, and finite element models. The book is largely based on the author's original work on refined theories of laminateci composite plates and shells, and analytical and finite element solutions he and his collaborators have developed over the last two decades. Some mathematical preliminaries, equations of anisotropie elasticity, and virtual

work principles and variational methods are reviewed in Chapters 1 through 3. A reader who has had a course in elasticity or energy and variational principles of mechanics may skip these chapters and go directly to Chapter 4, where certain terminology common to composite materials is introduced, followed by a discussion of 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 first-order shear deformation laminated plate theories is presented, and laminate stiffness characteristics of selected laminates are discussed. Chapter 6 includes applications of the classica! and first-order shear deformation theories to laminated beams and plate strips in cylindrical bending. Here analytical solutions are developed for bending, buckling, natural vibration, and transient response of simple beam and plate structures. Chapter 7 deals with the analysis of specially orthotropic rectangular laminates using the classica! laminated plate theory (CLPT). Here, the parametric effects of materiai anisotropy, lamination scheme, and plate aspect ratio on bending deflections and stresses, buckling loads, vibration frequencies, and transient response are discussed.

Preface to the First Edition

PREFACE TO THE FIRST EDITION XX!

Analytical solutions for bending, buckling, natural vibration, and transient response of rectangular laminates based on the N avier and L®vy solution approaches are presented in Chapters 8 and 9 for the classica! and f³rst-order shear deformation plate theories (FSDT), respectively. The Rayleigh-Ritz solutions are also discussed for laminates that do not admit the Navier solutions. Chapter 10 deals with finite element analysis of composite laminates. One-dimensional (for beams and plate strips) as well as two-dimensional (plates) finite element models based on CLPT and FSDT are discussed and numerica! examples are presented. Chapters 11 and 12 are devoted to higher-order (third-order) laminate theories

and layerwise theories, respectively. Analytical as well as finite element models are discussed. The materiai included in these chapters is up to date at the time of this writing. Finally, Chapter 13 is concerned about the geometrically nonlinear analysis of composite laminates. Displacement finite element models of laminateci plates with the von Karrnan nonlinearity are derived, and numerica! results are presented for some 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 theory and/or analysis of composite laminates. It can also be used for a course on stress analysis of laminateci composite plates. An introductory course on mechanics of composite materials may prove to be helpful but not necessary because a review of the basics is included in the first four chapters of this book. The first course may cover Chapters 1 through 8 or 9, and a second course may cover Chapters 8 through 13. The author wishes to thank ali his former doctoral students for their research

collaboration on the subject. In particular, Chapters 7 through 13 contain results of the research conducted by Drs. Ahmed Khdeir, Stephen Engelstad, Asghar Nosier, and Donald Robbins, Jr. on the development of theories, analytical solutions, and finite element analysis of equivalent single-layer and layerwise theories of composite laminates. The research of the author in composite materials was influenced by many researchers. The author wishes to thank Professor Charles W. Bert of the University of Oklahoma, Professor Robert M. Jones of the Virginia Polytechnic Institute and State 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 the author's pleasure to acknowledge the help of Mr. Praveen Grama, Mr. Dakshina Moorthy, and Mr. Govind Rengarajan for their help with the proofreading of the manuscript. The author is indebted to Dr. Filis Kokkinos for his dedication and innovative and creative production of the final artwork in this book. Indeed, without his imagination and hundreds of hours of effort the artwork would not have looked as 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 Air Force Office of Scientific Research (AFOSR), the U.S. Army Research Office (ARO), the National Aeronautics and Space Administration (NASA Lewis and NASA Langley), the U.S. National Science Foundation (NSF), and the Oscar S. Wyatt Chair in the Department of Mechanical Engineering at Texas A&M University. Without this support, it would not have been possible to contribute to the subject of this book. The author is also grateful to Professor G. P. Peterson, a colleague

xxii PREFACE TO THE FIRST EDITION

All that is not given is lost

J. N. Reddy College Station, Texas

and friend, for his encouragement and support of the author's professional activities at Texas A&M University. The writing of th³s book took thousands of hours over the last ten years. Most

of these hours carne from evenings and holidays that could have been devoted to family 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 understanding and support. During the long peri od 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. This book is dedicated to the memory of these individuals.

PREFACE TO THE FIRST EDITION xxiii

1.1 Fiber-Reinforced Composite Materials Composite materials consist of two or more materials which together produce desirable properties that cannot be achieved with any of the constituents alone. Fiber-reinforced composite materials, for example, contain high strength and high modulus fibers in a matrix materiai. Reinforced steel bars embedded in concrete provide an example of fiber-reinforced composites. In these composites, fibers are the principal load-carrying members, and the matrix material keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment ( e.g., moisture, humidity, etc.). It is known that fibers are stiffer and stronger than the same material

in bulk form, whereas matrix materials have their usual bulk-form properties. Geometrically, fibers have near crystal-sized diameter and a very high length-to- diameter ratio. Short fibers, called whiskers, paradoxically exhibit better structural properties than long fibers. To gain a full understanding of the behavior of fibers, matrix materials, agents that are used to enhance bonding between fibers and matrix, and other properties of fiber-reinforced materials, it is necessary to know certain aspects of materia} science. Since the present study is entirely devoted to mechanics aspects and analysis methods of fiber-reinforced composite materials, no attempt is made here to present basic materiai science aspects, such as the molecular structure or inter-atomic forces those hold the matter together. However, an abstract understanding of the materia} behavior is useful. Materials are studied at various levels: atomic level, nano-level, single-crystal

level, a group of crystals, and so on. For the purpose of gaining some insight into the materia} behavior, we consider a basic unit of materia! as one that has properties, such as the modulus, strength, thermal coefficient of expansion, electrical resistance, etc., whose magnitudes depend on the direction. The directional dependence of properties is a result of the inter-atomic bonds, which are "stronger" in one direction than in other directions. Materials are "processed" such that the basic units are aligned so that the desired property is maximized in a given direction. Fibers provide an example of such materials. When a property is maximized in one direction, it may be achieved at the expense of the same property in other directions and other properties in the same direction. When materials are processed such that the basic

Equations of Anisotropie Elasticity, Virtual Work Principles, and

Variational Methods

1

Figure 1.1.1: Basic blocks in the analysis of composite materials.

Damage/ Failure Theories

Analytical and Computational Methods

Structural Theories

Anisotropie Elasticity Equations

Analysis of Laminated

Composite Structures

units are randomly oriented, the resulting material tends to have the same value of the property, in an average statistical sense, in all directions. Such materials are called isotropie materials. A matrix material, which is made in bulk form, provides an example of isotropie materials. Material scientists are continuously striving to develop better materials for specific applications. The fibers and matrix materials used in composites are either metallic or non-rnetallic. The fiber materials in use are common metals like aluminum, copper, iron, nickel, steel, and titanium, and organic materials like glass, boron, and graphite materials. Fiber-reinforced composite materials for structural applications are often made

in the form of a thin layer, called lamina. A lamina is a macro unit of material whose material properties are determined through appropriate laboratory tests. Structural elements, such as bars, beams or plates are then formed by stacking the layers to achieve desired strength and stiffness. Fiber orientation in each lamina and stacking sequence of the layers can be chosen to achieve desired strength and stiffness for a specific application. It is the purpose of the present study to develop equations that describe appropriate kinematics of deformation, govern force equilibrium, and represent the material response of laminated structural elements. Analysis of structural elements made of laminated composite materials involves

several steps. As shown in Figure 1.1.1, the analysis requires a knowledge of anisotropie elasticity, structural theories (i.e., kinematics of deformation) of laminates, analytical or computational methods to determine solutions of the governing equations, and failure theories to predict modes of failures and to determine failure loads. A detailed study of the theoretical formulations and solutions of governing equations of laminated composite plate structures constitutes the objective of the present book.

2 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS

In the analytical description of physical phenomena, a coordinate system in the chosen frame of reference is introduced, and various physical quantities involved in the description are expressed in terms of measurements made in that system. The description thus depends upon the chosen coordinate system and may appear different in another type of coordinate system. The laws of nature, however, should be independent of the choice of a coordinate system, and we may seek to represent the law in a manner independent of a particular coordinate system. A way of doing this is provided by vector and tensor notation. When vector notation ³s used, a particular coordinate system need not be introduced. Consequently, use of vector notation in formulating natural laws leaves them invariant to coordinate transformations.

1.2.2 Vectors and Tensors

The quantities used to express physical laws can be classified into two classes, according to the information needed to specify them completely: scalars and nonscalars. The scalars are given by a single number. Nonscalar quantities require not only a magnitude specified, but also additional information, such as direction. Time, temperature, volume, and mass density provide examples of scalars. Displacement, temperature gradient, force, moment, and acceleration are examples of nonscalars. The term vector is used to imply a nonscalar that has magnitude and "direction"

and obeys the parallelogram law of vector addition and rules of scalar multiplication. Vector in modern mathematical analysis is an abstraction of the elementary notion of a physical vector, and it is "an element from a linear vector space." While the definition of a vector in abstract analysis does not require the vector to have a magnitude, in nearly ali cases of practical interest the vector is endowed with a magnitude. In this book, we need only vectors with magnitude. Some nonscalar quantities require the specification of magnitude and two directions. For example, the specification of stress requires not only a force, but also an area upon which the force acts. A stress is a second-order tensor. Sometimes a vector is referred to as a tensor of order one, and a tensor of order 2 is also called a dyad. First- and second-order tensors (i.e., vectors and dyads) will be of primary interest in the present study ( see [1-8] for additional details). We also encounter third-order and fourth-order tensors in the discussion of constitutive equations. A brief discussion of vectors and tensors is presented next.

1.2 Mathematical Preliminaries 1.2.1 Generai Comments

Following this general introduction, a review of vectors and tensors, integral relations, equations governing a deformable anisotropie medium, and virtual work principles and variational methods is presented, as they are needed in the sequel. Readers familiar with these topics can skip the remaining portion of this chapter and go directly to Chapter 2.

EQUATIONS OF ANISOTROPIC ELASTICITY 3

(x, y, z); Figure 1.2.1: A rectangular Cartesian coordinate system, (x1, x2, x3) (<h, 2,3) = (x,®y,®z) are the unit basis vectors.

A= A1®1 + A2®2 + A3e3 B = B1®1 + B2®2 + B3®3

where ¯, (i= 1, 2, 3) is the orthonormal basis, and Ai and Bi are the corresponding physical components (i.e., the components have the same physical dimensions as the vector).

(1.2.3) (1, 2,3) or (x,y,z)

For an orthonormal basis the vectors A and B can be written as

The familiar rectangular Cartesian coordinate system is shown in Figure 1.2.1. We shall always use a right-hand coordinate system. When the basis vectors are of unit lengths and mutually orthogonal, they are called orihonormal. In many situations an orthotiormal basis simplifies calculations. We denote an orthonormal Cartesian basis by

(1.2.2)

When the basis vectors of a coordinate system are constants, i.e., with fixed lengths and directions, the coordinate system is called a Cartesian coordinate system. The general Cartesian system is oblique. When the Cartesian system is orthogonal, it is called rectangular Cartesian. The Cartesian coordinates are denoted by

(1.2.1)

Often a specific coordinate system is chosen to express governing equations of a problem to facilitate their solution. Then the vector and tensor quantities are expressed in terms of their components in that coordinate system. For example, a vector A in a three-dimensional space may be expressed in terms of its components (a1,a2,a3) and basis vectors (e1,e2,e3) (e, are not necessarily unit vectors) as

Vectors

4 MECHANICS OF LAMINATED COMPOSITE PLATES ANO SHELLS

(1.2.9)

Differentiation of vector functions with respect to the coordinates is a common occurrence in mechanics. Most of the operations involve the "del operator," denoted by V'. In a rectangular Cartesian system it has the form

(1.2.8)

Further, the Kronecker delta and the permutation symbol are relateci by the identity, known as the E-O identity,

(1.2.7)

if i, j, k are in cyclic arder and not repeated (i =J j =J k) if i, j, k are not in cyclic order and not repeated (i =J j =J k) if any of i, i, k are repeated

Eijk := 1-:: o,

(1.2.6) if i= j if i =J j

where

(l.2.5a) (1.2.5b)

A. B = (Aiei). (Bjej) = AiBjDij = AiBi A x B = (Aiei) x (Bjej) = AiBjEijkek

The range of summation is always known in the context of the discussion. For example, in the present context the range of j, k and m is 1 to 3 because we are discussing vectors in a three-dimensional space. In an orthonormal basis the scalar produci ( also called the "dot product'') and

vector produci (also called the "cross product") can be expressed in the index form using the Kronecker delta symbol Dij and the alternating symbol (or permutation symbol) Eijk:

A- j _ k _ m - a ej - a ek - a em

(1.2.4) 3

A= Lajej = ajej j=l

The repeated index is a dummy index in the sense that any other symbol that is not already used in that expression can be employed:

where (e1,e2,e3) are basis vectors (not necessarily unit), can be expressed in the form

Summation Convention

It is convenient to abbreviate a summation of terms by understanding that a repeated index means summation over all values of that index. For example, the component form of vector A


Figure 1.2.2: Cylindrical coordinate system.

X

z

and all other derivatives of the base vectors are zero. For more on vector calculus, see Reddy and Rasmussen [5] and Reddy [6], among other references.

(1.2.13) (1.2.14)

(1.2.15)

x = r cos (), y = r sin (), z = z ; = cos () ex + sin () ey' ee = - sin () ex + cos () ey' e z = e z

Ber . () , () , , oee () , . () , , fJ() = - Sl Il ex+ cos ey = ee, f)() = - cos ex - sin ey = -er

is a scalar differential operator. Thus the del operator does not commute in this sense. The operation V' r/J(x) is called the gradient of a scalar function rjJ whereas V' x A(x) is called the curl of a vector function A. We have the following relations between the rectangular Cartesian coordinates

(x, y, z) and cylindrical coordinates (r, (), z) (see Figure 1.2.2):

(1.2.12)

whereas A Ŀ V'

(1.2.11)

lt is important to note that the del operator has some of the properties of a vector but it does not have them all, because it is an operator. For instance V' Ŀ A is a scalar, called the divergence of A,

(1.2.10)

or, in the summation convention, we have


Figure 1.2.3: (a) Force on an area element. (b) Tetrahedral element in Cartesian coordinates.

(b) (a)

fi

where .6. V is the volume of the tetrahedron, p the density, f the body force per unit mass, and a the acceleration. Since the total vector area of a closed surface is zero

(1.2.17)

We see that the stress vector is a point function of the unit normal ii which denotes the orientation of the surface .6.S. The component of t that is in the direction of ii is called the normal stress. The component of t that is normal to f³ is called a shear stress. Because of Newton's third law for action and reaction, we see that t(-ii) = -t(ii). Note that t(ii) is, in general, not in the direction of Il. It is useful to establish a relationship between t and Il. To do this we now set

up an infinitesimal tetrahedron in Cartesian coordinates as shown in Figure 1.2.3b. If -t1, -t2, -t3, and t denote the stress vectors in the outward directions on the faces of the infinitesimal tetrahedron whose areas are .6.S1, .6.S2, .6.S3, and .6.S, respectively, we have by Newton's second law for the mass inside the tetrahedron,

(1.2.16) (') .6.F(ii) t n = lim b.S-tO .6.S

Tensors

To introduce the concept of a second-order tensor, also called a dyad, we consider the equilibrium of an element of a continuum acted upon by forces. The surface force acting on a small element of area in a continuous medium depends not only on the magnitude of the area but also upon the orientation of the area. It is customary to denote the direction of a plane area by means of a unit vector drawn normal to that plane. To fix the direction of the normal, we assign a sense of travet along the contour of the boundary of the plane area in question. The direction of the normal is taken by convention as that in which a right-handed screw advances as it is rotated according to the sense of travel along the boundary curve or contour. Let the unit normal vector be given by f³. Then the area A can be denoted by A = Ai³. If we denote by .6.F(i³) the force on a small area i³.6.S located at the position r

(see Figure 1.2.3a), the stress vector can be defined as follows:


The component aij represents the stress (force per unit area) on an area perpendicular to the ith coordinate and in the jth coordinate direction (see Figure 1.2.4). The stress vector t represents the vectorial stress on an area perpendicular to the direction n. Equation (1.2.25) is known as the Cauchy stress formula, and (; is termed the Cauchy stress tensor.

(1.2.27)

for i = 1, 2, 3. Hence, the stress dyadic can be expressed in summation notation as

(1.2.26)

and the dependence of t on n has been explicitly displayed. It is useful to resolve the stress vectors t1, t2, and t3 into their orthogonal

components. We have

(1.2.25) (A) A ,_, t n = n Āa Thus, we have

(1.2.24)

The terms in the parenthesis are to be treated as a dyad³c, called stress dyadic or stress tensor 7l ( we will not use the "double arrow" notation for tensors after this discussi on):

(1.2.23)

It is now convenient to display the above equation as

(1.2.22)

In the limit when the tetrahedron shrinks to a point, Lih ____,O, we are left with

(1.2.21)

where Lih is the perpendicular distance from the origin to the slant face. Substitution of Eqs. (1.2.19) and (1.2.20) in (1.2.17) and dividing throughout by

LiS reduces it to

(1.2.20)

The volume of the element Li V can be expressed as

(1.2.19)

it follows that

(1.2.18) (see Problem 1.3),


(1.2.31) [

<P11 [] = <P21

<P:n

array:

This form is called the nonion form. Equation (1.2.30) illustrates that a dyad in three-dimensional space, or what we shall call a second-order tensor, has nine independent components in general, each component associated with a certain dyad pair. The components are thus said to be ordered. When the ordering is understood, the explicit writing of the dyads can be suppressed and the dyad is written as an

(1.2.30)

.p = <Pn ®1 ®1 + <P12®1 ®2 + </J13®1 e3 + <P21 ®2®1 + <P22®2®2 + </J23®2®3 + </J31 e3e1 + </J32e3e2 + </J33e3e3

We can display all of the components ij of a dyad ~ by letting the j index run to the right and the i index run downward:

(1.2.29)

(1.2.28)

It is clear that we have

One of the properties of a dyadic is defined by the dot product with a vector. For example, dot products of a second-order tensor .P with a vector A from the right and left are given, respectively, by

.p. A= (<1\jei®j). (Akek) = <PijA]EÁli

AĿ .P = (Akek) Ŀ (ijeieJ) = ijAieJ

Thus the dot operation with a vector produces another vector. The two operations in general produce different vectors. The transpose of a second-order tensor is defined as the result obtained by the interchange of the two basis vectors:

Figure 1.2.4: Notation used far the stress components in Cartesian rectangular coordinates.


Integrai Relations Relations between volume integrals and surface integrals of the gradient (V') of a scalar or a vector and divergence (V'Ŀ) of a vector are needed in the later chapters. We record them here for future reference and use. Let O denote a region in space surrounded by the surface r, and !et ds be a

differential element of the surface whose unit outward normai is denoted by n. Let dv be a differential volume element. Let 1jJ be a scalar function and A be a vector function defined over the region O. Then the following integrai identities hold (see Figure 1.2.5):

(1.2.36)

 : W = ( cPijij) : ( 1/Jmnm n) = cPij'l/Jmn(j Ā m)(i Ŀ n) = cPij'l/JmnDjmDin = cPnm'l/Jmn = cPij'l/Jji

If the components do not satisfy the above transformation law, then it is not a tensor. The double-dot product between tensors of second arder and higher arder is

encountered in mechanics. The double-dot product between two second-order tensors and '11 is defined as

(1.2.35)

where úij are called the direction cosines. Similarly, the components of a second- order tensor transform according to the rule

(1.2.34)

Here we have selected a rectangular Cartesian basis to represent the tensor. Tensors are sometimes defined by the transformation law for its components. For

example, a vector A has components Ai with respect to the rectangular Cartesian basis ( j , 2, 3); its components referred to another rectangular Cartesian basis ( ~, ~,~) are A:j. The two sets of components are related according to

(1.2.33) = ,!, Ākn ¯ kn Ā Ā Ā 'f/i] c.. i J ~

In the general scheme that is developed, vectors are called first-order tensors and dyads are called second-order tensore. Scalars are called zeroth-order tensore. The generalization to ihird-order tensore thus leads, or is derived from, triadics, or three vectors standing side by side. It follows that higher arder tensors are developed from polyads. An nth-order tensor can be expressed in a short form using the summation convention:

(1.2.32)

This representation is simpler than Eq. (1.2.30), but it is taken to mean the same. A unit second arder tensor I is defined by


(1.2.40a)

wherc * denotes an appropriate operation, i.e., gradient, divergencc or curi operation, and F is a scalar or vector function. Some additional integral relations can be derived from Eqs. (1.2.37) and (1.2.38).

Let A= \lr.p in Eq. (1.2.38a), where sp is a scalar function, and obtain

r \7. ('Vrp) dv= r \72r.p dv= i i³. (\lr.p) ds (vector form) i. i. Tr

(1.2.39)

In the above integral relations, .fr denotes the integral on the closed boundary r of the domain O, and the component forrns refer to the usual rectangular Cartesian coordinate system. Equations (1.2.37) and (1.2.38) are valid in two as well as three dimensions. Thc integral relations in Eqs. (1.2.37) and (1.2.38) can be expressed concisely in the single statement

(1.2.38b) ( cornponent form)

(1.2.38a) ( vector forrn) f \7 . A dv = jr i³ Ŀ A ds i. Jr

Divergence Theorem

(1.2.37b) r ~'ljJ dv = i ni'l/J ds ( component form) i, ox, Jr

(1.2.37a) r \l'ljJ dv = i frlj) ds (vector forrn) i; Jr

Gradient Theorem

Figure 1.2.5: A solid body with a surface norrnal vector n.

n


The objeetive of this seetion is to review the governing equations of a linear anisotropie elastie body. The equations governing the motion of a solid body ean be classified into four basie eategories:

( 1) Kinematies ( strain-displaeement equations) ( 2) Kinetics ( conservation of momenta) (3) Thermodynamics (first and seeond laws of thermodynamies) (4) Constitutive equations (stress-strain relations)

Kinematics is a study of the geometrie changes or deformation in a body, without the consideration of forces causing the deformation. Kinetics is the study of the static or dynamie equilibrium of forces and moments acting on a body. This leads to equations of motion as well as the symmetry of stress tensor in the absence of body moments. The thermodynamie principles are eoneerned with the conservation of energy and relations among heat, mechanical work, and thermodynamie properties of the body. The constitutive equations describe thermomechanical behavior of the materiai of the body, and they relate the dependent variables introduced in the kinetic description to those in the kinematic and thermodynamic descriptions. These equations are supplemented by appropriate boundary and initial eonditions of the problem. In the following seetions, an overview of the governing equations of an anisotropie

elastie body is presented. The strain-displacement relations, equations of motion, and the eonstitutive equations for an isothermal state (i.e., no ehange in the temperature of the body) are presented first. Subsequently, the thermodynamic principles are considered only to determine the temperature distribution in a solid body and to account for the effect of non-uniform temperature distribution on the strains. A solid body B is a set of materiai particles whieh can be identif³ed as having

one- to-one correspondence wi th the points of a regi on n of Euclidean point space R3.

1.3 Equations of Anisotropie Elasticity 1.3.1 Introduction

(1.2.41)

81.p ~ '7 (i . e ) On = Il Ŀ V i.p invariant torrn 81.p . = ni~ (reetangular Cartesian component form) UXi 8 IP fJ IP 81.p

= nx ox + ny oy + nz oz The integrai relations presented in this section are useful in developing the so-ealled weak forms of differential equations in conneetion with the Ritz method and finite element formulations of boundary value problems.

The quantity ft Ŀ V' <p is ealled the normai derivative of <p on the surface r, and is denoted by

(1.2.40b) { -0-21.p- dv= J ni_fJ_<p ds lo oxiOXi Jr Bx,

or, in eomponent form


(1.3.2) u = X - X or Ui = Xi - xi

1.3.2 Strain-Displacement Equations

The phrase deformation of a body refers to relative displacements and changes in the geometry experienced by the body. Referred to a rectangular Cartesian frame of reference ( X1 , X 2, X 3), every particle X in the body corresponds to a set of coordinates X = (X1, X2, X3). When the body is deformed under the action of external forces, the parti cl e X moves to a new position x = ( x1, x2, x3). The displacement of the particle X is given by

Often, the reference configuration cR is chosen to be the unstressed state of the body, i.e., cR = c0. The coordinates (X1, X2, X3) are called the materiol coordinates. In the spatial or Eulerian description of a body B, the motion is referred to the

current configuration C occupied by the body B. The spatial description focuses attention on a given region of space instead of on a given body of matter, and is the description most used in fluid mechanics, whereas in the Lagrangian description the coordinate system X is fixed on a given body of matter in its undeformed configuration, and its position x at any time is referred to the materia! coordinates Xi. Thus, during a motion of a body B, a representative particle X occupies a succession of points which together form a curve in Euclidean space. This curve is called the path of X and is given parametrically by Eq. (1.3.1).

(1.3.1)

The particles of B are identified by their time-dependent positions relative to the selected frame of reference. The simultaneous position of all material points of l3 at a fixed time is called a configuration of the structure. The analytical description of configurations at various times of a material body acted on by various loads results in a set of governing equations. Consider a deformable body B of known geometry, constitution, and loading.

Under given geometrie restrictions and loading, the body will undergo motion and/or deformation (i.e., geometrie changes within the body). If the applied loads are time dependent, the deformation of the body will be a function of time, i.e., the geometry of the body will change continuously with time. If the loads are applied slowly so that the deformation is only dependent on the loads, the body will take a definitive shape at the end of each load application. Whether the deformation is time dependent or not, the forces acting on the body will be in equilibrium at all times. Suppose that the body l3 under consideration at time t = O occupies a

configuration c0, in which a particle X of the body l3 occupies a position X. Note that X is the name of the particle that occupies the location X in the reference configuration. At time t > O, the body assumes a new configuration C and the particle X occupies the new position x. There are two commonly used descriptions of motion and deformation in

continuum mechanics. In the referential or Lagrangian description, the motion of a body B is referred to a reference configuration es. Thus, in the Lagrangian description the current coordinates (x1, x2, x3) are expressed in terms of the reference coordinates (X 1 , X 2, X 3) and time t as


Figure 1.3.1: Kinematics of deformation of a continuous medium.

C0 (time t =O) ParticleX (occupying position X) /cx2,X2 u C=C1

Co

A rigid-body motion is one in which all material particles of the body undergo the same linear and angular displacements. A deformable body is one in which the material particles can move relative to each other. The deformation (i.e., relative motion of material particles) of a deformable body can be determined only by considering the change of distance between any two arbitrary but infinitesimally close points of the body. Consider two neighboring material particles P and Q which occupy the positions

P : (X1, X2, X3) and Q : (X1 + dX1, X2 + dX2, X3 + dX3), respectively, in the undeformed configuration c0 of the body B. The particles are separated by the infinitesima! distance dS = J dXidXi (sum on i) in c0, and dX is the vector connecting the position of P to the position of Q. These two particles move to new places P and Q, respectively, in the deformed body (see Figure 1.3.1). Suppose that the positions of P and Q are (x1, x2, x3) and (x1 + dx, x2 + dx2, x3 + dx3), respectively. The two particles are now separated by the distance ds = J dxi dxi in the deformed configuration C, and dx is the vector connecting P to Q. The vector dx can be interpreted as the position occupied by the deformed material vector dX. When the material vector dX is small but finite, the line vector dx in general does not coincide exactly with the deformed position of dX, which lies along a curve in the deformed body. The deformation (or strains) in a body can be measured in a number of ways. Here we use the standard strain measure of solid mechanics, namely the Green-Lagrange strain E, which is defined such that it gives the change

(1.3.3)

If the displacement of every particle in the body is known. we can construct the current ( deformed) configuration C from the reference (or undeformed) configuration C0. In the Lagrangian description, the displacements are expressed in terms of the material coordinates xi' and we have


(1.3.9)

Note that the Green-Lagrange strain tensor is symmetric, E= ET (Eij = Eji)Ŀ The strain components defined in Eq. (1.3.8) are called finite strain components because no assumption concerning the smallness ( compared to unity) of the strains is made. The rectangular Cartesian component form is given by

(1.3.8)

E=~ [(I+ 'Vu) Ŀ(I+ 'Vuf - 1]

1 [ T T] = 2 'Vu + ('Vu) + 'Vu Ŀ (Vu)

Thus the Green (or Green-Lagrange) strain tensor E is given in terms of the displacement gradients as

( 1.3. 7)

2dX Ŀ E Ŀ dX = dx Ŀ dx - dX Ŀ dX = [dX Ŀ(I+ 'Vu)] Ŀ [dX Ŀ(I+ 'Vu)] - dX Ŀ dX = dX Ŀ(I+ 'Vu) Ŀ(I+ 'Vu)T Ŀ dX - dX Ŀ dX = dX Ŀ [(I+ 'Vu) Ŀ (I+ 'Vuf - I] Ŀ dX

where \7 denotes the gradient operator with respect to the materiai coordinates, X. Now the strain tensor or its components from Eqs. (1.3.4a,b) can be expressed in terms of the displacement vector or its components with the help of Eq. (1.3.6):

(1.3.6) dx= dX + dX Ŀ 'Vu = dX Ŀ(I+ 'Vu)

Since x is a function of X, its total differential is givcn by [using the chain rule of differentiation and Eq. (1.3.5)]

(1.3.5)

In Eq. (1.3.4b) and in the equations that follow, the summation convention on repeated indices is used, and the range of summation is 1 to 3. In order to express the strains in terms of the displacements, we use Eq. (1.3.2)

and write

(1.3.4b)

and in rectangular Cartesian component form we have

(1.3.4a) 2dX ĿEĿ dX = (ds)2 - (dS)2 =dxĿ dx - dX Ŀ dX

in the square of the length of the materia} vector dX


(1.3.15) e

E12 = 2a = 0.01 cm/cm

Then the Green-Lagrangian strains can be computed using Eq. (1.3.10). The only nonzero strain component is (e= 0.2cm and a= lOcm)

(1.3.14)

The displacements are

(1.3.13)

Example 1.3.1:

(a) A square block is deformed as shown by dotted lines in Figure 1.3.2a. Assuming that the body is very thin and the strains (due to the Poisson effect) associated with the thickness direction are negligible, we wish to determine the two-dimensional strains. A materiai particle which occupied position (X1, X2, X3) in the undeformed body takes the

position (x1,x2,x3) in the deformed body. The current coordinates of the materiai particle can be expressed in terms of its originai position as

(1.3.12)

8u1 8u2 ou3 8u1 8u2 ®U = ~ , E22 = ~ , ú33 = ~ , /'12 =: 2E12 = -- + --

uX1 uX2 uX3 OX2 OX1

8u1 8u3 8u2 8u3 /'13 = 2ú13 = ~ + ~ ' /'23 = 2ú23 = - + - uX3 ux1 OX3 8x2

(1.3.11) Eij = ~ (OUi + OUj) 2 OXj OXi

The explicit form of the infinitesimal strain components (1.3.11) ³s given by (rij denote the engineering shear strains)

If the displacement gradients are so small, [Vu] << 1, that their squares and products are negligible compared to IVul. Then the Green-Lagrange strain tensor reduces to the infinitesima[ strain tensor, E ~ E:

(1.3.10)

[ ( 8u1 )

2 + ( 8u2 )

2 + ( 8u3 )

2]

&X1 &X1 &X1

[ ( 8u1 )

2 + ( 8u2 )

2 + ( 8u3 )

2]

8X2 8X2 8X2

[ ( 8u1 )

2 + ( 8u2 )

2 + ( 8u3 )

2]

0X3 0X3 0X3

8u1 1 En = -- + - &X1 2

OU2 1 E22 = -- + -

&X2 2

OU3 1 E33 = -- + -

0X3 2

Explicit form of the six Cartesian components of strain are given by


This completes the kinematic description. In the coming chapters, we use only the linear strains and the von Karrnn nonlinear strains derived from Eq. (1.3.10).

The strain is nonlinear. The nonlinear part of the strain is 0.02 percent.

(1.3.18) e 1 (e)2 E11 = - + - - = (0.02 + 0.0002) cm/cm a 2 a

The only nonzero Lagrangian strain is

(U.17)

The displacements are

(1.3.16)

(b) Consider a square block, deformed as shown by dotted lines in Figure l.3.2b. The current coordinates of the materiai particle occupying position (X1,X2,X3) in the undeformed body can be expressed as

Figure 1.3.2: Undeformed and deformed configurations of a solid square block. (a) Pure shear deformation. (b) Pure extensional deformation.

X2,X2 X2,X2

j~

e I+- -+1 e I+-

T e:::> a

1 X1,X1 Xi,X1 - -i-- a -----+-/

(a)

X2,X2 X2,X2

j~ ' -+I e I+-

T e:::> a

1 I

X1,X1 I Xi,X1

j4- a-----+-/

(b)


where N is the unit normal to the undeformed area dA. The stress tensor P is called the first Piala-Kirchhaff stress tensor, and it gives the curreni farce per unit undefarmed area. The first Piola-Kirchhoff stress tensor is not symmetric.

(1.3.22) df = dA Ā P, where dA = dA N

where Cauchy's formula, t = (J' Å f³, is used. Expressing df in terms of a stress times the initial undeformed area dA requires

a new stress tensor P,

(1.3.21) df = t da= daĀ O', where da= da n

Stress at a point was introduced in Section 1.2 as a measure of farce per unit area. Equation (1.2.16) indicates that the stress vector at a point depends on the farce vector (its direction and magnitude) and the surface area. The surface area in turn depends on the orientation of the plane used to slice the body. It was shown that the state of stress at a point inside a body can be expressed in terms of stress vectors on three mutually perpendicular planes, say planes perpendicular to the rectangular coordinate axes by Cauchy's formula in Eq. (1.2.25). In the above discussion, stress vector t at a point in a deformed body ³s measured

as the farce per unit area in the deformed body. The area element .6s in the deformed body corresponds to an area element 6.S in the reference configuration, in much the same way x is the position of a materiai particle X in the deformed body whose position in the reference configuration was X. Thus the Cauchy stress iensor O' is defined to be the curreni farce per unit defarmed area:

1.3.4 Stress Measures

It should be noted that the strain compatibility equations are satisfied automatically when the strains are computed from a displacement field. Thus, one needs to verify the compatibility conditions only when the strains are computed from stresses that are in equilibrium.

(1.3.20)

for any i,j,m,n = 1,2,3. For the two-dimensional case, Eq, (1.3.19) reduces to the following single compatibility equation

02E11 + 02E22 _ 2 02E12 = Q ax~ axi OX10X2

(1.3.19)

1.3.3 Strain Compatibility Equations By definition, the components of the strain tensor can be computed from a differentiable displacement field using Eq. (1.3.8) or Eq. (1.3.11). However, if the six components of strain tensor are given and if we are required to find the three displacement components, the strains given should be such that a unique solution to the six differential equations relating the strains and displacements exists. The existence of a unique solution is guaranteed if the infinitesima! strain components satisfy the following six compatibility conditions:


For kinematically infinitesima! deformations, i.e., IVul << 1, we do not distinguish bctween x and X, between a and S and between E and E, and we use the first symbol of each pair. In much of this book we deal with kinematically infinitesima! deformations (i.e., linearizecl elasticity). The strain-displacement relations and the equations of motion in any coordinate

system can be obtained from the vector forrns in Eqs. (1.3.8), (1.3.11), (1.3.26a) and

(1.3.27a)

(1.3.27b)

\7 Ŀ a + f =O (vector form)

OUji + fi =O (Cartesian component form) OXj

where p is the density in the cleformed configuration and f is the body force vector (measured per unit volume). The equations of equilibrium are obtained by setting the time derivative term to zero:

(1.3.26b) (Cartesian component form)

(1.3.26a) ( vector form)

1.3.5 Equations of Motion The principle of conservation of linear momentum states that the rate of change of the total linear momcntum of a given continuous medium equals the vector sum of all the external forces acting on the body B, which initially occupied a configuration c0, providecl Newton's third law of action and reaction govcrns the internal forces. The principle leads to the following equations of motion:

Thus, the second Piola-Kirchhoff stress tensor gives thc iransjormed curreni farce per unit usuieformed area. The seconcl Piola-Kirchhoff stress tensor is symmetric whenever the Cauchy stress tensor is symmetric.

(1.3.25) dF = F-1 Ŀ df = p-l Ŀ (dA Ŀ P) = dA Ŀ P Ŀ F-T =: dA Ŀ S

and \7 is the gradient operator with respect to x. Analogous to the transformation between X and x, we can transform the force df on the deformed elemental area da to the force dF on tbc undeformed elemental area dA ( not to be confusecl between the force dF and deformation gradient tensor F)

(1.3.24) ax

where p-T = ox = \7X

and \7 o is the gradient operator with respcct to X. We also have

(1.3.23) ( {} )T dx=FĿdX=dXĿFT where F= 0~

=:(Vox)T

The second Piola-Kirchhoff stress tensor S is introduced as follows. First, we introduce the deforrnation gradient tensor F


(l.3.29)

Example 1.3.3:

Consider the cantilevered beam under an end load (see Figure 1.3.3). The bending moment about the xraxis at any distance x1 is given by M2 = P(L - xi). Then the stress component 0'11 can be calculated using the flexure stress formula from elementary strength of materials:

Thus, the first two equations of equilibrium are identically satisfied for any choice of const.ants, c1, c2, c3, and c4. The third equation of equilibrium is trivially satisfied.

and ali other components of stress are zero. We wish to determine if the stress field satisfies the equations of equilibrium in the presence of body forces, fi = O, f2 = -c1, and h = O. We assume that the body experienced only a small deformation. We have

Example 1.3.2:

Consider the following stress field in a body that is in equilibrium:

Note that the equations of motion or equilibrium contain three equations relating six stress components and therefore cannot be solved for all six components uniquely. Additional equations are required. These include the strain-displacement relations discussed in Section 1.3.2 and constitutive relations or stress-strain relations to be discussed in the next section.

(1.3.28b) +--> ~ ~ a = ei aij ej

This notation is meaningful and descriptive of the nature of the tensor; the notation indicates that the quantity is a dyad (i.e., having two base vectors) and it is symmetric:

(1.3.28a) a

Thus there are only six independent components of the Cauchy stress tensor. Since the Cauchy stress tensor is a second-order tensor and symmetric, we may write it with a "double arrow" notation as

(1.3.27a) by expressing a, f, u, and \7 in the chosen coordinate system. The vector forms of equations are invariant, i.e., independent of the choice of the coordinate system. The principle of conservation of angular momentum, in the absence of any

distributed body couples, leads to the symmetry of the stress tensor:


Figure 1.3.3: A cantilevered beam (i.e., fixed at one end and no support at the other end) under an end load.

(1.3.32)

c1h2 C2 = --8-

Thus the two-dimensional state of stress is given by

_ !1:f._ = O, g = O, or f = c2 and g = O dx1 Ŀ

Vanishing of a-13 at X3 = Ñh/2 gives

which imply that

df h --(--)+g=O dxi 2

df h ---+g=O dxi 2 '

The functions f and g can be deterrnined using the boundary conditions of the beam. Note that ai3 and a33 must be zero on the top and bottom surfaces of the bearn (i.e., at x3 = Ñh/2). Vanishing of a33 at X3 = Ñh/2 gives

(1.3.31)

Integration with respect to x3 yields

where f is a function of x1 only. The second equation of equilibrium is trivially satisfied. The third equation of equilibrium gives

(1.3.30)

Integration with respect to x3 gives

where 122 is area moment of inertia about the xraxis. Assuming a two-dimensional state of stress (with respect to the x1 and x3 coordinates) in the beam, we wish to determine the stress components cri3 and CT33 in the absence of body forces. Since the stress components cri2, CT22, and a-23 are assumed to be zero, the first equation of equilibrium yields


The kinematic relations and the mechanical and thermodynamic principles are applicable to any continuum irrespective of its physical constitution. Here we consider equations characterizing the individuai materiai and its reaction to applied loads. These equations are called the eonstitutive equations. Materials for which the constitutive behavior is only a function of the current

state of deformation are known as elastie. In the special case in which the work clone by the stresses during a deformation is dependent only on the initial state and the current configuration, the materiai is called hyperelastie. A materiai body is said to be homogeneous if the materiai properties are the same

throughout the body (i.e., independent of position). In a heterogeneous body, the materiai properties are a function of position. For example, a structure composed of severa! uniform thickness layers of different materials stacked on top of each other and bonded to each other is heterogeneous through the thickness. An anisotropie body is one that has different values of a materiai property in different directions at a point; i.e., materia! properties are direetion-dependent. An isotropie body is one for whieh every materia! property in ali direetions at a point is the same. An isotropie or anisotropie materiai ean be nonhomogeneous or homogeneous.

1.3.6 Generalized Hooke's Law

p p -S15-1 +O+ 2515-1 ,O: O

22 22

Thus the strains are compatible only if S15 =O, which is the case when the materia! is isotropie or orthotropic with respect to the problem coordinates.

we obtain

(1.3.34)

Substituting these strain components into the compatibility equation [see Eq. (1.3.20)],

(1.3.33)

Then

Eu = Su au + S13cr33 + S15cr13 E33 = S13cru + 5330-33 + S35CT1;3 E13 = S15au + 5350-33 + S55cr13

Since the stress field is derived from stress equilibrium equations, it is necessary to see if the strain compatibility condition in Eq. (1.3.20) is satisfied. Suppose that the strains E11, E13, and E33 are related to the stress components cru, cr13, and cr33 by the relations (see the next section for details)


®PUo ---- = CijkC OEi:J&ke

Since the arder of differentiation is arbitrary, fJ2U0/8Ei:J8Eke = 82U0/DEke&i.i, it fallows that Ci.ikf = Ckfi.i. This reduces the number of independent materiai stiffness components to 21. To show this wc express Eq. (1.3.35) in an alternate farm using single subscript notation far stresses and strains and two subscript notation far the

we have

(1.3.36)

where e is the faurth-order tensor of material parameters and is termed stiffness tensor. There are, in generai, 34 = 81 scalar cornponents of a faurth-order tcnsor. The numbcr of independent components of C are considcrably less because of the symmetry of CJ, symrnetry of E, and symmetry of C, as discussed next [6]. In the absence of body couples, the principle of conservation of angular

momentum requires the stress tensor to be symmctric, CJij = CJjiĿ Then it fallows from Eq. (1.3.35) that Cijke must be syrnmetric in the f³rst two subscripts. Hence the number of independent materiai stiffness components reduces to 6(3)2 = 54. Since the strain tensor is symmetric (by its definition), Eij = Eji, then cijkf must be symmetric in the last two subscripts as well, further reducing the nurnber of independent materiai stiffness components to 6 x 6 = 36. If we also assume that the materiai is hyperelastic, i.e., there exists a strain

energy density function Uo(E,;j) such that

(1.3.35)

A material body is said to be ideally elastic when, under isothermal conditions, the body recovers its origina! form completely upon removal of the forces causing defarmation, and there is a one-to-one rclationship between the state of stress and the state of strain in the current configuration. The constitutive equations described bere do not include creep at constant stress and stress relaxation at constant strain. Thus, the material coefficients that specify the constitutive relationship between the stress and strain components are assumed to be constant during the defarmation. This does not automatically imply that we neglect temperature effects on defarmation. We account far the thermal expansion of the materiai, which can produce strains or stresses as large as those produced by the applied mechanical forces. Here, we discuss the constitutive equations of linear elasticity (i.e., relations between stress and strain are linear) far the case of infinitesima! deformation (i.e., I V' u] < < 1). Hence, we will not distinguish between various rneasures of stress and strain, and use S >=::; o far the stress tensor and E >=::; E far strain tensor in the material description used in solid rnechanics. Thc linear constitutive model far infinitesimal defarmation is referred to as the generalized Hooke 's law. Suppose that the reference configuration has a ( resid ual) stress state of CJO. Then if the stress cornponents are assumed to be linear functions of the cornponents of strain, then the rnost generai forrn of the linear constitutive equations far infinitesima! deforrnations is


In the following discussion we assume that the reference configuration is stress free, cr? =O and strain free 10? =O.

where 8ij are the material compliance parameters with [8] = [C]-1 (the compliance tensor is the inverse of the stiffness tensor: S = c-1 ). In matrix form Eq, (1.3.39a) becomes

E1 Su S12 S13 S14 S1s s16 CT1 t::? E2 S21 822 823 824 82s 826 cr2 Eg ú3 831 832 S33 834 835 836 (T3 +

®g (1.3.39b) = S41 842 843 844 845 S46 Eo E4 CT4 4

E5 851 8s2 853 854 8ss 8s6 CT5 ®g E6 861 862 s63 s64 s6s s66 0"6 EO 6

(1.3.39a)

Now the coefficients Cij must be symmetric (Cij = Cji) by virtue ofthe assumption that the material is hyperelastic. Hence, we have 6+5+4+3+2+1 = 21 independent stiffness coefficients for the most general elastic material. We assume that the stress-strain relations (1.3.38a,b) are invertible. Thus, the

components of strain are related to the components of stress by

cr1 C11 C12 C13 C14 C15 c16 El ero 1 cr2 C21 C22 C23 C24 C25 c26 E2 (T~ CT3 C31 C32 C33 C34 C35 C36 E3 +

(T~ (1.3.38b) CT4 C41 C42 C43 C44 C45 c46 E4 cr2 CT5 Cs1 C52 C53 Cs4 Css Cs6 E5 ero 5 0"6 C61 C52 c63 c64 c6s c66 E6 ero 6

where summation on repeated subscripts is implied (now from 1 to 6). In matrix notation, Eq. (1.3.38a) can be written as

(1.3.38a)

It should be cautioned that the single subscript notation used for stresses and strains and the two-subscript components Cij render them non-tensor components (i.e., cri, Ei, and Cij do not transform like the components of a vector or tensor). The single subscript notation for stresses and strains is called the engineering notation or the Voigt-Kelvin notation. Equation (1.3.35) now takes the form

(1.3.37b) 11 ---+ 1 22 ---+ 2 33 ---+ 3 23 ---+ 4 13 ---+ 5 12 ---+ 6.

(1.3.37a) cr1 = cru, cr2 = cr22, cr3 = cr33, cr4 = cr23 , cr5 = cr13, CT6 = cr12 El = E11, E2 = E22, E3 = E33, E4 = 2E23, E5 = 2E13, E6 = 2E12

material stiffness coefficients:


or, in single-su bscri pt notation

When the elastic coefficients at a point have the same value for every pair of coordinate systems which are the mirror images of each other with respect to a plane, the material is called a monoclinic maierial. For example, let (x1, x2, x3) and (x~, x~, x3) be two coordinate systems, with the x1, x2-plane parallel to the piane of symmetry. Choose x3-axis such that x3 = -x3 (never mind about the left-handed coordinate system as it does not affect the discussion) so that one system is the mirror image of the other. The definitions and sign conventions of the stress and strain components show that

Monoclinic Materials

where fij are the direction cosines associated with the coordinate systems (x1, x2, x3) and ( x~, x~, x3), and C~jkl and Cpqrs are the components of the fourth-order tensor e in the primed and unprimed coordinate systems, respectively.

(1.3.40)

Materia! Symmetry

Further reduction in the number of independent stiffness (or compliance) parameters com es from the so-called materi al symmetry. Suppose that ( x1, x2, X3) denote the coordinate system with respect to which Eqs. (l.3.38a,b) and (l.3.39a,b) are defined. We shall call them materiai coordinate system. The coordinate system (x, y, z) used to write the equations of motion and strain-displacement equations will be called the problem coordinates to distinguish them from the material coordinate system. Note that the phrase "material coordinates" used in connection with the materiai description should not be confused with the present term. In the remaining discussion, we will use the material description for everything, but we may use one material coordinate system, say (x, y, z), to describe the kinematics as well as stress state in the body and another material coordinate system (x1, x2, x3) to describe the stress-strain behavior. Both are fixed in the body, and the two systems are oriented with respect to each other. When elastic material parameters at a point have the same values for every pair of coordinate systems that are mirror images of each other in a certain plane, that plane is called a maierial plane of symmetry (e.g., symmetry of internal structure due to crystallographic form, regular arrangement of fibers or molecules, etc.). We note that the symmetry under discussion is a directional property and not a positional property. Thus, a material may have certain elastic symmetry at every point of a materia} body the properties may vary from point to point. Positional dependence of material properties is what we called the inhomogeneity of the material, In the following we discuss various planes of symmetry and forms of associated

stress-strain relations. Note that use of the tensor components of stress and strain is necessary because the transformation laws of the form (1.2.35) are valid only for the tensor components. The fourth-order tensor, for example, transforms according to the formula


(1.3.44)

Cn C12 C13 O O O C12 C22 C23 O O O C13 C23 C33 O O O O O O C44 O O O O O O Css O o o o o o c66

Orthotropic Materials

When three mutually orthogonal planes of materia} symmetry exist, the number of elastic coefficients is reduced to 9 using arguments similar to those given for single materia} symmetry plane, and such materials are called orthoiropic. The stress- strain relations for an orthotropic materia} take the form

Note that monoclinic materials exhibit shear-extensional coupling; i.e., a shear strain can produce a normal stress; for example, C³11 = C16c6 = 2C16El2Ŀ Therefore, the principal axes of stress do not coincide with those of strain. The result in Eq. (1.3.42) can also be obtained using the following transformation

matrix (which converts the unprimed coordinate system to the primed one) in Eq. (1.3.40):

[L] = [~ ~ ~ l (or fu= .t'22 = 1, .t'33 = -1, eiJ =O fori i- j) (1.3.43) o o -1

C11 C12 C13 o o Cm C12 C22 C23 o o C26

[C] C13 C23 C33 o o C3(; (1.3.42) o o o C44 C45 o o o o C45 Css o Crn C26 C36 o o c6(i

while all their independent stress and strain components remain unchanged in value by the change from one coordinate system to the other. U sing the stress-strain relations of the form in Eq. (1.3.38b), we can write

Ci~ = Cuc~ + C12c; + C13c~ + C14E~ + C1sc; + Crnc~ C³1 = C11c1 + C12c2 + C13c3 - C14c4 - C15c5 + Crnc6

But we also have

C³1 = C11c1 + C12c2 + C13c3 + C14c4 + C15c5 + Crnc5 Note that the elastic parameters Cij are the same for the two coordinate systems because they are the mirror images in the plane of symmetry. From the above two equations (subtract one from the other) we arrive at

C14E4 + C15E5 = O for all values of E4 and Es The above equation holds only if C14 = O and C15 = O. Similar discussion with the two alternative expressions of the remaining stress components yield C24 = O and C25 = O; C34 = O and C35 = O; and C45 = O and Cs6 = O. Thus out of 21 material parameters, we only have 21 - 8 = 13 independent parameters, as indicated below


e u v21 = --

E22

Most often, the material properties are dctermined in a laboratory in terms of the engineering constants such as Young's modulus, shear modulus, and so on. These constants are measured using simple tests like uniaxial tension test or pure shear test. Because of their direct and obvious physical meaning, engineering constants are used in place of the more abstract stiffness coefficients Cij and compliance coefficients Sij. N ext we discuss how to relate the compliance coefficients 8ij to the engineering constants. One of the consequences of linearity (both kinematic and materia} linearizations)

is that the principle of superposition applies. That is, if the applied loads and geometrie constraints are independent of deformation, the sum of the displacements ( and hencc strains) produced by two sets of loads is equal to the displacements ( and strains) produced by the sum of the two sets of loads. In particular, the strains of the same type produced by the application of individual stress components can be superposed. For example, the extensional strain EW in the materia! coordinate direction x1 due to the stress au in the same direction is 0'11/ E1, where E1 denotes Young's modulus of the materiai in the x1 direction. Thc extensional strain f~;) due to the stress 0'22 applied in the x2 direction is -v210'22/ E2, where v21 is the Poisson ratio

(1.3.46)

where Sij are the compliance coefficients ([C] = [5]~1)

fl 811 812 81:1 o o o 0'1 f2 812 822 823 o o o 0'2 f:3 813 823 833 o o o 0'3 (1.3.45) f4 o o o 844 o o 0'4 f5 o o o o 855 o 0',5 f5 o o o o o S5G 0'5

Most sirnple mechanical-property characterization tests are performed with a known load or stress. Hence, it is convenient to write the inverse of relations in (1.3.44). The strain-stress relations of an orthotropic material are given by

~] o -1 o

o 1 o

o] [-1 O ; [L(2)] = O -1 o

o 1 o

The transformation matrices associated with the planes of symmetry are


(1.3.49) are

(1.3.48) u.: vĀĀ _31_ = _1!_ (no sum on i, j) e; e,

fori, j = 1, 2, 3. The 9 independent materia! coefficients for an orthotropic material

or, in short

V13 V32 V23

E1' E3 E2

where E1, E2, E3 are Young's moduli in 1, 2, and 3 material directions, respectively, vij is Poisson's ratio, defined as the ratio of transverse strain in the jth direction to the axial strain in the ith direction when stressed in the ith direction, and G23, G13, G12 are shear moduli in the 2-3, 1-3, and 1-2 planes, respectively. Since the compliance matrix [S] is the inverse of the stiffness matrix [C] and the inverse of a symmetric matrix is symmetric, it follows that the compliance matrix [ S] is also a symmetric matrix. This in turn irnplies that the following reciproca! relations hold [see Eq. (1.3.47)]:

1 _!:'.2..l_ - !:'.il o o o c1 Ei E2 E3 a1

-~ 1 _ _l:'.J.2. o o o c2 Ei E2 E3 a2 _!:'.l.3_ -~ 1 o o o ú3 Ei E2 E3 0"3 (1.3.47)

ú4 o o o 1 o o 0"4 G23

ú5 o o o o 1 o 0"5 G13

ú6 o o o o o 1 0"6 G12

Recall that 2Eij (i f. j) is the change in the right angle between two lines parallel to the x1 and x2 directions at a point, O"ij (i f. j) denotes the corresponding shear stress in the XiXj plane, and Gij (i f. j) are the shear moduli in the XiXj plane. Writing Eqs. (a)-(d) in matrix form, we obtain

(d) 0"12 0"13 0"23 2c12 = -G , 2c13 = -G , 2c23 = -G

12 13 23

(c) O"ll V13 1722 V23 0"33 ú33 = ---- - ---+-

E1 E2 E3

The simple shear tests with an orthotropic material give the results

(b)

where the direction of loading is denoted by the superscript. Similarly, we can write

(a)

and E2 is Young's modulus of the materiai in the x2 direction. Similarly, a33 produces a strain cg) equal to -v31a33/ E3. Hence, the total strain c11 due to the simultaneous application of all three normal stress components is


Figure 1.3.4: Distinction between v12 and v21.

(b)

(\)

--------- - - _:{u2

I I I I

lx1 I I aÅ I

I .. a ~1 l~uol I

u<ZJ a 2 )_

} -------- --~ (2) Xz

Lxl I U1 I I

) a

(a)

and the reciprocal relation (1.3.48) gives u~1) = ui2), which is the statement of Betti's reciprocity theorem (see Reddy [6]).

(1.3.52b)

(1.3.52a)

While it is obvious that El~) < Eg) if E1 > E2, we have no clue about the relative magnitudes of c:W and c:g). However, the displacements associated with the two loads are

(1.3.51)

It is important to note the difference, for example, between Vij and Vi! far i =/=- j for an orthotropic material [10]. For example the difference between v12 and v21 for an orthotropic material is illustrated in Figure 1.3.4 with two cases of uniaxial stress for a square element of length a. First a stress O" is applied in the x1-direction as shown in Figure l.3.4a. The resulting strains are

(1) _ O" (1) _ V12 ( ) E11 - - E22 - --(}" 1.3.50 E1 E1

where the direction of loading is denoted by the superscript and negative sign indicates compression. Next, the same value of stress is applied in the x2-direction as shown in Figure l.3.4b. The strains are


A qualitative understanding of the anisotropie behavior of a material ean be obtained by simple tension and shear tests [10]. Applieation of a normal stress to a reetangular bloek of isotropie or orthotropie material leads to only extension in the direetion of the applied stress and eontraetion perpendicular to it, whereas an anisotropie material experienees extension in the direetion of the applied normal stress, eontraetion perpendieular to it, as well as shearing strain (see Figure 1.3.5). Conversely, the applieation of a shearing stress to an anisotropie material eauses

V12 = 0.036, V13 = 0.25, V23 = 0.171

The matrix of elastic coefficients for the materiai can be calculated using Eq. (1.3.54) as

25.16 0.2063 0.1934 o o o 0.2063 4.8240 0.1304 o o o

[C]= 0.1934 0.1304 4.8320 o o o (msi) o o o 0.47 o o o o o o 1.2 o o o o o o 1.36

E1 = 25.1x106 psi , E2 = 4.8 x 106 psi , E3 = 0.75 x 106 psi

G12 = 1.36 x 106 psi , G13 = 1.2 x 106 psi , G23 = 0.47 x 106 psi

The materiai properties of graphite fabric-carbon matrix layers, which are characterized as orthotropic, are:

Example 1.3.4:

(1.3.54) !::i,. = 1 - V12V21 - V23V32 - V31V13 - 2v21V32V13 E1E2E3

Cn = 1 - V23V23 C12 = ll21 + V31 V23 V12 + V32V13 E2E3b..

, E2E3b.. E1E3b..

C13 = l/31 + V21 V32 V13 + V12V23 E2E3b.. E1E2b..

C22 = 1 - V13V31 C23 = V32 + V12V31 v23 + v21v13 E1E3!::i. ' E1E3!::i. E1E3!::i.

C33 = 1 - V12V21 C44 = G23 Css = G31 C66 = G12 E1E2!::i.

,

and using Eq. (1.3.46) the stiffness eoeffieients ean be expressed in terms of the engineering eonstants

(1.3.53)

1 Sn = E1'

1 S22 = E2,

1 S44 = -G ,

23

Comparing Eqs. (1.3.45) and (1.3.47), we note that


E1 = E2 = E3 = E, G12 = G1:3 = G2:3 = G, V12 = V2;3 = V13 = V ( 1.3.55) Consequently, Eqs. (1.3.44) and (1.3.47), in view of the relations (1.3.53), (1.3.54) and (1.3.55), take the form a1 l-v V V o o o E1 a2 V l-v V o o o E2 a3 V V l-v o o o ú3 =A o o o ~(1 - 2v) o o a4 ú4 a5 o o o o ~(1-2v) o ú5 a5 o o o o o ~(1-2v) ú5

(1.3.56)

shearing strain as well as normai strains. Normai stress applied to an orthotropic materiai at an angle to its principal material directions causes it to behave like an anisotropie materiai. The coupling between the two loading modes and the two deformation modes plays a significant role in the testing, analysis, and design of composite materials.

Isotropie Materials

When there exist no preferred directions in the materiai (i.e., the materiai has infinite number of planes of material symmetry), the number of independent elastic coefficients reduces to 2. Such materials are called isotropie. For isotropie materials we have

Figure 1.3.5: Deformation of orthotropic and anisotropie rectangular block under un³axial tension.

Anisotropie

Isotropie and

Orthotropic - - _,

r - - - I

Shear Stress Normai Stress


tr(c)I

(1.3.66)

(1.3.67) 1 1

Cij = E [(1 + v)uij - vukkoij], e = E [(1 + v)u - vtr(u)I]

E vE E vE Uij = 1 +V Cij + (1 + v)(l - 2v) CkkDij' a = 1 +V e + (1 + v)(l - 2v)

where K is the bulk modulus and Õ = G is the shear modulus. In view of the relations between the Lam® constants and engineering constants,

Eqs. (1.3.60) and (1.3.61) can be written in terms of engineering constants:

- 1 mean stress, a =:=3uii, dilatation, e= cii (1.3.63) 1

deviatoric stress, o' = a - iH, deviatoric strain, e' = c - 3tr(c) (1.3.64)

2 uii = (3>. + 2Õ)cii, O-= Ke, K = >. + "3Õ (1.3.65)

The following definitions and constitutive relations are of interest in the sequel:

(1.3.62) G=Õ ê

V=--- 2(Õ + >.)'

E= Õ(3>. + 2Õ) >. + Õ ,

We note the following relations between the Lam® constants >. and Õ and engineering constants E, v and G for anisotropie material [8]:

( 1.3.61)

The strain-stress relations are

(1.3.60)

where >. and Õ are called Lam® constants. Therefore, the stress-strain relation for the isotropie case takes the form

(1.3.59)

Alternatively, the stress-strain relations can be written in more compact form using the fact that a fourth-order isotropie tensor can be expressed as

(1.3.58)

(1.3.57)

c1 1 -v -V o o o u1 cz -V 1 -v o o o u2 c3 1 -v -V 1 o o o U3 = - c4 E o o o l+v o o U4 c5 o o o o l+v o U5 c6 o o o o o l+v 176

where A= E

(1 + v)(l - 2v)


(1.3.73)

Note that the reduced stiffnesses involve four independent materia! constants, E1, E2, v12, and G12. The transverse shear stresses are relateci to the transverse shear strains in an

orthotropic materia! by the relations

S22 E1 Q _ S12 V12E2 Qu = S S S2 ' 12 - s s 2 =

11 22 - 12 1 - V12V21 11 22 - 812 1 - V12V21 Su E2 1

(1.3.72) Q22 = s s 52 1 - V12V21 ' Q55 = - = G12 11 22 - 12 855

where the Qij, called the plane stress-reduced stiffnesses, are given by

(1.3.71)

The strain-stress relations (1.3. 70a) are inverted to obtain the stress-strain relations

(1.3.70b)

and the transverse normal strain is given by

(1.3.70a)

where a and (3 take the values of 1 and 2. Although (]"33 = O, E33 is not zero. The strain-stress relations of an orthotropic body in plane stress state can be

written as [see Eq, (1.3.47)]

(1.3.69)

A state of genemlized plane stress with respect to the x1x2-plane is defined to be one in which

Plane Stress-Reduced Constitutive Relations

1 1 Uo = 2cijkeEijEke = 2(]"ijEij

1 = 2 ( (]"11®ll + (]"22E22 + (]"33®33 + 2(]"12E12 + 2(]"13E13 + 2(]"23E23) (1.3.68)

The strain energy density for a linear isotropie materiai is given by


temperatures.

(1.3. 76) 8T

q = -k Ŀ \7T or qi = -kij - OXj

where k denotes the thermal conductivity iensor of order two. The negative sign in Eq. (1.3.51) indicates that heat fl.ows from higher temperatures to lower

where r is the total boundary enclosing the heat transfer region, r = rr u r q, I' 'r n r q = 0, hc is the convective heat transfer coefficient, Te is a reference (or sink) temperature for convective transfer, <in is the specified boundary fl.ux, and s denotes the position of a point on the boundary,

Thermoelasticity

The thermoelastic problem is governed by the strain-displacement equations of Section 1.3.4, equations of motion of Section 1.3.5, therrnodynamic equations of this section, and the constitutive equations to be given in this section. The constitutive equation of the thermal problem is the well known Fourier's heat conduction law, which states that heat flux is proportional to the gradient of temperature:

(1.3.75a)

(1.3.75b)

T = T(s, t) on rr nĀq+hc(T-Tc)=<Jn(s,t) on fq

(1.3.74) 8T .

PCvBt = -\7 Ŀ q + Q+ a: E

where T is the temperature, q is the heat fl.ux vector, Q is the internal heat generation ( measured per unit volume), p is the density, Cv is the specific heat at constant volume or constant strain, a is the stress tensor, and Ĉ is the strain rate iensor (or time rate of the strain tensor). Equation (1.3.74), termed the generalized heat conduction equation, is used to

determine the temperature distribution in the body. The viscous dissipation couples the thermal problem to the stress problem. Even when the viscous dissipation is neglected, the thermal problem is coupled to the stress problem through constitutive relations, as explained in the next section. The thermal problem for the solid requires the temperature or the heat fl.ux to

be specified on all parts of the boundary enclosing the heat transfer region as

Of the four principles of therrnodynamics, the first law of thermoclynamics and the second law of therrnodynamics are important in the study of deforrnable solids. The first law of thermodynamics, also known as the principle of conservation of energy, states that the time rate of change of the total energy is equal to the sum of the rate of work clone by applied forces and the change of heat content per unit time. The second law of thermodynamics places restrictions on the interconvertibility of heat and work clone. For irreversible processes, the second law states that the entropy production is positive. The therrnodynarnic principles can be expressed, in the Lagrangian description

of deforrnation of solid bodies, as

1.3. 7 Thermodynamic Principles


(1.3.81)

where I' = f1 Uf2, and f1 nf2 = (/J and quantities with a hat are specificd functions on the respective boundaries. The moisture-induced strains {E }M are given by

(1.3.80a)

(1.3.80b)

e = ®( s, t) on r 1 nĿq1={j_1(s,t) on f2

where D denotes the mass diffusitivity iensor of order two, Qf is the ftux vector, and cp f is the moisture source in the dornain. The negative sign in Eq. (1.3. 79b) indicatcs that moisture seeps from higher concentration to lower concentration. The boundary conditions involve specifying the moisture concentration or the ftux normal to the boundary:

(1.3.79a)

(1.3.79b)

8c at = - v . Qf + <P J Qf = -D Ŀ'Ve

Temperature and moisture concentration in f³ber-reinforced composites cause reductions of both strength and stiffness [15-18]. Therefore, it is important to determine the temperature and moisture concentration in composite laminates under given initial and boundary conditions. As described in the previous section, the heat conduction problem described by equations (1.3.74)--(1.3.76) can be used to determine the temperature field. The moisture concentration problem is mathematically similar to the heat

transfer problem. The moisture concentration e in a solid is described by Fick's second law:

Hygrothermal Elasticity

where Sijke are the elastic compliances, and CXij are the thermal coefficients of expansion and relateci to /3ij by /3ij = CijkC akeĿ

(1.3.78)

(1.3. 77b) 8\llo

IJij = -- = cijkc c:kc - /3ij e Ocij

where e= T - To, To is the reference temperature, ry is the entropy density, and /3ij are material coefficients. It is assumed that ry and IJij are initially zero. Equation ( 1.3. 77b) is known as the Duhamel-N eumann law for an anisotropie body. Inverting relations (1.3.77b), we obtain

such that

(1.3.77a)

\llo(®ij, T) = Ui, - TfT 1 p~ 2 = 2cijkC Eij EkC - /3ij Eije - 2To e

The constitutive equations of thermoelasticity are derived by assuming the existence of the Helmholtz free-energy function \ll0 = \ll0(®ij, T) (see [11~14])


(1.3.85a)

<])o(Eij, [i, T) = Uo - E Ŀ D - 'T]T 1

= 2,Cijkf Eij Ekf - Cijk Eij[k - /3ij Eij()

1 PCv 2 - -Eke EkEe - Pk[k() - -e 2 2To

where p is the vector of pyroelectric coefficients. The coupling between the mechanical, thermal, and electrical fields can be

established using thermodynamical principles and Maxwell's relations. Analogous to the strain energy function Uo for elasticity and the Helmholtz free-energy function '110 for thermoelasticity, we assume the existence of a function <])o

(1.3.84) fj,_p = p/j,_T

Note that dkij is symmetric with respect to indices i and j because of the symmetry of Eij (note that i,j, k = 1, 2, 3). The pyroelectic effect is another phenomenon that relates temperature changes

to polarization of a materiai. For a small temperature change /j,_T, the change in polarization vector fj,_p is given by

(1.3.83b)

where d is the third-order tensor of piezoelectric moduli. The inverse effect relates the electric field vector [ to the linear strain tensor E by

(1.3.83a)

where To and co are reference values from which the strains and stresses are measured. In view of the similarity between the thermal and moisture strains, we will use only thermal strains to show their contribution to governing equations in the sequel.

Electroelasticity

Electroelasticity deals with the phenomena caused by interactions between electric and mechanical fields. The piezoelectric effect is one such phenomenon, and it is concerned with the effect of the electric charge on the deformation [14- 16]. A laminated structure with piezoelectric laminae receives actuation through an applied electric field, and the piezoelectric laminae send electric signals that are used to measure the motion or deformation of the laminate. In these problems, the electric charge that is applied to actuate a structure provides an additional body force to the stress analysis problem, much the same way a temperature field induces a body force through thermal strains. The p³ezoelectric effect is descr³bed by the polarization vector P, wh³ch represents

the electric moment per unit volume or polarization charge per unit area. It is related to the stress tensor by the relat³on (see [14-17])

(1.3.82) {E} = [ S]{ o"} + { ar }( T - To) + {a M }(e - co)

where { aM} is the vector of coefficients of hygroscopic expansion. Thus, the hygrothermal strains have the same form as the thermal strains [see Eq, (1.3.76)]. The total strains are given by


This completes a review of the basic equations of solid mechanics. In the coming chapters reference is made to many of the equations presented here.

(l.3.90b) where

(l.3.90a)

This assumption allows us to write Eq. (1.3.88), in view of Eq. (1.3.87b ), as

(1.3.89) E = -\J'ljJ

It is often assumed that the electric field E is derivable from an electric scalar potential function 1/J:

(1.3.88)

Note that the range of summation in (l.3.87a-c) is different for different terms: i, j = 1, 2, Ŀ Ŀ Ŀ, 6; k, t' = 1, 2, 3. For the generai anisotropie material, there are 21 independent elastic constants, 18 piezoelectric constants, 6 dielectric constants, 3 pyroelectric constants, and 6 thermal expansion coefficients. Maxwell's equation governing the electric displacement vector is given by

(1.3.87a) (1.3.87b)

(1.3.87c)

cr; = C;jEj - e;kEk - /3iB

Dk = ekJ®J + EkeEe + Pke PCv

T/ = /3;i::; + PkEk +-e To

where Cijkf are the elastic moduli, eijk are the piezoelectric moduli, Eij are the dielectric constants, Pk are the pyroelectric constants, /3ij are the stress-temperature expansion coefficients, Cv is the specific heat per unit mass, and To is the reference temperature. In single-subscript notation, Eqs. (l.3.86a-c) can be expressed as

(l.3.86a) (l.3.86b)

(l.3.86c)

criJ = Cijke ese - e;jkEk - /3;je

Di; = eijk ®ij + EkeEe + Pke PCv

T/ = /3;j Eij + PkEk + To B

where crij are the components of the stress tensor, D; are the components of the electric displacement vector, and T/ is the entropy. Use of Eq. (1.3.85a) in Eq. (l.3.85b) gives the constitutive equations of a deformable piezoelectric medium:

(l.3.85b) o<Po crij =-a '

Eij

which is called the electric Gibbs free-energy function or enthalpy function, such that


From purely geometrica! considerations, a given mechanical system can take many possible configurations consistent with the geometrie constraints of the system. Of all the possible configurations, only one corresponds to the actual configuration, and it is this configuration that satisfies Newton's second law (i.e., equations of equilibrium or motion of the system). The set of configurations that satisfy the geometrie constraints but not necessarily Newton's second law is called the set of admissible configurations. These configurations are restricted to a neighborhood of the true configuration so that they are obtained from infinitesimal variations of the true configuration. During such variations, the geometrie constraints of the system are not violated and all the forces are fixed at their actual values. When a mechanical system experiences such variations in its configuration, it is said to undergo virtual displacements from its true or actual configuration. These displacements need not bave any relationship to the actual displacements that might occur due to a change in the applied loads. The displacements are called virtual because they are imagined to take piace (i.e., hypothetical) while the actual loads acting at their fixed values. The virtual displacements at the boundary points at which t.he geometrie conditions (or displacements) are specified, are necessarily zero. The work done by the actual forces moving through virtual displacements is

called virtual work. The virtual work done by actual forces F in a body Oo in moving through the virtual displacements bu is given by

bW = { F Ŀ bu dv (1.4.1) lno

1.4.2 Virtual Displacements and Virtual Work

In solid mechanics some of the laws of physics take several alternative forms. For example, the principle of conservation of linear momentum, which requires that the vector sum of all applied forces acting on a body be equal to the total time rate of momentum of the body, is known in mechanics as Newton's second law and it is also derivable from a variational principle. The use of Newton's laws to determine the governing equations of a structural problem requires isolation of a typical volume element of the structure with all its applied and reactive forces (i.e., the free-body diagram of the element). For complicated systems the procedure becomes more cumbersome and intractable. In addition, the type of boundary conditions to be used in conjunction with the derived equations is not always clear. In a variational approach, the governing equations are obtained by the principle of virtual displacements or by seeking the minimum of the total potential energy of the system. The variational approach, applicable to linear or nonlinear theories, is useful both in deriving governing equations and boundary conditions, and obtaining approximate solutions by variational methods. In the context of the present study, the principle of virtual displacements will be

used to derive the equations of motion of laminated plates. Hence, it is useful to study variational principles and methods (see Reddy [6] for additional details). We begin with the concepts of virtual displacements and forces.

1.4 Virtual Work Principles 1.4.1 Introduction


(1.4.4)

Thus, the total virtual work done by forces due to ali the stresses in a volume element ( that originally occupied the materiai element dv) in moving through their respective displacements is

Here Eij denote the strain components and O"ij the stress components. Similarly, the work done by the farce due to stress 0"12 in the body is

The work clone by the farce due to actual stress 0"11, for example, in moving through the virtual displacement 8u1 = 8cudx1 is

(1.4.3)

where ds denotes a surface element and T (Y denotes the portion of the boundary on which stresses are specified. The negative sign in Eq. (1.4.2) indicates that the work is performed on the body. It is understood that the displacements are specified on the remaining portion r11 = I' - I' (Y of the boundary I'. Therefore, the virtual displacements are zero on I' 11, irrespective of whether u is specified to be zero or not. For example, a bar fixed at one end (x = O) and subjected to an axial load at the other end ( x = L) can be imagined to have a virtual displacement bu( x), provided bu(O) = O, because the actual displacement is specified at x = O. Thus, one may select ou(x) =cx, where e is an arbitrary constant. Recall that the deformation of solid body acted upon by forces can be measured

in terms of strains and that the body experiences internal stresses. The forces associated with the stress field move the materiai particles through displacements corresponding to the strain field in the body, and hence work is clone. The work clone by these internal forces in moving through displacements of the materiai particles is called internal uiork. Note that the work clone on the body is responsible for the internal work stored in the body. The internal virtual work due to the virtual displacement bu can be computed as

follows. Suppose that an infinitesima! materiai element of volume dv = dx1dx2dx3 of the body experiences virtual strains &ij due to the virtual displacements Siu, where [see Eq. (1.3.12)]

(1.4.2) sv = - (fn0 f Ŀ ou dv + 1r a t Ŀ 8u ds)

where dv denotes the volume element dv = dx1dx2dx3 in the materiai body no. The external virtual work done due to virtual displacements 8u in a solid body

no subjected to body forces f per unit volume and surface tractions t per unit area of the boundary I' (Y is given by


(1.4.10) >:p = {)F s: fJF >: I u fJu ou + fJu' ou

Since Ou is small, terms involving squares and higher powers of Su can be neglected. We bave

(1.4.9) a F a F , 1 82 F 2 1 a2 F , 2 8F = - Ŀ Su. + - Ŀ ou + -- (ou) + -- (ou) + Ŀ .. fJu au' 2 8u2 2 8u12

The first variation of F is

(1.4.8)

The delta symbol t5 used in conjunction with virtual displacements and forces can be interpreted as an operator, called the variational operator. It is used to denote a variation (or change) in a given quantity; i.e., 8u denotes a variation in u. Thus t5 is an operator that produces virtual change or variation Su in a dependent variable u, in much the same way as dx denotes a change in x, and 8u is called the first variation of u. The operator proves to be very useful in constructing virtual work statements and deriving governing equations from virtual work principles, as will be shown shortly. There is an analogy between the variational operator t5 and the total differential

operator d. To see this consider a function F of the dependent variable u and its derivative u' = du/dx in one dimension. The total differential of F, for fixed x, is

1.4.3 Variational Operator and Euler Equations

In the present study we will not consider complementary energy principles.

(1.4.7a) (1.4.7b)

[Dajil,j + 8fi =O in Oo St; :::::;: 0Gjinj = 0 on I' O'

(1.4.6) 8U* = r Eij Oaij dv lo.o The expression in Eq. (1.4.6) is also known as the virtual complementary strain energy. The virtual forces ( 8 fi, oti) and virtual stresses ( Oaij) should be such that the stress equilibrium equations [see Eq. (1.3.27b)] and stress boundary conditions [see Eq. (1.2.25)] are satisfied:

Equation (1.4.5) is valid for any material body irrespective of its constitutive behavior. The expression in Eq, (1.4.5) is called the virtual strain energy of a deformable body. The internal virtual work done by virtual stresses 8aij in moving through the

actual strains Eij is

(1.4.5)

The total internal virtual work done is obtained by integrating the above expression over the entire volume of the body

8U = { aij &ij dv lo.o


(1.4.20) I(au) = a2I(u)

for all constants a and (3 and dependent variables u and v. A quadratic functional is one which satisfies the relation

(1.4.19) I(au + (3v) = aI(u) + (3I(v)

qualifies as a functional for all integrable and square-integrable functions u(x). Note that J(u) is a number whose value depends on the choice of u. A functional is said to be linear if

I(u) = .lL [au(x) + bu'(x) + cu"(x)] dx

For example, the integrai expression

(1.4.18) I:H---7R

Integrai expressions whose integrands are functions of dependent variables and their derivati ves are called functionals. Mathematically, a functional is a real number (or scalar) obtained by operating on functions (dependent variables) from a given set (or vector space). Thus, a functional I(-) is an operator which maps functions u of a vector space H into a real number I ( 1l) in the set of real numbers, R:

Functionals

where, for example, Du denotes the partial variation of G with respect to u.

(1.4.17)

where F1 = F1(u) and F2 = F2(u). If G = G(u,v,w) is a function of severa! dependent variables (and possibly their derivatives), the total variation is the sum of partial variations:

(1.4.11)

(1.4.12)

(1.4.13) (1.4.14)

(1.4.15)

(1.4.16)

8(\7u) = \7(8u)

fJ (l u dD) = .l Su dD 8 (F1 Ñ F2) = 8F1 Ñ 8F2 s (F1F2) = 8F1 F2 + F1 8F2 s (F1) = 6F1 _ r. (fJF;) F2 F2 F2

8 (F1t = n (F1)n-l 8F1

Since x is fixed during the variation of 'U to u+8u, we have dx= O in Eq. (1.4,8) and the analogy between 8F in Eq. (1.4.10) and dF in Eq. (1.4.8) becomes apparent: the variational operator, 8, is a differential operator with respect to the dependent variable, u. Indeed, the laws of variation of sums, products, ratios, powers, and so forth, are completely analogous to the corresponding laws of differentiation. The following properties of the variational operator should be noted:


(1.4.26) 8I =o

The necessary condition for the functional to have a minimum or maximum is (analogous to minima or maxima of functions) that its first variation be zero:

(1.4.25) I(u) = lb F(x,u,u') dx, u(a) = Ua, u(b) = Ub

In most of our study in this book, we shall be interested in the use of Eqs. (1.4.24a,b) because they provide the means to the determination of the governing equations and boundary conditions and their solution by the variational methods. Consider the question of finding the extremum (i.e., minimum or maximum) of

the functional

(1.4.24a)

(1.4.24b)

if lb G17 dx+ B(a)17(a) =O

then G = O in a < x < b and B (a) = O

Since an integrai of a positive function is positive, the above statement irnplies that G = O. A more generai statement of the fundamental lemma is as follows: If 17 is arbitrary in a ç: x <band 17(a) is arbitrary, then

holds for any arbitrary continuous function 17(x), for all x in (a, b), then it follows that G = O in (a, b). A mathematical proof of the lemma can be found in most books on variational calculus. A simple proof of the lemma follows. Since 17 is arbitrary, it can be replaced by G. We have

(1.4.23)

The fundamental lemma of calculus of variations can be stated as follows: for any integrable function G, if the statement

lb G Ā 17 dx= O

Fundamental Lemma of Variational Calculus

Thus, the variation of a functional can be readily calculated.

(1.4.22) Jb Jb Jb (a F a F ) 8I = 8 a F dx= a 8F dx= a au ou + au18u1 dx

where F is a function, in general, of x, u and du/dx = u', The first variation of the functional I is

(1.4.21) I( u) = lb F(x, u, u') dx

for all constants a and dependent variable u. The first variation of a functional J( u) of u (and its derivatives) can be calculated

using the definition in Eq. (1.4.10). For instance consider the functional J(u) defined in the interval (a, b)


Thus the necessary condition for I ( u) to be an extremum at u = u( x) is that u( x) be the solution of Eq. (1.4.28).

(1.4.28) G = f)F - !!:__ (8F) =O m a ç: x < b Bu dx ou'

which must hold for any bu in (a, b). In view of the fundamental lemma of calculus of variations ( ry = /5u), it follows that

b [8F d (oF)] o = r - - - --, Su dx l; ou dx ou

There are two parts to this expression: a varied quantity and its coefficient. The variable u that is subjected to variation is called the primary variable. The coefficient of the varied quantity, i.e., the expression next to Su in the boundary term, is called a secotuiaru variable. The product of the primary variable (or its variation) with the secondary variable often represents the work done (or virtual work done). The specification of the primary variable at a boundary point is termed the esseniial boundary condition, and the specification of the secondary variable ( 8F / ou') is called the natural boundary condition. In solid mechanics, these are known as the geometrie and farce boundary conditions, respectively. All admissible variations must satisfy the homogeneous form of the essential (or geometrie) boundary conditions: /5u(a) = O and bu(b) = O. Elsewhere, a < x < b, Su is arbitrary. Returning to Eq. (1.4.27), we note that the boundary terms drop out because of

the conditions on Su. We have

[8F] . /5u ou'

Let us first examine the boundary expression:

(1.4.27)

rb (aF er ') O= l; ou 8u + ou'8u dx

= jb (oF Du + 8F d8u) dx a ou ou' dx

= jb [8F - !!___ (8F)] 8u dx+ [8F -r a ou dx ou' Bu' a

Note that 8u' = 8(du/dx) = d(8u)/dx. We cannot use the fundamental lemma in the above equation because it is not in the form of Eq. (1.4.24). To recast the above equation in the form of Eq. (1.4.24), we integrate the second term by parts and obtain

lb (8F 8F ') o = a ou /5u + ou' 8u dx

Using Eq, (1.4.10) we obtain


But by Newton's second law, the vector sum of the forces acting on a body in static equilibrium is zero. This implies that the tota! virtual work, bU + bV, is equa! to zero. Thus, for a body in equilibrium the tota! virtual work done due to

(1.4.32)

The internal virtual work done bU is zero because a rigid body does not undergo any strains (hence virtual strains are zero). In addition, the virtual displacements bui, bu2, Ŀ Ŀ Ŀ, bun should all be the same, say bu, for a rigid body. Thus, we have

(1.4.31) n

bV = -[F1 Ŀbui+ F2 Ŀ bu2 + Ŀ Ŀ Ŀ + Fn Ā bun] = - LFi Ābui i=l

Recall that the virtual work due to virtual displacements is the work done by actual forces in displacing the body through virtual displacements that are consistent with the geometrie constraints. All applied forces are kept constant during the virtual displacements. Consider a rigid body acted upon by a set of applied forces F1, F2, ... F n, and suppose that the points of application of these forces are subjected to the virtual displacements bu1, bu2, Ŀ Ŀ Ŀ, bun, respectively. The virtual displacement bui has no relation to buj for i i- j. The external virtual work done by the virtual displacements is

1.4.4 Principle of Virtual Displacements

Both Eq. (1.4.30a) and Eq. (1.4.30b) are called the Euler-Lagrange equations. Note that the boundary conditions that are a part of the Euler-Lagrange equations always belong to the class of natural boundary conditions. Now we have all the necessary concepts and tools in place to study the principles

of virtual work. In the next section, we discuss the principle of virtual displacements and its special case, the principle of minimum total potential energy. Fora discussion of the principle of virtual forces and its special cases, consult Reddy [6].

(1.4.30b)

(1.4.30a) aF d (aF) OU - dx OU1 = O, a < X < b

([)F) =0 atx=b au'

Since Su is arbitrary in (a, b) and bu(b) is arbitrary, the above equation implies, in view of Eq. (1.4.28), that both the integral expression and the boundary term be zero separately:

(1.4.29) 1b [aF d (aF)] (aF) O= - - - --, Su dx+ -, bu(b) a au dx Bu au x=b

If u(a) = Ua and bu(b) is arbitrary (i.e., u(a) is specified but u is not specified at x = b), then bu(a) =O and we have from Eq. (1.4.27) the result


where the summation on repeated subscripts is implied. The Euler-Lagrange equations associated with the statement (1.4.35) of the

principle of virtual displacements are nothing but the equilibrium equations of the

(1.4.35)

where <J : Se denotes the "double dot product," Do is the volume of the undeformed body, and dv and ds denote the volume and surface elements of Do. Writing in terms of the Cartesian rectangular components, Eq. (1.4.34) takes the form

(1.4.34)

J ust as we derived the Euler-Lagrange equations associated with the statement 8I = O, we can derive them for the statement in Eq. (1.4.33). However, first we must identify 6U and 6V for a given problem. The principle of virtual work is independent of any constitutive law and applies to both elastic (linear and nonlinear) and inelastic continua. For a solid body, the external and internal virtual work expressions are given in

Eqs. (1.4.2) and (1.4.5), respectively. The principle can be expressed as

r (J : Se dv - r f . 6u dv - r t . 6u ds = o lno lno lra

(1.4.33) 6U +8V = 8W =O

virtual displacements is zero. This statement is known as the principle of virtual displacements. The principle also holds for continuous, deformable bodies, for which 8U is not zero. In this section, the principle of virtual displacements and its special case are described since they play an important role in the formulation of theories (e.g., plate theories) and their analysis by variational methods of approximation. Considera continuous body B in equilibrium under the action of body forces f and

surface tractions t. Let the reference configuration be the initial configuration c0, whose volume is denoted as Do. Suppose that over portion r.u of the total boundary r of the region Do the displacements are specified to be ½, and on portion r a the tractions are specified to be t. The boundary portions T u and r a are disjoint (i.e., do not overlap), and their sum is the tota} boundary r. Let u be the displacement vector corresponding to the equilibrium configuration of the body, and let <J and E be the associateci stress and strain tensors, respectively. The set of admissible configurations are defined by sufficiently differentiable functions that satisfy the geometrie boundary conditions: u = ½ on r uĀ If the body is in equilibrium, then of all admissible configurations, the actual one

corresponding to the equilibrium configuration makes the total virtual work clone zero. In order to determine the equations governing the equilibrium configuration e' we let the body experience a virtual displacement 8u from the true configuration C. The virtual displacements are arbitrary, continuous functions except that they satisfy the homogeneous form of geometrie boundary conditions; i.e., they must belong to the set of admissible variations. The principle of virtual displacements can be stated as: if a continuous body

is in equilibrium, the virtual work of all actual forces in moving through a virtual displacement is zero:


where ( u, v, w) are the displacements of a point (x, y, z) along the x, y and z coordinates, respectively, and (u0, w0) are the displacements of the point (x, O, O). Under the assumption of smallness of strains

(1.4.41) dw0 u=uo(x)-z-, v=O, w=wo(x) dx

Consider the bending of a beam of length L, Young's modulus E and moment of inertia I, and subjected to distributed axial force f(x) and transverse load q (see Figure 1.4.1). Under the assumption of small strains and displacements, we derive the governing differential equation of the beam using the Euler-Bernoulli hypotheses, which assumes that straight lines perpendicular to the beam axis before deformation remain (1) straight, (2) perpendicular to the tangent line to the beam axis, and (3) inextensible after deformation. These assumptions lead to the displacement field (see Figure 1.4.la)

Example 1.4.1: (Euler-Bernoulli beam theory) ------------------

Equations (1.4.39) and (1.4.40) are the Euler-Lagrange equations associated with the principle of virtual displacements for a body undergoing small deformation. The stress boundary conditions in Eq, (1.4.40) are the natural boundary conditions. The principle of virtual displacements is applicable to any continuous body with arbitrary constitutive behavior (i.e., elastic or inelastic).

(1.4.40)

(1.4.39)

Because the virtual displacements are arbitrary in no and on r (7, Eq. (1.4.38) yields the following equations [cf., Eq. (1.3.27b)]

®fo . __!:l + li = o in no OXj

(jijnj - ti = o on r (7

(1.4.38)

Since r = r u u r (7 and 8ui =o on r U) we have

(1.4.37)

Substituting &ij from the above equation into Eq. (1.4.35), and using the divergence theorem, Eq. (1.2.38), to transfer differentiation from Su; to its coefficient, one obtains ((jij = (jji)

(1.4.36) 1 8E = -(8u Ā + Su. Ā) iJ 2 i,J J,i '

3-D elasticity theory. Recall the strain-displacement equations from Eq. (1.3.11). The virtual strains &ij are relateci to the virtual displacements Su, by


(1.4.45a, b) d2M -- =q(x) dx2

dN - dx = f(:r),

Here A denotes the area of cross section. Equations (1.4.43b) and (l.4.43c) can be combined into the single equation so that Eqs. (1.4.43a-c) reduce to

(1.4.44) N(x) = (a,, dA, M(x) = ;Ŀ a,,xz dA, V(x) = ( axzdA .!A ĀA .!A

where N(x) is the net axial force, M(x) the bending moment, and V(x) the shear force, which are known as the stress resultants, and they are defined in terms of the stresses axx and axz on a cross section as (see Figure 1.4.lc)

(1.4.43c)

(1.4.43b)

(1.4.43a) dN - dx = f(x). dV

- - = q(x) dx

V- dM =0 dx

LFx =0: LFz =0:

First we derive the equilibrium equations using Newton's second law of motion. Summing the forces and moments on an element of the beam (see Figure 1.4.lb) gives the following equilibrium equations:

(1.4.42)

and rotations, the only nonzero strain is

Figure 1.4.1: Bending of beams. (a) Kinematics of deformation of an Euler- Bernoulli beam. (b) Equilibrium of a beam element. (e) Definitions (or internal equilibrium) of stress resultants.

(e) (b)

M(x)

N(~ j(x) ---q------------~(x)+ N(x)

V(x) V(x)+LIV(x)

q(x)

M(x)

N(~) fix) --------- ~---- .... -

V(x) V(x)+LIV(x)

(a)

z

dWo '~ax '

~u0ilwo ~--~l----un-z-a-x-


and the virtual work done by an axial point load PL in moving through 8uo(L) and a transverse point load FL in moving through the virtual displacement 8w0(L) is (see Figure 1.4.2)

ML (- d/5wo) dx L

The virtual work done by any applied point loads (and moments) must be added to 8V in Eq. (1.4.51b). For example, the virtual work done by the counterclockwise moment ML at x = L in rotating through the virtual rotati on d~',;;0 ( L) is

(1.4.51b) /5V = - lL (fl5uo + q8wo) dx

The virtual work done by the external distributed forces f(x) and q(x) in moving through the displacements 15uo and 8wo, respectively, is

(1.4.51a)

where all other stresses are assumed to be zero; i.e., the Euler-Bernoulli assumptions are invoked. The actual strain in the Euler-Bernoulli beam theory is given by Eq. (1.4.42). The virtual strain &xx is related to the virtual displacements (15uo,15wo) by &xx= (d/5uo/dx)-z(d215wo/dx2). Substituting this expression into (1.4.50), we obtain

(1.4.50)

Next, we derive the governing equations (l.4.45a,b) using the principle of virtual displacements. Note that for the problem at hand the only nonzero stress is CTxxĀ Hence, the internal virtual work done per unit length of the beam by the actual internal farce CTxx dA in moving through the virtual displacements DExx dx is given by CTxxdA Ā &xxdx. The tota! internal virtual work done is

(1.4.49) N Mz CTxx =A+ J,

where I is the moment of inertia about the axis of bending (y-axis) and z ³s the transverse coordinate. Note that the x-axis is taken through the geometrie centroid of the cross section so that JA zdA =O. Using the relations in Eq. (1.4.48) in Eq. (1.4.46), we obtain

(1.4.48) M El

du0 N d2w0 dx EA' dx2

or

(1.4.47b) r r (duo d2wo) d2wo M(x) =}A CTxxZ dA =E }A dX - z dx2 z dA =-El dx2

(l.4.47a) N(x) = 1 crxx dA = E~:o 1 dA = EA d;: First, note that

(1.4.46)

The stress resultants (N, M) can be related back to the stress CTxx using the linear elastic constitutive relation for an isotropie materiai as [see Eq. (1.4.42)]


Figure 1.4.2: A cantilever beam with distributed loads f and q, and concentrated loads PL, FL and ML at the right end.

q(x) uJO) =O Wo(O) =O w;,(O)=O

which are the same as those in Eqs. (l.4.45a,b).

( l.4.55b) 8w0:

( l.4.55a) dN - dx -f=O, O<x<L

d2M - dx2 - q = O , O < x < L

8u0:

First, consider the integrai expressions in (1.4.54). Since 15u0 and 15w0 are independent and arbitrary in O < x < L, we obtain the Euler equations

1L [ (-: - f) 15'U0 + (- ~:: - q) 6wo] dx+ [N(L) - PL]l5'Uo(L) - N(0)6'Uo(O) - [M(L) - ML] ( d~;o) x=L + M(O) ( d~;o) o:=D

+ [(:t=L -FL]15wo(L)-(~~)x=Ol5wo(0)=0 (1.4.54)

Note from the boundary terms that 'UQ, wo and dwo/dx are primary variables and N, dM/dx =V and M are the secondary variables of the problem. We have

lL [ ( dN ) ( d2 M ) ] [ d6wo dM ] L -- -f 6'U0 + --- - q 6wo dx+ N6'U0 - M-- + -6wo . 0 dx dx2 dx dx o

( dl5wo) - Nh -~ x=L - PL6'Uo(L) - FLtiwo(L) =o

To obtain the Euler-Lagrange equations associateci with the virtual work staternent (1.4.47), integrate the first terrn by parts once and the second term by parts twice and obtain

1L ( d6'Uo d26w0 ) ( d6w0) N-- - M--2- - f6'U0 - q6w0 dx - ML --- - PL6'U0(L) - FL6wo(L) =O 0 dx dx dx x zx L

(1.4.53)

The principle of virtual displacements states that if the beam is in equilibriurn we rnust have 6U + 6V =O or

(l.4.52)

'I'hus, the tota! external virtual work dono is


(1.4.62a) U = { Uo dv lno

where U is the internal strain energy functional

{ 8Uo dv= 8U Jrio The first integrai is equal to

(1.4.61b)

or, in component form,

(1.4.61a) f a;:o : 8E dv - [ { f . 8u dv + { t Ŀ 8u ds] = O }0,0 ae }0,0 Jr a

Equation (1.4.60) represents the constitutive equation of an hyperelastic materiai. The strain energy density Ui, is a single-valued function of strains at a point and is assumed to be positive definite. The statement of the principle of virtual displacements, Eq. (1.4.34), can be expressed in terms of the strain energy density Uo:

(1.4.60) 8Uo (J"ij = --

&ij or 8Uo

(}"=-- ¨e

A special case of the principle of virtual displacements that deals with linear as well as nonlinear elastic bodies is known as the principle of minimum total potential energy. For elastic bodies (in the absence of temperature variations) there exists a strain energy density function Uo such that

The Principle of Minimum Total Potential Energy

We note that Eqs. (1.4.55a) and (1.4.57) together define axial deformation, while Eqs. (1.4.55b), (1.4.58) and (1.4.59) describe bending deformation of the beam. Thesc sets of equations can be solved independently as N is only a function of u0 and Misa function of only wo [see Eq. (1.4.48)].

(1.4.59) M(L) ~ML= O, at x = L (dl5wo) : dx x=L

(1.4.57)

(1.4.58)

N(L)~PL=O, at x=L

(ddM) ~FL=O, at x=L X x=L

15uo(L):

15wo(L):

and they are arbitrary at x = L. Consequently, the second, fourth and sixth boundary expressions vanish, and we have the (natural) boundary conditions resulting from the virtual work principle:

(1.4.56) (dl5wo) = 0 dx x=O

/5u0(0) =O, 15wo(O) =O,

Next, consider the boundary expressions in (1.4.54). If the beam is fixed at x =O and subjected to forces P1, ML, and FL, the virtual displacements 15uo and 15wo must satisfy the conditions


O -1L (EAduo dinu, Eld2w0 d2bw0 ts e ) d. - --- + ----- - uuo -quwo x 0 dx dx dx2 dx2

. ( dbwo) - PL¸uo(L) - ML -~ x=L - FL¸wo(L)

The tota! potential energy principle requires that b(U + V) = O:

(1.4.67)

t' [EA (du0)2 El (d2w0)2 l Il= U +V= lo 2 dx + 2 dx2 - fuo - qwo dx

( dwo) - PLuo(L) - ML - dx r.=L - FLwo(L)

The tota! potential energy of the bearn is given by

(1.4.66) V= - [.lL (fuo + qwo)dx + PLuo(L) +Ah (- d;,~0) x=L + FLwo(L)]

where Eq. (1.4.48) is used to write the last expression for U. The work done by external applied loads i, q, lvh, PL and FL is

(1.4.65) U = ~ t' (Nduo - Md211:0) dx=~ t' [EA (duo )2 ç et (d2wo) 2] dx 2 lo d::r dx2 2 lo dx dx2

We consider the cantilever beam problem of Example 1.4.1 (see Figure 1.4.2). The minimum tota! potential energy principle requires usto construct the tota! potential energy (i.e., sum of the strain energy and potential energy due to appl³ed loads) of the bcam and set its first variation to zero to obtain the Euler-Lagrange equations of the functional. The tota! strain energy stored in the beam is

Example 1.4.2:

where u is the true solution and ii is any admissible displacement ficld. The equality holds only if u = ii.

(1.4.64) II(u) < II(ii)

The sum U + V = II is called the total potential energy of the elastic body. The statement in Eq. (1.4.63) is known as the principle of minimum total potential energy. It means that of all admissible displacements, those which satisfy the equilibrium equaiions make the total potential energy a minimum:

(1.4.63) bU + bV = b(U +V)= 811 =O

Then the principle of virtual work takes the form

(1.4.62b)

Suppose that there exists a potential V whose first variation is


{1L ( d2w0 d2v2 ) ( dv2) } +(3 EI-2---2 - qv2 dx - ML -- -FLv2(L) O dx dx dx x=L

Now, consider the second integrai and the boundary terms

(1.4. 70a)

For the example problem we have

(1.4.69a)

(1.4.69b) (dv2) _ 0 (IX :r.=0 -

w = wo + f3v2, (3 small, v2(0) =O,

u = uo + av1, a small, v1(0) =O

Equations (1.4.55a,b), and (1.4.57)-(1.4.59) are the same as above when N and Mare replaced in terms of u0 and wo using Eq. (1.4.47a,b), i.e., when the beam constitutive equations are used. The minimum property of the tota! potential energy can be established by considering an

arbitrary admissible displacement field, (u, w)

(1.4.68e)

(1.4.68d)

(1.4.68c)

(1.4.68b)

(1.4.68a) _ _!}_ (EA duo) - f =O , O < x < L dx dx

!!!_(Eld2w0)- =0 O<x<L dx2 dx2 q '

(EAduo) -PL=O dx x=L

(-Eld;::io) :r.=L - ML= 0

[- d~ ( Eld;~o) L=L - FL =o

Integration by parts of the first two terms, and use of Eq. (1.4.56) and the property that liuo and 8w0 are arbitrary both in (O, L) and at x = L, yields the Euler equations


(1.4.73a) (l.4.73b)

8u = O on S1 for all t bu(x, t1) = 8u(x, t2) =O far all x

where m is the mass, a the acceleration vector, and F is the resultant of all forces acting on the body. The actual path u = u(x, t) followed by a material particle in position x in the body is varied, consistent with kinematic ( essential) boundary conditions, to u + bu, where bu is the admissible variation (or virtual displacement) of the path. We suppose that the varied path differs from the actual path except at initial and final times, t1 and t2, respectively. Thus, an admissible variation bu satisfies the conditions,

(1.4.72) F-ma=O

Harnilton's Principle

Hamilton's principle is a generalization of the principle of virtual displacements to dynamics of systems. The principle assumes that the system under consideration is characterized by two energy functions; a kinetic energy K and a potential energy II. For deformable bodies, the energies can be expressed in terms of the dependent variables (which are functions of position) of the problem. Hamilton's principle may be considered as dynamics version of the principle of virtual displacements [6]. Newton's second law of motion applied to deformable bodies expresses the global

statement of the principle of conservation of linear momentum. However, it should be noted that Newton's second law of motion for continuous media is not suff³cient to determine its motion u = u(x, t); the kinematic conditions and constitutive equations discussed in the previous sections are needed to completely determine the motion. Newton's second law of motion for a continuous body can be written in genera}

terms as

and the equality holds only when ii= u0 and w = w0. Thus II(ii, w) is greater than II(u0, w0) when ii: i= wo and Āu i= u0, establishing the minimum character of the tota! potential energy of the beam, One may note that in this example, we considered axial deformation of a bar (set wo = O) as

well as pure bending of a beam (set uo =O). These equations are uncoupled for the case of srnall strains. The tota! potential energy is the minimum with rcspect to both uo and wo.

(1.4.71)

TI(u, Ŀw) = Il(u0, wo) + foL [ n2 E2A ( ~~:) 2 + (J2 ~I (~:i) 2] dx 2: II( uo, wo)

(1.4.70b)

= n { 1L [- :x ( EA dd~) - f J vi dx+ [ ( EA ~~ - PL) V1 L=L} + (3{ t' [ d22 (E!d2~0) - q] V2 dx+ [-E!d2~0 - A,h] (- dv2) }0 dx dx dx x=L dx x=L

+ [- d~ ( Eld:~O) - FL] x=L v2(L) The boundary terms at x =O are zero because of the conditions in Eq, (1.4.69a,b). Sincc (u0, w0) is the true solution of the problem, ali terms in Eq, (1.4. 70b) are zero. Thus, Eq, ( 1.4. 70a) becomes

EQUATIONS OF ANISOTROPIC ELASTJCITY 53

(1.4.78) 1t2 8 [K - (V+ U)]dt =O ti

Substituting Eqs. (1.4.77a,b) into Eq. (1.4.76), we obtain

(1.4.77b)

and that there exists a strain energy density function U0 = Uo(Eij) such that

(1.4.77a) 8V = - (i f Ŀ 8u dV + fs2 t Ŀ 8u dS)

In arriving at the expression in Eq. (1.4.76), integration-by-parts is used on the first term; the integrateci terms vanish because of the initial and final conditions in Eq. (1.4.73b). Equation (1.4.76) is known as the general forrn of Hamilton's principle far a continuous medium (conservative or not, and elastic or not). For an ideal elastic body, we recall from the previous discussions that the farces

f and t are conservative,

1t2 [1 au aou 1 ( ..... .....) j , J - p-a Ā-a dV+ fĀ8u-(}:0E dV+ tĀ8udSdt=O (1.4.76) t1 V t t V S2

or

1:2 {i p ~:~ Ā 8u dV - [i ( f Ŀ 8u - 7f : o?) dV + fs2 t Ŀ 8u dS] }dt =O where p is the mass density ( can be a function of position) of the medium. We have the result

(1.4.75) 1 a2u p- Ā8u dV V 8t2

where f is the body farce vector, t the specified surface traction vector, and 7f and E are the stress and strain tensors. The last term in Eq. (1.4.74) represents the virtual work of internal farces stored in the body. The strains oú are assumed to be compatible in the sense that the strain-displacement relations (1.3.11) are satisfied. The work clone by the inertia farce ma in moving through the virtual displacement bu is given by

(1.4.74) { f Ŀ 8u dV + { t Ŀ 8u dS - { 7J : o? dV lv l~ lv

where 81 denotes the portion of the boundary of the body where the displacement vector u is specified. Note that the scalar product of Eq. (1.4.72) with bu gives work clone at point x, because F, a, and u are vector functions of position (whereas the work is a scalar). Integration of the product aver the volume (and surface) of the body gives the total work clone by all points. The work done on the body at time t by the resultant farce in moving through

the virtual displacement bu is given by


whcrc c1 = 4/(3h2), Āu0 is the axial displacement, w0 the transvcrsc displacement, and <P the rotation of a point on thc ccntroidal axis x of the beam. The displaccment field is arrived by (a) relaxing the Eulcr -Bcrnoulli hypotheses to Jet the straight lines normai to the bcam axis bcfore deformation t.o become (cubie) curvcs with arbitrary slope at z =O, and (b) requiring thc transverse shear stress to vanish at the top and bottom of the beam. Thus, only restriction from the Euler-13ernoulli bearn thcory that is kept is w(.r, z, t) = w0(x, t) (j.e., transverse deflection is independent of the thickncss coordinate z). The displaccment field ( L4.82) accommodates quadratic variation of trans verse shear strain E,rz and shear stress o ç, through the beam height, as can be seen from the strains computed next. Now suppose that the bcam is subjected to distributed axial forcc f(x) and transverse load of

q(x, t) along the length of the bearn. Since we are prirnarily interestcd in deriving the equations of motion and the nature of the boundary conditions of the bearn that experiences a displacemcnt ficld of the form in Eq. (L4.82), wc will not considcr specific geometrie or force boundary conditions bere. The procedure to obtain the equations of motion and boundary condit.ions involves the

(1.4.82)

) _ ,. a ( 8wo) u(x, z, t - v.0(x, t) + up(,1,, t) - c1 z </i+ ¸x w(:r, z, t) = wo(x, t)

Example 1.4.3 ( Third-order beam theory) ----------------------

Consider the displaccmcnt field

Equations (1.4.81) are the Euler-Lagrange equations for an elastic body.

(1.4.81) t-t=O in V

82u . +-+ p~ - div O" - f = O ®)t2

where integration-by-parts, gradient theorerns, ami Eqs. (1.4.73a,b) were used in arriving at Eq. (1.4.80) frorn Eq. (1.4.78). Because 8u is arbitrary for t., t1 < t < t2, and for x in V and also on S2, it follows that

(1.4.80)

1t2 O= 8 L(u, 'Vu, ½) dt t1

1t2 J ( 32 u +-+ ) j ~ = [ p~-divO"-f Ā8udV+ ,(t-t)-8udS]dt t1 V ut 52

Equation(l.4. 78) represents Hamilton's principle for an elastic body (linear or nonlinear). Recall that the sum of the strain energy and potential energy of external forces, U + V, is called the total potential energy, II, of the body. For bodies involving no rnotion (i.e., forces are applied sufficiently slowly such that the motion is independent of time, and the inertia forces are negligible), Harnilton's principle (1.4.78) reduces to the principle of virtual displacements. The Euler-Lagrange equations associated with the Lagrangian, L = K - II,

(II= U +V) can be obtained from Eq. (1.4.78):

(1.4. 79) K = { {!_ au Ā au dV, U = fv. o, dV lv 2 at at Ā l,

where K and U are the kinetic and strain energies:


(1.4.85) { o, } 1h/Z { ] } R; = -h/2 z2 axz dz { s.; } lh/2 { 1 } Mxx = z axx dz , e.; -h/2 z3

where ali the terms involving [ Ŀ ] vanish on account of the assumption that ali variations and their derivatives are zero at t =O and t = T, and the new variables introduced in arriving at the last expression are defined as follows:

{T r r [ ( (O) (1) 3s: (3)) (" (O) 2>: (2))] dA O= lo lo 1 A axx bExx + ZbExx + Z uExx + axz U"fxz + Z U"fxz dxdt

-lor 1L 1p{[½o+z~-c1z3(~+88~0)] [t½0+z8~-c1z3(o~+8~~0)]

+ Wobwo }dAdxdt -1T 1L (j8uo + q8wo)dxdt

{T t' ( " (O) e (1) p r (3) Q s: (O) r (2)) d =lo lo NxxUExx + MxxVExx + xxVExx + xU"fxz + Rxu"fxz dx t

-1T 1L { Io½o8½o + [12~ - c1/4 (<ii+ 0:0)] o<ii + f8uo + qbwo} dxdt - lor 1L { -c1 [14~ - c1h ( ~ + 0:0) J ( b~ + 8~~0) + I0w0bw0} dxdt

and c2 = 4/h2. Note that "[x z = 2Exz is a quadratic function of z. Hence, axz = G"fxz is also quadratic in z. From the dynamic version of the principle of virtual displacements (i.e. Hamilton's principle)

we have

(1.4.83b)

(D)_Ouo (1)_8</J (3) __ (8</)+82wo) Exx - fu ' Exx - OX ' Exx - C1 OX 8:c2

(O) _ 8wo (2) _ ( 8wo) 'Yxz - </J + Dx , 'Yxz - -C2 </J + 8x

where

(1.4.83a) Exx =E~~ + ZE~;,i + z3E~~ 'Yxz = 'Y¨~) + z27;)

following steps: (i) compute the strains, (ii) compute the virtual energies required in Hamilton's principle, and (iii) use Hamilton's principle, derive the Euler-Lagrange equations of motion and identify the primary and secondary variables of the theory (which in turn help identify the nature of the boundary conditions). Although one can use the generai nonlinear strain-displacement relations, here we restrict the

development to small strains and displacements. The linear strains associated with the displacement field are


These equations are lower-order than those in Eqs, (1.4.87) and (1.4.88a,b).

(1.4.95)

(1.4.94)

(1.4.93)

(1.4.87) and (1.4.88a,b) by where K is the shear correction factor. A simplified third-order beam theory can be obtained frorn Eqs.

setting c1 =O (but not c2):

(1.4.92) Qx = K j~ O"xzdA

The primary and secondary variables of the Timoshenko beam theory are: ( uo, wo, <P) and (N"'x' Qx, lvlxx)Ā Note that the Timoshenko beam theory accounts for transverse shear strain rxz = r~z and hence Qx. In thc Timoshenko beam theory Qx is defined, in piace of the definition (1.4.85), by

(1.4.91 b)

(1.4.91a)

(1.4.90) 8Nxx f _ I 82uo 8x + - O 8t2 8Qx _I 82wo 8x + q - o 8t2

8Mxx _ Q _ I 82</J 8X X - 2 8t2

When c1 =O in Eq. (1.4.82), it corresponds to the displacement field of the Timoshenko beam theor. Thus, the equations of motion of the Timoshenko beam theory can be obtained directly frorn Eqs. (1.4.87) and (1.4.88a,b) by setting c1 = c2 =O:

(l.4.89)

The last line of Eq. (1.4.84) includes boundary terrns, which indicate that the prirnary variables of the theory are (those with the variational syrnbol) uo, wo, </J, and 8wo/8x. The corresponding secondary variables are the coefficients of Duo, Dwo, DrjJ, and fJDwo/8x:

(1.4.88b)

(1.4.88a)

Dw0:

(1.4.87) Duo:

Note that I; are zero for odd values of i (i.e., 11 = !3 = h =O). Thus, the Euler-Lagrange equations are

(1.4.86b)

(l.4.86a)


As noted in Section 1.4 the principle of virtual displacements gives the equilibrium equations as the Euler-Lagrange equations. These governing equations are in the form of differential equations that are not always solvable by exact methods of solution. There exists a number of approximate methods that can be used to solve differential equations ( e.g., finite-difference methods, the finite element method, etc.). The most direct methods are those which bypass the derivation of the Euler- Lagrange equations, and go directly from a variational staternent of the problem to the solution of the equations. One such direct method was proposed by Ritz [26]. The Ritz method is based on variational statements, such as those provided by the principles of virtual displacements or the minimum total potential energy, which are

1.5.2 The Ritz Method

In Section 1.4, we saw how virtual work and variational principles can be used to obtain governing differential equations and associated boundary conditions. Here we study the direct use of the variational principles in the solution of the underlying equations. The methods to be described here are known as the classical variational methods. In these methods, we seek an approximate solution to the problem in terms of adjustable parameters that are determined by suhstituting the assumed solution into a variational statement equivalent to the governing equations of the problem. Such solution methods are called direct methods because the approximate solutions are obtained directly by applying the same variational principle that was used to derive the governing (i.e., Euler-Lagrange) equations. The assumed solutions in the variational methods are in the form of a finite linear

combination of undetermined parameters with appropriately chosen functions. This amounts to representing a continuous function by a finite set of functions. Since the solution of a continuum problem in general cannot be represented by a finite set of functions, error is introduced into the solution. Therefore, the solution obtained is an approximation to the true solution of the equations describing a physical problem. As the number of linearly independent terms in the assumed solution is increased, the error in the approximation will be reduced, and the assumed solution converges to the exact solution. It should be understood that the equations governing a physical problem are

themselves approximate. The approximations are introduced by several sources, including the geometry, representation of specified loads ancl boundary conditions, and material behavior. Therefore, when one thinks of permissible error in an approximate solution, it is understood to be relative to exact solutions of the governing equations that inherently contain approximations. The variational methods of approximation to be described here are limited to the Ritz method. and the weighted-residual methods (e.g., the least-squares method, collocation method, and so on). The weighted-residual methods will be visited only briefly. Interested readers may consult the references at the end of the chapter for additional details [6].

1.5.1 Introduction

1.5 Variational Methods


where Il Ŀ Il denotes a norm in the vector space of functions u. The set { Zf!.i} is called the spanning set. A sequence of algebraic polynomials, for example, is complete if it contains terms of all degrees up to the highest degree (N).

(1.5.3) N

llu - L Cjl{>]ll <E j=l

Substitution of Eq, (1.5.1) into II(u) for u and the minimization of II(cj) results in a set of algebraic equations among the parameters Cj. In arder to ensure that the algebraic equations resulting from the Ritz procedure have a solution, and the approxirnate solution converges to the true solution of the problern as the number of pararneters N is increased, we must choose l{>j (j = 1, 2, 3, Ŀ Ŀ Ŀ, N) and Zf!o such that they meet the following requirements:

1. rp0 has the principal purpose of satisfying the specified essential (or geometrie) boundary conditions associateci with the variational formulation; rpo plays the role of particular solution. It should be the lowest arder possible for completeness.

2. Zf!.i (j = 1, 2, Ŀ Ŀ Ŀ , N) should satisfy the following three conditions: (a) be continuous as required in the variational staternent (i.e., l{>j should be such that it has a nonzero contribution to the virtual work statement);

(b) satisfy the homogeneous form of the specified essential boundary conditions; (e) the set {rpj} is linearly independent and complete. (1.5.2)

The completeness property is defined mathematically as follows. Given a function u and a real number E > O, the sequence { l{>j} is said to be complete if there exists an integer N (which depends on E ) and scalars c1, c2, Ŀ Ŀ Ŀ, CN such that

Properties of Approximation Functions

and then determine the parameters Cj by requiring that the principle of virtual displacements holds for the approximate solution, i.e., minimize Il(U N) with respect to c1, j = 1, 2, Ŀ Ŀ Ŀ, N. In Eq. (1.5.1) c1 denote undetermined parameters, and rpo and (f!j are the approximation functions, which are appropriately selected functions of position x. Equation (1.5.1) can be viewed as a representation of u in a finite component form; Cj are termed the Ritz coefficients. The selection of l{>.f is discussed next.

(1.5.1) N

u ~UN= LC.Jl{>j + rpo j=l

equivalent to the governing differential equations as well as the natural boundary conditions, and they are also known as the weak forms. The basic idea of the Ritz method is described here using the principle of virtual

displacements or the minimum total potential energy principle. In the Ritz method we approximate a dependent unknown (e.g., the displacement) u of a given problem by a finite linear combination of the form


Note that when the specified values are zero, i.e., ½ =O, there is no need to include cpo (or equivalently, <po = O); however, 'PJ are still required to satisfy the specified (homogeneous) essential boundary conditions. The conditions in Eq. (1.5.2) provide guidelines far selecting the coordinate

functions; they do not give any formula far generating the functions. As a general rule, coordinate functions should be selected from the admissible set, from the lowest arder to a desirable arder without missing any intermediate admissible terms in the

'P1(L) =O far j = 1,2,ĿĿĿ,N

N

½ = L Cjt.pj(L) + ½ j=l

and, therefore, it follows that I:f=l CJIPJ(L) =O. Since this condition must hold far any set of parameters Cj, it follows that

N UN(L) = L c1<p1(L) + <po(L)

j=l

The requirement on 'Pi to satisfy the homogeneous form of the specified essential boundary conditions follows from the approximation adopted in Eq. (1.5.1). Since UN= ½ and cpo = ½ at x = L, we have

cpo = ½ at x = L and 'Pi= O at x = L for i= 1, 2, Ŀ Ŀ Ŀ, N

The completeness property is essential far the convergence of the Ritz approximation (see Reddy [29], p. 262). Since the natural boundary conditions of the problem are included in the

variational statements, we require the Ritz approximation UN to satisfy only the specified essential boundary conditions of the problem. This is done by selecting 'Pi to satisfy the homogeneous form and <po to satisfy the actual form of the essential boundary conditions. For instance, if u is specified to be ½ on the boundary x = L, we require

holds only far all a1 = O. Thus no function is expressible as a linear combination of others in the set. For polynomial approximations functions, the linear independence and

completeness properties require 'PJ to be increasingly higher-order polynomials. For example, if cp1 is a linear polynomial, cp2 should be a quadratic polynomial, cp3 should be a cubie polynomial, and so on (but each 'PJ need not be complete by itself):

a1cp1 + a2cp2 + Ŀ Ŀ Ŀ + lXN'PN =O

Linear independence of a set of functions { 'Pj} refers to the property that there exists no trivia} relation among them; i.e., the relation


3. If the resulting algebraic equations are symmetric, one needs to compute only upper or lower diagonal elements in the coefficient matrix, [A]. The symmetry of the coefficient matrix depends on the variational statement of the problem.

4. If the variational (or virtual work) statement is nonlinear in u, then the resulting algebraic equations will also be nonlinear in the parameters Ci. To solve such nonlinear equations, a variety of numerical methods are available (e.g., Newton's method, the Newton-Raphson method, the Picard method), which will be discussed later in this book (see Chapter 13).

where Aij and bi are known coefficients that depend on the problem parameters ( e.g., geometry, materia} coefficients, and loads) and the approximation functions. These coefficients will be defined for each problem discussed in the sequel. Equations (1.5.5) are then solved for {e} and substituted back into Eq. (1.5.1) to obtain the N-parameter Ritz solution. Some genera} features of the Ritz method based on the principle of virtual

displacements are listed below:

1. If the approximate functions I.Pi are selected to satisfy the conditions in Eq. (1.5.2), the assumed approximation for the displacements converges to the true solution with an increase in the number of parameters (i.e., as N --+ oo). A mathematical proof of such an assertion can be found in [20-22, 29].

2. For increasing values of N, the previously computed coefficients A1 and bi of the algebraic equations (1.5.5) remain unchanged, provided the previously selected coordinate functions are not changed. One must add only the newly computed coefficients to the system of equations. Of course, Cj will be different for different values of N.

(1.5.5)

This gives N algebraic equations in the N coefficients (c1, c2, ... , CN)

(1.5.4)

Once the functions 'Po and 'Pi are selected, the parameters Cj in Eq. (1.5.1) are determined by requiring UN to minimize the total potential energy functional II (or satisfy the principle of virtual work) of the problem: 8II(U N) =O. Note that II(U N) is now a real-valued function of variables, c1, c2, Ŀ Ŀ Ŀ, CN. Hence minimization of the functional II(U N) is reduced to the minimization of a function of several variables:

Algebraic Equations for the Ritz Parameters

representation of UN(i.e., satisfy the completeness property). The function r.po has no other role to play than to satisfy specified (nonhomogeneous) essential boundary conditions; there are no continuity conditions on r.p0. Therefore, one should select the lowest order I.Po that satisfies the essential boundary conditions.


and 'Pi must be differentiable as required by the tota! potential energy functional in Eq. (1.4.67) of Example 1.4.2. Since there are two conditions to satisfy, we begin with 'Pl = a+ bx + cx2 and

(1.5.7) dip 'Pi(O) =O and dx' (O)= O

Since the specified essential boundary conditions are homogeneous, 'Po = O. Next , we must select 'Pi to satisfy the homogeneous form of the specified essential boundary conditions

(1.5.6) N

w0(x) ""'WN = L CJ'PJ +'Po j=l

The force (or natural) boundary conditions can be arbitrary. For example, the beam can be subjected to uniformly distributed transverse load q(x) = q0, concentrated point load F0, and moment Mo, as in Figure 1.4.2. The applied loads will have no bearing on the selection of 'Po and 'PiĀ The applied loads will enter the analysis through the expression for the external work done [see Eq. (1.4.52)], which will alter the expression for the coefficients F; of Eq. (1.5.5). An N-parameter Ritz approximation of the transverse deflection w0(x) is chosen in the form

dw0 w0(0) = O, dx (O) = O

Consider the cantilever beam shown in Figure 1.4.2. We consider the pure bending case (i.e., uo = O). We set up the coordinate system such that the origin is at the fixed end. For this case the geometrie (or essential) boundary conditions are

Example 1.5.1:

8. The Ritz method can be applied, in principle, to any physical problem that can be cast in a weak form - a form that is equivalent to the governing equations and natural boundary conditions of the problem. In particular, the Ritz method can be applied to all structural problems since a virtual work principle exists.

where UN denotes the N-parameter Ritz approximation of u obtained from the principle of virtual displacements or the principle of minimum total potential energy. It should be noted that the displacements obtained from the Ritz method based on the total complementary energy (maximum) principle provide the upper bound.

U1 < U2 < ... <UN< UMĀĀĀ < u(exact), for M > N

7. Since a continuous system is approximated by a finite number of coordinates (or degrees of freedom), the approximate system is less flexible than the actual system. Consequently, the displacements obtained using the principle of minimum total potential energy by the Ritz method converge to the exact displacements from below:

6. The equilibrium equations of the problem are satisfied only in the energy sense, not in the differential equation sense. Therefore the displacements obtained from the Ritz approximation, in general do not satisfy the equations of equilibrium pointwise, unless the solution converged to the exact solution.

5. Since the strains are computed from an approximate displacement field, the strains and stresses are generally less accurate than the displacement.


(1.5.12b)

N=2:

(1.5.12a)

where

A,J =El .lL j(j+ l)xJ-l Ŀi(i+ l)xi-ldx, b, = 1L q(x)xi+1dx+FLLi+l +.!Vh(i+ l)V (1.5.llb)

For onc- and two-pararnetcr approximations wc have the following equations: qoLa

N = 1: An = 4ElL, b1 = -3- + FLL2 + 2MLL qoL2 Fr.,L Mr.,

cl = 12El + 4El + 2EI , W (x) _ ( qoL4 FLL3 lvhL2) x2

1 - 12El + 4EI + 2El L2

(1.5.lla)

Thc ith equation in (1.5.10) has the form

O= ~~ = 1L {El [2c1 + 6c2x + Ŀ Ŀ Ŀ + N(N + l)cNXN-l] i(i + l)xi-l - q xi+1 }dx

- Fr.,L'+l - ML(i + l)V

=c1 [.lL 2ElĀi(i+l)xi-ldx] +cz [1L 6ElxĀi(i+l)x'-1dx] +ĿĿĿ

+ cN [.lL EIN(N + l)xN-li('i + l)x'-1dx] -1L q(x)xi+1dx - Fr.,Li+l - ML(i + l)Li N

= c1Ail + c2Ai2 + Ŀ Ŀ Ŀ + cNAiN - Fi = LA.;JCj - b;, (i= 1, 2, ... ,N) j=l

(1.5.10) arr arr acl = O, -8 = O,

C2

Using the tota! potential energy principlc, oll = O, which requires that Il be a minimum with respect to each of c1, c2, Ŀ Ŀ Ŀ, cN, we arrive at the conditions

(1.5.9)

Il(c1, c2, Ŀ Ŀ Ŀ, CN) = 1L { ~l [2c1 + 6c2x + Ŀ Ŀ Ŀ + N(N + l)cNxN-l] 2

- q(c1x2 + c2x3 + + CNXN+l) }dx

- FL[cix2 + czx3 + + CNXN+l]x=L - .ML[2c1x + 3czx2 + Ŀ Ŀ Ŀ + (N + l)cNxN]x=L

Substituting Eq. (1.5.8) into Eq. (1.4.67) we obt.ain Il as a function of the coefficients c1, c2, Ŀ CN:

(1.5.8)

T'he Ritz approximation becornes

WN = C1X2 + C2J:3 + Ŀ Ŀ Ŀ + CNXN+l

'PN= XN+l 'PI = :r2' 'P2 = x3' 'P:~ = x4'

determinc two of the three constants using Eq. (1.5.7). The third constant will rcmain arbitrary. Conditions (1.5.7) give a= b =O, and rp1(3:) = cx2. We can arbitrarily take e= 1. Using the same procedure, we can determinc i.p2, i.p3, etc. One may set the coefficients of lower order terms to zero, since they are already accounted in the preceding 'Pi:


The method based on this procedure is called, for obvious reason, a weighted-residual method.

(1.5.16b) k 1/JiRN(Ci, 'Pi,!) dx= O, i= 1, 2, Ŀ Ŀ Ŀ, N

(1.5.16a) RN = A (f. CJ<fJJ +<po) - f -/= O ]=l

be orthogonal to N linearly independent set of weight functions 'l/Ji:

where the parameters Cj are determined by requiring the residua! of the approximation

(1.5.15) N

UN= L CJ<PJ +<po j=l

r1 is the point X= o, r2 is the point X= L We seek a solution in the form

where A is a linear or nonlinear differential operator, u is the dependent variable, f is a given force term in the domain O, B1 and B2 are boundary operators associateci with essential and natural boundary conditions of the operator A, and u and fj are specified values on the portions I" 1 and f2 of the boundary r of the domain. An example of Eq. (1.5.14) is given by

(1.5.14)

A(u) =fin O

Consider an operator equation in the form

1.5.3 Weighted-Residual Methods

The two-parameter solution is exact for the case in which q0 = O. For q0 i= O, the solution is not exact for every x but the maximum deflection W2(L) coincides with the exact value wo(L). The three-parameter solution, with <jJ3 = x4, would be exact for this problem. If we were to choose trigonometrie functions for 'Pi, we may select the functions <p; ( x) =

1 - cos[(2i - 1)7rx/2L]. This particular choice would not give the exact solution for a finite value of N, because the applied load q0, when expanded in terms of <p;, would involve infinite number of terms. Thus, a proper choice of the coordinate functions is important in realizing the exact solution. Of course, both algebraic and trigonometrie functions would yield acceptable results with finite number of terms.

(1.5.13)

The exact solution is


(1.5.19b) Gij = { '!/JiA(cpj)dx, qi = { '!/Ji [! - A(ipo)] dx .J o .J o Note that Gij is not symmetric in general, even when '!/Ji ='{!i (Galerkin's method). It is symmetric when A is a linear operator and '!/Ji = A(cpi) (the least-squares method). It should be noted that in most problems of interest in solid mechanics, the

operator A is of the form that permits the use of integration by parts to transfer

where

(1.5.19a) N L GijCj - qi =e U, i= 1, 2, ... 'N j=l

and Eq. (1.5.16b) becomes

(1.5.18)

Galerkin's method: '!/Ji =<fii Least-squares method: '!/Ji = A( 'f!i) Collocation method: '!/Ji = 8(x - xi)

Here 8(-) denotes the Dirac delta function. The weighted-residual method in the general form (1.5.16b) (with '!/Ji -::/- 'f!i) is known as the Petrov-Galerkin method. Equation (1.5.16b) provides N linearly independent equations for the determination of the parameters CiĿ If A is a nonlinear operator, the resulting algebraic equations will be nonlinear. Whenever A is linear, we have

Å cp0 should satisfy all specified boundary conditions.

Å 'Pi should satisfy homogeneous form of all specified boundary conditions. (1.5.17) The variational statement referred to in Property 2a of (1.5.2) is now given in Eq. (1.5.16b). Properties in (1.5.17) are required because the boundary conditions, both essential and natural, are not included in Eq, (1.5.16b). Both properties now require 'f!i to be of higher order than those used in the Ritz method. On the other hand, '!/Ji can be any linearly independent set, such as { 1, x, Ŀ Ŀ Ŀ}, and no continuity requirements are placed on '!/Ji. Various special cases of the weighted-residual method differ from each other due

to the eh o ice of the weight function '!/Ji. The most commonly used weight functions are

The coordinate function 'Po and 'f!i in a weighted-residual method should satisfy the properties in Eq. (1.5.2), except that they should satisfy all specified boundary conditions:


N j; [L A(cpi)A(cpj)dx] Cj - L [A(cpi)f - A(ipi)A(cp0)] dx= O

Then from Eq. (1.5.20) we have

(1.5.21)

where RN is the residuai defined in Eq. (1.5.16a). Equation (1.5.20) provides N algebraic equations for the constants CiĿ First we note that the least-squares method is a special case of the weighted-

residual method for the weight function, 1/Ji = 2( 8RN I aci) [compare Eqs. (1.5.16b) and (1.5.20b)]. Therefore, the coordinate functions cpi should satisfy the same conditions as in the case of the weighted-residual method. Next, if the operator A in the governing equation is linear, the weight function 1/Ji becomes

(1.5.20b) 1 oRN . 2RN(Cj, cpj, !)---;:;--- dx= O, i= 1, 2, Ŀ Ŀ Ŀ, N

O oc;

or

(1.5.20a)

The least-squares method is a variational method in which the integral of the square of the residuai in the approximation of a given differential equation is minimized with respect to the parameters in the approximation:

Least-Squares Method

The Galerkin method is a special case of the Petrov-Galerkin method in which the coordinate functions and the weighted functions are the same ( cpi = 1/Ji). It constitutes a generalization of the Ritz method. When the governing equation has even arder of highest derivative, it is possible to construct a weak form of the equation, and use the Ritz method. If the Galerkin method is used in such cases, it would involve the use of higher-order coordinate functions and the solution of unsymmetric equations. The Ritz and Galerkin methods yield the same set of algebraic equations for the

following two cases: 1. The specified boundary conditions of the problem are all essential type, and therefore the requirements on cpi in both methods are the same.

2. The problem has both essential and natural boundary conditions, but the coordinate functions used in the Galerkin method are also used in the Ritz method.

The Galerkin Method

half of the differentiation to the weight functions 1/Ji and include natural boundary conditions in the integral statement (see Reddy [6]). For problems for which there exists a quadratic functional or a virtual work statement, the Ritz method is most suitable. The least-squares method is applicable to all types operators A but requires higher-order differentiability of cpi.


(1.5.26) At(u) + A(u) = J(x, t) Time-dependent problem

( 1.5.25) A(v,) - >-.C(u) =O Eiqenoalue problem

Eigenvalue and Time-Dependent Problems

It should be noted that if the problcm at hand is an eigenvalue problem or a tirne-dependent problern, the operator equation in Eq. (1.5.14) takes the following alternative forms:

Thus, the collocation method is a special case of the weighted-residual method (1.5.16b) with 'l/Ji(x) = b(x - xi). In the collocation method, one must choose as many colloeation points as there are undetcrmined parameters. In general, these points should be distributed uniformly in thc domain. Otherwise, ill-conditioned equations among Cj may result.

(1.5.24b) { b(x-xi)RN(x,{c},{<p},f) dx=O, (i= 1,2,ĿĿĿ,N) ln

which can be written, with the help of the Dirac delta function, as

(1.5.24a) RN(x'i, {e}, {<ti},!)= O, (i= 1, 2, Ŀ Ŀ Ŀ, N)

In the collocation method, we require the residua! to vanish at a selected nurnber of points xi in the domain:

Collocation Method

Note that the coefficient matrix is symmetric. The least-squares method requires higher-order coordinate functions than the Ritz method because the coefficient matrix Lij involves the same operator as in the original differential equation and no trading of differentiation can be achieved. For first-order differential equations the least-squares method yields a symmetric coefficient matrix, whereas the Ritz and Galerkin methods yield unsymmetric coefficient matrices. Note that in the least- squares method the boundary conditions can also be included in the functional. For example, consider Eq. (1.5.14). The least-squares functional is given by

(l.5.22b)

where

(1.5.22a) N

L LijCj - ti, =o, i= 1, 2, ... 'N j=l

or


which gives (for nonzero c1) ê = 50/12 = 4.167. If the same function is used for i.p1 in the one- parameter Ritz solution, we obtain the same result as in the one-parameter Galerkin solution.

(1.5.31)

The one-pararneter Galerkin's solution for the natural frequency can be computed using

(1.5.30) 'Pl (x) = 3x - 2x2

In a weighted-residual method, <p, must satisfy not only the condition 'Pl (O) = O but also the condition <p;(1) + 'Pi(l) =O. The lowest-order function that satisfies tho two conditions is

(1.5.29) du - +u =O at x = 1 dx

d2u - dx2 - Xu. = O, u(O) = O,

Consider the eigenvalue problem described by the equations

Application of the weighted-residual method to Eqs. (1.5.25) and (1.5.26) follows the same idea, i.e., Eq. (1.5.16b) holds. For additional details and examples, the reader may consult [6].

a ( au) A(u) = -- EAo- fJx Bç

(1.5.28) -p ()2u - !!__ (EAo au) = f(x, t) 8t2 Bè ax

where u denotes the axial displacement, p the density, E Young's modulus, Ao area of cross section, and f body force per unit length. In this case, we have

In Eq. (1.5.26) A is a spatial differential operator and At is a temporal differential operator. Examples of Eq. (1.5.26) are provided by the equations governing the axial motion of a bar:

d2u C(u) = -- dx2 >. = P, d2 ( d2u)

A(u) = dx2 El dx2 '

where u denotes the lateral deflection and P is the axial compressive load. The problem involves determining the value of P and mode shape u(x) such that the governing equation and certain end conditions of the beam are satisfied. The minimum value of P is called the criticai buckling load. Comparing Eq. (1.5.27) with Eq, (1.5.25), we note that

(1.5.27)

In Eq. (1.5.25), parameter >. is called the eigenvalue, which is to be determined along with the eigenvector u(x), and A and Care spatial differential operators. An example of the equation is provided by the buckling of a beam-column


U2(x) = c1 'P1 (x) + c2<p2(:i;) = c1(3x - 2x2) + c2(4x2 - 3x3) = 3c1x + (-2c1+4c2)x2 - 3c2x3

U2(x) ""c1<.p1(x) + c2<P2(x) = c1 (3x - 2x2) + c2(x + 2x2 - 2x3) = (3c1 + C2) X+ (-2C1 + 2c2) x2 - 2c2x3

The set { 'Pl, 'P2} is equivalent to the set { 'Pl, <P2}. Note that

(1.5.37b) 'P2(x) =a+ bx + cx2 + d:i:3 = x + 2x2 - 2x3 = <jj2(x) On the other hand, if we choose b = 1 and e= 2, we have d = -2, and 'P2 becomes

( l.5.37a) <p2(x) =a+ bx + cx2 + dx3 = 4x2 - 3x3

We can arbitrarily pick the values of band e, except that not both are equa! to zero (for obvious reasons). Thus we have infinite number of possibilities. If we pick b =O and e= 4, we have d = -3, and 'P2 becomes

and obtain

'P2(x) =a+ bx + cx2 + dx:l

where <p1(x) is given by Eq. (1.5.30). To determine <p2(x), we begin with a polynornial that is one degree higher than that used for <p1:

(1.5.36)

Neither root is closer to the exact value of 4.116. This indicates that the least-squares method with 1/J; = A(<p;) is perhaps more suitable than 1/J; = A(<p;) - .\C(<p;). Let us consider a two-pararneter weighted-residual solution to the problern

(1.5.35) >-1,2 = 2: Ñ ~ ,/445 ___, >-1 = 7.6825, >-2 = 0.6508 w hose roots are

(1.5.34)

r (d2 ) 0 = C1 J O d~l + ê<p1

(4 2 20 ) = -.\ - -.\ + 16 c1 5 3

and ,\ = 4.8. If we use 1j;1 = A( <.p1) - .\<p1, we obtain

(1.5.33)

which gives ,\ = 4. The one-parameter least-squares approximation with 1f;1 = A(<pi) gives

O= c1 'Pl (0.5) [ ( d:~l) lx=O 5 + ê<p1 (0.5)] or (-4 + .\) c1 =O (1.5.32)

For one-parameter collocation method with the collocation point at x 0.5, we obtain [<pi(0.5) = 1.0 and (d2<pifd:c2) = -4.0]


( 1 [50 35] x [28 21]) {c1} {()} 15 35 38 - 35 21 17 C2 = ()

and

jĿl d2ef;l 11 10 K11 = - cP1-d 2 dx= (3x - 2x2)(4)dx = ~

0 X O ~

11 d2ef; 11 7 K12 = - cP1----.:} dx= (3x - 2x2)(-8 + 18x)dx = -3 o dx o {I ~(/J {I 7

K21 = - lo cP2 dx} dx =l; (4x2 - 3x3)(4)dx = 3

li ~if; 11 ~ K22 = - cP2 d 22 dx= (4x2 - 3x3)(-8 + 18x)dx = - , O X O 15

11 11 4 M11 = cP14>1 dx = (3x - 2x2)(3x - 2x2)dx = ~ o o ~

11 li 3 M12 = cP14>2 dx= (3x - 2x2)(4x2 - 3x3)dx = - = M21 o . o 5 r {1 17

M22 =lo cP2cP2 dx= lo (4x2 - 3x3)(4x2 - 3x3)dx = 35

Evaluating the integrals, we obtain

d2'P2 = 8 - 18x dx2

d2cp1 = -4 dx2 '

First, for the choice of functions in Eqs. (1.5.30) and (1.5.37a), we have

11 d2 - 'Pi K;7 - - o 'Pi dx2 dx,

where

[K]{c} - >.[M]{c} ={O} In matrix form, we have

o = 11 't)i(:c )R dx= 11 'PI (x) [-c1 d;;:i1 - C2 d;;:.i1 - x ( C] 'PI + C2'P2)] dx

= K 11c1 + K 12C2 - >. (M11 c1 + M12c2)

O= 11 'P2(x)R dx= 11 'P2(x) [-ci d;;:i1 - c2 d;;:.i1 - x (c1'P1 + c24>2)] dx = K21c1 + K22c2 - >. (M21c1 + M22c2)

For the Galerkin method, we set the integrai of the weighted-residual to zero and obtain

(1.5.38)

as

Hence, either set will yield the same final solution for U2 (x) or >.. Using 'Pl from (1.5.30) and 'P2 from Eq. (1.5.37a), we compute the residuai of the approximation

Comparing the two relations we can show that


In this chapter a review of the linear and nonlinear strain-displacement relations, equations of motion in terms of stresses and displacements, compatibility conditions on strains, and linear constitutive equations of elasticity, thermoelasticity and electroelasticity is presented. Also, an introduction to the principle of virtual displacements and its special case, the principle of minimum total potential energy, is also presented. The virtual work principles provide a means far the derivation of the governing equations of structural systems, provided one can write the internal and external virtual work expressions far the system. They also yield the natural boundary conditions and give the forrn of the essential and natural boundary conditions. The last feature proves to be very helpful in the derivation of higher- order plate theories, as will be shown in the sequel. A brief but complete introduction to the Ritz method and weighted-residual methods (Galerkin, least-squares, and collocation methods) is also included in this chapter. The principle of virtual displacements will be used in this book to derive

governing equations of plates according to various theories, and the Ritz and Galerkin methods will be used to determine solutions of simple beam and plate problems. The ideas introduced in connection with classica! variational methods are also useful in the study of the finite element method (see Chapter 9). The single most difficult step in all classical variational methods is the selection

of the coordinate functions. The selection of coordinate functions becomes more difficult far problems with irregular domains or discontinuous data (i.e., loading or geometry). Further, the generation of coefficient matrices far the resulting algebraic equations cannot be automated far a class of problems that differ from each other only in the geometry of the domain, boundary conditions, or loading. These lirnitations of the classica! variational methods are overcome by the finite element method. In the finite element method, the domain is represented as an assemblage ( called mesh) of subdomains, called finite elements, that permit the construction of the approximation functions required in Ritz and Galerkin methods. Traditionally, the choice of the approximation functions in the finite element method is limited to algebraic polynomials, Recent trend in computational mechanics is to return to traditional variational methods that are rneshless and find ways to construct approximation functions far arbitrary domains [31-36]. The traditional finite element method is discussed in Chapter 9.

1.6 Summary

Clearly, the value of êi has improved over that computed using the one-parameter approximation. The exact value of the second eigenvalue is 24.139. If we were to use the collocation method, we may select x = 1/3 and x = 2/3 as the collocation

points, among other choices. We leave this as an exercise to the reader.

(1.5.40) ê1 = 4.121, ê2 = 25.479 which gives

( 1.5.39) 675 - 1332 ê + 315 ..\2 =O or 525 - 148..\ + 5..\2 =O 7 49

For nontrivial solution, c1 # O and c2 # O, we set the determinant of the coefficient matrix to zero to obtain the characteristic polynomial


Xz,X2 Xz,X2 eo

i--<

--- I eo

Ŀ1 Ŀ1 eo

Xi.Xi X1,X1 a a

(a) (b)

Figure Pl.6

1.2 Prove the following vector identities using the summation convention and the E - {j identity (1.2.8). In the first three identities A, B, C and D denote vectors:

(a) (A x B) x (C x D) = [AĿ (Cx D)]B - [B Ŀ(Cx D)]A (b) (A X B). (CX D) =(A. C)(B. D) - (A. D)(B. C)

(c) (A x B) Ŀ [(B x C) x (Cx A)]= [A. (B x C)]2 (d) (AB)T = (B)T(A)T, where A and Bare dyads

1.3 Use the integrai theorems to establish the following results:

(a) The tota! vector area of a closed surface is zero.

(b) Show that ~V= ~~S (see Figure l.2.3b). 1.4 Derive the following integrai identities:

(a)- r W; [__Ä___ (8U, + 8Uj )] df! = { 8w; (8ui + 8uj) df!- 1 Winj (8ui + 8uj) dr lo 8x1 8x1 8x; lo 8xj 8x1 8xi Jr 8x1 8xi

(b) l ('P'14'1j;- 'i72'P'i72'1j;) dn =i ['P :n ('12'1j;)- '72'1j; ~~]dr

where W; and u, are functions of position in n, and I' is the boundary of n. The summation convention on repeated subscripts is used.

1.5 If A is an arbitrary vector and is an arbitrary second-order tensor, show that

(a) (Ix A)Ŀ = A x , I = unit tensor (b) ( X A)T =-A X <1>T

1.6 Write the position of an arbitrary point (x1, x2, x3) in the deformed body (solid lines) in terms of its coordinates in the undeformed body (broken lines) and compute the nonlinear Lagrangian strains for the body shown in Figure Pl.6.

(e) Eijk = Ekij = Ejki = -Ejik = -Eikj = -Ekji

where Eijk is the permuiation symbol. Prove the following properties of D;j and Eijk:

(a) F;jDjk = F;k

(b) D;j D;j = D;; (e) EijkEijk = 6, (for i,j, k over a range of 1 to 3)

( d) EijkAiAj =o

1.1 The nine cross-product (or vector product) relations among the basis (1,2,3) can be expressed using the index notation as

Problems


(a) Sketch the deformed configuration of the body.

(b) Compute the components ofthe deformation gradient tensor F and its inverse (display them in matrix form).

(e) Compute the Green's strain tensor components (display them in matrix form).

1.11 Find the linear strains associated with the 2-D displacement field

_ Pxix2 PL2 (2 + v)P .3 ui - - 2EI + 2EIX2 + 6EI x2

Ph2(l+v) vP. ,.2 P 3 PL2. PLa wz = El (L - x1) + 2EIX1X2 + 6EJxl - 2EI X1 + 3EJ

where e.,, a, and b are constants. 1.10 Consider the uniform deformation of a square of side 2 units initially centered at X= (O, O).

The deformation is given by the mapping

1.9 Compute the nonlinear strain components E,J associated with the displacement field

X2,X2 Xz,X2 eo ......., e e

Ŀ1 ii Xi,Xi B X1,X1 a

a

(a) (b)

Figure Pl.8

1.8 Compute the axial strain in the Iine element AB and the shear strain at point O of the rectangular block shown in Figure Pl.8 using the engineering definitions.

X2,X2 X2,X2

eo

'1 b)

...

Parallel quadra tic

x1,X1 x1'X1 a a

(a) (b)

Figure Pl.7

1. 7 Write the position of an arbitrary point (x1, x2, x3) in the deformed body (solid lines) in terms of its coordinates in the undeformed body (broken lines) and compute the nonlinear Lagrangian strains for the body shown in Figure Pl.7.


~] MPa 9

-12 o [

12 [a)= ~

1.17 The components of a stress dyadic a at a point, referred to the rec:tangular Cartesian system (x1,x2,x3), are:

oarr ~Bare 8arz an - aee f _ 2ur or + r [)() + 8z + r + r - Po 8t2

E31 = E32 = E33 = Q

E11 = Ax~, E22 = Axi, 2e:12 = Bx1x2 where A and B are constants. (a) Determine the relation between A and B required for there to exist a continuous, single-valued displacement field that corresponds to this strain field.

(b) Determine the most generai form of the corresponding displacement field with the A and B from Part (a).

( c) Determine the specific corresponding displacement field that is fixed at the origin so that u = O and \7 x u = O when x = O.

1.16 Use the del operator ('V) and the dyadic form of a in the cylindrical coordinate system (r, (), z) to express the equations of motion (1.3.26a) in the cylindrical coordinate system:

1.15 Consider the Cartesian components of an infinitesima! strain field for an elastic body [8]:

where Eijk is the permutation symbol [see Eqs. (1.2.5b) and (1.2.7)] and Eij are the Cartesian components of the strain tensor. Hints: Begin with \7 x E and use the requirement ui,Jk = ui,kJ Ŀ

1.14 Show that in order to have a valid displacement field corresponding to a given infinitesima! strain tensor E, it must satisfy the compatibility relation

1 OUz Bue OUz ďUr OUz ČIJz = r 7fii + fu , Ezz = fu , "[r z = fu + Br

1 OUI] u; Eee = --- +- r (}() r

1 OUr Bue ue 10---+--- r- - r (}() or r

to compute the linear strain-displacement relations in the cylindrical coordinate system:

1.13 Use the clefinition (1.3.11) and the vector form ofthe displacement field and the del operator ('V) in the cylindrical coordinate system

where co, c1, Ŀ Ŀ Ŀ , c6 are constants.

U1 = -CoX1X2 + C1X2 + c2x3 + C4 U2 =~Co [v (x~ - xi)+ xi] + C4X3 + C5X1 + C6

where P, h, i/, and El are constants.

1.12 Find the linear strains associated with the 2-D displacement fielcl (u3 =O)

7 4 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS

Figure Pl.19

p 3

X T b

1 Beam (El, EA )

a

1.19-1.20 Write expressions for t.he tota! virtual work done, bW = 8U + 6V, for each of the beam structures shown in Figs. Pl.19 and Pl.20.

Figure Pl.18

Ur = 'Ug =O, v., = Uz(r)

(b) Fine! tbc relationship between the applied load P and displacernent b of the rigid core. (e) Determine the work clone by t.he load P.

Here tbe hollow cylinder represents tbc matrix around the fiber whilc the fiber is idealizcd as the rigicl core.

Find the following: (a) The stress vector acting on a piane perpcndicular to the vector 21 - 22 + ;i passing through the point. Here e; denote the basis vectors in (x1' x2, X;3) systcm.

(b ) The magnitude of the stress vector and the angle bctween the stress vector ami thc normai to the piane.

(e) The magnitudes of the normai and tangential components of the stress vcctor.

( d) Principal stresses. 1.18 The problcm of pulling a fiber imbeddcd in a matrix materia! can be idcalizcd (in the intcrcst

of gaining qualitative understanding of thc stress distributions at the fiber-matrix interfacc) as one of studying the following problern [8]: consider a hollow circular cylinder with outcr radius a, inner radius b, and lcngth L. T'ho out.er surface of thc hollow cylindcr is assumed to be fixed and its inner surface ideally bonded to a rigid circular cylindrical core of radius band length L, as shown in Fig. Pl.18. Suppose that an axial force F = Pis, is applied to the rigid core along its centroiclal axis.

(a) Find the axial displacement b of the rigid core by assurning thc following displaccment fiele! in the bollow cylinder:


u = u and V = V on I' 1' I' 1 + I' 2 = r

l(u,v) = L { ~ [ (c11~~ +c12~~f + (c12~~ +c22~~f + c33 (~~ + ~~f] - fiu - hv }dxdy- fr2 (t1u + t2v)ds

1.25

w0 = O, owo = O on the boundary I' on

l [ ( 2 )2 ( 2 )2 ( 2 )2 l D11 a wo D22 a wo a wo IT(wo) = !³ -2- 8x2 + -2- 8y2 + 2D12 8x8y - qwo dxdy

1.24

dw0 u0(0) =O, w0(0) =O, dx (O)= O

1L{ [ 2]2 (2 )2} EA duo 1 dwo El d wo II(uo,wo) =

0 2 dX + 2 (dx) + 2 dx2 dx

- Fowo(L) - Puo(L)

1.23

{L [El (d2 ) 2 k l IT(wo) =lo 2 d~o + 2wg - qw0 dx, w0(0) =O, w0(L) =O 1.22

1.21

Find the Euler-Lagrange equations and the natural boundary conditions associated with each of the functionals in Problems 1.21 through 1.25. The dependent variables are listed as the arguments of the functional. Ali other variables are not functions of the dependent variables.

Figure Pl.20


N

UN = L C³j sin i?rx sin j-rry ³,j=l

using the following N-parameter Galerkin approximation

-V2u = fo in a unit square, u. =O on the boundary

Use the least-squares method. Use the operator definition to be A= -(d2/dx2) to avoid increasing the degree of the characteristic polynomial for >..

1.34 Solve the Poisson equation

d2u -dx2=>.u, O<x<l; ti(O)=O, u(l)+u'(l)=O

that governs a cantilever beam on elastic foundation and subjected to linearly varying load (from zero at the free end to q0 at the fixed end). Take k = L = 1 and q0 = 3, and use algebraic polynomials.

1.33 Find the first two eigenvalues associated with the differential equation

1.31 Determine a two-parameter collocation solution of the cantilever beam problem in Example 1.5.1. Use collocation points x = L/2 and x = L.

1.32 Determine the one-parameter Galerkin solution of the equation

where ('uo, wo) denote the displacements of a point (x, y, O) along the x and z directions, respectively, r/>x denotes the rotation of a transverse normai about the y-axis, ami 1/Jx,ex,r/>z,1/!z, and ez are functions of X. Construct the tota! potential energy functional for the theory. Assume that the beam is subjected to a distributed load q(x) at thc top surface of the beam.

1.27 Give the approximation functions 'Pl and 'PO required in the (i) Ritz and (ii) weighted- residual methods to solve the following problems: (a) A bar fixed at the left end and connected to an axial elastic spring (spring constant.,

k) at the right end.

(b) A beam clamped at the left end and simply supported at the right end.

1.28 Considera uniform beam fixed atone end and supported by an elastic spring (spring constant k) in the vertical direction. Assume that the beam is loaded by uniformly distributed load qo. Determine a one-parameter Ritz solution using algebraic functions.

1.29 Use the tota! potential energy functional in Eq, (1.4.67) to determine a two-parameter Ritz solution of a simply supported beam subjected a transverse point load F0 at the center. You may use the symmetry about the center (x = L/2) of the beam to set up the solution.

1.30 Determine a two-parameter Galerkin solution of the cantilever beam problem in Example 1.5.1.

u(x, z) = uo(x) + z<f!x(x) + z21/Jx(x) + z30x(x) v(x,z) =O w(x, z) = wo(x) + z<f!z(x) + z21/Jz(x) + z:>e2(x)

1.26 Suppose that the tota! displacements ( u, v, w) along the three coordinate axes (x, y, z) in a laminated beam can be expressed as


1. Aris, R., Vectors, Tensore, and the Basic Equations of Fluid Mechanics, Prentice Hall, Englewood Cliffs, NJ (1962).

2. Hildebrand, F. I3., Methods of Applied Mathematics, Second Edition, Prentice-Hall, Englewood Cliffs, NJ (1965).

3. Jeffreys, H., Cartesian Tensors, Cambridge University Press, London, UK (1965).

4. Kreyszig, E., Advanced Engineering Mathematics, 6th Edition, .John Wiley, New York (1988). 5. Reddy, J. N. and Rasmussen, M. L., Advanced Engineering Analysis, John Wiley, New York,

1982; reprinted by Krieger, Melbourne, FL, 1990.

6. Reddy, J. N., Energy Principles and Variational Methods in Applied Mechanics, Second Edition, John Wiley, New York (2002).

7. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, NJ (1969).

8. Slaughter, W. S., The Linearized Theory of Elasticity, I3irkhiiuser, Boston, MA (2002).

9. Lekhnitskii, S. G., Theory o] Elasticity of an Anisotropie Elastic Body, Holden-Day, San Francisco, CA (1963).

10. Jones, R. M., Mechanics of Composite Materials, Second Edition, Taylor & Francis, Philadelphia, PA (1999).

11. Nowinski, J. L., Theory of Thermoelasticity with Applications, Sijthoff & Noordhoff, Alphen aan den Rijn, The Nctherlands (1978).

12. Carslaw, H. S. and Jaeger, .J. C., Conduction of Heat in Solids. Second Edition, Oxford University Press, London, UK (1959).

13. Jost, W., Diffusion in Solids, Liquuls, and Gases, Academic Press, New York (1952). 14. Tiersten, H. F., Linear Piezoelectric Plate Vibrations, Plenum, New York (1969).

15. Penfield, P., .Jr. and Hermann, A. H., Electrodynamics of Moving Media, Research Monograph No. 40, The M. I. T. Press, Cambridge, MA (1967).

16. Gandhi, M. V. and Thompson, B. S., Smart Materials and Siructures, Chapman & Hall, London, UK (1992).

17. Parton, V. Z. and Kudryavtsev, B. A., Engineering Mechanics of Composite Structures, CRC Press, Boca Raton, FL (1993).

18. Reddy, J. N. (Ed.), Mechanics of Composite Materials. Selected Works of Nicholas J. Pagano, Kluwer, The Netherlands (1994).

19. Lanczos, C., The Variational Principles of Mechanics, The University of Toronto Press, Toronto (1964).

20. Mikhlin, S. G., Variational Methods in Mathematical Physics, (translated from the 1957 Russian edition by T. Boddington) The MacMillan Company, New York (1964).

21. Mikhlin, S. G., The Problem of the Minimum of a Quadratic Functional (translated from the 1952 Russian edition by A. Feinstein), Holden-Day, San Francisco, CA (1965).

22. Mikhlin, S. G., An Advanced Course of Mathematical Physics, American Elsevier, New York (1970).

23. Kantorovitch, L. V. and Krylov, V. I., Approximate Meihods of Hiqher Analysis (translated by C. D. Benster), Noordhoff, The Netherlands (1958).

24. Galerkin, B. G., "Series-Solutions of Some Cases of Equilibrium of Elastic Beams and Plates" (in Russian), Vestn. Inshenernou., l, 897-903 (1915).

25. Galerkin, B. G., "Berechung der frei galagerten elliptschen Piatte auf I3iegung," Z Agnew. Math. Mech. (1923).

References for Additional Reading


3:~. Duartc, A. C., and Oden, .J. T., "An h-p Adaptive Method Using Clouds," Computer Methods in Applied Mechanics and Enqineerinq, 139, 2:37--262 (1996).

34. Licw, K. M., Huang, Y. Q., and Reddy, J. N., "A Hybrid Moving Least Squares ami Differcntial Quadrature (MLSDQ) Meshfree Method," Intemational Journal of Computatioruil Enqineerinq Sciencc, 3(1), 1-12 (2002).

35. Liew, K. M., T.Y. Ng, T. Y., Zhoa, X., Zou, G. P., and Reddy, J. N., "Harmonic Rcproducing Kernel Particle Method for Free Vibration Analysis of Rotating Cylindrical Shclls," Computer Methods in Applied Mechanics and Engineering, (to appear ).

:~6. Liew, K. M., Huang, Y. Q., and Reddy, .J. N., "Moving Least Square Differential Quadrature Method and Its Application to the Analysis of Shear Deforrnablc Plates," International Journal [or Numerical Methods iii Engineering, (to appear).

26. Ritz, W., "Uber eine neue Methode zur Losung gewisscr Variationsproblerne dcr mathernatischen Physik," J. Reine Angew. Math., 135, 1- 61 (1908).

27. Oden, J. T. and Reddy, J. N., Variational Methods in Theoretical Mechanics, Second Edition, Springer -Verlag, Berlin (1982).

28. Oden, .J. T. and Ripperger, E. A., Mechanics of Elastic Struciures, Second Edition, Hemisphere, New York (1981).

29. Reddy, .J. N., Applied Functional Analysis and VaTiational Metliods in Enqineerinq, McGraw-Ŀ Hill, New York (1986); reprinted by Krieger, Melbourne, FL (1992).

30. Washizu, K., Variational Methods in Elasticity arul Plasticity, Third Edition, Pergamon Prcss, New York (1982).

:n. Belytschko, T., Lu, Y., and Gu, L., "Element Free Galerkin Methods," Iniertiaiiotial .l ournal [or Numerical Methods in Enineerinq, 37, 229 256 (1994).

32. Melenk, J. M., and Babuska, I., "The Partition of Unity Finite Element Method: Basic Theory and Applications," Computer- Methods in Applied Mechanics and Enqineerinq, 139, 289-314 (1996).


Composite materials are those formed by combining two or more materials on a macroscopic scale such that they have better engineering properties than the conventional materials, for example, metals. Some of the properties that can be improved by forming a composite material are stiffness, strength, weight reduction, corrosion resistance, thermal properties, fatigue life, and wear resistance. Most man- made composite materials are made from two materials: a reinforcement materia! called jiber and a base material, called matrix material. Composite materials are commonly formed in three different types: (1) fibrous

composites, which consist of fibers of one material in a matrix material of another; (2) particulate composites, which are composed of macro size particles of one material in a matrix of another; and (3) laminated composites, which are made of layers of different materials, including composites of the first two types. The particles and matrix in particulate composites can be either metallic or nonmetallic. Thus, there exist four possible combinations: metallic in nonmetallic, nonmetallic in metallic, nonmetallic in nonmetallic, and metallic in metallic. The stiffness and strength of fibrous composites come from fibers which are

stiffer and stronger than the same material in bulk form. Shorter fibers, called whiskers, exhibit better strength and stiffness properties than long fibers. Whiskers are about 1to10 microns (i.e., micro inches orÕ in.) in diameter and 10 to 100 times as long. Fibers may be 5 microns to 0.005 inches. Some forms of graphite fibers are 5 to 10 microns in diameter, and they are handled as a bundle of several thousand fibers. The matrix materiai keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment. Matrix materials have their usual bulk-forrn properties whereas fibers have directionally dependent properties. The basic mechanism of load transfer between the matrix and a fiber can be

explained by considering a cylindrical bar of single fiber in a matrix material (see Figure 2.1.la). The load transfer between the matrix materiai and fiber takes place through shear stress. When the applied load P on the matrix is tensile, shear stress T develops on the outer surface of the fiber, and its magnitude decreases from a high value at the end of the fiber to zero at a distance from the end. The tensile stress a- in the fiber cross section has the apposite trend, starting from zero value at the end of the fiber to its maximum at a distance from the end. The two stresses together balance the applied load, P, on the matrix. The distance from the free end to the

2 .1 Basic Concepts and Terminology 2.1.1 Fibers and Matrix

2 Introduction to

Composite Materials

Figure 2.1.1: Load transfer and stress distributions in a single fiber embedded in a matrix material and subjected to an axial load.

(e)

Broken fiber

(b)

Springs representing the lateral restraint provided by the matrix

(~~

(a)

I.. ÅI Characteristic distance

point at which the normal stress attains its maximum and shear stress becomes zero is known as the characteristic distance. The pure tensile state continues along the rest of the fiber. When a compressive load is applied on the matrix, the stresses in the region of

characteristic length are reversed in sign; in the compressive region, i.e., rest of the fiber length, the fiber tends to buckle, much like a wire subjected to compressive load. At this stage, the matrix provides a lateral support to reduce the tendency of the fiber to buckle (Figure 2.1. l b). When a fiber is broken, the load carried by the fiber is transferred through shear stress to the neighboring two fibers (see Figure 2.1.lc), elevating the fiber axial stress level to a value of 1.50-.


Figure 2.1.2: Various types of fiber-reinforced composite laminae.

(d) Woven

(b) Bi-directional

(e) Discontinuous fiber

(a) Unidirectional

A lamina or ply is a typical sheet of composite material. It represents a fundamental building block. A fiber-reinforced lamina consists of many fibers embedded in a matrix material, which can be a metal like aluminum, or a nonmetal like thermoset or thermoplastic polymer. Often, coupling (chemical) agents and fillers are added to improve thc bonding between fibers and matrix material and increase toughness. The fibers can be continuous or discontinuous, woven, unidirectional, bidirectional, or randomly distributed (see Figure 2.1.2). Unidirectional fiber-reinforced laminae exhibit the highest strength and modulus in the direction of the fibers, but they have very low strength and modulus in the direction transverse to the fibers. A poor bonding between a fiber and matrix results in poor transverse properties and failures in the form of fiber pull out, fiber breakage, and fiber buckling. Discontinuous fiber-reinforced composites have lower strength and modulus than continuous fiber- reinforced composites. A laminate is a collection of laminae stacked to achieve the desired stiffness and

thickness. For exarnple, unidirectional fiber-reinforced laminae can be stacked so that the fibers in each lamina are oriented in the same or different directions (see Figure 2.1.3). The sequence of various orientations of a fiber-reinforced composite layer in a laminate is termed the lamination scheme or stacking sequence. The layers are usually bonded together with the same matrix material as that in a lamina. If a laminate has layers with fibers oriented at 30Á or 45Á, it can take shear loads. The lamination scheme and material properties of individual lamina provide an added fiexibility to designers to tailor the stiffness and strength of the laminate to match the structural stiffness and strength requirements.

INTRODUCTION TO COMPOSITE MATERIALS 83

2.1.2 Laminae and Laminates

Laminates made of fiber-reinforced composite materials also have disadvantages. Because of the mismatch of material properties between layers, the shear stresses produced between the layers, especially at the edges of a laminate, may cause delamination. Similarly, because of the mismatch of material properties between matrix and fiber, fiber debonding may take place. Also, during manufacturing of laminates, material defects such as interlaminar voids, delamination, incorrect orientation, damaged fibers, and variation in thickness may be introduced. It is impossible to eliminate manufacturing defects altogether; therefore, analysis and design methodologies must account for various mechanisms of failure. This book is devoted to the theoretical study of laminateci structures.

Determination of static, transient, vibration, and buckling characteristics of fiber- reinforced composite laminates with different lamination schemes, thicknesses, loads, and boundary conditions constitutes the major objective of the study. The theoretical concepts and analysis methods presented herein can help structural engineers in aerospace, civil, and mechanical engineering industries to select suitable materials and the number and orientations of fiber-reinforced laminae for the best performance in a particular application.

Figure 2.1.3: A laminate made up of laminae with different fiber orientations.

-0


where O"ij (O"i) are the stress components, Eij (ci) are the strain components, and Cij are the material coefficients, all referred to an orthogonal Cartesian coordinate system (x1, x2, x3). In general, there are 21 independent elastic constants for the most general hyperelastic material as discussed in detail in Section 1.3.6. When materials possess one or more planes of material symmetry, the number of independent elastic coefficients can be reduced. For materials with one plane of materiai symmetry, called monoclinic materials, there are only 13 independent parameters, and for materials with three mutually orthogonal planes of symmetry, called orthotropic materials, the number of material parameters is reduced to 9 in three-dimensional cases.

(2.2.1)

In this section we study the mechanical behavior of a typical fiber-reinforced composite lamina, which is the basic building block of a composite laminate. In formulating the constitutive equations of a lamina we assume that:

(1) a lamina is a continuum; i.e., no gaps or empty spaces exist.

(2) a lamina behaves as a linear elastic materiai.

The first assumption amounts to considering the macromechanical behavior of a lamina. If f³ber-rnatrix debonding and fiber breakage, for example, are to be included in the formulation of the constitutive equations of a lamina, then we must consider the micromechanics approach, which treats the constituent materials as continua and accounts for the mechanical behavior of the constituents and possibly their interactions. The second assumption implies that the generalized Hooke's law is valid. It should be noted that both assumptions can be removed if we were to develop micromechanical constitutive models for inelastic ( e.g., plastic, viscoelastic, etc.) behavior of a lamina. Composite materials are inherently heterogeneous from the microscopie point

of view. From the macroscopic point of view, wherein the materia! properties of a composite are derived from a weighted average of the constituent materials, fiber and matrix, composite materials are assumed to be homogeneous. The following discussion of constitutive equations is independent of whether the material is homogeneous or not, because the stress-strain relations hold for a typical point in the body. The generalized Hooke's law for an anisotropie materia! under isothermal

conditions is given in contraeteci notation [see Eq. (1.3.37a,b)J by

2.2 Constitutive Equations of a Lamina 2.2.1 Generalized Hooke's Law

In the remaining portion of this chapter, we study the mechanical behavior of a single lamina, treating it as an orthotropic, linear elastic continuum. The generalized Hooke's law is revisited (see Section 1.3.6) for an orthotropic material, the elastic coefficients of an orthotropic material are expressed in terms of engineering constants of a lamina, and the fiber-matrix interactions in a unidirectional lamina are discussed. Transformation of stresses, strains, and elasticity coefficients from the lamina material coordinates to the problem coordinates are also presented.


Figure 2.2.1: A unidirectional fiber-reinforced composite layer with the materia! coordinate system (xi, x2, x3) ( with the xi -ax³s oriented along the fi ber direction).

(2.2.2)

E1 = EjVJ + EmVm, EJEm E2---~---

- EjVm + EmVJ '

A unidirectional fiber-reinforced lamina is treated as an orthotropic material whose material symmetry planes are parallel and transverse to the fiber direction. The material coordinate axis x1 is taken to be parallel to the fiber, the x2-axis transverse to the fiber direction in the plane of the lamina, and the x3-axis is perpendicular to the plane of the lamina (see Figure 2.2.1). The orthotropic material properties of a lamina are obtained either by the theoretical approach or through suitable laboratory tests. The theoretical approach, called a micromechanics approach, used to determine

the engineering constants of a continuous fiber-reinforced composite material is based on the following assumptions:

1. Perfect bonding exists between fibers and matrix. 2. Fibers are parallel, and uniformly distributed throughout. 3. The matrix is free of voids or microcracks and initially in a stress-free state. 4. Both fibers and rnatrix are isotropie and obey Hooke's law. 5. The applied loads are either parallel or perpendicular to the fiber direction.

The moduli and Poisson's ratio of a fiber-reinforced material can be expressed in terms of the moduli, Poisson's ratios, and volume fractions of the constituents. To this end, let

EJ = modulus of the fiber; Em = modulus of the matrix VJ = Poisson's ratio of the fiber; Vrri = Poisson's ratio of the matrix VJ = fiber volume fraction; Vm = matrix volume fraction

Then it can be shown (see Problems 2.1 and 2.2) that the lamina engineering constants are given by

2.2.2 Characterization of a Unidirectional Lamina


Figure 2.2.2: Tests required for the mechanical characterization of a laminate.

! Xz Xz

4 p 111111111

p Å .. X1 X1

(a) (b)

~I p Å .. X1 X1

(e) (d)

' ;,rnin gages ~1 Ŀ~ -Ŀ V

(e)

X= Puit A

E2 Vf_t = Vl2 = -- ,

El p

Ec = E1 = -- ' AE1

Other micromechanics approaches use elasticity, as opposed to mechanics of materials approaches. Interested readers may consult Chapter 3 of Jones [3] and the references given there (also see [18-20]). The engineering parameters E1, E2, E3, G12, G1:3, G2:3, v12, vu, and v23 of an

orthotropic material can be deterrnined experimentally using an appropriate test specirnen made up of the material. At least four tests are required to determine the four constants E1, E2, E3 and G12 and the longitudinal strength X, transverse strength Y and shear strength S (and additional tests to determine G13 and G23). These are shown schematically in Figure 2.2.2a-d. For example, E1, v12 and X of a fiber-reinforced rnaterial are measured using a

uniaxial test shown in Figure 2.2.2a. The specimen consists of several layers of the material with fibers in each layer being aligned with the longitudinal direction. The specimen is then loaded along the longitudinal direction and strains along and perpendicular to the fiber directions are rneasured using strain gauges (see Figure 2.2.2e). By measuring the applied load P, the cross-sectional area A, the longitudinal strain Ec = E1 and transverse strain Et = E2, we can calculate

(2.2.3) G - EJ G - Em f - 2(1 + Vj) ' m - 2(1 + Vm)

where E1 is the longitudinal modulus, E2 is transverse rnodulus, v12 is the rnajor Poisson's ratio, and G12 is the shear rnodulus, and


* Units of E3 are msi, and the units of Á'l and Cl'.2 are 10-6 in./in.jÁF.

Table 2.2.2: Values of additional engineering constants for the materials listed in Table 2.2.1 *.

Materiai E3 Z/13 Z/23 Á'l Á'2

Aluminum 10.6 0.33 0.33 13.l 13.1 Copper 18.0 0.33 0.33 18.0 18.0 Steel 30.0 0.29 0.29 10.0 10.0 Gr.-Ep (AS) 1.3 0.30 0.49 1.0 30.0 Gr.-Ep (T) 1.5 0.22 0.49 -0.167 15.6 Gl.-Ep (1) 2.6 0.25 0.34 :3.5 11.4 Gl.-Ep (2) 1.3 0.26 0.34 4.8 12.3 Br.-Ep 3.0 0.25 0.25 2.5 8.0

*Moduli are in msi = million psi; 1 psi= 6,894.76 N/m2; Pa = N/m2; kPa = 103 Pa; MPa = 106 Pa; GPa = 109 Pa. t The following abbreviations are used for various materiai systems: Gr.-Ep (AS) = graphite-epoxy (AS/3501); Gr.-Ep (T) = graphite-epoxy (T300/934); Gl.-Ep = glass-epoxy; Br.-Ep = boron-epoxy.

Table 2.2.1: Values of the engineering constants for severa! materials*.

Materia!T E1 E2 G12 G13 G23 Z/12

Aluminum 10.6 10.6 3.38 3.38 3.38 0.33 Copper 18.0 18.0 6.39 6.39 6.39 0.33 Steel 30.0 30.0 11.24 11.24 11.24 0.29 Gr.-Ep (AS) 20.0 1.3 1.03 1.03 0.90 0.30 Gr.-Ep (T) 19.0 1.5 1.00 0.90 0.90 0.22 Gl.-Ep (1) 7.8 2.6 1.30 1.30 0.50 0.25 Gl.-Ep (2) 5.6 1.2 0.60 0.60 0.50 0.26 Br.-Ep 30.0 3.0 1.00 1.00 0.60 0.30

Tu1t S=TuJt= -- 27rr2h

where T is the applied torque, and r and h are the mean radius and thickness of the tube, respectively. The values of the engineering constants for several materials are presented in Tables 2.2.1 and 2.2.2.

1 1 ( 1 1 1 2vu) E1 = 4 Ee +Et+ Get - Ee

wherein Get is the only unknown. The shear strength S is determined from the test shown in Figure 2.2.2d:

The shear modulus is determined from the test shown in Figure 2.2.2c by measuring E1 = P / Ai::1, Ee, Et and vet, and using the transformation equation ( 4a) of Problem 2.3:

y = Pu1t A

E2 Vtf! = V12 = -- ,

E1

where Puit is the ultimate load (say, load at which the materia! reaches its elastic limit). Similarly, E2, v21 and Y can be determined from the test shown in Figure 2.2.2b:


(2.3.3)

Note that the inverse of [L] is equal to its transpose: [L]~1 = [LjT. The transformation relations (2.3.1) and (2.3.2) are also valid for the unit vectors

associated with the two coordinate systems:

(2.3.2) -sin e cose o {

x} [cose ; = si~e

The inverse of Eq. (2.3.1) is

(2.3.1) sin e cose o [

cose -s~ne

2.3 Transformation of Stresses and Strains 2.3.1 Coordinate Transforrnat.³ons

The constitutive relations (1.3.44) and (1.3.45) for an orthotropic material were written in terms of the stress and strain components that are referred to a coordinate system that coincides with the principal material coordinate system. The coordinate system used in the problem formulation, in general, does not coincide with the principal material coordinate system. Further, composite laminates have several layers, each with different orientation of their material coordinates with respect to the laminate coordinates. Thus, there is a need to establish transformation relations among stresses and strains in one coordinate system to the corresponding quantities in another coordinate system. These relations can be used to transform constitutive equations from the material coordinates of each layer to the coordinates used in the problem description. In forming f³at laminates, fiber-reinforced laminae are stacked with their x1x2-

planes parallel but each having its own fiber direction. If the z-coordinate of the problem is taken along the laminate thickness, the x3-coordinate of each lamina we will always coincide with the z-coordinate of the problem. Thus we have a special type of coordinate transformation between the materiai coordinates and the coordinates used in the problem description. Let (x, y, z) denote the coordinate system used to write the governing equations

of a laminate, and let (x1, x2, x3) be the principal material coordinates of a typical layer in the laminate such that x3-axis is parallel to the z-axis (i.e., the x1x2- plane and the xy-plane are parallel) and the x1-axis is oriented at an angle of +e counterclockwise (when looking down on the lamina) from the x-axis (see Figure 2.3.1). The coordinates of a materiai point in the two coordinate systems are related as follows (z = x3):


w here [ L] is the 3 x 3 matrix of direction cosines Pij.

(2.3.6a, b)

Then Eqs. (2.3.4) can be expressed in matrix form as

(2.3.5) axzl [all ayz , [a]m = a12 azz a13

and (i)m and (i)p are the orthonormal basis vectors in the materia} and problem coordinate systems, respectively. Note that the tensor transformation equations (2.3.4) hold among tensor components only. Equations (2.3.4) can be expressed in matrix forms. First, we introduce the 3 x 3 arrays of the stress components in the two coordinate systems:

where ( aij )m are the components of the stress tensor a in the materia} coordinates (x1, x2, x3), whereas (aij)p are the components of the same stress tensor a in the problem coordinates (x, y, z), and Pij are the direction cosines defined by

(2.3.4)

2.3.2 Transformation of Stress Components

Next we consider the relationship between the components of stress in (x, y, z) and (x1, x2, x3) coordinate systems. Let a denote the stress tensor, which has components an, a12, Ŀ Ŀ Ŀ, a33 in the materiai (m) coordinates (x1, x2, x3) and components axx,axy,ĀĀĀ,azz in the problem (p) coordinates (x,y,z). Since stress tensor is a second-order tensor, it transforms according to the formula

Figure 2.3.1: A lamina with materiai and problem coordinate systems.

X

Z=X3


or

lT22D..S h-(O"'"'"D..SsinB h)sinB+ (lTxyD..SsinB h)cosB-((TyyD..ScosB h)cosB

+ (axyD..S cos (} h) sin(}= O

The stress transformation equations (2.3.9) can be derived directly by considering the equilibrium of an element of the lamina (see Figure 2.3.2). Considcr a wcdgc clernent whose slant face is parallel to t.he fibers. Suppose that the thickness of the lamina is h., and the length of the slant face is D..S. Then the horizontal and verti ca] sidcs of thc wcdgcs are of lcngths D..S cos (} and D..S sin(}, respectivcly. The forccs acting on any face of the weclge are obtainecl by multiplying the stresses acting on the face with the area of the surface. Suppose that we wish to cletermine lT22 in terms of ( a xx, lTyy, a xy). Then by surnming ali forces

acting on the wedge along coordinate x2 (i.e., cquilibrium of forccs along x2) wc obtain

Exarnple 2.3.1:

The result in Eq. (2.3.9) can also be obtained from Eq, (2.3.7) by replacing () with -e.

(2.3.10) {CY}m = [R]{CY}p

CJ1 cos2 e sin2e o o o sin 2() CYxx CJ2 sin2 e cos2 e o o o - sin 2() CYyy CJ3 o o 1 o o o (J zz CJ4 o o o cos () - sin ®' o CYyz Uf> o o o sin e cos () o CYxz CJ5 - sin f cose sin ® cos ® o o o cos2 e - sin2 e CYxy

(2.3.9) or

The inverse relationship between { CJ }m and { CJ }p, Eq. (2.3.6a), is given by

CYxx cos2 o sin2 e o o o - sin2B CJ1 CYyy sin2 e cos2 o o o o sin 2() CJ2 (J zz o o 1 o o o CJ3

CYyz o o o cos () sin e o CJ4

CYxz o o o -sin() cose o CJ5

CYxy sin e cose - sin ® cos e o o o cos2 e - sin2 () CJ5 (2.3.7)

or {CY}p = [T]{CY}m (2.3.8)

Equation (2.3.6a) provides a means to convert stress components referred to the problem (laminate) coordinate system to those referred to the material (lamina) coordinate system, while Eq. (2.3.6b) allows computation of stress components referred to the problem coordinates in terms of stress components referred to the materiai coordinates. Equations (2.3.6a,b) hold for any generai coordinate transformation, and hence it holds for the special transformation in Eqs. (2.3.1). Carrying out the matrix multiplications in Eq. (2.3.6b), with [L] defined by

Eq. (2.3.1), and rearranging the equations in terms of the single-subscript stress components in (x, y, z) ami (x1, x2, x3) coordinate systems, we obtain


Consider a thin (i.e., the thickness is about one-tenth of the radius), f³lament-wound , closed cylindrical pressure vessel (see Figure 2.3.3). The vessel is of 63.5 cm (25 in.) internal diameter and pressurized to 1.379 MPa (200 psi). We wish to determine the shear and norma! forces per unit length of filament winding. Assume a filament winding angle of e = 53.125Á from the longitudinal axis of the pressure vessel, and use the following materia! properties, typical of graphite-epoxy materia!: E1 = 140 MPa (20.3 Msi), E2 = 10 MPa (1.45 Msi), G12 = 7 MPa (1.02 Msi), and v12 = 0.3. Note that MPa means mega (106) Pascal (Pa) and Pa = N/m2 (1 psi= 6,894.76 Pa).

Example 2.3.2:

Clearly, the expressions for a22 and a12 derived here are the same as those for a1 and a6, respectively, in Eq. (2.3.9). The stress component a11 can be determined in terms of (O'xx, O'yy, O'xy) by considering a wedge element whose slant face is perpendicular to the fibers (see Figure 2.3.2). By summing forces along the x- and y-coordinates we can obtain stresses a xx and a xy in terms of (au, a22, a12).

a12 = -axx sin e cose+ O"yy cose sin e+ O"xy( cos2 e - sin2 e) or

a12D.S h + (axx.6.S sin e h) cose+ (axy.6.S sin e h) sin e - (ayy.6.S cose h) sin e - (axy.6.Scose h)cose =O

Similarly, summing the forces along x1 coordinate, we obtain

Figure 2.3.2: A free-body diagram of a wedge element with stress components.

X

e Xz

y

e

y


(2.3.lla)

Since strains are also second-order tensor quantities, transfarmation equations derived far stresses, Eqs. (2.3.6a,b ), are also valid far tensor components of strains:

2.3.3 Transformation of Strain Components

0"11 = 0.2~89 (0.6)2 + 0.4~78 (0.8)2 = 0.3~90 MPa

0"22 = 0.2~89 (0.8)2 + 0.4~78 (0.6)2 = 0.2~77 MPa

= (0.4378 ~ 0.2189) O 6 O 8 = 0.1051 MP 0"12 h h X . X . h a

Thus the norma! and shear forces per unit length along the fiber-matrix interface are F22 = 0.2977 MN and F12 = 0.1051 MN, whereas the force per unit length in the fiber direction is F11 = 0.359 MN.

where p is internal pressure, D; is internal diameter, and h is thickness of the pressure vessel. We obtain

O"xx = 1.379 X 0.635 = 0.2189 MPa 1.379 X 0.635 = 0.4378 MP 4h h ' a YY = 2h h a

The shear stress O" xy is zero. Next we determine the shear stress along the fiber and the norma! stress in the fiber using the

transformation equations (2.3.9) or from the equations derived in Example 2.3.1. We obtain

pD; O"yy = 2h pD;

O"xx = 4h '

The equations of equilibrium of forces in a structure do not depend on the materia! properties. Hence, equations derived for the longitudinal (o-xx) and circumferential (o-yy) stresses in a thin- walled cylindrical pressure vessel are valid here:

Figure 2.3.3: A filament-wound cylindrical pressure vessel.

X

x,

X

y


c6 = 2c12 = O+ O+ 2Exy ( ~ - ~) = O.O

1 1 <::1 = c11 =O+ O+ 2Exy V2 V2 = 0.01 cm/cm

and the shear strain is given by

is

A square lamina of thickness h and planar dimension a is made of glass-epoxy materiai (E1 = 40 x 103 MPa, E2 = 10 x 103 MPa, G12 = 3.5 x 103 MPa, and v12 = 0.25). When the lamina is deformed as shown in Figure 2.3.4, we wish to determine the longitudinal strain in the fiber and shear strain at the center of the lamina. The fibers are oriented at 45Á to the horizontal. From Eq. (2.3.14), the only nonzero strain is Exy = 0.01. Hence, longitudinal strain in the fiber

Example 2.3.3:

(2.3.15)

We note that the transformation matrix [T] in Eq. (2.3.8) is the transpose of the square matrix in Eq. (2.3.14). Similarly, the transformation matrix in Eq, (2.3.13) is the transpose of the matrix [R] in Eq. (2.3.10):

Slight modification of the results in Eqs. (2.3.7) and (2.3.9) will yield the proper relations for the engineering components of strains. We have

Exx cos2 () sin2 () o o o - sin () cos () c1 Eyy sin2 o cos2 e o o o sin ® cos () E2

czz o o 1 o o o E3 (2.3.13) 2Eyz o o o cos () sin ® o E4 2Exz o o o - sin ®' cos e o c5 2Exy sin 2() - sin 2() o o o cos2 () - sin2 () c5

The inverse relation is given by

c1 cos2 () sin2 () o o o sin ® cos () Exx E2 sin2 () cos2 e o o o - sin e cos () Eyy c3 o o 1 o o o czz (2.3.14) E4 o o o cose -sin O o 2Eyz es o o o sin ® cos () o 2Exz c5 - sin 2() sin W o o o cos2 e - sin2 e 2Exy

(2.3.12)

Therefore, Eqs. (2.3.7) and (2.3.9) are valid for strains when the stress components are replaced with tensor components of strains from the two coordinate systems. However, the single-column formats in Eqs. (2.3.7) and (2.3.9) for stresses are not valid for single-column formats of strains because of the definition:

(2.3.llb)


E:r:r = 0.0963 X (0.6)2 + 1.45 X (0.8)2 - 0.3757 X 0.6 X 0.8 = 0.782 m/m Eyy = 0.0963 X (0.8)2 + 1.45 X (0.6)2 + 0.3757 X 0.6 X 0.8 = 0.764 m/m Exy = 2(0.0963 - 1.45) X (0.6) X 0.8 + 0.3757[(0.6)2 - (0.8)2] = -1.405

The strains in the (x,y) coordinates can be computed using Eq. (2.3.13):

®22 = _ o-n v12 + 0"22 = _ 17.95 x 0.3 + 14.885 = 1.45 m/m E1 E2 140 10

= 0-12 = 5.255 = 0.3757 E12 2G12 2 x 7

En = o-11 _ a22v12 = 17.95 _ 14.885 x 0.3 = 0.0963 m/m E1 E1 140 140

The strains in the materiai coordinates can be calculated using the strain-stress relations (1.3.47). Wc have (v2i/ E2 = V12/ Ei)

a-11 = 17.95 MPa, 0-22 = 14.885 MPa, 0-12 = 5.255 MPa

Suppose that the thickness of the cylindrical pressure vessel of Example 2.3.2 is h = 2 cm. Then the stress field in the materiai coordinates becomes

Example 2.3.4:

Figure 2.3.4: Deformation of a fiber-reinforced lamina.

y X1 y e= 0.2 cm

----

aI X X

a

a= 10 cm

,, ,,


Cn = Cn cos" () - 4C16 cos3 ()sin()+ 2( C12 + 2C55) cos2 () sin2 () - 4C25 cos () sin3 () + C22 sin4 ()

C12 = C12 cos4 () + 2( C16 - C25) cos3 ()sin()+ ( Cn + C22 - 4C55) cos2 () sin2 () + 2( C25 - C16) cos () sin3 () + C12 sin4 ()

C13 = C13 cos2 () - 2C35 cos ()sin() + C23 sin2 () C15 = C15 cos4 () + ( Cn - C12 - 2C55) cos3 ()sin()+ 3( C25 - C15) cos2 () sin2 ()

+ (2C55 + C12 - C22) cos () sin3 () - C25 sin" () C22 = C22 cos4 () + 4C25 cos3 ()sin()+ 2( C12 + 2C55) cos2 () sin2 ()

+ 4C15 cos () sin3 () + Cu sin" () C23 = C23 cos2 () + 2C35 cos ()sin() + C13 sin2 ()

Equation (2.3.17) is valid for generai constitutive matrix [C] (i.e., for orthotropic as well as anisotropie). Of course, [T] is the matrix based on the particular transformation (2.3.1) (rotation about a transverse normai to the lamina). Carrying out the matrix multiplications in (2.3.17) far the generai anisotropie

case, we obtain

(2.3.17) [C] = [Tl[Cl[T]T

where [C]m is the 6 x 6 materia! stiffness matrix [see Eq. (1.3.38a)] in the materiai coordinates and [T] is the transformation matrix defined in Eq, (2.3.8). Thus the transfarmed materiai stiffness matrix is given by ([C] = [C]p and [C] = [C]m)

(2.3.16)

However, the above equation involves five matrix multiplications with four-subscript material coefficients. Alternatively, the same result can be obtained by using the stress-strain and strain-stress relations (1.3.38a,b), and the stress and strain transformation equations in (2.3.8) and (2.3.15):

In farmulating the problem of a laminated structure, we must write the governing equations, with all their variables and coefficients, in the problem coordinates. In the previous section we discussed transfarmation of coordinates (which are also valid far displacements and farces), stresses, and strains. The only remaining quantities that need to be transfarmed from the materiai coordinate system to the problem coordinates are the materia! stiffnesses Cij and thermal coefficients of expansion <Yij.

The materiai stiffnesses Cij in their origina! form [see Eq. (1.3.35)] are the components of a fourth-order tensor. Hence, the tensor transformation law holds. The fourth-order elasticity tensor components Cijkf in the problem coordinates can be related to the components Cmnpq in the materia! coordinates by the tensor transfarmation law

2.3.4 Transformation of Materiai Coefficients


(2.3.20b)

Thus the compliance coefficients Sij referred to the (x, y, z) system are related to the compliance coefficients Sij in the material coordinates by ([S]p = [S] and [S]m = [S])

(2.3.20a) { E}p = [R]T {dm = [R]T ([S]m { o }m) = [R]T[S]m ([R]{ a }p) = [S]p{a}p

where the Cij are the transformed elastic coefficients referred to the (x, y, z) coordinate system, which are related to the elastic coefficients in the material coordinates e; by Eq. (2.3.18). Note that C14, C1s, C15, C24, C25, C26, C34, C35, C35, C45, C45, and C55 are zero for an orthotropic material. In order to relate compliance coefficients in the two coordinate systems, we use

the strain transformation equation in Eq. (2.3.15):

(2.3.19)

Exx Eyy Ezz 2Eyz 2Exz 2Exy

Cm c26 C35 o o C55

o o o o o o C44 C4s C4s Css o o

C11 C12 C13 C21 C22 C23 C31 C32 C33 o o o o o o C15 C25 C35

axx ayy azz ayz axz rYxy

When [C] is the matrix corresponding to an orthotropic material, it has the form shown in Eq, (1.3.44); then Eq. (2.3.16) has the explicit form [cf. Eq. (1.3.42) for monoclinic materials]

- 4 3 2 2 C25 = C25 cos e+ ( C12 - C22 + 2C55) cos e sin e+ 3( C15 - C25) cos e sin e + (Cn - C12 - 2C55) cose sin3 e - C15 sin" e

C33 = C33 C35 = (C13 - C23) cose sin e+ C35( cos2 e - sin2 l³) C55 = 2( C15 - C25) cos3 e sin e+ ( C11 + C22 - 2C12 - 2C55) cos2 e sin2 e

+ 2( C26 - Cm) cose sin3 e+ C55(cos4 e+ sin4 B) C44 = C44 cos2 e+ C5s sin2 e+ 2C4s cose sin e C4s = C45( cos2 e - sin2 B) + ( C55 - C44) cose sin e Cs5 = C55 cos2 e + C44 sin2 e - 2C4s cose sin e C14 = C14 cos3 e+ ( C15 - 2C46) cos2 e sin e+ ( C24 - 2C56) cose sin2 e+ C2s sin3 e C1s = C1s cos3 e - ( C14 + 2C55) cos2 e sin e+ ( C2s + 2C45) cose sin2 e - C24 sin" e C24 = C24 cos3 e+ ( C25 + 2C45) cos2 e sin e+ ( C14 + 2Cs5) cose sin2 e+ C15 sin3 e C2s = C2s cos3 e+ (2Cs6 - C24) cos2 e sin e+ ( C1s - 2C45) cose sin2 e - C14 sin3 e C34 = C34 cose + C3s sin e C3s = C3s cose - C34 sin e C45 = C45 cos3 e+ ( Cs5 + C14 - C24) cos2 e sin e+ ( C1s - C2s - C45) cose sin2 e

- C55 sin3 e Cs5 = Cs5 cos3 e+ ( C1s - C2s - C45) cos2 e sin e+ (C24 - C14 - Cs6) cose sin2 e

+ C45 sin3 e (2.3.18)


For an orthotropic material, the compliance matrix [8] has the form shown in Eq. (1.3.45), and the strain-stress relations in the problem coordinates are given by

®xx Su 812 S13 o o 516 CTxx

®yy 821 822 S13 o o 826 CTyy

®zz S31 S32 S33 o o S35 (T zz (2.3.22) 2®yz o o o S44 S45 o CTyz

2®xz o o o S45 S55 o CTxz

2®xy S15 S2a S35 o o 566 CTxy

811 = 511 cos4 e - 2516 cos3 e sin e+ (2512 + 555) cos2 e sin2 e - 2526 cose sin3 () + 522 sin" ()

812 = 512 cos4 e+ (516 - 525) cos3 e sin e+ (511 + 522 - 555) cos2 e sin2 e + (526 - 516) cos () sin3 e+ 512 sin? e

S13 = 513 cos2 e - 535 cose sin e + 523 sin2 e 516 = 516 cos4 e+ (2511 - 2512 - 555) cos3 ()sin()+ 3(526 - 515) cos2 e sin2 e

+ (566 + 2512 - 2522) cose sin3 () - 525 sin4 () S22 = 822 cos4 () + 2525 cos3 ()sin()+ (2512 + 555) cos2 () sin2 e

+ 2816 cos () sin3 () + 511 sin4 ()

S23 = 523 cos2 e+ 835 cose sin e+ 813 sin2 e 526 = 825 cos" e+ (2812 - 2522 + 855) cos3 e sin e+ 3(816 - 525) cos2 e sin2 e

+ (28u - 2812 - 855) cose sin3 e - 816 sin4 e S33 = 533 S35 = 2(513 - 823) cos ®' sin ® + 536(cos2 e - sin2 e) S55 = 855(cos2 () - sin2 ())2 + 4(816 - 825)(cos2 () - sin2 8) cos8sinB

+ 4(811 + 822 - 2512) cos2 () sin2 () S44 = 544 cos2 () + 2545 cos ()sin()+ 555 sin2 () S45 = 545(cos2 () - sin2 ()) + (855 - 844) cos ® sin ®' S55 = 855 cos2 e+ 544 sin2 e - 2545 cose sin e S14 = 514 cos3 e+ (515 - 845) cos2 e sin e+ (524 - 855) cose sirr' e+ 525 sin3 e S15 = 815 cos3 e - (814 + 555) cos2 e sin e+ (525 + 545) cose sin2 e - 824 sin3 e S24 = 524 cos3 e+ (525 + 845) cos2 ()sin()+ (814 + 555) cos () sin2 () + 515 sin3 () S25 = 525 cos3 e+ (-824 + 555) cos2 e sin e+ (815 - 845) cose sin2 e - 814 sin3 e S34 = 834 cos () + 535 sin ®' S35 = 835 cos () - 834 sin ®

S45 = (2814 - 2824 + 855) cos2 8 s³n ®' } (2815 - 2825 - 845) cos () sin2 () + 545 cos3 e - 855 sin3 ()

S55 = (2815 - 2825 - 845) cos2 e sin e+ (2824 - 2514 - 855) cose sin2 e + 855 cos3 () + 845 sin3 () (2.3.21)

Expanded form of the relations in Eq. (2.3.20b) is


Most laminates are typically thin and experience a plane state of stress ( see Section 1.3.6). For a lamina in the x1x2-plane, the transverse stress cornponents are a33, a13, and a2:3 (see Figure 2.4.1). Although these stress cornponents are srnall in cornparison to au, a22, and a12, they can induce failures because fiber-reinforced composite laminates are weak in the transvcrso direction (because the strength providing fibers are in the x1x2-plane). For this reason, the transverse shear stress components are not neglected in shear deformation theories. However, in most equivalent-single layer theories the transverse normai stress 0'33 is neglected. Then the constitutive equations rnust be modified to account for this fact.

2.4 Plane Stress Constitutive Relations

where all quantities are referred to the (x,y,z) coordinate system, and {ar} and { O'.M} are vectors of thermal and hygroscopic coefficients of expansion, respectively.

(2.3.24)

{a}1k) = [C](k) ({c}1k) _ {ar}~k)(T-To)-{(J'M}1k)(c-co))

{c}1k) = [5](k){a}1k) + {ar}1k)(T- To) + {txAf }1kl(c - co)

The same transformations hold for the coefficients of hygroscopic expansion. The transforrnation relations (2.3.18), (2.3.21), and (2.3.23) are valid for a rectangular coordinate system (x1, x2, x3) which is oriented at an angle e (in the xy-plane) from the (x, y, z) coordinate system (see Figure 2.3.1). The orientation angle e is measured counterclockwise from the x-axis to the x1 -axis. In sumrnary, Eq. (1.3.44) represents the stress-strain relations in the principal

materiai coordinates (x1,x2,x3), and Eq. (2.3.19) represents the stress-strain relations in the ( x, y, z) coordinate system. The materiai coefficients of the lamina in the (x, y, z) coordinate system are relateci to materiai coefficients in the materiai coordinates by Eq. (2.3.18). In general, for the kth layer of a laminate, the hygro-therrno-elastic stress-strain relations in the laminate coordinate system can be written as

(2.3.23)

O'.xx = au cos2 e + 0:22 sin2 e . 2 2

CYyy = çu sin 8 + 0'.22 COS 8 2axy = 2(a11 - n22)sin8cos8 2txxz = O, 2ayz = O, LXzz = tx33

Note that Eq. (2.3.22) relates stresses to strains in the problem coordinates while Eq. (1.3.45) relates the stresses to strains in the material coordinates. The thermal coefficients O'.ij are the components of a second-order tensor, and

therefore they transform like the strain components (because 0:5 = 20:12, and so on). In the context of the present study, only nonzero components of thermal expansion tensor are 0:11 = cq, 0:22 = 0:2, and 0:33 = 0:3. All other components are zero. Hence, following Eq. (2.3.7), we can write the transformation relations (0:0 = 0:12 = O, 0:5 = 0:13 = O, 0:4 = 0:2:~ = O)


where Q~Jl are the plane stress-reduced stiffnesses, e~J) are the piezoelectric moduli, and Eij are the dielectric constants of the kth lamina in its materiai coordinate system, ( O"i, Ei, Ei, Di) are the stress, strain, electric field, and electric displacement components, respectively, referred to the materiai coordinate system (x1, x2, x3), o:1 and 0:2 are the coefficients of thermal expansion along the x1 and x2 directions,

r r [Qu Q12 o r) rl ~ "I L\.T rk) [~ o e31 l p:r 0"2 = Q12 Q22 o E2 - 0:2 !:lT o e32 0"6 o o Q55 E6 o o [3

(2.4.1)

{ ~: I" = [ Qi4 o rk) { E4 rk) [ o e24 0rr Q55 E5 e15 o o] E2 (2.4.2) [3

rr rn:r [ C~J o o e15 orÅI E2 ['~I o ~Ŀ r { ~:} (k) o e24 o 0 E4 + E22

e32 o o 0 E5 o E33 C3 E5

(2.4.3)

The condition 0"33 = O results in the following thermoelastic constitutive equations for the kth layer that is characterized as an orthotropic lamina with piezoelectric effect:

Figure 2.4.1: A lamina in a piane state of stress.


O'.xy = ( a1 - a2) sin ®' cos () (2.4.9)

. 2{] 2() O:yy = 0:1 sm o + 0:2 cos ,

O'.xx, ayy, and O'.xy are the transformed thermal coefficients of expansion [see Eq. (2.3.23)]

Q11 = Q11 cos4 () + 2( Q12 + 2Q55) sin2 () cos2 () + Q22 sin4 () Q12 = (Qu + Q22 - 4Q55) sin2 () cos2 () + Q12(sin4 () + cos4 ()) Q22 = Qu sin4 () + 2(Q12 + 2Q55) sin2 () cos2 () + Q22 cos4 () Q15 = ( Qu - Q12 - 2Q55) sin ® cos3 () + ( Q12 - Q22 + 2Q55) sin3 () cos () Q25 = ( Qu - Q12 - 2Q55) sin3 () cos () + ( Q12 - Q22 + 2Q55) s³n ® cos3 () Qf³6 = ( Qu + Q22 - 2Q12 - 2Q55) sin2 () cos2 () + Q55(sin4 () + cos4 ()) ~ 2 2 Q44 = Q44 cos () + Q55 sin () Q45 = (Q55 -Q44)cosBsin() ~ 2 2 Q55 = Q55 cos () + Q44 sin () (2.4.8)

where 'ljJ denotes the scalar electric potential [see Eq. (1.3.89)] and

(2.4. 7) O

{

Q];_} ax &'lj; O ay J "" az

Exy

Eyy

o

q45] { Čyz} + [~14 Q55 Čxz e15

(2.4.6) ::: ~l{ Ä} O l ! ::: )- [ Exx O Čyz fxy e35 rxz o

Čxy

{ ~:} &'!f; &y 0P_ &z (2.4.5)

({ Exx} { O'.xx } ) [O O Eyy - O'.yy 6.T + o o Tuy 2a~ O O

Note that the reduced stiffnesses involve six independent engineering constants: E1, E2, v12, G12, G13, and G23. The transformed stress-strain relations of an orthotropic lamina in a plane state

of stress are ( the superscript k is omitted in the interest of brevity)

(2.4.4b) Q(k) - a(k) Q(k) - a(k) Q(k) - a(k) 66 - 12 ' 44 - 23 ' 55 - 13

(2.4.4a) (k)

(k) _ E2 Q22 - (k) (k)

1 - V12 V21

(k) (k) _ E1

Qll - (k) (k) ' 1 - V12 V21

respectively, and t::.T is the temperature increment from a reference state, t::.T = T - Ti; Recall from Eq. (1.3.72) that QlJ) are related to the engineering constants as follows:


(2.4.15)

(2.4.14)

(2.4.13)

(2.4.12)

(2.4.11)

The transformed coefficients for various angles of orientation are given below:

[48010 0.1728 o o

11 0.1728 25.11 o o

[Q]o=90 = ~ o 1.20 o msi o o 0.47 o o o

[8923 6.203 o o 50761 6.203 8.923 o o 5.076 [Q]o=45 = o o 0.835 0.365 o rnsi

o o 0.365 0.835 o 5.076 5.076 o o 7.390

[ 8923 6.203 o o -50701 6.203 8.923 o o -5.076

[Q]o=-45 = o o 0.835 -0.365 o msi o o -0.365 0.835 o -5.076 -5.076 o o 7.390

[ 1551 4.696 o o 70071 4.696 5.355 o o 1.785

[Q]e=30 = o o 0.6525 0.3161

5.~83

msi o o 0.3161 1.0175 7.007 1.785 o o

[

25.11 0.1728 o o 0.1728 4.8010 o o

[Q]o=O = 0 0 0.47 0 o o o 1.20 o o o o

G12 = 1.36 x 106 psi, G13 = 1.2 x 106 psi, G23 = 0.47 x 106 psi

V12 = 0.036, V13 = 0.25, Vz3 = 0.171 The matrix of piane stress-reduced elastic coefficients for the materiai can be calculated using Eqs. (2.4.4) and (2.4.8) for various values of (} as

E1 = 25.1 x 106 psi, E2 = 4.8 x 106 psi, E3 = 0.75 x 106 psi

The materiai properties of graphite fabric-carbon matrix layers are (see Example 1.3.4):

Example 2.4.1:

This completes the development of constitutive relations for an orthotropic lamina in a plane state of stress.

(2.4.10)

- 2() . 2() - . 2() 2() e31 = e31 cos + e32 sin , e32 = e31 sin + e32 cos , e35 = (e31- e32)sin()cos(), e14 = (e15 -e24)sin()cos() - 2() . 2() - 2() . 2() e24 = e24 cos + e15 sm , e15 = e15 cos + e24 sm e25 = ( e15 - e24) sin ® COS (), Exx = E11 COS2 () + E22 Sin2 () Eyy = E11 sin2 () + E22 COS2 (), Exy = ( E11 - E22) sin ® COS ()

and eij are the transformed piezoelectric moduli, and Exx, Exy, and Exy are transformed dielectric coefficients


(4a) ~x = S11 = 811 cos4 () + (2512 + S55) sin2 () cos2 () + S22 sin4 O

and so on. Thus, the equivalent modulus of elasticity Ex in the problem coordinates, far example, can be evaluated using the engineering constants in the materiai coordinate system:

(3) 1 - Vyx - 7/xy,x _ S- T/xy Y S- -E =Su, --E = 812, -- 16 --' = 26 x y Ex - ' Ey

Comparing Eq. (2) with Eq. (1), we note that

(2) 'I";,. x l { (T XX } Tfxy. y ---e;- (T yy _I_ Uxy Gx11

Vy.r. -E;

1 Ey

1'lxy,Ŀy Ey

{

Exx } [ i, E _ _ Vxy YY - Ex

2®xy ry~~x

where S;i are the transformed compliances defined in Eq. (2.3.21). Guided by the form of the strain-stress relations (1.3.47) in the materiai coordinates, we can write strain-stress relations in the problem coordinates as

(1)

2.3 (Apparent moduli of an orthotropic materiai) Note that the transformed materiai compliance matrix [S] is relatively full and is in the same form as that for a monoclinic materiai. For an orthotropic materiai, we have

2.2 Consider the composite lamina of Problem 2.1 but subjected to axial stress u2 alone, as shown in Fig. P2.2. Derive the result

X2

b b ~~ ~~

''""I~ (j 1 ''""I~ h,-- x, h,-- x, 0.5hmI 0.5hmI

L ~

Figure P2.1 Figure P2.2

2.1 Consider the composite lamina subjected to axial stress u1, as shown in Fig. P2.l below. Let E1, Vf and A1 denote Young's modulus, volume fraction and area of cross section of the fiber, and (Ern, Vrn, Am) be thc same quantities for the matrix. Assuming that piane sections remain piane during the deformation process and both matrix and fiber undergo the same Jongitudinal deformation ~x1, derive the law of mixtures,

Problems


2.6 (Continuation of Problem 2.3) Show that the coefficient of mutua] influence is zero at () = 0Á and () = 90Á.

1 cos4 () = S ( 3 + 4 cos W + cos 4())

sin4 () = ~(3 - 4cosW + cos-³®)

cos2 () sin2 () = ~ (1 - cos 4())

2.4 (Continuation of Problem 2.3) Derive an expression for Gxy in t.erms of E1, E2, v12, G12, and ().

2.5 (Continuation of Problem 2.3) Show that Gxy is a maximum for () = 45Á. Make use of the following trigonometrie identities:

(10)

(9)

Show that

(8)

The compliance 816 and 826 are related, by definition, to the coefficients T/xy,x and T/xy,y by

(7) for a;; =I- O and ali other stresses being zero

T/ij,i =characterizes shearing in the x;xi-plane caused by a normai stress in the X; -direction (i =I- j)

Physically, 816 represents the normai strain in the x-direction caused by the shear stress in the xy-plane, when ali other stresses are zero. Since 816 = 861, it also represents the shear strain in the xy-plane caused by the normai stress along the x-direction, when ali other stresses are zero. Guided by these observations, Lekhnitskii [4] introduced the following engineering constants, called coefficients of mutuai infiuence:

816 = (2 + 2v21 - -1-) sin ®' cos3 (J _ (2 + 2v21 - -1-) sin3 ()cos() (6) E1 E2 G12 E2 E2 G12

We note that the compliance 816, which was zero in the materiai coordinates, is contributed by 811, 812, 822, and 866:

(5)

Thus, the apparent compliance Su in the (x, y, z) coordinate system is contributed by the compliances Su, 812, 822, and 866 and the lamination angle ():

(4b) where


1] msi 0.392 o o 1.308 o o o 0.9 o o o 1.03 o o o

[

20.118 0.392

[Q] = o o o

2.12 Deterrnine the transformation matrix (i.e., direction cosines) relating the orthonormal basis vectors (e1,e2,e3) of the systern (x1'x2,x3) to the orthonormal basis (e't,e2,e~l) of the system (x~' x2, x3), when e; are given as follows: e~ is along the vector e1 - e2 + e3 and e2 is perpendicular to the piane 2x1 + 3x2 + x3 - 5 = O.

2.13 Verify the transformation relations for the piezoelectric moduli given in Eq, (2.4.10).

2.14 Consider a square, graphite-epoxy lamina of length 8 in., width 2 in., and thickness 0.005 in., and subjected to an axial load of 1000 lbs. Dctermine the transverse normai strain t:3. Assume that the load is applied parallel to the fibers, and use E1 = 20 rnsi, E2 = 1.3 msi, G12 = G13 = 1.03 msi, G23 = 0.9 rnsi, v12 = v1:3 = 0.3, and v23 = 0.49.

2.15 Compute the numerica! values of the recluced stiffnesses Q;j for the graphite-epoxy materiai of Problern 2.14. Ans:

1 . U1 = S (3Qu + 3Q22 + 2Q12 + 4Q56)

1 U2 = 2 (Q11 - Q22)

1 U3 = 8 (Qu + Q22 - 2Q12 - 4Q55)

1 U4 = 8 (Q11 + Q22 + 6Q12 - 4Q55)

where

Qn = U1 + U2 cos2B + U3cos48 Q12 = U4 - U3 cos48 Q22 = U1 - U2 cosW + U3 cos48 Ql6 = ~U2sin2B+U3sin48

Q25 = ~ U2 sin 28 - U3 sin 48 - 1 Q66 = 2 (U1 - U4) - U3cos48

2.11 Rewrite the transformation equations (2.4.8) as

S1 = (3Q11 + 3Q22 + 2Q12 + 4Q66) S2 = (Q12 - Q6G) S3 = (Q11 + Q22 + 2Q56) S4 = (Qn + Q22 + 2Q12)

2.7 (Continuation of Problem 2.3) Show that the moduli Ex (and Ey) varies between E1 and E2, but it can either exceed or get smaller than both E1 and E2.

2.8 (Continuation of Problem 2.3) Derive the expression for Ber in terms of E1, E2, v12, G12, a1, a2, and e for the nonisothermal case.

2.9 (Continuation of Problem 2.3) Derive the expression for G,ry in terrns of E1, E2, v12, G12, a1, a2, and e for the nonisotherrnal case.

2.10 Show that the following combinations of stiffness coefficients are invariant:


1. Ambartsumyan, S. A., Theory of Anisotropie Shells, NASA Report TT F-118 (1964).

2. Ambartsumyan, S. A., Theory of Anisotropie Plates, Izdat. Nauka, Moskva (1967), English translation by Technomic, Stamford, CT (1969).

3. Jones, R. M., Mechanics of Composite Materiale, Second Edition, Taylor & Francis, PA (1999). 4. Lekhnitskii, S. G., Theory of Elasticity of an Anisotropie Body, Mir Publishers, Moscow (1982). 5. Christensen, R. M., Mechanics of Composite Materiale, John Wiley, New York (1979).

6. Tsai, S. W. and Hahn, H. T., Introduction to Composite Materiale, Technomic, Lancaster, PA (1980).

7. Agarwal, B. D. and Broutman, L. J., Analysis and Performance of Fiber Composites, John Wiley, New York (1980).


9. Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials, Kluwer, The Netherlands (1986).

10. Mallick, P. K., Fiber-Reinforced Composites, Marce! Dekker, New York (1988).

11. Gibson, R. F., Principles of Composite Materiai Mechanics, McGraw-ĿHill, New York (1994).

12. Parton, V. Z. and Kudryavtsev, B. A., Engineering Mechanics of Composite Struciures, CRC Press, Boca Raton, FL (1993).


Also, compute the transformed thermal coefficients of expansion for () = 45Á.

The transformed coefficients for various angles of orientation are given below:

r 1006 3.02 o o

~] GPo

3.02 140.9 o o [Qle=90 = ~ o 7 o

o o 7 o o o

rĿĿ25 32.25 o o 3271 l 32.25 46.25 o o 32.71

[Q]e=45 = o o 7 o O GPa o o o 7 o 32.71 32.71 o o 36.23

3.02 o o o] 10.06 o o o O 7 O O GPa o o 7 o o o o 7 r

140.90 3.02

[Q] = o o o

G13 = 7 x 103 MPa, G23 = 7 x 103 MPa, v12 = 0.3 o:1 = 1.0 x 10-6 m/m/ÁK, a2 = 30 x 10-6 m/m/ÁK

Show that (1 GPa = 103 MPa = 109 Pa)

E1 = 140 x 103 MPa, E2 = 10 x 103 MPa, G12 = 7 x 103 MPa

2.16 The materiai properties of AS/3501 graphite-epoxy materiai layers are


13. Rcddy, J. N. (Ed.), Mechanics of Composite Materials, Selected Works of Nicholas J. Pagano, Kluwcr, The Netherlands (1994).

14. Adams, D. F. and Doncr, D. R., "Longitudinal Shear Loading of a Unidirectional Composite," Journal of Composite Materials, 1, 4-17 (1967).

15. Adams, D. F. and Doner, D. R., "Transverse Normai Loading of a Unidirectional Composite," Journal of Composite Maieriols, 1, 152-164 (1967).

16. Ishikawa, T., Koyama, K., and Kobayashi, S., "Thcrrnal Expansion Coefficients of Unidircctional Cornposites," Journal o] Composite Maieriols, 12, 153-168 (1978).

17. Halpin, J. C. and Tsai, S. W., "Effects of Environmental Factors on Composite Materials," AFML-TR-67-423, Air Force Flight Mechanics Laboratory, Dayton, OH (1969).

18. Tsai, S. W., Struciural Behavior of Composite Maierials, NASA CR-71, (1964). 19. Chamis, C. C. and Sendeckyj, G. P., "Critique on Theories Predicting Thermoelastic Properties of Fibrous Composites," Journal of Composite Materials, 332-358 (1968).

20. Zhang, G. and June, R. R., "An Analytical and Numerica! Study of Fiber Microbuckling," Composite Science and Technology, 51, 95-109 (1994).


Analyses of composite plates in the past have been based on one of the following approaches:

(1) Equivalent single-layer theories (2-D) (a) Classica! laminated pia te theory (b) Shear deformation laminated plate theories

(2) Three-dimensional elasticity theory (3-D) (a) Traditional 3-D elasticity formulations (b) Layerwise theories

(3) Multiple model methods (2-D and 3-D)

The equivalent single layer (ESL) plate theories are derived from the 3-D elasticity theory by making suitable assumptions concerning the kinematics of deformation or the stress state through the thickness of the laminate. These assumptions allow the reduction of a 3-D problem to a 2-D problem. In the three-dimensional elasticity theory or in a layerwise theory, each layer is modeled as a 3-D solid. In this chapter, we present the classica! plate theory and the first-order shear deformation plate theory as applied to laminated plates. Literature reviews and development of the governing equations of the third-order shear deformation plate theory and the layerwise theory will be presented in later chapters (see Chapters 11 and 12).

3.1.2 Classification of Structural Theories

Composite laminates are formed by stacking layers of different composite materials and/or fiber orientation. By construction, composite laminates have their planar dimensions one to two orders of magnitude larger than their thickness. Often laminates are used in applications that require membrane and bending strengths. Therefore, composite laminates are treated as plate elements. The objective of this chapter is to develop two commonly used laminate plate

theories, namely the classica! plate theory and the first-order shear deformation plate theory. To provi de a background for the theories discussed in this chapter, an overview of pertinent literature on laminate plate theories is included here.

3.1.1 Preliminary Comments 3.1 lntroduction

3 Classica! and First-Order Theories of Laminated Composite Plates

The resultants can be written in terms of 'Pi with the help of the assumed constitutive equations ( stress-strain relations) and strain-displacement relations. More complete development of this procedure is forthcoming in this chapter. The same approach is used when 'Pi denote stress components, except that the

basis of the derivation of the governing equations is the principle of virtual forces. In

(3.2.4) h

(m) !2 ( )m Rij = _il z aij dz 2

where h denotes the tota! thickness of the plate, and Do denotes the undeformed midplane of the plate, which is chosen as the reference piane. Since all functions are explicit in the thickness coordinate, the integration over plate thickness is carried out explicitly, reducing the problem to a two dimensionai one. Consequently, the Euler-Lagrange equations of Eq. (3.2.2) consist of differential equations involving the dependent variables cp{ (x, y, t) and thickness-averaged stress resultants, R~';):

(3.2.3) h

f (-) dV = j 2 f (-) dD dz lv al. - ~ ln0

where 8U, 8V, and 8K denote the virtual strain energy, virtual work clone by external applied forces, and the virtual kinetic energy, respectively. These quantities are determined in terms of actual stresses and virtual strains, which depend on the assumed displacement functions, 'Pi and their variations. For plate structures, laminated or not, the integration over the domain of the plate is represented as the (tensor) product of integration over the plane of the plate and integration over the thickness of the plate, because of the explicit nature of the assumed displacement field in the thickness coordinate:

(3.2.2) O = lor ( 8U + 8V - 8K) dt

(3.2.1) N

'Pi(x, y, z, t) = 2)z)1 cpHx, y, t) j=O

where 'Pi is the ith component of displacement or stress, (x, y) the in-piane coordinates, z the thickness coordinate, t the time, and 'Pi are functions to be determined. When 'Pi are displacements, then the equations governing cp{ are determined by

the principle of virtual displacements (or its dynamic version when time dependency is to be included; see Section 1.4):

The equivalent single layer laminated plate theories are those in which a heterogeneous laminated plate is treated as a statically equivalent single layer having a complex constitutive behavior, reducing the 3-D continuum problem to a 2-D problem. The ESL theories are developed by assuming the form of the displacement field or stress field as a linear combination of unknown functions and the thickness coordinate [1 - 13]:

3.2 An Overview of Laminated Plate Theories


(3.2.7)

where <Px and -</Jy denote rotations about the y and x axes, respectively. The FSDT extends the kinematics of the CLPT by including a gross transverse shear deformation in its kinematic assumptions; i.e., the transverse shear strain is assumed to be constant with respect to the thickness coordinate. Inclusion of this rudimentary form of shear deformation allows the normality restriction of the classical laminate theory to be relaxed. The first-order shear deformation theory requires shear correction factors (see [28-32]), which are difficult to determine for arbitrarily laminateci composite plate structures. The shear correction factors depend not only on the larnination and geometrie parameters, but also on the loading and boundary conditions. In both CLPT and FSDT, the plano-stress state assumption is used and plane-

stress reduced forrn of the constitutive law of Section 2.4 is used. In both theories the inextensibility and/or straightness of transversc norrnals can be removed. Such extensions lead to second- and higher-order theories of plates. Second- and higher-order ESL laminateci plate theories use higher-order

polynom³als [i.e., N > 1 in Eq. (3.2.1)] in the expansion of the displacement cornponents through the thickness of the laminate ( see [33-38], arnong rnany others). The higher-order theories introduce additional unknowns that are often difficult to interpret in physical terms. The second-order theory with transverse inextensibility is based on the displacernent field

u(x, y, z, t) = uo(x, y, t) + z<Px(x, y, t) + z2'1/Jx(x, y, t) v(x, y, z, t) = vo(x, y, t) + z</>y(x, y, t) + z21/Jy(x, y, t) w(x, y, z, t) = wo(x, y, t)

(3.2.6)

where (uo, vo, w0) are the displacement components along the (z', y, z) coordinate directions, respectively, of a point on the midplane (i.e., z =O). The displacement field (3.2.5) implies that straight lines normal to the xy-plane before deformation remain straight and normal to the midsurface after deformation. The Kirchhoff assumption arnounts to neglecting both transverse shear and transverse normal effects; i.e., deformation is due entirely to bending and in-plane stretching. The next theory in the hierarchy of ESL laminated plate theories is the first-order

shear deformation theory (or FSDT) [21-27], which is based on the displacement f³eld

u(x, y, z, t) = uo(x, y, t) + z</J:J:(x, y, t) v(x, y, z, t) = vo(x, y, t) + z</Jy(x, y, t) w(x, y, z, t) = wo(x, y, t)

(3.2.5)

the present book, the stress-based theories will not be developed. Readers interested in stress-based theories may consult thc book by Pane [14]. The simplest ESL laminateci plate theory is the classica[ laminated plate theory

(or CLPT) [15-20], which is an extension of the Kirchhoff ( classical) plate theory to laminated composite plates. It is based on the displacemcnt field

owo 1t(x, y, z, t) = uu(x, y, t) - z~

ux awo

v(x, y, z, t) = vo(x, y, t) - z ay

w(x, y, z, t) = wo(x, y, t)

CLASSICAL AND FIRST-ORDER THEORIES 111

3.3 The Classica! Laminated Plate Theory 3.3.1 Assumptions

The classica! laminated plate theory is an extension of the classica! plate theory to composite laminates. In the classical laminated plate theory ( CLPT) it is assumed t that the Kirchhoff hypothesis holds:

t An assumption is that which is necessary for the development of the mathematical model, whereas a restriction is not a necessary condition for the development of the theory.

The displacement field accommodates quadratic variation of transverse shear strains ( and hence stresses) and vanishing of transverse shear stresses on the top and bottom of a general laminate composed of monoclinic layers. Thus there is no need to use shear correction factors in a third-order theory. The third-order theories provide a slight increase in accuracy relative to the FSDT solution, at the expense of an increase in computational effort. Further, finite element models of third-order theories that satisfy the vanishing of transverse shear stresses on the bounding planes require continuity of the transverse deflection and its derivatives between elements. Complete derivations of the govrning equations of the third-order laminated plate theory and their solutions are presented in Chapter 11. In addition to their inherent simplicity and low computational cost, the ESL

models often provide a sufficiently accurate description of global response for thin to moderately thick laminates, e.g., gross deflections, critical buckling loads, and fundamental vibration frequencies and associated mode shapes. Of the ESL theories, the FSDT with transverse extensibility appears to provide the best compromise of solution accuracy, economy, and simplicity. However, the ESL models have limitations that prevent them from being used to solve the whole spectrum of composite laminate problems. First, the accuracy of the global response predicted by the ESL models deteriorates as the laminate becomes thicker. Second, the ESL models are often incapable of accurately describing the state of stress and strain at the ply level near geometrie and materiai discontinuities or near regions of intense loading - the areas where accurate stresses are needed most. In such cases, 3-D theories or multiple model approaches are required (see Chapter 12 for the layerwise theory and multiple model approaches). This completes an overview of various ESL theories. For additional discussion

and references, one may consult the review articles [40-43]. In the remaining sections of this chapter, we study the classica! and first-order shear deformation plate theories for laminated plates [44-52].

(3.2.8)

u(x, y, z, t) = ua(x, y, t) + z</>x(x, y, t) + z3 ( - 3!2) ( <l>x + 08:0)

v(x, y, z, t) = vo(x, y, t) + z</>y(x, y, t) + z3 ( - 3!2) ( </>y + 0;0) w(x, y, z, t) = wa(x, y, t)

The third-order laminated plate theory of Reddy [38,39] with transverse inextensibility is based on the displacement field


Figure 3.3.1: Coordinate system and layer numbering used fora laminateci plate.

h~ =zk+I -Zk z

Zk _ _ Zk+I

J. ZL - - - - " - .. ZL+I

= = = = = = =-l h. 2

h 2

y

Consider a plate of total thickness h composed of N orthotropic layers with the principal material coordinates ( xt, x~, x~) of the kth lamina oriented at an angle (h to the laminate coordinate, x. Although not necessary, it is convenient to take the xy-plane of the problem in the undeformed midplane Oo of the laminate (see Figure 3.3.1). The z-axis is taken positive downward from the midplane. The kth layer is locateci between the points z = Zk and z = Zk+l in the thickness direction.

3.3.2 Displacements and Strains

The first two assumptions imply that the transverse displacement is independent of the transverse (or thickness) coordinate and the trans verse nor mal strain E zz is zero. The third assumption results in zero transverse shear strains, Exz = O, Eyz = O.

(2) The transverse normals do not experience elongation (i.e., they are inextensible).

(3) The transverse normals rotate such that they remain perpendicular to the midsurface after deformation.

(1) Straight lines perpendicular to the midsurface (i.e., transverse normals) before deformation remain straight after deformation.

CLASSICAL ANO FIRST-ORDER THEORIES 113

where ( uo, vo, wo) are the displacements aiong the coordinate lines of a materiai point on the xy-piane. Note that the form of the d³splacernent f³eld (3.3.1) allows reduction of the 3-D problem to one of studying the deformation of the reference piane z =O (or midplane). Once the midplane displacements (uo, vo, wo) are known, the displacements of any arbitrary point (x, y, z) in the 3-D continuum can be determined using Eq. (3.3.2).

(3.3.2)

8wo u(x, y, z, t) = uo(x, y, t) - z ox

8wo v(x, y, z, t) = vo(x, y, t) - z oy

w(x, y, z, t) = wo(x, y, t)

where (x, y, z) are unit vectors along the (x, y, z) coordinates. Due to small strain and small displacement assumption, no distinction is made between the materia} coordinates and spatial coordinates, between the finite Green strain tensor and infinitesimal strain tensor, and between the second Piola-K³rchhoff stress tensor and the Cauchy stress tensor (see Chapter 1). The Kirchhoff hypothesis requires the displacements ( u, v, w) to be such that (see Figure 3.3.2)

(3.3.1)

The tota} domain Oo of the laminate is the tensor product of no X (-h/2, h/2). The boundary of Oo consists of top surface St(z = -h/2) and bottom surfaces Sb(Z = h/2), and the edge r = r X (-h/2, h/2) of the laminate. In general, T is a curved surface, with outward norma} ii = nxx + nyy. Different parts of the boundary r are subjected to, in generai, a combination of generalized forces and generalized displacements. A discussion of the boundary conditions is presented in the sequel. In formuiating the theory, we make certain assumptions or place restrictions, as

stated bere:

Å The layers are perfectly bonded together (assumption).

Å The materia} of each layer is linearly elastic and has three planes of material symmetry (i.e., orthotropic) (restriction).

Å Each layer is of uniform thickness ( restriction).

Å The strains and displacements are small ( restriction).

Å Tbe transverse shear stresses on the top and bottom surfaces of the laminate are zero ( restriction).

By the Kirchhoff assumptions, a materiai point occupying the position (x, y, z) in the undeformed laminate moves to the position ( x + u; y + v, z + w) in the deformed laminate, where (u, v, w) are the components of the total displacement vector u along the (x, y, z) coordinates. We bave


(3.3.3)

The strains associated with the displacement field (3.3.2) can be computed using either the nonlinear strain-displacement relations (1.3.10) or the linear strain- displacement relations (1.3.12). The nonlinear strains are given by

Figure 3.3.2: Undeformed and deformed geometries of an edge of a plate under the Kirchhoff assumptions.

z

Wo

J_

I I

T '_ awo I X ' ax I

'

X

y


(3.3.8)

where, for this special case of geometrie nonlinearity (i.e., small strains but moderate rotations), the notation cij is used in piace of Eij. The corresponding second Piola- Kirchhoff stresses will be denoted aij.

For the assumed displacement field in Eq. (3.3.2), ow/oz = O. In view of the assumptions in Eqs. (3.3.4)-(3.3.6), the strains in Eq. (3.3.7) reduce to

- ouo 1 ( owo) 2

82wo cxx - ax + 2 ax - z ax2 cx = ~ (fJuo + fJvo + fJwo fJwo) _ Z o2wo y 2 ay 8x ax 8y 8x8y

_ 8vo 1 ( 8wo) 2

82w0 cyy - f)y + 2 ay - z fJy2

1 ( 8wo awo) cxz = 2 - ax + ax =o

1 ( 8wo owo) cyz = 2 - f)y + f)y = O Czz =O

(3.3.7) 1 (f)v 8w) Bu: cyz = 2 f}z + f}y , czz = f}z

1 (fJu aw) av 1 (8w)2 cxz = 2 f)z + ox ' cyy = 8y + 2 oy

au 1 ( 8w ) 2 1 ( Bu av Bu: Bu: ) cxx = 8x + 2 8x ' cxy = 2 oy + 8x + ox 8y

and they should be included in the strain-displacement relations. Thus for small strains and moderate rotations cases the strain-displacement relations (3.3.3) take the form

(3.3.6) ( aw)

2 ( aw)

2 aw aw

Bè 8y ' 8x 8y

If the rotations aw0/8x and 8w0/ay of transverse normals are moderate (say 10Á- 150), then the following terms are small but not negligible compared to E:

(~~r, (~~r, (~~r, (~~) (~~), (~~) (~~), (~~) (~~) (~~r, (~~r. (~~r. (~~) (~~), (~~) (~~), (~~) (~~)

( ~:) ( ~:), ( ~;) ( ~:), ( ~:r (3.3.5)

then the small strain assumption implies that terms of the order E2 are negligible in the strains. Terms of order E2 are

(3.3.4)

If the components of the displacement gradients are of the order E, i.e.,

f)u 8u ov av aw = o ( ) fJx ' fJy ' fJx ' fJy ' OZ E


Figure 3.3.3: Variations of strains and stresses through layer and laminate thicknesses. (a) Variation of a typical in-plane strain. (b) Variation of corresponding stress.

(b) (a)

Exx X

z z z

In the classica! laminated plate theory, all three transverse strain components (Ezz, Exz, Eyz) are zero by definition. Fora laminate composed of orthotropic layers, with their x1x2-plane oriented arbitrarily with respect to the xy-plane (x3 = z),

3.3.3 Lamina Constitutive Relations

{ (O)} { ~ + ! (~)2 } { (1)} {-~ } Exx fJx 2 ox Exx 8x

O _ (O) _ 2 1 _ (1) _ o2w {e } - Eyy - ~ + ! (~) , {e } - Eyy - -w (o) ay 2 ay (l) 82

'"Yxy ~ + ~ + ~~ '"Yxy -2axao ~fu fu~ y (3.3.10)

(O) (O) (O) . (1) (1) (1)) where (Exx, Eyy, "(xy) are the membrane strains, and (E.xx, Eyy, "(xy are the fiexural (bending) strains, known as the curvatures. Once the displacements (uo, vo, wo) of the midplane are known, strains at any

point (x, y, z) in the plate can be computed using Eqs. (3.3.9) and (3.3.10). Note from Eq. (3.3.9) that all strain components vary linearly through the laminate thickness, and they are independent of the material variations through the laminate thickness (see Figure 3.3.3a). For a fixed value of z, the strains are, in general, nonlinear functions of x and y, and they depend on time t far dynamic problems.

(3.3.9) {

(O) } { (1) } Exx E.xx E.xx

{ Eyy } = E.~w + z E.~v '"Yxy 'Á"( ) 'Á"( )

Å 1xy 1xy

The strains in Eqs. (3.3.8) are called the van Kdrmdm strains, and the associated plate theory is termed the van Kdrrruiti plate theary. Note that the transverse strains (Exz, Eyz, Ezz) are identically zero in the classica! plate theory. The first three strains in Eq, (3.3.8) have the form


rxxr [Q" Q12 Q11;r wxx} rxx} ) O"yy Q12 Q22 Q26 Eyy - O'.yy 6.T O"xy Q15 Q25 Q55 rxy 2axy

rn o e31 l (k) { Ex } (k] o e32 Ey (3.3.12a) o e35 e,

(3.3.llb) E2 Q22 = , Q55 = G12

I - V12V21

Since the laminate is made of several orthotropic layers, with their material axes oriented arbitrarily with respect to the laminate coordinates, the constitutive equations of each layer must be transformed to the laminate coordinates ( x, y, z), as explained in Section 2.3. The stress-strain relations (3.3.lla) when transformed to the laminate coordinates ( x, y, z) relate the stresses (a xx, a yy, a xy) to the strains (Exx, Eyy, rxy) and components of the electric field vector (Ex, Ey, E2) in the laminate coordinates [see Eq. (2.4.5)]

where Q~;) are the plane stress-reduced stiffnesses and e~;) are the piezoelectric moduli of the kth lamina [cf., Eq. (2.4.4a,b)], (ai, Ei, Ei) are the stress, strain, and electric field components, respectively, referred to the material coordinate system (x1, x2, x3), 0:1 and 0:2 are the coefficients of thermal expansion along the x1 and x2 directions, respectively, and !:::J..T is the temperature increment from a reference state, 6.T = T-TrefĿ When piezoelectric effects are not present, the part containing the piezoelectric moduli e~;) should be omitted. The coefficients Q~;) are known in terms of the engineering constants of the kth layer:

(3.3.lla)

the transverse shear stresses ( O"xz, O"yz) are also zero. Since Ezz = O, the transverse normal stress a z z i although not zero identically, does not appear in the virtual work statement and hence in the equations of motion. Consequently, it amounts to neglecting the transverse normal stress. Thus we have, in theory, a case of both plane strain and plane stress. However, from practical considerations, a thin or moderately thick plate is in a state of plane stress because of thickness being small compared to the in-plane dimensions. Hence, the plane-stress reduced constitutive relations of Section 2.4 may be used. The linear constitutive relations for the kth orthotropic (piezoelectric) lamina in

the principal material coordinates of a lamina are


Dcxz =O, Dcyz =O, Dczz =O

As noted earlier, the transverse strains bxz, '"'fyz, c:22) are identically zero in the classical plate theory. Consequently, the transverse shear stresses (O"xz, O"yz) are zero fora laminate made of orthotropic layers if they are computed from the constitutive relations. The transverse norma} stress O" zz is not zero by the consti tu ti ve relation because of the Poisson effect. However, all three stress components do not enter the formulation because the virtual strain energy of these stresses is zero due to the fact that kinematically consistent virtual strains must be zero [see Eq. (3.3.8)]:

3.3.4 Equations of Motion

(3.3.14) {

(O) } { (1) } cxx - O'.xxTo C::J::r - O'.x:ETl O _ (O) 1 _ (1) {e: } - C:yy - ayyTo , {e: } - C:yy - ayyT1

(O) (1) '"'(xy - 20'.xyTo '"'fxy - 20'.xyTl

and the total strains are of the form in Eq. (3.3.9) with

(3.3.13) ~T = To(x, y, t) + zT1 (x, y, t)

Here () is the angle measured counterclockwise from the x-coordinate to the x1 - coordinate. Note that stresses are also linear through the thickness of each layer; however, they will have different linear variation in different material layers when Qi;) change from layer to layer (see Fig. 3.3.3b ). If we assume that the temperature increment varies linearly, consistent with the mechanical strains, we can write

(3.3.12d)

and eij are the transformed piezoelectric moduli

e31 = e31 cos2 () + e32 sin2 () e32 = e31 sin2 () + e32 cos2 () e35 = ( e31 - e32) sin e cos ()

(3.3.12c)

O'.xx = 0'.1 cos2 () + a2 sin2 () O'.yy = 0'.1 sin2 e + a2 cos2 e 20'.xy = 2( 0'.1 - 0:2) sin ®' cos ()

and O'.xx, O'.yy, and O'.xy are the transformed thermal coefficients of expansion [see Eq. (2.3.23)]

Qu = Qu cos4 e+ 2( Q12 + 2Q66) sin2 e cos2 e+ Q22 sin4 e Q12 = ( Qu + Q22 - 4Q55) sin2 e cos2 e+ Q12 (sin4 () + cos4 ()) Q22 = Qu sin" e+ 2( Q12 + 2Q66) sin2 e cos2 e + Q22 cos4 () Q16 = ( Qu - Q12 - 2Q66) sin e cos3 e+ ( Q12 - Q22 + 2Q55) sin3 e cose Q26 = ( Qu - Q12 - 2Q66) sin3 e cose+ ( Q12 - Q22 + 2Q66) sin() cos3 () Q66 = ( Qu + Q22 - 2Q12 - 2Q66) sin2 e cos2 () + Q66(sin4 () + cos4 e) (3.3.12b)

where


where % is the distributed farce at the bottom (z = h/2) of the laminate, qt is the distributed farce at the top (z = -h/2) of the laminate, ("nn, ď"ns, ď"nz) are the

(3.3.18)

(3.3.17)

(3.3.16)

h

8U = { 12h (axxDcxx + ayyOcyy + 2axyOcxy) dzdxdy lno -2 = 1 {1~ [a (&(O)+ z&(l)) +a (&(O) + z&(l)) h xx xx xx yy yy yy Do -2

+ (J xy ( D'YlV + ZO'Yw)] dz} dxdy

8V = - { [qb(x, y)ow(x, u, 73:_) + qt(x, y)ow(x, y, -~)] dxdy Jn0 2 2 h - 1r J_2!!. [annOUn + an80U8 + 0-nzOW] dzds

u 2

= - r {[qb(X, y) + qt(X, y)] Owo(x, y)} dxdy lno { 1~ [A ( at5wo) A ( aow0) - Jr" -~ ann buon - z-a;;: + Uns 8uo8 - z----a;-

+ O-nz8wo] dzds

r 1~ [(. awo) ( . aowo) 8K = Jrio -~Po uo - z ax 8uo - =s: + (ilo - z 8~0) ( 8vo - z a~~o) + wo8wo l dz dxdy

where the virtual strain energy 8U (volume integral of 8Uo), virtual work done by applied farces 8V, and the virtual kinetic energy 8K are given by

(3.3.15) O= lor (8U + 8V - 8K) dt

Whether the transverse stresses are accounted for or not in a theory, they are present in reality to keep the plate in equilibrium. In addition, these stress components may be specified on the boundary. Thus, the transverse stresses do not enter the virtual strain energy expression, but they must be accounted for in the boundary conditions and equilibrium of forces. Here, the governing equations are derived using the principle of virtual

displacements. In the derivations, we account far thermal (and hence, moisture) and piezoelectric effects only with the understanding that the materia} properties are independent of temperature and electric fields, and that the temperature T and electric field vector E are known functions of position (hence, 8T = O and 8[ = O). Thus temperature and electric fields enter the farmulation only through constitutive equations [see Eq. (3.3.12a)]. The dynamic version of the principle of virtual work [see Eq. (1.4.78)] is


Figure 3.3.4: Geometry of a laminateci plate with curved boundary.

------x

y z

specified stress components on the portion r a of the boundary r, (8uon, 8uos) are the virtual displacements along the normal and tangential directions, respectively, on the boundary r (see Figure 3.3.4), Po is the density of the plate material, and a superposed dot on a variable indicates its time derivative, ilo = 8uo/8t. Details of how (uon,Uos) and ((]"nn,(]"ns) are related to (uo,vo) and ((]"xx,(]"yy,(]"xy), respectively, will be presented shortly. The virtual displacements are zero on the portion of the boundary where the

corresponding actual displacements are specified. For time-dependent problems, the admissible virtual displacements must also vanish at time t = O and t = T [see Eq. ( 1.4. 73b)]. Sin ce we are interested in the governing differential equations and the form of the boundary conditions of the theory, we can assume that the stresses are specified on either a part or whole of the boundary. If a stress component is specified only on a part of the boundary, on the remaining part of the boundary the corresponding displacement must be known and hence the virtual displacement must be zero there, contributing nothing to the virtual work done. Substituting for t5U, t5V, and t5K from Eqs. (3.3.16)~(3.3.18) into the virtual work

statement in Eq. (3.3.15) and integrating through the thickness of the laminate, we obtain


Figure 3.3.5: Force and moment resultants on a plate element.

X ..

The quantities (Nxx, Nyy, Nxy) are called the in-plane farce resultants, and (Mxx, Myy, Mxy) are called the moment resultants (see Figure 3.3.5); Qn denotes the transverse force resultant, and (Io, Ii, h) are the mass moments of inertia. All stress resultants are measured per unit length (e.g., N, and Qi in lb/in. and Mi in lb-in/in.).

(3.3.20c)

(3.3.20b)

(3.3.20a) { Nxx} !!. { O"xx} { Afxx} !!!_ { O"xx} Nyy = [21:;_ "vu dz, Myy = _21:;_ (TYY z dz Nxy 2 O"xy Mxy 2 O"xy

{ if_nn} = J~ { ~nn} dz, { i[nn} = J~ { ~nn} z dz Nns _!!_ O"ns Mns _!!_ O"ns 2 2

{Io} i,_ { 1 } ~~ = [2~ :2 Po dz,

where q = % + qt is the total transverse load and


where a comma followed by subscripts denotes differentiation with respect to the subscripts: Nxx,x = Nxx/x, and so on. Note that both spatial and time integration-by-parts were used in arriving at the last expression. The terrns obtained

Substituting far the virtual strains from Eq. (3.3.21) into Eq. (3.3.19) and integrating by parts to relieve the virtual displacements (c5uo, Ovo, Owo) in Oo of any differentiation, so that we can use the fundamental lemma of variational calculus, we obtain

(3.3.21)

The virtual strains are known in terms of the virtual displacements in the same way as the true strains in terms of the true displacernents [see Eq. (3.3.10)]:


Dw0:

Duo:

(3.3.25)

Dv0:

The Euler-Lagrange equations of the theory are obtained by setting the coefficients of Duo, Dvo, and 8wo over Oo of Eq, (3.3.23) to zero separately:

(3.3.24b)

(3.3.24a)

where

O = laT { fo0 [ - ( Nxx,x + Nxy,y - Iouo +Ii 88~0) Duo

( 8 .. ) - Nxy,x + Nyy,y - Iovo + li 0:0 ovo

- ( Mxx,xx + 2Mxy,xy + Myy,yy +N(wo) + q

. 8uo avo 82wo 82wo) l - Iowa - Ii- -Ii-+ h-- + h-- 8wo dxdy Bè ay 8x2 8y2

+ lr(J" [ (Nxxnx + Nxyny) 8uo + (Nxynx + Nyyny) Dvo

+ ( Mxx,xnx + Mxy,ynx + lvfyy,yny + Mxy,xny + P( wo)

I .. 1 .. 1 8wo 1 8wo ) >: - 1Uonx - 1Vony + 2 OX nx + 2 Oy ny uWo 86wo 8Dwo] - (Mxxnx + Mxyny) 8-;;- - (Mxynx + Myyny) ---a:;} ds

- lr(J" ( NnnDUon + NnsDUos - u.; O~~O - Mns O~~~O + CJnDWo) ds }dt

(3.3.23)

in 00 but evaluated at t = O and t = T were set to zero because the virtual displacements are zero there. Collecting the coefficients of each of the virtual displacements (Duo, Dvo, Dwo)

together and noting that the virtual displacements are zero on r u, we obtain


(3.3.29b)

(3.3.29a)

(3.3.28b) { } [ 2 n2 2nxny l {;xx} ann nx y

ans - -nxny nxny n2 _ n2 YY x Y axy

Hence we have

{ Nnn} [ n; 2 2nxny ] rxx} ny Nyy Nns - -nxny nxny n2 -n2 X y Nxy

{ Mnn} [ n; n2 2nxny l { :xx } y Mns -nxny nxny n2 - n2 YY x y Mxy

We recognize that the coefficients of buon and buos in the right-hand side of the above equation are equal to Nnn and Nn8, respectively. This follows from the fact that the stresses (ann, ans) are relateci to (axx, ayy, axy) by the transformation in Eq. (2.3.9):

(Nxxnx + Nxyny) Duo+ (Nxynx + Nyyny) bvo = (Nxxnx + Nxyny) (nxDUn - nybus) + (Nxynx + Nyyny) (nyOUn + nxOUs) = ( Nxxn; + 2Nxynxny + Nyyn;) OUn + [(Nyy - Nxx) nxny + Nxy ( n; - n;) J OUs

(3.3.28a)

Now we can rewrite the boundary expressions in terms of ( uon, uos) and (wo,n, wo,8). We bave

(3.3.27b)

Similarly, the normal and tangential derivatives (wo,n, wo,s) are relateci to the derivatives ( wo,x, wo,y) by

(3.3.27a)

Therefore, the displacernents ( uon, Uos) are relateci to ( uo, vo) by

(3.3.26)

ex = cos e en - sin e es ey = sin e en + cose s , = ;

The terms involving ]z are called rotary inertia terms, and are often neglected in most books. The term can contribute to higher-order vibration or frequency modes. Next we obtain the boundary conditions of the theory from Eq, (3.3.23). In order

to collect the coefficients of the virtual displacements and their derivatives on the boundary, we should express (8uo,bvo) in terms of (buo,,,buo8). Ifthe unit outward normal vector f³ is oriented at an angle e from the x-axis, then its direction cosines are nx = cose and ny = sin e. Hence, the transformation between the coordinate system (n, s, r) and (x, y, z) is given by


(3.3.33a) j 88wo J 8Mns ] - Jr Mns-a;- ds =Jr ~8wo ds - [Mn8DWo r

The generalized displacements are specified on I' u, which constitutes the essential (or geometrie) boundary conditions. We note that the equations in Eq. (3.3.25) bave the total spatial differential arder

of eight. In other words, if the equations are expressed in terms of the displacements ( uo, vo, wo), they would contain second-order spatial derivatives of uo and vo and fourth-order spatial derivatives of wo. Hence, the classica! laminateci plate theory is said to be an eighth-order theory. This implies that there should be only eight boundary conditions, whereas Eq. (3.3.32) shows five essential and five natural boundary conditions, giving a total of ten boundary conditions. To eliminate this discrepancy, one integrates the tangential derivative term by parts to obtain the boundary term

(3.3.32)

8wo os

8wo on' primary variables:

secondary variables:

Thus the primary variables (i.e., generalized displacements) and secondary variables (i.e., generalized forces) of the theory are

(3.3.31b)

o; = ( Mxx,x + Mxy,y - li uo + t, 08~0) nx+

( Myy,y + Mxy,x - Iivo + h 08~0) ny + P(wo)

on fa, where

(3.3.31a) ' ' Mnn - Mnn = O , 1'1ns - Mns = O

The natural boundary conditions are then given by

(3.3.30)

O= laT lr,, [ (Nnn - Nnn) 8uon + (Nns - Nns) 8uos + ( Mxx,xnx + Mxy,ynx + Myy,yny + M:ry,xny + P( wo)

I .. I .. I 8wo I OWo Q' ) >: - 1 uonx - 1 vony + 2 ax nx + 2 oy ny - n uwo

( , ) 88wo ( , ) 88wo l - Mnn - Mnn ---a:;;: - Mns - Mns -a;- dsdt

In view of the above relations, the boundary integrals in Eq. (3.3.23) can be written as


{ Nxx} N r { O"xx} Nyy = L O"yy dz Nxy k=l Zk O"xy

3.3.5 Laminate Constitutive Equations

Here we derive the constitutive equations that relate the force and moment resultants in Eq. (3.3.20a) to the strains of a laminate. To this end, we assume that each layer is orthotropic with respect to its material symmetry lines and obeys Hooke's law; i.e., Eq, (3.3.12a) holds for the kth lamina in the problem coordinates. For the moment we consider the case in which the temperature and piezoelectric effects are not included. Although the strains are continuous through the thickness, stresses are not, due to the change in material coefficients through the thickness (i.e., each lamina). Hence, the integration of stresses through the laminate thickness requires lamina-wise integration. The force resultants are given by

where variables with superscript 'O' denotes values at time t =O. We note that both the displacement and velocities must be specified. This completes the basic development of the classica! laminated plate theory for

nonlinear and dynamic analyses. As a special case, one can obtain the equations of equilibrium from (3.3.25) by setting all terms involving time derivatives to zero. For linear analysis, we set N(wo) and P(wo) to zero, in addition to setting the nonlinear terms in the strain-displacement equations to zero. Equations (3.3.25) are applicable to linear and nonlinear elastic bodies, since the constitutive equations were not utilized in deriving the governing equations of motion.

(3.3.35) Ŀ ĿO Ŀ ĿO Ŀ ĿO Un= Un, Us = lls, WQ = Wo

o - o o Un= un, U8 - U8, Wo = w0

The initial conditions of the theory involve specifying the values of the displacements and their first derivatives with respect to time at t = O:

(3.3.34)

8wo un, U8, wo, -8 (essential)

ti

Nnn1 Nns, V11, u.; (natural)

which should be balanced by the applied force o; This boundary condition, Vn = o: is known as the Kirchhoff free-edge condition. The boundary conditions of the classica! laminated plate theory are

(3.3.33b) V, = Q + 8Mn.s n n as

The term in the square bracket is zero since the end points of a closed curve coincide. This terrn now must be added to Qn (because it is a coefficient of c5wo):


(3.3.41b)

(3.3.41a)

(3.3.40) { {N} }- [[A] {M} - [B]

where { E0} and { E1} are vectors of the membrane and bending strains defined in Eq. (3.3.10), and [A], [B], and [D] are the 3 x 3 symmetric matrices of laminate coefficients defined in Eqs. (3.3.38a,b). Values of the laminate stiffnesses for various stacking sequences will be presented in Section 3.5. For the nonisothermal case, the strains are given by Eq. (3.3.14) and the laminate

constitutive equations (39) become

(3.3.38b)

~ -(k) 1 ~ -(k) 2 2 Aij = ~ Qij (zk+1 - zk) , Bij = 2 ~ Qij (zk+I - zk)

k=l k=l

1 ~ -(k) 3 3 o., = 3 e: Qij (zk+l - zk) k=l

Note that Q's, and therefore A's, B's, and D's, are, in general, functions of position (x, y). Equations (3.3.36) and (3.3.37) can be written in a compact form as

{ {N}} _ [(A] [B]] { {®o}} ( ) { M} - [B] [D] { c1} 3.3.39

(3.3.38a)

or

where Aij are called extensional stiffnesses, Dij the bending stiffnesses, and Bij the bending-extensional coupling stiffnesses, which are defined in terms of the lamina stiffnesses Q~;) as

{Mxx} [Bn Myy = B12 Mxy Bw

{Mxx} N t=: { axx} Myy = L l, Oyy z dz Mxy k=l Zk Oxy

Q- l (k) { "(O) + Z"(l) } 16 ~xx ~xx (126 c1V + zc1.V z dz Q66 1W +z1W

{

(O) } { (1) } B15 Exx o., D12 D15 Exx B25] c1~ + [D12 D22 D25] cW (3.3.37) B66 'V(o) Dw D25 D55 'V(l)

tXY tXY

{

(1) } Exx cW (3.3.36) (1) "(xy


The equations of motion (3.3.25) can be expressed in terms of displacements ( uo, vo, wo) by substituting for the farce and moment resultants from Eqs. (3.3.43) and (3.3.44). In general, the laminate stiffnesses can be functions of position (x, y) (i.e., nonhomogeneous plates). For homogeneous laminates (i.e., for laminates with constant A's, B's, and D's), the equations of motion (3.3.25) take the form

(3.3.44)

(3.3.43)

{ ~ + 1(~)2 } ax 2 8x ~ + 1(~)2 ay 2 &y ~+~+~~ oy &x &x oy

{ ~} 8x fJ2wf

2ito &x&y

3.3.6 Equations of Motion in Terms of Displacements

The stress resultants (N's and M's) are related to the displacement gradients, temperature increment, and electric field. In the absence of the temperature and electric effects, the farce and moment resultants can be expressed in terms of the displacements ( uo, vo, wo) by the relations

Relations similar to Eqs. (3.3.41a, b) can be written for hygroscopic effects.

(3.3.42b)

(3.3.42a)

and {NP} and {MP} are the piezoelectric resultants


(3.3.46)

(3.3.45)


Suppose that a six-layer (Ñ60/0)s symmetric laminate is subjected to loads such t.hat t.he only nonzero strains at a point (x, y) are E,\,0J = Eo in./in. and E~,1) = i'i:o/in. Assume that layers are of thickness 0.005 in. with materiai properties E1 = 7.8 psi, E2 = 2.6 psi, G12 = G13 = 1.:3 psi, G23 = 0.5 psi, and v12 = 0.25. We wish to dcterrnine the state of stress (a,,",,cr1n1,cr,,,y) and force resultants in the laminate. The only nonzero strain is Exx = t:o + zKo- Hcnce, the stresses in kth lamina are givcn by

Example 3.3.2:

( :U. 48c)

(3.:3.48b) A (D2u0 Dw0 821110) A EPvo B 83w0 DN[y _I 82vo

16 -- + ---- + 55-- - 15-- - -- - o-- Dx2 x EJ:r2 8:r2 Dx3 Dx Dt2

(3.3.48a) (82 8 82 ) 82 83 aNT "2 8Ŀi A _..'.:Q wo ~ A ~ - B ~ - ~ - I u uo - I ~ 11 Dx2 + Dx 8x2 + 16 8x2 11 ax:l EJx - o 8t2 I EJxDt2

Ifa plate is infinitely long in one direction, the plate becorncs a plate strip. Consider a platc strip that has a finite dirnension along the x-axis and subjected to a trans verse load q(x) that is uniform at any section parallel to the x-axis. In such a case, the defiection wo and displacements (uo, 110) of the plate are functions of only x. Therefore, ali derivatives with respect to y are zero. In such cases, thc def³ected surface of the plate strip is cylindrical, and it is referred to as the cuiindrical bending. For this case, the goveming equations (3.3.45)---(3.3.47) reduce to

Example 3.3.1: ( Cylindrical Bending)

where N(w0) was defined in Eq, (3.3.24a). The nonlinear partial differential equations (3.3.45)-(3.3.47) can be simplified

for linear analyses, static analyses, and lamination schemes for which some of the stiffnesses ( Aij, Bij, Dij) are zero. These cases will be considered in the sequel. Once the displacements are determined by solving Eqs. (3.3.45) (3.3.47), analytically or numerically for a given problem, the strains and stresses in each lamina can be computed using Eqs. (3.3.10) and (3.3.12), respectively.

(3.3.47)


which indicate that rf>x and </>y are the rotations of a transverse normal about the y- and x-axes, respectively (see Figure 3.4.1). The notation that rf>x denotes the rotation of a transverse normal about the y-axis and </>y denotes the rotation about the x-axis may be confusing to some, and they do not follow the right-hand rule. However, the notation has been used extensively in the literature, and we will not

(3.4.2a)

where (ua, va, wa, rf>x, </>y) are unknown functions to be determined. As before, (ua, va, wa) denote the displacements of a point on the plane z =O. Note that

(3.4.1)

u(x, y, z, t) = ua(x, y, t) + zrf>x(x, y, t) v(x, y, z, t) = va(x, y, t) + z</>y(x, y, t) w(x, y, z, t) = wa(x, y, t)

3.4.1 Displacements and Strains

In the first-order shear deformation laminated plate theory (FSDT), the Kirchhoff hypothesis is relaxed by removing the third part; i.e., the transverse normals do not remain perpendicular to the midsurface after deformation (see Figure 3.4.1). This amounts to including transverse shear strains in the theory. The inextensibility of transverse normals requires that w not be a function of the thickness coordinate, z.

Under the same assumptions and restrictions as in the classical laminate theory, the displacement field of the first-order theory is of the form

3.4 The First-Order Laminated Plate Theory

{ z:: } = { ~} lb/in., {E:: } = { ~:!~~~} lb-in/in.

If Eo =O in./in. and Ko = 1.0 /in., we have

{ E:: } = { ~ } lb-in /In. { Nxx} { 144} z:: = 3~.3 lb/in.,

If Eo = 1000 x 10-6 in./in. and Ko =O, we have

{ Mxx } { 7.6306 } Myy = 3.1566 Ko lb-in/in. Mxy 0.7066 {

Nxx} { 0.1440} Z:: = o.og53 Eo X 106 lb/in.,

The stress resultants are given by

msi [

7.966 0.664 o l [Q]oo = 0.6064 2.655 O

o 1.3 msi,

[

3.215 1.431 Ñ0.707] [QlÑ600 = 1.431 5.871 Ñ1.593

Ñ0.707 Ñ1.593 2.068

where


_ Buo ~ ( Bwo ) 2

B</>x Exx - OX + 2 OX + Z OX '"Yx = (ouo + ovo + 8wo owo) + z (8</>x + 8</>y) y 8y ax ax oy 8y ¨è

The nonlinear strains associated with the displacement field (3.4.1) are obtained by using Eq. (3.4.1) in Eq. (3.3.7):

The quantities (uo, vo, wo, </>x, </>y) will be called the generalized displacements. For thin plates, i.e., when the plate in-plane characteristic dimension to thickness ratio is on the order 50 or greater, the rotation functions <Px and </>y should approach the respective slopes of the transverse deflection:

(3.4.2b) f3x = -</>y , /3y = </>x

depart from it. If (/3x, /3y) denote the rotations about the x and y axes, respectively, that follow the right-hand rule, then

Figure 3.4.1: Undeformed and deformed geometries of an edge of a plate under the assumptions of the first-order plate theory.

I -'- - - - -

Wo

ĿĿ.

z

X

y


where all variables were previously introduced [see Eqs. (3.3.16)-(3.3.18) and the paragraph following the equations].

(3.4.8) + wo8wo l dz dxdy

(3.4.6)

8U = 1 {!~ [(]' (&(O)+ z&(l)) + o (oE(O) + ZOE(l)) I XX xx xx yy yy yy flo -~

(s: (O) s: (1)) e (O) s: (O)] d7}d d + {J'xy U"fxy + ZU"fxy + {J'xzU'³'xz + {J'yzU"fyz ,~ X y

h

8V = - { [(qb + qt) 8wo] dxdy - { f 2h [0-nn (8un + z8</Jn) 100 Jr" -2 + 0-ns (8u8 + z8</Js) + 0-nzDWo] dzds (3.4.7)

8K= fo0J_~~Po[(½o+z~x) (8½o+z8~x) + (vo+z~v) (ov0+z8~y)

where the virtual strain energy 8U, virtual work done by applied forces 8V, and the virtual kinetic energy 8K are given by

(3.4.5) O= lor (8U + 8V - 8K) dt

The governing equations of the first-order theory will be derived using the dynamic version of the principle of virtual displacements:

3.4.2 Equations of Motion

(O) (1) ~ + 1.(~)2 O</Jx

rxx) Exx Exx ax 2 ax 8x (O) (1) ~ + 1.(~)2 8</;y Eyy Eyy Eyy 8y 2 8y &y (O) (1) 8wo +</J "/yz = "/yz +z "/yz 8y y +z o

"/xz (O) (1) 8wo + </J o "/xz "/xz OX X 8</Jx + 8</;y "/xy (O) (1) 8uo + avo + 8wo 8wo "/xy ')'xy 8y ax ax 8y 8y ax (3.4.4)

Note that the strains (Exx, Eyy, "/xy) are linear through the laminate thickness, while the transverse shear strains ( "!xz, '³'yz) are constant through the thickness of the laminate in the first-order laminated plate theory. Of course, the constant state of transverse shear strains through the laminate thickness is a gross approximation of the true stress field, which is at least quadratic through the thickness. The strains in Eq. (3.4.3) have the form

(3.4.3)

- OVo ~ ( OWo) 2

o</Jy Eyy - ay + 2 ay + z ay

owo owo "!xz = OX + <Px, "/yz = oy + </Jy, Ezz =o


This amounts to modifying the plate transverse shear stiffnesses. The factor K is computed such that the strain energy due to transverse shear stresses in Eq. (3.4. lOb) equals the strain energy due to the true transverse stresses predicted by the three-dimensional elasticity theory.

For example, consider a homogeneous beam with rectangular cross section, with width b and height h. The actual shear stress distribution through the thickness of the beam, from a course on mechanics of materials, is given by

(3.4.lOb) {Qx}=KJ~ {CJxz}dz Qy _fl CJyz 2

Since the transverse shear strains are represented as constant through the laminate thickness, it follows that the transverse shear stresses will also be constant. It is well known from elementary theory of homogeneous beams that the transverse shear stress varies parabolically through the beam thickness. In composite laminated beams and plates, the transverse shear stresses vary at least quadratically through layer thickness. This discrepancy between the actual stress state and the constant stress state predicted by the first-order theory is often corrected in computing the transverse shear force resultants (Qx, Qy) by multiplying the integrals in Eq. (3.4.lOa) with a parameter K, called shear correction coefficient:

Shear Correction Factors

The quantities ( Qx, Qy) are called the tromsoerse farce resulianis.

(3.4.lOa) { Qx} = J~ { CJxz} dz Qy _fl CJyz 2

where q = qb + qt, the stress resultants (Nxx, Nyy, Nxy, Mxx, Myy, Mxy) and the inertias (lo, Ii, h) are as defined in Eq. (3.3.20), (Nnn, Nns, Mnn, Mns) are as defined in Eq. (3.3.29a,b), and

O = foT { ku [NxxDE~~ + lvfrx&1~ + Nyy&~V + Myy&W + Nxy8riV + Mxy8ri~ + Qxhi~) + Qy8r~~) - q8wo - lo ( ½o8½o + iJo8iJo + wo8wo) - Ii ( ~x8½o + ~yďVo + 8~x½o + 8~yVO) - h ( ~xď~x + ~y8~y) J dxdy

- fra (NnnďUn + NnsďUs + Ēnnď</Jn + Ēnsď</Js + Qn8Wo) ds }dt (3.4.9)

Substituting for 8U, 8V, and 8K from Eqs. (3.4.6)-(3.4.8) into the virtual work statement in Eq. (3.4.5) and integrating through the thickness of the laminate, we obtain


6w0:

6u0:

6v0:

The Euler-Lagrange equations are obtained by setting the coefficients of 8uo, 8vo, 6wo, 8</Jx, and 6</Jy in Oo to zero separately:

(3.4.12)

where N(w0) and P(wo) were defined in Eq. (3.3.24), and the boundary expressions were arrived by expressing <Px and </Jy in terms of the normal and tangential rotations, (</Jn, <Ps):

O = foT fn0 [ - (Nxx,x + Nxy,y - Ioilo - Ii Jx) Duo

- ( Nxy,x + Nyy,y - Iovo - IiJY) 8vo

- ( Mxx,x + Mxy,y - Qx - hJx - t, ilo) O</Jx

- ( Mxy,x + Myy,y - Qy - hJy - /i Vo) 8</Jy

- ( Qx,x + Qy,y + N( wo) + q - Iowo) 6wo] dxdy

+ foT l [ (Nnn - Nnn) OUn + ( è; - Nns) Dus + ( e: - Qn) 8wo + ( u.; - Mnn) 8</Jn + ( Mns - Mns) Ocp8] dsdt (3.4.11)

The shear correction factor is the ratio of Uf to u;, which gives K = 5/6. The shear correction factor for a generai laminate depends on lamina properties and lamination scheme. Returning to the virtual work statement in Eq. (3.4.9), we substitute for

the virtual strains into Eq, (3.4.9) and integrate by parts to relieve the virtual generalized displacements (8uo, 8vo, 8wo, 8</Jx, 8</Jy) in Oo of any differentiation, so that we can use the fundamental lemma of variational calculus: we obtain

u: 1 { ( e )2 dA 3Q2 8 = 2G13 }A crxz = 5G13bh

J - 1 r ( J ) 2 - Q2 o; -2G13 }A crxz dA - 2G13bh

where Q is the transverse shear force. The transverse shear stress in the first-order theory is a constant, a"tz = Q /bh. The strain energies due to transverse shear stresses in the two theories are


(3.4.17c) e14 = (e15 - e24)sinecose, e24 = e24cos2e + e15sin2() e15 = e15 cos2 e + e24 sin2 e' e25 = ( e15 - e24) sin e cos ()

(3.4.17b)

- 2 2 Q44 = Q44 cos () + Q55 sin () Q45 = ( Q55 - Q44) cose sin e - 2 2 Q55 = Q44 sin e+ Q55 cos e

where [see Eq. (2.4.10)]

(3.4.17a) {<Jyz}(k) = [q44 <Jxz Q45

The laminate constitutive equations for the first-order theory are obtained using the lamina constitutive equations (3.3.12a) and the following relations:

3.4.3 Laminate Constitutive Equations

for all points in Oo.

(3.4.16) Un =u~, Us = u~, wo = w8, <Pn = </;~, <Ps = </J~ . ĿO . ĿO . ĿO . Ŀo . Ŀo Un =un, Us = us, Wo = Wo, </Jn = <Pn, <Ps = <Ps

Note that Qn defined in Eq. (3.4.14b) is the same as that defined in Eq. (3.3.31b). This follows from the last two equations of (3.4.13). The initial conditions of the theory involve specifying the values of the

displacements and their first derivatives with respect to time at t = O:

(3.4.15) primary variables: Un, U8, wo, </Jn, <Ps secondary variables: Nnn, Nns, o.; Mnn, u.;

(3.4.14b) Qn = Qxnx + Qyny + P(wo) Thus the primary and secondary variables of the theory are

where

(3.4.14a)

The natural boundary conditions are obtained by setting the coefficients of bun, bu8, bwo, b<Pn, and b<Ps on r to zero separately:

Nnn - Nnn = O , Nns - s; = O , Qn - o; = O

(3.4.13)

8Mxx 8Mxy _ Q _ I 82</Jx I 82uo ax + Oy X - 2 8t2 + l ®)t2 8Mxy 8Myy _ Q _I 82</;y I 82vo ax + ay y - 2 8t2 + 1 8t2

b</Jx :


(3.4.21)

{ ~ + 1(~)2 } Brn ¨o: 2 ax

B ~ + 1(~)2 26 ay 2 ay B,J '""' + !1!!l! + '""" ""'1 ay ax è ay

{

8<Px } Dm 7JX D ~ 26 ay D,J 84'. + a<1, ay ax

(3.4.20)

{ ~ + 1(~)2 } Al6 ¨è 2 ax

A avo+ 1(~)2 26 ay 2 ay A,J '""' + '""' + """' '""" ay ax ax ay

{

8<Px } Brn 7JX B 8ef;y 26 8y B,J a;. + op,, ay ax

[

Du + D12 Dm

{ Nxx} [An Nyy = A12 Nxy A16

When thermal and piezoelectric effects are not present, the stress resultants (N's and M's) are related to the generalized displacements (uo,vo,wo,</>x,</>y) by the relations

(3.4.19b) { (" }(k) ~t) ~: dz

and the piezoelectric forces Q{: and Q~ are defined by

(3.4.19a)

where the extensional stiffnesses A44, A45, and A55 are defined by

(3.4.18)

or

The laminate constitutive equations in Eqs. (3.3.36) and (3.3.37) are valid also for the first-order laminate theory. In addition, we have the following laminate constitutive equations:


(3.4.23)

A (fPuo Dwo D2wo) A ( D2vo Dwo 82wo)

11 --+---- + 12 --+---- + Dx2 Dx 8x2 8y8x /Jy /Jy/Jx

A ( 82uo 82vo 82wo 8wo Dwo 82wo)

16 --+--+----+--- + /Jy/Jx 8x2 8x2 8y 8x 8y8x

82 rPx a2 rPy ( 82 rPx 82 rPy ) Bu ax2 + Bi2 8y8x + Brn 8x8y + 8x2 + A ( 82uo UW() fPwo) A ( 82vo 8wo 82wo) 15 ~+~~ + 26 -;:)2+~-8 2 + oxou uX oxou uy uy y

A ( 82uo 82vo 82wo 8wo 8wo 82wo)

66 --+--+----+--- + ) 8y2 ax8y 8x8y 8y 8x 8y2

82 rPx 82 rPy ( 82 <Px 82 </Jy ) B15--+B25--+B55 --+--Ŀ - 8x8y 8y2 8y2 8y8x

(aN'[x 8N'{y) _ (8N!x 8Nfv) = l 82uo I 82</Jx ax + 8y ax + 8y o 8t2 + 1 8t2

The equations of motion (3.4.13) can be expressed in terms of displacements ( uo, vo, wo, <Px, r/Jy) by substituting for the force and moment resultants from Eqs, (3.4.20) (3.4.22). For homogeneous laminates, the equations of motion (3.4.13) take the form (including thermal and piezoelectric effects)

3.4.4 Equations of Motion in Terms of Displacements

(3.4.22) { o; } = K [ A44 A45 ] { ~ + rPy } Qx A45 A55 ~ + rPx

When thermal and piezoelectric effects are present, Eqs. (3.4.20) and (3.4.21) take the same form as Eq. (3.3.40), and Eq. (3.4.22) will contain the column of piezoelectric forces given in Eq. (3.4.18).


(3.4.26)

(3.4.25)

(3.4.24)


(3.4.32)

(3.4.31)

(3.4.30)

(3.4.29)

The linearized equations of motion for cylindrical bending according to the f³rst-order shear deformation theory are given by setting ali derivatives with respect to y in Eqs. (3.4.23)-(3.4.27):

Example 3.4.1:

Conversely, the relations in Eq. (3.4.28) can be used to derive the first-order theory from the classical plate theory via the penalty function method (see Chapter 10).

(3.4.28) <P __ owo y - ay and <P - - owo

X - OX

Equations (3.4.23)-(3.4.27) describe five second-order, nonlinear, partial differential equations in terms of the five generalized displacements. Hence, the first-order laminated plate theory is a tenth-order theory and there are ten boundary conditions, as stated earlier in Eqs. (3.4.14) and (3.4.15). Note that the displacement field of the classical plate theory can be obtained from that of the first-order theory by setting

(3.4.27)


Figure 3.5.1: A laminate with general stacking sequence.

z

h 2

X

h 2

A dose examination of the laminate stiffnesses defined in Eqs. (3.3.38) and (3.4.19a) show that their values depend on the material stiffnesses, layer thicknesses, and the lamination scheme. Symmetry or antisymmetry of the lamination scheme and material properties about the midplane of the laminate reduce some of the laminate stiffnesses to zero. The book by Jones [44] has an excellent discussion of the laminate stiffnesses for various types of laminated plates. In this section, we review selective lamination schemes for their laminate stiffness characteristics, Before we embark on the discussion of laminate stiffnesses, it is useful to introduce

the terminology and notation associated with special lamination schemes. The lamination scheme of a laminate will be denoted by (a/ f3 h / 6 /E/ Ŀ Ŀ Ŀ), w here a is the orientation of the first ply, f3 is the orientation of the second ply, and so on (see Figure 3.5.1). The plies are counted in the positive z direction (see Figure 3.3.1). Unless stated otherwise, this notation also implies that all layers are of the same thickness and made of the same materiai. A general laminate has layers of different orientations () where -90Á ::; () ::;

90Á. For example, (0/15/-35/45/90/--45) is a six-ply laminate. General angle-ply laminates (see Figure 3.5.2) have ply orientations of () and -{} where 0Á ::; () ::; 90Á, and with at least one layer having an orientation other than 0Á or 90Á. An example

3.5.1 Generai Discussion

3.5 Laminate Stiffnesses far Selected Laminates

(3.4.33)


Figure 3.5.3: A cross-ply laminated plate with the 0Á and 90Á layers.

z

Figure 3.5.2: A generai angle-ply laminate.

t1 t2 f1_

2 t3

tk X

f1_ 2

tL-1 tL

z

of angle-ply laminates is provided by (15/-30/0/90/45/-45). Crnss-ply laminates are those which have ply orientations of 0Á or 90Á (see Figure 3.5.3). An example of a cross-ply laminate is (0/90/90/0/0/90). For layers with 0Á or 90Á orientations, the layer stiffnesses Ql6, Q26, Q45 are zero. Hence, Al6 = A26 = A45 = Dl6 = D26 =O. When ply stacking sequence, material, and geometry (i.e., ply thicknesses) are

symmetric about the midplane of the laminate, the laminate is called a symmetric laminate (see Figure 3.5.4). For a symmetric laminate, the upper half through the laminate thickness is a mirror image of the lower half. The laminates (-- 45/ 45/ 45/-45)=(-45/ 45)8 and (45/-45/-45/45) = (45/-45)8, with all layers having the same thickness and materiai, are examples of a symmetric angle-ply laminate, (0/90/90/0) = (0/90)s is a symmetric cross-ply laminate, and (30/-45/0/90/90/0/- 45/30)=(30/-45/0/90)8 is a generai symmetric laminate.


Here we discuss some special cases of single-layered configurations and their stiffnesses. The special single layer plates discussed here include: isotropie, specially orthotropic (i.e., the principal materiai coordinates coincide with those of the plate), generally orthotropic (i.e., the principal material coordinates do noi coincide with those of the plate), and anisotropie. The bending-stretching coupling coefficients

3.5.2 Single-Layer Plates

Note that symmetric laminates are also denoted by displaying only the lamination scheme of the upper half. The symmetric laminate (-25/35/0/90/90/0/35/-25) is denoted as (-25/35/0/90)8Å An unsymmetric or asymmetric laminate is a laminate that is not symmetric.

An antisymmetric laminate is one whose lamination scheme is antisymmetric and material and thicknesses are symmetric about the midplane. Examples of antisymmetric angle-ply and cross-ply laminates are provided, respectively, by (- 30/30/-30/30/-30/30)= (-30/30)3 and (0/90/0/90/0/90)= (0/90)3. Laminate stiffnesses Aij depend on only on the thicknesses and stiffnesses of

the layers but not on their placement in the laminate. On the other hand, laminate stiffnesses Dij depend not only on the layer thickness and stiffnesses but also on their location relative to the midplane. For example, both (0/90)8 and (90/0)s laminates will have the same in-plane stiffnesses AijĿ However, (0/90)s laminate will have larger bending stiffnesses Dij about an axis perpendicular to the fiber direction than the (90/0)s laminate, because the 0Á layers are located farther from the midplane in the (0/90)s laminate. Both Aij and Dij are always positive. Laminate stiffnesses Bij

also depend on the layer thickness, stiffnesses and location relative to the midplane, and they can be negative, depending on the lamination scheme and the number of layers.

X

h 2

Figure 3.5.4: A symmetric laminate.

z


h tL-1=t2 2

tL =f 1

where Qij are the plane-stress-reduced stiffnesses, and they are given in terms of the engineering constants [see Eq, (3.3.llb)] as

(3.5.6)

Au = Quh, A12 = Q12h, A22 = Q22h ê66 = Q55h, ê44 = Q44h, ê55 = Q55h

D _ Quh3 D _ Q12h3 D _ Q22h3 D _ Q55h3 11 - 12 ' 12 - 12 ' 22 - 12 1 66 - 12

For a single specially orthotropic layer, the stiffnesses can be expressed in terms of the Qij and thickness h. The nonzero stiffnesses of Eqs. (3.3.38) and (3.4.19a) become

(3.5.5)

(3.5.4)

(3.5.3)

(3.5.2)

(3.5.1)

Single Specially Orthotropic Leyet

h h T T Ea j2 T T Ea j2 Nxx = NYY = ( ) sr dz, Mxx = Myy = ( ) sr, dz 1 - I/ _!]_ 1 - I/ _!]_

2 2

The nonzero thermal stress resultants {NT} and {MT} are given by

{ o; } = K ~ [ A11 Qx 2 O

o l { (l) } o :rv (lb-in/in.) 1-v D (1) 2 11 "(xy

} ] { 1f ~; } (lb-in) 11 'Yxz

{Mxx} [ D11 vD11 Myy = vu., oç Mxy O O

vA11 A11 o {

(O)} O Exx o ] E~V (lb /in.)

l-v A (O) 2 11 "(xy

E~ 1-v Du = 12(l _ v2) , D12 = vDu, D22 = Du, D66 = -2-Du

The plate constitutive equations for the classical and first-order theories become

Bij and the shear stiffnesses Arn, A25, D15, and D26 can be shown to be zero for all single-layer plates except for generally orthotropic and anisotropie single-layer plates. The units of N, and Mi, in the U.S. Customary System (USCS), are lb-in. and lb-in/in., respectively.

Single Isotropie Leyet

For a single isotropie layer with material constants E and v [G = 2(l~i) and thickness h, the nonzero laminate stiffnesses of Eqs. (3.3.38) and (3.4.19a) become

Eh 1-v 1-v A11 = l _ v2, A12 = vA11, A22 = Au, A55 = -2-Au, A44 = A55 = -2-Au


(3.5.16)

The thermal stress resultants for this case are given by

(3.5.15)

(3.5.14)

(3.5.13) rxx} [ An A12 { (O) } A16 l Exg Nyy = A12 A22 A26 E~J Nxy ê15 A25 ê55 1<0) xy

rxx} [D11 D12 { (1) } D"] <y Myy = D12 D22 D25 E~J Mxy D15 D26 D55 1<1) xy

{ Qy} = K [ ê44 { (O)} ê45] /yz Qx ê45 Ass 1(0) xz

The plate constitutive equations are

(3.5.12)

Single Generally Orthotropic Layer

For a single generally orthotropic layer (i.e., the principal material coordinates do not coincide with those of the plate), the stiffnesses can be expressed in terms of the transformed coefficients Qij and thickness h. The nonzero stiffnesses are (Bij =O)

- 3 - Qijh - - Aij = Qijh, Dij = ~ , A44 = hQ44, ê55 = hQ55

(3.5.llb)

(3.5.lla)

The nonzero thermal stress resultants are given by

(3.5.10)

(3.5.9)

(3.5.8) { Nxx} [Q11 Q12 0 ]{ (O)} Exx

Nyy = h Q12 Q22 (O) 0 Eyy Nxy O o Q55 'Y1V

{ Mxx } _ h' [ Qn Q12 o ]{ (J)} Exx

Q22 o (1) Myy - - Q12

Q55 ~rv Mxy 12 O o

The plate constitutive equations for the classica! and first-order theories become

(3.5.7)


[30189 0.755 o o n me; 0.755 3.019 o o [Q] = o o 0.6 o

o o o 1.5 o o o o 1.5

The transformed stiffness matrix [Q] for (} = 60Á is given by

['993 5.573 o o 31011 5.573 18.578 o o 8.664 [Q]50 = ~ o 1.275 0.390 o msi

o 0.390 0.825 o 3.101 8.664 o o 6.318

The piane stress-reducod elastic coefficient matrix in t.he materia! coordinates is

G23 = 0.6 x 106 psi, v12 = 0.25, v13 = 0.25, v2:3 = 0.25 (3.5.19)

The matrix of elastic coefficients for tho materia! is [see Eq. (1.3.44)]

30.508 1.017 1.017 o o o 1.017 3.234 0.834 o o o

[C]= 1.017 0.834 3.234 o o o msi o o o 0.6 o o o o o o 1.5 o o o o o o 1.5

E1 = 30 x 106 psi, E2 = E3 = 3 x 106 psi, G12 = G13 = 1.5 x 106 psi

The materiai properties of boron-epoxy materiai layers are

Example 3.5.1:

for i, j = 1, 2, 3, 4, 5 and 6 [see Eq. (2.4.3a)]. The plate constitutive equations are the same as in Eqs, (3.5.13)-(3.5.16) with the plate stiffnesses given by Eq. (3.5.18).

(3.5.18)

Fora single anisotropie layer, the stiffnesses are expressed in terms of the coefficients Cij and thickness h. The nonzero stiffnesses are (Bij =O)

rrx} [Q" Q12 Qrn l { axx } N~ = Q12 Q22 Q25 ayy Toh Nxy Qrn Q25 Q55 2a,r,y

{ MJ~} [Q" Q12 Qrn l { ~xx } T1h3 T - Q22 Q25 (3.5.17) Myy = Q12 yy 12

M'f;; Qrn Q25 Q55 2axy

Single Anisotropie Layer

A similar expression holds for { MT}. If the temperature increment is linear through the layer thickness, tlT = To + zT1,

the thermal stress resultants have the form


Figure 3.5.5: A symmetric cross-ply laminate.

z

When the materia! properties, locations, and lamination scheme are symmetric about the midplane, the laminate is called a symmetric laminate. Ifa laminate is not symmetric, it is said to be an unsymmetric laminate. Due to the symmetry of the layer material coefficients Q~), distances Zk, and thicknesses hk about the midplane of the laminate for every layer, the coupling stiffnesses Bij are zero for symmetric laminates (see Figure 3.5.5). The elimination of the coupling between bending and extension simplifies the governing equations. When the strain-displacement equations are linear, the equations governing the in-plane deformation can be uncoupled from those governing bending of symmetric laminates. Further, if there are no applied in-plane forces or displacements, the in-plane deformation (i.e., strains) will be zero, and only the bending equations must be analyzed. From production point of view, symmetric laminates do not have the tendency to twist from the thermally induced contractions that occur during cooling following the curing process.

3.5.3 Symmetric Laminates

{

Ctxx } { 6.625 } ayy = 3.875 x 10-6 in./in./ÁF 2axy 600 -4. 763

The transformed coefficients are

(3.5.20) a1 = 2.5 x 10-6 in./in./ÁF , a2 = 8.0 x 10-6 in./in./ÁF

The laminate stiffnesses A;1 and D;1 for i,j = 1, 2,6 may be computed using Eq. (3.5.12). The transverse shear stiffnesses A44,A45, and A55 are given by AiJ = Q;1h for i,j = 4,5. Suppose that the thermal coefficients of expansion of the materiai are


A laminate composed of multiple specially orthotropic layers that are symmetrically disposed, both from a material and geometrie properties standpoint, about the midplane of the laminate does not exhibit coupling between bending and extension

Symmetric Laminates with Multiple Specially Orthotropic Layers

(3.5.24)

and similar expression holds for { MT}. If ~T = To + zT1, then Eq, (3.5.23) can be written as

(3.5.23) >12] (k) { O:xx } ~T dz Q22 O:yy

The thermal stress resultants for this case are given by

(3.5.22)

where the laminate stiffnesses AiJ and Dij are defined by Eqs. (3.3.38) and (3.4.19a) with

(3.5.21c)

(3.5.2lb)

(3.5.21a) rxx} ~ [ A11 A12 0 ]{ (O) } fxx

A12 Au 0 (O) Nyy - A66 Erlf) Nxy o o rxy

{ M,,} [ o., D12 o]{ (l)} Exx

Myy = D12 o., o (1)

Mxy O o D55 ~!v { ~~} = K [ ~4 Q ] { (O)}

A55 ~!~)

Symmetric Laminates with Multiple Isotropie Layers

When isotropie layers of possibly different materiai properties and thicknesses are arranged symmetrically from both a geometrie and a materia! property standpoint, the resulting laminate will have the following laminate constitutive equations for the classica! or first-order theories:

The farce and moment resultants for a symmetric laminate, in general, have the same form as the generally orthotropic single-layer plates [see Eqs. (3.5.13}- (3.5.15)]. For certain special cases of symmetric laminates, the relations between strains and resultants can be further simplified, as explained next.


Laminates can be composed of generally orthotropic layers whose principal materia! directions are aligned with the laminate axes at an angle () degrees. If the thicknesses, locations, and material properties of the layers are symmetric about the midplane of the laminate, the coupling between bending and extension is zero, Bij = O, and the laminate constitutive equations are given by Eqs. (3.5.13)-(3.5.15). Note that the coupling between normal forces and shearing strain, shearing force and normal strains, normal moments and twist, and twisting moment and normal curvatures is not zero for these laminates (i.e., A15, A25, D15, and D25 are not zero). An example of a general symmetric laminate with generally orthotropic laminae is provided by (30/-603/155/-603/30), where the subscript denotes tbe number of layers of tbe same orientation and thickness, Regular symmetric angle-ply laminates are those that bave an odd number of

ortbotropic laminae of equal thicknesses and alternating orientations: (a/-a/a/- a/a/ Ā Ā Ā), 0Á <a< 90Á (see Figure 3.5.6). A generai symmetric angle-ply laminate has the form (()/(3/'Y/ Ŀ Ŀ Ŀ)8, wbere (),(3, and 'Y can take any values between -90Á and 90Á, and each layer can have any thickness, but they should be symmetrically placed about the midplane. It can be shown that the stiffnesses A15, A25, D15, and D25 of a regular symmetric angle-ply laminate are the largest when the number of layers N is equal to 3, and they decrease in proportion to 1/N as N increases. Thus, for symmetric angle-ply laminates with many layers, the values of A15, A25, D15, and D26 can be qui te small compared to other Aij and Dij. A laminate composed of multiple anisotropie layers that are symmetrically

disposed about the midplane of the laminate does not bave any stiffness simplification other than Bij = O, whicb holds for all symmetric laminates. Stiffnesses A15, A25, D15, and D25 are not zero, and they do not necessarily go to zero as the number of layers is increased.

Symmetric Laminates with Multiple Generally Orthotropic Layers

Such laminates are also called specially orthotropic laminates. The thermal stress resultants have the same form as those given in Eq, (3.5.23). A common example of specially orthotropic laminates is provided by the regular

symmetric cross-ply laminates, which consist of laminae of the same thickness and material properties but have their major principal material coordinates (i.e., x1 and x2) alternating at 0Á and 90Á to the laminate axes x and y: (0/90/0/90/ Ŀ Ŀ Ŀ). The regular symmetric cross-ply laminates necessarily contain an odd number of layers; otherwise, they are not symmetric. Of course, a generai symmetric cross-ply laminate can have either an even or odd number of layers: (0/90/0/90/90/0/90/0) or (0/90/90/0/0/90/90/0) (see Figure 3.5.5).

(3.5.25) Q-(k) - o Q-(k) - o -(k) k -(k) k -(k) k 16 - , 26 - , Q66 = G12, Q44 = G23, Q55 = G13

Ek1 k Ek Ek -(k) _ -(k) V21 1 -(k) 2 Qll - k k , Q12 = 1 k k , Q22 = k k 1 - V12V21 V V 1 V V . - 12 21 - 12 21

i.e., Bij = O. The laminate constitutive equations are again given by Eqs. (3.5.21a- c), where the laminate stiffnesses Aij and Dij are defined by Eqs. (3.3.38) and (3.4.19a) with


A44 = 1.05, A45 = O.O, A55 = 1.05

The transverse shear stiffnesses are (in 106 lb/in.)

~ ] 106 lb/in., [D] = [~:~~~ ~:~~~ ~ ] 106 lb-in. 1.5 o o 0.125

0.755 16.604 o [

16.604 [A]= o.;55

A symmetric cross-ply laminate (0/90/0/90), of boron-epoxy layers has the stiffnesses

{ Jvl.T } { 0} M~ = O lb-in./in. Mxy O {

NJ"x } { 57.241 } NYt = 50.307 106T0 lb/in., ç; -0.929

Thc thcrmal stress resultants are (To f= O, T1 = O)

A44 = 0.9938, A45 = -0.0151, A55 = 1.106:3

The transverse shear stiffnesses are (in 106 lb/in.)

0.468 l [ 1.683 0.303 0.409 l ~0.923 106 lb/in., [D] = 0.303 0.604 0.141 106 lb-in. 4.311 0.409 0.141 0.366 [

15.491 3.565 [A] = 3.565 12.095

0.468 -0.923

A generai symmetric laminate (30/0/90/~45)8 of tota! thickness 1 in. and made of boron-epoxy layers [see Eqs. (3.5.19) and (3.5.20) for materiai properties] has the following laminate stiffnesses:

Exarnple 3.5.2:

In general, symmetric laminates are preferred wherever they meet the application requirements. Symmetric laminates are much easier to analyze than general or unsymmetric laminates. Further, symmetric laminates do not have a tendency to twist due to thermally induced contractions that occur during cooling following the curing process.

Figure 3.5.6: A symmetric angle-ply laminate.

z


Although symmetric laminates are more desirable from an analysis standpoint, they may not meet the design requirements in some applications. For example, a heat shield receives heat from one side and thus requires nonsymmetric laminates to effectively shield the heat. Another example that requires coupling is provided by turbine blades with pretwist. Moreover, the shear stiffness of laminates can be increased by orienting the layers at angle to the laminate coordinates. The genera} class of antisymmetric laminates must have an even number of

orthotropic laminae if adjacent laminae have equal thicknesses and alternating orientations: ( () /-()), 0Á ::; () ::; 90Á. Due to the antisymmetry of the lamination

3.5.4 Antisymmetric Laminates

{Mxx} Myy = u.;

o l { e:~;J } 0 E(l)

D66 ³'![) o] { 0.1} { 1.7862} ~ ~:~ = o.og03 Ib-in./in. [

17.862 0.503 0.503 4.277 o o

Ali moments will be zero on account of the fact that there are no bending strains and the coupling stiffnesses B;1 are zero. Now suppose that the laminate is subjected to loads such that it experiences only nonzero strain

of e:~;J = 0.1. Hence, the only nonzero strain is Exx = e:~;J z. Then the force resultants are zero, and the moment resultants are given by

.. ~~~ ~ l { :~~} O O A55 ³'!~)

[0.3321 0.0151 o l { 1,000} { 332.1} 0.0151 0.3321 O O = 15.1 lb/in. o o 0.03 o o

{Nxx} Nyy = s.;

Considera symmetric laminate (0/90)s made of boron-epoxy layers of thickness 0.005 in. Suppose that the laminate is subjected to loads such that it experiences only nonzero strain of E~x = 103 Õ in./in. We wish to determine the forces and moment resultants. The only nonzero strain is Exx =E~~. Hence the force resultants in the laminate are given by

Exarnple 3.5.3:

A44 = 0.9375 x 106 lb/in., A45 =O.O lb/in., A55 = 1.1625 x 106 Ib/in.

The transverse shear stiffnesses are

[

14.379 6.376 o l [ 1.461 0.481 0.256 l [A] = 6.376 7.586 O 106 Ib/in., [D] = 0.481 0.470 0.126 106 lb-in.

o o 7.122 0.256 0.126 0.543

Note that the cross-ply laminate considered here is equivalent to (0/90/0/90/0/90/0) where ali layers except the middle layer having a thickness of h/8 and the middle layer (90) has a thickness of h/4; here h is the tota! thickness of the laminate. A symmetric angle-ply laminate (30/-30/45/-45)8 of boron-epoxy layers has the stiffnesses


- 2 h

X

h 2

Figure 3.5. 7: An antisymmetric laminate.

z

Similar expression holds for { MT}.

(3.5.27)

(3.5.26c)

The thermal force resultants are given by

rxx} [ A11 A12 CO)} [ e., B12 B10 l { E~2} O Exx

0 E(O) -t (1) Nyy = A12 A22 J ~rv B12 B22 B26 Eyy Nxy O o B15 B25 B66 (1)

/xy

(3.5.26a)

rxx} [ B11 B12 { (O) } D12 JJ { E~Q} B15 Exx o.. Myy = B12 B22 B26 l E~~ + [ D12 D22 (1)

Eyy Mxy B15 B25 B66 1(0) O o (1)

xy /xy

(3.5.26b)

scheme (see Figure 3.5.7) but symmetry of the thicknesses of each pair of layers, this class of antisymmetric laminates has the feature that Al6 = A26 = D15 = D25 = O. The coupling stiffnesses Bij are not all zero; they go to zero as the number of layers is increased. Foa general antisymmetric laminate, the relations between the stress resultants and the strains are given by


Figure 3.5.8: An antisymmetric cross-ply laminate.

z

rxx} [ Au A12 { (O) } o 0]{ E~~} 0 l E(6\ [ B11 Nyy = A12 A22 -Bn o (1) 0 Eyy + 0 Eyy è; O o A55 /'(o) O o o (1)

xy /'xy (3.5.29a)

rxx} ~ [B11 o Ql{ (O)} [ D11 D12 o ]{ (l)} o :~~ + Exx

-Bn D12 D22 o (1) Myy - O D55 ~m Mxy O o 0 (O) o o /'xy

(3.5.29b)

{ ~~} = K [ A~4 Q ] { (O)} A55 ~r~) (3.5.30)

The relations between the stress resultants and the strains are

(3.5.28) B22 = - Bu, and all other Bij = O

A special case of antisymmetric laminates are those which have an even number of orthotropic layers with principal material directions alternating at 0Á to 90Á to the laminate axes. Such laminates are called antisymmetric cross-ply laminates. Examples of antisymmetric cross-ply laminates are (0/90/0/90/ Ŀ Ŀ Ŀ) with all layers of the same thickness, and (0/90/90/0/0/90) with layers of the thicknesses (hi/h2/h3/h3/h2/h1). Note that for every 0Á layer of a given thickness and location, there is a 90Á layer of the same thickness and location on the other side of the midplane (see Figure 3.5.8). For these laminates, the coupling stiffnesses Bij have the properties

Antisymmetric Cross-ply Laminates

In the following pages, we discuss some special cases of the class of antisymmetric laminates described above (i.e., laminates that have an even number of orthotropic laminae, each pair having equal thicknesses and alternating orientations).


[17.281 5.172 o l [ o o -0.194]

[A] = 5.0172 7.093 O 106 lb/in., [B] = O O o.0067 106 lb o 5.917 -0.194 0.067

[D] = [~Ŀ~O~~ ~.~~! ~ ] 106 lb-in., { ~::} = { l.~50} 106 lb/in. O 0.125 A5.5 1.050

Note that if thc same 0Á and 90Á layers are positioned differently, say (0/90/90/0/90/0/0/90), t.hcn the coefficients B;J would vanish ( why?). An antisymmetric angle-ply laminate (~45/45/30/0/0/ 30(45/45) of boron-epoxy layers has

the laminate stiffnesses

o l [-0.849 o o] O 106 lb/in., [B] = O 0.849 O 106 lb 1.5 o o o [

16.604 0.755 [A] = o.;55 16.g04

Example 3.5.4:

A regular antisyrnmctric cross-ply laminate (0/90/0/90/0/90/0/90) of boron-epoxy layers has the laminate stiffnesses

For a fixed laminate thickness, the stiffnesses B15 and B25 goto zero as the number of layers in the laminate increases.

The relations between the stress resultants and the strains are

{ Nxx} [Au A12 0 ]{E~~} [ 0 o { (1) } B16 l Exf Nyy = A12 A22 (O) o B25 E1J (3.5.32) 0 Eyy + 0 Nxy o o A55 11V B15 B25 o (1)

/xy

rxx} [ o o { (O}} [ o., D12 { (1)} B15 Exx O Exx Myy = O o B26 l E\~ + D12 D22 0 l E;~ (3.5.33) M:ry B15 B25 0 (O) o o D55 (l) rxy /xy

{ Qy} = K [ A44 Q ] { (O)} Ass ~r0 (3.5.34)

Qx O

(3.5.31)

Antisymmetric Angle-ply Laminates

An antisymmetric angle-ply laminate has an even number of orthotropic layers with principal material directions alternating at e degrees to the laminate axes on one side of the midplane and corresponding equal thickness laminae oriented at -e degrees on the other side. When e = O, -e should be interpreted as 90Á or vice versa. A regular antisymmetric angle-ply laminate is one that has an even number of layers of equal thickness and material properties. An example is given by (-45/40/-15/15/-40/45). For antisymmetric angle-ply laminates without 90Á layers, the stiffnesses can be

simplified as

A regular ant'isymmetric cross-ply laminate is one that has an even number of layers of equal thickness and the same materia! properties and which have alternating 0Á and 90Á orientations. For these laminates, the coupling coefficient B11 approaches zero as the number of layers is increased.


O l { (O) } Exx 0 (O)

(Au - A12)/2 ~rv

For a generai balanced laminate, the laminate constitutive relations are not that much simpler than for a generai laminate. However, for a symmetric balanced laminate they are given by Eqs. (3.5.13)-(3.5.15) with A15 = A25 =O. Laminates consisting of three or more orthotropic laminae of identica! materiai

and thickness which are oriented at the same angle relative to adjacent laminae exhibit in-piane isotropy in the sense that Au = A22, A55 = (Au - A12)/2, and A15 = A25 = O. Such laminates are called quasi-isotropie laminates. Examples of quasi-isotropie laminates are provided by (90/45/0/-45) and (60/0/-60) (see Example 3.3.2). When the bending-stretching coupling coefficients are zero, the relations between force resultants and membrane strains are the same as those for isotropie plates. The stress resultants are given by

A laminate is said to be balanced if for every layer in the laminate there exists, somewhere in the laminate, another layer with identica! materiai and thickness but apposite fiber orientation. The two layers are not necessarily symmetrically locateci with respect to the midplane. Thus, the unsymmetric laminate (Ñ35/0)r =(35/- 35/0) as well as the symmetric laminate (Ñ35/0)8 are balanced laminates. The characteristic feature of any balanced laminate is that the in-piane shear stiffnesses Al6 and A26 are zero. The reason is that Q15 and Q25 from apposite orientations of the pair of layers are of opposite sign and therefore the net contribution from the pair to Al6 and A25 is zero:

3.5.5 Balanced and Quasi-Isotropie Laminates

{ MJx } { 48.113} M;fy = -48.113 106 lb-in./in. M'[y 121.78 {

NJx } { 5573.6 } Zf = 59~8.5 106 lb/in.,

{ A44} { 0.9938} A45 = O 1061b/in. A55 1.1063 [

1.470 0.303 o l [D] = 0.303 0.816 O 106 lb-in.,

o o 0.366

[ 15.491 3.565 o l [-0.425 o -0.842 l

[A] = 3.565 12.095 O 106 lb/in., [B] = O 0.425 -0.233 106 lb o o 4.311 -0.842 -0.233 o

[D] = [~:~~~ ~:~~~ ~ ] 106 lb-in., { ~::} = { 0.~81} 106 lb/in. O O 0.637 A55 1.219

A generai antisymmetric laminate (30/0/90/45)as = (30/0/90/-45/45/ 0/90/-30) of tota! thickness 1 in. and composed of boron-epoxy layers has the following laminate stiffnesses and thermal resultants:


(6b) where

(6a) { Nxx} = [An Mxx Bn

Note that the displacement field (1), hence the equations of equilibrium (3), contain those of the classica! (Euler-Bernoulli) beam theory (co = -1, c1 = O) and the first-order (Timoshenko) beam theory (c0 =O, c1=1).

3.3 ( Continuation of Problem 3.1) Assume linear elastic constitutive behavior and show that the laminateci beam's constitutive equations are given by

(5) s., = l <J"xx dA, Mxx = l ClxxZ dA, o. = l Clxz dA

where

(4) dw0 dx coMxx or

N,,,, or uo c1Mxx or (2b)

3.2 (Continuation of Problem 3.1) Use the principle of virtual displacements to derive the equations of equilibrium and the natural and essential boundary conditions associateci with the displacement field of Problem 3.1, when the beam is subjected to axial distributed load p(x) and transverse distributed load q(x). In particular, show that

dNxx + _ Q (IX P-

where

(2a)

where (u0,w0) denote the displacements of a point (x,y,0) along the x and z directions, respectively, and <P denotes the rotation of a transverse normai about the y-axis. Show that the nonzero linear strains are given by

(1)

u(x, z) = uo(x) + z [ co d~~o + ci <P(x)]

v(x,z) =O w(x,z) = wo(x)

3.1 Suppose that the displacements (u,v,w) along the three coordinate axes (x,y,z) in a laminated beam can be expressed as

Problems


3.7 Consider a single, orthotropic layer plate (Q45 = O), and assume that the materiai coordinates coincide with the plate coordinates. Compute the stresses ( a xx, a vv- a xy) using the constitutive equations of the first-order plate theory, and then use the equilibrium equations of the three-dimensional elasticity theory to deterrnine the transverse stresses ( a xz, Uyz, a zz) as a function of the thickness coordinate.

aw av aw 8x ' 8z - By

aw =0 au az , az

determine the functions (Fi, F2, F3) such that the Kirchhoff hypothesis holds:

u(x, y, z, t) = uo(x, y, t) + zF1 (x, y, t) v(x, y, z, t) = vo(x, y, t) + zF2(x, y, t) w(x, y, z, t) = wo(x, y, t) + zF3(x, y, t)

3.6 Starting with a linear distribution of the displacements through the laminate thickness in terms of unknown functions ( uo, vo. wo, F1 , F2, F3)

(2) (k)

8Ma/3 (Hl) (k) Qk _ -0-- + U 3 Zk+l - U 3 Zk - a - Q X13 a a

with z and integrate over the lamina thickness to obtain the third equation

(1)

3.5 (Continuation of Problem 3.4) Multiply the equilibrium equations

(6)

(5)

for k = 1,2,ĿĿĿ,N and a,{3 = 1,2 (x1 = x,x2 = y,x3 = z), where N is the tota! number of layers, and

(4)

(3)

where summation on repeated subscripts (a,{3 = 1,2) is implied. Integrate the equations over the thickness (zk,zk+I) with respect to z = x3 to obtain:

(2)

(1)

3.4 The 3-D equilibrium equations of a kth layer, in the absence of body forces, can be expressed in index notation as


These cquations bring out the bending-extensional coupling for laminates with nonzero [ B]. For example, when the bending strains are zero, the applied in-piane forces induce bending rnoments for laminates with nonzero coupling coefficients (B].

3.11 Show that if B;j =O (e.g., for symmetric laminates), the equation of motion governing the transverse deflection w0 in the classica) laminate theory is

{.=:o}= [AJ-1 ({N}- [B]{.=:l})

{M} = ([Bl[AJ-1) {N}- ([B][AJ-1[BJ - [DJ) {.=:1}

Next, multiply the equations of motion with z and integrate with respect to z over the interval (-h/2, h/2) and express the results in terms of the rnornent resultants defined in Eq. (3.3.20a).

3.10 Show that the membrane strains {.=:0} and the moment resultants {M} in the classica! or first- order laminateci plate theory can be expresscd in terms of force resultants {N} and bending strains {E: 1 } as

Integrate the above equations with respect to z over the interval (-h/2, h/2) and express the results in terms of the force resultants defined in Eq. (3.3.20a). Use the following boundary conditions:

Baxx Baxy Baxz B2u Bx + Di) + Bz = Po Bt2 Baxy Bayy Bayz B2v 7JX + By + Bz = Po Bt2 Baxz Bayz Bazz B2w Bx + By + az = Po Bt2

3.9 Consider the equations of rnotion of 3-D elasticity [see Eq. (1.3.26)] in the absence of body forccs:

Compute o, using the transverse shear stresses obtained in Problem 3. 7 from the three- dimensional elasticity, and equate it with U, to determine the shear correction coefficient, K.

ti, = ~io f: ( D"xz/~~) + D"yz/~~)) dzdxdy

_ 1 r 1~ [ ( (0))2 ( (0))2] - 2 Jno -~ Q55 lxz + Q44 Čyz dzdxdy

= 2~ r (i~ + ~~ ) dxdy Jn0 55 44

3.8 Considera single, orthotropic layer plate (Q45 =O), and assume that the materiai coordinates coincide with the plate coordinates. According to the first-order theory, the strain energy due to transverse shear stresses is given by


Determine the stiffnesses [A], [B], and [D] for the antisymmetric laminate (0/90) composed of equa! thickness (0.5 mm) layers.

3.19 Determine the stiffnesses [A], [B], and [D] for an antisymmetric laminate (-45/45) composed of equa! thickness (0.5 mm) layers of AS/3501 graphite-epoxy layers (see Problem 3.18 for the materiai properties).

3.20 If the laminate of Problem 3.18 is heated from 20Á to 90Á, determine the thermal forces and moments generateci in the laminate, if it were restrained from free expansion.

3.21 If the laminate in Problem 3.19 is made of four layers (-45/45/-45/45) of thickness 0.25 mm each, show that the stiffnesses [A] and [D] remain unchanged. Compare the stiffnesses B;j for the two laminates (do they increase or decrease in values?).

3.22 Suppose that a four-layer (0/90)s symmetric laminate is subjected to loads such that the only nonzero strain at a point (x, y) is E~~= 103Õ in./in. The materiai properties of a lamina are (typical of a graphite-epoxy materiai) E1 = 20 rnsi, E2 = 1.30 msi, G12 = 1.03 msi, v12 = 0.3. Assume that each layer is of thickness 0.005 in. Determine the state of stress ( a xx, CTyy, a xy) with respect to the laminate coordinates in each layer. Interpret the results you obtain in light of the assumed strains.

3.23 Compute the stains and stresses in the principal materiai coordinate system of each layer for the problem in Problem 3.22.

3.24 Compute the stress resultants N's and M's for the problem in Problem 3.22. 3.25 Repeat Problem 3.22 for the case in which the laminate is subjected to loads such that the

only nonzero strain at a point (x, y) is E~;) = (1/12) /in. 3.26 Compute the stains and stresses in the principal materiai coordinate system of each layer for

the problem in Problem 3.25.

3.27 Compute the stress resultants N's and M's for the problem in Problem 3.25. 3.28 Determine the displacement associateci with the assumed strain field in Problem 3.25.

3.29 Suppose that a six-layer (Ñ45/0)s symmetric laminate is subjected to loads such that the only nonzero strain at a point (x,y) is E~~= 103Õ in./in. The thickness and materiai properties of a lamina are the same as those listed in Problem 3.22. Deterrnine the state of stress (axx,ayy,axy) and farce resultants.

a1 = -0.3 x 10-6 m/m/Á K, 0.2 = 28 x 10-6 m/m/Á K

G13 = 7 x 103 MPa, G23 = 7 x 103 MPa, V12 = 0.3

E1 = 140 x 103 MPa, E2 = 10 x 103 MPa, G12 = 7 x 103 MPa

3.12 Show that fora generai laminate composed of multiple isotropie layers, the laminate stiffness A15,A25,B15,B25,D15, and D25 are zero, and that A22 = A11,B22 = B11, and D22 = Du.

3.13 Show that for a generai laminate composed of multiple specially orthotropic layers, the laminate stiffness A15, A25, B15, B25, D15, and D25 are zero.

3.14 Show that for antisymmetric laminates the stiffnesses, A16, A26, D16, and D26 are zero, and the coupling stiffnesses B;j are not zero.

3.15 Show that for antisymmetric cross-ply laminates, the coupling stiffnesses B;j bave the properties: B22 = -B11 and ali other B;j =O.

3.16 Show that for antisymmetric angle-ply laminateci plates, the following stiffnesses are zero: A15,A25,D15,D25,Bu,B22,B12, and B55.

3.17 Show that for laminates (a/(3/(3/a/{3/a/a/{3) where -90Á::; a::; 90Á and -90Á::; (3::; 90Á, coefficients B;j are zero.

3.18 The materiai properties of AS/3501 graphite-epoxy materiai layers are:


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2. Poisson, S. D., "Memoire sur l'equilibre et le mouvement des corps elastique," Mem. Acad. Sci., 8(2), 357-570 (1829).

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4. Ba.sset, A. B., "On the Extension and Flexure of Cylindrical and Spherical Thin Elastic Shells," Philosophical Transactions of the Royal Society, (London) Ser. A, 181 (6), 433-480 (1890).

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6. Reissner, E., "On the Theory of Bending of Elastic Plates," Journal of Mathematical Physics, 23, 184-191 (1944).

7. Reissner, E., "The Effect of Transverse Shear Deformation on the Bending of Elastic Platcs," Journal of Applied Mechanics, 12, 69-77 (1945).

8. Reissner, E., "Reflections on the Theory of Eia.stie Plates," Applied Mechanics Reviews, 38(11), 1453Ŀ-1464 (1985).

9. Boll®, E., "Contribution au Probleme Lineare de Flexion d'une Plaque Elastique," Bull. Tech. Suisse. Romande., 73, 281-285 and 293-298 (1947).

10. Hencky, H., "Uber die Berucksichtigung der Schubverzerrung in ebenen Platten," Jng. Arch., 16, 72-76 (1947).

11. Hildebrand, F. B., Reissner, E., and Thomas, G. B., "Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells," NACA TN-1833, Washington, D.C. (1949).

12. Mindlin, R. D., "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropie, Elastic Plates," Journal of Applied Mechanics, Transactions of ASME, 18, 31-38 (1951).

13. Vlasov, B. F., "Ob uravneniyakh teovii isgiba plastinok (On the Equations of the Theory of Bending of Plates)," Izu. Akd. Nauk SSR, OTN, 4, 102-109 (1958).

14. Pane, V., Theories of Elastic Plates, Noordhoff, Leyden, The Netherlands (1975). 15. Reissner, E. and Stavsky, Y., "Bending and Stretching of Certain Types of Aeolotropic Elastic Plates," Journal of Applied Mechanics, 28, 402-408 (1961).

16. Stavsky, Y., "Bending and Stretching of Laminated Aeolotropic Plates," Journal of Engineering Mechanics, ASCE, 87 (EM6), 31-56 (1961).

17. Dong, S. B., Pister, K. S., and Taylor, R. L., "On the Theory of Laminated Anisotropie Shells and Plates," Journal of Aeronautical Science, 29(8), 969-975 (1962).

18. Yang, P. C., Norris, C. H., and Stavsky, Y., "Elastic Wave Propagation in Heterogeneous Plates," International Journal of Solids and Structures, 2, 665-684 (1966).

19. Ambartsumyan, S. A., Theory of Anisotropie Plates, translated from Russian by T. Cheron, Technomic, Stamford, CT (1969).


3.30 Repeat Problem 3.29 for the case in which the laminate is subjected to loads such that the only nonzero strain at a point (x,y) is ®~~ = (1/12) /in.

3.31 Suppose that a three-layer (Ñ45/0) unsymmetric laminate is subjected to loads such that the only nonzero strain at a point (x, y) is ®~~) = 10-3 in./in. The thickness and materia! properties of a lamina are the same as those listed in Problem 3.22. Determine the state of stress (a xx, a vv- a xy) and stress resultants.


20. Whitney, J. M. and Leissa, A. W., "Analysis of Heterogeneous Anisotropie Plates," Journal of Applied Mechanics, 36(2), 261-266 (1969).

21. Whitney, J. M., "The Effect of Transverse Shear Deformation in the Bending of Laminated Plates," Journal of Composite Materials, 3, 534-547 (1969).

22. Whitney, J. M. and Pagano, N. J., "Shear Deformation in Heterogeneous Anisotropie Plates," Journal of Applied Mechanics, 37(4), 1031-1036 (1970).

23. Reissner, E., "A Consistent Treatment of Transverse Shear Deformations in Laminateci Anisotropie Plates," AIAA Journal, 10(5), 716-718 (1972).

24. Librescu, L., Elastostatics and Kinetics of Anisotropie and Heterogeneous Shell-Type Structures, Noordhoff, Leycien, The Netherlands (1975).

25. Reissner, E., "Note on the Effect of Transverse Shear Deformation in Laminateci Anisotropie Plates," Computer Methods in Applied Mechanics and Engineering, 20, 203-209 (1979).


27. Librescu, L. and Recidy, J. N., "A Criticai Review and Generalization of Transverse Shear Deformable Anisotropie Plate Theories," Euromech Colloquium 219, Kassel, Germany, Sept. 1986, Refined Dynamical Theories of Beams, Plates and Shells and Their Applications, I. Elishakoff and H. Irretier (Eds.), Springer-Verlag, Berlin, pp. 32-43 (1987).

28. Whitney, J. M., "Shear Correction Factors for Orthotropic Laminates Under Static Load," Journal of Applied Mechanics, 40(1), 302-304 (1973).

29. Bert, C. W., "Simplified Analysis of Static Shear Correction Factors for Beams of Non- Homogeneous Cross Section," Journal of Composite Materials, 7, 525-529 (1973).

30. Chow, T. S., "On the Propagation of Flexural Waves in an Orthotropic Laminated Plate and Its Response to an Impulsive Load," Journal of Composite Materials, 5, 306-319 (1971).

31. Srinivas, S. R., Joga Rao, C. V., and Rao, A. K., "An Exact Analysis for Vibration of Simply- Supported Homogeneous and Laminated Thick Rectangular Plat.es," Journal of Sound and Vibration, 12, 187-199 (1970).

32. Wittrick, W. H., "Analytical Three-Dimensional Elasticity Solutions to Some Plate Problems and Some Observations on Mindlirr's Plate Theory," International Journal of Solids and Structures, 23, 441-464 (1987).

33. Whitney, J. M. and Sun, C. T., "A Higher Order Theory for Extensional Motion of Laminated Composites," Journal of Sound and Vibration, 30, 85--97 (1973).

34. Sun, C. T. and Whitney, J. M., "Theories for the Dynamic Response of Laminateci Plates," AIAA Journal, 11(2), 178-183 (1973).

35. Lo, K. H., Christensen, R. M., and Wu, E. M., "A Higher Order Theory of Plate Deformation, Part 2; Laminated Plates," Journal of Applied Mechanics, 44, 66D-676 (1977).

36. Krishna Murty, A. V., "Higher Order Theory for Vibration of Thick Plates," AIAA Journal, 15(12), 1823-1824 (1977).

37. Murthy, M. V. V., "An lmproved Transverse Shear Deformation Theory for Laminated Anisotropie Plates," NASA Technical Paper 1903, 1-37 (1981).

38. Reddy, J. N., "A Simple Higher-Order Theory for Laminateci Composite Plates," Journal of Applied Mechanics, 51, 745-752 (1984).

39. Reddy, J. N., "A Generai Non-Linear Third-Order Theory of Plates with Moderate Thickness," International Journal of Non-Linear Mechanics, 25(6), 677-686 (L990).

40. Noor, A. K. and Burton, W. S., "Assessment of shear deformation theories for multilayered composite plates," Applied Mechanics Reviews, 42(1), 1-13 (1989).

41. Carrera, E., "An Assessment of Mixed and Classica! Theories on Global and Locai Response of Multilayered Orthotropic Plates," Composite Struciures, 50, 183-198 (2000).


49. Vasiliev, V. V., Mechanics of Composite Struciures. Translated from Russian by L. I. Man, Taylor and Francis, Washington, DC (1988).

50. Ochoa, O. O. and Reddy, J. N., Finite Elernent Analysis of Composite Laminates, Kluwcr, The Netherlands (1992).

51. Reddy, J. N. (Ed.), Mecluuiics of Composite Mtiterials. Selected Works o] Nicholas J. Pagano, Kluwer , The Netherlands (1994).

52. Reddy, J. N. and Miravetc, A., Practical Ana.lysis of Composite Lcminaies. CRC Press, Boca Raton, FL (1995).

47. Vinson, J. R. ancl Sierakowski, R. L., The Bchasnor of Siructures Composed of Composite Maierials, Kluwer, The Netherlancls (1986).

48. Whitney, J. M., Structural Analysis of Laminated Anisotropie Plaies, Technornic, Lancaster, PA (1987).

43. Carrera, E., "Theories and Finite Elernents for Multilayered, Anisotropie, Composite Plat.es and Shells," Arehives of Computational Methods in Engineering, 9(2), 87-140 (2002).

44. Jones, R. M., Mechanics of Composite Maieriols, Second Edition, Taylor and Francis, Philaclelphia, PA (1999).

45. Lekhnitskii, S. G., Anisotropie Plaies, Translated from Russian by S. W. Tsai and T. Cheron, Gordon and Breach, Newark, NJ (1968).

46. Ashton, J. E. and Whitney, J. M., Theoru of Laminated Plaies, Technornic, Starnford, CT (1970).

42. Carrera, E., "Developments, Ideas, and Evaluations Based upon Reissner's Mixed Variational Theorem in the Modeling of Multilayered Plates and Shells," Applied Mechanics Reoieuis, 54(4), 301-329 (2001).


Figure 4.1.1: Geometry of a laminated beam.

y z

z

There are two cases of laminated plates that can be treated as one-dimensional problems; i.e., the displacements are functions of just one coordinate: (1) laminated beams, and (2) cylindrical bending of laminated plate strips. When the width b (length along the y-axis) of a laminated plate is very small compared to the length along the x-axis and the lamination scheme, and loading is such that the displacements are functions of x only, the laminate is treated as a beam (see Figure 4.1. l ). In cylindrical bending, the laminated plate is assumed to be a plate strip that is very long along the y-axis and has a finite dimension a along the x-axis ( see Figure 4.1. 2). The transverse load q is assumed to be a function of x only. In such a case, the deftection wo and displacements ( uo, vo) of the plate are functions of only x, and all derivatives with respect to y are zero. The cylindrical bending problem is a plane strain problem, whereas the beam problem is a plane stress problem. In this chapter we develop exact analytical solutions for the two classes of

problems. An exact solution of a problem is one that satisfies the governing equations at every point of the domain and the boundary and initial conditions of the problem. A numerical solut³on ³s one that is obtained by satisfying the governing equations and boundary conditions of the problem in an approximate sense. The solutions obtained with any of the variational methods (see Chapter 1) and numerica} methods, such as the finite difference, finite element, and boundary element methods, are termed numerical solutions. An exact solution can be either

4.1 Introduction

One-Dimensional Analysis of Laminated Composite Plates

4

will be termed an analytical solution, although it is approximate because not all terms of the series (4.1.1) are included in (4.1.2). For all practical purposes, it is "exact." Due to their one-dimensional nature, analytical ~ exact as well as numerical

~ solutions can be developed for a number of laminated beams and plate strips. The analytical solutions presented here for simple problems serve as a basis for understanding the response. In addition, the results can serve as a reference for verification of computational methods designed to analyze more complicated problems.

n=l (4.1.2)

N

UN(x) = L an sin rixç

(4.1.1) CX)

u(x) = L an sin naiç n=l

where an are real numbers, is not a closed-form solution because the number of terms in the series is not finite. Since the series solution, in reality, is evaluated for a finite number of terms, it is, in a sense, approximate. The finite-sum series solution

seri es

closed-forrn or an infinite series. Closed-form solutions are those that can be expressed in terms of a finite number of terms. For example, u(x) = 2 - x + 3x2 + 4 sin rin x is a closed-forrn solution, whereas a solution in the form of a convergent

Figure 4.1.2: Geometry of a plate strip in cylindrical bending.

y

X

X

y


Equations (4.2.3a) indicate that the transverse deflection wo cannot be independent of the coordinate y due to the Poisson effect (Di2) and anisotropie shear coupling (Di6). These effects can be neglected only for long beams (i.e., when the length-to- width ratio is large). The length-to-width ratio for which the transverse deflection can be assumed to be independent of y is a function of the lamination scheme. For angle-ply laminates this ratio must be rather large to make the twisting curvature negligible.

( 4.2.3b)

D~l = (D22D55 - D25D25) ID* D~2 = (D15D25 - D12D55) ID* D~6 = (D12D25 - D22D15) ID* D* =DuD1 +D12D2 + D15D3 , D1 = D22D55 - D25D25 D2 =D15D25 - D12D55 , D3 = D12D25 - D22D15

where

( 4.2.3a)

where DiJ denote the elements of the inverse matrix of DijĀ In view of the assumption (4.2.1), we have

( 4.2.2b)

or, in inverse form, we have

(4.2.2a) {

lvfxx }- _ Myy - Mxy

everywhere in the beam. The classica! laminated plate theory constitutive equations for symmetric laminates, in the absence of in-plane forces, are given by [see Eqs. (3.3.44)]

(4.2.1) Myy = Mxy =O

Here we consider the bending of symmetrically laminated beams according to CLPT. For symmetric laminates, the equations for bending deflection are uncoupled from those of the stretching displacements. If the in-plane forces are zero, the in-plane displacements ( uo, vo) are zero, and the problem is reduced to one of solving for bending defiection and stresses. In deriving the laminated beam theory we assume that

4.2 Analysis of Laminated Beams Us³ng CLPT 4.2.1 Governing Equations

ONE-DIMENSIONAL ANALYSIS OF LAMINATED COMPOSITE PLATES 167

(4.2.9b) specify Force:

( 4.2.9a) owo Wo' OX

fJM Q = ox ' M

specify Geometrie:

The boundary conditions are of the form

(4.2.8c) h

q = bq, io= blo, Č2 = bh, Ii= b [: p(z)i dz (i= O, 1, 2) 2

where s: is the applied axial load, and

(4.2.8b)

or, for symmetrically laminated long beams, we have

( 4.2.8a)

where b is the width and h is the total thickness of the laminate. The equation of motion of laminated beams can be obtained directly from Eq.

(3.3.25) by setting all terms involving differentiation with respect to y to zero:

(4.2.7b) or Q = 81'VI ox

and the shear force and bending moments are related by

(4.2.7a) ®Pwo 8x2

and write Eq. ( 4.2.5) as

(4.2.6) Eb = __E_ = b bh3 M = bMxx, Q = bQx, xx h3D* I D* ' Iyy = 12

11 yy 11

In arder to cast Eq. (4.2.5) in the familiar form used in the classical Euler-Bernoulli beam theory, we introduce the quantities

(4.2.5) Then we can write

(4.2.4) wo = wo(x, t)

In the following derivations we assume that the laminated beam under consideration is long enough to make the effects of the Poisson ratio and shear coupling on the deflection negligible. Then the transverse deflection can be treated only as a function of coordinate x (along the length of the beam) and time t:


Calculation of Stresses

The in-piane stresses in the kth layer can be computed from the equations [see Eqs. (3.3.12a) and (4.2.2b)]

( 4.2. l l c) dwo =O dx Wo = 0,

Simply Supported :

Clamped:

Q = dM =O, M =O dx

wo =o' 111 =o Free:

The constants of integration, b1, b2, and c1 through c4, can be determined using the boundary conditions of the problem. The boundary conditions for various types of supports are defined below:

(4.2.llb)

The general solutions of Eqs. (4.2.lOa,b) are obtained by direct integration. We obtain from Eq. (4.2.lOa)

E~xfyywo(x) = - fox [fo17 M()d] dT/ + b1x + b2 (4.2.lla)

and from Eq. (4.2.lOb)

where q = bq. Equation (4.2.lOa) is the most eonvenient when it is possible to express the bending moment M in terms of the applied loads. For indeterminate beams, use of Eq. (4.2.lOb) is more convenient.

General Solutions

( 4.2. lOa, b) M

4.2.2 Bending

For statie bending without the axial farce, Nxx =O, Eqs. (4.2.7a) and (4.2.8b) take the form [cf., Eqs. (1.4.47b) and (1.4.45b); see Figure 1.4.1 for the sign convention]

Equations ( 4.2. 7)-( 4.2.9) are identieal, in forrn, to those of the Euler -Bernoulli beam theory of homogeneous, isotropie beams. Henee, the solutions available far defleetions of isotropie beams under various boundary eonditions ean be readily used far laminated beams by replaeing the modulus E with E~x and multiplying loads and mass inertias with b. Note that the rotary (or rotatory) inertia h is not neglected in Eqs. (4.2.8a-e).


where ((}~kJ,(}W,(}W) are known from Eq. (4.2.12), and c(k), rè, and H(k) are constants.

(4.2.14c) dz + H(k)

(4.2.14b) dz + p(k)

( 4.2.14a) dz + c(k) z ( (k) (k)) (}~~) = _ { O(}xx + O(}xy L, ox oy

(

(k) (k)) (k) = -1z O(}xy O(}yy (}yz ~ + ~

zk uX uy

jĀz ( ~ (k) ~ (k)) (k) = _ u(}xz u(}yz (}zz ~ + ~ Zk UX uy

(4.2.13)

O= 0(}xx + O(}xy + O(}xz ax oy oz

O= 0(}xy + O(}yy + 0(}yz ax oy az

O= 0(}xz + O(}yz + O(}zz ax ay az

Por each layer, these equations may be integrated with respect to z to obtain the interlaminar stresses within each layer (zk ::; z ::; Zk+1):

(4.2.12b)

(k)( ) M(x)z (-(k) * -(k) * -(k) * ) (}xx x, z = b Qll Du + Ql2 D12 + Ql6 D15 (k)( ) M(x)z (-(k) * -(k) * -(k) * ) (}yy x, z = b Q12 Dn + Q22 D12 + Q25 D15 (k)( ) M(x)z (-(k) * -(k) * -(k) * ) (}xy x, z = b Ql6 Dn + Q26 D12 + Q66 Dm

In general, the maximum stress does not occur at the top or bottom of a laminated beam. The maximum stress location through the beam thickness depends on the lamination scheme. As will be seen later in this section, the 0Á layers take the most axial stress. The stresses given by Eq. ( 4.2.12b) are approximate for the purpose of analyzing

laminated beams. They are not valid especially in the free-edge zone, where the stress state is three dimensional, The width of the edge zone is about the arder of the thickness of the beam. In the classica! beam theory, the interlaminar stresses ( o xz, a zz) are identically

zero when computed using the constitutive equations. However, these stresses do exist in reality, and they can be responsible for failures in composite laminates because of the relatively low shear and transverse normal strengths of materials used. Interlaminar stresses may be computed using the equilibrium equations of 3-D elasticity [see Eq. (1.3.27)]:

or

(4.2.12a) q16] (k) Q25 Q55


Figure 4.2.1: (a) Sign convention. (b) Equilibrium of interlaminar stresses in a laminated beam.

dM -Q =O dx

dQ +q =0 dx

M =-Eld2wo dx2

o: ĿĿĿĿĿĿĿĿĿĿ- (k) ! (k)

(b) O"zz f O"zx (k+l) (k)

O"zx = (J zx (k+l) (k+l) (k) (J zz

O"zz = (J zz (k+l) CE.P O"zx

~---x y

OJ³[Il)]]JE:=;;E=ĿĿĿ ~Ŀ~-Ŀ::::~Ŀ:: ... :.:~-- ..... E: ...... ~. :;::Ŀ ....... ~Ŀ-ĿĿ:Ŀ.EE:.Ŀ.:::.:EJĿ - ______.. .. X

z~~:)MÅI Q Q Sign convention

(a)

(k)( ) - (k+l)( ) (k)( ) - (k+l)( ) axz X, Zk+l - axz X, Zk+l , azz X, Zk+l - azz x, Zk+l

where Eqs. (4.2.6) and (4.2.7b) are used to replace dM/dx with Q = bQx, and G(k) and H(k) are the integration constants, which are evaluated using the boundary and interface continuity conditions. For layer 1, the constants should be such that O"u and a zz equal the shear and normal stresses at the bottom face of the laminate. For example, if the laminate bottom is stress free, we have G(1l = O and H(1l = O. The constants G(k) and H(k) for k = 2, 3, Ŀ Ŀ Ŀ are determined by requiring that a~~) and ai~l be continuous at the layer interfaces (see Figure 4.2.1):

( 2 2) (k) _ .. -(k) * -(k) * -(k) * z - zk (k) . O":cz(x,z)--Qx(x)(QuD11+Q12D12+Q16D16) 2

+G (4.2.15a)

O"(k)(x z) - - dQx (Q-(k) D* + Q-(k) D* + Q-(k) D* ) (z3 - z~) + H(k) (4 2 15b) zz ' - dx ll ll 12 12 16 16 6 Ŀ Ŀ

For beams, all variables are independent of y and v = O. Hence, derivatives with respect to y are zero. For example, from Eqs. (4.2.14a,c) and (4.2.12b), we obtain


(4.2.21) (k)( / ) _ Foaz (-(k) Å -(k) Å -(k) Å ) 17xx a 2, Z - -4- Qll D11 + Q12 D12 + Q16 D15

( 4.2.20) e: = F0ba3 xx 4bh3wc

The maximum in-piane stress axx occurs at x = a/2 (M(a/2) = F0ba/4)

This expression can be used to determine the modulus of the materiai in terms of the measured center deflection Wc, applied load Fo, and the geometrie parameters of the laminated beam in a three-point bend test:

( 4.2.19) F0ba3 Wmax = 48Eb J = Wc xx yy

The deflection is the maximum at x = a/2, which is given by

(4.2.18)

We obtain (c1 = Foba2 /16, c2 =O)

wo(O) =O, dwo (a/2) =O dx

The constants c1 and c2 are evaluated using the boundary conditions of the problem

(4.2.17) M( ) = (Fob)x for O< x < '!:. X 2 ' - -2

Substituting this expression into Eq. (4.2.lla) and evaluating the integrals, we obtain

Considera simply supported beam with a center point load (see Figure 4.2.2). This case is known as the three-point bending. The deflection is symmetric about the point x = a/2. The expression for the bending moment is

Example 4.2.1 (Simply supported beam):

Note from Eqs. (4.2.15a,b) that the transverse shear stress axz is quadratic and normal stress o- zz is cubie through the thickness of each lamina. The distributions are described by different functions in different layers but they are continuous across layers.

(4.2.16b) - (k) ( ) - O"zz x, Zk+l

nv+è __ dQx (Q(k) D* + Q(k) D* + Q(k) D* ) (z~+16- z~) + H(k) - dx 11 11 12 12 16 16

c(k+l) = -Qx(x) ( Qi~l Dii+ QW Di2 + Qi~) Di6) (z~+12- z~) + c(k) = o-i~l(x,zk+I) (4.2.16a)

This gives, for k = 1, 2, Ŀ Ŀ Ŀ, the result


(4.2.22) [

. ] 2 qoba4 x 2 x wo(x) = - - - 24E~xlyy e) e)

qobx4 x3 x2 E~Jyyw0(x) = ~ + c1 6 + c22 + c3x + c4

The constants c1 through c4 are evaluated using the boundary condit³ons of the half (because of the symmetry) or full beam. For the full beam case we have

wo(O) =O, wo(a) =O, d;~o (O) =O, d~o (a)= O

and for the half beam model we have dwo dwo a a dM d3w0 a

wo(O) =O, dx (O)= O, dx ( '2) =O, Q( '2) = ([X = -Exxlyy dx3 ( '2) =O

Either set of boundary conditions will yield the same solution. We obtain (c1 = -q0ba/2, c2 = q0ba2/12, c3 = c4 =O)

Consider a laminated beam, clamped at both ends, and subjected to uniformly distributed load acting downward, q = q0 (see Figure 4.2.3). The deflection is symmetric about the point x = a/2. We have from Eq. (4.2.llb) the result

Example 4.2.2 (Clamped beam):

Figure 4.2.2: Three-point bending of a laminated beam (see Figure 4.2.la for the sign convention).

Q(x) Ef) .

F. M(x) ~ 2 X

Fo I-----~ 2 X

fil 2

~ ~

z

~ ~

X


Expressions for the transverse defiection of laminated beams with simple supports, clamped edges, and clamped-free ( cantilever) supports and subjected to a trans verse point load or uniformly distributed load are presented in Table 4.2.l. The maximum deflections and bending moments are also listed (note that the loads are assumed to be applied in the downward direction). Recali that wo(x) is taken positive upward and M(x) is positive clockwise on the right end. When both point load and uniformly distributed load are applied simultaneously, the solution can be obtained by superposing (i.e., adding) the expressions corresponding to each load. Expressions for other boundary conditions can be found in textbooks on a first course in reformable solids. The effects of materiai properties and stacking sequence are accounted for through the bending stiffness Egxlyy = b/Dil' as can be seen from Eqs, (4.2.6) and (4.2.3b).

(4.2.24) (k)( ) _ qoa2z (Q(k) * Q(k) Å (k) Å ) <Jxx o, z - -~ 11 Dll + 12 D12 + Q16 D15

The maximum bending moment, and hence the maximum in-plane stress <Jxx, occurs at x = O,a:

( 4.2.23)

The deflection is the maximum at x = a/2, which is given by

Figure 4.2.3: Clamped beam under uniformly distributed load.

at x =a

Wo = d:Xo =O or

Wo=<l>x= O

Wo = d:Xo =0 or

Wo=<l>x=O

X

17 4 MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS

F0ba3 c1=-E, '

ixfyy

Superscript "e" refers to the center (at x = a/2), "a" to the end x =a, and "O" refers to x =O. The constants in the expressions for the deflection are defined as

J I++ I+ I+ f0 1.. a ÅI

1_1),~ia:r == -Ä- C2 M;;,ax = ic4

U niforrn load

Point load at free end 'W~~ia:J: == A cr Jvf,r;,ax = C3 ~------fo

1.. a

Å Fixed-Free

fi I I I I I I I t0 1.. a ÅI

WiÁnax = ~C2

M,9wx = -bc4 U niform load

1_1),~ru1;.c = 1Ä2 C1

M~wx = ftC3

Centrai point load

Å Fixed- Fixed

w;;wx = 3~4 C2

lv!~,"". = -f;c4 U niforrn load

a

u1;~ta:r: = iB C1

M.;~Ŀ"'' = - i C3

Å Hinged-Hinged Centrai point load

+F0

11!rnax and J\.1rrwx

Deflection, wo ( x) Laminated Bearn

Table 4.2.1: Transverse deflections of laminated composite bcams with various boundary conditions and subjected to point load or uniformly distributcd load (acting downward) according to the classica! beam theory.


by substituting wo = w6 + W, where w6 is the original equilibrium (prebuckling) deflection and W is the buckling deflection. Note that wg satisfies the equation

d4we d2we E~xlyy dx40 + bNxx(w[D dx20 =O

[The reader is asked to verify the result in Eq. (4.2.26).]

( 4.2.26) d4W bNÁ --+ xx dx4 E~Jyy

where W denotes the buckling deflection. Equation ( 4.2.26) is obtained from the nonlinear equilibrium equation

b d4wo d2wo Exxlyy dx4 + bNxx(wo) dx2 =O

4.2.3 Buckling A beam subjected to axial compressive load s.; = - Ngx remains straight but shortens as the load increases from zero to a certain magnitude. Ifa small additional axial or lateral disturbance applied to the beam keeps it in equilibrium, then the beam is said to be stable. If the small additional disturbance results in a large response and the beam does not return to its original equilibrium configuration, the beam is said to be unstable. The onset of instability is called buckling (see Figure 4.2.8). The magnitude of the compressive axial load at which the beam becomes unstable is termed the criticai buckling load. If the load is increased beyond this critical buckling load, it results in a large deflection and the beam seeks another equilibrium configuration. Thus, the load at which a beam becomes unstable is of practical importance in the design of structural elements. Here we determine criticai buckling loads for laminated straight beams. The equation governing buckling of laminateci beams is also given by Eq. (4.2.8b), wherein the applied transverse load and inertia terms are set to zero, and axial force is assumed to be unknown. In addition, the deflection is measured from onset of buckling, and it is termed buckling defiection.

. ' - o Settmg Nxx - -Nxx' q =O, and all inertia terms to zero in Eqs. (4.2.8b), we obtain the equation

The maximum norma! stress distribution in an orthotropic beam (with eight 0Á layers) is shown in the figures by dashed lines. It is clear the 0Á layer carries the most axial stress while the 90Á layer carries the least axial stress, in proportion to their axial stiffness. Figures 4.2.6 and 4.2.7 show the effect of stacking sequence on maximum transverse shear

stress, as predicted by Eq. (4.2.15a), for laminates (0/45/-45/90)s and (90/45/-45/0)s, respectively (Fo = 1.0, b = 0.2, a = 1.0, h = 0.1 ). The parabolic distribution of transverse shear stress through an orthotropic beam is shown in dashed lines for comparison. The maximum stress value is dependent on the stacking sequence and considerably different from that in a homogeneous beam.

( 4.2.25)

Figures 4.2.4 and 4.2.5 show the maximum normai stress distribution, as predicted by Eq. (4.2.12b), through the thickness of (0/45/-45/90)s (0Á corresponds to outer layers) and (90/45/- 45/0)s (90Á corresponds to outer layers) laminated beams, respectively, subjected to three-point bending (Fo = 1.0, b = 0.2, a = 1.0, h = 0.1). The following layer materia! properties are used (E2 = 1 msi):


I f

l> Orthotropic

(90/45/-45/0)8

I I

'

I I

100 o

I I

-0.5-+-~~~~~,...,.._~--.-...~~~~~~~~~-.-. -400 -300 -200 -100 o 100 200 300 400

Stress, -a xx (a I 2,z) Figure 4.2.5: Maximum normal stress, -<7xx(a/2, z), distribution through the

thickness of a symmetrically laminated (90/Ñ45/0)s beam.

I I

-0.4 -0.3 -0.2

NJ->:: ul 0.1 00. Q)

~ O.O ~ r -0.1

0.5 0.4 Fo

4 ! t: 0.3 a

0.2

Stress, -a xx (a I 2,z)

Figure 4.2.4: Maximum normal stress, -<7xx(a/2, z), distribution through the thickness of a symmetrically laminated (0/Ñ45/90)8 beam.

300 200

I

'

I I

-0.5-+~-+--~~-~~-.--~~-.--~~-.--~~-.--r-4

-300 -200 -100

I I

/'- Orthotropic

(0/45/-45/90

-0.4

-0.3

-0.2

O.O

Q)

~ 0.1 i:: ;-a s o ~ -0.1 00. Q)

~ o ~

NJ-S::: 0.2

0.3

o. 5 ---r-"T""""T-.--.-."T""""T-.--.-."T""""T-.--.-.~-.--.-.~-.-.-.-r->r-.--~ I I

0.4


Transverse shear stress,<J .rz (O, z)

Figure 4.2. 7: Variation of transverse shear stress (-O"xz) through the thickness of a symmetrically laminated 90/Ñ45/0)s beam subjected to three- point bending (see Figure 4.2.6 for data).

\

/\ Orthotropic 1

I

(90/45/-45/0)8

0.5

0.4

0.3 Nl..S:: a) 0.2 ...., al ~ ..... 0.1 '"C ~ o o O.O <.)

in u: Q,) -0.1 e ~

<.)

:.a -0.2 E-<

-0.3

-0.4

-0.5 o 2

Fo

4 t ~ a

4 6 8 10 12

Figure 4.2.6: Variation of transverse shear stress (-O"xz) through the thickness of a symmetrically laminated (0/Ñ45/90)8 beam subjected to three- point bending (Fo = 1.0, b = 0.2, a= 1.0, h = 0.1).

8 7 Transverse shear stress, o .rz (O, z)

6 5 4 3 2 1

(0/45/-45/90)8 -, \

Fo

4 t ~ a

0.5

0.4

Nl..S:: 0.3 Q,)" 0.2 ...., al i:: ;a 0.1 i... o o o.o <.)

u: u: Q,) -0.1 ~ ~

<.)

:.a -0.2 E-<

-0.3

-0.4

-0.5 o


The smallest value of N2x, which is given by the smallest value of êb, is the critical buckling load. The buckling shape (or mode) is given by W(x). In the following, we consider beams with different boundary conditions to determine êb and then the critical buckling load for each beam.

( 4.2.30)

and the constants cj , c2, c3, and c4 can be determined using the boundary condi tions of the beam. We are interested in determining the values of êb for which there exists a nonzero

solution W(x), i.e., when beam experiences defiection. Once such a êb is known (often there will be many), the buckling load is determined from Eq, (4.2.29):

(4.2.29) ,\2 _ bN~x b - Eb I ' xx yy

where

( 4.2.28)

The general solution of Eq. ( 4.2.27) is

(4.2.27)

Integrating Eq. (4.2.26) twice with respect to x, we obtain

Figure 4.2.8: Buckling of laminated beams under various edge conditions.

t X t X r Nx~ Nx~

è:

T a

------- ------- 1 z z

(a) (b) (e) Simply supported Clamped clamped Clamped free

beam beam beam


( 4.2.37)

iv(O) = 0: C2 + C4 = 0 iv' (0) = 0: C1êb + C3 = 0 iv (a) = O : c1 sin êba + c2 cos êba + c3a + c4 = O iv' (a) =O: c1êb cos êba - c2êb sin êba + c3 =O

We have

( 4.2.36b)

( 4.2.36a) wo(O) =O, d~o(O) =O, wo(a) =O, dwo (a)= 0 dx

which can be expressed as

iv(O) =O, ~'::(O)= O, iv(a) =O, div (a)= O dx

When the beam is fixed at both ends, the boundary conditions are


iv(x) = c1 sin 7rX a

and the buckling mode ( eigenfunction) associated with it is

( 4.2.35) N = (~)2 E~xlyy = (7r2) E~xh3 cr a b 12 a2

The criticai buckling load becomes ( n = 1)

( 4.2.34b) T"( ) . ntt-x vv X = C1 Sin -- , a

The buckling mode is

( 4.2.34a)

and the buckling load is given by

( 4.2.33) c1 sin êba =O implies that êba = nm , n = 1, 2, Ŀ Ŀ Ŀ

For a nontrivial solution, the condition

(4.2.32)

C2 + C4 = Q - c2>.~ =O which implies c2 =O, c4 =O c1 sin êba + c3a = O c1 sin >.ba = O which implies c3 = O

iv(O) =O: iv" (O)= O: iv(a) =O: iv" (a) =O:

We have

(4.2.31b) d2iv d2iv iv(O) =O, iv(a) =O, dx2 (O)= O, dx2 (a) =O

These boundary conditions imply

( 4.2.31a) wo(O) =O, wo(a) =O, Mxx(O) =O, Mxx(a) =O

For a simply supported beam, the boundary conditions are



( 4.2.41) 48a2

N = (~)2 E~xlyy cr 2a b

The critical buckling load is given by

( 4.2.40b) at x =a

at X= 0 dW =O dx W=O,

which are equivalent to

(4.2.40a) wo(O) =O, dd:o (O) =O, Qx(a) =O, Mxx(a) =O

Table 4.2.2 contains governing equations for >.b, with some typical values, and values of the constants c1, c2, c3, and c4 for several combinations of simply supported (hinged), clamped (fixed), and free-edge conditions. Por example, for the critical buckling load of a cantilever beam (i.e., fixed atone end and free at the other end), the boundary condi tions are

( 4.2.39)

n.; =(e: )2 ( E~b!yy) = (2:)2 ( E~b!yy) = (n32) ( e;2h3)

The solution of equation ( 4.2.38b ), known as the characteristic equation, gives the eigenvalues en = >.ba, and the buckling load is calculated from Eq. (4.2.30). Equations (4.2.38b) is a transcendental equation, i.e., nonlinear equation involving trigonometrie functions. A plot of the function f(en) = en sin e., + 2cosen - 2 against en shows that f(en) is zero at en = O, 6.2832( = 2n ), 8.9868, 12.5664( = 4n), 15.4505, 6n, Ā Ā Ā (>.2n-l a = 2nn). Hence, the criticai (i.e., smallest) buckling load is [see Eq. (4.2.30)]

(4.2.38b)

cos X,ç - 11 - sin >.ba 0 = I sin >.ba - >.0a

cos >.ba - 1

For a nontrivial solution, the determinant of the coefficient matrix of the above two equations must be zero (eigenvalue problem):

(4.2.38a) c1 (sin >.ba - >.ba) + c2 ( cos >.ba - 1) =0

c1 ( cos >.ba - 1) - c2 sin êba =0

Expressing these equations in terms of constants c1 and c2, we obtain


( 4.2.45) where

( 4.2.44)

( 4.2.43) b d4W , d2W 2, 2, d2W Exxlyy dx4 - bNxx dx2 = w Io W - w h dx2

Equation ( 4.2.43) has the general form

d4W d2W p dx4 + q dx2 - r W = O

In the absence of applied transverse load q, the governing equation ( 4.2.8b) reduces to

( 4.2.42) wo(x, t) = W(x)eiwt, i= yCI

For natural vibration, the solution is assumed to be periodic

4.2.4 Vibration

t SeeEq. (4.2.28): W(x)=c1sinêbx+c2cosêbx+c3x+c4. *For criticai buckling load, only the first (minimum) value of e= êa is needed.

Å Hinged-Fixed tan e., = en en = 4.493, 7.725, Ŀ Ŀ Ŀ

C1=1/enCOSen, C3 = -1 C2 = C4 = Q

sin e., =O en = n7r

C1 = C3 = Q C2 =I O, C4 =I o

ÅFree-Free

cosen =o en = (2n -1)7r/2

C} = C3 = Q C2 = -C4 =fa Q

~l====:::::i

Å Fixed- Free

en = 27r, 8.987, 47r, Ŀ Ŀ

Å Fixed-Fixed en sin en = 2(1 - cos en) c1=1/(sinen - en) C3 = -1/ên c2 = -c4 = 1/(cosen - 1)

sin e., =O en = T/,7r

Å Hinged-Hinged

Characteristic equation and values* of en = êna Constantst End conditions at

x =O and x =a

Table 4.2.2: Values of the constants and eigenvalues for buckling of laminated composite beams with various boundary conditions (,\2 bNgx/ E~xlyy = (en/a)2). The classical laminate theory is used.


In the following discussion beams with both ends simply supported or clamped are considered to illustrate the procedure to evaluate the constants c1 through c4, and more importantly, to determine ê so that Eqs, ( 4.2.46)-( 4.2.48) can be used to find w. The smallest frequency w is known as the fundamental frequency. For other boundary conditions, the reader is referred to Table 4.2.3. Far boundary conditions other than simply supported, one must salve a transcendental equation for en = êna.

( 4.2.50) w = ê2ao, ao =

It is clear from the first expression that rotary inertia decreases the frequency of natural vibration. If the rotary inertia is neglected, we have ê = Õ and

( 4.2.49) ( -2) b ( -2) _ h,\ _ 4 Exxlyy hÕ 1 - - -Õ - l+ - -

Io + h>-2 Io Io - hÕ2

The two expressions for w in Eqs. ( 4.2.48a, b) are the same and hence either one can be used to calculate the frequency once ê is known. When the applied axial load is zero, the frequency of vibration can be calculated

from

(4.2.48b)

( 4.2.48a)

Substituting for p, q, and r from Eq. (4,2.45) into Eq. (4.2.47a,b) and solving for w2, we obtain

(4.2.47b) ( 2pÕ,2 + q) 2 = q2 + 4pr or pÕ4 + qÕ2 - r = O

(4.2.47a)

and c1, c2, c3, and c4 are constants, which are to be determined using the boundary conditions. From Eqs, (4.2.46b), we have

(4.2.46b) ê = J 2~ ( q + V q2 + 4pr), Õ = J 2~ ( -q + V q2 + 4pr) ( 4.2.46a) W ( x) = c1 sin êx + c2 cos êx + C3 sinh Õx + c4 cosh Õx

The general solution of Eq. (4.2.44) is


(4.2.53c) n7r[f:.xx Wn ~ - -.-

a Io

Thus the effect of the axial tensile force s.. is to increase the natural frequencies. If we have a very flexible beam, say a cable under large tension, the second term under the radical in Eq. ( 4.2.53b) becomes very large in comparison with unity; if n is not large, we have

(4.2.53b)

1 + (=)2b. a Io

( 4.2.53a) 1

bNxx 1 + (=)2Eb I a xx YY

( n7r) 2 Wn = a ao

If the rotary inertia is neglected, we obtain

l bNxx + (=)2Eb I a xx YY

(n7r) 2 Wn = a ao

Substituting for ê from Eqs. (4.2.45) and (4.2.46a) into Eq. (4.2.48a), we obtain

(4.2.51)

( 4.2.52)

C2 = C3 = C4 = 0

c1 sin êa = O, which implies ê = nst a


Fora simply supported beam, the boundary conditions in Eq. (4.2.31b) give

t See Eq, (4.2.46a): W(x) = c1 sin Az + c2cos.êx + c3sinhÕx + c4coshÕx.

tan en = tanh en e,. = 3.927, 7.069, Ŀ Ŀ Ŀ

tan en = tanh en Cn = 3.927, 7.069, Ŀ Ŀ Ŀ

cos Cn cosh en - 1 = O Cn = 4.730, 7.853, Ŀ Ŀ Ŀ

cos en cosh en + 1 = O Cn = 1.875, 4.694, Ŀ Ŀ Ŀ

cos en cosh en - 1 = O Cn = 4.730, 7.853, Ŀ Ŀ Ŀ

c1 = 1 / sin en, c3 = -1 / sinh en C2 = C4 = 0

Å Hinged-Free

.Jh

c1 = 1 / sin Cn, C3 = 1 / sinh en C2 = C4 = 0

Å Hinged-Fixed

c1=c3=1/(sinen - s³nh e.) C2 = C4 = -1 / ( COS Cn - cosh Cn)

ÅFree-Free

ci = -c3 = 1/(sin Cn + sinh en) -c2 = c4 = 1/(cosen + cosh e.,')

Å Fixed-Free

c1=-c3=1/(sinen - sinh c.i ) -c2 = c4 = l/(cosen - cosh e.,')

Å Fixed- Fixed

sin e., =O Å Hinged-Hinged C1 i- o, C2 = C3 = C4 =o

Characteristic equation and values of en = êna Constantst End conditions at

x =O and x =a

Table 4.2.3: Values of the constants and eigenvalues for natural vibration of laminated composite beams with various boundary conditions (>.; = w~lo/ E~xlyy = (en/a)4). The classica! laminate theory without rotary inertia is used.


Maximum transverse deflections, critica! buckling loads, and fondamenta! natural frequencies of various laminated beams, according to the classica! beam theory, are presented in Table 4.2.4 for simply supported (hinged-hinged), clamped (fixed-fixed), and cantilever (ciamped-free) boundary conditions. In the case of bending, the point load is F0b, where F0 is the line load across the width of the beam (force/unit length), and the distributed line load along the length is qob, where qo is the intensity of the distributed load (force/unit square area). In Table 4.2.4, the first row corresponds to deftections due to point load F0, and the second row corresponds to deflcctions due to uniformly distributed load qo. Also, on the second and third rows, frequencies corresponding to a/ h = 100

( 4.2.60)

Equation ( 4.2.59) is satisfied for the following values of ê:

( 4.2.59) cos Xo cosh Ac - 1 =O

( 4.2.58) - 2 + 2 cos êa cosh Õa + ( ~ - ~) sin êa sinh Õa = O The solution of this nonlinear equation gives .ê and Õ. Then the natural frequency of vibration can be calculated from Eq. (4.2.48a) or (4.2.48b); if the applied axial farce is zero, Eq. (4.2.49) can be used to calculate the frequency of vibration. For natural vibration without rotatory inertia and applied in-piane farce (i.e., q = O in Eq.

(4.2.46b) and ê = Õ), Eq. (4.2.58) takes the simpler form

where relations (4.2.56) are used to eliminate c3 and c4. For nonzero c1 and c2, we require the determinant of the coefficient matrix of the above equations to vanish, which yields the characteristic polynomial

( 4.2.57) cos êa - cosh Õa ] { ci } { O } - sin êa - (X) sinh Õa c2 - O [

sin êa - ( ~) sinh Õa

cos >.a - cosh Õa

and the eigenvalue problem

( 4.2.56)


Fora beam clamped at both ends, the boundary conditions in Eq. (4.2.36) lead to

( 4.2.55)

Thus, rotatory inertia decreases frequencies of natural vibration. If the rotatory inertia is neglected, we obtain

(4.2.54) = (n:f JE~f:YY

1 ( n7r) 2 Wn = a ao

which are natural frequencies of a stretched laminateci cable. We also note from Eq. (4.2.53b) that frequencies of natural vibration decrease when a compressive force instead of a tensile force is acting on the beam. When IČfxx =O, we obtain from Eq. (4.2.53a)


Hinged-Hinged Clamped-Clamped Clamped-Free

Laminate w N w w R w iiJ R w

o 1.000 20.562 14.246 0.250 82.247 32.292 16.000 5.140 5.074 0.625 14.245 0.125 32.291 6.000 5.074

14.187 32.129 5.071

90 25.000 0.822 2.849 6.250 3.290 6.458 400.00 0.205 1.015 15.625 3.125 150.00

(0/90), 1.134 18.127 13.375 0.283 72.507 30.320 18.149 4.532 4.764 0.709 0.142 6.806

(90/0)s 6.239 3.296 5.703 1.560 13.183 12.929 99.821 0.824 2.032 3.899 0.780 37.433

(45/-45)s 14.308 1.437 3.766 3.577 5.748 8.537 228.93 0.359 1.341 8.942 1.788 85.847

Laminate A 1.607 12.790 11.236 0.402 51.162 25.469 25.721 3.197 4.002 1.005 0.201 9.645

Laminate B 2.801 7.341 8.512 0.700 29.366 19.296 44.813 1.835 3.032 1.751 0.350 16.805

Laminate C 7.945 2.588 5.054 1.986 10.351 11.456 127.13 0.647 1.800 4.966 0.993 47.673

Laminate A= (0/Ñ45/90)s, Laminate B = (45/0/-45/90)5, Laminate C = (90/Ñ45/0)s.

Table 4.2.4: Maximum transverse deflections, critica! buckling loads, and fundamental frequencies of laminated beams according to the classica! beam theory (Ei/ E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, V12 = 0.25).

The stiffness in a laminate is largest in the fiber direction because E1 > E2. Also, the bending stiffness increases with ( cube of) the distance of the oc layers from the midplane. Thus, the oc-laminated beam is stiffer in bending than the 9oc-laminated beam, and therefore, 0Á beam has smaller defiection and larger buckling load and natural frequencies when compared to the 90Á beam. Since the oc laminae are placed farther from the midplane in (0/90)s laminate, it has sma:ller defiection and larger buckling load and natural frequencies when compared to the (90/0)s beams. Similarly, due to the placement of the 0Á layers, laminate A is stiffer than laminate B, and laminate Bis stiffer than laminate C. Symmetric angle-ply laminated beams (B/-ĀB)s have the same stiffness characteristics as (-B/B)s, and they are less stiff compared to the symmetric cross-ply laminated beams.

( 4.2.61) N = NÁ a2/E h3 w = w a2 J I /E h3 xx 2 ' 1 o 2

and a/h = 10 are listed when rotary inertia is included. All other frequencies were computed by neglecting the rotary inertia. The following nondimensionalizations are used:


(4.3.4) wo = wo(x, t), <Px = <Px(x, t)

As in Section 4.2, we assume that Myy = Mxy = Qy = </>y =O and both wo and <Px are functions of only x and t:

(4.3.3) A* - A55 A* - A44 A* - A45 , A= A44A,55 - A45A45 44 - A ' 55 - A ' 45 - - A

where K is the shear correction coefficient, Di_j, (i,j = 1, 2, 6) denote the elements of the inverse of [D], and Ai_j, (i,j = 4, 5) denote elements of the inverse of [A]:

( 4.3.2b) { ~ + </>y } = ]._ [ A44 Dwo ,+. K A* 7fX +~x 45

( 4.3.2a)

or, in inverse form, we have

(4.3.lb)

(4.3.la)

Here we consider the bending of symmetrically laminated beams using the first- order shear deformation theory. When applied to beams, FSDT is known as the Timoshenko beam theory. The governing equations can be readily obtained from the results of Section 3.4. The laminate constitutive equations for symmetric laminates, in the absence of

in-plane forces, are given by [see Eqs, (3.4.21) and (3.4.22)]

4.3 Analysis of Laminated Beams U sing FSDT 4.3.1 Governing Equations

We note that for clamped-clamped and clamped-free beams, the calculation of natural frequencies require the solutions of transcendental equations for ê. For the case where rotary inertia is negligible, the roots of these equations are given in Table 4.2.3. To see the effect of rotary inertia, Eq. ( 4.2.58) were solved for ê and the frequencies were calculated. From the frequencies listed in rows 2 and 3 of Table 4.2.4, it is clear that the effect of rotary inertia on fondamenta! frequencies is negligible for small length-to-height ratios. Except for second and third rows, ali other frequencies listed in the table were calculated by neglecting the rotary inertia, in which case the values of >.1 givcn in Tablc 4.2.3 are applicablc.


Note that when the laminateci beam problem is such that the bending moment M(x) and Q(x) can be written readily in terms of known applied loads (like in statically determinate beam problems), Eq. (4.3.7a) can be utilized to determine <Px, and then wo can be determined using Eq. (4.3.7b). When M(x) and Q(x) cannot be expressed in terms of known loads, Eqs. (4.3.9a,b) are used to determine wo(x) and <f>x ( x). In the latter case, the following relations prove to be useful.

4.3.2 Bending

(4.3.9c) q = bq, io= bfo, i, = bh where

(4.3.9a, b)

Using Eq. (4.3.7) in Eq. (4.3.8), the equations of motion can be recast in terms of the displacement functions:

( 4.3.8b) 8Mxx _ Q _ I 82</>x ax X - 2 8t2

( 4.3.8a)

(4.3.7b) KG~zbh ( 88:0 + <Px) = Q(x), Q(x) = ia; G~z = A:5h

The equations of motion from Eq. (3.4.13) are

( 4.3. 7a) E~xfyy 8:: = M(x), M(x) = bMxx, E~x = D~2h3 11

or

(4.3.6) 8wo .+. _ A55Q ax +'!'X - K X

From Eqs. (4.3.2a,b) we have

(4.3.5b)

and the linear strain-displacement relations give

( 4.3.5a) u(x, z) = z<Px(x), w(x, z) = wo(x)

From Eq. (3.4.1) the displacement field takes the form (when the in-plane displacements uo and vo are zero)


wg(x) = Eg~Iyy [lox lo lo'l loÕ q((,)d(,dÕd71d®, - cr ~3 - c2 ~2 - C3X - c4]

wi)(x) = KG~zbh [- lox lo q(()d(,d®,+c1x] (4.3.14b)

where

(4.3.14a) wo(x) = wS(x) + w0(x)

where the constants of integration c1 through c4 can be determined using the boundary conditions of the beam. It is informative to note from Eq. (4.3.13) that the transverse deflection of the

Timoshenko beam theory consists of two parts, one due to pure bending and the other due to transverse shear:

wo(x) = - E~~Iyy [- lox lo lo'l loÕ q((,)d(,dÕd71d®, + c1 ~3 + c2 ~2 + c3x + c4]

+ KG~zbh [- lox lo q((,)d(,d®,+c1x] (4.3.13b)

Eg:,Iyy [- fox lo( fo'l q(E,)d®,d71d(, + ci ~2 + c2x + c3]

+ KG~zbh [- lox q(E,)d®,+c1] (4.3.13a)

dwo dx

(4.3.12b) E~xiyycPx(x) = - l" {( I" q(E,)d®,d71d(, + C1 x2 + C2X + C3 lo lo lo 2

Substituting for c/J(x) from Eq. (4.3.12b) into Eq, (4.3.11), we arrive at

(4.3.12a)

Substituting the result into Eq. (4.3.lOb) and integrating with respect to x, we obtain

(4.3.11) b (dwo ) r . KGxzbh dx + cPx = - lo q(E,)d®, + C1

Integrating Eq. (4.3.lOa) with respect to x, we obtain

(4.3.lOa, b)

For bending analysis, Eqs. ( 4.3.9a,b) reduce to


( 4.3. lSa) F0ba2 [i-4(::)2] + F0b 16Egxfyy a 2KGLbh

dw0 dx

It is interesting to note frorn Eq. (4.3.17) that the rotation function <Px(x) is the same as the slope -dwo/dx from the Euler-Bernoulli beam theory (i.e., <Px is independent of transverse shear stiffness). Consequently, the bending moment [see Eq. (4.3.7a)], and therefore the axial stress, is independent of shear deformation. In fact, <Px is independent of shear deformation for ali statically determinate bearns and indeterminate beams with symmetric boundary conditions and loading (see Wang [27]). However, for generai statically indeterminate beams, the rotation <Px will depend on the shear stiffness KG~zbh (see Problem 4.11). Substituting for <Px into Eq. (4.3.7b), we obtain

(4.3.17) [ (

'f )2] a 1-4 :..._ O<x<- a ' - - 2 F0ba2

<Px(x) = -16Eb I xx YY

and the solution becomes

By symmetry, u1 = uo + z</Jx is zero at x = a/2. This implies that <Px(a/2) =O. Hence

( ) Fob 2 </Jx X = 4Eb J X + C1 xx yy

Using Eq. (4.3.16) for M in Eq. (4.3.7a) and integrating with respect to x, we obtain

(4.3.16) M( ) = F0bx Q( ) = dM = Fob O < x < ~ X 2' X dx 2' - -2


Here we consider the three-point bending problem of Section 4.2 (see Figure 4.2.2). For this case, the bending moment [see Eq. (4.2.17)] and shear forces are

( 4.3.15) u<k)(x z) = Q-(k) A* Q(x) xz ' 55 55 b

The pure bending deflection w8 ( x) is the same as that derived in the classica! beam theory [cf., Eq. (4.2.llb)]. When the transverse shear stiffness is infinite, the shear deflection wg(x) goes to zero, and the Timoshenko beam theory solutions reduce to those of the classica! beam theory. In fact, one can establish exact relationships between the solutions of the Euler-Bernoulli beam solutions and Timoshenko beam solutions (see [27-29]). These relationships enable one to obtain the Timoshenko beam solutions from known classica! beam solutions for any set of boundary conditions (see Problems 4.33 and 4.36). The expressions for in-plane stresses of the Timoshenko beam theory remain the

same as those in the classica! beam theory [see Eq. (4.2.12b)]. The expressions given in Eqs. (4.2.15a,b) for transverse shear stresses derived from 3-D equilibrium are also valid for the present case. The transverse shear stress can also be computed via constitutive equation in

the Timoshenko beam theory. We have


( 4.:3.24)

which in turn irnply that

( 4.3.2:3) ( ) (a ) ( ) Q .. (~ ) = (Fob qoba) _ qoba = Fob U 0, Z = 0, U 2 > Z = 0, W o, Z =o, X 2 > Z 2 + 2 2 2

Consider a larninated beam fixed at both ends and subjected to uniformly distributed transverse load q0b as well as a point load F0b at the centcr, both acting downward. For this case, the boundary conditions are ( using half beam)

Example 4.3.2 (Clamped bearn):

Equation ( 4.3.22) shows that the effect of shear deforrnation is to increase the deflection. The contribution due to shear deformation to the deflection depends on the rnodulus ratio E~x/G~2 as well as the ratio of thickness to length h/ a. The effect of shear deformation is negligible for thin and long beams.

( 4.3.22)

where the constant of integration is found to be zero on account of the boundary condition w0(0) = O. Note that the first part (wg) is the sarne as that obtained in the classica! beam theory [cf., Eq. (4.2.18)]. The maximum deflection occurs at x = a/2 and it is given by

( 4.3.21) F0ba3

wo(x) = 48Eb I xx yy

However, dw8/dx = -4'x is zero at x = a/2. Integrating Eq. (4.3.18a) with respcct to x, we arrivo at the expression

( 4.3.20) dw0 a Fob dx (2) = 2KGLbh

Note from Eq. (4.3.18a) that, in contrast to the classica! beam theory, the slope dwo/dx at the center of the beam in the Timoshenko beam theory is nonzero. We have (Iyy = bh3 /12)

(4.3.19) dw0 _ dw0 dw3 _ dw0 ( ) _ dx - dx - dx - dx + 4'x X = /xz

Indeed, dw0/dx can be interpreted as the transverse shear strain [cf., Eq. (4.3.5b)]

( 4.3.18c) dw0 _ dwg dwiČ ---+- da: - dx dx

In light of Eq. ( 4.3.14a), the first part of Eq. ( 4.3.18a) can be viewed as the slope (or rotation) due to bending and the second one due to transverse shear strain:

(4.3.18b) dw8 dx

Let us denote the first expression in ( 4.3.18a) by


(4.3.28b)

( 4.3.28a)

For buckling analysis, the inertia terms and the applied transverse load q in Eqs. (4.3.9a,b) are set to zero to obtain the governing equations of buckling under compressive edge load ç: = -Ngx:

4.3.3 Buckling

Table 4.3.2 contains maximum transverse defiections w of various laminated beams according to the Timoshenko shear deformation beam theory. The effect of length-to-height (or thickness) ratios of the beam on the defiections can be seen from the results. Thin or long beams do not experience transverse shear strains. Clamped beams show the most difference in defiections due to transverse shear deformation (i.e., accounting for the transverse shear strain). The effect of shear deformation on maximum defiection can be seen from Figures 4.3.1 and 4.3.2, where the nondimensionalized maximum defiection, w = WmçxE2h3 /qoa4 (Fo = qoa), of a simply supported beam is plotted as a function of length-to-height ratio a/h for various laminated beams under a point load and uniformly distributed load, respectively. The materiai properties of a lamina are taken to be those in Eq. (4.2.25). The effect of shear deformation is more significant for beams with length-to-thickness ratios smaller than 10.

( 4.3.27b) S=4( E~~) (0.)2 KGxz a

Table 4.3.1 contains expressions for transverse defiections and maximum transverse defiections of laminated beams according to the first-order shear deformation theory. By comparison to the classica! theory (see Table 4.2.1), it is clear that the shear deformation increases the defiection.

where S is the positive parameter that characterizes the contribution due to the transverse shear strain to the displacement field

( 4.3.27a)

Qoba4 Qoa2 F0ba3 Foa Wrnax = 384Egxfyy + BKGLh + 192Egxfyy + 4KG~zh

[ Qoba4 Foba3 ]

= 384Eb I + 192Eb I (l + S) xx YY xx YY

The maximum defiection is at x = a/2 and is given by [cf., Eq. (4.2.23)]

(4.3.26)

( 4.3.25)

The solution is


at x =a

Uniform load ~ [6 (~)2 - 4 (;)3 + (~)4] + '?- [ ( 2 ~) - ( ~) 2] alllllllf0

I" a ÅI

Point load at free end lr [3 (~)2 - (;)3] +s1 (~)

Å Fixed-Free

at X=~

U niform load ~i [ ( ~) 2 - ( ~)] 2 +q- [(;) - (~)2] Jtttttttf0

'"' a ÅI

1.. a

Jkc1 +!-si at X=~

;m [3 (;)2 - 4 (~)3] +"Ñ(;)

Å Fixed-Fixed

at X=~

U niform load ~~ [ (;) - 2 (;) 3 + (;) 4] +q- [(~) - (~)2]

a

-fsc1+:!s1 at X=~

~ [3(~)-4(~):l] +-g.(;)

Centrai point load

tFo

Å Hinged-Hinged

Max. Deflection Deflection, wo(x) Laminateci Beam

Foba3 qoba4 Foba qoba2 c1 = Eb I , c2 = Eb I , s1 = KGh bh ' s2 = Gb bh xx yy xx yy xz xz

Table 4.3.1: Transverse defiections of laminateci composite beams with various boundary conditions and subjected to point load or uniformly distributed load (acting downward) according to the shear deformation theory.


( 4.3.33) ê2 = bN~x O ).2 E~xlyy

(1 bN2x ) Eb 1 or bNXX = (1 + >,2 E~xlyy) - KGLbh XX YY KG~zbh

and c1 through c4 are constants of integration, which must be evaluated using the boundary conditions.

where

( 4.3.32)

(4.3.31) b ( bN~x ) d4W 0 d2W Exxlyy 1 - KGLbh dx4 + bNxx dx2 =O

The general solution of Eq. (4.3.31) is W(x) = c1 sin êx + c2 cos Xz + c3x + C4

or

N ext differentiate Eq. ( 4.3.28b) with respect to x and substitute for dX /dx from Eq. ( 4.3.29) to obtain the result

( 4.3.30)

( 4.3.29)

Solving Eq. ( 4.3.28a) for dX /dx one obtains

b dX ( b 0) d2W KGxzbh dx = - KGxzbh - bNxx dx2 Integration with respect to x yields

b ( b o)dW KGxzbhX(x) = - KGxzbh - bNxx dx + K1

tThe first row of each laminate refers to nondimensionalized maximum defiections under point load (Fob) and the second one refers to maximum defiections under uniformly distributed load (qob). The deflection is nondimensionalized as w = W-max(E2h3 /qoa4) x 102 (Fo = qoa).


Table 4.3.2: Maximum transverse deflections of laminated beams according to the Timoshenko beam theory t (Ei/ E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, v12 = 0.25).


Laminate a 100 20 10 100 20 10 100 20 10 t:

o 1.001 1.150 1.600 0.256 0.400 0.850 16.02 16.60 18.40 0.628 0.700 0.925 0.128 0.200 0.425 6.01 6.30 7.20

90 25.015 25.375 26.500 6.265 6.625 7.750 400.00 401.50 406.00 15.633 15.813 16.375 3.132 3.312 3.875 150.00 150.75 153.00

(90/0)s 1.143 1.348 1.991 0.292 0.498 1.141 18.18 19.01 21.58 0.713 0.816 1.137 0.146 0.249 0.570 6.82 7.23 8.52

(45/ ~ 45)s 14.316 14.522 15.165 3.585 3.791 4.434 228.96 229.78 232.35 8.947 9.049 9.371 1.793 1.895 2.217 85.86 86.28 87.56

Figure 4.3.2: Transverse defiection ( w) versus length-to-thickness ratio (a/ h) of simply supported beams under uniformly distributed load.

10 20 30 40 50 60 70 80 90 100 Side-to-thickness ratio, a/h

0.12

0.11

0.10

0.09

I;: 0.08 i::Ŀ 0.07 .s ..., 0.06 u

Ili e:;:: Ili 0.05 Q 0.04

0.03

0.02

0.01

0.00 o

Figure 4.3.1: Transverse deflection ('w) versus length-to-thickness ratio (ajh) of simply supported beams under centcr point load.

0.20

0.18 _ Eoh1 w = w0(a/2)---1

0.16 F0aĀ

(45/-45)8 0.14

I;: ČFo :::::Ŀ 0.12 .s -Jf et- ..., 0.10 u a Ili e:;:: (90/-45/45/0)8 Ili

Q 0.08

0.06

0.04 (0/-45/45/90)8 (0/90)8

0.02

0.00 o 10 20 30 40 50 60 70 80 90 100

Side-to-thickness ratio, a/h


( 4.3.39) ( .>,2 Eb I )

2(cos.>.a-1) 1+ Kc1:b~ +>.a sin.>.a=O

and then setting the determinant of the resulting algebraic equations among c1 and c2 to zero, we obtain

1 l bN~x KGb bh xz

c2 + c4 = O, c1 sin .>.a+ c2 cos >.a+ c3a + c4 = O

(1 bN~x ) ' O - - KGLbh /\CJ - C3 =

- ( 1 - Kb~:xbh) ( .>.c1 cos >.a - .>.c2 sin .>.a) - c3 = O

Expressing c1 and c2 in terms of c3 and c4, noting that

In order to impose the boundary conditions on X, we use Eq. (4.3.30). The constant K1 appearing in Eq. (4.3.30) can be shown (see Problem 4.10) to be equa) to K1 = -c3(bN~x)Ā The boundary conditions yield

( 4.3.38) W(O) =O, W(a) =O, X(O) =O, X(a) =O

For a beam fixed at both ends, the boundary conditions are

Example 4.3.4 ( Clamped beam):

It is clear from the result in Eq. ( 4.3.37) that shear deformation has the effect of decreasing the buckling load [cf., Eq, (4.2.35)].

(4.3.37) 2 [ Eb I (" )2 l bN = e: I ('!!_) 1 - xx YY a cr xx YY 2 a KGLbh + Efxfyy ( ~)

The criticai buckling load is given by the minimum (n = 1)

(4.3.36)

( 4.3.35) sin >.a = O implies >.a = mr Substituting for .>. from Eq. ( 4.3.35) into Eq. ( 4.3.33) for N~x, we obtain

The boundary conditions in Eq. (4.3.34b) lead to the result c2 = c3 = c4 =O, and for c1 i= O the requirement

(4.3.34b) d2W d2W W(O) =O, W(a) =O, dx2 (O)= O, dx2 (a)= O

In view of Eq. (4.3.29), the above conditions are equivalent to

( 4.3.34a) dX dX W(O) =O, W(a) =O, dx (O)= O, dx (a)= O

Fora simply supported beam, the boundary conditions are [see Eq. (4.2.31a)]



(4.3.44)

and ci , c2, c3, and C4 are constants, which are to be determined using the boundary conditions. Note that we have

( 4.3.43b) x = J 2~ ( q + J q2 + 4pr) , Õ = J 2~ ( -q + J q2 + 4pr) where

(4.3.43a) W ( x) = c1 sin >.x + c2 cos >.x + c3 sinh Õx + c4 cosh Õx

The general solution of Eq. (4.3.42b) is

( 4.3.42c) ( b ') ( 2' ) b Exxfyy Iz ' 2 W Iz ' 2

p = Exxlyy, q = KGb bh +-;e-- Iow , r = 1 - KGbxzbh Iow xz Io

where

(4.3.42b) d4W d2W p-- +q-- -rW =O dx4 dx2

or

4 ( b A ) 2 ( 2A ) b d W Exxlyylo - 2 d W w Jz - 2 Exxlyy dx4 + KGLbh + Jz w dx2 - 1 - KG~zbh Iow W =O (4.3.42a)

Substitute the above result into the derivative of Eq. ( 4.3.40b) for dX /dx and obtain the result

( 4.3.41) b dX A 2 b d2W KGx2bh- = -Low W - KG bh-- dx xz dx2

We use the same procedure as before to eliminate X from Eqs. (4.3.40a,b). From Eq. (4.3.40a), we have

(4.3.40b)

( 4.3.40a)

For natural vibration, we assume that the applied axial force and transverse load are zero and that the motion is periodic. Equations (4.3.9a,b) take the form

4.3.4 Vibration

Once the value of .êa is determined by solving the nonlincar equation (4.3.39), the lmckling load can be readily determined from Eq. (4.3.33).


(4.3.51) -2 + 2 cos .\a cosh Õa + sin .\a sinh Õa ( ~~~ - ~~~ ) = O

Eliminating c2 and c4 from the above equations, and setting the determinant of the resulting equations arnong c1 and c2 to zero (fora nontrivial solution), we obtain

(4.3.50b) where

( 4.3.50a)

c2 + c4 = O, ci sin .\a + c2 cos .\a + c3 sinh Õa + c4 cosh Õa = O

Using Eq. (4.3.40a) and expression (4.3.43a) for W(x), dX/dx can be determined in terms of the constants c1 through c4, which then can be integrated with respect to x to obtain an expression for X. Using the boundary conditions in Eq. (4.3.38), we obtain

Example 4.3.6 ( Clamped beam):

Thus, shear deformation decreases the frequencies of natural vibration [see Eq. (4.2.55)].

( 4.3.49) (mr)2Eb J l _ a xx YY

KGLbh + ( '~7f )2 E~xIYY - (nn:)2 E~xIYY Wn -

a Čo

Substitution of x from Eq. ( 4.3.48) into Eq. ( 4.3.47) and the result into Eq. ( 4.3.46a,b) gives two frequencies for each value of .\. The fondamenta! frequency will come from Eq. (4.3.46a). When the rotary inertia is neglected, we obtain from Eq. (4.3.47) the result

(4.3.48) c1 sin .\a = O, which implies >.,, = mr a


Fora simply supported bearn, the boundary conditions in Eq. (4.3.34b) yield c2 = c3 = c4 =O and

(4.3.47) W2 = R, Q- = [l + ( E~xiyy ) ).2] R = (E~xiyy) ).4 Q ' K Gb bh ' i xz o

It can be shown that Q2 - 4PR > O (and PQ > O), and therefore the frequency given by the first equation is the smaller of the two values. When the rotary inertia is neglected, we have P =O and the frequency is given by

( 4.3.46)

Hence, there are two (sets of) roots of this equation (when Č2 i- O)

p = d~ ' Q = [1 + ( E~/yy + ~2) ).2], R = (E~:x/YY) ).4 (4.3.45b) K xzbh K xzbh Io Io

where

( 4.3.45a) Pw4 - Qw2 + R = O

Alternatively, Eq. ( 4.3.42a) can be written, with W given by Eq. ( 4.3.43), in terms of w as


The frequency equations ( 4.3.51) of the Timoshenko theory depend, for clamped- clamped and clamped-free boundary conditions, on the lamination scheme and geometrie parameters ( through Sij), whereas those of the classical laminate theory [see Eqs. (4.2.58) and (4.2.59)] are independent of the beam geometry or material properties. Thus, there are two different things that infiuence the frequencies in the Timoshenko theory: (i) the effect of transversc shear deformation [see Eqs. ( 4.3.47) and (4.3.49)], and (ii) the values of >.., which are governed by different equations than those of the classical theory (for clamped-clamped and clamped-free beams). The second effect is not significant, as can be seen from rows 3 and 5 of Table 4.3.3. Also, for clamped-clamped and clamped-free boundary conditions, the effect of rotary inertia on the frequencies is not as obvious as it was in the case of simply supported beams, where the rotary inertia would decrease the frequencies. From the results presented in Table 4.3.3, it appears that rotary inertia may actually increase the frequencies slightly. The effect of length-to-height (or thickness) ratios of the beam on critical

buckling loads N and fundamental frequencies w is shown in Figures 4.3.3 and 4.3.4, respectively, for various lamination schemes. The material properties used are those listed in Eq. (4.2.25). Transverse shear deformation has the effect of decreasing both buckling loads and natural frequencies. Thus, the classical laminate theory overpredicts buckling loads and natural frequencies. This is primarily due to the assumed infinite rigidity of the transverse normals in the classica! laminate theory. Note that the assumption does not yield a conservative result; i.e., if one designs a beam for buckling load based on the classica! laminate theory and if no safety factor is used, it will fail for a working load smaller than the critical buckling load. Once again we note that the relationships between the classica! beam theory

and the Timoshenko beam theory may be used determine the defiections, buckling loads and fundamental frequencies according to the Timoshenko beam theory from those of the Euler-Bernoulli beam theory [29]. Such relationships exist only for isotropie beams, and the reader may find it challenging to develop the relationships for bending, buckling and vibration of laminated beams (see Section 5.5 of [29]).

( 4.3.52)

Table 4.3.3 contains critical buckling loads and fundamental frequencies of various laminated beams according to the Timoshenko beam theory. The first row of each laminate refers to the nondimensionalized critical buckling load, the second row refers to nondimensionalized fundamental frequencies with rotary inertia, and the fourth row refers to fundamental frequencies without rotary inertia. The numbers in rows 3 and 5 refer to the fundamental frequencies calculated using the frequency equations of the classica! laminate theory (for the simply supported boundary conditions, the frequency equations are the same in both theories). The following nondimensionalizations are used:


(4.4.lb)

(4.4.la) ®Puo ®Pvo 83wo aN'fx 82u0 83w0 An ax2 + A16 8x2 - Bu 8x3 - ----a;;- =Io 8t2 - li 8xat2

82uo 82vo 83wo aN'[y 82vo A15 8x2 + A66 ax2 - B15 ox3 - ----a;;- = Io [)t2

Consider a laminated rectangular plate strip, and let the x and y coordinates be parallel to the edges of the strip. Suppose that the plate is long in the y-direction and has a finite dimension along the x-direction, and subjected to a transverse load q(x) that is uniform at any section parallel to the x-axis. In such a case, the deflection wo and displacements ( uo, vo) of the plate are functions of only x. Therefore, all derivatives with respect to y are zero, and the plate bends into a cylindrical surface. For this cylindrical bending problem (see Figure 4.1.2), the governing equations of motion according to the linear classica! laminate plate theory (CLPT) are given by [see Example 3.3.1; Eqs. (3.3.48)]

4.4.1 Governing Equations

4.4 Cylindrical Bending Using CLPT


Table 4.3.3: Critical buckling loads (N) and fundamental frequencies (w) of laminated beams according to the Timoshenko beam theory (Ei/ E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, V12 = 0.25).


Laminate a --t 100 20 10 100 20 10 100 20 10 t:

o N 20.461 18.304 13.768 80.655 55.070 27.656 5.134 4.987 4.576

w(iz t= O) 14.210 13.430 11.635 31.899 25.327 17.212 5.070 4.930 4.528 14.210 13.430 11.635 32.110 28.506 22.140 5.070 4.965 4.675

w(iz =O) 14.211 13.441 11.657 31.824 24.636 16.680 5.063 4.813 4.229 14.211 13.441 11.657 32.113 28.547 22.186 5.070 4.966 4.680

90 0.822 0.812 0.784 3.283 3.135 2.747 0.205 0.205 0.203

2.848 2.829 2.771 6.450 6.260 5.761 1.015 1.012 1.004 2.848 2.829 2.771 6.454 6.356 6.079 1.015 1.012 1.005

2.848 2.832 2.781 6.449 6.232 5.681 1.015 1.009 0.993 2.848 2.832 2.781 6.455 6.370 6.125 1.015 1.013 1.006

(90/0)s 18.015 15.689 11.179 70.748 44.716 20.800 4.525 4.362 3.922

13.334 12.434 10.488 29.857 22.672 14.837 4.758 4.594 4.132 13.334 12.434 10.488 30.106 26.041 19.504 4.759 4.636 4.307

( 45/-45)s 1.436 1.419 1.369 5.737 5.478 4.802 0.359 0.358 0.355

3.765 3.739 3.663 8.526 8.275 7.616 1.341 1.338 1.326 3.765 3.739 3.663 8.531 8.402 8.036 1.341 1.338 1.328

o 10 20 30 40 50 60 70 80 90 100 Side-to-thickness ratio, alh

Figure 4.3.4: Nondimensionalized fondamenta! frequency (w) versus length-to- thickness ratio (a/h) of simply supported beams.

<, (45/-45)8

(90/-45/45/0)8

(0/-45/45/90)8

<, (0/90)8

Orthotropic

16

14

12

18 >; 10 u i:: ai ::s 8 O' ai r... ~ 6

4

2

Figure 4.3.3: Nondimensionalized criticai buckling load (N) versus length-to- thickness ratio (a/h) of simply supported beams.

24

N = N~x(a2 I E1h3) /_ Orthotropic

20 /(0/90)8

I<; 16 "Ci' (0/-45/45/90)8 CIS o ~ 12 bi) i:: .,.., - ~ o ç: 2f ?!F No ::s 8 o:::i xx

a

4 (45/-45)8 è: (90/-45/45/0)8

o o 10 20 30 40 50 60 70 80 90 100



Note that e= o fora cross-ply laminate (A15 = Bl6 = Dl6 =O), and Vis identically zero unless N'[y is at least a linear function of x. If the in-plane inertias are neglected, Eq. ( 4.4. 2c) for wo is uncoupled from those

of uo and vo. In the absence of thermal forces and axial loads, Eq. (4.4.2c) will have the same form as Eq. (4.2.8b). Therefore, the solutions developed in Sections 4.2.2 through 4.2.4 are also valid for cylindrical bending with appropriate change of the coefficients.

( 4.4.2d) - - - B - C D = Dn - BuB - B15C, B = A , C = A

where

For a general lamination scheme, the three equations are fully coupled. In the case of cross-ply laminates, the second equation becomes uncoupled from the rest. In the general case, Eqs. (4.4.la-c) can be expressed in an alternative form by solving the first two equations for u" and v" and substituting the results into the third equation

(4.4.ld) L t':: 2 (k)

(Io, Ii, h) = ~ l., (1, z, z )p0 dz

where ç: is an applied axial load, and


( 4.4.6b)

(4.4.6a)

(4.4.5)

( 4.4.4c)

( 4.4.4b)

( 4.4.4a)

(4.4.3c)

( 4.4.3b)

( 4.4.3a)

Further integrations lead to

- B B=- D - e C=- D

BB BB G1 = D + A55, F; = D - A15,

BC BC G2 = D - A15, F2 = D + Au,

a1 = Bc1, b1 = ®c1

where

For static bending analysis, Eqs. ( 4.4.2a-c) reduce to

d2 d3 dNT dNT A~ = B ~ + A55 _______EE_ - A15 ~ dx2 dx3 dx dx d2 d3 dNT dNT A~ - C~ A ~ - A _______EE_ dx2 - dx3 + 11 dx 16 dx

D d4wo B d2 N'fx C d2 è; d2 M'fx dx4 = dx2 + ~ - dx2 + q

4.4.2 Bending

Equation ( 4.4.3c) governing wo is uncoupled from those governing ( uo, vo). Equation ( 4.4.3c) closely resernbles that for symmetrically laminated beams [see Eq. (4.2.lOb)]. While Eq. (4.4.3c) is valid for more general laminates (symmetric as well as nonsymmetric), it differs from Eq. (4.2.lOb) mainly in the bending stiffness term. Hence, much of the discussion presented in Section 4.2 on exact solutions applies to Eq. ( 4.4.3c). The limitation on the lamination scheme in cylindrical bending comes from the boundary conditions on all three displacements of the problem. When both edges are simply supported or clamped, exact solutions can be developed without any restrictions on the lamination scheme. For clamped-free laminated plate strips, satisfaction of the boundary conditions places a restriction on the lamination scheme, as will be seen shortly. Since Eq. ( 4.4.3c) is uncoupled from Eqs. ( 4.4.3a, b), it can be integrated, for

given thermal and mechanical loads, to obtain wo(x), and the result can be used in Eqs. (4.4.3a) and (4.4.3b) to determine uo(x) and vo(x):

d3 dNT dNT dMT lx D d~o = B dxx +e d xy - dxx + q() df, + C1

X" X X X O

A d2uo - B t" (C) dC G dNJx F dN'{y BdA{fx dx2 - lo q '> "' + 1 ~ + 1 ~ - --;;;;--- + a1 d2 1Āx dNT dNT dMT A~= ® q(f,) di;+ G2_______EE_ + F2~ - ®_____E!_ +bi dx2 o dx dx dx


(4.4.9c)

(4.4.9b)

( 4.4.9a)

where To and T1 are constants, then we have

(4.4.8) bi.T(x, z) = To + zT1

If the temperature distribution in the laminate is of the form

(4.4.7d)

Dwo(x) = fox { fo [117 (lo( q(Õ)dÕ) d(] d77} d + B fox (fo NJ'x(ry)dry) d

+ C fax (fa N~(ry)d77) d - fax (fa MJ'x(ry)dry) d

( 4.4. 7c)

and

(4.4.6c)


( 4.4.13b)

(4.4.13a) where

(4.4.12)

Fora plate strip with simply supported edges at x =O and x =a, the boundary conditions are (see Table 4.4.1)

Example 4.4.1 (Simply supported plate strip):

The constants of integration ai, bi, and Ci can be determined using the boundary conditions. The in-plane stresses in each layer can be computed using the constitutive

equations, and the transverse stresses can be determined using equilibrium equations of 3-D elasticity [see Eqs. (4.2.13) and (4.2.14)]. Fora cross-ply laminate the only nonzero strain is ®xxĀ

3

Auo(x) = Ĉqo ~ + G1 (AfTo + B[T1) x + F1 ( A[To + B[T1) x 2 -( T T) X - B B1 To + D1 T1 x + a1 2 + a2x + a3

- x3 x2 = Bqo5 + a12 + ii2x + a3 (4.4.lla) 3

Avo(x) = ďqo ~ + G2 ( AfTo + B[T1) x + F2 ( A[To + BT,T1) x 2 -( T T) X - C B1 To + D1 T1 x + b1 2 + b2x + b3

- x3 x2 - = Cqo6 + b1 2 + b2x + b3 (4.4.llb)

( ) x4 - ( T T ) x2 - ( T T ) x2 Dwo x = qo24 + B A1 To + B1 T1 2 + C A6 To + B6 T1 2 x2 x3 x2

- ( B[To + D[T1) 2 + ci 6 + c22 + c3x + c4 x4 x3 x2 = Qo24 + C1 6 + c22 + C3X + C4 (4.4.llc)

In addition, if q = qo, expressions in Eqs. ( 4.4. 7) become

(4.4.10)

where


(4.4.17e)

(4.4.17d)

(4.4.l 7c) Mxx= qo2a2 [ Gf - G)] Mxy = ~~a~ (B15B + B55C - D15A) [ (~)

2 - (~)]

Å T Å T T + B15Nxx - D15Mxx - Mxy

u.; = ~~a~ (E12E + E25C - D12A) [ G) 2 - ( D]

Å T Å T T + B12Nxx - D12Mxx - MYY

(4.4.l 7b)

( 4.4. l 7a)

where the constants a3 and b3 can be interpreted as rigid body displaccrnents. The constants can be determined by setting uo(O) =O and v0(0) =O, which give a3 = b3 =O. The stress resultants for any x are then given by substituting Eqs. (4.4.16) into Eqs. (4.4.13)

and (4.4.14):

( 4.4.16c)

(4.4.16b)

( 4.4.16a) B qoa3 [ (X) 3 (X) 2] . T 1io(x) = AD 12 2 a - 3 a + Nxx x + a3

C q0a3 [ (x)3 (x)2] Va (X) = AD 12 2 a - 3 a + b3 q0a4 [(x)4 (x)3 (x)] wa(x) = 24D a - 2 a + a + M~a2 [ (~)2 _ G)]

Since only the derivatives of u0 and v0 are specified at the boundary points, the solution for u0 and va can be determined only with an arbitrary constant (i.e., rigid body motion is not eliminated). Using boundary conditions (4.4.15) in Eq. (4.4.lla-c), we obtain

( 4.4.15b)

( 4.4.15a) dvo =O dx '

duo Å T WQ =O, --;[;;: = Nxx>

From Eqs, (4.4.12), (4.4.13a), and (4.4.14a) it follows that, for an arbitrary lamination scheme and dv0/dx =O, we must bave at x = O,a

( 4.4.14c) du0 dv0 d2w0 T Nfxy = B15-d + B55-d - D15-2- - Mxy

X X dx

(4.4.14b)

(4.4.14a)

( 4.4.13c)


Analytical solutions far beams under uniform transverse load with other boundary conditions may be obtained from Eqs. ( 4.4. lla-c). For loads other than uniformly distributed transverse load, one must use Eqs. (4.4.7a-d).

whereas the bending stiffness used in the bearn theory is E~xlyy = E~,,,bh3 /12. Thus, the difference is in the expression containing Poissons ratios, which is due to the piane strain assumption used in cylindrical bending compared to the piane stress assumption uscd in the bearn theory. Thc difference bctween the two solutions will be the most for laminates containing angle-ply layers, whcre v~Y can be very large.

( 4.4.23)

Note that when the bending-stretching coupling tcrms are zero (e.g., for syrnmetric laminates), the cylindrical bending and laminateci bcam solutions have the same form. The difference is only in the bending stiffness terrn. The bending stiffness D11 used in cylindrical bending is given by

( 4.4.22) AČ'J'xa2 ---- 8

5qoa4 (i Bin ) Wmao: = 384D + A D - B2

11 66 11 16

In the case of antisymmetric angle-ply larninates, we have A15 = A25 = B11 = B22 = B12 = B55 = Drn = D2f³ =O, [3 =O, C = B15/A55, and D = Du - Br6/A55. The maximum deflection becomes

(4.4.21) 5 4 ( B2 ) .-fT 2 - - -----2!t!__ 1 1 I - 11Å """ a Wmax - , D + 2 8 384 11 A11Du - B11

It can be shown that the expression Bu B + B15C is always positive. Therefore, it follows that the effect of the coupling is to increasc the maximum transverse deflection of the plate strip. For example, for antisymmetric cross-ply laminates, we have A15 = A25 = B15 = B25 = Drn = D25 =O, B = B11/A11, C =O, and D = D11 - Bii/A11. Thus the maximum deflection becomes

( 4.4.20) 5qua4 AČ'.[xa2 ------- 384D11 8 W.,na::1:

For syrnmetric laminates the coupling tcrms are zero, and the maximum deflection is given by

( 4.4.19) x:r[xa2 ---- 8

Hencc, the maximum deflection can be expressed in the form

~ = _1_ (D11) = _1_ (D + B11B + B15C) D Dn D Dn D

In arder to see thc effect of the bending-stretching coupling on the transverse deflection, the reciproca! of the bending stiffncss D [see Eq. ( 4.4.2d)] is expressed as

( 4.4.18)

The maximum transverse deflection occurs at x = a/2, and it is given by


( 4.4.27)

where (U, V, W) denote the displacements measured from the prebuckling equilibrium state. Equation ( 4.4.26), which is uncoupled from ( 4.4.24) and ( 4.4.25), can be

integrated twice with respect to x to obtain

( 4.4.26)

(4.4.25)

(4.4.24)

The equilibrium of the plate strip under the applied in-piane compressive load s.; = -s: can be obtained from Eqs. (4.4.2a-c) by omitting the inertia terms and thermal resultants

4.4.3 Buckling

Edge Condition CLPT FSDT

zt free Na=O Nxy=O Nxx=O Nxy=O

dMxx -O ~ĿĿ . Ŀ1-- Mxx=O Mxx=O a, =0 X dx -

zt roller wo=O duo -O Wo=O duo =0 dx - dx ,gĿĿĿ .. ĿĿ'"i-- X Nxx=O Mxx=O Nxx=O Mxx=O

zt simple support uo=O Wo=O uo=O Wo=O ~Ā~--x du0 duo -O dx =0 Mxx=O dx - Mxx=O

z t clamped uo=O v0=0 uo=O Vo =0

~ dwo =0 ..... ::: .. ~:...:....:~ .. = --X Wo=O wo=O <l>x =0 dx

Table 4.4.1: Boundary conditions in the classica! (CLPT) and first-order shear deformation (FSDT) theories of beams and plate strips. The boundary conditions on uo and v0 are only for laminated strips in cylindrical bending.


(4.4.36) wo(x, t) = W(x)eiwt, i= H where 12 = h - iu.. Fora periodic motion, we assume

(4.4.35)

For vibration in the absence of in-plane inertias, thermal forces, and transverse load, Eq. (4.4.2c) is reduced to

4.4.4 Vibration

(4.4.34) N = D 4;r2 (1- B11B + B15C) cr 11 a2 D11A

The smallest root of this equation is ê = 2;r, and the criticai buckling load becomes

( 4.4.33) êa sin êa + 2 cos êa - 2 = O equation

Thus the effect of the bending-extensional coupling is to decrease the criticai buckling load. Recali from Section 4.2.3 that when both edges are clamped, ê is determined by solving the

( 4.4.32) N = D ;r2 (1 _ B11B + B16C) cr lla2 DuA

( 4.4.31) sin Ac =sin(/%)= O, or N~x = D (na;r) 2

The criticai buckling load Ne, is given by (n = 1)

Use of the boundary conditions on W gives c2 = c3 = c4 =O and the result

(4.4.30) dU dV d2W W = O, dx = O, dx = O, dx2 = O

When the plate strip is simply supported at x =O, a, from Eq. ( 4.4.15a) we have

Example 4.4.2:

The three of the four constants c1, c2, c3, c4, and ,\ are determined using ( four) boundary conditions of the problem. Once ê is known, the buckling load can be determined using Eq. (4.4.29). The results of Section 4.2.3 are applicable here with b = 1 and E~xlyy = D. Here we consider only the case of simply supported boundary conditions for illustrative purposes.

( 4.4.29) ê 2 = N~x or NÁ = Dê 2 D xx

( 4.4.28) w (X) = C1 sin ,\x + C2 cos ,\x + C3X + C4 where c3 = Ki/ ,\2, c4 = K2/ ,\2, and

where K1 and K2 are constants. The general solution of Eq. ( 4.4.27) is


( 4.4.45)

Note that the rotary inertia has the effect of decreasing the natural frequency. When the rotary inertia is zero, we have

(4.4.44) 1

Example 4.4.3:

Fora simply supported plate strip, ên is given by ên = ':," and from Eq. (4.4.42) it follows that

(4.4.43) w = >.2 {i5 V t;

When rotary inertia is neglected, we have

( 4.4.42)

If the applied axial force is zero, the natural frequency of vibration, with rotary inertia included, is given by

( 4.4.41)

and c1, c2, c3, and c4 are integration constants, which are determined using the boundary conditions. For natural vibration without rotary inertia and applied axial load, the equation for ê = Õ reduces to

( 4.4.40) - 2 A 2 p = D, q = hw - Nxx, r = Iow

( 4.4.39)

where

(4.4.38) W ( x) = c1 sin >.x + c2 cos êx + c3 sinh Õx + c4 cosh Õx

Equation (4.4.35) has the same form as Eq. (4.2.43). Hence, ali of the results of Section 4.2.4 are applicable here with b = 1 (Io =Io, f2 = 12) and E~xlyy = D. We summarize the results here for completeness. The general solution of Eq. ( 4.4.37) is

( 4.4.37)

where w is the natural frequency of vibration. Then Eq. (4.4.35) becomes


Figure 4.4.1: Effect of rotary inertia on nondimensionalized fundamental frequency of a simply supported (-45/45) laminated plate strip.

4.90

(-45/45)

4.85

18 Fundamental mode, ro1 >;

u '1 Q) 4.80 ;::l

O' Q) ~ ~

4.75

Plate strip

4.70 o 10 20 30 40 50 60 70 80 90 100


Figure 4.4.1 contains a plot of the nondimensionalized fundamental frequency w = wa2 J Io/ E2h3 of a simply supported plate strip with rotary inertia versus length-to-thickness ratio, a/ h. For small values of a/ h, rotary inertia is more significant in reducing the frequency than for thin and long plate strips.

In genera!, the roots of the transcendental equation in ( 4.4.46) are not the same as those of Eq. (4.4.47). If one approximates Eq. (4.4.46) as (4.4.47) (i.e., ê ~ 11), the roots in Eq. ( 4.4.48) can be used to determine the natural frequencies of vibration with rotary inertia from Eq. (4.4.42). When rotary inertia is neglected, the frequencies are given by Eq. (4.4.43) with >.. as given in Eq. (4.4.48). The frequencies obtained from Eq. (4.4.42) with the values of >.. from Eq. ( 4.4.48) are only an approximation of the frequencies with rotary inertia.

(4.4.48) 1 .ê.1 a= 4.730, ê2a = 7.85~1, ê3a = 10.996, Ŀ Ŀ Ŀ, >..,,a~ (n + 2 )7r

The roots of Eq. (4.4.47) are

(4.4.47) cos >..a cosh êa - 1 = O

For natural vibration without rotary inertia, Eq. (4.4.46) takes the simpler form

( 4.4.46) -2 + 2cos.êacoshÕa + (~ - ~) sin>..asinhÕa =O

Fora plate strip clamped at both ends, ê must be determined from [see Eqs. (4.2.56)~(4.2.60)]


Laminate Hinged-Hinged Clamped-Clamped

w N w w N w o 0.623 20.613 14.205 0.125 82.453 32.169

90 15.586 0.824 2.841 3.117 3.298 6.434

(0/90/0) 0.646 19.880 13.950 0.129 79.521 31.592 (90/0/90) 8.251 1.557 3.905 1.650 6.230 8.842

(0/90) 3.321 3.869 6.154 0.664 15.476 13.937 (0/90)as 1.427 9.006 9.389 0.285 36.026 21.264

(0/90)s 0.708 18.140 13.326 0.142 72.558 30.177 (90/0)s 3.896 3.298 5.682 0.779 1:3.192 12.868

(-45/45) 5.396 2.382 4.828 1.079 9.526 10.935 (-45/45)as 2.570 5.000 6.996 0.514 20.003 15.845 (45/ - 45)s 2.188 5.873 7.583 0.437 23.495 17.172

Laminate A 4.035 3.185 5.584 0.807 12.740 12.645 Laminate B 0.897 14.316 11.838 0.179 57.264 26.809

(I)s = symmetric, (I)as = antisymmetric (four layers). Laminate A: (90/Ñ45/0),; Laminate B: (0/Ñ45/90)5Å

W = -Wmax(E2h3/qoa4) X 102, N = N~x(a2/E2h3), w = wa2)Io/E2h3.

Table 4.4.2: Maximum deflections ( w) under uniform load, critical buckling loads ( N), and fundamental frequencies ( w) of laminated plate strips according to the classica! laminate theory (Ei/ E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, v12 = 0.25).

Table 4.4.2 contains nondimensionalized maximum deflections, critical buckling loads, and fundamental natural frequencies of simply supported and clamped ( at both ends) laminated plate strips with various lamination schemes. Compared to laminated beams (see Table 4.2.4), laminated plates in cylindrical bending undergo smaller displacements and have larger buckling loads and frequencies. This is due to the Poisson effect discussed earlier. All of the frequencies listed in Table 4.4.2 are for the case where rotary inertia is included and a/h = 10. The (0/90/0) laminates have larger bending stiffness as well as axial stiffness compared to the (90/0/90) laminates. This is because there are two 0Á layers and they are placed farther from the midplane in the first laminate than in the second laminate. Hence, (0/90/0) laminates undergo smaller deflections and have larger buckling loads and natural frequencies. The (0/90)8 laminates have larger bending stiffness than the (90/0)s laminates; both have the same axial stiffness. The antisymmetric laminates have some of the Bij # O and thus are relatively flexible when compared to symmetric laminates.

Figures 4.4.2 and 4.4.3 show the effect of lamination angle on maximum deflections iiJ = -Wmax(E2h3 /qoa4), critical buckling load N, and fundamental frequency w of two--layer antisymmetric angle-ply (-() j()) plates. It should be noted that antisymmetric angle-ply laminates with more than two plies are stiffer, i.e., deflect less and carry more buckling load.


Buckling load

o 10 20 30 40 50 60 70 80 90 Lamination angle, 0

Figure 4.4.3: Nondimensionalized critical buckling load (N) and fundamental frequency (w) versus lamination angle (O) of a simply supported (-e I e) laminated plate strip in cylindrical bending ( CLPT).

(0/-0 ) laminated, simply supported plate strip

20

o 10 20 30 40 50 60 70 80 90 Lamination angle, 0

0.04

///\ Uniform load

Figure 4.4.2: Nondimensionalized maximum transverse defiection ( w) versus laminati on angle (e) of a simply supported (-e/ e) larninated plate strip in cylindrical bending (CLPT).

(0/-0 ) laminated, simply supported plate strip

Center point load

\

0.28

0.24

0.20 18 r::f 0.16 .s ..., '-' Cl ~ Cl 0.12 o

0.08


(4.5.4)

(4.5.3)

Next, we eliminate uo and vo from Eqs. (4.5.2a-c) by solving (4.5.2a) and (4.5.2b) for uo and vo in terms of <Px and substituting the result into Eq. (4.5.2c):

(82wo 8</>x) ~ 82wo 82wo

K A55 ax2 + EJx + Nxx ax2 + q = Io EJt2

82</>x (fJwo ) 82</>x D 8x2 - KA55 EJx + <!>x = h &t2

( 4.5.2d)

(4.5.2c)

( 4.5.2b)

( 4.5.2a)

For cylindrical bending we further assume that </>y = O everywhere, and omit Eq. (4.5.ld) from further consideration. For the purpose of developing analytical solutions, we neglect the in-plane inertia terms and assume that there are no thermal effects. Then Eqs. ( 4.5. la-e) are simplified to

82uo 82vo 82</>x 82</>x An 8x2 + Aw 8x2 + Bn 8x2 =Ii 8t2 82uo 82vo 82</>x

Aw EJx2 + A55 8x2 + B16 8x2 = O 82uo 82vo EP <!>x ( Dwo ) 82 <!>x

B11 ax2 + B15 ax2 + D11 ax2 - K A55 8x + </>x = h 8t2

( 82wo 8</>x) fJ ( 8wo) 82wo

K A55 fJx2 + fJx + fJx Nxx Dx + q = Io EJt2

(4.5.lc)

4.5.1 Governing Equations

In order to see the effect of shear deformation on bending deflections and buckling loads, we consider the equations of motion for cylindrical bending according to the first-order shear deformation theory (FSDT) [see Eqs. (3.4.23)-(3.4.27)]:

82uo EPvo 82</>x 82</>y 8NJx EPuo 82</>x Au 8x2 + Aw 8x2 + B11 8x2 + Bw EJx2 - =s:': Io [)t2 +Ii 8t2 (4.5.la)

EJ2uo EJ2vo 82</>x 82</>y 8N'[y 82vo 82</>y ( Aw EJx2 + A55 EJx2 + Bw EJx2 + B55 8x2 - a;;-= Io EJt2 +li 8t2 4.5.lb)

4.5 Cylindrical Bending Using FSDT


(4.5.10)

The maximum deftection occurs at x = a/2 and it is given by

(4.5.9)

(4.5.8) q0a3 [ (x)3 . (x)2 ] F0a2 [ (x)2] <Px(x)=-24D 4 a -5 a +l + 16D l-4 a q0a4 [(x)4 (x)3 (x)] q0a2 [(x) (x)2] wo(x)= 24D a -2 a + a +2KA55 a - a F0a3 [3 (x) 4 (x)3] F0a (x) + 48D a - a + 2K A55 a:


For a plate strip simply supported at both ends and subjected to uniforrnly distributed load q = q0 as well as a downward point load F0 at the center, we obtain

where the constants of integration c1 through c4 can be determined using the boundary conditions. The solutions developed are general in the sense that they are applicable to any symmetrically laminated beams. Next we illustrate the procedure to determine the constants for beams with both edges simply supported or clamped.

wo(x) = - ~ [- fox fo foT} fot' q(() d(dÕdrydE, + C1 ~3 + C2 ~2 + C3X + C4]

+ K~55 [-fax fof, q(()d(dE, +cix] (4.5.7)

and transverse deflection

(4.5.6) 1 [ r r r x2 l <Px(x) = D - lo lo lo q(() d(drydf, + c12 + c2x + es

Following the procedure of Section 4.3.2, we obtain [see Eqs. (4.3.12)-(4.3.14)] the generai solution for the rotation

( 4.5.5b)

( 4.5.5a) (d2wo d<Px)

K A55 dx2 + dx + q = O

d2</Jx (dwo ) D-- - K Ass - + <Px = O dx2 dx

4.5.2 Bending

For static analysis, Eqs. ( 4.5.3) and ( 4.5.4) reduce to

Equations (4.5.3) and (4.5.4) are similar to Eqs. (4.3.9a,b) for laminated beams, and therefore all developments of Section 4.3 would apply here.


(4.5.15)

Following the procedure of Section 4.3.3, we obtain

(4.5.14b)

(4.5.14a) (d2W dX) A d2W

KA55 dx2 + dx + Nxx dx2 =O

d2X (dW ) D--KA55 -+X =0 dx2 . dx

4.5.3 Buckling

For stability analysis, we set q =O , s.: = -N~x' and Io= h =O in Eqs. (4.5.3) and (4.5.4):

The determination of the shear correction coefficient K for laminated structures is still an unresolved issue. Values of K for various special cases are available in the literature (see [4-8]). The most commonly used value of K = 5/6 is based on homogeneous, isotropie plates (see Section 3.4), although K depends, in generai, on the lamination scheme, geometry, and materiai properties. Figure 4.5.1 shows the effect of shear deformation, shear correction coefficient, and lamination

scheme on nondimensionalized deflections w = Wrnax(E2h3 /qoa4) of simply supported, cross-ply (0/90) and angle-ply (45/--45) laminates under uniformly distributed load. The shear correction factor has little infiuence on the global response for the antisymmetric laminates analyzed. The effect of shear deformation is to increase the defiections, especially for a/ h ::; 10. Antisymmetric angle-ply laminates are relatively more fiexible than antisymmetric cross-ply laminates. Figure 4.5.2 contains plots of nondimensionalized maximum defiection versus length-to-height

ratio for two-layer antisymmetric cross-ply (0/90) and angle-ply (45/-45) laminates (K = 5/6) under uniformly distributed load and with simply supported edges as well as for clamped edges. For clamped boundary conditions, shear deformation is relatively more significant for a/h ::; 10. The effect of orthotropy on deflections is shown in Figure 4.5.3 (G12 = G13 = 0.5E2, G23 = 0.2E2, v12 = 0.25, and K = 5/6).

( 4.5.13)

The maximum defiection is given by

( q0a4 q0a2 F0a3 F0a )

Wmax = 384D + 8KA55 + 192D + 4KA55

( 4.5.12)

(4.5.11)

q0a3 [ (x)3 (x)2 (x)] rPx(x)=-12D 2 a -3 a + a - ~;2 [2 (~)2 - (~)]

[ ] 2 [ ]

qoa4 x 2 x qoa2 x x 2 wa(x) = 24D (a) - (a) + 2KA55 (a) - (a)

+ F0a3 [3 (::_)2 _ 4 (::_)3] + ~ (::_) 48D a a 2KA55 a

Exarnple 4.5.2 (Clamped beam):

Consider a laminated plate strip fixed at both ends and subjected to uniformly distributed transverse load q0 and a point load Fa at the center, both acting downward. For this case, the solution is given by


Figure 4.5.2: Transverse defiection (w) versus length-to-thickness ratio (a/h) of simply supported (SS) and clamped (CC) plate strips.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, a!h

_,.........(0/90), ss

_,.......-(-45/45), ss

SS = Simply supported at both ends CC = Clamped at both ends

0.09

0.08

0.07

l::S 0.06 i:::~ o 0.05 ..... ...., o Q)

!+::: 0.04 Q)

Cl 0.03

0.02

0.01

Figure 4.5.1: Transverse defiection (w) versus length-to-thickness ratio (a/h) of simply supported plate strips (K = 1.0, 5/6, 2/3).

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, a/h

0.03

(0/90)

(-45/45)

--K=I --- K= 516 ----K= 2/3


( 4.5.20)

For a simply supported plate strip, the criticai buckling load is given by

using the

(4.5.19)

(4.5.18)

(4.5.17)

(4.5.16)

Example 4.5.3:

No >..2D ê2 = ( ~~ ) or è: = (l >..2D)

l - KA~s D + KAss and c1 through q are constants of integration, which are evaluated boundary conditions.

where

W(x) = c1 sin êx + c2 cos êx + c3x + c4

( NÁ ) d4W d2W

D 1 - K 1:5 dx4 + è: dx2 = O

The general solution of Eq. (4.5.17) is

( ) ( è: ) dW X X = - 1 - -- - + K1 KA55 dx

Figure 4.5.3: The effect of material orthotropy and shear deformation on transverse defiections w of simply supported cross-ply (0/90) laminateci plate strips under uniformly distributed load.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, al h

0.09

0.08

0.07

I~ 0.06

s:f 0.05 .s ....,

<.) Q.)

!+:: 0.04 Q.)

Cl 0.03

0.02


(4.5.25a) W ( x) = c1 sin êx + c2 cos Xz + e:~ sinh Õx + c4 cosh Õx

The general solution of Eq. ( 4.5.24a) is

(4.5.24b) r = Iow2 IoD 2 q=--W KA55 '

p=D, where

( 4.5.24a) d4W d2W p--+q-- -rW =O dx4 dx2

Following the results of Section 4.3.4, we obtain

( 4.5.23b)

( 4.5.23a) d2X (dW ) 2 D-d 2 -KA55 -d, +x +Lz-c X=O X .r

(d2W dX) 2 K A55 -d 2 + -d + Iow W = O X X

where w is the natural frequency of vibration, and W(x) and X(x) are the mode shapes. Substitution of the above solution forms into Eqs. ( 4.5.3) and ( 4.5.4) yields [cf. Eq. (4.3.40a,b)]

wo(x,t) = W(x)eiwt, </>x(x,t) = X(x)eiwt, i= yC1

For a periodic motion, we assume solution in the form

4.5.4 Vibration

Figures 4.5.4 and 4.5.5 show the effect of shear deformation ami modulus ratio on nondirnensionalized criticai buckling loads N = N~:c(a2/E2h3) of two-layer antisyrnrnetric angle- ply (-45/45) and cross-ply (0/90) plate strips (Ei/E2 = 25, G12 = G1:i = 0.5E2, G23 = 0.2E2, v = 0.25, K = 5/6). In Figure 4.5.4 results are presented for sirnply supported as well as clamped boundary conditions. The effect of shear deformation is significant for a/ h ::; 10 in the case of sirnply supported boundary conditions, and a/ h ::; 20 in the case of clamped boundary conditions. The effect of shear deforrnation is more for rnaterials with larger modulus ratios (see Figure 4.5.5).

(4.5.22)

The roots of the equation are approximately the samo as for the case in which shear deformation is neglected [see Eq. (4.2.38b)J. The first root ofthc cquation is >.1 = 27r. Hence, the criticai buckling load is given by

(4.5.21) ( >.2D) 2 ( cos ,\a - 1) 1 + -A + êa sin êa = O K 55

Thus, the effect of the transverse shear deformation is to decrcase the buckling load. Omission of the transverse shear dcforrnation in the classica! theory arnounts to assuming infinite rigidity in the transvcrsc direction (i.e., A55 = G13 = oo}; hence, in the classica! laminate theory the structure is represented stiffer than it is. For a plate strip fixed at both ends, ,\ is governed by the equation


Figure 4.5.5: The effect of material orthotropy and shear deformation on critical buckling loads of simply supported cross-ply (0/90) laminated plate strips.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, al h

6

N = N':x(a2 I E2h3)

5 ~ - E11Ez-40

I~ 'ti (Il 4 3 bJJ E1/E2= 25 ~ ..... :g 3 ;:l

i::Q

2 \ E1/E2= 10

Figure 4.5.4: The effect of shear deformation on the critical buckling loads of simply supported (SS) and clamped (CC) cross-ply and angle-ply plate strips.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, a I h

16

14

12 N = N~x(a2 I E2h3) I~ "O 10 (Il o <, ....:i bi) 8

(-45/45), cc ~ ..... ~ SS = Simply supported at both ends u ;:l 6 CC = Clamped at both ends i::Q

4 /(0/90), ss

2 (-45/45), ss


Once the value of ê is known, frequencies of vibration can be determined from Eqs. (4.5.26a,b).

(4.5.31b)

( 4.5.31a) -2 + 2cosêacoshÕa + sinêasinhÕa (5522 - 5n) = (} 11 522

which is the same as in Eq. (4.4.45). Thus, the effect ofshear deformation is to reduce the frequency of natural vibration. For a laminateci strip with clamped edges, the following equation governs ê:

(4.5.30)

By neglecting the shear deformation (i.e., A55 = G13 = co ) we obtain the result

( 4.5.29)

Substitution of ê from Eq. ( 4.5.28) into Eq, ( 4.5.26a,b) gives two frequencies for each value of ê. The fondamenta! frequency will come from Eq, ( 4.5.26a). When the rotary inertia is neglected, we obtain from Eq. ( 4.5.27) the result

(4.5.28) sin Aa =O, or ên = mr a

For a simply supported plate strip, the boundary conditions give c2 = c3 = c4 = O, and Example 4.5.4:

( 4.5.27) 2 Q - [ ( D ) 2] (D) 4 w = R ' Q = l + K Ass ,\ ' R = Io ,\

When the rotary inertia is neglected, we have P = O and the frequency is given by

( 4.5.26b)

where

( 4.5.26a)

and c1, c2, c3, and c4 are integration constants. Use of the boundary conditions leads to the determination of three of the four constants, the fourth one being arbitrary, and an equation governing ,\ and Õ (see Section 4.3.4 for details). The frequencies w can be determined from

( 4.5.25b) x = V 2~ ( q + J q2 + 4pr)' Õ = V 2~ ( -q + J q2 + 4pr) where


4.6.2 Theoretical Formulation

Displacement and strain fields

Consider a symmetrically laminated beam of n layers. Suppose that two of the layers, namely, the mth and ( n - m + 1 )th layers, are made of magnetostrictive material, such as Terfenol-D particles embedded in a resin (see Figure 4.6). The remaining n - 2 layers can be made of any fiber-reinforced materials with varying fiber orientation e but symmetrically disposed about the mid-plane of the beam. We wish to study the problem of vibration suppression in these beams using the Euler-Bernoulli, Timoshenko, and Reddy third-order beam theories. To facilitate the development of all three theories in a unified manner, we introduce tracers whose values will yield the results for a particular theory [29].

The grains of certain materials consist of numerous small, randomly oriented magneti e domains that can rotate and align under the infi uence of an external electric or magnetic field. The electric (magnetic) orientation brings about internal strains in the materi al. This is known as the electrostriction. ( magnetostriction). For example, a commercially available magnetostrictive material Terfenol-D is an alloy of terbium, iron, and dysprosium. The use of Terfenol-D for vibration suppression has some advantages aver other smart materials, in particular, it has easy embedability into host materials, such as the modern carbon fiber- reinforced polymeric (CFRP) composites, without significantly affecting the structural integrity. Considerable effort is spent to understand the interaction between magnetostrictive layers and composite laminates and the feasibility of using magnetostrictive materials for active vibration suppression (see [30-32]). Although there have been important research efforts devoted to characterizing the properties of Terfenol-D materiai, fundamental information about variation in elasto-magnetic materia! properties is not available. Few studies [33-35] report experimental evidence of significant variation in materia! properties such as Young's modulus and magneto-mechanical coupling coefficient. Here we present a generalized beam theory that contains the classica! Euler-

Bernnoulli beam theory as well as the first-order and the third-order beam theories, and bring out the effects of materia! properties of a lamina, lamination scheme, and placement of the actuating layers on vibration suppression time.

4.6 Vibration Suppression in Beams 4.6.1 Introduction

Figures 4.5.6 and 4.5.7 show the effect of shear deformation and modulus ratio (Ei/E2) on nondimensionalized fundamental frequencies w = wa2 J Io/ E2h3 of two-layer antisymmetric angle- ply (-45/45) and cross-ply (0/90) plate strips (K = 5/6, Ei/ E2 = 25, G12 = G13 = 0.5E2, G23 = 0.2E2, v12 = 0.25). From Figure 4.5.6 it is clear that shear deformation effect in decreasing frequencies is felt for a/ h ::; 10 for simply supported boundary condit.ions, whereas for clamped boundary conditions the effect is felt for a/ h ::; 15. Also, the effect of shear deformation is more for materials with larger modulus ratio, as can be seen from the results of Figure 4.5.7.

222 MECHANICS OF LAMJNATED COMPOSITE PLATES AND SHELLS

Figure 4.5. 7: The effect of material orthotropy and shear deformation on fundamental frequencies of simply supported cross-ply (0/90) laminated plate strips.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio,a/ h

7 E1/E2=40

18 >, 6 '-' E1/E2= 25 i:: Q)

::i O' Q) ;... i:r.. 5

E1/E2= 10 4

Figure 4.5.6: The effect of shear deformation on the fundamental frequencies of simply supported and clamped cross-ply and angle-ply plate strips.

o 10 20 30 40 50 60 70 80 90 100 Length-to-thickness ratio, a I h

16

ro = roa\10 I E h3 112 (0/90), cc 14

12 (-45/45), cc

10 I~ SS = Simply supported at both ends o i:: 8 CC = Clamped at both ends Q)

::i O' (0190), ss Q) ;... i:r.. 6

4 (-45/45), ss

2


(4.6.2) C1 = 1, Co= 0

C1 = C3 = 0 Co= C3 = 0

Co= 1, C1=1, 4

C3 = 3h2'

Euler-Bernoulli beam theory (EBT): Timoshenko beam theory (TBT):

Reddy beam theory (RBT):

The displacement field ( 4.6.la) can be specialized to various beam theories as follows:

(4.6.lb)

where (u, v, w) are the displacement components along the (x, y, z) coordinate directions, respectively, wo is the transverse deflection of a point on the midplane (i.e., z = O), and </>(x, t) is the rotation of a transverse normai line. The functions fi(z) and h(z) are given by

(4.6.la)

Consider the displacement field

8wo 3 ( 8wo) u(x, y, z, t) = -zco ox + zci</> - z c3 </> + ox 8wo = fi(z) ox + h(z)</>(x, t)

v(x,y,z,t)=O w(x, y, z, t) = wo(x, t)

Figure 4.6.1: Layered composite beam with embedded actuating layers.

y

X

-0

Actuating layer #=="Il +0

0= 90Á

~~===~t-~~~~~Actuating layer


(4.6.7) s(m) = _l_ = _1_ E(m) Q(m)

and d(m) is the magneto-mechanical coupling coefficient, E(m) being the modulus of the magnetostrictive layer (e(m) = Q(m)d(ml).

where H is the magnetic field intensity, s(m) is the compliance of the mth magnetostrictive layer

(4.6.6) (T(m) = -1- (E - d(m) H) = Q(m)E - e(m) H xx S(m) xx - xx

The constitutive relation far an actuating (say, a magnetostrictive) layer is

(4.6.5) Q(k) -dk) 66 - 12 ' Q(k) _ c(k) Q(kJ _ cCk) 44 - 23 ' 55 - 13 '

Q-(k) - Q(k) cos2 e(k) + Q(k) sin2 e(k) 55 - 55 44

Q-(k) - Q(k) cos? e(k) + 2 (Q(k) + 2Q(k)) cos2 e(k) sin2 e(k) + Q(k) sin" e(k) 11 - 11 12 66 22

where

( 4.6.4)

The constitutive relations of the kth fiber-reinforced (structural) layer are

Constitutive relations

( 4.6.3b)

where

82wo 8</) 3 (8</) 82wo) = Z""(l) + Z3""(3) Exx = -zco ®Jx2 + zc1 ®Jx - z C3 ®Jx + ®Jx2 "'xx '-xx

"fxz = ( 1 - Co) 88:0 + C1 </> - 3c3z2 ( </> + ®J:xo) = "(~~) + Z2"(~~) ( 4.6.3a)

The non-zero linear strains are given by


(4.6.12) Mxx = ci Mxx - c3Pxx ' CJx = C1 Qx - 3c3Rx Mxx = -coMxx - c3Pxx , CJx = (1 - co)Qx - 3c3Rx

where all the terms involving [ Ŀ ]6 vanish on account of the assumption that all variations and their derivatives are zero at t = O and t = T. Various symbols introduced in Eq. (4.6.11) are defined as

and c(t) is the control gain.

Equations of motion

Using Hamilton's principle (or the dynamic version of the principle of virtual displacements), we obtain

(4.6.10) k - ne e - Jb2 + 4r2 e e

where ke is the coil constant, which can be expressed in terrns of the coil width be, coil radius re, and number of turns ne in the coil by

( 4.6.9) 8wo I(x, t) = c(t)Tt

and I ( t) is related to the velocity wo by ( 4.6.8) H(x, t) = kcI(x, t)

Velocity feedback conirol

Considering velocity proportional closed-loop feedback control, the magnetic field intensity H is expressed in terms of coil current I(x, t) as


4.6.3 Analytical Solution

First we write the equations of motion (4.6.16) and (4.6.17) in terms of the displacemcnt variables ( wo, </>) by expressing 1Vl3,x, Qx, Pxx, and R,, [see Eqs. (4.6.13a,b) and (4.6.12)]. We bave

(4.6.19)

( 4.6.18a)

(4.6.18b)

(4.6.17)

(4.6.16)

( 4.6.15b)

(4.6.15a)

(4.6.14)

(4.6.13b)

( 4.6.13a)

- 81\;fxx .. 8wo V =:Q. +--+K3,-1,+K1- x X OX , 'f' OX

where

PrimaryVariables : 8wo

</> Wo, ax ' SecondaryVariables : Vx, P.i:i;, 1t1xx

The primary and secondary variables of the formulations are

2- - 3 4 2 a Ivl3;;1; _ aQ x _ _ K ~ _ K a wo 1 a wo _ 0 8x2 ax q 3 8x8t2 1 8x28t2 + o 8t2 -

The cquations of motion are

and (K1, K2, Ka) are the mass inertias

{Mxx}- J { z } d _ {Due~~+ F11E~:~} { B} 8wo - 3 axx Z - (1) P) - -- Pxx A Z Ŀ FuExx + H11E;x E 8t

{ Q }

. { } { (O) (2) } x _ 1 dz _ A55')'xz + D55')'xz - 2 axz - (O) (2) u; JA Z D55')'xz + F55')'xz

where (Ivlxx, Qx, P.--c:c, Rx) denote the conventional and higher-ordcr stress resultants


where the coefficients Sij = s; and Mij = Mji are defined by

(4.6.23)

and substituting into Eqs. (4.6.20) and (4.6.21), we obtain

[~~: ~::] {i}+ [2~: 2::] { t}

(4.6.22) rin x rj>(x, t) = X(t) cos -

a n7rX

wo(x, t) = W(t) sin - , a

This completes the development of the governing equations in terms of the displacements ( wo, r/>). Of course, the equations can be specialized to any of the three theories. Here we discuss the Navier's solution of these equations for the case of simply

supported boundary conditions. Assuming solution of the form

(4.6.21)

(4.6.20)


( 4.6.27)

M23 = [-c3J4 + (c3)2h) (naK) M33 = h - 2c3J4 + (c3)2 h, 622 = -c3[ (na7f) 2

623 =O, 632 = (B - c3E) (n:), 633 =O

Reddy beam theory (RBT) (co =O, ci = 1, c3 = ~)

( 4.6.26)

Timoshenko beam theory (TBT) (co =O, c1 = 1, c3 =O)

(4.6.25) (n7f)4 (nK)2 (nK) 2 S22 = o.. ~ , M22 = h ~ +Io, 622 = -B ~

Equation ( 4.6.24) can be specialized to various theories as follows ( only non-zero coefficients are listed): Euler-Bernoulli beam theory (EBT) (co = 1, c1 = c3 =O)

( 4.6.24)

S23 = [-coc1D11 + CQC3F11 - ci c3F11 + ( c3)2 n., J ( na7f) 3

+ [(1- co)c1A55 - 3(1 - ca+ c1)c3D55 + 9(c3)2 Fss] c~7f) S33 = [ ( c1)2 o.. - 2c1c3F11 + ( c3)2 H11 J ( na7f) 2

+ ( c1 )2 A55 - 6c1 c3D55 + 9( c:3)2 Fs5

M22 = [(co)2 t, + 2coc3J4 + (c3)2 h) ( n:r +Io M23 = [-2coc1h + (co - c1)c3J4 + (c3)215] (na7f) ' 2 2 M33 = c1h - 2c1c3J4 + (c3) h

622 = - (coB + c3[) (naK) 2

62:3 =O, 632 = (c1B- c3E) (n:), 633 =O


Numerical studies were carried out to analyze damped natural frequencies, damping coefficients, and the vibration suppression time, using the three theories [29]. Different lamination schemes were used to show the infl.uence of the position of magnetostrictive layer from the neutra} axis on the vibration suppression time. A time ratio relation between the thickness of the layers and the distance to the neutra} axis of the laminated composite beam is also found. All values of the material and structural constants are indicated in the tables. The materia} properties used are the same as those used in [32]. The numerical values of various coefficients (namely, the inertial and

magnetostrictive coefficients) based on different lay-ups and materiai properties [CFRP, Graphite-Epoxy (AS), Glass-Epoxy and Boron-Epoxy] are listed in Tables 4.6.1 and 4.6.2. Table 4.6.2 also shows the damping coefficients and natural frequencies for different materials and lay-ups. The damping and frequency parameters for transverse modes n = 1 to n = 5 are shown in Table 4.6.3, and they are compared with the results obtained by Krishna Murty et al. [32] using the Euler-Bernoulli beam theory (EBT). There is some difference between the numerical

4.6.4 Numerica! Results

The actuation stress is (J'd = -EmdH.

(4.6.33) w0(x, O) =O, w0(x, O) = 1, ij>(x, O) =O, ~(x, O) =O

In arriving at the solution (4.6.32), the following initial conditions were used:

( 4.6.32) 1 t . . nnx wo(x, t) = - e-a smwdt sm -- Wd a

Equation ( 4.6.31) gives two sets of eigenvalues. A typical eigenvalue can be expressed as ê = -o: + iwd. The lowest imaginary part (wd) corresponds to the transverse motion, and we can write

( 4.6.31)

for the Timoshenko and third-order beam theories, where

( 4.6.30) ~231 =o S33

for the Euler-Bernoulli beam theory, and

(4.6.29)

and obtain, for non-trivial solution, the result

( 4.6.28)

For vibration control, we assume q = O and solution of the ordinary differential equations in Eq. ( 4.6.23) in the form


-a Ñ ½ldn ( rad/s) - mode 1 Lav-up Murtv et al EBT TBT RBT

[45/m/-45/0/90)s 4.60Ñ102.17 4.62Ñ102.15 4.62Ñ102. 12 4.62Ñ102.11 [m/Ñ45/0/90]s 5.90Ñ98.44 5.94+98.42 5.94+98.39 5.93+98.38

{m/90,Js 5.90Ñ64.65 5.94Ñ64.65 5.94Ñ64.64 5.94Ñ64.64 [m/O,]s 5.90Ñ143.58 5.94+143.57 5.93+ 143.49 5.93Ñ143.44

CFRP :En=l38.6 GPa E22=8.27 GPa G12=4.12 GPa G1.1=G23=0 .6 E22. V12=0.26, p=1824 kg.m-è Magoetostrictive layer : Em=26.5 Gpa, pm=9250 kg.rn-', dk=l.67xl0Ŀ8m/A, c(t).R,=104, Vm=O, a=I m

Tahle 4.6.4: Damping and Frequency Parameters a and cod for Various Lamination Schemes

-aÑ ½ldn (rad/s) - Lav-up [Ñ45/m/0/90]s Mode Murtv et al EBT TBT RBT 1 3.29Ñ104.88 3.30Ñ104.85 3.30Ñ104.82 3.30Ñ104.82 2 13.19Ñ419.50 13.20Ñ419.37 13.17Ñ418.90 13.16Ñ418.80 3 29. 70Ñ943.88 29.68Ñ943.40 29.53Ñ941.05 29.48Ñ940.52 4 52.86Ñ1678.83 52. 73Ñ1676. 72 52.27Ñ1669.32 52.10Ñ1667.68 5 82.59Ñ2621.87 82.34Ñ2619.02 81.22Ñ2601.04 80.80Ñ2597.09

CFRP: E11=138.6 Gpa, E2z=8.27 GPa, G12=4.12 GPa, G"=G2a=0.6 E22. v12=0.26, p=l824 kg.m- Magoetostrictive layer : Em=26.5 GPa, pm=9250 kg.m-> , dk=l.67xl0Ŀ8m/A, c(t).&=10 Ŀ Vm=O, a=l m

Tahle 4.6.3: Comparison of the Damping and Frequency Parameters a and cod as Predicted by Various Theories (see Reddy and Barbosa [30])

Materiai Lay-up Io r, oo-ÅJ 1, (lQ-9) r. -13 -ú (10ĿÅ) -aÑ Ctldn(rad/s) (10-14)

[Ñ45/m/0/90]s 33.092 2.461 2.907 4.508 22.128 1.438 3.30Ñ104.85 [45/m/-45/0/90]s 33.092 3.352 4.600 7.084 30.979 3.872 4.62Ñ102.15

CFRP {m/Ñ45/0/90]s 33.092 4.540 8.521 17.171 39.830 8.165 5.94Ñ98.42 {m/90,]s 33.092 4.540 8.521 17.171 39.830 8.165 5.94Ñ64.65 [m/O,]s 33.092 4.540 8.521 17.171 39.830 8.165 5.94Ñ143.57

Gr.-Ep [Ñ45/m/0/90 l s 30.100 2.196 2.471 3.696 22.128 1.438 3.63+ l13.06 Gl.-EP [Ñ45/m/0/90]s 33.700 2.514 2.995 4.674 22.128 1.438 3.24+85.54 Br.-Eo [Ñ45/m/0/90ls 34.100 2.550 3.054 4.782 22.128 1.438 3.20Ñ127.90

Table 4.6.2: Mass Inertias and Magnetostrictive Coefficients, and Parameters a and cod for V arious Laminates

Materiai Lav-up Dn(l03) Fn (10~) n., (10-1) A,. (109) Dss (102) Fso (10-3) [Ñ45/m/0/90ls 3.739 5.246 9.333 6.620 5.185 6.902

[45/m/-45/0/90ls 3.552 4.891 8.793 6.620 6.179 8.792 CFRP lm/Ñ45/0/90ls 3.303 4.069 6.679 6.620 7.506 13.168

[m/90,]s 1.432 2.567 5.063 6.620 7.506 13.168 [m/O,]s 7.015 7.927 11.189 6.620 7.506 13.168

Gr.-Ep (AS) [Ñ45/m/0/90 l s 3.954 5.629 10.053 7.974 6.399 8.881 Gl.-EP [Ñ45/m/0/90]s 2.535 3.700 6.589 7.614 6.173 8.384 Br.-Ep [Ñ45/m/0/90]s 5.730 8.259 14.865 7.066 5.634 7.569

CFRP: E11=138.6 GPa, E22=8.27 GPa, G13=G;,=0.6 E22. G.,=4.12 GPa , Vl2=0.26, p= 1824 kg.rn" Graphite-Epoxy (AS): En=137.9 GPa, E22=8.96 GPa, G12=G1a=7.10 GPa, G2a=6.21 GPa, v12=0.30, p=l450 kg.ms Glass-Epoxy: Eu=53.78 GPa, E22=17.93 GPa,G12=G13=8.96 GPa, G,,=3.45 GPa , V12=0.25, p=1900 kg.m-' Boron-Epoxy : Eu=206.9 GPa,E22=20.69 GPa, G12= G13=6.9 GPa, Gi:1=4.14 GPa, v12=0.30,p=l950 kg.m-'

Table 4.6.1: Coefficients for Different Lamination Schemes and Materials (from Reddy and Barbosa [30])


4.1 Consider a simply supported laminated beam under point loads F0 at x = a/4 and x = 3a/4 (the so-called four-point bending). Use the symmetry about x = a/2 to determine the deflection wo(x) using the classica! beam theory. (Ans: The maximum deflection is Wmax = 11Foa3 /384E~JyyĀ)

4.2 Determine the static deflection of a clamped laminated beam under uniformly distributed load qo and a point load F0 at the midspan using the classica! beam theory.

4.3 Show that the criticai buckling load of a clamped-free laminated beam using the classica! beam theory is given by

Problems

In this chapter analytical solutions are developed for laminated beams and plate strips in cylindrical bending using the classica! and first-order shear deformation theories. Analytical solutions are presented for static bending, natural vibration, and buckling problems under a number of boundary conditions. A unified formulation for laminated beams with embedded actuating layers is

presented. The formulation includes the Euler-Bernoulli, Timoshenko, and Reddy third-order beam theories as special cases. Analytical solution for the simply supported beam is presented to bring out the effects of the materiai properties of a lamina, lamination scheme, and placement of the actuating layers on vibration suppression. When closed-form solutions can be derived, they are preferred over the series

solutions. However, when exact closed-form solutions cannot be developed, the series solutions are the best alternative. When analytical solutions cannot be derived at all, numerica! solutions based on the finite element method (see Chapters 9 and 10) can be used to determine the solutions.

4.7 Closing Remarks

results predicted by the three theories only in the higher modes. Table 4.6.4 shows the influence of the position of the magnetostrictive layer in the z-direction and the influence of the lamination scheme in the damping and frequency parameters. The value of a increases when the magnetostrictive layer is located further away from the x-axis, indicating faster vibration suppression. The lay-up [m/904]8 represents the softest beam and the lay-up [m/04]8 the stiffest beam. A comparison of the fundamental transverse and axial modes, obtained using the

three theories show that there is no significant difference between the results. The uncontrolled and controlled motions at the midpoint of the beam, as predicted by RBT, are shown in Figures 4.6.2-4.6.5 for the first mode when the actuating layer (m) is placed at different distances from the midplane of the laminate. These figures show that the vibration suppression time decreases when the distance to the neutral axis is increased, and it remains nearly the same in the laminates with different stiffness. Figures 4.6.6 shows that the vibration suppression time decreases very rapidly for higher modes. Figure 4.6. 7 shows the controlled motion of the beam, as predicted by EBT and RBT, for mode n = 5. Clearly, the difference between the predictions of the two theories is not significant.


Figure 4.6.3: Comparison of uncontrolled and controlled maximum defiection ( at midpoint of the beam) for (45/m/~45/0/90)s laminate.

0.02 Uncontrolled

~ Controlled

5 .... 0.01 Å ~ Å' . ' . Å Å ~ ,, Å' Å' " ,. " ,, ..

Q) ,, Å' Å' " Å' " ,, " ,. ' " 8 Å' Å' ,. ,,

Ŀ: Å' Å' '' ,, ,, Q) '' '' '' ''

,, e) '' ': ro ': - e, 0.00 00 ..... -e Q) ' . ' '' 00 '' ' 'Å i-. " ' ,, 'Å Q) " ,, ,, 'Å > " ,, Å' ., 00

,, " .. .. ,, Å' 'Å ~ -0.01 Å . . . Å Å' 'Å ro i-. ~

-0.02 0.00 0.20 0.40 0.60 0.80 1.00

Time (s)

Figure 4.6.2: Comparison of uncontrolled and controlled maximum defiection (at midpoint of the beam) for (Ñ45/m/0/90)8 laminate.

0.02

Uncontrolled ~ Controlled 5 .... 0.01 I, . i:: " ,, ~ " n . . 'Å Q) .. ,, ,, :Ā ,, " "

,, 'Å 8 .. ,, " " " I

'Å Å' " " I ,, Å' ,,

" Q) Ŀ: ,, ,, e) ro - e, 0.00 00 ..... -e Q) ,, 00 ,, i-. ,. " I' ,, Q) Å' " " Å' ,, " > I' Å' .. " Å' ,, " Å' .. " 00 Ā: ~ . Ā: ,, . ~ -0.01 ro i-. ~

-0.02 0.00 0.20 0.40 0.60 0.80 1.00

Time (s)

233 ONE-DIMENSIONAL ANALYSIS OF LAMINATED COMPOSITE PLATES

Figure 4.6.5: Comparison of uncontrolled and controlled maximum defiection ( at midpoint of the beam) for (m/904)8 laminate.

Time (s)

1.00 0.80 0.60 0.40 0.20 0.00 -0.03

0.00

' I . . -0.01 I . I I I I I O I I I 11 I '. 1 Å Il ,1 1, I O I O '.

" .. 1' " ,1

11 " 1, 'Å .. ~Ŀ Ā: .. ,1 ~I " Å1 ~Ŀ -0.02

Uncontrolled Controlled

,. " ' " Å' .,

,1 .. ,. Å1

,1 ,1 .. IO .. ,1 .. 'I 'I .: 11 '' 11 . : Å I O I I O '

I ' O I I . I I I I .

] 0.02

0.03

Figure 4.6.4: Comparison of uncontrolled and controlled maximum defiection (at midpoint of the beam) for (m/Ñ45/0/90)8 laminate.

Time (s)

1.00 0.80 0.60 0.40 0.20 -0.02

0.00

I I '. I I o . . I ,. 'Å " I .. ,, I ,. 'Å " I .. " 11 .. 'Å "

Il .. " " 1, .. 'Å "

Il . . .. ~ Ā: 1; 1: Å ., V

Uncontrolled Controlled

0.02

] ..., 0.01 Å ~ " " a> .. " s " 11

" a> Il

<:.) (Il - o. 0.00 m ..... "O a> 11 u: Å1 11

1, .... 1, 11 11 a> 11 " lo > .. .. Å1 ,1 .. .. u: Ā: Å1 11 ~ -0.01 ' (Il .... ~

MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS 234

0.04 0.03 0.01 -0.0004 -+-rm~~m~~m~~mm~mm-+-

0.00

0.0004

:Ä ----------- EBT ..., 0.0002 RBT ~ <1) s <1) u cd - o. 0.0000 U) ..... "O <1) U) ;... <1) > U)

~ -0.0002 cd ;... E-i

0.02 Time (s)

Figure 4.6. 7: Controlled motion of the laminateci beam (Ñ45/m/0/90).s as predicted by EBT and RBT for mode ti = 5.

Time (s)

Figure 4.6.6: Controlled motion of the laminateci beam (Ñ45/m/0/90)8, as predicted by RBT, for modes n = 1 and ti = 2.

1.00 0.60 0.40 0.80 0.20 -0.010 --+..~~~~~~~~~~~~~~...+-

0.00

!i \I ~ J J v i ... \ , ...... Ŀ .. Ŀ Ŀ.Ŀ

Å, " 'Å " 'Å . 'Å 'Å :Ŀ

- . ,, ,,

mode n=2

mode n= L s '-' 1:1 0.005 <1) s <1) u cd - ~ 0.000 ..... "O <1) U) ;... <1) > U)

Ä -0.005 ~


(2) qoa3 efix(x) = - 48Eb I

xx yy

(1)

wa(x)= 48~~::YY [2Gt-C1:132:) (D3 +3(i;3a) Gf] + -s-i-b-~-:-b-h [ C1: 132:)

( ~) - 4 G f] [ s G / - 3 ( 51: 132:) ( ~ f + 6 C ; 3a) G)]

4.11 Show that the solution to the equations governing the bending of a hinged-fixed beam according to the Timoshenko beam theory, under uniformly distributed transverse load, is given by

Show that the generai solution of Eq. (3) is

(3)

(2)

(1) b ( b o ) dW KGxzbhX + KGxzbh - bNxx dx = K1 b dx o - ExxIYY dx - bNxx W - K1x + K2

Combine the above two equations to arrive at d2W ,>..2 dx2 + >..2W + bN2x (K1x + K2) =O

This equation shows that the deflection of the Timoshenko beam theory can be obtained from that of the classica! beam theory by replacing the load ij_ [see Eq. (4.2.lOb)] with a equivalent load given by the right-hand side of the above equation. Although the effect of shear deformation is zero when the load variation is linear or less, this effect will come through the boundary conditions.

4.10 Show that the equations governing the stability of a laminated beam according to the Timoshenko theory can be expressed as

4.7 Show that the characteristic equation governing natural vibration of a hinged-free laminated beam using the classica! beam theory, when rotary inertia is neglected, is the same as that for a clamped-hinged beam.

4.8 Derive the characteristic equation governing natural vibration of a clamped-hinged laminated beam using the classica! beam theory, when rotary inertia is not neglected.

4.9 Show that Eqs. (4.3.lOa,b) can be reduced to the single equation

e: I d4wo '( ) ( E~xiyy ) d2ij_ xx YY dx4 = q x - KGLbh dx2

4.6 Show that the characteristic equation governing natural vibration of a clamped-hinged laminated beam using the classica! beam theory, when rotary inertia is neglected, is

sin >..a cosh >..a - cos >..a sinh >..a = O

4.5 Show that the characteristic equation governing natural vibration of a clamped-free laminated beam using the classica! beam theory is given by

cos >..a cosh >..a + 1 = O

4.4 Show that the characteristic equation governing buckling of a clamped-hinged laminated beam using the classica! beam theory is given by

sin Xc - >..acos>..a =O


Use the laminate constitutive equations (4.4.13a), (4.4.13c), and (4.4.14a) to express the resulting Euler-Lagrange equations in terms of the displacements and the thermal stress resultants. These equations are a static versi on of those in Eqs. ( 4.4. l).

( d2w0) } + Mxx 8 - dx2 - q8wo dx

8W = r {N 8 [duo ~ (dwo)2] N 8 (dvo) J o xx dx + 2 dx + xy dx

4.16 Derive the equations of equilibrium for cylindrical bending using the principle of virtual displacements, 8W = O, where

(2)

511 = ê (f3fow2 + KG~zbh - -\2 E~xIYY) 522 = Õ (f3fow2 + KG~zbh + Õ2E~Jyy)

(1) 511 cos êa sinh Õa+ 522 sin êa cosh Õa= O

4.15 Show that the characteristic equation governing natural vibrations of a clamped-hinged beam according to the Timoshenko beam theory is given by

(2)

R1 = f3Čow2 + KG~zbh - -\2 E~xlyy, R2 = f3Čow2 + KG~zbh + /J,2 E~Jm1 51 = fow2 - -\2 KG~zbh , 52 = fow2 + Õ2 KG~zbh

ê ( ~~ + ~~ ~~) + Õ ( 1 - ~~ ~~) sin êa sinh Õa - ê ( 1 + ~~ ) cos êa cosh Õa = O ( 1)

4.13 Determine the criticai buckling load of a clamped-free laminated beam using the Timoshenko beam theory.

4.14 Show that the characteristic equation governing natural vibrations of a clamped-free beam according to the Timoshenko beam theory, when rotary inertia is neglected, is given by

(5) 1 1 bN2x KGb bh xz

In addition, note that

(1)

(2)

(3)

(4)

wa(O) =O gives c2 + c4 =O

<Px(O) =O gives ( 1 - :~:xbh) êc1 + C3 =O wo(a) =O gives c1 sin .êa + c2 cos êa + c3a + c4 =O

ddxx (a)= O gives ê2 ( 1 - Kb~:bh) (c1 sin .êa + c2 cos êa) =O

Ans: The boundary conditions give

' ' . ' (1 .ê2 E~xlyy) = O Aa cos Aa - sin Aa + KGLbh

4.12 Show that the characteristic equation governing the buckling load of a hinged-f³xed beam according to the Timoshenko beam theory is given by


to construct a one-parameter (for each variable) Ritz solution of (u0,v0,w0) fora simply supported plate strip. Use algebraic polynomials for the approximate functions. (Ans: a1 = -Bqoa2/12AD, b1 = -Cqoa2/I2AD, c1 = -qoa2/24D.)

2 ( 2 )2 l d w0 du0 dv0 D11 d wo --- (B11- + B16-) + -- -- -qw0 dx dx? dx dx 2 d:r;2

II( ) _ t [Au (duo)2 A duo dv0 A66 (dv0)2 uo,vo,wo - Jo 2 dx + 16 dx dx + 2 dx

to construct a one-parameter Ritz solution to determine the natural frequency of vibration, w, of a simply supported laminated beam with compressive load N~xĀ Use algebraic polynomials for the approximate functions. (Ans: w = (1/a)j(10/Io)[(12Egxlyy/a2b) - Ngx].)

4.20 Repeat Problem 4.19 for a laminated beam with clamped boundar condition at x =O and free at x =a (i.e., cantilever beam). (Ans: w = (1/a) (5/3Io)[(12Egxlyy/a2b) - 4Ngx].)

4.21 Use the tota! potential energy functional

1a [ ( 2 )2 2 l 1 b d wo 0 dwo 2 2 II(wo) = 2 0 ExxIYY dx2 - bNxx ( dx ) - Iobw w0 dx

where G" and H" are constants to be determined such that the stress boundary conditions on CJxz and r7zz at z = Ñh/2 and the stress continuity conditions at the interfaces are satisfied.

4.19 Use the total potential energy functional

k Q~l ( z3 z2 k k) '""' Ā <Tzz = D A115 -B112 +G Z +H ~ QmS!Ill³mX

and (b) the transverse stresses from the 3-D equations of equilibrium are given by

<Tk = (A11z-B11)Q~1 '""'Qm . xx D Z:: -2- sin o-sz

am 1n=l

where D = A11D11 - Bf1 and am = ':". The load q(x) is also expanded in sine series with coefficient Qm.

4.18 For the cylindrical bending problem of cross-ply plates (see Problem 4.17), show that (a) the stresses in the kth layer are given by

Show that the Navier solution of these equations for the simply supported boundary conditions is given by

4.17 Consider the equations of equilibrium of cross-ply laminates in cylindrical bending in the absence of thermal effects:


(b)

(Eld2wo) =Q2 di; x=O

- (Eld2wo) = Q4 dx

-rzx l,

!!._ (Eld2wo) = Q1, dx dx

x=O

_!!._ (E1d2wo) =Q3, dx dx

:1:=L

ami force boundary conditions

(a) dwo (L) _, --- - U4 dx

wherc w0(x) is the transverse defiection, rf>x is the rotation, D is the Hexural stiffness, S is the shear stiffncss, and N2x is the axial compressive load. Determine the criticai buckling load of a beam clamped at one end and sirnply supported at the other end. Use one-pararneter Rayleigh-Hitz approximation for each variable.

4.28 Consider a laminated beam of length L, flexural stiffness El =constant, and subjected to uniforrnly distributed transverse load q(x) = q0. Suppose t hat the beam is subjected to the following geometrie boundary conditions

to construct a one-pararneter (for each variable) Ritz solution to determine the natural frequency of vibration, w, of a sirnply supported plate strip with edge compressive load N2xĀ Use algebraic polynomials for the approximate functions. (Ans: w = (1/a)J(10/Io)[(12D/a2)- N23).)

4.25 Repeat Exercise 4.24 for a plate strip with clamped boundary condition at :r = O and free at x =a. (Ans: w = (1/a)j(20/3Io)[(3D/a,2)- NpJ.)

4.26 Repeat Exercise 4.25 for cylindrical bending of a plate strip using the first-order shear deformation theory but neglccting rotary inertia.

4.27 Consider the buckling of a uniform beam according to the Timoshenko bcam theory. The tota! potential energy functional for the problern can be written as

2 ( 2 )2 2 l d w0 (B duo B dv0) Du d wo low 2 ia: --- 11-- + 15- + -- --- - --11!0 (,,[, dx2 dx Ā di; 2 dx2 2

1<1 [ 2 2 o 2 Il( . ) _ Au (duo) A duo dvo A55 (rivo) Nxx (dwo) uo,vo,wo - -- -- + 15---- +-- -- - -- --

0 2 dx dx dx 2 dx 2 dx

to construct a one-parameter (for each variable) Ritz solution to determine the criticai buckling load N.,,. of a plate strip with clamped boundary conditions at x = O and free boundary conditions at :r = a. Use algebraic polynornials for the approximate functions. (Ans: Ne, = 3D/a2.)


Il( . )-la [Au (duo)2 A duo dvo A55 (dvo)2 è: (dw0)2 uo,vo,wo - -- -- + 15---- + -- -- - -- -- 0 2 dx dx dx 2 dx 2 dx

_d2wo(B dĀuo B dvo) Dn(d2wo)2]dĿ dx2 11 dx + 16 dx + 2 dx2 x

4.22 Repeat Problern 4.21 for a plate strip wit.h clarnped boundary conditions at x = O and free boundary conditions at x =a. (Ans: a1 = Bqoa2 /6AD, b1 = Cqoa2 /6AD, c1 = qoa2 /12D.)



(a)

Rewrite the constants a; in terms of the values of <!>x at x = O, x = 0.5L, and x = L and obtain

(Ans: The stiffness matrix [K] and force vector {q} are the same as those given in Section 10.2 for the Euler-Bernoulli beam element.)

4.30 Since Eq. ( d) of Problem 4.29 is valid for any boundary conditions, it can be used to determine solutions (which turn out to be exact) even for indeterminate beams. In particular, determine the displacement in the spring that supports the right end of a beam when the left end is fixed and the beam is subjected to uniformly distributed transverse load q0.

4.31 Equations ( 4.3.12b) and ( 4.3.13b) for <!>x and wo of the Timoshenko beam theory suggest that they can be approximated with quadratic and cubie polynomials

(d) [K]{u} = {q} + {Q}

(a) Define and evaluate the coefficients K;1 of the stiffness matrix and q; of the force vector when EI = constant and q(x) = qo, a constant, and (b) use the total potential energy principle to determine the four-parameter Ritz solution for the problem. In particular, show that

(e) 4 4 4

II(u1,uz,u3,u4) = LL ~K;jUiUj- L(q;u; +Qjuj) i=l j=l j=l

and express it in the form

(b) 1 t' (d2 )2 t' 4 II(wo) = 2 lo EI d;io dx - lo q(x)wo(x) dx - L Qjuj o o 1=1

into the tota! potential energy functional associated with the Euler-Bernoulli beam theory

(a) 4

wo(x) = 2.:::u]4?i(x) j=l

Define the functions 4);(x) (i= 1,2,3,4) that you derived. These functions can serve as the approximation functions for the Rayleigh-Ritz method (see the next exercise). (Ans: 4?i are the same as the Hermite cubie interpolation functions given in Section 10.2.)

4.29 ( Continuation of Problem 4.28) Substitute the approximation

(d)

and express the constants c1, c2, c3, and c4 in terms of u1, u2, u3, and u4 using the geometrie boundary conditions (a) and rewrite (e) in the form

(e)

Here (u1,u2) and (u3,u4) denote the transverse defiections and rotations (clockwise) at the left and right ends, respectively, and ( Q1, Q3) and ( Q2, Q 4) are the associated shear forces and bending moments at the same points. Note that u; and Q; are introduced into the formulation to have the convenience of specifying a geometrie or force boundary condition.

Assume Ritz approximation of the form (the exact solution of the homogeneous equation, EJd4w0/dx4 =O suggests this polynomial)


Show that the constants of integration in Problem 4.33 are given by

(1)

(2)

dwE w{f(O) = w{f (L) = d: (O)= M:X(L) =O

wif (O)= wif(L) = <PT(O) = M:f'x(L) =O

EBT:

TBT:

4.36 Consider bending of a beam of length L, clamped (or fixed) at the left end and simply supported at the right, and subjected to a uniformly distributed transverse load q0. The boundary conditions of the Euler-Bernoulli and Timoshenko beam theories for the problem are as follows:

where C1,C2,C3, and C4 are constants of integration, which are to be determined using the boundary conditions of the particular beam.

4.34 Show that for simply supported beams ali C; of Problem 4.33 are zero. 4.35 Show that for cantilevered beams ali C; except C4 = M!:,(O)Dxx/(AxzK.,) of Problem 4.33

are zero.

(2)

T - E - Dxx E ( Dxx . x3) DxxWo (x) - DxxWo (x) + AxzK., Mxx(x) + C1 AxzKs X - 6 x2

-C22-C3x-C4 dwE x2

DxxtPT(x) =-Dxx d: +Ci 2 + C2x + C3 Mfx(x) = M:'.,(x) +Cix+ C2, Q;;(x) = Q~(x) + C1

where K,, is the shear correction coefficient, and superscripts E and T on variables refer to the Euler-Bernoulli and Timoshenko beam theories. Show that

(la - e)

where P, (j = 1,2,3) are the moments corresponding to the rotations <Pj. Then use the tota! potential energy principle to derive the Ritz equations for the problem.

4.33 The deftection, bending moment, and shear force of the Timoshenko beam theory can be expressed in terms of the corresponding quantities of the Euler-Bernoull³ beam theory (see [27,28]). In order to establish these relationships, we use the following equations of the two theories:

(a)

4.32 Use Eq. (a) of Problem 4.31 and Eq. (a) of Exercise 4.29 to express the tota! potential energy functional in terms of Uj and <P j:

where <P1 = <Px(O) etc. Show that 1/Ji(x) (i= 1, 2, 3) are the quadratic Lagrange interpolation functions derived in Section 10.3.



2. Reddy, J. N. (Ed.), Mechanics of Composite Materials. Selected Works of Nicholas J. Pagano, Kluwer, The Netherlands (1994).

3. Whitney, J. M. , Structural Analysis of Laminated Anisotropie Plates, Technomic, Lancaster, PA (1987).

4. Reissner, E., "The Effect of Transverse Shear Deformation on the Bcnding of Elastic Plates," Journal of Applied Mechanics, 12, 69-77 (1945).

5. Mindlin, R. D., "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropie Elastic Plates," Journal of Applied Mechanics, Transactions of ASME, 18, 31--38 (1951).

6. Uflyand, Ya. S., "The Propagation of Waves in the Transverse Vibrations of Bars and Plates," Akad. Nauk SSSR. Prikl Mat. Mekh., 12, 287-300 (1948) (in Russian).

7. Yang, P. C., Norris, C. H., and Stavsky, Y., "Elastic Wave Propagation in Heterogeneous Plates," International Journal of Solids and Struciures, 2, 665-684 (1966).

8. Whitney, J. M., "Shear Correction Factors for Orthotropic Laminates Under Static Load," Journal of Applied Mechanics, 40(1), 302-304 (1973).

9. Brogan, W. L., Modern Control Theory, Prentice-Hall, Englewood Ciiffs, NJ (1985). 10. Franklin, J. N., Matrix Theory, Prentice-Hall, Englewood Cliffs, NJ (1968). 11. Goldberg, J. L. and Schwartz, A. J., Systems of Ordinary Differential Equations, An Introduction, Harper and Row, New York, 1972.

12. Khdeir, A. A. and Reddy, J. N., "Free Vibration of Cross-Ply Laminated Beams with Arbitrary Boundary Conditions," International Journal of Engineering Science, 32 (12), 1971-1980 (1994).

13. Pagano, N. J., "Exact Solutions for Composite Laminates in Cylindrical Bending," Journal of Composite Materials, 3, 398-411 (1969).

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15. Pagano, N. J. and Wang, A. S. D., "Further Study of Composite Laminates Under Cylindrical Bending," Journal of Composite Materials, 5, 521-528 (1971).

16. Weaver, W., Jr., Timoshenko, S. P., and Young, D. H., Vibration Problems in Engineering, Fifth Edition, John Wiley, New York (1990).

17. Clough, R. W. and Penzien, J., Dynamics of Struciures, McGraw--Hill, New York (1975). 18. Pipes, L. A. and Harvill, L. R., Applied Mathematics [or Engineers and Physicists, Third Edition, McGraw-Hill, New York (1970).

19. Timoshenko, S. P. and Gere, J. P., Theory of Elastic Stability, Second Edition, McGraw-Hill, New York (1959).

20. Timoshenko, S. P., "On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars," Philosophical Magazine, 41, 744--746 (1921).

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23. Levinson, M., "A New Rectangular Beam Theory," Journal o] Sound and Vibration, 74, 81--87 (1981).

24. Bickford, W. B., "A Consistent Higher Order Bearn Theory," Developments in Theoretical and Applied Mechanics, 11, 137-150 (1982).

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