Energy Principles and Variational Methods in Applied Mechanics

by J. N. Reddy

Overview

A comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics

This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates.

It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton’s principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method.

Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new material, including a new chapter devoted to the latest developments in functionally graded beams and plates.

• Offers clear and easy-to-follow descriptions of the concepts of work, energy, energy principles and variational methods
• Covers energy principles of solid and structural mechanics, traditional variational methods, the least-squares variational method, and the finite element, along with applications for each
• Provides an abundance of examples, in a problem-solving format, with descriptions of applications for equations derived in obtaining solutions to engineering structures
• Features end-of-the-chapter problems for course assignments, a Companion Website with a Solutions Manual, Instructor's Manual, figures, and more

Energy Principles and Variational Methods in Applied Mechanics, Third Edition is both a superb text/reference for engineering students in aerospace, civil, mechanical, and applied mechanics, and a valuable working resource for engineers in design and analysis in the aircraft, automobile, civil engineering, and shipbuilding industries.

Product Details

ISBN-13:	9781119087397
Publisher:	Wiley
Publication date:	07/21/2017
Sold by:	JOHN WILEY & SONS
Format:	eBook
Pages:	768
File size:	23 MB
Note:	This product may take a few minutes to download.

About the Author

J. N. REDDY, PhD, is a University Distinguished Professor and inaugural holder of the Oscar S. Wyatt Endowed Chair in Mechanical Engineering at Texas A&M University, College Station, TX. He has authored and coauthored several books, including Energy and Variational Methods in Applied Mechanics: Advanced Engineering Analysis (with M. L. Rasmussen), and A Mathematical Theory of Finite Elements (with J. T. Oden), both published by Wiley.

Read an Excerpt

Energy Principles and Variational Methods in Applied Mechanics

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By J. N. Reddy

John Wiley & Sons

ISBN: 0-471-17985-X

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Chapter One

INTRODUCTION

1.1 PRELIMINARY COMMENTS

The phrase "energy methods" in the present study refers to methods that make use of the total potential energy (i.e., strain energy and potential energy due to applied loads) of a system to obtain values of an unknown displacement or force, at a specific point of the system. These include Castigliano's theorems, unit-dummy-load and unit-dummy-displacement methods, and Betti's and Maxwell's theorems. These methods are often limited to the (exact) determination of generalized displacements or forces at fixed points in the structure; in most cases, they cannot be used to determine the complete solution (i.e., displacements and/or forces) as a function of position in the structure. The phrase "variational methods," on the other hand, refers to methods that make use of the variational principles, such as the principles of virtual work and the principle of minimum total potential energy, to determine approximate solutions as continuous functions of position in a body. In the classical sense, a variational principle has to do with the minimization or finding stationary values of a functional with respect to a set of undetermined parameters introduced in the assumed solution. The functional represents the total energy of the system in solid and structural mechanics problems, and in other problems it is simplyan integral representation of the governing equations. In all cases, the functional includes all the intrinsic features of the problem, such as the governing equations, boundary and/or initial conditions, and constraint conditions.

1.2 THE ROLE OF ENERGY METHODS AND VARIATIONAL PRINCIPLES

Variational principles have always played an important role in mechanics. Variational formulations can be useful in three related ways. First, many problems of mechanics are posed in terms of finding the extremum (i.e., minima or maxima) and thus, by their nature, can be formulated in terms of variational statements. Second, there are problems that can be formulated by other means, such as by vector mechanics (e.g., Newton's laws), but these can also be formulated by means of variational principles. Third, variational formulations form a powerful basis for obtaining approximate solutions to practical problems, many of which are intractable otherwise. The principle of minimum total potential energy, for example, can be regarded as a substitute for the equations of equilibrium of an elastic body, as well as a basis for the development of displacement finite element models that can be used to determine approximate displacement and stress fields in the body. Variational formulations can also serve to unify diverse fields, suggest new theories, and provide a powerful means for studying the existence and uniqueness of solutions to problems. In many cases they can also be used to establish upper and/or lower bounds on approximate solutions.

1.3 SOME HISTORICAL COMMENTS

In modern times, the term "variational formulation" applies to a wide spectrum of concepts having to do with weak, generalized, or direct variational formulations of boundary- and initial-value problems. Still, many of the essential features of variational methods remain the same as they were over 200 years ago when the first notions of variational calculus began to be formulated.

Although Archimedes (287-212 b.c.) is generally credited with the first to use work arguments in his study of levers, the most primitive ideas of variational theory (the minimum hypothesis) are present in the writings of the Greek philosopher Aristotle (384-322 b.c.), to be revived again by the Italian mathematician/engineer Galileo (1564-1642), and finally formulated into a Principle of Least Time by the French mathematician Fermat (1601-1665). The phrase virtual velocities was used by Jean Bernoulli in 1717 in his letter to Varignon (1654-1722). The development of early variational calculus, by which we mean the classical problems associated with minimizing certain functionals, had to await the works of Newton (1642-1727) and Leibniz (1646-1716). The earliest applications of such variational ideas included the classical isoperimetric problem of finding among closed curves of given length the one that encloses the greatest area, and Newton's problem of determining the solid of revolution of "minimum resistance." In 1696, Jean Bernoulli proposed the problem of the brachistochrone: among all curves connecting two points, find the curve traversed in the shortest time by a particle under the influence of gravity. It stood as a challenge to the mathematicians of their day to solve the problem using the rudimentary tools of analysis then available to them or whatever new ones they were capable of developing. Solutions to this problem were presented by some of the greatest mathematicians of the time: Leibniz, Jean Bernoulli's older brother Jacques Bernoulli, L'Hôpital, and Newton.

The first step toward developing a general method for solving variational problems was given by the Swiss genius Leonhard Euler (1707-1783) in 1732 when he presented a "general solution of the isoperimetric problem," although Maupertuis is credited with having put forward a law of minimal property of potential energy for stable equilibrium in his Mémoires de l'Académie des Sciences in 1740. It was in Euler's 1732 work and subsequent publication of the principle of least action (in his book Methodus inveniendi lineas curvas ...) in 1744 by Euler that variational concepts found a welcome and permanent home in mechanics. He developed all ideas surrounding the principle of minimum potential energy in his work on the Elastica, and he demonstrated the relationship between his variational equations and those governing the flexure and buckling of thin rods.

A great impetus to the development of variational mechanics began in the writings of Lagrange (1736-1813), first in his correspondence with Euler. Euler worked intensely in developing Lagrange's method, but delayed publishing his results until Lagrange's works were published in 1760 and 1761. Lagrange used d'Alembert's principle to convert dynamics to statics and then used the principle of virtual displacements to derive his famous equations governing the laws of dynamics in terms of kinetic and potential energy. Euler's work, together with Lagrange's Mécanique analytique of 1788, laid down the basis for the variational theory of dynamical systems. Further generalizations appeared in the fundamental work of Hamilton in 1834. Collectively, all these works have had a monumental impact on virtually every branch of mechanics.

A more solid mathematical basis for variational theory began to be developed in the eighteenth and early nineteenth century. Necessary conditions for the existence of "minimizing curves" of certain functionals were studied during this period, and we find among contributors of that era the familiar names of Legendre, Jacobi, and Weierstrass. Legendre gave criteria for distinguishing between maxima and minima in 1786, without considering criteria for existence, and Jacobi gave sufficient conditions for existence of extrema in 1837. Amore rigorous theory of existence of extrema was put together by Weierstrass, who, with Erdmann, established in 1865 conditions on extrema for variational problems involving corner behavior.

During the last half of the nineteenth century, the use of variational ideas was widespread among leaders in theoretical mechanics. We mention the works of Kirchhoff on plate theory, Lamé, Green, and Kelvin on elasticity, and the works of Betti, Maxwell, Castigliano, Menabrea, and Engesser for discrete structural systems. Lamé was the first in 1852 to prove a work equation, named after his colleague Clapeyron, for deformable bodies. Lamé's equation was used by Maxwell for the solution of redundant frame works using the unit-dummy-load technique. In 1875 Castigliano published an extremum version of this technique, but attributed the idea to Menabrea. A generalization of Castigliano's work is due to Engesser.

Among prominent contributors to the subject near the end of the nineteenth century and in the early years of the twentieth century, particularly in the area of variational methods of approximation and their applications to physical problems, were Rayleigh, Ritz, and Galerkin. Modern variational principles began in the 1950s with the works of Hellinger and Reissner on mixed variational principles for elasticity problems. A variety of generalizations of classical variational principles have appeared, and we shall not describe them here.

In closing this section, we note that a short historical account of early variational methods in mechanics can be found in the book of Lanczos and a brief review of certain aspects of the subject as it stood in the early 1950s can be found in the book of Truesdell and Toupin; additional information can be found in Smith's history of mathematics and in the historical treatises on mechanics by Mach, Dugas, and Timoshenko. Reference to much of the relevant contemporary literature can be found in the books by Washizu and Oden and Reddy. Additional historical papers and textbooks on variational methods are listed at the end of this chapter (see [17-56]).

1.4 PRESENT STUDY

The objective of the present study is to introduce energy methods and variational principles of solid and structural mechanics and to illustrate their use in the derivation and solution of the equations of applied mechanics, including plane elasticity, beams, frames, and plates. Of course, variational formulations and methods presented in this book are also applicable to problems outside solid mechanics. To equip the reader with the necessary mathematical tools and background from the theory of elasticity that are useful in the sequel, a review of vectors, matrices, tensors, and governing equations of elasticity are provided in the next two chapters. To keep the scope of the book within reasonable limits, only linear problems are considered. Although stability and vibration problems are introduced via examples and exercises, a detailed study of these topics is omitted.

In the following chapter we summarize the algebra and calculus of vectors and tensors. In Chapter 3 we give a brief review of the equations of solid mechanics, and in Chapter 4 we present the concepts of work and energy, energy principles, and Castigliano's theorems of structural mechanics. In Chapter 5 we present principles of virtual work, potential energy, and complementary energy. Chapter 6 is dedicated to Hamilton's principle for dynamical systems, and in Chapter 7 we introduce the Ritz, Galerkin, and weighted-residual methods. In Chapter 8, applications of variational methods to the formulation of plate bending theories and their solution by variational methods are presented. For the sake of completeness and comparison, analytical solutions of bending, vibration, and buckling of circular and rectangular plates are also presented. An introduction to the finite element method and its application to displacement finite element models of beams and plates is discussed in Chapter 9. The final chapter, Chapter 10, is devoted to the discussion of mixed variational principles, and mixed finite element models of beams and plates. To keep the scope of the book within reasonable limits, theory and analysis of shells is not included.

(Continues...)

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Table of Contents

About the Author xvii

About the Companion Website xix

Preface to the Third Edition xxi

Preface to the Second Edition xxiii

Preface to the First Edition xxv

1. Introduction and Mathematical Preliminaries 1

1.1 Introduction   1

1.1.1 Preliminary Comments 1

1.1.2 The Role of Energy Methods and Variational Principles 1

1.1.3 A Brief Review of Historical Developments 2

1.1.4 Preview 4

1.2 Vectors 5

1.2.1 Introduction 5

1.2.2 Definition of a Vector 6

1.2.3 Scalar and Vector Products 8

1.2.4 Components of a Vector 12

1.2.5 Summation Convention 13

1.2.6 Vector Calculus 17

1.2.7 Gradient, Divergence, and Curl Theorems 22

1.3 Tensors 26

1.3.1 Second-Order Tensors 26

1.3.2 General Properties of a Dyadic 29

1.3.3 Nonion Form and Matrix Representation of a Dyad 30

1.3.4 Eigenvectors Associated with Dyads 34

1.4 Summary 39

Problems 40

2. Review of Equations of Solid Mechanics 47

2.1 Introduction 47

2.1.1 Classification of Equations 47

2.1.2 Descriptions of Motion 48

2.2 Balance of Linear and Angular Momenta 50

2.2.1 Equations of Motion 50

2.2.2 Symmetry of Stress Tensors 54

2.3 Kinematics of Deformation 56

2.3.1 Green-Lagrange Strain Tensor 56

2.3.2 Strain Compatibility Equations 62

2.4 Constitutive Equations 65

2.4.1 Introduction 65

2.4.2 Generalized Hooke's Law 66

2.4.3 Plane Stress-Reduced Constitutive Relations 68

2.4.4 Thermoelastic Constitutive Relations 70

2.5 Theories of Straight Beams 71

2.5.1 Introduction 71

2.5.2 The Bernoulli-Euler Beam Theory 73

2.5.3 The Timoshenko Beam Theory 76

2.5.4 The von Ka^’rma^’n Theory of Beams 81

2.5.4.1 Preliminary Discussion 81

2.5.4.2 The Bernoulli-Euler Beam Theory 82

2.5.4.3 The Timoshenko Beam Theory 84

2.6 Summary 85

Problems 88

3. Work, Energy, and Variational Calculus 97

3.1 Concepts of Work and Energy 97

3.1.1 Preliminary Comments 97

3.1.2 External and Internal Work Done 98

3.2 Strain Energy and Complementary Strain Energy 102

3.2.1 General Development 102

3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 107

3.2.2.1 Stain energy density 107

3.2.2.2 Complementary stain energy density 108

3.2.3 Strain Energy and Complementary Strain Energy for Trusses 109

3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 114

3.2.5 Strain Energy and Complementary Strain Energy for Beams 117

3.2.5.1 The Bernoulli-Euler Beam Theory 117

3.2.5.2 The Timoshenko Beam Theory 119

3.3 Total Potential Energy and Total Complementary Energy 123

3.3.1 Introduction 123

3.3.2 Total Potential Energy of Beams 124

3.3.3 Total Complementary Energy of Beams 125

3.4 Virtual Work 126

3.4.1 Virtual Displacements 126

3.4.2 Virtual Forces 131

3.5 Calculus of Variations 135

3.5.1 The Variational Operator 135

3.5.2 Functionals 138

3.5.3 The First Variation of a Functional 139

3.5.4 Fundamental Lemma of Variational Calculus 140

3.5.5 Extremum of a Functional 141

3.5.6 The Euler Equations 143

3.5.7 Natural and Essential Boundary Conditions 146

3.5.8 Minimization of Functionals with Equality Constraints 151

3.5.8.1 The Lagrange Multiplier Method 151

3.5.8.2 The Penalty Function Method 153

3.6 Summary 156

Problems 159

4. Virtual Work and Energy Principles of Mechanics 167

4.1 Introduction 167

4.2 The Principle of Virtual Displacements 167

4.2.1 Rigid Bodies 167

4.2.2 Deformable Solids 168

4.2.3 Unit Dummy-Displacement Method 172

4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 179

4.3.1 The Principle of Minimum Total Potential Energy179

4.3.2 Castigliano's Theorem I 188

4.4 The Principle of Virtual Forces 196

4.4.1 Deformable Solids 196

4.4.2 Unit Dummy-Load Method 198

4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 204

4.5.1 The Principle of the Minimum total Complementary Potential Energy 204

4.5.2 Castigliano's Theorem II 206

4.6 Clapeyron's, Betti's, and Maxwell's Theorems 217

4.6.1 Principle of Superposition for Linear Problems 217

4.6.2 Clapeyron's Theorem 220

4.6.3 Types of Elasticity Problems and Uniqueness of Solutions 224

4.6.4 Betti's Reciprocity Theorem 226

4.6.5 Maxwell's Reciprocity Theorem 230

4.7 Summary 232

Problems 235

5. Dynamical Systems: Hamilton's Principle 243

5.1 Introduction 243

5.2 Hamilton's Principle for Discrete Systems 243

5.3 Hamilton's Principle for a Continuum 249

5.4 Hamilton's Principle for Constrained Systems 255

5.5 Rayleigh's Method 260

5.6 Summary 262

Problems 263

6. Direct Variational Methods 269

6.1 Introduction 269

6.2 Concepts from Functional Analysis 270

6.2.1 General Introduction 270

6.2.2 Linear Vector Spaces 271

6.2.3 Normed and Inner Product Spaces 276

6.2.3.1 Norm 276

6.2.3.2 Inner product 279

6.2.3.3 Orthogonality 280

6.2.4 Transformations, and Linear and Bilinear Forms 281

6.2.5 Minimum of a Quadratic Functional 282

6.3 The Ritz Method 287

6.3.1 Introduction 287

6.3.2 Description of the Method 288

6.3.3 Properties of Approximation Functions 293

6.3.3.1 Preliminary Comments 293

6.3.3.2 Boundary Conditions 293

6.3.3.3 Convergence 294

6.3.3.4 Completeness 294

6.3.3.5 Requirements on ɸ₀ and ɸ_i 295

6.3.4 General Features of the Ritz Method 299

6.3.5 Examples 300

6.3.6 The Ritz Method for General Boundary-Value Problems 323

6.3.6.1 Preliminary Comments 323

6.3.6.2 Weak Forms 323

6.3.6.3 Model Equation 1 324

6.3.6.4 Model Equation 2 328

6.3.6.5 Model Equation 3 330

6.3.6.6 Ritz Approximations 332

6.4 Weighted-Residual Methods 337

6.4.1 Introduction 337

6.4.2 The General Method of Weighted Residuals 339

6.4.3 The Galerkin Method 44

6.4.4 The Least-Squares Method 349

6.4.5 The Collocation Method 356

6.4.6 The Subdomain Method 359

6.4.7 Eigenvalue and Time-Dependent Problems 361

6.4.7.1 Eigenvalue Problems 361

6.4.7.2 Time-Dependent Problems 362

6.5 Summary 381

Problems 383

7. Theory and Analysis of Plates 391

7.1 Introduction 391

7.1.1 General Comments 391

7.1.2 An Overview of Plate Theories 393

7.1.2.1 The Classical Plate Theory 394

7.1.2.2 The First-Order Plate Theory 395

7.1.2.3 The Third-Order Plate Theory 396

7.1.2.4 Stress-Based Theories 397

7.2 The Classical Plate Theory 398

7.2.1 Governing Equations of Circular Plates 398

7.2.2 Analysis of Circular Plates 405

7.2.2.1 Analytical Solutions For Bending 405

7.2.2.2 Analytical Solutions For Buckling 411

7.2.2.3 Variational Solutions 414

7.2.3 Governing Equations in Rectangular Coordinates 427

7.2.4 Navier Solutions of Rectangular Plates 435

7.2.4.1 Bending 438

7.2.4.2 Natural Vibration 443

7.2.4.3 Buckling Analysis 445

7.2.4.4 Transient Analysis 447

7.2.5 Lévy Solutions of Rectangular Plates 449

7.2.6 Variational Solutions: Bending 454

7.2.7 Variational Solutions: Natural Vibration 470

7.2.8 Variational Solutions: Buckling 475

7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 475

7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 478

7.3 The First-Order Shear Deformation Plate Theory 486

7.3.1 Equations of Circular Plates 486

7.3.2 Exact Solutions of Axisymmetric Circular Plates 488

7.3.3 Equations of Plates in Rectangular Coordinates 492

7.3.4 Exact Solutions of Rectangular Plates 496

7.3.4.1 Bending Analysis 498

7.3.4.2 Natural Vibration 501

7.3.4.3 Buckling Analysis 502

7.3.5 Variational Solutions of Circular and Rectangular Plates 503

7.3.5.1 Axisymmetric Circular Plates 503

7.3.5.2 Rectangular Plates 505

7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 507

7.4.1 Beams 507

7.4.1.1 Governing Equations 508

7.4.1.2 Relationships Between BET and TBT 508

7.4.2 Circular Plates 512

7.4.3 Rectangular Plates 516

7.5 Summary 521

Problems 521

8. The Finite Element Method 527

8.1 Introduction 527

8.2 Finite Element Analysis of Straight Bars 529

8.2.1 Governing Equation 529

8.2.2 Representation of the Domain by Finite Elements 530

8.2.3 Weak Form over an Element 531

8.2.4 Approximation over an Element 532

8.2.5 Finite Element Equations 537

8.2.5.1 Linear Element 538

8.2.5.2 Quadratic Element 539

8.2.6 Assembly (Connectivity) of Elements 539

8.2.7 Imposition of Boundary Conditions 542

8.2.8 Postprocessing 543

8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 549

8.3.1 Governing Equation 549

8.3.2 Weak Form over an Element 549

8.3.3 Derivation of the Approximation Functions 550

8.3.4 Finite Element Model 552

8.3.5 Assembly of Element Equations 553

8.3.6 Imposition of Boundary Conditions 555

8.4 Finite Element Analysis of the Timoshenko Beam Theory 558

8.4.1 Governing Equations 558

8.4.2 Weak Forms 558

8.4.3 Finite Element Models 559

8.4.4 Reduced Integration Element (RIE) 559

8.4.5 Consistent Interpolation Element (CIE) 561

8.4.6 Superconvergent Element (SCE) 562

8.5 Finite Element Analysis of the Classical Plate Theory 565

8.5.1 Introduction 565

8.5.2 General Formulation 566

8.5.3 Conforming and Nonconforming Plate Elements 568

8.5.4 Fully Discretized Finite Element Models 569

8.5.4.1 Static Bending 569

8.5.4.2 Buckling 569

8.5.4.3 Natural Vibration 570

8.5.4.4 Transient Response 570

8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 574

8.6.1 Governing Equations and Weak Forms 574

8.6.2 Finite Element Approximations 576

8.6.3 Finite Element Model 577

8.6.4 Numerical Integration 579

8.6.5 Numerical Examples 582

8.6.5.1 Isotropic Plates 582

8.6.5.2 Laminated Plates 584

8.7 Summary 587

Problems 588

9. Mixed Variational and Finite Element Formulations 595

9.1 Introduction 595

9.1.1 General Comments 595

9.1.2 Mixed Variational Principles 595

9.1.3 Extremum and Stationary Behavior of Functionals 597

9.2 Stationary Variational Principles 599

9.2.1 Minimum Total Potential Energy 599

9.2.2 The Hellinger-Reissner Variational Principle 601

9.2.3 The Reissner Variational Principle 605

9.3 Variational Solutions Based on Mixed Formulations 606

9.4 Mixed Finite Element Models of Beams 610

9.4.1 The Bernoulli-Euler Beam Theory 610

9.4.1.1 Governing Equations And Weak Forms 610

9.4.1.2 Weak-Form Mixed Finite Element Model 610

9.4.1.3 Weighted-Residual Finite Element Models 613

9.4.2 The Timoshenko Beam Theory 615

9.4.2.1 Governing Equations 615

9.4.2.2 General Finite Element Model 615

9.4.2.3 ASD-LLCC Element 617

9.4.2.4 ASD-QLCC Element 617

9.4.2.5 ASD-HQLC Element 618

9.5 Mixed Finite Element Analysis of the Classical Plate Theory 620

9.5.1 Preliminary Comments 620

9.5.2 Mixed Model I 620

9.5.2.1 Governing Equations 620

9.5.2.2 Weak Forms 621

9.5.2.3 Finite Element Model 622

9.5.3 Mixed Model II 625

9.5.3.1 Governing Equations 625

9.5.3.2 Weak Forms 625

9.5.3.3 Finite Element Model 626

9.6 Summary 630

Problems 631

10. Analysis of Functionally Graded Beams and Plates 635

10.1 Introduction 635

10.2 Functionally Graded Beams 638

10.2.1 The Bernoulli-Euler Beam Theory 638

10.2.1.1 Displacement and strain fields  638

10.2.1.2 Equations of motion and boundary conditions 638

10.2.2 The Timoshenko Beam Theory 639

10.2.2.1 Displacement and strain fields 639

10.2.2.2 Equations of motion and boundary conditions 640

10.2.3 Equations of Motion in terms of Generalized Displacements 641

10.2.3.1 Constitutive Equations 641

10.2.3.2 Stress Resultants of BET 641

10.2.3.3 Stress Resultants of TBT 642

10.2.3.4 Equations of Motion of the BET 642

10.2.3.5 Equations of Motion of the TBT 642

10.2.4 Stiffiness Coefficients643

10.3 Functionally Graded Circular Plates 645

10.3.1 Introduction 645

10.3.2 Classical Plate Theory 646

10.3.2.1 Displacement and Strain Fields 646

10.3.2.2 Equations of Motion 646

10.3.3 First-Order Shear Deformation Theory 647

10.3.3.1 Displacement and Strain Fields 647

10.3.3.2 Equations of Motion 648

10.3.4 Plate Constitutive Relations 649

10.3.4.1 Classical Plate Theory 649

10.3.4.2 First-Order Plate Theory 649

10.4 A General Third-Order Plate Theory 650

10.4.1 Introduction 650

10.4.2 Displacements and Strains 651

10.4.3 Equations of Motion 653

10.4.4 Constitutive Relations 657

10.4.5 Specialization to Other Theories 658

10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 658

10.4.5.2 The Reddy Third-Order Plate Theory 661

10.4.5.3 The First-Order Plate Theory 663

10.4.5.4 The Classical Plate Theory 664

10.5 Navier's Solutions 664

10.5.1 Preliminary Comments 664

10.5.2 Analysis of Beams 665

10.5.2.1 Bernoulli-Euler Beams 665

10.5.2.2 Timoshenko Beams 667

10.5.2.3 Numerical Results 669

10.5.3 Analysis of Plates 671

10.5.3.1 Boundary Conditions 672

10.5.3.2 Expansions of Generalized Displacements 672

10.5.3.3 Bending Analysis 673

10.5.3.4 Free Vibration Analysis 676

10.5.3.5 Buckling Analysis 677

10.5.3.6 Numerical Results 679

10.6 Finite Element Models 681

10.6.1 Bending of Beams 681

10.6.1.1 Bernoulli-Euler Beam Theory 681

10.6.1.2 Timoshenko Beam Theory 683

10.6.2 Axisymmetric Bending of Circular Plates 684

10.6.2.1 Classical Plate Theory 681

10.6.2.2 First-Order Shear Deformation Plate Theory 686

10.6.3 Solution of Nonlinear Equations 688

10.6.3.1 Times approximation 688

10.6.3.2 Newton's Iteration Approach 688

10.6.3.3 Tangent Stiffiness Coefficients for the BET 690

10.6.3.4 Tangent Stiffiness Coefficients for the TBT 692

10.6.3.5 Tangent Stiffiness Coefficients for the CPT 693

10.6.3.6 Tangent Stiffiness Coefficients for the FSDT 693

10.6.4 Numerical Results for Beams and Circular Plates 694

10.6.4.1 Beams 694

10.6.4.2 Circular Plates 697

10.7 Summary 699

Problems 700

References 701

Answers to Most Problems 711

Index 723