Energy Principles and Variational Methods in Applied Mechanics

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.

Autorentext

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.

Inhalt
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 ...