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.

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.

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

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