Energy Principles andVariational Methods inApplied Mechanics

Energy Principles andVariational Methods inApplied Mechanics

Third Edition

J. N. ReddyTexas A&M University, USA

This edition first published 2017

c© 2017 John Wiley & Sons Ltd

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Contents

Preface to the Third EditionPreface to the Second Edition Preface to the First Edition

About the

1. Introduction and Mathematical Preliminaries . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Preliminary Comments . . . . . . . . . . . . . . . . 11.1.2 The Role of Energy Methods and Variational Principles . . 11.1.3 A Brief Review of Historical Developments . . . . . . . 21.1.4 Preview . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 51.2.2 Definition of a Vector . . . . . . . . . . . . . . . . . 61.2.3 Scalar and Vector Products . . . . . . . . . . . . . . 81.2.4 Components of a Vector . . . . . . . . . . . . . . . 121.2.5 Summation Convention . . . . . . . . . . . . . . . 131.2.6 Vector Calculus . . . . . . . . . . . . . . . . . . 171.2.7 Gradient, Divergence, and Curl Theorems . . . . . . . 22

1.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 Second-Order Tensors . . . . . . . . . . . . . . . . 261.3.2 General Properties of a Dyadic . . . . . . . . . . . . 291.3.3 Nonion Form and Matrix Representation of a Dyad . . . 301.3.4 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 . . . . . . . . . . . . . 472.1.2 Descriptions of Motion . . . . . . . . . . . . . . . 48

2.2 Balance of Linear and Angular Momenta . . . . . . . . . 50

2.2.1 Equations of Motion . . . . . . . . . . . . . . . . 502.2.2 Symmetry of Stress Tensors . . . . . . . . . . . . . 54

About the Author Companion Website

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viii CONTENTS

2.3 Kinematics of Deformation . . . . . . . . . . . . . . . . 56

2.3.1 Green–Lagrange Strain Tensor . . . . . . . . . . . . 562.3.2 Strain Compatibility Equations . . . . . . . . . . . 62

2.4 Constitutive Equations . . . . . . . . . . . . . . . . . 65

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . 652.4.2 Generalized Hooke’s Law . . . . . . . . . . . . . . 662.4.3 Plane Stress–Reduced Constitutive Relations . . . . . 682.4.4 Thermoelastic Constitutive Relations . . . . . . . . . 70

2.5 Theories of Straight Beams . . . . . . . . . . . . . . . . 71

2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . 712.5.2 The Bernoulli–Euler Beam Theory . . . . . . . . . . 732.5.3 The Timoshenko Beam Theory . . . . . . . . . . . 762.5.4 The von Karman Theory of Beams . . . . . . . . . . 81

2.5.4.1 Preliminary Discussion . . . . . . . . . . . . 812.5.4.2 The Bernoulli–Euler Beam Theory . . . . . . 822.5.4.3 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 . . . . . . . . . . . . . . . 973.1.2 External and Internal Work Done . . . . . . . . . . 98

3.2 Strain Energy and Complementary Strain Energy . . . . . . 102

3.2.1 General Development . . . . . . . . . . . . . . . . 1023.2.2 Expressions for Strain Energy and Complementary

Strain Energy Densities of Isotropic Linear Elastic Solids 107

3.2.2.1 Stain energy density . . . . . . . . . . . . . 1073.2.2.2 Complementary stain energy density . . . . . . 108

3.2.3 Strain Energy and Complementary Strain Energyfor Trusses . . . . . . . . . . . . . . . . . . . . 109

3.2.4 Strain Energy and Complementary Strain Energyfor Torsional Members . . . . . . . . . . . . . . . 114

3.2.5 Strain Energy and Complementary Strain Energyfor Beams . . . . . . . . . . . . . . . . . . . . . 117

3.2.5.1 The Bernoulli–Euler Beam Theory . . . . . . 1173.2.5.2 The Timoshenko Beam Theory . . . . . . . . 119

CONTENTS ix

3.3 Total Potential Energy and Total Complementary Energy . . 123

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . 1233.3.2 Total Potential Energy of Beams . . . . . . . . . . . 1243.3.3 Total Complementary Energy of Beams . . . . . . . . 125

3.4 Virtual Work . . . . . . . . . . . . . . . . . . . . . . 126

3.4.1 Virtual Displacements . . . . . . . . . . . . . . . 1263.4.2 Virtual Forces . . . . . . . . . . . . . . . . . . . 131

3.5 Calculus of Variations . . . . . . . . . . . . . . . . . . 135

3.5.1 The Variational Operator . . . . . . . . . . . . . . 1353.5.2 Functionals . . . . . . . . . . . . . . . . . . . . 1383.5.3 The First Variation of a Functional . . . . . . . . . . 1393.5.4 Fundamental Lemma of Variational Calculus . . . . . 1403.5.5 Extremum of a Functional . . . . . . . . . . . . . . 1413.5.6 The Euler Equations . . . . . . . . . . . . . . . . 1433.5.7 Natural and Essential Boundary Conditions . . . . . . 1463.5.8 Minimization of Functionals with Equality Constraints . 151

3.5.8.1 The Lagrange Multiplier Method . . . . . . . 1513.5.8.2 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 . . . . . . . . . . . . . . . . . . . . 1674.2.2 Deformable Solids . . . . . . . . . . . . . . . . . 1684.2.3 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 Energy . . . 1794.3.2 Castigliano’s Theorem I . . . . . . . . . . . . . . . 188

4.4 The Principle of Virtual Forces . . . . . . . . . . . . . . 196

4.4.1 Deformable Solids . . . . . . . . . . . . . . . . . 1964.4.2 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 ComplementaryPotential Energy . . . . . . . . . . . . . . . . . . 204

4.5.2 Castigliano’s Theorem II . . . . . . . . . . . . . . 206

x CONTENTS

4.6 Clapeyron’s, Betti’s, and Maxwell’s Theorems . . . . . . . 217

4.6.1 Principle of Superposition for Linear Problems . . . . . 2174.6.2 Clapeyron’s Theorem . . . . . . . . . . . . . . . . 2204.6.3 Types of Elasticity Problems and Uniqueness of Solutions 2244.6.4 Betti’s Reciprocity Theorem . . . . . . . . . . . . . 2264.6.5 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 . . . . . . . . . . . . . . . . 2706.2.2 Linear Vector Spaces . . . . . . . . . . . . . . . . 2716.2.3 Normed and Inner Product Spaces . . . . . . . . . . 276

6.2.3.1 Norm . . . . . . . . . . . . . . . . . . . 2766.2.3.2 Inner product . . . . . . . . . . . . . . . . 2796.2.3.3 Orthogonality . . . . . . . . . . . . . . . . 280

6.2.4 Transformations, and Linear and Bilinear Forms . . . . 2816.2.5 Minimum of a Quadratic Functional . . . . . . . . . 282

6.3 The Ritz Method . . . . . . . . . . . . . . . . . . . . 287

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . 2876.3.2 Description of the Method . . . . . . . . . . . . . . 2886.3.3 Properties of Approximation Functions . . . . . . . . 293

6.3.3.1 Preliminary Comments . . . . . . . . . . . . 2936.3.3.2 Boundary Conditions . . . . . . . . . . . . 2936.3.3.3 Convergence . . . . . . . . . . . . . . . . 2946.3.3.4 Completeness . . . . . . . . . . . . . . . 2946.3.3.5 Requirements on φ0 and φi . . . . . . . . . 295

6.3.4 General Features of the Ritz Method . . . . . . . . . 299

CONTENTS xi

6.3.5 Examples . . . . . . . . . . . . . . . . . . . . . 3006.3.6 The Ritz Method for General Boundary-Value Problems 323

6.3.6.1 Preliminary Comments . . . . . . . . . . . . 3236.3.6.2 Weak Forms . . . . . . . . . . . . . . . . 3236.3.6.3 Model Equation 1 . . . . . . . . . . . . . . 3246.3.6.4 Model Equation 2 . . . . . . . . . . . . . . 3286.3.6.5 Model Equation 3 . . . . . . . . . . . . . . 3306.3.6.6 Ritz Approximations . . . . . . . . . . . . . 332

6.4 Weighted-Residual Methods . . . . . . . . . . . . . . . 337

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . 3376.4.2 The General Method of Weighted Residuals . . . . . . 3396.4.3 The Galerkin Method . . . . . . . . . . . . . . . . 3446.4.4 The Least-Squares Method . . . . . . . . . . . . . 3496.4.5 The Collocation Method . . . . . . . . . . . . . . 3566.4.6 The Subdomain Method . . . . . . . . . . . . . . 3596.4.7 Eigenvalue and Time-Dependent Problems . . . . . . 361

6.4.7.1 Eigenvalue Problems . . . . . . . . . . . . . 3616.4.7.2 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 . . . . . . . . . . . . . . . . . 3917.1.2 An Overview of Plate Theories . . . . . . . . . . . . 393

7.1.2.1 The Classical Plate Theory . . . . . . . . . . 3947.1.2.2 The First-Order Plate Theory . . . . . . . . . 3957.1.2.3 The Third-Order Plate Theory . . . . . . . . 3967.1.2.4 Stress–Based Theories . . . . . . . . . . . . 397

7.2 The Classical Plate Theory . . . . . . . . . . . . . . . . 398

7.2.1 Governing Equations of Circular Plates . . . . . . . . 3987.2.2 Analysis of Circular Plates . . . . . . . . . . . . . 405

7.2.2.1 Analytical Solutions For Bending . . . . . . . 4057.2.2.2 Analytical Solutions For Buckling . . . . . . . 4117.2.2.3 Variational Solutions . . . . . . . . . . . . 414

7.2.3 Governing Equations in Rectangular Coordinates . . . . 4277.2.4 Navier Solutions of Rectangular Plates . . . . . . . . 435

7.2.4.1 Bending . . . . . . . . . . . . . . . . . . 4387.2.4.2 Natural Vibration . . . . . . . . . . . . . . 4437.2.4.3 Buckling Analysis . . . . . . . . . . . . . . 4457.2.4.4 Transient Analysis . . . . . . . . . . . . . 447

xii CONTENTS

7.2.5 Levy Solutions of Rectangular Plates . . . . . . . . . 4497.2.6 Variational Solutions: Bending . . . . . . . . . . . . 4547.2.7 Variational Solutions: Natural Vibration . . . . . . . 4707.2.8 Variational Solutions: Buckling . . . . . . . . . . . 475

7.2.8.1 Rectangular Plates Simply Supported along TwoOpposite Sides and Compressed in the DirectionPerpendicular to Those Sides . . . . . . . . . 475

7.2.8.2 Formulation for Rectangular Plates withArbitrary Boundary Conditions . . . . . . . . 478

7.3 The First-Order Shear Deformation Plate Theory . . . . . . 486

7.3.1 Equations of Circular Plates . . . . . . . . . . . . . 4867.3.2 Exact Solutions of Axisymmetric Circular Plates . . . . 4887.3.3 Equations of Plates in Rectangular Coordinates . . . . 4927.3.4 Exact Solutions of Rectangular Plates . . . . . . . . 496

7.3.4.1 Bending Analysis . . . . . . . . . . . . . . 4987.3.4.2 Natural Vibration . . . . . . . . . . . . . . 5017.3.4.3 Buckling Analysis . . . . . . . . . . . . . . 502

7.3.5 Variational Solutions of Circular and Rectangular Plates 503

7.3.5.1 Axisymmetric Circular Plates . . . . . . . . . 5037.3.5.2 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 . . . . . . . . . . . . 5087.4.1.2 Relationships Between BET and TBT . . . . . 508

7.4.2 Circular Plates . . . . . . . . . . . . . . . . . . . 5127.4.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5298.2.2 Representation of the Domain by Finite Elements . . . . . . . . 5308.2.3 Weak Form over an Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5318.2.4 Approximation over an Element . . . . . . . . . . . . . . . . . . . . . . . . . . 5328.2.5 Finite Element Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

8.2.5.1 Linear Element . . . . . . . . . . . . . . . 5388.2.5.2 Quadratic Element . . . . . . . . . . . . . 539

CONTENTS xiii

8.2.6 Assembly (Connectivity) of Elements . . . . . . . . . . . . . . . . . . . . . 5398.2.7 Imposition of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 5428.2.8 Postprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

8.3 Finite Element Analysis of the Bernoulli–Euler

Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

8.3.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5498.3.2 Weak Form over an Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5498.3.3 Derivation of the Approximation Functions . . . . . . . . . . . . . . . 5508.3.4 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5528.3.5 Assembly of Element Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 5538.3.6 Imposition of Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 555

8.4 Finite Element Analysis of the Timoshenko Beam Theory . . . . . . 558

8.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5588.4.2 Weak Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5588.4.3 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5598.4.4 Reduced Integration Element (RIE) . . . . . . . . . . . . . . . . . . . . . . 5598.4.5 Consistent Interpolation Element (CIE) . . . . . . . . . . . . . . . . . . 5618.4.6 Superconvergent Element (SCE) . . . . . . . . . . . . . . . . . . . . . . . . . . 562

8.5 Finite Element Analysis of the Classical Plate Theory . . . . . . . . . . 565

8.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5658.5.2 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5668.5.3 Conforming and Nonconforming Plate Elements . . . . . . . . . . 5688.5.4 Fully Discretized Finite Element Models . . . . . . . . . . . . . . . . . . 569

8.5.4.1 Static Bending . . . . . . . . . . . . . . . 5698.5.4.2 Buckling . . . . . . . . . . . . . . . . . . 5698.5.4.3 Natural Vibration . . . . . . . . . . . . . . 5708.5.4.4 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. . . . . . . . . . . . . . . . . . . .5748.6.2 Finite Element Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . .5768.6.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5778.6.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5798.6.5 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .582

8.6.5.1 Isotropic Plates . . . . . . . . . . . . . . . 5828.6.5.2 Laminated Plates . . . . . . . . . . . . . . 584

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

xiv CONTENTS

9. Mixed Variational and Finite Element Formulations . . . . . . . 595

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

9.1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5959.1.2 Mixed Variational Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5959.1.3 Extremum and Stationary Behavior of Functionals. . . . . . . .597

9.2 Stationary Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

9.2.1 Minimum Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . 5999.2.2 The Hellinger–Reissner Variational Principle . . . . . . . . . . . . . .6019.2.3 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 . . . . . . . . . . . 6109.4.1.2 Weak-Form Mixed Finite Element Model. . . . . . . . . .6109.4.1.3 Weighted-Residual Finite Element Models . . . . . . . . 613

9.4.2 The Timoshenko Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

9.4.2.1 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6159.4.2.2 General Finite Element Model . . . . . . . . . . . . . . . . . . . . 6159.4.2.3 ASD–LLCC Element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6179.4.2.4 ASD–QLCC Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6179.4.2.5 ASD–HQLC Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

9.5 Mixed Finite Element Analysis of the Classical Plate Theory . . . 620

9.5.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6209.5.2 Mixed Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

9.5.2.1 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6209.5.2.2 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.5.2.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

9.5.3 Mixed Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

9.5.3.1 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6259.5.3.2 Weak Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6259.5.3.3 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 . . . . . . . . . . . . . . . . . 63810.2.1.2 Equations of motion and boundary conditions . 638

CONTENTS xv

10.2.2 The Timoshenko Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . 639

10.2.2.1 Displacement and strain fields . . . . . . . . . . . . . . . . . 63910.2.2.2 Equations of motion and boundary conditions . 640

10.2.3 Equations of Motion in terms of GeneralizedDisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641

10.2.3.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . 64110.2.3.2 Stress Resultants of BET. . . . . . . . . . . . . . . . . . . . . .64110.2.3.3 Stress Resultants of TBT. . . . . . . . . . . . . . . . . . . . . .64210.2.3.4 Equations of Motion of the BET . . . . . . . . . . . . . . 64210.2.3.5 Equations of Motion of the TBT . . . . . . . . . . . . . . 642

10.2.4 Stiffness Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

10.3 Functionally Graded Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 645

10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64510.3.2 Classical Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

10.3.2.1 Displacement and Strain Fields . . . . . . . . . . . . . . . .64610.3.2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 646

10.3.3 First-Order Shear Deformation Theory . . . . . . . . . . . . . . . . . 647

10.3.3.1 Displacement and Strain Fields . . . . . . . . . . . . . . . .64710.3.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 648

10.3.4 Plate Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

10.3.4.1 Classical Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . 64910.3.4.2 First-Order Plate Theory. . . . . . . . . . . . . . . . . . . . . .649

10.4 A General Third-Order Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . 650

10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65010.4.2 Displacements and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65110.4.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65310.4.4 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65710.4.5 Specialization to Other Theories . . . . . . . . . . . . . . . . . . . . . . . 658

10.4.5.1 A General Third-Order Plate Theory withTraction-Free Top and Bottom Surfaces . . . . . . . 658

10.4.5.2 The Reddy Third-Order Plate Theory . . . . . . . . . 66110.4.5.3 The First-Order Plate Theory . . . . . . . . . . . . . . . . . 66310.4.5.4 The Classical Plate Theory . . . . . . . . . . . . . . . . . . . . 664

10.5 Navier’s Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

10.5.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66410.5.2 Analysis of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

10.5.2.1 Bernoulli–Euler Beams . . . . . . . . . . . . . . . . . . . . . . . . 66510.5.2.2 Timoshenko Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . .66710.5.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

xvi CONTENTS

10.5.3 Analysis of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .671

10.5.3.1 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . .67210.5.3.2 Expansions of Generalized Displacements . . . . . . 67210.5.3.3 Bending Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67310.5.3.4 Free Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . 67610.5.3.5 Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67710.5.3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

10.6 Finite Element Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

10.6.1 Bending of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

10.6.1.1 Bernoulli–Euler Beam Theory . . . . . . . . . . . . . . . . . 68110.6.1.2 Timoshenko Beam Theory. . . . . . . . . . . . . . . . . . . . .683

10.6.2 Axisymmetric Bending of Circular Plates . . . . . . . . . . . . . . 684

10.6.2.1 Classical Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . 68110.6.2.2 First-Order Shear Deformation Plate Theory . . 686

10.6.3 Solution of Nonlinear Equations. . . . . . . . . . . . . . . . . . . . . . . .688

10.6.3.1 Times approximation . . . . . . . . . . . . . . . . . . . . . . . . . 68810.6.3.2 Newton’s Iteration Approach . . . . . . . . . . . . . . . . . 68810.6.3.3 Tangent Stiffness Coefficients for the BET . . . . . 69010.6.3.4 Tangent Stiffness Coefficients for the TBT. . . . . 69210.6.3.5 Tangent Stiffness Coefficients for the CPT . . . . . 69310.6.3.6 Tangent Stiffness Coefficients for the FSDT . . . 693

10.6.4 Numerical Results for Beams and Circular Plates . . . . . . 694

10.6.4.1 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69410.6.4.2 Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

10.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .699

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .701

Answers to Most Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

About the Author

J. N. Reddy is a university distinguished professor, regents’ professor, andthe holder of Oscar S. Wyatt Endowed Chair in the Department of MechanicalEngineering at Texas A&M University. He is known worldwide for his significantcontributions to the field of applied mechanics through the authorship of widelyused textbooks on the linear and nonlinear finite element analysis, variationalmethods, composite materials and structures, applied functional analysis, andcontinuum mechanics.

Professor Reddy’s earlier research focused primarily on mathematics of finiteelements, variational principles of mechanics, shear deformation and layerwisetheories of laminated composite plates and shells, analysis of bimodular materi-als, modeling of geological and geophysical phenomena, penalty finite elementsfor flows of viscous incompressible fluids, and least-squares finite element mod-els of fluid flows and solid continua. His pioneering works on the development ofshear deformation theories (that bear his name in the literature as the Reddythird-order plate theory and the Reddy layerwise theory) have had a majorimpact and have led to new research developments and applications. The fi-nite element formulations and models he developed have been implemented incommercial software like Abaqus, NISA, and HyperXtrude. In recent years, hisresearch involved the development of 7- and 12-parameter shell theories, non-local and non-classical continuum mechanics problems, and problems involvingcouple stresses, surface stress effects, discrete fracture and flow, micropolarcohesive damage, and continuum plasticity of metals from considerations ofnon-equilibrium thermodynamics.

His most significant awards and honors are: the Worcester Reed WarnerMedal and Charles Russ Richards Memorial Award from the American Societyof Mechanical Engineers; Archie Higdon Distinguished Educator Award from theMechanics Division of the American Society for Engineering Education; NathanM. Newmark Medal and Raymond D. Mindlin Medal from the American Societyof Civil Engineers; Excellence in the Field of Composites and Distinguished Re-search Award from the American Society for Composites; Computational SolidMechanics award from the US Association for Computational Mechanics; TheIACM O.C. Zienkiewicz Award from the International Association of Com-putational Mechanics; William Prager Medal from the Society of EngineeringScience; and ASME Medal from the American Society of Mechanical Engineers.He is a member of the US National Academy of Engineering and a foreignfellow of the Indian National Academy of Engineering. He is a fellow of allprofessional societies of his research (e.g., AIAA, ASC, ASCE, ASME, AAM,USACM, IACM).

xviii CONTENTS

Professor Reddy is one of the original top 100 ISI Highly Cited Researchersin Engineering around the world, with over 21,000 citations and an h-indexof over 68 per Web of Science; the number of citations is over 52,200 with anh-index of 92 according to Google Scholar. He is the founding editor in chief ofMechanics of Advanced Materials and Structures and International Journal forComputational Methods in Engineering Science and Mechanics and coeditor ofInternational Journal of Structural Stability and Dynamics; he also serves onthe editorial boards of more than two dozen journals. A more complete resumewith links to journal papers can be found at http://mechanics.tamu.edu.

“What we have done for ourselves alone dies with us; what we have done for others and theworld, remains and is immortal.” – Albert Pike (American attorney, soldier, and writer)

Note: Quotes by various people included in this book were found at different web sites. Forexample, visit:

http://naturalscience.com/dsqhome.html,http://thinkexist.com/quotes/david hilbert/,http://www.yalescientific.org/2010/10/from-the-editor-imagination-in-science/,https://www.brainyquote.com/quotes/topics/topic science.html. https://www.wikipedia.org/.

This author is motivated to include the quotes at various places in his book for their wit andwisdom; The author cannot vouch for their accuracy.

About the Companion Website

Don’t forget to visit the companion website for this book:

www.wiley.com/go/reddy/applied mechanics EPVM

There you will find valuable material designed to enhance your learning,including:1. Solutions manual

Scan this QR code to visit the companion website

Preface to the Third Edition

The development of new material systems paved the way for the develop-ment of new devices and improved components of existing structures. Devel-opment of mathematical models and their numerical simulations are central todesign. While a select number of courses on elasticity and structural mechan-ics provide engineers with background to formulate a suitable mathematicalmodel using the laws of physics (such as the second law of Newton), refinedmathematical models require the tools of variational calculus to formulate andevaluate them. The latest commercial codes like COMSOL Multiphysics R© andFEniCS R© require the user to just give the weak forms of the governing equa-tions that they like to solve using the finite element method. It is in thisconnection that a course on energy principles and variational methods providesthe needed background.

Most books on energy principles and variational methods traditionally treatthe topics, not incorporating the latest developments and generalizing the clas-sical methods to a broader class of problems. This textbook is unique (i.e.,there is no parallel to this book in its class) because most available books in themarket are very old (some are even out of print) and do not treat topics thatare of current interest. In particular, this book provides a systematic coverageof old and new topics (e.g., finite elements, functionally graded structures, re-lationships between classical and refined theories) and it contains illustrativeexamples and problem sets that enable readers to test their understanding ofthe subject matter.

The third edition of Energy Principles and Variational Methods has thesame objective as its previous editions, namely, to facilitate an easy and thor-ough understanding of the concepts and tools necessary to develop variationalformulations and associated numerical evaluations. The book offers easy-to-understand treatment of the subject of energy principles and variational meth-ods. The new edition is extensively reorganized and contains a substantiallylarge amount of new material. In particular, Chapters 1 and 2 are combinedinto a single chapter (Chapter 1 in the new edition); Chapter 2 on the review ofequations of solid mechanics is an extensively revised version of Chapter 3 fromthe second edition; Chapters 3 to 9 in the new edition correspond to Chapters4 to 10 of the second edition but with additional explanations, examples, andexercise problems; Chapter 10 in this new edition is entirely new and devotedto functionally graded beams and plates. In general, all of the chapters of thethird edition contain additional explanations, detailed example problems, andfresh exercise problems. Thus the new edition more than replaces the previouseditions, and it is hoped that it is acquired by the library of every institutionof higher learning by serious structural analysts.

xxii PREFACE TO THE THIRD EDITION

Since the publication of the previous editions, many users of the book com-municated their comments and compliments as well as errors they found, forwhich the author thanks them. The author is grateful to the following pro-fessional colleagues for their friendship, encouragement, and support over theyears:

Marcilio Alves, University of Sao Paulo, BrazilMarco Amabili, McGill University, CanadaErasmo Carrera, University of Torino, ItalyAntonio Ferreira, University of Porto, PortugalSomnath Ghosh, Johns Hopkins UniversityAntonio Grimaldi, University of Rome II, ItalyYonggang Huang, Northwestern UniversityS. Kitipornchai, University of Queensland, AustraliaK. M. Liew, City University of Hong KongC. W. Lim, City University of Hong KongFranco Maceri, University of Rome II, ItalyCristovao Mota Soares, Technical University of Lisbon, PortugalAntonio Miravete, Zaragoza University, SpainGlaucio Paulino, Georgia Institute of TechnologyAmirtham Rajagopal, Indian Institute of Technology, Hyderabad, IndiaJani Romanoff, Aalto University, FinlandJose Roesset, Texas A&M UniversityDebasish Roy, Indian Institute of Science, Bangalore, IndiaElio Sacco, University of Cassino and Southern Lazio, ItalyE. C. N. Silva, University of Sao Paulo, BrazilArun Srinivasa, Texas A&M UniversityKaran Surana, University of KansasLiqun Tang, South China University of TechnologyC. M. Wang, National University of SingaporeY. B. Yang, National University of Taiwan

Drafts of the manuscript of this book prior to its publication were read bythe author’s doctoral students, who have made suggestions for improvements.In particular, the author wishes to thank the following former and current stu-dents (listed in alphabetical order): Ronald Averill, K. Chandrashekhara, PaulHeyliger, Filis Kokkinos, Filipa Moleiro, Asghar Nosier, Felix Palmerio, GregoryPayette, Grama Praveen, Donald Robbins, Jr., Samit Roy, Ginu Unnikrishnan,Vinu Unnikrishnan, Archana Arbind, Parisa Khodabakhshi, Jinseok Kim, andNamhee Kim. The author requests readers to send their comments and correc-tions to jnreddy@tamu.edu.

J. N. ReddyCollege Station, TX

March 2017

Preface to the Second Edition

The increasing use of numerical and computational methods in engineeringand applied sciences has shed new light on the importance of energy principlesand variational methods. The number of engineering courses that make useof energy principles and variational formulations and methods has also grownvery rapidly in recent years. In view of the increase in the use of variationalformulations and methods (including the finite element method), there is aneed to introduce the concepts of energy principles and variational methodsand their use in the formulation and solution of problems of mechanics to bothundergraduate and beginning graduate students. This book, an extensivelyrevised version of the author’s earlier book Energy and Variational Methods inApplied Mechanics, is intended for senior undergraduate students and beginninggraduate students of aerospace, civil, and mechanical engineering and appliedmechanics who have had a course in fundamental engineering and ordinary andpartial differential equations.

The book is organized into 10 chapters and is self-contained as far as thesubject matter is concerned. Chapter 1 presents a general introduction to thesubject of variational principles. Chapter 2 contains a brief review of the algebraand calculus of vectors and Cartesian tensors, whereas Chapter 3 reviews of thebasic equations of linear solid continuum mechanics, which will be frequentlyreferred to in subsequent chapters. Much of the material presented in Chapters1 to 3 can be assigned as a reading material, especially in a graduate class.

Chapter 4 deals with the concepts of work and energy and basic topics ofvariational calculus, including the Euler equations, fundamental lemma of calcu-lus of variations, essential and natural boundary conditions, and minimizationof functionals with and without equality constraints. Principles of virtual workand energy and energy methods of solid and structural mechanics are presentedin Chapter 5. Chapter 6 is devoted to a discussion of Hamilton’s principle fordynamical systems. Classical variational methods of approximation (e.g., themethods of Ritz, Galerkin, and Kantorovich) are presented in Chapter 7. All ofthe concepts and methods presented in Chapters 4 to 7 are illustrated using barsand beams although the methods discussed in Chapter 7 are readily applicableto field problems whose differential equations resemble those of bars and beams.Chapter 8 is dedicated to applications of the energy principles and variationalmethods developed in earlier chapters to circular and rectangular plates. In theinterest of completeness and for use as a reference for approximate solutions,exact solutions are also included. The finite element method is introduced inChapter 9, with applications to beams and plates. Displacement finite elementmodels of Euler–Bernoulli and Timoshenko beam theories and classical andfirst-order shear deformation plate theories are presented. A unified approach,more general than that found in most solid mechanics books, is used to intro-

xxiv PREFACE TO THE SECOND EDITION

duce the finite element method. As a result, the student can readily extend themethod to other subject areas of solid mechanics and other branches of engi-neering. Lastly, the mixed variational principles of Hellinger and Reissner forelasticity are derived in Chapter 10. Mixed variational formulations, includingmixed finite element models of beams and plates, are discussed.

Each chapter of the book contains many example problems and exercisesthat illustrate, test, and broaden the understanding of the topics covered. Alist of references, by no means complete or up to date, is also provided at theend of each chapter. Answers to selective problems are included at the end ofthe book.

The book is suitable as a textbook for a senior undergraduate course or afirst-year graduate course on energy principles and variational methods taughtin aerospace, civil, and mechanical engineering and applied mechanics depart-ments. To gain the most from the text, the student should have a senior under-graduate or first-year graduate standing in engineering. Some familiarity withbasic courses in differential equations, mechanics of materials, and dynamicswould also be helpful.

The author has professionally benefited from the works, encouragement, andsupport of many colleagues and students who have taught him how to explaincomplicated concepts in simple terms. While it is not possible to name all ofthem, without their help and support, it would not have been possible for theauthor to modestly contribute to the field of mechanics through his teaching,research, and writing. Special thanks are due to his teacher Professor J. T.Oden (University of Texas at Austin) and Professor C. W. Bert (University ofOklahoma, Norman) for their mentorship, advice, and support.

J. N. ReddyCollege Station, TX

August 2002

Preface to the First Edition

The increasing use of finite element methods in engineering and applied sci-ence has shed new light on the importance of energy and variational methods.The number of engineering courses and research papers that make use of vari-ational and energy methods has also grown very rapidly in recent years. Inview of the increased use of variational methods (including the finite elementmethod), there is a need to introduce the concepts of energy and variationalmethods and their use in the formulation and solution of equations of me-chanics to both undergraduate and beginning graduate students. This bookis intended for senior undergraduate students and beginning graduate studentsof aerospace, civil, and mechanical engineering and applied mechanics, whohave had a course in ordinary and partial differential equations. The text isorganized into four chapters. Chapter 1 is essentially a review, especially forgraduate students, of the equations of applied mechanics. Much of the chaptercan be assigned as reading material to the student. The equations of bars,beams, torsion, and plane elasticity presented in Section 1.7 are used to illus-trate concepts from energy and variational methods. Chapter 2 deals with thestudy of the basic topics from variational calculus, virtual work and energyprinciples, and energy methods of mechanics. The instructor can omit Section2.4 on stationary principles and Section 2.5 on Hamilton’s principle if he or shewants to cover all of Chapter 4. Classical variational methods of approximation(e.g., the methods of Ritz, Galerkin, and Kantorovich) and the finite elementmethod are introduced and illustrated in Chapter 3 via linear problems of sci-ence and engineering, especially solid mechanics. A unified approach, moregeneral than that found in most solid mechanics books, is used to introduce thevariational methods. As a result, the student can readily extend the methodsto other subject areas of solid mechanics and other branches of engineering.

The classical variational methods and the finite element method are put towork in Chapter 4 in the derivation and approximate solution of the governingequations of elastic plates and shells. In the interest of completeness, and for useas a reference for approximate solutions, exact solutions of plates and shells arealso included. Keeping in mind the current developments in composite materialstructures, a brief but reasonably complete discussion of laminated plates andshells is included in Sections 4.3 and 4.4. The book contains many example andexercise problems that illustrate, test, and broaden the understanding of thetopics covered. A long list of references, by no means complete or up to date,is provided in Bibliography at the end of the book.

The author wishes to acknowledge, with great pleasure and appreciation,the encouragement and support by Professor Daniel Frederick (Head, ESMDepartment at Virginia Tech) during the course of this writing and the skillfultyping of the manuscript by Vanessa McCoy. The author is also thankful to the

xxvi PREFACE TO THE FIRST EDITION

many students who, through their comments, contributed to the improvementof this book. Special thanks to K. Chandrashekhara, Glenn Creamer, C. F.Liu, and Paul Heyliger for their help in proofreading the galleys and pages andto Dr. Ozden Ochoa for constructive comments on the preliminary draft of themanuscript. It is a pleasure to acknowledge, with many thanks, the cooperationof the technical staff at Wiley, New York (Frank Cerra, Christina Mikulak, andLisa Morano).

J. N. ReddyBlacksburg, VA

June 1984

1

1

Introduction andMathematical Preliminaries

1.1 Introduction

1.1.1 Preliminary Comments

The phrase “energy principles” or “energy methods” in the present study refersto methods that make use of the total potential energy (i.e., strain energy andpotential energy due to applied loads) of a system to obtain values of an un-known displacement or force, at a specific point of the system. These includeCastigliano’s theorems, unit dummy load and unit dummy displacement meth-ods, and Betti’s and Maxwell’s theorems. These methods are often limited tothe (exact) determination of generalized displacements or forces at fixed pointsin the structure; in most cases, they cannot be used to determine the com-plete solution (i.e., displacements and/or forces) as a function of position inthe structure. The phrase “variational methods,” on the other hand, refers tomethods that make use of the variational principles, such as the principles ofvirtual work and the principle of minimum total potential energy, to determineapproximate solutions as continuous functions of position in a body. In the clas-sical sense, a variational principle has to do with the minimization or findingstationary values of a functional with respect to a set of undetermined param-eters introduced in the assumed solution. The functional represents the totalenergy of the system in solid and structural mechanics problems, and in otherproblems it is simply an integral representation of the governing equations. Inall cases, the functional includes all the intrinsic features of the problem, suchas the governing equations, boundary and/or initial conditions, and constraintconditions.

1.1.2 The Role of Energy Methods and Variational Principles

Variational principles have always played an important role in mechanics. Vari-ational formulations can be useful in three related ways. First, many problemsof mechanics are posed in terms of finding the extremum (i.e., minima or max-ima) and thus, by their nature, can be formulated in terms of variational state-

Energy Principles and Variational Methods in Applied Mechanics, Third Edition. J.N. Reddy. ©2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.Companion Website: www.wiley.com/go/reddy/applied_mechanics_EPVM

2 INTRODUCTION AND MATHEMATICAL PRELIMINARIES

ments. 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 formu-lated by means of variational principles. Third, variational formulations form apowerful basis for obtaining approximate solutions to practical problems, manyof which are intractable otherwise. The principle of minimum total potentialenergy, for example, can be regarded as a substitute to the equations of equilib-rium of an elastic body, as well as a basis for the development of displacementfinite element models that can be used to determine approximate displacementand stress fields in the body. Variational formulations can also serve to unifydiverse fields, suggest new theories, and provide a powerful means for studyingthe existence and uniqueness of solutions to problems. In many cases they canalso be used to establish upper and/or lower bounds on approximate solutions.

1.1.3 A Brief Review of Historical Developments

In modern times, the term “variational formulation” applies to a wide spectrumof concepts having to do with weak, generalized, or direct variational formu-lations of boundary- and initial-value problems. Still, many of the essentialfeatures of variational methods remain the same as they were over 200 yearsago when the first notions of variational calculus began to be formulated.1

Although Archimedes (287–212 B.C.) is generally credited as the first touse work arguments in his study of levers, the most primitive ideas of varia-tional theory (the minimum hypothesis) are present in the writings of the Greekphilosopher Aristotle (384–322 B.C.), to be revived again by the Italian math-ematician/engineer Galileo (1564–1642), and finally formulated into a principleof least time by the French mathematician Fermat (1601–1665). The phrasevirtual velocities was used by Jean Bernoulli in 1717 in his letter to Varignon(1654–1722). The development of early variational calculus, by which we meanthe classical problems associated with minimizing certain functionals, had toawait the works of Newton (1642–1727) and Leibniz (1646–1716). The earli-est applications of such variational ideas included the classical isoperimetricproblem of finding among closed curves of given length the one that enclosesthe greatest area, and Newton’s problem of determining the solid of revolu-tion of “minimum resistance.” In 1696, Jean Bernoulli proposed the problemof the brachistochrone: among all curves connecting two points, find the curvetraversed in the shortest time by a particle under the influence of gravity. Itstood as a challenge to the mathematicians of their day to solve the problemusing the rudimentary tools of analysis then available to them or whatever newones they were capable of developing. Solutions to this problem were presentedby some of the greatest mathematicians of the time: Leibniz, Jean Bernoulli’solder brother Jacques Bernoulli, L’Hopital, and Newton.

1Many of the developments came from European scientists, whose works appeared in theirnative language and were not accessible to the whole scientific community.