Energy Principles and Variational Methods in Applied Mechanics
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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 ¿0 and ¿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 Le¿vy 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
