Energy Principles and Variational Methods in Applied Mechanics

Contents Preface to the Third Edition Preface 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 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 About the Author Companion Website 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 Karman 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 Energy 179 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 phi 0 and phi 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 344 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 Levy 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 Stiffness Coefficients 643 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 Stiffness Coefficients for the BET. 690 10.6.3.4 Tangent Stiffness Coefficients for the TBT. 692 10.6.3.5 Tangent Stiffness Coefficients for the CPT 693 10.6.3.6 Tangent Stiffness Coefficients for the FSDT 693 10.6.4 Numerical Results for Beams and Circular Plates 694 10.6.4.1 Beams 694 10.6.4.2 Circular Plates 697 10.7 Summary. 699 Problems 700 References. 701 Answers to Most Problems. 711 Index 723