Energy Principles and Variational Methods in Applied Mechanics

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 xviiAbout the Companion Website xixPreface to the Third Edition xxiPreface to the Second Edition xxiiiPreface to the First Edition xxv1. Introduction and Mathematical Preliminaries 11.1 Introduction   11.1.1 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 41.2 Vectors 51.2.1 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 221.3 Tensors 261.3.1 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 341.4 Summary 39Problems 402. Review of Equations of Solid Mechanics 472.1 Introduction 472.1.1 Classification of Equations 472.1.2 Descriptions of Motion 482.2 Balance of Linear and Angular Momenta 502.2.1 Equations of Motion 502.2.2 Symmetry of Stress Tensors 542.3 Kinematics of Deformation 562.3.1 Green-Lagrange Strain Tensor 562.3.2 Strain Compatibility Equations 622.4 Constitutive Equations 652.4.1 Introduction 652.4.2 Generalized Hooke's Law 662.4.3 Plane Stress-Reduced Constitutive Relations 682.4.4 Thermoelastic Constitutive Relations 702.5 Theories of Straight Beams 712.5.1 Introduction 712.5.2 The Bernoulli-Euler Beam Theory 732.5.3 The Timoshenko Beam Theory 762.5.4 The von Ka’rma’n Theory of Beams 812.5.4.1 Preliminary Discussion 812.5.4.2 The Bernoulli-Euler Beam Theory 822.5.4.3 The Timoshenko Beam Theory 842.6 Summary 85Problems 883. Work, Energy, and Variational Calculus 973.1 Concepts of Work and Energy 973.1.1 Preliminary Comments 973.1.2 External and Internal Work Done 983.2 Strain Energy and Complementary Strain Energy 1023.2.1 General Development 1023.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 1073.2.2.1 Stain energy density 1073.2.2.2 Complementary stain energy density 1083.2.3 Strain Energy and Complementary Strain Energy for Trusses 1093.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 1143.2.5 Strain Energy and Complementary Strain Energy for Beams 1173.2.5.1 The Bernoulli-Euler Beam Theory 1173.2.5.2 The Timoshenko Beam Theory 1193.3 Total Potential Energy and Total Complementary Energy 1233.3.1 Introduction 1233.3.2 Total Potential Energy of Beams 1243.3.3 Total Complementary Energy of Beams 1253.4 Virtual Work 1263.4.1 Virtual Displacements 1263.4.2 Virtual Forces 1313.5 Calculus of Variations 1353.5.1 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 1513.5.8.1 The Lagrange Multiplier Method 1513.5.8.2 The Penalty Function Method 1533.6 Summary 156Problems 1594. Virtual Work and Energy Principles of Mechanics 1674.1 Introduction 1674.2 The Principle of Virtual Displacements 1674.2.1 Rigid Bodies 1674.2.2 Deformable Solids 1684.2.3 Unit Dummy-Displacement Method 1724.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 1794.3.1 The Principle of Minimum Total Potential Energy1794.3.2 Castigliano's Theorem I 1884.4 The Principle of Virtual Forces 1964.4.1 Deformable Solids 1964.4.2 Unit Dummy-Load Method 1984.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 2044.5.1 The Principle of the Minimum total Complementary Potential Energy 2044.5.2 Castigliano's Theorem II 2064.6 Clapeyron's, Betti's, and Maxwell's Theorems 2174.6.1 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 2304.7 Summary 232Problems 2355. Dynamical Systems: Hamilton's Principle 2435.1 Introduction 2435.2 Hamilton's Principle for Discrete Systems 2435.3 Hamilton's Principle for a Continuum 2495.4 Hamilton's Principle for Constrained Systems 2555.5 Rayleigh's Method 2605.6 Summary 262Problems 2636. Direct Variational Methods 2696.1 Introduction 2696.2 Concepts from Functional Analysis 2706.2.1 General Introduction 2706.2.2 Linear Vector Spaces 2716.2.3 Normed and Inner Product Spaces 2766.2.3.1 Norm 2766.2.3.2 Inner product 2796.2.3.3 Orthogonality 2806.2.4 Transformations, and Linear and Bilinear Forms 2816.2.5 Minimum of a Quadratic Functional 2826.3 The Ritz Method 2876.3.1 Introduction 2876.3.2 Description of the Method 2886.3.3 Properties of Approximation Functions 2936.3.3.1 Preliminary Comments 2936.3.3.2 Boundary Conditions 2936.3.3.3 Convergence 2946.3.3.4 Completeness 2946.3.3.5 Requirements on ɸ0 and ɸi 2956.3.4 General Features of the Ritz Method 2996.3.5 Examples 3006.3.6 The Ritz Method for General Boundary-Value Problems 3236.3.6.1 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 3326.4 Weighted-Residual Methods 3376.4.1 Introduction 3376.4.2 The General Method of Weighted Residuals 3396.4.3 The Galerkin Method 446.4.4 The Least-Squares Method 3496.4.5 The Collocation Method 3566.4.6 The Subdomain Method 3596.4.7 Eigenvalue and Time-Dependent Problems 3616.4.7.1 Eigenvalue Problems 3616.4.7.2 Time-Dependent Problems 3626.5 Summary 381Problems 3837. Theory and Analysis of Plates 3917.1 Introduction 3917.1.1 General Comments 3917.1.2 An Overview of Plate Theories 3937.1.2.1 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 3977.2 The Classical Plate Theory 3987.2.1 Governing Equations of Circular Plates 3987.2.2 Analysis of Circular Plates 4057.2.2.1 Analytical Solutions For Bending 4057.2.2.2 Analytical Solutions For Buckling 4117.2.2.3 Variational Solutions 4147.2.3 Governing Equations in Rectangular Coordinates 4277.2.4 Navier Solutions of Rectangular Plates 4357.2.4.1 Bending 4387.2.4.2 Natural Vibration 4437.2.4.3 Buckling Analysis 4457.2.4.4 Transient Analysis 4477.2.5 Lévy Solutions of Rectangular Plates 4497.2.6 Variational Solutions: Bending 4547.2.7 Variational Solutions: Natural Vibration 4707.2.8 Variational Solutions: Buckling 4757.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 4757.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 4787.3 The First-Order Shear Deformation Plate Theory 4867.3.1 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 4967.3.4.1 Bending Analysis 4987.3.4.2 Natural Vibration 5017.3.4.3 Buckling Analysis 5027.3.5 Variational Solutions of Circular and Rectangular Plates 5037.3.5.1 Axisymmetric Circular Plates 5037.3.5.2 Rectangular Plates 5057.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 5077.4.1 Beams 5077.4.1.1 Governing Equations 5087.4.1.2 Relationships Between BET and TBT 5087.4.2 Circular Plates 5127.4.3 Rectangular Plates 5167.5 Summary 521Problems 5218. The Finite Element Method 5278.1 Introduction 5278.2 Finite Element Analysis of Straight Bars 5298.2.1 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 5378.2.5.1 Linear Element 5388.2.5.2 Quadratic Element 5398.2.6 Assembly (Connectivity) of Elements 5398.2.7 Imposition of Boundary Conditions 5428.2.8 Postprocessing 5438.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 5498.3.1 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 5558.4 Finite Element Analysis of the Timoshenko Beam Theory 5588.4.1 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) 5628.5 Finite Element Analysis of the Classical Plate Theory 5658.5.1 Introduction 5658.5.2 General Formulation 5668.5.3 Conforming and Nonconforming Plate Elements 5688.5.4 Fully Discretized Finite Element Models 5698.5.4.1 Static Bending 5698.5.4.2 Buckling 5698.5.4.3 Natural Vibration 5708.5.4.4 Transient Response 5708.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 5748.6.1 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 5828.6.5.1 Isotropic Plates 5828.6.5.2 Laminated Plates 5848.7 Summary 587Problems 5889. Mixed Variational and Finite Element Formulations 5959.1 Introduction 5959.1.1 General Comments 5959.1.2 Mixed Variational Principles 5959.1.3 Extremum and Stationary Behavior of Functionals 5979.2 Stationary Variational Principles 5999.2.1 Minimum Total Potential Energy 5999.2.2 The Hellinger-Reissner Variational Principle 6019.2.3 The Reissner Variational Principle 6059.3 Variational Solutions Based on Mixed Formulations 6069.4 Mixed Finite Element Models of Beams 6109.4.1 The Bernoulli-Euler Beam Theory 6109.4.1.1 Governing Equations And Weak Forms 6109.4.1.2 Weak-Form Mixed Finite Element Model 6109.4.1.3 Weighted-Residual Finite Element Models 6139.4.2 The Timoshenko Beam Theory 6159.4.2.1 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 6189.5 Mixed Finite Element Analysis of the Classical Plate Theory 6209.5.1 Preliminary Comments 6209.5.2 Mixed Model I 6209.5.2.1 Governing Equations 6209.5.2.2 Weak Forms 6219.5.2.3 Finite Element Model 6229.5.3 Mixed Model II 6259.5.3.1 Governing Equations 6259.5.3.2 Weak Forms 6259.5.3.3 Finite Element Model 6269.6 Summary 630Problems 63110. Analysis of Functionally Graded Beams and Plates 63510.1 Introduction 63510.2 Functionally Graded Beams 63810.2.1 The Bernoulli-Euler Beam Theory 63810.2.1.1 Displacement and strain fields  63810.2.1.2 Equations of motion and boundary conditions 63810.2.2 The Timoshenko Beam Theory 63910.2.2.1 Displacement and strain fields 63910.2.2.2 Equations of motion and boundary conditions 64010.2.3 Equations of Motion in terms of Generalized Displacements 64110.2.3.1 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 64210.2.4 Stiffiness Coefficients64310.3 Functionally Graded Circular Plates 64510.3.1 Introduction 64510.3.2 Classical Plate Theory 64610.3.2.1 Displacement and Strain Fields 64610.3.2.2 Equations of Motion 64610.3.3 First-Order Shear Deformation Theory 64710.3.3.1 Displacement and Strain Fields 64710.3.3.2 Equations of Motion 64810.3.4 Plate Constitutive Relations 64910.3.4.1 Classical Plate Theory 64910.3.4.2 First-Order Plate Theory 64910.4 A General Third-Order Plate Theory 65010.4.1 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 65810.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 65810.4.5.2 The Reddy Third-Order Plate Theory 66110.4.5.3 The First-Order Plate Theory 66310.4.5.4 The Classical Plate Theory 66410.5 Navier's Solutions 66410.5.1 Preliminary Comments 66410.5.2 Analysis of Beams 66510.5.2.1 Bernoulli-Euler Beams 66510.5.2.2 Timoshenko Beams 66710.5.2.3 Numerical Results 66910.5.3 Analysis of Plates 67110.5.3.1 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 67910.6 Finite Element Models 68110.6.1 Bending of Beams 68110.6.1.1 Bernoulli-Euler Beam Theory 68110.6.1.2 Timoshenko Beam Theory 68310.6.2 Axisymmetric Bending of Circular Plates 68410.6.2.1 Classical Plate Theory 68110.6.2.2 First-Order Shear Deformation Plate Theory 68610.6.3 Solution of Nonlinear Equations 68810.6.3.1 Times approximation 68810.6.3.2 Newton's Iteration Approach 68810.6.3.3 Tangent Stiffiness Coefficients for the BET 69010.6.3.4 Tangent Stiffiness Coefficients for the TBT 69210.6.3.5 Tangent Stiffiness Coefficients for the CPT 69310.6.3.6 Tangent Stiffiness Coefficients for the FSDT 69310.6.4 Numerical Results for Beams and Circular Plates 69410.6.4.1 Beams 69410.6.4.2 Circular Plates 69710.7 Summary 699Problems 700References 701Answers to Most Problems 711Index 723