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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
The phrase "energy principles" or "energy methods" in the present study refers to methods that make use of the total potential energy (i.e., strain energy and potential energy due to applied loads) of a system to obtain values of an unknown displacement or force, at a specific point of the system. These include Castigliano's theorems, unit dummy load and unit dummy displacement methods, and Betti's and Maxwell's theorems. These methods are often limited to the (exact) determination of generalized displacements or forces at fixed points in the structure; in most cases, they cannot be used to determine the complete solution (i.e., displacements and/or forces) as a function of position in the structure. The phrase "variational methods," on the other hand, refers to methods that make use of the variational principles, such as the principles of virtual work and the principle of minimum total potential energy, to determine approximate solutions as continuous functions of position in a body. In the classical sense, a variational principle has to do with the minimization or finding stationary values of a functional with respect to a set of undetermined parameters introduced in the assumed solution. The functional represents the total energy of the system in solid and structural mechanics problems, and in other problems it is simply an integral representation of the governing equations. In all cases, the functional includes all the intrinsic features of the problem, such as the governing equations, boundary and/or initial conditions, and constraint conditions.
Variational principles have always played an important role in mechanics. Variational formulations can be useful in three related ways. First, many problems of mechanics are posed in terms of finding the extremum (i.e., minima or maxima) and thus, by their nature, can be formulated in terms of variational statements. 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 formulated by means of variational principles. Third, variational formulations form a powerful basis for obtaining approximate solutions to practical problems, many of which are intractable otherwise. The principle of minimum total potential energy, for example, can be regarded as a substitute to the equations of equilibrium of an elastic body, as well as a basis for the development of displacement finite element models that can be used to determine approximate displacement and stress fields in the body. Variational formulations can also serve to unify diverse fields, suggest new theories, and provide a powerful means for studying the existence and uniqueness of solutions to problems. In many cases they can also be used to establish upper and/or lower bounds on approximate solutions.
In modern times, the term "variational formulation" applies to a wide spectrum of concepts having to do with weak, generalized, or direct variational formulations of boundary- and initial-value problems. Still, many of the essential features of variational methods remain the same as they were over 200 years ago when the first notions of variational calculus began to be formulated.1
Although Archimedes (287-212 B.C.) is generally credited as the first to use work arguments in his study of levers, the most primitive ideas of variational theory (the minimum hypothesis) are present in the writings of the Greek philosopher Aristotle (384-322 B.C.), to be revived again by the Italian mathematician/engineer Galileo (1564-1642), and finally formulated into a principle of least time by the French mathematician Fermat (1601-1665). The phrase virtual velocities was used by Jean Bernoulli in 1717 in his letter to Varignon (1654-1722). The development of early variational calculus, by which we mean the classical problems associated with minimizing certain functionals, had to await the works of Newton (1642-1727) and Leibniz (1646-1716). The earliest applications of such variational ideas included the classical isoperimetric problem of finding among closed curves of given length the one that encloses the greatest area, and Newton's problem of determining the solid of revolution of "minimum resistance." In 1696, Jean Bernoulli proposed the problem of the brachistochrone: among all curves connecting two points, find the curve traversed in the shortest time by a particle under the influence of gravity. It stood as a challenge to the mathematicians of their day to solve the problem using the rudimentary tools of analysis then available to them or whatever new ones they were capable of developing. Solutions to this problem were presented by some of the greatest mathematicians of the time: Leibniz, Jean Bernoulli's older brother Jacques Bernoulli, L'Hopital, and Newton.
The first step toward developing a general method for solving variational problems was given by the Swiss genius Leonhard Euler (1707-1783) in 1732 when he presented a "general solution of the isoperimetric problem," although Maupertuis is credited to have put forward a law of minimal property of potential energy for stable equilibrium in his Mémoires de lÁcadémie des Sciences in 1740. It was in Euler's 1732 work and subsequent publication of the principle of least action (in his book Methodus inveniendi lineas curvas .) in 1744 that variational concepts found a welcome and permanent home in mechanics. He developed all ideas surrounding the principle of minimum potential energy in his work on the elastica, and he demonstrated the relationship between his variational equations and those governing the flexure and buckling of thin rods.
A great impetus to the development of variational mechanics began in the writings of Lagrange (1736-1813), first in his correspondence with Euler. Euler worked intensely in developing Lagrange's method but delayed publishing his results until Lagrange's works were published in 1760 and 1761. Lagrange used D'Alembert's principle to convert dynamics to statics and then used the principle of virtual displacements to derive his famous equations governing the laws of dynamics in terms of kinetic and potential energy. Euler's work, together with Lagrange's Mécanique analytique of 1788, laid down the basis for the variational theory of dynamical systems. Further generalizations appeared in the fundamental work of Hamilton in 1834. Collectively, all these works have had a monumental impact on virtually every branch of mechanics.
A more solid mathematical basis for variational theory began to be developed in the eighteenth and early nineteenth century. Necessary conditions for the existence of "minimizing curves" of certain functionals were studied during this period, and we find among contributors of that era the familiar names of Legendre, Jacobi, and Weierstrass. Legendre gave criteria for distinguishing between maxima and minima in 1786, and Jacobi gave sufficient conditions for existence of extrema in 1837. A more rigorous theory of existence of extrema was put together by Weierstrass, who established in 1865 the conditions on extrema for variational problems.
During the last half of the nineteenth century, the use of variational ideas was widespread among leaders in theoretical mechanics. We mention the works of Kirchhoff on plate theory; Lamé, Green, and Kelvin on elasticity; and the works of Betti, Maxwell, Castigliano, Menabrea, and Engesser on discrete structural systems. Lamé was the first in 1852 to prove a work equation, named after his colleague Claperon, for deformable bodies. Lamé's equation was used by Maxwell [1]2 to the solution of redundant frame-works using the unit dummy load technique. In 1875 Castigliano published an extremum version of this technique but attributed the idea to Menabrea. A generalization of Castigliano's work is due to Engesser [2].
Among the prominent contributors to the subject near the end of the nineteenth century and in the early years of the twentieth century, particularly in the area of variational methods of approximation and their applications, were Rayleigh [3], Ritz [4], and Galerkin [5]. Modern variational principles began in the works of Hellinger [6], Hu [7], and Reissner [8-10] on mixed variational principles for elasticity problems. A short historical account of early variational methods in mechanics can be found in the book of Lanczos [11] and Truesdell and Toupin [12]; additional information can be found in Dugas [13] and Timoshenko [14], and historical development of energetical principles in elastomechanics can be found in the paper by Oravas and McLean [15,16]. Reference to much of the relevant contemporary literature can be found in the books by Washizu [17] and Oden and Reddy [18]. Additional historical papers and textbooks on variational principles and methods can be found in [19-60].
The objective of the present book is to introduce energy methods and variational principles of solid and structural mechanics and to illustrate their use in the derivation and solution of the equations of applied mechanics, including plane elasticity, beams, frames, and plates. Of course, variational formulations and methods presented in this book are also...
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