Energy Methods in Applied Mechanics

by Henry L. Langhaar

Overview

This comprehensive work presents an integrated and modern treatment of the energy principles of mechanics. An introductory chapter on the general concepts and principles of mechanics is followed by discussions of applications to dynamics of rigid bodies, analysis of elastic frames, general elastic theory, the theory of plates and shells, the theory of buckling, and the theory of vibrations.
Geared toward advanced undergraduates and graduate students of engineering, the text's prerequisites include familiarity with the calculus of variations and vector theory as well as some knowledge of advanced strength of materials and the foundations of elasticity theory. The treatment is also suitable for independent study and reference purposes. In addition to a helpful Appendix on quadratic forms, this volume features a substantial section of answers to problems.

Product Details

ISBN-13:	9780486811130
Publisher:	Dover Publications
Publication date:	11/16/2016
Edition description:	First Edition, First
Pages:	368
Product dimensions:	6.00(w) x 16.50(h) x 0.80(d)

About the Author

Henry L. Langhaar (1909–92) was Professor of Theoretical and Applied Mechanics at the University of Illinois.

Table of Contents

Chapter 1 General Concepts and Principles of Mechanics 1

1-1 Mechanical Systems 2

1-2 Generalized Coordinates 6

1-3 Elementary Principles of Dynamics 8

1-4 First Law of Thermodynamics 11

1-5 Fourier's Inequality 13

1-6 The Principle of Virtual Work 14

1-7 Generalized Force 16

1-8 Potential Energy 18

1-9 Properties of Conservative Systems 23

1-10 Potential Energy of a System of Particles 24

1-11 Stability 29

Chapter 2 Elastic Beams and Frames 39

2-1 Strain Energy of Beams, Columns and Shafts 39

2-2 Beam Columns Analyzed by Fourier Series 44

2-3 Curved Beams 48

2-4 Pin-Jointed Trusses 52

2-5 Frames with Torsional and Flexural Members 58

Chapter 3 Methods of the Calculus of Variations 75

3-1 Cantilever Beam 75

3-2 Euler's Equation 78

3-3 Variational Notation 83

3-4 Special Forms of the Euler Equation 84

3-5 The Differential Equation of Beams 85

3-6 Curved Cantilever Beam 86

3-7 Isoperimetric Problems 89

3-8 Auxiliary Differential Equations 91

3-9 First Variation of a Double Integral 92

3-10 First Variation of a Triple Integral 97

3-11 The Rayleigh-Ritz Method 98

Chapter 4 Deformable Bodies 104

4-1 Deformation of a Body 104

4-2 Stress 109

4-3 Equations of Stress and Strain Referred to Orthogonal Curvilinear Coordinates 112

4-4 First Law of Thermodynamics Applied to a Deformation Process 115

4-5 Stress-Strain Relations of Elastic Bodies 117

4-6 Complementary Energy Density 119

4-7 Hookean Materials 121

4-8 Generalization of Castigliano's Theorem of Least Work 126

4-9 Reissner's Variational Theorem of Elasticity 130

4-10 Castigliano's Theorem on Deflections 133

4-11 Corollaries to Castigliano's Theorem 136

4-12 Castigliano's Theorem Applied to Trusses 138

4-13 Complementary Energy of Beams 144

4-14 Unit-Dummy-Load Method 146

4-15 Analysis of Statically Indeterminate Structures by the Unit-Dummy-Load Method 149

Chapter 5 Theory of Plates and Shells 159

5-1 The Von Kármán Theory of Flat Plates 159

5-2 Small-Deflection Theory of Plates 165

5-3 Boundary Conditions in the Classical Theory of Plates 168

5-4 Simply Supported Rectangular Plates 170

5-5 Shear Deformation of Plates 173

5-6 Geometry of Shells 177

5-7 Equilibrium of Shells 181

5-8 Strain Energy of Shells 187

5-9 Axially Symmetric Shells 191

5-10 Circular Plates 195

Chapter 6 Theory of Buckling 201

6-1 Introduction 201

6-2 Postbuckling Behavior of a Simple Column 203

6-3 Buckling of Conservative Systems with Enumerable Degrees of Freedom 206

6-4 General Principles of Buckling of Conservative Systems 210

6-5 Buckling of Simply Supported Compressed Rectangular Plates 214

6-6 Buckling of a Uniformly Compressed Circular Plate 216

6-7 Lateral Buckling of Beams 218

6-8 Torsional-Flexural Buckling of Columns 224

Chapter 7 Hamilton's Principle and the Equations of Lagrange and Hamilton 233

7-1 Kinetic Energy of a System with Finite Degrees of Freedom 233

7-2 Hamilton's Principle 234

7-3 Lagrange's Equations for Conservative Systems 239

7-4 Time-Dependent Constraints 241

7-5 The Hamiltonian Function 243

7-6 Hamilton's Equations 244

7-7 Lagrange's Equations for Nonconservative Systems 247

7-8 Kinematics of a Rigid Body 249

7-9 Euler's Dynamical Equations 253

7-10 Motion of an Ideal Top 255

7-11 The Gyroscope 259

Chapter 8 Theory of Vibrations 268

8-1 Systems with Two Degrees of Freedom 268

8-2 Vibrations of Undamped Systems with Finite Degrees of Freedom 272

8-3 Systems with Viscous Damping 277

8-4 Free Vibrations of a Beam with Clamped Ends 280

8-5 Orthogonality Properties of Mode Forms of a Beam 282

8-6 Vibrations of an Airplane Wing 285

8-7 Effects of Rotary Inertia and Shear Deformation on Vibrations of Beams 288

8-8 Free Vibrations of a Rectangular Elastic Plate 289

8-9 Spherical Pressure Waves in an Ideal Gas 292

8-10 Plane Progressive Gravity Waves in a Liquid 295

8-11 Wave Motion in Solids 299

Appendix On Quadratic Forms 308

A-1 Type of a Quadratic Form 308

A-2 Principal-Axis Theory of Quadratic Forms 313

A-3 Theory of Principal Axes of Strain 319

A-4 Simultaneous Transformation of Two Quadratic Forms to the Canonical Form 320

A-5 Quadratic Forms and Buckling Theory 324

Answers to Problems 329

Index 347