# A Variational Approach to Structural Analysis

ISBN: 978-0-471-42125-2

Oct 2002

424 pages

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\$132.99

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## Description

An insightful examination of the numerical methods used to develop finite element methods
A Variational Approach to Structural Analysis provides readers with the underpinnings of the finite element method (FEM) while highlighting the power and pitfalls of virtual methods. In an easy-to-follow, logical format, this book gives complete coverage of the principle of virtual work, complementary virtual work and energy methods, and static and dynamic stability concepts.
The first two chapters prepare the reader with preliminary material, introducing in detail the variational approach used in the book as well as reviewing the equilibrium and compatibility equations of mechanics. The next chapter, on virtual work, teaches how to use kinematical formulations for the determination of the required strain relationships for straight, curved, and thin walled beams. The chapters on complementary virtual work and energy methods are problem-solving chapters that incorporate Castigliano's first theorem, the Engesser-Crotti theorem, and the Galerkin method. In the final chapter, the reader is introduced to various geometric measures of strain and revisits straight, curved, and thin walled beams by examining them in a deformed geometry.
Based on nearly two decades of work on the development of the world's most used FEM code, A Variational Approach to Structural Analysis has been designed as a self-contained, single-source reference for mechanical, aerospace, and civil engineering professionals. The book's straightforward style also provides accessible instruction for graduate students in aeronautical, civil, mechanical, and engineering mechanics courses.

PREFACE xi

1 INTRODUCTION 1

2 PRELIMINARIES 7

2.1 Variational Notation 7

2.3 Integration by Parts 11

2.4 Stokes’s Theorem 13

2.5 Green’s Theorem in the Plane 15

2.7 Meaning of ∇2 19

2.8 Total Differentials 20

2.9 Legendre Transformation 21

2.10 Lagrange Multipliers 24

2.11 Differential Equations of Equilibrium 27

2.12 Strain-Displacement Relations 29

2.13 Compatibility Conditions of Strain 33

2.14 Thermodynamic Considerations 35

Problems 38

3 PRINCIPLE OF VIRTUAL WORK 40

3.1 Virtual Work Definition 40

3.2 Generalized Coordinates 41

3.3 Virtual Work of a Deformable Body 42

3.4 Thermal Stress, Initial Strain, and Initial Stress 47

3.5 Some Constitutive Relationships 48

3.6 Accounting for All Work 51

3.8 The Unit-Displacement Method 60

3.9 Finite Elements for Axial Members 65

3.10 Coordinate Transformations 71

3.11 Review of the Simple Beam Theory 74

3.12 Shear Stress in Simple Beams 92

3.13 Shear Deflection in Straight Beams 95

3.14 Beams with Initial Curvature 99

3.15 Thermal Strain Correction in Curved Beams 112

3.16 Shear and Radial Stress in Curved Beams 114

3.17 Thin Walled Beams of Open Section 121

3.18 Shear in Open Section Beams 147

3.19 Slope-Deflection Equations 155

3.20 Approximate Methods 165

Problems 173

4 COMPLEMENTARY VIRTUAL WORK 196

4.1 Complementary Virtual Work Definition 196

4.2 Complementary Virtual Work of a Deformable Body 197

4.3 Symmetry 210

4.4 The Unit Load Method 217

4.5 Force Elements 235

4.6 Generalized Force-Displacement Transformations 239

Problems 242

5 SOME ENERGY METHODS 261

5.1 Conservative Forces and Potential Functions 261

5.2 Stationary Potential Energy 271

5.3 Castigliano’s First Theorem 274

5.4 Complementary Energy 277

5.5 Stationary Complementary Potential Energy 280

5.6 Engesser-Crotti Theorem 282

5.7 Variational Statements 287

5.8 The Galerkin Method 290

5.9 Derived Variational Principles 300

Problems 305

6 SOME STATIC AND DYNAMIC STABILITY CONCEPTS 318

6.1 Linear-Stability Analysis 318

6.2 Geometric Measure of Strain 322

6.3 A Beam with Initial Curvature Revisited 330

6.4 Thin Walled Open Beams Revisited 337

6.5 Some Stability Concepts 349

6.6 Energy Criterion of Stability 350

6.7 Stiffness 353

6.8 Stiffening and Unstiffening Models 360

6.9 Bifurcation Analysis 369

6.10 Imperfection Analysis 372

6.11 Circulatory Dynamic Stability 377

6.12 Instationary Dynamic Stability 384

Problems 388

REFERENCES 396

INDEX 401