A variational approach to structural analysis

David V. Wallerstein0471395935, 9780471395935

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.

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Table of contents :
A VARIATIONAL APPROACH TO STRUCTURAL ANALYSIS……Page 3
CONTENTS……Page 9
PREFACE……Page 13
1 INTRODUCTION……Page 17
2.1 Variational Notation……Page 23
2.2 The Gradient……Page 26
2.3 Integration by Parts……Page 27
2.4 Stokes’s Theorem……Page 29
2.5 Green’s Theorem in the Plane……Page 31
2.6 Adjoint Equations……Page 32
2.7 Meaning of (2)……Page 35
2.8 Total Differentials……Page 36
2.9 Legendre Transformation……Page 37
2.10 Lagrange Multipliers……Page 40
2.11 Differential Equations of Equilibrium……Page 43
2.12 Strain-Displacement Relations……Page 45
2.13 Compatibility Conditions of Strain……Page 49
2.14 Thermodynamic Considerations……Page 51
Problems……Page 54
3.1 Virtual Work Definition……Page 56
3.2 Generalized Coordinates……Page 57
3.3 Virtual Work of a Deformable Body……Page 58
3.4 Thermal Stress, Initial Strain, and Initial Stress……Page 63
3.5 Some Constitutive Relationships……Page 64
3.6 Accounting for All Work……Page 67
3.7 Axially Loaded Members……Page 69
3.8 The Unit-Displacement Method……Page 76
3.9 Finite Elements for Axial Members……Page 81
3.10 Coordinate Transformations……Page 87
3.11 Review of the Simple Beam Theory……Page 90
3.12 Shear Stress in Simple Beams……Page 108
3.13 Shear Deflection in Straight Beams……Page 111
3.14 Beams with Initial Curvature……Page 115
3.15 Thermal Strain Correction in Curved Beams……Page 128
3.16 Shear and Radial Stress in Curved Beams……Page 130
3.17 Thin Walled Beams of Open Section……Page 137
3.18 Shear in Open Section Beams……Page 163
3.19 Slope-Deflection Equations……Page 171
3.20 Approximate Methods……Page 181
Problems……Page 189
4.1 Complementary Virtual Work Definition……Page 212
4.2 Complementary Virtual Work of a Deformable Body……Page 213
4.3 Symmetry……Page 226
4.4 The Unit Load Method……Page 233
4.5 Force Elements……Page 251
4.6 Generalized Force-Displacement Transformations……Page 255
Problems……Page 258
5.1 Conservative Forces and Potential Functions……Page 277
5.2 Stationary Potential Energy……Page 287
5.3 Castigliano’s First Theorem……Page 290
5.4 Complementary Energy……Page 293
5.5 Stationary Complementary Potential Energy……Page 296
5.6 Engesser-Crotti Theorem……Page 298
5.7 Variational Statements……Page 303
5.8 The Galerkin Method……Page 306
5.9 Derived Variational Principles……Page 316
Problems……Page 322
6.1 Linear-Stability Analysis……Page 334
6.2 Geometric Measure of Strain……Page 338
6.3 A Beam with Initial Curvature Revisited……Page 346
6.4 Thin Walled Open Beams Revisited……Page 353
6.5 Some Stability Concepts……Page 365
6.6 Energy Criterion of Stability……Page 366
6.7 Stiffness……Page 369
6.8 Stiffening and Unstiffening Models……Page 376
6.9 Bifurcation Analysis……Page 385
6.10 Imperfection Analysis……Page 388
6.11 Circulatory Dynamic Stability……Page 393
6.12 Instationary Dynamic Stability……Page 400
Problems……Page 404
REFERENCES……Page 412
INDEX……Page 417