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Nonlinear Beam and Cable Mechanics : Mathematical Modeling and Engineering Applications,9781848214217
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Nonlinear Beam and Cable Mechanics : Mathematical Modeling and Engineering Applications

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Nonlinear models of elastic and visco-elastic one-dimensional continuous structures (beams and cables) are formulated. Several models of increasing complexity are dealt with: straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warping-unsensitive/sensitive, prestressed/unprestressed beams, both in statics and dynamics. Typical engineering problems are solved via perturbation and/or numerical approaches, as Bifurcation & Stability under potential and/or tangential loads, Parametric Excitation, Nonlinear Dynamics, Aeroelasticity.

Table of Contents

Preface xi

Introduction xiii

List of Main Symbols xxiii

Chapter 1. A One-Dimensional Beam Metamodel 1

1.1. Models and metamodel 2

1.2. Internally unconstrained beams 3

1.3. Internally constrained beams 12

1.4. Internally unconstrained prestressed beams 24

1.5. Internally constrained prestressed beams 29

1.6. The variational formulation 33

1.7. Example: the linear Timoshenko beam 44

1.8. Summary 47

Chapter 2. Straight Beams 55

2.1. Kinematics 55

2.2. Dynamics 82

2.3. Constitutive law 102

2.4. The Fundamental Problem 114

2.5. The planar beam 122

2.6. Summary 129

Chapter 3. Curved Beams 133

3.1. The reference configuration and the initial curvature 133

3.2. The beam model in the 3D-space 137

3.3. The planar curved beam 152

3.4. Summary 160

Chapter 4. Internally Constrained Beams 163

4.1. Stiff beams and internal constraints 163

4.2. The general approach 166

4.3. The unshearable straight beam in 3D 168

4.4. The unshearable straight planar beam 177

4.5. The inextensible and unshearable straight beam in 3D 180

4.6. The inextensible and unshearable straight planar beam 183

4.7. The inextensible, unshearable and untwistable straight beam 190

4.8. The foil-beam 192

4.9. The shear–shear–torsional beam 193

4.10. The planar unshearable and inextensible curved beam 197

4.11. Summary 201

Chapter 5. Flexible Cables 205

5.1. Flexible cables as a limit of slender beams 205

5.2. Unprestressed cables 207

5.3. Prestressed cables 220

5.4. Shallow cables 230

5.5. Inextensible cables 235

5.6. Summary 240

Chapter 6. Stiff Cables 243

6.1. Motivations 243

6.2. Unprestressed stiff cables 246

6.3. Prestressed stiff cables 252

6.4. Reduced models 261

6.5. Inextensible stiff cables 264

6.6. Summary 269

Chapter 7. Locally-Deformable Thin-Walled Beams 271

7.1. Motivations 271

7.2. A one-dimensional direct model for double-symmetric TWB 273

7.3. A one-dimensional direct model for non-symmetric TWB 277

7.4. Identification strategy from 3D-models of TWB 284

7.5. A fiber-model of TWB 285

7.6. Warpable, cross-undeformable TWB 289

7.7. Unwarpable, cross-deformable, planar TWB 299

7.8. Summary 308

Chapter 8. Distortion-Constrained Thin-Walled Beams 311

8.1. Introduction 311

8.2. Internal constraints 312

8.3. The non-uniform torsion problem for bi-symmetric cross-sections 317

8.4. The general problem for warpable TWB 324

8.5. Cross-deformable planar TWB 328

8.6. Summary 332

Bibliography 335

Index 345

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