E-book

# Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells

ISBN: 978-1-118-99686-7
992 pages
October 2015

## Description

Comprehensively introduces linear and nonlinear structural analysis through mesh generation, solid mechanics and a new numerical methodology called c-type finite element method

• Takes a self-contained approach of including all the essential background materials such as differential geometry, mesh generation, tensor analysis with particular elaboration on rotation tensor, finite element methodology and numerical analysis for a thorough understanding of the topics
• Presents for the first time in closed form the geometric stiffness, the mass, the gyroscopic damping and the centrifugal stiffness matrices for beams, plates and shells
• Includes numerous examples and exercises
• Presents solutions for locking problems
See More

Acknowledgements xi

1 Introduction: Background and Motivation 1

1.1 What This Book Is All About 1

1.2 A Brief Historical Perspective 2

1.3 Symbiotic Structural Analysis 9

1.4 Linear Curved Beams and Arches 9

1.5 Geometrically Nonlinear Curved Beams and Arches 10

1.6 Geometrically Nonlinear Plates and Shells 11

1.7 Symmetry of the Tangent Operator: Nonlinear Beams and Shells 12

1.8 Road Map of the Book 14

References 15

Part I ESSENTIAL MATHEMATICS 19

2 Mathematical Preliminaries 21

2.1 Essential Preliminaries 21

2.2 Affine Space, Vectors and Barycentric Combination 33

2.3 Generalization: Euclidean to Riemannian Space 36

2.4 Where We Would Like to Go 40

3 Tensors 41

3.1 Introduction 41

3.2 Tensors as Linear Transformation 44

3.3 General Tensor Space 46

3.4 Tensor by Component Transformation Property 50

3.5 Special Tensors 57

3.6 Second-order Tensors 62

3.7 Calculus Tensor 74

3.8 Partial Derivatives of Tensors 74

3.9 Covariant or Absolute Derivative 75

3.10 Riemann–Christoffel Tensor: Ordered Differentiation 78

3.11 Partial (PD) and Covariant (C.D.) Derivatives of Tensors 79

3.12 Partial Derivatives of Scalar Functions of Tensors 80

3.13 Partial Derivatives of Tensor Functions of Tensors 81

3.14 Partial Derivatives of Parametric Functions of Tensors 81

3.15 Differential Operators 82

3.17 Divergence Operator: DIV or ∇∙ 84

3.18 Integral Transforms: Green–Gauss Theorems 87

3.19 Where We Would Like to Go 90

4 Rotation Tensor 91

4.1 Introduction 91

4.2 Cayley’s Representation 100

4.3 Rodrigues Parameters 107

4.4 Euler – Rodrigues Parameters 112

4.5 Hamilton’s Quaternions 115

4.6 Hamilton–Rodrigues Quaternion 119

4.7 Derivatives, Angular Velocity and Variations 125

Part II ESSENTIAL MESH GENERATION 133

5 Curves: Theory and Computation 135

5.1 Introduction 135

5.2 Affine Transformation and Ratios 136

5.3 Real Parametric Curves: Differential Geometry 139

5.4 Frenet–Serret Derivatives 145

5.5 Bernstein Polynomials 148

5.6 Non-rational Curves Bezier–Bernstein–de Casteljau 154

5.7 Composite Bezier–Bernstein Curves 181

5.8 Splines: Schoenberg B-spline Curves 185

5.9 Recursive Algorithm: de Boor–Cox Spline 195

5.10 Rational Bezier Curves: Conics and Splines 198

5.11 Composite Bezier Form: Quadratic and Cubic B-spline Curves 215

5.12 Curve Fitting: Interpolations 229

5.13 Where We Would Like to Go 245

6 Surfaces: Theory and Computation 247

6.1 Introduction 247

6.2 Real Parametric Surface: Differential Geometry 248

6.3 Gauss–Weingarten Formulas: Optimal Coordinate System 272

6.4 Cartesian Product Bernstein–Bezier Surfaces 280

6.5 Control Net Generation: Cartesian Product Surfaces 296

6.6 Composite Bezier Form: Quadratic and Cubic B-splines 300

6.7 Triangular Bezier–Bernstein Surfaces 306

Part III ESSENTIAL MECHANICS 323

7 Nonlinear Mechanics: A Lagrangian Approach 325

7.1 Introduction 325

7.2 Deformation Geometry: Strain Tensors 326

7.3 Balance Principles: Stress Tensors 337

7.4 Constitutive Theory: Hyperelastic Stress–Strain Relation 351

Part IV A NEW FINITE ELEMENT METHOD 365

8 C-type Finite Element Method 367

8.1 Introduction 367

8.2 Variational Formulations 369

8.3 Energy Precursor to Finite Element Method 386

8.4 c-type FEM: Linear Elasticity and Heat Conduction 402

8.5 Newton Iteration and Arc Length Constraint 438

Part V APPLICATIONS: LINEAR AND NONLINEAR 457

9 Application to Linear Problems and Locking Solutions 459

9.1 Introduction 459

9.2 c-type Truss and Bar Element 460

9.3 c-type Straight Beam Element 465

9.4 c-type Curved Beam Element 484

9.5 c-type Deep Beam: Plane Stress Element 498

9.6 c-type Solutions: Locking Problems 509

10 Nonlinear Beams 523

10.1 Introduction 523

10.2 Beam Geometry: Definition and Assumptions 530

10.3 Static and Dynamic Equations: Engineering Approach 534

10.4 Static and Dynamic Equations: Continuum Approach – 3D to 1D 539

10.5 Weak Form: Kinematic and Configuration Space 555

10.6 Admissible Virtual Space: Curvature, Velocity and Variation 560

10.7 Real Strain and Strain Rates from Weak Form 570

10.8 Component or Operational Vector Form 580

10.9 Covariant Derivatives of Component Vectors 587

10.10 Computational Equations of Motion: Component Vector Form 590

10.11 Computational Derivatives and Variations 596

10.12 Computational Virtual Work Equations 607

10.13 Computational Virtual Work Equations and Virtual Strains: Revisited 614

10.14 Computational Real Strains 627

10.15 Hyperelastic Material Property 630

10.16 Covariant Linearization of Virtual Work 639

10.17 Material Stiffness Matrix and Symmetry 655

10.18 Geometric Stiffness Matrix and Symmetry 658

11 Nonlinear Shell 721

11.1 Introduction 721

11.2 Shell Geometry: Definition and Assumptions 727

11.3 Static and Dynamic Equations: Continuum Approach – 3D to 2D 746

11.4 Static and Dynamic Equations: Continuum Approach – Revisited 763

11.5 Static and Dynamic Equations: Engineering Approach 771

11.6 Weak Form: Kinematic and Configuration Space 783

11.7 Admissible Virtual Space: Curvature, Velocity and Variation 788

11.8 Real Strain and Strain Rates from Weak Form 799

11.9 Component or Operational Vector Form 810

11.10 Covariant Derivatives of Component Vectors 817

11.11 Computational Equations of Motion: Component Vector Form 820

11.12 Computational Derivatives and Variations 830

11.13 Computational Virtual Work Equations 841

11.14 Computational Virtual Work Equations and Virtual Strains: Revisited 851

11.15 Computational Real Strains 861

11.16 Hyperelastic Material Property 864

11.17 Covariant Linearization of Virtual Work 877