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

Informationen zum Autor Debabrata Ray, Institute for Dynamic Response, Inc, USA For more than thirty years, Dr. Ray has been a consultant working on structural issues including the finite element method, mesh generation, computer -aided geometric design, soil-structure interaction for earthquake resistance, fluid-structure interaction, continuum-finite element synthesis for Nuclear Power Plant structures. His clients include General Electric and the Electric Power and Research Institute. He was previously the Vice President at the URS Corporation and is the Ex-Principal of the Institute for Dynamic Response, Inc. Klappentext 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 Zusammenfassung Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells introduces linear and nonlinear computational structural analysis through symbiosis among mesh generation, operational solid mechanics and a new numerical methodology called c-Type finite element method. Inhaltsverzeichnis 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 3.16 Gradient Operator: GRAD(¿) or ¿(¿) 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 Rea...

List of contents

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
 
3.16 Gradient Operator: GRAD(.) or nabla(.) 82
 
3.17 Divergence Operator: DIV or nabla. 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 NE

Product details

• Authors: D Ray, Debabrata Ray, Ray Debabrata
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 11.12.2015
• EAN: 9781118996959
• ISBN: 978-1-118-99695-9
• No. of pages: 992
• Subjects: Natural sciences, medicine, IT, technology > Technology > Mechanical engineering, production engineering; Mathematik, Maschinenbau, Mathematics, Baustatik, Mechanical Engineering, Bauingenieur- u. Bauwesen, Civil Engineering & Construction, Baustatik u. Baumechanik, Structural Theory & Structural Mechanics, Computational / Numerical Methods, Rechnergestützte / Numerische Verfahren im Maschinenbau, Mathematik in den Ingenieurwissenschaften, Applied Mathematics in Engineering, Nichtlineares System