Debabrata Ray

Computation of Nonlinear Structures (eBook, ePUB)

Extremely Large Elements for Frames, Plates and Shells

• Format: ePub

Produktbeschreibung

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

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 992
• Erscheinungstermin: 13. Oktober 2015
• Englisch
• ISBN-13: 9781118996867
• Artikelnr.: 43984517

Autorenporträt

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.

Inhaltsangabe

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 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
8.6 Gauss-Legendre Quadrature Formulas 446 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 10.19 c-type FE Formulation: Dynamic
Loading 673 10.20 c-type FE Implementation and Examples: Quasi-static
Loading 685 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 11.18 c-type FE Formulation: Dynamic Loading 891 11.19 c-type FE
Formulation: Quasi-static Loading 914 11.20 c-type FE Implementation and
Examples: Quasi-static Loading 930 Index 967