Computation of Nonlinear Structures

Debabrata Ray, Institute for Dynamic Response, Inc, USAFor 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.

• Acknowledgements xi1 Introduction: Background and Motivation 11.1 What This Book Is All About 11.2 A Brief Historical Perspective 21.3 Symbiotic Structural Analysis 91.4 Linear Curved Beams and Arches 91.5 Geometrically Nonlinear Curved Beams and Arches 101.6 Geometrically Nonlinear Plates and Shells 111.7 Symmetry of the Tangent Operator: Nonlinear Beams and Shells 121.8 Road Map of the Book 14References 15Part I ESSENTIAL MATHEMATICS 192 Mathematical Preliminaries 212.1 Essential Preliminaries 212.2 Affine Space, Vectors and Barycentric Combination 332.3 Generalization: Euclidean to Riemannian Space 362.4 Where We Would Like to Go 403 Tensors 413.1 Introduction 413.2 Tensors as Linear Transformation 443.3 General Tensor Space 463.4 Tensor by Component Transformation Property 503.5 Special Tensors 573.6 Second-order Tensors 623.7 Calculus Tensor 743.8 Partial Derivatives of Tensors 743.9 Covariant or Absolute Derivative 753.10 Riemann–Christoffel Tensor: Ordered Differentiation 783.11 Partial (PD) and Covariant (C.D.) Derivatives of Tensors 793.12 Partial Derivatives of Scalar Functions of Tensors 803.13 Partial Derivatives of Tensor Functions of Tensors 813.14 Partial Derivatives of Parametric Functions of Tensors 813.15 Differential Operators 823.16 Gradient Operator: GRAD(∙) or ∇(∙) 823.17 Divergence Operator: DIV or ∇∙ 843.18 Integral Transforms: Green–Gauss Theorems 873.19 Where We Would Like to Go 904 Rotation Tensor 914.1 Introduction 914.2 Cayley’s Representation 1004.3 Rodrigues Parameters 1074.4 Euler – Rodrigues Parameters 1124.5 Hamilton’s Quaternions 1154.6 Hamilton–Rodrigues Quaternion 1194.7 Derivatives, Angular Velocity and Variations 125Part II ESSENTIAL MESH GENERATION 1335 Curves: Theory and Computation 1355.1 Introduction 1355.2 Affine Transformation and Ratios 1365.3 Real Parametric Curves: Differential Geometry 1395.4 Frenet–Serret Derivatives 1455.5 Bernstein Polynomials 1485.6 Non-rational Curves Bezier–Bernstein–de Casteljau 1545.7 Composite Bezier–Bernstein Curves 1815.8 Splines: Schoenberg B-spline Curves 1855.9 Recursive Algorithm: de Boor–Cox Spline 1955.10 Rational Bezier Curves: Conics and Splines 1985.11 Composite Bezier Form: Quadratic and Cubic B-spline Curves 2155.12 Curve Fitting: Interpolations 2295.13 Where We Would Like to Go 2456 Surfaces: Theory and Computation 2476.1 Introduction 2476.2 Real Parametric Surface: Differential Geometry 2486.3 Gauss–Weingarten Formulas: Optimal Coordinate System 2726.4 Cartesian Product Bernstein–Bezier Surfaces 2806.5 Control Net Generation: Cartesian Product Surfaces 2966.6 Composite Bezier Form: Quadratic and Cubic B-splines 3006.7 Triangular Bezier–Bernstein Surfaces 306Part III ESSENTIAL MECHANICS 3237 Nonlinear Mechanics: A Lagrangian Approach 3257.1 Introduction 3257.2 Deformation Geometry: Strain Tensors 3267.3 Balance Principles: Stress Tensors 3377.4 Constitutive Theory: Hyperelastic Stress–Strain Relation 351Part IV A NEW FINITE ELEMENT METHOD 3658 C-type Finite Element Method 3678.1 Introduction 3678.2 Variational Formulations 3698.3 Energy Precursor to Finite Element Method 3868.4 c-type FEM: Linear Elasticity and Heat Conduction 4028.5 Newton Iteration and Arc Length Constraint 4388.6 Gauss–Legendre Quadrature Formulas 446Part V APPLICATIONS: LINEAR AND NONLINEAR 4579 Application to Linear Problems and Locking Solutions 4599.1 Introduction 4599.2 c-type Truss and Bar Element 4609.3 c-type Straight Beam Element 4659.4 c-type Curved Beam Element 4849.5 c-type Deep Beam: Plane Stress Element 4989.6 c-type Solutions: Locking Problems 50910 Nonlinear Beams 52310.1 Introduction 52310.2 Beam Geometry: Definition and Assumptions 53010.3 Static and Dynamic Equations: Engineering Approach 53410.4 Static and Dynamic Equations: Continuum Approach – 3D to 1D 53910.5 Weak Form: Kinematic and Configuration Space 55510.6 Admissible Virtual Space: Curvature, Velocity and Variation 56010.7 Real Strain and Strain Rates from Weak Form 57010.8 Component or Operational Vector Form 58010.9 Covariant Derivatives of Component Vectors 58710.10 Computational Equations of Motion: Component Vector Form 59010.11 Computational Derivatives and Variations 59610.12 Computational Virtual Work Equations 60710.13 Computational Virtual Work Equations and Virtual Strains: Revisited 61410.14 Computational Real Strains 62710.15 Hyperelastic Material Property 63010.16 Covariant Linearization of Virtual Work 63910.17 Material Stiffness Matrix and Symmetry 65510.18 Geometric Stiffness Matrix and Symmetry 65810.19 c-type FE Formulation: Dynamic Loading 67310.20 c-type FE Implementation and Examples: Quasi-static Loading 68511 Nonlinear Shell 72111.1 Introduction 72111.2 Shell Geometry: Definition and Assumptions 72711.3 Static and Dynamic Equations: Continuum Approach – 3D to 2D 74611.4 Static and Dynamic Equations: Continuum Approach – Revisited 76311.5 Static and Dynamic Equations: Engineering Approach 77111.6 Weak Form: Kinematic and Configuration Space 78311.7 Admissible Virtual Space: Curvature, Velocity and Variation 78811.8 Real Strain and Strain Rates from Weak Form 79911.9 Component or Operational Vector Form 81011.10 Covariant Derivatives of Component Vectors 81711.11 Computational Equations of Motion: Component Vector Form 82011.12 Computational Derivatives and Variations 83011.13 Computational Virtual Work Equations 84111.14 Computational Virtual Work Equations and Virtual Strains: Revisited 85111.15 Computational Real Strains 86111.16 Hyperelastic Material Property 86411.17 Covariant Linearization of Virtual Work 87711.18 c-type FE Formulation: Dynamic Loading 89111.19 c-type FE Formulation: Quasi-static Loading 91411.20 c-type FE Implementation and Examples: Quasi-static Loading 930Index 967

"Comprehensively introduces linear and nonlinear structural analysis through mesh generation, solid mechanics and a new numerical methodology called c-type finite element method." (Zentralblatt MATH 2016)