Computation of Nonlinear Structures E-Book

Debabrata Ray

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 1422

Veröffentlichungsjahr: 2015

Beschreibung

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

Leseprobe

COMPUTATION OF NONLINEAR STRUCTURES

EXTREMELY LARGE ELEMENTS FOR FRAMES, PLATES AND SHELLS

Debabrata Ray

PhD (Univ. of California, Berkeley) ME, BE (Bengal Engineering College, Shibpur) Principal, Institute for Dynamic Response

This edition first published 2016 © 2016 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data applied for.

A catalogue record for this book is available from the British Library.

ISBN: 9781118996959

To the memory of my dad & mom, Provanshu and Sushila Roy, with whose altruistic love it was nurtured

&

the best thing ever happened to me, my wife, Anjana Ray, M.D., with whose infinite patience and eternal support it blossomed

&

the budding analysts and researchers like my sons, Dipanjan Ray, PhD and Shonket Ray, PhD, to whom it is offered

Contents

Acknowledgements

1 Introduction: Background and Motivation

1.1 What This Book Is All About

1.2 A Brief Historical Perspective

1.3 Symbiotic Structural Analysis

1.4 Linear Curved Beams and Arches

1.5 Geometrically Nonlinear Curved Beams and Arches

1.6 Geometrically Nonlinear Plates and Shells

1.7 Symmetry of the Tangent Operator: Nonlinear Beams and Shells

1.8 Road Map of the Book

References

Part I ESSENTIAL MATHEMATICS

2 Mathematical Preliminaries

2.1 Essential Preliminaries

2.2 Affine Space, Vectors and Barycentric Combination

2.3 Generalization: Euclidean to Riemannian Space

2.4 Where We Would Like to Go

3 Tensors

3.1 Introduction

3.2 Tensors as Linear Transformation

3.3 General Tensor Space

3.4 Tensor by Component Transformation Property

3.5 Special Tensors

3.6 Second-order Tensors

3.7 Calculus Tensor

3.8 Partial Derivatives of Tensors

3.9 Covariant or Absolute Derivative

3.10 Riemann–Christoffel Tensor: Ordered Differentiation

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

3.12 Partial Derivatives of Scalar Functions of Tensors

3.13 Partial Derivatives of Tensor Functions of Tensors

3.14 Partial Derivatives of Parametric Functions of Tensors

3.15 Differential Operators

3.16 Gradient Operator:

GRAD

(•) or ∇(•)

3.17 Divergence Operator:

DIV

or ∇•

3.18 Integral Transforms: Green–Gauss Theorems

3.19 Where We Would Like to Go

4 Rotation Tensor

4.1 Introduction

4.2 Cayley's Representation

4.3 Rodrigues Parameters

4.4 Euler – Rodrigues Parameters

4.5 Hamilton's Quaternions

4.6 Hamilton–Rodrigues Quaternion

4.7 Derivatives, Angular Velocity and Variations

Part II ESSENTIAL MESH GENERATION

5 Curves: Theory and Computation

5.1 Introduction

5.2 Affine Transformation and Ratios

5.3 Real Parametric Curves: Differential Geometry

5.4 Frenet–Serret Derivatives

5.5 Bernstein Polynomials

5.6 Non-rational Curves Bezier–Bernstein–de Casteljau

5.7 Composite Bezier–Bernstein Curves

5.8 Splines: Schoenberg B-spline Curves

5.9 Recursive Algorithm: de Boor–Cox Spline

5.10 Rational Bezier Curves: Conics and Splines

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

5.12 Curve Fitting: Interpolations

5.13 Where We Would Like to Go

6 Surfaces: Theory and Computation

6.1 Introduction

6.2 Real Parametric Surface: Differential Geometry

6.3 Gauss–Weingarten Formulas: Optimal Coordinate System

6.4 Cartesian Product Bernstein–Bezier surfaces

6.5 Control Net Generation: Cartesian Product Surfaces

6.6 Composite Bezier Form: Quadratic and Cubic B-splines

6.7 Triangular Bezier–Bernstein Surfaces

Part III ESSENTIAL MECHANICS

7 Nonlinear Mechanics: A Lagrangian Approach

7.1 Introduction

7.2 Deformation Geometry: Strain Tensors

7.3 Balance Principles: Stress Tensors

7.4 Constitutive Theory: Hyperelastic Stress–Strain Relation

Part IV A NEW FINITE ELEMENT METHOD

8 C-type Finite Element Method

8.1 Introduction

8.2 Variational Formulations

8.3 Energy Precursor to Finite Element Method

8.4 c-type FEM: Linear Elasticity and Heat Conduction

8.5 Newton Iteration and Arc Length Constraint

8.6 Gauss–Legendre Quadrature Formulas

Part V APPLICATIONS: LINEAR AND NONLINEAR

9 Application to Linear Problems and Locking Solutions

9.1 Introduction

9.2 c-type Truss and Bar Element

9.3 c-type Straight Beam Element

9.4 c-type Curved Beam Element

9.5 c-type Deep Beam: Plane Stress Element

9.6 c-type Solutions: Locking Problems

10 Nonlinear Beams

10.1 Introduction

10.2 Beam Geometry: Definition and Assumptions

10.3 Static and Dynamic Equations: Engineering Approach

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

10.5 Weak Form: Kinematic and Configuration Space

10.6 Admissible Virtual Space: Curvature, Velocity and Variation

10.7 Real Strain and Strain Rates from Weak Form

10.8 Component or Operational Vector Form

10.9 Covariant Derivatives of Component Vectors

10.10 Computational Equations of Motion: Component Vector Form

10.11 Computational Derivatives and Variations

10.12 Computational Virtual Work Equations

10.13 Computational Virtual Work Equations and Virtual Strains: Revisited

10.14 Computational Real Strains

10.15 Hyperelastic Material Property

10.16 Covariant Linearization of Virtual Work

10.17 Material Stiffness Matrix and Symmetry

10.18 Geometric Stiffness Matrix and Symmetry

10.19 c-type FE Formulation: Dynamic Loading

10.20 c-type FE Implementation and Examples: Quasi-static Loading

11 Nonlinear Shell

11.1 Introduction

11.2 Shell Geometry: Definition and Assumptions

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

11.4 Static and Dynamic Equations: Continuum Approach – Revisited

11.5 Static and Dynamic Equations: Engineering Approach

11.6 Weak Form: Kinematic and Configuration Space

11.7 Admissible Virtual Space: Curvature, Velocity and Variation

11.8 Real Strain and Strain Rates from Weak Form

11.9 Component or Operational Vector Form

11.10 Covariant Derivatives of Component Vectors

11.11 Computational Equations of Motion: Component Vector Form

11.12 Computational Derivatives and Variations

11.13 Computational Virtual Work Equations

11.14 Computational Virtual Work Equations and Virtual Strains: Revisited

11.15 Computational Real Strains

11.16 Hyperelastic Material Property

11.17 Covariant Linearization of Virtual Work

11.18 c-type FE Formulation: Dynamic Loading

11.19 c-type FE Formulation: Quasi-static Loading

11.20 c-type FE Implementation and Examples: Quasi-static Loading

Index

EULA

Guide

Cover

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Acknowledgements

I would like to sincerely thank my friend, Anil K. Chopra, Johnson Professor of Structural Engineering, Department of Civil and Environmental Engineering, University of California, Berkeley, who introduced me to structural dynamics, for his encouragement and generous support in presenting my work to Eric Willner, Executive Commissioning Editor of John Wiley & Sons Ltd, who graciously agreed to publish the book, for which I am deeply appreciative. I must also thank Anne Hunt, Associate Commissioning Editor, Mechanical Engineering, for her unstinting support, Clive Lawson, Project Editor, Content Capture, Natural Sciences, Engineering & Stats, Professional Practice and Learning, Wiley. My special thanks for meticulous scrutiny of the manuscript to copy editor, Paul Beverley, LCGI, and Baljinder Kaur, Project Manager, Professional Publishing, Aptara.

During the writing of the manuscript for the book, I was paralysed as a result of a botched surgical procedure; however, from the initial period of rehabilitation to the date of this writing, I have been extremely fortunate to be surrounded by innumerable friends and well-wishers. I would like to thank them all; especially, my deep appreciation goes to my friend, Prof. Amitabha Basu of Dept. of South Asian Studies, University of California, Berkeley, and, to my care-givers, Levi Soler and John Viray.

Finally, however, the faults and mistakes, if any, are entirely mine.

DR August 31, 2014