1st Edition

# Computational Solid Mechanics Variational Formulation and High Order Approximation

By Marco L. Bittencourt Copyright 2015
680 Pages 423 B/W Illustrations
by CRC Press

680 Pages
by CRC Press

Also available as eBook on:

Presents a Systematic Approach for Modeling Mechanical Models Using Variational Formulation—Uses Real-World Examples and Applications of Mechanical Models

Utilizing material developed in a classroom setting and tested over a 12-year period, Computational Solid Mechanics: Variational Formulation and High-Order Approximation details an approach that establishes a logical sequence for the treatment of any mechanical problem. Incorporating variational formulation based on the principle of virtual work, this text considers various aspects of mechanical models, explores analytical mechanics and their variational principles, and presents model approximations using the finite element method. It introduces the basics of mechanics for one-, two-, and three-dimensional models, emphasizes the simplification aspects required in their formulation, and provides relevant applications.

Introduces Approximation Concepts Gradually throughout the Chapters

Organized into ten chapters, this text provides a clear separation of formulation and finite element approximation. It details standard procedures to formulate and approximate models, while at the same time illustrating their application via software. Chapter one provides a general introduction to variational formulation and an overview of the mechanical models to be presented in the other chapters. Chapter two uses the concepts on equilibrium that readers should have to introduce basic notions on kinematics, duality, virtual work, and the PVW. Chapters three to ten present mechanical models, approximation and applications to bars, shafts, beams, beams with shear, general two- and three-dimensional beams, solids, plane models, and generic torsion and plates.

Learn Theory Step by Step

In each chapter, the material profiles all aspects of a specific mechanical model, and uses the same sequence of steps for all models. The steps include kinematics, strain, rigid body deformation, internal loads, external loads, equilibrium, constitutive equations, and structural design.

The text uses MATLAB® scripts to calculate analytic and approximated solutions of the considered mechanical models.

Computational Solid Mechanics: Variational Formulation and High Order Approximation presents mechanical models, their main hypothesis, and applications, and is intended for graduate and undergraduate engineering students taking courses in solid mechanics.

INTRODUCTION

Initial Aspects

Bars

Shafts

Beams

Two-dimensional Problems

Plates

Linear Elastic Solids

EQUILIBRIUM OF PARTICLES AND RIGID BODIES

Introduction

Diagrammatic Conventions

Equilibrium of Particles

Equilibrium of Rigid Bodies

Principle of Virtual Power (PVP)

Some aspects about the definition of power

Problems

FORMULATION AND APPROXIMATION OF BARS

Introduction

Kinematics

Strain Measure

Rigid actions

Equilibrium

Material Behavior

Application of the Constitutive Equation

Design and Verification

Bars Subjected to Temperature Changes

Volume and Area Strain Measures

Summary of the Variational Formulation of Bars

Approximated Solution

Finite Element Method (FEM)

Analysis of Trusses

Problems

FORMULATION AND APPROXIMATION OF SHAFTS

Introduction

Kinematics

Strain Measure

Rigid Actions

Equilibrium

Material Behavior

Application of the Constitutive Equation

Design and Verification

Summary of the Variational Formulation of Shafts

Approximated Solution

Mathematical Aspects of the FEM

Local Coordinate Systems

One-dimensional Shape Functions

Mapping

Numerical Integration

Collocation Derivative

Problems

FORMULATION AND APPROXIMATION OF BEAMS IN BENDING

Introduction

Kinematics

Strain Measure

Rigid Actions

Equilibrium

Application of the Constitutive Equation

Design and Verification

Summary of the Variational Formulation for the Euler-Bernouilli Beam

Buckling of Columns

Euler Column

Approximation of the Euler-Bernouilli Beam

High Order Beam Element

Mathematical Aspects of the FEM

Problems

FORMULATION AND APPROXIMATION OF BEAM WITH SHEAR

Introduction

Kinematics

Strain Measure

Rigid Actions

Equilibrium

Application of the Constitutive Equation

Shear Stress Distribution

Design and Verification

Standardized Cross Section Shapes

Shear Center

Summary of the Variational Formulation of Beams with Shear

Energy Methods

Approximation of the Timoshenko Beam

Mathematical Aspects of the FEM

Problems

FORMULATION AND APPROXIMATION OF D AND D BEAMS

Introduction

Two-dimensional Beam

Three-dimensional Beam

BeamLab Program

Summary of the Variational Formulation of Beams

Approximation of Beams

Exercises

FORMULATION AND APPROXIMATION OF SOLIDS

Introduction

Kinematics

Strain Measures

Rigid Actions

Equilibrium

Generalized Hooke Law

Application of the Constitutive Equation

Formulation Employing Tensors

Verification of Linear Elastic Solids

Approximation of Linear Elastic Solids

Problems

FORMULATION AND APPROXIMATION OF PLANE STATE PROBLEMS

Plane Stress State

Plane Strain State

Compatibility Equations

Analytical Solutions for Plane Problems in Elasticity

Analytical Solutions for Problems in Three-dimensional Elasticity

Plane State Approximation

(hp)fem program

Twist of Generic Sections

Multi-dimensional Numerical Integration

Summary of the Variational Formulation of Mechanical Models

Problems

FORMULATION AND APPROXIMATION OF PLATES IN BENDING

Introduction

Kinematics

Strain Measures

Rigid Actions

Equilibrium

Application of the Constitutive Equation

Approximation of the Kirchhoff Plate

Exercises

References

### Biography

Marco L. Bittencourt has a PhD in mechanical engineering at the University of Campinas in Brazil. He was a post-doc at the University of Kentucky and a visiting professor at the Division of Applied Mathematics at Brown University. His main research area is computational mechanics with emphasis on non-linear structural mechanics, high-order finite elements, and high-performance computing applied to the design of engine components and more recently biomechanics.