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

    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

    Final comments

    Problems

    FORMULATION AND APPROXIMATION OF BARS

    Introduction

    Kinematics

    Strain Measure

    Rigid actions

    Determination of Internal Loads

    Determination of External Loads

    Equilibrium

    Material Behavior

    Application of the Constitutive Equation

    Design and Verification

    Bars Subjected to Temperature Changes

    Volume and Area Strain Measures

    Singularity Functions for External Loading Representation

    Summary of the Variational Formulation of Bars

    Approximated Solution

    Finite Element Method (FEM)

    Analysis of Trusses

    Final Comments

    Problems

    FORMULATION AND APPROXIMATION OF SHAFTS

    Introduction

    Kinematics

    Strain Measure

    Rigid Actions

    Determination of Internal Loads

    Determination of External Loads

    Equilibrium

    Material Behavior

    Application of the Constitutive Equation

    Design and Verification

    Singularity Functions for External Loading Representation

    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

    Final Comments

    Problems

    FORMULATION AND APPROXIMATION OF BEAMS IN BENDING

    Introduction

    Kinematics

    Strain Measure

    Rigid Actions

    Determination of Internal Loads

    Determination of External Loads

    Equilibrium

    Application of the Constitutive Equation

    Design and Verification

    Singularity Functions for External Loading Representation

    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

    Final Comments

    Problems

    FORMULATION AND APPROXIMATION OF BEAM WITH SHEAR

    Introduction

    Kinematics

    Strain Measure

    Rigid Actions

    Determination of Internal Loads

    Determination of External Loads

    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

    Final Comments

    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

    Final Comments

    Exercises

    FORMULATION AND APPROXIMATION OF SOLIDS

    Introduction

    Kinematics

    Strain Measures

    Rigid Actions

    Determination of Internal Loads

    Determination of External Loads

    Equilibrium

    Generalized Hooke Law

    Application of the Constitutive Equation

    Formulation Employing Tensors

    Verification of Linear Elastic Solids

    Approximation of Linear Elastic Solids

    Final Comments

    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

    Final Comments

    Problems

    FORMULATION AND APPROXIMATION OF PLATES IN BENDING

    Introduction

    Kinematics

    Strain Measures

    Rigid Actions

    Determination of Internal Loads

    Determination of External Loads

    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.