Cover
Half-title
Title page
Copyright information
Dedication
Contents
Detailed Contents
Preface
Overview
Highlights Pertaining to Computer Programs Provided
Highlights Pertaining to Book Chapters
Chapter 1: Energy and Galerkin Approaches
Chapter 2: Gaussian Elimination
Chapter 3: One-Dimensional Elasticity
Chapter 4: One-Dimensional Heat Conduction
Chapter 5: Trusses
Chapter 6: Two- and Three-Dimensional Elasticity
Chapter 7: Axisymmetric Problems in Elasticity
Chapter 8: Two- and Three-Dimensional Heat Conduction and Other Scalar Field Problems
Chapter 9: Elements in Bending
Chapter 10: Structural Vibration
Chapter 11: Miscellaneous Topics
Chapter 12: Preprocessing and Postprocessing
How to Use this Book
1 Energy and Galerkin Approaches
1.1 Introduction
1.2 Outline of Presentation
1.3 Overview of Types of Problems and Solution Approaches
1.3.1 Scalar Field Problems
1.3.2 Structural Mechanics
1.3.3 Solution Approaches
1.3.4 Decisions Involved in Defining the Problem
1.3.5 Consistent Units
1.4 Computer Programs
1.5 Energy Approach
1.5.1 Energy in a Spring System
1.5.2 Energy in a Rod Undergoing Axial Deformation
1.5.3 Energy in a Beam Undergoing Bending
1.5.4 Work Potential
1.5.5 Principle of Minimum Potential Energy
1.5.6 Hamilton's Principle
1.5.7 Rayleigh-Ritz Method
1.5.8 Rayleigh-Ritz Method with Hamilton's Principle
1.6 Galerkin's Method
1.7 Linearity, Symmetry, and Positive Definiteness
Computer Programs
Problems
2 Gaussian Elimination
2.1 Introduction
2.2 Basic Concepts in Matrix Algebra
2.2.1 Simultaneous Equations
Quadratic Forms
2.2.2 Positive Definiteness
2.2.3 Positive Semi-Definiteness
2.2.4 Diagonally Dominant Matrix
2.2.5 Eigenvalues and Eigenvectors
2.2.6 Norms
2.3 Gaussian Elimination
2.3.1 General Algorithm for Gaussian Elimination
2.4 LU Factorization
2.4.1 Forward and Backward Substitution
Multiple Right-Hand Sides
2.5 Pivoting
2.6 Symmetric, Banded Matrices
2.7 Cuthill-McKee Algorithm to Reduce Bandwidth
2.8 Skyline (Variable-Band) Matrices
2.8.1 Storage of a Symmetric Skyline Matrix
2.8.2 Gaussian Elimination with Column Reduction
Reduction of Profile Using the Reverse Cuthill-McKee Algorithm
2.8.3 Note on Pivoting with Ordering
2.9 General Sparse Symmetric Matrices
2.9.1 Pivoting Based on the Minimum Degree Algorithm
2.10 Conjugate Gradient Method for Equation Solving
2.10.1 Conjugate Gradient Algorithm
2.11 Frontal Method for FE Matrices
2.11.1 Connectivity and the Prefront Routine
2.11.2 Element Assembly and Consideration of Specified dof
2.11.3 Elimination of Completed dof
2.11.4 Backward Substitution
2.11.5 Consideration of Multipoint Constraints
2.12 Using In-Built MATLABยฎ Routines
Computer Programs
Problems
3 One-Dimensional Elasticity
3.1 Introduction
3.2 Equations of Elasticity in 1D
3.3 Finite Element Modeling
3.3.1 Element Division
3.3.2 Numbering Scheme
3.4 Local Coordinates, Shape Functions, [N] and [B] Matrices
3.4.1 ฮพ
- or Intrinsic Coordinates
3.4.2 Shape or Interpolating Functions N1 and N2
3.4.3 Displacement Interpolation
3.4.4 Admissibility Conditions
3.4.5 The Strain-Displacement Matrix [B]
3.5 Element Stiffness and Load Matrices from Potential Energy
3.6 Assembly of the Global Stiffness Matrix and Load Vector
3.7 SPC-Type Boundary Conditions
3.7.1 Elimination Technique
Summary: Elimination Technique
3.7.2 Decoupling Technique
3.7.3 Penalty Function Technique
Obtaining Stresses and Support Reactions
3.8 Boundary Conditions and Singularity of [K]
3.9 MPC-Type Boundary Conditions
3.9.1 Master-Slave Elimination Technique
3.9.2 Penalty Function Technique
3.10 Banded Assembly of Stiffness Matrix
3.11 Convergence Aspects and Adaptivity
3.11.1 Monotonic Convergence in Energy
3.11.2 Rate of Convergence
3.11.3 Adaptivity
Computer Programs
Problems
4 One-Dimensional Heat Conduction
4.1 Introduction
4.2 Governing Differential Equation
4.2.1 Boundary Conditions
4.3 Two-Node Linear Element
4.4 Galerkin's Approach
4.5 Handling SPC Boundary Conditions
4.5.1 BC T1 = T0 via Elimination Technique
4.5.2 BC T1 = T0 via Decoupling Technique
4.5.3 BC T1 = T0 via Penalty Function Technique
4.6 One-Dimensional Heat Transfer in Thin Fins
4.6.1 Derivation of Matrices via Galerkin's Approach
4.7 Convergence and Adaptivity
4.8 Transient Heat Conduction
4.8.1 Weak Form
4.8.2 Derivation of Matrices
4.9 A Discussion of Integration Techniques
4.9.1 Mesh Size
4.9.2 Explicit Methods
Pros of Explicit Methods
Cons of Explicit Methods
4.9.3 Implicit Methods
Pros of Implicit Methods
Cons of Implicit Methods
4.9.4 Mode Superposition
Computer Programs
Problems
5 Trusses
5.1 Introduction
5.2 Two-Dimensional Trusses
5.2.1 Local and Global Coordinate Systems
5.2.2 Formulas for Calculating and m
5.2.3 Element Stiffness Matrix
5.2.4 Stress Calculation
5.2.5 Temperature Load Vector
5.3 Three-Dimensional Trusses
5.4 Problem Modeling and Boundary Conditions
5.4.1 Incorrect BCs Causing Singularity
5.4.2 Symmetry
5.4.3 Anti-symmetry
5.4.4 Multipoint Constraints
5.5 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions
5.5.1 Assembly for Banded Solution
5.5.2 Assembly for Skyline Solution
Computer Programs
Problems
6 Two- and Three-Dimensional Elasticity
6.1 Introduction
6.2 Equations of Elasticity in 3D
6.2.1 Displacements and External Loads
6.2.2 Stresses and Equilibrium
6.2.3 Strain-Displacement Relations
6.2.4 Stress-Strain Relations
6.2.5 Compatibility Equations
6.2.6 Potential Energy and Work Potential
6.2.7 Weak Form Associated with Galerkin's Method
6.3 Equations of Elasticity in 2D
6.3.1 Plane Stress
6.3.2 Plane Strain
Compatibility Condition for Both Plane Stress and Plane Strain
6.4 Numbering Scheme, Element Library, Shape Functions, and Notation Used
6.4.1 Three-Node Linear Triangular Element
6.4.2 Four-Node Quadrilateral Element
6.4.3 Degenerate Quad
6.4.4 Nine-Node Quadrilateral
6.4.5 Eight-Node Quadrilateral
6.4.6 Six-Node Triangle
6.4.7 Four-Node Tetrahedral
6.4.8 Eight-Node Hexahedral
6.4.9 Admissibility Conditions
6.5 The Strain-Displacement Matrix [B]
6.5.1 Two-Dimensional Elements
6.5.2 Three-Dimensional Elements
6.6 Element Stiffness Matrix and Load Vectors
6.7 Numerical Integration
6.7.1 One-Dimensional Integrals
6.7.2 Two- and Three-Dimensional Integrals for Quadrilateral and Hexahedral Elements
6.7.3 Stiffness Integration
6.7.4 Numerical Integration for Triangular and Tetrahedral Elements
6.7.5 Stress Calculations
6.8 Modeling and Boundary Conditions
6.8.1 Rectangular Plate
6.8.2 Octagonal Pipe (Inclined Roller)
6.8.3 Rigid Connection
6.8.4 Part with Pin Connection Subject to an Impact Load
6.8.5 Three-Dimensional Model of a Cylinder on Bearings
6.8.6 Pyramid-Shaped Part
6.8.7 Some General Comments on Meshing
6.9 Assembly for Sparse Solvers
6.9.1 Banded Assembly
6.9.2 Skyline Assembly
6.9.3 Sparse Assembly and Solution via MATLABยฎ Routines
6.9.4 Frontal Assembly
6.10 Convergence Aspects and Adaptivity
6.10.1 Monotonic Convergence in Energy
Positive Definiteness
Completeness
Rate of Convergence
6.10.2 Adaptivity
6.10.3 Stress Singularities
Computer Programs
Problems
7 Axisymmetric Problems in Elasticity
7.1 Introduction
7.2 Equations of Elasticity in Axisymmetric Analysis
7.2.1 Displacements and External Loads
7.2.2 Stresses and Equilibrium
7.2.3 Strain-Displacement Relations
7.2.4 Stress-Strain Relations
7.2.5 Potential Energy and Work Potential
7.3 Derivation of Strain-Displacement Matrix [Ba]
7.4 Element Stiffness Matrix and Load Vectors
7.4.1 Formulas for Constant Body Force and Constant Traction on a Three-Node Triangular Element
7.5 Problem Modeling and Boundary Conditions
7.5.1 Arresting Rigid-Body Motion
7.5.2 Cylinder Subjected to Internal Pressure
7.5.3 Infinite Cylinder
7.5.4 Press-Fit on a Rigid Shaft
7.5.5 Press-Fit on an Elastic Shaft
7.5.6 Belleville Spring
7.5.7 Thermal Stress Problem
7.5.8 Notation Used in Commercial Programs
Computer Programs
Problems
8 Two- and Three-Dimensional Heat Conduction and Other Scalar Field Problems
8.1 Introduction
8.2 Steady-State Heat Transfer
8.2.1 Differential Equation
8.2.2 Boundary Conditions
8.2.3 Finite Element Interpolation for 2D Elements
8.2.4 Three-Node Triangular Element
8.2.5 Four-Node Quadrilateral Element
8.2.6 The [B] Matrix
8.2.7 Element Matrices Derived from Galerkin's Approach
8.2.8 Specific Expressions for the Three-Node Triangular Element
8.2.9 Preprocessing for Program Heat2D
8.2.10 Two-Dimensional Fins
8.3 Transient Heat Conduction, Including Axisymmetric Problems
8.4 Torsion, Potential Flow, Seepage, Electric and Magnetic Fields, Fluid Flow in Ducts, and Acoustics
8.4.1 Torsion Problem
8.4.2 Potential Flow
8.4.3 Seepage
8.4.4 Fluid Flow in Ducts
8.4.5 Electrical and Magnetic Field Problems
8.4.6 Acoustics
Boundary Conditions
One-Dimensional Acoustics
Computer Programs
Problems
9 Elements in Bending
9.1 Introduction
9.2 Beams
9.2.1 Potential Energy
9.2.2 Finite Element Formulation
9.2.3 Hermite Shape Functions
9.2.4 Element Stiffness Matrix
9.2.5 Load Vector
9.2.6 Shear Force and Bending Moment
9.3 Beams on Elastic Supports
9.4 Plane Frames
9.5 Three-Dimensional Frames
9.6 Modeling Aspects
9.6.1 Symmetry
9.6.2 Pin Joint
9.7 Thin Plates in Bending
9.7.1 Potential Energy
9.7.2 Nine-dof Triangular Element
9.7.3 Absence of a C1 Continuous Displacement Field for a Nine-dof Element
9.7.4 The DKT Element
9.7.5 Converting Distributed Loads to Equivalent Point Loads in the DKT Element
9.8 Folded Plates
9.9 Thermal Stresses
Computer Programs
Problems
10 Structural Vibration
10.1 Introduction
10.2 Formulation
10.2.1 Hamilton's Principle
10.2.2 Solid Body with Distributed Mass
10.2.3 Natural Frequencies and Mode Shapes: the Eigenvalue Problem
10.3 Consistent Element Mass Matrices
10.3.1 One-Dimensional or Bar Element
10.3.2 Truss Element
10.3.3 Three-Node Triangular or CST Element
10.3.4 Isoparametric Elements in 2D and 3D Elasticity
10.3.5 Tetrahedral Element
10.3.6 Hexahedral Element
10.3.7 Axisymmetric Triangular Element
10.3.8 Beam Element
10.3.9 Frame Element
10.3.10 DKT Plate Bending Element
Transverse Motion
In-Plane Motion
10.4 Lumped Element Mass Matrices
10.4.1 Technique 1
Bar Element
Truss Element
CST Element
Beam Element
DKT Plate Bending Element
10.4.2 Technique 2
Beam Element
Frame Element in Local Coordinate System
10.5 Forced Response Problems
10.5.1 Mode Superposition
10.5.2 Direct Integration Techniques
10.6 Guyan Reduction
10.7 Methods for Solving the Eigenvalue Problem
10.7.1 Properties of Eigenvectors
10.7.2 Characteristic Polynomial Method
10.7.3 Vector Iteration Methods
Inverse Iteration Method
Forward Iteration
10.7.4 Shifting
10.7.5 Orthogonal Space
10.7.6 Transformation Methods
10.7.7 Generalized Jacobi Method
10.7.8 Tridiagonalization and the Implicit Shift Approach
10.7.9 Bringing the Generalized Problem to the Standard Form
10.7.10 Tridiagonalization
10.7.11 Implicit Symmetric QR Step with Wilkinson Shift for Diagonalization
10.8 Buckling Analysis
Computer Programs
Problems
11 Miscellaneous Topics
11.1 Introduction
11.2 Linear Orthotropic Materials
11.2.1 Temperature Effects
11.3 Nonlinear Heat Conduction Problems in 1D
11.3.1 Temperature-Dependent Thermal Conductivity
11.3.2 Load Increment Loop
11.3.3 Nonlinear Elastic Material Under Small Strain
11.3.4 Physical Interpretation of the Newton-Raphson Method
11.4 Additional Nonlinear Problems
11.4.1 Elastoplastic Problem in 1D
11.4.2 Buckling Load Determination via Nonlinear Analysis
11.4.3 General Comments
Problems
12 Preprocessing and Postprocessing
12.1 Introduction
12.2 Mesh Generation for 2D Plane Problems
12.2.1 Region and Block Representation
12.2.2 Block Corner Nodes, Sides, and Subdivisions
12.2.3 Generation of Node Numbers
12.2.4 Generation of Coordinates and Connectivity
12.2.5 Examples of Mesh Generation
12.2.6 Mesh Plotting
12.2.7 Data Handling and Editing
12.3 Filling a 3D Region with Tetrahedral Elements
12.4 Postprocessing
12.4.1 Deformed Configuration and Mode Shape
12.4.2 Contour Plotting
12.4.3 Nodal Values from Known Constant Element Values for a Triangle
12.4.4 Least-Squares Fit for a Four-Node Quadrilateral
12.5 Conclusion
Computer Programs
Problems
Appendix List of Symbols and Computer Code Variables
Bibliography
Web Documents
Index