Introduction to Finite Elements in Engineering

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 MATLAB® 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 ξ - 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 MATLAB® 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 "Finite elements ("FE or FEA") is a numerical tool used for analyzing problems involving stress analysis, heat and fluid flow, resonance frequencies and mode shapes, etc. Irregular shaped domains, various materials can be incorporated. The book deals with a variety of topics in a manner that integrates theory, algorithms, modeling, and computer implementation. Many solved examples reinforce this pedagogy along with end-of-chapter problems, in-house source codes on multiple platforms, and a solutions manual for the instructor. Topics include energy and Galerkin approaches, equation solving with sparsity, elasticity, heat conduction and other scalar field problems, vibration and preand post- processing. The variety of topics dealt with enables the book to be used as a text in various engineering disciplines, at the senior-undergraduate or 1st year graduate level. The book can also serve as a learning resource for practicing engineers"-- Provided by publisher