A First Course in the Finite Element Method

Description

Gain a clear understanding of the basics of the finite element method (FEM) with this simple, direct, contemporary approach in Logan's A FIRST COURSE IN THE FINITE ELEMENT METHOD, Enhanced 6th Edition, SI Version. This unique presentation is written so you can easily comprehend content without the usual prerequisites, such as structural analysis. This book is ideal, whether you are a civil or mechanical engineering student primarily interested in stress analysis and heat transfer, or you need a foundation for applying FEM as a tool in solving practical physical problems. New and expanded real-world examples and problems demonstrate FEM applications in a variety of engineering and mathematical physics-related fields. Each chapter uses a consistent structure with step-by-step, worked-out examples, ideal for undergraduate or graduate-level study. A new WebAssign digital platform provides additional online resources to clarify concepts and assist you in completing assignments. Cover Contents Preface to the SI Edition Preface Digital Resources Notation Chapter 1: Introduction Chapter Objectives Prologue 1.1 Brief History 1.2 Introduction to Matrix Notation 1.3 Role of the Computer 1.4 General Steps of the Finite Element Method 1.5 Applications of the Finite Element Method 1.6 Advantages of the Finite Element Method 1.7 Computer Programs for the Finite Element Method References Problems Chapter 2: Introduction to the Stiffness (Displacement) Method Chapter Objectives Introduction 2.1 Definition of the Stiffness Matrix 2.2 Derivation of the Stiffness Matrix for a Spring Element 2.3 Example of a Spring Assemblage 2.4 Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) 2.5 Boundary Conditions 2.6 Potential Energy Approach to Derive Spring Element Equations Summary Equations References Problems Chapter 3: Development of Truss Equations Chapter Objectives Introduction 3.1 Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates 3.2 Selecting a Displacement Function in Step 2 of the Derivation of Stiffness Matrix for the One-Dimensional Bar Element 3.3 Transformation of Vectors in Two Dimensions 3.4 Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane 3.5 Computation of Stress for a Bar in the x - y Plane 3.6 Solution of a Plane Truss 3.7 Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space 3.8 Use of Symmetry in Structures 3.9 Inclined, or Skewed, Supports 3.10 Potential Energy Approach to Derive Bar Element Equations 3.11 Comparison of Finite Element Solution to Exact Solution for Bar 3.12 Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations 3.13 Other Residual Methods and Their Application to a One-Dimensional Bar Problem 3.14 Flowchart for Solution of Three-Dimensional Truss Problems 3.15 Computer Program Assisted Step-by-Step Solution for Truss Problem Summary Equations References Problems Chapter 4: Development of Beam Equations Chapter Objectives Introduction 4.1 Beam Stiffness 4.2 Example of Assemblage of Beam Stiffness Matrices 4.3 Examples of Beam Analysis Using the Direct Stiffness Method 4.4 Distributed Loading 4.5 Comparison of the Finite Element Solution to the Exact Solution for a Beam 4.6 Beam Element with Nodal Hinge 4.7 Potential Energy Approach to Derive Beam Element Equations 4.8 Galerkin's Method for Deriving Beam Element Equations Summary Equations References Problems Chapter 5: Frame and Grid Equations Chapter Objectives Introduction 5.1 Two-Dimensional Arbitrarily Oriented Beam Element 5.2 Rigid Plane Frame Examples 5.3 Inclined or Skewed Supports - Frame Element 5.4 Grid Equations 5.5 Beam Element Arbitrarily Oriented in Space 5.6 Concept of Substructure Analysis Summary Equations References Problems Chapter 6: Development of the Plane Stress and Plane Strain Stiffness Equations Chapter Objectives Introduction 6.1 Basic Concepts of Plane Stress and Plane Strain 6.2 Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations 6.3 Treatment of Body and Surface Forces 6.4 Explicit Expression for the Constant-Strain Triangle Stiffness Matrix 6.5 Finite Element Solution of a Plane Stress Problem 6.6 Rectangular Plane Element (Bilinear Rectangle, Q4) Summary Equations References Problems Chapter 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis Chapter Objectives Introduction 7.1 Finite Element Modeling 7.2 Equilibrium and Compatibility of Finite Element Results 7.3 Convergence of Solution and Mesh Refinement 7.4 Interpretation of Stresses 7.5 Flowchart for the Solution of Plane Stress/Strain Problems 7.6 Computer Program-Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems References Problems Chapter 8: Development of the Linear-Strain Triangle Equations Chapter Objectives Introduction 8.1 Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations 8.2 Example LST Stiffness Determination 8.3 Comparison of Elements Summary Equations References Problems Chapter 9: Axisymmetric Elements Chapter Objectives Introduction 9.1 Derivation of the Stiffness Matrix 9.2 Solution of an Axisymmetric Pressure Vessel 9.3 Applications of Axisymmetric Elements Summary Equations References Problems Chapter 10: Isoparametric Formulation Chapter Objectives Introduction 10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix 10.2 Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix 10.3 Newton-Cotes and Gaussian Quadrature 10.4 Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature 10.5 Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements) Summary Equations References Problems Chapter 11: Three-Dimensional Stress Analysis Chapter Objectives Introduction 11.1 Three-Dimensional Stress and Strain 11.2 Tetrahedral Element 11.3 Isoparametric Formulation and Hexahedral Element Summary Equations References Problems Chapter 12: Plate Bending Element Chapter Objectives Introduction 12.1 Basic Concepts of Plate Bending 12.2 Derivation of a Plate Bending Element Stiffness Matrix and Equations 12.3 Some Plate Element Numerical Comparisons 12.4 Computer Solutions for Plate Bending Problems Summary Equations References Problems Chapter 13: Heat Transfer and Mass Transport Chapter Objectives Introduction 13.1 Derivation of the Basic Differential Equation 13.2 Heat Transfer with Convection 13.3 Typical Units; Thermal Conductivities, K; and Heat Transfer Coefficients, h 13.4 One-Dimensional Finite Element Formulation Using a Variational Method 13.5 Two-Dimensional Finite Element Formulation 13.6 Line or Point Sources 13.7 Three-Dimensional Heat Transfer by the Finite Element Method 13.8 One-Dimensional Heat Transfer with Mass Transport 13.9 Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method 13.10 Flowchart and Examples of a Heat Transfer Program Summary Equations References Problems Chapter 14: Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics Chapter Objectives Introduction 14.1 Derivation of the Basic Differential Equations 14.2 One-Dimensional Finite Element Formulation 14.3 Two-Dimensional Finite Element Formulation 14.4 Flowchart and Example of a Fluid-Flow Program 14.5 Electrical Networks 14.6 Electrostatics Summary Equations References Problems Chapter 15: Thermal Stress Chapter Objectives Introduction 15.1 Formulation of the Thermal Stress Problem and Examples Summary Equations Reference Problems Chapter 16: Structural Dynamics and Time-Dependent Heat Transfer Chapter Objectives Introduction 16.1 Dynamics of a Spring-Mass System 16.2 Direct Derivation of the Bar Element Equations 16.3 Numerical Integration in Time 16.4 Natural Frequencies of a One-Dimensional Bar 16.5 Time-Dependent One-Dimensional Bar Analysis 16.6 Beam Element Mass Matrices and Natural Frequencies 16.7 Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices 16.8 Time-Dependent Heat Transfer 16.9 Computer Program Example Solutions for Structural Dynamics Summary Equations References Problems Appendix A: Matrix Algebra Appendix B: Methods for Solution of Simultaneous Linear Equations Appendix C: Equations from Elasticity Theory Appendix D: Equivalent Nodal Forces Appendix E: Principle of Virtual Work Appendix F: Geometric Properties of Structural Steel Wide-Flange Sections (W Shapes) Answers to Selected Problems Index