Table of Contents
1. INTRODUCTION. Prologue. Brief History. Introduction to Matrix Notation. Role of the Computer. General Steps of the Finite Element of Method. Applications of the Finite Element Methods. Advantages of the Finite Element Method. Computer Programs for the Finite Element Method. References. Problems. 2. INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD. Introduction. Definitions of the Stiffness Matrix. Derivation of the Stiffness Matrix for a Spring Element. Example of a Spring Assemblage. Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method). Boundary Conditions. Potential Energy Approach to Derive Spring Element Equations. References. Problems. 3. DEVELOPMENT OF TRUSS EQUATIONS. Introduction. Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates. Selecting Approximation Functions for Displacements. Transformation of Vectors in Two Dimensions. Global Stiffness Matrix. Computation of Stress for a Bar in the x-y Plane. Solution of a Plane Truss. Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space. Use of Symmetry in Structure. Inclined, or Skewed, Supports. Potential Energy Approach to Derive Bar Element Equations. Comparison of Finite Element Solution to Exact Solution for Bar. Galerkin's Residual Method and Its Application to a One-Dimensional Bar. References. Problems. 4. DEVELOPMENT OF BEAM EQUATIONS. Introduction. Beam Stiffness. Example of Assemblage of Beam Stiffness Matrices. Examples of Beam Analysis Using the Direct Stiffness Method. Distributed Loading. Comparison of Finite Element Solution to the Exact Solution for a Beam. Beam Element with Nodal Hinge. Potential Energy Approach to Derive Beam ElementEquations. Galerkin's Method for Deriving Beam Element Equations. References. Problems. 5. FRAME AND GRID EQUATIONS. Introduction. Two-Dimensional Arbitrarily Oriented Beam Element. Rigid Plane Frame Examples. Inclined or Skewed Supports-Frame Element. Grid Equations. Beam Element Arbitrarily Oriented in Space. Concepts of Substructure Analysis. References. Problems. 6. DEVELOPMENT OF THE PLANE STRESS AND PLANE STRAIN STIFFNESS EQUATIONS. Introduction. Basic Concepts of Plane Stress and Plane Strain. Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations. Treatment of Body and Surface Forces. Explicit Expression for the Constant-Strain Triangle Stiffness Matrix. Finite Element Solution of a Plane Stress Problem. References. Problems. 7. PRACTICAL CONSIDERATIONS IN MODELING; INTERPRETING RESULTS/ AND EXAMPLES OF PLANE STRESS/STRAIN ANALYSIS. Introduction. Finite Element Modeling. Equilibrium and Compatibility of Finite Element Results. Convergence of Solution. Interpretation of Stresses. Static Condensation. Flowchart for the Solution of Plane Stress Problems. Computer Program Results for Some Plane Stress/Strain Problems. References. Problems. 8. DEVELOPMENT OF THE LINEAR-STRAIN TRIANGLE EQUATIONS. Introduction. Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations. Example LST Stiffness Determination. Comparison of Elements. References. Problems. 9. AXISYMMETRIC ELEMENTS. Introduction. Derivation of the Stiffness Matrix. Solutions of an Axisymmetric Pressure Vessel. Applications of Axisymmetric Elements. References. Problems. 10. ISOPARAMETRIC FORMULATION. Introduction. Isoparametric Formulation of the Bar Element Stiffness Matrix. Rectangular Plane Stress Element. Isoparametric Formulation of the Plane Element Stiffness Matrix. Gaussian Quadrature (Numerical Integration). Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature. Higher-Order Shape Functions. References. Problems. 11. THREE-DIMENSIANAL STRESS ANALYSIS. Introduction. Three Dimensional Stress and Strain. Tetrahedral Element. Isoparametric Formulation. References. Problems. 12. PLATE BENDING ELEMENT. Introduction. Basic Concepts of Plate Bending. Derivation of a Plate Bending Element Stiffness Matrix and Equations. Some Plate Element Numerical Comparisons. Computer Solutions for a Plate Bending Problem. References. Problems. 13. HEAT TRANSFER AND MASS TRANSPORT. Introduction. Derivation of the Basic Differential Equation. Heat Transfer with Convection. Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h. One-Dimensional Finite Element Formulation Using a Variational Method. Two-Dimensional Finite Element Formulation. Line or Point Sources. One-Dimensional Heat Transfer with Mass Transport. Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method. Flowchart and Examples of Heat-Transfer Program. References. Problems. 14. FLUID FLOW. Introduction. Derivation of the Basic Differential Equations. One-Dimensional Finite Element Formulation. Two-Dimensional Finite Element Formulation. Flowchart and Example of a Fluid-Flow Program. References. Problems. 15. THERMAL STRESS. Introduction. Formulation of the Thermal Stress Problems and Examples. References. Problems. 16. STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER. Introduction. Dynamics of a Spring-Mass System. Direct Derivation of the Bar Element Equations. Numerical Integration in Time. Natural Frequencies of a One-Dimensional Bar. Time-Dependent One-Dimensional Bar Analysis. Beam Element Mass Matrices and Natural Frequencies. Truss, Plane Frame, Plane Stress/Strain, Axisymmetric, And Solid Element Mass Matrices. Time-Dependent Heat Transfer. Computer Program Example Solutions for Structural Dynamics. References. Problems. APPENDIX A. MATRIX ALGEBRA. Introduction. Definition of a Matrix. Matrix Operations. Cofactor or Adjoint Method to Determine the Inverse of a Matrix. Inverse of a Matrix by Row Reduction. References. Problems. APPENDIX B. METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS. Introduction. General Forms of the Equations. Uniqueness, Nonuniqueness, and Nonexistence of Solutions. Methods for Solving Linear Algebraic Equations. Banded-Symmetric Matrices, Bandwidth, Skyline and Wavefront Methods. References. Problems. APPENDIX C. EQUATIONS FROM ELASTICITY THEORY. Introduction. Differential Equations of Equilibrium. Strain/Displacement and Compatibility Equations. Stress/Strain Relationships. Reference. APPENDIX D. EQUIVALENT NODAL FORCES. Problems. APPENDIX E. PRINCIPLE OF VIRTUAL WORK. REFERENCES. ANSWER TO SELECTED PROBLEMS. INDEX.
