

A First Course in the Finite Element Method, SI Version
, 5th Edition
 Daryl L. Logan
 University of Wisconsin, Platteville
 ISBN10: 0495668273 ISBN13: 9780495668275
 954 Pages Paperbound
 © 2012
Published

Table of Contents

1. INTRODUCTION
Brief History. Introduction to Matrix Notation. Role of the Computer. General Steps of the Finite Element Method. Applications of the Finite Element Method. Advantages of the Finite Element Method. Computer Programs for the Finite Element Method.
2. INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD
Definition 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.
3. DEVELOPMENT OF TRUSS EQUATIONS
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 for Bar Arbitrarily Oriented in the Plane. Computation of Stress for a Bar in the xy Plane. Solution of a Plane Truss. Transformation Matrix and Stiffness Matrix for a Bar in ThreeDimensional 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 Use to Derive the OneDimensional Bar Element Equations. Other Residual Methods and Their Application to a OneDimensional Bar Problem. Flowchart for Solutions of ThreeDimensional Truss Problems. Computer Program Assisted StepbyStep Solution for Truss Problem.
4. DEVELOPMENT OF BEAM EQUATIONS
Beam Stiffness. Example of Assemblage of Beam Stiffness Matrices. Examples of Beam Analysis Using the Direct Stiffness Method. Distribution Loading. Comparison of the Finite Element Solution to the Exact Solution for a Beam. Beam Element with Nodal Hinge. Potential Energy Approach to Derive Beam Element Equations. Galerkin’s Method for Deriving Beam Element Equations.
5. FRAME AND GRID EQUATIONS
TwoDimensional Arbitrarily Oriented Beam Element. Rigid Plane Frame Examples. Inclined or Skewed Supports  Frame Element. Grid Equations. Beam Element Arbitrarily Oriented in Space. Concept of Substructure Analysis.
6. DEVELOPMENT OF THE PLANE STRESS AND STRAIN STIFFNESS EQUATIONS
Basic Concepts of Plane Stress and Plane Strain. Derivation of the ConstantStrain Triangular Element Stiffness Matrix and Equations. Treatment of Body and Surface Forces. Explicit Expression for the ConstantStrain Triangle Stiffness Matrix. Finite Element Solution of a Plane Stress Problem. Rectangular Plane Element (Bilinear Rectangle, Q4).
7. PRACTICAL CONSIDERATIONS IN MODELING: INTERPRETING RESULTS AND EXAMPELS OF PLANE STRESS/STRAIN ANALYSIS
Finite Element Modeling. Equilibrium and Compatibility of Finite Element Results. Convergence of Solution. Interpretation of Stresses. Static Condensation. Flowchart for the Solution of Plane StressStrain Problems. Computer Program Assisted StepbyStep Solution, Other Models, and Results for Plane StressStrain Problems.
8. DEVELOPMENT OF THE LINEARSTRAIN TRAINGLE EQUATIONS
Derivation of the LinearStrain Triangular Element Stiffness Matrix and Equations. Example of LST Stiffness Determination. Comparison of Elements.
9. AXISYMMETRIC ELEMENTS
Derivation of the Stiffness Matrix. Solution of an Axisymmetric Pressure Vessel. Applications of Axisymmetric Elements.
10. ISOPARAMETRIC FORMULATION
Isoparametric Formulation of the Bar Element Stiffness Matrix. Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix. NewtonCotes and Gaussian Quadrature. Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature. HigherOrder Shape Functions.
11. THREEDIMENSIONAL STRESS ANALYSIS
ThreeDimensional Stress and Strain. Tetrahedral Element. Isoparametric Formulation.
12. PLATE BENDING ELEMENT
Basic Concepts of Plate Bending. Derivation of a Plate Bending Element Stiffness Matrix and Equations. Some Plate Element Numerical Comparisons. Computer Solutions for Plate Bending Problems.
13. HEAT TRANSFER AND MASS TRANSPORT
Derivation of the Basic Differential Equation. Heat Transfer with Convection. Typical Units; Thermal Conductivities K; and HeatTransfer Coefficients, h. OneDimensional Finite Element Formulation Using a Variational Method. TwoDimensional Finite Element Formulation. Line or Point Sources. ThreeDimensional Heat Transfer by the Finite Element Method. OneDimensional Heat Transfer with Mass Transport. Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin’s Method. Flowchart and Examples of a HeatTransfer Program.
14. FLUID FLOW IN POROUS MEDIA AND THROUGH HYDRAULIC NETWORKS; AND ELECTRICAL NETWORKS AND ELECTROSTATICS
Derivation of the Basic Differential Equations. OneDimensional Finite Element Formulation. TwoDimensional Finite Element Formulation. Flowchart and Example of a FluidFlow Program. Electrical Networks. Electrostatics.
15. THERMAL STRESS
Formulation of the Thermal Stress Problem and Examples.
16. STRUCTURAL DYNAMICS AND TIMEDEPENDENT HEAT TRANSFER
Dynamics of a SpringMass System. Direct Derivation of the Bar Element Equations. Numerical Integration in Time. Natural Frequencies of a OneDimensional Bar. TimeDependent OneDimensional Bar Analysis. Beam Element Mass Matrices and Natural Frequencies. Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices. TimeDependent HeatTransfer. Computer Program Example Solutions for Structural Dynamics.
APPENDIX A – MATRIX ALGEBRA
Definition of a Matrix. Matrix Operations. Cofactor of Adjoint Method to Determine the Inverse of a Matrix. Inverse of a Matrix by Row Reduction. Properties of Stiffness Matrices.
APPENDIX B – METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS
Introduction. General Form of the Equations. Uniqueness, Nonuniqueness, and Nonexistence of Solution. Methods for Solving Linear Algebraic Equations. BandedSymmetric Matrices, Bandwidth, Skyline, and Wavefront Methods.
APPENDIX C – EQUATIONS FOR ELASTICITY THEORY
Introduction. Differential Equations of Equilibrium. Strain/Displacement and Compatibility Equations. StressStrain Relationships.
APPENDIX D – EQUIVALENT NODAL FORCES
APPENDIX E – PRINCIPLE OF VIRTUAL WORK
APPENDIX F – PROPERTIES OF STRUCTURAL STEEL AND ALUMINUM SHAPES
ANSWERS TO SELECTED PROBLEMS
INDEX