A First Course in the Finite Element Analysis provides a simple, basic approach to the finite element method that can be understood by both undergraduate and graduate students. It does not have the usual prerequisites (such as structural analysis) required by most available texts in this area. The book is written primarily as a basic learning tool for the undergraduate student in civil and mechanical engineering whose main interest is in stress analysis and heat transfer. The text is geared toward those who want to apply the finite element method as a tool to solve practical physical problems. This revised fourth edition includes the addition of a large number of new problems (including SI problems), an appendix for mechanical and thermal properties, and more civil applications.

Features

The book proceeds from basic to advanced topics and is suitable for a one or two-course sequence.
Each chapter is structured in a similar format. General principles are presented for each topic, followed by traditional applications of these principles, which are in turn followed by computer applications where relevant.
The principle of minimum potential energy and Galerkin’’s residual method are introduced at various stages as required to develop the equations needed for analysis.
Appendices include basic matrix algebra (used throughout the text), solutions methods for simultaneous equations, equations from elasticity theory, equivalent nodal forces, the principle of virtual work, and properties of structural steel and aluminum shapes.
Many worked examples appear throughout the text. These examples are solved "longhand" to illustrate how essential concepts are applied.

Chapter 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.

Chapter 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.

Chapter 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.

Chapter 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 Element Equations. Galerkin’’s Method for Deriving Beam Element Equations. References. Problems.

Chapter 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.

Chapter 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.

Chapter 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.

Chapter 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.

Chapter 9 - Axisymmetric Elements
Introduction. Derivation of the Stiffness Matrix. Solutions of an Axisymmetric Pressure Vessel. Applications of Axisymmetric Elements. References. Problems.

Chapter 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.

Chapter 11 - Three-Dimensional Stress Analysis
Introduction. Three Dimensional Stress and Strain. Tetrahedral Element. Isoparametric Formulation. References. Problems.

Chapter 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.

Chapter 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.

Chapter 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.

Chapter 15 - Thermal Stress
Introduction. Formulation of the Thermal Stress Problems and Examples. References. Problems.

Chapter 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.

Appendix F - Properties of Structural Steel and Aluminum Shapes

Answer to Selected Problems.
Index.