A First Course in the Finite Element Method, SI Version

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ISBN: 9780495668275 | 0495668273
Cover: Paperback
Copyright: 4/11/2011

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A FIRST COURSE IN THE FINITE ELEMENT METHOD provides a simple, basic approach to the course material that can be understood by both undergraduate and graduate students without the usual prerequisites (i.e. structural analysis). 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.

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.
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
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 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 Use to Derive the One-Dimensional Bar Element Equations
Other Residual Methods and Their Application to a One-Dimensional Bar Problem
Flowchart for Solutions of Three-Dimensional Truss Problems
Computer Program Assisted Step-by-Step Solution for Truss Problem
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
Frame And Grid Equations
Two-Dimensional 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
Development Of The Plane Stress And Strain Stiffness Equations
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
Rectangular Plane Element (Bilinear Rectangle, Q4)
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 Stress-Strain Problems
Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems
Development Of The Linear-Strain Traingle Equations
Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations
Example of LST Stiffness Determination
Comparison of Elements
Axisymmetric Elements
Derivation of the Stiffness Matrix
Solution of an Axisymmetric Pressure Vessel
Applications of Axisymmetric Elements
Isoparametric Formulation
Isoparametric Formulation of the Bar Element Stiffness Matrix
Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix
Newton-Cotes and Gaussian Quadrature
Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature
Higher-Order Shape Functions
Three-Dimensional Stress Analysis
Three-Dimensional Stress and Strain
Tetrahedral Element
Isoparametric Formulation
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
Heat Transfer And Mass Transport
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
Three-Dimensional Heat Transfer by the Finite Element Method
One-Dimensional Heat Transfer with Mass Transport
Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method
Flowchart and Examples of a Heat-Transfer Program
Fluid Flow In Porous Media And Through Hydraulic Networks; And Electrical Networks And Electrostatics
Derivation of the Basic Differential Equations
One-Dimensional Finite Element Formulation
Two-Dimensional Finite Element Formulation
Flowchart and Example of a Fluid-Flow Program
Electrical Networks
Electrostatics
Thermal Stress
Formulation of the Thermal Stress Problem and Examples
Structural Dynamics And Time-Dependent Heat Transfer
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, Plane Strain, Axisymmetric, and Solid Element Mass Matrices
Time-Dependent Heat-Transfer
Computer Program Example Solutions for Structural Dynamics
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
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
Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods
Equations For Elasticity Theory
Introduction
Differential Equations of Equilibrium
Strain/Displacement and Compatibility Equations
Stress-Strain Relationships
Equivalent Nodal Forces
Principle Of Virtual Work
Properties Of Structural Steel And Aluminum Shapes
Answers To Selected Problems
Index
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A First Course in the Finite Element Method, SI Version

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