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A First Course in the Finite Element Method 6th edition by Daryl Logan ISBN 1305635116 9781305635111

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ISBN 10: 1305635116
ISBN 13: 9781305635111
Author: Daryl Logan

Provide a simple, direct approach that highlights the basics with A FIRST COURSE IN THE FINITE ELEMENT METHOD, 6E. This unique book is written so both undergraduate and graduate students can easily comprehend the content without the usual prerequisites, such as structural analysis. The book is written primarily as a basic learning tool for the undergraduate students in civil and mechanical engineering who are primarily interested in stress analysis and heat transfer. The text offers ideal preparation for students who want to apply the finite element method as a tool to solve practical physical problems.

A First Course in the Finite Element Method 6th Table of contents:

Chapter 1: Introduction

  • 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

    • Direct Methods

    • Variational Methods

    • Weighted Residual Methods

    • Direct Equilibrium Method

    • Work or Energy Methods

    • Weighted Residuals Methods

  • 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

  • Problems

Chapter 2: Introduction to the Stiffness (Displacement) Method

  • 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

    • Homogeneous Boundary Conditions

    • Nonhomogeneous Boundary Conditions

  • 2.6. Potential Energy Approach to Derive Spring Element Equations

    • Summary Equations

  • Problems

Chapter 3: Development of Truss Equations

  • 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

    • Guidelines for Selecting Displacement Functions

  • 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

    • General Formulation

    • Bar Element Formulation

  • 3.13. Other Residual Methods and Their Application to a One-Dimensional Bar Problem

    • Collocation Method

    • Subdomain Method

    • Least Squares Method

    • Galerkin’s Method

  • 3.14. Flowchart for Solution of Three-Dimensional Truss Problems

  • 3.15. Computer Program Assisted Step-by-Step Solution for Truss Problem

    • Summary Equations

  • Problems

Chapter 4: Development of Beam Equations

  • 4.1. Beam Stiffness

    • Beam Stiffness Matrix Based on Euler-Bernoulli Beam Theory (Considering Bending Deformations Only)

    • Beam Stiffness Matrix Based on Timoshenko Beam Theory (Including Transverse Shear Deformation)

  • 4.2. Example of Assemblage of Beam Stiffness Matrices

  • 4.3. Examples of Beam Analysis Using the Direct Stiffness Method

  • 4.4. Distributed Loading

    • Work-Equivalence Method

    • Example of Load Replacement

    • General Formulation

  • 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

    • Galerkin’s Method for Deriving Beam Element Equations

  • Summary Equations

  • Problems

Chapter 5: Frame and Grid Equations

  • 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

    • Bending in x ′ – z ′ Plane

    • Bending in the x ′ – y ′ Plane

  • 5.6. Concept of Substructure Analysis

  • Summary Equations

  • References

  • Problems

Chapter 6: Development of the Plane Stress and Plane Strain Stiffness Equations

  • 6.1. Basic Concepts of Plane Stress and Plane Strain

    • Plane Stress

    • Plane Strain

    • Two-Dimensional State of Stress and Strain

  • 6.2. Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations

  • 6.3. Treatment of Body and Surface Forces

    • Body Forces

    • 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)

    • Numerical Comparison of CST to Q4 Element Models and Element Defects

  • Summary Equations

  • Problems

Chapter 7: Practical Considerations in Modeling; Interpreting Results; and Examples of Plane Stress/Strain Analysis

  • 7.1. Finite Element Modeling

    • General Considerations

    • Aspect Ratio and Element Shapes

    • Use of Symmetry

    • Natural Subdivisions at Discontinuities

    • Mesh Revision (Refinement) and Convergence and h, p, and r Methods of Refinement

    • Transition Triangles

    • Concentrated or Point Loads and Infinite Stress

    • Infinite Medium

    • Connecting (Mixing) Different Kinds of Elements

    • Checking the Model for Errors

    • Checking the Results and Typical Postprocessor Results

  • 7.2. Equilibrium and Compatibility of Finite Element Results

  • 7.3. Convergence of Solution and Mesh Refinement

    • Patch Test

    • Patch Test Example

  • 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

  • Problems

Chapter 8: Development of the Linear-Strain Triangle Equations

  • 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

  • Problems

Chapter 9: Axisymmetric Elements

  • 9.1. Derivation of the Stiffness Matrix

    • Distributed Body Forces

    • Surface Forces

  • 9.2. Solution of an Axisymmetric Pressure Vessel

  • 9.3. Applications of Axisymmetric Elements

  • Summary Equations

  • Problems

Chapter 10: Isoparametric Formulation

  • 10.1. Isoparametric Formulation of the Bar Element Stiffness Matrix

    • Body Forces

    • Surface Forces

  • 10.2. Isoparametric Formulation of the Plane Quadrilateral (Q4) Element Stiffness Matrix

    • Body Forces

    • Surface Forces

  • 10.3. Newton-Cotes and Gaussian Quadrature

    • Newton-Cotes Numerical Integration

    • Gaussian Quadrature

    • Two-Point Formula

  • 10.4. Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature

    • Evaluation of the Stiffness Matrix

    • Evaluation of Element Stresses

  • 10.5. Higher-Order Shape Functions (Including Q6, Q8, Q9, and Q12 Elements)

    • Linear Strain Bar

    • Improved Bilinear Quadratic (Q6)

    • Quadratic Rectangle (Q8 and Q9)

    • Cubic Rectangle (Q12)

  • Summary Equations

  • Problems

Chapter 11: Three-Dimensional Stress Analysis

  • 11.1. Three-Dimensional Stress and Strain

  • 11.2. Tetrahedral Element

    • Body Forces

    • Surface Forces

  • 11.3. Isoparametric Formulation and Hexahedral Element

    • Linear Hexahedral Element

    • Quadratic Hexahedral Element

  • Summary Equations

  • Problems

Chapter 12: Plate Bending Element

  • 12.1. Basic Concepts of Plate Bending

    • Basic Behavior of Geometry and Deformation

    • Kirchhoff Assumptions

    • Stress/Strain Relations

    • Potential Energy of a Plate

  • 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

  • Problems

Chapter 13: Heat Transfer and Mass Transport

  • 13.1. Derivation of the Basic Differential Equation

    • One-Dimensional Heat Conduction (without Convection)

    • Two-Dimensional Heat Conduction (Without Convection)

  • 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

  • Problems

Chapter 14: Fluid Flow in Porous Media and through Hydraulic Networks; and Electrical Networks and Electrostatics

  • 14.1. Derivation of the Basic Differential Equations

    • Fluid Flow through a Porous Medium

    • Fluid Flow in Pipes and Around Solid Bodies

  • 14.2. One-Dimensional Finite Element Formulation

    • Fluid Flow through Hydraulic Networks

  • 14.3. Two-Dimensional Finite Element Formulation

  • 14.4. Flowchart and Example of a Fluid-Flow Program

  • 14.5. Electrical Networks

  • 14.6. Electrostatics

    • Coulomb’s Law

    • Gauss’s Law

    • Poisson’s Equation

    • Dielectric Constants

    • Finite Element Formulation of a Two-Dimensional Triangle Element

  • Summary Equations

  • Problems

Chapter 15: Thermal Stress

  • 15.1. Formulation of the Thermal Stress Problem and Examples

    • One-Dimensional Bar

    • Two-Dimensional Plane Stress and Plane Strain

    • Axisymmetric Element

  • Summary Equations

  • Problems

Chapter 16: Structural Dynamics and Time-Dependent Heat Transfer

  • 16.1. Dynamics of a Spring-Mass System

  • 16.2. Direct Derivation of the Bar Element Equations

  • 16.3. Numerical Integration in Time

    • Central Difference Method

    • Newmark’s Method

    • Wilson’s Method

  • 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

    • Truss Element

    • Plane Frame Element

    • Plane Stress/Strain Element

    • Axisymmetric Element

    • Tetrahedral Solid Element

  • 16.8. Time-Dependent Heat Transfer

    • Numerical Time Integration

  • 16.9. Computer Program Example Solutions for Structural Dynamics

    • Damping

  • Summary Equations

  • Problems

Appendices

  • 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. Properties of Structural Steel Shapes

  • Conversion Factors U.S. Customary Units to SI Units

  • Properties of Plane Areas

  • Properties of Solids

  • Physical Properties in SI and USCS Units

  • Typical Properties of Selected Engineering Materials

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