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# Solution Manual for A First Course in the Finite Element Method 6th Edition by Daryl L. Logan

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#### Solution Manual for A First Course in the Finite Element Method 6th Edition by Daryl L. Logan

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

1305635116, 0357704746, 1337342408, 9781305635111, 9780357704745, 9781337342407, 9781337028769

## Contents

Preface

Acknowledgments

Notation

Chapter 1: Introduction

Chapter Objectives

Prologue

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

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

Reference

Problems

Chapter 2: Introduction to the Stiffness (Displacement) Method

Chapter Objectives

Introduction

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

2.6 Potential Energy Approach to Derive Spring Element Equations

Summary Equations

References

Problems

Chapter 3: Development of Truss Equations

Chapter Objectives

Introduction

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-Di

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

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

3.14 Flowchart for Solution of Three-Dimensional Truss Problems

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

Summary Equations

References

Problems

Chapter 4: Development of Beam Equations

Chapter Objectives

Introduction

4.1 Beam Stiffness

4.2 Example of Assemblage of Beam Stiffness Matrices

4.3 Examples of Beam Analysis Using the Direct Stiffness Method

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

4.8 Galerkin’s Method for Deriving Beam Element Equations

Summary Equations

References

Problems

Chapter 5: Frame and Grid Equations

Chapter Objectives

Introduction

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

5.6 Concept of Substructure Analysis

Summary Equations

References

Problems

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

Chapter Objectives

Introduction

6.1 Basic Concepts of Plane Stress and Plane Strain

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

6.3 Treatment of Body and 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)

Summary Equations

References

Problems

Chapter 7: Practical Considerationsin Modeling; Interpreting Results; and Examples of Plane Stress/S

Chapter Objectives

Introduction

7.1 Finite Element Modeling

7.2 Equilibrium and Compatibility of Finite Element Results

7.3 Convergence of Solution and Mesh Refinement

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/Stra

References

Problems

Chapter 8: Development of the Linear-Strain Triangle Equations

Chapter Objectives

Introduction

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

References

Problems

Chapter 9: Axisymmetric Elements

Chapter Objectives

Introduction

9.1 Derivation of the Stiffness Matrix

9.2 Solution of an Axisymmetric Pressure Vessel

9.3 Applications of Axisymmetric Elements

Summary Equations

References

Problems

Chapter 10: Isoparametric Formulation

Chapter Objectives

Introduction

10.1 Isoparametric Formulation of the Bar Element Stiffness Matrix

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

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

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

Summary Equations

References

Problems

Chapter 11: Three-Dimensional Stress Analysis

Chapter Objectives

Introduction

11.1 Three-Dimensional Stress and Strain

11.2 Tetrahedral Element

11.3 Isoparametric Formulation and Hexahedral Element

Summary Equations

References

Problems

Chapter 12: Plate Bending Element

Chapter Objectives

Introduction

12.1 Basic Concepts of Plate Bending

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

References

Problems

Chapter 13: Heat Transfer and Mass Transport

Chapter Objectives

Introduction

13.1 Derivation of the Basic Differential Equation

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

References

Problems

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

Chapter Objectives

Introduction

14.1 Derivation of the Basic Differential Equations

14.2 One-Dimensional Finite Element Formulation

14.3 Two-Dimensional Finite Element Formulation

14.4 Flowchart and Example of a Fluid-Flow Program

14.5 Electrical Networks

14.6 Electrostatics

Summary Equations

References

Problems

Chapter 15: Thermal Stress

Chapter Objectives

Introduction

15.1 Formulation of the Thermal Stress Problem and Examples

Summary Equations

Reference

Problems

Chapter 16: Structural Dynamics and Time-Dependent Heat Transfer

Chapter Objectives

Introduction

16.1 Dynamics of a Spring-Mass System

16.2 Direct Derivation of the Bar Element Equations

16.3 Numerical Integration in Time

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

16.8 Time-Dependent Heat Transfer

16.9 Computer Program Example Solutions for Structural Dynamics

Summary Equations

References

Problems

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