A First Course in the Finite Element Method

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ISBN: 9780534552985 | 0534552986
Cover: Hardcover
Copyright: 7/25/2006

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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.

Introduction

(27)

Prologue

(1)

Brief History

(2)

Introduction to Matrix Notation

(2)

Role of the Computer

(1)

General Steps of the Finite Element Method

(8)

Applications of the Finite Element Method

(4)

Advantages of the Finite Element Method

(4)

Computer Programs for the Finite Element Method

(5)

References

(3)

Problems

(1)

Introduction to the Stiffness (Displacement) Method

(37)

Introduction

(1)

Definition of the Stiffness Matrix

(1)

Derivation of the Stiffness Matrix for a Spring Element

(5)

Example of a Spring Assemblage

(3)

Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method)

(2)

Boundary Conditions

(13)

Potential Energy Approach to Derive Spring Element Equations

(13)

References

(1)

Problems

(4)

Development of Truss Equations

(86)

Introduction

(1)

Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates

(6)

Selecting Approximation Functions for Displacements

(3)

Transformation of Vectors in Two Dimensions

(3)

Global Stiffness Matrix

(4)

Computation of Stress for a Bar in the x-y Plane

(2)

Solution of a Plane Truss

(8)

Transformation, Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space

(8)

Use of Symmetry in Structure

100

(3)

Inclined, or Skewed, Supports

103

(6)

Potential Energy Approach to Derive Bar Element Equations

109

(11)

Comparison of Finite Element Solution to Exact Solution for Bar

120

(4)

Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations

124

(3)

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

127

(24)

References

132

(1)

Problems

132

(19)

Development of Beam Equations

151

(63)

Introduction

151

(1)

Beam Stiffness

152

(9)

Example of Assemblage of Beam Stiffness Matrices

161

(2)

Examples of Beam Analysis Using the Direct Stiffness Method

163

(12)

Distributed Loading

175

(13)

Comparison of the Finite Element Solution to the Exact Solution for a Beam

188

(6)

Beam Element with Nodal Hinge

194

(5)

Potential Energy Approach to Derive Beam Element Equations

199

(2)

Galerkin's Method for Deriving Beam Element Equations

201

(13)

References

203

(1)

Problems

204

(10)

Frame and Grid Equations

214

(90)

Introduction

214

(1)

Two-Dimensional Arbitrarily Oriented Beam Element

214

(4)

Rigid Plane Frame Examples

218

(19)

Inclined or Skewed Supports---Frame Element

237

(1)

Grid Equations

238

(17)

Beam Element Arbitrarily Oriented in Space

255

(14)

Concept of Substructure Analysis

269

(35)

References

275

(1)

Problems

275

(29)

Development of the Plane Stress and Plane Strain Stiffness Equations

304

(46)

Introduction

304

(1)

Basic Concepts of Plane Stress and Plane Strain

305

(5)

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

310

(14)

Treatment of Body and Surfaces Forces

324

(5)

Explicit Expression for the Constant-Strain Triangle Stiffness Matrix

329

(2)

Finite Element Solution of a Plane Stress Problem

331

(19)

References

342

(1)

Problems

343

(7)

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

350

(48)

Introduction

350

(1)

Finite Element Modeling

350

(13)

Equilibrium and Compatibility of Finite Element Results

363

(4)

Convergence of Solution

367

(1)

Interpretation of Stresses

368

(1)

Static Condensation

369

(5)

Flowchart for the Solution of Plane Stress/Strain Problems

374

(1)

Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress/Strain Problems

374

(24)

References

381

(1)

Problems

382

(16)

Development of the Linear-Strain Triangle Equations

398

(14)

Introduction

398

(1)

Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations

398

(5)

Example LST Stiffness Determination

403

(3)

Comparison of Elements

406

(6)

References

409

(1)

Problems

409

(3)

Axisymmetric Elements

412

(31)

Introduction

412

(1)

Derivation of the Stiffness Matrix

412

(10)

Solution of an Axisymmetric Pressure Vessel

422

(6)

Applications of Axisymmetric Elements

428

(15)

References

433

(1)

Problems

434

(9)

Isoparametric Formulation

443

(47)

Introduction

443

(1)

Isoparametric Formulation of the Bar Element Stiffness Matrix

444

(5)

Rectangular Plane Stress Element

449

(3)

Isoparametric Formulation of the Plane Element Stiffness Matrix

452

(11)

Gaussian and Newton-Cotes Quadrature (Numerical Integration)

463

(6)

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature

469

(6)

Higher-Order Shape Functions

475

(15)

References

484

(1)

Problems

484

(6)

Three-Dimensional Stress Analysis

490

(24)

Introduction

490

(1)

Three-Dimensional Stress and Strain

490

(3)

Tetrahedral Element

493

(8)

Isoparametric Formulation

501

(13)

References

508

(1)

Problems

509

(5)

Plate Bending Element

514

(20)

Introduction

514

(1)

Basic Concepts of Plate Bending

514

(5)

Derivation of a Plate Bending Element Stiffness Matrix and Equations

519

(4)

Some Plate Element Numerical Comparisons

523

(1)

Computer Solution for a Plate Bending Problem

524

(10)

References

528

(1)

Problems

529

(5)

Heat Transfer and Mass Transport

534

(59)

Introduction

534

(1)

Derivation of the Basic Differential Equation

535

(3)

Heat Transfer with Convection

538

(1)

Typical Units; Thermal Conductivities, K; and Heat-Transfer Coefficients, h

539

(1)

One-Dimensional Finite Element Formulation Using a Variational Method

540

(15)

Two-Dimensional Finite Element Formulation

555

(9)

Line or Point Sources

564

(2)

Three-Dimensional Heat Transfer Finite Element Formulation

566

(3)

One-Dimensional Heat Transfer with Mass Transport

569

(1)

Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method

569

(5)

Flowchart and Examples of a Heat-Transfer Program

574

(19)

References

577

(1)

Problems

577

(16)

Fluid Flow

593

(24)

Introduction

593

(1)

Derivation of the Basic Differential Equations

594

(4)

One-Dimensional Finite Element Formulation

598

(8)

Two-Dimensional Finite Element Formulation

606

(5)

Flowchart and Example of a Fluid-Flow Program

611

(6)

References

612

(1)

Problems

613

(4)

Thermal Stress

617

(30)

Introduction

617

(1)

Formulation of the Thermal Stress Problem and Examples

617

(30)

Reference

640

(1)

Problems

641

(6)

Structural Dynamics and Time-Dependent Heat Transfer

647

(61)

Introduction

647

(1)

Dynamics of a Spring-Mass System

647

(2)

Direct Derivation of the Bar Element Equations

649

(4)

Numerical Integration in Time

653

(12)

Natural Frequencies of a One-Dimensional Bar

665

(4)

Time-Dependent One-Dimensional Bar Analysis

669

(5)

Beam Element Mass Matrices and Natural Frequencies

674

(7)

Trusts, Plane Frame, Plane Stress/Strain Axisymmetric, and Solid Element Mass Matrices

681

(5)

Time-Dependent Heat Transfer

686

(7)

Computer Program Example Solutions for Structural Dynamics

693

(15)

References

702

(1)

Problems

702

(6)

Appendix A Matrix Algebra

708

(14)

Introduction

708

(1)

Definition of a Matrix

708

(1)

Matrix Operations

709

(7)

Cofactor or Adjoint Method to Determine the Inverse of a Matrix

716

(2)

Inverse of a Matrix by Row Reduction

718

(4)

References

720

(1)

Problems

720

(2)

Appendix B Methods for Solution of Simultaneous Linear Equations

722

(22)

Introduction

722

(1)

General Form of the Equations

722

(1)

Uniqueness, Nonuniqueness, and Nonexistence of Solution

723

(1)

Methods for Solving Linear Algebraic Equations

724

(11)

Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods

735

(9)

References

741

(1)

Problems

742

(2)

Appendix C Equations from Elasticity Theory

744

(8)

Introduction

744

(1)

Differential Equations of Equilibrium

744

(2)

Strain/Displacement and Compatibility Equations

746

(2)

Stress/Strain Relationships

748

(4)

Reference

751

(1)

Appendix D Equivalent Nodal Forces

752

(3)

Problems

752

(3)

Appendix E Principle of Virtual Work

755

(4)

References

758

(1)

Appendix F Properties of Structural Steel and Aluminum Shapes

759

(14)

Answers to Selected Problems

773

(26)

Index

799

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A First Course in the Finite Element Method

Summary

Table of Contents

Supplemental Materials