A First Course in the Finite Element Method

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ISBN: 9780534385170 | 0534385176
Cover: Hardcover
Copyright: 4/5/2001

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This third edition provides a simple, basic approach to the finite element method that can be understood by readers. It does not have the usual prerequisites (such as structural analysis) required by most available books in this area. The book is written primarily as a basic learning tool for civil and mechanical engineering readers whose main interest is in stress analysis and heat transfer. The book is geared toward those who want to apply the finite element method as a tool to solve practical physical problems.

Introduction

(25)

Prologue

(1)

Brief History

(1)

Introduction to Matrix Notation

(3)

Role of the Computer

(1)

General Steps of the Finite Element Method

(7)

Applications of the Finite Element Method

(5)

Advantages of the Finite Element Method

(1)

Computer Programs for the Finite Element Method

(7)

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

(74)

Introduction

(1)

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

(6)

Selecting Approximation Functions for Displacements

(2)

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

(7)

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

(5)

Use of Symmetry in Structure

(3)

Inclined, or Skewed, Supports

(6)

Potential Energy Approach to Derive Bar Element Equations

101

(11)

Comparison of Finite Element Solution to Exact Solution for Bar

112

(4)

Galerkin's Residual Method and Its Application to a One-Dimensional Bar

116

(21)

References

119

(1)

Problems

120

(17)

Development of Beam Equations

137

(51)

Introduction

137

(1)

Beam Stiffness

138

(5)

Example of Assemblage of Beam Stiffness Matrices

143

(2)

Examples of Beam Analysis Using the Direct Stiffness Method

145

(9)

Distributed Loading

154

(11)

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

165

(6)

Beam Element with Nodal Hinge

171

(5)

Potential Energy Approach to Derive Beam Element Equations

176

(3)

Galerkin's Method for Deriving Beam Element Equations

179

(9)

References

181

(1)

Problems

181

(7)

Frame and Grid Equations

188

(76)

Introduction

188

(1)

Two-Dimensional Arbitrarily Oriented Beam Element

188

(4)

Rigid Plane Frame Examples

192

(19)

Inclined or Skewed Supports-Frame Element

211

(1)

Grid Equations

212

(17)

Beam Element Arbitrarily Oriented in Space

229

(5)

Concept of Substructure Analysis

234

(30)

References

240

(1)

Problems

240

(24)

Development of the Plane Stress and Plane Strain Stiffness Equations

264

(43)

Introduction

264

(1)

Basic Concepts of Plane Stress and Plane Strain

265

(5)

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

270

(14)

Treatment of Body and Surface Forces

284

(5)

Explicit Expression for the Constant-Strain Triangle Stiffness Matrix

289

(2)

Finite Element Solution of a Plane Stress Problem

291

(16)

References

301

(1)

Problems

301

(6)

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

307

(37)

Introduction

307

(1)

Finite Element Modeling

308

(10)

Equilibrium and Compatibility of Finite Element Results

318

(2)

Convergence of Solution

320

(1)

Interpretation of Stresses

321

(2)

Static Condensation

323

(4)

Flowchart for the Solution of Plane Stress/Strain Problems

327

(1)

Computer Program Results for Some Plane Stress/Strain Problems

328

(16)

References

331

(1)

Problems

332

(12)

Development of the Linear-Strain Triangle Equations

344

(14)

Introduction

344

(1)

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

344

(5)

Example LST Stiffness Determination

349

(3)

Comparision of Element

352

(6)

References

354

(1)

Problems

355

(3)

Axisymmetric Elements

358

(28)

Introduction

358

(1)

Derivation of the Stiffness Matrix

358

(10)

Solution of an Axisymmetric Pressure Vessel

368

(8)

Applications of Axisymmetric Elements

376

(10)

References

380

(1)

Problems

381

(5)

Isoparametric Formulation

386

(35)

Introduction

386

(1)

Isoparametric Formulation of the Bar Element Stiffness Matrix

386

(6)

Rectangular Plane Stress Element

392

(3)

Isoparametric Formulation of the Plane Element Stiffness Matrix

395

(9)

Gaussian Quadrature (Numerical Integration)

404

(3)

Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature

407

(6)

Higher-Order Shape Functions

413

(8)

References

417

(1)

Problems

417

(4)

Three-Dimensional Stress Analysis

421

(20)

Introduction

421

(1)

Three-Dimensional Stress and Strain

421

(2)

Tetrahedral Element

423

(7)

Isoparametric Formulation

430

(11)

References

436

(1)

Problems

436

(5)

Plate Bending Element

441

(17)

Introduction

441

(1)

Basic Concepts of Plate Bending

441

(4)

Derivation of a Plate Bending Element Stiffness Matrix and Equations

445

(5)

Some Plate Element Numerical Comparisions

450

(2)

Computer Solution for a Plate Bending Problem

452

(6)

References

454

(1)

Problems

455

(3)

Heat Transfer and Mass Transport

458

(50)

Introduction

458

(1)

Derivation of the Basic Differential Equation

459

(3)

Heat Transfer with Convection

462

(1)

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

463

(1)

One-Dimensional Finite Element Formulation Using a Variational Method

464

(14)

Two-Dimensional Finite Element Formulation

478

(9)

Line or Point Sources

487

(3)

One-Dimensional Heat Transfer with Mass Transport

490

(1)

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

491

(4)

Flowchart and Examples of a Heat-Transfer Program

495

(13)

References

499

(1)

Problems

499

(9)

Fluid Flow

508

(24)

Introduction

508

(1)

Derivation of the Basic Differential Equations

508

(5)

One-Dimensional Finite Element Formulation

513

(8)

Two-Dimensional Finite Element Formulation

521

(5)

Flowchart and Example of a Fluid-Flow Program

526

(6)

References

527

(1)

Problems

528

(4)

Thermal Stress

532

(27)

Introduction

532

(1)

Formulation of the Thermal Stress Problem and Examples

532

(27)

References

553

(1)

Problems

554

(5)

Structural Dynamics and Time-Dependent Heat Transfer

559

(57)

Introduction

559

(1)

Dynamics of a Spring-Mass System

559

(2)

Direct Derivation of the Bar Element Equations

561

(4)

Numerical Integration in Time

565

(12)

Natural Frequencies of a One-Dimensional Bar

577

(4)

Time-Dependent One-Dimensional Bar Analysis

581

(5)

Beam Element Mass Matrices and Natural Frequencies

586

(5)

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

591

(4)

Time-Dependent Heat Transfer

595

(7)

Computer Program Example Solutions for Structural Dynamics

602

(14)

References

609

(1)

Problems

610

(6)

Appendix A Matrix Algebra

616

(14)

Introduction

616

(12)

A.1 Definition of a Matrix

616

(1)

A.2 Matrix Operations

617

(7)

A.3 Cofactor or Adjoint Method to Determine the Inverse of a Matrix

624

(2)

A.4 Inverse of a Matrix by Row Reduction

626

(2)

References

628

(1)

Problems

628

(2)

Appendix B Methods for Solution of Simultaneous Linear Equations

630

(22)

Introduction

630

(19)

B.1 General Form of the Equations

630

(1)

B.2 Uniqueness, Nonuniqueness, and Nonexistence of Solution

631

(1)

B.3 Methods for Solving Linear Algebraic Equations

632

(11)

B.4 Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods

643

(6)

References

649

(1)

Problems

650

(2)

Appendix C Equations from Elasticity Theory

652

(8)

Introduction

652

(7)

C.1 Differential Equations of Equilibrium

652

(2)

C.2 Strain/Displacement and Compatibility Equations

654

(2)

C.3 Stress/Strain Relationships

656

(3)

Reference

659

(1)

Appendix D Equivalent Nodal Forces

660

(3)

Problems

660

(3)

Appendix E Principle of Virtual Work

663

(4)

References

666

(1)

Answers to Selected Problems

667

(22)

Index

689

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A First Course in the Finite Element Method

Summary

Table of Contents

Supplemental Materials