Finite ElementAnalysis for

Design Engineers

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Finite ElementAnalysis for

Design Engineers

Paul M. Kurowski

Warrendale, Pa.

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Library of Congress Cataloging-in-Publication Data

Kurowski, Paul M.Finite element analysis for design engineers / Paul M. Kurowski.

p. cm.ISBN 0-7680-1140-X1. Finite element method. 2. Engineering design—Data processing.

I. Title.

TA347.F5K83 2004620'.0042—dc22

2004054717

SAE400 Commonwealth DriveWarrendale, PA 15096-0001 USAE-mail: CustomerService@sae.orgTel: 877-606-7323 (inside USA and Canada)

724-776-4970 (outside USA)Fax: 724-776-1615

Copyright © 2004 SAE International

ISBN 0-7680-1140-X

SAE Order No. R-349

Printed in the United States of America.

Disclaimer: Every effort has been made to provide an accurate text. The author and the manu-facturers shall not be held liable for any parts developed with this book or be held responsible forany inaccuracies or errors that may appear in the book.

Acknowledgments

I would like to thank Professor Barna Szabo and his team at EngineeringSoftware Research & Development, Inc. (ESRD) and Vlad Zila at GenexisDesign Inc. for their help and support. I also thank my aunt, Wanda Siwek,who inspired me to write this book, and my wife Elzbieta, for her encourage-ment that made this book possible.

Paul M. Kurowski

Table of Contents

Preface xi

Chapter 1 Introduction 11.1 What Is Finite Element Analysis? 11.2 What Is "FEA for Design Engineers?" 21.3 Note on Hands-On Exercises and Illustrations in This Book 3

Chapter 2 From CAD Model to Finite Element Analysis Results 52.1 Formulation of the Mathematical Model 52.2 Selecting the Numerical Method to Solve the

Mathematical Model 72.2.1 Numerical Methods in Engineering Analysis 72.2.2 Reasons for the Dominance of the Finite Element

Method 82.3 The Finite Element Model 9

2.3.1 Meshing 92.3.2 Formulation of Finite Element Equations 102.3.3 Errors in FEA Results 11

Chapter 3 Major Concepts of the Finite Element Model 133.1 Formulation of a Finite Element 13

3.1.1 Shape Functions, Degrees of Freedom, andElement Order 13

3.1.2 Requirements to Be Satisfied by Shape Functions 153.1.3 Artificial Constraints 153.1.4 The Choices of Discretization 17

3.2 Types of Finite Elements 183.2.1 Element Dimensionality 193.2.2 Analysis Dimensionality 203.2.3 Element Shape 213.2.4 Element Order and Element Type 213.2.5 Element Modeling Capabilities 22

Chapter 4 Controlling Discretization Errors 254.1 Convergence Process 25

4.1.1 h-Convergence Process 254.1.2 Convergence Error 294.1.3 Solution Error 294.1.4 p-Convergence Process 304.1.5 Variations in Convergence Processes 32

viii Finite Element Analysis for Design Engineers

4.2 Problems with Convergence 344.2.1 Stress Singularity 354.2.2 Displacement Singularity 38

4.3 Hands-On Exercises 424.3.1 Tensile Strip with a Circular Hole 424.3.2 L-Shaped Bracket 434.3.3 Spot-Welded Cantilever 45

Chapter 5 Finite Element Mesh 475.1 Meshing Techniques 47

5.1.1 Manual Meshing 475.1.2 Semi-Automatic Meshing 485.1.3 Automatic Meshing 48

5.2 Mesh Compatibility 505.2.1 Compatible Elements 505.2.2 Incompatible Elements 515.2.3 Forced Compatibility 52

5.3 Common Meshing Problems 525.3.1 Element Distortion 545.3.2 Not Enough Elements to Represent Model Stiffness 555.3.3 Incorrect Mapping to Geometry 565.3.4 Incorrect Conversion to Shell Model 57

5.4 Hands-On Exercises 585.4.1 Hollow Cantilever Bracket 585.4.2 Beam in Bending 59

Chapter 6 Modeling Process 616.1 Modeling Steps 61

6.1.1 Definition of the Objective of Analysis 626.1.2 Selection of the Units of Measurement 626.1.3 Geometry Creation 636.1.4 Defining Material Properties 646.1.5 Defining Boundary Conditions 64

6.2 Some Useful Modeling Techniques 656.2.1 Taking Advantage of Symmetry and Anti-Symmetry 656.2.2 Axial Symmetry 686.2.3 Realignment of Degrees of Freedom 69

6.3 Hands-On Exercises 716.3.1 Cantilever Bracket with Symmetry Boundary

Conditions 716.3.2 Cantilever Bracket with Anti-Symmetry

Boundary Conditions 716.3.3 Shaft in Torsion 726.3.4 Plate in Bending 73

Table of Contents ix

6.3.5 Ring in Bending 746.3.6 Pin-Supported Link 75

Chapter 7 Types of Finite Element Analysis 777.1 Thermal Analysis 77

7.1.1 Heat Flow Induced by Prescribed Temperatures 777.1.2 Heat Flow Induced by Heat Load and Convection 787.1.3 Modeling Considerations in Thermal Analysis 79

7.2 Nonlinear Analysis 807.2.1 Nonlinear Material 817.2.2 Nonlinear Geometry 847.2.3 Contact Stress Analysis 89

7.3 Modal Analysis 927.3.1 Differences Between Modal and Static Analyses 937.3.2 Interpretation of Displacement and Stress Results

in Modal Analysis 947.3.3 Modal Analysis with Rigid Body Modes 947.3.4 Importance of Supports in Modal Analysis 947.3.5 Applications of Modal Analysis 957.3.6 Pre-Stress Modal Analysis 977.3.7 Symmetry and Anti-Symmetry Boundary Conditions

in Modal Analysis 977.3.8 Convergence of Modal Frequencies Results 997.3.9 Meshing Considerations for Modal Analysis 99

7.4 Buckling Analysis 997.4.1 Buckling Load Factor 1007.4.2 Interpretation of Results of Buckling Analysis 100

7.5 Dynamic Analysis 1017.5.1 Modal Superposition Method 1017.5.2 Time Response Analysis 1027.5.3 Frequency Response Analysis 104

7.6 Hands-On Exercises 1067.6.1 Crossing Pipes 1067.6.2 Radiator 1077.6.3 Cantilever Bracket, Elasto-Plastic Material 1087.6.4 Slender Cantilever Beam in Bending 1097.6.5 Round Membrane Under Pressure 1107.6.6 Tuning Fork 1117.6.7 U-Shaped Support Bracket 1127.6.8 Unsupported and Hinge-Supported Beam 1137.6.9 Hollow Cantilever Bracket, Modes Separation 1147.6.10 Beam in Compression 115

x Finite Element Analysis for Design Engineers

Chapter 8 Design Optimization 1178.1 Topological Optimization 1178.2 Structural Optimization 117

8.2.1 Factors Defining an Optimization Process 1198.2.2 Sensitivity Studies 119

Chapter 9 Using Finite Element Analysis in the Design Process 1239.1 Differences Between CAD and FEA Geometries 123

9.1.1 Defeaturing 1249.1.2 Idealization 1259.1.3 Cleanup 1269.1.4 Common Meshing Problems 1289.1.5 Mesh Inadequacy 130

9.2 Integration of CAD and FEA Software 1319.2.1 Standalone FEA Software 1319.2.2 Integrated FEA Software 131

9.3 FEA Implementation 1329.3.1 Positioning of CAD and FEA Activities 1329.3.2 Personnel Training 1339.3.3 Software Selection Criteria 1369.3.4 Building Confidence in the FEA 1399.3.5 Expected Return on Investment 139

9.4 FEA Projects 1409.4.1 Major Steps in an FEA Project 1419.4.2 FEA Report 1449.4.3 Importance of Documentation and Backups 1469.4.4 Contracting Out FEA Services 1479.4.5 Common Errors in FEA Management 147

Chapter 10 Some Misconceptions and Frequently AskedQuestions 151

10.1 FEA Quiz 15110.2 Frequently Asked Questions About FEA 155

Chapter 11 Finite Element Analysis Resources 169

Chapter 12 Glossary of Terms 171

Index 177

About the Author 187

Preface

During the 50 years of its development, finite element analysis (FEA) evolvedfrom an exotic analysis method accessible only to specialized analysts into amainstream engineering tool. Phenomenal progress in computer hardwareand operating systems, combined with similar progress in computer-aideddesign (CAD), made FEA the tool of choice for design engineers, who nowuse it during the product design process.

Many books have been written about FEA. At one end of the spectrum, wefind books going deep into theory; at the other end of the spectrum, softwaremanuals teach how to use FEA software. There is very little FEA literaturethat takes a "middle ground" approach and specifically addresses the needs ofdesign engineers who use FEA as an everyday design tool.

This book attempts to fill that void by focusing on an understanding of FEAfundamentals, which are explained by simple and intuitive examples that canbe grasped by any engineer. This book also takes a practical approach, char-acteristic to the attitudes of design engineers, and offers readers an opportu-nity to try all discussed topics by solving downloadable exercises using eithertheir own FEA software or the commercial FEA software StressCheck®, whichis licensed for free to readers.