Finite element analysis: theory and application with ANSYS

_ F I N I T Ẹ

E L Ệ M Ệ N T

A N A L Y S Ì S

Theory

and Application

with ANSYS

Fourth Edition

FINITE ELEMENT ANALYSIS

FINITE ELEMENT ANALYSIS

Theory and Application with ANSYS

Fourth Edition

Saeed Moaveni

Minnesota State University, Mankato

PEARSON

B oston C olum bus Indianapolis New York San Francisco U p p e r Saddle R iver

A m sterd am C apeT ow n D ubai L ondon M adrid M ilan M unich Paris M o n treal T oronto

D elhi M exico City São Paulo Sydney H ong Kona Seoul Singapore Taipei Tokvo

Vice President and E ditorial D irector. ECS:

M arcia J. H orton

A cquisitions E ditor: S o rrin Dias

M arketina A ssistant: yort B ryant

Senior M anaaine E ditor: Scott D isanno

Production Proeram M anager: Clare R om eo

Production Project M anaeer: Jennifer Sargunar

D irecto r of O perations: N ick Sklitsis

O p eratio n s Specialist: Linda Sager

C over D esiener: B lack H orse Designs

Copyright © 2015,2008,2003,1999 by Pearson Higher Education. Intu, Upper Saddle River, NJ 07458- All rights

reserved. .\lan u factu red in the U n ited States of A m erica. This publication is p ro tected by Copyright and perm is￾sions should be o b tained ữ o m the publisher p rio r to any pro h ib ited rep ro d u ctio n . sto raae in a retrieval system . or

transm ission in anv form or bỵ any m eans. electronic. m echanical. photocopyíng. recordine. o r likew ise.To o btain

perm ission(s) to use m aterials from this vvork. please subm it a w ritten req u est to Pearson H iah er E d u catio n . Perm is￾sions D ep artm en t. O ne Lake Street. U p p er Saddle R iver. NJ 07458.

M any of the designations by m an u íactu rers and seller to distinauish their Products are claim ed as tradem arks. \\lie r e

those desienations ap p ear in this book. and the publisher was avvare of a trad em ark claim . the d esien atio n s have been

prin ted in initial caps or all caps.

M A TL A B is a reeistered trad em ark of The M ath\V orks. Inc.. 3 A pple Hill Drive. N atick. M A 01760-2098.

The au th o r and publisher of this book have used their best efforts in p rep arin a this b o o k .T h ese efforts include the de￾velopm ent. research. and testin s of theories and proeram s to determ in e th eir effectiveness. T he a u th o r and p u b lish er

m ake no \varrantv of anv kind. expressed o r im plied. with reeard to these pro eram s or the d o cu m en tatio n co n tain ed

in this book. The au th o r and publisher shall not be liable in anv event for incidental o r consequential d am aaes with. or

arisina out o t the íurnishina. pertorm ance. o r use of these proeram s.

Pearson E ducation Ltd.. L o n d o n

Pearson E ducation Sinaapore. Pte. Ltd

Pearson E ducation C anada. Inc.

Pearson E ducation —Japan

Pearson E ducation A ustralia PTY. Lim ited

Pearson E ducation N orth A sia. Ltd.. H ong K ong

Pearson E ducación de M exico. S.A. de C A ’

Pearson E ducation M alaysia. Pte. Ltd.

Pearson E ducation. Inc.. L p p er Saddle River. X e n Jerse\

Librarỵ of Congress Cataloging-in-Publication Data

M oaveni. Saeed.

Finite elem ent a n alv sís: th eo r\ and application with A N SYS Saeed M oaveni. — Fourth edition.

paees cm

Includes biblioaraphical reíeren ces and index.

ISB N -13: 978-0-13-384080-3

ĨSB N -10:0-13-384080-8

1. Finite elem ent m ethod —D ata processina. 2. A N SY S (C o m p u ter system ) I. Title.

TA347.F5M62 2015

620.001 ‘51825 — dc2?

2014002008

10 987654321

C over Photo: Im ages reproduced b\ perm ission o f

A X S Y S . Inc.

M anaaer. R ights and Perm issions: Rachel Youdelm an

Photo Perm ission C oordinator: Rachel Youdelm an

Im age Perm ission C oordinator: Paul Sarkis

Full-Ser\ice Project M anaeem ent: M ahalatchoum y

Saravanan. Jouve India

P rinter/B inder: E d n a rd s Brothers \ía llo \

C over Printer: P hoenix C olor/H agerstonn

PEARSON ISBN -13: 9^8-0-13-5S-iJjSj0-5

ISBN-10:

To memories o f m y m o th er and ỷather

Contents

Preíace xiii

Acknowledgments xvii

1 Introduction 1

1.1 Engineering Problems 2

1.2 Numerical M ethods 5

1.3 A Brief History of the Finite Elem ent M ethod and ANSYS

1.4 Basic Steps in the Finite Elem ent M ethod 6

1.5 Direct Formulation 8

1.6 Minimum Total Potential Energy Formulation 37

1.7 W eighted Residual Formulations 43

1.8 Veriíication of Results 48

1.9 Understanding the Problem 49

Summary 54

Reíerences 54

Problems 54

2 Matrix Algebra 66

2.1 Basic D eíinitions 66

2.2 Matrix Addition or Subtraction 69

2.3 Matrix Multiplication 69

2.4 Partitioning of a Matrix 73

2.5 Transpose of a Matrix 77

2.6 Determ inant of a Matrix 81

2.7 Solutions of Simultaneous Linear Equations 86

2.8 Inverse of a Matrix 94

2.9 Eigenvalues and Eigenvectors 98

2.10 Using MATLAB to Manipulate Matrices 102

2.11 Using Excel to Manipulate Matrices 106

Summary 120

Reíerences 121

Problems 121

3 Trusses 125

3.1 D eíinition of aTruss 125

3.2 Finite Elem ent Formulation 126

3.3 SpaceTrusses 151

viii Contents

3.4 Overview of the ANSYS Program 153

3.5 Examples Using ANSYS 161

3.6 Veriíication of Results 193

Summary 195

Reíerences 195

Problems 195

4 Axial Members, Beams, and Frames 205

4.1 Members Under Axia) Loading 205

4.2 Beams 213

4.3 Finite Element Formulation of Beams 218

4.4 Finite Element Formulation of Frames 234

4.5 Three-Dimensional Beam Elem ent 240

4.6 An Example Using ANSYS 242

4.7 Veriíication of Results 267

Summary 269

Reíerences 270

Problems 271

5 One-Dim ensional Elements 283

5.1 Linear Elements 283

5.2 Quadratic Elements 287

5.3 Cubic Elements 289

5.4 Global, Local, and Natural Coordinates 292

5.5 Isoparametric Elements 294

5.6 Numerical Integration: G auss-Legenđre Quadrature 296

5.7 Examples of One-Dim ensional Elem ents in ANSYS 301

Summary 301

Reíerences 301

Problems 301

6 Analysis of One-Dim ensional Problems 308

6.1 Heat Transíer Problems 308

6.2 A Fluid Mechanics Problem 327

6.3 An Example U sine ANSYS 331

6.4 Veriíication of Results 346

Summary 347

Rcíerences 347

Problems 348

7 Two-Dimensional Elements 351

7.1 Rectangular Elem ents 351

7.2 Quadratic Ouadrilateral Elem ents 355

Contents ix

7.3 Linear Triangular Elements 360

7.4 Quadratic Triangular Elements 365

7.5 Axisymmetric Elements 369

7.6 Isoparametric Elements 374

7.7 Two-Dimensional Integrals: G auss-Legendre Quadrature 377

7.8 Examples of Two-Dimensional Elem ents in ANSYS 378

Summary 379

Reíerences 379

Problems 380

8 More ANSYS 387

8.1 ANSYS Program 387

8.2 ANSYS Database and Files 388

8.3 Creating a Finite Elem ent Model with ANSYS: Preprocessing 390

8.4 /ỉ-M ethod Versus p-M ethod 404

8.5 Applying Boundary Conditions, Loads, and the Solution 404

8.6 Results of Your Finite Elem ent Model: Postprocessing 407

8.7 Selection Options 412

8.8 Graphics Capabilities 413

8.9 Error-Estimation Procedures 415

8.10 An Example Problem 417

Summary 431

Reíerences 432

9 Analysis of Two-Dimensional H eat Transíer Problems 433

9.1 General Conduction Problems 433

9.2 Formulation with Rectangular Elem ents 440

9.3 Formulation with Triangular Elements 451

9.4 Axisymmetric Formulation of Three-Dim ensional Problems 470

9.5 Unsteady HeatTransíer 477

9.6 Conduction Elements Used by ANSYS 487

9.7 Examples Using ANSYS 488

9.8 Veriíication of Results 528

Summary 528

Reíerences 530

Problems 530

10 Analysis o f Two-Dimensional Solid Mechanics Problems 542

10.1 Torsion of Members with Arbitrary Cross-Section Shape 542

10.2 Plane-Stress Formulation 558

10.3 Isoparametric Formulation: Using a Quadrilateral Elem ent 566

10.4 Axisymmetric Formulation 573

10.5 Basic Failure Theories 575

X Contents

10.6 Examples ưsing ANSYS 576

10.7 Veriíication of Results 598

Summary 598

Reíerences 600

Problems 600

11 Dynamic Problems 609

11.1 Review of Dynamics 609

11.2 Revievv of Vibration of Mechanical and Structural Systems 623

11.3 Lagrange’s Equations 640

11.4 Finite Elem ent Formulation of Axial Members 642

11.5 Finite Elem ent Formulation of Beam s and Frames 651

11.6 Examples Using ANSYS 665

Summary 684

R eíerences 684

Problems 684

12 Analysis o f Fluid Mechanics Problems 691

12.1 Direct Formulation of Flow Through Pipes 691

12.2 Ideal FIuid Flow 703

12.3 Groundwater Flow 709

12.4 Examples Using ANSYS 712

12.5 Veriíication of Results 733

Summary 734

Reíerences 735

Problems 736

13 Three-Dim ensional Elements 741

13.1 The Four-Node Tetrahedral Elem ent 741

13.2 Analysis of Three-Dim ensional Solid Problems Using Four-Node

Tetrahedral Elem ents 744

13.3 The Eight-Node Brick Elem ent 749

13.4 The Ten-Node Tetrahedral Elem ent 751

13.5 The Twenty-Node Brick Elem ent 752

13.6 Examples of Three-Dim ensional Elem ents in ANSYS 754

13.7 Basic Solid-M odeling Ideas 758

13.8 A Therm al Example Using ANSYS 769

13.9 A Structural Example Using ANSYS 786

Summary 799

Reíerences 799

Problems 799

Contents

14 Desỉgn and Material Selection 808

14.1 Engineering Design Process 809

14.2 Material Selection 812

14.3 Electrical, Mechanical, and Thermophysical Properties of Materials 813

14.4 Common Solid Engineering Materials 815

14.5 Som e Common Fluid Materials 822

Summary 824

Reíerences 824

Problems 824

15 Design Optimization 826

15.1 Introduction to Design Optimization 826

15.2 The Parametric Design Language of ANSYS 830

15.3 Examples of Batch Files 832

Summary 843

Reíerences 844

Problems 844

Appendix A Mechanỉcal Properties of Some Materials 845

Appendix B Thermophysical Properties o f Some Materials 849

Appendix c Properties of Common Line and Area Shapes 851

Appendix D Geometrical Properties o f Structural Steel Shapes 855

Appendix E Conversion Factors 859

Appendix F An Introduction to M ATLAB 861

Index 895

Preface

CHANGES IN THE FOURTH EDITION

The íourth edition, consisting of 15 chapters, includes a number of new additions and

changes that were incorporated in response to ANSYS revisions and suggestions and

requests made by proíessors, students, and proíessionals using the third edition of the

book.The major changes include:

• Explanation of the changes that were made in the A N SY S’s newest release

(Chapters 3 and 8)

• Explanation of new elem ent type capabilities (Chapters 3 ,4 ,6,8 through

13, and 15)

• A new comprehensive example problem that demonstrates the use of

BEAM 188 elem ent in modeling beam and frame problems (Chapter 4)

• M odiíication of twenty example problems to incorporate new ANSYS elem ent

types (Chapters 3 ,4 ,6,8 through 13, and 15)

• Eight new comprehensive example problems that show in great detail how

to use Excel to solve different types of íinite elem ent problems (Chapters 2

through 6 and 9 through 12)

• More detail on theory and expanded derivations

• Explanation of new MATLAB revisions in Appendix F

ORGANIZATION

There are many good textbooks already in existence that cover the theory of íinite

elem ent m ethods for advanced students. H owever, none of these books incorporate

A NSYS as an integral part of their materials to introduce finite elem ent m odeling

to undergraduate students and newcomers. In recent years, the use of íinite elem ent

analysis (F E A ) as a design tool has grown rapidly. Easy-to-use, com prehensive pack￾ages such as ANSYS, a general-purpose íinite elem ent Computer program, have

becom e com m on tools in the hands of design engineers. Uníortunately, many engi￾neers who lack the proper training or understanding of the underlying concepts have

been using these tools. This introductory book is written to assist engineering students

and practicing engineers new to the field of finite elem ent m odeling to gain a clear

understanding of the basic concepts.The text offers insight into the theoretical aspects

of FE A and also covers some practical aspects of modeling. Great care has been exer￾cised to avoid overwhelming students with theory, yet enough theoretical background

is offered to allow individuals to use ANSYS intelli^ently and effectively. ANSYS is an

xiii

integral part of this text. In each chapter, the relevant basic theory is discussed first and

demonstrated using simple problems with hand calculations. These problems are fol￾lowed by examples that are solved using ANSYS. Exercises in the text are also presented

in this manner. Some exercises require manual calculations, while others. more com plex

in nature. require the use of ANSYS. The simpler hand-calculation problems will en￾hance students' understanding of the concepts by encouraging them to go through the

necessary steps in a FEA. Design problems are also included at the end of Chapters 3,

4 ,6, and 9 through 14.

Various sources of error that can contribute to incorrect results are discussed.

A good engineer must alvvays find vvays to check the results. While experim ental test￾ing of models may be the best way, such testing may be expensive or time consuming.

Thereíore, vvhenever possible, throughout this text emphasis is placed on doing a "sanity

check” to verify one’s FEA. A section at the end of each appropriate chapter is devoted

to possible approaches for veriíying ANSYS results.

Another unique íeature of this book is that the last two chapters are devoted to

the introduction of design, material selection, optimization, and parametric program￾ming with ANSYS.

The book is organized into 15 chapters. Chapter 1 reviews basic ideas in finite

elem ent analysis. Common íormulations, such as direct, potential energy, and vveighted

residual methods, are discussed. Chapter 2 provides a com prehensive review of matrix

algebra. Chapter 3 deals with the analysis of trusses, because trusses offer econom i￾cal solutions to many engineering structural problems. An overview of the A N SY S

program is given in Chapter 3 so that students can begin to use A N SY S right away.

Finite elem ent íormulation of members under axial loading, beams, and íram es are

introduced in Chapter 4. Chapter 5 lays the íoundation for analysis of one-dim ensional

problem s by introducing one-dim ensional linear, quadratic, and cubic elem ents.

Global, local. and natural coordinate systems are also discussed in detail in Chapter 5.

An introduction to isoparametric íorm ulation and numerical integration by G auss￾Legendre íormulae is also presented in Chapter 5. Chapter 6 considers Galerkin for￾mulation of one-dimensional heat transíer and fluid problems. Two-dimensional linear

and higher order elem ents are introduced in Chapter 7. G auss-L egendre íorm ulae

for two-dimensional integrals are also presented in Chapter 7. In Chapter 8 the essen￾tial capabilities and the organization of the AN SY S program are covered. The basic

steps in creating and analyzing a model with ANSYS is discussed in detail. Chapter 9

includes the analysis of tvvo-dimensional heat transíer problems with a section devoted

to unsteady situations. Chapter 10 provides an analysis of torsion of noncircular shaíts

and plane stress problems. Dynamic problems are explored in Chapter 11. Revie\v of

dynamics and vibrations of mechanical and structural system s are also given in this

chapter. In Chapter 12. two-dimensional. ideal fluid-mechanics problems are analyzed.

Direct íormulation of the piping network problems and underground seepaee flow are

also discussed. Chapter 13 provides a discussion on three-dim ensional elem ents and

íormulations.This chapter also presents basic ideas regardina top-down and bottom-up

solid m odelins methods. The last two chapters of the book are devoted to desian and

xiv Preface

Preface XV

optimization ideas. Design process and material selection are explained in Chapter 14.

Design optimization ideas and parametric programming are discussed in Chapter 15.

Examples of ANSYS batch files are also given in Chapter 15. Each chapter begins by

stating the objectives and concludes by summarizing what the reader should have gained

from studying that chapter.

The examples that are solved using ANSYS show in great detail how to use ANSYS

to model and analyze a variety of engineering problems. Chapter 8 is also written such

that it can be taught right away if the instructor sees the need to start with ANSYS.

A brief review of appropriate íundamental principles in solid mechanics. heat trans￾fer, dynamics, and fluid mechanics is also provided throughout the book. Additionally,

when appropriate, students are warned about becoming too quick to generate íinite ele￾ment models for problems for which there exist simple analytical solutions. Mechanical

and thermophysical properties of some common materials used in engineering are given

in Appendices A and B. Appendices c and D give properties of common area shapes

and properties of structural Steel shapes, respectively. A com prehensive introduction to

M ATLAB is given in Appendix F.

Finally, a Web site at http://www.pearsonhighered.com/moaveni will be maintained

for the following purposes: (1) to share any changes in the upcoming versions of ANSYS;

(2) to share additional iníormation on upcoming text revisions; (3) to provide additional

homework problems and design problems; and (4) although I have done my best to

eliminate errors and mistakes, as is with most books, some errors may still exist. I will

post the corrections that are brought to my attention at the site. The Web site will be

accessible to all instructors and students.

Thank you for considering this book and I hope you erýoy the fourth edition.

Saeed Moaveni