Table Of Content/
Applied
Finite Element Analysis
for Engineers
FRANK L.STASA
Florida InstituteofTechnology
CBS Publishing -Japan Ltd.
NewYork Chicago San Francisco Philadelphia
Montreal Toronto London Sydney Tokyo
Mexico City Rio de Janeiro Madrid
-J/--t
'JCf 7l
'~~5:r2
To Frank W. andLena C. Stasa,
Ann R. Holt, r9~5 .
and to thememory of
John A. Holt
HRW
Series in
Mechanical
Engineering
L. S. Fletcher, Series Editor
F. L.Stasa APPLIED FINITE ELEMENT ANALYSIS FOR ENGINEERS
B.J. Torby ADVANCED DYNAMICS FOR ENGINEERS
Copyright © 1985CBS Publishing
All rights reserved
Address correspondenceto
383 Madison Avenue, New York. NY 10017
LibraryofCongress Cataloging inPublicationData
Stasa, Frank L. .
Applied finiteeledJent analysis forengineers.
(HRW series inmechanicalengineering)
Includes bibliographiesand index.
I. Finite elementmethod. I. Title. II. Series.
TA347.F5Sn 1985 . 620'.001'515353 85-742
ISBN 0'{)3'-%2737.{) (US';~llege Edition)
ISBN 0'{)3·910744·2(CBS International Edition)
This International Edition is not for sale in the United
States of America. its dependencies or Canada.
Printed in Japan. 1986
5678 038 987654321
CBS COLLEGEPUBLISHING
The Dryden Press
SaundersCollegePublishing
Preface
Writtenforsenior-yearundergraduatesand first-yeargraduatestudentswith solid back
grounds in differential and integral calculus, this book is oriented toward engineers and
applied mathematicians. A course in linearalgebra is helpful but not essential. Courses in
elasticity, heat transfer, and fluid mechanicsshouldfacilitate the student'sunderstandingof
the applications emphasized in this text. Overall, the author's approach represents a com
promise betweenthe purely mathematical and the purely applieddevelopments.
Stress or structural analysis isgiven about the same level of treatmentas thermal and
fluid flow analysis. However, this book is structured so that it may be used in courses in
which the applicationarea is strictly stress analysis or strictly thermal analysis. Moreover,
itisalso possibletouse this book withoutcoveringanyofthe materialrelatedtovariational
calculus and variational formulations. This strategy is made possible by emphasizing the
Galerkin weighted-residual method inthermal (and fluid flow) analysis and the principleof
virtualdisplacementsinstress analysis. Consequently,this bookshouldbeuseful toinstruc
tors teaching acourseon the finite elementmethodtoseniorundergraduatestudents. Ifthe
nonvariationalpath isdesired, the instructorshouldomit the followingsections:the last part
of Sec. 4-2, Sec. 4-3, parts of Sees, 4-4, and Sees. 4-8, 5-3, and 8-6.
The first two chapters are introductory in nature. Chapter 1containsa briefsurveyof
what thefiniteelementmethodis,aswellasabriefhistory ofthemethod.Chapter2contains
areviewofthenecessarymathematicalconceptsofmatrices, vectors,anddeterminants.The
reason for including this review in the text properwas toestablishacommongroundfrom
which the finite elementmethodcould be developed, regardlessof the specific background
ofthe engineering student.
Allaspectsofthefinite elementareexploredinChapter3bywayofoneofthesimplest
of allengineeringapplications-thetruss. The so-calleddirect approach isadoptedfor this
purpose and each step in the finite element solution process isgivenin full detail. For this
reason, all studentsmust be exposedto (and indeedshould master)Chapter3, which isthe
only structural analysis chapterthat must be covered by all users of this book. Both two
and three-dimensional trusses are covered, but the former is given broader, more compre
hensive treatment. The studentisalso introducedtocomputerprogrammingconceptsand a
two-dimensional truss program.
Chapter4 provides theg>neral framework.for the developmentof nearly all (nonstruc
tural)finite elementmodels. Here thestudentisintroducedtotheconceptsofgloballybased
approximations to the true solution of simple ordinary differential equations. Among the
methods covered are the Ritz, Rayleigh-Ritz (variational), point collocation, subdomain
collocation, least squares, and Galerkin methods. The last four of these methodscomprise
the class of approximate solution methods known as the weighted-residual methods. Both
the Rayleigh-Ritz (variational) and the Galerkin (weighted-residual) methods are extended
topiecewisecontinuousapproximationsand, hence, tothe finite elementmethoditself.The
student is introduced, of course, to the concept of shape functions at this point. Chapter4
concludes with an application in the thermal analysis area: a pin fin (a type of extended
iii
iv PREFACE
surface). Instructors may wish to have their students review Chapter I before proceeding
withthenextchapter.
Chapter 5 isdevoted specifically to the development of finite element models in the
stress (or structural) analysis area. Abriefreview ofsomeof themore importantconcepts
ofelasticity isprovidedandmaybeskippedbythosewhohavehadapriorgraduatecourse
on this subject. Depending on the instructor's preference, eitherof two developments may
be used: the principle of minimum potential energy (variational)or theprinciple of virtual
displacement with a simple one-dimensional application: the uniaxial stress member. The
resultsfromChapter 5areusedthroughoutChapter 7,theprimarystressapplicationchapter.
Chapter 6 is, ineffect, a catchall chapter, which contains essential material that has
notbeencovered adequately uptothispoint. Herethestudent isintroduced moreformally
totheconcept ofparameter functions (suchasdisplacement andtemperature functions)and
tothecompatibilityandcompletenessrequirementsthatthesefunctionsshouldsatisfy.Shape
functions (CD-continuous only) are derived and presented for the following types of one-,
two-, and three-dimensional elements: two-node lineal (I-D), three-node triangular (2-D),
four-noderectangular (2-D), four-nodetetrahedral(3-D), andeight-nodebrick(3-0). Local,
normalizedcoordinates, suchaslength, area, andvolumecoordinates, aswellasserendipity
coordinates, are introduced. Axisymmetric elements arealsopresented. Three simple inte
gration formulasaregiven intermsoflength, area, andvolumecoordinatesforintegrations
over lineal, triangular, and tetrahedral elements, respectively. Finally, analternative tothe
matrixinversiontechniqueisprovided, namely,theactivezoneequationsolver.Thismethod
is basedon triangular decomposition, forward elimination, and backward substitution, and
takes advantage of the banded, and often symmetric, nature of the assemblage stiffness
matrix. This method requires that the assemblage stiffness matrix be stored as a column
vector (instead of asquare matrix).
Chapter 7isthemainstressanalysisapplicationchapter. Amongthetopicscoveredare
thefollowing: two-dimensional stressanalysis(planestressandplanestrain), axisymmetric
stress analysis, three-dimensional stressanalysis, andtheanalysisofbeams. The notionof
substructuring and condensation is also introduced and working equations are developed.
The chapter concludes with a brief description of the development of a two-dimensional
stressanalysisprogram. The instructor isreferredtotheInstructor'sSolutionsmanualfora
listing(inFORTRAN)ofoneversionofthisprogram-calledprogramSTRESS(theuser's
manualtotheprogram isalso included intheInstructor's Solutionsmanual).
Chapter 8istheprincipalthermalandfluidflowanalysischapter, inwhichthefollowing
topics are covered: one-, two-, and three-dimensional thermal analysis, and axisymmetric
thermal analysis. Material on variational formulations intwo-dimensional problems ispro
vided, but this also may be skipped by those preferring the nonvariational path. Other
applicationareasinChapter 8includeconvectiveenergytransport,two-dimensionalpotential
flow, and two-dimensional incompressible fluid flow. The chapter concludes with a brief
descriptionofthedevelopmentofatwo-dimensional, steady-statethermalanalysisprogram.
Again the instructor is referred tothe Instructor's Solutions manual for a listing (in FOR
TRAN) ofone versionofthisprogram-calledHEAT(theuser's manualtotheprogramis
also included inthe Instructor's Solutions manual).
Chapter 9introduceshigher-orderelementsandnumericalintegration(quadrature). The
one-, two-,andthree-dimensionalelementsintroducedinChapter6areextendedtoquadratic
and cubic order. Subparametric, isoparametric, and superparametric elements are also in
troduced. Two- and three-dimensional isoparametric formulations are developed for the
quadrilateral andtriangularelementsaswellasforthebrickandtetrahedralelements. Special
formulas are presented, whichfacilitate greatlytheevaluationoftheintegralsthatnaturally
arise.
PREFACE V
Inconclusion, Chapter lOisdevotedtotransientthermalanalysisanddynamicstructural
analysis. Theconcept ofpartialdiscretization ispresented andappliedtostressanalysisand
thermal analysis. Lumped andconsistent capacitance andmass matrices arediscussed. So
lution methods are developed based on the finite element method itself (i.e., in time) and
on the finite difference method. The result istwo- and three-point recurrence schemes for
transient thermal analysis and dynamic structural analysis, respectively. The chapter con
cludes withabrief introduction tomodalanalysis.
Appendix A contains the material property data to be used in the problems, unless
otherwise noted in the problem statements. Appendix Bcontains a short user's manual to
the(two-dimensional)trussprogramalongwiththeprogramlisting(inFORTRAN).AppendixC
contains listings of subroutines ACTCOL and UACTCL (and function DOT), which have
been used with the written permission of McGraw-Hill and which appear in Professor
Zienkiewicz's thirdedition of The Finite Element Method. The length of the STRESS and
HEAT programs and their respective user's manuals precluded their inclusion in this text.
Instructors who would like to have copies of all of these programs on floppy disks are
encouraged towritetotheauthor. Thedisk formats available areIBM PC, AppleIIseries,
and 8-inch IBM3740 standard format for CP/M-based machines. *Other FORTRAN pro
grams that can beobtained on these disk formats include: beam analysis, aTurboPascal**
versionofthetwo-dimensionaltrussprogram, finanalysis, transientone-dimensional thermal
analysis, andtransient two-dimensional thermal analysis.
Thefollowing suggestions aremadetoinstructors teachingonthequarterandsemester
system.Forthoseteachingonthequartersystem:Chapters Ito4canbecoveredcomfortably
during thefallquarter. Thewinterquartercouldbedevotedtostressanalysiswithcoverage
ofChapters 5, 6, 7, 9, and 10(omitSees, 10-4, 10-7, and 10-8).The springquarter could
be devoted to thermal and fluid flow analysis with coverage of Chapters 6, 8, 9, and 10
(omit Sees. 10-3, 10-9, and 10-10). For students who take all three courses, some of the
material is necessarily repeated. The author has found this repetition not to bea problem,
because these students get afirmer grasp ofthe material thesecond timearound.
Forthoseteachingonthesemestersystem:Chapters Ito6couldbecoveredcomfortably
duringthefirstsemester, andChapters 7to10duringthesecondsemester. Thispaceallows
sufficientclasstimeforthediscussionofcomputerprogrammingtechniques. Ausefulproject
for the first course is to have the students modify the two-dimensional truss program in
Appendix B sothat itcould be usedinthree-dimensional truss applications.
The Instructor's Solutions manual contains a listing (in FORTRAN) of a two-dimen
sional, static stress analysis program (for Chapter 8). The manual is available from the
publisher uponproper writtenrequest. Theinstructors mayfinditusefultodistribute copies
ofthese programs totheirclasses.
The author wishes to thank all of the students who have used the original notes and
class-tested thetext manuscript. Theircomments andsuggestions weretaken seriously, and
theirwordsofencouragement willalways beremembered. Deservingofspecialrecognition
isJay A. Buckman, whodidanoutstanding job ofproofreading thepage proofs. Aspecial
debt of gratitude isowed toGeneral Herbert McChrystal, who first suggested this project.
Veryspecial thanks areextended toU. Shripathi Kamath, whohadthemonumental taskof
providing thesolutions manualtothetext, andtoMichael Weaverforconverting theFOR
TRAN version of the truss program into Pascal. The author isalso thankful to the editors
*CP/M isregistered trademark ofDigitalResearch, Inc.
**TurboPascal isatrademark of BorlandInternational.
vi PREFACE
JohnJ.Beck,LynnContrucci,andRachelHockettfortheirhelpandcooperationinpublishing
themanuscript. Finally, and mostimportant ofall, theauthorisonceagaindeeply indebted
tohiswife, Donna, whometiculously typedeverypageofthemanuscriptandwithoutwhose
encouragement this book would probably not have been completed, and to Lisa Ann, for
helping to put this project into proper perspective and for providing the perpetual light at
theend of tunnel.
Frank L. Stasa
CONTENTS
Preface iii
1 General Concepts 1
1·1 Introduction 1
1·2 What Isthe Finite Element Method? 3
1-3 Discretization 5
1-4 Relationship to the Finite-Difference Method 8
1-5 Advantages and Disadvantages of the Finite
Element Method 10
1-6 Brief History ofthe Finite Element Method 13
1-7 Remarks 15
References 16
2 Mathematical Preliminaries 19
2-1 Introduction 19
2-2 Definition of a Matrix 19
2-3 Equality of Two Matrices 21
2-4 Matrix Algebra 21
Matrix Addition and Subtraction 22
Matrix Multiplication 22
The Commutative, Distributive, and Associative
Properties 24
2-5 Transpose of a Matrix 24
2-6 Determinants, Minors, and Cofactors 25
vii
viii TABLE OFCONTENTS
2-7 Adjoint and Inverse of a Matrix 27
2-8 Matrix Partitioning 28
2-9 Definition of a Vector 29
2-10 Equality of Two Vectors 31
2-11 Length of aVector 31
2-12 Vector Algebra 32
2-13 Direction Cosines 33
2-14 Solution to Systems of Linear Algebraic
Equations 35
References 36
Problems 37
3 Truss Analysis: The Direct Approach 43
3-1 Introduction 43
3-2 Finite Element Formulation: The Direct
Approach 45
Discretization 45
ElementStiffnessRelationshipinLocalCoordinates 47
TransformationfromLocaltoGlobalCoordinates 49
Global Element Stiffness Relationship 51
Assemblage 55
Application of Loads 58
Application of Restraints on Nodal Displacements and
Solution 59
Computation of the Element Resultants 61
3-3 Application to a Specific Example 62
3-4 Summary of the Basic Steps 70
Step I: Discretization 70
Step 2: Determination of the Local Element
Characteristics 71
Step 3: Transformation of the Element
Characteristics 71
Step 4: Assemblage of the Global Element
Characteristics 72
Step5:ApplicationofthePrescribedDisplacements 72
Step 6: Solution 72
Step 7: Calculation of Element Resultants 72
3-5 Three-Dimensional Truss Formulation 73
3-6 Description of a Simple Computer Program:
TRUSS 79
Description of the Program 79
Mesh Generation 83
3-7 Remarks 87
References 90
Problems 90
TABLEOFCONTENTS ix
4 Variational and Weighted Residual
Formulations 101
4-1 Introduction 101
4-2 Some Approximate Solution Methods 102
General Concepts 103
The Ritz Method 104
The Variational Method 107
4-3 Variational Calculus: An Introduction 110
4-4 Some Additional Mathematics 112
Integration by Parts 113
Taylor's Series 113
The Differential of a Function 113
The Variation of a Functional 114
The Commutative Properties 114
Miscellaneous Rules of Variational Calculus 114
4-5 The Euler-Lagrange Equation, Geometric and
Natural Boundary Conditions 115
4-6 The Method of Weighted Residuals 121
General Concepts 121
Point Collocation 122
Subdomain Collocation 124
Least Squares 126
Galerkin 128
Comparison with the Exact Solution 129
4-7 More Mathematics 130
Derivative of a Matrix with Respect to a Scalar 131
Integral of a Matrix with Respect to a Scalar 131
Derivative of a Scalar Function with Respect to a
Vector 131
The Integral as a Sum of Other Integrals 133
4-8 The Rayleigh-Ritz Finite Element Method 133
4-9 The Galerkin Finite Element Method 142
4·10 Application: One-Dimensional Heat Transfer in a
Pin Fin 146
The Element Characteristics 147
Heat Removal Rate 149
Fin Efficiency 150
4-11 Remarks 155
References 157
Problems 157
5 General Approach to Structural Analysis 185
5-1 Introduction 185
5-2 Basic Concepts in Elasticity 186
