Essentials of the finite element method : for mechanical and structural engineers

484 Pages·2015·42.02 MB·English

by  Pavlou

Essentials of the Finite
Element Method
Essentials of the Finite
Element Method
For Mechanical and Structural
Engineers
Dimitrios G. Pavlou, PhD
Department of Mechanicaland StructuralEngineering and MaterialsScience,
University of Stavanger,Norway
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To my children, Evangelia and Georgios Pavlou, and to my wife,
Mina, for their love and patience.
Preface
Inlastdecades,“finiteelements”(FE)hasbecomeastandardcourseinstudyprogramsofmechanical
andcivilengineeringspecialtiesandtendstobeuniquetoengineeringdesignpractice.SinceFEMisa
numerical method, itsevolutionconcurswith theevolutionofdigitalcomputing technology.Today,
contemporary FEM software packages allow fast and accurate design of complex engineering prob-
lems pertaining in the fields of solid mechanics, heat transfer, fluid mechanics, and electrical
engineering.
TheaimofthisbookistoprovideBachelor(BSc)andMaster(MSc)ofSciencestudents,aswellas
professionalmechanicalandcivilengineers,withacompleteandunifiedcoverageoffiniteelementanal-
ysisandtodemonstratehowFEMcanbeprogrammed.Throughoutthetext,readersareshownstep-by-
stepdetailedFEanalysesofintegratedengineeringproblems.Forthispurpose,analyticmathematicsis
used for the development of stiffness matrices for widely used elements in mechanical and structural
engineeringpractice.Tohelpreaderstounderstandhowtheboundaryconditionscanbetakenintoac-
countintheprocedureoftheFEmodeling,aspecialtypeofstructuralmatrixequationincorporatingboth
globalstiffnessmatrixandsubmatricescontainingtheboundaryconditionsinaunifiedformisproposed.
AftercompletingFEMcourses,newengineerswouldbeexpectedtohaveadequateknowledgetouse
acommercialFEMprogramintheirfirstjob.Amongthetargetsofthebookistoassistreaderstoun-
derstandthearchitectureofFEMsoftware,aswellasitslimitations.SincecomputercodingtosolveFEM
problemsistherequiredbackgroundforsubsequentlearninghowtousecommercialpackagesoftware
intelligentlyandcritically,efforthasbeenmadetoteachreadershowtodeveloptheirownprogramsin
ordertounderstandthefundamentalabilitiesofcommercialpackages.Therefore,theaimsofthebook
are:(1)toprovidethetoolstohelpstudentsandengineerstobesoftwaredevelopers,and(2)toprovide
“how-to”knowledgeinrunningacommercialFEMprogram.Toachievethefirstaim,thebookusesa
contemporary computer-aided learning platform, called CALFEM, which adopts the facilities of the
well-knowncomputationalmatrixlaboratoryMATLAB.Forcommontypesofstructures(e.g.,trusses,
beams,frames,hybridstructures,structuraldynamicproblems,etc.),thebookprovidesthelogicalsteps
fordevelopinganFEMcomputeralgorithm.Intheofferedexamples,theanalysisoftherequiredsoftware
commands,thecomputercode,andthenumerical/graphicresultsarealsoexhibited.Toachievethesec-
ondaim,aself-learningon“how-to”runthewidelyusedcommercialFEMprogramANSYSisprovided.
Tothisend,mostoftheexamplestreatedwiththeCALFEM/MATLABplatformareusedforstep-by-
stepANSYSlearning.Therefore,theANSYSwindowsfordataentryareprovidedandtherequiredcom-
mandsaredescribedthroughacombinationoftextandgraphics.
ThisbookintroducesFEmethodologywithsimpleconcepts(e.g.,themethodofdirectequilibrium)
andprogressestomorecomplicatedprinciples(suchasvariationalmethods),allowingasmoothtran-
sitionof the reader todeeperknowledge ofthe method.
DimitriosG. Pavlou, PhD
Professor
Department of Mechanical and Structural Engineering
andMaterials Science, UniversityofStavanger, Norway
xiii
Acknowledgments
I would like to thank my students of the Department of Mechanical and Structural Engineering and
Materials Science for their useful comments during the lectures.
IamgratefultoProfessorIvarLangenandtoAssociateProfessorsHirpaLemuandOveMikkelsen
for their support during the preparation of my FEM courses. Thanks also to Overingeniør Adugna
Akessa for his help to implementthe computeralgorithms tothe FEM lab.
I express my deepest appreciation to the Head of the Department, Professor Per Skjerpe, for the
excellent working conditions allowing me towrite this book.
Sixreviewersprovidedimmenselyusefulcommentstoimprovethecontentofthisbook.Fortheir
time,I am deeply grateful.
ParticularthanksgotoAsiminaKechagiaforherpatienceinreadingthisbookandmakingvaluable
linguistic comments.
Inparticular,IwishtoexpressmyenormousappreciationtoJosephP.Hayton,PublisherofElse-
vier, for givingme the opportunitytodevelopthis book,and toKattie Washington,SeniorEditorial
ProjectManager,andtoChelseaJohnston,formerEditorialProjectManagerofElsevierfortheirvalu-
able help in issues regarding the publishingprocedure.
xv
CHAPTER
1
AN OVERVIEW OF THE FINITE
ELEMENT METHOD
1.1 WHAT ARE FINITE ELEMENTS?
Sincethedifferentialequationsdescribingthedisplacementfieldofastructurearedifficult(orimpos-
sible)tosolvebyanalyticalmethods,thedomainofthestructuralproblemcanbedividedintoalarge
numberof small subdomains,called finiteelements (FE). Thedisplacement field ofeach element is
approximatedbypolynomials,whichareinterpolatedwithrespecttoprescribedpoints(nodes)located
on the boundary (or within) the element. The polynomials are referred to as interpolation functions,
wherevariational or weighted residualmethods are applied todetermine the unknown nodal values.
1.2 WHY FINITE ELEMENT METHOD IS VERY POPULAR?
Theconceptofthefiniteelementmethod(FEM)wasdescribedin1956,whenTurneretal.usedpin-
jointedbarsandtriangularplatestocalculateaircraftstructures.However,asthemethodisbasedonthe
solutionofsystemsofalgebraicequationswithlargenumberofunknowns,inpastfewdecades,FEM
has become very populardue tothe development of high-speeddigitalcomputers.
After1980,newcommercialsoftwarepackagesweredeveloped,boostingtheapplicationofFEM
to structuralengineering,heat transfer, fluid mechanics, aerodynamics, andelectrostatics.
Among the pioneers who founded and developed FEM are Przemieniecki, Zienkiewicz and
Cheung,Gallagher, Argyris,etc.
1.3 MAIN ADVANTAGES OF FINITE ELEMENT METHOD
1. Analyzesproblemswithcomplexgeometry.
2. Analyzesproblemswithcomplexloading(point loads, pressure,inertial forces, thermalloading,
fluid-structure interactions, etc.).
3. Analyzesa wide variety ofengineering problems (structural engineering,heat transfer, fluid
mechanics,aerodynamics, and electrostatics).
1.4 MAIN DISADVANTAGES OF FINITE ELEMENT METHOD
1. FEM results are approximate. Their accuracy dependson the numberofelements, the type of
elements, the adopted assumptions, etc.
1
EssentialsoftheFiniteElementMethod.http://dx.doi.org/10.1016/B978-0-12-802386-0.00001-3
Copyright©2015ElsevierInc.Allrightsreserved.
2 CHAPTER 1 AN OVERVIEW OF THE FINITE ELEMENT METHOD
2. TheaccuracyoftheresultsofFEMdependsontheexperienceofthesoftwareuser,forexample,the
useofthewrongtypeordistortedelements,insufficientsupportstopreventallrigidbodymotions,
anddifferent units for the same quantity yields mistakes.
3. FEMhasinherenterrors(e.g.,thegeometryofthestructureisapproximate,thefielddeformation
isassumed to bea polynomial over the element, the computer carries only a finite number of
digits, the combination ofelements with very large stiffness differences yields numerical
difficulties).
1.5 WHAT IS STRUCTURAL MATRIX?
Structural matrix is a matrix correlating the forces and displacements in the nodal points of the ele-
ments. For a structural FE, the structural matrix contains the geometric and material behavior infor-
mation that indicated the resistance of the element to deformation when subjected to loading. The
primary characteristics of an FE are embodied in the element structural matrix. There are two types
ofstructural matrices:stiffness matrices, andtransfermatrices (Figure 1.1).
TakingintoaccountthenomenclatureofFigure1.2,thestiffnessandthetransfermatrixforasimple
beam are:
1.5.1 STIFFNESS MATRIX
2 3
12EJ 6EJ 12EJ 6EJ
2 3 66 ‘3 (cid:2) ‘2 (cid:2) ‘3 (cid:2) ‘2 772 3
6 7
664MVaa775¼6666 (cid:2)6‘E2J 4E‘J 6‘E2J 2E‘J 7777664wθaa775
MVbb 6664(cid:2)12‘E3J 6‘E2J 12‘E3J 6‘E2J 7775 wθbb
6EJ 2EJ 6EJ 4EJ
(cid:2)
‘2 ‘ ‘2 ‘
Structural
Matrices
Transfer Matrix
Stiffness Matrix
Relates the displacements
Relates the displacements at and forces of a node to the
the nodes to the forces displacements and forces of
another node
FIGURE1.1
Typesofstructuralmatrices.
Va V
M b
a ℓ
a b
w
a
w
b
θ
a
θ
b
FIGURE1.2
Nomenclatureofthenodalforcesanddisplacementsattheendsofabeam.
1.5.2 TRANSFER MATRIX
2 3 2 32 3
1 (cid:2)‘ (cid:2)‘3=6EJ (cid:2)‘2=2EJ
w w
664Vθaba775¼666400 10 ‘2=12EJ ‘=0EJ 7775664Vθaaa775
M M
b 0 0 0 1 a
1.6 WHATARETHESTEPSTOBEFOLLOWEDFORFINITEELEMENTMETHOD
ANALYSIS OF STRUCTURE?
1.
Discretize
the
structure
2.
7. Define the
Calculate element
stresses properties
Steps of
Finite
6. Element 3.
Solve the system Analysis Assemble
of algebraic the stiffness
equations matrices
5.
4.
Define
Apply the
boundary
loads
conditions
4 CHAPTER 1 AN OVERVIEW OF THE FINITE ELEMENT METHOD
1.6.1 STEP 1. DISCRETIZE OR MODEL THE STRUCTURE
The structure is divided into FEs. This step is one of the most crucial in determining the solution
accuracy ofthe problem.
1.6.2 STEP 2. DEFINE THE ELEMENT PROPERTIES
Atthisstep,theusermustdefinetheelementpropertiesandselectthetypesofFEsthatarethemost
suitable tomodelthe physical problem.
1.6.3 STEP 3. ASSEMBLE THE ELEMENT STRUCTURAL MATRICES
Thestructuralmatrixofanelementconsistsofcoefficientsthatcanbederived,forexample,fromequi-
librium. The structural matrix relates the nodal displacements to the applied forces at the nodes.
Assemblingoftheelementstructuralmatricesimpliesapplicationofequilibriumforthewholestructure.
1.6.4 STEP 4. APPLY THE LOADS
At this step, externally applied concentrated or uniform forces, moments, or ground motions are
provided.
1.6.5 STEP 5. DEFINE BOUNDARY CONDITIONS
Atthisstepthesupportconditionsmustbeprovided,thatis,severalnodaldisplacementsmustbesetto
knownvalues.
1.6.6 STEP 6. SOLVE THE SYSTEM OF LINEAR ALGEBRAIC EQUATIONS
The sequential application of the above steps leads to a system of simultaneous algebraic equations
where the nodal displacementsare usuallythe unknowns.
1.6.7 STEP 7. CALCULATE STRESSES
At the users discretion, the commercial programs can also calculate stresses, reactions, mode
shapes,etc.
1.7 WHAT ABOUT THE AVAILABLE SOFTWARE PACKAGES?
Some of the important FEM packages that are available today include Ansys, Abaqus, Nastran, and
Lusas. Their structure isbased on pre-processor, solution process, post-processor.
Pre-processor stage:data preparation takes place, that is, selection ofelements, selectionofma-
terial properties, discretization of the structure, definition of boundary conditions, and definition of
loadings.Withthesedata,thecomputeralgorithmcreatesthestructuralequationsforeveryelement.
Sincethedatainputtakesplaceduringthisstage,theuserinteractswiththesoftwareonlyduringthe
pre-processing step.