Table Of ContentLecture Notes in Applied and Computational Mechanics 100
Adnan Ibrahimbegovic
Rosa-Adela Mejia-Nava
Structural
Engineering
Models and Methods for Statics,
Instability and Inelasticity
Lecture Notes in Applied and Computational
Mechanics
Volume 100
SeriesEditors
PeterWriggers,InstitutfürKontinuumsmechanik,LeibnizUniversitätHannover,
Hannover,Niedersachsen,Germany
PeterEberhard,InstituteofEngineeringandComputationalMechanics,University
ofStuttgart,Stuttgart,Germany
This series aims to report new developments in applied and computational
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·
Adnan Ibrahimbegovic Rosa-Adela Mejia-Nava
Structural Engineering
Models and Methods for Statics, Instability
and Inelasticity
AdnanIbrahimbegovic Rosa-AdelaMejia-Nava
LaboratoryRobervalMecanique LaboratoryRobervalMecanique
UniversitedeTechnologieCompiegne UniversitedeTechnologieCompiegne
Compiegne,France Compiegne,France
ISSN 1613-7736 ISSN 1860-0816 (electronic)
LectureNotesinAppliedandComputationalMechanics
ISBN 978-3-031-23591-7 ISBN 978-3-031-23592-4 (eBook)
https://doi.org/10.1007/978-3-031-23592-4
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Preface
Therootsofthisbookgobacktomyengineeringeducationinex-Yougoslavia,where
during my studies till my master’s degree in 1984, I have learned from the most
active professors in structural mechanics in different scientific centers throughout
thecountry(BranislavVerbicandOgnjenJokanovicfromSarajevo,JosipDvornik
from Zagreb, Peter Fajfar and Miran Saje from Ljubljana and Milos Kojic from
Kragujevac). With the good fortune of winning a Fullbright Grant, I was able to
completemydoctoralstudiesattheUniversityofCaliforniaatBerkeley,from1986
to1989.TheUCBerkeley,ingeneral,andStructuralEngineering,Mechanicsand
MaterialsDivisioninparticular,providedanexcellentstudyandresearchenviron-
ment,withtheopportunitiestoexchangetheideaswithsomeextraordinarytalented
peoplefromallovertheworld.Thehighpointofmyeducationwasduringanother
coupleofyearsasapost-doc,whereIcouldstronglyinteractwithmythesismentors,
BerkeleyprofessorsEdwardL.WilsonandRobertL.Taylor,onverywidevarietyof
topics,contributingtosomeofmymosthighlycitedpapers.Thesamegoodfortune
was my subsequent research appointment from 1991 to 1994 at the Swiss Federal
InstituteofTechnologyinLausanneatStructuralandContinuumMechanicsLabo-
ratory, directed by François Frey, who granted me complete freedom to carry on
with further explorations. Further inspiration for the book contents came from my
subsequent appointments in France, first in 1994 at the Compiègne University of
Technology, where I was interacting with Jean-Louis Batoz and Gouri Dhatt, the
authorsofFrenchbooksonlinearstructuralmechanics,andthenin1999atEcole
Normale Supérieure of Cachan, where I was in contact with Pierre Ladevèze, the
authorofFrenchbookonnonlinearstructuralmechanics.Alltheseinteractionsatfive
differentuniversitiesinfourdifferentcountries,andmanyprivatediscussionswith
colleaguesandfriendslikeJürgenBathe,MikeCrisfieldandTedBelytschko,have
allowed me to enrich my understanding of structural mechanics in many different
aspects, including a great diversity of labels that experts of structural mechanics
attachtodifferentmodels(e.g.,sheardeformablebeamwasTimoshenkobeamfor
onesandHanckybeamforothers).Thediversityoflabelsanddiversityofmodelsare
perhapsthemostchallengingaspectwhenwritingabookonStructuralEngineering,
whichwetriedtohandleinthiswork.
vii
viii Preface
The work on the book started in 2021, when I won my renewal in prestigious
‘InstitutUniversitairedeFrance’(IUF),whichallowedmetokeepreducedteaching
load and have more free time for research. This has coincided with my return to
theCompiègne University of Technology withtheposition of Chair forComputa-
tionalMechanics,withthespecialroleofteachingcoursesinDoctoralSchooland
federatinginterdisciplinaryresearchefforts.Here,thestructuralmodelshaveproved
very helpful in constructing an accessible basis to experts in different domains of
engineering science that were participating in such projects, in order to build best
microscalerepresentation(asreplacementofatomisticmodels)inmulti-scalemodels
forvariousmulti-physicsphenomenawestudied.Thatwasthefirstmotivationfor
writingthisbook.Thesecondmotivationcomesfromtheintenttopreservetheknowl-
edgeaccumulatedbylong-termeffortsofexpertsinstructuralmechanics(withmany
ofthemretired,orgone)andprovideaprioriselectionofreducedmodelbymeansof
correspondingkinematichypothesesandconstraintsthatreducethecostcompared
tosolidorcontinuummechanicsapproach(e.g.,see[175–177]).Suchanapproach
is opposite of the currently active research on constructing such reduced model in
aposteriorifashion,byusingthestatisticalapproachofdataprocessingtoprovide
thereducedmodelintermsofProperOrthogonalDecomposition(POD)orProper
GeneralizedDecomposition(PGD),whichseemstoreducethemodelconstruction
skillstotheapplicationofartificialintelligencealgorithms.Ifirmlybelievethatthe
knowledgeofalternativemethodswithnaturalintelligenceofengineersshouldalso
bebroughttobearuponanycompletelysuccessfulsolutiontocomplexengineering
problems,andthateitherapproachshouldbetriedinseekingsuchoptimalresults.
Infact,wewouldliketoillustratewiththisbookperhapsthebestpossiblemanner
tocombinetheexpertiseinstructuralengineeringwithartificialintelligenceskills.
Namely,formodelsconstructionintermsofthebestchoiceofhypothesesforgiven
goal,oneshouldhavetheexpertiseinstructuralengineering,andformodelsselec-
tion in terms of validation and verification, one should also include the adequate
stochastictoolswithstatisticalmethodsandprobability(e.g.,see[212–215]).
Compiegne,France AdnanIbrahimbegovic
October2022 ProfessorClasseExceptionnelle
Contents
1 Introduction ................................................... 1
1.1 MotivationandObjectives ................................... 1
1.2 MainTopicsOutline ........................................ 5
1.3 FurtherStudiesRecommendations ............................ 11
1.4 SummaryofMainNotations ................................. 12
2 Truss Model: General Theorems and Methods of Force,
DisplacementandFiniteElements ............................... 15
2.1 TrussModel—StrongFormandWeakForm .................... 15
2.1.1 StrongorDifferentialForm:AnalyticSolution ........... 16
2.1.2 WeakorIntegralForm ................................ 20
2.2 GeneralTheoremsofStructuralMechanicsonTrussModel ...... 22
2.2.1 PrincipleofVirtualWork .............................. 22
2.2.2 PrincipleofComplementaryVirtualWork ............... 25
2.2.3 PrincipleofMinimumofTotalPotentialEnergy .......... 27
2.2.4 AppliedGeneralTheorems ............................ 31
2.3 Castigliano’sTheorems,ForceandDisplacementMethods ....... 34
2.3.1 Castigliano’sTheorems—StiffnessandFlexibility ........ 34
2.3.2 ForceandDisplacementMethods ...................... 37
2.4 FiniteElementMethodImplementationforTrussModel ......... 43
2.4.1 LocalorElementaryDescription ....................... 43
2.4.2 ConsistenceofFiniteElementApproximation ............ 48
2.4.3 EquivalentNodalExternalLoadVector ................. 49
2.4.4 HigherOrderFiniteElements .......................... 50
2.4.5 RoleofNumericalIntegration ......................... 52
2.4.6 FiniteElementAssemblyProcedure .................... 56
3 BeamModels:RefinementandReduction ........................ 59
3.1 ReducedModelsofSolidMechanics:PlanarBeamsofEuler,
TimoshenkoandReissner .................................... 59
3.1.1 Euler-BernoulliPlanarBeamModel .................... 59
ix
x Contents
3.1.2 Solid Mechanics Versus Beam Model Accuracy
forPlanarCantileverBeam ............................ 69
3.1.3 TimoshenkoPlanarBeamModel ....................... 72
3.1.4 BriefonReissnerPlanarBeamModel ................... 76
3.2 BeamModelRefinementandReduction ....................... 81
3.2.1 MethodofDirectStiffnessAssemblyfor3DBeam
Elements ........................................... 83
3.2.2 Beam Model Refinement: Flexibility Approach
forReducedModelinDeformationSpace ............... 87
3.2.3 BeamModelReduction:JointReleasesandLength
Invariance ........................................... 91
3.3 CurvedShallowBeamandNon-lockingFEInterpolations ........ 103
3.3.1 Two-DimensionalCurvedShallowBeam:Linear
Kinematics .......................................... 103
3.3.2 Non-lockingFiniteElementInterpolationforShallow
Beam .............................................. 106
3.3.3 IllustrativeNumericalExamplesandClosingRemarks .... 114
4 PlateModels:ValidationandVerification ......................... 119
4.1 FiniteElementsforAnalysisofThickandThinPlates ........... 119
4.1.1 Motivation: Timoshenko Beam Element Linked
Interpolations ....................................... 120
4.1.2 Reissner-MindlinPlateModelandFEDiscretization ...... 122
4.1.3 IllustrativeNumericalExamplesandClosingRemarks .... 132
4.2 DiscreteKirchhoffPlateElementExtensionwithIncompatible
Modes .................................................... 137
4.2.1 Reissner-Mindlin Plate Model and Enhanced FE
Interpolations ....................................... 138
4.2.2 IllustrativeNumericalExamplesandClosingRemarks .... 141
4.3 ValidationorModelAdaptivityforThickorThinPlates
BasedonEquilibratedBoundaryStressResultants .............. 144
4.3.1 ThickandThinPlateFiniteElementModels ............. 146
4.3.2 ModelAdaptivityforPlates ........................... 154
4.3.3 IllustrativeNumericalExamplesandClosingRemarks .... 163
4.4 Verification or Discrete Approximation Adaptivity
forDiscreteKirchhoffPlateFiniteElement .................... 173
4.4.1 KirchhoffPlateBendingModel ........................ 179
4.4.2 KirchhoffPlateFiniteElements ........................ 183
4.4.3 ErrorEstimatesforKirchhoffPlateElementsBased
UponEquilibratedBoundaryStressResultants ........... 190
4.4.4 ImplementationofEquilibratedElementBoundary
TractionsMethodForDKTPlateElement ............... 194
4.4.5 Examples on Error Indicators Comparison
andClosingRemarks ................................. 199
