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Lecture 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 mechanics-quickly,informallyandatahighlevel.Thisincludesthefieldsoffluid, solid and structural mechanics, dynamics and control, and related disciplines. The applied methods can be of analytical, numerical and computational nature. The seriesscopeincludesmonographs,professionalbooks,selectedcontributionsfrom specialized conferences or workshops, edited volumes, as well as outstanding advancedtextbooks. Indexed by EI-Compendex, SCOPUS, Zentralblatt Math, Ulrich’s, Current MathematicalPublications,MathematicalReviewsandMetaPress. · 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Toourfamilies 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

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