Table Of Content

Mathematical Models of Beams and Cables ToFiorella(A.L.) TomyparentsElio&Marisa(D.Z.) Mathematical Models of Beams and Cables Angelo Luongo Daniele Zulli Series Editor Noël Challamel Firstpublished2013inGreatBritainandtheUnitedStatesbyISTELtdandJohnWiley&Sons,Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentionedaddress: ISTELtd JohnWiley&Sons,Inc. 27-37StGeorge’sRoad 111RiverStreet LondonSW194EU Hoboken,NJ07030 UK USA www.iste.co.uk www.wiley.com ©ISTELtd2013 The rights of Angelo Luongo and Daniele Zulli to be identified as the author of this work have been assertedbytheminaccordancewiththeCopyright,DesignsandPatentsAct1988. LibraryofCongressControlNumber: 2013948060 BritishLibraryCataloguing-in-PublicationData ACIPrecordforthisbookisavailablefromtheBritishLibrary ISBN:978-1-84821-421-7 PrintedandboundinGreatBritainbyCPIGroup(UK)Ltd.,Croydon,SurreyCR04YY Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii ListofMainSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Chapter1.AOne-DimensionalBeamMetamodel . . . . . . . . . . . . . . . 1 1.1.Modelsandmetamodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.Internallyunconstrainedbeams . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3.Thehyperelasticlaw . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.4.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . 11 1.3.Internallyconstrainedbeams . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1.Themixedformulationfortheinternallyconstrainedbeam . . . . kinematicsandconstraints. . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2.Thedisplacementmethodfortheinternallyconstrainedbeam . . 18 1.4.Internallyunconstrainedprestressedbeams . . . . . . . . . . . . . . . . 24 1.4.1.Thenonlineartheory . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.2.Thelinearizedtheory . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.Internallyconstrainedprestressedbeams. . . . . . . . . . . . . . . . . . 29 1.5.1.Thenonlinearmixedformulation . . . . . . . . . . . . . . . . . . . 29 1.5.2.Thelinearizedmixedformulation . . . . . . . . . . . . . . . . . . 30 1.5.3.Thenonlineardisplacementformulation. . . . . . . . . . . . . . . 31 1.5.4.Thelinearizeddisplacementformulation . . . . . . . . . . . . . . 32 vi MathematicalModelsofBeamsandCables 1.6.Thevariationalformulation . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.1.Thetotalpotentialenergyprinciple. . . . . . . . . . . . . . . . . . 34 1.6.2.Unconstrainedbeams . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.6.3.Constrainedbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.6.4.Unconstrainedprestressedbeams. . . . . . . . . . . . . . . . . . . 39 1.6.5.Constrainedprestressedbeams . . . . . . . . . . . . . . . . . . . . 41 1.7.Example:thelinearTimoshenkobeam. . . . . . . . . . . . . . . . . . . 44 1.8.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter2.StraightBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.1.Thedisplacementandrotationfields . . . . . . . . . . . . . . . . . 55 2.1.2.Tacklingtherotationtensor . . . . . . . . . . . . . . . . . . . . . . 60 2.1.3.Thegeometricboundaryconditions . . . . . . . . . . . . . . . . . 63 2.1.4.Thestrainvector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.1.5.Thecurvaturevector . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.6.Thestrain–displacementrelationships . . . . . . . . . . . . . . . . 73 2.1.7.Thevelocityandspinfields . . . . . . . . . . . . . . . . . . . . . . 74 2.1.8.Thevelocitygradientsandstrain-rates . . . . . . . . . . . . . . . 78 2.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.2.1.Thebalanceofvirtualpowers . . . . . . . . . . . . . . . . . . . . . 83 2.2.2.Theinertialcontributions . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.3.Thebalanceofmomentum . . . . . . . . . . . . . . . . . . . . . . 91 2.2.4.Thescalarformsofthebalanceequations . . . . . . . . . . . . . . andboundaryconditions. . . . . . . . . . . . . . . . . . . . . . . . 97 2.2.5.TheLagrangianbalanceequations . . . . . . . . . . . . . . . . . . 99 2.3.Constitutivelaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.3.1.Thehyperelasticlaw . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.3.2.Identificationoftheelasticlawfroma3D-model. . . . . . . . . . 104 2.3.3.Homogenizationofbeam-likestructures. . . . . . . . . . . . . . . 110 2.3.4.Linearviscoelasticlaws . . . . . . . . . . . . . . . . . . . . . . . . 112 2.4.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.4.1.Exactequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.4.2.Thelinearizedtheoryforelasticprestressedbeams. . . . . . . . . 118 2.5.Theplanarbeam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.5.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.5.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.5.3.TheVirtualPowerPrinciple . . . . . . . . . . . . . . . . . . . . . . 126 2.5.4.Constitutivelaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.5.5.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . 127 2.6.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 TableofContents vii Chapter3.CurvedBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1.Thereferenceconfigurationandtheinitialcurvature . . . . . . . . . . . 133 3.2.Thebeammodelinthe3D-space . . . . . . . . . . . . . . . . . . . . . . 137 3.2.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.2.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3.2.3.Theelasticlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.2.4.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . 144 3.3.Theplanarcurvedbeam . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.3.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.3.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.3.3.TheVirtualPowerPrinciple . . . . . . . . . . . . . . . . . . . . . . 157 3.3.4.Constitutivelaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3.3.5.FundamentalProblem . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.4.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Chapter4.InternallyConstrainedBeams . . . . . . . . . . . . . . . . . . . . 163 4.1.Stiffbeamsandinternalconstraints . . . . . . . . . . . . . . . . . . . . 163 4.2.Thegeneralapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.3.Theunshearablestraightbeamin3D . . . . . . . . . . . . . . . . . . . . 168 4.3.1.Themixedformulation . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3.2.Thedisplacementformulation . . . . . . . . . . . . . . . . . . . . 172 4.4.Theunshearablestraightplanarbeam . . . . . . . . . . . . . . . . . . . 177 4.5.Theinextensibleandunshearablestraightbeamin3D . . . . . . . . . . 180 4.5.1.Hybridformulation:VersionI . . . . . . . . . . . . . . . . . . . . 181 4.5.2.Hybridformulation:VersionII . . . . . . . . . . . . . . . . . . . . 182 4.6.Theinextensibleandunshearablestraightplanarbeam. . . . . . . . . . 183 4.6.1.Hybridformulation:VersionI . . . . . . . . . . . . . . . . . . . . 183 4.6.2.Hybridformulation:VersionII . . . . . . . . . . . . . . . . . . . . 185 4.6.3.Themixedformulation . . . . . . . . . . . . . . . . . . . . . . . . 186 4.6.4.Thedirectcondensationoftheelasticaequilibriumequations . . . 188 4.7.Theinextensible,unshearableanduntwistablestraightbeam . . . . . . 190 4.8.Thefoil-beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.9.Theshear–shear–torsionalbeam . . . . . . . . . . . . . . . . . . . . . . 193 4.10.Theplanarunshearableandinextensiblecurvedbeam . . . . . . . . . 197 4.10.1.Thehybridformulation. . . . . . . . . . . . . . . . . . . . . . . . 197 4.10.2.Themixedformulation . . . . . . . . . . . . . . . . . . . . . . . . 200 4.11.Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter5.FlexibleCables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.1.Flexiblecablesasalimitofslenderbeams. . . . . . . . . . . . . . . . . 205 5.2.Unprestressedcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.2.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 viii MathematicalModelsofBeamsandCables 5.2.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.2.3.Constitutivelaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2.4.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . 218 5.3.Prestressedcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.3.1.Quasi-exactmodels . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.3.2.Thelinearizedtheory . . . . . . . . . . . . . . . . . . . . . . . . . 225 5.3.3.Tautstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.4.Shallowcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.4.1.Anapproximatednonlinearmodel . . . . . . . . . . . . . . . . . . 231 5.4.2.Anapproximatedlinearizedmodel . . . . . . . . . . . . . . . . . . 234 5.5.Inextensiblecables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.5.1.Inextensibleunprestressedcables. . . . . . . . . . . . . . . . . . . 236 5.5.2.Inextensibleprestressedcables . . . . . . . . . . . . . . . . . . . . 237 5.6.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Chapter6.StiffCables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.1.Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.2.Unprestressedstiffcables . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.2.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.2.2.Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.2.3.Theelasticlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.2.4.TheFundamentalProblem . . . . . . . . . . . . . . . . . . . . . . 251 6.3.Prestressedstiffcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.3.1.Nonlinearmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.3.2.Thelinearizedmodel. . . . . . . . . . . . . . . . . . . . . . . . . . 256 6.3.3.Tautstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.4.Reducedmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.4.1.Saggedcables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.4.2.Shallowcables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6.5.Inextensiblestiffcables . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.5.1.Unprestressedcables . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.5.2.Prestressedcables . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 6.5.3.Reducedmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.6.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Chapter7.Locally-DeformableThin-WalledBeams . . . . . . . . . . . . . 271 7.1.Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 7.2.Aone-dimensionaldirectmodelfordouble-symmetricTWB . . . . . . 273 7.3.Aone-dimensionaldirectmodelfornon-symmetricTWB . . . . . . . . 277 7.4.Identificationstrategyfrom3D-modelsofTWB . . . . . . . . . . . . . 284 7.5.Afiber-modelofTWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 7.6.Warpable,cross-undeformableTWB . . . . . . . . . . . . . . . . . . . . 289 7.6.1.Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

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Mathematical Models of Beams and Cables PDF

379 Pages·2013·2.147 MB·English
by  Angelo Luongo, Daniele Zulli
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