Plate and Shell Structures: Selected Analytical and Finite Element Solutions

420 Pages·2017·15.14 MB·English

by  Maria Radwańska

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PlateandShellStructures
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Plate and Shell Structures
SelectedAnalyticalandFiniteElementSolutions
MariaRadwan´ska
AnnaStankiewicz
AdamWosatko
(cid:2) JerzyPamin (cid:2)
CracowUniversityofTechnology,Poland
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Thiseditionfirstpublished2017
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vii
Contents
Preface xvii
Notation xix
Part1 Fundamentals:TheoryandModelling 1
1 GeneralInformation 3
1.1 Introduction 3
1.2 ReviewofTheoriesDescribingElasticPlatesandShells 6
(cid:2) 1.3 DescriptionofGeometryfor2DFormulation 9 (cid:2)
1.3.1 CoordinateSystems,MiddleSurface,CrossSection,PrincipalCoordinate
Lines 9
1.3.2 GeometryofMiddleSurface 10
1.3.3 GeometryofSurfaceEquidistantfromMiddleSurface 12
1.3.4 GeometryofSelectedSurfaces 13
1.3.4.1 SphericalSurface 13
1.3.4.2 CylindricalSurface 14
1.3.4.3 HyperbolicParaboloid 15
1.4 DefinitionsandAssumptionsfor2DFormulation 16
1.4.1 GeneralizedDisplacementsandStrainsConsistentwiththeKinematic
HypothesisofThree-ParameterKirchhoff–LoveShellTheory 16
1.4.2 GeneralizedDisplacementsandStrainsConsistentwiththeKinematic
HypothesisofFive-ParameterMindlin–ReissnerShellTheory 18
1.4.3 ForceandMomentResultantsRelatedtoMiddleSurface 18
1.4.4 GeneralizedStrainsinMiddleSurface 20
1.5 ClassificationofShellStructures 21
1.5.1 Curved,ShallowandFlatShellStructures 22
1.5.2 Thin,ModeratelyThick,ThickStructures 22
1.5.3 PlatesandShellswithDifferentStressDistributionsAlongThickness 23
1.5.4 RangeofValidityofGeometricallyLinearandNonlinearTheoriesforPlates
andShells 23
References 24
2 EquationsforTheoryofElasticityfor3DProblems 26
Reference 30
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3 EquationsofThinShellsAccordingtotheThree-Parameter
Kirchhoff–LoveTheory 31
3.1 GeneralEquationsforThinShells 31
3.2 SpecificationofLameParametersandPrincipalCurvatureRadiiforTypical
Surfaces 38
3.2.1 ShellsofRevolutioninaSphericalCoordinateSystem 39
3.2.2 ShellsofRevolutioninaCylindricalCoordinateSystem 40
3.2.3 ShallowShellsRepresentedbyRectangularProjection 41
3.2.4 FlatMembranesandPlatesinCartesianorPolarCoordinateSystem 41
3.3 TransitionfromGeneralShellEquationstoParticularCasesofPlatesand
Shells 42
3.3.1 EquationsofRectangularFlatMembranes 42
3.3.2 EquationsofRectangularPlatesinBending 43
3.3.3 EquationsofCylindricalShellsinanAxisymmetricMembrane-Bending
State 44
3.4 DisplacementEquationsforMulti-ParameterPlateandShellTheories 45
3.5 Remarks 47
References 47
4 GeneralInformationaboutModelsandComputationalAspects 48
4.1 AnalyticalApproachtoStatics,BucklingandFreeVibrations 49
4.1.1 StaticsofaThinPlateinBending 49
(cid:2) 4.1.2 BucklingofaPlate 50 (cid:2)
4.1.3 TransverseFreeVibrationsofaPlate 51
4.2 ApproximateApproachAccordingtotheFiniteDifferenceMethod 51
4.2.1 SetofAlgebraicEquationsforStaticsofaPlateinBending 52
4.2.2 SetofHomogeneousAlgebraicEquationsforPlateBuckling 53
4.2.3 SetofHomogeneousAlgebraicEquationsforTransverseFreeVibrationsofa
Plate 53
4.3 ComputationalAnalysisbyFiniteElementMethod 54
4.4 ComputationalModels–Summary 55
Reference 55
5 DescriptionofFiniteElementsforAnalysisofPlatesandShells 56
5.1 GeneralInformationonFiniteElements 56
5.2 DescriptionofSelectedFEs 58
5.2.1 FlatRectangularFour-NodeMembraneFE 58
5.2.2 ConformingRectangularFour-NodePlateBendingFE 60
5.2.3 NonconformingFlatThree-andFour-NodeFEsforThinShells 63
5.2.4 Two-DimensionalCurvedShellFEbasedontheKirchhoff–LoveThinShell
Theory 64
5.2.5 CurvedFEbasedontheMindlin–ReissnerModeratelyThickShell
Theory 64
5.2.6 DegeneratedShellFE 65
5.2.7 Three-DimensionalSolidFEforThickShells 65
5.2.8 GeometricallyOne-DimensionalFEforThinShellStructures 66
5.3 RemarksonDisplacement-basedFEFormulation 69
References 70
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Part2 Plates 73
6 FlatRectangularMembranes 75
6.1 Introduction 75
6.2 GoverningEquations 76
6.2.1 LocalFormulation 76
6.2.2 EquilibriumEquationsinTermsofIn-PlaneDisplacements 78
6.2.3 PrincipalMembraneForcesandtheirDirections 78
6.2.4 EquationsforaFlatMembraneFormulatedusingAiry’sStressFunction 79
6.2.5 GlobalFormulation 80
6.3 SquareMembraneunderUnidirectionalTension 81
6.3.1 AnalyticalSolution 81
6.3.2 AnalyticalSolutionwithAiry’sStressFunction 83
6.3.3 NumericalSolution 83
6.4 SquareMembraneunderUniformShear 83
6.4.1 AnalyticalSolution 83
6.4.2 FEMResults 84
6.5 PureIn-PlaneBendingofaSquareMembrane 85
6.6 CantileverBeamwithaLoadontheFreeSide 88
6.6.1 AnalyticalSolution 88
6.6.2 FEMResults 92
6.7 RectangularDeepBeams 94
(cid:2) 6.7.1 BeamsandDeepBeams 94 (cid:2)
6.7.2 SquareMembranewithaUniformLoadontheTopEdge,SupportedonTwo
PartsoftheBottomEdge–FDMandFEMResults 94
6.8 MembranewithVariableThicknessesorMaterialParameters 97
6.8.1 Introduction 97
6.8.2 MembranewithDifferentThicknessesinThreeSubdomains–FEM
Solution 97
6.8.3 MembranewithDifferentMaterialParametersinThreeSubdomains–FEM
Solution 99
References 101
7 CircularandAnnularMembranes 102
7.1 EquationsofMembranes–LocalandGlobalFormulation 102
7.2 EquationsfortheAxisymmetricMembraneState 104
7.3 AnnularMembrane 105
7.3.1 AnalyticalSolution 107
7.3.2 FEMSolution 108
References 109
8 RectangularPlatesunderBending 110
8.1 Introduction 110
8.2 EquationsfortheClassicalKirchhoff–LoveThinPlateTheory 110
8.2.1 AssumptionsandBasicRelations 110
8.2.2 EquilibriumEquationforaPlateExpressedbyMoments 116
8.2.3 DisplacementDifferentialEquationforaThinRectangularPlateAccordingto
theKirchhoff–LoveTheory 116
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8.2.4 GlobalFormulationforaKirchhoff–LoveThinPlate 117
8.3 DerivationofDisplacementEquationforaThinPlatefromthePrincipleof
MinimumPotentialEnergy 117
8.4 EquationforaPlateunderBendingRestingonaWinklerElastic
Foundation 118
8.5 EquationsofMindlin–ReissnerModeratelyThickPlateTheory 119
8.5.1 KinematicsandFundamentalRelationsforMindlin–ReissnerPlates 119
8.5.2 GlobalFormulationforModeratelyThickPlates 120
8.5.3 EquationsforMindlin–ReissnerModeratelyThickPlatesExpressedby
GeneralizedDisplacements 122
8.6 AnalyticalSolutionofaSinusoidallyLoadedRectangularPlate 122
8.7 AnalysisofPlatesunderBendingUsingExpansionsinDoubleorSingle
TrigonometricSeries 127
8.7.1 ApplicationofNavier’sMethod–DoubleTrigonometricSeries 127
8.7.2 IdeaofLevy’sMethod–SingleTrigonometricSeries 130
8.8 SimplySupportedorClampedSquarePlatewithUniformLoad 131
8.8.1 ResultsObtainedusingDTSMandFEMforaSimplySupportedPlate 132
8.8.2 ResultsObtainedusingSTSMandFEMforaClampedPlate 135
8.9 RectangularPlatewithaUniformLoadandVariousBoundary
Conditions–ComparisonofSTSMandFEMResults 135
8.10 UniformlyLoadedRectangularPlatewithClampedandFreeBoundary
Lines–ComparisonofSTSMandFEMResults 139
(cid:2) 8.11 ApproximateSolutiontoaPlateBendingProblemusingFDM 143 (cid:2)
8.11.1 IdeaofFDM 143
8.11.2 ApplicationofFDMtotheSolutionofaBendingProblemforaRectangular
Plate 144
8.11.3 SimplySupportedSquarePlatewithaUniformLoad 147
8.11.4 SimplySupportedUniformlyLoadedSquarePlateRestingona
One-ParameterElasticFoundation 149
8.12 ApproximateSolutiontoaBendingPlateProblemusingtheRitz
Method 151
8.12.1 IdeaoftheRitzMethod 151
8.12.2 SimplySupportedRectangularPlatewithaUniformLoad 152
8.13 PlatewithVariableThickness 153
8.13.1 DescriptionofDeformation 154
8.13.2 FEMResults 154
8.14 AnalysisofThinandModeratelyThickPlatesinBending 155
8.14.1 PreliminaryRemarks 155
8.14.2 SimplySupportedSquarePlatewithUniformLoad–AnalyticalandFEM
Results 156
8.14.3 SimplySupportedPlatewithaConcentratedCentralLoad–Analyticaland
FEMSolutions 157
References 159
9 CircularandAnnularPlatesunderBending 160
9.1 GeneralState 160
9.2 AxisymmetricState 162
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9.3 AnalyticalSolutionusingaTrigonometricSeriesExpansion 164
9.4 ClampedCircularPlatewithaUniformlyDistributedLoad 166
9.5 SimplySupportedCircularPlatewithaConcentratedCentralForce 169
9.6 SimplySupportedCircularPlatewithanAsymmetricDistributed
Load 171
9.6.1 AnalyticalSolution 172
9.6.2 FEMSolution 174
9.7 UniformlyLoadedAnnularPlatewithStaticandKinematicBoundary
Conditions 174
9.7.1 AnalyticalSolution 174
9.7.2 NumericalSolutionusingFEM 177
References 177
Part3 Shells 179
10 ShellsintheMembraneState 181
10.1 Introduction 181
10.2 GeneralMembraneStateinShellsofRevolution 182
10.3 AxisymmetricMembraneState 183
10.3.1 MembraneForcesforShellsDescribedinaSphericalCoordinate
System 184
(cid:2) 10.3.2 MembraneForcesforShellsDescribedinaCylindricalCoordinate (cid:2)
System 185
10.4 HemisphericalShell 186
10.4.1 ShellunderSelfWeight–AnalyticalSolution 186
10.4.2 ShellunderUniformPressure–AnalyticalSolution 187
10.4.3 SuspendedTankunderHydrostaticPressure–AnalyticalSolution 189
10.4.4 SupportedTankunderHydrostaticPressure–AnalyticalandFEM
Solutions 191
10.4.4.1 CaseI–uppershellpart𝜓 ∈[40∘,90∘] 192
10.4.4.2 CaseII–lowershellpart𝜓 ∈[0∘,40∘] 192
10.5 OpenConicalShellunderSelfWeight 193
10.6 CylindricalShell 195
10.6.1 EquationsoftheAxisymmetricMembraneState 195
10.6.2 CircularCylindricalShellunderSelfWeightandHydrostatic
Pressure–AnalyticalSolution 197
10.6.3 FEMResults 198
10.7 HemisphericalShellwithanAsymmetricWindAction 199
10.7.1 AnalyticalSolution 200
10.7.2 FEMResults 202
References 204
11 ShellsintheMembrane-BendingState 205
11.1 CylindricalShells 205
11.1.1 DescriptionofGeometry 205
11.1.2 EquationsoftheMembrane-BendingState 205
11.1.3 EquationsoftheAxisymmetricMembrane-BendingState 207
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11.1.4 EquationsoftheAxisymmetricMembraneState 208
11.1.5 AnalyticalSolutionfortheAxisymmetricMembrane-BendingState 209
11.1.6 LongCylindricalShellClampedontheBottomEdgeunderSelfWeightand
HydrostaticPressure–AnalyticalandFEMSolutions 211
11.1.6.1 StageI–analysisofthemembranestate 211
11.1.6.2 StageII–analysisofthemembrane-bendingstate 213
11.1.7 ShortCylindricalShellClampedontheBottomEdgeunderSelfWeightand
HydrostaticPressure–AnalyticalandFEMSolutions 218
11.2 SphericalShells 221
11.2.1 DescriptionofGeometry 221
11.2.2 EquationsfortheAxisymmetricMembrane-BendingState 221
11.2.3 ClampedorSimplySupportedSphericalShellunderSelfWeight–Analytical
andFEMSolutions 224
11.2.3.1 StageI–analysisofthemembranestate 226
11.2.3.2 StageII–analysisofthemembrane-bendingstate 226
11.3 CylindricalandSphericalShellsLoadedbyaUniformlyDistributedBoundary
MomentandHorizontalForce 229
11.3.1 CylindricalShell 229
11.3.1.1 CaseA–distributedboundarymomentM 229
11.3.1.2 CaseB–horizontalboundarytractionH 230
11.3.2 Sphericalshell 231
11.3.2.1 CaseA–distributedboundarymomentM 232
(cid:2) 11.3.2.2 CaseB–horizontalboundarytractionH 232 (cid:2)
11.4 CylindricalShellwithaSphericalCap–AnalyticalandNumerical
Solution 232
11.4.1 AnalyticalSolution 233
11.4.2 FEMSolution 236
11.5 GeneralCaseofDeformationofCylindricalShells 237
11.6 CylindricalShellwithaSemicircularCrossSectionunderSelf
Weight–AnalyticalSolutionofMembraneState 238
11.7 CylindricalScordelis-LoRoofintheMembrane-BendingState–Analytical
andNumericalSolution 242
11.8 Single-SpanClampedHorizontalCylindricalShellunderSelfWeight 246
11.8.1 AnalyticalSolution 249
11.8.1.1 StageI–analysisofthemembranestate 249
11.8.1.2 StageII–analysisofthemembrane-bendingstate 250
11.8.2 FEMSolution 252
References 254
12 ShallowShells 256
12.1 EquationsforShallowShells 256
12.2 Pucher’sEquationsforShallowShellsintheMembraneState 260
12.3 HyperbolicParaboloidwithRectangularProjection 262
12.3.1 DescriptionofGeometry 262
12.3.2 AnalyticalSolutionoftheMembraneState 263
12.4 RemarksonEngineeringApplications 266
References 267
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