Torsion of Thin Walled Structures

229 Pages·2022·4.278 MB·English

by  Krishnaiyengar Rajagopalan

Krishnaiyengar Rajagopalan
Torsion
of Thin
Walled
Structures
Torsion of Thin Walled Structures
Krishnaiyengar Rajagopalan
Torsion of Thin Walled
Structures
KrishnaiyengarRajagopalan
DepartmentofOceanEngineering
IndianInstituteofTechnologyMadras
Chennai,TamilNadu,India
ISBN978-981-16-7457-0 ISBN978-981-16-7458-7 (eBook)
https://doi.org/10.1007/978-981-16-7458-7
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Dedicatedto
ShriGuruvayoorappanSwamy
SrimadBhagavadGitaChapter18,Verse46
yatah. pravr.ittirbhu¯ta¯na¯m˙ yenasarvamidam˙
tatam
sva-karman.a¯ tamabhyarchyasiddhim˙
vindatima¯navah.
Manattainsthehighestperfectionby
worshippinghimwithhisownnaturalduties,
fromwhomallthisuniversespringsforthand
bywhomallthisuniverseispervaded
(Translationsource:https://www.holy-bha
gavad-gita.org/)
Preface
Thin-walledstructuresinwhichtorsionisdominantoccurinshipandaircraftstruc-
tures, shear walls and bridges, ladder frames of commercial automobile vehicles,
satellitesupportsystems,andthelike.Thesestructureswarpundertorsion,andsuch
warpingcomplicatestheanalysis.Theelucidationofwarpingpropertiesisbasicto
theunderstandingofthesubject,butthistopicisnotwelladdressedintheexisting
books.
Considerableattentionisthereforedevotedinthisbooktomakethissomewhat
esotericproblemexoteric,andtheauthorconsidersthisastheforteofthebook.
Reviewquestionsandproblemswithfullanswersfurtherembellishtheportrayal
ofthesubject.Classicalandfiniteelementtechniquesareusedintheanalysis,and
wherever possible,thesameexample issolvedbybothmethods sothatthereader
cancometogripswiththesubject.
Thesubjectisaclassicalaswellasamodernone.Thereferencesgivenattheend
ofthechapterswerefreelyconsulted,gleaned,andcirculatedinclassroomsituations.
Theauthorwishestothankthewritersofthesereferences.Nevertheless,effortshave
been made to present the classical concepts in lucid terms so as to dovetail them
with modern finite element concepts for emphasizing the importance of torsion in
thin-walledstructures.
Chennai,India KrishnaiyengarRajagopalan
[email protected]
vii
Contents
1 TorsionofThin-WalledStructures ............................... 1
1.1 Introduction .............................................. 1
1.2 PictureofSectionalStresses ................................ 3
1.3 DecouplingofBendingandWarping ......................... 8
1.4 HistoricalOverview ....................................... 10
1.5 ReviewProblems .......................................... 11
1.6 AnswerstoReviewProblems ............................... 13
Reference ...................................................... 16
2 St.VenantTorsion .............................................. 17
2.1 Introduction .............................................. 17
2.2 ThinRectangularSection ................................... 17
2.3 SingleCellThin-WalledSection ............................. 21
2.4 MulticellularSection ....................................... 24
2.5 StressConcentration ....................................... 27
2.6 BoxwithLatticeWalls ..................................... 27
2.7 CompositeCrossSection ................................... 31
2.8 BoundsforTorsionConstant ................................ 32
2.9 TorsionofMulticellularSectionConnectedbyaBaseCell ...... 33
2.10 ReviewProblems .......................................... 36
2.11 AnswerstoReviewProblems ............................... 36
References ..................................................... 43
3 WarpingPropertiesofThin-WalledSections ...................... 45
3.1 Introduction .............................................. 45
3.2 SectorialArea ............................................ 45
3.3 IntegralsInvolvingSectorialArea ............................ 46
3.4 PrincipalSectorialArea .................................... 49
3.5 TransformationsofUnitWarping ............................ 49
3.6 DecouplingEquations ...................................... 51
3.7 ContourWarpingofMulticellSections ....................... 57
3.8 ThicknessWarping ........................................ 57
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x Contents
3.9 SectorialShearFunction ................................... 58
3.10 IllustrativeExamples ....................................... 59
References ..................................................... 75
4 TheoriesofTorsion ............................................. 77
4.1 Introduction .............................................. 77
4.2 BasicAssumptions ........................................ 78
4.3 EquilibriumEquations ..................................... 78
4.4 Stress–StrainRelationships ................................. 80
4.5 CompatibilityEquations .................................... 81
4.5.1 WarpingDisplacements:OpenProfiles ................. 84
4.5.2 WarpingDisplacements:ClosedProfiles ............... 86
4.6 Stress–DisplacementExpressions ............................ 86
4.7 DifferentialEquationsofEquilibrium ........................ 87
4.8 Decoupling ............................................... 88
4.9 Benscoter’sTheory ........................................ 90
4.10 TheWagnerEffect ......................................... 94
4.11 ReviewQuestions ......................................... 95
4.12 AnswerstoReviewQuestions ............................... 95
Reference ...................................................... 96
5 AnalysisofThin-WalledStructures .............................. 97
5.1 Introduction .............................................. 97
5.2 St.VenantTorsion ......................................... 97
5.3 VLASOV(Warping)Torsion ................................ 102
5.4 MixedTorsion ............................................ 103
5.5 WarpingRestraints ........................................ 109
5.5.1 OtherWarpingRestraints ............................ 114
5.6 ReviewProblems .......................................... 115
5.7 AnswerstoReviewProblems ............................... 116
Reference ...................................................... 136
6 FiniteElementAnalysisofThin-WalledStructures ................ 137
6.1 Introduction .............................................. 137
6.2 StiffnessMatrix ........................................... 137
6.3 StructuralAnalysis ........................................ 139
6.4 ExpressionforStiffnessMatrix .............................. 140
6.5 St.VenantTorsionElement ................................. 141
6.6 ApplicationsofSt.VenantTorsionElement ................... 142
6.7 VlasovTorsionElement .................................... 147
6.8 MixedTorsionFiniteElement ............................... 149
6.9 Finite Element Method for Warping Properties
ofThin-WalledSections .................................... 158
6.10 ReviewProblems .......................................... 161
6.11 AnswerstoReviewProblems ............................... 162
Reference ...................................................... 173
Contents xi
7 StabilityAnalysisofThin-WalledStructuresinTorsion ............ 175
7.1 Introduction .............................................. 175
7.2 Flexural–TorsionalBuckling ................................ 176
7.2.1 StiffenersforTorsionalBuckling ...................... 187
7.3 LateralBucklingofThin-WalledBeams ...................... 188
7.4 BucklingUnderTorsionalLoadings .......................... 190
7.5 FiniteElementAnalysisofBucklingProblems ................ 190
References ..................................................... 196
8 PlasticTorsionofThin-WalledStructures ........................ 197
8.1 Introduction .............................................. 197
8.2 UltimateTorqueofaThinRectangle ......................... 197
8.3 UltimateSt.VenantTorqueofClosedSections ................ 198
8.4 Upper-BoundSolution ..................................... 201
8.5 MixedTorsion ............................................ 205
References ..................................................... 208
Appendix:AnalysisforTransverseShear ............................ 209
Index ............................................................. 221
About the Author
Dr. Krishnaiyengar Rajagopalan obtained his bachelor’s and master’s in Civil
EngineeringandStructuralEngineering,respectively,fromtheUniversityofMadras,
India.HepursuedhisPh.D.fromtheIndianInstituteofTechnology(IIT)Madrasfor
hisresearchonthefiniteelementbucklinganalysisofthinwalledshellstructures.
Dr.RajagopalanhasworkedasascientistwithCSIR,Indiaandasaprofessorwith
IITMadras.Heworkedextensivelyintheareasofthinwalledstructuresmainlyin
the classical and finite element analysis of ship structures, power plant structures,
storagestructures,calandria(nuclear)vessels,boilersupportingstructures,submarine
structures,steelandconcretebuildingstructuresandofsimilarilk.
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