Springback Analysis of Thin Tubes with Arbitrary Stress ... Journal of Mechanical Engineering by using Ramberg-Osgood stress-strain relation and deformation theory of plasticity. A
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Journal of Mechanical Engineering Vol. 8, No. 1, 57-76, 2011
A general theoretical method for determining springback of arbitrary shaped thin tubular section of materials having arbitrary stress-strain relationship under torsional loading is presented. The theoretical analysis has been compared with earlier analysis of tubular sections of particular materials and has been shown that the expressions obtained in the, work presented. The theoretical results also have been found to quit in agreement with the results obtained experimentally. It has been shown that when applied torque is kinetic loading or angle of twist given is kinetic loading, springback angle and residual angle of twist can directly
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be calculated from the shear stress-strain curve, avoiding any idealization/approximation for the same.
u, v, and w Small displacements in x, y, and z direction, respectivelyθ AngleoftwistperunitlengthofthetubeθR Residual angle of twist per unit lengthθ– Non-dimensionalized angle of twist per unit lengthθ–R Non-dimensionalized residual angle of twistγ Shearstrainτxz,τyz Shear stressG Modulus of rigidity
Stress function GradientT Twisting moment (torque)TO Twisting moment at yieldingT– Non-dimensionalized torqueξ Deflectionofmembraneσx Normal stressβ Anglemadebythelineoflengthρwiththetangenttothe contour elementq Constant (shear flow)ρ DistanceoftheelementfromtheoriginA Meanoftheareasenclosedbytheouterandinnerboundaries ofthecross-sectionofthetubeS Lengthofthecenterlineofringsectionofthetubet Thicknessofthetubeτo Yield shear stressγo Yield shear strainα Plasticmodulusofrigidityεx,εy Longitudinal strainμ Poisson’sratio
Introduction
A large variety of aerospace, automotive, appliances, structural and other industrial components aremade from thinwalled tubes.Theses tubesmay
Δ
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Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
beindifferentcross-sectionssuchascircular,square,rectangular, triangularetc. Such components are advantages from view-point of reduction in weight alongwithimprovedstructuralstrength.Tubesofdifferentcross-sectionshaveproducedbyformingthemetalsheetsbyplasticbendingandtorsionwiththeaidofpunchesanddies.Aftertheendofbendingprocessandontheremovaloftooling,thedimensionsofbendtubechangesduetoelasticrecoverybehavioroftubematerial.Thisphenomenoniscalledspringback.Duringtheprocessofbendingtheinternalstressesareinducedintubebuttheyarenotvanishesonunloading.Afterbendingtheextradosandintradosaresubjectedtobothtypeofresidual stresses i.e. tensile and compressive respectively. Such residual stresses produceanetinternalbendingmomentwhichcausesspringback.Springbackisaveryimportantfactorwhichhastobetakenintoconsideration,whiledesigningthe tool on loading a bodyplastically, particularly in sheetmetalworking.Springbackhastobeeliminatedorcorrectedattheinitialphasesofdesigntoachieve consistent and accurate dimensions of the final part. It is affecting the whole product development and manufacturing process in terms of production delay, higher rejection rate and increment in cost for tooling revisions.Allthese factors reduce customer satisfaction and loyalty for the final product. US automotiveindustriesalonebearalossofmorethan$50millionperyear[1].
Springbackisinfluencedbynumberofparametersbothfrommaterialandprocessselected.Thesesmaybematerial properties, sheet thickness, elastic modulus, Poisson’s coefficient, blankmaterial, punch and die radii, initialclearancefrictionconditions,binderforce,toolinggeometry,blankholderforce,blankholdergeometry,andgeometryofdrawbeads[2].The relationships that existbetweenspringbackandtheseparametersareextremelynonlinearwithmultiple interactions [3].Based on the part geometry and deformation area, differenttypesofspringbackinsheetmetalexist:bending,membrane,twistingandcombinedbendingandmembrane[4].The twistingor torsional typeofspringbackis the measure of elastic recovery of angle of twist on the removal ofappliedtorqueaftertwistingthesectionbeyondelasticlimit.Uneven elastic recovery in different directions is the main cause for this type.
Inthelast40years,numerousresearcheffortshavebeenmadetoreducespringbackbyeffectivelypredictinganddeterminingthecontrollingparametersinthesheetbendingoperationsthroughexperiments[5]-[19]andsimulation[20]-[27].Inalltheseworks,issuesrelatedtoaccuratepredictionandeffectivecompensationofspringbackfordifferenttypegeometriesarediscussed.Inrecentyears,muchattentionhasbeenplacedonspringbackoftubes[28]-[32].Mostof the researchers use simplified models and different stress-strain relationship infindingouttheamountofspringbackintubebendingoperations.Torsionalspringbackofbarsofdifferentcross-sectionshasbeenanalyzedbyDwivedietal.[33]-[38].Springbackofnarrowrectangularstripsoflinearworkhardeningmaterialsundertorsionalloading[33]-[34]andgeneralcross-sectionswiththetorsionalspringbackforwork-hardeningmaterialswereestimated[35]-[37]
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byusingRamberg-Osgood stress-strain relation and deformation theory ofplasticity.Anumericalschemebasedonfinitedifferenceapproximationwasused.Theelastic-plasticboundaryinthebarsofsquaresectionandL-sectionwerealsodeterminedinadditiontospringback.Theoreticalresultswereverifiedby experiments onmild steel bars. In a study [38], springback analysis ofnarrowrectangularbarintorsionfornon-linearwork-hardeningmaterialswasconsidered.Non-linearbehaviorofthematerialwasapproximatedbyassumingModifiedLudwicktypestress-strainrelationship.Theresultofbi-linearcase[33]wascomparedwithbyconsideringnon-linearbehaviorof thematerialoftubularbars.Itwasfoundthatexperimentalvalueshadexcellentmatchformaterialshavingnon-linearbehavior.Recently,Choubeyetal.[39]developedatheoreticalexpressionfortorsionalspringbackinthintubeswithnon-linearworkhardeningbehaviorofthematerialandcomparedthemwithexperimentalfindings for mild steel case.
The present work is concerned with a theoretical method of determining springbackinthintubeswitharbitrarysectionsofmaterialshavingarbitrarystress-strain relationship avoiding any approximation/idealization of the same. It has been shown that springback/residual angle of twist can be obtainedfrom the applied torque in kinetic loading or from the given angle of twist in kinematic loading directly using the shear stress-strain curve. Bi-linear stress-strain relationshiphas been considered as a particular case of zonegeneralbehavior and theexpressionsobtainedbyusingbi-linear relationship in thegeneralanalysishavebeenfoundtobeexactlythesameexpressionsobtainedearlierinreference[36].
Basic Theory
Consider a prismatic thin tube undergoing torsion and elastic deformation (Figure1).Letu,vandwbethesmalldisplacementsofthepoint(x,y,z)relativeto its initial position, in the x, y and z directions respectively. At a section, where zisconstant,thecross-sectionrotatesaboutthez-axisandso[38],[39].
Whenthestressesinducedbytorsionis intheplasticrange,fromequations(9) and (15)
(18)
Again since the relationship (17) is independent of the material properties andsinceithasbeenassumedthatthesmallstrainrelationships(2)amongstrainsand displacements are valid also in the plastic range relationship (7) remains valid for plastic torsion. So, from equations (17) and (18)
then assuming that unloading is elastic (Figure 5),
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Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
so,
(22)
also,
(23)
Incaseofbi-linearrelationship(Figure6)givenby[30]
(24)
Where,τ0andγ0areyieldshearstressandstrainrespectivelyandαistheplasticmodulusofrigiditygivenbytheslopeofthestressstrainforτ≥τ0. From theabove,γmaybeexpressedexplicitlyas
The above relationshipmaybe expressed in non-dimensionalized formas,
or
(29)
Asobtainedinreference[30].
Experiments
The shear stress-strain curve for a mild steel specimen was plotted from a tensilestress-straincurvebyusingtherelationshipsobtainedinAppendixA.Thintubesofsquare(25mm×25mm)sectionhavinglengthandthicknessof200mmand2mmrespectivelyweremadeandsubjectedtotorsiontestonAvery Torsion Testing machine, in which the angle of twist is measured with a Vernier scale, giving an accuracy upto 0.10. The angles of twist were noted in the elasto-plastic regime for different test pieces. Experimental points are shownalongwiththetheoreticalcurvesinFigure7.Detailsoftheexperimentsare given in Appendix B.
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Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
Figu
re 7
: Non
-dim
ensi
onal
ized
Ang
le o
f Tw
ist (θ– ) V
ersu
s Non
-dim
ensi
onal
ized
Res
idua
l Ang
le o
f Tw
ist (θ– R
) and
Non-dimensionalizedSpringbackAngleofTwist(θ– S)
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Results and Discussion
Ithasalreadybeenshownintheanalysisthatwhenstress-strainrelationshipisidealizedasbi-linearone,thetheoreticalexpressionsobtainedfromthegeneralexpressionareexactlythesameasthoseobtainedinreference[36].ItisalsoseenfromFigure7thattheexperimentalresultsobtainedareincloseagreementwith the results obtained bymethodobtained using the actual stress-straincurves. The differences in the theoretical and experimental values are due to theassumptionmadeintheanalysisthattherelationshipbetweendisplacementand strains for small deformation are also valid for the strain range under consideration.However,inspiteofthedifferencesthecloseagreementbetweenthevaluesobtainedtheoreticallyfromthestress-straincurveandthoseobtainedexperimentally confirms the validity of the theoretical analysis. The method has the advantage that no approximation or idealization of the stress-strain curve is needed to get the results theoretically.
Conclusions
Based on the results presented above the following conclusions have beendrawn:
1. Themethodprovidesvaluesofspringbackangleof twist/residualangleof twist to satisfactory level of accuracy and is in close agreement with experimental results.
2. No approximation/idealization of the stress-stain curve is needed.3. Notorqueversusangleoftwistcurveisdrawnforobtainingthevaluesof
[1] Wang,C. (2002).“An industrialoutlook forspringbackpredictability,measurementreliabilityandcompensationtechnology”,Proceedingsofthe 5th International Conference and Workshop on Numerical Simulation of 3DSheetFormingProcesses, (NUMISHEET2002). 597-604, JejuIsland, Korea, 2002.
[2] Gardiner, F. (1957). “The springback ofmetals”,Transactions of theASME, 79, 1-9.
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Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
[19] Kutt,L.M.,Nardiello,J.A.,Ogilvie,P.L.,Pifko,A.B.andPapazian,J.M. (1999). “Nonlinear finite element analysis of springback”,Communications in Numerical Methods in Engineering, 15(1), 33-42.
[20] Tekkaya,A.E. (2000). “State-of-the-art of simulation of sheetmetalforming”,JournalofMaterialsProcessingTechnology,103(1),4-22.
[24] Liu,W.,Liu,Q.,Ruan,F.,Liang,Z.andQiu,H.(2007).“Springbackprediction for sheetmetal forming based onGA–ANN technology”,JournalofMaterialsProcessingTechnology,187-188,227-231.
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Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
[28] Da-xin,E.,Hua-hui,H.,Xiao-yi,L.andRu-xin,N.(2009).“Experimentalstudy and finite element analysis of spring-back deformation in tubebending”.InternationalJournalofMinerals,MetallurgyandMaterials,16(2) 177-183.
[31] Dwivedi,J.P.,Sarkar,P.K.,Sohan,R.andTalukder,N.K.D.(1987).“Experimental aspects of torsional springback in rectangular strips”,JournaloftheInstitutionofEngineers(India).Vol.67(PartME-4).70-83.
[33] Dwivedi,J.P.,Upadhyay,P.C.andTalukder,N.K.D.(1992).“Springbackanalysis of torsionofL-sectionedbars ofwork-hardeningmaterials”,Computer&Structures,43(5),815-822.
[34] Dwivedi,J.P.,Upadhyay,P.C.andTalukder,N.K.D.(1992).“Parametricassessment of torsional springback inmembers ofwork-hardeningmaterials”,Computers&Structures,45(3),421-429.
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[35] Dwivedi, J. P., Shah, S.K.,Upadhyay, P.C. andTalukder,N.K.D.(2002).“Springbackanalysisofthinrectangularbarsofnon-linearwork-hardeningmaterials”,InternationalJournalofMechanicalSciences,44,1505-1519.
Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
APPENDIX-A
Though it is not easy to get the shear stress-strain relationship experimentally, itmaybederivedfromthestress-straincurvefromtensiletestandthevalueofGasfollows[5].
Figure8:DeformationofSquareElementabcdDueto Tension in the x-direction
LetasquareelementshowninFigureA1bedeformedtoduetotensioninx-direction.Letthestraininx-directionbe , then the maximum shear strain willbeatanangleπ⁄4withtheX-axix,i.e.alongthediagonalofthesquare.Theangleisdefinitelyequalto[(π⁄4)+(γ⁄2)].So,
(A1)
Takingμ=0.5inplasticstateofstressfromequation(A1)
or
or
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so
(A2)
(A3)
Whereτisthemaximumshearstressanditactsalong,sofromequations(A2) and (A3)
Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves
Figure A: Avery Torsion Testing Machine
Maximum Pointer Reset
Capacity Change Handle
Sliding Spindle
Straining Control Handwheel
Duouble Graduations
Totally Enclosed Gearing
Protractor and Vernier Reading to 1/10th Degree
Torque Arm
APENDIX-B
Experimental Procedure
The present work is concerned with experimentally quantifying the torque, springbackandresidualangleoftwistperunitlengthofthinmildsteeltubesofsquare(25mm×25mm)sectionhavinglengthandthicknessof200mmand2mmrespectively.FigureAshowsthepictorialviewof“AveryTorsionTestingMachine”.Thestrainwasappliedtothespecimenbywormandspurgearingwassoarrangedthatthefullloadmightbeappliedbyhandwithoutundueeffort.Theloadwastransmittedfromtheweighingspindlebymeansofahorizontaltorquearm,mountedonantifrictionbearings,whichtransmitteda vertical pull to the indicating unit. The load indicator was of self indicating cam-resistant type and the die carried two sets of graduations with pointer and chartedgetoedgetoavoidparallax.Capacitycouldbecontrolledbymeansofhand lever and a maximum load pointer mounted on a separate spindle provided arecordofbreakingloads.Forthepurposeoftestingandgrippingoftubes,aholderhadbeendesignedandfabricatedwhichcouldbefittedinfour-jawtypeself gripping chuck and attached with the face plate holder.
With the help of hand wheel, a small torque was applied to the specimen that was noted directly from the indicator and corresponding angle of twist was also noted with the help of Vernier and protractor fitted in the machine. The torque was gradually increased up to elasto-plastic regions and corresponding
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angle of twists were noted. Now the torque was released slowly to zero and the residualanglesoftwist(θR)ofthedeformedtubeswerenoted.Thesedeflectionreadingswereagaincheckedwiththehelpofcombinationsetjustaftersettingthe torque to zero.The procedurewas repeated for number of specimens.Springbackpercentageandresidualangletotwistperunitlengthinpercentagewere calculated for each specimen with the help of equations (24) and (25).