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

57

Journal of Mechanical Engineering Vol. 8, No. 1, 57-76, 2011

ISSN 1823-5514© 2011 Faculty of Mechanical Engineering, Universiti Teknologi MARA (UiTM), Malaysia.

Springback Analysis of Thin Tubes with Arbitrary Stress-Strain Curves

Mayank Gangwar*Department of Mechanical Engineering

Motilal Nehru National Institute of TechnologyAllahabad-211004, India

Vikas Kumar ChoubeyInstitute of Engineering & Rural Technology

(Engineering Degree Division) Allahabad-211002, India

J. P. DwivediDepartment of Mechanical Engineering

Institute of Technology Banaras Hindu University

Varanasi-221005, India

N. K. Das TalukderInstitute of Technology

Banaras Hindu UniversityVaranasi-221005, India

*Corresponding Author / Email: [email protected]

ABSTRACT

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

Artkl 4.indd 57 10/21/2011 10:14:28 AM

58

Journal of Mechanical Engineering

be calculated from the shear stress-strain curve, avoiding any idealization/approximation for the same.

Keywords: Metal forming, Springback, Torsional springback, Thin tubes, Arbitrary stress-strain curves

Nomenclature

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

Artkl 4.indd 58 10/21/2011 10:14:28 AM

59


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]

Artkl 4.indd 59 10/21/2011 10:14:28 AM

60


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].

u=–yzθ, v=xzθ, and w=θf (x,y) (1)

whereisθtheangleoftwistperunitlengthofthetube.Letitbeassumedthateveninplasticrangeconsideredinthisarticlethe

displacement of a point (u, v, w) are small enough so that the strain displacement relationships are in the elastic range.

Artkl 4.indd 60 10/21/2011 10:14:28 AM

61


(2)

are valid in the plastic range also. Letitbeassumedthatthematerialofthetubehasthearbitrarystressstrain

relationshipgivenby

τ=f(γ) (3)

which degenerates into the relationship

τ=Gγ (4)

intheelasticrange.Gbeingthemodulusofrigidity,fromequations(2)and(4)

(5)

If a stress function is taken such that

(6)

theequilibriumequationisgivenby

(7)

is automatically satisfied. Again from equation (6) the resultant shear stress is givenby

(8)

Figure1:GeometryoftheProblem

Artkl 4.indd 61 10/21/2011 10:14:31 AM

62


From equations (5) and (6)

(9)

Sincetheresultantshearstressτataboundaryisalongthetangenttotheboundary(Figure2)

(10)

i.e. =aconstantalongaboundary.SothedifferenceΔ of betweenthetwoboundariesofthetubeisconstantwhilegoingalongtheboundaryi.e.

≈|grad |t=q, aconstant (11)

wheretisthethicknessofthetubeatthepointconsidered.Sofromequation(8) and (11)

τ.t=constant (12)

whereτistheaverageshearstressandqisconstantalongthemeanboundaryis called shear flow.

Again considering an elementary mean contour length ds of the section (Figure3),thetangentialforceontheelementisqdsanditsmomentaboutanypoint O is

dT=qdsρsinβ (13)

whereρisthedistanceofthepointOfromtheelementandβistheanglemadebythelineoflengthρwiththetangenttothecontourelement[40].Forequilibriumofthetubetheappliedandtractivetorquesmustbeequali.e.,

(14)

Figure 2: Stresses on an Infinitesimal Element at the Boundary

Artkl 4.indd 62 10/21/2011 10:14:31 AM

63


But½ρsinβdsistheareaoftheshadedtriangleinFigure3.So,

T=2qA=2Atτ (15)

or

(16)

WhereAistheareaenclosedbythemeanboundary.

or

or

(17)

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)

(19)

Figure3:ResistingTorqueontheCross-sectionofaThinTube

Artkl 4.indd 63 10/21/2011 10:14:31 AM

64


whichshowsthatthefunctionalrelationshipbetween is same as

thatbetweenshearstressτandshearstrainasshowninFigure4.

Figure 5: Shear Stress-strain Curve with Elastic Unloading

Figure 4: Shear Stress-strain Curve

HenceforagiventorqueT,angleoftwistθandtheresidualangleoftwistθRcandirectlybeobtainedfromtheshearstress-straincurve.Whenthethicknesstofthetubeisconstantequation(19)becomes

(20)

However,iffromtherelationship(3)γcanbeexplicitlyexpressedas

γ=F(τ) (21)

then assuming that unloading is elastic (Figure 5),

Artkl 4.indd 64 10/21/2011 10:14:32 AM

65


so,

(22)

also,

(23)

Incaseofbi-linearrelationship(Figure6)givenby[30]

(24)

Where,τ0andγ0areyieldshearstressandstrainrespectivelyandαistheplasticmodulusofrigiditygivenbytheslopeofthestressstrainforτ≥τ0. From theabove,γmaybeexpressedexplicitlyas

So,τcorrespondingtoT,(T>T0)isgivenby

(25)

Fromequations(20).(22)and(23).TandθRmaybeexpressedas

Figure 6: Bi-linear Stress-strain Curve

Artkl 4.indd 65 10/21/2011 10:14:32 AM

66


(26)

writing,

(27)

(28)

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.

Artkl 4.indd 66 10/21/2011 10:14:32 AM

67


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)

Artkl 4.indd 67 10/21/2011 10:14:32 AM

68


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

springbackangleoftwist.4. Thevaluesofspringbackangleoftwistcanbeobtainedwithsamecase

whether the loading is kinematic or kinetic.

References

[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.

Artkl 4.indd 68 10/21/2011 10:14:32 AM

69


[3] Queener,C.andDeAngelis,R.J.(1968).“Elasticspringbackandresidualstressesinsheetmetalformedbybending”,TransactionsoftheASME,61, 757-768.

[4] Johnson,W.andSingh,A.N.(1980).“Springbackincircularblanks”,Metallurgia, 47, 275-280.

[5] El-Domiaty,A.andShabaik,A.(1984).“Bendingofworking-hardeningmetalsundertheinfluenceofaxialload”,JournalofMechanicalWorkingTechnology, 10, 57-66.

[6] Chu,C.(1986).“Elastic-plasticspringbackofsheetmetalssubjectedtocomplexplanestrainbendinghistories”,InternationalJournalofSolidsStructures, 22(10), 1071-1081.

[7] El-Megharbel,A.,El-Domiaty,A.andShaker,M.(1990).“Springbackandresidualstressesafterstretchbendingofworkingsheetmetal”,JournalofMaterialsProcessingTechnology,24,191-200.

[8] Chu, C. (1991). “The effect of restraining force on springback”,InternationalJournalofSolidsStructures,27(8),1035-1046.

[9] Schmoeckel,D.andBath,M. (1993).“Springbackreduction indraw-bendingprocessofsheetmetals”,AnnalsofCIRP,42(1),339-342.

[10] Tan,Z.,Li,W.andPersson,B.(1994).“Onanalysisandmeasurementofresidualstressinthebendingofsheetmetals”,InternationalJournalofMechanical Sciences, 36(5), 483-491.

[11] Morestin,F.,Boivin,M.andSilva,C.(1996).“Elastoplasticformulationusingakinematichardeningmodelforspringbackanalysisinsheetmetalforming”, Journal ofMaterials ProcessingTechnology,Vol. 56, 619-630.

[12] Zhang,Z.andHu,S.(1998).“Stressandresidualstressdistributionsinplanestrainbending”,InternationalJournalofMechanicalSciences,Vol.40(6), 543-553.

[13] Asnafi,N.(2000).“Springbackandfractureinv-dieairbendingofthickstainlesssteelsheets”,MaterialsandDesign,21,217-36.

Artkl 4.indd 69 10/21/2011 10:14:32 AM

70


[14] Kuwabara,T.(2007).“Advancesinexperimentsonmetalsheetsandtubesinsupportofconstitutivemodelingandformingsimulations”,InternationalJournalofPlasticity,23,385-419.

[15] Yi,H.K.,Kim,D.W.,VanTyne,C.J.andMoon,Y.H.(2008).“Analytical predictionofspringbackbasedonresidualdifferentialstrainduringsheetmetalbending”,Proc.IMechE,Vol.222(part2).PartC,J.MechanicalEngineering Science pages, 117-129.

[16] Bakhshi-Jooybari,M.,Rahmani,B.,Daeezadeh,V.andGorji,A.(2009).“Thestudyofspring-backofCK67steelsheetinV-dieandU-diebendingprocesses”,MaterialsandDesign,30,410-419.

[17] Wang,C.,Kinzel,G. andAltan,T. (1993). “Process simulation andspringback control in plane-strain sheet bending”, SheetMetal andStampingSymposium,SAESpecialPublications,944,45-54.

[18] Hsu,T.C.andShien, I.R. (1997).“Finiteelementmodelingofsheetformingprocesswithbendingeffects”,JournalofMaterialsProcessingTechnology, 63, 733-737.

[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.

[21] Esat,V.,Darendeliler,H.andGokler,M.I.(2002).“Finiteelementanalysisofspringbackinbendingofaluminiumsheets”,Materials&Design,23,223-229.

[22] Li,K. P.,Carden,W. P. andWagoner,R.H. (2002). “Simulation ofspringback”,InternationalJournalofMechanicalSciences,44(1),103-122.

[23] Ling,Y.E.,Lee,H.P.andCheok,B.T.(2005).“FiniteelementanalysisofspringbackinL-bendingofsheetmetal”,JournalofMaterialsProcessingTechnology, 168, 296-302.

[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.

Artkl 4.indd 70 10/21/2011 10:14:32 AM

71


[25] Al-Qureshi,H.A. (1999). “Elastic-plastic analysis of tube bending”,International Journal ofMachineTools andManufacturing”, 39, 87-104.

[26] Yang,H.andLin,Y. (2004).“Wrinklinganalysis for forming limitoftubebendingprocesses”, JournalofMaterialsProcessingTechnology,152, 363-369.

[27] Megharbel,A.,Nasser,G.A.andDomiaty,A.(2008).“Bendingoftubeand sectionmadeof strain-hardeningmaterials”, Journal ofMaterialsProcessingTechnology,203,372-80.

[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.

[29] Daxin,E.andLiu,Y.(2009).“Springbackandtime-dependentspringbackof1Cr18Ni9Tistainlesssteeltubesunderbending”,Materials&Design(Accepted). (oi:10.1016/j.matdes.2009.09.026).

[30] Dwivedi,J.P.,Singh,A.N.,Sohan,R.andTalukder,N.K.D.(1986).“Springbackanalysisintorsionofrectangularstrips”,InternationalJournalof Mechanical Sciences, 28(8), 505-515.

[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.

[32] Dwivedi,J.P.,Upadhyay,P.C.andTalukder,N.K.D.(1990).“Torsionalspringbackinsquaresectionbarsofnon-linearwork-hardeningmaterials”,InternationalJournalofMechanicalSciences,32(10),863-876.

[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.

Artkl 4.indd 71 10/21/2011 10:14:33 AM

72


[35] Dwivedi, J. P., Shah, S.K.,Upadhyay, P.C. andTalukder,N.K.D.(2002).“Springbackanalysisofthinrectangularbarsofnon-linearwork-hardeningmaterials”,InternationalJournalofMechanicalSciences,44,1505-1519.

[36] Choubey,V.,Gangwar,M.andDwivedi,J.P.,“Springbackanalysisintorsionofthintubes”,JournalofMechanicalEngineering,JMechE,UiTM(Comunicated).

[37] Chakrabarty, J. (2006). “Theory of plasticity”, 3rd Edition, Elsevier Butterworth-Heinemann, MA, USA.

[38] Timoshenko,S. andGoodier, J.N. (1987). “TheoryofElasticity”, 3rd

Edition, McGraw-Hill, NY, USA.

[39] Johnson,W. andMelIor, P. B. (1962). “Plasticity forMechanicalEngineers”,D.VanNostrand,London,UK.

[40] Talukder,N.K.D.andChakraborti,R.K.(1965).“Torsionofbarsunderplasticdeformations”.JournaloftheInstitutionofEngineers(India).PartME-2,XLV(3),67-71.

Artkl 4.indd 72 10/21/2011 10:14:33 AM

73


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

Artkl 4.indd 73 10/21/2011 10:14:33 AM

74


so

(A2)

(A3)

Whereτisthemaximumshearstressanditactsalong,sofromequations(A2) and (A3)

(A4)

So,theshearstress-straincurveintheelastoplasticregimecanbeobtainedfromtensilestress-straincurvebyusingtheaboverelationship.

TableA:MechanicalPropertiesofTubeMaterials

Modulusof Modulusof Yieldshear Plasticmodulus elasticity (E) rigidity (G) stress (t0) of rigidity (a) a/G (N / mm2) (N / mm2) (N / mm2) (N / mm2)

2.1×105 8.24×104 108 6.1×103 0.074

Artkl 4.indd 74 10/21/2011 10:14:33 AM

75


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

Artkl 4.indd 75 10/21/2011 10:14:34 AM

76


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).

Artkl 4.indd 76 10/21/2011 10:14:34 AM

Documents

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