K. Slimani , M. Zaaf , and H. Bendjama Simplified Modelling of Tandem Cold Rolling

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PACSnumbers:06.60.Vz,07.50.Tp,81.20.Hy,81.40.Ef,81.40.Lm,81.70.Bt,83.50.Uv

SimplifiedModellingofTandemColdRolling

K.Slimani^*,**,M.Zaaf^*,andH.Bendjama^**

*BadjiMokhtarUniversity, FacultyofEngineeringSciences,

LaboratoryofMetalMaterialsForming(LMF2M), B.P.12,23000Annaba,Algeria

**ResearchCenterinIndustrialTechnologiesCRTI,

P.O.Box64,Cheraga16014Algiers,Algeria

Inthis paper,acalculationtechnique forsolvingtheproblemofregulating interstandtension inatandemcoldrollingisproposed.Basedontheslices’

method,theproposedtechniquedevelopsacomputationalmodelforasingle stand,andthengeneralizesitforfivestands.Theeffectivenessofthistech- niqueisevaluatedusingexperimentaldataacquiredfromtandemrollingmill of IMETALsteelcomplexof El-Hadjar-Algeria.Bytakinginto accountthe elasticityoftherollsandusingNewton’smethod,thedevelopedmodelcanbe used to calculate successfully the tensions’ correction of the five stands.

Compared withtheLAM3software,theobtained resultsindicated thatthe proposed technique is effective and can be used to produce better perfor- manceoftandemcoldrolling.

Key words:modelling,tandemcoldrolling,slices’method,elasticity, New- ton’smethod.

Запропоновано методикурозрахункудлявирішеннязадачіреґулювання натягуміжклітямиутандемномухолодномувальцюванні.Наосновіме- тоди зрізів дана методика розвиває обчислювальний модель для однієї кліті,апотімузагальнюєйогодляп’ятьохклітей.Ефективністьцієїме- тодикиоцінюєтьсяпривикористанніекспериментальнихданих,одержа- нихнатандемномупрокатномустанікомплексуIMETALSteelEl-Hadjar- Algeria.БеручидоувагипружністьвальцівівикористовуючиНьютонову Correspondingauthor:KheireddineSlimani

E-mail:[email protected]

Citation:K.Slimani,M.Zaaf,andH.Bendjama,SimplifiedModellingofTandem ColdRolling,Metallofiz.NoveishieTekhnol.,40,No.11:1509–1520(2018), DOI:10.15407/mfint.40.11.1509.

Ôотокопирование разрешено только

в соответствии с лицензией НапечатановÓкраине.

1509

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методу,розроблениймодельможебутиуспішнозастосованимприобчис- ленні натягудляп’ятьохклітей.Óпорівняннізпрограмнимзабезпечен- нямLAM3одержанірезультатипоказали,щозапропонована методикає ефективною іможебутивикористаноюдляпідвищенняпродуктивности тандемногохолодноговальцювання.

Ключові слова: моделювання, тандемне холодне вальцювання, метода зрізів,пружність,Ньютоноваметода.

Предложена методикарасчётадлярешениязадачирегулированиянатя- жения междуклетями втандемной холодной прокатке(слитков содно- временнûмманипулированием).Наосновеметодасрезовданнаяметоди- каразвиваетвûчислительнуюмодельдляоднойклети,азатемобобщает её для пяти клетей.Эффективность этой методики оцениваетсяпри ис- пользовании экспериментальнûх даннûх, полученнûх на тандемном прокатном стане комплекса IMETAL Steel El-Hadjar-Algeria. Принимая вовниманиеупругостьвальцовииспользуяметодНьютона,разработан- наямодельможетбûтьуспешно примененапривûчислениинатяжения для пятиклетей.Посравнению спрограммнûм обеспечениемLAM3по- лученнûерезультатûпоказали,чтопредложеннаяметодикаэффективна иможетбûтьиспользованадляповûшенияпроизводительноститандем- нойхолоднойпрокатки.

Ключевые слова: моделирование, тандемная холодная прокатка, метод срезов,упругость,методНьютона.

(ReceivedFebruary12,2018;lastversion—October9,2018)

1.INTRODUCTION

Rolling mill processplays animportant role inassuringhigh perfor- mance, safety and reliability insteel industry.A typical rolling mill consistsofanumberofrollingstandsalignedinline.Itspurposeisto convertslabsofmetalintoastripofmetalofdesireddimensions.How- ever,interstandtensioncontrolremainsaproblemduetostronginter- actions ofrolling stands. So, theassuranceof ahigh product quality providesthechallengefortechnicalinnovationsinmodelling,automa- tionandcontrol.

Infact,tandemcoldrollingisahighlycomplex,nonlinearandmul- tivariableprocessbecauseitallowsalarge coiltoberolledintomulti- plestands,whichmakesitscontrolamajorchallengeforengineering.

Many researchers havepaid great attentiontoproduce anacceptable product by proposed basic controllers [1–3]. Other works have also beenproposed toofferimprovement androbustnesstodisturbanceof theprocess[4,5].

Themodellingofarollingprocesscanbeperformedforvariouspur- poses. Various models have been studied and developed to predict a rollingtorque,force,stressesandspeedsinordertosizethestandsand

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adjusttherollgapforoptimalrollingoperation[6,7].Thepreliminary knowledgeofthemechanicalrollingpropertiesandquantitiesisneces- sary toensure the optimal energy consumption. Actually, the Finite ElementMethod(FEM)iswidelyusedinthisfield;greatprogresshas been made to develop efficient models tostudy the propagation, i.e., theenlargementofthehot-rolledstripandtheedgeshapetoachievean optimized pass design [8–10]. Unfortunately, the models based on FEM require ahigh computationtime andpowerful machines, which makes itnot advisablefortheindustrialistswho mainlylookforeffi- cientandfastonlinemodelstopilottheiroperations.Toovercomethis difficulty,theslicesmethodisconsidered.Itisanimportantandeffi- cient method usedforcoldrollingof thinsheets[11–13].Its applica- tions incoldrolling havebeenrealizedtotake intoaccounttheshear, theelasticityofthebandandtheelasticityoftherolls[14–18].

Themainobjectiveofthisworkistodevelopamodelforcalculating thestressesofastandandtheireffectsonthenextstand.Thismodel allows correcting the interstand tensions, which represent a serious problem with tandem rolling. Based on the slices method and using Newton’stechnique,wecandevelopamodelforthefivestandsofthe process.Itisshownthattheobtainedmodelcanbeappliedeffectively for the correction of interstand tensions and obtaining acceptable speeds. Bycomparingwith LAM3, theresults demonstratetheeffec- tivenessoftheproposedmodel.

2.MATERIALSANDMETHODS 2.1.TandemColdRollingProcess

Tandem cold rolling process consists of a number of rolling stands alignedinline,wherethethicknessofasteelsheetisreducedbypass- ing itthrough rolling stands (see Fig. 1). This reduction causedby a

Fig.1.Typicalfivestandtandemcoldmill[2].

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very large compression forcebetween theworking rolland thesheet, enlarges the strip. There are different types of parameters to be ac- quired, with reliable instrumentation in tough operating conditions suchas:therolling force,theinterstandtension,thestrip thickness, andthespeeds[4,5].

ThespeedofarolledstripleavingastandNmustbeidenticaltothe entryspeedofthestandN+1.Ifthesetwospeedsdiffer,alongitudi- nal force, generally called interstand tension, will result inside the rolledstripcanintroducevariationsinthethicknessand/orthewidth ofthesheet,thusdeterioratingtheproductquality.

Toimprove theefficiency andthequalityoftheprocessoperation, wehavedevelopedanefficientmodel thatallowsustocontrolthein- terstandtensionsandthuscorrecttherotationspeeds.

2.2.ModellingMethod

Theproposedmodelisbasedontheslicesmethod.UnderMATLABen- vironment,thismethodiscomputedtakingintoaccounttheelasticity oftherolls.Inthefollowing,wecalculate,respectively,themathemat- ical models of one stand, five stands, and the corresponding correc- tionsonthespeeds.

Inthestandardrolling model,thestressandstrainonlydepend on thelongitudinalvariablex.Horizontalequilibriumandthehypothesis

1, _xz _xx dh

dx << ε << ε  reducetheKarmanequationasexpressedbelow[6, 11,13]:

1

1 3

( ) ,

d dh

h dx dx

σ = σ − σ − τ (1)

wherehisthehalfthicknessoftheslab,σ₁,σ₃andτdenotethelongitu- dinal(xdirection),normal(zdirection)andshearstresses,respectively.

This equilibriumequationhas tobesupplemented by twoconstitu- tiveequations:

–forthematerialbehaviourrelating(σ₁−σ₃)to ε₃ (orε₃intheplas- ticcase);

–forthefrictionlawrelatingtheshearstressτtothevelocityv.

Moreprecisely,theshearstressisassumedsmallandforaverywide sheetε₂=0,sothat

3

0 0 0 0 0 ,

0 0

ij

−ε 

 

ε =  

 ε 

 

1 2

3

0 0

0 0 ,

0 0

ij

σ 

 

σ =  σ 

 σ 

 

3 1

0

log h. ε = −ε = h

Theviscoplasticorplasticflowrulefirstgivesσ₂asafunctionofσ₁and

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σ₃fromtheplanestrainconditionandthen(σ₁−σ₃).Differentconsidera- tionscanbeusedsuchas:isotropicornot,plasticorviscoplastic,associ- ated flow rule or not. Similarly, thefriction lawcan betaken as: rate- independent;CoulomborTresca,orrate-dependent;Norton–Hoff[6,7].

Neglectingthehypothesisofrigidstandsandassumingthatthede- formedrollcanbelocallyassimilatedtoarollofradiusR′≥R,whereR′

is theHitchcock radius,itiscalculated bythefollowingformula [13, 14](J.H.Hitchcock,1935):

16(1 2)

1 ,

2 _e 2 _s R R F

bE h h

′=  + π− ν − 

where E is theYoung’s modulus, νthe Poisson’s ratio, Fthe rolling force,he theinputthickness,hs theoutputthicknessandbthewidthof theband.TherollingforceFiscalculatedby:

0 z .

F R d

−α

= σ

∫

′ θ

The flowchartof thecomputed modelbasedon slicesmethod is illus-

Fig.2.Frameworkoftheproposedmodel.

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trated inFig. 2. Wemustfirst determinethe neutralpointby firing method. Foragivenpositionoftheneutralpointxn,thelongitudinal stress σ₁ is calculated by integrating the Karman equation using Runge–Kutta method. Then, we adjustthe value of xn by the secant method,whichappearstocoincidewiththevalueσ₁.Itresultsaneffi- cientandfastmodelthatmakesiteasytotestdifferenthypotheses.

For atandemrolling processwithfive stands,which representour casestudy,thecalculationofeachstandinvolvesboundaryconditions [11];theimposedtractionsareassumedtobehomogeneousontheinlet andoutletsections,i.e.,t1(i)andt2(i).Obviously,

t2(i)=t1(i+1)withi=1,2,3,4, (2) wheret1 andt2 denotetheinputandtheoutputtensions.

Therefore,theboundarytensionsareimposedas:∆t₁(1)=∆t₂(5)=0.

Furthermore,theslicesmethodcalculatestheneutralpointofeach stand, which makes itpossible toobtain thespeeds v1(i)and v2(i).To achievethis,wemustrespectthefollowingcondition:

2( )i v1(i 1).

v = + (3)

We start an iterative calculation of the interstand (1), (2), which will then be applied to the other interstands. First, we calculate the valuesoft1(i)andt2(i),andthus,thecorrespondingcorrections∆tthat canbeappliedtoobtainanearrequiredprecision.Weassumethenob- tained,atacertainstage,theapproximationsv2(1)andv1(2),whichare thefunctionsoft1(2)andt2(2).UsingNewton’smethod,weobtain:

2 2

2 2 2 1 2 1 1

1 2

(1) (1)

(1) (1) (1) (1) (1) (2) (2),

(1) (1)

v v

v v v t t v v

t t

∂ ∂

+ ∆ = + ∆ + ∆ = + ∆

∂ ∂

1 1

1 1 1 1 2

1 2

(2) (2)

(2) (2) (2) (2) (2),

(2) (2)

v v

v v v t t

t t

∂ ∂

+ ∆ = + ∆ + ∆

∂ ∂

2(1) 2(1) [ 1(2) 1(2)] 0

v + ∆v − v + ∆v =

2 2 1 1

2 1 2 1 1 2

1 2 1 2

(1) (1) (2) (2)

(1) (1) (1) (2) (2) (2)

(1) (1) (2) (2)

v v v v

v t t v t t

t t t t

∂ ∂ ∂ ∂

+ ∆ + ∆ − − ∆ − ∆

∂ ∂ ∂ ∂

2 2 1 1

2 1 1 2 1 2

1 2 1 2

(1) (1) (2) (2)

(1) (2) (1) (1) (2) (2)

(1) (1) (2) (2)

v v v v

v v t t t t

t t t t

∂ ∂ ∂ ∂

− = + ∆ + ∆ − ∆ − ∆

∂ ∂ ∂ ∂

Knowingthat ∆t₁(1) =0 and ∆t₂(1)= ∆t₁(2),so:

2 1 1

2 1 2 2 2

2 1(2) 2(2)

(1) (2) (2)

(1) (2) (1) (1) (2)

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v v v

v v t t t

t t t

∂ ∂ ∂

− = ∆ − ∆ − ∆

∂ ∂ ∂

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2 1 1

2 1 2 2

2 1 2

(1) (2) (2)

(1) (2) (1) (2).

(1) (2) (2)

v v v

v v t t

t t t

∂ ∂  ∂

− =∂ − ∂ ∆ − ∂ ∆

By usingthesamecalculation steps fortheother interstands,(2)–

(3),(3)–(4),and(4)–(5),weobtainthefollowingequations:

2 1 1

1 2 2 2

2 1 2

(1) (2) (2)

(2) (1) (1) (2)

(1) (2) (2)

v v v

v v t t

t t t

∂ ∂  ∂ 

∂ − ∂ = ∂ − ∂ ∆ −∂ ∆ (4)

2 1 1 2

1 2 2 2 2

2 1 2 1

(2) (3) (3) (2)

(3) (2) (2) (3) (1),

(2) (3) (3) (2)

v v v v

v v t t t

t t t t

∂ ∂  ∂  ∂ 

∂ − ∂ = ∂ − ∂ ∆ −∂ ∆ +∂ ∆ (5)

2 1 1 2

1 2 2 2 2

2 1 2 1

(3) (4) (4) (3)

(4) (3) (3) (4) (2),

(3) (4) (4) (3)

v v v v

v v t t t

t t t t

∂ ∂  ∂  ∂ 

∂ − ∂ = ∂ − ∂ ∆ −∂ ∆ +∂ ∆ (6)

2 1 1

1 2 2 2

2 1 2

(4) (5) (2)

(5) (4) (4) (4).

(4) (5) (2)

v v v

v v t t

t t t

∂ ∂  ∂ 

∂ − ∂ = ∂ − ∂ ∆ −∂ ∆ (7) Inthematrixform ∆ =t M⁻¹∆v,thepreviousequationsbecome:

2 2 2 2

(2) (2) (3) (4)

∆

=

∆

 

 

∆

 

t t t t

2 1 1

2 1 2

2 2 1 1

1 2 1 2

2 2 1 1

1 2 1 1

1 2 1

(1) (2) (2)

0 0

(1) (2) (2)

(2) (2) (3) (3)

(2) (2) (3) (3) 0

(3) (3) (4) (4)

0 (3) (3) (4) (4)

(4) (4) (5)

0 0

(4) (4) (5)

 

 



=

∂ ∂ ∂

− −

∂ ∂ ∂

∂ ∂ −∂ ∂



∂ ∂ ∂ −∂

∂ ∂ −∂ −∂

∂ ∂ ∂ ∂

∂ ∂ −∂

∂ ∂ ∂





 

v v v

t t t

v v v v

t t t t

v v v v

t t t t

v v v

t t t

−1

 ×



1 2

(2) (1) (3) (2) (4) (3) . (5) (4)

v v

 

 

×

−

 

 

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The relation between velocity and tension makes it possible to determine theinfluencecoefficients: ∂v₁ ∂t₁, ∂v₂ ∂t₁, ∂v₁ ∂t₂, ∂v₂ ∂t₂,

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thatthenallowsdeducingthe∆taswellastherealtensions t′ = + ∆t t. 3.RESULTSANDDISCUSSION

In this section, the algorithm presented earlier will be demonstrated and discussed. We start with a validation of our single-stand model compared toLAM3 software. LAM3 is atool used for finite element thermomechanicalcalculations;itiscurrentlyappliedby ArcelorMit- talresearcherstosimulaterollingoperations.Thecomparisonismade usingthedataandresultspresentedin[13].Thedatasetofcoldrolling employedare:he =1.957mm,hs =1.370,k=0.4,σ_e=150MPa,σ_s=150 MPa,σ₀=600MPa,R=250mm,ω=4rad/s.

Figure 3 shows the calculation of the longitudinal and normal stresses ofcold rolling intheelastoplastic case. The obtainedresults showthatthecalculatedmodelisreliablecomparedtoLAM3.

After validating the obtained model, testing the model for five stands was established. The data set was collected from the tandem cold rolling of the IMETAL steel complex of El-Hadjar-Algeria. The validation of the proposed model is carried out using a steel coil DX51D(A9M)correspondingtothepresumedFePO2Ggrade.

Afterdeterminingofanoperatingpointusingatypicalproduction program, therolling millandstrip parameters usedinthis studyare presentedinTable1.

Exploitation of the experimentaldata must identify the Hollomon hardeninglaw (σ = εA ⁿ) [13].Therefore,itwasaquestionofidentify- ingAandn.Foreachstand,thecalculatedvaluesarelistedinTable2.

Fig.3. Stress of cold rolling in the elastoplastic case: a—longitudinal, b—

normal.

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Figure4shows thecalculationresults ofthe firststandbefore the regulation of the tension, the same calculations are repeated forthe otherstands.Thecalculatedspeedsbeforeregulationarepresentedin Table 3. It is clear that the speed of arolled strip leaving anearlier standisnot identical toits entryspeed intotheadjacent downstream stand. Our goal is therefore to determine for each stand, the corre- spondingtensioncorrectiontoobtainidenticalspeedsandtensionsbe- tweentheoutputofastandandtheinputofthenextstand.

Thestepstofollowtocorrectthespeedsare:

– build matrix C = v2 − v1 by the input and output speeds of each stand;

–buildmatrix(M)bytheinfluencecoefficientsofeachstand;

–replacethenewtensiont′=t+∆t,where∆tisthecorrectionoften- sions,ttheoldtensionandt′thenewtension.

Toensureproductquality,wemusttakeintoaccountthefollowing corrections:

(

^MPa

)

^41.2402^{64.3154 .}

238.5046 137.8939 t

 

 

∆ = −

− 

 

 

Figure 5 shows the calculation results after tension regulation. It simultaneouslygivestheevolutionofflowconstraints,normal,longi- tudinalandtangentialstressesaswellasslidingspeedsforeachstand.

TABLE2.Hollomonhardeningcoefficients.

Standnumber 1 2 3 4 5

A 663.3 753.4 807.2 843.2 894.5

n 0.125 0.064 0.027 0.024 0.022

TABLE1.Rollingparameters.

Stand h_input,mm h_output,mm R_cyl,mm Ω,rad/s T_e(tf) T_s(tf)

1 02 1.53 281 30.6 30 22

2 1.53 1.07 264 56.08 22 19

3 1.07 0.78 270 64.07 19 14

4 0.78 0.54 281 70.37 14 11

5 0.54 0.37 281 76.02 11 9

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Themainnotesare:

– thethicknessvariesfrom2mm(inputofthefirst stand)to0.37 mm(outputofthefifthstand);

–fortheflowstress,theflowofmaterialisremarkableatthebegin- ning of the reduction at the neutral point because the rolling force reachesitsmaximumatthispoint,thenbecomesalmostconstant,i.e., the dislocation densityreaches its saturationpointwhich is near the neutralpoint,thenabalanceiscreatedbetweenrestorationandannihi- lation according to thedeformation effect. Untilthe fifth stand, the curverepresentsalmostperfectplasticbehaviour:

Fig.4.Calculatedparametersinthefirststand.

TABLE3.Speedsvaluesbeforeregulation.

v₁,m/s 7.006 11,356 13,868 15,077 19.29 v₂,m/s 9.1511 16,237 19,024 21,777 28.15 v_n,m/s 8.531 14.805 17.298 19.7739 21.361

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– the longitudinal and normal stresses evolve in the same way by forming ahill atthe top corresponding to the neutral point(Vg = 0) wherethefrictionchangesdirection;

– the friction is calculated by Norton–Hoff law where the coeffi- cients are adjustmentfactors so that thecalculatedeffortis equal to themeasuredeffort;

–theslidingspeedisnegativeattheneutralpointinput(Vg=0)and changesdirectionuntiltheoutput.

FromthecalculatedvaluesoftherollingspeedslistedinTable4,it Fig.5.Calculatedparametersforfivestands.

TABLE4.Correctedspeeds.

v₁,m/s 7.9193 10.574 15.869 18.911 20.901 v₂,m/s 10.552 15.849 18.907 20.892 27.061

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is seen clearly that the output speedsof each stand is approximately identical to the input speed of the next stand. So, the calculation speedsobtainedbyourmodelshowthattherefinementsproposedmake itpossibletosolvetheproblem:₂v i( )=v i₁( +1 .)

4.CONCLUSION

Theworkpresentedinthispaperisfocusedontheimprovementofthe classical simplifiedmodelsofrollingprocess.Forthispurpose,asim- plified model that takes into account the plastic deformations of the plates with hardening, the elastic deformation of the rolls, and the regulationofthetensionsbetweenthestandshasbeendeveloped.Itis basedontheslices’methodwiththeconsiderationofrollelasticity.It resultsasimpleandeffectivemodelthatisvalidatedbyapplicationson realdataacquiredfromtandemcoldrollingmillofIMETALEl-Hadjar.

The obtained model cansuccessfully correct the interstand tensions.

Comparing with LAM3, the results of the proposed model have con- firmeditsreliabilitythatwerecommendittomanufacturers.

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