ANALYTICAL MODEL FOR PREDICTION OF FORCE ... Int. J. Mech. Eng. & Rob. Res. 2012 M K Chudasama and H...

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Int. J. Mech. Eng. & Rob. Res. 2012 M K Chudasama and H K Raval, 2012

ANALYTICAL MODEL FOR PREDICTION OF FORCEDURING 3-ROLLER MULTIPASS CONICAL

BENDING AND ITS EXPERIMENTALVERIFICATION

M K Chudasama1* and H K Raval2

*Corresponding Author: M K Chudasama,[email protected]

Conical sections and shells are produce using 3-roller conical bending machines. In this conicalbending process, the blank is given static bending by placing the blank over the bottom rollersand lowering the top roller. The rollers are rotated than to get roll bending action. Static bendingof the plate requires larger force and it is done in multiple stages to lower down the value ofrequired force [Gajjar et. al., 2008]. The total deflection of the top roller required is divided insteps to get the multipass bending. The machine is designed on the basis of maximum reactionforces during bending. In this paper an analytical model is proposed for the prediction of bendingforce during the multiple pass 3-roller conical bending. Multipass bending experiments arecarried out to validate the developed model. Experimental results do not match exactly with theanalytical results. So a correction factor for each pass has been found out and applied to theanalytical results. The corrected analytical results match with experimental results quitesatisfactorily. The model derived can be effectively used to study the effect of various parameterson the bending force and can be helpful to the researchers working in this area.

Keywords: 3-roller conical bending, Force prediction, Experimental validation, Internal bendingmoment, External bending moment

ISSN 2278 – 0149 www.ijmerr.comVol. 1, No. 3, October 2012

© 2012 IJMERR. All Rights Reserved

Int. J. Mech. Eng. & Rob. Res. 2012

1 Government Engineering College, Surat, Gujarat, India.2 S V National Institute of Technology, Surat, Gujarat, India.

Research Paper

INTRODUCTIONFor construction of various structures as wellas integral part of machines various conicalsections are widely used. Such conicalsections are manufactured by various methodsand 3-roller conical bending process is one

such process. It consists of two bottom rollersand a top roller. Metal plates with specifiedcontours are rolled without decrease inthickness to get the desired cone angle. Theplate undergoes plastic deformation and it iscold forming process and hence it has higher

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dimensional accuracy. 3-roller conical bendingprocess has four stages, (a) static bending,

(b) forward rolling, (c) backward rolling, and(d) unloading. In the first stage the plate is kept

Figure 1: 3-Roller Conical Bending Setup: Schematic Representation

LargerRadius

BottomRollers

W

TopRoller

ForwardPass Direction

Blank

Top Roller

Top RollerDisplacement, ‘U’

BottomRollers

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between top roller and bottom rollers asshown in Figure 1 and the top roller is givenvertical displacement to get the requiredbend. In next stage the bottom rollers aredriven using motors in forward direction to getthe roll bending of the plate. Similarly therollers are driven in reverse direction to getbetter dimensional accuracy of the finalproduct. The bent plate is than unloaded byraising the top roller.

Mechanics involved in the conical bendingis very complex. It can be observed that therewill be 3-dimensional force pattern at the rollerplate interface during the conical bending. Inthis paper an attempt is made to get theanalytical model for bending force predictionfor such complex mechanics.

Various researchers have worked for thedevelopment of mathematical models forcylindrical bending process (Wang et al.,1993). Hua et al. have proposed a method ofdetermining the plate internal bendingresistance at the top roll contact for multi-passfour-roll thin-plate bending operations (Huaet al., 1995). For continuous single-pass four-roll thin plate bending a model was proposedby Hua and Baines, considering theequilibrium of the internal and external bendingmoment at and about the plate-top roll contact(Baines et al., 1997). Lin et al. had consideredvarying radius of curvature for the platebetween the rollers and proposed amathematical model to simulate themechanics in a steady continuous bendingmode for four-roll thin plate bending process(Hua and Lin, 1999). Hua and Lin alsoinvestigated Influence of material strainhardening on the mechanics of steadycontinuous roll and edge-bending mode in

the four-roll plate bending process (Hua andLin, 2000).

For continuous multi-pass bending of conefrustum on 3-roller bending machines with non-compatible (cylindrical) rollers, Gandhi et al.had reported the formulation of springback andmachine setting parameters (Gandhi et al.,2008 and 2009). They incorporated the effectof change of flexural modulus during thedeformation in the formulation to study the effecton springback prediction. For plane strain flowof sheet metal subjected to strain rate effectsduring cyclic bending under tension Sanchezpresented an elastic–plastic mathematicalmodel (Sanchez, 2010). He also includedBauschinger factors in the model for stressreversal. Chudasama and Raval have reportedanalytical model of force prediction for singlepass in static bending stage during 3-rollerconical bending process (Chudasama andRaval, 2011).

The roll bending process is used for years,it can be observed from the literature reviewedthat conical bending process is untouchedarea as far as force prediction is concerned.Even in the industries the normal practice ofplate roller bending still heavily depends uponthe experience and the skill of the operator.Working to templates, or by trial and error, yetremains a common practice (Gandhi, 2009).For conical bending investigation related toconical bending as far as machine setting isconcerned is done by Gandhi et al. (2008 and2009), but few references are available toaddressed the problem of bending forceprediction for roll-bending so far as theknowledge of the authors is concerned.Considering this, in this paper an attempt ismade to develop force prediction model for

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multipass 3-roller conical bending processwhich will be useful to the researchers tounderstand the complex mechanics of theprocess.

ANALYTICAL MODELHua et. al. have concluded that it is very difficultto have a single mathematical model thattakes into account all the complexities of thebending process (Hua et al., 1995). As 3-rollerconical bending is also a bending process itis very difficult to get a mathematical modelfor of 3-roller conical bending process takingall the factors into consideration. A realisticsimplification is thus necessary.

So following simplifying assumptions havebeen made for formulation of the relation forforce prediction:

Assumptions• Plate is always having line contact with the

roller which is parallel to roller axis duringthe process.

• The forces acting during the bending arelarger than the self weight of the plate. Sothe self weight of the plate is neglected.

• The shift of the neutral plane is zero, i.e., itis considered to be at the center line of theplate thickness.

• Frictional force at the bottom roller and theplate interface is always tangent to the rollersurface.

• Rollers are assumed to be rigid. Rollermaterial and plate material is assumed tohave stable microstructure throughout thedeformation process.

• Deformation occurs under isothermalconditions and ‘E’, i.e., Modulus of Elasticityremains constant during the process.

• As cone angle considered being small.

• As cone angle considered being smallblanks for the cone frustum bending areselected such that ((w/t) > 8) and hence planestrain conditions maintained (Marciniak,1992; and Wang et al., 1993).

• Plane section remains plane, before andafter the bending. Blank thickness (t)remains constant during and after thebending.

• Baushinger effect is neglected. Blank ishaving uniform/constant radius of curvaturefor the supported length of the blankbetween two bottom rollers.

• Further simplifying assumptions arediscussed as and when required during theformulation.

Based on the above assumptions thebending force equation can be derived byequating the external bending momentrequired to bend the plate and internal bendingmoment induced in the plate.

External Bending Moment

In 3-roller conical bending process first step isbending of the plate by lowering down the toproller. In this step it is required to exert forceon the top roller to lower it, which will bend theplate. This vertical force exerted on the toproller will give the external bending momentrequired to bend the plate.

External bending moment over the plateexerted by the rollers can be derived byconsidering the geometry of the setup duringbending. The force pattern during conicalbending is complicate and needs to besimplified. During the bending process therewill be reaction forces on the bottom rollerbecause of the top roller load.

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In the derivation of external bendingmoment over the plate, forces in vertical planepassing through the three rollers andperpendicular to the top roller axis isconsidered. If axial forces on the bottom rollersare resolved, they will not have any componentalong the vertical plane, as bottom rollers areinclined only in horizontal plane as explainedearlier. So axial force on the bottom rollers willnot affect the derivation of the external bendingmoment derived earlier by Chudasama andRaval (2011). Reaction on the top roller willhave three components in mutuallyperpendicular directions. Axial force on the toproller will also not affect the external bending

moment as it is not inclined in the vertical plane.In the present analysis vertical component ofthe force on top roller, ‘P’, is considered forthe calculations.

Figure 2 gives schematic of blank and rollerarrangement for static bending. Equation forexternal bending moment can be derived asreported (Chudasama and Raval, 2011),

sincos

1

sin

sin1

2 1

1

ra

rxPMtotal

tan*tan 1 xUrx

Ur 1costan*sin1 ...(1)

Figure 2: Schematic of Blank and Roller Arrangement for Static Bending

a

x

P

U r1

It is to be stated here that external bendingmoment equation for multipass is similar to thesingle pass equation but the values of theparameters ‘U’ and ‘’ will be different formultipass.

Internal Bending Moment forMultipass

When the top roller is lowered down andexternal force is exerted to bend the plate, the

material of the plate will resist the deformationof the plate. There will be elastic as well asplastic deformation of the plate and hencethere will elastic as well plastic resistance tothe deforation. Combined effect of elastic andplastic resistance will give the internal bendingmoment and can be calculated as below:

The bending moment can be split into anelastic contribution and plastic contribution as

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discussed earlier and can be calculated byChudasama and Raval (2011).

Mtotal

= Melastic

+ Mplastic

...(2)

dyydyy xplasticxelastic 22 ...(3)

Hill’s non-quadratic yield criteria (Hill, 1979)for plane strain deformation in plastic regionfor isotropic material is,

xx 3

2and

3

2 ...(4)

where, F = Anisotropic constant,

x = Uniaxial stress,

= Effective stress,

= Effective strain,

x = Axial strain.

The power law material behaviourconsidering initial strain å0, is assumed in theplastic region.

So, nK 0...(5)

From this,

n

xx K

3

2

3

20

(Gandhi et al., 2009) ...(6)

n

xx K

3

2

2

30 ...(7)

For elastic region x =

x E, where E =

Elastic constant ...(8)

Considering plane strain bending, E isreplaced by E’ for plain strain conditions and

21

EE (where, = Poison’s Ratio)

...(9)

For deciding the limits of the integration,the elastic zone thickness is considered uptoa distance of y

ep from the neutral plane.

Inserting the values of x and E elastic and

plastic bending moment in Equation (3), weget,

epy

xtotal dyyE

M0 21

2

dyyKn

x

t

yep

3

2

2

32 0

2/

...(10)

For bending operation,

bR

E

y

...(11)

where, = bending stress, y = distance ofoutermost fibre from neutral axis, E = elasticityconstant, and R

b = bend radius.

From this

x

b

x

R

y

E

...(12)

In Single pass bending, initial radius of theblank will be infinite, i.e., flat plate is considered.But in case of multipass bending, blank willhave some initial radius except for first pass.So the radius of curvature for multipass will be

0

11

fRR , where R = final radius of the blank,

Rf0 = initial radius of the blank. Radius of

curvature is inverse of the bend radius.

So, bf RRR

B111

0

...(13)

ByR

y

E b

xx

...(14)

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Now considering only elastic moment,

epy

elastic dyyyBEM0

2 ...(15)

epy

elasticyBEM

0

3

32

...(16)

3

3

2epyBE ...(17)

Now considering plastic moment,

dyyKMn

x

t

yplastic

ep

3

2

2

32 0

2/

...(18)

dyyRR

yKt

y

n

foep

2/

0

11

3

2

2

32

...(19)

dyyByKt

y

n

ep

2/

03

23 ...(20)

From Equations (17) and (19)

3

3

2epTotal yBEM

2/

03

23

t

y

n

ep

dyyByK

...(21)

where E

Ryep

(Chudasama and Raval,

2011)

Expression for Bending Load P

Equating Equations (1) and (21), andrearranging,

3

3

2epTotal yBEM

2/

03

23

t

y

n

ep

dyyByK

sincos

1

sin

sin1

1

1

ra

rxP

tan*tan 1 xUrx

Ur 1costan*sin1 ...(22)

Xrarx

dyyByKyBE

P

t

y

n

epep

sincos1

sinsin

1

3

23

32

1

1

2/

03

...(23)

where, X = (x – tan (r1 – U) + (x * tan –

r1(sin * tan + cos – 1) – U))

Equation (23) gives the bending forcerequired to get the required bend radius formultipass conical bending. It is required tointegrate the equation for plastic bendingmoment. In the present case of integrationthere is no standard formula available for it.Also it is not possible to solve it by using anyconventional method like substitution. So it isto be evaluated using Numerical Method.

In Equation (23), there is no term whichreflects the effect of bottom roller inclinationon the bending force P. Bottom roller inclinationis set by setting the bending radius at front endand rear end on 3-roller conical bendingmachine. The relation between rolling radiusand bottom roller inclination is given inEquation (24) (Gandhi, 2009). If bottom roller

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inclination is zero than both the radius will besame, with no top roller inclination, thiscondition is of cylindrical bending.

FR

FR

RR

AA 2/1 sin

cos

1

2tan

...(24)

where, = bottom roller inclination,

AF, A

R = Center distance between bottom

rollers at front and rear end respectively

= Top roller inclination, in the present caseit is zero

= Cone angle

RF, R

R = Bending radius at the front end and

rear end respectively

To get the value of P, i.e., top roller load,from the above Equation (23) along with the

material properties and geometricalparameters of the machine setting, the valueof coefficient of friction at roller plate interfaceis required. Value of angle can be calculatedfrom the geometrical configurations of themachine setting.

EXPERIMENTATIONFor verification of the developed analyticalmodel for prediction of force during multipasscone frustum bending, experimentation iscarried out. Details of experimentation arediscussed in subsequent sections.

Experimental Setup

To validate the analytical models of forceprediction developed in earlier section it isrequired to do the experiments. 3-rollerconical bending machine is used to carry out

Figure 3: 3-Roller Conical Bending Machine Setup; and Arrangement of Load Cell toMeasure the Bending Force

(a)

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the bending force over the top roller. Theseload cells are attached to the online dataacquisition system which gives the forcevariation during the continuous roll bendingprocess.

Material Selection

Structural steel FE 410 WA is widely used forthe various structural applications and is easilyavailable in Indian commercial market. Hence,structural steel of material grade FE 410 WAas per IS 2062 (2006) is selected for conefrustum bending experiments. Mechanicalproperties and chemical composition of thestructural steel FE 410 WA is given inTables 1 and 2 respectively.

the experiments. It is shown in Figure 3. Thebottom rollers of the machine are driven bytwo independently driven electric motors.The rollers are attached with the motors usinguniversal joints as shown in Figure 3a.Bottom rollers can be set at angle as thespherical roller bearings are provided at theroller supports. The top roller can be lowereddown using power screws as the bearingblocks for top rollers can slide verticallyFigure 3b. The setup is facilitated with theload cells at the top roller as well bottom rollerbearings to measure the reaction in 3mutually perpendicular planes. Figure 3bshows the location of load cell to measure

Table 1: Mechanical Properties of Structural Steel Material Grade FE 410 WA(IS 2062, 2006)

<20 20-40 >40

410 250 240 230 23

Tensile Strength Minimum (MPa) % Elongation at Gauge Length 5.65, MinimumYield Stress, Minimum (MPa)

Figure 3 (Cont.)

Power Screw to Movethe Top Roller

Load Cell to Measurethe Bending Force

Top Roller BearingBlock (Can Slide

Vertically Up or Down)

Top Roller

(b)

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Blank Geometry

Plate thickness for the plate to be bend areselected as per the availability in thecommercial market. The plates were taken andthe thicknesses were measured over the lengthat 6 points on each side using vernier calliper.The average of the reading were taken as thethickness of the plates. The thickness ‘t’ of theplates are: 5.84 mm, 7.87 mm, 8.85 mm, 11.84mm and 13.97 mm. Blank dimensions aretaken from the earlier work done by Gandhiregarding parametric investigation of 3-rollerconical bending process (Gandhi, 2009).Generators are marked on the blank as perthe requirements of the experiments. Figure 4shows the photograph of the one of developedblanks (plate thickness ‘t’ = 5.84 mm, = 1.86°)used for the cone frustum bendingexperiments.

Experimental Procedure

As per the experimental planning presentedin this section, cone frustum bendingexperiments were performed for theverification of the developed analyticalmodels. First the bottom rollers are inclined

at required inclination by sliding the bearingblocks at required distance for the inclinationof = 1.86°. The blank is than loadedbetween top and bottom rollers. To performthe bending the top roller is than lowereddown at required distance for the first passusing the power screws. The bending forcesat the top roller bearing are measured withthe help of load cells and the data acquisitionsystem. Then bottom rollers are rotated inone direction for forming the radius over theentire length of the blank in the forward passfollowed by the reverse rotation of rollersduring the reverse pass. Reverse pass isnecessary for forming of the initial blank lengthto the desired radius which is not formedduring forward pass and for the uniformity ofthe radius. After reverse pass the plate isunloaded by raising the top roller upward.After completion of this first pass. Again theplate is kept between the rollers and the toproller is lowered further for second pass. Thedistance travelled by the top roller in secondpass will be more than the distance travelledin the first pass. Again forward and reverse

Table 2: Chemical Composition, Structural Steel of Material Grade FE 410 WA(IS 2062, 2006)

Carbon Manganese Sulphur Phosphorous Silicon Carbon Equivalent

0.23 ± 0.02 1.5 ± 0.05 0.05 ± 0.005 0.05 ± 0.005 0.40 ± 0.03 0.42

Note: In percentage maximum values.

Figure 4: Blank Before Bending (Thickness ‘t’ = 5.84 mm, = 1.86°)

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rolling is done and the cycle repeats. Inpresent case the final dimensions of bendingis achieved in 5 passes and the bending forcerequired during each pass is measured to getthe experimental bending force (Figure 5).

Experiments have been performed as perthe plan by different thickness plates. Data offorce required for the roll bending have beenacquired through the Data Acquisition System(DAS) as discussed earlier. Data has beenprocessed and the sample results for platethickness 0f 7.87 mm thick plate are shown inTable 3.

VALIDATION OF THEDEVELOPED ANALYTICALMODELAnalytical model for prediction of force during3-roller conical bending process for multipassstatic bending process have been developed inanalytical model section. Analytical calculationswere done using Simpson's 3/8th Rule fornumerical integration. Experiments of 3-rollerconical bending have been performed to verifythe analytical models and have been reported inexpermentation section. In this section developedbending force models are validated using thedata obtained from experiments.

Model for Multipass Bendig

Analytical model of force prediction formultipass 3-roller conical bending process hasbeen developed in analytical model section.The Equation (23) is rewritten below:

Xrarx

dyyByKyBE

P

t

y

n

epep

sincos1

sinsin

1

3

23

32

1

1

2/

03

where, X = (x – tan (r1 – U) + (x * tan –

r1(sin * tan + cos – 1) – U))

The values of various parameters involvedin the above equation are substituted and theanalytical results were obtained. They havebeen compared. There was large differencein the values of the analytical force andexperimental force. It can be due to thesimplifying assumptions made during thederivation as well the uncertainties in theexperimentation. But it was observed that thetrend of the experimental results was matchingwith the analytical results. So a correction factor

Figure 5: Plate After Bending

1. 20 899

2. 25 1007

3. 30 681

4. 35 931

5. 40 1176

Table 3: Sample Result Tablefor t = 7.87 mm for = 1.86°

PassNumber

Top Roller PositionU (mm)

Top RollerForce (N)

Data obtained through the experimentationis to be used for the validation of the developedanalytical model of force prediction, in thesubsequent section.

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was found out. The ratio of experimental valueand analytical value was taken for eachthickness. Than the average of the ratio fordifferent thicknesses were taken. The averagevalue of the ratio has been taken as thecorrection factor and applied to the analyticalresults. Graphs of Experimental force andcorrected analytical force have been plottedfor each thickness as shown in Figure 6.

It can be observed from the Figure 6a to6e that as thickness of the plate increases,the value of required bending force alsoincreases. It is reflected in both, correctedanalytical bending force as well asexperimental bending force. It is quite well

represents the actual case. It can also beobserved that corrected analytical results arehaving good agreement with the experimentalresults except for the first pass and 5.84 mmthickness in Figure 6a. That can be due tosome error occurred during theexperimentation or some unknown factor.Otherwise the error observed between theexperimental and corrected analytical resultsis in the range of ±20%, which is quitesatisfactory. So the analytical modeldeveloped here can be utilised for the forceprediction with the correction factor for eachpass. Though it is not giving the exact results,it can be further modified to give better results.

Figure 6: Variation of Bending Force P, with Respect to Number of Passesfor Plate Thickness

(a) t = 5.84 mm

(c) t = 8.85 mm

(b) t = 7.87 mm

(d) t = 11.84 mm

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CONCLUSIONAn analytical model for multipass 3-rollerconical bending process has been derived.External bending moment required to bend theplate has been equated to internal bendingmoment developed in the plate to get therequired formulation for the bending force ‘P’.To validate the developed analytical model,experimentation has been carried out. 3-rollerconical bending machine was set for particularvalue of bottom roller inclination. Differentthickness plates were bent using the 3-rollerconical bending setup and force required toget the required bend radius have beenmeasured. For the same geometrical as wellas material parameters, analytical bendingforce have been calculated using Equation(23). The value of the analytical bending forceand experimental force had large difference.This can be due to the simplifying assumptionsmade. It can be due to the uncertainties in theexperimentations also. But it was observedthat the trend of the analytical results wasmatching with the experimental results. So acorrection factor has been obtained for eachpass for different thicknesses. The correctionfactor was applied to the analytical results andcorrected analytical force results have beencompared with the experimental results. They

were in good agreement and the error foundto be ±20%.

Though the developed model is not givingthe exact results for the bending force, it canbe used satisfactorily with the correction factorfor each pass. Further the model can bemodified by removing the simplifyingassumptions like considering the varyingradius of curvature of bending instead ofconstant radius, considering axial forces duringbending, etc. The developed analytical modelcan be useful to the developers of the 3-rollerconical bending machines as well as theresearchers working in this area.

ACKNOWLEDGMENTThis research work is carried out with thefinancial support of Department of Scienceand Technology, Government of India, underSERC scheme.

REFERENCES1. Baines K, Hua M, Cole I M and Rao K P

(1997), “A Formulation for Determiningthe Single-Pass Mechanics of theContinuous Four-Roll Thin Plate BendingProcess”, Journal of MaterialsProcessing Technology , Vol. 67,pp. 189-194.

2. Chudasama M K and Raval H K (2011),“An Approximate Bending ForcePrediction for 3-Roller Conical BendingProcess”, Journal of Material Forming,DOI 10.1007/s12289-011-1087-y

3. Firat M (2007), “Computer Aided Analysisand Design of Sheet Metal FormingProcesses: Part II – DeformationResponse Modeling”, Materials andDesign, Vol. 28, pp. 1304-1310.

(e) t = 13.97 mm

Figure 6 (Cont.)

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4. Gandhi A H, Gajjar H V and Raval H K(2008), “Mathematical Modelling andFinite Element Simulation of Pre-BendingStage of Three-Roller Plate BendingProcess”, MSEC 2008, doi:10.1115/MSEC_ICMP2008-72454, pp. 617-625.

5. Gandhi A H (2009), “Investigation onMachine Setting Parameters for 3-RollerConical Bending Machine forSpringback”, Ph.D. Thesis, S V NationalInstitute of National Institute of Technology.

6. Gandhi A H, Shaikh A A and Raval H K(2009), “Formulation of Springback andMachine Setting Parameters for Multi-Pass Three-Roller Cone FrustumBending with Change of FlexuralModulus”, International Journal ofMaterial Forming, Vol. 2, pp. 45-57.

7. Hill R (1979), The Mathematical Theoryof Plasticity, Oxford University Press.

8. Hua M and Lin Y H (1999), “Large DeflectionAnalysis of Elastoplastic Plate in SteadyContinuous Four-Roll Bending Process”,International Journal of MechanicalSciences, Vol. 41, pp. 1461-1483.

9. Hua M, Sansome D H and Baines K(1995), “Mathematical Modeling of theInternal Bending Moment at the Top Roll

Contact in Multi-Pass Four-Roll Thin-PlateBending”, Journal of MaterialsProcessing Technology , Vol. 52,pp. 425-459.

10. Kim H, Nargundkar N and Altan T (2007),“Prediction of Bend Allowance andSpringback in Air Bending”, ASMETransactions, Vol. 129, pp. 342-351.

11. Lin Y H and Hua M (2000), “Influence ofStrain Hardening on Continuous PlateRoll-Bending Process”, InternationalJournal of Non-Linear Mechanics, Vol.35, pp. 883-896.

12. Marciniac Z, Duncan J L and Hu S J(1992), Mechanics of Sheet MetalForming, Butterworth-Heinemann.

13. Moreira L P and Ferron G (2004),“Influence of the Plasticity Model in SheetMetal Forming Simulations”, Journal ofMaterials Processing Technology ,Vols. 155-156, pp. 1596-1603.

14. Sanchez L R (2010), “Modeling ofSpringback, Strainrate and BauschingerEffects for Two-Dimensional Steady StateCyclic Flow of Sheet Metal Subjected toBending Under Tension”, InternationalJournal of Mechanical Sciences, Vol. 52,pp. 429-439.

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APPENDIX

w = Width of the blank in mm.

t = Thickness of the plate in mm.

M = Bending moment in N-m.

P = Vertical load at the top roller and bending plate interface in N.

a = Borizontal distance of the bottom roller centers in mm.

x = Balf the horizontal distance of the bottom roller centers in mm.

Q = Normal force exerted by the plate on the bottom roller at roller plate interface in N.

= Angle between frictional force and horizontal plane at the roller plate interface in radians.

U = Vertical distance travelled by the top roller for first stage of static bending in mm.

E = Young’s modulus in N/mm2.

K = Strength coefficient in N/mm2.

n = Strain hardening exponent.

r1

= Radius of bottom roller in mm.

R = Radius of curvature of the bent plate in mm.

y = Distance of fiber from neutral plane in mm.

= Coefficient of friction at roller plate interface.

= Strain.

= Stress in N/mm2.

= Poisson’s ratio.

yep

= Distance of the fiber upto which elasticity E is constant in mm.

= Curvature of the bend plate between bottom rollers, mm–1.

* = Strain at yield point.

E* = The ratio of modulus of elasticity to s.

te

= Thickness of elastic layer in mm.

0

= Strain of the strip mid-line.

= Effective strain.

= Effective stress.

= Bottom roller inclination.

AF, A

R= Center distance between bottom rollers at front and rear end respectively.

= Top roller inclination, in the present case it is zero.

= Cone angle.

RF, R

R= Bending radius at the front end and rear end respectively.

Nomenclatures

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