Bending is a common metal working process used in sheet-metal forming, such as parts ofautomobiles, aircraft and ships. In addition, bending is used in many sheet metal forming, suchas deep drawing and stamping processes. Bending process is usually followed by some elasticrecovery upon unloading, called spring-back. A static approach is described to determine newformulas for the bending force, the shift of neutral-axis and the value of spring-back, in V-diebending process. Due to friction force between the sheet-metal and the die edge, a variableneutral-axis shift is deduced in the elastic zone of the bending length. The amount of shift is,rapidly, decreased with the distance starting from the die edge, along the bending length. Basedon the reflexive action in elastic zone, and the residual stresses, in elastic-plastic zone, theamount of spring-back is derived. As a case study, the obtained expressions are applied tocompare the values of bending force and the spring-back in, previously, published experimentalstudy on V-die bending conducted on two different die-angles 90° and 120°. The comparisonshows that the values of bending force and the spring-back, obtained by the present work, are,in general slightly under-estimating.
INTRODUCTIONBending is the metal working process by whicha straight length is transformed into a curvedlength. During the bending operation, the outersurface of the sheet metal is in tension andthe inside surface is in compression. The strainin the bent material increases with decreasing
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1 Higher Technological Institute-Mechanical Engineering Department, PO Box.4, 6th of October-Giza-12585, Egypt.
the radius of curvature. Due to the friction forcebetween the sheet and the die, the neutral axismoves toward the inner surface. Amitabha andKumar (1995) stated that the position of neutralaxis depends on the radius and angle of bend.The neutral axis shift s towards the center ofcurvature, usually a shift of 5-10% of the
Research Paper
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
thickness is assumed for the calculation ofstrain and stock length. In the present study,an expression for determination of the locationof neutral axis is derived considering thecoefficient of friction.
In bending process the plastic deformationis followed by some elastic recovery uponunloading called spring-back. Spring-backdepends on the modulus of elasticity, strengthof material, sheet thickness, die radius andheat treatment of material. Precise predictionof sheet spring-back is very important in diedesign.
In bending production there are severaltechnical problems, such as prediction ofspring-back and punch load (bending force).In dealing with spring-back problem, threeapproaches have been commonly used:analytical methods, experimental methods andnumerical methods. Purely theoretical studiesor purely experimental analysis or both of themare encountered. Some researchers prepareinformation just for the sake of analyticalanalysis of solid mechanics, and some othersuse available finite element package programslike ABAQUS and ANSYS.
A survey of the previous research work onspring-back prediction and compensation indie manufacturing industry has beeninvestigated by Alfaid and Xiaoxing (2009).Based on Hill’s yielding criterion and planestrain condition, an analytical model for spring-back is proposed by Zhang et al. (2007), whichtakes into account the effects of contactpressure, the length of bending arm betweenthe punch and die, transverse stress, neutralsurface shifting and sheet thickness thinningon the sheet spring-back of V-die bending.
Florica et al. (2007) used Finite ElementMethod (FEM) to evaluate the spring-back,as well as the stress and strain state in thepart before and after the spring-back by usingABAQUS Standard. The Stribeck frictionmodel is investigated by Maziar and Zaidi(2009) to predict the spring-back behavior ofAA6061-T4 sheets. The amount of spring-back is predicted by Grizelj et al. (2010) fora high strength steel plate using FEM for withtwo different tool geometries for V-diebending.
The spring-back and spring-go phenomenain a V-die bending process has beeninvestigated by Thipprakmas (2010). Heinvestigated, also, the effects of processparameters, including radius and height of thepunch. Spring-back behavior was investigatedat various temperatures ranging from RT to300 °C and various rolling directions byperforming experimental tests and ANSYSFEA software (Barouzeh and Mondali, 2011aand 2011b). The results indicate that theamount of spring-back decreases andformability increases with increasing thetemperature. An algorithm for inverse spring-back modeling using bending theory and FEmodeling is presented by Behrouzi et al.(2008). Some researchers have suggestedthe bending force based on bending moment.Many parameters such as material propertiesand die profile affect the bending moment. Ananalytical approach cannot correctly describethe amount of moment in the bending process.Thus, die designers use a simple equation todetermine the bending force (Farsi andBehrooz, 2011). Thus, a simple andreasonable formula for bending force issuggested in the present work.
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
DETERMINATION OFBENDING FORCEConsidering the free body diagram of half thesheet metal, under punching, Figure 1, threeequilibrium equations can be written:
cossin1
1
2
PDM ...(4)
where = 2 is the die angle.
Frequently, the sheet material used for theexperiments is of a low carbon type, for whichthe stress-strain diagram may be consideredas elastic/perfect plastic, with yield stress
y,
Figure 2.
Figure 1: Free Body Diagramof the Half Sheet Metal
N – Q cos + Q sin = 0 ...(1)
02
cossin P
QQ ...(2)
0csc2
QD
QM ...(3)
where = (R + t) – (R – t) csc
The equilibrium Equations (1)-(3), can besolved for N, Q and M in terms of bending forceP, punch and die dimensions. The reactiveforce Q and the bending moment M areobtained as:
cossin2
PQ
Figure 2: Stress-Strain Diagramfor an Elasto-Plastic Material
The minimum value of the bending force Pis sufficient to make yielding of the sheet cross-section, under the punch tip. The stress
Figure 3: Stress Distributionat the Yielded Sheet Section
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
distribution along the sheet thickness of thissection is shown in Figure 3 with compressivestrain at upper half thickness and tensile stressat lower half.
The bending moment, at this section, isdefined as:
42wtM yu ...(5)
where,
y is the yield stress of the sheet material,
w is the sheet width.
t is the sheet thickness.
Introducing the expression of Mu Equation
(5), into Equation (4), the minimum bendingforce is obtained as;
cossin1
22
D
wtP y
...(6)
The term in prances depends on thecoefficient of friction and the punch angle . Inpublished literatures, this term is consideredas die opening factor, which equals 1.3 forrectangular section D.
Considering the bend length L, threedifferent zones of elastic/plastic intermittentcan be distinct, starting from position of thedie reactive force Q (Figure 4);
Elastic zone, AB with length a, which canbe determined from the following equation
Q
wta y
6
2 ...(7)
• Elastic/plastic zone, BC with length b whichcan be determined as:
aQ
Mb u ...(8)
• Narrow plastic zone, CD which increasesas the value of the punch force increasesover than the minimum value given byEquation (6)
VARIATION OF NEUTRALAXIS ALONG THE BENDINGLENGTHConsider a section in the sheet metal at adistance x from the die edge (0 x a) asshown in Figure 5, the normal force N andbending moment M are expressed as:
N = Q
2
tQQxM ...(9)
The normal stress, caused by the aboveactions, N and M can be written as:
ywt
M
wt
N2
12 ...(10)
Where 2
0t
y
At the neutral axis, the value of the normalstress equals to zero ( = 0). Substituting of Nand M in Equation (9) into Equation (10) theshift of the neutral axis from the center line ofthe sheet y is obtained in the form:
txt
y
26
...(11)
Instead of considering constant value ofneutral axis shift ratio (5%-15%) to thicknesst, estimated by different authors, Equation (11)indicates that the shift ratio of the neutral axishas its maximum value 16.7% at x = 0 anddecreases as x increases. Variation of theneutral axis along the bending length in elasticzone for different coefficient of friction is shownin Figure 6.
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
SPRING-BACK ANALYSISTo determine the amount of spring back in asheet bending process, the stress-distributions at different zones, shown in Figure7, are considered, in detail during the punchingprocess and after the punch release. In Figure7, the stress distributions over different
Figure 4: Elastic/Plastic Intermittent Figure 5: Location of Neutral Axis
Figure 6: Variation of Neutral Axis, Along the Bending Length in the ElasticZone (0 x a)
sections along the bending length duringpunching are detailed. In the elastic part a, allthe stresses are elastic, with linear variationalong the plate thickness. In the elastic-plasticpart, b, an outer increasing plastic region isdeveloped, due to the increase of the bendingmoment, while the core remains elastic. The
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
variation of the elastic core along the sheetthickness is not linear. The bending momentat a section x* may be expressed in terms ofthe limit elastic moment, M
y, as:
yMa
xxM
*1* ...(12)
where
6
2wtM y
y
...(13)
0 x* b ...(14)
The reacting bending moment caused bythe shown stress distribution, Figure 7, can bededuced to be
31
4
1*
22 wtxM y ...(15)
where t is the thickness of the elastic core,
Equating the bending moment expressionin Equation (12) with the reactive bendingmoment, given by Equation (15), the elasticcore coefficient, , can be expressed as:
a
x *1232 ...(16)
Considering bending moment expressions(5) and (13), the length of the elastic-plasticzone, b, is related to that of the elastic zone,a, as:
3
2
u
y
M
M
ba
a...(17)
Considering Equation (16), it may be notedthat:
at x* = 0, = 1 limit of elastic zone.
at x* = b = a/2, = 0 end of elastic-plasticzone.
Figure 7: Stress-Distributions Over Different Sections of the Bending Length
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
The main amount of spring-back is causeddue to the totally reversed stresses in theelastic zone, a, and the induced residualstresses in the elastic-plastic zone, b.
Spring-Back in Elastic Zone
The differential equation relating the deflection,y, with the bending moment M as functions ofbending length, x, is given as, (Farsi andBehrooz, 2011):
0
2
2
xEI
xM
dx
yd...(18)
The integration of Equation (18) leads todetermination of the slop angle (dy/dx) whichis the spring-back due to the reversed stressesin elastic zone, a. Hence, the elastic spring-back,
e, can be obtained as:
a
e dxxEI
xM0
2 ...(19)
where M(x) is expressed in Equation (9),
12
3wtxI is the second moment of cross-
section which may varied due to variation ofsheet width, w, or thickness, t.
Spring-Back in Elastic/Plastic Zone
The residual stresses over a section atdistance x*, in elastic-plastic zone, areobtained from superposition of existedstresses, shown in part b of Figure 8, and theelastically reversed stresses, due to releaseof bending force. The maximum value of thereversed stress is obtained, using Equation(12), as shown in Figure 9.
Figure 8: Residual Stresses in Elastic/Plastic Zone
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
Figure 9: Residual Stresses Over the Elastic Core in Elastic/Plastic Zone
a
xy
*1max ...(20)
It may be observed that the spring-back is,only, occurred in the elastic core with thicknesst. Where is given, as a function of x*, inEquation (16). The maximum value of theresidual stress at the elastic core is obtainedas:
a
xyres
*11max ...(21)
which may be expressed in terms of onevariable, as:
2
max 32
11 yres ...(22)
The spring-back, through the elastic-plasticzone, b, can be obtained as the followingexpression:
a
e dxxEI
xMp 0
**
*2
...(23)
where max
2
6
2* res
txM
12
*2tw
xI
Considering the expression (16), relating as function of x*, the integration on the righthand side of Equation (23) can beperformed, which lead to the determinationof spring-back in the elastic-plastic zone inthe form:
Et
bp
ye
2
3
...(24)
The total spring-back is obtained bysummation of expressions (19) and (23) as:
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
CONCLUSIONDepending on a statical approach, an elastic-plastic analysis is presented to determinegeneral expressions for bending force, theamount of shift of the neutral axis and thespring-back, in V-die bending process. Insteadof assuming constant amount of neutral axisshift of (5% to 15% of the sheet thickness), avariable expression is deduced, in the elasticzone of bending length. The shift amount
decreased, along the bending length, startingwith rapid decrease at the die edge. A bendingforce expression, depending on the die-angleand the coefficient of friction, is deduced. Thedeep analysis shows that the reflexive action,in elastic zone and the residual stresses, inelastic-plastic zone, have the main reasons forspring-back action. The obtained expressions,for bending force and spring-back, arecompared with published experimental results,
Spring-back = pe
e ...(25)
A CASE STUDYAs a case study, the present analysis forbending force and spring-back, is applied toV-bending of sheet metal whichexperimentally studied by Farsi and Behrooz(2011). The material properties, used inexperiments, are given in Table 1, which is
that of a low carbon type. The sheet metalsample, the punch and die dimensions aregiven in Table 2.
The given data in Tables 1 and 2 areintroduced into the expressions (6), (19) and(23) to get the values of bending force andspring-back. The obtained results are shownin Table 3, in comparison with thatexperimentally obtained by Farsi and Arezoo(Farsi and Behrooz, 2011).
Width (w) = 50 mm Tip Radius (R) = 1 mm Width = 18 mm
Thickness (t) = 0.95mm
Table 2: Sample, Punch and Die Dimensions
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Int. J. Mech. Eng. & Rob. Res. 2012 Manar Abdelhakim Eltantawie and Eldesoky Elsayed Elsoaly, 2012
(Farsi and Behrooz, 2011), on V-die bendingwith two different die- angles, 90° and 120°.The compared results are in a goodagreement, within 9% error.
REFERENCES1. Alfaid M F and Xiaoxing L (2009),
“Determination of Spring-Back in SheetMetal Forming”, The Annals of ‘Dunareade Jos’, pp. 129-134, University of Galati,Fascicle V, Technologies in MachineBuilding.
2. Amitabha G and Kumar M A (1995),“Manufacturing Science”, Affiliated East-West Press PVT LTD., New Delhi.
3. Barouzeh M R and Mondali M (2011a),“Spring-Back Investigation at Warm V-Bending Conditions by Numerical andExperimental Methods”, Proc.International Conference on Trends inMechanical and Industrial Engineering,pp. 185-190, Bangkok.
4. Barouzeh M R and Mondali M (2011b),“Numerical and ExperimentalInvestigation on Springback Behavior atWarm V-Bending Conditions”, WorldAcademy of Science, Engineering andTechnology, Vol. 60, pp. 285-290.
5. Behrouzi A, Shakeri M and Dariani B M(2008), “Inverse Analysis of Springback inSheet Metal Forming by Finite ElementMethod”, Proc. International Conferenceon Engineering Optimization, pp. 1-5, Riode Janeiro, Brazil.
6. Farsi M A and Behrooz A (2011), “BendingForce and Spring-Back in V-Die-Bendingof Perforated Sheet-Metal Components”,Journal of Brazil Society MechanicalScience and Engineering, Vol. 33,No. 1, pp. 45-51.
7. Florica M G, Gheorghe A, Lucian L andVasile A C (2007), “Spring BackPrediction of the Bending Process UsingFinite Element Simulation”, Proc. 7th
International Multdisciplinary Conference,pp. 261-266, Baia Mare, Romania.
8. Grizelj B, Cumin J and Grizelj D (2010),“Effect of Spring-Back and Spring-Forward in V-Die Bending of St1403Sheet Metal Plates”, Journal for Theoryand Application in MechanicalEngineering, Vol. 52, No. 2, pp. 181-186.
9. Maziar R and Zaidi M R (2009), “Effect ofFriction Models on Stress Distribution ofSheet Material During V-BendingProcess”, World Academy of Science,Engineering and Technology, Vol. 32,pp. 654-659.
10. Thipprakmas S (2010), “Finite ElementAnalysis on V-Die Bending Process”,Finite Element Analysis, pp. 407-428,ISBN: 978-953-307-123-7.
11. Zhang D, Cui Z, Chen Z and Ruan X(2007), “An Analytical Model forPredicting Sheet Springback AfterV-Bending”, Journal of Zhejiang UniversityScience A, Vol. 8, No. 2, pp. 237-244.
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