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Journal of Materials Processing Technology 190 (2007) 189–198

Simulation of springback variation in formingof advanced high strength steels

Peng Chen b, Muammer Koc a,∗a NSF I/UCR Center for Precision Forming (CPF) and Department of Mechanical Engineering,

Virginia Commonwealth University (VCU), Richmond, VA, USAb Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA

Received in revised form 23 February 2007; accepted 27 February 2007

bstract

Variations in the mechanical and dimensional properties of the incoming material, lubrication and other forming process parameters are theain causes of springback variation. Variation of springback prevents the applicability of the springback prediction and compensation techniques.ence, it leads to amplified variations and problems during assembly of the stamped components, in turn, resulting in quality issues. To predict theariation of springback and to improve the robustness of the forming process, variation simulation analysis could be adopted in the early designtage. Design of experiment (DOE) and finite element analysis (FEA) approach was used for the variation simulation and analysis of the springbackor advanced high strength steel (AHSS) parts. To avoid the issues caused by the deterministic FEA simulation, random number generation wassed to introduce uncertainties in DOE. This approach was, then, applied to investigate the effects of variations in material, blank holder force and

riction on the springback variation for an open-channel shaped part made of dual phase (DP) steel. This approach provides a rapid and accuratenderstanding of the influence of the random process variations on the springback variation of the formed part using FEA techniques eliminatinghe need for lengthy and costly physical experiments. 2007 Elsevier B.V. All rights reserved.

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eywords: Springback variation; Variation simulation; Design of experiment; F

. Introduction

During the past two decades, the automotive industry haseen in a quest to achieve low-mass vehicles due to governmentegulations and consumer demands for light, fuel-efficient andow-emission vehicles. In addition, requirements for improvedafety, performance, and comfort features continue puttingressure on the automotive manufacturers especially in an envi-onment where there are ever-increasing global competitionnd continuous cost reduction demands. As an alternative toluminum and magnesium alloys, advanced/ultra high strengthteels (AHSS) have been proven to be a proper choice of mate-ial for various body and structural automotive parts meeting the

equired low-mass, affordable cost and increased performanceequirements. However, wide application of high strength steeln many potential auto body and structural parts is still limited

∗ Corresponding author. Tel.: +1 804 827 7029.E-mail address: mkoc@vcu.edu (M. Koc).

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924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.jmatprotec.2007.02.046

element analysis; Computer experiment; Advanced high strength steel

ue to challenges in terms of formability, springback, joiningnd die life issues.

Springback is an elastically driven change in shape and formf a part upon unloading after a part is formed. The concernsbout springback and quality control grow among auto-makerss the use of high strength–low weight materials increases [1].arts made of AHSS demonstrate more springback than partsade of mild steel do. Moreover, the experience with forming

f AHSS materials is limited compared to mild steel. Variationf springback prevents the applicability and use of springbackrediction and compensation techniques. Hence, it leads tomplified variations and problems during assembly of compo-ents, and in turn, results in quality issues. Thus, understanding,ccurate characterization, prediction, control and reduction ofpringback and its variation have become very crucial in termsf decreasing development times and reducing scrap rate in mass

roduction to achieve cost effective fabrication of AHSS parts.

Numerous studies during the last 40 years have attempted toetermine the controlling factors in springback and find ways toeduce it. An early study by Baba and Tozawa [2] focused on

1 Proce

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90 P. Chen, M. Koc / Journal of Materials

he effect of stretching a sheet by a tensile force, during or afterending, in minimizing springback. Other studies investigatedhe role of process variables on springback. Zhang and Lee [3]howed the influence of blank holder force, elastic modulus,train hardening exponent, blank thickness and yield strengthn the magnitude of the final springback strain in a part. Gengnd Wagoner [4] studied the effects of plastic anisotropy and itsvolution in springback. They developed a constitutive equationor 6022-T4 aluminum alloy using a new anisotropic hardeningodel and proved that Barlat’s yield function is more accurate

han other yield functions in their case.Some researchers evaluated FE simulation procedures in

erms of their springback prediction accuracy. Mattiasson etl. [5], Wagoner et al. [6], Li et al. [7] and Lee and Yang [8]ound that FEM simulations of springback are much more sen-itive to numerical tolerances than forming simulations are. Lit al. [9] investigated the effects of element type on the spring-ack simulation. Yuen [10] and Tang [11] found that differentnloading scheme will affect the accuracy of the springbackrediction. Similarly, Focellese et al. [12] and Narasimhan andovell [13] pointed out that different integration scheme willlso influence the result of springback simulation. In 1999,ark et al. [14] and Valente and Traversa [15] attempted to

ink dynamic explicit simulations of forming operations to staticmplicit simulations of springback. It was proved that this tech-ique is very effective for the springback simulation. Li etl. [16] explored a variety of issues in the springback simu-ations. They concluded that (1) typical forming simulationsre acceptably accurate with 5–9 through-thickness integrationoints for shell/beam type elements, whereas springback anal-sis within 1% numerical error requires up to 51 points, andore typically 15–25 points, depending on R/t, sheet tension

nd friction coefficient. (2) More contact nodes are necessaryor accurate springback simulations than for forming simulation,pproximately one node per 5◦ of turn angle versus 10◦ recom-ended for forming. (3) Three-dimensional shell and non-linear

olid elements are preferred for springback prediction even forarge w/t ratios because of the presence of persistent anticlas-ic curvature. For R/t > 5.6, shell elements are preferred sinceolid elements are too computation-intensive. For R/t < 5.6, non-inear 3D solid elements are required for accurate springbackrediction.

Most of the research efforts on springback focused on theccurate prediction and compensation of springback. The issuef springback variation was seldom concerned. Moreover, nonef the studies in the area of springback prediction touchedpon the variation simulation of springback. As lead timesre shortened and materials of high strength–low weight aresed in manufacturing, a fundamental understanding of thepringback variation has become essential for accurate andapid design of tooling and processes in the early design stage.s far as the variation simulation is concerned, most avail-

ble references are related to the assembly processes [17–21].

n 1997, Liu and Hu [17] summarized two variation simula-ion approaches for sheet metal assembly. They are: (1) direct

onte-Carlo simulation, which is popular and good for all kindsf random process simulation, and (2) mechanistic variation

samF

ssing Technology 190 (2007) 189–198

imulation, which is especially designed for the assembly pro-ess (welding) simulation based on the assumption of linearlasticity.

Generally speaking, Monte-Carlo simulation is effective forny kind of non-linear process. However, since large number ofEA simulations is required (more than 100 for reliable results),

t is not practical and too costly to apply it for a complicatedase, such as metal forming. Therefore, some researchers [22]pplied DOE techniques in their FE simulation to effectivelyeduce the amount of simulations. However, since the FEA sim-lation is deterministic (no random error), replicate observationsrom running the simulation with the same inputs will be identi-al. Despite some similarities to physical experiments, the lackf random error makes FEA simulations different from physi-al experiments. In the absence of independent random errors,he rationale for least-squares fitting of a response surface is notlear [23]. The usual measures of uncertainty derived from least-quares residuals have no obvious statistical meaning [23,24].ccording to Welch et al. [25], in the presence of systematic

rror rather than random error, statistical testing is inappropri-te. For deterministic FEA simulation, some statistics, including-statistics, have no statistical meaning since they assume thebservations include an error term which has mean of zero andnon-zero standard deviation. Consequently, the use of step-ise regression for polynomial model fitting is not appropriate

ince it utilizes F-statistic value when adding/removing modelarameters [24].

In an effort to solve the above problem, McKay et al. [26]ntroduced Latin hypercube sampling which ensures that eachf the input variables has all portions of its range presented.acks et al. [23,27] proposed the design and analysis of com-uter (DACE) method to model the deterministic output as theealization of a stochastic process, thereby providing a statis-ical basis for designing experiments for efficient prediction.leijnen [28] suggested incorporating substantial random error

hrough random number generators. Therefore, it is natural toesign and analyze such stochastic simulation experiments usingtandard techniques for physical experiments. Some researcherse.g., Giunta et al. [29,30] and Venter et al. [31]), have alsomployed metamodeling techniques such as RSM for modelingeterministic computer experiments which contain numericaloise. This numerical noise is used as a surrogate for randomrror, thus allowing the standard least-squares approach to bepplied. However, the assumption of equating numerical noiseo random error is questionable.

In this work, random number generator was used to ensurehe correctness of the regression model. Assuming a Gaussianistribution, the uncertainty was introduced by random numbereneration for controlled factors at different level and uncon-rolled variables according to their mean and range. By usinghis method, the effects of variations in material (mechanicalroperties) and process (blank holder force and friction) onhe springback variation were investigated for an open-channel

haped part made of DP steel. The variations in stamping processre introduced in Section 2 of the paper. In Section 3, measure-ents of springback are defined. Section 4 is the validation of theE model using existing experimental results in the literature.

P. Chen, M. Koc / Journal of Materials Processing Technology 190 (2007) 189–198 191

variat

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•

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ftsowtfsas twisting, was not considered since it is negligible is this case.As there is no clear distinction to separate a cross-section curvefor individual measurement of springback angles and sidewallcurl, two assumptions deduced from the sample observations

Fig. 1. Springback

n the following section, variation analysis of an open-channelart made of advanced high strength steel is presented. The lastection is discussions and conclusions.

. Variations in stamping process

In this study, the objective is to understand and accuratelyredict the variation of springback in an open-channel drawingonsidering the variations of material and process as shown inig. 1.

Total variation of springback in the stamping process haseveral components. Generally, different variation componentsan be attributed to different sources [32]. The following are theajor categories of variation source (Fig. 2):

Part-to-part variation is also referred to as system-level vari-ation or inherent variation. It is the amount of variation thatcan be expected across consecutive parts produced by the pro-cess during a given run. It is caused by the random variationof all the uncontrolled (controllable and uncontrollable) pro-cess variables. In the variation simulation of this paper, blank

thickness was considered as the uncontrolled variable.Within batch variation is usually due to the variations of thecontrolled variables such as BHF, material property and fric-tion.

Fig. 2. Source of variation in a typical forming process.F[

ion and its sources.

Batch-to-batch variation represents the variability among theindividual batches, which is mostly caused by material vari-ation from batch to batch and the variation introduced bytooling setup.

. Measurement of springback for open-channelrawing

A schematic view of die, punch, blank and their dimensionsor open-channel drawing, which is used in the analyses forhis study, is shown in Fig. 3. Fig. 4 shows the formed part afterpringback. Three measurements, namely the springback of wallpening angle (β1), the springback of flange angle (β2) and side-all curl radius (ρ) shown in Fig. 5, were used to characterize the

otal springback considering only the cross-sectional shapes oformed parts obtained before and after the removal of tools. Thepringback in the direction orthogonal to the cross-section, such

ig. 3. A schematic view of tools and dimensions for open-channel drawing32].

192 P. Chen, M. Koc / Journal of Materials Proce

Fig. 4. Open-channel parts after drawing.

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Fig. 5. Illustration of springbacks.

re introduced for the springback measurement. Firstly, it isssumed that wall opening angle, flange closing angles and side-all curl vary independently. Secondly, the sidewall curl coulde approximated by a piece of circular arc.

Fig. 5 also shows the measurements placements (A–E). Twoeasurements were conducted before springback, namely the x

nd y coordinates of A and B, which is denoted as A0 and B0

n this work. They are used to compute the wall angle (θ01) and

ange angle (θ02) before springback. After springback, another

ve measurements were placed on A–E, which were used in thealculation of the wall angle (θ1), flange angle (θ2) and sidewallurl radius (ρ) after springback. To estimate the sidewall curladius, a curve fitting technique that employs three points (A–C)

t

ha

Fig. 6. Material propertie

ssing Technology 190 (2007) 189–198

o construct a circular arc is used. Eq. (1) lists all the equationseeded for the calculation of the β1, β2 and ρ.

θ01 = arccos

(o�x · A0 �B0

|ox| · ∣∣A0B0∣∣)

θ02 = arccos

(o�x · A0 �B0

|ox| · ∣∣A0B0∣∣)

θ1 = arccos

(o�x · A�B

|ox| · |AB|

)

θ2 = arccos

(A�B · E �D

|ED| · |AB|

)

β1 = θ1 − θ01

β2 = θ02 − θ2

xO =

x2B + y2

B − x2A − y2

A − ((yA − yB)/

(yC − yB))(x2C + y2

C − x2B − y2

B)

2[xB − xA + (xC − xB)((yA − yB)/(yC − yB))

]yO = x2

A + y2A − x2

B − y2B + 2xO(xB − xA)

2(yA − yB)

ρ =√

(xA − xO)2 + (yA − yO)2

(1)

. Finite element modeling and validation for thepen-channel drawing of AHSS

The simulation work for this study is based on the exper-mental results of Lee et al. [33]. Information about theeometry and dimensions of the tooling and blank are pre-ented in Fig. 3. The initial dimension of the blank sheet was00 mm (length) × 35 mm (width). Forming was carried out on150 tonnes double action hydraulic press with a punch speedf 1 mm/s, and the total punch stroke was 70 mm. Blank holderorce (BHF) was 2.5 kN. The blank material used was DP Steelith the material properties presented in Fig. 6 based on the

ensile tests by Lee et al. [34].Considering the geometric symmetry of the process, only

alf of the blank was simulated. The material was modeled asn elastic–plastic material with isotropic elasticity, using the

s of DP steel [33].

P. Chen, M. Koc / Journal of Materials Processing Technology 190 (2007) 189–198 193

Table 1Different FEA procedure used in simulation

Case

1 2 3 4 5 6 7 8 9

Element type Solid Solid Solid Solid Shell Shell Shell Shell ShellContact Soft Soft Soft Hard Soft Soft Hard Hard SoftForming analysis (dynamic) Implicit Implicit Implicit Implicit Implicit Implicit Explicit Explicit ExplicitST

HoRrt0df

Fa

Fa

Fr

itatn

pringback analysis (static) Implicit Implicit Implicithrough-thickness element numberor integration point

5 9 21

ill anisotropic yield criterion for the plasticity. The coefficientsf Hill yield criterion (R11 = 1.0, R22 = 1.01951, R33 = 1.00219,12 = 0.992318, R13 = 1.0, R23 = 1.0) were computed from the-values as presented in Fig. 6. The friction coefficient betweenools and the sheet blank was assumed to be constant and

.1. To determine the appropriate element type, contact con-itions, through-thickness element number and analysis typeor simulation using ABAQUS, nine combinations (as tabulated

ig. 7. Effects of different FEA procedure on the prediction of wall openingngle.

ig. 8. Effects of different FEA procedure on the prediction of flange closingngle.

ig. 9. Effects of different FEA procedure on the prediction of sidewall curladius.

pwssitt(i

TO

R

111111111122222222223

Implicit Implicit Implicit Implicit Implicit Implicit9 9 21 5 15 9

n Table 1) of different element type, different contact condi-ion, different through-thickness element number and differentnalysis type were tried in the simulation. The term, soft con-act, denotes exponential pressure-overclosure definition for theormal behavior between contacting surfaces.

The comparison of different simulation procedure for therediction of wall opening angle (θ1), flange angle (θ2) and side-all curl radius (ρ) is shown in Figs. 7–9. It was found that the

idewall curl is very sensitive to the contact condition used inimulation. Since the soft contact tends to soften the contact-ng surface, it actually depresses the sidewall curl, which is not

rue for advanced high strength steel. Among these combina-ions, case 4 (hard contact), case 7 (hard contact) and case 8hard contact) show a good match with the experiment resultsn all three springback measurements. Hence, hard contact is

able 2riginal experiment design

un order BHF (kN) Friction Material

1 13.75 0.15 1.12 13.75 0.1 13 13.75 0.15 0.94 2.5 0.15 15 25 0.1 1.16 2.5 0.1 1.17 25 0.15 18 2.5 0.15 19 2.5 0.1 1.10 13.75 0.05 0.91 25 0.05 12 2.5 0.1 0.93 2.5 0.05 14 13.75 0.1 15 13.75 0.1 16 13.75 0.05 0.97 2.5 0.1 0.98 13.75 0.1 19 25 0.05 10 13.75 0.15 0.91 25 0.1 0.92 13.75 0.05 1.13 13.75 0.1 14 25 0.15 15 13.75 0.15 1.16 25 0.1 0.97 13.75 0.05 1.18 13.75 0.1 19 2.5 0.05 10 25 0.1 1.1

194 P. Chen, M. Koc / Journal of Materials Proce

Table 3Assumed statistics of variables

Mean Range S.D.

Uncontrolled factorPart thickness 1.2 mm 1.18–1.22 mm 0.066667

Controlled factorBHF level-1 2.5 kN 2.4–2.6 0.033333BHF level-2 13.75 kN 13.65–13.85 0.033333BHF level-3 25 kN 24.9–25.1 0.033333

Friction level-1 0.05 0.04–0.06 0.003333Friction level-2 0.1 0.09–0.11 0.003333Friction level-3 0.15 0.14–0.16 0.003333

Material level-1 90% 88–92% 0.00667

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TR

R

111111111122222222223

Material level-2 100% 98–102% 0.00667Material level-3 110% 108–112% 0.00667

referred. It can be seen that element type and forming anal-sis type do not affect the accuracy of springback predictionuch. Therefore, to reduce the computation time, hard con-

act, shell element, explicit (dynamic) for forming and implicitstatic) for springback were used in further simulations. Dif-erent through-thickness integration points (5, 9, 15, and 21) of

hell elements were also tried in the simulation, which showed nouch influence on the prediction of springback. Therefore, nine

hrough-thickness integration points were used in the furtherimulations.

acir

able 4andomized (random number generation) experiment table and simulation results

un BHF (kN) Friction Material P

1 13.649243 0.1405 1.0819 12 13.755335 0.0904 0.9814 13 13.721143 0.1534 0.8811 14 2.3992 0.1512 0.9933 15 24.9033 0.0970 1.0923 16 2.5053 0.1005 1.0913 17 25.0027 0.1466 1.0063 18 2.4711 0.1521 0.9910 19 2.5291 0.1016 1.0999 10 13.779106 0.0404 0.8937 11 24.9970 0.0536 1.0160 12 2.5072 0.0958 0.8941 13 2.4983 0.0422 0.9933 14 13.757157 0.0995 1.0031 15 13.748318 0.0959 1.0025 16 13.737183 0.0555 0.9081 17 2.4872 0.0970 0.8889 18 13.791958 0.1035 1.0077 19 24.9737 0.0527 0.9922 10 13.780872 0.1515 0.9056 11 24.9978 0.0977 0.8982 12 13.772126 0.0484 1.1079 13 13.718735 0.0964 0.9919 14 24.9718 0.1483 1.0078 15 13.785849 0.1407 1.1077 16 25.0469 0.1037 0.9091 17 13.768493 0.0428 1.0955 18 13.751131 0.0960 0.9936 19 2.542 0.0480 1.0014 10 24.9880 0.1004 1.0918 1

ssing Technology 190 (2007) 189–198

. Variation simulation of springback and results

In this study, we only considered the “part-to-part” andwithin batch” variations. The variation simulation and anal-sis of the springback of DP steel part are described step by steps follows.

tep 1 (Design of experiment). BHF, material property and fric-ion were chosen as design factors. Box–Behnken RSM designith 2-replicate and 6-center-point was used for this 3-factor and-level experiment design. The levels of the material propertyre considered as 110, 100 and 90% of the stress–strain curven Fig. 6, which indicates the strength of the material. Table 2hows the original experiment table.

teps 2 and 3Random number generation of controlled andncontrolled variablesIt is assumed that most random processesonform to a Gaussian distribution. Moreover, irrespective ofhe parent distribution of the population, the distribution of theverage of random samples taken from the population tends toe normal as the sample size increases (Central Limit Theo-em). Therefore, once we know the mean and standard deviationS.D.) of a random process, we can generate a random number

ccording to its Gaussian distribution. According to the statisticshosen in Table 3, the original experiment table was random-zed as shown in Table 4. Fig. 10 is an illustration of the numberandomization.

art thickness β1 (◦) β2 (◦) ρ (mm)

1.8017 16.1242 12.3581 191.58701.9484 16.5589 12.1747 169.74801.9619 12.5694 9.4674 301.40501.9606 18.4420 11.0999 133.47301.9473 19.8823 13.4837 137.70052.0388 17.8932 11.0234 138.22842.0067 17.0396 11.4995 158.01872.0383 17.5935 10.8937 140.59592.0638 15.0679 11.1937 140.68411.9781 14.1995 11.0920 156.65611.9351 17.7311 12.9760 151.33122.0725 9.8603 10.0025 178.03881.8973 14.9086 9.9908 159.48712.0039 16.4520 12.2300 169.07992.0272 16.6234 12.2783 166.01321.9032 15.3990 11.3933 166.43672.0405 14.7347 9.3532 168.84451.9982 16.3857 12.2430 171.57951.9569 17.2571 12.6146 156.69092.0172 12.6997 9.7055 298.53002.0322 15.9928 11.2144 172.02371.9815 18.5080 13.6613 122.97331.8797 16.5546 12.2819 169.29091.9887 17.0759 11.5550 158.11652.0855 17.0869 13.9510 178.88331.8921 16.4418 11.4263 169.86232.0642 18.7483 14.9717 125.45642.0129 16.4303 12.1832 169.69011.9466 15.2362 10.1161 157.66202.0695 18.7641 14.3375 139.4873

P. Chen, M. Koc / Journal of Materials Processing Technology 190 (2007) 189–198 195

Fig. 10. Schematic diagram of the variable randomizations (random numbergeneration).

Table 5Recommended variable level for the minimum of β1, β2 and maximum of ρ

Material Friction BHF

Minimum β1 Low Low LowMM

Sa

Smtisuit

β

β

r

m

Table 7Recommended variable level for the minimum variance ofβ1,β2 andρ accordingto Monte-Carlo simulation

Material Friction BHF

Min[Var(β1)] – High MiddleMM

eittoswwtbTBrss

Suttf

5

ittasawlT

TS

VVV

inimum β2 Low Low Lowaximum ρ Low Low –

tep 4 (Simulation). Simulations were run according to Table 4nd the simulation results are shown in Table 4 as well.

tep 5 (Regression analysis). Eqs. (2)–(4) are the regressionodels of β1, β2, and ρ as functions of BHF, material and fric-

ion. The variables in these equations are coded (−1, 0, 1) factorsn the DOE. To investigate whether the factors’ effect on eachpringback is significant, analysis of variance (ANOVA) wassed. Factors with a P-value larger than 0.05 were considered asnsignificant and ignored in the regression model. For example,he main effect of friction on β1 is negligible:

1 = 16.5008 + 1.5867XMaterial + 0.7287XBHF

−0.8454XBHFXFriction (2)

2 = 12.2319 + 1.3329XMaterial + 0.9646XBHF

−0.3929XFriction − 0.7297X2BHF − 0.5529XBHFXFriction

(3)

= 169.234 − 8.384XMaterial + 18.482XMaterialXFriction (4)

The effect of each factor on each springback could be deter-ined by the sign of the corresponding coefficient in the above

5

e

able 6pringback variation

Material Friction

Low Middle High Low

ar(β1) 0.0667 0.0667 0.0667 0.5349ar(β2) 0.1146 0.1146 0.1146 0.4798ar(ρ) 37.419 0 37.419 73.712

in[Var(β2)] – High Lowin[Var(ρ)] Middle Middle –

quations. For instance, the coefficient of the XMaterial is +1.5867n Eq. (2), therefore, the bigger the material strength, the largerhe springback. Table 5 tabulates the optimal variable level forhe minimum of each springback. A more detailed indicationf the relationship between the springbacks and the factors ishown in Fig. 11. As in the parameter’s range studied in thisork, springback increases with BHF and friction, which agreesith the experimental observations [35]. Papeleux and Pon-

hot [35] reported that springback increases with small BHF,ut decreases as the BHF increases for large force values.his phenomenon can be explained by the fact that with lowHF, the punch induces mostly bending stresses in the mate-

ial, but as the blankholder holds the blank more severely, thetresses included by the punching phase become mostly tensiletresses.

tep 6 (Variation sensitivity analysis). Three methods weresed to analyze the effects of the factors on the variation ofhe springback. Finally, it was found that the springback varia-ion magnitude is too small in this case and not distinguishablerom the system noise.

.1. Monte-Carlo simulation

Monte-Carlo simulation was applied to Eqs. (2)–(4). Accord-ng to the parameter levels used in the DOE, it is assumedhat all factors have equal variance in the Monte-Carlo simula-ion, i.e., Var(XMaterial) = Var(XBHF) = Var(XFriction) = 0.32, withzero mean value for each factor (coded factors). Monte-Carlo

imulation was run 100 times for each situation (a specific factort a specific level). The corresponding variance of the springbackas recorded in Table 6. Table 7 summarizes the optimal variable

evel for the minimum variance of each springback according toable 6.

.2. Sensitivity analysis

According to Eqs. (2)–(4), the variance of β1, β2 and ρ arexpressed as Eqs. (5)–(7) via linearized sensitivity analysis.

BHF

Middle High Low Middle High

0.324 0.2563 0.3501 0.2571 0.32070.3276 0.2367 0.1866 0.1926 0.26567.1786 10.414 11.813 11.813 11.813

196 P. Chen, M. Koc / Journal of Materials Processing Technology 190 (2007) 189–198

plots

V

V

V

Fig. 11. Response surface

ar(β1) = [1.8861 − 1.6648XMaterial]2Var(XMaterial)

+[1.028 − 0.8454XFriction]2Var(XBHF)

+[0.8454XBHF]2Var(XFriction) (5)

ar(β2) = [1.3329 + 0.2899XBHF]2Var(XMaterial)

+[0.9646 − 1.4594XBHF + 0.2899XMaterial]2

×Var(XBHF) + [−0.3929 − 0.5528XBHF]2

×Var(XFriction) (6)

ostc

of (a) β1, (b) β2 and (c) ρ.

ar(r) = [−27.3 − 19.35XFriction + 27.194XMaterial]2

×Var(XMaterial) + [−54.444XBHF]2Var(XBHF)

+[22.745 − 19.35XMaterial + 19.82XFriction]2

×Var(XFriction) (7)

The variance of the response is determined by the variance

f each factor and the sensitivity coefficient (the quantity in thequare parentheses). To minimize the variance of the springback,he most efficient way is to minimize the sensitivity coeffi-ients in the equation. Table 8 tabulates the optimal variable

P. Chen, M. Koc / Journal of Materials Pr

Table 8Recommended variable level for the minimum variance ofβ1,β2 andρ accordingto sensitivity analysis

Material Friction BHF

Min[Var(β1)] High High MiddleMin[Var(β2)] High Middle HighMin[Var(ρ)] High Low Middle

Table 9Extracted data (β1) used in MINITAB for Taguchi analysis

BHF (kN) Friction Material S.D. (β1) Mean (β1)

13.75 0.15 1.1 0.68073 16.605613.75 0.1 1 0.09154 16.500813.75 0.15 0.9 0.09214 12.6346

2.5 0.15 1 0.59998 18.017825 0.1 1.1 0.79069 19.3232

2.5 0.1 1.1 1.99779 16.480625 0.15 1 0.02567 17.057813.75 0.05 0.9 0.84817 14.799325 0.05 1 0.33517 17.4941

2.5 0.1 0.9 3.44672 12.29752.5 0.05 1 0.23165 15.0724

21

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5 0.1 0.9 0.31749 16.21733.75 0.05 1.1 0.16992 18.6282

evel for the minimum springback variation suggested by Eqs.5)–(7), which does not agree with Table 7. This discrepancyas explained by the third method.

.3. Taguchi approach

Taguchi analysis was used to analyze the springback vari-tion. MINITAB, a statistical software, was used to analyzehe existing experiment results (Table 4). MINITAB can auto-

atically extract data (standard deviation and mean) from thevailable experimental observations. For example, Table 9 is thextracted data of β1 used in MINITAB for Taguchi analysis. InINITAB, the main effects of each design factor on the standard

eviations of the response are obtained via regression analysis,nd the significance of these effects were tested via analysisf variation (ANOVA) and F-tests. P-values (P) were used toetermine which of the effects in the model are statistically sig-ificant, which are compared with a �-level of 0.05. As shownn Table 10, none of the effects are significant, which indicates

hat the springback variation in this case is not distinguishablerom the system-level noise. In other words, the springbackariation is not controllable in this case. Therefore, the conclu-ions based on Monte-Carlo simulation and sensitivity analysis

able 10-tests for the standard deviation of each springback

P-value

S.D. (β1) S.D. (β2) S.D. (ρ)

aterial 0.283 0.088 0.853HF 0.098 0.648 0.747riction 0.635 0.495 0.542

ocessing Technology 190 (2007) 189–198 197

Tables 7 and 8) actually do not have any meaning becausehe springback variations are totally random and uncontrollablen this case. In other words, conclusions from both methodsre neither correct nor wrong. Since the system-level noisesere introduced by random number generation (Table 3) in our

omputer experiment, we can solve the problem by reducinghe standard deviations used in the random number generation.owever, this kind of adjustment would not be easy in reality,

ince the tuning of the system-level noise is usually impossiblen most cases.

. Conclusions

The effects of BHF, material and friction on springbacknd springback variation of DP steel channel have been ana-yzed parametrically using the FEA and DOE with randomumber generation (computer experiment). On the basis of theuantitative and qualitative analysis made herein, the followingonclusions could be drawn.

The sidewall curl is very sensitive to the contact condition inhe simulation; hard contact is preferred for high strength steel.

Springback variation in this case is not distinguishable fromhe system-level noise. Therefore, it is uncontrollable in thisase. In order to reduce springback variation, the standard devi-tions used for variable randomization has to be decreased;irtually, it means that a system-level adjustment of the pressas to be performed to reduce the part-to-part variation of thequipment. On the other hand, if the springback variation is largend uncontrollable, then the springback compensation techniqueas to be chosen with it in mind.

A methodology for the variation simulation of springbackas developed, which provides a rapid understanding of the

nfluence of the random process variations on the springbackariation of the formed part using FEA techniques eliminatinghe need for lengthy and costly physical experiments.

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