1. Journal of Materials Processing Technology 190 (2007) 189–198
Simulation of springback variation in forming
of 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, USA
b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA
Received in revised form 23 February 2007; accepted 27 February 2007
Abstract
Variations in the mechanical and dimensional properties of the incoming material, lubrication and other forming process parameters are the
main causes of springback variation. Variation of springback prevents the applicability of the springback prediction and compensation techniques.
Hence, it leads to amplified variations and problems during assembly of the stamped components, in turn, resulting in quality issues. To predict the
variation of springback and to improve the robustness of the forming process, variation simulation analysis could be adopted in the early design
stage. Design of experiment (DOE) and finite element analysis (FEA) approach was used for the variation simulation and analysis of the springback
for advanced high strength steel (AHSS) parts. To avoid the issues caused by the deterministic FEA simulation, random number generation was
used to introduce uncertainties in DOE. This approach was, then, applied to investigate the effects of variations in material, blank holder force and
friction on the springback variation for an open-channel shaped part made of dual phase (DP) steel. This approach provides a rapid and accurate
understanding of the influence of the random process variations on the springback variation of the formed part using FEA techniques eliminating
the need for lengthy and costly physical experiments.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Springback variation; Variation simulation; Design of experiment; Finite element analysis; Computer experiment; Advanced high strength steel
1. Introduction due to challenges in terms of formability, springback, joining
and die life issues.
During the past two decades, the automotive industry has Springback is an elastically driven change in shape and form
been in a quest to achieve low-mass vehicles due to government of a part upon unloading after a part is formed. The concerns
regulations and consumer demands for light, fuel-efficient and about springback and quality control grow among auto-makers
low-emission vehicles. In addition, requirements for improved as the use of high strength–low weight materials increases [1].
safety, performance, and comfort features continue putting Parts made of AHSS demonstrate more springback than parts
pressure on the automotive manufacturers especially in an envi- made of mild steel do. Moreover, the experience with forming
ronment where there are ever-increasing global competition of AHSS materials is limited compared to mild steel. Variation
and continuous cost reduction demands. As an alternative to of springback prevents the applicability and use of springback
aluminum and magnesium alloys, advanced/ultra high strength prediction and compensation techniques. Hence, it leads to
steels (AHSS) have been proven to be a proper choice of mate- amplified variations and problems during assembly of compo-
rial for various body and structural automotive parts meeting the nents, and in turn, results in quality issues. Thus, understanding,
required low-mass, affordable cost and increased performance accurate characterization, prediction, control and reduction of
requirements. However, wide application of high strength steel springback and its variation have become very crucial in terms
in many potential auto body and structural parts is still limited of decreasing development times and reducing scrap rate in mass
production to achieve cost effective fabrication of AHSS parts.
Numerous studies during the last 40 years have attempted to
∗ Corresponding author. Tel.: +1 804 827 7029. determine the controlling factors in springback and find ways to
E-mail address: mkoc@vcu.edu (M. Koc). ¸ reduce it. An early study by Baba and Tozawa [2] focused on
0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmatprotec.2007.02.046

2. 190 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c
the effect of stretching a sheet by a tensile force, during or after simulation, which is especially designed for the assembly pro-
bending, in minimizing springback. Other studies investigated cess (welding) simulation based on the assumption of linear
the role of process variables on springback. Zhang and Lee [3] elasticity.
showed the influence of blank holder force, elastic modulus, Generally speaking, Monte-Carlo simulation is effective for
strain hardening exponent, blank thickness and yield strength any kind of non-linear process. However, since large number of
on the magnitude of the final springback strain in a part. Geng FEA simulations is required (more than 100 for reliable results),
and Wagoner [4] studied the effects of plastic anisotropy and its it is not practical and too costly to apply it for a complicated
evolution in springback. They developed a constitutive equation case, such as metal forming. Therefore, some researchers [22]
for 6022-T4 aluminum alloy using a new anisotropic hardening applied DOE techniques in their FE simulation to effectively
model and proved that Barlat’s yield function is more accurate reduce the amount of simulations. However, since the FEA sim-
than other yield functions in their case. ulation is deterministic (no random error), replicate observations
Some researchers evaluated FE simulation procedures in from running the simulation with the same inputs will be identi-
terms of their springback prediction accuracy. Mattiasson et cal. Despite some similarities to physical experiments, the lack
al. [5], Wagoner et al. [6], Li et al. [7] and Lee and Yang [8] of random error makes FEA simulations different from physi-
found that FEM simulations of springback are much more sen- cal experiments. In the absence of independent random errors,
sitive to numerical tolerances than forming simulations are. Li the rationale for least-squares fitting of a response surface is not
et al. [9] investigated the effects of element type on the spring- clear [23]. The usual measures of uncertainty derived from least-
back simulation. Yuen [10] and Tang [11] found that different squares residuals have no obvious statistical meaning [23,24].
unloading scheme will affect the accuracy of the springback According to Welch et al. [25], in the presence of systematic
prediction. Similarly, Focellese et al. [12] and Narasimhan and error rather than random error, statistical testing is inappropri-
Lovell [13] pointed out that different integration scheme will ate. For deterministic FEA simulation, some statistics, including
also influence the result of springback simulation. In 1999, F-statistics, have no statistical meaning since they assume the
Park et al. [14] and Valente and Traversa [15] attempted to observations include an error term which has mean of zero and
link dynamic explicit simulations of forming operations to static a non-zero standard deviation. Consequently, the use of step-
implicit simulations of springback. It was proved that this tech- wise regression for polynomial model fitting is not appropriate
nique is very effective for the springback simulation. Li et since it utilizes F-statistic value when adding/removing model
al. [16] explored a variety of issues in the springback simu- parameters [24].
lations. They concluded that (1) typical forming simulations In an effort to solve the above problem, McKay et al. [26]
are acceptably accurate with 5–9 through-thickness integration introduced Latin hypercube sampling which ensures that each
points for shell/beam type elements, whereas springback anal- of the input variables has all portions of its range presented.
ysis within 1% numerical error requires up to 51 points, and Sacks et al. [23,27] proposed the design and analysis of com-
more typically 15–25 points, depending on R/t, sheet tension puter (DACE) method to model the deterministic output as the
and friction coefficient. (2) More contact nodes are necessary realization of a stochastic process, thereby providing a statis-
for accurate springback simulations than for forming simulation, tical basis for designing experiments for efficient prediction.
approximately one node per 5◦ of turn angle versus 10◦ recom- Kleijnen [28] suggested incorporating substantial random error
mended for forming. (3) Three-dimensional shell and non-linear through random number generators. Therefore, it is natural to
solid elements are preferred for springback prediction even for design and analyze such stochastic simulation experiments using
large w/t ratios because of the presence of persistent anticlas- standard techniques for physical experiments. Some researchers
tic curvature. For R/t > 5.6, shell elements are preferred since (e.g., Giunta et al. [29,30] and Venter et al. [31]), have also
solid elements are too computation-intensive. For R/t < 5.6, non- employed metamodeling techniques such as RSM for modeling
linear 3D solid elements are required for accurate springback deterministic computer experiments which contain numerical
prediction. noise. This numerical noise is used as a surrogate for random
Most of the research efforts on springback focused on the error, thus allowing the standard least-squares approach to be
accurate prediction and compensation of springback. The issue applied. However, the assumption of equating numerical noise
of springback variation was seldom concerned. Moreover, none to random error is questionable.
of the studies in the area of springback prediction touched In this work, random number generator was used to ensure
upon the variation simulation of springback. As lead times the correctness of the regression model. Assuming a Gaussian
are shortened and materials of high strength–low weight are distribution, the uncertainty was introduced by random number
used in manufacturing, a fundamental understanding of the generation for controlled factors at different level and uncon-
springback variation has become essential for accurate and trolled variables according to their mean and range. By using
rapid design of tooling and processes in the early design stage. this method, the effects of variations in material (mechanical
As far as the variation simulation is concerned, most avail- properties) and process (blank holder force and friction) on
able references are related to the assembly processes [17–21]. the springback variation were investigated for an open-channel
In 1997, Liu and Hu [17] summarized two variation simula- shaped part made of DP steel. The variations in stamping process
tion approaches for sheet metal assembly. They are: (1) direct are introduced in Section 2 of the paper. In Section 3, measure-
Monte-Carlo simulation, which is popular and good for all kinds ments of springback are defined. Section 4 is the validation of the
of random process simulation, and (2) mechanistic variation FE model using existing experimental results in the literature.

3. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c 191
Fig. 1. Springback variation and its sources.
In the following section, variation analysis of an open-channel • Batch-to-batch variation represents the variability among the
part made of advanced high strength steel is presented. The last individual batches, which is mostly caused by material vari-
section is discussions and conclusions. ation from batch to batch and the variation introduced by
tooling setup.
2. Variations in stamping process
3. Measurement of springback for open-channel
In this study, the objective is to understand and accurately drawing
predict the variation of springback in an open-channel drawing
considering the variations of material and process as shown in A schematic view of die, punch, blank and their dimensions
Fig. 1. for open-channel drawing, which is used in the analyses for
Total variation of springback in the stamping process has this study, is shown in Fig. 3. Fig. 4 shows the formed part after
several components. Generally, different variation components springback. Three measurements, namely the springback of wall
can be attributed to different sources [32]. The following are the opening angle (β1 ), the springback of flange angle (β2 ) and side-
major categories of variation source (Fig. 2): wall curl radius (ρ) shown in Fig. 5, were used to characterize the
total springback considering only the cross-sectional shapes of
formed parts obtained before and after the removal of tools. The
• Part-to-part variation is also referred to as system-level vari-
springback in the direction orthogonal to the cross-section, such
ation or inherent variation. It is the amount of variation that
as twisting, was not considered since it is negligible is this case.
can be expected across consecutive parts produced by the pro-
As there is no clear distinction to separate a cross-section curve
cess during a given run. It is caused by the random variation
for individual measurement of springback angles and sidewall
of all the uncontrolled (controllable and uncontrollable) pro-
curl, two assumptions deduced from the sample observations
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 the
controlled variables such as BHF, material property and fric-
tion.
Fig. 3. A schematic view of tools and dimensions for open-channel drawing
Fig. 2. Source of variation in a typical forming process. [32].

4. 192 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c
to construct a circular arc is used. Eq. (1) lists all the equations
needed for the calculation of the β1 , β2 and ρ.
ox · A0 B0
θ1 = arccos
0
|ox| · A0 B0
ox · A0 B0
θ2 = arccos
0
|ox| · A0 B0
Fig. 4. Open-channel parts after drawing. ox · AB
θ1 = arccos
|ox| · |AB|
AB · ED
θ2 = arccos
|ED| · |AB| (1)
β1 = θ1 − θ1
0
β2 = θ2 − θ2
0
xB + yB − xA − yA − ((yA − yB )/
2 2 2 2
(yC − yB ))(xC + yC − xB − yB )
2 2 2 2
xO =
2 xB − xA + (xC − xB )((yA − yB )/(yC − yB ))
xA + yA − xB − yB + 2xO (xB − xA )
2 2 2 2
yO =
2(yA − yB )
ρ= (xA − xO )2 + (yA − yO )2
4. Finite element modeling and validation for the
Fig. 5. Illustration of springbacks.
open-channel drawing of AHSS
are introduced for the springback measurement. Firstly, it is The simulation work for this study is based on the exper-
assumed that wall opening angle, flange closing angles and side- imental results of Lee et al. [33]. Information about the
wall curl vary independently. Secondly, the sidewall curl could geometry and dimensions of the tooling and blank are pre-
be approximated by a piece of circular arc. sented in Fig. 3. The initial dimension of the blank sheet was
Fig. 5 also shows the measurements placements (A–E). Two 300 mm (length) × 35 mm (width). Forming was carried out on
measurements were conducted before springback, namely the x a 150 tonnes double action hydraulic press with a punch speed
and y coordinates of A and B, which is denoted as A0 and B0 of 1 mm/s, and the total punch stroke was 70 mm. Blank holder
in this work. They are used to compute the wall angle (θ1 ) and0 force (BHF) was 2.5 kN. The blank material used was DP Steel
flange angle (θ2 0 ) before springback. After springback, another with the material properties presented in Fig. 6 based on the
five measurements were placed on A–E, which were used in the tensile tests by Lee et al. [34].
calculation of the wall angle (θ 1 ), flange angle (θ 2 ) and sidewall Considering the geometric symmetry of the process, only
curl radius (ρ) after springback. To estimate the sidewall curl half of the blank was simulated. The material was modeled as
radius, a curve fitting technique that employs three points (A–C) an elastic–plastic material with isotropic elasticity, using the
Fig. 6. Material properties of DP steel [33].

5. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c 193
Table 1
Different 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 Shell
Contact Soft Soft Soft Hard Soft Soft Hard Hard Soft
Forming analysis (dynamic) Implicit Implicit Implicit Implicit Implicit Implicit Explicit Explicit Explicit
Springback analysis (static) Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit
Through-thickness element number 5 9 21 9 9 21 5 15 9
or integration point
Hill anisotropic yield criterion for the plasticity. The coefficients in Table 1) of different element type, different contact condi-
of Hill yield criterion (R11 = 1.0, R22 = 1.01951, R33 = 1.00219, tion, different through-thickness element number and different
R12 = 0.992318, R13 = 1.0, R23 = 1.0) were computed from the analysis type were tried in the simulation. The term, soft con-
r-values as presented in Fig. 6. The friction coefficient between tact, denotes exponential pressure-overclosure definition for the
tools and the sheet blank was assumed to be constant and normal behavior between contacting surfaces.
0.1. To determine the appropriate element type, contact con- The comparison of different simulation procedure for the
ditions, through-thickness element number and analysis type prediction of wall opening angle (θ 1 ), flange angle (θ 2 ) and side-
for simulation using ABAQUS, nine combinations (as tabulated wall curl radius (ρ) is shown in Figs. 7–9. It was found that the
sidewall curl is very sensitive to the contact condition used in
simulation. Since the soft contact tends to soften the contact-
ing surface, it actually depresses the sidewall curl, which is not
true for advanced high strength steel. Among these combina-
tions, case 4 (hard contact), case 7 (hard contact) and case 8
(hard contact) show a good match with the experiment results
in all three springback measurements. Hence, hard contact is
Table 2
Original experiment design
Run order BHF (kN) Friction Material
Fig. 7. Effects of different FEA procedure on the prediction of wall opening
angle. 1 13.75 0.15 1.1
2 13.75 0.1 1
3 13.75 0.15 0.9
4 2.5 0.15 1
5 25 0.1 1.1
6 2.5 0.1 1.1
7 25 0.15 1
8 2.5 0.15 1
9 2.5 0.1 1.1
10 13.75 0.05 0.9
11 25 0.05 1
12 2.5 0.1 0.9
13 2.5 0.05 1
14 13.75 0.1 1
Fig. 8. Effects of different FEA procedure on the prediction of flange closing 15 13.75 0.1 1
angle. 16 13.75 0.05 0.9
17 2.5 0.1 0.9
18 13.75 0.1 1
19 25 0.05 1
20 13.75 0.15 0.9
21 25 0.1 0.9
22 13.75 0.05 1.1
23 13.75 0.1 1
24 25 0.15 1
25 13.75 0.15 1.1
26 25 0.1 0.9
27 13.75 0.05 1.1
28 13.75 0.1 1
29 2.5 0.05 1
Fig. 9. Effects of different FEA procedure on the prediction of sidewall curl 30 25 0.1 1.1
radius.

6. 194 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c
Table 3 5. Variation simulation of springback and results
Assumed statistics of variables
Mean Range S.D. In this study, we only considered the “part-to-part” and
Uncontrolled factor
“within batch” variations. The variation simulation and anal-
Part thickness 1.2 mm 1.18–1.22 mm 0.066667 ysis of the springback of DP steel part are described step by step
as follows.
Controlled factor
BHF level-1 2.5 kN 2.4–2.6 0.033333
BHF level-2 13.75 kN 13.65–13.85 0.033333 Step 1 (Design of experiment). BHF, material property and fric-
BHF level-3 25 kN 24.9–25.1 0.033333 tion were chosen as design factors. Box–Behnken RSM design
Friction level-1 0.05 0.04–0.06 0.003333 with 2-replicate and 6-center-point was used for this 3-factor and
Friction level-2 0.1 0.09–0.11 0.003333 3-level experiment design. The levels of the material property
Friction level-3 0.15 0.14–0.16 0.003333 are considered as 110, 100 and 90% of the stress–strain curve
Material level-1 90% 88–92% 0.00667 in Fig. 6, which indicates the strength of the material. Table 2
Material level-2 100% 98–102% 0.00667 shows the original experiment table.
Material level-3 110% 108–112% 0.00667
Steps 2 and 3Random number generation of controlled and
uncontrolled variablesIt is assumed that most random processes
preferred. It can be seen that element type and forming anal- conform to a Gaussian distribution. Moreover, irrespective of
ysis type do not affect the accuracy of springback prediction the parent distribution of the population, the distribution of the
much. Therefore, to reduce the computation time, hard con- average of random samples taken from the population tends to
tact, shell element, explicit (dynamic) for forming and implicit be normal as the sample size increases (Central Limit Theo-
(static) for springback were used in further simulations. Dif- rem). Therefore, once we know the mean and standard deviation
ferent through-thickness integration points (5, 9, 15, and 21) of (S.D.) of a random process, we can generate a random number
shell elements were also tried in the simulation, which showed no according to its Gaussian distribution. According to the statistics
much influence on the prediction of springback. Therefore, nine chosen in Table 3, the original experiment table was random-
through-thickness integration points were used in the further ized as shown in Table 4. Fig. 10 is an illustration of the number
simulations. randomization.
Table 4
Randomized (random number generation) experiment table and simulation results
Run BHF (kN) Friction Material Part thickness β1 (◦ ) β2 (◦ ) ρ (mm)
1 13.649243 0.1405 1.0819 11.8017 16.1242 12.3581 191.5870
2 13.755335 0.0904 0.9814 11.9484 16.5589 12.1747 169.7480
3 13.721143 0.1534 0.8811 11.9619 12.5694 9.4674 301.4050
4 2.3992 0.1512 0.9933 11.9606 18.4420 11.0999 133.4730
5 24.9033 0.0970 1.0923 11.9473 19.8823 13.4837 137.7005
6 2.5053 0.1005 1.0913 12.0388 17.8932 11.0234 138.2284
7 25.0027 0.1466 1.0063 12.0067 17.0396 11.4995 158.0187
8 2.4711 0.1521 0.9910 12.0383 17.5935 10.8937 140.5959
9 2.5291 0.1016 1.0999 12.0638 15.0679 11.1937 140.6841
10 13.779106 0.0404 0.8937 11.9781 14.1995 11.0920 156.6561
11 24.9970 0.0536 1.0160 11.9351 17.7311 12.9760 151.3312
12 2.5072 0.0958 0.8941 12.0725 9.8603 10.0025 178.0388
13 2.4983 0.0422 0.9933 11.8973 14.9086 9.9908 159.4871
14 13.757157 0.0995 1.0031 12.0039 16.4520 12.2300 169.0799
15 13.748318 0.0959 1.0025 12.0272 16.6234 12.2783 166.0132
16 13.737183 0.0555 0.9081 11.9032 15.3990 11.3933 166.4367
17 2.4872 0.0970 0.8889 12.0405 14.7347 9.3532 168.8445
18 13.791958 0.1035 1.0077 11.9982 16.3857 12.2430 171.5795
19 24.9737 0.0527 0.9922 11.9569 17.2571 12.6146 156.6909
20 13.780872 0.1515 0.9056 12.0172 12.6997 9.7055 298.5300
21 24.9978 0.0977 0.8982 12.0322 15.9928 11.2144 172.0237
22 13.772126 0.0484 1.1079 11.9815 18.5080 13.6613 122.9733
23 13.718735 0.0964 0.9919 11.8797 16.5546 12.2819 169.2909
24 24.9718 0.1483 1.0078 11.9887 17.0759 11.5550 158.1165
25 13.785849 0.1407 1.1077 12.0855 17.0869 13.9510 178.8833
26 25.0469 0.1037 0.9091 11.8921 16.4418 11.4263 169.8623
27 13.768493 0.0428 1.0955 12.0642 18.7483 14.9717 125.4564
28 13.751131 0.0960 0.9936 12.0129 16.4303 12.1832 169.6901
29 2.542 0.0480 1.0014 11.9466 15.2362 10.1161 157.6620
30 24.9880 0.1004 1.0918 12.0695 18.7641 14.3375 139.4873

7. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c 195
Table 7
Recommended variable level for the minimum variance of β1 , β2 and ρ according
to Monte-Carlo simulation
Material Friction BHF
Min[Var(β1 )] – High Middle
Min[Var(β2 )] – High Low
Min[Var(ρ)] Middle Middle –
equations. For instance, the coefficient of the XMaterial is +1.5867
in Eq. (2), therefore, the bigger the material strength, the larger
the springback. Table 5 tabulates the optimal variable level for
the minimum of each springback. A more detailed indication
Fig. 10. Schematic diagram of the variable randomizations (random number of the relationship between the springbacks and the factors is
generation). shown in Fig. 11. As in the parameter’s range studied in this
work, springback increases with BHF and friction, which agrees
Table 5 with the experimental observations [35]. Papeleux and Pon-
Recommended variable level for the minimum of β1 , β2 and maximum of ρ thot [35] reported that springback increases with small BHF,
Material Friction BHF but decreases as the BHF increases for large force values.
Minimum β1 Low Low Low
This phenomenon can be explained by the fact that with low
Minimum β2 Low Low Low BHF, the punch induces mostly bending stresses in the mate-
Maximum ρ Low Low – rial, but as the blankholder holds the blank more severely, the
stresses included by the punching phase become mostly tensile
stresses.
Step 4 (Simulation). Simulations were run according to Table 4
and the simulation results are shown in Table 4 as well. Step 6 (Variation sensitivity analysis). Three methods were
used to analyze the effects of the factors on the variation of
Step 5 (Regression analysis). Eqs. (2)–(4) are the regression the springback. Finally, it was found that the springback varia-
models of β1 , β2 , and ρ as functions of BHF, material and fric- tion magnitude is too small in this case and not distinguishable
tion. The variables in these equations are coded (−1, 0, 1) factors from the system noise.
in the DOE. To investigate whether the factors’ effect on each
springback is significant, analysis of variance (ANOVA) was
used. Factors with a P-value larger than 0.05 were considered as 5.1. Monte-Carlo simulation
insignificant and ignored in the regression model. For example,
Monte-Carlo simulation was applied to Eqs. (2)–(4). Accord-
the main effect of friction on β1 is negligible:
ing to the parameter levels used in the DOE, it is assumed
β1 = 16.5008 + 1.5867XMaterial + 0.7287XBHF that all factors have equal variance in the Monte-Carlo simula-
tion, i.e., Var(XMaterial ) = Var(XBHF ) = Var(XFriction ) = 0.32 , with
−0.8454XBHF XFriction (2) a zero mean value for each factor (coded factors). Monte-Carlo
simulation was run 100 times for each situation (a specific factor
β2 = 12.2319 + 1.3329XMaterial + 0.9646XBHF at a specific level). The corresponding variance of the springback
was recorded in Table 6. Table 7 summarizes the optimal variable
−0.3929XFriction − 0.7297XBHF − 0.5529XBHF XFriction
2
level for the minimum variance of each springback according to
(3) Table 6.
r = 169.234 − 8.384XMaterial + 18.482XMaterial XFriction (4) 5.2. Sensitivity analysis
The effect of each factor on each springback could be deter- According to Eqs. (2)–(4), the variance of β1 , β2 and ρ are
mined by the sign of the corresponding coefficient in the above expressed as Eqs. (5)–(7) via linearized sensitivity analysis.
Table 6
Springback variation
Material Friction BHF
Low Middle High Low Middle High Low Middle High
Var(β1 ) 0.0667 0.0667 0.0667 0.5349 0.324 0.2563 0.3501 0.2571 0.3207
Var(β2 ) 0.1146 0.1146 0.1146 0.4798 0.3276 0.2367 0.1866 0.1926 0.2656
Var(ρ) 37.419 0 37.419 73.712 7.1786 10.414 11.813 11.813 11.813

8. 196 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c
Fig. 11. Response surface plots of (a) β1 , (b) β2 and (c) ρ.
Var(r) = [−27.3 − 19.35XFriction + 27.194XMaterial ]2
Var(β1 ) = [1.8861 − 1.6648XMaterial ]2 Var(XMaterial )
×Var(XMaterial ) + [−54.444XBHF ]2 Var(XBHF )
+[1.028 − 0.8454XFriction ]2 Var(XBHF )
+[22.745 − 19.35XMaterial + 19.82XFriction ]2
+[0.8454XBHF ]2 Var(XFriction ) (5)
×Var(XFriction ) (7)
Var(β2 ) = [1.3329 + 0.2899XBHF ]2 Var(XMaterial )
The variance of the response is determined by the variance
+[0.9646 − 1.4594XBHF + 0.2899XMaterial ]2 of each factor and the sensitivity coefficient (the quantity in the
square parentheses). To minimize the variance of the springback,
×Var(XBHF ) + [−0.3929 − 0.5528XBHF ]2
the most efficient way is to minimize the sensitivity coeffi-
×Var(XFriction ) (6) cients in the equation. Table 8 tabulates the optimal variable

9. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198
c 197
Table 8 (Tables 7 and 8) actually do not have any meaning because
Recommended variable level for the minimum variance of β1 , β2 and ρ according the springback variations are totally random and uncontrollable
to sensitivity analysis
in this case. In other words, conclusions from both methods
Material Friction BHF are neither correct nor wrong. Since the system-level noises
Min[Var(β1 )] High High Middle were introduced by random number generation (Table 3) in our
Min[Var(β2 )] High Middle High computer experiment, we can solve the problem by reducing
Min[Var(ρ)] High Low Middle the standard deviations used in the random number generation.
However, this kind of adjustment would not be easy in reality,
since the tuning of the system-level noise is usually impossible
Table 9
Extracted data (β1 ) used in MINITAB for Taguchi analysis in most cases.
BHF (kN) Friction Material S.D. (β1 ) Mean (β1 )
6. Conclusions
13.75 0.15 1.1 0.68073 16.6056
13.75 0.1 1 0.09154 16.5008 The effects of BHF, material and friction on springback
13.75 0.15 0.9 0.09214 12.6346 and springback variation of DP steel channel have been ana-
2.5 0.15 1 0.59998 18.0178
25 0.1 1.1 0.79069 19.3232
lyzed parametrically using the FEA and DOE with random
2.5 0.1 1.1 1.99779 16.4806 number generation (computer experiment). On the basis of the
25 0.15 1 0.02567 17.0578 quantitative and qualitative analysis made herein, the following
13.75 0.05 0.9 0.84817 14.7993 conclusions could be drawn.
25 0.05 1 0.33517 17.4941 The sidewall curl is very sensitive to the contact condition in
2.5 0.1 0.9 3.44672 12.2975
2.5 0.05 1 0.23165 15.0724
the simulation; hard contact is preferred for high strength steel.
25 0.1 0.9 0.31749 16.2173 Springback variation in this case is not distinguishable from
13.75 0.05 1.1 0.16992 18.6282 the system-level noise. Therefore, it is uncontrollable in this
case. In order to reduce springback variation, the standard devi-
ations used for variable randomization has to be decreased;
level for the minimum springback variation suggested by Eqs. virtually, it means that a system-level adjustment of the press
(5)–(7), which does not agree with Table 7. This discrepancy has to be performed to reduce the part-to-part variation of the
was explained by the third method. equipment. On the other hand, if the springback variation is large
and uncontrollable, then the springback compensation technique
5.3. Taguchi approach has to be chosen with it in mind.
A methodology for the variation simulation of springback
Taguchi analysis was used to analyze the springback vari- was developed, which provides a rapid understanding of the
ation. MINITAB, a statistical software, was used to analyze influence of the random process variations on the springback
the existing experiment results (Table 4). MINITAB can auto- variation of the formed part using FEA techniques eliminating
matically extract data (standard deviation and mean) from the the need for lengthy and costly physical experiments.
available experimental observations. For example, Table 9 is the
extracted data of β1 used in MINITAB for Taguchi analysis. In References
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