Multi-regression modeling for springback effect on automotive body in white stamped parts

Materials and Design 46 (2013) 175–190

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Materials and Design

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Multi-regression modeling for springback effect on automotive bodyin white stamped parts

Nagur Aziz Kamal Bashah a,b, Norhamidi Muhamad a,⇑, Baba Md Deros a, Ahmad Zakaria b,Shaharum Ashari b, Achmed Mobin b, Mohd Safuan Mohd Abdul Lazat b

a Department of Mechanical and Materials Engineering, Faculty of Engineering and Build Environment, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysiab Institute of Product Design & Manufacturing, Universiti Kuala Lumpur, Jalan 7/91, Taman Shamelin Perkasa, 3.5 miles Cheras, 56100 Kuala Lumpur, Malaysia

a r t i c l e i n f o

Article history:Received 14 August 2012Accepted 7 October 2012Available online 23 October 2012

Keywords:SpringbackStamped partMulti-regression techniqueBody in white

0261-3069/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.matdes.2012.10.006

⇑ Corresponding author. Tel.: +60 03 8921 5331; faE-mail address: [email protected] (N. Muhama

a b s t r a c t

The use of high strength steel (HSS) materials in automotive body in white (BIW) stamped parts hasincreased the occurrence of springback after the forming process. Although HSS exhibits superiorstrength, weight reduction, and crash energy, it strongly influences springback impact on the sustainabledevelopment of BIW stamped parts. In this study, an empirical springback prediction model was synthe-sized based on the contemporary data sets of springback-prone components of automotive BIW stampedparts. Two different BIW stamped parts from an actual industrial stamping production line were selectedas pilot parts for this study. A statistical multi-regression (MR) analysis was used to model the springbackprediction effect by examining the sensitivity of springback input parameters on existing die geometry.The outputs represent the total springback values of the stamped parts. A total of 240 data from samplesof selected stamped parts were tabulated to synthesize the springback prediction model. The resultsshow that the MR models for the two parts were linear with the springback estimated errors betweenthe measured and predicted values between 0.5� and 3�, which is acceptable from an industrial view-point. The proposed MR models are capable of predicting the springback effect with minimal error byincorporating all possible variations that are inherent in the shop floor process.

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1. Introduction

Springback is a defect in sheet metals that commonly occursduring cold forming processes. Springback is characterized by theelastic recovery of the shape of a deformed part upon removal ofa load. Thus, the structure and appearance of the produced part un-dergo geometrical changes that deviate from the designed targetshape. Springback is a major geometrical variation issue in the me-tal forming of the inner and outer panels of automobiles because itcan strongly influence the quality of the final stamping and affectsubsequent assembly processes. In addition, springback has anindirect economic impact because assembly problems result in de-layed production, tooling revision costs, and rejection of unquali-fied parts.

Die design is one of the geometrical factors that contribute tospringback. Geometry correction on finished die is the currentstandard practice in various auto manufacturing industries, but iscostly and time consuming. This cost and time factors are due tothe several numbers of stamping iteration upon every die modifi-cation. Precisely in order to obtain corrected or optimized die it

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would take 5–10 times try-outs for capturing the die behavior pat-tern and its process. There are always uncertainties such as geom-etry parameters and human factors involved thus it affect the costof die modification. In addition to this, production and manufactur-ing team working time is also being compensated. Previous studiesobserved the influences of die design parameters, such as punchradius, die radius, sector, and bend angle, on springback [1]. In pro-ducing high-precision parts, the concern for springback is more fo-cused on the die design instead of on the degree of springbackdeviation on the part design [2]. In most cases, springback isgreatly dependent on die design parameters under minimal loadconditions and decreases as compression depth increases.

Numerous techniques are employed to decrease the amount ofspringback on geometrical defects via die design, as reported in lit-erature. These techniques can be divided into analytical methodand numerical simulation method. Analytical method, such asthose proposed by Marretta et al. [3], involve complex mathemat-ical formulae for the prevention of excessive part thinning and thecontrol of springback but limiting their application to simplegeometries. Behrouzi et al. [4] proposed a one step modificationof the die shape to compensate the springback error by producingthe target shape that can be obtained in one step calculation.Narayanasamy and Padmanabhan [5] investigate punch travel,

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176 N.A. Kamal Bashah et al. / Materials and Design 46 (2013) 175–190

punch radius, punch velocity and width of the sheet by comparingmulti-regression and neural network modeling. In general, Sanchez[6] recognize that experimental framework verification of the ana-lytical method is still required to improve the modeling of spring-back control.

On the other hand, numerical simulation method involves FEsimulation supported by empirical data provides ‘‘boundlessopportunities’’ for the behavior of springback in any complex alter-ation of die shape. Gan and Wagoner [7] develop the DisplacementAdjustment (DA) method based on iteratively comparing a targetpart shape with a FE simulation part shape to compensate spring-back error of the die surface. This method explains how the dis-placement vectors that define the direction opposite to thespringback error at each surface node are used to adjust iterativelythe trial die design until the target part shape is achieved. Later,Lan et al. [8] and Yang and Ruan [9] expand the ‘DA’ method withthe purpose to overcome the issue of the element nodes are irreg-ularly scattered after simulations. In their study [8], the smoothtool surface from offset scattered data points and the input of toolsurface is constructed into a commercial FE code for further simu-lation. Meanwhile, a modified ‘DA’ method namely ComprehensiveCompensation (CC) method [9] was developed to consider largerotation and displacement that may occur during springback.

In practice, despite the accuracy of FE simulations, minoradjustment of compensation method on the actual die design stillhas to be made to produce the intended parts. Normally, to capture100% accuracy of formed parts is very much difficult using FE sim-ulation as the inaccurate spring-back value will still exist. The rea-son being prediction thru FE simulations are mainly based oncertain parameters only but not on uncertainty parameters suchas un-uniformed thickness of blank and human error. Accordingto Hama et al. [10], numerical simulations are still not widely dem-onstrated in the actual formed product. Asgari et al. [11] found thatprediction by simulation could only achieve about 60% accuracy incapturing springback behavior. Most studies focused on a particu-lar approach for specific springback parameters, and their experi-ments and analyses were simple or arbitrarily modeled. Thus,causing difficulty in establishing the cause of discrepancies be-tween the magnitudes of springback obtained from simulationsand actual experiments [12].

The major issue in springback studies, namely, the influence ofspringback on actual stamped part compared with that on die sim-ulation, has not been investigated. Based on actual practice, the ac-tual die requires minor compensation to produce the intended partdesign. Thus, the actual stamping die will differ from the final sim-ulated die. The compensated die may be the crucial element insolving springback issues, as it has not been investigated by previ-ous studies. Therefore, a technique that can obtain data on

Table 1aSelected stamped parts dimension.

Part no. Parts name Stamped part dimensions

1 Side Member (SM)

2 Sill Rear Floor (SRF)

compensated die and to analyze springback accurately should bedeveloped.

This study mainly focuses on demonstrating the applicabilityand adequacy of multi-regression (MR) modeling for the springbackeffect analysis of data that were obtained from actual automotiveBIW stamped parts from an industrial stamping production line.In this paper, the relationship between selected die parametersand deviation of springback between actual stamped part and sim-ulated die is being established. The selected parts are common partsthat are highly likely to be integrated in future model development,and the proposed approach can be replicated thereafter.

2. Methodology

Contemporary data sets of springback-prone components thatwere obtained from actual automotive BIW stamped parts froman industrial stamping production line [1] were analyzed to devel-op an initial springback prediction model by using a statistical MRtechnique. Two different stamping parts correspond to the die sim-ulation parts, namely, Sill Rear Floor (SRF) and Side Member (SM),and were selected as pilot parts for this study. SRF and SM exhibitdifferent degrees of springback severity and have a comparativelynon-linear material, a complex shape design channel structurewith different draw and surface depths, draw bead, addendum,and reinforced features, as well as different ranges of springbackdegree deviation along their lengths. High strength steel (HSS)SPH 440 was used to manufacture the materials used in this study.Tables 1a and 1b illustrate the design and material properties ofthe selected parts, respectively.

Using a systematically random sampling strategy, four samplesof each selected part were collected from two batches of produc-tion lines that consist of 1500 pieces each; two samples were ob-tained from each batch. Using a FARO Arm scanner, the profilesof the sample parts were converted into virtual digitized data toobtain representative 3D geometry values. The digitized data werethen superimposed onto corresponding die simulation data byusing the GEOMAGIC software. Constraint points P1–P4 (seeFig. 1) were used to align the digitized stamped part over the diesimulation. P1 was fixed for translation in the X, Y, and Z directions,P2 was fixed for translation in the Y direction, and P3 and P4 wereboth fixed for translation in the Z direction [11]. To measurespringback and examine any deviations, the digitized profile wascross sectioned into 30 segments (see Fig. 2) and compared withthe die simulation (see Fig. 3). The amount of springback wasmeasured as the normal angle of the points of each part at 30cross-sections. Six die simulation parameters were identified andtheir order of significance was determined using the MR technique

Blank dimensions

Table 1bSPH440 material properties.

Thickness 1.2 mmCoefficient of friction 0.15Poisson ratio 0.0291Young modulus 205.8 GPaYield strength 360 MPaTensile strength 514 MPa

Fig. 3. Springback angular measurement.

N.A. Kamal Bashah et al. / Materials and Design 46 (2013) 175–190 177

(Fig. 4 and Table 2). Angular springback is defined as the anglemeasured between a stamped part under surface and its referencegeometry, which is the die simulation in this study [13]. Spring-back angle has two possible values, i.e., left (L) and right (R). Thesevalues can be equal when the geometry is symmetrical, but theyare not equal in most cases. The angle difference between the dig-itized stamped part (UA) and the surface of die simulation (UB) was

Fig. 1. Constra

Fig. 2. Cross sect

determined to calculate the springback angle, as shown in Fig. 3and by using Eq. (1). However, according to Carden et al. [14],the errors associated with UA and UB are much larger. Therefore,in this study, a summation of these two values was taken system-atically as the total springback response [15]. Eq. (2) was employed

int points.

ion process.

Fig. 4. Die parameters.

Table 2Die parameters.

No. Parameters Notations Units

1 Die radius – right dr-R mm2 Punch radius – right pr-R mm3 Die width dw mm4 Die depth dd mm5 Punch radius – left pr-L mm6 Die radius – left dr-L mm


to calculate the total springback angle (YT) on the left (L) and theright (R) sides.

Springback angle ðYÞ ¼ Stamped part ðUAÞ � Die ðUBÞ ð1Þ

Total springback angle ðYTÞ ¼ Y-Rightþ Y-Left

¼ ðUA �UBÞR þ ðUA �UBÞL ð2Þ

The data sampling for die variables (parameters) and the totalspringback (YT) from the selected parts are shown in Tables 3aand 3b. A total of 240 data from eight samples of selected stamped

Table 3aData sampling for SRF parts.

Cross section location Total springback response (YT)

Sample parts

YS1 YS2 YS3 YS4

1–10 3.34 2.33 4.00 5.192–20 5.62 3.53 3.13 1.113–30 2.00 �0.27 �0.17 0.744–40 1.61 0.75 �0.21 �0.085–50 1.46 1.93 1.06 1.666–60 3.07 2.32 2.79 3.047–70 3.77 3.71 2.06 2.608–80 5.24 4.65 3.72 3.899–90 5.36 6.01 4.36 4.60

10–100 4.22 4.26 3.37 2.9511–110 4.14 4.67 3.10 3.1812–120 3.62 7.08 5.60 6.0013–130 5.85 6.54 4.39 4.9414–140 4.91 5.43 4.60 4.3615–150 4.82 5.04 3.88 4.5916–160 5.19 5.93 3.51 3.8017–170 6.41 6.17 5.13 4.7618–180 5.76 5.93 5.66 5.0719–190 6.84 6.82 5.24 5.4720–200 1.56 �1.27 1.30 3.0921–210 3.03 2.45 2.98 2.6922–220 4.56 3.94 3.98 4.4423–230 4.22 3.54 2.40 3.1124–240 �4.33 1.60 3.58 3.1325–250 1.27 2.87 1.90 2.2326–260 1.22 2.16 1.01 0.5627–270 1.39 1.35 0.26 0.8928–280 1.80 0.45 0.06 0.7629–290 6.55 3.84 1.44 0.5930–300 2.72 1.89 1.86 1.40

parts (120 data for SRF and 120 data for SM) were tabulated for theMR analysis to produce a springback prediction model. Therefore,in this study, the four samples of each part were used to producethe MR model and one sample was selected randomly from thefour samples for validation. Finally, the accuracy of the proposedmodel was validated by measuring the predicted value againstthe measured one.

3. Multi-regression (MR) technique

In this paper, MR technique was used to examine the sensitivityof the independent variables (die simulation) that affect the depen-dent variable (springback angle) on selected sample parts corre-sponding to die simulation data. The levels of the independentvariables were adjusted to evaluate their interaction and effectson the dependency of a single dependent variable and to determinethe springback prediction ability of the model. The overall processis illustrated below (see Fig. 5).

Dependent and independent variables were assumed to have alinear relationship [16], which can be expressed using a mathemat-ical MR model as follows:

Y ¼ f ðXiÞ ð3Þ

f ðXiÞ ¼ aþX

biXi þ e or;

¼ aþ b1X1 þ b2X2 þ � � � þ bnXn þ eð4Þ

where Y is a dependent variable that represents the measured valueof the springback angle, Xi is an independent variable that repre-sents the value of the die simulation parameters, a denotes theintercept, bi corresponds to the slope of the linear relation betweenthe means of the dependent and independent variables, and e is therandom error with a mean of zero. To compare and normalize all

Die parameters

dr-R pr-R dw dd pr-L dr-L

7.94 5.00 230.00 44.15 6.30 9.877.94 5.00 197.10 43.61 6.37 8.863.86 5.11 173.26 43.51 6.93 4.013.01 5.08 155.44 44.99 7.89 3.933.02 5.12 148.69 45.48 6.61 4.013.65 5.13 146.80 45.80 7.65 4.022.99 5.20 145.48 46.56 8.09 3.942.99 5.18 147.08 47.20 8.03 3.983.00 5.18 146.36 47.78 8.03 4.013.03 5.19 142.61 48.29 7.00 3.963.04 5.18 145.89 48.89 8.04 3.973.04 5.18 144.69 49.37 8.04 4.013.04 6.58 142.33 49.82 8.10 3.993.01 5.18 143.53 50.24 8.11 3.963.01 5.19 143.37 50.08 8.11 3.963.01 5.19 143.67 50.08 8.11 3.963.01 5.18 143.91 50.24 8.11 3.963.04 6.58 141.68 49.82 8.10 3.993.04 5.18 145.05 49.36 8.04 4.013.04 5.18 145.87 48.89 8.01 3.973.03 5.19 142.98 48.29 7.00 3.963.00 5.18 145.72 47.79 8.03 4.012.99 5.18 146.69 47.20 8.03 3.982.99 5.20 145.66 46.55 9.33 3.943.65 5.13 146.62 45.95 7.65 4.023.02 5.12 148.78 45.44 6.61 4.013.01 5.08 154.55 45.02 7.98 3.933.86 5.11 170.75 43.51 6.93 4.017.94 5.00 196.11 43.61 6.37 8.867.94 5.00 229.05 44.15 6.30 9.87

Table 3bData sampling for SM parts.

Cross section location Total springback response (YT) Die parameters

Sample parts

YS1 YS2 YS3 YS4 dr-R pr-R dw dd pr-L dr-L

1–10 �1.32 �1.33 �1.81 �1.77 14.11 16.64 158.71 170.79 11.85 7.842–20 �0.27 �0.77 �0.65 �0.65 14.05 13.98 147.97 173.05 11.97 7.823–30 �3.05 �2.38 �2.09 �2.34 14.04 14.08 149.26 175.09 12.00 7.634–40 �3.21 �2.77 �2.23 �2.41 14.03 14.07 148.54 174.82 11.92 7.655–50 �1.36 �1.12 �0.76 �0.76 14.03 14.07 148.71 174.37 12.06 7.676–60 �4.25 �3.09 �3.01 �3.09 14.57 13.90 151.02 174.09 21.75 7.917–70 �3.22 �1.14 �0.69 �0.73 14.54 14.07 159.76 159.54 14.09 8.878–80 �3.04 �2.41 �2.93 �3.12 14.48 14.50 171.12 143.92 13.35 7.999–90 �2.65 �2.55 �2.64 �2.40 14.41 16.04 182.03 128.65 13.35 8.66

10–100 �3.07 �1.38 �2.08 �2.09 14.95 17.19 196.32 123.14 13.35 8.3411–110 �1.01 0.32 �0.31 �0.36 28.81 13.08 194.71 114.67 13.35 8.6112–120 �1.03 0.96 0.62 0.62 13.46 11.45 150.74 107.82 13.84 8.3713–130 �1.18 �1.46 0.43 �0.49 13.38 11.96 121.96 98.68 13.27 8.3614–140 2.10 1.96 2.15 2.08 14.05 13.10 109.92 97.25 12.94 8.2715–150 1.60 2.49 2.78 2.23 14.76 13.45 108.17 97.65 12.84 8.1316–160 2.63 2.77 2.90 2.12 7.88 13.78 108.69 96.56 12.20 7.6517–170 3.10 4.33 3.97 2.53 7.89 13.78 104.61 91.75 11.92 7.5518–180 4.15 4.39 3.95 2.84 7.94 10.39 100.47 83.28 10.83 7.4219–190 3.19 2.82 2.06 2.10 7.95 9.79 96.68 76.24 9.85 7.1220–200 3.03 3.07 3.80 3.10 5.00 4.93 92.91 74.55 9.80 7.1321–210 3.85 3.60 2.75 2.36 4.93 4.87 91.83 73.62 9.80 7.3822–220 3.14 3.26 3.09 2.52 7.87 9.80 93.06 72.70 9.80 8.1523–230 3.64 3.06 2.51 1.80 7.95 9.80 93.36 72.77 9.79 7.2024–240 2.27 2.38 1.63 0.93 7.95 9.80 93.70 71.85 9.79 7.2325–250 2.74 3.55 4.98 4.49 8.44 9.80 93.75 70.87 9.79 7.2326–260 0.71 1.36 0.77 0.77 7.66 9.80 94.11 70.91 9.79 7.3727–270 0.10 0.21 0.21 0.03 7.41 9.80 94.29 70.90 9.79 7.3328–280 0.22 0.32 0.03 �0.05 7.42 9.79 94.49 70.90 9.79 7.3429–290 0.06 �0.18 �0.48 �0.66 7.42 9.79 94.67 70.89 9.79 7.3930–300 0.84 0.42 0.08 �0.13 7.42 9.79 94.63 70.89 9.79 7.40

Fig. 5. MR analysis procedures.


relations between the dependent and independent variables, theMinitab 15 Statistical Analysis Software was used to analyze thesampling data and the variables’ responses, as well as to developthe initial springback prediction model. Thus, using Eq. (4), the ex-

pected or predicted springback model can be written with the errorterm e left unexplained as follows:

Y 0 ¼ f ðXiÞY 0 ¼ aþ b1dr-Lþ b2pr-Lþ b3dw þ b4dd þ b5dr-R þ b6pr-R

ð5Þ

where Y0 is the predicted (measured) springback value, i.e., Y (esti-mated from Xi).

To employ Eq. (5), each individual Xi data from Eq. (4) is multi-plied by the resultant bi value and the constant a is added to thesum, yielding Y 0, which is the springback predicted value of Y.The values of a and the bi’s from the given set data were mathemat-ically determined to minimize the variation between Y 0 and Y.Using such model, the actual die parameters value can be substi-tuted in the equation to predict the future springback value of Y0.

4. Results and discussion

In this study, the formulation of MR modeling for the spring-back prediction model was performed using two automotive BIWstamped parts, namely, SRF and SM. Four samples of each partwere collected based on a systematic random sampling strategyvia two batches of the respective production lines with 1500 pieceseach. Each batch consisted of two samples with 30 data points(sectional) that were strategically located across each sample. A to-tal of 240 data from eight sample parts (120 data for SRF and 120data for SM) were tabulated to create the springback predictionmodel. Minitab 15 with a significance level (a) of 5% [17,18] wasused for the analysis. The following assumptions were made in thisstudy:

(1) Production processes are statistically stable or in controlthroughout the study.


(2) The springback exhibited a minimal effect on the study sam-ples under different production runs/plans. The types ofeffect included, but were not limited to, the following:

a. Machine setting.b. Material quality.c. Die setting and alignment.

If the analysis reveals significant evidence that would proveotherwise, the production processes significantly contributes tothe springback effect. In such a case, the MR model would beinadequate in explaining the variations in data because no pro-cess parameters were included in the MR model study. A numberof iterations were performed on the original data sets to developthe final MR model for the two parts. Several points were ex-cluded from the analysis because of abnormalities found in thereadings. A close and thorough physical analysis of the actualparts at specific locations in relation to the data points yieldedthe following:

(1) The points in data points with low residual value are locatedclose to the addendum (design compensation).

(2) Several points with high residual value, which are abnormal-ities in the part formation, are observed based on sample tosample comparison of the same part.

Abnormalities in the readings are known to be due to specialcauses, and thus, excluding those data from the MR analysis wasan appropriate strategy to minimize the number of errors intro-duced into the model. SRF was left with 98 data and SM was leftwith 105 data from the original 120 data for each part.

4.1. Pre-MR analysis on SRF and SM data

Pre-analysis on both parts provided insights on the nature ofthe springback effect on the parts as well as on understandingthe underlying statistical nature of the data set. Pre-analysis con-ducted on the data also revealed significant differences in thespringback data among the samples caused by the instability ofthe processes. Figs. 6a and 6b show the sampling of the SRF andSM parts from the four samples against the location (sectional) ofthe data on the part. The drop line shows the magnitude of theinherent differences in the data on a specific location on the part.

Fig. 6a. Y-angular vs. dat

A longer drop line indicates larger differences (for example, loca-tions 2, 24, and 29 in Fig. 6a, and locations 7, 25, and 12 inFig. 6b). As can be seen in Figs. 6a and 6b, the differences betweensamples 1 and 2, and those between samples 3 and 4 were smallercompared with those between all other parts. The mean differ-ences between the batches on the other graph were likely to beseen because samples 1 and 2 came from batch 1 and samples 3and 4 came from batch 2. Figs. 7a and 7b show the ranges of differ-ences between the maximum reading minus the minimum readingfrom the four samples on a specific location on the part (maxij

�minij = range). For the SRF sample (Fig. 7a), the range for locationnumber 24 was the highest (7.91), followed by location number 29(5.96), and then location number 2 (4.51). For the SM sample(Fig. 7b), the range for location number 7 was the highest(2.527), followed by location number 25 (2.2416), and then loca-tion number 12 (1.986).

Comparing Figs. 8a and 8b, a difference in the magnitude ofrange between the two batches of the two parts was noted. Forexample, the maximum range for the SRF sample was 4.7 (seeFig. 8a for location 24), whereas that for the SM sample was 1.5(see Fig. 8b for location 25). Thus, the magnitude of range for theSR sample was almost triple of that for the SM sample. A closerobservation of the two graphs revealed that the difference in rangeoccurred in every location of both parts. Based on the comparisonabove, the analysis revealed that SM parts are more robust (consis-tent) to the springback effect compared with SRF parts. SM partshave a complex design, whereas SRF parts have a simple design.A complex design is more robust to the springback effect than asimple design.

The following procedure was conducted for the equality of thetwo samples’ mean tests:

(1) The normality of the data was tested using the Anderson–Darling normality test and by comparing the p-value witha known a. The data were normally distributed [19] whenthe p-value was higher than a.

(2) If the two data sets are significantly different from eachother in terms of mean value and variance, then MR analysiscan be separately conducted on the SRF and SM parts. Other-wise, if the two data sets are not significantly different fromeach other, then only one MR analysis can be conducted onboth parts.

a location (SRF part).

Fig. 6b. Y-angular vs. data location (SM part).

Fig. 7a. Range of Y-angular vs. data location (SRF part).

Fig. 7b. Range of Y-angular vs. data location (SM part).


Figs. 9a and 9b show the graphical plots of the summary of theAnderson–Darling normality test and the statistics of the two data-sets from the SRF and SM parts, respectively. The p-values for the

Anderson–Darling normality test were 0.360 and 0.005 for theSRF and SM parts, respectively. Based on these values, it can beconcluded that SRF data were normally distributed whereas SM

Fig. 8a. Batch’s average Y-angular vs. data location (SRF part).

Fig. 8b. Batch’s average Y-angular vs. data location (SM part).


data were not. The mean values for SRF and SM data were 3.2466and 0.482 with standard deviations of 2.005 and 2.297, respec-tively. These values show the statistical difference between thetwo sets of data and suggest that the two sets of data come fromdifferent populations. The pre-analysis provided some insights onthe nature of the springback effect on the parts. The pre-analysisshowed that the SRF part had a higher springback effect comparedthan the SM part. The SRF part was slightly more consistent thanthe SM part based on the variance values of 4.01 (for SRF) and5.28 (for SM), in which a smaller variance value indicates betterconsistency. These results indicate that the regression analysesfor the SRF and SM parts were conducted separately, thus develop-ing two regression models.

4.2. MR matrix plots analysis

Matrix plot analysis was conducted to determine the significanteffects of die variables on the springback angular (Y-angular) effect.Figs. 10a and 10b show the matrix plot analyses of each column ofdata against other selected columns. For the SRF sample (Fig. 10a),two out of six variables, namely, die depth (dd) and die width (dw),

slightly correlated with the springback effect. A probability to in-clude dd and dw as variables in the MR analysis existed at suchpoint. For the SM part (Fig. 10b), four out of six variables, namely,die radius-L (dr-L), punch radius-R (pr-R), die depth (dd), and diewidth (dw), showed correlation with the Y-angular effect. Correla-tion analysis between the Y-angular effect and the six die variableswas conducted to test the strength of the correlation between indi-vidual variables and their responses.

4.3. Correlation analysis

Tables 4a and 4b show the correlation for the lower triangle ofthe correlation matrix, including the p-values for the individualhypothesis tests with correlations of zero.

In the left columns of Tables 4a and 4b, the correlation analysistables show that the Y-angular effect was highly correlated with allthe die parameters with a significant p-value of 0.000 for bothparts, except for two variables in the SRF part, namely, die ra-dius-L (dr-L) and die radius-R (dr-R) with p-values of 0.636 and0.301, respectively. Moving from the left to the right columns,the test shows that the variables were also highly correlated with

Fig. 9a. Graphical summary of statistics for SRF data.

Fig. 9b. Graphical summary of statistics for SM data.


Fig. 10a. Matrix plots for SRF part.


one another with as their p-values for correlation were smallerthan the known a = 5% (0.05). These results are initial indicationsof potential multi-collinearity or the Variation Inflation Factor(VIF) effect among the predictors in the regression model [20].Based on the analysis for the SRF part, the initial conclusion wasto exclude the two predictors, namely, die radius-L (dr-L) and dieradius-R (dr-R), in the regression analysis because of the highly po-tential multicollinearity effect in the regression model. Furtheranalysis was conducted using the best subset regressions to deter-mine the effect of excluding the two variables or including all vari-ables, as well as the best variable combination in the regressionmodel.

4.4. Best subsets analysis

MR analysis with the best subset regression [5] was conductedto determine significant die variable effect on the springback angu-lar effect in the regression model. All possible variables from thesubsets were evaluated with one parameter, two parameters, andso on. Tables 3a and 3b show the best subset regressions for bothselected parts.

Each line of output in Tables 5a and 5b represents a differentmodel. Indicator ‘Vars’ represents the number of die variables inthe model. The variables present in the model were indicated bythe sign ‘

p’. For the SRF part (Table 5a), the three best subset mod-

els were reviewed, first, two of the four-variable models, and then,the six-variable model. The four-variable model (�) exhibited thehighest adjusted R2 (81.3%), the lowest Mallows’ Cp value (3.6),and the lowest S value (0.73639). On the other hand, the six-vari-able model with all the variables (��) demonstrated a higher Cp

value (7.0), a slightly higher S value (0.74196), and a slightly loweradjusted R2 (81.0%) than the four-variable model. The followingthree factors were considered in choosing the best MR model:

(1) High adjusted R2 [20].(2) Low Mallow Cp score [21]. The rule of thumb dictates that Cp

should be smaller than the total number of variables in themodel plus the constant. For instance, in the four-variablemodel, Cp should be lower than 5.

(3) The number of variables should be balanced between the leftand right sides of the part. Two of the variables, namely, dieradius and punch radius, were symmetrical. Therefore, hav-ing a die radius-L (dr-L) in the model without a die radius-R(dr-R) would result in an imbalance.

The four-variable model (��) was selected based on the afore-mentioned considerations supported by the correlation analysis(dr-L and dr-R did not significantly correlate with the Y-angular ef-fect). This model exhibited the second highest adjusted R2 at 81.2%,a lower Cp of 4.2, and a lower S of 0.73864 than the six-variablemodel. The difference was insignificant, although the hypothesistest scores were slightly higher compared with those of the four-variable model (�).

The two best subset models for the SM part (Table 5b) were re-viewed, first, the three-variable model and then, the six-variablemodel. The three-variable model (�) had the highest adjusted R2

(62.8%), the lowest Mallows’ Cp value (3.2), and the lowest S value(1.4012), whereas the six-variable model with all the predictors(��) had a higher Cp value (7.0), a slightly higher S value comparedwith the previous model (1.4060), and a slightly lower adjusted R2

Table 4aCorrelation for SRF part: Y-angular vs. die parameters.

Y-angular dr-L pr-L dr-R pr-R dd

dr-L �0.0440.636

pr-L 0.259 �0.6700.004 0.000

dr-R �0.095 0.984 �0.7050.301 0.000 0.000

pr-R 0.351 �0.253 0.313 �0.2750.000 0.000 0.001 0.002

dd 0.616 �0.538 0.651 �0.612 0.4540.000 0.000 0.000 0.000 0.000

dw �0.206 0.943 �0.714 0.943 �0.319 �0.6840.000 0.000 0.000 0.000 0.000 0.000

Table 4bCorrelation for SM part: Y-angular vs. die parameters.

Y-angular dr-L pr-L dr-R pr-R dd

dr-L �0.4600.000

pr-L �0.557 0.5940.000 0.000

dr-R �0.555 0.738 0.5800.000 0.000 0.000

pr-R �0.630 0.620 0.578 0.6610.000 0.000 0.001 0.000

dd �0.739 0.454 0.628 0.612 0.7350.000 0.000 0.000 0.000 0.000

dw �0.749 0.727 0.619 0.835 0.776 0.7690.000 0.000 0.000 0.000 0.000 0.000

Fig. 10b. Matrix plots for SM part.


(62.6%). A high S value in both models was an early indication ofthe amount of errors inherent in the regression model (S shouldbe kept as small a possible). The S value of the SM part was rela-tively double in magnitude compared with that of the SRF part.This result is an indication of a later ‘‘lack of fits’’ condition inthe regression model based on the knowledge gained on the SRFpart. Therefore, the six-variable model (��) was selected based onthe considerations highlighted above.

4.5. MR model of springback prediction

The results of the analysis indicate that the springback angulareffect for each selected part can be predicted and expressed usingthe single generic linear MR model as follows Tables 6a and 6b:

Based on the models above, the regression coefficients with cor-responding die variables were shown to be a measure of howstrongly each of the items interacts and influences the effect ofthe springback angle. The positive and negative values of theregression coefficients indicated that optimal die variables canminimize the opening distance of the springback effect [22]. Foreach die variable, the size of the effect that the variable was havingon springback angle response was based on the sizes of the regres-sion coefficients. Moreover, the positive or negative sign of thesecoefficients would give the final direction of the response. Matsuy-ama and Ohtsuka [2] found that the positive and negative signswould cause the springback effect to orient itself in the reversedirection of the normal springback, implying that regression coef-ficients can significantly reduce the springback effect. The esti-

Table 5aBest subsets regression for SRF part.

Vars R2 (%) R2ðadjÞ (%) Mallow Cp S dr-L pr-L dr-R pr-R dd dw

1 76.2 75.9 27.7 0.83257p

1 39.4 38.8 215.8 1.3328p

2 78.7 78.3 16.9 0.79428p p

2 78.7 78.2 17.1 0.79515p p

3 81.2 80.6 6.1 0.75036p p p

3 81.1 80.5 6.6 0.75210p p p

4� 82.1 81.3 3.6 0.73639p p p p

4�� 82.0 81.2 4.2 0.73864p p p p p

5 82.2 81.2 5.0 0.73792p p p p p

5 82.1 81.2 5.3 0.73913p p p p p

6�� 82.2 81.0 7.0 0.74196p p p p p p

Table 5bBest subsets regression for SM part.

Vars R2 (%) R2ðadjÞ (%) Mallow Cp S dr-L pr-L dr-R pr-R dd dw

1 56.1 55.7 23.6 1.5290p

1 54.7 54.3 28.1 1.5540p

2 62.6 62.0 4.8 1.4165p p

2 57.7 57.0 20.5 1.5073p p

3� 63.8 62.8 3.2 1.4012p p

3 63.1 62.1 5.5 1.4148p p p

4 64.2 62.9 3.9 1.3994p p p p

4 63.8 62.6 4.9 1.4055p p p p

5 64.5 62.9 5.0 1.3998p p p p p

5 64.2 62.6 5.9 1.4053p p p p p

6�� 64.5 62.6 7.0 1.4060p p p p p p

Table 6aThe MR model for SRF part.

Y0 = �37.6 + 0.5421pr-L + 0.0228dw + 0.6544dd + 0.4447pr-R (6)

Table 6bThe MR model for SM part.

Y0 = 0.14 + 0.161dr-L + 0.847pr-L � 0.059dw � 0.00297dd � 0.260dr-R � 0.013pr-R

(7)

Table 7bMR statistics output 1-SM part.

R2 73.6%

R2adj

72.0%

Standard error (S) 1.1495

Table 8aMR statistics output 2-SRF part.

Coefficients Standard error t-Stat p-Value VIF

a �37.62 2.632 �14.29 0.000b2 0.5421 0.1576 3.44 0.001 2.121b3 0.0228 0.004851 4.71 0.000 2.089b4 0.6544 0.04953 13.21 0.000 2.268b5 0.4447 0.2106 2.11 0.037 1.255

Table 8bMR statistics output 2-SM part.


mates gave better knowledge and understanding that die variablesare related and have direct and positive relationships with one an-other [23], significantly affecting the springback. Moreover, thecapability of the MR model to predict the springback effect wasalso directly and significantly affected. Nevertheless, giving a directphysical justification and interpretation of the proposed regressioncoefficients was not always possible. The regression coefficientvalues of a = �37.6 and 0.14 were the expected values of thespringback response in the proposed MR model because they cor-respond to a setting when all the values of the die variable were setto zero. In general, a does not have a direct, practical, and meaning-ful interpretation [16], which will be impossible to achieve [19]. Inconclusion, the significant effect of each die variable on the spring-back was identified and explained in this study.

Table 7aMR statistics output 1-SRF part.

R2 82.0%

R2adj

81.2%

Standard error (S) 0.73864

Tables 7a and 7b show the statistical results of the MR analysis.Both models displayed a low standard error (S) value, indicatingthat the proposed MR model predicted the springback angle re-sponse with minimal errors [24]. The proportion of variance inthe springback response, which can be explained using the die

Coefficients Standard error t-Stat p-Value VIF

a 0.1410 2.852 0.05 0.961b1 0.1605 0.5055 0.32 0.751 4.506b2 0.8469 0.1921 4.41 0.000 6.778b3 �0.0590 0.0081 �7.26 0.000 5.536b4 �0.0029 0.0062 �0.48 0.632 4.720b5 �0.0130 0.07432 �0.18 0.861 4.141b6 �0.2600 0.09625 �2.70 0.008 9.027


variables, gave a better fit for coefficient of determination value(R2) at 82% for SRF and 73.6% for SM parts. Only 18% of the SRF partand 26.4% of the SM part were unexplained because of factors orlimitations encountered in the MR models. The instabilities inthe manufacturing processes, as well as in the material and processparameters, are highly likely the causes of the differences in the re-sults. The R2 values of both parts indicated that the die variablescan explain the variations in the springback effect. The analysis

Table 9aThe ANOVA of the MR model-SRF part.

Source ofvariance

Degree offreedom (df)

Sum ofsquares (SS)

Meansquares(MS)

F p

Regression 4 230.623 57.656 105.68 0.000Residual 93 50.739 0.546Lack of fit 25 19.932 0.797 1.76 0.035Pure error 68 30.807 0.453

Total 97 281.362

Table 9bThe ANOVA of the MR model-SM part.

Source ofvariance

Degree offreedom (df)

Sum ofsquares (SS)

Meansquares(MS)

F p

Regression 6 360.486 60.081 45.47 0.000Residual 98 129.482 1.321Lack of fit 21 111.731 5.321 23.08 0.000pure error 77 17.751 0.231

Total 104 489.967

Fig. 11a. Residual plots fo

of the MR models for both parts exhibited a better fit of adjustedcoefficient of determination value (R2

adj) at 81.2% for SRF and72.0% for SM, indicating the significant relationship of the regres-sion coefficients in the die values, as well as the high variance thatmakes the models stronger. These R2

adj values indicate that themodels fit the data well. The results above show the strong and sig-nificant relationships of the predicted springback response (Y0)with die variable variance in the models.

p-Values for the estimated coefficients of the SRF show a signif-icant relationship with the springback angular effect because theywere approximately less than 0.05, as shown in Table 8a. More-over, three out of six variable coefficients of the SM part, namely,b2, b3, and b6, were significantly related to the springback response(Table 8b). VIF scores for all variables of both parts were low, indi-cating a low multicollinearity issue among the variables. The com-bination of variables that best minimizes the multicollinearityeffect in the model include the variables with moderate (<10) tolow (<5) VIF scores [20]. Such combination is crucial for evaluatingthe contribution of individual die variables to the springback re-sponse. The results above indicate the significant relationship be-tween the regression coefficients and the die variable variances[25], as well as the high variance that makes the model strongerand relatively adequate [5]. The MR models can be used to predictthe effect of the die variables on springback performance, with theconsideration that the values of the die variables are within theranges concerned. The relative significance of the die variablescan also be observed from the proposed coefficients, which areimportant parts of the models.

Analysis of variance (ANOVA) was used to determine the depen-dency of the springback response to selected die variables. Tables9a and 9b show the results of ANOVA, which supported the stronglinear relationships in the model. The p-values of the SRF and SMparts show that the model estimated using the regression proce-

r Y-angular SRF part.

Fig. 11b. Residual plots for Y-angular SM part.


dure is significant at an a-level of 0.05, i.e., confidence interval of95% [17,18]. Furthermore, the table shows the number of errorsin the model broken down into two portions, namely, Lack-of-Fitand Pure Error. Lack-of-Fit was used to test how well the model ex-plains the data, whereas Pure Error was used as a random variationinherited in the data. The p-value of Lack-of-Fit for SRF was 0.037,which is significant at an a-level of 0.05. On the contrary, the p-va-lue of Lack-of-Fit for SM was 0.000, which is significant at an a-le-vel of 0.05 and indicates a great significance of 0.000 for the overallregression relationship that is less than the level of significance ofa = 0.05. The presented MR models are important for predictingspringback because the probability of the p-value is less than thelevel of significance of a = 0.05 [26]. These results can be explainedfurther using graphical residual analysis.

The normal probability plotted diagram presents the measuredspringback response value ( red dotted line),1 which was a goodmatch and was close to the line that represents the predictedspringback response value (blue diagonal line), as shown in Figs.11a and 11b. As can be seen in the first three residual plots inFig. 11a, the data were randomly distributed (approximate to nor-mal distribution) and independent. No evidence of significant pat-terns was observed on any of the residual plots that violated theadequacy (goodness of fit) of the regression model. However, theStandardized residual versus Observation order final plot shows ashift in the data pattern. Observation numbers 35–49 have higherstandardized residual values compared to the rest. The issue orig-inated from sample number two because the data came from fourdifferent samples with 30 data sets from each sample (120 datasets in total). This also explains the significant p-value of Lack-of-Fit in the previous analysis. The difference is suspected to have

1 For interpretation of color in Figs. 11a and 11b, the reader is referred to the webversion of this article.

been caused by the significant variations among the four samplesused in this study. Moreover, as can be seen in the first residualplot in Fig. 11b, the data were randomly distributed and indepen-dent. In the second plot, evidence of slight distortion was observedin the standardized residual pattern, displaying a curved shape(quadratic). This analysis, reinforced by the fourth plot, shows anincreasing trend in the standardized residual pattern in the middleof the observation order of each sample from the first to the third(10–20, 40–50, and 70–80). It shows satisfactory results withstrong positive correlations [21,23] between the springbackresponse and the die variables involved. This also justifies the esti-mated springback value variations measured around the regressionline. Narayanasamy and Padmanabhan [5,27] reported such posi-tive effect when they applied the MR technique for springback pre-diction in the air bending process of interstitial free (IF) steel sheet.Firat et al. [28] also reported a positive effect when they used theMR technique in conjunction with numerical analysis for spring-back prediction in the stamping process design. The residual distri-bution concerning the predicted springback values has a lineartendency, clearly indicating that the proposed MR model is well fit-ted to the data.

4.6. Model evaluation for comparison of predicted and measuredspringback values

The performance adequacies of the proposed models for pre-dicting springback defects were evaluated using a comparisonanalysis between the measured springback and the predictedspringback responses. The results of the comparison analyses arepresented in Table 10.

The adequacies of the proposed models were evaluated bycomparing the predicted results and the measured values of thespringback angle (Table 10). The first column in Table 10 repre-

Table 10Comparison of measured and predicted springback values.

Cross sections Springback response (U�) Error (U�)

Measured (Y) Predicted (Y0)

SRF SM SRF SM SRF SM

1–10 5.19 �1.77 2.15 �2.316 3.03 0.552–20 1.11 �0.65 1.09 �1.545 0.01 0.893–30 0.74 �2.34 0.83 �1.627 0.09 0.724–40 �0.08 �2.41 1.91 �1.643 1.99 0.775–50 1.66 �0.76 1.39 �1.536 0.27 0.776–60 3.04 �3.09 2.13 6.442 0.91 9.537–70 2.60 �0.73 2.87 �0.361 0.27 0.378–80 3.89 �3.12 3.28 �1.744 0.61 1.379–90 4.60 �2.40 3.65 �2.235 0.95 0.17

10–100 2.95 �2.09 3.34 �3.269 0.39 1.1811–110 3.18 �0.36 4.36 �6.657 1.19 6.3012–120 6.00 0.62 4.65 0.344 1.36 0.2813–130 4.94 �0.49 5.54 1.603 0.61 2.0914–140 4.36 2.08 5.23 1.835 0.86 0.2515–150 4.59 2.23 5.12 1.636 0.54 0.6016–160 3.80 2.12 5.13 2.779 1.33 0.6617–170 4.76 2.53 5.24 2.771 0.48 0.2418–180 5.07 2.84 5.53 2.134 0.46 0.7119–190 5.47 2.10 4.65 1.500 0.82 0.6020–200 3.09 3.10 4.35 2.523 1.26 0.5721–210 2.69 2.36 3.35 2.647 0.65 0.2922–220 4.44 2.52 3.63 1.867 0.80 0.6523–230 3.11 1.80 3.27 1.674 0.16 0.1224–240 3.13 0.93 3.53 1.660 0.41 0.7325–250 2.23 4.49 2.22 1.530 0.01 2.9626–260 0.56 0.77 1.38 1.733 0.81 0.9727–270 0.89 0.03 1.96 1.781 1.06 1.7528–280 0.76 �0.05 0.78 1.769 0.02 1.8229–290 0.59 �0.66 1.07 1.766 0.48 2.4330–300 1.40 �0.13 2.13 1.773 0.74 1.90

Average error 0.75 1.41


sents 30 dice sections that were taken systematically from selectedstamped parts and die simulations. The second column representsthe springback measured value, and the third column representsthe theoretically predicted springback values calculated from theMR model presented in Eqs. (6) and (7). The fourth column repre-sents the error difference between measured and predicted values.The average error of the MR model for SRF the part was 0.75�,whereas the average error of the MR model for the SM part was1.41�. The estimated errors of the springback angle for the pre-dicted results of both models were between 0.5� and 3�. This rangecorresponds to the estimated errors for the springback angle be-tween 2� and 3� observed by Banu et al. [13] and those between0.9� and 2.1� documented by Naceur et al. [22] when comparingthe experimental results with the simulated results. Gosling et al.[29] reported a similar phenomenon in the springback angle rangeby using tool modification values varying from �1� to 1�, withflange angle values from 0.4� to 0.6� and stiffness modification val-ues from 0.1� to 0.5�. Firat et al. [30] predicted the springback angleby using FE analysis and regression models and compared the re-sults with the experimental data, obtaining an estimated error be-tween 0.3� and 3.4�. In certain published guidelines [31],recommendations and indications were made for specifying openwall angles of at least 6� for HSS-DP600 materials, 8� for HSS-DP800 materials, and 10� for HSS-DP1000 materials. The proposedMR models adequately predicted springback defects with an aver-age error of 0.75� and 1.41� by using prediction and measurementtechniques. The averages of the predicted results obtained from themodel are also reasonably approximate of and are close to the ac-tual springback measurements. Thus, the estimated error is accept-able for industrial standards [13]. The large errors in the datapoints were expected, except for the data points 3.03� of SRF and9.53� and 6.30� of SM. These errors necessitated further analysis,

which concluded that the large errors in the model were causedby the large variations inherited from the samples used in thestudy. The introduction of large variations in the data is causedby the unstable production processes of the parts used in thisstudy. This discrepancy is also caused by irregularities on thedice-sectioning of the sample parts, which caused by part anddie design complexity and deformations. deSouza and Rolfe [32]stated that variations in the incoming sheet material and fluctua-tions in the press setup are unavoidable in many stamping plants.Furthermore, Govik et al. [33] reported that part variation is to beanticipated in the part production process. The production of qual-ity parts is a challenge because these parts may vary from produc-tion line to production line. In addition, varying parts may beproduced with multiple features and dimensions. Marretta et al.[3] summarized that materials, tools, processes, and lubricantsare the potential sources of variations in the stamping process. Go-vik et al. [33] suggested that other parameters, such as materialand process, can drastically change the results. Although spring-back absolute values are generally underestimated, they show cor-rect tendencies [34]. The models at hand are needed for developingthe predictive model for springback effect because the materialand process parameters are beyond the scope of this study. Futurestudies can enhance the abilities of the proposed models by per-forming and supplying more data to the MR analysis.

5. Conclusions

Two MR models for springback prediction, including SRF andSM parts, were produced in this study. These models, with compar-ison data in Table 8, were produced to provide die designers arough estimate of the springback effect in relation to design vari-ables namely, die radius, punch radius, depth, and weight. Bothmodels have the capability to predict the springback responsebased on high R2 predicted values backed by the results of the val-idation procedure, which generated an angular error of less than 3�on average. This study also observed the consistency of the spring-back effect to be strongly related to part design, with the assump-tion that the process is stable. Complex designs were observed tohave a more robust, smaller, and consistent springback effect com-pared with simple designs. SRF parts resemble simple designs,whereas SM parts resemble complex designs. Thus, simple designsare less stable than complex designs in terms of springback,whereas complex designs are more stable or robust. This studywas conducted using the same materials and process parameters,and thus, the proposed model is only valid for parts under similarconditions. The selected parts are common parts with high proba-bilities of being carried forward into future model developments.In addition, the effectiveness of the MR model presented an excel-lent degree of approximation and correlation of springback re-sponses with the proposed die variables. Consequently, theinclusion of draw bead parameter and process parameter is ex-pected to improve the reliability of the proposed models. Theseparameters, as well as the effects of variations involved, are consid-ered to anticipate others parameters and to predict springback ofMR models accurately. The proposed models are also recom-mended to be further ‘‘experienced’’ in the MR analysis with largerdata, more stamped parts, and more design variables from differ-ent types of vehicles.

Acknowledgments

The authors would like to acknowledge the Ministry of Science,Technology, and Innovation (MOSTI) of Malaysia and PerusahanAutomobil Nasional Holdings Sdn. Bhd. (PROTON) for the financialsupport under the Technofund project (TF0608C073-17) of the


Computationally Optimized Fuel-Efficient Concept Car (COFEC2)and for the cooperation of all those concerned in this project,who assisted with valuable help and support.

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Multi-regression modeling for springback effect on automotive body in white stamped parts