Probabilistic Design in a Sheet Metal Stamping Process under Failure Analysis Thaweepat Buranathiti a , Jian Cao a *, Z. Cedric Xia b , Wei Chen a a Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208. b Ford Scientific Research Laboratory, Dearborn, Michigan 48121. * Corresponding author. Tel.: +1-847-467-1032; Fax: +1-847-491-3915; [email protected]Abstract. Sheet metal stamping processes have been widely implemented in many industries due to its repeatability and productivity. In general, the simulations for a sheet metal forming process involve nonlinearity, complex material behavior and tool-material interaction. Instabilities in terms of tearing and wrinkling are major concerns in many sheet metal stamping processes. In this work, a sheet metal stamping process of a mild steel for a wheelhouse used in automobile industry is studied by using an explicit nonlinear finite element code and incorporating failure analysis (tearing and wrinkling) and design under uncertainty. Margins of tearing and wrinkling are quantitatively defined via stress-based criteria for system-level design. The forming process utilizes drawbeads instead of using the blank holder force to restrain the blank. The main parameters of interest in this work are friction conditions, drawbead configurations, sheet metal properties, and numerical errors. A robust design model is created to conduct a probabilistic design, which is made possible for this complex engineering process via an efficient uncertainty propagation technique. The method called the weighted three-point-based method estimates the statistical characteristics (mean and variance) of the responses of interest (margins of failures), and provide a systematic approach in designing a sheet metal forming process under the framework of design under uncertainty. 1. INTRODUCTION Sheet metal forming processes have been widely used to fabricate a desired sheet metal product in many industries such as automotive, appliance, aerospace, and others. Productivity and high strength but light weight products are among strength of sheet metal forming processes. However, uncertainties due to uncontrollable conditions (e.g. metal suppliers, forming conditions, and numerical errors) play an important role and need to be taken into account in the design process. The design process considering uncertainties for sheet metal forming processes requires integration of engineering analysis, statistical analysis, and decision making. Design and optimization under uncertainty provides a means such that unacceptable failures are unlikely to occur with an acceptable confidence. Models for uncertainty propagation, an important part of the uncertainty analysis, are generally costly. Furthermore, the probabilistic model generally is even more costly to the extent of unaffordable when its corresponding deterministic simulation is already very expensive to model, such as simulations of sheet metal forming processes. In most engineering applications, mean and variance are the parameters of interest to define a probabilistic system. Uncertainty propagation techniques for statistical moments have evolved from analytical probabilistic models and a sampling-based approximation technique called Monte Carlo simulation (MCS). Analytical probabilistic models mostly are very difficult to develop. MCS, on the other hand, is a powerful approximation technique that relies on pseudo random numbers but it requires a large (and often impractical) computational effort. We use an approximate technique, called weighted three-point- based method, for mean and variance approximation proposed in Buranathiti, Cao and Chen (2004) and Buranathiti, Cao, Chen and Xia (2005) to conduct a probabilistic design. Failures in sheet metal fabrications generally come from three main phenomena: springback, tearing and wrinkling. In this stamping study our focus is on tearing and wrinkling. Tearing is a failure due to an excessive localized load leading to a local instability 867
Transcript
Probabilistic Design in a Sheet Metal Stamping Process under Failure Analysis
Thaweepat Buranathiti a, Jian Cao a*, Z. Cedric Xia b, Wei Chen a
a Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208. b Ford Scientific Research Laboratory, Dearborn, Michigan 48121.
Abstract. Sheet metal stamping processes have been widely implemented in many industries due to its repeatability and productivity. In general, the simulations for a sheet metal forming process involve nonlinearity, complex material behavior and tool-material interaction. Instabilities in terms of tearing and wrinkling are major concerns in many sheet metal stamping processes. In this work, a sheet metal stamping process of a mild steel for a wheelhouse used in automobile industry is studied by using an explicit nonlinear finite element code and incorporating failure analysis (tearing and wrinkling) and design under uncertainty. Margins of tearing and wrinkling are quantitatively defined via stress-based criteria for system-level design. The forming process utilizes drawbeads instead of using the blank holder force to restrain the blank. The main parameters of interest in this work are friction conditions, drawbead configurations, sheet metal properties, and numerical errors. A robust design model is created to conduct a probabilistic design, which is made possible for this complex engineering process via an efficient uncertainty propagation technique. The method called the weighted three-point-based method estimates the statistical characteristics (mean and variance) of the responses of interest (margins of failures), and provide a systematic approach in designing a sheet metal forming process under the framework of design under uncertainty.
1. INTRODUCTION
Sheet metal forming processes have been widely used to fabricate a desired sheet metal product in many industries such as automotive, appliance, aerospace, and others. Productivity and high strength but light weight products are among strength of sheet metal forming processes. However, uncertainties due to uncontrollable conditions (e.g. metal suppliers, forming conditions, and numerical errors) play an important role and need to be taken into account in the design process.
The design process considering uncertainties for sheet metal forming processes requires integration of engineering analysis, statistical analysis, and decision making. Design and optimization under uncertainty provides a means such that unacceptable failures are unlikely to occur with an acceptable confidence. Models for uncertainty propagation, an important part of the uncertainty analysis, are generally costly. Furthermore, the probabilistic model generally is even more costly to the extent of unaffordable when its corresponding deterministic simulation is already very
expensive to model, such as simulations of sheet metal forming processes.
In most engineering applications, mean and variance are the parameters of interest to define a probabilistic system. Uncertainty propagation techniques for statistical moments have evolved from analytical probabilistic models and a sampling-based approximation technique called Monte Carlo simulation (MCS). Analytical probabilistic models mostly are very difficult to develop. MCS, on the other hand, is a powerful approximation technique that relies on pseudo random numbers but it requires a large (and often impractical) computational effort. We use an approximate technique, called weighted three-point-based method, for mean and variance approximation proposed in Buranathiti, Cao and Chen (2004) and Buranathiti, Cao, Chen and Xia (2005) to conduct a probabilistic design.
Failures in sheet metal fabrications generally come from three main phenomena: springback, tearing and wrinkling. In this stamping study our focus is on tearing and wrinkling. Tearing is a failure due to an excessive localized load leading to a local instability
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causing the sheet metal to split (Swift, 1952; Keeler and Backofen, 1964; Marciniak and Kuczynski, 1967; Cao and Yao, 2002). Wrinkling is an unstable phenomenon mainly due to excessive in-plane compressive loads (Wang and Cao, 2000). These failure modes need consistent means of quantification so that further design is possible.
The main focus of this work is to create a framework of a robust design model that achieves a design specification in terms of failure free and takes into account uncertainty with an efficient effort for sheet metal forming processes. The paper is organized as follows: A robust design model for a wheelhouse stamping process is presented in Section 2. Quantification of margins of failures in both tearing and wrinkling is presented in Section 3. Statistical moment estimation via the weighted three-point-based method is briefly given in Section 4. An illustration example is presented and summarized in Section 5, followed by discussions and concluding remarks in Section 6.
2. ROBUST DESIGN MODEL FOR A WHEELHOUSE STAMPING PROCESS
2.1. Robust Design Model
The model for robust design is defined to maximize the total margin and to minimize the variance of the margin while the margins at each criterion are positive. The mathematical model can be written as follows:
Minimize *2
2
2*1f
jfj
f
fi
i jiw
Ww
WyΣ
⋅+
Μ
⋅−=
∑∑ σµ,
Subjected to 0)( ≤−−=ii fifi kg σµ ,
. Ω∈x
(1)
where is the objective function; is the ith constraint value; and are weighting factors for normalized means and normalized variances, respectively; wi is a weighting factor for the ith component; is the mean value of fi; is the
variance value of ; and are used to normalize the two aspects of the robust design objective; is the ith moment factor (corresponding to a minimum reliability) to capture the feasibility of a robust design constraint under uncertainty. In this model, we assume a better design has a larger safety margin against failure.
y ig
1W 2W
ifµ 2
ifσ
if*fΜ *2
fΣ
ik
2.2. A Wheelhouse Stamping Process
In this study, we focus on a design of restraining sources for forming a (1.0m × 0.5m) wheelhouse used
in automobile industry. We used drawbeads as a restraining tool. The main tooling setup for the wheelhouse forming is shown in Fig. 1.
FIGURE 1. The main tooling setup for a wheelhouse forming.
In this study, we use LSDYNA as the analysis tool for the finite element analysis. The material model used is Hill’48 (*MAT_037 in LSDYNA). The nominal material properties of the sheet metal are given as follows: density (ρ) = 7890 kg/m3; Young’s modulus (E) = 207 GPa; Poisson’s ratio (ν) = 0.28; initial yield stress (Y) = 172.78 MPa; Lankford value (R) = 1.95 (assume the normal anisotropy); Strength coefficient (K) = 551.4 MPa; and Exponent coefficient (n) = 0.2363.
All tools are assumed to be rigid bodies in this study like most cases in literature. The binder and the punch are moved to form a blank into a desired shape. In explicit codes, we increased the tooling speed from the real physical conditions in order to obtain an affordable computational time because of the conditionally stable structure of the code. The tooling speeds of the binder and the punch are set at around 4.8 m/s and 1.2 m/s, respectively, compared to about 100 mm/s in a physical stamping process.
3. QUANTIFICATION OF MARGINS OF FAILURES
Failure analysis in sheet metal forming is a very challenging task for sheet metal process designers. It is crucial to be able to identify any possible failure before conducting costly tryout experiments. In this wheelhouse stamping process, we focus failures in terms of tearing and wrinkling. In tearing prediction, we use stress-based forming limit diagrams (SFLD) as the prime analysis tool. In wrinkling prediction, an energy-based approach proposed in Wang and Cao (2000) is used to estimate the critical in-plane compressive stress.
3.1. Tearing Prediction Model
Die
Blank
Binder
Punch Outer drawbead
Inner drawbead
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Plastic instability relates to a condition of plastic deformation at a point that the deformation would continue under a failing load; the process would become unstable. A general technique for tearing prediction is to establish a forming limit diagram (FLD, Keeler and Backofen, 1964) that defines the boundary of uniform straining and the onset of local necking. However, there is a strong dependence of strain-based FLD on nonlinear strain path as shown in Fig. 2. This phenomenon leads to great difficulties in design processes.
FIGURE 2. Illustration of dependence of strain-based FLD from Cao and Yao (2002).
Works on stress-based forming limit diagrams (SFLDs) presented in Arrieux et al. (1982) and Stoughton (2000) have provided a detailed argument that SFLD is somewhat independent to strain paths. Reports in literature show that there is a small band of critical values for tearing corresponding to different loading paths in the stress space. The idea is that the conclusion from the tearing analysis can mostly be drawn from a single diagram.
In a design model, we need a quantitative measure or index rather than a qualitative measure giving only either fail or safe (e.g. zero or one). In this study, we define a margin of tearing, which represents the shortest distance between the limited and applied stress on the stress-based forming limit diagram as illustrated in Fig. 3.
0
100
200
300
400
500
600
700
0
Maj
or P
rinci
pal S
tres
s (M
Pa)
FIGURE
To bmargin vice verwritten
sigf =1
where is the SFLD and is the applied stress. The unit of is MPa.
criticalσ σ
ifIn-plane principal stresses at the last forming step
at particular critical elements in a FEA model are used to calculate the margin of tearing. We averaged the stress information through the sheet thickness. For the particular wheelhouse problem, we can have the potentially critical elements as shown in Fig. 4.
FIGURE 4. The location of potentially critical elements to tearing.
3.2. Wrinkling Prediction Model
Under in-plane compressive loads, sheet metals face a challenging phenomenon known as ‘wrinkling.’ The occurrence of wrinkles leads to possible appearance and assembly problems. For deformed sheet metals, Wang and Cao (2000) proposed an energy-based wrinkling criterion. The stability condition is expressed as
UT ∆≤∆ , (3) where T∆ is the work done by the in-plane membrane forces, and U∆ is the internal energy of the buckled plate.
To analyze wrinkling in sheet metal forming processes, we consider a model of a rectangular region (a×b) in an x-y plane. The region is under a uniform compressive load σx in the x-direction and a uniform l
Fai
100 200 300 400 500 600 700
Minor Pricipal Stress (MPa) 3. Illustration of margin of tearing.
e consistent, we define the positive value for when the applied stress is on the safe side and sa. An equivalently mathematical expression is as
( ) criticalcriticaln σσσσ −⋅− min , (2)
tenanyin-Safe
FIGloa
reg
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otentially critical elements
sile load σy in the y-direction. The region is free of surface contact. A schematic of a plate under an
plane biaxial loading condition is shown in Fig. 5.
URE 5. A schematic of a plate under an in-plane biaxial
ding.
For a straight side-wall model, the concerned ion can be simplified as a plate clamped at four
sides. At the critical point (A(7 = AJ), the criticalbuckling stress can be obtained as
(4)where Laj3cr is defined in details in Wang and Cao(2000), and m and n are buckling modes.
Similarly to the tearing analysis, we need aquantitative measure for the design model. The marginof wrinkling is defined by the difference between theapplied compressive stress and the criticalcompressive stress. Also, in order to be consistent, wedefine the margin to be positive when the appliedcompressive stress is less than the critical compressivestress as/2 = cr - <Jcritical, (5)where <Jcritical is defined in Eq. (4).
From one finite element analysis, we can define apotentially critical region that is risky to wrinkling asshown in Fig. 6. To calculate the margin of wrinkling,we use the in-plane principal stress from averaging thestresses through the sheet thickness. It should be notedthat the dimension of the potentially critical regionmay change from case to case.
FIGUwrink
AapprstatisCao,(200then
where $k is a corresponding weight to the k^observation point. The weight and location of theobservation points depend on the distribution of x. Asummary of optimized values of these parameters fornormal and uniform distributions is given in Table 1.
For the normal distribution, the most widely usedprobability density function, we transform the randomvariable x into the standard normal space z, i.e. z e (-00,00), by using the following standard normaltransformation
where // is the expected or mean value of x, and cr2 isthe variance of x. For uniform distributions, wetransform the random variable x into a master space z(i.e. z e [- l,l]) by using the following linear mappingtransformationz = 2 X~X^——I-T^'1' (8)
After the locations in the z space are selected, oneneeds to transform z back to the real space x.
Table 1. Summary of the parameters for the weightedthree-point-based method.Sampling (for normal
distributions) WeightSampling (for
uniformdistributions)
Weight
z t = -1.8257 ^=0.075 z,= -0.84517 ^=0.04667z2= 0.00000 (j)2= 0.90666z2= 0.0000
RE 6. A location of a potentially critical region toling.
4. STATISTICAL MOMENTESTIMATION VIA WEIGHTED
THREE-POINT-BASED METHOD
weighted three-point-based method is anoximation technique used in this study fortical moment estimation, proposed in Buranathiti, Chen (2004) and Buranathiti, Cao, Chen, Xia5). The total variance of a system response g is formulated as follows
(6)
= 0.04667z = +1.8257 z3= +0.84517
For mean estimation, we have the expressionsummarized as follows
E[g] * g(x) +^ j
where
a43pr[g] -
VAR[X]-
(9)
(10)
(11)
(12)
(13)
5. ILLUSTRATION EXAMPLE
In this section, we present the details of our robustdesign process and results. An initial set of parameters
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of interest is summarized in Table 2 as we begin withan initial screening process.
Table 2. An initial set of parameters of interest.____Upper Lower
Parameter Type Symbol Unit Initial limit limitFriction coeff. x ju - 0.10 0.25 0.05
Outer drawbead* x dbo N/mm 100.0 500.0 0.0Inner drawbead* x dbi N/mm 100.0 500.0 0.0
Y p Y MPa 172.78 189.78 155.78n p n - 0.23630.28630.1863E p E GPa 207.0 228.0 186.0
punch speed d ps m/s 1.0 5.0 0.5where x is a design variable, p is a design parameter,and d is a noise parameter.
The data from Table 2 is used to create a design ofexperiments (the one-factor-at-a-time strategy) for aninitial screening process. The results in terms ofmargins from the design of experiments are presentedin Table 3. According to Table 3, we can conclude thatthe Young's modulus (E) has insignificant impact tothe system and therefore is dropped to a constantprocess parameter. In addition, the Young's modulusis not considered to vary that much in reality. Also, theyield stress (Y) has small impact to the systemresponse and therefore is dropped to a constant processparameter. For the punch speed, we can observe that ithas some effect to the system response as we canconsider this parameter as the numerical noise.However, the numerical noise is not the focus in thisstudy and therefore is discarded.
Table 3. A set of margins from the sample simulationscorresponding to the initial screening design.#vlv2v3v4v5v6v7v8
/I29.6851.129.0080.2912.0833.5941.1521.60
/285.3571.55126.86-109.32260.90114.14-120.6389.27
y-2.30-2.45-2.720.58-5.46-2.951.59-2.22
#v9vlOvllv!2v!3v!4v!5
/l11.50151.0323.6520.1527.21110.0844.54
/270.6878.0142.5986.5928.9640.1976.73
y-1.64-4.58-1.32-2.13-1.12-3.01-2.43
^, W2J M>I, w2, are 1.0; M^ is 50.0; Z^ is750.0; and kt is 3.0 for the robust design model.
After we defined the design variables and processparameters of the system, we conducted a design ofexperiments based on the Latin hypercube design(Kalagnanam and Diwekar, 1997) for betterexploration as shown in Table 4. A set of simulationresults (fi and /2) according to the design is alsosummarized in Table 4. It should be noted that y inTables 3 and 4 does not take uncertainty into account.
Table 4. Samples from the Latin hypercube design.# ii dbo dbi n___fa____fa____y
After we studied the stamping process through twodesigns of experiments in Tables 3 and 4, we cancreate a surrogate model to approximate the responsefor further design under uncertainty. It should be notedthat we will only vary the design variables; not thedesign parameters. An initial set of design variablesand process parameters for the robust design modelsolving via sequential quadratic programming (SQP)in all cases in this study is summarized in Table 5.
Table 5. Summary of initial design variables, processcharacteristics and robust design model parameters.
Parameter Type Initial //m Distribution Std.Friction coeff.
First, we considered a case of the deterministicdesign by neglecting all the uncertainties in the model.After the optimization search, we have an optimal setof design variables as shown in Table 6.
Table 6. An optimal set of design variables of thedeterministic design model.______________
Parameter Symbol Value Response EstimateFriction coeff.
Outer drawbeadInner drawbead
n
Vdbodbin
0.10500.00100.000.2363
yfih
-5.459612.08
260.90
The deterministic solution was then put through auncertainty check following the standard deviations inTable 5, and it was found that the margin of failure fortearing violates the constraint by 25.20 MPa (or at18.17% of cases by a normal approximation). In otherwords, the deterministic model is too risky to beimplemented.
Then, we take uncertainties into account in thedesign optimization codes. By solving the designmodel with the weighted three-point-based method forstatistical moment estimations, we have an optimal setof design variables as summarized in Table 7.
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Table 7. Summary of design variables and systemresponses with the weighted three-point-based method.Parameter
//dbodbi
n
Solution0.05
308.6840.00
0.2363
ResponsejLlfi
jnf2af,cr/2
gi82y
Estimate§
46.36162.1415.458.020.00
-138.09-4.57
MCS ref.*47.67162.0715.137.79-2.27
-138.70-4.58
§obtained from our method in Section 4. *obtainedfrom the MCS with a large sample size (500,000samples) for the verification purpose.
A good agreement between Estimate and MCS refis obtained in Table 7. In addition to the comparison tothe deterministic design, we compare the design withthe implementation of MCS (with 50,000 samples)instead of the weighted three-point-based method forthe statistical moment estimation. The results of threedifferent MCS simulations with the same initialconditions are summarized in Table 8.
Table 8. Summary of optimal sets of design variablesand system responses via MCS.
ParameterV
dbodbin
Response/'/I»f*af,°hgi82y
Solution 10.0640.00168.570.2363
Estimate 196.7696.6713.6313.61-55.87-55.83-4.36
Solution20.0592.9840.00
0.2363Estimate2
59.2659.2814.0914.06-16.97-17.10-2.90
Solution30.17
170.0059.96
0.2363Estimate3
52.4752.5314.3514.27-9.42-9.71-2.65
As can be seen from the table, the results fromMCS are not robust due to numerical errors (i.e., quitedifferent solutions with totally different outcomes areobtained from multiple tries even though the initialvalues are the same). The sets of solution shown inTable 8 are merely a demonstration of the argument.In terms of computational time saving, to get thesolution in Table 7 only needs about 1 secondcompared to about 10 minutes in Table 8.
6. DISCUSSIONS AND CONCLUDINGREMARKS
The main focus in this study for robust design insheet metal stamping is how to consistently quantifythe margin of safety/failure (tearing and wrinkling)
and to efficiently take uncertainties into account tocreate a system-level robust design model. Awheelhouse stamping process is used as an example todemonstrate the design under uncertainty approachconstrained with failure analysis. The robust design fora wheelhouse stamping process is conducted tomaximize the total mean value of margins and tominimize the total variance of margins. We used theweighted three-point-based method as the uncertaintypropagation technique for statistical momentestimation to compare with results from othertechniques. We can see that the weighted three-point-based method offered a good solution that well agreeswith moments and responses from MCS in a moreefficient and a more robust manner. The feasibility ofthe implementation of design and optimization underuncertainty in sheet metal forming processesconstrained with failure analysis is presented in thiswork to encourage further implementation in otherindustrial cases.
ACKNOWLEDGMENTS
The support from NSF Grant (DMI-0084582) andFord University Research Program is deeplyappreciated.
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