Probabilistic Design in a Sheet Metal Stamping Process under Failure Analysis

Transcription

1 Probablstc Desg a Sheet Metal Stampg Process uder Falure Aalyss Thaweepat Buraatht a, Ja Cao a *, Z. Cedrc Xa b, We Che a a Departmet of Mechacal Egeerg, Northwester Uversty, Evasto, Illos b Ford Scetfc Research Laboratory, Dearbor, Mchga * Correspodg author. Tel.: ; Fax: ; jcao@orthwester.edu Abstract. Sheet metal stampg processes have bee wdely mplemeted may dustres due to ts repeatablty ad productvty. I geeral, the smulatos for a sheet metal formg process volve olearty, complex materal behavor ad tool-materal teracto. Istabltes terms of tearg ad wrklg are major cocers may sheet metal stampg processes. I ths work, a sheet metal stampg process of a mld steel for a wheelhouse used automoble dustry s studed by usg a explct olear fte elemet code ad corporatg falure aalyss (tearg ad wrklg) ad desg uder ucertaty. Margs of tearg ad wrklg are quattatvely defed va stress-based crtera for system-level desg. The formg process utlzes drawbeads stead of usg the blak holder force to restra the blak. The ma parameters of terest ths work are frcto codtos, drawbead cofguratos, sheet metal propertes, ad umercal errors. A robust desg model s created to coduct a probablstc desg, whch s made possble for ths complex egeerg process va a effcet ucertaty propagato techque. The method called the weghted three-pot-based method estmates the statstcal characterstcs (mea ad varace) of the resposes of terest (margs of falures), ad provde a systematc approach desgg a sheet metal formg process uder the framework of desg uder ucertaty. 1. INTRODUCTION Sheet metal formg processes have bee wdely used to fabrcate a desred sheet metal product may dustres such as automotve, applace, aerospace, ad others. Productvty ad hgh stregth but lght weght products are amog stregth of sheet metal formg processes. However, ucertates due to ucotrollable codtos (e.g. metal supplers, formg codtos, ad umercal errors) play a mportat role ad eed to be take to accout the desg process. The desg process cosderg ucertates for sheet metal formg processes requres tegrato of egeerg aalyss, statstcal aalyss, ad decso makg. Desg ad optmzato uder ucertaty provdes a meas such that uacceptable falures are ulkely to occur wth a acceptable cofdece. Models for ucertaty propagato, a mportat part of the ucertaty aalyss, are geerally costly. Furthermore, the probablstc model geerally s eve more costly to the extet of uaffordable whe ts correspodg determstc smulato s already very expesve to model, such as smulatos of sheet metal formg processes. I most egeerg applcatos, mea ad varace are the parameters of terest to defe a probablstc system. Ucertaty propagato techques for statstcal momets have evolved from aalytcal probablstc models ad a samplg-based approxmato techque called Mote Carlo smulato (MCS). Aalytcal probablstc models mostly are very dffcult to develop. MCS, o the other had, s a powerful approxmato techque that reles o pseudo radom umbers but t requres a large (ad ofte mpractcal) computatoal effort. We use a approxmate techque, called weghted three-potbased method, for mea ad varace approxmato proposed Buraatht, Cao ad Che (2004) ad Buraatht, Cao, Che ad Xa (2005) to coduct a probablstc desg. Falures sheet metal fabrcatos geerally come from three ma pheomea: sprgback, tearg ad wrklg. I ths stampg study our focus s o tearg ad wrklg. Tearg s a falure due to a excessve localzed load leadg to a local stablty 867

2 causg the sheet metal to splt (Swft, 1952; Keeler ad Backofe, 1964; Marcak ad Kuczysk, 1967; Cao ad Yao, 2002). Wrklg s a ustable pheomeo maly due to excessve -plae compressve loads (Wag ad Cao, 2000). These falure modes eed cosstet meas of quatfcato so that further desg s possble. The ma focus of ths work s to create a framework of a robust desg model that acheves a desg specfcato terms of falure free ad takes to accout ucertaty wth a effcet effort for sheet metal formg processes. The paper s orgazed as follows: A robust desg model for a wheelhouse stampg process s preseted Secto 2. Quatfcato of margs of falures both tearg ad wrklg s preseted Secto 3. Statstcal momet estmato va the weghted three-pot-based method s brefly gve Secto 4. A llustrato example s preseted ad summarzed Secto 5, followed by dscussos ad cocludg remarks Secto ROBUST DESIGN MODEL FOR A WHEELHOUSE STAMPING PROCESS 2.1. Robust Desg Model The model for robust desg s defed to maxmze the total marg ad to mmze the varace of the marg whle the margs at each crtero are postve. The mathematcal model ca be wrtte as follows: 2 w µ f w j σ f j j Mmze y = W1 + W * 2, 2* Μ f Σ f (1) Subjected to g ( µ k σ ) 0, = f f x Ω. where y s the objectve fucto; g s the th costrat value; W1 ad W2 are weghtg factors for ormalzed meas ad ormalzed varaces, respectvely; w s a weghtg factor for the th 2 compoet; s the mea value of f ; s the µ f * 2* varace value of ; ad are used to f Μ f ormalze the two aspects of the robust desg objectve; k s the th momet factor (correspodg to a mmum relablty) to capture the feasblty of a robust desg costrat uder ucertaty. I ths model, we assume a better desg has a larger safety marg agast falure. Σ f 2.2. A Wheelhouse Stampg Process σ f I ths study, we focus o a desg of restrag sources for formg a (1.0m 0.5m) wheelhouse used automoble dustry. We used drawbeads as a restrag tool. The ma toolg setup for the wheelhouse formg s show Fg. 1. Outer drawbead FIGURE 1. The ma toolg setup for a wheelhouse formg. I ths study, we use LSDYNA as the aalyss tool for the fte elemet aalyss. The materal model used s Hll 48 (*MAT_037 LSDYNA). The omal materal propertes of the sheet metal are gve as follows: desty (ρ) = 7890 kg/m 3 ; Youg s modulus (E) = 207 GPa; Posso s rato (ν) = 0.28; tal yeld stress (Y) = MPa; Lakford value (R) = 1.95 (assume the ormal asotropy); Stregth coeffcet (K) = MPa; ad Expoet coeffcet () =. All tools are assumed to be rgd bodes ths study lke most cases lterature. The bder ad the puch are moved to form a blak to a desred shape. I explct codes, we creased the toolg speed from the real physcal codtos order to obta a affordable computatoal tme because of the codtoally stable structure of the code. The toolg speeds of the bder ad the puch are set at aroud 4.8 m/s ad 1.2 m/s, respectvely, compared to about 100 mm/s a physcal stampg process. 3. QUANTIFICATION OF MARGINS OF FAILURES Falure aalyss sheet metal formg s a very challegg task for sheet metal process desgers. It s crucal to be able to detfy ay possble falure before coductg costly tryout expermets. I ths wheelhouse stampg process, we focus falures terms of tearg ad wrklg. I tearg predcto, we use stress-based formg lmt dagrams (SFLD) as the prme aalyss tool. I wrklg predcto, a eergy-based approach proposed Wag ad Cao (2000) s used to estmate the crtcal -plae compressve stress Tearg Predcto Model De Blak Bder Ier drawbead Puch 868

3 Plastc stablty relates to a codto of plastc deformato at a pot that the deformato would cotue uder a falg load; the process would become ustable. A geeral techque for tearg predcto s to establsh a formg lmt dagram (FLD, Keeler ad Backofe, 1964) that defes the boudary of uform strag ad the oset of local eckg. However, there s a strog depedece of stra-based FLD o olear stra path as show Fg. 2. Ths pheomeo leads to great dffcultes desg processes. ε at trasverse drecto FIGURE 2. Illustrato of depedece of stra-based FLD from Cao ad Yao (2002). Works o stress-based formg lmt dagrams (SFLDs) preseted Arreux et al. (1982) ad Stoughto (2000) have provded a detaled argumet that SFLD s somewhat depedet to stra paths. Reports lterature show that there s a small bad of crtcal values for tearg correspodg to dfferet loadg paths the stress space. The dea s that the cocluso from the tearg aalyss ca mostly be draw from a sgle dagram. I a desg model, we eed a quattatve measure or dex rather tha a qualtatve measure gvg oly ether fal or safe (e.g. zero or oe). I ths study, we defe a marg of tearg, whch represets the shortest dstace betwee the lmted ad appled stress o the stress-based formg lmt dagram as llustrated Fg. 3. Major Prcpal Stress (MPa) exp_al6111 Exp_mx_u_0.05 Exp_mx_u_0.095 Exp_mx_u_0.14 KB_mx_u_0.05 KB_mx_u_0.095 KB_mx_u_ ε at rollg drecto Fal Safe where σ s the SFLD ad σ s the appled stress. crtcal The ut of f s MPa. I-plae prcpal stresses at the last formg step at partcular crtcal elemets a FEA model are used to calculate the marg of tearg. We averaged the stress formato through the sheet thckess. For the partcular wheelhouse problem, we ca have the potetally crtcal elemets as show Fg. 4. Potetally crtcal elemets FIGURE 4. The locato of potetally crtcal elemets to tearg Wrklg Predcto Model Uder -plae compressve loads, sheet metals face a challegg pheomeo kow as wrklg. The occurrece of wrkles leads to possble appearace ad assembly problems. For deformed sheet metals, Wag ad Cao (2000) proposed a eergy-based wrklg crtero. The stablty codto s expressed as T U, (3) where T s the work doe by the -plae membrae forces, ad U s the teral eergy of the buckled plate. To aalyze wrklg sheet metal formg processes, we cosder a model of a rectagular rego (a b) a x-y plae. The rego s uder a uform compressve load σ x the x-drecto ad a uform tesle load σ y the y-drecto. The rego s free of ay surface cotact. A schematc of a plate uder a -plae baxal loadg codto s show Fg Mor Prcpal Stress (MPa) FIGURE 3. Illustrato of marg of tearg. To be cosstet, we defe the postve value for marg whe the appled stress s o the safe sde ad vce versa. A equvaletly mathematcal expresso s wrtte as f = sg ( σ σcrtcal ) m σ σcrtcal 1, (2) FIGURE 5. A schematc of a plate uder a -plae baxal loadg. For a straght sde-wall model, the cocered rego ca be smplfed as a plate clamped at four 869

4 sdes. At the crtcal pot (A(7 = AJ), the crtcal bucklg stress ca be obtaed as (4) where L aj3}cr s defed detals Wag ad Cao (2000), ad m ad are bucklg modes. Smlarly to the tearg aalyss, we eed a quattatve measure for the desg model. The marg of wrklg s defed by the dfferece betwee the appled compressve stress ad the crtcal compressve stress. Also, order to be cosstet, we defe the marg to be postve whe the appled compressve stress s less tha the crtcal compressve stress as / 2 = cr - <J crtcal, (5) where <J crtcal s defed Eq. (4). From oe fte elemet aalyss, we ca defe a potetally crtcal rego that s rsky to wrklg as show Fg. 6. To calculate the marg of wrklg, we use the -plae prcpal stress from averagg the stresses through the sheet thckess. It should be oted that the dmeso of the potetally crtcal rego may chage from case to case. where $ k s a correspodg weght to the k^ observato pot. The weght ad locato of the observato pots deped o the dstrbuto of x. A summary of optmzed values of these parameters for ormal ad uform dstrbutos s gve Table 1. For the ormal dstrbuto, the most wdely used probablty desty fucto, we trasform the radom varable x to the stadard ormal space z,.e. z e (- 00,00), by usg the followg stadard ormal trasformato where // s the expected or mea value of x, ad cr 2 s the varace of x. For uform dstrbutos, we trasform the radom varable x to a master space z (.e. z e [- l,l]) by usg the followg lear mappg trasformato z = 2 X ~ X^ I- T^' 1 ' (8) After the locatos the z space are selected, oe eeds to trasform z back to the real space x. Table 1. Summary of the parameters for the weghted three-pot-based method. Samplg (for ormal dstrbutos) Weght Samplg (for uform dstrbutos) Weght z t = ^=0.075 z,= ^= z 2 = z 2 = (j) 2 = z= z 3 = = For mea estmato, we have the expresso summarzed as follows FIGURE 6. A locato of a potetally crtcal rego to wrklg. 4. STATISTICAL MOMENT ESTIMATION VIA WEIGHTED THREE-POINT-BASED METHOD E[g] * g(x) + ^ j where a4 3pr [g] - (9) (10) (11) (12) A weghted three-pot-based method s a approxmato techque used ths study for statstcal momet estmato, proposed Buraatht, Cao, Che (2004) ad Buraatht, Cao, Che, Xa (2005). The total varace of a system respose g s the formulated as follows VAR[ X ]- 5. ILLUSTRATION EXAMPLE (13) (6) I ths secto, we preset the detals of our robust desg process ad results. A tal set of parameters 870

5 of terest s summarzed Table 2 as we beg wth a tal screeg process. Table 2. A tal set of parameters of terest. Upper Lower Parameter Type Symbol Ut Ital lmt lmt Frcto coeff. x ju Outer drawbead* x dbo N/mm Ier drawbead* x db N/mm Y p Y MPa p E p E GPa puch speed d ps m/s where x s a desg varable, p s a desg parameter, ad d s a ose parameter. The data from Table 2 s used to create a desg of expermets (the oe-factor-at-a-tme strategy) for a tal screeg process. The results terms of margs from the desg of expermets are preseted Table 3. Accordg to Table 3, we ca coclude that the Youg's modulus (E) has sgfcat mpact to the system ad therefore s dropped to a costat process parameter. I addto, the Youg's modulus s ot cosdered to vary that much realty. Also, the yeld stress (Y) has small mpact to the system respose ad therefore s dropped to a costat process parameter. For the puch speed, we ca observe that t has some effect to the system respose as we ca cosder ths parameter as the umercal ose. However, the umercal ose s ot the focus ths study ad therefore s dscarded. Table 3. A set of margs from the sample smulatos correspodg to the tal screeg desg. # /I /2 y # /l /2 y vl v v2 v3 v4 v5 v6 v7 v vlo vll v!2 v!3 v!4 v! ^, W 2J M>I, w 2, are 1.0; M^ s 50.0; Z^ s 750.0; ad k t s 3.0 for the robust desg model. After we defed the desg varables ad process parameters of the system, we coducted a desg of expermets based o the Lat hypercube desg (Kalagaam ad Dwekar, 1997) for better explorato as show Table 4. A set of smulato results (f ad / 2 ) accordg to the desg s also summarzed Table 4. It should be oted that y Tables 3 ad 4 does ot take ucertaty to accout. Table 4. Samples from the Lat hypercube desg. # dbo db fa fa y v O v202 v203 v204 v205 v206 v207 v208 v209 v210 v211 v212 v213 v214 v After we studed the stampg process through two desgs of expermets Tables 3 ad 4, we ca create a surrogate model to approxmate the respose for further desg uder ucertaty. It should be oted that we wll oly vary the desg varables; ot the desg parameters. A tal set of desg varables ad process parameters for the robust desg model solvg va sequetal quadratc programmg (SQP) all cases ths study s summarzed Table 5. Table 5. Summary of tal desg varables, process characterstcs ad robust desg model parameters. Parameter Type Ital // m Dstrbuto Std. Frcto coeff. Outer drawbead Ier drawbead x x x p ormal ormal 8.0 ormal 8.0 ormal 0.01 Frst, we cosdered a case of the determstc desg by eglectg all the ucertates the model. After the optmzato search, we have a optmal set of desg varables as show Table 6. Table 6. A optmal set of desg varables of the determstc desg model. Parameter Symbol Value Respose Estmate Frcto coeff. Outer drawbead Ier drawbead V dbo db y f h The determstc soluto was the put through a ucertaty check followg the stadard devatos Table 5, ad t was foud that the marg of falure for tearg volates the costrat by MPa (or at 18.17% of cases by a ormal approxmato). I other words, the determstc model s too rsky to be mplemeted. The, we take ucertates to accout the desg optmzato codes. By solvg the desg model wth the weghted three-pot-based method for statstcal momet estmatos, we have a optmal set of desg varables as summarzed Table

6 Table 7. Summary of desg varables ad system resposes wth the weghted three-pot-based method. Parameter // dbo db Soluto Respose jll f Estmate MCS ref.* j f2 a f, cr /2 g 82 y obtaed from our method Secto 4. *obtaed from the MCS wth a large sample sze (500,000 samples) for the verfcato purpose. A good agreemet betwee Estmate ad MCS ref s obtaed Table 7. I addto to the comparso to the determstc desg, we compare the desg wth the mplemetato of MCS (wth 50,000 samples) stead of the weghted three-pot-based method for the statstcal momet estmato. The results of three dfferet MCS smulatos wth the same tal codtos are summarzed Table 8. Table 8. Summary of optmal sets of desg varables ad system resposes va MCS. Parameter Soluto 1 Soluto2 Soluto3 V dbo db Respose Estmate Estmate Estmate3 /'/I»f* a f, h g 82 y As ca be see from the table, the results from MCS are ot robust due to umercal errors (.e., qute dfferet solutos wth totally dfferet outcomes are obtaed from multple tres eve though the tal values are the same). The sets of soluto show Table 8 are merely a demostrato of the argumet. I terms of computatoal tme savg, to get the soluto Table 7 oly eeds about 1 secod compared to about 10 mutes Table DISCUSSIONS AND CONCLUDING REMARKS The ma focus ths study for robust desg sheet metal stampg s how to cosstetly quatfy the marg of safety/falure (tearg ad wrklg) ad to effcetly take ucertates to accout to create a system-level robust desg model. A wheelhouse stampg process s used as a example to demostrate the desg uder ucertaty approach costraed wth falure aalyss. The robust desg for a wheelhouse stampg process s coducted to maxmze the total mea value of margs ad to mmze the total varace of margs. We used the weghted three-pot-based method as the ucertaty propagato techque for statstcal momet estmato to compare wth results from other techques. We ca see that the weghted three-potbased method offered a good soluto that well agrees wth momets ad resposes from MCS a more effcet ad a more robust maer. The feasblty of the mplemetato of desg ad optmzato uder ucertaty sheet metal formg processes costraed wth falure aalyss s preseted ths work to ecourage further mplemetato other dustral cases. ACKNOWLEDGMENTS The support from NSF Grat (DMI ) ad Ford Uversty Research Program s deeply apprecated.. 2. REFERENCES Arreux, R., Bedr, C., Bov, M., 1982, Determato of a trsc formg lmt stress dagram for sotropc metal sheets, Proceedgs of the 12 th Beal Cogress of the Iteratoal Deep Drawg Research Group (IDDRG), Italy, pp Buraatht, T., Cao, J., Che, W., 2004, Weghted three-potbased strategy for varace estmato, Proceedgs of the DETC2004, ASME 2004 Desg Egeerg Techcal Cofereces, Salt Lake Cty, Utah. 3. Buraatht, T., Cao, J., Che, W., Xa, Z.C., 2005, A weghted three-pot-based methodology for varace estmato, submtted to Egeerg Optmzato. 4. Cao, J., Yao, H., 2002, Predcto of formg lmt curves usg a asotropc yeld fucto wth prestra duced backstress, Iteratoal Joural of Plastcty, 18, 8, Kalagaam, J.R., ad Dwekar, U.M., 1997, A effcet samplg techque for off-le qualty cotrol, Techometrcs, 39,3, Keeler, S. P., Backofe, W.A., 1964, Plastc Istablty ad fracture sheets stretched over rgd puches, ASM Trasactos Quarterly, 56, Marcak, Z., Kuczysk, K., 1967, Lmt stras the processes of stretch-formg sheet metal, Iteratoal Joural of Mechacal Scece, 9, 9, Semt, S.L., Joas, J.J., 1984, Formablty ad Workablty of Metals: Plastc Istablty ad Flow Localzato, Amerca 9. Socety for Metals, 299p. Stoughto, T.B., 2000, A geeral formg lmt crtero for sheet metal formg, Iteratoal Joural of Mechacal Sceces, 42, 1, Swft, H.W., 1952, Plastc stablty uder plae stress, Joural of the Mechacs ad Physcs of Solds, 1, Wag, X., Cao, J., 2000, O the predcto of sde-wall wrklg sheet metal formg processes, Iteratoal Joural of Mechacal Sceces, 42, 12,