Optimum blank design of an automobile sub-frameJong-Yop Kim a ,Naksoo Kim a,*,Man-Sung Huh baDepartment of Mechanical Engineering,Sogang University,Shinsu-dong 1,Mapo-ku,Seoul 121-742,South KoreabHwa-shin Corporation,Young-chun,Kyung-buk,770-140,South KoreaReceived 17July 1998AbstractA roll-back method is proposed to predict the optimum initial blank shape in the sheet metal forming process.The method takes the difference between the ®nal deformed shape and the target contour shape into account.Based on the method,a computer program composed of a blank design module,an FE-analysis program and a mesh generation module is developed.The roll-back method is applied to the drawing of a square cup with the ¯ange of uniform size around its periphery,to con®rm its validity.Good agreement is recognized between the numerical results and the published results for initial blank shape and thickness strain distribution.The optimum blank shapes for two parts of an automobile sub-frame are designed.Both the thickness distribution and the level of punch load are improved with the designed blank.Also,the method is applied to design the weld line in a tailor-welded blank.It is concluded that the roll-back method is an effective and convenient method for an optimum blank shape design.#2000Elsevier Science S.A.All rights reserved.Keywords:Blank design;Sheet metal forming;Finite element method;Roll-back method1.IntroductionIt is important to determine the optimum blank shape of a sheet metalpart.However,because its deformation during the forming process is very complicated,it is not easy to design the optimum blank shape even by the skilled labor based on the experience of many years.Recently,computa-tional analysis for a complex automobile part has been able to be carried out easily due to improved computer perfor-mance and the numerical analysis technique.In the analysis process,all kinds of variables that affect the deformation should be considered.The optimum blank shape leads to the prevention of tearing,uniform thickness distribution and to the reduction of the press load during drawing.If the blank shape is designed optimally,the formability will be increased and the ®nal product will require the least amount of trimming at the end of theprocess.Therefore,it is desirable to design the blank shape with a uniform ¯ange of its periphery after deep drawing.Several numerical solutions for the deep drawing process of non-circular components have been reported.Hasek and Lange [1]gave an analytical solution to this problem usingthe slip-line ®eld-method with the assumption of plane-strain ¯ange deformation.Also,Jimma [2]and Karima [3]used the same method.V ogel and Lee [4]and Chen and Sowerby [5]developed ideal blank shapes by the method of plane-stress characteristics.Sowerby et al.[6]developed a geometric mapping method providing a trans-formation between a ¯at sheet and the ®nal surface.Majlessi and Lee [7,8]developed a multi-stage sheet metal forming analysis method.Chung and Richmond [9±12]determined ideal con®gurations for both the initial and the intermediate stages that are required to form a speci®ed ®nal shape using the ideal forming theory.Lee and Huh [13]introduced a three-dimensional multi-step inverse method for the optimum design of blank shapes.Toh and Kobayashi [14]developed arigid±plastic ®nite-element method for the drawing of general shapes based on membrane theory and ®nite-strain formulations.Zhaotao [15]used the boundary element method for a 2D potential problem to design optimum blank shapes.This paper presents an optimum design method of blank shapes for the square cup drawing process considering process variables.An optimum blank shape of square cup drawing was obtained using the proposed method.Also,it was applied to the deep drawing of an automobile sub-frame,and an optimum blank shape with a uniform ¯ange at its periphery weredetermined.Journal of Materials Processing Technology 101(200031±43*Corresponding author.Tel.: 82-2-705-8635;fax: 82-2-712-0799.E-mailaddress :nskim@ccs.sogang.ac.kr (Naksoo Kim0924-0136/00/$±see front matter #2000Elsevier Science S.A.All rights reserved.PII:S 0924-0136(9900436-72.Design of optimum blank shapeThe de®nition of the optimum blank shape is the mini-mization of the difference between the outer contour of the deformed blank and the target contour that indicates the residual ¯ange of uniform size around the periphery of the product.The target contour is generated from the outer contour of the product and determines an optimum blank shape using the results of ®nite-element simulation with the roll-back method.In the process of blank design the simula-tion is performed using an explicit ®nite-element software PAM-STAMP and the interface program is developed for con-necting the blank design module,the remeshing module,the post-processor module and the FE-analysis package.2.1.Roll-back method`The roll-back method starts by de®ning the target con-tour.After determining the length of the ¯ange that remains around the periphery of the product,the pro®le of the target contour is created by offsetting an equal distance from the outer contour of the product and its mesh system is gener-ated by beam elements.The process of blank design is illustrated in Fig.1.The mesh system of the prepared square blank for initial analysis is shown in Fig.1(a.After an analysis,the mesh system of the deformed blank and the target contour are shown in Fig.1(b.At the ¯ange of the deformed blank,a distinction is made between the interior ¯ange within the target contour and the exterior ¯ange out ofthe target contour.The ¯ange out of the target contour is the part that will be trimmed and the ¯ange within the target contour is the part which does not keep shape is due to the incompletion of the blank shape.Thus the modi®ed blank shape should be designed to take the shape of the outer contour of the product completely.The contour of themodi®ed blank shape using the roll-back method and the initial blank shape is shown in Fig.1(c.The mesh system of the modi®ed blank shape for FE-analysis is shown in Fig.1(d.The blank design method will be introduced in detail.The quarter of the deformed blank and the target contour are shown in Fig.2(a.According to the previous explanation,the remained ¯ange can be divided into the interior and the exterior ¯ange.The design process of region A is shown in Fig.2(b.In the mesh of the deformed blank a square grid IJKL on the target contour will be considered,and then the internal dividing point Q in will be calculated at the ratio of m tonFig.1.Illustrating the process of ®nding the optimum blank:(ainitial blankshape;(bdeformed blank and target contour;(croll-back blank and contour;(dmodi®ed blankshape.Fig.2.The roll-back process of a mesh located on the surface of the ¯ange:(aa mesh located on the surface of the ¯ange;(bregion A:residual drawing part out of target contour;(cregion B:residual drawing part inside the target contour.32J.-Y.Kim et al./Journal of Materials Processing Technology 101(200031±43between the node J and K.This point is mapped back into the mesh system of the initial blank.The internal dividing point Q H in is calculated at the ratio of m to n between the same node J H and K H.The following process is performed on the element of the deformed blank on the target contour.The describing point of the outer contour of themodi®ed blank shape can be calculated.If the coordinates of the nodes J and K areJ(x1,y1,K(x2,y2and the coordinates of the nodes J H and K H are J H x H1Y y H1 Y K H x H2Y y H2 ,the ratio of m to n ism X n JQJKX QKJK(1The coordinate of the internal dividing point Q H in can be expressed asQ H inmx H2 nx H1m nYmy H2 ny H1m n(2The design process of region B is shown in Fig.2(c.In the mesh of the deformed blank a square grid MNOP of which the outward edge crosses the target contour should be considered,and then the external dividing point Q out can be calculated at the ratio of m to n between nodes O and P.This point is mapped back into the mesh system of the initial blank.The external dividing point Q H out can be calculated at the ratio of m to nbetween the same nodes Q H and P H.If the coordinates of the nodes O and P areO(x1,y1,P(x2,y2and the coordinates of the O H and P H are O H x H1Y y H1 Y P H xH2Y y H2 ,the ratio of m to n ism X n OQOPX QPOP(3The coordinate of the external dividing point Q H out can be expressed asQ H outmx H2Ànx H1Ymy H2Àny H1(4The following process is performed on all the element of the deformed blank related on the target contour.The points describing the outer contour of modi®ed blank shape can be calculated.When all points of two cases are connected by the spline,the outer contour of modi®ed blank can be described.This process is shown in Fig.3.2.2.The development of the optimum blank design programTo optimize the initial blank shape,a design program was developed following the prescribing method and procedures. This program consists of the blank shaper designmodule, the mesh generation module and the post-processor module. The whole procedure is illustrated in Fig.4.To perform the design process of a blank shape,an interface module is needed.This module is developed to read the output®le of ®nite-element analysis and design the optimum blank shape and generate theinput®le.3.Designs of blank shape and application3.1.Blank design of a square cupTo verify the validity of the roll-back method,it is applied to the process of square cup deep drawing.Several numerical solutions of the deep drawing process for non-circular components have been reported recently.The pub-lished blank shapes by Lee and coworkers[16±18]are compared with the resultusing the roll-back method.The Fig.3.Flowchart of the blank design module.Fig.4.Flow chart of the main program.J.-Y.Kim et al./Journal of Materials Processing Technology101(200031±4333dimensions of the die and punch set for an analysis are shown in Fig.5.The material of the sheet metal is cold-rolled steel for an automobile part.The following are the material propertiesand process variables.Stress±strain relation:"s58X 78Â 0X 00003 "e0X 274 kgf a mm 2 ;Lankford value:"R 1X 679;initial blank size:160mm Â160mm square blank;initial thickness:t 0.69mm;friction coef®cient:m 0.123;and blank-holding force:4000kgf (1kgf 9.81N.The deformed shapes of the square cup obtained from the initial blank and the optimum blank are shown in Fig.6.Inthe present work the optimum blank shape for a square cup that is of 40mm height and 5mm width of ¯ange will be determined.Each modi®ed blank shape after the application of the roll-back method is illustrated in Fig.7.When an 160mm Â160mm square blank is used for an initial blank the outer contour of deformed blank is shown in Fig.7(a.A ®rst modi®ed blank shape can be calculated with the result of the initial square blank.An analysis result is shown inFig.7(b.The difference between the deformed shape and the target contour issigni®cant.If the blank design process is repeated several times the difference decreases and con-verges to zero.Hence a square cup with a uniform ¯ange at its periphery can be made.The comparison between the ®nal result and a published result is shown in Fig.8.In the transverse direction the optimum blank shape using the roll-back method is larger than the published result.The load±displacement curves in square cup drawing process with various initial blank shapes are shown in Fig.9.As the modi®cation is repeated,the gap of the load±displacement curves before and after iteration decreases.Thus after the third modi®cation the maximum value of the load becomes the mean value between that of the ®rst and second modi®cation.After three modi®cations the optimum blank shape is determined,then the result with the optimum blank shape is compared with results in the literature.The thickness strain distribution in the diagonal direction is shown in Fig.10(a,whilst the thickness strain distribution in the transverse direction is shown in Fig.10(b.In the thickness strain distribution the result using the roll-back method is slightly different from the published result,but the overall strain distributions are quite similar.It is thus veri®ed that the roll-back method is a useful approach in the design of optimum blank shapes.3.2.Blank design of the left member of a front sub-frameAn analysis for members of a box-type front sub-frame is performed.The left member is selected as one of the subjects for analysis because its shape is shallow but complex.Fig.11shows the manufacturing set-up as modeled for the numer-ical simulation.The left member requires a uniform ¯ange for the spot welding between the upper and the lower parts besides the improvement of formability.It is recommended that the length of uniform ¯ange is 30mm.The target contour is de®ned at the position which is 30mm from the outer contour of product and is shown in Fig.12.Its mesh system is generated by beam elements.The material of the sheet metal is SAPH38P,a hot-rolled steel for automobile parts.The following are the material properties and process variables.Stress±strain relationship:"s 629Â"e 0X 274(MPa;Lankford value:"R1X 030;initial thickness:t 2.3mm;friction coef®cient:m 0.1;blank holding pressure:1MPa.Fig.5.Geometrical description of the tooling for the deep drawing of a square cup (dimensions:mm.Fig.6.The deformed shape of square cups with FE-mesh geometry where the cup height is 40mm:(adeformed shape of the square cup obtained from the initialblank;(bdeformed shape of the square cup obtained from the optimum blank.34J.-Y.Kim et al./Journal of Materials Processing Technology 101(200031±43A hexagonal blank is used as the initial blank.After three modi®cations the optimum blank shape is determined.For this case,the load±displacement curves with various blank shapes are shown in Fig.13.The comparison of the initial ¯ange and the deformed ¯ange with various blank shapes is shown in Fig.14.As the modi®cation is repeated,the maximum punch load is reduced and the outer contour may be drawn to the target contour at the same time.The thickness distribution is improved step by step;the thickness distribution with various blank shapes being shown in Fig.15.The comparison between the optimum blank shape designed by the roll-back method and the blank shape for mass production is illustrated in Fig.16.The optimum blank shape shows curvature because the outer contour of the product and the ¯ow rate of the sheet metal are considered.However,the blank shape for mass production is simple and straight because the convenience of cutting is considered.To verify the result an initial blank cut by a laser-cutting machine was prepared.The ®nal shape drawn with the initial blank in the press shop isshownparison of the initial ¯ange shapes and the deformed ¯angeshapes:(ainitial square blank;(b®rst modi®ed blank;(csecond modi®ed blank;(dthird modi®edblank.parison of the initial blank contour between the roll-back method and Huh's method.J.-Y.Kim et al./Journal of Materials Processing Technology 101(200031±4335in Fig.17.It had a ¯ange of uniform size around its periphery.The thickness distribution at the position of four sections in the longitudinal direction of the left member was mea-sured.Fig.18shows a comparison of thickness between the computed results and the experimental results in each sec-tion.In section A,the thickness distribution has some error at the end of the ¯ange,whilst in sections B and C,the computed results are compatible with the experimental results.In section D,the computed results predicted that a split might happen,but the experimental cup did notsplit.Fig.9.Load±displacement curves in the square cup drawing process with various initial blankshapes.Fig.10.Thickness strain distribution in a square cup:(adiagonal direction;(btransversedirection.Fig.11.FE-model for a sub-frame left member.If the initial blank shape,the ®nal shape and thickness distribution are considered,the results predicted by the roll-back method has a good agreement with the experimental values.Therefore,as well as the roll-back method being applicable to a simple shape,it can be applied to a complex and large shape.3.3.Blank design of No.2member of front sub-frameAn analysis of No.2member is performed,with its deep and complex shape.Its optimum blank shape is designed using the roll-back method.Fig.19shows the manufacturing set-up as modeled for the numerical simulation.Because its drawing depth is very deep,eccentricity may occur due to the blank initial position or shape.Thus the target contour is de®ned at the position that is 40mm from the outer contour of product and it is shown in Fig.20.A square blank is used as the initial blank.After threemodi®cations the optimum blank shape isdetermined.Fig.12.Target contour for the leftmember.Fig.13.Load±displacement curves in the left member drawing process with various blankparison of the initial ¯ange shapes and the deformed ¯ange shapes:(ainitial blank;(b®rst modi®ed blank;(csecond modi®ed blank;(dthird modi®ed blank.Fig.15.Thickness distribution with various blank shapes(unit:mm:(ainitial blank;(b®rst modi®ed blank;(csecond modi®ed blank;(dthird modi®ed blank.parison of the initial blank shapes predicted by the roll-back method and those designed by skilled labor.For this case,load±displacement curves for various blank shapes are shown in Fig.21,whilst a comparison of the initial ¯ange and the deformed ¯ange with various blank shapes in shown in Fig.22.The thickness distribution with the initial shape is shown in Fig.23,whilst the thickness distribution with the optimum blank shape is shown in Fig.24.The thickness distribution of the side-wall and of the ®llet connecting the side-wall to the top isimproved.Fig.17.Left member drawn in the press shop with the initial blank predicted by the roll-backmethod.Fig.18.(aSections for measuring the thickness distribution.(b±eThickness distributions at sections A±D,respectively.3.4.Design of the welding line with TWB analysis of No.2memberAfter designing the optimum blank shape of No.2member,a tailor-welded blank is applied to this member.To reduce the weight of the sub-frame,structural analysis is performed.On the area where the stress intensity level is low,it is proposed to reduce the thickness locally.Therefore,it is required to design a tailor-welded blank that makes a speci®ed shape after deformation.When two sheet metals of different thickness are welded together,their metal ¯ow is different from that of sheet metal of the same thickness.Thus it is dif®cult to design the location of the weld line.In this simulation the weld line is designed by the use of the roll-back method and the welding line should be located at the speci®ed position after deformation:the speci®ed position is 120mm on both sides of the centerline.Thus the target line is de®ned and meshed by beam elements.The outer contour of TWB and the welding line are shown in Fig.25,and the results are shown in Figs.26and 27.The welding lines can be reached to the target line but,on the top of the blank that has the lower thickness,fracture may occur.This is the same as the result that in the deep drawing of a tailor-welded blank with different thickness,failure occurred at the ¯at bottom of the punch parallel to the weld line.This is due to the deformation not beingdis-Fig.19.FE-model for the sub-frame leftmember.Fig.20.Target contour for the No.2member.Fig.21.Load±displacement curves in the No.2member drawing process with various blank shapes.J.-Y. Kim et al. / Journal of Materials Processing Technology 101 (2000 31±43 41 Fig. 23. Thickness distribution with the initial blank shape (unit: mm: (a front view; (b rear view. Fig. 24. Thickness distribution with the optimum blank shape (unit: mm: (a front view; (b rear view. Fig. 22. Comparison of the initial ¯ange shapes and the deformed ¯ange shapes: (a initial blank; (b ®rst modi®ed blank; (c second modi®ed blank; (d third modi®ed blank. Fig. 25. Comparison of the weld line between the initial blank shape and the deformed blank shape.42 J.-Y. Kim et al. / Journal of Materials Processing Technology 101 (2000 31±43 4. Conclusions In this paper the roll-back method that designs an optimum blank shape is proposed. Based on the method, a computer program composed of a blank design module,an FE-analysis program and a mesh generation module is developed and it is applied to the deep drawing of a front sub-frame. The results of the present paper are summarized as follows: 1. To verify the validity of the proposed method it is applied to the deep drawing of a square cup. The outer contour may be drawn to the target contour. 2. The roll-back method is applied to the optimum blank design of a left member of an automobile sub-frame. The thickness distribution and the load level are improved. When the initial blank shape, the ®nal shape and thickness distribution are compared, the results predicted by the roll-back method have a good agreement with the experimental results. It is concluded that this method can be applied to the deep drawing of the complex automobile parts. 3. The analysis of No. 2 member with a tailor-welded blank is performed. The position of welding lines on the initial blank is designed. The roll-back method can be applied to the design of the welding line position. 4. In most cases, the edge of blank takes the shape of the target contour within a few iterations, which shows that the roll-back method is an effective and convenient method for an optimum blank shape design. Fig. 26. Deformed shape of No. 2 member with the tailor-welded blank. Fig. 27. Deformed shape of No. 2 member with the tailor-welded blank: (a front view; (b rear view. tributed uniformly, most of the stretching being concentrated on the side of the blank with lower strength. The process condition without fracture should be determined for the combination of the drawing depth and the two different thickness as shown in Fig.28. References [1] V.V. Hasek, K. Lange, Use of slip line ®eld method in deep drawing of large irregular shaped components, Proceedings of the Seventh NAMRC, Ann Arbor, MI, 1979, pp. 65±71. [2] T. Jimma, Deep drawing convex polygon shell researches on the deep drawing of sheet metal by the slip line theory. First report, Jpn. Soc. Tech. Plasticity 11 (116 (1970 653±670. [3] M. Karima, Blank development and tooling design for drawn parts using a modi®ed slip line ®eld based approach, ASME Trans. 11 (1989 345±350. [4] J.H. Vogel, D. Lee, An analysis method for deep drawing process design, Int. J. Mech. Sci. 32 (1990 891. [5] X. Chen, R. Sowerby, The development of ideas blank shapes by the method of plane stress characteristics, Int. J. Mech. Sci. 34 (2 (1992159±166. [6] R. Sowerby, J.L. Duncan, E. Chu, The modelling of sheet metal stamping, Int. J. Mech. Sci. 28 (7 (1986 415±430. [7] S.A. Majlessi, D. Lee, Further development of sheet metal forming analysis method, ASME Trans. 109 (1987 330±337. [8] S.A. Majlessi, D. Lee, Development of multistage sheet metal forming analysis method, J. Mater. Shap. Technol. 6 (1 (1988 41± 54. [9] K. Chung, O. Richmond, Ideal forming-I. Homogeneous deformation with minimum plastic work, Int. J. Mech. Sci. 34 (7 (1992 575±591. [10] K. Chung, O. Richmond, Ideal forming-II. Sheet forming with optimum deformation, Int. J. Mech. Sci. 34 (8 (1992 617±633. Fig. 28. Thickness distribution with the tailor-welded blank (unit: mm: (a front view; (b rear view.J.-Y. Kim et al. / Journal of Materials Processing Technology 101 (2000 31±43 [11] K. Chung, O. Richmond, Sheet forming process design based on ideal forming theory, Proceedings of the Fourth International Conference on NUMIFORM, 1992, pp. 455±460.[12] K. Chung, O. Richmond, The mechanics of ideal forming, ASME Trans. 61 (1994 176±181. [13] C.H. Lee, H. Huh, Blank design and strain prediction of automobile stamping parts by and inverse ®nite element approach, J. Mater. Process. Technol. 63 (1997 645±650. [14] C.H. Toh, S. Kobayashi, Deformation analysis and blank design in square cup drawing, Int. J. Mech. Tool Des. Res. 25 (1 (1985 15± 32. 43 [15] Z. Zhatao, L. Bingwen, Determination of blank shapes for drawing irregular cups using and electrical analogue methods, Int. J. Mech. Sci. 28 (8 (1986 499±503. [16] H. Huh, S.S. Han, Analysis of square cup deep drawing from two types of blanks with a modi®ed membrane ®nite element method, Trans. KSME 18 (10 (1994 2653±2663. [17] C.H. Lee, H. Huh, Blank design and strain prediction in sheet metal forming process, Trans. KSME A 20 (6 (1996 1810±1818. [18] C.H. Lee, H. Huh, Three-dimensional multi-step inverse analysis for optimum design of initial blank in sheet metal forming, Trans. KSME A 21 (12 (1997 2055±2067.。