PC-based blank design system for deep-drawing irregularly shaped prismatic shells with arbitrarily shape flange

Joumalof

Materials Processing Technology ELSEVIER

Journal of Materials Processing Teclmology 63 (1997) 89-94

PC-based blank design system for deep-drawing irregularly shaped prismatic shells with arbitrarily shaped flange Toshihiko KUWABARA, Wen-hua SI Department of Mechanical Systems Engineering, Tokyo University ofAgriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184, JAPAN Abstract A method for determining optimum blank shapes for the production of irregularly shaped prismatic shells with an arbitrarily shaped flange and stepped bottoms is proposed. All of the procedures for calculation are incorporated into a CAD system that can output the optimum blank shape in a few seconds using a personal computer. The calculation method is based on the slip-line field theory; a kinematically admissible velocity field for the flange region is determined from the theory and the calculated velocity field is used to conduct a backward calculation of the metal flow, from the designed flange shape to the initial blank shape. In the calculation it is assumed that the material is isotropic and that the thickness of the blank does not change during the deep-drawing process. A deepdrawing experiment is conducted for a square shell with six different flange shapes, using aluminum alloy sheet A5182-0 l.Omm thick and cold-rolled steel sheet SPCE O.5mm thick. Experimental flange shapes are in good agreement with the designed (predetermined) ones. The influence of material characteristics on the blank design is found to be relatively small. Keywords: sheet metalforming, deep-drawing, prismatic shell, flange shape, blank design, slip-line field theory, CAD

1. Introduction

In deep-drawing of irregularly shaped shells, the blank shape is an important factor. In order to improve the deep-drawability of the product and use the material most efficiently, the blank shape should be designed such that the deep-drawn shell becomes a finished product as it is without trimming of the blank edge. In this paper, such a blank shape is referred to as an optimum blank shape. Several numerical analysis methods for predicting optimum blank shapes have been proposed so far, as listed in Ref.[l], however, many of them are for shells with uniform wall heights, not for shells with arbitrarily shaped flanges. Jimma and one of the present authors [2] applied the slip-line field theory to the prediction of an optimum blank shape for an oil-pan-shaped shell produced by two drawing stages, and verified that the actual metal flow in the flange area was in good agreement with that predicted using the kinematically admissible velocity field calculated from the slip-line field theory. Later, this analysis method was successfully applied to irregularly shaped autobody parts by Ikura et al. [3]. The aim of this study is to refine the blank design method proposed in Ref. [2] and to develop PC-based software for determining optimum blank shapes for irregularly shaped prismatic shells with predetermined arbitrarily shaped flanges. As far as computational time and investment cost are concerned, this method is superior to those based on FEM [4,5] and yields a practical blank design system for use in sheet metal stamping shops. 0924-0136197/$15.oo@ 1997 Elsevier Science SA All rights reserved

PII S0924-0136(96)02605-2

2. Analytical Method 2.1. Basic assumptions (i) The blank is an isotropic, rigid-perfectly plastic material. (ii) The thickness of the blank does not change during the deepdrawing operation. (iii) The die profile radius, rd, and the punch profile radius, r p , are taken to be zero in the deformation analysis of the blank, while their actual values are considered in calculating the effective draw-height H (see eq. (2» . (iv) The material part that has flowed into the die cavity does not deform but moves vertically with punch speed, V, as a rigid body; accordingly, material elements have velocity vectors normal to the die cavity contour, the magnitude of which equals V. (v) The die cavity contour coincides with the minimum principal stress trajectory. 2.2. Kinematically admissible velocity field in the flange area in deep-drawing ofconvex polygonal prismatic shells The slip-line field as shown in Fig.l(a) can be constructed based on assumption (v). Jimma [6] has proven that given assumption (iv) and the equi-areal characteristics of the net of principal stress trajectories [7] the streak lines of material elements in the flange area coincide with the maximum principal stress trajectories. The Geiringer equations are then easily integrable and it is readily shown that the hodograph for tlle slipline field in Fig.l(a) consists of logarithmic spirals as shown in Fig.l(b) [8]. Hence the kinematically admissible velocity field for the flange area can be calculated using equation (l):

Toshihiko Kuwabara. Wen-Hua Si I Journal of Materials Processing Technology 63 (1997) 89-94

90

Die cavity contour v

(4) Select material elements on Poqo with some interval and let (Xo,Yo) denote the coordinates of these material elements. They become the starting points for calculation. (5) Calculate the magnitude Vo and the angle 120 of the velocity vector at (Xo,Yo) from equation (1). (6) Using the Vo and 120 obtained in (5), calculate the tlrst approximation of the coordinates, (Xl', f l '), at which the material elements on Poqo are located at the onset of the (N-I )-th drawstep.

)Y

p

Xl' = X o + Vo ·Llt· cos

12 0

f l' = f o + Vo ·Llt· sin 120

(i,i) (a)

(3)

(7) Calculate the VI' and a!, of the velocity vector at (XI',YI') from equation (1). Then substitute them into equation (4) to determine the coordinates, (XI,fl ), at which the material elements on poqo are located at the onset of the (N-I )-th drawstep.

VI" = ( Vo + VI' ) I 2 121"

=(

120

+

121') /

2

XI =Xo +vI"·Llt·cos 12 1"

f 1 = f o + VI" ·Llt· sin 12 1"

(i,i)

(9,9)

Q)(O,O)

(b)

Fig.! (a) Slip-line field in the flange area of a deep-drawn rectangular shell ( 5° net) and (b) the hodograph.

= v(O,O) = V,

Area I

v

Area II

v = v(O,i),

Area ill : v

a

a

Connecting the material elements at (Xj,f l ), we determine the contour of the flange PI ql which is to remain at the onset of the (N-I)-th draw-step. Thus (XI'yl) become the starting points for the next calculation step. (8) Repeating (4) to (7) N times, we obtain the initial (optimum) blank shape PNqN.

2.4. Some calculated examples

= 0°

Figure 3 shows the calculated blank shapes for a rectangular shell with an elliptic flange (a) and for a rectangular shell with a polygonal flange with rounded comers (b).

a = [5 x ijO

= v(0,9) = 0.456V,

(4)

= 45°

v = v (i, i), a = [5 x (j - i)r

Area IV : Area V : v = v(i, 9), a

= [5 x (9 -

i)r

Area VI : v=V·rclr; r=Ocp,a //OcP' where v(i,i) = Vexp(-n{i+i)/36);

(1)

Maximum principal stress trajectories \ _...\----/1 (Streak lines of material elements)

i,i=0~9.

2.3. Calculation of an optimum blank shape for a prismatic shell with an arbitrarily shapedflange (a) Input data TIle input data for this blank design system are the dimensions of the horizontal section of the shell to be drawn, draw-height of the shell h, punch profile radius rp , die profile radius rd, and the designed flange shape. From these input data, the computer calculates the optimum blank shape automatically. (b) Calculation Procedure (I) Calculate the effective draw-height H of the shell as H= h - 0.43 (rp + ra).

(2)

(2) Divide the whole deep-drawing process into N draw-steps, and assume that the draw-height of the shell increases by H/N for every draw-step. (3) Superimpose the contour of the designed flange shape Poqo on the slip-line field constructed as shown in Fig.2.

Die cavity contour

Fig.2 A schema for calculating the blank shape PNqN to give a rectangular shell with the designed flange Poqo.

Toshihiko Kuwabara. Wen-Hua Si I Journal of Materials Processing Technology 63 (1997) 89-94

Figure 4 shows the calculation procedure used to obtain a shell with stepped bottom having a flange of constant width (Fig.4a). It is assumed that the shell is fonned by two drawing stages: bottom I is fonned by the first drawing stage and bottom II by the second drawing stage. The calculation starts from the designed flange shape Poqo: first, the flange shape Prrqrr giving Poqo is detennined from the slip-line field constructed around the

second draw-die (Fig.4b), where prrqrr is the flange shape at the onset of the second drawing stage. Next, the flange shape P I q I giving Prrqrr is detennined from the slip-line field constructed around the first draw-die (Fig.4c), where P I q I is the initial blank shape at the onset of the first drawing stage. Therefore, P I q I is the optimum blank shape for the shell in Fig.4a.

I/_mnm")m ,'~RlO

I

-- --i-=-- - -- L _ o R30

i "

1"<1" /

I

R21'/-~~~~~~~~l~~~ ~

60 80

96

w

~ Fig.3 Examples of calculated blank shapes for rectangular shells with an elliptic flange (a) and with a polygonal flange with rounded comers (b). From inner to outer, draw-height II =30mm and 40mm ( r p = 0 mm, rd =0 mm ).

I o

]I

N

100 80

(b) The second draw

o

'D

(a)

91

0

00

(c) The first draw

Fig.4 Calculation Procedure to obtain a shell with stepped bottom having a flange of constant width.

92

Toshihiko Kuwahara, Wen-Hua Si I Journal of Materials Processing Technology 63 (1997) 89-94

3. Experimental

4. Results and discussions

The conditions for deep-drawing experiments are as follows: ·Press: hydraulic single-action press ·Blank holding force: 6.5kN constant ·Punch speed: 2.2mm/sec . Tool: square punch with side length L=60mm, comer radius rc =15mm, punch profile radius rp =6mm, die profile radii ra =5mm and lOmm, clearance 1.2mm. In order to examine the influence of the r-value on the flange deformation, materials with different r-values, i.e., sheet aluminum alloy A5182-0 (l.Omm thick) and AI-killed sheet steel SPCE (0.5mm thick) were used. The mechanical properties of these materials are listed in Table 1. The blanks were lubricated on both sides with Johnson Wax No.700 for A5182-0 and with machine oil #54. Figure 5 shows the designed flange shapes adopted in this study. The calculated blank shapes for the shells in Fig.5 are presented in Fig.6, where the increment of draw-height is taken to be 1mm for one calculation step and the starting points for calculation are taken on the contour of the designed flange shape with 1.0mm spaces.

Figure 7 compares the designed flange shapes with those of experimentally drawn shells. The shapes are generally in good agreement in the transverse directions of the shell, while in the diagonal directions the experimental flange' edges tend to remain somewhat outside the designed flange shape. Kuwabara [9] showed that this phenomenon was due to the delay of metal flow in the diagonal directions of the shell and that the delay occurred because the circumferential compressive strains of the material elements which comprised the straight side wall of the shell were less than the theoretically predicted values. It is noted that the difference between the results for A5182-0 and SPCE is quite small for all the flange shapes. This is possibly due to the relatively small draw-height of the shell: half of the side length of the punch. Therefore, the influence of material characteristics can be ignored when determining optimum blank shapes for prismatic shells, provided that the draw-height of the shell is about half of the representative dimension of the shell or less . 5. Method for correcting blank shapes In this section, a method for determining more accurate blank shapes is proposed, which involves a single trial-draw.

Table 1 Mechanical properties of materials. Material 00 A5182-0

*1

45° 90°

0'0.2

U B

/MPa

/MPa 264 268 262

122 126 127

c-value*2 n-value r-value /MPa 541 549 535

0.31 0.31 0.30

0.70 0.52 0.63

0° 146 552 2.27 293 0.26 149 537 SPCE 45° 286 0.25 1.77 533 90° 152 0.25 2.47 290 * 1 DirectlOn relatIve to the sheet rollmg dIrection of the material. *2 The uniaxial true stress-true strain curves of the materials are approximated by a = c&".

5.1. Calculation procedure (1) Superimpose the designed flange shape Poqo and the contour of the experimentally obtained flange afJ on the slip-line field as shown in Fig.8. (2) In Fig.8, the difference between Poqo and afJ indicates that the amount of metal flow into the die cavity is larger than the theoretical prediction in the sections where afJ is inside Poqo, while the reverse applies in the section where afJ is outside Poqo. From this consideration, the contour of an imaginary flange shape PoQo is constructed so that the gap betweenap and Poqo becomes equal to the gap between poqo and PoQo in the direction of the maximum principal stress trajectory at every point on Poqo.

105 Initial blank shape C>

0\

1

Po

Po a

(a)

Die cavity contour

(b)

(e)

(f)

Fig.5 Geometry of designed flange shapes.

Fig.8 A schema for calculating a corrected blank shape Pt!2N to produce a shell with the designed flange shape poqo.

ToshihikoKuwabara. Wen-Hua Si I Journal of Materials Processing Technology 63 (1997) 89-94

,

,

/-------r-----"\

//-----t------',\

r

I

/-------1'-----"\

_L_l_J_

)

I ' l, __ -l.--_~: _

(a)

93

r

I

)

_~: _-L

__ i_

(b)

(c)

,

,

y-----+----y

/cL-t---n",

I

~'

iii

-~-~-~-

--~'--

(f)

(e)

-- - - - - Die cavity contour

\p

l'

-~--

- - - Designed flange shape

- - - Calculated blank shape

Fig.6 Blank shapes calculated by the present method. From inner to outer, draw-height h=30mm, 40mm, and 50mm.

~~~-.:-:::---.-

~//

I

'\\

·1

-~-

-~-

I

_ --Ll-

rF:--~----~--~

",<~,

III/

,

Y

\)

I

J.._ rd=lOmm, h =50mm

,

~

JL

,

_1_

~

-i

_1_ N=IOmm, h =30mm (b)

(a)

(c)

A5l82-0

o rd =IOmm, h =30mm

h=30mm

h=40mm

h =50mm

h=40mm (f)

h =50mm

SPCE

rd =5mm, h =30mm (d)

h=30mm

- - Die cavity contour - - - - - - Designed flange shape - - - Experimental ( A5l82-0 ) - - - - Experimental ( SPCE ) Fig.7 Comparison between experimental results for A5l82-0 and SPCE. A5182-0 could be drawn only for h=30mm.

94

/, =

Toshihiko Kuwahara, Wen-Hua Si / Journal of Materials Processing Technology 63 (1997) 89-94

I

:

_ r

't

~ ,

;

I

~_

\

_~_

A5182-0

,

;

;

-, SPCE - - - Die cavity contour - - - - - - Designed flange shape

Corrected blank shape - - - - Initial blank shape

Fig.9 Comparison between initial blank shapes and corrected blank shapes ( rd=lOmm, h =30mm) .

(I , _1 _1 ~

/

I

'\

I

--~

- - - Die cavity contour

Designed flange shape

L_

~ ~---

_..i-

1

~.

Experimental (A5l82-0) - - - - Experimental (SPCE)

Fig. 10 Comparison between designed flange shapes and those experimentally obtained from the corrected blanks shown in Fig. 9.

(3) Select material elements on PoQo with some interval as the starting points for calculation. (4) Repeat the same calculation procedure (5)- (7) as described in subsection 2.3. The blank shape, PNQN, finally obtained is the corrected blank shape to produce the shell with the designed tlange shape Poqo .

after a single trial-draw they almost coincide. It has been found that the influence of material characteristics on the blank design is relatively small. Since this blank design system works on a personal computer and requires only a few seconds for computation, it appears to be a practical blank design tool in sheet metal stamping shops.

5.2. Experimental

References

Experimental conditions arc the same as those in section 3. Figure 9 compares the corrected blank shapes calculated from the experimental flange shapes in Fig.7 with the initial blank shapes.

[1] T. Jimma and T. Kuwabara, in PLASTICITY AND lvfODERN METAL-FORJMING TEGINOLOGY, ed. Blazynski, T.Z., Elsevier Science Publishers, (1989) 207. [2] T. Jimma and T. Kuwabara,l'roc. 2nd Int. Con! Technology ofPlasticity, Vol.II(1987) 1159. [3] S. lkura, A. Sueki, N. Nakagawa, K. lkemoto and M. Toda, TOYOTA Technical Review, 42 (1992) 75. (in Japanese) [4] Y.Q. Guo and lL. Batoz, Int. .I. NUn!. Methods Eng., 30 (1990) 1385. [5] H. Iseki and R. Sowerby, JSME Int. .I., 38 (1995) 473. [6] T. Jilmna,.1. Jpn. Soc. Tech. l'lasticif-V, 11(1970) 707. (in Japanese) [7] M.A. Sadowsky, .I. Appl. Mech., 8(1941) A-74. [8] J. Chakrabarty, The01Y of Plasticity, McGraw-Hill, New York, (1987) 534. [9] T. Kuwabara, Proc. 1991 Jpn. Spring Con! Techno!. Plasticity, (1991) 507. (in Japanese)

5.3. Results Figure 10 compares the designed flange shapes with those experimentally obtained using the corrected blank shapes in Fig.9. Fairly good agreement is obtained,which proves that the present method is effective in determining more accurate blank shapes. 6. Conclusions A method for detenllining optimum blank shapes for the production of irregularly shaped prismatic shells with arbitrarily shaped flanges and stepped bottoms has been proposed using the slip-line field theory. Experimental flange shapes are in relatively good agreement with the designed flange shapes, and