A simplified method for the simulation of cold-roll forming

Int. J. Mr,oh, ,~¢i. Vol. 35, No. 10, pp. 867-878, 1993 Printed in Great Britain.

0020-7403/93 $6.00 + ,00 ~ 1993 Pergamon Press Ltd

A SIMPLIFIED METHOD FOR THE SIMULATION OF COLD-ROLL FORMING G. NEFUSSI and P. GILORMINI Laboratoire de M6canique et Technologie, ENS de Cachan/CNRS/Universit6 Paris 6, 61 Avenue du Pr6sident Wilson, F-94230 Cachan, France

(Received 9 October 1992; and in revisedform 19 April 1993) Almtraet--A kinematical approach is proposed in this paper for predicting the optimal shape and the deformed length of a metal sheet during cold-roll forming, before the first roll stand. The middle surface of the sheet is described as a Coons patch depending on one geometrical parameter. A velocity field is then defined on this surface so that the plastic work rate depends only on this single geometrical parameter. Its minimization gives the optimal shape for a strain-hardening rigid-plastic material. This approach has been implemented on a workstation, and it allows a very fast simulation of the process. Moreover, this method can be extended to different shapes, and the whole process including several roll stands can also be calculated.

NOTATION A a~, a3 a.p b~p D~p 2e g., g~ g~p H i K I L M M' n p Pc u ~, u z, u 3 V0 V(M)

material constant local curvilinear basis at current point on the reference surface first fundamental tensor second fundamental tensor strain-rate tensor strip thickness local curvilinear basis in the shell metric tensor mean curvature Roman subscript (i = 1,2, 3) total curvature strip width deformed length current point on the reference surface current point in the shell hardening rate equivalent plastic strain power rate curvilinear coordinates of M' entrance velocity of the strip velocity at a current point on the reference surface Greek subscript (~, = !,2) F ~ Christoffei symbol ~r~ Cauchy stress tensor 6- yield stress of the material

INTRODUCTION

Cold-roll forming can be described as a process where a metal sheet is formed continuously in the transverse direction into a product with a desired cross-sectional profile by passing through a series o f pairs cf forming rolls. Many kinds of products with various crosssectional profiles and sizes are manufactured through cold-roll forming, but only circular arcs, V and trapezoidal sections are usually studied. Since the deformation process is quite complex and the curved surfaces are difficult to describe, cold-roll forming has been mostly developed o n an empirical basis and from experimental knowledge. Moreover, finite element approaches are not convenient for this three-dimensional problem, and most authors have used kinematical simplified methods. 867

868

G. NEFUSSIand P. GILORM1N!

In 1964, Masuda et al. [1] studied the forming of a strip with a circular cross-section and, neglecting membrane strains, obtained the roll force and torque. The deforming surface was approximated by a continuous series of circular arcs. For the case of a channel section, Bhattacharyya et al. [2, 3] considered a one-parameter velocity field and minimized the deformation energy for a rigid-plastic material. Bending was assumed to take place along the fold-line and was neglected in the flange. The deformation length between two roll stations was calculated in Ref. [2], and the roll load was given in Ref. [3]. Very important work has been achieved during the last two decades by Kiuchi and coworkers [ 4 7] who have shown, in their experimental and analytical papers, that all strain components must be considered, and not bending strains only. Their description of the surface for the forming of a circular profile uses sinusoidal functions depending on one parameter which is determined by minimizing the total strain energy. Finally, Walker and Pick [8] suggested the modelling of the sheet with a B-Spline tensorial surface for tube forming, and an approximation of the axial strain rate was obtained. A similar representation of complex surfaces has also been used by Bernadou and Lalanne [9] for elastic problems, but for application to finite elements. In the first part of this paper, we describe the middle sheet surface between two roll stands with a technique (Coons patch) used in computer-aided design. The shape of this surface depends on a few geometrical parameters, such as the deformed length between two rolls. A stationary velocity field, tangent to the surface, is defined on this Coons patch and the strain-rate tensor is calculated by adapting a shell theory where displacements are replaced by velocities. This leads to the second part where the dissipated power is calculated for a strain-hardening rigid-plastic material. Its minimization with respect to the geometrical parameters gives an optimal shape in this kinematical simplified approach. The results are shown in the last section, for a trapezoidal profile. This procedure can be repeated for each consecutive pair of rolls, so that the whole process can be described. DEFINITIONS OF THE SURFACE AND VELOCITY FIEt.D The sheet middle surface between two sets of rolls is described as a Coons patch in the present work. It is a curved quadrilateral surface (Fig. 1) defined in a parametric form by the following relation [10]: O M ( u l,u 2 ) = - F o ( u l ) F I ( u 2) O A - F o ( u I ) F o ( u 2) O B - F I ( U l ) F o ( u 2) OC - F l ( u l ) F l (u 2) O D

+ F0(U1) OMl(U 2) + Fo(U 2) OM2(u 1) + FIIul)OM3(u

2) + F1{u 2) OM4(u 1)

(1)

for a patch between the four curves OMl(ul), OM2 (u2), OM3(u 1) and OM4{u e) defined in a parametric form with u 1 and u z varying from 0 to 1, and containing the four corners OA, OB, OC and OD. The functions Fo and F1 are given by: Fo(u)=

- 6 u 5 + 15u 4 -

10u 3 + 1 and

Fl(u)=6u

~ - 15u 4 + 10u 3.

From the u 1 and u 2 curvilinear coordinates on the surface, a local non-orthonormal basis can be defined (Fig. 1): (ul, u2 ) a l ( u ~, u 2) _ t~OM au 1

and

fl2(U 1, u 2) - 0OM (u t, u 2) . ~u 2

(2)

It can be verified easily from Eqn (1) that the tangent in the u 2 direction along a u I = 0 edge of the Coons patch is: a l ( 0 , U2) = F o ( U 2 ) a l ( 0 , 0) + Fl(u2)al(0, 1)

(3)

and the same relation applies for a u I = 1 edge, and similarly for u 2 = 0 or 1 edges with al replaced by az. Thus, there will be no discontinuity of the surface and of the normals between two adjacent Coons patches, provided that the tangent vectors at the two common

Simulation of cold-roll forming

A

869

u2= 1

C

FIG. 1. Coons patch.

corners are continuous. Moreover, with this choice of Fo and F1, it can be shown that the curvatures are also continuous [10]. Complex profiles can be described with this technique, but application is restricted here to the case of a simple channel shape, to restrain the number of free geometrical parameters. For symmetry reasons, one-half of the sheet is considered (and, more precisely, only the deforming region in this half), and a single Coons patch will be used between two roll stands. Its shape is restricted by the following considerations (Fig. 2). First, the AB and CD sides are given by the shapes of the roll stands, which are prescribed. Then, the BC line is assumed to be straight, since it bounds a rigid zone (the bottom part of the channel, of uniform width). This line defines the extension of the area where plastic deformation occurs and its length L is the geometrical parameter to be calculated. Finally, the AD edge is free, and has to be determined in the analysis. Moreover, the length of the AB and CD arcs are supposed to be equal, and the AD edge is described as a fifth-order spline. This curve is completely defined by the positions of A and D, and by the tangents and curvatures at these points. The second step in our analysis consists of defining a stationary velocity field on the above surface. The most natural choice has been made, viz. taking the u 1 lines as trajectories and the u 2 coordinates as the successive positions of a given material line. This provides a field of velocities that are tangent to the surface, and such that V(M) = Cal(u ~, U2). C is a constant defined by considering the movement along the straight BC line at a constant speed: C = Vo/L where Vo is the overall velocity of the sheet (i.e. its velocity when it moves as a rigid body before the first roll stand). When all the n roll stands are considered in turn, various cases may appear for the definition of the n - 1 Coons patches, which are best described by going upstream. Since the sheet exits as a rigid body after the last roll stand, the velocity should be equal to Vo all along the last CD line. A property of the Coons patch that can be deduced from Eqn (1) is the uniformity of vector al all along a u 1 = 0 or 1 edge when the tangents a~ at the ends of the edge are equal. Therefore, the tangent to the AD spline at D must be al = V o / C , where C,_ ~ denotes the value of constant C mentioned above, for the (n - l)th Coons patch. Two possibilities may occur for the length L,_ 1 of the deformed zone, which appears in C._ i = Vo/L._ ~. If L._ 1 is smaller than the distance d._ 1 between roll stands n - 1 and n (Fig. 2b), then there exists a transverse rigid zone after roll stand n - 1, and the velocity should be equal to Vo all along the AB line. The tangent vector to the spline at point A should also be equal to Vo/C._ 1, consequently, since Eqn (3) ensures the desired uniformity of the velocity along the AB line in these conditions. Thus, the geometry and the velocity field for this patch are completely defined by the single parameter L._ 1. By contrast, if the plastically deformed zone extends from roll n - 1 to roll n (no transverse rigid zone, Fig. 2c), then L._ 1 must be equal to the given distance d._ 1 between the rolls, and the intensity of the velocity V._ 1 (parallel to V0) at point A is taken as the parameter entirely defining the geometry and the velocity field for the considered patch. Similar considerations apply to the preceding pairs of roll stands, up to roll number 1. The first type of reasoning (L._ 1 undetermined) applies to the first Coons patch (Fig. 2a), since the sheet is assumed to be flat and undeformed at an unbounded distance before the MS 35:10-E

870

G. NEFUSSIand P. GILORMINI

(a) D

."

~.A

,

7

'

.,,

(b) A

D •

J

/

,,'

(c) FIG.2. View of the deformed sheet: (a) before the first roll stand; (b) and (c) before the last roll stand.

first roll. It should be noted that a single parameter is required in this approach, for the descriptions of the surface and velocity field between each pair of roll stands, and that the velocity is continuous in the whole process. For the sake of clarity, the index corresponding to the Coons patch number is dropped in the next section, where the case of a Coons patch with L undetermined is considered.

CALCULATIONS OF THE ENERGY RATE The power dissipated by plastic deformation between two roll stands can be calculated from the strain-rate tensor deduced from the above velocity field, after it has been extended through the sheet thickness. For this purpose, an additional unit local vector must be

defined in a first step: a3(u 1, u 2) =

al(ul' u2) ^ a2(ul' u2) l ai(ui, u 2) ^ a2(u 1,u2) 1 '

(4)

where a3 is normal to al and a2. Since the sheet is thin, a material line initially normal to the reference surface is assumed to remain so at any subsequent time; this is analogous to the

Simulation of cold-roll forming

871

Kirchhoff-Love hypothesis on displacements in classical shell theory. Consequently, the velocity of a point M' of the sheet (on the local basis defined on the middle surface) is assumed to be given by [11]: V l ( M ') = Vt(M)

-- u3bl

VI(M),

VZ(M') = - u3b~ V'(M),

(5)

v3(m ') = 0, where u 3 defines the position of a point M' located along the normal at point M belonging to the reference surface. In Eqn (5): (6)

b~ = a~eb~a,

where bro is the second fundamental tensor: Oa~

b, a = ~ u~ "a3. Moreover: a ~r = a s" a ~,

(7)

by introducing the dual basis (with superscripts), given by aa.a~ = 3~, where 6 is the Kronecker delta. Summation over repeated indices is used throughout this paper; superscripts identify the contravariant components of a tensor, and subscripts its covariant components. In Eqn (5), use has also been made of the assumption V2(M) = VS(M) = 0 mentioned in the previous section. It should also be noted that VS(M ') has been supposed to be zero through the thickness, since this simplifies the subsequent calculations. As a consequence, plastic incompressibility cannot be obtained exactly from the above velocity field. However, the D] component of the strain-rate tensor will be deduced here directly from the components in the sheet plane by prescribing volume preservation, and it will be verified afterwards that this leads to small thickness variations compatible with the neglecting of V3(M'). The derivation of the components D} of the strain-rate tensor can be performed by taking partial derivatives of the velocity field with respect to the u i coordinates. The following intermediate components D~p are first calculated: Oil = [ a l l F ~ l + a12F211 - u 3 ( b l l ' 1)] VI(M), D22 = [aa2F~2 + a22F22 - uZ(b12.2 - b~F12 - b12F22 + b12F22 + b 2 2 F 2 0 ] V l ( M ) , 2D,2 = 2D21 = [a~tr~2 + a~2(rzl2 + r~a) + a22F~

-- u3(bll,2 + b12, 1 - bllI'12 - blzF22 -{- b121~l, + b22F21)] V~(M),

(8)

where: a,,p = a~ "ao

(9)

is the first fundamental tensor, and where the second-order terms in u 3 have been neglected, as usual. The Christoffel symbols F~a are necessary in surface theories when the derivatives of the components of a vector are calculated on a curved surface: t~a~ F,ra = ~ u~ "a r.

(10)

The D} components are finally given by: D~ = g~D~a ,

D3 = - D ~ - D 2 2 , 03l = D 3 = O 2 = D 3 = 0 ,

(11)

872

G. NkFUSSIand P. GILORMINI

where the incompressibility condition has been used to obtain D~ (it is recalled that the Greek indices take the values 1 and 2 only). The first components in Eqn (1 I) are calculated from Eqn (8), using the following set of relations: ~fl" ,q,,p = 6~. ¢/,p = g~." gp,

g, = a t -

(12}

u-~b~a~,

where the gp vectors are similar to the ap for a current point M' in the sheet. The last line in Eqn (11) is a direct consequence of the hypothesis that material lines normal to the reference surface remain so. All the derivations in this section are similar to those involved in classical thin shell theories [11], where small displacements are replaced formally by velocities. The local power rate dissipated by plastic deformation per unit volume at a current point in the sheet takes the following form for a yon Mises material: P(M')

=

= o-5o

=

tS,

!13i

where the following expression of the equivalent strain rate has been introduced, by using the incompressibility condition: 2 /5 = ~ [ ( D I ) x/3

2 + (D]) z + D~ D ] + D~D2] 12

{14)

and where 6- is the yield stress, related to the deviatoric part s~ of the stress tensor by:

The total power rate dissipated on the whole Coons patch and through the sheet thickness 2e, where plastic deformation occurs, is: P(M')(I - 2Hu 3 + K(u3j2)x/det(a~p) du I du 2 du 3 .

Pc = 1=0

{16)

• u2=O

In Eqn (16), the surface element in curvilinear coordinates has been used, and the mean and total curvatures have been introduced [11]: H = _~(b] + b ~ )

and

K = h{b 2 - b 2 h~i .'

{I7~

The calculation of Eqn (16) is performed numerically by Gaussian quadrature. Its minimum with respect to the geometrical parameter L is found by using Brent's method [12]. This optimization method only requires evaluations of the function itself and not of its derivatives. Strain hardening is introduced through a given 6(p) function. The values of the equivalent plastic strain p at the integration points are obtained with an additional Gaussian quadrature along the trajectories. It is noteworthy that the total power rate depends only on the chosen geometrical parameter (here the optimal deformed length L~ with this approach. It should be noted that the dynamic terms have been neglected in this analysis. Although the velocities may be quite large in roll forming, they lead to small accelerations: along any trajectory, both the curvature and the variations of the velocity magnitude are small. R ESU L]S This analysis has been applied to a metal strip of thickness 2e, and width I before the first roll stand. The exit profile is supposed to be trapezoidal and defined by the fold angle 0: the C D line is made of (i) a circular arc of radius 1 mm and angle 0; and fii) a straight line. The AD line, which is described here with a fifth-order spline, is assumed to have a zero curvature at point A, since the strip is flat at this point. At point D the radius of curvature of the sheet along the u I coordinate is given by the roll radius. The strip thickness was 0.6 mm, the exit roll radius 100 mm, and a power-hardening rule was used: 6" = Ap" with A = 1 and n = 0.2. The minimization leads to the optimal deformed length L, to the shape of the strip, to the velocity field and to the dissipated energy rate.

Simulation of cold-roll forming

873

First, calculations were performed for different widths and fold angles. Figure 3 presents the deformed shapes for three different angles (20 °, 40 ° and 60°); the corresponding values of the deformed length L are not much different from each other, so that L seems to be steady in the three cases. This is more obvious in Fig. 4, where it can be seen that the deformed length L is not very much affected by the exit angle 0, although L exhibits a maximum around 0 = 40 °. On the other hand, the deformed length L is almost proportional to the strip width 1 (Fig. 5). The variation would be exactly linear if the sheet thickness was also changed in the same proportion. Then, the strain-hardening parameter n was varied from 0 to 0.7. In Fig. 6, it is evident that the deformed length increases with n, which is satisfactory since hardening should lead to lower strain gradients and thus to a longer deformed zone. At last, different sheet

................ ]" F"'~"~e

FIG. 3. Calculated shapes of the sheet for different fold angles: (a) 20°; (b) 40°; and {c) 60 °, for I = 15ram.

874

G. NEFUSSI and P. GILOrMINI

25

~J 24

L l

23 0

I0

I

I

l

I

I

I

i

20

30

40

50

60

70

b(

o (deg) FIG. 4. Deformed length L (mm) for 1 = 15 mm and various fold angles 0 (degrees).

70

6O

50

40

30

.jJ

20

I0

0 0

.J

i

i

i

i

L

5

I0

15

20

25

30

35

I (mm) FIG. 5. Deformed length L (mm) for O = 30 and various sheet widths I (mm).

27

/.Ji

25

/'JJJ

23

/

21 ..-.I 19

i7

I

i i

L5

0

OI

i

i

~

i

i

]

02

03

04

05

06

oz

n

FIG. 6. Deformed length L (mm) for different values of the hardening parameter n.

875

Simulation of cold-roll forming

0

02

06

04

06

e Imm) FIG. 7. Power rate PC (lo-’ W) for various half-thicknesses e (mm).

0a

FIG. 8. Maps of the equivalent strain rate for I = 15 mm: (a) 20”; (b) 40”; and (c) 60”.

876

G. NEFUSSIand P. GILORMINI

thicknesses were considered (from 0.2 to 1 ram). The variation of the power rate is presented in Fig. 7. The curve is quite parabolic, as suggested by Kiuchi [7]. More precisely, it can be shown that the variation of Pc/e vs e is a straight line, so that it can be verified that the total p o w e r rate is a p p r o x i m a t e l y the sum of a quadratic term (the bending power ratei and a linear one (the m e m b r a n e power rate), with negligible coupling between the two terms. Moreover, after minimization of the plastic power rate, the velocity lield was also obtained. F o r instance, the maps of the equivalent strain rate are presented in Fig. 8 tor 0 = 20 °, 40 ° and 60 °. In each case, it is clear that high values are reached along the fold line. where curvatures are large, and along the free edge, near the exit roll, where m e m b r a n e strain is important. Of course, the larger the fold angle, the higher the strain rate values. The equivalent strains (Fig. 9) were obtained by integration along trajectories. I,ike the strata rates, they increase with the exit angle and show high values along the fold line and near the free edge. The relative thickness changes, deduced from the incompressibility condition, are shown in Fig. 10. O f course, there is a relatively large thickening along the fold. and a thinning near the free edge. In each case, the thickness variation is small, because of the choice of the velocity field. This is consistent with the hypothesis made above, where variations in e were neglected.

.............

/ .....~,,~ii~iii~iii~iii~iii~iiii~ii,~................... :~

//~

(a) 0.35

0.3 0.25

i¸¸ i

.....~ ~

0.2

~ : ~ i . l . i i i ~ l l l i l i i t~i~i'i ~iii~ i ~~!~i~ i ~i I

0.15

(b)

0. I iiii~iii

0.05

(cl FIG. 9. Maps of the equivalent strain for I = 15 mm: [a~ 20'; (b) 40; and tel 60.

Simulation of cold-roll forming

o.o9

877

(a)

0.07

0.05

0.03

0.01

-

(.

o,01

0.03

I FIG. 10. Relativethickness variation for I = 15 mm: (a) 20°; (b) 40°; and (c) 60°.

CONCLUSIONS In this paper, a kinematical approach has been proposed for the simulation of the cold-roll forming of rigid-plastic materials. This simplified method is based on a description of the sheet surface with Coons patches and adapts calculations used in classical shell theory. Thus, a finite element calculation, which would be quite difficult and computer-time consuming, is avoided. It should be stressed that only one parameter has been used to describe the complex shape of the sheet between two roll stands, the corresponding velocity field, and then the dissipated rate of power, so that minimization is very fast. Moreover, the membrane and bending effects were both taken into account and strain hardening as well. The proposed model has been applied here to a simple case: only one roll stand and one type of profile. However, the method is quite flexible, other shapes and more complex velocity fields can easily be introduced, if required. For instance, different profiles can be described by modifying only the equation of the edge M3, and additional parameters may be introduced by changing the equation of M,. Moreover, a series of roll stands can be

878

G. NEFUSSI and P. GILORMINI

considered as in the first section. In this case, a nonuniform yield stress must be introduced at each entrance profile, because of strain hardening in the previous steps. The main drawback of the present study is that only rigid-plastic behaviour has been considered, so that elastic springback cannot be deduced. The next step should include elasticity, but this will require a more complex analysis and is left for future work. Acknowledgements The authors wish to thank C. Karra and S. Quement for their numerical implementation of the model. Financial support for this research was provided by the Sollac Company, under Contract No~ 6561/'345061.

REFERENCES 1. M. MASUDA, T. MUROTA, T. JIMMA, T. TAMANOand T. AMAGAI, Fundamental research on the cold-roll forming of metal strips. Bull. JSME 7, 827 (1964}. 2. D. BHATTACHARYYA,P. D. SMITH,C. Y. YEE and I. F. COLLINS, The prediction of roll load in cold-roll forming. J. Mech. Working Technol. 9, 181 (1984). 3. D. BHATTACHARYYA,P. D. SMITH,S. K. THADAKAMALLAand I. F. COLLINS,The prediction of deformation length in cold-roll forming. J. Mech. Working Technol. 14, 363 (1987). 4. M. KIUCHI, Analytical study on cold-roll forming process, Report of the Institute of Industrial Science, University of Tokyo, Tokyo (1972). 5. H. SUZUKI, M. KIUCHI and S. NAKAJIMA,Experimental investigation on cold-roll forming process, I, Report of the Institute of Industrial Science, University of Tokyo, Tokyo (1972). 6. H. SUZUKI, M. KIUSHI, S. NAKAJIMA,S. ICHIDAYAMAand K. TAKADA,Experimental investigation on cold-roll forming process, II, Report of the Institute of Industrial Science, University of Tokyo, Tokyo (1976). 7. M. KIUCHI and T. KOUDABASHI,Automated design system of optimal roll profiles for cold-roll forming. Proc. 3rd Inst. Conf. on Rotary Metalworking Processes, Kyoto, p. 423 (1985). 8. T. R. WALKERand R. J. PICK, Approximation of the axial strains developed during the roll-forming process of ERW pipe. J. Mater. Process. Technol. 22, 29 (1990). 9. M. BERNADOUand B. LALANNE, Sur [' approximation des coques minces par des m6thodes "B-Splines" et 6l$ments finis. Acres du 3brae Colloque en Calcul des Structures, Bastia, p. 939 (1985). 10. S. A. COONS, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, MIT. Cambridge, MA (1967). l l. M. DIKMEN, Theory of thin elastic shells. Surveys and Reference Works in Mathematics, Vol. 8. Pitman, New York (1982). 12. W. H. PRESS, B. P. FLANNERY, S. A. TEUKOLSKY and W. T. VETTERLING, Numerical Recipes The Art oI Scientific Computing. Cambridge University Press, Cambridge, U.K. (1989).