Finite element analysis of roll-forming of thin sheet metal

J o u r n a l of

Materials Processing Technology ELSEVIER

J. Mater. Process. Technol. 45 (1994) 255-260

FINITE ELEMENT ANALYSIS OF ROLL-FORMING OF THIN SHEET METAL M. Brunet and S. Ronel Laboratoire de M~canique des Solides Institut National des Sciences Appliqu(es Bat. 304. 69621 Villeurbanne France Abstract In roll forming, flat strip is progressively deformed by successive sets of rotating rolls. For the purpose of studying this sheet forming process, a specific finite element method has been developed based on plane strain shell elements for cross-section analysis with elastic-plastic behaviour and combined with a 3D analysis of the sheet between two roll-stands. The contact and friction boundary conditions are imposed by moving shaped tools in order to form sections of great complexity.

1 .INTRODUCTION In roll forming process, a sheet metal is continuously and progressively tbrmed into a product with required cross-section by passing through a series of forming rolls arranged in tandem, as it is shown in Fig.(1) :

to form section of great complexity but it is required that the thickness should remain unchanged. The art of roll forming is to impart shape by forming the bends without deforming the linear elements sufficiently to cause sectional and longitudinal defects. A finite element approach is adapted to model the roll forming in order to analyze deformations of the sheet, therefore it has the functions to evaluate the designed roll profiles, to make necessary amendments and to attain optimization of them. The outlines of these simulation method are being explained in this paper. 2. CROSS-SECTION ANALYSIS

Y

Fig. (1) Roll forming process There are number of experimental and analytical studies of the roll forming process published /1/,/2/ but they are restricted to simple shapes, mainly circular or channel sections. This paper proposes a general numerical method for arbitrary cross sections where the roll forming practice is still largely based on empiricism and active experimentation By suitable roll design it is possible

The basic governing equations are given by the principle of virtual work at time t + At referred to the current configuration at time t witch can be written in the following way /3/: ~ [AS:~ + (o-+ AS):(v~u.~uv)]dv = v

R + [. At.au dst+ [ Af.au ds ~st / -sc c

0924-0136/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSD! 0924-0136(94)00191-X

(1)

256 where o- is the Cauchy stress tensor and txS the convected increment of Kirchhoff stress. This permits us to calculate a certain state of equilibrium based on the knowledge of the previous e q u ~ f i u m state together with the actual increments in the external loads on St or the prescribed displacements and contact forces on Sc. To avoid error accumulation and to permit unloading the new equilibrium state is calculated by an iterative Newton's method where the term:

It is well known that the so-called Reissner Mindlin thick shell element family in direct application to thin plate situations can induce locking. In the thin plate limit Kirchhoff' s assumptions of vanishing transverse shear strain must be satisfied. A procedure is to impose a shear strain field which satisfies the limit thin plate condition. For our two nodes element this is achieved by evaluating the shear strain ~, at the element midpoint. 3. B O U N D A R Y

CONDITIONS

R=[t.~u dst+ [f.6u ds c- ~o-- ~e dv (2) Jt

~c

is often called the residual which is expected to vanish in the increment. The finite element discretization has been made on plane strain shell element of Reissner-Mindlin type. Since independent rotational degrees of freedom are used only Co continuity is required and this by-passes the difficulties caused by the C1 requirement of the classic Kirchhoff theory. With these assumptions using linear shape functions and considering generalized plane strain conditions we derived a finite element with two nodes and one rotational and two translational degrees of freedom at each nodes. This element which allows for combined in-plane and bending behaviour is divided into a number of layers in order to capture the spread of plasticity over the thickness. For integration over the sheet thickness five layers are used although the number of layers can be extended up to 10 in particular for element on small tools radii. Since moderately thick sheets will be analysis the usual assumption of plane stress state in the thickness direction can be justified here. Then the thinnin.g effect due to large plastic strain is taken into account by an update procedure assuming plastic incompressibility in every time step .

In the present formulation of sheetmetal forming process it is assumed that the tool parts are completely rigid bodies whose relative movements are forced on a deformable elastic-plastic solid. It is worth noting that the moving tools not only translate as rigid bodies but can change their shapes continuously with time for roll-forming process (see figure 2 for example). Then we can use parametric functions of space and coordinates varying with time to describe the rigid moving surfaces : n

X(t) :

Z Nk(~) Xk(t) k--1 n

Z(t) = ~ Nk(g) Zk(t) k=l wheren = 2 or 3 .

Fig.(2) Forming of circular section

(3)

(4)

257 In an incremental approach of time step dt we determine an appropriate prescribed displacement where the nonpenetration inequality is used to just bring a or some material points to be incontact with the rigid surface "

the applied loads and to the stiffness matrix of the system such that:

du n - dgn+ dn-< 0 on S c (5) where du n is the increment of the normal

(11) and (12) From a mechanical or computational point of view it seems better in (11) and (12) to associate the local normal axes to the element side instead to the tool because displacements stresses and equilibrium are analyzed only for the deformable body. However this approach can give rise to problems especially if a rather coarse mesh is to be used but it is not really the case for our cross section analysis. The contact and friction boundary conditions are solved by a static implicit algorithm where we assume that in the incremental solution the response at time t has been calculated and that (i-l) iterations have been performed to calculate the solution at time t + r t . Then in each iteration the first step in our procedure is to update the pressure contact vectors with their material increment evaluated at the integration point of the contactor element sides. The decision on whether a contactor integration point is releasing or is in sticking or sliding condition is based on considering the total and relative magnitudes of these updated pressure contact forces as follows • If f --- 0 the point is assumed to have

displacement constrained by the given rigid surface motion dg n. To resolve this boundary constraint exterior penalty method:

we apply the

dfn= -kn(dU n- dgn) on Sc

(6)

with a very large penalty parameter k n > 0. The point will stay in contact as long as the resultant contact pressure tn updated by its material increment d f remains negative. We assume that there exists a Coulomb friction law so that no relative motion is observed if:

Ifnl

Ift I < v (7) Thus a same relation that (6) holds between the tangential increment of the friction force and the tangential increment of the displacement: dft-- -kt(du t- dgt) on Sc (8) When the state of stress of a contact point is such that: [ft I : ~ I f l and Idftl = vt I d f l (9) the point is said to be sliding and (8) gives the consistent tangential penalty parameter as • k t : vt I d f l / I ( d u t- dgt)l (10) The contact condition and the Coulomb s' friction law are satisfied in a integrated sense over each individual contact element side (upper and lower surface) consistent with the level of finite element shape functions used. The friction forces can then be determined using the quantifies available in the course of the iteration process and contribute via the penalty parameters to

A~t3=~kndgnn i+ktdgtj(~ij-ninj)] N/3 dS KeU~/3 = ~clk n n.n.+ .1 1 kt(~ij-nini)] No~ N/3 dS

experienced tension release and in this case the penalty parameters and the normal and tangential contact pressure are set to zero. If fn ~ 0 and I ftl < I fnl the point continues to stick or sticking contact conditions must be assumed by setting the penalty parameters to a large positive number which may be function of the local stiffness of the body. If f - 0 and Iftl ~ ~ If l then the

258 state of the point must be updated to sliding with :

sign=dft/Idftl and ~=~-l+sign uldfnl (13) where

-~t-lmust satisfy the sliding condition (iteration i-l). 4.

SOLUTION

previous

PROCEDURE

Conceptionally the roll forming process is a 3D. bending work of sheet metal which develops complicated 3D. curved surfaces with smooth or sharp comers (stiffeners). A given cross section of the sheet is numerically deformed continuously by tools from the i-th to the (i+l)-th roll-stand profile (see figure 2 for a circular tube and the following figure 3 for more complex profile) by the 2D. cross section analysis already explained. In order to take account of the transitions of the longitudinal strains we use a less r e ~ mesh of the 3D. curved surface between two roll-stand (figure 4 for example) with 4 nodes quadrilateral Mindlin shell elements which are under-integrated by lxl Gausspoint and hourglass control by matrix stabilization. Therefore, the same family of elements (Co Mindlin ) is used for the deformed surface analysis between two roll-stands. The boundary conditions are imposed incrementally by mean of connected nodes between the 2D. and 3D. shell meshes. The longitudinal displacements of the shell element nodes in each section i and i + 1 must be equal which leads to linearly constrained boundary conditions. In our program, a modification of the frontal solution technique for incorporating linearly constrained equations into the global matrix equations is applied. Computationally, our solution procedure for roll-forming can be viewed as a "master" 2D, cross section analysis with a "slave" 3D. shell analysis.

Both 2D. and 3D. analyses include the anisotropic Hill's model of initial anisotropy with isotropic and cinematic hardening. The extension to the complete large strain case requires an objective formulation of these classical small strain constitutive relations. This is simply achieved by using a rotating frame formalism where the stress strain and state variables tensors are replaced by their rotated tensors by an objective rotation R : tr- = R T . o- . R , e- - R T . c R (14) Since elastic stratus are assumed to be small it is convenient to ensure the objectivity of the hypoelastic law by writing it in the same rotating frame.In order to integrate the elasto-plastic relations the rotating frame formalism simply requires following the rotation R for all elements and making the appropriate rotation but taken in the state at the middle of the total displacement increment to evaluate the rotated tensors of strain and stress. We use the elastic predictor-radial corrector implicit algorithm in the rotating frame to integrate the constitutive relations and many authors have demonstrated the overall superiority of this method over others proposed algorithms. Notice that according to the rotating frame formalism the values are calculated based solely on the converged values at the end of the previous increment. The non-converged values at the previous global iteration within the current step play no explicit role in the stress evaluation. This scheme is path-independent and avoids numerical unloading. The last aspect demands that in an efficient solution procedure the elasto.-plastic stiffness matrix must be consistent with the implicit integration procedure of the constitutive rate equations in order to improve the rate of convergence. A derivation of this consistent modulus can be found in /3/ .

259 5. NUMERICAL EXAMPLES First and in order to analyze a spring back sensitive situation, the numerical model of the cross-section only has been used to simulate experimental plane strain deep drawings carried with blank-holder pressure as it had been demanded in the Numisheet'93 Conference benchmark tests (see /4/). By the above mentioned method of 2D. and 3D. analysis, numerical calculations were made about forming process for the product with a circular arc crosssectional profile and compared with experimental results carried out by Kiuchi and al. /1/. As an example of a more general case, the following figure shows some of the successive crosssectional deformed meshes limited to an half-wave of an industrial product.

Fig(4). Fig.(3) Complex sheet profiles

Example of 3D. deformed meshes between roll-stands

260

Cross section bottom strain e, 0.15

|

i

i

Calculus Experimental--c---

0.i

0.05

,ib w

-0.05

-0.1

-0.15

20

4o

60

so

Distance from Z axis

loo

120

mm

Fig.(5) Final cross-section bottom strain Fig. (5) shows the calculated and measured cross-section bottom strains on the final form profile (half wave) shown in Fig.(3). Experimentally, a square grid has been deposed on the sheet before rolling to measure the final surface strain, We have obtained a good agreement between calculated and measured strains. It can be seen that the optimization of the designed roll profiles is nearly obtained here since linear parts are slightly deformed. REFERENCES

1. Kiuchi M. and Koudabashi T. Automated design system of optimal roll profiles for cold roll forming. Proc. 3rd. Int. Conf. on Rotary Metal Working Processes K y o t o Japan 8-10 sept, (1985),423-436

2. Walker T.R. and Pick R.J. Approximation of the axial strains developed during the roll forming of ERW pipes. J. of Mat. Proc. Technology, 22. (1990),29-44 3. Brunet M. Some computational aspects in three dimensional and plane stress finite elastoplastic deformation problem. Engineering Analysis with Boundary Element. VoL 6 (1989) n °2 78-83 4. Brunet M and Sabourin F. Benchmark results of 2D draw bending. Numisheet'93 2nd. Int. Conf. Num. Sim. of 3D Sheet Metal Forming Processes. Isehara Japan 31 Aug.-2 Sep.,(1993), 639 and 647-652 5. Brunet, M. A finite element method for unilateral contact and friction problem involving finite strain and large displacement. Journal of Theoretical and Applied Mech. Vol.7 (1988),209-220