The optimum die profile for the cylindrical bending of plates

ELSEVIER

Journal of Materials ProcessingTechnology 70 (1997) 151- I55

The optimum die profile for the cylindrical bending of plates Suha Oral *, Haluk Darendeliler

Abstract

This paper presentsa methodologyfor the design of plate-forming dies in cylindrical bending using optimization techniques to reduce the cost of die production by reducing the trial-and-error procedure considerably in determiningthe final die geometry. The plate thickness is discretized by plane-strain finite-elements. The die is taken to be rigid and its profile is approximated by Bezier curves the control-point

coordinates of which are the design variables. The die

profile is varied

to minimize the difference

between the required shape and the shape of the bent plate, considering springback action. The unconstrained optimization method. A numerical exampleis presentedwhere the problem is solved by the BFGS (Broyden-Fletcher-Goldfarb-Shanno) optimum die profile is obtained for a plate bent into a quarter circle. 0 1997 Elsevier Science S.A. Keywords: Optimum die profile; Cylindrical bendmg

1. Introduction In general, the production of bending dies is based on a trial-and-error process where the dies are reproduced or modified a number of times to obtain the required shape. This is mainly due to the springback action and causes loss of time and labor, and increases the overall cost. In recent years, powerful computational techniques based on high-speed computers have been developed for the analysis of plasticity problems. The springback in forming can be determined with sufficient accuracy by the use of such techniques. This study has been motivated by the thought that the computational determination of the springback can bc used effectively to reduce the effort in die design by posing the die design as an optimization problem. This had led to an unconstrained optimization problem, the objective of which is to obtain a die geometry that minimizes the deviation of the plate after springback from the required shape. A literature survey revealed that an accurate determination of the optimum die shape based on mathematical programming as presented herein has not yet been attempted.

* Corresponding author: Tel.: f90 2101266.

In this study, the determination of the optimum due profile is invebtigated for the bending plates. For simplicity, only cylindrical bending is considered, which reduces the dimension of the problem. In the following sections, an elastic-plastic plane-strain finite-element and a contact element are formulated, the ot%imum design problem is posed and a numerical example is

2. Formulation The die and the plate shown in Fig. I. The die is

312 2101381; Paax:+90 312

0924-0136/97/517.00 0 1997 ElsevierScienceS.A. All rights reserved

Fig. I. A plate in cylinorical bending.

152

S. Oral, H. Darendeliler/Journal of Marerids Processing Technology 70 (1997) IS/ - I55

plate is bent under the action of hydrostatic pressure of intensity q. The cylindrical-bending operation is inde-

pendent of the z-axis, hence it can be reduced to a two-dimensional problem where the cross-section of the plate is discretized by plane-strain finite-elements and the die profile is represented by a plane curve. It is to be noted that a single mathematical function may not be sufticient to describe a complicated die profile and different functions may have to be used for different segments of the profile. Hence, care must bc taken to choose functions that can be blended smoothly. Beizier

(;-mpr:ai)

c, =~+mpd+_ ; C2=

-~+mPa;s;,-f(~-mp.:rl)(~-)nv~;,~a)

(8b) C, = 2 [mpr:.:,

c~=;+m/Xr;‘-~

c, = 2 [tnpc+r.,

2.1. The elastic-plastic

plane-strain

(Sd=C - ‘d - 26dTPd + SgrRg)

dV =

&=i dS

sY (1)

the vectors of velocities, velocity gradients and deformation rates, respectively. These can be expressed as:

where

u, g and d are

VT= [o,

UY]

i?= [%.X 0%” OJLA%.Yl ff = L

sj.+

s&Y+

%,I

Cl

c2

c3

c4

c5

[ sym and the material

C6

(mpal.a:,)2

1

W?

and E is the elasticity modulus, v is Poisson’s ratio, $, are the deviatoric stresses, d = ,/wi is the equivalent stress, 6, is Kronecker’s delta and m is equal to zero for elastic loading and unloading and unity for plastic loading. H is the slope of the Cauchy stress

versus logarithmic plastic-strain curve, which can be obtained from the foliowing relationship: (10)

(5)

and g can be interpolated

as:

d=N,q

(lla)

g=U

(1 lb)

where NJ and N, are appropriate shape matrices and q is the vector of nodal velocities. Then the element stiffness matrix k is obtained from Eq. (1) by the usual procedures a+: ’ - 2P)N,

+ N:RN,]

dV

(12)

as:

1

constants

(8e)

where:

k= jV [NZL’

c=

tnpn;a+npo jt:,.)]

where G(,B and n are constams to be determined from the simple tension test. During deformation, the material yields when the equivalent stress equals to yield strength uY. The strain-hardening of the material is taken into account by updating the value of a, in the

0 i

law can be written

(‘E-

a=x(P+p)”

(6)

The material

+;

(8d)

subsequent loadings and unloadings. By using the standard bi-linear shape functions [2], d

1 0

q (+1prr;aj)2

(8~)

(3) (4)

components:

br,

mpo;+n,~c:c:J]

(2)

where i is the surface traction rate vector, and P and R are the properly arranged matrices of Cauchy stress

rax

+ $ (‘E-

-i

@ire-e&tent

In this section a four-node, elastic-plastic, quadrilateral plane-strain finite-element based on the updated Lagrangian rate formulation [l] is described, this element being used in the discretization of the plate thickness. The principle of virtual work at the element level can be expressed as:

@a)

(7)

be defined

as:

The contact element used is shown in Fig. 2, where x and y are the global axes, the axes I and n being tangential and normal to the contact surface, respectively. A contact element is formed during the bending

153

where 0 < r < 1 and (x,, J+) are the coordinates of the control points. For complicated die profiles, it might be necessary to employ more than one Bezier curve, which are smoothly blended at the end points. The coordinates of the nodes on the die profiG are determined by specifying the coordinates of the Bezier control points.

I

2.4. Non-linear bending analysis x

Fig.2. Thecontact

element.

During the bending process, the plate is bent under the action of applied hydrostatic pressure and it comes into contact with the rigid die surface as bending progresses. By replacing the rate quantities with incremental quantities, assuming that the rates are kept constant within an incremental step, the problem is solved incrementally and the contact conditions are imposed by the penalty method as: (17)

process for each node on the die profile as the normal distance h between that node and the plate surface becomes less than a pm-defined value, E. The contact element formed has three nodes: node A is on the die

and nodes B and C are the closest nodes on the plate contact surface to the die node A. The plate is free to slide on the die but it is not allowed to penetrate into it. Hence the element level constraint condition can be written as:

w=o

(13)

e=[-sin$cosq%(l-$sinqSG-l)cos4isin)

-;cosq5 1 q:= br, 4 .A

(Ma) VXe

qs

“vc

q

(14b)

where a, b and 4 are the geometrical quantities as delined in Fig. 2, and the elements of qcare the velocities evaluated at the contact element nodes. For a rigid and stationary die, u, and u are zero. Then the element constraht

ma&

can be formed as:

k,=pcTc

(15)

where p is a penalty number. 2.3. Bezier curves In this study, the die profile is represented by Bezier curves [3]. A Sezier curve in the ?cy-plane can be defined as: x(r) = (1 - r))x, + 3r(l - rj2x2 + 3r*(l - r)q + r3.r, y(r) = (1 - r)‘y, + 3r(l - r)‘y, + 3r2(1 - r)y3 + r3y4 (16)

where K is the assembled stiffness matrix, s is the assembled constraint matrix. and AU and AF are the incremental nodal disD!acements and forces referred to

global After accomplishing the full and the die, the date is unloaded and the springback is determined. . 2.5. The design problem The objective of the present study is to determine a die profile such that a required shape is obtained after the springhack of the plate at the end of the unloading stage. In the present study, this is posed as an uncon~ strained minimization problem where the objective function to be minimized is a notm that defines the deviation of the plate after springback from the required shape. Hence the objective function, f, can be defined as:

f=

_;,&a, $_

(18)

where A4 is the number of plate nodes on the contact surface and d,,, is the noGal distance from the mth node to the required shape. The design variables are the coordinates of the control points of the Bezier curves used in defining the die profile. The unconstrained minimization problem is solved using the BFGS (Broyden-Fletcher-GoldfarbShanno) optimizer [4-71 in the ADS Automated Design Synthesis Package [8]. The detailed derivation of the method is given by Gill et al. [9]. Numerical experiments with the BFGS algorithm suggest that this algorithm is superior to all known variable-metric algorithms [IO].The gradients of the objective function with respect to the design variables are evaluated by overall finite differences. The bending analysis is an inner loop in the optimization process and yields the

nodal-point coordinates of the plate after springback that are used in constructing the OF’ tive function.

3. Nmnerial example A 2 mm thick rectangular plate with a free length of 250 mm is to be bent into a quarter circle of 100 mm radius. The material properties of the plate are: E = 200 GPa; Y= 0.25; ov = 225 MPa; a = 330 MPa; p = 0.15; and n = 0.2. In this example, a single Etexier curve is employed to represent the die profile, as shown in Fig. 3. The coordinates of the fourth control point are kept fized as (0,O). On the other hand, to satisfy the zero slope condition at the 6xed end of the p!ate, y, should also be zero. Hence there are five desim variablesfor which the initial design is x, = y, = x, =~OO mm and xs = yz = 75 mm. This closely resembles a quarter circle of 100 mm

radius. The rigid-die profile is divided into equal length segments using 200 nodes for the contact analysis. The plate is discretized by 5 x 80 plane-strain t’inite-elements. A uniform load with a rate of 1 kPa is applied on the plate surface. The penalty number p has been taken to be 10’s and the contact check has been done by taking c as 0.001 mm. The initial die profile and the contact surface of the deformed plate after unloading are shown together with the required shape in Fig. 4. The optimization history is given in Fig. 5. As showo, the initial error norm is 32.80 mm and it is minimized by 17 iterations to 2.26 mm. The optimum values of the

design variables are: x, = 87.57 mm; y, = 92.76 mm; x, = 100.53 mm; yZ = 69.50 mm; xs = 69.78 mm. The optimum die protile, the contact surface of the deformed plate after unloading and the required shape are shown in Fig. 6.

Fig.

4. The

initial

die design

and the plate after

springback.

4. Conchtsions this study an optimum design methodology has oronosed for the dies used in the cylindrical bending of plates. The die profile has been modeled by Bezier curves. the control-point coordinates of which have been taken as the design variables. The use of Bezier curves has the advantage of smooth blending in complicated die profiles which are defined by more than one segment, each of which requiring a different mathematical form. The plate thickness has been discretixed

In been

by elastic-plastic plane-strain finite-elements, and a triangular contact element has been employed. The numerical examde has demonstrated that the ootimum

die profile can be significantly different from-the required shape of the bent plate. Hence it can be concluded that the proposed methodology effectively to reduce the cost of die

can be used

production by reducing the trial-and-error process considerably in de-

X Iteration

Fig.

3. A Bezier

curve

and the control

points.

Fig.

5. Convergence

in the optimization

process

155

face, and

by modeling the blank with shell finite-elements usmg three-dimensional contact elements.

References [I] R.M. McMeeking, J.R. Rice, Finite clement formulations for problemsof large elastic-plastic formulation. Int. J. SolidsLruct. II (1975) 60-616. [Z] K.J. Bathe, Finite Element Proceduresin Engineering Analysis, Prent~cc-Hall,Em&wood Cliffs, NJ. 1982. 19 M. Berger.Computer Graphics with Pascal.The Benjamin-Cummings Publishing, CA, 1986. [4] C.G. Broyden,Thcconvergenceoraclassofdouble-mnkminim~lion algorithms.2.Thenewdgorilhm. J. Inst. Math. Appl.6(1970) 222-231. [5] R. Fletcher. A new approach to variable-metric algorithms, Comput. J. 13 (1970) 317-322. 161 V. Goldfarb, A family of variable-metric methods derived by variational means, Math. Comput. 24 (1970) 23-26. [7] D.F. Shanno. Conditioningofquasi-Newtonmethodsforfunction minimization. Math. Comput. 24 (1970) 647-656. [S] G.N. Vanderplaats. ADS-A Fortran Program for Automated Design Synthesis.vols. 2.01, Engineering Design Optimization, 1987. [9] PI. Gill. W. Murray. M.H. Wright, Practical Optimization, Academic Press,New York, 1981. [IO] J.E. DennisJr., J.J. More,Quasi-Newton methodsmocivationand theory, SIAM Rev. 19 (I) (1977) 46-89.