strep pel 93 suitability

8/9/2019 Strep Pel 93 Suitability

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Journal of Materials Processing Technology 36 (1993) 33~356 339Elsevier

Sui t ab i l i t y o f shee t bend ing mode l l ingt e c h n i q u e s i n C A P P a p p l i c a t i o n s

A.H. Streppel, L.J. de Vin, J. Brinkman and H.J.J. Kals

Laboratory of Production Engineering University of Twente P.O. Box 217 7500 AE Enschedethe Netherlands

(Received February 26, 1992; accepted July 7, 1992)

I n d u s t r ia l S u m m a r y

The use of CNC machine tools, together with decreasing lot sizes and stricter toleranceprescriptions, has led to changes in sheet-metal part manufacturing. In this paper, problemsintroduced by the difference between the actual material behaviour and the results obtainedfrom analytical models and FEM simulations are discussed against the background of therequired predictable accuracy in small-batch part manufacturing and FMS environments.The topics are limited to those relevant to bending along straight lines. Applications ofanalytical and empirical models in the area of CAPP and adaptive control are discussed.

Process planning for sheet bending, however, should not be treated as an individual task butmust be related to other process-planning tasks. An integrated CAPP system for sheet metal,PART-S, is introduced briefly.

1 . I n t r o d u c t i o n

Over the past few years , an increase d effort has been put into re searc h in thesheet-metal area, due part ly t ot he introdu ction of CNC machine tools for sheet-

metal man ufac turin g processes. General ly s imultaneous with this development,customer demands changed, leading to decreased lot s izes, increased productvariety and str icter tolerance prescript ions. As a resul t , a bet ter and predict-able process acc ura cy is required now ada ys and test products should no longerbe needed. This has led to research both on more trad i t ional processes such aspunching and bending and on relat ively new processes such as laser cut t ingand laser welding. Furthermore, the workload for process-planning depart-ments has increased (as has previously been the case for machining opera-tions). Thus, the need for CAPP systems arises also in the sheet-metal area.

Correspondence to: L.J. de Vin, Laboratory of Production Engineering, University of Twente,P.O. Box 217, 7500 AE Enschede, the Netherlands.

0924-0136/93/ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.


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340 A.H . S treppel et al ./ Shee t ben ding mo delling techniques

The overall accuracy of a sheet-metal product depends on the subsequentoperations it undergoes, the bending process having a profound influence onthis accuracy. With respect to the bending process, closed-die bending used to

be the most common way of bending along straight lines. However, attentionhas moved towards air-bending, as this is a more flexible method allowingdifferent products (with respect to angles, sheet thickness or material) to bebent without the need for frequent tool changes. Calculation of the correctpunch displacement, however, is not easy. Many effects influence the spring-back and, consequently, the required punch displacement. Although most ofthese effects have been known for a long time already, their influence is notalways easy to predict. Together wit h the inc reased ac cura cy demands, this isone of the reasons for the increased research effort on the bending process.Apart from analytical and experimental work, FEM simulations are used andattention has moved from description of individual effects towards the calcu-lation of punch displacement and adaptive control.

In order to benefit fully from technologic al know-how, the bending processmust not be treated as a separate operation. Process pla nning for bending mustbe imbedded in an integrated CAPP system, such a system including theselection of equipment, the de term inat ion of the bending sequence and instruc-tions for positioning. A systematic ana lysis of toleran ce prescript ions must beperformed initially. Individual bending operations may be selected usingmanufacturing rules in combination with the accuracy of preceding and fol-lowing operations (such as laser cutting, punching, laser welding and otherbending). Final ly, there is a strong relati onship wit h plann ing aspects since, insmall-batch manufacturing, different products are nested in one single sheet.

PART-S, a sheet metal CAPP system currently under development at theLaboratory of Production Engineering, offers this required integrated func-tionality. After completion, it may be used in CNC and FMS environmentsproducing sheet-metal components in small batches within a thickness rangeof 0.5 to 5 mm.

2 . M a t e r i a l b e h a v i o u r

Material behaviour is an important aspect in the understanding of thebending process. There are several models to describe the mate rial behaviour,but some phenomena, however, are difficult to model.

2 .1 . M o d e l l e d b e h a v i o u rThe behaviour of a material is described primarily by its stress-strain

relationshi p, graphic al repr esent atio n of relati onships being shown in Fig. 1.Two areas can be distinguished: an elastic region and a plastic region. The

stre ss-s trai n diagr am shown in Fig. 1 (a) represents a rigid ideal plastic mater-ial. There is no work hardening nor an elastic region. A simple model like thismay be used as a reference to describe the effects of an elastic region or work


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A.H . Streppel e t a l . / Sheet bending model l ing techniques 341

o &

eRigid-ideal plastic

EElas t ic - Elas t ic- l in eari d e a l p l a st ic w o r k h a r d e n i n g

e eW o r k h a r d e n i n g ' A c tu a l' c u r v e R i g i d - w o r kaccordingt o N a d a i h a r d e n i n g

F i g . 1. S t r e s s - s t r a i n d i a g r a m s .

h a r d e n i n g . A m a t e r i a l w i t h o u t w o r k h a r d e n i n g , b u t w i t h a n e l a s ti c r e g io n , isr e p r e s e n t e d b y t h e s t r e s s - s t r a i n d i a g r a m i n F ig . 1 (b). T h i s m o d e l m a y b e u s e d ,f o r i n s t a n c e , f o r r o u g h e s t i m a t i o n s o f s p r i n g - b a c k .

F i g u r e s l ( c ) a n d l ( d ) a r e e x a m p l e s o f s t r e s s - s t r a i n d i a g r a m s w i t hw o r k h a r d e n i n g . L i n e a r w o r k - h a r d e n i n g is r e p r e s e n t e d b y t h e d i a g r a mi n F ig . l ( c ), w h e r e a s w o r k h a r d e n i n g a c c o r d i n g t o N a d a i i s s h o w n i nF ig . l ( d ). T h e s e s t r e s s - s t r a i n r e l a t io n s h i p s a r e t h o s e u s e d m o s t c o m m o n l yt o d e s c r ib e t h e m a t e r i a l b e h a v i o u r i n a n a l y t i c a l m o d e l s. S o m e m a t e r i a l ss h o w a n a r e a w i t h o u t w o r k h a r d e n i n g a t t h e b e g i n n i n g o f t h e p l as t ic r e g io n ,a s i n F i g . 1 (e ), th i s e f f e c t u s u a l l y b e i n g n e g l e c t e d i n a n a l y t i c a l m o d e l s.M a t e r ia l b e h a v i o u r w i th o u t a n e l a st ic r e g io n b u t w i t h w o r k h a r d e n i n g isi l lu s t r a t e d b y F ig . l ( f ), s u c h s t r e s s - s t r a i n r e l a t i o n s h i p s b e i n g u s e f u l fo ra p p l i c a t i o n s f o r w h i c h e l a s t i c e f f e c ts a r e l e s s i m p o r t a n t , f o r i n s t a n c e i n

t h e p r e d i c t i o n o f p r o c e s s f o r ce s .

2.2. Actual material behaviourO n e o f t h e b i g g e s t p r o b l e m s i n t h e c a l c u l a t i o n o f p r o c e s s p a r a m e t e r s i s t h a t

i t i s d i ff ic u l t t o d e t e r m i n e t h e v a l u e s o f t h e r e l e v a n t m a t e r i a l p a r a m e t e r s . T h em o s t c o m m o n e x p e r i m e n t t o d e t e r m i n e t h e s e m a t e r i a l p a r a m e t e r s is t h e u n i-a x i a l t e n s i l e te s t . T h i s t e s t h a s t w o d i s a d v a n t a g e s : t h e r e i s n o p l a n e - s t r a i ns i tu a t i o n , a n d t h e d e f o r m a t i o n r a t io o c c u r r i n g i n s h e e t b e n d i n g is u s u a l l yl a r g e r t h a n t h e m a x i m u m r a t i o i n a u n i - a x i a l t e n s i l e t e s t . T h e M i y a u c h i s h e a rt e s t a l lo w s l a r g e r d e f o r m a t i o n r a ti o s , b u t i s u s e d l e s s c o m m o n l y [1 ]. T h e v a l u e

o f Yo u n g ' s m o d u l u s o f a m a t e r i a l i s u s u a l l y a s s u m e d t o b e fix ed : h o w e v e r ,Yo u n g ' s m o d u l u s m a y d e c r e a se s u b s t a n t ia l l y ( so m e 16 ) u n d e r d e f o r m a t i o n[ 2 ] a n d a s a r e s u l t , t h e s p r i n g - b a c k i n c r e a s e s .


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342 A.H. Streppel et al./Sheet bending modelling techniques

In general, the flow curve will depend on the stra in path followed [3]. Fora total reve rsal of the str ain path, the Baus chi nger effect occurs. As a sheet isproduced by rolling, this phenomenon occurs at the compressive side during

bending. Anisotropic material behaviour is caused also by the productionprocess of the sheet. Aniso tropy is expressed by means of the plastic st rainratio R [4]. More im portant in sheet bending is the phenome non that the ratioR may differ in different directions: the so-called planar anisotropy. As a re-sult, the m ateria l be haviour depends on the ori enta tion of the bending linewith respect to the rolling direction of the sheet.

The phenomena mentioned above may differ in importance and do notalways have a profound influence on the bending process. However, it maysometimes prove difficult to determine the correct values of the relevantmaterial parameters. This especially applies to production environments, dueto the lack o f testin g equi pment and time.

3. T h e o r y o f s h e e t b e n d i n g

3 . 1 . A s s u m p t i o n sThe bending process was first subjected to a thorough investigation by

Ludwik [5]. The as sumptions made by him and late r a dopted by othe rs [6,7] areas follows:

(i) the thickness and width of the sheet are uniform thr oug hout the sheet

and remain constant during the bending process. The sheet is bent bya pure moment, therefore the bending line is a circular arc;

(ii) the sheet has a large width/thickness ratio, so edge deformations arenegligible and a plane state of strain exists;

(iii) the material is homogeneous and isotropic and there are no residualstresses present prior to bending;

(iv) plane sections normal to the sheet surface and parallel to the bend axisremain plane;

(v) the sheet consists of many thin inde pend ent layers. Stress es perpen-dicular to the sheet surface and stresses parallel to the bend axis areneglected;

(vi) the stress stra in curves of a workpiec e in the tensile and compre ssiveregions are symmetric with respect to the origin;

(vii) from assumptions (i)-(vi) and volume constanc y, it follows tha t the sheetthickness remains constant during the bending process.

To satisfy these conditions it is required th at the width/t hickness ratio of thesheet and the bending radi us/th ickne ss ra tio are sufficiently large.

3 .2 . B e n d i n g m o m e n t a n d p ro c e s s f or c e

The strain s and stresses in the bendin g plane may be analyzed (Fig. 2) unde rthe conditi ons men tion ed in Sec tion 3.1. From assumptions (i) to (iv) it followsth at ~:= = 0 and ~ , G' and ~= are the prin cipa l stresses, whils t from as sump tion


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A . H . S t r e p p e l e t a l . / S h e e t b e n d i n g m o d e l l i n g t e c h n i q u e s 3 4 3

~x__ z . x >) / ~ , >

Fig. 2. Stresses and layers in a sheet.

(v) it follows th at ay and at may be neglecte d. T he ben ding m ome nt Mb (with Soas the sheet thickness and b as the width) may be calculated using:

Iso/ 2

M b = 2 ~x Y) by dyj o

The value of this momen t depends on the stress str ain r ela tions hip used. Incalculations, sometimes a unit bending moment (which is the bending momentrequi red to bend a sheet with a width and a thickne ss of I mm) is used. Thevalue of this unit bending moment depends on the stress -stra in r elationshipand the st rain in the oute r fibre (the ratio S o / 2 R m= ~xa). Severa l e xpr ess ion s fo rbending moment and process force have been derived, based on differentstres s-str ain relationships [8-10]. Usually, friction between the workpiece andthe die is taken into account.

The process force is usually calculate d based on the stress distribution in thebending zone. The bending moment is calculated, and as a resul t the reacti onforces between the sheet and the die may be determined. For this purpose, anapproxima tion of the bending geometr y based on a rigid-plastic model is usedoften (Figs. 1 (a) and 3). Thus, a process fo rc e-p unc h displac ement diagra mmay be determined taking into account the friction between the sheet and thedie, an example of such a diagram being given in Fig. 4. At some point, theprocess becomes that of closed-die bending and the process force rises veryquickly [11,12]. The final stage of closed-die bending is shown in Fig. 3, at thelower right. Referring to Fig. 3 and neglecting the friction force, the processforce may be c alculated from

2MbFp - L cosZ(~/2) L cos (~/2) = H I 2

The right-hand side equat ion gives an approximat ion which is valid only if thedie opening H is large with respect to radii R, and Rd. Nevertheless, H is

considered often to be a process param eter although the length of the freesection L is more important. The influence of friction is taken into a ccounteasily by modifying the process force equation.


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344 A.H. Streppel et al . /Sheet bending modelling techniques

. . . . . . . . . . . . . . . . . . . . . . . . . U

? ; . . . . . . . . . . . . . . . . . . . . . . . . . ? :

Geom etry (rigid-plastic)

Air-bending

Closed-die bending

F i g . 3 . B r a k e f o r m i n g .

[ Closed-die bending

Air-bending

D i s p l ~ . ~ m e n t

F i g . 4 . E x a m p l e o f a p r o c es s f o rc e p u n c h d i s p l a c e m e n t d i a g r am .

3 .3 . Spr ing-backTo d e s c r i b e s p r in g - b a c k u s i n g a n a l y t i c a l m o d e ls , t h e b e n d i n g m o m e n t i s

u s u a l l y s p l i t i n t o a n e l a s t i c a n d a p l a s t i c c o n t r i b u t i o n : M b = M el + M p l. B a s e d o ne l a s t i c i d e a l p l a s t i c m a t e r i a l , t h e s t r e s s e s b e f o r e a n d a f t e r s p r i n g - b a c k a r ei l l u s t r a t e d i n F i g . 2 . O t h e r s t r e s s - s t r a i n r e l a t i o n s h i p s l e a d t o d i f f e r e n t r e s i d u a ls t r e s s e s a n d d i f f e r e n t s p r i n g - b a c k a n g l e s [1 3,1 4]. H o w e v e r , c o m m o n t r e n d s a r et h a t s p r i n g - b a c k i n c r e a s e s w i th : (i) d e c r e a s i n g s h e e t th i c k n e s s ; ( ii) i n c r e a s i n gy i e l d s t r e n g t h ; ( iii) i n c r e a s i n g b e n d i n g r a d i u s ; a n d ( iv ) d e c r e a s i n g Yo u n g ' s

m o d u l u s . T h e s p r i n g - b a c k i s e x p r e s s e d o f t e n b y t h e r a t i o K , b e i n g t h e r a t i ob e t w e e n t h e b e n d a n g l e s a f t e r a n d b e f o r e s p r i n g - b a c k (K ~< 1).

T h e b e n d r a d i u s i s o f t e n a s su m e d t o be u n i f o r m t h r o u g h o u t t h e b e n d i n g z o n e( a s s u m p t i o n (i)). O t h e r p a r t s o f t h e s h e e t a r e a s s u m e d t o r e m a i n f la t a ft e rs p r i n g - b a c k ( so c a l le d c i r c u l a r s t r a i g h t m o d e l s ). H o w e v e r , f o r a c c u r a t e c a l c u -l a ti o n o f p r o d u c t a n g le s , b o t h b e f o r e a n d a f te r s p r in g - b a c k , t h e s e a s s u m p t i o n sa r e n o t v a l i d , s i n c e t h e b e n d r a d i u s v a r i e s a l o n g t h e b e n d i n g z o n e [ 9 , 1 5 ] . A sa r e s u l t , b e n d a n g l e s b e f o r e s p r i n g - b a c k ( a n d , c o n s e q u e n t l y, a f t e r s p r i n g - b a c k )a r e n o t e q u a l t o t h o s e c a l c u l a t e d f r o m a c i r c u l a r - s t r a i g h t m o d e l .

3 .4 . Le ng th o f the ben t sec tionA t t h e b e g i n n i n g o f t h e b e n d i n g p r o c e s s , a n e u t r a l l a y e r e x i s t s a t t h e m i d d l eo f t h e s h e e t , b u t th i s s i t u a t i o n c h a n g e s d u r i n g t h e b e n d i n g p r o c e s s . T h e l a y e r s


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A.H. Streppel et al./ Sheet ben ding mode lling techniques 345

with strains equal to zero and stresses equal to zero no longer coincide[6,8,9,12]. There are several layers, with changing positions as the bendingprocess continues (Fig. 2). The material between the outer radius Ra and the

initial middle layer Rm0 has been stretched. The material between Rm0 and Rghas first been compressed and la ter been stretched: the Ba uschi nger effect thusoccurs in t his zone. The to tal deforma tion of the layer ind icate d by Ru is equalto zero. The material between Rg and Ri has been compressed. Assuming thatthe sheet thickness is constant, the length of the l ayer with total deformationequal to zero (thus, the length of the unfolded bend) may be calculated. Therelative difference in length d L / L m L mis the length of the middle layer)increases as the bending radius-sheet thickness ratio, Rm/s , decreases. How-ever, the sheet thickness does not remain constant but decreases if workhard eni ng occurs [9,16]. This effect is stronger for larger work hard enin g andsmaller Rm/s ratios. As a result of this complication, calculation of the unfol-ded sheet length is usually based on experimental data.

4. F E M a p p r o a c h

In the development of theoretical models for the sheet-bending process,several assumptions h ave been made (see Section 3.1), but in ind ustri al sheet-bending processes these conditions are not satisfied. In fact, the process isstarted by shear forces instead of by a pure bending moment. Thus, a sheardeformation will occur over the sheet thickness. Because of this, the bendingmoment and the bend radius are not constant and it is not possible to predictthe geometry of the bent sheet sufficiently acc urat ely with simple analy tic almodels based on theory employing the assumptions mentioned above.

FEM simulation is not suitable for use in a CAPP system, due to thecomputing time needed. However, FEM simulation is a very useful toolin determining the influence of several parameters. Comparing the resultsof such simulations with theory and experiment, it could be possible todevelop a mathematical model for sheet bending that can be used in aCAPP system.

In 1980 Kobaya shi and Oh [17] analyzed shee t bending as a bulk-deformati onprocess. They assumed a process without friction and used two materialmodels: a rigid-plastic and an elastic-plastic. To calculate the spring-backangle in the case of a rigid-pla stic mat erial, unloa ding was also done accordingto elas tic-pl astic behavi our. The results of both methods correspond. Becauseof the lesser computing time, these authors prefer the rigid-plastic materialmodel. Other researc h on FEM simula tion has been done by Maki nouc hi [18],who also neglects friction. Inste ad of using a st anda rd program (available onthe market) he developed a special-purpose program for bending of (laminate)sheet metal, which gives the advantage of shorter computing time.

It is common practice to use a fine mesh in the sheet-punch contact area. Iffriction between the sheet and the die is taken into account, a fine mesh isneeded for the sheet-die contact area (Fig. 5) also. This friction is especially


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346 A.H. Streppel et al./Sheet bending modelling techniques

D i e / : ~ : ~ . i

Fig. 5. Exam ples of FE M meshes.

i m p o r t a n t i f s m a l l d ie - o p e n i n g s, a c c o r d i n g t o i n d u s t r i a l p r a c t i c e , a r e u s e d . Int h i s c a se , t h e c o n t a c t f o r c e s a r e v e r y h i g h .

M o r e w o r k o n F E M s i m u l a t i o n h a s b e e n r e p o r te d , b u t t h e r e s u l t s a r e n o ta l w a y s a p p l i c a b l e f o r t h e a i r -b e n d i n g o f s h e e t : th e r a t i o o f t h e d ie o p e n i n g t ot h e s h e e t t h i c k n e s s d o e s n o t r e s e m b l e i n d u s t r i a l p r a c t i c e , f r i c ti o n is n o t t a k e ni n t o a c c o u n t , o r th e r a t i o o f t h e s h e e t w i d t h t o t h e t h i c k n e s s is t o o s m a ll .

5 . E x p e r i m e n t a l v e r i f i c a t i o n

5.1. Bending theoryB e n d i n g e q u i p m e n t c o n s i s t in g o f t h r e e c y l in d r i c a l ro l ls w a s u s e d b y L u d w i k

[5], w h o f o u n d a d i f f e r e n c e o f 1 0 i n b e n d i n g m o m e n t b e t w e e n t h e o r e t i c a l a n de x p e r i m e n t a l r e s u l ts : t h i s is p r o b a b l y d u e t o t h e s h e e t s n o t b e i n g b e n t b y a p u r eb e n d i n g m o m e n t . W o l t e r [9] u s e d s p e c ia l e q u i p m e n t w h i c h a l l o w e d h i m t o b e n ds h e e ts w i th a p u r e m o m e n t , f i n d i n g go o d c o r r e s p o n d e n c e b e t w e e n t h e b e n d i n gm o m e n t n e e d e d in h is e x p e r i m e n t s a n d th e t h e o r e t i c a l b e n d in g m o m e n t . T h isis r e m a r k a b l e , a s h e n e g l e c t e d t h e s t r e s s a.., w h e r e a s , a c c o r d i n g t o T r e s c a a n dv o n M i se s , a : is e q u a l t o (ay + a x)/2 . O f t e n a u n i t b e n d i n g m o m e n t i s c a l c u l a t e du s i n g t h e f o l l o w i n g e q u a t i o n :

MbMu = bs

T h i s e q u a t i o n i s b a s e d o n t h e a s s u m p t i o n t h a t t h e s t r a i n d i s t r i b u t i o n iss y m m e t r i c a l a n d l i n e a r [1 9]: f o r s m a l l b e n d r a d i i e s p e c i a l l y, t h i s is n o t t h e c a s e[20]. A s a r e s u l t , d i f f e r e n t e x p e r i m e n t s m a y l e a d t o d i f f e r e n t u n i t b e n d i n gm o m e n t s f o r t h e s a m e m a t e r i a l. A s e c o n d a s s u m p t io n m a d e is t h a t t h e s h e a rf o r ce h a s n o i n f lu e n c e o n t h e u n i t b e n d i n g m o m e n t .

5.2. Plane strainTo z a w a [21] h a s d o n e w o r k i n t h e f ie ld o f t h e b e h a v i o u r o f s te e l u n d e r

d i f f e r e n t s t re s s c o n d i t i o n s a n d a n i s o t r o p y. A n e x a m p l e o f t h e d i f f e r e n c e be -t w e e n s t r e s s - s t r a i n c u r v e s fo r a u n i - a x i a l a n d a p la n e - s t r a i n s i t u a t i o n is g i v e nin F ig . 6 , l e f t .


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A.H . Streppel et al. / Sheet b ending mo delling techniques 347

0 4 o

I3O

2 c

lO

- Steel (0.12 C) .. . . . . . . . . . . . .

...... ~ 3 1 )

..... plan e swain (o=10 2¢0.38

I I I I I0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 O . 1 0

- - C

U

Yl

yFig. 6. Plane strain (left) and the Bauschinger effect (right).

5 .3 . P l a n a r a n i s o t r o p yO t h e r e x p e r i m e n t s b y To z a w a [ 22 ] c o n c e n t r a t e d o n p l a n a r a n i s o t r o p y, w h i c h

i s s u b s t a n t i a l f o r s t e e l u n d e r c o m p r e s s i v e o r t e n s i l e s tr e s s . H o w e v e r , a d i r e c tr e l a t i o n s h i p b e t w e e n p l a n a r a n i s o t r o p y u n d e r t h e s e c o n d i ti o n s a n d d u r in gb e n d i n g c a n n o t b e g iv e n , d u e to b e n d i n g t a k i n g p l a c e u n d e r p l a n e - s t r a i nc o n d i t io n s . A s a r e s u l t o f p l a n a r a n i s o t r o p y, d i ff e r e n c e s o f u p t o 1 6 m a y b ef o u n d i f t h e u n i t b e n d i n g m o m e n t i s d e te r m i n e d f o r b e n d i n g l in e s w i t h d i f f e r e n t

o r i e n t a t i o n s w i t h r e s p e c t t o t h e r o l l i n g d i r e c t i o n o f a sh e e t .

5 .4 . T h e B a u s c h i n g e r e ff ec tT h e B a u s c h i n g e r e f fe c t h a s b e e n i n v e s t i g a t e d t h o r o u g h l y. H o w e v e r , l it tl e

r e s e a r c h h a s b e e n c o n c e n t r a t e d o n t h is p h e n o m e n o n in r e la t io n t o s h e e tb e n d in g . A f l o w c u r v e o f a m a t e r i a l w i t h t h e B a u s c h i n g e r e f fe c t is s h o w n i nF i g . 6 ( r ig h t ), t h e m a t e r i a l d i s p l a y i n g a m i x e d b e h a v i o u r o f i s o t r o p i c y i e l d i n ga n d k i n e m a t i c h a r d e n i n g . R o l f e [ 23 ] h a s i n v e s t i g a t e d t h e B a u s c h i n g e r e f fe c tu s i n g t e s t - s p e c i m e n s t a k e n f r o m a b e n t s h e e t o f91, t h i c k n e s s (6 3.5 m m ) w i t hd i ff e r e n t o r i e n t a t i o n s ( t r a n s v e r s e a n d l o n g i t u d i n a l ) a n d f r o m b o t h c o m p r e s s i v ea n d t e n s i le r e g io n s . H i s e x p e r im e n t s s h o w e d a s u b s t a n t i a l B a u s c h i n g e r e f fe c tf o r c o l d - r o l l e d s t e e l : t h e y i e l d s t r e n g t h i n t h e c o m p r e s s i v e r e g i o n m a y b e a sm u c h a s 3 0 l o w e r t h a n t h e y i e ld s t r e n g t h i n t h e te n s i le r e g io n .

W e i n m a n n [ 24 ] u s e d t h i n s h e e t (0 .8 6 m m ) w h i c h h e b e n t c y c l i c a l ly w i t h a p u r em o m e n t . T h e B a u s c h i n g e r e f f e c t r e s u l t e d in a r e d u c i n g u n i t b e n d i n g m o m e n t e v e r yt im e t h e b e n d i n g m o m e n t w a s r e v er s e d. T h e B a u s c h i n g e r e f fe c t f a c t o r ( B E F ) w a su s e d t o q u a n t i f y t h e e f f e c t , t h i s f a c t o r b e i n g d e f i n e d a s ( s e e a l s o F i g . 6 , r i g h t ) :

B E F - Y 1 - IY21Y1

I n g e n e r a l , p r e - s t r a i n e d m a t e r i a l h a s a l a rg e r B E F f o r t h e f i rs t c y c l e b u tr e a c h e s a s t a b l e v a l u e a f t e r a f e w c y c l e s .


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348 A.H . Streppel et al./ Sheet bending modelling techniques

5 .5 . She e t l eng thT h e l e n g t h o f t h e b e n d i n g z o n e ( t h a t i s, t h e l e n g t h o f t h e l a y e r i n d i c a t e d b y

R u i n F ig . 2 ) is i m p o r t a n t f o r t h e c a l c u l a t i o n o f b l a n k d i m e n s i o n s . A s h a s b e e n

m e n t i o n e d b e f or e , t h e r e a r e n o s u i t a b le a n a l y t i c a l m o d e l s t o c a l c u l a t et h is l e n g t h w h e n t h e b e n d r a d i i a r e s m a l l (t h e i n i ti a l a s s u m p t i o n s m a d e i nS e c t i o n 3.1 a r e n o l o n g e r s a t is f ie d ) , w h i c h is th e c a s e f o r m a n y i n d u s t r i a lp r o d u c t s . T h e p r e s e n c e o f s h e a r f o r ce s a n d c o n t a c t f o r ce s a re a d d i t i o n a lc o m p l i c a t i o n s . F u r t h e r m o r e , i t h a s b e e n m e n t i o n e d t h a t t h e c u r v a t u r e o f t h eb e n d i n g z o n e is n o t c o n s t a n t . Tw o q u a l i t a t i v e e f f e c ts m a y b e m e n t i o n e d : (1 )w h e n t h e g e o m e t r y d i ff e rs m o r e s t r o n g l y f r o m a c i rc u l a r a r c , t h e l e n g t h o f t h eb e n d i n g z o n e is s m a l l e r , a n d ( 2) w h e n t h e c o n t a c t f o r c e s a r e l o w e r , t h e r a d i u sR . b e c o m e s l a r g e r a n d t h u s t h e l e n g t h o f t h e b e n d i n g z o n e i s l a rg e r .

T h e l e n g t h o f t h e b e n d i n g z o n e is u s u a l l y c a l c u l a t e d a c c o r d i n g t o t h e A S M o rD I N s t a n d a r d s [1 ,25]. B o t h s t a n d a r d s m a k e u s e o f e q u a t i o n s w i t h a c o r r e c t i o nf a c t o r b a se d o n e x p e r i m e n t a l d a t a .

T h e A S M s t a n d a r d u s e s t h e f o l lo w i n g e q u a t i o n s f o r s te e l:

L = tp + r 00 .01745 f l

w i t h :

p = r i 0 . 0 4 + 0 . 3 f o r r - ~ < lt t

0 00 + 0 4 for r : lw i t h fl i n d e g r e e s a n d r i, t a n d L a s s h o w n i n F i g . 7, l e ft . T h e D I N s t a n d a r d u s e st h e f o l l o w i n g e q u a t i o n s f o r s te e l :

sL = a + b + v , v = ~ ~ r i+ ~ k - 2 ( r i + s )

r i ) ksk - - 0 . 6 5 + 0 . 5 1 o g s ' n o t e t h a t r u = r ~ + ~ -

w i t h ~ i n d e g r e e s a n da, b, r~, s a s s h o w n i n F i g . 7 , r i g h t . L is t h e t o t a l l e n g t h o ft h e ( u n f o l d e d ) s h e e t .

I t i s s u g g e s t e d t h a t rl = r m i fr~/s i s s u f f i c ie n t l y l a rg e . A c c o r d i n g t o D I N t h i s i st h e c a s e f o r a r a t i o o f 5, w h e r e a s a c c o r d i n g t o th e A S M e q u a t i o n a r a t i o o f 3 .6 7i s s u f f i c i e n t .

J a h n k e [26] h a s c a r r i e d o u t r e s e a r c h i n t h e s a m e f ie ld . F o r a = 9 0 °, h ed e v e l o p e d a s i m p l e r e q u a t i o n t o c a l c u l a t e t h e l e n g t h v i n th e D I N e q u a t i o n :

v = (0.429ri - 1 .47 6s)

5 .6. M i n i m u m b e n d r a d i u sT h e m i n i m u m b e n d - r a d i u s i s l i m i t e d b y t h e s t r a i n i n t h e o u t e r f i br e s. F r a c -

t u r e o c c u r s w h e n t h i s s t r a i n i s a b o u t 1 .5 t i m e s t h e f r a c t u r e s t r a i n o f a u n i - a x i a l


11/18

A.H . Streppel e t a l . / Sheet bending mod el l ing techniques 349

A S M

2,

| D I N

jJ

\\

t r

I t I I s

Fig. 7. Sheet length according to ASM and DIN.

tension test, as was found by Jahnke [26]. For smaller bend angles and lowwidt h/th ickne ss ratios a larger s trai n is possibl~. When the bend is parallel tothe rolling direction or the edge quality is poor, fracture occurs at a lowerstrain. To avoid initial fractures, the strain should not exceed 20 to 25%, asmay be calculated from various standards such as DIN 6935 and 9003 [25,27].However, even for smaller deformations the surface r oughness will increase, asindic ated by Dautze nberg [28]. The limits men tioned should act as design rules.

5 . 7. S p r i n g - b a c kSpring-back is expressed often as the spring-back factor K, which equals

1 ÷ A~/~ (~ is the bend angle , As the spring-back). However, K is no t a m at er ia lparameter but depends on the bend angle and the ratio between the sheetthi ckn ess and t he bend radius. Schw ark [13] and Fhppl [15] found good corres-pondence between average experimental spring-back and calculated spring-back, the actu al spring-back, however, showing a large vari ation. It should berealised also th at the prediction of spring-back itself is not sufficient for CAPPapplications. For CAPP, correct prediction of the punch displacement is re-quired in order to obtain the required product angle.

5 . 8 . B a t c h v a r i a t i o nIt has been mentione d above that the actu al spring-back may vary consider-ably. This may be caused by variation in sheet thickness and yield strength.A thickness variation of 15 to 20% is allowed for thin cold-rolled sheet,accordi ng to DIN 1541 [29]. Even within one bat ch or one sheet, differences insheet thickness may occur. Furthermore, the yield strength may show vari-ations of 20%, as indicated in DIN 1623 [30]. Both the thickness and the yieldstre ngth have a profound influence on the spring-back. Thus, it will be under-stood that the control of spring-back is a serious problem.

5 .9 . C o n t a c t b e h a v i o u rSpring-back is influenced also by the d istr ibuti on of the cont act forces in the

wrap-around zone. In some cases, however, there is no wrap-around zone. In


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350 A.H . Streppel et al . / Sheet b ending mo delling techniques

the case of a small tool-radius in combination with a large die-opening, theradius may be greate r t han the tool radius (Fig. 8, left). Under the influence ofshear forces, the sheet ma y lose contact with the centre of the punch, as shown

in Fig. 8, right.In the case of air-bending, it will be understood that the product angle and

length of the bending zone will be influenced by these effects. When the finalstate of the bending process is that of closed-die bending, these effects mayresult in an unpredictable (or even negative) spring-back.

6. I n - p r o c e s s c o n t r o l

In order to achieve the required product accuracy, it is necessary to ensuretha t bend angles stay within certain - usually narrow tolerances.

6 .1 . P roce s se sWhen closed-die bending is selected as the ma nuf act uri ng process, rel ative ly

large variations of the product angle must be allowed: this is caused by theangle before spring-back being dete rmined uni quely by the selected tools. Ashas been seen, spring-back may vary considerably due to variations in sheetthickness and yield strength. Thus, the angle obtained after spring-back willvary accordingly.

When air-bending is selected and the variation in sheet thickness is nottaken into account, the deviation will be even larger, since the actual punch

displacement differs from the calc ulat ed value (which is based on the nominalsheet thickness). However, when it is possible to control the spring-back byadapting the punch displacement to the actual process conditions, then twoimpor tant result s are achieved. The first result is an adequa te control of thedeviation of the bend angle, regardless of variations in sheet thickness andyield strengt h, whil st the second result is that one set of tools can be used forthe manufacturing of different bend angles for a wide range of sheet thick-nesses and material qualities.

6.2. L o a d - u n l o a d m e t h o d

The loa d-un load method is based on the measure ment of the angle obt ainedunder loading and the spring-back angle after partial unloading. The equip-ment used is shown in Fig. 9. Instead of a conventional die, a die consisting of

Fig. 8. Special contact behaviour.


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A.H. Streppel et al./Sheet bending modelling techniques 351

Fig. 9. Special die.

rolls with flat faces is used. After the first bending step, the bent angle ismeasured. Secondly, the bending force is reduced to 50%: the spring-back ismeasured and the required angle under loading is calculated. Finally, the

product is bent until this latter angle is reached. This method has beeninvestigated by von Finckenstein [31], who indicates that it is important toknow the actual relationship between force-reduction and spring-back, whichis non-linear due to friction. A disadvantage of this method is that tool-costsare high and that the application range is limited. The method is used inindus try for the bending of long members.

6.3. Force displacement methodThis method is based on the measurement of the force-punch displacement

curve. Using this curve, it is possible to determine the sheet thic kness an d themechanical properties. Determination of the sheet thickness is relativelystraight-forward, as the thickness can be calculated from the displacement atthe point where the force starts to rise. In order to determine the mechanicalproperties of the sheet, comparison with experimental data or a model isneeded. Stelson [14,32] has analyzed this method, finding that the results maybe very sati sfact ory (angles to within 1 °) if a real isti c model is used and enoughdat a on the force-di splace ment curve are collected. Over-bending of the prod-uct tends to occur when work hardening is not included in the model, due tothe the oretical geometry and the actual geometry of the sheet near the wrap-around zone differing: as a result, the product is bent more than is intended.The large r spring-back does not compensate for this overbend error, especi allywhen a small die-opening is used. For large die-openings, the error may beover-compensated for. As for the problem of collecting sufficient data, Stelsonindicates that this may be a problem for large product angles. It may beexpected that this problem also arises whe n small die-openings are used (in thiscase the punch displacement is low also). Thus, in-process control has twoaspects: the modelling problem a nd the identif icati on problem.

6.4. Alternative methods

An alterna tive method is the measurement of both the angle and the force,provided tha t the r ight equipment is available for use in an industr ial environ-ment: in this case, the sheet t hickne ss can be calculated. The sheet is bent unti l


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352 A.H . Streppel et al./ Sheet b end ing m ode lling techniques

a given (safe) angle is reached. Mea sureme nt of the force is needed to cal cul atethe spring-back and the required angle under loading. The amount of data tha tmust be collected is relatively low.

When a simple, uniform bending test is available, an even simpler means ofin-process control is possible. Such a test should require simple equipment andthe a mou nt of dat a to be collected is low. In this case, the tes t can be performedin a production envi ronme nt using, for instance, two or three test-pieces tak enfrom each sheet. The results of this test are characteristic for the mechanicalproperties of the sheet. For each bend, only the sheet thickness must bedetermined in-process (the thickness can vary throughout a sheet).

7. C o m p u t e r a i d e d p r o c e s s p l a n n i n g ( C A P P )

7.1 . P r o c e ss p l a n n i n g t a s k sProcess-planning tasks rel ated to bending include the selection of equipment

(tools and press brakes), the calc ulat ion of unfolded blank dimensions and therequired punch displacement, and the de term inat ion of the bending sequence.These tasks are i nter rela ted in many ways, one obvious rela tions hip being therelationship between the die opening and the required punch displacement.The die opening also influences the conta ct be haviour of the sheet and therequired process force. Another important aspect is the relationship betweenthe bending sequence and the tools to be used [33]. To avoid collisions, the toolsselected must be within certain attribute ranges.

The determination of the bending sequence is related also to accuracyaspects. The actual accuracy required of bend angles is mostly not indicatedexplicitly by tolerances on angles, but must be calculated from tolerancesindicat ed on linear dimensions, such as the dista nce between holes. In general,the resulting admissible tolerance in bending depends also on the bendingsequence and the positioning strategy. Thus, by selecting the right bendingsequence and suita ble posit ioning edges, the admissible errors in bending maybe increased or are, at least, not unnecessarily restricted. This can, for in-stance, result in more products being able to be bent on press brakes with

stan dard equipment, thus relie ving the workload for press brakes equipped forin-process control.

7 .2 . PA RT-S : A n i n t e g r a t e d p r o c e s s - p l a n n i n g s y s t e mAs has been indicated, process planning for sheet-metal products is more

than just calculation of the required punch displacement (although this isindeed an important task). An integrated process-planning system currentlyunde r development at the aut hor s' Labor at ory is PART-S [34]. Its arc hit ect ure(Fig. 10) is based on t he ar ch it ec tu re of PART, which is a proces s-pl anningsystem for prismati c parts, presentl y in use in in dus try [35]. The process-

planning functions, however, are partly different. Characteristic for sheetmetal are the functions related to the generation of unfolded blank sizes andnesting. The unfolding function is needed because the outpu t from the design


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A.H. Streppel e t a l . / Sheet bending model l ing techniques 353

P l a n n i n g o f A c t i v i t i e s , R e s o u r c e s a n d Te c h n o l o g y - U s e rI

S h e e t m e t a l (PART-S) [ User InterfaceI

Supervisor

CI

I

FIG FR

I I

ModeUer

II Tuning

[ User Interface [

NITS LOB HPCC MM TS

A. ~

Manufacturing Database InterfaceI

D a t a b a s e

PC NC

Fig. 10. PART-S re fe re nc e dia gram.

PL

I

process will, in general, not incl ude a model of the blank. Nestin g funct ions arerequired, since, in small-batch pr oduct ion of sheet-meta l products, ma ny (oftendifferent) products are nested in one sheet. Nesting is not just a geometricalproblem but has strong relationships with planning aspects.

PART-S offers fully au toma ted facilities to analyze and man ipul ate bou nda ryrepresentation (B-rep) product models stored in the internal modeller. Theprocess pl an ni ng func ti ons in PART-S are:

CI - CAD Interface.This module offers the possibility to exchange productmodels with external CAD systems via STEP or neutral file format.

FR Feature recognition. This module extracts manufacturing features suchas bending lines or flat-pattern features from the product model.

FPG - Flat-pattern generator. A module which gene rates the unfolded blank

dimensions of the product and a B-rep repr esen tati on of this blank.MM - Selection of the mach ining methods.This function has to be performedat several levels, for instance the choice between punching/nibbling and lasercutting, but also the determination of punching sequences.

MTS - Selection o f ma chine tools.The selection of machine tools is one of thefirst decisions one would like to make, but in general, this is not possible, asmany factors influence this decision. In the case of a press brake for instance,the required back gauge facilities depend on decisions made later. The proce-dure is to reduce the number of candidate machine-tools stepwise.

PL - Capaci ty p lann ing.Flexible manufacturing requires a relatively short

planning horizon, which means that planning tasks such as handling priorityof orders, the allo cati on of jobs and tools, and load ba lanc ing mus t be integ-rated with technical process-planning.


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354 A.H. Streppel et al./ Sheet b ending mode lling techniques

LOB - L a y o u t o f b l a n k s .This function is characteristic for the small-batchproducti on of sheet-metal parts. In this type of production, usually severaldifferent products are nested in one sheet in order to reduce set-up times.

Material utilization is a possible optimization goal. More important, however,is that machining methods for the nested products must correspond and thatconstraints imposed by planning aspects are taken into account.

HPCC D e t e r m i n a t i o n o f h a n d l i n g , p o s i t i o n i n g a n d c l a m p i n g .Positioningand clamping for flat-pattern operations are relatively straight-forward. Thedeter minati on of handling and positioning for bending operations, however, isone of the majo r tasks of a sheet-metal CAPP system. It includes ev alua tionand conversion of tolerances, evaluat ion and subsequently selection of candi-date bending operations and positioning edges.

TS Tool se lec t ion. The appropriate tool-attribute ranges are determinedduring m ethod selection. Based on the availability of tools and commonattribute ranges, the best-fitting tools are selected.

PC -Determinat ion of p rocess cond i t ions .This can be relatively straight-forward for punching and nibbling, but is complicated for bending, lasercutting and laser welding.

NC N C a n d t e x t o u t p u t .This module generates NC files and operatorinstructions.

Each module is subdivided in a number of so-called phases: indepen-dent programs communicat ing with each other via the database. The executio nof a process- planning session is controlle d by a scenario in which the execu-tion sequ ence of the var ious phases is stored and is carried ou t by the supervi-sor. Scenarios are user definable. The use of scenarios allows users withdif fer ent views on pro cess pla nni ng to use PART-S in t hei r own company -tailored way.

8. C o n c l u d i n g r e m a r k s

As has been discussed, analytical models have some serious limitations.Under industrial conditions (small bend-radii, small die-openings) the assum-

ptions needed for analytical description of the bending process are no longervalid. However, analytical models may support certain decisions in the pro-cess-planning phase, for instance, the estimation of required punch force or theextrapolation and interpolation of empirical data. In the case of in-processcontrol, analytical models may be used to calculate a safe first-value for thepunch displacement.

A typical application of empirical models is the calcula tion of the unfoldedblank dimensions. Presently, these models do not take into account the influ-ence of die opening, yield streng th and the Baus chin ger effect. To improve theaccuracy of these models, further research on these topics is needed.

A typical problem related to the use of empirical models for the calculation ofthe required punch displacement is the necessity to collect numerous data fordifferent angles, tools, materials and sheet thicknesses.


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A.H. Streppel et al./ Sheet bending modelling techniques 355

F E M s i m u l a t i o n i s n o t s u i t a b l e f o r u s e in t h e p r o c e s s - p l a n n i n g p h a s e , d u e t ot h e c o m p u t e r t i m e n ee d e d . F u r t h e r m o r e , w h e n t h e c o m p u t i n g t im e c a n b er e d u c e d , t h e n e c e s s i t y t o h a v e t h e a c t u a l m a t e r i a l d a t a a v a i l a b l e i s s t i ll th e r e .

H o w e v e r , F E M s i m u l a t i o n i s a v e r y u s e fu l t o o l t o d e t e r m i n e t h e i n f l u e n c e o fs e v e r a l p a r a m e t e r s .

I n - p ro c e s s c o n t r o l i s a n a l t e r n a t i v e t o d e a l w i t h v a r i a t i o n o f m e c h a n i c a lp r o p e r t i e s a n d s h e e t t h i c k n e s s . T h i s a p p r o a c h , h o w e v e r , d o e s n o t e l i m i n a t e t h en e c e s s i t y f o r b o t h m o d e l l in g a n d i d e n t i f i c a t i o n o f m a t e r i a l b e h a v i o u r . I t i s n o ty e t s u i t a b l e f o r u s e in s m a l l -b a t c h m a n u f a c t u r i n g w i t h i t s t y p i c a l w i d e r a n g e o fp r o d u c t t y p e s . A s u g g e s t i o n f o r f u t u r e r e s e a r c h i n th i s f i el d i s t h e d e v e l o p m e n to f a n i n - p r o c e s s c o n t r o l m e t h o d r e q u i r i n g l e s s d a t a . I n g e n e r a l , i d e n t i f i c a t i o n o fm a t e r i a l p a r a m e t e r s i s d if fi cu l t . A s i m p l e te s t , to b e p e r f o r m e d i n a p r o d u c t i o ne n v i r o n m e n t , s h o u l d b e d e v e l o p e d t o de a l w i t h t h i s p r o b le m .

D u r i n g p r o c e s s p l a n n i n g , m a n y d e c i s i o n s h a v e t o be m a d e , m o s t o f t h e s ed e c i s io n s b e i n g i n t e r d e p e n d e n t . T h u s , t h e n e e d f o r i n t e g r a t e d C A P P i s e v i d e n t .T h e r e q u i r e d a c c u r a c y o f b e n d a n g l e s i s i n f lu e n c e d b y th e s e l e c te d b e n d i n gs e q u e n c e a n d p o s i t i o n i n g e d ge s . I n t e g r a t i o n o f d i f fe r e n t p r o c e s s - p l a n n i n g f un -c t io n s i n o n e s y s t e m a l l o w s t h i s i n f lu e n c e t o b e t a k e n i n t o a c c o u n t , w h i c hm e a n s t h a t t o l e r a n c e s a r e n o t u n n e c e s s a r i l y r e s t r i c t e d a n d t h e u s e o f a w i d e rr a n g e o f e q u i p m e n t ( e s p e c i a l l y s t a n d a r d t o o l s a n d p r e s s b r a k e s ) is p o s s ib l e . O nt h e o t h e r h a n d , s t a n d a r d s e q u e n c e s a n d h a n d l i n g p r o c e d u r e s c a n b e u s e d w h e nt h e a l t e r n a t i v e s d o n o t l e a d to s u b s t a n t i a l a d v a n t a g e s .

R e f e r e n c e s

[1] ASM,ASM-Metals Handbook 9th ed ., Am er ican So c ie ty for Meta l s , Meta l s P ark , OH,Vo l. 1, 1978, pp. 548 549, pp. 552-554; Vo l. 8, 1985, pp. 118 129, pp. 552 558 , pp. 564-569;Vol. 14, 1988, p. 12, pp. 523 525 , p. 531, pp. 541-543, p. 552, p. 629, p. 646, pp. 656 657,pp. 7 62-763 , p . 776, p~ 792, p. 794 , pp. 820 82 1, p. 825, p. 839, p. 846, pp. 880 -881, pp.887 894 , pp. 913 922.

[2] W. Lems, The chan ge o f Yo ung ' s mo dulus a f te r deform at ion a t low tem pera ture an d i t sreco ve ry, Ph.D. Disse r ta t ion, Delf t , 1963.

[3] W.H. S i l lekens , J .H. D autzen berg and J .A.G. Kals , S t ra in pa th dependen ce o f f lowcurves, Ann. CIRP 40(1) (1991) 255 258.

[4] ASTM, M echan ica l t es t ing , S tanda rd tes t method for p las t ic s t ra in ra t io R for shee tmeta l , Annual Book of AS TM Standards Section 3, Vol. 03.01, Des. E 517-81, 1987.

[5] P. Ludwik,Technologische Studie i~ber Blechbiegung (Sonderabdruck aus den Technis -chen Bl i i tt e rn) , Deu tschen Poly tec hnisch en V ere ins , BShmen, P rag , 1904.

[6] C. Bach,Elasticity and Strength Springer, Berlin, 1934.[7] A. Nadai,Plasticity of Materials Springer, Berlin, 1927.[8] R. Hill,The Mathematical Theory of Plasticity Cla ren do n Pre ss ~, Oxfo rd, 1950.[9] K.H. Wolter,Freies Biegen yon Blechen VD I-Forschung shef t Vol. 435 , Au sgabe B,

Bd. 18, D eu tsch er Inge nie ur-V erla g, Diisseldorf, 1952.[10] B. Zi inkler, Rech ne rische Erf assu ng der Vorg~inge beim Biegen im V-fSrmigen Gese nk,

Industrieanz. 73 (1966 ) 213 217.[11] M . Herm ann , M. Wi lhe lm an d P. Ke ck , Process s imu la t ion of m eta l - forming processes :

Some examples ,Ann. CIRP 40(1) (1991 ) 285 289.[12] K. Lange , Blechum formun g,Lehrbuch der Umformtechnik Bd. 3, Sp ringe r, Be rlin, 1975.


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356 A.H. S t reppe l e t a l . / Shee t bend ing mode l l ing t echn iques

[13] H.F. Schwark , R~tckfede rung an b i ldsam gebogenen Blechen ,Ph.D. Dissertation,T.H. Hannover, 1952.

[14] A. FSppl, Vor lesungen i~be r Techn i sche M echan ik ,Bd. 5, Leipzig, 1907, pp. 73-81.[15] K.A. Stelson, Real time identi ficat ion of workpiece-material cha racter isti cs from

measurements duri ng brakeforming, J . E n g . I n d ., Tr a n s . A S M E ,105 (1983) 45 53.[16] F. Proksa, Plastisches Biegen von Blechen, S t a h l b a u , 2 (1959) 29 36.[17] S. Kobayashi a nd S.I. Oh, F ini te elements anal ysis of plan e-s trai n sheet bendi ng, In t . J .

Mech. Sci . ,22 (1980) 583 595.[18] A. Makinou chi, Ela stic plastic stress anal ysis of bendi ng and hem ming of sheet metal,

in: N.B. Wang and S.C. Tang (Eds.), Compu te r-Mode l ing o f Shee t Meta l Form ing Pro -cesses, Meta ll urgica l Socie ty, 1985, pp. 161 176.

[19] G. Oehler, Biegen, Carl Hanser, Mfinchen, 1963.[20] G. Nicklas, D a s B i e g e n y o n B l e c h e n a n H a n d e i n e r B i e g u n g i n V- G e s e nk e n(Ph.D.

Dissertation), VDI, Dfisseldorf, 1976.[21] Y. Tozawa, Plas tic deformati on behavio r und er condit ions of combined stress, in: D.P.

Koistinen and N.M. Wang (Eds.), M e c h a n i c s o f S h ee t M e t a l F o r m i n g ,Plenum Press,London, 1978, pp. 81 110.[22] Y. Tozawa, N. Kan etake , A. Honda and H. Kudo, Pre dict ion of the bendi ng moment of

predeformed sheet from uniaxial test results, A n n . C I R P, 29(1) (1980) 163 166.[23] S.T. Rolfe, R.P. Ha ak and J.H. Gross, Effect of state-of-stress and yield crit eri on on the

Bauschinger effect, J . B a s i c E n g . , Tr a n s . A S M E ,(September 1968) 403-408.[24] K.J. W einma nn, A.H. Rosenberger , L.R. Sanchez an d B.T. von Turkov ich , The Baus-

ching er effect of sheet me tal un der cyclic reverse pure bend ing, A n n . C I R P,37(1) (1988)289 293.

[25] DIN 6935, Kalt bieg en von Flac herze ugnis sen aus Stahl, Biegehalbmess er klei nsteSchenkell/inge zul/issige Abweichungen der Winkel stel lungen Berechnu ng dergestreckten L/ingen, 1975.

[26] H. Jahnk e, R. Retzke and W. Weber, F e r t i g u n g s te c h n i k , U m f o r m e n u n d S c h n e i d e u ,VEB,Berlin, 1971.

[27] DIN 9003, Biegen yon Blechen, Pla tte n und B/indern aus Stah l u nd hochw armfest enLegierungen, Biegehalbmesser, Konst ruktion s-rich tlinie n, 1984.

[28] J.H. Dautzenberg and J.A.G. Kals, Surface roughness caused by metal forming, A n n .C I R P, 34(1) (1985), 477 479.

[29] DIN 1541, Kaltgewaltze s Brei tban d und Blech aus unl egi ert en St/ihlen, Masse, zul/issigeMass- u nd F ormab weic hungen , 1975.

[30] DIN 1623, Kaltgewaltzes Band und Blech aus weichen unlegierten St/ihlen, Gfitevor-schri ften, 1972.

[31] E. von Fin cke nst ein and R. Rothst ein, Ge senkbi egen von Blechen, Aut omatis che Rfick-federungskorrektur, I n d u s t r i e a n z . ,24 (1988) 19-20.

[32] K.A. Stelson, An adaptive pressbrake control for strain hardening materials, J . Eng .I n d . , Tr a n s . A S M E ,108 (1986) 127 132.

[33] M. Hoffmann, U. Geil~ler and M. Geiger, Compute r-aided ge ner ati on of bendi ngsequences for die -bending machines, J . Mater. Process . Technol . ,30 (1992) 1 12.

[34] L.J. de Vim J. de Vries , A.H. Streppel and H.J.J . Kals , PART-S, a CAPP sys tem for smallbatch m anufa cturi ng of sheet metal components, 2 4 th C I R P I n t. S e m i n a r o n M a n u f a c -tu r ing Sys tems ,Copenhagen, 1992, pp. 171 182.

[35] F.J.A.M. Houten, PART: A computer aided process pla nn ing system, Ph.D. Diss ertatio n,Enschede, 1991.

strep pel 93 suitability

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