- Email: [email protected]
Finite-element simulation of V-die bending: a comparison with experimental results
Journal of Materials Processing Technology 65 (1997) 52 -58
Finite-element simulation of V-die bending: a compariso experimental results Annika Nilsson *, Lars Melin, Claes Magnusson
Industrial summary A major problem in sheet bending is to compensate for springback. Analytical descriptions are not sufhciently general to accommodate influences from the material and the geometry due to the simplifications that have to be made. Practical experiments are still needed to be able to compensate for springback. The aim of this work has been to test the finite-element method on its abilities to predict springback for free bending in a V-die. The work shows that the finite-element method can be used to predict springback off-line. The true stress-true strain curve from a tensile test is used as the material description. Springback has been studied for eight different materials of varying thickness. The process has been simulated with the code Nike2d and the results from the simulations compared with those from experiments, good correlation between the simulations and the experiments being achieved. 0 1997 Elsevier Science S.A. Kq~or&
V-die bending; Springback; Finite-element simulation
The control system in an NC-controlled press brakes compensates for springback, but often not successfully, as has been shown by Tan [l]. The calculation algorithms are too simplified to work for all materials. Sidebottom and Gebhardt [2] say in their work that ‘one should not expect an analytical solution to predict springback with great accuracy because of the simplifying assumptions that have to be made and of the different behaviour of materials that can occur between different heats of material’. Oehler [12] has mapped empirically the springback of several common materials but these curves cannot handle variations in the material and there is no documentation for new materials. Common approximations in analytical models are: a simplified stress-strain relationship, no friction, the bending curve approximated with an arc (the inner side is often assumed to equal the punch radius), no thinning of the sheet occurs, no Bauchinger effect. Approximations can be used for some cases, but they are never general. One general way of solving the problem is to use the
* Correspondingauthor. Present address: MEFOS,
(FEM), where the material behaviour and the geometry can be input easily and factors such as the shape of the bending line and the displacement of the neutral layer are taken care of by the numerical code. Oh and Kobayashi [3] have used the FEM to simulate the bending process and compared a rigid-plastic material model to an elastic-plastic material model. Makinouchi [4] has simulated U-bending using elasticplastic deformation theory in rate form. Nagai and Makinouchi [S] have simulated a bending restriking process which decreases springback by adding extra plastic deformation in a U-bending process. Kim and Stelson [6] have used the FEM as a tool to identify material characteristics from force and displacement measurements. Shilling [7] has compared isotropic and a combined isotropic-kinematic hardening model using the FEM. The work presented in Refs. [3-71 is limited to one set of tools and/or one material and few if any comparisons to experiments are made to verify the results. The aim with the present work is to obtain a general picture of the reliablilty of the FEM when it is used as a package without any modifications done to the original code. Other ways to develop a general bending model are to use a combined empirical-analytical approach such finite-element
1. Background
Box 812, S-971
25 Luld. Sweden. 0924-0136/97/$17.000 1997 Elsevier Science S.A. All rights reserved SSDI 0924-0136(95)02241-4
method
Fig. I. Photograph
before and aitcr unloading
an X mm QStE340 shret. cmplu> itt,11a tool \vti!l I die M4th of 6-l mm and ;I punch radius of 6 mm.
as, for instance, that of Fleichcr [8] and Tan [l], where a number of bending tests is wsed to evaluate constants that can be applied in the analytical model. Andreen and Crawfoord [9] suggest a numerical code. developed especially for bending processes. to predict springback.
angles. The bending angle was measured before and after springback from photographs taken of the spezimen in the press-brake (Fig. I .), the angles being mesured directly from the negati\,es in a profile microscope.
2. Conditions for the study To be able to evaluate the accurac> 0;‘the FE.M the simulations have to be compared with experimental data. The conditions for the experiments sre described in Section 2.1 and the conditions for the simulations are described in Sections 2.2 and 2.3. The experiments were performed with a dry tool. To decide if friction needs to be included in the study. experiments were made with two lubrication situations. A comparison was made between a dry tool and when the contact zone between the sheet and the die is lubricated by layers of plastic film with oil between the layers to decrease the friction. The experiments gave no difference in springback between these samples. The effect of springback in the simulations are minor and therefore it was decided to neglect the friction in the springback simulations. This assumption also shortens the calculation time.
The experiments were perfomed in an NC-controlled press brake. Eight materials were bent to different
The cumputcr simulations were perfomcd with the implicit FEM code Nike2d. Four-node plane-strain elements were used in the simulations. The FEM model is shown in Fig. 2. Due to symmetry, only one half of the geometry needed to be modelled. The deformation was achieved by prescribing the displacement of the punch. which corresponds with how the deformation is achieved in reality. The BFGS method is used to solve the non-linear equilibrium equations. The definition of the angles and springback is shown in Fig. 3.
The information supplied from the vendor on the material properties is insufficent for a FEM simulation: for the present work the stress-strain curve from a tensile test is used; in the simulation this is recalculated into a true stress -true strain behaviour. The tensile curve is taken in the direction perpendicular to the bend, i.e. along the curvt:
54
A. Nilsson et al. /Journal of Materials Processhg Techology 65 (1997) 52-58
In the simulations and elastic-plastic isotropic hardening material model is used. The true stress-true strain material description is input in discrete form, interpolating between the points. The materials that have been used are two aluminium, one stainless steel, two low-carbon deep-drawing steels and three hot-rolled steels. A list of the materials used is shown in Table 1. The mechanical properties are taken from a tensile test.
3. Comparison between experiments and simulations The amount of springback depends on the bending angle. The elastic springback increases with increased
Table 1 List of the materials used and their properties _____ Material
Thickness (mm)
R, (0.2) (MPa)
R,,, CM!%)
A511
St06 St06 st12 STEEL 1” STEEL 2;’ QStE340 AK1304 AA6016-T4 AA5182
3 2 1.1 8.3 7.9 3.95 3 1.13 1.01
158 200. I 241.7 488.6 341 470 273 II4 94
310 323.4 326.1 595.4 412 540 637 217 192
49 (A”,,) 29.8 30.0 1858 24.8 23 56 (A,,,) 28 24
a The exact chemical composition has not been determined.
deformation in the bend, to the point where coining takes over and a fully plastic bend is achieved, when the springback drops. To evaluate the effect of the degree of deformation, springback is plotted versus bending angle for four of the test materials (see Section 3.1). The springback for all test materials has been compared at about the same degree of deformation. The results from this study are shown in Section 3.2. A comparison of the shape and thinning of one of the test materials is made in Section 3.3. 3. I. Springback springback
Fig. 2. FEM model bending of an 8 mm steel sheet in a 64 mm die. Due to symmetry, only one half the geometry needs to be modelled.
ns u function of bend angle uftei
To study whether the FEM can predict springback, generally different materials have been bent in various tools. In Figs. 4-7, predicted springback is compared with experimentally measured values, springback being plotted as a function of the bend angle after springback (as defined in Fig. 3.). In Figs. 4-7, good correlation between simulations and experiments is noted, the difference between the simulations and experiments being less than one degree, the discrepancy increasing with increasing deformation. 3.2. Springback ut a common degree of deformation Additional materials have been tested in various tools. Springback has then been analysed and compared with a common angle of about 90” for the different materials. Springback does not change much for bend angles lying close to each other (as shown in Section 3.1). The results are shown in Table 2 and Fig. 8. 3.3. Shape and thickness variations during bending
Fig. 3. Computer simulation of an 8 mm steel sheet bent in a 64 mm wide die. (Springback is defined as a,-+)
The shape of the bending curve is often approximated by an arc, although it has a much more complex shape. Wolter [lo] was one of the first who paid attention to the shape of lie bending line. A mathematical
0
bend Fig.
4.
Springback
vs. bend angle aft-r
angleafterspringbeck
unloading
for AA6016-T4. P= 1.1 mm.
II‘= 32 mm.
4
35
3
0 ir” F C 0
2
1,5
1
085
0 0
20
Fig. 5. Springback
40
vs.
bend angle after unloading for 906. r = 2 mm.
model taking this matter into account must contain some kind of mathematical description of the shape of the curve. Kahl [II] describes the shape of the curve
II‘=
32
mm.
with a cosine function. This description requires constants that uniquely define the function, which constants can be determined by empirical tests. One major
56
3.5
3
285
‘ii
2
% :: B K 185 4
1
0.5
0
20
60
40
1w
80
bend angle alter sprlngback
I = I mtn. IV = X mm.
Fig. 6. Springback vs. bend angle after unloading for AA5182,
4 4 4
2,5
4 0
0 p
2
f $
1,5
0
8
1
0.5
a
-l
0
20
40
60
60
100
band angle fdtw sprlngbrck
Fig. 7. Springback vs. bend angle after unloading for STEEL 2.
r = 7.9 mm. IV = 64 mm.
120
Table 2 Summary of the springhack rcsultc, for the test materials bent to -90” -~. _ Bend-case
I
Material
QStE340 QSlE340 QSt E.740 St06 S!!? AlSl 304 STEEL 1,’ STEEL 2,’ AAh016-T4 A.431 82 .____~~
2 3 4 5
6 7 8 9 IO
Sheet thickness
Die \
(mm)
(mm)
Mcu~ured springback (“)
8 3.9 3.9 2
h-8 ._ 1’ 64 8
2.h 2.3 6.0 0.Y
8
2.0
I I 3 x.5 7.9 1.13 1.01
3’ ._ 64 04 64 8
3.x 3.7 3.0 9,s 3.0
.-..___ ”The exact chemical composition has oat been dctcrmincd.
drawback with this method is that experiments are still required to be able to predict springback analytically. In the present experiments, the profile of the 8.3 mm steel sheet was measured in a coordin;:e measuring machine and the form of it was compared with computer simulations: this is shown in Fig. 4. The curvature of the bend corresponds well with what the simulations predict. The measurements showed a thickness rtduction at the bottom of the profile of 0.58 mm. In the FEM simulation with zero friction the reduction was
0.39 mm, whilst if a coefficient of friction of/l = 0.3 ~vas used the thickness reduction in the simulation became 0.51 mm.
The FEM can be used to predict springback for the tested materials. The method avoids many approximations that :vould be used in simplified analytical meth-
12
10
0 0
8 3 I 6
i6 P z % 4
Q 0
0
0
0 0
* 2
r]
0
0 0
1
2
3
4
5
6
7
a
BEND CASE NR.
Fig. 8. Springback in degrees for the materials listed in Table 2.
9
10
A. Nilsson et al. /Journal
58
-io
45
-10
-5
0
5
IO
of Materials Processing Technology 65 (1997) 52-58
15
al
mmfhmolu
Fig. 9. Comparison between the measured shape and the FEM simulation for an 8 mm steel sheet. The continuous line is the shape from the simulation when friction is neglected whilst the points correspond to the actual shape.
ods. The results correspond well with the experimental result fJr the materials tested, the deviation being roughly half a degree. However, due to variation in the material and the accuracy of the method, the error in measured values is estimated to be half a degree also. In this study, the effects of friction have been neglected, except in the case where the actual shape of the sheet is studied, where it is seen clearly that friction influences the thinning of the sheet in the deformed area. The effect on the springback, however, is very small. The discrepancies can be explained by the area of local plastic deformation below the punch not having been modelled in detail and by the plate sample having had a layer of paint that wore off during forming. From the results it can be concluded that, for most cases, the springback is slightly underestimated in the simulations. The discrepancies increase with increasing deformation since the deformed elements fail to provide
fully accurate results. The use of nine-node elements 01 remeshing can improve the results if a better accuracy is needed. The four-node element used in the simulation may slightly overestimate the stiffness of the sheet, which also is a source of error. The calculation time on a Sun sparcl -t machine is about lo- 15 min for a conventional tool with a die width of 8-10 times the sheet thickness. For the thin aluminium sheet in the 64 mm die the calculation time was more than 5 h, due to the larger punch stroke needed.
References VI Z.
Tan, B. Persson and C. Magnusson. An empirical model for controlling springback in V-die bending of sheet metals, analysis and modelling of plastic bending processes. Ph.D. Thesis, LuleA University of Technology, 1994, Paper B. VI O.M. Sidebottom and C.F. Gebhardt. E.up. Mrck., (Ott) (1979) 371-377. 131S.I. Oh and S. Kobayashi, ht. J. Mech. Sci., 22 (1980) 583-594. 141A. Makinouchi, Proc. Advanced Twimology of Plasticity. Tokyo. Sept. 1984, Vol. I, pp. 672-677. PI Y. Nagai and A. Makinouchi, in K. Lange (ed.). Proc. Advanced Technology of Plasticity, 1987, Proc. Secortd ht. Coaj: on Technotogy of Plast;:ity, Stuttgart, Germany, 24-28 Aug. 1987, Springer, Be&n. I61 S. Kim and K.A. Stelson, Trans. ASME, l/O (Au& (1988) 218-222. I71 R. Schilling, Lhder Bleclra Robe, 7 (1993) 29-38. PI J. Fleicher, Sheet Metats Tubes Sect., 8 (1989) 19-25. [91 0. Andreen and R. Crawfoord, Pro<. Adoawed Tcclmotogy of Plasticity, Tokyo, Sept. 1984. Vol. 1, pp. 593-598. WI K.H. Wollter. Freies Biegen von Blechen, VDI Forsckzmgs/wft 435, Diisseldorf, VDI Verlag, 1952. 11II K.W. Kahl, Untersuchungen zur Verbesserung der Form und Massgenauigkeit beim Biegen von Blechen, Fortsc/rr.-Ber. VDI Reihe 2. Nr. tt4, Diisseldorf, VDI Verlag, 1986. WI G. Oehler. Biegen, Carl Hanser, Munich, 1963, pp. 14-21.
Finite-element simulation of V-die bending: a compariso experimental results Annika Nilsson *, Lars Melin, Claes Magnusson
Industrial summary A major problem in sheet bending is to compensate for springback. Analytical descriptions are not sufhciently general to accommodate influences from the material and the geometry due to the simplifications that have to be made. Practical experiments are still needed to be able to compensate for springback. The aim of this work has been to test the finite-element method on its abilities to predict springback for free bending in a V-die. The work shows that the finite-element method can be used to predict springback off-line. The true stress-true strain curve from a tensile test is used as the material description. Springback has been studied for eight different materials of varying thickness. The process has been simulated with the code Nike2d and the results from the simulations compared with those from experiments, good correlation between the simulations and the experiments being achieved. 0 1997 Elsevier Science S.A. Kq~or&
V-die bending; Springback; Finite-element simulation
The control system in an NC-controlled press brakes compensates for springback, but often not successfully, as has been shown by Tan [l]. The calculation algorithms are too simplified to work for all materials. Sidebottom and Gebhardt [2] say in their work that ‘one should not expect an analytical solution to predict springback with great accuracy because of the simplifying assumptions that have to be made and of the different behaviour of materials that can occur between different heats of material’. Oehler [12] has mapped empirically the springback of several common materials but these curves cannot handle variations in the material and there is no documentation for new materials. Common approximations in analytical models are: a simplified stress-strain relationship, no friction, the bending curve approximated with an arc (the inner side is often assumed to equal the punch radius), no thinning of the sheet occurs, no Bauchinger effect. Approximations can be used for some cases, but they are never general. One general way of solving the problem is to use the
* Correspondingauthor. Present address: MEFOS,
(FEM), where the material behaviour and the geometry can be input easily and factors such as the shape of the bending line and the displacement of the neutral layer are taken care of by the numerical code. Oh and Kobayashi [3] have used the FEM to simulate the bending process and compared a rigid-plastic material model to an elastic-plastic material model. Makinouchi [4] has simulated U-bending using elasticplastic deformation theory in rate form. Nagai and Makinouchi [S] have simulated a bending restriking process which decreases springback by adding extra plastic deformation in a U-bending process. Kim and Stelson [6] have used the FEM as a tool to identify material characteristics from force and displacement measurements. Shilling [7] has compared isotropic and a combined isotropic-kinematic hardening model using the FEM. The work presented in Refs. [3-71 is limited to one set of tools and/or one material and few if any comparisons to experiments are made to verify the results. The aim with the present work is to obtain a general picture of the reliablilty of the FEM when it is used as a package without any modifications done to the original code. Other ways to develop a general bending model are to use a combined empirical-analytical approach such finite-element
1. Background
Box 812, S-971
25 Luld. Sweden. 0924-0136/97/$17.000 1997 Elsevier Science S.A. All rights reserved SSDI 0924-0136(95)02241-4
method
Fig. I. Photograph
before and aitcr unloading
an X mm QStE340 shret. cmplu> itt,11a tool \vti!l I die M4th of 6-l mm and ;I punch radius of 6 mm.
as, for instance, that of Fleichcr [8] and Tan [l], where a number of bending tests is wsed to evaluate constants that can be applied in the analytical model. Andreen and Crawfoord [9] suggest a numerical code. developed especially for bending processes. to predict springback.
angles. The bending angle was measured before and after springback from photographs taken of the spezimen in the press-brake (Fig. I .), the angles being mesured directly from the negati\,es in a profile microscope.
2. Conditions for the study To be able to evaluate the accurac> 0;‘the FE.M the simulations have to be compared with experimental data. The conditions for the experiments sre described in Section 2.1 and the conditions for the simulations are described in Sections 2.2 and 2.3. The experiments were performed with a dry tool. To decide if friction needs to be included in the study. experiments were made with two lubrication situations. A comparison was made between a dry tool and when the contact zone between the sheet and the die is lubricated by layers of plastic film with oil between the layers to decrease the friction. The experiments gave no difference in springback between these samples. The effect of springback in the simulations are minor and therefore it was decided to neglect the friction in the springback simulations. This assumption also shortens the calculation time.
The experiments were perfomed in an NC-controlled press brake. Eight materials were bent to different
The cumputcr simulations were perfomcd with the implicit FEM code Nike2d. Four-node plane-strain elements were used in the simulations. The FEM model is shown in Fig. 2. Due to symmetry, only one half of the geometry needed to be modelled. The deformation was achieved by prescribing the displacement of the punch. which corresponds with how the deformation is achieved in reality. The BFGS method is used to solve the non-linear equilibrium equations. The definition of the angles and springback is shown in Fig. 3.
The information supplied from the vendor on the material properties is insufficent for a FEM simulation: for the present work the stress-strain curve from a tensile test is used; in the simulation this is recalculated into a true stress -true strain behaviour. The tensile curve is taken in the direction perpendicular to the bend, i.e. along the curvt:
54
A. Nilsson et al. /Journal of Materials Processhg Techology 65 (1997) 52-58
In the simulations and elastic-plastic isotropic hardening material model is used. The true stress-true strain material description is input in discrete form, interpolating between the points. The materials that have been used are two aluminium, one stainless steel, two low-carbon deep-drawing steels and three hot-rolled steels. A list of the materials used is shown in Table 1. The mechanical properties are taken from a tensile test.
3. Comparison between experiments and simulations The amount of springback depends on the bending angle. The elastic springback increases with increased
Table 1 List of the materials used and their properties _____ Material
Thickness (mm)
R, (0.2) (MPa)
R,,, CM!%)
A511
St06 St06 st12 STEEL 1” STEEL 2;’ QStE340 AK1304 AA6016-T4 AA5182
3 2 1.1 8.3 7.9 3.95 3 1.13 1.01
158 200. I 241.7 488.6 341 470 273 II4 94
310 323.4 326.1 595.4 412 540 637 217 192
49 (A”,,) 29.8 30.0 1858 24.8 23 56 (A,,,) 28 24
a The exact chemical composition has not been determined.
deformation in the bend, to the point where coining takes over and a fully plastic bend is achieved, when the springback drops. To evaluate the effect of the degree of deformation, springback is plotted versus bending angle for four of the test materials (see Section 3.1). The springback for all test materials has been compared at about the same degree of deformation. The results from this study are shown in Section 3.2. A comparison of the shape and thinning of one of the test materials is made in Section 3.3. 3. I. Springback springback
Fig. 2. FEM model bending of an 8 mm steel sheet in a 64 mm die. Due to symmetry, only one half the geometry needs to be modelled.
ns u function of bend angle uftei
To study whether the FEM can predict springback, generally different materials have been bent in various tools. In Figs. 4-7, predicted springback is compared with experimentally measured values, springback being plotted as a function of the bend angle after springback (as defined in Fig. 3.). In Figs. 4-7, good correlation between simulations and experiments is noted, the difference between the simulations and experiments being less than one degree, the discrepancy increasing with increasing deformation. 3.2. Springback ut a common degree of deformation Additional materials have been tested in various tools. Springback has then been analysed and compared with a common angle of about 90” for the different materials. Springback does not change much for bend angles lying close to each other (as shown in Section 3.1). The results are shown in Table 2 and Fig. 8. 3.3. Shape and thickness variations during bending
Fig. 3. Computer simulation of an 8 mm steel sheet bent in a 64 mm wide die. (Springback is defined as a,-+)
The shape of the bending curve is often approximated by an arc, although it has a much more complex shape. Wolter [lo] was one of the first who paid attention to the shape of lie bending line. A mathematical
0
bend Fig.
4.
Springback
vs. bend angle aft-r
angleafterspringbeck
unloading
for AA6016-T4. P= 1.1 mm.
II‘= 32 mm.
4
35
3
0 ir” F C 0
2
1,5
1
085
0 0
20
Fig. 5. Springback
40
vs.
bend angle after unloading for 906. r = 2 mm.
model taking this matter into account must contain some kind of mathematical description of the shape of the curve. Kahl [II] describes the shape of the curve
II‘=
32
mm.
with a cosine function. This description requires constants that uniquely define the function, which constants can be determined by empirical tests. One major
56
3.5
3
285
‘ii
2
% :: B K 185 4
1
0.5
0
20
60
40
1w
80
bend angle alter sprlngback
I = I mtn. IV = X mm.
Fig. 6. Springback vs. bend angle after unloading for AA5182,
4 4 4
2,5
4 0
0 p
2
f $
1,5
0
8
1
0.5
a
-l
0
20
40
60
60
100
band angle fdtw sprlngbrck
Fig. 7. Springback vs. bend angle after unloading for STEEL 2.
r = 7.9 mm. IV = 64 mm.
120
Table 2 Summary of the springhack rcsultc, for the test materials bent to -90” -~. _ Bend-case
I
Material
QStE340 QSlE340 QSt E.740 St06 S!!? AlSl 304 STEEL 1,’ STEEL 2,’ AAh016-T4 A.431 82 .____~~
2 3 4 5
6 7 8 9 IO
Sheet thickness
Die \
(mm)
(mm)
Mcu~ured springback (“)
8 3.9 3.9 2
h-8 ._ 1’ 64 8
2.h 2.3 6.0 0.Y
8
2.0
I I 3 x.5 7.9 1.13 1.01
3’ ._ 64 04 64 8
3.x 3.7 3.0 9,s 3.0
.-..___ ”The exact chemical composition has oat been dctcrmincd.
drawback with this method is that experiments are still required to be able to predict springback analytically. In the present experiments, the profile of the 8.3 mm steel sheet was measured in a coordin;:e measuring machine and the form of it was compared with computer simulations: this is shown in Fig. 4. The curvature of the bend corresponds well with what the simulations predict. The measurements showed a thickness rtduction at the bottom of the profile of 0.58 mm. In the FEM simulation with zero friction the reduction was
0.39 mm, whilst if a coefficient of friction of/l = 0.3 ~vas used the thickness reduction in the simulation became 0.51 mm.
The FEM can be used to predict springback for the tested materials. The method avoids many approximations that :vould be used in simplified analytical meth-
12
10
0 0
8 3 I 6
i6 P z % 4
Q 0
0
0
0 0
* 2
r]
0
0 0
1
2
3
4
5
6
7
a
BEND CASE NR.
Fig. 8. Springback in degrees for the materials listed in Table 2.
9
10
A. Nilsson et al. /Journal
58
-io
45
-10
-5
0
5
IO
of Materials Processing Technology 65 (1997) 52-58
15
al
mmfhmolu
Fig. 9. Comparison between the measured shape and the FEM simulation for an 8 mm steel sheet. The continuous line is the shape from the simulation when friction is neglected whilst the points correspond to the actual shape.
ods. The results correspond well with the experimental result fJr the materials tested, the deviation being roughly half a degree. However, due to variation in the material and the accuracy of the method, the error in measured values is estimated to be half a degree also. In this study, the effects of friction have been neglected, except in the case where the actual shape of the sheet is studied, where it is seen clearly that friction influences the thinning of the sheet in the deformed area. The effect on the springback, however, is very small. The discrepancies can be explained by the area of local plastic deformation below the punch not having been modelled in detail and by the plate sample having had a layer of paint that wore off during forming. From the results it can be concluded that, for most cases, the springback is slightly underestimated in the simulations. The discrepancies increase with increasing deformation since the deformed elements fail to provide
fully accurate results. The use of nine-node elements 01 remeshing can improve the results if a better accuracy is needed. The four-node element used in the simulation may slightly overestimate the stiffness of the sheet, which also is a source of error. The calculation time on a Sun sparcl -t machine is about lo- 15 min for a conventional tool with a die width of 8-10 times the sheet thickness. For the thin aluminium sheet in the 64 mm die the calculation time was more than 5 h, due to the larger punch stroke needed.
References VI Z.
Tan, B. Persson and C. Magnusson. An empirical model for controlling springback in V-die bending of sheet metals, analysis and modelling of plastic bending processes. Ph.D. Thesis, LuleA University of Technology, 1994, Paper B. VI O.M. Sidebottom and C.F. Gebhardt. E.up. Mrck., (Ott) (1979) 371-377. 131S.I. Oh and S. Kobayashi, ht. J. Mech. Sci., 22 (1980) 583-594. 141A. Makinouchi, Proc. Advanced Twimology of Plasticity. Tokyo. Sept. 1984, Vol. I, pp. 672-677. PI Y. Nagai and A. Makinouchi, in K. Lange (ed.). Proc. Advanced Technology of Plasticity, 1987, Proc. Secortd ht. Coaj: on Technotogy of Plast;:ity, Stuttgart, Germany, 24-28 Aug. 1987, Springer, Be&n. I61 S. Kim and K.A. Stelson, Trans. ASME, l/O (Au& (1988) 218-222. I71 R. Schilling, Lhder Bleclra Robe, 7 (1993) 29-38. PI J. Fleicher, Sheet Metats Tubes Sect., 8 (1989) 19-25. [91 0. Andreen and R. Crawfoord, Pro<. Adoawed Tcclmotogy of Plasticity, Tokyo, Sept. 1984. Vol. 1, pp. 593-598. WI K.H. Wollter. Freies Biegen von Blechen, VDI Forsckzmgs/wft 435, Diisseldorf, VDI Verlag, 1952. 11II K.W. Kahl, Untersuchungen zur Verbesserung der Form und Massgenauigkeit beim Biegen von Blechen, Fortsc/rr.-Ber. VDI Reihe 2. Nr. tt4, Diisseldorf, VDI Verlag, 1986. WI G. Oehler. Biegen, Carl Hanser, Munich, 1963, pp. 14-21.