The reciprocal effects of bending and torsion on springback during 3D bending of profiles

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Procedia Engineering 00 (2017) 000–000

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Procedia Engineering 207 (2017) 2322–2327

International Conference on the Technology of Plasticity, ICTP 2017, 17-22 September 2017, Kingdom International Conference on theCambridge, TechnologyUnited of Plasticity, ICTP 2017, 17-22 September 2017, Cambridge, United Kingdom

The reciprocal effects of bending and torsion on springback during The reciprocal effects 3D of bending torsion on springback during bendingand of profiles 3D bending of profiles Daniel Staupendahl*, A. Erman Tekkaya Daniel Staupendahl*, A. Erman Tekkaya TU Dortmund University, Institute of Forming Technology and Lightweight Construction (IUL), Baroper 303, 44227 Dortmund, Germany TU Dortmund University, InstituteStr. of Forming Technology and Lightweight Construction (IUL), Baroper Str. 303, 44227 Dortmund, Germany

Abstract Abstract Profiles with circular cross-sections can be geometrically described by the shape of the bending line. To achieve 3D bending lines with kinematic processes, change of the bending plane of is needed, resulting vectors Profiles with circularbending cross-sections can abecontinuous geometrically described by the shape the bending line. in Tobending achieveforce 3D bending that change in direction accordingly. These force vectors generate a bending the forming zone thus,force longitudinal lines with kinematic bending processes, a continuous change of the bending moment plane is in needed, resulting in and, bending vectors tensile and in compressive stresses. ForThese profiles non-circular the in orientation of zone the cross-section along the that change direction accordingly. forcewith vectors generate cross-sections, a bending moment the forming and, thus, longitudinal bending linecompressive needs to be additionally Thisnon-circular can be achieved by applyingthe a specific torque to the bending process tensile and stresses. Forcontrolled. profiles with cross-sections, orientation of the cross-section alongand, the thus, introducing desired shear stresses into the forming zone. Up until now, this fundamental aspect of 3D profile bending has bending line needs to be additionally controlled. This can be achieved by applying a specific torque to the bending process and, not been regardeddesired in a coherent fashion. into To take account theUp reciprocal effects of the stressesaspect applied forming zone and thus, introducing shear stresses the into forming zone. until now, this fundamental of to 3Dthe profile bending has their effect on the bending moment and, To thus, oninto springback, a comprehensive analytical process applied model was setforming up. The zone model is not been regarded in a coherent fashion. take account the reciprocal effects of the stresses to the and validated investigations performed using the TSS profile analytical bending machine and comprehensive their effectby on experimental the bending moment and, thus, on springback, a comprehensive process model was set up. Thenumerical model is investigations. Analyses were performed during planeusing bending wellprofile as bending superposed The investigations validated by experimental investigations performed the asTSS bending machinewith and torsion. comprehensive numerical show that the applied bending and torque onlybending result inasstress but actuallywith alsotorsion. affect the of investigations. Analyses were force performed duringnot plane well superposition as bending superposed Thedevelopment investigations shear strains the bending cross-section thetorque profile. to theinlongitudinal strains, the strains from the show that theover applied forceof and notSimilar only result stress superposition but shear actually also decrease affect thelinearly development of intrados and extrados the profile to Considering this newlystrains, observed in thedecrease analytical processfrom model, shear strains over the of cross-section of the the neutral profile.axis. Similar to the longitudinal thebehavior shear strains linearly the the bending is intoaccordance experimental numerical results. intrados andmoment extradosprediction of the profile the neutralwith axis.the Considering thisand newly observed behavior in the analytical process model, the bending moment prediction is in accordance with the experimental and numerical results. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The under Authors. Published by Elsevier Ltd. the scientific committee of the International Conference on the Technology Peer-review responsibility of Elsevier © 2017 The Authors. Published by Ltd. Peer-review under responsibility of the scientific committee of the International Conference on the Technology of Plasticity. of Plasticity . responsibility of the scientific committee of the International Conference on the Technology Peer-review under Keywords: 3D profile bending; TSS bending; profiles; tubes; analytical model; stress superposition; torsion; of Plasticity . Keywords: 3D profile bending; TSS bending; profiles; tubes; analytical model; stress superposition; torsion;

* Corresponding author. Tel.: +49-231-755-7174; fax: +49-231-755-2489. address:author. [email protected] * E-mail Corresponding Tel.: +49-231-755-7174; fax: +49-231-755-2489.

E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. the scientific Peer-review 1877-7058 ©under 2017responsibility The Authors. of Published by Elseviercommittee Ltd. Plasticity . Peer-review under responsibility of the scientific committee

Plasticity.

of the International Conference on the Technology of of the International Conference on the Technology of

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the International Conference on the Technology of Plasticity. 10.1016/j.proeng.2017.10.1002

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1. Introduction The most widely used industrial kinematic bending process to produce 3D bent tubular structures is three-roll push bending. Extensive investigations have been done to produce stable process models that accurately predict part behavior during production. Hagenah et al. setup an FE-model for three-roll push bending, including machine stiffness to increase simulation accuracy [1]. An analytical process model of three-roll push bending for twodimensional bending contours was presented by Gerlach [2]. Building up on this knowledge, Engel [3] and Kersten [4] presented an analytical formulation that included machine stiffness for increased accuracy in two-dimensional bending. Plettke et al. [5] presented a mathematical description of bending contours based on Frenet-Serret formulations and also pointed out the deviation of applied tube rotation and resulting rotation angle of the bending plane. The significance of this variation they indicated by the introduction of the torsion adjustment coefficient. Vatter and Plettke [6] set up a detailed FE-model, capable of predicting three-dimensional bending contours. They performed detailed analyses, especially on the torsion adjustment coefficient. Their conclusion was that an empirical characteristic map is needed to describe the relation of the setting roll position, tube feed and tube rotation to the resulting curvature and tube rotation angle. They noticed a slight influence of the applied tube rotation on the curvature, but could not define a clear trend. Engel and Groth [7] performed similar investigations to analyze the torsion deformation that occurs during applied tube rotation and did notice the trend that curvatures do in fact increase slightly with increasing tube rotation. This, however, they could only observe at high curvatures. The first analytical model to predict the tube rotation needed to produce accurate three-dimensional curves was generated by Staupendahl et al. [8]. They were able to show, that during three-dimensional bending of tubes, meaning profiles with circular cross-sections, plastic torsion is negligible and that the main drivers for an offset of the applied tube rotation to the resulting rotation of the bending plane can be described solely by elastic tube deformation inside the machine during the bending process. A comprehensive analytical model was set up that describes the necessary tube rotation as a combination of geometrical tube rotation, static elastic tube deformation, and dynamic elastic tube deformation. Although Gerlach has shown that profiles with non-circular cross-sections can be bent with three-roll push bending to two-dimensional bending lines [2], bending of three-dimensional contours necessitates specialized machines as, for instance, the Hexabend, the TKS-Mewag, the machines by Nissin and J.Neu, and the TSS Bender. Up until now, the focus in all of the performed investigations has been the accurate description of in-plane bending [9,10]. As in three-roll push bending, three-dimensional bending was looked at as a sequence of single curvatures, located on different bending planes. This kind of mathematical formulation is acceptable as long as the cross-section of the target contour follows the rotation of the bending plane. However, in most of the cases during profile bending, the orientation of the cross-section does need to additionally be varied to produce the desired results. The best example for such a profile is a simple hand rail with a rectangular cross-section, where the top face is always supposed to point upwards. This fundamental aspect of 3D profile bending, the control of the orientation of the cross-section and the effect this control has on the bending force and, thus, on the resulting bending curvature, has until now been neglected in analytical process models and is addressed in the following work. Nomenclature dAy y ye, ymax a, b, n, m εep, εx γ σf,eq σx τxy Fb Mb L

infinitesimal segment of the profile cross-section distance from the neutral axis to the segment dAy distance from the neutral axis to the end of the elastic area, to the intrados/extrados of the profile Hockett-Sherby constants equivalent plastic strain, strain in longitudinal direction shear strain (with γ = 2εxy) equivalent flow stress longitudinal stress shear stress bending force bending moment length of the profile between front feeding roll and bending head

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LB RL, RU Φ r w, h, t

3

lever arm of the bending force loaded bending radius, unloaded bending radius loaded torsion angle – angle between α1 and α2 of the TSS Bender equivalent radius of middle axis of the profile wall to the center of gravity width, height, and wall-thickness of the profile

2. 3D profile bending setup and material used The kinematic 3D profile bending setup used in the investigations is the TSS Bender, a roll-based bending process developed at the IUL. The TSS Bender (TSS = Torque Superposed Spatial) is primarily made up of a rotatable transportation system and a bending head. The rotatable transportation system incorporates three feeding roll pairs that feed the profile forward into the bending head (c-axis). To bend two-dimensional contours with consecutive radii and spline curves, the bending head moves horizontally from side to side (x-axis). The bending head is equipped with a vertical rotation axis (τ-axis) to achieve a tangential run of the bending head relative to the profile. To bend three-dimensional contours, the transportation system rotates (α1-axis), resulting in a vertical force component being exerted by the bending head on the profile. This changes the angle of the initially horizontal bending plane and allows bending of arbitrary bending contours. To additionally control the orientation of the profile cross-section, the horizontal α2-axis is used. For the analysis of the process behavior the machine includes a force sensor, measuring the bending force component in the x-direction, and a sensor to measure the torque applied to the profile. The setup is shown in Fig. 1.

Fig. 1. The TSS bending process (top left) and TSS bending machine with included force and torque measurement systems

As profile geometry, a square cross-section was chosen with the size 40x40x2.5 mm. As material, soft annealed air hardening steel produced by Salzgitter Mannesmann Precision (MW700L Z1) was used. Since the material shows an elongated yield point, the flow curve was extrapolated using the Hockett-Sherby formulation, which actually allows an initial convex curve trend that closely matches the experimental data:

σ f ,eq =b − ( b − a ) ⋅ e

( )

− m⋅ e ep

n

with a = 418 MPa, b = 661.99 MPa, n = 1.6676, m = 70.802

(1)

The extrapolated flow curve was used in numerical simulations as well as in the analytical calculations. Investigations were performed on bending of profiles to the loaded radii 600, 800 and 1000 mm, and to the loaded torsion angles Φ = 0°, Φ = 11.25°, and Φ = 22.5°. The terms loaded radius and loaded torsion angle are used to

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describe the geometry that would theoretically be produced without springback, profile stiffness, and machine stiffness. Important for the analyses shown in section 3 is the knowledge about the target bending force, which could not directly be calculated from the experimental x-axis force measurements. The experimental data was rather used to calibrate the numerical simulation, which then provided the needed bending force data. 3. Effect of torsion on the bending process A simple and effective way to describe pure bending of profiles is by the elementary bending theory. Regarding the material used as isotropic and the strains over the cross-section as linear, the stress-distribution over the crosssection can easily be acquired via matching the flow curve of the material to the strains in the cross-section. Using this stress-distribution together with the area of the profile, segmented into infinitesimal area segments dA, the bending moment can be calculated. In the case of the TSS Bender, the distance between the bending head and the front feeding roll (lever arm of the bending force) can be used to calculate the bending force. The bending moment can additionally be used together with the second moment of inertia and the Young’s Modulus to calculate the radius after springback RU.

∫ σ (e ) ⋅ y ⋅ dA

= Mb

x

p e

and

y

Fb =

Mb LB

and

1 1 Mb = − RU RL EI

(2)

For pure torsion, the Saint Venant formulation can be used. For thin-walled profiles, this formulation can be combined with the formulation by Bredt, which considers a constant shear flow over the cross-section and describes the shear stress as a function of the area, enclosed by the middle axis of the profile wall. Using the torsion constant of a thin-walled rectangular cross-section the shear strain can be calculated as follows:

r L

γ = Φ=

( w − t )( h − t ) Φ ( w + h − 2t ) L

with L = length of the profile

(3)

However, using the assumption of a constant shear strain over the cross-section, which was also done by Zhang et al. to describe the interaction between torsion and bending during forming of thick-walled tube [11], did not lead to satisfactory results in the present case of profile bending. The predicted reduction of the bending force was about 300% higher than could be observed numerically. Looking at a numerical simulation of the TSS bending process performed in Abaqus Explicit, using the shell element formulation S4R and a mesh size of 3.4 mm over the crosssection and 3.5 mm over the length of the profile, it can be noticed that the shear strain, in fact, is not constant over the cross-section (Fig. 2). To check if this local variation of shear strain was not caused by contact stresses applied by the feeding apparatus, a simplified implicit FE model was used that allowed the simultaneous application of a bending moment and a torsional moment, without the influence of tool contact. In all of the simplified numerical experiments it was observed that the absolute number of the shear strain is maximum in the shell elements nearest to the intrados and extrados of the profile cross-section and minimal in the shell elements nearest to the neutral axis. Feeding rolls

Bending head

A A Tension

Shear Shearstrain strainγγ

Profile

0.02 0.01 0

0

37.5

75

112.5 A-A 0

R600 11.2511,25° Numerical CS PE12 Tension R600 22,5022,5° Numerical CS PE12 R600 11,25 11,25° Analytical Saint Venant Analytisch R600 22,5022,5° Analytical Saint Venant Analytisch

-0.01 -0.02

Length over middle axis of profile wall in mm Length over circumference in mm

Fig. 2. FE model used to analyze the process (left) and shear strain distribution over the cross-section (right)

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5

Using this knowledge, an enhanced analytical model was set up that describes the combined loading of bending and torsion in the forming zone. Assuming plain stress, the Levy Mises flow rule and the Von Mises yield criterion:

λ =

e x γ = ′ σ x′ τ xy

σ 2f ( y= ) σ x2 + 3τ xy2

(4)

and assuming linear strain paths, and equal strain in width and thickness, it follows that: 2

= e p e

 y  γ −γ  1  ( w − t )( h − t ) e x ln 1 +  Φ − ( ymax − y ) max e  with= ex +  3  ( w + h − 2t ) L ymax − ye   RL  2

(5)

and

σx =

σ 2f ,eq ( e ep ) γ −γ  4  1 ( w − t )( h − t ) Φ − ( ymax − y ) max e  1+  3  e x ( w + h − 2t ) L ymax − ye 

(6)

2

Using formulas 5 and 6 in combination with 1 and 2, the bending force can be calculated. Because the experimental setup only allowed the measurement of the x-component of the bending force, the explicit FE-model described above was used to extract the necessary data, as explained in section 2.

5000 4500 4000 3500 3000

6000 Bending force in N

6000

Numerical model

5500

4000 3500 600 mm 800 mm 1000 mm Loaded bending radius

22,5°

3500

11,25° 22,5°

11,25°

4000

0°

4500

0°

4500

Enhanced analytical model

5000

3000

5000

3000

600 mm 800 mm 1000 mm Loaded bending radius

Analytical model

5500

Length over middle axis of profile wall in mm

5500

Bending force in N

Bending force in N

6000

600 mm 800 mm 1000 mm Loaded bending radius

-300 80 70 60 50 40 30 20 10 0 -0.01

Shear stress in MPa -150 0 γ analytical Saint Venant γ analytical enhanced σ analytical Saint Venant σ analytical enhanced σ numerical

-0.005 Shear strain γ

0

Fig. 3. Numerical results of bending force (top left), analytical calculation of bending force by assuming a uniform shear strain distribution over the cross-section (top right), enhanced analytical calculation of bending force by assuming a linear decrease of the shear strain from the intrados/extrados to the neutral axis (bottom left), different shear stress distributions while bending a profile to RL = 600mm and Φ = 11.25°

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Fig. 3 shows the comparison of the bending force results from the numerical model to the results from the analytical model with a uniform shear strain distribution over the cross-section and the enhanced analytical model, which considers a linear decrease of the shear strain from the intrados and extrados to the neutral axis of the profile as observed in the numerical investigations. The bending force in both of the analytical models is higher than in the numerical model, because the elastic profile deformation in the feeding roll system as well as in between the front feeding roll and the bending head is not considered. Apart from this difference, the initial analytical model shows bending force reductions during applied torsion that are up to 300% higher than numerically calculated. This can be explained by the high deviation of the calculated shear stresses relative to the numerical results. The shear stresses calculated by the enhanced analytical model, on the other hand, are much closer to the numerical results, with larger deviations occurring only close to the neutral axis of the profile. With the introduction of the enhanced model the absolute error in bending force was reduced from 23% to 9%. 4. Conclusion Kinematic 3D profile bending processes provide the flexibility needed to cope with current demands of lightweight and individual design. Because kinematic bending processes do not use a bending form to shape a part, but rather variate machine axis movements, the accuracy of the produced product strongly depends on the process model used to generate the movement data. To be able to not only accurately bend two-dimensional but also threedimensional shapes, an enhanced analytical model was developed that considers the reciprocal effects of bending and torsion on the bending moment and, as a result, the bending force and the springback. In numerical analyses that were performed alongside experimental investigations, it was observed that the shear strain during the combined loading of bending and torsion is not uniform over the cross-section, but rather linearly decreases towards the neutral axis. By considering this effect in the enhanced analytical model, bending forces were calculated that are in accordance with targeted values, not only qualitatively but also quantitatively. With the future introduction of elastic profile behavior, it is expected that the small remaining error will be reduced even further. Acknowledgements The authors thank Prof. Dr. Peter Haupt for the enlightening discussions about the topics of space curves and stress superposition. References [1] H. Hagenah, D. Vipavc, R. Plettke, M. Merklein, Numerical Model of Tube Freeform Bending by Three-Roll-Push-Bending, 2nd Int. Conf. on Engineering Optimization, Lisbon, Portugal, 2010. [2] C. Gerlach, Ein Beitrag zur Herstellung definierter Freiformbiegegeometrien bei Rohren und Profilen [A Contribution to the Manufacturing of Tubes and Profiles with Free Form Bending Geometries], Shaker Verlag, Aachen 2010. [3] B. Engel, S. Kersten, Analytical Models to Improve the Three-Roll-Pushbending Process, Steel research international (2011), pp. 355-360. [4] S. Kersten, Prozessmodelle zum Drei-Rollen-Schubbiegen von Rohrprofilen [Process Models for Three-Roll Push Bending of Tubes], Shaker Verlag, Aachen 2013. [5] R. Plettke, P.H. Vatter, D. Vlpavc, Basics of Process Design for 3D Freeform Bending, Steel research international: 14th Int. Conf. on Metal Forming (2012), pp. 307-310. [6] P. H. Vatter, R. Plettke, Process model for the design of bent 3-dimensional free-form geometries for the three-roll-push-bending process, Procedia CIRP 7 (2013), pp. 240-245. [7] B. Engel, S. Groth, Analyse der Torsionsverformung beim Drei-Rollen-Schubbiegen von Rundrohren [Analysis of the Torsion Deformation during Three-Roll Push Bending of Tubes], Proc. of the 34th Verformungskundlichen Kolloquium der Montanuni. Leoben (2015), S.87-92 [8] D. Staupendahl, C. Becker, A.E. Tekkaya, The Impact of Torsion on the Bending Curve during 3D Bending of Thin-Walled Tubes – a Case Study on Forming Helices, Key Engineering Materials 651-653 (2015), pp. 1595-1601. [9] S. Chatti, M. Hermes, A.E. Tekkaya, M. Kleiner, The new TSS bending process: 3D bending of profiles with arbitrary cross-sections, CIRP Annals – Manufacturing Technology, 59/1 (2010), pp. 315-318. [10] M. Hudovernik, F. Kosel, D. Staupendahl, A.E. Tekkaya, K. Kuzman, Application of the bending theory on sqaure-hollow sections made from high strength steel with a changing angle of the bending plane, J. of Mat. Proc. Techn., Vol. 214, No. 11 (2014), pp. 2505-2513. [11] Z.K. Zhang, J.J. Wu, R.C. Guo, M.Z. Wang, F.F. Li, S.C. Guo, Y.A. Wang, W.P. Liu, A semi-analytical method for the springback prediction of thick-walled 3D tubes, Materials & Design 99 (2016), pp. 57-67.