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

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1 Available online at ScienceDirect Procedia Engineering 207 (2017) International Conference on the Technology of Plasticity, ICTP 2017, Setember 2017, Cambridge, United Kingdom The recirocal effects of bending and torsion on sringback during 3D bending of rofiles Daniel Stauendahl*, A. Erman Tekkaya TU Dortmund University, Institute of Forming Technology and Lightweight Construction (IUL), Baroer Str. 303, Dortmund, Germany Abstract Profiles with circular cross-sections can be geometrically described by the shae of the bending line. To achieve 3D bending lines with kinematic bending rocesses, a continuous change of the bending lane is needed, resulting in bending force vectors that change in direction accordingly. These force vectors generate a bending moment in the forming zone and, thus, longitudinal tensile and comressive stresses. For rofiles with non-circular cross-sections, the orientation of the cross-section along the bending line needs to be additionally controlled. This can be achieved by alying a secific torque to the bending rocess and, thus, introducing desired shear stresses into the forming zone. U until now, this fundamental asect of 3D rofile bending has not been regarded in a coherent fashion. To take into account the recirocal effects of the stresses alied to the forming zone and their effect on the bending moment and, thus, on sringback, a comrehensive analytical rocess model was set u. The model is validated by exerimental investigations erformed using the TSS rofile bending machine and comrehensive numerical investigations. Analyses were erformed during lane bending as well as bending suerosed with torsion. The investigations show that the alied bending force and torque not only result in stress suerosition but actually also affect the develoment of shear strains over the cross-section of the rofile. Similar to the longitudinal strains, the shear strains decrease linearly from the intrados and extrados of the rofile to the neutral axis. Considering this newly observed behavior in the analytical rocess model, the bending moment rediction is in accordance with the exerimental and numerical results The Authors. Published by Elsevier Ltd. Peer-review under resonsibility of the scientific committee of the International Conference on the Technology of Plasticity. Keywords: 3D rofile bending; TSS bending; rofiles; tubes; analytical model; stress suerosition; torsion; * Corresonding author. Tel.: ; fax: address: daniel.stauendahl@iul.tu-dortmund.de The Authors. Published by Elsevier Ltd. Peer-review under resonsibility of the scientific committee of the International Conference on the Technology of Plasticity /j.roeng

2 Daniel Stauendahl et al. / Procedia Engineering 207 (2017) Introduction The most widely used industrial kinematic bending rocess to roduce 3D bent tubular structures is three-roll ush bending. Extensive investigations have been done to roduce stable rocess models that accurately redict art behavior during roduction. Hagenah et al. setu an FE-model for three-roll ush bending, including machine stiffness to increase simulation accuracy [1]. An analytical rocess model of three-roll ush bending for twodimensional bending contours was resented by Gerlach [2]. Building u on this knowledge, Engel [3] and Kersten [4] resented an analytical formulation that included machine stiffness for increased accuracy in two-dimensional bending. Plettke et al. [5] resented a mathematical descrition of bending contours based on Frenet-Serret formulations and also ointed out the deviation of alied tube rotation and resulting rotation angle of the bending lane. The significance of this variation they indicated by the introduction of the torsion adjustment coefficient. Vatter and Plettke [6] set u a detailed FE-model, caable of redicting three-dimensional bending contours. They erformed detailed analyses, esecially on the torsion adjustment coefficient. Their conclusion was that an emirical characteristic ma is needed to describe the relation of the setting roll osition, tube feed and tube rotation to the resulting curvature and tube rotation angle. They noticed a slight influence of the alied tube rotation on the curvature, but could not define a clear trend. Engel and Groth [7] erformed similar investigations to analyze the torsion deformation that occurs during alied 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 redict the tube rotation needed to roduce accurate three-dimensional curves was generated by Stauendahl et al. [8]. They were able to show, that during three-dimensional bending of tubes, meaning rofiles with circular cross-sections, lastic torsion is negligible and that the main drivers for an offset of the alied tube rotation to the resulting rotation of the bending lane can be described solely by elastic tube deformation inside the machine during the bending rocess. A comrehensive analytical model was set u 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 rofiles with non-circular cross-sections can be bent with three-roll ush bending to two-dimensional bending lines [2], bending of three-dimensional contours necessitates secialized machines as, for instance, the Hexabend, the TKS-Mewag, the machines by Nissin and J.Neu, and the TSS Bender. U until now, the focus in all of the erformed investigations has been the accurate descrition of in-lane bending [9,10]. As in three-roll ush bending, three-dimensional bending was looked at as a sequence of single curvatures, located on different bending lanes. This kind of mathematical formulation is accetable as long as the cross-section of the target contour follows the rotation of the bending lane. However, in most of the cases during rofile bending, the orientation of the cross-section does need to additionally be varied to roduce the desired results. The best examle for such a rofile is a simle hand rail with a rectangular cross-section, where the to face is always suosed to oint uwards. This fundamental asect of 3D rofile 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 rocess models and is addressed in the following work. Nomenclature da y infinitesimal segment of the rofile cross-section y distance from the neutral axis to the segment da y y e, y max distance from the neutral axis to the end of the elastic area, to the intrados/extrados of the rofile a, b, n, m Hockett-Sherby constants ε e, ε x equivalent lastic strain, strain in longitudinal direction γ shear strain (with γ = 2ε xy ) σ f,eq equivalent flow stress σ x longitudinal stress τ xy shear stress F b bending force M b bending moment L length of the rofile between front feeding roll and bending head

3 2324 Daniel Stauendahl et al. / Procedia Engineering 207 (2017) L B lever arm of the bending force R L, R U loaded bending radius, unloaded bending radius Φ loaded torsion angle angle between α 1 and α 2 of the TSS Bender r equivalent radius of middle axis of the rofile wall to the center of gravity w, h, t width, height, and wall-thickness of the rofile 2. 3D rofile bending setu and material used The kinematic 3D rofile bending setu used in the investigations is the TSS Bender, a roll-based bending rocess develoed at the IUL. The TSS Bender (TSS = Torque Suerosed Satial) is rimarily made u of a rotatable transortation system and a bending head. The rotatable transortation system incororates three feeding roll airs that feed the rofile forward into the bending head (c-axis). To bend two-dimensional contours with consecutive radii and sline curves, the bending head moves horizontally from side to side (x-axis). The bending head is equied with a vertical rotation axis (τ-axis) to achieve a tangential run of the bending head relative to the rofile. To bend three-dimensional contours, the transortation system rotates (α 1 -axis), resulting in a vertical force comonent being exerted by the bending head on the rofile. This changes the angle of the initially horizontal bending lane and allows bending of arbitrary bending contours. To additionally control the orientation of the rofile cross-section, the horizontal α 2 -axis is used. For the analysis of the rocess behavior the machine includes a force sensor, measuring the bending force comonent in the x-direction, and a sensor to measure the torque alied to the rofile. The setu is shown in Fig. 1. Fig. 1. The TSS bending rocess (to left) and TSS bending machine with included force and torque measurement systems As rofile geometry, a square cross-section was chosen with the size 40x40x2.5 mm. As material, soft annealed air hardening steel roduced by Salzgitter Mannesmann Precision (MW700L Z1) was used. Since the material shows an elongated yield oint, the flow curve was extraolated using the Hockett-Sherby formulation, which actually allows an initial convex curve trend that closely matches the exerimental data: m ( e ) ( ) n e σ f,eq = b b a e with a = 418 MPa, b = MPa, n = , m = (1) The extraolated flow curve was used in numerical simulations as well as in the analytical calculations. Investigations were erformed on bending of rofiles to the loaded radii 600, 800 and 1000 mm, and to the loaded torsion angles Φ = 0, Φ = 11.25, and Φ = The terms loaded radius and loaded torsion angle are used to

4 Daniel Stauendahl et al. / Procedia Engineering 207 (2017) describe the geometry that would theoretically be roduced without sringback, rofile stiffness, and machine stiffness. Imortant for the analyses shown in section 3 is the knowledge about the target bending force, which could not directly be calculated from the exerimental x-axis force measurements. The exerimental data was rather used to calibrate the numerical simulation, which then rovided the needed bending force data. 3. Effect of torsion on the bending rocess A simle and effective way to describe ure bending of rofiles is by the elementary bending theory. Regarding the material used as isotroic 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 rofile, 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 sringback R U. M b ( e ) b σx y da and y Fb LB = M 1 1 M b = and = (2) R R EI For ure torsion, the Saint Venant formulation can be used. For thin-walled rofiles, 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 rofile wall. Using the torsion constant of a thin-walled rectangular cross-section the shear strain can be calculated as follows: ( w t)( h t) ( + 2 ) r γ = Φ= Φ with L = length of the rofile (3) L w h t L However, using the assumtion 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 resent case of rofile bending. The redicted reduction of the bending force was about 300% higher than could be observed numerically. Looking at a numerical simulation of the TSS bending rocess erformed in Abaqus Exlicit, 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 rofile, 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 alied by the feeding aaratus, a simlified imlicit FE model was used that allowed the simultaneous alication of a bending moment and a torsional moment, without the influence of tool contact. In all of the simlified numerical exeriments 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 rofile cross-section and minimal in the shell elements nearest to the neutral axis. U L Profile Feeding rolls Bending head A A Shear strain γ A-A CS R600 PE ,25 Numerical Tension CS R600 PE12 22,5022,5 Numerical Analytisch R600 11,2511,25 Analytical Saint Venant Analytisch R600 22,5022,5 Analytical Saint Venant Tension Length Length over over middle circumference axis of rofile in mm wall in mm Fig. 2. FE model used to analyze the rocess (left) and shear strain distribution over the cross-section (right)

5 2326 Daniel Stauendahl et al. / Procedia Engineering 207 (2017) Using this knowledge, an enhanced analytical model was set u that describes the combined loading of bending and torsion in the forming zone. Assuming lain stress, the Levy Mises flow rule and the Von Mises yield criterion: e γ ( y) λ x = = σ τ x xy σ = σ + 3τ (4) f x xy and assuming linear strain aths, and equal strain in width and thickness, it follows that: ( w t)( h t) ( ) 2 1 γmax γ e e = ex + Φ ( ymax y) 3 w+ h 2t L ymax ye 2 y with e x = ln 1+ RL (5) and σ = σ 2 f, eq ( w t)( h t) ( ) ( e ) x γmax γ e 1+ Φ ( ymax y) 3 e x w+ h 2 t L ymax ye (6) Using formulas 5 and 6 in combination with 1 and 2, the bending force can be calculated. Because the exerimental setu only allowed the measurement of the x-comonent of the bending force, the exlicit FE-model described above was used to extract the necessary data, as exlained in section 2. Bending force in N Numerical model 600 mm 800 mm 1000 mm Loaded bending radius Bending force in N Analytical model 600 mm 800 mm 1000 mm Loaded bending radius 0 11,25 22,5 Bending force in N Enhanced analytical model 600 mm 800 mm 1000 mm Loaded bending radius 0 11,25 22,5 Length over middle axis of rofile wall in mm Shear stress in MPa Shear strain γ γ analytical Saint Venant γ analytical enhanced σ analytical Saint Venant σ analytical enhanced σ numerical Fig. 3. Numerical results of bending force (to left), analytical calculation of bending force by assuming a uniform shear strain distribution over the cross-section (to 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 rofile to R L = 600mm and Φ = 11.25

6 Daniel Stauendahl et al. / Procedia Engineering 207 (2017) Fig. 3 shows the comarison 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 rofile 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 rofile deformation in the feeding roll system as well as in between the front feeding roll and the bending head is not considered. Aart from this difference, the initial analytical model shows bending force reductions during alied torsion that are u to 300% higher than numerically calculated. This can be exlained 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 rofile. With the introduction of the enhanced model the absolute error in bending force was reduced from 23% to 9%. 4. Conclusion Kinematic 3D rofile bending rocesses rovide the flexibility needed to coe with current demands of lightweight and individual design. Because kinematic bending rocesses do not use a bending form to shae a art, but rather variate machine axis movements, the accuracy of the roduced roduct strongly deends on the rocess model used to generate the movement data. To be able to not only accurately bend two-dimensional but also threedimensional shaes, an enhanced analytical model was develoed that considers the recirocal effects of bending and torsion on the bending moment and, as a result, the bending force and the sringback. In numerical analyses that were erformed alongside exerimental 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 rofile behavior, it is exected that the small remaining error will be reduced even further. Acknowledgements The authors thank Prof. Dr. Peter Haut for the enlightening discussions about the toics of sace curves and stress suerosition. References [1] H. Hagenah, D. Viavc, R. Plettke, M. Merklein, Numerical Model of Tube Freeform Bending by Three-Roll-Push-Bending, 2nd Int. Conf. on Engineering Otimization, Lisbon, Portugal, [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 [3] B. Engel, S. Kersten, Analytical Models to Imrove the Three-Roll-Pushbending Process, Steel research international (2011), [4] S. Kersten, Prozessmodelle zum Drei-Rollen-Schubbiegen von Rohrrofilen [Process Models for Three-Roll Push Bending of Tubes], Shaker Verlag, Aachen [5] R. Plettke, P.H. Vatter, D. Vlavc, Basics of Process Design for 3D Freeform Bending, Steel research international: 14th Int. Conf. on Metal Forming (2012), [6] P. H. Vatter, R. Plettke, Process model for the design of bent 3-dimensional free-form geometries for the three-roll-ush-bending rocess, Procedia CIRP 7 (2013), [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 34 th Verformungskundlichen Kolloquium der Montanuni. Leoben (2015), S [8] D. Stauendahl, C. Becker, A.E. Tekkaya, The Imact of Torsion on the Bending Curve during 3D Bending of Thin-Walled Tubes a Case Study on Forming Helices, Key Engineering Materials (2015), [9] S. Chatti, M. Hermes, A.E. Tekkaya, M. Kleiner, The new TSS bending rocess: 3D bending of rofiles with arbitrary cross-sections, CIRP Annals Manufacturing Technology, 59/1 (2010), [10] M. Hudovernik, F. Kosel, D. Stauendahl, A.E. Tekkaya, K. Kuzman, Alication of the bending theory on sqaure-hollow sections made from high strength steel with a changing angle of the bending lane, J. of Mat. Proc. Techn., Vol. 214, No. 11 (2014), [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 sringback rediction of thick-walled 3D tubes, Materials & Design 99 (2016),