Benefits of stress superposition in combined bendinglinear flow splitting process
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Abstract
Linear flow splitting is a multistation sheetbulk metal forming process which allows continuous production of bifurcated profiles without joining, lamination or external heating of sheet metal. This process induces high hydrostatic stresses in the forming zone which leads to an elevated formability of the workpiece material. The aim of this research is to bend linear flow split profiles in transverse direction in a continuous manner. This is achieved by combining the linear flow splitting process with a continuous bending process. An analytical and a numerical model are described in this paper which predict bending moments for different radii. Results from both models are validated with experimental results. It is found that combining the linear flow splitting with a bending process leads to a severe reduction in the bending moments due to superposition of stresses in the forming zone. The superposition maintains the cross sectional shape of the bent profiles.
Keywords
Linear flow splitting Finite element analysis Bending SpringbackNomenclature
 Symbol
unit Explanation
 a_{1f}
[mm] Width of Section 1
 a_{2f}
[mm] Width of Section 2
 a_{total}
[mm] Flange width
 b_{4el}
[mm] Height of Section 4
 b_{n}
[mm] Substitute for height
 R_{b}
[mm] Bending radius
 b_{total}
[mm] Total height (analytical)
 D_{SpR}
[mm] Diameter splitting roll
 D_{SuR}
[mm] Diameter supporting roll
 E
[N/mm^{2}] Young’s modulus
 E_{sub}
[N/mm^{2}] Substitute modulus
 h
[mm] Radius at supporting roll
 h_{n}
[mm] Substitute for height
 J_{z}
[mm^{4}] Second moment of area
 k_{f}
[N/mm^{2}] Flow stress
 l_{SuR}
[mm] Width of supporting roll
 M_{b}
[Nm] Bending moment
 R
[mm] Radius at splitting roll
 s_{0}
[mm] Web width
 s_{f}
[mm] Flange width
 y_{F}
[mm] Distance from neutral axis to boundary elastic plastic region
 y_{inc}
[mm] Incremental infeed
 y_{total}
[mm] Total infeed
 α
[°] Supporting roll angle
 α_{s}
[°] Splitting roll angle
Introduction
Demand for high strength steel profiles has increased strongly due to requirements to reduce fuel consumption by weight reduction [1]. This is indicated by a statistic of the fabrication of flat rolled steel within Germany which has quintupled since the 1950s [2]. Roll forming has been established in the manufacturing industry for numerous years because it fulfils the requirements of high volume production and low costs. The geometrical flexibility can be increased by preforms produced through the linear flow splitting process [2, 3].
Major challenges in bending high strength steel profiles result from the limited formability, the large and uncertain springback and the high tool loads. As a characteristic of linear flow splitting processes, a significant increase of strength is observed in the flanges of the profile [12]. Therefore, linear flow split profiles manufactured from high strength steel are expected to encounter higher springback and lower formability than conventional mild steel profiles. The objective of this study is to propose a method which will allow for bending linear flow split profiles with a constant crosssection in a continuous manner without any expensive tool change costs. Further objectives are to propose an analytical and a FE model to evaluate bending moments required to bend the profile and to investigate shape accuracy after bending.
State of the art
Linear flow splitting
Profile bending
Bending is one of the oldest and most frequently used forming processes. The bending of profiles allows an engineer to save space, produce aerodynamic advantageous shapes and even neglect expensive joints [13, 14, 15]. The varying requirements result in a wide range of different bending technologies. The bending processes with rotary as well as straight tool movement are divided into bending with formbound contour or kinematically defined bending contour and thermally induced bending processes [16].
This method is adapted to profile bending under ideal conditions regarding shape, strain and material [20].
Methods for bending linear flow split profiles
Requirements for bending linear flow split profiles
 1.
The bending process should allow continuous forming
 2.
Due to requirements of shape accuracy, the bending process should be easily adjustable to incorporate changes in the bending radius
 3.
The UFG properties in the flanges of linear flow split profiles should remain unchanged throughout the bending process
Profile bending procedures are categorized into two methods: Kinematic shaping and forming with shape defining dies. In kinematic shaping processes, the final shape of the profile is not determined by the shape of the bending tool. The final shape is rather produced by relative movements of tool and workpiece. Whereas, in forming with shape defining dies, the shape of the bending tool determines the final shape of the profile. In both procedures, the bending tool can possess a linear or rotary motion while bending a profile [19]. Due to fixed geometry of the bending tool in the forming with shape defining dies, the second requirement is violated. Thermally induced bending methods which are classified under the kinematic shaping procedure, have an influence on the microstructure of a material. Therefore, these bending methods can be excluded regarding the third requirement. As a result, only kinematic shaping bending procedures with rotary tool motion during bending satisfy all three requirements. Therefore, these methods are considered for preliminary investigations.
Preliminary investigations on different bending methods
A. Threeroll bending
In threerollbending, a profile is bent by three bending rolls after linear flow splitting process (Fig.4 b). The position of rolls can be adjusted according to the required radius. The process is analysed numerically for a bending radius, R _{ b } = 2312.5 mm and material RAWAEL 80. This bending radius results in a bending force of F _{ b } = 5600 N and a springback factor K = 0.68 (K = initial radius/ final radius) in the simulations. Although, buckling was not observed for aforementioned bending radius, there exists a high possibility of buckling under nonideal experimental conditions [24]. Also, this high bending force can lead to high springback which makes overbending compensation errorprone.
B. Local rolling
The purpose of this method is to thin a profile locally at a certain position on the outer side of the web. This results in local elongation of the thinned zone in the longitudinal direction. Because of these induced longitudinal strains on the outer part of the web, the profile is bent in transverse direction. To get the bending radius (R _{ b } = 2312.5 mm), the web thickness has to be reduced by 0.24 mm on the outer side and a local rolling force of F _{ r } = 220 kN is required. In real processes, this high force would lead to deflection of the tool system or enforce a very stiff system. The reduced thickness of the web caused by the local rolling is another disadvantage of this bending strategy.
C. Bending after straightening station
In this method, a bending roll is positioned after a straightening station. This bending roll is moved in transverse direction as per required bending radius (R _{ b } = 2312.5 mm). This method also leads to the same disadvantages like threeroll bending. The simulation results show that springback factor in this bending strategy is 0.64 whereas a bending force F _{ b } = 1000 N is exerted on the bending roll.
D. Bending after a linear flow splitting station
In this method, after linear flow splitting, the split profile is bent by a bending roll which can be moved in both transverse and longitudinal directions as per the designed bending radius. In a linear flow splitting station, high multiaxial stresses are applied in the forming zone. Therefore, the crosssection of the profile within the forming zone is deformed plastically. The bending roll after the last splitting station applies bending moment at the forming zone which is already deformed plastically. Therefore, a very small force on bending roll is exerted whereas very high forces are exerted on the linear flow splitting station (described in Section 7).
This method shows good results regarding the required bending force and the springback factor. In the numerical simulation, a bending force of only F _{ b } = 200 N is required to bend the targeted radius. The springback factor for this method is found to be K = 0.93. This shows that this bending strategy can lead to less springback than all three bending methods explained before. Therefore, this bending strategy is selected and evaluated in detail in this research work.
Tool system and experimental work

uncoiling of sheet material

manufacturing of bifurcations at the band edge by linear flow splitting

bending of the profile

cutting into desired sections
(a) Position of the bending tool, (b) Process parameters
a  
X (mm)  Y (mm)  Bending radius R _{ 1 } (mm) 
425.36  100  600 
546.01  100  1000 
644.46  100  1400 
729.74  100  1800 
806.06  100  2200 
b  
Process Parameter  Value  
Velocity of the metal sheet, V _{ sheet }  3 m/min  
Initial sheet width, b _{ initial }  50 mm  
Width of the profile before bending, b _{ n1 }  34 mm, 35 mm  
Incremental depth at the last station, y _{ inc }  1.5 mm, 2 mm  
Total incremental depth, y _{ total }  9.5 mm 
Numerical and analytical approach
Numerical model
A numerical model of the bending process is created with a FEsoftware package Marc Mentat 2012 as shown in Fig. 6. The model is created using PYTHON scripts to achieve consistency in the geometries and solved with an implicit quasistatic solver. In this model, the last linear flow splitting operation and the bending process are considered. The geometry of the profile before the last stand is measured by a laser triangulation system. This geometry is considered as an initial geometry for the profile. The process parameters for the numerical model are same as in experiments which are described in Table 1 (b).
The length of the sheet is taken as 2500 mm to achieve the steady state condition. To reduce the computational time, the end part of the sheet is fixed in X direction and rolls move across the sheet to retain the relative motion. Additional supporting roll sets are required to hold the profile in Ydirection. These rolls do not perform any forming. The bending roll is first moved in transverse and in longitudinal direction until it reaches a designed position for a specific bending radius (Fig. 5 (b)). After reaching the specific position, it is moved only in longitudinal direction until the end of the simulation. All rolls are free to rotate about their own axis due to friction between the sheet and the rolls.
Numerical investigation of the forming zone
At path 1, just before the profile width reduction, material on both sides is compressed. At this position the longitudinal strain distribution is almost symmetrical. The longitudinal strain distribution becomes asymmetrical after path 2 which is a characteristic of a bending process where the inner side is compressed and the outer side is stretched. Therefore, it can be stated that the bending of the profile evolves after path 2. At position of path 3 where the boundary of the plasticized zone ends, the longitudinal strain distribution at the outer side and inner side are almost similar to distribution at path 4. This confirms that the bending operation takes place entirely within the forming zone induced by the flow splitting station.
It can be seen that, the linear flow splitting process introduces high symmetrical longitudinal compressive stresses in the forming zone. Because of the bending operation this distribution becomes asymmetrical. The difference between longitudinal stresses for bent profiles and the straight linear flow split profile can be attributed to the bending process. However, these additional bending stresses are small compared to the equivalent stresses (1100 N/mm2) acting in the forming zone. To confirm this fact, the yield locus for a point on the outer edge of path 3 is plotted for linear flow splitting and combined bending with linear flow splitting as shown in Fig. 8 (b). For comparison, the stress state at the same position is also plotted in case of the three roll bending process (method A) which is described in Section 3. A. It can be seen that the stress state on the yield locus does not vary by a considerable margin when bending is combined with the linear flow splitting process. This difference is higher when three roll bending is considered. This proves that small changes in the stress state are sufficient to change strain components in the forming zone of the profile which also justifies lower magnitude of forces exerted on the bending roll.
Analytical model
In eq. 3, J _{ z } is moment of area for sections 1 and 4. To calculate individual bending moments for sections 1 to 4, b _{ n } and h _{ n } are substituted by a _{ 1f } , a _{ 2f } , s _{ 0 } (for sections 3 and 4) and s _{ f } (for sections 3 and 4), b _{ total }  b _{ 4el } , b _{ 4el } respectively where a _{ 1f } and a _{ 2f } are the widths of the flange in sections 1 and 2. The thickness of web is equal to the width of section 3, s _{ 0 } whereas s _{ f } is the thickness of the flange. The total width of the profile is b _{ total } whereas b _{ el } is the height of section 4 as shown in Fig. 9 (b).
The resulting cross section of the profile in the presented investigation has a width b _{ total } = 31 mm and a flange width of a _{ z,total } = 13 mm with the thickness of the web s _{ 0 } = 2 mm and the thickness of the flange s _{ f } = 1 mm. These dimensions are calculated from results obtained from experiments and numerical analysis. Other dimensions of the sections are dependent on the incremental infeed. The accurate choice of these dimensions is necessary to validate analytical with experimental results. From numerical analysis, it was found that widths of sections 1 and 2 are a _{ 1f } = 0.6 mm and a _{ 2f } = 10.4 mm respectively whereas the height of zone 4 (b _{ 4el }) is 8 mm for y _{ inc } = 1.5 mm. At an increased infeed of y _{ inc } = 2 mm, the plastic zone increases and it was found that the web is fully plasticised resulting in b _{ 4el } = 0 mm. Zone 1 also decreased with a _{ 1f } = 0.4 mm and a _{ 2f } = 10.6 mm. From tensile tests of RAWAEL 80, initial yield stress is determined as k _{ f } = 840 N/mm2.
Bending moment
A good correlation can be seen between the analytical model, the FEA and the experimental results. The curve for analytical results decreases with increasing infeed y _{ inc }. The increase in infeed and the plastic zone is described in the model with a decrease of size of zones 1 and 4 with corresponding increase of size of zones 2 and 3. As shown in the graphs, there is a variation of the experimental results displayed as error indicators. This is a result of the tumbling movement of the linear flow splitting stands which is described by Vucic [3].
Profile properties
The bending takes place in the forming zone of a linear flow splitting process. Therefore, any additional thickening of the profile due to bending is rolled out by the supporting rolls of the linear flow splitting station. Consequently, higher forces are exerted on rolls of the linear flow splitting station in the forming zone as shown in Fig. 11b.
Therefore, there is no change in the thickness of the web after bending process.
The variation of the bending radius also has no influence on the resulting angle between the flanges, the wing span of the flanges and the distance between the splitting centres. All these geometrical properties as well as the bending radius R _{ b } are found to be reproducible.
The surface roughness at the splitting centre is investigated. The variations of both measurements (0.2 μm) overlap, as the surface roughness of the bent profile is R _{ z } = 0.84 μm while the roughness of the straight profile is at R _{ z } = 1.16 μm. There is no difference in the results between outer and inner side of the bent profile. The inspection of the surface also showed no cracks at the surface of the profile.
Conclusion and outlook
The investigations show that the continuous bending of linear flow splitting profiles is possible with high flexibility to accommodate changes in the bending radius. The most promising method for bending linear flow split profiles is the superposition of linear flow splitting process with a bending process. Analytical and FE models are put forward for calculating the bending moment which are found to be beneficial in designing a tooling system.
The analytical model predicts the bending moment within the average experimental deviation of ΔM _{ b } ≈ 6 %, whereas the FE model predicts with maximum error of 1 %. Results show that the bending moment and resulting bending forces can be influenced by the choice of the incremental infeed in the linear flow splitting station.
The bent profile’s geometrical properties of the cross section are similar to a straight profile. Combining this fact with the reproducibility of the bending radii makes bending of linear flow splitting profiles suitable for continuous production.
Notes
Acknowledgment
The investigations presented in this paper are supported by the German Research Foundation (DFG). The authors would like to thank DFG (Die Deutsche Forschungsgemeinschaft) for funding the Collaborative Research Centre 666 “Integral sheet metal design with higher order bifurcations – Development, Production, Evaluation”.
Compliance with ethical standards
Funding
This study was funded by Deutsche Forschungsgemeinschaft (GZ: SFB666/2 T1).
Conflict of interest
The authors declare that they have no conflict of interest.
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