**1. Introduction**

The warping of profiles with an asymmetric cross-section regarding the force application axis during bending is a common issue in profile bending. The shift between the force application axis and the shear center induces torsion moments in the cross-section [1] (Figure 1a). Altering the position of the force application axis to the shear center either requires specialized tools or is not possible without changing the product geometry as the position of the shear center can be located outside of the profile cross-section [2]. However, it is possible to move the force application axis to the shear center through welded plate elements (see Figure 1b). Methods to counter warping include the superposition with counteracting torsion-inducing forces [3]. One example of this method is the torque superposed spatial bending (TSS) process [4]. In this process, additional torque is superposed during the bending process which suppresses profile warping. The disadvantage is the additional torque axis necessary. Other methods to suppress warping include the use of supporting elements [5]. In a patent by Kreye [6], plastic warping is preventable through geometrical restrictions by supporting the profile at the whole shell surface in the forming area (see Figure 1c). Support elements have also been used to suppress the formation of a secondary bending axis in the bending of asymmetric profiles through a flexible bending process [7]. In the aforementioned flexible bending process, the profile is guided through a pusher. The position of the pusher is adjustable in the profile length direction which changes the position of the forming zone.

**Citation:** Hoffmann, E.; Meya, R.; Tekkaya, A.E. Reduction of Warping in Kinematic L-Profile Bending Using Local Heating. *Metals* **2021**, *11*, 1146. https://doi.org/10.3390/met11071146

Academic Editor: José Valdemar Fernandes

Received: 24 June 2021 Accepted: 16 July 2021 Published: 20 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Countermeasures to reduce warping in profile bending profiles for the example of a U-profile. (**a**) profile crosssection with the shear center as cause for warping, (**b**) countermeasure: moving the position of force application by altering the profile geometry. (**c**) countermeasure: use of supporting elements.

Additionally, process control can be used to counteract geometric deviations in kinematic bending processes. Wang et al. [8] proposed a process control concept by asymmetric loading patterns in a four-roll bending process of Z-profiles. Through the asymmetric loading pattern, warping can be eliminated. In return, process control requires complex models and high computational efforts. The mentioned methods to decrease geometric deviations in profile bending reduce flexibility as either more tools, machine axes, or workpiecedependent models for process control are necessary. To reduce geometric deviations in kinematic profile bending, other methods are required to maximize flexibility.

To decrease the number of geometric deviations in bending processes without workpiece-dependent measures, bending processes have been supported by heating methods [9]. Park et al. [10] managed to employ a synchronous incremental forming and incremental heating process to bend hat-shaped profiles consisting of DP590 without buckling, collapse, or necking. Behne [11] describes a bending process with local induction heating and consequent cooling for tubes. The process manages to reduce thinning for bending radii of up to 2.5 times the diameter without a mandrel. Yanagimoto et al. [12] managed bending without springback for the v-bending of high-strength steel sheets. The reduction of springback can be attributed to the lowered bending moments due to thermal softening.

To use heating as a warping reduction measure, a flow stress gradient in the profile cross-section is to be reached through heating of one area of the cross-section, while the temperature of other areas remains approximately at room temperature. Local, crosssectional heating is a known concept to reduce geometrical deviations in profile products. The so called "heat-straightening" is used in bridge [13] or ship repair [14] to suppress deformations due to thermal or mechanical loads. In this process, heat is introduced into a deformed part. Thermal expansion of a heated part is restricted to build up compressive stresses. Upon cooling, the amount of plastic compression remains. Through this process, deformed parts can be restored to their original shape. This method is not integrated into manufacturing processes and the knowledge is mostly empirical. The development of laser bending evolved from this method. Laser bending uses heat generated by a laser to induce thermal stresses in a workpiece, which leads to the shaping of the part. Kraus [15] used a wedge-shaped heating strategy to realize the bending of rectangular tubes. The products produced in this process are not subject to springback. In the laser bending of tube parts, it is shown that the process reduces tube ovalization compared to mechanical bending without a mandrel [16]. While the laser bending process offers advantages compared to conventional bending, a complex heating strategy is necessary to produce bend profiles without distortion.

The state of the art shows various methods to decrease warping in the bending of profiles and methods to decrease geometrical deviations in bending by local heating. On the

one hand, these methods require special tools or computational efforts to reduce warping and springback. On the other hand, the displayed methods are not tested on profiles with asymmetry to the force application axis. In this work, a novel heating strategy by partial, cross-sectional heating and cooling to support a three-roll push-bending process is analyzed. The heating strategy aims to reduce profile warping and springback of the products. A gradient stiffness over a profile cross-section will influence the position of the shear center [17]. Additionally, through heating of the profile part, the flow stress and Young's modulus, and therefore the stiffness, will decrease. The decreased flow stress will also lead to lowered stresses in the profile in this area. Both of these effects will consequently influence the shear center position. As the flow stress gradient will also cause a shift in stress free fiber position, it is suspected that the shear center position and the stress-free fiber position are linked. Through the decreased flow stress, a springback reduction in the bending process is expected.

Three main hypotheses emerge from the observations in the literature:


To prove the defined hypotheses, L-profiles of S500MC are tested in a three-roll push bending process with partial-cross sectional heating and consequent water-jet cooling. The resulting unloaded profiles are evaluated for springback and warping and are compared to FEM results. With the geometrically validated FEM results in the unloaded state, the model in the continuous push-bending phase is used to describe the bending moment, warping, and position of stress-free fiber during the process. These results are compared to the developed analytical model for the continuous push-bending phase, which is able to predict the profile warping and bending moment.

In Table 1 the Nomenclature for this manuscript is presented.


**Table 1.** Nomenclature.


**Table 1.** *Cont.* ୧ୖ
