Research on Comparison of simulation and experiment in warpage defects in roll forming process of the cap-shaped part using a five-boundary condition forming angle distribution function

: Warpage is one of the main defects in multi-pass roll forming. It manifests as the horizontal deviation in the sheet flange part after forming, impacting the final quality of the sheet. In this manuscript, we focus on the cap-shaped part of a small section profile. Considering the lack of previous scientific guidance on the distribution of forming angles—angles that lead to warpage defects and other defects reducing the quality of these products—a five-boundary condition distribution function of forming angle is proposed. We propose a mechanism for warpage defects in roll bending of a cap-shaped part based on this five-boundary condition forming angle distribution function. Furthermore, we examine the effects of forming angle, sheet thickness, and material yield strength on the warpage defects by examining the designed fluctuation of the edge wave, ∆ z , and the maximum deviation of the profile curves of the flange edge, z' , to test this theory, roll bending experiments are compared to simulation.


Fig. 1 Roll forming process
The bending angle is the most important processing parameter in the roll forming process.
This bending angle is determined by production experience, and there is no strict theory to define the division of the bending angle. To optimize this process, the development of an effective bending angle division method is of value. To this end, Hirosh et al. [1] proposed that the edge deformation of the strip follows a cubic curve. This work shows that the horizontal plane projection trajectory at the end of the standing edge follows a cubic curve, proposing an ideal distribution of the bending angles. Bidabadi et al. have also studied this phenomenon, [2] [3] evaluating the longitudinal bending of roll forming in a symmetrical U-shaped cross section both experimentally and numerically. These results show that the most important parameter is the forming angle.
Furthermore, Han et al. [4] effectively utilized a B3-spline finite strip method to study the influence of molding parameters in cold rolling forming of channel section.
Warping is a common defect formed in multi-pass roll bending, as shown in Figure 2.
Specifically, Tehrani et al. [5] believed that warpage is caused by changes in longitudinal strain on the sheet. To this end, Luo Xiaoliang et al. [6] have found, through numerical simulation, that during the bending process of high-strength steel rolls, certain material properties decrease the likelihood of warping, including: a larger yield strength, an increased strengthening coefficient, and a larger thickness anisotropy coefficient. Han et al. [4] found that the longitudinal strain in roll bending increases with plate thickness, larger deformation, and other parameters. By establishing a numerical simulation model, Wang Zhenxiao et al. [7] found that the deformation path in flexible roll bending has a large impact on the generation of warping. Moreover, with an increase in the number of rolls and the distance between passes, the side wave effect is weakened. Through numerical simulation, Li Yu et al. [8] evaluated the distribution of changes in stress and strain. At the same time, orthogonal testing shows that the flange height, sheet metal thickness, yield strength of sheet metal, and the number of forming paths all impact the formation and size of warpage. In conclusion, the roll forming process requires further study, specifically in regards to the control and reduction of warping through technological parameters. Before completing our study, we utilized production experience and theoretical analysis to propose a successful forming angle distribution function based on a five-boundary condition. To this end, we constructed a curve equation of the projection trajectory for the contour section edge in the horizontal plane. Simultaneously, we prove that when the forming angle interval of the first third of the forming process is 30% × θ0 ≤ θN/3 ≤ 35% × θ0, the forming angle distribution is ideal [9][10].

Roll forming theory with five-boundary condition forming angle distribution function
Bending angle (forming angle) is the most important and challenging issue in roll forming design. A well designed division of bending angle reduces stress concentration and resulting defects, such as longitudinal bending, edge tearing, and springback. By utilizing production experience and related theories, Professor Konai Hiroshi from Japan has proposed a bending angle distribution formula for roller forming [1]. This work suggests that when the projection track of the horizontal plane of the edge end follows a cubic curve, the bending angle distribution of plate is the ideal.
This method of bending angle allocation is shown in Figure 3. As shown in Figure 3, assuming that the number of forming channels is N, the final bending angle of the vertical edge is θ0, the length of the vertical edge is H, and the bending angle of the ith vertical edge is θi, the expression of the cubic curve is as follows: This includes four-boundary condition equations: By combining Equation (1) and boundary conditions (2), the following can be obtained: When x = i and y i= Hcosθi, the relationship between bending angle and forming pass, 3) can be written as follows: By taking I = 1,2,3…N into Equation (4), the bending angle of each forming pass can be calculated. A bending angle distribution using a four-boundary condition model with nine passes is utilized in this study and shown in Table 1, where i is the iterative forming pass, θi is the bending angle, Δθi is the bending angle increment.

Simulation and optimization of forming angle distribution method based on fiveboundary condition
The bending angle allocation method based on four-boundary conditions has some regularity, resulting in a quantitative bending angle allocation, but it cannot guarantee that the distribution result is ideal as there are some limitations. Following practical production experience and statistical analysis, generally the bending angle for the first third of the process should not exceed 50% of the final bending angle, that is, /3 ≤ 50% × 0 . To this end, a fifth boundary condition, yN/3 = HcosθN/3, can be applied to the bending angle distribution function with a four-boundary condition, resulting in the construction of a bending angle distribution function with a fiveboundary condition.
Assuming that the projection trajectory of the vertical edge of the horizontal plane is part of the quadric curve, the expression for constructing the quadric curve is as follows: Thus, when = /3, and = cos /3 , the five-boundary conditions of Equation (5) are: According to former research [10], the optimal first third of the forming angle interval is 30% × 0 ≤ 3 ⁄ ≤ 35% × 0 . Further optimization of the forming angle in the distribution 33% × θ0，34% × θ0，35% × θ0 and i = 1, 2… N, Equation (7) can be used to obtain the following results for each forming angle under different θN/3, as shown in Table 2: Copra was used to simulate the roll bending forming process. First, according to the results in Table 2, developed using the five-boundary condition of roller bend forming process simulation, the main performance parameters of the material as shown in Table 3. Key shaping parameters entered into the software DataM Copra RF v2005 SR1 include: (1) the calculation party was set to DIN6935 (2) the simulation method was the Hauschild method (3) the number of racks was 10 (4) rack spacing was 500 mm, etc. The resulting simulation of the peak longitudinal strain is shown in Figure 4. Similarly, by utilizing the above parameters and Table 1  In order to analyze the simulation results of peak longitudinal strain, Figure .6 compares the peak longitudinal strain with the different boundary conditions. When θN/3 = 50% × θ0, the trends of the four-boundary condition and the five-boundary condition is consistent. However, for the four-boundary condition method, the maximum peak longitudinal strain occurs at the first pass, that is, at the bite stage of forming is too high, the deformation too large, and the forming angle too big, features that are not conducive to the bite of sheet metal. Among the four fold lines, the 33% × θ0 fold line is gentler than the other three fold lines, resulting in a distribution that is more reasonable. The above analysis shows that the method with four-boundary condition cannot guarantee the best forming angle distribution result, thus the five-boundary condition method obtains the optimal forming angle allocation result.
Based on these results, we selected θN/3 = 33% × θ0 for experimental analysis. By applying I = 1,2…, N into Equation (7), we obtained the distribution of the forming angles for each pass and the resulting distribution is shown in Table 4.

Abaqus Finite element simulation
This study uses the software Abaqus to simulate the sheet roll forming process. The mechanical properties of the two materials studied are shown in Table 5, yielding the stress-strain curve shown in Figure 7. Figure 8 shows an established finite element model of roll forming, of which the grid of the sheet is shown in Figure 9. To improve accuracy of this simulation, the mesh is further refined in the bending area.

Warpage measurement method of the cap-shaped part
The hat-shaped piece obtained by the experiment is shown in Figure 11, and the forming quality is different under different forming parameters. In order to experimentally verify the impact of different forming parameters on the numerical simulation in the previous section, it is necessary to evaluate the warpage fluctuation at the flange edge through the maximum deviation of the z-axis coordinates. This value is measured as discussed below.

Roll forming warpage evaluation standard
In the hat-shaped piece, the warpage reflects the differences between the vertical coordinates of the edge of the flange. Based on this, Figure 13 was developed to schematically show the warpage evaluation standard. To extract the node coordinates (xi, zi) of the edge, the average of zi was taken to determine ̅ , by determining the standard deviation of the difference between zi and ̅ , the z-coordinate of each node was calculated relative to the average ̅ . The maximum deviation is used as the evaluation index of warpage, and the calculation of the standard deviation is shown in Equation (11).
In the Equation, ∆z is the standard deviation of the difference between zi and ̅ of each node, ′ is the maximum deviation between the z-coordinate of each node and the average ̅ , N is the number of flange edge nodes, yi(I = 1,2,3…N) is the z-coordinate of each node, and ̅ is the average value of the z-coordinate of each node. In summary, the larger the value of ∆z and ′ , the greater the coordinate difference, indicating increased warpage.

Simulation analysis on the formation mechanism of warpage defects(Analysis of the formation mechanisms of edge wave)
Li Yu et al. [8] analyzed the stress and strain of thin-walled channel steel parts after roll forming to make the flange edge show plastic instability, producing a side wave. In contrast, our analysis of the stress and strain generated in the roll forming process of hat-shaped sheet metal parts explores the mechanisms of warpage in hat-shaped sheet metal parts.
Based on the numerical simulation cloud diagram in Section 2, it can be seen that there are specific warping defects at the area of the forming angle. The key difference from Section 2 is that warping mainly occurs at the flange edge, thus research should be shifted from the area of forming angle in Section 2 to the flange edge.
In order to match observation with analysis in accordance with the forming law of the symmetrical section profile, the 1.5 mm thick Q235 steel with the same parameters as in the previous section with the five-boundary condition is again subjected to numerical simulation of roll forming, and the new cloud diagram showing the stress distribution of hat-shaped piece after forming is shown in Figure 14.
It can be seen from Figure 14

Analysis of simulated and experimental results considering the influence of distribution methods on warpage defects
The simulated and experimental parameters in this section are shown in Table 6. This section focuses on evaluating the z-coordinate and the shift of ∆z after the sheet is fully formed.  Overall, the trend of the experimentally results is consistent with the simulated data, however the distribution of the simulated data fluctuates more. This difference between experiment and simulation is caused by the frictional forces on the roll, the manufacturing error of the equipment, and the error generated during the sheet feeding.
In order to show the influence of different bending angle distribution methods on the warpage defect, we compare the data collected through simulation and experimental and Equation (7), thereby calculating the warpage fluctuation, ∆z, under different forming angle distribution and the maximum deviation of z-axis coordinate ′ , as shown in Table 7 and Figure 19.  Table 7 and Figure 19 show that the warpage fluctuation, ∆z, and maximum deviation, ′ , produced by roll forming of the hat-shaped piece based on the five-boundary condition is smaller than the amount of warpage produced by the other bending angle distribution methods.
Through this comparison of experimental and simulation results, it can also be found that the warpage fluctuation and maximum deviation of the z-axis coordinate calculated from the experimental data are slightly larger than the simulated value. This result is due to the unavoidable errors in actual production. However, the trend remains consistent with the simulation results.

The influence of different sheet thicknesses on simulated and experimental results of warpage defects of hat-shaped parts
Based on the optimized five-boundary condition forming angle distribution method, the influence of sheet thickness (1.0 mm, 1.5 mm, and 2.0 mm) on the warpage defects is evaluated.
The material used in this study was also Q235 steel.
Once simulation and experiment were complete, the coordinates of each node (x, z) on the edge were measured and recorded. The contour curves of the longitudinal nodes are shown in  Table 8 and Figure 21 with their calculated warpage fluctuation, ∆z, and maximum deviation of z-axis coordinate, ′ , for different forming angle distributions.  Table 8 and Figure 21 show that when the plate thickness increases, the warpage fluctuation, ∆z, and the maximum deviation of the z-axis coordinate are significantly reduced; that is, the warpage fluctuation of the flange edge decreases with the increase of the plate thickness. The thicker the sheet, the less likely it is to warp. These experimental results are in good agreement with simulation. However, while increasing the thickness can help reduce the degree of warpage, it may also introduce defects such as springback or aggravation, thus in actual production, the thickness of the selected sheet should be chosen appropriately.

Influence of yield strength on the simulation and experimental results of warpage defects of hat-shaped parts
This section focuses on the five-boundary conditions of the forming angle distribution method when the sheet thickness is 1.5 mm. Select DP780 steel (σs = 578MPa) is compared to Q235 steel (σs = 235MPa) to study the effect of material yield strength on the warpage defect.
After the simulation and experiment were finished, we collected and measured the x and zcoordinates of each node at the edge of the hat-shaped piece and record the values. These values are compared in Figure 22.  Table 9 and Figure 23.  Table 9 and Figure 23 show the the warpage fluctuations in DP780 and Q235 to be 0.76 and 1.23 in simulation, respectively, and that the maximum deviation of the z-axis coordinate is 1.66 mm and 2.09 mm. However, the warpage fluctuations, ∆z, as calculated from the experimental data, are 1.16 and 1.54, while the maximum deviations of the z-coordinates are 2.22 mm and 3 mm. Thus, we see that the yield strength of the material has a great influence on warping defects and that the fluctuation of the flange edge decreases with increased yield strength. However, the forming quality of sheet materials with higher yield strength is not as good and as discussed in the previous section, the cap forming parts will have greater springback.

Conclusion
This manuscript examines the cap-shaped roll bending part of a small section profile that is common in industrial production. We proposed and optimized a five-boundary condition forming angle distribution function. For the first time, the forming angle, sheet thickness, and sheet yield strength were analyzed by ABAQUS finite element simulation study under the proposed forming angle distribution method. Three forming parameters were shown to impact warpage defects and roll forming experiments were completed and compared to simulations. The conclusions are as follows: (1) By using the ABAQUS software to simulate the stress and strain of the hat-shaped piece after roll forming, we found that the warping defect of the flange is caused by the uneven distribution of longitudinal stress and longitudinal plastic strain on the flange of the hat.
(2) Through use of the forming angle on the three distribution modes, the simulation of the flange edge warpage fluctuation, ∆z, and the maximum deviation of z-coordinates were be obtained.
The contour curve of the flange edge of the hat-shaped piece under the five-boundary condition angle distribution method is relatively smooth. This result is consistent with experiment. Thus, the amount of warpage under the five-boundary condition bending angle distribution method is less than the degree of warpage produced by the other two bending angle distribution methods.