A study of transverse bowing defect in cold roll-forming asymmetric corrugated channels

The transverse bowing greatly affects the accuracy of the roll-forming asymmetric corrugated channels (ACC). In order to control this defect, the paper first elucidates its mechanism of the production. Then, the transverse bowing is explored by finite element method (FEM) using ABAQUS 2016 software and the effects of its forming parameters are analyzed. Also, a linear regression model is built using Minitab 19 software to evaluate their effects, of which standard values are measured by a Pareto chart. It is observed that the number of forming channels has the greatest effect, followed by the roll gap and bending angle, and the friction coefficient has the least. Finally, to decrease the transverse bowing defect (TBD), the dominant forming parameters are optimized according to the evaluation results and the operating conditions. And the results of simulation and optimization are verified by the experiments, respectively. This research shows that the TBD can be greatly controlled through the optimization of three dominant forming parameters including the roll gap, bending angle, and inter-station distance for given ACC.


Introduction
Cold roll forming is a continuous metal forming process with high efficiency and precision for manufacturing strip profiles with constant cross section. The roll-formed sheet products have been widely used in the construction industry, such as roofing, wall cladding, concrete form-work, and fencing. The asymmetric corrugated channels (ACC) under study, as this type of product, are mainly for fabricating the aluminum sleeves. Compared with traditional competitive paper sleeves used for the supports of industrial roll products such as aluminum coils, the aluminum ones have the advantages of low cost, high load capacity, low collapse, recyclability, environmental protection, and so on [1]. And thus, they can be used as the supports for aluminum coils instead of paper sleeves and have wider promotion value and market development prospect (see Fig. 1).
However, the desired accuracy of ACC is difficult to be guaranteed due to the defects like twisting and bowing in roll-forming process, which inevitably affects the quality of aluminum sleeves. It has been regarded that the springback or the elastic recovery causes those above defects. For this reason, some scholars have carried out a series of related studies to control these defects. Poursina et al. [2] used BPANN and regression analysis to predict the longitudinal bow defects and then compared the prediction results with the simulation results. The results showed that BPANN analysis was more accurate. The research of Abvabi et al. [3] showed that the reduction of material elastic modulus can more accurately predict the springback of high-strength steel in the numerical simulation of profile forming. Guan et al. [4] in another study used the shear failure criterion to numerically simulate the fracture at the corner of U-shaped profile during cold bending, analyzed the stress-strain and damage evolution during the fracture process, and provided a reference for the study of fracture mechanism of cold bending. Murugesan et al. [5] studied the longitudinal bending and springback phenomenon of Al alloy with U-profile using digital image technology and finite element method. The results show that the longitudinal strain has an effect on the geometry and the springback phenomenon only appears at the end of the profile. With respect to the thickness reduction in the bending zone in free U-bending, Qian et al. [6] explored the influence of three main factors with numerical simulation and experimental through three different loading modes. Cha and Kim [7] found that the twisting and bowing behaviors of asymmetric channel section were most likely due to the inconsistent longitudinal strain in the web zone and flange region of the section. And the defect was significantly improved by applying a compression force in the web thickness direction after the sheet passed the final pass. Tehrani et al. [8] performed a study on the symmetrical channel section with local edge buckling phenomenon and analyzed the effect of the incremental bending angle of the adjacent roll stations on this defect. Zeng et al. [9] found that the optimized roll diameter and the forming angle increment were effective in avoiding the U-channel edge wave phenomenon through optimizing the design of each station with the response surface method, and the springback was also controlled to a minimum value. In another study, Wiebenga et al. [10] performed robust optimization of the cold roll-forming process to compensate for the product defects, of which effectiveness was proved by both experimental and numerical results. A parametric study of the product forming quality was performed by Bui and Ponthot [11], which proved that the material yield strength had a remarkable influence on the springback, while the roll station distance affected little.
In general, the interests of cold roll-forming defects so far have mainly focused on the single U-or V-shaped section. Due to their fewer channels, the defects mainly occur in the longitudinal direction, while the research on the defects is rarely involved in transverse direction. Especially for the corrugated channels under study, the roll force exerted on each channel is unequal due to their asymmetric channel width, which makes the transverse bowing defect (TBD) occur easily. Therefore, in view of the TBD of ACC, this study first elucidates its mechanism of the production in Section 2. Then, the finite element model, TBD, and non-uniform springback simulation are presented in Section 3. The effects of forming parameters are discussed in Section 4. The influence evaluation and the optimization of forming parameters and experimental verification are completed in Section 5. Finally, some valuable conclusions on how to control the TBD are provided in Section 6.

Transverse bowing mechanism of ACC
Different from longitudinal defect, the TBD usually occurs in the cross section of the roll-formed product and is caused mainly by transverse non-uniform springback. As shown in Fig. 2a, a large amount of bending deformation of the sheet is required to move the flat part in a roll-forming single channel. Naturally, the transverse tension is produced considerably due to the transverse stretching at the corners of the sheet section, which easily leads to the corners thinning or tearing under the roll force.
But for the roll-forming ACC, since the flat parts are reduced relatively, they only require less transverse tension during being pulled toward the center (see Fig. 2b). Correspondingly, the transverse tension at the corners is significantly reduced. When a final ACC is rolled into shape, the tension strain at the corners is variability for their asymmetrical channel width, where the strain of the middle channel is greater than that of the two sides. This means that the springback of the sheet middle area is smaller and thus, the transverse non-uniform springback inevitably appears, which makes the ACC produce TBD, correspondingly. And the greater the transverse non-uniform springback, the more significant the TBD of ACC.

Forming sequence and geometric model
Here, the roll-forming technology of ACC adopts a sequential forming method. Namely, the whole forming

Paper sleeve
Aluminum sleeve process is divided into four groups, and the middle channels are formed first, followed by the two sides, as shown in Fig. 3a. Unlike V-or U-shaped single channel, the ACC can be regarded as the composition of multiple cap sections, including narrow and wide channels, which are symmetrically distributed on both sides of the middle channel. Their geometric characteristics and dimensions are shown in Fig. 3b and Table 1, respectively.

Finite element model
This section is carried out by numerical simulation of the cold roll-forming process and springback of the ACC using ABAQUS 2016 software. The built finite element model is shown in Fig. 4, which mainly consists of sheet material and 13 roll stations. The aluminum sheet is 110 mm × 550 mm, of which mechanical properties are shown in Table 2. Due to the lower sheet feeding, the sheet forming is considered a quasi-static process. And the roll deformation can be neglected and treated as an analytic rigid body in the software [12]. Meanwhile, to improve calculation efficiency, the forming line velocity is set to 3 m/s between roll station distance 300 mm, the sheet is divided into 30,360 elements using a thin shell mesh (S4R) of size 0.5 mm × 4 mm, and 5 integration points are used in the thickness direction of each element. Moreover, to better simulate the actual production   with lubrication conditions, the Coulomb friction law is adopted and the friction coefficient between sheet and roll is set as 0.1, as suggested by Paralikas et al. [13].
Considering the asymmetric characteristic of the channel width, the simulation cannot be performed only by taking half of the ACC similar to the case of the single channel, which results in the addition of more boundary conditions. In order to eliminate the effect of excessive friction because of the roll speed difference, the lower roll is kept as the driving roll [14]. Meanwhile, since the calculation accuracy of the springback simulation in Abaqus/Standard is higher than that in Abaqus/Explicit [15], here, the standard algorithm is adopted as well. When the modeling results are imported into the Abaqus/Standard module, all the entities and related contacts are deleted except the sheet material. And only the URY degrees of freedom are retained for the upper and lower rolls. With respect to the sheet, only the URY and URZ degrees of freedom are restricted to avoid its end curling forward and deviation from the forming direction. During this period, all the degrees of freedom of the intermediate nodes of the profile are constrained, while the other regions of freedom are consistent with the description above in order to prevent insufficient springback and non-convergence. Then, the elastic recovery of the profile is analyzed based on the state at the end of quasi-static forming of the sheet.

Transverse bowing defect and non-uniform springback
According to Fig. 5, the TBD can be represented by the maximum angle θ at which the crest web deviates from the horizontal direction. Given that the angles θ 1 and θ 4 are always larger than θ 2 and θ 3 and the size of the TBD is measured only by the sum of θ 1 and θ 4 from the horizontal direction of the outermost two wave crest webs, thus, θ can be calculated by Eq. (1): where ∆Z denotes the displacement difference between the two ends of the wave web along the Z-axis direction and ∆Y is the displacement difference along the Y-axis direction. In order to improve the measurement accuracy of TBD, the present and after studies take the cross section of sheet after springback as the measurement path, which is M (M = 20 mm) from the sheet head (see Fig. 6). The corresponding transverse strain is illustrated in Fig. 7. As can be seen, the transverse tensile strain is not uniformly distributed in the width direction of ACC. There is a significant difference at the bending zone of each channel. The tensile strains of both lateral channels are significantly smaller than those of the middle channel. And the strains on the left side are larger than those on the right side because of their asymmetric channel width. This leads to the nonuniform springback of ACC, and thus, their bowing defect inevitably occurs, which is consistent with the transverse bowing mechanism of ACC above, as shown in Fig. 8.

Forming parameters
The stress-strain behavior of ACC is very complex under roll force [1], which hence leads to very complex defect types and their influencing factors. Here, the simulations are performed for forming variables only including the friction coefficient, inter-station distance, bending angle, roll gap, and number of forming channels according to the actual control requirements. And the simulation and after test values of the variables are provided in Table 3.

Friction coefficient
In cold roll-forming process, the friction is principally used to transmit the roll driving force to the sheet metal. In order to explore its effect on the TBD, the friction coefficient  between the sheet and roll is respectively set as 0.1, 0.15, and 0.2 according to the suggestion in literature [16]. The transverse bowing defects caused by springback under different friction coefficients are shown in Fig. 9a. Different from single channel, although there is a sliding arc surface between the roll and the sheet in roll forming, it is seen that the TBD does not change significantly with the friction coefficient, which means that the friction coefficient has little effect. However, to improve the surface smoothness of the product and to decrease sheet tearing, the friction coefficient should be as small as possible by adding lubricating oil in the actual production.

Bending angle
Since the TBD is not obvious in the middle region of ACC (see Fig. 8), here, the effects of the bending angles are only considered in the groups 2, 3, and 4. Correspondingly, the forming cases of Table 4 are used for the research. The transverse bowing defects under different bending angles of three cases are shown in Fig. 9b. As can be seen, the TBD of case 3 is dropped by 50% compared to case 1. This indicates that increasing the forming angle of the first pass of groups 2, 3, and 4 can effectively ameliorate the bowing phenomenon of ACC. This is mainly because as the bending angle increases, the contact region between the roll and sheet becomes larger. Accordingly, the contraction and slip of the sheet deformation in roll forming is further confined in the transverse section direction. However, the maximum peak strain of the ACC becomes large with the bending angle, which enlarges the risk of profile fracture [1].

Inter-station distance
The determination of the inter-station distance needs to define the length of sheet deformation zone between two passes. Based on the study of Bhattacharyya et al. [17], the minimum deformation zone length (L) can be calculated by Eq. (2): where a is sheet flange length and ∆θ and t are forming angle increment and material thickness, respectively.
To investigate the effect of inter-station distance, the distances 300, 350, and 400 mm are taken for consideration in    2  Case 3   1  1  33  33  33  2  60  60  60  3  81  81  81  4  90  90  90  2  5  43  50  57  6  75  75  75  7  90  90  90  3  8  43  50  57  9  75  75  75  10  90  90  90  4  11  43  50  57  12  75  75  75  13  90  90  90 this section, respectively, as shown in Fig. 10a. It is seen that the TBD decreases obviously with the increase of roll station distance. In other words, increasing the roll station distance can reduce the springback to a certain extent, which conforms to the results of previous numerical studies by Park and Anh [18]. It can be well-illustrated by this reason that with the increase of the inter-station distance, the peak strain of the sheet material gradually decreases. Correspondingly, the forming process tends to be stable. However, enlarging the roll station distance also means that the roll device will take up more space and increase its cost.

Roll gap
Roll gap is regarded as an important process parameter in the assembly of cold roll-forming machine, of which value has a significant impact on the quality of the final product. According to sheet width, roll gaps 0.25, 0.5, and 0.75 mm are used for simulation, respectively. As shown in Fig. 10b, the TBD of ACC can be significantly improved by enlarging the roll gap. This indicates that the transverse non-uniform springback decreases. Interestingly, the simulation result contradicts the conclusion of the previous research by Wiebenga et al. [10]. This is mainly because the previous study is limited to the single channel, of which springback becomes larger with the roll gap. But with respect to the multi-channel springback, the tension strain inhomogeneity between channels and their mutual interference in roll forming can be effectively decreased with the increase of the roll gap, and thus, their non-uniform springback is naturally improved.
It must be noted that the roll-forming aluminum sheet, as a kind of soft material, is more sensitive to the roll gap than the steel. Although the TBD of roll gap 0.75 mm (3t) is reduced by about 48.8% and 33.4% compared to that of roll gaps 0.25 mm (t) and 0.5 mm (2t), the dimensional accuracy of both the bending angle zone and web region of the formed section is greatly affected, and even a large error is produced. Conversely, if the dimensional accuracy is guaranteed under the roll gap 0.25 mm (t), it not only increases the difficulty of roll-forming device assembly, but also the ACC are prone to fracture failure in the bending region during the forming process. Therefore, the roll gap 0.5 mm (2t) becomes the best choice of three gaps.

Number of forming channels
In order to analyze the effect of forming channels on the TBD, the number of forming channels 3, 5, and 7 is used to investigate, respectively, and the corresponding roll flower patterns are shown in Fig. 11. Unexpectedly, the TBD of the product varies noticeably with the increase of the number of forming channels, as shown in Fig. 12a. It is calculated that the TBD of the forming channel 7 increases by about 74.6% relative to that of the channel 3. This is mainly because the increase of the number of forming channels enlarges the non-uniformity transverse springback of the product, correspondingly. The springback difference is more obvious between the edge area and the middle zone of the corrugated channel 7 especially under the asymmetric roll force, which more easily causes both sides' tearing, as shown in Fig. 12b.

Experimental setup
The cold roll-forming experimental setup for ACC mainly composed of roll stations, uncoiler, motor, and control box (see Fig. 13). The sheet is first unwound by the uncoiler, then is passed through the roll stations in succession until the final product is formed. And the inverter motor provides the driving force for the lower rolls during this process.

Verification of simulation results
In order to further verify the reliability of the FEM results, the simulation data by FEM and experimental results of the transverse bowing of ACC under the initial operation conditions (see Table 5) are compared. In addition, the relative error can be defined by Eq. (3): where e represents the relative error, D f means the simulation data, and D e represents the experimental results.
As shown in Fig. 14, the relative error e of the TBD is within the allowable range of 5%. This may be caused by the assembly error of the experimental device. For example, the roll gap is adjusted greater than sheet width 0.25 mm for avoiding sheet tearing in the experiment, which results in smaller flange height and springback, correspondingly. Therefore, the built FEM in Fig. 4 can be used for research of the TBD in roll-forming ACC.

Regression analysis
In order to further measure the effects of above forming parameters on TBD, a linear regression analysis is performed in this section using Minitab 19 software. And the regression equation can be expressed as Eq. (4).
where the values of FC, ID, RG, N, and BAC are shown in Table 3. Table 6 shows the results evaluated by Eq. (4), where P represents the influence degree of each forming parameter; T denotes the positive or negative correlation between the impacts of influencing parameters on the defect. It    can be seen that the value of R-square is 98.51%, which implies the high accuracy of the fitting curve. And thus, Eq. (4) can be used to evaluate the effects of forming parameters on TBD. When the P is greater than 0.05, the effect of parameter is negligible. And the smaller the P, the greater the impact of the parameter, while the larger the absolute value of T, the greater the effect.
To evaluate above forming parameters on the transverse bowing more accurately, their standardized effects are measured by a Pareto chart in Fig. 15. It can be seen that the standardized effect of the number of forming channel is the largest, which is 13.71, while the effect of the friction coefficient is the smallest, which is near − 2.66. Again, it further illustrates that the number of forming channels has prominent influence on the TBD, followed by the roll gap, bending angle, and inter-station distance, while the impact of friction coefficient is negligible, as agreed with the studies by Bidabadi et al. [19].

Optimization of dominant forming parameters
Based on the evaluation results of the regression model, the transverse bowing of ACC is affected mainly by the forming parameters of the channel number, roll gap, bending angle, and inter-station distance. Thus, to improve the TBD, the dominant forming parameters are optimized under the given number 7 of the forming channels. Here, the optimized bending angle case is adjusted from 1 to 3, the roll gap is corrected from 0.25 to 0.5 mm, and the inter-station distance is expanded from 300 to 400 mm. Figure 16 shows the profile of roll-forming ACC before and after parameter optimization. It can be seen that the quality of product has been significantly improved after parameter optimization.
In addition, the optimized θ decreases by about 88.5%, 89.9%, and 89.8% compared to their initial values for simulation, experiment, and regression prediction by calculation, respectively. And their relative errors are both within 10%, as shown in Fig. 17. This not only proves the reliability of   16 Asymmetric corrugated channels a before and b after optimization the regression model, but also indicates that the TBD can be controlled by optimizing the dominant forming parameters.

Conclusions
(1) The transverse bowing defect of ACC arises from nonuniform springback in the transverse direction after the sheet unloading by analyzing its mechanism of production. (2) Different from single channel, the transverse springback of ACC decreases with the roll gap increasing, mainly because its non-uniformity is greatly improved with the increase of channel number and roll gap. (3) Evaluated by linear regression equation, the number of forming channel has a significant effect on the transverse bowing defect of ACC, followed by the roll gap, bending angle, and inter-station distance, and the friction coefficient has the least. (4) For given ACC, the transverse bowing defect can be controlled by optimizing the dominant forming parameters of roll gap, bending angle, and inter-station distance.