Springback and longitudinal bow in chain-die forming U and hat channels

Springback and longitudinal bow are two major shape defects in gradual forming processes like chain-die forming and roll forming. In this study, the springback and longitudinal bow of AHSS in chain-die forming of hat and U profiles are investigated through experiment and finite element simulation. The disparity of springback along the longitudinal direction and longitudinal bow signifies complex deformation during chain-die forming. Based on finite element simulation of the chain-die forming, the gradual forming process realized by sequential die blocks causes bending and reverse-bending in the web area and redundant deformation of the sheet metal, which leads to nonuniform bending moment and accumulated stresses along the longitudinal direction. At the same time, the redundant deformation will also result in the longitudinal strain on the edge, while the downhill characteristic of chain-die forming reduces the maximum longitudinal strain on the edge and introduces longitudinal strain on the web. The nonuniform longitudinal strain distribution along the transversal direction causes disparate downward and upward bowing longitudinal bow for chain-die formed AHSS hat and U profiles. By the established model, the disparate springback and longitudinal bow behavior can be quantitatively evaluated, which is helpful to chain-die forming process design.


Introduction
Advanced high-strength steels (AHSS) have found more extensive application in automotive manufacturing for weight reduction and crash safety [1]. The development of highstrength steels posed great challenges to stamping, e.g., shear fracture when being draw-bended through a tight radius [2]. Gradual forming techniques, like roll forming and chain-die forming, became attractive to AHSS product manufacturing due to their incremental deformation nature that enables the forming of small radius [3]. Roll forming and flexible roll forming progressively deform a steel strip into products with required cross-sectional profiles by a series of rolls installed at the tandems along the longitudinal direction [4,5]; sheet metals are gradually formed through incremental bending, and thus a small bend radius could be realized [6]. In contrast with roll forming, chain-die forming [7] increases the forming length by enlarging the radii of the rolls to tens of meters, and realize gradual forming by a series of discrete die blocks attached to a rotational track board, as shown in Fig. 1 [8,9].
Channel products with constant and variable cross-sections can be formed by chain-die forming technique. Springback and longitudinal bow are two typical shape errors in gradual forming processes, especially for manufacturing AHSS products considering their large springback and complex material behaviors [10,11].
Springback, referring to the shape changes after removing forming loads due to elastic recovery, has been always a critical problem in forming AHSS sheets. Springback in stamping and roll forming has been extensively studied in the recent two decades. In stamping, the main characteristics of springback for AHSS products are the sidewall curl and large springback angle [12]. Badr et al. [13] compared the springback of roll-formed channels with V-die-bended ones and found roll forming leads to a lower level accumulated stresses and thus smaller springback. When a channel with a uniform crosssection is roll-formed, nonuniform springback along the longitudinal direction can be observed. Halmos [14] concluded that in the roll forming process, the amount of springback at the entry and exit ends of the pre-cut metal sheet is larger than that of the middle area, which is also known as end flare, referring to the outwards curling at the front and tail ends. The nonuniform springback engendered at the front and tail ends of roll-formed products was attributed to lack of constraints at two ends of the strip, i.e., not fully clamped by neighboring rolls. Bui and Ponthot [15] established a 3D finite element model to simulate a cold roll forming process. Both the experiments and numerical simulation signified that the front end of roll formed precut sheet metal shows flare out deformation. Wiebenga et al. [10] proposed a robust optimization method to compensate shape error and minimize the sensitivity to variation of material properties in the cold forming process. It was found springback angle varies along the longitudinal direction in roll forming DP780 V-section profile, and end flares were observed in both experiment and simulation. Abvabi et al. [16] studied the influence of residual stress on the longitudinal bow and end flare for the roll-formed DP780 V-section profile.
Longitudinal bow is defined as the vertical deviation of web area along the longitudinal direction and is also considered as an accompanied deformation of springback [17]. Many efforts have been made to study longitudinal bow in the cold roll forming processes and mainly focused on the relationship between the longitudinal bow and redundant longitudinal strain introduced in the forming processes. Jeong et al. [18] designed an under-rail product and simulated the roll forming process of a profile, and found that the flower pattern results in a larger longitudinal strain and also leads to a severe longitudinal bow. Shirani Bidabadi et al. [19] studied the longitudinal bow defects in cold roll-formed U profile, and found that the difference between longitudinal strain induced in the flange and web causes the longitudinal bow in cold roll formed products. Meanwhile, they proposed a downhill strategy as an effective method to eliminate the longitudinal bow. Woo et al. [11] studied the characteristics of the longitudinal bow in flexible roll-formed U profiles using three different blank sheets, and the longitudinal bow defects were ascribed to the transversal nonuniformity of the longitudinal strain. Badr et al. [20] studied the influence of two roll forming approaches, constant bend radius and constant arc-length methods, on springback and longitudinal bow of Ti-6Al-4V sheet. The results showed that the constant bend radius forming method leads to less longitudinal bow and less permanent longitudinal deformation in the edge. Park et al. [5] proposed an incremental counter forming (ICF) process to reduce the bowing defect in cold roll forming. The experimental results showed effectiveness of the proposed method.
A few studies have been carried out on simulation and experiments of the chain-die forming process. Sun et al. [21] studied the longitudinal strain development on the edge of the plate during U profiles chain-die forming processes and found that peak longitudinal strain varies inversely with the virtual roll radius. Li et al. [22] compared the chain-die forming and roll forming process of a U profile. Smaller longitudinal strain and less obvious longitudinal bow defects is achieved by chain-die forming due to a smoother transitional surface. Li et al. [8] proposed an analytical model for rapid prediction and compensation of the U profile chain-die forming process. Qian et al. [23] established a finite element model for the multi-stand chain-die forming process of a U channel and investigated the influence of material strength, sheet metal Fig. 1. Schematic diagram of the chain-die forming process [9] thickness, vertical forming gap, and effective virtual roll radius on springback for chain-die formed AHSS U profiles. Recent chain-die forming experiments showed that there is significant nonuniformity in springback along the longitudinal direction. Meanwhile, disparate longitudinal bow behavior was observed for hat and U profiles. In the previous studies on chain-die forming, the focus was mainly on the sectional springback and the development of longitudinal strain. However, the characteristic gradual deformation process and the cause of the longitudinally discrepant springback have not been well investigated. A more detailed research regarding the deformation mechanism is necessary for well understanding of chain-die forming process and to ensure sound products.
The objective of this work is to investigate the characteristic deformations and product shape defects in the AHSS chain-die forming, focusing on springback, and longitudinal bow. The rest of the paper is organized as follows. In Section 2, chain-die forming experiments are performed. In Section 3, a finite element model is established. In Section 4, the nonuniform springback along the longitudinal direction and the disparate longitudinal bow behaviors are analyzed and numerical models are verified with experimental results. In Section 5, three characteristic deformations in chain-die forming are discussed. Moreover, the cause of nonuniform springback and longitudinal bow are probed. Section 6 gives conclusions.

Experiments
Chain-die forming experiments on the hat and U profiles with different lengths and corner radii are conducted. Figure 2 shows one case of the hat and U products, with the desired bending angle equal to 90°for both sidewall and flange. The longitudinal length L is set to be 100 mm, 220 mm, and 420 mm, and the corner radius R is set to be 3.6 mm and 5.0 mm, respectively, as listed in Table 1.
The material considered in this study is a 1.2 mm thickness QP1180 (QP refers to quenching and partitioning) sheet acquired from Baosteel. The tensile and compression-tension (CT) tests were previously carried on this alloy [22]. The width of the specimen tested in the forming experiments is 132 mm for the hat profiles and 86 mm for the U profiles. Figure 3 shows the lab-scale chain-die forming machine, which mainly consists of track boards, rollers, chains, and die blocks [22]. The radius of the track board R v is 35 m. The discrete die blocks that are mounted to the chains are driven by the gear [22] and the linear velocity of the die blocks is set to 70 mm/s. Figure 3 also shows the resistant strain gage (BX120-5AA), the National Instruments (NI) cDQA-9171 and NI 9237 used to record the longitudinal strain evolution developed on the flange edge of the hat profile (middle position, 210 mm from the lead end for 420 mm-length hat profile). Two strain gages are firmly attached on both sides of the plate and the average value of the longitudinal strain for the two positions can be therefore obtained. The edge of the strain gage is about 1.5 mm from the flange edge. A reference test piece is added to improve accuracy. The ATOS Triple Scan noncontact structured blue light 3D scanner is used to acquire the geometry of the formed samples. Chain-die forming experiments are carried out at least three times to ensure repeatability.

Finite element modeling
A finite element model for the chain-die forming process is established by using ABAQUS, as shown in Fig. 4, and a half model is considered due to symmetry. The plate is assumed to have the same dimensions as in the experiments and discretized by 5 layers of the eight-node linear brick element with reduced integration (C3D8R). A fine mesh with a size of 0.55 mm is evenly assigned for the regions that experience bending and unbending deformation along the transversal direction. The mesh size for other areas is 2.66mm. Twelve pairs of die blocks with a length of 36 mm are used. The die blocks are modeled as discrete rigid surfaces by 4node 3-D bilinear rigid quadrilateral (R3D4) elements. The discrete die blocks are assembled on a track board with a radius of 35 m. In the gradual forming process, the sheet is first set stationary and the die blocks are then assigned with the angular velocity of 0.002 rad/s around the X-axis (opposite in direction for top and bottom die blocks). Note that, other degrees of freedom for the die blocks are fully constrained except the rotation around the X-axis. The surface-tosurface contacts are assigned between the sheet metal and the die blocks. The Coulomb friction model is employed and the friction coefficient between the sheet and die blocks is set to 0.1 [22]. The sheet metal moves forward driven by the friction force and is gradually deformed at the same time. After forming, the top and bottom die blocks are sequentially removed from the lead end to the tail in the subsequent springback stage. Finally, the formed sample is released by removing the final pair of die blocks until the accumulated stress/strain of the formed sample reach a steady state.
The von Mises yield criterion and Chaboche nonlinear kinematic hardening model with 3 back stress components are used to describe the hardening behavior of the QP1180 materials. The model can be described as where σ is the stress tensor; X is the back stress denoting the translation of yield surface; σ ′ and X ′ are deviator stress and deviator back stress respectively. The back stress X can be obtained by summing the three back stress components X i , as in Eq. (2), where i is the index of components. Back stress components X i can be obtained from integration on back stress increment rate componentẊ i , defined by Eq. (3), where c i and γ i are kinematic hardening constants respectively;ε p is plastic strain rate;ṗ is accumulated equivalent plastic strain rate.
Thus, the evolution of X i is described as [22] where ν is the flow direction; X i0 and ε p0 are the magnitude of X i and ε p ; ε p is the equivalent plastic strain. Meanwhile, the evolution of isotropic hardening is described as where φ is the size of yield surface in stress space; σ 0 is the initial yield stress; σ iso is the isotropic hardening stress, which can be expressed as [22] where Q is the saturation stress; b is the saturation rate of isotropic hardening stress; p is accumulated plastic strain. The coefficients from the previous study [22] are adopted and are listed in Table 2. A saturated elastic modulus of 180.5 GPa calculated from the LUL test [8] is used in this study.

Springback in chain-die forming
Chain-die formed hat and U test samples show different nonuniform springback characteristics. In this section, the springback nonuniformity along the longitudinal direction is quantitatively studied. Figure 5 illustrates the strategy for springback measurement. Along the longitudinal direction, two, three, and six cross-sections are selected to measure their configurations for the 100 mm-length, 220 mm-length, and 420 mm-length profiles, respectively. Section 1 is defined as 5 mm from the lead end of the part. Section 2 for 100 mm-length parts (Section 3 for 220 mm-length parts, and Section 6 for 420 mm-length parts) is defined at 5 mm from the tail end. The interval distances between the two neighbor sections are 90 mm, 105 mm, and 82 mm for 100 mm-length, 220 mm-length, and 420 mm-length profiles, respectively.
To quantitatively describe the springback characteristics of chain-die formed hat profiles, the following four springback indicators are employed here, as illustrated in Fig. 6(a). θ 1 the springback angle between web and sidewall. θ 2 the springback angle between sidewall and flange. ρ 1 the curvature of web curl. ρ 2 he curvature of the sidewall curl. Figure 7 compares the measured springback configurations of different cross-sections from the lead end to the tail end for the hat profile (L=420mm, R=5.0mm). Two distinctive features of springback, sidewall curl, and web curl are observed. Web curl is the curvature formed in the web and also refers to crossbow defects in the roll forming process [14]. As shown in Fig. 7, the sectional springback configuration varies along the longitudinal direction. The lead end exhibits a large springback, and the tail end has a curved web. With the increase of longitudinal length, the sidewall gradually flares further inward, and the web area tends to be more curved.

Springback of hat profile
The measured θ 1 , θ 2 , ρ 1, and ρ 2 with respect to the distance from the lead end for hat profiles of different dimensions are listed in Table 3. The variation of relative springback angle Δθ 1 and ρ 1 are shown in Fig. 8, where the relative springback angle Δθ 1 are defined as θ 1 -90, and the longitudinal distance from the lead section is normalized by the distance between the lead end and tail end of the profile. Figure 8(a) shows that for hat profiles, the springback angle θ 1 decreases with increasing distance from the lead end, and the web-curl curvature ρ 1 exhibits a different tendency, as shown in Fig. 8(b). By comparing θ 1 and ρ 1 between hat profiles with different Table 2 The fitted coefficients of the Chaboche model for QP1180 [22]  Note: c i and γ i (i=1,2,3) are kinematic hardening constants; Q is the saturation stress; b is the saturation rate of isotropic hardening stress lengths, it is found that with the increase of profile length, the deviation of the relative springback angle Δθ 1 and web-curl curvature ρ 1 between the lead and the tail end increases, as shown in Fig. 8. The nonuniform springback configurations shown in Fig. 7 are also confirmed by Fig. 8. Figure 8(a) also shows that the springback of the hat profiles with a 3.6 mm corner radius is smaller than that with a 5.0 mm corner radius, i.e., a tighter die radius generally results in a reduced springback angle [24]. Meanwhile, with the increase of the distance from the lead end, the variation of θ 2 and ρ 2 are not noticeable, in contrast with θ 1 and ρ 1 . Figure 9 compares the measured springback configurations of the cross-sections of interest for chain-die formed U profile (L = 420 mm, R = 3.6 mm). Similar to the hat profile, discrepancies are found between the springback angle for different cross-sections. However, the web is much flatter than the hat profile and does not show obvious variation along the longitudinal direction. Two springback indicators, θ 1 and ρ 1 , are adopted, and the measurements for different sections are shown in Fig. 10 and listed in Table 4. Figure 10(a) shows a downward trend for Δθ 1 . Meanwhile, the springback angles at the tail end for the 420 mm-length U profiles are relatively larger, which indicates an end flare. Figure 10(a) also shows that Δθ 1 decreases with increasing longitudinal distance from the lead end for U profiles with a length of 100 mm or 220 mm. For the U profile of 420 mm length, θ 1 decreases firstly and then increase, signifying a different nonuniform springback behavior compared with the hat profiles. Besides, the web-curl curvature ρ 1 and its variation for U profiles shows significant difference from the hat profiles, with no obvious discrepancies being found between the cross-sections measured, as shown in Figs. 9 and 10(b). Figure 11 compares the calculated springback configurations with experimental measurement of the 420 mm-length profiles, which exhibit the largest nonuniformity. Figure 11 shows that the calculated configurations of lead end and tail end are consistent with the measured springback configuration for both the hat and U profiles. The disparities between the lead end and tail end are well captured. The variation of relative springback angle Δθ 1 with respect to longitudinal distance from the lead end is presented to illustrate the nonuniform springback characteristics. The coordinates for the crosssections of interest are extracted and the relative springback angles are obtained following Fig. 6. Figures 12 and 13 compare the variation of calculated and measured relative springback angle Δθ 1 about longitudinal length for the hat and U profiles respectively. Figure 12 shows that the calculated Δθ 1 for hat profiles are generally smaller than experimental values. The deviation between experimental results and numerical results is denoted with <Δθ 1 >, which is defined as

Experimental verification
where Δθ 1i and Δθ′ 1i are measured and calculated Δθ 1 , respectively, for section i, i=1, 2, …,6. The numerical model underestimates the springback angle by 1.39°and 2.07°for the 420 mm-length hat profiles with a corner radius of 5.0 mm and 3.6 mm, respectively. Figure 12 also shows that the variation trend of the relative springback angle with longitudinal distance agrees well with experimental results for both profiles. Δθ 1 decreases with longitudinal length. Figure 13 shows that the deviation between calculated Δθ 1 and experimental results for chain-die formed U profiles are smaller than hat products, and the average deviations are 1.87°a nd 0.19°for the U profiles with a corner radius of 5.0 mm and 3.6 mm, respectively. Meanwhile, a similar variation of θ 1 with respect to longitudinal length is observed in both experimental and numerical results. Δθ 1 decreases firstly and then increases, resulting in a concave curve.

Longitudinal bow in chain-die forming
Chain-die formed hat and U test samples show disparate longitudinal bow behaviors. In this section, the experimental results of longitudinal bow are presented and the numerical model is verified. Figure 14 shows the measured longitudinal bows for chain-die formed 420 mm-length U and hat profiles. The extracted curves are rotated to make two ends of the profiles be aligned with each other. Figure 14 shows that chain-die formed U profiles show an upward longitudinal bow, while hat profiles have a downward longitudinal bow.  Figure 16 compares the measured and calculated maximum deviation of the longitudinal bow (ΔH in Fig. 14) for the hat and U profiles with different longitudinal lengths and corner radius and specific values are listed in Table 5. The calculated ΔH matches well with the experimental result for both trends and values, and for the 420 mm-length profiles, the maximum relative error is 16.21%. It can also be found that the maximum deviations of longitudinal bow always increase with the increase of profile length and decrease of corner radius, but the influence of the corner radius is limited.

Deformation analysis
Chain-die formed hat and U test samples show different nonuniform springback characteristics and disparate longitudinal bow behaviors. In this section, the characteristic forming mechanisms leading to the nonuniform springback and

Bending and reverse-bending deformation
The deformation of sheet metal in chain-die forming is characterized as a gradual forming process. Two representative cross-sections, corresponding to sections 1 and 6 in Fig. 5, are selected to illustrate the discrepant deformation process for different cross-sections along the longitudinal direction in the hat profile chain-die forming process, as shown in Fig.  17. Three sequential forming states are plotted and compared in terms of section configuration, where R L1 represents the web curl radius of the lead end at the forming state I, and R T2 denotes the radius of web curl for the tail end at the forming state II. R L1 is much larger than R T1 , indicating a curved web at the tail end. With the proceeding of forming, the curved web is gradually deformed into the desired geometry. Figure 17 shows that the web area near the lead end barely experiences redundant deformation, while the web of the tail end is subjected to bending and reverse-bending deformation. Figure 18 schematically depicts the bending and reversebending deformation process. Due to the gradual forming nature and the less constrained boundary condition in the chain-die forming process, the tail end is firstly deformed into a curved geometry with a radius of R 1 under the action of a bending moment M ∞→R 1 at the early stage of the forming process. Then the curved web is reversely bended into the desired geometry, i.e., a flat web, by taking advantage of a reverse-bending moment M R 1 →∞ . Finally, the formed part is released and a bending moment M ∞→R 2 is engendered to balance the reverse bending moment M R 1 →∞ , and the bending moment M ∞→R 2 results in an additionally inward springback. Due to the bending, reverse-bending, and subsequent inward springback, the web area near the tail end for hat profile is deformed complicatedly into a flare-in geometry with an obvious curvature, which often refers to the crossbow defect that occurs in roll forming processes.     Figures. 17 and 18 show that the bending and reversebending deformations at the web are responsible for small Δθ 1 and large ρ 1 at the tail end of the hat profiles observed in the experiment, especially in the case of channels with 420 mm length, where the reverse-bending effect is enhanced compared with 220 mm length and 100 mm length ones. A similar conclusion was drawn by Phanitwong and Thipprakmas [25] in their study on the springback of a Ubending process. The deformation for a single cross-section in the chain-die forming process shown in Fig. 18 is similar to a U-bending process, e.g., the inward springback is caused by the bending and reverse-bending deformation in the web area.
To quantitatively illustrate the discrepant bending process along forming direction, seven points located on the symmetrical line of the web (on the outer surface), P1 to P7, are selected and the logarithmic transversal strain evolutions are traced and presented in Fig. 19. With the increase of longitudinal distance, the maximum value of transversal strain increases, and the maximum transversal strain at P7 is three times as large as that at P1, indicating an enhanced bending deformation and a larger bending moment M ∞→R 2 depicted in Fig. 18. Therefore, the main reason for the nonuniform springback of the chain-die formed hat profile is the bending and reverse-bending deformation in web area in the gradual forming process. Besides, there is also a discrepancy in the time when the transversal strain reaches the maximum for different points. The maximum transversal strain turns to be delayed with the increase of longitudinal length, characterizing the gradual nature of chain-die forming.

Redundant deformation
The curvature of web curl for the chain-die formed U profiles shows no obvious discrepancies with respect to longitudinal length, as shown in Fig. 9. Meanwhile, the variation of springback angle θ 1 is different from that of hat profiles, and both ends of the chain-die formed U profiles of 420 mm length show a flare out deformation, indicated by a large springback angle θ 1 . Figure 20 shows the U profile chain-die forming process, where redundant deformation engenders because the edge of the plate is geometrically necessary to travel a longer distance compared with the central line [21]. The redundant deformation basically consists of four forming stages, represented by four dashed lines AB, CD, EF, and GH, respectively, where A, C, E, and G are points located at the edge of the plate, and B, D, F, and H are points at the corner. The plate is bended concavely at line AB and then experiences convex deformation at line CD. Finally, the deformed area is reversely bended through the line EF, due to geometry continuity, to the desired geometry represented by line GH. During these processes, longitudinal stress and in-plane shear stresses are introduced, and the redundant deformation is similar to the cold roll forming process described by [26].
The plate is gradually bended into the desired profiles in the chain-die forming process, and the accumulated longitudinal stress and shear stress will be released once the upper and lower die blocks are removed. Since the die blocks are engaged one pair after another, the accumulated stress is also released sequentially. Therefore, stresses at the lead end are released while the tail end can be still fully constrained, leading to the fact that the accumulated stress varies from lead end to tail end before removing corresponding die block pairs. Figure 21 illustrates the stress releasing process at the lead and tail ends for the U profile of 420 mm length, where the die blocks are removed in sequence. As shown in Fig. 21(a), at the lead end, the accumulated longitudinal stress is negative on the inner surface and positive outside before removing the die blocks, accompanied by a bending moment M z around the yaxis. The part subjected to tension at the end of the forming process tends to shrink after unloading, while the part subjected to compression tends to be expanded [27]. After removing the die blocks, the release of bending moment M z will introduce an opposite bending moment M' z , resulting in flaring out deformation. In contrast, at the tail end, as depicted in Fig.  21(b), the accumulated longitudinal stress is positive inside and negative outside, and the stress releasing leads to flaring in deformation.
In addition to the longitudinal stress, the in-plane shear stress also contributes to the nonuniform springback for chain-die formed U products. As depicted in Fig. 20, bending and reverse bending, which is dominant in the area of CDEF, will introduce in-plane shear stress in the transversallongitudinal direction, while the shear stress along the thickness-longitudinal direction is almost zero considering the small thickness [26]. The in-plane shear stress in transversal-longitudinal direction will lead to the twisting moments M zy and M yz [26], as depicted in Fig. 22(a). When removing the die blocks gradually, opposite twisting moments are introduced, resulting in flaring in deformation at the lead end and flaring out deformation at the tail end, as illustrated in Fig. 22(b).
In summary, redundant deformation causes the accumulated nonuniform longitudinal stress and in-plane shear stress, and release of the accumulated nonuniform longitudinal stress leads to flare out at the lead end and flare in at the tail end, while the release of the accumulated in-plane shear stress has an opposite effect. Their joint action determines the overall springback configuration.

Downhills deformation
In the chain-die forming process, the redundant deformation will result in the longitudinal strain on the edges, as shown in Fig. 20. However, the longitudinal strain developed on the web is less investigated. In this section, downhills in the single-pass chain-die forming process are discussed.
Downhill is usually used in roll forming of electricresistance-welded pipes [14] and U channels [28] to eliminate the nonuniform longitudinal strain, where the descent of the  pass line-height is optimized to minimize the edge stretch. In the downhill roll forming lines, the bottom of the section does not travel in a straight plane, and at the same time, the edge of the plate travel a short deformation distance compared to the regular forming line, which therefore engenders a least amount of redundant strain and stress [14]. The downhill strategy is one of the most important factors that affect the longitudinal bow in cold roll forming [19]. Qian et al. [23] studied the downhill in a multi-stand chain-die forming process, and it can also be classified as the downhills between forming passes, which is similar to the downhills in roll forming processes, usually achieved by elevation of forming stands. The downhill discussed in this study refers to the unique deformation in one single pass and is one of the inherent characteristics for chain-die forming. Figure 23 depicts a random forming posture in U profile chain-die forming process. The virtual rolls of chain-die forming consists of multiple discrete die blocks, and the gaps between the top and bottom die blocks are gradually reduced with synchronous motions of the dies [21]. It was proved that the strain paths of the points located on the edge of the plate are similar from the initial engaged position to the exit end [29]. The deformed configurations for different cross-sections along the longitudinal direction can be taken as those at successive roll-stands in roll forming. Comparing the deformed sections 1 to 4 shown in Fig. 23(b), it is found that a downhill phenomenon is naturally existing in the single-pass chain-die forming process. Fig. 10. The variation of a relative springback angle Δθ 1 and b web-curl curvature ρ 1 with regard to the distance from the lead end for U profiles Fig. 11. Comparison of the springback configurations between the lead end (Section 1) and tail end (Section 6) for a hat profile (L = 420 mm, R = 5.0 mm), and b U profile (L = 420 mm, R = 3.6 mm) The webs of the extracted cross-sections are further compared in Fig. 23(c) by aligning their symmetrical line and horizontal base. It is shown that the edge of the plate travels a vertical distance of ΔH 1 . However, in the chain-die forming process, as shown in Fig. 23(b), a downhill of ΔH 3 engenders, leading to a shorter increment of ΔH 2 . Paralikas et al. [28] proved that maximum longitudinal strain on the edges decreases with increasing downhill displacement. Thus, the existence of downhill has two significant effects. On the one hand, the longitudinal strains developed on the edges caused by redundant deformation are reduced, and on the other hand, longitudinal strains on the web are introduced. Figure 23 indicates that downhills are naturally existing in chain-die forming processes and result in longitudinal strain on the web. Chain-die forming machine consists of a pair of track boards which usually have a very large radius R v , as shown in Fig. 1. To give an insight to the influence of virtual roll radius on the maximum longitudinal strain on the web, the downhill in chain-die forming is further discussed here. The virtual roll radius shown in Fig. 4 is set to vary from 10000 mm to 45000 mm, with an interval of 5000 mm. The maximum longitudinal strain developed on the web for different virtual roll radii are compared in Fig. 24. It shows that for both hat and U profiles, the peak longitudinal strain on the web decreases slowly with the increase of virtual roll radius.   Figure 25 schematically depicts the downhill in single-pass chain-die forming process, where S is the stroke of tools. Once R v and S are given, a downhill displacement of DH is therefore determined as The angle between the trajectory of the plate and the horizontal line is defined as the downhill angle in the chain-die forming, and can be calculated as where L n is the nominal deformation distance and can be calculated as  Note: σ (%)-relative error between calculated and experimental results; <σ> (%)-average relative error between calculated and experimental results By substituting Eq. (2) and Eq. (4) to Eq. (3), the downhill angle θ can be calculated as The maximum longitudinal strain developed on the web can be reflected by the downhill angle θ. Equation (5) shows that θ increases with the decreasing of virtual roll radius R v . An increased θ signifies an enhanced downhill deformation, and will further lead to an increased maximum longitudinal strain on the web for both U and hat profiles, which is also confirmed by numerical results, as shown in Fig. 24. Therefore, downhill deformation is closely related to the longitudinal strain developed on the web.
It has been proved that chain-die forming can achieve a smaller redundant longitudinal strain on edge compared with cold roll forming processes, owing to the elongated deformation length [21] and smoother transitional surfaces [22]. This study further clarifies that another factor that contributes to the reduction of longitudinal strain is the downhill nature in chaindie forming.

Contrast between the hat and U profile
Three characteristic deformations in chain-die forming are discussed in previous section. In this part, the deformations of hat and U profiles are compared to investigate the effect of the flange on the chain-die forming process. Figure 8 shows that for hat profiles, Δθ 1 increases with length, and synchronous increase of ρ 1 indicates a strong reversebending effect. In contrast, for the U profiles, as shown in Fig. 10, ρ 1 does not remarkably vary with channel length. Thus, the reverse-bending effect is very limited for U profiles.  Figure 26 compares the chain-die forming processes of the hat and U profiles. In the hat profile chain-die forming process, the top die blocks prevent the plate from bending upward, while the sidewalls of U profiles can move freely when bending the corners. Thus, for the hat profile chain-die forming process, redundant deformation is depressed due to the constraints of top die blocks, which at the same time lead to an enhanced bending and reverse-bending deformation at the web area. Figure 27(a) compares the evolution of longitudinal stress developed at the edge points A 1 and A 2 (inner side) shown in Fig. 26 for hat and U profiles, respectively. The maximum accumulated stresses prior to springback developed on the edge of the hat profile are much less than those on the edge of the U profile, even though the sidewall of the hat profile is 10 mm larger than the U profile. Figure 27(b) shows the evolution of in-plane shear stress developed at points B 1 , B 2 , on the inner, outer surfaces of the sidewall for the hat profile, and the counterparts, points B' 1 and B' 2 , on the U profile. Figure 27(b) shows that the difference of shear stress magnitude between the inner and outer surfaces of the U profile is obviously much larger than those of the hat profile. In general, Fig. 27 signifies that the influence of redundant deformation is very limited for hat profiles compared with U profiles, and the nonuniform springback along longitudinal direction for hat profiles should be ascribed to the bending and reversebending deformation on the web. However, for the U profiles, redundant deformation is the major cause of nonuniform springback. Figure 28 compares the nonuniformity of longitudinal strain distribution for path A'-A (A 1 , A 2 , A' 1 , and A' 2 are shown in Fig. 26) for the moment during the forming process and after springback respectively. Figure 28(a) shows that during the U profile forming process, a tensile longitudinal strain is introduced on the edge, and a significant difference can be observed between point A' 2 and point A 2 . For the hat profile, both the edge and web experience tension, and the difference between edge and web is much smaller. More importantly, longitudinal plastic strain distribution along section A'-A tends to be different for hat and U profiles, as shown in Fig.  28(b). The decreased and increased longitudinal strain variation from the web to edge indicate a relatively elongated and compressed edge for U and hat profiles, respectively, which would lead to upward and downward longitudinal bow respectively.

Disparate longitudinal bow
As aforementioned, the longitudinal strain induced on the edges is limited in the hat profile chain-die forming process, corresponding to the limited redundant deformation due to the constraints of top die blocks. The longitudinal strain developed on the edge of the hat profile is much smaller than those for the U profile. Meanwhile, the downhill in the hat profile chain-die forming process introduces a relatively larger longitudinal strain on the web (as shown in Fig. 24). Furthermore, based on the comparison of the nonuniform longitudinal strain distribution along the path A'-A , as shown in Fig. 28, it is reasonable to expect a downwards longitudinal bow in chaindie formed hat profiles after springback.
The difference in longitudinal plastic strain between points A and A' (δε = ε nodeA -ε nodeA' ) for the hat and U profiles with different length and corner radius is presented in Fig. 29. It shows that δε remains positive for U and negative for hat profiles after springback, signifying elongation on edges and web, respectively. Therefore, the disparate nonuniformity of longitudinal strain for hat and U profiles should lead to a downward and upward longitudinal bow, respectively.
In general, two co-existing longitudinal strains determines the behavior of longitudinal bowing in chain-die forming processes, the longitudinal strain developed on the edges of the plate (ε edge ), caused by redundant deformation, and the longitudinal strain introduced on the web (ε web ) due to the downhill deformation, respectively. If the value of ε edge is greater than ε web , an upward longitudinal bow is expected; otherwise, there will be a downward longitudinal bow. The measured longitudinal bow behaviors (upward or downward, shown in Fig. 14) for both the U and hat profiles are consistent with the calculated longitudinal strain difference between edge and web. The experimental results confirm that the nonuniform distribution of the longitudinal strain caused by redundant deformation and downhill deformation is responsible for the disparate longitudinal bow behaviors in the chain-die forming. Since the downhill naturally exists in chain-die forming processes, and the longitudinal bows are ascribed to the nonuniformity of longitudinal strain, the longitudinal strain developed on the web and edges should be simultaneously taken into consideration in future process design of chain-die forming.

Conclusions
In this study, shape errors in chain-die forming of AHSS channels are numerically and experimentally studied. Hat and U profiles with different lengths are formed to illustrate the characteristic mechanism of chain-die forming and its effect on springback and longitudinal bow. Three characteristic deformations are discussed by FE simulation. Specifically, downhill in chain-die forming are qualitatively and quantitatively investigated, and the longitudinal strain developed on the edges and web are studied. The following conclusions can be drawn.
(1) Chain-die formed AHSS hat and U products exhibit different inhomogeneous springback along the longitudinal direction and disparate longitudinal bowing behavior. By  the established model, both the springback and longitudinal bow behavior for chain-die formed AHSS products can be determined. Future process design should take the nonuniformity in springback and longitudinal bow into account simultaneously. (2) Two major causes lead to the nonuniform springback in chain-die forming, the bending and reverse-bending deformation and the redundant deformation. Both effects result from the gradual forming process. For the hat profile, the influences of the redundant deformation are limited and the bending and reverse-bending deformation on the web is dominant. For U profile, nonuniform springback is mainly attributed to the redundant deformation.