Introduction

In roll forming, profiles with simple and complex geometries are produced by successively arranged rotating rolls in a continuous process [1]. Due to the process characteristics, redundant deformations such as longitudinal bending, longitudinal elongation and shear occur in addition to the transverse bending required for profile forming. A superposition of the redundant deformations to the transverse bending influences the shape of the product and the occurrence of profile defects, such as vertical bow, horizontal bow and twist as well as strip edge waviness [2]. During profiling, the strip edge travels a longer distance compared to the bending edge. When the longitudinal strain exceeds the elastic yield point, the mentioned profile defects occur. The formation and shape of these defects depend on the distribution of the longitudinal strain [1]. In particular, the difference in longitudinal strains between the flange and the bottom causes longitudinal bow of the profile [3, 4]. Qualitative relationships of strain distribution and specific profile defects are known from [1]. Quantitative relationships between the strains and defects have not yet been researched. Nevertheless, numerous studies are known from the literature, which show the influencing parameters on longitudinal strains and also on profile defects.

Numerical and experimental investigations known from the literature show that both material and process properties influence profile defects. Material properties such as yield strength are studied by Lindgren [5] and Han et al. [6]. While in studies of [5] the peak longitudinal strain decrease with increasing yield strength, [6] establishes an increase in the peak longitudinal strain.

Abeyrathna et al. [7] confirm that as yield strength increases, the longitudinal bow decreases and derive a relationship with measured process force [8]. For two steel grades with different yield strengths, [9] shows that the longitudinal strain peaks and longitudinal bow are lower for the material with the higher yield strength.

Other parameters influencing longitudinal strains are bending angle [10], inter stand distance [11], bending angle increment [12], sheet width [13], flange length [12, 14], roll diameter [12], sheet thickness [12, 14], and feed rate [14]. Bui and Ponthot [11] use METAFOR code to simulate roll forming of symmetrical profiles. The results show that the longitudinal strain increases as the inter stand distance decreases, whereas the roll velocity and friction have no effect. Zhu et al. state that as the flange length increases, the longitudinal strain initially increases and then decreases after a certain value [12]. For increasing thickness, the longitudinal strain increases, as well as for larger bending angle increments [12]. Abeyrathna et al. show in [14] that for large values, the inter stand distance has little effect on longitudinal strain and the longitudinal bow. For increasing flange lengths, the strain and bow decrease in [14]. These investigation results are partly in agreement with the results of [12] since there, too, a decrease of the longitudinal strains with increasing flange lengths can be observed.

Experimental studies with varying process parameters to improve process understanding and to investigate the effect of horizontal or vertical misalignment on longitudinal strains are conducted in [15]. In addition to numerical and experimental investigations, [16] performed artificial neural network and regression analysis to predict the maximum downward deflection of the exit profile based on numerical data. Asymmetric profiles exhibit horizontal bow and twist in addition to vertical bow during roll forming [17]. The above studies show that the material properties, the profile geometry and the process parameters influence both longitudinal strain and bow. It is difficult to determine a generally valid relationship between the influence of material properties, the profile geometry and process parameters and the formation of strains and defects.

The studies listed are mostly not based on an analytical relationship, but are empirically performed for basic profile shapes such as U- or V- cross sections. An analytical correlation of longitudinal strain and profile defects has not yet been reported. Models for calculating longitudinal edge strains for simple profile geometries are known from Bhattacharyya’s investigations in [18]. By minimizing the total plastic energy required for bending and elongation of U-sections, an expression for the forming length is derived that dependents on both geometrical (such as flange and thickness) and process parameters such as the bending angle increment. To confirm the model, experimental investigations with one and several-forming stands are performed to form a U-section [19]. Fong [4] formulates an equation for the peak longitudinal strain based on the minimization of plastic work during forming.

A distinction between the fundamental deformations occurring during roll forming, such as stretching and bending, transversal bending and shearing, as well as the derivation of equations for the respective strains is carried out in [20]. For the longitudinal strains, it is shown in [21] that they additionally depend on the bending angle progression in the forming process as well as on the diameter of the lower forming roll, which are taken into account in analytical models for U-channels by [22]. Similarly, based on the findings in [21], a model for determining the longitudinal strains is constructed in [23], this time for top-hat sections. Another study adopts an analytical model for longitudinal strains at the flange during flexible roll forming and shows that these here depend on the bending angle progression in the process. It is also shown that geometric profile defects can be reduced by controlling the bending angle progression.

The analytical models of these investigations depend on numerous material, component and also process variables. A general validity for these models is therefore not given. Since the process variables are often unknown in practice, the application of the models is complicated and restricted to simple profile geometries.

In order to obtain a generally valid model for the calculation of longitudinal strain that is independent of these input variables, the approach taken in this paper is to calculate longitudinal strain based on the profile defects.

It is well known from the literature that profile defects and longitudinal strains can be influenced by various material and process parameters. Although it is known that there has to be a relationship between the longitudinal strains and profile defects [1], there are no approaches so far that question this dependence and show the direct analytical relationship. This paper aims to show the relationship between longitudinal strains and profile defects. The model is generally valid and applicable for various material, component and process parameters. The analytical model for longitudinal strains shows the dependence on profile defects such as vertical and horizontal bow and twist. The general analytical model is applied to asymmetric hat profiles and validated with the help of numerical investigations. The numerical model is validated by experimental investigations.

Material and methods

Analytical model

This section presents an analytical model that can be used to calculate the longitudinal strain distribution of a roll-formed profile across its cross-section based on given values for bow and twist. The model is based on geometric relationships and beam theory. Figure 1 illustrates the basic procedure for the analytical calculation of the longitudinal strain of a roll-formed hat-profile, which is explained in more detail in the following sections. The model can also be used to predict the bow and twist of a profile for a given longitudinal strain distribution.

Fig. 1
figure 1

Procedure for the analytical calculation of strain

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The basic idea of the model is that cross sections that are parallel in the undeformed flat sheet are also parallel in the deformed state if the longitudinal strain distribution is constant over the cross section of the profile. If, on the other hand, the longitudinal strain distribution of the profile is inhomogeneous bows and twist will occur. Figure 2 shows the resulting position of the profile cross sections in the case of a uniform and a non-uniform longitudinal strain distribution over the profile cross section using the example of a U-profile. The longitudinal strain can be determined from the distance between the corner points of the cross sections according to the following Eq. (1):

$${\varepsilon }_{i}=\frac{{l}_{i}-{l}_{0}}{{l}_{0}}$$
(1)

\({\varepsilon }_{i}\) describes the longitudinal strain at the respective edge, \({l}_{0}\) the distance between two parallel cross sections on the flat sheet and \({l}_{i}\) the distance between the corner points of the profile cross sections. The longitudinal strain is therefore only dependent on the position of the cross sections relative to each other.

Fig. 2
figure 2

Comparison of uniform and non-uniform longitudinal strain distribution (left side: no bow, right side: vertically bowed profile)

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Thus, the challenge is to determine the position of the cross sections depending on given bowing and twisting curves. The following assumptions are made for this purpose:

  1. 1.

    plane cross sections remain plane even after forming

  2. 2.

    the cross-section geometry is invariant over the length of the profile

  3. 3.

    the profile bends around its main axis

  4. 4.

    the profile twists around its shear center point

To calculate the position of the cross sections, the cross-section geometry of the roll-formed profile is required in addition to the horizontal and vertical bow curves and the twisting curve. Based on beam theory, the center of gravity of the cross-section, its principal axis and the shear center point are determined. The shear center point is the point on the profile section where the application of loads does not cause a twisting of the profile. If loads are not applied to the shear center of a profile, the profile will twist around the shear center in addition to the bending caused by the load. The shear center point, the center of gravity and the principal axis are needed to determine a three-dimensional deformation from the individual uniaxial deformations such as the horizontal and vertical bow as well as the twist of the profile. Figure 3 shows how the three one-dimensional deformations can be superimposed and how bow or a twist of the profile affects the center of gravity and the shear center position of the profile. It can be seen that bending leads to an identical displacement of the center of gravity G, the shear center S and the profile cross section. A twist of the profile, on the other hand, only affects the position of the profile cross section and the center of gravity G, but not the position of the shear center point S. Consequently, the position of the shear center point is exclusively dependent on the bow of the profile.

Fig. 3
figure 3

Superposition of horizontal and vertical bow and twist

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This implies that the three-dimensional position of the shear center points in the longitudinal direction of the profile is the result of the superposition of the two-dimensional curves of the horizontal and vertical bow of the profile, as shown in Fig. 4.

Fig. 4
figure 4

Three-dimensional shear center curve, z is longitudinal direction of the profile

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To determine the position of the profile cross sections, equidistant points are defined on the three-dimensional shear center curve. Subsequently, the tangential vectors \({\overrightarrow{n}}_{i}\) to the shear center curve are determined at the previously defined equidistant points \({\overrightarrow{P}}_{i}\). To define the direction of the vector \({\overrightarrow{n}}_{i}\) at the point \({\overrightarrow{P}}_{i}\), a vector \({\overrightarrow{n}}_{\widehat{i}}\) is formed between the points \({\overrightarrow{P}}_{i-1}\) and \({\overrightarrow{P}}_{i+1}\), as shown in Fig. 5. This vector points in the direction of the tangent vector \({\overrightarrow{n}}_{i}\), assuming that the distance between the points on the shear center curve is small and equidistant.

Fig. 5
figure 5

Determination of the normal vectors at the curve of shear center point. Side view of a vertically bent profile

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This results in the following Eq. (2):

$${\overrightarrow{n}}_{i}\frac{||{\overrightarrow{n}}_{\widehat{i}}||}{||{\overrightarrow{n}}_{i}||}\approx {\overrightarrow{n}}_{\widehat{i}}= {\overrightarrow{P}}_{i+1}- {\overrightarrow{P}}_{i-1}$$
(2)

The vectors \({\overrightarrow{n}}_{i}\) serve as normal vectors of the planes in which the profile cross sections lie.

In the next step, for each cross-sectional plane, the angles of the associated normal vectors with respect to the xz-plane (horizontal bow) and the yz-plane (vertical bow) (see Fig. 3) are calculated according to the following Eqs. (3) and (4):

$${\alpha }_{{i}_{hor}}= -\mathrm{atan}\left(\frac{{n}_{{i}_{x}}}{{n}_{{i}_{z}}}\right)$$
(3)
$${\alpha }_{{i}_{vert}}= \mathrm{atan}\left(\frac{{n}_{{i}_{y}}}{{n}_{{i}_{z}}}\right)$$
(4)

Here \({\alpha }_{hor}\) is the angle with respect to the z axis in the xz-plane and \({\alpha }_{vert}\) is the angle with respect to the z axis in the yz-plane. \({n}_{{i}_{x}}\), \({n}_{{i}_{y}}\) and \({n}_{{i}_{z}}\) are the components of the vector \({\overrightarrow{n}}_{i}\). The twisting angle \({\alpha }_{{i}_{twist}}\) at cross section i can be determined directly from the twisting curve.

For each defined point on the shear center curve, the cross-section geometry is placed in the xy-plane of the coordinate system. The center of gravity of the profile cross-section is located at the origin of the coordinate system. The coordinates of corner points of the profile cross sections are given in the matrices \({\overline{\overline{C}}}_{{i}_{org}}\). In addition, the following rotation matrices are defined:

$${\overline{\overline{R}}}_{{i}_{x}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\left({\alpha }_{{i}_{hor}}\right)& -\mathrm{sin}\left({\alpha }_{{i}_{hor}}\right)\\ 0& \mathrm{sin}\left({\alpha }_{{i}_{hor}}\right)& \mathrm{cos}\left({\alpha }_{{i}_{hor}}\right)\end{array}\right]$$
(5)
$${\overline{\overline{R}}}_{{i}_{y}}=\left[\begin{array}{ccc}\mathrm{cos}\left({\alpha }_{{i}_{vert}}\right)& 0& \mathrm{sin}\left({\alpha }_{{i}_{vert}}\right)\\ 0& 1& 0\\ -\mathrm{sin}\left({\alpha }_{{i}_{vert}}\right)& 0& \mathrm{cos}\left({\alpha }_{{i}_{vert}}\right)\end{array}\right]$$
(6)
$${\overline{\overline{R}}}_{{i}_{z}}=\left[\begin{array}{ccc}\mathrm{cos}\left({\alpha }_{{i}_{twist}}\right)& -sin\left({\alpha }_{{i}_{twist}}\right)& 0\\ sin\left({\alpha }_{{i}_{twist}}\right)& cos\left({\alpha }_{{i}_{twist}}\right)& 0\\ 0& 0& 1\end{array}\right]$$
(7)
$${\overline{\overline{R}}}_{mainaxes}=\left[\begin{array}{ccc}\mathrm{cos}(-\gamma )& -\mathrm{sin}(-\gamma )& 0\\ sin(-\gamma )& cos(-\gamma )& 0\\ 0& 0& 1\end{array}\right]$$
(8)

\(\gamma\) is the angle between the center of gravity coordinate system and the principal axis system. Using the following Eq. (9), the cross-section geometries are first rotated into their principal axis system and then bent around their principal axis according to the previously defined angles.

$${\overline{\overline{C}}}_{{i}_{bendmainaxes}}= {\overline{\overline{R}}}_{{i}_{x}}*{\overline{\overline{R}}}_{{i}_{y}}*{\overline{\overline{R}}}_{mainaxes}*{\overline{\overline{C}}}_{{i}_{org}}$$
(9)

Subsequently, the cross sections are transferred back to the original coordinate system using (10):

$${\overline{\overline{C}}}_{{i}_{bend}}= {\overline{\overline{R}}}_{mainaxes}^{-1}*{\overline{\overline{C}}}_{{i}_{bendmainaxes}}$$
(10)

For twisting, the cross sections \({\overline{\overline{C}}}_{{i}_{bend}}\) are shifted so that the shear center points of the cross sections lie at the origin of the coordinate system. This allows the profile cross sections to be twisted around their shear center point according to the following Eq. (11):

$${\overline{\overline{C}}}_{{i}_{bendtwist}}= {\overline{\overline{R}}}_{{i}_{z}}*{\overline{\overline{C}}}_{{i}_{bend}}$$
(11)

Subsequently, the cross sections \({\overline{\overline{C}}}_{{i}_{bendtwist}}\) are shifted with their shear centers to the points \({\overrightarrow{P}}_{i}\) belonging to the cross sections on the shear center curve, resulting in the arrangement of the profile cross sections shown in Fig. 6.

Fig. 6
figure 6

Profile cross sections building the bent and twisted profile

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Using the profile cross sections positioned as described, the longitudinal strain at each corner point (including the band edges) j of the profile cross section i can be calculated according to Eq. (12):

$${\varepsilon }_{ij}=\frac{\left|{\overline{\overline{C}}}_{{ij}_{bendtwist}}-{{\overline{\overline{C}}}_{{ij}_{bendtwist}}}_{i-1}\right|-\left|{\overrightarrow{g}}_{i}-{\overrightarrow{g}}_{i-1}\right|}{\left|{\overrightarrow{g}}_{i}-{\overrightarrow{g}}_{i-1}\right|}$$
(12)

\({g}_{i}\) stands for the coordinates of the center of gravity of cross section i, \({g}_{i-1}\) for the coordinates of the center of gravity of cross section i-1.

The model can also be used to predict the bow and twist of a profile for a given longitudinal strain distribution. For this purpose, the described model is coupled with an optimization algorithm. The algorithm optimizes the values of the bow and twist so that the difference between the longitudinal strain distribution calculated with the model and the given longitudinal strain distribution is minimized.

Experimental setup

The analytical model is demonstrated using a roll-formed hat profile. For validation purposes, a hat profile with a final bending angle of 70° is bent incrementally from a flat sheet metal (length 2000 m, thickness 2.0 mm, width 180 mm, material DC04). The roll forming machine used for the experiments is type P450/4 manufactured by Voestalpine. The flower pattern used and the bending angle increments can be seen in Fig. 7. In the first three forming stages, only the inner flanges are bent with an inner bending radius of 3 mm. Beginning in forming stage number four, the outer flange is formed with an inner bending radius of 5 mm. To counteract springback, the inner flange is bent over in the fifth forming stage to 72°. The inter stand distance is 525 mm and the feed speed is 6 m/min. The six forming stands each consist of a lower roller with a minimum diameter of 200 mm and a split upper roller with a maximum diameter of 280 mm. Guiding rollers are available to guide the sheet into the first forming stand.

Fig. 7
figure 7

Flower pattern (a) and bending angles (b) of the hat profile

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Numerical model

The described sample process is simulated numerically with the software package Marc Mentat. The rollers are simulated as rigid bodies and the sheet metal as deformable part. The principle of kinematic reversal is applied, i.e. the rollers move over the sheet that is fixed at the front end at two nodes in the z-direction. The previously described values of the sample process are adopted. In order to represent the stands’ stiffness, the upper rollers have one degree of freedom in the y-direction and are supported by two springs with a stiffness of 31,500 N/mm each [24]. The process is simulated without friction, since according to [11] the influence of friction is negligible in roll forming simulations. In order to obtain a larger stationary range, the sheet ends are fixed in x- and y-direction to validate the analytical model. As soon as the sheet leaves the roll gap of the last roll forming stand, this boundary condition is released to allow deformations. Since only sheet metal plate operation is possible in the experiments, this boundary condition is omitted to validate the simulation.

The sheet is modelled as a solid with element type 7 as "8-node hexagonal full integration". The size of the elements is determined with a convergence analysis. All elements are 6 mm long and 1 mm thick. In the bending radii, the sheet with a width (x-direction) of 1.25 mm is meshed finer than in the web, inner and outer flange, where it is 6 mm wide. Between narrow and wide elements there are two elements with a width of 3.625 mm. An assumed strain formulation is selected which allows a better representation of the bending of the sheet [25]. Elastic–plastic isotropic material behavior is assumed with a Young's modulus 210 MPa, yield point of 178 MPa and a Poisson's ratio 0.3. The yield curve of the material is achieved by tensile tests. Mises’ yield criterion is used. Strain rate and thermal influences are neglected. The processes are assumed to be quasistatic.

In order to simulate asymmetric profiles, the sheet is displaced by 5 mm in the X-direction (difference of flange length of 10 mm, hat1) and by 10 mm (difference of flange length of 20 mm, hat2). The roll forming model, the flow curve and the asymmetric profile cross sections (hat1 and hat2) are shown in Fig. 8.

Fig. 8
figure 8

Roll forming model (a), flow curve of the used material (b) and the simulated asymmetric profile cross sections (c)

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Results and discussion

Experimental validation of the numerical simulations

To validate the numerical model, three identical experiments are carried out with an asymmetric profile with a difference of flange length of 10 mm (shift of 5 mm in the X-direction, see hat1 in Fig. 8). The measurements are obtained with the help of the described setup in Sect. 2.2. The profiles are measured with the 3D scanner GOM Atos 5 and the twist as well as horizontal and vertical bow are evaluated. In the numerical simulation, the boundary condition for fixing the ends of the sheet metal plate is omitted in order to better represent the sheet operation, which results in a smaller stationary area. The mean values of the three experiments and the corresponding simulation can be seen in Fig. 9.

Fig. 9
figure 9

Horizontal and vertical bow and twist of the simulated profile and the mean value of the experiments

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Twist and horizontal bow show good agreement. In general, even minimal displacements of the rolls due to set-up inaccuracies can have a considerable influence on the longitudinal strains and profile defects. The deviations in twist and horizontal bow from the center of the profile occur at 950 mm profile length, which corresponds to twice the inter stand distance. This suggests that the change is caused by a not exactly set-up forming stand. In addition, slight ripples in the inter stand distance (gray lines in Fig. 9) can be observed in the simulated curve of the twist. These could be caused by the run-in of the beginning of the profile into a new forming stand and the associated instationarities. The beginning of the profile in the feed direction is located in the graphs at a length of 2000 mm.

In the case of vertical bow, experiment and simulation agree in the course and direction of bow, but higher deviations up to 27% at the maximum can be observed. The vertical bow is caused by deviations of the roll position in vertical direction between the forming stations and horizontal bow by horizontal deviations. The horizontal roll position can be set accurately in the roll forming machine using spacer rings. The vertical roll position, on the other hand, is adjusted manually using spirit levels between the forming stations. Furthermore, the roll gap is adjusted with feeler gauges. Those inaccuracies can easily lead to larger deviations in the vertical bow, which can be seen in the experimental results. In addition, vertical deviations of the roller positions can occur due to different stand suspensions as a result of deviating loads in experiment and simulation. Reasons for this could be the neglect of friction in the numerical model or deviating contact conditions, e.g. due to roll flattening.

Validation of the analytical model

Adapted numerical simulations are used to validate the analytical model. In contrast to the experimentally validated model, in these simulations the sheet metal plate is fixed to the front center nodes in all directions until the start of the sheet metal reaches the final profiling stand. This eliminates the free bow of the profile during forming and approximates a steady-state process (profiling from coil) in which the sheet is retracted in all profiling stands. In the simulation used for validating the simulation in Sect. 3.1, those boundary conditions were not applied to simulate a sheet metal plate operation.

The bow maxima of the stationary region of the simulations and the twist maximum, as well as the cross-sectional geometry of the profile are used as input data of the analytical model. The bow and twist curve of the simulation with a difference of flange length of 10 mm (hat1 in Fig. 8) is shown in Fig. 10. The maximum value of the horizontal bow is 0.05199 mm, that of the vertical bow is 0.08341 mm and the maximum twist of the profile is -2.589° in the used stationary area with a length of 204 mm.

Fig. 10
figure 10

Bow and twist curve in the steady-state area of the simulation

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Figure 11 shows the longitudinal strain distribution over the profile cross-section calculated analytically with the aid of the above-mentioned values, as well as the longitudinal strain distribution determined from the simulation. The longitudinal strain curve from the simulation is an average of the longitudinal strain distribution of the individual cross sections in the stationary range.

Fig. 11
figure 11

Comparison of the analytically calculated longitudinal strain distribution over the profile cross-section with the longitudinal strain distribution determined from the simulation for a profile with a difference of flange length of 10 mm (hat1)

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It can be seen that the qualitative curves of the analytically calculated and the simulated longitudinal strain distribution agree well.

For quantification of the differences, the SSD, ASD, the mean value of the deviation of the simulated and the analytical strain distribution, and MAS values are introduced (see Fig. 11). The SSD is the value of the maximum variation of the longitudinal strain distribution of the simulation. ASD describes the same for the simulated strain. Those values are interesting as they are an indicator for the expected size of the profile defects and are significant values for possible straightening operations as shown in [26]. The mean value of the difference between the simulated longitudinal strain and the analytically calculated longitudinal strain is calculated to give a quantitative value of the global accuracy of the analytically calculated strain. The associated standard deviation allows an estimation of the variation of the deviation of the two longitudinal strain distributions along the profile cross section. MAS describes the local maximum difference of the analytically calculated and the simulated longitudinal strain distribution.

The SSD in Fig. 11 is 0.151% strain; the ASD is 0.165% strain. The deviation of SSD and ASD in relation to the SSD is 8.9%. The mean value of the difference between the simulated and the analytically calculated longitudinal strain is 0.0145% strain with a standard deviation of 0.0066% strain. The MAS is 0.023% strain.

The analytical model is additionally compared with the simulation data of a profile with a difference of flange length of 20 mm (hat2 in Fig. 8). The maximum value of the horizontal bow of the simulation is 0.06061 mm in the steady-state region with a length of 204 mm, the maximum value of the vertical bow is 0.03583 mm and the maximum twist is -3.599°. Figure 12 compares the analytically calculated and the simulated longitudinal strain distribution over the profile cross-section for a profile with a difference of flange length of 20 mm.

Fig. 12
figure 12

Comparison of the analytically calculated longitudinal strain distribution over the profile cross-section with the longitudinal strain distribution determined from the simulation for a profile with a difference of flange length of 20 mm (hat2)

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The quantitative curves of the longitudinal strain distributions agree well, with the analytically calculated strain generally being slightly higher than the strain of the simulation. The SSD is 0.157% strain, the ASD is 0.173% strain. The values differ by 10.3% in relation to the SSD. The mean value of the strain differences between the analytical and the simulated longitudinal strain distribution is 0.026% strain with a standard deviation of 0.004% strain. The MAS is 0.035% strain.

In addition to the results shown in Fig. 11 and Fig. 12, the analytical model is also tested on the data of 79 additional simulations of asymmetric hat profiles. The validation data are published as supplementary material. One part of these simulations was originally set up to investigate the influences of process parameters (such as roll diameter, material yield strength, inter stand distance and roll gap) on profile defects. The remaining simulations were used to investigate the suitability of different straightening strategies (e.g. shifting or twisting the last forming stand, reducing the roll gap and overbending and rebending the profile legs) for roll-formed hat profiles. All the simulations have different bow and twist behavior as well as different longitudinal strain distributions. The boundary conditions in some of the simulations are chosen differently than in the simulations presented above, which may result in the absence of a well-defined steady-state region.

The quantitative curves of the simulated and the calculated longitudinal strain distributions agree well. The mean value of the differences between SSD and ASD of all simulations is 0.025% strain with a standard deviation of 0.024% strain. The mean value of the mean value of the differences between the analytically calculated and the simulated longitudinal strain distributions of all simulations is 0.031% strain with a standard deviation of 0.036% strain.

There are various possible explanations for the deviations. Firstly, the profile geometries of simulation and analytical model are not exactly identical: In the analytical model, the profile is composed only of straight elements and thus neglects the bending radii of the simulated profile. In addition, the analytical model assumes ideal bending angles of the profile of 70°, which may deviate slightly from the bending angles of the simulated profile geometry due to springback effects. On the other hand, the analytical model assumes that the bowing and twisting as well as the strain curves are perfectly stationary, which does not apply perfectly to the curves of the simulated profiles despite the adjusted boundary conditions in the simulations shown in Fig. 11 and Fig. 12. An explanation for the shift between the analytically calculated and the simulated longitudinal strain curve is that the simulated profile can be stretched or compressed in longitudinal direction in addition to the longitudinal strains caused by the bending and twisting of the profile. This is particularly visible in the strain curves of the simulations where reduced roll gaps are used for straightening operations, resulting in additional longitudinal strains (see supplementary material). The analytical model presented in this paper does not include these influencing factors.

Validation of the inversed analytical model

The model described in this paper uses values for horizontal and vertical bow and twist to calculate the longitudinal strain distribution of the profile. By coupling the model with an optimization algorithm, it is possible to inverse the model and to fit the values for bow and twist of the profile for a given longitudinal strain distribution. During optimization, the algorithm tries to minimize the difference between the given longitudinal strain distribution and the longitudinal strain distribution calculated by the model by varying the values for bow and twist. Since the simulated longitudinal strain curves are shifted compared to the analytically calculated curves (see Fig. 12) and the cross sections are not exactly plane in the simulation, the optimization algorithm converges to values for bow and twist that do not correspond to the bow and twist in the simulation. Therefore, the inversed model does not work reliable for simulated longitudinal strain distributions.

Consequently, the analytically calculated longitudinal strain curve was used as the target strain curve for the validation of the inversed model. The values of the longitudinal strain curve were rounded to four, three, two and one significant digits to find out for what amount of significant digits the model does converge to the correct values of bow and twist. The maximum relative errors compared to the analytically calculated strains are ± 50% by rounding to one significant digit, ± 5% by using two significant digits, ± 0.5% for using three significant digits and ± 0.05% by rounding to four significant digits. The following tables show the results for the inversed model tested with the longitudinal strains of the profiles with a difference in flange length of 10 mm (hat1 in Fig. 8, strain distribution in Fig. 11) and 20 mm (hat2 in Fig. 8, strain distribution in Fig. 12). Absolute values less than 0.5% are marked with a white background, absolute values between 0.5% and 5% are marked in light gray and absolute values higher than 5% are marked in dark gray.

Table 1 and Table 2 show that the inversed model is able to find the target values of bow and twist with a difference of the target value lower than 5%, as long as the values of the given longitudinal strain distribution have at least two significant digits.

Table 1 Difference from target values of bow and twist for a profile with a difference of flange length of 10 mm
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Table 2 Difference from target values of bow and twist for a profile with a difference of flange length of 20 mm
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In addition to the results shown in Table 1 and Table 2, the inversed model is also tested on the data of the 79 other simulations of asymmetric hat profiles. The algorithm did not converge to the target values with not rounded values of the longitudinal strain distribution in 12 cases. The following table lists the mean values of the cases where the algorithm converges.

Table 3 confirms the assumption that the inversed model is able to find the target values of bow and twist with a mean difference of less than 5% if the values of longitudinal strain distribution are at least two significant digits.

Table 3 Mean values of the difference from target values of bow and twist for all cases with converging algorithm
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Conclusion and outlook

In this paper, an analytical model for calculating the longitudinal strains present in the profile cross-section is presented and validated with simulations and experiments. The comparison of simulation and experiments shows minor deviations, which are probably caused by small deviations in the setting of the roll-forming machine. Therefore, the simulation is considered validated. The validation of the analytical model shows that the analytically calculated longitudinal strain distributions qualitatively agree very well with the longitudinal strain distributions of the simulations. The maximal deviation between the analytically calculated longitudinal strain and the simulated longitudinal strain is less than 0.04% strain. The presented model can also be inversed to predict the values of bow and twist by a given longitudinal strain distribution. The results show that the inversed model is able to find the correct values for bow and twist with a deviation of less than 5%, as long as the values of the given longitudinal strain distribution have at least two significant digits. The model presented in this paper demonstrates for the first time the direct relationship between profile errors and the underlying longitudinal strain distribution.

The analytical model that calculates the longitudinal strain distribution over the profile cross section makes it possible to move from a purely experience-based straightening of roll-formed profiles to a knowledge-based setting of straighteners. It is to be used in the future as part of a controlled straightener for roll-formed products. The straightener will consist of different, separately adjustable rolls to enable local rolling of the profile. To control the process, the bow and twist of a hat profile will be measured inline as the profile emerges from the final roll stand. With the aid of the analytical model, the longitudinal strain distribution of the profile can be calculated. Based on the analytically calculated longitudinal strain, the control system of the straightener then specifies target values for the infeed of the roll gap of the straightener in order to homogenize the longitudinal strain distribution of the bowed and twisted profile by local rolling and thus straighten the profile. The model that predicts the bow and twist of a profile could be used to accelerate the design process of roll forming processes. Therefore, the existing analytical models for estimating the longitudinal edge strains have to be further developed to calculate the longitudinal strains in the whole profile cross section.