A Comprehensive Model to Compute Roll Gap Profile for a Rolling Mill Including Bearing and Mill Housing Deflections

Cold-rolled strips are widely used in various applications and industries, ranging from the automotive industry, where they are employed for the realization of body panels and other components, to the production of home appliances, such as refrigerators and dishwashers. Also, cold rolled strips are becoming increasingly widespread as building and construction materials.

Cold rolling is performed by using a couple of work rolls as main tools. Work roll deflection has to be controlled in order to obtain the desired roll gap profile which is directly related to the strip flatness. There are several types and configurations of cold roll machines, differing in the number and arrangement of support rolls employed for mitigating and controlling the deflection of the work rolls. Among these, cluster rolling mill is a special class, in which each of the two working rolls is backed up by a cluster of larger backup rolls. By this arrangement, roll deflection can be effectively controlled and smaller work rolls can be used.

The 20-high rolling mill is a particular realization of cluster roll mill [1]. In this design, each of the two work rolls is supported by a cluster of nine supporting rolls. Eight rows of backing rolls, four for each side, are fixed to the mill housing at several axial positions by supporting elements called “saddles”. This provides high rigidity throughout the entire length of the work rolls, ensuring minimization of the rolls’ deflection and allowing to achieve very tight tolerances. Twenty-high mills are used for high-speed production of cold-rolled strips for both ferrous and nonferrous metals.

The quality of the final product in a cold rolling process is primarily defined by the obtained shape of the strip. In cold rolling, the strip material cannot flow transversely with respect to the rolling direction. Consequently, any non-uniform elongation of the strip material in the rolling direction leads to flatness defects such as buckles and waves. As a result, the strip shape or profile is not flat but exhibits thickness variations that must be finely controlled during the process.

To reduce the thickness of the incoming coil, plastic deformations are induced in the material being rolled. This process is characterized by the large interaction forces between strip and work rolls. This leads to non-negligible deflections of the rolls, backup bearings, and mill housing. Consequently, a non-uniform gap results between the work rolls, causing the strips to be thinner on the edges and thicker in the central section. This non-uniform thickness across the width of the strip is associated with the occurrence of “profile” defects. Another equally important dimensional quality parameter of flat-rolled metals is the strip “flatness”, which is related to the variation in length across the strip width. Profile and flatness defects are strongly interconnected. Flatness defects occur in the form of waves and ripples, extending along the strip rolling direction. Waves develop due to the arising of longitudinal compressive stresses originating from non-uniform deformation along the strip width [2,3].

In recent years, shape defects in cold rolling have been extensively discussed in the literature [4–6]. Intensive research activities have been dedicated to accurately predicting the strip thickness profile and developing models able to simulate rolling mills’ behavior.

A critical point in the modeling of rolling mills is the presence of several non-linear contacts occurring among the rolls and between the working rolls and the strip. For modeling such contacts, elastic foundation elements have been introduced by Stone and Gray [7] for the prediction of strip crown in four-high rolling mills. In the presented model, the work roll is modeled as an Euler–Bernoulli beam and connected to elastic foundation elements representing strip and backup roll. More recently, Malik and Grandhi [8] extended this approach to cluster-type mill configurations. Each roll of the cluster is modeled by a Timoshenko beam. The rolls are then connected to each other by elastic foundation elements.

Shohet proposed the influence coefficient method to calculate the gap profile [9]. Rolls are axially discretized into several elements. The method assumes initial arbitrary force distributions between elements in contact and iteratively adjusts the force distribution to satisfy equilibrium and compatibility. Each element deformation is related to the force distribution between all elements in contact and the influence function is used to establish the relationship between the load and deformation. Gap profile is obtained by superimposing the deformation of all elements. The influence function method is widely adopted for four- and six-high mills as it ensures fast and accurate computation [10,11]. However, this approach is ill-suited to solve complex contact problems of rolls, as in the case of cluster mills. Still, few works in the literature attempt to adapt the traditional coefficient method to 20-high mills [12,13]. In Ref. [14], rolls deformation of a 20-high mill is predicted adopting the influence function method. Some simplifications are assumed, namely zero rolls crown, no first intermediate rolls lateral shifting and rigid backup rolls.

Other mathematical approaches are proposed in Refs. [15–17]. In Refs. [15,16], Guo defined the transfer matrix method for a fast evaluation of the mill behavior. In this method, the mill is modeled by means of a series of spring, beam, and gap elements. Pawelski and Teutsch [17] developed a model that combines analytical methods to the rolling direction and numerical approaches to the width direction. The model is able to simulate multi-pass cold rolling on two- and four-Hi Mills.

Given the availability of large-scale finite element method (FEM) commercial software and the improvement of computational technology, FEM models are being increasingly used for analyzing the behavior of cold rolling machines. Most recent research works present the implementation of three-dimensional finite element models of rolling mills [18,19]. Advantages of using large-scale FEM commercial software for rolling mill modeling include the possibility of modeling complex mill architectures and roll profiles, considering several loading types and boundary conditions, obtain a more precise description of the structural behavior of rolls.

In Ref. [2], a 3D finite element model is used to investigate the formation of shape defects in cold rolling. By analyzing the material flow and stresses in the strip, the relationship between shape defects and thickness profile variations is assessed. Zhou et al. [20] employed a commercial FE software to realize the 3D FEM model of a 20-high mill and evaluate high-order flatness defects under roll shifting and backup rolls’ adjustments.

Due to the complexity of the cluster-type mills architecture, large-scale finite element models involve a large number of elements. Moreover, modeling of several contact interactions is required, leading to non-linearities in the model. As a consequence, 3D finite element models of cluster-type mills require large computational times. To reduce the computational cost, most large-scale finite element models are simplified by assuming half, quarter, or one-eighth symmetry. In some cases, to further decrease the computational time, other simplifications are assumed, leading to worsened solution accuracy [13].

To the best of the authors’ knowledge, in no published model of cluster mills detailed models of backup bearings and mill housing stiffness are included. Backup bearings are critical elements of the 20Hi rolling mill. Their outer rings are in contact with the intermediate rolls and participate in the stiffness of the cluster. For this reason, high outer ring sections and rolling elements with large diameters are usually employed to reduce bearing deflection and increase their fatigue life [21]. The mill housing supports the roll cluster and withstands the roll separating forces. The action of these forces deflects the mill housing affecting the final strip profile [22]. Although monoblock housings, characterized by high stiffness moduli and reduced deformation with respect to other mill configurations [22], are usually employed for cluster mills, their deformation should be considered when computing the strip profile. Therefore, the precise modeling of the housing stiffness is of fundamental importance for cluster-type rolling mills as the 20-high mill.

In Ref. [23], a plane lumped parameters model for the computation of the dynamic response of a 20-high mill is developed. This model considers the backup bearing stiffness by means of concentrated springs connecting intermediate and backup rolls. The non-linear springs characteristic is analytically calculated by using theoretical models developed for standard rolling bearings [24] followed by a parameter-tuning phase by means of experimental tests done by the bearings’ manufacturer [23]. In some cases [8,12,25–27], backup rolls with variable diameters are employed to model the presence of the backup bearings. In these cases, only the geometrical effect of the bearings is considered. The diameters of the backup rolls are increased in correspondence with the bearings and a segmented contact is realized with the intermediate rolls. Rolling elements’ stiffness and outer rings’ deformations are not specifically modeled.

In this paper, a combined finite element–analytical model able to predict the roll gap profile in cold rolling processes and including backup bearings and mill housing stiffness is presented. The developed method is suitable to model any type of rolling mill and, in this paper, is applied to a 20-high rolling mill. The model significantly enhances prediction accuracy for mill working parameters, leading to an immediate improvement in the quality of the produced strip from the onset of the rolling process. For validation purposes, also a complete 3D FE model featuring backup bearings and mill housing is developed. By using the presented finite element–analytical model, the effects of bearing and mill housing stiffness on the strip profile are shown.

The remainder of the paper is organized as follows. The developed finite element–analytical model is presented in Sec. 2. Section 3 describes the full 3D finite element model of the rolling mill. In Sec. 4, the numerical and experimental validation of the finite element–analytical model is reported. Finally, the effect of backup bearing and mill housing stiffness variation on the strip profile is discussed in Sec. 5. Conclusions are drawn in Sec. 6.

In this section, a combined finite element and analytical model for the simulation of cluster-type roll mills is presented. The section view of a 20Hi roll mill is depicted in Fig. 1. In this figure, the names of the rolls and the corresponding abbreviations used throughout the paper are reported.

The 20Hi mill is characterized by working rolls of relatively small diameter, along with three rows of rolls with progressively increasing diameters. The rolls in the outer rows consist of a series of rolling bearings, referred to as backup bearings, with each bearing row mounted on a common shaft. Notably, only the backup bearing shafts are connected to the mill housing. Specifically, each backup bearing shaft is connected to the frame by using collars, known as saddles, positioned between the bearings and at the two extremities of the shaft. As a result of this segmented configuration, each backup bearing shaft has one additional saddle compared to the number of bearings it supports.

The relatively small diameter of the working rolls (WR) offers the advantage of controlling their deformed shape by manipulating the other rolls in the mill. In standard industrial applications, this is achieved by adjusting the positions of the saddles, which results in the deformation of the backup bearing roll shaft (BURsh) that subsequently transmits a non-uniform pressure to the WR. Typically, in practice, only the saddles of the top half of the mill are actuated, either all four of them or only the central two.

Another form of actuation is related to the intermediate rolls (IR1). These rolls can have a tapered variation in diameter at one of their extremities. By altering the slope of this taper and adjusting the axial position of the rolls, the pressure distribution at the extremities of the WR can be modified and the occurrence of the so-called “edge drop” can be effectively reduced [28]. It is important to note that the taper is consistent and on the operator side for the two IR1 rolls of the top half of the mill, while it is on the drive side for the two IR1 rolls of the bottom half of the mill.

Lastly, both the WR and the other intermediate rolls, usually the second intermediate idle rolls (IR2i), can have a crown curvature, typically parabolic, to further fine-tune and control the pressure distribution during the rolling process. This combination of various roll adjustments allows for precise control of the flatness and pressure distribution across the metal strip during rolling operations. However, due to the large number of parameters that can be tuned, accurate, and fast models able to predict the strip profile are necessary for an effective optimization of the mill.

The combined finite element and analytical model of the mill developed in this section aims to accurately predict the strip profile at a reduced computational cost. The model includes roll models, contact interactions between rolls, and the stiffness of support bearings and mill housing. Rolls are modeled as standard Timoshenko beam elements, which take into account the shear deformation. The models of the other components are described in the following subsections.

As the paper focuses specifically on the effect of bearings and mill housing stiffness, a simple constant line load is applied vertically to the work roll, spanning the entire width of the strip, as to simulate a uniform reduction of the strip from side to side. This assumption leads to nearly symmetric mill results, allowing for independent simulation of the top and bottom halves of the mill. However, this simplified assumption also limits the applicability of the model as refined analyses of the strip profile, such as edge drop prediction, or dynamic analyses require more accurate, and physically based, strip models. Models for edge drop computation [28] or dynamic chattering analyses [29] are available in the literature. These, or similar models, could be adapted and implemented in the model if needed. In such cases, interactions between the two halves of the mill should be considered, and the full mill needs to be simulated. A model similar to the one considered in this section can be found in Refs. [8,30,31]. With respect to the cited papers, the present model includes the model of the backup bearings and the mill housing deformation and employs a more refined contact model able to decouple the mesh of the different rolls in contact. Also, the contact model employs a reformulated Winkler foundation elements [32] for taking into account the actual shape of the rolls when crown or taper are present.

Winkler contact elements have been used for several years for the simulation of beams lying on a continuous elastic foundation [32]. Applications of such elements to non-linear stiffness foundations are also available [33], in case considering initial gaps [34].

In Refs. [8,30,31], the Winkler foundation is applied to define a distributed stiffness between two beams. Basically, both sides of the Winkler element are considered deformable and their deformation is described by the beam elements used to model the rolls. The nodes of the beam elements are shared with the foundation elements. By this formulation, the distributed stiffness due to the contact is well described, however, the same mesh has to be used for the two beams. In the case of mills with a large number of rolls, this constraint can lead to cumbersome meshes as any mesh refinement required on one roll is propagated to all other rolls. The formulation proposed in Ref. [31] also takes into account complex roll profiles; in such cases, it exploits a crown adjustment variable for contact force computation. Here, this approach is modified in the following aspects.

Winkler element mesh. Figure 2 depicts a pair of beams connected by Winkler foundation elements. In Fig. 2, x-direction corresponds to the direction parallel to the rolls axes, while the $y$-axis is defined by the line that connects the centers of the two rolls. The two beams have different lengths, numbers of nodes, and element sizes. The Winkler elements are attached to the axis of each beam and are defined in a local plane reference system, denoted as $xy$⁠, and situated in the plane containing the axes of the two beams. It is always possible to define this plane as in any mill configuration, rolls always have parallel axes.

The nodes of the beams and the nodes of the Winkler elements do not coincide. The nodes of the Winkler elements are constrained by suitable multi-point constraint (MPC) analytic conditions to remain on the axis of each beam, i.e., to have the same displacement and rotation of the corresponding point on the beams. Therefore, the nodes of the Winkler elements do not have any free degrees-of-freedom. The stiffness matrices of the elastic foundation can be computed considering the nodes of the 2D plane Winkler elements in the local reference systems, and then the components of the matrices can be rotated to match the 3D spatial configuration of the mill in the global reference system. Finally, the MPC constraints are applied and the components of the Winkler element stiffness matrices are projected to the pertinent components of the global stiffness matrix of the system.

The mesh approach depicted in Fig. 2 allows for independent meshes on the two beams and on the elastic foundation. In this way, when building the mill model, mesh refinements can be applied whenever needed without propagation to other rolls.

The contact stiffness $k(x)$ is computed by considering the Hertz contact model [35] and is set to zero where the rolls lose contact. As the contact stiffness is non-linear and varies with $x$⁠, the integrals in Eqs. (7) and (8) cannot be computed analytically, but a numerical Gauss integration is used. The contact force per unit of length can be directly computed at Gauss points in post-processing after the nodal displacements have been found. Moreover, as the contact stiffness depends on the actual contact gap overclosure, an iterative procedure is employed to solve the resulting non-linear finite element system and the global stiffness matrix is updated at each iteration until convergence is reached.

Referring to Fig. 1, the roll cluster is connected to the frame by saddles attached to the shaft of the backup bearings. The bearings are segmented to accommodate a specific number of saddles, typically with four to nine bearings present on each backup shaft. Outer and inner bearing rings are connected by cylindrical rolling elements. As pointed out in Ref. [21], to minimize their deflection, backup bearings feature a large thickness of the outer ring and large rolling elements’ diameters. While this arrangement is relatively stiff, some radial and rotational displacements exist between the inner and outer rings. To replicate the stiffness of the bearings, two rows of Winkler foundation elements, arranged at 90 deg, connect each outer ring with the corresponding inner ring. This configuration ensures the reproduction of both radial and rotational stiffness between the rings.

The model in Eq. (9) is derived under the assumption of bearings with rigid rings. This assumption is reasonable for conventional rolling bearing applications, where the outer ring is usually inserted into a support, enhancing its stiffness. However, in the present case, the outer ring is not encased in a support; rather, it is in direct contact with one or more rolls, sustaining high pressure concentrated in a small area. In this situation, the assumption of a rigid outer ring does not apply.

The top half of the mill housing of the considered mill is depicted in Fig. 4. The housing is made from steel. The top half of the mill housing is composed of half of the monoblock, realized by the two side frames and an upper slab, and the saddles supporting the four backup roll shafts. This complex structure can only be accurately modeled by using 3D finite elements. However, incorporating such a large 3D FE model into the finite element–analytical model of the mill would significantly increase computational time.

To address this, a Guyan reduction [36] is applied to the housing stiffness matrix, enabling the inclusion of the mill housing model without a perceivable increase in computational time. In matrix reduction or condensation, the size of the stiffness matrix is reduced through a proper mathematical procedure. The stiffness matrix components associated with nodes not connected to the rest of the model and not subjected to any loads are condensed within terms related to the remaining nodes through a coordinate transformation [36]. The degrees-of-freedom associated with these nodes are referred to as retained degrees-of-freedom. The resulting stiffness matrix has dimensions determined solely by the retained degrees-of-freedom, significantly expediting the inversion computation. In our case, only the nodes corresponding to the center of the saddles are retained, as they interface with the shaft of the backup bearings. As demonstrated in Ref. [36], matrix condensation does not introduce any approximation in static problems.

As only half of the mill is considered at a time, separate condensed matrices are computed for each of the two halves of the housing by using the FEA commercial software abaqus. The FE model of the upper half of the mill housing depicted in Fig. 4 features the upper half of the monoblock and the saddles connecting the housing and the BUR shafts A–D (refer to Fig. 1 for nomenclature). The saddles are rigidly connected to the frame by surface-to-surface tie connections. Interface nodes are created at the center of each saddle and connected to the corresponding saddle by rigid MPC. The three displacements in the $x$⁠, $y$⁠, and $z$ directions of these nodes constitute the retained degrees-of-freedom. For the simulated mill configuration, a total of 84 degrees-of-freedom have been retained; three for each central node of the seven saddles on each of the four BUR shafts. This corresponds to an 84 by 84 condensed stiffness matrix to be imported into the finite element–analytical model.

Lastly, the assembly is grounded by enforcing the symmetry with respect to the $xy$ plane of the nodes on the bottom surface of the half monoblock. To eliminate any residual rigid motions of the half housing, constraints are applied to the $x$ and $y$ translations and to the rotation around $z$ of a reference point located at the center of the top surface of the monoblock.

A 3D finite element model of the 20Hi rolling mill is created using the commercial software abaqus. The model allows a comprehensive understanding of the system’s behavior when the cold rolling forces are applied to the working rolls. The deformations computed by the 3D finite element model are used for a numerical validation of the combined finite element–analytical model of the rolling mill described in Sec. 2.

The developed 3D model of the rolling mill is depicted in Fig. 5. It features a quarter symmetry of the whole rolling mill, i.e., symmetry with respect to the horizontal median plane (⁠$xy$ plane in Fig. 5) and the vertical median plane (⁠$xz$ plane in Fig. 5). With reference to Fig. 5, the developed 3D numerical model includes the following parts (see Fig. 1 for roll nomenclature):

• right part of the top half of the mill housing;

• two backup shafts (BURsh C and D), including six backup bearings and seven saddles for each shaft;

• a driven roll (IR2d), only half of the shaft is modeled for symmetry;

• an idle roll (IR2i);

• a first intermediate roll (IR1);

• a WR, only half of the shaft is modeled for symmetry.

The FE model of the backup shafts features simplified models of the saddles and of the bearings mounted on the shaft. Saddles are modeled by means of hollow disks.

A simplified model of the backup bearings is developed to capture the behavior of the bearings at a reasonable computational cost. Bearing outer and inner rings are modeled by linear brick elements considering their actual geometry and material. Rolling elements are not modeled as solid elements; instead, they are replaced by non-linear spring elements connecting the corresponding nodes of the inner and outer rings at the contact regions of these elements. The equivalent stiffness of the spring elements reproduces the local stiffness of the rolling elements. The same model described in Eq. (10) is employed for defining the equivalent stiffness of the rolling element. The bearing model is shown in Fig. 6.

The top right quarter of the mill housing is included in the model by a reduced matrix approach. This approach has been chosen to reduce the total number of elements in the model. The reduced stiffness matrix has been computed with a similar approach to the one described in Sec. 2.3.

The remaining rolls are modeled by linear brick elements, taking into account their actual material (steel) and geometry, including taper and crown if present. Specifically, work and driven rolls have a parabolic crown profile of $100μm$ and $120μm$ respectively, while the first intermediate roll has a 2% taper starting 30 mm inside the strip edge.

The strip is replaced by a uniformly distributed vertical load applied to the bottom edge of the work roll over a length of 900 mm, equivalent to the strip width.

Frictionless normal contact is considered to model the interaction between rolls. A global mesh size of $5mm$ has been used for all parts. In all the contact areas, a mesh refinement down to 0.2 mm has been applied. The resulting FE model features over 2.7 million elements and over 3 million nodes. The computation has been performed on a high-performance server with an Intel Xeon Gold 6248 CPU with 20 cores and 100 GB of RAM, taking about 120 h.

In this section, the results of the combined finite element–analytical model presented in Sec. 2 are compared with the results of the complete 3D FE model of Sec. 3 and with experimental data.

The same geometrical configuration of the rolls described in Sec. 3 for the full 3D FE model has been implemented in the combined finite element–analytical model. The main parameters of the FE–analytical model are summarized in Table 1. A refinement of the mesh of the Winkler elements has been applied in proximity of the starting point of the taper. The computational time of the combined model has been about 20 s on a laptop equipped with an Intel i7 12700H CPU.

Table 2 shows the comparison between the displacements and internal forces at the connection points between saddles and mill housing for BUR C and BUR D computed by the full 3D FE model and by the combined FE–analytical model. In this table, the displacement differences are normalized with respect to the maximum computed displacement and the internal forces differences are normalized with respect to the applied load. Displacements and internal forces computed by both models show a fully satisfactory agreement, with differences less than 3% for displacements and well below 1% for forces.

The contact pressure distributions between roll contact pairs computed by the two models are shown in Fig. 7, while Figs. 8 and 9 depict the axis displacements of BUR C, BUR D, and work roll in vertical and lateral direction, respectively. Note that the lateral direction corresponds to the rolling direction. Referring to both contact pressures and roll displacements, the agreement between the two models is very good. Contact pressures show the same trend and very similar values for the two models. Also, the same contact areas are computed by the two models. Similarly, almost the same displacements are computed by the two models, with small differences only on the left side of the work roll in the vertical direction, outside the width of the strip. The difference in the computed work roll vertical displacements between the two models is between 0.2% in the central zone and 3% in correspondence of the strip edge. The difference in the displacements of the left end part of the WR is due to the presence of the taper on the IR1. The FEM model has difficulties in getting the slope change due to the mesh dimensions. This part of the WR is particularly sensitive to small errors in the slope change as is close to the end of the line load. Smaller meshes could improve the results, but are impracticable due to the complexity of the FEM model. The overall agreement of the two models is anyway satisfactory, also considering that this part of the WR is outside of the strip.

The gap profile computed by the combined finite element–analytical model described in Sec. 2 is compared with the gap measured in an actual operating condition of the rolling mill. For the computation of the gap profile, numerical simulations are carried out for both the top and bottom halves of the mill. Top and bottom halves have been simulated independently as explained in Sec. 2. Different parameters have been used for two halves to take into account the differences in the respective configurations. For the top half, backup bearing movements and crown control actuation have been considered, while for the bottom half no such actuation was considered as not present on the actual mill. Also, different frame models have been considered for the two halves, each frame model representing the considered half of the mill. From these simulations, the deformed shapes of the top and bottom working rolls are obtained. The gap profile is, then, computed as the difference between the two computed deformed shapes. For both the top and bottom halves of the mill, the contribution of the housing has been considered.

The comparison between the computed and measured gap profiles is depicted in Fig. 10, while the mill configuration data are reported in Table 3. The gap profile has been measured by the onboard flatness control system. The considered mill configuration comprises tapers on the first intermediate rolls, located on opposite sides at the top and bottom parts of the mill. Additionally, crown actuation is applied to the B and C backup rolls. The comparison between the measured and computed gaps shows that the model accurately approximates the gap, with a maximum difference between the two profiles on the order of $1μm$⁠. The reported profile does not include the edges of the strip, as the constant line load used in the model does not allow for a correct computation of the edge drop. A more realistic strip load model could potentially yield better agreement with the experimental data. However, incorporating such a model is beyond the scope of the paper. Also, the comparison between the two models could benefit from using a more precise measuring system to extract the experimental profile.

The combined analytical–finite element model presented in the paper is employed to assess the impact of mill housing and backup bearing deformations on the strip profile. Although in the literature these deformations are usually neglected [26], due to the high stiffness of the roll cluster, they may influence the resulting strip profile.

Figures 11 and 12 show the comparison between the strip profile obtained by considering the parameters of Table 3 and the profiles obtained by changing the mill housing and bearing stiffness, respectively. The two stiffness values have been modified by multiplying the reduced stiffness matrices and the bearing characteristics by a constant. In both cases, the stiffness variations have non-negligible effects on the gap profile. Referring to the mill housing stiffness, a change of 25% in this parameter results in a variation of the gap profile amplitude of about 10–15 $μm$ and also leads to a different profile shape. The bearing stiffness variation has similar effects, with a variation of about $5μm$ in the gap profile amplitude for a 25% change of the parameter.

Finally, Fig. 13 depicts the variation in the strip profile when the mill is equipped with a different monoblock housing design. The alternative mill housing features the same number of saddles, and the rolls are mounted in the same positions. However, the design and the location of the anchor points to the ground of the two frames is different. The figure shows that different mill housing actually change the obtained gap profile. Also, the comparison of the results of Figs. 11 and 13 shows that the range of variation of the stiffness considered in the previous analysis is reasonable.

The present paper has been devoted to the study of the strip profile for a 20-high rolling mill. A combined finite element–analytical model able to quickly simulate the deformation of the mill and compute the strip profile has been presented. The model features not only the roll cluster but also the mill housing and the bearings of the backup rolls, both generally neglected in the literature. The model relies on 3D Timoshenko beam elements and a modified Winkler’s foundation element to model the rolls deformation and contact interactions respectively. MPC constraints were employed to connect the nodes of the Winkler’s foundation elements with those of the rolls, allowing for a complete independent meshing of each roll and contact element. This results in a remarkable potential reduction of the number of elements with respect to the state-of-the-art approaches. Additionally, a modified formulation of the Winkler’s foundation element is adopted by including the initial gap function in the internal work formulation. In this way, the resulting gap force vector can be straightforwardly computed and assembled in the finite element system.

A 3D finite element model of the mill has been constructed considering the roll cluster, the mill housing, and the backup bearings. The finite element model has allowed for a numerical validation of the combined finite element–analytical model. The two models have shown a very good agreement of the internal forces between rolls and housing, contact pressure between rolls, and roll deformations. Experimental data referring to the strip profile measured on an actual mill have also been employed for the validation of the combined finite element–analytical model. The agreement between the measured and computed strip profile is satisfactory with maximum differences of the order of $1μm$⁠.

The combined finite element–analytical model has been employed to quantify the influence of the stiffness of the mill housing and of the backup bearings. It turned out that the stiffness of these components of the mill has a non-negligible effect on the strip profile. Specifically, a 25% variation of the stiffness of the mill housing leads to a change of approximately 10–15 $μm$ in the strip profile for the considered mill, while a similar variation of the stiffness of the bearings results in a change of about $5μm$⁠. Finally, when an alternative mill housing is considered for the mill, a comparable variation has been observed in the strip profile. The obtained results indicate that, although mill housing and backup bearings stiffness are mostly not considered in the literature, these parameters should be included in models for the computation of the strip profile of cold rolling mills.

Future developments should be directed toward a fully coupled model where also the strip needs to be modeled.

The authors wish to acknowledge Eng. Jacopo Grassino for the precious support provided during the research activity.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.