A Roll-Stack Contact Mechanics Model to Predict Strip Profile in Rolling Mills With Asymmetric, Continuously Variable Crown Rolls

For more than a century, researchers have attempted to model the problem of mutual contact between two approaching elastic bodies having varying geometries and loading conditions. As part of contact mechanics research into the deflections in metal rolling mills, Allwood summarized both challenges and available solution techniques associated with general elastic contact problems [1]. The main challenge arises from uncertainty regarding the specific contact conditions at various stages of loading. This leads to binary nonlinearities at unknown boundaries corresponding to the contact and noncontact domains for the two approaching bodies. The issue is particularly problematic when the elastic bodies have irregular geometries or orientations. Analytic solutions for quadratic geometries (specifically cylinders and spheres) with parallel surface normals were derived by Hertz using elastic half space assumptions [2]. Analytic solutions for many other regular contacting geometries, including those with both normal and tangential contact surface tractions, were documented, respectively, by Johnson [3] and Hill et al. [4]. Most of the analytical solutions with arbitrary loads rely on superposition of spatial influence functions, together with the elastic half space simplification by Timoshenko and Goodier [5]. Influence functions for finite geometries have also been developed for specific cases, including cylinder surface deformations [6,7]. However, as pointed out by Allwood et al. [8], numerical solutions become necessary when the approaching elastic bodies have irregular geometry, since the coupled relationship between the loading and the domains of contact cannot be directly established. Thus, discretization of influence function effects is required for the relevant continuous fields, including domain boundaries, contact tractions, and displacements. A common solution technique (applied for instance by Chiu and Hartnett [6], Ahmadi et al. [7], Kalker and van Randen [9]) is to assume an initial state of contact, then iteratively update progressive solutions until convergence is reached, and equilibrium, compatibility, and material constitutive requirements are satisfied. Today, multipurpose finite element (FE) methods that apply the foregoing procedure are widespread. However, they suffer the major disadvantage of requiring enormous numbers of elements to represent the class of contact interfaces characterized by lines (e.g., cylinders) and points (e.g., spheres) as the respective elastic bodies establish contact.

The above contact mechanics issues are especially problematic for predicting deflections occurring in metal rolling mills having complex, machined roll diameter profiles and roll bending devices designed to control geometric quality. Indeed, many modern hot and cold rolling mills are equipped with “continuously variable crown” (CVC) rolls having cubic diameter profiles that are designed to operate with roll shifting and bending mechanisms for controlling the thickness profile and flatness of the rolled metal strip. Wang et al. stated that there have been more than 150 rolling production lines commissioned with CVC roll shifting technology in the past 25 years [10].

Figures 1(a) and 1(b) show side and front views of a four-high rolling mill. Indicated are the strip, work-rolls (WR), and back-up rolls (BURs). Note that, when equipped with a work-roll shifting mechanism, the WR face length, L_s, is longer than the stationary back-up roll face length, L. This prevents roll-edge contact marks on the strip after shifting. Figures 1(c)–1(e) depict highly exaggerated cubic CVC profiles machined onto the WRs. The figures reveal the conceptual benefit of “CVC shifting” in being able to adjust the rolling force distribution and the resulting strip thickness profile. In the configuration shown, positive bidirectional shifting of the antisymmetric CVC WRs transfers a greater share of the contact force to the strip outer edges, further reducing thickness in those regions. Conversely, negative bidirectional shifting transfers a greater share of the rolling force and thickness reduction closer to the strip center [11,12]. Although not shown, roll bending jacks applied to the necks of the work-rolls, back-up rolls, or both are also frequently included to further adjust the work-roll gap and strip profile.

Such complex CVC rolling mills have not yet been modeled adequately for real-time machine use without imposing substantial simplifications, as explained in detail in the literature review of Sec. 2. The aim of this paper, therefore, is to introduce a new, highly efficient method for predicting roll deflections, contact forces, and corresponding strip thickness profile in any type of existing hot or cold CVC (or other) rolling mill. The method captures the antisymmetry of CVC mills, and is efficient enough to be adapted for real-time machine set-up calculations.

While previous investigators have applied both analytical and numerical methods to model rolling contact behaviors, to efficiently handle the problem size, simplifications that exploit half, quarter, or eighth symmetric models, or that reduce the contact problem from three dimensions to two, are common [11–14]. CVC mills, however, contain at most a single symmetry plane, the x-y plane in Fig. 1(a), but this is only if none of the rolls are “offset” in the z-direction, as is sometimes the case.

Published work on the characteristics of CVC mills is limited. Bald et al. developed the CVC mechanism for controlling strip flatness and thickness profile [15]. Hartung et al. subsequently patented a specific CVC roll profile to give antisymmetric contact force distribution between the work-rolls and strip [16]. Allwood applied the transport matrix method to study the binary nonlinearity of contact interference (loss of contact) of CVC rolls [8]. Guo also used a transport matrix method and found that the antisymmetric contact force distribution possibly generated “wedge” type strip thickness profiles [17]. John et al. created a wear model to predict the wear profile on the CVC rolls and adjacent rolls [18]. Wang et al. designed a CVC back-up roll profile to prevent the existing CVC work-roll profile from generating nonsymmetric roll wear [10].

Continuously variable crown shifting behavior in rolling mills poses a complex contact-mechanics problem involving binary nonlinearities due to changing contact conditions as a function of total mill loading, roll shifting, and roll bending. Three main challenges exist. First, because CVC roll geometries are irregular, with varying diameter along the roll face length, an analysis initiating with “points contact” can induce numerical instabilities. Second, loss of contact is typical, with the noncontacting regions varying between CVC rolls and adjacent rolls and strip. Third, the elastic–plastic contact problem with the rolled strip presents high computational cost. Accordingly, this paper presents a new, static contact-mechanics approach applicable not only to simulate CVC mills but also to numerous other problems involving near line-contact or point contact conditions. The new contact algorithm adopts the simplified, mixed finite element method, which integrates classic analytical solutions into elastic foundation elements to represent the near line or point contacting regions, and strip “moduli” foundations to model strip plastic behavior.

The paper is organized as follows: First, existing rolling models are introduced, together with their advantages and disadvantages regarding the CVC mill contact problem. Next, mathematical formulation of the new static CVC model is given. Verification of the model's predictions to some theoretically required results is then discussed. Following this, comparisons with large-scale commercial finite element solutions are provided. Finally, case study results are presented, which reveal the static contact behaviors of a four-high work-roll CVC mill, and a six-high intermediate-roll (IR) CVC mill when rolling high-strength, thin strip.

This section introduces existing metal rolling models, and among them, feasible models for CVC profile contact problem.

Malik previously discussed strip profile calculation and developments in roll-stack deformation modeling from the 1960s to 2000s [19]. The earliest model by Stone and Gray was a single beam on elastic foundation [20]. It required a uniform diameter beam (roll), whereas CVC mills have multiple rolls and complex machined diameter profiles. Shohet and Townsend created the influence coefficient (function) method [21,22]. Guo expanded this method with two major features [23]. First, applying Boussinesq's equation, he approximated roll indentation due to contact force. Second, the roll displacements accommodated shear deflection. These were important improvements in roll-stack deflection models since mills have large roll diameter-to-length ratios, necessitating shear deformation in the formulation. The influence coefficient method has been widely adapted to two-high, four-high, six-high, and cluster mills. However, the approach assumes adjacent bodies are in complete contact [11,23], but lost contact is typical for CVC rolls [10–12,15]. Nonphysics-based or statistical models provide an alternative approach. For instance, Hattori et al. applied neuro-fuzzy techniques for cold strip rolling [24,25]. Dixit and Dixit applied fuzzy set-theory to optimal pass reduction scheduling [26]. All of these nonphysics methods, however, rely heavily on prior knowledge databases, and are influenced by the quality of learning rules developed.

None of the above rolling models are suitable for the CVC mill contact problem. Three other methods either have been applied, or may be considered: the transport matrix method, large-scale FEM, and simplified-mixed FEM. Allwood applied the transport matrix method to efficiently treat binary nonlinearity of elastic contact, with particular application to CVC rolling mills [1]. An iterative solution to check for contact and noncontact regions was used, and comparisons were made for high, medium, and low rolling force on the contact regions and interference of the CVC rolls. Allwood also applied the model to a six-high rolling mill with WR and IR bending, as well as IR CVC shifting control mechanisms [27].

Guo and Malik extended the discrete transport matrix method to two-high cluster mills, which required small core memory, but they assumed the contact and noncontact regions remained constant for industry application [28]. However, Cook cautioned that discrete “contact spring” methods (such as the transport matrix method) can generate stiffness errors at critical edge nodes [29]. To achieve adequate results, the transport matrix method requires a dense mesh to represent CVC profiles and the nonlinear contact behavior [30].

While large-scale FEM is not practical for real-time rolling models in industry, it is useful in academic contact-mechanics research. Using commercial FEM, Cao et al. built a four-high work-roll CVC mill model [31]. Du et al. built a Six-high intermediate-roll CVC mill model [32]. Wang et al. used a two-dimensional variable thickness FEM model for a four-high mill with CVC profiles on both the WRs and BURs [10]. To reduce computational cost, most large-scale FEM models assume top/bottom symmetric roll contacts. The foregoing FEM models were simplified to include only top half sections. This simplification, however, fails to reveal actual CVC contact behavior since it produces incorrect contact force distribution over the strip width. Section 4.1 highlights the problem by comparing cases with and without the top section symmetry simplification. Linghu et al. built a full-section six-high IR CVC mill model, and tested several case studies involving WR bending, IR bending, and IR CVC shifting. A single simulation required 52 h using the following computing platform: Intel Core (TM) I7-2600 3.4 GHz CPU, 16 GB RAM, Windows 7 Professional 64-bit [33].

Malik developed the simplified-mixed FEM containing analytically derived continuous Winker elastic foundations coupled to Timoshenko beam finite elements [34]. To date, this method has only been applied to symmetric mill contact behaviors. With fixed contact regions, the model rapidly and noniteratively computes contact behavior in both vertical and cluster mills [35]. Iteration, however, is required for initially unknown contact domains; the method was used iteratively to design optimum parabolic WR crown and taper profiles on a four-high production cold mill [36].

Ginzburg noted the fact that, to satisfy compatibility requirements, the initial roll positions in any model with no existing strain energy must not contain initial roll profile interference [11,12]. This implies “points contact” only, which is problematic in the CVC profile contact problem because both axial and vertical locations of contact points between CVC rolls and adjacent rolls/strip are unknown and create numerical instabilities. Allwood et al. applied a zero potential energy approach to iteratively calculate initial roll positions [8]. While large-scale FEM accommodates initial point contact or interference between rolls, convergence instabilities usually occur if the difference between internal strain energy and external work is too large for the solution increment [37]. This “pre-adjustment” is termed the conventional or forward compatibility deformation calculation. In this paper, an alternative “backward compatibility” deformation calculation is introduced in Sec. 3.3. The paper's focus is on a new contact algorithm together with its incorporation into the simplified mixed finite element method. The resulting model readily accommodates asymmetric contact conditions while retaining iterative solution stability; it is, therefore, applicable to CVC mills with complex, antisymmetric profiles on works rolls or intermediate rolls shifted laterally during rolling.

The new static contact algorithm for CVC roll profiles is incorporated into the simplified mixed finite element approach. Bending and shear deformation are captured using Timoshenko beam element stiffness. Flattening contact is captured using Winker foundation elements, whose stiffness is derived using analytical solutions for the mechanical interference. This section provides the mathematical details.

In Malik's original simplified mixed finite element formulation, the performance of several classic analytical solutions for contact between a pair of cylinders was compared [34]. These included plane-strain solutions of Föppl [39], elastic half-space solutions of Johnson [3], and an alternative to Hertz by Matsubara et al. [40]. The comparison studies for three-dimensional contact between rolls indicated that the Föppl analytical contact stiffness provided best agreement with large-scale FEM. When one of the bodies is the strip, an equivalent foundation modulus based on rolling force per unit strip width, per unit thickness reduction is used in series with the roll elastic modulus [23,28,34,35].

Figure 2 is a flowchart illustrating the overall procedure to assemble and solve the global system of equations for the static, nonlinear rolling mill contact problem. The procedure applies to arbitrary mill stands, and specifically to those with complex CVC roll profiles. Aspects of the procedure critical to addressing the binary nonlinearities are discussed next; then details follow for applying the procedure to case studies with CVC roll profiles.

From Eq. (5) above, it is seen that element foundation matrices, $[K]F$⁠, depend on unit contact force $f(x)$⁠. Initially, however, $f(x)$ is unknown; therefore, it is first estimated using a parallel axis contact assumption between adjacent rolls, but with a nominal interference found such that the integral of $f(x)$ over the contact length L equilibrates applied load $P$⁠. Under this initial state, any detailed profiles of the contacting rolls change the local interference for a given nominal interference according to the parallel axis distance. Hence, using steps 3–8 of Fig. 2, $[K]F$ is updated iteratively using the contact force distribution and binary contact points from the prior solution until strip displacement convergence is achieved.

Figure 3 shows “classic” and “advanced” CVC roll diameter contours. From Eq. (8b), the advanced CVC roll diameter, $Da(x)$⁠, is calculated from the nominal diameter, $Dn$⁠, and an advanced CVC “crown” profile, $Ca(x)$⁠. Note that Eq. (8b) is defined such that $Da(0)=$$Dn$⁠. An expression for the classic CVC roll profile is in Eq. (8a)

While the computed total contact force, $Pc$⁠, is found from Eq. (13), it may not yet equilibrate the total applied rolling force, $P$⁠, for reasons discussed and addressed next.

Given that the flat contact surface problem independently satisfies equilibrium, compatibility, and constitutive requirements, the right integral for net crown effects must vanish. Thus, net contact force effects from crowns $C1(x)$ and $C2(x)$ should be zero. This requirement, however, requires strict compatibility in the definition of both the nominal diameters and crown equations in Eqs. (9a) and (9b).

Section 3 describes a roll-stack model for CVC mills that integrates a new contact algorithm into the simplified mixed finite element method. In this section, the model predictions are validated against theoretical special case solutions involving asymmetric rigid roll contact for a two-high mill. These theoretical examples also illustrate flaws in existing approaches that exploit symmetry between the upper and lower mill sections. After this, a comparison of the results from the new model with those from large-scale FEM is made for a four-high rolling mill with work-roll CVC shifting. Finally, a comparison of the model prediction with experimental strip profile data for a six-high CVC mill is made.

Tables 1 and 2 list dimensions, process parameters, and mechanical properties for the rolled strip and the 2- and 4-high mills. The strip thickness profile is assumed uniform prior to rolling so that C(x) is zero on the entry condition.

While several conventional modeling approaches described in Sec. 2 exploit symmetry between upper and lower roll sections (about x-z plane in Fig. 1), the theoretical examples below will illustrate that this results in incorrect contact force distributions when applying CVC roll profiles.

Figure 4 shows three theoretical examples for a two-high mill in which the initial roll gap profile between top and bottom WR is uniform (i.e., constant initial strip thickness). Case A has “flat” WRs. In case B, the top WR has a positive parabolic crown with $C(0)$  = 0.5 mm, and the bottom WR has negative parabolic crown with $C(0)$  = −0.5 mm. In case C, the top WR has a Classic CVC profile defined by Eq. (9a), while the bottom WR has the same CVC contour but with antisymmetric orientation. In cases B and C, the model developed will recalculate the strip center line using Eqs. (17)–(19). In these equations, subscripts on vertical positions, $y(x)$⁠, are defined as follows. First subscripts 1, 2, and 3 refer to the particular body (strip, upper WR, lower WR). The second subscripts c, u, l indicate center, upper, or lower surface of the body.

While the contact surface profiles indicated in Fig. 4 may only be ∼1% of the WR diameter, it is critical that the strip center line be recomputed when upper/lower symmetry does not exist. Indeed, if the rolls remained rigid during rolling, the contact force distributions in cases A, B, and C would theoretically be uniform, except near the strip outer edges. To verify that the new model predicts a uniform contact force profile, an elastic modulus $E$  = 10¹⁵GPa is assigned. With the two-high rolling parameters from Tables 1 and 2, the resulting contact interferences and force distributions for cases A, B, and C are shown in Figs. 5 and 6. As theoretically expected for near rigid rolls, Fig. 5 shows the contact inference at the upper and lower strip surfaces to be uniform at 2.131 mm.

Also, as theoretically expected, Fig. 6 shows uniform contact force (68.046 kN/mm) for cases A, B, and C, except near the strip edges, where the force drops slightly as the problem undergoes partial transition from plane strain to plane stress [29].

To highlight the problem with the commonly applied simplification of an upper/lower symmetric roll-stack model, Fig. 7 gives the contact force distributions for cases B and C without the strip center line recalculation via Eqs. (17)–(19). Note that for the classic CVC profile of case C, the top and bottom contact forces exhibit antisymmetric distribution, which violates total force equilibrium on the strip. Note also that the contact force is excessively influenced by the WR profiles, even when the roll gap is uniform. As a result, the upper/lower symmetry simplification represents a serious disadvantage in some existing CVC roll-stack deflection models.

Section 4.2 is separated into three parts. First, in Sec. 4.2.1, predictions using the new model for the four-high CVC mill (without roll shifting) are compared to predictions obtained using a large-scale commercial finite element package, ABAQUS^® 6.14. These comparisons include mesh density studies, and apply the same mill and strip parameters from Tables 1 and 2. In Sec. 4.2.2, model comparisons for positive and negative WR CVC shifting with the same four-high mill and strip are made. In Sec. 4.2.3, a comparison between the models in predicting binary contact nonlinearities for various CVC shift positions is presented. In cases of Sec. 4.2, the top CVC WR profile is the advanced contour defined earlier in Eq. (10b), and the bottom WR CVC contour is antisymmetric to the top WR. Both BURs have “flat” diameter profiles (no crown).

To reduce computational cost of the large-scale FE model, it exploits a true symmetry plane, as shown in Fig. 8. Roll neck portions are neglected in both the FE model and the new model. Table 3 lists the large-scale FE model parameters, including three tested mesh densities. Note that, for direct comparison with the new model, elastic–plastic behavior of the strip is represented by an equivalent linear foundation stiffness (for small deformation). Strip elastic constants are assigned values representing one-dimensional linear elastic deformation [35]. Hence, Poisson's ratio, v, is zero and an equivalent elastic modulus of 15,300 N/mm² is assigned to the strip, derived using required rolling force, strip width, entry thickness, reduction, and roll-bite contact arc length.

To achieve the thickness reduction in both models, a vertical displacement boundary condition of 12.7 mm is applied on each end of the top BUR. For the FE model, this displacement BC is applied to quarter circles on each top BUR end face. Corresponding end faces on the bottom BURs contain fixed vertical displacement BCs. The quarter-circle BC domains on the FE model avoid singularity issues associated with single node BCs, which induce excessive element distortion. For the new model, this issue does not apply. FE simulations are executed on a cluster at UT Dallas having 24 CPUs and 6 hosts with message passing interface parallel processing. Each simulation with the finest mesh (1.858 M elements with 0.4 M on the strip) requires about 3 h run time.

Figure 9 compares results of thickness strain predictions for the new model and for large-scale FE. Both models exhibit the same trend in strain prediction across the strip width. The new model predicts slightly greater overall strain than the FE model, but as the FE mesh is refined, the results become closer. Figure 10 shows the relative differences in strain prediction between the new model and large-scale FE. Note that all thickness strain differences range between −8.54% and 1.45%. If the outer 1.5% of each strip edge, which is often trimmed, is disregarded, the strain difference between the new model and FE Mesh 3 (finest) ranges from just −2% to 1.45%.

Figure 11 compares the vertical displacements of roll axes for the new model and FE model (finest mesh). Strong agreement is seen. Furthermore, both models predict a “roll tilting” phenomenon. Note that the left side of the top WR has greater negative vertical displacement than the right side. Conversely, the bottom WR exhibits an opposing tilting angle. As a result, a counterclockwise tilting of the WR pair occurs. This phenomenon is known to occur in industry and requires “corrective leveling” by mill operators.

This section again compares thickness strain predictions for the new model and large-scale FE, but with varying magnitudes of CVC work-roll shifting. Mesh 3 in Fig. 8 is used for the FE model.

Figure 12 clearly confirms the purpose of WR CVC shifting in influencing strip thickness strain distribution. Both models predict the same trends with respect to magnitude and direction of WR CVC shifting—as the top CVC WR is negatively shifted (see Figs. 1(d) and 1(e)) thickness strain in the strip central region is increased, while strain near the strip edges is decreased. Conversely, positive CVC WR shifting produces opposite effects on strain distribution to negative shifting.

Figure 13 shows percent differences between the two models for the strain predictions shown in Fig. 12. Note that negative shifting gives positive difference, and vice versa. From Figs. 12 and 13, it is seen that the new model predicts slightly greater influence of WR CVC shifting than does the large-scale FE simulation. Additional research is needed to understand why this is the case.

With aggressively machined roll diameter profiles characterized by CVC, loss of contact between adjacent rolls, which causes binary model nonlinearities, is a common occurrence.

Figure 14 illustrates contact force distributions for the large-scale FE model (top figure) and the new model (bottom figure) with no WR CVC shifting. Both models predict lost-contact regions between the CVC WRs and adjacent BURs. Lost contact locations for both models exist near the right end for the top WR and BUR contact interface, and near the left end for the bottom WR and BUR contact interface. The FE model predicts loss of contact at normalized lateral coordinates of ±1.88 while the new model predicts loss of contact at ±1.85. The difference of 0.03 in normalized locations is 7.62 mm, which is only 0.5% of the WR face length of 1524 mm. Antisymmetric characteristics of CVC mills are again noted, this time with respect to roll contact binary nonlinearities, since the loss-of-contact and full-contact instances occur at opposite lateral locations for both the top and bottom roll-stack sections.

As shown in Table 4, the starting position of lost contact between rolls for the large scale FE model and new model are similar. For all shifting positions (positive and negative), the top CVC WR remains in full contact with the top BUR at the left side of the mill; and loss of contact for the top roll-stack always initiates on the right side of the mill. This can be explained by the decreasing diameter at the right side of the top WR CVC profile (Figs. 1(c)–1(e)) combined with the induced counterclockwise roll-stack tilting observed earlier. Note that the lost-contact region increases as the top CVC WR is shifted from left to right. As seen in Table 4, loss-of-contact predictions for the new model strongly agree with large-scale FE. The greatest difference occurs with largest negative shifting.

Therefore, rolling force $P$ is estimated as 3530 kN in Linghu's paper (rolling pass 5).

Accordingly, Fig. 15 shows Linghu's measured strip entry and exit profiles, as well as the predicted exit profile of the new model. Both the measured and predicted exit profile crowns are reduced because of the −142.5 mm intermediate roll CVC shifting applied. In addition, both exit strip profiles have less differential edge thickness compared with the entry profile. The absolute difference in exit strip profile between the measured and predicted ranges from −22.441 μm to 4.734 μm, with the average magnitude of absolute difference at 7.539 μm. Despite this good agreement, the unknown actual rolling load and unknown applied positions of work-roll and intermediate-roll bending forces are likely sources of error.

Based on the agreement with large-scale FE simulation in predicting contact mechanics behavior on a four-high CVC mill, and the agreement with measured strip profile for a six-high IR CVC mill, the new model is now applied to study shifting effects on a six-high IR mill rolling thin 301 stainless steel. The six-high mill is a popular configuration for incorporating CVC shifting [26,29,31,32].

Table 5 lists information for the strip, which is annealed before rolling and assumed to have parabolic entry thickness crown of 0.051 mm, or 2% greater thickness at the center than the edges. Table 6 lists dimensions of the six-high mill depicted in Fig. 16. The CVC contour on the top IR is given by Eq. (24). The IR face length is L_s. WRs and BURs have “flat” diameter profile and stationary roll face length, L

Figure 16 shows the six-high mill to have three primary strip profile control mechanisms:

• (a)

  Work-roll bending (WRB)

• (b)

  IRB

• (c)

  Intermediate roll bidirectional shifting (IRS)

Three case studies (Table 7) are investigated next to illustrate the new model's ability to reveal characteristics of the nonlinear contact problem involving strip profile prediction. Case D includes no WRB, IRB, or IRS. Case E includes only negative IRS. Case F includes positive WRB, positive IRB, and negative IRS. Note that each of the profile control mechanism set-points in Table 7 is designed to help reduce the tendency to further increase strip crown during rolling that results from the mill's natural deflection behavior. A relative increase in strip crown will tend to create wavy edges on the rolled strip while too much decrease in the relative crown will create center buckles.

Case study results using the new model to predict strip thickness profile and thickness strain are shown in Fig. 17. Case D (no control) has steeper edge drop in the exit strip profile than cases E and F (left plot). Steeper edge drop is accompanied by greater edge strain (right plot), and would result in a wavy edge flatness condition. Case F has more uniform strip profile as well as more uniform strain, which implies that the flatness during rolling would not change as much as for cases D and E. In addition, while it is not directly visible in Fig. 17, all three cases have a differential in the edge thickness, defined by h(x = −1) minus h(x = 1). In industry, this differential is termed “wedge profile.” Case D has 0.333 μm wedge, case E has 0.391 μm wedge, but case F has a comparatively larger 5.120 μm wedge. The wedge thickness profile condition is induced by the IR CVC profile, but is magnified by the coupled roll bending control inputs even though they are symmetric. Wedge in CVC mills was mentioned by Guo and Malik [28], and will result in one-sided edge waves during rolling.

Figure 18 illustrates the axes' vertical displacements and contact force distributions for cases D–F. With IR shifting (cases E,F), the vertical displacements show roll-stack tilting. Case E exhibits a larger tilting of the WRs and IRs than cases D and F. This is clearly evidenced by rotation of the strip axis. The tilting phenomenon appears proportional to the magnitude of CVC shifting, but the tilting is not directly equivalent to wedge generation. Note also that the WR axis displacement becomes more nonlinear (M or W shaped) when CVC shifting is combined with roll bending control. This explains why case F has more pronounced strip wedge than the other cases. With or without roll bending control, IR axis displacement maintains a parabolic curve.

The contact force distribution on the strip relates directly to thickness strain, and less directly to exit strip profile (because of entry profile dependence). Case D has lower contact force in the strip center than the edges. Consequently, the exit strip profile increases in parabolic nature, again creating wavy edges. Case F has the most uniform contact force on the strip, which would generally produce more consistent strip flatness during rolling. Note that negative shifting of the top CVC IR distributes a greater portion of the total contact force toward the center (cases E and F). Positive WR and IR bending forces help resist contact forces from dropping too aggressively near the roll ends. In other words, it can reduce the size of the loss of contact region, or help prevent lost contact near the roll ends.

To better understand how control mechanisms impact the strip profile, this section includes a sensitivity study for the strip crown ratio change, $ΔCR$⁠. Strip entry crown, $C1$⁠, is defined as the difference between the center thickness, $H(0)$⁠, and the thickness at location $x$ along the strip width, $H(x).$ Figure 19 indicates standard practice of measuring $H(x)$ at 25 mm from the strip edges. By normalizing the crown by $H(0)$⁠, an entry strip crown ratio, $CR1$⁠, is defined. The strip crown ratio change $ΔCR$ is thus the difference between entry and exit sheet crown ratios, expressed by Eqs. (25)–(27). Ideally, to reduce flatness changes during rolling, $ΔCR$ should be near zero across the strip width.

Table 5 lists strip entry thickness $H$ as 2.576 mm, which is the nominal value. However, variation in $H$ between coils and along a coil length always exists. For instance, if $H$ were to increase by 1.55% (or 0.04 mm) to 2.616 mm, from Eq. (28), the crown ratio change $ΔCR$ will increase by 0.000832. To compensate for the increased crown ratio change using WRB, Eq. (28) suggests a decrease of 18. 5 kN to −0.7 kN of WRB. If IRS were instead applied as the control mechanism, about 176 mm of shifting in positive direction would be needed. While WRB control is much faster than IRS, if the mill operation requires some positive WRB to prevent slip, then IRS or IRB could be applied to compensate for the $ΔCR$ change.

Efficiently modeling the contact interferences coupled to roll-stack deflections in metal rolling mills is a major challenge for conventional prediction models, particularly for asymmetric CVC roll profiles. Accurate and numerically stable computation of the contact interference is prerequisite in calculating contact force distributions, roll deformation and indentation, metal strip/sheet thickness profiles, and flatness. This paper has therefore presented an efficient and accurate roll-stack deflection model capable of effectively handing the antisymmetric CVC contact conditions to better simulate the metal rolling process. The new roll-stack model can be used to generate appropriate mill set-ups, the successful implementation of which can improve overall efficiency of the flatness control system in achieving actuator set-points. The model can also contribute to mill design, identification of high productivity pass schedules, optimal machined CVC roll crowns for the product mix, and to give insights into flatness actuator response using the sensitivity matrices. The results in Secs. 4 and 5 bring the following major conclusions:

1. (1)

   The newly developed model shows that a full-section (nonsymmetric) approach is required to satisfy theoretical contact force distributions. This is demonstrated for uniform roll gap requirements of rigid rolls for symmetric and asymmetric roll profiles. Indeed, the top/bottom roll-stack symmetry simplification exploited by many models is shown to be inappropriate for CVC roll contact problems.

2. (2)

   Net crown effects can be separated within the model since they do not affect the total contact force. Moreover, if roll crowns are not defined with strict compatibility to nominal roll dimensions, static equilibrium requirements will be violated. The developed net crown adjustment approach assures equilibrium and prevents numerical instabilities associated with point contact at interfaces.

3. (3)

   The new model shows strong agreement with large-scale finite element simulations for the strip thickness strain, roll axes vertical displacement, and binary nonlinearity locations corresponding to contact and noncontact regions. In addition, the model predictions exhibit good agreement with the experimental measurements of strip profile on a six-high CVC mill.

4. (4)

   While CVC roll profiles significantly influence contact force distributions during roll shifting, the roll-stack is tilted due to the asymmetric force distribution. This phenomenon is exaggerated with the addition of roll bending control inputs.

Any opinions, findings, or conclusions expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

• Directorate for Engineering (Grant No. CMMI-1555531).

a =

contact arc length between work-roll and strip (mm)

b =

contact width in Hertz Contact theory (mm)

C =

roll and strip crown (mm)

C_a =

advanced CVC roll crown (mm)

C_adj =

roll adjustable crown (mm)

C_c =

classic CVC roll crown (mm)

CVC =

acronym for continuously variable crown

d =

distance between two adjacent rolls axes (mm)

D =

roll exact diameter (mm)

D_a =

advanced CVC roll exact diameter (mm)

D_c =

classic CVC roll exact diameter (mm)

D_n =

roll nominal diameter (mm)

E =

elastic modulus (GPa)

f =

Gauss point specific interface contact force (kN/mm)

F =

global load vector (kN, kN-mm)

h =

strip nominal exit thickness or gauge (mm)

H =

strip nominal entry thickness or gauge (mm)

i =

element number

I =

total contact interference between foundations (mm)

k =

elastic foundation moduli (kN/mm)

[K]_G=

global stiffness matrix

[K]_T =

Timoshenko beam contribution to global stiffness matrix

[K]_F =

elastic foundation contribution to global stiffness matrix

l_i =

element length (mm)

L =

stationary roll face length (mm)

L_s =

shifting roll face length (mm)

L_n =

roll neck length (mm)

[N] =

shape function for vertical and horizontal displacement

P =

applied total rolling force (kN)

P_c =

computed total contact force (kN)

r =

strip thickness reduction (mm)

R =

work-roll radius (mm)

$R′$ =

deformed work-roll radius (mm)

RF =

reaction force (N)

$sy$ =

strip yield strength (GPa)

u =

global displacement vector (mm, radians)

v =

vertical displacement (mm)

v =

Poisson's ratio

w =

horizontal displacement (mm)

w =

strip width (mm)

x =

normalized lateral direction coordinate

y =

vertical direction coordinate

z =

rolling direction coordinate

δ =

contact interference in Hertz contact theory (mm)

ε =

thickness strain

$σy$ =

strip vertical stress (GPa)