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3D finite element model for roll stack deformation coupled with a Multi-Slab model for
strip deformation for flat rolling simulation
Yukio Shigaki, Pierre Montmitonnet, and Jonatas M. Silva
Citation: AIP Conference Proceedings 1896, 190018 (2017);
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Published by the American Institute of Physics
3D Finite Element Model for Roll Stack Deformation
Coupled with a Multi-Slab Model for Strip Deformation for
Flat Rolling Simulation
Yukio Shigaki1, a), Pierre Montmitonnet2, b) and Jonatas M. Silva1
Centro Federal de Educação Tecnológica de Minas Gerais (CEFET-MG)
Av. Amazonas, 7675, Belo Horizonte, Minas Gerais, Brazil, CEP 30510-000
Centre de Mise en Forme des Matériaux (CEMEF), MINES Paristech, PSL Research University, CNRS UMR
7635, CS 10207 Rue Claude Daunesse, Sophia Antipolis Cedex, 06904, France
Corresponding author:
Abstract. Roll deformation is an extremely important problem in strip rolling, all the more as the strip is thinner. It
results in profile defects (the strip thickness varies in the transverse direction) and flatness defects (the strip exits the
rolling mill wavy). This becomes a more and more stringent issue for modern, harder steels such as Advanced High
Strength Steels (AHSS). Numerous compensation techniques are used, alone or in combination: roll grinding crown, 4or 6-Hi mills, cluster mills, with shiftable tapered rolls, CVC-shaped rolls (Continuously Variable Camber), pair-cross
stands, etc. Elaborating the correction strategy for a specific strip rolling operation requires a model of the action of these
profile and flatness actuators. A hybrid model was developed in which the roll stack deformation is modelled with a
commercial Finite Element Method (FEM), coupled with a Multi-Slab model for strip deformation. This new model
allows simulating virtually any type of rolling mill configuration at a reasonable cost (CVC, shiftable rolls, pair cross
stands, cluster mills) including cases of incoming strip with defects (crown or wedge), by virtue of the FEM generality
and versatility. The example taken here is a 6-high rolling mill with shiftable intermediate rolls (an anti-symmetric
configuration). Results show quick convergence of FEM l Multi-Slab iterations and good agreement with experiments.
Keywords. Flat rolling simulation, Roll Stack Model, Noncircular arc of contact
Flat metal products are widely used in automobile industry, domestic devices, electric appliances, ships,
packaging etc. These basic metal strips are produced through continuous casting followed by hot then cold rolling.
This manufacturing route has high productivity and gives good mechanical and surface finishing properties.
Customers demand tighter and tighter tolerances for these flat products, especially for their thickness uniformity
and perfect flatness. This is more and more difficult to obtain with harder and thinner materials (e.g. AHSS,
Advanced High Strength Steels) resulting in more elastic deformation of the roll stack – and a consequently more
difficult control of the abovementioned geometrical properties.
Roll deformation is an extremely important problem, being responsible for the profile imprinted on the rolled
strip. The relative profile of the strip should be, as close as possible, equal to the relative profile of the entering strip
in a rolling stand, preventing bad flatness.
Proceedings of the 20th International ESAFORM Conference on Material Forming
AIP Conf. Proc. 1896, 190018-1–190018-6;
Published by AIP Publishing. 978-0-7354-1580-5/$30.00
These better flat strips are produced in modern rolling mills with many controlling actuators, searching for a
uniform relative deformation concerning the strip thickness and, consequently, zero flatness defects. To master the
lateral distribution of the roll gap, these mills may use rolls with variable camber (CVC for instance), axially shifting
rolls (HC mill), vertical and horizontal roll bending systems, roll crossing systems, etc.
Many mathematical models are used to simulate these rolling mills in order to predict the rolling force and
torque for different operational conditions. These find use for a production set-up, for a better understanding of the
process, for production optimization, to predict strip geometrical characteristics, to develop a sensitivity study in
order to find simple rules to be used in a closed control loop, etc.
3D roll stack deformation models to be used for strip profile prediction appeared as early as 1965 by Stone and
Gray 1 using the elastic foundation method. Then the influence function method was introduced by Shohet and
Townsend 2 to integrate equations for both roll bending and flattening. At this stage, strip deformation was addressed
by variants of the Slab Method introduced in 1925 by von Karman 3,4,5.
The transverse variations of the roll load are an essential feature of the profile and flatness problem, not caught
by a single slab since the Slab Method is plane strain in the Rolling Direction (RD) / normal Direction (ND).
Therefore, coupling analytical roll deformation models similar to the abovementioned ones with a series of Slab
Models distributed in the transverse direction was proposed e.g. by Pawelski et al.6, and used e.g. by the MEFOS
team to develop the CROWN models 7.
In hot strip rolling, complex strip metal flow (e.g. widening or spread) suggests 3D FEM. Yanagimoto and
Kiuchi 8 coupled this with an hybrid 3D roll deformation model; bending was dealt with analytically by the beam
theory as above, but owing to the complexity and 3D character of roll flattening, they computed it by the FEM. For
the sake of computing time, Hacquin et al. preferred to couple FEM for the strip with a powerful fully analytical
thermo-elastic model of the roll stack deformation 9,10,11. Later on, for the sake of precision, Kim, Lee and Hwang 12
chose to use finite element method for both strip (or plate) and rolls, followed recently by other groups 13,14.
Montmitonnet 15 showed by comparing with experimental results from Yanagimoto and Kiuchi 8 that all three latter
approaches have satisfactory precision. The cost is however different: a full FEM model for a 6-high CVC rolling
mill 14 shows good results, but each run takes 52 hours on a computer with Intel Core ™ I7-2600@3.4GHz for CPU
processor, 16Gb RAM, Windows 7 professional 64-bit. The steady state model with semi-analytical roll deformation
model 9 would take less than 1 hour in similar cases.
In cold strip rolling, spread is not so much a problem and the “Multi-Slab” treatment of strip should be sufficient.
However roll deformation is large and to deal with all kinds of actuator configurations, 3D FEM is more versatile.
Therefore in this paper a hybrid model is presented, using 3D FEM for the roll stack modelling, while elastoplastic
strip deformation is calculated by the Multi-Slab method (two different versions for simple and for severe rolling
configurations). Avoiding a 3D FE elastoplastic model for the strip and using approximate symmetries reduces
dramatically the processing time without jeopardizing precision too much. Typical CPU times are circa 30 minutes
for a 6-High mill for 8 iterations for a Intel Xeon 3.4GHz with 6 cores, 32Gb RAM, Windows 7 professional 64-bit,
for stands with CVC rolls, roll shifting and pair-crossed rolls e.g.
This section presents briefly the Slab Method for calculation of roll separating forces (or rolling load). It first
describes the traditional Slab Method with Hitchcock’s hypothesis for roll deformation, valid for moderate reduction
on thicker strip, then a more precise model that takes into consideration the noncircular deformed profile of the rolls,
especially important for simulating hard and thin metal strips.
Slab Method with Hitchcock’s Hypothesis
Von Karman 3 introduced the vertical segments homogeneous compression concept during rolling in a plane
strain state, reducing it to a 2D geometrical but 1D mathematical problem. In addition, the contact surface between
roll and strip was assumed to remain circular, leading to an easier solution with minor errors for common cases. The
friction coefficient was chosen constant and the sheet and roll elasticity was neglected. Thus, an analytical solution
could easily be obtained. These simplifying assumptions were progressively relaxed for more precision. Hitchcock
investigated roll elastic deformation. Assuming the profile remains circular and applying Hertz’ contact theory, he
came out with a closed-formed formula for the deformed radius. Later on, Ford et al.17 introduced the contribution
of the elastic zones at the entry and exit of contact and allowed for strain hardening (e.g. Nascimento et al. 18).
This brought this method to a generally satisfactory degree of precision, while being easy and fast to solve on a
computer. One critical assumption however is the maintained circularity of the roll profile. Wiklund and Sandberg 19
present a “flattening risk factor” D = L/hi (L is the length of the arc of contact, hi the entrance thickness). When D >
10, severe flattening is expected so that circularity may not hold anymore. Authors still dispute if the estimation of L
in D should account or not for Hitchcock’s deformed radius (Lenard 20).
In this work, the Bland-Ford and Hitchcock model was programmed, named hereafter “BFH module”.
Slab Method with Noncircular Models for Roll Flattening
For highly loaded cases however, the abovementioned model lacks precision and even fails to converge. In such
cases, it is in fact necessary to take precisely into account the non-circularity of the deformed profile, which has a
strong impact on L and therefore on the stress profile. Based on this account, a computer program was developed
based on the paper by Le and Sutcliffe 21 for the elastoplastic deformation of the strip (including a plastic contained
region) and on Krimpelstätter 22 for calculating the real deformation of the roll. The balance of the forces applied to
all the sides of a slab inside the arc of contact between the strip and the roll is made, resulting in the classical
equation of equilibrium of Orowan. The latter is solved considering possible occurrence of five different situations
inside the contact arc: 1- an elastic entry zone; 2- a plastic slipping zone; 3- a plastic contained zone in which
sticking friction occurs; 4-an exit plastic zone; 5- an elastic exit zone. The program uses a friction regularization
method. Details can be found in Shigaki et al.23.
Half-thickness (mm)
Normal stress (MPa)
x (mm)
x (mm)
kVs (Vsliding/Vroll)
Tangential stress (MPa)
x (mm)
x (mm)
FIGURE 1. Results from the 2D Noncirc model.
a) Normal stresses, b) arc of contact, c) tangential stresses, and d) sliding velocity/roll’s velocity.
In the present work, this program called “Noncirc” is used for extreme cases with harder and thinner metal strips.
For example in the severe case of Tab. 1, Fig. 1a), b), c) and d) shows the normal pressure, the deformed profile of
the roll in contact with the strip, the friction stress and the relative velocity. x is the position along the rolling
direction. In Fig. 1b, the flat region between x=-7mm until x=4.8mm is the neutral zone where sticking occurs
(quasi-zero sliding velocity) and is typical of highly loaded rolling cases. This program was validated with
LAM3/TEC3 software at CEMEF.
TABLE 1. Parameters and values used for simulation in Noncirc and the results presented in Fig. 1.
Sheet thickness at entry (mm)
Sheet thickness at exit (mm)
Work roll radius (mm)
Strain hardening (MPa)
Front applied tension (MPa)
Back applied tension (MPa)
Friction coefficient
In this section the hybrid model proposed is presented, coupling the 3D FEM elastic roll stack model with N
Slab Models at different positions in the width direction, where N is the number of subdivisions of the rolls. The
latter provide the load to the former, which in turn feeds back the deformed roll shape. The algorithm of the 3D
FEM/Slab Method is iterative, its convergence is reached when the former strip profile is identical to the profile of
the last iteration to a given precision. The communication between the commercial FE software, Abaqus v. 6.14, and
the rolling load program, BFH or Noncirc, is done by a Python script. Here is a brief description of the method.
In the contact area with the strip, the work roll is partitioned into N rectangular patches for the load transfer. In
the center yi of each of them, a normal stress distribution p(x,yi) is applied. The patch length parallel to the rolling
direction (x) is based on a value of the contact arc length averaged over the width direction. It may be either kept
fixed all along the simulation, or updated after for each iteration; differences have been found negligible until now.
These patches must cover the whole width of the strip, with a refinement near the edges to provide a precise picture
of edge drop. Any possible variation of strip width (spread) is neglected since experimental results show it is less
than 0.5% for most cold rolling operations.
The simulation consists of the following steps:
Step 1- Based on a single Slab Model (BFH or Noncirc depending on the severity of the case), an initial guess of
the total rolling load is calculated, and this total load is applied uniformly on the work roll.
Step 2 - Running the FE software with this loading gives a new deformed profile of the work roll in the contact
region with the strip. It is assumed that this gives the transverse profile of exit strip thickness imprinted by the roll.
Step 3 - With the entering strip profile and the exiting thickness, BFH or Noncirc is run for each patch along the
width of the strip. This gives the updated normal load applied on the rolls by the deformation of the strip.
Step 4 – If the new profile is identical to the last, then stop. Otherwise go back to Step 2.
Further simplifications can be done to save computing time. In the example of a HC mill shown in Fig. 2 and
simulated hereafter, two approximate symmetries are implemented. First, only the upper stack is modelled; to
recover the strip thickness profile, the lower stack is reconstructed by anti-symmetry (the upper intermediate roll has
its taper and shifts to the right, the lower one to the left). Similarly it is assumed that the strip also has a plane of
symmetry formed by the rolling direction and the vertical axis. Secondly, as the FEM model is just in charge of
determining the exit strip thickness profile (not the detailed shape of the work roll in the whole bite), an approximate
upwind / downwind symmetry is assumed ( Fig. 2); this means that the load is applied vertically beneath the roll
whereas in reality, it is slightly shifted upwind (the bite is in majority on the entry side). If these assumptions are not
deemed acceptable, a more complete FEM model should be selected, there is no difficulty in principle.
Backup roll necks are considered as simply supported (their bearings are thus freely rotating around x).
The 6-High HC-Mill and the strip are described in Figure 2 and Table 2, and numerical results are compared
with experimental data in Figure 4. Input data and experimental strip thickness profile were provided by a major
Brazilian flat rolling industry. Figure 3 gives an idea of the convergence of the hybrid model. Correct convergence is
obtained after 5 iterations (CPU time: 20 minutes), it can still be improved slightly (beyond 8 iterations, the
evolution is less than 1 μm everywhere). Figure 4 shows good agreement of the strip profile with experimental data.
The experimental absolute crown is 32.7Pm (2.44% of strip thickness) vs 35.6Pm and 2.65% from the model. The
edge drop defect, shown by the minimum thickness of the point on the edge, also shows good agreement.
TABLE 2. Parameters and values used for simulation of the case.
Strip thickness at entry (mm)
Strip thickness at exit (mm)
Strip width (mm)
Work roll radius (mm)
Backup roll radius (mm)
Intermediate roll radius (mm)
Strain hardening (MPa)
Front applied tension (MPa)
Back applied tension (MPa)
Friction coefficient
Roll shift (mm)
FIGURE 2. FEM model used in the
Thickness (mm)
Thickness (mm)
iteration 1
iteration 2
iteration 3
iteration 4
iteration 5
Strip width (mm)
Strip width (mm)
FIGURE 4. 3D FEM/BFH model results and
FIGURE 3. 3D FEM/BFH model results for
experimental data.
a 6-High mill.
More simulations must be done in order to assess the precision of the present model and check the validity of the
assumptions (approximate symmetries etc.). But considering the flexibility of FEM for roll stack modeling and the
great speed to run it with Slab Methods (in this case, 20 minutes compared with full FEM model – 52 hours), this
new model is promising. A major advantage of this model is the capability to use noncircular arc models for hard
and thin strips, whereas such cases are practically impossible with full FEM because they would require extremely
fines 3D roll meshes with a tremendous CPU cost. This strategy allows e.g. antisymmetric stands to be addressed at
no extra cost compared with symmetric ones, or strips with wedge or other defects to be dealt with efficiently.
The authors wish to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Process
488027/2013-6, CEFET-MG and CEMEF for financial support with which the present investigation was possible,
and for financing the subscription fee, plane tickets and the hotel.
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