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Presented is a newcomputationalmethod for predicting the static cross-sectional thickness
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profile of rolled metal strip. Methods to model the strip profile and related flatness with
improved efficiency and accuracy remain central for achieving high quality flat-rolled prod-
ucts. The new method involves a novel combination of Timoshenko beam finite elements
with multiple coupled Winkler elastic foundations. It applies to simple mill configura-
tions, such as the common 4-high rolling mill, in addition to complex mill types, such
as the 20-high Sendzimir mill. The inherent benefits over traditional strip profile mod-
els include non-discrete elastic foundations, cubic displacement fields, rapid solution, and
mixed boundary conditions. The flexible nature of the model allows it to readily accommo-
date typical mechanisms used in industry to control strip profile, such as roll crowning,
roll bending, roll shifting, and roll crossing. Comparison of the predicted displacement
for a 4-high mill with that obtained using a large-scale finite element simulation pro-
vides validation of the presented strip profile calculation method for real-time industrial
applications.1. Introduction 5279
To remain viable in a highly competitive global market, met-
alsmanufacturersmust embrace newtechnologies to increase
the quality of rolled metal products, including steel, alu-
minum, titanium, copper, and brass. In response to this need,
presented is a new mathematical method to efficiently cal-
culate the cross-sectional thickness profile of rolled metal
strip, an important dimensional quality attribute of flat-rolled
metals. The new method is suitable for application with real-
time computer systems for predicting and controlling the strip
thickness profile.
1.1. Dimensional quality requirements for rolled metal
strip
The rolling operation is characterized by incidental elastic
deflection of the mill housing, rolls, bearings, and other com-
ponents occurring simultaneously with the elastic–plastic
deflection of the rolled strip. These combined deflections fre-
quently lead to a non-uniform reduction in the thickness
of the rolled strip, and hence a non-uniform final thick-
ness profile, as indicated for example in Fig. 1. Many factors
influence the evolution of a given strip profile, including the
mill configuration, operating loads, profile control devices,and the incoming strip profile from a prior rolling operation.
Although the general convex profile of Fig. 1 occursmost often,
other arbitrary profiles are possible, including concave types,
depending on the profile control settings and other parame-
ters.
A commonly usedmetric to quantify the strip profile is the
strip “crown”, denoted C(x). The strip crown is defined as the
difference between the thickness, H(0), at the strip center, and
the thickness, H(x), at an arbitrary location x, as indicated in
the following equation
C(x) = H(0) − H(x) (1)
Because of end-user requirements to evaluate the differ-
ence in strip thickness at the center and edges, respectively, it
is customary to calculate an average strip crown at a small
distance from the two edges of the strip. When this refer-
ence distance, indicated as “a”in Fig. 1, is 25mm, the resulting
crownmetric is known as “C25” crown. Control of the strip pro-
file usually requires maintaining a target relative strip crown,
CR(x),which is the ratio of the strip crown to a reference value,
typically the strip center thickness, H(0).
An equally important dimensional quality parameter of
flat-rolled metals is the strip “flatness” or “shape”. Whereas
crown is the variation in thickness across the strip width, flat-