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ness is the variation in length across the strip width. Crown
and flatness are strongly related since the plastic deforma-tion of the strip is incompressible and expansion of the strip
width is usually negligible, particularly in the cold rolling pro-
cess. Fig. 2 illustrates a certain type of flatness defect during
the rolling of austenitic stainless steel. This type of defect
is known as “center buckle”, since the center portion of the
stripwidth contains buckles due to excessive plastic deforma-
tion in the middle of the strip width. Other common flatness
manifestations include “wavy edges”, “quarter buckles”, and
“herringbones”, as depicted in Fig. 3.
1.2. Need for an efficient and flexible model
Manufacturers undertake variousmeasures to control the pro-
file and flatness and achieve the desired dimensional quality
of rolled metal strip. These measures may include employing
flatness control systems, designing optimal pass schedules,
or utilizing roll crowns. Since the foregoing measures benefit
from a rapid and accurate calculation of the strip thick-
ness profile, more efficient computational models provide a
competitive operational advantage. As discussed next, the
conventional strip profile calculationmethods do not compute
the strip thickness profile with adequate efficiency and accu-
racy. Furthermore, fewof the existing computationalmethods
are flexible enough to model cluster-type rolling mills such as
the 20-high Sendzimir mill shown in Fig. 4.may be classified broadly as either: the single-beamon elastic
foundationmethod (Stone and Gray, 1965); the influence coef-
ficient method (Shohet and Townsend, 1968, 1971; Hacquin
et al., 1994); the transport matrix method (Poplawski and
Seccombe, 1980; Guo, 1998); the pattern recognition/heuristics
methods (Hattori et al., 1993; Zhu et al., 1993; Jung and Im,
1997); and the large-scale finite element method (Eibe, 1984;
Chen and Zhou, 1987). Although each method has unique
advantages, none satisfies the combined requirements for an
efficient, accurate, and flexible model capable of simulating
complex mill configurations. Stone’s work studied the effects
of work roll bending and back-up roll bending to control strip
crown on 4-high rolling mills. In evaluating the effect of work
roll bending on strip profile, Stone modeled the work roll as
a single Euler–Bernoulli beam on a constant elastic founda-
tion that represented themutual flattening between the work
roll and the back-up roll. Hence, no independent shear or
bending deflection of the back-up roll was considered. Shohet
and Townsend’s influence coefficient method employs a dis-
cretized Green’s function to superpose the effects of multiple
point loads for the purpose of representing load distributions.
“Pointmatching” is utilized to satisfy equilibriumand compat-
ibility conditions at a finite number of discrete points along the
interfaces between the contacting rolls. Themethod assumes
initial arbitrary force distributions between contacting bodies
and uses an iterative procedure to adjust the force distribu-
tions to satisfy the pointmatching. Several improvements and
enhancements have been made to this popular method over
the nearly four decades that it has been used. For instance,
Kuhn and Weinstein (1970) modified the method to consider
the Poisson deflection due to axial bending stresses. Indenta-
tion flattening at the interface between the work roll and the
stripwas considered using Boussinesq’s theory by Kono (1983),
then by Tozawa (1984). Semi-empirical methods to model the
work roll and strip interaction were employed by Nakajima