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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
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