An airy function to rapidly predict stresses in wound metal strip having asymmetric thickness profile

With increased demand for thin gage flat metals, control of strip flatness or shape in cold rolling processes has become very important. To improve the flatness quality of cold rolled metal strip and sheet, this work provides a rapid method to predict the transient strains (or stresses) occurring during the rewinding of flat-rolled steels having problematic asymmetric strip thickness profile (or wedge). Flatness control systems, used to monitor and correct the distribution of stress across the width of rolled sheet, are unable to distinguish between stresses induced during rolling, and those caused when rewinding strip containing asymmetric thickness profile. The winding stresses, unless large enough to plastically deform the strip, vanish upon unwinding during subsequent operations such as stamping. Therefore, to help avoid strip flatness defects in thin strip containing wedge, a method is developed to separate the winding stress contribution from the overall stresses that are measured indirectly by flatness control systems. A fourth-order polynomial Airy function is developed to rapidly predict the in-plane stresses based on mandrel wrap number and spatial location on the strip. The Airy function is obtained by applying two-dimensional finite element analysis to study the transient in-plane stresses during rewinding at various numbers of mandrel wraps for a strip containing wedge profile. Three-dimensional finite element analysis is first employed, however, to show justification to a simplified two-dimensional problem described by the plane-stress Airy function. The two-dimensional finite element analysis provides insight as to how the in-plane stresses evolve, and allows determination of coefficients for the Airy function based upon model geometry and displacement boundary conditions. This approach differs from other methods that employ Fourier series to solve the biharmonic equations for an assumed two-dimensional problem. Finally, filtering of the winding stresses from flatness control system input signals is also discussed based on data taken from a rolling mill different to that used for model development.