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Therefore, an automated compensation process has been developed recently, in which hydraulic cylinder pushes back the rolls in proportion to the level of reaction force from the plate. FIGURE 5. Wedge Compensation. MATHEMATICAL MODEL During the leveling process, the plate touches the roll away from the roll center, mainly due to the defection of the plate – Fig.6. As the contact condition between the plate and rolls changes frequently any calculation model requires to analyze dynamic contact condition with friction effects. FIGURE 6. Roll / Plate Geometry. For a given contact condition between the plate and rolls, deflection of an arbitrary point on the plate can be precisely described by the following equations [2 (6) Due to changes in contact conditions, to evaluate deflection of the plate computation process has to follow three stages of calculations (and need to repeat for every new contact conditions): a) Contact reaction force. b) Bending of the plate due to the reaction. c) New contact position between the plate and a roll. FEM ANALYSIS Accurate modeling of contact conditions between rolls and a plate requires to precisely describe geometry of a roll. Typical approach to this task is to use 3D solid elements. However, a large number of solid elements used to describe rolls will result in a time consuming calculations. To deal with this problem authors utilized a simplified approach – Fig. 7. Rolls are described using beam elements, rigid links and contact segments [3].
A summary of FEM features of our models is presented below: a) Beam elements representing geometry of a roll; b) Contact segments on a surface of each roll; c) Rigid links connecting beam elements’ nodes and contact segments; d) Steel plate described by 3/D solid u/p elements with elastic-plastic material properties; e) Contact between rolls’ contact surface and plate; f) Frame structure is modeled by a grillage. Model of a roll can be compared with human body with flexible back bone (beam elements), rigid rib (rigid links), and skin (contact elements). FIGURE 12. Plastic Strain (3/4). FIGURE 13. Plastic Strain (4/4). CONCLUDING REMARKS a) Due to the deflection of rolls, roll leveling process creates a non-uniform plastic strain on the top and the bottom surfaces of the plate. b) The effect of roll deflection is successfully simulated using non-linear FEM code. c) The total load of a leveler frame – structure shows a very good agreement between numerical results and experimental observations. d) For a given shape distortion - simulation models make possible to effectively calculate technological parameters of a leveling process. e) A very good agreement is observed between plastic strain distribution obtained in numerical simulations and experiments. f) A unique approach to finite element model of a roll has been presented and successfully used in simulations. REFERENCES 1. Higo, T., Matsumoto, H., and Ogawa, S., “ Numerical Simulation for Coupled Analysis of Plate and Mill Deformation in Roller Leveler,” CAMP-ISIJ, Vol.16, 2003, pp. 388-391. 2. Kadota, K., and Maeda, R., “ A Method of Analysis of Curvature in Leveling Process – Numerical Study of Roller Leveling Process 1,” Journal of the JSTP vol.34, No.388 1993, pp.481- 486. 3. ADINA R&D,Inc., “ Theory and Modeling Guide Volume I : ADINA, Report ARD 02-7,” Sept. 2002.