# Reduced-Order Model Based on Neural Network of Roll Bending
## 3 Finite Element Model
### Simulation Results of the Roll Surface Shape  
Comparisons of the 2D and 3D FE models are shown below (Figure 4).  
Figure 4 illustrates the results of the calculations of the displacement, \(\Delta y\), of the surface of the work roll in the cross section of the band that exits from the rolls (shape of the deformed surface) with a rolling force of 35,000 (Figure 4a) and 17,500 kN (Figure 4b). Due to the fact that the bottom set of rolls is considered, these displacements are negative. Wider sheets cause less stress and deformation (flattening) on the surface of the roll. Narrow sheets give large displacements in the central part of the roll, especially in the contact zone with the band. Additionally, at the edges of the sheet, the deformation of the roll surface is rapidly decreasing, more for narrow sheets than for wide sheets. The reduced rolling  
Figure 3: The 2D FE mesh of two rolls (**a**) and side view of the 3D mesh (**b**).  
Figure 2: Sheet convexity and its components (contact with the lower roll); symbols described in the text.  
force (17,500 kN) has less influence on the elastic deformation of the rolls (Figure 4b). The sensitivity of displacement to changes in the width of the band is smaller with smaller loads.  
Figure 5 shows the influence of the anti-bending force on the shape of the roll surface. Figure 5a contains the results obtained for flat rolls, and comparison of Figures 4a and 5a shows that as a result of the applying of the anti-bending force, displacement of the surface of the roll becomes more uniform, but this effect is very small. Displacements closer to the edge of the roll decrease with a greater decrease in the anti-bending force than it does in the center of the roll. Figure 5a shows that applying an anti-bending force to rolls with convexity of \(c=0.5\) gives a flat shape for wide strips with a force of 35,000 kN. Analysis of the results shown in Figure 5b shows that the application of the same anti-bending force as the low rolling force (\(F=17\),500 kN) leads to a convex shape of the rolls, leading to concavity of the sheet, and gives a flat shape for narrower strips.  
The analysis of the results of the calculations shown in Figures 4 and 5 shows that the developed 2D FE model allows one to simulate the elastic deformation of the rolls for various conditions. The model takes into account the impact of various parameters, such as load of the rolls (rolling force), sheet width, convexity of both work and backup rolls, and anti-bending force. This model is also adapted for simulation tests in the field of deformations of rolls made of various materials, rolls that have variable properties in a  
Figure 4: Shape of the surface of the work roll in 2D (filled symbols) and 3D (empty symbols) simulations for sheet widths of 2000 (diamond) and 3500 (square) mm: **(a)** rolling force of \(F=35\),000 kN and roll convexity of \(c=0\) mm; **(b)** rolling force of \(F=17\),500 kN and roll convexity of \(c=0.5\) mm.  
Figure 5: Shape of the surface of the work roll in 2D (filled symbols) and 3D (empty symbols) simulations for sheet widths of 2000 (diamond) and 3500 (square) mm and anti-bending force of \(Q=2500\) kN: **(a)** rolling force of \(F=35\),000 kN and roll convexity of \(c=0\) mm; **(b)** rolling force of \(F=17\),500 kN and roll convexity of \(c=0.5\) mm.  
cross section, or rolls with axial holes. Generally, it can be stated that the program built on the basis of the FE model is very effective. The calculation time for one case on a PC is approximately 1 s. For comparison, the calculations for the 3D model using ABAQUS take from 1 to 2 min. The shape of the rolling sheet for both models is very similar. A slightly higher level of displacement obtained in the 3D model can be explained by an additional layer, taking into account the deformation of bearing.