- Original Article
- Open Access

# Modification of Roll Flattening Analytical Model Based on the Plane Assumption

- Tao Wang
^{1}, - Qing-Xue Huang
^{1}Email author, - Hong Xiao
^{2}and - Xiang-Dong Qi
^{2}

**31**:46

https://doi.org/10.1186/s10033-018-0246-1

© The Author(s) 2018

**Received:**10 April 2017**Accepted:**30 May 2018**Published:**19 June 2018

## Abstract

Roll flattening is an important component in the roll stack elastic deformation, which has important influence on controlling of the strip crown and flatness. Foppl formula and semi-infinite body model are the most popular analytical models in the roll flattening calculation. However, the roll flattening calculated by traditional flattening models has a great deviation from actual situation, especially near the barrel edges. Therefore, in order to improve the accuracy of roll flattening, a new model is proposed based on the elastic half plane theory. The calculation formulas of roll flattening are deduced respectively under the assumptions of plane strain and plane stress. Then, the two assumptions are combined through the method of introducing an transition coefficient, and the distribution rules of roll flattening for different rolling force, flattening width, roll length and roll diameter are analyzed by using the FEM analysis software Marc. Regarding the ratio of the length to roll end and the roll diameter as variable to fit the transition coefficient, the new model of roll flattening is established based on the elastic half plane theory. Finally, the transition coefficient is fitted to establish the model. Compared with the traditional models, the new model can effectively improve the calculation deviation in the roll end, which has important significance for accurate simulation of plate shape, especially for the distribution of rolling force between rolls.

## Keywords

- Roll flattening
- Plane strain
- Plane stress
- Transition coefficient
- FEM

## 1 Introduction

Many scholars have studied on this problem, Berger et al. [14] established a way for the roll flattening calculation considering the pressure gradients along the roll axis direction. Based on Berger’s theory, Hacquin et al. [15] proposed a semi-analytical model by coupling with the analytical model and FEM. Yu et al. [16–18] chosen Foppl formula as the roll flattening model and established a roll deformation model of 20-high mill. Coupled with the metal plastic deformation, the effects of work roll crown and the second intermediate non-drive roll crown on strip edge drop were analyzed. Jiang et al. [19, 20] chosen the semi-infinite body as the roll flattening model and proposed a roll deformation model in cold thin strip rolling with roll edge contact. Then, the mechanics of the cold strip rolling was analyzed based on the model. Zhou et al. [21, 22] examined the validity of the classical formulae by comparison of the results from different methods and modified the semi-infinite body model based on FEM. Xiao and Yuan et al. [23–27] analyzed the error of semi-infinite body model in roll calculation and proposed an analytical model in flat rolling by boundary integral equation method. Wang et al. [28] used the 3-D elastic-plastic FEM to simulate the thin strip rolling process of UCM cold rolling mill. In the model, a novel calculation approach is proposed to obtain the actuator efficiency factors of the flatness actuators in terms of intermediate roll bending, work roll bending and intermediate roll shifting. Du and Linghu et al. [29–31] developed a 3-D FEM simulation model of six-high CVC rolling process by using nonlinear elastic-plastic finite element method and the elastic flattening of rolls and the elastic workpiece are coupled as a whole in the model.

From the existing research results, it’s seen that the analytical method based on semi-infinite body model lack of theoretical support and the FEM cost considerable time [32]. For the geometry and stress situation of the work roll and back-up roll, the space problem can be simplified to plane problem approximately, which can effectively reduce the amount of calculation and meet the engineering precision requirement. In this paper, the roll flattening model is analyzed and modified based on the elastic half plane theory.

## 2 Formula Derivation Based on Elastic Half Plane Theory

*p*, which is balanced with shear stress. The sketch map of force analysis is shown as Figure 2.

*p*along the direction of flattening width 2

*b*is elliptical and according to the elastic theory [33], the stress component of the roll cross section along the directions of the

*X*axis and

*Y*axis in the rectangular coordinate system can be expressed as:

- (1)
Plane stress assumption (the axial stress is 0)

*Y*axis can be obtained by integration:

*OA*can be obtained by Integration from 0 to

*R*:

*OB*can be obtained by integration from −

*R*to 0:

- (2)
Plane strain assumption (the axial strain is 0)

For plane strain assumption, according to the elastic mechanics theory, the calculation formula can be obtained by replacing *E* to \( \frac{E}{{1 - \upsilon^{2} }} \) and replacing *υ* to \( \frac{\upsilon }{1 - \upsilon } \).

*OA*can be expressed as:

*OB*can be expressed as:

*OA*section. For the work roll, which bears the pressure on both sides, therefore, the roll flattening need be calculated not only in the bearing area (

*OA*section) caused by the pressure between rolls

*p*, but also in the nonbearing area (

*OB*section) caused by the rolling force

*q*. The total size of the flattening should be the sum of the work roll’ s and back-up roll’s.

## 3 Modification of Roll Flattening Model

### 3.1 Finite Element Simulation of Roll Flattening

In the roll flattening calculation, because the area of plane strain assumption and plane stress assumption is uncertain, the commercial finite element software MSC. Marc is employed to simulate the roll flattening and offer reference for the comprehensive application of the two assumptions. An example is illustrated for this, in which the roll diameter is 850 mm, the roll length is 1780 mm and the roll surface bears the uniform load 10 N/mm along the direction of roll length. In order to have higher comparability between the results calculated by finite element method and plane, the flattening width is calculated by Hertz formula firstly, and then according to the elliptic curve assumption of distribution rule, the grids in stress area are subdivided to make the finite element model and analytical model have same input conditions.

Comparison of roll flattening

Flattening size | The centre (minimum) (mm) | The end (maximum) (mm) |
---|---|---|

Based on plane assumption | 0.129 | 0.146 |

Calculated by FEM | 0.128 | 0.152 |

### 3.2 Introduction and Fitting Method of Transition Coefficient

*δ*is the size of roll flattening;

*δ*

_{0}is the size of roll flattening calculated by the plane strain assumption;

*δ*

_{1}is the size of roll flattening calculated by the plane stress assumption;

*x*is the distance from the roll end;

*s*is the length of roll edge area; and

*k*(

*x*) is the transition coefficient.

FEM simulation conditions of roll flattening

No. | Type | Value |
---|---|---|

1 | Pressure size (kN/mm) | 5/10/15 |

2 | Flattening width (mm) | 4/5.47/7 |

3 | Roll length (mm) | 1335/1780/2225 |

4 | Roll diameter (mm) | 700/850/1000 |

*x*and the roll diameter

*D*as variables, as shown in Figure 7.

*k*(

*x*) by taking

*x*/

*D*as variable, which can be expressed as

*a*is the fitting coefficient,

*a*= 4.8269 × 10

^{−5}.

## 4 Influence Analysis of Roll Flattening Model on Strip Shape Simulation

In roll stack deformation calculation, the modified flattening model can be used to calculate the flattening between rolls. Because the length of work roll is longer than strip width, it is reasonable to regard the work roll as a semi infinite body. To study the influence on strip shape calculated by different flattening models, an example of F1 stand is simulated by 3-D FEM Coupled Model [34].

From Figure 9(a) it can be seen that the size of roll flattening calculated by the modified model is larger than Foppl formula in the area of both ends which is suitable for plane stress assumption. As shown by the value, the extends range is about 10 μm which is about 4% of the average size of flattening, while the values in roll central are basically identical. From Figure 9(b) it can be seen that the pressure between rolls in the area of both ends becomes smaller since the flattening changing. As shown by the value, the reduction range is about 1 kN/mm which is more than 10% of the average pressure between rolls. From the Figure 9(c) and Figure 9(d) it can be seen that the strip thickness and front tension are basically identical. Although the modification of flattening model does not have obvious influence on the final results of strip shape, the influence of pressure between rolls should not be ignored. In view of the actual production, the excessive wear of back-up rolls often appears, so the modified flattening model is of important significance for the accurate calculation of pressure between rolls in the roll contour optimization.

## 5 Conclusions

- (1)
For the roll flattening, the change from the roll central to the edge can be regarded as a transformation from plane strain problem to plane stress problem. According to the elastic theory, the calculation formulas are deduced respectively under the assumptions of plane strain and plane stress.

- (2)
Through introducing the transition coefficient, the plane strain assumption and the plane stress assumption are combined. By using the FEM software, it can be seen that the different rolling force, different flattening width and different roll length have little influence on transition coefficient in a certain range, which can be ignored; while the different roll diameters influence transition coefficient obviously.

- (3)
Regarding the ratio of the length to roll end and the roll diameter as variable to fit the transition coefficient, the new model of roll flattening is established based on the elastic half plane theory.

- (4)
By contrast, the modified flattening model has little influence on the distribution of strip thickness and tension, but the influence of pressure distribution between rolls should not be ignored which is of important guiding significance to analyze the roll wear, and so on.

## Declarations

### Authors’ contributions

Q-XH was in charge of the whole trial; TW wrote the manuscript; HX and X-DQ assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

### Authors’ Information

Tao Wang, born in 1985, is currently a lecturer at *Taiyuan University of Technology, China*. He received his Ph.D. degree from *Yanshan University, China*, in 2013. His research interests include rolling process and equipment.

Qing-Xue Huang, born in 1960, is currently a professor at *Taiyuan University of Technology, China*. He received his PhD degree from *Yanshan University, China*, in 1999. His research interests include rolling process and equipment, composite material forming and mechanical design.

Hong Xiao, born in 1962, is currently a professor at *Yanshan University, China*. He received his Ph.D. degree from *Yanshan University, China*, in 1991. His research interests include computer simulation of rolling process and plastic processing theory.

Xiang-Dong Qi, born in 1970, is currently a professor at *Yanshan University, China*. He received his Ph.D. degree from *Yanshan University, China*, in 2002. His research interests include shape control theory, rolling process and equipment.

### Competing interests

The authors declare that they have no competing interests.

### Funding

Supported by Shanxi Provincial Science and Technology Major Project of China (Grant No. MC2016-01), Major Program of National Natural Science Foundation of China (Grant No. U1710254), and Natural Science Foundation of Shanxi Province (Grant No. 201701D221143).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Z C Lin, V H Lin. A study of radial thermal expansion of a work roll for three-dimensional strip deformation during rolling.
*Journal of Materials Processing Technology*, 1997, 70(1-3): 62-76.View ArticleGoogle Scholar - V Salganik. Mathematical modelling of roll load and deformation in a four-high strip mill.
*Journal of Materials Processing Technology*, 2002, 125(02): 695–699.View ArticleGoogle Scholar - R Nandan, R Rai, R Jayakanth, et al. Regulating crown and flatness during hot rolling: a multiobjective optimization study using genetic algorithms.
*Advanced Materials and Manufacturing Processes*, 2005, 20(3): 459-478.View ArticleGoogle Scholar - Y S Hu, X R Cheng, H X Li, et al. Roll deformation rule of varying contact length back-up roll under the condition of equal contact pressure between rolls.
*Research on Iron & Steel*, 2006, 34(3): 37-39. (in Chinese)Google Scholar - D H Zhang, J Z Cao, J J Xu, et al. Simplified weighted velocity field for prediction of hot strip rolling force by taking into account flattening of rolls.
*Journal of Iron and Steel Research, International*, 2014, 21(7): 637-643.View ArticleGoogle Scholar - R Li, Q Zhang, X Zhang, et al. Control method for steel strip roughness in two-stand temper mill rolling.
*Chinese Journal of Mechanical Engineering*, 2015, 28(3): 573-579.View ArticleGoogle Scholar - J Cao, X Chai, Y Li, et al. Integrated design of roll contours for strip edge drop and crown control in tandem cold rolling mills.
*Journal of Materials Processing Technology*, 2017, 252(7): 432-439.Google Scholar - H Liu, S Chen, Y Peng, et al. Strip layer method for analysis of the three-dimensional stresses and spread of large cylindrical shell rolling.
*Chinese Journal of Mechanical Engineering*, 2015, 28(3): 556-564.View ArticleGoogle Scholar - Y Zhao, C Xia, H Wang, et al. Analysis and numerical simulation of rolling contact between sphere and cone.
*Chinese Journal of Mechanical Engineering*, 2015, 28(3): 521-529.View ArticleGoogle Scholar - D Wang, H Liu, Liu J. Research and development trend of shape control for cold rolling strip.
*Chinese Journal of Mechanical Engineering*, 2017, 30(5): 1248-1261.Google Scholar - Y Tozawa, M Ueda. Analysis to obtain the pressure and distribution from the contour of deformed roll.
*Journal of the Japan Society for Technology of Plasticity*, 1970, 11: 29-37.Google Scholar - M C Chen, H Q Yu. Three-dimensional surface crack analysis of elastic half-space under rolling contact load.
*International Journal of Computational Methods*, 2011, 06(2): 317-332.View ArticleMATHGoogle Scholar - R J Roark.
*Formulas for stress and strain*. New York: McGraw-Hill, 1954.Google Scholar - B Berger, O Pawelski, P Funke. Elastic deformation of rolls in four-high rolling stands.
*Archiv fuer das Eisenhuettenwesen*, 1976, 47(6): 351-356.View ArticleGoogle Scholar - A Hacquin, P Montmitonnet, J P Guillerault. A three-dimensional semi-analytical model of rolling stand deformation with finite element validation.
*European Journal of Mechanics*-*A/Solids*, 1998, 17(1): 79-106.View ArticleMATHGoogle Scholar - H L Yu, X H Liu, G T Lee. Contact element method with two relative coordinations and its application of prediction for strip profile for a sendzimir mill.
*ISIJ International*, 2007, 47(7): 996-1005.View ArticleGoogle Scholar - H L Yu, X H Liu, G T Lee, et al. Numerical analysis of roll deflection for sendzimir mill.
*Journal of Manufacturing Science and Engineering*, 2008, 130(1): 0110161-0110167.View ArticleGoogle Scholar - H L Yu, X H Liu, G T Lee, et al. Numerical analysis of strip edge drop for sendzimir mill.
*Journal of Materials Processing Technology*, 2008, 208: 42-52.View ArticleGoogle Scholar - Z Y Jiang, A K Tieu. Contact mechanics and work roll wear in cold rolling of thin strip.
*Wear*, 2007, 263(7-12): 1447-1453..View ArticleGoogle Scholar - Z Y Jiang, D Wei, A K Tieu. Analysis of cold rolling of ultra thin strip.
*Journal of Materials Processing Technology*, 2009, 209(9): 4584-4589.View ArticleGoogle Scholar - S Zhou, P Funke, J Zhong. Influence of roll geometry and strip width on flattening in flat rolling.
*Steel Research International*, 1996, 67(5): 200-204.View ArticleGoogle Scholar - S Zhou, P Funke, J Zhong, et al. Modification of a classical formula for determination of roll flattening in flat rolling.
*Steel Research International*, 1996, 67(11): 491-494.View ArticleGoogle Scholar - H Xiao, Z W Yuan, T Wang. Roll flattening analytical model in flat rolling by boundary integral equation method.
*Journal of Iron and Steel Research, International*, 2013, 20(10): 39-45.View ArticleGoogle Scholar - Z W Yuan, H Xiao. Plate shape control theory and experiment for 20-high mill.
*Journal of Iron and Steel Research, International*, 2015, 22(11): 996-1001.View ArticleGoogle Scholar - Z W Yuan, H Xiao, H B Xie. Practice of improving roll deformation theory in strip rolling process based on boundary integral equation method.
*Metallurgical and Materials Transactions A*, 2014, 45(2): 1019-1026.View ArticleGoogle Scholar - Z W Yuan, H Xiao, H B Xie. 6-Hi mill deformation analysis based on boundary integral equation method.
*Journal of Mechanical Engineering*, 2013, 49(18): 125-131. (in Chinese)View ArticleGoogle Scholar - Z K Ren, Z W Yuan, H Xiao, et al. Plate shape control of work roll edge contact for 4-Hi mill.
*Journal of Yanshan University*, 2016, 40(1): 39-44. (in Chinese)Google Scholar - Q L Wang, J Sun, Y M Liu, et al. Analysis of symmetrical flatness actuator efficiencies for UCM cold rolling mill by 3D elastic–plastic FEM.
*The International Journal of Advanced Manufacturing Technology*, 2017, 92(1-4): 1371-1389.View ArticleGoogle Scholar - F S Du, T Xue, J N Sun. FEM analysis for roll deformation and flattening of six-high CVC mill.
*Journal of Yanshan University*, 2011, 35(5): 396-401. (in Chinese)Google Scholar - J N Sun, T Xue, F S Du. FEM Analysis of contact stress and crown control for six-high CVC mill.
*Iron & Steel*, 2012, 47(2): 49-52. (in Chinese)Google Scholar - K Z Linghu, Z Y Jiang, J W Zhao, et al. 3D FEM analysis of strip shape during multi-pass rolling in a 6-high CVC cold rolling mill.
*The International Journal of Advanced Manufacturing Technology*, 2014, 74(9-12): 1733-1745.View ArticleGoogle Scholar - M A Cavaliere, M B Goldschmit, E N Dvorkin. Finite element simulation of the steel plates hot rolling process.
*International Journal for Numerical Methods in Engineering*, 2001, 52(52): 1411-1430.View ArticleMATHGoogle Scholar - S Timoshenko.
*Theory of elasticity*. New York: McGraw-Hill, 1951.MATHGoogle Scholar - T Wang, H Xiao, T Y Zhao, et al. Improvement of 3-D FEM coupled model on strip crown and flatness in hot rolling.
*Journal of Iron and Steel Research, International*, 2012, 19(3): 14-19.View ArticleGoogle Scholar