Koji Yoshimura
Nippon Steel Corporation
Tokyo, Japan
Rajiv Shivpuri
Department of Industrial Systems Engineering
The Ohio State University
Columbus, Ohio
Kosuke Ishii
Department of Mechanical Engineering
Stanford University
Stanford, California
ABSTRACT
This study seeks a robust design of square-to-box rolling that leads to minimum variation in the geometric profile of rolled workpieces. Variations considered are the workpiece geometry, the roll diameter, and roll gap. The paper focuses on the variation of the workpiece width. The key is to define parameters that correctly reflect the physical characteristics of shape rolling. The study applies the Taguchi method to finite element simulation of rolling process using seven design and process parameters. Sensitivity analysis and comparison with empirical knowledge lead to design guidelines that reduce lateral spread variation in the square-to-box roll pass process.
1.0 INTRODUCTION
Quality issues in rolling include dimensional accuracy, surface quality, and inner defects in the finished part. Engineers seek a roll pass sequence that produces quality parts under conditions that may involve significant variation or noise. In actual practice, roll pass design is more of an art that depends on experimental knowledge of experts. To seek further quality improvements, this paper develops a systematic methodology for robust design of the rolling process.
The shape rolling process involves complex 3-D deformation that has attracted many researchers to develop empirical models. Helmi et al. (1968) conducted experiments on a wide range of geometric factors in flat rolling and developed a formula for prediction of the lateral spread ratio. El-Kalay et al. (1968) also addressed flat rolling and clarified the effect of friction between the rolls and material on lateral spread and roll separating force. In shape rolling, Shinokura et al. (1982) developed a formula for prediction of lateral spread in rod rolling. Numerical analysis of the rolling process has been the focus of many studies since the 1980's. Kiuchi et al. (1987) developed a three-dimensional finite element method to simulate shape rolling. Kim et al. (1991) simplified Kiuchi's method by the combination of 2-D FEM and the slab element method. Most of these methods predict the physical state of rolling accurately enough for industries to use them as a tool for process control and management.
Many researchers have studied the Taguchi method and its applications in mechanical design. Kackar (1985) and Hunter (1985) promoted the application of statistical experiments and parameter design to quality control. d'Entremont et al. (1988) adopted the concept of quality loss and developed a nonlinear programming code. Sundaresan et al. (1989, 1991) adapted Taguchi's method and incorporated a Sensitivity Index in the optimization procedure to seek a robust optimum. Another challenge in robust design is dealing with constraint uncertainties. A conventional actively constrained optimum may not be statistically feasible. Sundaresan et al. (1993) compared the efficiency of three different methods which incorporate variations in constraints. Yu et al. (1994) dealt with manufacturing errors that affect design variables with specific characteristics, Manufacturing Variation Patterns .
This study focuses on the robust optimization of the rod rolling pass design in a square-to-box process. Existing numerical methods enables engineers to visually observe the complex rolling process. These methods have benefited process control and management. However, few studies have addressed robust process design. This paper investigates this issue by combining the robust design technology with numerical simulation for the rolling process (Kim et al., 1991). Section 2 gives an overview of the design of the rolling process and identifies the key parameters, namely the workpiece spread. Section 3 defines the design, control, and noise parameters involved in square-to-box rolling. Section 4 contains the result of the simulation experiment that seeks the robust solution. Section 5 compares the result with empirical knowledge and develops a set of general design guidelines. Section 6 gives conclusions and future directions of research.
2.0 DESIGN OF THE ROLLING PROCESS
2.1 Overview of the Rolling Process
Figure 1 shows a typical seven-pass sequence for the rough-rolling train for a rod. The cross section of the initial material is square and the final rough product is round. The geometry at the seventh pass must be quite accurate, for example, 43.5+/-0.5 mm in diameter. In production, this geometry deviates between 42.0 mm and 45.0 mm. The deviation often results in inappropriate geometry in later stages (intermediate or finishing train), causing over or under filling. This variation is related to noise factors inherent in the process, such as variance in material geometry, roll diameter, friction between rolls and material, and roll wear.
Figure 1. Typical seven pass sequence of a rough rolling train
Let us consider flat rolling to understand the mechanics of the process (Figure 2). The stock enters the rolls at point 1 and exits at point 2. Initially the workpiece has width w1, thickness h1, cross-sectional area A1, and length L1. After rolling, it has width w2, thickness h2, cross-sectional area A2, and length L2. The reduction in cross-sectional area is expressed as:
r = [(A1- A2) / A1] x 100 (%) (1)
This reduction of area leads to elongation and spread. Elongation is expressed as the ratio of the final length to initial length.
* Elongation factor : E = L2/ L1 (2)
* Percentage elongation : e = [(L2- L1) / L1] x100 (%) (3)
Spread is due to lateral flow of material.
* Absolute spread : Sa = w2- w1 (4)
* Percentage spread : Sp = [(w2- w1) / w1] x100 (%) (5)
* Proportional spread : Sr= [(w2- w1) / (h1- h2)] x100 (%) (6)
Spread coefficient is expressed as:
S = [ln(w2- w1) / ln(h1- h2)] (7)
2.2 Factors Affecting Lateral Spread
Workpiece spread, or the change in workpiece width, is an important parameter in tracking rolling quality. The complexity of 3-D material flow prevents any simple prediction of the spread. Hence, most of the roll pass design is due to empirical formulae developed over the years. The factors affecting spread fall into three main categories:
Geometric factors: Geometric variables involved in the rolling process, such as initial width to thickness workpiece ratio, shape of the groove, and the roll diameter.
Frictional factors: State of surfaces of the rolls and of the material being rolled. The condition of the scale becomes another governing factor in the case of hot rolling.
Material factors: Factors affecting the yield stress of the material, i.e., strain rate in hot rolling, rolling temperature, and material composition.
Figure 2. Basic Principles of the Rolling Process
2.3 Geometric Factors
Geometry has such a pronounced affect on spread that many researchers have derived a variety of formulae. Of the several geometric parameters involved in hot rolling, the following factors exert a significant influence on spread (Roberts, 1983):
(1) the initial thickness (h1)
(2) the final thickness (h2)
(3) the initial width (w1)
(4) the work roll radius (R)
Helmi et al. (1968) conducted experiments covering a variety of ratios in flat slab rolling, including w1/h1, R/h1 and (h1- h2)/h1, and developed a formula:
(8)
where S is the spread coefficient defined in Equation (7).
Note that the factor w1/[R(h1- h2)]0.5 can be expressed as:
w1/[R(h1- h2)]0.5 = [h1/R]0.5 [h1/(h1- h2)]0.5 [w1/h1] (9)
which suggests that the pertinent geometric factors are w1/h1, R/h1 and (h1- h2)/h1.
Rod rolling is more complex than flat rolling making spread prediction more difficult. Spread calculations by Lendl (1948a, b), Moses (1959) and Shinokura et al. (1982) are still in practical use. These formulae consist of the geometric factors discussed above, i.e., w1/h1, R/h1 and (h1- h2)/h1. Discussions above lead to the following non-dimensional geometric parameters as being significant factors that affect workpiece spread: (1) the initial width / thickness ratio [w1/h1], (2) the roll radius / initial thickness ratio [R/h1], and (3) the reduction ratio [(h1- h2)/h1].
2.4 Frictional Factors
El-Kalay et al. (1968) concluded that smoother rolls provided a higher value for the spread coefficient. Later, the results of computer simulation by Sevenler et al. (1986) found that the effect of friction on lateral spread depends on geometric factors. Generally, under the condition of hot steel rolling without lubricant, the friction coefficient is m=0.7~1.0 (Altan et al., 1983), although the influence on spread is reportedly small.
2.5 Material Factors
Empirically, material factors have a small influence on spread. Sheppard et al. (1981) cited that material properties and strain rate had almost negligible effect. However, their formula suggests that flow stress, expressed by the temperature parameter, has some effect on lateral spread.
3.0 PARAMETERS FOR SQUARE-BOX DESIGN
3.1 Controllable (Design) Parameters
A hexagonal geometry as shown in Figure 3 can define a box caliber, using the following non-dimensional parameters:
(1) Roll Diameter / Material Height Ratio (2R/h1)
(2) Reduction Ratio (h2/h1)
(3) Bottom Width / Material Width Ratio (wb/w1)
(4) Corner Radius / Material Width Ratio (2r/w1)
(5) Relief Angle (a)
Figure 3. Geometry of a Box Caliber
3.2 Other Controllable Parameters
(1) Width/Height Ratio of Material (w1/h1). For billets, this value is usually 1. Since it is important in the discussion of spread, we include this factor as a controllable parameter so that the resulting guidelines retain the largest flexibility possible.
(2) Rolling Temperature (T ). We included this parameter here to investigate the influence of process condition.
3.3 Uncontrollable (Noise) Parameters
The first author's experience led to some observations.
(1) Deviation in Material Geometry (w1/h1): In pass #1, material geometry deviates because of breakdown-rolling allowance and billet treatment. Engineers control the tolerance of measurement on square billet cross section to within +/-2%.
(2) Deviation in Roll Radius (R): The rolls require re-grinding after a given number of products. Usually the effective diameter of a roll is 14~15% of the initial diameter, or +/-7% of the mean diameter. The initial diameter and minimum diameter depend on the characteristics of roll material and equipment.
(3) Deviation in Groove Height (h2): The groove height, deviates after set up because of the roll separating force. One must also account for roll wear. These two factors influence the correct adjustment of rolling condition. Although the deviation of h2 depends on equipment capacity (rigidity of rolls and housing), the range of deviation is approximately +/-2.5% of the initial setting of h2, including roll jumping, roll wear, and operator adjustment.
(4) Other Factors (frictional and material factors): We consider the frictional factor as negligible. El-Kalay et al. (1968) concluded that it affects the percentage spread by only 2%. This study adopted a constant frictional coefficient of m=0.8. Material factors were also neglected here for the same reason. Figure 4 shows no difference in flow stress between AISI1016 (low carbon steel) and AISI1095 (high carbon steel). The variation in rolling temperature, e.g. +/-50[[ring]]C, should not be significant. These factors may affect roll separating force. El-Kalay et al. (1968) concluded that friction affected roll separating force by 44%. The flow stress corresponding to the variance in rolling temperature (+/-50[[ring]]C) has influence on roll separating force. Hence, we assume that h2 includes the variations in frictional and material factors.
Figure 4. Flow Stress and Temperature (Altan, et al)
Figure 5. Geometry of Deformation
3.4 Quality Parameter
Here, we select the percentage spread Sp (%) as the quality output (response parameter):
Sp = [(w2 - w1) / w1]x100 (%) (5)
where w1 is the width of material before rolling, and w2 is the width after rolling as shown in Figure 5. Table 1 summarizes the parameters used in this study
Table 1. Parameters Defined for Experiments
Definition Symbol
1. Controllable (Design) Parameters
1.1 material width / material height w1/h1
1.2 roll diameter / material height 2R/h1
1.3 reduction ratio h2/h1
1.4 bottom width / material width ratio wb/w1
1.5 corner radius / material width ratio 2r/w1
1.6 relief angle a
1.7 rolling temperature T
2. Uncontrollable (Noise) Parameters
2.1 material width / material height [[Delta]](w1/h1)
2.2 roll diameter / material height [[Delta]](2R/h1)
2.3 reduction ratio [[Delta]](h2/h1)
3. Quality (Response) Parameters
3.1 percentage spread Sp = [(w2 - w1) / w1]x100 (%)
4.0 ROBUST DESIGN PROCEDURE
4.1 Design of Experiments
Inner Array (Design Parameters): Table 2 shows the controllable factors and the levels of each value. The deformation in this process seems so complex that there may be a non-linear correlation of spread with those factors, thus three-level experiments are appropriate. The nominal level of each factor represents the existing design. Table 4 is the orthogonal array for this case. Since no interaction is expected between those factors, we applied the L18 array.
Outer Array (Noise factors): Based on the discussion in section 3.3, Table 3 shows the noise parameters and the levels. Assuming the linear effects of noise on the quality parameter, we applied the 2-level experimental design for these parameters. Table 5 is the corresponding L4 orthogonal array (outer array).
Simulation Experiments: We conducted the 18x4 =72 experiments using the finite element computer code TASKS 2.0 (Kim et al, 1991, Lee et al, 1992). Since deformation is the only point of interest, we applied an isothermal simulation method.
Table 2 Levels of Controllable Factors
Factor Examined Levels
Low Nominal High
(A) 0.60 1.00 1.40
w1/h1
(B) 2.50 3.50 4.50
2R/h1
(C) 0.70 0.75 0.80
h2/h1
(D) 0.80 0.90 1.00
wb/w1
(E) 0.60 0.70 0.80
2r/w1
(F) a 30.0[[r 35.0[[r 40.0[[r
ing]] ing]] ing]]
(G) T 800[[ri 1000[[r 1200[[r
ng]]C ing]]C ing]]C
Table
3 Levels of Uncontrollable Factors
Factor Nomina Applied Levels
l
Low High
(dA) 0 - 2.0% + 2.0%
[[Delta]](w1
/h1)
(dB) 0 - 7.0% + 7.0%
[[Delta]](2R
/h1)
(dC) 0 - 2.5% + 2.5%
[[Delta]](h2
/h1)
Table 4 L18 Array for Controllable Factors
Run# A B C D E F G 1 L L L L L L L 2 L N N N N N N 3 L H H H H H H 4 N L L N N H H 5 N N N H H L L 6 N H H L L N N 7 H L N L H N H 8 H N H N L H L 9 H H L H N L N 10 L L H H N N L 11 L N L L H H N 12 L H N N L L H 13 N L N H L H N 14 N N H L N L H 15 N H L N H N L 16 H L H N H L N 17 H N L H L N H 18 H H N L N H LL: Low; N: Nominal; H: High;
Table 5. L4 Array for Uncontrollable Factors
Run# dA dB dC 1 L L L 2 L H H 3 H L H 4 H H L
4.2 Result of the Experiments
Table 6 shows the values of the percentage spread for the L18 experiments. Since the outer array for the noise parameters is L4, each run has four repetitions. ("s" is the standard deviation in each run . See section 4.4 for the definition of Dn).
Table 6 Results of the Experiments over L18 OA
Percentage Spread (%) Sp
1 2 3 4 mean s Dn
1 8.13 7.77 9.04 13.21 9.54 2.508 0.263
2 7.65 5.48 7.95 9.57 7.66 1.684 0.220
3 5.38 4.25 5.99 8.57 6.05 1.832 0.303
4 5.62 5.46 7.10 9.95 7.03 2.079 0.296
5 5.58 3.93 7.31 8.38 6.30 1.956 0.310
6 4.07 4.05 6.17 8.58 5.72 2.149 0.376
7 1.58 3.04 7.53 7.24 4.85 2.993 0.617
8 1.56 1.06 4.33 6.04 3.25 2.354 0.725
9 5.76 5.24 7.65 8.97 6.90 1.722 0.249
10 2.27 1.63 2.20 5.61 2.93 1.810 0.619
11 11.19 10.07 10.99 13.70 11.52 1.545 0.134
12 9.63 8.42 9.14 13.82 10.25 2.430 0.237
13 3.00 3.01 5.72 7.17 4.72 2.073 0.439
14 4.90 3.23 6.55 7.76 5.61 1.974 0.352
15 8.20 6.59 10.14 12.51 9.36 2.550 0.272
16 1.27 1.62 5.44 6.40 3.68 2.616 0.711
17 4.99 4.500 6.75 9.62 6.47 2.314 0.358
18 4.13 4.17 7.58 10.31 6.55 2.984 0.456
4.3
Analysis of VarianceTable 7 shows the analysis of variance, and we observe:
(1) w1/h1, 2R/h1, h2/h1, and wb/w1 have significant effect.
(2) The other parameters have very little effect.
Table 7. ANOVA Summary on Percentage Spread
Source d.f. S.S. M.S. F-valu
e
w1/h1 2 88.52 44.26 9.21**
2R/h1 2 50.43 25.22 5.25**
h2/h1 2 186.13 93.07 19.37*
*
wb/w1 2 39.25 19.63 4.09*
2r/w1 2 8.81 4.40 0.92
a 2 9.51 4.76 0.99
T 2 2.37 1.19 0.25
error 57 273.85 4.80
Total 71 658.87
d.f. : Degrees of Freedom S.S. : Sums of SquareM.S. : Mean Square (= S.S./d.f.)
** : At least 99% confidence
* : At least 95% confidence
4.4 Design for Consistent Lateral Spread
We are seeking the most robust design for the response Sp, i.e., minimum variance. Here, we define the normalized standard deviation Dn as follows ;
Dn = std. deviation (Sp) / mean (Sp) (10)
Figure 6 shows the plot of Dn with respect to each level of design parameters. The plotted value is the mean of Dn from all the experiments that used the particular levels of the parameter. These plots suggest the following.
(1) The spread is most sensitive to w1/h1, 2R/h1, and h2/h1 .
(2) Smaller w1/h1 , h2/h1 and larger 2R/h1 leads to robustness.
(3) The other four parameters do not play a significant role in robust design.
Figure 6(a). Dn versus Parameters and Their Levels
Figure 6(b). Dn versus Parameters and Their Levels
5.0 DISCUSSION
5.1 Comparison with Empirical Knowledge
The result in section 4.4 provides a clear direction for robust design. However, in practical application, there are additional constraints for each parameter.
Parameter w1/h1 has a significant effect on variance of spread and the smaller the better. However, an actual wire or bar mill uses billets only of w1/h1=1. The reasons for this are:
(a) Axis-symmetry of Products: Usually bar products have round or close-to-round shape. To obtain homogeneous deformation , material should have axis-symmetric cross section.
(b) Freedom in Direction: A square billet can be rolled in any direction, thus one does not need to orient the billets during treatment, storage, handling, or carriage in furnace.
(a) (b) (c)
Figure 7. Results of Non-uniform Deformation
(a) double-bulge (b) single-bulge (c) central defect and alligatoring
Mielnik (1991), p.378
Figure 8. Double-bulge Ratio versus [[Delta]]-value
Parameters 2R/h1 and h2/h1
(1) Deformation-zone Parameter ([[Delta]]): The following parameter dominates the homogeneity of deformation (Hosford et al., 1993).
[[Delta]]=h2/[R(h1- h2)]0.5 (11)
Excessive [[Delta]] causes the process to deform only the surface layer of the stock and causes inhomogeneous deformation (double-bulge) as shown in Figure 7(a). In an extreme case, central defects may occur (Figure 7(c)). Figure 8 shows the double-bulge ratio against [[Delta]] observed in this series of experiments. Here,
double-bulge ratio = (wmax-w2)/w2 (12)
where wmax is the maximum width after rolling and w2 is the width on the horizontal center line defined in Figure 5. Figure 8 suggests that double-bulge ratio rises dramatically when [[Delta]] is larger than 1.0, while it is negligible when [[Delta]] is smaller than 1.0.
(2) Angle of bite: The rolls cannot draw material when the friction is too small. Figure 9 shows the forces acting on any point along the arc of contact. Material of thickness h1, brought in contact with the rolls at point A, will be drawn into the rolls if the forward force Hf is larger than Hr, the backward force. Since Hf = Pfsin[[alpha]] and Hr= Prcos[[alpha]], the material will be drawn when Pfsin[[alpha]] <= Prcos[[alpha]], or Pf/Pr >= tan[[alpha]]. Since u = Pf/Pr, the criterion for the material to be drawn into rolls will be u >= tan[[alpha]]. Experts in roll pass design usually set [[alpha]], the angle of bite, at 22.5[[ring]] ~ 24[[ring]], which corresponds to u of 0.40 ~ 0.45 (Roberts, 1988).
Figure 9. Forces Acting on Material (Mielnik, 1991)
(3) Surface quality: The largest problem with rolling is surface quality. If the reduction ratio is too high, the free sides of the material will receive high strain, which may result in wrinkling. Although no quantitative limit exists, experts generally support an empirical limit of h2/h1>0.7.
Figure 10 shows the range of 2R/h1 and h2/h1 which satisfies the conditions discussed above. This figure suggests that
* h2/h1 must be greater than 0.7.
* 2R/h1 must be greater than 3.5.
* Larger h2/h1 requires larger rolls due to the [[Delta]] criterion.
These practical constraints give the conditional boundaries for 2R/h1 and h2/h1.
Figure 11 shows the plot of sn for each level of w1/h1, 2R/h1, and h2/h1 superimposed with the practical boundaries. The suggested values of the parameters are as follows:
(1) w1/h1 should be 1.0.
(2) 2R/h1 must be larger than 3.5, but has no significant effect in the range of 2R/h1>3.5. To reduce the facility and utility cost, a smaller size is preferred.
(3) h2/h1 must be larger than 0.70.
Figure
10. Acceptable Range of Roll Diameter and Reduction
Figure
11. Real Boundaries and Preferable Levels
Figure
12. Correlation of Dn to w2/wb
5.2 Robustness Trends
One can observe in Figure 6 or Figure 11 that the Dn value with respect to the three important parameters, w1/h1, 2R/h1, and h2/h1, is monotonic. In Figure 12, the horizontal axis, w2/wb, indicates how far the material has deformed into the edge part of the box groove. It clearly shows that larger w2/wb leads to smaller Dn. This trend is intuitively correct, since large w2/wb constrains the metal flow of the workpiece near the lateral edge, thus leading to consistent spread.
5.3 Robust Design Guidelines
The sensitivity analysis and the comparison with empirical knowledge provided a direction for robust design of a square-to-box roll pass to reduce variance in lateral spread. The general design guidelines for robustness, on the basis of Figures 6(a)(b) and Figure 11, are as follows:
(1) Figure 11 suggests that w1/h1, 2R/h1, and h2/h1 should be 1.0, 3.5, and 0.70, respectively.
(2) The parameter wb/w1 does not play a significant role in robust design. However, this parameter influences homogeneity of deformation. If too small, severe strain on the edge of material will cause wrinkling. If too large, smaller w2/wb will lead to inconsistent lateral spread as Figure 12 suggests. Hence, the general target should be wb/w1=1.0 .
(3) The parameter 2r/w1 has minimum effect on Dn. However, a small value may lead to local wrinkling, while a large value may lose robustness. Combined with the least value of Dn (Figure 6(b)), we recommend 2r/w1=0.7.
(3) Relief angle a should be smaller according to the trends in robust response, although the effect is small (Figure 6(b)). To prevent overfill, we recommend a=30[[ring]].
(4) Temperature T is a process factor rather than a design parameter. Figure 6(b) shows that T over 1000[[ring]]C does not affect robustness. Hence, if possible, the rolling temperature higher than 1000[[ring]]C is preferable.
Table 7 shows preferred design parameters and an example design of a square-to-box caliber.
Table 7 Preferable Parameters and Example of Design
Paramete Preferable Example of
r Value Design
w1/h1 1.0 w1=4.0",
h1=4.0"
2R/h1 3.5 R=14"
h2/h1 0.70 h2=0.28"
wb/w1 1.0 wb=4.0"
2r/w1 0.70 r=1.4"
a 30 a=30[[ring]]
( T >=1000 T=1000[[ring]]
C )
6.0 CONCLUSIONS AND FUTURE DIRECTION
The study sought a robust design of a square-to-box rolling pass that would lead to a minimum variance of the width of the rolled workpiece under geometric noise factors during the process. We applied the Taguchi method to a finite element simulation of the process. The study led to a general design guideline of a square-to-box roll pass in reducing lateral spread variation.
(1) The parameters w1/h1, 2R/h1, and h2/h1 are most important in robust design of a square-to-box rolling pass. These parameters are the dominant factors on the mean value and variance in lateral spread.
(2) The preferred designs are those with w1/h1=1.0, 2R/h1=3.5 and h2/h1=0.7. The other controllable parameters are not significant factors.
In this study, the simulation program TASKS executed the experiments. Hence, this study may only be as good as the accuracy of this software. However, past verification studies indicate that the program is quite accurate in simulating rolling processes (Kim et al, 1991, Lee et al, 1992). Coupled with the first author's ten years experience in the field, we are quite confident of our claims in this paper.
The authors intend to expand their target to the other patterns included in Figure 1, and to discuss robust design of rod rolling pass sequence in general.
ACKNOWLEDGMENTS
The authors express their sincere appreciation to Mr. Yasuhito Sasaki, Nippon Steel Corporation, for his support in practical background. The authors would also like to thank to Dr. Jyh-Cheng Yu and Dr. Wooyong Shin, former research associates of The Ohio State University, and Prof. Jun Yanagimoto, University of Tokyo, for their support and encouragement. The authors acknowledge the support received from the Engineering Research Center for Net Shape Manufacturing in the use of computer software TASKS. Many thanks to Mr. Kurt Beiter for proof reading the manuscript.
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