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Preform design in ring rolling processes by the three-dimensional finite element method
Int. J, Mach. Tools Manufact. Printed in Great Britain
Vol. 31, No. I, pp.139-151, 1991.
089~-6955/9153.00 + .INi Pergamon Press pie
PREFORM DESIGN IN RING ROLLING PROCESSES BY THE THREE-DIMENSIONAL FINITE ELEMENT METHOD* BEOM-SOO KANGt a n d SHIRO KOBAYASHIt (Received 8 June 1990)
Abstract--Preform design by the three-dimensional finite element method has been carried out for plain ring rolling and T-section profiled ring rolling. The application of the backward tracing scheme for preform design in ring rolling processes demonstrates, for the first time, the extension of the scheme into three dimensions. The preform design in plain ring rolling aims at obtaining a preform which simulates a final plain ring product with uniform axial height. Loading simulations were carried out for two progressively modified preforms derived from the result of loading simulations with a rectangular-shaped ring. Then, backward tracing was applied to obtain a final preform for the specified product configuration of uniform axial height. The preform in the T-section profiled ring rolling process was to be designed to simulate a final profiled ring product with complete filling in the groove and uniform axial height. The final preform shape for plain ring rolling was selected as a trial preform in T-section profiled ring rolling. A more satisfactory preform was obtained from the backward tracing results. It was shown by forward simulation that the final preform was good enough to satisfy the design criteria.
1.
INTRODUCTION
THIS STUDYof preform design in ring rolling processes is the last part of the research series, 'Preform Design in Metal Forming'. The series began with the paper by Park, Rebelo, and Kobayashi [1], in which a new concept, called 'backward tracing scheme', was introduced and applied to preform design in shell nosing. Approximate solutions for preform design in shell nosing and rolling were also obtained by Kobayashi [2, 3]. The backward tracing scheme was used for design of end shapes for eliminating crop loss in rolling [4]. Preform design in square and rectangular cup drawing was carried out by Toh, Kim and Kobayashi [5, 6]. The backward tracing scheme was applied to the design of a disk forging process where a uniformly deformed disk is required under the presence of friction at the die-workpiece interface [7], and the technique including temperature calculations into backward tracing was used for preform design in shell nosing at elevated temperatures [8]. Recently, preforms in blade forging as a twodimensional plane strain problem [9] and in closed-die forging of axisymmetric Hshaped cross-sections [10] have also been obtained using the scheme. The study of preform design in ring rolling processes, presented here, demonstrates, for the first time, the extension of the scheme into three dimensions. Since the first machine for producing seamless rings was built in Manchester in 1842, the ring rolling process has been widely used in industrial fields for manufacturing rings of desirable mechanical properties and good surface quality. As the process involves three-dimensional metal flow, and continuous change of radius and curvature of ring workpiece, designing of the ring rolling process by numerical techniques is complicated. In the past the ring rolling process has been analysed by various methods such as the slab method, the slip line method, the upper bound method, and the two-dimensional finite element method. Johnson et al. have published a large quantity of work on the topic which covers some experimental measurements of the roll torque and pressing load [11], and prediction of roll force and torque by the slip line method [12]. Mamalis et al. [13] carried out experiments for measurement of the pressure distribution along the contact length between roll and workpiece. Spread in ring rolling was analysed by *Part 11 of Preform Design in Metal Forming. fDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A. 139
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BEoM-Soo KANG and S. KOBAYASHI
Mamalis et al. with experimental work [14] and recently by Lugora and Bramley with Hill's general method [15]. The finite element method has been widely used for simulating and designing various metal forming processes such as forging, rolling and extrusion. However, the characteristics of three-dimensional unsteady metal flow and asymmetric geometrical configuration of the workpiece in the ring rolling process make even the numerical analysis difficult. Two-dimensional approaches using the finite element method have been reported previously but under limited conditions [16, 17]. Recently, Kim, Machida and Kobayashi have developed a computer program for the analysis of the ring rolling process by the three-dimensional finite element method using a unique updating procedure for deformation [18]. In this study the backward tracing scheme was incorporated into three-dimensional finite element code and applied to preform design in ring rolling. The main objective of design in plain ring rolling is to obtain a preform which satisfies the design criterion of uniform axial height in the final ring product. In the preform design of T-profiled ring rolling, complete filling in the groove is also a requirement as well as uniformity in the axial height. 2.
METHOD OF ANALYSIS
The preform design by the finite element method in metal forming usually requires not only forward loading simulation but also backward tracing of loading. The finite element method used in the present study is based on the rigid-viscoplastic formulation and general description of the method can be found elsewhere [19]. A ring rolling mill is schematically shown in Fig. 1 for plain ring rolling. A ring shaped blank is expanded to the desired final diameter and is formed to the desired cross-section. The radial set of rolls controls the radial thickness of the ring, while the axial set of rolls controls the height of the ring. The pressure roll is driven radially toward the driving roll and the main deformation of the workpiece occurs within the radial gap leading to increase of the ring diameter. Since geometrical changes of the workpiece are continuous, the procedure of updating deformation as a function of time is critical in analysing the process. Two mesh systems are used for the numerical analysis as shown in Fig. 2. The actual mesh system (AMS) is a mesh system fixed to the workpiece. It stores all information on the material of the workpiece, i.e. geometry of the workpiece (coordinates of the nodal points), history of plastic deformation (total effective strain), etc. The spatial mesh system (SMS) is a mesh system fixed in space but changing its geometry according to that of the AMS. The SMS keeps providing a velocity field for the given geometry by the finite element method. The velocities of the AMS nodes are obtained by interpolating those of the nodes in the SMS. From the interpolated velocities, the displacements in the AMS are calculated during a given time increment. In the SMS, the position vector of the nodal points in the circumferential direction is fixed in space and it changes its geometry in the radial and axial directions according to the shape
GUIDEROLL
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FIG. I. Ring rolling process.
Preform Design
141
AMS(ACTUALMESHSYSTEM) z
(a) x~(e)
~Y
(0
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•,=t--- fine
mesh in deforming
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SMS(SPACIALMESHSYSTEM) (b)
Fl6. 2. Mesh systems for numerical analysis of plain ring rolling. (a) AMS (actual mesh system). (b) SMS (spatial mesh system).
changes of the AMS. During the time step At, the velocity field in the SMS is assumed to be constant and the AMS passes or rotates through the SMS. In every time increment ddt which is smaller than At and used for updating the location of ring center, the velocities are interpolated from the SMS to the AMS. Similarly to forward simulation, the backward tracing method uses the finite element method. Backward tracing refers to the prediction of the part configuration at any stage in a deformation process, when the final part geometry and process conditions are given. The application of backward tracing is straightforward if the changes of the boundary conditions during a process are known. The boundary conditions for backward tracing are usually derived from the loading simulations of a trial preform. Even if the trial preform does not satisfy the design criteria completely, its boundary conditions during loading simulations could be used in the backward tracing simulations. Although three-dimensional deformation is involved in ring rolling, the change of boundary conditions is relatively simple because of the limited region of contact between the two rolls and the workpiece. This aspect of determining the boundary conditions during backward tracing is discussed in some detail in the following section. 3.
P R E F O R M D E S I G N IN P L A I N R I N G R O L L I N G
In commercial ring rolling processes, the edge shape of the ring is generally controlled by the use of edge rolls or flanges on the driving roll. But, in this numerical approach, no extra roll or equipment to control the ring edge is considered. When rolling rings which are unconstrained axially, spreading of the material in the axial direction occurs and an irregular and non-rectangular spread profile is developed, resulting in a characteristic of 'fishtail' appearance [14]. Thus, the design criterion of uniform axial height in the final ring product requires an appropriate preform based on a systemmatic process design. Loading simulations with rings of simple shape are the first trial to derive a preform. If the derived preform does not satisfy the design criterion, backward tracing simulations will be applied to obtain a more satisfactory preform.
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BEoM-Soo KANG and S. KOBAYASHI
The geometrical configuration of the plain ring rolling process is shown in Fig. 3. The ring workpiece is placed between a driving roll and a pressure roll. The pressure roll advances toward the driving roll while the ring is rotated by the driving roll. Axes of reference are taken as shown in the figure, with the origin at the initial contact point between the workpiece and the driving roll of the entry plane; x (or 0) in the circumferential direction, y (or r) in the thickness or radial direction, and z in the axial direction. The height, in this study, means the axial thickness of the ring workpiece. The reduction in thickness is defined as the ratio of the change of ring thickness to the original ring thickness in the radial direction.
Computational conditions For the finite element computations, both the driving roll and the pressure roll are assumed to be rigid. The friction between rolls and ring workpiece is expressed by the friction law of constant factor. In this problem, the value of the friction factor is assumed to be 0.5. The material of the ring workpiece is tellurium lead which is characterized as rate-sensitive and work-hardening. The relationship of the flow stress, total effective strain, and effective strain-rate of the material can be expressed as 6" = 2.24 (1.0 + 16.5~) °°512 ~o.22 (tone/in 2) where 6" is the effective stress, ~ is the effective strain, and ~ is the effective strain rate. The process conditions and workpiece dimensions are as follows: the driving roll radius is 4.5 in., the pressure roll radius is 1.375 in., the rotational speed of the driving roll is 31 rpm and the feedrate of the pressure roll is 0.05167 in/s. The outer diameter and inner diameter of the ring workpiece are 5.0 in. and 3.0 in., respectively. The axial height of the ring workpiece is 1.0 in. for the initial rectangular preform. As the preform design progresses, the axial height of preform varies due to the modification of the cross-sectional shape. One hundred and ninety two nodes and 80 eight-node elements of SMS are used to solve the three-dimensional plain ring rolling. To transform the solution data of SMS by interpolation into actual ring-shaped nodal values, an additional mesh system, AMS, which has 2160 nodes and 900 eight-node elements is used.
Forward loading simulation Knowledge of material flow involved in the process is quite helpful in preform design. Thus the loading simulations contribute to the determination of trial preform shapes. To derive a trial preform from the results of loading simulation of a rectangularshaped ring [18], a method used in industry was applied as shown in Fig. 4. The volumes
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143
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.5" 25" Original Rectangular Shape
41.0% Reduction
FIG. 4. Schematic representation for the determination of a preform from the results of loading simulations of a rectangular-sectioned ring.
of the inner spread and the outer spread from the minimum height are calculated from the result of loading simulation of rectangular-shaped preform at the reduction of 41.0% in thickness. The two volumes are removed from the original rectangular-shaped ring workpiece based on the assumption that the spread volume is distributed linearly on the cross-section in the radial direction from the point of minimum height. It is not expected that a preform derived by this method will satisfy the design criterion of uniform axial height in the final ring product, since the configuration of spread at 41.0% reduction is not linear along the axial edge. The results of loading simulations using the derived preform are shown in Fig. 5(a). The figure represents the 10 degree sector of the whole ring workpiece. Only half of the domain is shown in the figure because of its symmetry to the x-y plane. The deformation pattern of 40.1% thickness reduction shows that the central region of the ring undergoes the minimum material flow in the axial direction relative to the regions in contact with the rolls. It also shows that the maximum axial material flow occurs near the outside surface of the ring, and the axial material flow at the inner side is relatively small. The preform shape in Fig. 5(a) improves uniformity of the axial height in the final reduction. However, the preform shape needs further modification to satisfy the design criterion. The backward tracing scheme is not yet applicable at this stage because of the considerable change needed in rezoning. Thus, another trial preform was derived by the use of the same procedure for modification as applied before (see Fig. 4). Figure 5(b) shows the second trial preform and the deformed shape of the ring at 40% reduction in thickness. The deformation patterns are similar to those of the first trial preform except for the outside part of the ring that is in contact with the driving roll. The results of the loading simulation show that the uniformity of axial height is further improved in comparison with the first preform. Since the modification of the deformed shape in Fig. 5(b) to the configuration of uniform axial height is minor, simulation results of the second preform can be used for the application of the backward tracing scheme, following the same change of the boundary conditions.
Application of the backward tracing scheme Although, in ring rolling processes, the changes of boundary conditions during loading simulation may be relatively simple, the exact boundary conditions are difficult to reproduce during backward tracing simulation. The difficulty arises from the fact that the nodal points of separation or touching of the workpiece are not known exactly because of the updating procedure involving interpolations between SMS and AMS. To overcome this difficulty, the average boundary conditions at each revolution of the ring workpiece are utilized. The number of nodal points and the locations of the specific
144
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(a)
(b) FIG. 5. (a) Grid system in the first trial preform and deformed configuration at 40.1% thickness reduction. (b) Grid system in the second trial preform and deformed configuration at 40.0% thickness reduction.
nodes touching the driving roll and the pressure roll are known from the results of the loading simulation of the preform. Every revolution in the numerical analysis has several combinations of boundary conditions both for the driving roll and for the pressure roll. For each roll and each revolution, a dominant set of boundary conditions is chosen. The sets of contact angles for the driving roll and the pressure roll which are determined from contact nodal points are shown schematically in Fig. 6. The dominant sets of boundary conditions for each numerical revolution are shown in Table 1.
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The program for backward tracing has the same scheme as the loading program for updating with interpolations between AMS and SMS. For application of the backward tracing scheme, the final configuration is assumed to be a ring of uniform axial height and with the same volume as before modification. The construction of the final configuration is done with rezoning of the axial surface, i.e. changing some nodal coordinates in the axial direction in the deformed ring of 40% reduction of the second trial preform so that the axial height of the ring becomes uniform. The application of the technique is now straightforward, since the boundary conditions are specified at each revolution in the backward tracing simulation and the required final ring product is constructed. Figure 7(a) shows the modified final configuration for application of the backward tracing scheme. Figure 7(b) shows the deformed part of the ring workpiece and the circumferential cross-section at 26.7% reduction in thickness traced backward from 40% reduction in thickness shown in Fig. 7(a). The final preform geometry (0% reduction) obtained from the backward tracing simulation is shown in Fig. 7(c). The axial profile of the preform has a smooth configuration and slightly different from those of the first and second trial preforms. The final preform shape thus obtained should satisfy the design criterion of uniform axial height, if the boundary conditions used in the backward tracing simulation and those in the loading simulation are the same. However, complete satisfaction is not expected since the boundary conditions given during the backward tracing are decided by averaging the boundary conditions of the loading simulation of the second trial preform. In order to confirm that the preform resulting from backward tracing satisfies the final design condition, forward loading simulation is performed. The simulation result is shown in Fig. 8. The uniformity in axial height at the reduction of 40.9% is improved
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BEoM-Soo KANG and S. KOBAYASHI
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(a)
iti
(b)
(c)
I
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FIG. 7. Application of backward tracing scheme: (a) final shape rezoned for uniform axial height; (b) 26.7% thickness reduction; and (c) 0.0%.
far more than that with the second trial preform. Even if it is not perfectly uniform in height in the axial direction at the deformed configuration of 40.9% reduction, the straightness of the surface in the circumferential cross-section is considered to be satisfactory. 4.
PREFORM DESIGN IN A T-SECTION PROFILED RING ROLLING
Computational conditions The geometrical configuration of a T-section profiled ring is shown in Fig. 9. The mesh system and coordinate system for the finite element analysis are the same as the plain ring rolling case. The friction factor at the roll-workpiece interface is assumed to be 0.5 for both the main and the pressure rolls. The material properties of tellurium lead are used in the simulations, the same as the plain ring rolling case. As shown in the figure, the driving roll is plain, but the pressure roll has a groove. Process data of the two rolls and the workpiece are as follows. Driving roll: radius = 4.5 in.; rotational speed = 31 rpm. Pressure roll: radius = 1.375 in.; depth of groove = 0.1 in.; maximum width of groove = 0.3 in.; minimum width of groove = 0.24 in.; feedrate = 0.05167 in/s. Workpiece: inner diameter -5.0 in.; outer diameter = 3.0 in.
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y
Z
FIG. 8. Loading simulation of the final preform in plain ring rolling and its deformed configuration at 40.9% reduction.
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One hundred and ninety nodes and 110 eight-node elements of the SMS are used to solve the numerical problem. The solution data of the SMS are interpolated to store in the AMS which has 1710 nodes and 990 eight-node elements. The problem requires that an appropriate preform and a systematic procedure be obtained for producing a final profiled ring product with complete filling in the groove and uniform height in the axial direction. It should be noted, however, that some of the process conditions and requirement are not definite and are flexible for design. For example, friction at the roll-workpiece interface, and material properties are definitely given. However, the stock geometry, volume of the workpiece and axial height in the final ring are not definite. The fixed conditions for these quantities are merely the references for preform design.
Forward loading simulation The prerequisite to preform design using backward tracing scheme is to obtain first a close-to-final preform through forward loading simulations. If it is assumed that the groove in T-section profiled ring rolling does not have significant effect on the uniformity
148
BEOM-SooKANGand S. KOBAYASHI
of axial height, selection of the final preform shape in plain ring rolling for a trial preform is reasonable. Figure 10 shows the preform shape and the results of forward loading simulation. The two stages of loading simulation at 30.4% and 42.3% reduction in thickness are shown in the figure. No fishtail side spread is found at either the inner side or the outer side of the ring throughout the loading simulation. The axial material flow at the central part of the ring is relatively little in comparison with the flow at the outer side. Also, the material flow close to the outer side in the axial direction is much larger than the flow in the inner side as is true in the case of plain ring rolling. The irregular profile is formed at the axial edge face. As shown in the final stage, 42.3% reduction, the groove is filled completely. Thus, when using the preform, complete filling is achieved while the uniformity in axial height needs further improvement. Since the irregularity in the axial surface is not so significant, the backward tracing scheme can be applied to the results shown in Fig. 10 for obtaining the final preform.
Backward tracing simulation Since the filling is satisfied in foward loading simulation, particular attention should be paid to the uniformity in axial height in the final design using the backward tracing scheme. The configuration for starting backward tracing is obtained from the deformed T-section profiled ring at the reduction of 42.3% in thickness shown in Fig. 10(c) by modifying the axial height uniform. This is done in the same manner as for preparation of backward tracing in plain ring rolling. Figure ll(a) shows the modified deformed ring configuration and its circumferential cross-section. To control the geometry of the
(a)
(b)
FIG. 10. Preform shape and its deformed configurations at 30.4% and 42.3% reductions.
Preform Design
149
(b)
(c) FIG. 11. Backward tracing simulations of T-section profiled ring with uniform height: (a) final shape rezoned at 42.3% thickness reduction; (b) 16.7%; and (c) 0.0%.
workpiece during backward tracing, the average boundary conditions at each revolution of the ring workpiece are used as in the plain ring rolling case. Fig. ll(b) shows the deformed part of the ring workpiece and the circumferential cross-section at 16.7% reduction in thickness, and Fig. ll(c) is the final preform shape (corresponding to 0.0% reduction in thickness). The final configuration of 0.0% reduction in thickness derived from the backward tracing simulation does not have a flat profile along the groove. Since the profile along the pressure roll seems to have little effect on complete filling of the groove, the final preform was determined with the same configuration along the axial edge as the result of backward tracing, but with the flat inner surface that is in contact with the grooved pressure roll. The preform is shown in Fig. 12(a). In order to confirm that this preform satisfies the final design conditions, forward loading simulation was performed and the results are given in Fig. 12 (b) and (c). The uniformity in axial height at a reduction of 41.0% is improved far better than that of the previous preform, and considered to be satisfactory. It is also seen that complete filling of the groove is achieved. 5.
SUMMARY AND CONCLUDING REMARKS
The backward tracing scheme using the three-dimensional finite element method is applied to the preform design in ring rolling processes. The simplicity of the change of
150
BEOM-SooKANGand S. KOBAYASH!
(a)
(b)
(c) FIG. 12. (a) A final preformin the T-sectionprofiledring rollingand loadingsimulationresultsat (b) 26.7% and (c) 41.0% thickness reductions.
boundary conditions makes it possible to derive the boundary conditions to be used during backward tracing in the three-dimensional ring rolling problem. The main goal of the preform design in plain ring rolling is to obtain a preform which simulates a final plain ring product with uniform axial height. Loading simulations were carried out for the material properties of tellurium lead, starting with the first trial preform derived from the result of loading simulation with a rectangular-shaped ring. The second trial preform determined by the simulation results of the first trial preform showed the results close enough for application of the backward tracing scheme for the final design in view of the height uniformity in the axial direction and by providing the average boundary conditions each revolution of the ring workpiece. Then, backward tracing was performed to obtain an appropriate preform by starting from the specified configuration of uniform axial height. The final preform determined from the result of the backward tracing was used in loading simulation and showed satisfactory result in the uniformity of axial height. The final preform in the T-section profiled ring rolling was to be designed to simulate a final ring product which has complete filling in the groove and uniform axial height. The final preform shape in the plain ring rolling was selected as a trial preform in the T-section profiled ring rolling. The trial preform shape was modified on the basis of the results of the backward tracing simulation. It is shown that the final preform in the profiled ring rolling satisfied the design criteria for given material and friction condition.
Preform Design
151
An important conclusion is that the information obtained from the backward tracing simulation as well as from the forward loading simulation can be useful in arriving at satisfactory preform design in three-dimensional ring rolling processes. Once a close preform to the final condition by the forward simulation is found, the backward tracing scheme improves the preform shape to more satisfactory shape. It was demonstrated that in the three-dimensional finite element analysis of ring rolling, the use of the average boundary conditions obtained from loading simulations made it possible to apply backward tracing in obtaining the satisfactory final preform. Acknowledgements--The authors wish to thank the National Science Foundation for the grant DDM-8618965 under which the present study was possible, and CRAY Research Inc. for the University Research and Development Grant Program which supported computation with CRAY XM-P/25 supercomputer.
REFERENCES [1] J. J. PARK,N. REBELOand S. KOBAYASHI,A new approach to preform design in metal forming with the finite element method, Int. J. Mach. Tool Des. Res. 23, 71 (1983). [2] S. KOBAYASHI,Approximate solutions for preform design in shell nosing, Int. J. Mach. Tool Des. Res. 23, 111 0983). [3] S. KOBAYASHI,Approximate solutions for preform design in rolling, Int. J. Mach. Tool Des. Res. 24, 215 (1984). [4] S. M. HWANGand S. KOBAYASHI,Preform design in plane-strain rolling by the finite element method, Int. J. Mach. Tool Des. Res., 24, 253 (1984). [5] C. H. TOll and S. KOBAYASHI,Deformation analysis and blank design in square cup drawing, Int. J. Mach. Tool Des. Res. 25, 15 (1985). [6] N. KIM and S. KOBAYASltl,Blank design in rectangular cup drawing by an approximate method, Int. J. Mach. Tool Des. Res. 26, 125 (1986). [7] S. M. HWANGand S. KOBAYASHI,Preform design in disk forging, Int. J. Mach. Tool Des. Res. 26, 231 0986). [8] S. M. HWANGand S. KOBAYASHI,Preform design in shell nosing at elevated temperatures, Int. J. Mach. Tools Manufact. 27, 1-14 (1987). [9] B. S. KANG, and S. KOBAYASHI,Computer-aided preform design in forging of an airfoil section blade. lnt. J. Mach. Tools Manufact. 30, 43--52 (1990). [10] N. KIM and S. KoaAvAsm, Preform design in H-shaped cross sectional axisymmetric forging by the finite element method, Int. J. Mach. Tools Manufact. 30, 243 (1990). [l 1] W. JOHNSON, 1. MACLEODand G. NEEDHAM,An experimental investigation into the process of ring or metal tyre rolling, Int. J. mech. Sci. 10, 455 (1968). [12] J. B. HAWKYARD,W. JOHNSONand J. KIRKLAND,Analyses for roll force and torque in ring rolling with some supporting experiments, Int. J. mech. Sci. 15, 873 (1973). [13] A. G. MAMALIS,W. JOHNSON and J. B. HAWKYARD,Pressure distribution, roll force and torque in cold ring rolling, J. mech. Engng Sci. 18, 196 (1976). [14] A. G. MAMALIS,J. B. HAWKYARDand W. JOHNSON, Spread and flow patterns in ring rolling, Int. J. mech. Sci. 18, 11-16 (1976). [15] C. F. LUGORAand A. N. BRAMLEY,Analysis of spread in ring rolling, Int. J. mech. Sci. 29, 149-157 (1987). [16] D. Y. YANG and K. H. KIM, Rigid-plastic finite element analysis of plane strain ring rolling, Int. J. mech. Sci. 30, 571-580 (1988). [17] Y. MAEKAWA,T. HIRAI, T. KATAYAMAand J. B. HAWKYARD,Modelling and numerical analysis of cross rolling and profile ring rolling processes, Adv. Technol. Plastic. 2, 930 (1984). [18] N. KIM, S. MACHIDAand S. KOBAYASHI,Ring rolling process simulation by the three dimensional finite element method, Int. J. Mach. Tools Manufact. 30, 569-577 (1990). [19] S. KOaAYASm,S. I. OH and T. ALTAN,Metal Forming and the Finite Element Method. Oxford University Press, Oxford (1989).
Vol. 31, No. I, pp.139-151, 1991.
089~-6955/9153.00 + .INi Pergamon Press pie
PREFORM DESIGN IN RING ROLLING PROCESSES BY THE THREE-DIMENSIONAL FINITE ELEMENT METHOD* BEOM-SOO KANGt a n d SHIRO KOBAYASHIt (Received 8 June 1990)
Abstract--Preform design by the three-dimensional finite element method has been carried out for plain ring rolling and T-section profiled ring rolling. The application of the backward tracing scheme for preform design in ring rolling processes demonstrates, for the first time, the extension of the scheme into three dimensions. The preform design in plain ring rolling aims at obtaining a preform which simulates a final plain ring product with uniform axial height. Loading simulations were carried out for two progressively modified preforms derived from the result of loading simulations with a rectangular-shaped ring. Then, backward tracing was applied to obtain a final preform for the specified product configuration of uniform axial height. The preform in the T-section profiled ring rolling process was to be designed to simulate a final profiled ring product with complete filling in the groove and uniform axial height. The final preform shape for plain ring rolling was selected as a trial preform in T-section profiled ring rolling. A more satisfactory preform was obtained from the backward tracing results. It was shown by forward simulation that the final preform was good enough to satisfy the design criteria.
1.
INTRODUCTION
THIS STUDYof preform design in ring rolling processes is the last part of the research series, 'Preform Design in Metal Forming'. The series began with the paper by Park, Rebelo, and Kobayashi [1], in which a new concept, called 'backward tracing scheme', was introduced and applied to preform design in shell nosing. Approximate solutions for preform design in shell nosing and rolling were also obtained by Kobayashi [2, 3]. The backward tracing scheme was used for design of end shapes for eliminating crop loss in rolling [4]. Preform design in square and rectangular cup drawing was carried out by Toh, Kim and Kobayashi [5, 6]. The backward tracing scheme was applied to the design of a disk forging process where a uniformly deformed disk is required under the presence of friction at the die-workpiece interface [7], and the technique including temperature calculations into backward tracing was used for preform design in shell nosing at elevated temperatures [8]. Recently, preforms in blade forging as a twodimensional plane strain problem [9] and in closed-die forging of axisymmetric Hshaped cross-sections [10] have also been obtained using the scheme. The study of preform design in ring rolling processes, presented here, demonstrates, for the first time, the extension of the scheme into three dimensions. Since the first machine for producing seamless rings was built in Manchester in 1842, the ring rolling process has been widely used in industrial fields for manufacturing rings of desirable mechanical properties and good surface quality. As the process involves three-dimensional metal flow, and continuous change of radius and curvature of ring workpiece, designing of the ring rolling process by numerical techniques is complicated. In the past the ring rolling process has been analysed by various methods such as the slab method, the slip line method, the upper bound method, and the two-dimensional finite element method. Johnson et al. have published a large quantity of work on the topic which covers some experimental measurements of the roll torque and pressing load [11], and prediction of roll force and torque by the slip line method [12]. Mamalis et al. [13] carried out experiments for measurement of the pressure distribution along the contact length between roll and workpiece. Spread in ring rolling was analysed by *Part 11 of Preform Design in Metal Forming. fDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A. 139
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BEoM-Soo KANG and S. KOBAYASHI
Mamalis et al. with experimental work [14] and recently by Lugora and Bramley with Hill's general method [15]. The finite element method has been widely used for simulating and designing various metal forming processes such as forging, rolling and extrusion. However, the characteristics of three-dimensional unsteady metal flow and asymmetric geometrical configuration of the workpiece in the ring rolling process make even the numerical analysis difficult. Two-dimensional approaches using the finite element method have been reported previously but under limited conditions [16, 17]. Recently, Kim, Machida and Kobayashi have developed a computer program for the analysis of the ring rolling process by the three-dimensional finite element method using a unique updating procedure for deformation [18]. In this study the backward tracing scheme was incorporated into three-dimensional finite element code and applied to preform design in ring rolling. The main objective of design in plain ring rolling is to obtain a preform which satisfies the design criterion of uniform axial height in the final ring product. In the preform design of T-profiled ring rolling, complete filling in the groove is also a requirement as well as uniformity in the axial height. 2.
METHOD OF ANALYSIS
The preform design by the finite element method in metal forming usually requires not only forward loading simulation but also backward tracing of loading. The finite element method used in the present study is based on the rigid-viscoplastic formulation and general description of the method can be found elsewhere [19]. A ring rolling mill is schematically shown in Fig. 1 for plain ring rolling. A ring shaped blank is expanded to the desired final diameter and is formed to the desired cross-section. The radial set of rolls controls the radial thickness of the ring, while the axial set of rolls controls the height of the ring. The pressure roll is driven radially toward the driving roll and the main deformation of the workpiece occurs within the radial gap leading to increase of the ring diameter. Since geometrical changes of the workpiece are continuous, the procedure of updating deformation as a function of time is critical in analysing the process. Two mesh systems are used for the numerical analysis as shown in Fig. 2. The actual mesh system (AMS) is a mesh system fixed to the workpiece. It stores all information on the material of the workpiece, i.e. geometry of the workpiece (coordinates of the nodal points), history of plastic deformation (total effective strain), etc. The spatial mesh system (SMS) is a mesh system fixed in space but changing its geometry according to that of the AMS. The SMS keeps providing a velocity field for the given geometry by the finite element method. The velocities of the AMS nodes are obtained by interpolating those of the nodes in the SMS. From the interpolated velocities, the displacements in the AMS are calculated during a given time increment. In the SMS, the position vector of the nodal points in the circumferential direction is fixed in space and it changes its geometry in the radial and axial directions according to the shape
GUIDEROLL
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FIG. I. Ring rolling process.
Preform Design
141
AMS(ACTUALMESHSYSTEM) z
(a) x~(e)
~Y
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rough mesh in rigid zone
•,=t--- fine
mesh in deforming
zone
SMS(SPACIALMESHSYSTEM) (b)
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changes of the AMS. During the time step At, the velocity field in the SMS is assumed to be constant and the AMS passes or rotates through the SMS. In every time increment ddt which is smaller than At and used for updating the location of ring center, the velocities are interpolated from the SMS to the AMS. Similarly to forward simulation, the backward tracing method uses the finite element method. Backward tracing refers to the prediction of the part configuration at any stage in a deformation process, when the final part geometry and process conditions are given. The application of backward tracing is straightforward if the changes of the boundary conditions during a process are known. The boundary conditions for backward tracing are usually derived from the loading simulations of a trial preform. Even if the trial preform does not satisfy the design criteria completely, its boundary conditions during loading simulations could be used in the backward tracing simulations. Although three-dimensional deformation is involved in ring rolling, the change of boundary conditions is relatively simple because of the limited region of contact between the two rolls and the workpiece. This aspect of determining the boundary conditions during backward tracing is discussed in some detail in the following section. 3.
P R E F O R M D E S I G N IN P L A I N R I N G R O L L I N G
In commercial ring rolling processes, the edge shape of the ring is generally controlled by the use of edge rolls or flanges on the driving roll. But, in this numerical approach, no extra roll or equipment to control the ring edge is considered. When rolling rings which are unconstrained axially, spreading of the material in the axial direction occurs and an irregular and non-rectangular spread profile is developed, resulting in a characteristic of 'fishtail' appearance [14]. Thus, the design criterion of uniform axial height in the final ring product requires an appropriate preform based on a systemmatic process design. Loading simulations with rings of simple shape are the first trial to derive a preform. If the derived preform does not satisfy the design criterion, backward tracing simulations will be applied to obtain a more satisfactory preform.
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The geometrical configuration of the plain ring rolling process is shown in Fig. 3. The ring workpiece is placed between a driving roll and a pressure roll. The pressure roll advances toward the driving roll while the ring is rotated by the driving roll. Axes of reference are taken as shown in the figure, with the origin at the initial contact point between the workpiece and the driving roll of the entry plane; x (or 0) in the circumferential direction, y (or r) in the thickness or radial direction, and z in the axial direction. The height, in this study, means the axial thickness of the ring workpiece. The reduction in thickness is defined as the ratio of the change of ring thickness to the original ring thickness in the radial direction.
Computational conditions For the finite element computations, both the driving roll and the pressure roll are assumed to be rigid. The friction between rolls and ring workpiece is expressed by the friction law of constant factor. In this problem, the value of the friction factor is assumed to be 0.5. The material of the ring workpiece is tellurium lead which is characterized as rate-sensitive and work-hardening. The relationship of the flow stress, total effective strain, and effective strain-rate of the material can be expressed as 6" = 2.24 (1.0 + 16.5~) °°512 ~o.22 (tone/in 2) where 6" is the effective stress, ~ is the effective strain, and ~ is the effective strain rate. The process conditions and workpiece dimensions are as follows: the driving roll radius is 4.5 in., the pressure roll radius is 1.375 in., the rotational speed of the driving roll is 31 rpm and the feedrate of the pressure roll is 0.05167 in/s. The outer diameter and inner diameter of the ring workpiece are 5.0 in. and 3.0 in., respectively. The axial height of the ring workpiece is 1.0 in. for the initial rectangular preform. As the preform design progresses, the axial height of preform varies due to the modification of the cross-sectional shape. One hundred and ninety two nodes and 80 eight-node elements of SMS are used to solve the three-dimensional plain ring rolling. To transform the solution data of SMS by interpolation into actual ring-shaped nodal values, an additional mesh system, AMS, which has 2160 nodes and 900 eight-node elements is used.
Forward loading simulation Knowledge of material flow involved in the process is quite helpful in preform design. Thus the loading simulations contribute to the determination of trial preform shapes. To derive a trial preform from the results of loading simulation of a rectangularshaped ring [18], a method used in industry was applied as shown in Fig. 4. The volumes
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143
Same Volume
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41.0% Reduction
FIG. 4. Schematic representation for the determination of a preform from the results of loading simulations of a rectangular-sectioned ring.
of the inner spread and the outer spread from the minimum height are calculated from the result of loading simulation of rectangular-shaped preform at the reduction of 41.0% in thickness. The two volumes are removed from the original rectangular-shaped ring workpiece based on the assumption that the spread volume is distributed linearly on the cross-section in the radial direction from the point of minimum height. It is not expected that a preform derived by this method will satisfy the design criterion of uniform axial height in the final ring product, since the configuration of spread at 41.0% reduction is not linear along the axial edge. The results of loading simulations using the derived preform are shown in Fig. 5(a). The figure represents the 10 degree sector of the whole ring workpiece. Only half of the domain is shown in the figure because of its symmetry to the x-y plane. The deformation pattern of 40.1% thickness reduction shows that the central region of the ring undergoes the minimum material flow in the axial direction relative to the regions in contact with the rolls. It also shows that the maximum axial material flow occurs near the outside surface of the ring, and the axial material flow at the inner side is relatively small. The preform shape in Fig. 5(a) improves uniformity of the axial height in the final reduction. However, the preform shape needs further modification to satisfy the design criterion. The backward tracing scheme is not yet applicable at this stage because of the considerable change needed in rezoning. Thus, another trial preform was derived by the use of the same procedure for modification as applied before (see Fig. 4). Figure 5(b) shows the second trial preform and the deformed shape of the ring at 40% reduction in thickness. The deformation patterns are similar to those of the first trial preform except for the outside part of the ring that is in contact with the driving roll. The results of the loading simulation show that the uniformity of axial height is further improved in comparison with the first preform. Since the modification of the deformed shape in Fig. 5(b) to the configuration of uniform axial height is minor, simulation results of the second preform can be used for the application of the backward tracing scheme, following the same change of the boundary conditions.
Application of the backward tracing scheme Although, in ring rolling processes, the changes of boundary conditions during loading simulation may be relatively simple, the exact boundary conditions are difficult to reproduce during backward tracing simulation. The difficulty arises from the fact that the nodal points of separation or touching of the workpiece are not known exactly because of the updating procedure involving interpolations between SMS and AMS. To overcome this difficulty, the average boundary conditions at each revolution of the ring workpiece are utilized. The number of nodal points and the locations of the specific
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BEOM-Soo KANG and S. KOBAYASHI
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(b) FIG. 5. (a) Grid system in the first trial preform and deformed configuration at 40.1% thickness reduction. (b) Grid system in the second trial preform and deformed configuration at 40.0% thickness reduction.
nodes touching the driving roll and the pressure roll are known from the results of the loading simulation of the preform. Every revolution in the numerical analysis has several combinations of boundary conditions both for the driving roll and for the pressure roll. For each roll and each revolution, a dominant set of boundary conditions is chosen. The sets of contact angles for the driving roll and the pressure roll which are determined from contact nodal points are shown schematically in Fig. 6. The dominant sets of boundary conditions for each numerical revolution are shown in Table 1.
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The program for backward tracing has the same scheme as the loading program for updating with interpolations between AMS and SMS. For application of the backward tracing scheme, the final configuration is assumed to be a ring of uniform axial height and with the same volume as before modification. The construction of the final configuration is done with rezoning of the axial surface, i.e. changing some nodal coordinates in the axial direction in the deformed ring of 40% reduction of the second trial preform so that the axial height of the ring becomes uniform. The application of the technique is now straightforward, since the boundary conditions are specified at each revolution in the backward tracing simulation and the required final ring product is constructed. Figure 7(a) shows the modified final configuration for application of the backward tracing scheme. Figure 7(b) shows the deformed part of the ring workpiece and the circumferential cross-section at 26.7% reduction in thickness traced backward from 40% reduction in thickness shown in Fig. 7(a). The final preform geometry (0% reduction) obtained from the backward tracing simulation is shown in Fig. 7(c). The axial profile of the preform has a smooth configuration and slightly different from those of the first and second trial preforms. The final preform shape thus obtained should satisfy the design criterion of uniform axial height, if the boundary conditions used in the backward tracing simulation and those in the loading simulation are the same. However, complete satisfaction is not expected since the boundary conditions given during the backward tracing are decided by averaging the boundary conditions of the loading simulation of the second trial preform. In order to confirm that the preform resulting from backward tracing satisfies the final design condition, forward loading simulation is performed. The simulation result is shown in Fig. 8. The uniformity in axial height at the reduction of 40.9% is improved
146
BEoM-Soo KANG and S. KOBAYASHI
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far more than that with the second trial preform. Even if it is not perfectly uniform in height in the axial direction at the deformed configuration of 40.9% reduction, the straightness of the surface in the circumferential cross-section is considered to be satisfactory. 4.
PREFORM DESIGN IN A T-SECTION PROFILED RING ROLLING
Computational conditions The geometrical configuration of a T-section profiled ring is shown in Fig. 9. The mesh system and coordinate system for the finite element analysis are the same as the plain ring rolling case. The friction factor at the roll-workpiece interface is assumed to be 0.5 for both the main and the pressure rolls. The material properties of tellurium lead are used in the simulations, the same as the plain ring rolling case. As shown in the figure, the driving roll is plain, but the pressure roll has a groove. Process data of the two rolls and the workpiece are as follows. Driving roll: radius = 4.5 in.; rotational speed = 31 rpm. Pressure roll: radius = 1.375 in.; depth of groove = 0.1 in.; maximum width of groove = 0.3 in.; minimum width of groove = 0.24 in.; feedrate = 0.05167 in/s. Workpiece: inner diameter -5.0 in.; outer diameter = 3.0 in.
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y
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FIG. 8. Loading simulation of the final preform in plain ring rolling and its deformed configuration at 40.9% reduction.
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One hundred and ninety nodes and 110 eight-node elements of the SMS are used to solve the numerical problem. The solution data of the SMS are interpolated to store in the AMS which has 1710 nodes and 990 eight-node elements. The problem requires that an appropriate preform and a systematic procedure be obtained for producing a final profiled ring product with complete filling in the groove and uniform height in the axial direction. It should be noted, however, that some of the process conditions and requirement are not definite and are flexible for design. For example, friction at the roll-workpiece interface, and material properties are definitely given. However, the stock geometry, volume of the workpiece and axial height in the final ring are not definite. The fixed conditions for these quantities are merely the references for preform design.
Forward loading simulation The prerequisite to preform design using backward tracing scheme is to obtain first a close-to-final preform through forward loading simulations. If it is assumed that the groove in T-section profiled ring rolling does not have significant effect on the uniformity
148
BEOM-SooKANGand S. KOBAYASHI
of axial height, selection of the final preform shape in plain ring rolling for a trial preform is reasonable. Figure 10 shows the preform shape and the results of forward loading simulation. The two stages of loading simulation at 30.4% and 42.3% reduction in thickness are shown in the figure. No fishtail side spread is found at either the inner side or the outer side of the ring throughout the loading simulation. The axial material flow at the central part of the ring is relatively little in comparison with the flow at the outer side. Also, the material flow close to the outer side in the axial direction is much larger than the flow in the inner side as is true in the case of plain ring rolling. The irregular profile is formed at the axial edge face. As shown in the final stage, 42.3% reduction, the groove is filled completely. Thus, when using the preform, complete filling is achieved while the uniformity in axial height needs further improvement. Since the irregularity in the axial surface is not so significant, the backward tracing scheme can be applied to the results shown in Fig. 10 for obtaining the final preform.
Backward tracing simulation Since the filling is satisfied in foward loading simulation, particular attention should be paid to the uniformity in axial height in the final design using the backward tracing scheme. The configuration for starting backward tracing is obtained from the deformed T-section profiled ring at the reduction of 42.3% in thickness shown in Fig. 10(c) by modifying the axial height uniform. This is done in the same manner as for preparation of backward tracing in plain ring rolling. Figure ll(a) shows the modified deformed ring configuration and its circumferential cross-section. To control the geometry of the
(a)
(b)
FIG. 10. Preform shape and its deformed configurations at 30.4% and 42.3% reductions.
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149
(b)
(c) FIG. 11. Backward tracing simulations of T-section profiled ring with uniform height: (a) final shape rezoned at 42.3% thickness reduction; (b) 16.7%; and (c) 0.0%.
workpiece during backward tracing, the average boundary conditions at each revolution of the ring workpiece are used as in the plain ring rolling case. Fig. ll(b) shows the deformed part of the ring workpiece and the circumferential cross-section at 16.7% reduction in thickness, and Fig. ll(c) is the final preform shape (corresponding to 0.0% reduction in thickness). The final configuration of 0.0% reduction in thickness derived from the backward tracing simulation does not have a flat profile along the groove. Since the profile along the pressure roll seems to have little effect on complete filling of the groove, the final preform was determined with the same configuration along the axial edge as the result of backward tracing, but with the flat inner surface that is in contact with the grooved pressure roll. The preform is shown in Fig. 12(a). In order to confirm that this preform satisfies the final design conditions, forward loading simulation was performed and the results are given in Fig. 12 (b) and (c). The uniformity in axial height at a reduction of 41.0% is improved far better than that of the previous preform, and considered to be satisfactory. It is also seen that complete filling of the groove is achieved. 5.
SUMMARY AND CONCLUDING REMARKS
The backward tracing scheme using the three-dimensional finite element method is applied to the preform design in ring rolling processes. The simplicity of the change of
150
BEOM-SooKANGand S. KOBAYASH!
(a)
(b)
(c) FIG. 12. (a) A final preformin the T-sectionprofiledring rollingand loadingsimulationresultsat (b) 26.7% and (c) 41.0% thickness reductions.
boundary conditions makes it possible to derive the boundary conditions to be used during backward tracing in the three-dimensional ring rolling problem. The main goal of the preform design in plain ring rolling is to obtain a preform which simulates a final plain ring product with uniform axial height. Loading simulations were carried out for the material properties of tellurium lead, starting with the first trial preform derived from the result of loading simulation with a rectangular-shaped ring. The second trial preform determined by the simulation results of the first trial preform showed the results close enough for application of the backward tracing scheme for the final design in view of the height uniformity in the axial direction and by providing the average boundary conditions each revolution of the ring workpiece. Then, backward tracing was performed to obtain an appropriate preform by starting from the specified configuration of uniform axial height. The final preform determined from the result of the backward tracing was used in loading simulation and showed satisfactory result in the uniformity of axial height. The final preform in the T-section profiled ring rolling was to be designed to simulate a final ring product which has complete filling in the groove and uniform axial height. The final preform shape in the plain ring rolling was selected as a trial preform in the T-section profiled ring rolling. The trial preform shape was modified on the basis of the results of the backward tracing simulation. It is shown that the final preform in the profiled ring rolling satisfied the design criteria for given material and friction condition.
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An important conclusion is that the information obtained from the backward tracing simulation as well as from the forward loading simulation can be useful in arriving at satisfactory preform design in three-dimensional ring rolling processes. Once a close preform to the final condition by the forward simulation is found, the backward tracing scheme improves the preform shape to more satisfactory shape. It was demonstrated that in the three-dimensional finite element analysis of ring rolling, the use of the average boundary conditions obtained from loading simulations made it possible to apply backward tracing in obtaining the satisfactory final preform. Acknowledgements--The authors wish to thank the National Science Foundation for the grant DDM-8618965 under which the present study was possible, and CRAY Research Inc. for the University Research and Development Grant Program which supported computation with CRAY XM-P/25 supercomputer.
REFERENCES [1] J. J. PARK,N. REBELOand S. KOBAYASHI,A new approach to preform design in metal forming with the finite element method, Int. J. Mach. Tool Des. Res. 23, 71 (1983). [2] S. KOBAYASHI,Approximate solutions for preform design in shell nosing, Int. J. Mach. Tool Des. Res. 23, 111 0983). [3] S. KOBAYASHI,Approximate solutions for preform design in rolling, Int. J. Mach. Tool Des. Res. 24, 215 (1984). [4] S. M. HWANGand S. KOBAYASHI,Preform design in plane-strain rolling by the finite element method, Int. J. Mach. Tool Des. Res., 24, 253 (1984). [5] C. H. TOll and S. KOBAYASHI,Deformation analysis and blank design in square cup drawing, Int. J. Mach. Tool Des. Res. 25, 15 (1985). [6] N. KIM and S. KOBAYASltl,Blank design in rectangular cup drawing by an approximate method, Int. J. Mach. Tool Des. Res. 26, 125 (1986). [7] S. M. HWANGand S. KOBAYASHI,Preform design in disk forging, Int. J. Mach. Tool Des. Res. 26, 231 0986). [8] S. M. HWANGand S. KOBAYASHI,Preform design in shell nosing at elevated temperatures, Int. J. Mach. Tools Manufact. 27, 1-14 (1987). [9] B. S. KANG, and S. KOBAYASHI,Computer-aided preform design in forging of an airfoil section blade. lnt. J. Mach. Tools Manufact. 30, 43--52 (1990). [10] N. KIM and S. KoaAvAsm, Preform design in H-shaped cross sectional axisymmetric forging by the finite element method, Int. J. Mach. Tools Manufact. 30, 243 (1990). [l 1] W. JOHNSON, 1. MACLEODand G. NEEDHAM,An experimental investigation into the process of ring or metal tyre rolling, Int. J. mech. Sci. 10, 455 (1968). [12] J. B. HAWKYARD,W. JOHNSONand J. KIRKLAND,Analyses for roll force and torque in ring rolling with some supporting experiments, Int. J. mech. Sci. 15, 873 (1973). [13] A. G. MAMALIS,W. JOHNSON and J. B. HAWKYARD,Pressure distribution, roll force and torque in cold ring rolling, J. mech. Engng Sci. 18, 196 (1976). [14] A. G. MAMALIS,J. B. HAWKYARDand W. JOHNSON, Spread and flow patterns in ring rolling, Int. J. mech. Sci. 18, 11-16 (1976). [15] C. F. LUGORAand A. N. BRAMLEY,Analysis of spread in ring rolling, Int. J. mech. Sci. 29, 149-157 (1987). [16] D. Y. YANG and K. H. KIM, Rigid-plastic finite element analysis of plane strain ring rolling, Int. J. mech. Sci. 30, 571-580 (1988). [17] Y. MAEKAWA,T. HIRAI, T. KATAYAMAand J. B. HAWKYARD,Modelling and numerical analysis of cross rolling and profile ring rolling processes, Adv. Technol. Plastic. 2, 930 (1984). [18] N. KIM, S. MACHIDAand S. KOBAYASHI,Ring rolling process simulation by the three dimensional finite element method, Int. J. Mach. Tools Manufact. 30, 569-577 (1990). [19] S. KOaAYASm,S. I. OH and T. ALTAN,Metal Forming and the Finite Element Method. Oxford University Press, Oxford (1989).