Abstract:
A sheet rolling method uses a rolling mill ( 10 ) for rolling a sheet (x). The rolling mill ( 10 ) includes upper and lower backup rolls ( 13, 14 ), and a pair of work rolls ( 11, 12 ) respectively having different diameters and disposed between the upper and the lower backup roll ( 13, 14 ). A small-diameter work roll ( 11 ) of the pair of work rolls is disposed so that a rotational axis thereof is positioned on a mill center or a downstream side with respect to the mill center in a rolling direction, and the large-diameter work roll ( 12 ) is disposed so that a rotational axis thereof is positioned on a downstream side with respect to the rotational axis of the small-diameter work roll ( 11 ) in the rolling direction. Thus, mechanical load on the work rolls ( 11, 12 ) can be reduced even when a high rolling force is necessary for rolling a wide sheet.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of Invention 
     The present invention relates to a rolling mill provided with a pair of work rolls having different diameters, and a sheet rolling method employing the same rolling mill. 
     2. Description of Related Art 
     A conventional rolling mill is provided with upper and lower work rolls respectively having different diameters and supported by upper and lower backup rolls. In this type of rolling mill, the larger work roll, i.e., the work roll having a larger diameter, is driven by a motor or the like to roll a sheet. A rolling mill provided with work rolls having different diameters, sometimes called a differential rolling mill (as compared with ordinary rolling mills provided with work rolls of the same diameter) is able to roll a sheet at a high draft by a low rolling force, which is advantageous in manufacturing steel sheets by rolling. Since only a small rolling force is necessary, edge drop resulting from the flattening of the rolls can be suppressed and hence steel sheets having a small thickness deviation can be manufactured. 
     Generally, as shown in FIG. 8, working rolls  11 ′ and  12 ′ included in most rolling mills are shifted downstream by an offset e with respect to backup rolls  13 ′ and  14 ′. The work rolls are thus shifted downstream with respect to the backup rolls because a rolling mill in which work rolls are shifted downstream with respect to backup rolls is able to stabilize loading conditions for loading a rolled sheet more effectively than a rolling mill in which work rolls are shifted upstream with respect to backup rolls. 
     A related art is disclosed in JP-B No. 47421/1976. 
     Recently, hot rolling techniques for hot-rolling sheets are required to be capable of rolling sheets in a greater rolling width, i.e., the width of the rolled sheet, and in smaller thickness, and of rolling sheets at higher drafts. However, the diameter of the smaller work roll of the differential rolling mill is smaller and the mechanical strength of the smaller work roll is insufficient to meet the foregoing requirements. More specifically, a high stress is induced in necks, including stepped parts, at the joints of the body, which is used for rolling, of the smaller work roller and the journals, supported in bearings, of the smaller work roll. 
     Thus, the upper limit of the rolling width of steel sheets hot-rolled by differential rolling mills has been 4 ft (about 1200 mm). Even the differential rolling mill that needs a relatively low rolling force requires a high rolling force exceeding 3000 tons (3000 tf=2.94×10 7  N) when the rolling width exceeds 4 ft. An excessively high stress unbearable by the mechanical strength of the smaller work roll is thus induced in the necks of the smaller work roll. 
     The present invention is intended to meet the foregoing requirements required of rolling mills for hot-rolling sheets, including capability of rolling sheets in an increased width exceeding 4 ft by reducing mechanical load on work rolls. 
     SUMMARY OF THE INVENTION 
     A sheet rolling method according to a first aspect of the present invention includes: disposing a pair of work rolls respectively having different diameters between upper and lower backup rolls; and driving only the large-diameter work roll having the greater diameter for rolling to produce a sheet; wherein the small-diameter work roll having the smaller diameter is disposed so that a rotational axis of the small-diameter work roll is positioned on a mill center or a downstream side with respect to the mill center in a rolling direction, and the large-diameter work roll is disposed so that a rotational axis of the large-diameter work roll is positioned on a downstream side with respect to the rotational axis of the small-diameter work roll in the rolling direction. 
     Since the sheet rolling method does not shift both the two working rolls on the upstream side of the mill center plane including the center axes of the backup rolls with respect to the rolling direction, loading conditions for rolling the sheet is stabilized, and the sheet can be smoothly and continuously rolled. 
     This sheet rolling method is characterized in reducing mechanical load on the work rolls even when a high rolling force is necessary for rolling a wide sheet. 
     The ability of the sheet rolling method to reduce the mechanical load on the work rolls can be reasoned as follows. 
     When a rolling mill provided with two work rolls respectively having different diameters operates for rolling to produce a sheet, the following forces a) to c) are exerted on the journals, supported in bearings, of the smaller work roll having a smaller diameter: 
     a) A horizontal force acting downstream with respect to the rolling direction resulting from driving only the large-diameter work roll having the greater diameter and exerted on the small-diameter work roll by a sheet being rolled (Force SR 1  in FIG.  3 ); 
     b) A roll bender force acting on the work roll in a plane (vertical plane) perpendicular to the rolling direction (Force P B , not shown); and 
     c) A horizontal force equal to the difference between the horizontal components of vertical forces exerted on the small-diameter work roll by the backup roll and the large-diameter work roll (SB 1  and SD 1  shown in FIG. 2) (Force P mt  shown in FIG.  2 ). 
     These forces are exerted on the journals supported in the bearings to induce stresses in the necks of the small-diameter work roll. 
     Although all those forces are produced necessarily during the rolling operation, the magnitude (and the direction, in some cases) of the horizontal force (P mt ) produced by the forces (SB 1  and SD 1 ) is dependent on the dispositions of the large and the small-diameter work rolls relative to the backup rolls, represented by offsets. 
     According to an exemplary embodiment of the invention, the small-diameter and the large-diameter work rolls are disposed such that the offset of the axis of the large-diameter work roll with respect to the mill center plane is greater than the offset which could be zero in some cases of the small-diameter work roll with respect to the mill center plane. This arrangement provides for the horizontal component (SB 1 ) of the vertical force exerted by the large-diameter work roll on the small-diameter work roll and used to determine the horizontal force (P mt , the force c)) to be directed upstream with respect to the rolling direction. Consequently, the horizontal force (P mt , the force c)) is reduced. Since the direction of the horizontal component (SB 1 ) is opposite to the rolling direction, the horizontal force that acts on the small-diameter work roll, i.e., the resultant force acting on the small-diameter work roll, i.e., the sum of the horizontal force (SR 1 , the force a)) and the horizontal component (SB 1 ), is reduced. When the horizontal force is reduced, the mechanical load on the small-diameter work roll is reduced accordingly even if the vertical force, such as the force b), does not change. Consequently, a sheet having a big width and a small thickness can be produced and draft at which the sheet can be rolled by one rolling mill can be increased. 
     In the sheet rolling method according to the first aspect of the present invention, it is preferable that an offset e 1  by which the rotational axis of the small-diameter work roll is shifted from the mill center plane, and an offset e 2  by which the rotational axis of the large-diameter work roll is shifted from the rotational axis of the small-diameter work roll (refer to FIG. 1 for e 1  and e 2 ) meet inequalities: 
     
       
         0 mm≦ e   1  and 0 mm&lt; e   2 &lt;7 mm 
       
     
     when the small-diameter work roll has necks of a diameter of about 270 mm or below, and a rolling force of 3000 tons (3000 tf=2.94×10 7  N, 1 ton=1 tf=9800 N) or above is used for rolling work. 
     The diameter of the body of the small-diameter work roll having the necks of a diameter of 270 mm or below is limited by the relation of the small-diameter work roll with support means including bearings and is considerably small, such as about 400 mm or below. Since the small-diameter work roll has such a small diameter, the sheet can be rolled at a high draft by using a low rolling force. Consequently, edge drop in the sheet can be suppressed and advantages specific to differential rolling mills can be fully utilized. 
     As mentioned above, it is generally advantageous that both the small-diameter and the large-diameter work roll are shifted downstream of the rolling direction with respect to the mill center plane and that the offset of the large-diameter work roll is greater than the offset of the small-diameter work roll, namely, 
     
       
         0≦ e   1  and 0&lt; e   2 , 
       
     
     when a high rolling force is used. 
     However, since the smaller the diameter of the small-diameter work roll, the more effective are the advantages specific to differential rolling mills, and the higher the rolling force, the greater the possibility of increasing rolling width. Accordingly, the offsets e 1  and e 2  must satisfy inequalities: 
     
       
         0≦ e   1  and 0 mm&lt; e   2 &lt;7 mm 
       
     
     under rolling conditions, where the diameter of the necks of the small-diameter work roll is 270 mm or below and the rolling force is about 3000 tons or above (a proper roll bender force is also added) because the horizontal force of c) increases and a stress excessively high to the small-diameter work roll of a general material is induced in the necks of the small-diameter work roll when the diameter of the necks of the small-diameter work roll and the rolling force meet the foregoing conditions, if e 2 ≦0. An undesirable warping of the sheet called bowing, i.e., the upward warping of the leading edge of the sheet passed between the small-diameter and the large-diameter work roll, occurs if 7 mm≦e 2 . A sheet of a width (rolling width) on the order of 5 ft can be produced by hot-rolling a steel sheet when the rolling force is about 3000 tons or above. 
     A rolling mill according to a second aspect of the present invention includes: upper and lower backup rolls; and a pair of work rolls respectively having different diameters and disposed between the upper and the lower backup roll; wherein only the large-diameter work roll having the greater diameter is connected to a driving source, and wherein the small-diameter work roll having the smaller diameter is disposed so that a rotational axis of the small-diameter work roll is positioned on a mill center or a downstream side with respect to the mill center in a rolling direction, and the large-diameter work roll is disposed so that a rotational axis of the large-diameter is positioned on a downstream side with respect to the rotational axis of the small-diameter work roll in the rolling direction. 
     In the rolling mill according to the second aspect of the present invention, it is preferable that the small-diameter work roll has necks of a diameter of 270 mm or below, and an offset e 1  by which the axis of the small-diameter work roll is shifted from the mill center plane, and an offset e 2  by which the axis of the large-diameter work roll is shifted from the axis of the small-diameter work roll meet inequalities: 
     
       
         0 mm≦ e   1  and 0 mm&lt; e   2 &lt;7 mm. 
       
     
     Since the rolling mill is provided with the small-diameter work roll having the necks of a diameter of about 270 mm or below and a body of a considerably small diameter on the order of, for example, 400 mm, the rolling mill has characteristics specific to differential rolling mills and is capable of rolling a sheet at a high draft. Thus, the rolling mill is capable of producing steel sheets having uniform thickness effectively by rolling. 
     The rolling mill, in which the respective offsets e 1  and e 2  of the small-diameter and the large-diameter work roll satisfy the equalities: 
     
       
         0 ≦e   1  and 0 &lt;e   2 &lt;7 mm, 
       
     
     has excellent abilities a) to hot-roll a steel sheet having a width (rolling width) on the order of 5 ft by using a rolling force of 3000 tons or above, b) to limit stress induced in the necks of the small-diameter work roll below a level that is not dangerous to the small-diameter work roll formed of a general material even if such a high rolling force is used, and c) to prevent undesirable bowing of the sheet passed between the work rolls. 
     Preferably, in the sheet rolling mill according to the second aspect of the present invention, the small-diameter work roll has a core formed of a material having a tensile strength of 45 kgf/mm 2  or above (4.41×10 8  Pa), such as a nickel grain roll (cast high-alloy steel grain roll), a high-chromium alloy roll (high-chromium cast steel), high-speed steel roll (high-speed tool steel) or a forged high-speed steel roll. 
     When the small-diameter work roll is a nickel grain roll, a high-chromium alloy roll, a high-speed steel roll or a forged high-speed steel roll formed of a material having a tensile strength of 45 kgf/mm 2  or above, the rolling method according to the first aspect of the present invention can be advantageously carried out without being subject to restrictions, because a rolling force of abut 3000 tons or above can be exerted on the small-diameter work roll having the necks of a diameter of about 270 mm or above, and the small-diameter work roll formed of a material having a tensile strength of 45 kgf/mm 2  or above, which is higher than a maximum stress of about 40 kgf/mm 2  (3.92×10 8  Pa) that is expected to be induced in the small-diameter work roll when a roll bender force, which is comparatively low because the small-diameter work roll has a small diameter, does not have any problem in mechanical strength. Generally, high-speed steel rolls or forged high-speed steel rolls have a tensile strength of 80 kgf/mm 2  (7.84×10 8  Pa) or above and hence problems attributable to fatigue resulting from rotation involving repeated stress cycles can be easily avoided. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a typical view of one of the rolling mills  10  shown in FIG. 7 in a preferred embodiment according to the present invention; 
     FIG. 2 is a typical view of assistance in explaining horizontal forces exerted on rolls by vertical rolling force; 
     FIG. 3 is a typical view of assistance in explaining horizontal forces produced when only a large-diameter work roll  12  is driven for rotation; 
     FIG. 4 is a graph showing the dependence of resultant forces F 1  and F 2  exerted, respectively, on the work rolls  11  and  12  on the offset e 2  of the rotational axis of the large-diameter work roll  12  from the rotational axis of the small-diameter work roll  11 ; 
     FIG. 5 is a graph showing the dependence of stresses σ 1  and σ 2  induced, respectively, in the necks of the work rolls  11  and  12  on the offset e 2  of the rotational axis of the large-diameter work roll; 
     FIG. 6 is a front elevation of the small-diameter work roll; 
     FIG. 7 is a typical view of a sheet rolling mill for hot-rolling sheets; and 
     FIG. 8 is a typical view of a conventional rolling mill. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     FIGS. 1 to  7  show a preferred embodiment of the present invention. FIG. 1 is a typical side elevation of one of three downstream mills  10  in a back stage (downstream side) of a rolling line  1  shown in FIG.  7 . 
     The rolling line  1  for hot-rolling a steel sheet x is a tandem rolling line having six rolling mills  5  and  10  as shown in FIG.  7 . The three front rolling mills  5  in a front stage (upstream side) are ordinary four-high mills each having two work rolls  6  and  7  of the same diameter disposed one on top of the other, and upper and lower backup rolls  8  and  9  supporting the work rolls  6  and  7 . The three back rolling mills  10  in the back stage are so-called differential rolling mills each having an upper backup roll  13 , a lower backup roll  14  and a pair of work rolls  11  and  12  respectively having different diameters and disposed between the backup rolls  13  and  14 . Both the two work rolls  6  and  7  of each of the three front rolling mills  5  are driven for rotation, while only the lower work roll  12  of each of the three back rolling mills  10  in the back stage is driven for rotation because the required torque of the back rolling mills  10  is not high. 
     Referring to FIG. 1 showing the back rolling mill  10 , the diameter DW 1  of the small-diameter work roll  11  is 450 mm, the diameter DW 2  of the large-diameter work-roll  12  is 590 mm, the diameters DB of the backup rolls  13  and  14  are 1300 mm, Unless otherwise specified, the diameter of a roll is that of a part of the roll that comes into contact with the steel sheet x and the body of the adjacent roll. In the back rolling stand  10 , an offset e 1  of the rotational axis of the small-diameter work roll  11  from the mill center plane, i.e., the plane including the center axes of the backup rolls  13  and  14 , and an offset e 2  of the rotational axis of the large-diameter work roll  12  from the rotational axis of the small-diameter roll  11  are variable. In this embodiment, 
     
       
           e   1 =6 mm and  e   2 =4 mm. 
       
     
     The rolling line  1  hot-rolls a hot-rolled soft steel plate (SPHC, JIS) of 25 mm in thickness into a steel sheet of 1.2 mm in thickness and 1550 mm in width. The rolling line  1  operates on a pass schedule setting the thicknesses of the sheet at the respective exits of the front rolling mills  5  and the back rolling mills  10  to, for example, 10.97 mm, 5.12 mm, 3.46 mm, 2.22 mm, 1.49 mm and 1.17 mm, respectively. A roll bender force of 80 ton (P B1  and P B2 ) is exerted on each of chocks supporting the work rolls  11  and  12  of the rolling mills  5  and  10  to control the shape of the steel sheet x. 
     Generally, the rolling mills  5  and  10  need to exert considerably high rolling forces on the steel sheet when the rolling width is big. Mechanical measures must be incorporated into the back rolling mills  10  provided with the small-diameter work roll  11  in which an excessively high stress is liable to be induced when a high rolling force is used. Elaborate measures to withstand stress must be taken particularly for the fourth rolling mill  10  that uses a high rolling force higher than those used by the rest of the back rolling mills  10 , i.e., the uppermost one among the three back rolling mills  10  in the back stage. In the rolling line  1  shown in FIG. 7, the fourth rolling mill  10  uses a rolling force as high as 3000 tons. The offsets e 1  and e 2  in the fourth rolling mill  10 , i.e., a differential rolling mill, are determined so that an excessively high stress may not be induced in the small-diameter work roll  11  even a high rolling force is exerted to the small-diameter work roll  11 . 
     In a rolling mill in a comparative example, 
     
       
           e   1 =6 mm and  e   2 =0 mm. 
       
     
     The rolling mill  10  and the rolling mill in the comparative example will be compared, and the results of mechanical examination of the work rolls  11  and  12  will be explained hereinafter. 
     Stresses that may be induced in the small-diameter work roll  11  and the large-diameter work roll  12  are calculated in the following manner. Forces exerted on the work rolls  11  and  12  include: 
     a) horizontal forces SR 1  and SR 2  acting on the work rolls  11  and  13  in directions shown in FIG. 3, respectively, by the steel sheet x when only the large-diameter work roll  12  is driven for rotation, 
     b) roll bender forces P B1  and P B2  (80 tons, not indicated) acting on the work rolls  11  and  12  in a vertical plane perpendicular to the rolling direction, and 
     c) horizontal forces P mT  and P mB  acting on the work rolls  11  and  12 , respectively, when a rolling force (3000 tons for the fourth rolling mill  10 ) is exerted on the work rolls  11  and  12  through the backup rolls  13  and  14  in contact with the work rolls  11  and  12 , respectively. 
     Vertical forces acting on the work rolls  11  and  12  do not need to be considered because forces exerted on the working rolls  11  and  12  by the backup rolls  13  and  14  are balanced by forces exerted on the work rolls  11  and  12  by the steel sheet x. Referring to FIG. 6 showing the small-diameter work roll  11 , the forces a) to c) exerted on a body  11   b  included in the small-diameter work roll  11  are counterbalanced by reaction forces exerted by bearings, not shown, on journals  11   c . The magnitudes of the forces a) to c) will be examined supposing that forces acting in the rolling direction, i.e., the direction of the blank arrows in FIGS. 2 and 3, are positive forces. 
     
       
           SR   1   =P   R ·tan(α/ s )  (1) 
       
     
     
       
           SR   2   =SR   1   (2) 
       
     
     where P R  is rolling force, and α is center angle corresponding to a part, in contact with the steel sheet x, of the circumference of the small-diameter work roll  11  expressed by: 
     
       
         α=cos −1   [{DW   1 −2 DW   2   ΔH/ ( DW   1   +DW   2 )} /DW   1 ] 
       
     
     where ΔH is the difference (1.24 mm) between the thickness H 1  (3.46 mm) of the steel sheet x at the entrance of the fourth rolling mill  10 , and the thickness H 2  (2.22 mm) of the steel sheet x at the exit of the fourth rolling mill  10 . In both the rolling mill  10  in the embodiment and the rolling mill in the comparative example, α=4.53° from ΔH=1.24 mm, and hence, using Expressions (1) and (2), 
     SR 1 =118.7 tons, and 
     SR 2 =118.7 tons. 
     The forces c) shown in FIG. 2 are: 
     
       
           P   mT   =SB   =1   −SD   1   (3) 
       
     
     
       
           P   mB   =SD   2   +SB   1   (4) 
       
     
     where 
     
       
           SB   1   =P   R ·tan[sin −1 {2 e   2 /( DW   1   +DW   2 )}] 
       
     
     
       
           SD   1   =P   R ·tan[sin −1 {2 e   1 /( DB+DW   1 )}] 
       
     
     
       
           SD   2   =P   R ·tan[sin −1 {2( e   1   +e   2 )/( DB+DW   2 )}] 
       
     
     Since e 1 =6 mm and e 2 =4 mm in the rolling mill  10  in the embodiment, horizontal forces P mT  and P mB  are calculated by using Expressions (3) and (4). 
     
       
           P   mT   =SB   1   −SD   1 =23.1−20.6=2.5 (tons) 
       
     
     
       
           P   mB   =SD   2   +SB   1 =31.7+23.1=54.8 (tons) 
       
     
     Since e 1 =6 mm and e 2 =0 mm in the rolling mill in the comparative example, 
     
       
           P   mT   =SB   1   −SD   1 =0−20.6=−20.6 (tons) 
       
     
     
       
           P   mB   =SD   2   +SB   1 =10.0+0=19.0 (tons) 
       
     
     Total forces (the sum of the forces a) to c)) F 1  and F 2  that act, respectively, on the work rolls  11  and  12  are: 
     
       
           F   1 ={( SR   1   −P   mT ) 2   +PB   1   2 } 1/2   
       
     
     
       
           F   2 ={(− SR   2   +PmB ) 2   +PB   2   2 } 1/2   
       
     
     Thus, reaction forces corresponding to the forces F 1  and F 2  act on the journals  11   c  of the small-diameter work roll  11 , and those of the large-diameter work roll  12 . The values of the total forces F 1  and F 2  are converted into those in kgf (1 kgf=9.8 N) as follows. 
     F 1  and F 2  for the embodiment are: 
     F 1 =141,100 (kgf) and F 2 =102,400 (kgf). 
     F 1  and F 2  for the comparative example are: 
     F 1 =160,600 (kgf) and F 2 =127,800 (kgf). 
     Whereas the forces SR 1  and P B1  of the total force F 1  acting on the small-diameter work roll  11  are always positive, the force P mT =SB 1 −SD 1  is negative and the total force F 1  can be reduced when 
     
       
         2 e   2 /( DW   1   +DW   2 )&gt;2 e   1 /( DB+DW   1 ). 
       
     
     The rolling mill in the comparative example, in which e 1 =6 mm and e 2 =0 mm, is unable to satisfy this inequality, and hence the total force F 1  is high. 
     Since forces respectively corresponding to the total forces F 1  and F 2  are exerted on the necks  11   n  (FIG. 6) of the small-diameter work roll  11  and those of the large-diameter work roll  12 , bending moments M 1  and M 2  proportional to the lengths L 1  and L 2  between the necks and the centers of the corresponding journals are produced at the necks  11   n  of the small-diameter work roll  11  and those of the large-diameter work roll  12 . Consequently, bending stresses π 1  and π 2  are induced in the necks of the work rolls  11  and  12  according to the respective section moduli Z 1  and Z 2  of the work rolls  11  and  12  and a stress concentration factor α at the neck. Generally, M=F×L, Z=πD 3 /32 and σ=σ×M/Z, where D is diameter. L 1 =265 mm and D (diameter of the neck)=270 mm in the small-diameter work roll  11 , L 1 =265 mm and D (diameter of the neck)=270 mm in the large-diameter work roll  12 , and α is about 1.8. Therefore, in the rolling mill  10  in the embodiment, in which e 1 =6 mm and e 2 =4 mm, 
     σ 1 =34.8 kgf/mm 2 , and 
     σ 2 =15.9 kgf/mm 2 , 
     and in the rolling mill in the comparative example, in which e 1 =6 mm and e 2 =0 mm, 
     σ 1 =39.7 kgf/mm 2 , and 
     σ 2 =19.9 kgf/mm 2  (1 kgf/mm 2 =9.8×10 6  Pa). 
     FIGS. 4 and 5 are graphs showing the variation with the offset e 2  of the total forces F 1  and F 2  acting on the work rolls  11  and  12  and bending stresses σ 1  and σ 2  induced in the necks of the work rolls  11  and  12  when e 1  is 6 mm. The total force F 1  and the bending stress σ 1  decreases as the offset e 2  increases in both the work rolls  11  and  12 . 
     As obvious from FIG. 5, the stress σ 1  induced in the small-diameter work roll  11  exceeds 40 kgf/mm 2  when e 2 &lt;0 mm. Since the core  11   a  of an ordinary material, such as a nickel grain roll (a part of the body  11   b  of the work roll  11  excluding a surface skin as shown in FIG. 6) has problem in withstanding the stress σ 1  exceeding 40 kgf/mm 2 , it is preferable that e 2 &gt;0. 
     If e 2 &gt;7 mm, bowing of the steel sheet x, i.e., upward warping of the leading edge of the steel sheet x passed between the small-diameter work roll  11  and the large-diameter work roll  12  occurs and smooth rolling is impossible. 
     Therefore, the offset e 2  must meet an inequality: 
     
       
         0&lt; e   2 &lt;7 mm, 
       
     
     namely, a value in a not shaded region in FIG. 5, when e 1 =6 mm under the foregoing rolling conditions including the rolling force, the diameter of the work roll, the pass schedule, the roll bender force and such. 
     The present invention is applicable to rolling of sheets using a rolling mill provided with a pair of work rolls respectively having different diameters.