Tube rolling method and apparatus

According to the rolling method and the apparatus to be used for its execution of the invention, by rolling a tube to be rolled by using four rolls possessing roll grooves for forming a caliber in a shape of having a relief portion, cold reducing or cold stretch reducing is done continuously without causing wall thickness deviation, and by sizing the rolled tube material by a die disposed at the exit side, the dimensional precision and yield of rolled tubes may be enhanced by a small number of stands.

BACKGROUND 
1. Field of the Invention 
The present invention relates to a rolling method and apparatus for 
continuously reducing the outer diameter of a hollow tube of carbon steel, 
stainless steel, or the like in a cold state, and to a method and 
apparatus for die processing, in addition to continuous cold reducing or 
cold stretch reducing. 
2. Description of the Related Art 
Hot stretch reducing is known as a method of producing metal tubes. In this 
method, a plurality of stands having three rolls forming arcuate grooves 
are disposed in tandem, and a heated mother tube is passed through the 
stands, so that the outer diameter of the mother tube is continuously 
reduced. Since this method is hot rolling, it involves its problems in the 
dimensional precision of the products and surface quality, and it is 
expensive due to the need for a heating furnace and fuel and other 
expense. 
In producing of metal tubes, when a tube with small diameter of less than 
an inch is produced, generally, a hollow mother tube produced by hot 
rolling is acid-cleaned, lubricated, and then cold drawn by die or cold 
rolled by Pilger rolling mill. 
FIG. 1 is a schematic side view showing the constitution of an apparatus in 
case of producing tubes by the cold drawing method. In the drawing, 
numeral 21 designates a tube, and the tube 21 is inserted into a die 22 
having a circular hole. At the exit side of the die 22, a drawing machine 
23 is disposed at a specified interval., and it is designed to draw a 
small diameter tube to reduce it in diameter. At this time, a chuck 24 
disposed between the die 22 and the drawing machine 23 holds the small 
diameter tube. For this holding, as pretreatment of the drawing, a step 
for squeezing to reduce one end of the tube 21 is required. In drawing, a 
large tension is applied to the tube, but this tension must be limited to 
an extent that the mother tube may not be broken, and the reduction rate 
in one pass is limited, and when the total reduction rate becomes higher, 
moreover, the mother tube undergoes work-hardening and therefore 
intermediate annealing is required, which results in low yield and low 
working efficiency. 
In the latter method of cold rolling, on the other hand, a pair of rolls 
having grooves tapered along the circumference are used and the tube is 
reduced in diameter and processed by moving the rolls reciprocally while 
pressing down by holding the tube by the rolls. In this cold rolling 
method, the reduction rate of mother tube in one pass is greater than the 
former method, but the working efficiency is inferior because the rolls 
must be moved reciprocally and pressed down upon the tube. 
In producing small diameter tubes, the hot stretch reducing method 
mentioned above may be employed in certain cases, and the yield and 
working efficiency are notably enhanced by the hot stretch reducing 
method, but, as mentioned above, there are problems in the dimensional 
precision of products and surface quality. It also is expensive due to a 
need for a heating furnace and fuel and other expenses. 
Accordingly, as disclosed in the Japanese Patent Application Laid-open 
63-33105 (1988) and the collected papers of the 118th general meeting of 
Iron and Steel Society of Japan (CAMP-ISIJ, vol. 2, 1989, pp. 1494), the 
three-roll type cold stretch reducing method applying the hot rolling in 
cold rolling has been proposed. 
FIG. 2(a) is a schematic side view explaining the arrangement of stands of 
a stretch reducer, and FIG. 2(b) is a schematic front view explaining the 
arrangement of stands of the stretch reducer. In the drawings, numeral 25 
designates rolls, and a plurality of stands 26, 27, 28, . . . having three 
rolls 25, 25, 25 disposed at intervals of 120 degrees around pass line X 
are disposed in the pass line direction. The stands are arranged in 
tandem, by matching the calibers, varying the phase of the roll 
disposition of the adjacent stands by 60 degrees, and decreasing the 
caliber diameters gradually. In the final stand, a stand of round caliber 
is disposed. 
Stand calibers consist of round calibers and oval calibers. FIGS. 3(a), (b) 
are sectional views showing the calibers used in the three-roll stretch 
reducer, and specifically FIG. 3(a) shows the round caliber, and FIG. 
3(b), the oval caliber. The round caliber is a caliber composed of an arc 
R.sub.1 having the center in the caliber center, and the oval caliber is a 
caliber having another arc R.sub.2, with the center of the arc located on 
the center line of the roll gap, in the relief part of the caliber. 
Among stands having oval calibers and round calibers consisting of three 
rolls, while applying a tension to the tubes between stands by setting the 
peripheral velocity ratio of roll surface between the adjacent stands 
larger than the elongation rate of tubes in a single stand, the mother 
tube is continuously passed among stands to reduce it to desired outer 
diameter. 
In such three-roll cold stretch reducing method applying hot rolling in 
cold rolling, the tube wall thickness increases or decreases in the 
circumferential direction due to the reason mentioned below to cause 
so-called wall thickness deviation, and the inner sectional shape of the 
tube is deformed into a hexagonal form as shown in FIG. 4. That is, in hot 
stretch reducing, the friction coefficient of roll and mother tube is 0.3, 
and a sufficient tension is obtained among stands, and increase of wall 
thickness being a cause of wall thickness deviation is sufficiently 
suppressed, and deviation hardly occurs, but in cold stretch reducing, the 
friction coefficient is less than 0.1, being less than 1/3 of that of hot 
process, and sufficient tension cannot be obtained among the stands, and 
the increasing tendency of uneven wall thickness in the tube peripheral 
direction cannot be suppressed between the abutting portions of the roll 
groove bottom and roll groove edge. 
Besides, seizure of the mother tube to the roll is caused by slipping by 
reason that the tension among stands is increased, or overfilling of 
mother tube into the roll gap occurs by reason that specified tension is 
not obtained. 
To solve such problems, a method of rolling by setting the groove bottom 
diameter of the roll at 10 times or more of the outer diameter of the 
mother tube has been disclosed in the Japanese Patent Application 
Laid-open Hei. 4-4905. 
FIG. 5(a) and FIG. 5(b) are conceptual diagrams explaining the ratio of 
outer diameter of mother tube and diameter of roll groove bottom, and 
rolling condition of mother tube, being a front view of roll and a side 
view of roll, respectively. Referring to these drawings, the method 
disclosed in the Japanese Patent Application Laid-Open 4-4905 is 
explained. Three rolls 31, 32, 33 are disposed around the pass line of 
mother tube A whose distance from central axis C.sub.2 to the outer 
circumference is D.sub.2 /2, and the groove bottom radius of these rolls, 
that is the distance from the axial center C.sub.1 to the groove bottom is 
D.sub.1 /2. By using the rolls 31, 32, 33 whose D.sub.1 /D.sub.2 is 10 or 
more, the frictional force is enhanced, and the outer diameter of the 
mother tube A is continuously reduced in cold state. 
In the conventional method as mentioned above, incidentally, since the 
contact area of the roll and mother tube is increased by setting the roll 
groove diameter at more than 10 times the outer diameter of the mother 
tube, a sufficient frictional force can be obtained even in the cold 
stretch reducing method being low in the coefficient of friction, so that 
a necessary tension among rolls is obtained. However, increase of contact 
area gives rise to increase of rolling force, that is, rolling load, and 
the required power for rolling and torque increases, and the increase of 
roll groove bottom diameter causes to increase the roll volume and gives 
rise to a substantial enlargement of facility, and problems of economy and 
facility are involved, and moreover in the three-roll rolling method, 
overfilling of tube into roll gap is likely to occur, and the reduction 
per stand cannot be increased, and therefore the rolling efficiency is 
poor, the number of stands required for reducing a tube to specified outer 
diameter increases, and the facility becomes gigantic. 
SUMMARY 
The invention has been devised to solve these problems, and it is hence a 
primary object of the invention to provide a method of performing cold 
reducing continuously without causing wall thickness deviation by using 
four rolls, and an apparatus to be used for the execution thereof. It is 
another object of the invention to provide a method of improving the 
dimensional precision and yield of rolled tubes by a small number of 
stands, by sizing with a die disposed at the exit side by using four 
rolls, and an apparatus to be used for the execution thereof. 
The tube rolling method and rolling apparatus of the invention are 
characterized by the constitution in which plural stands comprising four 
rolls are constituted in tandem so as to be different in phase by about 45 
degrees from the pass line, and these rolls possess the roll grooves 
forming nearly circular calibers satisfying the following conditions. 
EQU a.sub.i &gt;b.sub.i 
EQU a.sub.i &lt;b.sub.i-1 
where a.sub.i : caliber radius of roll groove edge of i-th stand 
b.sub.i : caliber radius of roll groove center of i-th stand 
b.sub.i-1 : caliber radius of roll groove center of i-1-th stand 
Therefore, in the cold reducing with four rolls capable of reducing almost 
uniformly on the whole circumference, since the groove of the roll for 
forming the caliber is designed so that the radius of the groove edge part 
may be larger than the radius in the groove central part, a relief portion 
is formed in the caliber, and in this relief portion, therefore, 
overfilling of tube and formation of flaw on the tube surface are reduced, 
and the radius of the groove edge is set smaller than the radius of the 
middle part of the roll groove of the stand being one step up to the 
upstream side, and it is hence possible to reduce uniformly in the 
centripetal direction of tube axis in the groove center and edge of the 
groove, so that formation of wall thickness deviation may be suppressed. 
It is hence a feature of the tube rolling method and rolling apparatus of 
the invention to use rolls which have the relief portion and roll grooves 
forming nearly circular calibers satisfying the following condition. 
EQU 1.0&lt;a.sub.i /b.sub.i .ltoreq.1.050 
It is another feature to use rolls which form nearly circular calibers 
satisfying the following conditions, and possess roll grooves in a shape 
being specific in the radius of curvature. 
EQU 1.05b.sub.i .ltoreq.R.sub.i .ltoreq.1.20 b.sub.i 
where R.sub.i : radius of curvature of roll caliber of i-th stand 
Furthermore, it is also a feature to use rolls which possess roll grooves 
so as to form nearly circular calibers satisfying the following 
conditions. 
EQU 0.88.ltoreq.b.sub.i /b.sub.i-1 .ltoreq.0.95 
EQU 0.60.ltoreq.(b.sub.i-1 -a.sub.i)/(b.sub.i-1 -b.sub.i).ltoreq.0.90 
Therefore, overfilling of tube and formation of flaw on tube surface are 
further decreased, and it is possible to roll uniformly in the centripetal 
direction of the tube axis in the groove center and edge parts of the 
roll, so that formation of wall thickness deviation may be suppressed. 
The tube rolling apparatus of the invention is characterized by reducing 
the outer diameter of the tube to be rolled, by comprising rolls 
possessing calibers forming relief portions therein, with the roll groove 
bottom diameter being more than five times the outer diameter of the tube 
to be rolled. It is a feature of the rolling method to reduce the outer 
diameter of the tube to be rolled by 12% or less per stand, in addition to 
the above features. Therefore, with the rolls having a smaller diameter 
than in the constitution of three coils, cold rolling without corner 
squareness in inner sectional shape of the tube after rolling is achieved 
without causing roll-biting failure. Furthermore, in the rolling method 
with the reduction in outer diameter of 12% or less, it is possible to 
roll at a high reduction without causing slipping of roll or seizure of 
the tube to be rolled to the roll. Therefore, if the reduction per stand 
is set larger than in case of two-roll or three-roll type, over-filling of 
mother tube into the roll gap hardly occurs, so that the number of stands 
required for reducing the total outer diameter may be smaller. 
It is another feature of the tube rolling method of the invention to roll 
the tube by adjusting the peripheral speed of the rolls of each stand so 
that the acceleration ratio of the peripheral speed at the groove center 
of roll of the extreme downstream side stand to the peripheral speed at 
the groove center of the roll of the extreme upstream side stand may be 
1.0 to 1.8 times the reference acceleration ratio without action of 
tension on the stand tube. Therefore, a tension among stands can be 
obtained without slipping of the rolls, and the increase of wall thickness 
due to reduction of outer diameter of the tube to be rolled can be 
suppressed. 
It is a feature of the tube rolling method and rolling apparatus of the 
invention to comprise rolls having calibers forming relief portions, with 
a die disposed at the exit side of the extreme downstream side stand among 
the stands, thereby sizing the tube to be rolled which has been Foiled and 
reduced by the die. Therefore, the precision of the finishing dimension is 
enhanced. 
The tube rolling method of the invention can comprise a die, the reduction 
in outer diameter at the die being set at 0.5 to 5.0% during execution. 
Therefore, buckling of tube material may be prevented. 
In the tube rolling method and rolling apparatus of the invention, the 
distance L between the center of the nearly circular caliber of the 
extreme downstream side stand and the inlet of the die bearing portion 
satisfies the following relation. 
EQU L.ltoreq.6.times.[{d.sub.1.sup.4 -(d.sub.1 -2t).sup.4 }/(d.sub.1.sup.2 
-d.sub.2.sup.2)].sup.1/2 
where d.sub.1 : caliber diameter of roll of extreme downstream side stand 
t: wall thickness of tube to be rolled at the exit of extreme downstream 
side stand 
d.sub.2 : die diameter 
Therefore, slipping hardly occurs between the roll and tube, and seizure is 
prevented. 
The tube rolling method and rolling apparatus of the invention can be 
provided with a die, and with a pinch roll at the exit side of the die, 
and when the tail end of the tube to be rolled is stopped between the roll 
and the die, the tail end is pulled out by the pinch roll. Therefore, when 
the tail end of the tube is stopped just before the die, the pinch roll 
holds the tube and rotates so that the tube can be drawn out. 
In the tube rolling method and rolling apparatus of the invention, there is 
at least one detecting means for detecting the tail end of the tube to be 
rolled at the entrance of the plural stands or between the stands, and the 
pinch roll disposed at the exit side of the die is operated or stopped 
according to the result of the detecting means. Therefore, by judging the 
timing of stopping the tail end by the detecting means, the pinch roll 
holds the tube so that the tube may not be flawed. 
It is a further feature of the tube rolling method and rolling apparatus of 
the invention that a sized tube is conveyed by using tube conveying means 
disposed at the exit side of the die. The conveying speed of the tube 
conveying means is greater than the exit side speed of the die. Therefore, 
the rolled tube can be conveyed easily. 
The above and further objects and features of the invention will more fully 
be apparent from the following detailed description with accompanying 
drawings.

DETAILED DESCRIPTION 
The invention is described in detail below referring to the drawings 
showing the embodiments thereof. 
FIG. 6 is a front view showing a constitutional example of caliber of a 
cold reducer according to the invention, in which numerals 111, 112, 113, 
and 114 designate rolls. The rolls 111, 112, 113, 114 have grooves 111b, 
112b, 113b, 114b cut in their circumferential surfaces to form a caliber 
115, and at both sides of the rolls 111, 114 and at one side of the rolls 
112, 113, internal gears 116, 116, . . . are fixed individually so as to 
be mutually engaged on the circumferential surfaces. The rolls 111, 112, 
113, 114, and internal gears 116, 116, . . . are fixed to roll shafts 
111a, 112a, 113a, 114a disposed rotatably at the openings of roll housings 
110 having a cross opening, and by driving the roll shaft 111a projecting 
from the side wall of the roll housing 110, all rolls 111, 112, 113, 114 
can be driven simultaneously by the internal gears 116, 116, . . . 
FIG. 7 is a schematic diagram for explaining an arrangement of stands. The 
stands are arranged in tandem by matching the calibers 115, 115, . . . The 
rolls 111, 112, 113, 114 of the stands are shifted by 45 degrees in phase 
relatively to the pass line with respect to the rolls 111, 112, 113, 114 
of the upstream stands. 
In FIGS. 8(a), (b), (c), the state of the stress acting on the hollow 
mother tube in the roll gap by the number of rolls for forming the 
calibers is expressed in vectors in a schematic diagram, and FIG. 8(a) and 
FIG. 8(b) show the two-roll and three-roll types, and FIG. 8(c) shows the 
four-roll type of the invention. As clear from the diagrams, in case of 
two-roll type, the mother tube A tends to overfill into the roll gap due 
to the stress received in the roll gap direction, and in case of 
three-roll type, the stress in the peripheral direction from the 
centripetal direction of the tube axis acts in the roll gap, which may 
cause wall thickness deviation. In the four-roll type of the invention, 
the stress in the peripheral direction is suppressed in the roll gap, and 
a almost; uniform rolling is performed on the whole circumference. 
Accordingly, if the reduction per stand is set larger than in two-roll or 
three-roll type, overfilling of mother tube into the roll gap hardly 
occurs, so that the number of stands required for the reduction of total 
outer diameter can be decreased. 
Furthermore, at both edges of the groove cut in the circumference of the 
roll forming the calibers, in order to prevent overfilling of mother tube 
into the roll gap and formation of flaw on the mother tube at the groove 
edge, the caliber radius of the groove edge is set larger than the caliber 
radius of the groove bottom. This point is further described below. 
FIG. 9 is a schematic diagram for explaining the caliber shape according to 
the invention, in which i designates a roll of an i-th stand, and (i-1) 
designates a roll of an (i-1)-th stand one position closer to the upstream 
side. As mentioned above, the roll i and roll (i-1) differ in phase by 45 
degrees relatively to the pass line. The intersection of the caliber 
virtual line formed by the neighboring roll i and virtual line OY from 
caliber center O of the pass line to the roll gap middle part is P.sub.i, 
and the contact point of the caliber virtual line and groove middle part 
of roll i is Q.sub.i, then the OP.sub.i distance is the radius of the roll 
i to the gap middle part, and the OQ.sub.i distance is the radius b.sub.i 
to the groove bottom in the middle of the groove. 
Meanwhile, as shown in FIG. 9, the roll gap between adjacent rolls is 
present, but this roll gap is as small as about, for example, 0.1 to 0.2 
mm, and the roll edge part in the roll gap part is cut off and made with a 
small corner radius of about 0.1 to 0.2 mm, and substantially the OP.sub.I 
distance is equal to the radius a.sub.i of the roll i to the groove edge. 
Likewise, the radii of the roll (i-1) to the groove edge and to the groove 
bottom are respectively a.sub.i-1 and b.sub.i-1. At this time, the cold 
rolling apparatus of the invention employs the rolls which form such 
calibers that the relation between the radius b.sub.i to the groove bottom 
and the radius a.sub.i to the groove edge, and the relation between 
a.sub.i and b.sub.i-1 may be given in the following formula. 
EQU a.sub.i &gt;b.sub.i 
EQU a.sub.i /b.sub.i-1 &lt;1 (1) 
Rewriting formula (1) yields a.sub.i -b.sub.i-1 &lt;0, and the i-th caliber is 
a so-called side relief minus caliber having a minus difference between 
the radius of the roll to the groove edge and the radius of the roll (i-1) 
to the groove bottom. The side relief minus caliber provides the both 
groove edges of the caliber roll with relief portions, but the radius 
a.sub.i to the groove edge is set smaller than the radius b.sub.i-1 to the 
groove bottom one stand before, and by disposing stands with side relief 
minus calibers, it is possible to roll a tube in the centripetal direction 
of tube axis uniformly to the mother tube, in the roll groove bottom and 
groove edge. 
Furthermore, in the apparatus of the invention, the calibers of the stands 
are designed in the range given by formulas (2) and (3), using the above 
values of a.sub.i, b.sub.i and b.sub.i-1. 
EQU 0.88.ltoreq.b.sub.i /b.sub.i-1 .ltoreq.0.95 (2) 
EQU 0.66.ltoreq.(b.sub.i-1 -a.sub.i)/(b.sub.i-1 
-b.sub.i)=.alpha..ltoreq.0.90(3) 
In formula (2), b.sub.i /b.sub.i-1 being 0.88 and 0.95 suggests that the 
reduction in outer diameter per stand is 12% and 5%, and it means that 
overfilling of the mother tube into the roll gap occurs when the reduction 
in outer diameter exceeds 12%, and that the rolling with the reduction in 
outer diameter of less than 5% is not meaningful substantially except for 
the finishing stand, and hence the range is defined as specified above. 
Rewriting the formula (1) yields 0&lt;b.sub.i-1 -a.sub.i, and when .alpha. is 
defined in the range specified in the formula (3), it means an appropriate 
minus relief extent for the side relief minus caliber, and by the 
definition of formula (3) when .alpha.=1, it follows that a.sub.i 
=b.sub.i, and the caliber is a round caliber. To prevent wall thickness 
deviation, what is ideal is a complete uniform compressive processing from 
the whole circumference by the round caliber, but over-filling occurs in 
this case, and hence the upper limit of .alpha. is set at 0.90 at which 
overfilling does not occur. On the other hand, as the value of .alpha. 
becomes smaller, a.sub.i becomes larger for b.sub.i, and the caliber is 
changed from a round form to an oval form. When a.sub.i becomes too large 
for b.sub.i, wall thickness deviation is likely to occur even in rolling 
with four rolls, and the inner surface is squared, and therefore the lower 
limit is defined at 0.60 at which the inner surface of the mother tube may 
not be squared. 
The result of cold tube rolling of steel tube by the method and apparatus 
of the invention is compared with the reference case. The reference case, 
using four rolls, does not satisfy formula (1) and/or (2), (3). 
FIG. 10 and FIG. 11 are explanatory diagrams of the shape of the groove 
edge of the roll used in the embodiment, and two types were used, that is, 
the double radius type (DR type) having the center on the circumference of 
radius b.sub.i contacting with the groove bottom, and varying the angle 
.theta. formed by the groove edge contacting with the arc with radius 
2b.sub.i as shown in FIG. 10, and the single radius type (SR type) having 
the center on the line linking the center of the circle of radius b.sub.i 
contacting with the groove bottom and the lowest part of the groove, and 
varying the radius R.sub.i of the groove edge drawn by an arc of radius 
R.sub.i as shown in FIG. 11. The other conditions are as follows. 
______________________________________ 
Steel tube Material: Low carbon steel 
Dimensions: o 18 mm .times. 2 mmt 
Stand Quantity: 6 stands + finishing 
stand 
Nominal roll diameter: o 140 mm 
Lubricant Water-soluble oil 
______________________________________ 
However, the finishing stand has the caliber in the same dimensions as the 
sixth stand varied by 45 degrees in phase from the pass line. The nominal 
roll diameter is the distance between confronting roll shafts. 
Table 1 shows the result, the code of the rolling result in the column of 
overfilling in the table is .smallcircle. when overfilling does not occur, 
.times. when overfilling occurs, and .DELTA. when it occurs slightly, and 
in the column of interior squareness, it is .times. when the ratio of 
maximum value/minimum value of inside diameter of the steel tube after 
rolling is 1.15 or more, .DELTA. when 1.10 to 1.15, and .smallcircle. when 
1.10 or less. 
As clear from Table 1, in the embodiment, over-filling does not occur by 
using the rolls of caliber of either SR type or DR type, and wall 
thickness deviation is suppressed, and the inner surface of the steel tube 
after rolling is not squared. In the reference example, by contrast, when 
the reduction per stand is 13% and when .alpha. exceeds 0.90 in the 
formula (3), overfilling occurs, and when .alpha. is less than 0.60, 
squaring occurred in-the internal surface of steel tube after rolling. 
On the other hand, when rolling is performed by the three-roll type cold 
tube stretch reducing method with the reduction in outer diameter per 
stand being 10%, over-filling occurred. To prevent over-filling with the 
reduction in outer diameter being 10%, the side relief must be positive, 
and in this case squaring occurred after rolling. The upper limit of the 
reduction in outer diameter to be free from overfilling at minus side of 
side relief was 6%, but squaring after rolling could not be prevented with 
the outer diameter reduction being 6%. 
In this way, by setting in a proper range with respect to the radius of the 
middle and edge part of the roll grooves of the neighboring stands 
possessing four rolls, it is possible to roll in cold process without 
causing overfilling or squaring after rolling. 
In the single radius caliber, incidentally, the radius of curvature of the 
caliber is set somewhere between 1.05 times and 1.20 times of the caliber 
radius b.sub.i in the groove bottom. That is, the range of the offset 
extent e to the pass line center of the center of radius of curvature of 
the caliber is desired to be 
EQU 0.05b.sub.i .ltoreq.e.ltoreq.0.20 b.sub.i 
That is, when the offset extent e exceeds 0.20 b.sub.i, since the ratio of 
major radius and minor radius of the caliber is too large, formation of 
wall thickness deviation during rolling is avoided, and on the other hand, 
when the offset extent e is less than 0.05 b.sub.i, since the caliber 
shape is too close to the round circle, overfilling into the roll gap 
occurs while rolling. 
In the roll forming such single radius caliber, since the radius of 
curvature is constant, it is possible to cut the roll calibers by disk 
cutter with the roll assembled in the roll unit. Hence, assembling labor 
of roll unit is saved, and the roll caliber can be cut regardless of the 
fine adjustment in the roll width direction and assembling precision. 
Table 2 shows the result of judgement on the necessity of how much the 
ratio (a.sub.i /b.sub.i) or the major radius to the minor radius of the 
caliber must be close to 1, and in the same way as above, continuous 
rolling of seven stands was compared with outer diameter reduction per 
stand being 10%. In the column of internal surface squaring, in the same 
way as above, it is expressed by .times. when the ratio of maximum 
value/minimum value of the inside diameter of steel tube after rolling is 
1.15 or more, and .smallcircle. when 1.10 or less. As clear from Table 2, 
the ratio of the major radius to the minor radius of the caliber is 
desired to be 
EQU 1&lt;a.sub.i /b.sub.i .ltoreq.1.050. 
Thus, using the four-roll stands, it is possible to roll a tube uniformly 
on the whole circumference of the tube to be rolled by setting the radius 
to the groove bottom and edge of the roll to satisfy the above condition. 
Another embodiment of the invention is described below while referring to 
the accompanying drawings. 
FIG. 12(a) is a partial front view a stand for forming the pass line, and 
FIG. 12(b) is a partial side view of a roll for rolling the mother tube, 
and these are conceptual diagrams for explaining the ratio of the outer 
diameter of the mother tube and roll groove bottom diameter, and rolling 
condition of the mother tube in the invention. The constitution of the 
stands and arrangement of the stands are same as in the cold reducer shown 
in FIGS. 6 and 7, and same reference numerals are given to the 
corresponding parts, and their explanations are omitted. 
As shown in FIGS. 12(a), (b), in four rolls 211, 212, 213, 213 forming a 
caliber 215, the distance from the axial center c.sub.1 of each roll shaft 
not shown in the drawing to the roll groove bottom is D.sub.1 /2, and the 
distance from the center axis c.sub.2 of the mother tube A to the outer 
circumference is D.sub.2 /2. And the rolls 211, 212, 213, 214 having 
D.sub.1 so that the ratio D.sub.1 /D.sub.2 may be 5 or more are used outer 
diameter reduction per stand is 12% or less. 
In the invention, since it is possible to roll almost uniformly on the 
whole circumference of the mother tube A as mentioned above, it is not 
necessary to suppress the internal surface squaring after rolling by 
making use of the tension obtained by setting D.sub.1 /D.sub.2 at 10 or 
more as in the conventional case of three-roll type. In the invention, 
therefore, by setting the value of D.sub.1 /D.sub.2 at 5 or more as the 
minimum value causing no overfilling trouble of the mother tube A into the 
caliber, stable rolling is realized by four rolls. In the invention, by 
setting the value of D.sub.1 /D.sub.2 at 7 or more, the increase of tube 
wall thickness due to reduction of outer diameter can be suppressed. 
In the invention, the outer diameter reduction per stand can be set higher 
than before, but slipping of a roll occurs when the outer diameter 
reduction is set over 12%, and to prevent this, if D.sub.1 is increased, 
overfilling of the mother tube A into the roll edge occurs, and hence the 
upper limit of the outer diameter reduction is set at 12%. 
On the other hand, by increasing the roll speed of the exit side stand 
faster than the roll speed of the entrance side stand, the tension between 
the stands can be obtained. When four rolls are used, wall thickness 
deviation can be prevented without obtaining tension, but when tension is 
obtained, increase of tube wall thickness due to reduction of outer 
diameter can be suppressed, which is beneficial for producing of rolled 
tube. In the invention, accordingly, the ratio of the speed of the 
entrance side stand roll to the exit side stand roll is set between the 
reference speed ratio and 1.8 times the reference speed ratio, the 
reference speed ratio being the ratio of the speed when tension does not 
act between the stands. By gradually increasing the speed of each stand so 
that the ratio of the speed of the rolls of the both stands may remain 
within this range, it is possible to obtain a specified tension without 
slipping of rolls. 
The result of cold tube rolling of steel tube by the method of the 
invention is described below. 
FIG. 13 is an explanatory diagram of the shape of groove edge of a roll 
being used, and in the following numerical examples, as shown in FIG. 13, 
the roll of DR type with angle of 20 degrees formed by the groove edge 
contacting with the arc of radius of 2b.sub.i, having the center on the 
circumference of radius b.sub.i contacting with the groove bottom was 
used. (Numerical example 1) 
______________________________________ 
Steel tube Material: Low carbon steel 
Dimensions: o 95 16 mm .times. 2 mmt 
Stand Number: 5 stands 
Nominal roll diameter: o 20 mm 
First stand roll groove bottom 
diameter/mother tube diameter: 
D.sub.1 /D.sub.2 = approx. 6.6 
______________________________________ 
The nominal roll diameter is equal to the distance between confronting roll 
shafts. 
In such conditions, continuous rolling was conducted with the outer 
diameter reduction per stand being 8, 10, 12, and 14%. As a result, in 
case of the outer diameter reduction per stand being 8, 10, 12%, the inner 
surface was not squared after rolling, and overfilling of a mother tube 
into the roll groove edge was not found. However, the overfilling occurred 
with the outer diameter reduction per stand being 14%. (Numerical example 
2) 
______________________________________ 
Steel tube Material: Low carbon steel 
Wall thickness: 1.5 or 2.0 mmt 
Stand Number: 5 stands + finishing stand 
Nominal roll diameter: o 120 mm 
Outer diameter draft: Approx. 
10%/stand 
______________________________________ 
However, the finishing stand has a caliber being in the same size as the 
fifth stand and varied about 45 degrees in phase from the pass line. 
In such conditions, the steel tube outer diameter D.sub.2 (see FIG. 9) was 
set at .phi. 16, 18, 20, 22, 24 mm, and the corresponding roll groove 
bottom diameter D.sub.1 at .phi. 105.6, 103.8, 102.0, 100.2, 98.4 mm. As a 
result of defining the D.sub.1 /D.sub.2 ratio thus at 6.6, 5.8, 5.1, 4.6, 
4.1, when the D.sub.1 /D.sub.2 ratio was over 5.0, the steel tube was 
caught by the rolls regardless of the wall thickness of the steel tube, 
but at the D.sub.1 /D.sub.2 ratio of 5.0 or less, biting failure occurred 
or slipping between the tube surface and roll occurred. (Numerical example 
3) 
______________________________________ 
Steel tube Material: S50C 
Dimensions: o 16 mm .times. 2 mmt 
Stand Number: 5 stands + finishing stand 
Nominal roll diameter: o 120 mm 
First stand roll groove bottom 
diameter/mother 
tube outer diameter: D.sub.1 /D.sub.2 = 
approx. 6.6 
Reduction in outside diameter: 
Approx. 9%/stand 
Caliber pass schedule: o 14.7 .fwdarw. 
13.3 .fwdarw. 12.0 .fwdarw. 10.9 .fwdarw. 10.0 .fwdarw. 
10.0 mm 
(Dimension between roll groove 
bottoms) 
______________________________________ 
In such conditions, by setting the peripheral speed ratio of the roll in 
accordance with the rate of reduction in area of the rolled tube so as to 
be the reference speed ratio at which the tension between stands may be 
approximately 0, the peripheral speed of the roll of the sixth stand was 
1.5 times the peripheral speed of the roll of the first stand. In this 
case, the dimensions of the steel tube after rolling were .phi. 15 
mm.times.2.5 mm t. As a result of setting the peripheral speed of the roll 
of the sixth stand at 2.0, 2.4, 2.7, and 3.0 times the peripheral speed of 
the roll of the first stand, that is, 1.3, 1.6, 1.8, 2.0 times the 
reference speed ratio, respectively, the increase of tube wall thickness 
due to reduction of outer diameter could be suppressed up to 1.8 times of 
the reference speed ratio, but when exceeding 1.8 times of the reference 
speed ratio, slipping between the tube and roll surface occurs, and 
seizure of steel tube on the roll surface occurred. 
Incidentally, when the D.sub.1 /D.sub.2 ratio was varied in every stand as 
in numerical example 2, the nominal roll diameter may be set constant, or 
set in plural values. FIG. 14 and FIG. 15 are schematic diagrams for 
explaining the arrangement of stands having four rolls of the cold reducer 
according to the invention. In FIG. 14, the nominal roll diameter is set 
specifically, and in FIG. 15, the nominal roll diameter is set in two 
types. In the diagrams, 201, 202, . . . , 208, and 251, 252, . . . , 258 
are stands, and the mother tube of .phi. 20 mm is rolled in the sequence 
of stands No. 1, 2, 3, . . . , 8. In the diagrams, the rolls of the stands 
are arranged in the same direction, but actually they are shifted in phase 
by 45 degrees each in the neighboring stands. 
The radius D.sub.1 in the groove bottom of the roll possessed by the stand, 
the tube outer diameter D.sub.2 at the stand entrance side, and D.sub.1 
/D.sub.2 are shown in Table 3. 
As shown in Table 3, when the nominal roll diameter is set specifically, 
the D.sub.1 /D.sub.2 ratio is large in the latter half stands, and as the 
roll diameter becomes larger, the facility becomes unnecessarily larger, 
but by compatibility of stands and sharing of parts, the pass line can be 
formed by one frame. When the nominal roll diameter is set at .phi. 120 mm 
and .phi. 80 mm, a large difference is not found in D.sub.1 /D.sub.2, and 
since the nominal roll diameter is set small in the latter half stands, 
the roll diameter may be reduced in the latter half stands. 
Another embodiment of the invention is specifically described below by 
referring to the drawings. FIG. 16 is a schematic diagram showing the 
constitution of a tube rolling apparatus according to the invention, and 
FIG. 17 is a sectional view magnifying the structure at the downstream 
side of the rolling apparatus. In the illustrated rolling apparatus, nine 
stands 301, 302, . . . , 309 are disposed in tandem, and each stand is 
matched in the caliber individually, and rolls 311, 312, . . . , 319 of 
the stands are shifted in phase by 45 degrees relatively to the pass line 
with respect to the rolls of the upstream stands. 
Numeral 309 shows the extreme downstream side stand which is the finishing 
stand, and a die 322 held by a die holder 321 is fixed at the exit side of 
the extreme downstream side stand 309. A tube material 320 rolled by 
driving of a roll 319 composing the extreme downstream side stand 309 is 
guided by a guide 323 fixed at the entrance side of the die 322, and is 
inserted into the round hole in the die 322 to be sized to a specified 
dimension. 
At both ends of the groove cut in the peripheral surface of the roll for 
forming the caliber, the caliber radius of the groove edge is set larger 
than the caliber radius of the groove bottom. Hence, overfilling of the 
mother tube into the roll gap and flaw of tube at the groove edge can be 
prevented. Furthermore, as the roll forms the side relief minus caliber as 
mentioned above, it is possible to roll uniformly in the centripetal 
direction of tube axis on the mother tube in the roll groove bottom and 
groove edge. Such roll shape, caliber and driving method are same as in 
the foregoing embodiments, and the detailed description is omitted. 
Using such apparatus, tubes were produced in the following conditions. 
______________________________________ 
Steel tube Material: Low carbon steel 
Dimensions: o 18 mm .times. 2.0 mmt 
Stand Number: 8 stands + finishing stand 
Nominal roll diameter: o 140 mm 
Lubricant: Online application of 
water-soluble liquid 
Caliber pass schedule: o 18 .fwdarw. 
16.2 .fwdarw. 14.6 .fwdarw. 13.2 .fwdarw. 11.9 .fwdarw. 
10.8 .fwdarw. 9.7 .fwdarw. 8.8 .fwdarw. 8.0 .fwdarw. 8.0 
(for finishing) mm 
Tube material mean outer diameter at 
exit side of extreme downstream side 
stand: d.sub.1 
Die diameter: d.sub.2 
Maximum reduction in outer diameter 
r: 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 
Distance L from extreme downstream 
side stand: 80, 169, 240 mm 
______________________________________ 
However, the finishing stand has a caliber being in the same size as the 
eighth stand and varied about 45 degrees in phase from the pass line. 
The reduction in outer diameter by the die of the exit side of the extreme 
downstream side stand is determined in the following formula. 
EQU r=(d.sub.1 -d.sub.2)/d.sub.1 .times.100% (4) 
Supposing the distance from the caliber center of the extreme downstream 
side stand to the die entrance in the pass line direction to be L, the 
following formula is set as the criterion. 
EQU L.ltoreq.6.times.[{d.sub.1.sup.4 -(d.sub.1 -2t).sup.4 }/(d.sub.1.sup.2 
-d.sub.2.sup.2)].sup.1/2 (5) 
The result is shown in Table 4. 
In Table 4, ".smallcircle." designates absence of seizure or buckling, and 
".times." designates presence of seizure or buckling. The reference 
example employs four rolls, but does not satisfy formula (4) in which r 
exceeds 5% and/or formula (5). The conditional formula is the value of 
formula (5) determined in terms of the tube material mean outer diameter 
d.sub.1, die diameter d.sub.2, and tube material thickness t. 
As clear from Table 4, in this embodiment, a small diameter tube with less 
wall thickness deviation and higher outer diameter precision as compared 
with the reference example could be produced without causing seizure or 
buckling. On the outer surface of the tube material, it is enough to apply 
soluble oil only, and surface conditioning for lubrication required in the 
drawing method is not necessary. 
In the reference example with the reduction in outer diameter r of the die 
exceeding 5.0%, the extruding resistance by the die is high, and although 
buckling could be prevented by narrowing the distance of the roll of the 
extreme downstream side stand and the die, slip occurred at the upstream 
side of the second stand in the tube material tail portion, and the tube 
material was stopped between the rolls. As a result, seizure phenomenon of 
the tube material on the roll surface occurred. At the reduction in outer 
diameter r of 7%, seizure occurred in the die, and buckling could not be 
prevented, too. 
At the reduction in outer diameter r of 5.0%, without disposing the die 
near the roll of the extreme downstream side stand, when the distance L 
does not satisfy the conditional formula of (5), buckling occurred in the 
tube material between the die and roll, seizure occurred on the tube 
material by reducing the diameter at the die, or flaw was formed on the 
surface of tube material. 
At the reduction in outer diameter r of less than 0.5%, although not shown 
in Table 4, the tube material does not contact with the circumference of 
the die at several points, and uneven contact occurs. As a result, 
vertical streaks were often formed on the tube material. 
Furthermore, as reference example, by rolling at the reduction in outer 
diameter of 10% per stand by the three-roll cold stretch reducing method, 
overfilling occurred at the time of continuous rolling. To prevent this 
overfilling, it was attempted to form a relief by setting the caliber 
radius larger for both edges of groove by using the side relief plus 
caliber, but the wall thickness deviation occurred in the tube material 
because of the plus side relief, and the inner surface of the tube 
material was deformed into a hexagon. Incidentally, in case of setting of 
the reduction in outer diameter of 6% with the side relief minus caliber, 
overfilling could be prevented, but the inner surface of the tube material 
was also deformed into a hexagon. 
Considering these results, a tube can be produced with the same high outer 
diameter precision as in the drawing method, for example, the outer 
diameter precision of .phi. 10+-0.02 mm of the drawing method, and wall 
thickness precision, by sizing the tube material rolled by a four-roll 
continuous reducer by means of a die. If not sized by die, the outer 
diameter precision is about .phi. .+-.0.05 min. Besides, reduction rolling 
of high degree of processing of 50 to 70% by continuous rolling of one 
pass can be executed, so that continuous rolling can be performed very 
efficiently. Furthermore, since the roundness of outer diameter is 
enhanced by sizing, it is possible to roll without deviating the wall 
thickness even by setting the outer diameter reduction as 10% per stand. 
FIG. 18 is a schematic diagram showing the constitution of a tube rolling 
apparatus according to the invention. In the diagram, numeral 324 
designates a pinch roll, which is disposed at the further downstream side 
of a die 322. The pinch roll 324 comprises two rollers disposed oppositely 
across a tube material, and by the movement of these rollers, the distance 
between the rollers is variable. The pinch roll 324 receives a signal from 
a drive unit 326 having a timer function, and rotates the rollers after a 
specified time, and the distance between the two rollers is shortened at 
the same time to hold the tube material. 
At the entrance side of an extreme upstream side stand 311, a detecting 
device 325, which can be an optical sensor, is disposed. The detecting 
device 325 detects the tail end of the tube to be rolled, and gives a 
signal to a drive unit, while the drive unit 326 feeds signal to the pinch 
roll 324 after a specified time, that is, after the lapse of time until 
the tail end passes through the extreme upstream side stand and is 
positioned at the exit of the extreme downstream side stand. The other 
constitution is same as the apparatus shown in FIG. 16 and FIG. 17, and 
the description is omitted. 
For example, when producing a final tube in continuous process, the tail 
end may stop between the roll of the extreme downstream side stand and the 
die by the extruding resistance of the die. In such a case, when employing 
the apparatus in tile above constitution, at first, the tail end of the 
mother tube is inserted into the extreme upstream side stand 311, and it 
is detected by the detecting device 325, and its signal is fed into the 
drive unit 326. Just before the tail end stops after the specified time 
set in the drive unit 326, a signal is sent from the drive unit 326 into 
the pinch roll 324. The pinch roll 324 holds the tube material reduced in 
diameter and rotates it, and the stopped tail end is sent out. After 
sending out the tail end, the distance between the rollers of the pinch 
roll 324 is extended, and rotation is stopped. 
In this way, continuous production is possible without stopping the tail 
end of the mother tube. It may be also constituted to hold the tube 
material always with the pinch roll without using detecting device for 
sensing the tail end, but useless flaw may be avoided in the constitution 
with detecting means so as to send out only the minimum required limit of 
tail end. 
Meanwhile, the portion to be pulled out by the pinch roll 324 is the tail 
end portion of 100 to 200 mm for a tube material length of, for example, 5 
to 10 m. 
In this embodiment, the detecting device for detecting the tail end of the 
tube is one optical sensor, but this is not limitative, and an 
electrostatic proximity sensor may be used, or a plurality of detecting 
devices may be disposed at the stand entrance side. 
In continuous production, if the tail end is not stopped but is pushed out 
of the die as the front end of a succeeding tube catches up to be joined 
to the tail end of a preceding tube, or the tail end passes through the 
die only by the inertial force of the tube material as the rolling speed 
of the tube material is fast and the rolling reduction of the die is 
small, it is not necessary to use a pinch roll. 
When handling the tube material in coil form, the front end portion of a 
tube can be wound in a coil form on a take-up machine disposed at the exit 
side. In this case, pulling of the tail end from the die is performed by 
the torque of the take-up machine, and the pinch roll is not needed. 
FIG. 19 is a schematic diagram showing the constitution of a tube rolling 
apparatus according to the invention. In the diagram, numeral 322 
designates a die disposed in the bottom of the extreme downstream side 
stand 319, and a roller table 327 is disposed at the downstream side of 
this die 322. The roller table 327 comprises plural rollers disposed 
parallel rotatably, and the tube sized by the die 322 is moved on the 
upper surface of the roller table 327 and is conveyed. At this time, by 
setting the tube feed speed of the roller table 327 slightly faster than 
the tube rolling speed, the tube can be conveyed efficiently. The other 
constitution is same as the apparatus shown in FIG. 16 and FIG. 17, and 
the description is omitted. 
The tube rolling apparatus and rolling method of the invention described 
herein may be applied, for example, in producing process of welded tube. 
When producing a welded tube, it is necessary to change over the type of 
the welding forming roll in the pass series unit every time the finishing 
dimension varies, or change the roll conditions such as roll distance of 
plural types of rolls, and therefore it requires labor in assembling roll 
stands, or lowers the yield as the tail end and front end of continuous 
welded tube are out of the product standard. Accordingly by constitution 
so as to weld the tube to be welded in a specified size and insert tile 
welded tube into the extreme upstream side stand of the apparatus of the 
invention, the finishing dimension can be easily changed only by changing 
the arrangement of the roll units of the apparatus of the invention, 
without exchanging the welding forming rolls, so that the welded tubes may 
be produced at high yield. 
As described so far, in the tube rolling method and apparatus of the 
invention, since the radius of the edge portion is set longer than the 
radius of the middle part in the groove of the roll for forming the 
caliber, cold rolling can be performed without squaring the inner surface 
after rolling of tube, and products of high dimensional precision and 
surface quality can be produced, and the yield is enhanced at the same 
time. Besides, the reduction per stand can be set high, and the total 
number of stands can be decreased, and the facility cost is lowered. Thus, 
the invention offers outstanding effects. 
Since the diameter of the roll for the outer diameter of the tube to be 
rolled can be set smaller than in the three-roll type, the facility is 
smaller in scale and the facility cost is lower, and the stand interval 
can be shortened, thereby decreasing the number of rolled tubes going out 
of gauge, and the reduction in outer diameter of the tube can be raised, 
so that the total number of stands required for reducing the outer 
diameter to specified value may be decreased, and moreover a tension 
between stands can be obtained without slipping of rolls, thereby 
suppressing the increase of wall thickness due to reduction of outer 
diameter of the tube to be rolled. Thus, the invention offers the 
excellent effects. 
Furthermore, after rolling by using four rolls with the caliber with the 
radius of the middle part longer than the radius of the edge part in the 
roll groove, by sizing the outer diameter under slight reduction by the 
die fixed at the exit side of the extreme downstream side stand, a high 
roundness of outer diameter of the same level as in drawing method can be 
obtained without reducing process, so that tubes may be produced at high 
dimensional precision and high yield. By the sizing, still more, the 
roundness of outer diameter is high, and by installing one finishing stand 
in the final place, the diameter can be reduced to a desired size by a 
small number of stands, and it is not necessary to use different pass 
series for each finishing dimension. 
As the invention may be embodied in several forms without departing from 
the spirit of essential characteristics thereof, the present embodiments 
are therefore illustrative and not restrictive, since the scope of the 
invention is defined by the appended claims rather than by the description 
preceding them, and all changes that fall within metes and bounds of the 
claims, or equivalence of such metes and bounds thereof are therefore 
intended to be embraced by the claims. 
TABLE 1 
__________________________________________________________________________ 
reduc- rolling result 
test caliber tion b.sub.i 
a.sub.i over- 
interior 
No. type 
shape per stand 
b.sub.i-1 
b.sub.i-1 
.alpha. 
filling 
squareness 
__________________________________________________________________________ 
embodiment 
1 DR .theta. = 15.degree. 
10% 0.90 
0.916 
0.84 .largecircle. 
.largecircle. 
2 DR .theta. = 22.5.degree. 
10% 0.90 
0.935 
0.65 .largecircle. 
.largecircle. 
3 SR R = 1.1 r 
10% 0.90 
0.924 
0.76 .largecircle. 
.largecircle. 
4 SR R = 1.15 r 
10% 0.90 
0.935 
0.65 .largecircle. 
.largecircle. 
5 SR R = 1.2 r 
12% 0.88 
0.924 
0.63 .largecircle. 
.largecircle. 
reference 
example 
6 DR .theta. = 5.degree. 
10% 0.90 
0.902 
0.98 X .largecircle. 
7 DR .theta. = 10.degree. 
10% 0.90 
0.907 
0.93 .DELTA. 
.largecircle. 
8 SR R = 1.2 r 
10% 0.90 
0.945 
0.55 .largecircle. 
.DELTA. 
9 SR R = 1.4 r 
10% 0.90 
0.979 
0.21 .largecircle. 
.DELTA. 
10 SR R = 1.6 r 
10% 0.90 
1.007 
-0.07 
.largecircle. 
X 
11 SR R = 1.3 r 
7% 0.93 
0.996 
0.06 .largecircle. 
.DELTA. 
12 SR R = 1.4 r 
7% 0.93 
1.012 
-0.17 
.largecircle. 
X 
13 SR R = 1.2 r 
13% 0.87 
0.914 
0.66 X .largecircle. 
__________________________________________________________________________ 
TABLE 2 
______________________________________ 
interior 
e/b.sub.i a.sub.i /b.sub.i 
squareness 
______________________________________ 
0.10 1.027 .largecircle. 
0.15 1.039 .largecircle. 
0.20 1.050 .largecircle. 
0.25 1.061 X 
______________________________________ 
TABLE 3 
______________________________________ 
stand No. 
1 2 3 4 5 6 7 8 
______________________________________ 
specified 
nominal roll 
diameter 
D.sub.1 (mm) 
102 103.8 105.4 
106.9 
108.2 
109.4 
110.4 
110.4 
D.sub.2 (mm) 
20 18 16.2 14.6 13.1 11.8 10.6 9.6 
D.sub.1 /D.sub.2 
5.1 5.8 6.5 7.3 8.3 9.3 10.4 11.5 
nominal roll 
diameter 
at two type 
D.sub.1 (mm) 
102 103.8 105.4 
106.9 
68.2 69.4 70.4 70.4 
D.sub.2 (mm) 
20 18 16.2 14.6 13.1 11.8 10.6 9.6 
D.sub.1 /D.sub.2 
5.1 5.8 6.5 7.3 5.2 5.9 6.6 7.3 
______________________________________ 
TABLE 4 
______________________________________ 
calculated 
value accord- 
L ing to equa- 
r (%) (mm) tion(5) (mm) 
seizure 
buckling 
______________________________________ 
embodiment 
0.5 80 461 .largecircle. 
.largecircle. 
160 .largecircle. 
.largecircle. 
240 .largecircle. 
.largecircle. 
1.5 80 267 .largecircle. 
.largecircle. 
160 .largecircle. 
.largecircle. 
240 .largecircle. 
.largecircle. 
2.5 80 207 .largecircle. 
.largecircle. 
160 .largecircle. 
.largecircle. 
reference 240 .largecircle. 
X 
example 
embodiment 
3.5 80 176 .largecircle. 
.largecircle. 
160 .largecircle. 
.largecircle. 
reference 240 .largecircle. 
X 
example 
embodiment 
4.5 80 155 .largecircle. 
.largecircle. 
reference 160 .largecircle. 
X 
example 240 .largecircle. 
X 
5.5 80 141 X .largecircle. 
160 X X 
240 X X 
______________________________________