Note: Descriptions are shown in the official language in which they were submitted.
~53~
1 BACKGROUND OF THE INVENTION
The present invention relates to a rolling
mill incorporati.ng a pair of rolls displaceable in the
axial direction in accordance with the sheet width of
the rolled sheet to permit a shape control of the
rolled sheet and, more particularly, to the shape and
size of the axial end porti.on of such displaceable
rolls.
In recent years, there is an increasing
demand for an enhanced precision of thickness of the
rolled sheet and improved flatness (shape) of the same.
To cope with this demand, the present inventors
have already proposed in the specification of the United
States Patent No. 3818743 a rolling mill having inter-
mediate rolls disposed between an upper work roll and
an upper backup roll and between a lower work roll and
a lower backup roll, respectively, the intermediate
rolls being axially displaceable in opposite directions,
and a work roll bender.
In this rolling mill, the length of contact
between cooperating rolls is changed by the axial
displacement of the intermediate rolls to permit the
control of the deflection of the work roll. This
rolling mill, therefore, can remarkably improve the
quality of control of the shape of the rolled product,
~L~LS35~6
1 thanks to the com~ination of the axial displacement ofthe intermediate rolls and the operation of the work
roll bender. In addition, this rolling mill offers
various additional advantages such as improvements in
efficiency of the rolling equipment as a whole, rate
of operation of the rolling mill and enhanced yield of
the product, as well as saving of labour and energy.
In the rolling mill having axially displace-
able rolls of the kind described, rolls are arranged
in an asymetric manner with respect to the central axis
of the rolling mill, so that an asymmetric axial load
distribution is formed between the axially displaceable
rolls and the cooperating rolls contacting the latter.
In particular, the greatest load is produced at each
axial end portion of the axially displaceable rolls.
This problem is serious particularly when
the axially displaceable rol has end portions having
a stepped form, because in such a case an extremely
large stress concentration takes place in the portions
Of the roll surface near the stepped ends, because the
roll is abruptly released from the rolling load at the
stepped axial ends.
Thus, the axial end portions of the axially
displaceable roll suffer double disadvantages in
connection with the load as compared with the roll of
the conventional rolling mill, resulting in a shortened
life of the roll and/or generation of spalling.
In addition, in the event that the axially
~3586
l displaceable roll has stepped end portions, linear
surface flaws or scores are formed in the surface of
the adjacent roll. Such surface flow or score not
only shortens the life of the roll but also is trans-
ferred to the rolled product to seriously degrade thequality of the latter if it is formed within the span
or width of the rolled product.
; The rolls of rolling mills are usually made
of forged steel or cast steel. It is, therefore,
extremely difficult to overcome the above-described
problems by drastically enhancing the roll strength.
The use of expensive hard materials, needless to say,
uneconomically raises the cost of the roll.
Under these circumstances, it is an impor~ant
technical sub~ect to be achieved in the fieled of
rolling mill to ensure a high quality of the rolled
products by avoiding flaws or scores, while realizing
sufficient durability and anti-spalling characteristics
of the roll, using the conventional less expensive roll
materials.
The rolling mill of the kind described
inherently has a superior shape controllability. Thus,
it is also an important technical sub~ect to optimize
the shape and size of the axial end portion of the
axially displaceable roll so as to further improve or
at least to maintain the superior shape controllability.
The present inventors have made proposals,
on an assumption to provide the axial end portion of
-- 3 --
~153~
1 the axiaily displaceable roll with an arcuate profile,
to represent the radius of curvature of the arcuate
profile by a value of no dimension in relation to the
roll diameter. In this connection, a reference shall
be made to Japanese Patent Publication No. 16784/1978.
This proposal, however, provides a solution to a
problem concerning the determination of the starting
point of the axial end portion of the roll, i.e. the
junction between the cylindrical roll body portion and
the arcuate axial end portion.
If the load is applied to the rolls in such
a state that an axial end portion of the displaceable
roll contacts the lengthwise mid portion of the adjacent
roll, a flattening deformation is caused in the contact
regions of both rolls, so that the axial contact length
is increased as compared with that presented when there
is no load applied to the rolls. It is true that the
stress concentration and the scoring in the rolled
product can be avoided to some extent by adopting an
arcuate profile of radius R of curvature at the axial
end portions of the displaceable roll. However, if
the portion of the increased length due to application
of load has inadequate shape and size, the contact
region between two rolls is abruptly terminated so that
the problems experienced with the use of displaceable
roll having stepped axial end portions are encountered
even if the displaceable roll has axial end portions
of arcuate profile.
1~;3~
l The rolling mill of the type ~escribed permits
good rolling for varying rolling load and rolling
width. In fact 5 the rolling can be satisfactorily
performed even at such a high reduction ratio of about
50%. In the rolling operation at such a high reduction
ratio, the amount of flattening deformation between
the rolls is innegligibly large, and the above-
mentioned problems cannot be obviated solely by adopt-
ing arcuate profile of the axial end portions of
displaceable roll.
SUMMARY OF THE INVENTION
It is, therefore, an object of the invention
to provide a roll having axial end portions shaped and
sized to avoid, even if the length of contact region
between two contacting rolls is increased under the
application of the rolling load, undesirable stress
concentration, as well as generation o~ spalling and
score at the axial end portion of the roll to improve
the durability of the roll thereby to overcome the
above-described problems of the prior art, whlle
improving the shape controllability of the rolling
mill.
It is another object of the invention to
provide a roll which can eliminate the generation of
surface flaw or score in the contacting roll, thereby
to ensure a good quality of the rolled products.
It is still another object of the invention
-- 5 --
S3~36
1 to provide a ro]l capable of eliminating the aforemen-
tioned stress concentration, spalling and scoring under
the application of rolling load, even when conventional
less-expensive material such as forged steel, cast steel
or the like is used as the material of the roll thereby
to make it possible to overcome the problems of the
prior art without incurring a rise of the cost.
It is a further object of the invention to
provide a roll incorporated in a rolling mill adapted
to make a shape control of the rolled product by an
axial adjustment of axially displaceable roll or by
a combination of such an axial adjustment and an
adjustment of bending force of the work rolls having
high flexural rigidity, the roll having axial end
portions shaped and sized to permit tne roll to be
applied to a wide variety of size and use of the rolling
mill, e.g. rolling mill for aluminum, iron, hard metals
and so forth.
To this end, according to the invention,
there is provided a roll for use in a rolling mill
having upper and lower work rolls arranged in pair and
adapted to make contact with the material being rolled
to roll the latter, each work roll having a diameter
at least 15% of the roll barrel length thereof and a
high flexural rigidity, and a pair of displaceable
rolls disposed at one side or upper and lower sides
of said pair of work rolls in such a manner that a
plane defined by the axes of said displaceable rolls
-- o
1~5358.6
1 substantially or exactly coincides with a plane defined
by the axes of said work rolls and that said displace-
able rolls can be displa.ced in the axial direction in
accordance with the width of the rolled sheet thereby
to permit a shape control to control the shape of the
rolled product, characterized in that each displaceable
roll has an axial end porti.on having a diameter
decreasing gradually toward the axial extremity and
that the reduction in the radius of the axial end
portion within the axial region of 100 mm as measured
from the starting point of the axial end portion is at
least 0.3 mm.
In the rolling mill of the invention, the
end portion of the displaceable roll is suitably
]..ocated in relation to the widthwise end of the rolled.
material to perform a good shape control. Unfortunately~
however, there is a problem that the boundary between
the contacting region and non-contacting region of the
displaceable roll with t~e ad~acent roll is moved due
to a Hert~ flattening of the rolls when the rolling
load is actually applied. According to the invention,
even if the above-mentioned boundary is shifted this
problem is fairly overcome to ensure a good shape
control while avoiding the stress concentration and
generation of spalling and scoring of the roll, because,
in the rolling mill according to the invention, each
displaceable roll has an axial end portion which is
shaped in such a manner that a diameter gradually
-- 7 --
-
: -:
~;3~;1~66
1 reduces toward the axial outer extremit~ ~nd that the
reduction in radius in each axial end portion within
the axial region of 100 mm as measured from the
starting point of the axial end portion is at least
O.3 mm.
!rhese and other objects, as well as advan-
tageous features of the invention will become clear
from the following description of the preferr~d
embodiments taken in conjunction with the accompanying
drawings.
BRIEF DESCRIPTION OF THF DRAWINGS
Fig. 1 is a sectional view of a rolling mill
having axially displaceable rolls the axial end portions
of which being specifically shaped and sized in
accordance with the invention;
Fig. 2 is a side elevational view of the
rolling mill as shown in Fig. l;
Fig. 3 is an enlarged view of axial end
portion of the axially displaceable roll;
Figs. 4A, 4B and 4C are illustrations of
positional relationships of the rolls incorporated in
the rolling mill;
Fig. 5 is a chart showing the relationship
between the amount of Hertz flattening OI' rolls and
roll line load;
Fig. 6 is an illustration of deformation of
a roll of small diameter;
~53~6
1 Fig. 7 is a chart showing the relationship
between the sheet width and the rolled sheet and the
diameter of small work roll;
Figs. 8 and 9 are il~ustrations of other
shapes of the axial end portion of the axially
displaceable roll;
Fig. 10 is an illustration of load distribu-
tion on the roll;
and
Figs. 11 and 12 show different rolling mills
in which the axially displaceable rolls having axial
end portions specifically sized and shaped in accordance
with the invention are incorporated.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Preferred embodiments of the invention will
be described hereinunder.
Figs. 1 and 2 show a six high mill embodying
the invention having axially displaceable rolls the
axial end portions of which are shaped and sized in
accordance with the invention. More specifically, Fig.
1 is a sectional view of the rolling mill, while Fig. 2
is a side elevational view.
Upper and lower work rolls 2, 3 for rolling
the material 1 to be rolled in direct contact with the
latter are supported by metal chocks 5, ~ held in the
roll housing 4 at their both ends. The metal chocks
5, 5 in turn are carried by the inside of the left and
~53~8Çi
1 right projections 7, 8 provided on the roll hc,using 4
for free vertical adjustment. The projections 7, 8
incorporate hydraulic rams 9, 10 for effecting bending
of the upper and lower work rolls.
Upper and lower intermediate rolls 11, 12
arranged in pair and contacting the work rolls 2, 3,
respectively, are disposed such that their axes are
substantially in the same plane as those of the upper
and lower work rolls 2, 3, and are supported at their
both ends by metal chocks 13, 14. Each intermediate
roll has an axial end portion having an arcuate profile
and of a diameter gradually decreasing toward the axial
outer extremity. More specifically, the intermediate
rolls are arranged such that their arcuate axial end
portions are located at opposite sides of the rolling
mill. In other words, the arcuate axial end portion
of one intermediate roll is located at left side of the
path of the rolled material, while the arcuate axial
end portion of the other intermediate roll is located
at the right side of the same.
Upper and lower backup rolls 15, 16 are
arranged in a pair, in contact with the upper side of
the upper intermediate roll 11 and the lower side of
the lower intermediate roll 12, respectively, such
that the axes of these backup rolls are in the same
plane as the axes of the intermediate and work rolls.
These backup rolls 15~ 16 are supported at their both
ends by metal chocks 17, 18 provided in the roll
-- 10 --
11535;86
1 housing 4. A hydraulic ram 19 for efEecting the roll
reduction is connected to the lower side of the metal
chock 18 and is received by a cylinder 20. The metal
chocks 13, 14 for the intermediate rolls are received
by recesses 21, 22 of the metal chocks 17, 18 for backup
rolls, so as to permit the intermediate rolls 11, 12 to
be displaced in the upward and downward direction, as
well as in the axial direction.
The intermediate rolls 11, 12 are coupled,
through shafts 23, 24 connected to their one ends, with
means (not shown) for axially displacing the inter-
mediate rolls in opposite axial directions. The work
rolls 2, 3 are drivingly coupled with driving means
(not shown) through respective universal ~oints 25, 26
and drive shafts 27, 28.
In the rolling mill having the described
construction, the axial end portion of each intermediate
roll is axially adjusted in accordance with the width
of the sheet being rolled in such a manner that, for
example, the starting point of the axial end portion
of decreasing diameter is located to a position corre-
sponding to the widthwise end of the rolled sheet or
its vicinity. In consequence, the undesirable deflection
of the work roll, due to the load imposed by the backup
roll contacting therewith, is avoided to prevent
excessive rolling of the rolled sheet at both axial
ends of the work roll. In addition, the roll bending
effect is well performed by the hydraulic ram for
:
'
11~3~8S
1 bending, because each work roll is freed at its one
axial end from contact with the backup roll.
Hereinafter, a description will be made as
to the shape and size of the axial end portion of the
axially displaceable roll.
Fig. 3 is an enlarged view of the axial end
portion of the intermediate roll, in which XQ, R and
Ye represent, respectively, the axial length, radius
of curvature and radius reduction(relief) of the axial
end portion of the intermediate roll. Also, the
diameter of the cylindrical body portion and the point
at which the axial end portion starts are designated
with D and S, respectively. The direction of roll
axis is represented by x-axis, while the upward and
downward direction as viewed in this Figure is
represented by y-axis, with the crossing point of the
vertical line passing the starting point S and the
horizontal plane containing the intermediate roll
surface constituting the origin of the coordinate.
Figs. 4A to 4C show the positional relation-
ship of rolls. More specifically, Fig. 4A shows the
state in which no rolling load is imposed, Fig. 4B
~hows the state in which the roll end portion is in
contact with the cooperating roll over the entire
axial length thereof due to a rolling load imposed
thereon and Fig. 4C shows the state in which the
rolling load is imposed but the axial end portion makes
contact with the cooperating roll only at a part of
- 12 -
~3S86
1 axial length thereof.
In order to achieve the ob~ect of the
invention, it is necessary to satisfy the following
three requirements:
(1) To make sure that the axial end portion of
the axially displaceable roll does not make contact
over its entire axial length with the cooperating roll
even when under the presence of the rolling load as
shown in Fig. 4C, i.e. to have the minimum required
relief Yt
(2) To provide the axial end portion with a shape
and size which eliminate the problems concerning the
roll strength and scoring of the roll even when the
boundary between the contacting and non-contacting
axial regions of the axial end portion of the displace-
able roll is shifted due to the application of rolling
load.
(3) To provide the axial end portion with a
shape and size which ensure a higher shape control-
lability of the rolling mill, as will be describedlater in more detail.
In order to determine the relief amount Yt
as stated in item (1) above, it is necessary to obtain
the amount of deformation of rolls due to contact under
the presence of the load.
There are two kinds of deformation, one of
which is usually referred to as Hertz flattening.
Fig. 5 shows the theoretically obtained
: , '
1~5;~
1 relationship between the amount ~ of Hertz flattening
generated between two rolls 29, 30 and the line load p
(load per unit a.xial length of roll) imposed on the
roll. This relationship is t~eoretically defined by
the following equation.
~ = P~A (3 + ln ~A ~ ln p)
and A = 2(1 - v )
where,
E: Young's modulus
v: Poisson's ratio
Thus, t~e Hertz flattening amount can be
expressed by the following equation, if the sum (dl + d2)
of diameters of two rolls 29, 30 fall within the
practical range.
~ , 3 x lO p
where,
~: mm
p: Kg/mm
The roll line load adopted in actual rolling
mills usually falls within the following ranges:
(a) p = 200 to 500 Kg/mm: small-sized rolling mill,
rolling mill for aluminum, skin pass rolling
mill for iron
., :' '~
,
1 (b) p = 800 to 1000 ~g/mm: large-sized rolling mill,
rolling mill for hard material
Thus, the amount of Hertz flattening ~ is
calculated to be o.o6 to 0.15 mm and 0.24 to 0.3 mm
for the rolling mills belonging to the categories (a)
and (b). Therefore, in order to ensure that the contact
between the axial end portion of the displaceable roll
and the cooperating roll takes place only over a
portion of the axial end portion of the displaceable
roll by providing the relief amount Ye in only one of
these rolls, the relief amount Ye should be at least
0.3 mm.
Another factor which influences the roll
relief amount Ye is an increase in the length of
contact region between the rolls owing to the deflection
of rolls.
Namely, referring to Fig. 6, if the cooperat-
ing work roll 2 has a small diameter and low rigidity,
such a work roll makes a large deflection so that it
is necessary to provide a sufficiently large amount
of relief. In the case where the portion at which the
contact between two rolls is terminated is created by
the axial displacement of the displaceable roll as in
the case of the rolling mill of the invention, there
is a practical limit in increasing the relief amount
when the cooperating roll has a small diameter as in
the case of work roll.
Fig. 7 shows a relationship between the rolled
5~6
1 sheet width and the minimum diameter of ~ork roll a.,
obtained through a theoretical calculation on an
assumption that the diameters of the oackup roll and
the intermediate roll are 1400 mm and o50 mm, respec-
tively, and that the roll barrel length of the inter-
mediate roll is 1420 mm. It is necessary that the
work roll diameter D' has to be determined in relation
to the rolled sheet width B to satisfy the relationship
expressed by D' ~ 0.2B. This condition is generally
met by practical sizes of rolls. This relationship
expresses the limit for avoiding the so-called composite
elongation of the rolled material. In other words,
this relation determines the threshold value for
avoiding an abrupt deflection of the cooperating roll
at a portion of the latter where the support by the
displaceable roll is lost due to the axial displacement
of the latter.
Taking account of the meandering of the rolled
material during rolling, in the practical rolling mills,
the rolls having roll barrel length 100 to 150 mm
greater than the rolled sheet widt~ are employed. ~or
instance, for satisfactorily rolling a sheet having a
maximum width of 800 mm, the minimum required diameter
of the worl~ roll is 160 mm and the roll barrel length
is selected to be 900 to 950 mm. Thus, the ratio of
work roll diameter to the roll barrel length is 17 to
18%. Since the invention is applied to the rolling
mill having work roll of a high flexural rigidity and
L~ _
11~i35~36
1 the ratio of diameter to roll barrel length of wGrk
roll exceeding at least 15%, each work roll can be
supported by only one roll which is, in the arrangement
shown in ~ig. 1, the intermediate roll which is
disposed at each of upper and lower sides of the pair
of work rolls.
Therefore, provided that the diameter of the
cooperating roll is selected to be greater than the
above-mentioned minimum limit or threshold value, it
is not necessary to take into consideration the
expansion of the contact region attributable to the
abrupt deflection of the cooperating roll.
To sum up, it is required that the amount of
relief at the axial extremity or edge of the axially
displaceable roll is at least 0.3 mm in radius.
An explanation will be made hereinunder as
to the item (2) of the aforementioned requirements.
It is possible to preserve a non-contacting portion in
the axial end portion of the displaceable roll, if the
axial end portion has a relief amount in excess of 0.3
mm as stated above. In order to avoid the concentration
of stress to the boundary between the contacting and
non-contacting regions in the axial end portion of the
displaceable roll, as well as scoring in the cooperating
roll at the position of such a boundary, it is preferred
that the roll diameter of the axial end portion is
decreased toward the axial extremity or edge as
gradually as possible. When the axial end portion is
- 17 -
~i35;~;
1 formed in an arcuate shape. it is preferred that such
an arcuate axial end portion has a radius of curvature
in excess of 200 mm. Since a stress concentration
tends to occur at the starting point of the axial end
portion, i.e. at the boundary between the cylindrical
body portion and axial end portion of the displaceable
roll, it is preferrea that the axial end portion has
an arcuate profile of a radius of curvature of at
least 200 mm, more preferably between 300 and 4000 mm
to gradually decrease the roll diameter at such an
axial end portion.
Referring now to the item (3) of the afore-
mentioned requirements, although it is preferred to
reduce the diameter of axial end portion of the
displaceable roll as gradually as possiole to avoid
the stress concentration and scoring, a too small rate
of decrease in the roll diameter will cause a large
change of contact length between the axially displace-
able roll and the adjacent roll due to the action of
the rolling load which in turn hinders the precise
].ocation of the axial end portion of displaceable roll
in relation to the rolled material, resulting in an
insufficient shape controllabili.ty.
According to the results of studies made by
the present inventors, it has been made clear that, in
the large-size rolling mill having a roll line pressure
p of 800 to 1000 Kg~mm, the axial displacement of
the boundary between the contacting region and
- 18 -
li535t~
1 non-contacting region is preferab]y smaller than 10 mm.
A discussion will be made hereinunder as to the condition
for maintaining the axial displacement within the range
below the above-specified limit value.
Representing the axial displacements when the
roll line load p is 800 Kg/mm and 1000 Kg/mm, respec-
tively, by X2 and Xl, and assuming that the axial end
portion of the displaceable roll has an arcuate profile
of radius R of curvature for simplification of calcula-
tion, there exist the relationships expressed by the
following equations. As stated already, the amounts
of ~ertz flattening ~ are 0.24 mm and 0.3 mm, respec-
tively, when the roll line load p is 800 Kg/mm and
1000 Kg/mm.
X12
0 3 =
X 2 (X _ 10)2
0.24 2R 2R
From the above equations, it is derived that
the axial dlsplacement X1 is 94.7 mm. Thus, as a
standard, it is required to provide a relief amount Ye
in axial end portion of at least 0.3 mm in radius
within the region of 100 mm as measured from the star-
ring point of the axial end portion toward the axial
extremity or edge of the displaceable roll. Under the
presence of the rolling load, the boundary between the
contacting and non-contacting regions exist between
the starting point S of the axial end portion and the
- 19 -
.: :
~3~;~6
1 axial extremity or edge of the displace~ble roll. The
roll line load is decreased as such boundary is shifted
toward the axial extremity. It is, therefore, possible
to make the axial outer part of the axial end portion
have a radius R of curvature smaller than that at the
starting point of the axial end portion or to form such
an axial outer part by a straight line of a large
gradient. By so doing, it is possible to obtain the
smaller length between the starting point S of the axial
end portion and the point at which the radius reduction
of 0.3 mm is achieved.
For minimizing the axial length of the axial
end portion of the displaceable roll while avoiding
the inconveniences such as lack of strength at the
starting point S of the axial end portion, it is
suggested as a preferred embodiment that the arcuate
axial end portion of the displaceable roll has a radius
of curvature between 300 and 4000 mm. Although the
similar calculation is omitted, since the change of
rolling load for the same rolling mill is reduced, it
is desirable that the length between the starting point
S of the axial end portion of the displaceable roll and
the point at which the radius reduction of 0.3 mm or
greater is reached is selected to be smaller than
100 mm.
Figs. 8 and 9 show different forms of the
axial end portion of the displaceable roll. More
specifically, in the arrangement sho~n in Fig. 8,
- 20 -
~LS3~
1 the part of the axial end portion between the star~ing
point and the point at which the relief amount of 0.5 mm
is achieved has a radius of curvature of 5O00 mmR and
the part of the axial end portion beyond the above-
mentioned point is formed with a radius of curvatureof 500 mmR, the parts of 500 mmR and 5000 mmR being
connected smoothly.
On the other hand, Fig. 9 shows the form of
the axial end portion in which the axial outer part of
the axial end portion is relieved by a straight line.
The forms of roll end portion as shown in Figs. 8 and
9 offer an advantage that the axial length between the
starting point of the axial end portion and the axial
extremity or edge of the displaceable roll is diminished
to shorten the time required for grinding the axial end
portion of the displaceable roll, which is usually
troublesome and time consuming. In addition, it is
possible to obtain the large relief amount in radius
with a small axial length of the axial end portion~
e.g. an amount of Hertz flattening of 1 ~m or so
generated in the worst case such as a rolling accident.
In the foregoing description, the explanation
is focussed only specifically on the axial end portion
of the displaceable intermediate roll. It is clear,
however, that the axial end portion of the work roll
cooperating with the displaceable intermediate roll
makes a contact with the cylindrical body portion of
the latter, as a result of the axial adjustment of
~S~36
1 the displaceable roll, as shown in Fig. 10. In the
point of such a contact, the load distribution of roll
contact between the rolls is so small, as shown in Fig.
10, that no substantial problem is imposed concerning
the strength. However, in order to avoid the scoring
in the displaceable roll caused by the axial end portion
of the work roll, it is suggested that the work roll
has an axial end portion the diameter of which is
gradually decreased toward the axial extremity, e.~.
in an arcuate profile as shown in Fig. 10. Incidentally,
in Fig. 10 the mark P represents the rolling load.
Furthermore, although the invention has been
described specifically through a six high mill having
two intermediate rolls displaceable in opposite axial
directions and disposed between the upper work roll and
upper backup roll and between the lower work roll and
lower backup roll, this is not exclusive and the
invention is applicable to a four high mill as shown
in Fig. 11 in which backup rolls are axially displace-
able, a multi-stage mill as shown in Fig. 12 having
two intermediate rolls axially displaceable in opposite
directions and disposed between the upper work roll and
upper backup roll and various other types of rolling
mill.
The invention can be applied also to a
rolling mill incorporat ng rolls having a crown over
their entire axial length. In such a case, the point
at which the curvature of the crown or the taper
- 22 -
l~S3~8~
1 is abruptly changed is considered as being the starting
point of the axial end portion of roll.
