Proceedings of the Institution of Mechanical

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The Calculation of Roll Pressure in Hot and Cold Flat Rolling

E. Orowan
Proceedings of the Institution of Mechanical Engineers 1943 150: 140
DOI: 10.1243/PIME_PROC_1943_150_025_02

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140

The Calculation of Roll Pressure in Hot and

Cold Flat Rolling
By E. Orowan, Dr. hg.*
A numerical or graphical method is given for computing, in strip or plate rolling, the distribution
of roll pressure over the arc of contact and the quantities derived from this (e.g. the vertical roll
force, the torque, and the power consumption). The method avoids all mathematical approximations
previously used in the theoretical treatment of rolling, and permits any given variation of the yield
stress and of the coefficient of friction along the arc of contact to be taken into account. It can be
used, therefore, in both hot and cold rolling, provided that the basic physical quantities (yield stress
and coefficient of friction) are known.
The usual assumption that the deformation could be regarded as a locally homogeneous com-
pression has not been made, and the inhomogeneity of stress distribution has been taken into account
approximately by using results derived by Prandtl and NBdai from the Hencky treatment of two-
dimensional plastic deformation.
It is found that the discrepancy between the roll pressure distribution curves calculated from the
K h a n theory and those measured by Siebel and Lueg is due to the assumption in the theory that
the frictional drag between the rolls and the rolled stock is equal to the product of the roll pressure
and the coefficient of friction. If frictional effects are dominant, as in hot rolling, this product may
easily exceed the yield stress in shear which is the natural upper limit to the frictional drag, and then
static friction, instead of slipping, occurs. This has been taken into account in the present method,
and the calculated curves of roll pressure distribution show good agreement with the curves measured
by Siebel and Lueg.

( I ) Introduction. Beginning with the pioneer work of Siebel the material suffers work hardening ; and in hot rolling, because
(1924, 1925)t and Karman (1925), many attempts have been the yield stress depends on the rate of compression which varies
made during the last twenty years to obtain a satisfactory along the arc of contact.
method of calculating roll pressure and torque from geometrical ( d ) All detailed calculations were made with the assumption
and physical data of the rolls and of the rolled material. I n view of dry friction; only in a recent paper by Nadai (1939) are the
of the complexity of the problem, only the simplest case of flat alternative cases of purely viscous friction and of a constant
rolling without lateral spread has been treated; not until this frictional drag considered.
is thoroughly understood can section rolling and the effect of (e) All theories were based upon the assumption of a circular
spread be investigated. Even in dealing with flat rolling without arc of contact. As will be seen in section 33, p. 163, this excludes
spread, it was found necessary to use a number of more or less the treatment of thin strip or sheet rolling where roll flattening
drastic simplications, such as are given in paragraphs ( a ) - ( f ) is considerable and the arc of contact is far from being circular.
below. ( f ) Even with the above simplifications, the exact mathe-
(a) It was assumed that, for the purpose of the calculation, matical solution of the problem would have been far too com-
the rolled stock could be considered as consisting of thin vertical plicated for use in rolling mill practice. All authors, therefore,
segments perpendicular to the direction of rolling, with no shear resorted to mathematical approximations. It was in the nature of
stress but only a normal pressure acting between the neigh- these approximations that the various theories of rolling
bouring segments, and the plastic deformation of the segments differed ; the main physical simplifications of homogeneous
was assumed to be a homogeneous cornpression. I t was this deformation, dry slipping friction, constant yield stress, and
simplification upon which the first mathematical treatment of circular arc of contact, were used generally.$
rolling by Siebel and K h d n was based. It involved the assump- While thus all theories of rolling were based on a number of
tions that the vertical and horizontal pressures were constant more or less arbitrary approximations, there has been so far
within a segment, and that initially plane cross-sections only one experimental investigation available by which their
remained plane. results could be checked quantitatively. This is the work of
(b) Most authors assumed that the rolled stock was slipping Siebel and Lueg (1933) on the distribution of roll pressure over
on the rolls, i.e. that the friction was dynamic, not static. I n this the arc of contact. A comparison of these measurements with
case the frictional drag exerted by the rolls upon the rolled stock theoretical calculations, however, does not give directly the
was simply the normal roll pressure, multiplied by the coefficient informations required. I n cases of disagreement between theory
of friction (dry friction being assumed). and experiment (see, e.g. Fig. 3, p. 146) it is not clear which of
( c ) It was assumed that the yield stress (“compressive the numerous simplifying assumptions and approximations is
strength”) of the material remained constant during the pass. In responsible; on the other hand, agreement can be the result of
reality, it varies along the arc of contact : in cold rolling, because several errors cancelling each other. I n fact, Siebel’s theory with
The MS. of this report was originally received at the Institution its crude mathematical simplifications was often found to agree
cn 3rd May 1943; and in its revised form, as accepted by the with roll pressure measurements much better than the theory
Oouncil for publication, on 9th August 1943. put forward by Karman who, with the sameghysical assump-
The report gives an account of original research carried out for tions, had used far better mathematical approximations.
the Rolling Mill Research Subcommittee of the Iron and Steel T h e obvious way out of this deadlock was to discard as many
Industrial Council. Communications will be published later. simplifications as possible, and to work out a more accurate
* Cavendish Laboratory, Cambridge.
t An alphabetical list of references is given in Appendix 11,
p. 165.
*
Apart from an unsuccessful attempt by Ekelund to take into
account static friction (cf. section 9, p. 146).

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T A N D C O L D F L A T R O L L I N G 141
solution, irrespectively of whether it was simple enough for use were wrong, but also because the simplicity of its physical basis
in everyday rolling mill practice. The initial aim of the present makes it suitable for the explanation of the mathematical method.
investigation was, accordingly, to dispense with all mathe- In Part I11 the calculation of the roll torque, power consump-
matical approximations and with the assumption of a constant tion, and other quantities derived from the roll pressure distri-
yield stress. A graphical-numerical method of computation was bution will be treated, and in Part I V the question of how to
developed which, beside dispensing with mathematical approxi- obtain the yield stress experimentally, and how to use it for the
mations, permitted the direct use of experimentally determined calculation of the roll pressure distribution. The concluding
stress-strain curves (or, in hot rolling, of stress-compression rate Part V deals with the influence of roll flattening, and describes
curves). This method will be described in Part I of the present a simple experimental method of determining the shape of the
paper. Applied to Siebel and Lueg’s measured roll pressure flattened roll.
distribution curves, however, its results were not much better
than those of the older theories; the striking discrepancy shown (2) Practical Scope of the Present Theory. As mentioned,
in Fig. 3 remained. As in this case there was no appreciable roll previous theories of rolling aimed at methods of calculation
fianening, and no reason for assuming a complete breakdown simple enough for immediate use in rolling mill practice. T o
of the law of dry friction, only the assumptions of homogeneous achieve this, drastic simplifications were necessary which led
compression and of slipping friction could be responsible for not only to disagreement with experimental results, but to the
the failure. In fact, a simple consideration shows that slipping impossibility of tracing the source of the discrepancies among
could not have taken place over the whole arc of contact in the the numerous simplifying assumptions used. In order to clarify
experiment illustrated by Fig. 3. The highest value of the normal the situation, it is necessary to relax the demand for extreme
roll pressure, according to the measurements, was here nine simplicity, and to attempt, first, a sufficiently general and
times higher than the yield stress in shear, which was determined accurate treatment of the problem, without respect to whether
independently by compression tests. The coefficient of friction the method is simple enough for everyday use. This has been
was higher than 0.4; hence, the maximum frictional drag the basic idea of the present work. It should represent, not a tool
(tangential force per unit of surface area), calculated with the for use in the workshop, but a kind of “primary standard” from
assumption of slipping friction, would have been at least which simplified methods of calculation, valid for special cases
0.4~9 = 3.6 times greater than the yield stress in shear. This, of rolling, can be evolved, and on which existing simplified
of course, is impossible : one cannot exert upon the surface of methods can be tested. Such a simplified method has been
a body a frictional drag higher than its resistance to shear stress. developed in the meantime for hot flat rolling, with and without
If the frictional drag rises to the magnitude of the yield stress in spread; it goes far beyond previously used methods both in
shear, plastic shear will take place in the rolled stock, while the accuracy and in simplicity. This method will .be published in
surface will “stick” to the rolls with static friction. Ekelund due course.
(1933) attempted to take into account this circumstance ;instead Equally useful is the accurate method as a theoretical guide
of calculating when and where sticking would occur, however, in rolling mill research. A great amount of time, money, and
he made the assumption that sticking would always occur over ingenuity has been spent on rolling mill research, with com-
the whole exit side, and slipping over the whole entry side of the paratively moderate results : however great the number of
arc of contact. In reality, circumstances are different : if sticking experiments was, they could exhaust only a small fraction of the
occurs, it appears first at the neutral point, and the region of practically important combinations of roll diameter, thickness
sticking expands, with increasing coefficient of friction, towards ratio, reduction, coefficient of friction, stress-strain relationship,
the points of entry and exit. temperature, and speed. With a sufficiently general theory at
The last simplification to be discarded was the assumption of disposal, a small number of roll pressure distribution measure-
homogeneous compression. The plastic deformation between the ments would decide the accuracy of the theory, and indicate
rolls is fairly complicated, but, in the absence of lateral spread, which further improvements were necessary. After this, it only
it can be approximated with an accuracy sufficient for most remains to measure the two fundamental physical quantities
practical purposes by using results of Prandtl (1923) and needed for the calculation (the yield stress and the coefficient of
Nadai (1931). With the inhomogeneity of the plastic flow taken friction), and to develop simple approximate methods of cal-
into account, the differential equations of the roll pressure culation for special cases of rolling practice.
distribution were not more complicated than with the assump- At the moment, however, the most important use of the
tion of homogeneous compression ; they could be solved with accurate method is for deciding general questions of rolling mill
the same graphical method as used in Part I. The computation operation and design, such as :-
of the roll pressure distribution without the simplifying assump- What is the most economical magnitude of front and back
tions of homogeneous deformation and slipping friction will be tension in strip rolling?
treated in Part 11. The results agree extremely well with those Are large (e.g. 15-inch) or small (e.g. 2-inch) diameter working
measurements of Siebel and Lueg which can be used for such rolls preferable in strip rolling from the points of view of
comparison. These authors have published only two measure- economy and quality of the product?
ments for which both the yield stress curves of the material In given cases of rolling, are heavier or lighter drafts pre-
used, and the corrections for the finite thickness of the pressure ferable?
measuring pin (obtained from oscillograms of the time derivative Is it worth while to attempt a reduction of skin friction in
of the pressure distribution curve) have been given. Only these hot rolling?
two measurements could, therefore, be used for checking the
calculations. While this scantiness of the experimental basis The answers to these and many similar questions have
must be borne in mind, there is a circumstance that adds to the hitherto been a matter of personal belief and commercial
weight of the agreement shown in Figs. 18 and 19, p. 158-9. In advertisement. With the aid of the new method, they could be
the experiment represented by Fig. 19, sticking (static friction) calculated in many cases accurately enough for practical pur-
must have taken place over practically the whole arc of con- poses, and will be published in due course. For such applications
tact. In consequence, the value of the coefficient of friction, of the theory, obviously, it is irrelevant whether the work of
which could not be determined under the actual conditions of calculation takes a day or a week, if only the result is reliable.
rolling, did not appear in the calculations. This means that the
calculated curve in Fig. 19 does not contain any arbitrary para- P A R T I . G R A P H I C A L C A L C U L A T I O N OF T H E R O L L P R E S -
meter chosen so as to obtain agreement with the measured curve. SURE W I T H THE ASSUMPTIONS OF HOMOGENEOUS
The method described in Part I, based upon the assumptions COMPRESSION A N D S L I P P I N G FRICTION
of homogeneous compression and of slipping friction, is much (3) The Friction Hill. Let us consider a bar subdivided into
inferior to the method given in Part I1 even if sticking is absent, thin segments by planes perpendicular to the direction of rolling
and it is quite incorrect if sticking occurs; besides, its practical (this may be realized by scratching vertical lines on the sides
use is not simpler. Nevertheless, it has been incorporated in of the bar). As the bar passes between the rolls, the segments
the present paper, not only because it provided the means for become vertically compressed and at the same time warped.
finding out which of the simplifications of the older theories Owing to the compression, they expand longitudinally (in the

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142 T H E CALCULATION O F R O L L PRESSURE I N H O T A N D COLD F L A T R O L L I N G
Roll diameter.
direction of rolling) or laterally or in both directions; if the bar
is very wide (a plate), lateral spread is impossible, and the whole Roll efficiency (equation (86), p. 161).
expansion must be longitudinal. Owing to the longitudinal ex- Horizontal force per unit of width of the bar.
pansion, the vertical segments squeeze each other out from the Angular co-ordinare of the arc of contact.
roll gap ;this, however, is opposed by the friction between the Thickness ratio &ID.
bar and the rolls. I n consequence, a horizontal pressure arises Height of the rolled stock.
Auxiliary function defined by equation (75),
p. 160.
Yield stress in plane strain = ( 2 d / ? ) k * .
Yield stress in free homogeneous compression.
Yield stress, corrected for the inhomogeneity of
deformation.
Speed of compression.
Abbreviations, defined by equations (51) and (52),
p. 151.
P Coefficient of friction = tan a.
P Horizontal pressure.
P Total roll pressure (vertical roll force).
4 Vertical pressure.
R Roll radius.
S Normal roll pressure; normal pressure.
t Azimuthal pressure.
I’ Roll torque.
7 Shear stress.
V Mean horizontal velocity in a cross-section of the
bar.
V Volume of rhe material rolled per second.
W Defined by equation (45), p. 150.
W Pure work of rolling per unit of volume.
WO Work of free homogeneous plastic compression
per unit of volume.
X Distance from the plane of exit.
2 Defined by equation (19), p. 144.

Let u be the average horizontal velocity in a vertical plane BD

(Fig. I), h( = BD) the vertical distance between the rolls in this
plane, and V = wh the volume of the material passing through
the plane per unit of time. I n a steady state, V must be the
same for all vertical planes, so
V = v h = a c o n s t a n t . . . . . (1)
and thus u is inversely proportional to h. Let D = 2H be the roll
Fig. 1. Pressure Arising in Bar between Rolls +
diameter, ho the width of the roil gap, and the a n N a r c e
ordinate on the arc of contact in radians, counted from the plane
of exit. Then, according to Fig. 1, we have
in the bar, the magnitude of which, in a vertical plane BD
Fig. l), is such that it just overcomes the friction along h = h,+2R(1- cosq4) = ho+D(I- cosf$) . (2)
the surfaces AE! and CD and squeezes the material within
and v= . . v (3)
ABCD towards one end of the arc of contact (here it is AC). Ib+D(l-cos $1
In the neighbourhood of the plane of entry all vertical segments
are squeezed backwards, towards the plane of entry; in the On the other hand, the horizontal component of the cir-
neighbourhood of the plane of exit, they are squeezed forwards, cumferential speed of the rolls is
towards the plane of exit. Somewhere near the middle of the arc Vh = RwCOSf$ . . . . . . (4)
of contact there is a “neutral” plane (Fig. 1) which plays a role where w is their angulx speed.
analogous to that of a watershed: the material to its right is Thus, in general, the mean horizontal velocity v of the rolled
squeezed to the right, and that to its left is squeezed to the left. stock is not equal to the horizontal component Vh of the cir-
The horizontal pressure decreases with the distance from the cumferential velocity of the rolls in the same vertical plane.
neutral plane, because the areas AB and CD decrease and with This does not necessarily mean that the rolled stock must slip
them the frictional drag acting upon the volume ARCD. In the on the rolls; in reality, the horizontal velocity of the material is
absence of back and front tension, the horizontal force vanishes not constant over a vertical plane, and it may be e q d to Wh at
in the planes of entry and exit; it rises to a maximum in the the skin of the bar without its mean value ZJ being equal to Wh.
neutral plane. Its distribution over the arc of contact will be In the assumption of homogeneous compression (section 1,
given, therefore, by a hill-shaped curve denoted by F in Fig. 1. paragraph (a)), however, it is implied that the horizontal
This curve (or a similar w e representing the vertical pressure, velocity is constant over any vertical cross-section; from this
cf. Figs. 2,3, 18, and 19) is called the “friction hill”. it would follow that the rolled stock must slip on the rolls every-
where except in one vertical plane which, of course, must be
Alphabetical List of Symbols :- identical with the neutral plane. Thus, with the assumption of
+
Indices : refers to the exit side of the arc of contact ;
- ,, ,, entry side ,, ,, ,,
homogeneous compression, the neutral plane would be the only
“non-slip” plane. From this, conclusions have been drawn con-
o ,, ,, plane of exit; cerning the relationship between the forward slip and the co-
n ,, ,, neutral plane; efficient of friction (13. section 27, p. 160). Such conclusions,
m ,, ,, plane of entry. however, fall with the assumption of homogeneous compression.
a Abbreviation for @! (equation (9,section (17),
k ( 4 ) The Relationship between Friction and tbe Horizonral
p- 151). Force. Let f(x) be the total horizontal force, per unit width of
a Angle of friction. the bar, that acts across a vertical plane a t a distance x from the

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 T H E C A L C U L A T I O N OF ROLL PRESSURE I N HOT A N D COLD FLAT R O L L I N G 143
plane of exit. Taking f ( x ) as positive if it is a compressive force, plane, the lower between the neutral plane and the plane of
we consider a thin vertical segment of the bar between the entry). Thus we have*
vertical planes situated at x and at x+dx. The force exerted q=s(lFptan+) . . . . * (8)
upon this segment by the neighbouring parts of the bar is
Substituting this in equation (7), we obtain
f ( x ) - f ( x + d x ) = -gdx
dx df
4
= Dq cos 9
lFptan4
. . . (7A)

This force must be balanced by the horizontal components Using the angle of friction a instead of the coefficient of
of the frictional forces and of the normal pressure exerted by friction p = tan a, and remembering that
the rolls upon top and bottom of the segment. The top or bottom
surface per unit of width of the bar is du/cos 9; the resulting
(horizontal) force due to the normal pressure 5 of the rolls is,
therefore, we have = Dqcos+tan(+fa) . . . . (9)
dx d9
2s sin 4 -= 2s tan 4 dx As mentioned, the condition of plasticity will yield the second
cos 4
relationship between the unknown functionsf(+) and q(4).
If p is the coefficient of friction, and the assumption is introduced
that the bar slips on the rolls, the horizontal component of the (5) Thc Condition of Plasticity, and the Differential Equation
frictional drag acting upon top and bottom of the segment is of the Friction Hill. Let the material be subjected to a uniaxial
compressive stress, and let k* be the magnitude of the stress
dx necessary to produce plastic compression at a given speed. In
f 2 cos~9- = f2pdx
cos 4 the literature of rolling, k* is called the “compressive strength”
of the material. This expression, however, is not quite correct;
The positive sign applies where the friction acts upon the bar with the same right we could call the stress that produces plastic
in the direction of increasing x, i.e. between the plane of exit deformation with a given speed in a viscous liquid its “strength”.
and the neutral plane, and the negative sign between the neutral It is more sensible to use the word “strength” only for the stress
plane and the plane of entry. In what follows, we shall adhere to which definitely overcomes the resistance of the material and
the convention that, in the case of alternative signs, the upper produces fracture, and to call k* the yield strexs of the material,
one always applies on the exit side, and the lower one on the for a given state of strain hardening, a given temperature, and a
entry side of the friction hill. given speed of deformation.
The condition of equilibrium for.the segment between x and The state of stress in a bar between the rolls is not uni-
x+ak is axial; beside the vertical pressure q, a horizontal pressure p is
acting in the direction of rolling, and, unless there is free lateral
spread, there is also another horizontal pressure p’ parallel to
the roll axes. As mentioned in section (4), both q and p are
or -&=
df 2s(tanC+fp) . . . . . (5)
constant within a vertical segment of the bar if the assumption
of homogeneous compression is made ;then p can be calculated
by dividing the horizontal forcef by the height h of the segment.
We introduce the angle 9 instead of x as independent variable. so p =f/h . . . . . . (10)
We have (cf.Fig. 1)
x=Rsin$ . . . . . . (6) The third principal stress p’ vanishes only in the practically
unimportant case of free lateral spread, i.e. of a thin strip rolled
and hence
upon its edges. Its magnitude depends on the amount of the
lateral spread; it is easily found in the case where lateral spread
is prevented, as in plate and sheet rolling. With p’ = p, the
substituting this in equation (5) and using D = 2R,we obtain lateral spread would be, for reasons of symmetry, equal to the
spread in the direction of rolling; hence, p’ must be greater
than p. On the other hand, if p’ = q, then, again for reasons of
symmetry, the bar would suffer a lateral compression equal to
the vertical compression; hence, p‘ must be less than q. A mor?
This equation contains two unknown functions, i.e. the hori- detailed consideration (Nadai 1931, chap. 27) shows that p
zontal force f(+), and the normal pressure on the roll surface must be half-way between p and q, i.e. p’ = (p+q)/2.
s(#J). If we iind a second equation between these functions, we Thus, with the assumption of homogeneous compression made
can calculate both. We shall see in section (5) that the condition in section (4), we have in the case of vanishing lateral spread
of plasticity gives, in fact, the second relationship needed. three principal stresses, all compressive: q, p‘ = ( p + q ) / 2 , and
We have not used so far the assumption of homogeneous p. What is the condition of plastic yielding in this case? We can
deformation explicitly. Now we are introducing it by treating immediately find it in the simpler case when the principal
the vertical segments of the bar as if they were separated from stresses are q, p, and p (i.e. when two of the principal stresses
each other, and their vertical surfaces sliding without friction are equal). According to experience, a hydrostatic stress cannot
on the neighbouring segments, so that no shear stress can arise produce plastic deformation and, superposed to another stress
in them. In this case the vertical pressure is constant within each system, its influence on yielding is negligible. We can, therefore,
segment; it is one of the principal stresses, and the horizontal superpose the hydrostatic tensile stress -p, - p , -p upon
pressure is the other. We assume that there is no lateral spread; our system q, p, p, and then we obtain the stress system q-p,
as will be seen in section (3, the condition of plasticity in this 0,O which is a uniaxial compressive stress with the conditi In of
case is that the difference between the principal stresses must plasticity
be a constant. Hence,the horizontal stress must also be constant
within a vertical segment. We can introduce the vertical pressure
q - p = k* .
. . . . . (11)
q($) instead of the normal pressure s(4) into equation (7). The What is now the effect of a third principal stress of a magni-
vertical force in the segment, per unit of width of the bar, is tude between the greatest and the smallest principal stress?
qhr ;this must be equal to the vertical component 5 cos 4 - 1 dx * This relationship is a necessary consequence of the assumption
c0-3 9 of homogeneous compression and, therefore, we have to accept it in
= s d r of the normal pressure, plus the vertical component Part I of this report, where the aim is to draw conclusions from that
1 assumption and compare them with experimental data. It must be
~ p s i n $ - dx = ~p tan+ of the frictional drag (the emphasized, however, that equation (8) is not quite correct, because it
cos d takes no account of the shear stresses which, in general, cannot be
upper sign applying between the plane of exit and the neutral neglected (cf. section 9, p. 146).

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144 T H E C A L C U L A T I O N O F R O L L P R E S S U R E I N H O T A N D C O L D FLAT R O L L I N G
There are two theories which claim to answer this question. A(+) and B(4) are given functions of 4; the calculation of
The “maximum shear” theory of Tresca considers that the k(+) from geometrical data of the mill and from the stress-strain
third principal stress has no influence whatever, and that the curve or the stress-speed of compression curve will be treated
condition of plasticity under the stress system q, p’, p is simply in sections 28 and 29, p. 161.
equation (1 1). The theory of Muber-Mises, on the other hand, The formal solution of a differential equation of the type (13)
leads to a plasticity condition in which the third principal stress is simple. We consider f(+) as a product of two functions u(+)
plays an essential role; but in the special case of plane strain (no and v(d)* :-
spread in one direction; see section (12), p. 147) the Huber-
Mises condition differs from equation (11) only by a factor f(4)= u ( ~ ) v ( $ > . . . . . (15)
2/\/3Zl.15 on the right-hand side of the equation; it is By the substitution (15), equation (13) is transformed into
4 - p = 2/d3&* ff 1*15&*= k .
. (11A) dv du
-U--V-+UVA+B=O . . . (I3A)
Experiments by Lode (1925) indicate that the Huber-Mises d4 d4
theory is more correct than the maximum shear theory. One of the functions u and v can be fixed arbitrarily; we
From the point of view of the mathematics involved, there is define z i by the condition
no difference between the two conditions (11) and ( 1 1 ~ ) ;we
shall write the condition in the form (llA), remembering that
h has a value about 15 per cent higher than the yield stress K*
in uniaxial compression.
From (10) and (11A) follows or $=Au.
.
q =p+k =f/h+k . . . .(11B) The integral of this equation is
Putting this into equation (9), we obtain log, = SA(+)d4+ c
df/d+ = D ( f / h + k ) cos 4 tan ($*a) .
, (12)
or 24 = ule*P(j;14(4)d4) . . . . (17)
This is a differential equation for the functionf(+), which is the
only unknown function occurring in it. Integration of equation With regard to (16), expression ( 1 3 ~ reduces
) now to
(12) gives f(+) from which, by means of (10) and ( l l ~ ) ,the
horizontal pressure p and the vertical pressure q are obtained, dv
-u-+B =0
in their dependence upon 4. 4
This equation can be integrated directly :-
(6) Theoties of Rolling. As mentioned in the Introduction
(p. 140), all theoretical treatment of rolling has so far been based
upon the assumptions that the deformation is a homogeneous
compression of vertical segments of the bar; that the bar slips We introduce the abbreviation
on the rolls; and that the yield stress is constant. The various
theories of rolling, while accepting these simplifications, differ Zed) = 4, ;
A(4)d+) * . . * (19)
in additional mathematical approximations made in order to
obtain a solution of the differential equation (12) in simple and obtain, .by substituting the values of u and w from expres-
finite terms. We now make a brief review of the simplifications sions (17) and (18) in equation (15),
characteristic of the main theories.
To begin with, it seems that equation (12) in its strict form
has never been used. Tselikov (1936) was prevented only by a
slight error of differentiation from obtaining equation (12); but wherefl = ulvl is the value off for 9 =
in other papers it was usual to neglect the vertical component of
the frictional drag. This leads to (tan +& tan a) taking the O n the exit side of the arc of contact we have to take the upper
place of tan (+&a) on the right-hand side of equation (12). Most sign in expressions ( 1 4 ~ )and (14~),and integrate from the
authors, including Siebel and KAman, assumed that the angle of plane of exit inwards; thus the lower limit 41 of the integral is
contact was small, and put cos +-,I and sin +=tan qk+. here 40= 0. The integration constant f, in equation (20) is the
Of the different methods put forward, that of Siebel (1924) horizontal force per unit width of the bar in the plane of exit,
represents the roughest approximation; in it the vertical pres- and is positive if it is a compressive force. Hencef, equal tofo
sure q (whose variation along the arc of contact was to be cal- on the exit side, gives the front tension divided by the width of
culated) was assumed, for the purpose of the calculation, to be the bar and taken with a negative sign.
constant and equal to k. In this way the unknown function f(+) On the entry side the lower (negative) sign must be used in
disappears from the right-hand side of (12) and the equation can expressions ( 1 4 ~ and
) ( 1 4 ~ and
) ~ we have to integrate from the
be solved by one simple integration. plane of entry inwards, the lower limit of the integral being
Karmln’s method (1925) is a correct approximate solution dl = &. Since a is the gripping angle,4 is in most cases smaller
of the problem with the assumption that the contact angle is than a, and tan (+-a) is negative, except sometimes in the
small. immediate neighbourhood of the plane of entry. With
‘Tselikov’s simplification consists in replacing the arc of tan (4- u), the integrands in expressions (17) and (20) are
contact by two chords, one between the plane of entry and the negative ;the integrals, however, are positive because the upper
neutral plane, and the other between the neutral plane and the limit is less than the lower. I n graphical integration it is con-
plane of exit. venient to ignore this double change of sign by taking tan (a-+)
Ekelund’s (1933) method must be regarded as semi-empirical ; instead of the negative tan (9- a), and then interchanging the
it contains a great number of approximations, often very limits of all integrals on the entry side by regarding areas above
arbitrary, which are selected with a view to obtaining agree- the +-axis as positive. The integration constantfl on the entry
ment with the results of certain experiments on hot rolling of side is the back tension divided by the width of the bar, and
steel. taken with a negative sign.
Equation (20) solves the differential equation (12) by two
( 7 ) Solution of the Difierenrial Equation. Equation (12) is of integrations. Of the two integrals involved, JA(+)d4 could be
expressed in finite terms if roll flattening were neglected; but
the formula: are much too complicated to be of practical useful-
ness. The second integral contains &(+) which is given as an
where * The function w(4) will be used in this section only; it has nothing
10 do with the mean horizontal velocity Y introduced in section 2,
p. 141.

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T AND COLD FLAT R O L L I N G 145
contrast in Fig. 3 the measured curve with curves calculated by
Siebel’s, Kirman’s, and the present homogeneous graphical
method, for the value 0.4 of the coefficient of friction.

Experiment 1: Experiment 2 :
smooth rolls rough rolls

Material . Annealed aluminium

Roll diameter, mm.. . 180 180 Lueg’s measurement.
Initial thickness L,mm. . 2.0 2.0 -.- Homogeneous graphical method.
Final ,, ho, mm. 1.o 1.1 _ _ _ _ K a d n method.
Initialwidth,mm.. . 30.0 30.0 - - - - -- Siebelmethod.
Final 30.8 32.3
Angle of’EOn& & : 6” 5’ 5” 40’
Fig. 2 shows, for experiment 1 with smooth rolls, the curve
measured by Siebel and Lueg, the curve computed by the homo-
The yield stress of the material used has been determined by geneous graphical method described in the preceding section,
Siebel and Pomp (1927) in free homogeneous compression as a the Siebel curve (calculated with the mean value of the yield
function of the compressive strain. The yield stress to be used for stress over the arc of contact), and a curve calculated with the
our purpose is obtained by multiplying Siebel and Pomp’s values homogeneous graphical method but with the constant average
by 2 / d ? , because the lateral spread is very small (6.section 5, value of the yield stress : this curve is practically identical with
p. 143). Siebel and Lueg have measured the roll pressure distri- the Karman curve because the mathematical approximations
bution at different distances from the middle of the strip; for a made by Karmin are very good for small angles of contact such
comparison with the calculated curves, we have to use the curves as the present ones. The agreement between the curve calculated
measured at (or interpolated to) the middle of the strip, as they with the homogeneous graphical method and the measured curve
are nearest to the case of no lateral spread upon which our is fairly good. The absence of the sharp peak in the measured
calculations are based. There was no front or back tension curve is due to a circumstance that will be explained in sec-
applied in the two experiments, but, owing to the variation of tion (20) ;we shall see that the total roll pressure is not affected
the stress distribution from the middle to the sides of the strip, by this, and it is not worth while, therefore, to take it into
the roll pressure at the middle in the planes of entry and exit account. The Siebel method gives too low pressures; this is
was higher than would correspond to the values of the yield because in calculating the variation of the roll pressure it assumes
stress, in the absence of back and front tension. In other words, that, for the purpose of the calculation, the roll pressure can be
the sides of the strip exerted a (small negative) front and back regarded as approximately constant along the arc of contact.
tension upon its middle parts. This has been taken into account The Karmln method, which differs from the present homo-
in all calculations, both by the present homogeneous graphical geneous graphical method mainly by the assumption of a con-
method, by the methods used for comparison, and by the new stant yield stress, gives an increased roll pressure near the plane
method to be described in Part 11. of entry where the yield stress is much smaller than the mean
The coefficient of friction has not been measured by Siebel yield stress. In the present case this has comparatively little
and Lueg. For experiment 1 (smooth rolls), p = 0.14 was found influence upon the value of the total roll pressure; it causes,
to give best agreement with the measured curve. I n experiment 2 however, a considerable error in the roll torque, because the
(rough rolls) the coefficient of friction must have been higher largest lever arm for the roll torque occurs where the in-
than 0.4. We shall see in Part I1 that, under the circumstances accuracy of the method is highest (cf. section (24), p. 159).
of the experiment, the material must have stuck to the rolls Fig. 3 shows, for experiment 2 with rough rolls, the curves
along almost the whole arc of contact, and hence the value of p corresponding to those shown in Fig. 2; it is drawn on a smaller
did not appear at all in the calculation of the roll pressure. In this scale for the ordinate axis. There is a striking discrepancy
case all methods of calculation that are based on the assumption between the measured curve and all calculated curves, showing
of slipping friction fail completely; to give an idea of the that here where the material sticks to the rolls the methods that
magnitude of the errors to which their use may lead, we shall use the assumption of slipping friction fail completely. The

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146 THE CALCULATION O F R O L L PRESSURE I N HOT AND COLD FLAT R O L L I N G
per sq. mm. I n other words, the theories based on the‘assump-
tion of slipping friction would calculate in this case with a shear
stress of 36 kg. per sq. mm. at a point where the resistance of the
material against shear was only 10 kg. per sq. mm.
This contradiction is easily explained. Wherever the product
of the normal pressure s and the coefficient of friction exceeds
the yield stress in shear k / 2 , the friction between the rolls and
the rolled stock cannot be overcome because the material yields
at the shear stress k / 2 before the shear stress ps needed for over-
coming the friction is reached. I n the case where
p2k/2 ...... (21)
therefore, the stock does not slip on the rolls but is extruded
from the roll gap by plastic deformation. On the other hand, if
the product p is less than k / 2 , slip will take place; even in this
case, however, the extrusion of the compressed material from
the roll gap is partly done by plastic shear, because the shear
stress set up by the frictional drag contributes to yielding even
if it does not reach the magnitude of the yield stress.
The assumption of slipping must be the main cause of the
discrepancy between the measured and calculated curves in
Fig. 3 ; the comparative agreement in Fig. 2 indicates that the
error due to the assumption of homogeneous compression alone
is less serious. In what follows, we shall, nevertheless, dispense
with both simplifications and attempt to take into account the
real stress distribution in the rolled stock. This necessitates more
complicated calculations than the methods that simply neglect
the inhomogeneity of the stress distribution; once the general
solution of the problem has been worked out, however, the
numerical calculations with the more exact method are simpler
than with the assumption of homogeneous compression. There
is no reason, therefore, to use the less accurate method developed
in Part I, even if allowance is made for the possible occurrence
of sticking.
That the rolled stock must stick to the rolls under certain
circumstances has already been recognized by Ekelund (1928).
Instead of the correct condition (21) of sticking, however,
Fig. 3 . Stress Curves for Rough Rolls (p = 0.4) Ekelund assumed that the bar would always stick over the whole
Lueg’s measurement. exit side of the arc of contact (where it is pressed into a narrow-
- .- Homogeneous graphical method. ing gap) but would always slip over the whole entry side (where
- - - - Kirman method. it is pushed out of a widening gap). This error contributed to the
- _ - - - Siebel method. disagreement between the Ekelund formula and the measure-
ments of Siebel and Lueg, as shown in Table 3 (section (28),
p. 161). Ekelund’s method is not rcpresented in Figs. 2 and 3
Siebel method seems to give somewhat better agreement than because his formula gives only the total roll force but not the
the rest, because in it the erroncous physical assumptions are distribution of the roll pressurc over the arc of contact.
over-compensated by mathematical errors acting in the opposite Practical arguments against the assumption of slipping in hot
direction. rolling have been put forward by Sobolevsky (1935, 1936).

P A R T 11. C A L C U L A T I O N O F T H E R O L L P R E S S U R E W I T H -
OUT THE ASSUMPTIONS OF HOMOGENEOUS COM- (10) The Yield Stress in Shear. I n the preceding section a
PRESSION AND SLIPPING FRICTION theorem has been used according to which the yield stress in
shear is k / 2 if the yield stress in compression with inhibited
(9) The Shear Stress Produced by the Friction of the Rolls. lateral spread in one direction is k . This should be proved now.
We can easily trace the error that has led to the discrepancy According to a well-known theorem a shear stress is equivalent
shown in Fig. 3 between the measured roll pressure distribution to a numerically equal compressive stress and a numerically
curve and those calculated by methods based on the assumptions equal tensile stress in planes which are perpendicular to each
of homogeneous compression and slipping friction. According other and make angles of 45 deg. with the planes in which the
to measurements of Siebel and Pomp (1927), the yield stress shear stress is acting. Now, as we have seen in section (9,we
in compression of the material at the neutral point in this experi- can superpose a hydrostatic pressure upon any state of stress
ment was 20 kg. per sq. 111111. We shall see presently that the yield without influencing yielding. Superposing a hydrostatic pressure
stress in shear is numerically half of the yield stress in compres- of magnitude T upon the principal stresses T and - T (the third
sion; hence the maximum shear stress that could possibly exist principal stress being 0), we obtain the system of principal
in the material at the neutral point was 10 kg. per sq. mm. On stresses 27, T, 0. According to section ( 5 ) , this will produce
the other hand, the nornial roll pressure at the neutral point was yielding if 21 reaches the magnitude of the yield stress k in com-
90 kg. per sq. mm. according to the curve measured by Siebel pression with inhibited lateral spread in one direction. Hence
and Luep. Now the theories working with the assumption of the result : If k is the yield stress in compression with the lateral
slipping friction derived their results by calculating the frictional spread inhibited in one direction, then k / 2 is the yield stress in
drag upon the rolled stock as the product of the normal pressure pure shear.
and the coefficient of friction which, in this case, was not less
than 0.4. This would give a frictional drag of 0.4 x 90 = 36 kg.
per sq. mm. which would create a shear stress of equal magnitude (11) Equilibrium Condition for a Segment of the Rolled Stock.
in the surface layer of the rolled stock at the neutral point We generalize the consideration made in section (4) about the
where, according to the direct measurement, the maximum equilibrium of a thin vertical segment of the rolled stock. We
shear stress possible (the yield stress in shear) was only 10 kg. consider a thin segment bounded by cylindrical surfaces A, A‘

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T A N D C O L D F L A T R O L L I N G 147
(Fig. 4) of arbitrary shape; the lines in which the ends of a (12) The Plane Problem of Plasticity. T h e exact calculation
surface intersect the roll surfaces, however, should lie in the of the plastic deformation in a rolled bar is, at present, an almost
same vertical plane. That the surfaces are cylindrical means hopelessly difficult problem. Even in much simpler cases, and
that they are generated by moving a straight line so that it with the simplifying assumption made in the classical theory of
always r u n a i n s parallel to the roll axes. Letf($) be the resulting plasticity that there is a sharp yield point and no strain hardening,
horizontal force acting across a surface A which intersects the the calculation of the deformation is extremely complicated (see,
roll surfaces at angles d on the arc of contact, measured from the e.g. H. Geiringer (1937). What we need for calculating rol!
pressure, however, is the knowledge, not of the deformation,
but only of the stress distribution in the rolled stock. For-
tunately, a fundamental theorem of the classical theory of
plasticity, discovered by Hencky in 1923, states that the stress
distribution can be determined without any knowledge of the
deformations in the case of a “plane problem” where there is a
direction in the deformed body along which none of the variables
(stresses and strains) changes. This is the case in rolling if there
is no lateral spread: the displacements parallel to the roll axes
vanish then, and the stress distribution is the same in all planes
perpendicular to the roll axes.
It is easily seen why, in the case of a plane problem, the
stresses can be calculated without any knowledge of the defor-
mations. T h e state of stress at any point of the body is com-
pletely determined by three stress-components, e.g. by the
horizontal pressure, the vertical pressure, and the shear stress
in a vertical or horizontal plane (according to a well-known
theorem, the shear stresses in perpendicular planes are equal).
T h e stress distribution in the body, therefore, is determined
by three independent functions of the two coordinates x and y.
To determine these three functions (horizontal pressure, vertical
pressure, and shear stress), we have at disposal just three
equations: the equilibrium condition for a volume element in
the vertical direction (i.e. the condition that the vertical forces
Fig. 4. Equilibrium of Thin Segment of Rolled Stock acting upon the volume element must vanish); the equilibrium
condition in the horizontal direction; and the condition of
plasticity (6.section (3, p. 143). By means of these three equa-
plane of exit. Obviously,f($) is independent of the shape of the tions, the three stress-components are completely determined
surface A : if it were not the same for another surface B which without any reference to the deformations.
intersects the roll surfaces in the Same lines as A, the material T h e theorem of Hencky, of course, implies that there is a
enclosed between A and B would not be in equilibrium. sharp yield point, and that the yield stress (which occurs in the
f
Let now A’ intersect the roll surfaces at angles +d+, so that condition of plasticity (equation (11) or (1 1A), pp. 143-4) is con-
the total horizontal force acting across A‘ is (4+&). T h e stant. In reality, the yield stress changes along the arc of contact
horizontal forces acting across A and A‘ upon the segment owing to strain hardening (in cold rolling) or to its dependence
enclosed by them is then --d4. df T h e horizontal component of upon the rate of compression (in hot rolling). However, our
d$ present purpose with the Hencky theorem is merely to calculate
the normal roll pressure s exerts upon top and bottom of the the variation of stress within a cross-section of the rolled stock;
segment between A and A’ the horizontal force 2s sin $Rd$; we may assume that, in most practical cases, the yield stress will
the frictional drag which we denote by T, without making any be approximately constant in a surrounding of the cross-section
assumption as to the nature of the friction (slipping or sticking), comparable to its height, and that the distribution of stress over
acts upon top and bottom of the segment with a horizontal force the cross-section will not be considerably influenced by the
f 2 r cos $Rd+. Here again, according to our convention, the values of the yield stress at more remote parts of the arc of
upper sign hoid for the exit side and the lower sign for the entry contact.
side of the friction hill. The s u m of the three forces must vanish
in equilibrium ; we have, therefore, (13) The Stress Dism.bution in a Plastic Slab Compressed
( f l d d = D s s i n 4 & D r c o s + . . . (22) between Plates. Soon after Hencky published his theorem,
Prandtl (1923) found the mathemltical solution of a two-
We shall find tha: T can essily be expressed by s or by the yield
stress k, according to whether the bar slips or sticks; we can,
therefore, consider equation (22) as containing two unknown
functionsf($) and s(+) only. I n dealing with the corresponding
equation (7) in the homogeneous method of Part I, we first ex-
pressed s by the vertical pressure q, using equation (8) ;we then
replaced q by the horizontal pressure p from the condition of
plasticity ( l l ~ ) ,and finally eliminated p by making use of
p = f/k. It was this last step, however, where the assumption
of homogeneous compression came in. I n reality, p varies in
going from the surface of the bar to its interior, and thus the
relatianship p = j / h must be replaced by
$=f/h . . . . . (23)
where p is the mean value of the horizontal pressure over the
cms-section of the bar. O n the other hand, s and T on the right-
hand side of equation (22) are stresses at the surface; if they
are replaced by p , t h i s p will he the horizontal pressure at the
surface, not the mean horizontal pressure 6. Obviously, then, Fig. 5. Plastic Slab between Parallel Plates
we cannot replace s in equation (22) byf without knowing how
the stresses vary over the crass-section. We have to deal, there- dimensional case of plastic deformation from which the approxi-
fare, first with the problem of the stress distribution in the rolled mate distribution of stress over a cross-section of the rolled
stock. stock in flat rolling without spread can be derived.

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148 T H E CALCULATION OF R O L L PRESSURE I N H O T A N D COLD FLAT R O L L I N G
The case treated by Prandtl is that of a plastic slab compressed on the right-hand side, and
between two parallel plates (Fig. 5). The problem being a plane
one, we have to imagine the plastic slab and the compression c - = + ki L . . . . . (28) .
plates extending into infinity in both directions perpendicular to
the plane of the figure. Further, the solution refers to the peri- on the left-hand side of the slab. The vertical pressures 5 corre-
pheral regions of the plastic slab; that is to say, the centre of the sponding to these two solutions are shown as straight lines in
compressed slab must be assumed to be to the left outside the Fig. 6 ;they are drawn with full lines near the edges of the strip,
figure, so that the horizontal component of the direction of the and continued with dotted lines in the middle. However, it is
plastic flow is the same all over the region considered (in Fig. 5 easily seen that, for reasons of continuity, the two solutions
it is given by the horizontal arrow). We choose rectangular co- cannot be valid right to the vertical x = 0. Near the middle
ordinates x, y , with the origin in the middle between the com- of the strip there will be a disturbance which will be
pression plates, as shown in Fig. 6 . Let s(x,y) be the vertical discussed in section ( 2 0 ) ; in Fig. 6 its effect is indicated
and r(x,y) the horizontal pressure, 7 ( x , y ) the shear stress in schematically by the full lines for s, and the two wedge-shaped
vertical or horizontal planes, h the distance between the plates, areas where the stress distribution is different from that given
and k the yield stress in uniaxial constrained compression (6. by equations (24), (25), and (26) are shaded.
section (3, p. 143). According to Prandtl, the stress distribution I n section ( 1 1 ) we obtained the differential equation ( 2 2 )
is then given* by which contained the two unknown functions f and s. We could
not eliminate one of these functions without knowing how the
. . . (24) horizontal pressure varied over a vertical line in the rolled bar.
I n the simple case of a body compressed between parallel
k plates and sticking to them everywhere, Prandtl’s solution-
t = c+-x-kdl-4yz/hZ
h
. . . (25) equations (24) and (25)-gives the variation of the horizontal
pressure t over a vertical section x = a constant; we can write
k
and r= -,y. . . . . . . . . (26) this solution in the simple form

where c is a constant determined by boundary conditions.

t = s-kdl-4y*/hz . .. .
(29)
According to equations (24)-(26), the shear stress vanishes in by substituting the right-hand side of equation (24) for the first
the middle of the compressed slab, and the vertical pressure term on the right-hand side of equation (25). However, the
here exceeds the horizcntal pressure by the value of the yield simple case treated by Prandtl differs in several respects from
stress K ; in consequence, the plastic deformation takes place that of a material compressed between the rolls. The surfaces
here by vertical compression. At the compression plates, vertical of the rolls are neither plane nor parallel, and the material may
and horizontal pressures are equal, and the shear stress reaches slip on the rolls, instead of sticking to them. Of these points,
the value k / 2 of the yield stress in shear; thus the plastic defor- we shall see in the next Section how slipping can be taken into
mation at the surface of the slab is a pure shear parallel to the account approximately. The curvaNre of the rolls cannot affect
surface. The coefficient of friction must be high enough for the the relationship (29) considerably if the slope of the arc of
frictional drag k / 2 to be maintained; otherwise, its value does contact does not change materially over a length comparable
not matter and does not enter into the equations; the slab is with the height of the rolled stock. The last question is, then,
assumed to “stick” to the compression plates. how seriously equation (29) is affected if the compression
plates are not parallel, but inclined at a comparatively small
angle. This problem has been treated by Nadai (1931), pp. 232)
et seq.) who applied Prandtl’s mathematical method to polar
Y co-ordinates. Let the position of a point P (Fig. 7) be described

Fig. 6 . Pressures in Compressed Plastic Strip Fig. 7. Xon-parallel Compression Plates

As mentioned, the Prandtl solution applies only to the by the polar co-ordinates r, 8 with respect to the bisector of the
peripheral regions of a plastic slab where the horizontal direction compression plates (drawn with full lines) and to the apex of
of flow is the same everywhere. Thus, it would apply to the the angle made by the compression plates. Instead of the
peripheral regions of the compressed slab shown in Fig. 6 , the vertical pressure s, we use the azimuthal pressure which acts
centre of which is in the middle of the figure. Let 2L be the upon a surface element parallel to the radius vector t in a
width of this strip (in the x-direction). The vertical pressure direction perpendicular to r ; we retain the notation s for the
should vanish for x = &I,; according to equation ( 2 4 ) ; the azimuthal pressure. Instead of the horizontal pressure, we use
constant c must then have the value -he radial pressure which acts upon a surface element per-
k pendicular to r in the direction of r, and denote it by t . The
c + = --L
h
. . . . . . (27) azimuthal pressure is the normal pressure acting upon the
compression plates; in this respect, too, it plays the role which
the vertical pressure played in the case of parallel plates. Nadai
* Prandtl‘s solution is derived in A. Nadai’s “Plasticity”, pp. 221- has shown that the variation of the radial pressure along an arc
224 (New York, 1931, McGraw-Hill). The signs of some terms differ
from those in equations (24)-(26) because in the present paper, con- r = a constant is given by the same formula (29) as for parallel
trary. to the usual convention, compressive stresses are regarded as plates if we understand by t the radial pressure, by s the azi-
posiuve. muthal pressure, by y the co-ordinate angle 8, and by h the

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T AND COLD FLAT R O L L I N G 149
angle between the compression plates. In the case of rolling, the Equations (33) and (34) represent the stress distribution in a
angle corresponding to h is simply 24 ;we have, therefore plastic strip of thickness h at whose surface the shear stress is ps.
In the case of inclined compression plates we can proceed
. . (30) exactly in the same way, by calculating the semi-angle $*>$
which a fictitious wedge-shaped plastic body sucking to the
Similarly, the shear stress varies according to the same compression plates must have in order that the shear stress at
formula as in the Prandtl case; replacing in (26)y by 19and h the surface of a wedge of semi-angle 9, taken from its middle,
by 24, we get should be equal to p s < k / 2 . The semi-angle 4* is determined
from equation (31) :-
k
Finally, the azimuthal pressure s depends only on r , b u t not
P=-4 24*
on 8, just as the vertical pressure was constant along any vertical
in the Prandtl case. or $*=-$k
2P
.
. . . . . (35)
Thus the formula giving the variation of the radial pressure
and of the shear stress are the same as for parallel plates. There Replacing 4 in equation (31) by $*, we have
is, however, one reservation to be made. Nldai’s solution
applies only to the case in which the plastic material flows (36)
towards the apex of the angle made by the compression plates.
This is the case on the exit side of the arc of contact; on the and the approximate distribution of the radial stress t is obtained
entry side, however, the material flows backwards, out of by substituting 4* for 4 in equation (30)
the angle made by the plates. NQdai’s solution does not apply
to this case which could not be treated mathematically so far. (37)
We shall assume, however, that equations (30) and (31) give a
sufficient approximation on the entry side also. The azimuthal pressure J, of course, is independent of 8 and
varies only with r, in a way which is not of interest for our
present purpose.
(14) The Stress Distribution in the Case of Slipping Friction at As equations (36) and (37) are written, they apply to the case
the Compression Plates. The solutions of Prandtl and Nadai of slipping friction. If p is replaced by k / 2 (cf. equation (21)),
refer to the case when the material sticks to the plates and they go over into the corresponding equations (31) and (30), for
the shear stress in the surface layer has its maximum possible the case of sticking.
value k / 2 . If the material slips, the shear stress is less than
k / 2 , and is equal to the normal pressure s, multiplied by the ( 1 5 ) The Relationship between the Horizontal Force f and the
coefficient of friction p. Since the normal pressure increases as Normal Pressure s. We now return to the differential equation
we go from the edge of the strip towards its middle, the shear (22) which expresses the variation of the horizontal force f(4)
stress cannot be constant along any horizontal line y = a along the arc of contact in terms of the normal pressure ~ ( 4 ) .
constant, and thus in this case the Prandtl method loses its The information about the stress distribution in a compressed
mathematical basis. However, if the normal pressure does not plastic wedge, given by equations (30), (31), (36), and (37),
change too much within a distance from a cross-section com- enables us to calculate the horizontal force from the normal
parable with its height h, we may treat the problem of stress pressure at the compression plates, and thus to obtain from
distribution over the cross-section approximately by assuming equation (22) a differential equation containing only one un-
that the shear stress be approximately constant in the surround- known function which, for convenience, is chosen to be f(4).
ings, and that the stress distribution in the cross-section be the As we have seen in section ( 1 l ) , p. 1 4 6 , f ( 4 ) is the horizontal
same as if the shear stress would at every point of the surface component of the force, per unit of width of the bar, acting
of the strip have the same value as in the cross-section con- across any surface which joins the two straight lines parallel to
sidered. Then the case differs from that of Prandtl and Nadai the roll axes whose angular co-ordinates are 4. The traces of
only by the shear stress at the surface having a value ~ < k / 2 , two such lines are the points A and B in Fig. 8.
instead of being equal to k / 2 . For calculating the distribution of
the horizontal (or radial) pressure over a vertical, x = a constant
(or an arc r = a constant), we have to put the shear stress at
the surface equal to p, s being the vertical (or azimuthal)
pressure on the vertical (or on the arc).
The solution of this problem can be deduced from the solu-
tions of Prandtl and Nadai. Let us take Prandtl’s case of the
parallel plates. The magnitude of the shear stress decreases,
according to equation (26), as we go from the surface of the
strip towards its middle, and vanishes in the middle; we can,
therefore, cut out of the middle of the strip a thinner strip at
whose surface the shear stress has just the value p < k / 2 . Let h*
be the thickness of the thicker strip at whose surface the shear
stress is k / 2 , and h the thickness of the strip taken from its
middle at whose surface the shear stress has the value p. We
determine h* by substituting in equation (26), p for -7, h* for
h, and h/2 for y :-

or

The stress distribution in the strip of thickness h* is obtained Fig. 8. Illustrating Calculation of Horizontal Force in Bar
by substituting h* for h in equations (26) and (29):- between Cylindrical Rolls
. . (33) We have seen that the forcef(4) is the same for all surfaces
joining these two lines, and we can, therefore, choose the surface
so as to make the calculation off(+) convenient. We take
cylindrical surfaces whose side views (Fig. 8 ) are circular arcs

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150 THE CALCULATION O F ROLL PRESSURE I N HOT AND COLD FLAT R O L L I N G
intersecting the roll surfaces perpendicularly. We assume that pendent of 4; between 4 = 0 deg. and 4 = 30 deg. (which is
the variation along the arc AB of the shear stress and of the about the extreme range covered by practical rolling) the varia-
normal stress acting across the surface AB is approximately the tion of w, averaged for different values of a, is less than 2 in
same as if the plastic body were compressed between the planes 1,000, and its maximum variation (which occurs at u = 1) is
tangential to the rolls at the points A and B (shown by thin about 1 in 100. I n Fig. 9 two curves are shown, representing
full lines in Fig. 8). I n other words, we assume that the stress w(4,a) for 4 = 0 deg. and for 4 = 30 deg.
distribution in the bar over the surface AB is described by If the bar sticks to the rolls, the frictional drag has its maxi-
equations (36) and (37). mum possible value k / 2 . In this case we have to replace ps
The length of an element of the arc AB is rd9 where in all f o r m d z by k / 2 , i.e. we have to take for u the value 1.
r = h / 2 sin 4 (cf. Fig. 8). Hence the area of an element of the With vanishing 4, w(4,a) tends towards the value n/4 for u = 1,
surface AB, of unit width in the direction perpendicular to the as can be seen easily by calculating the integral (41) with cos @ l
plane of the figure is and sin +=$.
tlS = 2As md 4 9 . . . . . (38) With the abbreviations (40) and (41), equation (39) can be
written as
Now f(4)is the horizontal component of the force that acts jr(4) = hs-hkw(a) . . . . (42)
across the surface AB upon the shaded part of the bar. The where 9 as an independent variable of w is omitted in view of
element df off that acts across the surface element dS consists its negligible influence upon w.
of two parts, one representing the contribution of the radial The contribution f,.(+) to f ( 4 ) of the shear stress I acting in
pressure t , and the other that of the shear stress T. T h e first is the surface AE! (Fig. 8) is easily calculated. The horizontal
component of the shearing force acting in the surface element
t c o s 8 d S = tcos8- d8
2 sin 4
Substituting the value of f from equation (37) and integrating
from 9 = -4 to 19 = +4, we obtain the contributionf,($) o f t r sin 9dS = e8 sin 8 - h d 9 .
4 2 sin 4
to f(d', :-
apart from the sign. The sign can be ascertained by means of
Fig. 8. T h e shear stress acting upon the surface of the bar is
directed to the left on the entry side; it follows from the equili-
brium condition for a volume element subjected to shear stresses
(see, e.g. Timoshenko 1934) that the shear stress acting upon
the material on the shaded side of the surface AB must be
directed upwards and, therefore, its horizontal component must
be negative. By a similar consideration we find that the hori-
zontal component of the shear stress must be positive if the
where s could be taken out of the first integral because, accord- surface AB is on the exit side. Using again our convention on
ing to the preceding section, it does not depend on 8. alternative signs (upper sign for the exit side), we obtain, by
For the second integral, we introduce the abbreviations integration by parts
a = 2 p / k . . . . , * (40)

Hence we have
The quantity a = expresses the frictional drag ps as a
k/2
fraction of the yield stress in shear k/2. Slipping or sticking at
the rolls occurs according to whether a< 1 or u >1 (d. equation
(711).
I n the case of sticking, ps has to be replaced by K / 2 , and w(a)
becomes w( 1) = n / 4 :-

T h e contribution f7(+) of the shear stress can be neglected

in most cases where slipping friction occurs. I n hot rolling the
coefficient of friction is usually 0.4 or more, and under such
circumstances the bar sticks to the rolls practically along the
whole arc of contact. I n cold rolling, on the other hand, where
slipping may occur over extensive parts of the arc of contact,
the coefficient of friction is usually less than 0.20 or 0.25, and
the angle of contact, therefore, less than about 10-12 deg. With
these values of p and 4,the contribution hp(l/+-l/tan 4)of
fT(4)is only about 1 per cent of the product hs. In consequence,
j7(4)may be neglected in most practical cases of rolling where
slipping may occur, and we can use then the approximate
formula
f(4)= h ( s - k w ) . . . . . (-16)
instead of expression (45). I n hot rolling, where the cocficient
of friction and the angle of contact are often very great, we mav
retain expression (45) ;in cold rolling, however, expression (46j
can usually be applied even in regions of sticking.

Fig. 9. Graphical Representation of the Function w(4,a) (16) The Differential Equation of the Friction Hill. We can
express now s(4) byf(d), in the case of slipping friction by means
The function w(d,a) has been computed and is given of equation (46) :-
graphically in Fig. 9. It is not strictly, but is practically, inde- F = . f / h + k ~ . . . . . (47)

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 TI1E C A L C U L A T I O N O F R O L L P R E S S U R E I N H O T A N D C O L D F L A T R O L L I N G 151
and in the case of sticking by means of equation (45) :- section (7); the computation, however, is much simpler here
because A(+) does not depend on p and thus its integral can be
s =f/h+k[w(l)Ft(g-i)]1 1 - - - (48) given as a curve depending only on 9. The calculation of the roll
pressure involves, therefore, only one graphical integration ;
Equation (22) assumes, for slipping friction (T = p), the it will be described in section (18).
form With slipping friction, there is one additional complication.
df/d+ = Ds(sin+&pcos+) . . . P A ) Equation (49) is, from the mathematical viewpoint, very
and for sticking (7 = k/2) the form different from equation (12), because w in the second term on
the right-hand side depends on s(4) (cf. equation (40), p. 150)
k ~ c o s +.
df/d+= D s s i n ~ f D . . (228) and thus implicitly on f(4).Fortunately, however, the values of
w vary within very narrow limits (cf. Fig. 9), and this wiU
Substituting the value of s from expression (47) in equation enable us to solve equation (49) by a slightly modified graphical
(22A), we obtain the differential equation of the friction hill for method which is practically as simple and as accurate as the
slipplng friction :- method described in section (7).
df/d+ =$(sin +&p cos +)+DKw(sin+&p cos 4) . (49) (17) Determination of the Regions of Slipping and Sticking.
In the preceding sections we have distinguished between the
and similarly we obtain from equations (48) and (228) the cases of slipping and sticking,but so far we have not dealt with
differential equation of the friction hill in the case of sticking :- the question of how to recognize in practical calculation whether
df/d+ =fXD s i n + + D k [ { w F t ( $ - L ) ) s i n +&tm4] (50) at a given point of the arc of contact the material slips or sticks
to the rolls. According to equations (21) and (40), the criterion
tan d of slipping is a < l and that of sticking a > l . We have, there-
We introduce the abbreviations fore, to find the value of (I at given points of the arc of
contact. For this purpose, we can regard the horizontal force
f(4) as a known quantity; in the planes of entry or exit it is
simply the back or front tension per unit width of the bar,
and m-(# = [ w ( + , l ) + f ( $ - &+)]sin +-+ + .
cos (52) taken with a negative sign. As we solve the differential equations
(49) or (53) progressing from the ends of the arc of contact
With these we write equation (50) in the form towards its interior, we obtain further values of f(+).
Substituting in equation (46)the value of s taken from equa-
df/d+ = $ s i n + + D h . ... (53) tion (40), we get the equation

where m means m+(~$)on the exit side and m-(4) on the entry
(I
f = 2p
hk ---(a) . . . . . (46~)
side. mi($)and m-($) are given as curves in Fig. 10; the slight
dependence of w upon 4 has been taken into account in these This equation can be solved graphically in a very simple way.
curves.
Equations (49) and (53) play the same fundamental role in

"9

Fig. 10. Values of m+ and m- in Differential Equation of Fig. 11. Calculation of a Criterion for Deciding whether
the Friction Hill Material Sticks or Slips

the alternative cases of slipping and sticking as equation (12) We plot the curve w(a) over a as abscissa. As mentioned in
played in the homogeneous method treated in Part 1. Equa- section (15), we may regard w for most purposes as depending
tion (53) is of the same form upon a only (see Fig. 9). We plot the distance OA = f / h k
(Fig. 11) on the negative ordinate axis, and draw the parallel
-dfl@+A(+)f(+)+B($) = 0 . . (13) AB = 1 to the abscissa axis. I f f is negative, as in the case of
as equation (12), with the front or back tension, A will lie above 0. We consider A as
A(+) = (D/h) sin+ . . . . . (54) the origin of a new system of co-ordinates, with axes parallel to
the old ones ; then the ordinates of the curve PQ representing
and B(4) = Dkm . . . . .
. . (55) the dependence of w upon a in the new system are w+f/hk.
and it can be solved by the graphical method described in Equation ( 4 6 ~ is
) satisfied if these new ordinates are equal to

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152 THE CALCULATION OF ROLL PRESSURE I N HOT AND COLD FLAT R O L L I N G
a/2p. In the new system, the function a/2p of a is represented neutral angle. With the values of kw already calculated, equa-
by a straight line AC through the origin A, the tangent of the tion (46) gives the normal roll pressure ~(4).T h e technique of
angle BAC being 1/2p. Since AB = 1, we obtain the point C the computation will be described and a numerical example
by plotting the distance 1/2p vertically upwards from B. The given in section (22), p. 156.
value of a that satisfies equation ( 4 6 ~ therefore
) is the abscissa
of the point M in which the curve PQ and the line AC intersect.
If C coincides with Q, the value of a is 1 ;if C is above Q, a is (19) Solution of the Diflerential Equation in the Case of Siip-
less than 1 and we have slipping; if C is in or below Q, we have ping. With the abbreviations
sticking. As we proceed from the plane of entry or exit towards
the neutral plane,fincreases and A, B, and C move downwards.
F+($) = s i n + + p C O S ~. . . . (59)
If we had sticking in the plane of entry, the bar will stick all over F-(+) = s ~ ~ + - ~ c o .s $. . . (60)
the entry side of the friction hill, and the same holds for the exit we can write equation (49) as
side. If we had initially slipping, we may get sticking in the D
interior of the arc of contact iff rises high cnough for C to move df/d+ = f - F + D k w F
h
. . . . (61)
below Q. In the first case, the horizontal force and the normal
roll pressure in their dependence upon +
are obtained by where F means F+ on the exit side and F - on the entry side.
solving equation (53). I n the second case, we proceed from the 'This equation is of a type different from that of (13), because w
end of the arc of contact towards its interior by using the implicitly contains f(+). However, w varies little with f, and
differential equation (49) for slipping friction; if the value of this allows the application of a graphical method of integration
f(+) is reached at which a = 1, we continue the calculation by very similar to that used for solving equation (53). The values
means of equation (53). off are known in the planes of entry and exit; we can,therefore,
calculate .f for the interval points dl and d m (cf. section 22,
(18) Solution of the Differential Equation in the Case of Stick- p. 156) next to C0,= 0 and +,,,, from equation (61), assuming that
ing. We have seen in section (7) that the solution of a differential w can be approxlmately regarded as constant over the intervals
equation of the type to & - using in these intervals the values of w
0 to + I and #J,,~
derived from the front and back tension, and applying formula
(20). In this way we obtain the value.\ off in and $m- ,,and
calculate the corresponding values of w by means of the method
is described in section (17). Assuming again that the new values of
'u can be used over the next intervals 41++z and $m- I ++m- *, we
can calculatefin the points + 2 and 4m-2, and so on. The accuracy
of the method can be judged by comparing the value of us
In the case of equation (53) assumed for a point +x with the value corresponding to the

df/d+ = fi sin 4 +Dknr

calculated f(&). I n general, no second approximation will be
necessary; otherwise w(a) can be obtained from the first approxi-
mation to f(+) and used for the second. Thus the computation
D differs from that in sections (7) and (18) only by the integration
we have A($) = -i; sin (5; B(4) = Dkm . . . (56) being interrupted after every strip and by calculating from the
result the value of the coefficient B(4)for the next strip.
where again we have to use on the exit side B+ = Dkm+, and As with equation (53), the first integration-that of A(+)-
on the entry side B- = Dkm-. can be done m finite terms, and only the second (that of B / z )
With h = ho+D(l- C O S ~.) . . . . . (2) must be done graphically. We have here

The integral of the first term on the right-hand side has been
where y = ho/D . . . . . . . . . (5i) found in the preceding section (equations (58) to (58s); for the
is the gap ratio. second, we have
Obviously
Q
log& = sin+ d+ = b2(y+l-ccos+)
+1-cos+
$1

= log,
+
y 1 - cos 4
y+ 1 -cos
Tile function H(#J,y)is given in the graphs, Figs. 12a, 126,
and 12c. The thickness ratio y is chosen as abscissa, and H-
curves are drawn for a number of values of the parameter 4.
On the exit side we have to put = do = 0 and obtain This has been done because the valucs of 4 (the points by which
the arc of contact is subdivided into intervals) can be chosen
z+(+) = y+1--0s+ - _ h- . . . ( 5 8 ~ ; more or less arbitrarily; in calculating f,a vertical line is drawn
Y h0
at the given value of the gap ratio y, and the arc of contact
on the entry side = +,n and subdivided into intervals by points &, . . . represented
z-(& =
+l-cOs+ -
- -h . . (58B) among the curves on the chart.
y + 1-cos & h, The auxiliary function z(4) is given by
Nowf(4) can be computed from equation (20). For the exit y + 1 -cos
+pH . . .
#J

side, B + / z + = D k m + / z + -is integrated graphically with +o = 0 log, Z+(+) = loge

as the lower limit, and those values of 4 for whichf(+) is to be
Y
computed as the upper limit; then the amount of the front
tension per unit of width is subtracted, and the difference
multiplied by z + ; the result is f+(4). For the entry'side,
B - / z - = Dkm- / z - is integrated graphically with = +m as
the lower limit, and the back tension per unit of width sub-
tracted ;multiplicationbyz- givesf- (+). Bothf+ andf- are then
plotted over 4; the abscissa of the point of intersection is the

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 T H E C A L C U L A T I O N O F R O L L P R E S S U R E I N HOT A N D C O L D F L A T R O L L I N G 153
in view of equations (2) and (57); here H m is the value of H for the curve PQ, the end of the region of slipping has been reached
&; H vanishes for 4 0 = 0. The rest of the computation is done and the computation must be continued by means of the
as described above, by graphical integration of Blz progressing differential equation (53), as described in the preceding section.
interval by interval from the ends of the arc of contact inwards. In solving equation (53) by means of formula (20), of course, the
With each point &, a new value of w is obtained by the con- value off at the boundary of slipping and sticking must be taken
struction shown in Fig. 11j if the point M in this figure slips off as the constant f

(20) The Cause of the Rounded Peak of the Friction Hill :Slip
Cones of Unplastic Material at the Neutral Point. There would
be a discontinuity of the plastic flow along the y-axis in Figs. 12
if the stress field-equations (24), (25), (26), with (27) on
the right- and (28) on the left-hand side of the plastic strip-
would extend right to x = 0. Prandtl (1923) has assumed,
therefore, that the two plastic stress fields on the right- and
left-hand side of the y-axis are separated by two wedge-shaped
areas, indicated by shading in Fig. 6, in which the material
remains unplastic. The boundaries of the unplastic regions are
determined by the following considerations. In the plastic
region the material is extended in the direction of the major
principal stress, and contracted in the direction of the minor
principal stress. The amount of the strain must be opposite and
equal in the two directions, because no deformation takes place
perpendicularly to the plane of the figure, and the volume of
any element of the body must remain constant. Consequently,
there will be neither extension nor compression in the two
directions which bisect the angles between the two principal
stress axes in the plane of the figure. The two bisectors are the
traces of the only two planes not parallel to the plane of the
figure which remain undistorted in the course of the plastic
deformation; they are called the directions of principal shear.
Lines whose tangents are at every point parallel to a direction
of principal shear arc called slip lines; they are the traces of
surfaces which suaFer no distortion if the plastic deformation is
infinitesimally small. Now the boundary between a plastic and
an unplastic region must be an undistorted surface of the plastic
region ;otherwise there would be a discontinuity of the displace-
ments at the boundary. Hence, in the projection shown in
Fig. 6 the boundary of the unplastic region must consist

17 I8XIO'*
THICKNESS RAno, y

Fig. 120.
11
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154 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T A N D C O L D F L A T R O L L I N G
other in one point only which, for reasons of symmetry, must be
of slip lines. Furthermore, the unplastic region must include
the whole y-axis within the compressed body, as otherwise the centre of the figure. By these conditions the boundaries are
points of discontinuity would remain in the plastic region;completely determined; an easy calculation (Nidai 1923, p. 223)
shows that the slip lines are cycloids, and the boundaries of the
on the other hand, there can be no unplastic bridge of finite
two wedge-shaped unplastic regions shown in Fig. 6 are parts
width between the compression plates as otherwise they could
not approach. Hence, the unplastic regions must touch each of four cycloidal slip lines. The width of the unplastic wedges
is the distance between those two points of a cycloid which are
in the middle between the two compression plates; in the case
of sticking, this is (n/2+1) times the distance between the
The existence of unplastic wedges or (in the three-dimensional
case) of unplastic cones is a characteristic feature of compression
tests with plastic materials. They are usually called slip cones
because the deformation in their neighbourhood is prevalently
by shear (“slip”) parallel to the surface of the cone.
Qualitatively the same conditions will be observed in the
material between the rolls. There will be two unplastic wedges
around the neutral plane, touching each other along a line in
the nettral plane parallel to the roll axis. In the case of sticking,
the bases of the wedges will have an approximate width of
(7r/2+l)hn, where h, is the height of the bar in the neutral
plane. In the case of slipping the wedges are narrower, as can be
seen from section (14), p. 149. The stress distribution in the
plastic region is the same as if no unplastic wedges were present;
since the stress s(4) (which is nearly equal to the vertical
pressure) does not change considerably across the bar, the same
vertical pressures as would act upon the roll surface if no un-
plastic regions were present will now act upon the surface of
the unplastic wedges which can be regarded as protruding parts
of the rolls. This will, of course, not affect the total roll pressure
and the position of the neutral plane; but, if the distribution of
the roll pressure is measured by a pin embedded in the roll
surface, as in Siebel and Lueg’s experiments, the pressures
measured will be different from those acting upon the wedges.
Obviously, the purely elastic wedges act like pads between the
bar and the rolls, smoothing out the roll pressure distribution

I I I I
04
I 3 s 7 i
THICKNESS RATlO, y
Fig. 1%.

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 THE CALCULATION O F ROLL PRESSURE I N H O T AND COLD FLAT R O L L I N G 155
curve and rounding the peak of the friction hill. It would not be graphically. However, it is hardly worth while performing the
difficultto calculate this effect ;the problem is to find the stresses computation at present, because the presence of the unplastic
in an elastic body (consisting of the unplastic wedge and the wedges has little effect upon the calculated values of the roll
adjacent roll) upon whose surface (i.e. that of the wedge) force and of the torque. For this reason, the rounding of the
stresses of known distribution are acting. The mathematical peak has not been taken into account in the calculated roll
theory of this problem has been worked out by J. H. Michell pressure distribution curves. Its effect is indicated on the s-curves
(1902); by means of his results the problem can be solved in Fig. 6.
(21) Ex.periments on the Plastic Flow in Rolled Bars in the Case
of Complete Sticking. The boundary of the plastic region
towards the entry side shows up clearly in the photographs,
Figs. 13 and 14, of rolled laminated plasticine bars. These w u e
obtained from experiments the original purpose of which was
to show how a rolled bar is able to become longer even if its
surface cannot slip on the rolls (i.e. in the case of complete
sticking). Suppose that equidistant lines, perpendicular to the
direction of rolling, are scratched on top and bottom of the bar.
After rolling, the spacing of these lines will have mcreased in
the same ratio as the length of the bar; in other words, the top
and bottom surfaces will extend in the same ratio as the bar
itself. How is this possible if these surfaces stick to the rolls
and thus cannot expand while in contact with them? This
problem was formulated by Sir Lawrence Bragg at a dis-
cussion on the plastic flow in rolling; in order to elucidate the
matter, wide slabs built up of alternate layers of gray and white
plasticine were prepared and rolled between unpolished wooden
rolls. The coefficient of friction being higher than 1, practically
complete sticking must have occurred. The laminated plasticine
slabs were part-rolled, then withdrawn from the rolls and split
in the middle; Figs. 13 and 14 illustrate sections obtained in this
way. We see that the surface expands practically to its final
dimensions within a narrow region around the plane of entry.
In this region the inside parts of the slab are s t i l l undeformed;
only a thin surface layer has undergone a severe plastic shear.
As the slab moves along the arc of contact, the surface suffers
practically no further deformation; what happens is that the
plastically deformed zone extends more and more deeply into

I 3 7 9 II I3 IS 17 x 10'1
THICKNESS RATIO9 )'
Fig. 1%.

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156 THE CALCULATION OF ROLL PRESSURE I N H O T A N D COLD FLAT R O L L I N G
cannot be given here) of the conditions illustrated by this figure
explains many interesting phenomena observed in rolling.*
From Fig. 15 it would seem at first that our previous con-
siderations and calculations in which the material was assumed
to be in the plastic state over the whole arc of contact may be
unjustified if, as in this figure, only a fraction of the volume of
the bar between the rolls is actually in plastic flow. I n reality,
however, the stress distribution in the plastic region is com-
pletely determined by the differential equations of plastic flow,
together with the boundary conditions given by the shape and
movement of the roll surface, irrespectively of how far the plastic
regions extend to the right and left. On the other hand, we must
not forget that so far we possess no mathematical solution of the
problem of plastic deformation at the entry side (cf. section (13),
p. 149, last paragraph), where the material is squeezed backwards,
out of the roll gap. If the angle of contact is not too great, as can
be assumed in most practical cases of rolling, the use of the stress
Fig. 13. Rolled Laminated Plasticine Bar: Longitudinal distribution valid for parallel plates should represent a satis-
Section factory approximation; with large angles of contact, like those
in Figs. 13 and 14, however, the use of the Prandtl-NAdai solu-
the slab. An additional slight extension of the surface may occur tion may lead to considerable errors.
where the slab leaves the rolls. T h e boundary between the
plastically deformed and the undeformed parts of the slab is
(22) A Numerical Example. The Technique of the Calculation
of Roll Pressure Distribution Cumes. T h e computation begins
with subdividing the arc of contact into a number (e.g. 8 to 12)
of intervals by points +2, &-2, ...
&- The horizontal
forcef(4) and the normal roll*pressure s($) will be calculated in
the points +o = 0,
5(&)
+,, . .
+, and the values s(O), s(+,),
used for drawing the roll pressure distribution curve.
...
T h e intervals need not be equal; it is convenient to choose
them so that the values +2, etc., occur among the parameters
of the H-curves in Figs. 12a-12~~ pp. 153-5, if the values of H
are taken from such graphs.
We f i s t determine whether the bar sticks or slips in the
planes of entry and exit, by means of the construction shown in
Fig. 11. We then calculate the function z+(+) for the exit side
and z-(+) for the entry side. If the bar sticks in the plane of
exit (or entry), we obtain z+(+) (z-(+)) from equations ( 5 8 ~ or )
(58~).If it slips, equation (64~) or ( 6 4 ~ )must be used for the
exit side and (648)or (64~) for the entry side, with the values
Fig. 14. Rolled Laminated Plasticine Bar : Longitudinal of Htaken from the graphs (Figs. 12a-12c) or from equation (63).
Section, with View upon Surface of Contact T h e calculation is carried out for 40 = 0, 41, ... +me However,
z+ is needed only in the interval between the plane of exit and
the neutral plane, and z- between the neutral plane and the
easily discernible in Figs. 13 and 14; it is indicated by the line plane of entry. It is sufficient, therefore, to calculate z+ for
ABC in Fig. 15. This line corresponds to the cycloids mentioned values between +o = 0 and + z + # m y and z- between +$, and
in the previous section; its two branches are not cycloids dm; these limits correspond to the fact that the neutral plane is
somewhat nearer to the plane of exit than to the plane of entry.
With the values of z+ and z-, we can evaluate the integrals in
equation (20) for the exit side and entry side. The calculation
differs slightly according to whether the rolled stock sticks or
slips in the first point of the arc of contact (& = 0 or I#~). These
two possibilities are considered in subsections (i) and (ii) below.
(i) Zf it sticks in the plane of exit, we take B + ( $ ) from (56),
.
compute the values of B + / z + for the points 0, 41, . . of the
exit half of the arc of contact, and plot them as ordinates over
the +-axis. We determine the area between the B+ /z+-curve and
In the JI. Applied Mechanics, 1943, vol. 10, p. A-13) a very
interesting study of the strains in a rolled bar has been published by
C. W. MacGregor and L. F. Coffin, Jun. Owing to the careful de-
greasing of the rolls and the bar, the coefficient of friction must have
been fairly high, and some sticking may have occurred in the central
part of the arc of contact: comparison with the above experiments,
however, is not possible for the following reasons :-
(1) The width of the plasticine slabs was about five times their
final thickness, so that the deformation in their middle was approxi-
mately the same as the two-dimensional deformation in a very wide
Fig. 15. Schematical Representation of Deformation in the plate. MacGregor and Coffin, on the other hand, used copper bars
Rolled Plasticine Bar with square cross-section which represent no approximation to the
case of plane strain.
(2) The deformation at the sides of a bar or slab is markedly
different from that in the middle; Figs. 13 and 14 (see this page)
because here the material is compressed between cylindrical show, therefore, the interior of slabs split in the middle. MacGregor
rolls, instead of plane parallel plates. Beside the boundary ABC, and Coffin’s figures, on the other hand, refer to the deformation of nets
the boundaries of the unplastic wedges at the neutral point, and scratched on the side surfaces.
the boundary between plastic and unplastic regions at the exit (3) Owing to the softness of plasticine, the deformation in front of
side are also indicated in Fig. 15. A detailed discussion (which the rolls was hardly noticeable : with copper, it is very considerable.

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 THE C A L C U L A T I O N O F ROLL PRESSURE I N H O T A N D COLD FLAT R O L L I N G 157
the +-axis, in the strip between $0 = 0 p d . + p and multiply using both times the value wo of W , previously obtained by the
it by the scale factor necessary for convertmg I t mto the integral construction shown in Fig. 11. We compute the integral of
of B+/z+ between the limits 4 = 0 and +I. If one unit of the B + / z + over the first strip as above, subtract the front tension,
length of the abscissa axis represents i deg., and one unit of the and multiply the difference by z+(41),thus obtaining f+($]).
ordinate axis j units of B+/z+, then the scale factor is G/57*3, Withf+ (dl)we repeat the construction in Fig. 11 and determine
since 57.3 deg. = 1 radian. We subtract from the integral the w1 which we use for calculating B+(+J and B+(+,) from equa-
front tension per unit of width of the bar and multiply the tion (65). We integrate B+/z+ over the second strip, add the
difference by ~ + ( 4 According
~). to expression (20), the result is integral to the fist, compute f+ ( 4 2 > , and continue the procedure
f+(#~). In a similar manner we determine the integral of B+/z+ by determining the next value of w. The calculation is analogous
over the second strip, add it to the integral over the first strip, in the case when the bar slips in the plane of entry.
subtract the front tension, and multiply the difference by z+(42). As the curvature of the B/z-curve is not sharp, the strips
We thus obtain f + ( + 2 ) . By continuing t h i s procedure, we com- can be integrated graphically with sufficient accuracy simply by
putef+ in the points & of the exit half of the arc of contact. multiplying the ordinate of the curve in the middle of the strip
The calculation of the horizontal forcef-(+) for the entry side, by the width. It is not even necessary, to draw the curve: the
when the bar sticks in the plane of entry, is quite analogous. strip integrals can be obtained by multiplying the arithmetical
Instead of B + and z+, we take B- and z- ;we begin the integra- mean of the values of B / z at the boundaries of the strip by its
tion with the fist strip from the point of entry, subtract from width, expressed in radians. I n this way the introduction of scale
the integral the back tension per unit of width of the bar, factors becomes unnecessary.
multiply the difference by ~ - ( 4 , - ~ and ) , obtainf-($m- 11, etc. From f+ and f- we calculate the roll pressures+ for the exit
(ii) Zfit slips in the plane of exit, we compute B+(O)and B+(+,) side, and s- for the entry side of the friction hill, in the points
from d,, = 0, q5], etc. The intersection of the curves s+(d) .. . and s-($)
B+($) = Dkw(sin 4+p cos 4) ...
(65) determines the neutral angle +n.
2. CALCULATION
TABLE OF ROLLh s u R E S
-ROUGHROLLS(EXPERIMENTSIEBEL -
FOR LUE( - - OF AND

Valuesof + . 0" 0' 0'30' 1 1"O'

-
1" 30' 2" 0' 2" 30' 3" 0' 3"30' 4" 0' 4' 30' 5" 0' 5" 30' 5" 40'
-
log D/h = log 1 / ( ~ + 1 -
cos 4) (equation (2),
p.142) . 2.213 2.21c 2.203 2.190 2.172 2.151 2.126 2.097 2.068 2.036 2.003 1.970 1.959
k(+), kg. per sq. mm.
I (Fig. 17) . 27.1 27.0 264 26.6 25.9 25.1 24.0 22.8 21.4 19.8 17.0 12.7 10.4
IogW . 1.433 1.431 1.428 1425 1.413 1.400 1.380 1.358 1.330 1.297 1.230 1.104 1.017
_ _ -
Exit side:-
log z+ (equation (%A),
p. 152). ~.ooo 0.00: 0.011 0.024 0.041 0.063
logm+ (Fig. loj
logm+/z+ .
: 1.699
1.699
1.705
1.702
1.711
1.700
1.717
1693
1.723
i.682
1.727
1.664
log B+/Dz+= log km;
/z+ (equation (56),
p. 152).
Bf/Dz+ . . 1.132 1.132 1.128
13.6 13.6 13.4
1.118
13.1
1.095
12.4
1.064
11.6
0.118 0.117 0.116 0.111 0.105

0.043 9.161 0.278 0.394 0.505 0.610

9.633 1.20; 1.444 1.595 1.703 1.785
2,633 i . 2 1 ~ 1455 1.619 I.744 1.848
0446 1.42C 1658 1.809 1.916 1.999
7.1 26.3 45.5 64.4 82.4 99.8
- - -- - -
E n w ride:-
log 2- (equation (58B),
p. 152). 1.769 1.786 1.808 1.833 1.861 1.891 1.922 i.956 1.989 0~000
I
log m-1 (Fig. 10)
logm-/z-l .
log I B- 1Dz-l = log I
1.680
1.911
1.674
1.888
1.667
i.859
1.661
i.828
1653
1.792
1.646
I.755
1.640
1.718
1.631
1.675
i.624
1.635
1,621
1621
km-/z- I (equation
(56), p. 152). . 1.336 1.301 1.259 1-208 1.150 1.085 1.015 0.905
IB-/Dz-l . 21.7 20.0 18.2 16-1 14.1 12.2 10.4 8.0
0.739
5.5
0.638
4,3
0.182 0.166 0.149 0.132 0.115 0.099 0.080 0.059 0.014

1.092 0.910 0.744 0.595 0-463 0.348 0.249 0.169 0.1 10 0.096
0.038 1.959 1.872 1.775 1.666 1.542 '1.396 1.228 1.041 i.982
1.807 1.745 1.680 1.608 1.527 1.433 1.318 1.184 1.030 2.982
logf-/h . 1.997 1.917 1.831 1.734 1.624 1.501 1,354 1.187 1.000 0.941
f-/h, kg. per sq. mm. 99.3 82.6 67.8 54.2 $2.1 31.7 12.6 15.4 10.0 86
- -
kw 21.3 11.2 20.9 20.3 19.7 18.8 18.0 16.8 155 13.0 10.0 8.1
s+=f;/h+h kg. pe;
sq. mm. (equation (47),
p. 150 28.4 17.5 85.3 102.7 119-5
s-=f-/h+km : 120.2 102.9 87.5 73.0 50.1 48.5 38.1 28.4 zo.0 16.7
-- - -
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158 THE C A L C U L A T I O N O F R O L L P R E S S U R E I N H O T A N D C O L D F L A T R O L L I N G
Table 2 shows the calculation for experiment 2 (rough rolls) (23) Comparison m*th the Experiments of Sicbel and Lueg.
of Siebel and Lueg. Here the bar was sticking to the rolls along Fig. 18 shows the roll pressure distribution as measured by
practically the whole arc of contact. The table is self-explana- Siebel and Lueg in their experiment 1 (with smooth rolls), and the
tory; Fig. 16showstheauxiliaryfunctionsz+, z-, B+/Dz+,and curve calculated by the method described in the present report,
with p = 0.14; in t h i s case there was slipping over the whole arc
of contact. I n view of what has been said in section (20) about
the rounding of the peak of the friction hill, it appears that the
calculated curve agrees with the measured one within the limits
of the experimental errors. T h e slight discrepancy near the
point of entry is possibly due to an error of measurement, since
the measured curve rises here less rapidly than the k(4) curve.
Compared with the curve calculated with the homogeneous
method (Fig. 2), the curve obtained by the new method does not
show the shift towards the exit side, and the neutral angle
agrees much better with the measured one.
Fig. 19 shows the curve measured by Siebel and Lueg in their
experiment 2 (with rough rolls). This is a case of sticking along
the whole arc of contact, and the curves calculated with the
earlier methods (including the homogeneous graphical method
of Part I) were entirely different from the measured curve (cf.
Fig. 3,p. 146). The curve calculated with the new method almost
completely coincides with the measured curve, and the neutral
angles are equal. This success is particularly convincing because
in this case the coefficient of friction does not occur in the cal-
culation and, therefore, there is no constant whatever that
could be adjusted so as to obtain agreement with the measured
curve.
Although the experimental basis available for checking the
new method is very meagre, the excellent quantitative agreement
shown in Figs. 18 and 19 seems to indicate that the method
represents a satisfactory treatment of flat rolling. Further
measurements of the roll pressure distribution, similar to those

ANGULAR COORDINATE OFARC OF CONTACT, +EG.

Fig. 16. Graphical Representation of Auxiliary Functions

z+, z-, B+/Dz+, and B-/Dz-.

B-/Dz-. Fig. 17 shows the mean horizontal pressure f / h ; the

normal pressure s; the yield stress k = 1.15k* as given by Siebel
and Lueg, representing results of compression tests made by
-
Siebel and Pomp (1927); and the yield stress k, corrected for the

0 I 2 3 4 5
ANGULAR CO-ORDINATE OF ARC OF CONTACT, +DEG.
Fig. 18. Roll Pressure Distribution for Smooth Kol!~
( j=~ 0.14)
Lueg’s measurement.
ANGULAR CO-ORDINATEOF ARC OF CONTACT,+--DEG
--
. Calculated by the new method.

Fig. 17. Mean Horizontal Pressure, Normal Pressure, and of Siebel and Lueg, are desirable in order to give the theory
Yield Stress further trial ;measurements of the total roll pressure, of the roll
torque, or of the forward and backward slip, do not represent a
inhomogeneity of the deformation in a way that will be described severe enough test and give little information about the intrinsic
in section (30). truth and faults of the theory.

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T AND C O L D F L A T R O L L I N G 159
For sticking, T = k / 2 , and

where k, of course, is a function of 4. If k is approximately

constant along the arc of contact equation (68) simplifies to

D2 4 m
T = k -2( - 2 -4”) . . . . . . . . . . (68.4)

In the (rather infrequent) cases where K can be assumed to be

sufficiently constant, this formula may be useful for estimating
the neutral angle.
In the general case, both slipping and sticking may occur on
the arc of contact; the torque is then

where the boundary between the regions of slipping and sticking

is on the exit side and 4 2 on the entry side.
$he integrations must be done graphically; all quantities on
the right-hand sides of expressions (67), (68), and (69) are
known from the calculation of the roll pressure.
In the case of sticking, no knowledge of the roll pressure is
necessary for the calculation of the roll torque; the neuaal
angle, however, must be known very accurately. Usually, such
an accurate knowledge of $n can be obtained only by performing
the roll pressure calculation according to Part I1 of the present
report.
In the literature of rolling, the roll torque is usually calculated
as the moment of the vertical pressure with respect to the roll
axes (Siebel and Pomp 1930). If x is, as in Part I, the distance
from the plane of exit, and q(x) the vertical pressure, then qdx
is the vertical force in a column of thicknessdx and of unit width,
and xqdx is the torque of this force with respect to one of the
roll axes. The total torque Tv of the vertical pressure acting
uuon the two rolls is then
ANGULAR CO-ORDINATEOF ARC OF CONTACT. $-DEG. Tv = 21:xqdx . . ... (70)
Fig. 19. Roll Pressure Distribution for Rough Rolls (p = 0.4) It is often overlooked that T, is not strictly, and often not
Lueg’s measurement. even approximately, equal to the a d roll torque T, because
- --- - Calculated by the new method. it does not contain the contribution of the horizontal com-
ponent of the force acting upon the roll surface. The total
P A R T 1 1 1 . T H E C A L C U L A T I O N OF T H E R O L L T O R Q U E ,
horizontal force vanishes, of course, if there is no front and
POWER CONSUMPTION, ROLL EFFICIENCY, TOTAL back tension, but the lever a r m s of the horizontal forces with
R O L L PRESSURE, A N D S L I P
respect to the roll axes are greater on the exit side than on the
entry side, and thus the moments of the positive and negative
(24) Calculation of the Roll Torque. Let T again be the regions do not cancel. A detailed investigation (not reproduced
frictional drag per unit of area between the rolls and the bar. here) has shown that the contribution of the horizontal forces
The tangential force acting upon a surface element of a roll, of to the total torque, i.e. the quantity T - T,, is small if the angle
unit width in the direction of the axis and of the length Rd4 of contact is small; for angles greater than a few degrees,
along the circumference, is then T R ~and , the torque exerted however, it can amount to as much as 10 or 20 per cent of T
by this force is R+d+. The total torque acting upon one roll (see also section (30), p. 162). On the other hand a small error
is obtained by integrating this along the arc of contact. In in the neutral angle may cause considerable errors in the torque
evaluating the integral we have to take into account that the if one of the expressions (67) to (69) is used. Unless the com-
direction of the frictional drag changes at the neutral point. On putation of the neutral angle is done very carefully, therefore, it
the entry side of the friction hill the material slips backwards is safer to use expression (70), especially if the angle of contact
and its friction counteracts the movement of the rolls, while is small.
on the exit side it has a forward slip and helps the rolls. If we
call positive a torque that must be overcome by the driving
motor, we have to take the contributions to the torque as positive (25) The Power Consumption and the Pure Work of Rolling.
on the entry side, and negative on the exit side. It will be con- Let n r.p.m. be the speed of the rolls, and w = 2m/60 the
venient to consider T as an essentially positive quantity; the roll angular velocity. The power consumption of the two rolls, ex-
torque for the two rolls is then cluding the roll neck friction, is
W T= (2m/60)T . . . . .
(71)
per unit of width of the rolled bar, expressed by means of the
For slipping friction T = ps, and the torque is units of stress and of length used.
We now calculate the pure work of rolling per unit of volume
of the bar (in the “pure work” the losses due to roll neck friction
and those in the drive are not taken into account). The mean
if the coefficient of friction is constant. velocity un of the material in the neutral plane can be assumed

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160 THE CALCULATION OF R O L L PRESSURE I N H O T A N D COLD FLAT ROLLING
to be approximately equal to the circumferential speed of the torque has been calculated by equation (68), with the values of
rolls : the yield stress corrected for the inhomogeneity of the defor-
mation as will be described in section (30), p. 162. The corrected
yield stress k is shown in Fig. 17 as a function of 4. T o be as
independent as possible of arbitrary assumptions about the
The volume V of the material rolled per unit of time is coefficient of friction, the torque for the experiment with smooth
rolls has been obtained by first computing the torque of the
vertical forces from equation (70), and then subtracting the
torque of the horizontal forces. Details of the easy calculation
and the pure work of rolling W per unit of volume need not be given here. The work of compression W Ohas been
obtained from equation (77) with the values of k measured by
W = wT/V =
'p
1

Rh, cos $"

... (74) Siebel and Pomp (1927) in free homogeneous compression. The
result is :-
Experiment I (smooth rolls): 7, = 52 per cent.
(26) The Roll Eficimcy. The work W Onecessary for pro- Experiment 2 (rough rolls): 7,= 29 ,,
ducing a given reduction of the height of a bar is a minimum if
the reduction is achieved by free homogeneous compression These roll efficiencies are rather low, owing to the absence of
between frictionless compression plates. This work W o per lubrication and of front and back tension.
unit of volume, expressed as a fraction of the pure work of
rolling W,is the roll efficiency 7,:- (27) Forward and Backward Slip. The forward slip 0 is
defined as the difference between the mean velocity vo of the
7, = W,/W . . . . . (75) rolled stock in the plane of exit and the circumferential speed
T o calculate the roll efficiency, we must find the work W O Rw of the rolls, divided by the circumferential speed. If the
needed for free homogeneous compression. assumptions of homogeneous compression and slipping friction
Let u1 and hl be the initial area of the cross-section and the are made, a simple relationship between the forward slip and tbe
initial height of a cylindrical specimen, a and h the correspond- neutral angle can be derived. TIE neutral plane is defined as the
ing quantities at a moment during the compression, and uz and plane in which the frictional drag between the bar and the rolls
h2 after the compression. The volume of the cylinder is vanishes and the roll pressure reaches its maximum value. With
the assumption of slipping, this is the only plane in which no
C = a1hl = ah = a2hz . .
. , (76) slipping occurs; with the assumption of homogeneous com-
pression, the velocity of the material is equal to the horizontal
The force acting upon the cylinder at a moment during the component of the Circumferential speed of the rolls in this
compression is ka, and the work of compression is, with respect plane. If there is no lateral spread, we have
to (761,
. ..
V& = vnhn
Substituting the value of v,,from expression(72) above, we obtain
Hence the work of compression per unit of volume is vo = (hn/ho) Rw cos +my
and
- R ~-
W,= -J h
4
'kd(loge h) .. ..
(77) a = -v 0Rw - (hn/ho)C O S & - ~ (79) . .
For calculating W,,k(h) is plotted as ordinate over the corre- Similarly, the backward slip Y is the difference between the
sponding points logeh of the abscissa axis; the area between the circumferential speed of the rolls and the mean velocity vm of
resulting curve and the abscissa axis, multiplied by the scale the bar in the plane of entry, divided by the circumferential
factors used, gives Wo. In the special case where k = const., speed; with the above assumptions, we obtain

In cold rolling, k(h) is obtained directly from the stress-

strain curve. In hot rolling the complication arises that k is a
function of the speed of compression which varies as the material
moves along the arc of contact (6.section (31). Here the question
neglecting $n4 beside $n2 in the series for cos
well-known approximate formula
+", we obtain the
arises: Shall we calculate W ofor the same variation in time of
the speed of compression as takes place between the rolls, or
shall we calculate with a constant speed of compression, such
that the compression is completed during the time which the In the past, great importance was attached to formulz
material spends between the planes of entry and exit?The second (79)-(81). They were used for calculating the neutral angle from
alternative seems to be more reasonable, because it can be shown measurements of the forward or backward slip; from this was
that, unless K depends on the speed of compression in a very calculated the coefficient of friction, which was related to the
singular manner, the work needed for a given compression neutral angle in a comparatively simple manner in the theories
within a given time is a minimum when the speed of com- based upon the assumptions of homogeneous compression and
pression is kept constant. Thus W ois calculated from equation slipping. In the light of the results obtained in Part I1 of the
(77) with a constant speed of compression; how this constant present report, however, t h i s procedure appears more than
speed is calculated will be shown in section (31), p. 162. questionable. T o begin with, if the deformation is not a homo-
I t is of great practical importance to know the roll efficiency geneous compression of the vertical sections, the mean velocity
for different types of rolling. It depends on the value of 7, and of the material in the neutral plane is not strictly equal to
of the relative magnitude of the losses in the roll neck bearings the horizontal component of the circumferential roll speed.
whether there is much scope for improvements in rolling This, however, is far less important than the fact that sticking
technique, and if so, whether it is more important to reduce the must frequently occur in all types of rolling, and it usually
roll torque (and thus to improve 7,) or to reduce the bearing extends over most of the arc of contact in hot rolling. In such
friction losses by using better bearings or by lowering the vertical cases, values of the coefficient of friction obtained fkom forward
roll force (e.g. by better roll lubrication, by front or back tension, or backward slip are illusory; this is immediately obvious in the
or by using thinner rolls). It seems that no calculation of the roll case of complete sticking where p, apart from being high enough
efficiency has been published so far, and it is, therefore, of to produce sticking, has no further influence on the rolling
interest to calculate the roll efficiency for the two experiments process, and does not occur in the formulae that determine the
of Siebel and Lueg. For the experiment with rough rolls, the position of the neutral plane.

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 THE CALCULATION OF ROLL PRESSURE IN H O T A N D COLD FLAT R O L L I N G 161
(28) The Total Roll Pressure (Vertical Roll Force). We con- stress; for practical purposes it is often accurate enough to
sider a surface element of the roll, of circumferential length assume that at room temperature the yield stress depends only
Rd4, and of width 1 parallel to the roll axis. The normal on the amount of previous deformation and thus can be given
force acting upon the surface element is sRd4, and the tan- by a stress-strain curve.
gential force TRd4; in the case of slipping, T = p, and in that Hot rolling represents the opposite case. Here the tempera-
of sticking T = k/2. The vertical component of the normal ture is so high that, after a brief initial rise of stress, thermal
force is sRd4 cos 4; that of the tangential force FFrRd4 sin 4, softening continually annihilates the strain hardening produced;
if forces acting upwards are taken as positive. Hence the total thus hardening cannot accumulate, and the deformation takes
vertical force acting upon either roll per unit of width of the place at a constant stress if the speed of deformation is constant.
bar is We can easily see that in this case the stress will rise with in-
creasing speed of deformation, and the material will behave
more or less like a viscous liquid (apart from the presence of a
yield point). If during the time dt a specimen of height h is
where T has the value p in regions of slipping, and k / 2 in compressed by the amount dh, then the speed of deformation is
regions of sticking. For calculating P, we plot s cos $ F T sin 4
over 4, and integrate graphically. If the angle of contact is small
1 dh
. .
A=liTlt . . . . (83)
(less than 6-8 deg.), we may neglect the contribution of the
tangential force and calculate the total roll pressure from The deformation per unit of height during the time dt is hit,
and the increase 6k of the yield stress k is
6k = phdt . . . . . . (84)
The error of the approximate formula ( 8 2 ~will
) seldom exceed 10 where p is the “coefficient of strain hardening” which, of course,
per cent; t h i s accuracy is sufficientfor most practical purposes. in general depends on the stress and on the speed of defor-
Table 3 shows the values of P for the two experiments of mation. On the other hand, during the time dt the yield stress
Siebel and Lueg, as obtained from the measured roll pressure dis- diminishes, owing to thermal softening, by an amount which is
tribution curves, from the curves computed by the new method proportional to dt and, quite roughly, proportional to the strain
described in Part I1 of the present report, the homogeneous hardening present (i.e. to the difference between the yield stress
graphical, the corrected KhrmAn, and the Siebel methods, and k and the yield stress & in the absence of strain hardening).
the values of P calculated from Ekelund’s (1933) formula Thus, if E is the “coefficient of thermal softening”, thc decrease
dk of the stress during the time dt, due to thermal softening, is
dk = -€(k-&)dt .... . (85)
I n the stationary state &+dk = 0, or
where k is the mean value of the yield stress. The table shows
that the Ekelund formula is less accurate even than the Siebel k-KO = A ~ / c .... . . (86)
method. Its success in some cases of hot rolling must be attri- If P/E is constant, the increase of the yield stress is propor-
buted to the facts that it has been used in cormexion with values tional to the speed of deformation. If it depends on A, ( t h i s is the
of the yield stress arbitrarily determined so as to obtain agree- general case), k will be some more complicated function of A.
ment with experiments representing similar types of rolling, I t is important to realize that hot rolling is characterized not
and that the effect of friction upon the roll force in the case of simply by a high temperature, but by the circumstance that the
thick stock is comparatively small, so that even considerable temperature is high enough to annihilate continually the s t r a i n
errors in its calculation are only of minor inhence upon the hardening produced at the given speed of deformation. If the
value obtained for the roll force. speed of deformation is small, comparatively low temperatures
will produce hot rolling; on the other hand, extremely high
OF ROLLPRESSURE
3. VALUES
TABLE speeds of deformation may result in typical cold rolling at very
high temperatures. For a given speed range, the temperature at
which hot rolling begins is of the order of the recrystallization
Experiment 1 temperature. For different metals, the absolute recrystalliza-
(smooth rolls) tion temperature is, quite roughly, proportional to their absolute
temperature of melting Tm; if we call T / T mthe “corresponding
Method Error, Total temperature”, we may say that at similar speeds hot rolling
Total Error, begins, for different metals, roughly at the same corresponding
roll per cent roll percent
Jressure, pressure, temperature. For usual speeds of rolling, the lower limit of the
kg. per kg. per absolute temperature of hot rolling is about two-thirds to three-
sq. mm. sq. mm. quarters of the absolute melting temperature; for tin, the
~~
temperature of boiling water is sufEcient for hot rolling, and
with sodium or potassium hot rolling can be done at room
Lueg’s measurement . 358 - 508 - temperature. This circumstance can be used for studying hot
Newmethod .
Homogeneous graphicai
367 + 2.5 526 + 3.5 rolling at low temperatures with metals of low melting points,
method
Corrected
. K’&
353 - 1.4 933 +84
and thus getting rid of the experimental difficulties arising at
high temperatures.
method . 353 - 1.4 938 +85 On the other hand, hot rolling could be done with any metal
Siebelmethod . 293 -18 422 -17 at room temperature if the rate of deformation were sdciently
Ekelundmethod . 230 -36 410 -19 low. How low this rate must be, can be recognized directly from
the results of creep tests. Creep is a deformation that takes place
at a substantially constant stress; in its “primary” stage the rate
of deformation diminishes owing to work hardening, but in the
P A R T I V . T H E Y I E L D STRESS rND I T S V A R I A T I O N OVER subsequent “secondary” stage an equilibrium is reached
T H E ARC O F C O N T A C T between strain hardening and thermal softening, and the rate
(29 The Yield Stress in Hot and Cold Rolling. When de- of deformation settles down to a constant value. Thus the defor-
formed plastically at room temperature, commercial metals, in mation in the secondary stage of creep is of the same nature as
general, suffer s t r a i n hardening*: their yield stress increases in hot rolling; conversely, the deformation in hot rolling can be
with increasing amount of deformation. Variations of the speed considered as an extremely rapid creep at a high temperature
of deformation have a comparatively small effect upon the yield and high stress.
For our present purposes, the essential point is that in cold
An exception is, for example, annealed mild steel at the yield rolling we can regard the yield stress approximately as depending
point. only on the amount of deformation but not on its speed; in hot

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162 T H E C A L C U L A T I O N OF ROLL PRESSURE I N H O T A N D COLD FLAT R O L L I N G
rolling, on the other hand, we can assume that, approximately, k l -KO that would be produced by a homogeneous compression
the yield stress depends only on the rate but not on the amount of equal magnitude :-
of deformation. k2-ko = c(k1-Ko)
(30) The Variation of the Yield Stress along the Arc of Contact With this value :of c,* the corrected yield stress was calculated
in Cold Rolling. If the deformation of the rolled stock were a as
homogeneous compression of its cross-segments, the yield stress k = ko+c(k-ko) - . . . . .
(88)
at a given point of the arc of contact, where the height of the where k was the yield stress that would occur at a given point
stock is h, could be obtained by making a homogeneous com- of the arc of contact if the compression were homogeneous.
pression test* and determining the magnitude of the stress that
corresponds to the compressive strain (hm-h)/hm. In reality,
however, the deformation between the rolls is not homogeneous ; (31) The Variation of the Yield Stress along the Arc of Contact
the amount of cold work imparted to the material for a given in Hot Rolling. I n hot rolling the yield stress is, approximately,
reduction in height is greater than in homogeneous Compression, a function of the rate of deformation only. This function can be
and thus the yield stress along the arc of contact is higher than determined by homogeneous compression tests at elevated
would correspond to a homogeneous compression of magnitude temperatures; as in cold rolling, however, the results cannot be
(hm-h)/hm. T h e calculation of the actual plastic deformation applied directly to calculations of roll pressure, because the
between the rolls appears to be almost hopelessly complicated; deformation in rolling is not a homogeneous compression. T h e
even in the far simpler Prandtl case (cf. section (13), p. 147), with necessary correction cannot be determined here by ball in-
the assumption of a sharp yield point and no work hardening, dentation tests, as in cold rolling, but must be obtained by other
the calculation of the deformations demands heavy mathematical methods. T h e simplest way is to determine it from rolling
tools (Geiringer 1937). So the only practical way at present seems experiments in connexion with direct measurements of the yield
to be to determine empirically the correction that must be stress as a function of the rate of compression in homogeneous
applied to the yield stress calculated on the basis of homo- compression. This way is much easier here than in cold rolling,
geneous compression. This can be done, for example, in the because usually in hot rolling sticking takes place over most of
following way. We make a rolling test with a (sufficiently wide) the arc of contact, with the consequence that the coefficient of
strip, and interrupt it before the strip has passed through the friction does not appear as a second unknown physical quantity
rolls. We can then determine experimentally the variation of the in the calculation of roll pressure.
yield stress along the arc of contact, for instance, by ball When the correction for the inhomogeneity of the defor-
indentation tests. It is important to realize that tests made by mation has been determined, the yield stress can be found by
applying a given force to the ball and measuring the size of the applying this correction to the distribution of the yield stress
indentation (as in the Brine11 test) cannot be used for this pur- over the arc of contact calculated with the assumption of homo-
pose; the hardness number obtained in this way is a physically geneous compression. For this calculation, we need the know-
ill-defined quantity, reflecting the influences of the yield stress, ledge of the rate of compression as a fundon of According:to +.
coefficient of hardening, and of the geometry of the indentation. the definition equation (83), the rate of compression h is
e-<.:r
What we have to do is to increase slowly the force acting upon h dh x=l--l
the ball and determine its magnitude at the moment when the 11 dt
first trace of plastic indentation occurs. T h e yield stress can be where h is the height of the specimen, or of a body element
calculated then by the Hertz formula, and correlated with the
yield stress that would be observed at the same point if the com- within which the deformation can be regarded as homogeneous. dh
pression were homogeneous. In our case, h is the height of a vertical cross-segment, and --
In the cases calculated in Part I1 of the present report, the dt
corrected yield stress was obtained in the following way. From the decrease of h per unit of time as the material moves along
the measured roll pressure distribution the roll torque was the arc of contact. We have the identity
obtained as the sum of the torques of the vertical and horizontal
forces. The former is expressed by equation (70); the torque
Th of the horizontal forces is given by the formula
Here dhldd, represents the decrease of the height of the roll gap
Th = -2R2S:s[-*7 sin 24 cos24]dd, .. (87) with decreasing +,
and dx/dt is the mean velocity of the
2 __ material :-
where r is equal to p in the case of slipping, and to k / 2 in the rdx/dtI = v = V / h . .
. . [go) .
case of sticking. This formula is easily obtained by multiplying dh/dd, is obtained by differentiating equation (2) :-
the horizontal components of the forces acting upon a surface
element of the roll by the lever arm R cos 4, and integrating
dh/dd, = D O ~ + i 6 , (91).
over the arc of contact. The torque Th was only about 6-10 per and dd,/dx by differentiating equation (6) :-
cent of To;in the case of sticking, it was determined by a few
iT'
dd,/dx = 1/R cos 4 = 2 / D cos d,
successive approximations with gradually improved values of k Substituting the values from expressions (90), (91), and (92) in
, (92) .
(see, further, below). From the roll torque and the observed the identity (89), we have
neutral angle, the pure work of rolling was obtained by using
equation (74). The stress-strain curve of the material, deter- A = 2 V -tan 4 . . . .(93) .
mined by Siebel and Pomp (1927) by homogeneous compres- h2
sion tests, was then plotted by means of expression (77) as a Fig. 20 shows the variation of X along the arc of contact in the
curve showing the work of compression per unit of volume in case y = ho/D = 0.1, V = D2 per second, and 4, = 25 deg.;
its dependence upon the compressive strain. The pure work of h is plotted as a function of x / D , where x is the distance from the
rolling obtained from the torque was, of course, greater than the plane of exit. T h e rate of compression has here a maximum
work of homogeneous compression for the same amount of com- in the interior of the arc of contact; as the yield stress in hot
pression. Let ko be the yield stress before rolling, kl the yield rolling increases with A, it will also have a maximum there.
stress that would result after rolling if the compression were Measurements of the yield stress as a function of the compres-
homogeneous, and k2 the actual yield stress after rolling, cal- sion rate at elevated temperatures are, unfortunately, not
culated approximately as the stress that corresponds in homo- available at present. To give an example for the kind of variation
geneous compression to the same amount of work as that cal- of k that would follow from the function A(x/D) plotted in
culated from the roll torque by means of equation (74). Let Fig. 20, a compression rate curve derived by Trinks (1937) from
the increase k2-ko of the yield stress be c times the increase rolling experiments has been uscd. Trinks's curve shows how
* E.g. a cone compression test with the surfaces of the compression the yield stress would vary with the speed of rolling for a 0.45
plates sloping from the centre towards the circumference at an angle per cent carbon steel at 1,093 deg. C. if it were assumed constant
equal to the angle of friction (6.Siebel and Pomp (1927)). for each pass, and Kirman's method would be used. As both

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 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T AND COLD F L A T R O L L I N G 163
assumptions involved in this derivation of k()r)are far from being The problem of finding the flattened shape of the ro!ls in
justified in hot rolling, the resulting k(h)-curve is only a rough strip and sheet rolling is, from the mathematical point of view,
estimate. By using it, we obtain the variation of k along the arc that of linding the elastic deformation of an endless cylinder sub-
of contact shown in Fig. 20. However rough this way of esti- jected to surface forces that depend on the angle d onlv, but
mating k(h) is, it should give a true qualitative picture of how k not on the distance along the cylinder aixs. This problem has as
would vary along the arc of contact in the hot rolling of steel. comparatively simple general solution (Michell 1902) ; bv means
The main conclusion from the k-curve in Fig. 20 is that in hot of this, the shape of the flattened roll can be computed
rolling we are far less justified to assume k to be constant than numerically or graphically for any distribution of roll pressure.
in cold rolling. This is contrary to the usual view that in hot Such an exact calculation is at present far too complicated for
rolling k can be assumed constant because there is no work practical purposes; it is within reach, however, for the investi-
hardening. Further, we see that by using a mean value for k we gation of general problems in rolling mill research.
would under-estimate the roll pressure in the neighbourhood of The exact calculation of roll flattening can only be done by
the plane of entry where the lever arm of the vertical pressure is successive approximations, because it presupposes the know-
greatest. In consequence, we are bound to get too low values ledge of the roll pressure distribution which must be calculated
for the roll torque and power consumption if we calculate with a from the flattened shape of the rolls. First, an approximate
mean value of k determined so as to lead to satisfactory values distribution of roll pressure, based upon previous experience,
must be assumed, and used for a first calculation of the flattened
shape of the rolls. With this, the distribution of roll pressure
can be calculated more accurately; if the result agrees well
enough with the initially assumed distribution, no further step
is necessary. Otherwise, the obtained pressure distribution must
be used for re-calculating the flattened shape of the rolls, from
which a second approximation to the roll pressure distribution
can be computed, and so on.
If the flattened shape of the rolls is known or assumed, the cal-
culation of roll pressure, in principle, is the same as with un-
deformed rolls. In general, the flattened arc of contact will not
be circular, and thus the relationships between the thickness h
of the rolled stock, the distance x from the plane of exit, and the
angle 4 between the tangent to the arc of contact and the hori-
zontal plane will not be given by equations (2) and (6), but
must be read off the drawing of the flattened arc of contact. We
have then to use equation (5) instead of equation (7),and, in the
general case including both slipping and sticking, we have
L 2 s t a n 4 f 7 .
dx
. . . .
as the equilibrium condition for a cross-segment of the rolled
stock, instead of equation (22). Equations (47) and (48), ob-
viously, are unaltered, and thus the differential equation of the
friction hill is
df
dx
f
= 2-(tan$fp)+2kW(tan$fp)
h
. . . (49A)’
in the case of slipping, and
2) 4
d f = 2 - tfa n $ + k ( 2 w t a n $ F - f tan * -
(50.41‘
dx h 4
in the case of sticking. In these equations, the distance x from
the plane of exit is the independent variable j the corresponding
values of h and 4 must be taken from the drawing of the deformed
PROJECTED ARC OF CONTACT DIVIDED BY ROLL DIAMETER ( r / O )
arc of contact. Both differential equations are of the same type
Fig. 20. Variation of Rate of Compression along Arc of as the corresponding equations (49) and (50) without roll
Contact flattening, and they can be solved by the same method; both
integrations occurring in expressions (19) and (20), however,
for the roll force. Detailed calculations using the Trinks curves must be done graphically or numerically. Analogous modifi-
(not reproduced here) have shown that the error due to this cations are necessary for the calculation of the roll torque;
circumstance may amount to 20 per cent. or more in the case of details of this will not be given in the present paper. One im-
carbon steel. portant fact, however, must be mentioned. The high pressure a t
the peak of the friction hill produces a bump in the roll surface
PART V. T H E E L A S T I C D E F O R M A T I O N OF T H E ROLLS which may be so deep that there is a region in the middle of the
(32) The Calculation of the Roll Pressure Dism’bution with arc of contact where the rolled stock does not suffer plastic com-
Flattened Rolls. Under the pressure of the rolled stock, the pression (cf. section (36), p. 165). This possibility needs careful
rolls undergo elastic bending and also suffer local elastic defor- consideration in calculating the roll pressure distribution curve.
mations (“roll flattening”) from the forces acting over the
relatively small area of contact. As far as the calculation of the (33) The Flattened Arc of Contact is Not Circular. As we
roll pressure and roll torque is concerned, bending is less im- have seen in the preceding section, the calculation of rol1
portant because its effect upon the shape of the surfaces of flattening, in general, is a complicated problem. It would become
contact is more or less compensated in practice by an appropriate very simple, however, if the flattened arc of contact could be
convexity (“camber”) of the outlines of one or both rolls. Roll approximately regarded as circular, so that flattening would
flattening, however, is a factor of primary importance in thin alter only the effective radius of the rolls. According to Prescott
sheet and strip rolling. We shall see that the deformed arc of (1924) the flattened arc of contact would be approximately cir-
contact may be quite different from the arc that would corre- cular if the roll pressure were purely normal and its distribution
spond to undeformed rolls; in such cases any calculation that given by a semi-ellipse resting on one of the axes as a base.
does not take into account roll flattening is bound to give mis- Hitchcock (1935) devised a simple method, based on Prescott’s
leading results. formula, of calculating roll flattening. At first sight, it would

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164 T H E C A L C U L A T I O N O F R O L L PRESSURE I N H O T A N D C O L D FLAT R O L L I N G
seem that, for an approximate. calculation, a semi-ellipse could cases. The experiments, which completely confirm the above
satisfactorily replace the actual friction hill, and the Hitchcock conclusions, will be described in the next section.
method would represent a simple approximate solution of the
problem. We might think of using it by first assuming a plausible (34) Expmhents on Roll Flattening. For this investigation,a
value for the mean roll pressure (which has to be put into Hitch- rolling mill with steel rolls of 8 inches diameter, and electric
cock’s formula), determining the radius of the flattened arc of drive, was used. The mill was started and, after attaining a suit-
contact according to the Hitchcock method, and then cal- able speed, switched off. When it had slowed down sufficiently
culating the roll pressure distribution with the effective roll to ensure that it would stop with the rolled strip still between
radius thus obtained. If the mean pressure would agree with the the rolls, a previously cold rolled mild steel or brass strip of
initially assumed value, the solution would be satisfactory; about 5 1f inches in width and 4-8 inches in length was intro-
otherwise the calculaticn would be repeated with the calculated duced between the rolls. The mill stopped with the strip between
magnitude of the mean pressure, and this procedure continued the rolls ;at this moment, the upper roll was lifted and thus the
until the last obtained value of the mean pressure would agree
satisfactorily with the value assumed for the preceding cal-
culation of the radius of the arc of contact. This method was
tried for one of the last passes in rolling 1/100-inch tinplate with
rolls of 20 inches diameter; however, it was found that it did not
converge at all. In other words, the calculated mean roll pres-
sure was every time much (about 150 per cent) higher than the
one assumed for the calculation of the effective roll radius. A
closer consideration showed that the assumption of a circular
arc of contact must have been responsible for this failure. Let
the friction hill F in Fig. 21 represent the pressures needed at
different points of the arc of contact in order that plastic com-
pression may take place there, and let E be an elliptical distri-

f
d
W
I
Fig. 22. Measurements on Brass Strip in Experiments on
i

Roll Flattening : Specimen b

aIn
W
d
0
I

li 7 ‘3
I
PROJECTED ARC OF CONTACT-
Fig. 21. Requisite Pressures for Plastic Compression at
Different Points along Arc of Contact
I

3
0
ZI

.l-d++2
bution of roll pressure, with the same mean pressure as F,
producing a flattened arc of contact of circular shape. Obviously,
the elliptical pressure distribution cannot produce plastic com-
pression in the (vertically shaded) regions where the curve F is
above E. On the other hand, plastic compression must take place, ‘, 0
for purely geometrical reasons, at every point of the arc of con-
tact if this is circular: only where the arc is a horizontal line
could plastic deformation be absent. Thus the assumption of Fig. 23. Measurements on Brass Strip in Experiments on
plastic deformation in all regions of the arc of contact is in con- Roll Flattening : Specimen c1
tradiction to the assumption of an elliptical pressure distri-
bution which would produce a circular arc of contact. We may pressure released as quickly as possible, in order to prevent
expect that, in reality, the peak of the friction hill would produce avoidable creep. The shape of the arc of contact, impressed
a bump in the rolls around the neutral point which would reduce into the strip, was then determined by measuring the thickness
the magnitude of plastic compression in this region and thus of the strip point by point along the arc of contact* by means
would lower the peak of the friction hill ; with heavy pressures, of a thickness gauge whose dial was divided into 100 parts, each
the bump may be so deep that a part of the arc of contact would representing 1/10,000 inch. For the measurement the strip was
be horizontal and no plastic compression would take place here.
In order to investigate this problem, the shape of the arc of
* After t h i s manuscript was finished, Mr. E. C. Larke, of the
Research Department, I.C.I. (Metals), Ltd., informed the author
contact has been determined experimentally in a number of that he had made similar experiments (unpublished) several years ago.

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 THE C A L C U L A T I O N O F R O L L P R E S S U R E I N H O T A N D C O L D F L A T R O L L I N G 165
attached to a micrometer and moved 0.25 mm. after each unless the flattening is very slight. In Fig. 25 this bump is so
measurement in the direction of rolling between the contact strong that the arc of contact is horizontal in the neutral region;
balls (A inch diameter) of the gauge. Figs. 22-25 show the in consequence, the arc of contact was here subdivided into two
results of some measurements; half of the thickness of the strip effective parts near the points of entry and exit, and a “dead”
is plotted as it varies in the direction of rolling. Except in Fig. 24, part in the middle where no reduction of thickness was taking
where the strip was particularly thick, the abscissa axis is the place. This effect occurs much more strongly in Fig. 23 where
middle of the strip. The small circles represent actually measured the bump is so deep that the curve of the arc of contact has a
maximum in the middle. Whether or not this effect is a real one,
Cannot be recognized with certainty. Brass shows noticeable
creep during the couple of seconds between the stopping of the
/ mill and the release of the roll pressure; t h i s can be seen from
/ the depression below the level of the rolled sheet of the exit
part of the arc of contact. The creep may lead to an apparent
I- thickening of the sheet in the neutral zone which may not occur
during actual rolling. However, a genuine temporary thickening
of the sheet as it passes through the neutral zone is neither im-
possible nor improbable.
In order to suppress, as far as possible, the disturbing effect
of the creep, most experiments were made with strips initially
hardened by severe cold rolling. As can be seen from the
figures, the creep was in no case large enough to interfere with
the qualitative interpretation of the results.
;5 The necessity of using previously hardened strips could be
z
0
24-
avoided by means of an instantaneous pressure-release device.
This, for instance, could be similar to a dog-clutch with the two
f halves placed tooth on tooth between the upper roll neck bearing
a and the pressure screw ;the roU pressure would be released by
turning the two halves relative to each other until the teeth fell
E n- into the gaps.
According to Figs. 22-25, the flattened arc of contact cannot
be approximated by a circular arc except, perhaps, in the case
when the flattening is very small. In consequence, the Hitchcock
method cannot be used except when roll flattening represents a
small correction only.
Fig. 24. Measurements‘on Steel Strip in Experiments on (35) Experimental Determination of Roll Flattening in Rolling
Roll Flattening : Specimen f Mill Practice. The experiments described in the preceding
section have shown that, unfortunately, the calculation of roll
points and indicate the accuracy of the thickness measurement. flattening Cannot, in general, be based upon the assumption of a
The outlines of the roll in the undeformed state are indicated by circular arc of contact. On the other hand, they show a new way
the dotted lines ;these are ellipses because the vertical scale is of obtaining information about roll flattening which is simple
much larger than the horizontal. The ellipses are drawn so as enough to be used in rolling-mill practice and design. This con-
to go through the ends of the actual arc of contact. sists in directly measuring roll flattening by experiments such as
described in the previous section. The complete experiment,
including the thickness measurement, can be done in 15-20

‘1 minutes, and the result is far more reliable than any theoretical
calculation can be at present. The use of an instantaneous roll
pressure-release device, of course, is essential whenever the
rolled material shows appreciable creep.
(36) Thick Rolls versus Thin Rolls in Strip Rolling. The
results shown in Figs. 22-25 illuminate the question whether
thick or thin rolls are more to be favoured in strip rolling. If roll
flattening is not taken into account, and even more if a flattened
arc of contact of circular shape is assumed, theoretical cal-
culations lead to the result that the power consumption and, to
a much higher degree, the roll pressure increase rapidly with
increasing diameter of the working rolls. If such calculations
were justified in thin strip rolling, the 15-inch rolls of tandem
mills could not possibly compete with the thin rolls of Rohn or

I1
E2

0 ,
0 1
I
04
1 I
1 3
I
4
*2
I
I
I
b
IN@3
I
?Mfi
Steckel mills. This is contrary to experience, which shows that
each of these systems has its own field where it is at least not
much inferior to the others. This seeming contradiction between
theory and practice is explained by Figs. 22-25. If the roll
pressure is very high, as with thin strips and comparatively
thick working rolls, the bump at the neutral point becomes so
deep that no plastic compression takes place in the central part
of the arc of contact. In other words, the arc of contact breaks
Fig. 25. Measurements on Steel Strip in Experiments on up into three parts: a “dead” part in the middle, and two
Roll Flattening : Specimen s4 effective parts where the plastic compression of the rolled stock
takes place, near the planes of entry and exit. The effective parts
Figs. 22 and 23 represent brass strips, and Figs. 24 and 25 are short, and their radii of curvature may be much smaller than
mild steel strips in order of decreasing thickness, and hence, that of the undeformed roll; it is as if two thin rolls would work
in order of increasing roll pressure and roll flattening. They instead of a thick one. Thus, by means of its elasticity, the thick
show that the flattened arc of contact is not even approximately roll automatically avoids too long effective arcs of contact, with
circular; there is always a distinct bump in the neutral region, the accompanying excessive roll pressures. This does not mean,

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166 T H E CALCULATION O F R O L L P R E S S U R E I N H O T AND C O L D F L A T ROLLING
of course, that thick rolls are as good as thin ones with regard to [All Calculation of the Horizontal Force f + o n the Exit Side
oll pressure and power consumption : the effect of the bump in in the Cass of Slipping in the Plane of Exit. The value of z+(+)
the arc of contact is only to retard the rapid deterioration of or of loglo z+(+), according to whether slide rule or log-tables
efficiency with increasing roll diameter. A reliable answer to are used, is computed from
the question of how the performances of rolls of different dia-
meters compare needs a detailed investigation in every individual log, z+(+) = lo& y + l -
Y
+
cos + / m + , Y ) (6W
case, because roll pressure and power consumption are not the or from
only factors that must be taken into account.
Acknowledgements. This investigation was carried out in the h0
z+(+) = . .. ..
(64c)
Cavendish Laboratory on behalf of, and in collaboration with,
the Iron and Steel Industrial Research Council, for whose sup-
for = 0, +I, $2, . . .,
up to about &/2. Here y is the
thickness ratio ho/D; H(4, y ) is taken from Figs. 12a-12c or
port the author wishes to express his thanks. He also desires to from equation (63).
acknowledge encouragement and suggestions received from the
Rolling Mill Research Subcommittee, British Iron and Steel U(+)= Dns(-i
z + (4)
++p cos 4);. (94) ..
Federation. He is much indebted to Sir Lawrence Bragg,
F.R.S., and to Mr. G. M. Brown, past-chairman of the Rolling
Mill Research Subcommittee, British Iron and Steel Federa-
is computed for + 4
= 0 and = $1, with the value wo of w in the
point M previously obtained for the exit side by the construc-
tion, for valuable suggestions and information; to Mr. E. C. tion in Fig. 11. The values U(0) and U(C1)are used for com-
Evans, Secretary, Iron and Steel Industrial Research Council, puting
for initiating and sponsoring this work; to Professor G. I.
Taylor, F.R.S. ;Mr. P. D. Crowther and Mr. W. Hessenberg,
British Non-Ferrous Metals Research Association; Mr. E. C.
Larke, Research Department, I.C.I. (Metals), Ltd., and Dr. where the angle $1 is in degrees; +1/57*3stands for the width,
H. Lipson, Cavendish Laboratory, Cambridge, for helpful in radians,.of the first interval between +o = 0 and = +l.
The horlzontal force f+(+1) at $1 is computed from
+
criticism; to Professor R. S. Hutton and Mr. W. E. L. Brown,
M.A., for permission to use the rolling mill of the Goldsmiths’
Metallurgy Laboratory, University of Cambridge; and to Mr.
f+(#i> a z+(+i)[Jt($i)-GoI (96) .. .
K. J. Pascoe, B.A., for unflagging assistance, including a great Then +(+I) is plotted in Fig. 11 vertically from the origin 0,
part of the extensive computations. h(+l)k(+i)
upwards iff +($I) is negative (tensile), downwards if it is positive
(compressive). The construction of the point M is repeated
APPENDIX I and the value w1 of w in M obtained. The value of U(+,)is
computed from equation (94) with wI in the place of wo, and
then
A BRIEF S U M M A R Y OF T H E C O U R S E O F T H E
CALCULATION
(97)
Given : Roll diameter D (D/2 = roll radius R)
Initial height of the bar h,,, The horizontal force f + ($3in $2 is obtained as
Final height ,, ,, h~
Coefficient of friction p
f + ( + 3= z+(43EJ+-Gol (98) ...
where Z3+ means the s u m J+(+J+J+($2). This procedure is
Back tension per unit of width G, = -fm continued for the subsequent intervals up to the middle of the
Front ,, 3, Y, Go = -fo arc of contact, using the horizontal force f +(&- 1) for obtaining
Yield stress K(+), obtained from expermental data wK- by the construction in Fig. 11, then computing
and corrected for the inhomogeneity of the defor-
mation as described in sections (30) and (31). (99)
The angle of contact +,, is obtained from equation (2), p. 142 :-
and

The curve w(a) as given in Fig. 9, p. 150, is plotted, the specific The horizontal force f+(&) in the point +K is then found from
front tension Go = -fo (where fo is negative) divided by f+(+K) = z + ( + 3 [ ~ J + - G o l * -
(101) .
hok(O), and the quotient plotted from the origin vertically up- where zJ+is the s u m of alLJ+’s f r ~ r n J + ( $ ~toJ+($J.
)
wards (OA in Fig. 11, p. 151, where, however, the horizontal
forcef is assumed to be compressive (positive) and is, therefore, If, in the course of the calculation, the line AC ceases to
plotted downwards from 0).The horizontal AB is drawn, and intersect the curve w(a), the end of the region of slipping has
1/2p plotted vertically upwards from the point B whose abscissa been reached, and the calculation must be continued as under
is 1 (BC in Fig. 11). If the line AC intersects the curve w(a), [Bl], with the last obtained value of f+ instead of -Go in
the bar slips in the plane of exit; the exit side of the friction hill equation (106).
is then calculated as described below under [All. If AC passes
to the right of w(a) without intersecting it, the bar sticks to the [ A 3 Calculation of the Horizontal Force f- on the Entry Side
rolls in the plane of exit; course [Bl] below is then taken. in the Case of Slipping in the Plane of Entry. The calculation
The same construction is repeated with OA = - Gm . is similar to that for the exit side in all details, except that :-
hmk(+m) ’ (a) The points on the arc of contact in which the computation
according to whether AC intersects w(a) or not, the course [A21 is carried out are &,&,- I , 4m-2, . . ., down to about
or [B2] is followed for the calculation of the entry side of the +,,,/3. The computation starts with +,, instead of do = 0,
friction hill. and progresses downwards.
The interval from +o = 0 to 4 = $m is subdivided into 8-12
parts by points $1, .&, .. ., 4 ~ m - 2 , +m- 1, which should be
represented, if possible, among the parameters of the curves
(b) Instead of z+(+), z-(+) is used as given by

H(+, y ) in Figs. 12a-12~~ pp. 153-5, if the bar slips in the plane
of entry or exit; if it sticks at both places, any choice may be h
made. The intervals need not be equal. In what follows & will
be the general symbol for any of the values +o = 0, +1,+2, , . .,
or by z-(+) = h,
-e
/dHm-H)
. . . . (64~)

+In. where H m is the value of H in &.

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 T H E C A L C U L A T I O N O F R O L L P R E S S U R E I N H O T AND COLD F L A T R O L L I N G 167
( c ) All plus signs in expressions (94) and (99) change to of the ordinate a x i s j units of s(+), then the roll torque corre-
m i n u s :- sponding to the regions of slipping is
U(+)= *(Sin
z-mi
+-/A cos +I .. (102) T,I = g 22 ( L - - i + 57.3
)X ... (107)
(d) f - ( + K ) is obtained from +
The yield stress k(+) is plotted over in the regions of stick-
f - ( + x ) = z-(+J[zJ--Gml (103) .. ing. Let K+,be the area between the curve k(+) and the +-axis
in the interval between 4 = 0 and +n, and K- the area between
where ZJ- means the s u m of all J-’s fromJ-(+m-l) to
J- ( 6 ~ ) - 4. and dm.If i and j are, as before, the scale factors, the roll
torque corresponding to the regions of sticking is
[BZ] Calculation of ths Hmieontal Force f + on the Exit Side
in the Caseof Sticking in the Plane of Exit. The value of z+(+)
is computed from The total roll torque is]
a+(+) =
y+1-cos+ a-
Y
h
h0
...
(58~) T = T ~ I + T I ~i . . (109) . .
If, however, 4 does not exceed 6-8 deg., higher accuracy can
for 4 = 0, dl, $2, up to about +,,,/2.For the same values of 4 the be reached easier by using formula (70), p. 159, instead of (107),
quantities (108), or (109).

are computed with m+(# taken from Fig. (lo), and then the A P P E N D I X -1 Ij
quantities
REFERENCES

EKELUND,S. 1933 Steel, vol. 93 (August 21st and following

numbers) ; translated from Jernkontorets Ann. 1927,
The horizontal force f + ( & ) is then obtained from vol. 111, p. 39, and 1928, vol. 112, p. 67.
f+(+K) = z+(+K)[~~+-GoIZ * * (lo6) - GEIRINGER,H. 1937 “Fondements mathematiques de la
where 2J+is the s u m of all J+’s from J+(+J to J+(dK). thCorie des corps plastiques isotropes”, vol. 86 of
“Mimorial des sciences mathimatiques (Gauthier-
[BZ] Calculation of the Horizontal Force f - on the Entry Side Villars, Paris).
in the Case of Sticking in the Plane of Ently. The calculation is HITCHCOCK, J. 1935 “Roll Neck Bcarings” (A.S.M.E. Re-
similar to that for the exit side in all details, except that search Publication), Appendix I.
(a) The points on the arc of contact in which the calculation KARMAN, T. von. 1925 Zeitschrift f u r angewandte Mathc-
is carried out are +, +,,- 1,&-2,
9h/3. The computation begins at
. . .,
down to about
instead of 40 = 0,
matik und Mechanik, vol. 5, p. 139.
LODE,W. 1925 Zeitschrtft f u r angmandte Mathematik und
and piogresses downwards. Mechanik, vol. 5, p. 142. See also Proc. International
(b)
. _The value Congress in Applied Mechanics, Ziirich, 1926.
MCHELL, J. H. 1902 Proc. Lond. Mathematical SOC., vol. 34,
2-(+) = 7
+1-~0~$ h
r+ 1- cos 4n-hm
-- ..
(58B) p. 134. See also TIMOSHENKO, S., 1934 “Theory of
Elasticity” (McGraw-Hill, New York and London), I -

is used instead of z+(+). pp. 104kt seq.

(c) In equation (104), m- is used instead of m+, and-z- N ~ A IA., 1931 “Plasticity” (McGraw-Hill, New York and
instead of z+. London).
(d) The quantityf-(&) is obtained from [103], as in [MI. 1939 J I . Applied Mechamks, vol. 6, p. A-55.
PRANDn, L. 1923 Zeitschrtft fur angewandte Mathematik und
[C] Calculation of the Normal Roll Pressure s(+). According Mechanik, vol. 3, p. 401.
to equation (46), s(+) is obtained by adding k(+)w to f / h :- PRBSCOTT, J. 1924 “Apphed Elasticity” (Longmans, Green
s($) =L%+k(+)w . . . . and Company, London), p. 633.
Smm, E. 1924 Berichte des Walzwerksausschusses, Verein
deutscher Eisenhiittenleute, No. 37. See also Stahl und
where f(4)means f + (4) on the exit side and f- (4) on the entry Eisen, 1925, vol. 45, p. 1563.
side. I n regions of slipping, w has been obtained point for point S ~ B LE.,, and LUEG,W. 1933 Mitteilungen aus dem Kaiser
by the construction in Fig. 11, p. 15 1. In regions of sticking, w Wilhelm Institut f iir Eisenforschung, Diisseldorf, vol. 15,
. in the case of Sticking, 4 is very
has the constant value ~ / 4 If, p. 1. See also LUEG, W., Stahl und Eisen, 1933, vol. 53,
large and s not much greater than k, s(+) may be computed more part 1, p. 346.
accurately from [45]. S m a , E., and POMP,A. 1927 Mittdungen aus dem Kaiser
The neutral angle is determined as the abscissa of the point Wilhelm Institut f u r Eisenforschung, Diisseldorf, vol. 9,
of intersection of the curves s+ and s- j for the normal p. 157.
roll pressure is s+ j for +>+n, it is s-. S m n , E., and POMP,A. 1927 Mitteilungen aus dem Kaiser
Wilhelm Institut f ur Eisenforschung, Diisseldorf, vol. 9,
[ D ] Calculation of the Roll Torque. The curve s(+) is plotted
+
over in the regions of slipping, and the area between the curve
p. 157. 1930 Do., vol. 12, p. 149.
SOBOLEVSKY, N. A. 1935 Blast Furnace and Steel Plant, vol. 23,
and the abscissa axis computed. Let L+ be the area between pp. 685,763,850; 1936, vol. 24, pp. 149, 237,313,413.
4 = 0 and and L- the area between 4“ and &. If 1 unit of TRINKS, W. 1937 Blast Furnace and Steel Plant, vol. 25, p. 617.
the length of the abscissa axis represents i degrees, and 1 unit TSBLIKOV, A. T. 1936 Metallurg (Russian), No. 6, p. 61.

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