mechanical-properties-and-working-of-metals-and-alloys_compress

12.4 Deformation Zone in Rolling 539 O h an arbitrary angle formed by a very thin slice of the stock αψ in the deformed zone with the centreline of the rolls, i.e. XN Y with the exit plane; hence, the angle h may vary from 0 Y´ at the exit plane YY0 to a at the entry plane XX0: X´ N´ dh the angle formed at the roll centre by the thickness of the α ψ infinitely thin slice of the stock in the deformed zone; O´ a the angle of contact of stock formed by the entry plane XX0 at the roll-centre O or O0 with the centre-line of the x=0 rolls, OO0; x w the no-slip angle formed by the neutral plane NN0 at the roll-centre O or O0 with the centre-line of the rolls, OO0; R roll radius; x 0 at the exit plane; and x L (projected length of deformation zone) at the entry plane. Fig. 12.21 Neutral points N and N0 on the arc of contact, neutral plane To make the analysis simple, the following assumptions NN0; no-slip angle w; angle of contact a; entry zone XNN0X0; and exit are made: zone NYY0N0 1. The lateral spread is negligible that means the initial The angle between the neutral plane, NN0; and the cen- width of the stock, w; before rolling remains practically treline of the rolls, OO0; is called the ‘no-slip angle’. Hence, at the same value after rolling. the ‘no-slip angle’ is \\NOY ¼ \\N0O0Y0 ¼ w: 2. Elastic deformation of the roll owing to the forces exerted 12.5 Ekelund Expression for No-Slip Angle by the stock on the rolls is negligible, i.e. roll radius R (Ekelund 1933) remains constant. With reference to Fig. 12.22, consider a vertical thin element 3. Coulomb’s law of sliding friction holds good at the of metal located in the deformation zone in the roll gap at an contact surface between the rolls and the stock. In other arbitrary angular position h from the line, OO0; joining the words, roll centres. Let us define different symbols used in the derivation of expression for the no-slip angle: s ¼ lpr; where; s is the tangential frictional stress; l is the coefficient of friction and dθ O pr is the radial roll pressure: μ pr pr θ 4. The coefficient of friction l; is the same at all points on X pr Y the arc of contact. N μpr Y’ 5. The radial roll pressure pr between the stock and the rolls is uniform over the whole arc of contact. X’ N’ αψ 6. The algebraic sum of the horizontal components of all the forces exerted by the rolls on the stock is zero for O’ steady-state rolling. R In the entry zone, the horizontal component of the forces x=0 exerted by the rolls on the stock in the positive direction of x; x i.e. in a direction from the exit plane towards the entry plane, is Fig. 12.22 Radial roll pressure pr and tangential friction stress lpr Za acting on two vertical slabs—one in the entry zone XNN0X0 and another HEntry ¼ 2 pr sin h w R dh À 2 l pr cos h w R dh in the exit zone NYY0N0 w Za ¼ 2 w ðpr R sin h À l pr R cos hÞ dh w Since there is a reversal in the direction of the tangential frictional stress across the neutral plane, in the exit zone, the horizontal component of the forces exerted by the rolls

540 12 Rolling on the stock in the positive direction of x; i.e. in a direction Therefore, (12.19a) reduces to from the exit plane towards the entry plane, will be No-slip angle; w ¼ a À a2 ð12:19bÞ Zw 2 4l HExit ¼ 2 w ðpr R sin h þ l pr R cos hÞ dh Equations (12.19) are the formulae for no-slip angle, 0 which is only accurate when the draft is small compared to the stock thickness, because it is based on the assumption of The net horizontal force exerted by the rolls on the stock uniform radial roll pressure over the whole arc of contact and in the positive x-direction is: constant coefficient of friction. But the radial roll pressure is not constant; rather, it increases from the entry plane along the HTotal ¼ HEntry þ HExit arc of contact to a maximum value at the neutral plane and then decreases towards the exit plane. Further, there is a local Za Zw variation in the coefficient of friction from point to point along Or; HTotal ¼ 2 w pr R sin h dh þ 2 w l pr R cos h dh the arc of contact of the roll (Capus and Cockcroft 1961–62). 0 0 Za À 2 w l pr R cos h dh w Ekelund argues that this net horizontal force, HTotal; must be 12.6 Forward Slip equal to zero for steady-state rolling. According to him, if HTotal [ 0; the stock velocity will be decelerated and if The value by which the velocity of the stock leaving the rolls HTotal\\0; the stock velocity will be accelerated. exceeds their peripheral velocity is known as ‘forward slip’, which figures largely in rolling mill research. Forward slip, Thus, for steady static rolling, HTotal ¼ 0: SF; is determined as the ratio of the difference between the delivery velocity of stock and the peripheral velocity of the Za Zw ) 2w pr R sin hdh þ 2w lpr R cos hdh rolls to the peripheral velocity of the rolls: 0 0 SF ¼ v2 À vr ¼ v2 À 1 ð12:20aÞ vr vr Za À 2w lpr R cos hdh ¼ 0 w where 28 93 v2 the velocity of the stock leaving the rolls, i.e. the Za ><Zw Za dh>;=>57 velocity of the rolled product at the exit plane, 64 :> Or; 2 w pr R sin h dh þ l cos h dh À cos h ¼ 0 vr the peripheral velocity (or surface velocity) of the rolls and 0 0w As w ¼6 0; pr ¼6 0; and R ¼6 0; therefore; Most often forward slip is expressed in per cent. The importance of forward slip has been discussed subsequently ½À cos hŠ0a þ l ½sin hŠ0wÀl ½sin hŠaw¼ 0 in Sect. 12.6.3. Similarly, ‘backward slip’, SB; is defined as the value by Or; ð1 À cos aÞ þ l sin w À l ðsin a À sin wÞ ¼ 0 which the peripheral velocity of the rolls exceeds the Or; 2 sin2 a þ 2 l sin w À l sin a ¼ 0 velocity of the stock entering the rolls as follows: 2 SB ¼ vr À v1 ¼ 1 À v1 ð12:20bÞ Or; 2 l sin w ¼ l sin a À 2 sin2 a vr vr 2 where ) sin w ¼ sin a À sin2ða=2Þ ð12:19aÞ v1 the velocity of the stock entering the rolls, i.e. the 2l velocity of the stock at the entry plane. For small angles of contact, i.e. when a is very small and 12.6.1 Relation with No-Slip Angle expressed in radian, then sin a % a; sin2 a % a2 ; sin w % w; Ekelund derived an expression for forward slip in terms of 24 the no-slip angle w: This relationship is based on the fol- lowing assumptions:

12.6 Forward Slip 541 • There is negligible or no lateral spread. This assumption spread, i.e. the initial width of the stock before rolling will not be considered in the initial part of the following remains practically at the same value after rolling, then derivation. w1 ¼ ww ¼ w2 • Rolls are rigid, and no elastic deformation of the rolls occurs. Hence, from (12.21e) forward slip can be expressed as: • Initially plane vertical sections of the stock remain plane &' during rolling. 1 þ 2R ð1 À cos wÞ Froward slip; SF ¼ cos w h2 À1 Since equal volume of metals must pass a given point per ¼ cos w þ 2R cos wð1 À cos wÞ À 1 unit time, so it can be written as follows: h2 v1 h1 w1 ¼ vw hw ww ¼ v2 h2 w2 ð12:21aÞ ¼ 2R cos wð1 À cos wÞ À ð1 À cos wÞ h2  ¼ ð1 À cos wÞ 2R cos w À 1  where v1; h1; w1 velocity, height and width, respectively, of the  h2  vw; hw; ww stock at the entry plane; ¼ 2 sin2 w 2R cos w À 1 v2; h2; w2 velocity, height and width, respectively, of the 2 h2 stock at the neutral plane; and velocity, height and width, respectively, of the ð12:22aÞ stock at the exit plane. In the cold-rolling process, the angle of contact, a; is Since at the neutral point the stock velocity, vw, equals the usually small, and as the no-slip angle, w\\ða=2Þ, it may be horizontal component of the peripheral velocity (or surface assumed that sin w % w and cos w % 1, when the angle w is velocity) of the rolls, vr; expressed in radian. So, the above (12.22a) may be approximated as vw ¼ vr cos w ð12:21bÞ SF % w2  À  ¼ w2  À  ð12:22bÞ 2 2R 1 2 2R 1 h2 h2 In analogy with (12.6a), it can be written at the neutral 2 plane that Usually; roll radius; R ) 1; height of stock at roll0s exit plane; h2 R À hw À h2 ¼ R cos w; 2 or; hw À h2 ¼ 2 Rð1 À cos wÞ; i:e:; 2 R ) 1; h2 ) hw ¼ h2 þ 2 R ð1 À cos wÞ ð12:21cÞ Therefore, the approximate formula for forward slip turns out to be Combining (12.21a), (12.21b) and (12.21c), we can write v2 h2 w2 ¼ vr cos w f&h2 þ 2 R ð1 À cos wÞg'ww SF % w2 Â 2R ¼ R w2 ð12:22cÞ 2 h2 h2 Or; v2 ¼ ww cos w h2 þ 2 R ð1 À cos wÞ In practical calculations, this simplified formula, given by vr w2 h2 ' (12.22c), provides sufficiently accurate results. However, val- & ues of forward slip calculated from the relation derived by ¼ ww cos w 1 þ 2R ð1 À cos wÞ Ekelund appear to diverge from the experimental values for w2 h2 ð12:21dÞ rolling of thicker stocks. This may be due to the assumption that initially plane vertical sections of the stock remain plane during Now from (12.20a) and (12.21d), forward slip can be rolling, which is relatively less true for thick than for thin stocks. expressed as: However, forward slip in rolling depends on the radius of Froward slip; & ' the rolls and the outgoing thickness of the rolled product. Increasing the roll radius and/or decreasing the outgoing SF ¼ v2 À 1 ¼ ww cos w 1 þ 2R ð1 À cos wÞ À1 product thickness results in an increase in forward vr w2 h2 slip. Forward slip decreases parallel with the increase of lateral spread (deformation). Thus, forward slip depends on ð12:21eÞ the relationship between elongation and spread. When elongation is increased by decreasing spread, the speed of The forward slip given by (12.21e) takes into consider- the stock leaving the rolls is also increased. ation the effect of the work-piece width. Inasmuch as ww=w2\\1; forward slip decreases with an increase in spread. If it is assumed that there is negligible or no lateral

542 12 Rolling 12.6.2 Measurement of Forward Slip 12.6.3 Importance of Forward Slip The forward slip, SF; is usually measured by using the Forward slip is of vital importance in rolling for finding the ‘impression method’, as shown in Fig. 12.23. A roll is marked by two indentations spaced a distance ‘lr’ coefficient of friction between the roll surface and the apart. Suppose after rolling the distance between these dents created by the roll on the rolled product is l2: Due to forward work-piece being rolled. Forward slip being measured, the slip l2 [ lr; and the forward slip from (12.20a) is expressed as value of the no-slip angle, w; can be determined from the relationship, shown by (12.22c), i.e. SF ¼ ðR=h2Þ w2; from which we can write SF ¼ v2 À vr ¼ ðl2=tÞ À ðlr=tÞ ¼ l2 À lr ð12:20cÞ rffiffiffiffiffiffiffiffiffiffi ð12:24Þ vr lr =t lr w ¼ h2 SF R where The value of w being known, the value of the coefficient of friction, l; can be calculated from the relation given by t the time over which the effective rolling is carried out. (12.19b), as shown below: In hot rolling, where l2 on the stock is generally measured w ¼ a À a2 ; or; a2 ¼ a À w ¼ a À 2 w ; after the metal has cooled, account must be taken of the 2 4l 4l 2 2 change in the length of stock due to thermal expansion by the following expression: Or; 4l ¼ a 2 w ; a2 À2 l2 ¼ l0 ½1 þ a DTŠ ð12:23Þ ) l ¼ 2 a2 ¼ a2 ð12:25Þ a À 2wÞ À where 4 ð 2ð a 2wÞ l0 the distance separating dents on a cool stock; Forward slip is also important in continuous rolling with a the coefficient of thermal expansion; DT the temperature difference between the stock after hot several stands in series to avoid tearing of metal or looping of metal. If the entry velocity of stock into n th stand, say v1n; rolling and that after it has been cooled for exceeds the exit velocity of the stock from the preceding stand, i.e. ðn À 1Þth stand, say v2nÀ1; i.e. v1n [ v2nÀ1; then measurement. there will be development of tension, which will cause vr tearing of metal. On the other hand, if the reverse occurs, i.e. v1n\\vn2À1; then looping of metal will occur. Thus to avoid tearing or looping of metal, forward slip must be properly controlled and is ordinarily maintained between 3 and 6%. 12.7 Elastic Deformation of Rolls lr The reaction of the work-piece on each roll due to applica- tion of the rolling load is called the roll-separating force, which has the same value as that of the applied rolling load v2 but acts in a direction opposite. When the loads introduced by the roll separation forces are high, roll undergoes two l2 major types of elastic distortion, which are: • Elastic flattening of rolls in the regions where they con- tact the work-piece. • Elastic bending of rolls along their length, i.e. deflection of rolls. 12.7.1 Roll Flattening Under certain rolling conditions, the large pressure exerted by Fig. 12.23 Measurement of forward slip using ‘impression method’ the stock on the rolls causes elastic flattening of the regions of

12.7 Elastic Deformation of Rolls 543 the rolls in contact with the stock. Due to flattening, the radius Er Young’s modulus of the roll material. Er ¼ 207 GPa for steel rolls, Er ¼ 174 GPa for chilled-iron rolls and of curvature of the deformed arc of contact is increased from R Er ¼ 100:5 GPa for cast iron rolls. to R0; or still a higher value of R00; and so on. The extent of flattening depends on the magnitude of the reaction stress Equation (12.26) is known as Hitchcock equations for roll flattening. The deformed roll radius R0 or R00 cannot be exerted by the stock on the rolls and the elastic constants of determined directly from (12.26), since P or P0 is, respec- the rolls. In such cases—accepting the assumption that the tively, a function of R or R0; and to calculate a value of R0 or form of the contact arc remains circular—the method most R00; a trial-and-error procedure (Larke 1963) by successive commonly used to determine R0 or R00 or higher and higher approximations is required. This deformed roll radius R0 or R00 must be used to calculate rolling loads accurately in all cases. radii of curvature of deformed rolls is by means of an equation 12.7.2 Roll Deflection attributed to Hitchcock (Hitchcock 1935; Underwood 1950). Rolls are restrained at their ends, but their centre portion R0 ¼ 1 þ w Cr P ð12:26aÞ tends to bend or deflect longitudinally as the work-piece R ðh1 À h2Þ while being rolled tends to separate them. This results in variation of thickness over the width of the stock making the And centre thicker than the edges (crown) and leads to loss in shape of the product. Bending of straight cylindrical rolls R00 ¼ 1 þ Cr P0 ð12:26bÞ producing the thicker centre is shown in Fig. 12.24a. To R w ðh1 À h2Þ avoid this problem, the rolls are usually ground parallel to their axis in such a way that their diameters are slightly And so on, where greater at the centre than at the edges. This curvature on the roll diameter is known as camber, which is usually less than P the rolling load determined from the original radius, 0.5 mm in sheet metal rolling practice. Hence, the rolls P0 having ground convex camber or crown, as shown in W R; of the rolls; Fig. 12.24b, are used to correct for roll deflection. h1; h2 Figure 12.24c shows that rolls with convex camber still bend Cr the rolling load computed from the increased radius, under the roll-separating force, but the roll profile adjacent to R0; of the flattened rolls; the stock remains straight during rolling which gives a product of uniform thickness. One must be careful that rolls the width of the stock, which is assumed to remain having excess convex camber will result in the greater reduction of thickness at the centre than at the edges leading constant considering plane-strain rolling; to loss in shape of the product as before. the thickness of stock, respectively, before and after Formula for calculation of camber is based on the con- sideration of the bending of rolls as a thick, short beam rolling; simply supported at the ends. The deflection, dr; at the centre of the rolls thus contains two terms due to the bending the elastic deformation parameter of the roll material ÀÁ 16 1 À m2r ¼ ð12:27Þ p Er Cr ¼ 2:16 Â 10À11PaÀ1 for steel rolls, Cr ¼ 2:57 Â 10À11PaÀ1 for chilled-iron rolls, and Cr ¼ 4:45 Â 10À11PaÀ1 for cast iron rolls. In (12.27): mr Poisson’s ratio of the material of roll. For steel rolls, chilled-iron rolls and cast iron rolls, mr ¼ 0:35 (Harris 1983); (a) (b) (c) Fig. 12.24 a Bending of straight rolls producing thicker centre. b Rolls having convex camber. c Bending of rolls with convex camber, but maintaining straight roll gap

544 12 Rolling moment about midpoints of the neck bearings, and to the practical to grind the rolls with each change of stock size. shearing forces, and can be expressed as (Rowe 1977) A better method is to introduce hydraulic jacks onto the rolling necks which correct for roll deflection by actually Central deflection of rolls; dr ¼ kB P l3r þ kS P lr bending the rolls under various rolling conditions. Many Er Ir Gr Ar modern sheet mills are successfully using this technique to control the shape of the product. ð12:28Þ where P roll load or roll-separating force, 12.8 Simplified Assessment of Rolling Load lr effective length of each roll, Er; Gr elastic and shear moduli, respectively, of the roll The rolling load, P; can be determined from the multipli- material, Ir second moment of area of the roll cross-section cation of specific roll pressure, p; with the area of contact about a diameter, Ar cross-sectional area of the roll barrels, between the stock and the rolls, i.e. the area of deformation kB; kS constants, whose values depend on the relative dimensions of roll and stock. The typical values are zone. Again, the area of deformation zone can be considered kB $ 1:0; and kS $ 0:2; for a stock with width lr; and kB $ 0:5; and kS $ 0:1; for a stock with width to be the product of the length of the deformation zone, L; lr =2: and the width of the stock in the roll gap. If we assume lateral spread to be negligible, i.e. the initial width, w; of the stock before rolling remains practically at the same value after rolling, then rolling load can be written as: pffiffiffiffiffiffiffiffiffi ð12:29aÞ Rolling load; P ¼ p w L ¼ p w R Dh The above (12.28) will be strictly valid for one specified where L is given by the approximate (12.11b). For the rolling load and width of stock only. In addition, at the moment, if we neglect the friction, the specific roll pressure beginning of a run of rolling operation, the roll camber will p becomes equal to the flow stress of the material. Since no usually increase due to more temperature rise in the central portions which is heated more rapidly than the ends of the change in the width of the stock occurs during rolling, the rolls. This is known as thermal camber of rolls and most important in hot rolling. On the other hand, severity of wear, rolling may be considered to be a plane strain deformation which is the maximum at the centre, reduces the roll camber. and the specific roll pressure p will be equal to the plane To counteract these effects, suitable production planning can strain flow stress, r00: So, (12.29a) can be rewritten as: be made; one may start rolling with medium-width stock and a roll camber less than that required for full-width stock. As Rolling load under frictionless condition; ð12:29bÞ the roll-centre is heated up to develop thermal camber, wider pffiffiffiffiffiffiffiffiffi stock is rolled and the stock width is then reduced again as the roll-centre undergoes wear. However, thermal camber P ¼ r00 w R Dh can be controlled, particularly in hot rolling, by varying the amount of coolant (lubricant) flow at different locations on Since the contribution of friction is ignored, (12.29b) the rolls along their axial direction, thus minimizing the provides a lower limit for the rolling load. Examining a uneven temperature distribution. range of typical rolling passes, Orowan (Rowe 1977) has suggested the approximate rolling load under condition of Attempts to avoid or limit roll deflection by reducing the friction, providing an allowance of about 20% for the fric- rolling load have been made by the use of small work rolls, tion, i.e. considering p ¼ 1:2 r00 : as in four-high mills and cluster mills. But even with these mills, rolls still bend to a certain extent and it is required to Rolling load under frictional condition, after Orowan; camber the rolls, i.e. to make them convex shaped to pffiffiffiffiffiffiffiffiffi accommodate the roll deflection. However, the disadvantage of the roll-cambering procedure is that camber corrects for P ¼ 1:2 r00 w R Dh the deflection of the rolls to produce flat profile for one ð12:30Þ specified rolling load and width of stock only. To achieve a flat surface in cold rolling, the rolling load is maintained to a Considering the influence of friction an approximate constant value by adjusting front and back tension or inter- expression for the rolling load can be developed using the stand tension with multi-stand continuous rolling. Different analogy between plane-strain flat rolling and forging of amounts of roll deflection caused by different loads require rectangular plate of uniform thickness in plane strain con- different amounts of cambering or crowning, and it is not dition, as discussed in Sect. 11.7 of Chap. 11. So, similar to plane-strain forging, a friction hill is generated in plane-strain flat rolling, as shown in Fig. 12.25, which indicates the distribution of roll pressure (Al-Salehi et al. 1973) per unit width of stock along the arc of contact,

12.8 Simplified Assessment of Rolling Load 545 Pressure required to r00 À pffiffiÁ overcome friction the plane-strain average flow stress ¼ 2= À 3 pÂffiffiuÁni- axial homogeneous average flow stress ¼ 2p= ffiffiffiffi3ffiffiffiffiffir0; L the length of deformation zone in rolling % R Dh; Roll pressure l tQhe¼coðlefLfiÞc=iehn¼t of lCpouRffiffilffioffiDffimffiffihffiffib’=shfr.iction; and Combining (12.29a) and (12.31), we can get rolling load B for slipping friction as Homogeneous deformation p w pffiffiffiffiffiffiffiffiffi pressure in plane strain Rolling load; P ¼ p2ffiffi r0RDehQQÀ A ¼  pffiffiffiffiffiffiffiffiffi ! 3 1 R Dh w ð12:32Þ Entrance Exit Equations (12.31) and (12.32) are applicable to the con- plane plane Length of contact ditions where no back and front tensions are applied. But Fig. 12.25 Friction hill showing distribution of roll pressure in plane most often back tension at the entry plane and front tension strain along the arc of contact at the exit plane of the deformation zone are applied to the i.e. along the projected length of deformation zone. The stock in cold rolling in order to reduce the rolling load. pressure rises from the entry plane to a maximum value at the Hessenberg’s (Hessenberg and Sims 1951) data show that neutral point and then drops towards the exit plane. Mathe- matical treatments on rolling show that the pressure should the effectiveness in reducing the rolling load by back tension reach a sharp peak at neutral point, but in fact the pressure distribution does not come to a sharp peak at neutral point, is about twice of that by front tension. Introducing the effects indicating that the neutral point is not actually a point on the roll surface but an area. The total area under this friction hill of back and front tensions on the mean rolling pressure, curve is the rolling load per unit width of stock. The force required for overcoming frictional forces between the unit Avitzur (1964) wrote that width of stock and the roll is represented by the shaded area in Fig. 12.25, while ideal force required for deforming the p ¼ r00 À 1 ð2rB þ rF Þ!eQ À  ð12:33Þ unit width of stock in frictionless plane-strain homogeneous 3 Q 1 rolling is represented by the area under the dashed line AB. where Let us consider that low frictional condition exists at the contact surface between the rolls and the stock, such as in rB the back tension at the entry plane of the deformation cold rolling, where Coulomb’s law of sliding friction holds zone and good. The coefficient of friction for cold rolling (Roberts 1968) with lubricants varies from 0.05 to 0.1, whereas that rF the front tension at the exit plane of the deformation for hot rolling ranges from 0.2 up to the sticking condition. zone Then, using the analogy between plane-strain forging and plane-strain rolling, (11.30) may be applied to approximate Considering the effects of back and front tensions, rolling the mean rolling pressure, p; for slipping friction as follows: load for slipping friction can be written from (12.33) as follows: Rolling load; Þ!eQ 1wpffiRffiffiffiDffiffiffihffiffi pffiffiffiffiffiffiffiffiffi r00 À 1 ð2rB þ rF À P ¼ pw RDh ¼ 3 Q p2ffiffi 1 ! eQ À 1wpRffiffiffiffiDffiffiffihffiffi! 3 3 Q ¼ r0 À ð2rB þ rF Þ p ¼ r00 exp½ðl LÞ=hŠ À 1 ¼ r00 ðeQ À 1Þ ð12:31Þ ð12:34Þ ðl LÞ=h Q The coefficient of friction at elevated temperature may where be 0.2 and above, which is very difficult to estimate prop- erly. But the most common in hot rolling is sticking fric- h the mean thickness of the stock between the entry tional condition at the contact surface between the rolls and and the exit plane of the deformation zone in rolling the stock. Again using the analogy between plane-strain ¼ ðh1 þ h2Þ=2; in which h1 ¼ thickness of the stock at forging and plane-strain rolling, (11.45) may be used to the entry plane and h2 ¼ thickness of the stock at the approximate the mean rolling pressure, p; for sticking fric- exit plane; tion as follows:

546 12 Rolling  pffiffiffiffiffiffiffiffiffi! both in terms of temperature T; which is measured in °C, and L p2ffiffi RDh the ‘static’ mean yield stress r0 in kN=mm2; in terms of p ¼ r00 1þ 4h ¼ 3 r0 1þ 4h ð12:35Þ viscosity g and weight percentage of carbon, manganese and chromium contents of steel. These formulae are: Combining (12.29a) and (12.35), we can write for stick- ing friction Viscosity, g ¼ 1; 373 À 0:098 T (where T is measured in °C); Coefficient of friction, l ¼ 0:084 À 0:0004 T (for billet Rolling load; P ¼ p w pffiffiffiffiffiffiffiffiffi ¼ p2ffiffi r0 1þ pffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffi temperatures T above 700 °C); R Dh 3 R Dh w R Dh ‘Static’ mean yield stress, r0 ¼ 100 g ð1:4 þ C þ Mn þ 4 h 0:3 CrÞ kN=mm2; ð12:36Þ 12.8.1 Ekelund Equation for Rolling Load vr ¼ the peripheral velocity of rolls in mm=s: The rest (Ekelund 1933) terms have already been defined in relation to (12.37). It is usually desirable to determine the mean value of yield stress To calculate the load for cold rolling, the equation proposed corresponding to the temperature and strain rate prevailing at the time of hot rolling. by Ekelund based on the analysis of roll stresses is one of the 12.9 Theory of Rolling: Derivation earliest methods. Including the effects of roll flattening, of Differential Equation of Friction Hill Ekelund’s equation for cold rolling is: Using the analogy between forging and rolling, equations for P ¼ r00 pffiffiffiffiffiffiffiffiffiffiffi \" þ 1:6 pffiffiffiffiffiffiffiffiffiffiffi 1:2 # ð12:37aÞ the values of roll pressure in the deformation zone of rolling w R0 Dh 1 l R0 Dh À Dh were developed in Sect. 12.8. But this was a very simplified approach because the fact that the metal moves in rolling h1 þ h2 unlike forging and is gradually decreased in thickness was ignored. Derivation of an expression for roll pressure must If the stock is subjected to back tension rB and front take these factors into account. Prior to this derivation, it is required to develop the source equation on which most of the tension rF; Ekelund’s equation (12.37a) is modified to: rolling theories are based. P ¼ Àr00 À Á pffiffiffiffiffiffiffiffiffiffiffi \" þ 1:6 pffiffiffiffiffiffiffiffiffiffiffi 1:2 # ð12:37bÞ Almost all the theories of rolling, hot and cold, with the w rT R0 Dh 1 l R0 Dh À Dh exception of that put forward by Orowan (1943), have at their starting points the ‘differential equation of the friction h1 þ h2 hill’, as derived by Von Kármán (1925), using free-body equilibrium approach. The derivation of the differential where P ¼ rolling load with or without tension, w ¼ width equation is based on the following assumptions: of stock, r00 ¼ plane-strain average flow stress, R0 ¼ radius of elastically flattened rolls, Dh ¼ h1 À h2; h1 ¼ thickness of 1. Peripheral velocity of the rolls remains constant. stock before rolling, h2 ¼ thickness of stock after rolling, 2. Lateral spread is negligible; i.e., plane strain condition is l ¼ coefficient of friction, rT ¼ ð2 rB þ rFÞ=3; rB ¼ back tension and rF ¼ front tension. assumed. Therefore, stock width, w; (normal to the plane in Fig. 12.26) before rolling remains practically at the Rolling loads over a wide range of sizes and reductions same value after rolling. 3. Plane vertical sections before rolling remain plane during predicted by (12.37) appear close to the results obtained by and after rolling; i.e., homogeneous deformation is using more accurate and difficult theories. The Ekelund’s assumed and, hence, there is no redundant deformation. 4. Elastic deformation of the rolls owing to the forces method can be recommended for general purpose use in exerted by the stock on the rolls at the contact surface industrial practice particularly when factors such as coeffi- between the rolls and the stock is negligible. That means, the radius of the roll, R; does not change during rolling as cient of friction are not known accurately. the elastic flattening of the rolls is neglected. Ekelund’s formula to calculate load for hot rolling was 5. Coulomb’s law of sliding friction holds good at the contact surface between the rolls and the stock. In other words, actually first developed, in which the effect of strain rate on the mean yield stress of metal was taken into account. The rest of the equation is the same as for cold rolling, though the coefficient of friction will of course have a higher numerical value in hot rolling. Ekelund’s equation for hot rolling is: P ¼ ( þ 2 pffiffiffiffiffiffiffiffiffiffiffi) pffiffiffiffiffiffiffiffiffi \" 1:6 pffiffiffiffiffiffiffiffiffi 1:2 # w r0 vr g Dh=R R Dh 1þ l R Dh À Dh h1 þ h2 h1 þ h2 ð12:38Þ In (12.38), Ekelund gave empirical formulae for the ‘viscosity’ g of the hot steel, and the coefficient of friction l;

12.9 Theory of Rolling: Derivation of Differential … 547 (a) measure the local variation in friction coefficient accurately in rolling, although attempts to measure the dθ O local friction directly have been made (Siebel and Lueg ψ 1933a, b). If the coefficient of friction is small, the assumption of constant magnitude of l is valid, because it θ is found that loads predicted theoretically assuming con- Xα stant l generally agree with the experimental values for low coefficient of friction. NY 7. Constant circular arc of contact is maintained and neutral N´ Y´ point falls within the arc of contact. X´ x = 0 R´ x O´ The rolls, the stresses acting on two infinitesimally thin R slabs of metal stock, one on each side of the neutral plane, and (b) dθ pr the coordinate system are illustrated in Fig. 12.26. The thin μpr θ slabs are of infinitesimal thickness, dx; and located at an dh/2 pr arbitrary distance x from the exit plane, YY0; which is taken as R dx the origin of coordinate system, i.e. x ¼ 0; and x increases towards the entry plane, XX0: The heights of the two vertical μ pr faces that bound the thin slab at x and x þ dx are, respectively, h and h þ dh: The cross-section of the work-piece in θ Fig. 12.26 is identical with any other cross-section parallel to h it. The longitudinal stresses acting parallel to the x-direction ddxx on the two vertical faces of the thin slab are principal stresses σx + dσx σx σx and are assumed to be distributed uniformly on these faces. σx + dσx μ pr The longitudinal principal stress at a distance x is rx; which dh/2 θ pr becomes rx þ drx at the distance x þ dx; and they are assumed μ pr to be compressive in nature. The angle of contact of the stock Exit Zone pr formed by the entry plane at the roll-centres with the line Entry Zone joining the roll centres is a: The arbitrary angle formed at the roll-centres by the thin slab with the line joining the roll Fig. 12.26 a Two elements of strip in the deformation zone, one on centres is h: Since the angular position h is measured from the each side of the neutral plane NN0; and geometrical relationship for exit plane, so at the exit plane, h ¼ 0; and hence, h varies from those elements. The broken-line profile of roll along the deformation 0 at the exit plane to a at the entry plane of the stock. The angle zone shows the elastic flattening of roll under load causing to increase subtended by the thickness of slab at the centre of rolls is dh: the radius from R to R0: b Stresses acting on those two elemental strips in the entry as well as in the exit zone At any point of contact between the vertical thin slab and the roll surface having an area of R dh w; the stresses acting on s ¼ l pr; where, s is the tangential frictional stress, the slab due to each roll are the radial roll pressure pr; and the l is the coefficient of friction and tangential shearing friction stress s: Hence, the forces acting on the slab due to each roll are, respectively, pr R dh w; and pr is the radial roll pressure: s R dh w; and their horizontal components are, respectively, pr sin h R dh; and s cos h R dh w: The longitudinal forces 6. The coefficient of friction, l; is constant at all points on acting on two vertical faces of the slab in the horizontal for- the arc of contact. In reality, there is a local variation in the ward direction of rolling are ðrx þ drxÞðh þ dhÞw À rx h w: coefficient of friction from point to point along the arc of Hence for steady-state rolling, the horizontal equilibrium of contact of the roll (Capus and Cockcroft 1961–62). A constant coefficient of friction is forcefully assumed in forces exerted by the two rolls on the vertical thin slab of the all theories of rolling, because it is very difficult to stock in the entry zone gives ðrx þ drxÞðh þ dhÞw þ 2 s cos h R dh w ð12:39aÞ ¼ rx h w þ 2 pr sin h R dh w Or; rx h w þ rx w dh þ h w drx þ w drx dh þ 2 s cos h R dh w ¼ rx h w þ 2 pr sin h R dh w

548 12 Rolling where Equation (12.41b) is the famous Von Kármán differential equation of friction hill. In (12.41a) and (12.41b), the upper rx the longitudinal stress acting on the face of an negative sign ‘−’ applies for the entry zone and the lower arbitrary height h of the thin slab in the positive sign ‘+’ applies for the exit zone. The yielding rx þ drx deformation zone, the face being located at an criterion, assumed by Von Kármán, was arbitrary distance x from the exit plane, which is h assumed to lie at x ¼ 0; p À rx ¼ r00 ð12:42aÞ dh the longitudinal stress acting on the face of an pr arbitrary height h þ dh of the same thin slab, the where p is the vertical roll pressure (compressive), rx is the R face being located at an arbitrary distance x þ dx longitudinal compressive stress, and r00 is the plane-strain from the exit plane; deformation resistance (compressive) of the work-piece an arbitrary angular position of the slab from the line joining the roll centres; being homogenously deformed under plane strain condition. the angle subtended by the thickness of slab at the centre of rolls; Vertical equilibrium of forces exerted by the rolls on the the radial roll pressure; s = the tangential shearing friction stress; vertical thin slab of the stock in the entry as well as in the the roll radius; and w ¼ the width of the stock. exit zone gives p dx w ¼ pr cos h R dh w Æ s sin h R dh w ð12:43aÞ Or; p cos h R dh ¼ pr cos h R dh Æ s sin h R dh Now, after cancelling the term, rx h w; from both sides of In (12.43a), the upper positive sign, ‘+’ indicates the entry the equality symbol of (12.39a) and as w 6¼ 0; dividing zone and the lower negative sign ‘−’ indicates the exit zone. (12.39a) by w; we get As cos h R dh ¼6 0; dividing (12.43a) by cos h R dh; we get rx dh þ h drx þ drx dh þ 2 s cos h R dh ¼ 2 pr sin h R dh p ¼ pr Æ s tan h ð12:43bÞ ð12:39bÞ Since s ¼ l pr; assumed by Von Kármán, (12.43b) can be As the term, drx dh; is a multiplication of two small written as: quantities, the term drx dh may be neglected and (12.39b) may be given by p ¼ prð1 Æ l tan hÞ ð12:43cÞ With the help of (12.43c), the yielding criterion given by (12.42a) will become rx dh þ h drx ¼ 2 pr R sin h dh À 2 s R cos h dh; ð12:40aÞ prð1 Æ l tan hÞ À rx ¼ r00 ð12:42bÞ Or; dðh rxÞ ¼ 2 R ðpr sin h À s cos hÞ dh Again, the horizontal equilibrium of forces exerted by the 12.10 Bland and Ford Theory of Cold Rolling two rolls on a similar vertical thin slab of the stock in the exit zone gives an equation similar to (12.39a), but with the Almost all attempts so far made to derive an expression for frictional force in the opposite direction as follows: radial pressure in cold rolling are based commonly on the simplifying assumptions, on which the derivation of the Von ðrx þ drxÞðh þ dhÞw ¼ rx hw þ 2pr sin hRdhw ð12:39cÞ Kármán differential equation of friction hill is based. Using þ 2s cos hRdhw polar coordinates, Orowan (1943) developed the most accurate theory, but the theory proposed by Bland and Ford After similar simplification, (12.39c) reduces to was a far simpler one, which although is a little less accurate is considered in the following text. dðh rxÞ ¼ 2 R ðpr sin h þ s cos hÞ dh ð12:40bÞ Bland and Ford theory of cold rolling (Bland and Ford For the entry and exit zone, respectively, (12.40a) and 1948) has at its starting point Von Kármán differential (12.40b) are combined together as follows: equation of friction hill. Assumptions made by Bland and Ford in 1948 are: dðh rxÞ ¼ 2 R ðpr sin h Ç s cos hÞ dh ð12:41aÞ 1. In cold rolling, the angle of contact, a; is usually less than Since s ¼ l pr; assumed by Von Kármán, (12.41a) can be 8°; therefore, when radian is used to express the angles, it written as: follows that dðh rxÞ ¼ 2 R pr ðsin h Ç l cos hÞ dh ð12:41bÞ

12.10 Bland and Ford Theory of Cold Rolling 549 sin h % h; and cos h ¼ 1 À 2 sin2ðh=2Þ ¼ 1 À Àh2 Á stress r00: Further, the variation in the product h r00 is so =2 small that dðh r00 Þ may be considered to approach a negli- gible quantity, since r00 increases as h decreases. Hence, the % 1: above third assumption is usually reasonable for With the above assumption, the Von Kármán differential strain-hardened metal. But this assumption is not valid when (12.41b) is modified to: the rate of strain hardening is relatively high, for example dðh rxÞ ¼ 2 R pr ðh Ç lÞ dh ð12:44Þ during initial deformation stage of the work metal after In (12.44), the upper negative sign ‘−’ applies for the entry zone and the lower positive sign, ‘+’ applies for the exit annealing, or when the work metal is subjected to high back zone. tension because it reduces the variation of p=r00 over the arc of contact, which can be seen in (12.56a). For most rolling 2. Cold rolling occurs under usually low frictional condi- tions with the coefficient of friction, l 0:1: Since both l operations, in the second and subsequent deformation passes and h are small in cold rolling, so l tan h ( 1; and according to Bland and Ford, ‘l tan h’ is neglected in the after annealing, the accuracy of this approximation is satis- relationship between p and pr; given by (12.43c) as well as in the yielding criterion, given by (12.42b). Thus, factory leading to an error of only a few percent. (12.43c) is modified to: Equation (12.46) is the result of modification of the original Von Kármán differential equation of friction hill by the first two assumptions. Now, with the third assumption, made by Bland and Ford for cold rolling, (12.46) is further modified to: p % pr ð12:45aÞ  Since the radial roll pressure pr is the same as the vertical h r00 d p ¼ 2 R p ðh Ç lÞ dh; roll pressure p; as shown by (12.45a), the suffix may be r00   omitted and considered as the principal stress. Hence with pp the above assumption, the yielding criterion for cold rolling Or; h d r00 ¼ 2 R r00 ðh Ç lÞ dh ð12:47aÞ approximated by Bland and Ford, i.e. (12.42b), is modified to: Let y ¼ p=r00 ; then (12.47a) changes to pr À rx % p À rx ¼ r00 ð12:45bÞ h dy ¼ 2 R y ðh Ç lÞ dh; or; dy ¼ 2 R ðh Ç lÞ dh yh By substituting rx ¼ p À r00; from (12.45b), and pr % p ð12:47bÞ from (12.45a) into the modified Von Kármán differential Again, equation (12.44), we get h À h2 ¼ 2 R ð1 À cos hÞ; or; h ¼ h2 þ 4 R sin2ðh=2Þ ; È À r00ÁÉ ¼ 2 R p ðh Ç lÞ dh; dh pÀ ' & p where h2 ¼ the thickness of stock at the exit plane. Or; d h r00 r00 À 1 * The angle of contact, h, is very small in cold rolling and ¼ 2 R p ðh Ç lÞ dh;  sin h % h; therefore, h ¼ h2 þ 4 Rðh=2Þ2¼ h2 þ Rh2: Substi-    1 tuting for h into (12.47b) and integrating, we get: p 1 p À r00 Á ) h r00 À þ r00 À dh Z Z d r00 dy ¼ yZ 2 R ðh Ç lÞ dh ¼ 2 R p ð h Ç lÞ dh ð12:46Þ h2 þ R h2 Z 3. ÈInÀpth=er00tÁhiÀrd1aÉsdsuÀhmrp00tiÁon(, Bhlran00 ddÈaÀnpd=rF00oÁrdÀs1uÉggesatnedd that ¼ 2 R h dh Ç 2 l R dh to Z h2 þ R h2 h2 Zþ R h2 ÈsiÀmpp=lrif00yÁ Àh r1t00hÉeddÈÀÀhps=ror00luÁ00 tÁcioaÀnn of (12.46), the term ¼ 2 R h dh Ç 2lR 1 þ ÀRdhh2=h2Á the term bÉe ignored in comparison with h2 þ R h2 h2 1: ) ln y ¼ À þ R h2Á Ç 2 l R ln h2 rffiffiffiffih2 The reason behind such suggestion made by Bland and rffiffiffiffi R Ford is that under most circumstances, the variation in the  h2 tanÀ1 h roll pressure p with angular position in the deformation zone R þ Cc ð12:47cÞ is much greater than the variation in the plane strain flow h2 where Cc is an integration constant.

550 12 Rolling Substituting h2 þ Rh2 ¼ h; into (12.47c), we get: is h ¼ h2: Hence, from (12.49a) and (12.50a), we get at the entry plane: y rffiffiffiffiffi rffiffiffiffi  h R tanÀ1 R AcÀ h1 exp ðÀl H1Þ ¼ yÀ ¼ 1; ) AcÀ ¼ exp ðl H1Þ=h1 ln ¼ Ç2 l h2rffiffiffiffi h þ Cc; ð12:51aÞ  ) y ¼ h exp Ç2 l R tanÀ1 h2rffiffiffiffi ! h2 R h þ Cc ð12:47dÞ h2 where Let rffiffiffiffi rffiffiffiffi  rffiffiffiffi rffiffiffiffi  R tanÀ1 R H1 ¼ 2 h2 a ð12:47fÞ H  2 R tanÀ1 R h h2 h2 h2 ð12:47eÞ Since for the exit zone h ¼ 0; so from (12.47e) we get Hence, we can write from (12.47d): rffiffiffiffi rffiffiffiffi  R tanÀ1 R H2 ¼ 2 h2 0 ¼ 0: h2 y ¼ h exp ðÇl H þ CcÞ ¼ Ac h exp ðÇl H Þ ð12:48Þ where Ac ¼ expðCcÞ; is a new integration constant. Now, Hence at the exit plane, we get from (12.49b) and (12.48) can be separated into two equations—one with the (12.50b): upper negative sign ‘−’ for the entry zone and another with the lower positive sign ‘+’ for the exit zone, as given below: Acþ h2 exp ð þ l H2Þ ¼ Acþ h2 ¼ y þ ¼ 1; ) Acþ ¼ 1=h2 ð12:51bÞ For the entry zone: yÀ ¼ AÀc h exp ðÀl HÞ ð12:49aÞ Now, substituting the value of AÀc from (12.51a) into (12.49a) and the value of Acþ from (12.51b) into (12.49b), For the exit zone: y þ ¼ Acþ h exp ð þ l HÞ ð12:49bÞ we get The integration constants, AcÀ and Acþ , may be deter- For the entry zone: yÀ ¼ pÀ ¼ exp ðl H1Þ h exp ðÀl HÞ mined from the given stress conditions at the entry and exit r00 h1 planes. ¼ h exp fl ðH1 À HÞg h1 12.10.1 Cold Rolling with no External Tensions ð12:52aÞ If we assume that there is no back tension or front tension, For the exit zone: yþ ¼ pþ ¼ h exp ðl HÞ ð12:52bÞ then the longitudinal stresses at the entry and exit planes, r00 h2 which are free surfaces, must be zero, i.e. rx ¼ 0: As the plane strain homogenous deformation resistance of the Hence from (12.52), yÀ and y þ may be plotted against work-piece, r00; increases during cold rolling, it is assumed that r00 will increase gradually from a value of r00 1 at the the angular position h in the deformation zone, since H in entry plane, to a higher value of r002 at the exit plane. As (12.52) is a function of h: Again since y ¼ p=r00; this graph y ¼ p=r00 ; so at the entry plane, yÀ ¼ pÀ=r00 1; and at the exit provides the variation of roll pressure, p; as a function of plane, y þ ¼ p þ =r002: From the yielding criterion modified angular position h and is often referred to in rolling as the by Bland and Ford for cold rolling [see (12.45b)], it follows that friction hill because of its shape, which is shown in Fig. 12.27. The point at which the curves for yÀ or pÀ and At the entry plane: y þ or p þ intersect is considered as the location of neutral ð12:50aÞ point, whose angular position measured from the exit plane pÀ ¼ r001 þ rx ¼ r00 1; or; yÀ ¼ pÀ=r00 1 ¼ 1 is given by the no-slip angle, w; and which is shown at h ¼ w in Fig. 12.27. At the exit plane: The longitudinal stress, rx; at any angular position h from p þ ¼ r002 þ rx ¼ r00 2; or; y þ ¼ p þ =r002 ¼ 1 the line joining the roll centres can easily be determined ð12:50bÞ from (12.52), since from (12.45b) we get Let us assume that at the entry plane, where the angle of contact, h ¼ a; the stock thickness is h ¼ h1; and at the exit rx ¼ p À r00 ; or; rx=r00 ¼ p=r00 À 1 ¼ y À 1 ð12:53Þ plane, where the angle of contact, h ¼ 0; the stock thickness The longitudinal stresses are zero at the entry and the exit planes, but increases with h; i.e. with distance inwards from the entry as well as the exit planes due to the frictional contribution resulting in a longitudinal frictional hill. This

12.10 Bland and Ford Theory of Cold Rolling 551 Roll pressure p+ O p- αψ σB XN h1 Y h2 σF Y´ X´ N´ αψ O´ θ=α θ=Ψ θ= 0 x=0 Entry plane Angular coordinate θ Exit plane x Fig. 12.27 Friction hill showing schematically the variation of roll Fig. 12.28 Application of back tension, rB; and front tension, rF; pressure p as a function of angular position h and the location of the respectively, at the entry and exit sides of the stock neutral point, i.e. the no-slip angle w causes increasing resistance to the expansion of vertical ) yÀ ¼ pÀ ¼ r00 1 À rB ¼ 1 À rB ¼ y1 ðsay) ð12:54bÞ sections under vertical load. Thus, due to the presence of r001 r00 1 r00 1 longitudinal frictional hill, the rolling load required to pro- duce a given deformation is increased. At the exit plane: r00 2 ¼ p þ þ rF; or, p þ ¼ r002 À rF ð12:54cÞ 12.10.2 Cold Rolling with Back and Front Tensions ) yþ ¼ pþ ¼ r002 À rF ¼ 1 À rF ¼ y2 ðsay) ð12:54dÞ r00 2 r002 r002 Applications of ‘front tension’ and ‘back tension’ are quite common in cold-rolling operation. In a continuous rolling The integration constants, AcÀ and Acþ ; in (12.49) will mill arranged in tandem, where the work-piece is fed now be determined by inserting the boundary conditions for through several stands, it is usual to maintain an interstand tension in the work-piece to keep the work-piece straight and back and front tensions given by (12.54b) and (12.54d). also to improve the flatness and uniformity of thickness across the width of the rolled product. Even single-stand Hence, from (12.49a) and (12.54b), we get at the entry mills usually employ uncoiler and windup reel to apply, plane, where h ¼ a; and h ¼ h1: respectively, a back tension and a front tension. The appli- cation of such longitudinal tensile stress has the further AcÀ h1 exp ðÀl H1Þ ¼ yÀ ¼ y1; ð12:55aÞ advantage of reducing the rolling load, as already mentioned ) AcÀ ¼ ðy1=h1Þ exp ðl H1Þ in Sect. 12.8. This reduced rolling load further results in less wear of the rolls and improvement in roll-life. This reduction rffiffiffiffi rffiffiffiffi  rB of rolling load is evident from (12.56). R tanÀ1 R a ; and r00 1 where H1 ¼ 2 h2 h2 y1 ¼ 1 À : Figure 12.28 shows that tensile stresses are applied on the work-piece, with a value of rB; at the entry side of the stock, Again, from (12.49b) and (12.54d), we get at the exit known as ‘back tension’ and with a value of rF; at the exit plane, where h ¼ 0; and h ¼ h2: side of the stock, known as ‘front tension’. As the longitu- dinal stress rx is of compressive nature while the back and Acþ h2 exp ð þ l H2Þ ¼ Acþ h2 ¼ y þ ¼ y2; ð12:55bÞ front tensions are of tensile nature, so at the entry plane, ) Acþ ¼ y2=h2 rx ¼ ÀrB; and at the exit plane, rx ¼ ÀrF: From the yielding criterion modified by Bland and Ford for cold where rolling [see (12.45b)], it follows that rffiffiffiffi rffiffiffiffi  rF At the entry plane: r001 ¼ pÀ þ rB; or, pÀ ¼ r001 À rB R tanÀ1 R r002 ð12:54aÞ H2 ¼ 2 h2 0 ¼ 0; and y2 ¼ 1 À : h2 Now, substituting the value of AÀc from (12.55a) into (12.49a) and the value of Acþ from (12.55b) into (12.49b), we get

552 12 Rolling For the entry zone: yÀ ¼ pÀ ¼ y1 exp ðl H1Þ h exp ðÀl HÞ causing the no-slip angle to increase. The maximum possible r00  h1 reduction in height of the work-piece increases with the front rB h tension since the pulling force increases while the situation is ¼ 1 À r001 h1 exp fl ðH1 À HÞg reversed with the back tension. With the application of back and front tensions, the torque and mill power required in ð12:56aÞ rolling will also decrease since the rolling load decreases. For the exit zone: pþ  r00 y2 h rF h 12.10.3 No-Slip Angle in Cold Rolling yþ ¼ ¼ h2 exp ðl HÞ ¼ 1 À r002 h2 exp ðl HÞ ð12:56bÞ At the neutral point, where the angle of contact, h ¼ wðno-slip angleÞ; the friction hill curves for yÀ and y þ For most practical cold-rolling operations, equations intersect, i.e. y þ ¼ yÀ: If the back and front tensions are (12.56) yield reasonably accurate results, but rolling loads must be corrected for flattening of rolls by applying assumed to be present, then (12.56a) and (12.56b) can be (12.26). equated as follows, after putting the stock thickness at the Figure 12.29 shows the effects of back and front tensions neutral point, h ¼ hn; and setting H ¼ Hn; at h ¼ w: on the friction hill curve, i.e. on the distribution of roll pressure, p; over the projected length of contact, L; from the y2 hn expðl HnÞ ¼ y1 hn expfl ðH1 À HnÞg; plane of entrance, where h ¼ a; to the plane of exit, where h2 h1 h ¼ 0: The roll pressure can be reduced by either joint or individual application of back tension and front tension, as Or; exp ð2 l HnÞ ¼ y1 h2 expðl H1Þ; evident from (12.56). Since the application of back and/or  y2 h1 front tensions reduces the area under the friction hill curve,  so the rolling load is reduced. Depending on the relative y1 h2 y2 h1 magnitudes of applied back and front tensions, the neutral Or; 2 l Hn ¼ ln y2 h1 þ l H1 ¼ lH1 À ln y1 h2 point may shift causing the no-slip angle to change. The  application of only back tension not only reduces the rolling H1 1 y2 h1 load but also shifts the neutral point towards the exit plane, ) Hn ¼ 2 À 2l ln y1 h2 thereby reducing the no-slip angle. If the magnitude of only back tension is further increased, then along with the ð12:57aÞ increase in reduction of the rolling load, the neutral point will shift further towards the exit plane and ultimately reach From (12.47e): the exit plane at some critical high value of back tension, where the no-slip angle becomes zero. On the other hand, if rffiffiffiffi rffiffiffiffi  only front tension is applied the rolling load will decrease R Rw ; and the neutral point will shift towards the entry plane, thus Hn ¼ 2 rhffiffi2ffiffi tanÀ1 rhffi2ffiffiffi or; ! tan h2 Hn ¼ R w; R2 h2 Therefore, no-slip angle is rffiffiffiffi rffiffiffiffi ! w ¼ h2 tan h2 Hn R R2 ð12:57bÞ σB2> σB1 Back tension only No strip tension where the value of Hn will be given by (12.57a). Back and σB1 Front tension only 12.10.4 Cold-Rolling Load front tensions σB2 σF1 The appropriate values of the vertical roll pressure, p; Roll pressure σF2 applied to thin slab of area, w dx; is integrated over the projected length, L; of deformation zone to find the σF2> σF1 cold-rolling load, P. In this, the areas under the two arms of the pressure curves, i.e. the curves for yÀ or pÀ and y þ or θ=α Angular coordinate θ θ= 0 p þ in the friction hill, are integrated separately. Entry plane Exit plane For practical values of h and l; it has already been shown Fig. 12.29 Effect of back and front tensions on the distribution of roll in (12.45a) that the vertical roll pressure, p % pr (the radial pressure

12.10 Bland and Ford Theory of Cold Rolling 553 P Then, Ellis proposed that instead of variable r00 ; a steady value, i.e. an average value of plane strain deformation resistance or flow stress, r00 ; be used, which is given by: r00 ¼ R0Ra 0ar00dhdh ¼ Ra r00 dh ð12:58cÞ a 0 dθ O p Thus, it is renamed as ‘Bland, Ford and Ellis’ theory of θ cold rolling. Hence, if we consider the correction of the Y X rolling load due to elastic flattening of rolls, where roll radius increases from R to R0 as given by (12.26a), and Ellis’ X´ R Y´ suggestion given by (12.58c), where the mean flow stress, dx r00 , remains constant, then (12.58b) is modified to: O´ Cold-rolling load for deformed rolls, 23 ð12:59Þ Zw Za P0 ¼ w R0 r0046 y þ dh þ yÀ dh57 0w P 12.10.5 Cold-Rolling Torque Fig. 12.30 Application of vertical roll pressure p on an elemental The torque required for steady rolling will be the product of contact length of dx; where dx ¼ R cos h dh the net circumferential force in the deformation zone and the distance from the roll axis to the circumference, i.e. the roll roll pressure). Now with reference to Fig. 12.30, the radius, R: It is assumed that elastic flattening of rolls occurs cold-rolling load, P; is given by and the roll radius R increases to R0 according to (12.26a). In Fig. 12.31, consider a thin slice of metal located in the entry ZL Za ð12:58aÞ zone at an arbitrary angular position h from the line joining the P ¼ p w dx ¼ p w R cos h dh roll centres, where the thin slice makes an angle dh at the centre of rolls. If w is the stock width, the area of contact 00 between the rolls and the thin slice is w R0 dh: If pr is the radial Since h is very small, usually less than 8° in cold rolling, O and expressed in radian, we may assume that cos h % 1; in dθ (12.58a). Assuming plane-strain rolling, i.e. the stock width μ pr Pr θ w ¼ constant, and the roll radius R to be constant over the X Pr Y whole of the region of contact, the cold-rolling load from N μpr (12.58a) can be approximated as NY X Za P % w R p dh αψ 0 O´ R Za   Za p ¼ wR r00 r00 dh ¼ w R r00 y dh 20 0 3 Zw Za ¼ w R 64 r00 y þ dh þ r00 yÀ dh57 ð12:58bÞ 0w Equation (12.58b) proposed by Bland and Ford method Fig. 12.31 Friction stresses, lpr; exerted by the metal on the rolls in the entry as well as in the exit zone. Friction force in the entry zone did not produce good result, because plane strain deforma- tends to rotate the rolls in the opposite direction while that in the exit tion resistance, r00 ; varies from r001 at a contact angle h ¼ a zone assists to rotate the rolls in the forward direction of rolling to r002 at a contact angle, h ¼ 0:

554 12 Rolling roll pressure and l the coefficient of friction, each roll along integrals) and, therefore, do not give an accurate value of the arc of contact in the deformation zone experiences a torque. To find the difference between the two integrals in friction stress of l pr exerted by the metal. The friction force on rolls exerted by the thin slice at its contact with rolls in the (12.61a) and (12.61b), Ford suggested the following deriva- deformation zone is l pr w R0 dh: The friction force in the entry zone tends to rotate the rolls in the opposite direction, i.e. in tion starting from (12.44). If elastic flattening of rolls is con- the backward direction of rolling while the direction of friction sidered and R is replaced by R0 and pr % p is put in (12.44), we force is reversed in the exit zone and thus, the friction force in get d ðh rxÞ ¼ 2 R0 p ðh Ç lÞ dh; which is integrated between the exit zone assists to rotate the rolls in the forward direction limits considering back and front tensions as shown below: of rolling. In the exit zone, similar thin slab of metal along with the direction of friction stress of l pr exerted by the metal on h ¼ h1 20 13 the surface of roll has been shown in Fig. 12.31. Hence, the net circumferential force per roll in the deformation zone, rx ¼Z ÀrB Za Zw Za which is required to rotate each roll in the forward direction, is: dðh rxÞ ¼ 2 R0 64 p h dh þ l B@ p dh À p dhAC75; h ¼ h2 0 0w rx ¼ ÀrF 23 Or; Àh1 rB þ h2 rF Za Za Zw p dh57; 2 R0 l 46 Za Zw ¼ p h dh À p dh À l pr w R0 dh À l pr w R0 dh 0 w0 23 Za Zw Za w 20 3 ) l 64 p dh À p dh57 ¼ 1 Za Zw p h dh þ 2 R0 ðh1 rB À h2 rF Þ ¼ l w R0 64 pr dh À pr dh57 ð12:60Þ w0 0 w0 ð12:62Þ As l; w and R0; all remain constant during rolling, so they With the help of the above (12.62), (12.61a) changes to: are brought outside the integration sign. Although the rolls elastically deform to a larger radius R0; the distance from the For a pair of flattened rolls; ð12:63Þ surface to the centre of the roll will still be approximately Za equal to the original radius R; i.e. torque per roll = net cir- cumferential force per roll  R. According to (12.45a), MT ¼ 2 w R R0 p h dh þ w R ðh1 rB À h2 rFÞ considering that the radial roll pressure, pr % p (the vertical roll pressure), it follows from (12.60) that the torque for each 0 elastically flattened roll is: If we use Ford, Ellis and Bland method, then (12.63) takes the following form: 23 Za Zw For a pair of flattened rolls; Za l w R R064 p dh À p dh75 MT ¼ 2 w R R0 r00 y h dh þ w R ðh1 rB À h2 rFÞ w0 0 If MT ¼ total cold-rolling torque for a pair of rolls, then ð12:64Þ For a pair of elasti2cally flattened roll3s, ð12:61aÞ 12.10.6 Maximum Allowable Back Tension Za Zw Considering elastic flattening of rolls and replacing R by R0; MT ¼ 2 l w R R064 p dh À p dh75 w0 (12.57b) becomes: In case of undeformed rolls, R0 in (12.61a) will be rffiffiffiffi rffiffiffiffi ! replaced by R; and hence, total cold-rolling torque for a pair No-slip angle, w ¼ h2 tan h2 Hn ; R0 R0 2 of undeformed rolls is given by 23 1  Za Zw H1 2l y2 h1 ð12:61bÞ where Hn ¼ 2 À ln y1 h2 ; as given by (12.57a). MT ¼ 2 l w R264 p dh À p dh75 w0 When back tension, rB, is increased, the neutral plane, NN0; in Fig. 12.28, shifts towards the exit plane YY0 and if the Ford pointed out that the above (12.61a) and (12.61b) involve the difference between two nearly equal areas (the two value of the back tension, rB; is excessive then the neutral plane NN0 coincides with the exit plane YY0; which means the

12.10 Bland and Ford Theory of Cold Rolling 555 neutral angle, w ¼ 0: The effect of back tension on the neutral neutral point at the exit plane, but this method of estimating point and the neutral angle is evident from Fig. 12.29. If neutral angle, w ¼ 0; the exit zone vanishes and the whole l is laborious. From (12.58b), the rolling load per unit width deformation zone becomes the entry zone, where the roll velocity is greater than the stock velocity over the whole arc of the stock can be written as: of contact, and as a result, ‘skidding’ occurs. When skidding occurs, the rolls rotate but the stock does not move forward, Za ð12:58dÞ leading to unnecessary loss of energy. Therefore, skidding is P ¼ R p dh to be avoided at all cost, for which the neutral angle w must be w positive, i.e. greater than zero. Hence, 0 rffiffiffiffi rffiffiffiffi ! When the neutral point lies at the exit plane, the no-slip h2 h2 Hn angle w ¼ 0; and the frictional force acts only in the entry w [ 0; or; R0 tan R0 2 [ 0; when Hn [ 0; zone and entirely in one direction. The friction force on rolls exerted by the stock in the entry zone tends to rotate the rolls  in the backward direction of rolling. If MT is the total torque i.e.; H1 [ 1 ln y2 h1 y2 h1 for a pair of rolls, the torque per roll per unit width of the 2 2l y1 h2 ; or; exp ðl H1Þ [ y1 h2 ; stock, which is required to rotate the rolls in the forward direction of rolling, is: ) y1 [ y2 h1 exp ðÀl H1Þ ð12:65Þ MT =2 w ¼ circumferential force per roll per unit width h2 of stock in the entry zone 2 Za 3 As per Bland, Ford and Ellis, putting r001 ¼ r00 ; in Za (12.54b) and r002 ¼ r00; in (12.54d), we get y1 ¼  R ¼ 4R l p dh5  R ¼ l R2 p dh ¼ l R P 1 À rB=r00 ; and y2 ¼ 1 À rF=r00: Hence from (12.65), it w follows: 00 ð12:61cÞ rB  r00 rF h1 1 À [ 1 À r00 h2 exp ðÀl H1Þ; ) The coefficient of friction is:  rF h1 rB l ¼ MT Or; 1 À 1 À r00 h2 exp ðÀlH1Þ [ r00 ; 2RP ð12:67Þ Or, rB\\r00 À Àr00 À Á h1 exp ðÀl H1Þ; For actual measurement of l; the stock is subjected to a rF h2 ! back tension, which causes the neutral point to shift towards the exit plane and this back tension is increased gradually h1 h1 ! until the exit velocity of the stock equals the peripheral h2 h2 velocity of rolls making the forward slip zero, where the Or; rB\\r00 1 À exp ðÀl H1Þ þ exp ðÀl H1Þ rF neutral plane coincides with the exit plane. At this point, the torque and the rolling load at constant reduction and roll speed ð12:66aÞ are measured simultaneously to obtain l using (12.67). h1 !! h2 h1 ) ðrBÞmax¼ r00 1 À exp ðÀl H1Þ þ h2 exp ðÀl H1Þ rF ð12:66bÞ The above (12.66b) gives the formula for the maximum 12.11 Sims’ Theory of Hot Rolling allowable back tension. It is obvious that the maximum allowable back tension increases as the applied front tension The derivation given in Sect. 12.10 assumes that the coef- increases. ficient of friction is low and that the deformation is homo- geneous. Since the coefficient of friction in hot rolling is Now the question is: ‘can we increase the front tension high, the derivation in Sect. 12.10 leads to errors in calcu- indefinitely?’ lating the hot-rolling load and particularly large errors in the prediction of pressure distribution. Experimental finding by The applied front tension must not exceed the yield stress Siebel and Lueg (1933b) shows that in hot rolling, the of the work-piece in tension, so that the work-piece does not coefficient of friction is not constant at all points on the arc of tear under the action of the front tension. contact and the deformation is not homogeneous. The remaining important assumptions considered for hot rolling 12.10.7 Estimation of Friction Coefficient are: plane strain condition, constant circular arc of contact (Whitton and Ford 1955) and negligible elastic deformation of rolls. It is possible to estimate the coefficient of friction, l; when A number of methods to calculate the hot-rolling load the rolling reaches the limiting condition having location of have been developed over the years, but Sims’ theory (Sims

556 12 Rolling 1954) is frequently considered in hot rolling to determine the With the above assumptions, (12.68) can be written as rolling load. Von Kármán differential equation of friction hill follows: is the starting point of Sims’ theory. Considering flattening  r00 pr 2 of rolls due to elastic deformation and replacing the roll h drx þ rx dh ¼ 2 R0 h Ç dh ð12:70Þ radius, R by the radius of curvature of flattened rolls, R0, Von Karman differential equation, (12.41a) becomes: dðhrxÞ ¼ From (12.69b), 2R0ð pr sin h Ç s cos h Þ dh; where the upper negative sign dpr À drx ¼ 0; as r00 is constant: ) drx ¼ dpr ð12:71Þ ‘−’ applies for the entry zone and the lower positive sign, ‘+’ Again as applies for the exit zone. In hot rolling, the coefficient of friction is so high that h ¼ h2 þ R0 h2; ) dh ¼ 2 R0 h dh ð12:72Þ l pr ! r00 =2 ¼ k; where r00 is the plane-strain flow stress or deformation resistance, and k is the shearing yield stress. Under such circumstances, the stock surface in contact with With the help of (12.71) and (12.72), (12.70) is modified as follows: the rolls sticks to the roll surface, i.e. a condition of sticking friction prevails at the interface between the rolls and the h dpr þ rx 2 R0 h dh ¼ 2 R0 pr h dh Ç R0r00 dh; ð12:73Þ stock. Here, frictional stress, s ¼ r00 =2 ¼ k: In general, as Or, h dpr ¼ 2 R0 h ðpr À rxÞ dh Ç R0 r00 dh suggested by ‘Orowan’, s ¼ l pr; if l pr\\r00 =2; or s ¼ r00 =2; if l pr ! r00=2: Thus, assuming sticking frictional Now, using (12.69b) we get from (12.73): condition in hot rolling: p  dðh rxÞ ¼  sin h Ç r00  ð12:68Þ h dpr ¼ 2 R0 h r00 dh Ç R0 r00 dh 2 R0 pr 2 cos h dh ! 4 ð12:74Þ ð12:75Þ ) dpr ¼ R0 fðp hÞ=2g Æ 1 dh r00 h Orowan, using an analogy between rolling and plastic compression of a mass between rough non-parallel plates, Let pr=r00 ¼ y; since r00 is constant, for which Nadai (1931) had obtained a solution, suggested that a modified yielding criterion could be used for rolling as ) À =r00 Á ¼ dpr =r00 ¼ dy shown below: d pr pr À rx ¼ m r00 ð12:69aÞ Using (12.75) and substituting h ¼ h2 þ R0 h2 into (12.74), and then integrating we get: The value of m depends on the ratio l pr=k: When l ¼ 0; Z pZ 2 R0 h dh Z R0 dh the value of m of course is unity, but when the value of l is 4 Zh2 þ R0 h2 so high that l pr ¼ k; the value of m falls to p=4: Thus, dy ¼ ¼p 2 Ç h2 Zþ R0 h2 (12.68) is considered in Sims’ theory with the following h2 R0 1 yielding criterion applicable to hot-rolling conditions, where 4 R0 h dh Ç dh the replacement of m by the factor p=4 in (12.69a) is due to h2 þ R0 Orowan: þ R0 h2 h2 ) y ¼ rp4 ffilRffinffiffi0Àhta2nþÀ1R0rh2ffiRÁffiffiffi0 h2 Ç h2 h2 ðintegration pr À rx ¼ ðp=4Þ r00 ð12:69bÞ  þ Ch constant) h where pr is the radial roll pressure (compressive), rx is the ) For entry zone; longitudinal compressive stress, and r00 is the deformation rffiffiffiffi tanÀ1rffiRffiffiffi0  resistance (compressive) of the work-piece being homoge- yÀ ¼ p ln h À R0 h2 h þ ChÀ ð12:76aÞ 4 h2 nously deformed under plane strain condition. The assumptions in Sims’ theory are: 1. The plane strain flow stress r00 is constant over the whole And for exit zone; arc of contact. rffiffiffiffi rffiRffiffiffi0  yþ ¼ p ln h þ R0 tanÀ1 h þ Chþ ð12:76bÞ 2. Angle of contact, h; is small. 4 h2 h2 ) sin h % h; cos h % 1; and h ¼ h2 þ R0 h2: The integration constants, ChÀ and Chþ , are to be deter- mined from the following boundary conditions:

12.11 Sims’ Theory of Hot Rolling 557 Let us assume that at the entry plane, where the angle of 12.11.1 No-Slip Angle in Hot Rolling contact h ¼ a; the stock thickness is h ¼ h1; and at the exit plane, where the angle of contact, h ¼ 0; the stock thickness At the neutral point, where the angle of contact, h ¼ is h ¼ h2: Since the longitudinal stresses at the entry and exit planes, which are free surfaces, must be zero, i.e. rx ¼ 0; so w ðno-slip angleÞ; the friction hill curves for yÀ and y þ from the hot-rolling yielding criterion due to Orowan [see intersect, i.e. yÀ ¼ y þ : So, (12.79a) and (12.79b) can be (12.69b)], it follows that equated as follows, after putting the stock thickness at the neutral point, h ¼ hn; and setting H ¼ Hn; at h ¼ w: At the entry plane: prÀ ¼ p r00 ; ) yÀ ¼ prÀ ¼ p ð12:77aÞ p þ p  þ rffiffiffiffi tanÀ1 rffiRffiffiffi0  À tanÀ1rffiRffiffiffi0 ! 4 r00 4 ln hn R0 a w 4 4 h1  h2 rffiffiffiffi tanÀ1h2rffiRffiffiffi0  h2 ¼p R0 w p r00 ; prþ ¼p þ p hn þ prþ 4 ) yþ r00 4 ln At the exit plane: ¼ ¼ ð12:77bÞ 4 4 h2 h2 h2 The value of yÀ [(12.77a)], h ¼ a; and h ¼ h1; are sub- ) rffiffiffiffi tanÀ1rffiRffiffiffi0  R0 w stituted into (12.76a) and the integration constant for entry 2 h2   rh2ffiffiffiffi plane, ChÀ, is determined as follows: p h2 R0 tanÀ1rffiRffiffiffi0  ln a p p hp41lnÀhr1 þhffiRffiffi2ffi0rtahffiRnffiffi2Àffi0 t1anÀr1hffiRffiffi2ffi0raffihRffiffi2ffi0 ¼ 4 h1 þ h2 h2 ð12:80Þ 4 4 ) ¼ ln þ ChÀ ð12:78aÞ pffiFffiffiffioffiffiffirffiffiffieffiffilffiaffiffisffiffitffiiffifficffiffiaffiffilffily flattened roll radius R0; from (12.8b), a ¼ ChÀ ¼ p À  ðh1 À h2Þ=R0: Substituting this value of a into (12.80), we 4 a get Similarly, the value of y þ [(12.77b)], h ¼ 0; and h ¼ h2; 2 rffiffiffiffi tanÀ1rffiRffiffiffi0  R0 w are substituted into (12.76b) and the integration constant for exit plane, Chþ is determined as follows: h2 h2 rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi !  rffiffiffiffi R0 h1 À h2 ln h1 p ¼ p ln h2 þ Chþ ; ) Chþ ¼ p À p ln h2 ð12:78bÞ h2 R0 À p 4 4 4 4 ¼ R0 tanÀ1 4 h2 h2 tanÀ1rffiRffiffiffi0 The value of ChÀ from (12.78a) and that of Chþ from Or;  (12.78b) are substituted, respectively, into (12.76a) and w h2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi   (12.76b) to obtain the entry zone (12.79a) and the exit zone ¼ 1 tanÀ1 h1 À h2 p rhR20ffiffilffinffiffiffiffiffiffihhffiffi21ffiffiffi 2 rffiffiffiffi h2 À 8 (12.79b) as follows: p rffiffiffiffi tanÀ1rffiRffiffiffi0  h2 1 h1 À h2  R0 h R0 h2 For entry zone: yÀ ¼ ln h À ) No-slip angle, w ¼ tan 2 tanÀ1 4 h2 tanhÀ21rffiRffiffiffi0  # þ pÀ p rffiffiffiffi  Àp rffiffiffiffi p 4 4 ln h1 R0 a 8 h2 h1 þp þ R0 h2  rh2ffiffiffiffi hr2 ffiRffiffiffi0 ln h R0   ln a ) yÀ ¼ 4 tanÀ41rhffiRffiffi1ffi0 þ h2 tanÀ1 h2 À ! ð12:81Þ h h2 ð12:79aÞ For exit zone: yþ ¼ rffiffiffiffi tanÀ1 rffiRffiffiffi0ffi  12.11.2 Hot-Rolling Load p ln h þ R0 h 4 h2 h2 Similar to cold rolling, hot-rolling load is also given by (12.58b), but the only difference is that the plane strain flow pp stress r00 in hot rolling is not likely to increase from the entry þ À ln h2 to the exit plane, which occurs in cold rolling due to strain 4 4  hardening of metal. Hence, assuming r00 to remain constant in hot rolling over the whole arc of contact: ) yþ ¼ p þ p h ln 4 4 tanÀh12rffiRffiffiffi0 þ rffiffiffiffi  ð12:79bÞ h R0 h2 h2

558 12 Rolling Hot rolling load; Remember that for elastically flattened rolls, h ¼ 23 h2 þ R0 h2; in which h2 and RZ0 are constants, and that h ¼ h1 Za Zw Za at h ¼ a: Now, the p treated P ¼ w R r00 y dh ¼ w R r00 46 y þ dh þ yÀ dh57 a term 0 0w 40 ln h dh of (12.83) is with integration by parts, as follows: Now for elastically flattened rolls, from the exit zone Za ZZ & Z ' !a (12.79b): pp ln h dh À d ln h dh dh ln h dh ¼ dh 0 44 Zw Zw   rffiffiffiffi tanÀ1rffiRffiffiffi0 ! yþ dh ¼ p p ln h R0 h dh 0 Z   !a h ln h À 2 þ þ ¼p R0 h h dh 4 4 h2 h2 h2 4 h 00 2  0 3 p 4a dh5 pp p Zw 4 Za h2 þ R0 h2 À  h2 þ R0 h2 h2 ¼ 4 w À 4 w ln h2 þ 4 ln h dh ¼ ln h1 À 2 0 2 0 3 rffiffiffiffi Zw rffiRffiffiffi0  p Za Za dh 5 þ R0 h 4 h2 þ R0 h2 tanÀ1 dh ¼ 4a ln h1 À 2 dh þ 2 h2 h2 h2 0 0 0 ð12:82aÞ 2 Za 3 And from entry zone (12.79a) for elastically flattened ¼ p 4a ln h1 À 2 a þ 2 dh 5 rolls: 4 1 þ ðR0=h2Þ h2 p \" r0 ffiffiffiffi tanÀ1rffiRffiffiffi0 # 4 a h2 h2 a Za Za p p  h  rffiffiffiffi rffiRffiffiffi0  ¼ ln h1 À 2aþ2 R0 yÀ dh ¼ ln R0 a þ þ tanÀ1 4 4 h1 h2 h2 ð12:84aÞ ww tanÀ1rffiRffiffiffi0 rffiffiffiffiffi ! À R0 h dh R rffiRffiffiffi0  h2 h2 tanÀ1 h dh; Again, the term of (12.83) is trea- Za h2 p p p ¼ 4 ða À wÞ À 4 ða À wÞ ln h1 þ 4 ln hdh ted with integration by parts, as follows: rffiffiffiffi tanÀ1rffiRffiffiffi0  w Z tanÀ1rffiRffiffiffi0  tanÀ1rffiRffiffiffi0  Z R0 a h h þ ða À wÞ dh ¼ dh h2 Z &tanÀ1rffiRhffiffi2ffi0 h2 h2 À ' Z ! h dh dh rffiffiffiffi Za tanÀ1rffiRffiffiffi0  d R0 h dh À ð12:82bÞ dh ÀÀÀrrZrhhffiffiRRffiffirffiffi22ffiffi00ffiffiffih2ffihffiRhffi2ffi2Rffi02hZ012Zþh2hð2R2þhRþ10d=R0hhhR02h0dÞ2hhh22 h2 h2 rffiRffiffiffi0  w tanÀ1 h Now, addition of (12.82a) and (12.82b) gives: ¼ h tanÀ1rffiRhffiffiffi02h h dh ¼ h Zw Za pp p ¼ h tanÀ1rffihRffiffi2ffi0 ¼ h y þ dh þ yÀ dh ¼ 4 a À 4 w ln h2 À 4 ða À wÞ ln h1 ¼ h  h 0w rffiffiffiffi tanÀ1rffiRffiffiffi0  þ R0 ða À wÞ a tanÀ1rffihRffiffi2ffi0  h h2 h2 À1 h2 À þ R0 h2Á 2 R0 ln h2 Za tanÀ1rhffiRffiffi2ffi0 þ p ln hdh  À1 rffiffiffiffi h 2 h2 4 R0 ln h 0 2Zw rffiRffiffiffi0  Za tanÀ1rffiRffiffiffi0 3 h2 4 h2 h dh h2  þ rffiffiffiffi tanÀ1 À h dh75 ð12:84bÞ R0 h2 0w It is known that at the neutral plane, where the angle of contact, h ¼ w; (the no-slip angle), the stock thickness is ð12:83Þ h ¼ hn; at the entry plane, where h ¼ a; h ¼ h1; and at the exit plane, where h ¼ 0; h ¼ h2: Considering these, it fol- lows from (12.84b):

12.11 Sims’ Theory of Hot Rolling 559 23 23 rffiffiffiffi Zw tanÀ1rffiRffiffiffi0  Za tanÀ1rffiRffiffiffi0  rffiffiffiffi Zw tanÀ1rffiRffiffiffi0  Za tanÀ1rffiRffiffiffi0  R0 64 h dh h dh57 R0 46 h dh h dh75 À À h2 h2 h2 h2 h2 h2 0w 0 \"rffiffiffiffi w rffiffiffiffi taantaÀn1À1rrhffiRffiffi2ffi0ffiRffiwffiffi0 aÀÀ12rln ffiRhffiffinffi0 ! w tanÀ1 R0 rffiffiffiffi R0 1 rffiffiffiffi h2 & tanÀ1rffiRffiffiffi0ffia ¼ w þ 2 ln h2 ¼ R0 h2 R0 1 rh2ffiffiffiffi 2 tan À R0 h2 2 h2 w tanÀ1rffiRffiffiffi0  rffiffiffiffi  )# w Àp h2 h1 h2 ! h2 h2 h2 8 R0 ln h2 À 1 ln h1 þ 1 ln hn rffiffiffiffi tanÀ1rffiRffiffiffi0  pffiffiffiffiffiffiffiffiffi 2 2 R0 aþ h1h2 À a ln rffiffiffiffi tanÀ1rffiRffiffiffi0  rffiffiffiffi tanÀ1rffiRffiffiffi0  rffiffiffiffih2 rffiRffiffiffi0h2 hn  ¼ R0 w w À R0 a a R0 a ln h1 w 2 p hp2 ffiffiffiffiffiffiffiffiffi h2 h2 h2 tanÀ1 ¼ À w rh2ffiffiffiffi rh2ffiRffiffiffi0 4 pffiffiffihffiffi2ffiffiffiffi þ ln h1h2 R0  þ h1h2 hn tanÀ1 a À a ln h2 h2 hn ð12:84cÞ where hn ¼ h2 þ R0 w2: For elastically flattened roll radius ð12:84dÞ R0; we get from (12.8a), h1 À h2 ¼ R0 a2: Remembering this and substituting the value of w from (12.81) into (12.84c), Hence, with the help of (12.84a) and (12.84d), (12.83) is modified as follows: we get

560 12 Rolling Therefore, from (12.85), we get 12.11.3 Hot-Rolling Torque Hot-rolling load for elastically deformed rolls, P0 ¼ w R0 r00 Ra y dh Similar to cold rolling, the torque required for steady rolling will be the product of the net circumferential force in the 0 deformation zone and the distance from the roll axis to the circumference, i.e. the roll radius, R: It is assumed that ¼ w r00 R0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi \" pffiffiffiffi rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  elastic flattening of rolls occurs and the roll radius R h1 À h2 p pffiffiffiffiffiffiRffiffiffi0ffiffiffiffi h2 tanÀ1 h1 À h2 increases to R0 according to (12.26a). With reference to R0 R0 Fig. 12.31, the interfacial friction force on rolls exerted by a 2 h1 À h2 h2 thin slice of metal located at an arbitrary angular position h in the deformation zone and making an angle dh at the centre À rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ! of rolls will be s w R0 dh; where s is the tangential shearing R0 ln pffihffiffiffinffiffiffiffiffi À p friction stress and w the stock width. The friction force in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi entry zone tends to rotate the rolls in the opposite direction, w r00 R0 ðh1 À h2Þ ph1ffiffiffiÀffi h2 hr1 ffihffiffi2ffiffiffiffiffiffiffiffiffiffiffi4 i.e. in the backward direction of rolling, while the direction ppffiffiffiffiffiffihffiffi2ffiffiffiffiffi tanÀ1 h1 À h2 of friction force is reversed in the exit zone and thus, the ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi À2 rhhffiffi1ffi1ffiffiÀrffiRÀffiffiffi0ffihffiffihffiffi2ffiffiffi2ffiffiffilffiffinffiffiffiffiffipffiffihffihffiffi1ffinffiffihffiffi2ffi h2 friction force in the exit zone assists to rotate the rolls in the À forward direction of rolling. Hence, the net circumferential ! force per roll in the deformation zone, which is required to p rotate each roll in the forward direction, is: 4 ¼ w r00 L p h2 tanÀ1 h1 À h2 2 À h1 h2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2   ! p À R0 ln pffihffiffinffiffiffiffiffiffi À 4 h1 À h2 h1 h2 ð12:86aÞ ¼ w r00 L QS ð12:86bÞ Za Zw s w R0 dh À s w R0 dh Mean specific pressure for elastically flattened rolls is: w 02 3 pm ¼ rolling load ¼ P0 r00 Za Zw dh57 ð12:88Þ contacrt ffiaffiffirffiffieffiffiaffiffiffibffiffiffieffiffitweenrrollffiffiffiaffiffinffiffiffidffiffiffiffiffisffiffitock wL 2 64 ¼ w R0 dh À ¼Àrr00 hffip2ffiffi1ffiffiffiRÀffiffiffi0hffiffih1ffiffi2ffihÀl2nh2ptaffihhnffiffi1nffiÀffihffi1ffi2ffiffi h1 À h2 w0 Àp !h2 ¼ w R0 r00 ½ða À wÞ À wŠ 4 2 ¼ r00QS ð12:87aÞ where Since s ¼ k ¼ r00=2; and as r00 ; w and R0; all remain constant during rolling, so they are brought outside the rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  p h2 tanÀ1 h1 À h2 integration sign. Although the rolls elastically deform to a QS ¼ 2 rhffiffi1ffiffiffiÀffiffiffiffiffihffiffiffi2ffi  h2 larger radius R0; the distance from the surface to the centre of the roll will still be approximately equal to the original À R0 ln pffihffiffiffinffiffiffiffiffi À p ð12:87bÞ radius R; i.e. torque per roll ¼ net circumferential force per roll  R: If MT is the total hot-rolling torque for a pair of h1 À h2 h1 h2 4 rolls, it follows from (12.88) that the hot-rolling torque for a in which hn ¼ h2 þ R0 w2: pair of elastically flattened rolls is: In case of undeformed rolls, hot-rolling load, P; mean For a pair of flattened rolls; !  R ¼ w R R0 r00 ða À 2wÞ specific pressure, pm; L and QS can be obtained by replacing MT ¼2 w R0 r00 ða À 2 wÞ R0 with R in (12.86) and (12.87). 2 On the basis of Sims’ formulae for roll pressure distri- ð12:89Þ bution, Larke (1963) has computed the values of Q against percentage reduction in area for different ratios of R=h2; i.e. In case of undeformed rolls, R0 of (12.89) will be replaced roll radius to thickness of outgoing rolled product. The by R; and, hence, total hot-rolling torque for a pair of conclusion of Larke is that results obtained by Sims’ func- undeformed rolls is given by tion are somewhat more accurate than those obtained from MT ¼ w R2 r00 ða À 2wÞ ð12:90Þ either (12.38) proposed by Ekelund or a similar correcting function suggested by Orowan and Pascoe (1946).

12.11 Sims’ Theory of Hot Rolling 561 12.11.4 Limitations of Sims’ Theory length of deformation zone, L; to the speed of rolls, vr; i.e. L=vr: Hence, the average true strain rate will be: • Sims’ theory assumes that spread in negligible, i.e. plane strain condition prevails during rolling. Thus, this theory e_ ¼ lnðh1=h2Þ ¼ vr ln h1 ð12:91Þ is applicable only to hot rolling of wide strips, where L=vr L h2 spread is small. where h1 ¼ the thickness of stock before rolling and h2 ¼ • Sims’ theory assumes sinh % h; cos h % 1; etc. In other the thickness of stock after rolling, and the strain, e ¼ words, it assumes a small angle of contact. Thus, Sims’ theory is applicable only to hot rolling of wide strips plnðffihffiffiffi1ffiffi=ffiffihffiffi2ffiffiÞffiffi;ffiffiffiffitffihffiffieffi projected length of the aprcffiffioffiffiffifffiffifficffiffioffiffiffinffiffitffiffiaffifficffiffitffi,ffiffiffiLffiffiffiffi¼ffiffi where the angle of contact seldom exceeds 15°. R ðh1 À h2Þ; or more accurately, L ¼ R Dh À ðDh2=4Þ; • Sims’ theory also uses the yielding criterion pr À rx ¼ and the speed of rolls per second, vr ¼ 2 p R n; in which R ¼ ðp=4Þr00 ; which was proposed by Orowan on the basis of the roll radius and n ¼ revolutions of rolls per second, i.e. n a study by Nadai on plastic compression of a mass is expressed in Hz. between rough non-parallel plates. However, Alexander prefers the use of pr À rx ¼ r00 as yielding criterion. 12.12 Lever Arm Ratio, Roll Torque and Mill Power 12.11.5 Mean Strain Rate The work-piece is compressed by each roll with a rolling Apart from less well-defined frictional condition and inho- load equal to P: Similarly, the work-piece exerts a separating mogeneous deformation in hot rolling, the other difficulty is force on each roll with the same magnitude of rolling load; that the flow stress of metal in hot rolling depends upon both i.e., the roll-separating force is also equal in magnitude to P; temperature and strain rate (speed of rolls) as in other hot but it acts in a direction opposite to that of rolling load. Thus, working processes. It must be noted that the strain-rate the roll-separating force tends to rotate the rolls in a direction sensitivity of the flow stress (see Sect. 1.11.1 in Chap. 1) is opposite to the forward direction of rolling. Each roll is high at elevated temperature (it is negligible at room tem- therefore subjected to a turning moment ¼ P a; where P ¼ perature) and increases with increasing temperature. There- roll-separating force, and a ¼ the perpendicular distance fore, the flow stress of metal increases with increasing strain between the line of joining the roll centre and the line of rate or rolling speed at elevated temperature, resulting in the action of the resultant roll-separating force, known as the requirement of a higher specific pressure for rolling. Hence, lever arm (or, moment arm), as shown schematically in in calculating rolling load, one must obtain the flow stress of Fig. 12.32. An examination of the distribution of total roll- the material corresponding to the strain rate prevailing at the ing load over the arc of contact in the typical friction-hill time of hot rolling. Before the flow stress is used to calculate pressure distribution indicates that the resultant of the hot-rolling load, the strain rate in the actual rolling roll-separating force acts at a point lying between the point process must be calculated. of entry and the point of exit; i.e., the lever arm a is some fraction of projected length of arc of contact, L; which is For equal percentage of reduction, the strain rate of a denoted by k, and this fraction k is called the lever arm ratio. thinner work-piece will be much greater than that of a However, the motor must apply a torque ¼ P a to each roll thicker one. As the metal passes through the deformation to keep it rotating in the forward direction. zone, the strain rate varies, being the maximum at the entry plane and decreasing to zero at the exit plane. If it is desired Hence; torque for each work roll ¼ P a ¼ P ða=LÞ L ¼ P k L to obtain the deformation resistance of metal from the graphs produced by Alder and Phillips (1954), it is necessary to Since there are two work rolls, the torque (for a pair of calculate a mean strain rate in rolling. Since hot-rolling work rolls) is given by processes involve sticking friction, only the equation for mean strain rate in sticking friction has been given below. MT ¼ 2 P a ¼ 2 P k L ð12:92Þ The mean true strain rate, e_; in flat rolling with sticking where a ¼ the pleffiffivffiffiffieffiffirffiffiffiffiffiffiffiaffiffirffimffiffiffiffi,ffi L ¼ projected length of friction may be calculated by dividing the strain by the time deformation zone¼ R ðh1 À h2Þ; and the lever arm ratio required for a stock to undergo this strain in the deformation k ¼ a=L: A typical value of k is about 0.5 in hot rolling and zone of rolls. The time over which the effective rolling is about 0.45 for most cold rolling. If the roll radius is carried out can be approximated from the ratio of projected

562 12 Rolling k ¼ w R ÀR0 r00 ½ða=2ÁÞ À wŠ ¼ R R0 ½ða=2Þ À wŠ w r00 L QS L QS ðLÞ2 Torque, MT = 2Pa ¼ R R0 ½ða=2Þ À wŠ a QS R0 ðh1 À h2Þ R  ! P R ða=2Þ À w ð12:93bÞ ) Lever arm ratio; k ¼ h1 À h2 QS h1 h2 where P rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R p h2 tanÀ1 h1 À h2 QS ¼ 2 rhffiffi1ffiffiffiÀffiffiffiffiffihffiffi2ffiffi  h2 a À R0 ln pffihffiffinffiffiffiffiffiffi À p ð12:87bÞ h1 À h2 h1 h2 4 Fig. 12.32 Lever arm, 0a0; and roll torque Thus, the lever arm ratio k in hot rolling can be easily evaluated from (12.93b), if the thicknesses of the incoming increased from R to R0 due to elastic flattening of rolls, P will stock h1; and outgoing rolled product h2; the radii of unde- pbeffiffiffiffirffiffieffiffipffiffilffiffiaffifficffiffieffiffidffiffiffiffiffi by P0 and L will be calculated from formed rolls R; and elastically flattened rolls R0 are known, because other parameters in (12.93b) are functions of h1; h2; R0 ðh1 À h2Þ: and R or R0: Once the lever arm ratio k is evaluated by either of (12.93), the lever arm (or moment arm) a of elastically flat- tened rolls in hot rolling can easily be calculated from a ¼ k L: 12.12.2 Mill Power 12.12.1 Estimation of Lever Arm Ratio Power is usually supplied to a rolling mill by an electric from Sims’ Theory motor to overcome the mill torque and to drive the mill. The mill power supplies the energy for the deformation of metal, Based on the work of Sims, the lever arm ratio in hot rolling to overcome the frictional forces in the bearings and to can be determined, the calculation of which is straightfor- accommodate the energy losses in the motor and ward and given below. Let us consider elastic flattening of transmission. rolls where the radii of rolls have increased from R to R0: According to Sims’ theory of hot rolling, the torque for a Figure 12.32 shows that the resultant roll-separating force pair of elastically flattened work rolls given by (12.89) is P moves along the circumference of a circle equal to 2 p a; repeated below for convenience: during one revolution of a single roll. So, the work done to turn a single roll one revolution ¼ 2 p a P ¼ p MT [see MT ¼ w R R0 r00 ða À 2wÞ ð12:89Þ (12.92)], where MT is the torque for a pair of work rolls. Since two work rolls are involved in rolling, hence Equating (12.89) with (12.92) for MT ; considering replacement of P by P0 in (12.92) for elastically flattened For a pair of rolls, rolls, we get work per revolution ¼ 2 ð2 p a PÞ ¼ 2 p MT J ð12:94Þ 2 P0 k L ¼ w R R0 r00 ða À 2wÞ; a  where P is in newtons, a is in metres, and MT are in 2 w newtons-metres, i.e. in Joules. The power is defined as the ) Lever arm ratio; k ¼ w R R0 r00 À ð12:93aÞ work done per unit time and expressed in JsÀ1; i.e. in watts P0 L (W). If the rolls revolve at n Hz (revolutions per second), then the power (in watts) required to operate a pair of rolls is From (12.86b), hot-rolling load fopr effiffilffiffiaffiffisffitffiffiifficffiffiaffiffilffiffilffiyffiffiffiffiflffiffi attened given by rolls is: P0 ¼ w r00 L QS; where L ¼ R0 ðh1 À h2Þ: Now, substituting P0 into (12.93a), we get WR ¼ 4 p a P n ¼ 4 p a P ðN=60Þ ¼ 2 p MT ðN=60ÞW ð12:95Þ

12.12 Lever Arm Ratio, Roll Torque and Mill Power 563 where the roll speed N is measured in revolutions per min- friction hill and a corresponding increase in rolling load. Under plane strain condition as the stock width, w; does not ute. Equation (12.95) expresses the power needed to deform change and if the roll radius, R; and the draft, Dh; are kept constant, the lenpgffitffihffiffiffiffiffioffiffif the deformation zone, L; remains the metal as it travels through the deformation zone of constant as L ¼ R Dh; (12.11b), so the contact area ¼ w L; on which the frictional stress acts will also remain constant. rolling. In addition, the power is required to overcome the But as the ingoing stock thickness, h1; is decreased, the ingoing cross-sectional area of the metal ¼ w h1; will friction at each of the roll neck bearings, which support the decrease, resulting in an increase in the average longitudinal stress, which in turn increases the roll pressure and rolling rolls. load. Thus, with reduction in the stock thickness an increasingly high rolling load is required for a given draft, Assuming that each roll is supported by two bearings, the which is similar to the requirement of higher deformation load carried by each bearing will be equal to P=2; and the load for metal of greater hardness. So when a particular frictional drag caused by each bearing will be lb P=2; where thickness of the metal is reached, the rolling load becomes so lb is the coefficient of friction at the bearing. If db is the high that the rolls will deform more easily than the metal. bearing diameter (measured in metres), each bearing will require a torque equal to ðlb P=2Þ Â ðdb=2Þ ¼ lb P db=4: Any parameter that causes an increasingly high rolling Since there will be four bearings for a pair of rolls, so the load requirement for deformation of metal will limit the total torque required will be lb P db N m or J: Hence, the thickness of the ingoing stock. The main factors for total power (in watts) required for four bearings is given by increasing rolling load are increased mean flow stress or deformation resistance of the work-piece, r00 ; higher coeffi- WB ¼ lb P db x ¼ lb P db 2 p n ¼ lb P db ð2 p N=60ÞW cient of friction, l; larger area of contact between the ð12:96Þ work-piece and the rolls and lower elastic modulus of the rolls, Er; resulting in more elastic flattening of the rolls that where x ¼ the angular speed of rolls in rad=s: Usually, the causes an increased roll radius to R0, (see Sect. 12.7.1), and a power needed for the roll-neck bearings is small, because the corresponding increase in the contact area. Again, the area of typical values of the coefficient of friction of well-lubricated contact between the work-piece and the roll increases with bearings (Underwood 1943) range only from 0.002 to 0.01. the length of the deformation zone, L; which, in turn, increases witph ffiiffiffinffifficffiffirffiffiease of the roll radius, R; and/or the draft, Hence, the total power requirement for two rolls with four Dh; as L ¼ R Dh: Obviously, with the reduction of draft bearings will be and/or using smaller roll radius and/or with higher elastic modulus of the roll material, the load is reduced and a WT ¼ WR þ WB ¼ ðMT þ lb P dbÞ ð2 p N=60ÞW ð12:97Þ thinner rolled product can be achieved, but below a limiting thickness no further reduction is possible. In fact, the min- The energy losses in the motor and transmission may be imum or limiting thickness, hmin; below which the included in the overall power requirement of the mill by work-piece cannot be reduced further, is very nearly pro- factors representing their efficiencies, gm and gt respectively. portional to these following parameters: Hence, the overall power requirement of the mill will be WM ¼ WT ¼ 1 ðMT þ lb P dbÞ 2 pN W ð12:98Þ gm gt gm gt 60 12.13 Minimum Thickness in Rolling When a thin hard work-piece is rolled, it is found that it hmin / Cr l R r00 ð12:99aÞ cannot be reduced below a certain limiting thickness. Further rolling to bring thickness below the limiting one deforms the where Cr is the elastic deformation parameter of the rolls, as rolls heavily, while that work-piece remains plastically given by (12.27). Since Cr is inversely proportional to Er; undeformed. This may happen when the flow stress and the hardness of the work-piece becomes higher than that of the ) hmin / l R r00 ð12:99bÞ rolls, but it is yet to understand why the performance should Er be related to the thickness. In fact under perfectly frictionless condition, rolling could theoretically be continued till an where Er = elastic modulus of the roll material. infinitely thin rolled product is achieved. In such case, For steel rolls, a useful formula (Tong and Sachs 1957) to feeding the thin work-piece into the roll throat is of course necessary because the rolls would have no bite. The friction estimate the limiting thickness is: causes to increase the longitudinal stress, which in turn increases the roll pressure (see Sect. 12.10.1)—creating the hmin ¼ 0:035 l R r00 mm ð12:100Þ where R is measured in mm and r00 in kN=mm2:

564 12 Rolling Consequently, the following measures may be adopted to 1. The plain-strain yield stress or deformation resistance of achieve the thinnest possible rolled product: metal: • The metal to be used as a work-piece should be in an- nealed condition in order to obtain a low hardness and (i) The higher the deformation resistance of the metal, thereby a low flow stress or deformation resistance. Thus, the annealed steel is always preferred to the normalized the higher is the mean specific pressure, rolling steel. load, rolling torque and mill power required. • In order to reduce the rolling load, the interfacial coeffi- cient of friction should be as low as possible, for which (ii) As the mean deformation resistance, r00; of the polished rolls with a good lubricant should be used. metal increases, the limiting thickness ðhminÞ below • In order to reduce the length of the deformation zone and in turn to reduce the rolling load, small-diameter rolls which the work-piece ÀclanRnro00tÁ=bEe r reduced further should be used. increases, since hmin / ; (12.99b). • Use rolls having higher elastic modulus to improve the 2. Roll radius: angle of contact, a ¼ 2 sinÀ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi resistance to elastic flattening of the rolls themselves, (i) Since Dh=4 R which will reduce the area of contact between the work-piece and the rolls and thereby reduce the rolling (12.7), so for the same draft Dh; the larger the roll load. Thus, the rolls made of tungsten carbide that has a much higher elastic modulus than steel is preferable to radius R; the smaller is the angle of contact a; and use in industrial production. Although such rolls are very expensive, their industrial applications, especially in the the easier is the unaided entry of the work-piece Sendzimir mill (Sect. 12.2), have proved to be worthy. into the roll throat. • Apply front and back tensions to the work-piece in order to reduce the rolling load and, thus, the roll deformation (ii) Since maximum draft, Dhmax ¼ 4 R sin2 for a given pass. But applications of too much tension ðf =2Þ(12.17a), so for the same frictional condition, may be dangerous when the rolled product is very thin. the higher the roll radius R, the higher is the • Apart from adoption of all above measures, the rolled product may also be thinned down by pack rolling or maximum possible draft. rolling in pairs. However, it is usually not economical in production to roll the work-piece long before reaching (iii) Atrhmceacxo¼irndciðnohgm1 Àitnogh(21Þ2smt.ao1xc8=k)h,1st¼ihniccÂek4nRmesassxin,im2ðhuf1=m2¼ÞrÃeÂ=d4huR1c;tsioinon2r, the ultimate limit. ðf =2ފ=rmax; therefore for the same frictional 12.14 Factors Controlling Rolling condition and maximum reduction, a The main variables which control the rolling process are: large-diameter roll will permit a thicker 1. The plane-strain yield stress or deformation resistance of work-piece to enter the rolls than will a the material, r00 ¼ 2 k; where k is the shearing yield stress of the metal. small-diameter roll. 2. Roll radius, R: (iv) As the roll radius, R; increases, the limiting 3. Ingoing thickness of the work-piece, h1: 4. Reduction of the work-piece, r: thickness ðhminÞ below which the work-piece 5. Coefficient of friction between the rolls and the cÀalnRnort00Áb=eErre:dHuceendcef,urtthhienrneinrcgreaausgees, since hmin / work-piece, l: Usually, l varies from 0.05 to 0.1 for cold sheet can be rolling and from 0.2 up to sticking condition for hot rolling. (v) produced with small-diameter rolls. À w2Á=h2; so 6. Width of the work-piece, w: From (12.22c), forward slip, SF ¼ R 7. The presence of back tension, rB; and front tension, rF: 8. Rolling speed. for the same no-slip angle ðwÞ and outgoing thickness ðh2Þ of product, forward slip increases with increase in roll radius, provided no back tension or front tension is applied because tension changes the no-slip angle. (vi) The larger is the roll diameter, the higher is the mean specific pressure, rolling load, rolling torque and mill power required. 3. Ingoing thickness of the work-piece: R sin2ða=2ÞÃ=h1;  (i) the same roll )Sinac¼e re2dsuicntÀio1np, rffiðffi¼rffiffiffihffiðffiffi1hffiÞffi1ffi=ffiÀffiffi4ffiffihffiRffi2ffiÞ:=Sh1o¼fo4r radius and reduction, the higher the ingoing thick- ness of the work-piece, h1; the larger is the angle of contact, a; and the more difficult it is for the work-piece to enter the roll throat in an unaided rolling process.

12.14 Factors Controlling Rolling 565 (ii) ÂSÈince % mÉaximÃum reduction, % rmax ¼ power required. The width of the work-piece does not 4R sin2ðf =2Þ =h1  100; so the maximum affect the rolling pressure. possible per cent reduction is inversely propor- 7. Presence of back tension and front tension: tional to the ingoing thickness, h1; of the (a) Back tension: work-piece. Hence, the larger the ingoing thick- (i) Higher the back tension, the lesser will be the ness of the work-piece, the smaller is the maxi- exit velocity of the work-piece and the exit mum possible per cent reduction. zone area. Hence, higher is the back tension (iii) The lower the ingoing thickness of the work-piece, the lower is the no-slip angle, w: Thus, there is the higher is the mean specific pressure, rolling a maximum back tension, known as a critical load, rolling torque and mill power required. back tension for each rolling condition, at 4. Reduction of the work-piece: which w ¼ 0; and skidding of rolls occurs. (i) pSinffifficffiffieffiffiffiffiffiffiffiffitffihffiffiffieffiffiffi angle of contact, a ¼ 2 sinÀ1 (ii) As back tension increases, forward slip, SF; ðr h1Þ=4 R; so for the same roll radius and decreases SdFue¼toÀRdewc2rÁea=she2:inCtrhieticnaol-sbliapckantgelne-, ingoing thickness of the work-piece, the higher the w; since reduction, r; the greater is the angle of contact, a; sion, at which w ¼ 0; results in zero forward and the more difficult it is for the work-piece to slip. enter the roll throat in an unaided rolling process. (iii) Higher is the back tension, the lower is the (ii) If the reduction, r ¼ Dh=h1; or the draft, Dh; is mean specific pressure, rolling load, rolling reduced, the load will also be reduced and the torque and the mill power required, because thinner work-piece can be rolled, but a limit is from (12.54a), at the entry plane roll pressure pÀ ¼ r001 À rB where r00 1 is the plane-strain reached when no reduction at all is possible. deformation resistance of the work-piece at the (iii) The greater the reduction, the higher is the mean entry plane and rB is the back tension. (b) Front tension: specific pressure, rolling load, rolling torque and mill power required. (i) Higher the front tension, the greater will be the 5. Friction or coefficient of friction: exit velocity of the work-piece and the exit (i) Since maximum reduction, rmax ¼ Dhmax=h1 ¼ 4 zone area. Hence, higher is the front tension, ðR=h1Þ sin2ðf =2Þ; where, f ¼ tanÀ1 l; so for the same roll radius and ingoing thickness of the the larger is the no-slip angle, w: work-piece, the higher the coefficient of friction l; (ii) As the front tension increases, the forward the larger is the angle of bite ðamaxÞ; [since amax ¼ slip, SF; increases due to ÀiRncwre2aÁs=eh2 in the f Š; the maximum draft ðDhmaxÞ; and the maximum no-slip angle, w; since SF ¼ : possible reduction (rmax), in a pass. (iii) Higher is the front tension, the lower is the (ii) ÂS4inRcesin2ðifn=g2oÞi=nrgmax stock thickness, h1 ¼ mean specific pressure, rolling load, rolling therefore, for the same roll à torque and the mill power required, because ; from (12.54c), at the exit plane roll pressure radius, R; and maximum reduction, rmax; higher p þ ¼ r002 À rF where r002 is the plane-strain interface friction, l; will permit a thicker deformation resistance of the work-piece at the work-piece to enter the roll throat. exit plane and rF is the front tension. 8. Rolling Speed (iii) The higher is the coefficient of friction l; the greater is the mean specific pressure, rolling load, (i) As the rolling speed increases, the rate of pro- rolling torque and the mill power required. duction increases. (iv) As the coefficient of friction, l; increases, the (ii) As the rolling speed, i.e. the strain rate, increases, limiting thickness ðhminÞ below which the the deformation resistance of the stock increases work-piece cannot be reduced further increases, during hot rolling, since the strain-rate sensitivity, since hmin / ðl R r00Þ=Er: So to get a thinner m; of flow stress increases with temperature. gauge rolled product, cold rolling is adopted, Hence, the mean specific pressure, rolling load, because the coefficient of friction, l; for cold rolling torque and mill power required for rolling rolling is lower than that for hot rolling. increase when rolling speed increases in hot roll- 6. Width of the work-piece: ing. On the other hand, the deformation resistance The greater the width of the work-piece, the more is the of work-piece is only slightly affected by the contact area between the rolls and the work-piece and so, rolling speed or the strain rate in cold rolling, since the higher is the rolling load, rolling torque and the mill m is insignificant at cold working temperature.

566 12 Rolling Therefore, in cold rolling, rate of production can said to be soft or springy, while if it is small the mill is said be increased by increasing the rolling speed to be hard or rigid. This extension of housing will obviously without affecting the mean specific pressure, roll- influence the gauge of the rolled product, because the force ing load, rolling torque and mill power to any imparted by the entry of the stock stretches the mill and appreciable extent. causes the mill gap, which was set prior to feeding the stock (iii) The thickness of the rolled metal produced on a to be rolled, to increase. The setting of the rolls prior to the cold-rolling mill can be decreased appreciably by entry of the stock is called the passive roll gap, while the increasing the rolling speed. This is attributable to actual gap formed when the stock passes through is called the decrease in the coefficient of friction with the active roll gap. The difference between the active and increasing the speed of rolling (Sims and Arthur passive roll gaps is called the mill springback, which is 1952). related to a characteristic of the mill, known as mill modulus. 12.15 Gauge Control Mill modulus can be determined in the following way. A series of different pieces of work-pieces with constant The outgoing thickness of the rolled product or gauge is an widths are rolled through a constant passive roll gap (set important factor, which can be controlled by considering the prior to the entry of the stock), say g; and these yield dif- characteristic elastic and plastic curves for a rolling mill. The ferent rolling loads, which are measured. Either different elastic curve arises from the effect of rolling load on the metals or different initial gauges of the same metal can be elastic distortion of the rolling mill and known as mill used to vary the rolling loads. The thicknesses of different modulus graph. The effect of load on the plastic deformation pieces after rolling are measured to find the outgoing gauges of the work-piece in the roll gap provides the plastic that correspond to the active roll gaps. If sm denotes the mill deformation curve for the metal, which is essentially what ‘springback’, which is the difference between outgoing rol- would be obtained by the solution of rolling load equations led piece thickness, say h2; and the roll gap setting, i.e. the for rolling work-piece of a given initial thickness to various passive roll gap g; then final thicknesses. sm ¼ h2 À g ð12:101Þ Under the action of roll-separating force, the top of the mill housing is pushed upwards by the upper roll and the The mill springback increases almost linearly with the base is pushed downwards by the lower rolls. Therefore, a rolling load, say P; as shown in the plot of springback ðsmÞ tensile stress, which is obviously below the yield strength of versus load ðPÞ in Fig. 12.33a. The slope of this approxi- the cast steel normally used for housing, acts on the housing mately linear curve, i.e. sm=P; is known as mill modulus. The and causes an appreciable elastic extension. The extent of mill springback curve will vary slightly with the width of deformation depends upon (a) the roll-separating force, (b) the height of the housing and the cross-sectional area of work-piece being rolled. Since the passive roll gap g remains the housing. If the extent of extension is large, the mill is constant, so the active roll gap, i.e. the outgoing gauge, h2; increases almost linearly with the rolling load P: Fig- ure 12.33b shows a typical almost linear plot of the rolling load P against the outgoing gauge h2: If this approximately linear curve is extrapolated back to meet the gauge axis at a (a) (b) (c) Springback Rolling load Rolling load Rolling Load 0 g Thickness 0 Thickness h1 Fig. 12.33 a Mill springback versus rolling load curve. b Mill roll gap g: c Plastic curve showing the relationship between the modulus line showing the relationship between the outgoing gauges of outgoing gauges of the stock rolled from an initial thickness, h1; and the rolled piece and the resulting rolling loads for a constant passive the necessary rolling loads

12.15 Gauge Control 567 point where the rolling load P ¼ 0; that point will give the value of passive roll gap, g: Hence, the equation of the straight line in Fig. 12.33b, assuming m to be the slope of the curve, is: P À 0 ¼ mðh2 À gÞ or; h2 ¼ g þ ð1=mÞ P ð12:102Þ Rolling load P1 where 1=m is called the mill modulus. Therefore, mill modulus is 1 ¼ h2 À g ¼ sm ð12:103Þ mP P If the rolling load is expressed in N, and the units of the g h2 h1 active roll gap and the passive roll gap are mm, then the mill Thickness modulus will be expressed in mm=N: The mill modulus is an important factor required for setting the passive roll gap for a Fig. 12.34 Combination of mill modulus line and plastic curve. Their given pass. This is one of the reasons to determine rolling load with reasonable accuracy before rolling. intersection determines the actual thickness, h2; of metal produced for a given initial passive roll gap, g; and a given initial thickness of metal, h1: To achieve accurate gauge control for a particular metal being rolled, it is required to have the plastic deformation a given setting on a given mill. Thus, softer metal, better curve for the metal. This curve provides the relationship between the outgoing thickness h2 of the metal piece, rolled lubrication and the application of back or front tension yield from an initial thickness h1; and the necessary rolling load P: To obtain such a curve, a series of identical metal pieces of thinner rolled products, as discussed in Sect. 12.13. Let us constant initial thickness, h1; is rolled in a mill by reducing the initial setting of the roll gaps and the necessary rolling discuss the effect of applied tension, yield stress, friction and loads that vary in magnitude are measured. Figure 12.33c shows a typical curve, which is often called the plastic mill stiffness on the outgoing thicknesses of the rolled metal curve. The reason for the steep rise in the plastic curve for small values of thickness is the elastic flattening of rolls. while maintaining the incoming gauge of metal at the value of h1 and the passive roll gap at g: If there is an increase in To achieve a system of gauge control, it is required to plot tension, the plastic curve will move to the left without a graph by combining the mill modulus line given in changing the incoming gauge of metal from h1; as shown in Fig. 12.33b and the plastic curve given in Fig. 12.33c. Such Fig. 12.36. The outgoing gauge of the rolled metal will be a composite curve for a specific metal rolled in a given mill reduced from h2 to h2b; and the rolling load will be is shown in Fig. 12.34. The intersection of the mill modulus decreased from P1 to P4; for a passive roll gap initially set at line and the plastic curve determines the actual gauge pro- g: To maintain the outgoing gauge at a constant initial value duced for a given initial passive roll gap and a given initial of h2; instead of the reduced outgoing thickness h2b; the thickness of metal. Hence according to Fig. 12.34, in order passive roll gap would have to be opened to a value g3; to achieve an outgoing gauge h2 produced by rolling from an which would decrease the rolling load from P4 to P8: On the incoming gauge h1 of metal, the passive roll gap must be set other hand, Fig. 12.36 shows that the plastic curve with the at g; which produces a rolling load, say P1: same incoming gauge value of h1 will be raised if there is an If the incoming gauge of the metal is increased from h1 to increase in coefficient of friction, l; which may occur due to h1a; the plastic line will be moved to the right side along the gauge axis to start at h1a and the outgoing gauge of the rolled the breakdown of lubrication, or if there is an increase in the metal will be increased to h2a; for a passive roll gap initially plane strain yield stress, r00; which may occur due to the set at g; as shown in Fig. 12.35. This increase in the out- decrease in temperature. As a result, the thickness of out- going thickness increases the rolling load from P1 to P2: To going rolled product will be increased from h2 to h2c; for a avoid the production of off-gauge metal and maintain the passive roll gap initially set at g; and the rolling load will be outgoing gauge at a constant value of h2; the passive roll gap increased from P1 to P5: To maintain the outgoing gauge at a must be closed to a value g1; which further increases the constant initial value of h2; instead of the larger outgoing rolling load to P3: thickness h2c; the passive roll gap would have to be closed to a value g2; which would further increase the rolling load Factors which decrease the load necessary for rolling of from P5 to P7: The outgoing gauge of the rolled product is the metal reduce the outgoing gauge of the rolled product for also influenced by the mill stiffness. A stiffer mill, with stiffer mill modulus line for a passive roll gap initially set at g; will provide a less active roll gap and consequently a thinner final rolled product, h2d; resulting in an increase in rolling load from P1 to P6; as shown in Fig. 12.36.

568 Increase in μ or σ ´0 12 Rolling P3 P7 Stiffer mill P2 P5 P1Rolling Load P6 Rolling load PP14 Increase in P8 tension g1 g h2 h2α h1 h1α g2 g g3 h2d h2b h2 h2c h1 Thickness Thickness Fig. 12.35 Effects of varying ingoing stock thickness and the passive Fig. 12.36 Effects of applied tension, yield stress, friction and mill roll gap on the outgoing rolled-metal thickness stiffness on the outgoing thicknesses of the rolled metal The gauge of a rolled metal can vary across its width or with reduction in the stock thickness, so the active roll gap along its length. Normally, width variation in gauge is will increase and consequently the outgoing rolled product associated with shape control and this has been discussed in will be thicker. Correct outgoing gauge can be achieved if Sect. 12.7.2. Outgoing gauge variation along the length of the load on the rolls is reduced to its original value, which rolled metal arises mainly from variations in thickness and causes the active roll gap to return to its former size. To hardness in the ingoing work-piece. Factors that affect the achieve this, if horizontal tensile stresses (back tension, front outgoing gauge also include roll speed and lubricant which tension or both) are applied to the work-piece the stress influence the elastic flattening of the rolls, rolls which are not required to roll the metal, and therefore the rolling load, will precisely concentric with their bearings, and changes in decrease, as explained in Sect. 12.10.2. Resistance strain interstand tension, and rolling temperature. To control the gauges are either placed between the screwdowns and the outgoing gauge, in principle, it is possible to use any of the roll bearings or attached to the mill housing and used to parameters, which are ingoing stock thickness, passive roll continuously monitor the rolling load, so that variations in gap, yield stress, friction, mill stiffness and applied tension. roll-separating force, and therefore active roll gap, are The most obvious method is to change the passive roll gap, instantaneously detected. In response to electrical signals, which is generally applied in rolling heavy gauges. For this, corrections are rapidly made by adjusting the tension in the it is required to continuously measure the outgoing thickness work-piece through altering the speed of coiling-drum in of the rolled product by online sensors. The most commonly single-stand cold-rolling mills. Control of outgoing gauge used sensing devices are the flying micrometre and X-ray or through adjustment of the tension in the work-piece is a isotopes. The gauge systems measure the outgoing thickness more sensitive and rapid technique than control through the by monitoring the amount of radiation transmitted through adjustment in the setting of passive roll gap. Additional the rolled product. The values of outgoing gauges are fed advantage of gauge-control by adjusting the tension is that instantly to a device which adjusts the mill screws and the correction of outgoing gauge is achieved by reducing thereby correct the passive roll gap. rolling load, whereas in the passive roll gap adjustment method, the rolling load is increased. The disadvantage of The above method of gauge control is however not so the former method is that tension cannot be applied to the satisfactory if the incoming work-piece is very thin. As the work-piece during hot rolling. In continuous cold-rolling incoming work-piece becomes thinner and harder, the plastic mills, the gauge error detected by radiation following the curve becomes progressively steep and the intersection of first stand is usually feedback to adjust the passive roll gap the plastic curve and the mill modulus line becomes pro- setting on the first stand, while gauge control in subsequent gressively less dependent on variation of passive roll gap, stands is usually achieved by adjustment of the tension in the and thus, the outgoing gauges become progressively less work-piece through controlling the relative roll speed in responsive to the adjustment of the mill screws. Since the successive stands. rolling load and thereby the roll-separating force increase

12.16 Defects in Rolled Products 569 12.16 Defects in Rolled Products the centre, as shown in Fig. 12.38b. Hence, the centre may buckle; i.e., a wavy centre may form due to the compressive Flatness and uniform thickness over the width and along the stress, as shown in Fig. 12.38c, and edge cracking may length are two important aspects of a rolled product. For these, occur due to the tensile stress, as shown in Fig. 12.38d. a perfectly uniform roll gap must be maintained during rolling. In case of a non-uniform roll gap, the thickness of one edge Once the rolled product has lost its shape badly, which is will be reduced more than the other edge of the work-piece and generally considered to occur in the hot-rolling step, it can the thinner edge will elongate more since volume and width of never be recovered completely and must be scrapped. the work-piece remain constant during rolling. This results in Greatest shape problems occur during rolling of thin bowing of the rolled product. When rolls elastically bend work-piece, probably less than 0.25 mm, because slight under the roll-separating force, there will be lack of flatness in errors in the roll gap profile are magnified with reduction in the stock and wavy edges develop. The deflection of rolls thickness and thinner the section lesser is the resistance to causing the edges to become thinner than the centre is shown buckling. Minor shape problem may be rectified by applying in Fig. 12.37a. This in turn tends to elongate the edges more roller-levelling or stretcher-levelling operation. than the centre, but the edges are restrained by the centre portion from elongating freely since the work-piece is a con- Inhomogeneous deformation in the rolling direction of tinuous body. This results in the development of compressive the stock can also cause loss of flatness and shape. During stresses in the edges induced by the centre and tensile stresses rolling, the width of the stock tends to spread laterally in the in the centre due to the stretching by the edges, as shown in transverse direction, but this is resisted by transverse fric- Fig. 12.37b. The compression of edges in the rolling direction tional forces. Since frictional forces are higher towards the causes them to buckle; i.e., waviness results in the edges and centre of the stock, the lateral spread of the central region the flatness of the rolled product is lost, as shown in will be much less than that of the edge region. The thickness Fig. 12.37c. For relatively thick flat stocks, the residual ten- reduction in the centre mostly leads to an increase in its sion in the centre caused by the elongated edges could produce length, while one part of the thickness reduction at the edges short ‘zipper breaks’ or crack in the centre of the rolled pro- causes the lateral spread and the other part goes into an duct, as shown in Fig. 12.37d. To solve the above problem, the increase in the length, resulting in a shorter length at the rolls having ground convex camber (or crown) or the rolling edges than in the centre of the work-piece. Due to this, the mill equipped with hydraulic jack can be used, as discussed ends of the stock may be slightly rounded, as shown in earlier in Sect. 12.7.2. If the rolls are over-cambered, it will Fig. 12.39a. To maintain continuity between the centre and cause the centre to become thinner than the edges, as shown in edges, the centre is compressed while the edges are stretched Fig. 12.38a, and the stress pattern will be reversed; i.e., there in tension, which may lead to edge cracking, similar to that will be development of tension at the edges and compression at shown in Fig. 12.38d. Under severe conditions, the stress pattern shown in Fig. 12.39a may cause lengthwise splitting at the centre of the stock, as shown in Fig. 12.39b. Fig. 12.37 Possible effects of (a) (b) roll bending. Bent rolls cause a the edges to become thinner than the centre; b to develop residual stresses; c to produce wavy edges; and d short ‘zipper breaks’ or crack in the centre (c) (d)

570 (a) 12 Rolling Fig. 12.38 Possible effects of (b) over-cambering of rolls. Over-cambered rolls cause a the (d) centre to become thinner than the edges; b to develop residual stresses; c to produce wavy centre; and d edge cracking (c) Inhomogeneous deformation in the thickness direction to the projected length of deformation zone exceeds 2. With caused by either light reductions or heavy reductions can also heavy reductions, when the deformation zone extends up to lead to edge cracking (Polakowski 1949–1950). When a thick the centre of the stock, the lateral expansion will be more at work-piece is lightly reduced so that only its surface is the central portion than at the surfaces because of interfacial deformed, then it becomes wider at the surfaces than at the friction between the stock and the rolls. This will produce centre throughout its length, as shown in Fig. 12.40a. The barrelled edges as shown in Fig. 12.40b. Greater spread in overhanging edges of surface metal are not compressed the centre produces tension at the surfaces, whereas the centre directly in subsequent deformation through the rolls but are is left in compression. The secondary tensile stresses devel- compelled to increase in length by the neighbouring metal oped at the surfaces due to barrelling readily causes edge nearer to the centre. This stretching introduces high sec- cracking. This stress pattern also extends along the length of ondary tensile stresses at the edges, which causes edge the work-piece, and if there is any defect along the centre line cracking to occur throughout the length the work-piece of the stock, such as piping defect formed in the original cast similar to that shown in Fig. 12.38d. This type of edge ingot, fracture will take place there, as shown in Fig. 12.40c. deformation (Fig. 12.40a) leading to edge cracks has been This type of fracture is called alligatoring type of fracture, observed (Aleksandrov 1960) to occur during initial break- which is intensified if any bending of the work-piece occurs down of ingot in hot rolling where the ratio of stock thickness due to placement of one roll at a higher or lower position than the centreline of the roll gap. (a) (b) Edge cracking can be prevented or minimized by pre- Rolling venting a cumulative development of secondary tensile direction stresses caused by the barrelling of edges. To keep the edges straight, vertical edge rolls are employed commercially, or Fig. 12.39 Defects resulting from lateral spread. The developed the mill is equipped with edge-restraining bars, or the edges residual stress patterns cause (a) slightly rounded ends and (b) under are made straight by machining after each pass. To avoid severe conditions, lengthwise splitting at the centre excessive cracking in rolling of low-ductility materials, they are ‘canned’ on all sides with a material having a flow stress resembling to that of the stock. The canning material not only restrains the edges of the work-piece but also provides hydrostatic compression. Apart from cracks, there may be different kinds of defects which may be introduced during melting and casting of ingot or during rolling. Surface defects may result from impurities and inclusions in the material, such as dirt, rust, scale and

12.16 Defects in Rolled Products 571 (a) (b) (c) Rolling Rolling directon directon Fig. 12.40 Defects resulting from inhomogeneous deformation in the c Alligatoring type of fracture due to the presence of defect along the thickness direction. a Greater lateral expansion at the surfaces than at centre under conditions of heavy reduction. This defect is intensified the centre throughout the length under conditions of light reduction. due to bending of the work-piece resulting from improper placement of b Formation of barrelled edges under conditions of heavy reduction. one roll roll marks. Incomplete welding of blowholes and pipes each rolling mill and to sections rolled in it. Details on the formed during ingot solidification will cause internal roll pass design are beyond the scope of the text, and defects, such as fissures. Development of pearlite banding in attention will be only focussed on some relevant aspects. steels or non-metallic inclusions in the form of longitudinal stringers is related to melting and solidification practices. If 12.17.1 Types and Shapes of Passes these defects lead to laminations, then the strength in the thickness direction will be reduced drastically. One must be There are different types and shapes of roll passes. The careful to avoid formation of scratches in cold-rolled fin- following types of roll passes are known: ished products due to guides or defective rolls. The rolling lubricant must be easy to remove, and it must produce 1. Cogging-down or breakdown passes, which are used for harmless residue after burning, otherwise there may be dis- initial cogging and reducing the cross-sectional area of coloration or staining after heat treatment of rolled products. ingots or billets. 12.17 Roll Pass Design Fundamentals 2. Roughing passes, which are used to reduce the cross-sectional area of the rolled section and impart it the A plane surface or cylindrically bodied roll is used (in pairs) first shape. for rolling of strips, sheets and plates, while for other semifinished and finished products, grooves of suitable 3. Strand pass, which is the last pass but two and used to design are turned in the roll bodies. A desired shape prepare shape and size for the next penultimate pass. machined into the body of each of the top and bottom rolls is called a groove. The shape formed when the two grooves of 4. Leader pass. This is just before the last pass which two (mating) rolls are matched together is called the roll imparts the final size and shape to the rolled section. pass, i.e. two grooves (in two rolls) working jointly form a roll pass. Passes are located on the same vertical centre line. 5. Finishing pass. This is used to produce the final section. Each pass is separated from its neighbour by collars, as When designing the shape and calculating the dimen- shown in Fig. 12.41. The width of the collars is selected so sions of the finishing passes, corrections are made to take as to make the best use of barrel length, but their minimum into account the thermal expansion of the work-piece, the width is limited by their strength and also by the design irregularity of temperature distribution across the rolled features associated with the location of the roll fittings. shape section and the pass wear. Individual parts of a roll pass do not wear evenly because of mainly non-uniform A roll pass shapes stock at each passing. Several passes of deformation and cooling. This circumstance should be stock through rolls are required to achieve desired rolled taken into consideration when designing the finishing product. A system of successive passes that ensures the passes. rolling of adequately shaped sections from given stock is termed roll pass design. The concept of roll pass design Some common shapes of rolling passes are shown in involves a number of problems, such as the manufacture of a Fig. 12.42, which are: given section in a minimum number of passings, the obtaining of a specified surface quality, mechanization of • Box pass, rolling. These problems are solved in specific application to • Square pass, • Diamond pass, • Oval pass and • Round pass.

572 12 Rolling Fig. 12.41 Blooming mill rolls, Groove showing box passes Pass Pass Pass Drive Roll Collars end neck Roll barrel The difference between a square pass and a diamond Box pass series, as shown in Fig. 12.43, are used most pass is that a square pass is a square set on the corner with commonly in blooming mills for cogging of ingots. Also for apex angles varying from 90° 30′ to below 100°, whereas rolling of blooms to billets in billet mills, it is usual practice the diamond pass is a more open version of the square pass to commence the reduction with a few box passes, which with apex angles from 100° to 130°. Usually, the angles at helps in descaling of the stock. These series are widely used the apex of square passes are adopted to be 90° 30′ during for initial working of large and medium sections because of the finishing pass and 93° during the roughing passes. The adequate scaling, possibility to make several passings in the reason for having an apex angle more than 90° in a square pass and minimum weakening of rolls by shallow grooves. pass is explained below. Since the sharp corners of the However, box passes yield small reduction on the order of diamond are cooled more rapidly than the rest of its section, 5–15% and it is impossible to obtain geometrically correct the larger (horizontal) diagonal has a greater shrinkage square and rectangular sections with box passes because of during cooling. Since the stock from diamond pass is tilted large angle of sides. These difficulties have restricted the by 90° before feeding into the square pass, cooler angles of application of the box pass series to larger sections the diamond pass become the apices of the square pass. The exceeding 80–100 mm. Tilting of stock through 90° is dimensions of square, with an apex angle of 90°, rolled from required during rolling with the use of this series. such a diamond will be distorted. To prevent this distortion, the horizontal diagonal of the square pass is made somewhat Other shapes of rolling passes are used in roughing pas- longer by increasing the angle at the apex. ses, strand pass, leader pass and finishing pass to produce different kinds of rolled sections. The applications of these Fig. 12.42 Some common shapes of rolling passes (a) Box pass (b) Square pass (c) Diamond pass (d) Oval pass (e) Round pass

12.17 Roll Pass Design Fundamentals 573 Fig. 12.43 Roll-down box pass Intital stock Stock turned by 90º after each box pass series Reduced stock after Reduced product first box pass by box pass series passes will be considered in the rolling of billets to rods or (a) (b) square bars. Spass Roll parting line A line which divides the distance between axes (centre lines) of a pair of rolls into two equal parts is called the roll parting line. A roll pass is called closed if the parting line is outside the pass. If the parting line is within the boundaries of the pass outline, the pass is referred to as an open pass, as shown in Fig. 12.44. Fig. 12.44 Types of passes: a Open and b closed passes 12.17.2 Gap and Taper of Sides in Pass Under the pressure exerted by the stock, the working gap cross-section like strip around the roll. Passes tend to wear out in service and gradually lose their initial shape. This is between the rolls increases and all the parts of a work stand remedied by turning of rolls, which reduces the roll diame- ter, while the depth and the width of a pass are restored by in rolling undergo elastic deformation, which is known as tapering the side walls of the roll pass accordingly. Hence, the service life of the pass is increased. Therefore, one ‘springing’ of the stand. ‘Springing’ depends upon the always tries to increase the side taper wherever possible (Fig. 12.45). design and the size and material of the work stand compo- The greater the side taper (the angle of inclination of the nents and is taken into account in the calculation of roll side wall), u; the lesser is the reduction in diameter of rolls, D À D0; after they are reconditioned by turning, as shown gap. In order to prevent the rolls from contacting each other below with reference to Fig. 12.45: after the exit of the stock, the calculated value of the gap must exceed the ‘springing’ of the stand adequately. Prac- tically, the gap, as shown in Fig. 12.44, is spass ¼ 0:01 D þ 1mm; where D is the roll diameter. The side surfaces of the roll passes are made not exactly upright, but inclined at some angle to the longitudinal axis of tan u ¼ b ¼ 2b ; or; D À D0 ¼ 2 b À D0Þ=2 À D0 tan u rolls. The taper of the side walls (taper of sides), as shown in ðD D Fig. 12.45, is expressed either in per cent or as the tangent of ð12:105Þ the angle of inclination, u; of the pass side wall: ÂÀ Á À Á Ã ÂÀ Á À Á Ã where D ¼ the diameter of original roll, D0 ¼ the diameter wg maxÀ wg =2 wg maxÀ wg min of reconditioned roll, tan u ¼ min ¼ b ¼ the depth to which a pass is worn, and u ¼ the angle hg 2 hg of inclination of the side wall. ð12:104Þ where ðwgÞmaxandðwgÞmin = the maximum and the minimum 12.17.3 Pass Arrangement width of the groove (Fig. 12.45), A line along which the roll passes are arranged is called the hg = the height of the groove (Fig. 12.45). roll pass line. The line with respect to which the moments of The taper of sides insures the correct biting of the forces applied to a section being rolled by the top and the work-piece with respect to the centre line of the pass and bottom rolls are equal is termed the neutral (zero) line. If the prevents jamming of the work-piece by the side walls, as well as the danger of collaring of the work-piece of small

574 wg max 12 Rolling wg min Fig. 12.45 Effect of side taper ϕ of roll pass upon removal of roll ϕ material in roll turning bb hg Box pass D D D D neutral line matches with the roll parting line, the straight inequality in the exit speeds of the top and bottom parts of delivery of the work-piece is ensured. the work-piece, which thus bends towards the roll of the smaller diameter. When the diameter of the top roll is greater An unintentional bending of work-piece may occur due to than that of the bottom one, the rolling is said to be per- non-uniform temperature distribution across the work-piece formed with top pressure, while for the bottom roll of section, or an uneven and irregular wear of top and bottom greater diameter the bottom pressure acts in rolling. passes, or incorrect installation of guide arrangements. This unpredictable bending may cause collaring of the work-piece A bottom delivery guide is required to be used for rolling of small cross-section like strip around the roll and other with top pressure, while a top delivery guide is necessary troubles. To prevent such bending of the work-piece upon when rolling is carried out with bottom pressure. If there is leaving the rolls, conditions are specially created under neither top pressure nor bottom pressure or these are negli- which the work-piece bends in a definite direction, either gible, both the top and the bottom delivery guides are upwards or downwards. This makes it possible to ensure the required. Large cogging mills (blooming and slabbing mills) normal straight delivery of the work-piece by using a top or intended for heavy stock are provided with bottom pressure bottom delivery guide. Generally, to create such condition, to avoid the striking of the ends of the blooms or slabs the roll pass line is made misaligned with respect to the roll against roller tables. In case of rolling with bottom pressure parting line by rolling with different diameters of the top and in blooming mills, the difference between the diameters of the bottom rolls, as shown in Fig. 12.46. For the given the bottom and the top rolls, df ¼ Db À Dt; is usually equal rotational speed of rolls, the difference in diameters causes to 10–12 mm. Rolling in two-high section mills is usually Fig. 12.46 Position of pass line Top roll line in case of top pressure Cp Roll parting line Dt Pass line D/2 D Bottom roll line Db D D/2

12.17 Roll Pass Design Fundamentals 575 performed with top pressure, and in this case, the difference The oval–square pass sequence, as shown in Fig. 12.47, between the diameters of the top and the bottom rolls, df ¼ is primarily a high-reduction pass sequence. For a long time, Dt À Db; is 2–6 mm. this sequence has been widely applied as a system of breakdown passes with a maximum reduction and a mini- In case of rolling with top pressure the distance between mum number of passes. An oval pass yields up to about 42% the roll parting line and the roll pass line with respect to reduction in the square–oval reduction and a square pass Fig. 12.46 is: produces 30% reduction in the oval-square reduction. Thus, the oval and square passes combined give a maximum cp ¼ ðD=2Þ À ðDb=2Þ ð12:106Þ overall square to square reduction in area of 59%. Such Again from Fig. 12.46, high-reduction pass sequence is suitable for low-carbon steels in the early passes when the temperature is high. Db=2 ¼ D À ðDt=2Þ; or; ð12:107Þ When an oval stock is rolled in a square pass, it is necessary Db ¼ 2D À Dt ¼ 2 D À df À Db; or; to tightly clamp the oval stock inside guides. This has 2 Db ¼ 2 D À df ; practically prohibited the rolling of large section in the sequence. The shortcomings of this sequence are a consid- ÀÁ erable non-uniformity of square stock throughout the width ) Db ¼ D À df =2 of the oval pass. There is non-uniform wear of oval passes because they undergo intensive wear at points of maximum Combining (12.106) and (12.107), we get reductions on the stock. This results in pittings which are rolled in by subsequent working to produce surface defects À ÀÁ known as folds. Quality steels and alloys, which require D D df =2 ¼ df stringent quality, are particularly sensitive to this defect. cp ¼ 2 À 2 4 ð12:108Þ The diamond–square pass sequence, as shown in If there is neither top pressure nor bottom pressure, the Fig. 12.48, is a medium reduction pass sequence. The roll pass line would coincide with the roll parting line, i.e. square–diamond reduction caused by the diamond pass may cp ¼ 0 (see Fig. 12.46). The neutral line of forming passes, be from 20 to 30%, and the diamond–square reduction which usually runs through the centre of gravity of the pass caused by the square pass may be from 18 to 26%. It is often shape, must coincide with the roll pass line. In simple referred to as an all purpose design as it is suitable as a pass symmetrical shapes, the neutral line of forming pass coin- sequence for a wide range of billet, bar and rod products. cides with its axis of symmetry. Neither the top nor the This sequence lends itself to the manufacture of geometri- bottom pressure is applied during leader and final passes, cally regular squares and, thus, is widely used for producing because the difference between the effective roll diameters finished square sections. The corners of the square are well may cause to develop additional internal stresses in the filled with metal with the passes of the sequence, and hence, metal. there is a possibility for obtaining good intermediate squares. This sequence provides good stability of stock inside the In three-high mills, the roll passes are often made coupled; pass during rolling. The box pass series, however, provides a i.e., the groove cut into the middle roll is common for both better stability of stock inside the pass. The shortcomings of the top and bottom passes. Such an arrangement ensures this sequence are that their grooves are cut deeper into the better use of the roll barrels, especially in cogging stands with roll bodies (as compared to box passes) and these deep box (rectangular) passes, but its disadvantage is that it has an groove cuts weaken the rolls. Angles of square and diamond increased difference between the effective diameters of the sections form at same points of stock and thus cause a sharp rolls, thus giving high top or bottom pressure. local temperature drop which may result in cracks. This precludes the use of the sequence for rolling alloy steel. 12.17.4 Pass Sequences Used in Rolling Further, there is more intensive wear of the passes of this of Billets to Rods sequence due to uneven cooling of the corners of work-piece and its non-uniform deformation. However, when this Rods are usually rolled from billets, which has been obtained sequence is used as roughing passes, the pass corners are by rolling ingots or blooms with a few initial box passes. In rounded off and the pass is somewhat widened out to pro- the rolling of billets to rods, after the initial box passes, the vide free space for metal spreading. During the roughing common roughing pass sequence applied to billet in mer- passes, the width of the next pass in this series is usually chant mill is oval–square pass sequence or diamond–square taken to be much the same as the height of the preceding pass sequence followed by oval–round pass sequence. The pass, without considering rounded corners. different kinds of pass sequence are shown in Figs. 12.47, 12.48 and 12.49.

576 12 Rolling Fig. 12.47 Oval–square pass series No turning (a) Billet or square stock (b) Square stock placed on its flat side being fed into an oval pass The stock is then through 90º Turned through 45º (c) The oval shaped stock (d) The square stock being being fed into a square pass fed into an oval pass The stock is then turned through 90º Turned through 45º (e) The oval shaped stock (f) Square product of being fed into a square pass smaller cross-section The oval–round pass sequence, as shown in Fig. 12.49, is • The deformation of the round stock is fairly uniform a light reduction pass sequence. The general order of throughout the width of the oval pass. reductions is 20–30% by an oval pass in the round–oval reduction and 10–20% by a round pass in the oval–round • There is a uniform distribution of temperatures across the reduction. Since it is a light reduction sequence, it will section. require considerably more passes to achieve a given rod size from a given billet size. It is therefore frequently used in the • This sequence provides better conditions for preventing finishing train only. Apart from small reduction caused by laps and seams and for descaling. this sequence, its other disadvantage is that the filled ovals are not stable during rolling in round passes and the This ensures good quality of the products and makes the work-piece turns even in tight passes. To ensure the stability sequence suitable for rolling of high alloy steel and other of an oval work-piece in a round pass, the oval is made alloys. It is widely used in continuous small-section mills. underfilled. However, there are several advantages of this sequence, as given below: Let us consider the rolling of billets to rod. The flow diagram of pass sequences for getting rod from billet is • The smooth shape transition in these passes needs lower shown in Fig. 12.50. Since the final product is of round stresses. shape, the ultimate or final pass will of course be a round pass, but the leader pass must necessarily be an oval pass. • This sequence provides for an increased dimensional The majority of round products are rolled by the ‘guide– accuracy of round sections. round’ method in which a suitable oval is held by ‘guides’ with the major axis vertical (in a horizontal stand) and fed into the finishing round cross-section. The oval itself which

12.17 Roll Pass Design Fundamentals 577 Fig. 12.48 Diamond–square pass series The square stock is turned through 45º (a) Billet or square stock (b) Square stock placed on its diagonal being fed into a diamond pass The stock is then turned through 90º No turning (c) The diamond shaped stock (d) The square stock being being fed into a square pass fed into a diamond pass The stock is then turned through 90º (e) The diamond shaped stock Turned through 45º being fed into a square pass (f) Square product of smaller cross-section is called the ‘leader’ may be formed from a square or a 12.17.5 Pass Sequences Used in Rolling round which in its turn is produced in the ‘strand’ pass. of Billets to Square Bars The methods of getting to this strand pass are varied and may use box passes followed by diamond–square, oval– The most widely used pass sequences in the form of flow square or oval–round sequence, as shown in Fig. 12.50. diagram for rolling of billets to square bars is the one shown The actual pass sequence used depends on the material in Fig. 12.51. It consists of a square strand, diamond leader being rolled. and square finishing passes. Square bars are usually rolled

578 12 Rolling Fig. 12.49 Oval–round pass series No turning The stock is turned through 90º (a) Billet or square stock (b) Square stock placed on its flat (c) The oval shaped stock side being fed into an oval pass being fed into a round pass No turning The stock is turned through 90º (d) The round shaped stock (e) The oval shaped stock being fed into an oval pass being fed into a round pass No turning No turning The stock is turned through 90º (f) The round shaped stock (g) The oval shaped stock (h) Round product being fed into an oval pass being fed into a round pass desired cross-section from billets, which have been obtained by rolling ingots or 13 mm and with a wall thickness of less than 2 mm, blooms with a few initial box passes. These initial box whereas it is possible for cold processing to produce passes are usually followed by the diamond–square roughing seamless tubes and pipes with a diameter ranging from 1 to pass sequence (see Fig. 12.48) as a regular practice for the 150 mm and with a wall thickness as less as 0.1 mm. The rolling of billets to square bars, while an oval–square most important method of cold processing of seamless tubes roughing pass sequence (see Fig. 12.47) is used for squares is tube drawing, which is discussed in Chap. 14. In general, of the smaller sizes. the production of seamless pipe is more expensive than that of the welded one. Although welded pipes are cheaper than 12.18 Manufacture of Tubes and Pipes seamless ones, but the drawbacks of the welded pipes are the poor strength and corrosion resistance of seams. Metallic tubes and pipes may be produced by rolling, Attempts are made to minimize these shortcomings by cold extrusion, welding or a combination of these methods and rolling. mainly classified as welded or seamless based on the tech- nique of production. Welding produces a seam along the Welded tubes or pipes are produced by forming a strip length of the pipe or tube. The seamless tube or pipe is that and joining the edges of strip by hot forming, fusion or which does not have a welded seam. Seamless tube and pipe electric welding, etc. The starting material may be hot-rolled can be produced by using the method of hot extrusion or hot sheets or coiled stock. Sheet edges to be joined to form pipe rolling. The method of hot extrusion to produce tube is are descaled and mechanically processed to improve seam discussed in Chap. 13. Seamless tubes produced by either formation. Continuous furnace butt-welding may be used for hot rolling or hot extrusion may or may not be subjected to water and gas pipes. A coiled strip is uncoiled in a contin- further cold processing. Hot working methods are unable to uous device, then heated in a furnace and finally contour manufacture seamless tubes with a diameter of less than roll-formed into a pipe in several stands. At the end of forming, strip edges are forced against each other and butt-welded, after which the pipe is finished by passing

12.18 Manufacture of Tubes and Pipes 579 Fig. 12.50 Pass sequences for Billet getting round rod from billet A few Box passes Oval-square Strand Square Diamond–Square Oval–Square sequence pass sequence sequence Diamond– Oval–Square Oval–Round Square sequence sequence sequence Oval–Round sequence Strand Round pass Leader Oval pass Final Round pass through a reducing rolling mill. Thin-walled pipes with Billet better quality of weld can be manufactured by electric A few Box passes welding of edges after hot forming. After forming the pipe in a continuous hot-rolling mill of 5–12 stands, the edges are Diamond–Square Oval–Square heated by electric current and welded. After welding, the roughing pass roughing pass pipe may be sized or shaped into a square, an oval, etc., in a sequence continuous rolling mill and then cut to standard lengths by sequence saws. To join the longitudinal edges of tubes, high- frequency resistance welding and high-frequency induction Strand Square pass welding have also been introduced in recent years to improve the quality of welds for tube diameters less than Leader Diamond pass 25 mm. Edges are heated by stray currents in induction Final Square pass welding, and only a narrow zone near the edges is heated in resistance welding at a frequency of 400–500 kHz. Heated Fig. 12.51 Pass sequences for getting square bar from billet edges are then pressed together by rollers to cause welding. Using high-frequency welders, tube-wall thicknesses as thin as 0.13 mm and as thick as 19 mm are obtainable. After the welding operation, the tube is usually sized and then straightened. The application of laser welding has also begun. Powerful carbon dioxide (CO2) lasers have been used to weld the longitudinal edges of contour roll-formed stainless steel tube.

580 12 Rolling 12.18.1 Production of Seamless Tube and Pipe mill have already been described in Sect. 12.1.2 and shown by Hot Rolling in Fig. 12.2c. Figure 12.52 shows that deviation of roll axes by an angle b from the stock axis provides a tangential The process of rolling to manufacture seamless tube and pipe component of roll peripheral speed vy; which rotates the are generally more economical than extrusion. Commonly stock and an axial component vx; which causes the axial used starting materials for the production of seamless pipes advancement of the stock. This kind of rolling is often ter- and tubes are ingots, continuously cast or rolled billets with med helical. In the rotary piercing operation, when a round usually round cross-section. Centrifugally cast shells are solid billet is pushed to flow over a piercing mandrel at one rarely used as the starting materials. Large-diameter pipes end by means of two inclined rollers of the Mannesmann are manufactured by employing ingots as starting materials mill, the billet will alternately squeeze and bulge during for hot rolling, whereas continuously cast round starting rolling. The amount of inclination and speed of the rollers material is the most economic for the production of tubes at determine the feeding rate. The process of piercing is based the bottom end of the size range. The starting materials are on the fact that the simultaneous squeezing and rotating generally uniformly heated (e.g. temperature variations for action of the rolls deforms the round billet to an elliptical steel must not exceed 25 °C) in rotary hearth furnaces, shape under radial compression, and secondary tensile because uniform heating is an essential requirement for stresses develop at the centre of the elliptical billet initiating obtaining uniform wall thicknesses of tubes. Seamless pipes a crack in the centre. A further rotation of the deformed billet are produced in two stages and often finally finished in the causes the crack to open up and transform into a cavity third stage: which is finally shaped and sized by the piercing mandrel. If the piercing mandrel is pressed into the cavity these rolls can • First stage is piercing of heated starting material by hot produce a hollow shell, as shown in Fig. 12.2c. The outside rolling, in which usually a Mannesmann mill is used to diameter of the piercing mandrel is approximately equal to produce a hollow shell with larger diameter and wall the inside diameter of the desired shell. Thus, the inside thickness than the finished hot-worked tube or pipe. diameter of the outgoing shell is controlled by the piercing Piercing can also be performed in a press, which is mandrel, while the rolls control the outside diameter. The considered in Chap. 13. stand is provided with side rollers or guides to hold the stock. The force required to pierce a stock of same size is • The second stage is hot rolling of the hollow shell less in this mill than in hot extrusion press. The piercing mill (formed by piercing) into a pipe of a given diameter and may have roll barrel of more complicated shape than the wall thickness by means of a plug, a continuous or a Mannesmann mill. The Mannesmann mill is extensively pilger mill. The plug mill process is usually adopted for used for the rotary piercing of steel and copper billets. manufacture of seamless tubes of medium diameter up to about 150 mm, whereas the continuous mill process is The Assel elongator (Snee 1956) consists of three conical commonly used to produce seamless tubes of small driven rolls displaced by 120°, which are set skew with diameter up to about 100 mm. There are light, medium respect to the mill axis, as shown in Fig. 12.53. This has led and heavy pilger mills that are capable to produce seamless tube by hot rolling in a wide range with respect β vx to length (from 8 to 50 m), diameter (from 22 to 700 mm), wall thickness (from 2.25 to 13 mm) and also materials grade. Most pipe–rolling mills are equipped with heavy pilger mills. • In the final stage, very often the tubes manufactured by the plug, continuous or a pilger mill are further hot rolled in finishing (reeling, sizing, stretch-reducing, expanding) mills of various designs. It becomes sometimes necessary to perform rolling with intermediate heating operations. 12.18.1.1 Rotary Piercing vr vy The Mannesmann mill that uses the principle of helical rolling is widely used as rotary piercing mill. Tubes pro- Fig. 12.52 Tangential component of roll peripheral speed, vy; and an duced by rotary piercing rolls in a Mannesmann mill are of axial component, vx; for rolls whose axes deviate by an angle b from smaller sizes than those produced by piercing in a piercing the stock axis press. The shape and arrangement of rolls in Mannesmann

12.18 Manufacture of Tubes and Pipes 581 Fig. 12.53 a Assel elongating (a) (b) Roll mill; b roll disposition in three-roll arrangement Billet Axis·of·rolling 120º to the development of the three-roll piercing mill, as shown plug and is transformed by hot-rolling into a longer tube schematically in Fig. 12.54. This mill offers a higher rate of with definite wall thickness. To produce larger tubes, it is yield and produces more concentric pipes with smoother necessary for the stock to undergo a second operation on the interior and exterior surfaces than a comparable conven- rotary piercing rolls prior to feeding the shell into the plug tional two-roll Mannesmann mill. However, the success in mill. Most pipes are rolled in two passes with the plug mill, the operation of three-roll mills requires the provision for but the first pass provides the maximum draft. However, water cooling the plug bar holding the piercing plug and the three or more rolling passes are given to some metals that are development of higher quality material for the plug itself, difficult to deform. For second time rolling of the tube, the since the axial load on the plug and the bar is much higher in tube is delivered to the entry side of the mill through the a three-roll system than in a 2-roll system. In addition to the same roll gap by lifting the top roll to avoid further rolling of problem of tool wear, associated with the magnitude of the the tube during its return. To change the direction of load, the question of the stability of the plug bar in a reduction, the tube is rotated through 90° before the second three-roll system may also arise. roll-pass commences. The shell diameter is somewhat smaller than the width of the round roll pass because the pass As the Mannesmann mill does not provide sufficiently is made with larger radii at the roll parting line. large elongation and reduction of wall thickness, the pierced hollow thick-walled shell must undergo the following 12.18.1.3 Continuous Tube Rolling Mill hot-rolling processes for the production of final hot-worked The continuous tube rolling mill that uses the principle of pipe of desired shape and size. longitudinal rolling consists of several stands arranged in tandem, as shown in Fig. 12.56. Each stand consists of a 12.18.1.2 Plug Mill two-high non-reversing grooved rolls and the two grooves in The plug mill (Fig. 12.55) that uses the principle of longi- two (mating) rolls form a round pass in each stand which tudinal rolling is a two-high non-reversing mill with grooved ensures high dimensional accuracy of the tube. A special rolls. The two grooves in two (mating) rolls form a round pass. A mandrel consisting of a bar with a plug at its end is introduced into the roll pass. The hollow shell that emerges from the rotary piecing mill is fed to the round pass over the Fig. 12.54 Three-roll piercing mill Fig. 12.55 Plug rolling mill

582 12 Rolling Fig. 12.56 Continuous tube rolling mill pusher is used to insert a lubricated mandrel into the shell + that emerges from the rotary piecing mill. The outside diameter and the wall thickness of the tube are successively reduced by hot-rolling in the continuous mill while passing through the roll gap in each consecutive stand. The rolls in any two adjacent stands are arranged at right angles so that the direction of reduction of the tube is changed by 90° in the adjacent stand. Rolling speed in each successive stand can be increased by means of electric motor so that an interstand tension sufficient to stretch the pipe between stands is developed during rolling. This tension not only reduces the outside diameter of the hot pipe but also its wall thickness. When no reduction in wall thickness is desired, rolling is carried out without tension. After rolling, a stripper is used to withdraw the mandrel from the pipe. 12.18.1.4 Pilger Mill + One of the oldest methods of producing seamless pipes is the hot rolling of shells in pilger mills that use the principle of Fig. 12.57 Pilger mill rolls longitudinal rolling. It is a two-high non-reversing mill with grooved rolls. The pilger rolls constitute a round pass of feeder tilts the stock through 90° so that all the surface of the variable height and width, i.e. of varying cross-section, as stock is rolled over at least two times. This eliminates the shown in Fig. 12.57. Moreover, the pass is not completely slight oval shape of the pipe that may result from spreading. round, rather the sides of the pass is somewhat relieved to The mandrel is withdrawn after rolling. accommodate the spreading or bulging of the stock during rolling. 12.18.1.5 Finishing Mills (Reeling, Sizing, Stretch-Reducing and Expanding Mills) The shell being rolled on a mandrel is subjected to a reciprocal motion by means of a special device called feeder, 1. Reeling Mill lying towards the entrance of the working stand, as shown in Fig. 12.58. The shell is inserted between the rolls at the The principle of helical rolling is used by a reeling mill, moment when the varying cross-section of the pass consti- whose rolls are tilted at an angle of 6° to 7.5° with respect to tuted by the rotating rolls attains its maximum and the metal the axis of rolling, as shown in Fig. 12.59. The tube ceases to contact the rolls. During each revolution of the emerging from a plug mill is often rolled in the reeling mill rolls, the shell is forced to move in a direction opposite to over a mandrel in order to reduce variations in its wall that of its feed. The forward feed of the shell is made by the feeder when the roll groove cross-section increases again forming an idle pass. During each rotation, the rolls deform a shell section whose length corresponds to that of feed. During the combined advancing and feeding operation, the

12.18 Manufacture of Tubes and Pipes 583 Fig. 12.58 Schematic Variable profile Feeding mechanism representation of pilger rolling roll groove process Shell Piercing mandrel Variable profile roll groove Fig. 12.59 Reeling mill usually consists of 5–7 stands arranged in tandem, with each stand generally equipped with two or three rolls. A three–roll sizing mill is represented schematically in Fig. 12.60. The rolls in any two adjacent stands are arranged at right angles in case of two-roll stands and at 60° in case of three-roll stands. The round pass formed by the grooved rolls in the last stand has a true circular shape with the diameter of the pass exactly corresponding to that of the finished tube, whereas the round passes have a small taper in the rest of the stands. While passing through the roll gap from stand to stand, the diameter of the tube is reduced only to a small extent, but the wall thickness remains almost unchanged. thickness and diameter, burnish its inside and outside sur- 3. Stretch-Reducing Mill faces and eliminate the slight oval shape. The diameter of the reeling mandrel is usually 1–6 mm larger than the inside A stretch-reducing mill capable of reducing the tube diam- diameter of the tube delivered by the pilger mill. Therefore, eter by as much as 70% is often applied to hot-roll the tube the reeling mill causes the tube diameter to increase some- or pipe delivered by a plug, continuous or pilger mill without what by 3–9% with consequent reduction of 1–6% in the using a mandrel. Basically, the mill operates like the sizing length. one, but consists of a much greater number of stands, usually varying from 9 to 26, with each stand generally equipped 2. Sizing Mill with two, three or four rolls. The three- and four-roll reducing mills are more compact and provide a better surface A continuous sizing mill using the principle of longitudinal finish than the two-roll mills, but simultaneously they are rolling is often applied to hot-roll the tube or pipe delivered more complex and less convenient to adjust. A three-roll by a plug, continuous or pilger mill without using a mandrel stretch-reducing mill is represented schematically in in order to achieve the exact final dimensions. The mill Fig. 12.61. Fig. 12.60 Sizing mill

584 12 Rolling Fig. 12.61 Stretch-reducing mill In this mill, rolling speed in each successive stand is the tube and feed it forward over a large tapered mandrel. This increased by means of electric motor so that an interstand operation reduces the wall thickness and increases the diam- tension sufficient to stretch the pipe between stands is eter of the tube while its length does not alter substantially. developed during rolling. This tension not only reduces the outside diameter of the hot pipe but also its wall thickness. 12.19 Solved Problems 4. Expanding Mill 12.19.1. A pair of rolls is marked by two indentations spaced at a distance of 75 mm apart. After cold rolling a stock An expanding mill, also known as rotary-rolling mill, uses the through that pair of rolls, the distance between the dents on principle of helical rolling. This mill consists of a pair of the outgoing stock is found to be 78 mm. If the thickness of conical rolls whose axes are inclined to the axis of the pipe the stock before the pass is 1.8 mm and that after the pass is being rolled at angle of 60° in the horizontal plane and of 7.5° 1.2 mm, and the diameter of each roll is 100 mm, determine in the vertical plane, as represented schematically in the following: Fig. 12.62. In operation, the cone-shaped rolls bite and spin Fig. 12.62 Rotary-rolling mill

12.19 Solved Problems 585 (a) Coefficient of friction at the roll–stock interface, using Dhmax ¼ 4R sin2 tanÀ1 l ¼ Ekelund’s expression. ¼ 2 &tanÀ1ð0:127Þ'!2 4  50  sin mm (b) Maximum possible draft in an unaided rolling pass. 2 (c) Percentage of maximum reduction possible by that mill ¼ 0:7968 mm: without any spread. Solution (c) Since there is no spread, so the width of the stock, w; remains constant. Hence from (12.18), the percentage Two indentations marked on the roll are spaced at a distance of maximum reduction possible by that mill is: of lr ¼ 75 mm; and after rolling the distance between the dents created by the roll on the outgoing stock is l2 ¼ % rmax ¼ w D hmax  100 ¼ 0:7968  100 ¼ 44:27 %: 78 mm: Since l2 [ lr; so according to (12.20c) the forward w h1 1:8 slip is: SF ¼ l2 À lr ¼¼ 78 À 75 ¼ 0:04: 12.19.2. Based on the cold-rolling theory of Ford, Ellis and lr 75 Bland, calculate the maximum allowable back tension that Given that the roll radius is R ¼ 50 mm; the thickness of can be applied to a strip from the following data: the stock before the pass is h1 ¼ 1:8 mm; and that after the Strip thickness before the pass ¼ 2:0 mm; strip thickness pass is h2 ¼ 1:2 mm: Hence, according to (12.7) the initial angle of contact of the stock with roll is after the pass ¼ 1:5 mm; Roll diameter ¼ 200 mm; coefficient of friction at the rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 À h2 ¼ 2 sinÀ1 1:8 À 1:2 roll–strip interface ¼ 0:05; a ¼ 2 sinÀ1 4R 4  50 rad Plane–strain average deformation resistance of the strip ¼ 0:109599 rad: ¼ 500 MPa; Front tension applied to the strip ¼ 100 MPa: Since it is cold rolling and the angle of contact, a; is Solution small, so the no-slip angle, w; can be obtained from (12.22b) Given that the roll radius is R ¼ 100 mm; the thickness of the as follows: strip before the pass is h1 ¼ 2 mm; and that after the pass is h2 ¼ 1:5 mm: The coefficient of friction at the roll–strip sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi interface is l ¼ 0:05; the plane–strain average deformation resistance is r00 ¼ 500 MPa¼500 N=mm2; and the front w¼ fð2 2 SF À 1 ¼ 2  0:04 tension applied to the strip is rF ¼ 100 MPa ¼ 100 N=mm2: RÞ=h2g fð2  50Þ=1:2g À 1 rad According to (12.7), the initial angle of contact of the ¼ 0:03117146 rad: strip with roll is (a) The coefficient of friction, l; at the roll–stock interface rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be obtained from Ekelund’s expression given by h1 À h2 ¼ 2 sinÀ1 2 À 1:5 (12.19a) as follows: a ¼ 2 sinÀ1 4R 4  100 rad ¼ 0:0707 rad: l ¼ sin2ða=2Þ From (12.47f), ðsin a=2Þ À sin w rffiffiffiffi rffiffiffiffi  R tanÀ1 R ¼ ½sinð0:109599=2ފ2 H1 ¼ 2 hr2 ffiffiffiffiffiffiffi a ½fsinð0:109599Þg=2Š À sinð0:03117146Þ h2 \"rffiffiffiffiffiffiffi # ¼ 2:9999  10À3 ¼ 0:127: ¼2 100  tanÀ1 100 ð0:0707Þ ¼ 8:549: 23:5234  10À3 1:5 1:5 (b) According to (12.17a), the maximum possible draft in Hence from (12.66b), the maximum allowable back ten- an unaided rolling pass is sion that can be applied to the strip is:

586 12 Rolling !! The following procedure is required to find the value of 1 À h1 h1 Ra ðrBÞmax ¼ r00 h&2 exp ðÀlH1Þ þ h2 expðÀlH1Þ rF ' y dh: 2 ¼ 500  1 À 1:5  exp ðÀ0:05  8:549Þ 0 & '! As it is cold rolling having small angle of contact, (12.8b) 2 can be used to get the initial angle of contact of the strip with 1:5 þ  exp ðÀ0:05  8:549Þ Â 100 N=mm2 roll, which is: ¼ ½500  ð1 À 0:86956Þ þ 0:86956  100Š MPa rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 À h2 ¼ ð1:605 À 1:1325Þ Â 10À3 ¼ 152:176 MPa: a¼ R 0:125 ¼ 0:06148 rad: 12.19.3. A 100-mm-wide strip is cold reduced in thickness From (12.47f), from 1.605 to 1.1325 mm in one pass on a two-high mill having steel rolls operating at 150 rpm. The plane-strain rffiffiffiffi rffiffiffiffi  average flow stress of the strip is 545 MPa and the roll R tanÀ1 R diameter is 250 mm. The strip is subjected to a back tension H1 ¼ 2 a of 100 MPa and a front tension of 180 MPa. The coefficient hr2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihffiffi2ffiffiffiffiffiffi of friction at the roll–strip interface is 0.055. Assuming \"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # homogeneous deformation and neglecting elastic compres- sion and elastic recovery of the strip, determine the fol- ¼2 0:125  tanÀ1 0:125 ð0:06148Þ lowing on the basis of Ford, Ellis and Bland theory taking 1:1325  10À3 1:1325  10À3 elastic flattening of rolls into account, till the new roll radius does not change more than 1% of the previous one: ¼ 12:05: (a) Rolling load. For the entry zone of rolling, from (12.56a): (b) Rolling torque required for each work roll. (c) Lever arm ratio and lever arm or moment arm.  ! (d) Overall power requirement of the cold-rolling mill, if rB exp ðl H1Þ h exp ðÀl HÞ yÀ ¼ 1 À r00  10h61 exp ð0:055  12:05Þ ! the diameter of roll neck is half of that of the roll. ¼  À 100  106 1:605  10À3 Assume that the coefficient of friction at the roll neck is 0.01 and the overall efficiency of the power unit is 80%. 1 545 Solution Âh exp ðÀl HÞ ¼ 987 h : exp ðl HÞ Given in problem for the strip: Width is w ¼ 0:1 m; initial thickness is h1 ¼ 1:605  For an arbitrary contact angle of h: 10À3 m; thickness after rolling is h2 ¼ 1:1325  10À3 m; rffiffiffiffi rffiffiffiffi ! plane-strain average flow stress is r00 ¼ 545  106 N=m2; expðl HÞ ¼ exp 2 l R tanÀ1 R back tension is rB ¼ 100  106 N=m2; front tension is rF ¼ h 180  106 N=m2; coefficient of friction at the roll–strip \" h2 rffiffiffiffiffiffiffihffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi interface is l ¼ 0:055; and roll radius is R ¼ 0:125 m . 0:125 ¼ exp 2  0:055  1:1325  10À3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# 0:125  tanÀ1 1:1325  10À3 h ¼  tanÀ1ð10:506 à exp 1:156  hÞ For the exit zone of rolling, from (12.56b):  ! rF  1 h exp ðl HÞ (a) Cold-rolling load can be obtained from (12.59): yþ ¼ 1 À r00 h2 ¼  À 180 106 23  106  103 ! 1 545  1:1325 Zw Za Za  h exp ðl HÞ P ¼ w R r00 46 y þ dh þ yÀ dh75 ¼ w R r00 y dh ¼ 591:3 h exp ðl HÞ: 0w 0

12.19 Solved Problems 587 From (12.54b) and (12.54d): The radius of the elastically flattened rolls, R0; can be obtained from Hitchcock relation given by (12.26a), which is:  106 y1 ¼ 1 À rB ¼ 1 À 100  106 ¼ 0:8165; R0 Cr P r00 545 R ðh1 À h2Þ ¼ 1 þ w and y2 ¼ 1 À rF ¼ 1 À 180  106 ¼ 0:6697: where for steel rolls, Cr ¼ 2:16  10À11 PaÀ1 r00 545  106 Hence, At the neutral point, where the stock thickness is h ¼ hn; R0 ¼ 0:125  1 þ ð2:16  10À11Þ Â 342845:875 ! m from (12.57a): 0:1  ð1:605 À 1:1325Þ Â 10À3  ¼ 0:145 m: ¼ H1 À 1 ln y2 h1 Hn 2 2 l y1 h2 For elastically flattened rolls with R0 ¼ 0:145 m: ¼ 12:05 À 1  0:6697  1:605  10À3  From (12.8b), the new angle of contact is: 2 0:055 ln 0:8165  1:1325  10À3 2  rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1:605 À 1:1325Þ Â 10À3 ¼ 4:6568: a¼ h1 À h2 ¼ 0:145 R0 Hence, according to (12.57b), the no-slip angle is: ¼ 0:057084 rad: rffiffiffiffi rffiffiffiffi ! From (12.47f), w ¼ h2 tan h2 Hn R R2 rffiffiffiffi tanÀ1rffiRffiffiffi0  R0 a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! H1 ¼ 2 1:1325  10À3 1:1325  10À3 4:6568 hr2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihffiffi2ffiffiffiffiffiffi ¼ 0:125 tan 0:125 2 0:145 ¼2 1\":1r32ffiffi5ffiffiffiÂffiffiffiffiffi1ffiffi0ffiffiffiÀffiffi3ffiffiffiffiffiffiffiffiffiffiffi ¼ 0:021447 rad: # According to (12.8a), the stock thickness at an arbitrary  tanÀ1 0:145 ð0:057084Þ contact angle of h is: 1:1325  10À3 ¼ 12:98: h ¼ h2 þ R h2 ¼ 1:1325  10À3 þ 0:125 h2 m: From (12.56a), for the entry zone of rolling with H1 ¼ 12:98 is: To get arbitrarily the angular coordinate h; let us divide from the exit plane of rolling, where h ¼ 0; to the neutral  ! plane of rolling, where h ¼ w ¼ 0:021447 rad; into three rB exp ðl H1Þ h exp ðÀl HÞ nearly equal angles and from the neutral plane to the entry yÀ ¼ 1 À r00  10h61 exp ð0:055  12:98Þ ! plane of rolling, where h ¼ a ¼ 0:06148 rad; into six nearly ¼  À 100  106 1:605  10À3 equal angles, which are shown in the Table 12.2: 1 545 Hence, by summing up the values of y Dh from h ¼ 0 to h ¼ a ¼ 0:06148 rad from Table 12.2, we get Âh exp ðÀl HÞ ¼ 1038:8 h : exp ðl HÞ Za Xh¼a y dh ¼ y Dh ¼ 0:050326: With R0 ¼ 0:145 m; for an arbitrary contact angle of h: 0 h¼0 expðlHÞ ¼ exp rffiffiffiffi tanÀ1 rffiRffiffiffi0 ! R0 h Therefore, cold-rolling load of the strip is: rffiffiffiffihffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l \" h2 Za ¼ exp 2  0:055  0:145 P ¼ w R r00 y dh 1:1325  10À3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# 0:145  0 À 106Á à  tanÀ1 1:1325  10À3 h 0:1 545 0:050326 N ¼  0:125     à exp 1:245 hÞ ¼ 342845:875 N: ¼  tanÀ1ð11:315

Table 12.2 Intermediate steps to calculate rolling load for original radius of roll 588 12 Rolling Line No. Column No. 12 3 4 5 6 7 8 9 10 0.0143 0.021447 0.02813 0.0348 0.04148 0.04815 0.05483 0.06148 1 Angular coordinate, h; (radian) 0 0.0072 (N.P.) 1.1581 1.19 1.2314 1.2839 1.3476 1.4223 1.5083 1.605 2 h ¼ 1:1325  10À3 þ 0:125 h2ðmÞ 1.1325 1.139 Â10À3 Â10À3 Â10À3 Â10À3 Â10À3 Â10À3 Â10À3 Â10À3 1.1881 1.292 1.394 1.4996 1.6082 1.7184 1.8297 1.9405 Â10À3 Â10À3 0.8136 0.9091 − − − − − − − 0.9091 0.8719 0.845 0.8271 0.8169 0.8136 0.8163 3 expðl HÞ ¼ exp ½1:156 tanÀ1ð10:506 hފ 1.0 1.0912 0.8136 0.9091 0.8719 0.845 0.8271 0.8169 0.8136 0.8163 4 y þ ¼ 591:3 h exp ðl HÞ 0.6696 0.7349 0.7742 0.8613 0.8905 0.8585 0.836 0.822 0.8153 0.815 0.0071 0.007147 0.00668 0.00667 0.00668 0.00667 0.00668 0.00665 5 yÀ ¼ 987 h −− 0.0055 0.006156 0.00595 0.00573 0.00558 0.00548 0.00545 0.00542 exp ðl HÞ 6y 0.6696 0.7349 7 Consecutive means of values in line 6, y − 0.7023 8 Increments in radians, Dh − 0.0072 9 y Dh − 0.00506