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Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
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ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH. INTERNATIONAL. 2010. 17(3): 28-33
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector DENG Wei.
ZHAO De-wen.
QIN Xiao-rnei ,
GAO Xiu-hua ,
DU Lin-xiu ,
LIU Xiang-hua
(State Key Laboratory of Rolling and Automation. Northeastern University. Shenyang 110004. Liaoning. China) Abstract: A new linear integral method for bar hot rolling on roughing train was obtained. First. for plastic deformation energy rate. equivalent strain rate about Kobayashi's three-dimensional velocity field was expressed by two-dimensional strain rate vector; then. the two-dimensional strain rate vector was inverted into inner product and was integrated term by term. During those processes. boundary equation and mean value theorem were introduced; for friction and shear energy dissipation rate. definite integral was applied to the solution process. Sequentially. the total upper bound power was minimized. and the analytical expressions of rolling torque. separating force. and stress state factor were obtained. The calculated results by these expressions were compared with those of experimental values. The results show that the new linear integral method is available for bar rough rolling analysis and the calculated results by this method are a little higher than those of experimental ones. However. the maximum error between them is less than 10 %. Key words: bar rough rolling; linear integration; strain rate vector; roll separating force
Hot rolling. using shaped or grooved rolls. is one of the methods of producing bars and rods. Introduced from America. Morgan high-speed no-twist wire and rod mill is made up of 25 stands. Rough rolling unit refers to the first five stands of wire and rod continuous mill. In shape rolling. the deformation behavior of the billet is more complex than that of hot slabs and plates.•• Simplified one-dimensional models. derived for the calculation of rolling force and torque and mostly based on empirical modifications of the flat rolling theories. have been reported[I-3]. Since the quadratic equations of interface and free-edge surface (unconfined surface) of the deformation zone are complex. st udies on analytical solution of total energy rate for three-dimensional rolling are very difficulr-'". Although a comprehensive analysis was given by Kobayashi in 1975[f,] and afterward several finite element methods were rapidly developed[6-11]. there is still no analytical solution without simplifying above equations for the rolling. Therefore. a new linear integral method is applied to Kobayashi I s velocity fields. to obtain analytical solutions of total energy rate. roll separating force and
torque from Stand 1 to Stand 5 of the roughing train. and the calculated results by the expressions of analytical solutions are compared with those of experimental ones[12].
1
Velocity Field
The billet with dimensions of 115 mm X 115 mm is rolled through a pair of cylindrical rolls with radius of r(,. which enables the thickness of the billet to be reduced from Zh; to Zh, and the width increased from Zb; ( = 2h o ) to Zb«, The shapes and sizes of No.1 to No.3 square-oval-round grooves are shown in Fig. 1. The projected length of the contact arc between the roller and the billet is given by l , AS shown in Fig. 2 (a). Coordinate axes are taken with the origin at the midpoint of the entry cross-section of deformation zone. b, and h., Ch,') are half of spread and thickness when horizontal distance from the origin is .T. Owing to the symmetry of deformation zone. only a quarter is considered. as shown in Fig. 2 (b). For the three-dimensional rolling. Hill and Kobayashi gave the following velocity field r",13-'1J :
Foundation Item:ltem Sponsored by National Natural Science Foundation of China (50474015) Biography: DENG Wei0983-). Male. Doctor: E-mail: [email protected]: Received Date: July 21, 2008
• 29 •
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
(a)
150
110
130
40 (a) No.1 pass;
Fig. 1
U b'rY h", vr=IT' vy=vr b-:-' v==v, h z .r r where Ur-h b,», =hobo Va =h l b l VI = v,cosanhnb n I
hr=ro+h l
(b) No.2 pass;
(1)
...
-
plane in deforming zone. According to Cauchy equation, the strain rate components from Eqn, (1) are:
. [b'r h'rj' v." . v .r , e.z- = -VI -b +-h ' €y=bb r' €=='hh .r
jr~-Cl-:r)2
.r
~
br =b0 +Mx l 6.b=b l -bo ' v= are the velocity components in z , y , z directions; U is the flow volume per second; v, is the tangential velocity of the roll at groove bottom; an' h n , b; are angle, thickness, and width at neutral Vr
'
(c) No.3 pass.
The shapes and sizes of grooves for No.1 to No.3 pass
Vy
ry
J-
.T
J
Vi [~~" -~':~;' -2[ ~: ]2] v-
=
(2)
e.,> v... [h" ... _h'I~_21 h' ... ] 2] Z 2 hI h, i, h, It can be validated that Eqn. (1) and Eqn. (2) satisfy the velocity boundary condition, incompressibility, and steady-state rolling condition. (b) y
V.t
._._._.~._._._._._._.L._._._._._.
a ...
I
---~--.._.l
Fig.2
2
Deformation zone of the first stand (a) and half of spread in deforming zone (b)
Total Energy Rate
Nd =
By using the first variation principle[11 J of rigidplastic materials, the total energy rate function If> should be minimized. If>=Nd+Nr+N, (3) where, N d is the plastic deformation energy rate; N, is the friction energy rate; and N, is the shear energy dissipation rate.
La
idV=4 JtR.
f o fh, flo l. ,
Integration of plastic deformation energy rate Substituting
2E;, + 2E;' )J112
..
(j
= R. and ~ = [ ~ (~; +~; +~; + -
into the-following equation and integrating the plastic deformation energy rate[11], then
L
JEiJ€iJ dV=4 JtR.
0
(v;g2+v;H2y2+v;Fz2)1!2dxdydz=
0
4
JtR. J:' f:' J:, 1s1 Iso I
4
\I(2 3 RT fh 'f'> fl 0'
0'
0
+
cosrs ,so) dxdydz=
s. sodxdydz
where e> gvr i Hyvr j l z-o.k So =e] i+ezj +e3k
+
2. 1
x
(4)
Journal of Iron and Steel Research. International
• 30 •
I=~[h"r_h'rb'r_21~] Z] ,f[ hI
h r b,
(5)
hr
eo is the unit vector of e. Take cosine of projection about the unit vector of at axis .r , y, z directions as el , ez , e3' then the following equations can be obtained: _ dz _ dy el , ez , J dx z + dyz + dz z J dx z + dyz + dz z dz e3 = (6) z Jdx +dyz +dz z
e
Substituting Eqn. (5) and Eqn. (6) into Eqn. (4),
Vol. 17
nuity ~vr, and the roll surface equation is: z=h I =ro +h l -[r~ - ( l - X)2JI/Z Integrating Eqn. (9)[l5-16J leads to ~br~v, + Nfl =4mk {--t-O cos8-2cosan)+v,robl (8-
tan Z[ 2an)+Uroln h;
n
l
+a 4 2}
'It:
tanl: +
00)
~]
where Nfl is the friction energy rate on the contact surface between the roller groove bottom and the workpiece.
then
N'~4)tR. [ J:' J: [JJ+IV~?:dl~( +
HyvIdxdydz
]=
IzvIdxdydz
J1+[t]Z+[~~IZ J1+1:IZ+I~~]Z 4)tR e([I+Iz+I3)
(7)
Integrating Eqn. (7) by mean value theorem term by term, II' I z ' and 13 can be obtained as follows[15 J: II =,f[lU
J
Ml
b;
z
+1 ~h] hm
Z
~b~h 1/2
s,»;
lZ +~bz +~hz (b"b)zb"h 2(~b)3 I - lU l2h m lZbm lU ! z- 2,f[ Jlz+~b2 2,f[
=,f[lU!1
r
2,f[
(8)
2. 2
Surface integration of friction energy rate The friction energy rate on the interface between the roller and the work piece deduced by Kobayashi in 1975[5,11J is:
~vr
~v; + ~v; dzdy= 2kY~b
1 Ir~o= b o
u,
.,=0
Vz
I
vy
fh o
vo~b
~ydY=21
0
= ~ fh l - Vo tan8 d = _ Vo tan8 h 0 h z z 2 o o _ _ ltan8
.r = 0 -
--;:;;;-
Over the whole domain at entry section in Fig. 3, mean value theorem was introduced to integrate Eqn. (11) by definite integral to get an analytical result, N so = 4k
J:0 J:"
[Jv;+v;Jr=odzdy= B
(9)
I b~ IY I +
l
+
v, _ J1 (h ' r )Z] Z U bIh I 'fr is the friction shear stress. Noticing that along the tangential direction of the roll surface, 'fr is the eo-line vector with the tangential velocity disconti=U
s».
(1)
o
vo~b
0
Z
1
J
x
n
fl f ., 'frl~vrlds
where 'fr=mk 1
f:' L
where N. I is the shear energy dissipation rate on the exit cross-section. On the entry section, the tangential velocity discontinuity from Eqn, (1) is vo~b v ot an8 v, I.r=o r V z Ir=o =--h-- z
_I
lU!3
Nd=4)tRell,f[U!I+2~!Z+2~!3j
II
N. I = 4k
vy
Substituting II, I z ' and 13 into Eqn, (7) gives
Ni>
And shear energy dissipation rate on this section is
Z
~h
Jlz+~hz
u; 1 r=1 =~vz Ir=1 =0, v, 1 r=/ =~Vy 1 r~/ = ~b _~bVl y TVr Ir~/- " t t;
u;>
+~b~hz -2 ~h3] 2,f[_Ro-~h r»; lZh m 13 = ---'----='-----r;=;=::::;::;;=-------"-_ lU
Integral depending on shear energy dissipation rate The tangential velocity discontinuity at exit cross-section of deforming zone is
2. 3
O'---
I.. Fig. 3
..... y
~
bo
Integral domain of entry section
lZ tan' 8 1+~
2k U Sb
I
N oticmg h n= ro '0
(2)
where N,o is the shear energy dissipation rate on the entry cross-section.
Friction losses at sidewall of roller The tangential 'velocity discontinuity between sidewall of roller is
Xn
=v
r
-vrCOSa
(3)
N fz =4mk fn f'h
.n
'0
U= Cvrrob] +vrh] b, )cosa n
r
.n
[~-Vr I rdrda+ COSa
'0
[~-Vr]
on t h e entry an d exit Slid es as o
V r ::::::::
Vo +vr an d --2-
b, rovrcosza n-
'+
Vr /::;'br~ + Vr h l It:.bro ] cosc, sine, --1-
vrt:.br~ ( sine, - sin 3 an ) --10
0
hl b-1+ b.L an d dif --- , c=-1T a kimg a_vrdM
4mk r t:.r[ (vo + v r) lntan
~ +~]
I ~_+ ~ ] - 2 (v.+.~
-2vr(8-2an)]
(5)
Total energy rate and its minimization Substituting Eqn. (8), Eqn. (0), Eqn. (1), Eqn. (2),-and Eqn, (1&) into Eqn, (3), and considering N, = N,o + N,] and N, = N n + N fZ' then
N,] + N n + N fz =4 -!tR[I../2Ufj +
Ul f2+ Ul f3J+2kUM+2kUM 2../2 2../2 I I
1 +12 tan28+ Z M
t:.br~ Vr + 4mk { - 1 - • 0 cos8-2cosan)+vrrob] (8tan!
I !5.- ++ ~ an I
)+ o-, h ln ~ . 8 }+4mk rt:.r[(vo+ tan 1- !5.-+_j 4 2
"m •
lntanl
~ + ~ ] -2(v+vr) lntanl ~ +~
2v rC8-2an)]
m= [
HI
-rSr [2 Hroln i;
2. 5
q,=Nd+ N,o+
with respect to neutral angle an and setting it to be zero, the following equation can be obtained: dq, = JN d + IN,o + IN,] + JN n + JN f2 =0 (8) dan Jan Jan Jan Jan Jan Differentiating Eqn. (8) term by term and taking dU/ dan = N, the analytical expression of m is
V r ::::::::
Eqn. (4) becomes
vr)ln [
vr )
rr
2acosa n+ 3acos3an (17) Differentiating the total energy q, of Eqn. (6)
rdrd a (4) COSa where N fz is the friction energy rate on the contact surface between the roller groove sidewall and the workpiece. Assuming the velocity in the zone of slippage 4mk fO f'h
2a n
xn . + t:.9I"
dU = -abcsina n+acsin2a n-abcos2a ndan
v=(v] +vo)/2 r= Cr, +ro )/2 w is the angular velocity of the roll. The friction losses on the surface is,
n; =
ro cOSa n, bn = b0
r« rot:.b Ierentiating U with respect to neutral angle an' then
vr=WrO'
~Vr ,
I -
l-,
tsr> rh - ro
VI
+h
=l-rsina n,
2. 4
t:.VfZ where
• 31 •
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
1(6)
4lf] +l{z Vr-
(V+Vr)] cosc,
tan Z[ tan
+ 1{3] + H~~f4]
[
1C
4
-
t
+ an 2 _
8] '4 + '2 1C
-7- {2v r r ob]-
2t:.br~vrsinan I
--=----:-'------"-
2Uro } hmcosa n
(9)
Substituting an determined by Eqn, (19) into Eqn. (6), the minimum total energy rate q,min is obtained. Noticing that the average width of deforming zone is 13 = 2b.r , and the contact area is F = 2b r l , then the power required can be written as _ 1 fl _2pbr l z v r _ (20) ] -4pb .r l -I vrzdx-q,min o
r«
ro where, vrz=-vrsina, 15= 2b... 12v q,min, and pis avr erage unit pressure. Through q,min' the corresponding expressions of rolling torque CM), roll separating force (P), and stress effective factor (n.) are given by
P =15· F , M = z,;;'¥min' ro "" 15 rOq,min n, =R- = 2b lZ R r e x
Vr
(21) e
Or using Tselicov formula[17-]8] , q,min M= rOq,min n =2 (22) v r8Xbmt:.h' 2v r ' • R; where X is an arm factor, and for hot rolling, X= 0.3-0.6. 15=
Journal of Iron and Steel Research. International
• 32 •
3
owing to the uneven of reductions in passes. Therefore, it is necessary to increase the reduction of the second pass to balance the loading of each stand. I) Comparison between the calculated values and the measured ones According to Table 2. the comparisons of the calculated roll separating force and rolling torque with the measured ones[12] are shown in Fig. 4. Fig. 4 (a) indicates that the calculated rolling forces are slightly higher than the measured, for this is an upper bound method. But the errors are less than 10%. And Fig. 4 (b) shows that the calculated rolling torques approach to the measured values. Thus, it can be concluded that this new linear integration is available for bar rough rolling analysis. 2) Neutral angle and stress effective factor versus 1/ h for various values of m As shown in Fig. 5, neutral angle increases with the increase in friction when l/h=O. 8; when the
Discussion of Results
Morgan data from Stand 1 to Stand 5[12] are shown in Table 1. S80 high-quality carbon steel billet with dimensions of 115 mm X 115 mm was rolled through first strand of the roughing train at 1050 "C. The deformation resistance'P'' is given by ,
R. = 135. 4exp(3. 751-2. 974 T) •
"
O.034T+O.1089
I leO I
.
[1. 312'1 O~ 410.2181 -0. 78e]
(23)
d £ormation . temperature. h T = t+273 were 1 000 ; an d tz iIS the e
With the equations above, programming by C language and using golden section search method, the rolling torque and roll separating force are minimized with m = 1. O. The results are shown in Table 2. It shows that the rolling force and torque assignment among the first five stands are nonuniform, Table 1 Roll gap/
Vol. 17
Morgan data of rough stands
Height/ mm
Roll diameter/ mm
Motor speed/ Cr > min-I)
Gear ratio/
mm
Depth/ mm 32.5
85
460
283. 65
36. 14
0.1663 0.2132
Stand
Shape
1
Square
20
Exit speed/
%
(rn
>
S-l)
2
Oval
20
26
72
483
306. 27
34.071
3
Round
14
36.5
87
457
492. 33
34.679
O. 2978
4
Oval
14
17.5
49
480
335.92
18.774
O. 4322
5
Round
10
26
62
479
326.32
12.772
O. 5865
Table 2
Calculated results of roughing train
Stand
lRM
2RM
3RM
4RM
Reduction/ %
26. 1
15.3
40.7
43. 7
41. 7
1.2134
0.9783
1. 1869
J. 3912
J. 2832
".
5RM
l/h
0.80
0.87
J. 02
J. 50
J. 28
Rolling fo.rce/kN
677. 3
419.4
456. I
677.0
502. I
Rolling torque/(kN • m )
71. 2
39.0
62.0
41. 1
5 J. 6
Note: h= (h,+h o ) /2.
45
750 (a)
80 (b)
40
Z
35
i::
30
~
.£ ~50 1>.0 .5
25
==0
20
I:l:: 450
15 350 lRM
~ ~
0-
.5 >:
0
c::l
.g'"
QJ
I:l::
,-.,
70
~
60
5-
50
E
~
~
S
i
I:l::
40 Measured rolling torque
30
10 2RM
3RM
4RM
5RM lRM Pass number
2RM
3RM
4RM
Fig. 4 Relationship among the measured and calculated roll separating force and reduction (a) and rolling torque (b) and roll numbers
5RM
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
value of m is less than O. 5, the relation between an and m is nearly linear. The value of n, decreases at first and then increases with increasing the value of 1/ hand m , as shown in Fig. 6. When the value of 1/h is close to O. 9, n, reaches the minimum value. 0.18
.------------------:-1
References : [IJ [2J [3J
0.16 [4J 0.14 [5J
0.12 0.10,
[6J 0.08
_ _~_ _....__ 0.6 0.2 0.4 m
'--_~
o Fig. 5
____"_ _.......... 0.8 1.0
The relationship between neutral angle a. and friction factor m
[7J
[8J
1.5 . - - - - - - - - - - - - - - - - - - - , [9J 1.3 [10J
[IlJ
0.9 [12J
0.7 L . . . . . - - - - 7 - - - -.......- - -.......- - - ' 1.4 0.8 1.0 1.2 llh [l3J
Fig. 6
4
Relationship between stress effective factor n. and Ilh for various values of m
Conclusions
1) A new linear integral method is proposed for bar rolling analysis. The calculation during rough rolling shows that the calculated roll separating forces are a little higher than the measured ones, and the rolling torque values approach the measured ones. but the maximum error is less than 10%. 2) The reduction of the first five stands is nonuniform, which results in the uneven rolling force and torque among these stands. Therefore. it is necessary to increase the reduction of the second pass and to balance the loading of each stand. 3) The solution above is available for bar rolling in the roughing train, and after stand 6. the error percentage will be significant.
• 33 •
[14J
[15J
[16J
[l7J
[18J [19J
Khaikin BE. Kisilenko I A. Tranovskii I Y. Pressure of Metal on Rolls in Section Rolling [JJ. Steel USSR, 19710): 884. Arnold R R, Whitton P W. Spread and Roll Force in Rod Rolling [JJ. Metal Tech. 1975(2): 143. Shinokura T • Takai K. Mathematical Models of Roll Force and Torque in Steel Bar Rolling [JJ. Tetsu-to-Hagane , 1986, 72(4): 1870 (in Japanese). Ahmed Said. Lenard J G. Ragab A R. et aI. The Temperature. Roll Force and Roll Torque During Hot Bar Rolling [JJ. J Mater Process Technol , 1999.880-3): 147. Oh S I. Kobayashi S. An Approximate Method for a Three-Dimensional Analysis of Rolling [JJ. Int J Mech Sci. 1975. 17 (4): 295. Kazunori K. Tadao M. Toshihiko K. Flat-Rolling of RigidPerfectly Plastic Solid Bar by the Energy Method [JJ. J Jap Soc Tech Plasticity. 1980,21(231): 359 (in Japanese). Pittman J FT. Zienkiewicz 0 C, Wood R D, et aI. Numerical Analysis of Forming Processes [MJ. New York: John Wiley and Sons Ltd. 1984. Kim N. Lee S M. Shin W, et aI. Simulation of Square-to-Oval Single Pass Rolling Using a Computationally Effective Finite and Slab Element Method [JJ. J Eng Ind. 1992(14): 329. Park J J, Oh S I. Application of Three Dimensional Finite Element Analysis to Shape Rolling Processes [JJ. J Eng Ind. 1990 (112):36. Kim N, Kobayashi S. Allan T. Three-Dimensional Simulation of Gap Controlled Plate Rolling by the Finite Element Method [JJ. Int J Mach Tolls Manuf , 1990. 30( 2): 269. Montmitonnet P. Chenot J L. Bertrand-Corsini C. et aI. A Coupled Themomechanical Approach for Hot Rolling by a 3D Finite Element Method [JJ. J Eng Ind. 1992(14): 336. LI Chang-cheng . REN Yu-hui , ZHANG Li-Ien • et aI. Measurement and Study on 500 mm Rougher Mill of High Speed Untwist Rod Mill [JJ. Iron and Steel. 2002. 37< 2): 59 (in Chinese). ZHAO De-wen, WANG Guo-dong. BAI Guang-run. Theoretical Analysis of the Wire Drawing Through Two Roller Dies in Tandem [JJ. Science in China: Series A. 1993. 36 (5): 632. WANG Guo-dong, ZHAO De-wen. Modern Materiel Forming Mechanics [M]. Shenyang , Northeastern University Press. 2004 (in Chinese). ZHAO De-wen.' Mathematical Solution of Continuum Forming Force [M]. Shenyang , Northeastern University Press. 2003 (in Chinese). ZHAO De-wen, GUO Chang-wu , ,LIU Xiang-hua, Surface Integral of Three-Dimensional Velocity Field for Square Bar Drawing Through Conical Die [JJ. Trans Nonferrous Met Soc China. 1996. 6(3): 131. ZHAO De-wen. XIE Ying-jie , LIU Xiang-bua , et aI. ThreeDimensional Analysis of Rolling by Twin Shear Stress Yield Criterion [JJ. Journal of Iron and Steel Research. International. 2006, 13(6): 21. ZHAO Zhi-ye, Metal Forming Mechanics [MJ. Beijing: Metallurgicallndustry Press. 1987 (in Chinese). ZHAO Ii-hua , GUAN Ke-zhi. Resistance of Metal Plastic Deformation [MJ. Beijing: Mechanical Industry Press. 1989 (in Chinese).
ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH. INTERNATIONAL. 2010. 17(3): 28-33
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector DENG Wei.
ZHAO De-wen.
QIN Xiao-rnei ,
GAO Xiu-hua ,
DU Lin-xiu ,
LIU Xiang-hua
(State Key Laboratory of Rolling and Automation. Northeastern University. Shenyang 110004. Liaoning. China) Abstract: A new linear integral method for bar hot rolling on roughing train was obtained. First. for plastic deformation energy rate. equivalent strain rate about Kobayashi's three-dimensional velocity field was expressed by two-dimensional strain rate vector; then. the two-dimensional strain rate vector was inverted into inner product and was integrated term by term. During those processes. boundary equation and mean value theorem were introduced; for friction and shear energy dissipation rate. definite integral was applied to the solution process. Sequentially. the total upper bound power was minimized. and the analytical expressions of rolling torque. separating force. and stress state factor were obtained. The calculated results by these expressions were compared with those of experimental values. The results show that the new linear integral method is available for bar rough rolling analysis and the calculated results by this method are a little higher than those of experimental ones. However. the maximum error between them is less than 10 %. Key words: bar rough rolling; linear integration; strain rate vector; roll separating force
Hot rolling. using shaped or grooved rolls. is one of the methods of producing bars and rods. Introduced from America. Morgan high-speed no-twist wire and rod mill is made up of 25 stands. Rough rolling unit refers to the first five stands of wire and rod continuous mill. In shape rolling. the deformation behavior of the billet is more complex than that of hot slabs and plates.•• Simplified one-dimensional models. derived for the calculation of rolling force and torque and mostly based on empirical modifications of the flat rolling theories. have been reported[I-3]. Since the quadratic equations of interface and free-edge surface (unconfined surface) of the deformation zone are complex. st udies on analytical solution of total energy rate for three-dimensional rolling are very difficulr-'". Although a comprehensive analysis was given by Kobayashi in 1975[f,] and afterward several finite element methods were rapidly developed[6-11]. there is still no analytical solution without simplifying above equations for the rolling. Therefore. a new linear integral method is applied to Kobayashi I s velocity fields. to obtain analytical solutions of total energy rate. roll separating force and
torque from Stand 1 to Stand 5 of the roughing train. and the calculated results by the expressions of analytical solutions are compared with those of experimental ones[12].
1
Velocity Field
The billet with dimensions of 115 mm X 115 mm is rolled through a pair of cylindrical rolls with radius of r(,. which enables the thickness of the billet to be reduced from Zh; to Zh, and the width increased from Zb; ( = 2h o ) to Zb«, The shapes and sizes of No.1 to No.3 square-oval-round grooves are shown in Fig. 1. The projected length of the contact arc between the roller and the billet is given by l , AS shown in Fig. 2 (a). Coordinate axes are taken with the origin at the midpoint of the entry cross-section of deformation zone. b, and h., Ch,') are half of spread and thickness when horizontal distance from the origin is .T. Owing to the symmetry of deformation zone. only a quarter is considered. as shown in Fig. 2 (b). For the three-dimensional rolling. Hill and Kobayashi gave the following velocity field r",13-'1J :
Foundation Item:ltem Sponsored by National Natural Science Foundation of China (50474015) Biography: DENG Wei0983-). Male. Doctor: E-mail: [email protected]: Received Date: July 21, 2008
• 29 •
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
(a)
150
110
130
40 (a) No.1 pass;
Fig. 1
U b'rY h", vr=IT' vy=vr b-:-' v==v, h z .r r where Ur-h b,», =hobo Va =h l b l VI = v,cosanhnb n I
hr=ro+h l
(b) No.2 pass;
(1)
...
-
plane in deforming zone. According to Cauchy equation, the strain rate components from Eqn, (1) are:
. [b'r h'rj' v." . v .r , e.z- = -VI -b +-h ' €y=bb r' €=='hh .r
jr~-Cl-:r)2
.r
~
br =b0 +Mx l 6.b=b l -bo ' v= are the velocity components in z , y , z directions; U is the flow volume per second; v, is the tangential velocity of the roll at groove bottom; an' h n , b; are angle, thickness, and width at neutral Vr
'
(c) No.3 pass.
The shapes and sizes of grooves for No.1 to No.3 pass
Vy
ry
J-
.T
J
Vi [~~" -~':~;' -2[ ~: ]2] v-
=
(2)
e.,> v... [h" ... _h'I~_21 h' ... ] 2] Z 2 hI h, i, h, It can be validated that Eqn. (1) and Eqn. (2) satisfy the velocity boundary condition, incompressibility, and steady-state rolling condition. (b) y
V.t
._._._.~._._._._._._.L._._._._._.
a ...
I
---~--.._.l
Fig.2
2
Deformation zone of the first stand (a) and half of spread in deforming zone (b)
Total Energy Rate
Nd =
By using the first variation principle[11 J of rigidplastic materials, the total energy rate function If> should be minimized. If>=Nd+Nr+N, (3) where, N d is the plastic deformation energy rate; N, is the friction energy rate; and N, is the shear energy dissipation rate.
La
idV=4 JtR.
f o fh, flo l. ,
Integration of plastic deformation energy rate Substituting
2E;, + 2E;' )J112
..
(j
= R. and ~ = [ ~ (~; +~; +~; + -
into the-following equation and integrating the plastic deformation energy rate[11], then
L
JEiJ€iJ dV=4 JtR.
0
(v;g2+v;H2y2+v;Fz2)1!2dxdydz=
0
4
JtR. J:' f:' J:, 1s1 Iso I
4
\I(2 3 RT fh 'f'> fl 0'
0'
0
+
cosrs ,so) dxdydz=
s. sodxdydz
where e> gvr i Hyvr j l z-o.k So =e] i+ezj +e3k
+
2. 1
x
(4)
Journal of Iron and Steel Research. International
• 30 •
I=~[h"r_h'rb'r_21~] Z] ,f[ hI
h r b,
(5)
hr
eo is the unit vector of e. Take cosine of projection about the unit vector of at axis .r , y, z directions as el , ez , e3' then the following equations can be obtained: _ dz _ dy el , ez , J dx z + dyz + dz z J dx z + dyz + dz z dz e3 = (6) z Jdx +dyz +dz z
e
Substituting Eqn. (5) and Eqn. (6) into Eqn. (4),
Vol. 17
nuity ~vr, and the roll surface equation is: z=h I =ro +h l -[r~ - ( l - X)2JI/Z Integrating Eqn. (9)[l5-16J leads to ~br~v, + Nfl =4mk {--t-O cos8-2cosan)+v,robl (8-
tan Z[ 2an)+Uroln h;
n
l
+a 4 2}
'It:
tanl: +
00)
~]
where Nfl is the friction energy rate on the contact surface between the roller groove bottom and the workpiece.
then
N'~4)tR. [ J:' J: [JJ+IV~?:dl~( +
HyvIdxdydz
]=
IzvIdxdydz
J1+[t]Z+[~~IZ J1+1:IZ+I~~]Z 4)tR e([I+Iz+I3)
(7)
Integrating Eqn. (7) by mean value theorem term by term, II' I z ' and 13 can be obtained as follows[15 J: II =,f[lU
J
Ml
b;
z
+1 ~h] hm
Z
~b~h 1/2
s,»;
lZ +~bz +~hz (b"b)zb"h 2(~b)3 I - lU l2h m lZbm lU ! z- 2,f[ Jlz+~b2 2,f[
=,f[lU!1
r
2,f[
(8)
2. 2
Surface integration of friction energy rate The friction energy rate on the interface between the roller and the work piece deduced by Kobayashi in 1975[5,11J is:
~vr
~v; + ~v; dzdy= 2kY~b
1 Ir~o= b o
u,
.,=0
Vz
I
vy
fh o
vo~b
~ydY=21
0
= ~ fh l - Vo tan8 d = _ Vo tan8 h 0 h z z 2 o o _ _ ltan8
.r = 0 -
--;:;;;-
Over the whole domain at entry section in Fig. 3, mean value theorem was introduced to integrate Eqn. (11) by definite integral to get an analytical result, N so = 4k
J:0 J:"
[Jv;+v;Jr=odzdy= B
(9)
I b~ IY I +
l
+
v, _ J1 (h ' r )Z] Z U bIh I 'fr is the friction shear stress. Noticing that along the tangential direction of the roll surface, 'fr is the eo-line vector with the tangential velocity disconti=U
s».
(1)
o
vo~b
0
Z
1
J
x
n
fl f ., 'frl~vrlds
where 'fr=mk 1
f:' L
where N. I is the shear energy dissipation rate on the exit cross-section. On the entry section, the tangential velocity discontinuity from Eqn, (1) is vo~b v ot an8 v, I.r=o r V z Ir=o =--h-- z
_I
lU!3
Nd=4)tRell,f[U!I+2~!Z+2~!3j
II
N. I = 4k
vy
Substituting II, I z ' and 13 into Eqn, (7) gives
Ni>
And shear energy dissipation rate on this section is
Z
~h
Jlz+~hz
u; 1 r=1 =~vz Ir=1 =0, v, 1 r=/ =~Vy 1 r~/ = ~b _~bVl y TVr Ir~/- " t t;
u;>
+~b~hz -2 ~h3] 2,f[_Ro-~h r»; lZh m 13 = ---'----='-----r;=;=::::;::;;=-------"-_ lU
Integral depending on shear energy dissipation rate The tangential velocity discontinuity at exit cross-section of deforming zone is
2. 3
O'---
I.. Fig. 3
..... y
~
bo
Integral domain of entry section
lZ tan' 8 1+~
2k U Sb
I
N oticmg h n= ro '0
(2)
where N,o is the shear energy dissipation rate on the entry cross-section.
Friction losses at sidewall of roller The tangential 'velocity discontinuity between sidewall of roller is
Xn
=v
r
-vrCOSa
(3)
N fz =4mk fn f'h
.n
'0
U= Cvrrob] +vrh] b, )cosa n
r
.n
[~-Vr I rdrda+ COSa
'0
[~-Vr]
on t h e entry an d exit Slid es as o
V r ::::::::
Vo +vr an d --2-
b, rovrcosza n-
'+
Vr /::;'br~ + Vr h l It:.bro ] cosc, sine, --1-
vrt:.br~ ( sine, - sin 3 an ) --10
0
hl b-1+ b.L an d dif --- , c=-1T a kimg a_vrdM
4mk r t:.r[ (vo + v r) lntan
~ +~]
I ~_+ ~ ] - 2 (v.+.~
-2vr(8-2an)]
(5)
Total energy rate and its minimization Substituting Eqn. (8), Eqn. (0), Eqn. (1), Eqn. (2),-and Eqn, (1&) into Eqn, (3), and considering N, = N,o + N,] and N, = N n + N fZ' then
N,] + N n + N fz =4 -!tR[I../2Ufj +
Ul f2+ Ul f3J+2kUM+2kUM 2../2 2../2 I I
1 +12 tan28+ Z M
t:.br~ Vr + 4mk { - 1 - • 0 cos8-2cosan)+vrrob] (8tan!
I !5.- ++ ~ an I
)+ o-, h ln ~ . 8 }+4mk rt:.r[(vo+ tan 1- !5.-+_j 4 2
"m •
lntanl
~ + ~ ] -2(v+vr) lntanl ~ +~
2v rC8-2an)]
m= [
HI
-rSr [2 Hroln i;
2. 5
q,=Nd+ N,o+
with respect to neutral angle an and setting it to be zero, the following equation can be obtained: dq, = JN d + IN,o + IN,] + JN n + JN f2 =0 (8) dan Jan Jan Jan Jan Jan Differentiating Eqn. (8) term by term and taking dU/ dan = N, the analytical expression of m is
V r ::::::::
Eqn. (4) becomes
vr)ln [
vr )
rr
2acosa n+ 3acos3an (17) Differentiating the total energy q, of Eqn. (6)
rdrd a (4) COSa where N fz is the friction energy rate on the contact surface between the roller groove sidewall and the workpiece. Assuming the velocity in the zone of slippage 4mk fO f'h
2a n
xn . + t:.9I"
dU = -abcsina n+acsin2a n-abcos2a ndan
v=(v] +vo)/2 r= Cr, +ro )/2 w is the angular velocity of the roll. The friction losses on the surface is,
n; =
ro cOSa n, bn = b0
r« rot:.b Ierentiating U with respect to neutral angle an' then
vr=WrO'
~Vr ,
I -
l-,
tsr> rh - ro
VI
+h
=l-rsina n,
2. 4
t:.VfZ where
• 31 •
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
1(6)
4lf] +l{z Vr-
(V+Vr)] cosc,
tan Z[ tan
+ 1{3] + H~~f4]
[
1C
4
-
t
+ an 2 _
8] '4 + '2 1C
-7- {2v r r ob]-
2t:.br~vrsinan I
--=----:-'------"-
2Uro } hmcosa n
(9)
Substituting an determined by Eqn, (19) into Eqn. (6), the minimum total energy rate q,min is obtained. Noticing that the average width of deforming zone is 13 = 2b.r , and the contact area is F = 2b r l , then the power required can be written as _ 1 fl _2pbr l z v r _ (20) ] -4pb .r l -I vrzdx-q,min o
r«
ro where, vrz=-vrsina, 15= 2b... 12v q,min, and pis avr erage unit pressure. Through q,min' the corresponding expressions of rolling torque CM), roll separating force (P), and stress effective factor (n.) are given by
P =15· F , M = z,;;'¥min' ro "" 15 rOq,min n, =R- = 2b lZ R r e x
Vr
(21) e
Or using Tselicov formula[17-]8] , q,min M= rOq,min n =2 (22) v r8Xbmt:.h' 2v r ' • R; where X is an arm factor, and for hot rolling, X= 0.3-0.6. 15=
Journal of Iron and Steel Research. International
• 32 •
3
owing to the uneven of reductions in passes. Therefore, it is necessary to increase the reduction of the second pass to balance the loading of each stand. I) Comparison between the calculated values and the measured ones According to Table 2. the comparisons of the calculated roll separating force and rolling torque with the measured ones[12] are shown in Fig. 4. Fig. 4 (a) indicates that the calculated rolling forces are slightly higher than the measured, for this is an upper bound method. But the errors are less than 10%. And Fig. 4 (b) shows that the calculated rolling torques approach to the measured values. Thus, it can be concluded that this new linear integration is available for bar rough rolling analysis. 2) Neutral angle and stress effective factor versus 1/ h for various values of m As shown in Fig. 5, neutral angle increases with the increase in friction when l/h=O. 8; when the
Discussion of Results
Morgan data from Stand 1 to Stand 5[12] are shown in Table 1. S80 high-quality carbon steel billet with dimensions of 115 mm X 115 mm was rolled through first strand of the roughing train at 1050 "C. The deformation resistance'P'' is given by ,
R. = 135. 4exp(3. 751-2. 974 T) •
"
O.034T+O.1089
I leO I
.
[1. 312'1 O~ 410.2181 -0. 78e]
(23)
d £ormation . temperature. h T = t+273 were 1 000 ; an d tz iIS the e
With the equations above, programming by C language and using golden section search method, the rolling torque and roll separating force are minimized with m = 1. O. The results are shown in Table 2. It shows that the rolling force and torque assignment among the first five stands are nonuniform, Table 1 Roll gap/
Vol. 17
Morgan data of rough stands
Height/ mm
Roll diameter/ mm
Motor speed/ Cr > min-I)
Gear ratio/
mm
Depth/ mm 32.5
85
460
283. 65
36. 14
0.1663 0.2132
Stand
Shape
1
Square
20
Exit speed/
%
(rn
>
S-l)
2
Oval
20
26
72
483
306. 27
34.071
3
Round
14
36.5
87
457
492. 33
34.679
O. 2978
4
Oval
14
17.5
49
480
335.92
18.774
O. 4322
5
Round
10
26
62
479
326.32
12.772
O. 5865
Table 2
Calculated results of roughing train
Stand
lRM
2RM
3RM
4RM
Reduction/ %
26. 1
15.3
40.7
43. 7
41. 7
1.2134
0.9783
1. 1869
J. 3912
J. 2832
".
5RM
l/h
0.80
0.87
J. 02
J. 50
J. 28
Rolling fo.rce/kN
677. 3
419.4
456. I
677.0
502. I
Rolling torque/(kN • m )
71. 2
39.0
62.0
41. 1
5 J. 6
Note: h= (h,+h o ) /2.
45
750 (a)
80 (b)
40
Z
35
i::
30
~
.£ ~50 1>.0 .5
25
==0
20
I:l:: 450
15 350 lRM
~ ~
0-
.5 >:
0
c::l
.g'"
QJ
I:l::
,-.,
70
~
60
5-
50
E
~
~
S
i
I:l::
40 Measured rolling torque
30
10 2RM
3RM
4RM
5RM lRM Pass number
2RM
3RM
4RM
Fig. 4 Relationship among the measured and calculated roll separating force and reduction (a) and rolling torque (b) and roll numbers
5RM
Linear Integral Analysis of Bar Rough Rolling by Strain Rate Vector
Issue 3
value of m is less than O. 5, the relation between an and m is nearly linear. The value of n, decreases at first and then increases with increasing the value of 1/ hand m , as shown in Fig. 6. When the value of 1/h is close to O. 9, n, reaches the minimum value. 0.18
.------------------:-1
References : [IJ [2J [3J
0.16 [4J 0.14 [5J
0.12 0.10,
[6J 0.08
_ _~_ _....__ 0.6 0.2 0.4 m
'--_~
o Fig. 5
____"_ _.......... 0.8 1.0
The relationship between neutral angle a. and friction factor m
[7J
[8J
1.5 . - - - - - - - - - - - - - - - - - - - , [9J 1.3 [10J
[IlJ
0.9 [12J
0.7 L . . . . . - - - - 7 - - - -.......- - -.......- - - ' 1.4 0.8 1.0 1.2 llh [l3J
Fig. 6
4
Relationship between stress effective factor n. and Ilh for various values of m
Conclusions
1) A new linear integral method is proposed for bar rolling analysis. The calculation during rough rolling shows that the calculated roll separating forces are a little higher than the measured ones, and the rolling torque values approach the measured ones. but the maximum error is less than 10%. 2) The reduction of the first five stands is nonuniform, which results in the uneven rolling force and torque among these stands. Therefore. it is necessary to increase the reduction of the second pass and to balance the loading of each stand. 3) The solution above is available for bar rolling in the roughing train, and after stand 6. the error percentage will be significant.
• 33 •
[14J
[15J
[16J
[l7J
[18J [19J
Khaikin BE. Kisilenko I A. Tranovskii I Y. Pressure of Metal on Rolls in Section Rolling [JJ. Steel USSR, 19710): 884. Arnold R R, Whitton P W. Spread and Roll Force in Rod Rolling [JJ. Metal Tech. 1975(2): 143. Shinokura T • Takai K. Mathematical Models of Roll Force and Torque in Steel Bar Rolling [JJ. Tetsu-to-Hagane , 1986, 72(4): 1870 (in Japanese). Ahmed Said. Lenard J G. Ragab A R. et aI. The Temperature. Roll Force and Roll Torque During Hot Bar Rolling [JJ. J Mater Process Technol , 1999.880-3): 147. Oh S I. Kobayashi S. An Approximate Method for a Three-Dimensional Analysis of Rolling [JJ. Int J Mech Sci. 1975. 17 (4): 295. Kazunori K. Tadao M. Toshihiko K. Flat-Rolling of RigidPerfectly Plastic Solid Bar by the Energy Method [JJ. J Jap Soc Tech Plasticity. 1980,21(231): 359 (in Japanese). Pittman J FT. Zienkiewicz 0 C, Wood R D, et aI. Numerical Analysis of Forming Processes [MJ. New York: John Wiley and Sons Ltd. 1984. Kim N. Lee S M. Shin W, et aI. Simulation of Square-to-Oval Single Pass Rolling Using a Computationally Effective Finite and Slab Element Method [JJ. J Eng Ind. 1992(14): 329. Park J J, Oh S I. Application of Three Dimensional Finite Element Analysis to Shape Rolling Processes [JJ. J Eng Ind. 1990 (112):36. Kim N, Kobayashi S. Allan T. Three-Dimensional Simulation of Gap Controlled Plate Rolling by the Finite Element Method [JJ. Int J Mach Tolls Manuf , 1990. 30( 2): 269. Montmitonnet P. Chenot J L. Bertrand-Corsini C. et aI. A Coupled Themomechanical Approach for Hot Rolling by a 3D Finite Element Method [JJ. J Eng Ind. 1992(14): 336. LI Chang-cheng . REN Yu-hui , ZHANG Li-Ien • et aI. Measurement and Study on 500 mm Rougher Mill of High Speed Untwist Rod Mill [JJ. Iron and Steel. 2002. 37< 2): 59 (in Chinese). ZHAO De-wen, WANG Guo-dong. BAI Guang-run. Theoretical Analysis of the Wire Drawing Through Two Roller Dies in Tandem [JJ. Science in China: Series A. 1993. 36 (5): 632. WANG Guo-dong, ZHAO De-wen. Modern Materiel Forming Mechanics [M]. Shenyang , Northeastern University Press. 2004 (in Chinese). ZHAO De-wen.' Mathematical Solution of Continuum Forming Force [M]. Shenyang , Northeastern University Press. 2003 (in Chinese). ZHAO De-wen, GUO Chang-wu , ,LIU Xiang-hua, Surface Integral of Three-Dimensional Velocity Field for Square Bar Drawing Through Conical Die [JJ. Trans Nonferrous Met Soc China. 1996. 6(3): 131. ZHAO De-wen. XIE Ying-jie , LIU Xiang-bua , et aI. ThreeDimensional Analysis of Rolling by Twin Shear Stress Yield Criterion [JJ. Journal of Iron and Steel Research. International. 2006, 13(6): 21. ZHAO Zhi-ye, Metal Forming Mechanics [MJ. Beijing: Metallurgicallndustry Press. 1987 (in Chinese). ZHAO Ii-hua , GUAN Ke-zhi. Resistance of Metal Plastic Deformation [MJ. Beijing: Mechanical Industry Press. 1989 (in Chinese).