EKELUND'S FORMULA
10 input data (8+2):
H = entry thickness, mm
h = exit thickness, mm
B = entry width, mm
t = rolling temperature, ºC
v = exit speed, m/s
C = Carbon content %
Mn = Manganese content %
Cr = Chrome content %
plus
r = roll radius, mm
RM = roll material (steel=1; cast iron=0.8)
Calculated data:
f = coefficient of friction
m = effect of external friction
c = influence of speed on viscosity
e = viscosity of rolled metal, kg*s/mm^2
u = strain rate, 1/s
k = yield strength, kg/mm^2
KE = mean roll pressure (Ekelund), kg/mm^2
KE=(1+m)*(k+eu) <<<< FORMULA! (1)
where:
f=RM*(1.05-0.0005*t)
m={1.6*f*SQR[r*(H-h)]-1.2*(H-h)}/(H+h)
k=(14-0.01*t)*(1.4+C+Mn+0.3*Cr)
e=0.01*(14-0.01*t)*c
c=1; for v>6 SKF suggested that Table I be used
u=2*v*SQR[(H-h)/r]/(H+h)
v c
6 to 10 0.8
10 to 15 0.65 Table I
15 to 20 0.60 -------
To give the computer a continuous function, I suggested that
the following formula be used:
c=1.0942*EXP(-0.03*v) for v>3 (2)
c=1 for v<=3
Now let's see two examples, with c calculated from (2):
EXA: flat 200x40 reduced to 30 mm
EXB: square 40x40 reduced to 30 mm
In both examples, r=140, t=1000, v=14, C=1, Mn=1, Cr=1.
In both examples, roll material is pearlitic cast iron.
The results of the calculations are:
EXA EXB
f 0.4400 0.4400
m 0.2049 0.2049
k 14.800 14.800
c 0.7189 0.7189 (SKF: 0.65)
e 0.0288 0.0288 (SKF: 0.0260)
u 106.95 106.95
KE 21.544 21.544 (SKF: 21.183)
PHILOSOPHY: Ekelund's method is based on theoretical
evaluation of yield strength k (Formänderungsfestigkeit, in
German) and of friction between rolls and rolled stock.
Just one thing to notice: KE (Formänderungswiderstand, in
German) does not depend on bar width, i.e. B may be deleted
from the list of input data.
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IZZO'S FORMULA
11 input data (8+3):
First group of 8 data: same as Ekelund's
plus
d = roll diameter, mm
HSC = roll surface hardness, Shore C
KSC = roll material factor
The value for KSC can be found in the following table:
0 (Non-graphitic steel); 0.5 (Graphitic steel);
1 (Nodular pearlitic cast iron); 1.5 (Nodular acicular cast iron);
2 (Nodular martensitic cast iron); 2.5 (Chilled cast iron);
3 (Indefinite Chill cast iron); 3.5 (Hi-Chrome cast iron);
4 (Tungsten carbide).
In our examples EXA and EXB, d=280, HSC=60, KSC=1.
Calculated data: EXA EXB
hm = (H+h)/2 (average stock thickness), mm 35
ld = SQR[d*(H-h)/2] (projected contact length), mm 37.4166
r = (H-h)/H (fractional reduction) 0.25
R = 100*r (percentage reduction) 25
D = LN(B/H) 1.6094 0
KEM = (4.2+C+Mn+0.3*Cr)/4.77 1.3627
TH = 0.142+0.023*[(t-800)/200]^2.34 0.1650
T = 1.5-1000*v/d -48.5
S = 100*h/d 10.7143
O = [1-(2-KEM)^17]/2 0.4998
Kt = SRC roll pressure at 1000 ºC, kg/mm^2
Ks = SRC roll pressure, kg/mm^2
a1 = f(1000*v/d,t), effect of relative rolling speed
a2 = f(ld/hm), effect of the outer zones
a3 = f(B/H,R), effect of form factor
a4 = f(R,HSC,KSC), effect of roll material
a5 = f(h/d,KEM,t), effect of stock material
KI = mean roll pressure (Izzo), kg/mm^2
SRC stands for "Standard Rolling Conditions":
a1, a2, a3, a4 and a5 represent the "deviations"
from the SRC; in SRC each of them equals 1.
SRC are:
- 1000*v/d about 1.5 1/s
- ld/hm>=1
- B/H about 3
- Steel rolls with HSC about 40 ShC
- Rolled stock with C about 0.1%
(See "Iron and Steel International", Feb. 1974.)
KI=Ks*a1*a2*a3*a4*a5 <<<< FORMULA! (3)
where:
Ks=Kt*EXP(A)
Kt=(3+0.35*R)/S+10+0.025*R 11.7217
A=m*a*n*y
m=1.797/(S^0.12-0.517) 2.2125
a=(1000-t)/1000 0
n=t/1000-0.3 (for S<=0.6)
n=(t/2000+0.35)+(0.65-t/2000)*
SIN(300*S-270) (for S btw 0.6 and 1.2)
n=1 (for S>=1.2) 1
y=1.5*[(1100-t)/458]^[(900-t)/243.5]-
1.8*(S+2)^[(t-1000)/100] (for t<1000)
y=1 (for 1000<=t<=1100) 1
y=1.6*[(t-1000)/426.5]^[(t-1200)/127.3]-
4*(S+4)^[(1100-t)/100] (for t>1100)
A 0
EXP(A) 1
Ks=Kt*EXP(A) 11.7217
a1=1+TH*LN(1-T) (for 1000*v/d>=1.5) 1.6438
a1=1-TH*T-0.257*T^3+0.0547*T^5 (for 1000*v/d<1.5)
a2=1 (for ld/hm>=1) 1
a2=(ld/hm)^-0.4 (for ld/hm<1)
a3=0.797-[(60-R)/100]^3 (for D<=0) 0.7541
a3=0.797-[(60-R)/100]^3+0.247*D (for D btw 0 and 1)
a3=1.037-[(60-R)/100]^3+0.007*D (for D>=1) 1.0054
a4=1-0.5*(HSC-40)*(0.0001*R+KSC/1000) 0.965
a5=1+[(1200-t)/220]*O*[1-0.335*LN(S)] (for S<=14) 1.0934
a5=1+[(1200-t)/220]*O*(251/S^2.91) (for S>14)
KI=11.7217*1.6438*1*1.0054*0.965*1.0934 = 20.440 kg/mm^2 (EXA)
KI=11.7217*1.6438*1*0.7541*0.965*1.0934 = 15.331 kg/mm^2 (EXB)
PHILOSOPHY: Izzo's method is based on experimental values of
roll pressure (Formänderungswiderstand) in "standard rolling
conditions", corrected by a number of factors accounting for
"actual rolling conditions".
Another thing to notice: with formula (3) KI does depend on
bar width, which is confirmed by experience: a square (EXB)
is easier to roll than a flat (EXA).
--------
But wait a minute: few people know that there is also an
IZZO'S SIMPLIFIED FORMULA
The simplified formula coincides with formula (3), only with a
different expression of Ks. All other parameters are unchanged.
(See my paper in Archiv für das Eisenhüttenwesen, Februar 1976,
"Bemerkungen über ein Verfahren zur Errechnung des
Formänderungswiderstandes beim Warmwalzen").
The simplified expression of Ks is:
Ks=Kt*Z
In the original formula Kt is multiplied by EXP(A)=f(t,S);
in the simplified formula Kt is multiplied by Z=f(t), with
Z = 4.49 - 0.00349*t (for t<=1000 °C)
Z = 3.15 - 0.00215*t (for t>1000 °C)
Note that, for t = 1000 °C, the values of EXP(A) and Z coincide:
both are =1.
The simplified formula can be precious for manual calculations;
but it can only be used if two conditions exist:
1) S > 1
2) t >= 900 °C
We could try to apply the simplified formula to the above illustrated
example. But we would obtain no significant results, because, as we
said above, for t=1000 °C both formulae multiply Kt by 1.
We can try raising the temperature to t=1100 °C. In this case,
EXP(A)=0.8015 and Z=0.7850. Correspondingly, Ks=9.3949 kg/mm^2
(basic formula) and Ks=9.2015 kg/mm^2 (simplified formula).
However, from these results we should not infer that the simplified
formula underestimates rolling loads: in fact, for t=1125 °C,
EXP(A)=0.7068 and Z=0.7313.
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