Roll Pass Design Process

' • ' I I • . . _·, j ~ ... ··--· .,. •• ' ' y-> ----.. ~

INTRODUCT/ON - 113

Main factors affecting the rolling parameters

Geometrical factors

Rolling speed

Steel under rolling and

rolling temperature

Rolling rolls

Other factors

• ENTRY- EXIT SHAPES AND DIMENSIONS

_ _ ..,. • CONTACT LENGTH AND CONT ACT AREA • ENTITY OF REDUCTION (STRAIN)

•

• SPEED OF DEFORMA TION (STRAIN RATE)

• CHEMICAL COMPOSITION

• MECHANICHAL PROPERTIES • THERMAL PROPERTIES

• CHEMICAL COMPOSITION (mechanical- thermal properties) • ROLLS TEMPERATURE AND SURFACE WEAR

•DIAMETER

• FRICTION BETWEEN ROLLS AND STEEL UNDER ROLLING • SCALE THICKNESS ON THE BILLET SURFACE

• RECRYSTALLISATION AND GRAIN STRUCTURE EVOLUTION • TENSION BETWEEN THE ST ANDS

INTRODUCTION- 213

A good rolling mill simulation program must have

Mechanical

model

Thermal

model

Deformation

model

- - - - i ....

•Thern1o-mechanical properties for different steels •Yield stess model

• Rolling force - torque - power calculation models

Drop of temperature during the rolling process due to: • interstand cooling (radiation)

• water cooling effect • contact with the rolls • deformation heat

Calculation of grooves shape and dimensions on the stands: • spread model

• geometrical model

This document contains proprietary information of Danieli & _{C. S.p.A., not disclosable, not reproducible. All Rights Reserved.}

.,,., .. ,. ,.,_ ... ,, .... ,. '

INTRODUCTION- 313

IN ANYCASE

'

All the calculation models must be tuned

•USINGROL

•USINGFEM

YIELD STRESS -1114

THE STEEL YIELD STRESS IS FUNCTION OF:

• Chemical composition of the steel

• Strain entity of deformation

• Strain rate speed of deformation

• Temperature

YIELD STRESS- 2114

STRAIN DEFINITION

L-" /

,

, /

,

/ /

,

/ / 1 1 1 1

From the law of constancy of volume

V ₀=V ₁ --> h₀b₀1₀=h ₁b₁1₁

2

DEFINITIONS

- 1 \ dh - h, - ho h¡ dh

=

ln(h¡' rph

=

&h- -• h h ' ho h. ho ho o o - 1 b¡ db - b, - bo b¡~db

=

ln( b,) rpb

=

&b- -b _bo bo b bo b¡) o - 1 \. dl - ll -lo 11~dl l¡ &, - -lo rp,=

=

ln( ) l . • _{l lo} O /_{0 lo}

With the logarithmic stains it' s possible to ha ve a better description of the state of deformation

With the logarithmic stains it' s possible to follow a multiphase deformation as we find on the

rolling milis

YIELD STRESS- 3114

INFLUENCE OF STRAIN ON YIELD STRESS

Kan

hu

YIELD STRESS- 41 14

STRAIN RA TE DEFINITION

•dt: time spent to complete the deformation

•d<p: logarithmic strain

•vh:

_{speed of deformation in the h direction}

• (INST ANT ANEOUS)

dl

h

dl

h

, - 1

t '

1

t

dh

(jJ m - - (jJ

dl - -

==

1 Ho

lo lo

h

t

(AVERAGE)

From the law of conservation of volume:

V ₀=V₁ --> h₀b₀1₀=h₁b ₁1₁

(jJ~

+

cp;

+

(jJ~ ==o~ (jJ~

==

cp;

+

(jJ~

(jJ

~m

+

{jJ

l~

+

{jJ

~m

O

=>

{jJ

~m

, ,

==

l(j)lm

+

{jJ bm

YIELD STRESS- 5114

INFLUENCE OF STRAIN RATE ON YIELD STRESS

~h

~ti< ~t2

=>-.

~t 1

L1h

L1t:

_{time to}

complete the

deformation

The strain rate influence increase with the temperature so in hot rolling it assumes a great importance and takes to a sensible increase in the yield stress of the deformed

steel. At high rolling speeds (i.e. on the finishing passes of wire rod) it _{has a great}

injluence and it '_{s still difficult to find in literature good material properties curves for}

strain rafe values > 100 s-1.

YIELD STRESS- 6114

INFLUENCE OF TEMPERATURE ON YIELD STRESS

~h

T:

temperature

With an

increase

_{in temperature the yield stress decrease. The injluence of temperature} is very strong and affects also the strain rate influence. The temperature has also directs effects on the steel structure with phase changes and recrystallisation phenomena and

injluences al so the friction coefficient va fue ..

YIELD STRESS- 7/14

EXAMPLE of yield stress curve for a low carbon steel

: iji~tii~~~~~~~~~~~~~~~ ZD Z!l 210 :m 100 100 170 100 -1s:> ::t::1<Kl 13) 12) 110 100 00 00 70 00 cp=0.4 cp'=IS ~ ~~t;~ti==~==t===t=~t=:jc==dt2~

!Hl 1aD 1CID 1100 11ffi

T <ID 270 LOO 21l ~ ZD Z!l t-r- - + - - ...:..;,.~+----+--f---.:-..:...;. 210 :m 100 100 170 ~--+~~~---r~~~ S;2100 18) 1<Kl H-'---+---'-~f--4---,---4,...._,,...:_..,. ~~ <p =0.8 <p'=l5 13) +.4~~ ~:.,....,.+.-:--..,..,_._..., 12) 110 100 4+...:...-f-,...-..;...:; l--"'-4"-'-:...i...¡.,.,..._,.:._ ;.;,.._ -t-;;...,.__,.; ,-..--'-' ~..;...¡ 00 00 70 00 S) <Kl ~~~----~--~----4~--4-~~~~~~-+--~

ffi) 9l) _{!Hl 1aD 1CID 1100 11ffi}

T ~ ~~~~~~~~~~~~~~~~~~~~~~~ This docume LOO 2S) 240 ZD Z!l 210 :m 100 100 -170 ::t::100 18) 140 13) 12) 110 100 00 00 70 00 S) 40 700 T cp =0.4 cp'=-'5 11 1aD 1CID 1100 115) s Reserved.

YIELD STRESS- 8114 1 & = -pl E p cr

STRESS-STRAIN RELA TIONSHIP

1 1 : 1 1 1 : 1 1 & = -p2 E p

In the plastic range the deformed

metal flow in certain directions

under the actions of the applied

forces; the relationship between

stresses and strains in the three

main directions can be expressed

with the following expressions:

1

& =

-p3 E p

-The coefficient 1/m can be assumed to be equal to 0.5 (m=2)

-EP

is the modulus of plasticity

YIELD STRESS- 9/14

YIELD CRITERIA

~f _{in laboratory tests represents the stress at the yield point of metal under uniaxial strain}

conditions and in ideal contest (low influence of friction) and must be recalculated

according to the actual state of tension using the plasticity criteria (i.e. Von Mises, Nadai).

Por a plain state of strain with no lateral spread (as in flat rolling) whith

EP₂

=0 will be:

crccr₃

= 1.15*K

₀f

=K

Por

cr_2=cr3

the yield criterion become:

crccr3

=

_Kof

In a general case the expression for the yield criterion will be:

crt-cr3

ll*Kor

K

•11 varies between 1 and 1.155 depending on the deformation method (1.155 plane deformation)

• K=11~f _{is the constrained yield stress for the given method of working}

• Kof

_{is the stress at the yield point of metal under an uniaxial state of stress}

•

YIELD STRESS- JO/ 14

Resistance of metal to deformation real case

In an ideal situation there is no friction

between the working tools and the work piece. In

this case the stress produced by the vertical force gives rise to uniform distributed vertical forces over the whole cross section and the specimen can deform freely in the lateral direction and it is valid the precedent expression.

In the real case there is always an influence of friction

on the distribution of tensions

inside the specimen. Frictional forces on the surface of specimen restrict the lateral flow of metal and hence the pressure of the tool must overcome not only the constrained yield stress

11

*Kot

but al so the frictional forces.

A complete expression for the yield stress will be then the following:

-

~:

_{resistance of metal to deformation in the real case}

- 11K

₀f:

constrained yield stress for the given method of working

- Kr:

resistance of metal to flow that consider the influence of friction between the

compressive tool and the deformed specimen

YIELD STRESS -11114

Influence of friction

high plastic deformation region p ' 1 ' 1 1 1 ' 1 '

~m

--' 1 ' 1 . '

regions influenced by friction (small in laboratory tests)

compressive tool

neutral plane relative speed=O

• In the regions influenced by friction the plastic deformation is limited because friction has a contenitive effect on the plastic flow.

• The contenitive effect decreases if the distance between the pressed faces increases.

• The contenitive effect is at its maximum at the centre of the specimen ( on the neutral plan e)

• Kwm: _{mean value for the yield stress}

•

YIELD STRESS -12114

Influence of friction coefficient

~V=

llKor + Kr

' 1 ' ' 1 1 ' 1 ' 1 ' 1

K

-''' - - - w l

K

w2----' 1 ' 1 ' 1 ' 1

The region influenced by friction is smaller in 1 than in 2

YIELD STRESS -13114

Influence of the specimen heigth

K - - - w2

---

-

-

----flK o 1 o o 1 o 1 o 1 o 1 . ' o _o

'

_'

The region influenced by friction is smaller in 1 than in 2

YIELD STRESS -14114

Influence of the ratio h/b

~V=

TJK₀r+ Kr

---

_ _ _ K w l ----. ' 1 1 _' ' 1 ' • • _• '

_'

_'

--->

_... K <K ---':::: _{~1 ... ~2 -}

K <K

_{wl w2}

The region injluenced by friction is smaller in 1 than in 2

FORCE -117

ROLLING FORCE CALCULATION

rolling force

contact area

specific pressure

P=Fd*

(N)

(mm

2)

(N/mm

2)

There are a lot of formulae to determine Kwr _{in sorne of these all the effects describe befo re ( strain}

-strain rate- temperature- friction) are considered separately in others the things are more confused.

Only one thing is certain: none of them is true and none of them gives good results in all the situations .

•

a _{llil~§JSU!Jlli;.lli2!b_!Uill;Inru ~~::....!.!;~~~~!.!n~~~~ )...:~~~~}

~~-=-=-x _1, x_{2, ...• , X0 are sorne relevant geometrical and physical parameters characterising the rolling process;} for example: _{x 1} = Ld 1 hm

x₂ = D/hm

x₃ = T (temperature)

FORCE -217

Geometrical definitions

Bite angle:

h -h

h -h

!J.h

R

cosa

=

R -

0 1

=>

e os

a

=

1-

0 1

=

1

-w

w

2

2Rw

Dw

Contact length:

l

_d = 1 --- --- _{---~---~---- ---~---}V

Neutral plane:

Va h ---tli- n • •

At the neutral angle the peripheral roll speed is equal to the workpiece speed:

A

₀

*v

₀

=An *vn =A

₁

*v

₁

---> h

₀

*v

₀ =

hn *vn= h

₁

*v

₁

--->v

₁

>vn>v

₀

hn

=

Dw(l-cos8)+h

₁

vn

vrcos6

Forward slip:

V - V V S =

1

r=

1_1

f V V s = hn cosb" _ 1 f h 1 r r .

This document contains proprietary information of Danieli & C. S. p.A., not disclosable, not reproducible. All Rights Reserved.

V

FORCE -3/7

Kwr

in flat rolling plain deformation

In this case:

• Free spread and very small ~ W

•Ratio wlh very high (~30-40)

• Entry and exit geometry well known •11=1.15

• Pressure distribution on the width is constant

• 11

*Kof

_{can be considered constant along the}

contact length.

• The term ~ _{has not a constant distribution}

along the contact length with a maximum on the

neutral plane

• Kwr is calculated considering a mean value for

~ _{( with friction coefficient constant along the contact}

length). friction (Kr)

-1 o 1 o 1 o ~vr ho --- --- ; --- ----1-r- roll~g direction Wo --- ---··--~ R: roll radius h_{0 :} entry height h_1: exit height lct: contact length 1 o 1 o .--- _{----· neutral plane} 0 1 o o W¡ --- ---· w_{0 :} entry width w _1: exit width Fct: contact area

FORCE -4/7

Kwr

in long products rolling

(3-axial strain)

In general all the calculation formulae to calculate Kwr _{have been developed for flat rolling and}

must be corrected for long products rolling.

In this case are not satisfied the conditions seen before for flat rolling, in fact:

• NOT free spread in sorne cases, where there is a strong constraint of the lateral spread due to the grooved rolls, and almost free spread in other cases.

• Ratio w/h small compared to flat rolling (~1 _{also <1).}

• Entry and exit geometry is not well known. • 11 value not well defined (1 <11<1.15)

•

With these conditions the rolling

pressure varies not only along the

rolli

·

·

bu

spread direction .

•

This document contains proprietary information of Danieli & C. S.p.A., _{not disclosable, not reproducible. All Rights Reserved.}

w

FORCE- 5/7

Kwr

in long products rolling

(3-axial strain)

S

ummarizin for Ion

..

roducts:

• In the calculation of Kwr _{it' s necessaries to consider al so that the shapes of the entry and exit}

grooves doesn't have a constant height along the spread direction (equivalent rectangles). This assumption takes to errors on the calculation of strain and strain rate values and consequently on Kro.

• In the calculations is also necessaries to consideran average diameter (working diameter) because the height of the billet is not constant along the spread direction (Errors in Kwr and torque calculation).

• Another factor to consider is the relative change of shape from the entry to the exit plane (additional friction coefficient like for example in the Siebel's method).

• Also the determination of the contact area is much more complicated than in flat rolling and will introduce uncertainties on the final calculation of the rolling force.

FORCE- 6/7

Equivalent rectangles

(LONG PRODUCTS)

A ₀: entry area 1 -- --' 1 1 ' 1 ' 1 ' 1 ' 1 ' 1 ---·---1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 Womax hoeq=Aofw Omax hleq=A/w lmax ~Heq=hoeq -hleq ---- hüeq

~

W

eq=W lmax -Womax

A ₁: exit area 1 ' 1 ' 1 ' ' ' 1 ' 1 ' 1 ' 1 --- ---r--- ---₁ ' 1 ' 1 ' 1 ' ' ' 1 ' 1 ' 1 wlmax

( equi valent entry height)

(equivalent exit height)

(equivalent draught)

( equivalent spread)

FORCE- 717

hlmax

W orking diameter

(LONG PRODUCTS)

Dmin Dmax G --- - --- - ~--- ---1 ' 1 ' 1 ' 1 ' 1 1 ' 1 ' 1 ' 1

<

wlrnax

>

l

_deq --1 ' 1 ' 1 1 ' 1 ' 1 ' --- ---~---·

<

' 1 ' 1 ' 1 ' ' 1 ' 1 wleq

_>

D max: nominal diameter of the rolling rolls

G: gap between the rolling rolls

•

TORQUE -112

Rolling torque integral method

Elementar rolling torque: dT = -r ₈R2d<p

a

S

-·-·---- _{-·--- ---}

---a _rp"

Total rolling torque: T

=

2R 2 wrs drp- drp = 2R 2wr s(a- 2rpn)

rp/1 o

• 2 is for the 2 rolls

• w average width of the contact area

• <l>n neutral angle

This _{·on leads to · becaus} •

and rs _{it 's d{tficult to express too so usual/y it 's not used.}

TORQUE- 212

Rolling torque simplified method

V ery often in engineering calculations, the roll torque

is calculated with the assumption that the roll force

resultant acts at a distance X from the exit plane.

Total rolling torque:

T=2*P*x

p

R

--- --- ---

---If x is expressed as a fraction of the are of contact:

T 2*P*a*l

_d

1

o

1

o

a is the lever arm coefficient that is not a phisically determined quantity, but a coe.fficient usedfor a simpl{fied calculation procedure.

a is usually set =O. 5 or can be expressed as a function ofR/h ratio or as a .function of draugth

POWER -111

Calculation of rolling power

- T: rolling torque

- Dw:working diameter

-->

R=DJ2

- v: rolling speed

- ro: rolls speed

--->

ro=v/R (rad/s)

- rolls RPM:

N rol=

ro *60/2rc

=

60*v

1 2nR

Por the calculation of motor RPM:

Nmot =N rol *'t

1:

motor gear ratio

Total rolling power:

P

=

T*

_ro

=

T*2

rc

*Nrot

/60

MOTOR UTILISATION DIAGRAM

- + - - - -

---.--

-1

pavailable p max prol 1

o

prol 2

o

base RPM _maxRPM

TEMPERATURE -115

THERMAL FACTORS

One of the most important factors in hot rolling is the temperature of the work piece.

Inaccurate estimation of the billet temperature along the rolling train takes to errors in

estimated rolling loads and in the final product requirements ( quality and metallurgical

properties).

A good calculation model of the rolling parameters must ha ve a good thermal model

The principal factors affecting heat balance and thus the temperature

of the rolling billet are:

-heat conduction inside the workpiece

-radiation loss to the surrounding medium

'

-heat conduction to the rolls

-heat gained by work done in mechanical deformation

-heat loss due to rolls and guides cooling water

TEMPERATURE - 215

Heat conduction inside the workpiece

The temperature gradient inside the workpiece is a function of time and

can be calculated by solving the heat conduction equation:

k:

p:

e:

d:

at

thermal conductivity

density of the steel

speacific heat

thermal diffusivity

d==

k

pe

(W/m°C)

(kg/m

3)

(J/kgoC)

(m

2

/s)

This equation can be resolved applingfor example the finite difference method. The thermal properties of the steel are al! functions of the temperature itself.

TEMPERATURE -3/5

Radiation loss to the surrounding medium

The heat radiation loss is the most significant factor affecting the temperature drop

during hot rolling and can be resolved applying the equation of Stephan-Boltzman:

E:

S:

A:

p:

e:

ar

==

at

cVp

steel emissivity

Stephan-Boltzman constant

area of radiating surface

density of the steel

speacific heat

volume of the billet

billet temperature

ambient temperature

(-)

(J/m

2

s°K

4) (m2)

(kg/m

3)

(J/kg°C)

(m3) (oK4) (oK4)

The heat loss from the surface due to radiation is directly proportional to the emissivity of the steel that varies depending on temperature and amount of scale present on the surface. In addition to the radiation loss there is also heat loss due to natural convection

TEMPERATURE - 4/5 a:

p:

e:

Fct:

T:

Trol:

A:

v:

Heat conduction to the rolls

A*v*p*c

heat transfer coefficient

steel density

speacific heat

contact area

billet temperature

roll temperature

section area

·

rolling speed

(kW/m

2

_°C)

(kg/m

3)

(J/kg°C)

(m2) (oC) (oC) (m2)

(mis)

Heat supplied to the rolls is proportional to the contact area and temperature difference between the billet and the rolls, and also to the contact time.

The proportional factor is the a heat transfer coefficient of difficult determination and its

value depends by the amount of sea/e present on the two surfaces in contact, by the contact pressure, by rolls and billet thermal properties.

TEMPERATURE- 5/5

Heat gained by work done in mechanical deformation

~:

hl:

h2:

p:

e:

11:

K

_w

pe

steel resistance to deformation

entry heigth

exit heigth

steel density

speacific heat

efficiency

(N/mm

2)

(mm)

(mm)

(kg/m

3)

(J/kg°C)

(-)

The temperature gained due to mechanical deformation is a function of the yield stress of the rolled material and of the reduction ratio.

ANNEXJ -117

N cutral angle

At the neutral angle the peripheral roll speed is not only equal to the workpiece speed but also another phenomenon occurs, the frictional resistance forces change their direction and the resistance to deformation of the rolled steel reaches its maximum.

1 o 1 o 1 . V ---~---~--··-- _~ ---~---V '

N cutral plan e:

A

₀

*v

₀ =

An *vn

=A

₁

*v

1

---> h

0

*v

0 =

hn *vn

=

h

1

*v

1

--->v

1

>vn>v

0

hn

=

Dw(l-cos8)+h

₁

vn

vrcos8

•

ANNEXJ -217

Friction coefficient influence on neutral angle and yield stress

<

f=O

---.

contact length (ld) neutral plane position (fl

<

f2) ---> (hnl

<

hn2) ---> (82 > 81)---> (Sf2 > Sn) '

Jf

friction increases the position of the neutral plane moves toward the entry plane

(forward slip increase) and iffriction decreases the neutral angle decreases. For

¡-o

the neutral plane is on the exit plane.

ANNEXJ -317

Strain influence on neutral angle and yield

stress

1

- - - -

neutral plane

position

ANNEXJ -417

Roll diameter influence on neutral angle and yield stress

-- -- -- -- neutral plane position

ANNEXJ -517

Forward tension influence on neutral angle and yield stress

O" Al

a

Al <

a

A2 <

a

A3 O"A2 O"A3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

An increase of the forward tension (O") moves the neutral plane toward the entry plane.

38

ANNEXJ- 6/7

Backward tension influence on neutral angle and yield stress

1 1 1 1 1 1 1 1

An increase of the backward tension

(a-¡)

moves the neutral plane toward the exit plane

lf there is contemporary effect of backward and forward tension the relative diagram will be a combination of the two diagrams se en befo re.

ANNEXJ- 7/7

Yield stress function of

(ld/hm)

1 1 1 1 1 • 1 . ₁ • _• • _• 1 • . • • _• 1 • . • _• • . 1 · .. · .. 1 • 1 1

THE RATIO IN CREASE FOR INCREASING FRICTION

.,..-" ....

-~~ • ~~ \ ....

--,

.... - '

---~- ...

-

--

... .,.,..

--

··· .... . .... . ... -

.

-

...

--

-. . .. .... .- ... .... ....

---

___ .... . .. -- - - ... ··· ,.. ....

--

" ...

,

.. " ....

--

... .... .... . .. , , , - ··· ...

-

_ ....

-

··· .... - ... _ ....

-

....

-

.... ·· - _, ....

-

... . ... _, ... ....

-

··· -... ···

-

... :""' _ ....

-

··· .

--

- _... ... "'1 .... ~ 1 :.~~ -.... . ... - ...

....

-

.... _{LINE FOR HOMOGENEOUS}

DEFORMA TION ... " " . " " '!:::. ... " ·" • • • • • • • . . • 1 . • • • • •

LOW INFLUENCE OF INTERNAL FRICTION AND GREA T INFLUENCE

OF THE DEFORMA TION OF THE OUTERZONES

The ratio KjK₀

r

1 a1so for 1/hm~ l.

•

STRONGER INFLUENCE OF INTERNAL FRICTION AND NO INFLUENCE OF THE OUTER ZONES

For all the other conditions the ratio is KjK₀t> 1; in fact:

Kw = 11 K₀f + Kr for 1/hm> 1

~ = 11K₀f + Kr + Kext for 1/hm<l

Ld/Hm

The diagram that represents a state of inhomogeneous

deformation is the

combination between the two red dotted lines that

represents respective1y the effect of the extema1 zones

and of the intema1 friction on

the ratio KjKm.

For a homogeneous

deformation the ratio is ~ 1

(no friction and no shear

stresses and deformations).

ANNEX2 -116

Golovin-Tiagunov

Km:

steel resistance at

20°C

(function of

%

_C)

f: friction coefficient

(~1/3)

Ct: corrective coefficient for the rolling temperature from diagrams

Notes

•A too much simplified method

• For the calculation of

Km

only carbon content is considered

• It doesn' t take into consideration the strain rate

ANNEX2 -216

Tselikov

S/2

K

_w

1

K.r

_{0: steel resistance function of %C and temperature}

11=1.155

Eh: relative strain

8=2f*lct */L1h

-1

*n

₂

f: friction coefficient f=(l.05-0.0005*t) (see Ekelund)

n₂= 1 per 1/Hm~ 1 n₂=(L/Hm)-0.4 per L/Hm< 1 n3: tension coefficient cr A: front tension crB: back tension Notes

• A complete formula but difficult to interpret

• With the n2 coefficient it takes into consideration the effect for low lctfhm ratios • With the n3 coefficient it considers the tension effects

• For the calculation of

Km

it considers temperature and strain, but there is not a direct reference

to the strain rate effect

•The steel composition is characterised only by the %C content

ANNEX2 -316

Ekelund

K

_wr

-0.611h

h

1JV

11hl R

K¡o+---h

m m 11 : plasticity coefficient: 11 = 0.01(14-0.01 *t)*C₃

C₃: corrective coefficient of plasticity for the rolling speed

Kro: steel resistance function of o/oC and temperature Kro=(14-0.0lt)(l.4+C+Mn+0.3Cr)

f: friction coefficient f =C₁ *(1.05-0.000St) *C₂

C ₁: corrective coefficient for the rolling speed (introduced by Batchinov)

C₂: corrective coefficient for the roll material (introduced by others)

C₂= 1 for steel rolls

C₂=0.8 for hard cast iron rolls

Notes

•A formula difficult to interpret with a lot of corrective coefficients (introduced also by other authors).

• For the calculation of KfO there is not a direct reference to strain, the strain rate is considered into the plasticity terms, temperature is considered.

•The steel composition is characterised with a restricted number of chemical elements.

•However despite these disadvantages, the Ekelund formula gives good results; that' s why it is widely u sed.

1

ANNEX2 -416

Geleji

K

_wr

_l+C*

Km: steel resistance Km= 0.015*(1400-T)

C: coefficient that takes into consideration the effect of low lihm ratios

f: friction coefficient f =C ₁ *(1.05-0.00051) *C₂ (see Ekelund)

v: rolling speed

Notes

• A too

m u

eh simplified formula

•Por the calculation of

Km

only temperature is considered

•The second term represents the friction effect

• It

takes into consideration the effect for low ld/hm ratios