thermal stress

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146

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Calculation of thermal stress affecting strip flatness changeduring run-out table cooling in hot steel strip rolling

Xiaodong Wanga,b,∗, Quan Yangb, Anrui Heb

a Research Institute of Technology, Shougang Corporation, 68 Shijingshan Road, Shijingshan District, Beijing 100041, PR Chinab National Engineering Research Center for Advanced Rolling Technology, University of Science and Technology Beijing, 30 Xueyuan Road,Haidian District, Beijing 100083, PR China

a r t i c l e i n f o

Article history:

Received 26 October 2006

Received in revised form

21 November 2007

Accepted 17 December 2007

Keywords:

Hot rolled strip

Thermal stress

Run-out table

Flatness

ABAQUS

a b s t r a c t

To fulfill further reducing production costs at the meantime enhancing product quality is

always one of the objectives pursued by steel enterprises and researches. With increasing

demand on higher quality of hot rolled strip shape, strip flatness change developed during

cooling on the run-out table has received significant attention and should be considered in

online strip shape control model. Non-uniformity of transverse temperature distribution in

strip is the ultimate reason for the occurring of strip flatness defect after hot rolled flat strip

is cooled to room temperature. Based on the large amount of thermal image field measure-

ments on strip surface, transverse temperature distribution rules of strip were concluded.

A FE model was established to analyze the thermal stresses developed during the cooling of

hot rolled strip on the run-out table using finite element program ABAQUS. Analysis results

demonstrate that temperature drop within strip edge region will make strip flatness develop

to the trend of edge wave and are verified by the actual strip flatness observation on skin

pass rolling line. This viewpoint agrees well with the actual production condition. One com-

pensation control strategy, named slight center wave rolling, is proposed based on the FE

analysis work. The main idea is that steel strip is rolled with slight center wave at the exit of

the last stand of finishing mill to compensate the flatness change trend of edge wave during

the cooling.

ing upon the condition of temperature distribution profile

1. Introduction

Geometrical or dimensional attributes of hot rolled strip arevery important, since a highly waved or cambered strip mightcause damage to the equipment or retard the production.With increasing demand on higher strip dimensional toler-ances, maintaining a uniform strip crown and a flat shape
during hot rolling and consequently cooling has become oneof the most challenging technical tasks in the steel industry.After steel strip is hot rolled, it is then subjected to cooling
∗ Corresponding author at: Research Institute of Technology, Shougan100041, PR China. Tel.: +86 10 68845910.

E-mail address: x d [email protected] (X. Wang).0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.12.076

© 2007 Elsevier B.V. All rights reserved.

on the run-out table. Edge drop in strip is not evitable dur-ing hot rolling, and cooling conditions are different betweenstrip edge and strip center, so nonuniform temperature dis-tribution in the steel strip is common. The quality of the finalproduct, such as the metallurgical and mechanical propertiesand the flatness of the strip, may vary significantly depend-

g Corporation, 68 Shijingshan Road, Shijingshan District, Beijing

in strip. As a result of an uneven temperature distributionacross the width of the strip during the cooling process, sig-nificant thermal stress is generated. The large thermal stress

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ill cause local plastic deformation across the strip and con-equently introduces residual stress and strip flatness defect.any researches (Prieto et al., 2001; Liu, 2001; Sun et al., 2002;

erajzadeh, 2003) of heat transfer and metallurgical phenom-na during cooling of hot rolled strip have been done. In theontrast, the researches on how strip flatness changes duringooling are lack, and yet no consistent conclusion is reached.

To analyze the deformations of hot rolled product afterooling, precise investigation of thermal stress developed dur-ng the cooling is very important, and has received significantttention (Yoshida, 1984a,b; Boyadjiev et al., 2004a; Boyadjievt al., 2004b; Han et al., 2002; Suzuki and Isaka, 1998) fornhancing the final product quality. For hot rolled steel strip,he flatness observed at the exit of last stand of finishing milloes not stand for the final product flatness. Disturbances inhe strip flatness are caused by the creation of internal stressenerated from the results of nonuniform temperature distri-ution and uneven cooling. Four kinds of thermal gradients: (a)cross the width; (b) through the thickness on body; (c) throughhe thickness on head and tail end; (d) along the length, influ-ncing the strip flatness as different patterns of edge wavesr bows, were summarized by Ginzburg (1993) based on theorks of Guglielmetti et al. (1987). To the hot rolled thin strip,ecause heat conductions along strip thickness and length cane neglected, only the flatness defects resulted from nonuni-orm temperature distribution along strip width is considered.here are two contradictory viewpoints about how the stripatness changes during the cooling process after hot rolling.he first one is that the edge wave will happen after coolingf steel strip even if the flatness from after the rolling stage toefore the coiling stage is good or center wave. The represen-ative works include the researches having been carried outy Yoshida (Yoshida, 1984c; Yoshida et al., 1997), Guglielmettit al. (1987), Tani et al. (1998), Colas et al. (2004) and Cai (1998).he other one is that the center wave will happen after coolingf steel strip. The conclusion can be gotten from a preliminaryomputational analysis of the residual stress generated dur-ng cooling of hot rolled thin steel strip on the run-out tableerformed by Zhou et al. (2003). The result is that temperaturerop within strip edge region leads to the longitudinal stressistribution of tensile stress at strip edge and compressivetress at strip center, and this will lead to center wave flatnessefect. While in the research on deflections of the thermome-hanical controlled process (TMCP) plates conducted by Wangt al. (1996), temperature drop at strip edge will lead to cen-er wave flatness defect occurring at the first deflection stageuring or after cooling and before hot leveling while will leado edge wave flatness defect occurring at the second deflec-ion stage during the flattened plate is air cooled to ambientemperature. One consistent shape prediction simulator waseveloped by Ogai et al. (2004), in which the cooling coursesn the run-out table, at downcoiler and coil yard were covered.he calculation result acquired is that center wave occurs intrip during its cooling on the run-out table, and then cen-er wave weakens during coiling in the downcoiler, in the endenter wave disappears and edge wave appears during cool-
ng at the coil yard. That is to say, center wave may appearuring the cooling course of steel strip on the run-out tablerom the results obtained by Wang et al. (1996), Zhou et al.2003) and Ogai et al. (2004). This conclusion contradicts to the
h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146 131

results obtained by Yoshida (1984c), Guglielmetti et al. (1987)and Cai (1998) mentioned above. However, the new researchresult acquired by Zhou et al. (2007) shows that temperaturedifference between the edge and centre will make the steelstrip’s buckling change from the centre to the edge during itscooling on the run-out table.

For the purpose of exploring how the strip flatness changesduring and after cooling, thermal stress developed in stripduring cooling on the run-out table after finish rolling wasinvestigated based on a finite element model established usingABAQUS in this work. The FE model, referred the work ofZhongqing Zhou et al. (2003), has been established, and theresults of thermal stress were compared with those obtainedpreviously.

2. The meaning of this study

In terms of quality, nowadays flatness is not only one of themost important factors to consider in the hot coils production,but also one of the most difficult ones to broach. In thin andwide hot rolled steel strip, edge waves are often observed at theentry side of the skin pass mill after cooling even if the flatnessfrom after the rolling stage to before the coiling stage is well.When the edge wave becomes large, problems such as low-ering efficiency during skin pass rolling and imperfection inflatness after skin pass rolling will arise. If the edge wave is notprevented, the flatness control in the finishing mill becomesworthless (Yoshida, 1984c). Conventional means to solve thisproblem are improving temperature distribution uniformityalong transverse direction (Naoki and Hiroshi, 1995; Peregrinaet al., 2006) through controlling water flow rate distributionand edge masking amount. However, both the measures men-tioned above need modify and install new devices and theircontrol system. That will increase cost and affect normal pro-duction, and also has a certain difficulty to precisely controlwater flow rate and edge masking amount because there lackstemperature measurements of strip transverse direction online.

If the rules of transverse temperature distribution of stripat the exit of finishing mill and the entry of downcoiler aremastered through off-line measurements or temperature dis-tribution can be measured real time, the strip can be rolledto be a certain specific flatness at the exit of finishing mill tocompensate the flatness change which occurs during the cool-ing. This method may be one feasible solution to realize theaim of good final product flatness, and has the more attractiveadvantage of fewer investments in devices and modificationsthan conventional measures mentioned above. Strip flatnesschanges from finishing mill to cooled strip coil are generalizedas shown in Fig. 1. Final product flatness F is determined asfollows:

F = F1 + dF1 + dF2 (1)

where F1, dF1 and dF2 are flatness at exit of finishing mill, flat-ness change during cooling on the run-out table and flatnesschange during coiling process, respectively, IU. If dF1 and dF2are known, then F1 is the aim flatness value to be determined,

132 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146

s du
Fig. 1 – Influencing factors on steel strip flatnes
and can be expressed as

F1 = F − dF1 − dF2 (2)

To realize this method, measurements of transverse tem-perature distribution of steel strip and residual stress analysisare two key steps for determining dF1 and dF2.

From our knowledge, Ogai et al. (2004) is the first one whoconducted the research of online strip shape prediction simu-lator covering the cooling courses at run-out table, downcoilerand coil yard. However, there are no measures presented toprevent strip flatness defects in his research. One researchgroup of National Engineering Research Center for AdvancedRolling Technology at University of Science and TechnologyBeijing has invested great effort on study of automatic stripshape control model (Anrui et al., 2006; Wang et al., 2007) andits application in hot strip mill and also has acquired manyexcited achievements. However, main function of this stripshape control system is to solve the strip shape problem occur-ring at the stage of finish rolling. To enhance the final productquality, the cooling process of steel strip on the run-out tableshould be considered in the whole strip shape control system,as shown in Fig. 2. Determination of the aim flatness value isvery important for control effect and depends on the preci-
sion of thermal stress analysis. So this task of thermal stressanalysis has a great meaning to determine the aim flatness offinish rolling for better controlling final product flatness andthat is our main research work having been carried out.
Fig. 2 – Flow chart of strip shape co

ring hot rolling, cooling and coiling processes.

3. The thermal image measurements

To clearly analyze the influences of cooling after hot rollingon the strip flatness and to provide enough data for strip flat-ness compensation control during hot rolling in ASP1700 hotstrip mill of Anshan Iron and Steel Corporation, it is neces-sary to master the rules of temperature distribution along stripwidth at the exit of last stand of finishing mill and influencesof the laminar cooling on the temperature distribution on therun-out table. Two infrared thermal imagers were used to mea-sure the temperature at the exit of last stand of finishing milland the entry of the downcoiler, respectively, and the stripinformation such coil number, steel grade, gauge, end rollingtemperature and coiling temperature were recorded at thesame time. Large amount of measurements were conductedand recorded, and then processed, one example is shown inFig. 3.

From the processing and analyzing of strip thermal imagedata, the conclusions can be acquired as follows:

(1) The features of transverse temperature distribution ofstrip at the exit of the last stand finishing mill can besummarized as follows: (a) there is 40–80 ◦C temperature
decrease within approximately 150 mm from strip edge; (b)temperature decreases more slowly from center to edge inthin strip (strip thickness is less than 2.58 mm) than thatin thick strip (strip thickness is large than 4.50 mm); (c)
ntrol system for hot strip mill.

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146 133

age

(

Fig. 3 – One example of thermal im

within the strip center excluded 150 mm from the stripedge, temperature variation is as small as approximately10–20 ◦C in thick strip, while may reach 30 ◦C or above in
thin strip.
2) The features of transverse temperature distribution ofstrip at the entry of downcoiler can be summarized asfollows: (a) there is 20–40 ◦C slow temperature decrease

result and related processed data.

within approximately 150 mm from strip edge and a verysmall temperature variation generally less than 10 ◦Cwithin strip center excluded 150 mm from the strip edge in
thin steel strip (thickness is less than 2.58 mm); (b) there is20–30 ◦C slow temperature decrease within approximately150 mm from strip edge and a relatively even temperaturedistribution, which is 10 ◦C less than that of strip edge,

n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146

[Kuu KuT

KTu KTT

]{�u

�T

}=

{Ru

RT

}(6)

Table 1 – Thermal conductivity coefficient and specificheat depended on temperature

Temperature (◦C) Thermal conductivitycoefficient (W/(m K))

Specific heat(J/(kg K))

550 34.4 692600 31.8 735650 28.5 752700 27.2 866750 26.2 1021

134 j o u r n a l o f m a t e r i a l s p r o c e s s i

within strip center excluded 150 mm from the strip edgein thin steel strip (thickness is large than 4.5 mm).

(3) Empirically calculating, 1 ◦C temperature difference willlead to 1.05 × 10−5 fiber elongation difference which isalmost equal to 1 IU flatness value. This difference willresult 2.205 N mm−2 stress difference. If there is 20 ◦Ctemperature difference, 44 N mm−2 stress difference willgenerate, namely there are nearly 20 IU flatness change.So, it is necessary to compensate the aim flatness value atthe exit of the last stand of finishing mill.

4. Numerical model

This numerical analysis work was carried out on the basic ofthe real production processes and conditions in ASP1700 hotstrip mill of Anshan Iron and Steel Corporation. The run-outtable is 130 m long and the behavior of a strip with a certainlength was investigated. Analysis of thermal stress in the stripdeveloped during cooling was carried out using ABAQUS Ver-sion 6.5 (ABAQUS, 2005). A finite element model of the stripcooling process covering the full length of the run-out tablewill be computationally inefficient and costly. One segment ofstrip with 6 m length was chosen to be the research object. Inthe cooling process, the strip is transported on run-out table,while the cooling profile is stationary in time. For convenienceof calculation, it is assumed that the cooling process is mov-ing with a static strip. The steel grade of the strip selected tobe calculation object is HP295. The strip is cooled from 880 to620 ◦C on the run-out table.

4.1. Governing equations of the thermo-mechanicalmodel

The thermo-mechanical analysis involves solving the tran-sient heat transfer and thermal stress problem describing thetemperature and thermal stress variations with time in thestrip during cooling. During the cooling process, the temper-ature distribution in the strip can be calculated using thegoverning partial differential equation shown as follows:

∂

∂x

(�x(T)

∂T

∂x

)+ ∂

∂y

(�y(T)

∂T

∂y

)+ ∂

∂z

(�z(T)

∂T

∂z

)+ q

= �cp(T)∂T

∂t(3)

where � is the density, cp (T)is the temperature dependentspecific heat; �(T) is the temperature dependent thermalconductivity of the material with the subscripts x, y and z rep-resenting its components in three directions of width, lengthand thickness, respectively; T and t are temperature and time,respectively; q is the heat generation term representing theinternal heat source released due to phase of austenite trans-formation (Hawbolt et al., 1985)

q = �Hi�Xi (4)
�t
where �Hi is the amount of heat of transformation at temper-ature Ti and �Xi is the transformed fraction within the timeincrement �t.

Fig. 4 – Meshes of the computation model.

In order to find the spatial and temporal distribution oftemperature, T(x, y, t), Eq. (3) can be solved by employinga finite element discretization. After certain mathematicalmanipulations, the equations can be reduced to:

[Ce]{Te} + [Ke]{T} = {Re} (5)

where [Ce] is the elemental heat capacitance matrix and [Ke]is the elemental heat conduction matrix; the vector {Re} isthe heat flux (load) vector arising from internal heat genera-tion, specified surface heating and surface convection; {Te} isthe vector of temperature change with time and {Te} is thetemperature vector.

The transient response of the nonlinear system of equa-tions resulting from the assembly of the system of elementalequation, Eq. (5) is calculated using a step by step recurrencetechnique where temperatures are stored by ABAQUS at thenodal positions in a solution increment and then interpolatedto the integration point locations before solving the elementaldifferential equations.

For a fully coupled temperature–displacement analysis,ABAQUS solves a system of coupled equations represented byequation (ABAQUS, 2005) shown as follows

800 25.8 839850 25.5 742900 25.4 725950 25.5 718


Ft

wtmRta

4

Tbwdr3

4

Thi

ig. 5 – Thermal expansion coefficient depended onemperature.

here �T and �u are the corrections to the incrementalemperature and displacement, respectively; Kij are the sub-

atrices of the fully coupled stiffness matrices; and RT and

u are the thermal and mechanical residual vectors, respec-ively. These thermal and mechanical systems of equationsre solved simultaneously using the Newton’s method.

.2. Meshing of the physical model

he length, width and thickness of the strip segment toe computed are 6, 1200 and 4.0 mm, respectively. Modelas established in the rectangular coordinate system, andirection 1, 2 and 3 stand for length, width and thickness,espectively. Element type is C3D8T, and element size is0 mm. Meshes of the computation model is shown in Fig. 4.

.3. Determination of the physical parameters

he material physical parameters varying with temperatureave important effects on the results of computation. It

s necessary to correctly select the physical parameters for

Fig. 6 – Young’s modulus and Poisson’

Fig. 7 – Plastic characteristic curves depended ontemperature.

ensuring the computation precision of temperature and ther-mal stress. Specific heat, thermal conductivity coefficient andlinear expansion coefficient were provided by ASP1700 hotstrip mill of Anshan Iron and Steel Corporation as shown inTable 1 and Fig. 5, respectively. Young’s modulus and Poisson’sratio were determined according to the reference (Zhou et al.,2003) as shown in Fig. 6.

If stress is larger than material yield stress, materialhappens plastic deformation. In ABAQUS, material plasticcharacteristic curve depended on temperature is requiredbased on the relationship curve between true stress and truestrain as shown in Fig. 7.

4.4. Phase transformation model

There will happen phase transformation in the strip duringrun-out table cooling. The kinetics of the diffusional trans-formations of austenite to ferrite and pearlite have beendescribed for the isothermal condition by Avrami equation(Serajzadeh, 2003) as follows

X = 1 − exp (−btn) (7)

where b and n are material parameters that for steels, asshown in Table 2; t is the elapsed time from the beginningof the transformation.

s ratio depended on temperature.


rmal

the run-out table has been divided into two different kindsof cooling zones, namely water cooling and air cooling. Thewater cooling zone is treated as a whole part, and one aver-

Fig. 8 – (a) Structure of run-out table cooling and (b) the

4.5. Boundary conditions

To solve the temperature differential equation, it is very impor-tant to define boundary conditions accurately. The laminarcooling of the hot rolled strip on the run-out table is a rel-atively complicated process of heat transfer. The boundaryconditions can be generalized to be convection heat transferequation between the strip surface and cooling water and theradiative equation between the strip and the atmosphere. Theboundary conditions can be described as follows

⎧⎨⎩

−�∂T

∂n= hw1(T − Tw1)

−�∂T

∂n= hw2(T − Tw2) + �ε(T4 − T4

w2)(8)

where n is normal direction of strip’s surface; hw1 and hw2

are convection heat transfer coefficients on the strip sur-face between the cooling water and the air, respectively; �

is Stefan–Boltzmann constant, 5.6697 × 10−8 W m−2 K−4; ε isemissivity; Tw1 and Tw2 are temperatures of cooling water and
atmosphere, respectively.
Just after the exit of the last finishing stand, there existsa length of about 10 m where the strip is cooled in air, calledair cooling zone. Following this region, the strip enters into

Table 2 – b and n used in Avrami equation (Serajzadeh,2003)

Transformation type b(T) n

Austenite to ferrite 14.2 exp(− T−620

25.1

)0.7

Austenite to pearlite 65.3 exp(− T−595

4.2

)3.8

boundary conditions for water cooling and air cooling.

water cooling zone. A certain rows of water jets impinge onthe strip surface to take away the heat in the strip. From thelast bank of water cooling to the coiler, there also exits anair cooling zone. There will happen forced convection heattransfer and stable film boiling in two water flowing zonesof impinging zone and parallel flowing zone on the strip sur-face, respectively. Too much computation time is needed todefine the two kinds of heat transfer conditions very close toreality. Therefore, to make analysis simple, the total length of

Fig. 9 – Fit curve of end rolling temperature along striptransverse direction.


eratu

attri

drttetpd

Fig. 10 – Initial temp

ge heat flux referred to (Cox et al., 2001) is assumed duringhe analysis. The heat flux value can be adjusted to representhe running speed of the strip. The detailed structure of theun-out table and thermal boundary conditions can be seenn Fig. 8.

Mechanical boundary conditions also should be defineduring the thermal stress analysis of the strip cooled on theun-out table after hot rolling. Boundary Constraint condi-ions are defined during the establishment of FEM model usinghe ABAQUS. Full displacement constraint is applied to one
nd of the strip segment, while the other end is assumedo be free. Two strip edges are assumed to be simply sup-ort condition along y direction, namely normal to the rollingirection.
Fig. 11 – Relationship between initial flatness and i

re field of the strip.

4.6. Initial conditions

When steel strip leaves the last stand of finishing mill andbefore it enters into cooling zone, one initial temperature fieldand one initial stress distribution already exist. Both of theinitial temperature field and initial stress are not uniform.ABAQUS software provides many user subroutine programfunctions to fulfill loading uneven field variables.

Actual measurement data of strip temperature along trans-verse direction at the exit of the last stand finishing mill can
be fitted to a fourth order polynomial function of transversecoordinate using the least squares method (Sun et al., 2004).By means of user subroutine program UTEMP, initial tempera-ture field of fourth order polynomial function as shown in Fig. 9
nner stress: (a) edge wave and (b) center wave.


tial stress in strip, 20 IU flatness.
Fig. 12 – One example of ini
was defined. Because the thickness of strip to be analyzed isvery thin, uniform initial temperature distribution through thethickness direction of strip was assumed and defined in theanalysis. The result of initial temperature field loaded on thestrip can be seen in Fig. 10.

After finish rolling, strip will have a certain flatness.Namely, stress distribution already exists in the strip. To sim-ulate the initial flatness, initial stress distribution should beloaded. This initial stress distribution can be expressed usingrelative elongation difference along transverse direction of thestrip. As shown in Fig. 11, ε∗

z(x), x coordinate and z coordinaterepresent the relative elongation difference between centerand edge of strip, width and length directions of strip, respec-tively. Initial stress can be determined as follows:

�∗z (x) = −Eε∗

z(x) = E(εpm − ε

pz (x)) (9)

where E is Young’s modulus of material with end rolling tem-
perature (MPa); ε
pz (x) is the elongation of fiber at x coordinate

(m); εpm is the mean elongation of strip along transverse direc-

Fig. 13 – Coiling tension distribution along transversedirection.

Fig. 14 – Residual stress distribution with difference initialflatness (Cai, 1998).

tion, and can be calculated by the following equation.

εpm =

∫ b

−bε

pz (x)dx

2b(10)

where b is one half width of the strip (m). Based on thelarge amount of actual flatness gauge measurements, ε

pz (x) is

assumed to be a parabolic curve

εpz (x) = a0 + a1x + a2x2 (11)

where a0, a1, and a3 are coefficients of the function which aredetermined according to the initial IU distribution.

Same as the initial temperature, initial stress was loadedusing user subroutine program SIGINI, and the result of initialstress is shown as in Fig. 12.

4.7. Disposition of coiling tension

Hot rolled steel strip moves to downcoiler and will be woundinto coil after cooling on the run-out table. Before the head of


Fb

sWuiibpstttpstaum

�

w

5

5

Rcntawflcs

oov

Fig. 16 – Residual stress distribution pattern obtained byZhou et al. (2003).

Table 3 – Chemical composition of steel HP295 used insimulation (in wt%)

C 0.174Si 0.014Mn 0.9P 0.016

culation model of laminar cooling process and the chemicalcomposition is shown in Table 3.

ig. 15 – Initial temperature distribution pattern consideredy Zhou et al. (2003).

trip enters into downcoiler, there is no coiling tension in strip.hile coiling tension in strip should be considered during sim-

lation after strip head enters into downcoiler. Coiling tensions related to strip thickness profile and coil diameter, as shownn Fig. 13. Because of the strip crown, coiling tension distri-ution along transverse direction will vary during the coilingrocess. After steel strip is wound to coil, circumferential forcetill retains until uncoiling at room temperature because ofhe functions of the radial compression and friction betweenwo layers. This circumferential force distribution and its dis-ribution pattern also vary with the position of the layer. Theosition is closer to the outer layer of the steel coil, the ten-ion is larger at center while tension is smaller at edge. Inhis research, no coiling tension and coiling tension were sep-rately considered in computation and fulfilled by means ofser subroutine program DLOAD. The coiling tension is deter-ined by the following equations:

T(x) =

⎧⎪⎨⎪⎩

5�∗T/3 0 ≤ x < 0.4

5�∗T(0.8 − x)/1.2 0.4 ≤ x < 0.8

0 0.8 ≤ x < 1.0

(12)

here x is normalized coordinate, x = x/b; �∗T = 5.54 MPa.

. Calculation results and discussion

.1. Earlier computation results

esearches on residual stress analysis of steel strip duringooling after hot rolling are very lack, and the results still haveot reached consistency. The residual stress distribution pat-ern along transverse direction computed by Yoshida (1984c)nd Cai (1998) is that there is tensile stress at center parthile compressive stress at edge part of strip, namely the stripatness has the trend of developing to be edge wave duringooling of the strip because of temperature decrease withintrip edge. Fig. 14 is the results calculated by Cai (1998).

Zhou et al. (2003) also has performed numerical analysisf residual stress developed during cooling of hot rolled stripn run-out table. Initial temperature distribution along trans-erse direction considered in computation is shown in Fig. 15

S 0.011Al 0.02

and his result, as shown in Fig. 16, is contrary to that of Yoshida(1984c) and Cai (1998).

5.2. Result and analysis of this work

In this work, HP295 steel grade that was hot rolled at ASP1700hot strip mill of Anshan Iron and Steel Corporation wasselected to be the simulation object in the thermal stress cal-

Fig. 17 – Temperature trajectories of strip surfaces (stripspeed is 6.0 m/s and phase transformation is notconsidered).


Fig. 18 – Longitudinal stress distribution along transversedirection at different cooling stages without initial stress Fig. 19 – Longitudinal plastic strain distribution along

respectively. The second work step is water cooling process

and coiling tension.

Firstly, no initial stress and coiling tension were consideredduring the computation, and the residual stress induced onlyby uneven temperature distribution along transverse direc-tion and temperature decrease was acquired. And then, initialstress distribution was considered to simulate thermal stressin the strip with certain initial flatness before entering down-
coiler. At last, initial stress distribution and coiling tensionwere loaded at the same time to simulate thermal stress inthe strip after entering downcoiler. Computation procedure
Fig. 20 – Flatness measured at ex

transverse direction at different cooling stages withoutinitial stress and coiling tension.

is divided into three steps. The first step is air cooling pro-cess of 12.7 m distance from the last stand finishing mill F6to the first row of laminar water cooling pipes. Strip velocitywas assumed to be 6.0 m/s. Air temperature and its convec-tive coefficient were assumed to be 30 ◦C and 30 W m−2 K−1,

lasting 10 s, heat flux values on strip top surface and bottomsurface were set to be −2.4 × 107 and −1.8 × 107 W m−2, respec-tively. The atmosphere temperature was 28 ◦C. The third work

it of last stand finishing mill.

t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146 141

sew

ootbstw

asdFbwtdDftarciTewminebCn(l

hd

Fs

Fig. 22 – Longitudinal stress distribution along transversedirection at different cooling stages (with initial stress of60 IU flatness, without coiling tension).

Fig. 23 – Longitudinal stress distribution along transversedirection at different cooling stages (with initial stress of

j o u r n a l o f m a t e r i a l s p r o c e s s i n g

tep is also air cooling and the time is 4 s. The related param-ters for boundary conditions are same as those of the firstork step.

Precision of temperature computation is the foundationf thermal stress analysis. To prove consistency of the resultf temperature calculation with the product data, end rollingemperature and coiling temperature recorded by system cane used. Actually, only the temperature at the center of steeltrip is measured and controlled. The results of the tempera-ure trajectories on strip top and bottom surfaces agree wellith real industry process as shown in Fig. 17.

Because the transverse stress in steel strip is very smallnd the strip flatness is mainly attributed by longitudinaltress distribution along transverse direction, only longitu-inal stress and plastic strain are considered in this work.igs. 18 and 19 are longitudinal stress and plastic strain distri-utions along transverse direction at different cooling stagesithout initial stress and coiling tension, respectively. The

hermal stress in strip increases as the temperature of stripecreases rapidly after strip enters into water cooling zone.uring air cooling, thermal stress varies slowly. The reasons

or thermal stress generation are uneven temperature dis-ribution and high cooling rate. From the results mentionedbove, temperature decrease within strip edge leads to theesidual stress distribution pattern of tensile stress at stripenter region and compressive stress at strip edge region. Thats to say, the strip flatness develops to the trend of edge wave.emperature difference induced compressive stress at stripdge region reaches almost 80 MPa, and mainly concentratesithin the range of 120 mm to the strip edge. Plastic defor-ation happens in steel strip during water cooling stage, and

ncreases as temperature decreases. The change of strip flat-ess to edge wave is attributed by this plastic strain at thedge region of strip. The format of longitudinal stress distri-ution agrees well with that obtained by Yoshida (1984c) andai (1998), but is contrary to that of Zhou et al. (2003). The mag-itude order of the stress agrees well with that of Zhou et al.

2003) and Yoshida (1984c), but Cai’s (1998) result is obviously
arger.
It is difficult to accurately measure the residual stress inot rolled steel strip, so there is very little real measurementata to be used to prove the calculated results. To illustrate

ig. 21 – Strip flatness photo picture at the entry side of thekin pass mill.

20 IU flatness, without coiling tension).

that the strip flatness will move to edge wave, one exam-ple of large amount of actual observations is presented here.One set multi-channel profile gauge and one flatness gaugemade by IMS Corporation were installed behind F6 to measureand record strip’s crown and flatness real time during rollingin ASP1700 hot strip mill of Anshan Iron and Steel Corpora-tion. One flatness image of strip after finishing rolling can berecorded as shown in Fig. 20. From this picture, we can see thatthis strip has a perfect flatness at the exit of the last stand offinishing mill. Because there is no flatness gauge before down-coiler, to know real strip flatness after cooling, observationsand measurements at skin pass rolling line must be done.Fig. 21 is the real flatness image of the same strip in Fig. 20on skin pass rolling line. This strip has an obvious edge waveshape observed during uncoiling. From comparison betweenthe flatness picture recorded by the flatness gauge and the
real flatness image after cooling, strip flatness at exit of thelast stand finishing mill positively has the trend of changingto edge wave during the stage of cooling, and that is not ourrequired final product flatness.


Fig. 24 – Longitudinal stress distribution along transversedirection at different cooling stages (with initial stress of10 IU flatness, without coiling tension).

Fig. 25 – Longitudinal stress distribution along transverse


flatness change rules of strip head before strip enters into thedowncoiler.

Once the strip enters into the downcoiler, then coiling ten-sion is established in strip. Coiling tension also has effect on

direction at different cooling stages (with initial stress of−10 IU flatness, without coiling tension).

To better control final flatness of hot rolled strip product,finding an optimal flatness value at the exit of the last stand offinishing mill to compensate the influence of cooling processon strip flatness on the run-out table has a great meaning forfulfilling close loop control of strip flatness during hot rolling.Eight conditions of initial stress loaded on the basis of stripflatness at the exit of finishing mill ranging from 60 to −60 IUwere computed, and some calculation results of longitudinalstress distribution along transverse direction are listed fromFigs. 22–25. Magnitude and direction of initials stress havesome certain effects on the final pattern of longitudinal resid-ual stress distribution. Whether the residual stress increasesor decreases, direction relationship between initial stress andthermal stress plays a critical role. Thermal stress direction isdetermined by the format of temperature distribution. Fromthe rules concluded from the thermal image measurements
mentioned above, temperature decrease within strip edgeregion is affirmed. Therefore, thermal stress is compressiveat strip edge and tensile at strip center. Initial stress direc-tion of center wave is opposite to that of thermal stress, so
direction at different cooling stages (with initial stress of60 IU flatness and coiling tension described in Eq. (12)).

the residual stress will reduce on the condition of center waveand increase on the condition of edge wave. As shown in thefollowing Figs. 22–25, flatness defect of center wave at theexit of finishing mill will reduce within some certain degreeafter cooling on the run-out table. However, if flatness defectis edge wave, edge wave will be more severe. To compensatethe influences of thermal stress introduced during cooling onstrip flatness on the run-out table, strip should be rolled tobe some certain center wave shape. This method is economi-cal and relative simple because no equipment modification ordevices installation for ensuring even temperature distribu-tion are needed. From the computation results, the strip with4 mm in thickness and 1200 mm in width should be rolled tobe center wave within the range of 10–20 IU flatness, and thusthe product will have a perfect final flatness. The calculationresults in which no coiling tension was considered reflect the

Fig. 27 – Longitudinal stress distribution along transversedirection at different cooling stages (with initial stress of20 IU flatness and coiling tension described in Eq. (12)).


Fig. 28 – Longitudinal stress distribution along transversedirection at different cooling stages (with initial stress of1

rdag


0 IU flatness and coiling tension described in Eq. (12)).

esidual stress distribution in strip, and should be considereduring numerical analysis. Coiling tension was determinedccording to Eq. (12), and loaded through user subroutine pro-ram DLOAD. Calculation results of thermal stress in strip

Fig. 30 – The calculation flow chart of

direction at different cooling stages (with initial stress of−10 IU flatness and coiling tension described in Eq. (12)).

considering both initial stress and coiling tension are shown
as in Figs. 26–29. Because of steel strip crown, coiling tensionloaded in strip mainly concentrates only in the center part ofstrip width. When coiling tension is considered, stress in thecenter part of strip width has a small increase that almost
the subroutine program HETVAL.

n g t e c h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146

Fig. 32 – Longitudinal stress distribution along transversedirection at different cooling stages without initial stressand coiling tension.


equals to coiling tension loaded. The contribution result ofcoiling tension is consistent with the conclusion obtained byZhou et al. (2003). In the researches of Cai (1998) and Yoshida(1984c), no coiling tension was considered. The influence ofcoiling tension on strip flatness is little.

The results mentioned above were obtained through thecalculation in which the phase transformation was not con-sidered. Phase transformation will happen in the steel stripduring its cooling on the run-out table. The phase trans-formation latent heat will be released during the phasetransformation process, and has an influence on the tempera-ture profile in the steel. Therefore, the phase transformationsfrom austenite to ferrite and pearlite were also considered inthe thermal stress analysis in the steel strip during the cool-ing process. The subroutine program HETVAL was employedto calculate the phase transformation latent heat for realiz-ing the coupling of temperature and phase transformation.The detailed calculation flow chart of the subroutine programHETVAL is shown in Fig. 30.

Strip speed is another factor to be considered during thethermal stress analysis. At the end, the thermal stress gen-eration process of the strip, in which phase transformation isconsidered and strip runs with a higher speed of 12.0 m/s, wasanalyzed. No initial stress and coiling tension were consideredduring computation. Computation procedure was divided intofive steps to simulate the interrupted cooling style of the steelstrip. The first step lasting 1 second is also to consider the aircooling process of 12.7 m distance from the last stand finishingmill F6 to the first row of laminar water cooling pipes. The sec-ond work step is water cooling process lasting 5 s, and the heatflux values on strip top surface and bottom surface were setto be −5.6 × 107 and −4.9 × 107 W m−2, respectively. The thirdwork step is also air cooling and the time is 4.5 s. The fourthstep is water cooling procedure of 0.5 s which is the precisioncontrol segment of the coiling temperature, and the heat fluxvalues on strip top surface and bottom surface were set to be
−3.6 × 107 and −2.9 × 107 W m−2, respectively. The last step isair cooling before strip enters into downcoiler. For all the aircooling procedures, the atmosphere temperature, air temper-
Fig. 31 – Temperature trajectories of strip surfaces andmiddle plane (strip speed is 12.0 m/s and phasetransformation is considered).

Fig. 33 – Longitudinal stress distribution along transversedirection at different cooling stages (with 10 IU flatness of
initial stress, without coiling tension).
ature and its convective coefficient were assumed to be 28 ◦C,30 ◦C and 30 W m−2 K−1, respectively.

Fig. 31 is the temperature calculation results of top surface,bottom surface and middle plane of steel strip at its centerplace during its cooling on the run-out table. There is a greatdifference between Figs. 31 and 17. In Fig. 31, the tempera-ture on the surfaces, especially on the top surface, decreasemore rapidly due to the high cooling strength and has a cer-tain rebound, and then decreases again during water coolingprocedure of the steel strip. This kind of temperature reboundis mainly resulted by the function of phase transformationlatent heat. We can get the conclusion that it is reasonableto consider the phase transformation in steel strip during itscooling process.

Figs. 32–34 are the calculation results of longitudinal stressdistribution along transverse direction in the steel strip under
the conditions mentioned above including phase transfor-mation with three different initial flatness values of 0, 10and −10 IU. During the calculations, coiling tension was not

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c

Fig. 34 – Longitudinal stress distribution along transversedi

cutstsschinrotaamstst

6

(

(

(

r

irection at different cooling stages (with −10 IU flatness ofnitial stress, without coiling tension).

onsidered. The thermal stress distribution shape calculatednder the conditions mentioned above has no difference tohat having no phase transformation behavior and a lowpeed. That is to say the changing trend of strip flatness onhe run-out table is still to the edge wave defect. There areome numerical differences in the thermal stress results inteel strip with a higher running speed as shown in Figs. 32–34ompared to those not considering phase transformation andaving a low running speed on the run-out table at the same

nitial conditions as shown in Figs. 18, 24 and 25. There isearly 5 MPa compressive stress increase in the strip edgeegion while there is almost no change within the center partf the strip. The reason for this compressive stress increase ishat the strip with a more rapid speed has higher cooling ratet the edge region than that with a relative lower speed. Therelso another difference, namely the generating speed of ther-al stress is higher in the strip having a more rapid cooling

peed than that with lower cooling speed. That is to say thathe strip with a higher rolling speed will have a higher coolingpeed to be cooled to a certain coiling temperature, and thenhermal stress increases more rapidly in it.

. Conclusions

1) Large amount of measurements of strip thermal imageswere carried out at the exit of last stand of finishing milland at the entry of the downcoiler. Transverse temperaturedistribution rules of strip were concluded. There is somedegree temperature decrease within strip edge region dur-ing the whole cooling process on the run-out table.

2) The thermal stress developed in steel strip during cool-ing on the run-out table is attributed for the nonuniformtemperature distribution along transverse direction ofstrip and the high cooling rate. The thermal stress maycause local plastic deformation across the strip and con-
sequently introduces residual stress and strip flatnessdefect.
3) A numerical analysis of the thermal stress generatedduring cooling of hot rolled strip on the run-out table

h n o l o g y 2 0 7 ( 2 0 0 8 ) 130–146 145

was performed by means of the finite element programABAQUS. There is the thermal stress distribution patternof tensile longitudinal stress at strip center and com-pressive longitudinal stress at strip edge region for thetemperature drop in strip edge region. Plastic deformationoccurs in the strip edge region. It is reasonable to considerthe phase transformation in steel strip during its coolingprocess. The strip with a higher rolling speed will have ahigher cooling speed to be cooled to a certain coiling tem-perature, and then thermal stress increases more rapidlyin it.

(4) Temperature drop within strip edge region will make stripflatness develop to the trend of edge wave defect, andthat has been proved by the actual strip flatness observa-tions after cooling to ambient temperature. This viewpointagrees well with the actual production condition.

(5) To better control the final shape quality of hot rolled strip,one compensation control strategy named slight centerwave rolling is proposed based on the conclusion (4) inthis work. The main idea is that steel strip is rolled withslight center wave at the exit of the last stand of finish-ing mill to compensate the flatness change trend of edgewave occurring at cooling stage. This measurement hasthe advantages of small investment, easy implementedand convenience for troubleshooting, compared with themethods of realizing uniform transverse temperature dis-tribution in strip such as water flow rate control alongtransverse direction and edge masking on the run-outtable.

(6) From the calculation results of the research steel strip with1200 mm in width and 4 mm in thickness, the aim flatnessvalue at the exit of the last stand of finishing mill shouldbe determined within the range of 10–20 IU flatness.

(7) This FE analysis model is capable of providing the support-ing technology for fulfilling the end rolling strip flatnesscompensation control strategy in the online strip shapecontrol model.

Acknowledgements

The authors would like to thank Anshan Iron and Steel Corpo-ration for the support to this work. In addition, special thanksare extended to Dr. Lin Zhao, Xiaobo Guo (senior engineer) andLili Zhong (senior engineer) for their cooperation and assis-tance.

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