Analysis and Prediction of Overlapping Effect on Inherent Deformation During the Line Heating ...

18 Pages·2013·0.82 MB·English

Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
Analysis and Prediction of Overlapping Effect on Inherent
Deformation During the Line Heating Process
AdanVega,AlexandraCamaño,AmalyFong,SheriffRashed,
HidekazuMurakawa
Abstract: Theinherentdeformationproducedbytwoormoreoverlappedheat-
ing lines is not equal to that obtained when individual deformations are simply
added. Thus, the effect of overlapping on inherent deformation, need to be clari-
fiedinordertoachievefullunderstandingofthephenomenonofplateformingby
line heating. In this paper, a 3D thermal-elasto-plastic FEA has been performed
to study this problem in details. Using this FEA, the influence of previous heat-
ing on the inherent deformation of overlapped heating lines, considering various
conditions, is clarified. From the results of this study, a method to determine the
overlapping effect is developed. Using this method, the overlapping effect, for a
widerangeofplatethicknessandheatingconditions, iseasilyobtained, beenthis,
anotherimportantsteptowardtheautomationofthelineheatingprocess.
Keywords: Plateforming,lineheating,inherentdeformation,overlappedheating
lines,overlappingeffect,FEM.
1 Introduction
After more than five decades, the mechanism to create three-dimensional curved
platesusingthelineheatingprocessstillhasbeenanactiveresearchtopicinman-
ufacturing, especially in shipbuilding. Although, there is plenty of research done
bypeoplethataimtoexplaintheprocessofplateformingbylineheating,theyare
not successful due to a well known bottleneck; the relationship between applied
heat and final plate deformation is still not clear. As a result, the process of plate
formingbylineheatingisnotfullyautomated.
Manyresearchershavepresentedtheoriestoexplaintheproblem. Sawada,Tomita,
Osawa and Hashimoto (2007) developed a combustion model suited to the ther-
mal fluid simulation of line heating process, and showed that the measured and
estimated gas temperatures are in agreement. Shin and Woo (2003) proposed a
numerical procedure that comprised a non-combustion thermal fluid analysis and
148 Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
aniterativeheatconductionanalysistoestimatetheheattransferrateduringaspot
heatingtest. Morerecently,Chang,Liu,andChang(2005),LingandAtluri(2006),
Liu (2006), Liu, Liu, and Hong (2007) investigated the heat transfer problem in
lineheating. Osawaetal(2007)proposedanewtheoryofheattransmissionduring
line heating. They also developed a technique to identify the distribution of the
temperatureofthegasadjacenttotheplatesurfaceandthelocaloverallheattrans-
fer coefficient. The inherent strain method has become one of the most effective
methods to predict plate deformation (Ueda, Murakawa, Rashwan, Okumoto and
Kamic(1994)). Therelationshipbetweenbendingdeformation,heatingparameters
and plate thickness has been developed as empirical models (Masubichi, Imakita,
Miyashi, and Miyaki (1988)). Additional information, such as the influence of
strainhardening,edgeeffect,andsizeeffecthavealsobeenreportedinexperimen-
tal and numerical investigations (Cheng, Yao, Liu, Pratt and Fan (2005), Bao and
Yao (2001), Magee, Watkins, Steen, Calder, Sidhu and Kirby (1997)). In the last
decade,morepapershavebeenpublishedonlaserformationthanthoseonflameor
inductionforminghavebeenreported(Mucha,Hoffman,KalitaandMucha(1997),
Edwardon,Abed,Bartkowiak,DeardenandWatkins(2006).
More recently, the authors proposed a method considering influential factors such
as the geometry, the heating and cooling condition, the location of heating, and
so on, see Vega, Osawa, Rashed and Murakawa (2011), Vega (2009) and Vega,
Rashed,Tango,IshiyamaandMurakawa(2008).
This paper focuses on the case of heating lines applied in the same position as
others before, which is known as overlapped heating lines. First, a finite element
analysis developed before (See Vega, Osawa, Rashed and Murakawa (2011)), is
used to analyze the problem. Different circumstances are evaluated in detail. The
heatingcondition,aswellastheplategeometry,isvariedinordertocaptureevery
detail. Based on the results, the effect of overlapped heating lines is determined.
Furthermore, the overlapping effect is briefly introduced and its mathematical re-
lationship with heating conditions and plate geometry is presented. Finally, the
conclusionsarederivedofthisstudy.
2 Analysisoftheheatinduceddeformationduringlineheating
Thedeformationoftheplateisgivenintermsoftheinherentdeformation,whichis
definedastheintegrationoftheplasticstrainoverthecrosssectionoftheplateof
thickness h. Inherent deformations are classified into longitudinal shrinkage (δ ),
x
transverseshrinkage(δ ),longitudinalbending(θ )andtransversebending(θ )as
y x y
follows;
(cid:90)
δi= ε∗dydz/h (1)
x x
AnalysisandPredictionofOverlappingEffect 149
(cid:90)
δi= ε∗dydz/h (2)
y y
(cid:90)
θi = ε∗(z−h/2)/(h3/12)dydz (3)
x x
(cid:90)
θi= ε∗(z−h/2)/(h3/12)dydz (4)
y y
2.1 Numericalanalyses
An in house finite element code, based on the iterative substructure method, was
employed to achieve the thermal and mechanical analyses (Nishikawa, Serizawa
and Murakawa (2005)). In this FEA, the thermo-mechanical behavior of plate
forming by line heating is analyzed using uncoupled formulation. However, the
uncoupledformulationconsidersthecontributionofthetransienttemperaturefield
to stress through thermal expansion, as well as temperature-dependent thermo-
physicalandmechanicalproperties.
All analyses were carried out using rectangular flat plates of 2000x2000xh (mm).
Valuesofplatethickness(h)equalto20,30. 40and50mmwereused. Byholding
thelineheatingenergyconstant,theinputenergyperunitlengthalongtheheating
lineiskeptunchangedeventhoughpowerandspeedchange. Thehighesttemper-
ature on the surface in the heating area is kept at about 850˚C. Cooling is defined
relativetonaturalcoolinginair. Mildthermal,physicalandmechanicalproperties
of steel are considered with dependence on temperature and thermal conductivity,
specificheat,Young’smodulus,Poisonratioandyieldstress. Necessaryconstraints
are added to eliminate rigid body motion. The same finite element model used in
thethermalanalysisisemployedinmechanicalanalysis.
Theanalysisconditionsformostofthestudiedcasesareasfollows: speed(v)=3
mm/sec,Heatinput(Q)=3125J/mm. Forthosecaseswheretheheatinputand/or
thespeedoftheheatingsourcewerevaried,thenvaluesofheatinputfrom2000to
10000J/mmandofspeedfrom1to25mm/sec,werestudied. Othervariablesuse
duringthesimulationaswellmaterialpropertiesandotheraresimilartothoseused
inVegaetal(2011).
3 Inherentdeformationproducedbyoverlappedheatinglines
Let us consider a straight heating line applied along the length of the plate. This
heating line produces inherent deformation and residual stresses. An example of
the distribution of residual stress is shown in Figure 1. In this figure, the distribu-
tionofresidualstresses(σ andσ )attheheatedsurface,plottedalongtheheating
x y
line, is shown. After the heated area cools down to room temperature, a second
150 Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
(overlapped) heating line, under identical conditions, is applied. In this new situ-
ation, when the surface temperature is high, existing residual stresses may cause
plasticstraininaddition,tothatproducedbytheheatingcycle.
3000 350 700 1050 1400
300
x
150
150
y
]
a
P
M 0 0
[
 x
h
-150 z
-150
y L
W
-300
-300
0 350 700 1050 1400
L [mm]

Figure1: Residualstressesdistributionattheplatesurfacealongtheheatingline
Consequently,whentheplatecoolsdowntoroomtemperature,additionalinherent
deformationisproduced. Inordertoevaluatethisnewaddedinherentdeformation,
we compare the results obtained by simulation of overlapped heating lines with
thoseobtainedbysuperpositionoftheinherentdeformationproducedbyindividual
heatinglines. AnexampleofthiscomparisonisshowninFigure2. Asmaybeseen
inFigures2(a)and(b),atbothentryandexitedgesoftheplate,additionalinherent
deformation in y-direction is produced while, at the central region, it is slightly
reduced. Thereasonofthisdifferenceisfoundinthedistributionofresidualstress
iny-direction,showninFigure1. Asseeninthisfigure,compressiveresidualstress
iny-direction(σ )appearsatbothedgesoftheplatewhile,atthecentralregionof
y
theplate,tensileresidualstress(σ )isobserved.
y
AsshowninFigure2(c),thereisalmostnoadditionallongitudinalshrinkageatthe
central region of the plate produced by an overlapping heating line, if compared
with that produce by single heating. This is because the tensile residual stress
existing in the central region of the plate is large (close to the yield stress). Then,
whentheplateis heatedandcoolsagain, thenewplastic strainisalmostthesame
as that after previous heating. In the same way, at both edges of the plate, the
residual stress in x-direction (σ ) is small and, therefore, longitudinal shrinkage
x
of the overlapping heating line is only slightly influenced as may be observed in
the figure. The rise of longitudinal bending produced by overlapped heating lines
AnalysisandPredictionofOverlappingEffect 151
showninFigure2disduetotworeasons,firstisthetensileresidualstress(σ )as
x
explained above and the second is because the longitudinal bending is induced by
othercomponentsofinherentdeformation.
0.50 350 700 1050 1400 0.040 350 700 1050 1400
0.5 0.04
y (Superposing individual heating line)
0.25 0.25 0.03 y (Superposing individual heating line) 0.03
y (Simulation of overlapped heating line)
m] ns]
[my 0 0 [radiay0.02 y (Simulation of overlapped heating line) 0.02
-0.25 -0.25 0.01 0.01
-0.50 350 700 1050 140-0.50 00 350 700 1050 14000
L [mm] L [mm]

(a) (b)
0.30 350 700 1050 14000.3 0.0150 350 700 1050 14000.015
X (Superposing individual heating lines)
0.0075 x (Superposing individual heating line) 0.0075
0.15 0.15 s]
m] an
[mx 0 X (Simulation of overlapped heating) [radix 0 x (Simulation of overlapped heating line) 0
0
-0.0075
-0.0075
-0.150 350 700 1050 140-0.150 -0.0150 350 700 1050 140-0.0150
L [mm] L [mm]

(c) (d)
Figure 2: Comparison between results of inherent deformation produced by over-
lapped heating lines, and by superposing individual heating lines (a) Transverse
shrinkage, (b) Transverse bending, (c) Longitudinal shrinkage, and (d) Longitudi-
nalbending
4 Overlappingeffectoninherentdeformation
Residual stresses produced by former heating lines cause an increase or decrease
oftheabsoluteinherentdeformationproducedbyoverlappingasitisdemonstrated
already. Itisafactthatthedistributionofresidualstressesdependsontheapplied
heat,theplatethickness,andontheplatesize. Itisalsoafactthatresidualstresses
accumulate in the plate, therefore, their effect on inherent deformation, besides
the factors mentioned above, varies with the number of heating lines. In order to
152 Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
quantifythisdifferenceandtakeitintoaccountintheanalysisofplatedeformation,
we compare the inherent deformation produced by overlapping heating lines with
thatcalculatedbysuperposinginherentdeformationsproducedbythesameheating
lines if applied separately on a non stressed plate (Figure 2 for example). The
deferentialratioofthesetwoinherentdeformationsisnamedbytheauthorsasthe
overlappingeffect(SeeVega(2009)).
Accordingtotheauthors,theoverlappingeffecthasfourcomponents;overlapping
effect on longitudinal shrinkage (β ), overlapping effect on transverse shrinkage
x
(β ), overlapping effect on longitudinal bending (ζ ), and overlapping effect on
y x
transversebending(ζ ). Thesefourcomponentscanbedeterminedfromthetotal
y
inherentdeformation(obtainedfromthermo-elastic-plasticanalysisorexperiments
(δt,δt,θt andθt)),andtheinherentdeformationcalculatedbysuperposinginher-
x y x y
ent deformation of individual heating lines (simple addition (Σδi, Σδi, Σθi and
x y x
Σθi))asfollows;
y
∑δi −δt
β = xx xx (5)
xx ∑δi
xx
∑δi −δt
yy yy
β = (6)
yy ∑δi
yy
∑θi −θt
ζ = xx xx (7)
xx ∑θi
xx
∑θi −θt
yy yy
ζ = (8)
yy ∑θi
yy
Itisappropriatetomentionthattheoverlappingeffectvariesalongtheheatingline,
especiallyattheedgesoftheplate. However,forlargeplates,theoverlappingeffect
isalmostconstantalongmostpartoftheheatingline. Hereafter,positivevaluesof
overlappingeffectmeanreductionofinherentdeformation.
4.1 Variationoftheoverlappingeffectwithheatinput
Figure 3 compares the residual stresses in x-direction (σ ) produced by different
x
valuesofheatinput,plottedthroughtheplatethickness. Inthisfigure,itisclearly
observedthatresidualstressvarieswiththeappliedheat. Inthesameway,thefour
components of overlapping effect significantly vary with the heat input as shown
Figure 4. For small amounts of heat input, the overlapping effect on longitudi-
nalshrinkage,ontransverseshrinkageandthatontransversebendingislargerand
decreaseswiththeincreaseofheatinput. While,theoverlappingeffectonlongitu-
dinalbendingalmostlinearlyincreaseswiththeincreaseofheatinput.
AnalysisandPredictionofOverlappingEffect 153

20 0 50 100 150 200 250 300 350
0.2
Heat input
Q = 5.0 KJ/mm
10 Q = 4.096 KJ/mm
Q = 3.57 KJ/mm 0.1
]
m
m 0
[ 0
h
-10
-0.1
+20 mm
h
-20 mm
-20
-0.2
0 50 100 150 200 250 300 350
x [MPa]

Figure3: Distributionofresidualstresses(σ )throughplatethicknessfordifferent
x
heatinput
1.2 3 4 5 6 7 8 9
1.2
]
m
m
m/ 0.9 x
m 0.9
[
ct
e
eff 0.6 0.6
ng x
pi
p
erla 0.3 y 0.3
v
O
y
0
0
3 4 5 6 7 8 9
Q [KJ/mm]

Figure4: Variationofoverlappingeffectwithheatinput
154 Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
4.2 Variationofoverlappingeffectwithplatethickness
In the analyses performed to explain the variation of overlapping effect with heat
input, we use one value of plate thickness as a representative case. If plates with
different thicknesses are analyzed, the energy needed to achieve the plate defor-
mation, must be changed in order to get similar temperature at the plate surface.
As mentioned above, if the heat input varies, residual stress also varies (this may
be clearly observed in Figure 5, where the distribution of residual stress through
the plate thickness for different plate thickness is compared). Because of this, the
overlappingeffectvarieswithplatethicknessasmaybeobservedinFigure6.
200 50 100 150 200 250 300 3500.2 20-200 -100 0 100 2000.2
Plate thickness
h = 15 mm
10 h = 25 mm 0.1 10 0.1
h = 30 mm
m] h = 40 mm m] Plha t=e t1h5ic mknmess
h [m 0 0 h [m 0 hh == 2350 mmmm 0
h = 40 mm
-10 -10
-0.1 -0.1
-200 50 100 150 200 250 300 350-0.2 -2-0200 -100 0 100 200-0.2
x [MPa] x [MPa]

(a) (b)

Figure 5: Distribution of residual stresses for different plate thickness (a) σ and
x
(b)σ
y
Thevariationofresidualstressesthroughthethicknessalsoinfluencestheoverlap-
ping effect of overlapping heating lines applied over the opposite surfaces as may
be noted in Figure 7. It is clearly seen that for thin plates, the overlapping effect
on both, the longitudinal and the transverse components of inherent shrinkage of
heatinglinesappliedovertheoppositesurfaceisthesameasthoseobtainedwhen
both heating lines are applied over the same surface. On the other hand, for thick
plates,theoverlappingeffectofheatinglinesappliedovertheoppositesideissig-
nificantly smaller than those obtained when both heating lines, are applied over
the same surface. The reason of that is found in the distribution of residual stress
showninFigure5.
Figure 8 compares residual stresses at the heated surfaces of plates with different
length. Inthisfigure,itmaybeseenthatresidualstressesdecreasewithplatelength
up to approximately one meter, then for larger plates, it is almost constant. This
AnalysisandPredictionofOverlappingEffect 155
m] 1.4
1.
m x
m/
m 1
[ 1
ct
e
f
g ef 0.6 x 0.
n
pi
p
rla 0.2 y 0.
e
v
O y
-0.2
-0
10 20 30 40 50
h [mm]

Figure6: Variationofoverlappingeffectwithplatethickness
0.6
0.
]
m
m x (Same surface)
m/ 0.4
ct [m x (Opposite surface) 0.
e
f 0.2
f 0.
e
ng y (Same surface)
pi
ap 0 y (Opposite surface) 0
rl
e
v
O
-0.2
-0
10 20 30 40 50
h [mm]

Figure7: Variationofoverlappingeffectatdifferentplatesurface
156 Copyright©2013TechSciencePress CMES,vol.90,no.2,pp.147-163,2013
400
40
300 x
30
]
a
P
M 200 20
[

100 y
10
0
0
0 500 1000 1500 2000 2500 3000 350
L [mm]

Figure8: Residualstressesattheheatingsurfacefordifferentplatelength
is because in the case of small plates the edge effect significantly influences the
residualstressdistribution(seeFigure1forexample). Figure9showsthevariation
of overlapping effect with plate length. It may be observed that, for large plates,
theoverlappingeffectisnotdependingontheplatelength. Incaseofalargeplate,
the overlapping effect is only influenced by the applied heat and plate thickness.
Similarbehaviorisobservedwhenvaryingtheplatewidth.
4.3 Overlappingeffectofvariousoverlappingheatinglines
Figure 10 shows the residual stress at the heated surface after applying additional
overlappingheatinglines. Inthisfigure,itmaybeobservedthatcertainadditional
amountofresidualstressesaccumulateintheplateaftereachotheroverlappedheat-
ing. Becauseoftheincreaseofresidualstresses,inthosecaseswhenmorethantwo
heatinglinesareappliedoverthesamelocation,theoverlappingeffectincreasesas
showninFigure11. However,thisincreaseofoverlappingeffectbecomessmaller
withthenumberofoverlappingheatinglines.
5 Inherentdeformationdatabasesofoverlappedheatinglines
Inordertocreateaninherentdeformationdatabaseforoverlappedheatinglines,the
overlappingeffectisrelatedtotheheatingconditionsasshowninFigure12. Values
given in this figure can be used to make corrections to the inherent deformation
given in Figures 4, 6, 7, 9 and 11. In case of overlapped heating lines applied by
opposite surfaces, the consideration of overlapping effect is also needed. In that
case,iftheplatethicknessissmallerthan30mm,valuesgiveninFigure12canbe