Welding Formula

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IntroductionA durability assessment of a weld joint

needs the knowledge of residual stressesand distortion (Ref. 1). Thermal cycle his-tory during welding is a necessary input for

simulation of residual stresses. Many in-vestigators have studied heat flow duringarc welding analytically, numerically, andexperimentally (Refs. 26). Analyticalmethods developed are capable of com-puting temperature distribution with rea-sonable accuracy. An approximateanalytical solution for plate with finitethickness using Greens function and ef-fective heat source to compensate forNeumann boundary condition (Ref. 2) hasprovided a very efficient tool to quicklysimulate the transient thermal cycle dur-ing welding. This closed form solution has

a distinct advantage over a finite elementprocedure and doesnt need a tedious pro-cedure of modeling and discretization tobe followed. This solution can be extendedto multipass welding using the principle ofsuperposition; however, a closed form so-lution considering convection and radia-tion heat losses is very difficult to achieve.Neglecting convection and radiation heatloss for the weld pool gives relatively high

peak temperature values.Numerical techniques such as the finite

element method are used increasingly byresearchers particularly for complex weldgeometries but this requires tuning orcalibration of the heat source to get a so-

lution with acceptable accuracy. In thepresent research work, a method to extendan approximate analytical solution formultipass welding is proposed and a tem-perature correction term is derived to ac-count for convection and radiation heatloss from the weld. The authors have ver-ified the proposed analytical solution ex-perimentally and also with finite elementsimulation results obtained from

ABAQUS to an acceptable accuracy of90% for the problem being investigated.

Temperature Rise for Volume Heat Source

in Finite Body

The temperature rise during period t-

t for volume heat source in a finite body isgiven by

Gfin in Equation 1 is Greens function fora point heat source in the finite body thatsatisfies the Neumann boundary conditionof zero heat density (T/n = 0, wheren isthe normal direction) across its boundarysurfaces (Ref. 7). Finding an analytical so-lution for Gfinwould be almost an impos-sible task. An alternate approximateapproach to compensate for the Neumannboundary condition when dealing with a fi-nite body has been proposed by re-searchers (Ref. 2). In this approach, thesame Greens function for the point

source in an infinite body is used, but theheat source in an infinite body is replacedby the effective heat source Qeff(x', y', x', t')in the finite body. The effective heatsource produces the same amount of heatinput into the finite body as the originalheat source would in an infinite body.Using this approach, an approximate tem-perature field in a finite body subjected to

volume heat is estimated by Equation 2.

The effective heat source Qeff(x', y', x',t') in the finite body approximately com-pensates for the Neumann boundary con-dition when dealing with a finite body,

which enables the use of the same Greensfunction for the point source in an infinitebody.

Modeling Multipass Welding

The modeling of multipass welding is

more complex and difficult than single-pass welding due to repeated phase

T x y z t T Q x y z t

G

t

t

v

fi

c, , ,

', ', ', '

'( ) = 0

( )

nnx y z t x y z t dx dy dz dt , , , ; , , , ( ) (1)

T x y z t T Q x y z t

G

t

t

Vc

, , ,', ', ', '

'

inf

( ) = 0( )

xx y z t x y z t dxdy dz dt, , , ; , , , ( ) (2)

Analysis of Thermal Cycle duringMultipass Arc Welding

A new method is proposed to account for convection and radiation heat lossesfrom the surface during simulation of multipass welding thermal cycle

BY C. S. PATHAK, L. G. NAVALE, A. D. SAHASRABUDHE, AND M. J. RATHOD

KEYWORDS

Gas Tungsten Arc WeldingWeld Process Simulation

Approximate AnalyticalSolution

Finite Element MethodC. S. PATHAK, ([email protected]) iswith Sinhgad College of Engineering, Pune, India.L. G. NAVALE is associated with MES COE,

Pune, India. A. D. SHASRABUDHE and M. J.RATHOD are associated with CoE, Pune, India.

ABSTRACT

Convection and radiation heat loss from the plate surface during multipass gas tung-sten arc welding (GTAW) plays a very important role in deciding peak temperature.The heat losses from the surface can be efficiently incorporated in finite element for-mulation, but it is very difficult to derive an analytical expression for the same. A newmethod is proposed to account for convection and radiation heat losses from the sur-

face during simulation of a multipass welding thermal cycle. The proposed method findsthe temperature correction term for temperature distribution estimated using conduc-tion solution. Simulation results of an approximate analytical solution are compared

with experimental and finite element simulation results. Heat loss at thermocouple-plate junction due to contact conductance resistance induces error in temperature meas-urement. The temperature drop due to this effect at the thermocouple junction iscompensated by considering this loss.

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changes and annealing. Approximate an-alytical solution for multipass welding isnot yet reported. It is proposed in the pres-ent research work to use a method of su-perposition in time domain and changingabsolute time with differential time inEquation 2. Thus, the time corresponds tothe sum of welding and waiting time priorto beginning of the succeeding pass. Vari-ation in the amount of heat input in eachpass is taken into account. Dwell time isaccounted by converting real heat sourceat the end of the weld run, to fictitious

zero value heat source traveling beyondthe plate length.

To include the effect of variable ther-mal properties in an analytical expressionis a very complex and difficult task. There-fore for better accuracy, temperature-dependent thermal properties of the work-piece were updated while solving the inte-gral part of Equation 2 using thenumerical method.

Engineering Approach to Account for HeatLoss to Surrounding Surface

Heat loss occurs from the material sur-face during welding by both convection

and radiation. It is proposed to find a com-bined heat transfer coefficient to take intoaccount the convection and radiation heatloss. The surface heat flux due to this isthen found using the combined heat trans-fer coefficient. As an inverse problem, it isassumed that an instantaneous plane heatsource equal to heat loss is acting, andtemperature distribution due to this is cal-culated and then subtracted from the ear-lier calculated temperature field.Temperature correction done using thisapproach effectively compensates for con-

vection and radiation heat loss from theboundary. An empirical relationship pro-posed by Vinokurov,heff = 24.1 10

4

T1.61, is used quite frequently by many re-searchers (Refs. 6, 8, 9) to account for thecombined effects of radiation and convec-tion. Researchers (Ref. 10) have used sim-plified equations and terms for convectionand radiation loss for infinitely thin plate.However, estimation of quantum of heatloss due to convection and radiation fromthe weld plate surface has not been re-ported. In the present research work, thefollowing procedure is adopted to find a

combined temperature-dependent con-vective and radiation heat transfer coeffi-

cient. Convective heat transfer coefficientis estimated using Equation 3 (Ref. 11).

(3)Where,Nu is Nusselt number,kairis ther-mal conductivity of air, and Lc is charac-teristic length. Heat loss due to radiation isaccounted for by calculating the equiva-lent heat transfer coefficient hrad usingEquation 4.

(4)Where is emissivity of body surface, isStefan Boltzmann constant, Ts is plate sur-face temperature, and Ta is atmospherictemperature. This definition of the radia-tion heat transfer coefficient is analogousto convection in terms of a temperaturedifference (Ref. 11). Effective heat trans-fer coefficient considering combined con-

vection and radiation is calculated usingEquation 5.

(5)Figure 1 shows the comparison be-

hNu k

L

air

c

=

h T T T Trad s a s a= + +( )

2 2

h h heff rad

= +

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Fig. 1 Variation of combined heat loss coefficient with surface tempera-ture.

Fig. 2 Weldment specimen drawing.

Fig. 3 Simulated transient temperature time plot from code SANARC. Fig. 4 Finite element model of plate for thermal analysis.


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tween Vinokurovs solution and the pro-posed approach. The proposed methodgives higher values ofheff than calculatedfrom Vinokurovs solution, the differencebeing 1020 up to 400C.

For heat liberated to the surroundingarea from the surface, Qa is calculatedusing Equation 6.

(6)where A = area from which heat is beingliberated to the surrounding area (as-sumed to be 5% of the total area). The re-quired temperature correction, Tc, tocompensate for convection and radiationheat losses is calculated using Equation 7(Ref. 7).

(7)Where is thermal diffusivity,x' = x vt isdimension in moving coordinate system,and t is time. It should be noted that one-dimensional heat flow is assumed in theabove expression, which is quite realisticfor instantaneous plane heat source. Fi-nally, the corrected temperature field iscalculated as

T(x,y,z,t) = T(x,y,z,t) Tc (z,t) (8)

The computer program SANARC wasdeveloped to calculate the integral expres-sion in Equation 2 using Gauss quadraturenumerical technique for the double semi-ellipsoidal distributed heat source duringmultipass welding. For the same discretizedtime interval, temperature correction termfor convective and radiation heat losses wascalculated using Equation 7, and the tem-perature field was accordingly updated. Theresult of program SANARC developed toaccount for convection and radiation losses

using an engineering ap-proach was later compared

with finite element simulationresults obtained from the pro-gram ABAQUS and with anexperimentally measuredthermal cycle.

Case Study Welding of Du-plex Stainless Steel Plates

Duplex stainless steel has good weld-ability, and it can be easily welded by bothmanual and automatic gas tunsten arc weld-ing (GTAW), gas metal arc welding(GMAW), plasma arc welding (PAW),shielded metal arc welding (SMAW) withcovered electrodes, flux cored arc welding

(FCAW), and submerged arc welding(SAW). Although weldability of duplexstainless steel is good, an important precau-tion is to limit as much as possible the hold-ing time at an intermediate temperaturebetween 300 and 980C. Low thermal ex-pansion in duplex grades reduces distortionand residual stresses after welding. The so-lidification of the duplex alloy is not proneto hot cracking due to low impurity levels;however, it may occur under high-restraintconditions (Ref. 12). The weldment speci-men was prepared as per the drawing shownin Fig. 2.

Modeling Considerations

1) A combined convection and radia-tion boundary condition as calculatedfrom Equation 5 is used on the top surfaceand = 0.9 was assumed.

2) No forced convection was assumed,and the effect of gas diffusion in the weld

pool was not considered.3) Neumann boundary condition was as-sumed during calculation of initial (uncor-rected) temperature field.

4)Welding speed was assumed con-stant.

Welding Parameters

The geometric parameters for a doubleellipsoidal heat source used in simulationare listed in Table 1. The parameters men-tioned in Table 1 are in meters and basedon the specimen geometry and a weaving

Q h A Ts Taa eff

= ( ).

TQ

t

ec

a

x x

t

=

( )

2

2

4


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Table 1 Double Ellipsoidal Geometry

Parameters

Pass ah bh chf chb

1 0.001 0.002 0.012 0.022 0.002 0.002 0.012 0.023 0.003 0.002 0.01 0.024 0.0045 0.003 0.01 0.025 0.006 0.003 0.01 0.02

Table 2 Welding Parameters

Pass Interpass Heat Input perDwell Unit Length

s (HI)(a) kJ/mm

1 6.3222 13 6.4383 646 7.8354 97 8.9325 72 7.145

Total 36.672Avg. 7.335

(a) efficiency is not considered.

Fig. 5 Simulated transient temperature-time plot from FE codeABAQUS.

Fig. 6 Foam block support to thermocouple.


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pattern adopted while welding.Interpass dwell time and heat input

used during experimental investigations(Table 2) were used as an input for simu-lation. Higher interpass dwell time afterthe second pass was used to minimize theeffect of repeated heat exposures duringmultipass welding.

Material Properties

Material data, which change during a weldthermal cycle and during phase transforma-tions, are usually missing (Ref. 13) and are

one of the major fac-tors for inaccuracies insimulation (Ref. 14).Researchers (Refs.

1517) have studied theeffect of variable mate-rial properties on weld-ing simulation. Thetemperature-depen-dent physical proper-ties of duplex stainlesssteel, as stated in Table3, were used for thesimulation.

Sensor Location

The thermal cycle

was simulated for thethermocouple loca-tions mentioned in

Table 4. In this table, x is the distance fromthe weld start position, and y is the dis-tance from the weld center. These resultsare from the simulation program SAN-

ARC as shown in Fig. 3.The use and application of the pro-

posed analytical solution SANARC arementioned below.

1) Using simulated transient tempera-ture data and cooling rates, the austenite-ferrite phase balance can be estimated,

which is very useful information for thedual-phase alloy.

2) Heat-affected zone width can be es-

timated from temperature data for variouscombinations of net heat input and weldtorch travel speed, which will be useful indeciding these parameters.

3) The temperature distribution ob-tained will be useful to estimate the resid-ual stresses and any consequentdegradation in the mechanical propertiesof the welded joint.

Simulation of Welding ThermalCycle Using Finite ElementMethod

The finite element method is the mostwidely used simulation technique due to itsflexibility to adopt complex geometry andboundary conditions. Finite element code

ABAQUS is used to simulate transient ther-mal cycle during multipass welding.

ABAQUS has the capability to model solidbody heat conduction with general, temper-ature-dependent conductivity; internal en-ergy (including latent heat effects); andquite general convection and radiationboundary conditions. Energy is related totemperature in terms of a specific heat, ne-glecting coupling between mechanical andthermal behavior. Latent heat effects atphase changes are given separately in termsof solidus and liquidus temperatures. Whenlatent heat is given, it is assumed to be in ad-dition to the specific heat effect, and heatconduction is assumed to be governed bythe Fourier law.

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Fig. 7 Temperature drop due to thermal contact resistance. Fig. 8 Measured thermal cycle.

Table 3 Physical Properties of Duplex Stainless Steel

Temperature C 20 50 100 200 300Density (kg/m3) 7805Thermal conductivity 17 17 18 19 20(W/m.K)Specific heat J/kg C 450 450 500 530

Table 4 Thermocouple Location

Thermocouple x, mm y, mm(channel)

1 35 92 70 133 85 304 20 25

5 50 236 30 20

Fig. 9 Comparison of simulated and measured thermal cycles.


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Spatial Discretization and TimeIntegration

Standard Galerkin approach is used forvariational formulation of the energy bal-ance, together with the Fourier law. Thebackward difference method is used fortime integration time increment is calcu-lated using an automatic (self-adaptive)time-stepping algorithm. This is based ona user-supplied tolerance on the maximum

temperature change allowed in a time in-crement, and the increment is adjusted ac-cording to this parameter.

Finite Element Model Data

Diffusive first-order heat transfer ele-ment DC3D8 (8-node brick) is used fordiscretization of weld plate geometry,

which provides accurate solutions withlarge latent heats. Interpolation is definedin terms of the isoparametric element. Thefinite element model of plates to be

welded as shown in Fig. 4 consists of 9360,

eight-noded brick (hexahedral) elementswith filler material modeled in five distinctlayers to represent five weld passes. Aninitial temperature of 35C is assigned toall nodes of the model.

Procedure

Heat transfer procedure as defined inABAQUS is used to simulate the transientthermal cycle. Weld elements are deacti-

vated initially and successively added tosimulate the addition of material duringeach pass. Body flux is defined in the weld

filler element, while surface flux is givenon the slant face of the V-groove. Convec-tive heat transfer from the plate surface issimulated by considering combined con-

vective and radiation heat transfer coeffi-cient. A significant portion of heat isconvected and radiated to the surroundingarea from the molten weld pool becauseof strong convection currents and forcedconvection due to the flow of argon gasover the area. This heat is accounted forby tuning the heat input to achieve the fu-sion boundary temperature equal tosolidus temperature. The welding param-

eters as defined in Table 2 were used dur-ing the analysis.

Results

Simulation results of temperature-timethermal cycles for six distinct nodes fromABAQUS uncoupled transient heat trans-fer analysis are shown in Fig. 5. Distanceof the node from the weld centerline isgiven in the bracket next to the node num-ber. These results are obtained by re-questing history data output during theABAQUS simulation run. Since the solu-

tion domain is spatially discretized, history

data output for the temperature-time plotcannot be obtained exactly at every sensorlocation used in experimentation. There-fore, output of history data requested atthermocouple sensor locationy = 9mm isused for comparison purpose.

The Experimental Investigationsof Transient Thermal Cycle

To verify the simulated transient tem-perature profile as obtained in earlier sec-tion temperature histories, six points onthe plate were recorded during welding.

Temperature Sensing Device

Temperature measurement in the pres-ent work was done by using K-type(chromel-alumel) thermocouples. Theyare cheap and interchangeable, have stan-dard connectors, and can measure a widerange of temperatures. The transient tem-perature distribution during welding was

recorded using a precalibrated 12-channeldata acquisition system manufactured byYokogawa Corp. The system had a pro-grammable interface with in-built memoryalong with portable storage media func-tionality USB 2.0 data cards.

Test Rig Design and Fabrication

The test rig was to designed to meet thetwo requirements of easy and continuous

welding and simultaneous measurementof the temperatures using thermocouples.

Chromel-alumel thermocouples were

attached to the plate from the bottom tomeasure the temperature distribution atdifferent distances from the weld center-line. This was done to make the weldingoperation easier for the operator. Thetemperatures were measured at the mid-dle plane of the plate by drilling holes ofdiameters equal to that of the sensing rodsof the thermocouples, with sufficientclearance provided so that the sensinghead made positive contact with the plateat the correct measuring node. The datapoints were chosen to be staggered alongthe weld run so that at every instant the

sensor would ideally be able to see theheat source without any obstruction. Theaim of the experimentation to validate thesimulation program, hence the locationsof thermocouple, was selected so as not tofollow a particular pattern. Care was takento avoid arc strike and termination zone.The thermocouples were mounted onfoam blocks to ensure the required contactstiffness as shown in Fig. 6. This is morepractical than welding the thermocouplesto the plate for every experiment. Glass

wool insulation was provided around thethermocouples to protect them from radi-

ant heat.

The welding was carried out using 2209filler metal of diameter 1.6 mm, as per

AWS 1554.6 guidelines. The use of 2209filler metal ensured the deposited metal

was the correctly balanced duplex struc-ture because of increased nickel percent-age, which is a strong austenite former. A

weaving pattern was adopted for deposit-ing the filler metal. Cleaning and degreas-ing of the weld area was done prior to

welding and fine grinding was done to re-move weld spatter and undercut after

welding. The welding parameters used arelisted in Table 2

Thermal Contact Resistance

Thermocouple and plate (as seen inFig. 6) can be considered a composite sys-tem, as shown in Fig. 7, in which the tem-perature drops appreciably across theinterface between them. This temperaturedrop is attributed to the thermal contactresistance Rt,c (Ref. 18), and for a unitarea of the interface, the resistance is de-fined as

(9)The existence of a finite contact resist-

ance is primarily due to a surface rough-ness effect. Heat transfer is therefore dueto conduction across the actual contactarea and also due to conduction and/or ra-diation across the gaps. Thus the contactresistance consists of two parallel resist-ances, one due to the contact spots and theother due to the gaps. The contact area istypically small, and especially for roughsurfaces, the major contribution to the re-sistance is made by the gaps.

Surface contact resistance Rt,c is esti-mated to vary linearly from 30 to 300 m2

C/W (Ref. 18) in the temperature rangeof 35 to 700C over contact area definedby a circle of 0.25-mm radius. Heat trans-fer at the junction, estimated from weld-ing parameters and temperature sensed bythermocouples, was used to calculate tem-perature at the plate surface consideringloss due to thermal contact resistanceusing Equation 9. The corrected measuredthermal cycle still included unavoidableexperimental errors, such as locating thehole depth to which the thermocouple isin contact with plate.

Transient temperature during thewelding was measured at six locations asmentioned in Table 4, and the associatedthermal cycle is shown in Fig. 8. Thethermocouple number and its distancefrom the weld centerline (in bracket) ismentioned in the plot. Comparison ofthe corrected temperature at the platesurface with the simulated cycle is shownin Fig. 9 for sensor location 1 (y = 9mm)of the plate.

=

RT T

qt cs tc

z

,


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The data acquisition system was startedbefore welding and a small time lag wasobserved while recording temperaturedata as a result of inherent limitations ofhardware, hence for comparing the re-sults, simulated profiles were shifted ac-cordingly to accommodate these effects. Aclose match between simulated and exper-imental transient welding thermal cycleprofile can be observed in Fig. 9 with apeak temperature accuracy of over 90%.The accuracy achieved can be consideredsatisfactory given the complexity of theproblem and range of temperature.

Conclusions

An approximate analytical solutionas proposed by Nguyen et al. was extendedto simulate multipass welding using prin-ciples of superposition and fictitioussource method. A procedure based on anengineering approach was developed tofind temperature correction terms to ac-

count for convective and radiation heatlosses. The heat losses were accounted forwhile solving the integral expression intime domain. The proposed solution wasimplemented by developing the computerprogram SANARC, and marginal reduc-tion in peak temperature was achieved.

The finite element code ABAQUSwas used to simulate a transient thermalcycle with complex boundary conditions ofmoving heat source and temperature-de-pendent combined convective and radia-tive heat transfer. Addition of filler metaland latent heat effects were also consid-

ered in the program. Results obtainedfrom the finite element solution showhigher peak temperature than the approx-imate analytical solution.

Transient temperatures were recordedat the locations mentioned in Table 4 dur-ing welding using modern and rugged in-strumentation. The lower peak temperaturefor the spring-loaded thermocouple was due

to an insufficient contact between the ther-mocouple and the plate, which resulted insome loss of heat due to contact resistance.

Preflow and postflow of argon gas be-fore start of the weld pass and after com-pletion of the weld pass significantlyreduced the interpass temperature and in-terpass dwell time in the first two passes;however, both dwell time and interpasstemperature increased marginally in thelast two or three passes.

The proposed solution SANARC iscomputationally more efficient than the fi-nite element method and does not requirethe time-consuming process of discretiza-tion of domain. It is also useful in more ac-curate simulation of residual stresses andmicrostructure at less computational cost.

Acknowledgments

The authors express their sincerethanks to authorities of University of Punefor research grant BCUD/578. The au-thors are indebted to Mr. C. S. Kale of M/sSHOEI Finishing Pvt. Ltd. for providingfacility and instrumentation to carry outexperimental work.

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