Prediction of Welding Distortion and Residual Stress-2008-Thin-Plate

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Prediction of welding distortion and residualstress in a thin plate butt-welded joint

Article in Computational Materials Science August 2008

Impact Factor: 2.13 DOI: 10.1016/j.commatsci.2007.12.006

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Chongqing University

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Prediction of welding distortion and residual stress in athin plate butt-welded joint

Dean Deng a,*, Hidekazu Murakawa b

a Research Center of Computational Mechanics Inc., Togoshi NI-BLDG 7-1, Togoshi, Shinagawa-ku, Tokyo 142-0041, Japanb Joining and Welding Research Institute, Osaka University, 11-1, Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Received 1 October 2007; received in revised form 22 November 2007; accepted 4 December 2007Available online 29 January 2008

Abstract

In automotive industry, thin plate parts are commonly used. During assembling process, welding technology is usually employedbecause of high productivity. Welding distortion often occurs in thin plate welded structures due to relatively low stiffness. The distortioncauses problems not only in the assembling process but also in the final product quality. Therefore, prediction and reduction of weldingdeformation have become of critical importance. In this study, three-dimensional, thermo-elasticplastic, large deformation finite ele-ment method (FEM) is used to simulate welding distortion in a low carbon steel butt-welded joint with 1 mm thickness. To comparewith the large deformation theory, the small deformation theory is also used to simulate the welding deformation and welding residualsstress. Meanwhile, the characteristics of welding temperature field, plastic strain distribution and welding residual stress in thin weldedplates are also examined numerically. Experiments are also carried out to measure the welding distortion in the thin plate butt-welded

joint. By comparing the simulation results with the measurements, it is found that the results predicted by the thermo-elasticplastic,large deformation FEM match the experimental values well. Moreover, using the inherent strains obtained by the thermo-elasticplasticFEM, an elastic FEM is also employed to estimate welding deformation in the same butt-welded joint. Comparing the results simulatedby the elastic FEM with those predicted by the thermo-elasticplastic FEM, it is verified that the inherent strain method can effectivelypredict the welding deformation in the thin plate butt-welded joint with 1 mm thickness.2007 Elsevier B.V. All rights reserved.

PACS: 07.05.Tp; 47.11.Fg; 65.40.De; 81.20.Vj

Keywords: Welding distortion; Numerical simulation; Thin plate; Plastic strain; Finite element; Inherent strain; Nonlinear analysis

1. Introduction

Distortion in a welded structure is the result of the non-

uniform expansion and contraction of the weld and sur-rounding base material, caused by the heating and coolingcycle during welding process. Welding distortion has nega-tive effects on the accuracy of assembly, external appear-ance, and various strengths of the welded structures. Inmany cases, additional costs and schedule delays areincurred from straightening welding distortion. On the

other hand, increasingly, the design of engineering compo-nents and structures relies on the achievement of small tol-erance. For these reasons, prediction and control of

welding deformation have become of critical importance.In the past decades, a lot of experiments and numericalanalyses have been conducted for predicting welding dis-tortion, and a lot of fundamental knowledge has been alsoestablished[17]. However, there is very limited literaturedescribing the prediction and measurement of weldingdeformation in the thin plate welded structures especiallyfor these welded structure whose plate or wall thicknessis less than 3.0 mm. Recently, Liang et al. [810] haveestablished a number of databases of welding inherentdeformations for typical thin plate welded joint using

0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2007.12.006

* Corresponding author. Tel.: +81 3 3785 3033; fax: +81 3 3785 6066.E-mail address:[email protected](D. Deng).

www.elsevier.com/locate/commatsci

Available online at www.sciencedirect.com

Computational Materials Science 43 (2008) 353365
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experimental method and inverse analysis method, andsome meaningful achievements have been obtained.

In automotive industry the thin plate parts are com-monly employed, and a lot of thin plate parts are assembledby arc welding process because of high productivity. Due torelatively small stiffness, significant welding distortion often

occurs. To comprehensively understand the characteristicsof welding deformation in thin plate welded structure, it isnecessary to do further fundamental researches by meansof both experiment and numerical simulation.

In this study, three-dimensional, thermo-elasticplastic,large deformation finite element method (FEM) isemployed to simulate welding distortion and welding resid-ual stress in a low carbon steel butt-welded joint with 1 mmthickness. To compare with the large deformation theory,the small deformation theory is also used to simulate thewelding deformation and welding residuals stress. Mean-while, the characteristics of welding temperature field, plas-tic strain distribution and welding residual stress in thin

welded plates are also examined numerically. Experimentsare also carried out to verify the numerical simulationmethod. Moreover, using the inherent strains (plasticstrains) computed by the thermo-elasticplastic FEM, anelastic FEM, in which the large deformation is taken intoaccount, is also utilized to estimate the welding deforma-tion in the same butt-welded joint.

2. Experimental procedure

In this study, a simple experiment is carried out to mea-sure welding deformation in the thin plate butt-welded

joint. The butt-welded joint consists of two thin mild steelsheets. The dimension of each sheet is 100 mm 100 mm 1 mm. The welding method is gas metal arcwelding (GMAW). The shielding gas was 80%Ar +20%CO2. The welding wire is YGW16 [11]. The detailedwelding conditions are shown inTable 1.

To imitate the welding conditions used in automobileindustry, an initial gap between the two plates is set to be0.4 mm and a partially welding is performed in the joint.The length of welding line is about 60 mm. The specimensare welded without external constraints. In the experi-ments, three identical butt joints are performed. Fig. 1

shows the picture of a completed joint.After welding, the deflection at the center of the weldingline is measured using a simple method as shown in Fig. 2.In the experiments, a Vernier caliper is used to measure thevalues of h0 and h1. In this figure, w is the differencebetween h0and h1. The measured values (w) of these threejoints are 2.20 mm, 1.90 mm and 2.10 mm. Approximately,

w/2can be regard as the deflection at the center of the weld-ing line. The average value (w) of these three butt joints is2.067 mm. This means the average value of the deflection(w/2) at the welding line is 1.03 mm.

3. Prediction of temperature field, residual stress and

deformation using thermo-elasticplastic FEM

In this section, based on ABAQUS code [12] a sequentiallycoupled thermo-elasticplastic finite element computationalprocedure is developed to calculate temperature field, weldingresidual stress and welding deformations. The 3D finite ele-

ment analyses are performed to numerically study the weldingdistortion and welding residual stress in butt-welded thinplates. In order to capture thenonlinear geometrical behaviorsin a thin plate structure, the large deformation theory is incor-porated into thermo-elasticplastic FEM.

Fig. 3shows the finite element mesh model used in thesimulation. The dimensions of the finite element model

Table 1Welding conditions

Parameter Current(A)

Voltage(V)

Welding speed(mm/min)

Shielding gas flowrate (L/min)

Tip-to-workdistance (mm)

Wire diameter(mm)

Travelangle ()

Value 65.0 17.0 780.0 15.0 25.0 0.9 45.0

Fig. 1. Thin plate butt-welded joint.

h0 h1

w

Weld bead

Specimens

Block

Fig. 2. Schematic image of measuring the deflection in the butt-weldedjoint.

354 D. Deng, H. Murakawa / Computational Materials Science 43 (2008) 353365
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are the same as those of the experimental specimen. Ele-ment meshes are denser in the vicinity of the weld center-line, while the meshes become gradually coarser awayfrom the weld zone. The length of each element in the weld-ing direction is 2.5 mm. The number of division in thethickness direction is four. The total number of 8-nodebrick element is 4000.

In the present study, to consider the bead shape in theFE model, three dimensions of the weld bead obtained inthe experiments are measured. One is the height of weld

reinforcement; and the other two are the breadths betweenthe two toes on the upper surface and the bottom surface,respectively. In the FE model, the bead shape is roughlydetermined based on these three parameters. The beadshape is shown inFig. 3.

In the thermal analysis, the welding conditions areassumed to be the same as those used in the experiment.

3.1. Heat source and thermal analysis

Welding heat transfer analysis with given welding condi-tions is performed in the 3D thin plate model. In this step,

temperature histories at each element nodes are computedduring the welding process. 3D, 8-nodes, linear brick andheat elements (DC3D8) [12] are selected for the thermalanalysis. Temperature dependent physical properties ofthe mild carbon steel as shown inFig. 4 [13] are employedin heat transfer analysis. In this study, solid-state phasetransformation is neglected because the influence of phasetransformation on the welding deformation and weldingresidual stress is insignificant in the lower carbon steel[14].

During the welding, the governing equation for transientheat transfer analysis is given by:

qcoT

otx;y;z; t

r qx;y;z; t

Q

x;y;z; t

1

whereq is the density of the materials [g/mm3],c is the spe-cific heat capacity [J/(g C)], T is the current temperature[C], q is the heat flux vector [W/mm2], Q is the internalheat generation rate [W/mm3], x, y and z are the coordi-nates in the reference system [mm], t is the time [s], and$ is the spatial gradient operator.

The non-linear isotropic Fourier heat flux constitutiveequation is employed:

q krT 2wherekis the temperature-dependent thermal conductivity

[J/(mm s C)].In this study, the heat from the moving welding arc is

applied as a volumetric heat source with a double ellipsoi-dal distribution proposed by Goldak [15], which isexpressed by the following equations:

For the front heat source:

Qx0;y0;z0; t 6ffiffiffi

3p

ffQwa1bcp

ffiffiffip

p e3x02=a21 e3y02=b2 e3z02=c2 3

For the rear heat source:

Q

x0;y0;z0; t

6 ffiffiffi3p

frQw

a2bcpffiffiffipp e3x

02=a22 e3y0

2=b2 e3z02=c2

4

where x0, y0 and z0 are the local coordinates of the doubleellipsoid model aligned with the welded pipe; ffand fr areparameters which give the fraction of the heat depositedin the front and the rear parts, respectively. Because thetemperature gradient in the front leading part is steeperthan in the tailing edge, ff and fr are assumed to be 1.33and 0.67, respectively. Qwis the power of the welding heatsource. It can be calculated according to the welding cur-rent, the arc voltage and the arc efficiency. The arc effi-ciency g, is assumed to be 80% for the GMAW weldingprocess. The parameters a1, a2, b and c are related to the

characteristics of the welding heat source. The parameters

Fig. 3. Finite element model, gap and bead shape.

0.2

0.4

0.6

0.8

1

1.2

1.4

0 300 600 900 1200 1500

Thermal ConductivityDensitySpecificheat

ThermalCon

ductivity

Temperature (oC)

c (J/g/ C)

(102g/mm3)

k (101J/mm/s/ C)

Fig. 4. Temperature dependent thermal physical properties.

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of the heat source can be adjusted to create a desiredmelted zone according to the welding conditions. The val-ues of these parameters used in the present simulation aresummarized inTable 2.

When structural analysis FEM codes such as ABAQUSand MARC are used to simulate the temperature distribu-tion in welding process, the fluid flow and solidification ofmaterial in the weld pool cannot be directly consideredbecause the coupled problem between solid and liquid isnot involved in these software at present. However, theeffect of the fluid flow has significant effects on the temper-ature distribution and the shape of weld pool. If the effectof the fluid flow is neglected, the highest temperature inweld pool will be very high. According to the authorsexperience, the peak temperature in welding pool is higherthan 3000 C in some cases when the fluid flow effect isneglected. This phenomenon is much different from therealistic situation. Okagaito et al. [16] measured the sur-face temperature distribution on TIG weld pool inSUS304 steel. Their research suggests that the highest tem-perature on the molten pool surface is approximately1750 C. In this study, to consider the fluid flow an artifi-

cially increased thermal conductivity in the weld pool isused. The thermal conductivity is assumed to be twice aslarge as the value of room temperature for temperatureabove the melting point.

The thermal effects due to solidification of the weld poolare modeled by taking into account the latent heat forfusion. The value of the latent heat is 270 J/g[17]. The liq-uidus temperature TL and the solidus temperature TS areassumed to be 1500 C and 1450 C, respectively.

Heat losses (qc) due to convection are considered for allthe surfaces using Newtons law:

qc

hf

Tsur

T0

5

where hf is film coefficient for convection [W/(mm2 C)],Tsuris the surface temperature [C], andT0is ambient tem-perature [C].

In this study, a temperature-dependent film coefficient[18] is used, and the ambient temperature is assumed tobe 20 C.

Radiation heat losses (qr) are accounted for all the sur-faces by using StefanBoltzman law:

qr erT4sur T40 6where e is emissivity, r is StefanBoltzman constant forradiation.

In this study, the emissivity is assumed to be 0.2 [19,20].

The user-defined subroutines to ABAQUS code are uti-lized in the heat transfer analysis to model heat fluxes, con-vection and radiation boundary conditions.

3.2. Mechanical analysis

The same finite element models used in the thermal anal-yses are employed in mechanical analyses, except for theelement type and boundary conditions. The restraint con-ditions are shown inFig. 3by the arrows. The C3D8I ele-ment type [12] is used to simulate the stressstrain field.The analyses are conducted using the temperature historycalculated by the thermal analyses as the inputinformation.

For the mild steel, because phase transformation has aninsignificant effect on the welding residual stress and thedeformation, the total strain can therefore be decomposedinto three components as follows:

etotal ee ep eth 7The components on the right-hand side of Eq. (7) corre-spond to elastic, plastic and thermal strain, respectively.

The elastic strain is modeled using the isotropic Hookeslaw with temperature-dependent Youngs modulus andPoissons ratio. For the plastic strain component, a plasticmodel is employed with the following features: the VonMises yield surface and temperature-dependent materialproperties. Fig. 5 shows the temperature dependentmechanical properties[13]. Because the effect of work hard-ening is not significant in mild steel, it is neglected in thisstudy.

In this study, the thickness of the specimen is only 1 mm,so it can be expected that the geometrically nonlinear phe-nomenon probably occur during welding. To examine thedifference between the numerical results computed by the

Table 2Parameters of the heat source

Parameter Value (mm)

a1 2.0

a2 4.0b 1.2

c 1.0

0

50

100

150

200

250

300

350

0 300 600 900 1200 1500

Yield Strength (MPa)Young's Modulus (GMPa)

Thermal expansion coefficient (10-7/

oC)

Possion's ratio (10-2

)

MechanicalProp

erties

Temperature (oC)

Fig. 5. Temperature dependent mechanical properties.

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large deformation theory and that calculated by the smalldeformation theory, both the two methods are used to pre-dict welding distortion and welding residual stress. Thus, inthe mechanical analysis stage, two simulation cases (case Aand case B) are performed. The large deformation theory isconsidered in case A, while the small deformation theory is

used in case B.

3.3. Simulation results

3.3.1. Characteristics of welding temperature field

Fig. 6 shows the temperature histories at the center ofthe welding line. From this figure, it can be seen that thepeak temperature at the top surface of the weld pool isabout 1800 C. In the same figure, it can also be observedthat the difference between the peak temperature at thetop surface and that at the bottom surface is not signifi-cant. The thermal effects due to solidification of the weld

pool are considered in the FE model, so these two coolingcurves reflect the solidification phenomena of the weldpool. Except for the peak temperature, the temperature his-tory curves of the top surface and the bottom surface in thefusion zone have no difference.Fig. 7 shows the tempera-ture histories of the top surface and the bottom surfacein the heat-affected zone (HAZ) of the mid-section. Thedistance between this position and the weld centerline is3.2 mm. This figure indicates that both the top surfaceand the bottom surface in the HAZ have almost the samepeak temperature and the identical cooling rate. In the FEmodel, the plate thickness of the fusion zone is less than2 mm, and the plate thickness is only 1 mm in other parts.

It is the small plate thickness that resulted in an even tem-perature distribution through thickness during welding.

3.3.2. Welding residual stress

In the mechanics analysis, both the large deformationtheory and the small deformation theory are used to simu-

late welding residual stress and deformation in the butt-welded joint. Fig. 8 shows the longitudinal residual stressdistributions of the middle section, which are computedby the large deformation theory (case A). Fig. 9 showsthe longitudinal stress distributions of the middle section,which predicted by the small deformation theory (caseB). Due to a relatively large out-of-deformation are gener-ated after welding in case A, there is a difference betweenthe longitudinal stress of the top surface and that of thebottom surface. On the contrary, there is almost no differ-ence between the longitudinal stress of the top surface and

that of the bottom surface in case B.Fig. 10shows the transverse residual stress distributions

of the middle section predicted by case A. There is also asignificant difference between the transverse residual stressof the top surface and that of the bottom surface. It is very

0

500

1000

1500

2000

0 10 15 20 25

Top SurfaceBottom Surface

Temperature(oC)

Time (s)

5

Fig. 6. Temperature histories in the fusion zone.

0

200

400

600

800

1000

0 10 20 30 40 50

Top Bottom

Time (s)

Temperature

(oC)

Fig. 7. Temperature histories in the HAZ.

-300

-200

-100

0

100

200

300

400

500

0 50 100 150 200


LongitudinalStress(MPa)

Y-Coordinate (mm)

Fig. 8. Longitudinal residual stress in mid-section computed by Case A.

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clear that in the fusion zone and the HAZ the transverseresidual stress of the top surface is much larger than thatof the bottom surface. The difference results from the trans-

verse bending deformation. Fig. 11 shows the transverseresidual stress distributions of the middle section computedbye case B. This figure indicates that there is almost no dif-ference between the top surface and the bottom surface.

3.3.3. Welding deformation

Fig. 12shows the contours of the deflection distributioncomputed by the large deformation theory. From this fig-ure, it can be observed that a large longitudinal bendingand a transverse bending are produced after welding. Atthe two ends of the gaps, the maximum deflection is about1.7 mm.Fig. 13shows the contours of the deflection distri-

bution predicted by the small deformation theory (case B).

This figure also reflects that both longitudinal bending andtransverse bending are generated in the butt-welded joint.The deformation mode is similar to case A, however themagnitudes of deflection are much smaller than case A.Fig. 14 shows the deflection distributions of case A andcase B along the middle line (line AB), which is definedin Fig. 12. In this figure, point A shown in Fig. 12 isassumed to be the origin. The experimental measurement(deflection) at the center of the welding line is also plottedin the same figure. It is clear that the deflection at the centerof the welding line predicted by case A is much close to theexperimental value. From the same figure, it can be also

known that the experimental value is significantly largerthan the numerical result computed by case B. This infor-mation suggests that when welding deformation of a thinplate welded joint or structure is simulated numerically itis necessary to consider geometrically nonlinear phenome-non. Otherwise, it is probable to result in a very large error.

Fig. 15 shows the Y-directional displacement distribu-tion of the middle section predicted by the large deforma-tion theory. Comparing the transverse shrinkage of thetop surface with that of the bottom surface, it can beobserved that the former is smaller than the latter. How-ever, the difference is very small.Fig. 16shows theY-direc-

tional displacement distribution of the middle sectioncomputed by the small deformation theory. This figureindicates that the transverse shrinkage through thicknessis almost uniform. ComparingFig. 15withFig. 16, it canbe conclude that the Y-directional displacements predictedby case A are close to those computed by case B on thewhole.

3.3.4. Plastic strain distribution

Fig. 17shows the plastic strain distributions in the lon-gitudinal direction (welding direction) of the middle sec-tion. The plastic strain values are the average ones

through thickness predicted by the large deformation the-

-300

-200

-100

0

100

200

300

400

500

0 50 100 150 200


LongitudinalStress(MPa)

Y-Coordinate (mm)

Fig. 9. Longitudinal residual stress in mid-section simulated by Case B.

-100

0

100

200

300

400

0 50 100 150 200


Trans

verseStress(MPa)

Y-Coordinate (mm)

Fig. 10. Transverse residual stress in mid-section computed by case A.

-100

0

100

200

300

400

0 50 100 150 200


TransverseStress(MPa)

Y-Coordinate (mm)

Fig. 11. Transverse residual stress in mid-section simulated by Case B.

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ory and the small deformation theory. From this figure, itcan be observed that the plastic strain distributions pre-dicted by the two cases have no significant difference. Boththe range of the longitudinal plastic strain values and thedistribution shapes of these two cases are fairly close.

Fig. 18shows the plastic strain distributions in the trans-verse direction of the middle section computed by Case Aand Case B. This figure indicates that the transverse plasticstrain distributions of the two cases are quite similar.

FromFigs. 17 and 18, it is known that the range of the

longitudinal plastic strain distribution is larger than that of

the transverse plastic strain distribution. The transverseplastic strain range concentrates almost only in the fusionzone and the HAZ, while the longitudinal plastic strainrange distributes in a relatively large range. Generally,the range and magnitude of the plastic strain componentare mainly governed by the peak temperature and therestraint conditions [21]. During welding, because therestraint intensity in the longitudinal direction (weldingdirection) is larger than that in the transverse direction,the range of the longitudinal plastic strain is relatively

large.

Fig. 12. Deflection distribution of Case A.

Fig. 13. Deflection distribution of Case B.

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3.3.5. Discussions

Based on the simulation results, it is known that boththe longitudinal plastic strain and the transverse plasticstrain distribute narrowly in the fusion zone and its vicin-ity. For mild steel, the plastic strains are mainly governedby the thermo-mechanical behavior of the weld metal andthe base metal near the fusion zone during welding, sothe magnitudes and distributions predicted by the largedeformation theory and the small deformation theory arefairly similar. On the contrary, the final deformation ofthe thin plate butt-welded joint shows a significantly globalcharacteristic. It is very clear that even though the largedeformation theory and the small deformation theory pre-dict similar plastic strains in the butt-welded joint, how-

ever, the final welding deformations especially the out-of-

deformation (deflection) are significant different. The rea-son is that beyond the plastic strain zone the welding defor-mation is mainly governed by the elastic strain and thestraindisplacement relationship. By comparing with theexperiment, it is clear that the deflection at the center ofthe welding line predicted by the large deformation theoryis very close to the experimental measurement. The resultsimulated by the small deformation theory is much smallerthan the measurement. This indicates that when thethermo-elasticplastic FEM is used to predict the weldingdeformation in a thin plate structure the geometrically non-linear phenomenon should be carefully considered. It isvery interesting to note that even though the magnitudesof the welding deformation predicted by the two theoriesare much different, however, the deformation modes are

similar. BothFigs. 12 and 13show that the deflection dis-

0

0.5

1

1.5

0 50 100 150 200

Large deformation theorySmall deformation theoryExperiment

Deflection(m

m)

Y-Coordinate (mm)

Fig. 14. Deflection distributions along line AB.

-0.3

-0.2

-0.1

0

0.1

0 50 100 150 200


Y-Displacement(mm)

Y-Coordinate (mm)

Large deformation theory

Fig. 15. Y-directional displacement distributions of the middle section(Case A).

-0.3

-0.2

-0.1

0

0.1

0 50 100 150 200


Y-Displaceme

nt(mm)

Y-Coordinate (mm)

Small deformation theory

Fig. 16. Y-directional displacement distributions of the middle section(Case B).

-0.004

-0.003

-0.002

-0.001

0

60 80 100 120 140

Large deformation theorySmall deformation theoryL

ongitudinalPlasticStrain

Y-Coordinate (mm)

Fig. 17. Longitudinal plastic strain distribution of the middle section.

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tribution of the butt-welded joint has a saddle-shapedmode, which is generally opposite to that of a full weldingbutt-welded joint [13]. Because a partially welding is per-formed in the butt joint, the gaps remained at the two endscaused the saddle-shaped mode after welding.

Because relatively large longitudinal bending and trans-verse bending is predicted by the large deformation theory,both the longitudinal stress and the transverse stress of caseA have significant gradient distributions through thickness.On the contrary, a small out-of-deformation is simulated

by the small deformation theory, so either the longitudinalstress or the transverse stress of case B has almost no gra-dient through thickness in the butt-welded joint.

4. Prediction of welding deformation using inherent strain

method

4.1. Inherent strain

Although the thermo-elasticplastic FEM can be used tosimulate welding temperature field, welding residual stressand welding deformation, a very long computational timeis needed because the welding mechanical behavior ishighly nonlinear problem including material nonlinearity,geometrical nonlinearity and sometimes contact nonlinear-ity. Besides the thermo-elasticplastic FEM, elastic FEMbased on inherent strain theory[2126]can also be utilizedto predict welding deformation. Comparing with thethermo-elasticplastic FEM, only a very short computa-tional time is needed to complete the simulation even fora large and complex structure. Moreover, only the elasticmodulus and the Possions ratio at room temperature areused in the elastic FEM, and the temperature dependentmaterial properties are not needed. In this study, the inher-ent strain method is employed to simulate the welding

deformation in the thin plate butt-welded joint.

Based on experimental observations and theoreticalanalysis, it is found that the total welding distortion of aweld joint is mainly produced by four components, namelylongitudinal shrinkage (dx), transverse shrinkage (dy), lon-gitudinal bending (hx) and transverse bending (angular dis-tortion hy). The four fundamental deformation

components are also called inherent deformations [10].According to the numerical results obtained by thethermo-elasticplastic FEM, the four inherent deformationcomponents in a cross-section of the butt-welded joint canbe calculated using the following equations [811]:

dx 1h

Z epxdydz 8

dy 1h

Z epydydz 9

hx 12h3

Z epx z h=2dydz 10

hy12

h3Z e

p

yz h=2dydz 11where epx is the plastic strain in the welding direction (lon-gitudinal direction); epyis the plastic strain in the transversedirection; his the thickness of the plate, and z is the coor-dinate in the thickness direction.

The source induced welding deformation and weldingresidual stress is called inherent strain[22]. When the inher-ent deformations of a weld joint are known, they can betransferred into the corresponding inherent strain compo-nents [26]. When the inherent strains are introduced intoan elastic FEM, it is usually assumed that each component

has a uniform distribution along the welding line. Therefore,the average values of a weld joint are often used to approx-imately represent the inherent strains for the whole joint.

The average values of the four inherent deformations inthe butt-welded joint can be calculated using the followingformulae:

dx 1Lw

Z L0

dx dx 12

dy 1Lw

Z L0

dydx 13

hx

1

LwZ

L

0

hx dx

14

hy 1

Lw

Z L0

hydx 15

whereLwis the length of the welding line, Lis the length ofthe specimen.

4.2. Elastic finite element based on inherent strain

In the elastic FEM, the 4-node plate is used. For thinplate deformation, deflection w (x,y) is assumed to beequal to the deflection of mid-plane, w0(x,y). When large

deformation (geometrically nonlinear phenomenon) is con-

-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

80 90 100 110 120

Large deformation theorySmall deformation theory

TransversePlasticStrain

Y-Coordinate (mm)

Fig. 18. Longitudinal plastic strain distribution of the middle section.

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sidered, the straindisplacement relations can be defined asfollows:

ex eix ebx ou0

ox 1

2

ow0

ox

2" # zo

2w0

ox2

16

ey eiy eby ov0

oy1

2

ow0

oy 2" # zo

2w0

oy2 17

cxy cixy cbxy ou0

oy ov0

ox ow0

ox

ow0

oy

2zo

2w0

oxoy

18

whereu0,v0andw0are the displacement at mid-plane, ex,eyand cxy are total strains, and e

ix; e

iy and c

ixy are in-plane

strains, and ebx ; eby and c

bxy bending strains.

The curvature (jx) in a plane parallel to the xz planeand the curvature (jy) in a plane parallel to the yz planeand the twisting curvature (jxy), which represents thewarping of the xy plane, can be defined as follows:

jx o2w0ox2

19

jy o2w0

oy2 20

jxy o2w0

oxoy 21

Fig. 19 shows schematically a butt-welded joint. In thisjoint, the average longitudinal shrinkage ( dx) can be trans-formed into the in-plane strain component (ex) in longitu-dinal direction; the average transverse shrinkage ( dy) canbe changed into the in-plane strain component

ey

in

transverse direction; the average angular distortion hycan be converted into the curvaturejy along the y-axis;and the average longitudinal bending hx can be trans-formed into the curvaturejx along the x-axis. Theseinherent strain components are introduced into the elasticFEM as initial strains, and the total welding distortioncan be estimated through elastic finite element analysis pro-cedure[26]as shown inFig. 20.

InFig. 19, {e*}, {e}, {ee}, {f}, {u} and {r} are vectors ofinherent strain, total strain, elastic strain, equivalent nodalload, nodal displacement, and residual stress, and [B], [D],and [K] are straindisplacement matrix, elastic stressstrain

matrix and stiffness matrix, respectively.

4.3. Elastic finite element model

The elastic finite element model, the area introducedinherent strains, and the initial gaps are shown in Fig. 21.According to the simulation results computed by thethermo elasticplastic FEM the range of the longitudinal

x

yz

Welding line

Fig. 19. Butt-welded joint.

Input { }*

{ } [ ] { }fKu 1=

{ } [ ]{ }uB=

{ } { } { }* =e

{ } [ ]{ }eD =

{ } [ ] [ ]{ }dvDBfT

*=

Fig. 20. Analysis procedure of inherent strain method.

Fig. 21. The FE model, the area introduced inherent strains and the initial

gap.

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plastic strain is large than that of the transverse plasticstrain. In the elastic analysis, it is assumed that all the fourinherent strain components are introduced into the samearea. The four average inherent strain components aredetermined by the following equations:

ex

dx=Bis

22

ey dy=Bis 23hx hx=Bis 24hy hy=Bis 25whereBisis the breadth of the elements in which the inher-ent strains are introduced.

4.4. Simulation results and discussions

Fig. 22 shows the contour of final deflection distribu-tion. By comparing withFig. 11, it can be concluded that

the deflection distribution predicted by the elastic FEMmatches that computed by the thermo-elasticplasticFEM well.Fig. 23shows the deflection distribution alongthe line AB as defined inFig. 20. The corresponding deflec-tion distribution predicted by the thermo-elasticplastic isalso plotted in the same figure. It is clear that the differencebetween the two curves is very small.

Fig. 24 shows the Y-directional displacement distribu-tions along line AB predicted by the elastic FEM and thethermo-elasticplastic FEM. The average values of thetop surface and the bottom surface simulated by thethermo-elasticplastic FEM are plotted in this figure. Thisfigure means that the two distributed curves are quite sim-

ilar. Through carefully observation, it can be found that

the transverse shrinkage predicted by the thermo-elasticplastic FEM is slightly larger than that estimated by theelastic FEM. In the elastic FE model, the average trans-verse inherent strain of the whole joint is used. In thethermo-elasticplastic FE model, the transverse shrinkageshave non-uniform distribution along the welding line andthe maximum value is at the compared location (lineAB). These two factors are main reasons which resultedin a slightly difference between the two curves shown inFig. 24.

When the elastic FEM is used to predict welding defor-

mation, the attention is mainly paid to the total deforma-

Fig. 22. Deflection distribution predicted by elastic FEM.

-0.3

0

0.3

0.6

0.9

1.2

1.5

0 50 100 150 200

Elastic FEMThermo-elastic-plastic FEM

Deflection(mm)

Y-Coordinate (mm)

Fig. 23. Deflection distributions in the middle section.

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tion rather than the local deformation. From this view-point, it can be concluded that the transverse shrinkagepredicted by the large deformation elastic FEM matchesthat computed by the thermo-elasticplastic FEM well onthe whole.

FromFigs. 12, 22, 23 and 24, it can be concluded thatthe elastic FE model has accurately reproduced the weldingdeformation, which is simulated by the thermo-elasticplastic FE model. Based on the present study, it can beinferred that when the inherent strains of each joint

involved in a thin plate structure are known, the proposedelastic FEM can effectively predict welding deformation.Comparing the computational time, it is found that the

computational time to complete the elastic FE analysis isfar shorter than that used by the thermo-elasticplasticFE analysis. In the present study, the total computationaltime of both the thermal analysis and the thermo-mechan-ical analysis is approximately 12 hours for the thermo-elas-ticplastic FE model, whereas the computational time ofthe elastic FE model is shorter than 1 min. On the aspectof computational time, the elastic FEM has a significantadvantage over the thermo-elasticplastic FEM. Thus, thismethod is a promising alternative to predict welding defor-

mation for practical large thin-plate welded structures.

5. Conclusions

In this study, the thermo-elasticplastic FEM is used tosimulate the welding temperature field, residual stress anddistortion in a thin plate butt joint. Meanwhile, experi-ments are carried out to measure the welding deformation.Moreover, the elastic FEM based on inherent strain theoryis also utilized to simulate welding deformation in the samebutt joint. According to the numerical and experimental

results, the following conclusions can be drawn.

(1) Based on the simulation results, it is known that therealmost is no temperature gradient through thicknessin the thin plate butt joint during welding.

(2) Although the large deformation theory and the smalldeformation theory predict the similar plastic strains,however, the final welding deformations especially

the out-of-deformation computed by the two meth-ods are much different. By comparing with experi-ment, it is clear that the numerical result calculatedby the thermo-elasticplastic, large deformationFEM is in a good agreement with the measurement.Therefore, to precisely predict welding deformationin a thin plate structure, it is necessary to considerthe geometrically nonlinear problem.

(3) Because of relatively large longitudinal bending andtransverse bending deformations, both the longitudi-nal and the transverse residual stresses predicted bythe thermo-elasticplastic, large deformation FEMhave a gradient through thickness.

(4) The elastic FEM with considering large deformationcan be used to predict precisely welding deformationin the thin plate butt joint. Moreover, the computa-tional time is much shorter than that used in thethermo-elasticplastic FEM. For the automotiveindustry application, the elastic FEM based on theinherent strain theory is a promising method to pre-dict welding deformation.

References

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15 (1984) 299305.

-0.3

-0.2

-0.1

0

0.1

0 50 100 150 200

Elastic FEMThermo-elastic-plastic FEM

Y-D

irec

tiona

lDispla

cemen

t(mm

)

Y-Coordinate (mm)

Fig. 24. Y-directional displacement distributions in the middle section.

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