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Prediction of welding buckling distortion in a thin wallaluminum T joint

M. Asle Zaeem *, M.R. Nami, M.H. Kadivar

Department of Mechanical Engineering, Engineering School, Shiraz University, Zand Street, Shiraz 71348-51154, Iran

Received 21 October 2005; received in revised form 25 March 2006; accepted 29 March 2006

Abstract

In this paper, local and global welding buckling distortion of a thin wall aluminum T joint is investigated. A thermo-elastic–visco-plastic model is employed to determine longitudinal residual stresses; analysis of thermal model and elastic–viscoplastic (Anand) modelare uncoupled. Molten puddle motion (speed of welding) is modeled by using time dependent birth and death element method. Threedimensional nonlinear-transient heat flow analysis has been used to obtain the temperature distribution, and then by applying thermalresults and using three dimensional Anand elastic–viscoplastic model, stress and deformation distributions are obtained during weldingand after cooling. Local buckling is investigated by analyzing the history of stress and strain relations. Local buckling is assumed tooccur at a point if a small change in the magnitude of stress causes large deformation during of the welding process. By applying residualstresses on a structural model and using eigenvalue methods, global buckling instability of the welded structure is determined.� 2006 Elsevier B.V. All rights reserved.

Keywords: Welding; Finite element; Three dimensional; Anand elastic–viscoplastic model; Local buckling; Global buckling

1. Introduction

Welding in shipbuilding, nuclear reactors, petrochemicalplants, offshore structures and aerospace industries hasbeen employed at an increasing rate for its advantages indesign flexibility, cost saving, reduced overall weight andenhanced structural performance.

However, welding induces stresses which result intodistortion due to buckling, bowing and angular change[1]. Distortion can cause loss of dimensional control andstructural integrity and increase fabrication costs due topoor fit-up between components.

Therefore, validated methods for predicting welding dis-tortion and distortion mitigation techniques are desirable.

Many researchers, theoretically or experimentally havetried to predict the thermal and mechanical responses inthin plate weldments for finding ways to eliminate unfavor-

able large deformations. Tsai et al. [2] have studied weldingdistortion of a thin plate panel structure and the effect ofwelding sequence on panel distortion, local plate bendingand buckling. Mechaleris et al. [3,4] used 2D thermome-chanical welding process simulations to determine residualstresses and used a 3D structural model to determine buck-ling distortion after welding in a thin plate structure.

In this paper, local and global buckling in welding of athin wall aluminum T joint due to thermal stresses duringwelding is investigated during and after welding [5].

2. Welding simulation

The welding simulation involves two uncouple analysis.The effect of mechanical response is assumed to be negligi-ble on the thermal behavior, thus the temperature field issolved independently from the mechanical solution [3,6].To determine temperature history profiles a nonlineartransient thermal analysis is employed and a nonlineartransient mechanical analysis is used to determine stresshistory profiles.

0927-0256/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2006.03.016

* Corresponding author. Tel.: +98 262 4226192; fax: +98 711 6287508.E-mail address: mohsenzaeem@yahoo.com (M. Asle Zaeem).

www.elsevier.com/locate/commatsci

Computational Materials Science 38 (2007) 588–594

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The history dependent (transient) heat transfer probleminvolves an incremental scheme with several small timeincrements. The solution at a given time increment isobtained by using the solution at the previous time incre-ment as an initial condition. This problem is addressed indetail in Ref. [3].

The subsequent history dependent stress analysis isperformed by modeling the stress problem as a quasi-staticprocess in a Lagrangian frame. Similar to the heat transferanalysis, the numerical implementation of the quasi-staticanalysis involves an incremental scheme with several smallquasi-static increments. The solution at a given time inter-val is obtained by using the solution at the previous timeincrement as an initial condition. The temperature valuessolved in the previous thermal analysis are imported tothe mechanical analysis as loading.

Due to the nature of the welding process, additionalcomplexities are involved in simulation of welding, suchas temperature and history dependent material propertiesand large deformation in thin wall structures.

The finite element model of T weld joint is shown inFig. 1. This model is consists of about 12000 elements andfour-node tetrahedral elements are used to build this model.

2.1. Element rebirth technique (ERT)

In welding simulation for addition of welding materialin the numerical model during welding, the time dependentelement rebirth technique has been employed. A typical useof this feature to model assembly procedures is addingmaterials to the model at various loadsteps. Applying con-straints to the unused nodes in the model (nodes attachedto only deactivated elements) will prevent unwanted deflec-tions of the elements when they are later made alive [6,7].The deactivation of elements is done by reducing their con-tributions to stiffness matrices significantly and removingtheir contributions to load vectors completely.

2.2. Thermal analysis

Approximating a three-dimensional heat transfer analy-sis of a weld with a two-dimensional cross-sectional analy-

sis introduces an error in the computed temperatures [8].Kamala and Goldak [8] present a method for evaluatingthe error between 3D and 2D temperature fields. So toeliminate this type of error in this work we have used a3D heat transfer finite element model to simulate heattransfer during welding. The governing energy balanceequation for transient heat transfer analysis is given asfollows:

qCPdTdtðr; tÞ ¼ �rr � qðr; tÞ þ Qðr; tÞ

ðin the entire volume Vr of the materialÞ ð1Þ

q is the density of the body, CP is the specific heat, T is thetemperature, q is the heat flux vector, Q is the internal heatgeneration rate, t is the time, r is the coordinate in thereference configuration, and $r is the spatial gradientoperator.

The nonlinear isotropic Fourier heat flux constitutiverelation is enforced; using the temperature dependent ther-mal conductivity, k:

q ¼ �krrT : ð2ÞThe arc power was transferred into the model by use of themoving isothermal pool. The movement of the heat sourcein welding introduces a noticeable change in the shape ofisothermal surfaces. The arc power and its movement weremodeled in this work by assuming that the welded region isan isothermal melted pool and this pool has a constanttemperature and its location discretely changes with respectto time.

Convection and radiation boundary conditions areassigned to all free surfaces. The coefficients of thermalconvection and radiation for aluminum-1100 that wereused in this work are shown in Table 1 [9].

2.3. Anand elastic–viscoplastic model

A simple set of constitutive equations for large, isotro-pic, viscoplastic deformations but small elastic deforma-tions is the single-scalar internal variable model proposedby Anand and Brown [10]. The set of Anand constitutiveequations presented here accounts for the physical phe-nomena of strain-rate and temperature sensitivity, strainrate history effects, strain hardening and the restorationprocess of dynamic recovery.

Fig. 1. Detail and finite element mesh of T weld joint.

Table 1Coefficients of thermal convection and radiation [9]

T (�C) h (W/m2 �C) Emmisivity

20 2.5 0.03850 5.5 0.043150 6.7 0.047300 8.5 0.05450 8.2 0.053550 8.7 0.055650 9 0.057

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The Anand model is based on two physical ideas:

1. Plastic deformation due to dislocation motion is inher-ently rate-dependent. The rate effects (for a large rangeof interest) are due to thermal activation of various flowprocesses, and therefore plastic flow is also intrinsicallytemperature sensitive.

2. The instantaneous response of a material is determinedby its current state. The current state (which is producedby the entire past history of deformation undergone bythe material) may be assumed to be representable byrate equations.

There are two basic features in this Anand model [11].First, this model needs no explicit yield condition and noloading/unloading criterion. The plastic strain is assumedto take place at all nonzero stress values, although at lowstresses the rate of plastic flow may be immeasurably small.Second, this model employs a single scalar as an internalvariable to represent the isotropic resistance to plastic flowoffered by the internal state of the material. This internalvariable is denoted by s, which has the dimensions of stress,and is called deformation resistance. There are some rea-sonable considerations for the simplifications that only asingle scalar is used to characterize the internal structuralcharacteristics of a material. The internal variable repre-sents an averaged isotropic resistance to macroscopic plas-tic flow offered by the underlying isotropic strengtheningmechanisms such as dislocation density, solid solutionstrengthening, sub grain, and grain size effects, etc. Thedeformation resistance s is consequently proportional tothe equivalent stress r, that is,

r ¼ C � S; C < 0; ð3Þ

where C is a material parameter and constant in theconstant strain rate test, which is defined as:

C ¼ 1

nsinh�1 _ep

Aexp � Q

RT

� �� �m

; ð4Þ

where _ep is the inelastic strain rate, A is the pre-exponentialfactor, Q is the activation energy, m is the strain rate sen-sitivity, n is the multiplier of stress, R is the gas constant,and T is the absolute temperature, respectively.

The following functional form of the flow equation [11]was selected to exactly accommodate the strain rate depen-dence on the stress at constant structure:

_ep ¼ A exp � QRT

� �sinh n

rS

� �h i1=m: ð5Þ

Note that the internal state variable enters into the flowequation only as a ratio with the equivalent stress. Thetemperature dependence in Eq. (5) is incorporated via aclassical Arrhenius term.

The evolution equation for the internal variable s isassumed to be of the form:

_S ¼ gðr; S; T Þ_ep; ð6Þwhere the function g(r, s,T) is associated with dynamicprocesses that are strain hardening and dynamic recovery.A simple form of evolution equation of Eq. (6) was givenby Anand as follows:

_S ¼ h0 1� SS�

��������a

� sign 1� SS�

� �� �� _ep; a > 1; ð7Þ

S� ¼ S_ep

Aexp

QRT

� �� �n

: ð8Þ

In the above viscoplastic Anand model, there are ninematerial parameters: A, Q/R, n, m, h0, S, n, a, and S0. Theseparameters are displayed in Table 2. Material propertiesused in this paper are shown in Table 3.

3. Buckling distortion

Buckling is common and important distortion that oftenhappens in weldments. When thin plates are welded, com-pressive stresses occur in areas away from the weld and cancause buckling distortion. Where buckling distortionoccurs, the magnitude of distortion tends to be very large;furthermore, buckling instabilities reduce the structuralintegrity of welded structures.

Welding buckling distortion has been studied by manyresearchers. Most of those have developed methods to pre-dict and eliminate residual buckling distortion in weldedstructures. Investigation of buckling instabilities duringwelding is difficult to analysis, so most of works on buck-ling are limited after welding process in structures. Someresearch studies of buckling in weldments have employed

Table 2Material parameters for Anand model [10]

Parameter Meaning Value and unit

1 s0 Initial value of deformation resistance 36.6 MPa2 Q/R Q = activation energy 175.3 kJ/mole

R = universal gas content 8.314 kJ/(mole K)3 A Pre-exponential factor 1.91 · 107 (s�1)4 n Multiplier of stress 7.00 (dimensionless)5 m Strain rate sensitivity of stress 0.23348 (dimensionless)6 h0 Hardening/softening constant 1115.6 MPa7 S Coefficient for deformation resistance saturation value 18.9 MPa8 n Strain rate sensitivity of saturation (deformation resistance) value 0.07049 (dimensionless)9 a Strain rate sensitivity of hardening or softening 1.3 (dimensionless)

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restraints for preventing buckling distortion during weldingand after welding release restraints and investigate the glo-bal buckling that remains in weldments [3,4]. This type ofbuckling analysis cannot be used to investigate local buck-ling during welding process.

In studying welding distortion in thin-plated structures,it is important to first determine whether the distortion isbeing produce by buckling or bending. Buckling distortiondiffers from bending in that [1]:

1. There is more than one stable deformed shape.2. The amount of deformation in buckling distortion is

much greater.3. Buckling distortion happens very fast.

In this paper, local buckling is considered to happen at apoint if small changes in compressive stress cause very largedistortion in a small time increment, so by results of non-linear analyses we can determine when and where localbuckling probably can be observed.

To investigate global buckling, we use eigenvaluemethod to determine whether or not global buckling hap-pens. Using results of thermomechanical analysis, residuallongitudinal stress distribution is applied to a structuralmodel that can analyze buckling instabilities. The elasticinstability problem is defined as the generalized eigenvalueproblem in which K is the elastic stiffness matrix in theundeformed configuration [4,12]:

detðK þ kKGÞ ¼ 0; ð9Þwhere K is linear stress stiffness matrix, KG is the nonlinearstress stiffness matrix, and k is the eigenvalue.

ki ¼ðrcrÞirL

: ð10Þ

In the above equation, we applied longitudinal residualstresses that were determined by welding simulation, sorL is longitudinal residual stress and rcr is critical bucklingstress. We assumed that if structural analysis gives k1 > 1(k1 is the smallest eigenvalue), buckling distortion is not ap-

peared in weldment, and if k1 6 1 means longitudinal resid-ual stresses are bigger than critical buckling stresses, sobuckling distortion can appear in the weldment.

4. Results

Computed temperature distributions for three selectedtimes are shown in Fig. 2; in part (a) and (b), contours of

Table 3Material properties used in this paper

T (�C) Thermalconductivityk (W/m K)

Specific heatCp (J/kg K)

Coefficients ofthermal expansiona (10�6/�C)

Young’smodulusE (Gpa)

20 222 904 23.3 7250 230 930 23.6 –100 – – 24 70150 250 965 25 67200 260 – – –250 272 985 – 61.5300 – 980 26 –350 278 1040 – 53400 – 1100 26 45450 283 – – –500 285 – – 35550 – 1100 26 –600 320 – – 17650 400 – – 10

Fig. 2. Temperature distribution. (a) Time: 30 s after welding beginning(first pass). (b) Time: 160 s after welding beginning (second pass). (c) Time:300 s after welding beginning (during cooling).

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temperature are seen for a specific time in the first pass andin the second pass respectively. Fig. 2(c) shows temperaturedistribution for a specific time during the cooling of theweldment.

In the present work, an experiment was set up to inves-tigate buckling distortion due to welding. For welding, weused TIG equipment with argon gas shielding. The ACampere and voltage used for welding were 130 A and22 V and two pass welds were processed by the welder.To allow the heat transfer by convection and radiationfrom the bottom surface of base plate, two woodensupports were used in the experimental set up (Fig. 3).

In the numerical model, four test points on the T jointwere chosen to study how to recognize the happening ofthe local buckling distortion (Fig. 4). These points werechosen far from the weld zone (two points on the base plateand two points on the vertical plate) where the stresseshave compressive nature and may cause buckling distor-tion. The results show the ability of this simulation notonly to predict thermal stresses and deformation duringwelding but also to investigate local buckling of anyselected point.

The stress and deflection versus time for the first selectedtest point that is located on the vertical plate are shown inFig. 5. From the figure, it can be easily seen that the local

buckling has not happened since no jump exists in thedeflection diagram. In points no. 2 and no. 3, we see a jumpin the deflection diagrams without any noticeable changesin the stress diagrams simultaneously (Figs. 6 and 7), there-fore according to the previously mentioned facts, the localbuckling distortion are happened in the point no. 2, 122 safter the beginning of the welding and in the point no. 3,

Fig. 3. Wooden supports for experimental set up

Fig. 4. Four points on the T joint (dimensions in mm).

Fig. 5. Stress and deflection versus time for point no. 1.

Fig. 6. Stress and deflection versus time for point no. 2.

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py143 s after the welding begins. The magnitude of the jumpin the deflection for point no. 2 is 6.2 mm and for point no.3 is 5.3 mm. In Fig. 8, the stress and deflection versus timefor the forth point that is located far from the weld on thebase plate are shown and by the previous facts, local buck-

ling may not occur in this point. The local buckling ofpoints no. 2 and no. 3 are also shown in Fig. 9. Distribu-tion of longitudinal residual stresses is presented inFig. 10; as the figure demonstrates, in the base plate, tensilestresses occurred in the region near the weld and compres-sive stresses are produced in the region far from the weldzone. These residual compressive stresses cause globalbuckling distortion in the weldment (Fig. 11). From aneigenvalue buckling analysis, k is given 0.852, so the struc-ture is buckled. The calculated maximum deflections in the

Fig. 7. Stress and deflection versus time for point no. 3.

Fig. 8. Stress and deflection versus time for point no. 4.

Fig. 9. Longitudinal residual stresses and two local buckling points in anx–y plane section (deformation 2X).

Fig. 10. Contour of longitudinal residual stresses.

Fig. 11. Final deformation of T joint (3X) in the numerical modeling.

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base plate and in the vertical plate are 35.43 mm and16.78 mm respectively. In the experiment, buckling appearsand the maximum deflection measured 31.2 mm in the baseplate (Fig. 12) and 12.7 mm in the vertical plate (Fig. 13).

Bowing distortion in the bottom plate due to actual weld-ing process is also shown in Fig. 14.

5. Conclusions

In this paper welding buckling distortion is investigated.First for welding simulation, the thermal model is solvedand then by applying thermal results into the mechanicalmodel, stresses and deformations are determined. Secondby using the structural model and employing an eigenvaluebuckling analysis, the global buckling of the weldment isstudied.

In this research, because of using nonlinear transientapproach for both thermal and mechanical models, theresults may be more accurate. Decoupling of thermal anal-ysis from the mechanical analysis and using Anand consti-tutive equations for elastic–viscoplastic behaviors ofwelded structure reduce the computational time and costwithout noticeably decreasing the accuracy of results.Because of the ability of this model to give some informa-tion about thermomechanical responses of welded struc-tures during the welding, this model is useful to study thethermal and mechanical behaviors of structures duringthe welding process, and also is useful to investigate thelocal effects of welding in welded structures. The experi-mental results show the numerical model presented hereeffectively and efficiently predict the probability of bucklingdistortion in the weldments.

References

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[2] C.L. Tsai, S.C. Park, W.T. Cheng, Welding distortion of a thin-platepanel structure, Weld. J., Res. Suppl. 78 (1999) 156s–165s.

[3] P. Michaleris, A. DeBiccari, Prediction of welding distortion, Weld.J. 76 (4) (1997) 172–180.

[4] M.V. Deo, P. Michaleris, J. Sun, Prediction of buckling distortion ofwelded structures, Sci. Technol. Weld. Join. 8 (2003) 55–61.

[5] M. Asle Zaeem, Investigation of welding buckling distortion of thinwall plates (2005). M.Sc. thesis, Shiraz University, Iran.

[6] M. Asle Zaeem, M.R. Nami, Three dimensional finite elementsimulation of welding of a thick plate flange by using Anandviscoplastic model, ASME Int. Des. Eng. Technical Conf., California,USA (2005).

[7] M.R. Nami, M.H. Kadivar, K. Jafarpur, Three-dimensional thermalresponse of thick plate weldments: Effect of layer-wise and piecewisewelding, Model. Simul. Mater. Sci. Eng. 12 (2004) 731–743.

[8] V. Kamala, J.A. Goldak, Error due to two dimensional approxima-tions in heat transfer analysis of welds, Weld. J. 76 (9) (1993) 440s–446s.

[9] J.P. Holman, Heat Transfer, McGraw-Hill, Inc., 1999.[10] S.B. Brown, K.H. Kim, L. Anand, An internal variable constitutive

model for hot working of metals, Int. J. Plasticity 5 (1989) 95–130.[11] G.Z. Wang, Z.N. Cheng, K. Becker, J. Wilde, Applying Anand model

to represent the viscoplastic deformation behavior of solder alloys,J. Electron. Packaging 1 (123) (2001) 247–253.

[12] K.J. Bathe, Finite Element Procedures in Engineering Analysis,Prentice-Hall, Englewood Cliffs, New Jersey, 1982.

Fig. 12. Final deformation of the T joint due to experiment (bucklingdistortion in the base plate).

Fig. 13. Final deformation of the T joint due to experiment (bucklingdistortion in the vertical plate).

Fig. 14. Final deformation of the T joint due to experiment (bowingdistortion in the base plate).

594 M. Asle Zaeem et al. / Computational Materials Science 38 (2007) 588–594