Minimization of Welding Residual Stress and Distortion In

8/10/2019 Minimization of Welding Residual Stress and Distortion In

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Minimization of welding residual stress and distortion inlarge structures

P. Michaleris The Penn State University, University Park, PAJ. Dantzig and D. Tortorelli University of Illinois at Champaign Urbana, Urbana, IL

Abstract

Welding distortion in large structures is usually caused by buckling due to the residual stress. Incases where the design is fixed and minimum weld size requirements are in place, the thermaltensioning process has proven effective to reduce the welding residual stress below the criticallevel and eliminate buckling distortion. In this work, a systematic design approach usingconventional finite element analysis, analytic sensitivity analysis, and nonlinear programming isimplemented to investigate and optimize the thermal tensioning process.

1 Introduction

The trend in current design and manufacturing practice is to reduce product weight through theuse of weldable high strength materials in thin sections. However, use of thin section materialsincreases the susceptibility of a structure to buckling during manufacturing due to the weldingresidual stress. Buckling distortion can also degrade the product performance, increasemanufacturing cost due to the poor fit-up and the need for straightening, reduce structuralintegrity and cause excessive product rejection. Buckling distortion can be eliminated by either increasing the rigidity of the structure through improved designs or by reducing the weldingresidual stress through process modifications.

Over the past fifteen years, the finite element method has been used to predict distortion andresidual stress due to welding. Simulations of welding processes involve thermo-mechanicalfinite element analyses of the weld zone. Many investigators (Refs. 1- 5) have performedtransient nonlinear thermal analyses and small deformation quasi-static elasto-plastic analyses.Following such analyses, Michaleris and DeBiccari (Refs. 6-7) have demonstrated that thewelding residual stress can be accurately predicted and consequently applied as a pre-stress in a

buckling analysis of a structure.

Reducing the welding heat input and modifying the structural configuration reduces theoccurrence of buckling (Refs. 6-9). Design considerations, however, may impose limits on suchmodifications. In this case, new manufacturing techniques such as thermal tensioning can beused to eliminate buckling due to welding. The thermal tensioning technique for controllingwelding residual stress and distortion as discussed by Burak et al. (Refs. 10-11) involvesgenerating a tensile strain at the weld zone, prior to and during welding, by imposing a steadystate temperature differential. Recently, Michaleris and Sun (Ref. 12) used a thermo-elasto-

plastic model to demonstrate that the thermal tensioning process minimizes the welding residualstress by reducing the plastic strain accumulation during welding. However, generating a steadystate temperature differential prior to welding requires the use of heat sinks (cooling devices) thatare impractical, costly, and environmentally unfriendly. Therefore, the development of a thermaltensioning process (referred to here transient thermal tensioning ) that uses transient temperature


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differentials is desirable. Such transient differentials can be generated by a moving heat source.Effective implementation of this tensioning process requires determining the appropriateintensity, size and location of the heat source such to minimize the welding residual stress.Unfortunately, even for simplified geometries, a conventional parametric study for determiningthe appropriate process parameters is impractical as the number of process configurations is

prohibitively large. To resolve this problem we use, sensitivity analysis and optimization that has proven successful in determining optimum designs for linear (Refs. 13-14) and nonlinear (Refs.15-16) problems with large numbers of design variables.

In this work, a systematic design approach using conventional finite element analysis, analyticsensitivity analysis, and nonlinear programming is implemented to investigate and optimize thetransient thermal tensioning process.

2 Transient Thermal Tensioning Process

The transient thermal tensioning device investigated here consists of two heating bands travelingalong with the welding torches (Figure 1). Flame heating is investigated due to the low

acquisition and operational cost. Then, the design of a transient thermal tensioning device becomes the determination of the width of the heating bands ( d 1), length of the heating bands(d 2), offset from the first torch ( d 3), and offset from the weld centerline ( d 4) that minimize thewelding residual stress without degrading material performance with undesirable metallurgicaltransformations. A separate set of design variables ( d 1- d 4) must be determined for each weldheat input, panel and stiffener thickness combinations.

3 Optimization of Thermal Tensioning

A parametric investigation to identify the desired combination of the four design variables d 1 tod 4 would require numerous analyses. For example, if ten values of each design variable areconsidered, then ten thousand combinations will need to be investigated. Each combinationincludes the preparation, computation, and interpretation of ten thousand welding simulations.Furthermore, a discrete (fixed values for the design variables) parametric space will most likelymiss an optimum combination. Finally, this exercise will need to be repeated for each weld heatinput, panel and stiffener thickness combination. Such a parametric investigation is practicallyand economically unfeasible.

Rather than performing a parametric design analysis, the transient thermal tensioning process isinvestigated here by solving an optimization problem stated as follows:

Identify d i, i= 1 , n, such that:

(1)for:

(2)

),...,,(min 21 nd d d f

( ) ( )maxmin iii d d d 0),...,,( 21 n j d d d g


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)( iii ad f s+

where, f is the objective function which quantifies the welding residual stress, n is the number of design variables, d i are the design variables, ( d i)min and ( d i)max are the minimum and maximumvalues allowed for each of the design variables, and g j are a set of inequality constraints.

The problem defined by equations (1) and (2) is solved by the steepest descent method for simplicity (Ref. 17). In the steepest descent method, an initial selection of the design variables isiteratively modified until convergence is reached (Figure 2). At each iteration i, the objectivefunction gradient si is initially computed to check for convergence. If convergence is notachieved, a line search is performed in the - si direction to determine the a i that minimizes where:

The line search direction of equation (3) is determined by computing the derivative (sensitivity)of the objective function with respect to each design variable.

4 Numerical Approach

4.1 Analysis of the Forward problem

The residual stress is evaluated by performing a 2D transient heat conduction analysis followed by a quasi-static elasto-plastic analysis assuming generalized plane strain conditions. In thegeneralized plain condition, a model cross section is assumed to deform between two rigid

planes. This condition is acceptable for small sections (narrow panels). However for largesections (wide plates), a loading on one part of the cross section will generate finite strains over the entire section leading to inaccuracies. For example, Michaleris and DeBiccari, (Refs. 6-7)show that under the generalized condition the computed welding residual stresses extend tolocations away from the weld. However, experimental measurements indicate that, in largesections (wide panels) the residual stress gradually diminishes away from the weld (Ref. 8).

The analysis plane is set to a cross-section of the model perpendicular to the welding direction.The governing equations consist of the energy balance equation (which neglects the stress power so that the thermal analysis is independent from the mechanical analysis):

the mechanical equilibrium equation:

and the mechanical constitutive law:

(3)

(4)

(5)

0)( =

T k t h

r

0b! =+

),,,(21 n

i d f

d f

d f

= s


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where, r is the internal heat generation, h is the enthalpy, k is the thermal conductivity, t is time, is stress, C the elastic constitutive tensor, t , p, and th are the total, plastic and thermalexpansion strains, b is the body force, e p is the equivalent plastic strain, F is the plastic flow rule,and Y is the yield function. The system of equations (4-6) is solved using the finite elementmethod.

The welding heat input is modeled by a double ellipsoid body heat flux distribution (Ref. 3):

eabc

f Q36 =r c

)vt +3(Z +

b

3Y +

a

3X -b 2

2b

2b

2

2b

2

(7)

where, Qb is the welding heat input, is the welding efficiency (set here to 80%) X b, Y b, and Z bare the local coordinates of the double ellipsoid model, a is the weld width, b is weld penetration,c = a and f = 0.6 before the torch passes the analysis plane and c = 4a and f = 1.4 after the torch

passes the analysis plane, v is the torch travel speed, and t is the clock time.

The thermal tensioning heat source is applied on the top surface of the plate (Figure 1), and isdefined by an ellipsoid surface heat flux q:

(8)

where X b and Z b, are the local coordinates of the surface ellipsoid defines as follows:

14 5.0 d d x X s =

(10)

Q s is the flame heat flux per unit area, is the heating efficiency (set here to 30%), x is thehorizontal distance from the weld centerline, v is the welding torch travel speed, t is the clock time, and d 1 to d 4 are the design variables (i.e. the width of the heating bands, length of the

heating bands, offset from the first torch, and offset from the weld centerline), respectively.

Generalized plane strain conditions are assumed in the mechanical analysis to account for theout-of-plane expansion in the model. The out-of-plane strain z is assumed to have a linear distribution over the analysis plane:

(6)

(11)

( )T e F e p p p ,, !" =

=th pt """C!

T eY p ,,0 !

eQ

=q(x) d Z

d

X 3- s s s

2

+

22

2

21

34

23 5.0 d d vt Z s +=

x y z y xe +=


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where e is the out-of-plane strain at the coordinate origin and x and y are the strain variations inthe y and x axes respectively.

4.2 Sensitivity Analysis

The use of optimization drastically improves the ability to effectively design the thermaltensioning process. However, optimization requires the computation of the derivative(sensitivity) of the residual stress with respect to the design variables. The computationalapproach of the sensitivity analysis is analogous to the approach presented Michaleris et al. (Ref.15).

The gradient of the objective function is computed using the direct differentiation method (Ref.15). First, the objective function f is expressed as a generalized response function of thetemperature, displacement, and plastic strain field:

Then, the design derivative of the generalized function is computed by chain rule differentiationof equation (12):

The sensitivities of the temperature T , displacement u , and plastic strain p are computed bydifferentiating the governing equations (4-6) which are now written in discretized residual formas:

Differentiating and rearranging equations (14) yields the following expressions for thesensitivities:

The inverse operators in equations (15) are equivalent to the consistent tangent operators that arecomputed and inverted in the forward problem (Ref. 15), therefore the numerical overhead of

(12)

(13)

(14)

(15)

+

+

=

iii p

i

p

d T

T d d d HHu

uH

"

H"1

),,( pT f f "u=

i

p

piii d f

d f

d T

T f

d f

+

+

= "

"

uu

0W =)(T

+

=

i p pii p pi d d T

T d d H

"

H"

R R R uH

"

H"

R uR u

111

=

ii d T d T WW

1

0u"R =),,( T p

0u"H =),,( T p


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computing the design sensitivities is a very small fraction of that required for the forward problem.

5 Computational Results

The proposed design algorithm is implemented to determine the optimum transient thermal

tensioning device for welding a 4 x 4 (101.6 x 101.6mm) stiffener with 3/16 (4.76 mm) thick flange and web on a 3/16 (4.76 mm) thick, 2 (0.6 m) wide plate with 3/16 (4.76 mm) filletwelds. Both plate and stiffener are made of AH36 steel. This configuration is identical to theone selected by Michaleris and Sun (Ref. 12) in their investigation of steady state thermaltensioning. The 2D finite element mesh used in this study is illustrated in Figure 3. Radiationand convection boundary conditions are assigned for all free surfaces. A temperature dependentfree convection coefficient is used here and is plotted on Figure 4. The emissivity is set to 0.2.

The material properties and free surface boundary conditions are identical to those discussed inMichaleris and DeBiccari (Ref. 7). The temperature dependent thermal conductivity K, andspecific heat C p are plotted on Figure 4. The latent heat of fusion is set to 247 kJ/kg/ C and the

density to 7.86 x 103

kg/m3

. Elastic-plastic material response is assumed with isotropic work hardening. Figures 5 and 6 illustrate the temperature dependent mechanical properties.

The welding heat input Qb is set to 6682 J (25.7V, 260 A) for the first torch and 6180.5 J (26.3V, 235 A) for the second. The thermal tensioning heat input is Q s set to 8.3e-4 J/in

2 (0.5 J/mm 2).The welding travel speed is 24.72 ipm (10.5 mm/s). The thermal forward problem andsensitivity analyses are performed in an enhanced version of the commercial code FIDAP (Ref.18), while the mechanical forward problem and sensitivity analyses are performed in aFORTRAN finite element code developed by the authors.

The bounds of the design variables are selected based on practical limitations (Table 1). The

initial design selected for the optimization consists of two square heaters located 2 away fromthe weld centerline, and 5 ahead of the first welding torch.

Table 1. Bounds of design variables

Designvariable

Minimum(in.)

Maximum(in.)

d1 1 12d2 1 12d3 0 12d4 2 4

The computed longitudinal welding residual stress for the initial selection of design variables isillustrated in Figure 7. The weld region is under yield level tension which agrees with theconventional welding results of Michaleris and DeBiccari (Refs. 6-7). The compressive residualstress at the plates edge is 8.8 ksi (60.7 MPa) which is slightly higher than the conventionalwelding case where the compressive stress is 8.062 ksi (55.9 MPa). Thus, the initial set of design variables has an adverse effect on the residual stress.


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The following two sections describe the optimization results for two choices of the objectivefunction.

5.1.1 Minimum Stress at the Plates Edge

The objective function is set to the absolute value of the longitudinal residual stress of theelement located at the edge of the plate (location A in Figure 3). After six iterations, the

optimization converges towards the heating elements that are illustrated in Figure 8. The heater width ( d 1) is 8.72 (221.6 mm), the heater length ( d 2) is 8.19 (208 mm), the offset form firsttorch ( d 3) 1.3 (33.2 mm), and the offset from the weld centerline ( d 4) is 3.6 (90.5 mm). Thecorresponding longitudinal residual stress is illustrated in Figure 9. As seen in the figure, theresidual stress at the plates edge is negligible, however, a band of tensile residual stress isgenerated next to plates edge. This is caused by the plastification of the plate due to excessive

pre-heat.

To avoid optimization solutions that may plastify the plate, the optimization problem can bemodified by:

1) Imposing constraints to limit the peak temperature over the application region of the heating bands

2) Imposing constraints to limit the plastic strain over the application region of the heating bands

3) Modifying the objective function to include the effects of residual stress in applicationregion of the heating bands

The following section explores the option 3 above.

5.2 Minimum Sum of Squares Stress on the Plate

The objective function is selected as the sum of squares of the residual stress of all the elementson the plate located two inches away from the weld centerline (location B in Figure 3).

An optimum design is reached after ten design iterations resulting to an approximately zero valueobjective function, which corresponds to negligible residual stress over the plate two inches awayfrom the weld centerline. The optimal heating elements are illustrated in Figure 10 where it isseen that the heat width ( d 1) expands over the entire plate width and the heater length ( d 2) is7.35 (186.7 mm). A zero offset form the first torch ( d 3) is required and the minimum allowableoffset from the weld centerline ( d 4) is computed. This result is attributed to the thermal diffusionfrom both heating pads and welding torches.

The computed residual stress using the optimum heater configuration of Figure 10 is illustratedin Figure 11. The residual stress at the weld region is tensile with magnitude of about half theroom temperature yield stress. A small compressive stress region exists around the weld to

balance the tension of the weld. The residual stress over the plate two inches away from the weldis negligible. Figure 12 illustrates the computed temperature history at seven points (P1 thoughP7) located on the bottom of the plate (Figure 3). As seen in the figure, the peak temperature inthe plate is 530 oC. This temperature does not cause adverse metallurgical transformations on the


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material. However, if high temperatures were found to cause undesirable transformations on thematerial, constraints on peak temperature could be added in the optimization set up (equation (2))to exclude such heating configurations from the solution.

6 Discussion

This study demonstrates that numerical optimization and finite element analysis can be combinedto minimize residual stress and distortion using the thermal tensioning process. The generalized plane condition has been used in this study to minimize model size and computational time. Incomparisons with experimental measurements in narrow panels (2 wide), the generalized planestrain condition has given accurate results for modeling conventional welding (Ref. 7) andwelding with thermal tensioning (Ref. 12). However, additional numerical and experimentalresearch is needed to verify the general applicability of the generalized plane strain condition for modeling welding under thermal tensioning, especially for wide panels. In wide panels,inaccuracies may be introduced due the fact that in the generalized plane strain condition aloading on one part of the cross section generates finite strains over the entire section. Therefore,the restraint caused by a wide panel will be overestimated and therefore it will artificially reduce

the tension generated by the heaters. Using the generalized plane condition, a wide panel isexpected to require heaters extending over the entire panel width. The numerical approach presented here can be expanded to 3D finite element formulations to investigate wide panels.

7 Conclusions

The numerical investigation presented here demonstrates the effectiveness of using numericalanalysis and optimization to design the thermal tensioning process. The transient thermaltensioning using no cooling and localized heating can produce panels with zero residual stress onthe plate. Such panels will have no buckling distortion (Refs.6-7). They will also have improvedstructural integrity. Furthermore, they will have negligent longitudinal shrinking. Thus they willfacilitate the implementation of a neat cut manufacturing approach.

The computational approach can easily accommodate design or material limitations byintroducing constraints on the optimization set up. For example, constraints on peak temperatures can ensure that heating does not cause adverse metallurgical transformations. Theapproach can also be extended to 3D finite element formulations to investigate wide size panels.

8 Acknowledgment

This work was funded by the Navy Joining Center, Columbus, Ohio. The United States

Government, Navy Joining Center, and Edison Welding Institute make no warranties and assumeno legal liability or responsibility for the accuracy, completeness, or usefulness of the informationdisclosed in this report. Reference to commercial products, processes, or services does notconstitute or imply endorsement or recommendation.


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9 References

1. S.B. Brown and H. Song, Implications of Three-Dimensional Numerical Simulations of Welding of Large Structures, Welding Research Supplement, Welding Journal , 71(2):55s-62s (1992)

2. A. Chakravarti, L.M. Malik, and J. Goldak, Prediction of Distortion and Residual Stresses inPanel Welds. Symposium on Computer Modeling of Fabrication Processes and Constitutive

Behavior of Metals . Ottawa, Ontario, pp. 547-561 (1986)

3. J. Goldak, A. Chakravarti, and M. Bibby, A new finite element model for welding heatsources. Metallurgical Transactions B 15B:299-305 (1984)

4. L. Karlsson, M. Jonsson, L.E. Lindgren, M. Nasstrom, and L. Trovie, Residual Stressses andDeformations in a Welded Thin-Walled Pipe. ASME Pressure Vessels and Piping Division(Publication) PVP Weld Residual Stresses and PlasticDeformation Jul 23-27 1989 v 173

Honolulu, HI, (1989).

5. A.R. Ortega, J.F. Lathrop, R.E. Corderman, E.A. Fuchs, B.V. Hess, K.W. Mahin, A.F.Giamei, Analysis of buckling distortion in bead-on-plate Ti 6-4, Proceedings of the 1995 7thConference on Modeling of Casting, Welding and Advanced Solidification Processes, Sep10-15, 249-256 (1995)

6. P. Michaleris, and A. DeBiccari, A Predictive Technique for Buckling Analysis of ThinSection Panels due to Welding, Journal of Ship Production , 12(4): 269-275 (1996)

7. P. Michaleris, and A. DeBiccari, Prediction of Welding Distortion, Welding Journal , 76,

172-s-181-s, (1997)

8. K. Masubuchi, Analysis of Welded Structures . Oxford, Pergamon Press, (1980)

9. K. Terai, Study on prevention of welding deformation in thin-skin plate structures . KawasakiTechnical Review, no. 61, pp. 61-66 (1978)

10. Ya.I. Burak, L.P. Besedina, Ya.P. Romanchuk, A.A. Kazimirov, and V.P. Morgun,Controlling the Longintudinal Plastic Shrinkage of Metal during Welding, Avt. Svarka,1977, No.3, pp.27-29 (1977)

11. Ya. I. Burak, Ya.P. Romanchuk, A.A. Kazimirov, and V.P. Morgun, Selection of theOptimum Fields for Preheating Plates before Welding, Avt. Svarka , 1979, No.5, pp.5-9(1979)

12. P. Michaleris, and X. Sun. Finite Element Analysis of Thermal Tensioning TechniquesMitigating Weld Buckling Distortion, Welding Journal , 76(11): 451-457s, (1997)


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13. K. Dems, and Z. Mroz, Variational Approach to Sensitivity Analysis in Thermoelasticity, Journal of Thermal Stress , 10, 283-306 (1987)

14. Haug, E.J., Choi, K.K., and Komkov, V., Design Sensitivity Analysis of Structural Systems .Academic Press, New York. (1986).

15. P. Michaleris, D.A. Tortorelli, and C.A. Vidal, Analysis and Optimization of WeaklyCoupled Thermo-Elasto-Plastic Systems with Applications to Weldment Design,

International Journal for Numerical Methods in Engineering , 38, 1259-1285, (1995)

16. D.A. Tortorelli, M.M. Tiller, and J.A. Dantzig, Optimal Design of Advanced ParabolicSystems-Part I. Fixed Spatial Domain with Applications to Process Optimization , Computer

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18. FIDAP, Fluent, Inc, Nashua, NH.


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Figure 1. Welding with the transient thermal tensioning process


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Figure 2. Illustration of steepest descent method in a 2D design space.

d 1

d 2 1

2

3

s1

s2s3

1 > f 2 > f 3


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Figure 3. Finite element mesh.

A

B

P7 P1P2P3P4P5P6


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Figure 4. Thermal properties of AH36 steel (Michaleris and DeBiccari, 1997).


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Figure 5. Mechanical properties of AH36 steel (Michaleris and DeBiccari, 1997).


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Figure 6. Yield strength of AH36 steel (Michaleris and DeBiccari, 1997).


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Figure 7. Longitudinal residual stress for initial design (MPa).


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Figure 8. Configuration for minimum stress on plates edge (dimensions in mm).


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Figure 9. Longitudinal residual stress for minimum stress on the plates edge (MPa).


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Figure 10. Configuration for minimum sum of squares stress on plate (dimensions in mm).


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Figure 11. Longitudinal residual stress for minimum sum of squares on plate (MPa).


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Figure 12. Computed temperature history for minimum sum of squares on plate.

Minimization of Welding Residual Stress and Distortion In

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