IJEMS 18(6) 425-434

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INDIAN J. ENG. MATER. SCI., DECEMBER 2011426

gauges, even if more sophisticated methods, such as

holography and Moire have also been applied. The

accuracy and resolution of these methods can be

improved by means of numerical calibrationtechniques5. Thermal stress analysis plays an

important role in evaluation of residual stress, and of

distortion, as well as microstructure modeling of

welded joints and structures. Heat transfer analysis

provides the thermal history in the welded joints,

which will be utilized in stress analysis to determine

the residual stress fields.

The first classical solutions for the heat sources

were developed by Rosenthal back in 1941 and laterby Rykalin and others in the 1950s to obtain transient

temperature of welded plates. Nguyen6 has made

extensive survey and presented various analyticalsolutions for a number of stationary and moving heat

sources in semi-infinite body, thick plate, fillet joint,cylinders, sphere and cone and their application in

weld-pool simulation, residual stress and distortion

calculations, microstructure modeling and

optimization of multi-pass welded components. These

solutions are obtained with an assumption that onlyconduction is playing a major part in the thermal

analysis of welds. In the welding process, the fusion

zone (FZ) and the heat affected zone (HAZ) regions

experience high temperatures, which cause phase

transformations and alterations in the mechanicalproperties of the welded metal.

Lindgren7 discussed in detail the various issues

involved in the development of material models used

for residual stress analysis. Duranton et al.8 have

computed distortions and residual stresses through 3Dfinite element simulations of multi-pass welding of a

316 L stainless steel. Murugan and Narayanan9 have

performed the finite element simulation of three-

dimensional transient residual stresses in a T-joint and

a contour method was used to experimentally validate

the numerical results. The finite element simulation of

temperature field and residual stresses of butt-weldedplates have been performed over the last two

decades10,11

. Ueda and Yuan12,13

and Mochizuki and

Hattori14

have used the inherent strain method to

calculate the residual stresses. Dong15

has developed a

model utilizing the element birth and death technique

to simulate the metal deposition that is valid in the

case with no thermal effects of the sudden larger

temperature variation. Fanous et al.16

have introduced

a new technique of element movement for full 3D

simulation of the welding process.

Stamenkovic and Vasovic17

have performed a 3Dfinite element welding simulation and predicted weld-

induced residual stresses in butt welding of two

similar carbon steel plates. The welding simulationwas considered as a sequentially coupled thermo-

mechanical analysis and the element birth and deathtechnique was employed for the simulation of filler

metal deposition. The finite element analysis results

are found to be very close to the experimental results.

Tahami et al.18

have examined the thickness effect on

the residual stress states in butt-welded 2.25Cr1Mosteel plates. Finite element analysis results show that

by increasing the plate thickness, the residual stresses

increase and the residual stress affected zone becomes

larger. The longitudinal residual stress in weld axis

changes from compressive to tensile by increasing the

plate thickness. Tahami and Sorkhabi.19 have also

studied the effect of the welding-electrode speed

using birth and death of finite elements. They have

shown that use of the 3D and transient model will

lead to more accurate and realistic results which are

well compared with the test data. Accurate and

reliable residual stress prediction and measurements

are essential for structural integrity assessment of

components containing residual stresses.

Commercially available finite element method (FEM)

packages such as ABAQUS, ANSYS, NASTRAN

and MARC can be used for welding thermal elastic

plastic stresses and distortion analyses. Finite element

simulation of residual stresses due to welding

involves in general many phenomena, e.g., nonlinear

temperature dependent material behavior, 3D nature

of weld-pool and the welding processes and micro

structural phase transformation. The most powerful

strategy to reduce the cost of thermo-mechanical

simulations of welding has been to reduce the

dimension of the problem from three to two or one.

Two-dimensional models can often be useful when

evaluating strains and stresses and it is always a good

practice to start with simple model in initial

evaluations20.

Different geometric models, viz., 2D-X, 2D-P, 3D-

shell and 3D-solid are being used in the thermo-

mechanical simulations of welding. 2D-X ignores the

heat conduction in the welding direction. This

corresponds to plane strain, generalized plane

deformation and axisymmetric models. The 2D-X

models can be useful to obtain a residual stress state.

They can also be used for very accurate modeling

when studying stress-strain behavior near the weld

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JEYAKUMAR et al.:RESIDUAL STRESS EVALUATION IN BUTT-WELDED STEEL PLATES 427

with very small elements and time steps. 2D-Passumes constant temperature over the weld plate

thickness. The model is a plane stress model where

the heat source is moving in the plane of the model.Two-dimensional plane stress, 2D-P, is useful when

the in-plane deformations are of major concern.Multi-pass welds can be accommodated

approximately by changing thickness of elements.

3D-shell can have varying assumptions about possible

temperature variation over the thickness are possible.

The total bending strain varies linearly over thethickness according to shell theory. 3D-shell models

are useful when simulating the welding of thin-walled

structures in order to obtain the overall deformation

behavior and stresses. It is possible to combine them

with 3D solids near the weld for better representation

of thermal and mechanical fields. In general,

industrial applications require 3D models using shell

and/or solid elements.

Despite the simplification by excluding various

effects, welding simulation is still CPU time

demanding and complex. Hence, simplified 2D

welding simulation procedures are required in order to

reduce the complexity and thus maintain the accuracyof the residual stress predictions. In this paper, the

finite element analysis of residual stresses in butt

welding of steel plates is performed utilizing the

ANSYS with plane stress model. The present analysisresults are found to be in good agreement with theexisting complex 3D finite element analysis results

and experiments.

Welding SimulationIn general, the thermal history of welded joints can

be predicted by heat transfer analysis. Subsequently,

the calculated thermal history can be used for thermal

stress analysis to determine the residual stress fields

in the welded joints. During welding processes, heat

can be transmitted by conduction, convection and

radiation. For welding processes where an electric arcis used as the welding source, heat conduction

through the metal body is the major mode of heat

transfer and heat convection is less significant as far

as the temperature field in the welded body is

concerned. The partial differential equation for

transient heat conduction18

is

( ) f T t

T c +∇∇=

∂

∂κ ρ . …(1)

Where the density ( p), specific heat (c) and thethermal conductivity (k ) are dependent on temperature

(T ). t is the time and f represents the additional heat-

generation function in the body.The heat flux vector,

T q ∇−=

…(2)

The enthalpy is related to the temperature by

∫=T

T ref

d c H τ τ )( …(3)

which implies that

dT

dH c = …(4)

From Eqs (1) and (4), one can write the apparent heat

capacity equation in the form

( ) f T t

H +∇∇=

∂

∂κ ρ . …(5)

The heat conduction equation together with initial

and boundary conditions, defines the problem to besolved. Simple boundary conditions are prescribed

temperature or prescribed heat flux. The surface heat

flux, qn is defined as positive when directed in the

outward normal direction. It is zero in the case of an

isolated, adiabatic boundary. Convective and radiation

heat losses are more complex boundary conditions for

the outward flux. Then the surface heat flux depends

on the temperature of the body and the surrounding

and is written as16

( )( )

n c ref 4 4

f b ref

T

q T.n h T Tne s T T

∂= −κ ∇ = −κ = − +

∂

−

…(6)

where the first term is convective heat loss and hc is

the heat transfer coefficient. The second term is the

heat loss due to radiation and sb is Stefan-Boltzmann’s

constant and e f is the emissivity factor. The second

term is a nonlinear boundary condition. The total heat

loss in Eq. (6) can be written in a format more

convenient for finite element implementation as

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( )( ){ }( ) ( )

2 2n c f b ref ref

ref eff ref

q h e s T T T T

T T h T T

= + + +

− = −

…(7)

The effective heat transfer coefficient (heff ) is acombination of both the convection and radiation

coefficients:

( )3223

ref ref ref b f ceff T T T T T T sehh ++++= …(8)

The most important parameter to determine thetemperature distribution in the welded components is

the heat input. This heat quantity is the output from aparticular heat source used to fabricate the welded

joints. In all the welding processes, heat source

provides the required energy and causes localized

high temperature spot. In arc-welding with constant

voltage (V ) and amperage ( I ), the efficiency of the

heat source would be6

I V

Q

t I V

t Q S

weld

weld S ==η …(9)

where QS is the heat generating rate and t weld is thewelding time and η is the thermal efficiency. The

Gaussian-distributed heat source (see Fig. 1) can be

used to simulate the welding heat source to give a

better prediction of the temperature field near the

source center. The Gaussian heat source is used to

simulate the welding-arc, where the heat source

density, q(x, y) at an arbitrary point (x, y) is

represented by6

( ) ( )2

0 exp, r k q y xq −= …(10)

Here 0q is the maximum value of the heat source

density. The distance r in Eq. (10) is the distance from

the center point of the heat source to the point forwhich the heat flux is being calculated. The

coefficient ‘k ’ determines the concentration of the

heat source. It is also known as the distribution

parameter representing the width of the Gaussian

distribution curve. Higher value of k corresponds to amore concentrated heat source. From the heat

equilibrium condition

( ) ( )( )

20

S 20

q x, y dxdy q exp( kx )dxQ

exp( ky )dy qk

∞

−∞∞ ∞

−∞ −∞ ∞

−∞

= −∫= ∫ ∫ π

− =∫

…(11)

If the heat density at br r = drops to only 5% of the

maximum heat density, i.e.,

)exp(05.02

00 br k qq −= …(12)

The heat input parameter (k ) can be evaluated from

the heat source radius as

222

39957.2)05.0(ln

bbb r r r k ≈=−= …(13)

Using Eqs (11) and (13) in Eq. (10), one can write

−=

2

2

23exp

3),(

bb

S

r

r

r

Q y xq

π …(14)

The distribution of q(x, y) in Eq. (14) represents 95%

of the total heat QS when applied within a circle with

radius r b. The distance, ( ) 22 y x xr

h

+−= ;

)( 0t t v xh −= ; and v is the welding speed.

The time between the onset of welding and the end

of the cooling to ambient temperature can be divided

into sufficiently small intervals so that the

temperature and thermal stresses for each interval

may be regarded as constant. Since the temperature

Fig. 1–Gaussian distributed heat flux, q

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change is rapid at the beginning of welding andcontinuously decreases with time, the time step

increment should be sufficiently small at the

beginning of the weld and relatively large as the timeincreases. The calculation of welding residual stresses

is usually based on the temperature distribution andthe thermal stress increment ∆σ (= E α∆T ) is calculated

from the incremental thermal strain α∆T . Here α is the

thermal expansion and E is the Young’s modulus. The

residual stresses arise not only from the welding

shrinkage but also from the surface quenching (rapidcooling of the weld surface layers) and phase

transformation (transformation of austenite during the

cooling cycle).

The calculation starts with time t=0 and the thermal

stress is calculated for the initial temperature

distribution of the welded components. At the next

time step, the thermal stress increment is added to the

initial stress at step t=0. The magnitude of the

cumulative thermal stress is limited to the yield

strength of the material at actual temperatures. It

should be noted that at each step, the forces caused by

the induced thermal stresses must be in equilibrium.

This procedure is repeated until the last step at which

the thermal stress is that at ambient temperature, i.e.,

the residual stress. This numerical procedure for

residual stress evaluation involves adding together the

incremental thermal stresses, previous thermal

stresses and the equilibrium stresses.

The equilibrium and compatibility equations are18

( ) ( ) 023 ,,, =+−++ ikik kk iT uu α µ λ µ λ µ …(15)

0,,,, =−−+ ik jl jlik ijklklij ε ε ε ε …(16)

Here, the Lame’s coefficients are

)21()1( ν ν

ν λ

−+=

E …(17)

)1(2 ν µ

+=

E …(18)

α is the thermal expansion; uis are

displacements; ( )i j jiij

uu ,,2

1+=ε , is the strain; E is

the Young’s modulus; and ν is the Poisson’s ratio.

These equations, together with the defined boundary

conditions provide the residual stress field in thewelded joints. The term ‘simulation’ is often used

synonymously with modeling, but there are

differences in meaning

20

. A simulation should imitatethe internal processes and not merely the result of the

thing being simulated. This gives an association into asimulation as a model that imitates the evolution in

time of a studied process. For example, a simplified

model directly giving residual stresses due to a

welding procedure will not qualify as a welding

simulation. However, the term ‘simulation’ is oftenused to denote the actual computation. Simulation

errors will then be those errors related to the solution

of nonlinear equations as well as the time stepping

procedure.

Butt-Welded Steel PlatesIn general, the thermal analysis is straightforward

compared with the mechanical analysis. The

mechanical properties are more difficult to obtain than

the thermal properties, especially at high

temperatures, and they contribute to the numerical

problems in the solution process7,20. Many analyses

use a cut-off temperature above which no changes in

the mechanical properties are accounted for21

. It

serves as an upper limit of the temperature in the

mechanical analysis. Tekriwal and Mazumder22

varied

the cut-off temperature up to the melting temperature.

The residual transverse stress was overestimated by

2-15% when the cut-off temperature was lowered. It

should be noted that all material models need to have

good thermo-elasto-plastic properties up to the cut-off

temperature.

Fig. 2–Butt-welded 2.25Cr1Mo steel plates

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Butt-welded joint of 2.25Cr1Mo low-alloy-ferritic steel plate

Residual stress analysis has been carried out in a

butt-welded joint of 2.25 Cr 1 Mo low-alloy-ferritic

steel plate (300 × 72 × 6 mm) as shown in Fig. 2.

The commercial finite element code ANSYS hasbeen used to carry out the thermal and mechanical

analyses. For thermal analysis, 2D element Plane 77

is used. It is an 8 node thermal solid (8 node

quadrilateral element) with single degree of freedom

having temperature at each node. Generally,temperature around the arc is higher than the melting

temperature of materials and drops sharply in regionsaway from weld pool. In high temperature gradient

regions of FZ and HAZ, more refined mesh close to

weld line is essential for obtaining accurate

temperature field. For structural analysis, 2D element

Plane 82 is used. It is an 8 node structural solid

(8 node quadrilateral element) having two degrees of

freedom at each node, translation in the nodal x and y

directions.

The overall input of heat flux QS is evaluated from

Eq. (9) specifying the arc efficiency, η=0.7; arc

Fig. 3–Temperature dependent properties of2.25Cr1Mo low-alloy-ferritic steel18

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voltage, V =30 V; and the current, I =200 A. The radial

heat flux distribution in Eq. (14) is considered on thetop surface of the weldment. The heat density drops to5% of its maximum value at r = r b. In the present

analysis, r b. is set to 3 mm. When the value of r is less

than or equal to, r b the heat flux is calculated

according to the Eq. (14). Otherwise, the heat load is

set to zero. Due to symmetry, only half of the weld

and plate were modeled.

Figure 3 shows temperature dependent thermal and

mechanical properties of 2.25Cr1Mo low alloy steel18.

Filler weld material is assumed to have the same

chemical composition of the parent material. The

melting temperature of the filler material is 1783 K. A

cut-off temperature (T cut-off ) is set to 1073 K. Thematerial properties at T cut-off are specified in the

regions where the temperature is higher than T cut-off .

For convective and radiative heat losses, the constants

in the complex boundary conditions for the outward

flux in Eq. (6) are: Stefan-Boltzmann constant,

S b = 5.67 X 10-8

W / m2K

4; convection coefficient,

hc = 15 W / m2K ; and the emissivity factor, e f = 0.2.

To obtain thermal history, transient, non–linear

thermal problem is solved using temperature

dependent thermal properties and considering heat

conduction, convective and radiative boundary

conditions. Thermal stress analysis is performedspecifying the temperature distribution, temperature

dependent mechanical properties and symmetry

boundary conditions to obtain the transient and

residual stress fields. In thermal analysis the heat flux

is specified in 3172 time steps. It takes 6000 s to cooldown from the maximum temperature to ambient

(room) temperature. Since load steps are too many,

Ansys Parametric Design Language (APDL) has been

adopted to perform both thermal and structural

analyses.

Fig. 4–Variation of temperature from weld center line to the edgeof the 2.25Cr1Mo steel plate along its length direction

Fig. 5–Residual stress σ x (MPa) distribution from the weld centreline to the edge of the 2.25Cr1Mo steel plate along its lengthdirection

Fig. 6–Residual stresses (MPa) and strains at mid-sectionperpendicular to weld of the 2.25Cr1Mo steel plate along itslength direction

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Figure 4 shows the temperature variation from the

weld center line to the edge of the plate along

y-direction (that is along the length of the plate). The

results indicate that the plate is undergoing significanttemperature variation. At the beginning, the

temperature reduction in the area close to the weld

axis shows the quenching effect. Figure 5 shows a

comparison of the residual stress (σ x) distribution

perpendicular to the weld of 2.25Cr1Mo steel plateobtained from the present 2D plane stress analysis and

3D finite element analysis19. The 2D analysis result

varies from 257 MPa (tensile) to -181 MPa

(compressive) and reaches to zero is in good

agreement with those obtained from 3D FEA results.

Figure 6 gives the stresses and strains at mid section

perpendicular to the weld of the 2.25Cr1Mo steel

plate.

Butt-welded joint of ASTM36 steel plate

Following the above welding simulation studies,

the butt-weld joint of ASTM36 steel plate (200 × 100

× 3mm) is examined. Figure 7 gives the temperature

dependent properties of ASTM36 steel17

. Figure 8shows the temperature distribution at mid section

perpendicular to the weld. Figure 9 shows a

comparison of the residual stress (σ x) distribution

perpendicular to the weld of the ASTM36 steel plate

obtained from the present 2D plane stress analysis and

Fig. 7–Temperature dependent properties ofASTM36 steel17

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3D finite element analysis and test results17

. It can beseen from Fig. 9, tensile stresses were developed in

the weld zone. These tensile stresses gradually

decrease in the transverse direction away from theweld center line and become compressive towards the

edge of the plate. The peak tensile residual stress

estimates from the present 2D FEA is in good

agreement with those obtained from 3D FEA and

experimental results17

. Figure 10 gives the stresses

and strains at mid section perpendicular to the weld of

the ASTM36 steel plate.

ConclusionsFinite element analysis has been carried out

utilizing the commercial software package ANSYSwith plane stress model to estimate the residual

stresses in the butt-welded 2.25Cr1Mo low-alloy

ferritic steel plates and also ASTM36 steel plates. The

2D plane stress models in the present study for the

thin butt-welded steel plates neglect stress variations

through the thickness and provide an averaged value

of the stress components. The present analysis results

are found to be in good agreement with the existing

complex 3D finite element analysis results and

experiments. 2D welding simulations are found to

reduce the complexity of the problem and adequate

for several design purposes in providing theinformation regarding the criticality of residual stress

in weld joints.

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Fig. 8– Variation of temperature from weld center line to the edgeof the ASTM36 steel plate along its length direction

Fig. 9–Comparison of Residual stress, σ x (MPa) distribution from

the weld centre line to the edge of the ASTM36 steel plate alongits length direction

Fig. 10–Residual stresses and strains at mid-section perpendicular

to the weld of the ASTM36 steel plate along its length direction

8/14/2019 IJEMS 18(6) 425-434



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IJEMS 18(6) 425-434