12

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

Driven by the development of powerful and inexpensive

computers, the field of computer aided engineering emerged. It provides

predictive tools as well as insights into complex engineering processes.

Hence, engineers working in many different application areas demand

numerical simulation tools for their investigations which some years ago were

only accessible by experiments. Modeling of engineering problems leads in

many cases to ordinary and partial differential equations which often are of

nonlinear nature. A powerful tool to solve these differential equations is the

finite element method which was developed over the last 50 years (Peter

Wriggers 2008). Arc welding is one of the most versatile and widely used

manufacturing processes for the fabrication of complex, built-up, metallic

structures. Though welding process is used in many strategic sectors, the

design threat in using welding is that welded structures tend to distort from

their planned size and shape and also have high magnitude residual stress

fields. Analytical methods are not effective to predict the transient thermal

cycles experienced during welding of structures and the residual stresses and

distortions after welding.

2.2 FINITE ELEMENT METHOD

Finite element analysis, also called the finite element method, is a

method for numerical solution of field problems. A field problem requires the

13

determination of spatial distribution of one or more dependent variables.

Mathematically, a field problem is described by differential equations or by

an integral expression. Either description may be used to formulate finite

elements which can be visualized as small pieces of a structure (Cook et al

2003). The elements are connected at points called „nodes‟. The assemblage

of elements is called a finite element model or structure. The particular

arrangement of elements is called a mesh. Numerically, a finite element mesh

is represented by a system of algebraic equations to be solved for unknowns

at nodes. Finite element modelling is the process of preparing a computational

model. It decides about the significant features of the actual problem that can

be incorporated in the model. Simulation is the prediction of the intended

output of the computational model (Lindgren 2007).

2.3 FINITE ELEMENT METHOD IN ARC WELDING

SIMULATION

The important aspect in arc welding simulation is the heat

generation process. Since arc welding involves complex non-linear multi-

physical interactions between the source of the fusion and the local and global

thermo-mechanical responses of the components being welded, research

works with regard to welding simulation have been undertaken with varying

scopes for the past few decades. Welding simulation mainly consists of

transient thermal simulation to predict temperature histories and distributions

and subsequently the non-linear thermo-mechanical simulation to predict the

residual stress fields and distortion in the welded structures.

2.3.1 Computation of Transient Thermal Cycles

The main aim of transient thermal simulation is to capture the

complex transient thermal cycles involved in arc welding of components. As

is needed for any finite element analysis, the governing differential

equation (2.1) is provided by Fourier law of heat conduction

14

x y z

T T T Tk k k q c

x x y y z z t

(2.1)

where T is the temperature, kx, ky and kz are the thermal conductivities in x, y

and z directions respectively, q is the internal heat generation, c is the specific

heat capacity, ρ is the material density and t is the time.

The equation (2.1) can be easily derived on the basis of Fourier‟s

law of heat conduction and the law of energy conservation. In order to

formulate the problem of heat conduction in a solid body, the initial and

boundary conditions need to be specified. The appropriate initial condition

would be the initial temperature in the welding applications and is usually

isothermal: T(x,y,z,0) =T0, where T0 can be considered equal to the prevalent

ambient temperature. The boundary conditions represent the law of

interaction of the surfaces of the weld specimen with the ambience. The

boundary conditions in welding are the convective (qc) and radiative (qrad)

heat flux losses from the surfaces of the welded plate given by:

qc = h (T - To) (2.2)

qrad = ε σ (T4 – To

4) (2.3)

where h is the convective heat transfer coefficient, To is the ambient

temperature, ε is the emissivity and σ is the Stefan-Boltzman constant.

Some of the essential elements in the simulation of a welding

process are as follows:

i) Modeling of Heat Source

ii) Modeling of Filler material addition

iii) Modeling of Phase transformations

15

2.3.1.1 Modeling of heat source

Selection of the most appropriate model for the heat source of the

welding process becomes the crucial step in the simulation. Different types of

heat source models have been assumed by researchers, depending upon the

scopes of their works. Goldak and Akhlaghi (2005) give a comprehensive

account of various generations of heat source models used by many

researchers over the years. The earliest assumption of a moving point heat

source by Rosenthal (1946) suffered from serious setbacks such as infinite

temperature at the heat source and insensitivity of the material properties to

temperatures (Myers et al 1967). Pavelic et al (1969) suggested that the heat

source should be distributed and they proposed a Gaussian distribution of flux

deposited on the surface of the work piece. While Pavelic‟s „disc‟ model is

certainly a significant step forward, some authors have suggested that the heat

should be distributed throughout the molten zone to reflect more accurately

the digging action of the arc (Goldak and Akhlaghi 2005). These models

account for heat distributions in the surface of the welded specimen. Some

authors have suggested that the heat should be distributed throughout the

molten zone to reflect more effectively the digging action of the arc (Paley

and Hibbert 1975) and (Goldak and Akhlaghi 2005). But these models do not

account for the length of the molten pool. Goldak et al (1984) proposed a

three dimensional double ellipsoidal configuration for the heat source as

shown in Figure 2.1. They reported that the double ellipsoidal heat source

model was a more realistic and flexible than any other model yet proposed for

weld heat sources. They also added that both shallow and deep penetration

welds could be accommodated as well as asymmetrical situations.

16

(2.4)

(2.5)

Figure 2.1 Goldak’s heat source model (Goldak et al 1984)

The double ellipsoidal configuration refers to the size and shape of

the solid-liquid interface recognized by the melting point isotherm (Goldak

and Akhlaghi 2005). Goldak et al (1984) reported that the most accuracy was

obtained when the ellipsoidal size and shape was equal to that of the weld

pool. The non-dimensional system suggested by Christensen et al (1965)

could be used to estimate the ellipsoidal parameters. The mathematical

expressions for Goldak‟s double ellipsoidal heat source for both the front and

the rear ellipsoids are given as follows:

2 2 2

2 2 21

x y z3

a b cff

1

6 3f Qq e

abc

2 2 2

2 2 22

x y z3

a b crr

2

6 3f Q q e

abc

where qf and qr are the power densities in the front and rear ellipsoids

respectively, ff and fr are the respective fractions of heat, Q is the arc heat

transferred to the weld plate and a, b, c1 and c2 are the heat source

parameters.

c2

c1

17

2.3.1.2 Modeling of filler material addition

Another important aspect in the simulation of welding process is

the modeling of addition of the filler material during welding. This particular

aspect is dealt with by making use of element „birth and death‟ feature

available in many standard commercial finite element software packages, like

ANSYS. According to this feature, in order to model the filler material

addition, the volume of the weld metal which is to be filled, is initially

generated and meshed along with base metal elements. The elements

representing the filler material are then „killed‟ or in other words, deactivated.

When the heat source is near or at the time of filler material addition, these

elements are immediately given „birth‟ or activated to take part in the solution

(Brickstad and Josefson 1998, Hong et al 1998)

2.3.1.3 Modeling of phase transformations

As the fusion welding of material results involves steep

temperature rise to the material‟s melting point and even more, change of

phase takes place at the liquidus. Hence, it is necessary to incorporate the

latent heats associated with phase changes. Some researchers ignored this in

their modeling of welding process. But some of other researchers considered

the latent heats by changing the specific heat of the weld material at the

temperatures when the phase change took place (Frewin and Scot 1999, Cho

and Kim 2001). Goldak et al (1986) described another technique in which the

nodal temperatures at each time step were compared with the melting

temperature and if the nodal value exceeded the melting point, then nodal

temperature was fixed at the melting point and the excess heat level was

calculated corresponding to a mass associated to a node. This process was

repeated until the excess heat input was equal to the latent heat of the

material. The enthalpy (H) of the weld material is calculated as follows

(ANSYS 2002 and 2005):

18

H c(T)dT (2.6)

where ρ is the material density and T is the temperature. Then the enthalpies

at different temperatures are input in the model.

2.3.2 Transient Thermal Simulation of Welding

Jaroslav Mackerle (1996, 2002) gives a review of published papers

dealing with finite element methods applied in the area of welding processes

during two different periods, 1976 - 1996 and 1996 - 2001 respectively under

various topics as given below:

i) General Modeling of welding processes

ii) Modeling of specific welding processes

iii) Influence of geometrical parameters

iv) Heat transfer and fluid flow in welds

v) Residual stresses and deformations in welds

vi) Fracture mechanics and welding

vii) Fatigue of welded structures

viii) Destructive and nondestructive evaluation of weldments and

cracks

ix) Welded tubular joints, pipes and pressure vessels /

components

x) Welds in plates and other structures / components.

Since computational expense of FEM is greatly dependent on the

processing capability of computers as regards memory and speed, many of the

early research works in welding were limited to the computation of

19

temperature histories and distributions in the welded component with many

assumptions.

Tekriwal and Mazumder (1988) obtained the thermal histories of a

butt joint produced by GMAW process and analyzed it using a 3D finite

element model. The effect of phase change was ignored. The sizes of the heat

affected zone and the molten meltal zone were numerically predicted and

compared with the experimental results.

Kamala and Goldak (1993) developed a method for evaluation of

the errors involved in the approximation of a 3D heat transfer analysis of a

weld into a 2D cross-sectional analysis. It was reported that the errors in the

temperature fields obtained in the 2D analysis could be reduced by modifying

the true power density distribution function.

Ravichandran (2003) studied the thermal cycles experienced by a

pulsed gas tungsten arc welded pipes. The thermal cycles at various locations

and the temperature distribution for different time intervals were studied by

using 2D finite element method with Gaussian heat input. Surface losses were

considered in a combined manner. The effect of phase change was not

considered. It was reported that the temperatures in the weld pool showed

fluctuations in the case of pulsed welding which died out after the crossing of

the arc. Similarly, the effect of pulsing was reported to be felt up to a distance

of 5 to 10 mm from the weld centre line beyond which the fluctuations in the

thermal cycles are absent.

Siva Prasad and Sankara Narayanan (1996) developed a 2D

transient adaptive mesh to obtain the temperature distribution at the arc with

the objective of reducing the nodal degrees of freedom. Gaussian heat input

model was considered. A fine mesh around the arc and a coarse mesh in other

20

regions were used. Latent heat effects were included. It was claimed that

considerable reduction in the computation time was achieved.

Ravichandran et al (1995) modeled the thermal cycles during the

circumferential arc welding of components with cylindrical and spherical

shapes, by using a bilinear degenerative shell element adapted for the thermal

analysis. The analysis was conducted for the cases of butt welding of a thin

cylindrical pipe to a thin cylindrical pipe, a thin spherical end to a thin

spherical end and a thin spherical end to a thin cylindrical pipe. In all the

analyses, the thickness of the components, the diameter of the components

and the heat input were kept constant. Gaussian heat input model was

considered. The effect of latent heat was considered as proposed by Goldak

et al (1986)

Ravichandran (1998) developed a 2D finite element model to

predict the thermal cycles involved in butt welding during plasma arc

welding and experimentally compared the cycles at three specified locations

in the transverse direction of welding for three values of welding process

parameters such as welding current, arc voltage and welding speed. Gaussian

distribution of heat input was assumed. Surface heat losses like convection

and radiation and latent heats were included in the model. It was reported that

there was good agreement between the predicted and the experimental results.

As it was a 2D FE model, the heat flow in the perpendicular direction to the

plane of analysis could not be included.

Frewin and Scott (1999) developed a three dimensional finite

element model of the heat flow during pulsed laser beam welding, with many

assumptions. Temperature profiles and the dimensions of fusion and heat

affected zones in a AISI 1006 steel plates were calculated. Convective flow of

heat was neglected in their work.

21

Murugan et al (1999) investigated the temperature distribution

during bead on plate welding using manual metal arc welding experimentally.

A three-dimensional computer model based on control volume method was

developed to predict the temperature distribution in the heat affected zone and

in the base metal region in low carbon steel plates of thicknesses 6 and 12

mm. An assumed artificially high conductivity value was used to compensate

for weld pool convective heat transfer. The liquid-solid phase change and

associated latent heat were modeled using an artificial heat flow method. A

temperature of 1750 °C was applied over the control volume which

represented the weld bead. The distributive nature of the heat source was not

considered. But still, it was reported that there was good agreement between

the predicted and the experimentally measured temperature histories at

specified locations. Addition of mass of filler material was ignored in their

analysis.

Ohring and Lugt (1999) presented a numerical simulation of

transient, three dimensional GMA weld pool with a mushy zone, using a finite

difference technique with boundary-fitted coordinate scheme. The addition of

molten material was modeled by an impacting liquid metal spray on the weld

pool, with evaporation and latent heat absorption for boiling being computed

at the weld pool surface. Mass loss due to boiling was assumed negligible.

The material properties were assumed to be independent of temperatures.

Murugan et al (2000) developed a computer model based on the

control volume method to predict the temperature distribution in a multipass

weld of 12 mm thick stainless steel joined by manual metal arc welding

process. The addition of filler material in their model was considered by

continuously introducing new control volumes during the course of

computation to simulate the progressive deposition of weld metal in the V

22

groove. Other assumptions which were same as their earlier work (Murugan

et al 1999) were also considered.

Bonifaz (2000) presented a 2D finite element model to calculate the

transient thermal histories involved in fusion welding and calculated the sizes

of fusion and heat affected zones. The effect of introducing the melting

efficiency into the energy input rate to account for dilution was also studied,

using both Gaussian and ellipsoidal power density distribution functions.

Yang et al (2000) modeled macro and microstructural features in

gas tungsten arc welded titanimum, on the basis of a combination of transport

phenomena and phase transformation theory. A transient, three-dimensional,

turbulent heat transfer and fluid flow model was developed to calculate the

temperature and velocity fields, thermal cycles and the shape and size of the

fusion zone.

Cho and Kim (2001) performed a heat flow analysis using a 2D

finite element model to compute the bead shape in gas metal arc welding of

horizontal fillet joint. Welding current, arc voltage, welding speed and

effective arc radius were considered as process parameters for the analysis of

weld bead shape. Latent heat effect was included by changing the value of

specific heat at temperatures of phase transformation.

Nguyen et al (2004) presented analytical solutions for the transient

temperature field of the semi-infinite body subjected to 3D power density

moving heat sources such as semi-ellipsoidal and double ellipsoidal heat

sources. It was assumed in their work that there was no convective heat flow

through the upper and lower surface of the plate. Most importantly,

temperature dependent properties of the weld material were not taken into

account. They reported good agreements between the predicted transient

23

temperatures and the measured ones at various points in bead-on-plate weld

specimens.

Benyounis et al (2005a) developed mathematical models for the

prediction of heat input, penetration, width of fusion and heat affected zones

in terms of the welding process parameters such as the laser power, welding

speed and focused position in laser butt welding of medium carbon steel. The

mathematical models were optimized by response surface methodology

(Benyounis et al 2005b).

Erdal Karadeniz et al (2005) discussed the effects of three welding

parameters such as welding current, arc voltage and welding speed on depth

of penetration in Erdemir 6842 steel having 2.5 mm thickness welded by

robotic GMAW. With each parameter at three levels, the effects were studied

by conducting 27 experiments as per full factorial design. It was concluded

that the effect of welding current was approximately 2.5 times greater than

that of arc voltage and welding speed on penetration.

Gery et al (2005) analyzed the effects of welding speed, energy

input and heat source distribution on temperature variations in a butt joint by

developing 3D and 2D finite element models. The influence of heat source

parameters on the fusion zone and heat affected zone boundaries were also

studied. The material properties at elevated temperatures were assumed by

multiplying a factor with the room temperature material property.

Han GuoMing et al (2007) studied the distribution of the

temperature field in laser welding based on stainless steel 304 sheet. Gaussian

distribution of heat input was assumed. Latent heat was considered by

calculating the thermal enthalpy of the material at the temperature of phase

transition. The depth of penetration, the width of penetration, the height of

weld waist and the width of weld waist were predicted and compared with the

24

corresponding experimentally measured values. But the details of the

experimental measurement of these effects were not furnished. In addition,

the reported error in the prediction of the depth of penetration was as high as

27%.

2.3.3 Computation of Transient Stress Fields in Weldments

Simulations which deal with the mechanical effects of welding

should involve the computations of the thermal as well as mechanical fields.

The main issues involved in the thermo-mechanical simulation of welding are

large deformation effects and material modeling.

2.3.3.1 Material modeling

The modeling of the material behavior is a challenge as the

deformation mechanisms vary widely in the large temperature range

considered. Lindgren (2001) presents a detailed description of various aspects

in material modeling for simulation of welding. Publications presenting finite

element simulations of the mechanical effects of welding appeared in the

early 1970s, and simulations are currently only used in applications where

safety aspects are very important (like aerospace and nuclear power plants) or

when a large economic gain can be achieved (Lindgren 2007). It is also

mentioned that the simplest and the most common approach is to ignore the

microstructure change and assume that the material properties depend only on

temperatures.

2.3.3.2 Large deformation effects

The welding process causes severe and visible deformations.

Geometric nonlinearity arises when deformations are large enough to alter the

distribution or orientation of applied loads, or the orientation of internal

25

resisting forces and moments. Large deformations may deform a mesh so

greatly that well-shaped elements become poorly shaped (Cook et al 2003).

Hence, the simulation should incorporate large deformation effects and

strains. The large deformations and the use of small elements can cause these

elements to be severely distorted. This problem can be overcome by the use of

fine mesh and small time steps (Cook et al 2003).

It is assumed that the deformation can be decomposed into a

number of components. The increments in total strain are computed from the

incremental displacements during a non-linear finite element analysis. The

elastic part of the strain gives the stresses, and there are a number of inelastic

strain components that can be accounted for. The inelastic components of the

total strain rate are the plastic strain rate, the viscoplastic strain rate, the creep

strain rate, the thermal strain rate consisting of thermal expansion and volume

changes due to phase transformations and the transformation plasticity strain

rate. A welding simulation must at least account for elastic strains, thermal

strains and one more inelastic strain component in order to give residual

stresses. The plastic, viscoplastic and creep strain are all of the same nature

(Lindgren 2007).

The mechanical analysis requires much more time due to more

unknowns per node than in the thermal analysis. Furthermore, it is much more

non-linear due to the mechanical material behavior. The mechanical

properties are more difficult to obtain than the thermal properties, especially

at high temperatures, and they contribute to numerical problems in the

solution process. The high-temperature mechanical behavior is modeled in an

approximate way due to several factors: experimental data is scarce, too soft

material causes numerical problems and it is found that approximations

introduced do not significantly influence the resultant residual stresses. Many

analyses use a cut-off temperature above which no changes in the mechanical

26

material properties are accounted for. It serves as an upper limit of the

temperature in the mechanical analysis. Ueda (1985) assumed that the

material did not have any stiffness above 700 °C. This was called as the

mechanical rigidity recovery temperature, above which the Young‟s modulus

was set to zero. Tekriwal and Mazumder (1991) varied the cut-off

temperature from 600 °C up to the melting point. The residual transverse

stress was overestimated by 2 to 15 % when the cut-off temperature was

lowered.

The behavior of a material in the plastic range is generally

specified by yield criterion, flow rule and hardening rule. Yield criterion

specifies the stress level at which yield is initiated. The flow rule determines

the direction of plastic strain. In associative flow rule, it is assumed that

plastic straining occurs in a direction normal to the yield surface. Associated

flow rule is normally followed for ductile metals. Nonassociated rules are

better suited to soil and granular materials (Cook et al 2003). Hardening rule

is necessary to specify how the yield surface changes due to progressive

yielding so that the stress states for subsequent yielding can be established.

Hardening can be modeled as isotropic or as kinematic, either separately or in

combination. Isotropic hardening can be represented by plastic work per unit

volume which describes the growth of the yield surface. Kinematic hardening

can be represented by translation of the yield surface in stress space. Different

forms of yield criterion, flow rule and hardening rule are used for different

materials. The rules that work well for copper do not work well for concrete.

The observed behavior of commonly used metals is predicted fairly well by

the von Mises yield criterion and its associated flow rule (Cook et al 2003).

The model that has been used most widely for rate-independent plasticity is

the von Mises yield criterion as given in equation (2.7) together with the

associated flow rule (Lindgren 2001). Thus, the plastic strains are

incompressible and are not dependent on the hydrostatic part of the stresses.

27

The flow rule states that the plastic flow is orthogonal to the yield surface.

Isotropic work hardening has been assumed by many researchers in welding

simulation.

1

22 2 2 2 2 2

e x y y z z x xy yz xz

1- - - 6

2

(2.7)

where e is the effective stress, x, y and z are the principal stresses in x, y

and z directions respectively and xy, yz and xz are the shear stresses in xy, yz

and xz planes respectively.

2.3.4 Non-Linear Transient Thermo-Mechanical Simulation of

Welding

Tekriwal and Mazumder (1991) presented a three dimensional

transient thermo mechanical analysis for gas metal arc welding process to

predict transient and residual stresses in the mild steel weld. A rate

independent plastic model with kinematic hardening was assumed to

characterize the metal behavior. Von Mises yield criterion and an associated

flow rule were used to determine the onset of yielding and the amount of

incremental plastic strain. A cut off temperature of 600 °C was considered.

For validating the model, displacement and transient strains were measured

and the agreement between the computed and experimental values was

reported to be qualitatively good.

Mahin et al (1991) predicted thermal history and residual elastic

strain distribution in gas tungsten arc welds in a circular 304L specimen,

using 2D finite element model. Their predictions were compared with the

experimentally obtained transient temperature field around the weld and with

the residual elastic strain distributions in the as arc-welded specimen, using

neutron diffraction technique.

28

Brown and Song (1992) examined the interaction between the

structure and the weld during welding processes by two- and three-

dimensional finite element models of a ring-stiffened cylinder. The

deformation process was assumed to be rate independent. Isotropic strain

hardening was used.

Shim et al (1992) developed a 2D finite element model for

predicting residual stress field in thickness direction of mild steel butt weld. A

generalized plane model for the finite element formulation was considered.

Stair-stepped mesh was used to numerically represent the individual weld

beads in the fusion zone. A ramped heat input was used to avoid numerical

instability and to approximate the effect of a moving heat source. The

parameters of the material modeling in the stress analysis were not specified.

Jones et al (1993) discussed the characterization of the effects of

welding parameters upon the deformations and residual stresses produced by

circular welds on a thin plate, by using plane stress finite element model.

Displacements in the specimen were measured with the help of fiducial

marks. The effects of the factors such as heat sinking, preheating and

geometry on the deformations and residual stresses in the work piece were

studied.

Canas et al (1996) considered a plane stress model to study the

effect of the strain hardening and the temperature dependent material

properties on the residual stresses in welded Al-5083-O alloy plates. The

effects due to phase changes were not considered. Surface effects such as

convection and radiation were adjusted in the arc efficiency. Bessel function

of second kind was considered in the heat source model.

Ravichandran (2002) studied the residual stress field in a dissimilar

weldment between carbon steel and stainless steel plates using 2D finite

29

element simulation. GMAW process was employed and the residual stresses

at various locations in the weldment and the thermal stress distribution for

various time intervals were also predicted. In the elasto plastic modeling, von

Mises‟ yield criterion and the associated flow rule were considered. It was

reported that the peak tensile residual stress was close to the yield strength of

the weld materials at the room temperature.

Ravichandran (1997), also presented the thermo elasto plastic

simulation of an edge welded plate of dimensions 1200 mm x 100 mm x 20

mm, using 2-D finite element method. Transient thermal history and

longitudinal bending distortion in the welded plate were predicted. A cut off

temperature of 750 °C was used. In the elasto plastic analysis, von Mises

yield criterion and the associated flow rule were considered for modeling the

plastic behavior of the material.

Brickstad and Josefson (1998) employed 2D axisymmetric models

to numerically simulate a series of multi-pass circumferential butt-welds of

stainless steel pipe up to 40 mm thick in the non-linear thermo mechanical

finite element analysis. “Element birth” was used to represent the laying of

weld beads to avoid any displacement or strain mismatch at the nodes

connecting the weld metal elements to those of the base materials.

Hong et al (1998) developed 2D generalized plane strain model

with a five-pass weld in a mild steel plate and an axisymmetric model with a

six-pass girth weld in a mild steel pipe to compute residual stresses. Element

rebirth technique was incorporated to model the multipass weld metal

deposition. It was reported that predicted residual stress results were

insensitive to the detailed heat input parameters and initial temperature

settings used for deposited weld passes.

30

Dong and Zhang (1999) discussed the general residual stress

characteristics associated with mismatched welds in two specific cases of

butt-weld and multipass girth weld, using a 2D generalized plane strain finite

element model.

Bachorski et al (1999) predicted distortion in a butt joint by FEM

using a method called “shrinkage volume approach”. It was assumed in their

work that the linear thermal contraction was the main driving force for

distortion, without the need of calculating transient temperature field and

microstructural changes.

Ravichandran (2000) developed a 2D generalized plane strain finite

element analysis of longitudinal bending distortion in a fillet welded tee beam

of length 1500 mm. A cut off temperature of 750 °C was used. The bending

distortion in the beam was computed, based on the displacements obtained in

the finite element model. Experiments were conducted to validate the

transient thermal histories as well as transverse displacements at select

locations in the carbon steel beam.

Sun (2000) developed a procedure to perform 2D incrementally

coupled thermo mechanical finite element analysis to simulate the resistance

spot welding process of aluminum alloys. The incremental changes in sheet-

deformed shape, contact area and current density profile as well as large

deformation effects were taken into account. It was reported that the

interfacial contact behaviour in the form of contact area change during

welding time played a crucial role in the nugget formation process in welding

aluminum alloys.

Son et al (2000) developed an empirical formula for weld induced

angular distortion, in terms of welding parameters such as heat input and plate

thickness, using an infinite laminated plate theory to consider an ellipsoidal

31

cylindrical inclusion with eigen strain. Temperature dependent material

properties were not considered.

Fricke et al (2001) performed finite element simulation of

circumferential welding of DN100 x 6.3 mm austenitic pipe for the prediction

of residual stress field. The details of parameters of material modeling were

not given.

Lars Borjesson and Lindgren (2001) performed a 2D fully coupled

thermal, metallurgical and mechanical finite element simulation for the

calculation of residual stresses in a multipass butt welding of steel plates of

thickness 0.2 m. Macro material properties were used by considering the

temperature dependent properties of each phase in linear mixture rules.

Temperature dependent plasticity was assumed using von Mises yield

criterion and the associated flow rule. Linear isotropic hardening was also

assumed. Uncertainty in the measurements of residual stresses was reported to

be ± 20 MPa for ideal conditions.

Park et al (2002) investigated the effects of mechanical constraints

on angular distortion of butt as well as fillet joints of various thicknesses

obtained by flux cored arc welding process. For this, equivalent bending

moments due to welding were applied for the measurement of angular

deformation, without carrying out full elastic plastic simulation.

Cho and Kim (2002) presented a 2D finite element model to

investigate residual stress field in both medium and low carbon steels by

incorporating metallurgical phase transformation. It was reported that the

residual stress field was not significantly affected by the phase

transformations in low carbon steel.

32

Tso-Liang Teng et al (2003) analyzed the thermo-mechanical

behaviour and evaluated the residual stresses with various types of welding

sequence in single-pass, multi-pass butt welded plates and circular patch

welds, using a 2D finite element model. SAW process parameters such as arc

voltage, welding current and welding speed were considered. The effect of

weld pass was also included in the analysis. No description of heat source was

given. The material was assumed to follow the von Mises yield criterion and

the associated flow rule. Phase transformation effects were not incorporated.

Chen and Kovacevic (2003) performed thermo mechanical analysis

using finite element method to investigate the thermal impact and evolution of

the stresses in the weld by considering the mechanical effect of the tool in the

friction stir butt-welded aluminum alloy 6061-T6. Multilinear strain

hardening effects were incorporated in the mechanical model.

Duranton et al (2004) developed a 3D finite element simulation of

multipass welding of 316L stainless steel pipe involving 13 weld passes to

predict residual stresses and distortions. Adaptive mesh refinements were

adopted to reduce the mesh density in the regions experiencing low thermal

gradients and presented a procedure to interpolate the results between the

different meshes. It was concluded that though 2D model gave results close to

the ones obtained in 3D model, the latter was much more realistic especially

for the stress estimation.

Lee et al (2004) calculated the residual stress in stainless steel weld

by bead flush method which involved the experimental determination of eigen

strains during the removal of weld reinforcement. The experimental results

were compared with 2D finite element predictions. Only the elastic analysis

was performed for the computation of residual stress, without complete elasto

plastic simulation.

33

Cho et al (2004) conducted 2D finite element transient heat flow

analysis in conjunction with a coupled thermo mechanical analysis to

investigate the residual stress distribution after welding and after a post weld

heat treatment in a 12-pass K-groove weld joint in 56 mm thick plate and a

9-pass V-groove weld joint in 32 mm thick plate. It was reported that

reduction of the maximum of the residual stress was achieved after the post

weld heat treatment. No description of parameters of material modeling in the

thermo mechanical analysis was specified.

Jung and Tsai (2004) investigated the effect of external restraints

and thermal management techniques such as heat sinking and gas tungsten arc

preheating, on the relationship between cumulative plastic strains and angular

distortion in fillet welded T-joints in the plate of 3.2 mm thickness, by

“plasticity-based distortion” analysis. It was reported that external restraints

reduced the bend-up angular distortion induced by the transverse cumulative

plastic strain and that the higher restraint produced lesser angular distortion.

Thermo elastic plastic analysis was not done.

Masao Toyoda and Masahito Mochizuki (2004) developed

numerical simulation methods of coupling transient thermal analysis,

microstructure and stress-strain fields to investigate the effect of heat input

and interpass temperature in multipass weld joint of beam-to-column

connections on the strength and fracture. It was concluded that the welding

conditions had strong influence on the joint performance.

Larsson et al (2005) modeled residual stress in a gas tungsten arc

welded Haynes® 25 cylinder by 2D axisymmetric finite element model. The

outer diameter and thickness of the cylinder was 34.7 mm and 3.3 mm

respectively. The residual stresses were also measured using neutron

diffraction method. A heat sink fixture used in the welding procedure was

included in the finite element model. A coupled thermo mechanical analysis

34

accounting for large deformations was performed using a staggered approach.

The material was modeled as isotropic and the plastic deformation was

assumed to be described by the von Mises yield criterion and the associated

flow rule. The strain hardening modulus was assumed to be linear isotropic.

No other phase changes were accounted for. A low thermal resistance was

used at the interface between the fixture and the specimen to allow for

additional cooling of the specimen during contact.

Paolo Ferro et al (2005) presented a numerical model of electron

beam welding of Inconel 706 in a butt joint by using two different power

density distribution functions for the simulation of the nail shape of the fusion

zone. It was reported that the 3D thermal and residual stress field was strongly

influenced by the shape of the fusion zone. Their predictions of residual

stresses were compared with the experimental values obtained using X-ray

diffraction technique.

Abid and Siddique (2005) presented a 3D sequentially coupled

non-linear transient thermo-mechanical analysis to investigate the effect of

tack weld positions and root gap on welding distortion and residual stresses in

a carbon steel pipe-flange joint by considering various angular positions for

the placement of tack welds. It was reported that the axial displacement and

tilt of the flange face were strongly dependent on the tack weld orientation

and weakly dependent on the root gap. No strain hardening was assumed in

the elasto plasto analysis.

Mollicone et al (2006) developed different thermo-elastic-plastic

computational models to simplify the thermo-elastic-plastic characteristics of

the GMAW process in a butt joint, starting from transient temperature field

input and leading to outputs of angular deformation and a contraction stress

field. A number of assumptions were made to develop these computationally

efficient models for the prediction of angular distortions. A simplified

35

empirical model of angular distortion in terms of relative depth of penetration

of the weld, the relative width of the fusion zone on the surface and a

geometric parameter depending on the shape of the fusion zone was also used.

The specific heat input rate did not figure in the empirical model.

Dean Deng et al (2007) performed finite element thermo elasto

plastic simulations to estimate the deformations of different typical welded

joints found in a large welded structure. Then these deformations of the

individual joints were imposed as initial strains to compute the distortion of

the structure by only elastic analysis. There was no mention about the

sequence of welding of the individual joints in the overall structure.

Camilleri et al (2007) presented different computationally efficient

models to suit industrial applications for the prediction of welding distortions

in butt as well as fillet joints. These computationally efficient models were

highly approximate in respect of the elasto plastic longitudinal and transverse

thermal strains developed by the transient temperature fields.

Jijin Xu et al (2008) used finite element method based on the

inherent strain theory to simulate welding distortion in multipass submerged

arc girth-butt welded pipes, without the full thermo elasto plastic simulation.

Welding current, arc voltage and welding speed were treated as welding

process parameters. The wire feed rate was converted to the corresponding

welding current, based on the linear relationship between them. Distortions

were predicted by using inherent strain method which did not involve

complete elasto plastic analysis.

Chang and Lee (2009) performed finite element analysis to predict

residual stresses in a T-joint fillet welds made of similar and dissimilar steels

using flux cored arc welding process. The length of the weld, the width of the

flange and height of the web were considered to be 600 mm, 500 mm and 120

36

mm respectively. The plate thickness was 15 mm for the flange and 19 mm

for the web. The fillet welds on both sides of the web were assumed to be

simultaneously laid down under the same welding conditions. This was done

so as to make use of the symmetry of the weld specimen.

Long et al (2009) investigated distortions and residual stresses

induced in butt joint of plates joined by metal inert gas welding. With the

Goldak‟s double ellipsoidal heat source model, temperature variations, fusion

zone and heat affected zone as well as longitudinal and transverse shrinkage,

angular distortion and residual stresses were predicted. It was reported that the

welding speed and the plate thickness had considerable influence on welding

distortion and residual stresses.

2.4 EXPERIMENTAL DETERMINATION OF RESIDUAL

STRESSES

Dieter Radaj (1992) deals with different methods to determine

residual stresses induced as a result of welding process in a comprehensive

manner. Theoretical analyses and computational models involve certain

assumptions to simplify the calculation efforts. Hence, it is essential to

examine to what extent the predictions of those models coincide with the

reality determined by experiments. The method of measurement of residual

stresses in a weldment is broadly classified into two types which are presented

below

i) Non-destructive residual stress measurements

ii) Destructive residual stress measurements

Residual stresses or strains are measured non-destructively, for

example, by means of the X-ray method. X-rays are diffracted by the

crystal lattices and produce interference phenomena, from which it is

37

possible to draw conclusions relating to the interplanar spacing of the lattice.

The load stress or residual stress is determined from the change in the

interplanar spacing compared to the free stress state. Residual stresses or

strains are also non-destructively measured by the neutron diffraction method.

Neutrons are scattered by the atomic nuclei. Hence, neutrons penetrate far

deeper than

X-rays. Thus stresses or strains can be measured in the interior of the

components. Further non-destructive residual stress measurements are the

ultrasonic method and the magnetostriction method. Measurements are

performed on the basis of the speed dependence of ultrasound on the stress

state. The measurements performed are echo time measurements with two

transversal waves, which are orthogonally polarized. In the magnetostriction

or Barkhausen noise method, the stress state is deduced from the value of the

local magnetization restraint.

The general principle of destructive residual stress measurement is

based on the assumption that the material is elastic. The elongation or

shortening of a small measuring base on the surface of the component is

determined while the component is subjected to loading or unloading. The

measurement requires to be performed along at least three directions in order

to completely determine the biaxial stress state. The strains result from the

measured displacements, by relating them to the length of the measuring base;

the stresses from the strains by means of Hooke‟s law. Resistance strain

gauges, detachable strain gauges and photoelastic surface layers are used

primarily for such measurements.

It has been recognized that residual stresses in welds are difficult to

measure using diffraction techniques such as X-rays, synchrotron X-rays and

neutron diffraction, as there are limitations owing to the presence of

microstructural gradients, dissimilar material combinations and thickness

(Zhang et al 2004). A relatively new method, called Contour method, though

38

destructive in nature, was proposed by Prime (2001). It enables a 2D residual

stress map to be evaluated on a plane of interest. The theory of the contour

method is based on a variation of Bueckner‟s elastic superposition principle.

The method was numerically verified by 2D finite element simulation and

experimentally validated on a bent steel beam having a known residual stress

distribution.

Zhang et al (2004) presented measurements of the cross-sectional

residual stress profile in a 2024 aluminum alloy variable-polarity plasma arc

weld using the contour method. Finite element modeling was carried out to

calculate the stress field, based on the measurement of the surface contour

produced by relaxation of the pre-existing stress field in the welded

component.

2.5 COMMERCIAL FINITE ELEMENT SOFTWARES

Commercial finite element softwares play an important role in

solving many welding related problems. In early research works, finite

element codes were written to compute transient thermal and stress fields in a

weldment, using popular programming languages such as „C‟, „C++‟,

„FORTRAN‟ etc. While using these programming languages for writing finite

element codes, certain difficulties are experienced by the researchers

especially in the description of complex shapes and in post processing the

results. With the advent of powerful CAD packages and meshing algorithms,

popular general purpose standard finite element softwares such as ABAQUS,

ANSYS, MARC, NASTRAN etc have been introduced and they are being

increasingly used by the welding researchers for the past few decades.

„ANSYS‟ is the shortened term obtained from „SYStem ANalysis‟.

It contains many bench mark tests drawn from a variety of resources such as

NAFEMS (National Agency for Finite Element Methods and Standards),

39

based in the United Kingdom, to validate the performance of elements under

distorted or irregular shapes, different meshing schemes, different loading

conditions, various solution algorithms, energy norms etc.

ANSYS programme is organized into different processors such as

described below:

i) Preprocessor

ii) Solution Processor

iii) Post Processor

iv) Time History post processor.

While the geometry of the weld joint specimen, element type,

appropriate material properties, meshing patterns, boundary condition and

load application can be dealt with at the preprocessor level, the type of

solution (steady state or transient or modal or harmonic etc), solver type, other

solution options etc can be specified at the solution processor. After obtaining

solution of the model, the distributions of either temperature or heat fluxes or

nodal displacements (distortions), principal strains or stresses in the model

can be viewed in the postprocessor. The variation of any appropriate quantity

of interest with time at a specified location in the model can be obtained in the

time history post processor. ANSYS program has been used to compute the

temperature distributions, temperature histories, thermal strains and residual

stresses in weldments (Abid and Siddique 2005, Nnaji et al 2004, Cho et al

2004, Frewin and Scott 1999, Han GuoMing et al 2007).

2.6 MATHEMATICAL MODELING AND OPTIMIZATION OF

WELDING PROCESS

Optimization is the act of obtaining the best result under given

circumstances. In design, construction and maintenance of any engineering

40

system, engineers have to take many technological and managerial decisions

at several stages. The ultimate goal of all such decisions is either to minimize

the effort required or to maximize the desired benefit. Since the effort

required or to maximize the benefit desired as a function of decision

variables, which is called the mathematical model. In respect of welding, the

objective of mathematical modeling of welding process would be to identify

the set of welding input conditions to optimize the desired responses. The

obvious responses with regard to welding include weld bead geometry,

residual stresses, distortions, etc., which affect the quality of weldments.

Various approaches are employed for obtaining mathematical or predictive

models in terms of welding process parameters. The most significant among

them are the regression analysis and artificial neural networks.

2.6.1 Regression Analysis

Regression analysis involves the planned conduct of experiments

with the process parameters at various factor levels. Then the experimental

data are fit into a polynomial equation (2.8) which involves the terms to

account for main effects, interaction effects and non-linearity between the

process parameters. The resulting equation is then checked for adequacy and

significance, based on the analysis of variance (ANOVA).

k k k2

i 1 i,j 1 i 1i j

Y (2.8)

o i i ij i j ii ib b X b X X b X

where Y is the response, bi, bii and bj are the coefficients, Xi are the factors or

parameters and k is number of levels of the factors.

Gunaraj and Murugan (2000) developed individual mathematical

models for the responses of penetration, reinforcement, bead width, area of

penetration, area of reinforcement, percentage of dilution and weld bead

41

volume in SAW of pipes. Experiments as per DOE were conducted using a

central composite rotatable design and statistical concepts to develop and

validate the mathematical models. The optimum SAW process variables such

as welding voltage, wire feed rate, welding speed and nozzle-to-plate distance

were also obtained.

Kim and Rhee (2001), Kim et al (2002) used multiple regression

method and artificial neural network concepts to develop linear as well as

curvilinear models and to predict weld bead height in terms of number of

pass, welding speed, arc current and welding voltage.

Murugan and Gunaraj (2005) developed mathematical models for

submerged arc welding of pipes using five level factorial techniques to predict

three critical dimensions of the weld bead geometry and shape relationships.

After statistically checking the adequacy of the models, the main and

interaction effects of the process variables on bead geometry and shape

factors were also investigated.

Kanjilal et al (2005) developed rotatable designs based on

statistical experiments for mixtures to predict the combined effect of flux

mixture and welding parameters on submerged arc weld metal chemical

composition and mechanical properties. Bead-on-plate welds were deposited

on low carbon steel plates at different flux compositions and welding

parameter combinations.

Benyounis et al (2005a, 2005b) developed and optimized

mathematical models for weld penetration, heat input, width of weld zone and

width of the heat affected zone in a butt joint of medium carbon steel using

DOE and response surface methodology.

42

2.6.2 Artificial Neural Networks

Artificial neural networks (ANN) are non-linear mapping systems

that consist of simple processors, which are called neurons, linked by

weighted connections. Each neuron has inputs and generates an output that

can be seen as the reflection of local information that is stored in connections.

The output signal of a neuron is fed to other neurons as input signals through

interconnections. A neural network consists of at least three layers, i.e. input,

hidden and output layers as shown in Figure 2.2. The back propagation

training algorithm is commonly used to train the neural network.

Figure 2.2 Typical Configuration of an artificial neural network

Jeng et al (2002) presented back propagation and learning vector

quantization networks to optimize the laser welding parameters such as the

laser power, focused spot size, welding speed, focused position, welding gap

and the alignment of the laser beam with the centre of the welding gap for

achieving optimized focused position welding quality.

43

Lightfoot et al (2005) developed a neural network model to study

the factors affecting the distortion of 6- to 8-mm thick D and DH 36 grade

steel plates, with the data experimentally obtained from welding trials and

subsequent measurements of distortion. Based on the neural network model,

the sensitivity analysis was carried out to identify key factors which

influenced the distortion.

Hsien-Yu Tseng (2006) applied general regression neural network

concepts to approximately obtain the relationship between welding

parameters such as welding current, electrode force, welding time and sheet

thickness and the failure load. Genetic algorithm was used to optimize the

welding parameters to achieve maximum load carrying capacity, using the

trained ANN model as the objective function.

Kumar and Debroy (2007) described methods to determine

multiple sets of welding variables that were capable of producing target weld

geometry in a realistic time frame by coupling a genetic algorithm with a

neural network model of gas metal arc fillet welding trained with the results

of heat transfer and fluid flow models.

2.6.3 Optimization of Welding Process using Different Techniques

Kadivar et al (2000) used genetic algorithm along with a 2D finite

element thermo mechanical model to determine an optimum welding

sequence. The thermo mechanical model was employed to estimate the values

of the objective function to be used in the genetic algorithm. The circular

welding along the inner circumference of a circular disc specimen of

thickness 2 mm was modeled.

Kim and Rhee (2001) proposed a method to decide near-optimal

settings of the welding process parameters using a genetic algorithm, through

44

experiments but without a model between input and output variables. The

bead height and the depth of penetration were considered as output variables

and the root opening, wire feed rate, welding voltage and welding speed were

considered as input variables.

Tso-Liang Teng et al (2003) performed thermo elasto-plastic

analysis using finite element technique to analyse the thermo mechanical

behaviour and evaluate the residual stresses with various types of welding

sequences like progressive welding, back step welding and jump welding in

single-pass, multi-pass butt-welded and circular patch welding of plates.

Nnaji et al (2004) investigated and optimized the welding sequence

of a sub-assembly composed of thin wall extruded aluminum alloy beams by

a 2D finite element model. Pre-estimated angular shrinkages were applied for

each welding step without conducting non-linear transient analysis. Different

criteria such as overall deformation and weighted deformation with emphasis

on certain critical area were considered for the purpose of minimization of

deformation.

Correia et al (2004) optimized the process parameters namely,

welding voltage, wire feed speed and welding speed of GMAW on the basis

of deposition efficiency, bead width, depth of penetration and reinforcement

within the experimental region, by using genetic algorithm.

Voutchkov et al (2005) used surrogate models for optimization of

welding sequence in the welding of the tail bearing housing, a component

used in most gas turbines. The component which was made of Inconel 718,

was used to assist in mounting the engine to the air craft body. The structural

details are reported to be the outer ring, the inner ring and the vanes. By

dividing the fillet weld between a vane and the inner ring, into three sub-

welds on either side, determination of the optimum welding sequence

45

involving the minimum distortion was attempted. “Surrogate models” were

used for determining the optimum welding sequence from among the 27

distortion values obtained from finite element simulation carried out based on

the concepts of design of experiments.

Pankaj Biswas and Mandal (2007) developed a 3D finite element

model to estimate thermal history and resulting distortion and to study the

effect of welding sequence on the distortion pattern and its magnitude in

fabrication of orthogonally stiffened plate panels. Distortions in the

fabrication were predicted, by assuming three predetermined welding

sequences.

It is evident from the literature survey that finite element simulation

of the residual stresses in a T-joint has been attempted by few researchers,

Ravichandran (2000), Cho and Kim (2001), Jung and Tsai (2004), Camillery

et al (2007), Chang and Lee (2009). Ravichandran (2000) used only 2D finite

element generalized plane strain model to compute the bending distortion of a

T-joint. But optimization of process parameters was not attempted in his

work. Cho and Kim (2001) did only the transient thermal simulation by

considering weld bead shape. Park et al (2002) studied the effects of the

mechanical constraints on the angular distortion of a T-joint without

conducting thermo-mechanical finite element simulation. Jung and Tsai

(2004) studied the effect of distortion control plans on the angular distortion

in a fillet weld. But they did not optimize process parameters. Voutchkov et al

(2005) dealt with the optimization of welding sequences to minimize the weld

distortion in a fillet joint by considering only a “surrogate models”. Moreover

only a few sequences were considered for optimization. Camilleri et al (2007)

proposed different finite element models for fillet welds with the objective of

reducing computational time to suit industrial application, but optimization of

welding process parameters was not carried out. Chang and Lee (2009) dealt

with 2-D finite element modeling of T-joint by considering similar and

46

dissimilar materials for the joint and it was assumed that the fillet welds were

laid on both sides of the web simultaneously to utilize symmetry.

Hence, an attempt was made in the research to develop a 3D finite

element model to predict residual stress and distortion for the purpose of

optimization of GMAW process parameters and weld sequences by

considering as many sequences as possible.