Managing distortion in welded

structures using FEM

MEHDI GHANADI

Master of Science Thesis in Lightweight Structures Dept of Aeronautical and Vehicle Engineering

Stockholm, Sweden 2013

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MASTER THESIS PROJECT

Managing distortion in welded structures using FEM

Student:

Mehdi Ghanadi

Supervisor:

Dr. Zuheir Barsoum

A master thesis report written in collaboration with

Department of Aeronautical and Vehicle Engineering

Division of Light Weight Structures

Royal Institute of Technology

And

Volvo Construction Equipment Company

August 2013

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Table of Contents

1 Introduction ……………………………………………………………………. 5

2 Literature review………………………………………………………………… 5

3 Inherent strain …………………………………………………………………... 7

3.1 Inherent strain definition …………………………………………………… 7

3.2 Thermal history of inherent strain ………………………………………….. 8

3.3 Inherent strain distribution ………………………………………………… 10

3.3.1 Estimating formula for inherent strain distribution ……………........ 10

3.3.2 Determination of inherent strain zone from stress distribution ……. 11

4 Inherent Deformation …………………………………………………………. 12

5 Geometry and welding parameters …………………………………………….. 17

6 FEM Analysis ………………………………………………………………….. 18

6.1 Material properties …………………………………………………………. 19

6.2 Thermal analysis …………………………………………………………… 19

6.3 Mechanical analysis ………………………………………………………… 20

6.4 Large and small deformation theory ……………………………………….. 21

6.5 Elastic FEM analysis ………………………………………………………. 22

6.5.1 Inherent strain approach …………………………………………… 22

6.5.2 Inherent deformation approach ……………………………………. 23

6.5.3 Shrinkage force approach ………………………………………….. 24

7 Discussion ……………………………………………………………………… 26

8 Conclusion ……………………………………………………………………… 27

9 Recommendation for future work ……………………………………………… 28

10 References …………………………………………………………………….. 29

Appendix A: Experimental formula for inherent strain distribution …………………... 35

Appendix B: Material Properties of S355 …………………………………………….. 36

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List of Figures

Figure 1: Displacement State under (a) Standard state (b) Stress State(c) Stress Released

State………………………………………………………………………….……… 7

Figure 2: Restrained condition in general case …………………………………….. 9

Figure 3: Temperature history, stress and strain of bar C under different thermal

condition……………………………………………………………………………. 9

Figure 4: Inherent strain distribution by FEM…………………………….……….. 11

Figure 5: Inherent strain zone estimation by residual stress distribution…………… 12

Figure 6: Classification of welding distortion …………………………………….. 12

Figure 7: Inherent deformation components ………………………………………. 13

Figure 8: Inherent deformation components ………………………………………. 13

Figure 9: Inherent Strain Zone …………………………………………………….. 15

Figure 10: Longitudinal and Transverse deformation …………………………….. 15

Figure 11: Tendon(shrinkage) Force ………………………………………………. 17

Figure 12: T-joint: geometry and boundary condition …………………………….. 18

Figure 13: Temperature profile at weld toe ……………………………………….. 20

Figure 14: Thin plate ……………………………………………………………… 21

Figure 15: Inherent deformation and inherent bending distribution in T-joint …… 23

Figure 16: Inherent force distribution along inherent strain zone ………………… 24

Figure 17: Transverse shrinkage force ……………………………………………. 25

Figure 18: Longitudinal and transverse deflection by elastic and elastic plastic FEM … 25

Figure 19: Isometric and top view of a T-Joint ……………………….…………… 35

Figure 20: Material Thermal and Mechanical Properties…………………………… 36     

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List of Tables

Tabel 1 : Process of production of inherent strain ………..…………………………….. 10

Tabel 2 : Inherent strain distribution by FEM and experimental formula ……………… 10

Tabel 3 : Welding parameters …………………………….…….…..………………….. 18

Tabel 4 : Inherent deformation components by three inherent based methods ………... 26

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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1-Introduction

Welding as high productive joining method is widely employed in automotive, aerospace and

shipbuilding industries. In practice, welding distortion brings about undesirable effects on

production accuracy, appearance and strength of welded components. Thus, in order to

increase the productivity and decrease the cost of the product, prediction and analysis of

welding deformation are key factors in industrial context.

Distortion of a structure can be measured experimentally; whilst in case of large or complex

structures it is expensive and also time consuming. Numerical analysis is then performed

using finite element method (FEM) that reduces the cost; however, in case of large welded

structure and considering extremely nonlinear mechanical behavior of welding the

computational expense incurred which must be cut through elastic analysis. In this sense, the

residual plastic strain, namely inherent strain as a source of residual stress and welding

distortion should be analyzed.

The purpose of this study is to detail the prediction procedure of deformation in welded

structure by elastic finite element modeling using inherent strain method. As a matter of fact

inherent strain as an inelastic permanent strain or residual plastic strain, which exists in

vicinity of fusion zone, is responsible for welding deformation and residual stresses.

Comparing with elastic plastic analysis, inherent strain method has less computing time

however the state of welding may be not investigated in detail; Furthermore, appropriate

assumption of inherent strain region and determining the accurate values of inherent

deformations in each typical joint bring about some limitations. On the other hand, just the

elastic modulus and Poisson’s ratio at room temperature is used in elastic FEM, and there is

no need for temperature material properties. So, at the present time, thermal elastic plastic

finite element method can be used to predict residual stresses and welding deformations in

small or medium structures but for large components elastic FEM is promising method.

2-Literature review

In search of welding distortion many studies were carried out in past decades [1-7]. For small

and medium welded structures some investigations have been developed, using elastic plastic

FEM [8]. However, for large components this method is computationally burdensome that

leads to the purpose for elastic FEM, for which the inherent strain method is used. For seeking

inherent strain properties, Ueda et al. [9] developed an estimating formula and also introduced

6  

some series functions to predict inherent strain zone. By using the inherent strain theory,

Murakawa et al. [10] introduced the elastic finite element method. In this sense, Murakawa

also developed the in-house code JWRIAN to simulate the welding deformation during all

process of welding assembly of a welded structure. Thereafter the elastic method has become

of great interest especially in shipbuilding industries where computational time is a key factor

in analyzing of large welded structures. Furthermore, thin plate welded structures with low

stiffness property bring about more severe problems in welding distortion. In this sense, by

using elastic FEM and applying inherent strain as initial strain, the welding distortion of

asymmetric curved block [11] and also plate bending in ship hull curved plate [12] were

analyzed. Meanwhile, Nair [13] also studied about principal bending and twisting deformation

in ship structures. Considering parametric studies, Murakwa[14,15] also investigated the

influence of welding sequence and restrained condition. In their research work contact

condition is also analyzed by introducing interface element in fusion zone. LUO and his co-

workers [16] proposed a theoretical formula to calculate inherent strain distribution and

thereby deformation, considering the effects of high temperature and restrained condition.

Significant reduction in computational time is also of great interest. In this sense, Murakawa

[17, 18] performed iterative substructure method(ISM) on a bogie beam and also developed

an efficient three dimensional approach for residual stress estimation, using block dumping

method. In line with time simplification, Bachorski [19] and Deng [20] also introduced

shrinkage volume method, as contraction force, but limited to special materials. Saenz and his

co-workers [21-31] did a lot of effort in field of inherent deformation. In their research works,

inherent deformations and influential parameters, like plate dimension, side and edge effects

were completely studied. Wang [32] also analyzed buckling mode, as most critical welding

deformation for thin plate structures using inherent deformation. In order for inherent

deformation to be determined, an experimental approach namely inverse analysis was also

undertaken [33-36]. This method is reliable as it allows for measurement on some small

selected points to be directed on a path to a more accurate result. Shrinkage force method is

also another applicable approach in welding deformation analysis. Many efforts have been

done in different working areas like shipbuilding industries [37] and also in railway vehicles,

applied on bogie frame [38]. Recently, Asifa khuram et al [39] applied equivalent load to

calculate welding deformation of a butt welded joint, using integration of inherent strain

component along welding line. Good agreement with the experimental results was achieved in

their results.

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In general each of the previous works has been focused on one of the inherent strain based

approaches. However, deciding on choosing the proper methods leads to compare them that

are the main focus in this study. So in this sense, elastic FEM based on three inherent strain

based methods found in the literature, is analyzed and the resulted welding distortions are

compared. This is based on thermo elastic plastic modeling, which makes use of block

dumping approach in mechanical analysis to predict inherent strain in longitudinal and

transverse directions.

3-Inherent strain

3-1- Inherent strain definition

In order to describe the inherent strain we can consider a body under three different states. In

first case, name as standard state, there are neither external forces nor internal stresses. In the

second case, the body is under residual stress state and in the last one there is stress released

condition in which the elements are separated.

As stated by small deformation theory and considering , and ∗ as distance between two

nodes in standard state, stress state and stress release state respectively. The strain among

these states can be defined as follow [40]:

1

∗∗

2

(a)  (b (c) 

Figure 1.Displacement State under (a) Standard state (b) Stress State   (c) Stress Released State

8  

∗

3

In above formulas, , ∗ and are total strain, inherent strain and elastic strain respectively.

So we can say:

∗ 4

Thus the inherent strain is responsible for deformation (inherent deformation) and residual

stresses in the body. Besides, total strain consists of some strain components which produce

with different process. These strains can be explained by:

5

Where , , and represent plastic strain, thermal strain, creep strain and phase

transformation strain correspondingly. So, from the expression of inherent strain the above

equation is written as below:

∗ 6

It should be noted that the thermal strain disappear when temperature comes back to room

temperature. Besides, the solid-state phase transformation strain itself causes two other kind

of strains name as transformation strain resulted by volumetric strain and transformation

plasticity strain. For low carbon steel, because of small dilation resulted by martensitic

transformation and high transformation temperature, the phase transformation has less effect

on residual stresses and deformation so in this case the phase transformation strain can be

neglected. But, for medium carbon steel because of large dilation and low transformation

temperature range, phase transformation strain should be taken into account [20, 41].

3-2- Thermal history of inherent strain

Temperature distribution is classified into high temperature zone near the heat source and low

temperature field in the vicinity area. Assuming Young’s modulus E and yield stress ,

mechanical behavior of welded structure can be simulated by three different bars named as

central bar C and two side bars S representing the high temperature region and adjacent area

respectively [33,10].

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Figure 2. Restrained condition in general case

Noting that the surrounding zone (bar S) plays as a constrained role, like elastic spring, for the

high temperature area (bar C). As a general case that is shown in above figure, bar C is

restricted by bar S via movable rigid body condition.

Fig.3 is used to draw conclusions about when inherent strain is produced. The diagram plots

stress and strain versus temperature. Magnitude of maximum temperature and also yield

temperatures, and , are influential, and their importance is seen in the diagram.

Further to above graph, in process one 0 → → 0 , which represents low

temperature heating, plastic strain and residual stress are not created. However, during second

process (0 → 2 → 0 , residual stress which produced in cooling step is

resulted by residual plastic strain ∆ , or inherent strain, induced in heating stage. As

indicates in process three 0 → 2 → 0 residual plastic strains are induced

both in heating and cooling steps ∆ , ∆ , thus then inherent strain is summation of these

two residual plastic strains ∗ ∆ ∆ . To sum up, correlation among plastic strain,

Figure 3.Temperature history, stress and strain of bar C under different thermal condition

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inherent strain, inherent deformation ∆ ∗ ∗ and residual stress is described in below

table:

Table 1.Process of production of inherent strain 

Heating Process Low Temp. Medium Temp. High Temp.

2 2

∆ 0

∆ 0 0 2

∗ 0

∆ ∗ 0

0

3-3- Inherent strain distribution

In principle the distribution zone and magnitude of the inherent strain component are

governed by the highest heating temperature and the constraint conditions [14]. Two methods

are studied in this section, one based on experimental formula and the other is corresponding

to residual stress distribution.

3-3-1-Estimating formula for inherent strain distribution

Based on average temperature rise,T with respect to heat input, and specific heat, , Yukio

Ueda et al.[42] developed an experimental formula to determine the magnitude and

distribution of inherent strain zone. In their study, the effect of structural geometry and its

material property is also taken into account. As may be seen in below table, for a T-fillet joint,

the width of inherent strain distribution, and magnitude of inherent strain, ∗ , are

calculated by proposed formula (Appendix A). As it is seen, experimental formula and results

by FEM are in good agreement.

Table 2.Inherent strain distribution by FEM and experimental formula 

Weld 

type  

By F.E.M  By Formula 

  ∗    ∗ 

T‐Joint  108.6   44  2.06 10   42  2.8 10  

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Fig.4 shows the inherent strain distribution in middle cross section of a T-joint. As can be

seen for T-joint from almost 44 mm away from center line, the inherent strain is almost zero.

Noting that, inherent strain region is varied along welding line. So as a good estimation the

average value of inherent strain zone should be calculated.

3-3-2-Determinationofinherentstrainzonefromstressdistribution

Inherent strain zone may also be determined by residual stress distribution. As illustrated in

Fig.5, in case of thin plate welded structures, inherent strain is dispersed through the whole

thickness so it thus becomes evident that just the width of inherent strain zone is important,

however for thick plates, strain distribution inside the thickness should also be taken into

account [9].  As shown in Fig.5, by comparing the residual stress distribution with inherent

strain distribution, Fig.4, the inherent strain zone can also be determined.

Figure 4.Inherent strain distribution by FEM

10

12  

   

3-Welding deformations

4-Inherent Deformation

Basically there are six types of welding deformation namely as: longitudinal shrinkage,

transverse shrinkage, longitudinal bending, transverse bending (angular distortion), rotational

distortion and buckling distortion [43].

 

Figure 6.Classification of welding distortion

Figure 5.Inherent strain zone estimation by residual stress distribution 

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The first four deformations also called inherent deformations and the last two are resulted

from these inherent deformations.

In order to analyze the inherent deformation, first by using a simple method the transverse

shrinkage, longitudinal shrinkage and angular distortion are described. In an attempt to do

that, as shown in Fig.8 a simple T-joint welded structure is considered. So the transverse

shrinkage can be calculated by making the difference between displacements of lines A and B

in transverse direction [44].

7

Where:S: Transverse shrinkage; , : Displacement of line A and B in transverse

direction.

Figure 8.Inherent deformation components 

Figure 7.Inherent deformation components

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Similarly by following equation the angular distortion can be calculated through deflection

distribution.

arcsin 8

arcsin 9

Where is angular deflection and , and are displacement of line C, D and E

respectively. is also displacement of line B in transverse direction.

Longitudinal shrinkage is also expressed by shrinkage force in longitudinal direction name as

tendon force. Tendon force is defined with respect to the inherent deformation by below

formula:

∗ 10

Where ∗ represents the inherent strain in longitudinal (welding) direction. So in order to

transform inherent deformation into inherent strain the equations are analyzed on a fillet

welding joint. In this sense the relation between heat input and longitudinal inherent strain is

calculated by following formula [45,46]:

22

22

11

2 2 12  

∗ .. . .

13  

∗ .. . .

14  

WhereQ , ε∗ and h are heat input, longitudinal inherent strain and thickness respectively.

Indexes ‘bs’ corresponds to base plate and ‘st’ is related to stiffener. Q and Q are heat input

for left side and right side of the baseplate. Accordingly the transverse inherent strain is

calculated by following equation:

15  

∗2

∗ 15  

The amount of curvature is also calculated based on angular distortion. So there is:

∗

2

16  

 

 

 

 

 

 

 

 

For elements in inherent strain zone, one can explain the distribution of longitudinal and

transverse shrinkages by their corresponding inherent deformations. Following figure shows

this relation on deformed rectangular element.

∗ ∗ 

∗ ∗ 

x 

z                                    z(welding direction)

 

x 

∗ ∗  

a 

b 

 

Figure 9.Inherent Strain Zone

Figure 10.Longitudinal and Transverse deformation

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For thin plate structures longitudinal shrinkage, transverse shrinkage and angular distortion

are mainly produced by longitudinal and transverse inherent strains ∗ and ∗. By constitutive

equation longitudinal and transverse inherent stresses can also be calculated as follow:

∗1

∗ ∗ 17

∗1

∗ ∗ 18

The deformation of the plate is expressed using the inherent deformation method which

consists of integration of inherent strain over the cross section of the plate and perpendicular

to welding line. The inherent deformation can be divided into four components; Longitudinal

shrinkage ( ∗), transverse shrinkage ( ∗), longitudinal bending ( ∗), and transverse bending

( ∗), that are defined by the following equations [47, 48]:

∗ 1 ∗ 19

∗ 1 ∗ 20

∗ 1 ∗ 21

∗ 1 ∗ 22

In which means plate thickness and moment inertia (I) defines as bellow:

112

23

It is supposed that inherent strain components are uniformly distributed along welding line so

one can use the average values of inherent deformations which are calculated as follow:

∗ 1 ∗ 24

∗ 1 ∗ 25

17  

∗ 1 ∗ 26

∗ 1 ∗ 27

In above formulas and are specimen length and weld length respectively; ∗, ∗ , ∗ and ∗ are also average values of corresponding inherent deformations. Longitudinal inherent

shrinkage can also be expressed by longitudinal shrinkage force ( ∗) named as tendon force.

∗ ∗ ∗ ∗ 28

Other components of Inherent force and moments in longitudinal and transverse directions

can also be defined as follow:

∗ ∗ ∗ ∗ 29

∗∗ 30

∗ ∗ 31

5-Geometry and welding parameters

The method of experiment is to conduct welding distortion, where T-fillet joint is used. The

specimen which made of S355 steel, both for base material and filler, represents a good

similarity with large structure. The temperature dependent thermo-mechanical properties of

S355 are found from Bhatti and Barsoum [49]. The dimensions and boundary condition are

shown in Fig.12. As can be seen, for both base plate and stiffener the thickness of 8mm is

chosen. Besides, during welding the baseplate has been clamped in one side and in 60 mm

from center of the baseplate. The welding parameters are detailed in table.3.Welding is

T 

L

 

Figure 11.Tendon(shrinkage) Force

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performed in such a way that firstly the stiffener is tack welded to baseplate. Then the

complete weld is performed by TA-1400 robot consists of LNM MoNiVa 1.2mm filler wire.

Welding order has been conducted according to direction shown in Fig.12. In so doing, the

opposite side of clamped area is welded and thereafter cooled to the ambient temperature. In

next step, by keeping the same welding direction and weld start/stop location the clamped side

is welded and then the whole structure is cooled to ambient temperature.

Table 3.Welding parameters 

Throat

(mm)

Current

(I)

Voltage

(V)

Weld length

(mm)

Travel Speed

(mm/s)

Weld passes

T-Joint 4 390 30 300 13.3 1

 

 

 

 

 

 

 

 

 

6-FEM Analysis

In order to calculate the deformation in welded structure a three dimensional finite element

model has been employed by using ANSYS. Both models are meshed with 8-node solid 70

linear elements. The FE mesh was created in a way to give high accuracy, that it was finer as

it approached the heat affected zone (HAZ). A finer mesh increases accuracy, but the

computational expense increases must be cut through an educated estimation of the HAZ

Figure 12.T‐joint: geometry and boundary condition

19  

region. Thus a mesh density gradient can shorten the time while maintaining accuracy.

Thereafter by using thermal mechanical uncoupled formulation, thermal and mechanical

behavior of structure is analyzed. In fact, insignificant dimensional changes in welding

procedure and small mechanical work comparing with thermal energy resulted by weld are

the main reasons to use uncoupled formulation. Therefore thermal problem is analyzed

independently from mechanical part, however, using temperature dependent thermo physical

and mechanical properties make the formula as a combination of transient temperature field

and stress analysis.

6-1-Material Properties

The temperature dependent thermo physical and mechanical properties of S355 are obtained

from [13]. By using hook’s law and considering young modulus and poisons ratio that vary

with temperature the elastic strain is calculated. In order to calculate the plastic strain, the

material properties are modeled in such a way that bilinear hardening model is governed and

von misses stress as yielding criteria is also assumed.

6-2-Thermal Analysis

Deposition of filler material is investigated both in thermal and mechanical analysis to get

temperature and residual stress distribution respectively. In so doing, lumping method as a

highly time efficient heating technique is used [50]. Noting that, during welding just small

part of structure is molten and the rest are in solid condition. So in the heating region, thermal

process is analyzed and then the resulting temperature is dispersed over the rest of structure.

The rate of heat input is calculated based on arc current (I), arc voltage (U), welding speed

( ) and number of weld passes (n) by following formula [37]:

q 32

In above formula q is heat input per unit length j mm and η is the welding efficiency.

Correspondingly the total heat input is explained as follow [51]:

Q Q ηUI 33

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Where Q is volumetric heat flux which is defined as bellow:

Qq

VηUI 34

In above formula V is the volume of block.

According to Newton’s law the heat loss resulted by convection is calculated as below [52]:

q h T T 35

In above equation h , T , T represent temperature dependent film coefficient, surface

temperature of weld joint and ambient temperature. It should be noted that in current study the

heat loss due to radiation is ignored. Generally there are three approaches to analyze the

deposition of filler material name as gradual weld bead deposition, rapid dumping and

lumping or block dumping [49]. In the current study lumping method as a highly time

efficient heating method is used. According to the welding sequence that is described in

section 5, the temperature profile at weld toe has been shown in Fig.13. As it can be seen the

maximum temperature of 1420 causes inherent strain distribution at vicinity of welding

area.

6-3-Mechanical Analysis

By thermal analysis the temperature distribution and its history is calculated. Thereafter the

temperature history is applied as a thermal load in mechanical elastic plastic analysis. In the

mechanical part, by keeping the same mesh as thermal analysis just element type is changed

to solid 185, which allow for three degree of freedom on each of its 8 nodes. As a matter of

fact when the deflection of plates becomes more than certain amount then the linear theory is

Figure 13. Temperature profile at weld toe

21  

not validated. Thus then with consideration of the large deformation theory, that causes

accurate results [44], the mechanical simulation has been performed. In this sense, the total

strain is composed of elastic strain ( ), plastic strain ( ) and thermal strain ( ):

The thermal strain disappears when temperature comes back to room temperature.

6-4-Large and Small deformation Theory

When the deflection of plates becomes more than certain amount then the linear theory is not

validated. In this condition the results show that the deflection of member is more than its

length which is incorrect. So by using large deflection theory the differential equation with

non-linear term must be solved. In order to describe this non linearity the equation related to

strain and displacement is investigated. For thin plates it is supposed that, total deflection

, is equal to deflection of mid-plane, , .So according to Mindlin plate theory

following formula is written[53].As can be seen in the below equations the first part related to

elastic strain and the second part represents non-linear behavior.

12

37

12

38

2 39

In above formula index ‘0’,’i’ and ‘b’ represent mid-plane, in-plane and bending conditions

respectively , and are strain in x and y directions; represent shear strain on x-y

x 

y z 

Figure 14.Thin plate

22  

planes and u, v and w are displacements in x, y and z respectively. The curvature of plate is

also defined by following formulas:

40

41

42

Where and are curvature in plane parallel to x-z plane and y-z plane correspondingly

and as twisting curvature which shows the warping of x-y plane.

6-5-Elastic FEM Analysis

Elastic FEM may also be analyzed to reproduce the welding distortion. In so doing, just the

elastic modulus and Poisson’s ratio at room temperature are used, and there is no need for

temperature material properties. Thus by removing the nonlinear items, the computational

time decreases significantly. T-joint is meshed with shell 281 element where is more refined

in inherent strain zone. In elastic FEM three different inherent strain based approaches can be

introduced, which initial load condition is the main difference amongst them. In first case, the

inherent strain components are directly applied on inherent strain zone and then the elastic

FEM is carried out. Whilst, in the other two methods, firstly the inherent strain is integrated

over welding line then the results in the form of displacement and force, namely inherent

deformation and shrinkage force respectively, applied on inherent strain zone.

6-5-1-Inherent strain approach

The longitudinal and transverse inherent strain is calculated by elastic plastic analysis. As a

good approximation, the average value of inherent strain distribution is calculated along the

welding line and then the result is considered as inherent strain zone. Thereafter by equation

(44) the strain components are integrated over thickness (t) and the results applies on each

element in inherent strain zone. Therefore, by considering large deformation theory the

welding deformation is calculated by elastic FEM [9]. Noting that according to equation (43)

the magnitude of total strain consists of bending ( ) and membrane ( ) components. In this

sense, the in-plane and bending items causes in plane strain and curvature respectively.

23  

6-5-2-Inherent deformation approach

As illustrated in fig.15, in inherent deformation approach, the average value of inherent

deformation components along welding line is calculated by equation (24-27) and the results

in the form of initial displacements apply on inherent strain zone and thereafter elastic FEM

carried out [32].

Figure 15. Inherent deformation and inherent bending distribution in T‐joint

24  

6-5-3-Shrinkage force approach

Welding distortion can also be calculated by applying equivalent force and moments. In this

method, namely shrinkage force approach, by using equations (28-31) the longitudinal and

transverse inherent strains are integrated over cross section of welding line. Thereafter

according to fig.16 the transverse shrinkage force ( and transverse moment ( ) are

applied on inherent strain zone. Since the analysis is performed on flat plates, then the

longitudinal shrinkage force ( ) and longitudinal moment ( ) should just applied on the

nodes located at both ends of the plate [37-39, 54-56]. Noting that in order to get the loading

per node, the force should be divided by the number of nodes that exists in inherent strain

zone. Thereafter the nodal load has to be applied on each node and then carry out the elastic

analysis.

In search of shrinkage force method some experimental formulas are also introduced. As a

case in point according to White’s parametric study the longitudinal shrinkage force may also

be calculated by welding properties as below:

Where w is welding speed and η represents as arc efficiency. According to welding

parameters and also equation (45) the longitudinal shrinkage force is calculated as 123 KN

[8]. As can be seen in fig.17 and by using equation (29), the transverse shrinkage force is

calculated along the welding line.

Figure 16.Inherent force distribution along inherent strain zone

25  

Distortion components calculated by three inherent strain based methods are compared in

fig.18 and table.4. It is seen that, considering the constraint conditions, the maximum

transverse bending and longitudinal bending occurs in front edge and side edge of the model

respectively. Table.4 shows the other two deformation components, as longitudinal and

transverse shrinkages. The outcomes are also compared with elastic plastic results and

experimental data.

 

Table 4.Inherent deformation components by three inherent based methods 

T‐Joint  Elastic Plastic  Elastic 

    Inherent Strain  Inherent Deformation  Shrinkage Force 

Longitudinal Shrinkage (mm)  0.622  0.324  0.40  0.41 

Transverse Shrinkage    (mm)  0.412  0.247  0.422  0.724 

Angular Distortion  (rad)  0.0107  0.0126  0.0139  0.0139 

Figure 18.Longitudinal and transverse deflection by elastic and elastic plastic FEM 

Figure 17.Transverse shrinkage force

26  

7- Discussion

Thermo elastic plastic FEM and elastic FEM based on inherent strain theory are analyzed on

T-fillet joint and the derived deformations studied amongst that. Comparing these two

methods, nonlinear material properties in elastic plastic FEM leads to increasing the

computational time which is approximately 5 hours that is much longer than elastic FEM

which is less than 1 minute.

The residual stress and inherent strain both are predicted by block dumping and gradual weld

bead deposition methods have the same trends. Similar results are obtained in [49].Since less

computational time is the main objectives of inherent strain approach, so in this study the

block dumping as an efficient heating method has been chosen.

The requirement to determine inherent strain distribution is essential to perform elastic FEM.

Existence of such leads to purpose of inherent strain diagram, from which we can identify the

average value of inherent strain zone along welding line. Unsymmetrical distribution of

inherent strain, fig.4, is mainly resulted by constrained condition. The longitudinal and

transverse inherent strains, having been plotted on diagram, shows the maximum inherent

strain zone in the middle cross section of plate where maximum thermal gradient occurred. In

T-fillet joint the magnitude of transverse inherent strain is less than longitudinal component,

which corresponds to different constrained conditions in both models. It is also found that the

inherent strain zone, having been calculated by FE, is in good agreement with experimental

formulas. To precisely predict the welding deformation, one should make the average of

inherent strain region along welding line.

Inferences particular to the welding deformation are drawn from the inherent deformation and

shrinkage force graphs, which include inherent displacements, bending moments and

shrinkage force distribution respectively. Same as previous works [43], [47] in this study also

the sharp variation of data at entrance and exit of plate length, fig.15 and fig.17, namely edge

effect, causes undesirable results. Thus then, the average value of inherent deformation and

shrinkage force is calculated by removing some data located near to the plate edges.

For elastic plastic method, contact element has been defined between baseplate and stiffener

and also baseplate and the ground. This condition leads to introducing another nonlinear item

which increases the computational time. But, the results show that contact elements defined

27  

between baseplate and ground prevents deflection through the ground that causes the out of

plane distortion to be increased.

Considering the out of plane deformation, Fig.18 and table.4, it can be seen that compared to

inherent deformation and shrinkage force methods, the inherent strain approach causes more

accurate results. This is an expected consequence considering the integration step in the first

two methods that leads to not properly estimation of the inherent strain distribution.

Bending deformation caused by residual stress gradient across the plate thickness effects on

angular distortion. In T-joint the bending moment is a more influential factor in out of plane

deformation rather than two other inherent components, longitudinal and transverse

shrinkages.

As shown in table.4, the transverse shrinkage and longitudinal shrinkage calculated in elastic

plastic simulation are in good agreement with same items calculated by inherent deformation

and longitudinal force methods respectively. It is because of the initial load condition as initial

displacement and force that are directly applied on inherent strain zone. Besides, it should be

noted that the restrained condition is also an important item where different constrained

condition may causes different result.

8-Conclusion

In this study the thermo elastic plastic FEM is performed to simulate the inherent strain and

welding distortion in a T-fillet joint. Thereafter, the elastic FEM based on inherent strain

approach and with different loading conditions, as initial strain, inherent deformation and

shrinkage force is simulated. Further to FEM and experimental results the following

conclusions are outlined:

Elastic FEM presents a reliable method by which to predict deformation, as well as

whether an elastic plastic method has been attained. In this sense, predicted welding distortion

in elastic FEM is in quantitative and qualitative agreement with elastic plastic method , i.e.,

out of plane deflection and angular distortion in both models. Besides, in elastic FEM the

computational time decreases significantly, which is almost the same for all three approaches.

In case of elastic plastic analysis the contact condition, especially between baseplate and

ground has influential role in welding deformation.

28  

Out of plane deflection is well predicted by applying strain as initial load.

For in-plane deformation, the inherent deformation is reliable method to detect transverse

shrinkage. Both inherent deformation and shrinkage force methods are suitable to predict

longitudinal shrinkage.

9-Recommendation for future work

For effective application of elastic FEM in prediction of welding distortion the additional

works are suggested as below:

1-The elastic FEM performed in this thesis needs to be improved by parametric study. In

order to do so, the restrained conditions should be analyzed in detail. Furthermore, the relation

between bending deformation, heating parameters and plate thickness should also be

developed in the analysis.

2-In order to decrease the computational time, the iterative substructure method (ISM) can

also be implemented in elastic plastic FEM.

3-Inverse analysis based on inherent strain theory is also suggested to be performed. In this

experimental procedure the magnitude and distribution of residual stress is estimated as a

function of inherent strain.

29  

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35  

Appendix A :Experimental formula for inherent strain distribution

The T-joint studied in this section is shown in below figure:

   

 

Average temperature rise, T is expressed by following formula:

TQρcA

47

In which Q, ρ, c, A are heat input, density, specific heat and cross-sectional area respectively.

Welding eccentricity β , as a dimensionless parameter shows the flexibility against to

longitudinal bending deformation due to welding, is expressed with respect to I, y, A as inertia

moment of transverse cross section, distance between weld center to the neutral axis of cross

section and cross section respectively.

βy AI 48

The width ( W )of distribution zone of inherent strain is calculated as below:

W0.484αE

cρ 2h h σ 49

In above formula, α, h , h denoted as the coefficient of linear expansion, thickness of

stiffener and base plate and σ shows yield stress of base plate.

The magnitude of inherent strain at weld metal related to W can be calculated as:

ε∗σE 50

Figure 19.Isometric and top view of a T‐Joint 

36  

Therefore by defining normalized factors a and b, the width and magnitude of inherent strain

are expressed as follow:

W aW 51

ε∗ bε∗ 52

In above formula, aand b are calculated as below:

a 1 0.27αET 1 β

σ 53

b 1 0.27αETσ

54

Appendix B: Material Thermal and Mechanical Properties of S355

Material thermal and mechanical properties of S335 is described in following graphs:

Figure 20.Material Thermal and Mechanical Properties

TRITA-AVE 2013:44

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