**2.2. Welding distortions**

588 Numerical Simulation – From Theory to Industry

 <sup>2</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> 1 . *y b*

(6)

*<sup>y</sup> <sup>e</sup> b*

**Figure 3.** Typical distributions of residual stress in plate butt joints (source: Masubuchi, 1980)

**Figure 4.** Typical residual stresses in welded structural profiles (source: Masubuchi, 1980)

near the welds and compressive in area away from the welds.

Fig. 4a shows residual stresses produced in welded T-shape and the residual stresses distributions. As can be further seen, high tensile residual stresses parallel to the axis are produced in areas near the weld in section away from the end of the column. In addition, stresses in the flange are tensile near the weld and compressive away from the weld. The tensile stresses near the upper edge of web are due to longitudinal bending distortion caused by longitudinal shrinkage. Furthermore, Figs. 4b and 4c show the typical distribution of residual stress in an H-shape and a box shape, respectively, particularly the distributions of residual stresses parallel to the weld line, in which the residual stresses are tensile in areas

*x m*

 

Distortion is closely related to the amount of residual stress and the degree of joint restraint during welding process. The correlation between distortion and residual stress is illustrated in Fig. 5. As rule of thumb, the welded joint with lower degree of restraint has an advantage due to less residual stress but it tends to get higher distortion. Conversely, the welded joint with higher degree of restraint has less distortion but it will further result in higher residual stress.

**Figure 5.** Welding residual stress and distortion correlation (source: Bette, 1999)

**Figure 6.** Three basic dimensional changes during welding (source: AWS Welding Handbook, 1987)

There are three basic dimensional changes during welding process with which we can easily understand the mechanism of distortion, namely:


In actual structures, the welding distortions are frequently more complex than these basic distortions or taking place with some conditions. For examples, pure transverse or longitudinal shrinkage will only take place when the following conditions apply, i. e. thickness of member is large enough and centre of gravity of the welds is in line with the neutral axis of the components. When it is not the case, the rotational deformations such as the angular, bending and buckling distortion may be happened.

The empirical formula to estimate the quantity of transverse shrinkage of carbon and low alloy steel butt welds can be found in American Welding Society (AWS) Welding Handbook (1987) as follows:

$$S = 0.2 \frac{A\_w}{t} 0.05d.\tag{7}$$

3D Finite Element Simulation of T-Joint Fillet Weld:

Effect of Various Welding Sequences on the Residual Stresses and Distortions 591

(10)

Handbook, 1987) proposed a formula to estimate the longitudinal shrinkage of butt joint as

<sup>3</sup> <sup>7</sup> 10 . *C IL <sup>L</sup> t*

**Figure 7.** Angular change in T-joint fillet weld, (A) free restrained stiffeners, (B) restrained stiffeners

The primary source of angular change is due to non-uniform of transverse shrinkage in thickness direction. Fig. 7a shows angular change of the free restraint T-joint fillet weld. When the stiffeners are prevented from moving, a wavy distortion occurs as can be seen in Fig. 7b. Masubuchi et al., 1956 (as cited in AWS Welding Handbook, 1987) established a relationship between angular change and distortion at fillet weld using a rigid frame

0.25 0.5 . *<sup>x</sup>*

*L L*

*x* is distance from centreline of frame to the point at which *δ* is measured, Fig. 7b.

 2

 

(11)

follows:

where:

*ΔL* is longitudinal shrinkage, in. or mm,

analysis in the following expression:

*L* is length of stiffener spacing,

where:

*δ* is distortion,

is angular change,

*C*1 is 12 or 305 when using unit in. or mm, respectively.

*I* is welding current, A, *L* isweld length, in. or mm, *t* is plate thickness, in. or mm,

where:


In fillet weld, the amount of transverse shrinkage is less than that happened in butt weld. The transverse shrinkage in fillet weld may be expressed by the following formulas found in AWS Welding Handbook (1987):

For T-joint with two continuous fillet welds:

$$\mathbf{S} = \mathbf{C}\_1 \left( \frac{D\_f}{t\_b} \right). \tag{8}$$

where:


$$\mathbf{S} = \mathbf{C}\_2 \left( \frac{\mathbf{D}\_f}{t} \right). \tag{9}$$

where:


Compared to transverse shrinkage, the quantity of longitudinal shrinkage for butt joint is much less, approximately 1/1000 of the weld length. King, 1944 (as cited in AWS Welding Handbook, 1987) proposed a formula to estimate the longitudinal shrinkage of butt joint as follows:

$$
\Delta L = \frac{C\_3 \, IL}{t} \text{10}^{-7}.\tag{10}
$$

where:

590 Numerical Simulation – From Theory to Industry

*S* is transverse shrinkage, in,

AWS Welding Handbook (1987):

*S* is transverse shrinkage, in. or mm, *D*f is fillet leg length, in. or mm,

*t* is bottom plate thickness, in. or mm,

*S* is transverse shrinkage, in. or mm, *D*f is fillet leg length, in. or mm, *t* is plate thickness, in. or mm,

*C*1 is 0. 04 or 1. 02 when using unit in. or mm, respectively.

*C*2 is 0. 06 or 1. 52 when using unit in. or mm, respectively.

For T-joint with two continuous fillet welds:

*t* is thickness of plate, in, *d* is root opening, in.

*A*w is cross sectional area of weld, in2,

(1987) as follows:

where:

where:

where:

thickness of member is large enough and centre of gravity of the welds is in line with the neutral axis of the components. When it is not the case, the rotational deformations such as

The empirical formula to estimate the quantity of transverse shrinkage of carbon and low alloy steel butt welds can be found in American Welding Society (AWS) Welding Handbook

0.2 0.05 . *Aw S d*

In fillet weld, the amount of transverse shrinkage is less than that happened in butt weld. The transverse shrinkage in fillet weld may be expressed by the following formulas found in

> <sup>1</sup> . *<sup>f</sup> b*

*t* 

<sup>2</sup> . *Df S C t* 

Compared to transverse shrinkage, the quantity of longitudinal shrinkage for butt joint is much less, approximately 1/1000 of the weld length. King, 1944 (as cited in AWS Welding

*S C*

For lap joint with two fillet welds (the thickness of two plates are equal):

*D*

*<sup>t</sup>* (7)

(8)

(9)

the angular, bending and buckling distortion may be happened.


**Figure 7.** Angular change in T-joint fillet weld, (A) free restrained stiffeners, (B) restrained stiffeners

The primary source of angular change is due to non-uniform of transverse shrinkage in thickness direction. Fig. 7a shows angular change of the free restraint T-joint fillet weld. When the stiffeners are prevented from moving, a wavy distortion occurs as can be seen in Fig. 7b. Masubuchi et al., 1956 (as cited in AWS Welding Handbook, 1987) established a relationship between angular change and distortion at fillet weld using a rigid frame analysis in the following expression:

$$\frac{\delta}{L} = 0.25 \phi - \left[ \left( \frac{\chi}{L} \right) - 0.5 \right]^2 \phi. \tag{11}$$

where:


To summary this section, many factors affect the welding process, thus the produced residual stresses and distortions, such as types of material, types of welded joints, structure thickness, joint restraint, heat input as well as welding sequence, which is the subject of the present study.

### **2.3. Thermal and Mechanical Finite Element Equations**

The corresponding finite element equations of thermal and mechanical are obtained by choosing a form of interpolation function representing the variation of the field variables, namely temperature, *T* and displacement, *U*, within the corresponding finite elements of the structural model and by applying further the weighted-residual or variational argument to the mathematical models. Furthermore, with imposing the boundary and initial conditions, the discritized equations obtained are solved by finite element techniques through which the approximated solution over the finite element model considered could then be obtained.

The thermal finite element equation including boundary condition may be written as follows:

$$\begin{Bmatrix} \mathbf{C} \\ \end{Bmatrix} \begin{Bmatrix} \mathbf{\dot{T}} \\ \end{Bmatrix} + \begin{bmatrix} \mathbf{K} \\ \end{bmatrix} \begin{Bmatrix} \mathbf{T} \\ \end{Bmatrix} = \begin{Bmatrix} \mathbf{F}\_{\mathbf{T}} \\ \end{Bmatrix}, \tag{12}$$

3D Finite Element Simulation of T-Joint Fillet Weld:

Effect of Various Welding Sequences on the Residual Stresses and Distortions 593

*<sup>i</sup> U T* (16)

<sup>V</sup> dV, (17)

<sup>V</sup> dV, (18)

ep e p D DD . (20)

<sup>1</sup> <sup>Δ</sup> <sup>i</sup> <sup>i</sup> <sup>U</sup> *U U* . (21)

<sup>1</sup> <sup>Δ</sup> <sup>i</sup> <sup>i</sup> *σ σσ* , (22)

ep th <sup>Δ</sup> D B <sup>Δ</sup> C M <sup>Δ</sup> *<sup>σ</sup> U T* . (23)

S V R pd S fd V, (19)

analysis, the mechanical finite element equation may be written in the form of incremental

<sup>111</sup> K1 2 <sup>Δ</sup> <sup>K</sup> <sup>Δ</sup> iii R R,

ep K BD B <sup>1</sup> T

th K BC M <sup>2</sup> T

N N T T

The vector of nodal displacement at the next step of analysis, i+1{U} could be obtained from:

Furthermore, the updated condition of stress in the structure could be obtained from the

Commonly, the iterative method of Newton-Raphson is employed in the finite element solver to solve the nonlinear equations. For further treatment, see (Bathe, 1996). Note also

as:

in which:

where:

{∆*U*} is the incremental of nodal displacement, {∆*T*} is the incremental of nodal temperature, [B] is the matrix of strain-displacement, [De] is the matrix of elastic stiffness, [Dp] is the matrix of plastic stiffness, [Cth] is the matrix of thermal stiffness, [M] is the temperature shape function, {*p*} is the vector of traction or surface force,

{*f*} is the vector of body force, and *i* is the current step of analysis.

following stress-strain relation:

in which:

$$\mathbb{E}\left[\mathbf{C}\right] = \int\_{\mathcal{V}} \mathbf{c} \mathbf{c} \left[\mathbf{N}\right]^{\mathrm{T}} \left[\mathbf{N}\right] \mathrm{d}\mathbf{V}\_{\mathrm{v}} \tag{13}$$

$$\mathbf{k}\begin{bmatrix} \mathbf{K} \end{bmatrix} = \int\_{\mathcal{V}} \mathbf{k} \begin{bmatrix} \mathbf{B} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} \mathbf{B} \end{bmatrix} \mathrm{d}\mathbf{V} + \int\_{\mathcal{S}} \mathbf{h}\_{\mathrm{f}} \begin{bmatrix} \mathbf{N} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} \mathbf{N} \end{bmatrix} \mathrm{d}\mathbf{S}, \tag{14}$$

$$\mathbf{Q}\begin{Bmatrix}\mathbf{F}\_{\mathbf{T}}\end{Bmatrix} = \int\_{\mathcal{V}} \mathbf{Q} \begin{bmatrix} \mathbf{N} \end{bmatrix}^{\mathrm{T}} \mathrm{d}\mathbf{V} \; + \ \int\_{\mathcal{S}} \mathbf{h}\_{\mathbf{f}} \mathbf{T}\_{\mathrm{ref}} \begin{bmatrix} \mathbf{N} \end{bmatrix}^{\mathrm{T}} \mathrm{d}\mathbf{S}.\tag{15}$$

where:


The results of temperature distribution and history obtained from Eq. (12) are then inserted into the mechanical model in the form of thermal load. Incorporating the elasto-plasticity analysis, the mechanical finite element equation may be written in the form of incremental as:

$$\overset{\text{i}+1}{\text{I}}\left[\text{K}\_{1}\right]\left\{\Delta U\right\}-\overset{\text{i}+1}{\text{I}}\left[\text{K}\_{2}\right]\left\{\Delta T\right\}=\overset{\text{i}+1}{\text{I}}\left\{\text{R}\right\}-\overset{\text{i}}{\text{I}}\left\{\text{R}\right\},\tag{16}$$

in which:

592 Numerical Simulation – From Theory to Industry

present study.

follows:

in which:

where:

*ρ* is the density (kg/m3), *c* is the specific heat (J/kg. K), *k* isthe conductivity (W/m. K),

*hf* is the convective heat transfer coefficient (W/m2. K),

[N] is the matrix of element shape functions, [B] is the matrix of shape functions derivative, and

{*T*} is the vector of nodal temperature.

*Q* isthe rate of internal heat generation per unit volume (W/m3),

To summary this section, many factors affect the welding process, thus the produced residual stresses and distortions, such as types of material, types of welded joints, structure thickness, joint restraint, heat input as well as welding sequence, which is the subject of the

The corresponding finite element equations of thermal and mechanical are obtained by choosing a form of interpolation function representing the variation of the field variables, namely temperature, *T* and displacement, *U*, within the corresponding finite elements of the structural model and by applying further the weighted-residual or variational argument to the mathematical models. Furthermore, with imposing the boundary and initial conditions, the discritized equations obtained are solved by finite element techniques through which the approximated solution over the finite element model considered could then be obtained.

The thermal finite element equation including boundary condition may be written as

C K . <sup>T</sup> T T F,

C NN

K BB NN <sup>f</sup>

N N f ref

V S F Q dV h T dS.

The results of temperature distribution and history obtained from Eq. (12) are then inserted into the mechanical model in the form of thermal load. Incorporating the elasto-plasticity

V

V S

T

T T

T T

(12)

<sup>ρ</sup>c dV, (13)

k dV h dS, (14)

<sup>T</sup> (15)

**2.3. Thermal and Mechanical Finite Element Equations** 

$$
\begin{bmatrix} \mathbf{K}\_1 \ \end{bmatrix} = \int\_{\mathcal{V}} \begin{bmatrix} \mathbf{B} \end{bmatrix}^T \begin{bmatrix} \mathbf{D}^{\mathrm{op}} \end{bmatrix} \begin{bmatrix} \mathbf{B} \end{bmatrix} \mathrm{d}\mathbf{V}\_{\prime} \tag{17}
$$

$$
\begin{bmatrix} \mathbf{K}\_2 \end{bmatrix} = \int\_{\mathcal{V}} \begin{bmatrix} \mathbf{B} \end{bmatrix}^T \begin{bmatrix} \mathbf{C}^{\text{th}} \end{bmatrix} \begin{bmatrix} \mathbf{M} \end{bmatrix} \mathbf{dV}\_{\prime} \tag{18}
$$

$$\mathbf{S}\begin{Bmatrix}\mathbf{R}\end{Bmatrix} = \int\_{\mathbf{S}} \begin{bmatrix} \mathbf{N} \end{bmatrix}^{\mathrm{T}} \begin{Bmatrix} \mathbf{p} \end{Bmatrix} \mathrm{d}\mathbf{S} + \int\_{\mathbf{V}} \begin{bmatrix} \mathbf{N} \end{bmatrix}^{\mathrm{T}} \begin{Bmatrix} \mathbf{f} \end{Bmatrix} \mathrm{d}\mathbf{V},\tag{19}$$

$$
\left[\mathbf{D}^{\mathrm{ep}}\right] = \left[\mathbf{D}^{\mathrm{e}}\right] + \left[\mathbf{D}^{\mathrm{p}}\right].\tag{20}
$$

where:

{∆*U*} is the incremental of nodal displacement,

{∆*T*} is the incremental of nodal temperature,

[B] is the matrix of strain-displacement,

[De] is the matrix of elastic stiffness,

[Dp] is the matrix of plastic stiffness,

[Cth] is the matrix of thermal stiffness,

[M] is the temperature shape function,

{*p*} is the vector of traction or surface force,

{*f*} is the vector of body force, and

*i* is the current step of analysis.

The vector of nodal displacement at the next step of analysis, i+1{U} could be obtained from:

$$\mathbf{^i}^{i+1}\{\mathbf{U}\} = \mathbf{^i}\{U\} + \{\Delta U\}.\tag{21}$$

Furthermore, the updated condition of stress in the structure could be obtained from the following stress-strain relation:

<sup>1</sup> <sup>Δ</sup> <sup>i</sup> <sup>i</sup> *σ σσ* , (22)

$$
\begin{bmatrix} \Delta \sigma \end{bmatrix} = \begin{bmatrix} \mathbf{D}^{\mathrm{op}} \end{bmatrix} \begin{bmatrix} \mathbf{B} \end{bmatrix} \begin{bmatrix} \Delta U \end{bmatrix} \begin{aligned} \mathbf{J} & \begin{bmatrix} \mathbf{C}^{\mathrm{th}} \end{bmatrix} \begin{bmatrix} \mathbf{M} \end{bmatrix} \begin{bmatrix} \Delta T \end{bmatrix} . \end{aligned} \tag{23}
$$

Commonly, the iterative method of Newton-Raphson is employed in the finite element solver to solve the nonlinear equations. For further treatment, see (Bathe, 1996). Note also

that from the thermal analysis results, the updated stress and displacement conditions are now obtained.

3D Finite Element Simulation of T-Joint Fillet Weld:

Effect of Various Welding Sequences on the Residual Stresses and Distortions 595

In the present study, a thermal elasto-plastic finite element procedure was employed to simulate the thermo-mechanical response of welding problem. In the procedure, two sequenced thermal and mechanical analyses were carried out independently (uncoupled) to

A transient thermal analysis of heat conduction was carried out in the first step to obtain temperature distribution histories over the structural model. In the thermal analysis, the welding heat input, *Q*a was calculated according to Masubuchi (1980) and the arc efficiency, *η*a for GTAW was assumed to be 0. 60 (Grong, 1994). Also, the values of convective heat transfer coefficient, *h*f and reference temperature were taken, respectively, to be 15 W/m2. K

In the next step, a structural analysis was carried out to now obtain the mechanical response of the structural model, where the temperature history obtained from the first step was employed as a thermal load in the analysis. The material model of elasto-plastic based on the von Mises yield criterion and isotropic strain hardening rule was chosen, in which its response over the history was determined by the temperature-dependent material properties inputted. The boundary condition or constraint on the structural model needs

Fig. 9 represents the mesh of T-joint fillet weld employed in this study along with the position of constraint assigned on the finite element model. The total number of nodes and elements utilized for the 3D model were 3654 and 2961, respectively. The analyses were implemented in ANSYS environment utilizing the element type of SOLID70 for the thermal

**Figure 9.** (a) Geometry of T-joint fillet welds, (b) Mesh of T-joint fillet weld along with its constraint

obtain the total or desired response of the welding structure modelled.

**3. 2. Finite element simulation of welding** 

and 25°C (298. 15 K).

position.

also to be assigned accordingly.

analysis and that of SOLID45 for the structural analysis.
