*2.1.1. Residual stress due to mismatch*

The residual stress mechanism due to mismatch may be simply illustrated in Fig. 1. Consider three carbon-steel bars of equal length and cross section connected together with two rigid blocks at the ends. The middle bar is heated up to 600oC and then cooled to room temperature while no applied heating on the other two bars. Since the expansion of the

**Figure 1.** Illustration of residual stress mechanism in welding (source: Masubuchi, 1980)

middle bar is restricted by other bars, compressive stress is encountered at the middle bar and the two side bars are subjected to opposite tensile stress. The compressive stress on middle bar, increases in linear elastic manner when it is heated (AB curve) until the yield stress of material in particular temperature reached, then plastic deformation is encountered which affects in decreasing compressive stress (BC curve). During cooling stage, the stress sign in middle bar is dramatically changed from compressive to tension stress and increases in linear elastic way (CD curve) up to the yield stress at point D. Then, non-linear plastic behaviour takes place (DE curve) in room temperature resulting in a tensile residual stress in the middle bar and contrary a compressive residual stress in both side bars which are equal to one-half of tensile stress in the middle bar.

#### *2.1.2. Residual stress due to uneven distribution of non-elastic strains*

When a metal bar is subjected to a uniform heat, it produces a uniform expansion lead to no thermal stresses. However, when it is subjected to non-uniform heat as the case of welding, thermal stresses and strains will be formed. Residual stress field in plane stress condition (*σ*<sup>z</sup> = 0) can be expressed by the following formulas:

Elastic and plastic strains:

584 Numerical Simulation – From Theory to Industry

environment.

sections.

**2. Theoretical background** 

**2.1. Basic mechanism of welding residual stresses** 

after welding and cooling down to room temperature.

composed by plastic and thermal strains.

*2.1.1. Residual stress due to mismatch* 

welds made of similar and dissimilar steels.

techniques. Further, effect of welding sequences on residual stresses of multi-pass buttwelds and circular patch welds was also investigated by Teng et al. (2003). Moreover, Chang & Lee (2009) performed the finite element analysis of the residual stresses in T-joint fillet

The present study extends the previous work of Teng et al. (2001) and focuses on numerical simulation of welding sequence effect on temperature distribution, residual stresses and distortions of T-joint fillet welds. Several welding sequences were considered and the resulted distribution of welding temperature, longitudinal and transverse residual stresses and angular distortions were simulated utilizing three dimensional finite element models. Four welding sequences considered were one direction welding, contrary direction welding, welding from centre of one side and welding from centres of two sides. Further, a welding sequence producing the smallest residual stress, distortion as well as distortion difference between both flanges was then investigated. The numerical simulation was done in ANSYS

Basic mechanisms of welding residual stress and distortion together with the finite element formulations used in the 3D numerical simulation are described in the following sub-

Complex heating and cooling cycles encountered in weldments lead to transient thermal stresses and incompatible strains produced in region near the weld. After heat cycles of welding diminished, the incompatible strains remain and provoking locked stresses or frequently termed as welding residual stresses. In general, term of residual stress deal with those remaining stress in a structure even though no external load applied (Masubuchi, 1980). Several terms having similar meaning with residual stress were found in some literatures, namely: internal stress, initial stress, inherent stress, reaction stress, lock-in stress, etc. In term of welding process, residual stress are the remaining internal stresses

There are two basic mechanisms to explain how residual stress produced by welding process, namely: the structural mismatch and the uneven distribution of non-elastic strain

The residual stress mechanism due to mismatch may be simply illustrated in Fig. 1. Consider three carbon-steel bars of equal length and cross section connected together with two rigid blocks at the ends. The middle bar is heated up to 600oC and then cooled to room temperature while no applied heating on the other two bars. Since the expansion of the

$$\begin{aligned} \boldsymbol{\varepsilon}\_{\boldsymbol{x}} &= \boldsymbol{\varepsilon}\_{\boldsymbol{x}}\prime + \boldsymbol{\varepsilon}\_{\boldsymbol{x}}\prime \\ \boldsymbol{\varepsilon}\_{\boldsymbol{y}} &= \boldsymbol{\varepsilon}\_{\boldsymbol{y}}\prime + \boldsymbol{\varepsilon}\_{\boldsymbol{y}}\prime \\ \boldsymbol{\gamma}\_{\boldsymbol{xy}} &= \boldsymbol{\gamma}\_{\boldsymbol{xy}}\prime + \boldsymbol{\gamma}\_{\boldsymbol{xy}}\prime \end{aligned} \tag{1}$$

where:

, , *x y xy* is components of the total strain, , , *x y xy* is components of the elastic strains, , , *x y xy* is components of the plastic strains.

Relationships of stress vs. elastic strain by Hooke's law:

$$\begin{aligned} \varepsilon\_{\boldsymbol{x}}\prime &= \frac{1}{E} (\sigma\_{\boldsymbol{x}} - \nu \sigma\_{\boldsymbol{y}})\prime \\ \varepsilon\_{\boldsymbol{y}}\prime &= \frac{1}{E} (\sigma\_{\boldsymbol{y}} - \nu \sigma\_{\boldsymbol{x}})\prime \\ \gamma\_{\boldsymbol{xy}}\prime &= \frac{1}{G} \sigma\_{\boldsymbol{xy}}\prime \end{aligned} \tag{2}$$

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

Effect of Various Welding Sequences on the Residual Stresses and Distortions 587

**Figure 2.** Schematic illustrations of heat cycles in welding and residual stress results (source:

away from weld line, whilst at weldment has zero stress due to metal melted (Fig. 2c. 2). Section C-C which is located at some distance behind welding arc is subjected to moderate heat (Fig. 2b. 3) due to cooling stage started in this section in which the condition at this section is similar to those CD curve in Fig. 1. Some distance far away from heat source, cooling down into room temperature is achieved which results in residual stresses in similar

Furthermore, typical distributions of butt joints in plate are presented in Fig. 3. Components of residual stress are categorized into transverse and longitudinal, designated as *σ*x and *σ*yrespectively (Fig. 3a). Across the weldline, tensile residual stress in longitudinal direction parallel to the weldline is found in the weldment region and compressive residual stresses occur in the others region away from weldline (Fig. 3b). Transverse residual stresses distributions along weldline are typically compressive part in the ends of plate, otherwise are tensile part with magnitude of stresses is lower than longitudinal residual stress (Fig. 3c). Masubuchi & Martin (Masubuchi, 1980) have developed the distribution of longitudinal

Masubuchi, 1980)

way to those in the end of DE curve in Fig. 1.

residual stress *σ*x which can be estimated as follows:

The stress must satisfy the equilibrium conditions:

$$\begin{aligned} \frac{\partial \sigma\_x}{\partial x} + \frac{\partial \tau\_{xy}}{\partial y} &= 0, \\ \frac{\partial \sigma\_{xy}}{\partial x} + \frac{\partial \sigma\_y}{\partial y} &= 0. \end{aligned} \tag{3}$$

The total strain must satisfy the conditions of compatibility:

$$
\left[\frac{\partial^2 \mathcal{E}\_x'}{\partial y^2} + \frac{\partial^2 \mathcal{E}\_y'}{\partial \mathbf{x}^2} - \frac{\partial^2 \mathcal{V}\_{xy}'}{\partial \mathbf{x} \partial y}\right] + \left[\frac{\partial^2 \mathcal{E}\_x''}{\partial y^2} + \frac{\partial^2 \mathcal{E}\_y''}{\partial \mathbf{x}^2} - \frac{\partial^2 \mathcal{V}\_{xy}''}{\partial \mathbf{x} \partial y}\right] = 0. \tag{4}
$$

The second term of Eq. (4), which is called the incompatibility term, *R,* is determined by plastic strain. When the value of R is not zero, thus residual stresses will exist in the weld joint.

$$R = -\left[\frac{\partial^2 \varepsilon''\_{\chi}}{\partial y^2} + \frac{\partial^2 \varepsilon''\_{y}}{\partial \mathbf{x}^2} - \frac{\partial^2 \gamma''\_{xy}}{\partial \mathbf{x} \partial y}\right].\tag{5}$$

More realistic illustration of the residual stress mechanisms during welding in typical plate joints is shown in Fig. 2. Welding bead is made along *x*-axis on the plate. Welding is carried out by moving the welding arc at speed *v*, and presently it is located at the origin O, as illustrated in Fig. 2a. Temperature distributions along particular points at weldline are shown in Fig. 2b, while stress resulted in the respect points are shown in Fig. 2c.

Along point A-A which is located ahead of the welding arc is not affected by heat yet. Section B-B experiences highest heat distribution (Fig. 2b. 2) which results in compressive stresses at just besides of weldline and surrounded by opposite tensile stresses in the side far

is components of the total strain,

is components of the elastic strains,

is components of the plastic strains.

The stress must satisfy the equilibrium conditions:

The total strain must satisfy the conditions of compatibility:

 

shown in Fig. 2b, while stress resulted in the respect points are shown in Fig. 2c.

Relationships of stress vs. elastic strain by Hooke's law:

<sup>1</sup> ,

0,

(3)

0.

 

  (2)

(4)

(5)

<sup>1</sup> ,

<sup>1</sup> .

 

*xy x*

 

2 2 2 2 0. *y xy y xy x x yx yx x y x y*

 

*xy y x y*

*x y*

*xy xy*

 

 

2 2 2 2 2 2

The second term of Eq. (4), which is called the incompatibility term, *R,* is determined by plastic strain. When the value of R is not zero, thus residual stresses will exist in the weld

> 2 2 2 2 2 . *y xy <sup>x</sup> <sup>R</sup> y x x y*

 

More realistic illustration of the residual stress mechanisms during welding in typical plate joints is shown in Fig. 2. Welding bead is made along *x*-axis on the plate. Welding is carried out by moving the welding arc at speed *v*, and presently it is located at the origin O, as illustrated in Fig. 2a. Temperature distributions along particular points at weldline are

Along point A-A which is located ahead of the welding arc is not affected by heat yet. Section B-B experiences highest heat distribution (Fig. 2b. 2) which results in compressive stresses at just besides of weldline and surrounded by opposite tensile stresses in the side far

*G*

*E*

*E*

*y yx*

 

*x xy*

 

where:

, , *x y xy* 

, , *x y xy* 

, , *x y xy* 

joint.

**Figure 2.** Schematic illustrations of heat cycles in welding and residual stress results (source: Masubuchi, 1980)

away from weld line, whilst at weldment has zero stress due to metal melted (Fig. 2c. 2). Section C-C which is located at some distance behind welding arc is subjected to moderate heat (Fig. 2b. 3) due to cooling stage started in this section in which the condition at this section is similar to those CD curve in Fig. 1. Some distance far away from heat source, cooling down into room temperature is achieved which results in residual stresses in similar way to those in the end of DE curve in Fig. 1.

Furthermore, typical distributions of butt joints in plate are presented in Fig. 3. Components of residual stress are categorized into transverse and longitudinal, designated as *σ*x and *σ*yrespectively (Fig. 3a). Across the weldline, tensile residual stress in longitudinal direction parallel to the weldline is found in the weldment region and compressive residual stresses occur in the others region away from weldline (Fig. 3b). Transverse residual stresses distributions along weldline are typically compressive part in the ends of plate, otherwise are tensile part with magnitude of stresses is lower than longitudinal residual stress (Fig. 3c). Masubuchi & Martin (Masubuchi, 1980) have developed the distribution of longitudinal residual stress *σ*x which can be estimated as follows:

(6)

**2.2. Welding distortions** 

stress.

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

Effect of Various Welding Sequences on the Residual Stresses and Distortions 589

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

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

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

 Transverse shrinkage, Fig. 6A, is a distortion perpendicular to the weld line Longitudinal shrinkage, Fig. 6B, is a distortion parallel to the weld line

transverse shrinkage is not uniform in the thickness direction

understand the mechanism of distortion, namely:

There are three basic dimensional changes during welding process with which we can easily

 Angular change, in butt joint and T joint fillet weld, as shown in Figs. 6C and 6D, respectively, deformation in rotation form around the weld. It happens when the

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.

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

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