
Abstract
The main configuration of ship construction consists of standard and fabricated stiffening members, such as Tsections, which are commonly used in shipbuilding. During the welding process, the nonuniform heating and rapid cooling lead to welding imperfections such as outofplane distortion and residual stresses. Owing to these imperfections, the fabricated structural members may not attain their design load, and removing these imperfections will require extra manhours. The present work investigated controlling these imperfections at both the design and fabrication stages. A typical fabricated Tgirder was selected to investigate the problem of these imperfections using doublesided welding. A numerical simulation based on finite element modeling (FEM) was used to investigate the effects of geometrical properties and welding sequence on the magnitude of the welding imperfections of the Tgirder. The FEM results were validated with the experimental measurements of a doublesided fillet weld. Regarding the design stage, the optimum geometry of the fabricated Tgirder was determined based on the minimum steel weight and outofplane distortion. Furthermore, regarding the fabrication stage, a parametric study with two variables (geometrical properties and welding sequence) was conducted to determine the optimum geometry and welding sequence based on the minimum welding outofplane distortion. Increasing the flange thickness and reducing the breadth while keeping the Tgirder section modulus constant reduced the Tgirder weight and outofplane distortion. Noncontinuous welding produced a significant reduction in the outofplane distortion, while an insignificant increase in the compressive residual stress occurred.Article Highlights• Numerical simulation of welding process using uncoupled thermoelastoplastic fnite element method.• Efect of diferent geometrical confgurations of Tgirders with the same section modulus on weldinginduced imperfections and weight.• Efect of diferent welding sequences on weldinginduced imperfections.• The optimum design of the fabricated Tgirder and the optimum welding sequence were selected based on the minimum outofplane distortion and weight. 
1 Introduction
Welding technology is widely used in shipbuilding because of its high productivity. The expansion and contraction of the weld metal and adjacent base metal during the heating and cooling cycles result in welding imperfections such as distortions and weldinginduced residual stresses. These imperfections lead to additional cost of rework, misalignment, and more time to straighten the affected structures to be ready for assembly; also, the imperfections alter the structural strength. The threedimensional finite element modeling (FEM) technique has been used by several researchers to predict the thermal history, weldinginduced residual stresses, and outofplane distortion of a doublesided fillet weld, and then the results are compared with the experimental measurements.
Deng et al. (2007a) defined welding deformation in a filletwelded joint through numerical simulation and compared the results with the experimental measurements. Deng et al. (2007b) proposed a simple and efficient method to estimate the inherent deformation of large plate structures. Biswas et al. (2010) developed a numerical elastoplastic thermomechanical model for predicting the thermal history and resulting angular distortions of submerged arc–welded doublesided fillet joints. Tonković et al. (2012) used FEM to predict temperature, residual stresses, and distortion and then compared the results with the experimental data. Suman et al. (2016) applied FEM to model arc welding and predicted the welding distortion in butt and fillet weld joints. Gu et al. (2017) proposed a prediction approach to estimate welding distortions based on the local displacement in the weld plastic zone. Bai et al. (2017) predicted the welding deformation and residual stress of stiffened plates via numerical simulation and compared the results with the experimental data. Fu et al. (2014) investigated the welding residual stress and distortion in Tjoint welds under various mechanical boundary conditions. Chen et al. (2014) developed a threedimensional finite element model of a plate subjected to a doubleellipsoidal moving heat source to simulate a welding process using ANSYS and predict the temperature field, distortions, and residual stresses induced by butt welding. Chen and Guedes Soares (2016a) conducted threedimensional finiteelement thermoelastoplastic analyses to predict the effects of plate configurations on weldinduced deformations and the strength of filletwelded plates. Also, Chen and Guedes Soares (2016b) experimentally and numerically investigated the effect of the welding sequence on temperature distribution, distortions, and residual stress on stiffened plates. Chen et al. (2018) calculated the residual stress in buttwelded steel plates using numerical simulations and validated the results through Xray diffraction measurements. The effects of the boundary conditions, element size, and plate width were also investigated. Hashemzadeh et al. (2018) numerically and experimentally investigated the effect of multipass welding on the weldinginduced residual stresses in steel plates. Chen and Guedes Soares (2018) investigated the compressive axial ultimate strength of filletwelded steelplated ship structures subjected to uniaxial compression. To calculate the residual stresses in the welded plates, thermoelastoplastic finite element analysis was used to fit an idealized model of residual stress distribution.
Chen and Guedes Soares (2021) used nonlinear thermoelastoplastic finite element models to evaluate the temperature distribution, weldinginduced distortions, and residual stress in stiffened plates of higher aspect ratio to determine the presence of any significant effect of welding along the longitudinal direction. Hammad et al. (2021) experimentally and numerically investigated the hybrid laser arc welding process and the influence of the welding sequence on the manufacturing of stiffened flat panels. Mahapatra et al. (2007) studied the effect of constraints in the form of tack welds to minimize angular distortion in singlesided fillet welds. The proper positioning of tacks was found to have a significant effect on controlling the angular distortion. Chen et al. (2015a) investigated the effects of thickness, welding sequences, and finite element (FE) size on welding imperfections but did not consider the case of noncontinuous welding. Fu et al. (2016) investigated the effects of welding sequence on residual stresses and distortions in Tjoint welds. Kumanan and Vaghela (2017) investigated the effect of weld direction on distortion in a combined buttandfillet joint using FE analysis. Biswas et al. (2011) studied the effect of the welding sequence on the resulting deformation pattern and magnitude in a large stiffened panel for two different cases. Pandey et al. (2016) described the effect of changing the welding direction on distortion minimization in submerged arc–welded doublesided fillet joints. Venkatkumar et al. (2018) used a threedimensional FE numerical technique to study the effect of five different welding heat inputs on temperature, residual stresses, and distortion in buttwelded plates. Seo et al. (2018) investigated a thermal distortion mechanism and the effect of welding sequence on panel distortion during assembly. Liang and Deng (2018a) studied the influence of external restraint on welding distortion in three different thinplate steelwelded structures.
Liang and Deng (2018b) investigated the influences of heat input, welding sequence, and external restrain on twisting distortion in an asymmetrical curved stiffened panel. Perić et al. (2018) analyzed the effects of different preheat temperatures and the interpass time on longitudinal residual stress fields and structure deflection. Seleš et al. (2018) investigated an efficient FE procedure for the prediction of weldinginduced residual stresses and distortion in large structures. Lostado Lorza et al. (2017) presented a methodology to determine the most appropriate parameters for modeling thermomechanical behavior in welded joints through the FE technique. This methodology was based on the combined use of a support vector machine to predict critical features of the process and genetic algorithm (GA) with multiobjective functions to adjust the variables that define the FE models. Lostado Lorza et al. (2018) proposed a method based on experimental data and GA with multiobjective functions to determine the most appropriate parameters for the FE modeling of the thermomechanical behavior of butt joint single Vgroove weld created through onepass gas metal arc welding. Lostado Lorza et al. (2015) showed how a combination of FEM, GA, and regression trees for predicting the weld bead geometry according to the input parameters might be used to design and optimize complex welded products. Martinez et al. (2017) applied plasticstrainrange memorization based on a timeindependent cyclic plasticity theory for FE models of butt joints with a single Vgroove manufactured via gas metal arc welding.
In the present study, a parametric study was conducted to investigate the effect of geometrical properties and welding sequence on a fabricated Tgirder to minimize welding imperfections. Imperfections such as outofplane distortion and weldinginduced residual stresses may be controlled in both the design and the fabrication stages. First, regarding the design stage, the influence of the fabricated Tgirder geometrical properties on the welding imperfections was investigated. Second, regarding the fabrication stage, a parametric study with two parameters (geometrical properties and welding sequence) was conducted to determine the optimum geometry and the optimum welding sequence on the basis of minimum welding outofplane distortions. A remarkable reduction in both the maximum outofplane distortion and the Tgirder weight was achieved.
2 Verification of FE Model
2.1 Mechanical and Thermal Properties
Threedimensional thermoelastoplastic FEM for predicting weldinginduced residual stresses and distortions was performed. The numerical modeling of welding and the experimental validation of the FE model were conducted for the estimation of thermal history, outofplane distortions, and weldinginduced residual stress. The flow chart shown in Figure 1 illustrates the working procedure in both design and fabrication stages.
An FE model of two filletwelded plates was selected (Tonković et al. 2012; Seleš et al. 2018). This model was a typical Tgirder commonly fabricated in shipyards, such as girders in decks and bottoms. The geometrical configuration and boundary conditions are shown in Figure 2. The model was assumed to be a doublesided fillet weld. The first weld pass was taken to start from one edge to the other continuously, with a welding speed of 400 mm/min. A period of 214 s between the two welding passes was considered. The first and the second passes had the same welding parameters (Table 1).
Filler diameter (mm) 1.2 Shield gas composite 82% Ar, 18% CO_{2} Welding current, I (A) 270 Arc voltage, U (V) 29 Welding speed (mm/min) 400 Leg length (mm) 7 2.2 FE Mathematical Model
The dimensions of the model were as follows: L = 500 mm, B = 300 mm, H = 300 mm, and t = 10 mm. Within the appropriate computational effort, the mesh density shown in Figure 3 provided satisfactory results compared with the experimental measurements. The elements DC3D8 and C3D8 were used to discretize the geometry during the thermal analysis and stress analysis stages, respectively, using the Abaqus (2014). The mesh consisted of 19 851 finite elements with a mesh size of 2 mm at the weld bead to 7 mm at the model edges.
As the distance from the weld bead increased, the element size increased. The weld bead was divided into 150 element sets, which defined the discrete weld bead chunks (Seleš et al. 2018). Figure 4 shows the mechanical and thermal properties of st52–3 steel (Seleš et al. 2018; Pilipenko 2001).
The element birthanddeath technique with constant heat flux over the given set of finite elements, representing the weld bead chunk, was used to simulate the filler metal deposition. Afterward, the mechanical analysis was performed without the filler deposition. To correctly apply the welding speed, the length of each set was 6.667 mm in the weld direction, and the heating step was 1 s long. A uniform distribution of heat flux per volume of welded chunk, with a magnitude of 4 × 10^{10} W/m^{3}, was applied as shown in Figure 5 (Chen et al. 2015b; Seleš et al. 2018).
Nonlinear thermoelastoplastic FEM was performed to predict welding residual stresses and outofplane distortions. First, a thermal analysis stage was performed to predict the temperature history of the plate panel. A volumetric heat source was applied through the welding line. Thermal load induced by transient welding temperature fields was applied to the model. In this analysis, the mathematical model for nonlinear transient heat transfer is as follows (Perić et al. 2014):
$$ \frac{\partial }{\partial x}\left({k}_x\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left({k}_y\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left({k}_z\frac{\partial T}{\partial z}\right)+Q=\rho C\frac{\partial T}{\partial t} $$ (1) The general solution of Eq. (1) is obtained by introducing the initial and boundary conditions.
The initial conditions are as follows:
$$ T\left(x, y, z, 0\right)={T}_{0}\left(x, y, z\right) $$ (2) The thermal boundary conditions are as follows:
$$ \left(k_x\frac{\partial T}{\partial x}N_x+k_y\frac{\partial T}{\partial y}N_y+k_z\frac{\partial T}{\partial z}N_z\right)+q_s+h_c\left(TT_\infty\right)+h_r\left(TT_r\right)=0 $$ (3) Radiation heat losses are dominant near the weld and can be expressed by Eq. (4):
$$ {h}_{\mathrm{r}}=\sigma \varepsilon F\left({T}^{2}+{T}_{\mathrm{r}}^{2}\right)\left(T+{T}_{\mathrm{r}}\right) $$ (4) where σ is the Stefan–Boltzmann constant that is equal to 5.67 × 10^{−8} J/(m^{2}K^{4}s).
The total heat input applied to the weld volume is given as follows:
$$ Q=\frac{\eta UI}{{V}_{\mathrm{H}}} $$ (5) 2.2.1 Assumptions Made in the Numerical Analysis
1) All the relevant properties of the steel except density were considered as functions of temperature, while density was considered to be constant with the temperature change during the welding process.
2) Linear Newtonian convection cooling was considered for all surfaces.
3) A constant convection coefficient of 10 W/m^{2}K was considered for the web, flange, and weld bead shown in Figure 6.
4) The surface emissivity coefficient (ε) was equal to 0.9 for the flange (Face_Film_F), web (Face_Film_W), and weld bead (Face_Film_B) (Figure 6).
5) An arc efficiency (η) rate of 85% was considered for other losses.
6) To fix plates together before starting the welding process, tack welding was performed, but in the numerical models, the influences of the initial gap and the arrangement of the tack welds were not considered.
Figure 6 shows the thermal contact between the web and the weld bead (Contact_W), the flange and the weld bead (Contact_F), and the web and the flange (Contact_W_F).
2.2.2 Structural Analysis
A thermomechanical nonlinear elastoplastic analysis was performed. The stress–strain relationship used in the current study is as follows (Biswas et al. 2010):
$$ \sigma =D{\varepsilon }^{\mathrm{e}} $$ (6) where
$$ \boldsymbol\varepsilon^{\bf e}=\boldsymbol\varepsilon\boldsymbol\varepsilon^{\bf t} $$ (7) and
$$ \varepsilon^{\mathrm t}=\Delta T\left[\alpha_{x\;}\alpha_y\;\alpha_z\;000\right]^{\mathrm T} $$ (8) where $ \Delta T={T}_{n}{T}_{\infty } $, and T_{n} is the instant temperature at the considered point. Considering the plastic strains, Eq. (7) can be written as
$$ \boldsymbol\varepsilon^{\mathrm e}=\boldsymbol\varepsilon\boldsymbol\varepsilon^t\boldsymbol\varepsilon^P $$ (9) The temperatures created by the welding of the doublesided fillet joint obtained from the thermal analysis were indicated as the thermal loadings in the structural analysis.
2.3 Verification of FEM Model
2.3.1 Prediction of Thermal Results
In the experimental model, the surface temperature field was captured during the welding process using an infrared thermographic camera and thermocouples at 290 s and 403 s, respectively, after the beginning of the welding process (Seleš et al. 2018; Tonković et al. 2012). The locations of thermocouples are shown in Figure 7.
Figures 8 and 9 compare the temperature profiles obtained using the FE model and the experimental profiles. The temperature was measured 6 mm inside the flange. Figure 8 compares the results of temperature obtained via FEM and using thermocouples during the welding process. Figure 9 shows the relationship between the temperature profile and the position along line AD at 290 s and 403 s after the start of the welding process. From these figures, the FEM results agree well with the experimental measurements.
2.3.2 Mechanical Results
The outofplane distortion in the ydirection along line AG after the completion of the welding and cooling processes is shown in Figure 10. The numerical results of the current work are validated with experimental measurements Perić et al. (2014) and numerical results by Seleš et al. (2018). The FEM results match well with the experimental measurements. Figure 11 shows the distortion contour of the flange; there was a positive displacement at the flange edge and a negative displacement at the weld position.
Figure 12 shows the distortion pattern at the flange edge and weld position along lines BC and EF, respectively. The displacement was 0 at points C and B, because of the boundary condition, and was maximum (0.3 mm) at the flange midspan (point A). At the weld position along line EF, the maximum magnitude of distortion was 2.5 mm downward and the displacement was almost constant along the length of the model.
Figure 13 compares the FEM results of the current work, experimental measurements Perić et al. (2014), and numerical results by Seleš et al. (2018) for the weldinginduced residual stresses along line AG at the bottom surface. The tensile residual stress reached a magnitude of 340 MPa in the zdirection, which is close to the yield stress of the material. To obtain equilibrium, the corresponding compressive residual stress with a magnitude of 75 MPa in the zdirection was developed away from the welding line, near to the flange edges.
3 Effect of TGirder Geometrical Properties
As previously mentioned, during the welding process, outofplane distortion and welding residual stresses are initiated in the Tgirder. The presence of distortion depends on several parameters such as welding current, voltage, welding speed, geometrical properties, welding sequence, and restraints applied to the job while welding.
The welding imperfections may often make the structure unable to sustain its intended design load. Moreover, the existence of these imperfections will increase the production time necessary for fabricating the Tgirder, owing to the extra time consumed for corrective works such as straightening, adjusting, and trimming processes.
Furthermore, the geometrical properties of the Tgirder were varied to minimize the welding imperfections.
The flow chart shown in Figure 14 illustrates the suggested process during the variation of the Tgirder geometrical properties.
As previously mentioned, the typical fabricated Tgirder with a constant section modulus was studied. Four models of different geometrical properties but the same section modulus were selected. The leg length of the four models was determined based on the AWS d1.1 minimum weld size (AWS 2010). The boundary conditions and the welding sequence for each model were assumed to be the same as those in Sect. 2.
Figure 15 shows the cross section of the four different models of the fabricated Tgirder with different flange dimensions and length (L) of 700 mm. For convenience, case 1 was selected as the standard case.
Table 2 presents the geometrical properties, section modulus, and weight of each model, considering the weight of the weld line. The Tgirder of case 4 had 2% less weight than the Tgirder of case 1.
Cases Web (mm) Flange (mm) Section modulus at attached plate (cm^{3}) Girder weight (kg/m) AWS minimum leg length (mm) Weight of doublesided weld line (kg/m) Total weight (kg/m) 1 490 × 10 300 × 10 630.597 61.62 5 0.195 61.815 2 490 × 10 246 × 12 630.597 61.2456 5 0.195 61.4406 3 490 × 10 206 × 14 630.597 60.7152 6 0.2808 60.996 4 490 × 10 177 × 16 630.597 60.3096 6 0.2808 60.5904 Figure 16 demonstrates the distribution of the vertical deflection in the ydirection across the flange midspan for the four cases. As the flange thickness increased, the maximum value of outofplane distortion at the flange midspan decreased. However, an insignificant increase in the outofplane distortion occurred at the flange edges. This increase was due to the reduction in the flange width, which makes the flange edges close to the weld heataffected zone, and thus, the edges can be strongly influenced by the welding heat source.
Figure 17 shows the distribution of the outofplane distortion at the flange edge (along line CB) and the weld position (along line EF). The deflection curved upward at the edge and then decreased, reaching its maximum magnitude at the weld position, and this was because of the boundary condition at the flange corners.
The deflection at the weld position (along line EF) had a maximum value of 3.2 mm at point B, which was higher than the value at point C (2 mm), because the welding process was started at point B, with unidirectional welding.
Figure 18 shows the maximum magnitude of outofplane distortion at the flange edge and the weld position for the four cases. Case 4 had the minimum deflection between the weld position and the flange edge. Figure 19 shows the distribution of longitudinal residual stress at the flange midspan for the midsurface along line AG for the four cases. The tensile and compressive residual stress magnitudes of the four cases were almost the same. Moreover, the region of the tensile residual stress shrank as the flange width reduced. According to these analyses, case 4 had the optimum geometrical properties, with the minimum weight and outofplane distortion.
4 Effect of Welding Sequences
A parametric study with two variables (geometrical properties and welding sequence) was conducted to achieve the minimum outofplane distortion. The main objective was to select the optimum geometrical properties and the optimum welding sequence. Five different welding sequences were applied to the four cases of geometrical properties. The flow chart shown in Figure 20 illustrates the suggested process.
Figure 21 shows five different welding sequences explained as follows:
SEQ 1 is a doublesided unidirectional continuous welding from one end to the other end.
SEQ 2 is a doublesided unidirectional continuous welding from one end to the midlength.
SEQ 3 is a doublesided unidirectional continuous welding from the midlength to the end.
SEQ 4 is a doublesided unidirectional noncontinuous weld from one end to the other end.
SEQ 5 is a doublesided unidirectional noncontinuous weld from the midlength to the end.
Figure 22 shows the vertical deformation in the ydirection across the flange midspans along line AG for the five welding sequences, which were applied to the four cases. A comparison of the deflection patterns for the sequences showed that SEQ 4 and SEQ 5 had a remarkable reduction in distortion magnitude at the weld position, with an insignificant increase at the flange edge compared with other sequences. Moreover, SEQ 2 showed a significant reduction in the magnitude of outofplane distortion at the flange midspan; however, it showed a higher distortion at the flange edges, because of the temperature difference between the weld position and the flange edges at the end of the heating cycle; also, starting the cooling cycle with this temperature difference creates a huge contraction that results in larger distortion at the flange edges. A comparison of the four cases shows that case 4 and SEQ 5 yielded the minimum outofplane distortion, and thus, they are the optimum design and sequence, respectively.
Figure 23 shows the maximum magnitude of outofplane distortion at each sequence for the four cases at the flange edge along line CB and at the weld position along line EF. From the graphs, case 4 had lower distortion than the other cases. Figure 24 shows the distribution of the weldinginduced residual stress at the flange midspan for the midsurface along line AG in the welding direction for the five welding sequences.
The results show that the tensile residual stress magnitudes for the five sequences were almost the same. However, there were differences in the compressive residual stress distribution. Among the welding sequences, SEQ 5 attained the minimum distortion value but resulted in a higher compressive residual stress at the flange midspan, because the welding started at the flange midspan for both directions. A comparison of the four cases shows that the cases all had the same patterns and resulted in almost the same maximum tensile and compressive residual stresses.
5 Conclusions
The outofplane distortion and weldinginduced residual stresses of a fabricated Tgirder were simulated by performing uncoupled thermoelastoplastic analysis through FEM. The geometrical properties and welding sequence of the Tgirder were varied to investigate the minimization of the steel weight and outofplane distortion. The main conclusions of this study are as follows:
1) Increasing the flange thickness and reducing the flange breadth while keeping the Tgirder section modulus constant reduced the steel weight and the outofplane distortion at the flange midspan.
2) The magnitude and distribution of welding residual stresses showed no remarkable change with the change in the Tgirder geometrical properties.
3) A parametric study with two variables (geometrical properties and welding sequence) was conducted, and case 4 and SEQ 5 were the optimum design and sequence, respectively.
4) Noncontinuous welding sequence produced the following:
● A significant reduction in the outofplane distortion at the flange welding position.
● Insignificant increase in the outofplane distortion at the flange edges.
5) Noncontinuous (from inside, moving outward) welding sequence produced a higher compressive residual stress.
Applying the suggested procedures in the design and fabrication stages will reduce the reworking and production costs of fabricated Tgirders.
Nomenclature C: Specifc heat capacity; F: Confguration factor; h_{c}: Convection heat transfer coefcient; h_{r}: Radiation heat transfer coefcient; I: Current; k_{x}: Thermal conductivities in the xdirection; k_{y} : Thermal conductivities in the ydirection; k_{z}: Thermal conductivities in the zdirection; N_{x}, N_{y}, N_{z}: Direction cosines normal to the boundary; Q: Heat generation; q_{s}: Boundary heat fux; T: Current temperature; t: Time; T_{r}: Temperature of radiation heat source; T_{∞}: Surrounding temperature; U: Arc voltage; V_{H}: Volume of activated weld bead element; ρ: Density; ε: Efective emissivity; η: Arc efciency

Figure 5 Heat source model (Chen et al. 2015b)
Figure 8 Temperature profile comparison between experimental results obtained using thermocouples (Seleš et al. 2018) and FEM results
Figure 9 Relationship between temperature and the position along line AD (Tonković et al. 2012). a At 290 s. b At 403 s
Table 1 Welding parameters
Filler diameter (mm) 1.2 Shield gas composite 82% Ar, 18% CO_{2} Welding current, I (A) 270 Arc voltage, U (V) 29 Welding speed (mm/min) 400 Leg length (mm) 7 Table 2 Geometric properties of four different models
Cases Web (mm) Flange (mm) Section modulus at attached plate (cm^{3}) Girder weight (kg/m) AWS minimum leg length (mm) Weight of doublesided weld line (kg/m) Total weight (kg/m) 1 490 × 10 300 × 10 630.597 61.62 5 0.195 61.815 2 490 × 10 246 × 12 630.597 61.2456 5 0.195 61.4406 3 490 × 10 206 × 14 630.597 60.7152 6 0.2808 60.996 4 490 × 10 177 × 16 630.597 60.3096 6 0.2808 60.5904 
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