1. Introduction
To construct ships, which are mainly composed of steel materials, welding is generally employed in heavy industries. However, the large welding temperature range, involving extremely high heating to cooling, inevitably produces local shrinkages, such as longitudinal shrinkage, transverse shrinkage, and angular distortion, as illustrated in Fig. 1 (Woo et al., 2019).
These local shrinkages result in unexpected misalignments between welded structures and can compromise the structural safety. To constrain these welding displacements, various methods have been employed. In particular, use of specific welding techniques is a basic method used to mitigate welding displacements without additionally introducing constraint tools such as clamps and strongbacks.
Backstep welding and skipstep welding are widely used in heavy industries to alter the welding temperature mechanism. In these methods, a welding line is divided into several parts that are differently ordered according to the welder’s preference for mitigating welding displacement. In the backstep welding technique, the general progression of welding would be from left to right, but each bead segment is deposited from right to left as Fig. 2. To perform skipstep welding, a welder makes a series of intermittent bead segments then finally fills in the gaps between the as Fig. 2. Additionally, a newly introduced welding technique called bothstep welding is proposed herein as seen Fig. 2. In the present study, these welding techniques are numerically simulated to validate the most efficient method to mitigate welding displacement.
Previously, most studies of welding displacement using numerical simulations employed the thermal elasticplastic method (TEPM) to predict welding displacement precisely. However, this numerical method requires long calculation times to simulate even simple welding cases. Consequently, most of the recent studies on welding deformation using numerical simulations employ inherent strain theory in the elastic finite element method (EFEM). Luo et al. (1997) introduced inherent strain theory by considering the initial strain along welding lines to reflect the local shrinkages during progression of the welding process in the EFEM. Liang et al. (2005) validated the application of inherent strain theory in the EFEM to predict welding displacements by comparisons to experiment data. Deng et al. (2007) investigated the reliable applicability of inherent strain theory to the EFEM to analyze the welding process using a comparison with experiments results. Deng et al. (2004) developed the interface element method to reflect the misalignment between welded parts and nonwelded parts during numerical simulations. The Japan Shipbuilding Research Association (2000) proposed equations for predicting inherent strain and deformation of HT50 steel based on a measured experimental database using arc butt welding. As discussed above, inherent strain theory has been sufficiently proved to enable precise and economical analysis of welding displacements.
In the present study, HT50 steel was introduced to structures to be able to apply equations proposed by the Japan Shipbuilding Research Association (2000) to predict inherent strain and displacements. Based on several simulation results, the optimal welding technique for minimizing welding displacements to build a general ship grillage structure is then proposed.
2. Numerical approach for welding simulation
2.1 Inherent deformation and inherent strain
Based on the measured experimental database using arc butt welding (as shown in Table 1), the Japan Shipbuilding Research Association (2000) proposed equations as Eqs 13 to be able to calculate the amount of inherent displacements of HT50 steel (Table 2) such as transverse shrinkage S , longitudinal shrinkage F_{T} and angular distortion θ as Fig. 3. The amount of heat input Q^{*} decides the magnitude of welding displacement in these equations. Satoh et al. (1976) demonstrated the relationship between the input heat Q^{*} and the net heat Q_{n} with the thickness h of the welded steel plate as Q^{*} = Q_{n}/h^{2}.
Here,

S : transverse shrinkage [mm]

θ : angular deformation [mm]

S_{0}: transverse shrinkage at a welding length of 200 mm [mm]

θ_{0}: angular deformation at a welding length of 200 mm [mm]

${C}_{t}\left(L\right)$ : welding length compensation coefficient for lateral shrinkage

${C}_{a}\left(L\right)$ : welding length compensation coefficient for angular deformation

F_{t} : vertical contraction force [N]

L : welding length [mm]

v: Poisson’s ratio

Q_{n} : net heat input [J/mm]

h: plate thickness [mm]
Japan Shipbuilding Research Association (2000) derived Eqs 47 to calculate inherent strain. In the application of inherent strain in EFEM, the inherent strain is uniformly assigned along the welding lines which have same width at their each side (Ueda et al., 1993).
Here,

${\in}_{l}^{*}$ : inherent strain of longitudinal shrinkage

${\in}_{t}^{*}$ : inherent strain of transverse shrinkage

${\in}_{a}^{*}$ : inherent strain of angular distortion

k: magnitude of the curvature of the deformation

a: linear expansion coefficient [1/k]

${\sigma}_{y}$ : yield stress [MPa]
2.2 Interface element
In the sequential simulation, the definition of the relationship between welded structures and nonwelded structures is highly important to analyze welding displacements precisely. To reflect this phenomenon in the numerical simulation, the interface element is assigned along the welding line. Fig. 4 shows that the material properties of the interface element change depending on the stress state of the center of the interface elements prior to processing welding (Woo et al., 2019).
2.3 Multipoint constraint function
During finite element analysis, the multipoint constraint function (MPC) is applied for connecting different nodes with the same displacement at the same position. Fig. 5 (Woo et al., 2019) shows that, by using the MPC, nodes 1 and 2, which originally were in free movement from each other, are now connected to each other, and they can be defined as attached elements. The MPC can change the state of one node to work as a master or slave. In this study, tack welding (temporarily attaching plates) is employed to model the initial analysis model. As shown in Fig. 5, nodes a and b are initially connected using the MPC as tack welding. In addition, nodes (c, d), (e, f) and (g, h) install the MPC to be activated at an assigned welding order.
3. Analysis model
Two general plane plates (100 × 200 × 10 mm) are selected (see Fig. 6), to validate the efficiency of three welding techniques (backstep, skipstep and bothstep welding). HT50 steel was used to the material of the plane plate, and its properties are given in Table 2. The two general plane plates were welded to each other along the red line shown in Fig. 6.
To demonstrate the most efficient welding technique to reduce welding displacements in ships, a general ships grillage structure (2000 × 3000 × 10 mm) was introduced (see Fig. 7) (Woo et al., 2019). Table 3 provides the detailed dimensions of the structure. In this model, cross points at both ends of the welding line were tack welded prior to the complete connection. Arc butt weld is employed herein, and the condition of all welding lines is assumed constant, as given in Table 1.
In Fig. 7, the optimal welding sequence of all welding lines is schematically drawn as circled numbers on the general ship grillage structure. Woo et al. (2019) proposed the optimal welding sequence for the structure without the application of a welding technique.
4. Efficient welding technique
4.1 Results and discussion of the general plane plate
The representative value to discuss with respect to the efficiency of welding techniques to mitigate welding displacements is the average of the zaxis distance of all bottom plate nodes as Eq 8. Additionally, zaxis displacement distribution curves were measured at three lines (M1, M2, and M3) to discuss the magnitude of the angular displacement of the plane plate in Fig. 8.
Here,

A_{displacement} : zaxis displacement average (mm)

z_{k}: zaxis displacement of a node of a bottom plate

n: total number of nodes of a bottom plate
Figs. 9  12 clearly show the effect on the reduction of welding displacements of the plane plate in comparison to the result of welding displacements without employing an welding technique. In Fig. 9, compared to the value of A_{displacement} without a welding technique, the backstep, skipstep and bothstep techniques reduce by 8.1 %, 15.9 %, and 19.5 %, respectively. Hence, these welding techniques undoubtedly affect the mitigation of welding displacements of the total area of the general plane plate. Among these welding techniques, bothstep welding is the most efficient method. In Figs. 10  12, one can see that the zaxis displacement distributions curves of M1, M2, and M3 of the bothstep welding technique closely resemble a flat line. In other words, the welding technique greatly reduces the angular distortion of the welding process.
Compared to the results of the backstep welding technique, the other two welding techniques (skipstep and bothstep) exhibit higher efficiency for the reduction of welding distortion. According to these results, the time gap between the next bead segments to be cooled after finishing the previous bead segment is an important factor for mitigating welding displacements.
4.2 Results and discussion of the general ship grillage structure
The zaxis displacement distribution curves were measured at two lines to discuss the magnitude of the angular displacement of the plane plate, as seen Fig. 13.
Fig. 14 validates the effect of welding techniques during processing the optimal welding sequence on reducing welding displacements of the general ship grillage structure. In the terms of reductions of A_{displacement} from the case without employing a welding technique, the backstep, skipstep, and bothstep welding techniques reduce A_{displacement} by 5.8 %, 29.7 %, and 38.9 %, respectively. Compared to A_{displacement} of the skipstep and bothstep welding technique, the backstep welding technique is relatively inefficient.
Fig. 15 shows the zaxis displacement distribution of the general ship grillage structure using different welding techniques. In Fig. 15, severely buckled displacement is noted at the center of the general ship grillage structure in the case without using a welding technique. Similarly to what is seen in Fig. 14, Fig. 15 shows that the skipstep and bothstep welding techniques highly reduce the buckled displacements of the center of the general ship grillage structure.
The zaxis displacement distribution curves along T and L are plotted in Figs. 16 and 17, respectively. These clearly show the difference in the efficacies of welding techniques. The bothstep welding technique exhibits the lowest welding displacements in both zaxis displacement distributions along T and L. In Fig. 16, compared to the zaxis displacement distribution curve without a welding technique, the backstep welding technique successfully mitigates limited sections (100  500 mm, 1500  1900 mm / yaxis dimension); however, the section of the center (500  1500 mm / yaxis dimension) shows an the increase of zaxis displacement. In other words, the backstep welding technique is not always helpful for the mitigation of welding displacement of a welded structure. Compared to the result of the application of the backstep welding technique to the general ship grillage structure, the other two welding techniques (skipstep and bothstep) offer higher efficiency for the reduction of welding displacements of the welded structure. Hence, the time gap between the next bead segment to be cooled after finishing the previous bead segment is highly crucial for controling the welding heat mechanism, which leads to welding displacements.
5. Conclusion
In the present study, the most efficient welding technique for minimizing welding displacements in ship constructions was proposed. Several cases were numerically simulated using inherent strain theory, the interfaceelement method, and the MPC to analyze the effect of three welding techniques (backstep, skipstep, and bothstep) on the reduction of welding displacements. This study contributes to understanding of the efficiency of introduced welding techniques on the reduction of welding displacements. The conclusions are as follow:

Employing welding techniques obviously reduces A_{displacement} (8.1 ~ 19.5 %) of the plane plate structure. The bothstep welding technique is the most efficient method for the reduction of A_{displacement}.

Compared to the efficiency of the backstep welding technique, the skipstep and bothstep welding techniques exhibit much higher efficiency to mitigate welding displacements of the plane plate structure. In other words, the time gap between the next bead segment to be cooled after finishing the previous bead segment is an important factor for mitigating welding displacements.

The introduction of welding techniques to welding the general ship grillage structure improves the optimal welding sequence for minimizing welding displacements. In the application of welding techniques, the bothstep welding technique leads to the highest reduction in the severely buckled welding displacements of the general ship grillage structure.

Although use of the backstep welding technique cam reduce A_{displacement} (5.8 ~ 38.9 %) of the general ship grillage, inspection of the zaxis displacement curve of T shows that it produces an increase of zaxis displacement at some specific sections. Therefore, this method could lead to a negative effect on some parts of welding displacements of the welded structure.

To verify the method proposed in this study clearly, an actual experiment on a smallsized mockup is planned in a future study.