# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 2 Methods
### Bead-on-Plate Welding Experiment  
We conducted an experiment of bead-on-plate welding to verify the validity of Equations (1) to (8) while considering various assumptions. Low-carbon steel AH32 was used as the base metal, and E81T1-K2C flux cored wire with 1.2 mm in diameter and meeting the American Welding Society (AWS) Standards was used as welding material. The shielding gas was 100% CO\({}_{2}\). Welding was performed at two trisection lines in a 1500 \(\times\) 1000 mm specimen with different thicknesses without constraints, as shown in Figure 2. The thicknesses of the bead-on-plate welding specimen were determined in consideration of the bead-formed thickness generated during the multilayer welding. The width was larger than the length along the welding direction to minimize the influence of longitudinal distortion. The specimen size and measurement method were adopted, as in the previous study [10]. Constant current and voltage were maintained, but the welding speed was varied to achieve different heat input conditions. Table 1 lists the experimental conditions. We performed multi-pass welding up to two layers to increase the measurement accuracy of the bead cross-sectional area and height. Welding was performed twice per condition, and the angular distortion was measured for each pass of the first layer. For angular distortion, the average values were obtained at three points (A1, A2, A3 or B1, B2, B3 in Figure 2) while using a digital angle gauge. Metal cubes were used to facilitate measurements. Figure 3 shows the bead cross-section after welding. The bead cross-sectional area stacked on the base metal was obtained by calculating the number of pixels. The maximum and minimum bead heights were obtained by creating curves parallel to the deformed geometry of the base metal, with their average being considered to be the bead height. Figure 4 shows the measured angular distortion for varying thicknesses of the base metal. In a base metal with constant thickness, the angular distortion linearly increases with the number of passes. As the base metal thickness increases, the angular distortion reduces and, hence, the angular distortion per pass is constant at a given thickness. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Thickness (mm)** & **Voltage (V)** & **Current (A)** & **Travel Speed (cm/min)** \\ \hline
11.5, 12, 19.5, 45 & 29 & 285 & 25, 30, 40, 60 \\
28 & 29 & 285 & 25, 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Welding conditions for bead-on-plate welding experiment. Figure 3: Magnified image of welded sample for welding at 285 A, 29 V, and 30 cm/min  
Figure 2: Bead-on-plate welding and measurement (unit: millimeters) [10]. Figure 5 shows the measured bead cross-sectional area and square of bead height per pass, according to the heat input. As the heat input increases, the bead area and height proportionally increase. We assumed that the bead heights that were obtained during multilayer welding were the same as those measured in this experiment under the same conditions. Figure 6 shows the relationship between the ratio of bead area to the square of base metal thickness and angular distortion. The angular distortion increases proportionally to the ratio, but decreases after a threshold, which was found to be approximately 0.375. The threshold seems to occur as the center position of the contracted load becomes closer to the neutral axis. The data above the threshold produce low measurement accuracy of angular distortion due to in-plane shrinkage, and they exhibit nonlinearity [2,3]. If the center position of the shrinkage load agrees with the neutral axis, the angular distortion becomes zero, and shrinkage reaches its maximum. Reverse angular distortion might occur if it is located below the neutral axis. The reason is that if the thickness under the beads is not sufficient, the temperature gradient can be reversed, depending on the ambient cooling conditions.