# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 3 Results and Discussion
### Applications  
We applied the proposed prediction method to a plate with 50 mm in thickness. The current and voltage were set to 280 A and 30 V, respectively, and the groove angle and root gap were set to 35\({}^{\circ}\) and 5.7 mm, respectively. The thickness and angle were measured at each layer. The other experimental conditions were the same as those for the V groove experiment (Section 3.1). For travel speed of 25 cm/min, the initial thickness and angular distortion were 13 mm and 1.9\({}^{\circ}\), respectively, upon the completion of the second layer. The measured angular distortion after welding of the third layer was 3.35\({}^{\circ}\). For a welding speed of 60 cm/min, the initial thickness and angular distortion were 13 mm and 0.15\({}^{\circ}\) after third layer with four passes, respectively, and the measured distortion at the next layer was 1.23\({}^{\circ}\). In this case, the change of the angular distortion was so small at the beginning that the measurement was started from the third layer. In these two cases, \(K\) was directly calculated while using Equation (13), because the results before and after welding one layer were known, with the respective values being 1.14 and 1.47. Figure 12 shows the prediction curves, including subsequent layers. No strongback constraint was considered in both specimens, but the calculated degree of constraint was above 1.0. Hence,  
Figure 11: Comparison of experimental and predicted values under constraints for welding conditions of 285 A, 29 V, and 30 cm/min (ST\(x\)xt: strongback with \(xx\) mm thickness). Figure 10: Comparison of experimental and predicted values in free (unconstrained) and constrained states for welding conditions of 285 A, 29 V, and 30 cm/min (ST10t: strongback with 10 mm thickness). some constraints were applied to the welding joint. Specifically, the constraining effect by self-weight seems to have acted on the welding joint. The 50 mm thick test specimen was 1.8 times heavier than the 28 mm thick test plate above. Moreover, the thick temporary pieces that were attached to the starting and ending points of welding may have partially contributed to the increase in degree of constraint. A small heat input due to fast welding exhibited a higher degree of constraint than a large heat input. The constraining effect by self-weight seems to have been larger when the amount of weld deposition was smaller due to the smaller heat input. [PERSON] [18] defined the degree of constraint as the ratio of stiffness of the surrounding structure to sum the stiffness of the welding joint and the surrounding structure through the bar-spring model, which is commonly used as a welding mechanism. In summary, when the bead is small due to small heat input, the degree of constraint increases by the relatively small stiffness of the welding joint when compared to the stiffness of the surrounding structure, as confirmed in this study. The two prediction curves in Figure 12 were obtained by using the initial angular distortion, thickness, and degree of constraint. Although the curves suitably agree with the experiment results, the measured values showed smaller distortion than the predicted values over a bead that formed a thickness of approximately 40 mm for a travel speed of 25 cm/min. The prediction curves were calculated from experimental data before and after welding only one layer. Consequently, even small measurement fluctuations may cause a large error for estimating the degree of constraint. All the experimental data of angular distortion in Figures 10-12 show fluctuations, possibly by measurement errors or differences in the amount of weld deposition due to arc changes during welding. Thus, the prediction accuracy can be improved by acquiring more measurement points. The X groove, or double-V groove, has a symmetrical shape with two V grooves about the weld root.