# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 3 Results and Discussion  
### Validation of Estimation  
We conducted V butt welding experiments to examine whether the predictions that are based on Equation (11) agree with experimental values. The size of the specimen was \(500\times 1000\) mm and its thickness was 28 mm, as shown in Figure 9. Temporary hexahedron members were fit-up by tack welding at points where welding started and ended. The joint groove had a 4 mm root gap and a 50\({}^{\circ}\) groove angle. Multi-pass welding was then performed, and the thickness and angular distortion were measured at each layer. One strongback was installed in the center of the specimen, and two at each end where transient heat transfer and deformation occurs. The experiments were performed  
Figure 8: Prediction curve of angular distortion based on Equation (11).  
with and without strongback, while the other conditions were the same as those for the bead-on-plate welding experiment.  
In this experiment, the thickness that was generated up to the second layer was adopted as initial value \(h_{0}\), and the corresponding angular distortion was set as initial value \(\theta_{0}\). They were determined by measurement. The degree of constraint \(K\) is calculated from the angular distortion of the second measurement, but, in this section, we derived the \(K\) value that best matches all of the measured data for the verification of the equation. Table 2 lists the parameters for prediction by Equation (11). Figures 10 and 11 show the prediction curves and experiment results. In Figure 10, the value of \(K\) in the free (unconstrained) condition was calculated to be 1.09 (close to 1.0). We minimized the contribution to the restraint increase, as temporary pieces were tack-welded at both ends and the grooves were cut in the same way as specimens. When constrained by a 10 mm strongback, \(K\) was calcucated to 10.5. Figure 11 shows the results for different strongback thicknesses. The degree of constraint \(K\) was 13.5 for a 20 mm strongback. Although the thickness of the strongback doubled, the degree of constraint increased by 1.3 times. When compared to the case with no strongback, the change in angular distortion decreased to 1/11 for 10 mm strongback and 1/13 for 20 mm. Hence, we found that the fixture itself had a greater effect on the increase in the degree of constraint than the fixture thickness.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**No Strongback**} & **Strongback** & **Strongback** \\  & & **10t** & **20t** \\ \hline \(\theta_{0}\) & 0.05 & 0.3 & 0.1 \\ \(h_{0}\) & 8.7 & 10 & 9 \\ K & 1.09 & 10.5 & 13.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input parameters for angular distortion estimation.  
Figure 9: Specimen for V butt welding experiments (unit: millimeters).  
These results indicate that once the appropriate degree of constraint is determined from the values measured at a layer after initial thickness, the subsequent angular distortion can be accurately predicted. Thus, we verified the applicability of Equation (11) regarding the constraint effects.