# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 2 Methods
### Multilayer Welding Considering Degree of Constraint  
We derived the relationship between the bead area and angular distortion through a mechanical approach and experiments, as shown in the previous sections. We also aimed to obtain the analytic solutions for practical butt welding joints. As multilayer welding is a repetitive process, the welding conditions do not notably change across layers and, hence, we assumed a uniform amount of weld deposition. As the accumulation of beads increases the thickness, it complicates distortion across layers. The bead area and thickness for a heat input can be calculated while using the relation that is shown in Figure 5. Thus, the angular distortion can be predicted if the geometry of the welding joint is known. The total angular distortion at a layer can be obtained from the total bead area, as the angular distortion linearly increases with the number of passes at a given thickness (Figure 4). Equation (9) is obtained by applying Equation (8) to a V groove welding joint, as shown in Figure 7.  
\[\theta_{\text{total}}=\theta_{0}+\sum_{i=1}^{n+1}\Delta\theta_{i}=\theta_{0}+ \sum_{i=1}^{n+1}\frac{C}{Kt^{2}}\Delta A_{i}\ =\ \theta_{0}+\sum_{i=1}^{n+1}\frac{C}{Kt^{2}}\Big{|}y_{\text{R}}\big{|}\Delta z\ +\sum_{i=1}^{n+1}\frac{C}{Kt^{2}}\Big{|}y_{\text{L}}\big{|}\Delta z \tag{9}\]  
where \(\theta_{\text{total}}\): total angular distortion, \(\theta_{0}\): initial angular distortion, \(\Delta\theta_{i}\): angular distortion by layer \(i\), \(i\): layer number after initial thickness, \(n\): total number of layers, \(t\): thickness, \(\Delta A_{i}\): area of layer \(i\), \(y_{\text{R}}\), \(y_{\text{L}}\): linear equations of right and left bevel groove, respectively, \(\Delta z\): thickness of layer, heat height. In Equation (9) with Figure 7, \(\theta_{0}\) and \(h_{0}\) are the initial angular distortion and initial thickness, respectively. The initial angular distortion is the angle at the initial thickness, and the values should be determined after welding at least two layers. In typical weld groove joints, rather than welding onto the base material, the accumulation of beads (layers) creates the thickness. The angular distortion at the first and second layers cannot be accurately predicted, because the layers belong to conditions above the threshold, as shown in Figure 6. In addition, some distortions occur during initial setting and fit-up. Hence, the initial thickness and its angular distortion should be considered to include these errors, but the purely predicted angular distortion excludes the initial angular distortion. Layer number \(i\) is zero at the initial angular distortion, and \(n\) denotes the total number of layers stacked up to the thickness of the base material. This thickness is obtained by subtracting the initial one from that of the base material and dividing it by the bead height. \(n+1\) was set by adding a layer, as the surface of the final bead layer must be higher than the base metal thickness. The width of a layer,  
Figure 6: Relationship between angular distortion and ratio of bead area to square of thickness. \(\big{|}y_{L}\big{|}+\big{|}y_{R}\big{|}\), represents the length from the left to right level surfaces, as shown in Figure 7. The width can be geometrically obtained from the linear equation of each level surface in the \(y\)-\(z\) plane. The bead area of layer \(i\) is then obtained by multiplying the width by layer height \(\Delta z\). The angular distortion after the initial one can be calculated, as follows:  
\[ \begin{split}\sum\limits_{i=1}^{n+1}\frac{C}{K^{2}}\big{(}\big{|} y_{\text{L}}+y_{\text{R}}\big{|}\big{)}\Delta z\\ =\xi\sum\limits_{i=1}^{n+1}\frac{C}{K^{2}}\big{[}\tan\theta_{ \text{R}}\cdot(z-f)+g_{L}+\tan\theta_{\text{L}}\cdot(z-f)+g_{\text{R}}\big{]} \Delta z\end{split}  \tag{10}\]  
where \(h_{0}\): initial thickness, \(y_{\text{R}}=\tan\theta_{\text{R}}\cdot(z-f)+g_{\text{R}}\), \(y_{\text{L}}=\tan\theta_{\text{L}}\cdot(z-f)+g_{\text{L}}\), \(t=z\).