# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 2 Methods
### Multilayer Welding Considering Degree of Constraint
Substituting Equation (10) into Equation (9), the total angular distortion of a V groove can be obtained, as follows:  
\[\theta_{\text{total}}=\theta_{0}+\frac{C}{K}\sum\limits_{i=1}^{n+1}\frac{[ \tan(\theta_{\text{R}})_{i}+\tan(\theta_{\text{L}})_{i}]\cdot[(h_{0}+(i-1) \Delta z-f)+g_{\text{L}}+g_{\text{R}}]}{\big{(}h_{0}+(i-1)\Delta z\big{)}^{2} }\cdot\Delta z \tag{11}\]  
where \(z=h_{0}+(i-1)\Delta z\). Equation (11) represents the sum of the initial measured angular distortion and the predicted angular distortions at all subsequent layers. Angular distortion decreases as constant \(C\), root gap, and level angle decrease, and the degree of constraint \(K\), initial thickness, and root face increase. The angular distortion during V groove welding depends on geometric parameters, except for two factors, namely, bead height \(\Delta z\), which is determined by the heat input (Figure 5), and constant \(C\), which is experimentally determined (Figure 6). Figure 8 shows the angular distortion prediction curves that were obtained from Equation (11). The position of the starting point of the curve and its slope must be determined for a prediction curve to be determined. The starting point is obtained from initial thickness \(h_{0}\) and initial angular distortion \(\theta_{0}\), whereas the slope is obtained from degree of constraint \(K\), which requires another point after the starting point. In Figure 8, the initial thicknesses of curves A and B are \(x2\) and \(x1\), and their initial angular distortions are \(y1\) and \(y2\), respectively. Although curves A and B have different initial angular distortions and thicknesses, they are almost parallel with the initial and final differences in angular distortion being the same. In practice, this results from the distortion by fit-up, presetting, and welding up to the second layer. Curves B and C have the same starting point, but different slopes. The slope of the curve becomes steeper as the degree of constrain reduces and the root gap or level angle increases with decreasing root face. Under the same geometry, curve C has a lower degree of constraint \(K\) than curve A or B.  
Figure 7: Geometry of V groove welding joint (\(f\): root face, \(g_{\text{L}},g_{\text{R}}\): left and right root gaps, respectively, \(h_{\text{n}}\): thickness of base metal, \(h_{0}\): initial thickness, \(\theta_{\text{L}},\ \theta_{\text{R}}\): left and right level angles, respectively). In addition, we considered that the angular distortion of each layer reduces the level angle. Hence, the amount of deposition decreases at the subsequent layer, which further decreases the angular distortion. The predicted angular distortion and level angle at the previous layer must be subtracted to obtain the level angle at a layer, as shown in Equation (12). This is used as the level angle term in Equation (11). \[(\theta_{R})_{i}=(\theta_{R})_{i-1}-(\Delta\theta_{R})_{i-1},\ (\theta_{L})_{i}=( \theta_{L})_{i-1}-(\Delta\theta_{L})_{i-1} \tag{12}\]  
where \((\theta_{R})_{i}\): right level angle at layer \(i\), \((\Delta\theta_{R})_{i-1}\): angular distortion at layer \(i-1\), \((\theta_{L})_{i}\): left level angle at layer \(i\), \((\Delta\theta_{L})_{i-1}\): angular distortion at layer \(i-1\). Equation (11) for predicting the angular distortion has three unknown variables, namely, \(\theta_{0}\), \(h_{0}\), and \(K\). Initial thickness \(h_{0}\) and corresponding angular distortion \(\theta_{0}\) can be determined from measurements at the desired layer, but at least two layers should be stacked. The degree of constraint \(K\) can be obtained by measuring the angular distortion of the additionally accumulated layer after the first measurement (the starting point in Figure 8). For calculation, we increase \(K\) by 0.01 from 1.0 and adopt the value that agrees with the measurement after two or more layers have been accumulated after the first measurement.