# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 2 Methods  
### Mechanics of Welding Distortion  
Arc welding uses thermal energy that is converted from electrical energy. Once the molten wire is placed on a base metal, its heat is transferred to the base, and the weld pool cools down, with the base metal undergoing both heating and cooling. The shrinkage of the weld pool is determined by subtracting the tensile plastic strain that is induced by surrounding constraints from the thermal strain during cooling. On the other hand, the shrinkage of the base metal is determined by subtracting the tensile plastic strain that is caused by cooling from the compressive plastic strain induced by surrounding constraints during heating. The constraining effect disappears if the elastic region is removed from the cooled welding joint, and the final contracted strain in the pure plasticity region, known as inherent strain [18], can be obtained. Physically, the inherent strain corresponds to subtracting the elastic strain from the total strain and it equals the sum of the plastic strain and thermal strain if the strain due to phase transformation is negligible, as in low-carbon steel. As the inherent strain at the welding joint includes plasticity, the plastic region is identical to the region where the inherent strain is distributed, and the inherent strain region can be considered as the plastic region. Angular distortion occurs by the inconsistency between the location of contraction and the neutral axis of the plate. In this study, we aimed to predict angular distortion under constraints due to a connection to surrounding structures. Therefore, a mathematical model for the contraction of the welding joint was devised and experimentally verified. Figure 1 shows the free body diagram of the area where bending occurs due to weld-induced contraction. The figure depicts the force that is caused by contraction in the plastic region, reaction force, and geometrical dimensions. The inherent strain that causes contraction in the bead and base metal is assumed to have a rectangular shape with a uniform distribution. Force \(F\) represents the force by contraction, \(F_{\mathrm{r}}\) is its reaction force and \(M_{\mathrm{r}}\) is the reaction moment by self-equilibrium of the plate without any constraint or external force. When the target structure is connected to other structures and both ends are constrained, \(M_{\mathrm{c}}\) is the resulting reaction moment. Thus, the reaction moment that is caused by the equilibrium of force due to the contraction of the welding joint is expressed as the sum of \(M_{\mathrm{r}}\) and \(M_{\mathrm{C}}\). As we address angular distortion, only the constraints that are caused by the moment in the out-of-plane direction are considered, disregarding the compressive and tensile loads in the in-plane direction. Although the in-plane reaction force at the constraining sides might affect the angular distortion of the welding joint, it is negligible with respect to out-of-plane moment \(M_{\mathrm{c}}\). If no constraint by surrounding structures exists, \(M_{\mathrm{c}}\) becomes zero, and only the reaction moment by the self-equilibrium, \(M_{\mathrm{r}}\), should be considered. From Figure 1, the equilibrium equations can be expressed, as follows:  
\[\sum F=F_{r}-P=F_{r}-F=0 \tag{1}\]  
\[\sum M=M_{r}+M_{c}-F_{r}z=0 \tag{2}\]  
The load \(F\) due to shrinkage and the reaction force \(F_{r}\) against the plastic region beyond the yield strength are in equilibrium. Weighting factor \(w\) is adopted for the yield strength of the welding joint. The inherent strain that is generated by welding is inhomogeneous in terms of distribution and size due to heat transfer distribution, strain hardening, dilution with welding materials, degree of constraint, and residual stress, but it is higher than the yield strength of the material. Therefore, we introduce the weighting factor to quantify these uncertainties. The ratio of the reaction moment by constraints, \(M_{c}\), to the pure reaction moment by the plate, \(M_{r}\), is defined as \(K-1\).