# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 1 Introduction  
Welding large structures is essential in constructing ships, heavy machinery, nuclear power plants, platforms, among others. For structures whose mechanical parts require precision work, including rack and pinion jacking systems and large platforms as representative examples, strict criteria should be satisfied during construction. For structures supporting compressive loads, the presence of initial out-of-plane deformation can cause buckling and lead to collapse in the worst case. Deformation in the welding of thick structures is largely due to transverse angular distortion, but is controllable. Therefore, it is necessary to predict and minimize the distortion before construction. The application of welding technology to industries has been derived in many studies on the relationships between welding conditions and deformation. [PERSON] [1] proposed a formula for predicting the angular distortion through a bead-on-plate experiment. The obtained angular distortion was proportional to the heat input and melting efficiency, and inversely proportional to the square of thickness. The heat input is expressed as the product of current and voltage divided by the travel speed. Moreover, [PERSON] found that, when penetration in the base metal exceeds 0.6 of its thickness, the base metal under the fusion region fails to provide sufficient constraints, and angular distortion rapidly decreases. [PERSON] and [PERSON] [2] analyzed the relationships between material, heat input, thickness, and deformation. They theoretically determined that the angular distortion is proportional to the heat input and inversely proportional to the square of thickness, and then experimentally verified these results. A common conclusion of the abovementioned studies is that angular distortion first increases in proportion to the division of heat input by the square of thickness, and it gradually decreases after a threshold. Many researchers verified this relationship afterwards, being widely applied across industries. [PERSON] and [PERSON] [3] recently found that this relationship differs, depending on the welding process, and that the factor that actually affects angular distortion is the mechanical meltingregion, whose temperature corresponds to that at which the stiffness of the material sharply decreases. Through finite element analysis, they also found that angular distortion is proportional to the width and depth of the region and inversely proportional to the square of thickness, and then confirmed the applicability of these relationships to various types of welding processes. Besides studies on basic principles, angular distortion has been investigated during multilayer welding. [PERSON] and [PERSON] [4] determined that angular distortion, which is small at the beginning, increases in the middle section, and then decreases again during multilayer welding of a double-V groove in a ring-type structure. They investigated the effect of the groove shape on the angular change and found the optimal groove height ratio. In addition, they found that back chipping causes a small deformation. [PERSON] and [PERSON] [5] found that angular distortion during multilayer welding depends on the specific deposited heat and derived a formula for predicting deformation from the number of beads based on an existing bead-on-plate welding formulation. They concluded that the number of beads affecting the distortion a function of thickness, heat input, level angle, wire density, and specific deposited heat. [PERSON] et al. [6] calculated the angular distortion during the multi-pass welding of stainless steel in the vacuum and cryostat vessel of a fusion reactor by introducing the effective bending rigidity, and proposed a method for minimizing deformation during X groove welding. [PERSON] and [PERSON] [7] derived an analytical formula for estimating angular distortion during V groove multilayer welding while using inherent strain calculation.