# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 1 Introduction
In addition, they proposed a simple method for predicting the final distortion by considering the measured weld deposition rate. [PERSON] et al. [8] proposed a method for predicting the angular distortion of V groove welding by using curve fitting from the experimental results of butt multi-pass welding. They derived relationships between heat input, geometry, and distortion. [PERSON] et al. [9] conducted a simplified distortion analysis by applying an inherent strain database to improve the efficiency of multi-pass welding analysis and considered the effect of residual stress on existing beads. [PERSON] [10] aimed to apply the characteristics of angular distortion during bead-on-plate welding to multilayer butt welding joints and then proposed a numerical approach while using the geometric principle of weld groove. Consequently, the prediction of angular distortion per pass without using finite element analysis was achieved. [PERSON] et al. [11] introduced the concept of offset and constraint to develop a geometric based algorithm. The angular distortion of 145 mm thick X-groove joint welding was predicted and compared with the finite element analysis results. The optimum welding sequence was provided through the assessment procedure they proposed. Most of the structures are subject to unknown constraints that are caused by gravity or surrounding structures. Various studies on the degree of constraint and distortion have been conducted to consider these phenomena. [PERSON] [12] obtained the constraint strength in terms of thickness, free span of the plate, and experimental constant when both ends are fixed and welded at the middle. A function for estimating angular distortion was derived by dividing the existing angular formula by the restraint factor. [PERSON] and [PERSON] [13] proposed a method for calculating the degree of constraint through computer analysis for various joint configurations. The method might be used for predicting deformation and cold cracking. [PERSON] et al. [14] verified the deformation reduction according to the jig constraint position and pitch through experiments and theoretical analysis, and they addressed longitudinal and transverse shrinkage as well as angular distortion, which were significantly reduced. [PERSON] et al. [15] showed the deformation of SUS304 steel according to the number of jigs using finite element analysis and proposed the optimal location of jig fixtures. [PERSON] et al. [16] derived the constraint coefficient through various welding experiments and analyses, obtaining the actual degree of constraint of a ship block and devising the optimal welding sequence. [PERSON] et al. [17] proposed an equivalent strain method while using inherent strain containing functions of the degree of constraint to improve the efficiency of welding deformation analysis for actual ship fabrication, as experimentally verified. Overall, studies on multilayer welding and the degree of constraint have provided substantial advances. However, few tools have been reported for efficiently predicting angular distortion during multilayer welding while considering the degree of constraint. In fact, it is difficult to reflect unknown constraints, although they may be designed in advance through laboratory experiments. Finite element analysis has been the mainstream for predicting angular distortion, but its long computational time ranges from a few hours to days, because thermo-elastic-plastic analysis must be conducted for over a few hundred passes. Moreover, the results rely on the experience or intuition of researchers and specialists if analytical methods are not available. In this paper, we propose a method for predicting the angular distortion of a structure under unknown constraints during multilayer welding through a mathematical approach. To this end, a formula on the degree of constraint is derived by analyzing the distortion behavior of bead-on-plate welding. In addition, an analytical formula is derived by applying the first formulation to multilayer welding joint geometry.