# Thermal Strain-Based Simplified Prediction of Thermal Deformation Caused by Flame Bending
## 1 Introduction  
In the shipbuilding industry, flame bending is an economical method to form the complex curved hull plate located on the bow or stern of the ship. As intelligent manufacturing technology using information and communications technology (ICT) is being developed, the shipbuilding industry is also expanding its intelligent technology. These technologies are applied to various ship production processes, but the forming of curved plates of the hull is still difficult to automate. When multiple flame bending condition is applied, it is difficult to predict the strain according to the heating line. Therefore, a method for predicting the thermal deformation by the flaming bending should be developed in advance. To predict the thermal deformation of the steel plate by flame bending, a finite element method considering both heat transfer and thermal deformation has been proposed by many studies [1; 2; 3; 4]. Nevertheless, predicting the thermal deformation by multi-heating lines is a difficult task. Although rezoning techniques have been adapted to reduce the computational time and the speed of the computer has also increased, it still takes several hours [5]. Therefore, the classical thermo-elasto-plastic analysis, which calculates the thermal deformation caused by the heat transfer by flame, is not suitable for a real-time prediction system. In order to overcome these limitations, the equivalent load method based on the magnitude of inherent strain with shell elements was introduced by [PERSON] et al. [6]. This method assumes that the area heated above the critical temperature is a heat-affectedzone (HAZ) with residual deformation and performs elastic analysis by calculating the equivalent loads that can simulate the corresponding residual strain. When applying equivalent loads, it is complicated to define the rigidity or anisotropy of the plate. In addition, the definition of direction and magnitude of load for the automated strain analysis model is difficult because the heating line is replaced with a bending moment and shrinkage force as a boundary condition. [PERSON] et al. [7] suggested an inherent strain-calculating method considering the phase transformation by flame. This method assumes that the plate has different material properties when it is heated and cooled. They considered that when the martensite volumetric expansion was generated, the tensile stress in the heated region, which affected the magnitude of the inherent strain in the cooling phase, was released. The strain as direct boundary (SDB) method, based on inherent strain, was introduced to overcome the disadvantages of the equivalent load method [6; 7; 8; 9]. This method applies the inherent strain as the thermal expansion coefficient of a shell element, and the virtual temperature gradient, which can simulate the residual strain, is applied as the boundary condition, so equivalent thermal strain is naturally generated. The inherent strain region is bound by the transformation temperatures. The temperature is the lower transformation temperature, where austenite begins to form during heating of the steel. This paper discusses both inherent strain and thermal strain-based approaches to predict the deformation of the shell forming by flame bending (or line heating). Taking into account the equivalent loads assumed from both inherent strain and thermal strain, deformation is predicted by the finite element analysis using shell elements. In addition, an elastic plastic finite element model based on the thermal strain is suggested by [PERSON] [8]. In the flame bending process, multiple line heating is common. This suggested method made it possible to reflect the effects of the previous heating lines. The thermal strain can be given as a boundary condition by appropriately determining the heat-affected zone. In general, the sectional shape of the HAZ can be measured through welding experiments to determine the heat-affected zone [10]. [PERSON] et al. [6] suggested that the HAZ can be determined from the phase transformation temperature.