# Effect of Plate Curvature on Heat Source Distribution in Induction Line Heating for Plate Forming
## 4 Results and Analysis  
### Effect of Initial Curvature  
After the experimental test verified the FEM calculation method, the method was applied to investigate the influence of plate curvature on the heat source distribution. This paper focuses on the outer plate of the ship hull. According to the investigation in the shipyard, the curvatures of the ship hull plate are mostly in the range of 0 m-1 to 1 m-1. The initial curvature \(K\left(\text{m}^{-1}\right)\) in the electromagnetic models was in the range of 0 to 1 and increased by increments of 0.25. The calculation schemes are shown in Table 2. The coil in Figure 2 was used, the power was 65 kW, and the moving velocity was 5 mm/s. The highest temperature was about 600 \({}^{\circ}\)C. A square plate, 1 m \(\times\) 1 m, was modeled in this calculation, and the thickness was set to 20 mm and 16 mm. Figure 8: Deflection variation in the Z direction: (**a**) view section \(S_{B}-S_{B}\); (**b**) view section \(S_{L}-S_{L}\). Figure 7: Deformation of the plate: (**a**) FEM calculation result; (**b**) experimentally measured result. The heat source distribution with different initial curvatures was calculated. The normalized results are shown in Figure 9. We chose a cross section vertical to the symmetry axis (y = 0) \(S_{1}\)-\(S_{1}\) to compare the distribution of heat generation (see Figure 10). The heat generation is concentrated in the area under the coil, and the shape of the area is similar to the coil. As a result of the influence of the core on the magnetic field distribution, the Joule heat on the side of the axis is always higher than at the center. It is clear that when the curvature increased, the Joule heat generation near the symmetry axis was reduced because the air gap increased. The heat generation on the side of the workpiece does not change. Under the conditions of an initial curvature \(K\) = 1, the Joule heat generation on the center of the workpiece is reduced by about 21%. We performed the five heat source models on the heat transfer and mechanical analysis, and the plastic strain distribution and displacement were calculated. Figure 11 shows the plastic strain on the bottom surface (without the inductor side) of the plate along the section \(S_{B}\)-\(S_{B}\), which is vertical to the heating line, and the thickness is 20 mm. The strain was divided into in-plane strain and out-of-plane strain according to Equations (12) and (13). The strain components in the X and Y directions are shown. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Initial Curvature** & \multirow{2}{*}{**Coil Width (m)**} & **Air Gap** & \multirow{2}{*}{**Power (kW)**} & **Velocity** \\  & & & & **Increment (mm)** & & **(mm/s)** \\ \hline case 1 & 0 & 0.22 & 0 & 65 & 5 \\ case 2 & 0.25 & 0.22 & 1.51 & 65 & 5 \\ case 3 & 0.5 & 0.22 & 3.03 & 65 & 5 \\ case 4 & 0.75 & 0.22 & 4.55 & 65 & 5 \\ case 5 & 1 & 0.22 & 6.07 & 65 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation schemes. Figure 9: Normalized Joule heat generation. With the curvature of the plate increasing, the out-of-plane strain in the heating area becomes less homogeneous--lower in the center and higher on the side. This occurs because the air gap between the coil and plate increased, and the heat generation was reduced. The relative change of the out-of-plane strain in the X direction is the greatest, and the relative change of the in-plane strain in the X direction is the smallest. In the Y direction, due to the heat generation decreasing in the center of the heat line, both the in-plane strain and the out-of-plane strain are reduced. The absolute change of strain in the X direction is smaller than the components in the Y direction. Table 3 shows the maximum relative variation of the plastic strain compared with the curvature \(K=0\), and it can be seen that the relative change of strain in the X direction is less than the \(K=0.5\).