## 3 Results and Discussion
### Residual Stress Distribution Characteristics According to Sheet Thickness  
The magnitude of the residual stress distributed within the nugget varied depending on the nugget size. Additionally, the characteristics of the residual stress distribution within the nugget were examined in relation to the sheet thickness. It is believed that the stress distributed in the z-direction plays the most significant role in crack opening, and therefore, the stress distribution in this direction was specifically investigated. The sheet thicknesses selected for this study, which are commonly used in car bodies, were 1.6 mm, 1.4 mm, and 1.2 mm. Numerical analysis was performed using the same method for all three models. The results presented in Figure 12 illustrate the residual stress distribution  
Figure 11: Residual stress distribution in the z-direction (notch opening direction) for each nugget size: (**a**) \(3\sqrt{t}\), (**b**) \(3.5\sqrt{t}\), (**c**) \(4\sqrt{t}\), and (**d**) \(4.5\sqrt{t}\). in the z-direction (crack opening direction) from the nugget center to the nugget edge for nugget sizes of \(3\sqrt{t}\), \(3.5\sqrt{t}\), \(4\sqrt{t}\), and \(4.5\sqrt{t}\), according to the three sheet thicknesses. There was no significant difference in the maximum stress at the nugget center across the varying sheet thicknesses, but the residual stress distribution at the nugget edge exhibited some differences. Compressive residual stress was observed at the nugget edge starting from a nugget size of \(4\sqrt{t}\) for a sheet thickness of 1.6 mm and from a nugget size of \(4.5\sqrt{t}\) for 1.4 mm. For a sheet thickness of 1.2 mm, tensile stress was observed at the nugget edge for all nugget sizes. The tensile stress at the nugget edge decreased as the nugget diameter increased, with an almost uniform stress distribution throughout the nugget observed for the \(4.5\sqrt{t}\) nugget. As the sheet thickness decreased, the impact of spring back caused by the spacer became more pronounced compared to when the sheet thickness was higher. This is likely due to the reduction in compressive stress at the nugget edge, which is caused by the indentation of the electrode edge. When the nugget size is \(3\sqrt{t}\) with a sheet thickness of 1.6 mm, the joint strength is low due to the smaller nugget diameter. Consequently, despite the compressive stress at the nugget edge resulting from the electrode indentation, the tensile stress caused by the spring back effect becomes dominant, leading to a larger tensile stress distribution at the nugget edge compared to the nugget center. However, when the nugget size exceeds a certain diameter (e.g., \(4\sqrt{t}\) and \(4.5\sqrt{t}\)), the increased stiffness of the nugget makes the impact of the compressive stress caused by the electrode indentation more significant than that of the spring back caused by the spacer. This results in the distribution of compressive stress at the nugget edge. The examination of the stress distribution at the nugget, considering both sheet thickness and nugget diameter, revealed that tensile stress is consistently distributed at the nugget edge for the \(3\sqrt{t}\) nugget size, regardless of sheet thickness. If the part is subjected to a tensile load, it will be more susceptible to fracture. On the other hand, when the nugget size increases, a higher sheet thickness proves to be beneficial in ensuring fracture resistance and safety. Figure 12: Variation of opening stress(Z-direction) with sheet thickness. Welding Residual Stress Distribution Characteristics According to the Presence/Absence of the Spacer  
To investigate the impact of the tensile force induced by spring back on the nugget when spot welding is performed with a spacer inserted between the sheets, the residual stress distribution characteristics were examined based on the presence or absence of the spacer. The spacer used had a thickness of 1.5 mm, and a nugget size of \(3\sqrt{t}\) was selected for the analysis.