## 2 Materials and Methods
### Method for Examining Residual Stress Distribution Characteristics of Spot Weld Joints
#### 2.2.2 Construction of FEM Analysis System for Examining Residual Stress  
The experimental method has limitations in assessing residual stress within the nugget due to its small size. To overcome these challenges, a FEM analysis system was constructed to examine the residual stress characteristics inside the nugget numerically. To ensure the validity of the FEM analysis model, the residual stress characteristics obtained numerically were compared with those measured on the specimen surface. Simufact Welding ver. 2022.0.1, a commercial software program [25], was employed for the analysis. The finite  
Figure 4: Measurement positions for surface residual stress using each method: (**a**) X-ray method; (**b**) hole-drilling method.  
element model was numerically analyzed using the axisymmetric model as shown in Figure 5. Figure 5 illustrates the boundary conditions for the FE analysis. The displacements in the x-, y-, and z-directions were fully constrained at the connection point between the sheet and spacer, while the remaining parts were analyzed in a free state to minimize the influence of residual stress caused by constraints. In the analysis of the temperature field and stress field, the commercial software of Simufact Welding was used to perform the electrical-thermal stress coupling analysis. The electrical, thermal, and mechanical properties used the material property data of the high-strength steel plate. In addition, the material properties of the weld were divided into two types, martensite (base material) and austenite, and the effects of the phase transformation of austenite and martensite during cooling were considered. The sheet thickness was 1.6 mm, and the total specimen length was 100 mm. The nugget shape used in the FEM analysis was modeled to match the nugget size of the macro section of the actual spot weld joint, as shown in Figure 6a. Figure 6b illustrates the temperature distribution cross-section for the \(3\sqrt{t}\) nugget shape, enabling a comparison between the modeled nugget shape and the actual weld during the spot weld FE analysis. The minimum element size in the nugget was set to 0.1 mm \(\times\) 0.1 mm to accurately capture the residual stress distribution characteristics.  
The mechanical properties required for welding analysis, including temperature-dependent physical properties, were derived using JMatPro ver. 8.0 [26], a commercial software program that predicts high-temperature material properties based on chemical composition. Figure 7 presents the input parameter values used in the analysis. Figure 7a illustrates thermal properties, including thermal conductivity, specific heat capacity, and density as functions of temperature. Figure 7b depicts mechanical properties, such as the flow stress curve, Young's modulus, and thermal expansion coefficient relative to temperature. Figure 7c shows electromagnetic properties, including electrical resistivity and conductivity as functions of temperature. The validity of the residual stress analysis system developed in this study was confirmed through comparisons between the FEM analysis results and the experimentally measured residual stress values at corresponding positions on the specimen surface.  
Figure 5: Numerical analysis model for calculating welding residual stress in SPOT weld joint.  
Figure 6: Real macro section of SPOT weld and finite element model of the residual stress test specimen: (**a**) real macro section; (**b**) numerical macro section.