## 2 Methodology
### Strength Evaluation of Frame  
1. Evaluation of the static strength  
Due to the various complex stresses that the locomotive bogie frame is subjected to, strength theory can be adopted to determine whether the material is damaged. The materials used in this article are elastic-plastic materials, so the fourth strength theory is used for static strength verification and evaluation. According to the fourth strength theory, the conditions for the material structure of the frame to not fail under any stress state are as follows:  
\[\sigma_{e}=\sqrt{\frac{1}{2}\Big{[}\left(\sigma_{1}-\sigma_{2}\right)^{2}+ \left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2} \Big{]}}\leq[\sigma] \tag{1}\]  
In Equation (1), \(\sigma_{e}\) is the von Mises equivalent stress; \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the first, second, and third principal stresses, respectively; \([\sigma]\) is the allowable stress, where \([\sigma]=\sigma_{s}/n\); \(\sigma_{s}\) is the yield strength of the material; and \(n\) is the safety factor. The yield limit \(\sigma\) of the frame material is 355 MPa and the allowable safety factor \(S\) of the base metal of the bogie frame is 1.65, so the allowable stress on the frame is 215.15 MPa.  
2. Evaluation of the fatigue strength based on Goodman diagrams  
To assess the fatigue strength of frames, Goodman diagrams of fatigue strength curves recommended by the International Union of Railways (UIC) were adopted.  
The methods and criterions for evaluating the fatigue performance are described as follows:  
Step 1: Goodman diagrams of fatigue limit are drawn. A Goodman diagram is composed of an enclosed octagon ABCDEFGH (Figure 8) and coordinates for the inflection points are listed in Table 3. Therein, \(\sigma_{-1N}\) is the fatigue strength in symmetrical cycling under the given fatigue life N; \(\sigma\) denotes the ultimate tensile strength; \(\sigma\) is the tensile yield limit; and \(\sigma\) is the compressive yield limit.  
Figure 8: Schematic diagram for drawing a Goodman diagram of fatigue limit.  
Step 2: The maximum principal stress \(\sigma_{\text{max}}\), minimum principal stress \(\sigma_{\text{min}}\), mean stress \(\sigma\), and stress amplitude \(\sigma_{a}\) under operational loads are calculated.  
The minimum and maximum working stresses, \(\sigma_{\text{min}}\) and \(\sigma_{\text{max}}\), respectively, are calculated under 27 combinations of operational loads (Appendix A, Tables A3-A6). Figure 9 shows the method of determination of \(\sigma_{\text{max}}\) and \(\sigma_{\text{min}}\), which mainly involves the following steps: (a) the value and direction of principal stresses at each node on welds under different loading conditions are determined; (b) the direction of the maximum principal stress at each node on welds under all load conditions is set as the basic stress distribution direction, and the corresponding value is the calculated maximum principal stress \(\sigma_{\text{max}}\); and (c) principal stresses under other load conditions are transformed in the direction of the determined direction of the maximum principal stress and the minimum stress value is ascertained as the calculated minimum principal stress \(\sigma_{\text{min}}\).  
The mean stress \(\sigma_{m}\) and stress amplitude \(\sigma\) are calculated using Equation (2).  
\[\sigma_{m}=\frac{\sigma_{\text{min}}+\sigma_{\text{max}}}{2}\ \ \ \sigma_{a}=\frac{\sigma_{\text{max}}-\sigma_{\text{min}}}{2} \tag{2}\]  
Step 3: If the mean stress \(\sigma_{\text{m}}\), maximum principal stress \(\sigma_{\text{max}}\), and minimum principal stress \(\sigma_{\text{min}}\) of the bogie frame are all within the enclosed outline (ABCDEFGHA) in the Goodman diagram of fatigue limit of the materials, the fatigue strength of the frame meets the requirements.