## 2 Theoretical Background  
### Multiaxial Stress Criteria  
The head car body is affected not only by the aerodynamic load, but also by the brake friction, vibration, and other alternating mechanical loads during operation. Thus, the head car body's stress distribution is a multiaxial stress state, which cannot be directly used for fatigue life assessment. Therefore, the stress component in a global coordinate system needs to be converted into a local coordinate system to obtain relevant fatigue parameters. That is, the stress components need to be evaluated at the weld position, which includes the axial force parallel to and perpendicular to the weld direction (\(\sigma_{\parallel}\) and \(\sigma_{\perp}\)) and the shear stress (\(\tau_{\parallel}\)), as shown in Figure 1:  
The stress components of the node in the global coordinate system include normal stress components \(\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)\) and shear stress components \(\left(\tau_{xy},\tau_{yz},\tau_{xz}\right)\). The transformation formula for the node stress components to a different coordinate system is as follows:  
\[\sigma_{\vec{r}^{\prime}k^{\prime}}=\alpha_{\vec{r}j}\alpha_{k^{\prime}k} \sigma_{jk} \tag{1}\]  
\[\sigma_{\vec{r}^{\prime}}=\alpha_{\vec{r}^{\prime}1}^{2}\sigma_{1}+\alpha_{ \vec{r}^{\prime}2}^{2}\sigma_{2}+\alpha_{\vec{r}^{\prime}3}^{2}\sigma_{3} \tag{2}\]  
where \(\sigma_{\vec{r}^{\prime}k^{\prime}}\) is the second-order stress tensor in the local coordinate system, \(\alpha_{\vec{r}j}\) and \(\alpha_{k^{\prime}k}\) are the direction cosine between the corresponding directions of the two coordinate systems. \(\sigma_{\vec{r}^{\prime}}\) is the principal stress in the local coordinate system. After the transformation, the normal stress and corresponding shear stress in the local coordinate system at the critical position of the weld were extracted. Then, the utilization degree \(\left(a_{\perp},\ a_{\parallel},\ a_{\tau}\right)\) in the local coordinate system were evaluated according to Equations (3)-(5). Fatigue calculations should be conducted for each stress component:  
\[a_{\perp}=|\frac{\sigma_{\perp}}{\left[\sigma_{\perp}\right]}|\leq 1 \tag{3}\]  
\[a_{\parallel}=|\frac{\sigma_{\parallel}}{\left[\sigma_{\parallel}\right]}|\leq 1 \tag{4}\]  
\[a_{\tau}=|\frac{\tau_{\parallel}}{\left[\tau_{\parallel}\right]}|\leq 1 \tag{5}\]  
In addition, the influence of multiaxial stress satisfies Equation (6):  
\[\alpha_{v}=\sqrt{\left(a_{\perp}\right)^{2}+\left(a_{\parallel}\right)^{2}+f_ {v}\cdot\left(a_{\perp}\right)\cdot\left(a_{\parallel}\right)+\left(a_{\tau} \right)^{2}}\leq 1 \tag{6}\]  
Figure 1: Stress components decomposition of welded joints in the local coordinate system. The fatigue strength of the welding site is independent of the base metal alloy and only depends on the stress ratios \(R_{\sigma}\) and \(R_{\tau}\). The stress ratio is equal to the minimum stress divided by the maximum stress. The notch condition curve index \(x\) is given in the appendix of DVS1608. In addition, the values of medium stress sensitivity for the medium stress effect and internal stress effect are used for welding joints of aluminium alloys in railway vehicle manufacturing. The allowable fatigue stress amplitude of axial stress \(\sigma_{\perp}\) and \(\sigma_{\parallel}\) are obtained through Equations (7)-(10) [22]. Interval 1: \(R_{\sigma}>1\)  
\[\left[\sigma_{(R)}\right]=54\cdot 1.04^{-x} \tag{7}\]  
Interval 2: \(R_{\sigma}\leq 0\)  
\[\left[\sigma_{(R)}\right]=46\cdot 1.04^{-x}\left(\frac{1}{1+M_{\sigma}\frac{1+R _{\sigma}}{1-R_{\sigma}}}\right) \tag{8}\]  
Interval 3: \(0<R_{\sigma}<0.5\)  
\[\sigma_{\vec{r}^{\prime}k^{\prime}}=\alpha_{\vec{r}j}\alpha_{k^{\prime}k} \sigma_{jk} \tag{1}\]0  
Interval 4: \(0.5\leq R_{\sigma}<1\)  
\[\sigma_{\vec{r}^{\prime}k^{\prime}}=\alpha_{\vec{r}j}\alpha_{k^{\prime}k} \sigma_{jk} \tag{1}\]1  
Note that there is no case for \(R_{\sigma}=1\), since it refers to a quasi-static stress state (the maximum stress equals the minimum stress).