# A New Method for Fatigue Evaluation of Titanium Alloy Welded Structures
## 2 The Principle of Notch Equivalent Stress and the Proposal of a New Method for Life Prediction  
### The Principle of Notch Equivalent Stress  
According to different actual needs, different joint structures are required. Welded joints can be divided into Butt joint, Corner joint, Edge joint, Lap joint, and Tee joint. The transition forms of different joint welding positions have the most obvious impact on the stress concentration degree of the structure. When the welded structure bears the load, defects and cracks of different degrees will usually appear at the stress concentration position, so focus on the research, as shown in Figure 1.  
According to the difference in the relative displacement between the upper and lower surfaces of the crack, the crack can be divided into three basic forms, which are mainly open cracks. In order to study the stress distribution near the open crack, [PERSON] and [PERSON] [20] obtained the stress distribution of the open crack at the weld toe perpendicular to the crack direction based on the linear elasticity theory, as shown in Equation (1).  
\[\sigma_{\text{y}}=\frac{x^{\lambda_{1}-1}}{\sqrt{2\pi}}K_{\text{I}}[(1+\lambda_ {1})\cos(1-\lambda_{1})\vartheta+\chi_{1}(1-\lambda_{1})\,\cos(1+\lambda_{1}) \vartheta]/[(1+\lambda_{1})+\chi_{1}(1-\lambda_{1})] \tag{1}\]  
where \(x\) is the distance from the weld; \(K_{\text{I}}=\sigma_{\text{n}}\cdot k_{1}\cdot t^{1-\lambda_{1}}\), \(k_{1}=f(\,h,t,L\,)\), \(\sigma_{\text{n}}\) is the loading stress level, and the magnitude is consistent with the nominal stress; \(t\) is the thickness of the loaded plate, \(L\) is the thickness of the non-loaded plate, and \(h\) is the size of the welding leg, as shown in Figure 2; \(\vartheta\) is the angle from the corner bisector to the stress analysis line, \(\lambda_{1}\) and \(\chi_{1}\) are the dimensionless parameters corresponding to different transition angles at the welding position, as shown in Table 1.  
Figure 1: Critical locations with stress singularities [19]. **(a)** Titanium alloy bogie frame structure. **(b)** Titanium alloy bogie side member structure.  
Figure 2: Structure edges analyzed.  
Simplify Equation (1) to get Equation (2) as follows:  
\[\sigma_{\text{y}}=\frac{\sigma_{\text{n}}}{\sqrt{2\pi}}\cdot\frac{1}{x^{p}}\cdot \left[C(\alpha,\theta)\cdot f(h,t,L)^{\frac{1}{p}}\cdot t\right]^{p} \tag{2}\]  
where \(C(\alpha,\theta)\) is only related to the shape and size of the structure, and \(C(\alpha,\theta)=\left[\frac{(1+\lambda_{1})\cos(1-\lambda_{1})\theta+\chi_{ \text{X}}(1-\lambda_{1})\cos(1+\lambda_{1})\theta}{(1+\lambda_{1})+\chi_{ \text{X}}(1-\lambda_{1})}\right]^{\frac{1}{p}}\), \(p\) is only related to the angle of the structural welding transition position, and its magnitude is shown in Figure 3.  
[PERSON] and [PERSON] [22] determined the exact solution of the stress distribution at the crack tip. When the transition angle of the welding position is close to 0, the corner is regarded as a crack. When the transition angle of the welding position is at other angles, according to the calculation formula of the stress distribution near the crack, the stress distribution near the weld can be obtained by analogy. [PERSON] et al. [18], based on the exact solution of the stress distribution at the crack tip deduced by [PERSON] and [PERSON] [22], after generalization, taking the stress concentration factor into account, obtained a general form suitable for other angles, proposed the concept of \"singular strength _as_\", and determined the notch equivalent stress function, as shown in Equation (3).  
\[\sigma_{\text{y}}=\frac{\sigma_{\text{n}}\cdot(as+x)}{\left[\left[2^{\frac{1}{ p}}\cdot as^{(\frac{1}{p}-1)}\cdot x\right]^{q}+x^{\frac{q}{p}}\right]^{\frac{p}{ p}}} \tag{3}\]  
where \(as=C(\alpha,\theta)\cdot f(h,t,L)^{\frac{1}{p}}\cdot t\cdot\pi^{-\frac{1}{2p}}\), \(q=3p-0.5\).