## 3 Calculation of Equivalent Stress  
Based on the linear Miner's theory, the calculation formula of equivalent stress can be derived as follows:  
\[D=\sum\frac{n_{i}}{N_{fi}}=\frac{N_{T}}{N_{\text{eq}}} \tag{7}\]  
\[N_{\text{eq}}=\frac{C}{\sigma_{\text{eq}}^{m}} \tag{8}\]  
where \(n_{i}\) is the number of graded stress cycles; \(N_{T}\) is the number of cycles required; \(N_{\text{eq}}\) is the equivalent number of cycles; \(C\) is the material fatigue parameters; \(\sigma_{\text{eq}}\) is the equivalent stress; \(m\) is the reverse slope of the S-N curve. The calculation of equivalent stress can be deduced as follows:  
\[\sigma_{\text{eq}}=\left[\frac{\sum(n_{i}\sigma_{i}^{m})}{N_{T}}\right]^{ \frac{1}{m}} \tag{9}\]  
where \(\sigma_{i}\) is various stress levels as the range of stress variation. Miner's cumulative damage is independent of the loading sequence, which is similar to the calculation of equivalent stress according to Equation (9). However, many experiments have shown that the damage is related to the loading sequence. When the loading sequence of first a low load and then a high load is adopted, the low load has a certain training effect, and the fatigue life can be increased. When the sequence of first a high load and then a low load is adopted, the high load has the damaging effect, the crack will form in advance, and the low load in the later period will make the crack propagate and shorten the fatigue life. Therefore, in the fatigue design, the loading sequence effects should be considered, and the design should be carried out according to the nonlinear cumulative damage model. The M-H model can well consider the loading sequence effects. Combined with Equation (7) and the cumulative damage model shown in Equation (6), the calculation formula based on nonlinear cumulative damage can be achieved after loading classification as follows [18]:\[D=\sum D_{i}=\left\{\left[\left(\frac{n_{1}}{N_{f1}}\right)^{(\frac{N_{f1}}{N_{f2}} )^{0.4}}+\frac{n_{2}}{N_{f2}}\right]^{(\frac{N_{f2}}{N_{f3}})^{0.4}}+\cdots+ \frac{n_{i-1}}{N_{fi}}\right\}^{(\frac{N_{f(i-1)}}{N_{fi}})^{0.4}}+\frac{n_{i} }{N_{fi}}=\frac{N_{T}}{N_{\rm eq}} \tag{10}\]  
Correspondingly, the calculation formula for the equivalent stress \(N_{i}\sigma_{i}^{m}=C\) at all levels can be expressed through the following derivation of Equation (8):  
\[\left(\sigma_{\rm eq}\right)^{m}=\frac{C}{N_{T}}\left\{\left[\left(\frac{n_{1 }\sigma_{1}^{m}}{C}\right)^{(\frac{N_{f1}}{N_{f2}})^{0.4}}+\frac{n_{2}\sigma_ {2}^{m}}{C}\right]^{(\frac{N_{f2}}{N_{f3}})^{0.4}}+\cdots+\frac{n_{i-1}\sigma _{i-1}^{m}}{C}\right\}^{(\frac{N_{f(i-1)}}{N_{fi}})^{0.4}}+\frac{n_{i}\sigma_ {i}^{m}}{C}\right. \tag{11}\]  
Due to \(N_{fi}=C/\sigma_{i}^{m}\), Equation (11) can be expressed as  
\[\left(\sigma_{\rm eq}\right)^{m}=\frac{C}{N_{T}}\left\{\left[\left(\frac{n_{ 1}\sigma_{1}^{m}}{C}\right)^{(\frac{\sigma_{1}}{\sigma_{1}})^{0.4}}+\frac{n_{ 2}\sigma_{2}^{m}}{C}\right]^{(\frac{\sigma_{1}}{\sigma_{2}})^{0.4}}+\cdots+ \frac{n_{i-1}\sigma_{i-1}^{m}}{C}\right\}^{(\frac{N_{f(i-1)}}{N_{fi}})^{0.4}}+ \frac{n_{i}\sigma_{i}^{m}}{C}\right. \tag{12}\]  
The final equivalent stress calculation formula can be obtained after sorting out Equation (12).