## 2 Fatigue Analysis Methods  
The relationship between load and fatigue life needs to be established in order to estimate the fatigue life of materials or structures. The relationship curve between the cyclic load and its corresponding number of cycles to failure obtained through the fatigue test is called the \(S\)-\(N\) curve, as shown in Figure 2, which is given as follows:  
\[NS^{m}=K, \tag{1}\]  
where \(N\) is the number of cycles to failure for stress amplitude \(S\); and \(m\) and \(K\) are material parameters to be determined. Then, the logarithms on both sides of Equation (1) are taken as follows, in order to obtain the \(S\)-\(N\) curve in logarithmic form:  
\[\lg N=a+b\cdot\lg S, \tag{2}\]  
where \(a\) and \(b\) are constants to be determined, which are taken from reference [27].  
The stresses used for fatigue analysis are classified as nominal stresses and hot spot stresses. The nominal stresses in a component can be derived by using classical theories. If the positions for fatigue analysis are located in the stress concentration area caused by the global shape of the structure, the stress concentration factors of the global geometric shape must be considered. Therefore, the local stresses of the positions for fatigue analysis are given as follows:  
\[\sigma_{local}=SCF\sigma_{nominal}, \tag{3}\]  
where \(SCF\) stands for stress concentration factor.  
The hot spot stresses are the geometric stresses generated in consideration of the details of components. Generally, local hot spot stresses can be calculated by using a detailed structure or component model that is established by using finite element software and choosing reasonable boundary conditions; hot spot stress is obtained through loading analysis for fatigue calculation.  
According to the assumption of linear cumulative damage, the amount of accumulated damage at a given level of stress is proportional to the number of cycles; this means that the fatigue damage of a material or structure is determined by a relation of the proportional damage (cycle ratio \(n_{i}/N_{i}\)) at various load levels (multi-level loading). The fatigue damage under variable-amplitude repeated loads could be predicted by the following equation [28]:  
\[D=\sum_{i=1}^{k}D_{i}=\sum_{i=1}^{k}\frac{n_{i}}{N_{i}}, \tag{4}\]  
Figure 2: \(S\)–\(N\) curve of the materials.  
according to Equation (1), where \(n_{i}\) is the number of cycles with amplitude \(S_{i}\), and \(N_{i}\) is the number of cycles to failure for stress amplitude \(S_{i}\). When \(D\) = 1, fatigue failure will take place.  
The S-\(N\) curve is obtained from fatigue tests when the mean value of stress cycle \(S_{m}\) = 0. The mean stress value also affects the cumulative fatigue damage. When the mean stress \(S_{m}\) < 0, the fatigue life increases; when \(S_{m}\) > 0, the fatigue life decreases. Given that the mean value \(S_{m}\) is typically non-zero, the following Goodman method is used to provide an equivalent stress amplitude with zero mean level:  
\[S_{a}=S_{-1}(1-\frac{S_{m}}{S_{u}}), \tag{5}\]  
where \(S_{a}\) is the stress amplitude at non-zero mean value, \(S_{-1}\) is the stress amplitude at the zero-mean value, \(S_{m}\) is the average stress, and \(S_{u}\) is the tensile strength of the material.  
The rainflow counting method [29] is widely used in the study of structural fatigue performance. The stress-time or load-time history is counted, one by one, and divided into independent cycles in order to calculate the fatigue life, or to compile the load spectrum of fatigue tests. The occurrence times of each stress amplitude and their corresponding average stresses under a certain stress-time history can be obtained by using this method. After all stress amplitudes and stress mean values are counted, the equivalent stress amplitudes are provided by the Goodman method. Then, the accumulated fatigue damage under a certain stress-time history can be obtained by using the material \(S\)-\(N\) curve and linear cumulative fatigue theory.