## 4 Solution of Fatigue Life of Pipeline under Combined Excitation under OTM Frame
### Estimation of S-N Curves Based on the Strength Limit and Yield Limit  
After acquiring the stress data from the pipeline, it becomes imperative to establish the S-N curve, which depicts the relationship between stress amplitude (or maximum stress) characterizing cyclic load stress levels and the cycle times characterizing fatigue life. Figure 15: Dynamic stress simulation results of pipeline structure under combined excitation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Meaning of Parameters** \\ \hline Time\_Step\_Ratio & 0.2 & key time step long step scale \\ Simulation\_Time & 0.0016 (s) & total time simulated \\ Dump & 20 & number of visualization result files \\ Search\_Range & 1.5 & neighborhood search range control parameters \\ Beta & 1.4 & shape control coefficient of shape function for \\  & & local maximum entropy \\ Material\_Type & ESCAAS\_MAT\_J2\_ & elastic-plastic material model \\  & POWERLAW & critical energy release rate \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation parameters related to dynamic stress in pipelines under OTM simulation of composite excitation. Consequently, by referring to the material's S-N curve, one can determine the fatigue life of the pipeline under compound excitation. The fatigue strength at \(10^{3}\) cycles is typically determined by employing the material's strength limit approximation, which is subsequently adjusted to incorporate factors such as reliability. The fatigue strength at \(10^{3}\) cycles, denoted as \(S_{100.R}\), depends on the specific loading conditions and a correction factor accounting for reliability:  
\[S_{1000,R}=S_{1000}*C_{R} \tag{11}\]  
where \(S_{1000}\) is the fatigue strength of the standard test article at \(10^{3}\) cycles under different loading conditions; \(C_{R}\) is the reliability correction factor. Tests show that \(S_{1000}\) is a strength limit approximately equal to \(75\%\) at axial loading \(\sigma_{b}\):  
\[S_{1000}=0.75\sigma_{b} \tag{12}\]  
The fatigue limit \(\sigma_{-1}\) is based on the yield limit of materials [39], and it can be determined using empirical formulas, such as  
\[ \begin{split}&\sigma_{-1}=a\sigma_{s}+bE+c\\ & a=0.139e^{-0.302K_{t}}R^{2}+0.435e^{-0.395K_{t}}R+0.836e^{-0.480 K_{t}}\\ & b=\left(0.473e^{-0.577K_{t}}R^{2}+1.479e^{-0.670K_{t}}R+2.842e^{ -0.755K_{t}}\right)*10^{-3}\\ &\text{c}=12.0K_{t}R+28.2K_{t}-82.8R-191\end{split}  \tag{13}\]  
where \(E\) is [PERSON]'s modulus in MPa; \(\sigma_{s}\) is the yield strength of the material, in MPa; \(K_{t}\) is the stress concentration factor; and R is the cyclic excitation stress ratio. The strength assessment method discussed in this paper is based on the results of structural dynamic simulations, utilizing an actual structure-specific simulation model. We incorporate the stress concentration factor into the simulation; hence, there is no need for redundant consideration of its influence here. Additionally, through the implementation of an average stress correction model, asymmetric cyclic excitation can be equivalently transformed into symmetric cyclic excitation with a corresponding amplitude. Consequently, under \(K_{t}=1\) and R = \(-1\) conditions, it becomes feasible to calculate the \(\sigma_{-1}\) value. The cycles corresponding to the fatigue limit can be conservatively estimated as those corresponding to the airframe's service life, i.e.,  
\[N_{f}=3600*AF*f_{max} \tag{14}\]  
In the formula, \(f_{\text{max}}\) is the maximum value of the frequency of the fluid pulsating excitation and the random vibration excitation, in Hz. The power spectrum of the engine random excitation spans a frequency range of 15 to 2000 Hz, while the fluid pulsation excitation occurs at a specific frequency of 630 Hz. Therefore, considering these factors, the value of \(f_{\text{max}}\) is determined as 2000 Hz. AF represents the aircraft body's operational lifespan, measured in hours. The lifespan of the aircraft body encompasses two primary aspects: durability and calendar life.