## 3 Multiscale Numerical Model
### Multiaxial Fatigue-Damage  
As an example, the load case of C2 is used to reveal the progress of multiaxial fatigue-life assessment in detail. To validate the critical plane where the normal stress reaches a maximum value, the distribution of the major principal stress in the joining region was first calculated and obtained under load case C2. Then, the position of the maximum value of the major principal stress was selected as the fatigue-potential point where the critical plane is located. Figure 6 presents the curve of the major principal stress at this point; the distribution of the major principal stress at peak time is shown in the figure. The critical-plane parameters, such as the spatial location and the angles, were assessed and evaluated by analyzing the stress composition along three axial directions. Afterwards, the curve of the shearing stress on the critical plane was calculated and obtained, which is shown in Figure 7.  
Figure 4: Calculation results of first four dynamical modes. Figure 5: Monitoring and numerical curves of bending movement. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Frequency (Hz)**} & \multirow{2}{*}{**Difference Rate (\%)**} \\ \cline{2-3}  & **Field Calculation** & & **Numerical Simulation** \\ \hline \(1\) & \(0.508\) & \(0.5057\) & \(0.455\) \\ \(2\) & \(0.512\) & \(0.5101\) & \(-0.391\) \\ \(3\) & \(0.5581\) & \(0.5602\) & \(-0.376\) \\ \(4\) & \(0.965\) & \(0.9570\) & \(0.834\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of dynamic frequencies under the first four modes. The rain-flow counting method [26] was applied to process the above two stress curves; and the stress-amplitude spectra--including various stress amplitudes corresponding to their cycles--of the major principal stress and corresponding shearing stress were obtained. By substituting the above two stress amplitudes into Equations (1)-(3), the corresponding fatigue life in the form of the stress cycling number was calculated and obtained. The strength grade of steel of hangers applied was Q420, for which the tensile strength can be as great as 520 MPa. Thus, the relative parameters were validated by referring to a similar steel material [22] in these three equations, which are presented in Table 4. Finally, the spectrum of fatigue damage was calculated by dividing the number of stress cycles by the corresponding fatigue life, and the total fatigue damage was calculated by summing them. The traffic flow of eight loading cases in 2015 was resolved by the monitoring system and shown in Table 5. By applying these loading cases to the fatigue calculation model, the  
Figure 6: Distribution of major principle stress. Figure 7: Curves of nominal and shearing stress on the critical plane; (**a**) normal stress; (**b**) shearing stress. candidate train is similar to that of passenger trains, but the design loading capacity is as high as 1200 kN per carriage. The action of freight trains was the same as for passenger trains, whereas the live load applied was 1200 kN per carriage. The ratio of freight trains in the total traffic flow was applied as the parameter to investigate the influences of freight trains on fatigue damage and life. Another simplification was made--the traffic flow under the above eight cases was assumed to be consistent; and the ratio of the freight trains in each case was assumed to be the same. In this section, four ratios, i.e., 0%, 10%, 20% and 30%, were assumed to calculate the fatigue damage under eight different cases, where the annual growth rate of traffic flow was assumed to be 10%. Figure 10 presents curves of accumulated fatigue damage under these three ratios. The fatigue life was obtained and validated by accumulated fatigue damage arriving at 1.0. Therefore, the fatigue lives under these four ratios were 138, 73, 70 and 68 years. The comparison shows that the joining of freight trains in the traffic flow accelerates the accumulation of fatigue damage, greatly decreasing the fatigue life.