## 4 Numerical Results and Discussion  
### Comparison of the Influence of Up-Arch and Settlement on Vehicle Operational Safety  
For the changes in track irregularities caused by pier settlement and deck variation, assuming the existence of a stochastic variable set \(\Psi\), the track irregularities can be determined by a mapping relationship \(g(\Psi)\). The \(g(\Psi)\) can be divided into two parts: one is the track irregularity caused by foundation deformation \(g_{1}(\Psi)\). There are many reasons for foundation deformation in ballastless track jointless track on bridge, such as pier settlement, creep camber, deformation caused by temperature, cracking of slab, failure of CA mortar layer, etc. Due to too many influencing factors, the mapping relationship between the foundation deformation and the track irregularity caused by it is not discussed in this paper. Using the method in Refs. [27, 28], we described \(g_{1}(\Psi)\) in the form of sine or cosine function. The other component of \(g(\Psi)\) is the stochastic irregularity \(g_{2}(\Psi)\) of the original track. The deformation amplitude of the latter is generally smaller than that of the former. For the track irregularity \(g_{1}(\Psi)\), the main determining factors are the length of the deformation section and the amplitude of the irregularity caused by the deformation, which correspond to the wavelength and amplitude of the sine or cosine function, respectively. The track irregularity \(g_{2}(\Psi)\), after the mapping relationship has been determined, still has stochasticity. Since there are stochastic differences in irregularity amplitude even for the same wavelength in the track stochastic irregularity, the effect of infrastructure distortion on track stochastic irregularity is summarized in the stochasticity of track stochastic irregularity. After superimposing \(g_{1}(\Psi)\) and \(g_{2}(\Psi)\) on each other, the track irregularities sample \(z(x)\) can be expressed as  
\[z(x)=\left\{ \begin{array}{ll}\frac{1}{2}a(1-\cos\omega x)+z(x)&(x_{b}\leq x \leq x_{e})\\ z(x)&(\text{else})\end{array} \right. \tag{9}\]  
where the spatial frequency \(\omega=2\pi/L\), \(L\) is the wavelength, \(a\) is the amplitude, \([x_{b},x_{e}]\) is the track irregularity superposition range. Due to the diverse forms of track distortions, the length and the amplitude of the track irregularity's influence varies. Studies have shown that in the medium-wave band, at speeds of 300-400 km/h, the most unfavorable wavelengths of irregularity are located at 10-20 m [33]. Here, the harmonic wavelength of 10 m was considered; assuming the foundation settlement was 12 mm, using the Chinese ballastless track irregularity spectrum in [56], and the stochastic irregularity wavelength of 2-120 m, the single sample of track irregularity was calculated as shown in Figure 8.  
Figure 7: CDF comparison of the car coaxial accelerations at different distances. By using the method of stochastic analysis of a nonlinear vehicle-track-bridge coupled system based on the generalized probability density evolution theory described in this paper, the CRH380A high-speed train was selected as the vehicle model; the probability density evolution process of the reduction rate of wheel load and the acceleration of the vehicle was calculated by considering the travel speed of 350 km/h and the irregularity in Figure 8 as the input excitation, as shown in Figure 9. The reliability of vehicle operating safety with the reduction rate of wheel load of 0.8 as the limit value, and the reliability of vehicle operation safety under such condition was 0.9685. The maximum possible vehicle acceleration and reduction rate of wheel load in different settlement amplitudes is shown in Figure 10. The safety limits of vehicle acceleration and reduction rate of wheel load were 0.13 g (1.27 m/s\({}^{2}\)) and 0.8, respectively. Obviously, the reduction rate of wheel load was more stringent as a standard, so it was uniformly selected as the operation safety and reliability index.