# Large Deformation Settlement Series Solution and Numerical Simulation for Underpass Project Track
## 5 Results and Discussion
### Comparison Analysis between Measured and Numerical Solutions of Roadbed Settlement
Letting the maximum settlement value of the roadbed remain unchanged, a very strongly correlated Peck roadbed settlement curve can be determined by changing the size of the settlement trough, which can be obtained by fitting the following:  
\[S(x)=-68.96\times e^{-\frac{x^{2}}{2\times 18^{2}}} \tag{28}\]  
\[S(x)=-73.52\times e^{-\frac{x^{2}}{2\times 14\,5^{2}}} \tag{29}\]  
Figure 11: Roadbed 7 settlement curve. For comparative analysis, the cosine function formula of the track surface settlement deformation caused by uneven settlement of the roadbed can be used to calculate the track surface settlement under the condition that the rail sleeper is not hanging as proposed by the literature [31]. The formula is as follows:  
\[w(x)=-\frac{\delta}{2}\bigg{(}1+\cos\frac{2\pi}{l}x\bigg{)} \tag{30}\]  
where \(\delta\) is the wave amplitude, mm; \(l\) is the wavelength, m.  
\[l_{2}=l_{1}+\frac{2(b^{\prime}+h^{\prime})}{\tan\varphi} \tag{31}\]  
where \(l_{1}\) is the wavelength of the full-wave cosine type roadbed uneven settlement, m; \(l_{2}\) is the wavelength of the full-wave cosine type track settlement slot, m; \(\varphi\) is the diffusion angle of ballast deformation; \(b^{\prime}\) is the thickness of ballast, m; \(h^{\prime}\) is the height of rail sleeper, m.  
\[\delta_{2}=\frac{\delta_{1}l_{1}}{l_{2}} \tag{32}\]  
where \(\delta_{1}\) is the wave amplitude of the full-wave cosine type roadbed uneven settlement, mm; \(\delta_{2}\) is the wave amplitude of the full-wave cosine type track settlement slot, mm. According to the results of the model test in the literature [31], the deformation diffusion angle of the roadbed settlement transferred to the rail surface after the action of ballast and rail is 25\({}^{\circ}\), i.e., \(\varphi=25^{\circ}\). According to the actual situation of this project, the ballast thickness is 0.35 m, the rail sleeper height is 0.28 m, the wavelength \(l_{1}\) of uneven settlement of the roadbed is 120 m, the wave amplitude of uneven settlement of the roadbed 4 and roadbed 7 are 68.96 mm and 73.52 mm respectively. Using the Formula (31), it can be found that the wavelength \(l_{2}\) of rail surface settlement trough is 122.7 m, using (32) the rail 4 and rail 7 can be found The wave amplitudes of the track surface settlement groove are 67.44 mm and 71.9 mm respectively, and the result obtained from literature [31] is called the cosine function solution. Therefore, the cosine function settlement curves of track 4 and track 7 are as follows:  
\[w(x)=-\frac{67.44}{2}\bigg{(}1+\cos\frac{2\pi}{122.7}x\bigg{)} \tag{33}\]  
\[w(x)=-\frac{71.9}{2}\bigg{(}1+\cos\frac{2\pi}{122.7}x\bigg{)} \tag{34}\]  
Note: The solution calculated by the theoretical model of the orbit in this paper is the series solution, the solution obtained by the model test in the literature [31] is the cosine function solution, the settlement of the orbit in the numerical simulation results is the numerical solution. The numerical solution, the series solution, and the cosine function solution are three different results of the solution of the orbit settlement curve. The measured value, numerical solution, the series solution, and the cosine function solution are plotted as curves as shown in Figures 12 and 13. From Figures 12 and 13, the following can be seen  
The measured value of the settlement is smaller than the numerical solution; the main reason is that the railroad daily maintenance needs to adjust the track to ensure the safety of traffic, each adjustment makes the track restore to the initial horizontal position, that is, by letting the beam settle to zero, means the site monitoring does not monitor all the displacement changes of the beam; the measured value of the beam cannot fully reflect the settlement displacement of the track.