## 4 Analysis of Deformation Characteristics of Loess Tunnel
### Analysis of Scanning Results for Long-Term Initial Support Deformation
#### 4.2.2 Time Evolution Law of Initial Support Deformation in the Excavation Stage  
Three representative sections of YK280+212, YK280+216, and YK280+220 were selected for analysis, and the subsidence of the vault in the point cloud data was drawn as a scatter diagram. In order to study the time deformation law of loess tunnels, regression analysis was carried out on the time deformation characteristics of the vault subsidence in the three sections according to the point cloud data results. The selected regression function [3] is:  
\[\mu=A-Be^{-Ct} \tag{3}\]  
where \(\mu\) is the deformation (mm), \(t\) is the time (d), and \(A\), \(B\), and \(C\) are the regression coefficients.  
The results of the deformation regression analysis are shown in Table 3. According to Table 4, the regression curve of the vault subsidence amount with time and the curve of the corresponding deformation rate with time distribution are made, as shown in Figures  
From the deformation regression analysis results of the long-term scanning data in Table 4, it can be seen that the correlation coefficients of each regression function are all greater than 0.95, which indicates that the change of the vault deformation of the three selected sections with time conforms to the change law of the exponential function. The exponential function is a bounded function that can be used to predict the final deformation of the surrounding rock vault shown in Table 4. By comparing and analyzing the measured value of the subsidence of the vault in 30 days and the predicted value of the final subsidence, although the subsidence of each section of the vault within 30 days tends to be stable, there will be different degrees of increase in the future. In actual engineering, the slowly increasing deformation of the vault in the later stage must not be ignored. In addition, the application of the regression function also provides a certain reference for the design of the deformation reserve in the early stage of excavation.  
Figure 19: Deformation analysis diagram of the vault of the YK280+216 section of the tunnel.  
Figure 20: Deformation analysis diagram of the vault of the YK280+220 section of the tunnel.  
Figures 18-20 show that the tunnel's initial support deformation and deformation rate are closely related to the excavation sequence and time. After each step of excavation, the subsidence of the vault will show the characteristics of rapid deformation in the early stage and a gradual decrease in the deformation rate in the later stage. During the whole monitoring period, the overall trend of the tunnel deformation rate gradually decreased and finally stabilized.  
There are three stages of characteristics for the overall deformation of the section. The first stage is the rapid growth stage. In this stage, the deformation increases rapidly with the excavation sequence, and the deformation accounts for about 70-80% of the total deformation. The duration of deformation is related to the progress of the excavation and generally ends 3-5 days after the excavation of the core soil of the lower step. The deformation characteristics of this stage require that we do advance support measures and do quick excavation, quick support, and quick closure during construction to control the occurrence of large deformation as soon as possible. The second stage is the continuous growth stage. The deformation of this stage continues to grow slowly with time, and the deformation accounts for about 20-30% of the total deformation. In the case of fast excavation and fast support, the deformation growth continued to the time of the invert's excavation and support. At this stage, the excavation support and backfilling of the inverted arch should be carried out as soon as possible to reduce the duration of the successive growth stage. The third stage is the slow growth stage. The deformation is still growing slowly, and the secondary lining should be applied as soon as possible.