## 3 Analysis and Reliability Verification of an Engineering Case
### Analysis of Theoretical Calculation Results
#### 3.2.2 Accuracy Analysis of the Simplified Method  
The horizontal additional stress caused by grouting includes \(\sigma_{x1-2}\) (Equation (7)) and \(\sigma_{x3}\) (Equation (8)). The calculation of the additional stress \(\sigma_{x3}\) requires a more complex and highly time-consuming code in MATLAB(r), which seriously affects the application and promotion of this method. Therefore, a simplified calculation of the compensated level of horizontal displacement of the tunnel without considering \(\sigma_{x3}\) was performed. The comparison was also made between the results obtained by the simplified calculation and the original method.  
Figure 7 compares the compensated level of horizontal displacement of the tunnel calculated by the original calculation method with that estimated by the simplified method. First, the distribution of the horizontal displacement of the tunnel estimated by the simplified method was similar to the one calculated by the original method; both represented a normal distribution and a basically similar influence range. Second, the whole distribution of the horizontal displacement of the tunnel estimated by the simplified method was below the one calculated by the original method, that is, the simplified method estimated a smaller horizontal displacement. The maximum difference between the horizontal displacement of the tunnel predicted by the two methods occurred at \(y=0\), that is, in the center of the tunnel. The simplified method estimated the maximum horizontal displacement in the center of the tunnel at 2.65 mm, which was 82.3% of that calculated by the original method. Third, comparing the distributions in the range of \(y=-10\) m to \(y=10\) m revealed that the horizontal displacement values estimated by the simplified method were all about 79.8-82.3% of those calculated by the original method. Therefore, in the main influence  
Figure 6: Comparison of the compensated level of horizontal displacement of the tunnel calculated by the proposed method with that by the measured data.  
zone, that is, in the range of \(y=-10\) m to \(y=10\) m, the horizontal displacement estimated by the simplified method was about 80% of that determined by the original method. On the whole, the value estimated by the simplified method was smaller than that predicted by the original method.  
In order to improve accuracy of the simplified method, its result was corrected by multiplying it with by a correction coefficient (\(q\)). A reference value of 1.25 was used for the correction coefficient based on the above study. As shown in Figure 7, the distribution of the horizontal displacement of the tunnel determined by the simplified method after correction was basically consistent with the distribution calculated by the original method. The maximum horizontal displacement in the center of the tunnel determined by the revised simplified method and the original method was 3.31 and 3.22 mm, respectively, leading to a difference of only 0.09 mm. Thus, the revised simplified method satisfies the calculation accuracy requirements and can balance calculation efficiency against result accuracy, so it is more suitable for promotion and application. In order to facilitate calculations, the revised simplified method is uniformly used for calculations and analysis in the following sections.