## 3 Example Analysis and Verification
### Comparative Analysis of Statistical Characteristics of Surrounding Rock Stress in Composite Strata  
The validity and rationality of the improved Bierbaumer formula in composite strata are verified. However, considering the variability and discreteness of the geotechnical materials, the statistical characteristics of the surrounding rock stresses in composite strata are calculated by the improved Bierbaumer method, and the other formulas are used to analyze the sensitivities of the soil parameters. It is assumed that the burial depth of the tunnel is 10 m and the span is 10 m. The combination of class VI surrounding rock in the upper layer and class V surrounding rock in the lower layer is discussed. The thickness of class VI surrounding rock is set at 2.5 m, 5 m and 7.5 m, respectively, as shown in Figure 9. In the same way, the Monte Carlo method is used, and the improved Bierbaumer Formula with \(m=1.5\) is used to obtain the statistical characteristics of different calculation results of surrounding rock stresses, as shown in Figures 10-12.  
Figure 10: Statistical characteristics of surrounding rock stress in different layers. (**a**) Statistical characteristics of the surrounding rock stresses for 2.5 m thick class VI surrounding rock. (**b**) Statistical characteristics of the surrounding rock stresses for 5 m thick class VI surrounding rock. (**c**) Statistical characteristics of the surrounding rock stresses for 7.5 m thick class VI surrounding rock.  
Through variance analysis, the mean values and standard deviations of the surrounding rock stresses calculated by individual formulas are significantly different. In terms of the mean values of the surrounding rock stresses, total soil column formula \(>\) Xie Jiajie formula \(>\) Bierbaumer formula \(>\) improved Bierbaumer formula \(>\) Terzaghi formula. With an increase in the layer thickness of class VI surrounding rock, the mean values of the surrounding rock stresses using the improved Bierbaumer formula, [PERSON] formula, Xie Jiajie formula and the total soil column formula with \(m=1.5\) decrease, and those of the Terzaghi law are contrary, which is consistent with the above conclusion. The mean values of the improved [PERSON] formula's surrounding rock stresses are between those of the Terzaghi formula and the Bierbaumer formula, which shows that the formula is statistically reasonable.  
From Table 6, the bilateral 95% confidence intervals of individual formulas are studied. The confidence interval width of the improved [PERSON] formula decreases with increases in the class VI surrounding rock thickness, which is the same as those obtained by [PERSON], Terzaghi, Xie Jiajie and total soil column formulas. The widths of the confidence intervals follow the order: [PERSON] \(>\) improved Bierbaumer formula \(>\) [PERSON] formula \(>\) total soil column formula \(>\) Xie Jiajie formula. That is to say, the width of the confidence interval using the improved [PERSON] formula is between the values obtained by the [PERSON] formula and the Terzaghi formula, which also shows the rationality of the formula in statistical significance.  
Figure 11: Mean values of the surrounding rock stresses in different layers.  
Figure 12: Coefficients of variation using individual formulas in different layers.