## 3 Example Analysis and Verification
When the burial depth of the tunnel is 25 m, the mean values of the surrounding rock stresses calculated by individual formulas follow the order: total soil column formula > [PERSON] formula > \(m=1.5>m=1.2>m=1.0>\) Terzaghi formula. As for the dispersion degree of data, through the comparison of the coefficients of variation in Figure 7, it can be concluded that the dispersion degree of the improved Bierbaumer formula is between that of the Xie Jiajie formula and the Terzaghi formula, and the dispersion degree increases with an increase in the burial depth. When the burial depth reaches a certain value, with an increases in the stress and surrounding rock level (the geological engineering conditions become worse), the calculated results of the improved Bierbaumer formula are larger than those for \(m=1.0\), which is due to the adoption of the nonlinear criterion correction result. For further discussion, the formulas' parameter sensitivities in different cases are analyzed with standardized regression coefficient beta as shown in Figure 8.  
In the case of the same burial depth, the calculation results of the surrounding rock stresses of class VI surrounding rocks are less than those of class V surrounding rocks by the improved Bierbaumer formula, the Xie Jiajie formula and the total soil column formula, which shows that the surrounding rock gravity is an important index affecting the surrounding rock stresses of shallow tunnels with the greatest impact. For the Terzaghi formula, mechanical parameters, such as cohesion and friction angle, may affect the calculation results more than the above formulas, and more attention should be paid to the self-bearing capacities of surrounding rocks in soft and shallow-buried strata. The Terzaghi formula is more suitable for shallow tunnels with better geological engineering conditions. The improved Bierbaumer formula (including the theory of principle), the Xie Jiajie formula and the total soil column formula are suitable for shallow tunnels with poor geological engineering conditions. This can be seen from the parameters' sensitivity in the improved Bierbaumer formula corresponding to different m values, i.e., the improved Bierbaumer formula improves the deficiency of the original Bierbaumer formula when the friction angle exceeds 30\({}^{\circ}\). Figure 7: Analysis of coefficient of variation. Figure 8: Standard regression coefficients using different formulas for different burial depths. (**a**) Standard regression coefficients at 5 m burial depth. (**b**) Standard regression coefficients at 15 m burial depth. (**c**) Standard regression coefficients at 25 m burial depth. According to the results of the confidence analysis in Table 5, the confidence interval widths of different formulas for class V and VI surrounding rocks are ranked in the following order: Terzaghi formula > \(m=1.5>m=1.2>m=1.0>\) total soil column formula > Xie Jiajie formula. The confidence interval width of the improved Bierbaumer formula is between the commonly used formulas and increases with an increase in the burial depth, which shows that its statistical law is the same as other formulas.