## 2 Project Background and Calculation Method
### Calculation Method
#### 2.2.1 Methods for Determining Rock Mass Parameters in Areas Affected by Dynamic and Static Loads  
The Hoek-Brown criterion is an empirical guideline for determining the mechanical parameters of a rock mass and includes the GSI (Geological Strength Index) and the rock disturbance coefficient, \(D\).  
The expression for the Hoek-Brown empirical criterion [37] is:  
\[\sigma_{1}=\sigma_{3}+\sigma_{ci}[m_{b}(\sigma_{3}/\sigma_{ci})+s]^{n} \tag{1}\]  
Figure 1: Spatial location map of the underpass tunnel project.  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses; \(\sigma_{ci}\) is the uniaxial compressive strength of the rock mass; and \(m_{b}\), \(s\), and \(a\) are the semi-empirical parameters of rock properties as a function of the Geological Strength Index (GSI) [38]:  
\[m_{b}=m_{i}\mathrm{e}^{\left[\mathrm{(GSI-100)}/\left(28-14D\right)\right]} \tag{2}\]  
\[s=\mathrm{e}^{\left[\mathrm{(GSI-100)}/\left(9-3D\right)\right]} \tag{3}\]  
\[a=1/2+(\mathrm{e}^{-\mathrm{GSI/15}}-\mathrm{e}^{-20/3})/6 \tag{4}\]  
\[\mathrm{GSI}=1.5\mathrm{]cond}_{89}+\mathrm{RQD}/2 \tag{5}\]  
In the formula, \(\mathrm{Jcond}_{89}\) is the state of the joints, RQD is the quality of the rock, GSI is the Geological Strength Index, and these three values are in the range of 5\(\sim\)100; \(m_{i}\), the rock material constant obtained from the triaxial compression test of intact rock, can also be determined according to the type of rock, the structure, by taking a value in the range of 5\(\sim\)40; \(D\) is the weakening factor for the consideration of the rock body by anthropogenic or natural factors of perturbation, taking a value in the range of 0\(\sim\)1.  
1. Determination of modulus of elasticity  
The modulus of elasticity is an important parameter describing the strength of a material and the modulus of elasticity, \(E_{m}\), (GPa) of a rock is calculated as:  
\[E_{m}=(1-0.5D)\cdot(\sigma_{ci}/100)^{0.5}\cdot 10^{\left[\mathrm{(GSI-10)}/40 \right]},\sigma_{ci}\leq 100\ \mathrm{MPa} \tag{6}\]  
\[E_{m}=(1-0.5D)\cdot 10^{\left[\mathrm{(GSI-10)}/40\right]},\sigma_{ci}>100\ \mathrm{MPa} \tag{7}\]
2. Determination of shear strength parameters  
The shear strength parameters are cohesion, \(c\), and friction angle, \(\varphi\)[37]. The expressions for the equivalent cohesion, \(c^{\prime}\), and equivalent friction angle, \(\varphi^{\prime}\), of the rock adopted using the Hoek-Brown criterion are, respectively:  
\[c^{\prime}=\frac{\sigma_{ci}[(1+2a)s+(1-a)\,m_{b}\sigma_{3n}^{\prime}](s+m_{b }\sigma_{3n}^{\prime})^{a-1}}{(1+a)(2+a)\sqrt{1+\left[6\,am_{b}\left(s+m_{b} \sigma_{3n}^{\prime}\right)^{a-1}\right]/\left[(1+a)(2+a)\right]}} \tag{8}\]  
\[\varphi^{\prime}=\arcsin[\frac{6 am_{b}(s+m_{b}\sigma_{3n}^{\prime})^{a-1}}{2 (1+a)(2+a)+6 am_{b}\left(s+m_{b}\sigma_{3n}^{\prime}\right)^{a-1}}] \tag{9}\]  
where \(\sigma_{3n}^{\prime}=\sigma_{3\mathrm{max}}^{\prime}/\sigma_{ci}\), and for deep tunnels \(\sigma_{3\mathrm{max}}^{\prime}\) can be determined using the following equation:  
\[\sigma_{3\mathrm{max}}^{\prime}/\sigma_{cm}^{\prime}=0.47(\sigma_{cm}^{\prime }/\gamma H)^{-0.94} \tag{10}\]  
where \(\gamma\) is the bulk weight of the rock mass; \(H\) is the burial depth of the tunnel, and when the horizontal stress is higher than the vertical stress, \(\gamma H\) in Equation (10) is replaced by the value of the horizontal stress; and \(\sigma_{cm}^{\prime}\) is the strength of the rock mass, which is calculated using the formula:  
\[\sigma_{cm}^{\prime}=\sigma_{ci}\frac{\left[m_{b}+4s-a\left(m_{b}-8s\right) \right]\left(0.25\,m_{b}+s\right)^{a-1}}{2(1+a)(2+a)} \tag{11}\]
3. Disturbance factor of rock mass for tunneling \(D\)  
The disturbance factor for the rock mass is usually taken empirically, as shown in Table 1.