# Mechanical Response Law and Parameter Influence Analysis of Karst Tunnel Dynamic Excavation
## 2 Basic Theory  
The safety thickness of rock strata for outburst prevention between the tunnel and the cavern, which is the minimum distance between the tunnel and the cavern, can be analyzed by reasonably simplifying rock strata for outburst prevention into an elastic beam plate [28]. Based on the strength theory, the formula for calculating the minimum safety thickness of the anti-surge rock column is derived by structural mechanics. To facilitate the study, the following assumptions are made for karst tunnels:  
* Ignore the arching effect of rock strata for outburst prevention and simplify rock strata for outburst prevention to a horizontal yielding elastic beam;* The surrounding rock has good integrity, no obvious joints and cracks, and can be regarded as a homogeneous, continuous, isotropic elastic beam plate that can withstand tensile and compressive stresses;
* The pressure of karst filler, water pressure and self-weight of the rock beam are regarded as homogeneous loads acting directly on rock strata. Simplify the rock strata between the tunnel and the cave as a solid support beam. The loads on the rock strata are mainly self-weight load of rock strata (\(\gamma h\)), gravity loading of karst fill (\(p_{s}\)), water pressure (\(p_{w}\)), and ground stress (\(q\)). Figure 1 shows the simplified karst tunnel mechanics model. According to the theory of structural mechanics, when a solid supported beam is subjected to a uniform load, the maximum bending moment and the maximum shear force of the beam are calculated in Equations (1) and (2):  
\[M=\frac{1}{12}Fl^{2} \tag{1}\]  
\[Q=\frac{1}{2}Fl \tag{2}\]  
Then, the load is substituted in Equation (1). When the cavity is above the tunnel, the maximum bending moment and maximum shear force are calculated in Equations (3) and (4):  
\[M=\frac{1}{12}(p_{s}+p_{w}+q+\gamma h)BL^{2} \tag{3}\]  
\[Q=\frac{1}{2}(p_{s}+p_{w}+q+\gamma h)BL \tag{4}\]  
where \(M\) is the maximum bending moment on the rock beam, \(Q\) is the maximum shear force on the rock beam, \(p_{s}\) is the gravity loading of karst fill, \(p_{w}\) is the mean cavern water pressure, \(q\) is the equivalent vertical mean pressure converted from the surrounding rock pressure, \(\gamma\) is the heaviness of the rock, \(h\) is the safety thickness of the rock beam, \(B\) is the width of the rock beam and \(L\) is the length of the rock beam. When the cavity is below the tunnel, the maximum bending moment and maximum shear force are calculated in Equations (5) and (6):  
\[M=\frac{1}{12}(p_{s}+p_{w}-q-\gamma h)BL^{2} \tag{5}\]  
\[Q=\frac{1}{2}(p_{s}+p_{w}-q-\gamma h)BL \tag{6}\]  
The equivalent vertical mean pressure \(q\) is calculated in Equations (7) and (8):  
\[q=0.45\times 2^{S-1}\gamma\omega \tag{7}\]  
Figure 1: Simplified mechanical model of tunnel and cave: (**a**) Upper cave; (**b**) Lower cave. \[\omega=1+i(L-5) \tag{8}\]  
where \(S\) is the surrounding rock level; \(\omega\) is the coefficient of span, when \(L<5\), \(i=0.2\), when \(5\leq L<14\), \(i=0.1\). For rectangular section beams, their flexural strength is verified in Equation (9) and their shear strength is verified in Equation (10):  
\[\frac{6M}{Bh^{2}}\leq[\sigma] \tag{9}\]  
\[Q=\frac{1}{2}Fl \tag{2}\]0  
where \([\sigma]\) is the flexural strength of the rock and \([\tau]\) is the shear strength of the rock. Substituting Equations (3) and (4) into Equations (9) and (10), respectively, the safe distance between the upper cavity and the tunnel is calculated in Equations (11) and (12):  
\[Q=\frac{1}{2}Fl \tag{2}\]1  
\[Q=\frac{1}{2}Fl \tag{2}\]2  
Take the larger of the calculated values \(h_{1}\) and \(h_{2}\) of Equations (11) and (12) as the safe distance between the tunnel and the upper cavity.