## 2 The Establishment of the Improved Bierbaumer Formula  
The nonlinear M-C criterion can be expressed in the plane \(\tau-\sigma_{n}\) as follows [29]:  
\[\tau=c_{o}(1+\frac{\sigma_{n}}{\sigma_{t}})^{\frac{1}{m}} \tag{1}\]  
where \(m\) is the nonlinear coefficient whose value is greater than or equal to 1; \(\tau\) is the shear stress at yield; \(\sigma_{n}\) is the normal stress; \(\sigma_{t}\) is the uniaxial tensile strength of the rock-soil interface; and \(c_{o}\) is the initial cohesion of the rock-soil interface. When \(m=1\), the above formula degenerates into the linear M-C criterion. \[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]  
Therefore, the internal friction angle of the rock-soil interface is:  
\[\tan\varphi=\frac{c}{\sigma_{t}} \tag{3}\]  
In this paper, the tangent method is used for equivalence, as shown in Figure 1.  
In Figure 1, the tangent equation of point A can be expressed as:  
\[\tau=c_{t}+\sigma_{n}\tan\varphi_{t} \tag{4}\]  
where \(\varphi_{t}\) is the angle between the tangent and the normal stress axis; \(c_{t}\) is the intercept on the shear stress axis. The slope of each point can be obtained by deriving Equation (1) as:  
\[\tan\varphi_{t}=\frac{\mathrm{d}\tau}{\mathrm{d}\sigma_{n}}=\frac{c_{o}}{m \sigma_{t}}\bigg{(}1+\frac{\sigma_{n}}{\sigma_{t}}\bigg{)}^{\frac{1-m}{m}} \tag{5}\]  
The simultaneous Equations (1) and (4) [29] can be obtained as follows:  
\[c_{t}=\frac{m-1}{m}c_{o}\bigg{(}\frac{m\sigma_{t}\tan\varphi_{t}}{c_{o}}\bigg{)} ^{\frac{1}{1-m}}+\sigma_{t}\tan\varphi_{t} \tag{6}\]  
According to Equations (3) and (6), it can be concluded that:  
\[c_{t}=\frac{m-1}{m}c_{o}\bigg{(}\frac{m\tan\varphi_{t}}{\tan\varphi}\bigg{)}^ {\frac{1}{1-m}}+\frac{c_{o}\tan\varphi_{t}}{\tan\varphi} \tag{7}\]  
Figure 1: Tangent equivalent nonlinear M-C criterion. In consideration of the change in \(\tan\varphi_{t}\) with \(\sigma_{n}\) in the calculation process, the initial tangent method with \(\sigma_{n}=0\) is used for equivalence as:  
\[\tan\varphi_{t}=\frac{c_{0}}{m\sigma_{t}}=\frac{\tan\varphi}{m} \tag{8}\]  
According to the conclusions of reference [30], when the \(m\) value is large, the error is large, so this paper studies when the \(m\) value is 1.0-1.5. According to the actual tunnel failure, [PERSON] revised the whole soil column theory and put forward the Bierbaumer calculation formula, as shown in \[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]0  
Considering the tangent equivalent internal friction angle, the formula is given as follows:  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]1  
where \(a\) in Equation (9) is half of the tunnel span. According to the loose body theory, the friction in the original formula is replaced by the tangent equivalent friction as:  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]2  
In Equation (10), \(\sigma_{z}\) is the Rankine active soil stress:  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]3  
Integrating Equation (10) along the direction of depth, the lateral resistance of the rock column side is:  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]4  
where  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]5  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]6 Theoretical calculation diagram of the soil column considering the friction and cohesion. \[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]7  
Therefore, according to the rock column theory, the stress of surrounding rock acting on the top of the tunnel is:  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]8  
where  
\[\tau=c+\sigma_{n}\tan\varphi \tag{2}\]9  
When the nonlinear coefficient \(m=1\), the improved theoretical formula for the rock column degenerates to the original one. In practical engineering, composite strata are common, which are simplified as horizontal strata, as shown in \[\tan\varphi=\frac{c}{\sigma_{t}} \tag{3}\]0  
In calculating the surrounding rock stresses of the composite horizontal strata, the Bierbaumer formula often makes a weighted average for the physical and mechanical parameters of the rock-soil interface, but there is a certain deviation associated with the calculation of the frictional resistance.