## 3 Derivation of the New Modified Fenner Formula from Joint Strength  
The influence parameters are shown in Figure 1. The surrounding rock of the loess adopts peak strength, which is expressed by the shear strength index.  
Stress and Plastic Radius of Plastic Zone in Surrounding Rock of the Tunnel Based on Joint Strength 3.1.1.  
Formula of Stress in Plastic Zone of Surrounding Rock of the Tunnel Based on Joint Strength  
The surrounding rock of the plastic zone (\(a\leq r\leq R\)) is in the state of ultimate equilibrium. If the compressive stress is positive, then the major principal stress \(\sigma_{1}\) is tangential stress \(\sigma_{\rm{\Theta p}}\) and the minor principal stress \(\sigma_{3}\) is the radial stress \(\sigma_{\rm{rp}}\) in the plastic zone. Introducing them into Formula (12) yields  
\[\frac{\sigma_{\rm{\Theta p}}-\sigma_{\rm{rp}}}{2}=\sqrt{\left[\left(\frac{\sigma _{\rm{\Theta p}}+\sigma_{\rm{rp}}}{2}\right)\sin\varphi+c\cos\varphi\right]^{2}- \beta^{2}} \tag{13}\]  
where \(\sigma_{\rm{rp}}\) is radial stress in the plastic zone and \(\sigma_{\rm{\Theta p}}\) is tangential stress in the plastic zone.  
To simplify the calculation, we let  
\[\eta=\left(\frac{\sigma_{\rm{\Theta p}}+\sigma_{\rm{rp}}}{2}\right)\sin\varphi+ c\cos\varphi \tag{14}\]  
Introducing Formula (14) into Formula (13) yields  
\[\frac{\sigma_{\rm{\Theta p}}-\sigma_{\rm{rp}}}{2}=\sqrt{\eta^{2}-\beta^{2}} \tag{15}\]  
With simultaneous solution of Formulas (14) and (15), the formulas of \(\sigma_{\rm{rp}}\) and \(\sigma_{\rm{\Theta p}}\) based on joint strength can be obtained with \(\eta\) as a parameter.  
\[\left.\begin{array}{c}\sigma_{\rm{rp}}=\frac{\eta-c\cos\varphi}{\sin\varphi} -\sqrt{\eta^{2}-\beta^{2}}\\ \sigma_{\rm{\Theta p}}=\frac{\eta-c\cos\varphi}{\sin\varphi}+\sqrt{\eta^{2}- \beta^{2}}\end{array}\right\} \tag{16}\]  
The derivative can be obtained by Formula (16).  
\[\frac{{\rm{d}}\sigma_{\rm{rp}}}{{\rm{d}}r}=\left(\frac{1}{\sin\varphi}-\frac{ \eta}{\sqrt{\eta^{2}-\beta^{2}}}\right)\frac{{\rm{d}}\eta}{{\rm{d}}r} \tag{17}\]  
Introducing Formulas (16) and (17) into Formula (1) yields  
\[\left(\frac{1}{\sin\varphi}-\frac{\eta}{\sqrt{\eta^{2}-\beta^{2}}}\right) \frac{{\rm{d}}\eta}{{\rm{d}}r}-\frac{2\sqrt{\eta^{2}-\beta^{2}}}{r}=0 \tag{18}\]  
After separating the variables in Formula (18), Formula (19) can be obtained by the integral.  
\[\ln\frac{\left(\eta+\sqrt{\eta^{2}-\beta^{2}}\right)^{\frac{1}{\sin\varphi}} }{\sqrt{\eta^{2}-\beta^{2}}}-\ln r^{2}=C \tag{19}\]  
where \(C\) is constant of integration.  
When \(r\) equals \(a\), \(\sigma_{\rm{rp}}=\sigma_{\rm{a}}\), \(\eta=\eta_{0}\). Introducing them into Formulas (16) and (19) yields  
\[\eta_{0}=\frac{c+\sigma_{\rm{a}}\tan\!\varphi}{\cos\varphi}+\sqrt{\left(c+ \sigma_{\rm{a}}\tan\varphi\right)^{2}-\beta^{2}}\tan\varphi \tag{20}\]  
\[C=\ln\frac{\left(\eta_{0}+\sqrt{\eta_{0}^{2}-\beta^{2}}\right)^{\frac{1}{\sin \varphi}}}{\sqrt{\eta_{0}^{2}-\beta^{2}}}-\ln a^{2} \tag{21}\]  
By introducing Formula (21) into Formula (19), the equation between \(\eta\) and \(r\) can be obtained.  
\[\left(\frac{\eta+\sqrt{\eta^{2}-\beta^{2}}}{\eta_{0}+\sqrt{\eta_{0}^{2}-\beta ^{2}}}\right)^{\frac{1}{\sin\varphi}}\sqrt{\frac{\eta_{0}^{2}-\beta^{2}}{\sqrt {\eta^{2}-\beta^{2}}}}=\frac{r^{2}}{a^{2}} \tag{22}\]Formulas (16) and (22) show that the expressions of the tangential stress \(\sigma_{\rm op}\), the radial stress \(\sigma_{\rm rp}\) in the plastic zone and the vector diameter \(r\) of the calculated point regard \(\eta\) as a variable parameter based on the joint strength. By introducing the vector diameter \(r\) into Formula (22), the value of \(\eta\) the calculated point can be obtained. Introducing \(\eta\) into Formula (16) can determine the corresponding tangential stress \(\sigma_{\rm op}\) and radial stress \(\sigma_{\rm rp}\) in the plastic zone.