## 4 Modeling Assumptions and Analysis of Soil Nail Support Reinforcement Theory
### Establishment of Theoretical Model II and Deduction of Simplified Formula
According to the classical flow law of plastic mechanics, it is known that when the plastic potential energy surface has the same shape as the yield surface, i.e., if \(g=f\), the flow law is associated with the yield condition using the following equation:  
\[d\varepsilon_{n}{}^{p}=d\lambda.\partial f/\partial\sigma_{n};\;d\gamma^{p}=d \lambda.\partial f/\partial\tau \tag{15}\]  
Based on the yield surface equation, which can be solved by combining it with Equation (15) above, we obtain  
\[\partial f/\partial\sigma_{n}=-\tan\varphi;\;\partial f/\partial\tau=1 \tag{16}\]  
At this point, it follows that \(d\varepsilon_{n}{}^{p}/d\gamma^{p}=-\tan\varphi\). The energy consumption rate per unit volume on the rupture surface of a soil nail-supported structure is given by \(w=\tau.\gamma^{p}+\sigma_{n}\varepsilon_{n}{}^{p}\). Substituting Equation (16) and the relation \(\tau=c+\sigma_{n}\tan\varphi\) into the equation \(w=\tau.\gamma^{p}+\sigma_{n}\varepsilon_{n}{}^{p}\), we obtain  
\[w=\tau.\gamma^{p}+\sigma_{n}\varepsilon_{n}{}^{p}=(c+\sigma_{n}\tan\varphi) \gamma^{p}+\sigma_{n}.\varepsilon_{n}{}^{p}=c\gamma^{p}-\sigma_{n}.\varepsilon _{n}{}^{p}+\sigma_{n}.\varepsilon_{n}{}^{p}=c\gamma^{p} \tag{17}\]  
From the above calculations, it is clear that the relationship between the energy dissipation rate and frictional consumption of the soil is not reflected when both positive and shear stresses are present at the rupture surface of the soil nail-supported structure. The primary reason for this is that the frictional depletion between the soils counteracts the shear dilatation depletion that exists between and within the soils. 1. Power \(W_{i}\), consumed by the internal forces along the rupture surface in a soil nailing support structure, while taking into account the additional cohesive forces due to the action of soil nails, is given by Figure 8.0 (18) At this point, \[ \begin{array}{ll}W_{i}&=\int dw_{i}=\int\overline{c}v^{\tau}dl\\ &=\overline{c}hv_{1}.\cos\varphi+\overline{c}(H-h)v_{2}\cos\varphi/\sin\theta +\overline{c}(H-h)(\cot\theta-\cot\beta)v_{12}\cos\varphi\\ &=\overline{c}hv_{0}+\overline{c}(H-h)v_{0}/\sin^{2}\theta+\overline{c}(H-h) (\cot\theta-\cot\beta)v_{0}\cot\theta\end{array} \] (19)
2. When work is done by the soil under self-weight at an assumed velocity field rate, power We is given as \[ \begin{array}{ll}W_{e}&=\frac{1}{2}\gamma(H-h)^{2}(\cot\theta-\cot\beta)v_ {2}.\sin(\theta-\varphi)\\ &+\frac{1}{2}\gamma h.v_{0}[(H-h)\cot\theta-(H-h)\cot\beta+(H-h)\cot\theta-H \cot\beta]\\ &=\frac{\gamma(H-h)^{3}(cot\theta-\cot\beta)v_{0}\sin(\theta-\varphi)}{2\cos \varphi\sin\theta}\\ &+\frac{1}{2}\gamma h.v_{0}[2(H-h)\cot\theta-(2H-h)\cot\beta]\end{array} \] (20) Through assuming \(\theta=45^{\circ}+\varphi/2\) and combining the above with \(h=2\overline{c}/\gamma\tan(45^{\circ}-\varphi/2)\), according to plasticity theory, the following can be derived when \(W_{i}=W_{e}\): \[ \begin{array}{ll}\frac{2c^{2}+\gamma H\cot\beta}{\gamma H\tan^{2}(45^{ \circ}-\frac{\varphi}{2})-2\pi\tan(45^{\circ}-\frac{\varphi}{2})}\\ =\frac{\overline{c}(3\sin\varphi-2)}{\cos\varphi}+\overline{c}.\cot\beta+ \frac{\overline{c}.\cot\beta}{1-\sin\varphi}-\frac{\overline{c}.\cot\beta}{ \tan(45^{\circ}-\frac{\varphi}{2})}+\gamma H\Big{[}\frac{1}{2(1+\sin\varphi) }-\frac{\cot\beta}{2\cos\varphi}\Big{]}\end{array} \] (21)From Equation (21), the critical height of the soil nailing support structure when damage occurs can be given by Equation (22):  
\[H=\frac{\overline{c}/\gamma}{\tan(45^{\circ}-\frac{\varphi}{2})}\frac{\left[3(1- \sin\varphi)-2\cot\beta\cos\varphi+\sqrt{kp}(1+\sin\varphi)\cot\beta\right]+ \sqrt{r^{2}-4 st}}{1-\cot\beta\sqrt{kp}} \tag{22}\]  
It can be derived from Equation (22) that the critical height \(H\) at which damage occurs to the soil nailing support structure is a function of the angle of slope \(\beta\) and the angle of internal friction \(\varphi\).