## 4 Modeling Assumptions and Analysis of Soil Nail Support Reinforcement Theory  
### Establishment of Theoretical Model I and Method Verification  
In soil mechanics and geotechnical foundations, plasticity theory has always played an important role. Prime examples include the ultimate bearing capacity of foundations, soil slope stability, and soil pressure theories that are established based on the limit analysis method of plasticity theory.  
First, five assumptions are made about theoretical model I: (1) the slope of the soil nailing support structure represents a plane strain problem; (2) the slope is assumed to be that of homogeneous cohesive soil, and the soil beneath the slope base is stable; (3) the potential slip plane is an inclined plane (Figure 2a,b), (4) the groundwater level is below  
Figure 5: Effect of reinforcement on strength. (**a**) Effect of reinforcement spacing. (**b**) Effect of reinforcement strength.  
Figure 6: Effect of confining pressure on strength.  
the bottom of the slope or there is no groundwater; and (5) there is no overloading on the ground. Based on these assumptions, the soil body under the action of self-weight is divided into two zones--the damage zone and the stress zone--as shown in Figure 7. The stress zone is like a rigid body sliding along the rupture surface with a certain velocity \(v_{1}\). At this point, both the direction of the sliding speed and the angle of the rupture surface remain constant (\(\varphi\)).  
It should be noted that the following theoretical analysis is derived using continuum mechanics equations (including equilibrium equations, constitutive equations, and kinematic equations). Among them, the equilibrium equations are applicable to static or quasi-static problems. In the kinematic equations, the deformation is assumed to be continuous and small, and the constitutive relationship of the material is assumed to be elastoplastic.  
According to the upper-limit theorem of plasticity and the conservation of energy, the power of the work done by the soil under its own weight is equal to the power consumed on the slip surface owing to the action of the cohesive force \(\overline{c}\) (including the increase after the reinforcement).  
\[W_{p^{\prime}}=\frac{1}{2}\gamma H^{2}(\cot\theta-\cot\beta).V_{1}\sin(\theta -\varphi),\ BC_{p^{\prime}}=\overline{c}\frac{H}{\sin\theta}V_{1}\cos\varphi \tag{8}\]  
When \(W_{p^{\prime}}=BC_{p^{\prime}}\), we obtain  
\[H=\frac{2\overline{c}}{\gamma}.\frac{\cos\varphi}{\sin\theta\sin(\theta- \varphi)(\cot\theta-\cot\beta)} \tag{9}\]  
To reach its full potential and to reach the limit state, the critical height of the soil nailing support structure H is considered a function of the inclination angle (\(\theta\)) of the slip-cracking surface, in order to determine the extreme value of H when \(\frac{\partial H}{\partial\theta}=0\):  
\[\sin(\beta-\theta)\sin(2\theta-\varphi)=\sin(\theta-\varphi)\sin\beta \tag{10}\]  
From Equation (10), \(\beta-\theta=\theta-\varphi\), and the inclination of its slip fracture surface is \(\theta=(\beta+\varphi)/2\).  
When the slip fracture surface form is assumed to be an inclined plane, it is not only consistent with the slip fracture surface angle (as shown in Figure 2b, when \(\beta=90^{\circ}\)), but also consistent with the slip fracture surface angle (as shown in Figure 2a). Furthermore, it is consistent with the rupture surface form deduced from [PERSON]'s formula ([PERSON] [37]; [PERSON] and [PERSON] [38]; [PERSON] et al. [39]; [PERSON] [40]). This shows that the calculation of the plane fracture surface by this method is fairly reasonable.