## 4 Modeling Assumptions and Analysis of Soil Nail Support Reinforcement Theory
### Establishment of Theoretical Model II and Deduction of Simplified Formula  
Based on the validation of the rationality of the theoretical method and [PERSON]'s research on the limit analysis of a reinforced soil-retaining wall (Li [41]), five  
Figure 7: The rupture surface of the soil nailing support structure is assumed to be an inclined plane. assumptions are made: (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 surface is the folded line plane; (4) the groundwater level is below the bottom of the slope or there is no groundwater; and (5) there is no overloading on the ground. It should be noted that except for the fourth point, the rest of the points are described in the same manner as in the previous assumptions. It should be noted that the following theoretical analysis is still carried out using the equations of continuum mechanics (including equilibrium equations, constitutive equations, and geometric equations). Among them, the equilibrium equations are applicable to static or quasi-static problems. In the geometric equations, it is assumed that the deformation is continuous and small, and the material's constitutive relationship is assumed to be that of an elastoplastic material. On the basis of the above assumptions, the soil body under the action of self-weight is divided into a range of damage zones and different stress zones as shown in Figure 8. Where in Figure 8, \(\beta\) was the toe of the slope, L was the horizontal length of the failure surface at the slope crest, and H was the depth of the vertical failure surface (Figure 8a). Each stress zone slides along the respective rupture surfaces with different velocities like a rigid body, and the direction of the sliding velocity is at an angle \(\varphi\) to the rupture surface (\(\varphi\) is the angle of internal friction). At the same time, each velocity component consists of a closed vector triangle. According to the related literature ([PERSON] and [PERSON] [42]; [PERSON] [43]), owing to the existence of cohesion (\(c\)), the cohesive soil body in the active limit state exhibits tension cracks in a certain depth range at its top and at the depth of the cracks, as indicated by Equation (5). \[h=\frac{2\overline{c}}{\gamma\tan(45^{\circ}-\frac{\overline{\varphi}}{2})} \tag{11}\]  
According to the plastic limit theory mentioned by [PERSON] ([PERSON] [43]), the ultimate load is the smallest among all the loads corresponding to the allowable plastic deformation displacement rate field (Figure 8b). At this point, the soil nailing support structure will break down, and the work done by the soil's self-weight will be theoretically equal to the total energy expended in plastic deformation. From the geometric relations of the closed vector triangle in Figure 8c, we obtain  
\[V_{1}=V_{0}/\cos\varphi \tag{12}\]  
\[V_{2}=V_{0}/(\cos\varphi.\sin\theta) \tag{13}\]  
\[V_{12}=V_{0}.\tan\theta/\cos\varphi \tag{14}\]  
Here, \(V_{0},V_{1},V_{2}\), and \(V_{12}\) are the rates of plastic deformation displacement in the vector triangle, and \(\beta\), \(\varphi\), and \(\theta\) are the soil slope angle, internal friction angle, and rupture angle, respectively. Figure 8: Assumed rupture surface form for soil nail support. According to \"Elasticity and Plasticity\" by [PERSON] ([PERSON] et al. [44]), the M-C criterion is used as the yield condition in general geotechnical engineering. At this point, the yield surface equation is given as \(f=\tau-c-\sigma_{n}\) tan \(\varphi\).