# Finite Element Modeling of the Soil-Nailing Process in Nailed-Soil Slopes
## 3 FE Model of Nailed-Soil Slopes  
### Types of Elements  
Finite element formulation-based PLAXIS 2D software is employed to model the laboratory set-up of the soil-nailing process of various stages, as explained in the previous section. Finite element idealization of the soil-nailing process considering the plane strain conditions is made to simulate the laboratory model, i.e., to model the sand soil, facing elements (panels of Perspex), steel nails, foundation and boundary conditions. Table 4 presents the soil slope geometry parameters, and Figure 2 shows the laboratory model with a dimensional sketch of the nailed-soil slope used in the finite element modeling. Triangular and line elements were employed to model the sand, Perspex material and steel reinforcement. The soil materials were modeled using fifteen-node triangle elements. They provided an interpolation of the fourth order for displacements, The numerical integration comprises the 12 gauss points. The fifteen-node triangle element, as shown in Figure 3a, is a very precise element that can be used to yield quality results. The nails in the finite element model are simulated using a 2-node elastic spring element with constant spring stiffness (normal stiffness). Slope facing and foundation can be modeled in PLAXIS software by beam elements. The beam elements have three degrees of freedom per node, two translation degrees of freedom (u\({}_{\mathrm{x}}\), u\({}_{\mathrm{y}}\)) and one rotational degree of freedom (rotation in the x-y plane, \(\varphi_{2}\)). When 15-node elements are employed for soil medium, then each beam element is defined by 5 nodes, as shown in Figure 3b. PLAXIS code allows for beam deflections due to shearing as well as bending. In addition, the element can change length when an axial force is applied. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Geometry** & **A (mm)** & **B (mm)** & **X (mm)** & **Z (mm)** & **Y (mm)** & **Sh = Sv (mm)** & **Slope Angle** (\(\mathbf{\Theta^{0}}\))** & **H = Ln (mm)** \\ \hline Value & 405 & 150 & 225 & 146 & 834 & 280 & 40 \\ \hline \hline \end{tabular}  
* A—width of soil beyond the footing; B—footing width; X—distance of footing from crest Y—width of soil beyond slope toe; Z—base width of soil below the slope; \(\mathbf{\theta}\)—angle of the slope; \(\mathbf{\delta}\)—nail inclination; Ln—length of rail; H—height of excavation; S—vertical spacing of rail; S\({}_{\mathrm{b}}\)—horizontal spacing of rail.  
\end{table} 
Table 4: Nailed-soil slope geometry parameters in the laboratory model [36]. Figure 3: Nodes and stress points in various elements. (**a**) 15-node triangle element. (**b**) 5-node beam element. (**c**) 10-node interface element. Figure 2: Laboratory model and schematic sketch of the nailed-soil slope [36]. Beams can become plastic if a prescribed maximum bending moment or maximum axial force is reached. The most important parameters for slope facing and foundation elements are the flexural rigidity (bending stiffness, EI) and the axial stiffness (EA), as given in Table 1. The shear deformation is also considered, and the equivalent plate thickness \(d_{eq}\) and shear stiffness of the plate is determined as given below. \[\text{Equivalent plate thickness},d_{eq}=\sqrt{12\frac{EI}{EA}}\, \tag{1}\]  
\[\text{Shear stiffness}=\frac{5 EA}{12(1+\
u)}=\frac{5E\big{(}d_{eq}\big{)}}{12(1 +\
u)}, \tag{2}\]  
Interface elements are used between soil and nail, facing and foundation to simulate the interaction between structures and the soil. The roughness of the interaction between the soil and the structure elements is simulated by selecting an appropriate value for the factor for strength reduction in the interface (R\({}_{\text{inter}}\)). The factor relates the interface strength (slope friction and adhesion) to the soil strength (friction angle and cohesion) and the coulomb criterion is used to distinguish between elastic behavior, where small displacements can occur within the interface, and plastic behavior, where permanent slip may occur.