# Finite Element Modeling of the Soil-Nailing Process in Nailed-Soil Slopes
## 3 FE Model of Nailed-Soil Slopes
### Constitutive Models of Soil  
The ability to accurately reflect the field conditions depends mainly on the ability of the constitutive model to represent the soil behavior. The nonlinear elastic (hyperbolic) model, Hardening-Soil (HS) model, is used to simulate the sandy soil. Stress-strain behavior of soil becomes nonlinear, particularly at failure condition. [PERSON] and [PERSON] [39] have presented a procedure to model the soil behavior by varying the soil modulus, in which the stress-strain curve is hyperbolic and the soil modulus is a function of confining stress and the shear stress experienced by the soil. Figure 5 shows the nonlinear stress-strain behavior and different soil modules. When a soil is subjected to zero deviator stress (i.e., \(\sigma_{1}-\sigma_{3}=0\)), its stress-strain behavior is modeled using the initial modulus (\(E_{i}\)) given by the following relation.  
\[E_{i}=K_{L}P_{a}\left[\frac{\sigma_{3}}{P_{a}}\right]^{n}\,, \tag{6}\]  
where \(E_{i}\) is the initial tangent modulus, \(K_{L}\) is the loading modulus numbers, \(P_{a}\) is atmospheric pressure, \(\sigma_{3}\) is the confining stress and n is the exponent for defining the influence of the confining pressure on the initial modulus.  
When soil is subjected to stress higher than it has previously experienced, the soil constitutive behavior is governed by the tangent modulus (\(E_{t}\)) given by the following equation.  
\[E_{t}=\left[1-\frac{R_{f}(\sigma_{1}-\sigma_{3})(1-sin\phi)}{2c(cos\phi)+2 \sigma_{3}sin\phi}\right]E_{i}\,, \tag{7}\]  
where \(\phi\) is the angle of internal friction, \(c\) is the cohesive strength of soil, \(R_{f}\) is the ratio between the asymptote to the hyperbolic curve and maximum deviator stresses = (\(\sigma_{1}-\sigma_{3})_{t}/(\sigma_{1}-\sigma_{3})_{\text{ult}}\) and \(\sigma_{1}\) and \(\sigma_{3}\) are the major and minor principal stress, respectively.  
When a soil is unloaded from a higher shear stress state, the nonlinear model uses the unloading-reloading modulus (\(E_{ur}\)) and can be calculated as follows, similar to the initial modulus (\(E_{i}\)).  
\[E_{ur}=K_{ur}P_{a}\left[\frac{\sigma_{3}}{P_{a}}\right]^{n}\,, \tag{8}\]  
In the Hardening-Soil model (HS-model), the soil stiffness is described by using three different input stiffnesses, the triaxial loading stiffness (\(E_{50}\)), the triaxial unloading stiffness (\(E_{ur}\)) and the Oedometer loading stiffness (\(E_{\text{o}at}\)). The average values for various soil types are \((E_{ur})^{\text{ref}}\approx 3(E_{50})^{\text{ref}}\) and \((E_{\text{o}at})^{\text{ref}}\approx(E_{50})^{\text{ref}}\). The HS-model accounts for stress-dependency of the stiffness module. All three input stiffnesses are related to a reference stress, usually taken as 100 kN/m\({}^{2}\). The hyperbolic part of the stress-strain curve can be defined by using a single secant modulus, as shown in Figure 5, as an additional input parameter, \(E_{50}\)  
Figure 5: Stress–strain curve for soil material.  
In contrast to \(E_{50}\), which determines the magnitude of both the elastic and plastic strains, \(E_{ur}\) determines the soil behavior under unloading and reloading [40].  
Both the average primary loading modulus \(E_{50}\) and the unloading modulus \(E_{ur}\) are stress level-dependent.  
\[E_{50} = (E_{50})^{ref}\left[\frac{\sigma_{3}+c(cot\phi)}{p^{ref}+c(cot\phi )}\right]^{m}, \tag{9}\] \[E_{ur} = (E_{ur})^{ref}\left[\frac{\sigma_{3}+c(cot\phi)}{p^{ref}+c(cot\phi )}\right]^{m}, \tag{10}\]  
where \((E_{50})^{ref}\) is the secant stiffness in a standard drained triaxial test, \((E_{ur})^{ref}\) is unloading-reloading stiffness, \(p_{ref}\) is reference stress for stiffness and m is the power for stress-level dependency of stiffness (m = 0.5 for sand [40]). Table 5 shows the material model parameters for the three types of sand considered in the present study.