## 3 Toe Load Analysis
### Mechanical Model of the GRS Segmental Wall
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline _Facing block_ & \\ Density \(\rho_{\text{b}}\) (kg/m\({}^{3}\)) & 2200 \\ _Backfill and foundation_ & \\ Density \(\rho\) (kg/m\({}^{3}\)) & 2000 \\ Friction angle \(\varphi\) (\({}^{\circ}\)) & 40 \\ Dilation angle \(\psi\) (\({}^{\circ}\)) & 10 \\ Elastic modulus \(E\) (MPa) & 40 \\ _Reinforcement_ & \\ Axial stiffness (kN/m) & 725 \\ Ultimate tensile strength \(T\) (kN/m) & 115 \\ Soil-reinforcement interface friction angle \(\varphi_{\text{sr}}\) (\({}^{\circ}\)) & 30 \\ _Facing block–leveling pad interface_ & \\ Interface friction angle \(\varphi_{\text{bp}}\) (\({}^{\circ}\)) & Variable \\ Interface shear stiffness \(K_{\text{sbp}}\) (MPa/m) & Variable \\ _Leveling pad–foundation soil interface_ & \\ Interface cohesion \(c_{\text{pf}}\) (kPa) & 1 \\ Interface friction angle \(\varphi_{\text{pf}}\)(\({}^{\circ}\)) & 40 \\ _Block–block interface_ & \\ Interface cohesion \(c_{\text{bb}}\) (kPa) & 46 \\ Interface friction angle \(\varphi_{\text{bb}}\) (\({}^{\circ}\)) & 57 \\ _Block–backfill interface_ & \\ Interface normal stress \(K_{\text{sbs}}\) (MPa/m) & 100 \\ Interface shear stiffness \(K_{\text{sbs}}\) (MPa/m) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter values for baseline model. Figure 8: Comparison of connection loads in SR-18 walls with and without toe embedment soil. becomes exposed as a result of scour and soil erosion at the wall toe, the original leveling pad-foundation soil interface can be regarded as the facing block-leveling pad interface of a wall with the height of an added leveling pad, and this increase in wall height (usually 0.2 m) can be ignored. Hence, only the restraint condition at the facing block-leveling pad interface needs to be investigated. In this paper, the term \"toe interface\" refers to the interface between the facing block and the leveling pad. The selected values for the toe interface friction angle are presented in Table 5. A change in the interface friction angle influenced the interface shear stiffness, as shown in Figure 4. The interface shear stiffness values corresponding to the selected interface friction angles, determined via Equations (1) and (2), are also shown in Table 5.  
Figure 9 shows the loads carried by the wall toe and reinforcement layers and their sums for the different toe interface friction angles. The figure also shows the active earth pressure on the wall facing, calculated using the Coulomb earth pressure equation:  
\[E_{a}=\frac{1}{2}K_{a}\gamma H^{2} \tag{4}\]  
where \(E_{a}\) = the active earth pressure, \(K_{a}\) = the active earth pressure coefficient, \(\gamma\) = the unit weight of the backfill, and \(H\) = the wall height. \(K_{a}\) is expressed as the following:  
\[K_{a}=\frac{\cos^{2}(\varphi+\omega)}{\cos^{3}\omega\cdot\left(1+\frac{\sin \varphi}{\cos\omega}\right)^{2}} \tag{5}\]  
where \(\varphi\) = the friction angle of the backfill and \(\omega\) = the facing batter. Equation (5) is a reductive Coulomb active earth pressure coefficient, which is recommended by FHWA standard [15] for the internal stability analysis of GRS walls. The plane strain friction angle was used for the soil models in the numerical simulations of this paper, whereas a triaxial friction angle should be introduced for \(K_{a}\)[28]. Thus, a triaxial value of 38\({}^{\circ}\) was used for \(\varphi\) in Equation (5), according to the relationship between the plane strain and triaxial friction angles proposed by [PERSON] and [PERSON] [42]. It can be seen that the sum of the reinforcement and toe loads (i.e., \(\Sigma T_{\mathrm{con}}+F_{\mathrm{t}}\)) is closer to the calculated active earth pressure \(E_{a}\) than the total reinforcement connection load \(\Sigma T_{\mathrm{con}}\), but it is still smaller than \(E_{a}\).