# Stability of a Compacted Sand Slope Model Subject to
## 3 Numerical Study
### Validation of the Log-Spiral Curve of Observed Failure Surfaces  
Referring to [17; 18], in the framework of the yield design theory (commonly called limit analysis), the ultimate load can be associated to failure mechanisms inducing discontinuities of velocities on assumed failure surfaces described by the log-spiral curve defined by Equation (2):  
\[\text{r}=\text{r}_{0}\exp{(\delta\text{ t}\text{g}\varphi)}. \tag{2}\]  
Figure 24 shows the drawn log-spiral curve characterized by the rotation about the center \(\Omega\) from the initial radius \(\text{r}_{0}=\Omega\Lambda\) to the current radius \(\text{r}=\Omega\)Z located at angle  
Figure 23: Total displacement \(\text{u}_{\text{z}}\) and failure surface for the numerical model (M9). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **P\({}_{\text{ult}}\) (kN)** \\ **Experimental** \\ \end{tabular} } & \multicolumn{4}{c}{**Relative Differences (\%)**} \\ \cline{2-5}  & **P\({}_{\text{ult}}\) (kN) MC** & **P\({}_{\text{ult}}\) (kN) HSM** & \
\begin{tabular}{c} **Experimental—** \\ **MC** \\ \end{tabular} &
\begin{tabular}{c} **Experimental—** \\ **HSM** \\ \end{tabular} \\ \hline
10.0 (M1) & 9.35 & 4.0 & 10.9 & 61.9 \\ \hline
1.57 (M2) & 1.50 & 1.33 & 4.5 & 15.3 \\ \hline
0.50 (M3) & 0.51 & 0.41 & 2.0 & 18.0 \\ \hline
0.85 (M4) & 0.83 & 0.55 & 2.5 & 35.3 \\ \hline
0.86 (M5) & 0.94 & 0.61 & 8.1 & 29.1 \\ \hline
2.80 (M6) & 3.70 & 2.70 & 32.1 & 3.6 \\ \hline
1.87 (M7) & 1.87 & 1.66 & 0.0 & 15.5 \\ \hline
10.0 (M8) & 0.74 & 0.61 & 1.3 & 18.7 \\ \hline
1.57 (M9) & 0.90 & 0.64 & 5.3 & 34.7 \\ \hline
0.50 (M10) & 1.04 & 0.75 & 2.8 & 29.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Relative differences between the recorded failure loads and numerical predictions by the Mohr-Coulomb and hardening soil models. Figure 23: Relative differences between the recorded failure loads and numerical predictions by the Mohr-Coulomb and hardening soil models. 8. The main characteristic is the following: along this curve, at any point \(Z\), the formed angle between the tangent to the curve and the perpendicular to the current radius vector (\(\Omega Z\)) equals the friction angle \(\varphi\) of the material. As a result, the final radius of the drawn log-spiral curve results from the rotation, over angle \(\theta\), is \(\tau_{f}=\Omega B=\tau_{0}\) exp (\(\theta\) tg\(\varphi\)). Regarding the observed failure surfaces for models M3, M4 and M5, Table 6 summarizes the geometrical parameters of the best-fitted log-spiral curves. Equations of these curves consider the starting point of observed failure surfaces close to the rigid plate, and the final point is located on the slope side. The approximated location of rotation center \(\Omega\) fulfills the condition \(\Omega B>\Omega A\). Figure 25 compares the theoretical log-spiral curve obtained by Equation (2) and the observed failure surface for model M5 as reproduced from the drawn dashed curve in Figure 11. The good agreement between the two curves confirms that the assumed circular surface for frictional material does not prevail. Furthermore, the current attributed term in soil mechanics \"sliding surface\" is not meaningful since the discontinuity of velocity along the tangent of the log-spiral forms makes the theoretical angle equal to the frictional angle of the soil as clearly shown in Figure 25.