# Stability of a Compacted Sand Slope Model Subject to
## 3 Numerical Study
### Analytical Prediction of the Failure Load of Tested Models M3, M4 and M5  
Adopting the kinematic approach of the yield design theory ([PERSON], 1990), the log-spiral failure surface corresponds to a surface of discontinuity of velocity that results from the rotation of the slope portion formed between the crest, the slope facing and the failure surface, and is subject to the rotation velocity \(\omega\) around the center \(\Omega\).  
Therefore, along the tangent to the log-spiral curve (AB), the discontinuity of velocity between the rotating slope portion and the rest of the ground slope is described in Equation (3):  
\[\underline{V}=\omega\,\mathbf{r}\,\mathbf{e}_{t}, \tag{3}\]  
where \(\mathbf{e}_{t}\) denotes the tangent unit vector to the log-spiral curve.  
Equation (4) traduces the compatibility condition for the kinematic admissible constructed velocity field obtained by Equation (3):  
\[\mathrm{U}_{z0}=\mathrm{V}_{0}\cos\theta_{0}. \tag{4}\]  
\(\mathrm{U}_{z0}\) is the downward velocity displacement of the rigid plate ensuring the applied loading at the crest slope. From Equation (3), the velocity displacement corresponding to the initial radius \(\mathrm{r}_{0}\) is  
\[\mathrm{V}_{0}=\omega.\mathrm{r}_{0}. \tag{5}\]  
Then, according to the kinematic theorem of YDT, one obtains an upper bound solution of the failure load from the condition traducing that the work of external forces is bound by the maximum resisting work developed over the surface of discontinuity of velocity (AB). The considered velocity field (movement of rigid block) does not induce deformation within the sand slope for which the gravity is neglected; thus, it comes  
\[\mathrm{P}_{\mathrm{ult}}\leq\frac{\mathrm{C}.\cot g\phi}{2.\cos\theta_{0}}. \mathrm{r}_{0}\Big{[}\mathrm{e}^{2\beta.\mathrm{tg}\phi}-1\Big{]}. \tag{6}\]  
Equation (6) provides the analytical upper-bound solution of the exact failure load value denoted by \(\mathrm{P}_{\mathrm{ult}}\). In plane strain condition, the predicted value of this failure load from Equation (6) is assessed per lineal meter (kN/1m).  
Validation of the analytical upper-bound predictions of the failure load summarized in Table 6 (e.g., plane strain condition) needs to be multiplied by the geometrical ratio (1/4)  
Figure 25: Validation of the analytical log–spiral curve by the observed failure surface for model M5.  
between the width of the rigid plate B = 0.25 m (Figure 1) and the lineal meter. As such, one can compare the experimental and numerical values of the failure loads displayed in Tables 2 and 4 and the predicted values in Table 7.  
Table 7 summarizes the analytical predicted failure loads for models M3, M4 and M5, and the experimental and numerical values corresponding to the Mohr-Coulomb models since it complies with Mohr-Coulomb strength criterion adopted in the YDT framework.  
Comparison of the obtained results for the M3 and M5 models, it appears logical that the upper-bound value of failure load is higher than the experimental and numerical results. However, the analytical prediction of failure load is closer to the numerical prediction. Meanwhile, for model M3, there is a significant difference (approximately 100%) in the analytical prediction compared to the experimental value and numerical prediction. This result may justify the fact that the considered log-spiral curve does not optimize, at best, the upper-bound solution of the failure load. Regarding model M4, the relative differences between the predicted values (numerical and analytical) and the recorded experimental values are not significant; thus, they provide a good estimate of the exact failure load. Hence, one can deduce that the numerical model and the analytical plane strain model are comparable.