## 5 Discussion  
In this study, it is verified that considering the fractured area and its progressive modeling at the dam-foundation interface is crucial in the sliding safety evaluation of a concrete gravity dam. In the numerical analysis study, the conventional formulation of the SSF, in which the fractured area is not represented at all, shows extremely overestimated sliding safety judgements when a dam is subjected to strong earthquakes. This SSF formulation results in about five times safer side values than those evaluated by the newly suggested SSF formulation with initial and progressive fractures, which is based on a more realistic fracture-progress model. The so-called improved SSF formulation, with the fixed fractured area and simplified computation of the vertical force at the fractured area, also shows more than three times safer side values than those by the suggested SSF formulation. Consequently, it can be observed that those existing SSF formulations could provide wrongfully safe judgements on sliding stability checks, which implies that those SSFs possibly exaggerate the sliding safety of dams.  
From the numerical simulation, it is shown that the proposed contact-sliding model can reproduce fracture propagation along the dam-foundation interface, which is expected during strong earthquake ground motions, including the two artificial earthquakes used in the present numerical study. In the simulation results, the progressive fracture propagation realistically starts and stops according to the intensity of ground motions. It was observed in the simulation that the concrete gravity dam, subjected to a strong earthquake, slides monotonically toward the downstream due to the reservoir hydrostatic pressure. It is also noted that the fracture rapidly propagates through the entire interface immediately after the dam starts to slide. From these observations, it can reasonably be deduced that it is essential to use an appropriate contact-sliding model for the fracture propagation in an accurate evaluation of the sliding safety for concrete gravity dams. Otherwise, the shear resistance capacity against sliding can become significantly overestimated, as aforementioned in this section. The newly suggested SSF with the proposed contact-sliding model for the dam-foundation interface shows more accurate and conservative sliding safety evaluation results compared to the two existing conventional SSF formulations, since the degradation of the overall shear strength along the dam-foundation interface can be realistically represented in the dam-foundation model and its SSF evaluation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Artificial**} & \multicolumn{5}{c}{**SSF**} \\ \cline{2-6}  & **Case 1** & **Case 2** & **Case 3** & **Case 2/Case 1 (\%)** & **Case 3/Case 1 (\%)** \\ \hline I & 2.070 & 1.548 & 0.413 & 74.78 & 19.95 \\ II & 2.227 & 1.658 & 0.442 & 74.50 & 19.85 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Minimum SSF values for test cases and their ratios.  
Although, in this study, the uplift pressure acting along the interface fracture is ignored to avoid complexity, its effects on the SSF evaluation can be significant. The uplift force reduces the vertical forces, which results in weakening the frictional shear resistance forces. Additionally, it can cause mode I fracture propagation with other upward forces acting on the dam body. Since the dynamic distribution of the uplift pressure and its effect on the mode I and/or mixed mode fracture of a concrete dam-foundation interface are quite different from the static one, they remain undergoing research topics, and their impacts on the SSF evaluation should be addressed in a future study.