# Analysis of the Failure Area of the Slope Using the Slip Line Method
## 4 Conclusions  
This study conducted failure behavior analyses of single ground and heterogeneous ground slopes using the slip line method. The analysis results through the slip line method and the strength reduction method of the finite element analysis were compared, and the following conclusions were drawn.  
1. The slip line method showed a clear virtual failure surface regardless of the ground strength and slope. In addition, it is possible to calculate the internal stress of the coordinates according to the coordinates (x, y) expressed in the Cartesian coordinate system.
2. In the case of finite element limit analysis, the plastic area analyzed by the strength reduction method was similar to the analysis result of the slip line method. However, it should be noted that the strength reduction method reduces the strength of the entire ground properties. Therefore, the plastic area was over-analyzed because the analysis was performed using the strength reduction method in the absence of stress singularities such as load.
3. As a result of analyzing the failure area of a single ground, the slip line method increased the scale of the failure surface as the slope of the soft ground (alluvium, sedimentary layer) increased. In the case of finite element limit analysis, it was confirmed that there was no significant difference in the plastic region depending on the properties of the ground and the slope angle.
4. As a result of the analysis of the failure area of the heterogeneous ground, the slip line method showed that the shape of the failure surface changed according to the change of the ground characteristic value, and the failure area and the virtual failure surface were clearly expressed regardless of the ground condition and slope angle. In addition, in the case of heterogeneous ground, the upper ground was regarded as a surcharge load and applied to the lower ground for analysis, showing a difference from the analysis results of a single ground.  
In the case of finite element limit analysis, it can be seen that the failure surface is discontinuously connected to the ground boundary layer. In addition, when the difference in soil properties between the upper ground and the lower ground was the greatest, the failure area was limited to the upper ground. This is because the upper ground with the lowest ground properties failed before the lower ground because the strength reduction method was applied without external load. It should be noted that the plastic area was over-analyzed by considering the shape of the slope as a stress singular point, because the analysis was performed using the strength reduction method in the absence of stress singularity such as load.
6. The slip line method analyzes the plastic area based on boundary conditions. Since the Riemann problem applied to the slope analysis sets the slip line as the boundary condition, the boundary condition can change. Because of these features, there is a risk that the plastic area and failure surface may be analyzed to be larger than the shape of the slope. Therefore, it is thought that a combination of boundary conditions according to the structure or geotechnical problem is necessary.
7. The slip line method can determine the virtual failure surface, failure area (active area, transition area, passive area), and internal stress if only the basic boundary conditions and ground characteristics are given. In addition, in this paper, it was shown that analysis of heterogeneous ground is possible. Through these features, the applicability and development potential of the slip line method to various geotechnical problems or structures were confirmed.  
Investigation, [PERSON] ([PERSON]); Writing--original draft, [PERSON] ([PERSON]); Writing--review & editing, [PERSON] All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.