# Analysis of the Failure Area of the Slope Using the Slip Line Method
## 3 Analysis of Failure Area
### Analysis Condition of Slope Failure Area  
For the analysis of the failure area of a heterogeneous ground slope using the slip line method, the ground consists of soft ground (alluvium, sedimentary) and hard ground (weathered rock, soft rock); the physical properties of the ground are shown in Table 1.  
Figure 6: Schematic diagram of limit analysis applied to plastic area analysis.  
Figure 7: Schematic diagram of plastic area analysis using the slip line method.  
In this paper, to emphasize the features of the slip line method that can analyze the failure area under any conditions, the conditions in which it is difficult for failure to occur were set as follows.  
The slope angle was very gentle (\(10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\)), and only self-weight was considered without external load. Failure begins with stress singularity. However, there is no stress singularity when eliminating external loads to analyze the failure area under conditions where it is difficult failure for to occur. Therefore, a virtual stress singular point was arbitrarily assumed at 4 m to the left of the starting point of the slope so that the failure area could be analyzed.  
In this paper, the load condition was not set; therefore, the failure was induced through the strength reduction method. In addition, the strength reduction method does not assume the location of the critical failure surface, and there is no need to divide the slope at the critical failure surface into intercepts. Since the strength reduction method shares these features with the slip line method, it is expected that the difference in the analysis of the failure area could be compared; therefore, it was used as a comparison target.  
As an analysis condition of the strength reduction method, a mesh was created with very fine triangular elements, and the boundary conditions were set as lower hinges (vertical and horizontal fixed); left and right rollers (horizontal fixed) as shown in Figure 8.  
In this study, for the analysis of the slip line method, a virtual slope was analyzed rather than an actual slope. Due to the limitations of the Riemann boundary problem that was applied to analyze the failure area of the slope, the analysis conditions were set as follows.  
The initial boundary condition consisted of flat ground and slopes, no external load (\(p=0\)) was assumed, the \(\theta\) value was set arbitrarily, and the convergence value of \(\theta\) was presented through a recurrence formula.  
Moreover, to analyze the failure area of the heterogeneous ground slope, the slope was divided into upper and lower ground, and the slope angle was expressed as \(\delta\) (\(10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\)). The upper ground of the slope was set at 1.5 m, and the lower ground varied based on the failure analysis. This is shown in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Ground Property** & **Alluvium** & **Sedimentary Soil** & **Weathered Rock** & **Soft Rock** \\ \hline Cohesion & 15 & 17.5 & 50 & 100 \\ (c, kPa) & 17 & 18.5 & 21 & 24 \\ Unit Weight & 17 & 18.5 & 21 & 24 \\ (\(\gamma\), \(\mathrm{N}\mathrm{/}\mathrm{m}^{3}\)) & & & & \\ Inner friction angle & 20 & 31 & 33 & 35.5 \\ (\(\beta\),