# Stability Numbers for Unsupported Conical Excavations in Multi-Layered Cohesive Soils
## 3 Results and Discussion  
The variation of the stability number (\(N\)) for a range of values of the parameters (\(H\)/\(B\), \(T\)/\(H\), \(D\)/\(H\), \(c_{\text{u}1}\)/\(c_{\text{u}2}\), and \(\beta\)) was obtained from finite element (FE) analyses. Before showing a full set of results for different parametric combinations, a typical set of graphical results is given to explain the trends of the slip surface. Figure 6 shows the total deviatoric strain according to the different ranges of the thickness ratio (\(T\)/\(H\) = 1.2, 1.6, 2.0). All results were performed with only one soil layer on the rigid base. When the excavation surface is closer to the rigid base, the stability number increased due to the slip surface  
Figure 5: Comparison with published data (**a**) comparison with [PERSON] and [PERSON] [6] and (**b**) comparison with [PERSON] and [PERSON] [5]. through the rigid base. [PERSON] and [PERSON] [5] have represented the influence of the thickness ratio (\(T/H\)), which is \(T/H>2.02\) is the deep circle, \(2.02>T/H\geq 1.54\) is the base circle, \(1.54>T/H\geq 1.37\) is the toe circle, and \(1.37>T/H\) are the slope circles. As shown in Figure 6, it was confirmed that the slip surface was changed from deep circle to toe circle and stability number (\(N\)) was increased as decreasing the thickness ratio. Figure 6a shows the deep circle, which is critical failure surface outcrops to the left of the toe, but is not reached to the strong layer below. Figure 6b shows the base circle, which the slip surface tangent to tangent to the strong layer below. As an influence of the rigid layer, the stability number \(N\) was raised. Figure 6c shows the toe circle, which is the slip surface all passes through the toe and is tangent to the rigid layer below. The effect of the width ratio (\(H/B\)) is studied by [PERSON] and [PERSON] [27], [PERSON] and [PERSON] [6], and [PERSON] and [PERSON] [28]. The magnitudes of the stability numbers reduce further with decreases in the width ratio (\(H/B\)). In order to confirm the effect of width ratio (\(H/B\)), which is the excavation height to radius at the bottom of excavation, three cases, which was one soil layer (\(c_{n1/}c_{n2}=1.0\), \(D/H=\infty\)) on the rigid base, were shown in Figure 7. Figure 7a shows the case of the width ratio (\(H/B\)) being 0.5, which was a small value. It was represented that the slip surface was a deep circle, which included the entire slip surface due to wide radius of circular excavation. As the width ratio (\(H/B\)) increased, the slip surface was affected by the radius of circular excavation (Figure 7b). It was leading to increasing the stability number (Figure 7c). Figure 6: Total deviatoric strain according to \(T/H\), where \(H/B=0.5\), \(c_{n1/}c_{n2}=1\), \(\beta=15^{\circ}\) (**a**) \(T/H=2.0\), (**b**) \(T/H=1.6\), and (**c**) \(T/H=1.2\). To examine the effect of the strength difference between two layers, the typical geometry was chosen as \(T/H=2\), \(H/B=0.5\), \(\beta=30^{\circ}\), and \(D/H=0.75\). Figure 10 shows the total deviatoric strain according to strength difference between two layers (c\({}_{\rm{\tiny{4}}{\rm{\tiny{1}}}}\)/c\({}_{\rm{\tiny{4}}{\rm{\tiny{2}}}}\)). It is revealed that the types of critical failure circles change according to the relative two layers strength. Figure 10a shows the shape of the slip surface which is toe circle on top layer due to the lower layer was relatively strong. As strength ratio (c\({}_{\rm{\tiny{4}}{\rm{\tiny{1}}}}\)/cu\({}_{\rm{\tiny{2}}}\)) increases, it is confirmed that slip surface moves to the lower layer (Figure 10b-d). The results obtained from the analysis have been presented to give the slope stability numbers. Figure 11 shows the effect of the total thickness of the soil layer to the rigid body and strength difference between two layers on the slope stability number.