# Stability Numbers for Unsupported Conical Excavations in Multi-Layered Cohesive Soils
## 2 Materials and Methods  
### Problem Definition  
[PERSON] and [PERSON] [5] identified the failure regions, which are deep circle, base circle, toe circle, and slope circle by using the finite-element method (FEM). The deep circle is defined that the critical failure surface outcrops to the outside of the toe, but does not reach the strong layer below. The base circle is defined that the critical failure surfaces are base circles and tangent to the strong layer below. The strong layer starts to influence the stability number (\(N\)) causing it to rise gradually as the depth ratio is reduced. The toe circle is defined as the critical circles that all pass through the toe and are tangent to the strong layer below. The slope circles are defined as the critical circles outcrop on the slope and are tangent to the strong layer. Figure 1 shows the different types of critical failure circles. Figure 1a shows the deep circle failure surface, Figure 1b shows the base circle, which is tangent to the strong layer below, Figure 1c shows a toe circle in which the critical circles all pass through the toe and are tangent to the strong layer below, and Figure 1d shows the slope circle, which is the critical circles outcrop on the slope and is tangent to the strong layer. Figure 1: Different types of critical failure circles: (**a**) deep circle, (**b**) base circle, (**c**) toe circle, and (**d**) slope circle (modified from [PERSON] and [PERSON] [5]). The soil layers can be more varied at the excavation. Figure 2 shows the case with two layers on a rigid base. Figure 2a shows the case where the top soil layer was smaller than the excavation depth. Figure 2b shows when excavation depth was located on the boundary of two layers. Figure 2c shows the case where the boundary of two layers was behind the excavation depth. The problem definition is shown in Figure 3, where (1) \(H\) is the excavation depth, (2) \(B\) is the radius of circular excavation, (3) \(T\) is the thickness of total soil layers, (4) \(D\) is the depth of the relative top soil layer, and (5) \(\beta\) is the angle of inclination. The geometry analyzed with two purely cohesive layered soils, where \(H\)/\(B\) is the width ratio, which is the excavation height to radius at the bottom of excavation, \(T\)/\(H\) is the thickness ratio, which is the ratio of the excavation height to the thickness of soil layer ratio of the thickness of total soil layers to excavation depth, \(D\)/\(H\) is the depth ratio, which is relative top layer thickness to excavation depth, \(c_{\text{uI}}\)/\(c_{\text{uI}2}\) is the strength ratio, which is the top layer strength to second layer strength, and \(\beta\) is the angle of inclination. In order to assess the variation of the stability number (\(N\)) the valuable parameters (\(H\)/\(B\), \(T\)/\(H\), \(D\)/\(H\), \(c_{\text{uI}}\)/\(c_{\text{uI}2}\), and \(\beta\)) were changed. The variation of parameters is summarized in Table 1.  
The stability numbers are needed as convenient tools that provide an easier way to determine the factor of safety for an excavation. The stability number (\(N\)) is significantly affected by the angle of inclination, slope height and thickness of total soil layer [23]. Figure 3: Problem notation of an unsupported conical excavation. Figure 2: A schematic diagram of depth ratio variation (**a**) \(D\)/\(H\) < 1, (**b**) \(D\)/\(H\) = 1, and (**c**) \(D\)/\(H\) > 1. The stability number \(N\) is proportional to the factor of safety (\(FS\)) as shown in Equation (1)  
\[FS=N\frac{\rho}{\gamma} \tag{1}\]  
where \(\rho\) is the gradient of strength increase with depth \(z\) and \(\gamma\) is the saturated unit weight. [PERSON] and [PERSON] [23] introduced a dimensionless strength gradient parameter \(M\), defined as  
\[M=\frac{c_{u0}}{\rho H} \tag{2}\]  
where, \(c_{u0}\) is the strength at the crest level (\(z=0\)), \(\rho\) is the gradient of strength increase with depth \(z\), and \(H\) is the slope height (or excavation depth).