As discussed in Mechanics of Slope Stability, the most widely used approach for assessing both 2-dimensional and 3-dimensional slope stability is the Limit Equilibrium Method (LEM), i.e., assessing the stability of a slope via the estimation of the associated Factor of Safety (FoS). The Limit Equilibrium approaches that consider the equilibrium of the entire soil mass are known as the Single Free-Body Procedures.
An infinite slope with an inclination (β°) is assumed. The stability conditions of a typical cross-section of the slope above a potential failure surface are assessed. The forces acting on this cross-section as well as the force polygon are depicted in Figure 1.The weight of the analyzed section with height Η and width Β is calculated as:
where γ refers to the unit weight (kN/m3) of the soil. Given the inclination of the slope (β), the weight is analyzed into two components, as shown in Figure 1.
The vertical reaction acting on the section from the underlying soil is equal to N, while the maximum shear stress that can be developed along the failure plane is:
where φ is the friction angle of the soil.
The driving force of the slope movement is the component of the weight parallel to the potential failure surface (T), while the resisting force is the shear strength of the ground (S). Hence:
Equation  indicates that, in an infinite slope analysis, the FoS does not depend on the depth of the failure surface but is a function of the friction angle and the angle of the slope. In particular, the slope is stable for every β < φ, and unstable β > φ. Therefore, the maximum slope inclination that a cohesionless soil can be obtain in nature while being stable is equal to its friction angle.
The same slope described above with an angle of β° consists of a cohesive soil and seepage is allowed. The presence of pore-water pressures as well as the cohesion of the soil change the stability conditions as shown in Figure 2.
where H is the height of the slice. In case that the seepage flow height is smaller than H, the corresponding depth of flow is utilized. The total pore-water force acting on the base of the slice is:
The driving force T is derived as shown in Equation , however, the weight W is calculated using the saturated unit weight as:
Since the ground is cohesive, the resisting force consists of two components, the one deriving from the soil cohesion and a second one deriving from its frictional strength:
where B/cos(β) is the total length of the base of the slice.
Therefore, the FoS is calculated as:
When doing these analyses the assumption is always that there is ductile or fully Plastic Shear failure. Not strain softening.
Definition: Factor of Safety: The factor by which shear strength must be divided to render the slope barely stable or be in “equilibrium”.
The Free-body diagram in this case, is as shown.
As τ assumed to be purely cohesive with cohesion c (or Su in the case of undrained strength).
For force equilibrium (T is shear force):
Let’s examine this equation:
When β=0 or β=α the FS=∞ which makes sense!
The minimum FS, FSmin occurs when sinβ∙sin(α-β)=max
By taking the derivative we can calculate that the FSmin happens when β=α/2
so for β=α/2
This (equation ) is a very simple expression to remember.
Note that the planar failure becomes less likely (has higher FS), when the slopes are flatter, so in general the calculation is valid for slopes with angles α> 53°.
Research conducted in Sweden in the beginning of the 20th century found that the geometry of many landslides resembles circular arcs. Such failure surfaces are known as rotational slides.
For a given circular failure surface, the weight forces that act on the ground’s center of gravity apply a driving moment around the center of rotation (which is the center of the circle that defines the failure surface). The resisting moment derives from the shear resistance that this circular failure surface can develop multiplied by the radius R of the circle.
The Swedish slip circle is the simplest of the circular methods which assumes undrained conditions for the soil and may apply to short-term problems (usually at the end of a construction project). For such conditions, the undrained shear strength of the ground cu is used (c=cu and φ=0°). To fully understand the mechanism of failure, the driving and resisting moments are illustrated in Figure 7.The FoS is derived by dividing the resisting and the driving moments as:
Where L is the length of the failure surface, R is the radius of the circle, and D is the moment arc of the soil mass weight.
It should be noted that several failure surfaces should be investigated to find the one resulting in the lowest FoS and assess the stability of the slope.
Craig, R.F. (2004). Craig's Soil Mechanics (7th ed.). CRC Press. doi.org/10.4324/9780203494103
Murthy, V. N. S. (2003). Geotechnical engineering: Principles and practices of soil mechanics and foundation engineering. New York: Marcel Dekke
Samtani, N.C, Nowatzki, E.A. (2006). Soils and Foundations Reference Manual Volume 1. U.S. Department of Transportation, Federal Highway Administration, Washington, D.C. 20590
WG/WLI, 1994. A suggested method for reporting landslide causes. Bull. Int. Assoc. Eng. Geol. 50 (1), 71e74.
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