These analyses are used for cases where a planar failure surface can be assumed, which often represents a discontinuity zone. A failure surface is assumed to be located at a certain depth below the ground level, running parallel to the slope.
The expression for the factor of safety F$ derives from the limit equilibrium analysis of a soil block; depending on the case, it takes the following form:
Seepage parallel to the slope, with the water level below the ground profile:
FS = (tanφ/tanβ)·(1 – (γw /γ) + (γw·zw) / (γ·z))
Seepage parallel to the slope, water table at ground level:
FS = (tanφ/tanβ)·(1 – (γw /γ))
Unsaturated slope or vertical seepage (α =90°)
FS = (tanφ/tanβ)
From which it follows that a slope is in a state of limit equilibrium (F = 1) when its inclination angle is equal to the shear strength angle.
Infinite slope in purely cohesive soils:
FS =2·cu/(γ·z·cosβ·sinβ)
Where:
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φ: shear strength angle
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β: soil slope inclination
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γw: unit weight of water
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γ: saturated unit weight of the soil
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zw: depth of the water level
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z: depth of the planar failure surface
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cu: undrained shear strength