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Stability Analysis of Slopes

The quantitative determination of the stability of slopes is necessary in a number of engineering activities, such as:

  1. the design of earth dams and embankments,
  2. the analysis of stability of natural slopes,
  3. analysis of the stability of excavated slopes,
  4. analysis of the deepseated failure of foundations and retaining walls.

Quite a number of techniques are available for these analyses

Stability of Slopes | Soil Mechanics - Civil Engineering (CE)

Factor of Safety

The factor of safety is commonly thought of as the ratio of the maximum load or stress that a soil can sustain to the actual load or stress that is applied. Referring to Fig. given below the factor of safety F, with respect to strength, may be expressed as follows:

Stability of Slopes | Soil Mechanics - Civil Engineering (CE)

where τff is the maximum shear stress that the soil can sustain at the value of normal stress of σn, τ is the actual shear stress applied to the soil.
Above equation may be expressed in a slightly different form as follows:
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
Two other factors of safety which are occasionally used are the factor of safety with respect to cohesion, Fc, and the factor of safety with respect to friction, Fφ.
The factor of safety with respect to cohesion may be defined as the ratio between the actual cohesion and the cohesion required for stability when the frictional component of strength is fully mobilised. This may be expressed as follows:
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
The factor of safety with respect to friction, Fφ, may be defined as the ratio of the tangent of the angle of shearing resistance of the soil to the tangent of the mobilised angle of shearing resistance of the soil when the cohesive component of strength is fully mobilised.
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
A further factor of safety which is sometimes used is FH, the factor of safety with respect to height. This is defined as the ratio between the maximum height of a slope to the actual height of a slope and may be expressed as follows:
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
where, Cm = Mobilized Cohesion
m = Mobilized Friction Angle
Cm = C / Fs and tan∅m = tan∅ / Fs
Factor of Safety w.r.t. Cohesion (fc)
and Fc = C / Cm
where, Hc = Critical depth
H = Actual depth
Stability of Slopes | Soil Mechanics - Civil Engineering (CE)

Stability Analysis of Infinite Slopes

  1. Cohesionless dry soil/dry sand
    W = yz cosβ
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)Stability of Slopes | Soil Mechanics - Civil Engineering (CE)where,
    τ = Developed shear stress or mobilized shear stress
    σn = Normal stress.
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    where,
    Fs = Factor of safety against sliding
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE) 
    For safety of Slopes
    β < φ

    Fs > 1Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
  2. Seepage taking place and the water table is parallel to the slope in Cohesionless SoilStability of Slopes | Soil Mechanics - Civil Engineering (CE)h = Height of water table above the failure surface.
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    φ’ is effective friction angle
    γ-avg. total unit weight of soil above the slip surface upto ground level.
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
  3. If water table is at ground level: i.e.,
    h = z
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
  4. Infinite Slope of Purely Cohesive Soil
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    Fc = Hc/H
    Here H = z = depth of slice/cut.
    At Critical Stage Fc = 1
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    where, Sη = Stability Number.
  5. C-∅ Soil in Infinite Slope
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
  6. Taylor's stability no.
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    (for cohesive soil)
    Max. theoretical value of stability no. = 0.5
    Max. practical value is = 0.261
    Sη = [tanβ - tan∅]cos2β (for C-∅ soils)

Stability Analysis of Finite Slopes

  1. Fellinious Method
    (For Purely Cohesive Soil)
    (i) (F = Cr2θ / we) where, F = Factor of safety
    r = Radius of rupture curve
    I = Length of rupture curveStability of Slopes | Soil Mechanics - Civil Engineering (CE)(ii) F = Cr2θ1/ weStability of Slopes | Soil Mechanics - Civil Engineering (CE)F = Factor of safety it tension cracks has developed.
    zc = 2C / γ
  2. Swedish Circle Method
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    where, F = Factor of safetyStability of Slopes | Soil Mechanics - Civil Engineering (CE)
  3. Friction Circle Method
    Fc = C / Cm
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
  4. Taylor's Stability Method (C-∅ soil)
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    In case of submerged slope γ should be used instead of γ and if slope is saturated by capillary flow they γsat should be used instead of γ.
    Stability of Slopes | Soil Mechanics - Civil Engineering (CE)
    where ∅w = weight friction angle. 
The document Stability of Slopes | Soil Mechanics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Soil Mechanics.
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