Earth Pressure and Slope Stability

Retaining walls hold back soil; slopes must not slide. Both are designed using the soil's shear strength.

Earth pressure coefficients

• At rest (wall does not move): K₀ ≈ 1 − sin φ' (Jaky).
• Active (wall moves away; soil expands): K_a = (1 − sin φ)/(1 + sin φ) = tan²(45° − φ/2).
• Passive (wall pushed into soil): K_p = (1 + sin φ)/(1 − sin φ) = 1/K_a.
• K_a < K₀ < K_p.

Rankine's theory (smooth vertical wall, horizontal backfill)

For a cohesionless backfill of height H:

• Active pressure at depth z: p_a = K_a γ z.
• Total active thrust P_a = ½ K_a γ H², acting at H/3 from the base.
• Uniform surcharge q: adds K_a q at every depth (a rectangle, thrust at H/2).
• Water table in the backfill: use γ' below the water table and add the full water pressure γ_w z_w.

Cohesive soils: p_a = K_a γ z − 2c√K_a. Tension near the top creates a tension crack of depth z_c = 2c/(γ√K_a). An unsupported vertical cut can stand to about 2z_c = 4c/(γ√K_a).

Coulomb's theory includes wall friction δ and sloping walls and backfills, using a trial wedge.

Stability of slopes

Infinite slope (slope angle β):

• Dry cohesionless soil: FS = tan φ/tan β.
• Seepage parallel to the slope (water at the surface): FS = (γ'/γ_sat)(tan φ/tan β), roughly half the dry value.

Finite slopes:

• Swedish (slip circle) method for φ = 0 clays: FS = c_u L_arc R/(W x̄).
• Method of slices (Fellenius, Bishop) for c–φ soils.
• Taylor's stability number S_n = c/(F γ H); critical height H_c = c/(γ S_n).

Worked example

A 6 m smooth wall retains dry sand with γ = 18 kN/m³ and φ = 30°.

• K_a = 1/3. P_a = ½ × (1/3) × 18 × 36 = 108 kN/m, at 2 m above the base.

Practice