Shear Strength of Soils

Soils fail in shear. Bearing capacity, earth pressure and slope stability all depend on the shear strength parameters c and φ.

Mohr–Coulomb criterion

τ_f = c' + σ' tan φ' (in terms of effective stresses)

• Clean sand: c' = 0. Normally consolidated clay: c' ≈ 0 in drained terms.
• The failure plane makes an angle 45° + φ/2 with the major principal plane.
• Relation between principal stresses at failure: σ₁ = σ₃ tan²(45° + φ/2) + 2c tan(45° + φ/2).

Laboratory tests

Direct shear test: the failure plane is forced to be horizontal; quick and suited to sands. Drainage cannot be controlled well.

Triaxial test (cylindrical sample, cell pressure σ₃, deviator stress σ_d = σ₁ − σ₃):

Unconfined compression test (σ₃ = 0): for saturated clay, c_u = q_u/2.

Vane shear test (field, soft clays): c_u = T/(πD²(H/2 + D/6)).

Sensitivity S_t = undisturbed strength / remoulded strength. Above 8: quick clays.

Pore pressure parameters (Skempton)

Δu = B[Δσ₃ + A(Δσ₁ − Δσ₃)]

• B = 1 for fully saturated soil.
• A at failure: positive for normally consolidated clays (they contract), negative for heavily over-consolidated clays and dense sands (they dilate).

Sands

• Dense sand peaks and then softens (dilates); loose sand reaches its strength slowly (contracts).
• Saturated loose sand under rapid cyclic loading can liquefy (σ' → 0).

Worked example

A CD triaxial test on sand (c' = 0) fails at σ₃ = 100 kPa and σ₁ = 300 kPa.

• sin φ' = (σ₁ − σ₃)/(σ₁ + σ₃) = 200/400 = 0.5 → φ' = 30°.
• Failure plane at 45° + 15° = 60° to the major principal plane.

Practice