Geotechnical Engineering

Syllabus

• Soil mechanics: origin of soils, three-phase system, index properties, classification, permeability, seepage and flow nets, effective stress, compaction, consolidation, shear strength.
• Foundation engineering: sub-surface investigation, earth pressure theories, slope stability, stress distribution, bearing capacity, settlement, pile foundations.

Key concepts and formulas

Phase relationships (e void ratio, n porosity, w water content, G specific gravity, S degree of saturation):

• n = e/(1 + e), and wG = Se
• Dry unit weight γ_d = Gγ_w/(1 + e); bulk unit weight γ = (G + Se)γ_w/(1 + e); γ_d = γ/(1 + w)
• Relative density D_r = (e_max − e)/(e_max − e_min)
• Plasticity index PI = LL − PL

Permeability and seepage. Darcy's law v = ki. Flow through a flow net: q = kHN_f/N_d per unit width. Critical hydraulic gradient i_c = (G − 1)/(1 + e), at which quick sand occurs.

Effective stress: σ′ = σ − u. Below the water table, σ′ = γ′z with submerged unit weight γ′ = γ_sat − γ_w.

Consolidation.

• Settlement of a normally consolidated clay: S = C_cH/(1 + e₀) · log₁₀(σ′_f/σ′₀)
• Time factor T_v = c_vt/d², where d is the drainage path (half the layer thickness for double drainage). For U < 60%, T_v = (π/4)U².

Shear strength: τ = c + σ′ tan φ (Mohr-Coulomb). In a triaxial test, σ₁ = σ₃ tan²(45° + φ/2) + 2c tan(45° + φ/2).

Earth pressure (Rankine). K_a = (1 − sin φ)/(1 + sin φ), K_p = 1/K_a. Active thrust on a smooth vertical wall of height H in dry cohesionless soil: P_a = ½K_aγH², acting at H/3 from the base.

Bearing capacity (Terzaghi). Strip footing: q_u = cN_c + qN_q + 0.5γBN_γ. Square: 1.3cN_c + qN_q + 0.4γBN_γ. Circular: 1.3cN_c + qN_q + 0.3γBN_γ. For a purely cohesive soil (φ = 0), N_c = 5.7, N_q = 1, N_γ = 0.

Stress distribution (Boussinesq). Under a point load Q, directly below it at depth z: σ_z = 3Q/(2πz²) ≈ 0.4775Q/z².

Slopes. Infinite slope in dry cohesionless soil: FS = tan φ/tan β.

Worked example

A soil sample has G = 2.7, w = 18% and S = 90%. Find e, n, the dry and bulk unit weights, and the critical hydraulic gradient. Then find the consolidation settlement of a 4 m clay layer with C_c = 0.3, e₀ = 0.9, when the effective stress rises from 100 to 160 kPa.

import math

G, w, S, gw = 2.7, 0.18, 0.90, 9.81
e = w * G / S
print(f"e = {e:.3f}, n = {e / (1 + e):.3f}")
print(f"gamma_d = {G * gw / (1 + e):.3f} kN/m3, gamma = {(G + S * e) * gw / (1 + e):.3f} kN/m3")
print(f"i_c = {(G - 1) / (1 + e):.3f}")

Cc, H, e0 = 0.3, 4.0, 0.9
s = Cc * H / (1 + e0) * math.log10(160 / 100)
print(f"settlement = {s * 1000:.1f} mm")
ka = (1 - math.sin(math.radians(30))) / (1 + math.sin(math.radians(30)))
print(f"Ka(30 deg) = {ka:.4f}, Pa for H = 5 m, gamma = 18: {0.5 * ka * 18 * 25:.1f} kN/m")

e = 0.540, n = 0.351
gamma_d = 17.199 kN/m3, gamma = 20.295 kN/m3
i_c = 1.104
settlement = 128.9 mm
Ka(30 deg) = 0.3333, Pa for H = 5 m, gamma = 18: 75.0 kN/m

Practice questions