5.2 Soil Mechanics: Effective Stress, Consolidation & Shear Strength

The Principle of Effective Stress

Terzaghi's effective-stress principle is the foundation of geotechnical engineering: all measurable effects of a change in stress — compression, distortion, and shear strength — are governed by the effective stress carried by the soil skeleton, not the total stress.

σ' = σ − u

where σ' = effective (intergranular) stress, σ = total vertical stress, and u = pore water pressure. Below the water table, u is hydrostatic: u = γ_w·z_w (γ_w = 62.4 lb/ft³). Total stress at depth is the summed unit-weight × thickness of overlying layers.

Because water cannot carry shear, increasing pore pressure (e.g., rapid loading of clay) reduces σ' and weakens the soil. This is why excess pore pressure controls short-term stability. The relationship lives in the Geotechnical chapter of the FE Reference Handbook.

Darcy's Law, Permeability & Seepage

Groundwater moves through soil pores by laminar flow described by Darcy's law:

q = k·i·A, with discharge (seepage) velocity v = k·i

• q = flow rate (ft³/s), k = hydraulic conductivity / coefficient of permeability (ft/s or cm/s), A = gross cross-sectional area.
• i = hydraulic gradient = Δh/L (head loss over flow length, dimensionless).
• The seepage velocity through actual pore channels is v_s = v/n (divide by porosity).

Typical k values: clean gravel ~1–100 cm/s; clean sand ~10⁻³–1; silt ~10⁻⁵; clay <10⁻⁷ cm/s — a span of ~10 orders of magnitude. A common trap is reporting discharge velocity v where seepage velocity v_s is asked. Seepage drives uplift, piping, and quicksand when the upward gradient approaches the critical gradient i_cr = γ'/γ_w (≈1 for typical soils).

Consolidation Settlement

Saturated clays settle slowly as load squeezes pore water out — primary consolidation. For a normally consolidated (NC) clay:

ΔH = (Cc·H₀/(1+e₀))·log₁₀(σ'_f/σ'_0)

• Cc = compression index (slope of e–log σ' curve), H₀ = clay layer thickness, e₀ = initial void ratio.
• σ'_0 = initial effective stress at layer mid-depth; σ'_f = σ'_0 + Δσ' after loading.

For overconsolidated (OC) clay loaded below its preconsolidation pressure σ'_p, swap Cc for the smaller recompression index Cr; if loading crosses σ'_p, use both terms. The time for consolidation uses T_v = c_v·t/H_dr² with the consolidation coefficient c_v and drainage path H_dr (half-layer if double-drained). Settlement magnitude and rate are separate calculations — don't conflate them.

Mohr–Coulomb Shear Strength & a Worked Example

Soil fails in shear per the Mohr–Coulomb criterion:

τ_f = c + σ'·tanφ

where τ_f = shear strength on the failure plane, c = cohesion, φ = angle of internal friction, and σ' = effective normal stress. Sands are typically c = 0 (φ ≈ 28–40°); soft clays in undrained loading behave as φ = 0 with strength = undrained shear strength s_u = c.

• Drained (long-term, slow loading): use effective parameters c', φ'.
• Undrained (rapid loading of clay): use total-stress φ = 0, c = s_u.

Worked effective-stress example: A sand layer has γ = 120 lb/ft³ and the water table is at the surface. At depth z = 10 ft:

• σ = 120 × 10 = 1200 psf.
• u = 62.4 × 10 = 624 psf.
• σ' = 1200 − 624 = 576 psf.
• With φ' = 32°, c' = 0: τ_f = 576·tan32° = 360 psf. Using total stress (1200 psf) would overstate strength by ~2×.

Stress Distribution, Time Rate & Common Traps

Applied surface loads spread with depth, so the Δσ used in settlement is not the surface pressure. The FE Reference Handbook gives Boussinesq influence factors and the simple 2:1 method, where a footing load Q on a B×L footing spreads at 2 vertical to 1 horizontal: Δσ = Q/[(B+z)(L+z)] at depth z. Always evaluate Δσ at the mid-depth of the compressible layer.

The rate of consolidation depends on the degree of consolidation U and the dimensionless time factor T_v:

With T_v = c_v·t/H_dr², halving the drainage path quarters the time (t ∝ H_dr²). Double-drained layers use H_dr = H/2; single-drained use H_dr = H.

The most penalized FE Civil traps in soil mechanics: (1) forgetting to subtract pore pressure to get σ'; (2) using total surface pressure instead of depth-attenuated Δσ; (3) applying drained c', φ' to a rapid (undrained) clay loading; and (4) mixing γ_w units (62.4 lb/ft³ vs. 9.81 kN/m³). Each is a one-line slip that changes the answer by a factor of two or more.

** Most exam problems give a stratified profile, so build σ' depth by depth: total stress σ at any depth is the running sum of (unit weight × thickness) of every layer above, using moist γ above the water table and saturated γ_sat below it. Pore pressure u = γ_w × (depth below the water table). Then σ' = σ − u at that depth. A perched or artesian condition shifts u and must be read carefully. Equivalently, work directly in effective terms with buoyant unit weight γ' = γ_sat − γ_w below the water table — both routes give the same σ', and choosing the buoyant shortcut avoids a subtraction error.

This σ' is the value that enters Mohr–Coulomb strength and the consolidation σ'_0, tying all of soil mechanics back to the one effective-stress principle.