Why Settlement Governs Foundation Design

In many soft-soil conditions, bearing capacity is adequate but settlement controls foundation size and type. A classic example: a footing on soft Bay Mud in the San Francisco Bay Area may have adequate bearing capacity at a modest pressure but will settle 200 mm over 20 years if load is not limited. Understanding consolidation theory is therefore essential to predicting long-term performance.

Terzaghi One-Dimensional Consolidation Theory

Terzaghi's theory (1925) models consolidation as the dissipation of excess pore water pressure (Δu) from a saturated clay layer under an incremental load. Assumptions include: fully saturated soil, one-dimensional drainage and strain, constant coefficients Cv and mv, and Darcy's Law for seepage. The governing equation is:

∂u/∂t = Cv · ∂²u/∂z²

where Cv = k / (mv · γw) is the coefficient of consolidation (m²/year), k is hydraulic conductivity, mv is coefficient of volume compressibility, and γw = 9.81 kN/m³. Cv is determined from laboratory oedometer tests (ASTM D2435) by fitting the time-settlement curve using the Casagrande log-time method or the Taylor square-root-of-time method.

Compression Parameters from the e-log(σ'v) Curve

The oedometer test produces a void ratio vs log effective stress plot. Key parameters:

  • Compression index Cc: slope of the virgin compression (normal consolidation) line; Cc ≈ 0.009(LL − 10) for remolded clays (Terzaghi and Peck); for intact clays, Cc ranges from 0.1 (stiff) to 1.0+ (very soft, organic).
  • Recompression index Cr: slope of the reload/unload line; Cr ≈ Cc/5 to Cc/10 typically; critical for overconsolidated clays where most settlement occurs on the Cr portion.
  • Preconsolidation pressure Pc (σ'p): determined by Casagrande's construction on the e-log-σ' curve; represents the maximum past effective stress the soil experienced. OCR = Pc / σ'v0.

For OCR > 1 (overconsolidated), the soil is stiffer than normally consolidated (OCR = 1) soil of the same void ratio.

Calculating Final Primary Consolidation Settlement

For a clay layer of thickness H with double drainage (drained top and bottom), final settlement Sc depends on stress state:

Case 1 — Normally consolidated (σ'v0 = Pc, OCR = 1):

Sc = [Cc / (1 + e0)] · H · log[(σ'v0 + Δσ'v) / σ'v0]

Case 2 — Overconsolidated, Δσ keeps soil on recompression line (σ'v0 + Δσ ≤ Pc):

Sc = [Cr / (1 + e0)] · H · log[(σ'v0 + Δσ'v) / σ'v0]

Case 3 — Overconsolidated, Δσ crosses Pc:

Sc = [Cr / (1 + e0)] · H · log[Pc / σ'v0] + [Cc / (1 + e0)] · H · log[(σ'v0 + Δσ'v) / Pc]

where e0 is initial void ratio, σ'v0 is initial effective vertical stress at mid-layer, and Δσ'v is the stress increase from the applied load (computed via Boussinesq or 2:1 stress distribution).

Time Rate of Consolidation

The degree of consolidation U% at time t is governed by the dimensionless time factor:

Tv = Cv · t / Hdr²

where Hdr = H/2 for double drainage, H for single drainage. From Terzaghi's solution:

  • U = 50%: Tv ≈ 0.197
  • U = 90%: Tv ≈ 0.848
  • U = 95%: Tv ≈ 1.129

Approximations: For U ≤ 60%, U = 2√(Tv/π); for U > 60%, U = 1 − 10^(−(Tv + 0.085)/0.933).

Settlement at time t: s(t) = U(t) × Sc_final. This allows plotting a time-settlement curve and predicting when a structure will reach acceptable deformation limits.

Secondary Compression

After excess pore pressures dissipate (end of primary consolidation), creep continues at a rate defined by the secondary compression index Cα:

Ss = [Cα / (1 + ep)] · H · log(t2 / t1)

where ep is void ratio at end of primary consolidation. Cα/Cc ≈ 0.04–0.06 for inorganic clays; 0.05–0.07 for organic clays; up to 0.1 for peats. Secondary compression dominates long-term performance in organic soils and is a primary design concern for embankments over peat.

Differential Settlement Limits

Total settlement can often be accommodated; differential settlement between columns or bearing points controls structural distress. ASCE 7 and IBC refer to angular distortion β = δ/L, where δ is differential settlement and L is span between points:

  • β ≤ 1/150: threshold for structural damage to load-bearing walls
  • β ≤ 1/300: structural cracking in frame buildings; common design limit
  • β ≤ 1/500: recommended for sensitive equipment, overhead cranes, or brick cladding
  • β ≤ 1/1000: for machine foundations requiring high precision

IBC Section 1806 allows bearing pressures to be increased by 1/3 for wind and seismic load combinations, but settlement must still be checked under sustained loads. For driven pile or drilled shaft design on compressible soils, differential settlement between pile groups is typically limited to 25 mm (1 in).

See Bearing Capacity for the companion strength analysis and Deep Foundations for cases where settlement must be bypassed to a deeper stratum.