Consolidation

Overview

Consolidation describes the gradual volume decrease of saturated fine-grained soils when subjected to an increase in total stress. The additional load initially raises pore water pressure; as water drains and pore pressure dissipates, the effective stress in the soil skeleton increases and settlement occurs.

Design focus: predict the magnitude, rate and distribution of settlement to judge serviceability performance (total settlement, differential settlement and angular distortion). Behaviour in low-plasticity silts and very fine sands may deviate from ideal 1D assumptions and often requires particular care.

Primary vs Secondary Consolidation

Primary Consolidation

Settlement caused by dissipation of excess pore water pressure. Controlled mainly by soil permeability, drainage path length and coefficient of consolidation c_v.

Approaches completion as excess pore pressure tends to zero (for design, often taken as U ≈ 90–95%).
Idealised using Terzaghi’s 1D consolidation theory.
Dominant in saturated clays and silts under new load.

Secondary Consolidation (Creep)

Additional settlement that continues after most primary consolidation is complete. Caused by creep and rearrangement of soil particles under approximately constant effective stress.

Often approximated as linear with log time using C_α.
Important for organic clays, peats and highly plastic clays.
Begins before primary is fully complete and can continue for decades, depending on stress level and soil type.

Settlement–time curve: primary consolidation gives a curved portion that flattens out; secondary consolidation continues as an approximately straight line on a log-time plot. The separation is an idealisation – the two processes overlap in reality.

Terzaghi’s One-Dimensional Consolidation Theory

For one-dimensional vertical flow and deformation, Terzaghi derived a diffusion-type equation for excess pore water pressure u(z,t):

∂u/∂t = c_v ∂²u/∂z²

u – excess pore water pressure (kPa)
t – time (s, days, years)
z – depth coordinate (m)
c_v – coefficient of vertical consolidation (m²/s)

The solution is usually expressed using a non-dimensional time factor T_v = c_v t / H_dr², where H_dr is the drainage path length (H_dr = H for single drainage; H_dr = H/2 for double drainage), and the degree of consolidation U:

For small T_v, U ≈ (2/√π) √T_v
For larger T_v, standard curves or tables are used.

Terzaghi’s theory assumes 1D strain, small strains, constant layer thickness and constant soil properties (k, c_v, compressibility). Layering, non-linear compressibility, anisotropy and complex drainage boundaries often require staged loading assessments or numerical analysis for more reliable predictions.

Key Parameters

Compressibility

m_v – coefficient of volume compressibility (m²/kN or 1/kPa).
C_c, C_r – compression and recompression indices from e–log σ′ (base 10).
OCR – overconsolidation ratio, typically OCR = σ′_p / σ′₀, where σ′_p is preconsolidation pressure.

Derived from oedometer tests; strongly dependent on sampling quality, disturbance and stress history. Disturbance can blur σ′_p and reduce apparent stiffness.

Rate Parameters

c_v – vertical coefficient of consolidation for primary consolidation.
C_α – secondary compression index (log-time, usually base 10).
H_dr – drainage path length (single vs double drainage).

Back-analysed from lab data (t₅₀, t₉₀) or field monitoring. Field values often differ from laboratory values due to scale effects, disturbance and anisotropy; horizontal consolidation (c_h) may control when vertical drains are used.

Design Applications

Magnitude of Primary Settlement

For normally consolidated clay, the primary consolidation settlement S_c beneath a layer of thickness H is often estimated using the e–log σ′ method (log to base 10):

S_c = H · (C_c / (1 + e₀)) · log₁₀(σ′_f / σ′₀)

e₀ – initial void ratio.
σ′₀ – initial vertical effective stress.
σ′_f – final vertical effective stress after loading.

For overconsolidated clays, settlement is typically split into recompression (using C_r up to σ′_p) and virgin compression (using C_c beyond σ′_p).

Time to Reach a Given Settlement

For a target degree of consolidation U, obtain the corresponding T_v from consolidation charts or correlations, then solve:

t = T_v · H_dr² / c_v

Choice of H_dr depends on drainage boundary conditions (single vs double drainage).
Significant staged loading is better treated using incremental analysis with updated stress states and, if appropriate, stress-dependent parameters.

Secondary Compression

Secondary settlement between times t₁ and t₂ is commonly expressed as (log to base 10):

S_s = C_α · H · log₁₀(t₂ / t₁)

The ratio C_α/C_c is sometimes used to transfer data between stress levels, particularly for organic soils, but both C_α and this ratio may vary with stress level and soil type.

Preloading, Surcharging & Vertical Drains

Preloading / Surcharging: apply temporary fill to accelerate consolidation and “use up” a portion of long-term settlement before service loads are applied.
Vertical Drains (PVDs, sand drains): shorten drainage paths and increase the apparent consolidation rate. Design is based on radial consolidation theory, usually with a horizontal coefficient of consolidation c_h, and must consider smear, drain spacing and anisotropy.
Vacuum Consolidation: apply negative pore pressure to increase effective stress without excessive fill height; performance depends on achieving and maintaining target vacuum levels in the field.

Ground improvement design is typically verified by field instrumentation (settlement plates, piezometers, inclinometers) and back-analysis of c_v, c_h and C_α. Field back-analysis often provides more reliable parameters than laboratory data alone.

Lab Testing & Field Monitoring

Oedometer Testing

Determine e–log σ′ curve (base 10), C_c, C_r, preconsolidation pressure σ′_p.
Extract c_v from t–√t (Taylor) or log t (Casagrande) methods; load duration influences derived c_v and apparent secondary behaviour.
Assess C_α from long-duration stages, where applicable, recognising that C_α can vary with stress level.

Field Monitoring

Settlement plates / extensometers to track vertical movements.
Piezometers to monitor pore pressure dissipation and, where relevant, vacuum levels.
Back-analysis to refine c_v, c_h, C_α and confirm or adjust design assumptions.

Numerical Modelling

1D column models: finite difference or finite element approaches implementing Terzaghi consolidation with staged loading.
2D/3D coupled analysis: Biot consolidation in commercial FE codes to capture soil–structure interaction, non-uniform loading and complex stratigraphy; useful for embankments, rafts and excavations.
Advanced constitutive models: creep and time-dependent behaviour (e.g. soft soil creep models) to better capture secondary consolidation and stress-dependent stiffness.

Parameters for advanced models must be calibrated against both laboratory and field data; default values are not acceptable for design. For large settlements, large-strain formulations may be required, as small-strain models can underestimate movements.

Worked Example (Sketch)

Given: 6 m thick normally consolidated clay layer beneath a raft foundation. Initial vertical effective stress at mid-depth σ′₀ = 80 kPa. Additional stress from structure Δσ′ = 60 kPa. From oedometer tests: e₀ = 0.9, C_c = 0.25, c_v = 1.0×10⁻⁷ m²/s. Double drainage (permeable top and bottom).

Final effective stress:
σ′_f = σ′₀ + Δσ′ = 80 + 60 = 140 kPa.
Primary consolidation settlement (NC clay):
Using log base 10:
S_c = H · (C_c / (1 + e₀)) · log₁₀(σ′_f / σ′₀) = 6.0 · (0.25 / 1.9) · log₁₀(140/80)
With 0.25/1.9 ≈ 0.132 and log₁₀(140/80) ≈ 0.24, this gives:
S_c ≈ 0.19 m ≈ 190 mm.
This magnitude would typically be critical for building serviceability and would need careful consideration.
Time factor for 90% consolidation:
For double drainage, take H_dr = H/2 = 3.0 m. For U = 90%, use typical value T_v ≈ 0.848.
Time to 90% consolidation:
t₉₀ = T_v · H_dr² / c_v = 0.848 · (3.0)² / (1.0×10⁻⁷) ≈ 7.6×10⁷ s
which is approximately 2.4 years. Actual field times may be faster or slower depending on true c_v/c_h, drainage conditions and disturbance.
Secondary settlement (if needed):
Apply an appropriate C_α and chosen time window (e.g. 1–30 years) using
S_s = C_α · H · log₁₀(t₂ / t₁), recognising that C_α is stress- and soil-type dependent.

Values are illustrative and based on simplified assumptions. Replace with project-specific stress profiles, parameter values, groundwater conditions and applicable standards for design.

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