Compressibility and Consolidation

Compressibility and Consolidation

$Learning Objectives$

Identify the three main types of soil settlement: immediate, primary consolidation, and secondary compression.
Explain one-dimensional consolidation theory and the concept of preconsolidation pressure.
Calculate primary consolidation settlement for normally consolidated and overconsolidated clays.
Calculate secondary compression (creep) settlement.
Analyze the time rate of consolidation and determine the coefficient of consolidation ( $C_v$ ).

Consolidation is the time-dependent settlement of saturated fine-grained soils (clays/silts) resulting from the expulsion of water from the soil pores. It is a critical consideration for the design of structures on soft ground.

$Consolidation$

The process of time-dependent settlement in saturated fine-grained soils due to the expulsion of water from the pore spaces under an applied load.

Types of Settlement

$Settlement Components$

Immediate (Elastic) Settlement ( $S_e$ ): Occurs rapidly in all soils upon load application. Dominated by elastic deformation.
Primary Consolidation Settlement ( $S_c$ ): Occurs over time as excess pore water pressure dissipates. Only significant in saturated clays.
Secondary Compression (Creep) ( $S_s$ ): Occurs after primary consolidation is complete (after excess pore pressure has fully dissipated). Caused by the slow, viscous plastic readjustment of clay particles.

One-Dimensional Consolidation Theory

Terzaghi's one-dimensional consolidation theory models how water is squeezed out of a saturated clay layer. It assumes that the soil is completely saturated, both the water and soil grains are incompressible, Darcy's Law is valid, and both the applied stress and the resulting flow of water occur only in the vertical direction.

$\sigma'_c$

The maximum effective vertical stress that a soil element has ever sustained in the past.

$Normally Consolidated (NC) Soil$

A soil whose current effective overburden stress is the maximum effective stress it has ever experienced in its history ( $\sigma'_o = \sigma'_c$ ).

$Overconsolidated (OC) Soil$

A soil whose current effective overburden stress is less than its historical maximum effective stress, typically due to historical erosion, glacial retreat, or removal of past structures ( $\sigma'_o < \sigma'_c$ ).

$Overconsolidation and Stress History$

The stress history of a soil dramatically affects its compressibility. When a soil is normally consolidated, it settles significantly under new loads because it is being compressed to a state it has never reached before. When a soil is overconsolidated, it is much stiffer and settles less, because the soil structure has already been pre-compressed by past loads.

Overconsolidation Ratio

Ratio of past maximum effective stress to current effective stress; indicates whether a clay is normally consolidated (OCR=1) or overconsolidated (OCR>1).

$OCR = \frac{\sigma'_c}{\sigma'_o}$

Variables

Symbol	Description	Unit
$OCR$	Overconsolidation Ratio	-
$\sigma'_c$	Preconsolidation pressure (past maximum effective stress)	-
$\sigma'_o$	In-situ effective vertical stress	-

$Overconsolidation Ratio Ranges$

NC Soil: $OCR = 1$
OC Soil: $OCR > 1$

$Primary Consolidation Settlement Evaluation$

The calculation of primary consolidation settlement depends entirely on the soil's stress history. By determining whether the soil is Normally Consolidated or Overconsolidated, engineers choose the appropriate formula.

$Normally Consolidated Clay$

For a normally consolidated clay, use the Compression Index ( $C_c$ ) derived from the virgin compression line.

Primary Consolidation (NC Clay)

Settlement of a normally consolidated clay layer due to a stress increase; uses the Compression Index derived from the virgin compression line.

S_c = \frac{C_c H}{1+e_o} \log_{10} \left( \frac{\sigma'_o + \Delta \sigma}{\sigma'_o} \right)

Variables

Symbol	Description	Unit
$S_c$	Primary consolidation settlement	-
$C_c$	Compression Index (\approx 0.009(LL-10))	-
$H$	Thickness of clay layer	-
$e_o$	Initial void ratio	-
$\sigma'_o$	Initial effective vertical stress	-
$\Delta \sigma$	Increase in vertical stress at the center of the layer	-

$Overconsolidated Clay: Case 1 (Remains OC)$

When the final stress remains below the preconsolidation pressure ( $\sigma'_o + \Delta \sigma \le \sigma'_c$ ), use the Recompression Index ( $C_r$ ).

Primary Consolidation (OC Clay, Case 1)

Settlement for an overconsolidated clay where the final stress remains below the preconsolidation pressure; uses the smaller Recompression Index.

S_c = \frac{C_r H}{1+e_o} \log_{10} \left( \frac{\sigma'_o + \Delta \sigma}{\sigma'_o} \right)

Variables

Symbol	Description	Unit
$S_c$	Primary consolidation settlement	-
$C_r$	Recompression (Swell) Index	-
$H$	Thickness of clay layer	-
$e_o$	Initial void ratio	-
$\sigma'_o$	Initial effective vertical stress	-
$\Delta \sigma$	Increase in vertical stress	-

$Recompression Index Value$

$C_r$ is typically 1/5 to 1/10 of $C_c$ .

$Overconsolidated Clay: Case 2 (Becomes NC)$

When the final stress exceeds the preconsolidation pressure ( $\sigma'_o + \Delta \sigma > \sigma'_c$ ), the settlement is computed in two parts: recompression and virgin compression.

Primary Consolidation (OC Clay, Case 2)

Settlement for an overconsolidated clay that crosses the preconsolidation pressure; combines recompression and virgin compression ranges.

S_c = \frac{C_r H}{1+e_o} \log_{10} \left( \frac{\sigma'_c}{\sigma'_o} \right) + \frac{C_c H}{1+e_o} \log_{10} \left( \frac{\sigma'_o + \Delta \sigma}{\sigma'_c} \right)

Variables

Symbol	Description	Unit
$S_c$	Primary consolidation settlement	-
$C_r$	Recompression Index	-
$C_c$	Compression Index	-
$H$	Thickness of clay layer	-
$e_o$	Initial void ratio	-
$\sigma'_o$	Initial effective vertical stress	-
$\sigma'_c$	Preconsolidation pressure	-
$\Delta \sigma$	Increase in vertical stress	-

$Secondary Compression (Creep) Settlement$

Secondary compression continues indefinitely at a logarithmic rate after primary consolidation ends at time $t_p$ . It is highly significant in highly organic soils and peats.

Secondary Compression Settlement

Long-term creep settlement that continues after primary consolidation ends; particularly significant for organic soils and peats.

S_s = C_\alpha H_p \log_{10} \left( \frac{t}{t_p} \right)

Variables

Symbol	Description	Unit
$S_s$	Secondary compression settlement	-
$C_\alpha$	Secondary Compression Index	-
$H_p$	Thickness of the clay layer at the end of primary consolidation	-
$t_p$	Time at which primary consolidation is complete (e.g., U = 100%)	-
$t$	Time for which secondary settlement is being calculated (t > t_p)	-

Interactive Consolidation Simulation

$Interactive Simulation$

Explore how soil properties ( $C_v$ ) and layer thickness ( $H$ ) affect the rate of consolidation settlement over time.

Interactive Consolidation Lab

Clay Layer Thickness,

H

(10 m)

Drainage Condition

Double (Sand Top & Bottom)Single (Rock Bottom)

Install Prefabricated Vertical Drains (PVDs)

Time to 90% Consolidation ( $U=90\\%$ )

0.00 years

Excellent! Construction can proceed quickly.

Sand / Fill (Drainage)

SOFT CLAY

Sand (Drainage)

Time Rate of Consolidation

While settlement calculations determine how much a soil will compress, the time rate of consolidation determines how fast that settlement will happen. The time required for a certain percentage of consolidation to occur depends heavily on the permeability and compressibility of the soil, as well as the length of the drainage path.

$T_v$

The time factor is a dimensionless parameter that relates consolidation time, permeability, compressibility, and drainage path length.

Time Factor

Dimensionless time parameter that governs the rate of primary consolidation; combines permeability, compressibility, drainage path, and time.

T_v = \frac{C_v t}{H_{dr}^2}

Variables

Symbol	Description	Unit
$T_v$	Time factor (dimensionless)	-
$C_v$	Coefficient of consolidation	m²/year
$t$	Time	-
$H_{dr}$	Length of the longest drainage path	-

$Drainage Path Length$

Double Drainage: Sand layers above and below clay ( $H_{dr} = H/2$ )
Single Drainage: Impervious rock below clay ( $H_{dr} = H$ )

$U$

The percentage of primary consolidation that has occurred at time $t$ .

Time Factor vs. Degree of Consolidation

Relates the time factor to the degree of consolidation achieved at a given time; uses two approximate curve-fit equations for U<60% and U>60%.

T_v = \begin{cases} \frac{\pi}{4} \left( \frac{U\%}{100} \right)^2 & \text{for } U < 60\% \\ 1.781 - 0.933 \log_{10}(100 - U\%) & \text{for } U > 60\% \end{cases}

Variables

Symbol	Description	Unit
$T_v$	Time factor	-
$U\%$	Degree of consolidation in percent	-

$Determining Cv from Lab Data$

Because $C_v$ dictates how fast a building will settle, determining it accurately from an Oedometer lab test is critical. There are two standard graphical methods:

Casagrande's Logarithm-of-Time Method: Uses the settlement vs. $\log(time)$ curve. It identifies the $t_{50}$ point (time for 50% consolidation) graphically. $C_v$ is calculated using $T_{50} = 0.197$ .
Taylor's Square-Root-of-Time Method: Uses the settlement vs. $\sqrt{time}$ curve. It identifies the $t_{90}$ point (time for 90% consolidation) by drawing a secant line with 1.15 times the initial slope. $C_v$ is calculated using $T_{90} = 0.848$ .

Use this workflow to select the stress-history settlement branch, estimate primary settlement and its time rate, and include secondary compression when it matters to the design period. Parameters and serviceability limits remain project-specific.

Consolidation Settlement and Time Assessment

Choose the primary-settlement branch from stress history, then assess the time rate from drainage conditions and include secondary compression where relevant.

Use a consistent effective-stress basis and project loading sequence. Settlement parameters come from representative tests and the applicable stress range. Terzaghi one-dimensional consolidation theory assumes saturated soil, small strains, and specified drainage/boundary conditions; practical completion is project-defined rather than exactly 100% consolidation.

Start → Review stratigraphy, compressible thickness, loads, and test data; Review stratigraphy, compressible thickness, loads, and test data → Stress history and drainage boundaries resolved?; Stress history and drainage boundaries resolved? — Yes → Estimate initial and final effective stress by layer; Stress history and drainage boundaries resolved? — No → Hold: obtain data or resolve loading/drainage model; Estimate initial and final effective stress by layer → Current stress state normally consolidated?; Current stress state normally consolidated? — Yes → Use virgin-compression range from current stress; Current stress state normally consolidated? — No → Final stress exceeds preconsolidation stress?; Final stress exceeds preconsolidation stress? — Yes → Use recompression then virgin-compression ranges; Final stress exceeds preconsolidation stress? — No → Use recompression range only; Use recompression then virgin-compression ranges → Sum primary settlement across compressible layers; Use virgin-compression range from current stress → Sum primary settlement across compressible layers; Use recompression range only → Sum primary settlement across compressible layers; Sum primary settlement across compressible layers → Select drainage path and estimate time using Cv and Tv; Select drainage path and estimate time using Cv and Tv → Secondary compression relevant to design period?; Secondary compression relevant to design period? — Yes → Estimate secondary settlement over stated time interval; Secondary compression relevant to design period? — No → Exclude secondary term with stated basis; Estimate secondary settlement over stated time interval → Combine applicable settlement components and compare criteria; Exclude secondary term with stated basis → Combine applicable settlement components and compare criteria; Combine applicable settlement components and compare criteria → Predicted response meets project serviceability criteria?; Predicted response meets project serviceability criteria? — Yes → Record parameters, uncertainty, and time basis; Predicted response meets project serviceability criteria? — No → Revise ground model, loading, foundation, or improvement; Revise ground model, loading, foundation, or improvement → Review stratigraphy, compressible thickness, loads, and test data; Record parameters, uncertainty, and time basis → End

Start: terminator
Review stratigraphy, compressible thickness, loads, and test data: process
Stress history and drainage boundaries resolved?: decision
Hold: obtain data or resolve loading/drainage model: terminator
Estimate initial and final effective stress by layer: process
Current stress state normally consolidated?: decision
Use recompression then virgin-compression ranges: process
Use virgin-compression range from current stress: process
Final stress exceeds preconsolidation stress?: decision
Use recompression range only: process
Sum primary settlement across compressible layers: process
Select drainage path and estimate time using Cv and Tv: process
Secondary compression relevant to design period?: decision
Estimate secondary settlement over stated time interval: process
Exclude secondary term with stated basis: process
Combine applicable settlement components and compare criteria: process
Predicted response meets project serviceability criteria?: decision
Revise ground model, loading, foundation, or improvement: process
Record parameters, uncertainty, and time basis: document
End: terminator

Key Takeaways

Consolidation is the expulsion of water from saturated clay pores under load, leading to settlement over time.
Normally Consolidated (NC) soils settle significantly more than Overconsolidated (OC) soils for the same load increase.
Preconsolidation Pressure ( $\sigma'_c$ ) is the memory of the maximum past stress.
The Coefficient of Consolidation ( $C_v$ ) governs the time rate of settlement, evaluated using the Casagrande ( $t_{50}$ ) or Taylor ( $t_{90}$ ) graphical methods.
Double drainage speeds up consolidation by a factor of 4 compared to single drainage.
Secondary Compression (Creep) is the continuous, plastic readjustment of clay particles that occurs after pore pressures have fully dissipated.

$References$

USACE EM 1110-1-1904: Settlement Analysis

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Learning Objectives

Consolidation

Types of Settlement

Settlement Components

One-Dimensional Consolidation Theory

Preconsolidation Pressure (σc′\sigma'_cσc′​)

Normally Consolidated (NC) Soil

Overconsolidated (OC) Soil

Overconsolidation and Stress History

Overconsolidation Ratio

Overconsolidation Ratio Ranges

Primary Consolidation Settlement Evaluation

Normally Consolidated Clay

Primary Consolidation (NC Clay)

Overconsolidated Clay: Case 1 (Remains OC)

Primary Consolidation (OC Clay, Case 1)

Recompression Index Value

Overconsolidated Clay: Case 2 (Becomes NC)

Primary Consolidation (OC Clay, Case 2)

Secondary Compression (Creep) Settlement

Secondary Compression Settlement

Interactive Consolidation Simulation

Interactive Simulation

Interactive Consolidation Lab

Time to 90% Consolidation (U=90U=90\\%U=90)

Time Rate of Consolidation

Time Factor (TvT_vTv​)

Time Factor

Drainage Path Length

Degree of Consolidation (UUU)

Time Factor vs. Degree of Consolidation

Determining Cv from Lab Data

References

$Learning Objectives$

$Consolidation$

$Settlement Components$

$\sigma'_c$

$Normally Consolidated (NC) Soil$

$Overconsolidated (OC) Soil$

$Overconsolidation and Stress History$

$Overconsolidation Ratio Ranges$

$Primary Consolidation Settlement Evaluation$

$Normally Consolidated Clay$

$Overconsolidated Clay: Case 1 (Remains OC)$

$Recompression Index Value$

$Overconsolidated Clay: Case 2 (Becomes NC)$

$Secondary Compression (Creep) Settlement$

$Interactive Simulation$

Time to 90% Consolidation ( $U=90\\%$ )

$T_v$

$Drainage Path Length$

$U$

$Determining Cv from Lab Data$

$References$