Compressibility and Consolidation

Compressibility and Consolidation

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.

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 theory assumes flow occurs only in the vertical direction.

Preconsolidation Pressure ( $\sigma'_c$ )

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

Normally Consolidated (NC): Present effective stress equals the past maximum pressure ( $\sigma'_o = \sigma'_c$ ).
Overconsolidated (OC): Present effective stress is less than the past maximum pressure ( $\sigma'_o < \sigma'_c$ ).

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
$O C R$	Overconsolidation Ratio	-
$\sigma'_c$	Preconsolidation pressure (past maximum effective stress)	-
$\sigma'_o$	In-situ effective vertical stress	-

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

Settlement Formulas (Primary Consolidation)

Normally Consolidated Clay:

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: $\sigma'_o + \Delta \sigma \le \sigma'_c$ (Remains OC)

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	-

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

Case 2: $\sigma'_o + \Delta \sigma > \sigma'_c$ (becomes NC)

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

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

The time required for a certain percentage of consolidation to occur depends on the permeability and compressibility of the soil.

Time Factor ( $T_{v}$ )

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	-

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

Degree of Consolidation ( $U$ ): 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 $T_{90} = 0.848$ .

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.

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Types of Settlement

Settlement Components

One-Dimensional Consolidation Theory

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

Overconsolidation Ratio

Settlement Formulas (Primary Consolidation)

Primary Consolidation (NC Clay)

Primary Consolidation (OC Clay, Case 1)

Primary Consolidation (OC Clay, Case 2)

Secondary Compression (Creep) Settlement

Secondary Compression Settlement

Interactive Consolidation Simulation

Interactive Consolidation Lab

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

Time Rate of Consolidation

Time Factor (TvT_vTv​)

Time Factor

Time Factor vs. Degree of Consolidation

Determining Cv from Lab Data

Preconsolidation Pressure ( $\sigma'_c$ )

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

Time Factor ( $T_{v}$ )