Consolidation of Soils | Terzaghi's One-Dimensional Consolidation Theory

Consolidation is the process of gradual compression/settlement of soil due to the expulsion of pore water under steady pressure. In this blog, we try to cover the important points to note regarding consolidation.

Concept of Consolidation

What exactly is the consolidation of soils

For loads encountered in geotechnical engineering, it is generally considered that soil grains and pore water are incompressible. Therefore, the volume decrease of soil under stress can be attributed to the expulsion of pore water alone. This is exactly what a consolidation process is i.e., the expulsion of pore water.

Can consolidation occur in fine-grained soils

The expulsion of pore water is faster and is generally considered to keep pace with construction in fine-grained soils as it is more permeable. Therefore, the consolidation of fine-grained soils requires no attention. But, in clays expulsion of pore water takes more time as it is less permeable. Therefore, consolidation of clays can happen long after the construction has been completed and requires attention.

Compaction vs Consolidation of Soil

Compaction is just the process of mechanical compression resulting in a reduction of pore air and consequent densification of soil. Unlike compaction, consolidation is the expulsion of pore water. Though pore water is expelled, the soil remains fully saturated before, during and even after the consolidation process. Water content alone decreases during the consolidation process. For more details check this out.

Necessary conditions for consolidation to occur

• For consolidation to occur, there needs to be a way for pore water to come out i.e., it requires drainage. Drainage could be a layer of sand present above or below or both above and below a fully saturated clay deposit. Based on the presence of the sand layer, there are two drainage systems i.e., single drainage and double drainage, which will be covered shortly.

• Consolidation occurs only when there is an increase in stress experienced by the soil. This could happen due to the addition of a new footing which increases the effective stress experienced by the layer. As effective stress varies with depth within the saturated clay layer, average effective stress is taken as the increased effective stress at the middle of the layer for all further computations.

Now that we have understood what consolidation actually is, the difference between compaction and consolidation, and the requirement for consolidation to occur, let us now understand the basic equation to compute consolidation.

Let us consider a fully saturated clay layer that is subjected to increased effective stress. The below image shows the changes it undergoes with time.

Primary consolidation

The primary settlement, consolidation settlement, and primary consolidation all are necessarily the same term and is represented as Sprimary (ΔH). Expression for the same is derived below.

ΔH = H1 - H2

ΔH/H1 = (ΔH * A)/(H1 * A) = ΔV/V1 = (V1 - V2)/V1, sub V1 and V2

ΔH/H1 = ((Vs + Vv1) - (Vs + Vv2)) / (Vs + Vv1) = (Vv1 - Vv2) / (Vs + Vv1),

dividing each term by Vs,

ΔH/H1 = ((Vv1/Vs) - (Vv2/Vs)) / ((Vs/Vs) + (Vv1/Vs)), as we know void ratio (e) = Vv/Vs,

ΔH/H1 = (e1 - e2) / (1 - e1) = Δe / (1 + e1)

ΔH = H * (Δe / (1 + e0))

where,

H - total thickness of clay layer

e0 - initial void ratio = wnat * Gs, wnat - natural water content

Terzaghi published his one-dimensional consolidation theory to find the primary settlement soils and the same is discussed further.

Technical Terms to understand Consolidation

1. Co-efficient of Compressibility (av)

It is defined as a decrease in the void ratio per unit increase in effective stress applied on the soil. Its unit is m^2/kN. According to Dr Terzaghi, the value of av is constant during the consolidation of one type of soil.

av = (-) Δe/Δ𝜎'

2. Compression Index (Cc)

It is defined as the slope of the linear portion of the graph plotted between the void ratio on the y-axis and log(base10)𝜎' on the x-axis. It has no units.

Cc = Δe / log((𝜎o' + Δ𝜎') / 𝜎o')

According to Terzaghi and Peck, the compression index is totally dependent on the liquid limit of soil and the moulding condition of the soil.

Cc = 0.009 * (Ll - 10%), for natural soil

Cc = 0.007 * (Ll - 10%), for re-moulded soil,

where,

Ll - liquid limit of the clay in %

3. Coefficient of Volume Change (mv)

It is defined as the volumetric strain per unit increase in effective stress. It is the reciprocal of the bulk modulus of soil which is constant for a given soil. Its unit is m^2/kN. It is also called volume compressibility.

mv = ev/Δ𝜎' = (Δv/vo) / Δ𝜎', ev - volumetric strain,

mv = (Δe/(1+eo)) / Δ𝜎' = (Δe/Δ𝜎')* 1/(1+eo)

mv = av / (1+eo)

Primary Consolidation (Sp)

Let us now re-write the basic consolidation formula i.e., ΔH = H * (Δe / (1 + e0)), in terms of compression index and coefficient of volume change.

Primary Consolidation (Sp) in terms of Compression Index (Cc)

Sp = H * (Δe / (1 + e0))

Cc = Δe / log(𝜎o' + Δ𝜎'/𝜎o')

Δe = Cc * log((𝜎o' + Δ𝜎') / 𝜎o'), substituting this Δe in Sp formula,

Sp = (H * Cc * log((𝜎o' + Δ𝜎') / 𝜎o')) / (1 + eo), same is given in a pictorial form below,

Primary consolidation

where,

𝜎o' - initial effective stress experienced by the soil layer (at the middle of the layer),

Δ𝜎' - increase in effective stress experienced by the soil layer (at the middle of the layer),

H - initial thickness of the soil layer,

eo - initial void ratio.

Primary Consolidation (Sp) in terms of Volume Compressibility (mv)

Sp = mv * H * Δ𝜎',

where,

mv - volume compressibility,

H - initial thickness of the layer,

Δ𝜎' - increase in effective stress experienced by the soil layer (at the middle of the layer).

An increase in effective stress (Δ𝜎') due to footings can be calculated using Bossineq's solution or Westergaurd's solution or 2:1 method unless otherwise specifically mentioned in the question. To know more about stress distribution in soil refer here. Below mentioned are formulas to calculate the same using Bossineq and 2:1 method.

Bossineq's formula

Δ𝜎' = (Q/Z^2) * Kb,

where,

Q - load on the footing (kN),

Z - depth of the layer from the base of the footing, (note that this depth is between the base of the footing and centre of the layer under consideration),

Kb - Bossineq coefficient

2:1 method

Δ𝜎' = (q * L * B ) / ((L+Z)*(B+Z)),

where,

q - load on the footing (kN),

L - length of the footing,

B - breadth of the footing,

Z - depth of the layer from the base of the footing, (note that this depth is between the base of the footing and centre of the layer under consideration)

In case of newly placed fills

Δ𝜎' = 𝛾fill * tfill

where,

𝛾fill - the unit weight of the fill material (in kN/m^3),

tfill - thickness of the fill (in m).

Terzaghi's One-Dimensional Consolidation Theory

Assumptions in Terzaghi's One-Dimensional Theory

• Soil is homogeneous and isotropic

• The soil is completely saturated

• The soil grains and water are incompressible

• Load is applied in the vertical direction only

• Drainage of pore water is one-dimensional i.e., in the direction of the load

• Primary consolidation is totally governed by the expulsion of pore water i.e., at the initial stage pore water pressure (uw) is 100% and consolidation is 0%. At the final stage, pore water pressure (uw) is 0% and consolidation becomes 100%

• The soil mass remains saturated after primary consolidation

• The coefficient of permeability of the soil is constant

• The coefficient of compressibility (av) and the coefficient of volume change (mv) are constant for a given type of clay

• The time lag of consolidation occurs mainly due to the low permeability of clays

• Darcy law is valid

Equation of One-Dimensional Consolidation Theory

Terzaghi considered continuity equation, coefficient of compressibility, coefficient of volume change in developing the equation of 1-D theory as given below.

∂uw / ∂t = (K / (mv * γw)) * (∂^2uw/∂z^2), for better understanding the same formula is given in pictorial form.

One-D consolidation equation

where,

uw - pore water pressure,

t - time of primary consolidation,

k - coefficient of permeability of clay,

mv - coefficient of volume change,

z - effective drainage path.

Coefficient of Consolidation (Cv)

Cv = k / (mv * γw), substituting Cv in 1-D equation,

∂uw / ∂t = Cv * (∂^2uw/∂z^2)

Time of Consolidation (t)

The time of consolidation according to Terzaghi is given by,

Tv = (Cv * t) / d^2,

where,

Tv - time factor - a dimensionless quantity that is totally dependent on the degree of consolidation

Tv = (π/4) * (U/100)^2, for U less than or equal to 60%

Tv = -0.9332*log(1 - (U/100)) - 0.0851, for U greater than 60%

where,

U - degree of consolidation = (St / Sf) * 100,

St - settlement at any time,

Sf - total primary settlement of the soil.

Cv - coefficient of consolidation,

t - time of primary consolidation,

d - length of drainage path. It is dependent on the number of drainages as given below,

In the case of a single layer of sand (i.e., single drainage) the max distance travelled by the pore water is equal to the total thickness of the clay layer i.e., d = H

In the case of two layers of sand at the top and bottom (i.e., double drainage), the maximum distance travelled by the pore water is equal to half of the total thickness of the layer i.e.,

d = H/2

Single drainage and double drainage

Note

• In case of double drainage, the time of consolidation gets reduced to 1/4th of the time required in case of single drainage (i.e., t double drainage = (1/4) * t single drainage)

• The total consolidation will remain the same in the case of both single and double drainage.

• In the case of 100% consolidation, Tv becomes infinity and eventually, the time required for consolidation (t) also becomes infinity. This means that 100% consolidation can never be achieved.

Over Consolidation Ratio

It is the ratio of pre-consolidation pressure experienced in the past to the effective stress action at present on the soil.

OCR = 𝜎' (past) / 𝜎' (present)

Based on OCR, soils (clays) are classified as,

OCR > 1, Over consolidated clays

OCR = 1, Normally consolidated clays

OCR < 1, Under consolidated clays

Practise problem

Two practise problems based on primary consolidation and Terzaghi's 1-D theory are given below. Practise them and get clarified with the concept.

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