Consolidation and Compressibility Complete Study Notes

Compression and Consolidation of Soils

When a soil layer is subjected to vertical stress, volume change can take place through rearrangement of soil grains, and some amount of grain fracture may also take place. The volume of soil grains remains constant, so change in total volume is due to change in volume of water. In saturated soils, this can happen only if water is pushed out of the voids. The movement of water takes time and is controlled by the permeability of the soil and the locations of free draining boundary surfaces.

It is necessary to determine both the magnitude of volume change (or the settlement) and the time required for the volume change to occur. The magnitude of settlement is dependent on the magnitude of applied stress, thickness of the soil layer, and the compressibility of the soil.

When soil is loaded undrained, the pore pressure increases. As the excess pore pressure dissipates and water leaves the soil, settlement takes place. This process takes time, and the rate of settlement decreases over time. In coarse soils (sands and gravels), volume change occurs immediately as pore pressures are dissipated rapidly due to high permeability. In fine soils (silts and clays), slow seepage occurs due to low permeability.

Components of Total Settlement
The total settlement of a loaded soil has three components: Elastic settlement, primary consolidation, and secondary compression.

Primary consolidation is the major component and it can be reasonably estimated. A general theory for consolidation, incorporating three-dimensional flow is complicated and only applicable to a very limited range of problems in geotechnical engineering. For the vast majority of practical settlement problems, it is sufficient to consider that both seepage and strain take place in one direction only, as one-dimensional consolidation in the vertical direction. 

Compressibility Characteristics

Soils are often subjected to uniform loading over large areas, such as from wide foundations, fills or embankments. Under such conditions, the soil which is remote from the edges of the loaded area undergoes vertical strain, but no horizontal strain. Thus, the settlement occurs only in one-dimension.

The compressibility of soils under one-dimensional compression can be described from the decrease in the volume of voids with the increase of effective stress. This relation of void ratio and effective stress can be depicted either as an arithmetic plot or a semi-log plot.

In the arithmetic plot as shown, as the soil compresses, for the same increase of effective stress Ds', the void ratio reduces by a smaller magnitude, from De₁ to De₂. This is on account of an increasingly denser packing of the soil particles as the pore water is forced out. In fine soils, a much longer time is required for the pore water to escape, as compared to coarse soils.

It can be said that the compressibility of a soil decreases as the effective stress increases. This can be represented by the slope of the void ratio – effective stress relation, which is called the coefficient of compressibility, a_v.

For a small range of effective stress, 

The -ve sign is introduced to make a_v a positive parameter.

If e₀ is the initial void ratio of the consolidating layer, another useful parameter is the coefficient of volume compressibility, m_v, which is expressed as

It represents the compression of the soil, per unit original thickness, due to a unit increase of pressure. 

NC & OC Clays

OP corresponds to initial loading of the soil. PQ corresponds to unloading of the soil. QFR corresponds to a reloading of the soil. Upon reloading beyond P, the soil continues along the path that it would have followed if loaded from O to R continuously.

The preconsolidation stress, s'_pc, is defined to be the maximum effective stress experienced by the soil. This stress is identified in comparison with the effective stress in its present state. For soil at state Q or F, this would correspond to the effective stress at point P.

If the current effective stress, s', is equal (note that it cannot be greater than) to the preconsolidation stress, then the deposit is said to be normally consolidated (NC). If the current effective stress is less than the preconsolidation stress, then the soil is said to be over-consolidated (OC).

It may be seen that for the same increase in effective stress, the change in void ratio is much less for an overconsolidated soil (from e₀ to e_f), than it would have been for a normally consolidated soil as in path OP. In unloading, the soil swells but the increase in volume is much less than the initial decrease in volume for the same stress difference.

The distance from the normal consolidation line has an important influence on soil behaviour. This is described numerically by the overconsolidation ratio (OCR), which is defined as the ratio of the preconsolidation stress to the current effective stress.

Note that when the soil is normally consolidated, OCR = 1

Settlements will generally be much smaller for structures built on overconsolidated soils. Most soils are overconsolidated to some degree. This can be due to shrinking and swelling of the soil on drying and rewetting, changes in ground water levels, and unloading due to erosion of overlying strata.

For NC clays, the plot of void ratio versus log of effective stress can be approximated to a straight line, and the slope of this line is indicated by a parameter termed as compression index, C_c.

Estimation of Preconsolidation Stress

It is possible to determine the preconsolidation stress that the soil had experienced. The soil sample is to be loaded in the laboratory so as to obtain the void ratio - effective stress relationship. Empirical procedures are used to estimate the preconsolidation stress, the most widely used being Casagrande's construction which is illustrated.

The steps in the construction are:

•  Draw the graph using an appropriate scale.

•  Determine the point of maximum curvature A.

•  At A, draw a tangent AB to the curve.

•  At A, draw a horizontal line AC.

•  Draw the extension ED of the straight line portion of the curve.

•  Where the line ED cuts the bisector AF of angle CAB, that point corresponds to the pre-consolidation stress.

Coefficient of Compression (C_c)

A.

B. 

For undisturbed soil of medium sensitivity.

W_L = % liquid limit.

C. 

For remolded soil of low sensitivity

D.

For undisturbed soil of medium sensitivity e_{o = Initial void} ratio

E.

For remoulded soil of low sensitivity.

C_c = 1.15(e₀-0.35)

F.

C_c = 0.115w where, w = Water content

Over consolidation ratio

O.C.R > 1 For over consolidated soil.

O.C.R = 1 For normally consolidated soil.

O.C.R < 1 For under consolidated soil.

Differential Equation of 1-D Consolidation

where, u = Excess pore pressure.

 = Rate of change of pore pressure

C_v = Coefficient of consolidation

 = Rate of change of pore pressure with depth.

Coefficient of volume compressibility  where, e₀ = Initial void ratio

m_v = Coefficient of volume compressibility

Compression modulus

 where, E_c =Compression modulus.

Degree of consolidation

(i) 

 where,

%U = % degree of consolidation.

U = Excess pore pressure at any stage.

U₁ =  = Initial excess pore pressure

at 

at 

(ii)

  where,

e_f = Void ratio at 100% consolidation.

i.e. of t = ∞

e = Void ratio at time 't'

e₀ = Initial void ratio i.e., at t = 0

(iii)  where,

ΔH = Final total settlement at the end of completion of primary consolidation i.e.,

at t = ∞

Δh = Settlement occurred at any time 't'.

Time factor

 where, T_V = Time factor

C_V = Coeff. of consolidation in cm²/sec.

d = Length of drainage path

t = Time in 'sec'

 For 2-way drainage

d = H₀ For one-way drainage.

where, H₀ = Depth of soil sample.

Some cases

(i)  if u ≤ 60% T₅₀ = 0.196

(ii)  if u > 60%

Method to find 'C_v'

(i) Square Root of Time Fitting Method

 where,

T₉₀ = Time factor at 90% consolidation

t₉₀ = Time at 90% consolidation

d = Length of drainage path.

(ii) Logarithm of Time Fitting Method

 where, T₅₀ = Time factor at 50% consolidation

t₅₀ = Time of 50% consolidation.

Compression Ratio

(i) Initial Compression Ratio

where, R_i = Initial reading of dial gauge.

R₀ = Reading of dial gauge at 0% consolidation.

R_f = Final reading of dial gauge after secondary consolidation.

(ii) Primary Consolidation Ratio

where, R₁₀₀ = Reading of dial gauge at 100% primary consolidation.

(iii) Secondary Consolidation Ratio

Total Settlement

S = S_i + S_p + S_s where, S_i = Initial settlement

S_p = Primary settlement

S_s = Secondary settlement

(i) Initial Settlement

For cohesionless soil.

where, 

where, C_r = Static one resistance in kN/m²

H₀ = Depth of soil sample  For cohesive soil.

where, I_t = Shape factor or influence factor

A = Area.

(ii) Primary Settlement

 = Settlement for over consolidated stage

 = Settlement for normally consolidation stage

(ii) Secondary Settlement

where, 

H­₁₀₀ = Thickness of soil after 100% primary consolidation.

e₁₀₀ = Void ratio after 100% primary consolidation.

t₂ = Average time after t₁ in which secondary consolidation is calculated.

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