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Consolidation

 Based on part of the GeotechniCAL reference package
by Prof. John Atkinson, City University, London
When soil is loaded undrained, the pore pressures increase. Then, under site conditions, the excess pore pressures dissipate and water leaves the soil, resulting in consolidation settlement. This process takes time, and the rate of settlement decreases over time.

The amount of settlement which occurs in a given time depends on the

  1. permeability of the soil
  2. length of the drainage path
  3. compressibility of the soil
If soil is unloaded (e.g. by excavation) the excess pore pressures may be negative.
 
 

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The process of consolidation and settlement

In coarse soils (sands and gravels) any volume change resulting from a change in loading occurs immediately; increases in pore pressures are dissipated rapidly due to high permeability. This is called drained loading.
In fine soils (silts and clays) - with low permeabilities - the soil is undrained as the load is applied. Slow seepage occurs and the excess pore pressures dissipate slowly, consolidation settlement occurs.
The rate of volume change diminishes with time; about one-half of the total consolidation settlement occurs in one-tenth of the total time.
 
 

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The basic consolidation process and terminology

Consider a site on clay soil with initial steady-state groundwater conditions. An embankment is built, the loading is undrained: the pore pressure in the soil increases, seepage flow and therefore volume changes commences. As consolidation takes place, settlement occurs, and continues at a decreasing rate until steady-state conditions are regained.

Click on the buttons to see the sequence of loading and pore pressure changes.




Terms and symbols
Seepage refers to the flow of groundwater in a saturated soil.
q = rate of seepage flow
Excess pore pressure ( )
is the difference between the current pore pressure (u) and the steady state pore pressure (uo).
= u - uo
Hydraulic gradient (i)
is the difference in total head between two points in the soil.
Permeability or the coefficient of permeability (k)
relates to flow in a given direction, i.e. along a given drainage path.

 

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One-dimensional consolidation

A general theory for consolidation, incorporating three-dimensional flow vectors 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 strains take place in one direction only; this usually being vertical.

One-dimensional consolidation specifically occurs when there is no lateral strain, e.g. in theoedometer test

One-dimensional consolidation can be assumed to be occurring under wide foundations.

 

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One-dimensional consolidation theory

A simple one-dimensional consolidation model consists of rectilinear element of soil subject to vertical changes in loading and through which vertical (only) seepage flow is taking place.

There are three variables:

  1. the excess pore pressure ()
  2. the depth of the element in the layer (z)
  3. the time elapsed since application of the loading (t)
 
 

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Mathematical model and equation

Consider the element of consolidating soil. In time dt:
· the seepage flow is dq
(q = A k i = A k dh/dz)
· the change in excess pressure is

· the thickness changes by
dH = -mv dzds´
It can be shown that the basic equation for one-dimensional consolidation is:

By defining the coefficient of consolidation as


this can be written:


 
 

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Isochrones

Solutions to the one-dimensional consolidation equation can be obtained by plotting the variation of  with the depth in the layer at given elapsed times. The resulting curves are called isochrones. (Gk. iso = equal; kronos = time)

The figure shows a set of supposed standpipes inserted into a consolidating layer.
Before loading, the pore pressure in the drain is zero. At the base of each standpipe there is some initial pore pressure u= uo, the excess pore pressure  = 0.

Immediately after the loading is applied the standpipes will each show an initial excess pore pressure of i, thereafter the excess pore pressure will dissipate.

Click on the following time intervals to observe the changes in  across the thickness of the layer with time.
  1. Before loading = 0
  2. Initial (after loading) when time = 0 = Ds
  3. 0 < time < tc
  4. time = tc (still no change at the bottom)
  5. tc < time < t¥
  6. Finally at time = ¥
Adjacent to the drain (at the top) the excess pore pressure drops to zero almost immediately
At the bottom of the layer the dissipation is quite slow.  

 

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Some properties of isochrones

The gradient of an isochrone is related to the hydraulic gradient (i):

At the drainage surface, isochrones are steepest and  = 0.
At the impermeable (k = 0) base the seepage velocity is zero since V = ki; the isochrones will therefore be at 90° to the impermeable boundary.
Between two isochrones the change in thickness in time dt, i.e. (t2 - t1), is dH = -mv dz d,
where dz.d is the shaded area.
Thus, the settlement at the surface of the layer is given by:
r = DH = m´v area OAB
 
 

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Terzaghi's solution

The basic equation is

(z,t) is excess pore pressure at depth z after time t.
The solution depends on the boundary conditions:
The general solution is obtained for an overall (average) degree of consolidation using non-dimensional factors.
 
 

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General solution

The following non-dimensional factors are used in order to obtain a solution:

· Degree of consolidation at depth z

· Time factor

· Drainage path ratio

The differential equation can now be written as:

If the excess pore pressure is uniform with depth, the solution is:

Putting Ut = rt/r¥ = average degree of consolidation in the layer at time t:


 
 

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Drainage path length

During consolidation water escapes from the soil to the surface or to a permeable sub-surface layer above or below (where = 0). The rate of consolidation depends on the longest path taken by a drop of water. The length of this longest path is the drainage path length, d. Typical cases are:
An open layer, a permeable layer both above and below (d = H/2)
A half-closed layer, a permeable layer either above or below (d = H)
Vertical sand drains, horizontal drainage (d = L/2)

 

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Solution using parabolic isochrones

Isochrones can be approximated to parabolas, affording reasonably accurate solutions to the differential equation for one-dimensional consolidation. Solutions must be obtained for two separate, but adjoining, cases:
· When the elapsed time (t) is less than the critical time (tc)
· When the elapsed time (t) is greater than the critical time (tc)
The critical time is the time that must elapse before the excess pore pressures at the impermeable base first begin to dissipate.
 
 

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Solution for t < tc case

Putting time factor

and average degree of consolidation,

the general solution is

This is valid for 0 < t < tc
At t = tc, n = H = 
Giving

and

Ut = 0.3333
 
 

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Solution for t > tc case

Putting time factor

and average degree of consolidation,

the general solution is

This is valid for tc < t < t¥
At t = tc, n = H = 
Giving

and

Ut = 0.3333
 
 

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The oedometer test

The one-dimensional compression and swelling characteristics of a soil may be measured in the laboratory using the oedometer test (from the Greek: oidema = a swelling).
A cylindrical specimen of soil enclosed in a metal ring is subjected to a series of increasing static loads, while changes in thickness are recorded against time.
From the changes in thickness at the end of each load stage the compressibility of the soil may be observed, and parameters measured such as Compression Index (Cc) and Coefficient of Volume Compressibility (mv).
From the changes in thickness recorded against time during a load stage the rate of consolidation may be observed and the coefficient of consolidation (cv) measured.
The test is fully detailed in BS 1377.

 

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Apparatus and procedure

The saturated specimen is usually 75 mm diameter and 15-20 mm thick, enclosed in a circular metal ring and sandwiched between porous stones.

Vertical static load increments are applied at regular time intervals (e.g. 12, 24, 48 hr.). The load is doubled with each increment up to the required maximum (e.g. 25, 50, 100, 200, 400, 800 kPa). During each load stage thickness changes are recorded against time.

After full consolidation is reached under the final load, the loads are removed (in one or several stages - to a low nominal value close to zero) and the specimen allowed to swell, after which the specimen is removed and its thickness and water content determined. With a porous stone both above and below the soil specimen the drainage will be two-way (i.e. an open layer in which the drainage path length, d = H/2)

 

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Determination of cv from test results

The recorded thickness changes during one of the load stages in an oedometer test are used to evaluate the coefficient of consolidation (cv).
The procedure involves plotting thickness changes (i.e. settlement) against a suitable function of time [either Ötime or log(time)] and then fitting to this the theoretical Tv:Ut curve.
In this way known intercepts of Tv:Ut are located from which cv may be calculated.
 
 

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The Root-Time method

The first portion of the curve of settlement against Ötime is approximately a straight line. The U0 (Ut = 0) point is located at the intercept with the Ut axis. A second point is required: suppose this is U90/Öt90 (point C). The location of this point depends on the equation for the curved portion [See curve fitting methods: Terzaghi or parabolic isochrones]. Once U90 has been located other values follow since the Ut axis scale is linear. The coefficient of consolidation is therefore:

where d = drainage path length
[d = H for one-way drainage, d = H/2 for two-way drainage]
Other appropriate time-interval values could be used:
e.g. U50, ÖT50, Öt50 , etc.
 
 

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Curve fitting based on Terzaghi's equation

From Terzaghi’s analysis, the straight-line portion is:
For 0 < Ut < 0.6,

On the straight line:
ÖT90 = AB = 0.9 x Ö(p/4) = 0.7976
On the curved portion:
ÖT90 = AC = Ö0.848 = 0.9209
Thus, a line drawn through points O and C has abscissae 1.15 times greater than those of the straight line (OB). [0.9209/0.7976 = 1.15]
After the laboratory results curve has been plotted, line OB is drawn, followed by line OC: this crosses the laboratory curve at point (ÖT90,U90) and locates Öt90

The coefficient of consolidation is therefore:


 
 

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Curve fitting based on parabolic isochrones

From the parabola equation the straight-line portion is:
For 0 < Ut < 0.333,

On the straight line:
ÖT90 = AB = 0.9 x Ö(3/4) = 0.7794
On the curved portion:
ÖT90 = AC = Ö0.716 = 0.8462
Thus, a line drawn through points O and C has abscissae 1.086 times greater than those of the straight line (OB). [0.8462/0.7794 = 1.086]
After the laboratory results curve has been plotted, line OB is drawn, followed by line OC: this crosses the laboratory curve at point (ÖT90,U90) and locates Öt90

The coefficient of consolidation is therefore:


 
 

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The Log-Time method

An alternative to the Root-Time method, that is particularly useful when there is significant secondary compression (creep). The Uo point is located by selected two points on the curve for which the times (t) are in the ratio 1:4, e.g. 1 min and 4 min; or 2 min and 8 min.; the vertical intervals AP and PQ will be equal.
The U100 point can be located in the final part of the curve flattens sufficiently (i.e. no secondary compression). When there is significant secondary compression, U100 may be located at the intercept of straight line drawn through the middle and final portions of the curve.
Now U50 and log t50 can be located.
The coefficient of consolidation is therefore:


 
 

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Calculation of settlement times

After the coefficient of consolidation (cv) has been determined from laboratory data calculations are possible for site settlements. It is important to note that cv is not a constant, but varies with both the level of stress and degree of consolidation. For practical site settlement calculations, however, it is sufficiently accurate to measure cv relative to the loading range applicable on site and then assume this value to be approximately constant for all degrees of consolidation (except for very low values).
The basic equation used is:

where d = drainage path length
[d = H for one-way drainage, d = H/2 for two-way drainage]
Tv and t are coupled to a given degree of consolidation
 
 

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Prediction of time for given settlement

Example

The final consolidation settlement of a layer of clay 5.0 m thick is calculated to be 280mm. The coefficient of consolidation for the loading range is 0.955 mm²/min. There is two-way drainage, upward and downward. Calculate the time required for (a) 90% consolidation settlement, (b) a settlement of 100 mm.

(a) Drainage path length, d = 5.0/2 = 2.50 m = 2500 mm
For U90, T90 = 0.848. Then

(b) For 100 mm settlement, Ut = 100/280 = 0.357
and since Ut < 0.6, Tv = 0.357² x p/4 = 0.100
Then time for 100mm settlement


 
 

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Prediction of settlement amount at given time

Example

A layer of clay has a thickness of 4.0 m and drains both upward and downward. A laboratory test has yielded a coefficient of consolidation for the appropriate loading range of 0.675 mm²/min. The final consolidation settlement has been calculated to be 120mm. Provide estimates of the consolidation settlement that may be expected 1yr, 2yr, 5yr and 10yr after construction.

Drainage path length, d = 2000 mm
When Ut < 0.6, use Ut = Ö(4Tv/p)
When Ut > 0.6,

cv = 0.645 mm²/min = 928.8 mm²/day
 
time t
(years)		 time t
(days)		 Tv
= cvt/d²		 Ut
(<0.6)		 Ut
(>0.6)		 rc (mm)
at time t
1 365 0.0848 0.328   39
2 730 0.1965 0.465   56
5 1825 0.4238 0.735 0.715 86
10 3650 0.8475   0.900 108
23.6 8613 2.0   0.994 119

 

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Reliability for design purposes

Laboratory measurements of stress-strain parameters (Cc, Cs, mv) are generally acceptable, provided sampling quality is good, e.g. minimal disturbance, valid representation of strata, maintenance of structure and water content, careful preparation, etc.
Measurements of strain/time relationships (cv) and permeability (k) are not so reliable.
Observed rates of settlement are generally greater than values based on oedometer test results.
Reliability is compromised by factors such anisotropy (e.g. silt/sand layers, varves, fissures, etc), presence of roots, organic matter and voids, and also the effects of secondary compression.
Loads are not often applied instantaneously, and so due allowance should be for the gradual application of loading.
 
 

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Secondary compression or creep

In some soils (especially recent organic soils) one-dimensional compression continues under constant loading after all of the excess pore pressure has dissipated, i.e. after primary consolidation has ceased - this is called secondary compression or creep.
It is generally thought that creep is due to changes in soil structure, although no reliable theory has been proposed as yet.
It is likely that some creep is occurring due primary consolidation, affecting the linearity of the r/Ötime curve and thus making the accurate prediction of settlement difficult and possibly unreliable.
For practical purposes, the Log-Time plot (described elsewhere) can be used to estimate a coefficient of secondary compression (Ca).
 
 

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Coefficient of secondary compression

The amount of secondary compression is the settlement occurring after t100, i.e. after full dissipation of excess pressures
= ra (or sa).
The r/log t curve after t100 can be approximated to a straight line, the slope of which gives the coefficient of secondary compression (Ca).
The slope of the laboratory curve is measured over one log-time cycle, e.g.1000 to 10000 mins.


 
 

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Overconsolidation due to creep

Creep (secondary compression) is basically similar to compaction, except it takes place slowly.
The result of creep is a change in volume (also water content and void ratio).
The soil is in effect further consolidated, and therefore if unloaded is left overconsolidated.
The phenomenon of overconsolidation due to creep is noticeable in soft clays.