Consolidation

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10. ANALYSIS OF CONSOLIDATION CONS OLIDATION 10.1 Introduction: the consolidation consolidatio n process

From the response of soils under one-dimensional conditions it is apparent that when the effective stress increases there will be a tendency for the soil to compress. However, when a load is applied to a saturat saturated ed soil soil speci specime men n this this compre compressi ssion on does does not occur occur imme immedi diate ately ly.. This This beha behavi viou ourr is a cons conseq eque uenc ncee of the the soil soil cons consti titu tuen ents, ts, the the skel skelet etal al mate materi rial al and and pore pore water water,, bein being g almo almost st incom incompress pressib ible le compared to the soil element; element; deformation deformation can only only take place place by water being being squeeed squeee d out of the voids. voids . This can only occur at a finite rate and so initially when the soil is loaded it undergoes no volume change. !nder one dimensional conditions this implies that there can be no vertical strain and thus no change in vertical effective stress. For "-# conditions we have ε  = ε v =

− ∆e

"+ e

=

% log &σ ′F ' σ ′$ (

&"(

"+ e

Hence if εv ) * then ∆e ) * and σ+F ) σ+$. hen the load is first applied the total stress increases, but as shown above for "-# conditions there can be no instantaneous change in vertical effective stress, this implies that the pore pressure must increase by eactly the same amount amount as the total stress s tress as ∆σ+

) ∆σ - ∆u

&/(

0ubsequently there will be flow from regions of higher ecess pore pressure to regions of lower  ecess pore pressure, the ecess pore pressures will dissipate, the effective stress will change and the soil will deform &consolidate( &cons olidate( with time. This is shown schematically in Fig. ".

Excess Pore Pressure

Total Stress

Time

Effective Stress

Time

Settlement

Time

Time

Fig. Variation total stress and pore with time Fig. 1 1ariation a riation o!ofstress" stre ss" pore pressure pres sure andpressure settle#ent $ith ti#e

/

10.% Deri&ation o! the e'uation o! consolidation !or 1(D conditions

$f we assume that the pore fluid and soil skeleton are incompressible, then 1olume decrease of the soil ) 1olume of pore fluid which flows out and thus 2ate of volume decrease of soil ) 2ate at which pore fluid flows out $n deriving the equations governing consolidation we will consider only one-dimensional conditions with purely vertical soil movements and water flows. The solutions obtained will only be strictly relevant to the vertical consolidation of relatively thin soil layers occurring as a result of etensive uniform loading. &This is a common situation(. 3 similar approach can be followed for more general loading but the resulting equations can only be solved numerically.

v(z,t)

0oil element

v(z

z, t)

z

v z z

v

Fig. % Flo$ o! pore !luid into an ele#ent o! soil Fig. / Flow of pore fluid into an element of soil

2eferring to Fig / it can be seen that =

The rate at which water enters the element

The rate of volume decrease of the element

&v & z  + ∆ z , t ( −

=

)

∂  v ∂   z 

∆ z  A

∂  ε v ∂  t

v & z , t ((  A

∆ z  A

and thus ∂  v

=

∂  ε v

∂   z 

where

v ) the pore fluid velocity, εv ) the element volume strain, 3 ) the cross sectional area of the element.

∂ 

t

&4(

4

$t will also be assumed that #arcy5s law holds and thus that v

=

− k v

∂h

&6(

∂

$n applying #arcy5s law it is only the velocity due to the consolidation process that is of interest, and consequently the head in &6( is the excess head due to the consolidation process &not the total head(. The excess head is related to the excess pore water pressure by h

=

u

&7(

γ w

 8ote that the elevation is not involved in &7( because it only relates the ecess heads and water   pressures. From &4(, &6( and &7( it follows that

∂ ∂u 9 k v : = ∂ ∂



∂ε v ∂t

&(

$f it is also assumed that the soil element responds elastically to a change in effective stress then

ε v = m v σ ′e

&<(

where

with

σe

)

σe

) )

u

)

the increase in pore water pressure over the original value &ecess pore water pressure(

mv

)

the coefficient of volume decrease,

the change in effective stress from the original value σe u

&=(

the increase in total stress over the original value

and

The value of mv must be determined over the appropriate effective stress range because it depends on the mean effective stress. This can be seen by considering the relation between voids ratio and effective stress

e = 3 − % log"* σ ′ and hence de

= −

% dσ ′ /.4 σ ′

now

εv = −

% dσ ′ ∆e = = m v dσ ′ "+ e /.4 &" + e ( σ ′

&>(

Thus mv depends on both voids ratio e, and effective stress, σ+. %ombination of equations &(, &<( and &=( leads to the equation of consolidation

∂ k v ∂u 9 : = ∂ γ w ∂

mv9

∂u ∂σ e − : ∂t ∂t

&"*(

6

The equation of consolidation must be solved sub?ect to certain boundary conditions and initial conditions 10.) *oundar+ Conditions

3t a boundary where the soil is free to drain the pore water pressure will be constant and will not change during consolidation. For such a boundary the excess pore water pressure will be ero. u)*

at a permeable boundary

&""a(

3t an impermeable boundary the pore water velocity perpendicular to the boundary will be ero and thus from #arcy5s law ∂u ∂

=

*

at an impermeable boundary

&""b(

10., Initial Conditions

3t the instant of loading there is no volume strain and thus no change in vertical effective stress. 3t this instant the excess pore water pressure will be equal to the initial increase in total stress.

σ e  at the instant of loading.

u

&"/(

10.- The 'uation o! Consolidation !or a /o#ogeneous Soil

$f the soil layer being considered is homogeneous then equation &"*( becomes

∂/u ∂u ∂σ e − cv / = ∂ ∂t t ∂

&"4(

where

cv

=

k v m v γ w

is called the coefficient of consolidation.

The coefficient of consolidation &cv( can be estimated using the oedometer apparatus as can the coefficient of volume decrease &mv(. The procedure to do this will be discussed in the laboratory classes. $t is difficult &time consuming( to measure the permeability of clays &k v( and so the value of   permeability is usually inferred from the values of cv and mv . 10. Anal+tic Solutions to the e'uations o! consolidation

10.6.1 Two-way drainage Fig. 4 represents a layer of clay of thickness /H sub?ected to a uniform surface stress q applied at time t ) * and held constant thereafter. The clay layer is free to drain at both its top and bottom  boundaries. This is called two-way drainage.

7

Uniformly distributed surcharge q

Z

2H

Fig. ) /o#ogeneous cla+ la+er !ree to drain !ro# oth upper and lo$er oundaries Fig  Homogeneous !aturated "lay #ayer  free to drain at Upper and #ower $oundaries

The increase in stress through out the layer and does not vary with time and so

σe

=

q

@quation &"*( therefore becomes /

cv

∂  u

=

/

∂  u

&"6a(

∂  t 

∂   z 

The clay layer is free to drain at its upper and lower boundary and so u ) * when  ) * for t A *

&"6b(

u ) * when  ) /H for t A *

&"6c(

$nitially the ecess pore pressure will ?ust match the increase in total stress so that there will be no instantaneous volume change and thus u ) q when t ) * for * B  B /H.

&"6d(

$t can be shown that the solution of equations &"6 a,b,c,d( is ∞

=

u

/q

∑α n

n=*

where

αn

) &n D E(

C

)

Tv

)

 H

"

sin &α n C ( e

− α /n Tv

&"7(

π

, a dimensionless distance

cv t H/

, a dimensionless time

 8otice that H which occurs in both dimensionless quantities is the maimum drainage path length.



The settlement of the soil layer can be determined by summing the vertical &) volume( strains, giving

0

= =

/H

∫ ε v d

* /H

&"a(

∫ m v &q − u(d *

and the variation of settlement with time can be obtained by substituting in equation &"7( which gives the variation of u with time and depth. ∞ sin α C  −α n e = ∫  m v q " − /∑ αn * *  /H

0

/ n

Tv

  d 

giving ∞ e−α T   0 = m v q / H " − / ∑ /  α = n *   n /

n

v

&"b(

 8oting that the final settlement of the layer, 0∞ ) mv q /H the settlement may be written 0 0∞

∞ e −α T   = " − / ∑ /  α n * =   n /

= !

n

v

&"c(

where ! is known as the degree of settlement The variation of ecess pore pressure within the layer is shown in Figure 6 &also in data sheets(.

'

Z&*+H

%&'.(

1

'.)

'.

'.2 '.1

2 '.'

'.)

1.'

u+q Fig. , ariation o! e2cess pore pressure $ith depth

The lines on Figure 6 represent the variation of pore pressure with depth at different nondimensionalised times &T(. These lines are known as isochrones. $t can be seen that initially the ecess pore pressure is constant &u'q ) "( throughout the layer. ith time the pore water flows from the interior of the layer to the drainage boundaries, and the ecess pore pressures dissipate until after  a very long time there are no ecess pore pressures.

<

The variation of settlement with time is most conveniently plotted in the form of the degree of  settlement &!( versus dimensionless time Tv, and this is illustrated in Fig. 7 &also in data sheets(

-imensionless %ime %v 1',1 1',2

1',

1

1'

'.'' /elation of degree of  settlement and time '.2)

U

'.)'

'.)

1.'' Fig. - Degree o! settle#ent &ersus di#ensionless ti#e

There are several useful approimations for the degree of settlement, vi 6 Tv

=

!

π

& Tv ≤ *./(

&"<( !

"−

= π

/

e

/

− π Tv ' 6

& Tv > *../(

alternatively Fig. 7 may be used. $t is worth remembering that ! ) *.7 when T v ) *."><.

10.6.2 One-way drainage Fig.  represents a layer of clay of thickness H sub?ected to a uniform surface stress q applied at time t ) * and held constant thereafter. The clay layer is free to drain at its top boundary but is unable to drain at its base. This is called one way drainage.

= Uniformly distributed surcharge q

Z

H

$mpermeable base

Fig.  /o#ogeneous saturated cla+ la+er on an i#per#eale ase Fig 0

Homogeneous !aturated "lay#ayer resting on an impermeable base

The increase in stress through out the layer and does not vary with time and so

σe

=

q @quation &( therefore becomes /

cv

∂  u /

∂   z 

=

∂  u

&"=a(

∂  t 

The clay layer is free to drain at its upper boundary and as before u

=

at the lower boundary ∂u = ∂

*  when

 ) * for t A *

*  when  ) H for t A *

&"=b( &"=c(

$nitially the ecess pore pressure will ?ust match the increase in total stress so that there will be no instantaneous volume change and thus u ) q when t ) * for * B  B H.

&"=d(

2eference to figure 6 reveals that solution &"7( also satisfies the condition ∂u ∂

=

*  when  ) H for t A *

and is thus also the solution for one way drainage & the two way drainage problem can be viewed as two one-way drainage problems back to back5(. Further eamination reveals that although the epression for final settlement differs for the two cases the epression for degree of settlement is  precisely the same.

>

2a#ple ( Calculation o! settle#ent at a gi&en ti#e

Figure < shows a soil profile, it can be assumed that the sand and gravel are far more permeable than the clay and so consolidation in them will have occurred instantaneously.

Gravel Clay

4m

Final settlement&1''mm  c v&'.m2+year 

5m

Final settlement&'mm  c v&'.)m2+year 

Sand Clay

Imermea!le

Fig.Fig 4 La+ered soil deposit < layered 0oil #eposit

$t is assumed that the final settlement has for each of the clay layers has been determined by the methods described in the previous sections and that their values are as indicated on figure <. $t is required to find the settlement after " year  (a) Settleent o! t"e u##er $ayer $n layer " there is two way drainage and so the drainage path H ) /m.

Tv

=

c vt H/

=

*.6 × " //

=

*."

!sing Figure 7 it can be seen that ! ) *.4 and thus the settlement of layer " ) "** × *.4 ) 4mm (%) Settleent o! t"e lower $ayer  $n layer / there is one way drainage and so the drainage path H ) 7m.

Tv

=

cv t H/

=

*.7

×"

7/

=

*.*/

!sing Figure 7 it can be seen that ! ) *." and thus the settlement of layer / ) 6* × *.4 ) .6mm The total settlement after " year is thus ) 4 D .6 ) 6/.6mm 2a#ple ( 3se o! scaling

"*

3n oedometer specimen reaches 7*G settlement after / minutes. $f the specimen is "* mm thick  calculate the time for 7*G settlement of a "* m thick layer under conditions of one-way drainage. $n order that the test may be carried out as quickly as possible oedometer tests are normally conducted with two way drainage and thus the drainage path in the oedometer ) 7mm ) *.**7m. For the oedometer test

Tv

=

cvt H/

cv

=

×/

=

*.**7 /

=**** c v

For the clay layer the drainage path is "*m

Tv

=

cv t H/

=

cv × t

cv × t

=

"* /

"**

0ince the degree of settlement for the two case is the same the two values of the dimensionless time, Tv are equal and so

=**** c v

=

cv t "**

thus

t

= =****** min

= "7./ years

2a#ple ( Calculation o! the coe!!icient o! consolidation

The data in the previous eample can be used to calculate c v. The dimensionless time for 7*G consolidation is Tv ) *.">< &from Figure 7 T v ≈ *./( thus *."><

=

=**** c v

thus

cv

=

/.6;/7 ×"*

−;

m/ min

=

"./>6

m/ year 

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