H

δδδδ c

∆∆∆∆σσσσz

′

σσσσz0′

σσσσz0′

}σσσσz f′

� Soil Mechanics Review of Consolidation

VoidsVoids

Solids

Vv = eVs

Vs

∆∆∆∆ e

Vv = (e - ∆∆∆∆ e)Vs

Vs Solids

∆∆∆∆σσσσz

′

σσσσz0′

σσσσz0′

}σσσσz f′

Before After

σσσσ0

Before Loading

××××u0

•

Point, P

σσσσ0 + ∆σ∆σ∆σ∆σ

Immediately After Loading

××××u0+∆∆∆∆u

•

Point, P

σσσσ0 + ∆σ∆σ∆σ∆σ

σσσσ0 + ∆σ∆σ∆σ∆σ

Shortly after LoadingNo settlement

Long after LoadingSettlement Complete

××××u0+∆∆∆∆u

××××u0

σσσσ0 + ∆σ∆σ∆σ∆σ

Settlement

� Distortion Settlement (Immediate)

� Consolidation (Time Dependent)

� Secondary Compression time

Settlement

Laboratory Consolidation Test

Test Results

Test ResultsConsolidation Plot-Idealized

Normally and Over-Consolidated Soils

czo σσ ′=′ ….. Normally consolidated

czo σσ ′<′ ….. Over consolidated

czo σσ ′>′ ….. Under consolidated

When soil is loaded to a stress level greater than it ever ‘experienced’ in the past, the soil structure is no longer able to sustain the increased load, and start to breakdown.

� Preconsolidation Pressure - Pc:� Maximum pressure experienced by soil in the past

� Normal Consolidation: OCR = 1� when the preconsolidation pressure is equal to the existing effective vertical overburden pressure Pc = p’o

� present effective overburden pressure is the maximum pressure that soil has been subjected in the past

p’o

Pc

Effective Consolidation Stress p’o

Void ratio e

Pc

� Over Consolidation: OCR > 1� when the preconsolidation pressure is greater than the existing effective vertical overburden pressure Pc > p’o

� present effective overburden pressure is less than that which soil has been subjected in the past

� It also said soil is in preconsolidated condition

� OCR (over consolidation ratio) =

� Under Consolidation: OCR < 1� when the preconsolidation pressure is less than the existing effective vertical overburden pressure Pc < p’o,

� e.g : recently deposited soil geologically or manually.

Pc = p’o

'

c

o

P

p

Pc > p’o

p’o

Pc

p’o

Pc < p’o

Pc

Over-Consolidation Margin & Over-consolidation Ratio

zccm σσσ ′−′=′ ….. Over-consolidation Margin

zo

cOCRσ

σ

′

′= ….. Over consolidation ratio

Settlement PredictionsN.C. Clays

′

′

+=∑

00

log1 z

zfcc H

e

C

σ

σδ

Settlement Calculation:Normally consolidated clay

L H s eorε

∆ ∆ ∆= = =

soil + water

voids

solids

Hf

∆∆∆∆H = s

voids

solids

∆∆∆∆e

ef

1

Ho

eo

1

H =

2.4

2.6

2.8

1 2 1 2

22 1

1

2

1

1

1

'log ' log ' log 'log

'

'log

1 '

v

o o o o

o v o

o

c

o

oc c

o

L H s eor

L H H e

es H H

e

e e e eeC

pp p p

p

H pS C

e p

ε

ε

∆ ∆ ∆= = =

+

∆= =

+

− −−∆= = =

∆ −

=+

1

1.2

1.4

1.6

1.8

2

2.2

2.4

1 10 100

Pressure, p (log scale)

Vo

id r

ati

o,

e

Cc

1

Effective Consolidation Stress p’o

Pc

� For normally consolidated clay

� For layered normally consolidated clay:

'H p p +∆=∑

Settlement Calculation (cont’d):

'log

1 '

'log

'

o oc c

o o

oc ce o

o

H p pS C or

e p

p pS C H

p

+ ∆=

+

+ ∆=

p’1 = p’o, and p’2 include the additional stress ∆p applied by the structure

when computing settlement using percentage vertical strain vs log effective pressure

� In overconsolidated clay

� Cr is the slope of rebound curve (swell index); Cr ≈ 20% to 10% Cc

'log

1 '

o oc r

o o

H p pS C

e p

+∆=

+

'log

1 '

o oc c

o o

H p pS C

e p

+∆=

+ ∑

'log log

1 ' 1 '

o c o oc r c

o o o o

H P H p pS C C

e p e p

+ ∆= +

+ +

p’1 = p’o, and p’2 =po+∆p < Pc

p’1 = p’o, and p’2 =po+∆p > Pc

Time Rate of Consolidation

2%

0 to 60%,4 100

60%, 1.781 0.933log(100 %)

tt

tt v

t v

SU

S

Ufor U T

for U T U

π

=

= =

> = − −

2

2 or

t v

v v drv v v

dr v

t T HT c t

H c= =

� Ut = average degree of consolidation (%)

� St = settlement of the layer at time t

� S = ultimate settlement of the layer due to primary consolidation

� Tv = time factor

� Hdr= average longest drainage path during consolidation

� cv = coefficient of consolidation

� tv = time for consolidation

Example

� Calculate the settlement due to primary con-solidation for 5m clay layer due to a surcharge of 50kPa

2m

Sand

Sand50% saturation

Ground water table

Surcharge = 50kPa

surcharge of 50kPa applied at the ground level. The clay is normally consolidated.

� Calculate the time rate of settlement when cv is given as 0.85m2/yr

5m

5m

Rock

SandGs=2.65, e=0.7

ClayCc=0.45, eo=0.9γsat=15kPa

� Submerged unit weight of clay

� So

Solution

� Calculation of Average effective Overburden Pressure (po)� The moist unit weight of sand

above the ground water table( )2.65 0.5 0.7 9.81

1 1 0.7

s w wsand

G Sr e

e

γ γγ

+ ⋅ ⋅⋅ + ⋅ ⋅= =

+ +

( )

o sand sand clay

' '

15 9.81 5.19 kPa

5p' 2 +3 ' + '

2

clay sat clay wγ γ γ

γ γ γ

= −

= − =

= ⋅ ⋅ ⋅

� Calculation of Settlement

� Submerged unit weight of sand below the ground water table

( )

( )

( )' '

1

1 1

2.65 1 9.819.516kPa

1 0.7

sand sat sand w

s ws w ww

GG e

e e

γ γ γ

γγ γγ

= −

−⋅ + ⋅= − =

+ +

− ⋅= =

+

1 1 0.7

22.21kPa

e+ +

=

o sand sand clay2

52 22.21+3 9.516+ 5.19 85.94 kPa

2= ⋅ ⋅ ⋅ =

'log

1 '

2.5 85.94 500.45 log

1 0.9 85.94

0.592m 0.199m 0.792m 0.8m

o oc c

o o

H p pS C

e p

+ ∆=

+

+=

+

= + = �

Time rate of settlement

0.520.240.07130

0.230.160.03120

0.060.080.00810

t (yr)St(m)

TvUavg

0

0.1

0 1 2 3 4 5 6 7 8 9

Time (yr)

8.550.761.16395

∞0.80∞100

6.240.720.84890

4.170.640.56780

2.960.560.40370

2.110.480.28760

1.450.400.19750

0.930.320.12640

0.520.240.07130 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sett

ele

men

t (m

)