time rate consolidation

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L.18-19-Time Rate of Consolidation

CIVE 431 SOIL MECHANICS & LAB

Fall 2014-15

When a saturated clay is loaded externally,

saturated clay

GL

The water is squeezed out of the clay over a long time (due to low permeability of the clay).

Consolidation

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Region of highexcess waterpressure

Region of lowexcess waterpressure

Flow

The consolidation process is the process of the dissipation of the excess pore pressures that are generated on load application because water cannot freely drain from the void space.

The Consolidation Process

TotalStress

Time

Time

ExcessPorePressure

The Consolidation Process

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EffectiveStress

Time

Settlement

Time

The Consolidation Process

PlanArea A

ElevationzRate at which waterleaves the element

v

zzA

1. Water flow (due to consolidation)vz Flow In

vv

zzz

z

Flow Out

Terzaghi’s Consolidation Theory

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v

tzA

Rate of volume decrease

2. Deformation of soil element (due to change in effective stress)

PlanArea A

Elevationz

Terzaghi’s Consolidation Theory

Rate at which waterleaves the element

Rate of volume decreaseof soil element

=v

zzA

v

tzA

v

z v

tStorage Equation

Assume: Soil particles and water incompressible

Terzaghi’s Consolidation Theory

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z

hkv

Assume Darcy’s law

3. Flow of water (due to consolidation)

Note that because only flow due to consolidation is of interest, the head is the excess head, and this is related to the excess pore pressure by

hu

w

Terzaghi’s Consolidation Theory

Elastic response vv m

Assume soil behaves elastically

4. Stress, strain relation for soil

Coefficient of CompressibilityVolumetric Strain

Terzaghi’s Consolidation Theory

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v

z v

tStorage Equation

z

hkv

Darcy’s law

Elastic response vv m

+

+

Terzaghi’s Consolidation Theory

δt

δum

δz

uδ

γ

kv2

2

w

δt

δu

δz

uδc

2

2

v

wvv γm

kc Where = Coefficient of Consolidation

Terzaghi’s Consolidation Theory

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Uniformly distributed surcharge q

2HZ Homogeneous Saturated Clay Layer freeto drain at Upper and Lower Boundaries

Terzaghi’s Consolidation Theory (2-Way Drainage)

Boundary Conditions

Initial Condition

cu

z

u

tv

2

2

u = 0 when z = 2H for t > 0

u = 0 when z = 0 for t > 0

u = q when t = 0 for 0 < z < 2H

Terzaghi’s Consolidation Theory (2-Way Drainage)

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u q Z

where

and

Zz

H

Tc t

H

nn

nTv

n

vv

21

1

2

0

2

2

sin( )e

(n )

Time Factor

Terzaghi’s Consolidation Theory (2-Way Drainage)

T=0.8 0.5 0.3 0.2 0.1

0

1

20.0 0.5 1.0

Z=z/H

u/q

Variation of Excess pore pressure with depth

Terzaghi’s Consolidation Theory (2-Way Drainage)

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S

SU Tv

nTv

n

e

1 2

2

20

( )

Terzaghi’s Consolidation Theory (2-Way Drainage)

10-3 10-2 10-1 1 10

Dimensionless Time Tv

0.00

0.25

0.50

0.75

1.00

U

Relation of degree ofsettlement and time

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UT

T

U e T

vv

Tvv

40 2

18

0 22

2 4

( . )

( . )/

Approximate Expressions for Degree of Consolidation

Uniformly distributed surcharge q

HZ Homogeneous saturated clay layerresting on an impermeable base

Impermeable

Terzaghi’s Consolidation Theory (1-Way Drainage)

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Boundary Conditions

Initial Condition

cu

z

u

tv

2

2

when z = H for t > 0

u = 0 when z = 0 for t > 0

u = q when t = 0 for 0 < z < H

u

z 0

Terzaghi’s Consolidation Theory (1-Way Drainage)

T=0.8 0.5 0.3 0.2 0.1

0

1

20.0 0.5 1.0

Z=z/H

u/qVariation of Excess pore pressure with depth

Solution is identical to that for 2 way drainage. Note that the maximum drainage path length is H.

Terzaghi’s Consolidation Theory (1-Way Drainage)

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Gravel

4mClay

Clay

Sand

5m

Impermeable

Clay

Final settlement=100mmcv=0.4m2/year

Soil Profile

Final settlement=40mmcv=0.5m2/year

Example - Settlement Calculation

For the upper layer:

Now using Figure with Tv = 0.1

T vc v t

H

x

20 1

2 20 1

.4.

Example - Settlement Calculation (T = 1 Year)

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10-3 10-2 10-1 1 10

Dimensionless Time Tv

0.00

0.25

0.50

0.75

1.00

U

Relation of degree ofsettlement and time

For the upper layer

Now using Figure with Tv = 0.1

U = 0.36so

S1 = 100 x 0.36 = 36mm

T vc v t

H

x

20 1

2 20 1

.4.

Example - Settlement Calculation (T = 1 Year)

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For the lower layer:

Now using Figure with Tv = 0.02

T vc v t

H

x

20 5 1

5 20 02

..

Example - Settlement Calculation (T = 1 Year)

10-3 10-2 10-1 1 10

Dimensionless Time Tv

0.00

0.25

0.50

0.75

1.00

U

Relation of degree ofsettlement and time

0.02 0.05

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For the lower layer:

Now using Figure with Tv = 0.02

U = 0.16so

S2 = 40 x 0.16 = 6.4 mm

T vc v t

H

x

20 5 1

5 20 02

..

Example - Settlement Calculation (T = 1 Year)

Total Settlement at Surface = S1+S2 = 36mm + 6.4m ~43mm

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Stresses within Soil Mass

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2 to 1 Method of Finding Stress Increase under the CL of Footing

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Consolidation Settlement Calculation

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