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Time Rate Consolidation
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11/19/2014
1
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
© 2
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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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Tho
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