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L.18-19-Time Rate of
Consolidation
CIVE 431
SOIL MECHANICS & LAB
Fall 2014-15
Consolidation
When a saturated clay is loaded externally,
GL
saturated clay
The water is squeezed out of the clay over a
long time (due to low permeability of the clay).
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The Consolidation Process
Region of high
excess water
pressure
Flow
Region of low
excess water
pressure
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
Total
Stress
Time
Excess
Pore
Pressure
Time
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The Consolidation Process
Effective
Stress
Time
Settlement
Time
Terzaghis Consolidation Theory
1. Water flow (due to consolidation)
vz
Flow In
Rate at which water
leaves the element
Elevation
v z
vz
z
z
Plan
Area A
v
zA
z
Flow Out
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Terzaghis Consolidation Theory
2. Deformation of soil element (due to change in
effective stress)
Elevation
Rate of volume decrease
v
zA
t
Plan
Area A
Terzaghis Consolidation Theory
Assume: Soil particles and water incompressible
Rate at which water
leaves the element
v
zA
z
Storage Equation
Rate of volume decrease
of soil element
v
zA
t
v
z
v
t
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Terzaghis Consolidation Theory
3. Flow of water (due to consolidation)
Assume Darcys law
h
z
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
u
w
Terzaghis Consolidation Theory
4. Stress, strain relation for soil
Assume soil behaves elastically
Elastic response
Volumetric Strain
mv
Coefficient of Compressibility
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Terzaghis Consolidation Theory
v
Storage Equation
+
Darcys law
+
Elastic response
v
t
v k
h
z
v mv
Terzaghis Consolidation Theory
k 2u
u
m
v
w z2
t
2 u u
cv 2
z
t
Where c v
k
= Coefficient of Consolidation
mvw
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Terzaghis Consolidation Theory
(2-Way Drainage)
Uniformly distributed surcharge q
Homogeneous Saturated Clay Layer free
to drain at Upper and Lower Boundaries
2H
Terzaghis Consolidation Theory
(2-Way Drainage)
cv
2u
z 2
u
t
Boundary Conditions
u = 0 when z = 0 for t > 0
u = 0 when z = 2H for t > 0
Initial Condition
u = q when t = 0 for 0 < z < 2H
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Terzaghis Consolidation Theory
(2-Way Drainage)
2q
0
2
1
sin( n Z)e nTv
n
where
n
1
(n )
2
Tv
z
H
cv t
and
H2
Time Factor
Terzaghis Consolidation Theory
(2-Way Drainage)
Z=z/H
T=0.8
0.5
0.3
0.2
0.1
2
0.0
0.5
1.0
u/q
Variation of Excess pore pressure with depth
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Terzaghis Consolidation Theory
(2-Way Drainage)
S
S
U( Tv )
1 2
0
2n Tv
2n
Dimensionless Time Tv
0.00
10-3
10-2
10-1
10
Relation of degree of
settlement and time
0.25
0.50
0.75
1.00
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Approximate Expressions for
Degree of Consolidation
4Tv
U
U 1
8
2
(Tv 0.2)
2T / 4
v
(Tv 0.2)
Terzaghis Consolidation Theory
(1-Way Drainage)
Uniformly distributed surcharge q
Homogeneous saturated clay layer
resting on an impermeable base
Impermeable
10
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Terzaghis Consolidation Theory
(1-Way Drainage)
cv
2u
z 2
u
t
Boundary Conditions
u=0
when z = 0 for t > 0
u
z
when z = H for t > 0
Initial Condition
u = q when t = 0 for 0 < z < H
Terzaghis Consolidation Theory
(1-Way Drainage)
Z=z/H
T=0.8
0.5
0.3
0.2
0.1
2
0.0
0.5
1.0
u/q
Variation of Excess pore pressure with depth
Solution is identical to that for 2 way drainage. Note that
the maximum drainage path length is H.
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Example - Settlement Calculation
Soil Profile
Gravel
4m
Clay
Final settlement=100mm
cv=0.4m2/year
Sand
Clay Clay
Final settlement=40mm
cv=0.5m2/year
5m
Impermeable
Example - Settlement Calculation
(T = 1 Year)
For the upper layer:
Tv
cvt
H2
0 .4 x 1
22
0 .1
Now using Figure with Tv = 0.1
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Dimensionless Time Tv
0.00
10-3
10-1
10-2
10
Relation of degree of
settlement and time
0.25
0.50
0.75
1.00
Example - Settlement Calculation
(T = 1 Year)
For the upper layer
cvt
Tv
H2
0 .4 x 1
22
0 .1
Now using Figure with Tv = 0.1
U = 0.36
so
S1 = 100 x 0.36 = 36mm
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Example - Settlement Calculation
(T = 1 Year)
For the lower layer:
Tv
cvt
H2
0 .5 x 1
52
0 . 02
Now using Figure with Tv = 0.02
Dimensionless Time Tv
0.00
10-3
10-2
10-1
10
Relation of degree of
settlement and time
0.25
0.50
0.75
1.00
0.02 0.05
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Example - Settlement Calculation
(T = 1 Year)
For the lower layer:
Tv
cvt
H2
0 .5 x 1
52
0 . 02
Now using Figure with Tv = 0.02
U = 0.16
so
S2 = 40 x 0.16 = 6.4 mm
Total Settlement at Surface = S1+S2 = 36mm + 6.4m ~43mm
2004 Brooks/Cole Publishing / Thomson Learning
Stresses within Soil Mass
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2004 Brooks/Cole Publishing / Thomson Learning
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2004 Brooks/Cole Publishing / Thomson Learning
2 to 1 Method of Finding Stress Increase
under the CL of Footing
Consolidation
Settlement
Calculation
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