Download - Kuliah Konsolidasi 1
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Consolidation
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Skeletal Material(incompressible)
Pore water(incompressible)
Voids
Solid
Initial State
The consolidation process
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Skeletal Material(incompressible)
Pore water(incompressible)
Voids
Solid
Voids
Solid
Initial State Deformed State
The consolidation process
Water
+
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Deformation of saturated soil occurs by reduction of pore space & the squeezing out of pore water. The water can only escape through the pores which for fine-grained soils are very small
The consolidation process
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Deformation of saturated soil occurs by reduction of pore space & the squeezing out of pore water. The water can only escape through the pores which for fine-grained soils are very small
The consolidation process
Effective soil skeleton “spring”
water
water squeezed out
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The consolidation process
water
Instantaneously no water can flow, and hence there can be no change in volume.
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The consolidation process
water
Instantaneously no water can flow, and hence there can be no change in volume.
For 1-D conditions this means
zz = v = = 0 (1)
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The consolidation process
water
Instantaneously no water can flow, and hence there can be no change in volume.
For 1-D conditions this means
zz = v = = 0 (1)
and hence ´ = 0 instantaneously
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The consolidation process
water
From the principle of effective stress we have
´ + u (2)
and thus instantaneously we must have
u
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Region of highexcess water pressure
Region of lowexcess water pressure
Flow
The consolidation process
The consolidation process is the process of the dissipation of the excess pore pressures that occur on load application because water cannot freely drain from the void space.
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TotalStress
Time
The consolidation process
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TotalStress
Time
Time
ExcessPorePressure
The consolidation process
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EffectiveStress
Time
The consolidation process
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EffectiveStress
Time
Settlement
Time
The consolidation process
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vv
zzz
z
PlanArea A
Elevation
vz
z
Derivation of consolidation governing equation
1. Water flow (due to consolidation)
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vv
zzz
z
PlanArea A
Elevation
vz
zRate at which waterleaves the element
v
zzA
Derivation of consolidation governing equation
1. Water flow (due to consolidation)
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v
tzA
Rate of volume decrease
Derivation of consolidation governing equation
2. Deformation of soil element (due to change in effective stress)
PlanArea A
Elevationz
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Rate at which waterleaves the element
Rate of volume decreaseof soil element =
v
zzA
v
tzA
Derivation of consolidation governing equation
Assume: Soil particles and water incompressible
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Rate at which waterleaves the element
Rate of volume decreaseof soil element =
v
zzA
v
tzA
v
z v
t(3)Storage Equation
Derivation of consolidation governing equation
Assume: Soil particles and water incompressible
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v kh
zv
Assume Darcy’s law
(4)
Derivation of consolidation governing equation
3. Flow of water (due to consolidation)
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v kh
zv
Assume Darcy’s law
(4)
Derivation of consolidation governing equation
3. Flow of water (due to consolidation)
Note that because only flows due to consolidation are of interest the head is the excess head, and this is related to the excess pore pressure by
hu
w
(5)
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Elastic response v v em (7)
Assume soil behaves elastically
Derivation of consolidation governing equation
4. Stress, strain relation for soil
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Elastic response v v em (7)
Assume soil behaves elastically
Derivation of consolidation governing equation
4. Stress, strain relation for soil
Note that mv has to be chosen with care. It is not a universal soil constant. For 1-D conditions it can be shown that
(9)
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Derivation of consolidation governing equation
5. Principle of effective stress
Note that these are changes in stress due to consolidation
(8)
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v
z v
t (3)Storage Equation
v kh
zv Darcy’s law (4)
Elastic response v v em (7)
+
+
Derivation of consolidation governing equation
5. Principle of effective stress
Note that these are changes in stress due to consolidation
(8)
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Equation of 1-D Consolidation
z
k u
zm
u
t tv
wv
e[ ] [ ] (10)
Derivation of consolidation governing equation
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Very Permeable
Very Impermeable
At a very permeable boundary
u = 0
At a very impermeable boundarySaturated Clay
u
z 0
Solution of consolidation equation
1. Boundary conditions
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At the instantof loading
u e
Solution of consolidation equation
2. Initial conditions (1-D)
TotalStressChange
Time
Time
ExcessPorePressure
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(10)
Solution of consolidation equation
3. Homogeneous soil
z
k u
zm
u
t tv
wv
e[ ] [ ]
(13)
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cv is called the coefficient of consolidation
Solution of consolidation equation
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cv is called the coefficient of consolidation
cv has units L2/T and can be estimated from an oedometer test.The procedure will be explained in the laboratory sessions.
Solution of consolidation equation
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cv is called the coefficient of consolidation
cv has units L2/T and can be estimated from an oedometer test.The procedure will be explained in the laboratory sessions.
The coefficient of volume decrease mv can be measuredfrom the oedometer test.
Solution of consolidation equation
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cv is called the coefficient of consolidation
cv has units L2/T and can be estimated from an oedometer test.The procedure will be explained in the laboratory sessions.
The coefficient of volume decrease mv can be measuredfrom the oedometer test.
The value of kv is difficult to measure directly for clays butcan be inferred from the expression for cv.
Solution of consolidation equation
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Uniformly distributed surcharge q
2HZ Homogeneous Saturated Clay Layer freeto drain at Upper and Lower Boundaries
Solution of consolidation equation for 2 way drainage
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Governing Equation
cu
z
u
tv
2
2 (14a)
Solution of consolidation equation for 2 way drainage
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Governing Equation
Boundary Conditions
cu
z
u
tv
2
2
u = 0 when z = 2H for t > 0
u = 0 when z = 0 for t > 0
(14a)
(14 b,c)
Solution of consolidation equation for 2 way drainage
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Governing Equation
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
(14a)
(14 b,c)
(14d)
Solution of consolidation equation for 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 )
Solution
(15)
Solution of consolidation equation for 2 way drainage
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T=0.8 0.5 0.3 0.2 0.1
0
1
2
0.0 0.5 1.0
Z=z/H
u/q
Variation of Excess pore pressure with depth
Solution of consolidation equation for 2 way drainage
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Calculation of settlement
S vdzH
0
2
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Calculation of settlement
S vdzH
mv e u dzH
0
2
0
2( )
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Calculation of settlement
S vdzH
mv e u dzH
fromwhich it can be shown
S
SU Tv
nTv
n
e
0
2
0
2
1 2
2
20
( )
( )
(16c)
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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
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Approximate Expressions for Degree of Settlement
UT
T
U e T
vv
Tvv
40 2
18
0 22
2 4
( . )
( . )/
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Uniformly distributed surcharge q
HZ Homogeneous saturated clay layerresting on an impermeable base
Impermeable
Solution of consolidation equation for 1 way drainage
Impermeable
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Governing Equation
cu
z
u
tv
2
2 (18a)
Solution of consolidation equation for 1 way drainage
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Governing Equation
Boundary Conditions
cu
z
u
tv
2
2
u=0 when z = H for t > 0
u = 0 when z = 0 for t > 0
(18a)
(18b,c)u
z 0
Solution of consolidation equation for 1 way drainage
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Governing Equation
Boundary Conditions
Initial Condition
cu
z
u
tv
2
2
u=0 when z = H for t > 0
u = 0 when z = 0 for t > 0
u = q when t = 0 for 0 < z < H
(18a)
(18b,c)
(18d)
u
z 0
Solution of consolidation equation for 1 way drainage
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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
Solution of consolidation equation for 1 way drainage
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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
Solution of consolidation equation for 1 way drainage
Solution is identical to that for 2 way drainage. Note that the maximum drainage path length is identical.
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Gravel
4mClay
Clay
Sand
5m
Impermeable
Clay
Final settlement=100mm cv=0.4m2/year
Soil Profile
Final settlement=40mm cv=0.5m2/year
Example 1: Calculation of settlement at a given time
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For the upper layer
Now using Figure 5 with Tv = 0.1
Example 1: Calculation of settlement at a given time
T vc v t
H
20 1
2 20 1
.4.
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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
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For the upper layer
Now using Figure 5 with Tv = 0.1
U = 0.36so
S = 100 0.36 = 36mm
Example 1: Calculation of settlement at a given time
T vc v t
H
20 1
2 20 1
.4.
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For the lower layer
Now using Figure 5 with Tv = 0.02
Example 1: Calculation of settlement at a given time
T vc v t
H
20 5 1
5 20 02
..
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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
0.02 0.05
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For the lower layer
Now using Figure 5 with Tv = 0.02
U = 0.16so
S = 40 0.6 = 6.4 mm
Example 1: Calculation of settlement at a given time
T vc v t
H
20 5 1
5 20 02
..
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Example 2: Scaling
Oedometer U=0.5 after 2 minutes. 2 way drainage, H = 5 mm
Calculate time for U= 0.5 for 10 m thick layer of the same clay, 1 way drainage
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Example 2: Scaling
Oedometer U=0.5 after 2 minutes. 2 way drainage, H = 5 mm
Calculate time for U= 0.5 for 10 m thick layer of the same clay, 1 way drainage
Oedometer Tc t
H
ccv
v vv
2 2
2
0 00580000
.
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Example 2: Scaling
Oedometer U=0.5 after 2 minutes. 2 way drainage, H = 5 mm
Calculate time for U= 0.5 for 10 m thick layer of the same clay, 1 way drainage
Oedometer
Soil layer
Tc t
H
ccv
v vv
2 2
2
0 00580000
.
Tc t
H
c t c tv
v v v
2 210 100
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Example 2: Scaling
Oedometer U=0.5 after 2 minutes. 2 way drainage, H = 5 mm
Calculate time for U= 0.5 for 10 m thick layer of the same clay, 1 way drainage
Oedometer
Soil layer
Tv (oedometer) = Tv (soil layer)
hence t = 80000000 mins = 15.2 years
Tc t
H
ccv
v vv
2 2
2
0 00580000
.
Tc t
H
c t c tv
v v v
2 210 100
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