Soil Mechanics A

1
Consolidation
2
The consolidation process
Pore water (incompressible)
Voids
Skeletal Material (incompressible)
Solid
Initial State
3
The consolidation process
Ds
Pore water (incompressible)
Voids
Voids
Skeletal Material (incompressible)
Solid
Solid

Water
Initial State
Deformed State
4
The consolidation process
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
5
The consolidation process
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
water squeezed out
water
Effective soil skeleton spring
6
The consolidation process
Instantaneously no water can flow, and hence
there can be no change in volume.
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The consolidation process
Instantaneously no water can flow, and hence
there can be no change in volume. For 1-D
conditions this means Dezz Dev
0 (1)
8
The consolidation process
Instantaneously no water can flow, and hence
there can be no change in volume. For 1-D
conditions this means Dezz Dev
0
(1) and hence Ds 0 instantaneously
9
The consolidation process
From the principle of effective stress we have
Ds Ds Du
(2) and thus
instantaneously we must have
Ds Du
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The consolidation process
Region of high excess water pressure
Region of low excess water pressure
Flow
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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The consolidation process
Total Stress
Time
12
The consolidation process
Total Stress
Time
Excess Pore Pressure
Time
13
The consolidation process
Effective Stress
Time
14
The consolidation process
Effective Stress
Time
Settlement
Time
15
Derivation of consolidation governing equation
1. Water flow (due to consolidation)
Elevation
Plan Area A
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Derivation of consolidation governing equation
1. Water flow (due to consolidation)
Rate at which water leaves the element
Elevation
Plan Area A
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Derivation of consolidation governing equation
2. Deformation of soil element (due to change in
effective stress)
Rate of volume decrease
Elevation
Plan Area A
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Derivation of consolidation governing equation
Assume Soil particles and water incompressible
Rate of volume decrease of soil element

Rate at which water leaves the element

19
Derivation of consolidation governing equation
Assume Soil particles and water incompressible
Rate of volume decrease of soil element

Rate at which water leaves the element

Storage Equation
(3)
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Derivation of consolidation governing equation
3. Flow of water (due to consolidation)
Assume Darcys law
(4)
21
Derivation of consolidation governing equation
3. Flow of water (due to consolidation)
Assume Darcys law
(4)
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
(5)
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Derivation of consolidation governing equation
4. Stress, strain relation for soil
Assume soil behaves elastically
(7)
Elastic response
23
Derivation of consolidation governing equation
4. Stress, strain relation for soil
Assume soil behaves elastically
(7)
Elastic response
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)
24
Derivation of consolidation governing equation
5. Principle of effective stress
(8)
Note that these are changes in stress due to
consolidation
25
Derivation of consolidation governing equation
5. Principle of effective stress
(8)
Note that these are changes in stress due to
consolidation


(7)
Elastic response
26
Derivation of consolidation governing equation
Equation of 1-D Consolidation
(10)
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Solution of consolidation equation
1. Boundary conditions
At a very permeable boundary
u 0 At a very impermeable boundary
Very Permeable
Saturated Clay
Very Impermeable
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Solution of consolidation equation
2. Initial conditions (1-D)
Total Stress Change
Time
Excess Pore Pressure
Time
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Solution of consolidation equation
3. Homogeneous soil
(10)
(13)
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Solution of consolidation equation
cv is called the coefficient of
consolidation
31
Solution of consolidation equation
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.
32
Solution of consolidation equation
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
measured from the oedometer test.
33
Solution of consolidation equation
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
measured from the oedometer test. The value of
kv is difficult to measure directly for clays
but can be inferred from the expression for cv.
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Solution of consolidation equation for 2 way
drainage
Uniformly distributed surcharge q
Z
Homogeneous Saturated Clay Layer free to drain
at Upper and Lower Boundaries
2H
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Solution of consolidation equation for 2 way
drainage
Governing Equation
(14a)
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Solution of consolidation equation for 2 way
drainage
Governing Equation
(14a)
Boundary Conditions
u 0 when z 0 for t gt 0
(14 b,c)
u 0 when z 2H for t gt 0
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Solution of consolidation equation for 2 way
drainage
Governing Equation
(14a)
Boundary Conditions
u 0 when z 0 for t gt 0
(14 b,c)
u 0 when z 2H for t gt 0
Initial Condition
u q when t 0 for 0 lt z lt 2H
(14d)
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Solution of consolidation equation for 2 way
drainage
Solution
(15)
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Solution of consolidation equation for 2 way
drainage
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Calculation of settlement
H
2
S
dz

e
ò
v
0
41
Calculation of settlement
H
2
S
dz

e
ò
v
0
H
2
m
u
dz

-
s
(
)
ò
v
e
0
42
Calculation of settlement
H
2
S
dz

e
ò
v
0
H
2
m
u
dz

-
s
(
)
ò
v
e
0
S
U
T

(
)
v
S

2
T
-
a
n
v
e


-
å
1
2
(16c)
2
a
0
n
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Approximate Expressions for Degree of Settlement
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Solution of consolidation equation for 1 way
drainage
Uniformly distributed surcharge q
Z
Homogeneous saturated clay layer resting on an
impermeable base
H
Impermeable
Impermeable
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Solution of consolidation equation for 1 way
drainage
Governing Equation
(18a)
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Solution of consolidation equation for 1 way
drainage
Governing Equation
(18a)
Boundary Conditions
u 0 when z 0 for t gt 0
(18b,c)
u0 when z H for t gt 0
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Solution of consolidation equation for 1 way
drainage
Governing Equation
(18a)
Boundary Conditions
u 0 when z 0 for t gt 0
(18b,c)
u0 when z H for t gt 0
Initial Condition
u q when t 0 for 0 lt z lt H
(18d)
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Solution of consolidation equation for 1 way
drainage
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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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Example 1 Calculation of settlement at a given
time
Soil Profile
4m
Final settlement100mm cv0.4m2/year
Clay
Sand
Final settlement40mm cv0.5m2/year
Clay
5m
Clay
Impermeable
52
Example 1 Calculation of settlement at a given
time
For the upper layer
c
t

0
1
.4
v
T



0
1
.
v
2
2
H
2
Now using Figure 5 with Tv 0.1
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Example 1 Calculation of settlement at a given
time
For the upper layer
c
t

0
1
.4
v
T



0
1
.
v
2
2
H
2
Now using Figure 5 with Tv 0.1 U
0.36 so S 100 0.36 36mm
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Example 1 Calculation of settlement at a given
time
For the lower layer
c
t

0
5
1
.
v
T



0
02
.
v
2
2
H
5
Now using Figure 5 with Tv 0.02
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0.02
0.05
57
Example 1 Calculation of settlement at a given
time
For the lower layer
c
t

0
5
1
.
v
T



0
02
.
v
2
2
H
5
Now using Figure 5 with Tv 0.02 U
0.16 so S 40 0.6 6.4 mm
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Example 2 Scaling
Oedometer U0.5 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 U0.5 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
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Example 2 Scaling
Oedometer U0.5 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
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Example 2 Scaling
Oedometer U0.5 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
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