Title: Soil Mechanics A
1Consolidation
2The consolidation process
Pore water (incompressible)
Voids
Skeletal Material (incompressible)
Solid
Initial State
3The consolidation process
Ds
Pore water (incompressible)
Voids
Voids
Skeletal Material (incompressible)
Solid
Solid
Water
Initial State
Deformed State
4The 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
5The 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
6The consolidation process
Instantaneously no water can flow, and hence
there can be no change in volume.
7The 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)
8The 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
9The consolidation process
From the principle of effective stress we have
Ds Ds Du
(2) and thus
instantaneously we must have
Ds Du
10The 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.
11The consolidation process
Total Stress
Time
12The consolidation process
Total Stress
Time
Excess Pore Pressure
Time
13The consolidation process
Effective Stress
Time
14The consolidation process
Effective Stress
Time
Settlement
Time
15Derivation of consolidation governing equation
1. Water flow (due to consolidation)
Elevation
Plan Area A
16Derivation of consolidation governing equation
1. Water flow (due to consolidation)
Rate at which water leaves the element
Elevation
Plan Area A
17Derivation of consolidation governing equation
2. Deformation of soil element (due to change in
effective stress)
Rate of volume decrease
Elevation
Plan Area A
18Derivation of consolidation governing equation
Assume Soil particles and water incompressible
Rate of volume decrease of soil element
Rate at which water leaves the element
19Derivation 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)
20Derivation of consolidation governing equation
3. Flow of water (due to consolidation)
Assume Darcys law
(4)
21Derivation 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)
22Derivation of consolidation governing equation
4. Stress, strain relation for soil
Assume soil behaves elastically
(7)
Elastic response
23Derivation 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)
24Derivation of consolidation governing equation
5. Principle of effective stress
(8)
Note that these are changes in stress due to
consolidation
25Derivation of consolidation governing equation
5. Principle of effective stress
(8)
Note that these are changes in stress due to
consolidation
(7)
Elastic response
26Derivation of consolidation governing equation
Equation of 1-D Consolidation
(10)
27Solution of consolidation equation
1. Boundary conditions
At a very permeable boundary
u 0 At a very impermeable boundary
Very Permeable
Saturated Clay
Very Impermeable
28Solution of consolidation equation
2. Initial conditions (1-D)
Total Stress Change
Time
Excess Pore Pressure
Time
29Solution of consolidation equation
3. Homogeneous soil
(10)
(13)
30Solution of consolidation equation
cv is called the coefficient of
consolidation
31Solution 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.
32Solution 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.
33Solution 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.
34Solution 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
35Solution of consolidation equation for 2 way
drainage
Governing Equation
(14a)
36Solution 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
37Solution 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)
38Solution of consolidation equation for 2 way
drainage
Solution
(15)
39Solution of consolidation equation for 2 way
drainage
40Calculation of settlement
H
2
S
dz
e
ò
v
0
41Calculation of settlement
H
2
S
dz
e
ò
v
0
H
2
m
u
dz
-
s
(
)
ò
v
e
0
42Calculation 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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44Approximate Expressions for Degree of Settlement
45Solution of consolidation equation for 1 way
drainage
Uniformly distributed surcharge q
Z
Homogeneous saturated clay layer resting on an
impermeable base
H
Impermeable
Impermeable
46Solution of consolidation equation for 1 way
drainage
Governing Equation
(18a)
47Solution 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
48Solution 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)
49Solution of consolidation equation for 1 way
drainage
50Solution 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.
51Example 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
52Example 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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54Example 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
55Example 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
560.02
0.05
57Example 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
58Example 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
59Example 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
60Example 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
61Example 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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