7 compressibilty of soils

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College of Engineering
Civil Engineering Department
Iraq-Ramadi
Asst. Prof. Khalid R. Mahmood (PhD.) 290
Compressibilty of Soils
H
Deformation of soil grains
Expulsion of water
Relocation of soil particles

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Topics
Introduction
Immediate Settlement
Consolidation Settlement (Primary Consolidation)
Secondary Compression (Secondary consolidation)
Settlement
Time Rate of Consolidation
Methods for Accelerating Consolidation Settlement

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Introduction
Soil deformation may occur by change in:
a) Stress
b) Water content
c) Soil mass
d) Temperature
The compression is caused by
a) Deformation of soil particles
b) Relocation of soil particles
c) Expulsion of water or air from the voids

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Types of settlement:
a) Immediate (Elastic) Settlement e
b) Consolidation Settlement (primary consolidation) c
c) Secondary Compression (Consolidation) Settlement s
Thus, the total settlement will be sceT

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Immediate (Elastic) Settlement
Due elastic deformation of soil grains without any change in
moisture content
It is usually small and occurs directly after the application of a
load.
The magnitude of the contact settlement will depend on the
flexibility of the foundation and the type of material on which
it is resting, these distributions are true if E is constant with
depth.

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all the previous relationship discussed in previous chapter
were based on the following assumptions:-
a) The load is applied at the ground surface
Flexible
Rigid
Sand Clay
Contact pressure
distribution
Contact pressure
distribution
Contact pressure
distribution
Contact pressure
distribution
Settlement profile Settlement profile
Settlement profile Settlement profile

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b) The loaded area is flexible.
c)The soil medium is homogenous, elastic, isotropic,
and extends to a great depth.
Relations for Immediate Settlement Calculation
p
s
s
e I
E
B
2
1
Schleicher (1926)

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Improved Relationship for Immediate Settlement
Mayne and Polous (1999) present an improved relationship for
calculating e taking into account –
- rigidity of the foundation
- depth of embedment of the foundation
- the increase in modulus of elasticity E with depth
- the location of rigid layers at limited depth
EFG
o
s
ee III
E
B
2
1
BL
Be
4
for rectangular footing
DiameterBe for circular footing

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Consolidation Settlement (Primary Consolidation)
Fundamentals of Consolidation
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, while for coarse-grained soils
are large enough for the process to occur immediately after
the application of load.

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Skeletal Material
(incompressible)
Pore water
(Incompressible)
Voids
Solid
Voids
Solid
Initial State Deformed State
Water
+
Time
dependent
process

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Spring model
water
Effective soil skeleton “spring”
Water squeezed
out
u
P = P/A
u < u = 0u = 0

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Total Stress
Time
Time
Excess Pore
Pressure

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Effective Stress
Time
Settlement
Time

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Conclusions
Especially in low permeability soils (silts and clays) settlement is
delayed by the need to squeeze the water out of the soil
Consolidation is the process of gradual transfer of an applied load
from the pore water to the soil structure as pore water is squeezed
out of the voids.
The amount of water that escapes depends on the size of the load
and compressibility of the soil.
The rate at which it escapes depends on the coefficient of
permeability, thickness, and compressibility of the soil.
Consider a clay layer with thickness H subjected to an instantaneous
increase of total stress .

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Ground water table
Total stress increase Pore water pressure increase Effective stress increase
At time t = 0
At time 0 < t <
At time t =
u = /
= 0H
Depth
Sand
Sand
u = 0 /
=
/
> 0u <

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One-Dimensional Laboratory Consolidation Test
It was suggested by Terzaghi.
Consolidometer (Oedometer)
pn applied by a lever arm
H
Porous Stone
Porous Stone
Soil Sample
Dia. 64 mm(2.5 in)
H 25 mm (1 in)
Metal Ring
Sample
Water

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p1
p2
p3
pn
H
H1
H2
H3
Hn
Each load increment is kept
for 24 hrs, after that, the
load is usually doubled
The compression
deformation is
measured by a
micrometer dial
gauge
h
hn
Settlement (deformation) vs. time
U% = 100%
Time at
U%=100%
t50%
U% = 0
H d50
d100
d0
Stage I-Initial compression
Stage II-Primary consolidation
Stage III-Secondary Compression
Time (log scale)
Soil
Time, min Deformation, mm
o
¼
½
1
2
5
10
20
30
1 hr
2 hr
……..
24 hr
Cv & k are determined from this relationship
At each time point:
Record the dial indicator reading
At the end of each load increment:
Record the dial reading at completion of primary consolidation for each load (usually 24 hours)
At the unloading of the specimen:
Remove load in decrements, recording dial indicator readings
Conduct a water content test on specimen after unloading is complete

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Void Ratio-Pressure (e-log
1. calculate the height of solids, Hs
ws
s
s
AG
W
H Prove it.
2. calculate initial height of voids, Hv
sHHH
3. calculate initial void ratio, eo
s
v
s
v
v
v
o
H
H
AH
AH
V
V
e
4. for the 1st
incremental loading, 1 ( A
P1
) which causes a
deformation H1
sH
H
e 1
1

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5. calculate new void ratio after consolidation caused by 1
11 eee o
6. for the next loading, 2 ( A
PP 21
), which causes additional
deformation H2, the void ratio at the end of consolidation is
sH
H
ee 2
12
Note:- at the end of consolidation = /
Plot the corresponding e with /
on semi-logarithmic
paper. The typical shape of e-log /
will be as shown in
the figure.

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Change in void ratio vs. vertical effective stress
Void Ratio
Log /
Cc & Cs are determined from this relationship
Cc = Compression Index
Cs (Cr) = swelling (rebound) Index
Cc
Cs

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Normally Consolidated and Overconsolidated Clays
O
P
RCO ..
Log /
/
O
/
P
At past
p = max. preconsolidation pressure
At past
p = max. preconsolidation pressure
At present
o = Effective overburden pressure
At present
o = Effective overburden pressure
Normally Consolidated
Overconsolidate
d
po
po
May be removed due geologic processes or human processes
Relationship is linear and stepper when the effective stresses
is exceeding maximum past preconsolidation pressure

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Effect of Disturbance on Void Ratio-Pressure Relationship
Usually e-log /
founded by performing consolidation test
on undisturbed sample or remolded sample, does not
reflects the field (virgin) compression curve.
This difference is attribute to disturbance due to-
Handling and transferring samples into consolidation
cells.
Sampling and stress relief

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0.4eo
0.4eo
Laboratory compression curve
Field (virgin) compression curve, Slope =
Cc
Laboratory rebound curve
Slope = Cs (Cr)
p
/
= o
/
Laboratory compression curve
eo
Field (virgin) compression curve
Slope = Cc
eo
p
/
o
/
N.C. CLAY
O.C. CLAY

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Calculation of Settlement from One-Dimensional Primary
Consolidation

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o
c
o
sc
oo
o
s
svvvo
cco
e
e
H
e
AH
eeVAV
e
AH
e
V
V
eVVVVV
AAHHAVVV
1
1
11
)(
1
1

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N.C. CLAY
O.C. CLAY
)log(
11
)log(]log)[log(
log
o
o
o
c
o
c
o
o
coocc
H
e
C
e
e
H
CCe
e
C
eo
p
/
o
/
p
/
= o
/
o
/
+ /
eo
e
eo
p
/
o
/
o
/
+ /
)log(
11
log(]log)[log(
log
o
o
o
s
o
c
o
o
sooss
H
e
C
e
e
H
CCe
e
C
If o
/
+ / /
p
If o
/
+ / /
p
)log()log(
11
)log()log(
]log)[log(]log[log21
p
o
c
o
p
s
oo
c
p
o
c
o
p
s
pocops
CC
e
H
e
e
H
CC
CCeee

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Compression Index (Cc) and Swell Index (Cs)
Several correlations were suggested for Cc besides other eq.
and in most cases cs CtoC
10
1
5
1
Cc = 0.009(LL – 10) undisturbed clays LL= liquid limit
Cc = 0.007(LL – 7) remolded clays LL= liquid limit
=
o
/
+ /

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Secondary Compression (Consolidation) Settlement
Secondary compression settlement is a form of soil creep that
is largely controlled by the rate at which the skeleton of
compressible soils, particularly clays, silts, and peats, can yield
and compress.
Secondary compression is often conveniently identified to
follow primary consolidation when excess pore fluid pressure
can no longer be measured; however, both processes may occur
simultaneously.
Also referred to as the “secular effect”

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1
212 logloglog
t
t
e
tt
e
C
pe
C
C
1
1
2
log
t
t
HCs
Type of soil C
O.C clays 0.001 or less
N.C clays 0.005 to 0.03
Organic soil 0.04 or more
Linear relationship
e
t1 t2
ef

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Time Rate of Consolidation
Terzaghi (1925) proposed the first theory to consider the rate
of one-dimensional consolidation for saturated clay soils.
Assumptions:-
1. The clay-water system is homogenous.
2. Saturation is complete.
3. Compressibility of water is negligible
4. The flow of water is in one direction only (direction of
compression)
5. Darcy’
s law is valid

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To begin, consider a very small element of soil being subjected
to one-dimensional consolidation in the z-direction.
H=2Hdr
Depth
Sand
Sand
w
u
h
vz dxdy
A=dxdy
V=dxdydz
dx
dy
dz
(vz+( vz z)dz) dxdy

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Volume of pore fluid which flows out = Volume decrease of the
soil
and thus
Rate at which pore fluid flows out = Rate of volume decrease of
soil
t
V
dxdydz
z
v
t
V
dxdyvdz
z
v
v
z
z
z
z ])[(

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It will also be assumed that Darcy’s law holds and thus that
z
uk
z
h
kkiv
w
z
w
u
h
u = excess pore water pressure caused by the increase
of stress
t
V
dxdydzz
uk
w
1
2
2
During consolidation
t
V
e
t
e
V
t
V
t
eVV
t
V
t
V s
s
sssv )(
but 0
t
Vs

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t
e
V
t
V
s but oo
s
e
dxdydz
e
V
V
11
t
e
e
dxdydz
t
V
o1
t
e
ez
uk
ow 1
1
2
2
But uaae vv )(
av = coefficient of compressibility
t
u
m
t
u
e
a
z
uk
v
o
v
w 12
2
where o
v
v
e
a
m
1
mv = coefficient of volume compressibility
/
e

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or 2
2
z
u
c
t
u
v
where
o
v
w
vw
v
e
a
k
m
k
c
1
cv = coefficient of consolidation has units L2
/T
k = coefficient of permeability
This equation is the basic differential equation of Terzaghi,
s
consolidation theory and can be solved with the following
boundary conditions

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z = 0 u = 0 at a permeable boundary
z = 2Hdr u = 0 at a permeable boundary
t = 0 u = uo= initial excess pore water pressure
The solutions yields
vTM
m dr
o
e
H
MZ
M
u
u
2
0
)(sin
2
Where
)12(
2
mM

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drH
z
Z
, a depth factor dimensionless number
2
dr
v
v
H
tc
T
, a time factor is a nondimensoinal number
Because consolidation progresses by the dissipation of excess
pore water pressure, the degree of consolidation at distance z at
any time t is

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o
z
o
zo
z
u
u
u
uu
U 1
uz = excess pore pressure at time t
The variation of excess pore pressure within the layer is shown
in Figure below

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The average degree of consolidation for the entire depth of the
clay layer at any time t can be written as
m
m
TM
o
H
z
dr
c
tc v
dr
e
Mu
dzu
H
U
0
2
2
0)( 22
1
2
1
1
U = average degree of consolidation
c(t) = settlement of the layer at time t
c = ultimate (final) primary consolidation settlement

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The U – Tv relationship is represented in the figure below for the
case where uo is uniform for the entire depth of the
consolidating layer.

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The values of Tv and their corresponding average U for the case
presented above may also be approximately by the following
relationship:
For U = 0 to 60%
2
100
%
4
U
Tv
For U > 60% %)100log(933.0781.1 UTv
The following Table gives the variation Tv – U according to the
above equations

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Coefficient of Consolidation
Logarithm of Time method
It is particularly useful when there is significant secondary
compression (creep). The do point is located by selected two points
on the curve for which the times (t) are in the ratio 1:4, e.g. 1 min
and 4 min; or 2 min and 8 min.; the vertical intervals DB and BC
will be equal. The d100 point can be located in the final part of the
curve flattens sufficiently (i.e. no secondary compression). When
there is significant secondary compression, d100 may be located at
the intercept of straight line drawn through the middle and final
portions of the curve. Now d50 and log t50 can be located.
The coefficient of consolidation is therefore:

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50
2
2
50
50
197.0
t
H
c
H
tc
T dr
v
dr
v

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Square Root of Time method
After the laboratory results curve has been plotted, line AB is drawn,
followed by line AC in such a way that OBOC 15.1 : AC crosses the
laboratory curve at point D and locates 90t
The coefficient of consolidation is therefore:
90
2
2
90
90
848.0
t
H
c
H
tc
T dr
v
dr
v

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Hperbola method
It gives good results for U =60% - 90%.
The results are plotted on H
t
- t, then identify the straight portion bc of the
curve and project back to point d, and determine the intercept D, then
determine the slope m of bc
The coefficient of consolidation is therefore: D
mH
c dr
v
2
3.0

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Earl Stage log-t method
It gives the highest value while the conventional log-t method gives
the lowest value, this is due to the contribution of the lower part of
the consolidation curve in the conventional log-t method that means
the secondary compression plays a role in the value of cv, while in
this method cv obtained from the early stage log-t method, which
gives more realistic values of the fieldwork.
Follow the same steps in log-t method to locate do, draw a
horizontal line DE through do, then draw tangent through the point
of inflection F, the tangent intersects line DE at point G, determine
the corresponding time t corresponding to G, which is the time at U
= 22.14%, The coefficient of consolidation is therefore:

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14.22
2
0385.0
t
H
c dr
v

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Calculation of Consolidation Settlement under a Foundation
For limited area foundations (circular, square and rectangular), the
increase of effective stress ( /
) decrease with depth as shown in figure
below which can be estimated as described before in previous chapter.
Estimate /
o and /
av at the middle of the clay layer, then use the
previous equations in to determine final consolidation settlement.

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Using Simpson’
s rule
6
4 bmt
av
/
o /
m
/
b
/
t

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Alternative approach
Simply divide the clay layer to a number of sub layers, and
then estimate c for each sub layer taking into account effective
overburden pressure and an increase in effective stress at the
middle of each sub layer, then get the summation of
settlements of the sub layers to get the final consolidation of
the clay layer.

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Rate of consolidation
It is important to determine c – time relationship, which can
be helpful in estimating the differential settlement between
adjacent footings if the drainage condition at one footing
differs from the other.
c
tc
U
)(

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Examples
The following results were obtained from an oedometer test on
a specimen of saturated clay:
Pressure (kN/m2
) 27 54 107 214 429 214 107 54
Void ratio 1.243 1.217 1.144 1.068 0.994 1.001 1.012 1.024
A layer of this clay 8m thick lies below a 4m depth of sand, the
water table being at the surface. The saturated unit weight for
both soils is 19kN/m3
. A 4m depth of fill of unit weight 21
kN/m3
is placed on the sand over an extensive area. Determine
the final settlement due to consolidation of the clay. If the fill
were to be removed some time after the completion of

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consolidation, what heave would eventually take place due to
swelling of the clay?
H
e
ee
o
o
c
1
1
Appropriate values of e are obtained from e-log / drawn from
the result. The clay will be divided into four sub-layers, hence
H =2000 mm.

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Assuming the fill in pervious example is dumped very rapidly,
what would be the value of excess pore water pressure at the
centre of the clay layer after a period of 3 years? Assume Two-
way drainage condition and the value of cv is 2.4m2
/year.

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2
/2.35
58.0
84
1
mkNu
u
u
uu
U
z
z
o
zo
z

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In an oedometer test a specimen of saturated clay 19mm thick
reaches 50% consolidation in 20 min. How long would it take
a layer of this clay 5m thick to reach the same degree of
consolidation under the same stress and drainage conditions?
How long would it take the layer to reach 30% consolidation?

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7 compressibilty of soils

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