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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
4. COMPRESSIBILITY OF SOILS 4.1 INTRODUCTION Foundations of
structures usually rest on soil. Due to the additional stress
introduced by a foundation or due to other loadings, a soil mass
will compress. This compression is a sum of different and distinct
processes:
a. Deformation of soil particles b. Relocation of soil
particles, and c. Expulsion of water or air from the void
spaces
In general, soil settlement caused by different loadings can be
categorized into three categories: i. Immediate settlement (or
elastic settlement) Such a settlement is caused by the elastic
deformation of dry soil and of moist and saturated soils without
any change in the moisture content. Immediate settlement
calculations are generally based on equations derived from the
theory of elasticity. ii. Primary consolidation settlement The
volume of saturated soils under pressure will decrease due to the
expulsion of water from the void spaces. The associated settlement
is known as primary consolidation settlement. It is usually the
predominant type of settlement in fine grained soils, especially in
clays. iii. Secondary consolidation settlement This type of
settlement is observed in saturated cohesive soils and is the
result of the plastic adjustment of soil fabrics. If follows
primary consolidation settlement when soil is under a constant
effective stress. The total settlement of a foundation can then be
given by
ST = Sc + Ss + Se where ST = total settlement Sc = primary
consolidation settlement Ss = secondary consolidation settlement Se
= immediate settlement The total settlement obtained in this
fashion must be within acceptable limits for any structure. For
most practical purposes, the magnitude of the elastic settlement
can be ignored compared to the consolidation settlements.
Discussion here shall hence be limited to consolidation settlement.
But it must be noted that the volume change due to expulsion of air
is treated in geotechnical engineering under the title compaction
characteristics of soils. Prior to discussion of soil settlements,
discussing the nature of stresses in soils must be presented. It is
due to stresses that soils undergo settlements.
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
4.2 EFFECTIVE STRESS Stresses in a soil mass may be caused by
the self weight of the soil, and also by the external loads which
may be applied to the soil. The pattern of stresses caused by the
external loads is usually a complicated one, and the same may be
the case for stresses induced by self weight of soil. However,
there is one common situation in which the self weight of the soil
gives rise to a very simple pattern of stresses-that is when the
ground surface is horizontal and the nature of soil does not vary
significantly in the horizontal direction. The stress caused in
such a situation at a point in a soil mass is called the geostatic
stress. However, if external loading contributes to this stress,
the stress will simply be called as the total stress. In the
geostatic situation, there cannot occur any shear stresses upon the
vertical and horizontal planes within the soil. Hence, the vertical
geostatic stress, say σz, may be computed simply by considering the
weight of soil and weight of water above that depth.
In the figure above a soil column is standing with no seepage
allowed. The total stress at the elevation of point A can be
obtained from
( ) satAw HHH γγσ −+= This total stress can be divided into two
parts
1. Effective stress or intergranular stress, σ’ – the sum of the
vertical components of the forces developed at the points of
contact of the solid particles per unit cross-sectional area of the
soil mass.
2. Pore water pressure or neutral stress, u – a portion carried
by water in the continuous void spaces. This portion acts with
equal intensity in all directions.
A
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
The total stress can thus be rewritten as u+= 'σσ
or u−= σσ '
Substituting the values for the total stress and the pore water
pressure into this equation
Hence, it can be easily stated that the effective stress at a
point in a soil mass will be equal to its submerged unit weight
multiplied by the depth of soil mass up to the considered point.
The effective stress principle is arguably the most important
principle in geotechnical engineering. The compressibility and
shearing resistance of a soil depend to a great extent on the
effective stress. Thus the concept of effective stress is
significant in solving geotechnical engineering problems, such as
lateral earth pressure on retaining structures, the load bearing
capacity and settlement of foundations, and the stability of earth
slopes. 4.3 CONSOLIDATION SETTLEMENT When a clayey soil is
subjected to loading, elastic settlement occurs immediately. If the
loading continues, then its volume goes on compressing gradually
due to removal of water from the pores. This major compression,
which goes on for months and years, is the consolidation, and
strictly called the primary consolidation. During this process, the
transfer of load from the soil pores to the soil grains takes
place. Due to this gradual transfer of the excess hydrostatic
pressure from the water to the soil solids, a change in the initial
structure of the soil mass (plastic adjustment of soil fabric) may
occur, which may cause some compression, called secondary
compression or secondary consolidation. The time-dependent
deformation of saturated clayey soil can best be understood by
considering a simple model that consists of cylinder with a spring
at its center and filled with water. Three cases may be
observed.
a. Place a load, P, on the spring and keep the valve closed.
There will be no deformation of the spring in this case. The excess
hydrostatic pressure at this time can be given as
APu =∆
b. If the valve is opened, the water will flow outward. This
flow will be accompanied by a reduction of the excess hydrostatic
pressure and an increase in the compression of the spring. If Ps
and Pw denote portions of P carried by the spring and the water
respectively, after valve opening
Ps > 0 and Pw < P, i.e. APu
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
c. Finally the excess hydrostatic pressure dissipates, and the
system will reach a state of equilibrium, and thus Ps = P and Pw =
0 These equations mean that after the excess pore water pressure
dissipates the additional stress P will be carried by the soil
skeleton, i.e. 'σσ =∆
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
One-dimensional laboratory consolidation test The process of
consolidation settlement is studied in the laboratory by the
one-dimensional consolidation test. This test is performed in a
consolidometer (sometimes referred to as an oedometer). The
schematic diagram of a consolidometer is shown below.
The soil specimen is placed inside a metal ring with two porous
stones, one at eh top of the specimen and another at the bottom.
The specimens are usually 64mm in diameter and 25mm thick. The load
on the specimen is applied through a lever arm, and compression is
measured by a micrometer dial gauge. The specimen is kept under
water during the test. Each load is usually kept for 24 hours.
After that, the load is usually doubled, which doubles the pressure
on the specimen, and the compression measurement is continued. At
the end of the test, the dry weight of the specimen is determined.
From load and deformation values, a plot of deformation versus time
for a given load increment is prepared.
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
Form this plot; we can observe three distinct stages, which may
be described as: Stage I Initial compression, which is caused
mostly by preloading Stage II Primary consolidation Stage III
Secondary consolidation After time-deformation plots for various
loadings are obtained in the laboratory, it is necessary to study
the change in the void ratio of the specimen with pressure. The
procedure is described as follows:
1. Calculate the height of solids, Hs, in the specimen using the
equation
Solid
Specimen area =A
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
2. Calculate the initial height of voids as
Hv = H – Hs 3. Calculate the initial void ratio, eo, of the
specimen
4. For the first incremental loading, σ1, which causes a
deformation ∆H1, calculate the change in
void ratio as
sHHe 11
∆=∆
5. Calculate the new void ratio after consolidation caused by
the pressure increment as e1 = eo – ∆e1
For the next loading, σ2 (that equals the cumulative load per
unit area of the specimen), which causes additional deformation
∆H2, the void ratio at the end of consolidation can be calculated
as
sHHee 212
∆−=
At this time, total stress σ2 = effective stress, σ’2.
Proceeding in a similar manner, one can obtain the void ratios at
the end of the consolidation for all load increments. The effective
stress and the corresponding void ratios (e) at the end of
consolidation are plotted on semi logarithmic graph paper. The
typical shape of such a plot is shown below.
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
This figure shows that the upper part of the e-log σ’ plot is
somewhat curved with a gentle slope, followed by a linear
relationship for the void ratio with log σ’ having a steeper slope.
This is an important phenomenon that required further explanation.
A soil in the field at some depth has been subjected to a certain
maximum effective past pressure in its geologic history. This
maximum effective past pressure may be equal to or less than the
existing effective overburden pressure at the time of sampling. The
reduction of effective pressure in the field may be caused by
natural geologic processes or human activities. During sampling,
the existing effective overburden pressure is also released, which
results in some expansion. When this specimen is subjected to a
consolidation test, a small amount of compression (i.e. a small
change in void ratio) will occur when the effective pressure
applied is less than the maximum effective overburden pressure in
the field to which the soil has been subjected to in the past. When
the effective pressure on the specimen becomes greater than the
maximum effective past pressure, the change in void ratio is much
larger, and the e-log σ’ relationship is practically linear with a
steeper slope. This relationship can be verified in the lab by
loading the specimen to exceed the maximum effective overburden
pressure, and then unloading and reloading again. The e-log σ’ plot
for such cases is shown below, in which cd represents the unloading
and dfg represents the reloading process.
This discussion leads to two basic definitions of clay based on
stress history:
1. Normally consolidated clays, whose present effective
overburden pressure is the maximum pressure that the soil was
subjected to in the past
2. Overconsolidated clays, whose present effective overburden
pressure is less than that which the soil experienced in the past.
The maximum effective past pressure is called the preconsolidation
pressure.
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
The preconsolidation pressure can be obtained from the e-log σ’
plot following the procedure suggested by Casagrande.
i. By visual inspection, establish point a, at which the e-log
σ’ plot has a minimum radius of curvature
ii. Draw horizontal line ab iii. Draw the line ac tangent to the
curve at a iv. Draw the line ad, which is the bisector of the angle
bac v. Project the straight-line portion gh of the plot back to
intersect line ad at f. The abscissa of point f
is the preconsolidation pressure, σ’c
The overconsolidation ratio (OCR) for a soil can now be defined
as
''
σσ cOCR =
where σ’c = preconsolidation pressure of a specimen σ’ = present
effective vertical pressure Settlement calculation from
one-dimensional primary consolidation
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
Let us consider a saturated soil layer of thickness H and cross
sectional area A under an existing average effective overburden
pressure σ’o. Because of an increase of effective pressure, ∆σ’,
let the primary consolidation settlement be Sc. Thus the change in
volume can be given by ∆V = Vo – V1 = HA – (H –Sc) A = ScA But the
volume change is due to the change in the volume of voids, ∆Vv ∆Vv
= ∆eVs But, the volume of solids
oo
os e
AHe
VV+
=+
=11
Thus
∆V = ScA = ∆eVs = eeAH
o
∆+1
or
Sc = oe
eH+∆
1
For normally consolidated clays that exhibit a linear e-log σ’
relationship ∆e = Cc [ log (σ’o + ∆σ’) - log σ’ ] Where Cc is the
slope of the e-log σ’ plot and is defined as the compression index.
Substitution of this value into the equation of Sc yields
Sc = ⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆++ o
o
o
c
eHC
'''log
1 σσσ
In overconsolidated clays, for σ’o + ∆σ’ ≤ σ’c, the slope of the
plot is defined as the swell index, Cs. Hence,
Solid
Soil
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
∆e = Cs [ log (σ’o + ∆σ’) - log σ’ ] The settlement would then
be
Sc = ⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆++ o
o
o
s
eHC
'''log
1 σσσ
If σ’o + ∆σ’ > σ’c, then
Sc = ⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆++
+⎟⎟⎠
⎞⎜⎜⎝
⎛+ c
o
o
c
o
c
o
s
eHC
eHC
'''log
1''log
1 σσσ
σσ
However, if the e-log σ’ curve is available, one can simply pick
∆e off the plot for the appropriate range of pressures. This number
may be substituted directly into the equation of Sc. Several
empirical relations have also been suggested for Cc and Cs. The
common formulae for clays are the following. Cc = 0.009 (LL-10)
Cs = sGLL
⎥⎦⎤
⎢⎣⎡
100(%)0463.0
SECONDARY CONSOLIDATION SETTLEMENT We have noted earlier that at
the end of primary consolidation settlement some settlement is
observed due to the plastic adjustment of soil fabrics. This stage
of consolidation is called secondary consolidation. During this
phase the plot of deformation against the log of time is
practically linear. The variation of the void ratio, e, with the
time t for a given load increment is shown below.
From this figure, the secondary compression index, Cα, can be
defined as
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Soil Mechanics-I(CENG-2202) Chapter 4 : Compressibility of
soils
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Department of Civil Engineering, Faculty of Technology
Addis Ababa University
⎟⎠⎞⎜
⎝⎛∆
=−∆
=
1
212 logloglog tte
tteCα
The magnitude of the secondary consolidation can be calculated
as
⎟⎠⎞⎜
⎝⎛=
1
2log' ttHCSs α
where Pe
CC+
=1
' αα
eP = void ratio at the end of primary consolidation H =
thickness of clay layer The values of C’α as observed in various
natural deposits are as follows:
• Overconsolidated clays = 0.001 or less • Normally consolidated
clays = 0.005 to 0.03 • Organic soil = 0.04 or more
Secondary consolidation settlement is more important than
primary consolidation settlement in organic and highly compressible
inorganic soils. In overconsolidated inorganic clays, the secondary
compression index is very small and of less practical
significance.