Consider a soil element:whose volume = V and Porosity = n
V, n
in equilibrium under principal stresses:σ1
σ3
σ2
with a pore water pressure:
u0
Then, the element is subjected to an increase in stress in all 3 directions of σ3 which
produces an increase in pore pressure of u3
V, n
σ1
σ3
σ2
u0
σ1 + σ3
σ2 + σ3
σ3 + σ3 u0 +u3
As a result, the effective stress in each direction increases by σ3 -u3
V, n
σ1 + σ3
σ2 + σ3
σ3 + σ3 u0 +u3
σ3 - u3
σ3 - u3
σ3 - u3
This increase in effective stress reduces the volume of the soil skeleton and the
pore space.
Soil Skeleton
V, n
σ3 - u3
σ3 - u3
σ3 - u3
33sss ΔuΔσVCΔV where:
Vss = the change in volume of the soil skeleton caused by an increase in the cell pressure, σ3
Cs = the compressibility of the soil skeleton under an isotropic effective stress increment;
i.e., the fraction of volume reduction per kPa increase in cell pressure
Pore Space (Voids)
V, n
σ3 - u3
σ3 - u3
σ3 - u3
3VVPS ΔuVCΔV where:
VPS = the volume reduction in pore space caused by a change in the pore pressure, u3
CV = the compressibility of the pore fluid; i.e., the fraction of volume reduction per kPa increase in pore pressure
Since: nVVV Then: 3u nVCΔV VPS
Assuming:
V, n
σ3 - u3
σ3 - u3
σ3 - u3
1. the soil particles are incompressible2. no drainage of the pore fluid
Therefore, the reduction in soil skeleton volume must equal the
reduction in volume of pore space
or:
Therefore: 3V33S nVΔCΔuΔσVC u
S
V
33
CC
n1
1ΔσΔu
The value:
V, n
σ3 - u3
σ3 - u3
σ3 - u3
is called the pore pressure coefficient, B
If the void space is completely saturated, Cv = 0 and B = 1
This is illustrated on Figure 4.26 in the text.
So, u3 = B σ3
S
V
CC
n1
1
When soils are partially saturated, Cv > 0 and B < 1In an undrained triaxial test, B is estimated by increasing the cell pressure by σ3 and measuring the resulting
change in pore pressure, u3 so that:
3
3
ΔσΔu
B
What happens if the element is subjected to an increase in axial (major principal) stress of σ1
which produces an increase in pore pressure of u1
V, n
σ1
σ3
σ2
u0
σ1 + σ1
σ2 - u1
σ3 - u1 u0 +u1
As a result, the effective stress in each of the minor directions increases by -u1
V, n
σ1 + σ1
σ2 - u1
σ3 -u1 u0 +u1
σ1 - u1
-u1
- u1
This change in effective stress also changes the volume of the soil skeleton and the pore
space.
If we assume for a minute that soil is an elastic material, then the Volume change of
the soil skeleton can be expressed from elastic theory:
V, n
- u1
- u3
σ1 - u1
11s31
ss 3ΔΔσVCΔV u
As before, the change in volume of the pore space:
Again, if the soil particles are incompressible and no drainage of the pore fluid, then:
1u nVCΔV VPS
1V11S31 nVΔC3ΔΔσVC uu
Since soils are NOT elastic, this is rewritten as:
where A is a pore pressure coefficient to be determined by experiment
u1 = AB σ1 or
A value of A for a fully saturated soil can be determined by measuring the pore water pressure during the
application of the deviator stress in an undrained triaxial test
For different values of σ1 during the test, u1 is measured, although the values at failure are of particular
interest: saturation for 1 BΔσΔu
A1
1 ,
V, n
- u1
- u3
σ1 - u1
11
S
V
1 BΔ31Δσ
CC
n1
131Δu
or:
ABA where ΔσAΔu 11
For lightly overconsolidated clays, 0 < A < 0.5
Figure 4.28 in the text illustrates the variation of A with OCR (Overconsolidation Ratio).
In highly compressible soils (normally consolidated clays), A ranges between 0.5 and 1.0
For heavily overconsolidated clays, A may lie between -0.5 & 0
V, n
- u1
- u3
σ1 - u1
Normally Consolidated
Lightly Over-Consolidated
stress effective currentstress effective historical maximum
OCR
Heavily Over-Consolidated
The third pore pressure coefficient is determined from the response, u to a
combination of the effects of increasing both the cell pressure, σ3 and the axial stress (σ1 -
σ3) or deviator stress.From the two previous effects: u = u3 + u1
u = B[σ3+A(σ1-σ3)]
or:
If we divide through by σ1
u3 + u1
=
u3 = Bσ3
u1 = BA(σ1-σ3){
1
3
1
3
1 ΔσΔσ
1AΔσΔσ
BΔσΔu
1
3
1 ΔσΔσ
1A)(11BΔσΔu
or: BΔσΔu
1
The third pore pressure coefficient is not a constant but depends on σ3 and σ1
With no movement of water (undrained) and no change in water table level during subsequent consolidation, u = initial excess pore water pressure in fully saturated
soils.Testing under Back
Pressure
BΔσΔu
1
When a sample of saturated clay is extracted from the ground, it can swell thereby decreasing Sr as it breathes
in airThe pore pressure can be raised artificially (in sync with σ3) to a datum value for excess pore water pressure and
then the sample can be allowed to consolidate back to the in situ conditions (saturation, pore water pressure).
This process allows the calculation of the pore pressure coefficient, B.
Values of B 0.95 are considered to represent saturation.
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