1. Pore Pressure Coefficients
in equilibrium under principal stresses:
whose volume = V and Porosity = n
Consider a soil element:
with a pore water pressure:
Pore Pressure Coefficients σ1 σ3 u0
V, n σ2

2. Pore Pressure Coefficient, B
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 σ1
σ1 + σ3 σ3
σ3 + σ3
u0 +u3 u0
V, n σ2
σ2 + σ3

3. Pore Pressure Coefficient, B
This increase in effective stress reduces the volume of the soil skeleton and the pore space.
As a result, the effective stress in each direction increases by σ3 -u3
σ1 + σ3
σ3 - u3
σ3 - u3
σ3 + σ3
u0 +u3
V, n
σ3 - u3
σ2 + σ3

4. Pore Pressure Coefficient, B
Soil Skeleton
σ3 - u3
V, n
σ3 - u3
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

5. Pore Pressure Coefficient, B
Pore Space (Voids)
σ3 - u3
V, n
σ3 - u3
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:
Then:

6. Pore Pressure Coefficient, B
Assuming:
the soil particles are incompressible
no drainage of the pore fluid
σ3 - u3
Therefore, the reduction in soil skeleton volume must equal the reduction in volume of pore space
V, n
σ3 - u3
Therefore:
or:

7. Pore Pressure Coefficient, B
The value:
σ3 - u3
V, n
σ3 - u3
is called the pore pressure coefficient, B
So, u3 = B σ3
If the void space is completely saturated, Cv = 0 and B = 1
In 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:
When soils are partially saturated, Cv > 0 and B < 1
This is illustrated on Figure 4.26 in the text.

8. Pore Pressure Coefficient, A
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 σ1
σ1 + σ1 σ3
σ3 - u1
u0 +u1 u0
V, n σ2
σ2 - u1

9. Pore Pressure Coefficients
This change in effective stress also changes the volume of the soil skeleton and the pore space.
As a result, the effective stress in each of the minor directions increases by -u1
σ1 + σ1
σ1 - u1
- u1
σ3 -u1
u0 +u1
V, n
-u1
σ2 - u1

10. Pore Pressure Coefficients
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:
- u1
V, n
- u3
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:

11. Pore Pressure Coefficient, A
V, n
- u3
Since soils are NOT elastic, this is rewritten as:
u1 = AB σ1 or
For different values of σ1 during the test, u1 is measured, although the values at failure are of particular interest:
where A is a pore pressure coefficient to be determined by experiment
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

12. Pore Pressure Coefficient, A
Normally Consolidated
σ1 - u1
Figure 4.28 in the text illustrates the variation of A with OCR (Overconsolidation Ratio).
Lightly Over-Consolidated
- u1
V, n
Heavily Over-Consolidated
- u3
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
For lightly overconsolidated clays, 0 < A < 0.5

13. Pore Pressure Coefficient, B
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
If we divide through by σ1
u3 = Bσ3 {
u3 + u1 =
u = B[σ3+A(σ1-σ3)] {
u1 = BA(σ1-σ3)
or:
or:

14. Pore Pressure Coefficient, B
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
This process allows the calculation of the pore pressure coefficient, B.
When a sample of saturated clay is extracted from the ground, it can swell thereby decreasing Sr as it breathes in air
The 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).
Values of B  0.95 are considered to represent saturation.