Two approaches used in determining the relationship between pore pressure parameters

Introduction

A soil usually comprises of solid particles, soil water and soil air. When the soil is saturated, it comprises of solid particles and soil water. If the soil is subjected to load, the soil witnesses total stress which comprises of effective stresses (on solid particles) and pore water pressure. if the load is sustained, the pore water pressure dissipates from the pore spaces and all the load is transferred to the solid particles. This is where the Terzaghi’s effective stress principle (formulated in 1936) finds much relevance.

The Principle states thus: when stress is applied to a porous material, it is opposed by the fluid pressure filling the pores in the material

To determine the effective stress of a soil, it is necessary to know the pore water pressure of the soil because

Effective stress (σ¹) = total stress (σ) – pore water pressure (u)

i.e. σ¹ = σ – u

In the field, this pore water pressure is usually measured by installing piezometers. But in some cases, it is usually very difficult to install piezometers to measure the pore water pressure directly in the field. This shortcoming prompted Skempton, A.W. (1954) to develop a theoretical method that uses some parameters, A and B, to express the response of pore pressure due to changes in the total stresses under undrained conditions.

This because in the conventional triaxial test, the specimen is first isotropically consolidated under a pressure of Δσ₃, then it is sheared under a deviator stress of (Δσ₁ – Δσ₃).

The pore pressure parameters are derived under isotropic consolidation which is represented by parameters B and under deviatoric stress which is represented by parameter A.

There exists also a relationship between these two parameters which expresses the wholistic pore pressure effect on a soil subjected to both isotropic consolidation (cell pressure) and deviatoric stress (additional stresses).

Mathematically,

B = Δu_a/Δσ₃ , where Δu_a = change in pore pressure due to cell pressure.

A = Δu_d/(Δσ₁ – Δσ₃), where Δu_d = change in pore pressure due to deviator stress.

Δσ₁ – Δσ₃ = change in deviator stress

Approach 1

Parameter B

The increase in effective stress in a test

Δσ₃ = Δσ₃ – Δu_a

If Cs is the compressibility of the soil stratum, the volume change in the soil structure

ΔV_s = -C_s. V (Δσ – Δu_a)

Where V is the original volume of the sample. If C_w is the compressibility of the fluid (air and water) in the voids and if ‘n’ is the porosity of the soil, then the change in volume in the space is ΔV_w = -C_w . nV. Δu_a

But the two changes in value are identical.

Hence, Δu_a/Δσ₃ = B = 1/ (1 + (nC_v/C_s))

Parameter A

The increase in effective stress in a test as a result of increase in deviator stress (σ₁ – σ₃) is

Δσ₁¹ = (Δσ₁ –  Δσ₃) – Δu_d

The volume change of the soil structure under the increment of deviator stress is

ΔV_s = -C_s. V.1/3 (Δσ₁¹ + 2Δσ₃¹)

Or ΔV_s = -C_s. V.1/3 [(Δσ₁ + Δσ₃) – 3Δu_d]

The volume change in the void space is

ΔV_w = -C_v . nV. Δu_d

The two volume changes are identical and hence

Δu_d = 1/ (1 + (nC_v/C_s)) . 1/3 (Δσ₁ + Δσ₃)

Or Δu_d = B . 1/3 (Δσ₁ + Δσ₃)

Since the bahaviour of soil is not in accordance with elastic theory, Δu_d = B . A (Δσ₁ + Δσ₃) because A = 1/3 for a perfectly elastic material.

Generally, Δ_u = Δu_a + Δu_d

Δu_a = B.Δσ₃; Δu_d = B.A (Δσ₁ – Δσ₃)

Δu = B.Δσ₃ + B.A (Δσ₁ – Δσ₃)

Δu = B [Δσ₃ + A (Δσ₁ – Δσ₃)]

Approach 2

Consider a cube of saturated isotropic clay of volume, V (Figure 1)

Figure 1: Saturated isotropic clay cube

If the clay is subjected to an increase of Δσ₁, there will be a resulting decrease in volume and an increase in Δu in the pore water pressure. The increase in the effective principal stresses will be,

Δσ₁¹ = Δσ₁ – Δu

Δσ₂¹ = Δσ₂ – Δu

Δσ₃¹ = Δσ₃ – Δu

And the decrease in volume will be,

-Δu = V. ((1-2µ)/E) . {Δσ₁¹ + Δσ₂¹+ Δσ₃¹}

Where E = modulus of elasticity and µ = Poisson’s ratio

The decrease in volume will be due to a decrease in the volume of the voids and if no drainage occurs,

-ΔV = n . V . C_w . Δu

Where ‘n’ = porosity of the soil

Equating the two values for -ΔV,

n . V . C_w . Δu = V. ((1-2µ)/E) . {Δσ₁¹ + Δσ₂¹+ Δσ₃¹}

In the triaxial test, Δσ₃ = Δσ₂ and compressibility of the soil skeleton C_s is taken as:

(3 (1-2µ))/E [assuming an elastic material]

n . C_w . Δu = C_s/3 {Δσ₁ – Δu + Δσ₃ – Δu + Δσ₃ – Δu}

Therefore, n . (C_v/C_s) . Δu = (Δσ₁/3) + (2Δσ₃/3) – Δu

Δu (((nC_v)/C_s) + 1) = Δσ₃ + 1/3 (Δσ₁ – Δσ₃)

Δu = [1/(((nC_v) / C_s) + 1)]. [Δσ₃ + 1/3 (Δσ₁ – Δσ₃)]

Δu = B [Δσ₃ + A (Δσ₁ – Δσ₃)]

Note

1. In a fully saturated soil, the compressibility of the pore water (C_v) is negligible compared to that of the soil mass (C_s). Therefore, the ratio C_v/C_s tends to zero and parameter B becomes equal to unity (1).
2. In partially saturated soil, the compressibility of the air in the voids is high. The ratio C_v/C_s is greater than unity and parameter B is less than unity.
3. Parameter B varies with degree of saturation and stress change.
4. Parameter A varies with soil, its stress history and the applied deviator stress.

Uses/Applications of pore pressure parameters

In some construction works such as earth embankment or earth dam over soft clay deposits, if the rate of construction is such that pore water pressure induced in the foundation soil cannot be dissipated, undrained condition prevails. If the pore pressure developed is excessive, the shear strength of the foundation which is dependent on effective stresses decrease, thereby endangering the stability of the foundation.

Prediction of pore pressure changes with increase in the total stresses due to increase in the height of the embankment may be done using the pore pressure parameters. This would help to ensure the stability of the foundation.

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