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Effective Stresses Calculation in Undrained Materials

RS2 offers an undrained effective stress analysis for soil layers in combination with effective strength parameters. RS2 uses the effective stress and strength and stress parameters to model undrained behaviors. Pore pressure development is critical in identifying the correct effective stress path which leads to a realistic failure value of undrained shear strength (cu or su).

1. Undrained Effective Stress Calculation

When the undrained behaviour is selected for the material, special elements developed in RS2 will be automatically selected to account for the numerical stability when dealing with incompressible material.

Pore pressure in a soil body can usually be attributed to the presence of water and contributes to the total stress level within the soil. Terzaghi’s principle (see also 1.5) states that total stresses (σ) are divided into effective stresses (σ’), pore pressure (p), and pore water pressure (pw). Note that water does not sustain shear stress, therefore total shear stress is equal to effective shear stress.

σ = σ’ + mp (1.1a)


p = αSepw (1.1b)

m =

σxx = σ’xx + αSep w (1.1c)

σyy = σ’yy + αSep w (1.1d)

σzz = σ’zz + αSep w (1.1e)

σxy = σ’xy (1.1f)

σyz = σ’yz (1.1g)

σzx = σ’zx (1.1h)

Where α = Biot’s pore pressure coefficient and Se = effective degree of saturation. For incompressible grains, Biot’s pore pressure coefficient becomes α = 1. Additional details about compressible grains and compressible solid material (α < 1) are provided in the Biot’s coefficient section below.

Pore pressure is the product αSepw and is termed p in RS2. An additional distinction must be made between steady state pore stress (psteady) and excess pore stress (pexcess).

pw = psteady + pexcess (1.2)

Steady state pore pressures (psteady) are determined based on phreatic levels or by a groundwater flow calculation; this is considered input data. During plastic calculations, excess pore pressures are generated for undrained material behaviour or during a consolidation analysis. The following equations can be used to describe undrained material behaviour and related excess pore pressure calculations:

The time derivative of the steady state component is equal to zero, therefore:

Spw = Spexcess (1.3)

Inverting Hooke’s law gives the following:

Inverting Hooke’s law gives the following = Inverting Hooke’s law gives the following Inverting Hooke’s law gives the following (1.4)

Substituting in Equation 1.1 gives the following:

Result of substituting eq 1.1 = Result of substituting eq 1.1 Result of substituting eq 1.1 (1.5)

In cases with slightly compressible water, excess pore pressure can be defined using the following:

Equation 1.6a (1.6a)
Equation 1.6b (1.6b)
Equation 1.6c (1.6c)

Kw represents the bulk modulus of the water, Ks represents the bulk modulus of the solid material, Cw represents the compressibility of the water, Cs is the compressibility of the solid material, and n is the porosity of the soil.

Equation 1.7 (1.7)

e0 is the initial void ratio, as defined in the soil properties. The inverted form of Hooke’s law may be written in terms of the total stress rates and the undrained parameters Eu and vu:

Equation 1.8 (1.8)


Equation 1.9 Equation 1.9 Equation 1.9 (1.9)

where B represents Skempton’s B-parameter.

In RS2 undrained models, the effective parameters G and v’ are converted into their undrained counterparts Eu and Vu (see Equation 1.9 above).

To obtain realistic results, the bulk modulus of the water must be higher than the bulk modulus of the soil skeleton (Kw >> nK’). This can be achieved by requiring that v’ ≤ 0.35.

2. Skempton B-Parameter

When using undrained drainage conditions, RS2 employs an implicit undrained bulk modulus (Ku) for the soil in its entirety (soil skeleton and water). The following equations are used to differentiate between total stress rates, effective stress rates, and rates of excess pore pressure:

Total stress: Total Stress - Equation 2.1a (2.1a)
Excess pore pressure: Excess Pore pressure - equation 2.1b (2.1b)
Effective stress: Effective Stress - equation 2.1c (2.1c)

Undrained effective stiffness parameters should be entered with material properties (define E’ and v’, not Eu and Vu). Hooke’s law of elasticity is used to automatically calculate the undrained bulk modulus:

Equation 2.2 (2.2)

When the user manually inputs Skempton’s B-parameter and Biot’s pore pressure coefficient, vu is calculated using the following equation:

equation 2.3 (2.3)

The value of Skempton’s B-parameter is calculated using the ratio of the bulk stiffnesses of the soil skeleton and the pore fluid:

Equation 2.4 (2.4)

Where Kw = 2 * 106 kN/m2.

The following equation is used to calculate the rate of excess pore pressure using the (small) volumetric strain rate:

equation 2.5 (2.5)

The special undrained behaviour option in RS2 requires calculations that using effective stiffness parameters, with a distinction between effective stresses and excess pore pressures. The calculations may not completely address shear induced effective pore pressures.

This analysis requires effective soil parameters and is therefore very useful when these parameters are available. However, with soft soil projects, accurate effective soil parameter data may not be accessible. Therefore, in situ and laboratory tests may be performed to obtain undrained soil parameter data. In these cases, Hooke’s law can be used to convert measured values of undrained Young’s moduli into effective Young’s moduli:

Equation 2.6 (2.6)

This direct conversion from measured to effective values is not possible for advanced models. In these advanced cases, it is suggested to estimate the effective stress parameter from the measured parameter, then perform an undrained test to verify the resulting undrained stiffness (and adapt the estimated effective stiffness value if needed). The Soil test facility (Reference Manual) may be used in these circumstances.

3. Biot Pore Pressure Coefficient, α

Generally, the compressibility of the soil skeleton will be greater than that of the individual grains of soil in a mass; therefore, deformations of individual soil grains can be disregarded. In cases with deep soil layers at high pressures, the stiffness of the soil/rock matrix will approach that of the material of the soil/rock grains; in these cases, the compressibility of the solid material must be considered. This affects the division of total stress into effective stress and pore pressure. In cases with compressible solid material, Terzaghi’s effective stress can be defined using the following equation:

Equation 3.1 (3.1)

Where α is Biot’s pore pressure coefficient, Se is the effective degree of saturation, m is a vector (with unity values for normal components and 0-values for shear components), and pw is the pore water pressure. The following equation shows the definition of Biot’s pore pressure coefficient:

Equation 3.2 (3.2)

Where K’ is the effective bulk modulus for the soil matrix and Ks is the bulk modulus of the solid material. Note that for an incompressible solid material (Ks = ∞), Terzaghi’s original stress definition holds true. Lower values of α indicate that for given values of total stress and pore water pressure, the resulting effective stress is higher than that of an incompressible solid material (α = 1).

The value of Biot’s pore pressure coefficient is automatically calculated by RS2, but the value may be changed manually by the user.

The following information for undrained behavior in RS2 can be found here.

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