Unsaturated Zone Calculations

Unsaturated Zone Calculations section under Strength tab provides extension to the conventional geomechanical constitutive models to accommodate for the unsaturated soil behavior. RS3 provides two approaches, including Unsaturated Shear Strength and Single Effective Stress. In short, the Unsaturated Shear Strength provides direct control over how shear strength evolves with changes in pore pressure, whereas Single Effective Stress approach modifies the effective stress formulation by applying a suction (or degree of saturation)-related coefficient to the pore pressure term. Barcelona Basic model (Strength > Failure Criterion > Softening / Hardening > Barcelona Basic) is a constitutive model based on the Cam-Clay type effective stress model for the behavior of unsaturated clay-type soils.

Unsaturated Shear Strength

The Unsaturated Shear Strength models, including Fredlund (1978) and Vanapalli (1996), are provided under Unsaturated Zone Calculation section. These models are extensions of the Mohr–Coulomb criterion, enhancing its ability to represent the development of shear strength in relation to soil moisture conditions. They are only accessible when the Mohr–Coulomb criterion is selected and the Material Type is set to Plastic.

Fredlund (1978)

Suggested by Fredlund et al (1978), the unsaturated shear strength is calculated as:

whereis the effective saturated cohesion,is the total stress,is the net normal stress on the plane of failure at failure,is the matric suction,is the pore-air pressure,is the pore water pressure,is the effective saturated angle of friction, andis the angle of friction accounting for matric suction contribution to shear strength. Note that the Unsaturated Shear Strength Angle is usually not a well-known quantity. To obtain an appreciation of its importance, a parametric study can be carried out, in which the Unsaturated Shear Strength Angle is varied between 0 and the Friction Angle of the material.

Air Entry Value

Air Entry value is defined in Fredlund and Rahardjo (1993) as the matric suction value that must be exceeded before air recedes into the soil pores. The Air Entry Value is also referred to as the "displacement pressure" in petroleum engineering or the "bubbling pressure" in ceramics engineering.

In terms of strength, it is the point at which the strength of the soil deviates from its saturated effective Mohr-Coulomb behavior. If matric suction exceeds the air entry value, the strength is assumed to follow a linear envelope with as the friction angle. Thus RS3 uses a bilinear strength envelope where:

• and are used with positive pore pressures, and with negative pore pressures (matric suction) up to the air entry value
• is used when negative pore pressure exceeds the air entry value.

The following figure (after Fredlund (2000) ) illustrates the effect of the air entry value. For matric suction less than the air entry value, the saturated effective stress friction angle is used. For matric suction greater than the air entry value, the unsaturated shear strength angle, is used.

Maximum Negative Pore Water Pressure for Unsaturated Shear Strength

This is an optional parameter to set the threshold impact of suction for the effective stress calculation of soil above the water table. The essence of this parameter is different from the one under Hydraulics tab or Project Settings > Groundwater tab. The one under Hydraulics tab is the limitation applying to the in effective stress principle (' = - ) and the one under Fredlund (1978) model sets the limitation to the same parameter in shear strength formula ().

Vanapalli (1996)

Proposed by Vanapalli et al (1996), the unsaturated shear strength can be represented using either the degree of saturation or the water content.

With degree of saturation,

whereis the residual degree of saturation andis degree of saturation.

With water content,

whereis the water content,is the residual water content, andis the saturated water content. The water content or degree of saturation values are to be inputted on the Hydraulic tab of the Define Material Properties dialog.

Single Effective Stress

The Single Effective Stress calculation approach is available for all failure criteria models except for the Barcelona model. According to Bishop (1959), the single effective stress for unsaturated soils is written as:

Where is the effective stress, is the total stress, is the surface fraction coefficient, is the pore-air pressure, and is the pore water pressure. In practice, is omitted from the equation, thus the effective stress will be calculated as:

where is the suction, and is the unit matrix.

In RS3, the value ofcan be determined by nine different methods. Note that=1 for saturated soils when the Unsaturated Zone Calculations is off. For more information, see the “Soil Behaviors in Unsaturated Zones” theory manual for explanation on each method below:

1. Bishop (1959)
2. Tabular Values – with respect to suction
   • Users define a table ofvs. matric suction values.
3. Tabular Values – with respect to degree of saturation
   • Users define a table ofvs. degree of saturation ( ) values.
4. Tabular Values – with respect to effective degree of saturation
   • Users define a table of vs. effective degree of saturation () values.
5. Gudehus (1995)
6. Khalili (2004)
7. Bolzon (1996)
8. Aitchison (1961)
9. Kohgo (1993)

Use Cutoff

The Use Cutoff option is subjected to the Single Effective Stress approach only, intended for limiting the suction (). In the calculation based on the equation above, an upper limit cutoff value can be set for the term. The default input is -100 kPa.

Barcelona Basic

The Barcelona Basic model is based on the Cam-Clay type effective stress model for the behavior of unsaturated clay-type soils. This constitutive model can be adopted for a material property by selecting Failure Criterion > Softening/Hardening > Barcelona Basic under Strength tab. Alternative to single effective stress, the effect of suction on mechanical behaviors can be directly interpreted through this constitutive model that account for the matric suction.

In such cases, the effective stress for fully saturated zone will be calculated as usual, however, in the unsaturated zone, the total stress will be employed () and the unsaturated behavior will be considered in the constitutive model.