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Liner Type: Standard Beam

The Standard Beam liner type can be used to model a liner which has flexural rigidity (i.e. resistance to bending), such as a simple shotcrete or concrete liner. A Standard Beam liner is made up of "beam elements" which can respond to flexural, axial (compressive or tensile) and shear loads.

The Standard Beam liner type can also be used to model more complex liners by using the Area and Moment of Inertia option to define the liner cross-section. Alternatively you can use the Reinforced Concrete liner type which is specially designed for this purpose.

NOTE: see the Standard Beam versus Reinforced Concrete topic for a discussion of the differences between these two liner types in RS2.

To define the properties of a Standard Beam liner:

  1. Select Define Liners Define Liners icon from the toolbar or the Properties menu.
  2. Select Liner Type = Standard Beam.
  3. Enter the properties described below.

Geometry

For a Standard Beam liner there are two ways of defining the liner cross-section: Thickness or Area and Moment of Inertia.

Thickness

If your liner has a constant cross-section, and uniform properties (e.g. a shotcrete layer), then you can simply define the liner Thickness. The liner axial and flexural properties will then be based on a cross-sectional area = Thickness x 1 (unit width).

Area and Moment of Inertia

If your liner has a more complex cross-section (e.g. reinforced concrete, or a steel-set and shotcrete liner), then you can define the liner cross-section by entering an equivalent Area and Moment of Inertia. These values must be normalized per unit width of excavation.

NOTE: two-component liners can also be modeled using the Reinforced Concrete liner type.

Elastic Properties

For a Standard Beam liner, you must define the following Elastic Properties:

  • Young's Modulus
  • Poisson's Ratio

Liners are assumed to have isotropic Elastic Properties, therefore only a single Young’s Modulus and a single Poisson’s Ratio can be entered.

Strength Parameters

For a Standard Beam liner, the following Strength Parameter options are available:

Material Type: Elastic

If the liner Material Type = Elastic, then strength parameters are not considered. The liner will only respond elastically to loading, and there will be no upper limit to the loading which can be sustained by the liner.

Material Type: Plastic

If the liner Material Type = Plastic, then you may enter peak and residual compressive and tensile strength parameters, which will be used in the RS2 analysis. Liners defined as Plastic will yield if the peak strength is reached. The plasticity calculation uses the "layering" method presented in Owen and Hinton (1986), in which the cross-section of the beam is partitioned into layers. If the stress in a given layer exceeds the peak strength (compressive or tensile), the layer yields.

Include Weight in Analysis

By default in RS2, the forces due to the weight of a liner are NOT included in the finite element analysis (i.e. the liner has zero weight).

If your Liner has a substantial thickness (e.g. a thick concrete or shotcrete liner), you may wish to account for the Liner weight in the stress analysis. To do this, select the Include Weight in Analysis check box, and enter the Unit Weight of the Liner material.

RS2 will then use the Liner Geometry (Thickness or Area) and the Liner Unit Weight, to determine the weight of the liner (i.e. the weight of each beam element), and include the resulting forces in the finite element analysis.

Sliding Gap

The Sliding Gap option is a specialized option for modeling a type of steel-set support system which is sometimes used in extreme ground squeezing conditions. This type of support system incorporates one (or more) freely sliding joint(s), which allow the steel-set ring to freely close in the circumferential direction, until a pre-determined gap has been closed. After the gap closes, the steel-set ring "locks" and axial load can be transmitted through the liner.

NOTE: the Sliding Gap option must be used in conjunction with staged modeling in RS2. A sliding gap liner is NOT applicable for a single stage model.

To enable the Sliding Gap option, select the check box and enter a value of Strain at Locking.

Strain at Locking

In RS2 the length of the sliding gap is specified in terms of an equivalent (circumferential) strain in the liner. The strain is averaged over the entire length of liner being considered (i.e. the sliding gap does not have a physical location on the RS2 model, the effect of the gap is averaged over the entire liner). The value of Strain at Locking depends on the length of liner you are considering. Example: for a fully lined, circular tunnel of 5 meter diameter, and a total Sliding Gap = 1 meter, the Strain at Locking = 1/pi*5 * 100% = 6.4 %. If more than one sliding gap exists, then add up the total gap length, and divide by the total liner length, to determine the Strain at Locking.

The implementation of the sliding gap is as follows:

  • At the installation stage of a sliding gap liner, the axial stiffness and axial force in the liner is always ZERO, regardless of the liner strain (i.e. the gap is never closed at the installation stage). Note: this is true even if you specify Strain at Locking = 0.
  • After each stage, RS2 checks the liner axial strain. If the strain in the liner is less than the Strain at Locking, then the liner will have zero axial stiffness (and therefore zero axial force) in the NEXT stage.
  • If the axial strain in the liner is greater than or equal to the Strain at Locking, then the gap will close or "lock" at the NEXT stage, and the axial stiffness will be set to that defined by the user (EA/L). The liner will generally have axial forces in it after the NEXT stage.

So for the sliding gap liner to work correctly, you must have multiple stages with the liner installed and in general relax the boundary stresses (using tractions or core material softening) over a number of stages.

NOTE: the flexural behaviour of the liner is NOT directly influenced by the sliding gap option. The liner will provide resistance to bending moment, regardless of the status of the sliding gap.

Beam Element Formulation

A Standard Beam liner is made up of a series of "beam elements", corresponding to the edges of finite elements. For a Standard Beam liner you may select one of two different Beam Formulations:

Timoshenko

The Timoshenko beam formulation accounts for transverse shear deformation effects.

Bernoulli

This is the classical Euler-Bernoulli beam formulation, which does not account for transverse shear deformation.

The Timoshenko Beam Formulation is recommended if you are using finite elements with midside nodes (i.e. 6-noded triangles or 8-noded quadrilaterals), since in this case RS2 will automatically use 3-noded Timoshenko beam elements, resulting in displacement compatibility between the finite elements and the beam elements. Bernoulli beam elements are always 2-noded, and give less accurate results when used with 6-noded triangles or 8-noded quadrilaterals.

Pre-Tensioning

The Pre-Tensioning option allows you to use the Standard Beam liner type to model a support element which is pre-tensioned (e.g. cable truss).

To define a Pre-Tensioning force for a Standard Beam liner, select the Pre-Tensioning force check box and enter a value.

Stage Liner Properties

The properties of a Standard Beam liner can be modified at different stages of a multi-stage model, by using the Stage Liner Properties option in the Define Liner Properties dialog. This could be used, for example, to model the increase in strength and stiffness of shotcrete or concrete liners after initial placement.

Most of the parameters entered in the Define Liner Properties dialog, can be increased or decreased by user-defined factors at different stages. For details about staging liner properties, see the Stage Material Properties topic, as the general procedure for staging properties is the same.

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