General

Double lining is a kind of tunnel structure composed of an outer primary lining (segmental or shotcrete) and an inner cast-in-place concrete secondary lining. As the double lining structure has many advantages, in recent years its application has become more frequent.

Figure 1: Composite line sketch

In PLAXIS 2D it is possible to simulate a double lining in one of the following ways:

1. Model the primary lining as a solid (soil) element and the secondary lining as a plate element
2. Model the primary lining as a plate element and the secondary lining as a solid (soil) element
3. Model both primary and secondary linings as solid (soil) elements. It is possible to do that by using the “thick lining” option inside the Tunnel Designer. Different materials, e.g. one representing concrete and one representing shotcrete, can be assigned to the linings.

Note that it is not possible to model the two linings with overlapping plate elements. Plate elements are one-dimensional linear elements and therefore they cannot share more than one node. Consequently, modelling both primary and secondary lining with plate elements is not feasible.

Table 1: Methods used for the modelling of a composite liner

	Method 1		Method 2		Method 3		Method 4
	Soil Element	Plate Element	Soil Element	Plate Element	Soil Element	Plate Element	Soil Element	Plate Element
Primary Lining	✓			✓	✓			⨯
Secondary Lining		✓	✓		✓			⨯

Depending on the thickness of each lining and their expected geometric effects, one of the above methods can be selected. In general, meshing of thin clusters might prove problematic, therefore it is advised to model thin tunnel linings as plate elements instead of soil elements.

In most cases, an interface is also needed to simulate the interaction between the two linings and/or the cross permeability, if applicable. For instance, in some tunnels, a water-proofing membrane is installed between the primary and the secondary lining. This membrane acts as a slip joint and can be simulated as an interface between the two linings, with a user-defined interface and cross permeability. See also:

(+) Permeability in interfaces - PLAXIS | SOILVISION Wiki - Bentley Communities

Create thick lining

To create a soil element representing one (or both) of the tunnel linings, the use of thick lining is necessary. This option, available in the Tunnel Designer, creates concentric circles, parallel lines, and parallel curves, by offsetting a geometric object at a specified distance. To define a thick tunnel lining:

• Click the Generate thick lining button in the side toolbar in the Subsections tabsheet of the Tunnel Designer. The Generate thick lining window pops up.
• Specify the thickness of the lining in the appearing window. The thickness is defined as offset to the outline of the tunnel geometry. A positive or negative value of thickness indicates whether the object is to be offset outside or inside of the original tunnel contour, respectively. Note that when the tunnel is included in the geometry model, a subcluster will be created for the thick lining. A proper material can be assigned to it, as it is done for the soil clusters.

Figure 2: Generate thick lining

Output

PLAXIS 2D Output program provides a tool which can:

• automatically and accurately detect the centerline of any (curved) solid shape following the arcs and radii of the shape and
• compute the forces of on that shape.

This functionality is called Structural forces in Volume Plates and it is available in the toolbar of the Output program next to the Curves Manager icon. For more information please refer to Chapter 9 (Results available in Output program) in the PLAXIS 2D – Reference Manual.

When using this feature, it is possible to visualize structural forces (bending moments M, shear forces Q and axial forces N) in a regular structure (rectangular or tapered) that is composed of soil elements. Stresses are calculated at the stress points of the soil elements and the structural forces are then integrated from the stress points around the section cut of the centerline (see also below at section: “Cross-Section Results in Soil Elements”). Therefore, this process can be highly mesh-dependent and sufficient mesh discretization must be ensured to get accurate results.

Figure 3: Structural forces in volume plates

General considerations

Difference in modelling methods

The two different approaches (volume/volume and plate/volume methods) may yield slightly different results. This is because the plate is modelled at different locations in each case. Specifically, for the case of a plate/volume combination (Fig. 4a and 4b), the plate is modelled at the boundary (inner or outer side) of the adjacent lining and not at the centerline of the lining, which it represents.

Consequently, each of the three composite liner cases (Fig. 4a, b and c) will have a different center of gravity and as a result also a different flexural rigidity, EI. Therefore, special attention is needed when calculating the equivalent Area moment of Inertia, I.

Figure 4: Different modelling methods

Given that a plate is a line element where its stiffness properties, EA and EI, are concentrated along that line, we could think of a plate/volume combination being similar to the concept of a composite T-beam (see Fig. 5).

Figure 5: Equivalent representation of a plate / volume element approach

Moreover, when modelling a tunnel lining with a plate element, its shear deformation should also be taken into consideration. As described in PLAXIS 2D – Material Manual (see “Structural Behaviour”), the material behaviour in plate elements is defined by the following relationship between structural forces and strains:

The modified shear strain γ* takes into account the shear strain γ and some additional terms in order to give a more accurate approximation of the problem. Based on elementary transverse shear (Timoshenko Beam) theory and considering the assumption that the plate has a rectangular cross-section, the shear stiffness is determined using a shear correction factor equal to κ = 5/6 (0.833). In the case of modelling a solid wall, this will give the correct shear deformation. However, in the case of steel profile elements, the computed shear deformation may be too large.

You can check this by judging the value of d_eq, which can be computed as √(12EI/EA). For steel profile elements, d_eq should be at least of the order of a factor 10 times smaller than the length of the plate to ensure negligible shear deformations.

Cross-section results in soil elements

By definition, a cross-section represents a distribution of a quantity along a line. However, the distribution of quantities in cross-sections is obtained from the interpolation of stress-point data and may be less accurate than data presented in the 2D model. This means that the results of a cross-section are highly affected by the mesh density and its distance from the adjacent stress points.

The bigger the distance between the stress points and the cross-section line is, the bigger the interpolation and gradient error is.  When defining your cross-section line, you should make sure that the line "intersects" or is at least close to as many stress points as possible. You can turn on the visualization of these stress points by pressing Ctrl+I keyboard shortcut in Output.

The “dummy plate” approach

Although Structural forces in Volume Plates (see PLAXIS 2D - Reference Manual is an easy and fast way to extract the structural forces of a tunnel liner modelled as a soil cluster, as mentioned before, this method is dependent on the mesh density. An alternative approach for extracting the structural forces from a liner modelled as a solid (soil) element is that of the “dummy plate”. The strategy behind this approach involves the inclusion of a weightless plate element of low stiffness along the neutral axis of the tunnel lining to monitor the strain and curvatures changes in the considered composite section (see Fig. 6).

Figure 6: "Dummy plate” approach

While using the technique of the dummy plate, please make sure that you consider the following:

• The dummy plate is located at the neutral axis of the solid element of the structure (usually the middle of the soil cluster);
• The plate should not have any influence on the deformation of the lining. This means that the plate element should be weightless (the soil elements already contain the weight) and have a low stiffness (see below);
• There should not be an interface between the plate elements and the solid (soil) cluster. This ensures that the nodes between the dummy plate and the surrounding lining are fully connected (tight bonding - no relative movement takes place).

As mentioned above, the dummy plate has no influence on the deformation of the tunnel lining which encompasses the plate. This means that the strain, ε and curvature, κ of the dummy plate will be equal to that of the tunnel lining cluster. However, since the stiffness of the dummy plate (EA and EI) is factored down by a large factor (e.g. 10³ - 10⁵ times), the calculated forces should be scaled back up with the same factor. This entails some result post-processing which is summed up below: