1. Introduction

The lining construction is often regarded as the final construction stage on the mechanical aspect. Then, the lining is installed after the ground movement induced by excavation and other loads converges. So, one can suppose the self weight is the sole load for the tunnel lining. However, the other potential loads are considered in structural aspects that confirm the stability and deformation and member forces of the lining. Here, the modeling technique for the various loads will be dealt such as self-weight, water pressure and earth pressure. However, the water pressure modeling is skipped here because the input method is very similar to the rock/earth pressure case.

There is a circular tunnel lining whose center is located at 20m depth from the ground surface. The assumed rock release height is 10m high from the center of tunnel.

　Figure 1. Conceptual Sketch

2.Modeling condition

2.1 Material Property

The material properties of concrete lining and ground are summarized as below:

Young's modulus of concrete, E = 2 ▲ 10⁷ KPa

Thickness of lining, T = 30cm = 0.3m

Unit weight of reinforced concrete = 24.525 KN/m³

Ground reaction spring constant per unit area, K = 1 ▲ 10⁶ KN/m/m²

Nodal spring constant, k = 871600 KN/m for element length 0.8716 m

Tensile strength of ground, T = zero.

Nodal spring constants are obtained by multiplying the K by the contributing area of shell elements to the node.

Figure 2. Complete Mesh in PENTMESH

2.2 Loading condition

The self weight is applied on the negative y-direction on the global coordinate system. The selft weight is estimated by multiplying the unit weight of reinforced concrete with the gravity multiplier (here, -1.0 used for (-)y direction). The earth pressure is defined as two independent components of vertical and horizontal pressures on the global coordinate system. (The water pressure is defined as normal pressure to the frame elements. The normal pressure to each element should be defined on the local coordinate system where each element has the different orientation.)

3. Analysis

The ground reaction springs are used to model the ground and the interface between lining and ground. Due to bending effect of frame/shell elements, many spring elements are expected to experience both the compressive and tensile forces. But, the reaction spring elements as a interface have zero stiffness when they are subjected to the tensile stress. This interface characteristic is possible by implementing the tensile strength. Here, the zero tensile strength is used. Because the stiffness may be changed along the calculation results, the nature of this analysis is non-linear and iterative.

• Analysis control

The number of load increments is 50 and the maximum number of iteration is 50 for the nonlinear analysis due to the elastic-plastic spring elements. In order to verify the quality of the nonlinear analysis, the tensile spring elements in the nonlinear analysis results should be removed in the model and the linear analysis is performed, and then compared with the nonlinear analysis result. The following dialog box shows the control parameters for the non-linear analysis (see more details in the reference manual, Chapter 4).

The number of load increments and the maximum number of iterations are usually altered for better result as a case study. The term, "Overshot Displacement Limit" in the dialog box means the relative displacement ratio limit between the iterations during the nonlinear analysis.

Figure 3. Menu: FEM / Analysis Control

• Analysis stages

The following figure lists the analysis stages. Here, you can add or modify the analysis stages, and then you can assign these stages to the layer data.

Figure 4. FEM | Stage configuration menu

The load cases are summarized in the following table:

• Self-weight load

In the following figure, the list box of the lower part of the dialog box shows the input layer list. The layer means the 3D input data that consists of 2D data (sub-layer) and additional data. To input correctly, you should select the all the elements first, select the slices, and fill the edit boxes in "Data" button. Finally, press the "Append" button. If the program asks the stage, choose the Load case 1.

Then, the self weight will be applied in the Load case 1?

Figure 5. Dialog box for Gravity Load Input

Figure 6. Gravity factor dialog box by Data button

(means the load factor and the direction)

• Surface load

The following figure shows the input dialog box for the surface loads. Here, three earth pressure components will be defined, i.e., the left earth pressure, the right earth pressure and the top earth pressure.

The three sub-layers are Lframe, Rframe and Upframe, respectively.

Figure 7. Dialog box for Earth Pressure Loading

Figure 8. Left earth pressure for the left part of the lining

Figure 9. Left part of the lining (thick solid line)

Figure 10. Right earth pressure for the right part of the lining

Figure 11. Right part of the lining (thick solid line)

Figure 12. Vertical earth pressure for the upper part of the lining

Figure 13. Upper part of the lining (thick solid line)

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