5      Case studies

5.1 Stability analysis of a natural cave (Bisetti & al. 2000)

A decision of the authorities of the canton of Jura in Switzerland to investigate the possibility of rehabilitating the caves of the former lime mines of Saint Ursanne as a convention and concert hall required a complete safety reassessment of the caves which is described below. The age of the caves indicates that overall stability is not questionable; local instabilities are however identifiable on inspection. These local instabilities are related to weakness planes in the rock structure and indicate that chunks of rock could be mobilized in local failure mechanisms. A safety assessment study was conducted by visual inspection and numerical simulation with finite element code Z_Soil.PC, comparing results obtained using different modeling and constitutive assumptions.

5.1.1     Contact interfaces

Contact interfaces are needed for this analysis. Contact elements implement a Mohr-Coulomb type constitutive behavior with a tensile cut-off. They can accommodate opening discontinuities. In the elastic range, penalty stiffnesses are evaluated from adjacent elements in order to implement appropriate normal  and tangential behavior.

 Figure 13. 3D mesh

5.1.2   Model 1

The mesh and dimensions are defined in Figures 13 and 14. The initial state of stress is defined by Equation 20 (with s_y the vertical stress).

            (20)

The rock matrix material is initially isotropic and elastic, later elastoplastic. Discontinuities are neglected. This model serves as a reference for later computations. Results are shown in the form of iso-surfaces on a 3D view and in sections B-B and C-C. Figures 15 to17 show horizontal stresses. Figure 18 shows the failure mode.

Figure 14. 2D section A-A.

Figure 15. 3D upside-down view of horizontal stresses (s_x).

Figure 16. 2D section B-B of horizontal stresses (s_x).

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Figure 17 : 3D upside-down view of horizontal stresses (s_z).

A thick plate type behavior is clearly identifiable from the tensile stresses present at the cave's roof (notice that view is upside down). Tensile stresses are of the same order of magnitude in both horizontal directions and remain far below the tensile strength. The massif shows no initial plastification. The failure mode is obtained from a stability analysis, as described above, and yields a safety factor of  24, corresponding to the collapse of the pillars.

Figure 18. Failure mode.

5.1.3   Model 2

Mesh and initial state remain the same. The rock matrix is initially isotropic and elastic, later elastopastic.

Figure 19. Horizontal stratigraphic discontinuities.

                Figure 20. 2D section B-B of horizontal stresses

                         Figure 21. Failure mode.

 

Horizontal stratigraphic discontinuities are introduced via contact elements. Their geometry and position are illustrated in Figure 19.

Comparison of results from models 1 and 2 are presented in a similar manner, they call for the following remarks: the plate behavior of the calcareous layers is easily identifiable in Figure 20. At each joint, traction stresses appear at the lower edge, which, however, remain below tensile strength.

The massif again shows no initial plastification. Failure occurs with a safety factor of 9 and the associated mechanism corresponds to a separation of the first layer at the cavity's ceiling (Fig. 21).

5.1.4  Model 3

 

Mesh and initial state remain unchanged. Stratigraphic discontinuities are introduced via contact elements, as in model 2. Two families of tectonic discontinuities are introduced via a multilaminate material. Both are vertical and parallel to sections B-B and C-C (Fig 22).

     Figure 22. Tectonic discontinuities direction (section A-A).

   Figure 16. 2D section B-B of horizontal stresses (s_x).

   Figure 23. 2D section B-B of horizontal stresses (s_x)

 

Results call for the following comments: the plate behavior of calcareous layers is recognizable as in model 2. However, horizontal stresses are influenced by the presence of tectonic joints and reach the strength values in most exposed zones (Fig.23). Plastification of the massif is identifiable in the central roof area. Failure corresponds to a collapse of the roof plate with separation of all layers above the cavity (Fig. 24). A safety factor of 4 is computed.

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     Figure 24. Failure mode (section C-C).

5.1.5  Conclusion

 

The analyses show that the safety of the massif considered as a homogeneous Hoek-Brown type material is high. Taking discontinuities into account  reduces the safety factor significantly. Two types of  models were used in the simulation of discontinuities in this analysis: contact elements and multilaminate material. Both support the introduction of failure characteristics with preferential directions. Contact elements also withstand the opening of  discontinuities, they require meshes which match the contact surfaces, making parametric studies difficult. It appears in the present study that such elements are perfectly appropriate for the simulation of horizontal stratigraphic discontinuities. Multilaminate material is better suited for the simulation of densely distributed discontinuities. These limitations are sometimes incompatible with the simulation of the actual geology. In the particular case of a dense distribution of  microcracks coupled with a strong confinement, this type of model appears to be the most appropriate for the simulation of families of discontinuities.

In the first performed analysis stratigraphic and tectonic discontinuities are ignored, the medium is assumed to be of  Hoek-Brown type, characterised by tensile and compressive strength. The analysis results in failure of the supporting pillars, with an associated safety factor of about 24.

In the second analysis, stratigraphic discontinuities are modelled explicitely with appropriate meshing.  Results indicate the separation of a  surface layer in the cave's ceiling, with an associated safety factor of about 9. In the third analysis, stratigraphic discontinuities are modeled explicitely with contact elements and tectonic discontinuities are modelled  via lamina. Depending on material data and geometry of discontinuities, a collapse of the roof plate with separation of all layers above the cavity or a global shear failure of the roof plate is observed. An estimated critical safety factor of 4 is found.

These different analyses made it possible to identify the relative influence of different rock characteristics in the safety assessment and the most critical potential failure mechanisms; precise rehabilitation measures could be proposed to the site owner. The analysis illustrates the important role that elasto-plastic numerical simulations can play in a reliable safety evaluation of natural caves in layered media.