1. Reference problem#
1.1. Geometry#
It is a tunnel with a circular section, covered by a concrete ring, which is excavated in a mass of soil. Both materials are assumed to be linear elastic.
1.2. Material properties#
The materials are linear elastic.
1.2.1. Sol#
\({E}_{s}\mathrm{=}4\mathit{GPa}\)
\({\nu }_{s}\mathrm{=}\mathrm{0,3}\)
1.2.2. Béton#
\({E}_{b}\mathrm{=}20\mathit{GPa}\)
\({\nu }_{b}\mathrm{=}\mathrm{0,2}\)
1.3. Initial conditions, boundary conditions and loads#
The stresses in the massif are initially assumed to be isotropic \(({\sigma }_{\mathit{xx}}\mathrm{=}{\sigma }_{\mathit{yy}}\mathrm{=}{\sigma }_{\mathit{zz}}\mathrm{=}{\sigma }_{0})\). The method used to simulate the excavation and installation of the support is the so-called « convergence-confinement » method presented for example in [bib1] and [bib2].
The basic principle is based on a reduction in the nodal reactions generated at the edge of the future gallery by the initial state of stress. This operation is referred to as « deconfinement ». When the end of lockdown has reached the value that corresponds to the site conditions that we want to model, we proceed to simulate the installation of the concrete support at the edge of the gallery.
Boundary conditions and loading are summarized in the following table. The phases correspond to those in the diagram above, the edges are composed with the nodes identified on the diagram in paragraph [§3.1] and in brackets the name of the mesh or node groups in the .comm file).
Edges |
Phase 1 |
Phase 2 |
Phase 3 |
Phase 4 |
|
\(\mathit{N0N1}\) (low_no1) |
\(\mathrm{DY}=0\) |
|
|||
\(\mathrm{N1N2}\) (bas_bet) |
\(\mathrm{DY}=0\) |
|
\(\mathrm{DY}=0\) |
||
\(\mathrm{N2N3}\) (no_bas2) |
\(\mathrm{DY}=0\) |
|
|
|
|
\(\mathrm{N3N4}\) (right_name) |
\(\mathrm{DX}=0\) |
|
|
|
|
\(\mathrm{N4N5}\) (my_top) |
\({\sigma }_{\mathit{yy}}\mathrm{=}\mathrm{-}5\mathit{MPa}\) |
|
|
|
|
\(\mathrm{N5N6}\) (no_left2) |
\(\mathit{DX}\mathrm{=}0\) |
|
|
|
|
\(\mathrm{N6N7}\) (no_left_bet) |
\(\mathrm{DX}=0\) |
|
\(\mathrm{DX}=0\) |
||
\(\mathrm{N7N0}\) (no_left1) |
\(\mathrm{DX}=0\) |
|
|||
\(\mathrm{N6N2}\) (edge) |
Nodal reactions corresponding to the end of lockdown |
||||
\(\mathrm{N7N1}\) |
Free |