Benchmark solution
=====================


Calculation method
-----------------


Behaviour of the massif
~~~~~~~~~~~~~~~~~~~~~~~~~

Let :math:`\lambda` be the deconfinement rate, which represents the relative position of the tunnel section in question in relation to the waist front. In the "convergence - confinement" method, the future excavated ground is replaced by an equivalent stress tensor, whose intensity is reduced via :math:`\lambda` to simulate the digging and the distance from the waist front.


.. image:: images/10006466000069F000003ACFEB3BA202A5EEAD9E.svg
    :width: 350
    :height: 270


.. _RefImage_10006466000069F000003ACFEB3BA202A5EEAD9E.svg:

The solution of the problem is therefore similar to that of the infinitely thick tube loaded by an internal pressure of magnitude :math:`(1\mathrm{-}\lambda ){\sigma }_{0}` and by an external pressure of magnitude :math:`{\sigma }_{0}` (see [:ref:`bib3 <bib3>`] for the details of the calculations).

The radial and orthoradial stresses as well as the radial displacement at the wall of the tunnel in an elastic medium subjected to a deconfinement rate :math:`\lambda` are as follows

:math:`\mathrm{\{}\begin{array}{c}{\sigma }_{R}\mathrm{=}(1\mathrm{-}\frac{\lambda \mathrm{\cdot }{R}^{2}}{{r}^{2}}){\sigma }^{0}\\ {\sigma }_{\theta }\mathrm{=}(1+\frac{\lambda \mathrm{\cdot }{R}^{2}}{{r}^{2}}){\sigma }^{0}\\ {U}_{R}\mathrm{=}\lambda \frac{{R}^{2}}{r}\mathrm{\cdot }\frac{{\sigma }^{0}}{\mathrm{2G}}\end{array}`

:math:`G` is the shear modulus given by the following relationship: :math:`G\mathrm{=}\frac{E}{2(1+\nu )}`.


Support behavior
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The support will oppose the natural convergence movement of the tunnel and thus apply artificial confinement to the rock.

Let :math:`{K}_{s}` be the stiffness of the support, it is given by the following relationship if we consider that the support is comparable to a thin tube (:math:`{\nu }_{b}` is the Poisson's ratio of concrete):

:math:`{K}_{s}\mathrm{=}\frac{{E}_{b}\mathrm{\cdot }e}{(1\mathrm{-}{\nu }_{b}^{2})\mathrm{\cdot }R}`

If :math:`{k}_{s}\mathrm{=}\frac{{K}_{s}}{2\mathrm{\cdot }G}` represents the relative stiffness of the concrete in relation to the mass and :math:`{\lambda }_{d}` the deconfinement rate when the support is put in place, then the radial and orthoradial stresses as well as the radial displacement in the wall are given by [bia1]:

:math:`\mathrm{\{}\begin{array}{c}{\sigma }_{R}\mathrm{=}\frac{{k}_{s}}{1+{k}_{s}}(1\mathrm{-}{\lambda }_{d}){\sigma }_{0}\\ {\sigma }_{\theta }\mathrm{=}\frac{{k}_{s}}{1+{k}_{s}}(1+{\lambda }_{d}){\sigma }_{0}\\ {U}_{R}\mathrm{=}\frac{1+{\lambda }_{d}\mathrm{\cdot }{k}_{s}}{1+{k}_{s}}\mathrm{\cdot }\frac{{\sigma }_{0}}{\mathrm{2G}}\mathrm{\cdot }R\end{array}`


Reference quantities and results
-----------------------------------

The following quantities are tested at the level of the wall at points :math:`A` and :math:`B` of the figure in paragraph :ref:`1.1 <Ref27564029>`, at the moment when the lockdown is complete:


1. radial stress: :math:`{\sigma }_{\mathit{yy}}` in :math:`A` or :math:`{\sigma }_{\mathit{zz}}` in :math:`B`;


2. orthoradial stress: :math:`{\sigma }_{\mathit{zz}}` in :math:`A` or :math:`{\sigma }_{\mathit{yy}}` in :math:`B`;


3. radial displacement: :math:`{u}_{y}` in :math:`A` or :math:`{u}_{z}` in :math:`B`.


Uncertainty about the solution
----------------------------

None. Exact analytical result.


Bibliographical references
---------------------------


1. The calculation of tunnels by the convergence-confinement method, M. Panet, Presses de l'ENPC 1995


2. How do you simulate tunneling with*Code_Aster*? Principle of the method, implementation and validation, A. Courtois, A. Courtois, R., R. Saidani, P. Sémété, note EDF HT-25/02/045/A - 2002


3. Continuum mechanics, volume 2, J. Salençon, Ed. Ellipses - 1988