Bibliography ============= [:ref:`1 <1>`] D. BERNAUD and G. ROUSSET: The "new implicit method" for the study of the sizing of tunnels, Revue Française de Géotechnique no. 60, pp 5-26, - 1992 [:ref:`2 <2>`] P. CATEL: Aval du Cycle — Bure site — Site de Bure — Sheet 13 — Convergence-confinement method, note EDF TEGG EFT GG/00 168 A — 2000 [:ref:`3 <3>`] A. GIRAUD: Thermo-Hydro-Mechanical Couplings in poorly permeable porous media: application to deep clays, thesis from ENPC — 1993 [:ref:`4 <4>`] D. LE BOULCH: Comparison of the THM 3D and 2D models of a storage structure with the *Code_Aster*, report Ajilon Technologies CÉnergys 01-A — 2002 [:ref:`5 <5>`] M. PANET: The calculation of tunnels by the convergence-confinement method, Presses de l'ENPC — 1995 [:ref:`6 <6>`] N. SELLALI, C. CHAVANT and G. DEBRUYNE: Hydroplastic modeling of the excavation of an underground gallery with the *Code_Aster*, note EDF MMN HI-74/00/009/A — 2000 [:ref:`7 <7>`] N. SELLALI, C. CHAVANT and G. DEBRUYNE: THM modeling of an underground storage structure with the *Code_Aster*, note EDF MMN HI-74/01/014/A — 2001 .. _Ref19616171: **Analytical formulas to apply the convergence-confinement method in the case of an elastic and linear rock mass and support** The medium is assumed to be isotropic linear elastic and subject to an initial stress field that is also isotropic (:math:`{K}_{0}\mathrm{=}1`). **Radial stress, orthoradial stress and radial displacement at the tunnel wall in an elastic medium subject to a deconfinement rate** :math:`\mathrm{\lambda }` :math:`\{\begin{array}{}{\sigma }_{R}=(1-\frac{\lambda \text{.}{R}^{2}}{{r}^{2}})\text{.}{\sigma }^{0}\\ {\sigma }_{\mathrm{\theta }}=(1+\frac{\lambda \text{.}{R}^{2}}{{r}^{2}})\text{.}{\sigma }^{0}\\ {U}_{R}=\lambda \cdot \frac{{R}^{2}}{r}\cdot \frac{{\sigma }^{0}}{\mathrm{2G}}\end{array}` :math:`G` is given by the following relationship: :math:`G=\frac{E}{2\cdot (1+\mathrm{\nu })}` **Support behavior:** .. _OLE_LINK1: 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 thick or thin tube (:math:`{\nu }_{b}` is the Poisson's ratio of concrete): :math:`{K}_{s}\mathrm{=}\mathrm{\{}\begin{array}{cc}\frac{{E}_{b}\mathrm{\cdot }e}{(1\mathrm{-}{\nu }_{b}^{2})\mathrm{\cdot }R}& \text{si}R>\text{10}\mathrm{\cdot }e\\ \frac{{E}_{b}\mathrm{\cdot }({R}_{e}^{2}\mathrm{-}{R}_{i}^{2})}{(1+{\nu }_{b})\mathrm{\cdot }\left[(1\mathrm{-}2\mathrm{\cdot }{\nu }_{b})\mathrm{\cdot }{R}_{e}^{2}+{R}_{i}^{2}\right]}& \text{si}R\mathrm{\le }\text{10}\mathrm{\cdot }e\end{array}` Let :math:`{P}_{s}` be the confinement pressure defined in the following figure .. image:: images/10000000000000D8000000D450B1828F16CCE455.png :width: 2.2437in :height: 2.2083in .. _RefImage_10000000000000D8000000D450B1828F16CCE455.png: So we have: :math:`{P}_{s}\cdot R={\mathrm{\sigma }}_{b}\cdot e` If :math:`{k}_{s}=\frac{{K}_{s}}{2\cdot G}` represents the relative stiffness and :math:`{\lambda }_{d}` the deconfinement rate when the support is put in place, then the support pressure and the radial displacement in the wall are given by: :math:`\{\begin{array}{}{P}_{s}=\frac{{k}_{s}}{1+{k}_{s}}\cdot (1-{\lambda }_{d})\cdot {\sigma }^{0}\\ {U}_{R}=\frac{1+{\lambda }_{d}\cdot {k}_{s}}{1+{k}_{s}}\cdot \frac{{\sigma }^{0}}{2\cdot G}\cdot R\end{array}` .. _Ref19609717: **Summary flow chart on methods to simulate an excavation** **Notes** The names of the objects are those of the command files shown in the following appendices. :math:`\mathit{SNL}` means STAT_NON_LINE; :math:`\mathit{CC}` means CREA_CHAMP; :math:`\mathit{CL}` means boundary conditions +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |**Step 1: Constraint initialization** | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |:math:`I`: :math:`\mathit{SNL1}` with the desired load of its own weight or pressure and a material with a possibly fictitious Poisson's ratio |:math:`\mathit{II}`: assignment by command :math:`\mathit{CC}` of the desired field | | + + + .. image:: images/100006640000330E000036E1BEE86A9932F73C61.svg + | | | :width: 86 | + + + :height: 93 + | | | | + + + + | | | | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |**Step 2: Retrieving nodal reactions at the edge of the future gallery** | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |:math:`\mathit{CC}` to extract the constraints from :math:`\mathit{SNL1}` :math:`\mathit{SNL}2` with :math:`\mathit{CL}` on the object :math:`\mathit{BORD}` in DIDI Retrieving reactions |:math:`\mathit{SNL}1` with :math:`\mathit{CL}` on object :math:`\mathit{BORD}` in DIDIsur model :math:`\mathit{SOL}` Retrieving reactions |:math:`\mathit{SNL}1` with :math:`\mathit{CL}` on object :math:`\mathit{BORD}` in DIDIsur model SOL_REST Retrieving reactions | | + + + + .. image:: images/10000AF40000330E000036E1AC88AA4AB507C89B.svg + | | | | :width: 86 | + + + + :height: 93 + | | | | | + + + + + | | | | | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |**Step 3: End of lockdown** | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |:math:`\mathit{SNL}3` with the loading of the nodal reaction vector and a "soft" material in place of the "void" | |:math:`\mathit{SNL}2` (model :math:`\mathit{SOL}`) with the loading of the nodal reactions vector and a "soft" material in place of the "void"|:math:`\mathit{SNL}2` (a single material and model SOL_REST) with loading the vector of nodal reactions | | + + .. image:: images/10000F020000330E000042406790260678448080.svg + + + .. image:: images/10000E5C0000330E00003927B6CE8CD3EE0A5227.svg + | | :width: 86 | | | :width: 94 | + + :height: 93 + + + :height: 102 + | | | | | | + + + + + + | | | | | | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |**Step 4: Installing the support** | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ |:math:`\mathit{SNL}4` with 3 materials: rock, concrete and void (:math:`A` method) to complete the end of lockdown | |:math:`\mathit{SNL}3` with 3 materials: rock, concrete and void (:math:`A` method) to complete the end of lockdown |:math:`\mathit{CC}` to extract the results of :math:`\mathit{SNL}2` (:math:`\mathit{SNL}3`) on a model (SOL_REST + BETON) for intermediate calculation (Combination of fields :math:`\mathit{CC}` :math:`\mathit{SNL}4`) to complete the end of lockdown| | + + .. image:: images/10000F020000330E000042406790260678448080.svg + + + .. image:: images/100010AA00003B6E00003DB4E648CE7A8111549E.svg + | | :width: 86 | | | :width: 94 | + + :height: 93 + + + :height: 102 + | | | | | | + + + + + + | | | | | | +-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+ **Comparison of the constraints obtained by numerical calculation and by the analytical solution** **Unsupported tunnel case** .. image:: images/Object_61.svg :width: 86 :height: 93 .. _RefImage_Object_61.svg: **Case of supported tunnels (from 50% of deconfinement)** .. image:: images/1000020100000330000002138BA5B794EB2BBB91.png :width: 94 :height: 102 .. _refImage_1000020100000330000002138 BA5B794EB2BBB91 .png: