2. Benchmark solution#

2.1. Calculation method#

2.1.1. Behaviour of the massif#

Let \(\lambda\) 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 \(\lambda\) to simulate the digging and the distance from the waist front.

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The solution of the problem is therefore similar to that of the infinitely thick tube loaded by an internal pressure of magnitude \((1\mathrm{-}\lambda ){\sigma }_{0}\) and by an external pressure of magnitude \({\sigma }_{0}\) (see [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 \(\lambda\) are as follows

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

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

2.1.2. Support behavior#

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

Let \({K}_{s}\) 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 (\({\nu }_{b}\) is the Poisson’s ratio of concrete):

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

If \({k}_{s}\mathrm{=}\frac{{K}_{s}}{2\mathrm{\cdot }G}\) represents the relative stiffness of the concrete in relation to the mass and \({\lambda }_{d}\) 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]:

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

2.2. Reference quantities and results#

The following quantities are tested at the level of the wall at points \(A\) and \(B\) of the figure in paragraph 1.1, at the moment when the lockdown is complete:

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

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

  3. radial displacement: \({u}_{y}\) in \(A\) or \({u}_{z}\) in \(B\).

2.3. Uncertainty about the solution#

None. Exact analytical result.

2.4. 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