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Semi-Analytical Solution for Stresses and Displacements in a Tunnel Excavated in Transversely Isotropic Formation with Non-Linear Behavior

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Abstract

A semi-analytical solution based on the transfer matrix technique is proposed to analyze the stresses and displacements in a two-dimensional circular opening excavated in transversely isotropic formation with non-linear behavior. A non-isotropic far field can be accounted for and the process of excavation is simulated by progressive reduction of the internal radial stress. A hyperbolic stress–strain law is proposed to take into account the non-linear behavior of the rock. The model contains seven independent parameters corresponding to the five elastic constants of an elastic material with transverse isotropy and to the friction coefficient and cohesion along the parallel joints (weakness planes). This approach is based on the discretization of the space into concentric rings. It requires the establishment of elementary solutions corresponding to the stress and displacement fields inside each ring for given conditions at its boundaries. These solutions, based on complex variable theory, are obtained in the form of infinite series. The appropriate number of terms to be kept for acceptable approximation is discussed. This non-linear model is applied to back analyze the convergence measurements of Saint-Martin-la-Porte access gallery. Short-term and long-term ground parameters are evaluated.

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Abbreviations

2D:

Elasticity with transverse isotropy

E h :

Young modulus in horizontal direction (plane of isotropy)

E v :

Young modulus in vertical direction

νvh :

Poisson’s ratio for the effect of vertical stress on the horizontal strain

νhv :

Poisson’s ratio for the effect of horizontal stress on the vertical strain

νh :

Poisson’s ratio for the effect of horizontal stress on the horizontal strain

G vh :

Shear modulus in vertical plane

S ij :

(i, j = 1, 2) compliance matrix (Eq. (3))

ϕ:

Airy stress function

\( \sigma_{x}^{\infty } \), \( \sigma_{y}^{\infty } \) :

Principal stresses along the x and y axes before excavation

Δp x , Δp y :

Unloading stress increments applied at the tunnel wall for describing the excavation process

2D:

Non-linear elasticity for stratified rock

C w :

Cohesion of the parallel joints

φ:

Friction angle along the parallel joints

\( \mu_{w} = \tan \varphi \) :

Friction coefficient along the parallel joints

\( \tilde{\sigma }_{c} \) :

Uniaxial strength in the direction \( \tilde{\beta } = \pi /4 + \varphi /2 \)

γ:

Deviatoric strain (Eq. (41))

\( S_{ij}^{0} \) :

(i, j = 1, 2) Initial tangent compliance matrix at zero deformation

\( S_{ij} = S_{ij}^{0} \left( {1 + \alpha \gamma } \right)^{2} \) :

Tangent compliance matrix at deformation γ

N :

Number of concentric circular rings for the discretization of the domain

R 0, R N :

Inner and outer radius of the discretized domain

c r :

Rate of geometric progression for the discretization of the domain

r i−1, r i :

Inner and outer radius, respectively, of the i − th ring

N i :

Odal point of the i − th ring with polar coordinates \( \left( {\left( {r_{i - 1} + r_{i} } \right)/2,\;\pi /4 - \varphi /2} \right) \)

T :

Characteristic time for time-dependent properties of the ground

X :

Length related to the distance of influence of the face

C ∞x :

‘Instantaneous’ convergence

C total = C ∞x :

(1 + m) total (long term) convergence

C h, C v :

Computed convergence along the major and minor axes of the deformation ellipse, respectively

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Acknowledgments

The authors would like to thank Lyon-Turin Ferroviaire (LTF) for providing data on Saint-Martin-la-Porte access gallery.

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Authors and Affiliations

  1. Centre d’Etudes des Tunnels, Bron, France

    The Manh Vu & Didier Subrin

  2. UR Navier-CERMES, École des Ponts Paris Tech, IFSTTAR, CNRS, Université Paris-Est, Marne-la-Vallée Cedex, France

    The Manh Vu & Jean Sulem

  3. Lyon-Turin-Ferroviaire SAS, Chambéry, France

    Nathalie Monin

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  1. The Manh Vu
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  2. Jean Sulem
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  4. Nathalie Monin
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Correspondence to Jean Sulem.

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Vu, T.M., Sulem, J., Subrin, D. et al. Semi-Analytical Solution for Stresses and Displacements in a Tunnel Excavated in Transversely Isotropic Formation with Non-Linear Behavior. Rock Mech Rock Eng 46, 213–229 (2013). https://doi.org/10.1007/s00603-012-0296-0

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  • DOI: https://doi.org/10.1007/s00603-012-0296-0

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