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Stresses and Displacements in Steel-Lined Pressure Tunnels and Shafts in Anisotropic Rock Under Quasi-Static Internal Water Pressure

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Abstract

Steel-lined pressure tunnels and shafts are constructed to convey water from reservoirs to hydroelectric power plants. They are multilayer structures made of a steel liner, a cracked backfill concrete layer, a cracked or loosened near-field rock zone and a sound far-field rock zone. Designers often assume isotropic behavior of the far-field rock, considering the most unfavorable rock mass elastic modulus measured in situ, and a quasi-static internal water pressure. Such a conventional model is thus axisymmetrical and has an analytical solution for stresses and displacements. However, rock masses often have an anisotropic behavior and such isotropic assumption is usually conservative in terms of quasi-static maximum stresses in the steel liner. In this work, the stresses and displacements in steel-lined pressure tunnels and shafts in anisotropic rock mass are studied by means of the finite element method. A quasi-static internal water pressure is considered. The materials are considered linear elastic, and tied contact is assumed between the layers. The constitutive models used for the rock mass and the cracked layers are presented and the practical ranges of variation of the parameters are discussed. An extensive systematic parametric study is performed and stresses and displacements in the steel liner and in the far-field rock mass are presented. Finally, correction factors are derived to be included in the axisymmetrical solution which allow a rapid estimate of the maximum stresses in the steel liners of pressure tunnels and shafts in anisotropic rock.

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Abbreviations

E, \(E'\) :

Elastic moduli of a transversely isotropic rock

\(E_c\), \(E_{\rm crm}\), \(E_{\rm rm}\), \(E_s\) :

Elastic moduli of the backfill concrete, the near-field rock, the isotropic far-field rock and the steel liner, respectively

\(E_\theta\) :

Elastic modulus in the tangential direction in polar coordinates

G, \(G'\) :

Shear moduli of a transversely isotropic rock

\(G'_{S-V}\) :

Empirical cross-shear modulus of a transversely isotropic rock according to Saint-Venant

\(G_{\theta r}\), \(G_{\theta z}\) :

Shear moduli in polar coordinates

\(p_c\), \(p_{c, {\rm corr}}\), \(p_{\rm crm}\), \(p_{\rm rm}\) :

Pressures transmitted at radii \(r_c\) (and its correction), \(r_{\rm crm}\) and \(r_{\rm rm}\), respectively

\(p_i\) :

Quasi-static internal water pressure

\(r_c\), \(r_{\rm crm}\), \(r_i\), \(r_{\rm rm}\) :

Internal radii of the backfill concrete, the near-field rock, the steel liner and the far-field rock, respectively

\(t_c\), \(t_{\rm crm}\), \(t_s\) :

Thicknesses of the backfill concrete, the near-field rock and the steel liner, respectively

\(u_r^{c}\), \(u_r^{\rm crm}\), \(u_r^{\rm rm}\), \(u_r^{s}\) :

Radial displacements of the backfill concrete, the near-field rock, the far-field rock and the steel liner, respectively

\(\hat{u}_r^s\), \(\hat{u}_{r,{\rm max}}^s\), \(\hat{u}_{r,{\rm min}}^s\) :

Normalized radial displacements in the steel liner, and the maximum and minimum values, respectively

\(u_{r,{\rm iso}}^s\), \(u_{r,{\rm aniso}}^s\) :

Radial displacements in the steel liner considering isotropic and anisotropic rock, respectively

\(\gamma _{xy}\), \(\gamma _{xz}\), \(\gamma _{yz}\) :

Shear strains in Cartesian coordinates

\(\Delta r_0\) :

Initial gap between the steel liner and the backfill concrete

\(\epsilon _{x}\), \(\epsilon _{y}\), \(\epsilon _{z}\) :

Strains in Cartesian coordinates

\(\theta\) :

Angle in polar coordinates

\(\nu\), \(\nu '\) :

Poisson’s ratios of a transversely isotropic rock

\(\nu _c\), \(\nu _{\rm crm}\), \(\nu _{\rm rm}\), \(\nu _s\) :

Poisson’s ratios of the backfill concrete, the near-field rock, the isotropic far-field rock and the steel liner, respectively

\(\nu _{\theta r}\), \(\nu _{\theta z}\) :

Poisson’s ratio in polar coordinates

\(\sigma _{1}^s\), \(\sigma _{2}^s\), \(\sigma _{3}^s\) :

Principal stresses in the steel liner

\(\sigma _{1,{\rm iso}}^s\), \(\sigma _{1,{\rm aniso}}^s\) :

Major principal stresses in the steel liner considering isotropic and anisotropic rocks, respectively

\(\sigma _{1,{\rm iso}}^{\rm rm}\), \(\sigma _{1,{\rm aniso}}^{\rm rm}\), \(\sigma _{3,{\rm iso}}^{\rm rm}\), \(\sigma _{3,{\rm aniso}}^{\rm rm}\), \(\hat{\sigma }_{1,{\rm iso}}^{\rm rm}\), \(\hat{\sigma }_{1,{\rm aniso}}^{\rm rm}\) :

Major and minor principal stresses in the far-field rock considering isotropic and anisotropic rocks, respectively, and their normalized values for the major principal stresses

\(\hat{\sigma }_{1,{\rm max}}^{s}\), \(\hat{\sigma }_{1}^{s}\) :

Normalized maximum and major principal stresses in the steel liner, respectively

\(\hat{\sigma }_{1}^{\rm rm}\), \(\hat{\sigma }_{1,{\rm max}}^{\rm rm}\), \(\hat{\sigma }_{1,{\rm min}}^{\rm rm}\) :

Normalized major principal stresses in the far-field rock, and their maximum and minimum values, respectively

\(\hat{\sigma }_{3}^{\rm rm}\), \(\hat{\sigma }_{3,{\rm max}}^{\rm rm}\) :

Normalized minor principal stresses in the far-field rock and their maximum value, respectively

\(\hat{\sigma }_{1,{\rm num}}^s\), \(\hat{\sigma }_{{\rm eq,num}}^s\) :

Normalized numerical major principal and equivalent stresses in anisotropic rock

\(\sigma _{1,{\rm corr}}^s\), \(\sigma _{{\rm eq},{\rm corr}}^s\), \(\hat{\sigma }_{1,{\rm corr}}^s\), \(\hat{\sigma }_{{\rm eq},{\rm corr}}^s\), \(\sigma _{1,{\rm corr}}^{\rm rm}\), \(\hat{\sigma }_{1,{\rm corr}}^{\rm rm}\) :

Corrected maximum major principal and equivalent stresses in the steel liner and in the far-field rock, and their normalized values, respectively

\(\sigma _{1,{\rm int}}^s\), \(\sigma _{1,{\rm ext}}^s\) :

Major principal stresses at the internal and external fibers, respectively

\(\sigma _{1,{\rm max}}^s\), \(\sigma _{{\rm eq},{\rm max}}^s\) :

Maximum major principal and equivalent stresses in the steel liner

\(\sigma _{2,{\rm corr}}^s\) :

Corrected intermediate principal stress in the steel liner, corresponding to the corrected major principal stress

\(\sigma _{{\rm eq}}^s\) :

Equivalent stress in the steel liner

\(\sigma _{i}^{s}\), \(\sigma _{i}^{c}\), \(\sigma _{i}^{\rm crm}\), \(\sigma _{i}^{\rm rm}\) :

Stresses in the steel liner, the backfill concrete, the near-field rock, the isotropic far-field rock and the steel liner, respectively, along the i-coordinate

\(\sigma _{x}\), \(\sigma _{y}\), \(\sigma _{z}\) :

Stresses in Cartesian coordinates

\(\tau _{xy}\), \(\tau _{xz}\), \(\tau _{yz}\) :

Shear stresses in Cartesian coordinates

\(X_i\) :

Dimensionless parameters

\(\alpha _i\) :

Free coefficients

FE:

Finite element

FEM:

Finite element method

HSS:

High-strength steel

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Acknowledgments

This study is part of the consortium HydroNet 2: Modern methodologies for design, manufacturing and operation of hydropower plants, a research project funded by the Swiss Competence Center Energy and Mobility (CCEM-CH). The authors acknowledge the contributions to the article by Dr. Pedro Manso from the Swiss Competence Center for Energy Research Supply of Electricity (SCCER-SoE).

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  1. Laboratory of Hydraulic Constructions (LCH), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 18, 1015, Lausanne, Switzerland

    Alexandre J. Pachoud & Anton J. Schleiss

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  1. Alexandre J. Pachoud
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  2. Anton J. Schleiss
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Correspondence to Alexandre J. Pachoud.

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Pachoud, A.J., Schleiss, A.J. Stresses and Displacements in Steel-Lined Pressure Tunnels and Shafts in Anisotropic Rock Under Quasi-Static Internal Water Pressure. Rock Mech Rock Eng 49, 1263–1287 (2016). https://doi.org/10.1007/s00603-015-0813-z

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