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

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 farfield 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.Keywords Steel liner Á Anisotropy Á Transversely isotropic rock Á Pressure tunnels and shafts Á Finite element method Á Water pressure List of symbolsLatin characters E, E 0 Elastic moduli of a transversely isotropic rock E c , E crm , E rm , E s Elastic moduli of the backfill concrete, the near-field rock, the isotropic far-field rock and the steel liner, respectively E h Elastic modulus in the tangential direction in polar coordinates G, G 0 Shear moduli of a transversely isotropic rock G 0 SÀV Empirical cross-shear modulus of a transversely isotropic rock according to Saint-Venant G hr , G hz Shear moduli in polar coordinates p c , p c;corr , p crm , p rm Pressures transmitted at radii r c (and its correction), r crm and r rm , respectively p i Quasi-static internal water pressure r c , r crm , r i , r rm Internal radii of the backfill concrete, the near-field rock, the steel liner and the far-field rock, respectively t c , t crm , t s Thicknesses of the backfill concrete, the near-field rock and the steel liner, respectively u c r , u crm r , u rm r , u s r Radial displacements of the backfill concrete, the near-field rock, the farfield rock and the steel liner, respectively û s r ,û s r;max ,û s r;min Normalized radial displacements in the steel liner, and the maximum and minimum values, respectively u s r;iso , u s r;aniso Radial displacements in the steel liner considering isotropic and anisotropic rock, respectively Greek symbols c xy , c xz , c yz Shear st...

Full stress and displacement fields for steel-lined deep pressure tunnels in transversely anisotropic rock

“…The problem solved is defined in Figure 2 that, as mentioned earlier, is identical to that considered by Pachoud and Schleiss (2015), with few exceptions that expand the scope of the investigation. It consists of a deep tunnel with circular cross section with internal radius ri, subjected to an internal water pressure pi, excavated in a transversely anisotropic rock.…”

“…The concrete backfill and the damaged rock are assumed to be sufficiently cracked or damaged such that no tangential or shear stresses can be transmitted through them, which is a common assumption (e.g. Schleiss, 1988;Pachoud and Schleiss, 2015).…”

“…Given the assumption of no shear and tangential stresses and after integration of the radial strains, one gets the radial displacements U crm r as: Note that equation (19) satisfies compatibility of radial displacements at the concretedamaged rock interface. Note also that equation (19) can be used when the damaged rock is isotropic (which was the assumption by Pachoud and Schleiss, 2015) by making Ex = Ey = Ecrm, νxz = νyx = νcrm and Gxy= Ecrm/2(1+νcrm), which results in β1 = (1-ν 2 crm)/Ecrm and β2 = β3 = 0.…”

“…Analytical solutions are available that account for these assumptions and are included in the next section for completeness. Pachoud and Schleiss (2015) provided a number of correction factors to the analytical solution to expand it to cases where the intact rock is transversely anisotropic. The factors were obtained after an extensive numerical analysis using a Finite Element Method and a genetic algorithm to minimize errors between the corrected analytical solution and the numerical results.…”

“…The corrected factors were obtained to estimate the major principal stress in the liner and in the intact rock. This paper builds on the work from Pachoud and Schleiss (2015) and provides a full analytical solution for stresses and displacements for the steel liner, concrete, damaged rock and intact rock. It also expands the range of cases, as the solution accounts for the anisotropy of the damaged rock and does not have the limitation that the existing solution has of assuming that the shear modulus of the rock is equivalent to the empirical relation of Saint-Venant.…”