Vol.:(0123456789) 1 3 Rock Mechanics and Rock Engineering https://doi.org/10.1007/s00603-020-02157-5 ORIGINAL PAPER Analytical Solution for a Deep Circular Tunnel in Anisotropic Ground and Anisotropic Geostatic Stresses Osvaldo P. M. Vitali 1  · Tarcisio B. Celestino 2  · Antonio Bobet 1 Received: 20 December 2019 / Accepted: 21 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020 Abstract Rock masses that have a well-defned structure may also present a remarked in situ stress anisotropy; thus, the misalignment of a tunnel with the geostatic principal stresses and/or with the principal axes of material anisotropy is very likely. Analyti- cal solutions for tunnels in transversely anisotropic rock available in the literature assume alignment of the tunnel with the geostatic principal stresses and with one of the principal directions of the material anisotropy (i.e. 2D plane strain condition). Such assumption is quite restrictive. In this paper, a new analytical formulation for circular deep tunnels in fully anisotropic rock is presented. It provides the full stress and displacement felds around a tunnel misaligned with the geostatic principal stresses or with the directions of material anisotropy. The analytical solution has been verifed by comparing its predictions with results from 3D FEM modelling, for a number of scenarios with increasing complexity. A parametric analysis has been conducted to investigate the interplay that exists between the orientation of the axis of the tunnel and the directions of the principal geostatic stresses and/or the directions of material anisotropy. Keywords Tunnel · Tunnel misalignment · Geostatic stress anisotropy · Rock anisotropy · Analytical solution List of Symbols Ψ Angle between tunnel axis and major horizon- tal principal stress z Complex variable, x + iμy σ v , Vertical stress σ h Minor principal horizontal stress σ H Major principal horizontal stress r, θ Polar coordinates in the xy-plane X, Y, Z Coordinate system attached to the tunnel, with Z-axis parallel to the tunnel axis σ xx,f Far-feld horizontal stress normal to the tunnel axis σ yy,f Far-feld vertical stress normal to the tunnel axis σ zz,f Far-feld axial stress parallel to the tunnel axis τ zx,f , τ zy,f Far-feld out-of-plane shear stresses τ xy,f Far-feld in-plane shear stresses σ θθ Tangential stress τ θz Tangential axial shear stress u θ Tangential displacement u r Radial displacement u z Axial displacement G Shear modulus E Young modulus ν Poisson ratio r 0 Tunnel radius 1 Introduction Rock masses that have a complex origin may also present a marked stress and material anisotropy. The literature is rich in measurements of in situ stresses (Brown and Hoek 1978; Evans et al. 1989; Gysel 1975; Haimson et al. 2003; Martin and Kaiser 1996; McGarr and Gay 1978; Park et al. 2014; Perras et al. 2015; Souček et al. 2017; Wileveau et al. 2007; Zhang et al.. 2017; Zhao et al. 2015) and anisotropic rock properties (Amadei et al. 1987; Batugin and Nirenburg 1972; Exadaktylos 2001; Park and Min 2015; Worotnicki 1993) that show that anisotropy is expected in rock. When a tunnel is excavated in a rock mass, misalignment of the tunnel axis with the geostatic principal stresses and/or with the principal directions of material anisotropy is very likely. * Osvaldo P. M. Vitali ovitali@purdue.edu 1 Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA 2 Sao Carlos School of Engineering, University of Sao Paulo, São Carlos, SP 13566, Brazil