Analytical solution for tunnels not aligned with geostatic principal stress directions

It is well-known that rock masses may present marked stress anisotropy. However, most of the tunnel analyses (numerical and analytical) assume the tunnel axis aligned with one of the principal stress directions. When this is not the case, axial shear stresses appear, which then are neglected, as it is done in all analytical solutions available for tunnel analysis. Existing solutions may consider advanced nonlinear ground behavior (i.e. elastic-brittle-plastic with e.g. Hoek and Brown failure criteria), linear-elastic ground with transversely anisotropic properties, seismic loading, groundwater and support, etc., but all consider that the axis of the tunnel aligns with one of the principal far-field stresses. This is also what is generally assumed when conducting more sophisticated, three dimensional numerical analyses. In this paper, an analytical solution to calculate the stresses and displacements induced by far-field axial shear stresses is presented. Solutions for supported and unsupported tunnels are provided. The proposed analytical solution can be combined with the classical Kirsch and Einstein-Schwartz solutions to determine the complete stress and displacement fields around the tunnel. Further, the effects of stress anisotropy are discussed.