A semi-analytical approach for the stress and displacement around lined circular tunnels at shallow depths

Supports are typically installed to maintain the stability of tunnels in rock. The supporting forces and their spatial distributions are important quantities in tunnel and lining design. Analytical solutions can provide rapid predictions of the mechanical responses for the entire ground and lining; however, the rock-lining interaction under realistic conditions in shallow tunnel construction is still not fully considered in the current analytical research due to mathematical difficulties. This study provides a semi-analytical approach to accurately address the elastic stress and displacement around shallow lined tunnels induced by tunnel excavation and lining installation with full consideration of the rock-lining interaction and the lining thickness, stiffness and installation time. Based on the complex variable theory and conformal mappings, the stress and displacement of the rock and lining are expressed by potentials in the form of Laurent series. A few coefficients in these series are determined, and the relationships among the other coefficients are established by the boundary conditions. The optimisation model is then proposed by satisfying the compatibility conditions at several points along the rock-lining interface with minimum error, by which all the coefficients in the potentials of the rock and lining are finally addressed. The semi-analytical approach is verified by the good agreement between the results of the semi-analytical and numerical models under the same assumptions. A parametric study is performed to investigate the influence of the key parameters on the stress and displacement of the ground and lining. The proposed semi-analytical approach can rapidly and precisely provide the mechanical field of rock and lining, and the code can be provided to interested readers.