Abstract: The shield tunnel is typically simplified as an infinite beam with two free ends in existing analytical models, which are used to calculate the longitudinal deformation of the underlying shield tunnel induced by the excavation of a foundation pit. However, the applicability of those analytical models is limited due to the simplification. The current study is aimed at estimating analytically the longitudinal deformation of the underlying shield tunnel induced by the excavation of a foundation pit adjacent to the station (working shaft). The constraint on the shield tunnel generated by the joint between the station (working shaft) and the tunnel is treated as a rotation spring with the rotation stiffness of K and a vertical rod support. The Winkler foundation–Timoshenko beam model for calculating the longitudinal deformation of the shield tunnel adjacent to the station (working shaft) induced by the foundation pit excavation is proposed. The finite difference solution of the proposed model is strictly derived based on the basic principles of the force method. The reliability and applicability of the proposed analytical model are verified via the comparison with the finite element numerical solution of one-dimensional elastic foundation beam model and the global finite element simulation results of the longitudinal deformation of the underlying tunnel induced by the excavation of a foundation pit adjacent to the station. The parametric studies indicate the following conclusions. (i) The longitudinal deformation and internal forces of the shield tunnel are significantly influenced by the rotation stiffness, K, of the joint between the station (working shaft) and the tunnel. The internal forces and the longitudinal deformation (i.e. rotation angle) at the end of the tunnel increase and decreases nonlinearly with a increasing K, respectively. In addition, when the flexible connection is adopted at the joint between the station (working well) and tunnel, the working performance of the shield tunnel at the joint can be better guaranteed. (ii) The constraint effect of the joint on the end of the tunnel is non-negligible, when the distance from the center of the foundation pit to the station-tunnel joint ranges from 4 to 5 times the width of the pit along the tunnel axis. In this condition, the proposed analytical model should be adopted to evaluate the longitudinal working performance of the tunnel. (iii) The influence of the overlying foundation pit excavation on the underlying tunnel mainly exerts within 2 times the length of the pit perpendicular to the tunnel axis away from the center of the pit.

Key words: shield tunnel, foundation pit excavation, flexible boundary, longitudinal deformation, analytical solution