Simplified solutions for longitudinal deformation of shield tunnels considering influences of circumferential joints

In order to reflect the opening and dislocation between adjacent rings of a shield tunnel subjected to external loads, an analytical solution for the longitudinal deformation of the shield tunnel is proposed which considering the effects of circumferential joints. Firstly, a simplified longitudinal beam-spring shield tunnel model is introduced to simulate the longitudinal deformation of the shield tunnel. The Timoshenko short beam is used to consider the deformation of the segmental ring. The rotation and shearing springs are used to simulate the rotation and dislocation of the circumferential joints, respectively. Secondly, the finite difference equation of the longitudinal beam-spring model resting on the elastic foundation is established to solve the discontinuous deformation of the circumferential joint-segmental ring. The formula for the longitudinal deformation of the existing shield tunnel under external loads is further derived. Finally, the solutions for the longitudinal deformation of the existing shield tunnel associated with overcrossing tunneling and undercrossing tunneling are established, respectively. The proposed method is verified with new tunnel over-crossing and under-crossing case histories and previous theoretical methods. The results show that the predicted results by the present solution are consistent with those by the Timoshenko continuous beam model and cooperative deformation model and the field measurements, but the dislocations obtained by the present solution are slightly lower than those by the Timoshenko continuous beam model and the cooperative deformation model. The proposed method takes the effects of the circumferential joints into consideration, and it leads to the predicted displacement curve of the tunnel, which is neither smooth nor continuous. It is found that the rigid displacement mainly occurs in the segmental rings, while the rotation and dislocation occur in the circumferential joints. The previous methods always give continuous displacement curves of tunnels, which cannot truly reflect the real rotation and dislocation of joints.