Longitudinal Deformation of Deep Shield Tunnels Caused by Upper Load Reduction

Above-crossing excavations may cause uplift damages on existing shield tunnels. Therefore, to accurately calculate the deformation of shield tunnels is very necessary for geotechnical engineers. At present, the single-sided elastic foundation beam model is usually used in longitudinal deformation calculations for shield tunnels, which overestimates the uplift of deep shield tunnels. Because of the existence of the ground arch, deep shield tunnels are subjected to two-sided foundation reaction forces. Therefore, this paper proposes a partial missing double-sided elastic foundation beam model and the related fourth-order partial differential equations. In this model, the shield tunnel is subjected to double Winkler foundation springs and is simply considered a Euler–Bernoulli beam. A two-stage analysis method is used to solve the problem. First, the vertical unloading stress due to the above-crossing tunnelling at the tunnel location is calculated through Mindlin’s solution. Second, the deformation response of the beam subjected to an unloading stress is calculated by the finite difference method. Two engineering cases are used to verify the research. The results indicate that the proposed model is more accurate than traditional models in predicting the maximum uplift value, which is basically consistent with the observations. Due to the existence of segment staggering, the longitudinal influence range of the calculation by two models is larger than the actual measurement.