Semi-Analytical Method for the Response of Existing Tunnels to Tunneling Considering the Tunnel–Soil Interaction Based on the Modified Gaussian Function

The behavior response of an existing shield tunnel to under-cross tunneling is fundamentally governed by the tunnel–soil interaction. In this study, the existing tunnel is simplified as a single-variable Timoshenko beam to address the shear locking issue of the conventional Timoshenko beam. An elastic continuum solution, which can be degenerated into the Winkler–Timoshenko model, is established by considering the tunnel–soil interaction to evaluate the existing tunnel’s response to underlying tunneling. Meanwhile, greenfield settlement is described using a modified Gaussian function to fit practical engineering cases. The joint opening and segmental dislocation are also quantified. The applicability of the proposed method is validated by two reported engineering cases, where measured greenfield settlements are used to verify the modified Peck formula. Key parameters, including the ground loss rate, intersection angle, tunnel–soil stiffness factor, and vertical clearance, are discussed. The results show that the proposed method can provide references for predicting the potential diseases of existing tunnels affected by new tunnel excavation.