Longitudinal response of shield tunnels under crossing and asymmetric loading: An improved model considering coupled axial-bending-shear interactions

Abstract

Shield tunnels are inherently discontinuous structures, whereas conventional longitudinal beam models generally idealize them as continuous systems and therefore cannot adequately capture the mechanical interactions at segment joints. To address this limitation, this paper proposes an improved longitudinal Timoshenko short beam spring model, LTSBSM, for shield tunnels. In the proposed model, each segment ring is represented by a Timoshenko short beam to account for shear deformation within the ring, while rotational, tensile, compressive, and shear springs are introduced at the inter-ring joints to characterize the coupled tension, compression, bending, and shear behavior, as well as the associated deformation discontinuities. Based on this framework, governing equations are established for overcrossing, undercrossing, and asymmetric thrust conditions, and a 10-point hybrid finite difference template is developed for the discontinuous tunnel joint system. The proposed model is then validated using three representative engineering cases. In the case studies, the joint-related parameters are first calibrated using a reference foundation elastic coefficient expression, and alternative empirical expressions are then compared on the same basis. The results show that the proposed model can capture both global tunnel deformation and local joint opening and dislocation under different loading conditions, while preserving the discontinuous longitudinal response of segmental shield tunnels. Different foundation elastic coefficient expressions generally produce similar spatial response patterns but noticeable differences in peak magnitude and influence range. Within the investigated range, increasing the soil elastic modulus E_s, from 15 MPa to 60 MPa, reduces the peak tunnel heave from approximately 4.29 mm to 0.98 mm, whereas the scaling coefficients for joint rotational and shear stiffnesses mainly regulate the distribution of local joint opening and dislocation through rotational and shear restraint.

Keywords

modified longitudinal beam-spring model / segment joint / overcrossing tunneling / undercrossing tunneling / asymmetric thrust

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