Analytical approach for longitudinal deformation of shield tunnels considering bending-shear-torsional effects of circumferential joints

The longitudinal bending and shear deformations of existing shield tunnel induced by asymmetrical disturbances are normally accompanied by the cross-sectional torsion, and the longitudinal bending moment can result in the nonlinear torsional effect of circumferential joints. However, most existing soil-tunnel analysis methods simplify the shield tunnel as a one-dimensional continuous beam with constant stiffness lying on the elastic foundations, failing to consider its twisting motion. A novel analytical approach is therefore proposed in this work to better investigate the longitudinal bending-shear-torsional behaviors of in-service shield tunnel due to asymmetrical external loads. In this method, the existing shield tunnel is extended as a sequence of torsional Timoshenko short beams joined by linear rotational springs, linear shear springs and nonlinear torsional springs along its longitudinal direction. Simultaneously, the tangential and normal Winkler soil springs are evenly distributed on tunnel profile to emulate the soil responses to existing shield tunnel. The longitudinal displacement discontinuities at circumferential joints of existing shield tunnel are then solved using the state space method and the finite difference solution. In addition, the formulas of joint spring stiffness as the decisive parameters adopted in proposed approach are deduced. The effectiveness of the present solution is ultimately examined through two well-documented case histories respectively associated with up-crossing tunnelling and adjacent excavation, as well as comparison with previous analytical solutions. The impacts of several dominant factors on surface surcharge induced-tunnel longitudinal deformations are also further discussed, encompassing the joint rotational stiffness, longitudinal reinforced length of tunnel and load-tunnel relative location. The research implied that the proposed solution can satisfactorily evaluate the longitudinal discontinuous distributions of vertical displacement and cross-section torsion of shield tunnel induced by asymmetrical loads, in which the deformations of tunnel lining rings are mainly represented by rigid motion. The joint torsional property of shield tunnel is highly sensitive to its longitudinal bending moment, which makes a significant positive impact on the reduction of tunnel cross-section torsion.