Abstract: Current theoretical researches on shield tunneling induced existing metro tunnels deformation generally simplify the existing tunnels as Euler-Bernoulli beams without joints, neglecting the tunnel shear deformation and the local stiffness reduction caused by joints. Additionally, the existing foundation model theory rarely incorporates the coordination between lining cross-section and soil deformation. The study utilizes the Loganathan and Polous method to establish the shield tunneling-induced soil greenfield. To account for the local stiffness reduction, the paper then treats the existing tunnel as a Timoshenko beam, with the tunnel stiffness function varying along the axis. By combining the energy principle and the Mindlin solution, the paper stablishes the tunnel displacement equation considering the coordinated constraint between tunnel and soil. The soil greenfield is applied to the tunnel, resulting in longitudinal deformation, which is solved by Fourier series. Finally, the Fourier energy variational solution is verified by three sets of engineering data, achieving good consistency. To conduct parameter analysis, the relative effects among factors such as bending stiffness, shear stiffness, joint action, soil parameters and space relationship are also adopted. Analysis results indicate that the theoretical solution considering the coordinated constraints of cross-section aligns more closely with the field data. Tunnel displacement results obtained without considering the coordination constraints tend to be larger. The local stiffness weakening effect of joint will cause significant sudden variations in tunnel bending moments along the axis. The increase in both bending and shear stiffness coefficients leads to a decrease in the bending moment of the tunnel structure. Changes in shear stiffness coefficient from low to high would result in a three-stage process of the tunnel deformation mode, which is from bending controlled stage to bending-shear mixed stage and then to shear controlled stage. As the tunnel deformation mode enters the shear-controlled stage, the effect of bending stiffness coefficients gradually weakens. An increase in the joint coefficient generally causes a decrease in the bending moment of the tunnel, but the impact is slight as the bending coefficient increases.

Key words: shield tunnel, cross-sectional coordinated deformation, joint stiffness reduction, tunnel shear deformation, Timoshenko beam, Fourier energy variation