A Variational Approach to Analyze the Settlement of Existing Tunnels Caused by Ground Surcharge

This paper presents a variational approach to assess the settlement of the operational shield tunnels resulting from surface loading. The vertical additional force on the tunnel, induced by the surcharge, was computed using the Boussinesq solution. The structural behavior of the tunnel was modeled using the Timoshenko beam theory, which accounts for both bending and shear deformation mechanisms. Furthermore, the two-parameter Pasternak foundation model, which accounts for the continuity of foundation deformation, was used to model the interaction between the tunnel and the surrounding ground. A finite Fourier series was employed to approximate the vertical displacement and cross-sectional rotation angle of the tunnel. By conducting work and energy analyses, the energy balance equations for the tunnel and the soil were obtained. The governing equations were then formulated according to the minimum potential energy principle. The displacement and cross-sectional rotation angle of the tunnel were then expressed through the variational method. The accuracy of the proposed method was validated by comparison with in situ measurement data, confirming its effectiveness in predicting tunnel responses under a ground surcharge. Finally, a parametric study was conducted to evaluate the impact of various parameters on the settlement of the shield tunnel.