Optimal design method for S-shaped tunnels crossing above existing shield tunnels based on theoretical solutions of longitudinal deformation

In densely populated urban areas, constructing an S-shaped tunnel over an existing shield tunnel involves the spatial superposition of three crossing configurations within a short distance, inducing a highly nonuniform unloading stress field and complex longitudinal deformation in the existing tunnel. However, existing analytical models are limited to single crossing configurations and cannot handle such multi‑segment combined scenarios, nor do they provide evaluation frameworks or design optimisation methodologies specifically addressing such complex crossing configurations. This study proposes a two‑stage theoretical framework: in the first stage, Mindlin’s solution is employed to derive vertical additional stresses by establishing local coordinate systems for each crossing segment; in the second stage, the existing shield tunnel is idealised as a Timoshenko beam on a Pasternak foundation, with the governing equations solved via the finite difference method. This segmented strategy enables clear separation of the independent contributions from each crossing configuration. Validation against a real engineering case in Hohhot shows that the proposed method predicts a maximum uplift of 2.65 mm, compared with the measured value of 2.50 mm, yielding a relative error of 6.0%. Parametric analysis reveals that the oblique segment contributes 48.1% to the total deformation, the above‑crossing segment 40.0%, and the lateral parallel segment only 11.9%. Following the parametric analysis, a comprehensive risk index is introduced for safety assessment. Based on this index, the established design optimisation pathway indicates that high‑risk zones are associated with large excavation areas, long skew segments, and low foundation stiffness, whereas low‑risk zones correspond to high foundation stiffness and low unloading intensity. Accordingly, priority should be given to controlling excavation area and skew segment length in design.