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, including lateral parallel crossing, oblique crossing, and parallel above-crossing, within a short distance. This superposition induces a highly nonuniform unloading stress field and complex longitudinal deformation responses in the existing tunnel. However, existing analytical models are limited to single crossing configurations and cannot handle scenarios with coexisting multiple segments where geometric parameters are independently adjustable. This study proposes a theoretical framework for predicting and optimizing the longitudinal deformation of existing shield tunnels induced by S-shaped tunnel excavation. The analysis employs a two-stage method. In the first stage, vertical additional stress fields due to excavation unloading are derived based on Mindlin's solution. Local coordinate systems are established for each of the three crossing segments, and coordinate transformations enable the continuous superposition of additional stresses along the existing tunnel axis. In the second stage, the existing shield tunnel is simplified as a Timoshenko beam resting on a Pasternak two-parameter foundation, with the governing differential equations for generalized displacements solved using the finite difference method. This segmented solution strategy allows clear separation of the independent contributions from each crossing configuration: cumulative asymmetric disturbances from the oblique segment, direct unloading effects from the above-crossing segment, and far-field attenuation characteristics from the lateral parallel segment. Model validation is performed using a real engineering case in Hohhot, China. The results show that the proposed method predicts a maximum uplift of 2.65 mm, which aligns well with the measured value of 2.50 mm, yielding a relative error of only 6.0%. Moreover, the method outperforms the degraded Pasternak-Euler-Bernoulli model in full-segment prediction accuracy. Comprehensive parametric analysis reveals that the oblique crossing segment contributes 48.1% to the total deformation, the above-crossing segment 40.0%, and the lateral parallel segment only 11.9%. This finding indicates that effective deformation control relies not solely on suppressing peak values in the above-crossing segment but primarily on managing cumulative disturbances in the oblique segment. The study further introduces a comprehensive risk index that integrates longitudinal deformation range, maximum deformation slope, and coordination coefficient for safety assessment. Based on this index, a design optimization pathway is established: prioritize control of excavation area and oblique segment length, supplemented by ground reinforcement, enhanced counter-pressure, increased crossing angle, and repositioning of the oblique segment to shift the comprehensive risk of the existing tunnel from sensitive to controllable zones.