Analytical solution for deformation of an existing tunnel induced by shield tunnelling overcrossing in complex strata

This study investigates the influence of new tunnel construction which traverses above existing shield tunnels on the vertical displacement behavior in complex strata. Through the introduction of state vectors for soil layers, the displacement and stress propagation between different strata are formulated via integral transforms and matrix derivation. The layered characteristics of natural soils are effectively captured, and an extended Mindlin solution applicable to layered ground conditions is established.The existing tunnel is formulated as a Timoshenko beam interacting with a Pasternak foundation. By introducing a reduction factor for segmental joints and soil–tunnel coupling parameters, a governing equation for longitudinal displacement incorporating stiffness degradation is established and numerically solved using the finite difference method.Comparative analyses were conducted against traditional Euler–Bernoulli–Winkler models, with validation was performed using field monitoring data and finite element analysis (FEA) results. Conventional soil–tunnel interaction models based on the Mindlin solution tend to overestimate the vertical displacement of existing tunnels in layered foundations. In contrast, the proposed approach yields predictions with smaller errors. Validation against measured data shows that the proposed method reduces the prediction error for maximum displacement from 0.54 mm (conventional method) to 0.20 mm. Overall, the Timoshenko–Pasternak model, integrated with the elastic layered theory, provides more accurate predictions of existing tunnel deformation in layered strata compared to traditional beam–foundation combination models.