Deformation parameters for the Bi-Elliptical Displacement-Controlled Method and related tunnel deformation modes

This study investigates the deformation characteristics of shield tunnel cross-sections using a step-by-step three-dimensional finite element method and proposes recommended parameters for the Bi-Elliptical Displacement-Controlled Method (Bi-Elliptical DCM). The advanced Hardening Soil model with small-strain stiffness was adopted to simulate soil behaviour, and model reliability was validated using the Gaussian settlement trough expression and Kirsch solution. More than 700 numerical cases were analysed considering different soil conditions, tunnel diameters, buried depths, and tunnel volume losses. The results show that tunnel deformation is primarily controlled by soil characteristics, followed by tunnel diameter and buried depth. In coarse-grained soils, tunnel cross-sections exhibit pronounced bi-elliptical deformation, especially for large and deep tunnels, whereas fine-grained soils generally show circular contraction patterns. Based on the numerical results, reference charts for the Bi-Elliptical DCM parameters n 1 , n 2 and β are proposed to enable rapid prediction of tunnelling-induced deformation. The findings improve understanding of shield tunnel deformation mechanisms and provide practical guidance for applying the Bi-Elliptical DCM in engineering projects.