Previous theoretical studies on the deformation of shield tunnels induced by foundation pit excavation generally consider the stratum as a linear elastic body, which seldom take the irregular construction boundary into account. Meanwhile, Curved beam theory and Timoshenko beam theory are less applied in the study of tunnels. This paper provides an analytical method to predict the displacements of small curved tunnels caused by deep excavation with time effects. Firstly, by introducing the fractional derivative Merchant model, a mechanical approach is proposed for analyzing the structural deformation of neighboring tunnels induced by foundation pit excavation. The parameters of viscoelastic soils are further derived in the Laplace domain based on time variability properties. Secondly, the additional stress field on existing small curvature tunnels is solved with theory of viscoelastic Mindlin solution and load reduction in foundation pits. Moreover, a deformation calculation model for curved shield tunnels is established by applying Pasternak foundation and Timoshenko beam theory. The time domain solutions for the radial and vertical deformations of small curvature tunnels are then derived by finite difference method along with Laplace positive and inverse transforms. In addition, the engineering measured data and three-dimensional numerical simulation solutions are compared with the analytical solution to verify relatively accuracy. Finally, sensitivity analyses are performed for parameters such as the buried depth of tunnels, minimum clear distance, fractional order, excavation method and creep time.

With the increasing construction of engineering structures on soft soils, accurately assessing their consolidation behavior has become crucial. To address this, Terzaghi's one-dimensional consolidation model was revisited. The elastic behavior of soil skeleton was modified by incorporating viscous effects using the fractional derivative Merchant model (FDMM), while the linear Darcy's law governing flux-pressure relations was extended by introducing time memory formalism through the fractional derivative Darcy model (FDDM). The governing equation is derived by incorporating the resulting constitutive behavior of both the soil skeleton and water flow into the Terzaghi's formulation of the poroelasticity problem. The proposed rheological consolidation model is solved by a forward time-centered space scheme (FTCS). After verifying the numerical procedure with published data, the influence of parameters on both the average degree of settlement and the pressure was comprehensively studied.