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.

来源平台：APPLIED MATHEMATICAL MODELLING