Analytical solution for longitudinal deformation of shield tunnel considering nonlinear rotational effects of circumferential joints

The longitudinal axial force and the yield of joint bolts can result in significant nonlinear variations in rotational stiffness of circumferential joints of shield tunnel subjected to external loading. The existing analytical methods associated with longitudinal deformation of shield tunnel often simplify the shield tunnel as an equivalent continuous long beam with constant bending stiffness, which are challenging to reflect the nonlinear rotational effects of circumferential joints. Firstly, taking the transverse performance and bolt elastic-plastic behaviors of shield tunnel into account, the expressions of joint rotational stiffness under weak tensile bending, pure bending, and compressive bending conditions are derived based on the strict elliptic parametric equation of tunnel cross-section, respectively. Secondly, the shield tunnel is modeled as a series of Timoshenko short beams connected by nonlinear rotational springs and linear shear springs along its longitudinal direction, meanwhile the soil-tunnel interaction is simulated using normal and tangential Winkler foundation springs, which are evenly distributed on the outer wall of tunnel. Then, the state space method is used to obtain the longitudinal discontinuous displacement solution of shield tunnel under external loads, and the iterative solution process associated with longitudinal deformation of shield tunnel under axial pressure is proposed. Finally, the proposed method is validated by comparing existing theoretical methods and engineering measured results associated with nearby excavation-induced longitudinal deformation of shield tunnel, and the parametric analyses are also carried out to explore the impacts of longitudinal axial force and transverse performance on surface surcharge-induced longitudinal deformation of shield tunnel based on an engineering case. The results show that the yield of longitudinal bolts in shield tunnels can significantly increase joint opening and tensile area between adjacent rings; the application of axial pressure can prominently reduce the longitudinal displacement, joint opening, and shearing dislocation of shield tunnel induced by upper excavation. The parametric analyses show that as the longitudinal axial force increases from tension to compression, the tunnel maximum longitudinal displacement and maximum rotational angle decrease nonlinearly at first increased and then reduced rate, and the maximum joint opening diminish approximately linearly. The reduction of tunnel lateral stiffness can lead to significant growth in tunnel longitudinal displacement and joint opening, as well as a slight diminution in shearing dislocation.