Calculation model of longitudinal nonlinear equivalent bending stiffness of shield tunnel considering its transverse performance

The longitudinal equivalent bending stiffness (LEBS) of the shield tunnel is a key parameter when the longitudinal structural analyses of the shield tunnel are conducted by the equivalent continuous model. There will be a significantly nonlinear characteristics on the LBSE if the longitudinal axial force and the bending moment are loaded simultaneously. In this paper, based on the existing theories of the nonlinear bending stiffness of the shield tunnel, a calculation model of the longitudinal nonlinear equivalent bending stiffness of the shield tunnel incorporating its transverse performance has been proposed where the transverse deformation characteristics are considered. The LEBSs of the shield tunnel are derived strictly based on the elliptical integral when the circumferential joint of the shield is closed completely, semi open, and open fully respectively due to the coupling action of the longitudinal axial force and bending moment. The criteria of the critical axial force and the critical bending moment to distinguish the three bending modes are deduced with the equations about the neutral axial position. Then the reliability of the proposed model is validated via the results calculated by the existing analytical model, the model test data and the numerical results. After that, the analyses of the influences of the transverse performance of the shield tunnel on its longitudinal stiffness are carried out using the verified model when the shield tunnel suffers the coupling action of the longitudinal axial force and bending moment. Finally, the calculation errors of the simplified deductions are discussed. In addition, a coupled analysis method for the transverse and longitudinal deformations of a shield tunnel induced by additional load is developed based on the established analytical model. The results show that the analytical derivations of the proposed model are accurate and reliable. The LEBS of the shield tunnel is closely related to its transverse stiffness, and there is a positive correlation between the two. The LEBS of the shield tunnel is improved dramatically with the increase of the transverse bending stiffness if there is a compressive-bending state. The matching relationship between the longitudinal and transverse stiffnesses of the shield tunnel subjected to the coupling actions of the longitudinal axial force and bending moment is established with the proposed model, and a bridge to implement the longitudinal and transverse coupling analyses on the structures of the shield tunnel is also built by the proposed model.