Improved longitudinal seismic deformation method of shield tunnels based on the iteration of the nonlinear stiffness of ring joints

Shield tunnels have undergone some prominent damage at joints due to the special structural composition of reinforced concrete segments that are connected by bolts. The generalized longitudinal seismic deformation method can achieve a better balance between the accuracy and efficiency of the calculation than the traditional seismic deformation method by using the time history of ground displacement of free-field as the input for the longitudinal beam-spring model of shield tunnels to determine the seismic responses of shield tunnels. However, the bending and axial stiffness of ring joints in this method simply are assumed to be constant values, which ignores the nonlinearity of the ring joints due to their discrete structures. In this paper, we demonstrate the inherent nonlinearity of ring joints by analytical derivation and obtain the stiffness of the 3D surfaces of ring joints under different loadings. Then, we propose an iterative algorithm for the stiffness of ring joints based on their internal forces, and this algorithm is integrated with generalized longitudinal seismic deformation method to form an improved method. The validity of the proposed method was verified by comparing the results of the longitudinal internal forces, the opening of joints, and the stresses on the joints obtained by generalized method and the improved method based on a Xiamen metro shield tunnel. Compared to the generalized methods, the results of the improved method indicated that the maximum joint opening, the compressive stress on the concrete, and the tensile stress on the bolts increased significantly, and these results demonstrated that the former methods of seismic design is unsafe.