Longitudinal Nonlinear Equivalent Continuous Model for Shield Tunnel under Coupling of Longitudinal Axial Force and Bending Moment

The longitudinal equivalent continuous model generally only studies the stiffness of shield tunnels under longitudinal bending moments, considering it a constant. However, in actual engineering, shield tunnels are exposed to complex environments where seismic events, uneven settlement, etc., may cause simultaneous axial forces and bending moments between segmental rings, necessitating consideration of the longitudinal stiffness of shield tunnels under coupled axial force and bending moment effects. Therefore, based on the influence of different axial forces and bending moments on the separation effect between segmental rings, this study establishes a longitudinal nonlinear equivalent continuous model. Using Guangzhou Metro Line 18 as a case study background, a numerical model of segment ring-bolt is established for comparative analysis. The results show that the contact states between segmental rings can be classified into three modes: completely separated, completely in contact, and partially in contact. Longitudinal bending stiffness remains constant in modes 1 and 2 but decreases with decreasing e in mode 3. The numerically simulated φ−e curves are consistent with the theoretical results. At the special point e0, the numerical simulation result is −57.27° compared to the theoretical result of −59.66°; at point eφ0 (−0.3036), the numerical simulation result is close to 0°. The longitudinal bending stiffness curve shows an overall decreasing trend. When e≤−2r, which corresponds to mode 2, the longitudinal bending stiffness remains constant at πr3Ect. As the longitudinal axial pressure decreases, the longitudinal bending stiffness continues to decrease when −2r≤e≤eφ0. When the longitudinal axial pressure decreases to 0, then the tensile force gradually increases (eφ0≤e≤2r). −2r≤e≤2r belongs to mode 3, and the equivalent bending stiffness is 2(1+sinφ)r3EctA4′−A3′er. As tension continues to increase, when e≥2r, the stiffness no longer decreases, and the longitudinal bending stiffness is πr3Ectu+1, which belongs to mode 1. The overall trend of the tensile and compressive stiffness curves is an inverse proportional function, with the middle mutation point at φ=0, i.e., eφ0=−4u(2+u)πr≈−0.3036. The findings of this study can provide a basis for the rational calculation of longitudinal forces in shield tunnels in engineering applications.