Closed-form solution for the dynamic response of deep-buried loess tunnels under SV-wave excitation

Due to the inherently loose structure and low shear strength of loess, loess tunnels are particularly vulnerable to damage from surrounding deformation under seismic excitation. In this work, the pseudo-static method is adopted to transform the SV-wave excitation into equivalent far-field shear stresses and derives a closed-form solution for the internal force response of tunnel linings with explicit consideration of the flexibility ratio. Based on the analytical results, the internal force response of tunnel lining is examined, and the seismic load transfer mechanism within loess tunnel structural systems is elucidated. The results indicate that the closed-form solution exhibits good agreement with the simplified numerical solutions and theoretical solutions reported in the literature with respect to both variation trends and internal force distribution. Increasing seismic intensity strengthens the dynamic interaction between the loess surrounding ground and lining structure, thereby amplifying the internal force response of the lining. A reduction in loess surrounding ground stiffness diminishes the capacity for seismic load transfer and decreases the confinement of the lining structure, resulting in a gradual decrease in axial force response and an increase in bending moment. Under identical surrounding ground conditions, increasing the flexibility ratio leads to a gradual reduction in internal force response. When the flexibility ratio is below 50, greater lining thickness increases axial force, whereas beyond 100 it decreases, While the bending moment increases monotonically with lining thickness. With decreasing stiffness of the loess surrounding ground, the predicted internal force response of the lining tends to be systematically overestimated. Under identical surrounding ground conditions, an increase in flexibility ratio reduces the axial force response error but progressively increases the bending moment error. Moreover, greater lining thickness further amplifies the prediction errors. The results of this work provide a theoretical foundation and methodological guidance for preliminary seismic design of loess tunnels.