Linear analytical model for the seismic response of a straight-jointed shield tunnel

In this study, the segment joint and ring joint of the lining of the straight-jointed tunnel are simplified as an equivalent open cylindrical shell with a small central angle and an equivalent closed cylindrical shell with a small width, and the lining of the straight-jointed tunnel is thus simplified as an equivalent continuous periodic shell (ECPS) described by the cylindrical shell theory. Since the ECPS lining is periodic along the longitudinal direction, the tunnel-soil system can thus be treated as a periodic system, which is referred to as the periodic tunnel-soil system (PTSS) in this study. Based on the proposed ECPS lining model, an analytical method for the tunnel with the ECPS lining (ECPS tunnel) under seismic waves is established in this study. By employing the periodicity condition for the PTSS as well as the wave function expansion method, the representation for the wave field in the soil is established. With the aforementioned cylindrical shell theory and Fourier series expansion method, the convolution type constitutive relation for the ECPS lining and the Fourier space equations of motion are derived. By using the soil-lining continuity condition and aforementioned formulations for the ECPS lining and soil, the coupled Fourier space equations of motion for the PTSS are established, with which the response of the ECPS lining and scattered waves in the soil can be determined.