Analytical analysis of a circular tunnel with the segmental lining under harmonic plane waves

In this study, based on the Fourier series expansion and wave function expansion methods, an analytical method for a circular tunnel with segmental lining under seismic waves is established. The surrounding soil of the tunnel is assumed to be a linear elastic medium. The lining of the tunnel is assumed to be composed of several segments and joints. Each joint is simplified as a portion of a cylindrical shell with a small central angle. The segments and joints together thus constitute an equivalent continuous shell (ECS) lining which is described by the cylindrical shell theory. To develop the analytical method, the wave function expansion method is employed to establish the representation for the scattered wave field in the soil. Using the cylindrical shell theory and expanding the quantities and parameters of the ECS lining into Fourier series along the circumferential direction together with introducing the Fourier space convolution-type constitutive relation for the ECS lining, the differential equations for the ECS displacements with variable coefficients are reduced to a linear system of equations for the Fourier components of the ECS displacements. Combining the linear system of equations for the ECS displacements with the representation for the wave field in the soil yields the coupled linear system of equations for the ECS lining and soil. With the proposed analytical method, some results for the response of the tunnel to harmonic seismic waves are presented.