Analytical Model For A Lined Tunnel In The Unsaturated Soil Under Seismic Waves

The underground tunnel is one of the most important infrastructures in modern cities,and its seismic safety is an important guarantee for the economic development of modern cities.At present,the theoretical research on the seismic response of the underground tunnel at home and abroad is not comprehensive.Therefore,this paper intends to establish an analytical model of the seismic response for a circular lined tunnel in various mediums（such as homogeneous unsaturated soil,layered linear elastic soil and layered unsaturated soil）.The research results of this paper can provide a theoretical basis for the seismic design of the underground tunnel.The main research contents of this paper are as follows:（1）An analytical model of the seismic response for a circular lined tunnel in half-space homogeneous unsaturated soil under plane P₁ waves or plane SV waves.Firstly,the total wave fields presented in this model are analyzed and decomposed into the following three parts:the free wave field and the scattered wave field in the half-space unsaturated soil as well as the transmitted wave field in the lining of the tunnel.Then,based on the wave function expansion method,the expressions of the displacement,stress,pore water pressure and pore gas pressure corresponding to the three sub-wave fields are presented.In terms of the above expressions,the boundary value problem of the model is established,and the solution to this problem is obtained by truncating the infinite series related to the wave potential functions.Finally,the obtained solution is validated,and the effect of the saturation of unsaturated soil and the incident angle of seismic waves on the dynamic stress concentration factor along the inner and outer surfaces of the lining is investigated.The calculated results indicate that the saturation of the unsaturated soil has no significant influence on the dynamic stress concentration factor along the inner and outer surfaces of the lining,but the incident angle of seismic waves has a significant influence on it.（2）An analytical model of the seismic response for a circular lined tunnel in half-space layered linear elastic soil under plane P waves or plane SV waves.Firstly,the total wave fields presented in this model are analyzed and decomposed into the following three parts,namely,the free wave field and the scattered wave field in the layered linear elastic foundation as well as the transmitted wave field in the lining of the tunnel;then,based on the wave function expansion method and the exact stiffness method,the expressions of the displacement and stress components corresponding to the three sub-wave fields are presented;in terms of the above expressions,the boundary value problem of the model is established,and the solution to this problem is obtained by truncating the infinite series related to the wave potential functions.The established analytical model is validated by comparing it with known solutions,and the effect of the softer and stiffer interlayers of layered foundation and the incident angle of seismic waves on the dynamic stress concentration factor along the inner and outer surfaces of the lining is investigated;the calculated results indicate that both the softer and stiffer interlayers of layered foundation and the incident angle of seismic waves have significant influences on the dynamic stress concentration factor along the inner and outer surfaces of the lining.（3）An analytical model of the seismic response for a circular lined tunnel in half-space layered unsaturated soil under plane P₁waves or plane SV waves.Firstly,the total wave fields presented in this model are decomposed into the following three parts,namely,the free wave field and the scattered wave field in the layered unsaturated foundation as well as the transmitted wave field in the lining of the tunnel;then,based on the wave function expansion method and the exact stiffness method,the expressions of the displacement,stress,pore water pressure and pore gas pressure corresponding to the three sub-wave fields are presented;in terms of the above dynamic response expressions,the boundary value problem of the model is established,and the solution to this problem is obtained by truncating the infinite series related to the wave potential functions.The established model is validated by comparing it with known solutions;based on the established model,the effect of the saturation of unsaturated soil and the softer and stiffer interlayers of layered foundation as well as the incident angle of seismic waves on the dynamic stress concentration factor along the inner and outer surfaces of the lining is investigated;the calculated results indicate that the saturation of unsaturated soil has no significant influence on the dynamic stress concentration factor along the inner and outer surfaces of the lining,but both the softer and stiffer interlayers of layered foundation and the incident angle of seismic waves have significant influences on it.