Complex Variable Solution For A Non-circular Tunnel In An Elastic Half-plane

Shallow tunnels are widely used in municipal engineering,railway engineering and hydraulic and hydro-power engineering,in the field of solving shallow tunnel by analytic method,it is often simplified as an elastic-plane problem,considering the influence of the ground surface,the problem is actually an elastic half-plane problem with a cavity,and Verruijt is the pioneer to solve the problem by the complex variable method,he abtained the solution for a circular tunnel in an elastic half-plane.In many cases,however,we will meet a non-circular excavation in engineering.In this paper,the solution for a non-circular tunnel in an elastic half-plane is abtained by the complex variable method.A mapping function,which maps the region of a non-circular tunnel in a half-plane to the circular ring on the image plane,is proposed,and the method for solving the coefficients of mapping function is given.Based on this,an analytical solution for a non-circular tunnel excavated in an elastic homogeneous half-plane is obtained,which considers gravity and different lateral pressure coefficients.The solution is deduced using the complex variable method,the basic equations for the analytic functions are established using the stress boundary conditions at the ground surface and the edges of the tunnel and the analytical functions are solved by the power series method.In the solving process,the unblance force deduced by the excavation and it's distribution along tunnel edge are considered,an expression of the unblance is obtained.Because of the action of the unblance,the displacement will usually be unbounded at infinitely in solutions,but in the region of the tunnel excavation area and part of the ground surface,the differences between the displacement solution and really displacement are just a rigid displacement of the half-plane,therefore,the deformation of tunnel and the ground surface are ture.To elaborate the solution process clearly and verify the correctness of the solution,a shallow quasi-ellipse tunnel was analyzed as a computational example,in the process of solving,the Fourier series was used.Finally,the results obtained using the presented analytical method are compared with that obtained by the numerical software ANSYS.