Complex Variable Function Method Of Stress And Displacement Of Tunnel Cross Section In A Half Space

In recent years,with the rapid development of urban constructions in road tunnel,rail transit and railway tunnel,people pay great attention to the concept and basis of design of the tunnel,and the stress and deformation analysis of tunnel surrounding rock has become an important part of the tunnel design.Many studies indicate that the geometrical section shape and depth of the underground structure have a great influence on the stress distribution of surrounding rock and lining,while have a big limitation to the current level of development of rock mechanics.We can obtain the theoretical analytic solution for stress of surrounding rock by taking deep buried tunnel as infinite body whose section shapes are regular,such as circular or oval shape,while irregular section shape and shallow buried circular tunnel lack of accurate theoretical analytical solution to the stress distribution.In this paper,the analytical solutions of deep buried tunnel and shallow buried circular tunnel are solved and verified by literature review,theoretical analysis and numerical simulation and equivalent radius method is verified by complex variable function method.The main research work is as follows;(1)On the basis of a large number of references and the basic equations of elastic mechanics including equilibrium differential equation,geometric equation,constitutive equation,etc,the biharmonic equation is obtained by mathematical method and the complex function of the stress potential function is given according to biharmonic equation.The stress components and the displacement components are expressed in the potential function of complex variable according to their relationship among stress components,strain components and stress potential function,and then the displacement boundary condition is obtained by complex variable function.In this paper,we give a mapping function that can transform a complex boundary region into a simple boundary region,and then the theoretical basis of the complex variable function method for solving the half plane problem is given.(2)Using the basic theory of complex variable function,the deep buried tunnels of common section shapes are analyzed.We solve K-M function and give the general procedure for solving the complex function of tunnel section with arbitrary shape through the mapping function which transforms arbitrary boundary conditions to unit circle outer boundary and a series of mathematical operations,and give analytical expressions of stress of several common surrounding rock of the deep buried tunnel,such as round,oval,rectangular the straight wall arch and so on.The validity of the method is verified by using the finite element software to simulate all kinds of common tunnel section.(3)We obtain the approximate solution of deep buried tunnel by the equivalent radius method that using equivalent radius method and transforming the arbitrary shape boundary into standard circular section,the stress components of the surrounding rock are obtained by Lame solution.Considering the changes of parameters of the tunnel section shape,we solve the stress components which section shapes are circle,ellipse,rectangle and straight arm arch by the equivalent radius method.We take the circle shape as an example to verify the accuracy of complex function solution,the accurate solutions of oval,rectangular and straight arm arch given by complex variable function verify the precision of the equivalent radius and compare their difference between solutions of stress of surrounding rock.The results show that there is a close relationship among similarity between the solution of the equivalent radius and the exact solution and the cross-section shapes and geometric parameters of the tunnel,and the solution of equivalent radius is used for calculation of underground caverns,the ellipses and rectangles whose ratio of length to width is nearly 1 can be used for approximate calculation.(4)In order to research the self supporting capacity of surrounding rock under the shotcrete support,the complex variable function solution of the shallow tunnel under the given deformation is solved.Using the method of complex variable function and assuming that the liner and surrounding rock are linear elastic materials,the plane elastic solution of tunnel surrounding rock and lining under the given displacement deformation of shallow buried lining circular tunnel is analyzed.The semi-infinite plane with tunnel of circular section lining is transformed into plane single ring domain and another surrounding rock of ring domain by two conformal transformation formula,then complex potentials in the ring domain and the circular domain are expanded into two Laurent series.Based on the convergence of the series expansion of the boundary conditions in two circular domains,the stress component of the surrounding rock and the lining of the shallow buried tunnel under radial deformation is obtained.Using ABAQUS to verify the solution of complex variable function,we prove the validity of the complex variable function method through the change trend and stress.