Theoretical Solution Of Surrounding Rock Deformation And Stress Of Shallow Circular Tunnel Based On Complex Variable Function Method

In recent years,with the rapid development of urbanization in China,urban rail transit has become the primary choice to solve urban traffic problems with its less occupied urban land resources,no damage to the surface buildings,and the ability to transport a large number of passengers quickly and effectively.In the urban construction of subway tunnels,because the depth of tunnel is shallow,it is often vulnerable to the load of dense ground buildings.Because a semi infinite plane with holes is a semi infinite multi connected domain,it is difficult to analyze the stress and deformation of the surrounding rock of a shallow tunnel by the analytical method.At present,there is a lack of theoretical research on the deformation and stress of shallow buried circular tunnel under the action of surface loads and surrounding forces.Based on the complex variable function theory of elasticity,the mechanics problem of shallow buried circular tunnel under the action of surface load and surrounding force is studied.By using the principle of conformal transformation of complex function,the semi infinite plane containing holes is mapped to a unit concentric ring,and the analytic function of the complex function is expanded into the Laurent Series in the circle domain by the analytic property of the complex function in the circle domain.In order to find the Laurent series coefficient of the analytic function in the mapping domain,the boundary conditions of the surface and the hole stress are treated by the approximation principle of the complex function and the method of Fourier series transformation.The correctness and adaptability of the theory is verified by comparing the theoretical results with the data of finite element numerical simulation and measured data.Finally,the elastoplastic complex function solution of a shallow buried tunnel under uniformly distributed load is studied by the conformal mapping and the elastic analytical solution of the thick wall circular ring loaded uniformly distributed load.The main research contents and results include:(1)Based on the complex variable function theory of elasticity,the complex function solution of shallow buried circular tunnel under vertical load is studied.First,by using the conformal transformation principle of complex function,the semi infinite plane containing holes is mapped to a unit concentric ring and the analytic function is expanded into the Laurent Series in the circle domain by combining the analytic properties of the complex function in the circle domain.Then the surface boundary condition is treated by the approximation principle of complex functions and the Fourier series transformation ofcomplex functions.Finally,the theoretical calculation is compared with the numerical simulation results to verify the accuracy of the theory,and the effect of the range of ground load on the stress distribution and displacement of the surrounding rock is analyzed.(2)Based on the deformation boundary condition of Park deformation mode of tunnel lining and the condition of coordinated deformation between surrounding rock and lining,the complex function solution in the lining domain under the arbitrary inner boundary conditions and the complex function solution in the surrounding rock region under any tunnel surface force are given respectively.The correctness of the theoretical derivation is verified by the comparison between theoretical calculation and numerical simulation.And the measured data of two tunnels are compared with the theoretical calculation in this paper to verify the applicability of the theory.(3)Based on the conformal mapping principle of complex variable functions,the mechanical problems of the semi infinite plane under the action of the support pressure and the surface load are mapped to the uniform distribution of the internal and external boundary of the unit concentric ring.The elastoplastic stress distribution and the scope of the plastic zone in the mapping domain can be obtained by combined with the elastic analytical solution of elastic mechanics about the uniformly distributed load of thick wall rings and Mohr-Coulomb yield criterion.Finally,the elastoplastic stress distribution and the plastic zone of the semi infinite plane containing holes with uniformly distributed load on the surface are obtained by mapping function.The validity of this paper’s theoretical calculation is verified by the comparison between the theoretical calculation of this paper and the result of MASSINAS’s research.The calculation method of this paper can quickly estimate the stress distribution in the plastic zone and surrounding of the shallow tunnel,and provides an estimation method for the elastoplastic range and stress of the surrounding rock in tunnel construction.